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The Bridge & Structural Engineer ING - IABSE

Indian National Group of the International Association for Bridge and Structural Engineering

Contents :

Volume 45, Number 1 : March 2015

Editorial • From the desk of Chairman, Editorial Board : Mr. Alok Bhowmick • From the desk of Guest Editor : Prof. Sudhir K Jain

Special Topic : Earthquake Resistant Design of Structures 1. Normalized Response Spectrum of Ground Motion Praveen K. Malhotra

1

3. Shear-Moment Interaction in Confined Masonry Walls: Is it Worth Considering? J.J. Perez Gavilan, O. Cardel J.

19

4. Use of Confined Masonry for Improved Seismic Safety of Buildings in India Sudhir K. Jain, Svetlana Brzev, Durgesh C. Rai

29

5. Seismic Provisions for Bridges in Indian Standards-Past Practices and Recent Developments A.K. Banerjee, Alok Bhowmick

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6. Cyclic Performance of Asymmetric Friction Connections with Grade 10.9 Bolts J. Borzouie, G.A. MacRae, J.G. Chase, G.W. Rodgers, R. Xie, Jose Chanchi Golondrino, G.C. Clifton

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7. Vulnerability Analysis of Buildings for Seismic Risk Assessment: A Review Hemant B. Kaushik, Trishna Choudhury

63

8. Seismic Vulnerability Analysis of Bridge Pier Designed with different Codal Provisions Shivang Shekhar, Dr. Pankaj Agarwal 9. Retrofitting of Damaged RC Frame using Metallic Yielding Damper Romanbabu M. Oinam, Dr. Dipti Ranjan Sahoo 10. Efficient Generation of Statistically Consistent Demand Vectors for Seismic Performance Assessment  Dr. Dhiman Basu, Andrew Whittaker

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Contents

2. Geotechnical Impacts of Earthquake-Related Mass Movements: Some Lessons Learnt from the 2011 Sikkim Earthquake 12 Debasis Roy, Alpa Sheth, Nagendra Kola 

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Volume 45 Number 1 March 2015  i


The Bridge & Structural Engineer JOURNAL OF THE INDIAN NATIONAL GROUP OF THE INTERNATIONAL ASSOCIATION OF BRIDGE & STRUCTURAL ENGINEERING

June 2015 Issue of the Journal will be a Special issue with focus on

STRENGTHENING, REPAIR & REHABILITATION OF STRUCTURES FOLLOWING TYPE OF STRUCTURES ARE TO BE COVERED : 1. 2. 3. 4. 5. 6. 7.

Highway & Railway Bridges, Urban Transportation Structures Irrigation Structures, Marine Structures, Industrial Buildings Residential & Commercial Buildings Structures affected by natural disasters (Earthquake, Floods, Cyclones) Fire affected structures Tunnels, Pipe Lines, Water Mains Any other topics of Interest

The Bridge & Structural Engineer JOURNAL OF THE INDIAN NATIONAL GROUP OF THE INTERNATIONAL ASSOCIATION OF BRIDGE & STRUCTURAL ENGINEERING

September 2015 Issue of the Journal will be a Special Issue with focus on

AESTHETICS OF STRUCTURES SALIENT TOPICS TO BE COVERED ARE : 1.     Architecture and Aesthetics in general & state of the art 2.     Aesthetics of Structures other than Bridges 3.     Aesthetics of Bridges 4.     Aesthetics and Heritage Structures 5.     Aesthetics attributes and quantification Those interested to contribute Technical Papers on above themes shall submit the abstract by 7th August 2015 and full paper by 15th August 2015 in a prescribed format, at email id : ingiabse@bol.net.in, ingiabse@hotmail.com

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The Bridge and Structural Engineer


B&SE: The Bridge and Structural Engineer, is a quarterly journal published by ING-IABSE. It is one of the oldest and the foremost structural engineering Journal of its kind and repute in India. It was founded way back in 1957 and since then the journal is relentlessly disseminating latest technological progress in the spheres of structural engineering and bridging the gap between professionals and academics. Articles in this journal are written by practicing engineers as well as academia from around the world.

Disclaimer : All material published in this B&SE journal undergoes peer review to ensure fair balance, objectivity, independence and relevance. The Contents of this journal are however contributions of individual authors and reflect their independent opinions. Neither the members of the editorial board, nor its publishers will be liable for any direct, indirect, consequential, special, exemplary, or other damages arising from any misrepresentation in the papers. The advertisers & the advertisement in this Journal have no influence on editorial content or presentation. The posting of particular advertisement in this journal does not imply endorsement of the product or the company selling them by ING-IABSE, the B&SE Journal or its Editors.

Editorial Board Chair : Alok Bhowmick, Managing Director, B&S Engineering Consultants Pvt. Ltd., Noida

Members : Mahesh Tandon, Managing Director, Tandon Consultants Pvt. Ltd., New Delhi A K Banerjee, Former Member (Tech) NHAI, New Delhi Harshavardhan Subbarao, Chairman & MD, Construma Consultancy Pvt. Ltd., Mumbai Nirmalya Bandyopadhyay, Director, STUP Consultants Pvt. Ltd., New Delhi Jose Kurian, Chief Engineer, DTTDC Ltd., New Delhi S C Mehrotra, Chief Executive, Mehro Consultants, New Delhi

Advisors : A D Narain, Former DG (RD) & Additional Secretary to the GOI N K Sinha, Former DG (RD) & Special Secretary to the GOI G Sharan, Former DG (RD) & Special Secretary to the GOI A V Sinha, Former DG (RD) & Special Secretary to the GOI S K Puri, Former DG (RD) & Special Secretary to the GOI

Front Cover :

R P Indoria, Former DG (RD) & Special Secretary to the GOI S S Chakraborty, Chairman, CES (I ) Pvt. Ltd., New Delhi

Picture shows damaged multi-storeyed building in Bhuj District from Gujarat Earthquake, which occurred on 26th January 2001. The Photo clearly shows insufficient connection between the RC elevator core and rest of the building, leading to the under utilization of the lateral strength and stiffness of the elevator core.

B C Roy, Senior Executive Director, JACOBS-CES, Gurgaon

The devastating Gujarat Earthquake affected about 20 districts, killed around 20,000 people (including 18 in south eastern Pakistan), injured another 167,000 and destroyed nearly 400,000 homes.

Submission of Papers : All editorial communications should be addressed to Chairman, Editorial Board of Indian National Group of the IABSE, IDA Building, Ground Floor, Jamnagar House, Shahjahan Road, New Delhi – 110011.

• Price: ` 500

Published : Quarterly : March, June, September and December Publisher : ING-IABSE C/o Secretary, Indian National Group of the IABSE IDA Building, Ground Floor (Room Nos. 11 and 12) Jamnagar House, Shahjahan Road New Delhi-110011, India Telefax: 91+011+23388132 Phone: 91+011+23386724 E-mail: ingiabse@bol.net.in, ingiabse@hotmail.com, secy.ingiabse@bol.net.in

Advertising: All enquiries and correspondence in connection with advertising and the Equipments/Materials and Industry News Sections, should be addressed to Shri RK Pandey, Secretary, Indian National Group of the IABSE, IDA Building, Ground Floor, Jamnagar House, Shahjahan Road, New Delhi-110011.

The Bridge and Structural Engineer

Journal of the Indian National Group of the International Association for Bridge & Structural Engineering

March 2015

The Bridge & Structural Engineer, March 2015

The Bridge and Structural Engineer

Volume 45 Number 1 March 2015  iii


From the Desk of Chairman, Editorial Board

On April 2013, I began my journey as Chairman, Editorial Board of ‘The Bridge & Structural Engineer’, with emotions ranging from fear of failure to deep respect for the office of Editor, to pleasure at the idea of serving the ING-IABSE in such a prominent role. The Indian National Group of the IABSE has been my home for more than 30 years now and I am truly humbled to fill this role of Chairman, Editorial Board. I am highly indebted and thankful to many of my colleagues within the editorial board and outside, who came forward and helped me immensely in many ways (i,e. by contributing papers, by guest editing, by acting as referee, by advertising in the journal). This support base is the backbone and strength of the Editorial Team and I on behalf of the entire team, convey my sincere thanks to all those who are contributing towards the improvement of this journal and towards the progress of ING-IABSE. This issue of the journal is focused on the theme of “Earthquake Resistant Design of Structures". Earthquakes are natural phenomenon and their occurrence is beyond human control but their results are manmade. The theme of this issue is very topical in light of the recent devastating earthquake in Nepal and northern part of Bihar on 25th April 2015, killing about 9,000 people and damaging properties, which is estimated

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to be about $10 billion, according to the Nepal government—nearly half of its gross domestic product (GDP) of $19.2 billion !! In India, more than 60% of the geographical area of the country is under the threat of moderate to severe seismic hazard. A large stock of construction in India is currently standing vulnerable to strong earthquake shaking. The problem of vulnerability is further increased by the wave of nation re-building exercise, which is being taken up by successive central and state governments, with extensive infrastructure development all through the country. The current status of earthquake engineering education in India needs a whole hog improvement. The earthquake resistant design practice in the industry is also varied. There are three code making agencies (namely BIS, IRC and IRS) in India developing national codes on EQ resistant designs for practicing engineers. These codes are not in sync with each other resulting in structures with very large difference in safety margins. All these codes and standards needs to be consistent and require significant up gradation, in order to bring them at par with the International Codes and Standards. Sustainable development in the country can be possible only when it is ensured that the existing The Bridge and Structural Engineer


flock of structures are made safe as per current standards and more importantly, the future construction is made earthquake resistant. The engineering community has a major role to play in this fast movement towards sustainable growth of the nation. Considering the above scenario, a strong need was felt by the editorial board of ING-IABSE for dissemination of current knowledge and for exchange of recent experiences gained by the practicing engineers as well as research institutions in the field of earthquake resistant design. Our Guest Editor for this issue is none other than Prof. Sudhir K Jain, who is an outstanding academician, accomplished leader in academic administration and above all a world renowned expert in the field of earthquake engineering. I

The Bridge and Structural Engineer

am highly indebted to Prof. Sudhir K Jain for accepting the challenge of guest editing this journal despite his extremely busy schedule. I am also thankful to Asst. Prof. Dhiman Basu of IIT, Gandhinagar for his tremendous help to Prof. Jain in ensuring that all the papers published in this issue are appropriately peer reviewed. I am sure, the papers in this journal will reveal the present state of knowledge, practice and research activities that is going on in various parts of the world on earthquake resistant design. It is hoped that readers will find the information of value.

ALOK BHOWMICK

Volume 45 Number 1 March 2015  v


From the Desk of Guest Editor

Seismic Safety Challenge in India India (and Indian subcontinent) suffers from moderate to large sized earthquakes fairly regularly and the number of deaths caused by such earthquakes in the recent decades has been unacceptably high. For instance, about 9,000 deaths in 2015 Nepal (magnitude 7.8) earthquake, 100,000 deaths (including those in India and Pakistan) in the 2005 Kashmir (magnitude 7.5) earthquake, 13,805 deaths in 2001 Bhuj (magnitude 7.7) earthquake and 7,928 deaths in 1993 Latur (magnitude 6.2) earthquake will be considered far beyond what is acceptable in the modern times. Typically, earthquakes of magnitude 6.5 to 7.0 cause less than 100 deaths in California and that provides a benchmark with which we need to view vulnerability of Indian constructions. Indian engineers have not only embraced the practice of earthquake engineering but have also been instrumental in developing codes and maps more than half a century ago. Yet the present day building performance leaves much to be desired. Clearly, we need to do a lot more than what we have done to ensure safety of our built environment. Fortunately, in the recent years we have not had a strong earthquake near a densely populated

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area. However, should that happen, the country may see a disaster much worse than what we have seen to date. It will not only cause huge number of deaths, but will also substantially affect the economy and the development agenda of the region and the country. All of these facts, together with the rapidly growing housing stock in our cities clearly underline the urgency for us to do more towards seismic safety of our new constructions. Seismic safety of built environment is not just an engineering problem; it requires complex multidisciplinary interventions. Engineers, architects, economists, sociologists, political and public policy leaders have to work together with a single goal. However, we must recognize that in the end, safety of a building depends on how it has been designed and built, regardless of what laws or regulations may have been in place. In that sense, of all the disciplines involved, engineering becomes the most critical. As engineering professionals, it is expected that (a) we acquire the expertise and update ourselves regularly with developments in the field and (b) we never become party to design and constructions that do not meet the highest possible seismic and engineering standards of quality.

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Earthquake engineering is a new discipline that has only emerged in the last 60 years. Until recently, most engineering and architectural colleges in India did not teach principles of earthquake engineering at the undergraduate and the postgraduate levels. Fortunately, tremendous capacity building in this domain has been achieved as a result of the activities of National Information Centre of Earthquake Engineering (www.nicee.org) and the National Programme on Earthquake Engineering Education (NPEEE). A very substantial number of Indian professional engineers, architects and academics are now familiar with safe earthquake engineering practices. However, this field continues to evolve and there is plenty that needs to be done for constantly updating oneself with developments in earthquake engineering. Of course, implementation of codes and ensuring quality in design and construction practice remains a huge challenge in Indian construction

The Bridge and Structural Engineer

industry. This requires a lot of work in the country towards (a) establishing a system of competence-based licensing of engineers and (b) effective enforcement of codes by the municipal authorities. These two actions are likely to yield the maximum benefits towards seismic risk reduction. It is heartening to see this journal announce a special issue on the theme of earthquake engineering. The editorial effort has been to put more emphasis on sourcing application oriented papers for this issue. It is hoped that the practicing civil and structural engineers will find this issue of particular interest as would professionals of other related disciplines.

SUDHIR K JAIN

Volume 45 Number 1 March 2015  vii


Brief Profile of Prof. Sudhir K Jain Dr. Sudhir Jain is an active academic in Earthquake Engineering and a passionate academic administrator. He obtained a Bachelor of Engineering degree from the University of Roorkee and Masters and Doctoral degrees from the California Institute of Technology, Pasadena. He has been on the faculty of IIT Kanpur since 1984 and is currently on leave to the new Indian Institute of Technology Gandhinagar (IITGN) in Ahmedabad to shoulder additional responsibilities of its first Director since June 2009. He initiated Earthquake Engineering activities at IIT Kanpur in 1984. He set up and developed the National Information Centre of Earthquake Engineering (NICEE) at IITK. He also developed the National Programme on Earthquake Engineering Education (NPEEE), supported by the Government of India. He has carried out post-earthquake surveys of most of the disastrous earthquakes in India since 1988 and disseminated “Learnings From Earthquakes” through publications and seminars. Dr. Jain has provided comprehensive earthquake engineering consultancy for major bridge projects, buried pipelines for petrochemicals, concrete dams and the like to numerous organizations. He has contributed significantly to the development of Indian seismic codes, conducted numerous short courses and seminars on seismic design for practicing engineers and college teachers, which have been extremely popular in the country. Dr. Jain is currently President of the Board of Directors of the International Association for Earthquake Engineering and is Fellow of the Indian National Academy of Engineering.

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Normalized Response Spectrum of Ground Motion

Praveen K. MALHOTRA StrongMotions Inc. Praveen.Malhotra@StrongMotions.com www.StrongMotions.com Praveen K. Malhotra, Ph.D., P.E. is a Principal at StrongMotions Inc. in the Boston Area. He has more than 25 years of experience in practice and research of structural and geotechnical earthquake engineering, including teaching multiple short-courses throughout the United States and abroad. He specializes in transparent assessment and cost-effective mitigation of risk. He provides consulting services related to hazard analysis, risk analysis and performance-based engineering. He is often consulted to peer-review major projects.

Summary The peak ground acceleration PGA, peak ground velocity PGV and peak ground displacement PGD are the fundamental strong-motion parameters (intensity measures). The response spectrum of ground motion relative to PGA, PGV and PGD is known as the normalized response spectrum (NRS). In the past, the number of records has not been sufficient to conclusively establish the shape of the NRS and to study its sensitivity to various factors. In this study, the NRS is derived from 13,192 strong-motion records. It is shown that the shape of the NRS is sensitive only to the normalized velocity PGVn = PGV/(PGA•PGD)1/2. For the same PGVn, earthquake magnitude, distance and local soil conditions have insignificant effect on the NRS. For the same PGVn, the direction of motion (horizontal or vertical) has insignificant effect on the NRS. The shape of the NRS is rooted in structural dynamics; thus it should be preserved in predicting ground motions for future earthquakes. It is found that the latest empirical ground motion prediction equations do not always preserve the shape of the NRS. Therefore, it is recommended that ground motion prediction models should only be developed for PGA, PGV and PGD and that the response spectra for various damping ratios should be generated from PGA, PGV and PGD by using the NRS. The Bridge and Structural Engineer

Keywords: strong-motion, response spectrum, normalized response spectrum, ground motion prediction equations, damping. 

Introduction The response spectrum provides valuable information regarding the ground motion. It allows an engineer to estimate forces and deformations in structures due to ground shaking. The spectral values at short-, intermediate- and long-periods correlate well with PGA, PGV and PGD, respectively. Numerous studies1-6 have shown that a smooth response spectrum of ground motion can be constructed from PGA, PGV and PGD. The response spectrum relative to PGA, PGV and PGD is known as the normalized response spectrum (NRS). The attractiveness of the NRS is that it reduces the number of intensity measures to just three: PGA, PGV and PGD. In the past, the ground motion records have not been sufficient to conclusively establish the shape of the NRS and to study its sensitivity to various factors. In this study 13,192 strong-motion records are used to establish the NRS and to examine its sensitivity to earthquake magnitude, distance, local soil conditions and the direction of motion. This study also examines the shape of the response spectra generated by latest ground motion prediction equations (GMPE). Volume 45 Number 1 March 2015  1


Strong-Motion Data The strong-motion data from 599 earthquakes in seismically active regions around are world7 are used in this study. The same data have been used to develop GMPE for shallow crustal earthquakes in seismically active regions.8-12 The data consist of 21,335 mostly tri-axial ground motions. The earthquake magnitude ranges from M 3 to M 7.9, closest distance R ranges from 50 m to 1,533 km and the average shear-wave velocity in top 30 m of the site (VS30) ranges from 94 m/s to 2,100 m/s.7 The response spectra of horizontal and vertical ground motions were read from ‘flatfiles’ released by the Pacific Earthquake Engineering Research (PEER) Center.7 The response spectra in PEER flat-files were for natural periods between 0 and 20 s. The histories of ground motion in three directions were not available for this study. Nearly 40% of the response spectra in PEER flat-files could not be used because they did not show the correct asymptotic behavior in the long-period (low-frequency) region. This will become clear later in this paper.

Rotated-Median Response Spectrum The horizontal response spectra in PEER flat-files7 are rotated-median. The definition of rotated-median response spectrum is discussed here. The horizontal ground motion at a site is typically recorded in two orthogonal directions, say north and east. The responses of a single-degree-of-freedom (SDOF) system can be computed in two orthogonal directions by solving the equation of motion. From the responses in two orthogonal directions, the responses in numerous other horizontal directions can be calculated by using vector-geometry. The median (50th percentile) of the calculated responses in numerous horizontal directions gives the rotatedmedian response spectrum; it is usually denoted as RotD50.13 Figure 1 shows the processed acceleration histories in two orthogonal horizontal directions at strong-motion station CSMIP #68206 during the 2014 magnitude M 6 South Napa Earthquake in California.14 The PGA in two horizontal directions are indicated in Figure 1. The rotated-median PGA is 0.693 g, where g = 9.81 m/s2 = acceleration due to gravity. Figure 2 shows the processed velocity histories in two horizontal directions. The PGV in two horizontal directions are indicated in Figure 2. The rotated-median PGV is 2  Volume 45

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18.5 cm/s. Figure 3 shows the processed displacement histories in two horizontal directions. The PGD in two horizontal directions are indicated in Figure 3. The rotated-median PGD is 1.39 cm.

Fig. 1: Processed acceleration histories at strong-motion station CSMIP # 68206 during the 2014 M 6 South Napa Earthquake in California14

Figure 4 shows a tripartite plot of the 5% damping rotated-median response spectrum. In this plot, the peak pseudo-velocity PPV is shown along the vertical axis; the peak pseudo-acceleration PPA is shown along the -450 (counter-clockwise) axis; the peak deformation PD is shown along the 450 (clockwise) axis; and the natural period T is shown along the horizontal axis. PPA, PPV, PD and period T are related to each other as follows:5,15 PPA × T/(2π) = PPV = PD×2π/T

(1)

Note in Figure 4 that the PPA at short-periods approaches PGA and the PD at long-periods approaches PGD. This response spectrum was generated by simultaneously using the processed acceleration, velocity and displacement histories (Figures 1, 2 and 3).16 Therefore, it shows the correct asymptotic behavior at both short- and long-periods.15

Fig. 2: Processed velocity histories at strong-motion station CSMIP # 68206 during the 2014 M 6 South Napa Earthquake in California14

Fig. 3: Processed displacement histories at strong-motion station CSMIP # 68206 during the 2014 M 6 South Napa Earthquake in California14

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A normalized plot of the response spectrum is obtained in two steps:6 1. The natural period along the horizontal axis is divided by the central period Tc to obtain the normalized period Tn = T/Tc. This causes the PGA to become PGA•Tc and the PGD to become PGD/Tc. The PGV is not affected by normalizing the period.

Fig. 4: Five percent of critical damping rotated-median response spectrum of horizontal ground motion at strongmotion station CSMIP # 68206ofduring theMotion 2014 M 6 South Normalized Response Spectrum Ground Normalized Response Spectrum of Ground Motion Napa Earthquake

2. The peak pseudo-velocity PPV along the vertical axis is divided by (PGA•PGD)1/2 to obtain the normalized peak pseudo-velocity PPVn = PPV/ (PGA•PGD)1/2. This causes the PGA to become PGA•Tc/(PGA•PGD)1/2 = 2π, PGD to become PGD/Tc/(PGA•PGD)1/2 = 1/(2π) and PGV to become PGVn = PGV/(PGA•PGD)1/2.

Figure 5 shows the normalized plot of the 5% damping response spectrum shown in Figure 4. The normalized plot emphasizes the shape of the response spectrum From PGVparameters and PGD,can twobeadditional parameters , PGV and PGD, twoPGA, additional determined,which provide further , PGV and PGD, two additional parameters can be determined,which provide further relative to PGA, PGV and PGD. Next, the smooth can be determined, insight into velocity o the ground motion. These are thewhich centralprovide period Tfurther normalized c, and the o the ground motion. Thesemotion. are the central , and theperiod normalized shape of the NRS will be generated by considering the ground These period are theTccentral Tc velocity numerous ground motion records. 6 and the normalized velocity PGV : Normalized Response Spectrum Spectrum Normalized Response Normalized Response Spectrum

T� � ���PGD/PGA T� � ���PGD/PGA

n

PGV� � PGV/√PGA · PGD PGV� � PGV/√PGA · PGD

(2)

(2)

(2) Smooth Normalized Response Spectrum

(3)

(3)5% damping response spectra in PEER flat-file7 The(3) were normalized with respect to the PGA, PGV and A ground motion is composed of many different motion is composed of many different frequencies (or periods). Tc can be thought of as PGD motion is composed of many different frequencies (or periods). T can be thought of asvalues in the PEER flat-file. Figure 6 shows the of as frequencies (or periods). Tc can be thought c period of the ground motion.66PGVn is indicative of the band-width of frequencies 6 plots of 21,335 NRS. Note that for many of these period of thethe ground motion. PGVof indicative the band-width PGVnofisfrequencies central period groundof motion. n isthe a ground motion.66A higher value of PGVn implies a narrower band-width.PGV n response spectra, the peak deformation PD does not a ground motion. A higher value of PGVn of implies a narrower band-width.PGV indicative of the band-width frequencies present in n alue of 1 when the ground motion 6consists of a single frequency of period Tc. A higher PGVn of implies ground alue of 1 whenathe groundmotion. motion consists of avalue single of frequency period aTc. approach PGD at long-periods. In other words, many of the response spectra in PEER flat-file do not show narrower band-width. PGVn reaches a value of 1 when the correct asymptotic behavior at long-periods. This the ground motion consists of a single frequency of period Tc.

Fig. 5: Normalized of thespectrum responseshown spectrum shown 4. in Figure 5. Normalized plot of the plot response in Figure Fig. 6: Normalized plots of 21,335 five-percent damping Figure 5. Normalized plot of the response spectrum shown in Figure 4. Figure 4 response spectra in PEER flat-file7 6 zed plot of the response spectrum is obtained in two steps:6 zed plot of the response spectrum is obtained in two steps: tural period along the horizontal axis is divided by the central period Tc to obtain the tural period along the horizontal axis is divided by the central period Tc to obtain the Bridge Structural Volume 45 Number 1 March 2015  3 ized period TThe causes the PGA Engineer to becomePGA·Tc, and the PGD to n = T/T c.Thisand ized period Tn = T/Tc.This causes the PGA to becomePGA·Tc, and the PGD to e PGD/Tc. The PGV is not affected by normalizing the period. e PGD/Tc. The PGV is not affected by normalizing the period.

trongMotions Inc. ●www.StrongMotions.com● (781) 363-3003 ● 4/6/15 ● Page 8 of 29 trongMotions Inc. ●www.StrongMotions.com● (781) 363-3003 ● 4/6/15 ● Page 8 of 29


is because the response spectra were generated from acceleration histories with low-frequency ‘noise’. Had they been generated from acceleration, velocity and displacement histories simultaneously,14 they would have shown the correct asymptotic behavior at both short- and long-periods. The response spectra which do not show the correct asymptotic behavior at long-periods were removed from the data set so that the shape of the smooth NRS is not corrupted by the long-period ‘noise’. The undesirable response spectra were removed with the help of the 30% damping response spectra in PEER flat-file because 30% damping peak deformation PD approaches peak ground displacement PGD at a much faster rate than the 5% damping PD. Response spectra for which 30% damping PD(20 s) > 1.05 PGD or < 0.95 PGD were removed from the data set. This reduced the number of ‘clean’ usable response spectra to 13,192. Figure 7 shows the normalized plots of

13,192 ‘clean’ response spectra. These response spectra show the correct asymptotic behavior at both short - periods and long-periods. The NRS in Figure 7 have the same normalized PGA = 2π and the same normalized PGD = 1/(2π), but they have different normalized velocities PGVn, ranging from 0.13 to 1.1. The NRS were sorted by their normalized velocity PGVn and placed in different bins with specified normalized velocities. Figure 8 shows 380 NRS with median normalized velocity of PGVn = 0.4. The median (50th percentile) of these 380 NRS is shown by thick red lines in Figure 9; this is the smooth 5% damping NRS for PGVn = 0.4. The uncertainty in normalized values is indicated by thin blue lines representing 16th and 84th percentile values. Figure 9 also shows maximum spectral values relative to PGA, PGV and PGD. The maximum pseudo-acceleration is 2.46 PGA; maximum pseudo-velocity is 1.82 PGV; and maximum deformation is 1.86 PGD. For PGVn = 0.4, the maximum dynamic amplifications relative to PGA, PGV and PGD are 2.46, 1.82 and 1.86, respectively.

Fig. 7: Normalized plots of 13,192‘clean’ 5% of critical damping response spectra in PEER flat-file

Fig. 9: Smooth 5% damping median, 16th and 84th percentile NRS for PGVn = 0.4

Fig. 8: Normalized plots of 380 5% damping response spectra with median PGVn = 0.4

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Smooth NRS were also generated for PGVn = 0.6 and PGVn = 0.8. They are shown in Figures 10 and 11, respectively. For PGVn = 0.6, the maximum dynamic amplifications relative to PGA, PGV and PGD are 2.7, 2.26 and 2.1, respectively (Figure 10). For PGVn = 0.8, the maximum dynamic amplifications relative to PGA, PGV and PGD are 2.78, 2.67 and 2.39, respectively (Figure 11). With increase in the normalized velocity PGVn, the dynamic amplifications increase but the response spectrum becomes narrower. This is expected because higher The Bridge and Structural Engineer


values of PGVn imply narrower band of frequencies in ground motion leading to resonance-like response of the single-degree-of-freedom system.

Fig. 12: Smooth 5% damping median NRS for PGVn = 0.4, 0.6 and 0.8

Fig. 10: Smooth 5% damping median, 16th and 84th percentile NRS for PGVn = 0.6

Fig. 11: Smooth 5% damping median, 16th and 84th percentile NRS for PGVn = 0.8

Notice in Figures 9, 10 and 11 that the uncertainty is highest in the velocity-sensitive region of the normalized spectrum. As PGVn increases, the velocity-sensitive region becomes smaller and the uncertainty reduces. Figure 12 compares the median NRS for three different values of PGVn = 0.4, 0.6 and 0.8.

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Parametric Study The smooth NRS in Figure 12 were generated from ground motions due to different magnitude earthquakes, at different distances and at different soil conditions. The effects of magnitude, distance and local shear-wave velocity on the smooth NRS are examined in this section. To determine if the magnitude of the earthquake has any significant effect on the shape of the NRS, the data were sorted in terms of the earthquake magnitude and split into two nearly equal parts. The NRS were separately generated from each part. They are shown in Figure 13. Note that the smooth NRS are not significantly affected by the earthquake magnitude. It is more likely for a bigger earthquake to generate ground motions with wider frequency-band (or smaller normalized velocity PGVn), but for the same PGVn, the shape of the NRS is not significantly affected by the earthquake magnitude. The effect of magnitude on the NRS is further diminished by the fact that the site-specific response spectrum is not determined by a single magnitude earthquake but by earthquakes of many different magnitudes. Therefore, it is considered appropriate to ignore the effect of magnitude on the NRS.

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Fig. 13: Effect of magnitude M on the smooth NRS for 5% damping

Next, the data were sorted in terms of the distance R and split into two nearly equal parts. The NRS were generated from each part. They are shown in Figure 14. Note that the smooth NRS are not significantly affected by the distance from the source of the earthquake. Although, distance from the source can affect the frequency-band (or PGVn), for the same PGVn, distance has insignificant effect on the shape of the NRS. The effect of distance on the NRS is further diminished by the fact that the site-specific response spectrum is determined by earthquakes that could occur at various distances from the site. Therefore, it is considered appropriate to ignore the effect of distance on the NRS.

Fig. 15: Effect of VS30 on the smooth NRS for 5% damping

Next, the data were sorted in terms of the average shear-wave velocity in top 30 m VS30 and split into two nearly equal parts. The normalized response spectra were generated from each part. They are shown in Figure 15. Note that the smooth NRS are not significantly affected by the average shear-wave velocity VS30. Although, local soil conditions can affect the frequency-band (or PGVn), for the same PGVn, local soils have insignificant effect on the shape of the NRS. In this section the median NRS are shown to depend only on the normalized velocity PGVn; they do not additionally depend on the earthquake magnitude, distance and local shear-wave velocity. Due to space limitation, the results for 16th and 84th percentile NRS could not be shown in this section. But those NRS also depend only on the normalized velocity PGVn; they do not additionally depend on the earthquake magnitude, distance and local shear-wave velocity.

Smooth NRS of Vertical Ground Motion

Fig. 14: Effect of distance R on the smooth NRS for 5% damping

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The smooth NRS were also generated for vertical ground motions. In Figure 16, the NRS of horizontal and vertical motions are compared with each other. Note that there is no significant difference between the shapes of NRS of horizontal and vertical motions. This should not be misunderstood. Only the NRS of horizontal and vertical motions are nearly identical. Since PGA, PGV and PGD of vertical motion are significantly different from those of horizontal motion, the actual response spectrum of vertical motion is The Bridge and Structural Engineer


generally quite different from that of horizontal motion. Response spectra of vertical motions tend to have wider frequency band (smaller PGVn), but for the same PGVn, the NRS of vertical and horizontal motions are almost identical.

Figure 20 shows a plot between damping and PPVn for three selected values of normalized period T/Tc = 0.3, 1 and 3. The damping along the horizontal axis is shown on a logarithmic scale and PPVn along the vertical axis is shown on a linear scale. The open circles in Figure 20 are the actual values of PPVn for different values of damping. The straight-lines in Figure 20 pass through the PPVn values for 0.5, 5 and 20% damping. Figure 20 shows that the PPVn values for damping other than 0.5, 5 and 20% can be obtained by assuming a piece-wise linear relationship between log(ζ) and PPVn.

Fig. 16: Smooth NRS of horizontal and vertical ground motions for 5% damping

Smooth NRS for Different Values of Damping Smooth NRS were generated for different values of damping. Figures 17, 18 and 19 show the effect of damping on the smooth NRS of horizontal ground motion for PGVn = 0.4, 0.6 and 0.8, respectively. Table 1 lists the smooth NRS for PGVn = 0.4, 0.6 and 0.8 and damping ζ = 0.5, 5 and 20% of critical. NRS for any other value of PGVn between 0.4 and 0.8 can be generated through interpolation. NRS for PGVn = 0.4 can also be used for PGVn< 0.4. NRS for PGVn = 0.8 can also be used for PGVn> 0.8.

Fig. 18: Effect of damping on NRS of horizontal ground motion for PGVn = 0.6

Fig. 17: Effect of damping on NRS of horizontal ground motion for PGVn = 0.4

Fig. 19: Effect of damping on NRS of horizontal ground motion for PGVn = 0.8

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Volume 45 Number 1 March 2015  7


Fig. 20: Normalized peak pseudo-velocity PPVn versus damping for three different values of normalized period T/Tc and normalized velocity PGVn = 0.6

Table 1. Normalized response spectra of horizontal motion for different damping ζ and normalized velocities. Damping values are percentage of critical Tn = T/Tc

PPVn = PPV/(PGD•PGA)1/2 PGVn = 0.4 0.5%

ζ = 5%

PGVn = 0.6 20%

0.5%

ζ = 5%

PGVn = 0.8 20%

0.5%

ζ = 5%

20%

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.0147

0.0149

0.0149

0.0149

0.0149

0.0149

0.0149

0.0148

0.0148

0.0148

0.0215

0.0222

0.0222

0.0221

0.0218

0.0218

0.0218

0.0218

0.0218

0.0218

0.0316

0.0341

0.0332

0.0329

0.0323

0.0323

0.0322

0.0322

0.0322

0.0321

0.0464

0.0600

0.0523

0.0500

0.0478

0.0478

0.0476

0.0473

0.0473

0.0472

0.0681

0.134

0.0915

0.0792

0.0731

0.0718

0.0711

0.0703

0.0701

0.0697

0.1

0.331

0.183

0.130

0.131

0.115

0.110

0.111

0.106

0.105

0.147

0.642

0.329

0.208

0.324

0.217

0.180

0.219

0.175

0.162

0.215

1.026

0.529

0.310

0.811

0.454

0.302

0.550

0.338

0.267

0.316

1.323

0.688

0.385

1.56

0.828

0.48

1.25

0.68

0.45

0.464

1.357

0.727

0.401

2.30

1.20

0.65

2.29

1.22

0.72

0.681

1.242

0.673

0.392

2.55

1.35

0.74

3.52

1.87

1.02

1

1.148

0.632

0.365

2.45

1.31

0.73

3.96

2.12

1.13

1.47

1.068

0.599

0.347

2.09

1.17

0.67

2.90

1.63

0.95

2.15

0.952

0.562

0.333

1.60

0.940

0.55

1.78

1.07

0.68

3.16

0.794

0.509

0.313

1.05

0.661

0.41

0.94

0.62

0.44

4.64

0.567

0.395

0.259

0.573

0.391

0.280

0.462

0.337

0.272

6.81

0.351

0.264

0.187

0.282

0.221

0.178

0.228

0.190

0.170

10

0.189

0.158

0.121

0.137

0.125

0.114

0.119

0.113

0.109

14.7

0.0987

0.0918

0.0777

0.0778

0.0760

0.0732

0.0728

0.0721

0.0712

21.5

0.0573

0.0553

0.0504

0.0492

0.0490

0.0483

0.0478

0.0478

0.0476

31.6

0.0361

0.0354

0.0335

0.0327

0.0326

0.0325

0.0322

0.0322

0.0322

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The Bridge and Structural Engineer


Tn = T/Tc

PPVn = PPV/(PGD•PGA)1/2 PGVn = 0.4 0.5%

ζ = 5%

PGVn = 0.6 20%

0.5%

ζ = 5%

PGVn = 0.8 20%

0.5%

ζ = 5%

20%

46.4

0.0235

0.0233

0.0225

0.0220

0.0220

0.0220

0.0218

0.0218

0.0218

68.1

0.0156

0.0156

0.0152

0.0149

0.0149

0.0149

0.0148

0.0148

0.0148

100

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

Response Spectra of Predicted Ground Motions The response spectra of predicted ground motions should be consistent with the normalized response spectrum (NRS) derived from recorded ground motions. Figure 21 shows a plot of the predicted 5% damping median response spectrum of ground motion at a ‘firm-rock’ site due to a magnitude M 6.8 earthquake at distance R = 10 km. This response spectrum is an average of the response spectra predicted by 5 GMPE.8-12 Figure 21 also shows the predicted median values of PGA, PGV and PGD. PGA is average of 5 GMPE. PGV is average of 4 GMPE because one of the equations20 does not predict PGV. PGD is inferred from the spectral values at long periods since none of the GMPE8-12 predicts PGD directly due to lack of confidence in low-frequency content of recorded ground motions. PGD was assumed equal to the maximum spectral deformation divided by 2, on the basis of Figures 9 and 10.

The shape of the response spectrum in Figure 21 is not consistent with the shape of the NRS derived from recorded ground motions. Specifically, the amplification of pseudo-velocities relative to peak ground velocity is too low. Next, a 5% damping response spectrum is generated from PGA, PGV and PGD by using the following steps: 1. The central period of the predicted ground motion is calculated from Equation 2 to be Tc = 2π•(0.0783/0.2025)1/2 = 1.125 s.

Fig. 22: Comparison between the response spectrum derived from GMPE (Figure 21) and the smooth response spectrum derived from PGA, PGV and PGD using the NRS

2. The normalized velocity of the ground motion is calculated from Equation 3 to be PGVn = 2.443/ (0.2025×0.0783)1/2 = 0.46.

Fig. 21: Five-percent damping predicted median response spectrum of horizontal ground motion at a ‘firm-rock’ site due to a magnitude M 6.8 earthquake at 10 km distance

The Bridge and Structural Engineer

3. From Table 1, the normalized peak pseudovelocities PPVn are read corresponding to PGVn = 0.4 and PGVn = 0.6. The values for PGVn = 0.46 are obtained by linear-interpolation between the values for PGVn = 0.4 and 0.6. The normalized pseudo-velocities PPVn are multiplied by (PGA•PGD)1/2 = 0.437 m/s to obtain pseudovelocities PPV. Volume 45 Number 1 March 2015  9


4. The normalized periods are read from the 1st column of Table 1; they are multiplied by Tc = 1.125 s to obtain natural periods T. In Figure 22 the response spectrum derived from the NRS (using PGA, PGV and PGD) is compared with the response spectrum derived from GMPE. The difference between the two response spectra is quite significant in the velocity- and displacement-sensitive regions. It is apparent that the response spectrum derived from the GMPE8-12 is not consistent with the recorded ground motions. Similar results were also obtained for several other earthquake scenarios defined by magnitude, distance and average shearwave velocity. There are at least two possible reasons for the incorrect shapes of predicted response spectra: 

Many of the response spectra in the database, used for developing empirical GMPE, do not show the correct asymptotic behavior in the long-period range (Figure 6). This might have corrupted the shape of the predicted response spectrum. The form of the empirical GMPE is unsuitable to capture the physics of a SDOF system responding to seismic ground shaking.

In view of the unreasonable shape of the predicted response spectrum, it is recommended that the empirical GMPE should only be developed to predict PGA, PGV and PGD and the response spectrum should be generated from the predicted values of PGA, PGV and PGD by using the NRS.

4. For the same PGVn, the damping adjustment factors depend only on the normalized period. They do not depend on the earthquake magnitude, distance and local soil conditions. 5. The shape of the NRS is rooted in structural dynamics; hence it should be preserved in predicting ground motions for future earthquakes. The response spectra derived from the latest GMPE8-12 are inconsistent with the recorded ground motions. 6. It is strongly recommended that prediction models should only be developed for PGA, PGV and PGD. Response spectrum for any damping ratio should be generated from PGA, PGV and PGD by using the NRS.

References 1.

Newmark, N. M. and W. J. Hall (1969). Seismic design criteria for nuclear reactor facilities, in Proc. 4th World Conf. on Earthquake Eng., Santiago,Chile, B-4, 37–50.

2.

Mohraz, B., W. J. Hall and N. M. Newmark (1972). A study of vertical and horizontal earthquake spectra, AEC Report WASH-1255, Nathan M. Newmark Consulting Engineering Services, Urbana, Illinois.

3.

Hall, W. J., B. Mohraz and N. M. Newmark (1975). Statistical studies of vertical and horizontal earthquake spectra, Nathan M. Newmark Consulting Engineering Services, Urbana, Illinois.

4.

Mohraz, B. (1976). A study of earthquake response spectra for different geological conditions, Bull. Seism. Soc. Am. 66(3), 915– 935.

5.

Newmark, N. M. and W. J. Hall (1982). Earthquake Spectra and Design, Earthquake Engineering Research Institute, Oakland, California.

6.

Malhotra, P. K. (2006). Smooth spectra of horizontal and vertical ground motions. Bull. Seism. Soc. Am., 96(2), 506–518.

7.

Seyhan, E., Stewart, J. P., Ancheta, T. D., Darragh, R. B., and Graves, R. W. (2014). NGAWest2 site database, Earthquake Spectra, 30(3),

Conclusions 1. The peak values of ground acceleration, velocity and displacement (PGA, PGV and PGD) are the fundamental ground motion parameters. Response spectrum for any damping ratio can be generated from PGA, PGV and PGD by using the normalized response spectrum (NRS) discussed in this paper. 2. The shape of the NRS is sensitive only to the normalized velocity PGVn = PGV/ (PGA•PGD)1/2. For the same PGVn, earthquake magnitude, distance and local soil conditions have insignificant effect on the NRS. 3. For the same PGVn, the direction of motion (horizontal or vertical) has insignificant effect on the NRS. 10  Volume 45

Number 1 March 2015

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1007–1024, Earthquake Engineering Research Institute.

4th edition, Prentice-Hall International Series in Civil Engineering and Engineering Mechanics.

8.

Abrahamson, N. A., Silva, W. J. and Kamai, R. (2014). Summary of the ASK14 ground motion relation for active crustal regions, Earthquake Spectra, 30(3), 1025–1055, Earthquake Engineering Research Institute.

16. Malhotra, P. K. (2001). Response spectrum of incompatible acceleration, velocity and displacement histories. J. Earthquake Eng. Struct. Dyn.,30(2),279-286.

9.

Boore, D. M. Stewart, J. P., Seyhan, E., and Atkinson, G. M. (2014). NGA-West2 equations for predicting PGA, PGV and 5% damped PSA for shallow crustal earthquakes, Earthquake Spectra, 30(3), 1057–1085, Earthquake Engineering Research Institute.

10. Campbell, K. W., and Bozorgnia, Y. (2014). NGA-West 2 ground motion model for the average horizontal components of PGA, PGV, and 5% damped linear acceleration response spectra, Earthquake Spectra, 30(3), 1087–1115, Earthquake Engineering Research Institute.

Nomenclature ζ

= Viscous damping (percent of critical)

g

= Acceleration due to gravity (9.81 m/s2)

GMPE = Ground motion prediction equation(s) M

= Moment magnitude of earthquake

NRS

= Normalized response spectrum

PD

= Peak deformation

PEER = Pacific Earthquake Research Center

Engineering

PGA

= Peak ground acceleration

11 Chiou, B. S.-J., and Youngs, R. R. (2014). Update of the Chiou and Youngs NGA model for the average horizontal component of peak ground motion and response spectra, Earthquake Spectra, 30(3), 1117–1153, Earthquake Engineering Research Institute.

PGD

= Peak ground displacement

PGV

= Peak ground velocity

PPA

= Peak pseudo-acceleration

12. Idriss, I. M. (2014). An NGA-West2 empirical model for estimating the horizontal spectral values generated by shallow crustal earthquakes, Earthquake Spectra, 30(3), 1155–1177, Earthquake Engineering Research Institute.

PPV

= Peak pseudo-velocity

PPVn

= PPV/(PGA•PGD)1/2 = normalized peak pseudo-velocity

R

= Distance from the source of earthquake

13. Boore, D. M., 2010. Orientation-independent, non geometric-mean measures of seismic intensity from two horizontal components of motion, Bull. Seismo. Soc. Am. 100, 1830–1835.

SDOF = Single-degree-of-freedom

14. CESMD (2014). South Napa Earthquake of 24 August 2014, Center for Engineering Strong Motion Data, http://strongmotioncenter.org/, December 12, 2014. 15. Chopra, A. K. (2011). Dynamics of structures,

The Bridge and Structural Engineer

PGVn = PGV/(PGA•PGD)1/2 = normalized peak ground velocity (Equation 3)

T

= Natural period of vibration

Tc

= Central period (Equation 2)

Tn

= T/Tc = normalized period

VS30

= Average shear-wave velocity in top 30 m (100 ft)

of

ground

motion

Volume 45 Number 1 March 2015  11


GEOTECHNICAL IMPACTS OF EARTHQUAKE-RELATED MASS MOVEMENTS: SOME LESSONS LEARNT FROM THE 2011 SIKKIM EARTHQUAKE

Debasis ROY Professor Department of Civil Engineering, IIT Kharagpur, WB 721302 debasis@civil.iitkgp.ernet.in

Alpa SHETH Managing Director VMS Consultants Private Limited, Mumbai 400021 alpa_sheth@vakilmehtasheth.com

Nagendra KOLA PhD Research Scholar IIT Kharagpur, WB 721302, INDIA kola_nagendra@yahoo.in

Debasis Roy, a 1985 B. Tech (Honors) from BHU, has an MS degree from University of Idaho (1992), USA and a PhD from the University of British Columbia, Canada (1997). He practiced as a Professional Engineer in India and Canada for over ten years mainly in projects on seismic hazard and risk and geotechnical earthquake engineering before joining IIT Kharagpur in 2002.

Alpa Sheth is the Managing Director of VMS Consultants Pvt. Ltd, a structural engineering practice in Mumbai. She is a Visiting Professor at Kamla Raheja Vidyanidhi Institute of Architecture and INAE Visiting Professor at IIT Madras. She has been a Seismic Advisor to Gujarat State Disaster Management Authority for many years and is on several Code Committees of the Bureau of Indian Standards. She holds a Master of Engineering from University of California, Berkeley.

Nagendra Kola received his Bachelor’s degree in Civil Engineering in 2011 form VR Siddhartha Engineering College, Vijayawada and Master’s degree in Geotechnical Engineering from NIT Rourkela in 2014. Presently he is a Research Scholar at IIT Kharagpur.

Abstract A magnitude (Mw) 6.9 earthquake of September 18, 2011, cantered in the Sikkim-Nepal border area, affected large portions of Sikkim, West Bengal states of India and adjoining Nepal, Bhutan, China and Bangladesh. Thin population density and absence of infrastructure facilities in the epicentral region, existence of few recording instruments within the affected area and a wide variation in the ground motion parameters because of topographic effects made precise engineering assessment of the event difficult. Regardless, geotechnical impacts were largely attributable to seismically-triggered permanent ground deformations, slope movements, landslides, rock falls and over reliance on isolated shallow foundations. Many of the affected facilities were constructed on pre-existing landslide or rock 12  Volume 45

Number 1 March 2015

fall tracks or on top of steep slopes leaving them vulnerable to reactivated mass movements, seismic ground motion amplification and permanent ground deformation. This paper summarizes the lessons learnt during the aftermath of the September 18, 2011 earthquake that may lead to the development of appropriate zoning and facility development strategies in seismically active Himalayan region. Key Words : Landslides, anisotropy, slope, rock fall, reactivation, development zoning

1.

Introduction

The magnitude (Mw) 6.9 earthquake of September 18, 2011 that affected large portions of Sikkim, West Bengal states of India and adjoining Nepal, Bhutan, China and Bangladesh leading to the The Bridge and Structural Engineer


more than 111 fatalities and estimated property and infrastructure damages exceeding Rupees 100 billion. Characterizations of the seismic parameters pertaining to the event are far from precise. For instance, the epicentre locations estimated by the India Meteorological Department (IMD) and United States Geological Survey (USGS) differ from each other by several tens of kilometres (Figure 1a). Estimates of focal depth also vary wildly: from 10 km (IMD) to 50 km (Ravi Kumar et al., 2012). Existence of only a few recording instruments within the affected area and a wide variation of ground motion parameters due to topographic effects made precise characterization of the event difficult. Geotechnical impacts of the event were largely attributable to seismically-triggered permanent ground deformations, slope movements,

landslides, rock falls and over reliance on isolated shallow foundations. Lessons learnt from the impacts of September 18, 2011 earthquake summarized in this article could be useful in developing appropriate zoning and facility development strategies in seismically active Himalayan region.

2.

Spatial Distribution of Observed Mass Movement Incidents

Several mass movement events ranging from shallow rockslides, deep seated hill slope failure, mudslides and rock fall affected structures and communities in Sikkim and Himalayan and Sub-Himalayan West Bengal. More than 1000 features related to such incidents have identified from satellite images

Fig. 1: Seismic parameters of September 18, 2011 earthquake, surface geology and inventory of mass movement incidents in Sikkim and Darjeeling

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immediately after the September 18, 2011 earthquake (Martha et al., 2014; Chakraborty et al., 2014) and from post-earthquake reconnaissance missions (e.g., Barua et al., 2011; Chourasia et al., 2011). Considering the Himalayan foothills within the state of West Bengal received precipitations exceeding 100 mm over a 2-day period immediately prior to the earthquake, the mass movements noted to have taken place immediately after September 18, 2011 in West Bengal may have been rainfall-related rather than seismogenic. The daily rainfall over a 1-week period before the earthquake in the upper reaches of Sikkim state were, for the most part, below 50 mm. While such intensities of precipitation could have led to saturation of near-surface soil and rock masses, the mass movement events identified immediately after the earthquake may actually have been triggered by earthquake shaking. Seismic and geological controls and qualitative periodicity of these events are discussed in the following subsections. 1.

Influence of seismic parameters on spatial density of mass movements

Inventory of mass movements noted immediately after September 18, 2011 identified from satellite imagery (Martha et al., 2014; Chakraborty et al., 2014) has been presented in Figure 1a along with surface geology of Sikkim (after Geological Survey of India, 2012) and Himalayan and sub-Himalayan West Bengal (after Geological Survey of India, 2013). Epicentral locations estimated by United States Geologic Survey (USGS) and India Meteorology Department (IMD) and MSK isoseismals estimated by Sharma et al. (2013) from a post-earthquake damage survey are also included in the figure. Figure 1 reveals dense clusters of mass movement incidents within Proterozoic migmatite, gneiss and schist particularly where earthquake intensity exceeded MSK VIII. September 18, 2011 mass movements ascribable to Sikkim earthquake were also absent where ground motion intensities were MSK VI or less. Figure 1a also indicates that in many areas with similar surface geology and similar ground shaking intensities spatial densities of September 18, 2011 mass movements were remarkably different. 2.

Influence of geology and surface topography on density of mass movements

Association of September 18, 2011 mass movements 14â&#x20AC;&#x192; Volume 45

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and drainage features is evident from Figure 1b. The association appears to have resulted from steepness of the slopes that overlook the streams in mountainous terrains and subsurface saturation levels. It is also of interest that the number of movements triggered on one of the stream banks was remarkably larger compared to those affecting the opposite bank for most of the drainage features (Table 1 and Figure 2). Table 1: Skewed landslide distribution on stream banks Stream(a)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Number of landslides: (left bank, right bank) 98, 37 2, 0 25, 4 0, 0 5, 0 1, 0 0, 6 23, 1 0, 1 5, 3 3, 1 2, 0 1, 1 11, 6 20, 13 11, 3 8, 0 16, 3 4, 2 3, 1 3, 0 1, 2 4, 1 6, 0 16,12 6, 0 1, 0 2, 3 2, 0 0, 3

Stream(a)

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59

Number of landslides: (left bank, right bank) 2, 0 2, 1 9, 0 1, 2 1, 0 3, 0 1, 0 2, 6 4, 1 20, 4 3, 1 3, 2 1, 1 1, 0 10, 2 0, 2 33, 27 26, 21 5, 1 2, 5 16, 9 1, 3 3, 1 14, 1 13, 0 4, 1 6, 2 6, 0 2, 3

Note. a. For stream number see Figure 1

Such a skewed distribution may have resulted from adverse dip directions and anisotropic shear strength The Bridge and Structural Engineer


behaviour often observed for gneissic, schistosic and phyllitic rock masses prevalent within the study area. Structural control on hill slope instability is illustrated with the damages suffered by Tashiling Secretariat building in Gangtok during September 18, 2011 earthquake because of slope instability (Figure 2). The area overlooks steep southerly and easterly slopes and is underlain by gneissic and schistosic bedrock capped with a shallow soil layer. The easterly slope is known to have been affected by – one during the earthquake of September 18, 2011 – whereas the northerly slope exhibited no instability. As shown in Figure 3, unlike the southerly slope, the dip direction

high to very high landslide hazard in the same study were not affected by mass movements during the earthquake (Figure 5).

Fig. 4: BIS (1998) landslide hazard zoning and September 18, 2011 mass movement locations for Upper Dzongu

Fig. 2: Earthquake-related and older mass movement locations in Upper Dzongu area

Fig. 5: Comparison of BIS (1998) landslide hazard zoning of Sikkim and observed September 18, 2011 mass movement locations Fig. 3: Tashiling Secretariat building damage

is adverse for the easterly slope. Consequently, influx of rainwater also had an adverse effect on the stability of the easterly slope. Marginal instability of the easterly slope during the earthquake appears to be the cause of the damage suffered by Tashiling Secretariat building, along with the inappropriately heavy cladding on its facade. Qualitative landslide hazard zoning developed by Sikkim State Disaster Mitigation Authority (SSDMA), www.ssdma.nic.in/ resources/publications/Sikkim HRVA. pdf, based on BIS (1998) guidelines reveals a reasonable agreement with observed mass movement density for drainage streams 15 and 18 in the Upper Dzongu area; one of the worst affected areas of the September 18, 2011 earthquake (Figure 4). However, many areas assigned The Bridge and Structural Engineer

3.

Reactivation of mass movements

Older mass movements identified from satellite imagery and from other compilations (Sarkar and Kanungo, 2004; Kanungo et al., 2008; Chatterjee, 2010; Starkel, 2010; Mandal and Maiti, 2015) show a spatial distribution similar to that of September 18, 2011 events (Figure 1a). Mass movement distribution in the Upper Dzongu area, which was affected severely during the September 18, 2011 earthquake, suffered from several instances of similar instabilities on earlier occasions as well (Figure 2). It therefore appears several of the earthquake-related mass movements could have been reactivation of older instabilities. Figure 2 data however indicate that the number of September 18, 2011 instabilities far outstrip that of older mass movements, although the Volume 45 Number 1 March 2015  15


observation could be misleading because published resources pertaining to older mass movement events used in this study cover different time windows and different focus areas within the region shown in Figure 1. Consequently, the inventory of older events is not comprehensive. Figure 6 shows the damages suffered by Chungthang health centre on account of rock fall. Incidentally, the facility was developed on the pre existing rock fall track; the boulder seen on the right of the photograph was deposited in an old rock fall event.

Fig. 6: Chungthang health centre rock fall damage

3.

Structural damages due to permanent ground deformation

Structural damage in Sikkim and the Darjeeling district of West Bengal were to a large part due to permanent ground deformations. Affected structures were supported on shallow, unconnected foundations (isolated, spread footings) often constructed within fill placed behind unanchored retaining walls on or atop

steep hill slopes. The retaining walls usually less than 2-m in height and constructed with reinforced cement concrete or timber crib or even unreinforced dry rubble masonry. Down slope sliding of gravity retaining walls and settlement of fill appeared to have been the main cause of damages. Buildings overlooking steep slopes were also affected by topographically amplified ground motion and permanent ground movements. Figure 7 illustrates such damages suffered by three unreinforced masonry buildings. Damages suffered by several buildings of the Tashiling Secretariat complex illustrated earlier were largely ascribable to permanent ground deformations. Necessity to account for estimated earthquakerelated permanent ground deformation in ensuring structural performance is well known in earthquake engineering community. Although simple procedures for estimating the displacements for foundations and slopes are available in the literature (e.g., Richards et al., 1993; Singh et al., 2007), such an approach is yet to be adopted in Indian codes and guidelines for earthquake design. These methodologies require site-specific ground motion estimates that depend on local seismicity as well as topographic amplification. It should be noted that the structural design code used at present in India, IS 1893 Part 1 (BIS, 2002) for earthquake loads is silent about ground motion amplification because of undulating surface topography. Topographic amplifications could lead to 50 % increase in earthquake ground motion amplitude (European Committee for Standardization, 2004).

Fig. 7: Building damage due to retaining wall movement and topographic amplification

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4.

Some Lessons Learnt

Sikkim earthquake of September 18, 2011 caused widespread damages to facilities in the states of Sikkim and West Bengal of India, Nepal, Bhutan and Bangladesh. Insights derived from site response and structural performance during Sikkim earthquake are likely be pertinent in other Himalayan locations with similar geologic setting as well. Structural and property damages caused by the September 18, 2011 earthquake were largely due to mass movements. Nearly 1000 incidents of mass movements triggered during an earthquake on September 18, 2011 affected a widespread area of the state of Sikkim, where earthquake intensities exceeded V on the MSK scale. About 100 incidents of mass movements were also noted at several locations of the Himalayan foothills in the state of West Bengal, although these events may have been rainfall-related. Incidents of mass movements in Sikkim controlled to a great extent by geologic structures affected areas known to have been affected by past incidents of mass movements. A zoning strategy to avoid known pathways of past mass movements could therefore have prevented a large proportion of facility damage following the earthquake. Many locations marked as areas of high hazard in the SSDMA landslide zoning of Sikkim based on BIS (1998) did not suffer predicted damages during the September 18, 2011 earthquake. Structural damages during the September 18, 2011 earthquake were often related to permanent ground deformations and inappropriate choice of foundation systems. Structures supported on isolated shallow footings constructed within backfill placed behind inadequately engineered unanchored retaining walls overlooking steep slopes were particularly vulnerable to damages due to differential movement related to permanent ground deformation developing during the earthquake of 2011. Buildings constructed on known rock fall corridors also suffered heavy damages at many locations because of coseismic rock fall events. Losses suffered during the 2011 Sikkim Earthquake could therefore have been minimized via appropriate zoning methodology. Avoidance of facility development on known mass movement corridors could, for instance, have prevented property damage to a great extent. Construction of structures on hill slopes could have been mandated to be designed for The Bridge and Structural Engineer

permanent ground deformations estimated for local seismicity as well as topographic effects.

5.

Conclusions

Lessons learnt during the September 18, 2011 Sikkim earthquake makes it apparent that risks of earthquakerelated damages to residential structures in Himalayan sites could be largely managed by adopting a simple development strategy aiming to avoid zones known to have been affected by mass movements in the past. BIS (1998) guidelines on landslide hazard zoning appear to lead to conservative inferences. They should therefore be calibrated against a comprehensive database of past mass movements such as that used in this study. Bylaws and structural design guidelines for hilly areas should be modified to adopt a displacementbased design approach and avoid foundation systems, e.g., isolated, shallow spread footings and building upon terraces constructed behind unanchored nonengineered retaining walls.

References 1.

Barua, I., Bhushan, S., Chaurasia, A., Deb, S.K., Murthy, C.V.R., Paul, D.K., Roy, D., Seth, A.R., Singh, M., and Sinha, R. 2011. 18 September 2011 Sikkim Earthquake: Post-earthquake Reconnaissance Report (for Sikkim). National Disaster Mitigation Authority (NDMA), New Delhi.

2.

Bureau of Indian Standards. 1998. Preparation of landslide hazard zonation maps in mountainous terrains â&#x20AC;&#x201C; guidelines. Part 2: Macro-zonation. IS14496 (Part 2): 1998. Bureau of Indian Standards, New Delhi.

3.

Chakraborty, I., Ghosh, S., Bhattacharya, D., and Bora, A. 2014. Earthquake induced landslides in the Sikkim-Darjeeling Himalayas â&#x20AC;&#x201C; an aftermath of the 18th September 2011 Sikkim earthquake. Report, Engineering Geology Division, Geological Survey of India, Kolkata

4.

Chourasia, A.P., Jaiswal. A., Joshi, B., Murty, C.V.R., Pachauri, A.K., Pande, P., Pareep Kumar, R., P.K., Ray, C., and Singh, B. 2011. 18 September 2011 Sikkim Earthquake: Postearthquake Reconnaissance Report - Darjeeling. NDMA, New Delhi.

5.

European Committee of Standardization. 2004. Volume 45 Number 1 March 2015â&#x20AC;&#x192; 17


Eurocode 8: Design of structures for earthquake resistance Part 5: Foundations, retaining structures and geotechnical aspects. Brussels. 6.

7.

8.

9.

Geological Survey of India. 2013. Geology and mineral resources of West Bengal. Miscellaneous Publication No. 30, Part 1. 3rd Ed. Geological Survey of India, Kolkata. Geological Survey of India. 2012. Geology and mineral resources of Sikkim. Miscellaneous Publication No. 30, Part XIX. Geological Survey of India, Kolkata. Kanungo, D.P., Arora, M.K., Gupta, R.P., and Sarkar, S. 2008. Landslide risk assessment using concepts of danger pixels and fuzzy set theory in Darjeeling Himalayas. Landslides, 5, 407-416. Mandal, S., and Maiti, R. 2015. Geo-spatial variability of physiographic parameters and landslide potentiality. Chapter 2 in Semiquantitative Approaches for Landslide Assessment and Prediction, Springer Natural Hazards, 57-93.

10. Martha, T.R., Babu Govindharaj, K., and Vinod Kumar, K. 2014. Damage and geological assessment of the 18 September 2011 Mw 6.9 earthquake in Sikkim, India using very high resolution satellite data. Geoscience Frontiers,

18â&#x20AC;&#x192; Volume 45

Number 1 March 2015

Elsevier (in press). 11. Ravi Kumar, M., Hazarika, P., Srihari Prasad, G., Singh, A., and Saha, S. 2012. Tectonic implications of the September 2011 Sikkim earthquake and its aftershocks. Current Science, 102(5), 788-792. 12. Richards, R., Elms, D.G., and Budhu, M. 1993. Seismic bearing capacity and settlements of foundations. Journal of Geotechnical Engineering, 119(4): 662-674. 13. Sarkar, S. and Kanungo, D.P. 2004. An integrated approach for landslide susceptibility mapping using remote sensing and GIS. Photogrammetric Engineering & Remote Sensing, 70(5), 617-625 14. Sharma, M.L., Sinvhal, A., Singh, Y., and Maheshwari, B.K. 2013. Damage Survey Report for Sikkim Earthquake of 18 September 2011. Seismological Research Letters, 84(1), 49 56. 15. Singh, R., Roy, D., and Das, D. 2007. A correlation for permanent earthquake-induced deformation of earth embankments. Engineering Geology, 90, Elsevier, 174-185. 16. Starkel, L. 2010. Ambootia landslide valley â&#x20AC;&#x201C; evolution, relaxation and prediction (Darjeeling Himalaya). Studia Geomorphologica CarpathoBalcanica, 154, 113-133.

The Bridge and Structural Engineer


Shear-Moment Interaction inConfined Masonry Walls: is it Worth Considering?

J. J. Perez GAVILAN Researcher in Applied Mechanics, Instituto de Ingeniería, Universidad Nacional Autónoma de México C.U., Av. Universidad 3000, CP 04510, México DF. jperezgavilane@iingen.unam.mx Received his Bachelor degree in Civil Engineering in 1985 and his Master Degree in Structural Engineering in 1987 from the Faculty of Engineering, UNAM. Received his PhD in Computational Mechanics from the London University, UK in 2001. Teaches the postgraduate courses in Advanced Mechanics and the Design of Masonry Structures. He is chair of the Mexico City code for the Design and Construction of Masonry Structures Committee.

O. CARDEL J. Master student, Instituto de Ingeniería, UNAM.

Received his bachelor degree in Civil Engineering in 2011 Universidad Veracruzana, Xalapa, Ver.

Abstract

1.

Overturning moment in confined masonry walls has shown to affect their shear cracking strength. Based on experimental results, a new strength equation was proposed that includes the effect of aspect ratio and moment, increasing cracking strength for squat walls (H/L<1) and reducing it with normalized flexural moment on top the wall β=2Ma/VH when in single curvature. However, the range of β and its correlation to aspect ratio was unknown. Eighteen typical buildings modelled with the Wide Column Method were analysed and the value of β calculated for each wall in their first storey. Its correlation with aspect ratio was then investigated. Results show that cases for which the combined values of aspect ratio and normalized moment that lead to a considerable reduction of cracking strength are not rare, supporting rationale for considering the shearmoment interaction.

Confined masonry (CM) is a construction system in which masonry walls are a built first usually with a running bond arrangement; later, vertical and horizontal reinforced concrete confining elements called tie-columns and tie-beams are cast in place, around the four sides of the walls. Tie-columns and tiebeams are intended to confine the masonry panel, thus enhancing wall deformation capacity and connectivity with other walls and floor diaphragms. Confining elements differ from typical concrete frame structures in that they tend to have smaller cross-sectional dimensions (Meli et al., 2011), making the walls load bearing elements. To increase its shear strength and displacement capacity, horizontal reinforcement may be embedded in the mortar joints and anchored in the tie-columns (Aguilar et al., 1996).

Keywords: Confined masonry, shear strength, shearmoment interaction, curvature, height/length aspect ratio, bending moment, seismic design. The Bridge and Structural Engineer

Introduction

Confined masonry is extensively used in many countries in Latin America, including México, Peru, Argentina and others, the Middle East (Iran, Algeria), some European countries (Italy, Slovenia, Serbia), South Asia (Indonesia) and China. Recently in India Volume 45 Number 1 March 2015  19


a very conscious effort to adapt CM to their design and construction practice is in place and for the first time a major university campus is being constructed using this system (Jain et al. 2014). Observed past performance of CM buildings confirms that well built structures survived major earthquakes without collapse and in most cases without significant damage (Meli et al., 2011, Asinari 2007). Extensive tests have demonstrated that when built and detailed adequately, walls may have reasonable levels of displacement capacity and of shear, flexural and axial resistance (Alcocer, 1995). Nowadays, CM is used for relatively low-rise buildings; in México, up to five storeys and in other countries restricted to one-or two-storey houses (NSR/10 210). However, a policy change in Mexico that favours vertical development of the cities, particularly buildings for housing, is rising the pressure for the design and construction of taller masonry buildings, specially confined masonry buildings, capable of resisting intense earthquake actions. Many aspects of the design of CM in Mexico are now being revised in order to have a more robust code that is not limited to low-rise structures. Recently, the effect of aspect ratio on shear strength of CM walls was studied (Pérez Gavilán et al., 2013) and the shear-moment interaction was investigated through full scaled specimens subjected to constant axial force and reversible cyclic application of shear forces and corresponding bending moments (Manzano et al., 2013). Based on those investigations, a new equation for shear strength design of CM walls was proposed that includes both the effect of aspect ratio, which increases the cracking strength of masonry as the aspect ratio reduces and a reduction in the shear capacity due to increase in the aspect ratio and normalized moment on top of the wall. Normalization of the moment on top of the wall lead to the parameter β=2Ma/VH, where Ma is the moment on top of the wall, V the shear force and H the storey height. Although the effect of normalized moment was well established, the range of β itself is unknown for typical masonry buildings, leaving the question of whether a more complex expression was justified for design. To address this issue, several masonry buildings of the type used in Mexico, especially in Mexico City, were analyzed using the Wide Column Method. Internal 20  Volume 45

Number 1 March 2015

forces and basic geometrical properties were gathered for first storey masonry walls and the parameter β was calculated for each load combination including earthquake loading. The minimum and the maximum values were registered. With the combination of aspect ratio and β, the reduction of cracking strength was estimated. A brief review of the interaction problem is presented and the proposed new equation for the Mexican code explained, showing the expected reductions in the cracking strength due to shear-moment interaction. The results from the analyses are presented next so that a clear idea of the convenience of considering the interaction effect may be drawn.

2.

Background Studies

The Mexican masonry code (NTCM 2004) shear strength equation for CM walls, without horizontal reinforcement is given by Eq. 1 VmR=FR (0.5v* AT+0.3P)≤FR 1.5v* AT (1) where v* is the shear strength of the masonry as obtained from diagonal compression tests, AT is the gross cross-sectional area of the wall including tiecolumns, P is the axial force on the wall and FR is the strength reduction factor. The right most term limits the amount of axial load that may increase the wall strength. The equation was calibrated with many fullscaled tests of walls, usually with a height-to-length (H/L) aspect ratio equal to one and with different types of masonry units and mortar (Alcocer et al., 1995; Meli, 1973). The effect of aspect ratio and the bending moment on top of the wall are not considered. The contribution of horizontal reinforcement will not be discussed here. The format of Eq. (1) is standard, the first term is related to masonry tensile strength capacity and the second term is the added strength due to the axial load in the wall. Unlike many code equations that estimate the maximum strength, Eq. (1) predicts the cracking strength of the wall. This decision allowed for conservative and more consistent estimations, as fewer variables are involved and elastic behaviour may be used to describe the wall behaviour up to the cracking point. For example, the longitudinal reinforcement and shear strength of the tie-columns do not significantly affect the wall’s cracking strength (also known as diagonal tension shear strength). The The Bridge and Structural Engineer


The rightmost term16/9 axial force on the wall and �� is the strength reduction NZS factor. 2004 7/9 of sh Table 1. Masonry code parameters for a linear variation limits the amount of axial load that may increase theUBC wallSDstrength. The equation was7/3 1997 of 4/3 function the aspect ratio. calibrated with many full-scaled tests of walls, usually with a height-to-length (H/L) Code � Table 1.mortar Masonry code parameters for a linear varia aspect ratio equal to one and with different of masonry units and Fortypes single�storey structures, the total moment � � ��, thus the shear span rati MSJCfunction SD 2011 16/9 of the aspect ratio. (Alcocer et al., 1995; Meli, 1973). The effect of aspect ratio and the bending moment �/�� � ��/�� � �/�,that is, equal to the �/� aspect ratio For structur CSAreinforcement 2004 2 of the wall. Code study was performed � on top ofstrength the wallisare not than considered. The contribution horizontal maximum larger the cracking strength More recently, search with severalofstoreys, the span ratio may be interpreted as antoeffective aspe7 NZSshear 2004are 16/9 will not be discussed here. MSJC SD 2011 16/9 ratio �� /� withverify an effective �/�. but only for walls with aspect ratios H/L≤1 (Perez the variation shear strength with UBC SDheight 1997 �of 7/3 aspect ratio 4 � � CSA 2004 2 Gavilan et al., 2013). With a database containing and it arrived at a similar conclusion (Perez Gavilan The format of Eq. (1) is standard, the first term is related to masonryofCM tensile strength NZS 16/9 Based on observations tests made2004 by the several Alvarez (1996) For single�storey structures, totalauthors, moment � �7/3 ��, thus concluded the shear results from wallsterm testedisfrom different countries capacity and many the second the added strength due to the axial load in the wall. UBC SD 1997 et al., 2013). In that study however, the factor was that cracking�/�� strength should increase as the aspect ratiofratio reduces. MoreFo � ��/�� � �/�,that is, equal to the �/� aspect of the wall. Unlike many code equations that estimate maximum Eq. (1) predicts therelated a back-bone model was proposed (Riahithe etrecently,aresearch al., 2009).strength, applied tostoreys, the terms tothe the masonry strength study was performed to verify variation ofinterpreted shear strength with with several the shear span ratio may be as an effe cracking strength of the wall. This decision allowed conservative more For single�storey structures, theGavilan total moment � � ��, thus ratio and itand arrived at an aand similar conclusion (Perez et al., 2013). In that The results indicate that the maximum aspect shear stressfor the contribution due to axial load, as shown in ratio � /� with effective height � � �/�. � � consistent estimations, as fewer variables arestudy involved and elastic behaviour may be to�the �/�� ��/�� �/�,that is, equaltotothe themasonry �/� aspect ratio of t however, theEq.(3) factor � was�applied terms related strength is, to ondescribe average,the 1.3 wall timesbehaviour the cracking strength of the point. used up to the cracking For example, the with several storeys, the shear span ratio may be interpreted and the contribution due to axial load,ofCM as shown in Eq.(3) Based on tests made by several authors, Alvarez (1996) c panels, which is an indication the contribution of longitudinal reinforcement and shearof strength of the tie-columns do observations not significantly ratio �� /� with an effective height �� � �/�. that cracking strength should increase as the aspect ratio reduce affecttie-columns the wall’sand cracking strength they (alsoprovide knownforasthediagonal tension shear the confinement (3) (3) stren � � � �� �� � �� study recently,aresearch was performed to verify the variation of shear � � strength).The maximum strength is larger than the cracking strength but on only for Based observations ofCM tests made by several authors, Alvare seismic performance of the walls. aspect ratio and it arrived at a similar conclusion (Perez Gavilan et al., 2013). walls with aspect ratios �/� � 1 (�erez �avilan et al., 2013). With database strength should increase as the aspect thata cracking rati study however, the factor � was applied to the terms related to the masonry st The shear strength forsquat walls with �/� � 1 was estimated relative to the strength shear model strength for squat withtoH/L<1 wasvariation of sh containing results from many walls testedfrom different countries a The back-bone recently,aresearch study was walls performed verify the and the contribution due to axial load, as shown in Eq.(3) of square walls (�/� � 1).The strength of square walls is well calibrated and can be 3. Aspect Ratio was proposed (Riahi et al.,2009). The results indicate that the maximum shear stress aspect ratio and ittoarrived at a similar conclusion (Perez Gavilan et estimated relative the strength of square wallsbased (H/ on withwhich Eq 1. A indication variation strength was calculated is, on average, 1.3 times the cracking strengthpredicted of the panels, is an ofinthethe study however, factor � was applied to the terms related to the m L=1). The strength square walls well calibrated ( considerations the relative flexural squat and ��of �stiffness �to� �� ��the � is It is generally accepted shear strength of to � �of the contribution of tie-columns andthat the the confinement they related provide for the seismic and the contribution due axial load, ascantilever shown in Eq.(3) square walls, Eq.(4) was developed for estimating � value as follows and can be predicted with Eq 1. A variation in the reinforced masonry walls increases with a decrease in performance of the walls. The shear forsquat with �/� strength was walls calculated based considerations �� � � 1�was � on ���estimated � �� � relative to the stren the aspect ratio. Most codes consider this variation as strength �� � 3� �10� � 1).The (4) 3��10�strength of square walls (�/� of square walls is well calibrated and can 3. Aspect Ratio � related � to� the flexural stiffness of the cantilever a linear function of the shear span ratio M/VL, which with �A�relative �� � predicted Eq 1. variation in the strength was calculated based o 100� � 60� 9 It is generally accepted that the shear strength of reinforced masonry The wallsshear increases strength forsquat walls with was �/� � 1 was estimated to squat and walls, Eq.(4) for relative is issued an effective aspect Theconsiderations general related to square the relative flexural stiffnessdeveloped of the cantilever squat and with a decrease in theas aspect ratio. Most codesratio. consider this variation as a linear of square walls (�/� � 1).The strength of square walls is well calibrated walls, Eq.(4) was developed for�is estimating � valueofaselasticity follows where �an �/�,� � �/�, �is shear modulus, and the modulus estimating f value as function of the shear spanshear ratio�/��,which isused�as effective aspect ratio. The form of the strength equations can besquare stated as predicted with Eq 1. Afollows variation in the strength for was calculated masonry. The expression can be simplified using as a typical value � � 2/10, and general form (MSJC of the shear strength equations can be stated as (MSJC SD 2011, SD 2011, CSA2004, NZS2004, Davis 2008,considerations related to the relative flexural stiffness of the cantilever � �10� � 3��10� � � 3� (4) considering the effective square aspect ratio,as follows SA2004, NZS2004, Davis 2008, Anderson 1992) walls, Eq.(4) for estimating � value as follows (4) � � was�developed Anderson 1992) � � 100� � � 60� � � 9 � � � (2) 1.55 � 0.2�10� � 3��10� �� � 3� (2) � �� � �� � � � �� � �� � �� �1 �� � modulus, where � �� �/�,� �where �/�, �is shear �is the� modulus of elasticity for �� �� �is w=H/L, η=G/E, G and � �shear � 60�modulus � � 9 and E is �� 100� �� masonry. The expression can be simplified using as a typical value � � by (5) � � 2/10, and �1 � 0.69 � 0.2of�elasticity �1.69 masonry affected aeffective where strength Vm is the the modulus e �� is the masonry affected by a strength factor which is a linear of � considering the function aspect ratio,as�follows for masonry. The expression �where � �/�,� � �/�, shear modulus, and �is the modulus of elastici factor which a linear function of shear ratio span ratio �/�� with � andis� constants, �� the strength duespan to � the axial�load ���is using canonbe simplified as a typical value η=2/10, masonry.1 The expression can be simplified using as a typical value � � 2/10 � 1 � all and �� the M/VL contribution reinforcement. values for � � and �toin several strength due with aofand b constants, The Vp the � and considering the effective aspect ratio, as follows considering the effective aspect ratio,as � follows 1.55 0.2 � re shown in Table 1. load on the wall and V the contribution of � the axial � s case of reinforced masonry, Similar to the shear strength ��with a�decrease in �� increases reinforcement. The values for a and (5) 0.2 � � 1 � 0.2 the aspect ratio.b in several codes� � � �1.69 � 0.69 � � � 1.55 � � are shown in Table 1. � �� � �� 1 �� � �1 �1.69 � 0.69� � (5) �1 � 0.2 � � (5) � � �   Table 1: Masonry code parameters for a linear �� � to theofcase shear 1 strength increases � 1with a decrease in variation of shear strength asSimilar a function theof reinforced masonry, � � the aspect ratio.

aspect ratio

Code

MSJC SD 2011

a

16/9

Similar to the case of reinforced masonry, shear strength increases with a decrease in the aspect b ratio. Similar to the case of reinforced masonry, shear

7/9

CSA 2004

2

1

NZS 2004

16/9

7/9

strength increases with a decrease in the aspect ratio.

4.

Shear-Moment Interaction

Table 1 showed that according to most international codes, masonry shear strength increases when the UBC SD 1997 7/3 4/3 4. Shear-Moment Interaction shear span ratio M/VL≤1. Peru's code (E.070 2006) Table 1 showed that according to most international codes, masonry shear stre For single-storey structures, the total moment M=VH, is anthe interesting exception, where design increaseswhen shear span ratio ���� � 1.the �eru�s codeshear (E.0�0 2006) is thus the shear span ratio is M/VL=VH/VL=H/L, that interesting exception, design shearas strengthis strength iswherethe presented in Eq.(6) follows presented in Eq.(6) as follo is, equal to the H/L aspect ratio of the wall. For VL structures with several storeys, the shear span ratio (6) �� � 0.5��� � � 0.23� � � � � 1�3 � � � 1 (6) M may be interpreted as an effective aspect ratio He/L with an effective height He=M/V. The equation was obtained from a numerical et al., study 1992). A total of 15 The equation was obtainedstudy from(Zeballos a numerical were analysed using et theal., Finite Element Method. Thewalls walls were consisted of one, (Zeballos 1992). A total of 15 Based on observations of CM tests made walls by several two,or three panels separated by tie-columns, and the height ranged from one to five analysed using the Finite Element Method. The walls authors, Alvarez (1996) concluded that cracking storeys. The walls were subjected to lateral loads with a triangular variation over wall consisted of one, two, or three panelsshear separated by different strength should increase as the aspect ratio reduces. height. Lateral forces were scaled to ensureequal average stressesfor

wall lengths. The variation of the principal tensile stress is shown in Figure 1, together with the variation of �.It is assumed that the cracking shear strength is directly stress. 45 Number 1 March 2015  21 The Bridge and Structural Engineer associated with the maximum tensileVolume The results showed an increase in tensile stressfor shorter and taller walls. �onsequently,� reduces the wall shear strength for increasing aspect ratios. The effect of wall height refers to increasing bending moment on the wall. It is also evident that the effect of bending moment is more significant in slender walls than in squat walls.


tie-columns and the height ranged from one to five storeys. The walls were subjected to lateral loads with a triangular variation over wall height. Lateral forces were scaled to ensure equal average shear stresses for different wall lengths. The variation of the principal tensile stress is shown in Figure 1, together with the variation of α. It is assumed that the cracking shear strength is directly associated with the maximum tensile stress. The results showed an increase in tensile stress for shorter and taller walls. Consequently, α reduces the wall shear strength for increasing aspect ratios. The effect of wall height refers to increasing bending moment on the wall. It is also evident that the effect of bending moment is more significant in slender walls than in squat walls.

Fig. 1: The effect of tensile stress and aspect ratio: a) maximum tensile stress for walls with one, two and three panels (M1, M2 and M3), points reflect different wall heights and b) a variation of α versus shear span ratio

A possible explanation for the decrease in cracking strength due to the bending moment on top the wall was presented by (Manzano et al., 2013). It was assumed that first tensile cracks in the wall depend on the level of lateral deformation, irrespective of whether the deformation was caused by a lateral force a bending moment or a combination of both. Based on this simple assumption, a cracking strength (Vc) equation was derived and compared with the nominal cracking strength (Vn) calculated using Eq. (1) with FR=1, as shown in Eq. (7) �� 1 � � ����� �� 1 � �

(7)

���� ��

Fig. 2: Predicted shear strength reduction (α) as a fraction of the nominal value for different values of normalized moment β, material η and aspect ratio w: a) α versusw and b) α versus β (Manzano et al., 2013)

When the wall is subjected to double curvature, the cracking strength is expected to increase in comparison to the nominal strength (Eq. 1). An experimental program was carried out to verify validity of Eq. (7) for single curvature walls. The results were in good agreement with the predicted values. Three pairs of walls were tested. In the first and second pair, walls had an aspect ratio H/L=1. The first wall of the pair was tested without moment on top of the wall, while the second wall of the pair was subjected to the maximum predefined bending moment value (Ma). The third pair had H/L=1.54. The axial force was kept constant, while lateral force and flexural moment were applied using cyclic loading protocol. A detailed description of the load sequence is explained in (Manzano et al., 2013). Table 2 shows the predicted and experimental values for shear strength reductions.   Table 2: Shear-moment interaction results Wall

type

H/L (w)

η

Vc kN

Ma kN m

Vc/Vn calc

Vc/Vn exp

M1-1 Clay, solid M1-2 Traditional

1 1

0.299 0.237

143.44 118.61

– 176.52

0.71

0.77

M2-1 M2-2

Clay extruded

1 1

0.142 0.157

174.32 69.81

– 637.43

0.35

0.40

M3-1 M3-2

Clay extruded

1.54 0.111 1.54 0.123

108.38 79.35

166.71 254.97

0.70 0.52 0.74(+)

0.73

Note: 0.66=0.46/0.7 should be equal to 0.73=79.35/108.38

(7) Instead of using the nominal value (Eq. 1), Vn was

taken as the shear strength without bending moment When meansdisplacement that Ma causes a lateralopposite from the first wall of the pair. Both walls M3-1 and that �M a lateral in the direction �� � 0it means a<0 it � causes isplacement produced by the in lateral �. In thatopposite case the wall in double displacement theforce direction to isthe M3-2 were tested with applied bending moment, so re. When� � displacement �1 the wall is produced fixed (rotation onlateral top of force the wall is restrained), by the V. In that the V value is not defined as in the first two pairs; � 0 indicates single curvature.The predictions obtainedusing Eq. (7) are n case the wall is in double curvature. When β = –1 the however, the estimation of the shear reduction for n Figure 2. wall is fixed (rotation on top of the wall is restrained), M3-1 and M3-2 can be calculated. The quotient of while β > 0 indicates single curvature. The predictions the shear strengths obtained experimentally should obtained using Eq. (7) are shown in Figure 2. be equal to the quotient of the estimated reduction of 22  Volume 45

Number 1 March 2015

The Bridge and Structural Engineer


Traditional Clay extruded Clay extruded

1 0.237 118.61 176.52 0.35 0.40 extruded M2-2 1 0.157 69.81 637.43 1 0.142 174.32 1.54 0.111 108.38 166.71 0.70 M3-1 Clay 0.35 0.40 1 M3-2 0.157 0.123 79.35 314.79 0.46 extruded69.81 1.54 637.43 1.54 0.111 108.38 166.71 0.70 0.66(+) 0.73 1.54 0.123 79.35 314.79 0.46 Note: 0.66=0.46/0.7 should be equal to 0.73=79.35/108.38 0.66(+) 0.73 6=0.46/0.7 should be�nstead equal toof 0.73=79.35/108.38 shear strengths forthethose walls. areas therelative to thewithout nominal using nominal valueThose (�q. 1),quotients � was taken shear strength

strength given by Eq. (1), is � bending wall2.of the pair. Both walls M3-1mainly and M3-2 were tested a function of the aspect ratio and the value of reported inmoment the lastfrom rowthe of first Table of using the nominal value (�q. 1), ��moment, was taken as � the shear strength without with applied bending so the � value is not defined as in the first two pairs; normalized moment β. g moment from however, the first wall of the pair.ofBoth wallsreduction M3-1 andfor M3-2 were the estimation the shear M3-1 and tested M3-2 can be calculated.

Shear plied bending5. moment, so Strength the �� value is not Interaction defined as inexperimentally the first two should pairs; be equal to the The quotient of the shearwith strengths obtained To investigate the possible values of the normalized quotient of shear the estimated reduction of shear strengths walls. Those quotients er, the estimation of the reduction for M3-1 and M3-2 canfor bethose calculated. β in real-life structures, eighteen blocks out The results presented in previous sections may be reported in theobtained last row of Table 2. otient of the are shear strengths experimentally should be equal to moment the of eight buildings were analysed. Figure 4 shows the integrated into a single equation to walls. estimate the t of the estimated reduction of shear strengths for those Those quotients 5.row Shear Strength Interaction orted in the last of Table 2. with plan of each building and a few dimensions were cracking strength of CM walls considering the effect The results presented in previous sections may be integrated into a singleequationto to provide information on the plan size. The estimate the cracking strength of CM moment walls considering aspect ratio and ofwith aspect ratio and bending on top the of effect the of added Shear Strength Interaction bending moment on top of the wall. The proposed equation(8) is presented number next of storeys varied from three to six. In some wall. The proposed equation(8) is presented next ults presented in previous sections may be integrated into a singleequationto e the cracking strength of CM walls considering and a building consisted of several blocks and �� the effect �of aspect ratio cases, � � �wall. � � � � �is1.5� � (8) (8) moment on top�of The �proposed presented ��the � ��0.5� � � 0.3��equation(8) � � �� �next each block was analysed separately. Also, in case of ��

repetitive modules in a block, the study was limited to �� inis parenthesis where the in first term is the from nominal where the parenthesis the nominal strength Eq. (1), � is(8)the factor �first term � � � 1.5� ��� � � � ��0.5� � � � 0.3�� � � � � � �� � � � obtained strength from Eq.(5) substituting �/�� .This change carefulfrom only one of those modules. Each block frombyEq. (1), f is�/� the for factor obtained from needsthea walls � of in Figure 4, with the number of storeys explanation, and a few comments are given below. The term �� /�� reflects the is effect shown Eq. (5) by substituting H/L for M/VL. This change the bending on topisofthe thenominal wall, where he first term in moment parenthesis strength from Eq. (1), � is the factor indicated in the brackets. The structural system for a careful explanation and .This a fewchange comments d from Eq.(5)needs by substituting �/� for �/�� needsarea careful consisted of CM and reinforced concrete �� /H � ��reflects 2M of the theeffectallofbuildings below.areThe tion, and a fewgiven comments given below. � /�effect � reflects a The k � term �the (9) �term � � walls. Reinforced concrete walls are identified on the � 3 ding moment on top of the wall, where bending moment on top of� the wall, where plans shown in Figure 4 with a slightly thicker line

� �� � ��/�� are flexural and shear stiffness is a characteristic height,�� � 3��/� 2 �� � �and � and (9) different colour. However, those walls were not �� � andI is moment � of inertia (9) components,� is the shear factor, for the wall section. 3 �� considered in this study. Very long walls were divided

is a ofcharacteristic height, kf=3EI/H Two different effects bending moment were described above,and i) forkwalls with 1, smaller segments for the analysis purpose. v=GA/ κH�/� � into � height,� � 3��/� and ��aspect � ��/�� are flexural and effect shearofstiffness � yaracteristic including in the � factor the effective ratio �/��and ii) the are flexural and shear stiffness components, κ is the shearents,�interaction is the shear andI is moment of inertia for the wall section. oment (�factor, � /�� ). In principle the effect of shear-moment interaction should be shear factorratio. andTo I is moment of inertia wallmoment lied to walls with any aspect avoid considering the effectfor of the bending section. eferent on squat walls (�/� � 1�moment the aspect ratio instead ofabove, the effective aspectwith ratio�/� in �is effects of bending were described i) for walls � 1, It wasin verified (Perezthe Gavilan et al. aspect 2014), that effect of the on uding the � infactor effective ratiothe�/��and ii) interaction the effectterm of sheardifferent of by bending moment with �/� � 1(� isTwo consistent with theeffects effect caused using �/�� in � for were walls with interaction � /�� ). In principle the effect of shear-moment interaction should be described above, for walls with H/L<1, byof including to walls with any aspect ratio. Toi)avoid considering the effect bending moment n squat walls (�/� � 1� the aspect insteadaspect of the effective aspectand ratio in �is in the f factor the ratio effective ratio M/VL was verified in (Perez Gavilan et al. 2014), that the effect of the interaction term on ii) the effect of shear-moment interaction (M /H ). ith �/� � 1 is consistent with the effect caused by using �/�� in �aforkwalls with 3

In principle the effect of shear-moment interaction should be applied to walls with any aspect ratio. To avoid considering the effect of bending moment twice on squat walls (H/L<1) the aspect ratio instead of the effective aspect ratio in f is used. It was verified in (Perez Gavilan et al. 2014), that the effect of the interaction term on walls with H/L<1 is consistent with the effect caused by using M/VL in f for walls with H/L<1.

Figure 3 shows the quotient of nominal shear strength, Vm, evaluated using the new equation (8) and the nominal strength, Vn obtained using Eq. (1). The curves shown in red and blue show the quotient using the new equation (8) for two values of the normalized moment β=0.5 and β=1.0. The case for β=0 is shown with a black line.

6.

Building Analysis

As mentioned in previous sections, the reduction in shear strength caused by shear-moment interaction, The Bridge and Structural Engineer

Fig. 3: Shear strength quotient considering the effects of aspect ratio and shear-moment interaction

Each structure had one or two underground levels for car parking. To leave space for car circulation and parking at underground levels, most walls were not taken down to the foundation level. Instead, they were supported by deep beams at the ground floor level. For the purpose of this study, all walls were considered fixed at the ground floor level. Statically equivalent lateral forces corresponding to the first vibration mode were applied at each storey level in order to create internal forces with proper signs to determine the curvature in the walls. Internal forces and normalized moment β in each wall at the first storey level were calculated for each load combination. The maximum and minimum values of normalized moment were recorded for each wall. Volume 45 Number 1 March 2015  23


The studied buildings were located in different parts of Mexico City. The seismic zone, number of storeys and natural vibration period in each horizontal directions for each block are summarized in Table 3.

Fig. 4: Analysed structures

24â&#x20AC;&#x192; Volume 45

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The Bridge and Structural Engineer


The design acceleration response spectrum is defined in Eq. (10)

ign acceleration The design acceleration response spectrum responseisspectrum defined inisEq.(10) defined in Eq.(10)

7.

Results

The main results are presented in Figure 5, where normalized moment β is plotted against the aspect � � �� � � �� ratio of the walls. The walls of each block studied are (10) �� � � � ���� � � � �� (10) identified with a different colour. (10) � � �� � � �� Walls with lager values of aspect ratio (H/L>1) have a where a is the design pseudo acceleration divided by clear tendency to act in double curvature, as indicated � is the where design � is the pseudoacceleration design pseudoacceleration divided by divided the acceleration by the acceleration of gravity, �of � gravity, �� and T are characteristic the acceleration of gravity, T �� are characteristic are and characteristic periods, � isperiods, known� is as known the seismic as seismic coefficient, � is an by �the is an negative sign of the normalized moment. A a b thecoefficient, nt, and exponent, � is theand modal � isperiod the modal for the period structure. structure. periods, c is known as for thethe seismic coefficient, r is an value of β=–1 indicates that the wall have its upper exponent and T is the modal period for the structure. end restricted for rotation (fixed). Walls with H/ .Characteristics Table 3.Characteristics of building blocksanalysed of building blocksanalysed in this studyin this study L<1 Seismic Block Seismic N �� ��� N�� ��� �� ���Block �� ��� Seismic Block Seismic N �� ��� N�� ����� ��� (squat �� ��� walls) tend to act in single curvature � � � � �� � ���������� � �� � �� � �� �� ��� ��� �� � �� � ��� � ��� � � �

Zone Storey Zone Storey C1 Agricola IIId C1 6 IIId 0.35 6 0.36 C2 Agricola IIId C2 5 IIId 0.32 5 0.27 C3 Agricola IIId C3 6 IIId 0.36 6 0.31 Block 1 Calle IIId5 C1 5 IIId 0.20 5 0.30 2 Calle IIId5 C2 4 IIId 0.29 4 0.17 A GoyaI 4 CA 3 I 0.30 3 0.23 B GoyaI 4 CB Agricola 3 I 0.32 C1 3 0.22 A GoyaI 5 CA 3 I 0.33 3 0.23 C GoyaI 5 CC 3 I 0.29 3 0.23

Zone Storey Zone Storey 0.35Goya 0.36 5 CF 3:Goya I5 CF 3 I 0.29 3 0.22 0.29 0.22analysed in this Table Characteristics of building blocks 0.32 San Angel 0.27 C1 San Angel I C1 4 I 0.32 4 0.37 0.32 0.37 0.36 San Angel 0.31 C2 San Angel I C2 4 I 0.32 4 0.37 0.32 0.37 Seismic NI Storey Ty (s) Block Seismic 0.20 San Angel 0.30 C3 San Angel C3 4 ITx (s) 0.33 4 0.37 0.33 0.37 Zone Zone 0.29 Sirio 0.17 CA Sirio IIIbCA 6 IIIb 0.22 6 0.26 0.22 0.26 0.30 Sirio 0.23 CB Sirio IIIbCB 6 IIIb 0.30 6 0.23 0.30 0.23 0.32 Suiza 0.22 C1 Suiza IIIaC16 5 IIIa0.35 0.37 5 0.28 0.28 5 IIId 0.360.37 Goya I 0.33 Suiza 0.23 C2 Suiza IIIaC2 5 IIIa 0.40 5 0.29 0.40 0.29 CF 0.29 Uxmal 0.23 Uxmal IIIb 4 IIIb 0.19 4 0.14 0.19 0.14

Agricola C2 IIId 5 0.32 0.27 San spectrumparameters Design spectrumparameters for various for zones various in Mexico zonesCityare in Mexico summarized Cityare summarized in Table in Table Angel C1 4.

Agricola C3 IIId 6 0.36 0.31 San Table 4.Design Table spectrum 4.Design spectrum parametersparameters for for Angel C2 seismic zones seismic in Mexico zonesCity in Mexico City �5 �� � Zone Zone �� � �� �IIId � �� �� Calle 5 �C1 0.20 0.30 San I 0.16 I 0.040.16 0.200.04 1.350.20 1.001.35 1.00 Angel C3 II 0.32 II 0.080.32 0.200.08 1.350.20 1.331.35 1.33 IIICalle a 50.4 C2IIIa 0.100.4 0.530.10 IIId1.800.53 2.00 41.80 2.00 0.29 0.17 Sirio CA 0.45IIIb 0.110.45 0.850.11 3.000.85 2.003.00 2.00 IIIb 4 CA 34.20 2.00 0.30 0.23 Sirio CB IIIGoya IIIc 0.100.40 1.250.10I 4.201.25 2.00 0.40 c IIId 0.30IIId 0.100.30 0.850.10 4.200.85 2.004.20 2.00 Goya 4 CB I 3 0.32 0.22 Suiza C1 Results 7. Results Goya 5 CA Inormalized 0.23 Suiza C2 in resultsare The main presented resultsare in presented Figure 5,inwhere Figure 5, where 3normalized moment0.33 � ismoment plotted � is plotted the against aspect ratio the aspect of the ratio walls.The of thewalls walls.The of each walls blockstudied of each blockstudied are identified arewith identified with Goya 5 CC I 3 0.29 0.23 Uxmal ent colour. a different colour.

study

N Storey

Tx (s)

Ty (s)

3

0.29

0.22

I

4

0.32

0.37

I

4

0.32

0.37

I

4

0.33

0.37

IIIb

6

0.22

0.26

IIIb

6

0.30

0.23

IIIa

5

0.37

0.28

IIIa

5

0.40

0.29

IIIb

4

0.19

0.14

with �alls lager with values lager of aspect values ratio of aspect (��� � ratio 1) have (���for a� clear 1)various have tendency a zones cleartotendency act Design spectrum parameters in in to act in curvature, double as curvature, indicatedasbyindicated the negative by thesign negative of the sign normalized of the normalized moment. Amoment. A City are summarized in end Table 4. end for f� � value �1 indicates of � Mexico � �1that indicates the wall that have the its wallupper have its upper restricted restricted rotationfor rotation �alls (fixed). with ��� �alls�with 1 (squat ��� walls) � 1 (squat tend walls) to act in tend single to act curvature. in single curvature.

Table 4: Design spectrum parameters for seismic zones in Mexico City c

a0

Ta

Tb

r

I

0.16

0.04

0.20

1.35

1.00

II

0.32

0.08

0.20

1.35

1.33

IIIa

0.4

0.10

0.53

1.80

2.00

IIIb

0.45

0.11

0.85

3.00

2.00

IIIc

0.40

0.10

1.25

4.20

2.00

IIId

0.30

0.10

0.85

4.20

2.00

Zone

The Bridge and Structural Engineer

Fig. 5: Scatter plot of normalized moment against aspect ratio

Volume 45 Number 1 March 2015  25


with H/L<1. The effect of aspect ratio is included in the nominal strength such that only the effect of interaction is considered. Figure 6 shows the quotient for different β and aspect ratio values. The quotient is equal to Vc/Vn for walls with H/L>1, since f=1 for H/L>1. For example, consider a wall with H/L=0.7 and β=1.5. The reduction in nominal shear strength measured with the quotient is 21.7%; however when β=2, the reduction is 27%.

Fig. 6: Double and single curvature in short and squat walls

Figure 6 illustrates a squat and a short wall coupled by a horizontal floor element. For the squat wall, the horizontal element cannot provide enough restriction to rotation of the wall’s upper end to produce double curvature, while this is possible in case of short walls. This phenomenon can be explained through bending moments. Long squat walls are subjected to large bending moments due to their large stiffness. The direction of those bending moments cannot be reversed by the horizontal floor, but this is possible for shorter walls. For that reason, long squat walls tend to act in single curvature while short walls act in a double curvature. Figure 3 shows that the effect of bending moment tends to be less important as the wall aspect ratio reduces and it tends to be more significant for short walls with higher aspect ratios. However, short walls with H/L≥1.5 tend to be in double curvature, as seen in Figure 5, with few exceptions. These exceptions are characterized by a considerable shear strength reduction due to the presence of bending moment on top of the wall. For example, shear strength reduction for a wall with H/L=1.6 and β=0.5 is 20% and for a wall with H/L=1.70 and β=0.8, the reduction is 28%. Many walls with 1<H/L<1.5 act in single curvature, but in most cases normalized moment β<0.5. The expected shear strength reduction is smaller for these walls; for example, for a wall with H/L=1.2 and β=0.5 the expected reduction is 15.5. Walls with aspect ratio 0.5<H/L<1 are characterized by a wide range of β values. The quotient Vc/(f . Vn) can be used to explain the effect of shear-moment interaction for walls 26  Volume 45

Number 1 March 2015

Fig. 6: Reduction of nominal shear strength due to shearmoment interaction, for different values of normalized moment β and aspect ratio (H/L)

8.

Conclusions

Several buildings designed and constructed in Mexico City were analysed in this study. The aspect ratio, along with the maximum and minimum values of normalized bending moment were recorded for each wall and load combination. The following conclusions can be drawn based on the results of the study: 

The observed range of normalized moment values is -1≤β≤2.5. The observed range is representative of three- to six-storey structures. Larger values can be expected for structures with a larger height (number of storeys). Walls with aspect ratio H/L>1.5 tend to act in double curvature. The implication of this result is that, for those walls, in theory, shear-moment interaction may result in an increase in shear strength. Due to the absence of experimental evidence to confirm such prediction, the design shear strength estimation for this case is conservative, no increase in strength is considered. Most walls with aspect ratios in the range 1<H/ L<1.5 have normalized moments -1≤β≤0.5. When The Bridge and Structural Engineer


the normalized moment β>0, a moderate reduction of shear strength is expected as compared with the nominal value (on the order of 15%). 

Walls with aspect ratios in the range 0.5<H/L<1 are characterized by a wide range of β values. The cases with large values (β>1.5) can have significant shear strength reductions on the order of 20 to 30%.

It appears that the assessment of shear-moment interaction is advisable, although in most cases the effect is moderate (less than 15%). However, cases with large reductions (greater than 25%) are not rare for walls with aspect ratio in the range 0.5<H/L<1. Exceptional cases were found over the whole range of wall aspect ratios where large normalized moments may produce significant shear strength reductions. The assessment of shear-moment interaction is a design tool that may be used to detect such cases.

for Advanced Studies in Reduction of Seismic Risk, ROSE School, Pavia, Italy. 4.

Aguilar, G., R. Meli, R. Diaz, R. Vazquezdel-Mercado (1996), Influence of Horizontal Reinforcement on the Behavior of Confined Masonry Walls, Proceedings of the Eleventh World Conference on Earthquake Engineering, Acapulco, Mexico, Paper no. 1380.

5.

Alcocer, S.M., R. MeIi (1995), Test Program on the Seismic Behavior of Confined Masonry Structures, The Masonry Society Journal, The Masonry Society, 12, 2, 68-76.

6. CSA. 2004. S304.1-04 Design of Masonry Structures. Mississauga, Canadian Standard Association; Ontario, Canada. 7.

Further studies are necessary to determine under what conditions walls may be subjected to large normalized moment (β) values. Preliminary results show that walls perpendicular to the structure’s boundary and having one of its edges in the building’s perimeter tend to have larger values of β.

Alvarez, J.J. (1996) Some Topics of the Seismic Behaviour of Confined Masonry Structures, Proceedings of the 11th World Conference on Earthquake Engineering, Acapulco, México, paper 180.

8.

Manzano, J J Pérez Gavilán, (2013) Shear Moment Interaction in Confined Masonry Walls, Paper 323, 12th Canadian Masonry Symposium Vancouver, British Columbia.

Acknowledgements

9.

Pérez Gavilán, J. J., L. E. Flores, A. Manzano, (2014) A new shear strength design equation for confined masonry walls: proposal to the Mexican code, Tenth U.S. National Conference on Earthquake Engineering, 10 NCEE Anchorage, Alaska, paper 1162.

The second author thanks the National Council for Science and Technology (CONACYT) of Mexico’s scholarship. Many thanks are given to the company Jean Ingenieros and to its director M.I. Raul Jean Perrilliat and Chief engineer Arturo Rodriguez for providing us with all the necessary information of the buildings used in this investigation.

References 1.

Davis, C. L. (2008), Evaluation of Design Provisions for In-plane Shear in Masonry Walls, Master of Science Thesis, Department of Civil Engineering, Washington State University, Washington, USA.

2. Roberto Meli, Svetlana Brzev, et al, (2011), Seismic Design Guide, Earthquake Engineering Research Institute, Oakland, California, USA. 3.

Asinari M.(2007), Buildings with Structural Masonry Walls Connected to Tie-columns and Bond-beams, Master Thesis, European School

The Bridge and Structural Engineer

10. Pérez Gavilán, J. J., L. E. Flores, S. M. Alcocer, (2014), An Experimental Study of Confined Masonry Walls with varying Aspect Ratio, Earthquake Spectra, http://dx.doi. org/10.1193/090712EQS284M 11. Riahi Z., K. J. Elwood and S. Alcocer (2009), Backbone Model for Confined Masonry Walls for Performance-Based Seismic Design, Journal of Structural Engineering, ASCE, DOI: 10.1061/ (ASCE) ST.1943-541X.0000012, 135: 644-654. 12. Sudhir K. Jain, Dhiman Basu, Indrajit Ghosh, Durgesh C. Rai, Svetlana Brzev and Laxmi Kant Bhargava, (2014), Application of confined masonry in a major project in India, Tenth U.S. National Conference on Earthquake Engineering, Anchorage, Alaska. Volume 45 Number 1 March 2015  27


13. Anderson, D. L., & Priestley, M. J., (1992). In plane strength of masonry walls. Proc. 6th Canadian Masonry Symposium, Saskatchewan, Sask., Canada, pp. 223−234. 14. E.070 (2006), “Norma Técnicas E.070 Abañilería”, Servicio Nacional de Normalización Capacitación e Investigación para la Industria de la Construcción (SENSICO), Perú. 15. NTCM (2004), Normas Técnicas Complementarias para el Diseño y Construcción de Estructuras de Mampostería, Gobierno del DF, México 2004. 16. NSR-10 (2010), “Reglamento Colombiano de Construcción Sismo Resistente”, Bogotá, D.C., Colombia, Ministerio de Ambiente, Vivienda y Desarrollo Territorial.

Concrete Masonry Structures”, Wellington: Standards Association of New Zealand. 18. MSJC SD (2011), TMS 402-11/ACI, 530-11/ ASCE 5-11, “Building Code Requirements and Specification for Masonry Structures”, Masonry Standards Joint Committee. 19. UBC SD (1997) “Uniform Building Code”, Whittier, California: International Conference of Building Officials 20. Zeballos, A., A. San Bartolomé y A. Muñoz, (1992), Efectos de la esbeltez sobre la resistencia a fuerza cortante de los muros de albañilería confinada. Analisis por elementos finitos, Blog de Angel San Bartolomé, Pontificia Universidad Católica de Perú, http://blog.pucp. edu.pe/media/688/20070504-Esbeltez%20-%20 Elementos%20 finitos.pdf.

17. NZS (2004), (4230:2004) “Design of Reinforced

28  Volume 45

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The Bridge and Structural Engineer


Use of Confined Masonry for Improved Seismic Safety of Buildings in India

Sudhir K. JAIN Professor and Director, Indian Institute of Technology Gandhinagar, INDIA

Svetlana BRZEV Faculty British Columbia Institute of Technology, Vancouver, CANADA sbrzev@bcit.ca

Durgesh C. RAI Professor Indian Institute of Technology Kanpur, Uttar Pradesh, INDIA dcrai@iitk.ac.in

Prof. Sudhir Jain holds a Bachelor of Engineering degree from the University of Roorkee and Masters and Doctoral degrees from the California Institute of Technology, Pasadena, USA. He has been on the faculty of IIT Kanpur since 1984 and since 2009 he has served as Director of the Indian Institute of Technology Gandhinagar. He was elected to the Board of Directors of the International Association for Earthquake Engineering in 2000 and is currently its President. He was elected Fellow of the Indian National Academy of Engineering in 2003 and was conferred Life Membership by the New Zealand Society for Earthquake Engineering in 2013. He has published more than 150 scholarly papers.

Dr. Brzev holds Bachelor’s and Master’s degrees in Civil Engineering from the University of Belgrade, Serbia and a Ph.D. degree in Earthquake Engineering from the Department of Earthquake Engineering, University of Roorkee. She has 30 years of combined research, consulting and teaching experience in structural and earthquake engineering. She is a member of the Technical Committee CSA S304 responsible for developing Canadian standard on design of masonry structures and serves on the Board of Directors of the US-based Masonry Society. Dr. Brzev serves as the chair of Confined Masonry Network (www.confinedmasonry. org) and is a past Vice-President of the Earthquake Engineering Research Institute.

Dr. Durgesh Rai received a Ph.D. degree from the University of Michigan, Ann Arbor (1996) and prior to joining IIT Kanpur, he was on the faculty of the Department of Earthquake Engineering at IIT Roorkee (1997-2001). His research interests are in design and behaviour of structures under earthquake loads, experimental investigations, supplemental damping, seismic rehabilitation, masonry structures and seismic design codes. He received the 2000 Shah Family Innovation Prize from the Earthquake Engineering Research Institute (USA) and is elected Fellow of Indian National Academy of Engineering (2010). He has published over 130 peerreviewed papers in journals and conferences in the area of structural and earthquake engineering.

skjain@iitgn.ac.in

Summary Good seismic performance of modern confined masonry construction practiced in many countries relies on two key features, namely confinement and bond between masonry walls and reinforced concrete confining elements that enclose these walls. A few initiatives have been recently launched to promote confined masonry construction and revive its application in India based on its proven record of good seismic performance. As a result of these The Bridge and Structural Engineer

initiatives, the first large scale application of modern confined masonry construction in India is currently in progress. Master plan of the permanent campus of Indian Institute of Technology Gandhinagar, a fully residential campus on 400 acres of land, envisaged the construction of 36 confined masonry buildings, including three- and four-storey faculty and staff residences. This paper describes the campus development project, including the design process and the challenges faced during design and construction. Volume 45 Number 1 March 2015  29


Keywords: brick masonry, confined masonry, seismic design, masonry compressive strength, masonry shear stress, masonry modulus of elasticity

1.

Introduction

In many developing countries such as India, significant death toll during earthquakes is caused by poor performance of unreinforced masonry (URM) buildings. Further, quite often reinforced concrete (RC) buildings in such countries leave much to be desired in terms of design and construction quality, making them highly vulnerable to strong ground shaking. In such a scenario, confined masonry can be an effective construction technology in view of the following: a) same materials are used as are prevalent in the country, that is, concrete, masonry and steel and b) it only requires nominal care in design and construction and yet performs very well in earthquakes. In recent years, authors of this paper have been actively engaged in advocating for application of confined masonry in India and other countries. A few initiatives have been taken in India, mostly related to strategic meetings, publications, research studies and training courses. In January 2008, National Information Centre of Earthquake Engineering (NICEE) [1] at the Indian Institute of Technology Kanpur (IITK), with support from the Earthquake Engineering Research Institute (EERI) and the World Seismic Safety Initiative (WSSI), organized an international workshop on confined masonry coordinated by Prof. Sudhir K. Jain, where 19 experts from 8 countries discussed and debated the issues related to applications and promotion of confined masonry in countries where this technology is not practiced. The Workshop resulted in creation of Confined Masonry Network [2], which builds on experience and resources from countries in which confined masonry construction has been practiced (e.g. Blondet [3]). Subsequently, in April 2011 Indian Institute of Technology Gandhinagar (IITGN), together with IIT Kanpur and Buildings and Materials Technology Promotion Council (BMPTC), organized an International Workshop on Confined Masonry. Around 15 invitees from Canada, USA, New Zealand, Peru and India participated in the workshop. More recently, in February 2014 another confined masonry workshop was hosted by the Safety Centre at IIT Gandhinagar. The workshop was co30  Volume 45

Number 1 March 2015

sponsored by the EERI and NICEE and it brought together about 20 experts from India, Mexico, USA and Canada who discussed a path forward for confined masonry in India. As a result of the workshop, a team of masonry experts from India, Mexico and Canada is currently developing a guideline and code provisions for seismic design of engineered confined masonry buildings in India. Several publications have been developed and published in India, mostly through NICEE, to create awareness of and promote application of confined masonry in Indian subcontinent, as discussed by Rai and Jain [4]. In 2005, Dr. S. Brzev spent several weeks at IIT Kanpur to develop a monograph “EarthquakeResistant Confined Masonry Construction” that was published by the NICEE [5] (in English and Hindi). NICEE also published a monograph on confined masonry construction for builders authored by Tom Schacher [6]. BMTPC funded a project at IITK to popularize the use of confined masonry in India, including the development of two Earthquake Tips on confined masonry. Earthquake Tips is an extremely popular NICEE publication authored by Dr. C.V.R. Murty [7] that describes concepts of earthquake-resistant design in a simple, lucid and graphical manner. Dr. C.V.R. Murty has also led the development of a guideline for non-engineered confined masonry construction of social housing in India that was recently published by Gujarat State Disaster Management Authority [8]. A few notable research projects have been undertaken in India since the inception of the confined masonry initiative. A comprehensive experimental study on inplane and out-of-plane seismic response of confined masonry walls was performed by Dr. D.C. Rai and his team at the IITK ([9], [10]) and a study on seismic response of a scaled model of confined masonry building was performed at the Central Building Research Institute, Roorkee [11]. A five-day short course on seismic design of reinforced and confined masonry buildings held at IITGN in February 2014 presented a rational approach for seismic design of engineered confined masonry buildings for the first time in India. The course was attended by about 60 Indian academics, practicing engineers and students. Development of a fully-residential 400-acre academic campus of IITGN in the State of Gujarat, India has provided an excellent opportunity for the first largeThe Bridge and Structural Engineer


scale formal deployment of confined masonry in India. This paper outlines the design and implementation processes related to this project.

2.

Indian Seismic and Construction Scenario

A major part of Indian territory, particularly in northern region close to the Himalayan mountain range, is located in areas of high seismic risk. India is divided into four seismic zones, namely, Zones II through V that are associated with increasing intensity of ground shaking [12]. Approximately 60 % of the land area of the country falls in Zone III or above. A few Indian mega-cities, including Delhi, Mumbai and Calcutta, are located in regions of moderate to high seismic risk. A significant population growth in India resulted in ongoing strong demand for safe and affordable housing throughout the country. Unfortunately, Indian construction practice is largely populated by unskilled workers and most buildings are nonengineered (constructed without input provided by qualified engineers and architects). Furthermore, the mechanisms for building code enforcement are not in place and there is no licensure process for engineers at a level similar to those in developed countries. As a result, human and economic losses in past earthquakes have been unacceptably high. In January 2001, magnitude 7.7 Bhuj earthquake struck the Kutch area of Gujarat and caused huge human and economic losses: death toll was 13,805 and over 167,000 people were injured, while the estimated economic loss was approximately US$ 5 billion. Both older nonengineered masonry dwellings and modern reinforced concrete (RC) apartment buildings were affected by the earthquake. Ahmedabad city (population 5.5 million), located at about 220 km away from the epicentre, experienced shaking intensity VII on MSK scale, which was consistent with Ahmedabad’s location in seismic Zone III. Despite that, 130 RC frame buildings in Ahmedabad collapsed leading to a death toll of 805. All these buildings were "engineered", that is, technical professionals such as architects and structural engineers were involved in their design and construction. Unfortunately, the quality of construction and understanding of the seismic design philosophy might be questionable even though the buildings were constructed in the formal sector. There are significant challenges related The Bridge and Structural Engineer

to RC construction practices in India and many new RC buildings are seismically vulnerable due to inadequate design and/or construction practices. The 2001 Bhuj earthquake has clearly demonstrated the need for inherently robust building technologies that ensure good seismic performance and reduce death toll even when adequate engineering input may not be available [13].

3.

Confined Masonry Construction Technology

Confined masonry provides a viable and seismically safer alternative to poorly built RC buildings and seismically vulnerable URM buildings which are widespread in India. This construction technology has evolved over the last 100 years through an informal process based on its satisfactory performance in past earthquakes. Confined masonry has been practiced in countries and regions of high seismic risk, including Latin America, Mediterranean Europe, Middle East and South Asia. It has been used for both engineered and non-engineered construction and its applications range from one- or two-storey single-family dwellings to six-storey apartment buildings. Engineering design provisions for confined masonry buildings are included in building codes of several countries, including Mexico, Peru, Chile, Eurocode in Europe, etc. Key structural components of a confined masonry building are (see Figure 1a): i) masonry walls transfer both lateral and gravity loads from the floor and roof slabs down to the foundations; ii) horizontal and vertical RC confining elements (tie-beams and tie-columns) - provide confinement to masonry walls and protect them from collapse, even during major earthquakes; iii) RC floor and roof slabs – distribute gravity and lateral load to the walls; iv) RC plinth band - transfers the loads from walls to the foundation system and reduces differential settlement; and v) foundation – transfers the load to the underlying soil. In a confined masonry panel, the masonry wall is constructed first and vertical RC tie-columns are then cast. A storey-high masonry wall panel is usually constructed in two 1.2 to 1.5 m lifts. Once the wall construction is completed up to the full storey soffit level, RC tie-beams are constructed atop the walls and the concrete is cast monolithically with the floor slab (see Figure 2). There are specific rules regarding placement and spacing of RC confining elements in a confined masonry building. For example, RC tiecolumns should be provided at wall intersections, Volume 45 Number 1 March 2015  31


door and window openings, free ends of the walls and at intermediate locations in long walls (usually at a maximum of 4 m spacing), as shown in Figure 1b. Good seismic performance of confined masonry construction can be achieved, provided that a satisfactory quality of construction is ensured. Furthermore, confined masonry construction essentially combines two construction technologies, namely, masonry and RC, which are prevalent in Indian construction practice. It should be noted that masonry construction currently accounts for more than 90 % of the Indian building stock.

4.

IIT Gandhinagar Campus Development Project

Along with few other new universities, IIT Gandhinagar (IITGN) became a part of the IIT system in the academic year 2008-09. Since its inception in 2008, IITGN has been housed in the premises of Vishwakarma Government Engineering College in Chandkheda, Ahmedabad. In July 2012, the Government of Gujarat provided a piece of land (approximate area 163 Hectares) on the banks of Sabarmati River at Palaj village, Gandhinagar District, for setting up the IITGN permanent campus. Campus development plan envisages the fully-residential new campus housing 2,400 students and the associated faculty and staff (Phase 1) and it is eventually expected to host about 6,000 students (see Figure 3). Phase 1A of the campus development (to be completed in mid 2015), comprises the construction of academic buildings, student hostels for 1,200 students, faculty and staff residences and the related infrastructure. The academic area includes about 48,000 m2 builtup area comprising of classrooms, laboratories and offices. There are six four-storey student hostels, with the total built-up area of 36,000 m2 and 30 threestorey buildings with 270 apartments in total housing the faculty and staff (the total built-up area of about 51,000 m2). Student hostels and faculty and staff housing were ideal candidates for the adoption of confined masonry technology, in terms of building height, small room size and a significant amount of walls relative to floor area (wall density ratio). An additional project feature was the use of Fly Ash Lime Gypsum (FALG) bricks for masonry construction. FALG bricks utilized fly ash available from nearby thermal plants and 32â&#x20AC;&#x192; Volume 45

Number 1 March 2015

were chosen as an environmentally sustainable and structurally sound alternative to traditional burnt clay bricks. To supplement the significant brick demand, a plant for manufacturing FALG bricks (capacity 65,000 bricks/day) was set up at the construction site. Since this was the first large-scale field application of confined masonry in India, the project team was faced with several challenges which were successfully resolved during construction. For a detailed overview of confined masonry construction process at the IITGN campus refer to Jain et al. [14], [15].

5.

Design of Confined Masonry Buildings

At the time of design, Indian seismic design standards did not contain provisions related to confined masonry construction, therefore the design team used EERI guidelines for design of low-rise confined masonry buildings [18] and other resources ([5], [6]), with due modifications related to site seismicity and material properties. Design of reinforced concrete components conformed to the IS 456 standard requirements [19]. 5.1 Design Approach Confined masonry is a loadbearing wall structural system, wherein gravity load is sustained by walls only and contribution of RC tie-columns in gravity load sharing was ignored. Allowable compressive stresses for masonry walls were computed considering slenderness effects and load eccentricity per IS 1905. A plan view of a typical confined masonry building block in the IITGN Campus is shown in Figure 4. The block is of irregular shape and was divided into three segments by means of seismic joints. Lateral load resistance is provided by confined masonry walls. It was assumed that shear failure mechanism governs in the wall design. Seismic capacity of a confined masonry building was evaluated through wall density index, which is defined as the ratio of wall area to the floor area of the building. EERI guidelines [18] use the Simplified Method to recommend the minimum wall density ratio based on the type of masonry units, site seismicity, soil conditions and the building height and mass. For this project, masonry units were solid bricks, the site is located in Zone III of India, which could be classified as moderate seismicity (Peak Ground Acceleration less than 0.25g) and the soil is of compacted granular type. For these parameters, the guidelines recommend a minimum wall density The Bridge and Structural Engineer


of 1 and 2% for one- and two-storey buildings, respectively. In line with that, minimum wall density of 1% per storey was used for this project (e.g. 3 and 4 % wall density for three- and four-storey buildings respectively). Only walls confined on all sides were considered for wall density calculations. Note that walls with height/length ratio of 1.5 or higher and walls with large openings were ignored in wall density calculations, hence these walls were not considered to contribute to seismic capacity of the building. Seismic base shear force was calculated according to IS 1893 [12] and subsequently distributed up the building height. Importance factor (I) of 1.0 and the response reduction factor (R) of 2.5 were used in the design. As a reference, R factor of 1.5 is prescribed for unreinforced masonry construction, 3.0 for ordinary RC moment resisting frames and 5.0 for special RC moment resisting frames. Average shear stresses in walls at the ground storey level were calculated by dividing the base shear force by the wall cross-sectional area (same area as used for wall density calculations). Seismic forces in the walls were increased by 15% to account for amplification due to torsional effects. The resulting shear stresses were compared with the allowable shear stress determined through material testing. It was assumed that overturning moment in a confined masonry wall panel is resisted through axial tension and compression in RC tie-columns. 5.2 Masonry Material Properties The project was masonry-intensive and required a significant brick supply on the order of 100,000 bricks/ day. The required material properties were a brick compressive strength of 9 MPa and the corresponding water absorption ratio up to 14 %. A preliminary survey showed that traditional burnt clay bricks manufactured in the Gandhinagar and Ahmedabad districts and nearby areas generally did not possess the required characteristics. This section describes results of an experimental study on mechanical properties of masonry materials performed at IIT Kanpur [19]. Three sets of bricks of two types of units (burnt clay and FALG) manufactured in Ahmedabad were tested to determine their mechanical properties. To determine masonry compressive strength, masonry prisms were fabricated in the laboratory using two different mortar mix proportions: i) 1:1:6 cement: lime: sand mix and ii) 1:4 cement: sand mix. Table 1 The Bridge and Structural Engineer

summarizes the tests performed on three different sets of bricks (one set of burnt clay and two sets of FALG bricks) and relevant testing standards ([20] to [23]). The clay bricks had an average compressive strength of 5.4 MPa, while this value was 5.3 MPa and 9.1 MPa, respectively, for the 1st and 2nd set of FALG bricks. The water absorption value is higher for the FALG bricks at about 25% and should not be used for the below grade masonry work. Burnt clay bricks were preferred due to lower water absorption value of about 15%. The porosity and saturation coefficient is also higher for FALG bricks, which alongwith water absorption provides some indication of durability of units. No significant differences were noticed in the tensile strength of units. The 2nd set of FALG bricks and clay bricks were found to be more suitable for the general masonry work. Table 2 summarizes the test results of masonry assemblages and a set of recommended design values compatible with the Allowable Stress Design Method according to IS 1905 [14] which were used in the design. Figure 5 shows the test setup and specimens for various tests. For clay bricks, the observed average value for prism compressive strength was 3.9 MPa and 3.8 MPa for 1:1:6 and 1:4 mortars, respectively. For 1st set of FALG bricks, these values were 3.03 MPa and 3.57 MPa for 1:1:6 and 1:4 mortars, respectively. For 2nd set of FALG bricks, higher compressive strength values were obtained: 7.3 MPa and 6.8 MPa, for 1:1:6 and 1:4 mortars, respectively. Cement: lime: sand mortar has resulted in higher prism strengths for clay bricks and 2nd set of FALG bricks and was recommended for use in all masonry work. The basic compressive strength for design as per IS 1905 was taken as 0.25 times the prism strength, fm. The reduction factor of 0.25 suggested in IS 1905 adequately accounts for uncertainty in test results, variability in the masonry work and desired level of safety. Using the chord modulus definition for Em, the average values of elastic modulus were computed. As expected, large variation in these values was noted even when expressed as multiples of masonry compressive strength, fm. The average Em/ fm values of clay brick masonry for both mortar types were found to be on the lower side of the typical values (409 for cement: lime: sand and 333 for cement: sand mortar). However, for FALG bricks, the average Em/ Volume 45 Number 1 March 2015â&#x20AC;&#x192; 33


fm values ranged from 707 to 924 for both types of bricks and mortars. Considering uncertainty in test results, the expected elastic modulus for the masonry can be taken as 0.75 times the observed lower bound values irrespective of mortar type for each brick type. Thus recommended Em values were 250 fm for clay brick masonry and 550 fm for FALG brick masonry. The bed-joint shear strength tests primarily provide the evaluation of cohesion and friction factor in a Coulomb friction type of relation. Since no significant differences were noted across various types of masonry investigated, one relation was proposed for all types of masonry. Using a reduction factor of 0.5 to account for uncertainty in results and for general factor of safety, the following expression is recommended for permissible shear stress, vmb, against bed joint shear failure: vmb = 0.1+0.35 σ

5.3 Design and Construction Challenges Considering that this is the first reported application of confined masonry in India, it is no surprise that the project team initially faced several design and construction challenges. The team faced several design challenges, such as: 1.

The architectural team was not familiar with the features of confined masonry buildings in terms of layout and planning. The buildings were originally designed following planning concepts for RC frame construction, therefore the design had to be modified to accommodate confined masonry.

2.

The structural designers were not familiar with analysis and design approaches related to confined masonry structures. This was due to the fact that the civil engineering curriculum at most colleges and universities in India does not include design of masonry buildings.

3.

Areas surrounding staircases were not suitable for confined masonry construction. Therefore, those areas were treated as RC frame systems and were isolated from the adjacent confined masonry construction by means of expansion joints (seismic gaps). Expansion joints were also used in a few buildings with complex plan shapes to create simple rectangular segments and minimize torsional effects.

(MPa)

where σ (MPa) is overburden pressure due to dead loads calculated over net cross-sectional area of the wall. In order to consider other shear failure modes for brick masonry, the permissible masonry shear stress, vm, can be determined from the following expression, which is similar to that prescribed by IS 1905 but modified for the bed-joint strength observed during this investigation: vm = minimum of (0.5 MPa, 0.1+0.35 σ, 0.125√ fm) (MPa) where fm denotes masonry compressive strength (MPa). The masonry tensile bond strength largely depends on the mortar type and the highest values were observed for FALG brick masonry in cement: lime: sand mortar. Adopting a reduction factor of 0.5 to arrive at permissible values, a value of 0.07 MPa was suggested for cement: lime: sand mortar and 0.05 MPa for the cement: sand mortar. However, in case of clay bricks, a value of 0.05 MPa was recommended for both mortar types. Foundations below the plinth level were constructed using conventional masonry, that is, burnt clay bricks in 1:4 cement:sand mortar with the minimum brick compressive strength of 5.0 MPa. FALG bricks in 1:1:6 cement:lime:sand mortar with the specified brick compressive strength of 9.0 MPa were used for 34  Volume 45

above grade masonry construction. The bulk quantity of fly ash was available from the nearby thermal plants and good quality lime was procured from the neighboring Rajasthan state (Hydrated Lime Class ‘C ‘conforming to IS 712 standard [24] in the form of a fine dry powder). Concrete grade M25 (25 MPa characteristic compressive strength) conforming to IS 456 standard [25] was used throughout the project.

Number 1 March 2015

The team also faced the following general construction challenges: 1.

The most significant challenge was explaining the concept of confined masonry to construction workers. It was essential to explain the differences between confined masonry and RC frame construction. RC frame construction is common in India and workers are familiar with it, but it was difficult to teach them a construction The Bridge and Structural Engineer


technology that looks similar and uses the same materials, but structurally behaves quite differently. It was also challenging to explain some confined masonry construction features (e.g. toothing) that are critical for seismic safety of these buildings. To ensure satisfactory construction quality, training camps for masons were organized at the construction site. 2.

3.

Masonry construction is labour-intensive. It was challenging to recruit both the required number of unskilled and skilled labourers. The total required number of workers varied. On average 1,400 workers per day were required continuously for the first eight months. The project required a significant brick supply and locally available burnt clay bricks generally did not meet the project specifications. Therefore, a plant for manufacturing FALG bricks had to be set up at the construction site.

In spite of these challenges, the construction progressed at a satisfactory pace and the quality improved over time. Figure 6 shows student hostel buildings close to the completion.

6.

Acknowledgments This project would not be possible without cooperation and contribution of all members of the design and construction teams, the Central Public Works Department (CPWD) team at the IITGN Permanent Campus site and the financial support by the Government of India.

7.

1. NICEE. National Information Centre of Earthquake Engineering, Indian Institute of Technology, Kanpur, India (www.nicee.org). 2.

EERI. Confined Masonry Network, Earthquake Engineering Research Institute, Oakland, CA (www.confinedmasonry.org).

3.

Blondet, M. (2005). Construction and Maintenance of Masonry Houses for Masons and Craftsmen. Pontificia Universidad Católica del Perú, Lima, Peru (www.world-housing.net/ tutorials).

4.

Rai, D.C. and Jain, S.K. (2010). NICEE's Role in Promoting Confined Masonry as an Appropriate Technology for Building Construction in India. Proceedings of the Ninth U.S. National and Tenth Canadian Conference on Earthquake Engineering, Paper No. 1689, Toronto, Canada.

5.

Brzev, S. (2008). Earthquake-Resistant Confined Masonry Construction. National Information Centre of Earthquake Engineering, Kanpur, India (www.nicee.org).

6.

Schacher, T. (2009). Confined Masonry for One and Two Storey Buildings in Low-Tech Environments - A Guidebook for Technicians and Artisans, National Information Centre of Earthquake Engineering: Kanpur, India (www. nicee.org).

7.

Murty, C.V.R. (2005). Earthquake Tips Learning Earthquake Design and Construction. Indian Institute of Technology Kanpur and Building Materials and Technology Promotion Council, National Information Centre of Earthquake Engineering, India (www.nicee. org).

8.

Murty, C.V.R. et al. (2013). Build a Safe House with Confined Masonry. Gujarat State Disaster

Concluding Remarks

The ultimate goal of the confined masonry initiative is to develop technical expertise required for design of safe and durable masonry buildings in India through educational activities and field applications. The first large-scale field application of confined masonry construction in India is hoped to popularize confined masonry in India and have a far-reaching impact on the construction industry in the country. It is acknowledged that a broader application of confined masonry is unlikely unless engineering codes and handbooks on structural and seismic design of this technology are available. A team of masonry experts from India, Mexico and Canada is currently developing code provisions and companion guidelines for seismic analysis and design of engineered confined masonry buildings in India, which are expected to facilitate its broader application in the country. Finally, a comprehensive educational initiative is needed to incorporate fundamentals of confined masonry design and construction into engineering and architectural programs at Indian colleges and universities and continuing education courses for practicing engineers and architects. The Bridge and Structural Engineer

References

Volume 45 Number 1 March 2015  35


Management Authority, Government of Gujarat, Gandhinagar, Gujarat, India. 9.

Komaraneni S, Rai DC, Singhal V. (2011). Seismic Behavior of Framed Masonry Panels with Prior Damage When Subjected to Out-ofPlane Loading. Earthquake Spectra, 27 (4), pp. 1077–1103.

10. Singhal, V. IS 712. and Rai, D.C. (2014). Role of Toothing on In-Plane and Out-Of-Plane Behavior of Confined Masonry Walls, Journal of Structural Engineering, American Society of Civil Engineering, 140(9), pp. 850-865. 11. Chourasia A. (2013). Influential Aspects on Seismic Performance of Confined Masonry Construction. Natural Science, 5 (8A1), pp. 5662. 12. BIS (2007). Indian Standard - Criteria for Earthquake-Resistant Design of Structures - Part 1, IS 1893 (Part 1):2002, Fifth Revision, Bureau of Indian Standards, New Delhi, India. 13. Jain, S.K. (2005). The Indian Earthquake Problem, Current Science, 89(9), pp. 1464-1466. 14. Jain, S.K., Basu, D., Ghosh, I., Rai, D.C., Brzev, S. and Bhargava, L.K. (2014). Application of Confined Masonry in a Major Project in India, Proceedings of the 10th National Conference on Earthquake Engineering, Earthquake Engineering Research Institute, Anchorage, AK, USA. 15. Jain, S.K., Brzev, S., Bhargava, L.K., Basu, D., Ghosh, I., Rai, D.C., and Ghaisas, K.V. (2015). First Confined Masonry for Residential Construction, Indian Institute of Technology Gandhinagar, India. 16. BIS (1987). Indian Standard - Code of Practice for Structural Use of Unreinforced Masonry, IS 1905:1987, Bureau of Indian Standards, New Delhi, India (reaffirmed in 2002).

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17. BIS (2013). Indian Standard – Earthquake Resistant Design and Construction of Buildings – Code of Practice (Third Revision), IS 4326:2013, Bureau of Indian Standards: New Delhi, India. 18. EERI (2011). Seismic Design Guide for LowRise Confined Masonry Buildings. Earthquake Engineering Research Institute and International Association for Earthquake Engineering: Oakland, CA, USA (www.confinedmasonry. org). 19. BIS (2000). Indian Standard - Plain and Reinforced Concrete – Code of Practice, IS 456:2000, Bureau of Indian Standards, New Delhi, India. 20. Rai, D.C. (2013). Report on Evaluation of Brick Masonry Design Parameters for IITGN Buildings, Department of Civil Engineering, Indian Institute of Technology Kanpur, India. 21. BIS (1984). Indian Standard - Specification for Building Limes, IS 712:1984, Bureau of Indian Standards, New Delhi, India. 22. BIS (1992). Indian Standard- Common Burnt Clay Building Bricks- Specification, IS 1077: 1992, Bureau of Indian Standards, New Delhi, India. 23. BIS (1992). Indian Standard -Methods of Tests of Burnt Clay Building Bricks, IS 3495:1992, Part 1-Determination of Compressive Strength, Part 2-Determination of Water Absorption, Part 3-Determination of Efflorescence, Bureau of Indian Standards, New Delhi, India. 24. ASTM Standard (2012). Standard Test Methods for Sampling and Testing Brick and Structural Clay Tile, ASTM C67-12, ASTM International, West Conshohocken, PA. 25. Khalaf, F.M. (2005). New Test for Determination of Masonry Tensile Strength, J. Mater. Civ. Eng., 17 (6), 725-732.

The Bridge and Structural Engineer


Table 1: Summary of tests performed on bricks Test

Dimensions (mm) Real density (g/cc) Apparent density (g/cc) Porosity (%) Water absorption (%) Water absorption (C) (%) Water absorption (B) (%) Saturation coefficient (C/B ratio) IRA (kg/min/m2) Comp. strength (MPa) Tensile Strength Flat Position (MPa) On Edge Efflorescence

Test Method No. of tests

IS 1077 :1992 IS 3495 : 1992 (1) ASTM C67-12 ASTM C67-12 ASTM C67-12 ASTM C 67-12 IS 3495 : 1992 (2) ASTM C 1006-07 ASTM C 1006-07 IS 3495 : 1992 (3)

20 5 5 5 10 5 5 5 10 10 5 5 5

Clay Bricks Avg. Values [%COV] 225 x 105 x 73 2.30 [0.7] 1.74 [1.0] 24.43 [3.8] 14.98 [6.6] 14.03 [4.7] 17.16 [4.1] 0.82 [1.1] 4.36 [18.3] 5.40 [17.7] 0.27 [20.3] 0.49 [24.2] Nil

FALG Bricks FALG Bricks (1st Set) (2nd Set) Avg. Values Avg. Values [%COV] [%COV] 226 x 100 x 76 227 x 108 x 78 2.49 [0.4] 2.59 [3.5] 1.52 [4.4] 1.60 [1.3] 38.93 [7.0] 38.26 [7.2] 24.38 [8.8] 25.41 [6.9] 25.68 [11.1] 23.94 [8.3] 28.21 [11.5] 27.26 [8.8] 0.91 [1.5] 0.88 [1.0] 4.44 [16.3] 1.59 [18.3] 5.30 [24.2] 9.10 [7.6] 0.43 [13.9] 0.44 [24.8] 0.45 [14.2] 0.68 [11.9] Nil Nil

Table 2: Masonry Material Properties Masonry Type

Clay bricks in 1:1:6 cement: lime: sand mortar Clay bricks in 1:4 cement: sand mortar Fly ash bricks (1st set) in 1:1:6 mortar Fly ash bricks (1st set) in 1:4 mortar Fly ash bricks (2nd set) in 1:1:6 mortar Fly ash bricks (2nd set) in 1:4 mortar

(a)

Compressive Strength of Masonry fm (MPa) 3.9 3.8 3.0 3.6 7.6 6.8

Basic Elastic Compressive Modulus of Stress Masonry (MPa) Em (MPa) 0.97 250 fm 0.95 0.76 0.89 550 fm 1.83 1.71

Permissible Shear Stress

Tensile Strength

vm (MPa)

(MPa)

Minimum of the following: a) 0.5 MPa, b) 0.1+0.35σ c) 0.125√ fm

0.05 0.07 0.05 0.07 0.05

(b)

Fig. 1: Confined masonry building: a) key components and b) a sample floor plan showing RC tie-column layout (Brzev, 2008)

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(a)

(b)

Fig. 2: Construction details: a) wall construction in progress and b) formwork for RC tie-columns (photos taken at the IIT Gandhinagar Palaj Campus)

Fig. 3: Master plan of the IITGN permanent campus, Palaj Village (HCP, 2014)

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The Bridge and Structural Engineer


Fig. 4: Floor plan of a typical confined masonry building block, IITGN Palaj Campus (Source: Vastu Shilpa Consultants)

(b)

(a)

(c)

Fig. 5: Tests on masonry assemblages: a) a 5-brick prism compression test; b) bed-joint shear strength test and c) tensile bond strength test

Fig. 6: Student hostels at the IITGN Palaj Campus nearing completion (October 2014)

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Volume 45 Number 1 March 2015â&#x20AC;&#x192; 39


SEISMIC PROVISIONS FOR BRIDGES IN INDIAN STANDARDS- PAST PRACTICES AND RECENT DEVELOPMENTS

A K BANERJEE Former Member (Technical), National Highways Authority of India ak_banerjee@hotmail.com

Alok BHOWMICK Managing Director, B&S Engineering Consultants Pvt. Ltd., Delhi bsec.ab@gmail.com

Mr. A. K. Banerjee graduated in Civil Engineering from Calcutta University in 1963 and later did his post graduation from IIT, Delhi. After a brief stint of two years in West Bengal State PWD, he joined Min. of Road Transport & Highways in 1965 and rose through various ranks to became Chief Engineer in 1997. In 2002, he joined NHAI as Member (Tech) and retired from this post in 2003. Since then, he had been associated with the Consulting Firms for more than a decade and is currently associated as Advisor to a Private Construction / Concession Company. During his entire career, Mr. Banerjee has been responsible for planning, design and supervision of several major road and bridge projects, as also repair and rehabilitation of some major bridges in the country. He has been a Member of various Technical Committees of IRC, including Bridges Specifications & Standards (BSS) Committee and is also the Convenor of Loads & Stresses Committee dealing with IRC:6. He is also a Member of the Managing Committee and Executive Committee of ING-IABSE.

Mr. Alok Bhowmick, born 1959, graduated in Civil Engineering from Delhi University in 1981 and did his post graduation from IIT, Delhi in 1992. Mr Bhowmick has made significant contributions in the field of structural engineering both within and outside his organization by sharing his expertise and experience. He is an active member of several technical committees of Indian Roads Congress, including Bridge Specification & Standards (BSS) Committee. He is Vice Chairman of ING-IABSE and Chairman of the Editorial Board for the journal "The Bridge and Structural Engineer" published by ING-IABSE. He is also a Fellow and Governing Council member of Consulting Engineers Association of India (CEAI) as well as Indian Association of Structural Engineers (IAStructE).

Abstract Evolution of seismic design provisions in Indian Standards over the last fifty years have been reviewed in this paper with an emphasis to the estimation of design seismic force. BIS code (IS 1893), Indian Roads Congress (IRC) standard for highway bridges Indian Railway Standards (IRS) and RDSO Guidelines for railway bridges are compared. Design parameters for comparison include the zone factor / peak ground acceleration, importance factor, local soil condition, design spectra and response reduction factor. The paper also highlights the ongoing work being carried out by the IRC code committees in further improving 40â&#x20AC;&#x192; Volume 45

Number 1 March 2015

the seismic provisions of the code. Recent changes adopted in the seismic design of railway bridges are also compared with the requirement for similar road bridges.

1.

Introduction

Based on geological and tectonic setting, Indian subcontinent is divided into three distinct parts, namely, Himalayan region, Indo-Gangetic Plain and Peninsular India. While the seismic catalog is available reporting the past seismic events triggered in Indian continent, the database is neither rich nor reliable before 18th century. Table 1 presents some The Bridge and Structural Engineer


of the major events and one may count seven events greater than magnitude 8 in last 200 years. This statistics is enough, perhaps in loose sense, to argue about the strong seismicity of Indian subcontinent. Over the years, dissemination of research output at the national and international levels and increasing awareness of socio-economic loss in some of the major events triggered in Indian subcontinent and elsewhere, have resulted in guidelines and standards for seismic design. Two essential components of the state of the art seismic design are the estimation of seismic hazard consistent with the design life and socio-economic importance of structure and its design against multiple performance objectives. Indian seismic design codes have not yet reached the international standard. This paper is aimed to discuss the evolution of seismic design provisions in Indian standards, especially, that related to the bridges. Bureau of Indian Standards (BIS) code on earthquake design, IS 1893, provides general seismic design requirements and guidelines, whereas other standards are also available and enforced in the seismic design of bridges, for example, standards published from Indian Road Congress (IRC-6) and Indian Railways (IRS Bridge Rules). IS 1893 was first published in 1962 and subsequently revised in 1966, 1970, 1975, 1984 and 2002 (BIS, 1962; 1966; 1970; 1975; 1984; 2002;). In its fifth revision, IS 1893 was split into five parts and to date, only three parts have been published (namely Part-1(1), Part-3 and Part-4). The other two parts dealing with (a) Liquid retaining tanks and (b) dams and embankments are yet to be published. The first IRC bridge code (i.e. IRC: 6) was published in December, 1958(2) and reprinted in 1962 and 1963. The second revision of IRC:6 (1964 edition)(3) included all the amendments, additions and alterations made by the Bridges Specifications and Standards (BSS) Committee. Third revision of the Code (in metric units) was published in 1966(4). Though a number of amendments were introduced in between, it took nearly 34 years to bring out the fourth revision in the year 2000. Fifth revision of the Code was published in the year 2010(5) and the revised edition of the Code, in its present form, was brought out in the year 2014(6). The first publication of ‘Bridge Rules’ specifying the design seismic load for railway bridges was published by Indian Railways in the year 1941(7). This was revised in 1964 and subsequently 48 correction slips The Bridge and Structural Engineer

have been issued by RDSO till date. IITK-RDSO jointly published a guideline on seismic design of Railway Bridges in November 2010(8). Indian Railways felt the need for bringing out a guideline which can substitute the outdated IS:1893-1984(9) for bridges and therefore came out with this guideline in the interim. This guideline was recently revised by RDSO (January, 2015). There are considerable differences in the provisions of this new RDSO guideline for Railway Bridges and those in IS:1893 (Part 3) published by BIS. This paper first briefly discusses the evolution of IS 1893 over the years. Next, evolution of seismic design provisions in other national standards (i.e. IRC, IRS, RDSO) exclusively for bridges is summarized. Finally, the paper concludes with the issues currently under discussion and expected to appear in next revision of the standards.

2

Evolution of Indian Standard IS 1893

IS 1893, ‘Recommendations for earthquake resistant design of structures’, was released in 1962 and revised first time in 1966. The entire country was divided into seven seismic zones (0 to VI) and associated maps are presented in Figure 1 (a and b). Seismic zoning was based on the isoseismals of the past earthquakes. Intensity measured in MMI scale less than or equal to V was assigned to zone 0; VI to 1; and so on. Koyna region, which at that time was located in zone 1, experienced an M6.5 earthquake with maximum shaking intensity VIII in MMI scale. This incident brought out a significant revision of seismic zone map that was included in IS 1893 in 1970. In this revision, zone 0 was merged with zone-I while zone-VI was merged with zone-V (Figure 1c). Also included in 1970 revision of IS 1893 was the more rational approach for design of buildings and substructures of bridges. Third revision of the standard was brought out in 1975 with a few changes including, a) zone factors on a more rational basis; b) importance factors; c) building’s flexibility in design force calculation; d) modal combination rule; and e) new clauses for hydro-dynamic force. Clauses on concrete and masonry dams were also modified. Based on the experience gained in 14 years, the fourth revision was published in 1984. The revision included the concept of performance factor depending on the structural framing system and on the ductility of construction Volume 45 Number 1 March 2015  41


for buildings, even though the performance factor was not introduced for other structures such as bridges and water tanks. 1993 Latur earthquake (M6.4 with maximum intensity IX in MMI scale) occurred in Zone I and thereby requiring significant revision of seismic zoning map. Fifth revision of IS 1893 was brought out in 2002 and it was decided to split up the standard into five parts in order to enable more frequent changes in the code. Part 1 contains provisions that are general in nature and applicable to all structures. Also, it contains provisions that are specific to buildings only. Part 2 presents the requirements related to the liquid retaining tanks both types, namely, elevated and ground supported. Part 3 is to discuss the requirements related to the bridges and retaining walls. Seismic resilience of industrial and stack like structures are addressed in Part 4. Part 5 is considered for dams and embankments. Unless stated otherwise, the provisions in Parts 2 to 5 should be read necessarily in conjunction with the general provisions in Part 1. Further, pending finalization of Parts 2 to 5 of IS 1893, provisions of Part 1 would be read along with the relevant clauses of IS 1893 : 1984 for structures other than buildings. The following are the major and important modifications made in the fifth revision released in 2002: a) The seismic zone map is revised with only four zones, instead of five (Figure 1d). Erstwhile Zone I has been merged to Zone II, b) Seismic zone factors have been changed with more realistic values of effective peak ground acceleration considering Maximum Considered Earthquake (MCE) in each seismic zone, c) Response spectra are now specified for three types of founding strata, namely rock and hard soil, medium soil and soft soil, d) Empirical expression for estimating the fundamental natural period of multi-storeyed buildings has been revised, e) The ‘response reduction factor’ is included in place of the earlier performance factor, f) Design base shear is restricted to a minimum through empirically specifying the natural period, g) The soil-foundation system factor is dropped by restricting the differential settlements in severe seismic zones and h) Provision of torsional eccentricity has been modified.

3.

Summary of IS 1893 Provisions for Bridges

A brief comparison of the seismic design provisions 42  Volume 45

Number 1 March 2015

of bridges specified in IS 1893 (Part 3)-2014(14) and 1984 version of the IS 1893 is present below. 1.

Scope of the work and seismic zoning

1984 version did not elaborate the scope and addressed five seismic zones (I-V). Horizontal seismic coefficients recommended were 0.01, 0.02, 0.04, 0.05 and 0.08 for the Zone I-V, respectively. Recent version presents a well-defined scope that includes highways, railways, pedestrian bridges and aqueducts. Zone I and II are clubbed and zone factors are specified as 0.10, 0.16, 0.24 and 0.36 for Zone II –V, respectively. 2. Applicability 1984 version recommends that i) Slab, pipe and box culverts need not be designed for seismic forces in any seismic zone; ii) Masonry and plain concrete arch bridges with span more than 10 m shall not be built in Zone IV and Zone V; iii) Bridges of total length not more than 60 m and individual span not more than 15 m need not be designed for earthquake forces, other than in Zone IV and V. Recent version introduces following additional points: i) Concept of ‘Regular’ Bridges and ‘Special and Irregular’ Bridges; ii) Site specific spectra for bridges with span greater than 150 m and for special bridges (where soil conditions are poor) in seismic zone IV and V. 3.

Method of analysis

1984 version provided two analysis methods, namely, seismic coefficient method and response spectrum method of analysis. Parameters involved in the estimation of design seismic force includes Basic Seismic Coefficient (αo), Seismic Zone Factor (Fo), Importance Factor (I) and soilfoundation factor (β). Response spectrum analysis was recommended only for bridges with complex geometry such as, suspension bridge, bascule bridge, cable stayed bridge, horizontally curved girder bridge and reinforced concrete arch or steel arch bridge. Response spectrum analysis was also required for bridges with pier height more than 30 m and span length more than 120 m. Table 2 presents the recommended values for soil-foundation factor. Recent version recommends two additional methods of analysis, namely, time history and pushover. Further, the soil-foundation factor is dropped and the effect of soil is accounted for in the design response The Bridge and Structural Engineer


spectrum. Detailed dynamic studies are envisaged for all special and major bridges. Empirical formula for calculating natural period is revised. Importance factor remains the same whereas response reduction factor is introduced. Effect of damping is also included in the design response spectrum (for example, 2%, 5% and 10% of critical). Capacity design principle is recommended for the design of connection, substructure and foundation. 4.

Live load consideration

with 30% of seismic force in other two orthogonal direction. 8.

Seismic detailing

No special detailing for seismic loading was recommended in the 1984 version whereas the recent version included the followings: i) Provision for antidislodging of girders; ii) Use of vertical holding down device, stoppers, restrainers and horizontal linkage elements and iii) Provisions for minimum seating width of superstructure over substructure.

1984 version recommended 50% live load for seismic load calculation in case of railway bridges and 25% for highway bridges. However, gravity load calculation should include 100% live load regardless of the case. Recent version considers 20% live load for the highway bridges and no live load for the rural roads and 30% live load for railway bridges without impact.

Only cohesion-less soil was considered in the 1984 version, which is extended to incorporate C-Ф soil in the recent version.

4.

Development of Seismic Provisions in IRC Code

5.

1.

First IRC Bridge Code IRC:6-1958 (Section-II)

Vertical Seismic Force

Both versions recommend consideration of vertical seismic force in Zone IV and V. 1984 version specified a seismic coefficient which is half of that for the horizontal ground motion. In other words, design spectra for the vertical ground motion is half of that for the horizontal ground motion. This factor is modified to 0.67 in the recent version, which also imposes a separate calculation of natural time period specific to the vertical vibration. 6.

Ductility and overstrength

1984 version did not account for the ductility and overstrength in the seismic design of bridges. Recent version has brought out these explicitly by introducing the response reduction factor, ‘R’. Different 'R' factors have been proposed for different components of the bridge, depending upon the redundancy, expected ductility and over-strength in them. ‘R’ factor is recommended to be ‘1’ for the design of foundation and 0.8 for the design of bearing. 7.

Directional combination

Simultaneous application of vertical and unidirectional horizontal seismic force was recommended in the 1984 version of the code whereas, recent version recommends the 100-30-30 rule for directional combination, in which, 100% seismic force in any particular direction is to be considered simultaneously The Bridge and Structural Engineer

9.

Earth pressure

The first IRC bridge code covering seismic provisions was published in the year 1958. Outline of the first draft was originally published in 1946 in the form of a paper(10) and the paper was discussed at the Indian Roads Congress session at Jaipur held in 1946. Clause 222 of this code pertained to ‘Seismic Force’ and the zone map as shown in Figure 2 was introduced. The subcontinent was divided into three seismic zones based on expected seismic intensity or degree of damage, namely, liable to minor damage or nil, liable to moderate damage and liable to severe damage. The code did not enforce seismic design in the first zone whereas, recommended a design seismic force as 5% and 10% of the seismic weight (dead load and (full) live load) in the second and third zones, respectively. Also identified is a set of isolated pockets (clause 222.3), namely, the Epicentral tracts, wherein the design seismic force was left to the discretion of structural designer. Note that there was a sense of deciding the site-specific design force in epicentral tracts and this may be considered as a precursor of site-specific seismic hazard / spectra. Clause 222.4 recommended the direction of seismic force to be such that maximized the resultant stresses in the member under consideration. This may be considered as the first step of accounting for the directional combination, such as 100-30 rule. Further, seismic and wind forces were not recommended to Volume 45 Number 1 March 2015  43


be considered simultaneously. Another important recommendation specified in this early version of IRC code was not to reduce the seismic force on account of the buoyancy, which recognized that seismic force depends on mass (and not weight) of the element. Nevertheless, effect of flexibility of structure is not identified in seismic design. 2.

Second and Third Revision of IRC: 6 -1964 & 1966

Second and Third revision of the IRC: 6 were published in the years 1964 and 1966, respectively. There was no modification in seismic provisions in these revisions of the code (except for change in unit system from FPS to Metric) and the same regional seismic map of 1958 was continued. Recall that IS 1893 was released in 1962 and revised first time in 1966. Associated seismic zone maps, as presented in Figures 1a and 1b, may be compared with that of IRC (Figure 2). Note the seven seismic zones in IS 1893 against three zones in IRC code. Clearly, the IRC code committee did not take cognizance of work on seismic zone map over the years. In 1980, the seismic zoning map of IS: 1893 was introduced for the first time in foundation code IRC:78-1980(11). This was later reproduced in IRC: 6 in September 1981 reprint. Apart from zone map, also adopted are the importance factor, soil-foundation coefficient etc. For example, horizontal seismic coefficients recommended are 0.01, 0.02, 0.04, 0.05 and 0.08 for the Zone I-V, respectively. Table 2 presents the adopted factor to account for the variability in soilfoundation systems. Important factor is considered as 1.5 for important bridges and 1.0 for all other cases. 3.

Fourth Revision of IRC:6 -2000 and Amendment in 2003

There was no modification in seismic provision in this revision of the code, when it was first published in the year 2000. However after the Bhuj Earthquake in January, 2001 followed by the publication of IS:1893 (Part-1) – 2002 with significant upward revision in the seismic provisions, IRC code was revised and upgraded in line with the BIS Code. Broadly, following were the major changes made in the Seismic Clause, introduced in 2003 as interim provisions, as compared to the previous version of the Code: a) The seismic zone map was revised in 44  Volume 45

Number 1 March 2015

line with IS: 1893 (Part1) – 2002 with seismic zone factors (PGA) introduced, in line with BIS code [PGA (in g) of 0.1, 0.16, 0.24 and 0.36 are considered for the Zone II-V, respectively]; b) Design force calculation was based on spectral acceleration, which accounted for the flexibility of the structure; c) Response spectrum is specified for three types of founding strata (i,e. Rock or Hard Soil, Medium Soil and Soft Soil; see Table 3 for details); d) Response reduction factor was introduced but assumed to take 2.5 regardless of the member; e) Ductile detailing was recommended in line with the provisions of IS 13920; and f) Also recommended are the provisions of Seismic stoppers, STU, Base Isolation, Seismic Fuse etc. 4.

Fifth Revision of IRC: 6 - 2010

Fifth Revision of IRC: 6 was brought out in the year 2010, wherein the following changes were incorporated: a) Response reduction factors were rationalized and made variable (ranging from 1.0 to 4.0) depending upon the type of substructure and bearings (Table 4); b) Criteria are outlined so as to ascertain the importance of bridges and Importance factor was rationalized based on different categories, for example, Normal, Important and Large critical bridges etc. (Table 5); c) Directional combination is specified through 100-30-30 rule; d) Reduced live load was suggested during the seismic event; and e) Provisions are made in calculation of spectral ordinate for damping other than 5%. 5.

Sixth Revision of IRC: 6 - 2014

Consequent upon the publication of Limit State Concrete Code IRC: 112 in the year 2011 and drafting of IS: 1893 Part 3 for Bridges undertaken by BIS, there was a need to further revise the Fifth Revision of IRC: 6 based on limit state approach and taking into account the related development in IRC:112, which was largely influenced by Eurocode. For example, the design philosophy of Eurocode is to achieve (with appropriate reliability) the non-collapse requirement, under ULS and IRC decided to follow the same philosophy and hence, requiring a revisit of the safety margin under seismic code. The revised edition of the Code, i.e. the Sixth Revision published in January, 2014 incorporated following amendments to the Fifth Revision of IRC: 6: a) Response Spectrum is modified in line with the IS:1893 spectra; b) Response The Bridge and Structural Engineer


Reduction Factors further rationalized (Table 6); c) Cracked moment of inertia is allowed for time period calculations in case of RCC structures considering the limit state design approach; d) Reference to IRC 112 is given in place of IS: 13920 for ductile detailing; e) Seismic load for design of foundation should be taken as 1.35 (1.25) times the force transmitted by RCC (steel) substructure.

5.

Development of Seismic Clauses in Railway Codes

1.

First Publication of Bridge Rules: 1941(7)

The first Railway Code, Bridge Rules, was published in the year 1941 and the seismic provision was laid down in Clause 27: “Seismic load need only be taken into account if the local conditions require it and the allowance for horizontal acceleration shall depend on such local conditions but shall not exceed one-eighth of gravity. Seismic load need not be allowed for in standard designs. Wind and seismic loads shall not be assumed to act simultaneously on a structure.” 2.

First Revision of Bridge Rules in 1964

First revision of Bridge Rule was published in the year 1964 and seismic provisions were recommended in Clause 2.12 that has two sub-clauses. “Sub-Clause 2.12.1: Seismic load need only be taken into account if the local conditions require it and the allowance for horizontal acceleration shall depend on such local conditions but shall not exceed 0.12 of gravity. Seismic load need not be allowed in standard designs except in cases given in Clause 2.12.2. Sub-Clause 2.12.2: In normal cases seismic effect need not be combined with live load; but where local conditions require it, a combination of seismic effect, live load, longitudinal forces and other co-existent forces should be taken in the design. Zone map is not recommended to calculate design seismic force in both original and the first revision of Bridge Rule. 3.

Incorporating Correction Slips

Bridge Rules has not been revised since 1964. However, a number of correction slips have been incorporated as interim provisions in subsequent The Bridge and Structural Engineer

reprints. Below is a summary of the reprints since the first revisions. Reprinted in 1986 (Incorporating Correction Slips 1-15) Seismic zoning map from IS:1893 was used for the first time in IRS code. Basic horizontal seismic coefficient was considered as 0.01, 0.02, 0.04, 0.05 and 0.08. Correction Slip No 28 dated July, 2002 Seismic forces to be considered in case of bridges of overall length more than 60 m or spans more than 15 m in Zone I, II and III and regardless of span if located in Zone IV and V. It is worth noting here that while IS:1893 (Part 1):2002 was published in June 2002, with significant changes in seismic zoning map and peak ground acceleration, the Railway Standards continued with the old practice of specifying seismic coefficient. Correction Slip No 42 dated October, 2009 Number of seismic zones was changed from five to four conforming to IS:1893 (Part 1):2002 but seismic coefficient method was continued. Correction Slip No 44 dated February, 2014 Seismically preferred and not preferred configurations of bridge are included. Also included are the provisions for vertical holding down device, horizontal linkages and minimum seating width requirements. 4.

IITK-RDSO Guideline published in 2010(8)

This guideline was developed for Indian Railways at IIT Kanpur and was published in the year 2010. This was substantially different in design philosophy and in design approach as compared to the prevailing IRS code at that time. The guideline included commentary of the codal clauses as well as worked out examples. Substantive changes included: a) Seismic Zones and Response Spectrum per IS:1893 (Part-1):2002; b) Flexibility of the Bridge depending upon natural period; c) Response Reduction Factor to allow inelastic behavior; d) Directional combination; e) Introducing detailed analysis such as Response Spectrum, Time History and Pushover; f) Assessment of liquefaction; g) Provisions for seismic isolation devices; and h) Post earthquake operations and inspection. Response Volume 45 Number 1 March 2015  45


Reduction Factors in this guideline was much lower as compared to the ‘R’ factor given in IRC:6 prevailing at the time of publication of this document (i,e. in 2010), leading to conservative design forces. This guideline was implemented in several projects by Delhi Metro Rail Corporation (DMRC) in metro projects, subsequent to its publication in 2010. 5.

Revised RDSO Guideline published in 2015(13)

RDSO has come out with the revised guideline in January, 2015 wherein the R factors were modified, as it was felt by RDSO that the design forces on structure as per IITK-RDSO guideline were too conservative. Also the live load factors in load combination were changed in the amended guideline. With the issuance of this new guideline, the earlier IITK-RDSO guideline stood withdrawn. Interesting to note that, by the time the new RDSO guideline is published, IRC:6 revised the ‘R’ factors in the code by lowering the values (thereby increasing the seismic forces). Around the same time, IS:1893 (Part 3) : 2014 appears with ‘R’ factors much lower than even the IITK-RDSO guidelines.

6.

Comparison of Design Seismic Force

Two river bridges with natural periods 0.5 sec and 1 sec, located in Delhi on medium soil (Type II) are considered with an importance factor of 1.5. RC circular pier on well foundation is chosen in both cases. Design seismic force (fraction of seismic weight) is calculated using different versions of IS 1893, different versions of IRC codes and the railway bridge provisions and presented in Table 7. It may be recalled that until recently all three bridge codes (IS code, IRC code and the RDSO provisions) did not account for flexibility of the bridge in calculating design seismic force. Hence, the seismic coefficient as per IS code ranged from 0.05 in 1962 to 0.09 in 1984. With consideration of bridge flexibility in 2015 version, the coefficient is 0.12 for T=0.5 and is 0.065 for T=1.0. The coefficient for the IRC was around 0.10 until the year 2000 when it became 0.18 for the rigid structure (T=0.5) and 0.098 for the bridge with T=1.0 sec. In later versions (2010 and 2014) of IRC, the coefficient has reduced considerably: 0.09 for rigid structure (T=0.5 sec) and 0.049 sec for bridge with T=1.0 sec. The coefficient for railway bridges in early years was to not exceed about 0.12; it became 0.09 in 1986. The IITK-RDSO guidelines provided 46  Volume 45

Number 1 March 2015

coefficient of 0.18 for the rigid (T=0.5 sec) structure and 0.098 for the bridge with natural period T=1.0 sec. The more recent RDSO Guidelines provides coefficient of 0.113 for rigid (T=0.5 sec) structure and 0.061 for bridge with T=1.0 sec. It is obvious that after the Indian bridge codes started to consider flexibility of bridge in computing design seismic force (something that is very logical and is done worldwide for a long period), the design force for the more flexible structures has reduced considerably. Further, there has been quite a bit of variation in the design force in different versions in the recent years; this is a reflection of the fact that the code committees are trying to balance the need for safety and rationality on one side and avoiding significant increase in design force on the other side. It may however be mentioned that design seismic coefficient is only one aspect of the seismic design. The actual design, both in terms of safety and the cost, depends not only on the design coefficient but on many other factors, such as a) method of design (limit state versus permissible stress design), b) the load factors and permissible stresses used in the design and c) how natural period is calculated (for instance, if moment of inertia is being calculated based on cracked section of the reinforced concrete member, consideration of live load), etc.

7.

The Way Forward and Work in Progress in IRC

Seismic provisions in BIS, IRC and IRS codes need to be further updated. The seismic provisions in IRC Bridge Code Section II (IRC: 6) relevant to the design of road bridges have undergone substantial changes recently to align these with the Limit State Method of design adopted in the IRC Concrete Bridge Code IRC: 112-2011. Clause No. 219 in the latest IRC: 6-2014 is a somewhat shorter version of the much detailed seismic provisions (Longer Version) being developed incorporating the modern thinking on the subject with scope for much detailed analysis to assess the strength of material and ductility. A few amendments in the seismic clause no. 219 in IRC: 6-2014 are also being made to further clarify / amplify the provisions, which have since been approved by IRC Council and the same has been printed in May 2015 edition of the journal ‘Indian Highways’ published by IRC. The Bridge and Structural Engineer


Longer Version of Seismic Clause, which might be brought out as “Guidelines for Seismic Design of Highway Bridges” in line with similar AASHTO publication is one of the present mandates of the IRC committee. Towards this, Guidelines are proposed comprising of various chapters on (a) Design Philosophy; (b) Design Methods; (c) Detailing etc., besides introducing an Explanatory Handbook for use by designers.

8.

of practice for road bridges (Section II) (second revision) - loads & stresses’, Indian Roads Congress, New Delhi 4.

IRC-6 (1966). ‘Standard specification and code of practice for road bridges (Section II) (third revision) - loads & stresses’, Indian Roads Congress, New Delhi

5.

IRC-6 (2010).‘Standard specification and code of practice for road bridges (section ii) (fifth revision) - loads & stresses’, Indian Roads Congress, New Delhi

6.

IRC-6 (2014). ‘Standard specification and code of practice for road bridges (Section II) - loads & stresses (revised edition)’, Indian Roads Congress, New Delhi

7.

BRIDGE RULES (1941). ‘Bridge Rules’, Ministry of Railways, New Delhi

8.

IITK-RDSO GUIDELINES (2010). ‘IITK RDSO guidelines on seismic design of railway bridges: provisions with commentary and explanatory examples’, RDSO, Lucknow.

9.

IS 1893 (1984)‘Criteria for earthquake resistant design of structures (fourth revision)’, Bureau of Indian Standards, New Delhi

Conclusions

Seismic provisions of first IRC Bridge Code, published in the year 1958 have undergone significant modifications during the last five decades. Continuous revision of the Seismic Clauses in IRC: 6 has been necessary to bring up the same at par with International Standards and also conforming to the Limit State Method of design adopted in the IRC Concrete Bridge Code IRC: 112-2011 and relevant BIS publication IS: 1893 – Part 3. The major aspects covered in the latest IRC: 6-2014 pertain to inclusion of appropriate map of seismic zones, as per IS 1893, introduction of Response Reduction factor, Importance Factors, Limit State Method of design and provision for Ductile Detailing etc. IRC is expected to finalize the Longer Version of Seismic Clause in the form of detailed Guidelines in near future. Evolution of seismic provisions for railway bridges since its first publication (1941) and first revision (1964) to publication of RDSO guidelines (2015) for seismic design of railway bridges has been substantial. As things stand today, there are significant variations in seismic provisions for bridges in IS 1893, IRC 6 and Railway codes, which needs to be narrowed down.

9.

References

1.

IS 1893-Part 1 (2002).‘Criteria for earthquake resistant design of structures: Part-1,general provisions and buildings (fifth revision)’, Bureau of Indian Standards, New Delhi

2.

IRC-6 (1958).‘Standard specification and code of practice for road bridges (Section II) - loads & stresses’, Indian Roads Congress, New Delhi.

3.

IRC-6 (1964). ‘Standard specification and code

The Bridge and Structural Engineer

10. Paper No.112 (1946). ‘Standard specification and code of practice for road bridges (Section II)’, Journal of IRC, Vol X. 11. IRC-78 (1980). ‘Standard specification and code of practice for road bridges (Section VII) (First Published) – Foundations & Substructure’, Indian Roads Congress, New Delhi 12. BRIDGE RULES (1964). ‘Bridge Rules’, Ministry of Railways, New Delhi 13. RDSO GUIDELINES (2015). ‘RDSO Guidelines on seismic design of railway bridges, published by bridge & structures directorate’, RDSO, Lucknow. 14. IS 1893-Part 3 (2014). ‘Criteria for Earthquake Resistant Design of Structures; Part-3 Bridges & Retaining Walls’, Bureau of Indian Standards, New Delhi

Volume 45 Number 1 March 2015  47


Table 1: Significant earthquakes in and around India S. No.

Epicentre

Date

Lat

(oN)

Region

Long (oE)

Magnitude

1

1819 Jun 16

24.0

70.0

KUTCH, GUJARAT

8.0

2

1869 Jan 10

24.5

92.5

NEAR CACHAR, ASSAM

7.5

3

1885 May 30

34.1

74.8

SOPORE, JAMMU & KASHMIR

7.0

4

1897 Jun 12

25.9

91.0

SHILLONG PLATEAU

8.7

5

1905 Apr 4

32.3

76.3

KANGRA, HIMACHAL PRADESH

8.0

6

1918 JUL 8

24.5

91.0

SRIMANGAL, ASSAM

7.6

7

1930 Jul 3

25.8

90.2

DHUBRI, ASSAM

7.1

8

1934 Jan 15

26.6

86.8

BIHAR-NEPAL BORDER

8.3

9

1941 Jun 26

12.4

92.5

ANDAMAN ISLANDS

8.1

10

1943 Oct 23

26.8

94.0

ASSAM

7.2

11

1950 Aug 15

28.5

96.7

ARUNACHAL PRADESH-CHINA BORDER

8.5

12

1956 Jul 21

23.3

70.2

ANJAR, GUJARAT

7.0

13

1967 Dec 11

17.4

73.7

KOYNA, MAHARASHTRA

6.5

14

1975 Jan 19

32.4

78.5

KINNAUR, HIMACHAL PRADESH

6.2

15

1988 Aug 6

25.1

95.1

MANIPUR-MYANMAR BORDER

6.6

16

1988 Aug 21

26.7

86.6

BIHAR-NEPAL BORDER

6.4

17

1991 Oct 20

30.7

78.9

UTTARKASHI, UTTRAKHAND

6.6

18

1993 Sep 30

18.1

76.6

LATUR-OSMANABAD, MAHRASHTRA

6.3

19

1997 May 22

23.1

80.1

JABALPUR, M.P

6.0

20

1999 Mar 29

30.4

79.4

CHAMOLI, UTTARAKHAND

6.8

21

2001 Jan 26

23.4

70.3

BHUJ, GUJARAT

7.7

22

2004 Dec 26

3.3

96.1

OFF WEST COAST OF SUMATRA

9.3

23

2005 Oct 8

34.5

73.1

MUZAFFARABAD

7.6

24

2011 Sep 18

27.8

88.1

SIKKIM-NEPAL BORDER

6.9

Table 2: Soil-foundation system factor as per IS 1893- 1970, 1975 and 1984 Soil Type

Values of β for Bearing Piles Friction Piles, Resting on Type Combined or Isolated I Soils or Raft RCC Footings with Tie Foundations Beams

Isolated RCC Footing without Tie Beams or Unreinforced Strip Foundations

Well Foundations

I – Hard Soils

1.0

1.0

1.0

1.0

II – Medium Soils

1.0

1.0

1.2

1.2

III – Soft Soils

1.0

1.2

1.5

1.5

48  Volume 45

Number 1 March 2015

The Bridge and Structural Engineer


Table 3: Effect of soil conditions on spectral accelerations as per IS 1893 (Part-1)-2002 For Type I Rocky or Hard Soil Sites: These are the soil having standard penetration value, N > 30.

For Type II Medium Soil Sites: These are the soils with 10 < N < 30.

For Type III Soft Soil Sites: These are all soils with N < 10.

Table 4: Response reduction factor as per IRC-6: 2010 Bridge Component

R with ductile detailing

R without ductile detailing

Superstructure

N.A

2.0

Substructure (i) Masonry/PCC piers, abutments (ii) RCC short plate piers where plastic hinge cannot develop in direction of length and RCC abutments (iii) RCC long piers where hinges can develop (iv) Column (v) Beams of RCC portal frames supporting bearings

– 3.0 4.0 4.0 1.0 2.0

1.0 2.5 3.3 3.3 1.0 2.0

Bearings Connectors and Stoppers (Reaction blocks) Those restraining dislodgement or drifting away of bridge elements.

When connectors and stoppers are designed to withstand seismic forces primarily, R value shall be taken as 1.0 When connectors and stoppers are designed as additional safety measures in the event of failure of bearings, R value specified in Table 8 for appropriate substructure shall be adopted.

Table 5: Importance factor as per IRC 6: 2010 Seismic Class

Illustrative Examples

Normal bridges

All bridges except those mentioned in other classes

Important bridges

a) b) c) d)

Large critical bridges in all Seismic a) Zones b)

Importance Factor ‘I’ 1

River bridges and flyovers inside cities Bridges on National and State Highways Bridges serving traffic near ports and other centers of economic activities Bridges crossing Railway lines

1.2

Long bridge more than 1km length across perennial rivers and creeks Bridge for which alternative routes are not available

1.5

NOTE: While checking for seismic effects during construction, the importance factor of 1.0 should be considered for all bridges in all zones.

The Bridge and Structural Engineer

Volume 45 Number 1 March 2015  49


Table 6: Response reduction factor as per IRC-6: 2014 Sl. Bridge Components No.

‘R’ with Ductile Detailing 2.0

‘R’ without Ductile Detailing (for Bridges in Zone II only) 1.0

1.0 1.0

1.0 1.0

3.0

2.5

3.0

2.5

Column

4.0

3.0

RCC beam

3.0

2.0

PSC beam

1.0

1.0

Steel Framed Construction

3.0

2.5

Steel Cantilever Pier

1.5

1.0

1.0 1.0

1.0 1.0

1

Superstructure of integral / Semi integral bridge / Framed bridges Other types of Superstructure, including precast segmental constructions Superstructure

2

(i) Masonry / PCC Piers, Abutments (ii) RCC wall piers and abutments in longitudinal direction (where hinges can develop) (iii) RCC wall piers and abutments in direction (where hinges can develop) (iv) RCC Single Column (v) RCC/PSC Frames

3 4

Bearings and Construction (see note v also) Stoppers (Reduction Blocks) Those restraining dislodgement or drifting away of bridge elements. (see Note (vi) also)

Notes below Table

(i) Those parts of the structural elements of foundations which are not in contract with soil and transferring load to it, are treated as part of sub-structure element. (ii) Response reduction factor is not to be applied for calculation on displacements of elements of bridge and for bridge as a whole. (iii) When elastomeric bearings are used to transmit horizontal seismic forces, the response reduction factor (R) shall be taken as 1.5 for RCC substructure and as 1.0 for masonry and PCC substructure. (iv) Ductile detailing is mandatory for piers of bridges located in seismic zones III, IV and V when adopted for bridges in seismic zone II, for which “R value with ductile detailing” as given in Table 9 shall be used. (v) Bearings and connections shall be designed to resist the lesser of the following forces, i.e., (a) design seismic forces obtained by using the response reduction factors given in Table 9 and (b) forces developed due to over strength moment when hinge is formed in the substructure (vi) When connectors and stoppers are designed as additional safety measures in the event of failure of bearings, R value specified in Table 9 for appropriate substructure shall be adopted.

Table 7: Variation in seismic coefficient (Ah) for design of bridges as per various Indian seismic codes

(considering a river bridge located in Delhi, in medium soil (Type II), supported on well foundation, with RC circular pier)

Eq. seismic coefficient, Ah Code / Guideline Salient features of the code Time period Time period T = 0.5 sec T = 1.0 sec BIS Code : IS 1893 (applicable for highway / railway bridges) 1 IS 1893-1962 0.050 0.050 First BIS Code; Seven seismic zones in the map Ah= 0.05 for Delhi. Delhi was in Zone III in 1962 2 IS 1893-1966 0.060 0.060 Zoning map revised. Ah = 0.06 for Delhi, which is in Zone IV 3 IS 1893-1970 0.060 0.060 Seismic zoning map revised. Five seismic zones in the map. Soil foundation factor introduced α0= 0.05, β = 1.2, Ah = 0.06

50  Volume 45

Number 1 March 2015

The Bridge and Structural Engineer


4

IS 1893-1975 & 1984

0.090

0.090

5

IS 1893 (Part 3)2015

0.120

0.065

1

2

Response spectrum method and Importance factor introduced in code. α0= 0.05, β = 1.2, I = 1.5, Ah = 0.09 Recently published code with very stringent provisions. Z = 0.24, I = 1.2, Sa/g = 2.5 (1.36) for T = 0.5 (1.0) sec. respectively, R = 3.0 for Reinforced Concrete pier with ductile detailing cantilever type, wall type;; Ah,0.5 = 0.120; Ah,1.0 = 0.065

IRC code : IRC 6 (applicable for highway bridges only) IRC 6 - 1958, 0.100 0.100 Published before BIS Code in 1958. Country divided 1964, 1966 (first & into 4 regions for seismic vulnerability. Subsequent second revision) revisions of code in 1964 & 1966 did not take cognisance of seismic provisions of BIS code, until 1979. Ah = g/10 = 0.1 IRC 6 - 1966 (with 0.090 0.090 Zone factor, Importance factor and Soil factor amendment in introduced in line with prevailing IS 1893; α= 0.05, 1981) I= 1.5, β = 1.2

3

IRC 6 - 2000 (fourth revision)

0.090

0.090

4

IRC 6 - 2000 (fourth revision with amendment in 2003)

0.180

0.098

5

IRC 6 - 2010 (fifth revision)

0.09

0.049

6

IRC 6 - 2014 (revised edition with amendments up to January 2015)

0.090

0.049

Ah = 0.09 (considering medium soil with well foundation) no change in seismic provisions of the code after 1981 Amendments brought in line with provisions of IS:1893 (Part I) - 2002 Z = 0.24, I = 1.5, Sa/g = 2.5 (1.36) for T = 0.5 (1.0) sec. respectively, R = 2.5 Ah,0.5 = 0.18 ; Ah,1.0 = 0.098 'R' factors and I.F are rationalised; Additional margin kept for foundation design (=1.25) to cover for the possible higher forces transmitted by substructure arising out of its overstrength. Z = 0.24, I = 1.2, Sa/g = 2.5 (1.36) for T = 0.5 (1.0) sec. respectively, R = 4 for RCC long piers with ductile detailing where hinges can develop; Ah,0.5 = 0.09 ; Ah,1.0 = 0.049 Z = 0.24, I= 1.2, Sa/g = 2.5 (1.36) for T = 0.5 (1.0) sec. respectively, R = 4 for RCC long piers with ductile detailing where hinges can develop ; Ah,0.5 = 0.09 ; Ah,1.0 = 0.049

IRS Code and IITK/RDSO Guideline / RDSO Guideline (applicable for railway bridges only) 1

Bridge Rules, 1941

2

Bridge Rules, 1964

3

Bridge Rules, 1986

Not exceeding 0.125 Not exceeding 0.12 0.09

The Bridge and Structural Engineer

Not exceeding 0.125 Not exceeding 0.12 0.09

Zone map was not recommended

Zone map was not recommended

Response spectrum method and Importance factor introduced in code. α0= 0.05, β = 1.2, I = 1.5 Ah = 0.09 Volume 45 Number 1 March 2015  51


4.

IITK/RDSO Guideline - 2010 (published in November 2010)

0.18

0.098

5

RDSO Guideline - 2015 (published in January 2015)

0.113

0.061

IS 1893: 1962

IS 1893: 1970

Published at the behest of RDSO, necessitated due to absence of BIS code on bridges since publication of IS 1893 (part I) - 2002 with revised seismic map and revised zone factors. Z = 0.24, I = 1.5, Sa/g = 2.5 (1.36) for T = 0.5 (1.0) sec. respectively, R = 2.5 for RCC piers with ductile detailing â&#x20AC;&#x201C; Single Column; Ah,0.5 = 0.18 ; Ah,1.0 = 0.098 RDSO modified the IITK/RDSO Guideline and diluted the 'R' factors in line with the IRC codes. Z = 0.24, I = 1.5, Sa/g = 2.5 (1.36) for T = 0.5 (1.0) sec. respectively, R = 4 for long RCC Piers where hinges can develop; Ah,0.5 = 0.113 ; Ah,1.0 = 0.061

IS 1893: 1966

IS 1893: 1984

Fig. 1: Evolution of seismic zone map in IS 1893

Fig. 2: First ever seismic zone map of India: documented in IRC-6: 1958

52â&#x20AC;&#x192; Volume 45

Number 1 March 2015

The Bridge and Structural Engineer


Cyclic Performance of Asymmetric Friction Connections with Grade 10.9 Bolts Jamaledin BORZOUIE Jamaledin Borzouie received his Civil Engineering B.S. from IKIU in 2007. He graduated from the International Institute of Earthquake Engineering (IIEES) with a Master of Earthquake Engineering in 2009. Then he joint to the school retrofitting office of Iran and worked as a structural engineer for 3 years. He started his PhD about “Low Damage Steel Base Connections” at the University of Canterbury in 2012.

Gregory MACRAE Gregory MacRae received his Bachelors degree in 1984 and PhD in 1989 from the University of Canterbury. He has worked as a design and research engineer in New Zealand, Japan and Seattle and has spent shorter times at Stanford, National Taiwan University and IIT Gandhinagar. He is a member of NZ loadings and steel code committees, is the seismic and a senior advisor to the board of the World Seismic Safety Initiative and NZ representative to the International Association of Earthquake Engineering.

J.G. CHASE Professor Chase received his Mechanical Engineering B.S. from Case Western Reserve University in 1986. His M.S. and PhD were at Stanford University in 1991 and 1996. He spent 6 years at General Motors and a further 5 years in Silicon Valley, including Xerox PARC, GN ReSound, Hughes Space and Communications and Infineon Technologies, before coming to the University of Canterbury in 2000. Research interests include: control, physiological systems, structural dynamics and vibrations and modeling. Dr. Chase has over 1000 journal and conference papers and 13 patents. He is also the Director of the UC Mechatronics degree program, co-founder of two venture funded start ups and a Fellow of the Royal Society of NZ (FRSNZ) and IPENZ (FIPENZ).

Geoff RODGERS Geoff Rodgers graduated with a Bachelor of Engineering (Honours) in 2005 and a PhD in Mechanical Engineering from the University of Canterbury in Christchurch, New Zealand in 2009. He then completed a postdoc in biomedical engineering through the University of Otago Christchurch medical school. He is now a senior lecturer in Mechanical engineering at the University of Canterbury. His research interest are in damage-free structural design and energy dissipation devices, structural control and health monitoring.

Robin XIE Robin Xie received his Bachelor degree in civil engineering in 2012 from University of Canterbury. He is currently completing a Masters degree in Earthquake engineering at University of Canterbury under the supervision of Associate Professor Gregory MacRae. His topic of research is Low Damage Braces using Friction Connections.

The Bridge and Structural Engineer

Volume 45 Number 1 March 2015  53


Jose Chanchi GOLONDRINO Jose Chanchi Golondrino received his Civil Engineering B.S. from University of Cauca-Colombia in 1997, and his Master degree from University of Andes-Colombia in 2002. He has been lecturing several structures subjects since 2004 to date at National University of Colombia, where he is Associate Professor in the Civil Engineering Department. Currently he is completing his PhD at University of Canterbury - New Zealand under the supervision of Professor Gregory MacRae and Professor Geoffrey Chase. His research topics involve friction connections, application of friction connections as dissipaters in different structural systems, and low damage dissipaters for seismic areas. jcchanchigo@unal.edu.co Charles CLIFTON University of Auckland, New Zealand Charles Clifton graduated from the University of Canterbury with a Batchelor of Civil Engineering (Hons) in 1978 and a Master of Civil Engineering in 1979. From 1979 to 1981 he worked for a major New Zealand consulting engineering firm, then from 1981 to 1983 for a joint UK/Saudi Arabian consulting engineering firm in London. In 1983 Charles joined the New Zealand Heavy Engineering Research Association (HERA) first as Structural Engineer then as Senior Structural Engineer. A long and productive collaboration with Auckland University while at HERA saw several innovations researched, developed and adopted by the profession and also the award of his PhD in 2005. Since 2008 he has been at the Dept of Civil and Environmental Engineering, University of Auckland, specialising in structural steel and composite engineering.

Abstract A number of structures around the world are using friction connections, rather than relying on yielding, to dissipate energy during earthquake shaking. The resulting structures sustain relatively low damage and only the bolts and shims associated with the sliding mechanism, may require replacement. This paper describes experimental testing of asymmetric friction connections (AFCs) using 2 M16 Grade 10.9 bolts with higher strength, but less ductility, compared to Grade 8.8 bolts. It is shown that the Grade 10.9 bolts increased the sliding force byup to 1.5 times. However, the higher localized stress on the sliding surfaces due to the Grade 10.9 bolts caused separation of material from the surface, which further increased the tensile force in the AFC bolts. For sliding displacements greater than~50 mm some Grade 10.9 bolts fractured. Recommendations are made for plate minimum thickness to prevent excessive localized bearing stress and minimum threads length within the grip length to mitigate the possibility of fracture.

1.

Introduction

Most building design standards aim to preserve life, while allowing significant structural damage, non54  Volume 45

Number 1 March 2015

structural damage and economic loss during design level earthquake shaking. The minimum requirements of such standards result in “damage-prone” buildings that may require replacement after a major event. Some newer buildings are being designed using “lowdamage” techniques. Such buildings are expected to experience lower levels of damage so that they can be returned to use very soon after the shaking stops. A low damage structure can be obtained by designing the building to remain elastic, by appropriately using seismic isolators, or other special energy absorbing elements that sustain very little damage or are easily replaced. Some of these energy absorbing devices, which have been the topic of significant research, are friction devices, yielding devices and lead dissipative devices. Research and experimental tests on lowdamage construction in steel frame structures has concentrated on beam-to-column moment resisting joints [Christopoulos et al. (2002), Clifton (2005), Christopoulos et al. (2002), MacRae et al. (2010), Rodgers et al. (2007), Rodgers et al. (2012), Mander et al. (2009)], braces [Chanchí et al. (2014)) and base connections (Borzouie et al. (2014)] for moment frames, braced frames or rocking frames. Examples of application include those reported by Gledhill et al. (2008) and Latham et al. (2013). The Bridge and Structural Engineer


Slotted Plate

High Strength Bolts F

F Shims Cap Plate

Fig. 1: AFC configuration

The sliding force is dependent on both the shim-steel plate friction coefficient and the normal force on the surfaces. The normal forces are controlled by the bolt tension force. Bolt strength and deformation capacity affect the ability of the bolt to provide reliable normal forces over large sliding displacements. Chanchí et al. (2012a) indicate that the most reliable friction sliding may be obtained if the hardnesses of the materials either side of the sliding interface are very different. For this reason, high hardness shims are generally used. Previous tests at the University of Canterbury and Auckland used Grade 8.8 bolts to provide interface clamping forces [Khoo et al. (2012), Chanchí et al. (2012b)]. In the NZ steel standard (Clause 4.2.4.1.2c in NZS3404: 2009), the length of threaded bolt within the grip length is required to be at least 5 threads for a bolt length up to and including 4 times the bolt diameter, 10 threads for a bolt length exceeding 8 times the bolt diameter and 7 threads for intermediate bolt lengths. This requirement, which is not in many overseas codes, was developed based on bolt fractures during proof-loading on construction sites. The Bridge and Structural Engineer

They were not developed considering any sliding friction issues. It was considered that using higher strength bolts with greater clamping forces may be desirable because they result in more economical sliding connections. Such connections would require a lower number and/ or size of bolt, which in turn would also reduce the length of the friction connection. However, there are concerns that the applied compressive force would not cause a uniform stress over the sliding surfaces, but may result in higher localized bearing stresses near the bolts, especially if the plate thicknesses are small. This may result in more degradation of the sliding surface especially near the bolt locations. In addition, there are concerns that the requirements of NZS3404 Clause 4.2.4.1.2c for the required number of threads in the grip length, may not be appropriate for higher strength, but less ductile, bolts. Chanchí et al. (2014) conducted 2 cyclic tests using Grade 8.8 bolts with length of 120 mm and shank length of 80 mm. The number of the threads within the grip length was 7. Also, 21 cycles up to +/- 90 mm were applied and there was no bolt fracture and little connection strength degradation. The displacement regime for Test 1 was then reapplied after the connection had cooled several hours after Test 1 without retightening of the bolts and with the same plates as in Test 1. The shims, cap plate and slotted plate were changed for Test 2. Again the second run was conducted without retightening of the bolts in Test 2. The average of sliding force is 75 kN for both runs of Test 1 and 72.5 kN for Test 2 as shown in Figure 2.

––1st run ----2nd run

One friction connection subject to significant research is the asymmetric friction connection (AFC) (Clifton (2005), MacRae et al. (2010)). As shown in Figure 1, this connection consists of 3 plates. The top (grey) structural steel plate is placed above as lotted structural steel plate which is in turn placed above a cap plate. Shims are placed between the top plate and the slotted plate and may also be placed between the slotted plate and the floating (or cap) plate. If a shim is not placed beside the cap plate, the cap plate itself may be made of shim material to obtain dependable sliding resistance. The floating cap plate is connected to the subassembly by means of the high strength bolts. Because sliding does not initiate at the same displacement either side of the slotted plate, it is referred to as an asymmetric friction connection (AFC).

Test 1

Volume 45 Number 1 March 2015  55


– Are special plate thickness appropriate for Grade 10.9 bolts?

2.

requirements

Methodology

2.1. Details of AFC, Test setup and Testing Plan

Test 2 Fig. 2: Force-Displacement Plot on Grade 8.8 (Chanchí et al., 2014)

From the standard nominal bolt properties in Table 1, it may be seen that the expected increase in initial sliding strength for Grade 10.9 bolts above that for Grade 8.8 bolts, estimated from the proof load ratio, is 830kN/600kN = 1.38. In addition, the ultimate bolt elongation of Grade 10.9 bolts is 9%/12% = 0.75 times that of Grade 8.8 bolts indicating that the stronger bolts are likely to fracture at a lower strain. Hence, there is a normal trade-off between sliding force based on a bolt strength and reduced ductility. Table 1: Standard Bolt Properties Grade 8.8 and 10.9 according to Bickford (2007) Grade 8.8 10.9

Proof stress 600 830

Yield stress 660 940

Ultimate stress 830 1040

Elongation 12% 9%

Thus there is a need to address these concerns related to the use of higher strength bolts in sliding connections. To address this need Grade 10.9 bolts, which have nominal yield and ultimate strengths 42% and 25% greater than that for Grade 8.8 bolts, respectively and a nominal ultimate strain capacity which is only 75% of that of a Grade 8.8 bolts, answers are sought to the following questions:

In the test configuration, a brace was placed horizontally and the AFC was placed between the brace and a moving support as shown in Figures 3 and 4 following Chanchí et al. (2014). The load cell and moving bracket were restricted against out-ofplane movement. The length of the 250 PFC Grade 300 brace was 2860 mm. A 16x200x300 bearing plate is welded to the inside of the PFC-section brace to increase brace buckling resistance. The two 200 mm slotted holes were cut in the slotted plate attached to the moving bracket allowing 100 mm sliding in each direction. This sliding length is appropriate for a friction brace, rather than to a sliding hinge joint type connection (MacRae et al. 2010) where much smaller displacements are generally expected. The total thickness of the plates held together by the 2M16 120 mm long Grade 10.9 bolts was 8 mm + 2 x 6 mm + 32 mm + 16 mm+ 16 mm = 84 mm. The total grip, including washers, was 93 mm. The bolt shank length is 80 mm and the threaded length is 40 mm. The number of threads within the grip length was 7, which satisfied the code requirement for a bolt length between 4 and 10 times the bolt diameter. All plates were Grade 300 steel. The 6 mm shims were Bisalloy 500 (2012) with a Rockwell hardness of 500 and the AFC bolts were Grade 10.9 bolts with actual ultimate and yielding stress of 1121 MPa and 955 MPa, respectively. Four linear potentiometers with 25 mm stroke were attached between strong floor and the brace to monitor brace out-of-plane deformation as shown in Figure 3. Horizontal movements of the brace and AFC were monitored by a rotary pot attached to the fixed bracket at the end of the brace. Horizontal force was monitored by the load cell attached to the actuator.

– How do AFC connections with Grade 10.9 bolts perform? – What minimum threaded length is appropriate to limit the possibility of Grade 8.8 bolt fracture to be similar to that for Grade 8.8 bolts? 56  Volume 45

Number 1 March 2015

Fig. 3: Test setup

The Bridge and Structural Engineer


45% of the slot length as shown in Figure 5. Three cycles at each level of displacement were applied as shown.

Longitudinal configuration

Fig. 5: Test loading regime

2.3. AFC Bolt Tightening

According to New Zealand steel structures standard (NZS3404 (2007)) all bolts in a friction connection According to New Zealand steel standardbolt (NZS3404 (2007)) all bolts i must be tightened at least tostructures the minimum tension specified in the standard. Thisto value is commonly connection must be tightened at least the minimum bolt tension specified in th referred to as the proof load.

Section A-A Fig. 4: Tested AFC detail

This value is commonly referred to as the proof load.

Five tests were conducted as described in Table 2. In this table, a new number is used when the element is replaced. In the first test, all of the plates (slotted plate, shims and bearing plate) and bolts were new. However, for the remaining tests (Tests 2-5) the bearing plate was not changed as it was welded to the brace. The slotted plate, shims and bolts were changed for Test 2. For Test 3 only the bolts were changed from Test 2. Test 4 was carried out without retightening of the bolt or changing of any plates to study the AFC performance without retightening the Grade 10.9 bolts. In the last test, all bolts and shims were replaced. Table 2: Tests Conducted Test

Five simple tests were conducted to obtain the torque and nut rotation for the bolt elongation corresponding to the proof load, ∆Bolt, computed based on the bolt axial stiffness in conducted Equationto obtain 1 following Bickford Five simple tests were the torque and nut rotationfor the bolt (2007). Here, Ntf is the proof load of the bolt, Lsh is corresponding to the proof load, ΔBolt, computed based on the bolt axial the shank length plus half of the bolt head thickness, AEquation is the1 following shank area, Lth is the threaded theof the bolt, Lsh i the proofofload Bickford (2007). Here, Ntf islength sh bolt from the nut up to the shank plus half of the nut length plus half of the bolt head thickness, Ash is the shank area, Lth is the threade thickness and Ath is the bolt tension area. For the total the bolt from of the 91 nut mm, up to the half of the nut thickness, grip length theshank M16plus Grade 10.9 bolts were and Ath is the tightened by applying torques increasing from 40 N.m area.For the total grip length of 91mm,the M16 Grade 10.9 bolts were tightened b to 550 N.m. At each step, torque the bolt elongation torques increasing from N.m to 550and N.m.At each step,attorque and turn-of-nut were40recorded the values a boltthe bolt elongatio elongation of 0.37 corresponded to the proof corresponded of-nut were recorded andmm thevalues at a bolt elongation of 0.37mm load, defined by ΔBolt: load, defined by ΔBolt:

Slotted Plate

Shims

Bolts Bearing Bolt plate retightened

1

1

1

1

1

Yes

2

2

2

2

1

Yes

3

2

2

3

1

Yes

4

2

2

3

1

No

5

2

3

4

1

Yes

2.2. Test Regime The displacement regime was applied with a constant velocity of 3 mm/s. It consisted of 21 cycles with amplitudes between 3 mm and 90 mm. This value is

∆���� �

��� ��� ��� 1�0 � 10� 81 26.5 � �� � � � 0.�7 �� (1) � � � ��� ��� 200 � 10� 201 157

2.4. Analytical Predictions of Behaviour 2.5. Analytical Predictions of Behaviour

The AFC connection initial nominal sliding force, Fs, was estimated using Equation 2 (MacRae et al. The AFC connection initial nominal sliding force, Fs,was estimated (2010)):

using

(MacRae et al. (2010)):

F��S �=�n×μ×μ×N = 10�.2 109.2��kN tf = � μ � η � ��� � 2×0.21×2×130 2 � 0.21 � 2 � 1�0 �

(2)

Where nnisisnumber of the boltsbolts in thein AFC, is the μeffective Where number of the the μAFC, is the

friction coeffic

Bisalloy500-steel interface taken as 0.21 according to Chanchí et al. (2012a), η is

The Bridge and Structural Engineer

Volume Number March 2015  a strength 57 of shear planes, and Ntf is the45proof load per1bolt. For design, reduction 10


effective friction coefficient for the Bisalloy 500-steel interface taken as 0.21 according to Chanchí et al. (2012a), η is the number of shear planes and Ntf is the proof load per bolt. For design, a strength reduction factor, φ, of 0.70 is generally used acknowledging the uncertainty associated with friction connections. Based on recommendations used in construction (Leslie et al. (2013)) an overstrength factor of 1.4 is reasonable. This factor is affected by factors including variability of bolt tightening and different sliding surfaces.

3.

Results

3.1. Torque, Nut Rotation and Bolt Elongation The median torque and nut rotation corresponding to the bolt proof load bolt elongation of 0.37 mm are 370 N. m and 185 degrees respectively according to Figure 6. This result is similar to the NZS 3404 (2007)

Torque- Bolt elongation

Nut rotation- Bolt elongation 58  Volume 45

Number 1 March 2015

Fig. 6: Nut rotation, torque and bolt elongation relation

Clause 15.2.5.2 recommendation that the nut rotation from the snugtight condition, for a bolt with the length of over 4 diameters but not exceeding 8 diameters, is ½ turn. The equivalent torque and nut rotation for Grade 8.8 bolts tested the same way were 330 N.m and 550 degrees respectively (Chanchí et al. (2012a)). 3.2.

Sliding Tests

The hysteresis loop for Test 1 is shown in Figure 7, where positive force indicates compression. The hysteresis loops are not square because of the bolthole oversize. The bolts move on an angle allowing so there is approximately a 10 mm slack area before the bolt carries increased load. Initial tension and compression sliding forces were 123kN and 101kN respectively giving an average initial sliding force of 112 kN. This value is close to the nominal value of 109kN computed in Equation 2 for Grade 10.9 bolts, shown as the solid line. Also, the average sliding force of AFC with Grade 10.9 was 1.5 times of AFC with Grade 8.8 bolts. According to Chanchí et al. (2014), the tensile/compressive strength difference occurs due to bending in the non-clamped zone of the slotted plate and it is caused by the AFC asymmetry. In the third cycle to 48 mm displacement, the force was increased to the maximum value of 163kN, as shown in Figure 7a and one of the bolts fractured. This is 91% greater than the strength of 85 kN for the Grade 8.8 bolts reported by Chanchí et al. (2014). The increased sliding force is possibly due to larger particulates separating from between the sliding surfaces due to the increased force. These particulates can be observed on the degraded slotted plate at the end of the test that is shown in Figure 8. No overlap of bearing area was observed in the slotted plate and the stress was distributed with 31 degrees angle according to the observed degradation on the plate. As a result of the increased localised bearing stress causing material degradation, bolt elongation occurred causing higher sliding stresses until the bolt fractured. Bolt elongation occurs due to the separation of small steel particles during the material degradation as a result of the high localized stresses. At larger displacement levels, a greater number of small particles, or bigger particles may separate. The theoretical elongation of the Grade 10.9 bolts over the grip length at proof load was 0.37 mm according

The Bridge and Structural Engineer


to Equation 1. Here the strain at proof load over the threaded length, εproof , is 0.414%. The maximum bolt deformation before fracture, Δub, may be estimated assuming all inelastic deformation occurs in threaded length of the bolt from the nut up to the shank plus half of the nut thickness, Lth, as shown in Equation 3. Here, εult is the maximum bolt elongation in Table 1. For Grade 8.8 bolts, Δub = 3.1 mm.

a. Test 1. New

∆ub = (εult – εproof ) × Lth = (0.09-0.00414) × 26.5 = 2.3 mm (3) Bolts, slotted plate and shims were replaced after Test 2 and the average initial sliding force was 107 kN. The sliding force dropped to 70% of the initial sliding force at 24 mm displacement, as shown in Figure 7b possibly as a result of permanent bolt elongation reducing the compressive test. No bolt fracture was observed at the end of this test. In Test 3, only the bolts were replaced and they were tightened with a torque equal to 370 Nm to reach the proof load according to Figure 6a. This test is similar to Test 2, with new bolts, except the shims were reused. Degradation of plates caused different levels of sliding force for the same level of drift as shown in Figure 7c. The minimum of the maximum sliding forces occurred in the last cycle and was 27% of that in the first cycle. At 48 mm displacement, the sliding force was 36% of that in the previous cycles possibly due to bolt necking and nonlinear deformation. This behaviour was not observed for the AFC using Grade 8.8 bolts except when the assembly materials were significantly worn causing instabilities in the behaviour of the AFC.

b. Test 2. New plate, bolts, shims

c. Test 3. New bolts

d. Test 4. Bolts not retightened The Bridge and Structural Engineer

Volume 45 Number 1 March 2015  59


test. One of the bolts fractured in the second cycle to 90 mm displacement and the other bolt was highly deformed as shown in Figure 9.

e. Test 5. New shims and bolts

Fig. 7: Force-Displacement Plot for AFC with Grade 10.9 bolts

Fig. 8: Degraded plate at the end of Test 1

Test 4, evaluating the effect of previous cyclic loading on a joint, was conducted several hours after Test 3, so the joint had first cooled and the test was run without retightening or replacing of plates and bolts. It may be seen from the force-displacement curve in Figure 7d that the sliding force was less than that of the previous tests, presumably as a result of bolt length increase due to inelastic deformation. Such deformation reduces the interface force. During these additional cycles, one of the bolts fractured at 49 mm displacement. The sliding force of Test 4 is about 70% of Test 3 for displacements up to 48 mm. For Grade 8.8 bolts the sliding force of repeated tests was 90% of a test with new elements (Chanchí et al. (2014)). In the final test, Test 5, both bolts and shims were replaced but the same slotted plate was used. The average initial strength of 85kN was only 77% of the nominal value from Equation 2 and from Test 1, as shown in Figure 7e. This result occurred because 42 cycles were conducted on the slotted plate before this

60  Volume 45

Number 1 March 2015

Fig. 9: Deformed and fractured bolts at the end of Test 5

The median sliding force for Tests 1 to 5 in the first cycle to different displacement levels is shown in Figure 10 and the dashed line shows the calculated nominal sliding force from Equation 2. Generally, as sliding length is increased there is a reduction in the sliding force, because of the bolt moving on an angle and greater MPV interaction that also can caused bolt nonlinearity. Also, the sliding force in each test (rather than Test 4 that bolts were not retightened) is reduced due to reduction of friction coefficient since the slotted plate was not changed from Test 2-5. The maximum reduction of friction coefficient in Test 2 of the AFC with Grade 8.8 bolts was 0.9 of the friction coefficient in test 1 as reported by Chanchí et al. (2014). The trend of test 4 is slightly different to those of the other tests. After 48 mm of displacement, the sliding force for test 4 increases, whereas the sliding forces for other tests generally decrease with displacement. This behaviour could be caused by the debris being stuck between plates, causing extra force to be required to achieve sliding.

Fig. 10: Test 1-5 Sliding force at Different Displacements

The Bridge and Structural Engineer


0

20

40 60 Displacement, mm

80

100

Figure 10: Test 1-5Sliding force at Different Displacements

4.

Recommendations

For more than one hole, the bearing area is that 17 Figure above minus the stress overlap area. From 11, it looks as though the greatest stress should be on the area between the slotted holes, as the stress areas overlap here. For thicker plates, the average stress on the sliding interfaces from each bolt decreases due to the spread of stress with distance. Since the total compressive force is the same, the sliding compressive force is similar. However, thicker plates can also increase the likelihood of an area of stress overlap where the average stress is n times that from one bolt, where n is the number of bolts contributing to the stress there. It is recommended that the spacing of bolts be such that there is no stress overlap area. It recommended that the peak stress, at the

interface area, is the minimum plate thickness to prevent the overlap area since increasing I. A minimum thickness plate is required to prevent 4.0. Recommendations thickness of the plates can prevent the overlap excessive localized bearing stress that can result area fromstress bolts. I. Aminimum thickness plate is required to prevent excessive localized bearing that can in sliding surface deterioration. This thickness II. the Since the and majority depends on thesurface bolt strength andThis sizethickness of the bolt result in sliding deterioration. depends on bolt strength size of of inelastic deformation occurs in the threaded length of bolt in the grip head. The bearing area, Ab, of the slotted plate the bolt head. The bearing area, Ab, of the slotted plate from bolt tensile force is given: length, the NZ standard should be modified for from bolt tensile force is given: Grade 10.9 bolts considering that the ultimate � � � ����� � �� � � elongation of Grade 10.9 bolts is 75% that of (4) �� � (4) 4 Grade 8.8 bolts (Bickford (2007). Therefore, deff = dhead + 1.15 × (twasher + tshim + tplate) (5) the length of threaded Grade 10.9 bolt within ���� � ����� � ���� � �������� � ����� � ������ � (5) the grip length is required to be 1/0.75 = 1.33 Where dh is hole size, deff is effective diameter times that of the Grade 8.8 bolts. This is, at least at the plate surface. Assuming the stress is 7 threads for a bolt length up to and including distributed with 30 degrees angle, dhead is 4 times the bolt diameter, 14 threads for a bolt diameter of bolt head that can be considered Where dh is hole size, deff is effective diameter at the plate surface. Assuming the stress is length exceeding 8 times the bolt diameter and as the longest dimension, twasher and tshim is distributed with 30 degrees of bolt head that can be considered as the head is diameter 10 threads for intermediate bolt lengths. thickness of the washerangle, and dshim respectively, thicknesst of the plates that are between tlongest plate is dimension, and shim is Therespectively, number oftplate threads may need to be greater washer, and tshim is thickness of the washer, III. the slotted plate and bolt (cap plate or bearing for shorter bolts since they have a lower elastic plate and brace web). deformations during the same bolt tightening

Fig. 11: Localized bearing stress

The Bridge and Structural Engineer

than longer bolts.

IV. Retightened bolts should not be used since the level of sliding force is considerably reduced.

5.

Conclusion

This paper describes the experimental testing of an asymmetric friction connection (AFC) with Grade 10.9 bolts under cyclic loading to displacements in one direction as large as 90 mm. It was shown that: a) Some of the bolted connections with Grade 10.9 bolts suffered from fracture and there was considerable wear on the sliding surfaces. Fracture did not occur to the same extent in similar tests using Grade 8.8 bolts. b) Since the nominal deformation capacity of Grade 10.9 bolts is 75% of that for Grade 8.8 bolts, the threaded length for these bolts should be at least 1/0.75 = 1.33 times that required for Grade 8.8 bolts. c) Since the nominal strength of Grade 10.9 bolts is approximately 38% greater than that of Grade 8.8 bolts, the thickness of plates required to limit localized bearing stress and material scour during sliding should be increased. Calculation of bearing stress needs to consider the effect of the holes either side of the sliding interfaces. Dispersion angle of 31 degrees was obtained from the experiments. Volume 45 Number 1 March 2015  61


Acknowledgment The authors would like to acknowledge the MBIE Natural Hazards Research Platform (NHRP) for its support of this study as part of the Composite Solutions project. All opinions expressed remain those of the authors.

References 1.

BICKFORD, J. 2007. An Introduction to the Design and Behavior of Bolted Joints, fourth Edition.

2.

BISALLOY 2012. Bisplate Technical Guide. Unanderra, Australia.

3.

BORZOUIE , J., MACRAE, G. A., CHASE, G. J. & CLIFTON, C. G. 2014. Experimental studies on cyclic behaviour of steel base plate connections considering anchor bolts post tensioning. NZSEE. Auckland, New Zealand.

4.

CHANCHÍ, J. C., MACRAE, G. A., CHASE, J. G., RODGERS, G. W. & CLIFTON, C. G. 2012a. Behaviour of Asymmetrical Friction Connections using different shim materials. NZSEE Conference.

5.

CHANCHÍ, J. C., MACRAE, G. A., CHASE, J. G., RODGERS, G. W. & CLIFTON, C. G. 2012b. Clamping Force Effects on the Behaviour of Asymmetrical Friction Connections. 15 WCEE. Lisbon.

6.

CHANCHÍ , J. C., XIE, R., MACRAE, G., CHASE, G., RODGERS, G. & CLIFTON, C. 2014. Low-damage braces using Asymmetrical Friction Connections (AFC). NZSEE. Auckland, New Zealand

7.

8.

CHRISTOPOULOS, C., FILIATRAULT, A., UANG, C. & FOLZ, B. 2002. Posttensioned Energy Dissipating Connections for MomentResisting Steel Frames. Journal of Structural Engineering, 128, 1111-1120. CLIFTON, C. G. 2005. SEMI-RIGID JOINTS FOR MOMENT.RESISTING STEEL FRAMED SEISMIC-RESISTING SYSTEMS. PhD, University of Auckland.

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Number 1 March 2015

9.

GLEDHILL, S., SIDWELL, G. & BELL, D. K. 2008. The Damage Avoidance Design of tall steel frame buildings - Fairlie Terrace Student Accommodation Project, Victoria University of Wellington. NZSEE

10. KHOO, H.-H., CLIFTON, C., BUTTERWORTH, J., MACRAE, G. & FERGUSON, G. 2012. Influence of steel shim hardness on the Sliding Hinge Joint performance. Journal of Constructional Steel Research, 72, 119-129. 11. LATHAM, D., REAY, A. & PAMPANIN, S. 2013. Kilmore Street Medical Centre: application of a post-tensioned steel rocking system. Proc., Steel Innovations Conf. 12. LESLIE, B., GLEDHILL, S. & MOGHADDASSI, M. 2013. Concentric Braced Frames with AFC Connections–A Designers View. 13. MACRAE, G. A., CLIFTON, G. C., MACKINVEN, H., MAGO, N., BUTTERWORTH, J. & PAMPANIN, S. 2010. The Sliding Hinge Joint Moment Connection. Bulletin of the New Zealand Society for Earthquake Engineering, 43, 10. 14. MANDER, T., RODGERS, G., CHASE, J., MANDER, J., MACRAE, G. & DHAKAL, R. 2009. Damage Avoidance Design Steel BeamColumn Moment Connection Using High-Forceto-Volume Dissipators. Journal of Structural Engineering, 135, 1390-1397. 15. NZS3404 2007. Steel structures standard. Part1. New Zealand. 16. RODGERS, G., SOLBERG, K., MANDER, J., CHASE, J., BRADLEY, B. & DHAKAL, R. 2012. High-Force-to-Volume Seismic Dissipators Embedded in a Jointed Precast Concrete Frame. Journal of Structural Engineering, 138, 375-386. 17. RODGERS, G. W., CHASE, J. G., MANDER, J. B., LEACH, N. C. & DENMEAD, C. S. 2007. Experimental Development, Tradeoff Analysis and Design Implementation of High Force-ToVolume Damping Technology. NZSEE Bulletin, 40, 35-48.

The Bridge and Structural Engineer


Vulnerability Analysis of Buildings for Seismic Risk Assessment: A Review

Hemant B KAUSHIK Associate Professor Department of Civil Engineering Indian Institute of Technology Guwahati Guwahati 781039, Assam, India hemantbk@iitg.ac.in

Trishna CHOUDHURY PhD Scholar Department of Civil Engineering Indian Institute of Technology Guwahati Guwahati 781039, Assam, India c.trishna@iitg.ernet.in

Hemant B. Kaushik received his Bachelors and Masters degrees in Civil/Structural Engineering from VNIT Nagpur. He was awarded his Doctoral degree in 2006 from IIT Kanpur after which he joined IIT Guwahati as a faculty member in 2007. His research interests include earthquake resistant design, nonlinear behavior of structures,seismic vulnerability assessment and retrofitting of structures. He received INAE Young Engineer Award in 2010 for his contribution in earthquake/structural engineering. He is Associate Scribe of "Earthquake Engineering Practice" a quarterly periodical published by NICEE at IIT Kanpur. He carried out post-earthquake reconnaissance studies after several earthquakes in India and abroad (including 2015 Nepal earthquakes). He has published several research papers in national and international journals and conferences on different research areas.

Trishna Choudhury received her Bachelor degree in Civil Engineering in 2010 from Assam Engineering College, Guwahati. She served as a Lecturer in Royal School of Engineering and Technology, Guwahati, for a period of two years from 2010 to 2012. She is currently pursuing her Doctoral studies in the Department of Civil Engineering at IIT Guwahati. Her research interests include seismic vulnerability assessment of structures, earthquake resistant design of structures and reliability based analysis of structures.

Summary Earthquakes cannot be predicted, however, structures can be designed to resist the earthquake loads safely. Still huge structural, monetary and human losses are reported every year due to effects of earthquakes across the world. This is because of large number of uncertainties associated with assessment of seismic hazard, material properties, analysis methods, design and detailing, structural capacity, workmanship, quality of construction, etc. Therefore, it is of utmost importance to estimate seismic risk associated with exposed elements (structure, human, surrounding) in order to plan mitigation strategies in the event of a damaging earthquake. Seismic risk is a function The Bridge and Structural Engineer

of hazard, vulnerability and exposure. In this paper, widely used methods for assessment of all primary components of seismic risk estimation (seismic hazard, building performance, seismic fragility and seismic vulnerability) have been reviewed. The review will expose the reader to available methods, with their pros and cons and complexities involved, for seismic vulnerability and risk estimation. Keywords : Seismic performance, uncertainty, fragility, vulnerability, risk

1.

damage,

Introduction

Seismic vulnerability is defined as the degree of loss Volume 45 Number 1 March 2015â&#x20AC;&#x192; 63


to a given element at risk, or set of such elements, resulting from an earthquake of a given magnitude or intensity (Shah 1984). This degree of loss/damage to a set of exposure (building) in an area depends on its seismic performance. It is thus essential to determine the seismic performance of buildings (or building stocks) under a given hazard (earthquake) to assess their seismic vulnerability and associated risk. Seismic risk is defined as the probability that social or economic consequences of earthquakes will exceed specified values at a site, at several sites, or in an area, during a specified exposure time (Shah 1984). Relevant past studies undertaken in order to develop a method for seismic vulnerability and risk estimation are presented in this paper. Vulnerability analysis is a complex process involving several components (Fig 1) each of which is reviewed in this paper with a view to develop an understanding for identifying the vulnerable building typologies in an area or an entire region. FEMAP-58-1 (2012) is by far one of the most comprehensive documents that provide a general methodology and recommends procedures to assess the probable seismic performance of individual buildings subjected to future earthquakes. The methodology assesses the likelihood that structural and non-structural components of building will be damaged by earth quake shaking. Further, it estimates the potential casualties, repair and replacement costs, repair time, etc., that could occur because of such damage. With such details, necessary steps can be undertaken towards developing an integrated disaster and risk management system thereby reducing the social and economic consequences of earthquake.

Fig. 1: Basic components of seismic risk assessment

2.

Seismic Hazard

Seismic hazard estimation is the first basic step in vulnerability assessment at a site. Seismic hazard is defined as any physical phenomenon (e.g., ground shaking, ground failure) associated with an 64â&#x20AC;&#x192; Volume 45

Number 1 March 2015

earthquake that may adversely affect human activities (Shah 1984). Seismic hazard assessment is a broad area of research involving people from different fields, viz., geologists, seismologists and engineers, who quantifies this physical phenomenon at any site or region of interest. Seismic hazard may be analyzed either deterministically, when a particular earthquake scenario is assumed, or probabilistically, in which uncertainties are explicitly considered (Kramer 1996). Deterministic seismic hazard analysis (DSHA), makes use of discrete, single valued events to arrive at scenario-like descriptions of earthquake hazard. Firstly, seismic sources near the site are identified and the shortest source-site distances are estimated. For each source, a ground motion parameter (e.g., PGA, PGV, etc.) is determined using attenuation relationships and the one producing the highest level of shaking is selected as the controlling earthquake at the site. The hazard at the site is usually defined in terms of the ground motion parameters estimated or predicted for the controlling earthquake. While the procedure and calculations involved are simple, its implementation in practice involves numerous difficult judgments regarding selection of input parameters. Indication of how likely the earthquake will occur during the design lifetime of the structure remains unknown in DSHA. Thus, there exists lack of explicit consideration of uncertainties at all the steps in DSHA. Still because of its simplicity, DSHA is used in development of design response spectra in several seismic design codes including the Indian Seismic Code (BIS 2002). Probabilistic seismic hazard analysis (PSHA) is a framework that allows identification and quantification of uncertainties associated with every step of seismic hazard assessment and provides a better picture of the seismic hazard. PSHA involves two separate models: a seismicity model describing geographical distribution of event sources and distribution of magnitudes; and an attenuation model describing the effect at any given site as a function of magnitude, source to site distance and ground class (Dowrick 2009). Unlike DSHA, earthquake sources in PSHA are explicitly defined as being of uniform earthquake potential. Instead of selecting one or more controlling earthquakes, each source is characterized by an earthquake probability distribution or recurrence relationship. Distances from all possible locations within that source to the site The Bridge and Structural Engineer


are considered and their associated probabilities of occurrence are taken into account. Using attenuation relationships, families of ground motion curves are generated for different distances. Finally, seismic hazard is defined at the site using curves showing the probability of exceeding different levels of ground motion (e.g., PGA) at the site during a specified period. DSHA method is simpler compared to PSHA, but it does not treat uncertainties related to characterisation of earthquake sources well. On the other hand, PSHA procedure requires rigorous probability computations to be carried out for explicit considerations of uncertainties. One needs to carry out extensive probabilistic analysis for determining probabilistic hazard curves. PSHA is necessarily required to be carried out for estimation of seismic risk. Hence choice of a method depends on the scope and purpose of work and availability of data. Depending upon the importance of the structure, either established design response spectrum available in seismic codes are used as seismic hazard (for regular structures) or strong ground motion characterization is done for the site using any of the methods mentioned above (e.g., for important structure). In addition, FEMA-P-58-1 (2012) provides general procedures for earthquake hazard characterization for use in the simplified analysis procedure. Simplified analysis, which can be used only for regular, symmetric and short buildings that are not expected to undergo large lateral deformations, uses linear structural models and the lateral yield strength of the structure to estimate median values of response parameters. If PSHA is used, one need not separately consider the uncertainty in demand (βD) in fragility analysis (see Section 6). Once the hazard is defined quantitatively, next step is to classify the exposed building into groups for seismic performance assessment.

3.

Building Inventory

In an area of interest some sample of buildings, which represent the prevalent building stock are selected for vulnerability assessment. This requires standard and systematic inventories that classify the buildings according to their type, occupancy and function so that realistic estimates of capacity, seismic risk, loss, etc., can be made. Usually parameters important for vulnerability assessment are considered in classifying The Bridge and Structural Engineer

the buildings, some of which include structural material, e.g., reinforced concrete (RC), steel; structural system, e.g., framed system, unreinforced masonry (URM); number of stories; age (e.g., precode or post code designed), etc. Comprehensive classification of buildings is available in PAGER (Jaiswal et al. 2010) and GEM (Brzev et al. 2013). These classifications are however, country specific and are limited by paucity of available building data. These data need to be updated considering wide variety of building typologies and construction material used in different countries. Building inventory data aids in assessing vulnerability through observed damage from past earthquakes or using detailed seismic evaluation procedures where the mathematical model of the building is developed to evaluate its capacity. Capacity is estimated through nonlinear analyses of the building model, which is further used in detail seismic performance assessment as discussed in the next section. Though analytical procedures are preferred, but accuracy depends on several factors, such as, modelling aspects, analysis type, nonlinear material properties, etc., from where uncertainty in capacity (βC), (refer Section 6) arises. Estimation of uncertainty associated with seismic capacity for different building inventory is still a broad area of research.

4.

Seismic Performance Assessment

Seismic performance can be assessed with different levels of computational efficiency depending on available data and resources. Detailed nonlinear analyses (static or dynamic) are required to be carried out for assessing seismic capacity and performance of structures for a given hazard (or demand). The present state-of-the-art for performance assessment is described in FEMA P-695 (2009). The procedures in FEMA P-695 are based on the concept of incremental dynamic analysis (IDA), which is a computationally intensive procedure that is rarely practical in nonresearch applications. It is a common practice to use simple nonlinear static analysis procedures or pushover analysis (PO) for performance evaluation of structure using Capacity Spectrum Method (ATC 40, 1996), Displacement Coefficient Method (FEMA 2000), or the N2 method (Fajfar 2000) as discussed below. Development of seismic performance assessment methodologies using most simplistic but accurate method of analysis is the current trend of research. Volume 45 Number 1 March 2015â&#x20AC;&#x192; 65


Most recently, FEMA-P-58-1 (2012) provided seismic assessment methods using both incremental dynamic analysis and simplified nonlinear analysis procedures.

procedure and commented that deformation obtained by ATC 40 method is underestimated for any value of natural time period. It is significantly low in acceleration sensitive regions of response spectra.

Capacity Spectrum Method (CSM) CSM, originally developed by Freeman (1998), requires construction of damped elastic response spectrum (or demand spectrum) and capacity spectrum in acceleration displacement response spectrum (ADRS) format using Eq. 1 (Fig. 2). Sa = V / W

α1 and Sd = Δ roof / ( PF1 *φ1,roof )

(Eq.1)

where, α1 and PF1 are respectively the modal mass coefficient and participation factors for the first natural mode of the structure and ø1,roof is the roof level amplitude of the first mode. Intersection of capacity and demand spectrum curves gives the performance point (PP) for the structure for a given hazard (Fig. 3). ATC 40 suggests three procedures (A, B, C) for using CSM; selection of anyone of which is a personal choice. The basic methodology of procedures A, B and C is same, but differs in the assumption of post yield stiffness for deriving the bilinear capacity curve and in computational effort required. Determination of PP is an iterative procedure. An initial trial PP is determined from equal displacement approximation and compared with the PP obtained from intersection of capacity spectrum and demand spectrum effectively damped for nonlinearity using the concept of effective viscous damping (βeff). This procedure is iterated until PP obtained at a particular iteration is approximately equal (± 5% variation) to the PP obtained in the previous iteration. Although the CSM is simple, its basic disadvantage is that the lateral force distribution for performing nonlinear static analysis of buildings is assumed to be the first mode of vibration only. Other higher dominant modes are not considered. Also the PP obtained is dependent on elastic response spectra (rather than inelastic response spectrum) effectively damped for non-linearity using the concept of effective viscous damping. To overcome such drawbacks, modified procedures are proposed in literature, e.g., Fajfar (2000) introduced the concept of inelastic demand spectra using ductility dependent reduction factors. Chopra and Goel (1999a, 1999b) also introduced the idea of constant ductility dependent inelastic demand diagram as a modification for existing ATC 40 66  Volume 45

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Fig. 2: Steps in Capacity Spectrum Method (ATC 40, 1996): (a) Construction of demand spectrum and (b) Construction of capacity spectrum

Fig. 3: Plot representing capacity curve and demand curve (red) in ADRS format; performance point (PP) as obtained in CSM (ATC 40, 1996) and associated damage state thresholds and quantification as per Lagomarsino and Giovinazzi (2006).

Displacement Coefficient Method (DCM) DCM is a direct method as compared to CSM that modifies linear elastic response of the building (roof displacement) with the help of several coefficients (C0, C1, C2 and C3) to calculate a target displacement (δt) given by Eq. 2 (FEMA 2000). Unlike CSM, conversion of capacity and demand into ADRS is not required in DCM. δt = C0C1C2C3 Sa

Te2 4π 2

(Eq. 2)

C0 relates spectral displacement to likely building roof displacement and C1, C2, C3 take into account inelastic displacement, hysteresis effect and P-delta effects, respectively. This δt may be considered to be equivalent to the PP evaluated in CSM. The DCM is generally not preferred since the target displacement is heavily dependent on empirical relations for estimation of the coefficients. The Bridge and Structural Engineer


N2 Method The N2 method (Fajfar and Gašperšič 1996 and Fajfar 2000) is a non-iterative method and is based on ductility dependent inelastic response spectrum rather than on equivalent nonlinear response spectrum as in ATC 40 (1996). Originally, the method was developed for symmetric systems that oscillate predominantly in the first mode. Later on, it was extended to consider asymmetric buildings and buildings where higher modes are dominant. This method provides the user with graphical visualization of the performance of structure, though the graphical step is not necessary to obtain the seismic demand. The method starts with determining the elastic response spectrum in acceleration displacement (A-D) format. Next, the base shear (Vb)–roof displacement (Dt) relationship for a multi degree of freedom (MDOF) is obtained from nonlinear static (pushover) analysis. For this, a displacement shape is assumed and lateral load distribution is determined based on the assumed displacement shape. Thus, load and displacement are mutually dependent in N2 method, which is not the case in ATC 40. The Vb - Dt relationship obtained for the MDOF system is transformed to single degree of freedom (SDOF) system by dividing Vb and Dt by modal participation factor. The SDOF Vb- Dt relationship is approximated as elasto-plastic force deformation relationship and finally converted to A-D format which is called capacity diagram. The original method is applicable for those systems in which strain hardening has no practical influence on the results, so a simple zero postyield stiffness is assumed. For a given demand curve (or elastic response spectrum), the seismic demand on the SDOF model is determined as the intersection of inelastic demand curve in A-D format and the SDOF capacity diagram. The inelastic demand curve is obtained in two steps. First, the reduction factor due to ductility (Rµ) is determined as the ratio of elastic to inelastic accelerations where the parameters are obtained using equal displacement rule. Next, the ductility factor (µ) is determined using Eq. 3 in order to obtain the inelastic displacement demand (Sd) from elastic displacement demand (Sde) as in Eq 4.

Sd =

μ Rμ

Sde

(Eq. 4)

Here, TC is the characteristic period of the ground motion and T* is the elastic period of the idealized bilinear equivalent SDOF system. The above relations represent that the demand spectrum is reduced with Rµ if T* falls in acceleration sensitive region of the demand curve, otherwise reduced with µ if T* falls in velocity sensitive region. Although, the N2 method is simple, the estimates of inelastic displacements obtained for short period range are found to be less accurate. Moreover, the simplistic equations used for obtaining the inelastic demand spectra (in N2) is inappropriate for near fault ground motions, or soft soil conditions, or for hysteresis loops with significant strength deterioration. In contrast to N2 method, the CSM method considers a damping modification factor to enable simulation of imperfect hysteresis loops. In addition to these three common methods, researchers have developed other methods for performance assessment, e.g., modified CSM (Chopra and Goel, 2000). FEMA 440 (2005) evaluates the accuracy of CSM and DCM for estimation of displacement demand on inelastic SDOF systems and suggests improvements over both the methods. It proposes simplified expressions for different coefficients used in DCM and modified expressions for evaluation of effective natural period and effective damping for CSM while taking care of the ductility demand. Interestingly, only a minor difference was observed in the PP obtained using these different methods, when applied to regular frames with minor strain hardening in capacity curves. This shows that any of these methods can be used for performance assessment of regular symmetric buildings without compromising accuracy.

The seismic evaluation procedures described in this section give deterministic estimates of structural performance for a given hazard. A probabilistic estimate of building performance (or fragility) requires the knowledge of damage and its occurrence = μ Rμ ; T * ≥ TC ⎫ in buildings and uncertainty involved in analyses. ⎪ (Eq. 3) ⎬ TC These concepts are discussed briefly in the sections + 1 ; T * < TC ⎪ μ =( Rμ − 1) T* ⎭ to follow. The Bridge and Structural Engineer

Volume 45 Number 1 March 2015  67


5.

Damage and its Evaluation

Strength of the structural members eventually degrades as the structural material undergoes degradation (such as cracking in concrete, yielding of steel, concrete creep, fatigue, etc.). The damage in structure is a continuous process, where intermediate stages can be observed starting from no damage to the final collapse in the structure. The stages of damage can be quantified in the capacity curve (or force displacement curve) of the structure, each individual stage being known as damage state. Each damage state varies within a range, the initial point of which is called threshold damage (Fig.3). Threshold represents the value of a particular damage state, which has 50% chance of being exceeded. The quantification of damage and its threshold may be done separately for structural as well as non-structural members in terms of structural displacement or inter-storey drift (ISD). For example, Borzi et al. (2008) classified limit states based on inter-storey rotation capacity; HAZUS (2003) defines the median value of spectral displacement

( S d,Sds ) in terms of drift ratio at the threshold of structural damage state (δR,Sds ) as: Sd ,Sds = δ R,Sds .α2.h  . Here, α2 is the fraction of the building (roof) height at the location of pushover mode displacement and h is the typical roof height of the model building type of interest. Most basic classification was proposed by ATC 20 (1989) in which three damage states were defined based on occupancy criteria (namely, Inspected, Restricted Entry and Unsafe). ATC 40 (1996) and several other researchers, on the other hand, defines the damage states using a combination of structural and non-structural building performance levels. Details of damage states proposed in past literature are given in Table 1. Rossetto and Elnashai (2003) compared various damage states available in literature and proposed damage states for various RC structures. One set of damage states proposed for general RC structures has been updated in this paper, considering recent advances for a combination of structural and non-structural damages.

Table 1: Damage state classification of buildings proposed in past literature (modified from Rossetto and Elnashai 2003)

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Damage thresholds are estimated for a given structural system using relevant literature (or using experimental results, if available), to determine the damage level that is likely to occur for such structural systems, given a hazard (e.g., in Fig. 3 the structural performance falls in between slight and moderate damage levels). This likelihood or the conditional probability of exceedence of a damage level is commonly referred to as fragility. Fragility for a range of hazard level is determined as a function of some intensity measure and plotted as continuous curves, known as fragility curves (discussed later). Although several literatures (as shown in Table 1) exist for theoretical quantification of damage state thresholds, a clear cut separation of different damage states is not feasible. Theoretical values are mostly dependent on yield and ultimate displacement of the structure, accurate estimation of which is difficult and involves large uncertainty. Again the damage states are dependent on the type of structure and damage state definitions will be different for different building typologies; such independent quantification of damage states for different structural typologies rarely exist. Thus, there is also an existing scope for establishing damage states for different typologies experimentally. Uncertainty is inherent in fragility analysis of buildings, assessment of which is discussed in the next section.

6.

Uncertainty Analysis

Uncertainties are associated in every step of probabilistic analysis, whether it is hazard analysis (as mentioned in Section 2), capacity determination (Section 3), damage states (Section 5) or fragility/ vulnerability assessment. Structural reliability approach is one of the important frameworks to deal with uncertainty in performance assessment of structural systems. Different sources of uncertainties were pointed out in past literature and derivation and generalised estimated values of different uncertainties for different building typologies are readily available for use in probabilistic seismic risk assessment methodologies (e.g., HAZUS 2003, FEMA-P-58-1 2012). However, one should be careful about use of these direct values, since they are derived for specific set of data. Kwon and Elnashai (2006), Fatemeh et al. (2010), Meslem and D’ Ayala (2013), Meslem et al. (2014) and D’Ayala et al. (2014) investigated the effect of uncertainty in vulnerability analysis. Total The Bridge and Structural Engineer

uncertainty (βT) is estimated as the sum of individual uncertainty components (βi) considered in analysis (Eq. 5). βT =

∑β

2 i

(Eq. 5)

HAZUS (2003) gives a similar model (Eq.6) for determining the value of log normal standard deviation describing variability for structural damage state (βSds). β Sds = ( CONV[βC ,β D ,Sd ,Sds ]) + ( β M ( Sds ) ) 2

2

(Eq. 6)

Here, βC and βD represent the lognormal standard deviation parameters that describe the variability of the capacity curve and demand spectrum, respectively, βM(Sds) is the lognormal standard deviation parameter that describes the uncertainty in the estimate of the median value of the threshold of structural damage state (ds). It is clear from Eq. 5 and Eq. 6 that uncertainty estimation requires large number of parametric analysis to be carried out. The other alternative is to use specified values in prevalent documents; for example, HAZUS (2003) and FEMA-P-58-1 (2012) give a detailed explanation on various uncertainties and also recommend dispersion values that can be used in the vulnerability assessment. Clearly, there is a huge scope for development of methods for reasonable estimation of these uncertainties in seismic vulnerability analysis. It may be done using the data available in existing literature or by carrying out detailed experimental studies involving different building typologies.

7.

Seismic Fragility Assessment

In a probabilistic seismic risk assessment of any structure, it is of utmost importance to know the probability of failure (Pf) of different components or the global failure of the structure. This can be obtained from the seismic fragility assessment of the structure. Fragility curves represent a continuous relationship between probabilities of exceeding a particular level of damage state versus any parameter describing earthquake intensity measure. A fragility curve for a particular damage state is obtained by computing the conditional probabilities of reaching or exceeding that damage state for a given deterministic estimates of spectral response (e.g., Sd). This conditional probability can be expressed as in Eq.7 (HAZUS 2003). Volume 45 Number 1 March 2015  69


⎡ 1 ⎛ S P[ds|Sd ]=Φ ⎢ ln ⎜ d ⎜ ⎣⎢ β dsi ⎝ Sd ,ds

⎞⎤ ⎟⎟ ⎥ ⎠ ⎦⎥

(Eq. 7)

P[ds|Sd] represents the conditional probability of being in, or exceeding, a particular damage state (ds), given the spectral displacement at the performance point (Sd),Ф is the standard normal cumulative distribution function and βdsi is the normalised standard deviation of the natural logarithm of the displacement (damage) threshold ( d,ds) indicating uncertainties in capacity estimation, damage levels, modelling errors and ground shaking. Fig. 4 shows fragility curves obtained for a range of total uncertainty(βT), for a 3 bay, 4 storied RC bare frame. The dashed line represents the probability of damage with no uncertainty. Spectral displacement at 50 % probability value represents the damage state median value (damage threshold).As the value of βT increases the curve flattens, implying a wider range of demands over which there is a significant probability that the damage state will initiate or exceed. It can be inferred from Fig. 4 that after reaching an uncertainty of about 0.7, the difference in estimation of P[ds|Sd] for a particular intensity measure is not significant. On the other hand, the effect of uncertainty on collapse fragility curve was studied by Zareian and Krawinkler (2007) with different confidence level in median value of the collapse capacity and found that uncertainties have a significant effect on the collapse fragility. More research work is required to be carried out in order to get more insight on influence of uncertainty on vulnerability assessment.

Fig 4: Effect of total uncertainty (βT) on the construction of fragility curves

Fragility or failure probability can also be represented for discrete intensity measures rather than a continuous 70  Volume 45

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form or curve in the form of Damage Probability Matrix (DPM). DPM specifies the discrete probability (pik) of reaching or exceeding a particular damage state (dsi) for a given ground motion intensity (yk). This conditional probability is defined as in Eq. 8. = pik P= [D dsi = | Y yk ]

(Eq. 8)

where D and Y are any parameter representing damage and ground motion intensity in the building. Each of the discrete damage probabilities is written in a matrix form for different yk values and hence, the name Damage Probability Matrix. Whitman et al. (1973) were the first to have systematically compiled statistics on both structural and non-structural damage to various building typologies from damage experiences after 1971 San Fernando earthquake. ATC 13 (ATC 1985) provided the concept of DPM based on expert opinion. Eleftheriadou and Karabinis (2011) also provided empirical DPM using a simple relation between PGA and macro-seismic intensity. Singhal and Kiremidjian (1996) presented a method to estimate DPM based on nonlinear dynamic analysis of structures rather than based on empirical data. For any given value of spectral response, discrete damage state probabilities are calculated as the difference of the cumulative probabilities of reaching, or exceeding, successive damage states. If the probability of complete damage is represented as P[C] = P[C/Y], the discrete damage probabilities of different damage states – none (N), slight (S), moderate (M), extensive (E) and complete (C) can be expressed as: P[E] = P[E/Y] – P[C/Y], P[M] = P[M/Y] – P[E/Y], P[S] = P[S/Y] – P[M/Y], P[N] = 1 – P[S/Y]. These discrete probabilities are represented graphically in Fig. 5 (b) as probability histograms obtained from fragility curves of Fig 5(a) for Sd of 0.07 m. Difference between consecutive damage states are shown as different damage grades (dg) in Fig. 5.

(a)

(b)

Fig 5: (a) Fragility curve representing the continuous and discrete probabilities, (b) Discrete damage probabilities represented as histogram for Sd of 0.07 m

The Bridge and Structural Engineer


To assess uncertainty and explore variability in building performance, this process would ideally involve performing a large number of structural analyses, using a large suite of input ground motions and analytical models with properties that have been randomly varied. Such an approach would be impractical for implementation in practice. Therefore, FEMA-P-58-1 (2012) suggests a Monte Carlo procedure to assess a range of possible outcomes given a limited set of inputs. This procedure involves performing limited suites of analyses to derive a statistical distribution of demands from a series of building response states for a particular intensity of motion. From this distribution, statistically consistent demand sets are generated representing a large number of possible building response states. These demand sets, together with fragility and consequence functions, are used to determine a building damage state and compute consequences associated with that damage.

8.

Seismic Vulnerability Assessment

Once the fragility curves or DPM are developed, the next obvious step is to assess the seismic vulnerability. Fragility and vulnerability are often used interchangeably. However, vulnerability is different from fragility since some loss estimation (structural, monetary, life, etc.) is associated with vulnerability estimation. Different methods of seismic vulnerability assessment can be broadly categorized into: empirical, analytical, judgemental and hybrid methods (Lang 2002, Calvi et al. 2006). In addition, Rapid Visual Screening (RVS) of buildings is often carried out to quickly assess health of buildings. Empirical Vulnerability Assessment Empirical vulnerability assessment is based on quantitative or qualitative analysis of post-earthquake damage data and its distribution. Empirical vulnerability functions can be constructed directly from observations of damage and losses due to past earthquakes collected over sites affected by different intensities of ground motion. This method has been used quite commonly in the past and empirical vulnerability curves have been derived (Sabetta et al. 1998, Orsini 1999, Rossetto and Elnashai 2003, Colombi et al. 2008, Jaiswal et al. 2011, Lang et al. 2012 and Rossetto et al. 2013). The major problem with empirical studies is that these are generally based The Bridge and Structural Engineer

on limited damage data observed in the field covering only a limited range of earthquake magnitudes (or intensities) and is limited to the locality for which the damage data is extracted. Vulnerability curves derived using this approach will be used for the particular location only or any other location with similar site characteristics. However, if empirical vulnerability curve is based on large reliable data, it represents real damage incorporating all the factors present at the site, which are otherwise difficult to model. Analytical Vulnerability Assessment Borzi et al. (2008) and Haldar and Singh (2009) derived analytical vulnerability curves as the probability of accidence of limit state with respect to ground motion intensity. Karbassi and Nollet (2013) gave useful methods for analytical vulnerability assessment for URM buildings for which incremental dynamic analysis using applied element method was carried out. D’Ayala et al. (2014) defined vulnerability curves as conversion of physical damage (as obtained from fragility analysis) into monetary loss, given a measure of ground motion intensity. This analytical transformation is done using the probability relation of Eq. 9 E(C > c | im) =

n

∑ E(C > c | ds ).P(ds | im) i

i =0

i

(Eq. 9)

where, n is the number of damage states considered, P(dsi|im) is the fragility term for a damage state (dsi) and intensity measure (im); E(C > c | dsi) is the complementary cumulative distribution of the loss (cost) given dsi; E(C > c | im) is the complementary cumulative distribution of the loss (cost) given a level of im. The distribution of loss can then be used to estimate the building repair cost given a damage state (dsi) threshold, using Eq. 10 (D’Ayala et al. 2014). E (C | dsi ) = E ( LabCost | Area _ dsi ) + E( MatCost | Area _ dsi )

(Eq. 10)

where, E(LabCost| Area_dsi) is the local labour cost in the considered region (cost per percentage of damaged area) given dsi; E(MatCost | Area_dsi) is the local material cost in the considered region (cost per percentage of damaged area) given dsi. However, as defined by Shah (1984) vulnerability signifies the degree of loss, given a hazard and thus, it can also be defined in terms of structural loss, human casualties, etc. Advantage of analytical vulnerability assessment is that it gives an opportunity to consider multi-variant Volume 45 Number 1 March 2015  71


modeling of behavior of structures for real conditions based on detailed numerical analysis. Judgemental Vulnerability Assessment When the use of either empirical or analytical method is not feasible, opinions from experts are sought based on their expertise in vulnerability assessment. A group of experts from related field are asked to provide estimates of mean and standard deviation of the building damage distribution based on their experience of observing building performance in past earthquakes. The same disadvantages as in empirical method exist here also; in addition, the individual opinion of experts is mostly biased. Vulnerability expressed in the form of DPM, based on expert judgement and opinions were first introduced in ATC 13 (ATC 1985), which has been implemented in risk and loss study by Cardona and Yamin (1997) and Fah et al. (2001). Hybrid Vulnerability Assessment

structural vulnerability assessment procedure.

9.

Seismic Risk Estimation

A natural output of seismic vulnerability analysis of buildings is seismic risk. As defined in Shah (1984), risk is the probability that social or economic consequences of earthquake (losses of lives, persons injured, property damaged and economic activity disrupted) will equal or exceed specified values at a site, at several sites, or in an area, during a specified exposure time. Mathematically, risk is a product of hazard, vulnerability and exposure time of elements. Several additional details, e.g., building occupancy details, population density at different times, age of buildings, code compliance, different costs, etc., are required in order to estimate seismic risk associated with the building during a specified time. The losses per scenario expressed against annual probability give the risk curve (McGuire 2004). It is widely accepted that, seismic hazard curve is linear in log-log scale and capacity curve is log-normally distributed. Under such assumptions, probability of failure is calculated as in Eq.11 and Eq. 12, which are known as risk equations.

Hybrid methods generally combine the advantages of empirical and analytical methods; analytical estimates of performance are updated using empirical studies in hybrid method. Kappos et al. (2006) 1 PF = γ (a * )(a * / cˆ ) K exp[ (K H β c ) 2 ] (Eq. 11) derived vulnerability curves in terms of the PGA as 2 well as spectral displacement and further calibrated 1 ⎡ ⎤ PF γ (a* )(f s )− K exp ⎢− x p KH βc + (KH βc )2 ⎥   ( Eq. 12) = the curves using the damage database for several RC 2 ⎣ ⎦ and URM buildings in Greece. where, γ(a*)is annual frequency of exceedence of design amplitude a*, fs is a factor of safety, ĉ ismedian Rapid Vulnerability Assessment structural capacity, KH is slope of the hazard curve in log-log space, βc is log normal standard deviation of Often, rapid assessment of vulnerability in an area is the structural capacity curve, xp is number of standard done through visual inspection of the buildings and deviation corresponding to the high confidence of by providing a performance score using predefined low probability of failure (HCLPF), for a normal forms based on adequate seismic resistance features distribution. Traditionally, these equations have been visible in the building. This process is known as used to estimate risk in terms of stating only whether Rapid Visual Screening (RVS), since one can cover the structure is “safe” or “failed”. For estimating risk large numbers of buildings within a short period of associated with seismic performance levels, same time. FEMA-154 (Rojahn and Scawthorn 2002) equations apply with βc = βds (logarithmic standard provides the detailed procedure for carrying out RVS deviation of spectral displacement for damage of buildings. Recently, Jain et al. (2010) proposed state, ds). a statistical model for determining the performance score based on multiple linear regressions, where Seismic vulnerability and risk estimation methodologies are formulated by various past several vulnerability parameters (such as presence researchers for easy to use purpose. SELENA (Molina or absence of basement, re-entrant corners, short et al. 2010) is an open source tool that provides for columns, etc.) were considered. Although this method deterministic, probabilistic as well as real time seismic of vulnerability assessment is fast, it is limited for the risk analysis and loss estimation. Comprehensive inspected area only and is mostly a post-earthquake H

H

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The Bridge and Structural Engineer


Approach to Probabilistic Risk Assessment (CAPRA) (Marulanda et al. 2013) is another open source probabilistic seismic risk assessment tool.

10. Concluding Remarks Seismic vulnerability and risk assessment of buildings is extremely important in seismic-prone regions for developing strengthening strategies and post-earthquake disaster management plans. Methodologies available in literature on seismic vulnerability and risk assessment of buildings have been reviewed in the paper. The procedure is quite complex as it requires huge amount of data pertaining to building typologies and materials and large computational efforts for dealing with uncertainties, modeling, etc. Various methods used in the past to quantify all major components required in vulnerability/risk assessment have been compared using a step-by-step approach and pros and cons of various methodologies have been discussed. The first step is the assessment of seismic hazard (deterministic or probabilistic) at the site from which the seismic demand curve (response spectrum) is obtained. At present, there is lack of site-specific ground hazard maps and ground motion data in different regions of India. Existing buildings at the site are grouped into categories (building inventories) and structural capacity is estimated for representative building stock. Seismic fragility curves are then developed for different damage states taking into account uncertainty associated with every step of analysis. Building inventories of different regions need to be updated so that seismic vulnerability can be estimated on regional basis by combining loss probability with the seismic fragility. It is also equally important to update the analytical vulnerability curves with observed damage data during past earthquakes. This may lead to development of region-wise vulnerability that may further assist in estimation of seismic risk due to particular hazard for a given area and for a certain reference period. Finally, such data is essentially required for development of seismic risk map for that region.

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Kyriakides, N., and Silva, V. (2014). "Seismic vulnerability assessment of historical masonry structural systems". Engineering Structures, 6263, pp: 118–134. 3.

ATC (1985).“Earthquake Damage Evaluation Data for California, Report ATC-13”, Applied Technology Council, Redwood City, CA.

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Brzev, S., Scawthorn, C., Charleson, A.W., Allen, L., Greene, M., Jaiswal, K., and Silva, V. (2013). "GEM Building Taxonomy Version 2.0". GEM Technical Report 2013-02, 180 pp., GEM Foundation, Pavia, Italy.

10. Calvi, G.M., Pinho, R., Magenes, G., Bommer, J.J., and Crowley, H. (2006). "Development of Seismic Vulnerability Assessment Methodologies Over The Past 30 Years". ISET Journal of Earthquake Technology, Vol. 43, Issue 472, pp: 75–104. 11. Cardona, O. D., & Yamin, L. E. (1997). “Seismic microzonation and estimation of earthquake loss scenarios: integrated risk mitigation project of Bogotá, Colombia”. Earthquake Spectra, EERI, Volume 45 Number 1 March 2015  73


Vol. 13, Issue 4, pp: 795-814. 12. Chopra, A.K. and Goel, R.K., (1999a). "A report on Capacity-Demand-Diagram Methods for Estimating Seismic Deformation of Inelastic Structures: SDF Systems". Pacific Earthquake Engineering Research Center. University of California, Berkeley. 13. Chopra, A.K. and Goel, R.K., (1999b). "Capacity-Demand-Diagram Methods Based on Inelastic Design Spectrum". Earthquake Spectra, EERI, Vol. 15, Issue 4, pp: 637–656. 14. Colombi, M., Borzi, B., Crowley, H., Onida, M., Meroni, F., & Pinho, R. (2008). Deriving vulnerability curves using Italian earthquake damage data. Bulletin of Earthquake Engineering, Vol. 6, Issue 3, pp: 485-504. 15. D’Ayala D., Meslem A., Vamvatsikos D., Porter, K., Rossetto T., Crowley H., Silva V. (2014). “Guidelines for Analytical Vulnerability Assessment of low-mid-rise Buildings – Methodology”, GEM Technical Report, GEM Foundation, Pavia, Italy. 16. Dowrick, D. J. (2009).“Earthquake resistant design and risk reduction”. John Wiley & Sons. 17. Eleftheriadou, A.K. and Karabinis, A.I. (2011). "Development of damage probability matrices based on Greek earthquake damage data". Earthquake Engineering and Engineering Vibration, Vol. 10, Issue 1, pp: 129–141. 18. EMS 98 (1998)."European Macroseismic Scale 1998". European Seismological Commission. Luxembourg, Joseph Beffort, Helfent-Bertrange. 19. Fäh, D., Kind, F., Lang, K., & Giardini, D. (2001). “Earthquake scenarios for the city of Basel”. Soil Dynamics and Earthquake Engineering, Vol. 21, Issue 5, pp: 405-413. 20. Fajfar, P, and Gašperšič, P. (1996). "The N2 method for the seismic damage analysis of RC buildings." Earthquake Engineering & Structural Dynamics, Vol. 25, Issue 1, pp: 31-46. 21. Fajfar P. (2000). "A Nonlinear Analysis Method for Performance Based Seismic Design". Earthquake Spectra, EERI, Vol. 16, Issue 3, pp: 573–592. 22. Fatemeh, J., Iervolino, I. and Manfredi, G. 74  Volume 45

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(2010). "Structural modeling uncertainties and their influence on seismic assessment of existing RC structures". Structural Safety, Vol. 32, Issue 3, pp: 220-228. 23. FEMA 273 (1997). "NEHRP Guidelines For The Seismic Rehabilitation Of Buildings". Federal Emergency Management Agency, Washington, D.C. 24. FEMA 356 (2000). "Prestandard and commentary for the seismic rehabilitation of buildings". Federal Emergency Management Agency, Washington, D.C. 25. FEMA 440 (2005). "Improvement of Nonlinear Static Seismic Analysis Procedures". Federal Emergency Management Agency, Washington, D.C. 26. FEMA-P-58-1 (2012). “Seismic Performance Assessment of Buildings, Volume 1 – Methodology”. Federal Emergency Management Agency, Washington, D.C. 27. FEMA P-695 (2009). “Quantification of Building Seismic Performance Factors”. Federal Emergency Management Agency, Washington, D.C. 28. Freeman, S.A. (1998). "The Capacity Spectrum Method as a Tool for Seismic Design". Proceedings of the 11th European conference on earthquake engineering, pp: 6–11. 29. Haldar, P and Singh, Y. (2009). "Seismic performance and vulnerability of Indian code designed RC frame buildings." ISET Journal of Earthquake Technology, Vol. 46, Issue 1, pp: 29-45. 30. HAZUS (2003). “Multi-hazard Loss Estimation Methodology, HAZUS-MH MR4 Technical Manual”, FEMA-NIBS (Federal Emergency Management Agency - National Institute of Building Sciences), Washington, D.C. 31. Jain, S.K., Mitra, K., Kumar, M., and Shah, M. (2010). "A Proposed Rapid Visual Screening Procedure for Seismic Evaluation of RC-Frame Buildings in India". Earthquake Spectra, EERI, Vol. 26, Issue 3, pp: 709–729. 32. Jaiswal, K., Wald, D., and Porter, K. (2010).

The Bridge and Structural Engineer


"A Global Building Inventory for Earthquake Loss Estimation and Risk Management". Earthquake Spectra, EERI, Vol. 26, Issue 3, pp: 731–748. 33. Jaiswal, K., Wald, D., & D’Ayala, D. (2011). “Developing empirical collapse fragility functions for global building types”. Earthquake Spectra, EERI, Vol. 27, Issue 3, pp: 775-795. 34. Kappos, A. J., Panagopoulos, G., Panagiotopoulos, C., & Penelis, G. (2006). “A hybrid method for the vulnerability assessment of R/C and URM buildings”. Bulletin of Earthquake Engineering, Vol. 4, Issue 4, pp: 391-413. 35. Kappos, A. J., and Panagopoulos, G. (2010). “Fragility curves for reinforced concrete buildings in Greece”. Structure and Infrastructure Engineering, Vol. 6, Issue: 1-2, pp: 39-53. 36. Karbassi, A. and Nollet, M. J. (2013). “Performance-Based Seismic Vulnerability Evaluation of Masonry Buildings Using Applied Element Method in a Nonlinear Dynamic-Based Analytical Procedure”. Earthquake Spectra, EERI, Vol. 29, Issue 2, pp: 399-426. 37. Kramer, S. L. (1996) “Geotechnical Earthquake Engineering”. Prentice Hall, New Jersey. 38. Kwon, O. S., & Elnashai, A. (2006). “The effect of material and ground motion uncertainty on the seismic vulnerability curves of RC structure”. Engineering Structures, Vol. 28, Issue 2, pp: 289-303. 39. Lagomarsino, S. and Giovinazzi, S. (2006). “Macroseismic and mechanical models for the vulnerability and damage assessment of current buildings”. Bulletin of Earthquake Engineering, Vol 4, Issue 4, pp: 415–443. 40. Lang, D. H., Singh, Y. and Prasad, J. S. R. (2012). "Comparing empirical and analytical estimates of earthquake loss assessment studies for the city of Dehradun, India." Earthquake Spectra, EERI, Vol. 28, Issue 2, pp: 595-619. 41. Lang, K. (2002). “Seismic vulnerability of existing buildings”. PhD thesis. Institute of The Bridge and Structural Engineer

Structural Engineering, Swiss Federal Institute of Technology, Zürich, Switzerland. 42. Marulanda, M.C., Carreño, M.L., Cardona, O.D., Ordaz, M.G., and Barbat, A.H. (2013). "Probabilistic earthquake risk assessment using CAPRA: application to the city of Barcelona, Spain". Natural Hazards, Vol. 69, Issue 1, pp: 59–84. 43. McGuire, R. K. (2004). “Seismic hazard and risk analysis”. Earthquake Engineering Research Institute. 44. Meslem, A. and D’ Ayala, D. (2013). “Investigation into analytical vulnerability curves derivation aspects considering modelling uncertainty for infilled RC buildings”. In M. Papadrakakis, V. Papadopoulos, & V. Plevris (Eds.), 4th ECOMAS thematic conference on computational methods in structural dynamics and earthquake engineering COMPDYN 2013, Kos Island, Greece, 12–14 June 2013. 45. Molina, S., Lang, D.H., and Lindholm, C.D. (2010). "SELENA – An open-source tool for seismic risk and loss assessment using a logic tree computation procedure". Computers & Geosciences, Vol. 36, Issue 3, pp: 257–269. 46. Orsini, G. (1999). “A Model for Buildings' Vulnerability Assessment Using the Parameterless Scale of Seismic Intensity (PSI)”. Earthquake Spectra, EERI, Vol. 15, No. 3, pp. 463-483. 47. Rojahn, C. and Scawthorn, C. (2002). "FEMA154, Rapid Visual Screening of Buildings for Potential Seismic Hazards: A Handbook". Federal Emergency Management Agency. Washington, D.C. 48. Rossetto, T. and Elnashai, A. (2003). "Derivation of vulnerability functions for European-type RC structures based on observational data." Engineering structures, Vol. 25, Issue 10, pp:1241-1263. 49. Rossetto T., Ioannou, I., and D.N., G. (2013). "Existing Empirical Fragility and Vulnerability Relationships: Compendium and Guide for Selection". GEM Technical Report 2013-X, GEM Foundation, Pavia, Italy. Volume 45 Number 1 March 2015  75


50. Sabetta, F., Goretti, A. and Lucantoni, A. (1998). “Empirical Fragility Curves from Damage Surveys and Estimated Strong Ground Motion”, Proceedings of the 11th European Conference on Earthquake Engineering, Paris, France, pp. 1-11. 51. Shah, H. C. (1984). "Glossary of terms for probabilistic seismic-risk and hazard analysis." Earthquake spectra, EERI, Vol. 1, Issue 1, pp: 33-40. 52. Singhal, A. and Kiremidjian, A.S., (1996). "Method For Probabilistic Evaluation Of Seismic Structural Damage". Journal of Structural

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Engineering, ASCE, 122 (12), pp: 1459–1467. 53. Whitman, R. V., Reed, J.W., and Tien, H.S. (1973). "Earthquake Damage Probability Matrices". Proceedings of 5th World Conference on Earthquake Engineering, Vol. 2, pp: 2531– 2540. 54. Zareian, F., & Krawinkler, H. (2007). “Assessment of probability of collapse and design for collapse safety”. Earthquake Engineering & Structural Dynamics, Vol. 36, Issue 13, pp: 1901-1914.

The Bridge and Structural Engineer


SEISMIC VULNERABILITY ANALYSIS OF BRIDGE PIER DESIGNED WITH DIFFERENT CODAL PROVISIONS

Shivang SHEKHAR Post-graduate student, Department of Earthquake Engineering, IIT Roorkee, Roorkee - 247667 shivang12287@gmail.com

Pankaj AGARWAL Professor, Department of Earthquake Engineering, IIT Roorkee, Roorkee - 247667 panagfeq@gmail.com

Shivang Shekhar obtained his Bachelor of Engineering in Civil Engineering from Jadavpur University, Kolkata in the year 2010. He worked as Structural Design Engineer in TATA Consulting Engineers Ltd from 2010-2013. Afterwards, he completed M. Tech programme in Earthquake Engineering with specialization in Structural Dynamic from IIT Roorkee in the year 2015. At present, he is perusing Ph. D from the Civil Engineering Department of IIT Bombay.

Dr. Pankaj Agarwal is presently working as Professor at the Department of Earthquake Engineering, Indian Institute of Technology Roorkee. His current research interests are earthquake resistant design of masonry and RC structures, post damage assessment survey of earthquake-affected areas, cyclic testing of structures, seismic instrumentation in multistoried building, health monitoring and damage detection in buildings. Dr. Pankaj has co-authored one text book on “Earthquake Resistant Design of Structures” published by Prentice-Hall of India, New Delhi, 2006. He has also published a number of research papers in national and international journals.

Abstract Failure of bridge piers in the past major earthquakes has greatly influenced the seismic design practices at the national and international levels and has enforced to update the seismic design provisions. The present study highlights the importance of confining reinforcement at the plastic hinge region of piers under different criteria. The seismic design provisions of plastic hinge region given by the major international codes are selected and compared with the Indian code of practices. Seismic vulnerability of the bridge piers under confined and unconfined conditions is carried out by developing fragility curves using incremental dynamic analysis for different levels of peak ground accelerations.

Introduction Highway bridges are in the category of post earthquake importance type of structures which The Bridge and Structural Engineer

are expected to remain functional even after major earthquakes. The damage in bridges not only causes loss of life of people but also affects relief and restoration activities after the earthquakes. The extensive damage of bridges in past earthquakes and lesson learnt from the behaviour of bridges in such events result in updating of seismic codes throughout the world. Seismic behaviour of bridges is entirely different from other structures like buildings because of a little or no redundancy in the structural systems. As a result there is no mobilization of alternative load paths and the consequence is failure of one structural element or connections that may cause collapse of the bridge (Priestley et. al., 1996). As per the seismic design philosophy of codes of bridges, the dissipation of energy for the post yield behaviour is through the formation of plastic hinges in piers rather than the other components like superstructure or foundations. The failure or damage Volume 45 Number 1 March 2015  77


of superstructure leads to disruption of traffic and yielding at foundation leads to in accessibility of retrofitting after earthquakes. Therefore, piers are the only component in which yielding is allowed to dissipate the earthquake energy that is responsible for inelastic actions. This inelastic action of the bridge pier should be in ductile mode by avoiding the brittle mode of failure. The past damage reports identify the number of brittle modes of failure in bridge piers that may be mainly due to inadequate transverse reinforcement. The un-confinement of pier leads to crushing of concrete core and buckling of longitudinal reinforcement resulting in rapid strength degradation. It causes inability of the pier to sustain even the gravity load. Pre-mature termination of longitudinal reinforcement, in-adequate anchorage of flexural reinforcement in cap beam and foundation and in-adequate lap splices are also the cause of brittle mode of failure that limits the curvature ductility. Most of these brittle mode failures can be easily avoided by providing the confinement reinforcement in plastic hinge region on the basis of core confinement, anti-buckling of longitudinal reinforcement and shear failure. Therefore, ductility provisions in seismic design codes of bridges around the world are concentrated on the above criteria. In this study, seismic design provisions based on confinement are given by the ATC-32 (1996), NZS 3101(2006),AASTHO (2012) and Euro code (BS EN 1998 -1&2, 2004, 2005) are reviewed and compared with the provisions given in Indian code IS 1893 (Part 3): 2012.

Brief Review of Seismic Design Provisions for Confinement in Bridge Pier Failure of bridge piers in the past earthquakes underlines the importance of confining reinforcement in plastic hinge region of the pier because it enhances strength and ductility, the two main attributes of earthquake resistant design. Confining reinforcement in the form of closed hoop ties or spirals serves a threefold purpose i.e. a)

Confines the core of the concrete of bridge pier that is responsible to sustain the flexure stain and thereby enhance its strength and deformation characteristics

b)

Controls the buckling of longitudinal reinforcing bars

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c)

Provides adequate shear strength

There are different recommendations for minimum amount of transverse reinforcement in plastic hinge zones given by different International as well as Indian codes. These provisions are briefly summarized in Table 1 to 3 based on confinement, anti-buckling and shear resistance. Table 1: Seismic design provisions for the minimum amount of confinement reinforcement in plastic hinge region as per different codes Codes ATC-32

Transverse reinforcement based on core confinement

ρs is volumetric area of transverse reinforcement, f'ce is the expected strength of concrete, fye is the expected yield strength of steel, Ag is the gross area of the section, ρl is the longitudinal reinforcement ratio, P is column axial compressive load.

AAHSTO

Remarks-The equation is based on geometric properties of the section, material properties of concrete and reinforcement, column axial load as well as longitudinal reinforcement ratio.

ρs is volumetric area of transverse reinforcement, f’c = specified compressive strength of concrete, fyt is the specified yield strength of transverse steel, Ag is the gross area of the section, Ac is the core area of member measured to the outside of confinement. Remarks-The first equation depends on geometric properties of the bridge section, material properties of concrete and reinforcement and it ensure that the core of the column/pier can sustain axial load after exterior cover has spalled off of the section. The second equation is given for large pier for which Ag/Ac ratio is close to unity. This ensures minimum amount of volumetric ratio of circular hoop enabling adequate flexural capacity in plastic hinge regionor yielding region.

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Table 2: Minimum amount of transverse reinforcement on the basis of anti-buckling of longitudinal reinforcement as per different codes

EUROCODE (BS EN 1998 -1&2)

Codes ω,wd,c is the mechanical reinforcement ratio, Ac is the gross concrete area, Acc is the confined concrete area of the section, λ is a factor, ηk = (NEd/Acfck) is the axial compression ratio, ρl is the longitudinal reinforcement ratio, fyd is design value of yield strength of steel, fcd is the design value of concrete compressive strength, ρw is the transverse reinforcement ratio = 4Asp/DspS

NZS 3101

Remarks- The equation is based on geometric properties of the section of pier, material properties of concrete and reinforcement, column axial load as well as longitudinal reinforcement ratio.

ρs is volumetric area of transverse reinforcement, ρt = Ast/Ag, m = fy/(0.85fc’), Ag is the gross area of the section, Ac is the core area of pier measured to the centre of confinement, f’c = specified compressive strength of concrete, fyt is the lower characteristic yield strength of transverse steel, φ is the strength reduction factor, N* is column axial compressive load.

IS 1893 (Part3)

Remarks - All parameters such as material and geometric property, axial load and longitudinal reinforcement ratio are taken into account. Additional factor strength reduction factor (Φ)is introduced for piers which are not protected by capacity design.

Ash is area of cross-section of hoop, S is the spacing of hoops, Dk is the diameter of core measured to the outside of hoop, fck is the characteristic strength of concrete, fy is the yield stress of transverse steel, Ag is the gross area of section, Ac is the concrete core. Remarks - The equation is based on geometric properties of the section of pier, material properties of concrete and reinforcement. The effect of axial load is not considered.

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Confining reinforcement on the basis of anti-buckling of longitudinal reinforcement

ATC-32

ρs is volumetric area of transverse reinforcement, nb is the number of longitudinal bars contained by circular hoop, fu is the ultimate strength of reinforcement, fy is the yield strength of reinforcement, dbl is the diameter of longitudinal reinforcement. NZS 3101 ρs is volumetric area of transverse reinforcement, Ast is the total area of longitudinal reinforcement, fyt is the lower characteristic yield strength of transverse steel, fys is the lower characteristic yield strength of longitudinal reinforcement, d’’ is the depth of concrete core measured to the centre of confinement, db is the diameter of reinforcing bar. AASHTO s ≤ [6db, D/4, 4 inch (101.6 mm)] s is the spacing in mm, db is diameter of longitudinal reinforcement, D is minimum dimension of member. EUROCODE (BS EN 1998 -1&2) ftk and fyk is the characteristic tensile strength and characteristic yield strength of transverse reinforcement respectively, dbl is the diameter of longitudinal bars, At is the area of tie leg in mm2, sL is the spacing of the legs in m, Σ As is the sum of the areas of longitudinal bars restrained by one tie in mm2, fys and fyt is the yield strength of longitudinal bars and tie respectively. IS 1893 (Part 3):

s ≤ 6db

Volume 45 Number 1 March 2015  79


Table 3: Amount of transverse reinforcement on the basis of shear resistance as per different codes Codes

Transverse reinforcement on the basis of Shear resistance

AASHTO

Vu is the factored shear force, Vn is the nominal shear strength, φ is the capacity reduction factor for shear = 0.85, Vc is the nominal shear strength provided by the core concrete, Vs is the shear resistance provided by transverse reinforcement, Nu is axial compressive load, Ag is the gross area of section, fc’ specified compressive strength of concrete, Av is the area of shear reinforcement, fyh is the yield stress of transverse reinforcement, s is the spacing of tie, d is the distance from the extreme compression fibre to the centroid of farthest tension steel. NZS 3101

V* is the design shear action at the section derived from ultimate limit state, Vn is the nominal shear strength, Ka is a constant depends on aggregate size, Kn is a factor for axial load, ρw is the effective area of flexural tension reinforcement, fc’ is the specified compressive strength of concrete, Ah is the area of shear reinforcement, fyt is the yield stress of transverse reinforcement, s is the spacing of tie, d’’ is the distance between centreline of peripheral tie. ATC-32

Vn is the nominal shear strength, φ is the capacity reduction factor for shear, Vc is the nominal shear strength provided by the core concrete, Vs is the shear resistance provided by transverse reinforcement, K = 0.5 for end region, Pe is axial compressive load, Ag is the gross area of section, fc’ specified compressive strength of concrete, Ash is the area of shear reinforcement, fyh is

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the yield stress of transverse reinforcement, s is the spacing of tie, D’’ is the distance between centreline of peripheral tie IS 1893 Vu is the factored shear force, Vc is the (Part 3): nominal shear strength provided by the core concrete, Vs is the shear resistance provided by transverse reinforcement, Pu is axial compressive load, Ag is the gross area of section, Ag is the gross area of section, fck specified compressive strength of concrete, Asv is the area of shear reinforcement, fy is the yield stress of transverse reinforcement, s is the spacing of tie, τc is the nominal shear stress of concrete which depends on various factors such as grade of concrete and percentage of tension reinforcement.

Numerical Application of Seismic Design Provisions in Bridge Pier A numerical study is carried out to illustrate the effect of confining reinforcement on the performance of bridge and is simultaneously compared with the amount of reinforcement given by the different codes. The adequacy of confining reinforcement provisions is compared by considering a typical 2 lane, 4 span simply supported bridge having span length of 25 m. The cross section details of the bridge pier and superstructure are shown in Figure 1. The nominal concrete strength for substructure and superstructure is M 35 and M 40 respectively. The dead weight of superstructure is 105.2 kN/m which produces gravity dead load of 2630 kN on the pier. The pier is also designed as per IS 456: 2000 for conventional load i.e. dead load and live load. The lateral seismic forces are determined as per IS 1893 (Part 3): 2002 by assuming that the bridge lies in seismic zone V, with importance factor (I) of 1.2 on hard soil sites. The percentage of longitudinal reinforcement of bridge pier for conventional design as well as for seismic design is also shown in Figure 1. It also shows the details of confining reinforcement provided in plastic hinge zone as per different codes. Table 4 shows details of transverse reinforcement as per different codes based on maximum core-confinement, antibuckling and shear resistance.

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Table 4: Details of transverse reinforcement Confinement CODES

Antibuckling

Shear Spacing Provided

Ď s

Spacing reqd

Spacing reqd

Vc (kN)

Vs (kN)

Spacing reqd

AASHTO

0.0081

92

101.6

2162

Vc>Vn

Min

90

EUROCODE

0.0078

95

160

1875

Vc>Vn

Min

90

ATC-32

0.0076

100

103

1325

217

Min

100

IS 1893

0.0076

100

192

1350

158

Min

100

NZS3101

0.0063

120

153

1561

Vc>Vn

Min

120

The shear demand (factored shear force Vu) in considered numerical example is 1312 kN. The transverse reinforcement in the plastic hinge region of bridge pier is governed by confinement of core reinforcement.

Fig. 1: Cross -section details of bridge pier

Moment Curvature and Pushover Analysis The moment-curvature analysis of the given section under two conditions i.e. unconfined (conventional design) as well as confined state (seismic design by the various codes) is carried out in SAP 2000. Mander, 1998 confined model is used for confined concrete and Park and Ann, 1985 steel model is used for the reinforcement to determine the moment -curvature relationship as shown in Figure 2. The curvature ductility of pier under the confined condition is nearly 14 while it becomes 6.5 in unconfined condition.

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Fig. 2: Moment- curvature relationships and Pushover Curve of bridge pier

Nonlinear static pushover analysis is also carried out on the 2D model of the bridge to predict the lateral response of bridge pier by considering nonlinearity of material and geometry. Non-linearity in pier is modelled with lumped plasticity model at the centre of plastic hinge region at fixed support. In the considered numerical example, the transverse reinforcement in the plastic hinge region of bridge pier is governed by minimum spacing requirement of the confining reinforcement. There is no significant difference in minimum requirements for amount and spacing of transverse reinforcement of the bridge pier by the Volume 45 Number 1 March 2015â&#x20AC;&#x192; 81


provisions given in different codes. The idealized bi-linear pushover curves also reflect the same performance. However the capacity of conventional pier is significantly lower than the confined piers as shown in Figure 2. Thus for further analysis now bridge piers are differentiated on the basis of two cases i.e. conventionally designed (un-confined) analysis as case 1 and seismically designed (confined) analysis as case 2.

Table 5: Modal Properties of Bridge Analysis Case 1

Analysis Case 2

Direction

Degree of freedom

Transverse

Longitudinal

0.00

Transverse

0.68

0.68

Rotational

0.59

0.59

Longitudinal Longitudinal

Modal Mass Time Modal Mass Time Participation Period Participation Period factor factor

0.69

0.36

0.42

0.00

0.61

3D Analytical Model of Bridge

Transverse

0.00

0.00

The 3D finite element modelling of the bridge is carried out for the fragility analysis of the bridge. Superstructure is modelled as elastic beam element (spine modelling) which passes through the centroid of the cross-section and is connected using rigid links. Piers are modelled as beam-column elements and P-M2-M3 hinges are assigned at the ends of the piers. The expansion joint is modelled using gap element for compression. The bearings are modelled by using two nodes with zero length link element. In analysis of case 1, rocker and roller type steel bearings are used at two end of the span. In analysis of case 2, elastomeric bearing is used which is also modelled by using link element. Effective stiffness is used to consider the nonlinear behaviour of elastomeric bearing. The bridge is longitudinally free up to the maximum elastomer flexibility but is blocked transversely by concrete shear keys i.e. elastomeric pads do not displace in the transverse direction. Foundation is assumed as rigid support to pier. The complete finite element model of the bridge is shown in Figure 3. The modal analysis results of the bridge for both analysis cases and results are listed in Table 5. The bridge longitudinal vibration consists of mainly longitudinal deformation but transverse vibration consists of significant rotational component along with transverse deformation.

Rotational

0.05

0.01

Fig. 3: 3D Finite element model of bridge

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0.36

1.58

Fragility Analysis of Bridge Fragility curves are widely used to represent the seismic vulnerability of structure (C) under varying seismic hazard or demand (D). It describes the probability of exceeding a damage state as a function of ground motion intensity measure (IM) parameter. Fragility = P(D ≥ C / IM) (1) In this study, incremental dynamic analysis is used to develop fragility curves. This method requires the development of a finite element model of bridge, the definition of various damage states of bridge components and series of nonlinear time history analysis for different ground motion records to obtain the maximum response quantities. Here, the overall bridge vulnerability is evaluated with respect to the vulnerability of the piers only and the damage state is defined in terms of Park and Ang, 1985 damage model expressed as; (2) Where µd is the relative displacement ductility and µu is the ultimate ductility, Qy is the calculated yield strength, dE is the incremental hysteric energy absorbed in one cycle and β is the cyclic strength degradation factor taken as 0.15 for fairly stable hysteric energy behaviour (Gilmore et al. 2004). Total five damage states viz. None, Slight, Moderate, Extensive and Collapse are categorised for bridge system as defined by Ghobarah et al. (1997). The limit state of each damage state is given in Table 6.

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Table 6: Damage Limit state for different damage levels Damage Index

Damage Level

Definition

0.00 < DI ≤ 0.14

I

No Damage

0.14 < DI ≤ 0.40

II

Slight Damage

0.40 < DI ≤ 0.60

III

Moderate Damage

0.60 < DI < 1

IV

Extensive Damage

DI ≥ 1

V

Collapse

Characterization of Ground Motion In the present study, earthquake records of relatively larger magnitudes in range of 6.5-7.6 as recorded on firm soil are used. PGA is used as intensity measure of ground motion records. The provisions of FEMA P695 (FEMA, 2009) is followed to select these time histories. In total ten far field ground motions are selected which meet the above criteria from PEER NGA database. These time history records are normalized and scaled (GM scaled) as described in FEMA P695 (2009).The raw ground motion data (GMraw) is scaled using total three factors. The relation is described below.

direction are shown below.

Fig. 5 Acceleration Response Spectra of Normalized and Scaled GM

GMscaled = GMraw x NM x GS x HF (3) Where NM is Normalization Factor, GS is Global Scale Factor calculated as 2.7 and HF is Hazard Level ranging from 0.1g, 0.2g, 0.3g and so on. Figure 5 shows comparison between IS 1893 (2002) design spectrum with mean spectrum of 10 selected ground motions normalized and scaled to 1.0g hazard level.

Fragility Curves of Bridge A multi Incremental Dynamic Analysis (IDA)is performed on the 3D finite element model of bridge under scaled ground motion records (GMscaled) to different Intensity Measure (IM) levels represented by Peak Ground Acceleration (PGA) to obtain the entire range of behaviour i.e. slight to collapse. The relative displacement (Δtop) at the top of pier is calculated with corresponding displacement ductility µd = Δ/Δy and Damage Index is defined in Equation 2. IDA curves are plotted with respect to chosen IM and EDP (PGA vs Damage Index) for each ground motion. The IDA curves for both the analysis cases 1 & 2 in transverse The Bridge and Structural Engineer

Fig. 6: PGA (g) vs Damage Index in transverse direction of bridge under case 1 and 2

A measure of structural damage in terms of damage states viz. slight, moderate, extreme and collapse is obtained from IDA under both the cases for all ground motions. The data is assembled and Probabilistic Seismic Demand Model (PSDM) is generated for all the results. A linear regression analysis is performed by taking natural logarithm of the intensity measure i.e. PGA as abscissa and natural logarithm of Engineering Demand Parameter (EDP) i.e. Damage Index as ordinate. The coefficients a and bare obtained for each damage state from regression Volume 45 Number 1 March 2015  83


analysis (Figure 7). The values a and b are used to find median value of intensity measure for each limit state using Equation 4.

Fig. 7: PSDM for analysis case 1 & 2 in transverse direction

(4)

to each damage state. The standard deviation of set of Intensity measure (PGA) used in the calculation gives the variability in demand.

Fig. 8: Fragility curves for bridge pier under case 1 (unconfined) and case 2 (confined) in transverse direction

Conclusions Where, Sc is the median of the chosen limit state of damage calculated as the median of array of damage parameters corresponding to each damage state. The fragility curves are developed using Equation 5 (5) Where represent damage state variability calculated using Equation 6. (6) Where βc represent variability in capacity for each damage state and βd / IM represent variability in demand. Both capacity and demand are assumed to follow lognormal distribution. The variability in capacity is calculated as standard deviation of Engineering Demand Parameter (EDP) corresponding

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A numerical study based on confining reinforcement requirement at the plastic hinge region of bridge pier proposed in different country codes is compared by carrying out moment curvature and non-linear static pushover analysis. Numerical example considered here results in a case that the transverse reinforcement in the plastic hinge region of bridge pier is governed by minimum spacing requirement of the confining reinforcement. As a result, there is no significant difference in minimum requirements for amount and spacing of transverse reinforcement of the bridge pier by the provisions given in different codes. The idealized bi-linear pushover curves also reflect the same performance. However the capacity of conventional pier is significantly lower than the confined piers. The fragility curves of the bridge piers are derived by using incremental dynamic analysis and it also shows that conventionally designed bridge pier is more vulnerable as compared to bridge The Bridge and Structural Engineer


pier designed with seismic provisions related to confinement in codes. The conventional designed pier in MCE condition corresponds to PGA of 0.36g has the highest probability (40% to 50%) of extensive damage that reduced to only 4%in case of seismic designed (confined) pier.

References 1.

2.

3.

4.

American Association of State Highway and Transportation Officials (2012), “AASHTO LRFD bridge design specifications”, AASHTO, Washington, D.C. ATC-32 (1996), “Improved seismic design criteria for California bridges: provisional recommendations”, Applied Technology Council, Redwood City, California, USA ATC-40 (1996), “Seismic evaluation and retrofit of concrete buildings - Volume 1”,Applied Technology Council, Redwood City, California, USA BS EN 1998-1 (2004) – “Eurocode 8 – Design of structure for earthquake resistance – Part 1: General Rules”.

5.

BS EN 1998-2 (2005) – “Eurocode 8 – Design of structure for earthquake resistance – Part 2: Bridges”.

6.

CSI SAP2000 V14.2.4, “Linear and non-Linear static and dynamic analysis and design of three dimensional structures analysis reference manual”. Computers and Structures, Inc., Berkeley, USA.

7.

Federal Emergency Management Agency (FEMA). Quantification of Building Seismic Performance Factors. Report No. FEMA-P695, Washington, D.C (2009).

8.

Ghobarah, A., Aly, N.M. and El-Attar, M., (1997). “Performance level criteria and

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evaluation” Proceedings of the International Workshop on Seismic Design Methodologies for the next Generation of Codes. Balkema: Rotterdam, pp 207–215. 9.

HAZUS (1999), “Earthquake loss estimation methodology HAZUS99 Service Release 2 Technical Manual”, Federal Emergency Management Agency (FEMA), Washington, USA

10. IRC: 112 -2011, “Code of Practice for Concrete Road Bridges”. 11. IS:456 -2000, “Plain and Reinforced Concrete Code of Practice”, Fourth Revision.BIS, New Delhi 12. IS: 1893 (Part 3): 2012 “Indian standard criteria for earthquake resistant design of structures – bridges and retaining walls”, BIS, New Delhi 13. Mander. J, Priestley. M and Park. R (1988), “Theoretical stress strain model for confined concrete”, Journal of Structural Engineering, Volume 114. No. 8. 14. NZS 3101:2006, “Concrete Structures Standard, Part 1- The design of concrete structures; Part 2 -Commentary on the design of concrete structures”, Standards Association of New Zealand, Wellington, New Zealand. 15. Park, Young-Ji and Ang, Alfredo H.S. (1985), “Mechanistic seismic damage model for reinforced concrete”, Journal of Structural Engineering, 111(4). 16. Terán-Gilmore, A. and Jirsa, J.O., [2004]. “The Concept of Cumulative Ductility Strength Spectra and its use within Performance-Based Seismic Design”, ISET Journal of Earthquake Technology, Paper No. 446, Vol. 41, No. 1, 183200.

Volume 45 Number 1 March 2015  85


RETROFITTING OF DAMAGED RC FRAME USING METALLIC YIELDING DAMPER

Romanbabu M. OINAM

Dr. Dipti Ranjan SAHOO Assistant Professor Department of Civil Engineering Indian Institute of Technology Delhi New Delhi, INDIA drsahoo@civil.iitd.ac.in

Romanbabu M. Oinam received his B. E. (Civil Engineering) degree from Manipur Institute of Technology (M.I.T) Takyelpat, Imphal and M. Tech. in Earthquake Engineering from National Institute of Technology (NIT) Silchar. He is currently pursuing PhD at the Department of Civil Engineering, IIT Delhi. His research interests are Reinforced concrete structures, Masonry infill, Earthquake resistant design of building; Retrofitting of damage RC buildings; Energy dissipating devices, Performance based design of structures and Large-scale testing of structures

Dr. Dipti Ranjan Sahoo received his PhD in Civil Engineering from Indian Institute of Technology (IIT) Kanpur, India. He is currently working as Assistant Professor in the Department of Civil Engineering, IIT Delhi, India. His research interests are Evaluation of seismic behaviour of RC, Masonry and Steel structures, Fibre reinforced concrete, Seismic strengthening of structures using supplemental devices, Large-scale testing of structures and Performance-based seismic design. He is a recipient of Young Engineer Awards from Indian National Academy of Engineering (INAE), Department of Science and Technology (DST), Department of Atomic Energy (DAE) and Institute of Engineers India (IEI).

Research Scholar Department of Civil Engineering Indian Institute of Technology Delhi New Delhi, INDIA romaniitd@gmail.com

Abstract In this study, the performance of a reduced scale single-bay single-story RC frame is experimentally investigated under a constant gravity load and gradually-increased reversed-cyclic loading. Steel fiber reinforced concrete (SFRC) is used at the beam-column joints and at the expected plastic hinge locations of the test frame to improve their damage tolerance. The test specimen sustained cyclic excursion of 4.5% drift ratio with multiple cracking and severe damages in the beam and columns. Later, the damaged specimen is retrofitted using a combined yielding metallic damper (CMD) capable of dissipating the input energy through hysteresis. CMD, as a supplemental passive device, consists of a series of steel plates oriented along as well as normal 86â&#x20AC;&#x192; Volume 45

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to directions of loading so as to cause yielding of plates in both flexure and shear. Test results showed that the retrofitted frame exhibited the higher lateral strength, stiffness and energy dissipation as compared to the original RC frame. Further, the retrofitted specimen and the proposed connection scheme performed well till a cyclic excursion of 6.0% drift level. Keywords: RC frame; steel fiber reinforced concrete; plastic hinges; energy dissipation; metallic yielding devices

Introduction Reinforced concrete (RC) buildings designed only for gravity loadings are vulnerable to severe damages or complete collapse during earthquakes. Seismic performance of these deficient RC frames largely The Bridge and Structural Engineer


depend on the amount and detailing of longitudinal and transverse reinforcements as well as building configurations (Kaplan and Salih, 2011). The failure of columns is mostly controlled by their axial loadbending moment interaction or axial load-shear force interaction behavior resulting the formation of flexure and/or shear plastic hinges near the joint regions (Paulay and Priestley 1992). The absence of capacity design required in order to ensure strong-column/ weak-beam system induce the soft-story collapse of the RC frames. In a moment-resisting frame, the maximum bending moment and shear force demand is usually noticed near the beam-column joints under the lateral loading. Thus, high concentration of (shear) forces in the joint regions may lead to the severe damages/complete collapse of the RC framed structures. The key parameters that control the seismic performance of the RC frames are (i) lap splicing of longitudinal reinforcement, (ii) anchorage detailing of reinforcement bars, (iii) amount of longitudinal and transverse steel in main members and (iv) amount of transverse reinforcement in the joint regions (Jain and Uma, 2006; Moehle, 2000). At the local level, the brittle failure of panel zone is also expected within the beam-column joint subassemblies of the RC frame (Pampanin et al., 2002). The addition of randomly oriented discontinuous steel fibers into the concrete mix significantly improves the ductility, toughness and post-cracking tensile resistance of concrete (Hannant, 1978). A large number of small-sized (micro) steel fibers in the concrete matrix arrest the further propagation of micro-cracks existing in the RC members. Highperformance concrete using steel fibers in the concrete mixes can be used at these critical locations in order to improve the damage tolerance of the RC members. Past experimental studies have shown that steel fiber reinforced concrete (SFRC) also enhances the shear strength, flexural strength and ductility of structural members (Oinam et al., 2014). Several strengthening techniques have been developed to improve the seismic performance of the existing deficient RC buildings. Commonly used conventional techniques are concrete jacketing (Vandoros and Dritsos, 2008), steel jacketing (Nagaprasad et al., 2009), steel caging (Campione, 2013; Sahoo and Rai, 2010) and composite jacketing (Xiao et al., 1999). While these conventional techniques improve the lateral strength and stiffness of the RC frames, it is expected that The Bridge and Structural Engineer

the strengthened members may undergo damages in the event of a major earthquake which may require major intervention for repair and retrofitting. Other techniques of seismic strengthening of the existing undamaged/damaged RC frames include the use of passive energy dissipating devices for the reduction of seismic demand on the primary structures. These supplemental devices dissipate/absorb the input seismic energy based on the principle of friction, viscosity, or metallic yielding. Metallic yielding devices are believed to be economical and most effective in supplemental energy dissipation because of relatively simpler fabrication and installation. The metallic (steel/aluminum) plates are allowed to yield under the cyclic lateral loads resulting hysteretic energy dissipation. Depending on the orientation of plates, these devices are commonly termed as (i) Added Damping and Stiffness (ADAS) devices in which a series of steel plates of either X or triangular shapes undergo flexural yielding (Alehashem et al., 2008; Tsai and Hong, 1992) or (ii) shear links in which the web plates undergo yielding under the in-plane shear (Sahoo and Rai, 2010). Recently, Taraithia et al. (2013) developed combined-yielding metallic dampers (CMDs) by combining the flexural and shear plates as shown in Fig. 1(a). CMDs consist of X-shaped end (flexure) plates and rectangular web (shear) plate bounded by two base plates at top and bottom. The upper base plate of the CMD is allowed to move along with the structure to which they are connected, whereas the bottom base plate is held in position so as to provide unyielding (rigid) support. Fig. 1(b) shows the state of CMDs at 20% drift ratio level in the component testing. Slow-cyclic testing of the CMDs showed the stable hysteretic response with moderate reduction in the post-peak lateral load resisting capacity as shown in Fig. 1(c). While the rectangular plate has been used as web shear plate, the end flexural plates have been profile cut in the form of X-configuration. This has been carried out to maximize the utilization of material in flexural yielding. The end plates are connected to the top and bottom base plates by means of welding connections, thus acting as fixed-end beams. Under the action of lateral cyclic loading, plastic hinging of end plates is formed at the ends. The X-shaped configuration of plates help in yielding of the entire plate, thus maximizing the energy dissipation through material hysteresis. Volume 45 Number 1 March 2015â&#x20AC;&#x192; 87


(a)

(b)

(c)

Fig. 1: (a) Component testing of CMD, (b) Damaged CMD after test, (c) Hysteretic behavior of CMD (Taraithia et al., 2013)

The percentage of longitudinal reinforcement in the columns was 3.1% with 6 mm diameter stirrups at constant center-to-center spacing of 175 mm. Similarly, 6 mm diameter stirrups at a spacing of 50 mm were used near the joint regions of the beams, whereas the spacing of stirrups was increased 100 mm at their mid-span regions.

Research Objectives This study is focused on the cyclic performance of a non-ductile RC frame with SFRC at the critical locations and the evaluation of the effectiveness of CMDs as retrofitting technique. The primary objectives of the study are (i) to investigate the hysteretic response and failure mechanism of the RC frame with SFRC at the joint regions and (ii) to evaluate the improvement in the cyclic response of damaged RC frame retrofitted with CMDs. The main parameters evaluated in this study are lateral load-resisting capacity, lateral stiffness, damage potential and energy dissipation potential of the original (undamaged) and retrofitted RC frames. An experimental investigation has been carried out in on a reduced-scale RC frame in this study as discussed in the following sections.

Experimental Program A reduced-scale (0.4:1) single-bay single-story RC frame of a prototype five-story structure was chosen as a test frame in this study. All geometric dimensions, percentage of steel reinforcement, spacing of transverse steel, etc. in the test frame were determined using a scale factor of 0.4 on the respective values for the prototype frame. The overall height and width of the test frame were taken as 2100 mm and 3200 mm, respectively. Crosssection of the column and beam were 160x160 mm and 160x180 mm, respectively. The overall depth of the beam include 50 mm thick monolithic slab of 500 mm wide (Oinam and Sahoo, 2015). The details of dimensions and reinforcement detailing of the test specimen are shown in Fig. 2(a). The detailing of reinforcement in the test frame was carried out as per Indian Standard (IS-456, 2000) provisions. The values of clear cover to the main reinforcement bars in the footing, column, beam and slabs were 24 mm, 16 mm, 10 mm and 8 mm, respectively. 88â&#x20AC;&#x192; Volume 45

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(a)

(b)

Fig. 2: (a) Reinforcement detailing of test frame, (b) Regions of SFRC mixes used in the specimen

Original (undamaged) Specimen Fig. 2(b) shows the shaded regions where the SFRC mix was used in the test frame. It can be seen that these critical regions essentially represent the area of high bending moments and shear forces in the frame. The extent of critical regions was decided based on the expected length of plastic hinges (lp) as given by the following expression (Priestley and Pauley, 1992):

lp = 0.08l + 0.022db fy

(1)

Where, lp= plastic hinge length, l = length of beam, db= diameter of main bar, fy= yield strength of main bar. Accordingly, the plastic hinge length, lp was found to be 1.6 times of column depth. Hence, steel fiber matrix was provided 1.5 times column depth from the joint regions. Plain concrete of characteristic compressive strength of 25 MPa was used to cast the test frame. The target strength of concrete was computed as 31.6 MPa as per Indian standard (IS10262 2009) provisions. Ordinary Portland cement (OPC) was used in this mix design with Zone-II fine aggregates. Water-cement ratio of concrete mix was kept as 0.46 with a compaction factor of 0.9. End-hook steel fibers of 60 mm long and 0.75 mm diameter (i.e. aspect ratio=80) were used in the SFRC mix. The specified tensile yield strength of steel fibers was 1100 MPa. Based on the past studies (e.g., Sahoo and Sharma, 2014), fiber content of 1.0% in the SFRC mixes has been considered in this study. To get a better workability of SFRC mix, super plasticizer of 0.5% by volume was used into the concrete mix. The Bridge and Structural Engineer


Retrofitted Specimen The original test frame was subjected to graduallyincreasing reversed-cyclic displacements up to 4.5% drift ratio. At the end of the cyclic test, the specimen suffered some damages in the columns and beam (discussed later). The same frame was later retrofitted using CMD as passive energy dissipating device. The main objective was to improve the lateral strength, stiffness and energy dissipation potential of the damaged RC specimen. In this study, a CMD consisting of two X-shaped steel plates as flexure plates and one rectangular plate as web shear plate was used for retrofitting of the damaged RC frame. Flexure and shear plates of the CMD were oriented in the transverse and longitudinal directions to the cyclic lateral loading direction, respectively. A gap of 12.5 mm was left intentionally between the edges of flexure and shear plates in order to allow unconstrained flexural deformation of the end plates. Table 1 summarizes the sizes of various plates of the CMD used in this study. The flexure plates were welded to the base plates of 12 mm thick at both the sides. Since the web shear plate was relatively thinner than the flexure plates, the direct welding connection between the base and web plates was not carried out. Instead, 12 mm thick shear tabs were attached to the inner faces of the both base plates by means welding connection. Shear tabs and web shear plate were connected using sufficient numbers of 8 mm size high-strength bolts. CMD was installed on the test frame with the help of a chevron bracing system. Bottom base plate of CMD was fixed on top of the brace, while the upper base plate was connected at the bottom of beam using 12 mm size high-strength bolts. Table 1: Dimensions of various components of CMD Components

Length (mm)

Width (mm)

Thickness (mm)

Flexural plates

150

200

6.0

Shear plate

200

200

3.2

Base plates

600

500

12.0

The Bridge and Structural Engineer

Fig. 3: Retrofitted RC specimen showing the arrangement of CMD and test set-up

The details of retrofitted test frame installed with CMD at the beam level are shown in Fig. 3. The base plate of CMD was attached to a chevron bracing system providing unyielding support. Other ends of braces were attached to the column bases by means of the steel collars fabricated using four Indian Standard rolled angles ISA 50x50x6 and four numbers of 12 mm thick mild steel plates of 250 mm long and 160 mm wide. Two Indian Standard square hollow sections of 113.5 x 113.5 x 5.4 mm size were used as braces. The axial (tension/compression) capacities of these braces were significantly larger than the ultimate lateral capacity of the CMD. Thus, braces were not intended to undergo inelastic deformation under the cyclic lateral loading. Test Set-up and Loading History Servo-controlled hydraulic actuator of 250 kN capacity and stroke length of 250 mm was used to apply the lateral cyclic loading at the beam level of the test frame. The footing of test frame was rigidly held to the laboratory strong floor along its length. In order to avoid the out-of-plane movement of the specimens, side supports were used along with the roller sat the slab-level for facilitating smooth inplane displacement of the test frames under cyclic loading. Indian Standard rolled angle sections (ISA 60x60x8) were provided at the junction of slab and beam and were properly anchored with the slab for about three-fourth length of beam. The actuator was later attached to these angle sections and thus, the force exerted by the actuator was distributed more uniformly on slab of the test frame (Fig. 3). A constant gravity load of 7.2 kN was applied at the slab level by using concrete cubes. Displacement history (Fig. 4) as per ACI Committee 374.1-05 (ACI et al. 2006) recommendations was chosen for the cyclic testing of the test specimens. Each major drift cycle was Volume 45 Number 1 March 2015â&#x20AC;&#x192; 89


repeated for three times followed by one minor drift cycle. The cyclic excursions were gradually increased till 6% drift ratio. For the original RC frame, the maximum value of cyclic displacement was applied up to 4.5% drift level owing to its instability due to severe cracking in the beam and columns.

Fig. 4: Displacement history

Results And Discussion The main parameters investigated are (i) overall behavior, (ii) hysteretic response, (iii) lateral strength, (iv) lateral stiffness and (iv) energy dissipation and damping response of both undamaged and retrofitted specimens. These parameters are discussed in detail in the following sections.

of the fiber yielding and pull-out noticed during the testing of the original specimen. No major cracks were noticed in the columns unlike the RC frame. The connection between CMD and beam performed effectively at the low drift levels. At the 1.0% drift level, a relative sliding between the CMD and beam was noticed because of the widening of the earlier cracks at the anchor bolt locations in the beams. Fig. 6 shows damages noticed in the retrofitted frame after being subjected to cyclic excursion of 6%. Initially, several minor-cracks started appearing at the connection region, which were further widened at the large drift levels. Spelling of beam concrete at the anchored regions induced a high shear force on the bolts resulting their failure and loss of lateral resistance of the CMD as noticed by a sudden drop in hysteretic behavior (discussed in the following section). At 3.5% drift level, longitudinal reinforcement of beam was clearly visible from outside. At 4.5% drift level, the damper got separated from beam resulting high shear demand on the column as noticed by the diagonal cracks on both columns. At 6.0% drift level, major shear cracks were noticed diagonally in the column concrete. Fig. 7 shows the initial and final configurations of the CMD in the retrofitted frame. No major instability of the CMD was noted till the end of the testing.

Overall Behavior The undamaged RC specimen did not show any visible damages in the smaller drift levels of cyclic loading. Few minor cracks were noticed in the columns at 0.75% drift level. At the higher drift levels, a number of minor cracks were noticed in the SFRC regions at the column bases indicating fiber bridging action. At 1.0% drift level, major cracks were noticed on the column faces and these cracks were widened to 2.0 mm at 4.5% drift level. As shown in Fig. 5, major cracks were noticed in the columns of the RC specimen away from the SFRC regions. Some flexural cracks were also noticed in the mid-span regions of beam. In addition, multiple cracks were noticed in the joint regions of the specimen as shown in the figure. The damaged RC frame was then repaired by replacing the damaged cover concrete and retrofitted using CMD. In case of the retrofitted specimen, it was expected that SFRC would not provide any appreciable resistance to the lateral loading because 90â&#x20AC;&#x192; Volume 45 Number 1 March 2015

Fig. 5: Details of cracking and damages observed in various members of the original frame

Fig. 6: Details of cracking and damage observed in various members of the retrofitted frame

The failure of shear plate of the combined yielding damper (CMD) at higher drift level should not be considered as limited effectiveness damping system. In fact, the design of the damper has been carried out in an intention to yield and subsequent failure of the shear plate so as to dissipate the input seismic energy The Bridge and Structural Engineer


through metallic hysteresis. It is expected that the shear plate being relatively stiffer as compared to the end (flexure) plate would carry the large amount of lateral force acting on the frame. Once shear plate failed at the higher drift levels, the flexural plates would continue to carry the lateral cyclic force helping in dissipating the seismic energy. In this study, the metallic damper was connected to the retrofitted frame using 8 nos. of high-strength 12 mm bolts inserted into bolt holes in the beam for an anchored depth of 100 mm. The bolt holes were later grouted using epoxy to provide perfect bond between the bolts and beam concrete. Because of the limited beam width available in the RC frame, the premature failure of anchored bolts was noticed due to the inadequate bearing strength of available beam concrete. In order to avoid this failure, localized beam strengthening needs to be carried out so that the proper shear transfer between the RC frame and damper can be achieved.

(a)

(b)

Fig. 7: (a) Initial configuration of CMD in the retrofitted frame, (b) Damaged beam due to anchorage failure of CMD

Hysteresis Response Fig. 8 shows the lateral load vs. displacement (hysteretic) response of both specimens. The yield drift values of the original RC and the retrofitted frames were 0.75% (11.2 mm) and 0.5% (7.5 mm), respectively. The maximum loads resisted by the original RC specimen were 72.2 kN and -77.6 kN in the pull and push directions at 4.5% drift level, respectively. The retrofitted frame showed a maximum lateral resistance of 124.5 kN and -126.8 kN in the respective directions at 2.75 % drift level. This shows an increase of about 70% in the lateral load-resisting capacity due to the proposed retrofitting strategy. Because of the anchorage failure of CMD, a sudden drop in the lateral resistance was noticed in the hysteretic response of the retrofitted frame. Although the energy dissipation potential of the CMD could not be fully utilized, the proposed retrofitting scheme enhanced the lateral strength The Bridge and Structural Engineer

and deformability appreciably. Further, the pinching effect in the hysteretic response was reduced in case of the retrofitted frame as compared to the original RC frame.

(a)

(b)

Fig. 8: Hysteresis response of (a) original (undamaged) and (b) retrofitted frame

Backbone Curves The peak values of lateral loads resisted by both specimens at various drift levels are shown in Fig. 9(a).The backbone curve of the original RC frame was nearly bi-linear. As expected, the lateral resistance of the retrofitted frame was relatively higher at each drift level as compared to the original RC frame. The maximum load-resisting capacity of the retrofitted frame was observed at 2.75 % drift followed by the strength-degradation due to the failure of connections showing nearly a tri-linear backbone response. However, both these curves were symmetrical in the push and pull directions. Hence, in order to improve the lateral strength of the strengthened frame, a proper connection between the CMD and the beam should be designed. Thus, the proposed strengthening technique is capable of enhancing the lateral load carrying capacity at the higher drift levels once the cracks were initiated in the members. Fig. 9(b) shows the variation of lateral stiffness of both specimens with the drift levels. The initial lateral stiffness of the retrofitted frame was increased by 30% as compared to the original frame. The reduction in the lateral stiffness was more gradual for the retrofitted frame. The maximum lateral stiffness of the RC frame was 3.0 kN/mm, whereas the maximum value of lateral stiffness of the retrofitted frame was found to be 4.4 kN/mm. At 2% drift level, a maximum difference in the lateral stiffness was noted between the original and the retrofitted frames. As expected, the lateral stiffness of the retrofitted frame was nearly same as the original frame at the higher drift levels Volume 45 Number 1 March 2015â&#x20AC;&#x192; 91


(>4.5%) because of the failure of attachment of CMD with the frame. The installation of braces in a frame significantly increases the initial frame stiffness where the braces are directly connected to the beam and thus, share a large portion of lateral force acting on the frame. In this case, the braces are attached in series with the metallic damper that is connected directly to the RC frame. Further, the brace sections are so chosen such that these braces remain elastic for the entire range of applied drift cycles, thus providing an unyielding rigid-like support for the metallic damper. As a result, the total lateral force the retrofitted frame is shared largely by the metallic damper as compared to the RC frame. It is worth mentioning that the lateral stiffness of the as-built (original) frame at the end of 4.5% drift cycle was nearly 1.1 kN/mm, whereas the initial lateral stiffness of the retrofitted frame was nearly 3.5 kN/mm indicating an increase in initial lateral stiffness of frame by nearly 220%.

(a)

At drift level of 4.5%, the original frame dissipated a cumulative hysteretic energy of 5.62 kNm, while the retrofitted frame dissipated about 14.3 kNm of hysteretic energy. This showed that the retrofitted frame dissipated about 2.6 times the energy dissipated by the original frame. The maximum cumulative energy dissipated by the retrofitted frame was 17.6 kNm at 6% drift level. Using the hysteretic energy dissipation data, equivalent viscous damping potential of the test specimens was computed at each cyclic excursion level. The following equation recommended by FEMA 356 (2000) was used to estimate the equivalent viscous damping. β eq =

Eloop 1 2π K ( D + − D − )2 eq

(2)

Where βeq =Equivalent viscous damping and Eloop =Dissipated energy per hysteretic loop at a drift cycle, Keq = Effective stiffness of the loop and D = Peak displacement values in positive and negative directions of cyclic loading. Table 2 shows the comparison of equivalent viscous damping for both

(b)

Fig. 9: Comparison of (a) backbone curves and (b) lateral stiffness of both frames

Energy Dissipationand Viscous Damping The dissipation of input seismic energy is extremely essential in order to avoid the complete collapse of the structures. The energy dissipation potential of a structural system can be estimated as the area enclosed under its hysteretic response. The cumulative energy dissipated by the both specimens at each drift level was calculated from the average area enclosed under the hysteresis loops of three cycles. Fig. 10 shows the comparison of energy dissipated by the both frames. As expected, a significant difference was observed in the energy dissipation potential of the frames. This difference was noted from the initial drift cycles since the lateral load resisting capacity of the retrofitted frame was higher than that of the original frame. 92  Volume 45 Number 1 March 2015

Fig. 10: Comparison of energy dissipation of both frames

the frames. The equivalent viscous damping values for the original and retrofitted frames at 0.2% drift level were 6.17% and 8.46%, while at 4.5% drift level these values were increased to 17.80% and 38.87%, respectively. It indicates that the damping capacity of the RC frame was enhanced by 54.21% due to the proposed retrofitting technique.

Conclusions The following conclusions can be drawn from this study: The Bridge and Structural Engineer


Lateral strength and energy dissipation potential of the retrofitted frame is significantly higher than the original RC specimen. However, a premature failure of the connection between the combined metallic yielding damper and beam can be expected if the damaged beam was not strengthened at the damper location.

The retrofitted frame exhibited a higher lateral load resistance compared to the original SFRC frame. The energy dissipation potential of the damaged frame can be improved by 2.5 times if the CMD is used as passive energy dissipation device in the damaged frame.

This study showed that the proposed strengthening technique can be adopted to improve the seismic performance of a damaged RC structure located in low-to-medium intensity seismic regions in practice. Table 2: Computation of equivalent damping potential of the test frames

Drift level (%)

Energy dissipated, Eloop (kNmm)

Effective stiffness, Keq (kN/mm)

8.96

16.12

2.89

3.59

6.17

2.

Alehashem, S. M. S., Keyhani, A., and Pourmohammad, H. (2008). “Behavior and Performance of Structures Equipped With ADAS & TADAS Dampers (a Comparison with Conventional Structures).” The 14th World Conference Earthquake Engineering, Beijing, China.

3.

Campione, G. (2013). “RC Columns Strengthened with Steel Angles and Battens : Experimental Results and Design Procedure.” Practice Periodical on Structural Design and Construction, ASCE, 18 (February), 1–11.

4.

FEMA-356 (2000). Prestandard and Commentary for the Seismic Rehabilitation of Buildings. Federal Emergency Management Agency, Washington, DC.

5.

Hannant, D. J. (1978). Fibre Cements and Fibre Concretes. Wiley-Interscience, New York, 219.

6.

IS-10262. (2009). “Concrete Mix ProportioningGuidelines.” Bureau of Indian Standards, New Delhi.

7.

IS-456. (2000). “Plain and Reinforced Concrete – Code of Practice.” Bureau of Indian Standards, New Delhi.

8.

Jain, S. K., and Uma, S. R. (2006). “Seismic Design of Beam Column Joints in RC Momentresisting Frames- Review of Codes.” Structural Engineering and Mechanics, 23(5), 579–597.

9.

Kaplan, H., and Salih, Y. (2011). “Seismic Strengthening of Reinforced Concrete Buildings.” Earthquake-Resistant Structures - Design, Assessment and Rehabilitation, A. Moustafa, ed., Turkey, 407–428.

8.46

0.25

21.57

41.90

2.84

3.74

9.21

13.55

43.50

97.47

2.94

4.34

9.18

13.81

0.50

89.07

221.24

3.01

4.25

8.91

15.54

0.75

195.76

442.40

2.66

3.81

9.69

15.37

1.00

354.30

1086.73

2.41

3.77

10.82

21.86

1.40

671.50

2196.37

2.13

3.98

11.78

20.81

1.75

1030.23

3854.33

1.93

4.10

12.16

23.21

2.20

1562.57

6345.33

1.77

3.81

13.13

25.82

2.75

2369.67

9346.00

1.58

3.20

14.18

30.78

3.50

3661.00

11920.00

1.36

2.06

15.68

37.14

4.50

5616.00

14294.33

1.11

1.40

17.80

38.87

17585.33

ACI, Committee 374.1-05 (2006). “Acceptance Criteria for Moment Frames based on Structural Testing and Commentary- An ACI Standard.” American Concrete Institute, Farmington Hills, Michigan.

ßeq (%)

0.35

6.00

1.

Equivalent damping,

Original Retrofitted Original Retrofitted Original Retrofitted 0.20

References

0.99

35.87

Acknowledgments The funding received from Department of Science and Technology (DST) and Government of India (GOI) in carrying out this research is greatly acknowledged. Authors are thankful to the staff members of Heavy Structures Laboratory, Department of Civil Engineering, IIT Delhi for their support in conducting the experimental investigation. The Bridge and Structural Engineer

10. Moehle, J. P. (2000). “State of Research on Seismic Retrofit of Concrete Building Structures in the US.” US-Japan Symposium and Workshop on Seismic Retrofit of Concrete Structures— State of Research and Practice. Volume 45 Number 1 March 2015  93


11. Nagaprasad, P., Sahoo, D. R., and Rai, D. C. (2009). “Seismic Strengthening of RC Columns using External Steel Cage.” Earthquake Engineering & Structural Dynamics, 38(April), 1563–1586. 12. Oinam, R. M., and Sahoo, D. R. (2015). “Enhancement of Lateral Capacity of Damaged Non-ductile RC Frame using Combinedyielding Metallic Damper.” The Indian Concrete Journal, 89(1), 80–86. 13. Oinam, R. M., Sahoo, D. R., and Sindhu, R. (2014). “Cyclic Response of Non-ductile RC Frame with Steel Fibers at Beam-column Joints and Plastic Hinge Regions.” Journal of Earthquake Engineering, 18(6), 908–928. 14. Pampanin, S., Calvi, G. M., and Moratti, M. (2002). “Seismic Behaviour of RC Beamcolumn Joints Designed for Gravity Loads.” 12th European Conference on Earthquake Engineering, 1–10. 15. Paulay, T., and Priestley, M. (1992). Seismic Design Of Reinforced Concrete And Masonry Buildings. John Wiley & Sons, Inc., New York, 744. 16. Sahoo, D. R., and Rai, D. C. (2010). “Seismic Strengthening of Non-ductile Reinforced Concrete Frames using Aluminum Shear Links

94  Volume 45 Number 1 March 2015

as Energy-dissipation Devices.” Engineering Structures, Elsevier Ltd, 32(11), 3548–3557. 17. Sahoo, D. R., and Sharma, A. (2014). “Effect of Steel Fiber Content on Behavior of Concrete Beams with and without Stirrups.” American Concrete Institute Structural Journal, 111(5), 1157–1166. 18. Taraithia, S. S., Sahoo, D. R., and Madan, A. (2013). “Experimental Study of Combined Yielding Metallic Passive Devices for Enhanced Energy Dissipation of Structures.” The Pacific Structural Steel Conference, Singapore, 8–11. 19. Tsai, K. C., and Hong, C. P. (1992). “Steel Triangular Plate Energy Absorber for Earthquake Resistant Building.” 1st World Congress on Constructional Steel Design, Mexico. 20. Vandoros, K. G., and Dritsos, S. E. (2008). “Concrete Jacket Construction Detail Effectiveness When Strengthening RC columns.” Construction and Building Materials, 22(3), 264–276. 21. Xiao, Y., Wu, H., and Martin, G. R. (1999). “Prefabricated Composite Jacketing of RC Columns for Enhanced Shear Strength.” Journal of Structural Engineering, ASCE, 125 (March), 255–264.

The Bridge and Structural Engineer


Efficient Generation of Statistically Consistent Demand Vectors for Seismic Performance Assessment

Dhiman BASU Assistant Professor Department of Civil Engineering, IIT Gandhinagar, Ahmedabad dbasu@iitgn.ac.in Dr. Dhiman Basu is a faculty member in the discipline of civil engineering at Indian Institute of Technology Gandhinagar. His research interest includes rotational seismology, ground motion characterization, supplemental damping, irregular buildings and confined masonry system. Dr. Basu received BE (civil Engg) from Jadavpur University (1995), MTech (Structural Engg) from IIT Kanpur (1998) and Phd (Structural Engg) from SUNY Buffalo (2012). He also held a faculty position in GB Panth University (1999-2002) and a research scientist position in Structural Engineering Research Centre-Chennai (2002-2008).

Andrew WHITTAKER Professor and Chair Department of Civil, Structural and Environment Engineering State University of New York Buffalo Andrew Whittaker is Professor and Chair in the Department of Civil, Structural and Environmental Engineering at the University at Buffalo and serves as the Director of MCEER. He is a registered civil and structural engineer in the State of California. Whittaker served as the Vice-President and President of the Consortium of Universities for Research in Earthquake Engineering (www.curee.org) from 2003 to 2011 and on the Board of Directors of the Earthquake Engineering Research Institute (www.eeri.org) and the World Seismic Safety Initiative from 2008 to 2010. Currently, he is a member of the Advisory Board for the Southern California Earthquake Center. Whittaker made significant contributions to the first generation of tools for performance based earthquake engineering (FEMA 273/274, 19921997) and led the structural engineering team that developed the second generation of these tools (FEMA P58, 2000-2013). Whittaker serves on a number of national committees including ASCE 4, ASCE 7 and ASCE 43 and ACI 349. His research interests are broad and include earthquake and blast engineering of buildings, long-span bridges and nuclear structures. The US National Science Foundation, US Department of Energy, US Nuclear Regulatory Commission, US Federal Highway Administration and Canadian Nuclear Safety Commission fund his research. He consults to federal agencies, regulators, consultancies, contractors and utilities in the United States, Canada, United Kingdom, Europe and Asia.

Abstract Monte Carlo analysis underpins some modern seismic performance and risk assessment methodologies. Such analysis requires thousands to millions of simulations to generate stable distributions of loss. The Bridge and Structural Engineer

The generation of large numbers of simulations by nonlinear response-history analysis is impractical because of a lack of appropriate ground motion recordings and computational expense. An algorithm Volume 45 Number 1 March 2015â&#x20AC;&#x192; 95


based on spectral value decomposition is proposed to transform results of a small number of responsehistory analyses to a much larger space to enable Monte Carlo simulation. The algorithm includes an option to accelerate the recovery of the statistics underlying the results of the response-history analysis.

1.

Seismic Performance Assessment

and

Risk

Tools for seismic performance assessment of buildings and mission-critical infrastructure are used to predict distributions of loss and mean annual frequencies of specific losses and unacceptable performance. Loss for buildings is measured in many different ways, including repair cost (a direct loss), business interruption (an indirect loss), casualties and deaths. Unacceptable performance for a safety-related nuclear structure is generally measured by core melt or large dose of radiation release. Seismic performance (risk) assessment tools have been available in the nuclear industry for more than 30 years. Traditional risk assessment for nuclear structures has involved integrating a plant level seismic fragility curve over a seismic hazard curve to compute the mean annual frequency of unacceptable performance. Huang et al. [1, 2] summarizes these procedures. Seismic performance assessment of buildings is in its infancy. The first generation tools for seismic performance assessment of buildings focused on engineering parameters and did not seek to express performance in terms of loss. These tools were developed in the mid 1990s and have not substantially changed since. The latest versions of these tools are presented in ASCE 41 [3]. Second generation tools for performance-based earthquake engineering of buildings have been under development since the late 1990s, with work initially spearheaded by the Pacific Earthquake Engineering Research Center (PEER). A companion effort, funded by the Federal Emergency Management Agency (FEMA) and managed by the Applied Technology Council (ATC) sought to operationalize the research products from PEER and other researchers in the field, has developed Guidelines for the seismic performance assessment of buildings[4]. The Guidelines provide a framework for seismic performance assessment that involves the following five steps: 1) define a model of the building, 96â&#x20AC;&#x192; Volume 45 Number 1 March 2015

including all components that can be damaged, 2) develop fragility curves for all components that can be damaged and for which there are consequences that will contribute to loss, 3) generate ground motion records for each intensity of earthquake shaking considered, 4) perform response-history analysis of the model to estimate distributions of earthquake demand on each component and 5) perform Monte Carlo analysis to transform earthquake demand to loss through the use of fragility and consequence functions. A key to this framework is the development of statistically consistent demand vectors for the Monte Carlo simulations because response-history analysis is performed for best estimate models using a limited number of sets of ground motion records: 7 to 11 as noted in the Guidelines. The process used to generate statistically consistent vectors is described below. It is needed because hundreds to thousands of simulations are required to generate stable loss curves. The direct calculation of thousands to millions of demand vectors by nonlinear response-history analysis is not practical because of a lack of suitable ground motion records for such analysis and the computational expense of running a very large number of nonlinear analyses. Yang et al. [5] are credited with the development of the process included in the Guidelines. 1.1 Procedures to generate demand vectors for Monte Carlo Analysis A limited number, m, of response-history analyses are performed for ground motions scaled to a specified intensity. For each analysis, the peak absolute value of each demand parameter (e.g., third story drift, fourth floor acceleration) is assembled into a row vector with n entries, where n is the number of demand parameters. The m row vectors are catenated to form a m x n matrix, [X], where each column presents m values of one demand parameter. These entries in [X] are assumed to be jointly lognormal and so the natural logs of the entries, matrix [Y] are jointly normal. The entries in [Y] can be characterized by a vector of means, {mY}, a n x n correlation coefficient matrix, [RYY] and a diagonal matrix of standard deviations, [DY]. This diagonal matrix of standard deviations does not include variability due to modeling uncertainty and ground motion uncertainty and both were added to the diagonal entries of [DY] in the Guidelines using square-root-sum-of-the-squares. The Guidelines have assumed the means and correlation coefficients are The Bridge and Structural Engineer


unchanged by these other sources of uncertainty. Figure 1 illustrate the Yang et al. process for generating statistically consistent demand vectors. Objective is to generate a matrix of demand vectors [Z] with the same statistical characteristics as [Y] but which is much larger to enable Monte Carlo simulation. This process uses a set of vectors, one for each demand parameter, of uncorrelated standard normal random variables (zero mean and a unit standard deviation), [U] and a linear transformation and a linear translation:

The technical bases for the chol and cholcov routines are presented in the Matlab documentation [7] and are not repeated here. The cholalgorithm is stable only if the associated matrix (correlation matrix [AYY] or the related covariance matrix [∑] = [DY][RYY][DY]) is not rank deficient whereas cholcov is stable for rank deficient matrices. Appendix A provides a proof that the covariance matrix will be rank deficient when n ≥ m.

The goal of this paper is to present an alternate algorithm to Yang et al. process, which can better [Z]T = [a][U]T + [B] = [DY][LY][U]T + [MY] (1) recover the underlying statistics of [Z] with fewer simulations. The alternative algorithm is developed based Spectral Value Decomposition (SVD) and 1n 1 1. Read [ X ] , select OPT=on 0 for no acceleration (see above) and 1 for acceleration X X MY,DY, RYY can also be used with the cholcov.

W

exp

Z

2

2. Compute [Y ] = ln [ X ] and mean μY

3

U 3. Compute [G ] using Eq (3)

4

4. Generate a M × n matrix ⎡⎢U ⎤⎥ using a standard normal distribution, where M and n are t Algorithm ⎣ ⎦

5

2.

SVD Algorithm and its Mathematical Basis

number of simulations and number of demand parameters, respectively.

The following nine steps describe the SVD algorithm

6 5.demand Compute the correlation and the square root of the variance matrix of the column vectors Fig. 1: Generation of vectors of correlated for possible coding by others. A Matlab code is parameters [5] l Appendix B. G 7  them as R in and [σ G ] , respectively. [G ] and denote provided

( )

Matrices [A] and [B] are derived in Yang et al. [5] 1. Read [X], select OPT= 0 for no acceleration (see l = [ A ][λ ][ A ]T the value decomposition, R (G and are not repeated here. The matrix 8 6.[LUsing ) Gcor Gcor Gcor Y] is spectral above) and 1 for acceleration Cholesky decomposition [6] of [RYY] and all other 2 T ⎡G ⎤ = ⎡U ⎤ [λ ]1[Y] 9  7. If OPT=1, then 2. Compute = In mean μ ] [σ[X] terms are defined above. Gcor[ X ][ A G ] .and ⎢⎣ Read ⎥⎦ Read 1 ⎢⎣1  ⎥⎦ 1.1. select 00for selectOPT= OPT= forno noacceleration acceleration(see (seeabove) above)and and11for fo [ X ], ,Gcor 1 

1. Read [ X ] , select OPT= 0 for no acceleration (see above) and 1 for acceleration

1 1. Read [ X ] , select OPT= 0 for no acceleration (see above) and 1 for acceleration 3. Compute [G] using Eq (3) 10  algorithm If OPT=2, The original implementation of this usedthen 2  2  2.2. Compute Compute[Y[Y] = meanμμY Y ] =lnln[ X [ X] and ] andmean ln [ X 2  2. mean μY ] = mean ] and the chol subroutine [7][Compute performing the Yfor = ln 2  in 2. Matlab Compute μ ]11  [i)X []Yand Y 4. Generate a Mxn matrix using a standard calculate the covariance matr subtract the sample mean from each column of ⎡⎢U ⎤⎥ and ⎣ ⎦ G 3 3  3.3. Compute using Eq (3) [ ] G Compute using Eq (3) [ ] decomposition. Early studies using the algorithm normal distribution, where M and n are the G 3  3. Compute using Eq (3) [ ] Compute [G ] using Eq (3)l were reported in3 the 3.50% draft Guidelines number of simulations and number of demand 12  Σusing U an ⎡ ⎤ 4 ⎤ 4.4. Generate Generateaa MM××nn matrix matrix ⎣⎢U⎡⎢U⎦⎥ ⎤⎥using usingaastandard standardnormal normaldistribution distributio ⎡4  ⎣ M ⎦ where example that involved demand parameters 11 n are the ×⎡U n ⎤matrix 4  4. aGenerate a M and Ustandard using anormal standard normal distribution, and M n M × n matrix 4  4. 7Generate using a distribution, where and are the parameters, respectively. ⎢ ⎥ ⎣ ⎦ ⎣⎢ ⎦⎥ T ground motions, namely, n = 7 and m = 11. Results value l 5 5  number ofofdemand parameters, numberofofsimulations simulations and][number demand parameters,respecti respect 13  ii) using spectral decomposition, Σ U = [ Aand λnumber 5  number of simulations and number of demand parameters, respectively. cov Uand covthe U ][ Asquare covU ] 5. Compute the correlation root of 5  number of simulations and number of demand parameters, respectively. reported in Appendix G of that document showed the 6 6  5.5. Compute Computethe thecorrelation correlationand andthe thesquare squareroot rootofofthe thevariance variancematri matr the matrix of of theTthe column vectors in [G] 6  5.if the Compute the correlation and the square matrix of the column in −1 2of thematrix 1variance 2 5. Compute correlation and the⎡ square root ofvariance theroot variance column vectors in vectors algorithm yielded6 robust results the simulation space ⎤ ⎡ ⎤ 14  iii) calculate ⎢G ⎥ = ⎢U ⎥ [ AcovU ][ λcovU ] [ λGcor ] [ AGcor ] [σll G ][σ denote them R﴾Ĝ﴿ ⎣ ⎦ 7  ⎣l7 ⎦ and[G them asasRand R(G G anddenote denoteas them and respectively. ))and [G] ]]and [σrespectively. GG] ,] ,respectively. G] [,σ ( was large, with best results, as measured by accurate l G R G σ 7  and denote them as and , respectively. [ ] [ ( ) G σ G ] , respectively. 7  [G ] and denote them as R (G ) and [ recovery of statistical properties, being achieved for⎡⎢Y ⎤⎥ by6. Using R﴾Ĝ﴿ = 15  8. Compute adding backspectral the mean: value Y i = Gi decomposition, + μY {1} . T ll = ⎣ ⎦ 8 8  6.6. Using RR(G value decomposition, ][λGcor ][ AGcor ]T T G)) =[ A AGcor Using spectral value decomposition, [ ( l spectral Gcor ][ λ Gcor ][ A Gcor ] T]T [A ][λ ][A 1000s of simulations. Huang et al. [1] drew a similar l R G = A λ A 8  6. Using spectral value decomposition, ][ ] Gcor ][ Gcor ] λ) Gcor[ ][Gcor AGcor 8  6. Using spectral value decomposition, R (G ) =Gcor [ A ( ][Gcor ⎤. 16  and 9. 11 Generate demand vectors Gcor by taking theGcor exponential of ⎡⎢Y conclusion for 3 demand parameters ground ⎡G ⎤ ⎤==⎡U⎡U⎤ [⎤λλ ⎣]1 12⎦⎥2[ AA ]T T[σσ ] . . ⎡ 7. 9  7. If OPT=1, then 1If 27.OPT T = 1, then 9  If OPT=1, then G [ ] [ Gcor Gcor Gcor ] [ GG ] 1 2⎡U ⎤ λ T ⎣⎢ ⎢⎣ ⎦⎥ ⎥⎦ ⎣⎢ ⎢⎣ ⎦⎥ ⎥⎦ Gcor 7. If OPT=1, [ Gcor ]σ G[]AGcor ] [σ G ] . ⎡G ⎤ =then ⎡U ⎤ [λ⎡⎣⎢G ⎤⎦⎥ = motions. Subsequent for building structures 9  7.studies If 9  OPT=1, then A . ⎢ ⎥ ] [ ] [ ⎣ ⎦ Gcor Gcor ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ If OPT = 2, then 10  IfIfOPT=2, have shown that stable distributions loss can beBasis 10  OPT=2,then then 17  ofMathematical 10  If OPT=2, then 10  If OPT=2, then achieved with fewer than 1000 simulations i) subtract the sample mean from each column 11  mean 11  i)i) subtract subtract the the sample sample mean from from each each column column ofof ⎡⎢⎣U⎡⎢U⎤⎥⎦ ⎤⎥ and and calcul calcu ⎡Uis⎤ presented ⎣ ⎦ 11  i) subtract the sample mean from each column of and calculate the covariance ⎡ ⎤ 18  A mathematical proof for the SVD algorithm in what follows. The matrix uniqueness of t 11  i) subtract the sample mean from each column of of ⎢U ⎥ and and calculate thethecovariance matrix ⎢ ⎦⎥ calculate covariance matrix ⎣ ⎣ ⎦ Zareian [8] observed that the chol algorithm was

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19 solution is not guaranteed because ΣΣ UU the computation involves the generation of random numbe 12  l unstable if the number of12 demand parameters equal 12  l Σ U 12  Σ U and, motions, even though spectral unique,value by definition, the associated left and right ba ii) values usingare spectral decomposition, to or greater than the number of20 ground TT ll 13  ii) using spectral value decomposition, ΣΣ UU ==[ A 21  vectors are not. 13  However, uniqueness is T not an Tissue for loss or decomposition, AcovcovUU][][λλcovcovUU][][AAcovcov namely, n ≥ m. He proposed replacing the Matlab [computations l spectral value UU] ]ri l ii) using 13  ii) using spectral value decomposition, Σ U = A λ A [ ][ ][ ] using spectral value decomposition, Σ U = [ AcovU ][λcovU ][covAUcovU ]covU covU chol subroutine13 withii) the cholcov22 subroutine. The procedure differs from the Yang et al. algorithm assessment.The in two ways: 1) the factorizati ⎡G ⎡ ⎡U⎤ [⎤ AA ][λλ ]−−1 12 2[λλ ]1 12 2[ AA ]T T[σσ ] ⎡G⎤⎥ ⎤= 14  iii) 1 2−iii) calculate 1iii) 2 calculate 12 T=⎢U 14  calculate [ [ ][ ] cov UU ][ cov UU ] Gcor cholcov subroutine is stable for both m > n and n ≥ m. ⎢ ⎥ cov cov Gcor ] [ Gcor ⎡ ⎤ ⎡ ⎤ − 1 2 T ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ on ⎣G ]⎦ which is more 23  robustGcorthanGG the Choles 14  iii) ⎡G calculate U ⎥ [ AcovU ][λλcovmatrix ]σ [ AGcor ]⎦ [σSVD, ⎤ = ⎡U of ⎤ A⎢Gthe U ] A[ λis Gcor based 14  iii) calculate ⎥=λ ⎢correlation ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ [ ⎣ cov⎦U ][ ⎣ cov⎦ U ] [ Gcor ] [ Gcor ] [ G ] 15  15  8.8. Compute Compute⎡⎢⎣Y⎡⎢Y⎤⎥⎦ ⎤⎥ by byadding addingback backthe themean: mean:YYi i==GGi i++μμY Y{1{1}}. . ⎡Y ⎤ by adding back ⎦ μY {1} . 15  8. ⎡Compute the mean: Yi = G⎣.i + ⎤ by adding 15  8. Compute Y back the mean: Y = G + μ 1 ⎢ ⎥ i { } ⎣ ⎦ i Y ⎣⎢ ⎦⎥ The Bridge and Structural Engineer Volume 455  Number 1 March 2015  97 ⎡ ⎤

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16  by 16  9.9. Generate Generatedemand demandvectors vectors bytaking takingthe theexponential exponentialofof⎢⎣Y⎡⎢Y⎥⎦ ⎤⎥. . ⎡ ⎤ ⎣ ⎦ 16  9. demand Generate demand by exponential taking the exponential 9. Generate vectors by vectors taking the of ⎡⎢Y ⎤⎥ . of ⎢⎣Y ⎥⎦ . ⎣ ⎦

17

17 Mathematical Basis Mathematical Basis

17 17  Mathematical MathematicalBasis Basis


1 2 ⎡U ⎤ and calculate the covariance . G nthat 17 matrix Define that n mean from each column ]G= [.G.1GG 2] .such ] such l lofDefine Define such that [G[G ] =1 G [G[G ] = [G1 G 2 . . G17 n ] 17  ⎣⎢ ⎦⎥ [Gthen G] = Define that [ 12  Σ 10  U 17  n 21 . . 2G n ] such 12  Σ U If OPT=2, 17  Define [G ] = [G1 G 2 . . G n ] such that 18 ⎡ ⎤ Gi = Yi − μYi {1} and calculate the covariance matrix 11  i) subtract the sample mean from each U T l G =AYof 18  column l iμ i1− 13  ii)13 using spectral value decomposition, Σ18 U18 = AG − G μYi {1}decomposition, 18  (3) ][Yλμ[icov ][ { }U ⎣⎢Uμ] Y][i⎦⎥ A{1cov} U ]T GΣi[ = Yi Ui= − 1 (3) { } cov U cov Y i = Yi −value ii) using spectral U = A λ Yi covU i][ cov Gi = Yi − μYi {1} 18  (3) l T l =where Yi − μYμi Y{is 1}isthe 1  19 Giwhere ΣU U of columnofof[Y[Y] and{1}{1} themean mean of the i th column is m×1 vector o ecomposition, Σ12  U = [⎡Acov −1 2 12 T ] and ][⎡Uλ⎡cov U ][ AcovU ] th 12 T ⎤[ A i 14  iii)14 calculate G ⎤⎥ = λGcor]−]1 2 [[λA σAG where ⎤= ⎡U][⎤λ[ cov ] [ ] [ ] m×1of is the mean of and vector ofMatrix ones. μ {G1vector }]T 1isvector th the i column of [Y ] l thGcor19  cov U U lG=T[G th iii) calculate G A λ σ ][ ] [ ] [ ] ⎢ ⎢ ⎥ Y l =Matrix i mean ⎣ ⎣⎢ ⎦ ⎦⎥μYi ⎣⎢ is ⎦⎥ thecovmean U cov U the Gcor Gcor Gand Y 1 Y 1 m× m× 1 19  where is the of the column of and is ones. 19 ⎣ ⎦ where of column of is vector of ones. Matrix i i G G = μ [ ] { } [ ] { } [ Y 1 m× 1 19  where is the mean of the column of and is of ones. Matrix i G T μ [ ] { } [ ] x 1 vector of ones. Matrix represents the is m Y l = [G ] l 19  where isTthe mean of the i th column of [Y ] andYi {1}i is m×1 vector of ones. Matrix G −1 2 1 2 μYi T the random variable Y with a zero mean. The statistical prope 20  represents l l , fo λcovU ] 8. [8. Compute λGcor [σ Gadding withaalzero zero mean. statistical random ⎡Yusing ⎤] by 13 ] [ A ii)Gcor spectral value Σ U mean: = [.AμcovU{1][}λthe ][ AcovUlvariable ]variable covU ][15  Yla zero 20  represents with mean. TheThe statistical properties of G back thedecomposition, mean: Y20  1}+ ⎡]Y ⎤ bythe i =lGithe covUrandom {zero l , example, by adding lof, for adding back theback mean: 15 Compute 8. 20  Compute Y+i μ=Yai G .variable ⎢⎣ ⎥⎦ represents Yl properties Y represents the with statistical properties G for exam random variable with mean. Thevariable statistical ofmean. GThe , forThe example, Y with 20  represents the mean. statistical properties of G i random Yi random ⎢⎣ ⎥⎦ variable l with la,zero l l Y 20  represents the random a zero mean. The statistical properties of G for example, for example, the covariance matrix properties of Σ G and the correlation 21  the covariance matrix R G are . l matrix l are −1 2 21  1 2the T ΣtheG R G matrix and the correlation matrix ng back Y i 21  = Gicalculate + μYcovariance {1} .⎡G ⎤ =by⎡U ⎡YA⎤covariance l are l are lexponential l ⎤[ A Σ l l 16  the9.mean: Generate demand taking the of . ⎡ ⎤ 14  iii) λ λ σ i vectors and correlation matrix are Σ G R G G R G 21  the covariance matrix and the correlation matrix the matrix and the correlation matrix ][ ] [ ] [ ] [ ] Σ G R G 21  the covariance matrix and the correlation matrix are cov U cov U Gcor Gcor G 16  9. Generate demand the exponential l ⎣⎢ and lof are ⎣⎢ Σ⎦⎥ vectors ⎦⎥ by ⎢⎣Y ⎥⎦ . G G 21  9. the covariance thetaking correlation matrix⎣⎢ R⎦⎥the Generate matrix demand vectors by taking l = 1 [G ]T [G ] = 1 GG l lT 1  Σ lG 11 1 1 l l T TT T T ⎡ ⎤ tors by taking the15 exponential of Y . T ⎡ ⎤ l llll 1 1 m − m − ]T]T]T[G ] ]=]== 111 GG exponential of 8. Compute + μ Σ1 G .lG= = 111 [[G G G lGG ⎣⎢ ⎦⎥ ⎣⎢Y ⎦⎥ .by adding back the mean: Y i = Gi 1  lG lGG T (4) [[[G [[[G 1 1 1  Yi ΣT{ΣΣ}G == G ==1mm GG ] ] 1 1 m − − l l l 17  Mathematical Basis Basis T [G 1 1 m − − 1 1 T 1 l lT T 1  Σ G = G G GG 17  Mathematical ] ] 1 1 T l l l 1 1 − − m m l 1 1 m − m − l l l T G= = GG [GG]] [[GG]l]== 1 m1−GG Σ1 GT = G1] l[G = GG 6  1[− ] 1 1  ΣΣ G m [ l 1  Σ G = G G = GG [ ] [ ] 1 1 m − m − ⎡Y ⎤ l. m −1 −1 l m −−11 T m −12  of R 6  m −1 1 Mathematical Basisdemand vectors by taking the exponential T 16  1 9.l lGenerate   G = σ Σ G σ 1 1 m − m − ⎢ ⎥ [ ] [ ] 6   6   6   ⎣ l⎦ The 1−1 (4) uniqueness = GG proof for the SVD algorithm is presented  in what follows. G−1−1 l G− ] = A mathematical 18 [G ] A[G mathematical the lfollows. l[σ[of 18  proof for the SVD algorithm 1Σ ΣG R G == 2 2 in what [σσ[σ[σG ]G−]−]−1The m −1 m −1 σ lG lG G ]G−− ]−1 11 of the lG luniqueness   6 is presented     algorithm 1ΣΣ G = G σ (5) 2 1 RRRG [ ] = σ 2  [ ] [ ] l l G G A mathematical proof for the SVD is G G − − 1   = [[σσ G ]−]−11numbers 1 generation −of 1 [σrandom l = [involves lR [G solution isisnot guaranteed because the The computation involves the−] 2  numbers lG l [σ ]−1 G] Σ lG l R σ Σ G σ 2  19  17 solution is not guaranteed because the computation the generation of random ] R G = Σ G 2  − 1 r the 19  SVD algorithm presented in what follows. uniqueness of the R G = σ Σ G σ 2  [G ] ⎡ [σ2 ⎤ ] Σ G G l GG= l Mathematical Basis GG] 2  R presented in what follows. The uniqueness of[ the ] a diagonalG matrix of variances. G = [σ ][σ [σ]Gis −1 −1 3 associated where l 20  G G basis ⎢⎣σ ⎥⎦ andleft 20  and, even though spectral values are unique, by definition, the left right 2G =d[σbecause Σ G σ (5) ⎡ ⎤ 2 ] [ ] and, even though spectral values are unique, by definition, the associated and basis matrix a adiagonal the solution computation the generation of random numbers 3 3  where G G ⎡σ ⎤= [σ[σ[σG ][G][σ][σσG ]G]]isisisright 2 = where⎢⎣⎡σ diagonal matrixof ofvariances. variances. is involves not guaranteed because the computation where diagonal matrix variances. 3 3  where aaadiagonal matrix ofof σ⎡⎣⎢⎡σσ σ[σGG][][σσGG] ]isis ⎣⎢ GG2G2⎥⎦⎤⎦⎥G⎤⎦⎥= ⎦⎥= [ ⎤ ⎢ 3  where is diagonal matrix ofvariances. variances. ⎣ = diagonal matrix of variances. where 2 21  vectors are not. However, uniqueness is not an issue for loss computations or 18 vectors A mathematical proof for the SVD algorithm is presented in what follows. The uniqueness of the 2 ⎤risk G G G 2 ⎢ ⎥ ⎡ ⎤ ⎡ 21  are not. However, uniqueness is not an issue for loss computations or risk ⎣ ⎦ 3  where is a diagonal matrix of variances. σ ⎤ = [σ ][σσGG]] is ⎡a3 diagonal ⎡σ wherematrix al values are unique, bythe definition, the associated and basis [σdiagonal involves generation of randomleft numbers and, even 3  right where variances. G ][σ G ] is a diagonal matrix of variances. ⎢⎣σ ]G ⎥⎦is=of T l ⎢σ G2⎡ ⎤⎥ = ⎣⎢ ⎢⎣ GG⎦⎥ ⎥⎦=3 [σ GG][where a matrix of variances. σ σ [ ][ G the Gsimulated random variable of M realizations with a) ⎣=1)⎢ways: ⎦ T⎤⎥ T be Gtwo G 4  twoLet 22  procedure from thefrom Yang etYang al. in ways: factorization 19 assessment.The solution is not guaranteed because the computation involves of random numbers lthe though spectral values are unique, bythe definition, procedure differs etrisk al.the algorithm in 1) the factorization ofdiffers variances. σ G2 ⎤⎥⎦ = σ22  l= generation [σ G ][assessment.The ⎡G⎣⎡ ⎤ the ever, uniqueness not anmatrix issue for loss computations oralgorithm T⎦T⎤be be G ] is aisdiagonal G 4 4  Let the simulated random variable with lG simulated random variable of Let l = G Let be thethe simulated random variableof ofM realizations witha) a)ta Mrealizations ⎡ ⎤ T ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ l G = G 4  Let be the simulated random variable realizations with ⎢ ⎥ G = G 4  Let be the simulated random variable ofof realizations with a)a) ta MM ⎣ ⎦ T ⎡ ⎤ T ⎢ ⎥ 23  of23 associated the correlation matrix is based on SVD, which is more robust than the Cholesky ⎢ ⎥ 20 of and, even though spectral values are unique, by definition, the associated left and right basis left and right basis vectors are not. However, T l G = G ⎣ ⎦ 4  Let be the simulated random variable of realizations with a) M l ⎣ ⎦ lG = correlation is based on Let SVD, which more robust than the Cholesky ⎡G⎤ ⎤ 5  ⎡G ⎤target targetcovariance matrix (or, alternately, targetvariances andofmeans, ameans, targetcorr e differs from the Yangthe et al. algorithm matrix in two ways: 1) the realizations with a) and b)with avariable target beisthe the simulated random variable of means realizations with a) target target and M realizations G = 4 ⎡⎣⎢ ⎤⎦⎥Trandom Let be the simulated random realizati M l = ⎡⎢G 4 4  factorization Let G be simulated variable of a) and M ⎢ ⎥ ⎢ ⎥ ⎣ (or, ⎦ (or,alternately, targetcovariance matrix targetvariances and Gfor = ⎢Gloss Let the simulated random variable of M realizations with a) t ⎣ ⎣ or ⎦⎥ ⎦5 4  targetcovariance matrix alternately, targetvariances andaaaatargetcorre targetcorr T uniqueness is not an issue for loss computations ⎥ be computations vectors However, uniqueness is not issue or risk 5 5  targetcovariance targetcovariance matrix (or, l alternately, targetvariances and targetcorre 5 an matrix alternately, and targetcorre ⎤ bebased covariance (or,(or, alternately, target variances and ⎡G ⎤targetvariances rix⎡⎢Gis on21 SVD, whichare is not. more robust than the = the simulated random variable of M realizations withCholesky a) target means, and b) a⎣ ⎦ matrix 5  targetcovariance matrix (or, alternately, targetvariances and a targetcorr M × n 6  targetstatistics are those of G . Here is a matrix and the mean ⎥ ⎣ ⎦ targetcovariance matrix (or, (or, alternately, alternately, targetvariances and a targetcorrelation targetcorrelation matrix). To targetcovariance targetvariances and l l. matrix risk assessment. The procedure differs5 5 5from Yang6 6 algorithm ⎣⎢⎡ target ⎦⎥⎤isalternately, ⎡ (or, ⎤and matrix targetvariances matrix). Th   targetcovariance n nmatrix targetstatistics are those of(or, G Here and the 22  assessment.The procedure differs from thethe Yang two ways: 1)matrix). the atargetcovariance targetin5 correlation statistics are M× × targetstatistics are those offactorization .These Here⎣⎢⎡G G isaaaaaM matrix and themean mean lGalternately, 5  et al. 5  matrix targetvariances and a targetcorre ⎡ ⎤ lG ⎥ ⎤ ⎦ M × n 6  targetstatistics are those of . Here G is matrix and the mean ⎢ ⎥ M × n is 6  targetstatistics are those of G . Here G matrix and the mean o ⎣ ⎦ l ⎡ ⎤ ⎢ ⎥ ⎢ ⎥   M × n 6  targetstatistics are those of G . Here G matrix and the mean is a ⎣ ⎦ et al. two ways: 1) factorization ofmatrix). ⎣ ⎢⎣ ⎦ ⎥⎦ lthe ovariance matrix (or, alternately, in targetvariances andthe a targetcorrelation These l ..targetstatistics ⎡G⎤ ⎤ is aare   algorithm ⎡mean ⎤ iseach lG ⎡G M × n 6  targetstatistics targetstatistics arethose those of Here than matrix and the of each vector is Here matrix and mean of of M × n 6  those of G . Here G matrix an a M × n 6  are those of G . Here matrix and the mean of each vector is zzo is a 23  of the correlation matrix is based on SVD, which is more robust the Cholesky 7  This step is performed by generating the required set of independent rando l . Here ⎡G ⎤ is a M ×⎣⎢ n matrix 5  ⎦⎥ of G ⎦⎥ targetstatistics are those and the mean ⎣⎢ ⎣⎢ ⎦⎥ by the correlation matrix is based on SVD, which is7 6 7  This ⎢⎣ the ⎥⎦required step isis performed generating the setsetofofindependent random l This step performed by generating required independent rando ⎡ ⎤ vector is zero. This step is performed by generating atistics are those of G . Here ⎢G ⎥ is a M × n matrix and the mean of each7 vector is step zero. This stepisisperformed performedby bygenerating generatingthe therequired requiredset setofofindependent independentrandom rando This ⎣ ⎦ the Cholesky decomposition and 2) 7  l a li⎡ 7  This step is performed bywith generating the and required set of independent rando l more robust than 8  the of the distribution zero unit variance followed U= = This step step isis performed performed bysame generating the required setmean of independent random variables 7  This step is performed byindependent generating the required set of by indep required set of independent random variables U 7 7  This by generating the required set of random variables the same distribution with zero mean and unit variance followed byby a alinl⎢⎣U This step is performed by generating the required set of independent rando 5   8 7 8  of of the same distribution with zero mean and unit variance followed T l a procedure is provided to accelerate the convergence 8  of the same distribution with zero mean and unit variance followed by a li 8  of the same distribution with zero mean and unit variance followed by a ⎡Usame ⎤ of the 9  U the=the form same distribution with and ep is performed by generating the required set of independent random variables 8  of distribution with zero mean andzero unit mean variance followed by alin li   ⎢⎣ no ⎥⎦ 8 acceleration 8  of the same distribution with zero mean and unit variance followed by a linear transformatio of the same distribution with zero mean and unit variance foll 9  the form 8  of the same distribution with zero mean and unit variance followed by a linear transformatio X 1  1. Read , select OPT= 0 for (see above) and 1 for acceleration [ ] 9  the form 1  of1.theRead select OPT=and 0 forcorrelation no acceleration (see above) 1 for acceleration [ X ] , variances means, coefficients 9  and the form of the same distribution with zero mean and unit variance followed by a li 9 8  the form variance followed by a linear transformation of 9  unit the form ame distribution mean and unit followed by aaform linear transformation of the form 9  the form to with the zero target values to variance enable the use smaller l 9 9  of the the form 9 ] andl the form 10  Gmean = [T ][μJY l]Ull 2  2. 2  2. Compute [Y ] = ln [ X ] and mean μY Compute [Y ] = ln [ X l 10  [TT[T][ J][J]JU]lU number of simulations. (Whether the target values m lllG== 10  G = ]llU 10  10  GG l [G ] using l10  G==[T[[T][][][JJ]U Ul l ] l l 3  3. Compute Eq (3) matter, which is beyond the 3  are3. accurate Computeis[Ganother using Eq (3) ] 10  G G= =[[TT][][JJ]U ]U 10  l G = [T ][ J ]U (6) l 10  10  G = T J U [ ][ ] T J n × n matrices representing the linear transformation 11  Here and are [ ] [ ] scope of this study.) The two developments, SVD and T J n × n 11  Here and are matrices representing the linear (6) [ ] [ ] ][ J ]Ul T J n × 11  Here and are matrices representing the linear transformation [ ] [ ] ⎡ ⎤ Here [T] and [J] nxn matrices representing thetransformation ⎡U ⎤ using T] ]and J ] are ×nnnnare 11  Here anda[ [Jstandard are matrices representing the linear nnn× ntransformation × n11  Here matrices representing transformation Generate a M matrix normal distribution, wherethe and are the Mlinear 4  4. Generate a M × n matrix 4  a standard normal distribution, and the ⎢U ⎦⎥[T[using ⎢⎣ The ⎥⎦ 4. use T ]where ×correlated nare 11  Here and [ J]M are matrices representing the linear transformation [ ] ⎣ acceleration, are independent. of Here SVD is an 12  random variables to the random variables. Using the property of T J n × n 11  and are matrices representing the linear transformation from the indepen [ ] T J n × n 11  Here and are matrices representing linear [ ] [ ] J n × n 11  Here [[T ]]and are matrices representing the linear transformation from the independ linear transformation from the independent random [ ] 12  random variables to the correlated random variables. Using the property oftra 12  random variables to the correlated random variables. Using the property o T J n × n 11  Here and are matrices representing the linear transformation [ ] [ ] 12  random variables the correlated randomvariables. variables.Using Usingthe theproperty property of cholcov andand the5 number procedure to accelerate 12  random variables toto the random number of simulations and of demand parameters, respectively. 13 independent thenumber covariance matrix iscorrelated given by 5  alternative numbertoofrepresenting simulations of demand parameters, respectively. 12  random variables tocorrelated the correlated randomvariables. variables. Using the propertyofof n× n matrices the linear transformation from the variables to the random Using ] and [ J ] are 12  random variables to the correlated random variables. Using the property of linear transforma 12  random variables to correlated randomofvariables. Using th 13  covariance matrix isisgiven 12  random variables to the correlated random variables. Using the property linear 13  the the covariance matrix givenby bythe convergence could also be used with cholcov. 13  the covariance matrix given by 12  random to the correlated Using the transformat property 13  the covariance isis by 6  the 5.square Compute correlation and thevariables square root of the variance matrixvariables. of the column vectors in of 6  5. Compute the correlation and root the of the variance matrix of thematrix column vectors in random property of linear transformation, the covariance 13  the the covariance matrix isgiven given by T variables to the correlated random variables. Using the 13  property of linear transformation, 13  the thecovariance covariancematrix matrixis isgiven given by covariance 13  1by the matrix is given by l T 1 T l l l ⎡ by ⎤ ⎡Gis⎤ = the lcovariance given byGGT T = [TJ ] Σ U [TJT] The matrix [X] is transformed intoσG ]the [Y] space13  14  matrix =is given T l and 1 matrix l l ll l ll GlGl=and T denote them as ΣRΣ G ,⎣⎢⎡G respectively. T ⎦⎥⎤ M1111−1GG 7  is given , respectively. ⎤ T⎦⎥T⎤ T⎡G⎣⎢⎡ ⎤ = [Gby ] and denote them as R 7 G [ G ]and M1[1σ1−G1⎡⎢⎡]G ariance matrix 14  [TJ lGG lU[TJ l lGG lllT T== lU 14  Σ G G G TJ]]Σ ΣU = = = [ ] [TJ]]T]T]T ⎡ ⎤ ⎡ ⎤ T⎣⎢⎡ G ⎥ ⎥ ⎤ ⎤ 1 1 ⎣ ⎦ ⎦ 14  Σ G G TJ Σ TJ = = as shown in Figure 1. It is transformed again into l l lT[TJ ⎢ ⎥ ⎢ ⎥ [ ] [ 14  Σ G G G GG TJ Σ U = = = T [ ⎣ ⎦ ⎣ ⎦ 1 1 M M − − ⎡ ⎤ ⎡ ⎤ T T T ⎢ ⎥ ⎢ ⎥ 1 1 M M − − 1 1 ⎢ ⎥ ⎢ ⎥ T T 1 1 T 14  l l l l ⎣ ⎦ ⎣ ⎦ Σ G G G GG TJ Σ U TJ = = = [ ] GG ll [ ] l ⎣ ll⎦ ⎥⎦T ⎣ = 1 1−1Σ1lGG l = 1 ⎡ ⎡G⎤ ⎤ ⎡ ⎡Gl⎤ ⎤ =14 MM l − ⎢⎣ [⎦[TJ ⎥⎦TJ]M 14  Σ− U1⎤⎥11[[TJ ]ΣM G⎤ = Gl⎤⎥]T]T= T 1⎡⎢G T ] Σ U [TJ ] l⎡⎣⎢TJ l = [TJ 14  T ⎥= ΣΣ GG G G U = 1⎡⎢⎣l M− M −GG − ⎢ ⎥ ⎢ ⎡ ⎤ another space, [G], such that all of the demand l ⎣ ⎦ ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎦ 14  Σ G G G GG TJ Σ U TJ = = = T [ ] [ ] ⎣ ⎦ ⎣ ⎦ R G = A λ A 8  6. Using spectral value decomposition, 1 1 M M − − ][ Gcor R G = [ AGcor M spectral valueldecomposition, −][1 Gcor ][λ− ] MM−−11the 1 ][ AGcor (7) 15  Accordingly, matrix is ]M −1 1 ⎡ ⎤ T 8 ⎡ ⎤ 6. Using 1 ll ⎢⎣ ⎥⎦correlation ⎢⎣ [ M ⎥⎦ Gcor T Gcor 1 is − Mmatrix (7) the G ⎥ ⎢Gparameter GG Σ U a[TJmean = = = [TJ ]have ] 15  Accordingly, correlation vectors of zero. Matrix [G] ⎢ ⎥ 15  Accordingly, the correlation matrix is ⎣ ⎦ ⎣ ⎦ 15  Accordingly, the correlation matrix is M −1 M −1 15  Accordingly, the correlation matrix isis 12 T correlation 12 T the Accordingly, matrix is ⎡Ucorrelation ⎤Accordingly, ⎡G ⎤ a=larger ⎡U ⎤ λ9  dimensional then expanded space 7. AIf OPT=1, then ⎡⎢G ⎤⎥15  Amatrix =the ] the [σiscorrelation ] . l thematrix 15  Accordingly, 9  is 7. If OPT=1, theninto T matrix is ] [σAccordingly, Accordingly, correlation Gcor Gcor G] . −1 isG −1 ⎥ [λGcor l ]15 [ ⎡matrix ⎣ 15  ⎦ the ⎣⎢ correlation ⎦Accordingly, ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ [ Gcor ] [ 15  correlation matrix is ⎤−the ⎡ ⎤ T R G σ TJ Σ U σ TJ = 16  T l l ⎡ ⎣ G⎤ −⎦−11−11[ ] ll ⎡ ⎣ G⎤ −−⎦−11−11[ ]TT (8) such that theis mean, variance and correlation matrix ingly, the correlation matrix ⎤ [TJ ⎤ [TJ lG==⎣⎡σ⎡ ⎡σ lU ⎣⎡σ⎡ ⎡σ lG lU RG 16  R [TJ]] ]]ΣΣTΣTΣU [TJ]] ]] T 10  If OPT=2, then 16  G ⎦⎤G⎤ −1[TJ G ⎦⎤G⎤ −1[TJ 10  If OPT=2, then l l R σ σ = 16  R G σ TJ U σ TJ = 16  ⎣ ⎦ ⎣ ⎦ T [ [ are preserved. This is achieved by generating the l R G ll = ⎣ ⎣⎡⎡σGGG⎦ ⎦⎤⎦−−⎤ −111[lTJ ] Σ⎡ U⎤ −⎣1⎣⎡⎣σGGG⎦ ⎦⎤⎦−1 [TJl] T ⎡ ⎤ −1 −−11 lG ⎡σ 16  TJ]] ΣΣ U16 ⎡⎡⎣σ σ TJ= = ⎡σ 16  RR G ⎤ ⎤⎦ [[TJ ⎤⎤R⎦ [G [ ] σ TJ Σ U σ TJ [ ] [ ] l l U TJ = 16  ] G G G G ⎣ ⎣ ⎡ ⎤ ⎡σ ⎤and ⎣ of ⎤ ⎦ mean G ⎦ column ⎣ ⎡UG16  ⎣σthe Rcalculate G Ua⎦ diagonal =2 ⎤each i) each subtract the of sample from U covariance matrix [TJ ] Σis [TJcalculate ] ⎣ of⎦ variances. 11  required i) subtract the mean11 from column covariance matrix ofT sample independent random variables −1 −set 1 ⎡σ and, matrix ofthevariances. l ⎢⎣ ⎥⎦17 and and, a diagonal ⎣⎢ ⎣ ⎦⎥G ⎦ matrix ⎢⎣ (8) ⎥⎦⎣ =⎡G⎡⎣⎦σ G⎤ ⎡⎤⎦ ⎡⎣σ G⎤ ⎤⎦ is 2G = ⎡⎣σ G ⎤⎦ [TJ ] Σ U ⎡⎣σ G ⎤⎦ [TJ ] ⎡ ⎤ 2= 17  and, is a diagonal matrix of variances. σ σ σ ⎡ ⎤ ⎤ ⎡σ ⎤ is a diagonal matrix of variances. 2 2⎥ ⎤ = 17  and, and,⎢⎣⎡ ⎡σ⎣⎢σ of the same distribution with zero mean and unit G G ⎦⎤G⎤⎣⎡ ⎡σ ⎣⎡ ⎡σ⎡⎣σ 17  and, diagonalmatrix matrixofofvariances. variances. 17  aaadiagonal σ⎦ σ ⎣ GGG⎦⎤⎦G⎤⎦⎤⎦isis ⎡σGG2⎦⎤⎦⎥G⎦⎥= ⎤⎦⎥= l l GG⎦ ⎦⎣⎤ ⎣⎡now 17  We and,⎣⎢σ⎣⎢consider of variances. without and with =⎣σ⎣⎡⎣σ 2 ⎤ is two 22⎤ ⎤ =⎡ ⎡σ ⎤ ⎤⎡ ⎡σ ⎤ ⎤ is⎣⎢ a17  ⎥ ⎡ 12 transformation. Σ U Gdiagonal G ⎦ ⎣matrix G ⎡ ⎡σdiagonal ⎤ ⎡σcases, ⎤ ismatrix 12  variance Σ Ufollowed by a linear ⎦ ⎦ 17  and, of variances. σ ⎡ and, a diagonal matrix of variances. σ = 17  and, ⎢σ is a diagonal matrix of variances. σ σ 2 G G ⎢ ⎥ G G G ⎣⎣ 18  ⎦ ⎣ ⎦ ⎣⎣ ⎢ GG⎦⎥ ⎦⎥= 17  ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ G ⎦ ⎣ and, G ⎦ We consider now two cases, without and with acceleration, to fully r is a diagonal matrix of variances. σ σ σ = acceleration, recover the underlying statistics. Gto G⎦ ⎦ ⎣ fully ⎣⎢ G ⎦⎥ ⎣ now 2⎤ 18  We consider two cases, without and with acceleration, = ⎡⎣σ G ⎤⎦ ⎡⎣σ G ⎤⎦ is a diagonal matrix of variances. 18  We consider now two cases, without and with acceleration,to tofully fullyre r G ⎦⎥ 18  We consider now two cases, without and with acceleration, fully 18  We consider cases, without and with acceleration, toto fully rere T Tconsider now l two 19  1  statistics. lspectral value 18  We now two cases, without and with acceleration, to fully r Case 1: Largenumber of simulations,no acceleration 13  ii) using decomposition, Σ U = A λ A [ ][ ][ ] 13  Proof ii) using spectral value decomposition, Σ U = A λ A ][ ] 1  Case 1: Largenumber of simulations,no acceleration 18  We We[ consider consider now two cases, without and with acceleration, to fully recover the underl cov U cov U cov U 18  We consider now two cases, without and with acceleration covU ][ 19  cov U cov U statistics. 18  now two cases, without and with acceleration, to fully recover the underly Case 1: Largenumber of simulations, acceleration 19  statistics. statistics. 19  statistics. 18  We consider now two cases, without andnowith acceleration, to fully re 19  19  statistics. Letcases, [Y] = [Y1 Y2and ..Ywith m x n matrix represents the nsider now two without to fully recover the underlying 19  statistics. 19  statistics. 7   n], a acceleration, −1 2 12 T −1 2 19  1statistics. 2 T ⎡ ⎤ ⎡ ⎤ ⎤ = ⎡U ⎤ [ A ][14  G =[σ U 19 A statistics. λcase, ][  λcovFor ] this [case, [ AUlis approximately ] is[isσapproximately ] 14  realizations iii) calculateof⎡⎢G λGcorvariable this ]iii) [calculate ] [ AGcor 7an   identity U Gcor ] l G approximately n . Ac an identity matrix of order ⎥⎦ covU λcovUrandom ⎣⎢ ⎦⎥ ] such ⎣⎢ G⎦⎥ ][ 2 covU2 For ⎣ an⎥⎦ n ⎢⎣dimensional n . Accor For this case, Σ UΣ Gcor an identity of order 777 7   matrix s.     7     7   matrix of order. Accordingly, that 7  Eq (8) leads7to     15  the 8.mean: Compute Yi⎤⎥ +byμYadding the mean: 15  8. Compute ⎡⎢Y ⎤⎥ by adding back Y  i = ⎡⎢⎣G {1} . back   Y i = Gi + μYi {1} . i ⎦ ⎣ ⎦ T 7   T T⎤   T ⎡ T T l lR G 1  Yl = ⎢Yl1 Yl2 . Yln ⎥ = [Y ] (2) =][[LL]T][ L ] R G = L [ ⎡ ⎤ 16  9. Generate demand vectors by taking the exponential of ⎡⎢Y ⎤⎥ . ⎦ vectors by taking the exponential of ⎢Y ⎥ . 16  9. ⎣ Generate demand ⎣ ⎦ ⎣3  ⎦ 3  (9) − 1 Define [G]=]G1 G2 ..Gn] such that −1 ⎤ ⎡ σ L = TJ [ ] [ ] ⎡ ⎤ [ L ] = ⎣σ G ⎦⎣ G[TJ ⎦ ] 17  Mathematical Basis (3) = Y − μ {1} Basis 1 17  G Mathematical

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lR = G[ K=][ K R G [ K] ][ K ] 20 are and,unique, even though spectral the values are unique, by definition, and, even though spectral values by definition, associated left and right basis the associated left and right basis 5  5  −1 ⎞ ⎛ ⎛⎜1 1 −1 ⎞ ⎟⎟ ]or ⎟⎟⎟issue K⎜⎜] = ⎜computations ][]σ Gan vectors is are not not.anHowever, is ][[σG not vectors are not. However,21 uniqueness issue for risk for loss computations or risk G [ Kuniqueness ][=loss [ G ⎜ ⎟⎠ ⎠ ⎜⎝ m⎝ −m 1 −1 22  from assessment.The procedure differs from the 1) Yang al. algorithm in two ways: 1) the factorization assessment.The procedure differs the Yang et al. algorithm in two ways: the et factorization


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13 R G = ⎣σ G ⎦ [T ] ⎜[ J ] Σ U [ J ] ⎟ ⎣σ G ⎦ [T ] TT ll ⎝ ⎠ RR GG = =[[LL][][LL]] T TT −1 −1 ll T⎞ l l (9) ⎡σ ⎤ [T ] ⎛⎜[ J ] Σ U ⎡σ ⎤ [T ] ⎟ 3 3  R G = [ L ][ L ]T (9) 13  R G = J [ ] ⎜ G G ⎟ R G = [ L ][−−1L1 ] ⎣ ⎦ ⎝ ⎠ ⎣ ⎦ 3  (9) [[LL]]== ⎡⎣σ⎡⎣σGG⎤⎦−−⎤⎦ 11 [TJ [TJ]] (9) (9)⎛ 3  T⎞ l (9) ⎡ ⎤⎤ −1[TJ ] 14  Next we select [ J ] such that ⎜⎜[ J ] Σ U [ J ] ⎟⎟ becomes an identity matri [[LL]]= ⎝ ⎠ σGGG ⎦⎦ [TJ ] = ⎣⎡⎣σ ⎛ T⎞ l ⎟ X 1  1. Read , select OPT= 0 no acceleration (seematri abov [that] ⎜⎜[ J ] Σ U [ J ] ⎟ for J ] such Next we select such becomes an [[J] Eq (4) andEq Eq (5), correlation matrix the14  becomes an identity Next we select 4 4  Using Using (5), the correlation matrix of original set be as UsingEq Eq(4) (4)and and Eq (5), thethe correlation matrix ofthe theof original data setmay may beexpressed expressed asthat 15  data possible when ⎝ ⎠ identity matrix of order. This is possible when data set may be expressed as 4  original Using (4) Eq (5), the matrix data set be as rrelation of theEq original data set may be expressed as of 4  original Using Eq (4)setand and Eqbe (5), the correlation correlation matrix of the the original original data set may maywhen be2 expressed expressed as 2. Compute [Y ] = ln [ X ] and mean μY matrix ofmatrix the data may expressed as 15  possible TT −1 2 T lG l= RR G =[[KK]] [[KK]] 16  [ J ] = [ λcovU ] [ AcovU ] (16) T ll = [ K ]TTT [ K ] l [G ] using Eq (3) T R G [ L ][−3 L1]2T 3. Compute T 5 5  R (10) (10) l RG G = l [= ⎛⎛ [ K11] [ K ] ⎞ −−11⎞ 16  J = λ A R G = [ L ][ L ] [ ] ] [ ] Accordingly, Eq (15) is reduced to R G =cov[ LU ][ L ] covU (10)17  KK]]= GG][][σσGG]] ⎞⎟⎟⎟⎟⎟⎟ 1  =⎛⎜⎜⎜⎜ [ [ [ [ 5  (10) (10) (15)Generate is reduced to 11− 5  (10) −1 Eq 4. M × n matrix ⎡⎢U ⎤⎥ using a standard norma a 1  (10) 1  Accordingly, −11[G ][σ ]−−−111⎟⎟⎞⎠⎠ ⎜⎜⎛⎝⎜⎝⎜ mm ⎡σ ⎤ −4  ⎣−1 ⎦ [[KK ]]= T L = T [ ] [ ] l T 1 T G l G l ⎟ = ⎜⎝ m ⎣⎡σ[[GLL⎦⎤][][LL[]T] ] l −1 [G ][σTG ] ⎠⎟l R G = T[lL ][ L ] ⎡σ ⎤ [T ] T RL ]G G = R = L = [ ] = [ ⎜⎝R G = L L [ ][ ] G R G = L L G ⎣ ⎦ [ ][ ] ⎠ = [ L ][ L ] 17  1  Accordingly, is reduced to R G m −1 ⎣ ⎦ 5 Eq , select OPT= 0 for no acceleration (see 1 for [ X (15) ] (17) number of simulations and number of above) demandand parame (17) 1  1  TT 1. Read 1  1  (17) 1  (17) −1 lG (17) l isisisa1 a−symmetric −11 − a symmetric matrix and can be 1symmetric 6 6  Since Since matrix and can be expressed as , it is possible to show that K K Since RR G matrix and can be expressed as , it is possible to show that [[ K][Since ]] the ][LL[[]]K= ⎤ [T ] −1 ⎡ ⎤⎤simulated ⎡σ = [[TT]] 5.[Y Compute ⎤ T[T ]L = [⎡Lσ] =⎤ −[1L⎡⎣σT] = the mean correlation andsame the square root ofmatrix the vaa 2  TT2  have the correlation ⎡σ ⎤ [T ] = [RL ]G GG⎦⎦ 6  ll ⎡⎣σ ⎣⎣σis = ln [ X ]variables Compute and μY the [ ]], ititis⎣ismatrix [ Gand ] ⎦ to ]random Ga⎦ symmetric T [ K ] ,2.it G 6  be Since is can be expressed as possible show K [ ] ⎣ ⎦ G possible TT 2  Since random variable ⎦ 2  Since the simulated variables have thesimulated same correlation matrix atrix and can expressed as possible show that K K [ ] [ to show that all expressed as 6  Since matrix and can be expressed as [ K ] [ K ] , it is possible to torandom show that that can be expressed as [R is possible to show that K ]G[ Kis ] , ait symmetric l the lG l are 3  decomposition data set,the andsimulated from Eq (17) andasEq it may shown that R (11), G 7 equating[G[ L denote them and ] ]and [σ be ]L, ]respectively. RRnon-negative. G 7 7  all By value Since random variables have same allthe theeigenvalues eigenvalues arenon-negative. non-negative. Byspectral spectral value decomposition 3  (17) data set, equating from Eq (17 the eigenvalues ofofof are By spectral value G ] using 3. Compute Eq (3) [ L] from 3  3  data set, and equating Eq and Eqand (11), it mayG [be shown that [ l 2  Since the simulated random variables have the same correlation matrix as that of the original 2 correlation Since the the simulated random variables have the same correlation matrix a 2  Since simulated random have the same correlation matrix R G 7  By all eigenvalues of are non-negative. By spectral value decomposition l random 2  the Since the 2 simulated variables have therandom same matrix assame that ofthat the variables original non-negative. spectral value decomposition correlation matrix as of the original data set and 2  Since the simulated variables have the correlation matrix as that of the original Since the simulated random variables have the same correlation matrix as that of the original R G 7  all the eigenvalues of are non-negative. By spectral value decomposition 9   ative. By spectral value decomposition 1  1. Read [ X ] , select OPT= 0 for 1no acceleration (see above) and 1 for acceleration decomposition l = [ A ][λ 2 ⎡value Rbe G 8 equating 6. Using spectral decomposition,  [data L] from 3  data and Eq (17)4  and Eq (11), itfrom may be shown that [Eq × nfrom 4. Generate a M matrix U ⎤⎥Eq using a standard normal distribution Gcor equating [L] Eq (17) and (11), it may be L ⎡σand ⎤ [that 3  set, and equating from Eq (17) and Eq (11), may be shown that G L 3  data set, and Eq (17) and Eq (11), itit may [ ] [ ] TT[ L] set, TTEqequating T A λ = 4  ] ][ ] 3  data set, and equating from (17) and (11), it may be shown 1 2 shown that ⎢ l 1 2 l L 3  data set, and equating from Eq (17) Eq (11), it may be shown that Gcor Gcor ⎣ ⎦ [ ] L Git⎦⎤ may be shown that 3  data set, and equating from Eq (17) and Eq (11), [ ] ⎣ ⎡ ⎤ RR GG = λ A = L L =[[AAGcor λ A = L L 9   ][ ][ ] [ ][ ] ][ ][ ] [ ][ ] ⎡ T σ A λ = 4  [ ] [ ][ ] Gcor Gcor Gcor Gcor Gcor T σ A λ = 4  [ ] [ ][ ] Gcor Gcor Gcor Gcor G ⎣ ⎦ G ⎦[ X ] and mean μ shown ln 2  2. Compute ⎣ that T ll = [ A ][λ ][ A ]TTT = [ L ][ L ]TTT 8 8  R G (11) Y (11) 12 T   [Y ] = ⎡G ⎤ = ⎡Uof⎤ [demand 5  number and parameters, respecti L ][ L ] Gcor ][ λGcor Gcor Gcor 1122][ AGcor 9  of 7. simulations If11OPT=1, thennumber λGcor ] [ A RLLG = [ AGcor Gcor ] [σ G ] . 12 Gcor Gcor Gcor ] = [ L ][ L ] (11) ⎢ ⎥ ⎢ ⎥ 2 2 = A λ ⎣ ⎦ ⎣ ⎦ 8  (11) = A λ [ ] [ ][ ] [ ] [ ][ ] 1 2] = ⎡σ ⎤ [ A (11) Gcor Gcor Gcor ⎡using ⎡σ ⎤⎤[[AAGcor 12 4 λ [T (18) ][ ] 1λ 2 Gcor 8  (11) T σ λ = 4  T λ = 4  [ ] ][ ] [ ] ][ ] 11 22 ][ (18) ⎡σ Gcor ⎤ 4  Gcor (11) ⎡ ⎤ G Gcor Gcor G Gcor 3  3. Compute Eq (3) T A = 4  (18) [ ] ⎣ ⎦ [ ] [ ] ⎡ ⎤ G G 5  6  Substituting, Eq (18) and Eq (16) Eq (6), and (18) 5.⎣⎣ Compute the correlation and into the square roottaking of thetranspose variance matri ⎦⎦ ] =][λ⎣σGcor [ A ][λGcor ] σ G ⎦ [ A[TGcor = ⎣4  (18) 2 [T ] Gcor G ⎦ ]1Gcor [[LL]]= G ⎦ ] Gcor Gcor ][⎣ λGcor Gcor 10  Eq (18) If OPT=2, = [[ A AGcor 5  (18) transpose and Eq (16) 5  Substituting, and Eq then (16) Substituting, into Eq (6), Eq taking and Gcor ][ λGcor ] Substituting, Eqdenote (18) and Eq (16) Eq]distribution, (6), taking where M and l into ⎡Uthe ⎤ using 9 9  Since Since variables have the correlation matrix as that of original Since the the simulated simulated random random variables have the4 same same correlation matrix as that of the original simulated random variables have the same G R G σ 7  and them as and , respectively. [ ] [ M × n 4. Generate a matrix a standard normal ⎡ ⎤ T G 6  [σ G ] = ⎣σ G ⎦11  , it may that ⎡ l T ⎤from T sample mean i) subtract the each ⎣⎢ be ⎦⎥ shown ⎡G ⎤taking T column of ⎢U l Eq ⎡G ⎤and = 5  (18) Substituting, Eqoriginal (18) and Eq (16) into transpose l taking transpose and noting andnoting it may 6  be shown thatand σ 5  Substituting, Eq (18) and Eq (16) into Eq (6), transpose and ⎡G ⎤ and [into l ] =Eq 9  the Since the simulated random variables have the same correlation as of the original 5  Substituting, Eq (18) and (16) transpose 6  ittaking may be that = ⎡⎣σ(6), ⎣ [σmatrix ]Eq ⎡and ⎤ be ⎢G ⎥may ⎡G ariables correlation matrix that of the = G 5  same Substituting, Eq and Eq (16) into (6), taking transpose noting G matrix as that the original data set and ⎣σ⎤ G(6), ⎦ ,and GEq ⎣ ⎦ = G 5 ]as Substituting, Eq (18) Eq (16) into (6), taking transpose and ⎦ ,that 9  correlation Since theand simulated random variables have the same matrix as that of theshown original GG =noting Substituting, (18) and Eq (16) into Eq (6), taking transpose and ⎢ noting ⎥ andand 10  data set, by equating from Eq (9) and Eq (11), itcorrelation can be that have the have same correlation matrix as that the original L 10  data set, and by5  equating from Eq (9) and EqEq (11), itand can beshown shown that [[Lof ]of ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ T ⎣⎢ ⎦⎥ l respectively. 5  numbershown of simulations and number of demand parameters, that R G = A λ A 8  6.thatUsing spectral−1value decomposition, [ Gcor ][ Gcor ][ Gcor ] l by equating [L] from (9) and Eqmay (11), it can be be shown LL[σ data set, and equating Eq itit can [Eq ] from 2 12 T ⎡Eq ⎤(9) om EqEq (9)10  and it Eq (11), it canby ⎡σ⎤ [ A ⎤ ,, itit may ⎤12  Σ shown U 6  that , ⎡it and be(11), shown that6  ⎤ ]= ⎡U⎡σ 10  data set, and by equating from Eq and Eq (11), can bethat shown that 6  may be be shown that σthe = [be [[⎡⎢σG ⎡ be ⎤ shown G ]= ) and (11), can be shown that 7  λGcorthat G= ⎤ shown GG ]= ]1 2root [ AGcor⎡of]T⎤ the [σ G⎡ variance ]⎤ 6  shown that ⎣σ ⎦(9) [σthat −1 2 12 ⎤[,σit GG⎦⎦ covU ][ λand 6 ]⎡σ , it may be shown ⎣ σ −1 2 [square ⎣ covU ]the ] 6  5. Compute correlation column may be that G ] = ⎣σ G6  ⎥ ⎢ ⎥ [σ G112]2= G ⎦ , it may ⎣ ⎦ ⎣ ⎦ ⎡ ⎤ ⎡ ⎤ G shown ⎣ ⎦ λcovU ] of[λthe ⎣ G⎦ ][matrix 7  G ⎥ = ⎢U ⎥ [ AcovU ][λcovU ] [λGcor ] 7 [ AGcor [σ ⎣⎢GU] (19) Gcor ] [ AGco T covU ⎢G]1⎦⎥ 2= ⎥[ A ⎢ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎡ ⎤ ⎡ ⎤ λ λ 11  (12) 11  [TJ (12) ][ ] [TJ]]==⎡⎣σ⎡⎣σGG⎤⎦ ⎤⎦[[AAGcor ][ ] 9  7. If OPT=1, then G = ⎢U ⎥ [λGcor ] [ AGcor ] [σ G ] . Gcor Gcor Gcor 11 22 l and l ⎣⎢ ⎦⎥ (12) ⎦ R preserves G 7  denote them [1G2 ] and [σ G⎣ 11]the −1 2 T 12 −11 22 spectral λλGcor − 2,2 respectively. TT decomposition, 11  [TJ ] = ⎡⎣⎡σ GG ⎤⎦⎤[ AGcor ][ ] 13  as ii) using value Σ U = [ AcovU ][λc ⎡ ⎤ ⎡ ⎤ This procedure mean, variance and (12) − 1 2 1 2 T Gcor Gcor ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ − 1 2 1 2 T 7  G = U A λ λ A σ (19) σ A = 11  [TJ (12) [ ][ [ ] [ ] −1cov 2 U] 1[ 2 Gcor ] 7  T GcorThis [ ][ ] 7  G = U A λ λ A σ 8  procedure preserves the mean, variance and correlation matrix regard ⎤ ⎡ ⎤ G = U A λ λ A σ (12) [ ][ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] (12) cov U G ⎡ ⎤ ⎡ ⎤ Gcor Gcor ⎢ ⎥ ⎢ ⎥ cov U cov U Gcor Gcor G ⎡ ⎤ ⎡ ⎤ G cov U cov U Gcor Gcor G 7  ] ⎡⎢G U A λ λ A σ = (19) ] ⎦ A[ G ⎢][⎣⎢ A⎦⎥Gcor ⎣U ⎣Gcor⎦=] λU[ [Gcor A ] [λ][λG ] ] A [λ ]8 [σ]10  σ (19) ] preserves ⎦⎥ procedure ⎣⎢]⎣⎢ [If ⎦⎥⎦⎥ GOPT=2, =U 7  then 8 of the Thissimulation procedure preserves the mean, v G ⎣This the mean, variance and(19) correlation matrix regard ⎣ ⎣ ⎥⎦ ⎦ ⎢⎣ ⎥⎦ [7 covU ⎣⎢][G ⎦⎥cov TT matrix ⎣⎢ ⎦⎥ [ ⎣⎢covU⎦⎥ ][ ⎣⎢ cov⎦⎥U ] covU[ Gcorcov]U [ GcorGcor correlation regardless size if T l l ⎡G ⎤ ⎤ nand l ⎡G> −1 2 12 T ⎡ ⎤ ⎡ ⎤ M 9  size if . = G 12  Substituting (12) into Eq (6), taking the transpose, and noting , it = G σ σ = 12  Substituting Substituting Eq Eq (12) into Eq (6), taking the transpose, and noting and , it σ σ = R G = A λ A [ ] 8  6. Using spectral value decomposition, [ ] [ ][ ][ ] ⎡G9 ⎤ = ⎡U ⎤ [Gcor Eq (12) into Eq (6), taking the transpose 9  M Gcalculate G Gcor [ λ ⎢⎣subtract ⎥T14  ⎢⎣ > ⎥ n . iii) Aif λ>column AGcor ]calcul ][Gcor [σ G ⎣ ⎣ GG⎦ ⎦ mean size T cov UM covnU.] ll i) size ifn.M > 11  the sample each ofGcor⎡⎢U] ⎤⎥ [ and l Eq ⎡and ⎤⎤⎦TT⎦ correlation ⎣⎢regardless ⎦⎥ from ⎣⎢variance ⎦⎥ of ⎡σthe ⎤ , mean, TTinto ⎡This ⎤ (6), = G G 12  Substituting Eq (12) taking the noting and it σσGGof =matrix 8  procedure preserves theand mean, variance the simulation [ ] ⎡σtranspose, ⎤ , be 8  This procedure preserves the mean, variance and correlation matrix regard l ⎣ ⎦ 8  This procedure preserves and correlation matrix regard ⎡ = G G 6), taking the transpose, and noting and it σ = [ ] ⎡ ⎤ ⎢ ⎥ G ⎡ ⎤ G 8  This procedure preserves the mean, variance and correlation matrix regardless the simulation ⎣ ⎦ = G G 12  Substituting Eq (12) into Eq (6), taking the transpose, and noting and , it ⎡ ⎤ σ = G and noting and , it may shown ⎣ ⎦ [ ] This procedure the mean,and variance matrix regardless of the simulation G preserves g the transpose, G= and noting andprocedure 8 ⎢G ⎥ This mean, variance correlation matrix of the simulation [σ⎦⎥ GG ] = G ⎣⎢ 8  ⎢⎣ ⎥⎦ ⎤ ⎡ ⎤correlation G⎦ 1 2⎣regardless T 13  that 13  may maybe beshown shown that ⎣σ GGpreserves ⎦ , it ⎣ ⎦the ⎣ ⎦ ⎡M ⎤> ⎡] be ⎤[σmay 9  7. If10  OPT=1, then U computed, = . be computed Once, is by Ythe [λGcor ] ⎡⎢Y[ A⎤⎥ may ] adding Once, G⎡⎣⎢G is computed, computed by adding back asi { by back the mean: 15  8. Compute Gi + μ Gcor Y G 9  size if M > n . i =mean M > n ⎥ ⎢ ⎥ 9  size if . l n 9  size if . that ⎦ ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ⎡ ⎤ M > n 9  size if . 13  may be shown that ⎣10 be ⎦ computed Σ⎦⎤⎥ isUcomputed, ⎣⎡⎢Y ⎦⎤⎥ may Once, ⎢Gby isadding computed, Y ⎤⎥ may beasYco n . if M > n . 9  size if 9 M >size 10  12  Once, ⎣⎡⎢Gback back ⎡⎣⎢the mean 13  may be shown that ⎥ ⎣ ⎦ ⎦ adding the mean as ⎣ ⎦ ⎣ ⎦ 10  IfTOPT=2, then ⎡G⎤ ⎤= ⎡U⎤ ⎤ λλ ]1]122[[AA ]T]T[σ 14  (13) 14  ⎡G =⎡U (13) ]] (13) [σGGdecomposition l = Gcor 16  9. Generate demand vectors by taking the exponential of Gcor ⎡ ⎤ ⎡ ⎤ R G L L ⎦⎤⎥ ⎦⎥ ⎡⎣⎢ ⎣⎢ ⎤⎦⎥ ⎦⎥[[ Gcor to the cholcov⎡⎣⎢ ⎣⎢algorithm ifGcor the Cholesky T [ ][ ] ⎡ ⎤ ⎡ 1 2 T ⎡ ⎤ ⎡ 1 2 T 10  Once, G ⎥ algorithm isbe computed, Yby⎥ Cholesky may be computed by adding back the as computed l =back 10  Once, G isas computed, Y⎤⎥][⎤⎥mean may be computed byladding adding back back the mean mean as asT ⎡identical ⎤1 2is[ Acomputed, ⎡YOnce, ⎤ ⎣⎢may 10  Once, G is computed, YL may be (20) ⎡ ⎤ ⎡ ⎤ T [the 11  Y = G + μ 1 ⎢ ⎡ ⎤ ⎡ ⎤ iby { } R G L 10  Once, G computed adding back the mean i using 1  This form is to cholcov if the decomposition 14  G = U λ σ (13) [ ] column ⎢ ⎥ ⎢ [ ] ] ] ⎢ ⎥ ⎢ ⎦ ⎣ ⎦ 10  G is computed, Y may be computed by adding the mean asof ⎡UΣby Y ⎣ ⎦ ⎣ ⎦ ⎡ ⎤ ⎡ ⎤ 13  ii) spectral value decomposition, =calculate ⎣ ⎦ ⎣ ⎦ 10  Once, G is computed, Y may be computed adding back the mean as (13) Gcor Gcor G [ AcovU ][λthe Gcor Gcor ] [σ G ] ⎣⎢ ⎦⎥ i mean from each ⎢ ]⎦⎥ [ AGcor ⎢ ⎥ ⎢ ⎥ cov U ][ A covU ] 14  This (13) ⎣ the covaria 11  i) subtract the sample ⎣⎢⎣⎢G⎦⎥⎦⎥ =form ⎣⎢⎣⎢U ⎦⎥⎦⎥ [λGcor ⎢ ⎥ ⎢ ⎥ (13) Yi = G⎣⎢ i +⎤⎦⎥ U μand ⎣ ⎦ ⎣ ⎦ T 11  11  Y i = G i + μYi {1} l G⎣the ⎦ cholcov algorithm ⎣ ⎦ Yi {1} is identical to if the 1  This form is identical to the cholcov algorithm T 1 2 if the T Cholesky decomposition R G = [ L ][ L ] T 1Tvector of ones. T [A whereR{1} is [aL1 ][2Mathematical ML ]x inThis the form Cholesky decomposition is the ]v algorithm T equivalent l 17  l to=if[[λLthe Gcor Gcor ] in G = 1  is identical todecomposition the [cholcov algorithm decomposition −1 2 12 T decomposition performed, RCholesky G L]]Cholesky the Cholesky ][ 2  if Cholesky performed, where decomposition is equivalent tocalculate the Basis L11  λGcor AGcor ] in Ythe [ ] [ ⎡ ⎤ =]⎡Uin⎤ [of l 14  iii) G A λcovU ] [λGcor ] [ AGcor (20) i = G i + μY {1} ] [σ(20) 11  Y = G + μ 1 1 11  Y = G + μ 1 12  where is a vector ones. M × 1 { } ii { { } ii } cov U ][(20) G] 12  Σ U T 1 2 T ⎢ ⎥ ⎢ ⎥ Y Y i 11  Y μ 1 i =G { } i + i ⎣ ⎦ ⎣ ⎦ T i 11  Y = G + μ 1 i {1} decomposition { } i Yi L 11  Y = G + μ (20) i i Y 1 12  where is a vector of ones. M × 1 in the Cholesky is where [T] { } 2  performed, where in the Cholesky decomposition is equivalent to in the Y λ A 1 12  where is a vector of ones. This completes the simulation of the multivariate i M × 1 position. [ ] [ ] [ ] { } i 8   8   Gcor Gcor T 12 T 12 T 3  spectral T 18  proofmean, for theRvariance SVD 2  in[λthe to Cholesky is to normal inACholesky the l algorithm AGcor 1/2 leskyperformed, decomposition isvalue inidentical the [ Lequivalent ] todecomposition. [λGcor8.] [Compute [λGcor ] decomposition [ ATGcorin] isthe     where ⎡Ymathematical ⎤ by L ] is presented 88   equivalent to the cholcov if ]the decomposition spectral value equivalent variable with specified 15 algorithm adding back the mean:GY i==[ L G][i and 8  Gcor] 1  [AThis Gcor]form T+ μYi {1} . l ⎢ ⎥ value decomposition. 8  3  spectral ⎦vector 1}ones. where {of is a M ×1 vector of ones. 13  ii) using spectral Σones. U Acovmultivariate λcovU ][ Acov 12  wherecompletes is aM of ones. Msolution ×⎣11vector 11}} is 12  where of × [choose  12  acceleration {{value Yl 13  This the simulation of=the normal variableinvol U ][because U ]computation where {1}12  is a 12  vector M ×112  19 adecomposition, not guaranteed the r3 of simulations,with is a M of ones. 1 vector decomposition. {11}vector correlation matrix. Theisuser can to accelerate   decomposition. where a M of×Tones. × {1} iswhere spectral This completes simulation ofYl th 4  value Case 2: Small number of simulations,with acceleration 12 T the 13  This completes the simulation13 of the multivariate normal variable ⎤ 2  performed, where [ L ] in the14  Cholesky equivalent theof ⎡Yby A [λGcor ]the [Irrespective ] toinunique, 16  9.decomposition Generate demand vectorsThe byto taking exponential Gcorare variance and user choose accelerate reco the recovery ofand, theiseven underlying 20 correlation spectral values definiti ⎢⎣ ⎥⎦ .the −matrix. 1 2 though 1 2 statistics. Tcan 4  Case 2: Small number of simulations,with acceleration ⎡ ⎤ ⎡ ⎤ 14  variance and correlation matrix. 14  variance and correlation matrix. The user can choose to accelerate theThe reco = 14  iii) calculate G U A λ λ A σ [ ][ ] [ ] [ ] [ ] l Case 2: Small number of simulations, with covU covU variable Gcor Y with Gcor specified G ll u ⎢ ⎥ ⎢ ⎥ 13  This completes the simulation of the multivariate normal mean, ⎣ ⎦ ⎣ ⎦ l 4  Case 2: Small number of simulations,with acceleration 13  This completes the simulation of the the multivariate normal variable of the number of normal simulations, the SVDmean, algorithm s,with acceleration 13  This completes the simulation of multivariate normal lsimulations, nthe ill be considerably from identity matrix . First, we 15  statistics. Irrespective of the of the SVD algorithm lnumber 3  13  spectral value decomposition. 13  different This completes the simulation of order the multivariate normal variable specified mean, l the 21 Y with vectors are not. However, uniqueness isvariable not anYYret is Y This completes simulation of the multivariate variable with specified mean, Y 13  This completes the simulation of the multivariate normal variable with specified 5  acceleration In this case Σ U will be considerably different from the 15  identity matrix Irrespective of order n . First, we 15  statistics. Irrespective of the numberreo statistics. of the number of simulations, the SVD algorithm returns results similar to the cholcov routine if the 17  Mathematical Basis 14  variance and correlation matrix. The user can choose to accelerate the recovery of the underlying ⎡Ycholcov ⎤ bythe l 14  variance and correlation matrix. The user can choose to accelerate accelerate the reco 14  variance and correlation matrix. The can choose to 16  the routine acceleration option selected. For aetthe small n 15  8. the Compute adding back mean: Yprocedure G μdiffers 1}not . from i =user {of 14  variance matrix. The user canfrom choose to accelerate recovery of the underlying 22  assessment.The the Yang al.recov algo variance and correlation matrix. The user can choose accelerate the recovery the underlying nifto i +underlying Yi is 5  In case Σ and U correlation will be14  considerably different identity of order . the First, we16  variance and correlation matrix. The user can choose to accelerate the recovery ofthe the ⎣⎢matrix ⎦⎥ acceleration cholcov routine if theFor acceleration l In this l case 14 l will 16  acceleration the cholcov routine ifisthe acceleration option is not selected. a small n option not selected. For a small number 4  Case 2: Small number of simulations,with this be considerably different from U n mean from and compute Σ U . Assuming the number of simulation to 5  In this case Σ U will be considerably different from the identity matrix of order . First, we n ably different from the identityIrrespective matrix order . Irrespective First, we 15  ofstatistics. of the ofstatistics. simulations, the SVD algorithm returns results tosome l lnumber l number T 15 number statistics. Irrespective ofSVD the number of simulations, the SVD algorithm re 15  Irrespective of the number of simulations, the SVD algorithm ret 17  the use of the acceleration option may appear tosimilar violate basic rule 15  statistics. of Irrespective the of the SVD algorithm returns results similar to the correlation matrix isresults based on SVD, which 15  statistics. the of simulations, the algorithm returns similar to 6  ifsubtract the sample from U of. Assuming the ofofalgorithm simulation tothe 15  mean statistics. of] simulations, the Σnumber of simulations, the23  SVD returns results to RU Gand = [compute L][Irrespective L ov algorithm the Cholesky decomposition A number mathematical proof SVD algorithm presented in basic what follo 17  the usesimilar acceleration option ma of simulations, the usefor of the acceleration option 17  18  the use of the acceleration option may appear to⎤⎥ is some rule 16  l9. the Generate demand vectors by taking the exponential ofof⎡⎢Ythe . violate the identity matrix16  of order n. First, we subtract l ⎣ not ⎦ selected. the cholcov routine if the option issmall notroutine selected. Forsimulations, a small number ofis 16  the the cholcov routine the acceleration option issimulations, not selected. For For aa small small nn 16  cholcov ifif the acceleration option U 6  subtract sample mean from and compute Σ is Uacceleration . Assuming the ofFor simulation to 16 lparameters, thethe cholcov routine if acceleration not selected. anumber number 16  the cholcov routine if the acceleration option isisto not selected. For abasic small number ofstatistics simulations, loption 16  cholcov routine if lthe acceleration option isFor not selected. aaof small number of the simulations, > nthe umber of demand that is,the ,compute spectral value decomposition lM may appear violate some rules 18  instead of performing linear transformation onof simulated standardthe norma 19  solution not guaranteed because computation involves gene 1this 2 TΣ ΣU and . Assuming sample mean from n 5  In case U will be considerably different from the identity matrix of order . First, we U 6  subtract the sample mean from and compute . Assuming the number of simulation to and compute Σ U . Assuming the number of simulation to M > n 7  be greater than the number of demand parameters, that is, , spectral value decomposition 18  instead of performing a linear transfor olesky decomposition is equivalent17  to [ λthe the 18  may instead of a linear transformation onto standard [ Aoption ] inmay of acceleration option appear toperforming violate some basic rules ofappear statistics because 17  the use ofinstead the acceleration option may appear tosimulated violate some rule 17  the use the acceleration option may violate some basic rule Gcor ]use Gcorthe 5basic   norma 17  the use of the acceleration appear to violate some basic rules of statistics because because of performing a linear transformation 17  the use of the acceleration option may appear to violate some basic rules of statistics because 17  the use of acceleration option may appear to violate some basic rules of statistics because lthe asso the number of simulation to be greater than the number and, even though spectral valuessimulated are unique, by definition, 17  that Mathematical Basis M >20  ntransformation 7  greater is, 19  , spectral valueis decomposition   performed V on other random variables . Here 8  be shows that than the number of demand parameters, l other l l on lperformed simulated standard normal variables ,, aaislinear M > nfrom 7  be greaterof than the M number of demand parameters, that is, mean , transformation spectral value decomposition l normal 19  transformation performed on > n , spectral nd parameters, that is, decomposition 19  transformation is other simulated random variables .for Here 18 6  value instead of performing a linear on simulated variables llinear that is, Mtransformation >ofsample n,performing spectral 18  instead ofnormal performing linear onU simulated standard norma l ofon 18  instead of performing aa linear simulated standard Utransformation the and Σare Uvariables . standard Assuming the number U on 18 demand instead of18  performing onvalue standard , transformation atransformation linear 21  vectors not. However, uniqueness is not issueV normal lo 8  shows that parameters, U , aanto 18 asubtract instead asimulated linear on simulated standard normal linear Uvariables instead oflinear performing a linear transformation oncompute simulated standard normal variables , a simulation linear T transformation is performed on other simulated 18  A mathematical proof for the SVD algorithm is presented in what follows. The unique 8  shows that decomposition shows that l l (14) T [ AcovU ] ll in two l l l variables 22  assessment.The procedure from Yang al. algorithm V Vsimulated transformation is performedrandom on other simulated . Here is linearly related toto l to l the 10 V 19  transformation is performed on other simulated random variables Here lvariables ldiffers 19  transformation is on other random variables .. Here 9  Σ = [ AcovU ][λcovU ][19  (14) ns,with acceleration ] be greater   to etrelated V .random V performed 19 U transformation isAcov performed onisother Here isis, linearly related U19  random variables ..the Here is linearly related > ncomputation 7  thansimulated the is number ofsimulated demand that , Vspectral value Vis .linearly V related transformation performed onvariables simulated random Here isdecomposition linearly to Vof VM 19  transformation performed on other random variables Here 19  solution isother notparameters, guaranteed because involves the generation random T l l l l 10     (14) 9  Σ U = [ AcovUl][λcov A (14) U 1  (after subtracting the sample mean) such that V = J U where ][ covU ] [ 23  matrix is basedsuch on that SVD,where which [ is] more ro l of subtracting lthe correlation sample mean) T U l l sample mean) such that (after l [ J ]the   1  ][ AU (after subtracting V  = [ Jsuch where computed the from 8  shows the that ]U U (after subtracting the sample mean) that Vvalues = where computed the associated left and 9 form Σ of U = (14) [ J ] isfrom [ J ]Ulisare 20  and, even though spectral unique, by definition, the (14) [ A(14) covU ][ λ covUpossible covU ] because M 1  > n Eq is guarantees that all the l10   is thevalue spectral of 102   from rably different from the identity matrix of order .(14) First, theform form (14) is possible Mcomputed > 10  Factorization Factorization ofofthe of of Eqn Eq iswepossible all decomposition the value decomposition spectral of Σ U10 (after subtracting the sample 10because   10   that 10n  guarantees 21 l vectors are   l not. However, uniqueness is notmean) an issue for loss computation   (after subtracting the sample per Eq (16). 5   2  spectral value decomposition of Σ U (after subtracting the sample mean) per Eq (16). The 2  spectral value decomposition of Σ U (after subtracting the sample mean) per Eq (16). The   M >of T of n guarantees theisof eigenvalues lEq all M > n guarantees that all the 10  because Factorization the ofthat (14) possible because   form are non-negative. Accordingly, the simulated lthe 9 correlation Σ U  =matrix [ Aall ][λbecause ]the> correlation covUAccordingly, cov22  U ][ Acovassessment.The UM l m (14)  procedure l because eigenvalues of U are non-negative. matrix of the differs from et al.Σalgorithm ways:Um 1) the fa n guarantees 0 and thenon-negative. form ofn guarantees Eq (14) issimulation possible that the simulated MΣ > (14)Factorization is11 possible that the The transformation [J] isthe that andwhere 3  all transformation is such that V = Σ U ∞in two and [ J such ] Yang Accordingly, the correlation compute Σ U of .are Assuming the number of to ∞ is the l m l m 11  eigenvalues of Σ U are non-negative. Accordingly, the correlation matrix of the simulated l ] is of m U ∞ is the transformation23  that correlation Σ where V = Σ Um andthe where simulated standard normal ∞ is simulated standard normal variables the matrix issimulated based on SVD,normal which is more robust than the iven by [from Eq(8)] J 3 is such 3 l the transformation that Σ the V[ Jgiven = Σsuch Uby and where is the standard matrix simulated random variables is ∞ 1  of the Σ of Ucorrelation are non-negative. correlation matrix of U the∞4 simulated variables for an infinite number of realizations (or simulations).T ative.eigenvalues Accordingly, matrix of] Accordingly, theEq(8)] simulated 12  random variables is given by[[from M > n and parameters,[from that is, , spectral value decomposition M > n guarantees 10  Factorization of the form of Eq for (14) isinfinite possible because that all the number of realizations (orwhich simulations). 4  Eq(8)] variables for an infinite number of an realizations (or simulations).The degreefundamental to the of statistics can be m 12  randomEq(8)] variables is given by [from 5  acceleration option violates rules T 4  variables for an infinite number of realizations (or simulations).The degree to which the − 1 The degree to which the acceleration option violates ⎞ T l l 2  random variables is given by [from Eq(8)] T Eq(8)] 5   ⎡ ⎤ 5  acceleration option violates fundamental rules of statistics can be measured by testing of the −1 of Σ U (15) J ] Σ U [ J ] ⎟⎟ ⎣σ G ⎦ l [T ] ⎡ ⎤ −1 ⎛ 11  l eigenvalues l T⎞ are non-negative. Accordingly, the correlation matrixto of the simulated ⎡ ⎤ V 6  distribution of with respect the standard normal ⎟ ⎜ 13  ⎠ R G = 5 ⎣σ G ⎦acceleration T J Σ U J σ T (15) [ ] ⎜⎝[ ] option [ ] ⎟⎠violates ] T (15)   rules of statistics fundamental of statistics by distribution. Suc can berules measured by testingcan of be the measured ⎣ G ⎦ −1 [ fundamental l −1 ⎛ T⎞ l l V 6  distribution of with respect to the standard normal distribution. Such testing does not require ⎡ ⎤ ⎡ ⎤ ⎟ 7  the original data set to be known, assuming M > n . 13  R G = ⎛⎣σ G ⎦ [T ] ⎜⎜[ JT] Σ (15) [−J1 ] ⎟⎠ T⎣σvariables [T ] is given by [from Eq(8)] −1 G⎦ −1 ⎞⎟ ⎡ U random lT l (14) 3  Σ U [⎝J ] 12  σ GVl⎤⎦ 7  (15)testing does not require σ G ⎤⎦ R[TG (15) [T ] the with respect todata theset standard normal distribution. ] ⎛= ⎡⎣σ Gl⎤⎦ [T T]6 ⎞⎜⎜⎝[ J ]distribution M > n . Such original to be known, assuming ⎟⎠ ⎣of 8  3.0 Evaluation of results generated by the cholcov and SVD algorith n . This is uch that ⎜⎜[ J ] Σ U [ J ] ⎟⎟ becomes an identity⎛ matrixl of order T⎞ and Volume 45 isNumber 1 March 2015  99 ⎝ The n. the J ]Structural Jto] ΣbeUknown, n . This 14  Next Bridge we7 ⎠select suchdata thatset becomes M an>identity matrix of order [original [ J−1] ⎟⎟⎠ assuming ⎜⎜⎝⎛8 l[Engineer −1 the T ⎛ ⎞ T 3.0⎡l Evaluation of results generated by cholcov and SVD algorithms l ⎞ T ⎤ J ]the ⎟ ⎡σ ⎤ matrix ⎜⎜[ J ] Σ U an 13  thatR ⎜G = σ T J T (15) is evaluated usi [ ] [ ] [ ] ⎟ J J Σ U n Next we select such becomes identity of order . This isof the cholcov and SVD algorithms M > n (14) isT14  possible because guarantees that all [ ] [ ] [ G G 9  The performance ⎟ ⎟ ⎠ ⎣ ⎦ ⎛ ⎞⎟ l ⎝⎜ T ⎞⎟ ⎣ ⎦ ⎠ ⎝ l ⎜ J8  J ] ⎟The nalgorithms 4  U Next select such identity matrix ofSVD order . This is Σ n .becomes [an ] identity [9 results becomes matrix order This is an by [ J ]15  3.0 that Evaluation thecholcov cholcov and SVD possible when ⎟⎠we ⎜⎝[ J ] ΣofUof performance of the and algorithms is evaluated usingbuilding results ofwith analysis of ademand parameters (3 s 10  sample three-story seven ⎠ generated gative. Accordingly, thewhen correlation matrix of the simulated 15  possible T 10  sample three-story building with seven demand parameters (3 story drifts and 4 floor 11  accelerations). The base line study involved a maximum of 11 ground m ⎛ T⎞ lis evaluated (16) −1 2 T 5  possible when of the SVDthat algorithms usingan results of analysis of order a ⎜⎜[ J ] Σ U n . This is we cholcov select [ Jand identity matrix of ] such [ J ] ⎟⎟ becomes 16  [ J ] = [ λ 9  ] The (16) [ A performance ] 14  Next

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namely, and with acceleration is next testing the distribution of Vl =with J ] the (after subtracting the of sample mean) such that where [to is computed from without the [ J ]Ul respect

considered for illustration. Note that proposed standard normal distribution. Such testing does not l ectral value decomposition of Σ U data (after the sample mean) peralgorithm Eq (16). The without acceleration is same as the Yang require the original setsubtracting to be known, assuming. et al. process, if the Cholesky decomposition exists m l m and otherwise, U ∞ is the nsformation [ J3. that Σ V =of Σ U and where normal its modification using cholcov. Even ] is such Evaluation results generated bysimulated the standard ∞ though the power of proposed algorithm lies in cholcov and SVD algorithms riables for an infinite number of realizations (or simulations).The degree its to which the acceleration component, similitude of without The performance of the cholcov and SVD algorithms celeration option violates fundamental rules of statistics can be measured by acceleration testing of thepart to cholcov is first illustrated. Tables is evaluated using results of analysis of a sample 8 and 9 present the results for 100 and 1,000,000 l stribution of V three-story with respectbuilding to the standard Such testing does not require with normal seven distribution. demand parameters simulations, respectively, with 11 ground motions. (3 story drifts and 4 floorMaccelerations). The base line > n. e original data set to be known, assuming These results should be compared with results in study involved a maximum of 11 ground motions. Tables 5 and 6, respectively. For example, entries 0 Evaluation of results generated by the cholcov and SVD algorithms at (5,3) and (5,1) in Table 8 are 0.07 (0.25 with Table 1 presents benchmark data that are used later for anda 2.16 (5.38 with cholcov), respectively. of comparing results. The matrix of dataresults is ofcholcov) e performancethe of purpose the cholcov and SVD algorithms is evaluated using analysis of The accuracy of the SVD results is comparable to denoted as X. The demand parameters are assumed to mple three-story building with seven demand parameters (3 story drifts and 4 floor that achieved with the cholcov algorithm. Table 10 be jointly lognormal. The mean and variance for each celerations). The base line study involved a maximum of 11 ground motions. presents the results for 3 ground motions (G1 to G3) demand parameter from matrix are presented in Table with 100 simulations. Comparison of these data with 2. The mean and variance for each demand parameter ble 1 presents benchmark data that are used later for the purpose of comparing results. The Table 7 indicates that SVD algorithm also works for a computed for matrix [Y]=in [X] are presented in Table atrix of data is denoted as X . The demand parameters are assumed to be jointly lognormal. covariance matrix. Therefore, proposed rank-deficient 3. The correlation matrix, [RYY], computed for [Y] is e mean and variance for each demand Table 2.without The acceleration leads to the comparable presented in Table 4. parameter from matrix X are presented inmethod results as those from the existing method (modified ean and variance for each demand parameter computed for matrix [Y ] = ln [ X ] are presented in Table 5 presents the results of computations using by cholcov). cholcovmatrix, and [100 simulations. table presents RYY ] , computed ble 3. The correlation for [YThe in Table 4. ] is presented The SVD algorithm can also be used for a negative the ratio of cholcov to benchmark values of means definite covariance matrix by assigning zero to those space) and correlation (in simulations. the ble 5 presents(in thethe results of computations using coefficients cholcov and 100 The table spectral values for which the left and right spectral Y space). The correlation coefficients are seen to esents the ratio of cholcov to benchmark values of means (in the Y space) and correlation shapes differ only by sign. The cholcov algorithm will vary significantly from the target values because efficients (in the Y space). The correlation coefficients are seen to vary significantly from thesuch a case as it requires the covariance not work in of the small number of simulations. Minimum and get values because of the entries small number of simulations. and(5, maximum entries matrix to are be necessarily positive semi-definite. The maximum are noted as 0.25 (5,Minimum 3) and 5.38 context of the present paper, however, does not ted as 0.25 (5,1), 3) and 5.38 (5, 1),against respectively, againstof the1.0. target of 1.0. associated respectively, the target Note thatNote that explore this associated benchmark correlation coefficients also contributing to thedistinction. nchmark correlation coefficients are also small (close to zero) andare hence,

() () ( )

smallHowever, (close tothezero) and hence, the served higher ratio. ratio noted for (4, 1)contributing is 0.74 and thetocorrelation coefficient is We now consider the results of analysis using

observed higher ratio. However, the ratio noted for the sizes. acceleration component of the proposed SVD 375, which describes the limitation of existing procedure with smaller simulation This

(4, 1) is 0.74 and the correlation coefficient is 0.375, algorithm. We repeated the analysis above for a nclusion is verified by analysis using 1,000,000 simulations and the results presented in Table which describes the limitation of existing procedure

number of combinations of ground motions with 100, 10,000 and 1,000,000 simulations. Table 11 presents results. For every analysis, the means and correlation coefficients are equal to the target values, regardless of the number of simulations. Clearly, target statistics are completely recovered regardless of the number of simulations. This acceleration technique is the fundamental contribution in the present paper.

show that the statistical parameters convergesizes. to the target as the number of simulations with smaller simulation This values conclusion is

verified by analysis using 1,000,000 simulations 11 and the results presented in Table 6 show that the statistical parameters converge to the target values as the number of simulations is increased, which is an expected result. Three ground motions G1 to G3 are next considered and the results are presented in Table 7 for the 100 simulations. This is a case with rank deficient covariance matrix and the results in Table 7 indicate that the cholcov routine works for this case. Note the ratio of correlation coefficient to its benchmark value (0.375) at (4,1) is 1.49. Proposed SVD algorithm which has two components, 100  Volume 45

Number 1 March 2015

4.

Conclusions

A procedure to generate statistically consistent demand vectors for seismic performance (risk) assessment is developed. The proposed procedure has two distinct features. First, it utilizes the spectral The Bridge and Structural Engineer


value decomposition of the covarience/correlation matrix, which is more robust than the conventional Cholesky type decomposition. Second, the procedure introduces an (optional) acceleration technique in the convergence of the statistics of the simulated demand parameters to the target statistics. Without the acceleration technique, the proposed procedure is mathematically identical to the conventional cholcov algorithm, provided the Cholesky decomposition exists. Therefore, the use of spectral value decomposition is viable alternative to the cholcov routine. With acceleration technique, the target statistics are recovered in the proposed procedure regardless of the simulation size. The proposed acceleration technique can also be used with cholcov routine. Therefore, fundamental contribution in this paper is the acceleration technique and SVD is shown as an alternative to the Cholesky Decomposition.

5.

Yang, T. Y., Moehle, J., Stojadinovic, B. and Der Kiureghian, A. (2009). "Performance evaluation of structural systems: theory and implementation." Journal of Structural Engineering, ASCE, 135 (10): 1146-1154.

6.

Wilkinson, J. H. (1965). The algebraic eigenvalue problem, Oxford Science Publication.

7.

Mathworks, Ed. (2008). MATLAB 8—The language of technical computing, Natick, MA.

8.

Zareian, F. (2010). “Personal communication.”

Table 1: Demand matrix, X, from response-history analysis δ1 (%) δ2 (%) δ3 (%) δa1 (g) a2 (g)

a3 (g)

a4 (g)

G1

1.26

1.45

1.71

0.54

0.87

0.88

0.65

G2

1.41

2.05

2.43

0.55

0.87

0.77

0.78

Acknowledgments

G3

1.37

1.96

2.63

0.75

1.04

0.89

0.81

The study described in this paper was partially funded by the Applied Technology Council in support of the ATC-58 project on seismic performance assessment of buildings. This support is gratefully acknowledged. The study benefited greatly from comments received from Professor Jack Baker of Stanford University and Professor Farzin Zareian of UC Irvine.

G4

0.97

1.87

2.74

0.55

0.92

1.12

0.75

G5

0.94

1.8

2.02

0.40

0.77

0.74

0.64

G6

1.73

2.55

2.46

0.45

0.57

0.45

0.59

G7

1.05

2.15

2.26

0.38

0.59

0.49

0.52

G8

1.40

1.67

2.1

0.73

1.50

1.34

0.83

G9

1.59

1.76

2.01

0.59

0.94

0.81

0.72

Reference

G10

0.83

1.68

2.25

0.53

1.00

0.9

0.74

G11

0.96

1.83

2.25

0.49

0.90

0.81

0.64

1.

2.

3.

4.

Huang, Y.-N., Whittaker, A. S. and Luco, N. (2008). "Performance assessment of conventional and base-isolated nuclear power plants for earthquake and blast loadings." MCEER-08-0019, Multidisciplinary Center for Earthquake Engineering Research, SUNY, Buffalo, NY. Huang, Y.-N., Whittaker, A. S. and Luco, N. (2011). "A seismic risk assessment procedure for nuclearpower plants, (I) methodology." Nuclear Engineering and Design, 241: 3996-4003. ASCE (2006). "Seismic rehabilitation of existing buildings, American Society of Civil Engineers, Reston, VA," Standard ASCE/SEI 41-06. ATC (2012). "Guidelines for the Seismic Performance Assessment of Buildings," Report No ATC-58-1, Applied Technology Council, Redwood City, CA.

The Bridge and Structural Engineer

Table 2: Demand matrix statistics δ1 (%)

δ2 (%)

δ3 (%)

a1 (g)

a2 (g)

a3 (g)

a4 (g)

μx

1.2282 1.8882 2.2600 0.5418 0.9064 0.8364 0.6973

σx

0.0878 0.0849 0.0885 0.0139 0.0615 0.0627 0.0094

Table 3: Demand matrix statistics δ1 (%) δ2 (%) δ3 (%) a1 (g)

a2 (g)

a3 (g)

a4 (g)

μx

0.179

0.625

0.807

-0.634

-0.131

-0.222

-0.370

σx

0.059

0.022

0.018

0.046

0.070

0.0993 0.021

Table 4: Benchmark correlation matrix, 1.000

0.339

-0.019

0.375

-0.022

-0.193

0.145

0.339

1.000

0.656

-0.353

-0.646

-0.723

-0.376

-0.019

0.656

1.000

0.136

-0.094

-0.066

0.220

Volume 45 Number 1 March 2015  101


0.375

-0.353

0.136

1.000

0.839

0.731

0.881

1.0010 1.0000 1.0099 1.3168 1.3732 0.8603 1.0090

-0.022

-0.646

-0.094

0.839

1.000

0.934

0.863

1.0183 1.0099 1.0000 1.1286 1.1479 0.8145 1.0000

-0.193

-0.723

-0.066

0.731

0.934

1.000

0.820

1.4873 1.3168 1.1286 1.0000 1.0002 0.8754 1.1338

0.145

-0.376

0.220

0.881

0.863

0.820

1.000

1.5998 1.3732 1.1479 1.0002 1.0000 0.8808 1.1539

Table 5: cholcov algorithm using 11 ground motions, 100 simulations Demand parameters δ1

δ2

δ3

a1

a2

a3

a4

0.8824 0.8603 0.8145 0.8754 0.8808 1.0000 0.8166 1.0170 1.0090 1.0000 1.1338 1.1539 0.8166 1.0000

Table 8: SVD algorithm using 11 ground motions, 100 simulations, no acceleration Demand parameters

Ratio of means 1.0347 1.0054 1.0010 1.0032 1.0215 1.0410 1.0004

δ1

δ2

δ3

a1

a2

a3

a4

Ratio of correlation coefficients

Ratio of means

1.0000 1.2520 0.6032 0.7389 5.3804 1.4156 0.4426

0.9910 0.9897 0.9759 1.0313 1.0558 1.0311 1.0452

1.2520 1.0000 0.9645 0.8050 0.9612 1.0051 0.9401

Ratio of correlation coefficients

0.6032 0.9645 1.0000 1.7816 0.2477 0.7994 1.2546

1.0000 0.9031 1.3382 1.0446 2.1557 0.8650 1.1866

0.7389 0.8050 1.7816 1.0000 1.0010 0.9940 1.0031

0.9031 1.0000 1.0596 0.8851 0.8649 0.9325 0.7957

5.3804 0.9612 0.2477 1.0010 1.0000 1.0129 0.9842

1.3382 1.0596 1.0000 1.0812 0.0676 0.6748 1.1370

1.4156 1.0051 0.7994 0.9940 1.0129 1.0000 0.9888

1.0446 0.8851 1.0812 1.0000 0.9865 0.9892 0.9771

0.4426 0.9401 1.2546 1.0031 0.9842 0.9888 1.0000

2.1557 0.8649 0.0676 0.9865 1.0000 0.9857 0.9805

Table 6: cholcov algorithm using 11 ground motions, 1,000,000 simulations Demand parameters δ1

δ2

δ3

a1

a2

a3

a4

0.8650 0.9325 0.6748 0.9892 0.9857 1.0000 0.9916 1.1866 0.7957 1.1370 0.9771 0.9805 0.9916 1.0000

Table 9: SVD algorithm using 11 ground motions, 1,000,000 simulations, no acceleration Demand parameters

Ratio of means 1.0003 1.0000 0.9999 1.0000 1.0003 1.0003 1.0001

δ1

δ2

δ3

a1

a2

a3

a4

Ratio of correlation coefficients

Ratio of means

1.0000 1.0008 1.0599 0.9985 0.9982 1.0028 0.9979

1.0010 0.9999 0.9996 1.0003 1.0017 1.0016 1.0006

1.0008 1.0000 0.9990 1.0001 0.9997 0.9992 0.9997

Ratio of correlation coefficients

1.0599 0.9990 1.0000 0.9997 0.9941 0.9776 1.0013

1.0000 1.0006 1.0533 0.9964 1.0520 1.0049 0.9849

0.9985 1.0001 0.9997 1.0000 1.0003 1.0001 1.0000

1.0006 1.0000 0.9985 1.0002 1.0000 1.0001 1.0016

0.9982 0.9997 0.9941 1.0003 1.0000 0.9999 1.0002

1.0533 0.9985 1.0000 1.0078 0.9893 0.9849 1.0032

1.0028 0.9992 0.9776 1.0001 0.9999 1.0000 1.0002

0.9964 1.0002 1.0078 1.0000 0.9999 1.0000 0.9996

0.9979 0.9997 1.0013 1.0000 1.0002 1.0002 1.0000

1.0520 1.0000 0.9893 0.9999 1.0000 0.9999 1.0002

Table 7: cholcov algorithm using 3 ground motions, 100 simulations Demand parameters δ1

δ2

δ3

a1

a2

a3

a4

1.0049 1.0001 0.9849 1.0000 0.9999 1.0000 1.0004 0.9849 1.0016 1.0032 0.9996 1.0002 1.0004 1.0000

Table 10: SVD algorithm using 3 ground motions, 100 simulations, no acceleration Demand parameters

Ratio of means 0.9982 0.9901 0.9885 1.0086 1.0161 0.9986 1.0081

δ1

δ2

δ3

a1

a2

a3

a4

Ratio of correlation coefficients

Ratio of means

1.0000 1.0010 1.0183 1.4873 1.5998 0.8824 1.0170

1.0198 1.0319 1.0261 0.9889 0.9674 1.0318 0.9638

102  Volume 45

Number 1 March 2015

The Bridge and Structural Engineer


Ratio of correlation coefficients 1.0000 1.0013 1.0089 0.8330 0.7768 1.1076 1.0086 1.0013 1.0000 1.0030 0.8629 0.8277 1.1572 1.0028 1.0089 1.0030 1.0000 0.9152 0.8981 1.3930 1.0000 0.8330 0.8629 0.9152 1.0000 0.9996 0.9343 0.9132 0.7768 0.8277 0.8981 0.9996 1.0000 0.9512 0.8955 1.1076 1.1572 1.3930 0.9343 0.9512 1.0000 1.3752 1.0086 1.0028 1.0000 0.9132 0.8955 1.3752 1.0000

Table 11: SVD algorithm, with acceleration Demand parameters δ1

δ2

δ3

a1

a2

a3

a4

Ratio of means 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 Ratio of correlation coefficients

will have the same eigenvalues except the product, which is of higher order will have an additional |m–n| zero eigenvalues. Consider a specific case, namely [A]=[B]T. The matrix product [B]T [B] can be considered as the covariance matrix with some scale factor and will have (m– n) zero eigenvalues if n>m. Moreover, when the determinant of the covariance matrix will be zero and one of the eigenvalues will be zero. Accordingly, the covariance matrix is also rank deficient if. Combining this property with the mathematical proof presented above, the covariance matrix will be rank deficient when n ≥ m.

APPENDIX B: MATLAB Coding of the SVD Algorithm clear all

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1 close all 1  2  1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 2  %OPT=0;% no acceleration 1 3  3  1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 4  2  OPT=1;% with acceleration 4  1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 3 5  5  %Develop underlying statistics of the 4 6  1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 6  7  5  history analysis 7  1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 8  6  8  1  %step-1 7  1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 2 9  APPENDIX A: Rank Deficient Covariance Matrices 8  9  APPENDIX A: Rank Deficient Covariance%X=load('DP.txt'); Matrices 3  4  A:APPENDIX Rank basis Deficient Covariance 9  APPENDIX A: Rank Covariance Matrices m ( n and m are the number of 10  We provide the mathematical for theDeficient statement that if n ≥[mrow,ncol]=size(X); 5  We provide the mathematical basis for the statement that if n ≥ m ( n and m are the number of 10  Matrices Msize=1000000;% simulation size 6  11  demand parameters and ground motions, respectively) the covariance matrix is rank deficient. 11  parameters and groundbasis motions, theifcovariance matrix is rank deficient. n ≥ m ( n and 10  We provide the mathematical for therespectively) statement that the number of m are 7  demand tic; We theanmathematical for the statement 12  We provide first present identity of the basis form [9]: 8  12  first present an identity of themotions, form [9]:respectively) the covariance matrix is rank deficient. 11  We demand parameters and ground

that if (and are the number of demand parameters and

of the response

%step-2

ground motions, respectively) the covariance matrix APPENDIX A: Rank Deficient Covariance Matrices We present [9]: ⎤ ⎡ the form ⎡ an identity ⎤ of ⎡ first Y=log(X); A I o ⎤ ⎡⎤⎥ ⎢ μ I A ⎤ ⎥ ⎡ ⎢ μ I ⎤ ⎥ ⎡ ⎢ A an identity 13 is (A-1) =⎢ ⎢ μ Ipresent rank We⎥ ⎥ first of the I odeficient. ⎥⎥ ⎢⎥ ⎢ μ I A ⎥ ⎢ ⎢ −B 2 13  (A-1) = μ I ⎥ ⎢ B μ I ⎥ ⎥ ⎢ ⎢ o μ2 I − BA ⎥ ⎥ mu=(mean(Y)); ⎢ ⎢⎣ −Bprovide n ≥ m 10  form We basis for the statement that if ( and are the number of n m ⎤ ⎡ μ I ⎥⎥⎤ ⎢⎢the ⎦ ⎡⎦ ⎣ μBImathematical ⎤ ⎦ ⎣ μ I ⎡⎣⎢ I [6]: o μ I − BA ⎥ ⎢ A A ⎥⎦⎥ ⎣⎢ μ I o ⎦⎥ ⎣⎢ ⎦⎥ ⎢ 13  (A-1) = RY=corrcoef(Y); ⎥ ⎢ ⎢ ⎥ ⎢ −B μparameters 11  demand the covariance matrix is rank deficient. I ⎥⎥ ⎢ B μand I ⎥ ground μ 2 I − BArespectively) ⎥ ⎢ o motions, ⎢⎣ ⎡ μ I −A ⎦ ⎣⎤ ⎡ μ I A ⎦ ⎤ ⎣ ⎡ μ 2 I − AB o ⎦ ⎤ %step-3 ⎡ an identity ⎤ ⎡of 12  We first present ⎢ the form [9]: ⎤ ⎥ 14  ⎡⎢ ⎢⎢ μ I −A ⎤⎥ ⎢⎥⎥ ⎢⎢ μ I A ⎥ ⎥⎥ =⎢ ⎢ μ 2 I − AB o ⎥ ⎥ (A-2) 14  (A-2) G=Y-ones(mrow,ncol)*diag(mu); B μ I ⎥⎤ ⎥ (A-1) I ⎥ ⎢⎥ ⎢ B μ I ⎥ = ⎥ ⎢⎢ ⎢ ⎢⎣ o ⎦ ⎡⎢⎣ μoI −A μ I ⎥ I ⎥⎦⎤ ⎢⎣⎡⎦ ⎣ μBI μAI ⎥⎦⎤ ⎦ ⎢⎣⎡⎢ ⎣ μ 2 I B − AB o ⎤⎦⎥ ⎥⎤ ⎡ ⎥ ⎢⎡ ⎢⎡ %step-4 14  (A-2) A ⎥ o ⎥⎤ ⎢⎢ μ I A ⎥⎥ = ⎢⎢ μ I ⎢ I 13  (A-1) B 2 μ I ⎥⎥⎥ I ⎥⎦⎥ ⎢⎣ B μ I ⎥⎦ = ⎢⎣ ⎢⎣ o ⎦ (A-2) 15  If−B the second Eq− (A-1) Eq (A-2)] is denoted Q , then from Eq (A-1): μ I matrix colU=ncol; B μon I ⎥left⎢ side BA [and o ofofμ ⎥[andEq 15  If⎢⎣ the second⎥⎦ ⎢⎣matrix on left EqI (A-1) (A-2)] is denoted Q , then from Eq (A-1): ⎦ side ⎦ ⎣ 9  12 

U_star=randn (Msize,colU);

If the thesecond second matrix on left Eq [and (A-1) Eqis denoted Q , then from Eq (A-1): 15 If matrix on left side side of Eqof (A-1) Eq[and (A-2)] 16  (A-3) μn n det(Q ) =⎤ ⎡ μn n det ( μ2 2 I⎤ − BA ) ⎤ ⎡ 2 %step-5 ⎡ (A-2)] is denoted , then from Eq (A-1): 16  (A-3) μ μdet( ) = μ⎢ μdet BA )μ I − AB o ⎥ I ( μA I − I Q −A ⎢ ⎥ ⎥ ⎢ 14  (A-2) ⎥ ⎥=⎢ ⎥⎢ ⎢ RG=corrcoef(G); μI ⎥ ⎢ ) B ⎥⎦ BA 16  (A-3) μ⎢⎣ n odet(QI) =⎥⎦ μ⎢⎣ n B det ( μ 2II − (A-3) ⎦ ⎣ 17  and similarly from Eq (A-2): 17  and similarly from Eq (A-2): Var_G=var(G); 15  17  18  18 

and similarly from Eq (A-2):

n similarly 2 left side of Eq (A-1) [and Eq (A-2)] is denoted Ifμthe second Q , then from Eq (A-1): and from Sigma_G=diag(sqrt (Var_G)); (A-4) det( Q ) = matrix μ n detEq μ(A-2): I − AB ) (on (A-4) μ n det(Q ) = μ n det ( μ 2 I − AB ) (A-4)

%step-6

nn 18 (A-4) μ nn det( Q ) = μproducts det ( μ 22 I [A][B] AB ) and [B][A] will therefore 16  The (A-3) − BA matrix [A_Gcor,LAM_Gcor,B_Gcor]=svd(RG); 19  The matrix products [ A][ B ] and [ B ][ A] will therefore have the same eigenvalues. Moreover, if A B B A 19  The matrix products and will therefore have the same eigenvalues. Moreover, if [ ][ ] [ ][ ] have the same eigenvalues. Moreover, if [A] is of size

20 17  19  20 

sqrt_LAM_Gcor=zeros (ncol,ncol);

isand of size and [ B ][ A] will have the same m size m×and n then [xA]mismatrix [ B ]misthen [ A][ B ] and nand [B] isn×of [A][B] [B][A] from Aand products and will therefore Moreover, if ][ B ][nx ][ofAsize ]size A] similarly B ] is[ Bof of size and the same n×Eq m [(A-2): m× n thenhave [The [ A][ Bthe ] same[ Beigenvalues. ][ A] will have

21 [eigenvalues product, of higher zero A] is of sizeexcept 20  and of sizeis m× will have m− the nsame n× mthe n thenorder [ B ] is which [ A][ Bwill ] andhave [ B ][anA] additional 21  zero 18  eigenvalues μ n det(Q ) = except μ n det (the μ 2 I product, − AB ) which is of higher order will have an additional m− n(A-4) The Bridge and Structural Engineer Volume 45 Number 1 March 2015  103 22  eigenvalues eigenvalues. 21  22  eigenvalues. except the product, which is of higher order will have an additional m− n zero 19  22 

The matrix products [ A][ B ] and [ B ][ A] will therefore have the same eigenvalues. Moreover, if eigenvalues.

20

[ A] is of size

n× m and [ B ] is of size m× n then [ A][ B ] and [ B ][ A] will have the same 18


for j=1:ncol

%step-7.3

sqrt_LAM_Gcor(j,j)=sqrt (LAM_Gcor(j,j));

G_BAR=U_star*A_covU*inv_sqrt_LAM_ covU*sqrt_LAM_Gcor*A_Gcor'*Sigma_G;

end

end

%step-7 if OPT==0

end

G_BAR=U_star*sqrt_LAM_Gcor*A_ Gcor'*Sigma_G;

%step-8 Y_BAR=G_BAR+ones (Msize,ncol)* diag(mu);

else

%step-9

if OPT==1

W=exp (Y_BAR);

%step-7.1

Time_Elapsed=toc;

mu_U_star=mean(U_star);

save correlated_DP_svd_eff.txt W -ascii -double -tabs;

U_star=U_star-ones(Msize,colU)*diag (mu_U_star); %step-7.2 covU=cov(U_star); [A_covU,LAM_covU,B_covU]=svd(covU); inv_sqrt_LAM_covU=zeros(ncol,ncol); for j=1:ncol inv_sqrt_LAM_covU(j,j)=1/sqrt(LAM_covU(j,j));

%Check results mu_Y_BAR=mean (Y_BAR); RY_BAR=corrcoef (Y_BAR); G_8=mu_Y_BAR./mu % for Table G-8 G_9=RY_BAR./RY %for Table G-9 G_10=var (Y_BAR)./var(Y)

end

104â&#x20AC;&#x192; Volume 45

Number 1 March 2015

The Bridge and Structural Engineer


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106â&#x20AC;&#x192; Volume 45

Number 1 March 2015

The Bridge and Structural Engineer


The Bridge and Structural Engineer

Volume 45 Number 1 March 2015â&#x20AC;&#x192; 107


FORTHCOMING EVENT OF ING-IABSE The Indian National Group of the International Association for Bridge and Structural Engineering (ING-IABSE) in association with Govt of Telangana, R&B is organising two day Workshop on “Code of Practice for Concrete Road Bridges: IRC:112” on 30th and 31st October 2015 at Hyderabad. Programme of the Workshop is as under:

PROGRAMME Friday, October 30, 2015 0830 – 0930 0930 – 1000

Registration Inauguration

1000 – 1030

High-Tea

1030 – 1100 1100 – 1130 1130 – 1215 1215 – 1300 1300 – 1330

Overview & Scope Basis of Design Material Properties and their Design Values Actions and their Combinations Discussions for Session-1

1330 – 1415

Lunch

Session-1

Session-2

1415 – 1500 1500 – 1530

Analysis ULS of Linear Elements for Bending and Axial Forces

1530 – 1600

Tea

1600 – 1630 1630 – 1715 1715 – 1745

ULS of Two and Three Dimensional Elements for Out of Plane and In-Plane Loading Effects Serviceability Limit State Discussions for Session-2

Saturday, October 31, 2015

Session-3

0930 – 1100

ULS of Shear, Punching Shear and Torsion

1100 – 1130

Tea

1130 – 1200 1200 – 1230 1230 – 1315 1315 – 1330

ULS of Induced Deformations Concrete Shell Elements Prestressing Systems Discussions for Session-3

1330 – 1415

Lunch

1415 – 1500

Durability and Deterioration of Concrete Structures

1500 – 1530

Tea

1530 – 1615 1615 – 1645 1645 – 1730

Detailing Requirements Including Ductility Detailing Discussions for Session-4 Valedictory Session

108  Volume 45

Number 1 March 2015

Session-4

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The Bridge & Structural Engineer  

The Bridge & Structural Engineer Vol 45, Number 1, March 2015

The Bridge & Structural Engineer  

The Bridge & Structural Engineer Vol 45, Number 1, March 2015