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Engineering & Applied Sciences IN THIS ISSUE A Novel Finite Element Model for Annulus Fibrosus Tissue Engineering Using Homogenization Techniques Relevance Vector Machines for Earthquake Response Spectra Influence of Carbon in Iron on Characteristics of Surface Modification

by EDM in Liquid Nitrogen Establishing empirical relations to predict grain size and hardness of pulsed current micro plasma arc welded SS 304L sheets

Volume 1 Issue 1 (January 2012) ISSN 2229-1652 eISSN 2229-1660

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Cyclic Elastoplastic Large Displacement Analysis and Stability Evaluation of Steel Tubular Braces SAFARILAB: A Rugged and Reliable Optical Imaging System Characterization Set-up for Industrial Environment

2012 American Transactions on Engineering & Applied Sciences.

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International Editorial Board

Editor-in-Chief

Executive Editor

Associate Professor, South Dakota State University, USA

Associate Professor, Thammasat University, THAILAND

Zhong Hu, PhD

Boonsap Witchayangkoon, PhD

Associate Editors:

Associate Professor Dr. Ahmad Sanusi Hassan (Universiti Sains Malaysia ) Associate Prof. Dr.Vijay K. Goyal (University of Puerto Rico, Mayaguez) Associate Professor Dr. Narin Watanakul (Thammasat University, Thailand ) Assistant Research Professor Dr.Apichai Tuanyok (Northern Arizona University, USA) Associate Professor Dr. Kurt B. Wurm (New Mexico State University, USA ) Associate Prof. Dr. Jirarat Teeravaraprug (Thammasat University, Thailand) Dr. H. Mustafa Palancıoğlu (Erciyes University, Turkey )

Editorial Research Board Members

Professor Dr. Nellore S. Venkataraman (University of Puerto Rico, Mayaguez USA) Professor Dr. Marino Lupi (Università di Pisa, Italy) Professor Dr.Martin Tajmar (Dresden University of Technology, German ) Professor Dr. Gianni Caligiana (University of Bologna, Italy ) Professor Dr. Paolo Bassi ( Universita' di Bologna, Italy ) Associate Prof. Dr. Jale Tezcan (Southern Illinois University Carbondale, USA) Associate Prof. Dr. Burachat Chatveera (Thammasat University, Thailand) Associate Prof. Dr. Pietro Croce (University of Pisa, Italy) Associate Prof. Dr. Iraj H.P. Mamaghani (University of North Dakota, USA) Associate Prof. Dr. Wanchai Pijitrojana (Thammasat University, Thailand) Associate Prof. Dr. Nurak Grisadanurak (Thammasat University, Thailand ) Associate Prof.Dr. Montalee Sasananan (Thammasat University, Thailand ) Associate Prof. Dr. Gabriella Caroti (Università di Pisa, Italy) Associate Prof. Dr. Arti Ahluwalia (Università di Pisa, Italy) Assistant Prof. Dr. Malee Santikunaporn (Thammasat University, Thailand) Assistant Prof. Dr. Xi Lin (Boston University, USA ) Assistant Prof. Dr.Jie Cheng (University of Hawaii at Hilo, USA) Assistant Prof. Dr. Jeremiah Neubert (University of North Dakota, USA) Assistant Prof. Dr. Didem Ozevin (University of Illinois at Chicago, USA) Assistant Prof. Dr. Deepak Gupta (Southeast Missouri State University, USA) Assistant Prof. Dr. Xingmao (Samuel) Ma (Southern Illinois University Carbondale, USA) Assistant Prof. Dr. Aree Taylor (Thammasat University, Thailand) Assistant.Prof. Dr.Wuthichai Wongthatsanekorn (Thammasat University, Thailand ) Assistant Prof. Dr. Rasim Guldiken (University of South Florida, USA) Assistant Prof. Dr. Jaruek Teerawong (Khon Kaen University, Thailand) Assistant Prof. Dr. Luis A Montejo Valencia (University of Puerto Rico at Mayaguez) Assistant Prof. Dr. Ying Deng (University of South Dakota, USA) Assistant Prof. Dr. Apiwat Muttamara (Thammasat University, Thailand) Assistant Prof. Dr. Yang Deng (Montclair State University USA) Assistant Prof. Dr. Polacco Giovanni (Università di PISA, Italy) Dr. Monchai Pruekwilailert (Thammasat University, Thailand ) Dr. Piya Techateerawat (Thammasat University, Thailand ) Scientific and Technical Committee & Editorial Review Board on Engineering and Applied Sciences Dr. Yong Li (Research Associate, University of Missouri-Kansas City, USA) Dr. Ali H. Al-Jameel (University of Mosul, IRAQ) Dr. MENG GUO (Research Scientist, University of Michigan, Ann Arbor) Dr. Mohammad Hadi Dehghani Tafti (Tehran University of Medical Sciences)

American Transactions on Engineering & Applied Sciences

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FEATURE PEER-REVIEWED ARTICLES for Vol.1 No.1 (January 2012) A Novel Finite Element Model for Annulus Fibrosus Tissue Engineering Using Homogenization Techniques

1

Relevance Vector Machines for Earthquake Response Spectra

25

Influence of Carbon in Iron on Characteristics of Surface Modification by EDM in Liquid Nitrogen

41

Establishing empirical relations to predict grain size and hardness of pulsed current micro plasma arc welded SS 304L sheets

57

Cyclic Elastoplastic Large Displacement Analysis and Stability Evaluation of Steel Tubular Braces

75

SAFARILAB: A Rugged and Reliable Optical Imaging System Characterization Set-up for Industrial Environment

91

Contact & Office: Associate Professor Dr. Zhong Hu (Editor-in-Chief), CEH 222, Box 2219 Mechanical Engineering Department, College of Engineering, Center for Accelerated Applications at the Nanoscale and PhotoActivated Nanostructured Systems, South Dakota Materials Evaluation and Testing Laboratory (METLab), South Dakota State University, Brookings, SD 57007 Tel: 1-(605) 688-4817 Fax: 1-(605) 688-5878 Zhong.Hu@sdstate.edu, Editor@TuEngr.com Postal Paid in USA.

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2012 American Transactions on Engineering & Applied Sciences

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A Novel Finite Element Model for Annulus Fibrosus Tissue Engineering Using Homogenization Techniques a

b

b

Tyler S. Remund , Trevor J. Layh , Todd M. Rosenboom , a

Laura A. Koepsell , Ying Deng a b

a*

, and Zhong Hu

b*

Department of Biomedical Engineering Faculty of Engineering, University of South Dakota, USA Department of Mechanical Engineering Faculty of Engineering, South Dakota State University, USA

ARTICLEINFO

Article history: Received September 06, 2011 Received in revised form Accepted September 24, 2011 Available online: September 25, 2011

Keywords: Finite Element Method Annulus Fibrosus Tissue Engineering Homogenization

A B S T RA C T

In this work, a novel finite element model using the mechanical homogenization techniques of the human annulus fibrosus (AF) is proposed to accurately predict relevant moduli of the AF lamella for tissue engineering application. A general formulation for AF homogenization was laid out with appropriate boundary conditions. The geometry of the fibre and matrix were laid out in such a way as to properly mimic the native annulus fibrosus tissue’s various, location-dependent geometrical and histological states. The mechanical properties of the annulus fibrosus calculated with this model were then compared with the results obtained from the literature for native tissue.

Circumferential, axial, radial, and shear moduli were all in agreement with the values found in literature. This study helps to better understand the anisotropic nature of the annulus fibrosus tissue, and possibly could be used to predict the structure-function relationship of a tissue-engineered AF. 2012 American Transactions on Engineering and Applied Sciences.

*Corresponding authors (Y.Deng). Tel/Fax: +1-605-367-7775/+1-605-367-7836. E-mail addresses: ying.deng@usd.edu. (Z.Hu). Tel/Fax: +1-605-688-4817/+1-605-688-5878.

E-mail address: Zhong.hu@sdstate.edu.

2012. American Transactions on Engineering & Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660 Online Available at http://TUENGR.COM/ATEAS/V01/01-23.pdf

1

1. Introduction The annulus fibrosus (AF) is an annular cartilage in the intervertebral disc (IVD) that aids in supporting the structure of the spinal column. It experiences complex, multi-directional loads during normal physiological functioning. To compensate for the complex loading experienced, the AF exhibits anisotropic behavior, in which fibrous collagen bundles that are strong in tension, run in various angles in an intersecting, crossing pattern which helps to absorb the loadings. (Wu and Yao 1976) The layers of the AF are composed of fibrous collagen fibrils that are oriented in such a way that the angles rotate from Âą 28 degrees relative to the transverse axis of the spine in the outer AF (OAF) to Âą 44 degrees relative to the transverse axis of the spine in the inner AF (IAF). (Hickey and Hukins 1980; Cassidy, Hiltner et al. 1989; Marchand and Ahmed 1990).

The approach that homogenization offers to deal with anisotropic materials includes averaging the directionally-dependent mechanical properties in what is called a representative volume elements (RVE). These RVE are averages of the directionally- and spatially-dependent material properties. When summed over the volume of the material, they can be very useful in describing the macroscopic mechanical properties of materials with complex microstructures.

(Bensoussan A 1978; Sanchez-Palencia E 1987; Jones RM 1999) Homogenization has been applied to address some of the shortcomings of structural finite element analysis (FEA) models that utilized truss and cable elements (Shirazi-Adl 1989; Shirazi-Adl 1994; Gilbertson, Goel et al.

1995; Goel, Monroe et al. 1995; Lu, Hutton et al. 1998; Lee, Kim et al. 2000; Natarajan, Andersson et al. 2002) and fiber-reinforced strain energy models (Wu and Yao 1976; Klisch and Lotz 1999; Eberlein R 2000; Elliott and Setton 2000; Elliott and Setton 2001) for modeling the AF. Homogenization has also been used to describe biological tissues such as trabecular bone (Hollister, Fyhrie et al. 1991), articular cartilage (Schwartz, Leo et al. 1994; Wu and Herzog 2002) and AF. (Yin and Elliott 2005).

The mechanical complexity of the AF has posed substantial problems for engineers attempting to model the system. To date, the circumferential modulus and axial modulus have been predicted accurately, but the predicted shear modulus has been consistently two orders of magnitude high. An explanation proposed in a recent paper (Yin and Elliott 2005), which offered a novel homogenization model for the AF, is that the high magnitude prediction for shear 2

Tyler S. Remund, Trevor J. Layh, Todd M. Rosenboom, L. A. Koepsell, Y. Deng, Z. Hu

modulus can be explained by the fact that the models assume the tissue to be firmly anchored in surrounding tissue, whereas the experimentally measured tissue is removed from its surrounding tissue. This removal of the sample from surrounding tissue releases the fibers near the edge, which prevents a portion of the fiber stretch component from being included as a part of the overall shear measurement.

The purpose of this paper was to establish a novel method for modeling the AF using FEA and homogenization theory that predicts the circumferential-, axial-, and radial- modulus accurately while also predicting a shear modulus that accurately represents that of the experimentally measured tissue.

A general formulation for annulus fibrosus lamellar

homogenization was laid out. Appropriate changes to the boundary conditions as well as the geometry of the structural fibres was made to accommodate the measurements of the mechanical properties under various annulus fibrosus volume fractions and orientations.

The specific

changes in the three dimensional location and orientation of the cylindrical, crossing fibers within the matrix was taken into account. And the mechanical properties of the human AF by modeling were compared with the results obtained in the literatures for the native tissues.

2. Mathematical Model The general homogenization formulation used here was applied to the AF before. (Yin and Elliott 2005) In the homogenization approach volumetric averaging is used to arrive at the general formulation.

(Sanchez-Palencia 1987; Bendsoe 1995; Jones RM 1999)

The

homogenization formula is created by averaging material properties for a material that is assumed to be linear elastic over discrete, volumetric segments. The overall material is assumed to have inhomogeneous properties throughout the entire volume. So, the average material properties can be calculated by multiplying the inhomogeneous, localized material properties c by the independent strain rates u, in independent strain states α , β , over the volume of the tissue Ω like in Eq. (1).

Cα , β =

1 α β ui , j uk , l dΩ Ω Ω∫

*Corresponding authors (Y.Deng). Tel/Fax: +1-605-367-7775/+1-605-367-7836. E-mail addresses: ying.deng@usd.edu. (Z.Hu). Tel/Fax: +1-605-688-4817/+1-605-688-5878.

E-mail address: Zhong.hu@sdstate.edu.

2012. American Transactions on Engineering & Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660 Online Available at http://TUENGR.COM/ATEAS/V01/01-23.pdf

(1)

3

Cα , β : overall average material properties ci , j ,k ,l : non-homogeneous material properties ui, j :

independent strain rates

α , β : independent strain rates Ω:

volume

The stiffness tensor Eq. (2) rotates around a certain angle, α , in both the positive and negative direction.

This tensor thus rotates the average material properties to simulate the

direction of the AF collagenous fibers. This angle, α , is measured from the midline, θ , and it changes with spatial location.

C α = RT C ⋅ R

(2)

C∞:

average elasticity tensor for two lamellae

R:

rotation tensor

The elasticity tensor of two, combined lamella Eq. (3) rotated at the same angle, α , in opposite directions .

C

+ / −α

C + α + C −α = 2

(3)

There are four in-plane material properties: C11 , C22 , C12 , and C66 that are calculated for a single lamella. They are arranged in matrix notation, like in Eq. (4).

C11 C12 C = C12 C22  0 0

0  0  C66 

(4)

And the values for C11 , C22 , C12 , and C66 can be calculated from the system of equations shown in Eq. (5) using the height of the fiber portion of the segment ρ , the elastic modulus of the fiber and matrix E f , E m respectively and the Poisson ratio of the fiber and matrix υ f , υ m respectively: 4

Tyler S. Remund, Trevor J. Layh, Todd M. Rosenboom, L. A. Koepsell, Y. Deng, Z. Hu

C11 =

C12 = C 22 =

C66 =

ρE f 1 −ν f

2

2 ( (1 − ρ )Emν 2 + 1 − ρ )Em ρE fν f + − −

(ρν

(

1 −ν m

f

2

1 −ν f

2

1 −ν m

2

(ρν + (1 − ρ )ν ) E

(

f

)

m

2

(

m

Ef

)

ρEm 1 − ν f + (1 − ρ ) 1 − ν m 2 E f 2

+ (1 − ρ )ν m )E m E f

)

ρE m 1 −ν f 2 + (1 − ρ )(1 −ν m2 )E f

(

)

Em E f

(

)

ρE m 1 −ν f 2 + (1 − ρ ) 1 −ν m 2 E f Em E f 1 2 ρEm (1 + ν f ) + (1 − ρ )(1 + ν m )E f

ρ:

height of the fiber

Ef :

elastic modulus of the fiber

Em :

elastic modulus of the matrix

vf :

Poisson ratio of the fiber

vm :

Poisson ratio of the matrix

(5)

Taken together, this system of equations accurately modeled the AF in the existing model.

(Yin and Elliott 2005) It addressed many of the shortcomings of structural truss and cable models and of strain energy models. However it did predict a shear modulus that was two orders of magnitude higher than native tissue.

2.1 Model from the literature The homogenization model for the AF created by Yin et al. accurately predicted most of the important mechanical properties of the AF tissue. But it did not make accurate shear modulus predictions. As a matter of fact, the predictions from this model were two orders of magnitude higher than the measurements reported in the literature. In this section we will detail some aspects of the published model that may contribute to the unnaturally high modulus prediction.

*Corresponding authors (Y.Deng). Tel/Fax: +1-605-367-7775/+1-605-367-7836. E-mail addresses: ying.deng@usd.edu. (Z.Hu). Tel/Fax: +1-605-688-4817/+1-605-688-5878.

E-mail address: Zhong.hu@sdstate.edu.

2012. American Transactions on Engineering & Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660 Online Available at http://TUENGR.COM/ATEAS/V01/01-23.pdf

5

2.1.1 Fiber angle and fiber volume fraction The first two important geometric considerations are the volumetric ratio of fiber to matrix fiber volume fraction (FVF) within the RVE and the fiber angle. (Table 1) (Ohshima, Tsuji et al.

1989; Lu, Hutton et al. 1998) These ratios are used extensively in the calculations. Both the FVF and the fiber angle vary by which lamina they are located in. But the finite element method is a great tool for taking these variabilities into account. The original model used fiber angles in the range of 15 to 45 degrees. It also used FVFs in the range of 0 to 0.3. These ranges were used first in parametric studies in order to better understand how the fiber angle and FVF affect the various relevant moduli. Also, beings fiber angle, and to a lesser extent FVF, can be determined experimentally, the parametric studies helped in determining some of the more difficult to elucidate material properties of the collagen fibers and the proteoglycan matrix.

2.1.2 Fiber configuration The second important geometric consideration is the 3D arrangement of the fibers and matrix within the composite RVE. In the original formulation, (Yin and Elliott 2005) they assumed the two fiber populations to be within a single continuous material and not layered as in native tissue structure. (Sanchez-Palencia 1987) 2.1.3 Boundary conditions The final important consideration is the boundary conditions applied to the RVE. The boundary condition for the tensile case can be seen in Figure 1. A similar boundary condition for the tensile case was applied to the proposed model. But when they set the boundary conditions for the shear case, they fixed the edges along both the θ - and z- axis when they applied a shear along z = 1 and θ = 1 . (Sanchez-Palencia 1987) The proposed model has adopted a boundary condition from (K. Sivaji Babu 2008), It constrains the rz-surface at θ = 0 and applies a shear to the rz surface at θ = 1 . (K. Sivaji Babu 2008) This boundary condition can be visualized in Figure 2. Taken together, these geometric considerations allow the proposed model of the AF tissue’s mechanical behavior to be accurate.

2.2 Proposed model changes Changes to the original model are proposed here. They include changes to the fiber angle and FVF in order to bring them closer to the physiological range.

Changes in the fiber

configuration were proposed in order to more closely mimic the native state of the tissue where 6

Tyler S. Remund, Trevor J. Layh, Todd M. Rosenboom, L. A. Koepsell, Y. Deng, Z. Hu

the crossing collagen fibers are separated by a section of proteoglycan matrix, whereas in the original model they were welded together in the shape of an ‘X’. The final change made to the original model was in the applied boundary conditions.

2.2.1 Fiber angle and fiber volume fraction The ranges for this study were based loosely on the values used for the original study. In this simulation graphs of circumferential-, axial-, and radial- modulus as well as shear modulus against fiber volume fraction at fiber angles of 20, 25, 30, and 35 degrees were generated.

Graphs were also generated for axial- and circumferential- modulus as well as shear modulus against varying fiber angle at fiber volume fractions of 0.05, 0.1, 0.15, 0.2, 0.25, and 0.3. The angles of collagen in native tissue range from 24.5-36.3 degrees to the transverse plane with an average of 29.6 degrees.

2.2.2 Fiber configuration In this paper it is assumed that the fiber populations are layered and separated by matrix material. The three dimensional geometric arrangement for this fiber and matrix composite is shown in Figure 1 as a RVE along with the tensile case’s boundary conditions. corresponding RVE for the shear case is shown in Figure 2.

The

With the material being a

composite, it is important to assign dimensions to repeating components within the RVE. The width of the segment, which is denoted by c in Eq. (6) was set to be equal to 13 times the radius, r, of the fiber when the number of fibers, n, within the RVE is 4. This means that the distance between fibers is the equivalent of one radius. The length of b is dependent on the fiber angle α and the length of a. Eq. (7) The length of a was derived from looking at the ratio of total fiber volume to total segment volume. A number of new variables are introduced in the derivation of a Eq. (8). So a can be derived from Eq. (9) by substitution of Eq. (10) and then rearranging.

c = 13 ⋅ r

(6)

b = a ⋅ tan (α )

(7)

4π ⋅ r 2 a= ρ ⋅ c ⋅ sin (α )

(8)

*Corresponding authors (Y.Deng). Tel/Fax: +1-605-367-7775/+1-605-367-7836. E-mail addresses: ying.deng@usd.edu. (Z.Hu). Tel/Fax: +1-605-688-4817/+1-605-688-5878.

E-mail address: Zhong.hu@sdstate.edu.

2012. American Transactions on Engineering & Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660 Online Available at http://TUENGR.COM/ATEAS/V01/01-23.pdf

7

Figure 1: Meshed 3D geometric representation of matrix and fiber orientation along with coordinate system, dimensions, and tensile boundary conditions.

8

Tyler S. Remund, Trevor J. Layh, Todd M. Rosenboom, L. A. Koepsell, Y. Deng, Z. Hu

Figure 2: Meshed 3D geometric representation of composite RVE along with corresponding axes, dimensions, and shear boundary conditions.

ρ=

V fiber VRVE

=

π ⋅n⋅lf ⋅r2 a ⋅b⋅c

l f = a 1 + tan 2 (α )

(9) (10)

After substituting, making use of a trigonometric identity, and rearranging, the simplified formula for a, becomes clear.

So to equally space the four fibers along the c edge from each other and also the edge of the matrix, the length d was derived as given by Eq. (11). It makes use of the idea that when there *Corresponding authors (Y.Deng). Tel/Fax: +1-605-367-7775/+1-605-367-7836. E-mail addresses: ying.deng@usd.edu. (Z.Hu). Tel/Fax: +1-605-688-4817/+1-605-688-5878.

E-mail address: Zhong.hu@sdstate.edu.

2012. American Transactions on Engineering & Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660 Online Available at http://TUENGR.COM/ATEAS/V01/01-23.pdf

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are four fibers within the RVE, that there are five equal divisions of width. d=

2⋅n⋅c⋅r +r 5

(11)

a : width of the representative volume element b : height of the representative volume element

c : length of the representative volume element d : distance between fibers

n : number of fibers in the representative volume element r : radius of the fibers

α : angle between fibers.

So by putting the above equations into the prototype code, a master program code was developed that is useful for predicting the various moduli at each variation of fiber angle and FVF. 2.2.3 Boundary conditions The original paper had fixed boundary conditions along two adjoining faces of the RVE and applied shear on the two opposite faces of the RVE. In the proposed model one face has fixed boundary conditions, and the opposite face has an applied shear. These changes taken together make for a model that predicts all moduli, including the shear modulus, accurately.

3. Material Properties It is also important to assign material properties to the parameters that remain constant regardless of where they are measured throughout the AF. The elastic modulus and Poisson ratio for the collagen fibers and proteoglycan matrix can be assigned specific values. For modeling the varying conditions of the AF tissue, laminae, and IVD, the parameters were chosen based on the literature of past numerical models of the AF, and in some cases, direct measurements of the 10

Tyler S. Remund, Trevor J. Layh, Todd M. Rosenboom, L. A. Koepsell, Y. Deng, Z. Hu

tissues. An elastic modulus of 500 MPa and a Poisson’s Ratio of 0.35 were adopted for the collagen fibers (Goel, Monroe et al. 1995; Lu, Hutton et al. 1998), while an elastic modulus of 0.8 Mpa (Lee, Kim et al. 2000; Elliott and Setton 2001) and a Poisson’s Ratio of 0.45 (ShiraziAdl, Shrivastava et al. 1984; Goel, Monroe et al. 1995; Tohgo and Kawaguchi 2005) were assigned to the proteoglycan matrix. Fiber volume fractions and fiber angles were varied over ranges found in previous homogenization.

4. Results The first input parameter from the lamina that is varied in order to investigate the effect on the various moduli is the FVF.

The FVF is varied from 0.05 to 0.3, which are normal

physiological ranges. (Table 1) Table 1 gives estimates for the cross-sectional area of the AF, FVF of the AF, and fiber angle. Each are estimated for the corresponding lamella. Of course these parameters are variable throughout the AF. But this list was compiled for the original model, so it was used here for ease of comparison. There are also more than six lamellar layers in the AF, but six is a reasonable approximation.

Table 1: Annulus fibrosus cross-sectional area for each of the lamina layers, collagen fiber volume fraction for each of the lamina layers, and fiber orientation angle as reported in the literatures. These values were inserted into the proposed formulation.

Lamina Layer Annulus fibrosus cross sectional area Collagen fiber volume fraction Fiber angle

Inner

2nd

3rd

4th

5th

Outer

References

0.06

0.11

0.163

0.22

0.2662

0.195

(Lu, Hutton et al.

1998)

0.05

0.09

0.13

0.17

0.2

0.23

(Yin and Elliott 2005) (Lu, Hutton et al.

1998)

Annulus Fiber orientation average: 29.6 (range 24.5-36.3)

*Corresponding authors (Y.Deng). Tel/Fax: +1-605-367-7775/+1-605-367-7836. E-mail addresses: ying.deng@usd.edu. (Z.Hu). Tel/Fax: +1-605-688-4817/+1-605-688-5878.

E-mail address: Zhong.hu@sdstate.edu.

2012. American Transactions on Engineering & Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660 Online Available at http://TUENGR.COM/ATEAS/V01/01-23.pdf

11

Figure 3 looks at how the circumferential modulus varies with varying FVF and fiber angle.

At a fiber angle of 20 degrees the circumferential modulus varies from 7 Mpa at a FVF of 0.05 to 26 Mpa at a FVF of 0.3. At a fiber angle of 35 degrees the circumferential modulus varies from 2 Mpa at a FVF of 0.05 to 17 Mpa at a FVF of 0.3.

Figure 3: Circumferential modulus vs. fiber volume fraction at various fiber angles.

Figure 4 takes a look at how the axial modulus varies with FVF and fiber angle. The axial modulus at a fiber angle of 20 degrees varies from 1 Mpa at a FVF of 0.05 to 4 Mpa at a FVF of 0.3. It also varies from 1 Mpa at a FVF of 0.05 to 9 Mpa at a FVF of 0.3 when the fiber angle is 35 degrees.

12

Tyler S. Remund, Trevor J. Layh, Todd M. Rosenboom, L. A. Koepsell, Y. Deng, Z. Hu

Figure 4: Axial modulus vs. fiber volume fraction at various fiber angles.

Figure 5: Shear modulus vs. fiber volume fraction at various fiber angles. *Corresponding authors (Y.Deng). Tel/Fax: +1-605-367-7775/+1-605-367-7836. E-mail addresses: ying.deng@usd.edu. (Z.Hu). Tel/Fax: +1-605-688-4817/+1-605-688-5878.

E-mail address: Zhong.hu@sdstate.edu.

2012. American Transactions on Engineering & Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660 Online Available at http://TUENGR.COM/ATEAS/V01/01-23.pdf

13

In Figure 5 the shear modulus is evaluated against fiber volume fraction at various fiber angles. The shear modulus, at a fiber angle of 20 degrees, was 0.1 Mpa at a FVF of 0.05 and was 0.6 Mpa at a FVF of 0.3. The shear modulus, at a fiber angle of 35 degrees, was 0.3 Mpa at a FVF of 0.05 and was 1.2 Mpa at a FVF of 0.3.

Figure 6 shows that the radial modulus seemed to depend very little on fiber angle. But it also shows that radial modulus increases linearly with increasing FVF from 0 Mpa at a FVF of 0.05 to 1.6 Mpa at a FVF of 0.3.

Figure 6: Radial modulus vs. fiber volume fraction at various fiber angles.

The next input parameter from the lamina that is varied in order to investigate the effect on the various moduli is the fiber angle. The physiologically-relevant range of fiber angles is roughly 20 to 35 degrees (Table 1).

In Figure 7 the circumferential modulus at a FVF of 0.05 varies from 7 Mpa at a fiber angle of 20 degrees to 2 Mpa at a fiber angle of 35 degrees, and at a FVF of 0.3 it varies from 25 Mpa at a fiber angle of 20 degrees to 16 Mpa at a fiber angle of 35 degrees.

14

Tyler S. Remund, Trevor J. Layh, Todd M. Rosenboom, L. A. Koepsell, Y. Deng, Z. Hu

Figure 7: Circumferential modulus vs. fiber angle at various fiber volume fractions.

Figure 8: Axial modulus vs. fiber angle at various fiber volume fractions.

In Figure 8 the axial modulus at a FVF of 0.05 is 1 Mpa, and at a FVF of 0.3 it varies from *Corresponding authors (Y.Deng). Tel/Fax: +1-605-367-7775/+1-605-367-7836. E-mail addresses: ying.deng@usd.edu. (Z.Hu). Tel/Fax: +1-605-688-4817/+1-605-688-5878.

E-mail address: Zhong.hu@sdstate.edu.

2012. American Transactions on Engineering & Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660 Online Available at http://TUENGR.COM/ATEAS/V01/01-23.pdf

15

3.5 Mpa at a fiber angle of 20 degrees to 9 Mpa at a fiber angle of 35 degrees.

In Figure 9 the shear modulus at a FVF of 0.05 varies from 0.6 Mpa at a fiber angle of 20 degrees to 1.2 Mpa at a fiber angle of 35 degrees, and at a FVF of 0.3 it varies from 0.1 Mpa at a fiber angle of 20 degrees to 0.2 Mpa at a fiber angle of 35 degrees.

Figure 9: Shear modulus vs. fiber angle at various fiber volume fractions.

Table 2: Values predicted by the model in both range form and real case calculations as compared to the corresponding values of circumferential-, axial-, radial-, and shear- modulus measured experimentally as found in the literature.

16

Modulus (Mpa)

Modeling Ranges Fα[20-30] FVF [0.05-0.30]

Real Case

Circumferential Modulus

1.92≤E≤25.35

7.09

Axial Modulus

0.91≤E≤9.09

2.12

Radial Modulus

1.10≤E≤1.57

1.34

Shear Modulus

0.08≤G≤1.20

0.16

Experimental 18±14 (Elliott and Setton 2001) 0.7±0.8 (Acaroglu, Iatridis et al. 1995) (Ebara, Iatridis et al. 1996) (Elliott and Setton 2001) 0.1 (Iatridis, Kumar et al. 1999)

Tyler S. Remund, Trevor J. Layh, Todd M. Rosenboom, L. A. Koepsell, Y. Deng, Z. Hu

The changes to the moduli are mostly linear. But while the axial- and shear- moduli (Figures 8-9) increase with increasing fiber angle, the circumferential modulus (Figure 7) decreases with increasing fiber angle (Table 2).

While modeling ranges allow us to evaluate the effect of changing the input parameters such as fiber angle and fiber volume fraction on the various mechanical characteristics of the tissue, they don’t allow us to compare our model to the real case. Table 2 shows the ranges of the moduli predicted by the model accompanied by the modulus predicted when the input parameters used were what was assumed to be found in the human body. These values were then compared to experimentally measured values found in literature.

5. Discussion Here comparisons between the proposed model and existing homogenization model, as well as the experimentally measured data from the literature, will be made. It is worth repeating that in the 3D homogenization models, the fibres of the AF are modelled as truss or cable elements that are strong in tension but not capable of resisting compression or bending moment. This holds true for both the proposed as well as the existing homogenization model. Also, the surfaces of the fiber and matrix that come into contact with each other are ‘glued’ as if the surfaces that those two features share are actually one in the same. So the interface is a blend and there is no slippage between the components at their respective interfaces.

An explanation would be in order for how the ‘real case’ moduli (Table 2) were calculated.

The fiber angle in the native tissue varies not only from lamella-to-lamella, but also within each lamella. So an average fiber angle of 29.6 degrees was taken from the literature (Lu, Hutton et al.

1998). Fiber volume fraction is also variable, so a weighted FVF was used. To arrive at this weighted FVF, an approximate FVF from each lamella was considered (Yin and Elliott 2005) along with the cross sectional area of the corresponding lamella (Lu, Hutton et al. 1998). Using these parameters, calculations were made for the moduli for each of the lamella. Then the moduli were weighted based on the cross-sectional areas (Table 1) of the various lamellas relative to the overall cross sectional area. Once the weighting factors were multiplied by the modulus for that specific lamella, the various weighted moduli were summed to come to an actual modulus. *Corresponding authors (Y.Deng). Tel/Fax: +1-605-367-7775/+1-605-367-7836. E-mail addresses: ying.deng@usd.edu. (Z.Hu). Tel/Fax: +1-605-688-4817/+1-605-688-5878.

E-mail address: Zhong.hu@sdstate.edu.

2012. American Transactions on Engineering & Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660 Online Available at http://TUENGR.COM/ATEAS/V01/01-23.pdf

17

The existing model has a circumferential modulus in the 11 MPa range, an axial modulus of around 2 MPa, and a shear modulus of around 18 MPa. Conversely, the proposed model had a circumferential modulus of about 7 MPa, an axial modulus of about 2 MPa, and a shear modulus of around 0.5 MPa.

The experimentally measured values for these parameters are a

circumferential modulus in the range of 4-32 MPa, an axial modulus in the range of 0.1-1.5 MPa, and a shear modulus of 0.1 MPa. (Table 2).

While there is agreement between the various models and the experimentally-measured values from literature when it comes to tensile moduli, the models uniformly disagree with the experimentally measured data from the literature when it comes to the shear modulus. The shear modulus is over two orders of magnitude higher in the models than in the experimentally measured data from the literature. The author suggested that this is because the tissue has to be removed from its surroundings to be measured experimentally. (Yin and Elliott 2005) This frees up the ends of the fibers so there is fiber sliding but not fiber stretching contributing to overall shear measurements. Whereas the nature of the models can have more realistic in vivo boundary conditions, so the tissue can experience both fiber stretch and fiber sliding in its shear measurement. Conversely, the proposed model will more accurately emulate the former.

In this study, a homogenization model of the AF was revised to address the discrepancy between the shear modulus prediction in the previously proposed model and the experimental data of human AF tissue. The original model had a shear modulus two orders of magnitude higher than that of the experimental values for native AF tissue. It was suggested that the shear was lower in the experimental values, because the pieces of AF tissue were removed from their native surroundings. This causes the fibers of the tissue near the edges to not be anchored into the surrounding tissue. So the stretch of the tissue’s fibers may not have been contributing to shear measurements.

Here is suggested a model that gives accurate accounts of the shear

modulus in the AF tissue while not sacrificing modulus predictions in the circumferential-, axial-, and radial-directions.

Several significant changes have been made to the reported model (Yin and Elliott 2005) to address the discrepancy between the shear modulus in the model and that experimentally measured in the native tissue. The first change made to the model was the arrangement of the 18

Tyler S. Remund, Trevor J. Layh, Todd M. Rosenboom, L. A. Koepsell, Y. Deng, Z. Hu

fibers and matrix within the RVE. In both this model and the original, there are four fibers. In the original model there are two fibers on each opposing face. The two crossing fibers are in the same plane, so they are in effect welded together. One of the changes made to this model is in the geometrical layout of the fibers. The alternating fibers are separated in space and by matrix material. This separation of the fibers allows them to slide against each other. Once the arrangement of the fibers and the matrix were changed, the shear modulus prediction was decreased. But it had decreased to a level much smaller than that of the native tissue value. The value the model had predicted was actually 10 −12 MPa. This is much, much smaller than the value tested in native tissue of roughly 0.1 MPa. So a literature search was performed to try to find alternative approaches to improving shear predictions in homogenization models. The paper that was found called for changing the boundary conditions. In the original model, two adjoining sides of the RVE are constrained, and the opposing two sides of the RVE have the shear loadings applied. This model has one side constrained at a time. The opposing side of the RVE has the shear loading applied. This has brought the shear modulus prediction much closer to that tested in the native AF tissue. And while the original model is likely more accurate for 3D predictions as the tissue is in the IVD in vivo, if the aim is to develop a model that more accurately predicts the mechanical properties of a resected piece of AF tissue as is measured in the literature, then boundary conditions used in the proposed model are more applicable. This is because the boundary conditions in the proposed model allow for the fibres to slide more freely, avoiding incorporating fiber stretch, and resulting in significantly lower shear measurements.

This model is important in understanding the mechanics of the AF, especially when tissue samples are resected from the greater IVD.

It can be useful for better understanding disc

degeneration and for improving approaches to designing functional tissue engineered constructs.

It can help in understanding disc degeneration as the process is usually characterized by a degradation of the proteoglycan matrix. Through the alteration of the matrix, disc degradation can be modeled accurately. Also, more appropriate benchmarks for the design of functional tissue engineered constructs can be set through the better understanding of the interaction of the AF subcomponents that this model provides. *Corresponding authors (Y.Deng). Tel/Fax: +1-605-367-7775/+1-605-367-7836. E-mail addresses: ying.deng@usd.edu. (Z.Hu). Tel/Fax: +1-605-688-4817/+1-605-688-5878.

E-mail address: Zhong.hu@sdstate.edu.

2012. American Transactions on Engineering & Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660 Online Available at http://TUENGR.COM/ATEAS/V01/01-23.pdf

19

It should be noted that this model, like those proposed in the past, does not take interlamellar interactions into account.

To this point, it has not been determined if the interlamellar

interactions and interweaving, that have been observed in the literature, are of mechanical significance.

6. Conclusion In summary, this study established a novel approach to an existing homogenization model. It more closely models the anisotropic AF tissue’s in-plane shear modulus as if it were excised from the IVD. It did this while still making accurate predictions of circumferential-, axial-, and radial- moduli.

The lower shear stress predictions were more in line with experimental

measurements than past models. The model also elucidates the relationship between FVF, fiber angle, and composite mechanical properties.

The proposed model will also help to better

understand the structure-function relationship for future work with disc degeneration and functional tissue engineering.

7. Acknowledgements This research was partially supported by the joint Biomedical Engineering (BME) Program between the University of South Dakota and the South Dakota School of Mines and Technology.

The authors would also acknowledge the South Dakota Board of Regents Competitive Research Grant Award (No. SDBOR/USD 2011-10-07) for the financial support.

8. References Acaroglu, E. R., J. C. Iatridis, et al. (1995). "Degeneration and aging affect the tensile behavior of human lumbar anulus fibrosus." Spine (Phila Pa 1976) 20(24): 2690-2701.

Bendsoe (1995). "Optimization of structural topology, shape, and material." Berlin.

Bensoussan A, L. J., Papanicolaou G. (1978). Asymptomatic Analysis for Periodic Structures.

North Holland, Amsterdam.

Cassidy, J. J., A. Hiltner, et al. (1989). "Hierarchical structure of the intervertebral disc." Connect Tissue Res 23(1): 75-88.

Ebara, S., J. C. Iatridis, et al. (1996). "Tensile properties of nondegenerate human lumbar anulus 20

Tyler S. Remund, Trevor J. Layh, Todd M. Rosenboom, L. A. Koepsell, Y. Deng, Z. Hu

fibrosus." Spine 21(4): 452-461.

Eberlein R, H. G., Schulze-Bauer CAJ (2000). "An anisotropic model for annulus tissue and enhanced finite element analyses of intact lumbar bodies." Computational Methods in Biomechanics and Biomedical Engineering: 1-20.

Elliott, D. M. and L. A. Setton (2000). "A linear material model for fiber-induced anisotropy of the anulus fibrosus." J Biomech Eng 122(2): 173-179.

Elliott, D. M. and L. A. Setton (2001). "Anisotropic and inhomogeneous tensile behavior of the human anulus fibrosus: experimental measurement and material model predictions." J Biomech Eng 123(3): 256-263.

Gilbertson, L. G., V. K. Goel, et al. (1995). "Finite element methods in spine biomechanics research." Crit Rev Biomed Eng 23(5-6): 411-473.

Goel, V. K., B. T. Monroe, et al. (1995). "Interlaminar shear stresses and laminae separation in a disc. Finite element analysis of the L3-L4 motion segment subjected to axial compressive loads." Spine (Phila Pa 1976) 20(6): 689-698.

Hickey, D. S. and D. W. Hukins (1980). "X-ray diffraction studies of the arrangement of collagenous fibres in human fetal intervertebral disc." J Anat 131(Pt 1): 81-90.

Hollister, S. J., D. P. Fyhrie, et al. (1991). "Application of homogenization theory to the study of trabecular bone mechanics." J Biomech 24(9): 825-839.

Iatridis, J. C., S. Kumar, et al. (1999). "Shear mechanical properties of human lumbar annulus fibrosus." J Orthop Res 17(5): 732-737.

Jones RM (1999). Mechanics of Composite Materials. London, England, Taylor and Francis.

K. Sivaji Babu, K. M. R., V. Rama Chandra Raju, V. Bala Krishna Murthy, and MSR Niranjan Kumar (2008). "Prediction of Shear Moduli of Hybrid FRP Composite with Fiber-Matrix Interface Debond." International Journal of Mechanics and Solids 3(2): 147-156.

Klisch, S. M. and J. C. Lotz (1999). "Application of a fiber-reinforced continuum theory to multiple deformations of the annulus fibrosus." J Biomech 32(10): 1027-1036.

Lee, C. K., Y. E. Kim, et al. (2000). "Impact response of the intervertebral disc in a finite-element model." Spine (Phila Pa 1976) 25(19): 2431-2439.

Lu, Y. M., W. C. Hutton, et al. (1998). "The effect of fluid loss on the viscoelastic behavior of the *Corresponding authors (Y.Deng). Tel/Fax: +1-605-367-7775/+1-605-367-7836. E-mail addresses: ying.deng@usd.edu. (Z.Hu). Tel/Fax: +1-605-688-4817/+1-605-688-5878.

E-mail address: Zhong.hu@sdstate.edu.

2012. American Transactions on Engineering & Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660 Online Available at http://TUENGR.COM/ATEAS/V01/01-23.pdf

21

lumbar intervertebral disc in compression." J Biomech Eng 120(1): 48-54.

Marchand, F. and A. M. Ahmed (1990). "Investigation of the laminate structure of lumbar disc anulus fibrosus." Spine (Phila Pa 1976) 15(5): 402-410.

Natarajan, R. N., G. B. Andersson, et al. (2002). "Effect of annular incision type on the change in biomechanical properties in a herniated lumbar intervertebral disc." J Biomech Eng 124(2): 229-236.

Ohshima, H., H. Tsuji, et al. (1989). "Water diffusion pathway, swelling pressure, and biomechanical properties of the intervertebral disc during compression load." Spine (Phila Pa 1976) 14(11): 1234-1244.

Sanchez-Palencia E, Z. A. (1987). Homogenization Techniques for Composite Media. Verlag, Berlin, Springer.

Sanchez-Palencia, E. Z. A. (1987). Homogenization techniques for composite media. Berlin, Springer Verlag.

Schwartz, M. H., P. H. Leo, et al. (1994). "A microstructural model for the elastic response of articular cartilage." J Biomech 27(7): 865-873.

Shirazi-Adl, A. (1989). "On the fibre composite material models of disc annulus--comparison of predicted stresses." J Biomech 22(4): 357-365.

Shirazi-Adl, A. (1994). "Nonlinear stress analysis of the whole lumbar spine in torsion-mechanics of facet articulation." J Biomech 27(3): 289-299.

Shirazi-Adl, S. A., S. C. Shrivastava, et al. (1984). "Stress analysis of the lumbar disc-body unit in compression. A three-dimensional nonlinear finite element study." Spine (Phila Pa 1976) 9(2): 120-134.

Tohgo, K. and T. Kawaguchi (2005). "Influence of material composition on mechanical properties and fracture behavior of ceramic-metal composites." Advances in Fracture and Strength, Pts 1- 4 297-300: 1516-1521.

Wu, H. C. and R. F. Yao (1976). "Mechanical behavior of the human annulus fibrosus." J Biomech 9(1): 1-7.

Wu, J. Z. and W. Herzog (2002). "Elastic anisotropy of articular cartilage is associated with the microstructures of collagen fibers and chondrocytes." Journal of Biomechanics 35(7): 931-942.

Yin, L. Z. and D. M. Elliott (2005). "A homogenization model of the annulus fibrosus." Journal of Biomechanics 38(8): 1674-1684.

22

Tyler S. Remund, Trevor J. Layh, Todd M. Rosenboom, L. A. Koepsell, Y. Deng, Z. Hu

Tyler S. Remund is a PhD candidate in the Biomedical Engineering Department at the University of South Dakota. He holds a BS in Mechanical Engineering from South Dakota State University. He is interested in tissue engineering of the annulus fibrosus.

Trevor J. Layh holds a BS in Mechanical Engineering from South Dakota State University. After graduation he was accepted into the Department of Defense SMART Scholarship for Service Program in August 2010, Trevor is now employed by the Naval Surface Warfare Center Dahlgren Division in Dahlgren, VA as a Test Engineer.

Todd M. Rosenboom holds a BS in Mechanical Engineering from South Dakota State University. He currently works as an application engineer for Malloy Electric in Sioux Falls, SD.

Laura A. Koepsell holds a PhD in Biomedical Engineering and a BS in Chemistry, both from the University of South Dakota. She is a Postdoctoral Research Associate at the University of Nebraska Medical Center Department of Orthopedics and Nano-Biotechnology. She is interested in cellular adhesion, growth, and differentiation of mesenchymal stem cells on titanium dioxide nanocrystalline surfaces. She is trying to better understand any inflammatory responses evoked by these surfaces and to evaluate the expression patterns and levels of adhesion and extracellular matrix-related molecules present (particularly fibronectin).

Dr. Ying Deng received her Ph.D. from Huazhong University of Science and Technology in 2001.

She then completed a post-doctoral fellowship at Tsinghua University and a second postdoctoral fellowship at Rice University. In 2008, Dr. Deng joined the faculty of the University of South Dakota at Sioux Falls where she is currently assistant Professor of Biomedical Engineering. She has authored over 15 scientific publications in the biomedical engineering area.

Dr. Zhong Hu is an Associate Professor of Mechanical Engineering at South Dakota State University, Brookings, South Dakota, USA. He has about 70 publications in the journals and conferences in the areas of Nanotechnology and nanoscale modeling by quantum mechanical/molecular dynamics (QM/MD); Development of renewable energy (including photovoltaics, wind energy and energy storage material); Mechanical strength evaluation and failure prediction by finite element analysis (FEA) and nondestructive engineering (NDE); Design and optimization of advanced materials (such as biomaterials, carbon nanotube, polymer and composites). He has been worked on many projects funded by DoD, NSF RII/EPSCoR, NSF/IGERT, NASA EPSCoR, etc.

Peer Review: This article has been internationally peer-reviewed and accepted for publication according to the guidelines given at the journal’s website.

*Corresponding authors (Y.Deng). Tel/Fax: +1-605-367-7775/+1-605-367-7836. E-mail addresses: ying.deng@usd.edu. (Z.Hu). Tel/Fax: +1-605-688-4817/+1-605-688-5878.

E-mail address: Zhong.hu@sdstate.edu.

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Relevance Vector Machines for Earthquake Response Spectra Jale Tezcan

a*

, Qiang Cheng

b

a

Department of Civil and Environmental Engineering, Southern Illinois University Carbondale, Carbondale, IL 62901, USA b Department of Computer Science, Southern Illinois University Carbondale, Carbondale, IL 62901, USA ARTICLEINFO

Article history: Received 23 August 2011 Received in revised form 23 September 2011 Accepted 26 September 2011 Available online 26 September 2011

Keywords: Response spectrum Ground motion Supervised learning Bayesian regression Relevance Vector Machines

A B S T RA C T

This study uses Relevance Vector Machine (RVM) regression to develop a probabilistic model for the average horizontal component of 5%-damped earthquake response spectra. Unlike conventional models, the proposed approach does not require a functional form, and constructs the model based on a set predictive variables and a set of representative ground motion records. The RVM uses Bayesian inference to determine the confidence intervals, instead of estimating them from the mean squared errors on the training set. An example application using three predictive variables (magnitude, distance and fault mechanism) is presented for sites with shear wave velocities ranging from 450 m/s to 900 m/s.

The predictions from the proposed model are compared to an existing parametric model. The results demonstrate the validity of the proposed model, and suggest that it can be used as an alternative to the conventional ground motion models. Future studies will investigate the effect of additional predictive variables on the predictive performance of the model.

2012 American Transactions on Engineering & Applied Sciences.

*Corresponding author ( J. Tezcan). Tel/Fax: +001-618-4536125. E-mail address: jale@siu.edu.

2012. American Transactions on Engineering & Applied Sciences.

Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660.

Online Available at http://TUENGR.COM/ATEAS/V01/25-39.pdf

25

1. Introduction Reliable prediction of ground motions from future earthquakes is one of the primary challenges in seismic hazard assessment. Conventional ground motion models are based on parametric regression, which requires a fixed functional form for the predictive model. Because the mechanisms governing ground motion processes are not fully understood, identification of the mathematical form of the underlying function is a challenge. Once a functional form is selected, the model is fit to the data and the model coefficients minimizing the mean squared errors between the model and the data are determined. This approach, when the selected mathematical form does not accurately represent the actual input-output relationship, is susceptible to overfitting. Indeed, using a sufficiently complex model, one can achieve a perfect fit to the training data, regardless of the selected mathematical form.

However, a perfect fit to the

training data does not indicate the predictive performance of the model for new data.

Kernel regression offers a convenient way to perform regression without a fixed parametric form, or any knowledge of the underlying probability distribution. A special form of kernel regression, called the Support Vector Regression (SVR) (Drucker et al., 1997) is characterized by its compact representation and its high generalization performance. In SVR, the training data is first transformed into a high dimensional kernel space, and linear regression is performed on the transformed data. The resulting model is a linear combination of nonlinear kernel functions evaluated at a subset of the training input. Combination weights are determined by minimizing a penalized residual function. The SVR has proved successful in many studies since its introduction in 1997. The effectiveness of SVR in ground motion modeling has been recently demonstrated (Tezcan and Cheng, 2011), (Tezcan et al., 2010). A well-known weakness of the SVR is the lack of probabilistic outputs. Although the confidence intervals can be constructed using the mean-squared errors, similar to the approach used in conventional ground motion models, the posterior probabilities, which produce the most reliable estimate of prediction intervals, are not given. The lack of probabilistic outputs in the SVR formulation has motivated the development of a new kernel regression model called Relevance Vector Machine (RVM) (Tipping, 2000) which operates in a Bayesian framework.

To overcome the limitations of parametric regression while obtaining probabilistic 26

Jale Tezcan and Qiang Cheng

predictions, this paper proposes a new ground motion model based on the RVM regression.

Unlike standard ground motion models, which make point estimates of the optimal value of the weights by minimizing the fitting error, the RVM model treats the model coefficients as random variables with independent variances and attempts to find the model that maximizes the likelihood of the observations. This approach offers two main advantages over the conventional ground motion models.

First, the prediction uncertainty is explicitly determined using Bayesian

inference, as opposed to being estimated from the mean squared errors. Second, the complexity of the RVM model is controlled by assigning suitable prior distributions over the model coefficients, which reduces the overfit susceptibility of the model.

The rest of the paper is organized as follows. In Section 2, the RVM regression algorithm is described. Section 3 is devoted to the construction of ground motion model. Starting with the description of the ground motion data and the predictive and target variables, the training results are presented, and the prediction procedure for new data is described. Section 4 demonstrates computational results and compares the RVM predictions to an existing empirical parametric model. Section 5 concludes the paper by presenting the main conclusions of this study, and discusses the advantages and limitations of the proposed method.

2. The RVM Regression Algorithm Given a set of input vectors đ?‘Ľđ?‘– , đ?‘– = 1: đ?‘ and corresponding real-valued targets đ?‘Ąđ?‘– , the

regression task is to estimate the underlying input-output relationship. Using kernel representation

(Smola and SchĂślkopf, 2004), the regression function can be written as a linear combination of a set of nonlinear kernel functions: đ?‘

đ?‘“(đ?‘Ľ) = ďż˝ đ?‘¤đ?‘– đ??ž(đ?‘Ľ, đ?‘Ľđ?‘– ) + đ?‘¤0

(1)

đ?‘–=1

where đ?‘¤đ?‘– , đ?‘– = 1 ‌ đ?‘ are the combination weights and đ?‘¤0 is the bias term. *Corresponding author ( J. Tezcan). Tel/Fax: +001-618-4536125. E-mail address: jale@siu.edu.

2012. American Transactions on Engineering & Applied Sciences.

Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660.

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27

This study uses the radial basis function (RBF) kernel: đ??ž(đ?‘Ľđ?‘– , đ?‘Ľđ?‘— ) , where

2

= đ?‘’ −đ?›žďż˝đ?‘Ľđ?‘– −đ?‘Ľđ?‘—ďż˝ ,

�>0

(2)

� is the width parameter controlling the trade-off between model accuracy and

complexity. In this study, the width parameter has been determined using cross-validation.

Assuming independent

noise samples

from

a zero-mean

Gaussian

distribution,

i.e., đ?‘›đ?‘– ~đ?’Š(0, đ?œŽđ?‘›2 ), the target values can be written as: đ?‘Ąđ?‘– = đ?‘“(đ?‘Ľđ?‘– ) + đ?‘›đ?‘–

đ?‘– = 1, ‌ , đ?‘ .

(3)

Recast in matrix from, Equation (3) becomes: đ?‘Ą = ÎŚw + đ?‘›,

(4)

where đ?‘Ą = (đ?‘Ą1 , ‌ , đ?‘Ąđ?‘ )đ?‘‡ , đ?‘¤ = (đ?‘¤0 , ‌ , đ?‘¤đ?‘ )đ?‘‡ , and ÎŚ is an đ?‘ Ă— đ?‘ + 1 basis matrix with đ?›ˇđ?‘–1 = 1

and đ?›ˇđ?‘–đ?‘— = đ??žďż˝đ?‘Ľđ?‘– , đ?‘Ľđ?‘—−1 ďż˝. The likelihood of the entire set, assuming independent observations is given by:

đ?‘?(đ?‘Ą|đ?‘¤, đ?œŽđ?‘›2 )

=

đ?‘ (2đ?œ‹đ?œŽđ?‘›2 )− 2

đ?‘’

−

1 ‖đ?‘Ąâˆ’đ?›ˇđ?œ‡â€–2 2đ?œŽđ?‘›2 .

(5)

where đ?œ‡ = (đ?œ‡0 , ‌ , đ?œ‡đ?‘ )đ?‘‡ is the vector containing the mean values of the combination weights.

To control the complexity of the model, a zero-mean Gaussian prior is used where each weight is assigned a different variance (MacKay, 1992): đ?‘

đ?‘?(đ?‘¤|đ?›ź) = ďż˝ đ?’Š(0, 1/đ?›źđ?‘– ). đ?‘–=0

28

Jale Tezcan and Qiang Cheng

(6)

In Eq. (6), đ?›ź = (đ?›ź0 , ‌ , đ?›źđ?‘ ) where 1/đ?›źđ?‘– is the variance of đ?‘¤đ?‘– . The posterior distribution

of the weights is obtained as:

đ?‘?(đ?‘¤|đ?‘Ą, đ?›ź, đ?œŽđ?‘›2 ) = (2đ?œ‹)−

đ?‘ +1 2

1

1

đ?‘‡ đ??ś −1 (đ?‘¤âˆ’đ?œ‡)

|đ??ś|−2 đ?‘’ −2(đ?‘¤âˆ’đ?œ‡)

.

(7)

where the mean vector đ?œ‡ and covariance matrix đ??ś are: đ?œ‡ = đ?œŽđ?‘›âˆ’2 đ??ś đ?›ˇđ?‘‡ đ?‘Ą

(8)

đ??ś = [đ?œŽđ?‘›âˆ’2 đ?›ˇđ?‘‡ đ?›ˇ + đ??´ ]−1

(9)

with đ?›ź0 : đ??´=ďż˝ â‹Ž 0

‌ �1 ‌

‌

0

â‹ą â‹Ž â‹Ż đ?›źđ?‘

ďż˝.

(10)

The marginal likelihood of the dataset can be determined by integrating out the weights (MacKay, 1992) as follows: đ?‘

1

1 đ?‘‡ −1 đ??ť đ?‘Ą

đ?‘?(đ?‘Ą|đ?›ź, đ?œŽđ?‘›2 ) = (2đ?œ‹)− 2 |đ??ť|−2 đ?‘’ −2đ?‘Ą

(11)

where đ??ť = đ?œŽđ?‘›2 đ??źđ?‘ + đ?›ˇđ??´âˆ’1 đ?›ˇđ?‘‡ and đ??źđ?‘ is the identity matrix of size đ?‘ . Ideal Bayesian inference requires defining prior distributions over đ?›ź and đ?œŽđ?‘›2 , followed by marginalization. This process,

however, will not result in a closed form solution. Instead, the đ?›źđ?‘– and đ?œŽđ?‘›2 values maximizing Eq. (11) can be found iteratively as follows (MacKay, 1992):

*Corresponding author ( J. Tezcan). Tel/Fax: +001-618-4536125. E-mail address: jale@siu.edu.

2012. American Transactions on Engineering & Applied Sciences.

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29

(đ?›źđ?‘– )đ?‘›đ?‘’đ?‘¤ = (đ?œŽđ?‘›2 )đ?‘›đ?‘’đ?‘¤ =

1 − đ?›źđ?‘– đ??śđ?‘–đ?‘– đ?œ‡đ?‘–2

(12)

‖đ?‘Ą − đ?›ˇđ?œ‡â€–2 . đ?‘ − ∑ (1 − đ?›źđ?‘– đ??śđ?‘–đ?‘– )

(13)

Because the nominator in Eq.(12) is a positive number with a maximum value of 1, an ��

value tending to infinity implies that the posterior distribution of �� is infinitely peaked at zero,

i.e. �� = 0. As a consequence, the corresponding kernel function can be removed from the model. The procedure for determining the weights and the noise variance can be summarized as follows: 1) Select a width parameter of the kernel function and form the basis matrix Ό.

2) Initialize đ?›ź = (đ?›ź0 , ‌ , đ?›źđ?‘ ) and đ?œŽđ?‘›2 .

3) Compute matrix đ??´ using Eq.(10).

4) Compute the covariance matrix đ??ś using Eq.(9).

5) Compute the mean vector đ?œ‡ using Eq.(8).

6) Update đ?›ź and đ?œŽđ?‘›2 using Eq.(12) and Eq.(13).

7) If

đ?›źđ?‘– → ∞, set đ?‘¤đ?‘– = 0 and remove the corresponding column in ÎŚ.

8) Go back to step 3 until convergence.

9) Set the remaining weights equal to đ?œ‡ .

The training input points corresponding to the remaining nonzero weights are called the “relevance vectorsâ€?. After the weights and the noise variance are determined, the predictive mean for a new input đ?‘Ľâˆ— can be found as follows: đ?‘“(đ?‘Ľâˆ— ) = đ?‘¤ đ?‘‡ ÎŚâˆ— .

In Eq.(14) ÎŚâˆ— = [1 đ??ž(x ∗ , r1 ) đ??ž(x ∗ , r2 ) ‌ đ??ž(x ∗ , rNr )]T

relevance vectors.

30

Jale Tezcan and Qiang Cheng

(14)

where

(r1 , r2 ‌ , rNr ) are the

The total predictive variance can be found by adding the noise variance to the uncertainty due to the variance of the weights, as follows: đ?œŽâˆ— 2 = đ?œŽđ?‘›2 + ÎŚâˆ—T CÎŚâˆ— .

(15)

3. Construction of the Ground Motion Model In this section, RVM regression algorithm will be used to construct a ground motion model. In Section 4, the resulting model will be compared to an existing parametric model by Idriss (Idriss, 2008), which will be referred to as “I08 model� in this paper. To enable a fair comparison, the dataset and the predictive variables of I08 model have been adopted in this study. The RVM algorithm is independent of the size of the predictive variable set; additional variables can be introduced the set of predictive variables can be customized to specific applications.

3.1 Ground Motion Data The ground motion records used in the training have been obtained from the PEER-NGA database (PEER, 2007). Consistent with the I08 model, a total of 942 free-field records have been selected using the following criteria:

•

Shear wave velocity at the top 30 m (đ?‘‰đ?‘ 30 ) ranging from 450 m/s to 900 m/s,

•

Closest distance between the station and rupture surface (R) less than 200 km.

•

Magnitude larger than 4.5,

Detailed information regarding these records can be found in the paper by Idriss (Idriss, 2008).

3.2 Predictive and Target Variables The predictive variable set includes moment magnitude (M), natural logarithm of the closest distance between the station and the rupture surface in kilometers (đ?’?đ?’?đ?‘…) and fault mechanism (F).

Idriss finds that with the shear wave velocity (đ?‘˝đ?’”đ?&#x;‘đ?&#x;Ž ) constrained to 450 m/s- 900 m/s range, it has *Corresponding author ( J. Tezcan). Tel/Fax: +001-618-4536125. E-mail address: jale@siu.edu.

2012. American Transactions on Engineering & Applied Sciences.

Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660.

Online Available at http://TUENGR.COM/ATEAS/V01/25-39.pdf

31

negligible effect on spectral values up to 1 second. Therefore, đ?‘˝đ?’”đ?&#x;‘đ?&#x;Ž was not used as a predictive

variable. Following the convention used in I08 model, earthquakes that have been assigned a fault mechanism type 0 and 1 in the PEER database were merged to a single, “strike-slipâ€? group, while the rest were considered to be representative of “reverseâ€? events. In the RVM model, strike-slip and reverse earthquakes are assigned đ??š = −1 representing ith record has the following form: đ?‘Ľđ?‘– = [đ?‘€đ?‘–

����

and đ??š = 1, respectively.

The input vector

đ??šđ?‘– ].

(16)

A set of eight vibration periods (�� = 8) ranging from 0.01 second to 4 seconds was used in

the RVM model. The output for the ith record for the vibration period �� is defined as: �� = ���(�� )

for

� = 1 to ��.

(17)

In Equation (17), ��� is the natural logarithm of the average horizontal component of 5%-

damped pseudo-acceleration response spectrum. The spectral values(�) represent the median value of the geometric mean of the two horizontal components,

computed using non-redundant

rotations between 0 and 90 degrees (Boore, 2006).

3.3 Training of the RVM Regression Model As a pre-processing step, � and ��� values were linearly scaled to [-1 1] to achieve

uniformity between the ranges of the predictive variables. There is no need to scale the fault

mechanism identifier (đ??š) as it was already defined to take either -1 or 1. Because kernel functions use Euclidean distances between pairs of input vectors, such scaling will help prevent numerical problems due to large variations between the ranges of the values that variables can take. In the ground motion data used in this study, the ranges of the predictive variables are 4.53 ≤ đ?‘€ ≤ 7.68 , and 0.32 đ?‘˜đ?‘š ≤ đ?‘… ≤ 199.27 đ?‘˜đ?‘š.

following form:

32

Jale Tezcan and Qiang Cheng

Therefore, input scaling takes the

đ?‘Ľâˆ— = ďż˝

2đ?‘€âˆ— − 12.21 2đ?‘™đ?‘›đ?‘… ∗ − 4.16 ∗ , , đ??š ďż˝.

3.15 6.44

(18)

The optimal value of the kernel width parameter (�) for each vibration period was

determined using 10-fold cross validation (Webb, 2002). In 10-fold cross validation, the training data is randomly partitioned into 10 subsets of equal size; and the model is trained using 9 subsets, and the remaining subset is used to compute the validation error. This process is repeated 10 times, each time with a different validation subset, and the average validation error for a particular đ?›ž is computed. By computing the average validation error over a range of possible đ?›ž values, the optimal đ?›ž with the smallest average validation error is determined. The resulting đ?›ž values for each period are listed in Table 1, along with the standard deviation of noise (đ?œŽđ?‘› ), the mean value of

the constant term (đ?‘Š0 ) and the number of relevance vectors. The relevance vectors and the combination weights (đ?‘Šđ?‘– ) are listed in Table 2.

After the RVM models, one for each vibration period, were trained, standardized residuals were computed. Figure 1 shows the distribution of the standardized residuals, corresponding to T=1 second, with respect to đ?‘´, đ?‘š and đ?‘˝đ?’”đ?&#x;‘đ?&#x;Ž . The residual distribution patterns for other periods were similar, not indicating any systematic bias.

Table 1: Kernel width parameter (đ?›ž), logarithmic standard deviation of noise (đ?œŽđ?‘› ), mean value of the bias term(đ?‘Š0 ) and the number of relevance vectors (đ?‘ đ?‘&#x; ), for each period.

T (sec) 0.01 0.05 0.10 0.20 0.50 1.00 2.00 4.00

�

0.23 0.32 0.13 0.15 0.25 0.36 0.28 0.26

đ?œŽđ?‘›

0.633 0.666 0.718 0.661 0.695 0.748 0.869 0.983

đ?‘Šđ?‘œ

-3.069 -0.664 0.002 -15.042 -8.359 -4.670 -6.0548 -7.794

*Corresponding author ( J. Tezcan). Tel/Fax: +001-618-4536125. E-mail address: jale@siu.edu.

2012. American Transactions on Engineering & Applied Sciences.

Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660.

Online Available at http://TUENGR.COM/ATEAS/V01/25-39.pdf

đ?‘ đ?‘&#x; 7 7 7 6 7 5 5 5

33

Figure 1: Standardized residuals for T=1 second.

Table 2: Mean values of the combination weights (đ?‘Šđ?‘– ) and the relevance vectors (đ?‘Ľđ?‘– ) T=0.01 s.

i

Wi

1 2 3 4 5 6 7

13.258 15.393 0.4861 -5.073 -4.275 -14.173 -8.086

T=0.05 s.

ri [-0.1937 [0.5238 [ 0.8921 [0.9619 [0.9619 [-0.2889 [ 0.0603

0.2676 -0.2268 0.9414 -1.0000 -0.6751 0.7862 0.9789

-1] 1] -1] 1] 1] -1] 1]

i

Wi

ri

1 2 3 4 5 6 7

-6.177 6.355 28.555 -7.930 -0.402 -12.622 -16.194

[0.7905 -0.4227 1] [-0.3841 -0.1783 -1] [0.5238 0.5856 1] [-0.5111 0.7896 -1] [0.7460 -0.4021 -1] [0.9619 0.9545 1] [0.0603 0.9789 1]

T=0.1 s. i

Wi

1 2 3 4 5 6 7

64.423 -6.991 -36.297 15.875 -5.599 -17.361 -25.799 34

T=0.2 s. ri

[0.4159 [ 0.9619 [0.9619 [1.0000 [-0.3143 [ 0.6508 [-0.1302

-0.1499 0.9545 -1.0000 0.4559 0.0809 0.9961 0.9056

1] 1] 1] -1] 1] -1] 1]

Jale Tezcan and Qiang Cheng

i

Wi

ri

1 2 3 4 5 6

29.569 2.293 35.440 5.7412 3.5036 -48.496

[-0.8921 -0.0837 -1] [0.7905 -0.4227 1] [0.8921 0.6543 -1] [0.9619 -1.0000 1] [-0.8222 0.1385 1] [0.0603 0.4955 -1]

Table 2 (continued).

T=0.5 s.

T=1.0 s.

i

Wi

ri

i

Wi

1 2 3 4 5 6 7

6.4551 12.825 0.0283 -0.806 8.4335 -0.089 -12.9

[0.7905 -0.4227 1] [-0.2317 -0.2931 -1] [-0.7714 0.1214 1] [ 0.8921 -0.0318 -1] [0.8921 0.9414 -1] [ 0.9619 0.9545 1] [ 0.0603 0.5786 -1]

1 2 3 4 5

1.9699 4.8873 -4.1425 -3.9593 3.7352

T=2.0 s.

ri [0.7905 [0.0540 [-0.7524 [-0.7651 [-0.1302

-0.4227 -0.2785 0.7892 0.8672 -0.0121

1] -1] 1] -1] 1]

T=4.0 s.

i

Wi

ri

i

Wi

1 2 3 4 5

7.3574 4.5548 3.0086 -6.4695 -5.3630

[-0.2317 -0.2931 -1] [-0.0730 0.4691 1] [ 0.9619 -1.0000 1] [-1.0000 0.5142 -1] [-0.7524 0.7892 1]

1 2 3 4 5

0.4747 11.936 6.8109 -5.6050 -10.180

ri [0.7460 -0.4021 [0.7460 0.5118 [0.3714 -0.0296 [-0.7524 0.7892 [0.3778 1.0000

-1] -1] 1] 1] -1]

3.4 Prediction Phase After training, the spectral values for a new input vector đ?‘Ľ = [đ?‘€, đ?‘™đ?‘›đ?‘…, đ??š ] can be determined

as follows:

1. Scale the input to the range [-1 1] using Eq. (18); 2. Construct the basis vector ÎŚâˆ— = [1 đ??ž(đ?‘Ľ ∗ , đ?‘&#x;1 ) đ??ž(đ?‘Ľ ∗ , đ?‘&#x;2 ) ‌ đ??ž(đ?‘Ľ ∗ , đ?‘&#x;đ?‘ đ?‘&#x; )]T using the relevance vectors from Table 2 and the kernel width parameter from Table 1; 3. Determine the median value of đ?‘™đ?‘›đ?‘† using Eq.(14);

4. Obtain the standard deviation of the noise from Table 1. Total uncertainty, if needed, can be determined using Eq.(15).

4. Computational Results The RVM model was tested using different magnitude, distance and fault mechanisms, and the results were compared to the I08 model. Figure 2 shows the median spectral acceleration at T=1 *Corresponding author ( J. Tezcan). Tel/Fax: +001-618-4536125. E-mail address: jale@siu.edu.

2012. American Transactions on Engineering & Applied Sciences.

Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660.

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35

second, along with the 16th and 84th percentile values (Âąđ?œŽđ?‘› bounds) for strike-slip faults, for M=5 (left) and M=7 (right). The circles in the figure show the spectral values from earthquakes

with the same fault mechanism and within Âą0.25 magnitude units. Figure 3 shows the same

information for reverse faults. For periods about 1 second and longer, it was observed that the median estimates from the RVM model were generally lower than those from the I08 model. At very short distances, within ~20 km of the source, RVM estimates were higher for M=7, for both strike-slip and reverse faulting earthquakes.

Figure 2: Median ÂąĎƒ bounds for spectral acceleration at T=1 second, strike-slip faults.

Figure 3: Median ÂąĎƒ bounds for spectral acceleration at T=1 second, reverse faults.

36

Jale Tezcan and Qiang Cheng

Figure 4 presents the results for vibration period T=0.2 second, for strike-slip earthquakes.

The results for the reverse faulting earthquakes were similar. For shorter vibration periods, and M=7, RVM estimates were lower than those from the I08 model. For M=5, however, RVM predictions equaled or exceed the I08 predictions. Regarding the variation about the median (noise variance), the predictions from the two models were in general agreement for all vibration periods.

Figure 4: Median ÂąĎƒ bounds for spectral acceleration at T=0.2 second, strike-slip faults.

5. Conclusion This paper proposes an RVM-based model for the average horizontal component of earthquake response spectra. Given a set of predictive variable set, and a set of ground motion records, the RVM model predicts the most likely spectral values in addition to its variability. An example application has been presented where the predictions from the RVM model have been compared to an existing, parametric ground motion model. The results demonstrate the validity of the proposed model, and suggest that it can be used as an alternative to the conventional ground motion models.

The RVM model offers the following advantages over its conventional counterparts: (1) There is no need to select a fixed functional form. By determining the optimal variances associated with *Corresponding author ( J. Tezcan). Tel/Fax: +001-618-4536125. E-mail address: jale@siu.edu.

2012. American Transactions on Engineering & Applied Sciences.

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37

the weights, the RVM automatically detects the most plausible model; (2) The resulting RVM model has a simple mathematical structure (weighted average of exponential basis functions), and is based on a small number of samples that carry the most relevant information. Samples that are not well supported by the evidence (as measured by the increase in the marginal likelihood) are automatically pruned. (3) Because the model complexity is controlled during the training stage, the RVM has lower risk of over-fitting.

One limitation of the proposed approach is that the resulting model may be difficult to interpret. Because the RVM is not a physical model, it does not allow any user-defined, physical constraints, not allowing extension of the model to scenarios not represented in the training data set. However, in our opinion, this does not constitute a shortcoming, considering that the reliability such practice is questionable in any regression model. Another potential limitation is that the RVM requires a user-defined kernel width parameter, which does not have a very clear intuitive meaning, especially when working with high dimensional input vectors. However, the optimal value of the kernel width parameter can be determined using cross-validation, as has been done in this study.

Future studies will investigate the effect of using additional predictive variables on the performance of the model.

6. Acknowledgements This material is based in part upon work supported by the National Science Foundation under Grant Number CMMI-1100735.

7. References Boore, D.M., J. Watson-Lamprey, and N.A. Abrahamson. (2006). Orientation-independent measures of ground motion. Bulletin of the Seismological Society of America, 96(4A), 1502-1511.

Bozorgnia, Y. and K. W. Campbell. (2004). The vertical-to-horizontal response spectral ratio and tentative procedures for developing simplified V/H and vertical design spectra. Journal of Earthquake Engineering, 8(2), 175-207.

Campbell, K. W. and Y. Bozorgnia. (2003). Updated Near-Source Ground-Motion (Attenuation) Relations for the Horizontal and Vertical Components of Peak Ground Acceleration and Acceleration Response Spectra. Bulletin of the Seismological Society of America, 93(1), 314-331.

38

Jale Tezcan and Qiang Cheng

Drucker, H., C. J. C. Burges, L. Kaufman, A. Smola and V. Vapnik. (1997). Support vector regression machines, Advances in Neural Information Processing Systems 9, MIT Press.

Idriss, I. M. (2008). An NGA empirical model for estimating the horizontal spectral values generated by shallow crustal earthquakes. Earthquake spectra, 24(1), 217-242.

MacKay, D. J. C. (1992). Bayesian interpolation. Neural computation, 4(3), 415-447.

MacKay, D. J. C. (1992). The evidence framework applied to classification networks. Neural Computation, 4(5), 720-736.

PEER. (2007). PEER-NGA Database. http://peer.berkeley.edu/nga/index.html.

Smola, A. J. and B. Schölkopf. (2004). A tutorial on support vector regression. Statistics and Computing, 14(3), 199-222.

Tezcan, J. and Q. Cheng. (2011). A Nonparametric Characterization of Vertical Ground Motion Effects. Earthquake Engineering and Structural Dynamics (in print).

Tezcan, J., Q. Cheng and L. Hill. (2010). Response Spectrum Estimation using Support Vector Machines, 5th International Conference on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics, San Diego, CA. Tipping, M. (2000). The relevance vector machine. Advances in Neural Information Processing Systems MIT Press.

Webb, A. (2002). Statistical pattern recognition, New York, John Wiley and Sons.

Dr.Jale Tezcan is an Associate Professor in the Department of Civil and Environmental Engineering at Southern Illinois University Carbondale. She earned her Ph.D. from Rice University, Houston, TX in 2005. Dr.Tezcan’s research interests include earthquake engineering, material characterization, and numerical methods.

Dr.Qiang Cheng is an Assistant Professor in the Department of Computer Science at Southern Illinois University Carbondale. He earned his Ph.D. from the University of Illinois at Urbana Champaign, IL in 2002. Dr.Cheng’s research interests include pattern recognition, machine learning and signal processing.

Peer Review: This article has been internationally peer-reviewed and accepted for publication according to the guidelines given at the journal’s website.

*Corresponding author ( J. Tezcan). Tel/Fax: +001-618-4536125. E-mail address: jale@siu.edu.

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Influence of Carbon in Iron on Characteristics of Surface Modification by EDM in Liquid Nitrogen Apiwat Muttamara

a*

, Yasushi Fukuzawa

b

a

Department of Industrial Engineering Faculty of Engineering, Thammasat University, THAILAND Department of Mechanical Engineering Faculty of Engineering, Nagaoka University of Technology, JAPAN b

ARTICLEINFO

Article history: Received 23 August 2011 Received in revised form 23 September 2011 Accepted 26 September 2011 Available online 26 September 2011

Keywords: EDM, Surface modification Titanium nitride, Liquid nitrogen.

A B S T RA C T

Many surface modification technologies have been proposed and carried out practically by CVD, PVD et.al. Carbonized layer has been made using EDM method. In this paper, to make the nitride layer by EDM some new trials were carried out using a titanium electrode in liquid nitrogen. Experiments were carried out on carbon steel (S45C), pure iron and cast iron. TiN can be obtained on EDMed surface. Moreover, TiCN can be found on cast iron and steel (S45C)

by XRD investigation. To confirm the fabrication mechanisms of modified layer on the steel, the following experimental factors were investigated by EDS. 2012 American Transactions on Engineering & Applied Sciences.

1. Introduction Many surface modification technologies have been proposed and carried out practically by CVD, PVD et.al. Surface modification by EDM have been succeeded to make the modified layer *Corresponding author (A.Muttamara). Tel/Fax: +66-2-5643001 Ext.3189. E-mail 2012 American Transactions on Engineering & address: mapiwat@engr.tu.ac.th Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660 Online available at http://TUENGR.COM/ATEAS/V01/41-55.pdf

41

i.e. TiC, Si, WC etc. on the work piece by EDM method (N.Saito et.al.,1993). In this method, the carbon element that is supplied from the dissolution phenomena of working oil during discharges reacts with the electrode element of Titanium. When the compacted powder body used as an electrode, TiC products piled up easily on the steel surface. On the other hand, the surface modified TiN can be achieved with titanium electrode in liquid nitrogen. (Muttamara et al.,2002). Biing Hwa Yan et al., 2005, carried out EDM in urea solution in water with Ti electrode and obtained TiN machined surface. It is interesting that carbon come off by reverse diffusion from the workpiece to the recast layer (Marash et al., 1965). Therefore, the surface modified TiN and TiCN layers have attracted interest for workpiece materials which have high carbon content such as carbon steel and cast iron. Although hardness of TiN layer is lower than TiC layer but friction co-efficiency of TiN layer is quite stable and quite low. In this paper, a new modification method of nitride modified layer on steels by EDM in liquid nitrogen using a titanium electrode is proposed.

2. Experimental procedure Figure 1 shows the illustrated experimental set up. The machining was carried out in liquid nitrogen on carbon-steel (S45C), pure iron and cast iron. Cylindrical Ti solid was applied as an electrode. Table 2 shows chemical composition of S45C. Table 3 shows chemical composition of pure iron and cast iron. The discharge waveforms were observed with a current monitor to analyze the discharge phenomena on this machining.

Control circuit

Current Detector Oscilloscope

Electrical power source

Vessel Ti electrode Liquid Nitrogen

Ground

Workpiece

Figure 1: Experimental Set up for EDM in liquid nitrogen.

42

Apiwat Muttamara, and Yasushi Fukuzawa

Table 1 : Properties - PVD coating Datasheet.

Coating

Colour

Key Characteristics

Material

Hardness

Maximum

Friction

(Vickers)

Working

Coef

Temperature

(on dry steel)

TiN

Gold

Good general purpose

2300

600C

0.4

TiC

Grey

High hardness

3500+

TBD

>0.1

TiCN

Blue Gray

High hardness, good

3000

400C

0.4

Perple

wear resistance, enhanced toughness

Table 2 : Chemical composition of S45C (mass%) C

Si

Mn

P

S

Fe

0.45

0.2

0.77

0.17

0.25

Bal.

Table 3 : Chemical composition of pure iron and cast iron (%) Workpiece

C (%)

Si (%)

Fe

Pure iron

<0.005

0

Bal.

Cast Iron

2.11-4.5

3.5

Bal.

Table 4 : The experiment conditions Parameters Polarity (Electrode)

Values -

Current (A)

10, 47

On-time (Îźs)

32,512

Duty factor (%)

11,50

Open circuit voltage, ui (V)

220

Water pressure (kg/cm2)

40

Spindle speed (rpm)

500

*Corresponding author (A.Muttamara). Tel/Fax: +66-2-5643001 Ext.3189. E-mail 2012 American Transactions on Engineering & address: mapiwat@engr.tu.ac.th Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660 Online available at http://TUENGR.COM/ATEAS/V01/41-55.pdf

43

The machining characteristics are estimated in terms of surface roughness, Vickerâ&#x20AC;&#x2122;s hardness, surface layer thickness, X-ray diffraction pattern, EPMA and EDS analysis. The machining conditions are shown in Table 4. The special vessel was designed by polystyrene material for the machining in the liquid nitrogen.

3. Results and discussions At room temperature, liquid nitrogen holds as a boiling state in the vessel. It is known that when the discharge occurs in boiled working medium, the machining phenomena are affected by the bubble generation and the a few discharges contribute to the machining state. Further, exploding the vapor bubble and causing the molten metal to difficult be expelled from the workpiece so that only piling process occurs without machining process. To investigate the pulse discharges in liquid nitrogen, discharge waveforms were observed. Figure 2 shows the discharge waveforms in liquid nitrogen. The detailed waveforms were indicated as A` and A line in Figure 2, are shown in Figure 3. The experiments of EDM were performed on the surface of S45C. Machining conditions were as follows: negative polarity, ie=10A, te=32Âľs, D.F.=50%. There are 4 types waveforms: (a) normal, (b) short, (c) concentrate, (d) short eliminated current. Due to liquid nitrogen holds as a boiling state, therefore EDMed in liquid nitrogen requires a time to break down into ionic (charged) fragments, allowing an electrical current to pass from electrode to workpiece.

This region was named as an ignition delay time. Many shorts and concentrate discharges occurred in this process. It can be explained that the sludge was made by the gathering debris phenomena in the gap space during the short circuit and piled on the machined surface during ignition delay time.

When the electrode touches the workpiece through the sludge, the concentration of discharge pulse and short circuit occurs. It assumed that the surface modified layer was fabricated by these special discharge phenomena. When short occurs in EDM, it tends to continue long time such as several 100ms from several 10ms. To solve the problem, our EDM system automatically lunches eliminate current to the process (Goto A.et al.,1998). As Figure shows, during off-time it is checked whether gap is short, next pulse is eliminated.

44

Apiwat Muttamara, and Yasushi Fukuzawa

Voltage

Ignition delay time

A’ A

80V 0V

Current

Eliminate current 1

Short

0 Time 0.5ms/div Figure 2: Discharge waveforms in liquid nitrogen.

Voltage

80V.

Current

Discharge voltage (ue)= 15V

10

A’

Concentrate Normal

A

0V.

0A 50 µs/div Figure 3: Normal discharge and concentrate discharge in period A`– A.

3.1 Effect of electrode polarity On the normal EDM, the positive (+) electrode polarity is chosen for the machining (Janmanee P. and Muttamara A.,2011). On the contrary, the negative polarity (-) often uses for the modified technology (N.Saito et.al.,1993), (Muttamara et al.,2004), and also machines for insulating ceramic materials (Muttamara et al.,2009-2010).

These experiments were done under the

*Corresponding author (A.Muttamara). Tel/Fax: +66-2-5643001 Ext.3189. E-mail 2012 American Transactions on Engineering & address: mapiwat@engr.tu.ac.th Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660 Online available at http://TUENGR.COM/ATEAS/V01/41-55.pdf

45

following machining conditions: ie=10A, te=32µs, D.F.=11.1% with Ti solid electrode ofφ5mm.

Figure 4 shows the shape of electrical discharge marks from a single pulse electrical discharge experiment in which all other conditions are identical, and only the polarities are changed.

a) Positive

b) Negative

Figure 4: Single crater created by a) positive and b) negative polarity.

It can be seen that in the case of negative polarity, large amounts of the melted electrode implant to the workpiece. In comparison, in case of positive, a relatively clean surface crater is formed. Judging from the result, the negative polarity was selected.

3.2 EDM on S45C To study characteristics of modified layer, the cross sectional of nitride product modified layer on S45C was observed by laser microscope and EPMA analysis. Figure 5 (a),(b) and (c) show the cross sectional EDMed surface by laser microscope, EPMA map analysis and EPMA line analysis of cross sectional EDMed surface, respectively. The golden colored layer could be observed on the machined surface. The characteristics of the modified layer were investigated by the micro-hardness Vickers using a load of 10gf and the EPMA analysis. Figure 6 shows micro-hardness distribution on the cross section of modified layers with solid and semi-sintered. (ie=10A., te=32µs, D.F.=11.11%). On the machining of Ti solid electrode, there were three areas: (1) nearest surface region, 0-50 µm, the hardness reached to 1300HV that corresponded almost to the same value of other report (table 3.1), (2) thermal affected region, 50-100 µm: similar hardness of martensite structure of 800HV, (3) original substrate region: over 100 µm. On the contrary, the hardness of region (1) became the same value, 800HV at region (2) on the machining of semi-sintered electrode.

46

Apiwat Muttamara, and Yasushi Fukuzawa

EPMA analysis of Ti, N and C, was carried out on the cross sectional modified surface. The distribution of Ti and N element was divided to three regions same as Figure 5. The distribution of Ti and N element was detected from region (1) to (2). It indicated that the region composed with the thermal affected structure of substrate and the diffused TiN products. In the (1) and (2) region, the higher carbon element was observed than matrix regardless no supplying source around discharge circumstances. Because carbon was observed on the modified layer on S45C. It was thought that carbon come off by reverse diffusion (Barash et al.,1965).

Line analysis Area Modified layer

40

a) Modified layer on S45C in liquid nitrogen

b) EPMA Map analysis Modified layer on S45C in liquid nitrogen continue Figure 5 on next page *Corresponding author (A.Muttamara). Tel/Fax: +66-2-5643001 Ext.3189. E-mail 2012 American Transactions on Engineering & address: mapiwat@engr.tu.ac.th Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660 Online available at http://TUENGR.COM/ATEAS/V01/41-55.pdf

47

Int (Count) Ti N 1600 240

C 40

800

20

120

T

Modified layer

N C

0

50

100

150

Distance from top surface (µm)

250

c) EPMA Line analysis Figure 5: Cross sectional image of TiN layer on S45C by a) Laser Microscope b) EPMA Map analysis c) EPMA Line analysis.

Hardness (HV)

Modified layer 1400 1200 1000 800 600 400 200

0

50

100

150

200

Distance from top surface (µm) Figure 6 : Relationship between micro-hardness Vickers against the cross section of modified layer on S45C.

3.3 EDM on Pure iron Pure Iron does not contain carbon (less than 0.005%). The concentration of substances on the cross section of modified surface on pure iron were carried out with Ti solid.

Figure 7 shows

cross sectional SEM of EDMed surface on pure iron compared with EPMA results. Figure 8 shows the sectional micro-hardness measurements of modified surface. The thickness of modified layer is 100 µm as same as the modified layer on S45C. From the sectional micro-hardness result,

48

Apiwat Muttamara, and Yasushi Fukuzawa

hardness of modified surface is 600-800 HV. The hardness of modified layer on Fe is lower than that on S45C. This is considered that carbon in the material of S45C affect to the compound of modified layer.

Int (Count) Ti

N

C

600

800

20

300

400

10

0

100 µm Line analysis

Ti

Modified layer

N

C 100

200

Distance from top surface (µm)

a) SEM of TiN layer on pure iron

b) EPMA Line Analysis

Figure 7: Cross sectional TiN layer on Fe a) SEM and b) EPMA Line Analysis

Modified layer

Hardness (HV)

1000 800 600 400 200 0

100

200

300

400

500

Distance from top surface (µm) Figure 8: Micro-hardness distribution (EDM Conditions; ie=47A, te=256µs, D.F.= 11.1%).

3.4 EDM on cast iron Cast iron was used to confirm (reverse) diffusion of carbon. In this experiment, discharge current (ie)=47A, discharge duration (te)=256µs, (D.F.)=11.11%, were selected for EDMed condition. Figure 9 shows cross sectional SEM of EDMed surface on cast iron compared with EPMA results. Figure 10 shows the sectional micro-hardness measurements of modified surface. *Corresponding author (A.Muttamara). Tel/Fax: +66-2-5643001 Ext.3189. E-mail 2012 American Transactions on Engineering & address: mapiwat@engr.tu.ac.th Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660 Online available at http://TUENGR.COM/ATEAS/V01/41-55.pdf

49

Int (Count) Ti

N

C

600 800 20

Modified layer

Ti

300 400 10 N

C 0

100 µm Line analysis

100

200

Distance from top surface (µm)

a) SEM of TiN layer on cast iron

b) EPMA Line Analysis

Figure 9: Cross sectional TiN layer on cast iron a) SEM and b) EPMA Line Analysis Modified layer Hardness (HV)

1000 800 600 400 200

0

100

200

300

400

500

Distance from top surface (µm) Figure 10: Micro-hardness distribution (EDM Conditions; ie=47A, te=256µs, D.F.= 11.1%)

The C and N elements concentrations are measured on the modified layer, distance of the generation of C and N elements are 250 µm of modified layer as can seen from the Figure 9. First, it should be noticed that system experiment was decarburizing. So carbon on modified layer should come from the precipitated graphite in the cast iron. However, we cannot see clearly on EDS result of carbon. Etching was done on cross section surface of cast iron as shown in Figure 11.

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Apiwat Muttamara, and Yasushi Fukuzawa

HAZ

(1)

(2)

(1)

(2)

20 Âľm

20 Âľm

Figure 11: SEM micrographs of etched cross section surface of cast iron The low part represents the base material, the central part in the curve mark represents the base material that effect from heat affected zone (HAZ), and carbon diffused zone. The modified layer was generated irregularly. The dendritic parts in substrate are graphite exist in the form of flakes. It is pointed out that some areas inside close line carbon are depressed. The large scale of structure *Corresponding author (A.Muttamara). Tel/Fax: +66-2-5643001 Ext.3189. E-mail 2012 American Transactions on Engineering & address: mapiwat@engr.tu.ac.th Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660 Online available at http://TUENGR.COM/ATEAS/V01/41-55.pdf

51

under modified surface is shown in Figure 11 (a) Also the structure of normal graphite in cast iron is shown in Figure 11 (b). The presences of graphite in HAZ (a) are different from normal content (b). Therefore, it is considered that precipitated carbon diffuses by discharges or the changing of

Micro hardness (HV)

structure of case iron.

2000

Modified layer

HAZ + Diffused zone

1500

Matrix

1000 50

200

400

600

800

Depth Depth below below surfacesurface (mm) (Âľm)

Figure 12: Section hardness of machined surface before and after annealing To investigate effect of carbon and HAZ on the hardness, the hardness was evaluated on cross sectional of cast iron. Figure 12 shows the sectional hardness measurement of modified surface on cast iron. It can be considered that the machined surface is covered with TiN and TiCN layer. The hardness of modified layer is about 1450 Hv. On HAZ region, the hardness decreases gradually according to the distance from the surface. It reaches to the hardness of matrix cast iron through that of requenched region. Some hardness regions on HAZ are below the hardness of matrix region, it is considered that the coming off of carbon effects to the hardness of that region. HAZ.

3.5 X ray-diffraction (XRD) analysis As mentioned above, the some modified layer could be adhered on the work piece by EDM in liquid nitrogen. To confirm the layer composition X ray-diffraction (XRD) pattern was investigated for the EDMed surface with Ti solid electrode. Figure 13 shows the result of XRD on EDMed surface on S45C compared with EDMed surface on pure iron and cast iron. The peak of TiN and TiCN are very near. From the EPMA results and the hardness results, it indicates that the EDMed surface on S45C and cast iron are composed of TiN and TiCN. On the other hand, only TiN layer was observed on the EDMed surface of pure iron.

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Apiwat Muttamara, and Yasushi Fukuzawa

(CPS)

1000

TiCN

500

Fe Fe

TiN

TiCN TiN

TiN TiCN Fe

30

40

50

60

Diffraction angle 2θ (Cu Kα)

TiN TiN 70

80

a) EDMed surface on S45C ( CPS)

1000 1

Fe

Fe

TiN

500

TiN

TiN

30

Fe

40

50

60

Diffraction angle 2θ (Cu Kα)

TiN TiN

70

80

b) EDMed surface on pure iron (CPS)

1000

TiCN

500

TiCN TiN 30

Fe Fe

TiN TiN Fe TiCN 40

50

60

TiN

TiN

70

80

Diffraction angle 2θ (Cu Kα)

c) EDMed surface on cast iron Figure 13: X-ray diffraction patterns obtained from the EDMed layer in liquid nitrogen by solid Ti on a) S45C , b) pure iron and c) cast iron.

*Corresponding author (A.Muttamara). Tel/Fax: +66-2-5643001 Ext.3189. E-mail 2012 American Transactions on Engineering & address: mapiwat@engr.tu.ac.th Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660 Online available at http://TUENGR.COM/ATEAS/V01/41-55.pdf

53

4. Conclusion A new EDM surface modification method was tried in liquid nitrogen on S45C steel in various conditions. The results were summarized as follows: (1) In liquid nitrogen, machining process is not obtained, but the TiN products adhere on the work piece.

(2) Ti and N element diffused from nearest surface to the thermal affected zone.

(3) Discharge causes carbon migration from deeper layers of the substrate.

(4) TiCN modified layer could be generated on carbon steel and cast iron because carbon from substrate diffused to modified layer and reacted with nitride product of modified layer.

5. Acknowledgement

The authors are grateful to Faculty of Engineering, Thammasat University, the National

Research Council of Thailand (NRCT), the Thailand Research Fund (TRF) and the National Research University Project of Thailand Office of Higher Education Commission for the research funds and T. Klaykaow for carrying out this work.

6. References Barash, M.M.(1965). Effect of EDM on the surface properties of tool and die steels. Metals engineering quarterly, 5, (4), 48-51.

Biing H.Y., Tsai H.C., Huang F. Y. (2005).The effect in EDM of a dielectric of a urea solution in water on modifying the surface of titanium. International Journal of Machine Tools and Manufacture, 45, (2), 194-200.

Fredriksson G., and Hogmark S., (1995). Influence of dielectric temperature in EDM of hot worked tool steel. Surface Engineering, 11, (4), 324â&#x20AC;&#x201C;330.

Goto A., T. Magara, T. Moro, H. Miyake, N. Saito, N. Mohri.(1997). Formation of hard layer on metallic material by EDM. Proceedings of the ISEM-12, 271â&#x20AC;&#x201C;278.

Goto, A., Yuzawa, T., Magara, T., and Kobayashi, K. (1998). Study on Deterioration of Machining Performance by EDMed Sludge and its Prevention. IJEM, 3,1-6.

Mohri N., Fukusima Y., Fukuzawa Y., Tani T., and. Saito N.(2003). Layer Generation Process on Work-piece in Electrical Discharge Machining, Annals of the CIRP, 52(1),161-164.

Mohri, N., Saito, N., and Tsunekawa, Y. (1993). Metal Surface Modification by EDM with Composite Electrode. Annals of the CIRP, 42, (1) 219-222.

54

Apiwat Muttamara, and Yasushi Fukuzawa

Muttamara A., Fukuzawa Y., Mohri N., and Tani T. (2009). Effect of electrode Materials on EDM of Alumina. Journal of Materials Processing Technology, 209, 2545-2552.

Muttamara A., Janmanee P., and Fukuzawa Y.(2010). A Study of Micro–EDM on Silicon Nitride Using Electrode Materials. International Transaction Journal of Engineering, Management, & Applied Sciences & Technologies. 1(1), 1-7.

Janmanee P., and Muttamara A.(2011). A Study of hole drilling on Stainless Steel AISI 431 by EDM Using Brass Tube Electrode. International Transaction Journal of Engineering, Management, & Applied Sciences & Technologies. 2(4), 471-481.

Muttamara A., Fukuzawa Y., and Mohri N.(2002). A New Surface Modification Technology on Steel using EDM, Journal of Australian Ceramic Society (38), 2,125-129.

Dr.Apiwat Muttamara is an Assistant Professor of Department of Industrial Engineering at Thammasat University. He received his B.Eng. from Kasetsart University and the D.Eng. in Materials Science from Nagaoka University of Technology, Japan. Dr. Muttamara is interested involve Electrical Discharge Machining of insulating materials.

Yasushi FUKUZAWA is Professor of Material Science and Engineering group in Department of Mechanical Engineering at Nagaoka University of Technology, Japan. Prof. Dr. Fukuzawa’s fields are material processing and treatment.

Peer Review: This article has been internationally peer-reviewed and accepted for publication according to the guidelines given at the journal’s website.

*Corresponding author (A.Muttamara). Tel/Fax: +66-2-5643001 Ext.3189. E-mail 2012 American Transactions on Engineering & address: mapiwat@engr.tu.ac.th Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660 Online available at http://TUENGR.COM/ATEAS/V01/41-55.pdf

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2012 American Transactions on Engineering & Applied Sciences

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Establishing Empirical Relations to Predict Grain Size and Hardness of Pulsed Current Micro Plasma Arc Welded SS 304L Sheets Kondapalli Siva Prasad Damera Nageswara Rao

c

a*

b

, Chalamalasetti Srinivasa Rao , and

a

Department of Mechanical Engineering, Anil Neerukonda Institute of Technology and Sciences, Visakhapatnam, INDIA b Department of Mechanical Engineering, Andhra University,Visakhapatnam, INDIA c Centurion University of Technology & Management, Odisha, INDIA ARTICLEINFO

Article history: Received 23 August 2011 Received in revised form 01 December 2011 Accepted 25 December 2011 Available online 26 December 2011

Keywords: Pulsed current micro plasma arc welding, SS304L, grain size, hardness, Design of Experiments, ANOVA.

A B S T RA C T

SS 304L, an austenitic Chromium-Nickel stainless steel offering the optimum combination of corrosion resistance, strength and ductility, is favorable for many mechanical components. The low carbon content reduces susceptibility to carbide precipitation during welding. In case of single pass welding of thinner section of this alloy, pulsed current micro plasma arc welding was found beneficial due to its advantages over the conventional continuous current process.

The paper focuses on developing mathematical models to predict grain size and hardness of pulsed current micro plasma arc welded SS304L joints. Four factors, five level, central composite rotatable design matrix is used to optimize the number of experiments. The mathematical models have been developed by response surface method. The adequacy of the models is checked by ANOVA technique. By using the developed mathematical models, grain size and hardness of the joints can be predicted with 99% confidence level.

Contour plots are drawn to study the interaction effect of pulsed current micro plasma arc welding parameters on fusion zone grain size and hardness of SS304L steel.

2012 American Transactions on Engineering & Applied Sciences.

*Corresponding author ( Kondapalli Siva Prasad). Tel/Fax: +91-9849212391. E-mail address: kspanits@gmail.com.

2012. American Transactions on Engineering & Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660. Online Available at http://TUENGR.COM/ATEAS/V01/57-74.pdf

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1. Introduction In welding processes, the input parameters have greater influence on the mechanical properties of the weld joints. By varying the input process parameters, the output could be changed with significant variation in their mechanical properties. Accordingly, welding is usually selected to get a welded joint with excellent mechanical properties. To determine these welding combinations that would lead to excellent mechanical properties, different methods and approaches have been used.

Various optimization methods can be applied to define the desired output variables through developing mathematical models to specify the relationship between the input parameters and output variables. One of the most widely used methods to solve this problem is response surface methodology (RSM), in which the unknown mechanism with an appropriate empirical model is approximated, being the function of representing a response surface method Welding thin sheets is quite different from welding thick sections, because during welding of thin sheets many problems are experienced. These problems are usually linked with heat input.

Fusion welding generally involves joining of metals by application of heat for melting of metals to be joined. Almost all the conventional arc welding processes offer high heat input, which in turn leads to various problems such as burn through or melt trough, distortion, porosity, buckling warping and twisting of welded sheets, grain coarsening , evaporation of useful elements present in coating of the sheets, joint gap variation during welding, fume generation form coated sheets etc.

Use of proper welding process, procedure and technique is one tool to address this issue (Balasubramanian et.al, 2010). Micro Plasma arc Welding (MPAW) is a good process for joining thin sheet, but it suffers high equipment cost compared to GTAW. However it is more economical when compare with Laser Beam welding and Electron Beam Welding processes.

Pulsed current MPAW involves cycling the welding current at selected regular frequency. The maximum current is selected to give adequate penetration and bead contour, while the minimum is set at a level sufficient to maintain a stable arc (Balasubramanian et.al, 2006 and Madusudhana et.al, 1997). This permits arc energy to be used effectively to fuse a spot of controlled dimensions in a short time producing the weld as a series of overlapping nuggets. By contrast, in constant current welding, the heat required to melt the base material is supplied only during the peak current pulses allowing the heat to dissipate into the base material leading to narrower heat affected zone 58

Kondapalli Siva Prasad, Ch.Srinivasa Rao, and D.Nageswara Rao

(HAZ). Advantages include improved bead contours, greater tolerance to heat sink variations, lower heat input requirements, reduced residual stresses and distortion, refinement of fusion zone microstructure and reduced with of HAZ. There are four independent parameters that influence the process are peak current, back current, pulse and pulse width.

From the literature review (Zhang and Niu, 2000, Sheng-Chai Chi and LI-Chang Hsu, 2001, Hsiao et.al, 2008, Siva et.al, 2008, Lakshinarayana et.al, 2008, Balasubramanian et.al, 2009, Srimath and Muragan, 2011) it is understood that in most of the works reported the effect of welding current, arc voltage, welding speed, wire feed rate, magnitude of ion gas flow, torch stand-off, plasma gas flow rate on weld quality characteristics like front melting width, back melting width, weld reinforcement, welding groove root penetration, welding groove width, front-side undercut are considered. However much effort was not made to develop mathematical models to predict the same especially when welding thin sheets in a flat position. Hence an attempt is made to correlate important pulsed current MPAW process parameters to grain size and hardness of the weld joints by developing mathematical models by using statistical tools such as design of experiments, analysis of variance and regression analysis.

2. Literature review on Response Surface Method Response Surface Method or commonly known as RSM is an anthology of statistical and mathematical methods that are helpful in generating improved methods and optimizing a welding process. RSM is more frequently used in analyzing the relationships and the influences of input parameters on the responses. The method was introduced by G. E. P. Box and K. B. Wilson in 1951. The main idea of RSM is to use a set of designed experiments to obtain an optimal response.

Box and Wilson used first-degree polynomial model to obtain DOE through RSM and acknowledged that the model is only an approximation and is easy to estimate and apply, even when little information is known about the process. Response Surface Regression method is an assortment of mathematical and statistical techniques useful for modeling and analyzing experiments in which a response variable is influenced by several independent variables. It explores the relationships between several independent variables and one or more response *Corresponding author ( Kondapalli Siva Prasad). Tel/Fax: +91-9849212391. E-mail address: kspanits@gmail.com.

2012. American Transactions on Engineering & Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660. Online Available at http://TUENGR.COM/ATEAS/V01/57-74.pdf

59

variables; the response variable can be graphically viewed as a function of the process variables (or independent variables) and this graphical perspective of the problem has led to the term Response Surface Method (Myers and Montgomery, 2002). RSM is applied to fit the acquired model to the desired model when random factors are present and it may fit linear or quadratic models to describe the response in terms of the independent variables and then search for the optimal settings for the independent variables by performing an optimization step. According to (Clurkin and Rosen, 2002), the RSM was constructed to check the model part accuracy which uses the build time as function of the process variables and other parameters. According to (Asiabanpour et.al, 2006) developed the regression model that describes the relationship between the factors and the composite desirability. RSM also improves the analystâ&#x20AC;&#x2122;s understanding of the sensitivity between independent and dependent variables (Bauer et.al, 1999). With RSM, the relationship between the independent variables and the responses can be quantified (Kechagias, 2007).

RSM is an

experimental strategy and have been employed by research and development personnel in the industry, with considerable success in a wide variety of situations to obtain solutions for complicated problems.

The following two designs are widely used for fitting a quadratic model in RSM.

2.1 Central Composite Designs Central composite designs (CCDs), also known as Box-Wilson designs, are appropriate for calibrating the full quadratic models described in Response Surface Models. There are three types of CCDs, namely, circumscribed, inscribed and faced. The geometry of CCDâ&#x20AC;&#x2122;s is shown in the Figure 1.

Figure 1: Circumscribed, inscribed and faced designs.

60

Kondapalli Siva Prasad, Ch.Srinivasa Rao, and D.Nageswara Rao

Each design consists of a factorial design (the corners of a cube) together with center and star points that allow estimation of second-order effects. For a full quadratic model with n factors, CCDs have enough design points to estimate the (n+2)(n+1)/2 coefficients in a full quadratic model with n factors.

The type of CCD used (the position of the factorial and star points) is determined by the number of factors and by the desired properties of the design.

Table 1 summarizes some

important properties. A design is rotatable if the prediction variance depends only on the distance of the design point from the center of the design.

Table 1: Comparison of CCDâ&#x20AC;&#x2122;s.

Design

Rotatable

Accuracy of Estimates

Yes

Factor Uses Levels Points Outside Âą1 5 Yes

Circumscribed (CCC) Inscribed (CCI) Faced (CCF)

Yes

5

No

Good over central subset of design space

No

3

No

Fair over entire design space; poor for pure quadratic coefficients

Good over entire design space

2.2 Box-Behnken Designs Box-Behnken designs (Figure 2) are used to calibrate full quadratic models. These are rotatable and for a small number of factors (four or less), require fewer runs than CCDs. By avoiding the corners of the design space, they allow experimenters to work around extreme factor combinations. Like an inscribed CCD, however, extremes are then poorly estimated.

Figure 2: Box-Behnken design *Corresponding author ( Kondapalli Siva Prasad). Tel/Fax: +91-9849212391. E-mail address: kspanits@gmail.com.

2012. American Transactions on Engineering & Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660. Online Available at http://TUENGR.COM/ATEAS/V01/57-74.pdf

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3. Experimental Procedure Austenitic stainless steel (SS304L) sheets of 100 x 150 x 0.25mm are welded autogenously with square butt joint without edge preparation. The chemical composition of SS304L stainless steel sheet is given in Table 2. High purity argon gas (99.99%) is used as a shielding gas and a trailing gas right after welding to prevent absorption of oxygen and nitrogen from the atmosphere.

The welding has been carried out under the welding conditions presented in Table 3. From the literature (Balasubramaniam et.al, 2007, Balasubramaniam et.al, 2008, Balasubramaniam et.al, 2009, Balasubramaniam et.al, 2010) it is understood that in pulsed current arc welding processes, four important factors namely peak current, back current, pulse and pulse width are dominating over other factors. In the present work the above four factors of pulsed current MPAW are chosen and their values are presented in Table 4. A large number of trail experiments were carried out using 0.25mm thick SS304L sheets to find out the feasible working limits of pulsed current MPAW process parameters. Due to wide range of factors, it has been decided to use four factors, five levels, rotatable central composite design matrix to perform the number of experiments for investigation. Table 5 indicates the 31 set of coded conditions used to form the design matrix. The first sixteen experimental conditions (rows) have been formed for main effects. The next eight experimental conditions are called as corner points and the last seven experimental conditions are known as center points. The method of designing such matrix is dealt elsewhere (Montgomery, 1991, Box et.al,1978). For the convenience of recording and processing the experimental data, the upper and lower levels of the factors are coded as +2 and -2, respectively and the coded values of any intermediate levels can be calculated by using Equation (1) (Ravindra and Parmar, 1987).

Xi = 2[2X-(Xmax + Xmin)] / (Xmax â&#x20AC;&#x201C; Xmin)

(1)

Where Xi is the required coded value of a parameter X. The X is any value of the parameter from Xmin to Xmax, where Xmin is the lower limit of the parameter and Xmax is the upper limit of the parameter.

Table 2: Chemical composition of SS304L (weight %).

C 0.021

62

Si 0.35

Mn 1.27

P 0.030

S Cr 0.001 18.10

Ni 8.02

Mo --

Kondapalli Siva Prasad, Ch.Srinivasa Rao, and D.Nageswara Rao

Ti --

N 0.053

Table 3: Welding conditions.

Power source

Secheron Micro Plasma Arc Machine (Model: PLASMAFIX 50E) DCEN Pulse mode 2% thoriated tungsten electrode 1mm Argon and Hydrogen 6 Lpm Argon 0.4 Lpm Argon 0.4 Lpm 1mm 1mm 260mm/min Vertical Automatic

Polarity Mode of operation Electrode Electrode Diameter Plasma gas Plasma gas flow rate Shielding gas Shielding gas flow rate Purging gas Purging gas flow rate Copper Nozzle diameter Nozzle to plate distance Welding speed Torch Position Operation type

Table 4: Important factors and their levels.

SI No 1 2 3 4

Input Factor Peak Current Back Current Pulse Pulse width

Units Amps Amps Noâ&#x20AC;&#x2122;s/sec %

-2 6 3 20 30

-1 6.5 3.5 30 40

Levels 0 7 4 40 50

+1 7.5 4.5 50 60

+2 8 5 60 70

4. Recording the responses 4.1 Measurement of grain size Three metallurgical samples are cut from each joint, with the first sample being located at 25mm behind the trailing edge of the crater at the end of the weld and mounted using Bakelite.

Sample preparation and mounting is done as per ASTM E 3-1 standard. The samples are surface grounded using 120 grit size belt with the help of belt grinder, polished using grade 1/0 (245 mesh size), grade 2/0( 425 mesh size) and grade 3/0 (515 mesh size) sand paper. The specimens are further polished by using aluminum oxide initially and the by utilizing diamond paste and velvet *Corresponding author ( Kondapalli Siva Prasad). Tel/Fax: +91-9849212391. E-mail address: kspanits@gmail.com.

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63

cloth in a polishing machine. The polished specimens are etched by using 10% Oxalic acid solution to reveal the microstructure as per ASTM E407. Micrographs are taken using metallurgical microscope (Make: Carl Zeiss, Model: Axiovert 40MAT) at 100X magnification. The micrographs of parent metal zone and weld fusion zone are shown in Figures 3 and 4.

Table 5: Design matrix and experimental results.

SI No Peak Current Back current (Amps) (Amps) 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 31

-1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0

-1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 0 0 -2 2 0 0 0 0 0 0 0 0 0 0 0

Pulse (No/sec) -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 0 0 0 0 -2 2 0 0 0 0 0 0 0 0 0

Pulse width (%) -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 -2 2 0 0 0 0 0 0 0

Grain Size (Micons) 20.812 30.226 21.508 27.536 27.323 25.206 25.994 23.491 26.290 29.835 20.605 27.764 30.095 26.109 27.385 25.013 20.788 25.830 31.663 27.263 25.270 26.030 24.626 26.626 24.845 24.845 20.145 24.845 20.045 24.845 20.445

Hardness (VHN) 198 190 199 193 193 195 195 197 194 190 200 193 190 194 193 195 196 195 188 193 195 194 195 194 196 196 200 195 201 195 198

Grain size of parent metal and weld joint is measured by using Scanning Electron Microscope (Make: INCA Penta FETx3, Model:7573). Figure 5 and Figure 6 indicates the measurement of grain size for parent metal zone and weld fusion zone. Average values of grain size are presented in Table 5.

64

Kondapalli Siva Prasad, Ch.Srinivasa Rao, and D.Nageswara Rao

Figure 3: Microstructure of parent metal zone

Figure 5: Grain size of parent metal.

Figure 4: Microstructure of weld fusion zone.

Figure 6: Grain size of weld fusion zone.

The grain size at the weld fusion zone is smaller than parent metal zone, which indicates sound weld joint.

4.2 Measurement of hardness Vickersâ&#x20AC;&#x2122;s micro hardness testing machine (Make: METSUZAWA CO LTD, JAPAN, Model: MMT-X7) was used to measure the hardness at the weld fusion zone by applying a load of 0.5Kg as per ASTM E384. Average values of three samples of each test case are presented in Table 5.

5. Developing mathematical models In most RSM problems (Cochran and Cox, 1957, Barker, 1985, Montgomery,1991, Gardiner and Gettinby,1998), the form of the relationship between the response (Y) and the independent variables is unknown. Thus the first step in RSM is to find a suitable approximation for the true *Corresponding author ( Kondapalli Siva Prasad). Tel/Fax: +91-9849212391. E-mail address: kspanits@gmail.com.

2012. American Transactions on Engineering & Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660. Online Available at http://TUENGR.COM/ATEAS/V01/57-74.pdf

65

functional relationship between the response and the set of independent variables.

Usually, a low order polynomial is some region of the independent variables is employed. If the response is well modeled by a linear function of the independent variables then the approximating function in the first order model.

Y = bo+∑bi xi +∈

(2)

If interaction terms are added to main effects or first order model, then we have a model capable of representing some curvature in the response function.

Y = bo+∑bi xi + ∑∑bijxixj+∈

(3)

The curvature, of course, results from the twisting of the plane induced by the interaction term βijxixj Table 6: Estimated Regression Coefficients for grain size.

Term

Constant Peak Current Back Current Pulse Pulse Width Peak Current*Peak Current Back Current*Back Current Pulse*Pulse Pulse Width*Pulse Width Peak Current*Back Current Peak Current*Pulse Peak Current*Pulse Width Back Current*Pulse Back Current*Pulse Width Pulse*Pulse Width

S = 1.707

Coef

22.8593 1.0522 -1.0583 0.3150 0.6250 0.1020 1.6405 0.6873 0.6813 0.0910 -2.3203 -0.4047 0.1813 -0.4078 0.1360

SE Coef 0.6453 0.3485 0.3485 0.3485 0.3485 0.3193 0.3193 0.3193 0.3193 0.4268 0.4268 0.4268 0.4268 0.4268 0.4268

R-Sq = 84.2%

T

35.424 3.019 -3.037 0.904 1.793 0.320 5.138 2.153 2.134 0.213 -5.436 -0.948 0.425 -0.955 0.319

P

0.000 0.008 0.008 0.379 0.092 0.753 0.000 0.047 0.049 0.834 0.000 0.357 0.677 0.354 0.754

R-Sq(adj) = 70.4%

Remarks

Significant Significant Significant Insignificant Insignificant Insignificant Significant Insignificant Insignificant Insignificant Significant Insignificant Insignificant Insignificant Insignificant

There are going to be situations where the curvature in the response function is not adequately modeled by Equation-3. In such cases, a logical model to consider is 66

Kondapalli Siva Prasad, Ch.Srinivasa Rao, and D.Nageswara Rao

Y = bo+∑bi xi +∑biixi2 + ∑∑bijxixj+∈

(4)

Where bii repesent pure second order or quadratic effects. Equation 4 is a second order response surface model.

Using MINITAB 14 statistical software package, the significant coefficients were determined and final models are developed using significant coefficients to estimate grain size and hardness values of weld joint. The details of estimation of regression coefficients for grain size and hardness are presented in Tables 6 and 7.

Table 7: Estimated Regression Coefficients for hardness.

Term

Constant Peak Current Back Current Pulse Pulse Width Peak Current*Peak Current Back Current*Back Current Pulse*Pulse Pulse Width*Pulse Width Peak Current*Back Current Peak Current*Pulse Peak Current*Pulse Width Back Current*Pulse Back Current*Pulse Width Pulse*Pulse Width

S = 1.696

Coef

197.286 -0.708 1.292 -0.292 -0.542 -0.353 -1.603 -0.603 -0.603 -0.188 2.188 0.312 -0.313 0.313 -0.313

SE Coef 0.6410 0.3462 0.3462 0.3462 0.3462 0.3171 0.3171 0.3171 0.3171 0.4240 0.4240 0.4240 0.4240 0.4240 0.4240

R-Sq = 83.2%

T

307.801 -2.046 3.731 -0.843 -1.565 -1.112 -5.054 -1.900 -1.900 -0.442 5.160 0.737 -0.737 0.737 -0.737

P

0.000 0.058 0.002 0.412 0.137 0.283 0.000 0.076 0.076 0.664 0.000 0.472 0.472 0.472 0.472

Remarks

Significant Insignificant Significant Insignificant Insignificant Insignificant Significant Insignificant Insignificant Insignificant Significant Insignificant Insignificant Insignificant Insignificant

R-Sq(adj) = 68.5%

The final mathematical models are given in terms of grain size and hardness as below: Grain Size (G) G = 22.859+1.052X1-1.058X2+0.315X3+0.625X4+1.640X22-2.320X1X3 *Corresponding author ( Kondapalli Siva Prasad). Tel/Fax: +91-9849212391. E-mail address: kspanits@gmail.com.

2012. American Transactions on Engineering & Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660. Online Available at http://TUENGR.COM/ATEAS/V01/57-74.pdf

(5)

67

Hardness (H) H = 197.286-0.708X1+1.292X2-0.292X3-0.542X4-1.603X22+2.188X1X3

(6)

Where X1, X2, X3 and X4 are the coded values of peak current, back current, pulse and pulse width.

Table 8: ANOVA test results for grain size and hardness.

Source

Regression Linear Square Interaction Residual Error Lack-of-Fit Pure Error Total

Source

Regression Linear Square Interaction Residual Error Lack-of-Fit Pure Error Total

DF 14 4 4 6 16 10 6 30

DF 14 4 4 6 16 10 6 30

Grain Size Seq SS Adj SS

249.023 65.207 91.443 92.372 46.639 9.750 36.889 295.661

249.023 65.207 91.443 92.372 46.639 9.750 36.889

228.18 61.17 83.64 83.38 46.01 10.58 35.43 274.19

228.18 61.17 83.64 83.38 46.01 10.58 35.43

Hardness Seq SS Adj SS

Adj MS 17.7873 16.3018 22.8608 15.3954 2.9149 0.9750 6.1481

Adj MS 16.299 15.292 20.910 13.896 2.876 1.058 5.905

F

P

6.10 5.59 7.84 5.28

0.000 0.005 0.001 0.004

0.16

0.994

F

P

5.67 5.32 7.27 4.83

0.001 0.006 0.002 0.005

0.18

0.991

Table value of Fisher’s ratio is 7.87 for 99% confidence level

Where DF =Degrees of Freedom, SS=Sum of Squares, F=Fisher’s ratio

6. Checking the adequacy of the developed models The adequacy of the developed models was tested using the analysis of variance technique (ANOVA). As per this technique, if the calculated value of the Fratio of the developed model is less than the standard Fratio (from F-table) value at a desired level of confidence (say 99%), then the model is said to be adequate within the confidence limit. ANOVA test results are presented in Table 8 for all the models. From the table it is understood that the developed mathematical models are found to be adequate at 99% confidence level. Coefficient of determination ‘ R2 ’ is used to find how close the predicted and experimental values lie. The value of ‘ R2 ’ for the above 68

Kondapalli Siva Prasad, Ch.Srinivasa Rao, and D.Nageswara Rao

developed models is found to be about 0.84, which indicates good correlation exists between the experimental values and predicted values.

Figures 7 and 8 indicate the scatter plots for grain size and hardness of the weld joint and reveals that the actual and predicted values are close to each other with in the specified limits.

To validate the developed models further, one has to conduct validation tests and check for repeatability. However in the present paper confirmation test results are not implemented.

Scatterplot of Hardness

Scatterplot of Grain Size 202

32

200

30

198 Actual

Actual

28 26 24

196 194 192

22

190 188

20 20

22

24

26 Predicted

28

30

189.0

32

Figure 7: Scatter plot of Grain Size

Peak Current

196.5

198.0

199.5

Back Current

196 194

27.5

192

25.0

190

22.5

188 6.0

6.5

7.0 Pulse

7.5

8.0

3.0

3.5

4.0 4.5 Pulse Width

5.0

6.0

6.5

7.0 Pulse

7.5

8.0

3.0

3.5

20

30

40

50

60

30

40

4.0 4.5 Pulse Width

5.0

196

30.0

194

27.5

192

25.0

190

22.5 20.0

193.5 195.0 Predicted

Main Effects Plot for Hardness

Back Current

30.0

20.0

192.0

Figure 8: Scatter plot of Hardness

Main Effects Plot for Grain Size Peak Current

190.5

188 20

30

40

50

60

30

40

50

60

Figure 9: Variation of grain size.

70

50

60

Figure: 10 Variation of hardness.

*Corresponding author ( Kondapalli Siva Prasad). Tel/Fax: +91-9849212391. E-mail address: kspanits@gmail.com.

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69

70

7. Effect of process variable on output responses 7.1 Main effect The variation of grain size and hardness of SS304L welds with pulsed current MPAW input process parameters are presented in Figures 9 and 10.

From Figures 9 and 10 it is clearly understood that grain size and hardness are inversely proportional, i.e. smaller the grain size, higher the hardness of the weld joint.

7.2 Interaction effects Contour plots play a very important role in the study of the response surface. By generating contour plots using software (MINITAB14) for response surface analysis, the optimum is located by characterizing the shape of the surface. If the counter patterning of circular shaped counters occurs, it tends to suggest the independence of factor effects; while elliptical contours may indicate factor interaction. Figures 11a and 11b represent the contour plots for grain size and Figures 11a and 11b represents the contour plots for hardness.

From the contour plots, the interaction effect between the input process parameters and output response can be clearly analysed.

Contour Plot of Grain Size vs Back Current, Peak Current 5.0 24

Pulse Width

Back Current

22

25.5

30 6.5

7.0

50

40

28

26

7.5

Hold Values Peak Current 7 Back Current 4

27.0

60

4.0

3.5

28.5

25.5

26

4.5

3.0 6.0

Contour Plot of Grain Size vs Pulse Width, Pulse 70

Hold Values 40 Pulse Pulse Width 50

28

8.0

Peak Current

Figure 10a: Contour plot of Grain Size (Peak current, Back current)

30 20

24.0 30

40

50

60

Pulse

Figure 10b: Contour plot of Grain Size (Pulse, Pulse width)

From Figures 10a and 10b it is understood that the grain size is more sensitive to changes in pulse and pulse width than to changes in peak current and back current. Also from Figure 10a, the grain size is more sensitive to changes in peak current than changes in pulse and pulse width.

70

Kondapalli Siva Prasad, Ch.Srinivasa Rao, and D.Nageswara Rao

Contour Plot of Hardness vs Back Current, Peak Current 5.0

Contour Plot of Hardness vs Pulse Width, Pulse 70

Hold Values Pulse 40 Pulse Width 50

194

4.0 196

194

3.5

194

50

40 192

3.0 6.0

196

60

Pulse Width

Back Current

4.5

Hold Values Peak Current 7 Back Current 4

192

6.5

190 7.0

7.5

8.0

Peak Current

Figure 11a: Contour plot of Hardness (Peak current, Back current)

30 20

30

40

50

60

Pulse

Figure 11b: Contour plot of Hardness (Pulse, Pulse width)

From Figures 11a and 11b it is understood that the hardness is more sensitive to changes in pulse and pulse width than to changes in peak current and back current. Also from Figure 11a, the hardness is more sensitive to changes in peak current than changes in pulse and pulse width.

From the contour plots of grain size and hardness, it is understood that peak current and pulse plays a major role in deciding the grain size and hardness of the weld joint. The decrease in hardness is the result of the increased input heat associated with the use of higher peak current. The formation of coarse grains in the fusion zone is responsible for the lower hardness of the weld joints. Also increase in heat input results in slow cooling rate, which also contributes to longer time for grain coarsening. The increase in hardness is because of grain refinement at fusion zone caused by using pulsing current.

8. Conclusions Empirical relations are developed to predict grain size and hardness of pulsed current micro plasma arc welded SS304L sheets using response surface method. The developed model can be effectively used to predict grain size and hardness of pulsed current micro plasma arc welded joints at 99% confidence level. Contour plots are drawn and analysed that grain size and hardness are more sensitive to peak current and pulse. Peak current is most important parameter as it affects the grain size which signifies the hardness of weld joint. The decrease in hardness is because of *Corresponding author ( Kondapalli Siva Prasad). Tel/Fax: +91-9849212391. E-mail address: kspanits@gmail.com.

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71

formation of coarse grains in the fusion zone. Increase in peak current increases the heat input which results in slow cooling rate, which also contributes to longer time for grain coarsening.

Pulsing current helps to increase the hardness by refining the grains at the fusion zone. The mathematical models are developed considering only four factors and five levels (peak current, back current, pulse and pulse width). However one may consider more number of factors and their levels to improve the mathematical model.

9 Acknowledgments The authors would like to thank Shri. R.Gopla Krishnan, Director, M/s Metallic Bellows (I) Pvt Ltd, Chennai, India for his support to carry out experimentation work.

9 References Asiabanpour. B, Khoshnevis. B, and Palmer. K, (2006), Development of a rapid prototyping system using response surface methodology. Journal of Quality and Reliability Engineering International, 22,No.8, p.919.

Bauer.W.K, Parnell. S.G and Meyers. A.D, (1999), Response Surface Methodology as a SensitivityAnalysis Tool in Decision Analysis. Journal of Multi-Criteria decision Analysis, 8, p.162.

Balasubramaniam.M, Jayabalan.V, Balasubramaniam.V,(2007), Response surface approach to optimize the pulsed current gas tungsten arc welding parameters of Ti-6Al-4V titanium alloy, METALS and MATERIALS International, 13, No.4,p.335.

Balasubramaniam.M, Jayabalan.V, Balasubramaniam.V, (2008), A mathematical model to predict impact toughness of pulsed current gas tungsten arc welded titanium alloy, Int J Adv Manuf Technol, 35, p.852.

Balasubramaniam.M, Jayabalan.V, Balasubramaniam.V, (2008), Optimizing pulsed current parameters t o minimize corrosion rate in gas tungsten arc welde titanium alloy, Int J Adv Mnauf Technol, 39, p.474.

Balasubramaniam.M, Jayabalan.V, Balasubramaniam.V, (2009), Prediction and optimization of pulsed current gas tungsten arc welding process parameters to obtain sound weld pool geometry in titanium alloy using lexicographic method, JMEPEG,18,p.871.

Balasubramanian.M, Jayabalan.V, Balasubramanian.V,(2010) Effect of process parameters of pulsed current tungsten inert gas welding on weld pool geometry of titanium welds, Acta Metall.Sin.(Engl. Lett.),23, No.4,p. 312.

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Kondapalli Siva Prasad, Ch.Srinivasa Rao, and D.Nageswara Rao

Balasubramanian.V, Lakshminarayanan.A.K, Varahamoorthy.R and Babu.S, (2009), Application of Response Surface Methodolody to Prediction of Dilution in Plasma Transferred Arc Hardfacing of Stainless Steel on Carbon Steel , Science Direct, 16, No.1,p.44.

Balasubramanian.B, Jayabalan.V, Balasubramanian.V,(2006)Optimizing the Pulsed Current Gas Tungsten Arc Welding Parameters, J Mater Sci Technol, 22,No.6, p.821.

Barker T B, Quality by experimental design,(1985), ASQC Quality Press, Marcel Dekker.

Box G EP, Hunter W H, Hunter J S,(1978), Statistics for experiments [M], New York: John Wiley and Sons, p.112.

Cochran W G, Cox G M, (1957), Experimental Designs, John Wiley and Sons Inc, London.

Gardiner W P, Gettinby G, (1998), Experimental design techniques in statistical practice, Horwood press, Chichester.

Hsiao.Y.F, Tarng.Y.S, and Wang. J,(2008), Huang Optimization of Plasma Arc Welding Parameters by Using the Taguchi Method with the Grey Relational Analysis, Journal of Materials and Manufacturing Processes, 23,p.51.

Kechagias. J, (2007), An experiment investigation of the surface roughness of parts produced by LOM process. Rapid Prototyping Journal, 13,No.1, p.17.

LakshinarayanaA.K, Balasubramanian.V, Varahamoorthy.R and Babu.S, (2008), Predicted the Dilution of Plasma Transferred Arc Hardfacing of Stellite on Carbon Steel using Response Surface Methodology, Metals and Materials International, 14, No.6,p.779.

Madusudhana Reddy G, Gokhale A A, Prasad Rao K, (1997), Weld microstructure refinement in a 1441 grade aluminium-lithium alloy, Journal of Material Science, 32, No.5, p.4117.

Montgomery D.C,(1991), Design and analysis of experiments ,3rd Edition, New York, John Wiley and Sons,p.291.

Myers, R., and Montgomery. D, (2002), Response Surface Methodology, 2nd ed. Wiley: New York.

Mc Clurkin. J.E and Rosen, D.W, (2002), Computer-aided build style decision support for stereo lithography. Rapid Prototyping Journal, 4, No.1, p. 4.

Ravindra J, Parmar R S, (1987),Mathematical model to predict weld bead geometry for flux cored arc welding , Journal of Metal Construction, p.45.

Siva.K, Muragan.N, Logesh.R,(2008),Optimization of weld bead geometry in Plasma transferred arc hardfacing austenitic stainless steel plates using genetic algorithm, Int J Adv Manuf Technol, Volume 41, Numbers 1-2,p.24. *Corresponding author ( Kondapalli Siva Prasad). Tel/Fax: +91-9849212391. E-mail address: kspanits@gmail.com.

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Sheng-Chai Chi, LI-Chang Hsu , (2001),A fuzzy Radial Basis Function Neural Network for Predicting Multiple Quality characteristics of Plasma Arc Welding, IEEE,0-7803-7078-3,No.01,p.2807.

Srimanth.N and Murugan.N , (2011), Prediction and Optimisation of Weld Bead Geometry of Plasma Transferred Arc Hardfaced Valve Seat Rings, European Journal of Scientific Research, 51, No2,p.285.

Zhang.D.K and Niu.J.T ,(2000),Application of Artificial Neural Network modeling to Plasma Arc Welding of Aluminum alloys, Journal of Advanced Metallurgical Sciences, 13, No.1, p.194.

K.Siva Prasad is an Assistant Professor of Department of Mechanical Engineering at Anil Neerukonda Institute of Technology and Sciences, Visakhapatnam, India. He received his bachelor degree from Osmania University, India and master degree from JNTU, Hyderabad, India. He is also a part time scholar at Andhra University. He is a member of various professional bodies like ISTE, FPSI, ISHRAE etc. His area of research is micro welding processes.

Dr. Ch.Srinivasa Rao is an Associate Professor in the Mechanical Engineering Department at Andhra University, Visakhapatnam, India. He obtained his PhD degree from Andhra University, Visakhapatnam, India. He has published his research papers in various International Journals and conferences proceedings. He is a member of various professional bodies like ISTE, IE etc. His area of interest is manufacturing sciences, rapid prototyping and robotics.

Professor Dr. D.Nageswara Rao is now Vice Chancellor, Centurion University of Technology & Management, Odisha, INDIA. He obtained his PhD degree from Indian Institute of Technology Delhi, India. He was the coordinator for Centre for Nanotechnology at Andhra University. He has successfully completed various projects sponsored by DST, UGC, AICTE, NRB etc. His area of research is manufacturing sciences and nanotechnology.

Peer Review: This article has been internationally peer-reviewed and accepted for publication according to the guidelines given at the journalâ&#x20AC;&#x2122;s website.

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Kondapalli Siva Prasad, Ch.Srinivasa Rao, and D.Nageswara Rao

2012 American Transactions on Engineering & Applied Sciences

American Transactions on Engineering & Applied Sciences http://TuEngr.com/ATEAS

Cyclic Elastoplastic Large Displacement Analysis and Stability Evaluation of Steel Tubular Braces Iraj H. P. Mamaghani a

a*

Department of Civil Engineering, School of Engineering and Mines, University of North Dakota, USA.

ARTICLEINFO Article history: Received 23 August 2011 Accepted 9 January 2012 Available online 18 January 2012

Keywords: Cyclic, Elastoplastic, Large displacement, Analysis, Stability, Steel, Tubular, Brace, Finite-Element.

A B S T RA C T

This paper deals with the cyclic elastoplastic large displacement analysis and stability evaluation of steel tubular braces subjected to axial tension and compression. The inelastic cyclic performance of cold-formed steel braces made of circular hollow sections is examined through finite element analysis using the commercial computer program ABAQUS. First some of the most important parameters considered in the practical design and ductility evaluation of steel braces of tubular sections are presented. Then the details of finite element modeling and numerical analysis are described. Later the accuracy of the analytical model employed in the analysis is substantiated by comparing the analytical results with the available test data in the literature. Finally the effects of some important structural and material parameters on cyclic inelastic behavior of steel tubular braces are discussed and evaluated.

2012 American Transactions on Engineering & Applied Sciences.

1. Introduction Steel braced frames are one of the most commonly used structural systems because of their structural efficiency in providing significant lateral strength and stiffness. The steel braces *Corresponding author (Iraj H.P. Mamaghani). Tel: +1-701-777 3563, Fax: +1-701-777 3782. E-mail address: iraj.mamaghani@engr.und.edu.

2012. American Transactions on Engineering & Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660 Online Available at http://TUENGR.COM/ATEAS/V01/75-90.pdf

75

contribute to seismic energy dissipation by deforming inelastically during an earthquake. The use of this type of construction indeed avoids the brittle fractures found in beam-to-column connections in moment-resisting steel frames that occurred in the Northridge earthquake in 1994 and the Kobe earthquake in 1995 (ASCE, 2000; IGNTSDSS , 1996). However, careful design of steel braced frames is necessary to avoid possible catastrophic failure by brace rupture in the event of severe seismic loading. The current capacity design procedure adopted in most seismic design steel specifications (AISC, 1997; CAN-CSA S16.1, 1989), for concentrically braced frames requires yielding in the braces as primary members, whereas the secondary members of the frame should remain elastic and hence carry forces induced by the yielding members. The transition from current perspective seismic codes to performance-based design specifications requires accurate predictions of inelastic limit states up to structural collapse.

The cyclic behavior of steel brace members is complex due to the influence of various parameters such as material nonlinearity, structural nonlinearity, boundary condition, and loading history. The material nonlinearity includes structural steel characteristics such as residual stresses, yield plateau, strain hardening and Bauschinger effect. The structural nonlinearity includes parameters such as brace slenderness, cross-section slenderness, width-to-thickness ratio of the cross-sectionâ&#x20AC;&#x2122;s component elements (or radius-to-thickness ratio of circular hollow sections), and initial out-of-straightness of the brace. This complex behavior results in various physical phenomena, such as yielding in tension, buckling in compression, postbuckling deterioration of compressive load capacity, deterioration of axial stiffness with cycling, and low- cycle fatigue fractures at plastic regions.

Steel braces can be designed to resist only tensile forces, or to resist both tensile and compressive axial forces.

Recent earthquakes and experiments have shown that the

tension-compression braces provide better performance under cyclic loading (during an earthquake) as compared with the tension-only braces having almost no compressive strength (IGNTSDSS, 1996). Under severe earthquakes, the braces are subjected to cyclic axial forces and they are allowed to undergo compression buckling or tensile yield to dissipate the imposed energy while columns and collector beams respond elastically. Therefore, understanding the behavior of the bracing members under idealized cyclic loading is an important step in the careful design of steel braced frames.

76

Iraj H.P. Mamaghani

This paper deals with the inelastic cyclic analysis of steel tubular braces. The most important parameters considered in the practical design and ductility evaluation of steel braces of tubular sections are presented. The cyclic performance of steel tubular braces is examined through finite element analysis using the computer program ABAQUS (2005). The accuracy of the analytical model employed in the analysis is substantiated by comparing the analytical results with the available test data in the literature. The effects of some important structural and material parameters on inelastic cyclic behavior of steel braces are discussed and evaluated.

2. Brace Parameters Energy absorption through hysteretic damping is one of the great interests in seismic design, because it can reduce the amplitude of seismic response, and thereby reduce the ductility demand on the structure. Steel braces are very effective structural members and are widely used as energy dissipaters in skeletal buildings and offshore structures under extreme loading conditions such as severe earthquake and wave motion. They also minimize story drift of high-rise buildings for possible moderate earthquakes during their lifetime.

The most important parameters considered in the practical design and ductility evaluation of steel braces of tubular sections are section slenderness λs (Mamaghani, et. al., 1996a, 1996b, 1997; Mamaghani, 2005, 2008) and slenderness ratio of the member λc (AISC, 1997, 1999). While the former influences local buckling of the section, the latter controls the overal stability. They are given by:

λs =

= λs

λc =

σy 1b 3(1 −ν 2 ) π t E

σy d 3(1 −ν 2 ) 2t E

( for box sec tion )

( for circular sec tion )

1 KL σ y π r E

*Corresponding author (Iraj H.P. Mamaghani). Tel: +1-701-777 3563, Fax: +1-701-777 3782. E-mail address: iraj.mamaghani@engr.und.edu.

2012. American Transactions on Engineering & Applied Sciences. Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660 Online Available at http://TUENGR.COM/ATEAS/V01/75-90.pdf

(1)

(2)

(3)

77

where, b = flange width of a box section; t = plate thickness of the cross-section elements; σ y = measured yield stress; E = Young’s modulus; ν = Poisson’s ratio; d = outer diameter of

the circular section; K = effective length factor; L = measured length of the brace; and r = radius of gyration of the cross section. It is worth noting that the section slenderness, λs , represents the width-thickness ratio parameter of the flange plate for a box section and the diameter-thickness ratio of a circular hollow section for a given material.

The limiting diameter-thickness ratio specified in AISC (1997) for plastic design of circular hollow sections is d / t = 0.045E / σ y . This d / t limit can be converted to a limiting slenderness parameter for a compact element according to Equation 2. The corresponding value of λs , considering υ = 0.3 for structural steels, is:

λ= s

σy 0.045 E 3(1 − 0.32 ) = 0.037 2σ y E

(4)

This implies that when λs ≤ 0.037 , no local buckling occurs before the cross-section attains full plastic capacity.

The limiting width-thickness ratio specified in AISC (1997) for

non-compact circular hollow sections is d / t = 0.11E / σ y which corresponds to

λs = 0.09 .

The ductility behavior of the circular hollow section braces is significantly sensitive to λs when it is less than 0.09.

The maximum member slenderness limits specified in AISC (1997) for

special concentrically braced frames (SCBF) and ordinary concentrically braced frames are λc = 1.87 ( KL / r ≤ 1000 / σ y ) and λc = 1.35 ( KL / r ≤ 720 / σ y ), respectively. SCBF are

expected to withstand significant inelastic deformation when subjected to the force resulting from the motion of the design earthquake. SCBF have increased ductility due to lesser strength degradation when compression braces buckle.

3. Numerical Method Steel braces are vulnerable to damage caused by local and overall interaction buckling during a major earthquake. A sound understanding of the inelastic behavior of steel braces is important in developing a rational seismic design methodology and ductility evaluation of steel braced frame structures.

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Iraj H.P. Mamaghani

An accurate cyclic analysis of braced frames requires precise methods to predict the cyclic inelastic large-deflection response of the braces. This has been a subject of intensive research and a variety of analytical methods have been developed to simulate the hysteretic behavior of braces over the past few decades. The main research approaches used for the cyclic analysis of braces may be classified as: (1) empirical models, (2) plastic-hinge models, and (3) elastoplastic finite element models (Mamaghani et al., 1996a). The more accurate models were based on the finite element method considering geometric and material nonlinearities. This method is generally applicable to many types of problems, and it requires only the member geometry and material properties (constitutive law) to be defined.

3.1 Finite Element Method The finite element analysis is carried out by using the commercial computer program ABAQUS. The shell element S4R is used in modeling the brace member (ABAQUS, 2005). The S4R element is a three-dimensional, double-curved, four-node shell element with six degrees of freedom per node that uses bilinear interpolation. Because the S4R element contains only one sample point while five layers are assumed across the thickness, the spread of plasticity is considered through both the thickness and plane of the element. This shell element, which uses reduced integration, is applicable to both thin and thick shells, and can be used for finite strain applications.

In the analysis, both material and geometrical nonlinearities are considered. For large displacement analysis, the elements are formulated in the current configuration, using current nodal positions. Elements therefore distort from their original shapes as the deformation increases.

The stiffness matrix of the element is obtained from the variational principle of virtual work. The modified Newton-Raphson iteration technique coupled with the displacement control method is used in the analysis (Zienkiewicz, 1977). The displacement convergence criterion is adopted and the convergence tolerance is taken as 10-5. The details of elastoplastic large-displacement formulation and solution scheme are reported in the work by the author (Mamaghani, 1996).

*Corresponding author (Iraj H.P. Mamaghani). Tel: +1-701-777 3563, Fax: +1-701-777 3782. E-mail address: iraj.mamaghani@engr.und.edu.

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3.2 Analytical Modeling A series of numerical studies on the cyclic behavior of steel braces are carried out using the numerical finite element method described in the previous section, and the results are compared with the experiments. The results for three typical examples, S7A, S7B, and S7C (Elchalakani et al., 2003), presented hereafter are intended to verify the accuracy of the numerical method. These specimens are subjected to three loading histories in order to better understand the cyclic behavior of cold-formed circular hollow-section braces. The details of the test can be found in Elchalakani et al. ( 2003).

Figure 1: Analyzed circular hollow section steel brace and initial imperfection.

Table 1: Properties of the analyzed braces.

Test Number Specimen Shape

Ag (mm2) L (mm)

λs

λc

S7A

CHS 139.7x3.5

1498

2820

0.06

S7B

CHS 139.7x3.5

1498

2820

S7C

CHS 139.7x3.5

1498

2820

δ y (mm)

Py (kN)

0.4

5.34

568

0.06

0.4

5.34

568

0.06

0.4

5.34

568

The shape and dimensions of the analyzed braces are given in Table 1. For comparison, the selected brace parameters ( λc = 0.4 and λs = 0.06 ) are kept the same. These parameters represent a non-compact member having inelastic behavior. The analyzed fixed-end tubular braces subjected to cyclic concentric axial loading are modeled as shown in the Figure 1. An initial imperfection of  πx  γ x =γ 0 sin    L 

(5)

is assumed in the analysis, where the initial deflection at midspan of the member γ 0 is taken as the measured value of L/3160 during the test.

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Iraj H.P. Mamaghani

σ ( MPa )

σu =451

σy = 379 Stress

Est = 0.02 E

E = 200 GPa

εy

ε

Strain

Figure 2: Tri-linear stress-strain model for steel.

3.3 Material Model The analyzed cold-formed circular hollow sections are AS 1163 grade C350L0 (equivalent to ASTM A500 tubes) with the yield stress of σ u = 451

σ y = 379 MPa

and the ultimate tensile strength of

MPa. In the analysis, the material nonlinearity is accounted for by using the kinematic

hardening rule. Figure 2 shows the tri-linear stress-strain material model adopted in the analysis.

The Young modulus of elasticity of the steel is assumed to be E = 200 GPa. The strain hardening modulus is assumed to be 2 percent of the initial Young modulus ( E

st

= 0.02E ).

Node 1088

30 elements on perimeter

Node 1104

Figure 3: Meshing details and boundary conditions. *Corresponding author (Iraj H.P. Mamaghani). Tel: +1-701-777 3563, Fax: +1-701-777 3782. E-mail address: iraj.mamaghani@engr.und.edu.

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3.4 Cyclic Loading History In the analysis three cyclic loading histories are applied. The first loading history is a large compression-tension monocycle with a maximum normalized displacement amplitude m = δ max / δ y , where δ max is the maximum displacement in the compression-half cycle at load

reversal and section

= Py L / EA δ y ε= yL

Py = σ y A

is the yield displacement corresponding to the squash load of cross

(A = area of the cross-section; σ y = yield stress; L= the length of the brace). The

large amplitude used in the monocycle is applied to examine the inelastic response of the brace when subjected to a very large seismic demand during a possible near-field excitation (Krawinkler et al., 2000). The second loading history is a uniform increase of displacement amplitude up to failure with the maximum normalized displacement amplitudes of m = 1, 2, 3, …, where each amplitude is repeated only once.

In the third loading history, a uniform increase of the

displacement is used similar to the second loading history except that the oscillations are repeated three times at each amplitude ( m = 1, 2, 3, …, etc.).

3.5 Finite Element Meshing and Boundary Conditions The details of the finite-element meshing pattern adopted in the analysis of hollow circular sections are shown in Figure 3. The brace is subdivided into a total number of 2100 shell elements (70 elements along the brace length and 30 elements in the circumferential direction). A finer mesh pattern is used at the center and the ends of the brace, where large deformation is expected, as shown in Figure 3. In the analysis, the left end of the brace is fully fixed and the right end is modeled as a guided support to apply axial displacement, as shown in Figure 3. The axial load, P, and vertical deflection at midspan, V, are obtained from analysis.

4. Numerical Results 4.1 Example 1 The first example is concerned with the analysis of the brace S7A, which has a nominal length of 2820 mm, a member slenderness parameter of λc = 0.4 and a section slenderness of λs = 0.06 (Table 1). These parameters represent a non-compact member having inelastic behavior.

This brace is subjected to a large compression-tension monocycle with maximum normalized displacement amplitude of

82

m = 18.24 (the

Iraj H.P. Mamaghani

first loading history) to examine the inelastic response of

the brace under a very large seismic demand. This value is larger than the upper limit for m = 10 , which is likely to occur in a near-source excitation (Krawinkler et al., 2000). In order to check the effects of mesh density and loading increment (loading time steps) on the inealstic cyclic behavior of the brace, three analyses are carried out on this brace. The first analysis, designated as the original analysis, uses the original meshing pattern shown in Figure 3 with a total number of 2100 shell elements. The second analysis, designated as the mesh-increment analysis, uses a finer mesh density at the central segment and at the ends of the brace by doubling the mesh number in these regions with a total number of 3300 shell elements. The third analysis, designated as the step-increment analysis, utilizes the original meshing but doubling the time step by reducing the displacement increment to half of that used in the original analysis. Figures 4a and 4b compare the normalized axial load P / Py -axial deformation δ / δ y hysteresis loop obtained from the experiment and analyses. With reference to these figures, the following observations can be made: 1. The initial stiffness and buckling load capacity are slightly lower in the experiment than those predicted by the analyses using various mesh sizes and loading incremens. This may be due to the experimental boundary conditions (unavoidable rotation at the fix-ends) and the assumed initial imperfection in the analysis. In the analysis the cross-section out-of-straightness and residual stress are not accounted for. It is worth noting that the previous research by the author indicates that the initial residual stresses and initial section imperfections significantly decrease the initial stiffness and initial buckling load capacity and have almost no effect on the subsequent cyclic behavior of the member (Mamaghani et al., 1996a, Banno et al., 1998).

2. Under compressive load, the overall buckling was followed by local buckling at the center and brace ends. From Figures 4a and 4b, it can be observed that the overall shape of the predicted hysteresis loop is significantly closer to the experiment.

3. Under tension load, the behavior of the brace is well predicted up to

δ / δ y = 9.3 , where there

is a sharp decrease in predicted tensile strength beyond this displacement. The observed discrepancy between experimental and analytical results when the specimen is stretched beyond

δ / δ y = 9.3

might be due to the formation of a plastic hinge at the member

midspan under combined biaxial hoop stress and axial stress. By further stretching the *Corresponding author (Iraj H.P. Mamaghani). Tel: +1-701-777 3563, Fax: +1-701-777 3782. E-mail address: iraj.mamaghani@engr.und.edu.

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member, the spread of plasticity fully covered the whole cross section at midspan and extended on both sides of this section, leading to the reduction of load carrying capacity, see Figure 5.

1.5 1.5

1 1

0.5 0.5

0 -20

0 -20

-10

0

10

-10

0

10

20

20

δ/δy

-0.5

δ/δy

-0.5

Original analysis -1

P/Py

Analysis

-1.5

(a) Axial load versus axial displacement.

Mesh-increased Test

(b) Effects of mesh density and load steps.

1.5

S7A

1.5

1

1

0.5

0.5

0 -300

v (mm)

-200

-100

0

100

-0.5 Node 1088

0 -50

-40 Local buckling progress (mm)

-30

-20

-10

0

10

-0.5

-1

Node 1104

-1.5

P/Py

-400

-1

-1.5

(c) Deflection at the top face (Node 1088)

P/Py

-1.5

Test

Step-increased

P/Py

-1

(d) Local buckling progress at midspan

and bottom face (Node 1104) of the cross-section at midspan.

Figure 4: Comparison between experimental and predicted hysteretic loop for brace S7A. 4. The results in Figure 4(b) show that the increase in time step and use of fine mesh do not have significant effects on the overall predicted behavior except for a slight improvement in postbuckling behavior where the predicted results closely fit the test results. Under tensile loading beyond the δ / δ y = 9.3 , the predicted tensile load capacity drops slightly earlier

84

Iraj H.P. Mamaghani

for the analysis using fine mesh as compared with the other analyses. This is because the spread of plasticity and formation of the plastic hinge takes place faster for the fine mesh model.

Figure 4(c) shows normalized axial load

P / Py

versus vertical deflection V, at the top face

(Node 1088) and bottom face (Node 1104) of the cross-section at the midspan of the member (Figure 3), obtained from the analysis. The results in this figure show that the relative vertical deflection at the top and bottom faces of the cross-section at midspan increases as the member undergoes large axial deformation. The difference between the vertical displacements of the top face and bottom face at midspan indicates the progress of local buckling, which is plotted in Figure 4(d). Figure 5 shows the deformation of the specimen at the end of compression load and tension stretching. Under compression load, the overall buckling was followed by local buckling at the center and brace ends. A smooth kink formed at midspan of the brace under compression load. A semi-elephant-foot (an outward folding mechanism) was formed at the fixed ends of the brace, as shown in Figure 5. During the tensile stretching, the brace suffered excessive stretching at the midspan because of the development of a plastic hinge caused by a very large accumulation in local deformation. This represents a tear-through-failure mode, as the specimen exhibited during the test (Elchalakani et al., 2003). These observed behaviors under compression and tension loads are reflected in the normalized load-displacement hysteretic loop shown in Figure 4.

4.2 Example 2 The second example is concerned with the analysis of the brace S7B, which has a nominal length of 2820 mm, a member slenderness of

Îťc = 0.4 and

a section slenderness of

Îťs = 0.06 (Table

1). This brace is subjected to a uniform increase of displacement amplitude up to failure with the maximum normalized displacement amplitudes of m = 1, 2, 3, â&#x20AC;Ś, where each amplitude is repeated only once (the second loading history). The original meshing pattern shown in Figure 3, with a total number of 2100 shell elements, is utilized in the analysis.

*Corresponding author (Iraj H.P. Mamaghani). Tel: +1-701-777 3563, Fax: +1-701-777 3782. E-mail address: iraj.mamaghani@engr.und.edu.

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Figure 5: Deformed configuration of brace S7A at the final stage of compression and tension cyclic

loading.

Figure 6(a) compares the normalized axial load

P / Py

-axial deformation δ / δ y hysteresis

loops obtained from the experiment and analysis. Figure 6(b) shows the normalized axial load P / Py

versus vertical deflection V, at the midspan of the member (Figure 3), obtained from the

analysis. Comparison between hysteresis loops in Figure 6(a) shows that there is a relatively good agreement between analytical results and experiments. An observed small discrepancy between experimental and analytical hysteresis loops is that the predicted cyclic load capacities in compression direction of loading are slightly higher than those of the experiment. The possible reasons are: (a) the tri-linear kinematic hardening rule adopted in the analysis does not accurately consider the reduction of the elastic range due to plastic deformation (Bauschinger effect). In this model the size of the elastic range is taken to be constant which does not represent the actual behavior of structural steel (Mamaghani et al. 1995; Shen et al., 1995). More accurate results can be obtained from analysis using a cyclic constitutive law representing the more realistic behavior of the material; (b) the brace fixed-end boundary conditions may have shown some degree of flexibility during the tests, which is not considered in the analysis; and (c) in the analysis the cross-sectionâ&#x20AC;&#x2122;s out of straighness and residual stresses, which affect the initial buckling load, are not considered.

86

Iraj H.P. Mamaghani

1.5

1

1.5

0.5

1

0.5

0 -4

0

4

8

δ/δy

-0.5

0 -200

-150

-100

-50

V (mm)

-1.5

P/Py

-1

0 -0.5

Test

-1

Analysis

Analysi

P/Py

-8

-1.5

(a)

(b)

Figure 6: Comparison between experimental and predicted hysteretic loop for brace S7B. Figure 6(b) shows that there is a residual midspan deflection at the end of tensioning in each cycle. The residual deflection of the brace at the end of the previous tensioning has a large effect on the buckling capacity and subsequent cyclic behavior. Figure 6(b) shows the progress of residual midspan deflection due to cycling obtained from analysis. In spite of large progress in buckling, the buckling load does not decrease significantly due to cyclic strain hardening.

4.3 Example 3 The third example is concerned with the analysis of the brace S7C, which has a nominal length of 2820 mm, a member slenderness of

λc = 0.4 and

a section slenderness of λs

= 0.06 (Table

1).

This brace is subjected to a uniform increase in displacement amplitude up to failure with the maximum normalized displacement amplitudes of m = 1, 2, 3, …, where each amplitude is repeated three times (the third loading history). The original meshing pattern as shown in Figure 3, with a total number of 2100 shell elements, is utilized in the analysis.

Figure 7(a) compares the normalized axial load P / P -axial deformation δ / δ y hysteresis loop y

obtained from experiment and analysis. Figure 7(b) shows the normalized axial load

P / Py

versus

vertical deflection V, at the midspan of the member (Figure 3), obtained from the analysis.

Comparison between hysteresis loops in Figure 7(a) shows there is a relatively good agreement *Corresponding author (Iraj H.P. Mamaghani). Tel: +1-701-777 3563, Fax: +1-701-777 3782. E-mail address: iraj.mamaghani@engr.und.edu.

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between analytical results and experiments. These results indicate that the numerical method and finite element modeling employed in the numerical analysis can predict with a reasonable degree of accuracy the experimentally observed cyclic behavior of axially loaded fixed-end steel braces of circular hollow sections.

1.5 1.5

1 1

0.5 0.5

0 -4

0

4

8

δ/δy

-0.5

-150

-100

V (mm)

P/Py

(a)

-50

0 -0.5

Test

-1

-1.5

0 -200

-1

Analysis

Analysis

P/Py

-8

-1.5

(b)

Figure 7: Comparison between experimental and predicted hysteretic loop for brace S7C.

5. Conclusions This paper dealt with the inelastic cyclic elastoplastic finite-element analysis and stability (strength and ductility) evaluation of steel tubular braces subjected to axial tension and compression. The most important parameters considered in the practical seismic design and ductility evaluation of steel braces of tubular sections, such as brace slenderness, cross-section slenderness, material behavior, and loading history, were presented. The elastoplastic cyclic performance of cold-formed steel braces of circular hollow sections was examined through finite-element analysis using the commercial computer program ABAQUS and employing a tri-linear kinematic strain hardening model to account for material nonlinearity. The details of finite element modeling and numerical analysis were described. The accuracy of the analytical model employed in the analysis was substantiated by comparing the analytical results with the available test data in the literature. The effects of some important structural, material, and loading history parameters on cyclic inelastic behavior of steel braces were discussed and evaluated with reference to the experimental and analytical results. It has been shown that the numerical method and finite element modeling employed in the numerical analysis can predict with a reasonable

88

Iraj H.P. Mamaghani

degree of accuracy the experimentally observed cyclic behavior of axially loaded fixed-end steel braces of circular hollow sections.

6. References ABAQUS / Standard Userâ&#x20AC;&#x2122;s Manual. (2005). Ver. 6.5, Hibbitt, Karlsson and Sorensen, Inc. Amrican Institute of Steel Constructions (AISC-LRFD). (1999). Load and resistance factor design specification for structural steel buildings, 3rd Edition, Chicago.

Amrican Institute of Steel Constructions (AISC). (1997). Seismic provisions for structural steel buildings, Chicago, Illinois.

ASCE. (2000). Steel moment frames after Northridge. J. Struct. Eng., 126(1) (special issue).

Banno, S., Mamaghani, I. H.P., Usami, T., and Mizuno, E. (1998). Cyclic elastoplastic large deflection analysis of thin steel plates. Journal of Engineering Mechanics, ASCE, USA,Vol. 124, No. 4, pp. 363-370.

Canadian Standards Associations (CAN-CSA S16.1). (1989). Steel structures for buildings, limit state design.

Elchalakani, M., Zhao, X. L., Grzebieta, R. (2003). Test of cold-formed circular tubular braces under cyclic axial loading. J. of Struct. Eng., ASCE, 129(4), pp. 507-514.

Interim Guidelines and New Technologies for Seismic Design of Steel Structures (IGNTSDSS).

(1996). In T., Usami (eds), Committee on New Technology for Steel Structures, Japan Society of Civil Engineers (JSCE), Japan,(in Japanese).

Krawinkler, R., Akshay, G., Medina, R., and Luco, M. (2000). Development of loading histories for testing of steel-to-beam assemblies. Report prepared for SAC Steel Project, Dept. of Civil and Environmental Engineering, Stanford University.

Mamaghani, I.H.P. (2008). Seismic Design and Ductility Evaluation of Thin-Walled Steel Bridge Piers of Box Sections, Transportation Research Record: Journal of the Transportation Research Board, Volume 2050, pp. 137-142.

Mamaghani, I.H.P. (2005). Seismic performance evaluation of thin-walled steel tubular columns, Structural Stability, Structural Stability Research Council, Montreal, Quebec, Canada, pp.1-10.

Mamaghani, I.H.P. (1996). Cyclic elastoplastic behavior of steel structures: theory and experiments. Doctoral Dissertation, Department of Civil Engineering, Nagoya University, *Corresponding author (Iraj H.P. Mamaghani). Tel: +1-701-777 3563, Fax: +1-701-777 3782. E-mail address: iraj.mamaghani@engr.und.edu.

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Nagoya, Japan.

Mamaghani, I.H.P., Usami, T., and Mizuno, E. (1996a). Inelastic large deflection analysis of structural steel members under cyclic loading. Engineering Structures, UK, Elsevier Science, 18(9), pp. 659-668.

Mamaghani, I.H.P., Usami, T., and Mizuno, E. (1996b). Cyclic elastoplastic large displacement behavior of steel compression members. Journal of Structural Engineering, JSCE, Japan, Vol. 42A, pp. 135-145.

Mamaghani, I.H.P., Usami, T., and Mizuno, E. (1997). Hysteretic behavior of compact steel box beam- columns. Journal of Structural Engineering, JSCE, Japan, Vol. 43A, pp. 187-194.

Mamaghani, I.H.P., Shen, C., Mizuno, E., and Usami, T. (1995). Cyclic behavior of structural steels. I: experiments. Journal of Engineering Mechanics, ASCE, USA, Vol.121, No.11, pp. 1158-1164.

Shen, C., Mamaghani, I.H.P., Mizuno, E.,and Usami, T. (1995). Cyclic behavior of structural steels. II: theory. Journal of Engineering Mechanics, ASCE, USA, Vol.121, No.11, pp.

1165-1172.

Zienkiewicz, O.C. (1977). The finite element method. 3rd Ed., McGraw-Hill, New York.

Iraj H.P. Mamaghani is an Associate Professor of Civil Engineering at University of North Dakota. He received his B.Sc. in Civil Engineering from Istanbul Technical University with Honors in 1989. He continued his Master and PhD studies at University of Nagoya, Japan, where he obtained his Master and Doctor of Engineering degrees in Civil Engineering. Dr. Mamaghani has published several papers in professional journals and in conference proceedings. Dr. Mamaghani works in the area of civil engineering, with emphasis on structural mechanics and structural engineering. He focuses on cyclic elastoplastic material modeling, structural stability, seismic design, advanced finite element analysis and ductility evaluation of steel and composite (concrete-filled steel tubular) structures.

Peer Review: This article has been internationally peer-reviewed and accepted for publication according to the guidelines given at the journalâ&#x20AC;&#x2122;s website.

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Iraj H.P. Mamaghani

2012 2011American AmericanTransactions Transactionson onEngineering Engineering&&Applied AppliedSciences Sciences.

American Transactions on Engineering & Applied Sciences http://TuEngr.com/ATEAS

SAFARILAB: A RUGGED AND RELIABLE OPTICAL IMAGING SYSTEM CHARACTERIZATION SET-UP FOR INDUSTRIAL ENVIRONMENT a

a

b,d

b

Luisa De Marco , Anna Guagliumi , Marco Gnan Bassam Hallal

a,c

, Federico Canini , and Paolo Bassi

,

a*

a

Dipartimento di Elettronica, Informatica e Sistemistica, Alma Mater Studiorum - University of Bologna, Viale del Risorgimento 2, I 40136 Bologna, Italy b Datalogic Scanning Group s.r.l.,Via San Vitalino 13, I 40012 Lippo di Calderara di Reno BO, Italy c Present address: Datalogic Scanning Group s.r.l.,Via San Vitalino 13, I 40012 Lippo di Calderara di Reno BO, Italy d Present address: Khatod s.r.l., Via Monfalcone 41, I 20092 Cinisello Balsamo MI, Italy ARTICLEINFO

Article history: Received 15 December 2011 Accepted 20 January 2012 Available online 21 January 2012

Keywords: Incoherent imaging systems, Characterization, Spatial Frequency Response, Modulation Transfer Function, ISO 12233.

A B S T RA C T

This work proposes a rugged and reliable set-up to characterize incoherently illuminated optical imaging systems. It complies with the ISO 12233 standard and is suitable for industrial applications. The standard has been implemented taking advantage of the allowed degrees of freedom to optimize noise robustness. Key points of the algorithm are automatization of the whole measurement procedure and customization, to comply with specific needs.

Experimental results confirming the successful implementation are presented and discussed.

2012 American Transactions on Engineering & Applied Sciences.

*Corresponding author (Paolo Bassi). Tel +39-051-2093050 Fax: +39-051-2093053 E-mail address: paolo.bassi@unibo.it.

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91

1. Introduction Theoretical and experimental characterization of optical imaging systems is important in many classical applications, such as, for example, photography or microscopy, but also in emerging ones, such as those belonging to the biological and medical fields. Theoretical analysis of imaging systems can be done using linear system theory concepts both for coherent and incoherent illumination [Goodman]. Much scientific and technical literature has been devoted to the problem of their characterization (see, for example [Coltman, Park, Boreman 1995, Estribeau, Ducharme]).

On the whole, the possibility to use linear system analysis, greatly simplifies the task, since one can develop approaches in the spatial coordinate system, using the system impulse response, referred to as Point Spread Function (PSF), or in the spatial frequency domain, using the Spatial Frequency Response (SFR).

Application of these approaches in the industrial environment introduces a further challenge, related to the need of rugged and versatile tools that may withstand hostile environments and to the need for adapting the measurement set-up to different applications. This generally leads to the definition of International Standards referring to specific problems. In this paper we will refer to the ISO 12233 standard on Photography, Electronic Still Picture Cameras, Resolution Measurements [ISO 12233], which considers optical systems with incoherent illumination.

Many implementations of this standard are available. They can be freely distributed (for example as a plugin for IMAGEJ [ImageJ], based on the Java language, MITRE [Mitre] based on the C language, and SFRMAT [SFRmat], based on Matlab) or commercially distributed (for example IMATEST [Imatest] and QUICKMTF [QuickMTF]). Since the standard describes only guidelines for both the measurement conditions and the algorithm that calculates the SFR, it leaves degrees of freedom over its actual implementation that may impact on the overall results. In particular, the freely available packages are certainly easily customizable, but may be poorly documented and then should be anyway tested in depth before use, for example to determine their accuracy. On the contrary, commercially distributed ones allow no control on their algorithms and therefore it is impossible to customize them for specific needs, such as integrating them in portable devices.

In this paper, we present SAFARILAB (SFR measurement for a LAB environment), our

92

Luisa De Marco, Anna Guagliumi, Marco Gnan, Bassam Hallal, Federico Canini, Paolo Bassi

implementation of the ISO 12233 standard that, further than complying with the standard and guaranteeing versatility also in an embedded environment, addresses various critical points in its implementation and use, such as accuracy of the results and robustness to noise. In this way it allows the correspondence of the measured features to the designed ones and becomes rugged and reliable, therefore suitable for an industrial environment.

The paper is organized as follows: section 2 describes the developed software tool in detail.

Then its performance is illustrated and compared with other available software using synthetic (section 3) and experimental (section 4) images. After assessing the features of the software with respect to noise (section 5), results of a case study are demonstrated in section 6. Finally, conclusions are drawn in section 7.

2. Algorithm for SFR evaluation The ISO 12233 Standard concerns the measurement of the SFR of an incoherent imaging system. In the case of incoherent illumination, the principles describing linear systems can be applied simply using field intensities instead of field amplitudes [Goodman]. Measurement results, however, include not only the response of the optical part of the system to be characterized but also those of the other elements of the imaging chain, such as the detector (characterized by the finite pixel size of the used CCD camera) and the electronics [Boreman 2001]. The overall system SFR may be expressed as: SFR(đ?&#x2018;&#x201C;x , đ?&#x2018;&#x201C;y ) = đ??şđ?&#x2018;&#x153; (đ?&#x2018;&#x201C;x , đ?&#x2018;&#x201C;y ) â&#x2039;&#x2026; đ??şđ?&#x2018;&#x2018; (đ?&#x2018;&#x201C;x , đ?&#x2018;&#x201C;y ) â&#x2039;&#x2026; đ??şđ?&#x2018;&#x2019; (đ?&#x2018;&#x201C;x , đ?&#x2018;&#x201C;y )

(1)

where đ??şđ?&#x2018;&#x153; is the contribution of the optical part of the system, đ??şđ?&#x2018;&#x2018; is related to the detector and đ??şđ?&#x2018;&#x2019;

represents the electronic post-processing. If only one of these contributions is needed, for example the optical one, the others have to be evaluated and eliminated. This issue will be addressed later comparing measurements and simulations.

Rather than using a point source for direct PSF measurement, the ISO 12233 standard suggests

*Corresponding author (Paolo Bassi). Tel +39-051-2093050 Fax: +39-051-2093053 E-mail address: paolo.bassi@unibo.it.

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to evaluate the SFR of an imaging system using a knife-edge source, thus reducing the analysis to a 1D problem. This method has proven to combine flexibility and simplicity, which are important features in any standard definition.

The corresponding system response is named Edge Spread Function (ESF) and is related to the SFR by derivative and Fourier transform operations: đ?&#x2018;&#x2018;

SFR(đ?&#x2018;&#x201C;x ) = â&#x201E;ą{đ??żđ?&#x2018;&#x2020;đ??š(đ?&#x2018;Ľ)} = â&#x201E;ą ďż˝đ?&#x2018;&#x2018;đ?&#x2018;Ľ đ??¸đ?&#x2018;&#x2020;đ??š(đ?&#x2018;Ľ)ďż˝

(2)

where LSF stands for Line Spread Function and is the response of the system to a line source.

The standard recommends the use of an edge slanted at 5° with respect to the relevant coordinate axis in order to allow for the super-resolution of the ESF during the post-processing of the detected image, as it will be explained below, thus reducing the aliasing effects created by the sensor sampling.

The following algorithm steps, implemented by SAFARILAB, include those recommended by the standard ISO 12233 and complete them by focusing on noise and sampling issues important for the accuracy of the SFR evaluation: 1. Image acquisition and averaging. A number of images is acquired and averaged as to reduce the amount of noise. The average image is then cropped as to select the Region Of Interest (ROI) that contains the slanted edge to be processed in the following steps.

2. Estimation of the edge slope. The slope of the slanted edge is recovered with high precision via a linear regression operation on the so-called centroids, the peaks of the derivatives of each image line (LSF). To reduce noise effects, the LSF of each line is tapered using an asymmetrical Hamming window before centroids determination. This operation is repeated until the angle estimation achieves a precision of 10-6 degrees.

3. Edge super-resolution. By considering the image lines as independent sampled repetitions of the same ESF, an oversampled ESF is created. As it is shown in Figure 1, the information on the edge slope allows to find the relative offset among each ESF line, which in turn allows to

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use them to create a super-resolved ESF with irregular sampling. Then, groups of data points are averaged to obtain the oversampled ESF having regular sampling with the desired super-resolution factor. This parameter, that the standard suggests to be 4, can be adjusted in our implementation.

Figure 1: Creation of the oversampled ESF. 4. LSF calculation as the numerical derivative of the oversampled ESF, multiplied by an asymmetrical Hamming window.

5. SFR calculation as the Fourier transform of the LSF. 6. SFR averaging. A number of SFRs are obtained by repeating the previous steps and are averaged to improve the accuracy of the estimation.

Within these steps many parameters (such as the number of images on which to average the input to the tool, the super-resolution factor and the number of SFR calculations) are related to the *Corresponding author (Paolo Bassi). Tel +39-051-2093050 Fax: +39-051-2093053 E-mail address: paolo.bassi@unibo.it.

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accuracy of the extraction of the SFR. It is important that their values are correctly set.

In the remainder of the paper, the behavior of SAFARILAB is demonstrated. Initially accuracy tests are shown in which the software is used on numerically created images (synthetic) and on images obtained by real optical systems. In both cases, its results are compared to those of two available tools: IMAGEJ and IMATEST. Finally, an example of use in a real test case is given.

3. SFR extraction from synthetic images The initial verification of the performance of SAFARILAB is done using as its input a numerically created image, in which the black-to-white transition follows an analytically known function to allow comparison of obtained results with exact, reference, ones. The arctangent function has been used since it satisfies the following requirements: â&#x20AC;˘

It has the same qualitative shape of a real edge.

â&#x20AC;˘

The analytical expression of the convolution with a rectangular window (to include the sampling effect) is known: +â&#x2C6;&#x17E;

đ?&#x2018;Ľ

đ?&#x153;?

đ?&#x2018;&#x2021;/2

arctan(đ?&#x2018;Ľ) â&#x2039;&#x2020; rect ďż˝đ?&#x2018;&#x2021;ďż˝ = â&#x2C6;Ťâ&#x2C6;&#x2019;â&#x2C6;&#x17E; arctan(đ?&#x2018;Ľ â&#x2C6;&#x2019; đ?&#x153;?) đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018;?đ?&#x2018;Ą ďż˝đ?&#x2018;&#x2021;ďż˝ đ?&#x2018;&#x2018;đ?&#x153;? = â&#x2C6;Ťâ&#x2C6;&#x2019;đ?&#x2018;&#x2021;/2 arctan(x â&#x2C6;&#x2019; Ď&#x201E;)dĎ&#x201E; = đ?&#x2018;&#x2021;

đ?&#x2018;&#x2021;

đ?&#x2018;&#x2021;

đ?&#x2018;&#x2021;

= â&#x2C6;&#x2019; ďż˝đ?&#x2018;Ľ â&#x2C6;&#x2019; 2ďż˝ arctan ďż˝đ?&#x2018;Ľ â&#x2C6;&#x2019; 2ďż˝ + ďż˝đ?&#x2018;Ľ + 2ďż˝ arctan ďż˝đ?&#x2018;Ľ + 2ďż˝ + 1

đ?&#x2018;&#x2021; 2

1

đ?&#x2018;&#x2021; 2

+ 2 ln ��đ?&#x2018;Ľ â&#x2C6;&#x2019; 2ďż˝ + 1ďż˝ â&#x2C6;&#x2019; 2 ln ��đ?&#x2018;Ľ + 2ďż˝ + 1ďż˝.

â&#x20AC;˘

The derivative of this convolution is also analytically known: đ?&#x2018;&#x2018;

đ?&#x2018;&#x2018;đ?&#x2018;Ľ

â&#x20AC;˘

(3)

đ?&#x2018;Ľ

ďż˝arctan(đ?&#x2018;Ľ) â&#x2039;&#x2020; rect ďż˝đ?&#x2018;&#x2021;�� =

1

x2 +1

đ?&#x2018;Ľ

â&#x2039;&#x2020; rect ďż˝đ?&#x2018;&#x2021;ďż˝.

(4)

The analytical expression of the Fourier transform of this derivative, which is the SFR, exists: 1

đ?&#x2018;Ľ

â&#x201E;ą ďż˝x2 +1 â&#x2039;&#x2020; rect ďż˝đ?&#x2018;&#x2021;�� = đ?&#x153;&#x2039;đ?&#x2018;&#x2019; â&#x2C6;&#x2019;|2đ?&#x153;&#x2039;đ?&#x2018;&#x201C;đ?&#x2018;Ľ | â&#x2039;&#x2026; sinc(đ?&#x2018;&#x201C;đ?&#x2018;Ľ đ?&#x2018;&#x2021;).

(5)

To make the image more realistic, we also added shot noise, as it is typically the most relevant

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noise contribution given by photo-sensors. Noise is modeled as a Poisson process with mean value and standard deviation dependent on the signal.

Supposing to use 8 bit A/D conversion, a black and white image has pixels with values ranging from 0 to 255. In order to simulate measurements with large contrast and zero-mean noise, synthetic slanted-edge images have been created assuming a linear Opto-Electronic Conversion Function (OECF) leading to an image with grey levels in the range 30-220.

Evaluated SFRs are shown in Figure 2 using logarithmic plots. Spatial frequencies are normalized (as in the rest of the paper) to half of the reciprocal of the sampling step (i.e. the Nyquist frequency). The red curve is the theoretical SFR, while the other three are obtained using different software: IMATEST (black line), IMAGEJ (blue line) and SAFARILAB (green line).

Figure 2: Test with synthetic images with additional noise and contrast of 30-220 (red: theoretical; black: IMATEST; blue: IMAGEJ; green: SAFARILAB).

The three tools show similar degree of accuracy, with IMAGEJ performing slightly differently from the other two. In all cases, the presence of noise produces small oscillations in the SFRs as compared to the theoretical curve when the normalized frequency is greater than 0.5 and the SFR is lower than 0.02. The slight mismatch between the results obtained using IMATEST and IMAGEJ is directly related to the degrees of freedom allowed by the standard. The good agreement between these results and those obtained by SAFARILAB confirms the correctness of our algorithms. The next sections will show how the freedom left by the standard can be exploited to improve the *Corresponding author (Paolo Bassi). Tel +39-051-2093050 Fax: +39-051-2093053 E-mail address: paolo.bassi@unibo.it.

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quality of the provided results. Evidence will be given by experimental results.

4. SFR extraction on real images Real images are acquired using the set-up shown in Figure 3 (left). A slanted-edge transmission mask, made of chrome deposited on glass (right side of Figure 3), is uniformly illuminated by an incoherent source obtained by cascading two integrating spheres and a LED source. The image of the edge is acquired by the imaging system to be characterized, consisting of a lens, a photo-detector and its relevant driver electronics.

Figure 3: Measurement set-up scheme (left) and Slanted Edge target (right).

Figure 4 shows the SFR extracted from slanted-edge images taken with a distance of 190 mm between the mask and camera. The packages applied to experimental results always behave similarly: SAFARILAB (green plot) and IMATEST (black plot) results almost superimpose, while IMAGEJ (blue plot) is slightly different.

Figure 4: SFR obtained with experimental images taken at 190 mm from the target (black: IMATEST; blue: IMAGEJ; green: SAFARILAB).

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Instead of using directly the SFR, often the performance of an optical system is given by its resolution on the object plane. The minimum feature size R that the system can resolve is inversely proportional to the spatial frequency at which the SFR reaches a threshold value, depending on system requirements. The formula to evaluate the minimum resolution is the following one: đ?&#x2018;&#x2026;=

1

(6)

2đ?&#x2018;&#x201C;đ?&#x2018;Ąâ&#x201E;&#x17D; đ?&#x2018;&#x20AC;

where đ?&#x2018;&#x201C;đ?&#x2018;Ąâ&#x201E;&#x17D; is the threshold frequency and M is the optical system magnification. In the following, đ?&#x2018;&#x201C;đ?&#x2018;Ąâ&#x201E;&#x17D; will be normalized with respect to the Nyquist frequency.

Since the measured SFR includes the response of the optics together with that of the electronic

part of the system (equation (1)), the simulated SFRs were modified as include these contributions.

The effect of the sensor sampling was considered the predominant one and was modeled as a rect function in the spatial domain or, equivalently, a sinc function in the spatial frequency domain.

Figure 5: Comparison between designed (blue and cyan) and measured (red) performance.

Resolution is normalized to the Nyquist frequency of the sensor.

Figure 5 shows an example of application of the procedure. The measured resolution of the camera as a function of the camera-target distance (red curve) is compared to that of the lens (cyan curve) and to that of the system composed by the lens and the sensor (blue curve). As it will be better explained in the next section, in order to remove noise effects the measured resolution is *Corresponding author (Paolo Bassi). Tel +39-051-2093050 Fax: +39-051-2093053 E-mail address: paolo.bassi@unibo.it.

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calculated by averaging 20 SFRs. The relevant uncertainty bars are also reported. The curve that represents the lens and the sensor system is just within the error bars of the measured resolution curve, confirming the quality of the results. With reference to Equation (1), Figure 5 also shows that the electronics has a negligible effect on the overall system performance.

5. Noise reduction The noise in the image entering the processing chain introduces uncertainty to the SFR extraction. In order to improve the accuracy, an average SFR is found by repeating the extraction process. Figure 6 (left) shows a set of 20 SFR curves (green lines) calculated from 20 different slanted-edge images together with their average (solid red line) and average ±3σ(f) (dashed red lines), where σ(f) is the standard deviation as a function of the spatial frequency f. The image noise causes noise-like oscillations into the SFR so that averaging appears as a necessary step for a reliable evaluation of the SFR.

Figure 6: Noise effects on SAFARILAB responses (see text for details).

The influence of different noise levels has also been considered by changing the analog gain (G) and the integration time (τ) of the camera: lower gain and higher integration time correspond to a noise reduction. In the right part of Figure 6, the red curves are obtained with analog gain G = 4 and integration time τ = 8 ms (the same conditions of Figure 5-right), whereas the blue curves are obtained with G = 1 and τ = 32 ms. The solid curves represent the average SFR (calculated as the average of 50 SFR), whereas the dashed curves represent the average SFR ±3σ(f). As expected, the uncertainty is greater in the first case since noise is larger. Averaging 50 SFRs, SAFARILAB is able

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to recover the same mean value in both cases, thus demonstrating a good insensitivity to random noise effects.

However, it is sometimes impossible to control the analog gain and the integration time in the needed way. To overcome this limit and obtain the same effect of noise reduction without acting on the camera parameters, SAFARILAB may also make image averaging as the very first step of its algorithm. The number of images to be averaged should be chosen in relation with the number of SFR evaluations for the calculation of the average, since both impacts on the overall execution time.

6. Characterization of a system using SAFARILAB After the description of SAFARILAB capabilities evidencing the original way in which noise contributions can be filtered differently depending on the experimental situation, in this section its versatility will be demonstrated on an optical system containing also non-classical components designed to achieve Extended Depth of Field (EDoF). Among the possible ways of getting this result, see for example [McLeod, Kolodziejczyk, Davidson, Andersen, Iemmi], we adopted the so called “Wavefront Coding” approach [Dowski], based on the insertion of a Phase Mask (PM) into the optical system. Among the proposed solutions, see for example [Dowski, Sauceda, Sherif, Caron, Zhou], we have chosen the cubic one [Dowski] since it has also a solid theoretical background which may be useful in simulations. The cubic profile is described by: z = α(x 3 + y 3 )

(6)

where x and y are the planar spatial coordinates and α is a cubic coefficient.

In the following, the SFR of a lens with and without a cubic phase mask is characterized using

SAFARILAB. The match between designed and measured performance will confirm the reliability of the characterization tool itself.

The experimental curves are obtained by SAFARILAB as the average of 10 SFRs, each calculated on the average of 10 images. Also in this case the sensor sampling effect is included in the designed SFRs before comparing them with the measured ones. *Corresponding author (Paolo Bassi). Tel +39-051-2093050 Fax: +39-051-2093053 E-mail address: paolo.bassi@unibo.it.

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Figure 7 qualitatively shows the behavior of the system with (red curves) and without (blue curves) the phase mask. The plot on the left represents the designed SFR of the system for different object distances: the cubic mask provides a good invariance of the SFR, which corresponds to an extension of the DoF. The same behavior is shown also in the plot on the right obtained characterizing the real system with SAFARILAB.

Figure 7: Simulated (left) and experimental (right) SFRs for the lens with (red lines) and without (blue lines) the cubic phase mask.

Figure 8: Simulated (blue) and measured (red) results with (dashed) and without (solid) phase mask. The object distance increases moving from the leftmost to the rightmost plot.

A more detailed view of the agreement between the expected behavior and the obtained one is given in Figure 8. The designed SFRs of the system with and without the phase mask at increasing object distances are compared to the ones measured with SaFaRiLAB. The excellent agreement between simulated and experimental curves confirms how our tool is accurate and useful in the design and analysis stage of an optical system.

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7. Conclusion A tool named SAFARILAB has been proposed. It evaluates the Spatial Frequency Response of an optical system complying with the ISO 12233 standard, the reference standard for this kind of measurements. Its performance has been first successfully compared with those of other available software dedicated to this task. Then, an experimental set-up has been realized to perform the measurements and the results show excellent behavior in terms of repeatability. The robustness to noise, obtained taking advantage of the degrees of freedom left by the standard, has also been evidenced. Finally, its reliability in a practical case has been proven comparing the measured optical SFRs and the designed ones in a more complex case where also a cubic phase mask is present. SAFARILAB can then be proposed as a valid tool for industrial environments and embedded systems, where noise robustness and versatility are both a concern.

8. Acknowledgements This work has been developed in the framework of a research contract between DEIS and Datalogic Scanning Group.

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Boreman, G. D. and S. Yang. (1995). Modulation Transfer-Function measurement using 3-bar and 4-bar targets. Applied Optics, 34(34), 8050-8052.

Boreman, G. D. (2001). Modulation Transfer Function in Optical and Electro-Optical Systems.

SPIE Press.

Caron, N. and Y. Sheng. (2008). Polynomial phase masks for extending the depth of field of a microscope. Applied Optics, 47(22), 39-43.

Coltman, J. W. (1954). The Specification of Imaging Properties by Response to a Sine Wave Input. Journal of the Optical Society of America, 44(6), 468-471.

Davidson, N., A. Friesem and E. Hasman. (1991). Holographic axilens: high resolution and long focal depth. Optics Letters, 16(5), 523-525.

Dowski, E. R. and W. T. Cathey. (1995). Extended depth of field through wave-front coding. *Corresponding author (Paolo Bassi). Tel +39-051-2093050 Fax: +39-051-2093053 E-mail address: paolo.bassi@unibo.it.

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Applied Optics, 34(11), 1859-1866.

Ducharme, A. D. and S. P. Temple. (2008). Improved aperture for modulation transfer function measurement of detector arrays beyond the Nyquist frequency. Optical Engineering, 49(7), 093601.

Estribeau, M. and P. Magnan. (2004). Fast MTF measurement of CMOS imagers using ISO 12233 slanted-edge methodology. Proceedings of SPIE - Volume 5251 - Detectors and Associated Signal Processing, 243-252.

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Depth of focus increase by multiplexing programmable diffractive lenses. Optics Express, 14(22), 10207-10219.

IMAGEJ (Image Processing and Analysis in Java) available at: http://rsbweb.nih.gov/ij/.

IMATEST software, available at: http://www.imatest.com/products/software/master.

ISO 12233 Standard (Photography - Electronic Still Picture Cameras - Resolution Measurements), http://www.iso.org/iso/catalogue_detail.htm?csnumber=33715. available at: Kolodziejczyk, A., S. Bará, Z. Jaroszewicz and M. Sypek. (1990). The Light Sword Optical Element - a New Diffraction Structure with Extended Depth of Focus. Journal of Modern Optics, 37(8), 1283-1286.

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QUICK MTF, available at http://www.quickmtf.com/.

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Sauceda, A. and J. Ojeda-Castañeda. (2004). High focal depth with fractional-power wave fronts. Optics Letters, 29(6), 560-562.

Sherif, S. S., W. T. Cathey and E. R. Dowski. (2004). Phase plate to extend the depth of field of incoherent hybrid imaging systems. Applied Optics, 43(13), 2709-2721.

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Zhou, F., G. Li, H. Zhang and D. Wang. (2009). Rational phase mask to extend the depth of field in optical-digital hybrid imaging systems. Optics Letters, 34(3), 380-382.

Luisa De Marco received the B.Sc. and the M.Sc. degrees in Telecommunications Engineering from the University of Padova in 2006 and from the Alma Mater Studiorum – University of Bologna in 2008, respectively. She is currently working towards the PhD with a scholarship funded by Datalogic Scanning s.r.l. at the Dipartimento di Elettronica, Informatica e Sistemistica (DEIS) of the Alma Mater Studiorum – University of Bologna. In the framework of her doctoral research, she spent 6 months working at the Friedrich-Alexander University of Erlangen, Germany. Her main interest is the optical design of imaging and non-imaging systems for automatic identification and portable devices.

Anna Guagliumi received her M.Sc. degree (honors) in Telecommunication Engineering from the Alma Mater Studiorum – University of Bologna in 2010. Since 2011 she has been a PhD student at the Dipartimento di Elettronica, Informatica e Sistemistica (DEIS) of the Alma Mater Studiorum – University of Bologna. Her main interests are the design and characterization of optical structures, especially with respect to free space systems.

Marco Gnan received the M.Sc. degree (honors) in Telecommunication Engineering from the Alma Mater Studiorum – University of Bologna. From 2003 he worked towards the PhD in Optoelectronics at the University of Glasgow, which he achieved in 2007. Until February 2008 he was with the Department of Engineering at the University of Ferrara as a research assistant. Until December 2011 he was with the Dipartimento di Elettronica, Informatica e Sistemistica of the University of Bologna as a senior research assistant. Since January 2012 he has been with the R&D group of Datalogic. His interests include experimental work on integrated photonic devices and their numerical modeling, free space optics and incoherent imaging systems.

Bassam Hallal received the M.Sc. degree in Electronic Engineering (honors) from the Politecnico di Bari in 1997 extending the Knoesen-Gaylord-Moharam algorithm for the analysis of the hybrid guided modes to the uniaxial dielectric multilayered waveguides. In 1998 he received a Master in Optical Technologies from the AILUN (Associazione Istituzione Libera Università Nuorese). He spent the years 1999-2011 in Datalogic working on diffraction-free beams synthesis, pure refractive aiming systems, lens design for imaging systems, CCD/CMOS sensors characterization, and free-form lens design. Since December 2011 he has been with Khatod s.r.l. as the Optical R&D Supervisor.

Federico Canini received the M.Sc. degree with honors in Electronic Engineering from the Alma Mater Studiorum – University of Bologna in 1994. In 1995 he started his activity in Datalogic covering many engineering roles, developing new products based on imaging technologies and producing several patents. He currently is an R&D manager at Datalogic IP Tech and is responsible for the electronic and the optics group, and the relationship with academic partners.

Paolo Bassi is full professor of Electromagnetic fields at Dipartimento di Elettronica, Informatica e Sistemistica (DEIS) of the Alma Mater Studiorum – University of Bologna. He graduated with honors in Electronic Engineering in 1975. He is now the responsible of the optics group of DEIS. His research interests cover guided and free space application of optics, addressed both theoretically and experimentally.

Peer Review: This article has been internationally peer-reviewed and accepted for publication according to the guidelines given at the journal’s website. *Corresponding author (Paolo Bassi). Tel +39-051-2093050 Fax: +39-051-2093053 E-mail address: paolo.bassi@unibo.it.

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