Structural Control

August 10, 2017 | Author: Chalo Roberts | Category: Strength Of Materials, Beam (Structure), Bending, Plasticity (Physics), Column
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CONTENT 

CHAPTER 1 – PROJECT DESCRIPTION ..................................................................................... 1  1 

Introduction ........................................................................................................................... 1  1.1 

Description of the Building Structure ............................................................................ 1 

1.2 

Modeling Assumptions .................................................................................................. 3 

1.3 

Member Properties ......................................................................................................... 4 

1.4 

Curvature and ductility capacity .................................................................................... 7 

1.5 

Dynamic characteristics of the original structure .......................................................... 9 

1.6 

Pushover Analyses ......................................................................................................... 2 

CHAPTER 2 – DESIGN GROUND MOTIONS ........................................................................... 10  1 

Retrieval and analysis of Design Ground motions .............................................................. 10 



Response Spectra................................................................................................................. 11 

CHAPTER 3 - ANALYSIS OF THE ORIGINAL BUILDING .................................................... 13  1 

Introduction ......................................................................................................................... 13 



Performance of the existing structure.................................................................................. 13  2.1 

Energy balance ............................................................................................................. 13 

2.2 

Plastic Hinging Distribution ........................................................................................ 16 

2.3 

Inter-story peak and residual drifts ................................................................................ 4 

2.4 

Peak Acceleration .......................................................................................................... 7 

2.5 

Performance evaluation ............................................................................................... 10 

CHAPTER 4 - HYSTERETIC DAMPERS ................................................................................... 14  1 

Description .......................................................................................................................... 14 



Procedure to calculate the optimum activation load ........................................................... 15 



Fourier Spectra .................................................................................................................... 16 



Preliminary design............................................................................................................... 17 



Intermediate design ............................................................................................................. 20 



Final design ......................................................................................................................... 28  6.1 

Energy Balance ............................................................................................................ 29 

6.2 

Plastic hinging distribution .......................................................................................... 30 

6.3 

Peak and Residual Inter-Story Drifts ........................................................................... 32  i

STRUCTURAL CONTROL PROJECT

6.4  7 

TEAM 4

Accelerations................................................................................................................ 34 

Flow Chart for Hysteretic dampers optimum design .......................................................... 37 

CHAPTER 5 - VISCOUS DAMPERS .......................................................................................... 38  1 

Description .......................................................................................................................... 38 



Procedures to calculate the damping coefficients ............................................................... 39 



Modeling of dampers .......................................................................................................... 40 



Validation of the Damper element ...................................................................................... 42 



Preliminary design............................................................................................................... 44  5.1 

Stiffness proportional approach ................................................................................... 48 

5.2 

Constant damping approach ......................................................................................... 49 

5.3 

First mode proportional damping................................................................................. 50 



Intermediate design ............................................................................................................. 51 



Final Design ........................................................................................................................ 54 



7.1 

Energy Balance ............................................................................................................ 54 

7.2 

Hinge Distribution ....................................................................................................... 55 

7.3 

Peak and Residual Inter-Story Drifts ........................................................................... 58 

7.4 

Accelerations................................................................................................................ 60 

Flow chart for viscous dampers optimum design................................................................ 63 

CHAPTER 6 - BASE ISOLATION ............................................................................................... 64  1 

Description .......................................................................................................................... 64 



Preliminary Design .............................................................................................................. 68 



Intermediate design ............................................................................................................. 71 



Final Design ........................................................................................................................ 76 



Flow chart for optimum design of base isolation ................................................................ 86 

CHAPTER 7 - OPTIMUM DESIGN and NEAR-FAULT GROUND MOTION performance.... 87  1 

Optimum retrofit strategy .................................................................................................... 87 



Performance under near-fault ground motion ..................................................................... 88  2.1 

Near-Fault Ground Motion .......................................................................................... 88 

2.2 

Assessment of the existing structure under near fault ground motion ......................... 89 

2.3 

Retrofitted building performance under near fault ground motion .............................. 92 

APPENDIX A – RESULTS ANALYSIS WITH VBA SCRIPT ................................................... 97  APPENDIX B – COMPOSITE SECTION .................................................................................... 98  APPENDIX C: PEER REVIEW LETTERS .................................................................................. 99  REFERENCES ............................................................................................................................. 101  ii

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List of Tables Table 1: Design gravity loads _____________________________________________________________________ 3  Table 2: Material properties ______________________________________________________________________ 5  Table 3: Geometric and Elastic Member Properties  ___________________________________________________ 6  Table 4: Description of the frame members __________________________________________________________ 6  Table 5: Column axial load – moment interaction _____________________________________________________ 7  Table 6: Plastic Curvature of each element for a plastic Rotation limit   __________________________________ 8  Table 7: Curvature ductility capacity at failure  _______________________________________________________ 9  Table 8: Frequencies and periods __________________________________________________________________ 9  Table 9: Mass participation ratios _________________________________________________________________ 9  Table 10: Lateral Load Distribution, ASCE 41 _________________________________________________________ 3  Table 11: Lateral Load Distribution, Linear vertical ____________________________________________________ 3  Table 12: Lateral Load Distribution, New Zealand Code ________________________________________________ 3  Table 13: Fraction of Input Energy Absorbed. _______________________________________________________ 15  Table 14: Peak Absorbed Energy. _________________________________________________________________ 16  Table 15: Energy Balance Error. __________________________________________________________________ 16  Table 16: Maximum plastic rotations for LA‐02 ground motion __________________________________________ 3  Table 17: Maximum plastic rotations for LA‐07 ground motion.  _________________________________________ 4  Table 18: Maximum plastic rotations for LA‐16 ground motion.  _________________________________________ 3  Table 19: Reponse limits for different performance category ___________________________________________ 12  Table 20: Performance Indexes for design ground motions  ____________________________________________ 13  Table 21: Parameters __________________________________________________________________________ 19  Table 22: Parameters __________________________________________________________________________ 22  Table 23: Maximum plastic rotations for LA‐02 ground motion.  ________________________________________ 31  Table 24: Maximum plastic rotations for LA‐16 ground motion.  ________________________________________ 31  Table 25: Performance Indexes for structure retrofitted with hysteretic dampers compared to the original  performance.  ________________________________________________________________________________ 36  Table 26: Validation of damper element ___________________________________________________________ 43  Table 27: Summary of story stiffness ______________________________________________________________ 47  Table 28: Stiffness proportional Approach damping coefficients  ________________________________________ 51  Table 29: Constant Damping Approach damping coefficients  __________________________________________ 51  Table 30: First Mode proportional Approach damping coefficients  ______________________________________ 51  Table 31: Maximum plastic rotations for LA‐02 ground motion.  ________________________________________ 56  Table 32: Maximum plastic rotations for LA‐07 ground motion _________________________________________ 56  Table 33: Maximum plastic rotations for LA‐16 ground motion _________________________________________ 57  Table 34: Performance Indices for structure retrofitted with viscous dampers compared to existing building  performance.  ________________________________________________________________________________ 62  Table 35: Preliminary Design results  ______________________________________________________________ 70  Table 36: Summary of parameters to be studied in intermediate design __________________________________ 71  Table 37: Summary of Design Parameters __________________________________________________________ 75  Table 38: Performance Indexes for structure retrofitted with base isolation compared with existing performance.  85  Table 39: Summary of various retrofit options  ______________________________________________________ 87  Table 40: Performance level category _____________________________________________________________ 88  Table 41: Maximum plastic rotations for Near Fault ground motion in existing structure. ____________________ 90  Table 42: Performance Indexes of existing structure for near fault ground motion.  _________________________ 92  Table 43: Performance of existing and retrofitted structure for the near fault ground motion _________________ 96 

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List of figures Figure 1: Plan view, building to be retrofitted ________________________________________________________ 2  Figure 2: Elevation view Axis A – E.  ________________________________________________________________ 2  Figure 3: Bi‐Linear Moment‐Curvature Model.  _______________________________________________________ 3  Figure 4: Strength Degradation Model for Welded Beam‐Column Connections. _____________________________ 4  Figure 5: Elevation view with position of all nodes and members. ________________________________________ 5  Figure 6: Axial Load – Bending Moment Interaction Diagram  ___________________________________________ 7  Figure 7: Mode Shapes of the structure _____________________________________________________________ 2  Figure 8: Pushovers curves _______________________________________________________________________ 4  Figure 9: Top floor lateral displacement vs. time ______________________________________________________ 4  Figure 10: Deflected Shape ASCE 41 Pushover, Plastic Hinge Location _____________________________________ 5  Figure 11: Pushover Curve ASCE 41 ________________________________________________________________ 5  Figure 12: Base Shear vs. Time, ASCE 41. ____________________________________________________________ 6  Figure 13: Moment vs. Time, beam member 58 end 2  _________________________________________________ 6  Figure 14: Moment vs. time – Column member 23 ____________________________________________________ 7  Figure 15: Moment vs. Curvature, Member 58 end 2  __________________________________________________ 7  Figure 16: 1st Floor Beam Failure at end 1  __________________________________________________________ 8  Figure 17: 1st Floor Beam Failure at end 2  __________________________________________________________ 8  Figure 18: Bottom Storey Columns Failure at end 1  ___________________________________________________ 8  Figure 19: Bottom Storey Columns Failure at end 2  ___________________________________________________ 9  Figure 20: LA‐02 Ground Motion _________________________________________________________________ 10  Figure 21: LA‐07 Ground Motion _________________________________________________________________ 10  Figure 22: LA‐16 Ground Motion _________________________________________________________________ 11  Figure 23: Absolute Acceleration Response Spectra for 5% Damping _____________________________________ 11  Figure 24: Relative Velocity Response Spectrum _____________________________________________________ 12  Figure 25: Relative Displacement Response Spectrum  ________________________________________________ 12  Figure 26: Energy Components LA‐02  _____________________________________________________________ 14  Figure 27: Energy Components LA‐07  _____________________________________________________________ 14  Figure 28: Energy Components LA‐16. _____________________________________________________________ 14  Figure 29: Distribution of Plastic Hinges for LA‐02  ___________________________________________________ 17  Figure 30: Distribution of Plastic Hinges for LA‐07  ___________________________________________________ 17  Figure 31: Distribution of Plastic Hinges for LA‐16  ___________________________________________________ 17  Figure 32: Inter‐story Drift – Time History LA‐02 ______________________________________________________ 4  Figure 33: Inter‐story Drift – Time History LA‐07 ______________________________________________________ 5  Figure 34: Inter‐story drift – Time History LA‐16 ______________________________________________________ 5  Figure 35: Peak inter‐story drifts for LA‐02, LA‐07 and LA‐16.  ___________________________________________ 6  Figure 36: Peak inter‐story drifts. __________________________________________________________________ 6  Figure 37: Residual inter‐story drifts for LA‐02, LA‐07 and LA‐16 _________________________________________ 7  Figure 38: Residual inter‐story drifts for LA‐02, LA‐07 and LA‐16 _________________________________________ 7  Figure 39: Acceleration time ‐ history for LA‐02  ______________________________________________________ 8  Figure 40: Acceleration time ‐ history for LA‐07  ______________________________________________________ 8  Figure 41: Acceleration time ‐ history for LA‐16  ______________________________________________________ 9  Figure 42: Peak acceleration for LA‐02, LA‐07 and LA‐16 _______________________________________________ 9  Figure 43: Peak total accelerations  _______________________________________________________________ 10  Figure 44: Performance Levels (FEMA 273) _________________________________________________________ 11  Figure 45: Locations of added bracing and hysteretic dampers (Configuration‐C1) __________________________ 14  Figure 46: Elasto‐Plastic Hysteresis _______________________________________________________________ 15  Figure 47: Fourier Spectra  ______________________________________________________________________ 16  iv

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Figure 48: Preliminary design ____________________________________________________________________ 20  Figure 49: Sections ____________________________________________________________________________ 21  Figure 50: Alternative retrofit scheme considered in the analyses (Configuration‐C2) ________________________ 21  Figure 51: Optimum size study ___________________________________________________________________ 22  Figure 52: Optimum size study ___________________________________________________________________ 23  Figure 53: Optimum activation load study for HSS406‐C1  _____________________________________________ 24  Figure 54: Optimum activation load study for HSS304‐C1  _____________________________________________ 24  Figure 55: Optimum activation load study for HSS406&304‐C1 _________________________________________ 25  Figure 56: Optimum Design _____________________________________________________________________ 25  Figure 57: Optimum activation load study for HSS406&304C2 (1/1)  _____________________________________ 26  Figure 58: Optimum activation load study for HSS406&304C2 (2/1)  _____________________________________ 26  Figure 59: Optimum activation load study for HSS406&304C2 (3/1)  _____________________________________ 27  Figure 60: Optimum activation load study for HSS406&304C2 (4/1)  _____________________________________ 27  Figure 61: Optimum activation load study for HSS406&304C2 (1st Mode proportional)  ______________________ 28  Figure 62: Optimum Design _____________________________________________________________________ 28  Figure 63: Energy Components LA‐02  _____________________________________________________________ 29  Figure 64: Energy Components LA‐07  _____________________________________________________________ 29  Figure 65: Energy Components LA‐16  _____________________________________________________________ 30  Figure 66: Distribution of plastic hinges for LA‐02 and LA‐16.  __________________________________________ 30  Figure 67: Inter‐story drift time history motion La‐02.  ________________________________________________ 32  Figure 68: Inter‐story drift time history motion La‐07.  ________________________________________________ 33  Figure 69: Inter‐story drift time history motion La‐16.  ________________________________________________ 33  Figure 70: Peak inter‐story drifts for LA‐02, 07 and 16  ________________________________________________ 33  Figure 71: Comparison of peak inter‐story drifts _____________________________________________________ 33  Figure 72: Residual inter‐story drifts  ______________________________________________________________ 34  Figure 73: Comparison of residual inter‐story drifts  __________________________________________________ 34  Figure 74: Acceleration history of motion LA‐02. _____________________________________________________ 34  Figure 75: Acceleration history of motion LA‐07. _____________________________________________________ 35  Figure 76: Acceleration history of motion LA‐16. _____________________________________________________ 35  Figure 77: Comparison of total peak accelerations.  __________________________________________________ 36  Figure 78: Flow Chart for hysteretic dampers optimum design __________________________________________ 37  Figure 79: Location of added bracing and viscous dampers ____________________________________________ 38  Figure 80: Hysteretic Behavior of Viscous Dampers  __________________________________________________ 39  Figure 81: Plot showing comparison among viscous damping and Rayleigh damping ________________________ 41  Figure 82: Model View _________________________________________________________________________ 42  Figure 83: Displacement time history  _____________________________________________________________ 42  Figure 84: Spring and viscous damper forces ________________________________________________________ 43  Figure 85: Spring and viscous damping force  _______________________________________________________ 43  Figure 86: Spectra accelerations for LA2 under different damping ratios __________________________________ 44  Figure 87: Spectra accelerations for LA7 under different damping ratios __________________________________ 45  Figure 88: Spectra accelerations for LA16 under different damping ratios _________________________________ 45  Figure 89: Spectra displacements for LA2 under different damping ratios _________________________________ 46  Figure 90: Spectra displacements for LA7 under different damping ratios _________________________________ 46  Figure 91: Spectra displacements for LA16 under different damping ratios ________________________________ 47  Figure 92: Optimum damping comparison (Stiffness Approach) _________________________________________ 52  Figure 93: Optimum damping comparison (Constant damping Approach)  ________________________________ 52  Figure 94: Optimum damping comparison (First Mode proportional Approach) ____________________________ 53  Figure 95: Optimum damping approach  ___________________________________________________________ 53  Figure 96: Energy Components LA‐02. _____________________________________________________________ 54  Figure 97: Energy Components LA‐07. _____________________________________________________________ 54  Figure 98: Energy Components LA‐16  _____________________________________________________________ 55  Figure 99: Distribution of plastic hinges for LA‐02. ___________________________________________________ 55  Figure 100: Distribution of plastic hinges for LA‐07 and LA16. __________________________________________ 56  v

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Figure 101: Inter‐story drift time history motion La‐02.  _______________________________________________ 58  Figure 102: Inter‐story drift time history motion La‐07.  _______________________________________________ 58  Figure 103: Inter‐story drift time history motion La‐16 ________________________________________________ 59  Figure 104: Peak inter‐story drifts for LA‐02, 07 and 16 _______________________________________________ 59  Figure 105: Comparison of peak inter‐story drifts ____________________________________________________ 59  Figure 106: Residual inter‐story drifts for LA‐02, 07 and 16. ____________________________________________ 60  Figure 107: Comparison of residual inter‐story drifts. _________________________________________________ 60  Figure 108: Acceleration history of motion LA‐02. ____________________________________________________ 60  Figure 109: Acceleration history of motion LA‐07. ____________________________________________________ 61  Figure 110: Acceleration history of motion LA‐16. ____________________________________________________ 61  Figure 111: Comparison of total peak accelerations.  _________________________________________________ 62  Figure 112: Flow chart for optimum design for viscous dampers ________________________________________ 63  Figure 113: Modelling of Building Structure with Lead‐Rubber Base‐Isolation System  _______________________ 64  Figure 114: Components of Lead‐Rubber base isolation _______________________________________________ 65  Figure 115: Lead‐Rubber Bi‐Linear Model __________________________________________________________ 65  Figure 116: Spectral Displacement corresponding to effective period of the equivalent system ________________ 70  Figure 117: Optimum Fy study for k1=30kN/mm  ____________________________________________________ 72  Figure 118: Optimum Fy study for k1=45kN/mm  ____________________________________________________ 72  Figure 119: Optimum Fy study for k1=65kN/mm  ____________________________________________________ 73  Figure 120: Optimum Design ____________________________________________________________________ 73  Figure 121: Energy components time history for LA‐02. _______________________________________________ 76  Figure 122: Energy components time history for LA‐07. _______________________________________________ 76  Figure 123: Energy components time history for LA‐16. _______________________________________________ 77  Figure 124: Abscense of plastic hinges for LA‐02, 07 and 16. ___________________________________________ 77  Figure 125: : Interstory drift time history for LA‐02.  __________________________________________________ 78  Figure 126: Displacement time history for Bearings in Base isolation system LA‐02. _________________________ 78  Figure 127: Interstory drift time history for LA‐07.  ___________________________________________________ 79  Figure 128: Displacement time history for Bearings in Base isolation system LA‐07. _________________________ 79  Figure 129: Interstory drift time history for LA‐16.  ___________________________________________________ 80  Figure 130: Displacement time history for Bearings in Base isolation system LA‐16. _________________________ 80  Figure 131: Peak Inter‐storey drifts for Retrofitted structure. ___________________________________________ 81  Figure 132: Comparison of Peak inter‐storey drifts.  __________________________________________________ 81  Figure 133: Residual Inter‐storey drifts for Retrofitted structure. ________________________________________ 81  Figure 134: Comparison of residual inter‐storey drifts. ________________________________________________ 81  Figure 135: Acceleration time history for LA‐02. _____________________________________________________ 82  Figure 136: Acceleration time history for bearings in base isolation LA‐02. ________________________________ 82  Figure 137: Acceleration time history for LA‐07. _____________________________________________________ 83  Figure 138: Acceleration time history for bearings in base isolation LA‐07. ________________________________ 83  Figure 139: Acceleration time history for LA‐16. _____________________________________________________ 84  Figure 140: Acceleration time history for bearings in base isolation LA‐16. ________________________________ 84  Figure 141: Comparison of peak accelerations. ______________________________________________________ 85  Figure 142: Flow chart for optimum design of base isolation systems ____________________________________ 86  Figure 143: Near fault ground motion horizontal component___________________________________________ 88  Figure 144: Energy components time history for Near Fault Ground motion.  ______________________________ 89  Figure 145: Distribution of plastic hinges for Existing Structure. _________________________________________ 89  Figure 146: Inter storey ‐ drifts time history for Near Fault ground motion. ________________________________ 91  Figure 147: Acceleration time history for Near Fault ground motion. _____________________________________ 91  Figure 148: Energy components time history for Retrofitted structure. ___________________________________ 92  Figure 149: Inter storey ‐ drifts time history for retrofitted structure.  ____________________________________ 93  Figure 150: Displacement time history for Bearings in Base isolation system. ______________________________ 93  Figure 151: Acceleration time history for retrofitted structure.  _________________________________________ 94  Figure 152: Acceleration time history for bearings in base isolation. _____________________________________ 94  Figure 153: Comparison of peak inter‐storey drifts.  __________________________________________________ 95  vi

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Figure 154: Comparison of residual inter‐storey drifts. ________________________________________________ 95  Figure 155: Comparison of peak accelerations. ______________________________________________________ 95  Figure 156: Performance of the existing building compared to the optimum retrofit strategy _________________ 96  Figure 157: Details for the composite section _______________________________________________________ 98 

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CHAPTER 1 – PROJECT DESCRIPTION

1

Introduction

In recent years, the necessity to raise the structural performance of existing seismic-deficient structures under earthquake events has led to a better understanding and implementation of structural retrofit. In most cases life-safety and the financial savings could be achieved after retrofitting an existing structure. As a consequence, it is of vital importance to convince the building’s owners to have their buildings evaluated by a structural engineer who could assess the retrofitting necessity. Several devices with inelastic behavior have been introduced in order to protect structures against dynamics effects. These devices reduce the displacement demand over the structure through their capacity to venture into the plastic range. Additionally, devices to isolate the structure from the ground motions have been used as well. The objective of this work is to assess the seismic performance of the building studied by Tsai and Popov (1988) and retrofit it utilizing different devices. 1.1

Description of the Building Structure

The building is a six-storey steel structure with rectangular configuration in plan and in elevation (Figure 1). Structured employing W steel shapes and shear connection in all axes except in the moment frames in grids A and E. The building is located in a seismic Zone 4 with soil type S2 and was designed according to the 1994 UBC code requirements. The overall building floor area is approximately 4,816 m2 (including the ground floor) and a roof area of approximately 803 m2. With an inter-storey height of 3.810 meters except for the ground level 5.486 meters. The main seismic resisting systems in north-south direction are steel moment frames in grids A and E over the building height (Figure 2). The retrofitting strategies will be implemented in these moment frames. Different dissipation devices configuration will be assessed to determine the most efficient retrofitting solution in this structure.

1

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TEAM 4 NO RTH

A

B

C

D

E 4

7.315

3

7.315

2

7.315

1 9.144

9.144

9.144

9.144

36.576

Figure 1: Plan view, building to be retrofitted 1

2

3

4

W 24 x 104 W 24 x 104 W 27 x 146 W 27 x 146

5.486

W 30 x 173

3.810

W 30 x 99

W 30 x 99

W 30 x 173

3.810

W 14 x 109

24.536

W 14 x 109

3.810

W 14 x 159

3.810

W 14 x 159

3.810

W 14 x 193

W 24 x 76

W 14 x 193

21.945

W 24 x 76

W 27 x 94

W 27 x 94

Gravity Columns

Figure 2: Elevation view Axis A – E. 2

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1.2

TEAM 4

Modeling Assumptions

The building was designed with the 1994 UBC code requirements. The design gravity loads are presented in Table 1: Design gravity loads, wind loads are based on the wind speed of 113 km/h and an exposure type B. Table 1: Design gravity loads Dead Load Roof 3.8 kPA Floor 4.5 kPA Exterior Cladding 1.7 kPA

Live Load Roof Floor

1.0 kPA 3.8 kPA

All seismic/dynamic analyses are performed using the nonlinear dynamic analysis computer program RUAUMOKO (Carr 1998). One moment frame was modeled by 2D model due to the symmetry in the structure and it will resist half of the lateral load applied to the building in the north-south direction. The model includes an exterior moment-resisting frame with one gravity column which supports the total gravity loads acting on the interior columns to avoid the additional P-delta effect on the moment frame columns. At each floor, the frame is constrained to experience the same lateral deformation. The columns are fixed at the ground level, except the gravity column that is assumed pinned at the base and at each level. The slab participation as a composite beam is not included. The inelastic response is concentrated in plastic hinges that could form at both ends of the frame members. These plastic hinges are assigned a bi-linear hysteretic behavior with a curvature strain-hardening ratio of 0.02 (Figure 3), and their length is set equal to 90% of the associated member depth. The plastic resistance at the hinges is based on expected yield strength of 290 MPa. Bilinear Moment Curvature Model

M 9

r=2%

1.2Mp

0.02EI 1

Bending Moment

Mp

p

4.5

EI

(p=0.03 rad)

1

0 0

y

ult

9



Curvature

Figure 3: Bi-Linear Moment-Curvature Model. An axial load-moment interaction, as per LRFD 1993 (AISC 1993), is considered for the columns of the structure. Rigid-end offsets are specified at the end of the frame members to account for the actual size of the members at the joints. The panel zones of the beam-column connections are 3

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assumed to be stiff and strong enough to avoid any panel shear deformation and yielding under strong earthquakes. All hysteretic energy must be dissipated through plastic hinging in the beams and the columns. Gravity loads acting on the frame during the earthquake are the roof and floor dead loads, the weight of the exterior walls, and a portion of the floor live load (0.7 kPa). P-delta effects are accounted for in the analyses. Rayleigh damping of 5% based on the first two elastic modes of vibration of the structure is assigned. All analyses are performed at a time-step increment of 0.002 s. To capture the brittle failure of the welded beam-to-column connections, the flexural strength degradation model shown in Figure 4, is introduced at the ends of the beam and column elements. The strength degradation begins at a curvature ductility of 11.0. At a curvature ductility of 11.55, the strength reduces 1% of the yield moment. Strength Degradation Model

Multiplier on Yield Moment

1

0.8

0.6

0.4

0.2

0.01 0

0

5

10

11 11.55

15

Curvature Ductility

Figure 4: Strength Degradation Model for Welded Beam-Column Connections. 1.3

Member Properties

For the moment-resisting frame, the section properties are identical for both grid A and E. The two dimensional model contains 66 members (60 for frame and 6 for gravity columns) and 53 joints listed as shown in Figure 5: Elevation view with position of all nodes and members..

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STRUCTURAL CONTROL PROJECT 25

1

2

TEAM 4 27

26

4

29

28

1

10

33

32

12

35

39

20

41

10

43

25

45

28

46

42

12

47

30

48

24

63

31

32

64

16

14

16

62

23

29

27

13

22

11

44

26

15

21

8

61

8

40

19

9

36

7

38

18

14

13

6

37

17

7

4

34

11

5

30

3

2 31

9

6

5

3

15 49

33

34

51

50

36

53

52

35

17

54

18

55

21

45

44

43

56

57

46

58

59

48

66

24

51

50

40

65

47

60

23

22

49

39

20

19

42

41

38

37

52

53

Figure 5: Elevation view with position of all nodes and members. 1.3.1

Material Properties

The building structure was built with mild steel grade A36 for all members and the basic elastic properties for this material are defined in Table 2. Table 2: Material properties Modulus of Elasticity

200

Shear Modulus Yield Stress

G σ

.

77 GPa 290 MPa.

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TEAM 4

Geometric and Elastic Member Properties

The model includes 24 different member sections in order to represent the columns and beams in the frame. Each of this section has the properties defined in Table 3. Table 3: Geometric and Elastic Member Properties Member Type

Section

1, 2 3,4

W14x109 W24x104

5,6

W14x159

7,8

W27x146

9, 10

W14x193

11, 12

W30x173

13 - 16 17 - 20 21 - 24

W24x76 W27x94 W30x99

Member No. 1, 4, 5, 8 2, 3, 6, 7 9, 12, 13, 16 10, 11, 14, 15 17, 20, 21, 24 18, 19, 22, 23 25 - 36 37 - 48 49 - 60

lp

D

A

I

My

Ny

(mm)

(mm)

(mm2)

(mm4)

(KN-mm)

(KN)

328 550

364 611

20645 19742

5.16E+8 1.29E+9

8.22E+5 1.22E+6

5987 5725

343

381

30129

7.91E+8

1.20E+6

8737

626

696

27678

2.34E+9

1.95E+6

8027

354

393

36645

9.99E+8

1.47E+6

10627

696

773

32774

3.41E+9

2.56E+6

9505

547 616 678

608 684 753

14452 17871 18774

8.74E+8 1.36E+9 1.66E+9

8.34E+5 1.15E+6 1.28E+6

-

Where lp is the Plastic Hinge Length (mm), D the member depth (mm), A the cross sectional area (mm2), I the moment of inertia of the section (mm4), My the yield bending moment (kN-mm) and Ny, Yield Axial Force (kN). The section assignment for each of the columns and beam in the model is presented in Table 4. Table 4: Description of the frame members Member No.

Description

Section

1, 4, 5, 8 2, 3, 6, 7 9, 12, 13, 16 10, 11, 14, 15 17, 20, 21, 24 18, 19, 22, 23 25 - 36 37 - 48 49 - 60

Column Column Column Column Column Column Beam Beam Beam

W14X109 W24X104 W14X159 W27X146 W14X193 W30X173 W24X76 W27X94 W30X99

Section Type 1,2 3,4 5,6 7,8 9, 10, 11, 12 13 - 16 17 - 20 21 - 24

The axial load-moment interaction diagram were calculated for each of the column members and plotted in the Figure 6 with the respective coordinates listed in Table 5.

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-15000

Axial load (kN)

-10000

-5000

0 Sections 1,2 Sections 3,4 Sections 5,6 Sections 7,8 Sections 9,10 Sections 11,12

5000

10000

15000 0

5E+5

1E+6

1.5E+6

2E+6

2.5E+6

3E+6

Bending moment (kN-mm)

Figure 6: Axial Load – Bending Moment Interaction Diagram Table 5: Column axial load – moment interaction N (kN) M (kN-mm) N (kN) Section 3,4 M (kN-mm) N (kN) Section 5,6 M (kN-mm) N (kN) Section 7, 8 M (kN-mm) N (kN) Section 9, 10 M (kN-mm) N (kN) Section 11, 12 M (kN-mm) Section 1, 2

1.4

-5987 0.0 -5725 0.0 -8737 0.0 -8027 0.0 -10627 0.0 -9505 0.0

-1198 8.21E+5 -1145 1.24E+6 -1748 1.23E+6 -1605 1.97E+6 -2125 1.52E+6 -1901 2.59E+06

0.0 9.12E+5 0.0 1.37E+6 0.0 1.36E+6 0.0 2.19E+6 0.0 1.69E+6 0.0 2.88E+6

0.0 9.12E+5 0.0 1.37E+6 0.0 1.36E+6 0.0 2.19E+6 0.0 1.69E+6 0.0 2.88E+6

5987 0.0 5725 0.0 8737 0.0 8027 0.0 10627 0.0 9505 0.0

Curvature and ductility capacity

For all members of the structure building the moment curvature relationship and the failure criteria is described in section 1.4.1 and 1.4.2 respectively. 1.4.1

Moment Curvature Relationship

In the building structure, all members (beams and columns) were assigned a bi-linear momentcurvature relationship described by Figure 3. For each member it is possible to verify that the plastic curvature ∅ corresponds to a plastic rotation limit θ 0.03 rad. where in order to

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calculate this plastic rotation it is necessary first to compute the yielding curvature ϕ and based on this value to calculate the plastic curvature ϕ . The yielding curvature is defined by the following expression: My   ϕ EI The plastic curvature is defined as: 0.2  ϕ   ϕ 0.02  The ultimate capacity can readily be found from the figure above as:  ϕ

ϕ

 ϕ

Finally in order to find the plastic rotation of the members, the assumption that is considered is that a length of 90% of the depth of the cross section was assumed as a plastic hinge length therefore rotation and curvature are related through the following relationship. θ

 ϕ l

In the Table 6: Plastic Curvature of each element for a plastic Rotation limit θp are summarized the values for plastic rotation of all elements. Table 6: Plastic Curvature of each element for a plastic Rotation limit θp Member Section Type 1,2 W14x109

lp

My

I

ϕy

ϕp

ϕu

θp

(mm)

(KN-mm)

(mm4)

(rad/mm)

(rad/mm)

(rad/mm)

(rad.)

328

8.22E+5

5.16.E+8

7.97E-6

7.97E-5

8.76E-5

0.0261

3,4

W24x104

550

1.22E+6

1.29.E+9

4.75E-6

4.75E-5

5.22E-5

0.0261

5,6 7,8 9, 10, 11, 12 13 - 16 17 - 20 21 - 24

W14x159 W27x146 W14x193 W30x173 W24x76 W27x94 W30x99

343 626 354 696 547 616 678

1.20E+6 1.95E+6 1.47E+6 2.56E+6 8.34E+5 1.15E+6 1.28E+6

7.91.E+8 2.34.E+9 9.99.E+8 3.41.E+9 8.74.E+8 1.36.E+9 1.66.E+9

7.61E-6 4.17E-6 7.38E-6 3.75E-6 4.77E-6 4.24E-6 3.85E-6

7.61E-5 4.17E-5 7.38E-5 3.75E-5 4.77E-5 4.24E-5 3.85E-5

8.37E-5 4.58E-5 8.12E-5 4.13E-5 5.25E-5 4.66E-5 4.24E-5

0.0261 0.0261 0.0261 0.0261 0.0261 0.0261 0.0261

The rotation θp for all the members are less than the limit of 0.03 rad. 1.4.2

Strength Degradation Model

The strength degradation model for all structural members states that the strength degradation begins at a curvature ductility of 11.0 as shown in Figure 4. Once plastic rotations reach the plastic limit (θ 0.03 rad the corresponding moments and curvatures can be found only by clearing the value of ϕ from equation 1.4. 8

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ϕ

 

θ l

In accordance with Figure 3 the strength degradation should begin at a ductility ratio value of 11.0 in which the ductility ratio ( ) is defined by:  

ϕ ϕ

Table 7: Curvature ductility capacity at failure Member Type 1,2 3,4 5,6 7,8 9, 10, 11, 12 13 - 16 17 - 20 21 - 24

Section W14x109 W24x104 W14x159 W27x146 W14x193 W30x173 W24x76 W27x94 W30x99

lp

My

ϕp

ϕu

(KN-mm)

(mm4)

ϕy

(mm)

I

(rad/mm)

(rad/mm)

(rad/mm)

328 550 343 626 354 696 547 616 678

8.22E+5 1.22E+6 1.20E+6 1.95E+6 1.47E+6 2.56E+6 8.34E+5 1.15E+6 1.28E+6

5.16E+8 1.29E+9 7.91E+8 2.34E+9 9.99E+8 3.41E+9 8.74E+8 1.36E+9 1.66E+9

7.97E-6 4.75E-6 7.61E-6 4.17E-6 7.38E-6 3.75E-6 4.77E-6 4.24E-6 3.85E-6

9.15E-5 5.45E-5 8.75E-5 4.79E-5 8.47E-5 4.31E-5 5.48E-5 4.87E-5 4.42E-5

9.94E-5 5.93E-5 9.51E-5 5.21E-5 9.21E-5 4.69E-5 5.96E-5 5.29E-5 4.81E-5

∆M/My

μ

0.23 0.23 0.23 0.23 0.23 0.23 0.23 0.23 0.23

11 11 11 11 11 11 11 11 11

Therefore, with the plastic rotation of 0.03 rad, the curvature ductility at failure is 11. 1.5

Dynamic characteristics of the original structure

The dynamic characteristics of the building were calculated for the first 5 periods of vibration of the structure (Table 8). The mode shapes for the frame were plotted in Figure 7. From the dynamic analysis, it can be seen that the first three modes capture the dynamic behavior of the building adequately as shown in Table 9 through the mass participation (99%). Table 8: Frequencies and periods MODE 1 2 3 4 5

Frequency

Period

(Hz)

(s)

0.77 2.20 4.04 6.41 9.00

1.30 0.45 0.25 0.16 0.11

Table 9: Mass participation ratios MODE 1 2 3 4 5

% Mass 87 96 99 100 100

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Mode 4

Mode 3

Mode 2

Mode 1

Mode 5

25

25

25

25

25

20

20

20

20

20

15

15

15

15

15

10

10

10

10

10

5

5

5

5

5

0

0 0

0.25

0.5

0.75

1

-1

-0.5

0

0.5

1

0 -1

0 -0.5

0

0.5

1

0 -1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

Figure 7: Mode Shapes of the structure 1.6

Pushover Analyses

To assess the performance of the structure before carrying out more advanced analysis methods as non-linear time history analysis, a pushover analysis was performed to identify maximum lateral force capacity of the building and potential yield zones in members by statically increasing lateral load on the structure to collapse. Pushover analysis results are generally dependent on the applied load distribution given to the structural model. Consequently, three lateral load distributions along the height of the building were considered based on: (1) ASCE 41; (2) The first mode response of the building structure in free vibration and (3) New Zealand Code with 92% of the base shear distributed linearly according to inter-story height and 8% added to the top floor. 1.6.1

ASCE 41 lateral load pattern

For this case the fundamental mode of vibration of the structure is T = 1.30 s and k=1.4 in the equation 1.8 (ASCE 41). Load distribution over height with total base shear of 1kN is shown in Table 10.

Fx  CvxV Cvx 

wx hxk n

wh i 1

k i i

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Table 10: Lateral Load Distribution, ASCE 41

floor 6 floor 5 floor 4 floor 3 floor 2 floor 1 Total shear

1.6.2

Weight

H

(kN)

(m)

1815.5 2514.8 2514.8 2514.8 2514.8 2599.1

3.81 3.81 3.81 3.81 3.81 5.486

Elevation Distribution (m) 24.536 0.248 20.726 0.272 16.916 0.205 13.106 0.143 9.296 0.088 5.486 0.044 1.000

Linear vertical distribution

The linear load distribution along the height of the building used for another pushover load pattern is shown in Table 11. Table 11: Lateral Load Distribution, Linear vertical

floor 6 floor 5 floor 4 floor 3 floor 2 floor 1 Sum

1.6.3

Weight

H

(kN)

(m)

1815.53 2514.8 2514.8 2514.8 2514.8 2599.13

3.81 3.81 3.81 3.81 3.81 5.486

Elevation (m) 24.536 20.726 16.916 13.106 9.296 5.486

Distribution 0.272 0.230 0.188 0.146 0.103 0.061 1.000

New Zealand Code

According to New zeland code a 92% of base shear distributes linear according to heights, 8% added to the top floor. Table 12 shows load values for each floor level. Table 12: Lateral Load Distribution, New Zealand Code

floor 6 floor 5 floor 4 floor 3 floor 2 floor 1 Sum

Weight (kN) 1815.53 2514.8 2514.8 2514.8 2514.8 2599.13

H (m) 3.81 3.81 3.81 3.81 3.81 5.486

Elevation (m) 24.536 20.726 16.916 13.106 9.296 5.486

Distribution 0.331 0.212 0.173 0.134 0.095 0.056 1.000 3

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For each of the three different load patterns for pushover analysis, the corresponding curves are plotted in Figure 8 indicating the failure point for the structure. Figure 9 shows the steady increase of top floor lateral displacement versus time, which indicates that static pushover load increase is achieved, no dynamic effects is present. 3500 3250

Base shear (kN)

3000 2750

475.14, 3269.2

2500 2250

486.35, 3212.1

2000 1750

511.84, 3172

1500 1250

ASCE 41 Linear NZ code

1000 750 500 250 0 0

50

100

150

200 250 300 350 400 Top floor lateral displacement (mm)

450

500

550

9

10

Displacement (mm)

Figure 8: Pushovers curves 1500 1400 1300 1200 1100 1000 900 800 700 600 500 400 300 200 100 0 0

1

2

3

4

5 Time (s)

6

7

8

Figure 9: Top floor lateral displacement vs. time The plastic hinge locations are seen at the bottom part of the columns and most of the first 4 story beams (Figure 10: Deflected Shape ASCE 41 Pushover, Plastic Hinge Location)Figure 10. The structure fails at 8.8 sec. according to ASCE 41 load pattern (Figure 11).

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Figure 10: Deflected Shape ASCE 41 Pushover, Plastic Hinge Location For Pushover curve for ASCE 41 load pattern, the first and second yield points are indicated in Figure 11. In the same fashion the plot of base shear in time indicating first and second yield point in Figure 12. 3500 3250 3000

Base shear (kN)

2750 2500

(486.35, 3212.1) Second yield (104.77, 2163.6)

2250 2000 1750

First yield (100.11, 2074.3)

1500 1250 1000 750 500 250 0 0

50

100

150

200 250 300 350 400 Top floor lateral displacement (mm)

450

500

550

Figure 11: Pushover Curve ASCE 41

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4000 3500

Max base shear of 3212.1 (kN) at 8.8 (s)

Base shear (kN)

3000 2500 First yield (5.2, 2074.3)

2000

Second yield (5.5, 2163.6)

1500 1000 500 0 0

1

2

3

4

5 Time (s)

6

7

8

9

10

Figure 12: Base Shear vs. Time, ASCE 41. The first yield corresponding to ASCE 41 pushover curve occured at the first floor midspan beam member 58 at 5.2 sec. The second yield occurred in the first floor interior column member 23 at the bottom end at 5.5s. The moment–time and moment–curvature relations for beam member 58 and column 23 were plotted in Figure 13 and Figure 15, respectively with the yield point and failure point indicated. -1800 Yield at 5.2 (s) M= -1478.8

Bending moment (kN-m)

-1600 -1400 Fail at 8.9 (s) M= -1779.8

-1200 -1000 -800 -600 -400 -200 0 0

1

2

3

4

5 Time (s)

6

7

8

9

10

Figure 13: Moment vs. Time, beam member 58 end 2

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3300000 Yields at 5.5 (s) M= 2642700

3000000

Bending moment (kN-m)

2700000 2400000 2100000 Fails at 8.8 (s) M= 3174800

1800000 1500000 1200000 900000 600000 300000 0 -300000 0

1

2

3

4

5 Time (s)

6

7

8

9

10

Figure 14: Moment vs. time – Column member 23 -1800

Bending moment (kN-m)

-1600 -4.9133E-5, -1779.8

-1400 -4.4517E-6, -1478.8

-1200 -1000 -800 -600 -400 -200 0 0

-1E-5

-2E-5

-3E-5

-4E-5

-5E-5

-6E-5

Curvature (rad)

Figure 15: Moment vs. Curvature, Member 58 end 2 First failure in the building occurred at 8.8 sec. in column member 23, which yielded second during the pushover. Moment in building members versus time are plotted in Figure 16, Figure 17, Figure 18 and Figure 19, for the each of the beam ends connected to the columns in the first floor in order to identify the failure instant.

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1800000 1600000

Moment-End 1 (kN-mm)

1400000 1200000 1000000 800000 600000

Members Member 55 Member 57 Member 59

400000 200000 0 -200000 0

1

2

3

4

5

6

7

8

9

10

9

10

9

10

Time(s)

Figure 16: 1st Floor Beam Failure at end 1 0 -200000

Moment-End 2 (kN-mm)

-400000 Members Member 56 Member 58 Member 60

-600000 -800000 -1000000 -1200000

8.9

-1400000 -1600000 -1800000 0

1

2

3

4

5

6

7

8

Time(s)

Figure 17: 1st Floor Beam Failure at end 2 600000 300000

Moment-End 1 (kN-mm)

0 -300000 -600000 -900000 -1200000 -1500000 Members Member 21 Member 22 Member 23 Member 24

-1800000 -2100000 -2400000

9.0

-2700000 -3000000 0

1

2

3

4

5

6

7

8

Time(s)

Figure 18: Bottom Storey Columns Failure at end 1

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3300000 3000000

Moment-End 2 (kN-mm)

2700000

Members Member 21 Member 22 Member 23 Member 24

2400000 2100000

8.8

1800000 1500000 1200000 900000 600000 300000 0 -300000 0

1

2

3

4

5

6

7

8

9

10

Time(s)

Figure 19: Bottom Storey Columns Failure at end 2

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CHAPTER 2 – DESIGN GROUND MOTIONS

1

Retrieval and analysis of Design Ground motions

A seismic assessment for this building is based on a non-linear time history dynamic analysis. Three historical recording for ground motions in Los Angeles region, are used in the analysis mentioned early. The ground motions were scaled to match 10% probability of exceedance in 50 years corresponding to a design based earthquake based on current building code. The first accelerogram (Figure 20) corresponds to the fault parallel component of the Imperial Valley 1940 “El Centro” earthquake with a peak ground acceleration of 0.6757g and is designated as LA-02 record. The second ground motion (Figure 27) corresponds to the fault normal component of Landers Earthquake designated as LA-07 record. The third accelerogram (Figure 28) is taken as fault parallel component from the 1994 Northridge Earthquake designated as LA16 record with a peak ground acceleration of 0.58g. 0.8 Peak acc. 0.6757187 (g) at 2.12 (s)

Acceleration (g)

0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 0

10

20

30

40

50

60

70

80

60

70

80

Time (s)

Figure 20: LA-02 Ground Motion 0.8

Acceleration (g)

0.6 0.4 0.2 0 -0.2 -0.4

Peak acc. -0.4209786 (g) at 16.08 (s)

-0.6 -0.8 0

10

20

30

40

50

Time (s)

Figure 21: LA-07 Ground Motion

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0.8

Acceleration (g)

0.6 0.4 0.2 0 -0.2 -0.4 Peak acc. -0.5795923 (g) at 2.705 (s)

-0.6 -0.8 0

10

20

30

40

50

60

70

80

Time (s)

Figure 22: LA-16 Ground Motion

2

Response Spectra

Using signal analysis programs as ‘SeismoSignal’ (Seismosoft) and ‘Nspectral’ (University of Buffalo) to determine: the response spectrum for absolute acceleration for 5% damping (Figure 23), relative velocity (Figure 24) and relative displacement (Figure 25) for each of the ground motions. 2 1.8

La02 La07 La16

1.4

Fundamental period

Absolute acceleration (g)

1.6

1.2 1 0.8 0.6 0.4 0.2 0 0

0.25

0.5

0.75

1

1.25 Time (s)

1.5

1.75

2

2.25

2.5

Figure 23: Absolute Acceleration Response Spectra for 5% Damping

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2.5 2.25

La02 La07 La16

1.75

Fundamental period

Relative velocity (m/s)

2

1.5 1.25 1 0.75 0.5 0.25 0 0

0.25

0.5

0.75

1

1.25 Time (s)

1.5

1.75

2

2.25

2.5

2.25

2.5

Figure 24: Relative Velocity Response Spectrum

La02 La07 La16

1 Fundamental period

Relative displacement (m)

1.25

0.75

0.5

0.25

0 0

0.25

0.5

0.75

1

1.25 Time (s)

1.5

1.75

2

Figure 25: Relative Displacement Response Spectrum In the acceleration response spectra (Figure 23) can be noted that records LA-07 has the lower response of the set of ground motions and with high frequencies content. In the same fashion, record LA-02 has high frequencies content but with almost the double in spectral acceleration values that LA-07 in the same range of frequencies. However that difference between the two records is not accentuated for relative displacement response. Although record LA-16 (Figure 22) is a short duration ground motion, it has a wide range of frequencies content. Moreover, the maximum velocity and displacement response is greater that for records LA-02 and LA-07 for almost all frequencies.

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CHAPTER 3 - ANALYSIS OF THE ORIGINAL BUILDING

1

Introduction

The objective of this chapter is to evaluate the seismic response of the original building structure under each of the three design ground motions considered in chapter 2. The computer program RUAUMOKO and the post-processor DYNAPLOT were used to evaluate the performance of the original building structure. For each analysis of the building under ground motions, four output quantities are extracted to assess the existing building performance. They includes energy quantities, member curvature ductility, peak and residual interstory drifts and total floor accelerations.

2 2.1

Performance of the existing structure Energy balance

Plots of the time history energy components are shown in Figure 26 to Figure 28 (LA-02, LA-07 and LA16). In Figure 26 to Figure 28 five energy curves can be distinguished, three of them are the internal energy components, kinetic, viscous damping and absorbed (strain) energy, the fourth curve is the total energy and the last curve represents the input energy. The absorbed energy represents the total amount of energy that the structure has absorbed either through elastic or unrecoverable inelastic deformations of its elements and can be defined by the following equation: E t E t E t Where E is the elastic strain energy and E the Energy dissipated through hysteretic damping of the structural elements which depends on the hysteretic relation of each structural member. In the program RUAUMOKO it must be noted that the sum of the internal energy components in the static analysis is not equal to the total energy computed by the program (Applied work done) due to the applied work done is the product of the loads and the displacements and the internal strain energy is one half of the product of the elastic forces and the displacements. Figure 26 to Figure 28 show that an energy balance between the input energy and the sum of the internal energy components (kinetic, damping, strain) is achieved.

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3E+6

Energy (kN-mm)

2.4E+6

1.8E+6

1.2E+6 Kinetic Energy Viscous Damping Strain Energy Total Energy Input Energy

6E+5

0 0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

Time (sec.)

Figure 26: Energy Components LA-02 1E+6

Energy (kN-mm)

8E+5

6E+5

4E+5 Kinetic Energy Viscous Damping Strain Energy Total Energy Input Energy

2E+5

0 0

10

20

30

40

50

60

70

80

90

100

Time (sec.)

Figure 27: Energy Components LA-07 3E+6

Energy (kN-mm)

2.4E+6

1.8E+6

1.2E+6

Kinetic Energy Viscous Damping Strain Energy Total Energy Input Energy

6E+5

0 0

5

10

15

20

25

30

Time (sec.)

Figure 28: Energy Components LA-16. 14

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It can be seen that the input energy from LA02 and LA16 are equal and more than three times the input energy from LA07. Among three motions, LA16 excites larger kinematic energy at the beginning of the record; this is due to the long pause in the acceleration motion. Although the energy time histories generated for the three ground motions varies considerably from one to another, each energy component exhibits a particular pattern, for example the kinetic energy oscillates from zero (maximum deflections) to positive peaks (initial undeformed position). The energy dissipated by viscous damping always increases with time for the three ground motions reaching its maximum value for LA-02 and the lowest value for LA-07. For the absorbed energy two components can be distinguished, the first of them is the recoverable elastic energy which is represented by oscillations out of phase with the kinetic energy and the second one is the non-recoverable component represented by sudden shifts towards positive values due to the inelastic actions that occur in time. The strain energy curve E (green curve) as was mentioned previously is the total amount of energy that the structure has absorbed either through elastic straining or unrecoverable inelastic deformations and the peak value of this curve during an earthquake represents the largest demand on structural members. For each one of the ground motions the fraction of input energy absorbed by the building structure is shown in the Table 13. Table 13: Fraction of Input Energy Absorbed. Ground Motions LA - 02 LA - 07 LA - 16

Absorbed Energy (kN-m) 1152.7 314.21 1930.70

Total Energy (kN-m) 2942.7 912.61 2859.7

Fraction

Percentage (%)

0.392 0.344 0.675

39.19 34.43 67.51

According to Table 13 it can be observed that the structure absorbs more energy for the LA-16 ground motion with a considerable difference compared with the other two ground motions. The peak values of the absorbed energy for the three ground motions are detailed in Table 14.

15

STRUCTURAL CONTROL PROJECT

TEAM 4

Table 14: Peak Absorbed Energy. Ground Motions

Peak Absorbed Energy (kN-m)

LA - 02 LA - 07 LA - 16

1241.7 433.7 2000.70

The maximum difference in percentage between the input energy and the internal energy components is computed in Table 15 for the three ground motions considered. This indicates that the energy balance is achieved in the program. Table 15: Energy Balance Error.

2.2

Ground Motions

EBE %

LA - 02 LA - 07 LA - 16

0.16 0.13 0.08

Plastic Hinging Distribution

Figure 29 to Figure 31 shows the distribution of the plastic hinges due to the three motions considered and Table 16 to Table 18 provides the maximum curvature ductility demand and the maximum plastic rotation for each yielding member. It is important to mention that program RUAUMOKO list members with ductility ratios greater than 1 ( 1 ). The following convention was used: Bidirectional hinging in beams and columns Unidirectional hinging in beams Unidirectional hinging in columns In the case of unidirectional hinging, the dark side of the plastic hinge indicates the side where the plasticization on the member is occurring. It is shown that for LA-07 the maximum curvature ductility (μ 4.831) and the maximum plastic rotation (θp 0.013 rad) are the lowest in comparison to LA-02 and LA-16 which indicates that LA-07 induces the minor inelastic action to the members.

16

STRUCTURAL CONTROL PROJECT

TEAM 4

On the other hand the inelastic action produced by LA-16 is the greatest among the three ground motions producing a maximum curvature ductility of μ 8.494 and a maximum plastic rotation of θp 0.022 rad. For LA-02 the maximum values for ductility and plastic rotations are μ 5.710 and θp 0.015 rad., respectively. It is clear that LA-16 causes the most severe damage to the members in the structure, however for this motion none of the structural members reaches plastic rotations of θp 0.03 rad., the limit rotation established as the failure criterion for the elements. Clearly the ground motion LA-16 produces the most severe damage. As shown in Figure 29 and Figure 30 the hinging distribution for LA-02 and LA-07 is predominantly unidirectional while for LA-16 (Figure 31) is bidirectional.

Figure 29: Distribution of Plastic Hinges for LA-02

Figure 30: Distribution of Plastic Hinges for LA-07

Figure 31: Distribution of Plastic Hinges for LA-16

17

STRUCTURAL CONTROL PROJECT

TEAM 4

Table 16: Maximum plastic rotations for LA-02 ground motion Hinge Member Prop.

Type

Lp (mm)

ϕp

Ductility Lp(Int) μ

θp

1

21

10

Column

328

7.98E-05

3.995

131.0

0.0105

2

22

12

Column

328

7.98E-05

5.496

180.3

0.0144

3

23

12

Column

328

7.98E-05

5.413

177.5

0.0142

4

24

10

Column

328

7.98E-05

3.409

111.8

0.0089

5

31

13

Beam

547

4.78E-05

1.343

73.5

0.0035

6

32

14

Beam

547

4.78E-05

1.878

102.7

0.0049

7

33

15

Beam

547

4.78E-05

1.421

77.7

0.0037

8

34

14

Beam

547

4.78E-05

2.291

125.3

0.0060

9

35

15

Beam

547

4.78E-05

1.237

67.7

0.0032

10

36

16

Beam

547

4.78E-05

1.971

107.8

0.0052

11

37

17

Beam

616

4.24E-05

2.351

144.8

0.0061

12

38

18

Beam

616

4.24E-05

3.475

214.1

0.0091

13

39

19

Beam

616

4.24E-05

3.087

190.2

0.0081

14

40

18

Beam

616

4.24E-05

3.809

234.6

0.0099

15

41

19

Beam

616

4.24E-05

2.801

172.5

0.0073

16

42

20

Beam

616

4.24E-05

3.158

194.5

0.0082

17

43

17

Beam

616

4.24E-05

3.360

207.0

0.0088

18

44

18

Beam

616

4.24E-05

4.148

255.5

0.0108

19

45

19

Beam

616

4.24E-05

3.718

229.0

0.0097

20

46

18

Beam

616

4.24E-05

4.458

274.6

0.0116

21

47

19

Beam

616

4.24E-05

3.458

213.0

0.0090

22

48

20

Beam

616

4.24E-05

4.175

257.2

0.0109

23

49

21

Beam

678

3.84E-05

3.949

267.7

0.0103

24

50

22

Beam

678

3.84E-05

4.806

325.8

0.0125

25

51

23

Beam

678

3.84E-05

4.555

308.8

0.0119

26

52

22

Beam

678

3.84E-05

5.132

347.9

0.0134

27

53

23

Beam

678

3.84E-05

4.252

288.3

0.0111

28

54

24

Beam

678

3.84E-05

4.621

313.3

0.0120

29

55

21

Beam

678

3.84E-05

4.852

329.0

0.0126

30

56

22

Beam

678

3.84E-05

5.443

369.0

0.0142

31

57

23

Beam

678

3.84E-05

5.146

348.9

0.0134

32

58

22

Beam

678

3.84E-05

5.710

387.1

0.0149

33

59

23

Beam

678

3.84E-05

4.863

329.7

0.0127

34

60

24

Beam

678

3.84E-05

5.461

370.3

0.0142

Where the 8th column (Lp) is the plastic length of the hinge. 3

STRUCTURAL CONTROL PROJECT

TEAM 4

Table 17: Maximum plastic rotations for LA-07 ground motion. Hinge Member Prop.

Type

Lp (mm)

ϕp

Ductility μ

Lp(Int)

θp

1

22

12

Column

328

7.984E-05

-4.378

-143.6

-0.0115

2

23

12

Column

328

7.984E-05

-4.409

-144.6

-0.0115

3

39

19

Beam

616

4.24E-05

1.423

87.7

0.0037

4

41

19

Beam

616

4.24E-05

1.030

63.4

0.0027

5

43

17

Beam

616

4.24E-05

2.313

142.5

0.0060

6

44

18

Beam

616

4.24E-05

1.637

100.8

0.0043

7

45

19

Beam

616

4.24E-05

2.559

157.6

0.0067

8

46

18

Beam

616

4.24E-05

1.842

113.5

0.0048

9

47

19

Beam

616

4.24E-05

2.342

144.3

0.0061

10

48

20

Beam

616

4.24E-05

1.451

89.4

0.0038

11

49

21

Beam

678

3.844E-05

3.426

232.3

0.0089

12

50

22

Beam

678

3.844E-05

3.050

206.8

0.0079

13

51

23

Beam

678

3.844E-05

3.887

263.5

0.0101

14

52

22

Beam

678

3.844E-05

3.302

223.9

0.0086

15

53

23

Beam

678

3.844E-05

3.616

245.2

0.0094

16

54

24

Beam

678

3.844E-05

2.760

187.1

0.0072

17

55

21

Beam

678

3.844E-05

4.627

313.7

0.0121

18

56

22

Beam

678

3.844E-05

4.009

271.8

0.0104

19

57

23

Beam

678

3.844E-05

4.831

327.5

0.0126

20

58

22

Beam

678

3.844E-05

4.263

289.0

0.0111

21

59

23

Beam

678

3.844E-05

4.603

312.1

0.0120

22

60

24

Beam

678

3.844E-05

4.025

272.9

0.0105

4

STRUCTURAL CONTROL PROJECT

TEAM 4

Table 18: Maximum plastic rotations for LA-16 ground motion. Hinge Member Prop.

Type

Lp (mm)

ϕp

Ductility Lp(Int) μ

θp

1

21

10

Column

328

7.98E-05

7.679

251.9

0.0201

2

22

12

Column

328

7.98E-05

8.485

278.3

0.0222

3

23

12

Column

328

7.98E-05

8.394

275.3

0.0220

4

24

10

Column

328

7.98E-05

5.810

190.6

0.0152

5

32

14

Beam

547

4.78E-05

1.185

64.8

0.0031

6

33

15

Beam

547

4.78E-05

1.398

76.5

0.0037

7

34

14

Beam

547

4.78E-05

1.611

88.1

0.0042

8

36

16

Beam

547

4.78E-05

1.199

65.6

0.0031

9

37

17

Beam

616

4.24E-05

1.116

68.7

0.0029

10

38

18

Beam

616

4.24E-05

1.604

98.8

0.0042

11

39

19

Beam

616

4.24E-05

1.554

95.7

0.0041

12

40

18

Beam

616

4.24E-05

1.889

116.4

0.0049

13

41

19

Beam

616

4.24E-05

1.164

71.7

0.0030

14

42

20

Beam

616

4.24E-05

1.552

95.6

0.0041

15

43

17

Beam

616

4.24E-05

3.009

185.4

0.0079

16

44

18

Beam

616

4.24E-05

3.758

231.5

0.0098

17

45

19

Beam

616

4.24E-05

3.325

204.8

0.0087

18

46

18

Beam

616

4.24E-05

4.031

248.3

0.0105

19

47

19

Beam

616

4.24E-05

3.056

188.2

0.0080

20

48

20

Beam

616

4.24E-05

3.818

235.2

0.0100

21

49

21

Beam

678

3.84E-05

5.349

362.7

0.0139

22

50

22

Beam

678

3.84E-05

6.169

418.3

0.0161

23

51

23

Beam

678

3.84E-05

5.939

402.7

0.0155

24

52

22

Beam

678

3.84E-05

6.516

441.8

0.0170

25

53

23

Beam

678

3.84E-05

5.603

379.9

0.0146

26

54

24

Beam

678

3.84E-05

6.035

409.2

0.0157

27

55

21

Beam

678

3.84E-05

7.488

507.7

0.0195

28

56

22

Beam

678

3.84E-05

8.115

550.2

0.0212

29

57

23

Beam

678

3.84E-05

7.923

537.2

0.0207

30

58

22

Beam

678

3.84E-05

8.494

575.9

0.0221

31

59

23

Beam

678

3.84E-05

7.518

509.7

0.0196

32

60

24

Beam

678

3.84E-05

8.098

549.0

0.0211

3

STRUCTURAL CONTROL PROJECT

2.3

TEAM 4

Inter-story peak and residual drifts

The Inter-story peak and residual drift ratios are very important indicators of the structural damage in the building, Figure 32 to Figure 38 show the inter-story drift time history, the peak inter-storey drift and the residual inter-storey drift for each one of the three ground motions considered. Figure 32 to Figure 34 show the Inter-story peak drift time history of each floor for each motion. The maximum inter-story drift for the three ground motions occurs in the first floor which is justified due to larger height of the first floor producing the soft story mechanism. 120 110 100

2nd floor peak 62.74 (mm)

90

3rd floor peak 57.8 (mm) 4th floor peak 51.95 (mm)

80

Inter-storey drift (mm)

1st floor 2nd floor 3rd floor 4th floor 5th floor Roof

1st floor peak 98.94 (mm)

5th floor peak 39.56 (mm)

70

1st floor residual 39.26 (mm)

60

2nd floor residual 25.49 (mm)

50

3rd floor residual 23.31 (mm)

40 30 20 10 0 4th floor residual 21.58 (mm)

-10

5th floor residual 10.72 (mm)

-20

Roof residual 2.726 (mm)

Roof peak 26.13 (mm)

-30 0

10

20

30

40

50

60

70

80

90

Time (s)

Figure 32: Inter-story Drift – Time History LA-02

4

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TEAM 4

50 Roof peak -18.7 (mm)

40

5th floor peak -25.54 (mm)

30

Roof residual -0.2766 (mm) 4th foor peak -29.66 (mm)

20

5th floor residual -1.574 (mm) 4th floor residual -7.142 (mm)

Inter-storey drift (mm)

10 0 -10 -20 -30 -40 -50 1st floor 2nd floor 3rd floor 4th floor 5th floor Roof

-60 -70

3rd floor residual -17.56 (mm) 2nd floor residual -25.88 (mm)

3rd floor peak -41.85 (mm) 2nd floor peak -52.13 (mm)

1st floor residual -37.57 (mm)

-80 1st floor peak -83.38 (mm) -90 0

10

20

30

40

50

60

70

80

90

100

Time (s)

Inter-storey drift (mm)

Figure 33: Inter-story Drift – Time History LA-07 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 -10 -20 -30 -40 -50 -60 -70 -80

1st floor 2nd floor 3rd floor 4th floor 5th floor Roof

1st floor peak 148.1 (mm) 2nd floor peak 84.37 (mm) 3rd floor peak 63.91 (mm) 4th floor peak 39.26 (mm) 5th floor peak 29.71 (mm)

1st floor residual 19.84 (mm)

Roof peak 23.03 (mm)

2nd floor residual 12.72 (mm) 3rd floor residual 11.03 (mm)

4th floor residual 5.82 (mm) 5th floor residual 2.227 (mm) Roof residual 0.6811 (mm)

0

10

20

30

Time (s)

Figure 34: Inter-story drift – Time History LA-16 In Figure 35 the peak values of the Inter-story drifts are shown for each ground motion although the maximum values do not occur at the same time. This figure is an envelope of the inter-story drifts. From the three ground motions, LA-16 produces the maximum peak inter-story drift (first floor). 5

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TEAM 4

Peak Inter-storey Drifts Peak Inter-storey Drifts Peak Inter-storey Drifts Motion 02

Motion 07

25

Motion 16

25

25

20

20

2.5% drift

2.5% drift

15

10

5

2.5% drift

Height (m.)

Height (m.)

Height (m.)

20

15

10

15

10

5

5

0.7% drift

0.7% drift

0.7% drift

0

0 0

50

100

150

0 0

Displacement (mm.)

50

100

150

Displacement (mm.)

0

50

100

150

Displacement (mm.)

Figure 35: Peak inter-story drifts for LA-02, LA-07 and LA-16. It is also shown that in terms of inter-story drifts LA-16 governs from the first floor until the third floor and from the 4th to the top floor LA-02 is predominant over the other two motions. LA-07 does not exceed 1.5% drift in any of the floors. The inter-story drifts of the three ground motions superimposed are shown in Figure 36. From this graphic, as was mentioned before, LA-16 and LA-02 are the motions that govern this parameter. Peak Inter-storey Drifts 25

Height (m.)

20

Inter Storey Drifts LA-02 LA-07 LA-16 2.5% Drift (LS) 0.7% Drift (IO)

15

10

5

0 0

50

100

150

Displacement (mm.)

Figure 36: Peak inter-story drifts.

6

STRUCTURAL CONTROL PROJECT

TEAM 4

Residual Inter-story drifts for each ground motions is presented in Figure 37 in which the maximum residual Inter-story drift occurs for the first ground motion (LA-02) in the first floor (soft story mechanism). y y y Motion 07

Motion 16 25

20

20

20

15

15

15

1% drift 10

5

Height (m)

25

Height (m)

Height (m)

Motion 02 25

1% drift 10

5

0

5

0 0

50

100

150

1% drift

10

0 0

Displacement (mm)

50

100

150

Displacement (mm)

0

50

100

150

Displacement (mm)

Figure 37: Residual inter-story drifts for LA-02, LA-07 and LA-16 In terms of residual inter-story drifts, the first ground motion (LA-02) produces the maximum values in almost all the floors. In Figure 38 is shown that LA-16 produces the lowest values for this parameter. y 25

Height (m)

20

15

Residual Inter-Storey Drifts LA-02 LA-07 LA-16 1% Drift (LS)

10

5

0 0

50

100

150

Displacement (mm)

Figure 38: Residual inter-story drifts for LA-02, LA-07 and LA-16 2.4

Peak Acceleration

The peak absolute floor accelerations are also significant indicators for assessing the performance of non-structural components in buildings. Figure 39 to Figure 43 show the total acceleration time histories and the peak acceleration at each floor of the building for each ground motion. 7

STRUCTURAL CONTROL PROJECT

TEAM 4

As can be seen from these figures, in all cases the largest total acceleration is at the top floor. 1 Roof peak 0.9504587 0.8

4th floor peak 0.5923547

0.6

Total acceleration (g)

1st floor 2nd floor 3rd floor 4th floor 5th floor Roof

1st floor peak 0.5825688

0.4 0.2 0 -0.2 -0.4 -0.6 5th floor peak -0.4714577 -0.8

3rd floor peak -0.4893986 2nd floor peak -0.5550459

-1 0

10

20

30

40

50

60

70

80

90

Time (s)

Figure 39: Acceleration time - history for LA-02 1 1st floor 2nd floor 3rd floor 4th floor 5th floor Roof

Roof peak 0.6092762

0.8

5th floor peak 0.3884811

Total acceleration (g)

0.6

4th floor peak 0.3823649 3rd floor peak 0.2832824

0.4

2nd floor peak 0.2697248 1st floor peak 0.2770642

0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0

10

20

30

40

50

60

70

80

90

100

Time (s)

Figure 40: Acceleration time - history for LA-07

8

STRUCTURAL CONTROL PROJECT

TEAM 4

1 1st floor 2nd floor 3rd floor 4th floor 5th floor Roof

3r floor peak 0.3584098

0.8

2nd floor peak 0.4027523

Total acceleration (g)

0.6

1st floor peak 0.3646279

0.4 0.2 0 -0.2 -0.4 -0.6

4th floor peak -0.5123344 5th floor peak -0.5197757

-0.8

Roof peak -0.7851172 -1 0

5

10

15

20

25

30

Time (s)

Figure 41: Acceleration time - history for LA-16 In terms of accelerations the most critical ground motion appears to be LA-02 with the greatest accelerations at almost all floors except for the 5th floor (Figure 42 and Figure 43) in which the acceleration produced by LA-16 is the maximum one. The peak acceleration for LA-02 in the top floor reaches a value of 0.95g. Motion 07

Motion 16 25

20

20

20

15

15

15

10

5

Height (m)

25

Height (m)

Height (m)

Motion 02 25

10

5

0

5

0 0

0.25

0.5

0.75

Acceleration (g)

1

10

0 0

0.25

0.5

0.75

Acceleration (g)

1

0

0.25

0.5

0.75

1

Acceleration (g)

Figure 42: Peak acceleration for LA-02, LA-07 and LA-16

9

STRUCTURAL CONTROL PROJECT

TEAM 4 25

Height (m)

20

15

Peak Acceleration LA-02 LA-07 LA-16

10

5

0 0

0.25

0.5

0.75

1

Acceleration (g)

Figure 43: Peak total accelerations It can be concluded then that the LA-02 ground motion dominates in terms of accelerations while the LA-16 (1st, 2nd and 3th floor) and LA-02 (4th, 5th and 6th floor) prevails in terms of peak interstorey drifts. 2.5

Performance evaluation

In this section a global measuring tool, called a performance index (PI), is introduced to quantify numerically the performance of a building. It takes into account the effects of important response quantities; including member ductility, peak inter-story drifts, maximum residual drifts and peak acceleration. The objective of the performance index is to help the owner of the building understand the overall performance of the building under the design earthquake motions 2.5.1

Performance index

The performance index considered is based on FEMA 274 guidelines. This PI will measure the global structural performance level of the building. Each of the building performance levels defined in FEMA 274 document correlate with a combination of both structural and nonstructural parameters that may be expected. Figure 44 (adapted from FEMA 274) shows the different performance levels considered for a ductile structure.

10

STRUCTURAL CONTROL PROJECT

TEAM 4

Figure 44: Performance Levels (FEMA 273) The variables considered to characterize the performance in the PI formulation are as follows:  The maximum inter-story drift ∆.  The maximum residual inter-story drift ∆ .  The maximum floor acceleration  .  The maximum curvature ductility occurred in a beam  .  The maximum curvature ductility occurred in a column  . The numerical expression for the PI is defined as follows:  ∆  μ  μ ∆ w .    w .     w∆ .   w∆ .  μ  μ  ∆  ∆ PI %   1   w     w     w∆    w∆     w

 w .

a  a

. 100

Where μ and μ are maximum curvature ductilities of column and beam.  ∆ and  ∆ is the maximum values of the peak inter-story drift and residual drifts. And a is the peak total floor acceleration. These values represent response of the building under each ground motion. μ and μ are the limits for curvature ductility of column and beam.  ∆ and  ∆ is the limits for maximum values of the peak inter-story drift and residual drifts. And  a is the limit for peak total floor acceleration. These values are the worst case limit that a building is expected to have. w , w , w∆ , w∆ and w are assigned weights to the performance variables, representing the importance of the quantity towards the overall performance and design targets. These weights add up to 100 and have been distributed with the aim of penalizing the most critical variable.

11

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Based on FEMA 274, the collapse prevention is chosen to be the lower bound limit for the structure. Therefore,  ∆ and  ∆ will be of 5%. Values of μ and μ are taken to be 11, the maximum ductility a member can reach in this project. FEMA 274 does not limit the maximum floor acceleration so a reasonable upper bound value for this structure is set to be 1 . The weight for each quantity is chosen based on its importance in the performance of the building. There variables considered most important are peak drift, column ductility and acceleration. The weights for these variables are 30, 25 and 25 respectively. For beam ductility and residual drift, the weights are 10 and 10. The reason for put more weight into acceleration is from the fact that the existing structure is hospital building, which hosts medical equipment sensitive to acceleration. Therefore, the PI will be in the form of  μ  μ 10. 11    25. 11   PI %   1  

∆   5% 100

 30.

10.

∆ 5%

a  25. 1g

. 100

The value of PI can be from any negative value to 100%, which is ideal value that a structure only can get close to. PI equal to zero means that the structure is at the collapse limit in an overall sense. A negative value of PI would mean the structure collapses. Two thread holes are defined in this PI scale, corresponding to the Immediate Occupancy (IO) and Life Safety (LS) limits. According to FEMA 274, limits for peak and residual drifts are 0.7% and 0% for IO; and 2.5% and 1% for LS. Limits for other variables are chosen and presented in Table 19. Table 19: Reponse limit for different performance categories Variables  μ  μ ∆ ∆ a PI

Limit for performance category LS IO 45% of 11 20% of 11 45% of 11 20% of 11 2.5% 0.7% 1.0% 0.75g 45

0% 0.5g 65

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According to FEMA 274, the existing structure performance is from the collapse prevention limit to life safety, with maximum peak drift is higher than 2.5% under LA16. Based on the proposed performance index formulation, the PI values will be calculated from those response quantities for each ground motion and presented in Table 20. Table 20: Performance Indexes for design ground motions Ground Motion LA-02 LA-07 LA-16

 μ

μ

∆ (%)

∆ (%)

a(g)

PI

5.71 4.83 8.49

5.50 4.41 8.49

1.80 1.52 2.70

0.72 0.68 0.36

0.95 0.61 0.79

46% 60% 36%

The PI for the structure will be the smallest among the PI for each ground motion, which will be 36% corresponding to LA16 motion. This index is within the [0;45] range, which means the structure passes the collapse prevention limit but stays below life safety limit.

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CHAPTER 4 - HYSTERETIC DAMPERS

1

Description

The objective of this phase was to retrofit the original building with hysteretic dampers for the different ground motions considered (LA2, LA7 and LA16). It was shown in the previous phase of the project that collapse is not reached under the considered ground motions; nonetheless this approach is intended to improve the seismic performance of the building rather than prevent collapse. The retrofit strategy for the structure consists on introducing chevron braces at each moment resisting frame and installing hysteretic dampers at one end of the bracing members as shown in Figure 45. This retrofit scheme was selected because it minimizes the levels of intervention (i.e. only the middle bay will be affected when installing dampers and braces). For the final design other hysteretic damper locations will be studied.

Figure 45: Locations of added bracing and hysteretic dampers (Configuration-C1) The bracing members were designed to sustain the activation load assigned to the hysteretic dampers. This system dissipates energy through the elasto-plastic hysteretic behavior shown in Figure 46.

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Figure 46: Elasto-Plastic Hysteresis For this retrofit, it was specified that hollow steel sections (HSS) must be used for the cross braces. In addition, as the braces would be installed to the existing building, brace forces induced by dead loads were ignored in the analysis and design. In order to improve the behavior of hysteretic dampers in the structure composite sections composed by HSS sections will be also considered. The methods used to determine the slip load are based on design procedures provided by Christopoulos and Filiatrault (2006) as discussed below. The computer program RUAUMOKO and its post-processor DYNAPLOT were used to perform nonlinear time history dynamic analysis in order to completely estimate the response of the building structure and select an optimum solution. Finally a comparison between the optimum design configuration and the original building will be presented in terms of energy balance, plastic hinge distribution, envelopes of peak and residual inter-story drifts and envelopes of peak absolute floor accelerations. The merits of the optimum solution in terms of performance indices will also be discussed.

2

Procedure to calculate the optimum activation load

The first step in the design of structures equipped with hysteretic dampers is the estimation of the optimum parameters for the dampers. These parameters are the activation load “Fa” and the bracing stiffness. Christopoulos and Filiatrault (2006) found that the optimal use of hysteretic dampers will occurs when the addition of these devices to a system produce additional supplemental damping along with a modification of the dynamics properties of the system that optimizes the use of the added damper. Otherwise, the system will behave either as an unbraced frame or as a fully braced frame. The selection of the cross sections for the diagonal braces is based on the recommendation by Filiatrault and Cherry (1988), which is expressed as: 0.40 15

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Where Tb is the natural period of the fully braced structure and Tu is the natural period of the unbraced structure. Furthermore, based on parametric studies it was determined that the optimum value of the activation load “Fopt” of the hysteretic damper that minimizes the amplitude of the response at any forcing frequency is given by: a T T Q , g T T

F W

Where W is the seismic weight of the structure, ag is the peak ground acceleration, g is the acceleration of gravity, Tg is the period of the ground motion and Q is a singled valued function. The Q function depends on the Tg/Tu ratio and will be presented in the preliminary design. The equation shown above reveals that the optimum activation load of a hysteretic damper depends on the frequency and amplitude of the ground motion and is not strictly a structural property. Moreover, it shows that the optimum activation load is linearly proportional to the peak ground acceleration.

3

Fourier Spectra

For determining the predominant period of the design ground motions the Fast Fourier transform (FFT), which is an efficient method to compute the Discrete Fourier Transform (DFT) was used. This analysis was performed in the software SeismoSignal by inputting our design ground motions and running the FFT analysis. 0.6

Tg=0.68s

0.54

Fourier Amplitude

0.48 0.42

Fourier Spectra LA2 LA7 LA16

Tg=1.28 Tg=0.73s

0.36 0.3 0.24 0.18 0.12 0.06 0 0.1

0.2

0.3

0.4 0.5

0.7

1

2

3

4

5

6 7 8 910

20

30

Frequency (Hz)

Figure 47: Fourier Spectra 16

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Once the analysis is completed and the data is converted into a frequency domain format, the peaks corresponding to the highest values o Fourier amplitudes were selected. The plot shown above represents the decoupling of the equations of motion for single DOFs and the peaks represent the predominant frequencies for each of the design ground motions. Predominant periods corresponding to each ground motion are also shown in Figure 47.

4

Preliminary design

As mentioned above, the best response of hysteretically damped structures occurs for small values of Tb/Tu, which corresponds to large diagonal braces. Therefore the diagonal cross-braces were chosen with the largest possible cross-sectional area within the limits imposed by architecture, cost and availability of material. As a first trial an HSS406x406x15.9 section and an HSS304X304X15.9 section were selected among the largest possible sections according the AISC –provisions. The cross-braces were used along the six stories of the building. A spreadsheet in MathCAD was used to calculate the optimum activation loads at each damper for the proposed cross sections and for the different ground motions. Calculations corresponding to the ground motion LA2 and section HSS406x406x15.9 are shown in the following page. The idea is to get a felling on which are the best braces configuration and member cross section to take into account for the optimum activation load study to be carried out in the intermediate design.

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W  14475 kN

(Total seismic weight of the structure)

Nf  6

(Number of floors)

Tb  0.615 s

(Fundamental period of the braced structures)

Tu  1.304 s

(Fundamental period of the unbraced structure)

Tg  0.683 s

(Predominant period of the design ground motion LA2)

Tb Tu Tg Tu

 0.472

 0.524

ag  0.676 g

Q 

TEAM 4

Tg

 ( 1.24 Nf  0.31) 

(Design Peak Ground Acceleration) Tb

 1.04 Nf  0.43 if 0 

Tg

1

Tu   Tb  Tg Tg Tg  ( 0.01 Nf  0.02)   1.25 Nf  0.32   ( 0.002  0.002 Nf )  1.04 Nf  0.42 if 1 Tu  Tu Tu Tu  Tu

(Unknown single valued function)

Q  1.579

Vo  Q

ag g

Tu

W

Vo  15451.9 kN

(Optimum activation shear)

i  1  Nf

  56.3

Vs  i

1 2

 

Vo 

  Nf 

 180

  46.2

 180

(Story base shear, uniformly distributed)

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 1287.7  1287.7   1287.7  Vs   kN  1287.7  1287.7    1287.7 i  2  Nf

Vs Fa  1

1

2 cos ( )

Fa  1160.4kN  1

Vs Fa  i

i

2 cos (  )

(Optimum activation load for each damper)

 1160.4  930.2    930.2   Fa   kN  930.2   930.2     930.2 

Likewise calculations were performed for ground motions LA7 and LA16 for determining the activation loads (See Table 21). In order to assess the performance of this initial proposed configuration performance indeces were calculated and compared with the PI for the existing building (See Figure 48). Table 21: Parameters

The PI corresponding to the IO and LS performance levels are included in all the plots hereinafter to have an idea of the performance of the proposed retrofit scheme in comparison to these thresholds. This performance levels are shown in dot gray lines and the values corresponding to this levels were calculated in Chapter 3.

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Figure 48: Preliminary design The proposed brace size member from the previous Figure seems to work fine for ground motions LA2 and LA7, with an approximate increase on performance of about 30% with respect of the existing structure. Nonetheless for ground motion LA16 not significant improvement was found. This fact can be justified by arguing that for LA16 the predominant period of the ground motion is high producing high activation forces that will eventually prevent the damper to activate and contribute to the energy dissipation. In the intermediate design based on a comparative study aiming to improve the performance of our building, the optimum activation loads and brace sections will be found.

5

Intermediate design

In the preliminary design the members size and the activation forces were intuitively chosen based on the recommendation of Tb/Tu = 0.4 and the design procedures provided by Christopoulos and Filiatrault (2006). In order to optimize the design we proceed to perform multiple analyses in RUAMOKO but this time considering different cross sections and braces configurations. Regarding the members size we tried to get closer to the recommended ratio of Tb/Tu = 0.4 by proposing a composite section capable of increase the brace stiffness without violating the design specifications that states that hollow shape section are to be considered in the design. Details on this composite section are presented in Appendix B. For this approach the activation load corresponding to each configuration were still calculated based on the procedure suggested by Christopoulos and Filiatrault (2006). Figure 49 shows a sketch of the proposed cross sections to be considered in the analyses. 20

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Figure 49: Sections A second approach was followed by proposing an alternative configuration which presents frame braces in the three bays of the first floor (See Figure 50). For this particular configuration the way the optimum activation shear was redistributed in height was also studied.

Figure 50: Alternative retrofit scheme considered in the analyses (Configuration-C2) A total of 4 brace cross sections were used in the two different brace configurations just shown in Figure 45 and Figure 50). The HSS406x406x15.9 and HSS304x304x15.9 section were used as described in the preliminary design but in this part were compared with the composite sections and evaluated in the two different configurations (C1 and C2). A total of 21 analyses were performed since each alternative had to be evaluated for each of the three design ground motions specified. (See Table 22 for reference) 21

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It is important to mention that when pursuing an optimum design, performance indices were used to make a comparison between the different alternatives. These indices were presented in Chapter 3. Excel macros were used to get the relevant values used to compute the performance indices and accelerate the design process. Batch files were also created in RUAMOKO to efficiently get the relevant results regarding our performances indices. Appendix A shows a detailed explanation on how the macro works. Table 22: Parameters

The highlighted values in the previous table were inputted in the RUAMOKO files when defining the elasto-plastic hysteretic loop shown in Figure 46. A plot summarizing the performances indices obtained for each of the proposed configurations and the relevant member sizes are shown in the figure below.

Figure 51: Optimum size study 22

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It can be seen in the previous figure that in overall all the proposed alternatives reached performances indices higher than the existing building. The higher PI corresponds to the composite section HSS406&304 in the brace configuration C2. The same results are presented in the figure below but this time in terms of Tb/Tu. It can be inferred from this graph that the closer we get to 0.4 the higher the PI is.

Figure 52: Optimum size study

The previous results give us an idea on which sections and brace configurations should be considered for the final design. Nonetheless a study on the optimum activations loads will be performed in order to optimize the design. This optimum activation study will be considered among the brace sizes and configuration that performed better in the previous comparison. The selected configurations to be studied are the HSS406-C1, HSS406&304-C1 and HSS406&304C2. (See Figure 45, Figure 46 and Figure 50 for details on the configurations and cross sections selected). For this study the optimum activation load were evaluated in the range of 200kN to 2000kN based on previous calculations (See Table 22). Analyses in RUAMOKO were performed for each configuration under study for activation load increments of 200kN. In total 30 analysis were run per proposed configuration and the optimum activation load corresponds to the maximum PI value for the most critical earthquake (In this case LA16).

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Figure 53: Optimum activation load study for HSS406-C1

Figure 54: Optimum activation load study for HSS304-C1

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Figure 55: Optimum activation load study for HSS406&304-C1 The following plot summarizes the three previous analyses, and shows the higher PI obtained for each configuration. It is important to state that these performances indices correspond to different activation loads as shown in the previous plots.

Figure 56: Optimum Design

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Even though the HSS406&304-C2 configuration reached the higher performance, we assumed for this case a uniform load redistribution along the height of the building, even though the second configuration present higher stiffness in the first floor with respect to the other floors. In order to deal with this uncertainty the way the forces were redistributed was studied. In each of the plots presented below the ratio shown in brackets corresponds to the ratio of the first floor activation load to the other floors activation loads. A triangular distribution of the activation loads based on the first mode was also considered.

Figure 57: Optimum activation load study for HSS406&304C2 (1/1)

Figure 58: Optimum activation load study for HSS406&304C2 (2/1) 26

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Figure 59: Optimum activation load study for HSS406&304C2 (3/1)

Figure 60: Optimum activation load study for HSS406&304C2 (4/1)

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Figure 61: Optimum activation load study for HSS406&304C2 (1st Mode proportional)

Figure 62: Optimum Design

6

Final design

Based on the parametric study performed in the preliminary and intermediate design the HSS406&304 composite sections in the bracing configuration C2 was proved to be the more optimum in terms of performance. 28

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Energy Balance

A significant increase on the strain energy due to the friction damper is noted in the following figures compared to existing structure. In the existing structure the strain energy is due to the formation of plastic hinges which lead to the damage of the building. Time History Energy Components LA - 02 3.6E+6

Energy (kN-mm)

3E+6

2.4E+6

1.8E+6

1.2E+6 Kinetic Energy Viscous Damping Strain Energy Total Energy Input Energy

6E+5

0 0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

Time (sec.)

Figure 63: Energy Components LA-02 Time History Energy Components LA - 07 1.2E+6

Energy (kN-mm)

1E+6

8E+5

6E+5

4E+5 Kinetic Energy Viscous Damping Strain Energy Total Energy Input Energy

2E+5

0 0

10

20

30

40

50

60

70

80

90

100

Time (sec.)

Figure 64: Energy Components LA-07

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Time History Energy Components LA - 16 3E+6

Energy (kN-mm)

2.4E+6

1.8E+6

1.2E+6

Kinetic Energy Viscous Damping Strain Energy Total Energy Input Energy

6E+5

0 0

5

10

15

20

25

30

Time (sec.)

Figure 65: Energy Components LA-16 6.2

Plastic hinging distribution

The number of plastic hinges in the systems was significantly reduced. For ground motion LA7 no hinges were formed and LA16 still present the most number of hinges among our ground motions.

Figure 66: Distribution of plastic hinges for LA-02 and LA-16.

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Table 23: Maximum plastic rotations for LA-02 ground motion. Hinge Member Prop.

Type

Lp (mm)

ϕp

Ductility Lp(Int) μ

θp

1

50

22

Beam

678

3.84E-05

1.185

80.343

0.0031

2

52

22

Beam

678

3.84E-05

1.055

71.529

0.0027

3

54

24

Beam

678

3.84E-05

1.156

78.377

0.0030

Table 24: Maximum plastic rotations for LA-16 ground motion. Hinge

Member

Prop.

Type

Lp (mm)

ϕp

Ductility Lp(Int) μ

1

21

10

Column

328

7.98E-05

1.487

48.8

0.004

2

22

12

Column

328

7.98E-05

3.537

116.0

0.009

3

23

12

Column

328

7.98E-05

3.005

98.6

0.008

4

24

10

Column

328

7.98E-05

1.441

47.3

0.004

5

38

18

Beam

616

4.24E-05

1.088

67.0

0.003

6

43

17

Beam

616

4.24E-05

2.366

145.7

0.006

7

44

18

Beam

616

4.24E-05

3.218

198.2

0.008

8

45

19

Beam

616

4.24E-05

2.401

147.9

0.006

9

46

18

Beam

616

4.24E-05

2.99

184.2

0.008

10

47

19

Beam

616

4.24E-05

2.402

148.0

0.006

11

48

20

Beam

616

4.24E-05

3.21

197.7

0.008

12

49

21

Beam

678

3.84E-05

3.676

249.2

0.010

13

50

22

Beam

678

3.84E-05

4.509

305.7

0.012

14

51

23

Beam

678

3.84E-05

3.95

267.8

0.010

15

52

22

Beam

678

3.84E-05

4.542

307.9

0.012

16

53

23

Beam

678

3.84E-05

3.954

268.1

0.010

17

54

24

Beam

678

3.84E-05

4.343

294.5

0.011

18

55

21

Beam

678

3.84E-05

3.675

249.2

0.010

19

56

22

Beam

678

3.84E-05

4.267

289.3

0.011

20

57

23

Beam

678

3.84E-05

3.748

254.1

0.010

21

58

22

Beam

678

3.84E-05

4.232

286.9

0.011

22

59

23

Beam

678

3.84E-05

3.59

243.4

0.009

23

60

24

Beam

678

3.84E-05

4.279

290.1

0.011

θp

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Peak and Residual Inter-Story Drifts

Inter-Storey drift - Time history - Motion 02 25 1st floor peak 22.4 (mm)

20

1st floor 2nd floor 3rd floor 4th floor 5th floor Roof

3rd floor peak 21.9 (mm)

Inter-Storey drift (mm)

15

2nd floor peak 20.3 (mm)

10

1st floor residual 3.47 (mm)

2nd floor residual 1.15 (mm) 3rd floor residual 0.424 (mm)

5 0

Roof residual -0.107 (mm)

-5

5th floor residual -0.187 (mm) -10

Roof peak -6.44 (mm) 5th floor peak -15 (mm)

-15

4th floor residual -0.208 (mm)

4th floor peak -18.3 (mm)

-20 0

10

20

30

40

50

60

70

80

90

Time (s)

Figure 67: Inter-story drift time history motion La-02.

Inter-Storey drift - Time History - Motion 07 15 12

Inter - Storey drift (mm)

9 6

1st floor residual 3.41 (mm)

3 0 2nd floor residual 0.403 (mm)

-3

4th floor residual 0.524 (mm) Roof peak -3.33 (mm)

-6

3rd floor residual -0.26 (mm)

5th floor peak -9.63 (mm) 4th floor peak -13.7 (mm) 2nd floor peak -16.3 (mm) 3rd floor peak -16.9 (mm)

-9 -12 -15

1st floor peak -17.6 (mm)

Roof residual 0.0895 (mm) 1st floor 2nd floor 3rd floor 5th floor residual -0.307 (mm) 4th floor 5th floor Roof

-18 0

10

20

30

40

50

60

70

80

90

100

Time (s)

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Figure 68: Inter-story drift time history motion La-07.

Inter-Storey drift - Time History - Motion 16 70

Inter - Storey drift (mm)

1st floor peak 66.1 (mm) 60

2nd floor peak 55 (mm)

50

3rd floor peak 51.6 (mm)

40

4th floor peak 33.1 (mm) 5th floor peak 18.8 (mm)

1st floor 2nd floor 3rd floor 4th floor 5th floor Roof

1st floor residual 28.2 (mm) 2nd floor residual 21.4 (mm)

30 20 10 0 3rd floor residual 19.1 (mm)

-10

4th floor residual 9.85 (mm)

-20

Roof peak 7.93 (mm)

5th floor residual 1.93 (mm) Roof residual -1.39 (mm)

-30 -40 0

3

6

9

12

15

18

21

24

27

30

Time (s)

Figure 69: Inter-story drift time history motion La-16.

Peak Inter-storey Drifts

Comparison of peak inter-storey drifts

25

25 20

Inter Storey Drifts LA-02 Retrof. Struc. LA-07 Retrof. Struc. LA-16 Retrof. Struc. 2.5% Drift (LS) 0.7% Drift (IO)

15

10

Height (m.)

Height (m.)

20

Inter Storey Drifts LA-02 Retrof. Struc. LA-07 Retrof. Struc. LA-16 Retrof. Struc. LA-02 Orig. Struc. LA-07 Orig. Struc. LA-16 Orig. Struc.

15

10

5

5 0 0

50

100

150

Displacement (mm.)

0 -150

-50

50

150

Displacement (mm.)

Figure 70: Peak inter-story drifts for LA-02, 07 and 16

Figure 71: Comparison of peak inter-story drifts

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Comparison of residual inter-storey drifts

Residual Inter-Storey Drifts 25

25

20

15

Height (m)

Height (m)

20

Residual Inter-Storey Drifts LA-02 Retrof. Struc. LA-07 Retrof. Struc. LA-16 Retrof. Struc. 1% Drift (LS)

10

15

10

5

5

0 -60

0 0

15

30

45

60

-30

0

30

60

Displacement (mm)

Displacement (mm)

Figure 73: Comparison of residual inter-story drifts

Figure 72: Residual inter-story drifts

6.4

Residual Inter-Storey Drifts LA-02 Retrof. Struc. LA-07 Retrof. Struc. LA-16 Retrof. Struc. LA-02 Orig. Struc. LA-07 Orig. Struc. LA-16 Orig. Struc.

Accelerations

Acceleration History - Motion 02 0.8 1st floor peak 0.607 Roof peak 0.522 5th floor peak 0.494

Total Acceleration (g)

0.6

2nd floor peak 0.416 4th floor peak 0.364

0.4

3rd floor peak 0.366 0.2

0 1st floor 2nd floor 3rd floor 4th floor 5th floor Roof

-0.2

-0.4

-0.6 0

10

20

30

40

50

60

70

80

90

Time (s)

Figure 74: Acceleration history of motion LA-02.

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Acceleration History - Motion 07 0.4 0.3

Roof peak 0.389 2nd floor peak 0.311 3rd floor peak 0.298 5th floor peak 0.307

0.2

Total Acceleration (g)

1st floor 2nd floor 3rd floor 4th floor 5th floor Roof

0.1 0 -0.1 -0.2

4th floor peak -0.304 1st floor peak -0.34

-0.3 -0.4 0

10

20

30

40

50

60

70

80

90

100

Time (s)

Figure 75: Acceleration history of motion LA-07.

Acceleration History - Motion 16 0.6 1st floor 2nd floor 3rd floor 4th floor 5th floor Roof

0.5 0.4

Total Acceleration (g)

0.3 0.2 0.1 0 -0.1 -0.2 3rd floor peak -0.461 2nd floor peak -0.484 4th floor peak -0.528 5th foor peak -0.541 Roof peak -0.59 1st floor peak -0.599

-0.3 -0.4 -0.5 -0.6 -0.7 0

5

10

15

20

25

30

Time (s)

Figure 76: Acceleration history of motion LA-16.

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Comparison of total peak accelerations 25

Height (m)

20

15

Peak Acceleration LA-02 Retrof. Struc. LA-07 Retrof. Struc. LA-16 Retrof. Struc. LA-02 Orig. Struc. LA-07 Orig. Struc. LA-16 Orig. Struc.

10

5

0 -1

-0.5

0

0.5

1

Acceleration (g)

Figure 77: Comparison of total peak accelerations. We can notice on the table below that the performance of the structure was increased from 36% to 63%. We can also notice that ductility ratios for LA16 were reduced from 8.5 to 4.5 and the acceleration was slightly reduced from 0.79g to 0.60g. Table 25: Performance Indexes for structure retrofitted with hysteretic dampers compared to the original performance. Ground Motion LA-02 LA-07 LA-16

Ground Motion LA-02 LA-07 LA-16

 μ

μ

∆ (%)

∆ (%)

a(g)

PI

5.71 4.83 8.49

5.50 4.41 8.49

1.80 1.52 2.70

0.72 0.68 0.36

0.95 0.61 0.79

46% 60% 36%

 μ

μ

∆ (%)

∆ (%)

a(g)

PI

1.19 1.00 4.54

1.00 0.00 3.54

0.58 0.44 1.44

0.06 0.06 0.56

0.61 0.39 0.60

77.90% 86.58% 63.07%

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Flow Chart for Hysteretic dampers optimum design

A flow chart summarizing the procedure to obtain the optimum design is shown below. As is described in Appendix A a macro were used to get the relevant information used to computed the performance indices which allowed us to perform a parametric study in terms of optimum size, optimum activation load and optimum distribution of the dampers.

Figure 78: Flow Chart for hysteretic dampers optimum design

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CHAPTER 5 - VISCOUS DAMPERS

1

Description

The retrofit strategy for the structure consists on introducing chevron braced frame in the middle bay of each moment resisting frame and installing viscous-type energy dissipating devices at one end of the bracing members, as shown in Figure 79. The bracing members must be designed to sustain the maximum load developed by the viscous damper. Brace forces induced by gravity loads will be ignored in the design of the bracing and viscous energy dissipating systems, as the braces would be installed to the existing building and that live loads will have a negligible effect on the bracing members.

Figure 79: Location of added bracing and viscous dampers The retrofit system considered incorporates at one end of the bracing members, viscous damper connections with an axial force linearly proportional to the relative velocity between ends. This system exhibits the elliptical hysteretic behavior shown in Figure 80. The behavior of the damper element will be proven when referring to the DAMPER element in RUAMOKO and the validation process of this element presented at the end of the chapter.

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Figure 80: Hysteretic Behavior of Viscous Dampers

2

Procedures to calculate the damping coefficients

Viscous dampers were installed at all floors. The retrofit procedure included the calculation of the damping constants of the viscous dampers as well as their distribution along the height of the building. The first approach used to determine the target viscous damping constants was achieved by providing damping constant for each floor level proportional to the lateral inter-story stiffness of the story at which the damper is to be placed. By imposing the damping constants to be proportional to the inter-story lateral stiffness of the structure, this ensures that classical normal modes will be maintained (Christopoulos and Filiatrault, 2006). kˆ0T1 2 Where k0 are the spring constants, T1 the fundamental period of the building and CL are the damping coefficients. CL 

The drawback of this procedure is that the dampers will be different at each floor level, which would cause a construction issue. Secondly, a preliminary analysis for the most significant earthquake for the building showed that the structure would have a highly nonlinear behavior due to the large amount of plastic hinges occurring in the structure and the achievement of a linear response under the specific earthquake would not be possible. A second approach was implemented by using the same damping coefficient for all the floor of the building with the assumption that the building will behave mostly in the first mode of vibration. For this method, the fundamental natural period of the existing building and the target damping ratio(s) of the building retrofitted with viscous dampers, and the maximum inter-story drifts are needed. With these three parameters known, the calculation of the damping constant can be determined for a given time history and damping level desired using the equation below.

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CL 

1T1  ki i2 i 1

Nd

   i2 cos 2  i i 1

Where  is the inclination angle of the damper, Nd is the number of dampers, k the spring constants and d is the inter-story drift. A third approach was proposed by modifying the previous equation and assuming a damping coefficient distribution proportional to the first mode of vibration.

CLi   i CL Where d is the inter-story drift of the normalized first mode of vibration. In the above equation it is considered as non-dimensional. Nd

Evd   i 1

2 2 ( i CL ) i2 cos 2  i T1 N

1 f Ees   ki i2 2 i 1

1 

Evd 4 Ees Nf

CL 

1T1  ki i2 i 1

Nd

   i3 cos 2  i i 1

For the complete design protocol used the different approaches, refer to the MathCAD calculations on the preliminary design.

3

Modeling of dampers

Before modeling the damper and in order to obtain the corresponding trial value of the fundamental period, fictitious spring elements were modeled in RUAMOKO as brace elements.

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The spring constants were determined as shown in Table 27 and the trial period needed to correct the stiffness was computed. This procedure was followed in the first approach only. Having calculated the damping constants to be used in the model for the three approaches we proceeded to model the dampers. For the modeling of the dampers phantom nodes were placed directly on the node at mid-span of the beam in the middle bay of the structure. The phantom nodes were located at the same coordinates than the existing nodes, but have different degrees of freedom. The reason for using phantom nodes is to eliminate the effect of gravity dead loads on the damper as this is a retrofit of an existing building and these loads are already supported by the existing structure. These nodes are locked to the horizontal component of the node it is connected to but have different y-axis displacements and z-axis rotations. Then the damping constants were assigned to these damper elements. For the first approach were the damping constants varies along the height of the building multiple properties were defined. In Figure 81 history displacements for the viscous damping using the different methods were plotted in conjunction with the Rayleigh damping. A good correlation was found among the plots, proving a good estimate of the damping coefficient and validating the behavior of the dampers.

Displacement time history - Motion 02 125

Top floor displacement (mm)

100 75 50 25 0 -25 -50

Rayleigh Damping- 35% Viscous Dampers (Stiffness proportion approach) Viscous Dampers (Drift proportion approach) Viscous Dampers (Constant coeff. approach)

-75 -100 -125 0

10

20

30

40

50

60

70

80

90

Time (s)

Figure 81: Plot showing comparison among viscous damping and Rayleigh damping

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Validation of the Damper element

The dashpot element used in RUAMOKO was verified to ensure proper damping. A simplified model was proposed for the verification as seen in the figure below. The spring and dashpot elements were given values of stiffness (K=64 kN/mm), a damping coefficient (C=5 kN-s/mm) and a mass of 1kN-s2/mm. The system was forced in motion by imposing a sinusoidal acceleration excitation of ü(t) = 3200(sin8t).

Figure 82: Model View The displacement time history for the node with attached mass was plotted, see Figure 83. Lateral diplacement 80 60

Displacement (mm)

40 20 0 -20 -40 -60 -80 0

3

6

9

12

15

18

21

24

27

30

Time (s)

Figure 83: Displacement time history Figure 85 shows a displacement vs. force plot for the damper element under validation. From this graph and following equations present in Figure 84 the values of K and C were calculated. See Table 26 for reference.

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Figure 84: Spring and viscous damper forces

Spring and viscous damping force 8000 79.808, 5182.5

6000 3196

Force (kN)

4000 2000 0 -2000 -4000 -3183.3 -6000 -8000 -80

-79.825, -5074 -60

-40

-20

0

20

40

60

80

Displacement (mm)

Figure 85: Spring and viscous damping force The results obtained in this validation process for the stiffness and damping coefficient match the values assumed in the analysis. Therefore the DAMPER elements are proved to adequately respond and were implemented in the RUAMOKO model. Damping coefficients were calculated in the preliminary design for the different approaches. Table 26: Validation of damper element Assumed K=64 kN/mm C= 5 kN-s/mm

Obtained K=63.9 kN/mm C=4.9 kN-s/mm

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Preliminary design

The first step in the design process of the viscous dampers is to determine the target damping (ζ1) of the building for a desired performance level. Prior to any addition of supplemental damping elements in a building acceleration and displacement response spectra were developed for damping ratios ranging between 5% and 35% (See Figure 86 to Figure 91). Using these spectra as a tool, it was determined that target damping ratios of 10%, 20% and 30% provided logical target damping ratios for the design iterations to determine the optimal design of the linear viscous dampers. Previous research, caps damping at 35%, typically this level of damping is the maximum that can be achieved economically with currently available viscous dampers (Christopoulos and Filiatrault, 2006).

2

Response Aceleration (g)

1.75 1.5 LA2 Response Spectra 5% 10% 20% 30% 35%

1.25 1 0.75 0.46g 0.5 0.25

0.28g 0.27g

0 0

0.4

0.8

1.2

1.6

2

2.4

2.8

3.2

Period

Figure 86: Spectra accelerations for LA2 under different damping ratios

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1.2

Response Acceleration (g)

1.05 0.9

LA7 Response Spectra 5% 10% 20% 30% 35%

0.75 0.6 0.39g

0.45 0.3 0.26g

0.15

0.25g

0 0

0.4

0.8

1.2

1.6

2

2.4

2.8

3.2

Period

Figure 87: Spectra accelerations for LA7 under different damping ratios 2

Response Acceleration (g)

1.75 1.5

LA16 Response Spectra 5% 10% 20% 30% 35%

1.25 0.99g

1 0.75 0.5 0.60g

0.59g

0.25 0 0

0.4

0.8

1.2

1.6

2

2.4

2.8

3.2

Period

Figure 88: Spectra accelerations for LA16 under different damping ratios

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64

Response displacement (cm)

56

LA2 Response Spectra 5% 10% 20% 30% 35%

48 40 32

19.4cm

24 16 8

9.9cm

9.1cm 0 0

0.4

0.8

1.2

1.6

2

2.4

2.8

3.2

Period

Figure 89: Spectra displacements for LA2 under different damping ratios

80

Response displacement (cm)

70 LA7 Response Spectra 5% 10% 20% 30% 35%

60 50 40 30

16.0cm 9.0cm

20 10

8.3cm 0 0

0.4

0.8

1.2

1.6

2

2.4

2.8

3.2

Period

Figure 90: Spectra displacements for LA7 under different damping ratios

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64

Response displacement (cm)

56

LA16 Response Spectra 5% 10% 20% 30% 35%

48 40

41.0cm

32 24

20.0cm

16

18.5cm

8 0 0

0.4

0.8

1.2

1.6

2

2.4

2.8

3.2

Period

Figure 91: Spectra displacements for LA16 under different damping ratios It is shown in Figure 86 to Figure 91 that not much reduction in terms of spectral displacements and spectra acceleration is achieved by using 35% damping. A 30% damping was chosen as the maximum criteria condition in order to remain well under the threshold limit for economic factors as well as to limit the force demand in the damper braces. Then the required fundamental period of the fictitiously braced structure is computed.

T 1t arg et 

T1 2   1

Having defined the target fundamental period, we proceed to compute the inter-story drifts in order to compute the inter-story stiffness needed for all the approaches. For this purpose a pair of 1000kN forces was applied at opposite direction at each floor. Table 27 shows a summary of the stiffness calculated for each floor and the braces stiffness calculated at each floor. Section 5.1 and section 5.2 on this chapter shows MathCAD worksheets used to calculate the damping coefficients using both the stiffness and the energy approach. For both calculations the stiffness highlighted in gray on Table 27 were used. Table 27: Summary of story stiffness

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TEAM 4

Stiffness proportional approach

T1  1.304 s

(Fundamental period of un-braced structure)

1  0.30

(Assumed damping ratio)

T1target 

T1 2 1  1

(Target fundamental period of the structured braced with the fictitious springs)

T1target  1.031s

 70.72   85.67    123.25 kN  ko    137.47 mm  164.2     95.04  T1tr  0.950 s

kfinal 

ko

(Trial value of the fundamental period of the fictitious braced structure obtained with first trial spring constants, from Ruamoko)

 T1target 2  T1tr2   1  T1target 2  T12   

 57   68    98  kN kfinal     110  mm  131     76  CL 

(Fictitious spring constants at proposed locations, from Ruamoko and proportional to drifts)

kfinal T1 2 

(Final spring constants)

(Damping coefficient of each viscous damper)

 12   14    20 kN s CL      23  mm  27     16  48

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TEAM 4

Constant damping approach

Nd  6

(Number of dampers)

Nf  6

(Number of floors)

T1  1.304 s

(Fundamental period of umbraced structure)

1  0.30

(Assumed damping ratio)

 97.95   118.65   170.71 kN  k    190.40 mm  227.43    105.44

(Fictitious spring constants at proposed locations, from Ruamoko and proportional to drifts)

 0.806   0.806    0.806      0.806   0.806     0.983 

(Inclination angle of the dampers)

 0.09   0.14    0.15      0.18   0.18     0.27 

(Inter-story drift at the storey where the damper is located )

Nf

1  T1



i1

CL 

k   2  i  i 

Nd

2  

2 2  i cos  i 

i1

CL  22.628s 

kN mm

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TEAM 4

First mode proportional damping

Nd  6

(Number of dampers)

Nf  6

(Number of floors)

T1  1.304 s

(Fundamental period of umbraced structure)

1  0.30

(Assumed damping ratio)

 97.95   118.65   170.71 kN  k    190.40 mm  227.43    105.44

(Fictitious spring constants at proposed locations, from Ruamoko and proportional to drifts)

 0.806   0.806    0.806      0.806   0.806     0.983 

 0.09   0.14    0.15      0.18   0.18     0.27  Nf

1  T1



i1

CL 

(Inclination angle of the damper is defined on the left and Inter-story drift at the storey where the damper is located is defined on the right)

k   2  i  i 

Nd

2  

3 2  i cos  i 

i1

CL  117.556s 

kN mm

CLfm CL 

 10.6   16.5    17.6  kN  CLfm  s  21.2  mm  21.2     31.7  50

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TEAM 4

Intermediate design

In the intermediate design performance indices were calculated for the target damping under consideration (30%) and compared with different damping ratios ranging from 10% to 45%. The MathCAD worksheet shown in section 5 in this chapter was used for different target damping ratios. Table 28 and Table 29 and Table 30 show a summary of the damping constants obtained for the desired damping ratio using the different approaches. Table 28: Stiffness proportional Approach damping coefficients

Table 29: Constant Damping Approach damping coefficients

Table 30: First Mode proportional Approach damping coefficients

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Based on the figures presented below we can infer that there is not significant improvement in terms of performance for damping ratios higher than 30%. It is for this reason that for the optimum design just performance indices corresponding to 30% damping will be compared as seen in Figure 95.

Figure 92: Optimum damping comparison (Stiffness Approach)

Figure 93: Optimum damping comparison (Constant damping Approach)

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Figure 94: Optimum damping comparison (First Mode proportional Approach)

Figure 95: Optimum damping approach It is clear that the first mode proportional method reaches the higher performance of the building. Therefore this method is chosen for the optimum design.

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TEAM 4

Final Design

Third method design was chosen with 30% damping ratio and the response parameters under the three ground motions are presented below. 7.1

Energy Balance

For all the ground motions the energy absorbed by the viscous dampers is almost the same as the input energy. Time History Energy Components LA - 02 3.6E+6

Energy (kN-mm)

3E+6

2.4E+6

1.8E+6

1.2E+6 Kinetic Energy Viscous Damping Strain Energy Total Energy Input Energy

6E+5

0 0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

Time (sec.)

Figure 96: Energy Components LA-02.

Time History Energy Components LA - 07 1.2E+6

Energy (kN-mm)

1E+6

8E+5

6E+5

4E+5 Kinetic Energy Viscous Damping Strain Energy Total Energy Input Energy

2E+5

0 0

10

20

30

40

50

60

70

80

90

100

Time (sec.)

Figure 97: Energy Components LA-07.

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Time History Energy Components LA - 16 3E+6

Energy (kN-mm)

2.4E+6

1.8E+6

1.2E+6

Kinetic Energy Viscous Damping Strain Energy Total Energy Input Energy

6E+5

0 0

5

10

15

20

25

30

Time (sec.)

Figure 98: Energy Components LA-16

7.2

Hinge Distribution

The number of hinges was considerably reduced compared to the existing structure performance. Sketches presenting the hinge formation are presented below for each ground motion .

Figure 99: Distribution of plastic hinges for LA-02.

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Table 31: Maximum plastic rotations for LA-02 ground motion. Hinge

Member

Prop.

Type

Lp (mm)

ϕp

Ductility Lp(Int) μ

1

22

12

Column

328

7.98E-05

1.173

38.474 0.0031

2

23

12

Column

328

7.98E-05

1.636

53.661 0.0043

3

56

22

Beam

678

3.84E-05

1.155

78.309 0.0030

4

57

23

Beam

678

3.84E-05

1.262

85.5636 0.0033

5

58

22

Beam

678

3.84E-05

1.210

82.038 0.0032

6

59

23

Beam

678

3.84E-05

1.192

80.8176 0.0031

θp

Figure 100: Distribution of plastic hinges for LA-07 and LA16.

Table 32: Maximum plastic rotations for LA-07 ground motion Hinge

Member

Prop.

Type

Lp (mm)

ϕp

1

57

23

Beam

678

3.84E-05

Ductility Lp(Int) μ 1.061

θp

71.936 0.0028

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Table 33: Maximum plastic rotations for LA-16 ground motion Hinge

Member

Prop.

Type

Lp (mm)

1

19

11

Column

696

2

21

3

Column

3

22

3

4

23

5 6 7 8

ϕp

Ductility Lp(Int) μ

θp

1.628

113.3

0.0004

328

3.7557E-06 7.98E-05

1.424

46.7

0.0037

Column

328

7.98E-05

3.817

125.2

0.0100

1

Column

328

7.98E-05

2.681

87.9

0.0070

24

2

Column

328

7.98E-05

1.417

46.5

0.0037

38

4

Beam

616

4.24E-05

1.117

68.8

0.0029

4

Beam

616

4.24E-05

1.326

81.7

0.0035

2

Beam

616

4.24E-05

2.158

132.9

0.0056

616

4.24E-05

1.345

82.9

0.0035

43 44

9

45

5

Beam

10

46

7

Beam

616

4.24E-05

2.049

126.2

0.0054

11

47

7

Beam

616

4.24E-05

1.491

91.8

0.0039

12

48

5

Beam

616

4.24E-05

2.13

131.2

0.0056

13

49

6

Beam

678

3.84E-05

2.009

136.2

0.0052

14

50

8

Beam

678

3.84E-05

2.801

189.9

0.0073

8

Beam

678

3.84E-05

2.289

155.2

0.0060

6

Beam

678

3.84E-05

2.953

200.2

0.0077

678

3.84E-05

2.332

158.1

0.0061

15 16

51 52

17

53

9

Beam

18

54

11

Beam

678

3.84E-05

2.647

179.5

0.0069

19

55

11

Beam

678

3.84E-05

2.55

172.9

0.0066

20

56

9

Beam

678

3.84E-05

3.196

216.7

0.0083

21

57

10

Beam

678

3.84E-05

2.653

179.9

0.0069

22

58

12

Beam

678

3.84E-05

2.303

156.1

0.0060

12

Beam

678

3.84E-05

2.193

148.7

0.0057

10

Beam

678

3.84E-05

3.097

23 24

59 60

209.9766 0.0081

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TEAM 4

Peak and Residual Inter-Story Drifts

Inter-Storey drift - Time history - Motion 02 35 30

1st floor 2nd floor 3rd floor 4th floor 5th floor Roof

1st floor peak 30.3 (mm)

25

Inter-Storey drift (mm)

20 15 10

1st floor residual 2.81 (mm)

5 0 -5

Roof residual 0.00111 (mm)

-10

Roof peak -6.6 (mm)

-15

5th floor peak -11.8 (mm)

-20

4th floor peak -15.1 (mm)

4th floor residual 0.00142 (mm) 3rd floor residual 0.0108 (mm) 5th floor residual 0.00132 (mm)

3rd floor peak -18.5 (mm)

-25

2nd floor peak -19.5 (mm)

2nd floor residual 0.0339 (mm)

-30 0

10

20

30

40

50

60

70

80

90

Time (s)

Figure 101: Inter-story drift time history motion La-02.

Inter-Storey drift - Time History - Motion 07 20 1st floor 2nd floor 3rd floor 4th floor 5th floor Roof

15

Inter - Storey drift (mm)

10 5

5th floor residual -0.000901 (mm) 3rd floor residual -0.00891 (mm)

0 -5 -10 Roof peak -4.53 (mm) 5th floor peak -9.67 (mm) 4th floor peak -13.1 (mm) 3rd floor peak -16.6 (mm) 2nd floor peak -17.9 (mm) 1st floor peak -27 (mm)

-15 -20 -25 -30 0

10

20

30

40

50

2nd floor residual -0.0394 (mm) Roof residual -0.000684 (mm) 4th floor residual -0.00162 (mm) 1st floor residual -0.876 (mm)

60

70

80

90

100

Time (s)

Figure 102: Inter-story drift time history motion La-07. 58

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Inter-Storey drift - Time History - Motion 16 60 1st floor peak 58.8 (mm) 2nd floor peak 38.8 (mm) 3rd floor peak 35.2 (mm)

50

Inter - Storey drift (mm)

40

1st floor 2nd floor 3rd floor 3rd floor residual 10.3 (mm) 4th floor 5th floor 2nd floor residual 12.2 (mm) Roof 1st floor residual 11.6 (mm)

4th floor peak 26.6 (mm) 30

5th floor peak 19.4 (mm)

20 10 0 -10 4th floor residual 4.6 (mm)

-20

Roof residual 0.143 (mm) Roof peak 9.71 (mm)

-30

5th floor residual 0.884 (mm)

-40 -50 0

3

6

9

12

15

18

21

24

27

30

Time (s)

Figure 103: Inter-story drift time history motion La-16

Comparison of peak

Peak Inter-storey Drifts

inter-storey drifts

25

25 20

15

10

Height (m.)

Height (m.)

20 Inter Storey Drifts LA-02 Retrof. Struc. LA-07 Retrof. Struc. LA-16 Retrof. Struc. 2.5% Drift (LS) 0.7% Drift (IO)

Inter Storey Drifts LA-02 Retrof. Struc. LA-07 Retrof. Struc. LA-16 Retrof. Struc. LA-02 Orig. Struc. LA-07 Orig. Struc. LA-16 Orig. Struc.

15

10

5

5 0 0

50

100

150

Displacement (mm.)

0 -150

-50

50

150

Displacement (mm.)

Figure 104: Peak inter-story drifts for LA-02, 07 and 16

Figure 105: Comparison of peak inter-story drifts

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Residual Inter-Storey Drifts

Comparison of residual inter-storey drifts

25

25 20

15

Residual Inter-Storey Drifts LA-02 Retrof. Struc. LA-07 Retrof. Struc. LA-16 Retrof. Struc. 1% Drift (LS)

10

Height (m)

Height (m)

20

15

Residual Inter-Storey Drifts LA-02 Orig. Struc. LA-07 Orig. Struc. LA-16 Orig. Struc. LA-02 Retrof. Struc. LA-07 Retrof. Struc. LA-16 Retrof. Struc.

10

5

5 0 0

15

30

45

60

0 -60

Displacement (mm)

-30

0

30

60

Displacement (mm)

Figure 106: Residual inter-story drifts for LA02, 07 and 16. 7.4

Figure 107: Comparison of residual inter-story drifts.

Accelerations

Acceleration History - Motion 02 0.8 Roof peak 0.64 1st floor peak 0.47 5th floor peak 0.468 2nd floor peak 0.385 4th floor peak 0.35 3rd floor peak 0.343

Total Acceleration (g)

0.6

0.4

0.2

0 1st floor 2nd floor 3rd floor 4th floor 5th floor Roof

-0.2

-0.4

-0.6 0

10

20

30

40

50

60

70

80

90

Time (s)

Figure 108: Acceleration history of motion LA-02.

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Acceleration History - Motion 07 0.35

5th floor peak 0.295

0.25

4th floor peak 0.26

0.2

Total Acceleration (g)

1st floor 2nd floor 3rd floor 4th floor 5th floor Roof

Roof peak 0.324

0.3

3rd floor peak 0.243

0.15

2nd floor peak 0.238

0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25

1st floor peak -0.264

-0.3 0

10

20

30

40

50

60

70

80

90

100

Time (s)

Figure 109: Acceleration history of motion LA-07.

Acceleration History - Motion 16 0.6 1st floor 2nd floor 3rd floor 4th floor 5th floor Roof

0.5 0.4

Total Acceleration (g)

0.3 0.2 0.1 0 -0.1 -0.2 2nd floor peak -0.491 1st floor peak -0.503

-0.3

3rd floor peak -0.515 4th floor peak -0.564 5th floor peak -0.62 Roof peak -0.673

-0.4 -0.5 -0.6 -0.7 0

5

10

15

20

25

30

Time (s)

Figure 110: Acceleration history of motion LA-16.

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Comparison of total peak accelerations 25

Height (m)

20

15

Peak Acceleration LA-02 Orig. Struc. LA-07 Orig. Struc. LA-16 Orig. Struc. LA-02 Retrof. Struc. LA-07 Retrof. Struc. LA-16 Retrof. Struc.

10

5

0 -1

-0.5

0

0.5

1

Acceleration (g)

Figure 111: Comparison of total peak accelerations. Significant ductility reduction is achieved when implementing viscous dampers. We can also notice Peak drift were reduced and overall the acceleration are reduced for all the motions. The performance of the structure improved from 36% to 65%. Summary of results are presented Table 34.

Table 34: Performance Indices for structure retrofitted with viscous dampers compared to existing building performance. Ground Motion LA-02 LA-07 LA-16 Ground Motion LA-02 LA-07 LA-16

 μ

μ

∆ (%)

∆ (%)

a(g)

PI

5.71 4.83 8.49

5.50 4.41 8.49

1.80 1.52 2.70

0.72 0.68 0.36

0.95 0.61 0.79

46% 60% 36%

 μ

μ

∆ (%)

∆ (%)

a(g)

PI

1.19 1.00 3.01

1.58 0.00 3.75

0.54 0.47 1.05

0.04 0.01 0.31

0.63 0.32 0.69

76% 88% 65%

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Flow chart for viscous dampers optimum design

A flow chart describing the procedure that was followed to achieve the optimum design is presented below.

Figure 112: Flow chart for optimum design for viscous dampers

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CHAPTER 6 - BASE ISOLATION

1

Description

The retrofit strategy for the structure consists on introducing lead-rubber bearings at the base of the structure as shown in Figure 113. For this purpose, it will be assumed that a large foundation mat supports the building and that retrofit work will be required to introduce a link-frame between the columns and this mat. The isolators will be installed between this link-frame and the top surface of the mat. For modeling purposes, it will be assumed that all bearings operate in parallel and the complete base isolation system will be modeled as a single horizontal bilinear spring at the base of the structure.

Figure 113: Modelling of Building Structure with Lead-Rubber Base-Isolation System The base isolation system used for this retrofit strategy is lead-rubber bearings. These isolation elements are comprised of two distinct components; a laminated rubber bearing and a lead core as seen in Figure 114.

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Figure 114: Components of Lead-Rubber base isolation The first component of this type of isolation system is the laminated rubber bearing which is the primary mechanism of the isolation system and consists of thin layers of rubber and steel shim plates laminated together in an alternating pattern as shown in the figure above. The physics behind the use of laminated rubber bearings for base isolation is that the lateral stiffness of the bearings is significantly less than that of the supported structure. Consequently, the objective of the use of a base isolation system is to provide a shift of the structure’s fundamental natural period out of the frequency range at which most buildings are more vulnerable to damage due to the affects of ground motion during a seismic event. The second component of a lead-rubber bearing isolator is the lead core plug. The stiffness of the laminated rubber bearing is low, providing little damping by itself and as a result is susceptible to large lateral displacements. The lead core element is introduced to compensate for this by providing an element to increase damping as well as to dissipate hysteretic energy. To model the Lead-rubber bearings in our RUAMOKO model we used a non-linear spring with a bi-linear hysteretic model. The bi-linear hysteretic model is shown in Figure 115.

Figure 115: Lead-Rubber Bi-Linear Model

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If the mechanical properties based on experimental tests are not known, the determination of the mechanical properties requires an iterative approach for the preliminary bearing design. The three parameters that define the bi-linear model are k1, k2 and Fy. Where k1 is the combined elastic stiffness of the laminated rubber and the lead core assemblage, k2 is the post-yield stiffness equal to the stiffness of only the laminated rubber, and Fy is the yield force at which the lead core starts to yield. For modeling of the base isolators in RUAMOKO, a fixed node was introduced at the ground level. The horizontal degrees of freedom at the base of all the column nodes at the ground level were released. A non-linear spring element was connected to the base node of one of the exterior bay columns. Next, the base nodes at the remainder of the ground level columns were slaved to horizontal degree of freedom of the aforementioned column. This is shown schematically in Figure 113. The determination of the hysteretic model of the isolators is an iterative approach. The preliminary approach was used to determine the bi-linear properties using a MathCAD worksheet developed linking the assumptions listed below. 

Overlap factor was fixed to 0.6. 

A'  0.6 Ar 

The diameter of the bearing was fixed based on the following recommendation: Db 



xb 0.8(1

A' ) Ar

The thickness of the rubber was computed based on the following recommendation:   

tr  

Db 4S

The shape factor was considered in the following interval 10  S  20



The total height of the rubber layers was set to remain in the following range:  

Db 2 Db  hr  3 3 Db  hiso 66

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tr  2ts



The number of isolator was fixed to 6 to ensure enough redundancy.

nisolators  6 

The plug diameter was contained in the following range

1 1 Db  Dp  Db 3 6 When considering the assumptions stated above to determine the optimum k1 and Fy of the base isolation system the procedure is simplified. By limiting the range at which hr will be evaluated we can limit at the same time the total elastic stiffness of the system since this is proportional to k2 as follows.

k2 

GrAr hr

k 1  10k 2

By limiting the diameter of the plug we are also bounding the yield strength range since Fy it is proportional to the area of the plug as shown in the equation below.

Fy   pyAp (1 

GrAr ) GpAp

As a result of limiting the rubber height and the plug diameter we were able to establish a range in which k1 and Fy can be evaluated to get the better performance. The base isolation properties that were given for the design of the lead-rubber bearings for this project required the maximum lateral displacement of each bearing to not exceed 300 mm. An iterative procedure was carried out in the MathCAD worksheet presented in the following page by assuming values of k1 and calculating an equivalent stiffness of the system. Having the equivalent stiffness we were able to calculate the equivalent period of the system and the equivalent damping as well. Then for the most critical spectral displacement spectrum (LA7), a spectral displacement corresponding to the equivalent period of the system was obtained. Different values of k1 were given in conjunction with the assumptions listed above to ensure that the spectral displacement equal the desired lateral displacement of the bearing (300mm).

xb  S D 67

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Preliminary Design

Gr  1 MPa

(Rubber Shear Modulus)

Gp  150 MPa

(Lead shear Modulus)

kr  2000 MPa

(Rubber compression modulus)

py  10 MPa

(Lead Shear Yield Strength)

xb  300 mm

(Maximum seismic displacement of lead-rubber bearings)

OverlapFactor  0.6

(Maximum overlap factor for individual bearings, A1/A)

S  12

(Shape factor for individual bearings)

 BldgWtTotal  30950kN

(Total Weight of structure)

v  4.5

(Short-term failure strain of the rubber)

w  0.4 v  1.8

(Allowable shear strain under gravity load

Dbearing  

   937.5 mm  0.8( 1  OverlapFactor )  xb

(Rounded rubber bearing)

Dbearing  940 mm Ar   



tr  



Dbearing 

Dbearing 4 S

2

2

(Rubber cross area)

   693978mm 

2

(Diameter of the rubber bearing)

  19.583 mm  

(Rubber thickness per layer) (Rounded rubber thickness)

tr  20 mm 2

A1  OverlapFactor  Ar  416387 mm

(Overlap area)

Wmax  A1 Gr S w  8994 kN

(Maximum allowable vertical load)

nisolators  

(Number of isolators)



nisolators  6

BldgWtTotal Wmax

  3.441  

(Rounded number of isolators)

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nlayers  20

(Number of rubber layers)

hr  nlayers  tr  400 mm

(Total rubber height)

Dplug  

(Lead plug diameter, taken between 1/3 and 1/6 of Dbearing)

  313.333mm   

Dbearing



3

(Rounded lead plug diameter)

Dplug  315 mm Ap    



k2 

Dplug 

hr

2

  34636 mm 

3

Gr Ar

2

 1.735

mm

2

2 Fy

 xb

teff  2   Fy k1

beff 

kN mm

 1177.877kN 

 k2  1 



keff  17.477

dy 

 52.048

nisolators  Fy

keff  

(Lead rubber approximation of the lateral elastic stiffness)

mm

  392.626kN   Gp Ap 

nisolators  k1

TFy 

kN

Gr Ar



Tk1 

(Lead rubber Post-yield stiffness)

kN

k1  10 k2  17.3494

Fy  py  Ap   1 

(Area of the lead plug)

Fy

 nisolators 

k1 xb 

(Yield force of the bearing) (Total elastic stiffness of the system) (Total yield force of the system) (Equivalent Stiffness of the system)

kN mm

BldgWtTotal keff g

 2.67s

(Equivalent period of the system)

 22.63 mm

2 TFy ( xb  dy ) 2

 0.132

(Equivalent damping of the system)

  keff xb

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640

Spectral Displacement (mm)

560 480 LA7 13.2%

400

300mm 320 240 160 80 0 0

0.5

1

1.5

2

2.5

3

3.5

4

Period (s)

Figure 116: Spectral Displacement corresponding to effective period of the equivalent system The parameters for which the spectral displacement matched the maximum lateral displacements are presented below: Table 35: Preliminary Design results Parameter k1 k2 Fy

Value 52 kN/mm 5.2 kN/mm 1178 kN

Having defined the Bi-linear Rubber-Lead parameters to be used for the equivalent non-linear spring element in RUAMOKO, time history analyses were performed for the three design ground motions. As expected from the spectral displacements plots the motion LA7 governed the design since it presented the higher displacements. These displacements were below the 300mm. limit and consequently the preliminary design was satisfied. It was previously stated that the values of k1 and Fy can be limited at certain range of application by following the listed assumptions presented above. It is for this reason that optimum values of these parameters will be seeks in the intermediate design with the aim of finding the optimum design parameters that meet the maximum lateral displacement and give at the same time the higher performance indices.

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Intermediate design

Same assumptions that were used in the preliminary design will be followed. The idea is to limit the range at which hr and Dp can be evaluated as shown in the listed assumptions in the preliminary design. By doing this the range at which k1 and Fy are evaluated can be contained and evaluated. Three values of k1 will be considered for the analyses. Two values corresponding to the upper and lower bound ok k1 and one intermediate point ok k1. The lower and upper bound were obtained by assuming different hr values contained in the following range boundaries as previously stated.

Db 2 Db  hr  3 3 Then for each value of k1 the optimum Fy was studied. Since for our design Fy is primarily conditioned to the area of the plug, and we limited the diameter of the plug to:

1 1 Db  Dp  Db 3 6 Therefore the upper and lower bound of Fy can also be established. This range of application was found to vary from 750kN to 2500kN. Performances indices were calculated for increments of 250kN in the range of 750kN to 2500kN for each of the three assumed values of k1. Table 36: Summary of parameters to be studied in intermediate design

In Table 36 the different values of k1 and the range of application of Fy is shown. Figure 117, Figure 118 and Figure 119 shows the performance indices obtained for each run.

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Figure 117: Optimum Fy study for k1=30kN/mm

Figure 118: Optimum Fy study for k1=45kN/mm

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Figure 119: Optimum Fy study for k1=65kN/mm From Figure 120 we can observe that the highest performance is obtained when k1 equals 30kN/mm and from Figure 117 it is evident that this happens when Fy is 750 kN. This optimum configuration reached a performance index of 81%.

Figure 120: Optimum Design

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MathCAD calculations to obtain the optimum parameters just presented are shown below. Gr  1 MPa

(Rubber Shear Modulus)

Gp  150 MPa

(Lead shear Modulus)

kr  2000 MPa

(Rubber compression modulus)

py  10 MPa

(Lead Shear Yield Strength)

xb  300 mm

(Maximum seismic displacement of lead-rubber bearings)

OverlapFactor  0.6

(Maximum overlap factor for individual bearings, A1/A)

S  13

(Shape factor for individual bearings)

BldgWtTotal  30950kN

(Total Weight of structure)

v  4.5

(Short-term failure strain of the rubber)

w  0.4 v  1.8

(Allowable shear strain under gravity load)

Dbearing  

  937.5 mm   0.8( 1  OverlapFactor )  xb

(Rounded rubber bearing)

Dbearing  940 mm Ar   



tr  



Dbearing 

Dbearing 4 S

2

2

(Rubber cross area)

   693978mm 

2

(Diameter of the rubber bearing)

  18.077 mm  

(Rubber thickness per layer) (Rounded rubber thickness)

tr  20 mm 2

A1  OverlapFactor  Ar  416387 mm

(Overlap area)

Wmax  A1 Gr S w  9743.4kN 

(Maximum allowable vertical load)

nisolators  

(Number of isolators)



BldgWtTotal Wmax

  3.176  

nisolators  6

(Rounded number of isolators)

nlayers  34

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hr  nlayers  tr  680 mm

(Total rubber height)

Dplug  

(Lead plug diameter, taken with 1/3 and 1/6 of Dbearing)



Dbearing 

   156.667mm 

6

Dplug  161 mm Ap    



k2 

Dplug 

Gr Ar hr

(Rounded lead plug diameter) 2

2

  20358 mm 

2

 1.021

kN

Fy  py  Ap   1 



Tk1 

TFy 

kN

(Lead rubber approximation of the lateral elastic stiffness)

mm

  249.848kN   Gp Ap  Gr Ar

nisolators  k1 2 nisolators  Fy 2

(Lead rubber Post-yield stiffness)

mm

k1  10 k2  10.2056

(Area of the lead plug)

 30.6

kN mm

(Yield force of the bearing)

(Total elastic stiffness of the system) (Total yield force of the system)

 750 kN

A summary of the optimum design is shown in Table 37. Table 37: Summary of Design Parameters Parameter Diameter of the bearing Diameter of the plug Rubber thickness / number of layers Shape factor Number of isolators

Value 940 mm 160 mm 20 mm / 34 13 6

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Final Design

We can see form the energy plots that in overall the input energy was reduced by half. This will reduce significantly the demand on the structure. The strain energy is mostly due to the yielding of the plug. Time History Energy Components LA - 02 1.8E+6

Energy (kN-mm)

1.5E+6

1.2E+6

9E+5

6E+5 Kinetic Energy Viscous Damping Strain Energy Total Energy Input Energy

3E+5

0 0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

Time (sec.)

Figure 121: Energy components time history for LA-02.

Time History Energy Components LA - 07 2.1E+6

1.8E+6

Energy (kN-mm)

1.5E+6

1.2E+6

9E+5

6E+5

Kinetic Energy Viscous Damping Strain Energy Total Energy Input Energy

3E+5

0 0

10

20

30

40

50

60

70

80

90

100

Time (sec.)

Figure 122: Energy components time history for LA-07. 76

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Time History Energy Components LA - 16 1.8E+6

Energy (kN-mm)

1.5E+6

1.2E+6

9E+5

6E+5 Kinetic Energy Viscous Damping Strain Energy Total Energy Input Energy

3E+5

0 0

5

10

15

20

25

30

Time (sec.)

Figure 123: Energy components time history for LA-16. No hinges were reported on the RUAMOKO output for the optimum base isolation design

Figure 124: Abscense of plastic hinges for LA-02, 07 and 16.

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Inter-Storey drift - Time history - Motion 02 15 2nd floor peak -10.1 (mm)

12 9

Inter-Storey drift (mm)

1st floor 2nd floor 3rd floor 4th floor 5th floor Roof

3rd floor peak -12.6 (mm)

6 3 0 -3 1st floor residual -0.0349 (mm) Roof residual -0.00949 (mm)

-6 -9

5th floor residual -0.0154 (mm)

-12

Roof peak -12.4 (mm) 4th floor peak -13.7 (mm) 1st floor peak -13.9 (mm)

-15 -18 0

5th floor peak -16.4 (mm) 10 20 30

3rd floor residual -0.0209 (mm) 2nd floor residual -0.0226 (mm) 4th floor residual -0.0177 (mm) 40

50

60

70

80

90

80

90

Time (s)

Figure 125: : Interstory drift time history for LA-02.

Inter-Storey drift - Time history - Motion 02 180 150

Inter-Storey drift (mm)

120 90 60 30 0 -30 -60 -90 -120 -150 0

10

20

30

40

50

60

70

Time (s)

Figure 126: Displacement time history for Bearings in Base isolation system LA-02.

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Inter-Storey drift - Time History - Motion 07 15 1st floor peak 14.1 (mm) 12.5 10

Inter - Storey drift (mm)

1st floor 2nd floor 3rd floor 4th floor 5th floor Roof

5th floor peak 13.6 (mm) 4th floor peak 12 (mm) 3rd floor peak 11.4 (mm)

7.5 5

2nd floor residual -0.0306 (mm) 1st floor residual -0.0473 (mm)

2.5 0 -2.5

Roof residual -0.0128 (mm) 5th floor residual -0.0209 (mm) 4th floor residual -0.0239 (mm) 3rd floor residual -0.0283 (mm)

-5 -7.5 Roof peak 10.1 (mm)

-10

2nd floor peak 9.72 (mm)

-12.5 -15 0

10

20

30

40

50

60

70

80

90

100

90

100

Time (s)

Figure 127: Interstory drift time history for LA-07.

Inter-Storey drift - Time history - Motion 07 200

Inter-Storey drift (mm)

150 100 50 0 -50 -100 -150 -200 0

10

20

30

40

50

60

70

80

Time (s)

Figure 128: Displacement time history for Bearings in Base isolation system LA-07.

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Inter-Storey drift - Time History - Motion 16 18

4th floor peak 13 (mm)

12

Inter - Storey drift (mm)

1st floor 2nd floor 3rd floor 4th floor 5th floor Roof

5th floor peak 15.7 (mm)

15

Roof peak 12.1 (mm)

9 6 3 0 -3

2nd floor residual -0.101 (mm)

-6

3rd floor residual -0.0929 (mm) 1st floor residual -0.157 (mm) 5th floor residual -0.0676 (mm)

-9 2nd floor peak -11.5 (mm) 3rd floor peak -11.9 (mm)

-12 -15

4th floor residual -0.078 (mm) Roof residual -0.0412 (mm)

1st floor peak -16.2 (mm)

-18 0

3

6

9

12

15

18

21

24

27

30

27

30

Time (s)

Figure 129: Interstory drift time history for LA-16.

Inter-Storey drift - Time history - Motion 16 250 200

Inter-Storey drift (mm)

150 100 50 0 -50 -100 -150 -200 -250 0

3

6

9

12

15

18

21

24

Time (s)

Figure 130: Displacement time history for Bearings in Base isolation system LA-16. 80

STRUCTURAL CONTROL PROJECT

Peak Inter-storey Drifts

TEAM 4

Comparison of peak inter-storey drifts

25

25 20

15

10

Height (m.)

Height (m.)

20 Inter Storey Drifts LA-02 Retrof. Struc. LA-07 Retrof. Struc. LA-16 Retrof. Struc. 2.5% Drift (LS) 0.7% Drift (IO)

Inter Storey Drifts LA-02 Orig. Struc. LA-07 Orig. Struc. LA-16 Orig. Struc. LA-02 Retrof. Struc. LA-07 Retrof. Struc. LA-16 Retrof. Struc.

15

10

5

5 0 0

50

100

150

0 -150

Displacement (mm.)

-50

50

150

Displacement (mm.)

Figure 131: Peak Inter-storey drifts for Retrofitted structure.

Residual Inter-Storey Drifts

Figure 132: Comparison of Peak inter-storey drifts.

Comparison of residual inter-storey drifts

25

25 20

15

Residual Inter-Storey Drifts 1% Drift (LS) LA-02 Retrof. Struc. LA-07 Retrof. Struc. LA-16 Retrof. Struc.

10

Height (m)

Height (m)

20

15

Residual Inter-Storey Drifts LA-02 Orig. Struc. LA-07 Orig. Struc. LA-16 Orig. Struc. LA-02 Retrof. Struc. LA-07 Retrof. Struc. LA-16 Retrof. Struc.

10

5

5 0 0

15

30

45

60

Displacement (mm)

0 -60

-30

0

30

60

Displacement (mm)

Figure 133: Residual Inter-storey drifts for Retrofitted structure.

Figure 134: Comparison of residual inter-storey drifts.

81

STRUCTURAL CONTROL PROJECT

TEAM 4

Acceleration History - Motion 02 0.45

Total Acceleration (g)

Roof peak 0.414 0.35

1st floor peak 0.263

0.25

2nd floor peak 0.264 3rd floor peak 0.172 4th floor peak 0.162

0.15 0.05 -0.05 1st floor 2nd floor 3rd floor 4th floor 5th floor Roof

-0.15 5th floor peak -0.232

-0.25 -0.35 0

10

20

30

40

50

60

70

80

90

70

80

90

Time (s)

Figure 135: Acceleration time history for LA-02.

Acceleration History - Motion 02 0.35 0.3 0.25

Top bearing peak 0.339

Total Acceleration (g)

0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25 -0.3 0

10

20

30

40

50

60

Time (s)

Figure 136: Acceleration time history for bearings in base isolation LA-02. 82

STRUCTURAL CONTROL PROJECT

TEAM 4

Acceleration History - Motion 07 0.2 0.15

Total Acceleration (g)

0.1 0.05 0 -0.05 -0.1 1st floor 2nd floor 3rd floor 4th floor 5th floor Roof

4th floor peak -0.135 3rd floor peak -0.159

-0.15 -0.2 -0.25

2nd floor peak -0.172 5th floor peak -0.179

-0.3

1st floor peak -0.185 Roof peak -0.326

-0.35 0

10

20

30

40

50

60

70

80

90

100

80

90

100

Time (s)

Figure 137: Acceleration time history for LA-07.

Acceleration History - Motion 07 0.2 0.15

Total Acceleration (g)

0.1 0.05 0 -0.05 -0.1 -0.15 Top bearing peak -0.217 -0.2 -0.25 0

10

20

30

40

50

60

70

Time (s)

Figure 138: Acceleration time history for bearings in base isolation LA-07.

83

STRUCTURAL CONTROL PROJECT

TEAM 4

Acceleration History - Motion 16 0.6 0.5 0.4

Total Acceleration (g)

0.3 2nd floor peak 0.181

0.2 0.1 0 -0.1

1st floor 2nd floor 3rd floor 4th floor 5th floor Roof

3rd floor peak -0.185 1st floor peak -0.211 4th floor peak -0.213 5th floor peak -0.234 Roof peak -0.423

-0.2 -0.3 -0.4 -0.5 -0.6 -0.7 0

5

10

15

20

25

30

Time (s)

Figure 139: Acceleration time history for LA-16.

Acceleration History - Motion 16 0.25 0.2

Total Acceleration (g)

0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 Top bearing peak -0.284 -0.25 -0.3 0

3

6

9

12

15

18

21

24

27

30

Time (s)

Figure 140: Acceleration time history for bearings in base isolation LA-16.

84

STRUCTURAL CONTROL PROJECT

TEAM 4

Comparison of total peak accelerations 25

Height (m)

20

15

Peak Acceleration LA-02 Orig. Struc. LA-07 Orig. Struc. LA-16 Orig. Struc. LA-02 Retrof. Struc. LA-07 Retrof. Struc. LA-16 Retrof. Struc.

10

5

0 -1

-0.5

0

0.5

1

Acceleration (g)

Figure 141: Comparison of peak accelerations. The building behaves in the elastic range, there is no plastic hinging formation, and therefore the residual drifts in the superstructure were zero. The building performance increases from 36% to 81%.

Table 38: Performance Indexes for structure retrofitted with base isolation compared with existing performance. Existing LA-02 LA-07 LA-16

 μ 5.71 4.83 8.49

μ 5.50 4.41 8.49

∆ (%) 1.80 1.52 2.70

∆ (%) 0.72 0.68 0.36

a(g) 0.95 0.61 0.79

PI 46% 60% 36%

Retrofitted LA-02 LA-07 LA-16

 μ
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