Dynamic Analysis in a Building

POLO TERRITORIALE DI LECCO

Dynamic Analysis in a building

Part 1: Preliminary screening and finding the fundamental vibration modes

Buildings in Seismic Areas Prof. r. Paolucci

Master of Science in Architectural Engineering

A.Y. 2017/2018

Renan Prandini April, 2018

Contents PART 1: PRELIMINARY SCREENING AND FINDING THE FUNDAMENTAL VIBRATION MODES ......................................... 0 1

ENUNCIATION OF THE PROBLEM .........................................................................................................3

2

OBJECTIVES ...............................................................................................................................................4

3

PROCEDURES .............................................................................................................................................4 3.1

Fundamental Period calculation proposed by the Eurocode: ........................................................... 4

3.2

Rayleigh method ................................................................................................................................ 4 3.2.1

Obtaining displacement profiles ................................................................................................... 5

3.2.2

X-Direction frame displacement ................................................................................................... 6

3.2.3

Y-Direction frame displacement.................................................................................................... 6

3.3

2D SeismoStruct Eigenvalue analysis ................................................................................................ 8

3.4

3D SeismoStruct Eigenvalue Analysis ................................................................................................ 9 3.4.1

Analysis with L-walls .................................................................................................................... 9

3.4.2

Analysis without L-walls ............................................................................................................. 10

3.5

Results.............................................................................................................................................. 10

3.6

Conclusions ...................................................................................................................................... 12

PART 2: CALCULATING THE DESIGN ACTIONS THROUGH RESPONSE SPECTRUM ANALYSIS .......................................... 13 4

OBJECTIVES .............................................................................................................................................14

5

METHODOLOGY ......................................................................................................................................14

6

DESIGN RESPONSE SPECTRUM ...........................................................................................................14 6.1

Fase 1: Seismic Activities of the Location ........................................................................................ 14

6.2

Fase 2: Design Service Life ............................................................................................................... 15

6.3

Fase 3: Design Actions ..................................................................................................................... 15

7

SEISMOSTRUCT ANALYSIS ..................................................................................................................16

8

RESULTS ....................................................................................................................................................16 8.1

8.2

Identifying Critical Elements ............................................................................................................ 17 8.1.1

NO.WALL,SLV,X ......................................................................................................................... 17

8.1.2

NO.WALL,SLV,Y ......................................................................................................................... 18

8.1.3

WALL,SLD,X ............................................................................................................................... 19

8.1.4

WALL,SLD,Y ............................................................................................................................... 20

8.1.5

WALL,SLV,X ............................................................................................................................... 21

8.1.6

WALL,SLV,Y ................................................................................................................................ 22

8.1.7

Action Diagrams ......................................................................................................................... 22

8.1.8

Base Actions ................................................................................................................................ 27

Conclusion ....................................................................................................................................... 27

Page |1 PART 3: ASSESSING THE SEISMIC PERFORMANCE OF AN EXISTING BUILDING ................................................................ 28 9

OBJECTIVES .............................................................................................................................................29

10 METHODOLOGY ......................................................................................................................................29 11 HYPOTHESIS.............................................................................................................................................29 11.1 Non-linear material laws ................................................................................................................. 29 11.1.1

Exisiting concrete: Mander et al. non linear concrete model ..................................................... 29

11.1.2

Steel: Bilinear Steel Model .......................................................................................................... 29

11.1.3

New concrete: Mander et al. non linear concrete model ............................................................ 30

11.1.4

Steel: Bilinear Steel Model .......................................................................................................... 30

11.2 Structural Element class types and reinforcement configurations ................................................. 30 11.2.1

Retangular RC Columns.............................................................................................................. 30

11.2.2

Retangular RC Beams ................................................................................................................. 30

11.2.3

L-Shaped Shear Walls ................................................................................................................. 31

11.2.4

Mesh configurations .................................................................................................................... 31

11.3 Incremental loading pattern (triangular-modal) ............................................................................. 31 11.4 Elastic design spectra....................................................................................................................... 32 11.5 Performance limit values chord rotation and shear Force capacity ............................................... 32 12 RESULTS ....................................................................................................................................................34 12.1 Capacitance Curves.......................................................................................................................... 34 12.1.1

NO.WALL-X ................................................................................................................................ 34

12.1.2

WALL-X ....................................................................................................................................... 34

12.1.3

NO.WALL-Y ................................................................................................................................ 35

12.1.4

WALL-Y ....................................................................................................................................... 35

12.2 Failure of Elements .......................................................................................................................... 36 13 POST-ANALYSIS ......................................................................................................................................38 13.1 Total Base Actions ........................................................................................................................... 38 13.2 Displacements ................................................................................................................................. 38 13.3 Comparison: Response Spectrum Analysis vs. Static Pushover Analysis ........................................ 39 13.3.1

Displacements ............................................................................................................................. 39

13.3.2

Action Diagrams ......................................................................................................................... 39

13.3.3

Total Base Actions....................................................................................................................... 40

13.4 Wall introduction ............................................................................................................................. 40 13.4.1

X-Direction .................................................................................................................................. 42

13.4.2

Y-Direction .................................................................................................................................. 42

14 ADDITIONAL STUDIES ...........................................................................................................................43

Page |2 14.1 Eigenvalue analysis comparison ...................................................................................................... 43 14.2 Transversal Reinforcement.............................................................................................................. 43 15 CONCLUSION ...........................................................................................................................................43 16 BIBLIOGRAPHY .......................................................................................................................................43 Figures Index Figure 1: Dimensions of the existing building........................................................................................................ 3 Figure 2: Section properties of the central pillars and Beam elements rigid offset settings ................................... 5 Figure 3: X-direction deformed frame and horizontal displacement profile in the model with shear walls .......... 6 Figure 4: Y-direction deformed frame and horizontal displacement profile in the model with shear walls .......... 6 Figure 5. Displacement profiles in the triangular-modal load pattern for the X and Y directions ......................... 7 Figure 6: 1st, 2nd, 3rd and 4th, respectively, fundamental vibration modes with L-walls at the central columns. .... 9 Figure 7: 1st, 2nd, 3rd and 4th, respectively, fundamental vibration modes without L-walls at the central columns. ............................................................................................................................................................................... 10 Figure 8. Comparison of different calculation methods of the 1st fundamental period in the frame model ......... 11 Figure 9: Comparison of different calculation methods of the 1st fundamental period in the model with shear walls ............................................................................................................................................................................... 11 Figure 10: Periods for each fundamental vibration modes comparison with and without intervention. .............. 12 Figure 11. Response Design Spectrum ................................................................................................................. 15 Figure 12. Main action diagrams in the central columns along the X-direction analysis ..................................... 23 Figure 13. Horizontal Displacements along the X-Direction ............................................................................... 24 Figure 14. Main action diagrams in the central columns along the Y-direction analysis ..................................... 25 Figure 15. Horizontal Displacements along the Y-Direction ............................................................................... 26 Figure 16. Summary of base actions along X and Y analysis .............................................................................. 27 Figure 17. Elastic Design Spectra ......................................................................................................................... 32 Figure 18. Total base actions on the Static Pushover Analysis ............................................................................ 38 Table Index Table 1. Rayleigh method calculation log for the X-direction frame ..................................................................... 5 Table 2. Alterations in the mass elements .............................................................................................................. 8 Table 3: Mass check on the 2D analysis ................................................................................................................. 8 Table 4: Effective Modal Mass percentages ........................................................................................................... 9 Table 5: Effective Modal Mass percentages ......................................................................................................... 10 Table 6: Results for X and Y direction frames ..................................................................................................... 10 Table 7. Parameters for the Response Design Spectra according to the NTC-08 ................................................ 15 Table 8: Service Life in earthquake design........................................................................................................... 15 Table 9. Simulation Cases Index .......................................................................................................................... 16

Page |3

1 Enunciation of the problem The objective of the study is to analyze the characteristics and performance of a 1960’s style 6-story reinforced concrete building which is under re-design for residential use located in the downtown of Seismotown.

The plan and side views of the existing building is shown in Figure 1 (plan view: on the left, side view: on the right). The dimensions are in meters, all beams and columns are geometrically identical (beams: 0.6m x 0.4m).

Figure 1: Dimensions of the existing building

Structural team of the design office has decided to modify the staircase detailing of the structure, such that the support walls, highlighted in red, become major lateral load carrying members.

Based on the samples extracted from the structure, it has been concluded that the modulus of elasticity of concrete members can be taken as 25 GPa. Distributed floor mass that is needed to be included within the calculations is mfloor=1.024 t/m2. Distribution of mass to the supporting beams is already provided in the SeismoStruct project file, present in Beep repository.

You are asked to provide your professional feedback on the preliminary restoration work. In case you would find something critical, the design company is willing to modify the design details.

Page |4

2 Objectives • •

•

Estimate the fundamental vibration period based on the empirical equation proposed for reinforced concrete frame structures; Calculate the fundamental vibration period by considering 2D frames extracted from the 3D building (use the middle frame for Y-direction and second frame on the right for X-direction, See Figure 2); Analyze the 3D model in SeismoStruct and obtain all significant vibration periods with associated shapes and percent mass participations.

Report your findings to particularly highlight: • •

•

Fundamental vibration periods obtained by different methods and indicate accuracy of the estimates by providing some clarifications, The mode shapes associated with the periods. Point out any unusual mode shape (such as plan torsion). Support the visual output with the percent mass participation and comment on along which direction structure is behaving more irregular.

In your final report/presentation, compare the results of the structure with and without L-walls.

3 Procedures 3.1

FUNDAMENTAL PERIOD CALCULATION PROPOSED BY THE EUROCODE:

The period calculation proposed by the Eurocode 8 is empirical and relate to the total height (H) of the structure, given by the following formula: (I) Where:

đ?‘‡đ?‘‡0 = đ??śđ??ś1 đ??ťđ??ť 0.75 [đ?‘ đ?‘ ]

T0 is the first fundamental vibration period C1 is a coefficient depending on the structural material used on the structure. For reinforced concrete structures, it is equivalent to 0.075.

3.2

RAYLEIGH METHOD According to the Rayleigh method, the first fundamental vibration period is given by the formula (2) (II) Where:

2 ∑đ?‘›đ?‘› đ?‘–đ?‘–=1 đ?‘šđ?‘šđ?‘–đ?‘– .đ?‘‘đ?‘‘đ?‘–đ?‘–

đ?‘‡đ?‘‡1 = 2đ?œ‹đ?œ‹ďż˝ ∑đ?‘›đ?‘›

đ?‘–đ?‘–=1 đ??šđ??šđ?‘–đ?‘– .đ?‘‘đ?‘‘đ?‘–đ?‘–

mi: mass of the i-th story located at the height zi; di: horizontal displacement of the i-th story due to the lateral force distribution F; Fi: lateral force acting on the i-th story (đ??šđ??š = ∑ đ??šđ??šđ?‘–đ?‘– )

Page |5 As the seismostruct software is available, we decided to use the numerical structural analysis to find the displacement values. The total lateral force F was assumed as 1000 kN. Table 1 shows the calculation log for the calculation of the first period with the displacement profile taken from the Seismostruct model.

Table 1. Rayleigh method calculation log for the X-direction frame storey hs,I [m] hi [m] mi [t] mihi [t.m] F1[kN] di [m] midi²[t.m²] Fidi [kN.m] 1 4.1 4.1 43.008 176.33 53.88 0.015 0.009 0.79 2 4.1 8.2 43.008 352.67 107.75 0.039 0.07 4.24 3 3.1 11.3 43.008 485.99 148.49 0.056 0.13 8.27 4 3.1 14.4 43.008 619.32 189.22 0.069 0.21 13.14 5 3.1 17.5 43.008 752.64 229.96 0.080 0.28 18.49 6 3.1 20.6 43.008 885.96 270.70 0.1 0.34 23.96 Sum

258.048

3272.91

1.03

1000.00 T [s]

3.2.1

68.90 0.769

Obtaining displacement profiles

As the 3D model was given without the staircase alterations, the sections of the central pillars were modified, properly adjusting the beam elements connected to them by adding rigid offsets, so they do not overlap and cause irregularities on the model.

Figure 2: Section properties of the central pillars and Beam elements rigid offset settings

With the new modifications done, the next step was to extract both the X and Y frames off the 3D model in order to input the displacement profiles in the Rayleigh Method. For the Y-direction frame, this was done by deleting all the elements connected to the nodes located in the plane with coordinates X = 0 and X=10. Likewise, the X-direction frame was done by leaving only the elements within the plane Y=5.1.

By applying the Fi forces found in table 1 for each respective floor, the following displacements profile were found:

Page |6 3.2.2

X-Direction frame displacement X-Direction Displacement profile

Horizontal Displacement

20

15

10

5

0

0

0.025

0.05

0.075

0.1

Horizontal Displacement Linearized

Real

Figure 3: X-direction deformed frame and horizontal displacement profile in the model with shear walls

3.2.3

Y-Direction frame displacement

Vertical coordinate of the node

Y-Direction Displacement profile 20

15

10

5

0

0

0.025

0.05

0.075

Horizontal Displacement Linearized

Real

Figure 4: Y-direction deformed frame and horizontal displacement profile in the model with shear walls

0.1

Page |7

Y-Direction Displacement profile

21

21

16

16

Vertical coordinate of the node

Vertical coordinate of the node

X-Direction Displacement profile

11

6

1 0 -4

0.05

0.1

0.15

Horizontal Displacement no.walls

walls

0.2

11

6

1

0.25

0 -4

0.05

0.1

0.15

0.2

0.25

Horizontal Displacement no.walls

walls

Figure 5. Displacement profiles in the triangular-modal load pattern for the X and Y directions

By comparing both graphs in the previous page it can be said that the introduction of walls provides a more regular behavior to the building, since, with the introduction of the shear wall, the displacements in both dimensions are comparable. The inverse is also true: the differential displacements along the x and y directions without the wall may reveal an irregular behavior, which represents a vulnerability in the building.

Page |8 3.3

2D SEISMOSTRUCT EIGENVALUE ANALYSIS

In order to compare the calculation methods, we analyze the same frames with the Jacobi-Ritz transformation method used by the SeismoStruct software, analyzing the vibration and mass participation in the respective frames i.e. Frame X with mass participation along X,Z and RY directions, with 10 fundamental periods analyzed in the X direction.

It is important to notice that the mass found on the software did not correspond to the approximation used on the Rayleigh method calculation, so the masses were modified on the model to compare to provide the same hypothesis 1 as the other methods.

Y-direction

X-direction

Table 2. Alterations in the mass elements Original Model Mass Length ID [m] 101 4.65 102 5.35 sum

10

Mass ID 120 220 320 420

Length [m] 5.1 3.3 5.5 2.54

sum

10

Mass per length [t/m] 2.2700 2.6830

Mass per length [t/m] 2.6110 1.8780 2.1230 0.8200

Mass [t] 10.556 14.354

Modified Model Mass Length ID [m] 101 4.65 102 5.35

Mass [t] 19.999 23.009

24.910

sum

10

43.008

Mass [t] 13.316 6.197 11.677 2.083

Mass ID 120 220 320 420

Length [m] 5.1 3.3 5.5 2.54

Mass [t] 26.112 16.896 28.160 13.005

24.910

sum

16.44

84.173

Mass per length [t/m] 4.3008 4.3008

Mass per length [t/m] 5.1200 5.1200 5.1200 5.1200

Given this, the results for the first fundamental period and the respective mass participations:

Walls

Frame

Table 3: Mass check on the 2D analysis T0 [s]

Participation of the 1st period on the Horizontal displacement

Effective Cumulative mass on the 5th vibration period

X-direction

1.261

89.78%

258.05

258.05

OK

Y-direction

0.910

88.41%

505.03

505.04

OK

X-direction

0.771

84.26%

257.98

258.05

OK

Y-direction

0.832

87.48%

505.01

505.04

OK

1

floor.

Theoretical Calculation total mass check

The modified mass sum corresponds to the area of influence cause by the weight of the slabs on each

Page |9 3.4

3D SEISMOSTRUCT EIGENVALUE ANALYSIS

pillars.

The analysis was carried out by using the 3D model provided, with and without the L-profiles in the center 3.4.1

Analysis with L-walls

Figure 6: 1st, 2nd, 3rd and 4th, respectively, fundamental vibration modes with L-walls at the central columns.

Table 4: Effective Modal Mass percentages Mode 1 2 3 4 10 11

Period 0.834 0.803 0.699 0.242 0.102 0.087 Sum

Ux 73.15% 9.26% 2.86% 5.84% 0.07% 0.13%

Uy 1.48% 47.07% 39.62% 0.52% 0.01% 0.17%

91%

Uz 0.01% 0.00% 0.00% 0.06% 28.19% 35.94%

89%

Rx 0.10% 3.69% 3.10% 3.24% 2.53% 10.67%

64%

23%

Ry 9.27% 1.24% 0.38% 35.53% 7.03% 0.20% 54%

Rz 11.16% 31.38% 46.01% 1.36% 0.00% 0.00% 90%

From the results shown the movement of the structure in its main fundamental vibration modes is characterized by the torsion of the structure around the columns COL12, COL1 and COL3, respectively for the 1st and 4th, 2nd and 3rd vibration modes, caused by irregularities in the X-Y planes. The biggest fraction of the mass participation on the Z direction correspond from the 10th to 15th modes, which was predictable since it is a regular structure in height and so produces a stiffer response, that is, with lower periods according to the equation (III)

(III)

đ?‘‡đ?‘‡ = 2đ?œ‹đ?œ‹ďż˝

đ?‘šđ?‘š đ?‘˜đ?‘˜

Regarding the mass check, it was verified, since the theoretical mass (area of the slab*specific weight*number of storeys) of 924.72 t corresponds to the final cumulative mass participation of 918.78 t given by the analysis results. Further discussion must be made for the consideration of the influence area of the slab in each beam element, so the model correlates with the purposed model.

P a g e | 10 3.4.2

Analysis without L-walls

Figure 7: 1st, 2nd, 3rd and 4th, respectively, fundamental vibration modes without L-walls at the central columns.

Table 5: Effective Modal Mass percentages Mode 1 2 3 4 11 14

Period 1.097 0.826 0.724 0.323 0.109 0.077

Ux 84.30% 1.93% 2.55% 6.27% 0.00% 0.00%

Sum

Uy 0.03% 56.74% 31.80% 0.00% 0.00% 0.00%

95%

Uz 0.00% 0.00% 0.00% 0.02% 59.86% 21.86%

89%

Rx 0.00% 4.26% 2.35% 0.02% 0.00% 0.00%

82%

Ry 7.24% 0.30% 0.40% 53.02% 0.22% 3.34% 7%

Rz 4.88% 29.61% 54.46% 0.18% 0.00% 0.00%

65%

89%

The first fundamental mode in the model without the L-sectioned columns has a bigger period in respect to the previous model, meaning that the building has a more flexible behavior. This movement relates more to a single DOF oscillation since the torsion effect is less relevant. The mass check is also verified, where the cumulative mass participation is of 921.23 tons.

3.5

RESULTS By applying the methodology exposed on item 1.3, the following fundamental periods were found: Table 6: Results for X and Y direction frames Method T eurocode [s] T rayleigh,linear [s] T rayleigh, real [s] T 2D,Jacobi-Ritz [s] T 3D,Jacobi-Ritz [s]

Without Walls X-direction Y-direction 0.725 0.725 (-34%) (-34%) 1.193 0.852 (9%) (-22%) 1.281 0.906 (17%) (-17%) 1.261 0.91 (15%) (-17%)

With Walls X-direction Y-direction 0.725 0.725 (-13%) (-13%) 0.763 0.782 (-9%) (-6%) 0.785 0.827 (-6%) (-1%) 0.771 0.832 (-8%) (0%)

1.097

0.834

P a g e | 11 1.281

1.193

T eurocode [s]

1.097 0.910

0.906

0.852

0.725 0.725

1.261

T rayleigh,linear T rayleigh, real [s] T 2D,Jacobi-Ritz [s] [s] x-no.wall

Y-direction

T 3D,Jacobi-Ritz [s]

3D

Figure 8. Comparison of different calculation methods of the 1st fundamental period in the frame model

0.725

0.725

T eurocode [s]

0.763

0.785

0.782

T rayleigh,linear [s]

0.827

0.771

T rayleigh, real [s]

X-direction

Y-direction

0.832

T 2D,Jacobi-Ritz [s]

0.834

T 3D,Jacobi-Ritz [s]

3D

Figure 9: Comparison of different calculation methods of the 1st fundamental period in the model with shear walls

A summary of the results is provided, where it can be observed that the overall behavior of the building without the introduction of the shear walls is an intermediate situation among the x and y direction, while its insertion produces a more regular behavior in both directions.

P a g e | 12 1.200

Period [s]

1.000 0.800 0.600 0.400 0.200 0.000

1

2

3

4

10

11

14

Fundamental vibration modes

without L-sections

with L-sections

Figure 10: Periods for each fundamental vibration modes comparison with and without intervention.

Another important consideration is that the L-sectioned columns provide a stiffer behavior mainly in the 1 period, which is generally a good characteristic, but depends on the typology of the earthquake phenomenon the capacity of handling bigger stresses in the foundations and the behavior of the soil. st

3.6

CONCLUSIONS

The Rayleigh method was found to be adequate for calculating the period of vibration of 2D frames when put in comparison to numerical calculation methods such as the Jacobi-Ritz method. Although the first vibration period in the building is characterized by a major torsion movement on the z direction (due to geometrical irregularities in the building) and translation on the x direction, the 3D model numerical analysis did not converge to the model with the 2D analysis at the same direction. Although the fundamental period found in the model without the shear walls is between the both analyzed directions, the main movement (torsion) was not analyzed.

In any case, the analysis was beneficial to increase the awareness of such aspects in the earthquake engineering design.

Further analysis such as calculation of stresses and analysis of the structure must be made in order to define if the intervention is acceptable or not.

POLO TERRITORIALE DI LECCO

Dynamic Analysis in a building

Part 2: Calculating the design actions through response spectrum analysis

Buildings in Seismic Areas Prof. r. Paolucci

Master of Science in Architectural Engineering

A.Y. 2017/2018

Renan Prandini May, 2018

P a g e | 14

4 Objectives By referring to Italian National Technical Construction Code (NTC, 2008), our task is to carry out two different response spectrum analyses as specified below.

Case 1. Design spectrum corresponding to TR=50 years (probability of 63% of exceedance in 50 years (SLD), q=1).

Case 2. Design spectrum corresponding to TR=475 years (probability of 10% exceedance in 50 years (SLV), q=2.5) The location is Norcia, Perugia (Umbria), is a site class C (medium stiff soil) and topography class is T1 (flat).

Given these parameters and the geometrical model of the building, we need to find out appropriate design actions for each one of the cases, so the structural check can be further carried out and introduce new conceptual measures to mitigate the seismic effects on the building, if needed

5 Methodology The analysis will be carried out using the Design Response Spectrum according to the Italian National Technical Construction Code (NTC, 2008). In order to simplify the amount of calculations, and search for the available data near the site of the construction, the “Spettri-NTCver.1.0.3� calculation sheet was given as a material for the calculation of the Response Spectrum, according to the Italian standards, by the input of key parameters of the site and the building.

The Response Design Spectrum was put into SeismoStruct in both main horizontal directions of the building to carry out structural analysis of the components of the building. Only the most loaded elements such as base nodes and central columns were reported so the general behavior of the building could be observed and without extensive analysis of each individual element.

6 Design Response Spectrum 6.1

FASE 1: SEISMIC ACTIVITIES OF THE LOCATION

This fase corresponds to the geographical location on the site, where an extensive geographical database (provided by the Code) on seismic parameters is interpolated by Ruled Surface interpolation between the four nearest database neighbors. The nearest database neighbor of Norcia is within 1.25 km of the center of the city, which means it is a good interpolation. The following parameters were found in this interpolation:

P a g e | 15

Table 7. Parameters for the Response Design Spectra according to the NTC-08 TR [years] 30 50 72 101 140 201 475 975 2475

6.2

ag [g]

F0 [-]

Tc* [s]

0.078

2.388

0.272

0.103

2.322

0.279

0.122

2.294

0.286

0.142

2.282

0.292

0.162

2.296

0.298

0.186

2.323

0.312

0.255

2.376

0.335

0.327

2.408

0.353

0.447

2.458

0.373

FASE 2: DESIGN SERVICE LIFE

We are considering a normal building, of regular importance with no public functions and no dangerous content for the environment, so according to the standard, this building should provide a service life (and also a reference life) of at least 50 years. This gives us the following return periods and parameters: Table 8: Service Life in earthquake design Case 1 2

6.3

Design SLD SLV

TR [years]

PVR

ag [g]

F0 [-]

Tc* [s]

50

64%

0.103

2.322

0.279

475

10%

0.255

2.376

0.335

FASE 3: DESIGN ACTIONS

Considering a site class C (medium stiff soil), topography class is T1 (flat), 5% Damping coefficient (typical of concrete structures), q=1 for case 1 and q=2.5 for case 2, we can obtain the following Response Design Spectra.

0.9

SLD

0.8

SLV

0.7

Sa(g)

0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.5

1

1.5

2

T [s]

2.5

Figure 11. Response Design Spectrum

3

3.5

4

P a g e | 16 The construction of this graph is done according to the NTC-08, Eqs 3.2.5, 3.2.6, 3.2.3.5, 3.2.8, 3.2.7, 3.2.9, 3.2.4.

7 Seismostruct Analysis For each of the two Design Response Spectrum graphs, we import them into two different cases (two main directions, X and Y) and run a Response Spectrum Analysis with parameters compatible with the seismic regulations (Damping Ratio = 5% and Minimum Effective Modal Mass = 0.1%). The structural configuration, materials, masses and sections were refined previously on the first part of the assignment.

The same steps described are done to the model with and without the shear walls. A case summary is provided: Table 9. Simulation Cases Index CASE INDEX

SHEAR WALL

DIRECTION

DESIGN

WALL,SLD,X WALL,SLD,Y WALL,SLV,X WALL,SLV,Y NO.WALL,SLD,X NO.WALL,SLD,Y NO.WALL,SLV,X NO.WALL,SLV,Y

WALL WALL WALL WALL NO.WALL NO.WALL NO.WALL NO.WALL

X Y X Y X Y X Y

SLD SLD SLV SLV SLD SLD SLV SLV

8 Results The results of the simulations are presented as superposed Moments and Shear Graphs along the axis no.

8 which correspond to the central columns of the building. An overview of moment and shear diagrams are provided in order to identify the most critical columns and beams in it`s most relevant directions, that is, for a Response Spectrum excitation in the X directions, the most relevant moment direction is in My and the shear in the Vx direction. The most column is identified by a red rectangle and then showed in detail in succession.

A quantitative comparison of shear force (V)-height (z) and Moment (M)-height diagrams are presented for the support walls or columns on the central axis for each SLD, SLV analysis for the case with the presence or absence of shear walls, for each of the main (X and Y) directions.

The critical elements identified along the X and Y direction analysis were mainly the ones associated with the geometrical asymmetry. In the X direction, the most critical identified elements are the COL 10 and BEAM 202, although the beams seem to have a more distributed behavior in the model without the shear walls, with a maximum bending moment between 600 and 700 kNm. However, for the Y direction the simulation showed that the most loaded column was also affected by the irregularity in the X axis, producing bending moments of over 800kNm in COL11, although the order of magnitude didn’t change much for the beams.

The response behavior of the central columns (were shear walls will be located) is the same for different cases but with different order of magnitude. The introduction of the support walls results in a concentration of loads in these elements, which need to be carefully designed.

P a g e | 17 8.1

IDENTIFYING CRITICAL ELEMENTS

The identification of most critical elements was done by analyzing all the columns and beams separately, since they have different section properties. Although the introduction of the shear walls are supposed to handle a bigger load, a quantitative measure of acceptable limits for these elements have not been analyzed, so there is no criteria for defining the most critical elements but for the visual output from the main expected direction diagrams 2 in the software.

8.1.1

NO.WALL,SLV,X

COL 10

2

BEAM 202

i.e. for a X-direction ground motion the expected actions correspond to My and Vx.

P a g e | 18 8.1.2

NO.WALL,SLV,Y

COL 11

BEAM 301

P a g e | 19 8.1.3

WALL,SLD,X

BEAM101 M2

V3

P a g e | 20 COL 8 M2

8.1.4

WALL,SLD,Y

M3

V2

V3

P a g e | 21 Col 9, col 8

BEAM202

8.1.5

WALL,SLV,X

P a g e | 22 8.1.6

WALL,SLV,Y

It can be seen that the introduction of shear walls made a positive effect on the elements near to the asymmetry of the building, as the critical elements migrate from the irregularities on the building to the reinforced and more stiff elements, although it still produces significant effect on the column 11 (same magnitude of the shearwalls).

8.1.7

Action Diagrams

A series of action diagrams throughout the whole central axis of the building is provided. Significant increase of stresses is observed with the introduction of the support walls for both cases in both directions, but the horizontal displacements have shown to be smaller, which is intuitive, since we are introducing more stiff elements on the model.

P a g e | 23

20

18

18

18

18

16

16

16

16

14

14

14

14

12

12

12

12

10

10

Height [m]

20

Height [m]

20

Height [m]

20

10

10

0

0

0

0

1200

My [kN.m]

Mt [kN.m]

Figure 12. Main action diagrams in the central columns along the X-direction analysis

WALL,SLV

1

1

NO.WALL,SLV 8

6

WALL,SLD

4

2

0

20 18 16 14 12 10 8 6 4 2 0 NO.WALL,SLD

1000

Vx [kN]

800

600

400

200

0

Mx [kN.m]

4000

2

2000

2

0

2

6000

2

4000

4

2000

4

0

4

-2000

4

800

6

600

6

400

6

200

6

0

8

-200

8

-400

8

-600

8

H ei g ht [ m ]

Height [m]

8.1.7.1 X-Direction Analysis

P a g e | 24 NO.WALL,SLD,X NO.WALL,SLV,X WALL,SLD,X WALL,SLV,X

The main actions on this case occurs on the My and Vx, which were expected of a horizontal excitation in the x direction. The moment in Mx also presents considerable values that may not be neglected. In general, the actions tend to decrease in higher floors in a linear way, with the exception of the Vx diagram on the WALL,SLV direction case that is irregular in height. Another notable point is the My diagram in the ground floor column which possesses around 4000 kN.m on the base.

20

The Mt diagram is shown to an abnormal increase on torsion in the SLV case with shear walls, that may suggest errors in modeling or else a not foreseen motion of the building.

18 16

Height [m]

14 12 10 8 6 4 2 0 0.14

0.12

0.10

0.08

0.06

0.04

0.02

0.00

Displacements [m]

Figure 13. Horizontal Displacements along the X-Direction

P a g e | 25

Hei ght [m ]

8.1.7.2 Y-Direction Analysis 20 18 16 14 12 10 8 6 4 2 0

NO.WALL,SLD,Y 0

NO.WALL,SLv,Y 400

200

600

M [kN

]

WALL,SLD,Y 800

WALL,SLV,Y 1000

20

20

18

18

18

18

16

16

16

16

14

14

14

14

12

12

12

12

Height [m]

Height [m]

10

10

10

8 6 4 2 0

Mx [kN.m]

2000

Figure 14. Main action diagrams in the central columns along the Y-direction analysis

1000

My [kN.m]

0

1500

500

-500

-1500

Vx [kN]

10

-1000

0

600

0 500

0

400

2

300

2

200

2

100

4

0

4

600

4

500

6

400

6

300

6

200

8

100

8

0

8

Vy [kN]

Height [m]

20

Height [m]

20

1200

P a g e | 26 NO.WALL,SLD,Y

NO.WALL,SLV,Y

WALL,SLD,Y

WALL,SLV,Y

That happens due to the introduction of stiffer elements than the regular existing ones, which causes the concentration of loads in the shear wall.

20 18 16 14

Height [m]

In this case the expected output would be that predominant actions would occur in the Vy and Mx diagrams, however, the diagrams show significant participation of the My and Vx action forces throughout the whole height in the building with the introduction of the shear walls.

12 10 8 6 4 2 0 0.09

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0.00

Displacements [m]

Figure 15. Horizontal Displacements along the YDirection

P a g e | 27 8.1.8

Base Actions

The total base shear is given by the sum of applicant forces in all the base nodes, which presented an increase in both primary and secondary directions for all the cases, although for the Y direction analysis, this increase is not relevant. The most critical change occurs at the Base Bending moment when inserting the shear wall in the SLV design case, with an increase of +95.9% in respect to the model without supporting walls.

Base Bending Moment [kN.m]

Base Shear [kN] 4500 4000 3500 3000 2500 2000 1500 1000 500 0

14000 12000 10000 8000 6000 4000 2000 0

Primary direction Secondary direction Chart Title Figure 16. Summary of base actions along X and Y analysis

8.2

CONCLUSION

The insertion of shear walls on this model give a stiffer global behavior of the building, producing more intense action forces on the base and attributing a stiffer behavior of the building, that is, less plastic deformations.

However, this measure needs a careful design of the shear walls since a severe concentration of loads occurs towards this new element, which needs to be verified. This verification will occur in part 3 of the assignment.

POLO TERRITORIALE DI LECCO

Dynamic Analysis in a building

Part 3: Assessing the seismic performance of an existing building

Buildings in Seismic Areas Prof. r. Paolucci

Master of Science in Architectural Engineering

A.Y. 2017/2018

Renan Prandini May 2018

P a g e | 29

9 Objectives In order to assess the current performance of a given irregular structural model regarding seismic actions, a Static Pushover Analysis is carried out using SeismoStruct 2016 software according to the Italian Earthquake Code (NTC-08). Previous analysis have been carried out in the previous parts of the assignment, so few modelling parameters will be details such as the hypothesis assumed.

10 Methodology The analysis was carried out by SeismoStruct by defining criteria stablished on the NTC-08. The structural model was modified in the previous parts of the assignments. The report is organized by firstly introducing the assumptions made on the model such as material non-linearity on modelling, section properties and reinforcement patterns for each typology of section. The results of capacity and ideal curves, most critical structural elements, displacement profiles, main direction action diagrams and total base actions are provided and discussed. From the results reported, a comparison with the Response Spectrum Analysis is provided. Further analysis such as changing reinforcement parameters, natural periods according to the idealized curve and uncracked concrete modeling is carried out for a more detailed analysis of the building. A conclusion for the final assessment of the building is given, discussing the performance of the implementation of shear walls in the existing building.

11 Hypothesis 11.1

NON-LINEAR MATERIAL LAWS

11.1.1 Exisiting concrete: Mander et al. non linear concrete model • • • • •

Mean compressive strength: 25 MPa Mean Tensile Strength: 0 MPa Modulus of Elasticity (E) = 25 GPa Strain at peak stress: 0.2% Specific weight: 0 kN/m³ (Mass is introduced separately)

11.1.2 Steel: Bilinear Steel Model • • • • • • •

Simpler model, rough approximation of the non-linear behavior of steel.

E = 200 GPa YS = 200 MPa Strain Hardening Parameter: 0.005 Fracture/Buckling Strain: 0.1 Specific Weight = 0 kN/m³ Lower-bound strength value: 173913.043 kPa

P a g e | 30 11.1.3 New concrete: Mander et al. non linear concrete model • • • • •

Mean compressive strength: 30 MPa Mean Tensile Strength: 0 MPa Modulus of Elasticity (E) = 25 GPa Strain at peak stress: 0.2% Specific weight: 0 kN/m³ (Mass is introduced separately)

11.1.4 Steel: Bilinear Steel Model o Simpler model, rough approximation of the non-linear behavior of steel. o E = 200 GPa o YS = 500 MPa o Strain Hardening Parameter: 0.005 o Fracture/Buckling Strain: 0.1 o Specific Weight = 0 kN/m³ o Lower-bound strength value 3: 173913.043 kPa 11.2 STRUCTURAL ELEMENT CLASS TYPES AND REINFORCEMENT CONFIGURATIONS 11.2.1 Retangular RC Columns Height Width Cover thickness Longitudinal Reinforcment Pattern Transversal Reinforcement (perimetral) Non-confined (existing) concrete

70 cm 40 cm 2.5 cm 3@14mm in all sides 8mm at each 20 cm

11.2.2 Retangular RC Beams Height Width Cover thickness Longitudinal Reinforcement Pattern Transversal reinforcement Confined (existing) concrete

3

60 cm 40 cm 2.5 cm corners +2@12mm top and bottom +1@12mm left and right 8mm at each 20 cm 2 middle stirrups placed vertically

This parameter is possibly wrong, but since there was no fail involved in this material, plus the value is in favor of security.

P a g e | 31 11.2.3 L-Shaped Shear Walls Flange width Flange height Web width Web Height Web eccentricity Cover thickness Longitudinal Reinforcement Transversal Reinforcement Note: support walls are symmetrical

Confined (new) concrete

140 cm 25 cm 40 cm 45 0.1 cm 2.5 cm 10@14mm (corners) 8@14mm (top/bottom flange) 2@14mm (top/bottom web) 12mm every 10 cm 7 along height stirrups 3 along width stirrups

11.2.4 Mesh configurations

11.3 INCREMENTAL LOADING PATTERN (TRIANGULAR-MODAL) The incremental loading pattern used for the same triangular-modal pattern used in the Rayleigh calculation method, which corresponds to the mass of each slab multiplied by the height of the slab and divided by the total mass. The force is equally distributed through the nodes of the respective slab, paying special attention to the extreme nodes, which correspond to half the force. A scheme is presented for the calculation of the force in one slab: F/2 F

F

F/2

Where: • •

•

ďż˝ đ??šđ??šđ?‘–đ?‘– = đ?‘€đ?‘€đ?‘–đ?‘– . â„Žđ?‘–đ?‘–

∑ đ??šđ??šđ?‘–đ?‘– is the sum of the forces acting in the slab đ?‘€đ?‘€đ?‘–đ?‘– is the mass of the slab, given by the total area of the slab multiplied by 1.024 t/m² â„Žđ?‘–đ?‘– is the height of the storey

P a g e | 32 11.4 ELASTIC DESIGN SPECTRA By using the same parameters of the Part 2 we found out the same Elastic Design Spectra according to the Italian Earthquake Code for the Limit State of Damage Limitation (SLD), Limit State of Life Safety (SLV) and Limit State of Collapse Prevention (SLC) cases:

Figure 17. Elastic Design Spectra

These Limit States are defined as follows: Limit State of Damage Limitation (SLD): The building as a whole, including structural, non-structural elements and equipment, takes damage that does not compromise significantly the capacity and does not put users at risk, remaining immediately usable, with possible interruptions of the use of mechanical installations.

Limit State of Life Safety (SLV): The building suffers severe damage or collapse of the non-structural components and installations, and significant damage of the structural components. It sustains a significant loss of stiffness against horizontal actions but preserves a part of the lateral strength and stiffness.

Limit State of Collapse Prevention (SLC): The building suffers severe damage or collapse of the nonstructural components and installations, and very serious damage of the structural components. It still retains a margin of safety for actions loads, and a small margin of safety against collapse due to horizontal actions.

11.5 PERFORMANCE LIMIT VALUES CHORD ROTATION AND SHEAR FORCE CAPACITY The chord rotation limits are defined as following:

Although the Italian Earthquake Code recommend the chord rotation for the SLV limit state equal to 0.75θu, a more conservative value was adopted. The rest of the values were calculated according to the standard.

A brief explanation of the criteria calculation is given:

P a g e | 33

Where: θγαφωρν γel = 1.5 per gli elementi primari ed 1.0 per gli elementi secondari (come definiti al § 7.2.3 delle NTC); LV è la luce di taglio;

h è l’altezza della sezione; ν/(Ac fc) è lo sforzo assiale normalizzato di compressione agente su tutta la sezione Ac ω = As.fy /Ac.fc e ω′ = A′s.fy / Ac.fc percentuali meccaniche di armatura longitudinale in trazione e compressione rispettivamente (nelle pareti tutta l’armatura longitudinale d’anima è da includere nella percentuale in trazione); fc , fy e fyw sono la resistenza a compressione del calcestruzzo e la resistenza a snervamento dell’acciaio, longitudinale e trasversale, ottenute come media delle prove eseguite in sito, eventualmente corrette sulla base di fonti aggiuntive di informazione, divise per il fattore di confidenza appropriato in relazione al Livello di Conoscenza raggiunto; ρsx = Asx.bw.sh la percentuale di armatura trasversale (sh = interasse delle staffe nella zona critica);

ρd è la percentuale di eventuali armature diagonali in ciascuna direzione;

α è un fattore di efficienza del confinamento dato da:

(bo and ho are the dimensions of the confined core, bi distance between longitudinal rebars.

For walls, the value of the expression (C8A.6.1) should be divided by 1.6 according to the standards, but in this case, they are treated as columns.

For all the existing columns the effectiveness confinement factor is adopted as 0 as the current layout of the reinforcement is not known. Although the beams and L-shaped walls account for this confinement.

The calculation of the chord rotation limit on the collapse prevention state is given by:

(C8A.6.1)

P a g e | 34

12 Results 12.1 CAPACITANCE CURVES In order to analyze the performance of the building in each case according to the NTC-08, the idealized curve was put into comparison with the real capacity curve, to find out it the demands are met. The following table summarizes the performance results: 12.1.1 NO.WALL-X Capacity Curve

SLD

Idealized Curve

2000

SLV

SLC

20

1800

15

1400 1200

SLV

SLD

1000

957.07

922.45

800

947.70

Height [ m]

Base Shear [kN]

1600

SLC 861.90

10

600

5

400 200

0 0.08

0.1

0.12

0.14

0.16

0.18

Displacement [m]

0.20

0.06

0.15

0.04

0.10

0.02

0.05

0

0.00

0

Horizontal Displacement [m]

12.1.2 WALL-X Capacity Curve 2000

1865

SLV

20

SLC 1883

1824

1800

15

1600

SLD, 1550

1400

Height [ m]

Base Shear [kN]

Idealized Curve

1345

1200

10

1000 800

5

600 400 200 0

0 0

0.02

0.04

0.06

0.08

0.1

0.12

Displacement [m]

0.14

0.16

0.18

0

0.05

0.1

0.15

0.2

Horizontal Displacement [m] SLV

SLV

SLC

P a g e | 35 12.1.3 NO.WALL-Y Capacity Curve

Idealized Curve

20

2000 1800 1497

SLD

SLV

SLC

1446

1400 1200

15 1466

Height [ m]

Base Shear [kN]

1600

1206

1000 800

10

5

600 400

0

200 0

0.00

0.02

0.04

0.06

0.08

0.10

0.00 0.02 0.04 0.06 0.08

Horizontal Displacement [m]

Displacement [m]

SLD

SLV

SLC

12.1.4 WALL-Y

Capacity Curve

Idealized Curve

2000

SLV 1714

Base Shear [kN]

1600

SLC 1774 15

Height [ m]

1800

20

1700

1400 1200

SLD, 1142

1000

1130

800

10

5

600 400

0

200 0

0

0.02

0.04

0.06

Displacement [m]

0.08

0.1

0

0.02 0.04 0.06 0.08

Horizontal Displacement [m] SLD

SLV

SLC

P a g e | 36 12.2 FAILURE OF ELEMENTS A sheet with the failure of the elements found in all the analysis is given: Exceeded Criteria

SHEAR

Shear Walls?

NO

Load Factor

1.488

Static Pushover Direction

Y

Failed Element(s)

BEAM420

Exceeded Criteria

SLD

Shear Walls?

NO

Load Factor

1.470

Static Pushover Direction

Y

Failed Element(s)

COL1, COL 1004

Exceeded Criteria

SLD

Shear Walls?

YES

Load Factor

0.9356

Static Pushover Direction

X

Failed Element(s)

COL1002

P a g e | 37

Exceeded Criteria

SLD

Shear Walls?

YES

Load Factor

1.665

Static Pushover Direction

X

Failed Element(s)

COL1006, BEAM102, BEAM1102

Exceeded Criteria

SLD

Shear Walls?

YES

Load Factor

1.6333

Static Pushover Direction

Y

Failed Element(s)

COL1, COL2, COL3, COL4, COL1004

P a g e | 38

13 Post-analysis 13.1 TOTAL BASE ACTIONS

Primary Base Shear 2000 1800 1600 1400 1200 1000 800 600 400 200 0

Primary Base Bending Moment +100.0% +90.0% +80.0% +70.0% +60.0% +50.0% +40.0% +30.0% +19.1% +20.0% +10.0% +0.0%

+95.9%

+42.8%

+3.4%

8000 7000 6000 5000 4000 3000 2000 1000 0

+250.0%

+218.3%

+200.0% +150.0%

+134.2%

+100.0% +29.9% +0.4%

+50.0% +0.0%

Figure 18. Total base actions on the Static Pushover Analysis

It can be observed an increase of 95.9% on shear and 218.3% on bending moment regarding the model without the shear wall in the X-direction, SLV analysis. Secondary direction values were not relevant since they were much less than the main ones with exception to the NO.WALL-Y,SLV and WALL-Y,SLV cases, where they represented, respectively, 30% and 12% of the main value.

These results were according to what expected since the elements introduced increased the stiffness of the building, and so the reaction forces are bigger.

13.2 DISPLACEMENTS On both directions, the introduction of the walls represents a lower ductility of the structure.

Y-DIRECTION SPO ANALYSIS

20

20

15

15

Height [ m]

Height [m]

X-DIRECTION SPO ANALYSIS

10

5

0 0.000

10

5

0.025

0.050

0.075

0.100

0.125

0.150

0 0.000

0.025

Horizontal Displacement [m] WALL,SLV

0.050

0.075

0.100

0.125

Horizontal Displacement [m] NO.WALL,SLD WALL,SLV [m] Horizontal Displacement

Y DIRECTION SPO ANALYSIS

NO.WALL,SLV

0.150

P a g e | 39 13.3 COMPARISON: RESPONSE SPECTRUM ANALYSIS VS. STATIC PUSHOVER ANALYSIS By comparing both the Response Spectrum analysis made on Part 2 and the Static Pushover Analysis regarding their displacements profiles, main direction shear actions and bending moments diagrams, and the total base shear and base moment as well as displacement profiles, it was found that the RSA model produces more conservative results regarding the ductile performance of the structure.

13.3.1 Displacements

Y-DIRECTION ANALYSIS

20

20

15

15

Height [ m]

Height [ m]

X-DIRECTION ANALYSIS

10

5

10

5

0 0.000

0.025

0.050

0.075

0.100

0.125

0 0.000

0.150

0.025

Horizontal Displacement [m] RSA,NO.WALL,SLD RSA,WALL,SLD

0.050

0.075

0.100

0.125

Horizontal Displacement [m]

SPO,NO.WALL,SLD RSA,NO.WALL,SLV Horizontal Displacement [m] SPO,WALL,SLD RSA,WALL,SLV Y-DIRECTION ANALYSIS

SPO,NO.WALL,SLV SPO,WALL,SLV

20

15

15

Height [m]

20

10

10

500

RSA,SLV,WALL

5000

4000

3000

SPO,SLV,WALL

400

]]

My [kN.m] 300

[

200

RSA,SLV,NO.WALL 100

0

-20

-10

SPO,SLV,NO.WALL

2000

500

450

400

350

300

250

200

150

100

50

0

Vx [kN]

1000

0 0

0

-1000

5

-2000

5

H ei gh t […

Height [m]

13.3.2 Action Diagrams

20 15 10 5 0

0.150

P a g e | 40 13.3.3 Total Base Actions

Primary Base Bending Moment

Primary Base Shear 14000

4500 4000 3500 3000 2500 2000 1500 1000 500 0

12000 10000 8000 6000 4000 2000 0

SPO,X

RSA,X

SPO,Y

RSA,Y

13.4 WALL INTRODUCTION The following graph shows the performance of the structure with and without the shear wall along the x direction, that is, the most critical.

P a g e | 41 Capacity Curve,Wall

Idealized Curve,Wall

2000

Idealized Curve,No Wall

1865.20

1883.15

SLV

SLC

1824.38

1800 1600

SLD, 1549.78

1400

Base Shear [kN]

Capacity Curve,No Wall

1345.12

1200 1000

SLD

SLV

922.45

947.70

SLC

957.07

800

861.90

600 400 200 0

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Displacement [m]

In broad lines, it can be concluded that the introduction of the wall provides more stiffness and strength to the model while reducing its ductility, characteristics that are typical of the shear wall systems The action diagrams show that the strength at the base with the introduction of walls is highly increased (up to 300% in the SLV case), and may cause large deformations at the foundations, that need to be carefully studied. Furthermore, an unexpected high concentration of moments in the second story was found during the ydirection pushover analysis.

P a g e | 42

20

20

15

15

Height [m]

Height [m]

13.4.1 X-Direction

10

10

5

5

0

0

H e i g h

SLV,WALL

20

15

15

Height [m]

20

10

10

5

5

0

0

SLV,WALL 2

2

1

SLV,NO.WALL

600

400

200

0

-200

-400

-600

Vy [kN] 1

5

SLD,WALL

0

-5

20 15 10 5 0 SLD,NO.WALL

-800

-1000

350

300

250

200

150

100

50

0

Vy [kN] H e ig h t…

Height [m]

13.4.2 Y-Direction

2500

SLV,NO.WALL

2000

My [kNm]

1500

1000

500

SLD,WALL

0

20 15 10 5 0 SLD,NO.WALL

-500

-1000

500

450

400

350

300

250

200

150

100

50

0

Vx [kN]

P a g e | 43

14 Additional Studies 14.1 EIGENVALUE ANALYSIS COMPARISON

1st Fundamental Period [s]

SPO,X

Rayleigh,X

SPO,Y

Rayleigh,Y

Eigenvalue

1.097

1.080 0.910

0.739 0.770

0.832 0.834 0.809 0.771 0.741

Without Walls

By calculating the vibration periods according to the idealized pushover curves a good correlation with the eigenvalue analysis was found for the X direction, that is, the most critical.

With Walls

14.2 TRANSVERSAL REINFORCEMENT By performing an analysis with less stirrup reinforcement (φ8mm every 20cm) on the L-shaped walls, it was found that the shear limit criteria was reached on the base of the L-Shaped column (COL5) with a loading factor of 1.42, before the SLD criteria (1.67). The shear limit in this case was of 430 kN. In comparison to the model with a stirrup configuration of φ12mm every 10cm, the shear limit of 1930 kN was never reached, although the maximum demand found for this same element is 614 kN, that is, it is super dimensioned. Nevertheless, in practical application the stirrups vary throughout the span of the columns/beams so structural material is not wasted.

The shear failure of elements must be avoided in any case, since it is characterized by a brittle failure, which is undesirable for safety purposes and may cause sudden global collapse of the structure.

15 Conclusion An extensive seismic analysis was carried out in the purposed building. The effects of the introduction of shear walls have been analyzed, reducing the seismic vulnerability of the object in study. A comparison between the Response Spectrum and the Static Pushover methods have been given in order to perform a double check towards the seismic actions design, where the results have shown to be coherent. The comparison with the standards has also shown to be beneficial to raise the awareness of simplified approaches.

The introduction of walls has shown to provide a stiffer behavior with less ductility of the building regarding the current state, that would be enough for satisfying the SLV condition.

The criteria for defining the most critical beams on the building with the assumptions made was well performed during the SPO analysis but no structural check was carried out during the RSA analysis.

16 Bibliography Norme Tecniche per le Costruzioni DM 14 gennaio 2008 + circolari esplicative Lessons of Buildings in Seismic Areas by Prof. R. Paolucci & Tutoring by A. Guiney Ozcebe. Politecnico di Milano, Polo territoriale di Lecco, A.Y. 2017/18