Structural Control
Short Description
Download Structural Control...
Description
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
2
Response Spectra................................................................................................................. 11
CHAPTER 3 - ANALYSIS OF THE ORIGINAL BUILDING .................................................... 13 1
Introduction ......................................................................................................................... 13
2
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
2
Procedure to calculate the optimum activation load ........................................................... 15
3
Fourier Spectra .................................................................................................................... 16
4
Preliminary design............................................................................................................... 17
5
Intermediate design ............................................................................................................. 20
6
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
2
Procedures to calculate the damping coefficients ............................................................... 39
3
Modeling of dampers .......................................................................................................... 40
4
Validation of the Damper element ...................................................................................... 42
5
Preliminary design............................................................................................................... 44 5.1
Stiffness proportional approach ................................................................................... 48
5.2
Constant damping approach ......................................................................................... 49
5.3
First mode proportional damping................................................................................. 50
6
Intermediate design ............................................................................................................. 51
7
Final Design ........................................................................................................................ 54
8
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
2
Preliminary Design .............................................................................................................. 68
3
Intermediate design ............................................................................................................. 71
4
Final Design ........................................................................................................................ 76
5
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
2
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
STRUCTURAL CONTROL PROJECT
TEAM 4
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
iii
STRUCTURAL CONTROL PROJECT
TEAM 4
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
STRUCTURAL CONTROL PROJECT
TEAM 4
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
STRUCTURAL CONTROL PROJECT
TEAM 4
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
STRUCTURAL CONTROL PROJECT
TEAM 4
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
vii
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
STRUCTURAL CONTROL PROJECT
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
STRUCTURAL CONTROL PROJECT
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
STRUCTURAL CONTROL PROJECT
TEAM 4
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..
4
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.
5
STRUCTURAL CONTROL PROJECT
1.3.2
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.
6
STRUCTURAL CONTROL PROJECT
TEAM 4
-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
7
STRUCTURAL CONTROL PROJECT
TEAM 4
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
STRUCTURAL CONTROL PROJECT
TEAM 4
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
9
STRUCTURAL CONTROL PROJECT
TEAM 4
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
2
STRUCTURAL CONTROL PROJECT
TEAM 4
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
STRUCTURAL CONTROL PROJECT
TEAM 4
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).
4
STRUCTURAL CONTROL PROJECT
TEAM 4
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
5
STRUCTURAL CONTROL PROJECT
TEAM 4
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
6
STRUCTURAL CONTROL PROJECT
TEAM 4
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.
7
STRUCTURAL CONTROL PROJECT
TEAM 4
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
8
STRUCTURAL CONTROL PROJECT
TEAM 4
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
9
STRUCTURAL CONTROL PROJECT
TEAM 4
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
10
STRUCTURAL CONTROL PROJECT
TEAM 4
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
11
STRUCTURAL CONTROL PROJECT
TEAM 4
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.
12
STRUCTURAL CONTROL PROJECT
TEAM 4
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.
13
STRUCTURAL CONTROL PROJECT
TEAM 4
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
STRUCTURAL CONTROL PROJECT
TEAM 4
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
STRUCTURAL CONTROL PROJECT
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
STRUCTURAL CONTROL PROJECT
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
STRUCTURAL CONTROL PROJECT
TEAM 4
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
12
STRUCTURAL CONTROL PROJECT
TEAM 4
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.
13
STRUCTURAL CONTROL PROJECT
TEAM 4
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.
14
STRUCTURAL CONTROL PROJECT
TEAM 4
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
STRUCTURAL CONTROL PROJECT
TEAM 4
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
STRUCTURAL CONTROL PROJECT
TEAM 4
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.
17
STRUCTURAL CONTROL PROJECT
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)
18
STRUCTURAL CONTROL PROJECT
TEAM 4
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.
19
STRUCTURAL CONTROL PROJECT
TEAM 4
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
STRUCTURAL CONTROL PROJECT
TEAM 4
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
STRUCTURAL CONTROL PROJECT
TEAM 4
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
STRUCTURAL CONTROL PROJECT
TEAM 4
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).
23
STRUCTURAL CONTROL PROJECT
TEAM 4
Figure 53: Optimum activation load study for HSS406-C1
Figure 54: Optimum activation load study for HSS304-C1
24
STRUCTURAL CONTROL PROJECT
TEAM 4
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
25
STRUCTURAL CONTROL PROJECT
TEAM 4
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
STRUCTURAL CONTROL PROJECT
TEAM 4
Figure 59: Optimum activation load study for HSS406&304C2 (3/1)
Figure 60: Optimum activation load study for HSS406&304C2 (4/1)
27
STRUCTURAL CONTROL PROJECT
TEAM 4
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
STRUCTURAL CONTROL PROJECT
6.1
TEAM 4
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
29
STRUCTURAL CONTROL PROJECT
TEAM 4
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.
30
STRUCTURAL CONTROL PROJECT
TEAM 4
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
31
STRUCTURAL CONTROL PROJECT
6.3
TEAM 4
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)
32
STRUCTURAL CONTROL PROJECT
TEAM 4
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
33
STRUCTURAL CONTROL PROJECT
TEAM 4
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.
34
STRUCTURAL CONTROL PROJECT
TEAM 4
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.
35
STRUCTURAL CONTROL PROJECT
TEAM 4
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%
36
STRUCTURAL CONTROL PROJECT
7
TEAM 4
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
37
STRUCTURAL CONTROL PROJECT
TEAM 4
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.
38
STRUCTURAL CONTROL PROJECT
TEAM 4
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.
39
STRUCTURAL CONTROL PROJECT
TEAM 4 Nf
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.
40
STRUCTURAL CONTROL PROJECT
TEAM 4
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
41
STRUCTURAL CONTROL PROJECT
4
TEAM 4
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.
42
STRUCTURAL CONTROL PROJECT
TEAM 4
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
43
STRUCTURAL CONTROL PROJECT
5
TEAM 4
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
44
STRUCTURAL CONTROL PROJECT
TEAM 4
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
45
STRUCTURAL CONTROL PROJECT
TEAM 4
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
46
STRUCTURAL CONTROL PROJECT
TEAM 4
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
47
STRUCTURAL CONTROL PROJECT
5.1
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
STRUCTURAL CONTROL PROJECT
5.2
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
i1
CL
k 2 i i
Nd
2
2 2 i cos i
i1
CL 22.628s
kN mm
49
STRUCTURAL CONTROL PROJECT
5.3
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
i1
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
i1
CL 117.556s
kN mm
CLfm CL
10.6 16.5 17.6 kN CLfm s 21.2 mm 21.2 31.7 50
STRUCTURAL CONTROL PROJECT
6
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
51
STRUCTURAL CONTROL PROJECT
TEAM 4
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)
52
STRUCTURAL CONTROL PROJECT
TEAM 4
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.
53
STRUCTURAL CONTROL PROJECT
7
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.
54
STRUCTURAL CONTROL PROJECT
TEAM 4
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.
55
STRUCTURAL CONTROL PROJECT
TEAM 4
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
56
STRUCTURAL CONTROL PROJECT
TEAM 4
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
57
STRUCTURAL CONTROL PROJECT
7.3
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
STRUCTURAL CONTROL PROJECT
TEAM 4
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
59
STRUCTURAL CONTROL PROJECT
TEAM 4
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.
60
STRUCTURAL CONTROL PROJECT
TEAM 4
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.
61
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 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%
62
STRUCTURAL CONTROL PROJECT
8
TEAM 4
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
63
STRUCTURAL CONTROL PROJECT
TEAM 4
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.
64
STRUCTURAL CONTROL PROJECT
TEAM 4
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
65
STRUCTURAL CONTROL PROJECT
TEAM 4
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
STRUCTURAL CONTROL PROJECT
TEAM 4
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
STRUCTURAL CONTROL PROJECT
2
TEAM 4
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)
68
STRUCTURAL CONTROL PROJECT
TEAM 4
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
69
STRUCTURAL CONTROL PROJECT
TEAM 4
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.
70
STRUCTURAL CONTROL PROJECT
3
TEAM 4
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.
71
STRUCTURAL CONTROL PROJECT
TEAM 4
Figure 117: Optimum Fy study for k1=30kN/mm
Figure 118: Optimum Fy study for k1=45kN/mm
72
STRUCTURAL CONTROL PROJECT
TEAM 4
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
73
STRUCTURAL CONTROL PROJECT
TEAM 4
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
(Number of rubber layers) 74
STRUCTURAL CONTROL PROJECT
TEAM 4
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
75
STRUCTURAL CONTROL PROJECT
4
TEAM 4
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
STRUCTURAL CONTROL PROJECT
TEAM 4
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.
77
STRUCTURAL CONTROL PROJECT
TEAM 4
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.
78
STRUCTURAL CONTROL PROJECT
TEAM 4
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.
79
STRUCTURAL CONTROL PROJECT
TEAM 4
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
View more...
Comments
