List of Tables Table 1: Design gravity loads _____________________________________________________________________ 3 Table 2: Material properties ______________________________________________________________________ 5 Table 3: Geometric and Elastic Member Properties ___________________________________________________ 6 Table 4: Description of the frame members __________________________________________________________ 6 Table 5: Column axial load – moment interaction _____________________________________________________ 7 Table 6: Plastic Curvature of each element for a plastic Rotation limit __________________________________ 8 Table 7: Curvature ductility capacity at failure _______________________________________________________ 9 Table 8: Frequencies and periods __________________________________________________________________ 9 Table 9: Mass participation ratios _________________________________________________________________ 9 Table 10: Lateral Load Distribution, ASCE 41 _________________________________________________________ 3 Table 11: Lateral Load Distribution, Linear vertical ____________________________________________________ 3 Table 12: Lateral Load Distribution, New Zealand Code ________________________________________________ 3 Table 13: Fraction of Input Energy Absorbed. _______________________________________________________ 15 Table 14: Peak Absorbed Energy. _________________________________________________________________ 16 Table 15: Energy Balance Error. __________________________________________________________________ 16 Table 16: Maximum plastic rotations for LA‐02 ground motion __________________________________________ 3 Table 17: Maximum plastic rotations for LA‐07 ground motion. _________________________________________ 4 Table 18: Maximum plastic rotations for LA‐16 ground motion. _________________________________________ 3 Table 19: Reponse limits for different performance category ___________________________________________ 12 Table 20: Performance Indexes for design ground motions ____________________________________________ 13 Table 21: Parameters __________________________________________________________________________ 19 Table 22: Parameters __________________________________________________________________________ 22 Table 23: Maximum plastic rotations for LA‐02 ground motion. ________________________________________ 31 Table 24: Maximum plastic rotations for LA‐16 ground motion. ________________________________________ 31 Table 25: Performance Indexes for structure retrofitted with hysteretic dampers compared to the original performance. ________________________________________________________________________________ 36 Table 26: Validation of damper element ___________________________________________________________ 43 Table 27: Summary of story stiffness ______________________________________________________________ 47 Table 28: Stiffness proportional Approach damping coefficients ________________________________________ 51 Table 29: Constant Damping Approach damping coefficients __________________________________________ 51 Table 30: First Mode proportional Approach damping coefficients ______________________________________ 51 Table 31: Maximum plastic rotations for LA‐02 ground motion. ________________________________________ 56 Table 32: Maximum plastic rotations for LA‐07 ground motion _________________________________________ 56 Table 33: Maximum plastic rotations for LA‐16 ground motion _________________________________________ 57 Table 34: Performance Indices for structure retrofitted with viscous dampers compared to existing building performance. ________________________________________________________________________________ 62 Table 35: Preliminary Design results ______________________________________________________________ 70 Table 36: Summary of parameters to be studied in intermediate design __________________________________ 71 Table 37: Summary of Design Parameters __________________________________________________________ 75 Table 38: Performance Indexes for structure retrofitted with base isolation compared with existing performance. 85 Table 39: Summary of various retrofit options ______________________________________________________ 87 Table 40: Performance level category _____________________________________________________________ 88 Table 41: Maximum plastic rotations for Near Fault ground motion in existing structure. ____________________ 90 Table 42: Performance Indexes of existing structure for near fault ground motion. _________________________ 92 Table 43: Performance of existing and retrofitted structure for the near fault ground motion _________________ 96
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List of figures Figure 1: Plan view, building to be retrofitted ________________________________________________________ 2 Figure 2: Elevation view Axis A – E. ________________________________________________________________ 2 Figure 3: Bi‐Linear Moment‐Curvature Model. _______________________________________________________ 3 Figure 4: Strength Degradation Model for Welded Beam‐Column Connections. _____________________________ 4 Figure 5: Elevation view with position of all nodes and members. ________________________________________ 5 Figure 6: Axial Load – Bending Moment Interaction Diagram ___________________________________________ 7 Figure 7: Mode Shapes of the structure _____________________________________________________________ 2 Figure 8: Pushovers curves _______________________________________________________________________ 4 Figure 9: Top floor lateral displacement vs. time ______________________________________________________ 4 Figure 10: Deflected Shape ASCE 41 Pushover, Plastic Hinge Location _____________________________________ 5 Figure 11: Pushover Curve ASCE 41 ________________________________________________________________ 5 Figure 12: Base Shear vs. Time, ASCE 41. ____________________________________________________________ 6 Figure 13: Moment vs. Time, beam member 58 end 2 _________________________________________________ 6 Figure 14: Moment vs. time – Column member 23 ____________________________________________________ 7 Figure 15: Moment vs. Curvature, Member 58 end 2 __________________________________________________ 7 Figure 16: 1st Floor Beam Failure at end 1 __________________________________________________________ 8 Figure 17: 1st Floor Beam Failure at end 2 __________________________________________________________ 8 Figure 18: Bottom Storey Columns Failure at end 1 ___________________________________________________ 8 Figure 19: Bottom Storey Columns Failure at end 2 ___________________________________________________ 9 Figure 20: LA‐02 Ground Motion _________________________________________________________________ 10 Figure 21: LA‐07 Ground Motion _________________________________________________________________ 10 Figure 22: LA‐16 Ground Motion _________________________________________________________________ 11 Figure 23: Absolute Acceleration Response Spectra for 5% Damping _____________________________________ 11 Figure 24: Relative Velocity Response Spectrum _____________________________________________________ 12 Figure 25: Relative Displacement Response Spectrum ________________________________________________ 12 Figure 26: Energy Components LA‐02 _____________________________________________________________ 14 Figure 27: Energy Components LA‐07 _____________________________________________________________ 14 Figure 28: Energy Components LA‐16. _____________________________________________________________ 14 Figure 29: Distribution of Plastic Hinges for LA‐02 ___________________________________________________ 17 Figure 30: Distribution of Plastic Hinges for LA‐07 ___________________________________________________ 17 Figure 31: Distribution of Plastic Hinges for LA‐16 ___________________________________________________ 17 Figure 32: Inter‐story Drift – Time History LA‐02 ______________________________________________________ 4 Figure 33: Inter‐story Drift – Time History LA‐07 ______________________________________________________ 5 Figure 34: Inter‐story drift – Time History LA‐16 ______________________________________________________ 5 Figure 35: Peak inter‐story drifts for LA‐02, LA‐07 and LA‐16. ___________________________________________ 6 Figure 36: Peak inter‐story drifts. __________________________________________________________________ 6 Figure 37: Residual inter‐story drifts for LA‐02, LA‐07 and LA‐16 _________________________________________ 7 Figure 38: Residual inter‐story drifts for LA‐02, LA‐07 and LA‐16 _________________________________________ 7 Figure 39: Acceleration time ‐ history for LA‐02 ______________________________________________________ 8 Figure 40: Acceleration time ‐ history for LA‐07 ______________________________________________________ 8 Figure 41: Acceleration time ‐ history for LA‐16 ______________________________________________________ 9 Figure 42: Peak acceleration for LA‐02, LA‐07 and LA‐16 _______________________________________________ 9 Figure 43: Peak total accelerations _______________________________________________________________ 10 Figure 44: Performance Levels (FEMA 273) _________________________________________________________ 11 Figure 45: Locations of added bracing and hysteretic dampers (Configuration‐C1) __________________________ 14 Figure 46: Elasto‐Plastic Hysteresis _______________________________________________________________ 15 Figure 47: Fourier Spectra ______________________________________________________________________ 16
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Figure 48: Preliminary design ____________________________________________________________________ 20 Figure 49: Sections ____________________________________________________________________________ 21 Figure 50: Alternative retrofit scheme considered in the analyses (Configuration‐C2) ________________________ 21 Figure 51: Optimum size study ___________________________________________________________________ 22 Figure 52: Optimum size study ___________________________________________________________________ 23 Figure 53: Optimum activation load study for HSS406‐C1 _____________________________________________ 24 Figure 54: Optimum activation load study for HSS304‐C1 _____________________________________________ 24 Figure 55: Optimum activation load study for HSS406&304‐C1 _________________________________________ 25 Figure 56: Optimum Design _____________________________________________________________________ 25 Figure 57: Optimum activation load study for HSS406&304C2 (1/1) _____________________________________ 26 Figure 58: Optimum activation load study for HSS406&304C2 (2/1) _____________________________________ 26 Figure 59: Optimum activation load study for HSS406&304C2 (3/1) _____________________________________ 27 Figure 60: Optimum activation load study for HSS406&304C2 (4/1) _____________________________________ 27 Figure 61: Optimum activation load study for HSS406&304C2 (1st Mode proportional) ______________________ 28 Figure 62: Optimum Design _____________________________________________________________________ 28 Figure 63: Energy Components LA‐02 _____________________________________________________________ 29 Figure 64: Energy Components LA‐07 _____________________________________________________________ 29 Figure 65: Energy Components LA‐16 _____________________________________________________________ 30 Figure 66: Distribution of plastic hinges for LA‐02 and LA‐16. __________________________________________ 30 Figure 67: Inter‐story drift time history motion La‐02. ________________________________________________ 32 Figure 68: Inter‐story drift time history motion La‐07. ________________________________________________ 33 Figure 69: Inter‐story drift time history motion La‐16. ________________________________________________ 33 Figure 70: Peak inter‐story drifts for LA‐02, 07 and 16 ________________________________________________ 33 Figure 71: Comparison of peak inter‐story drifts _____________________________________________________ 33 Figure 72: Residual inter‐story drifts ______________________________________________________________ 34 Figure 73: Comparison of residual inter‐story drifts __________________________________________________ 34 Figure 74: Acceleration history of motion LA‐02. _____________________________________________________ 34 Figure 75: Acceleration history of motion LA‐07. _____________________________________________________ 35 Figure 76: Acceleration history of motion LA‐16. _____________________________________________________ 35 Figure 77: Comparison of total peak accelerations. __________________________________________________ 36 Figure 78: Flow Chart for hysteretic dampers optimum design __________________________________________ 37 Figure 79: Location of added bracing and viscous dampers ____________________________________________ 38 Figure 80: Hysteretic Behavior of Viscous Dampers __________________________________________________ 39 Figure 81: Plot showing comparison among viscous damping and Rayleigh damping ________________________ 41 Figure 82: Model View _________________________________________________________________________ 42 Figure 83: Displacement time history _____________________________________________________________ 42 Figure 84: Spring and viscous damper forces ________________________________________________________ 43 Figure 85: Spring and viscous damping force _______________________________________________________ 43 Figure 86: Spectra accelerations for LA2 under different damping ratios __________________________________ 44 Figure 87: Spectra accelerations for LA7 under different damping ratios __________________________________ 45 Figure 88: Spectra accelerations for LA16 under different damping ratios _________________________________ 45 Figure 89: Spectra displacements for LA2 under different damping ratios _________________________________ 46 Figure 90: Spectra displacements for LA7 under different damping ratios _________________________________ 46 Figure 91: Spectra displacements for LA16 under different damping ratios ________________________________ 47 Figure 92: Optimum damping comparison (Stiffness Approach) _________________________________________ 52 Figure 93: Optimum damping comparison (Constant damping Approach) ________________________________ 52 Figure 94: Optimum damping comparison (First Mode proportional Approach) ____________________________ 53 Figure 95: Optimum damping approach ___________________________________________________________ 53 Figure 96: Energy Components LA‐02. _____________________________________________________________ 54 Figure 97: Energy Components LA‐07. _____________________________________________________________ 54 Figure 98: Energy Components LA‐16 _____________________________________________________________ 55 Figure 99: Distribution of plastic hinges for LA‐02. ___________________________________________________ 55 Figure 100: Distribution of plastic hinges for LA‐07 and LA16. __________________________________________ 56
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Figure 101: Inter‐story drift time history motion La‐02. _______________________________________________ 58 Figure 102: Inter‐story drift time history motion La‐07. _______________________________________________ 58 Figure 103: Inter‐story drift time history motion La‐16 ________________________________________________ 59 Figure 104: Peak inter‐story drifts for LA‐02, 07 and 16 _______________________________________________ 59 Figure 105: Comparison of peak inter‐story drifts ____________________________________________________ 59 Figure 106: Residual inter‐story drifts for LA‐02, 07 and 16. ____________________________________________ 60 Figure 107: Comparison of residual inter‐story drifts. _________________________________________________ 60 Figure 108: Acceleration history of motion LA‐02. ____________________________________________________ 60 Figure 109: Acceleration history of motion LA‐07. ____________________________________________________ 61 Figure 110: Acceleration history of motion LA‐16. ____________________________________________________ 61 Figure 111: Comparison of total peak accelerations. _________________________________________________ 62 Figure 112: Flow chart for optimum design for viscous dampers ________________________________________ 63 Figure 113: Modelling of Building Structure with Lead‐Rubber Base‐Isolation System _______________________ 64 Figure 114: Components of Lead‐Rubber base isolation _______________________________________________ 65 Figure 115: Lead‐Rubber Bi‐Linear Model __________________________________________________________ 65 Figure 116: Spectral Displacement corresponding to effective period of the equivalent system ________________ 70 Figure 117: Optimum Fy study for k1=30kN/mm ____________________________________________________ 72 Figure 118: Optimum Fy study for k1=45kN/mm ____________________________________________________ 72 Figure 119: Optimum Fy study for k1=65kN/mm ____________________________________________________ 73 Figure 120: Optimum Design ____________________________________________________________________ 73 Figure 121: Energy components time history for LA‐02. _______________________________________________ 76 Figure 122: Energy components time history for LA‐07. _______________________________________________ 76 Figure 123: Energy components time history for LA‐16. _______________________________________________ 77 Figure 124: Abscense of plastic hinges for LA‐02, 07 and 16. ___________________________________________ 77 Figure 125: : Interstory drift time history for LA‐02. __________________________________________________ 78 Figure 126: Displacement time history for Bearings in Base isolation system LA‐02. _________________________ 78 Figure 127: Interstory drift time history for LA‐07. ___________________________________________________ 79 Figure 128: Displacement time history for Bearings in Base isolation system LA‐07. _________________________ 79 Figure 129: Interstory drift time history for LA‐16. ___________________________________________________ 80 Figure 130: Displacement time history for Bearings in Base isolation system LA‐16. _________________________ 80 Figure 131: Peak Inter‐storey drifts for Retrofitted structure. ___________________________________________ 81 Figure 132: Comparison of Peak inter‐storey drifts. __________________________________________________ 81 Figure 133: Residual Inter‐storey drifts for Retrofitted structure. ________________________________________ 81 Figure 134: Comparison of residual inter‐storey drifts. ________________________________________________ 81 Figure 135: Acceleration time history for LA‐02. _____________________________________________________ 82 Figure 136: Acceleration time history for bearings in base isolation LA‐02. ________________________________ 82 Figure 137: Acceleration time history for LA‐07. _____________________________________________________ 83 Figure 138: Acceleration time history for bearings in base isolation LA‐07. ________________________________ 83 Figure 139: Acceleration time history for LA‐16. _____________________________________________________ 84 Figure 140: Acceleration time history for bearings in base isolation LA‐16. ________________________________ 84 Figure 141: Comparison of peak accelerations. ______________________________________________________ 85 Figure 142: Flow chart for optimum design of base isolation systems ____________________________________ 86 Figure 143: Near fault ground motion horizontal component___________________________________________ 88 Figure 144: Energy components time history for Near Fault Ground motion. ______________________________ 89 Figure 145: Distribution of plastic hinges for Existing Structure. _________________________________________ 89 Figure 146: Inter storey ‐ drifts time history for Near Fault ground motion. ________________________________ 91 Figure 147: Acceleration time history for Near Fault ground motion. _____________________________________ 91 Figure 148: Energy components time history for Retrofitted structure. ___________________________________ 92 Figure 149: Inter storey ‐ drifts time history for retrofitted structure. ____________________________________ 93 Figure 150: Displacement time history for Bearings in Base isolation system. ______________________________ 93 Figure 151: Acceleration time history for retrofitted structure. _________________________________________ 94 Figure 152: Acceleration time history for bearings in base isolation. _____________________________________ 94 Figure 153: Comparison of peak inter‐storey drifts. __________________________________________________ 95
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Figure 154: Comparison of residual inter‐storey drifts. ________________________________________________ 95 Figure 155: Comparison of peak accelerations. ______________________________________________________ 95 Figure 156: Performance of the existing building compared to the optimum retrofit strategy _________________ 96 Figure 157: Details for the composite section _______________________________________________________ 98
1
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.
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Figure 1: Plan view, building to be retrofitted
Figure 2: Elevation view Axis A – E.
NORTH
7.315
21.945
9.144 9.144 9.144 9.144
36.576
B C D E
4
3
2
1
A
7.315
7.315
5.486
3.810
3.810
3.810
3.810
3.810
24.536
1 2 3 4
W 24 x 76
W 24 x 76
W 27 x 94
W 30 x 99
W 30 x 99
W 27 x 94
W 3
0 x
173
W 3
0 x
173
W 2
7 x
146
W 2
4 x
104
W 2
7 x
146
W 2
4 x
104
W 1
4 x
193
W 1
4 x
109
W 1
4 x
159
W 1
4 x
193
W 1
4 x
159
W 1
4 x
109
Gravity Columns
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1.2 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 Live Load Roof Floor Exterior Cladding
3.8 kPA Roof 1.0 kPA 3.8 kPA 4.5 kPA Floor
1.7 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.
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
Curvature
Ben
din
g M
omen
t
Bilinear Moment Curvature Model
0 9
0
4.5
9
y ult
Mp
1.2Mpr=2%
M
(p=0.03 rad)1
EI
10.02EI
p
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assumed to be stiff and strong enough to avoid any panel shear deformation and yielding under strong earthquakes. All hysteretic energy must be dissipated through plastic hinging in the beams and the columns. Gravity loads acting on the frame during the earthquake are the roof and floor dead loads, the weight of the exterior walls, and a portion of the floor live load (0.7 kPa). P-delta effects are accounted for in the analyses. Rayleigh damping of 5% based on the first two elastic modes of vibration of the structure is assigned. All analyses are performed at a time-step increment of 0.002 s. To capture the brittle failure of the welded beam-to-column connections, the flexural strength degradation model shown in Figure 4, is introduced at the ends of the beam and column elements. The strength degradation begins at a curvature ductility of 11.0. At a curvature ductility of 11.55, the strength reduces 1% of the yield moment.
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..
Curvature Ductility
Mu
ltip
lier
on Y
ield
Mom
ent
Strength Degradation Model
0 5 10 15
0
0.2
0.4
0.6
0.8
1
11 11.55
0.01
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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 G 77 GPa
Yield Stress σ 290 MPa.
12
34
56
7
9 1011
1213
14 15
1718
1920
2122 23
2526
2728
2930 31
3334
3536
3738 39
4142 43 44 45 46 47
49 50 51 52
8
16
24
32
40
48
53
1 2 3 4
5 6 7 8
9 10 11 12
13 14
15
16
17
18
1920
21 22 23 24
25 26 27 28 29 30
31 32 33 34 35 36
37 38 39 40 41 42
43 44 45 46 47 48
49 50 51 52 53 54
55 56 57 58 59 60
61
62
63
64
65
66
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1.3.2 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.
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.
The axial load-moment interaction diagram were calculated for each of the column members and plotted in the Figure 6 with the respective coordinates listed in Table 5.
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Figure 6: Axial Load – Bending Moment Interaction Diagram
Table 5: Column axial load – moment interaction
Section 1, 2 N (kN) -5987 -1198 0.0 0.0 5987
M (kN-mm) 0.0 8.21E+5 9.12E+5 9.12E+5 0.0
Section 3,4 N (kN) -5725 -1145 0.0 0.0 5725
M (kN-mm) 0.0 1.24E+6 1.37E+6 1.37E+6 0.0
Section 5,6 N (kN) -8737 -1748 0.0 0.0 8737
M (kN-mm) 0.0 1.23E+6 1.36E+6 1.36E+6 0.0
Section 7, 8 N (kN) -8027 -1605 0.0 0.0 8027
M (kN-mm) 0.0 1.97E+6 2.19E+6 2.19E+6 0.0
Section 9, 10 N (kN) -10627 -2125 0.0 0.0 10627
M (kN-mm) 0.0 1.52E+6 1.69E+6 1.69E+6 0.0
Section 11, 12 N (kN) -9505 -1901 0.0 0.0 9505
M (kN-mm) 0.0 2.59E+06 2.88E+6 2.88E+6 0.0
1.4 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 moment-curvature 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
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:
ϕ My
EI
The plastic curvature is defined as:
ϕ 0.2
0.02 ϕ
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
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.
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ϕ θ
l
In accordance with Figure 3 the strength degradation should begin at a ductility ratio value of 11.0 in which the ductility ratio ( ) is defined by:
21 - 24 W30x99 678 1.28E+6 1.66E+9 3.85E-6 4.42E-5 4.81E-5 0.23 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 Frequency (Hz)
Period (s)
1 0.77 1.30
2 2.20 0.45
3 4.04 0.25
4 6.41 0.16
5 9.00 0.11
Table 9: Mass participation ratios
MODE % Mass
1 87
2 96
3 99
4 100
5 100
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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.
x vxF C V
1
kx x
vx nk
i ii
w hC
w h
Mode 1
0 0.25 0.5 0.75 10
5
10
15
20
25
Mode 2
-1 -0.5 0 0.5 10
5
10
15
20
25
Mode 3
-1 -0.5 0 0.5 10
5
10
15
20
25
Mode 4
-1 -0.5 0 0.5 10
5
10
15
20
25
Mode 5
-1 -0.5 0 0.5 10
5
10
15
20
25
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Table 10: Lateral Load Distribution, ASCE 41
Weight
(kN) H
(m) Elevation
(m) Distribution
floor 6 1815.5 3.81 24.536 0.248
floor 5 2514.8 3.81 20.726 0.272
floor 4 2514.8 3.81 16.916 0.205
floor 3 2514.8 3.81 13.106 0.143
floor 2 2514.8 3.81 9.296 0.088
floor 1 2599.1 5.486 5.486 0.044
Total shear 1.000
1.6.2 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
Weight
(kN) H
(m) Elevation
(m) Distribution
floor 6 1815.53 3.81 24.536 0.272
floor 5 2514.8 3.81 20.726 0.230
floor 4 2514.8 3.81 16.916 0.188
floor 3 2514.8 3.81 13.106 0.146
floor 2 2514.8 3.81 9.296 0.103
floor 1 2599.13 5.486 5.486 0.061
Sum 1.000
1.6.3 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
Weight
(kN) H
(m) Elevation
(m) Distribution
floor 6 1815.53 3.81 24.536 0.331
floor 5 2514.8 3.81 20.726 0.212
floor 4 2514.8 3.81 16.916 0.173
floor 3 2514.8 3.81 13.106 0.134
floor 2 2514.8 3.81 9.296 0.095
floor 1 2599.13 5.486 5.486 0.056
Sum 1.000
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For each of the three different load patterns for pushover analysis, the corresponding curves are plotted in Figure 8 indicating the failure point for the structure. Figure 9 shows the steady increase of top floor lateral displacement versus time, which indicates that static pushover load increase is achieved, no dynamic effects is present.
Figure 8: Pushovers curves
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).
Top floor lateral displacement (mm)
Bas
e sh
ear
(kN
)
0 50 100 150 200 250 300 350 400 450 500 5500
250
500
750
1000
1250
1500
1750
2000
2250
2500
2750
3000
3250
3500
511.84, 3172
486.35, 3212.1
475.14, 3269.2
ASCE 41LinearNZ code
Time (s)
Dis
plac
emen
t (m
m)
0 1 2 3 4 5 6 7 8 9 100
100
200
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
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Figure 10: Deflected Shape ASCE 41 Pushover, Plastic Hinge Location
For Pushover curve for ASCE 41 load pattern, the first and second yield points are indicated in Figure 11. In the same fashion the plot of base shear in time indicating first and second yield point in Figure 12.
Figure 11: Pushover Curve ASCE 41
Top floor lateral displacement (mm)
Ba
se s
hea
r (k
N)
0 50 100 150 200 250 300 350 400 450 500 5500
250
500
750
1000
1250
1500
1750
2000
2250
2500
2750
3000
3250
3500
(486.35, 3212.1)
First yield (100.11, 2074.3)
Second yield (104.77, 2163.6)
STRUCTURAL CONTROL PROJECT TEAM 4
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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.
Figure 13: Moment vs. Time, beam member 58 end 2
Time (s)
Ba
se s
hea
r (k
N)
0 1 2 3 4 5 6 7 8 9 100
500
1000
1500
2000
2500
3000
3500
4000
Max base shear of 3212.1 (kN) at 8.8 (s)
First yield (5.2, 2074.3)Second yield (5.5, 2163.6)
Time (s)
Ben
ding
mo
men
t (kN
-m)
0 1 2 3 4 5 6 7 8 9 100
-200
-400
-600
-800
-1000
-1200
-1400
-1600
-1800
Fail at 8.9 (s) M= -1779.8
Yield at 5.2 (s) M= -1478.8
STRUCTURAL CONTROL PROJECT TEAM 4
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Figure 14: Moment vs. time – Column member 23
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.
Time (s)
Ben
din
g m
omen
t (kN
-m)
0 1 2 3 4 5 6 7 8 9 10-300000
0
300000
600000
900000
1200000
1500000
1800000
2100000
2400000
2700000
3000000
3300000Yields at 5.5 (s) M= 2642700
Fails at 8.8 (s) M= 3174800
Curvature (rad)
Be
ndin
g m
om
en
t (kN
-m)
0 -1E-5 -2E-5 -3E-5 -4E-5 -5E-5 -6E-50
-200
-400
-600
-800
-1000
-1200
-1400
-1600
-1800
-4.9133E-5, -1779.8
-4.4517E-6, -1478.8
STRUCTURAL CONTROL PROJECT TEAM 4
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Figure 16: 1st Floor Beam Failure at end 1
Figure 17: 1st Floor Beam Failure at end 2
Figure 18: Bottom Storey Columns Failure at end 1
Time(s)
Mom
ent-
End
1 (
kN-m
m)
0 1 2 3 4 5 6 7 8 9 10-200000
0
200000
400000
600000
800000
1000000
1200000
1400000
1600000
1800000
MembersMember 55Member 57Member 59
Time(s)
Mo
me
nt-E
nd
2 (
kN-m
m)
0 1 2 3 4 5 6 7 8 9 10-1800000
-1600000
-1400000
-1200000
-1000000
-800000
-600000
-400000
-200000
0
8.9
MembersMember 56Member 58Member 60
Time(s)
Mo
me
nt-E
nd
1 (
kN-m
m)
0 1 2 3 4 5 6 7 8 9 10-3000000
-2700000
-2400000
-2100000
-1800000
-1500000
-1200000
-900000
-600000
-300000
0
300000
600000
9.0
MembersMember 21Member 22Member 23Member 24
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Figure 19: Bottom Storey Columns Failure at end 2
Time(s)
Mom
ent-
End
2 (
kN-m
m)
0 1 2 3 4 5 6 7 8 9 10-300000
0
300000
600000
900000
1200000
1500000
1800000
2100000
2400000
2700000
3000000
3300000
8.8
MembersMember 21Member 22Member 23Member 24
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CHAPTER 2 – DESIGN GROUND MOTIONS
1 Retrieval and analysis of Design Ground motions
A seismic assessment for this building is based on a non-linear time history dynamic analysis. Three historical recording for ground motions in Los Angeles region, are used in the analysis mentioned early. The ground motions were scaled to match 10% probability of exceedance in 50 years corresponding to a design based earthquake based on current building code. The first accelerogram (Figure 20) corresponds to the fault parallel component of the Imperial Valley 1940 “El Centro” earthquake with a peak ground acceleration of 0.6757g and is designated as LA-02 record. The second ground motion (Figure 27) corresponds to the fault normal component of Landers Earthquake designated as LA-07 record. The third accelerogram (Figure 28) is taken as fault parallel component from the 1994 Northridge Earthquake designated as LA16 record with a peak ground acceleration of 0.58g.
Figure 20: LA-02 Ground Motion
Figure 21: LA-07 Ground Motion
Time (s)
Acc
eler
atio
n (
g)
0 10 20 30 40 50 60 70 80-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8Peak acc. 0.6757187 (g) at 2.12 (s)
Time (s)
Acc
eler
atio
n (
g)
0 10 20 30 40 50 60 70 80-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Peak acc. -0.4209786 (g) at 16.08 (s)
STRUCTURAL CONTROL PROJECT TEAM 4
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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.
Figure 23: Absolute Acceleration Response Spectra for 5% Damping
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.
Time (s)
Rel
ativ
e ve
loci
ty (
m/s
)
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.50
0.25
0.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
Fun
dam
enta
l per
iod
La02La07La16
Time (s)
Rel
ativ
e di
spla
cem
ent (
m)
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.50
0.25
0.5
0.75
1
1.25
Fun
dam
enta
l per
iod
La02La07La16
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CHAPTER 3 - ANALYSIS OF THE ORIGINAL BUILDING
1 Introduction
The objective of this chapter is to evaluate the seismic response of the original building structure under each of the three design ground motions considered in chapter 2. The computer program RUAUMOKO and the post-processor DYNAPLOT were used to evaluate the performance of the original building structure. For each analysis of the building under ground motions, four output quantities are extracted to assess the existing building performance. They includes energy quantities, member curvature ductility, peak and residual interstory drifts and total floor accelerations.
2 Performance of the existing structure
2.1 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.
Kinetic EnergyViscous DampingStrain EnergyTotal EnergyInput Energy
Time (sec.)
Ene
rgy
(kN
-mm
)
0 10 20 30 40 50 60 70 80 90 1000
2E+5
4E+5
6E+5
8E+5
1E+6
Kinetic EnergyViscous DampingStrain EnergyTotal EnergyInput Energy
Time (sec.)
Ene
rgy
(kN
-mm
)
0 5 10 15 20 25 300
6E+5
1.2E+6
1.8E+6
2.4E+6
3E+6
Kinetic EnergyViscous DampingStrain EnergyTotal EnergyInput Energy
STRUCTURAL CONTROL PROJECT TEAM 4
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It can be seen that the input energy from LA02 and LA16 are equal and more than three times the input energy from LA07. Among three motions, LA16 excites larger kinematic energy at the beginning of the record; this is due to the long pause in the acceleration motion. Although the energy time histories generated for the three ground motions varies considerably from one to another, each energy component exhibits a particular pattern, for example the kinetic energy oscillates from zero (maximum deflections) to positive peaks (initial undeformed position). The energy dissipated by viscous damping always increases with time for the three ground motions reaching its maximum value for LA-02 and the lowest value for LA-07. For the absorbed energy two components can be distinguished, the first of them is the recoverable elastic energy which is represented by oscillations out of phase with the kinetic energy and the second one is the non-recoverable component represented by sudden shifts towards positive values due to the inelastic actions that occur in time. The strain energy curve E (green curve) as was mentioned previously is the total amount of energy that the structure has absorbed either through elastic straining or unrecoverable inelastic deformations and the peak value of this curve during an earthquake represents the largest demand on structural members. For each one of the ground motions the fraction of input energy absorbed by the building structure is shown in the Table 13.
Table 13: Fraction of Input Energy Absorbed.
Ground Motions
Absorbed Energy (kN-m)
Total Energy (kN-m)
FractionPercentage
(%)
LA - 02 1152.7 2942.7 0.392 39.19 LA - 07 314.21 912.61 0.344 34.43 LA - 16 1930.70 2859.7 0.675 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.
STRUCTURAL CONTROL PROJECT TEAM 4
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Table 14: Peak Absorbed Energy.
Ground Motions
Peak Absorbed Energy (kN-m)
LA - 02 1241.7 LA - 07 433.7 LA - 16 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.
Ground Motions
EBE %
LA - 02 0.16 LA - 07 0.13 LA - 16 0.08
2.2 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.
STRUCTURAL CONTROL PROJECT TEAM 4
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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
STRUCTURAL CONTROL PROJECT TEAM 4
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Table 16: Maximum plastic rotations for LA-02 ground motion
Hinge Member Prop. Type Lp
(mm) ϕp
Ductility μ
Lp(Int) 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.
STRUCTURAL CONTROL PROJECT TEAM 4
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Table 17: Maximum plastic rotations for LA-07 ground motion.
Table 18: Maximum plastic rotations for LA-16 ground motion.
Hinge Member Prop. Type Lp
(mm) ϕp
Ductility μ
Lp(Int) 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
STRUCTURAL CONTROL PROJECT TEAM 4
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2.3 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.
Figure 32: Inter-story Drift – Time History LA-02
Time (s)
Inte
r-st
ore
y d
rift
(m
m)
0 10 20 30 40 50 60 70 80 90-30
-20
-10
0
10
20
30
40
50
60
70
80
90
100
110
120
1st floor peak 98.94 (mm)
1st floor residual 39.26 (mm)
2nd floor peak 62.74 (mm)
2nd floor residual 25.49 (mm)
3rd floor peak 57.8 (mm)
3rd floor residual 23.31 (mm)
4th floor peak 51.95 (mm)
4th floor residual 21.58 (mm)
5th floor peak 39.56 (mm)
5th floor residual 10.72 (mm)
Roof peak 26.13 (mm) Roof residual 2.726 (mm)
1st floor2nd floor3rd floor4th floor5th floorRoof
STRUCTURAL CONTROL PROJECT TEAM 4
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Figure 33: Inter-story Drift – Time History LA-07
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).
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.
Figure 36: Peak inter-story drifts.
Displacement (mm.)
Hei
ght (
m.)
Peak Inter-storey DriftsMotion 02
0 50 100 1500
5
10
15
20
25
2.5% drift
0.7% drift
Displacement (mm.)
Hei
ght (
m.)
Peak Inter-storey DriftsMotion 07
0 50 100 1500
5
10
15
20
25
2.5% drift
0.7% drift
Displacement (mm.)
Hei
ght (
m.)
Peak Inter-storey DriftsMotion 16
0 50 100 1500
5
10
15
20
25
2.5% drift
0.7% drift
Displacement (mm.)
Hei
ght (
m.)
Peak Inter-storey Drifts
0 50 100 1500
5
10
15
20
25
Inter Storey DriftsLA-02LA-07LA-162.5% Drift (LS)0.7% Drift (IO)
STRUCTURAL CONTROL PROJECT TEAM 4
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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).
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.
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.
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.
Figure 42: Peak acceleration for LA-02, LA-07 and LA-16
Time (s)
To
tal a
cce
lera
tio
n (
g)
0 5 10 15 20 25 30-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1st floor peak 0.3646279
2nd floor peak 0.4027523
3r floor peak 0.3584098
4th floor peak -0.5123344
5th floor peak -0.5197757
Roof peak -0.7851172
1st floor2nd floor3rd floor4th floor5th floorRoof
Acceleration (g)
Hei
ght
(m
)
Motion 02
0 0.25 0.5 0.75 10
5
10
15
20
25
Acceleration (g)
Hei
ght
(m
)
Motion 07
0 0.25 0.5 0.75 10
5
10
15
20
25
Acceleration (g)
Hei
ght
(m
)
Motion 16
0 0.25 0.5 0.75 10
5
10
15
20
25
STRUCTURAL CONTROL PROJECT TEAM 4
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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 inter-storey 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.
Acceleration (g)
Hei
gh
t (m
)
0 0.25 0.5 0.75 10
5
10
15
20
25
Peak AccelerationLA-02LA-07LA-16
STRUCTURAL CONTROL PROJECT TEAM 4
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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:
PI % 1
w . μ μ
w . μ μ
w∆ . ∆ ∆
w∆ .∆ ∆
w .a a
w w w∆ w∆ w. 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.
STRUCTURAL CONTROL PROJECT TEAM 4
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Based on FEMA 274, the collapse prevention is chosen to be the lower bound limit for the structure. Therefore, ∆ and ∆ will be of 5%. Values of μ and μ are taken to be 11, the
maximum ductility a member can reach in this project. FEMA 274 does not limit the maximum floor acceleration so a reasonable upper bound value for this structure is set to be 1 . The weight for each quantity is chosen based on its importance in the performance of the building. There variables considered most important are peak drift, column ductility and acceleration. The weights for these variables are 30, 25 and 25 respectively. For beam ductility and residual drift, the weights are 10 and 10. The reason for put more weight into acceleration is from the fact that the existing structure is hospital building, which hosts medical equipment sensitive to acceleration. Therefore, the PI will be in the form of
PI % 1 10.
μ11 25.
μ11 30.
∆5%
10.∆5%
25.a1g
100. 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 Limit for performance category LS IO
μ 45% of 11 20% of 11
μ 45% of 11 20% of 11
∆ 2.5% 0.7%
∆ 1.0% 0%
a 0.75g 0.5g PI 45 65
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According to FEMA 274, the existing structure performance is from the collapse prevention limit to life safety, with maximum peak drift is higher than 2.5% under LA16. Based on the proposed performance index formulation, the PI values will be calculated from those response quantities for each ground motion and presented in Table 20.
Table 20: Performance Indexes for design ground motions
The PI for the structure will be the smallest among the PI for each ground motion, which will be 36% corresponding to LA16 motion. This index is within the [0;45] range, which means the structure passes the collapse prevention limit but stays below life safety limit.
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CHAPTER 4 - HYSTERETIC DAMPERS
1 Description
The objective of this phase was to retrofit the original building with hysteretic dampers for the different ground motions considered (LA2, LA7 and LA16). It was shown in the previous phase of the project that collapse is not reached under the considered ground motions; nonetheless this approach is intended to improve the seismic performance of the building rather than prevent collapse. The retrofit strategy for the structure consists on introducing chevron braces at each moment resisting frame and installing hysteretic dampers at one end of the bracing members as shown in Figure 45. This retrofit scheme was selected because it minimizes the levels of intervention (i.e. only the middle bay will be affected when installing dampers and braces). For the final design other hysteretic damper locations will be studied.
Figure 45: Locations of added bracing and hysteretic dampers (Configuration-C1)
The bracing members were designed to sustain the activation load assigned to the hysteretic dampers. This system dissipates energy through the elasto-plastic hysteretic behavior shown in Figure 46.
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Figure 46: Elasto-Plastic Hysteresis
For this retrofit, it was specified that hollow steel sections (HSS) must be used for the cross braces. In addition, as the braces would be installed to the existing building, brace forces induced by dead loads were ignored in the analysis and design. In order to improve the behavior of hysteretic dampers in the structure composite sections composed by HSS sections will be also considered. The methods used to determine the slip load are based on design procedures provided by Christopoulos and Filiatrault (2006) as discussed below. The computer program RUAUMOKO and its post-processor DYNAPLOT were used to perform nonlinear time history dynamic analysis in order to completely estimate the response of the building structure and select an optimum solution. Finally a comparison between the optimum design configuration and the original building will be presented in terms of energy balance, plastic hinge distribution, envelopes of peak and residual inter-story drifts and envelopes of peak absolute floor accelerations. The merits of the optimum solution in terms of performance indices will also be discussed.
2 Procedure to calculate the optimum activation load
The first step in the design of structures equipped with hysteretic dampers is the estimation of the optimum parameters for the dampers. These parameters are the activation load “Fa” and the bracing stiffness. Christopoulos and Filiatrault (2006) found that the optimal use of hysteretic dampers will occurs when the addition of these devices to a system produce additional supplemental damping along with a modification of the dynamics properties of the system that optimizes the use of the added damper. Otherwise, the system will behave either as an unbraced frame or as a fully braced frame. The selection of the cross sections for the diagonal braces is based on the recommendation by Filiatrault and Cherry (1988), which is expressed as:
0.40
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Where Tb is the natural period of the fully braced structure and Tu is the natural period of the unbraced structure. Furthermore, based on parametric studies it was determined that the optimum value of the activation load “Fopt” of the hysteretic damper that minimizes the amplitude of the response at any forcing frequency is given by:
F
W
a
gQ
T
T,T
T
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.
Once the analysis is completed and the data is converted into a frequency domain format, the peaks corresponding to the highest values o Fourier amplitudes were selected. The plot shown above represents the decoupling of the equations of motion for single DOFs and the peaks represent the predominant frequencies for each of the design ground motions. Predominant periods corresponding to each ground motion are also shown in Figure 47.
4 Preliminary design As mentioned above, the best response of hysteretically damped structures occurs for small values of Tb/Tu, which corresponds to large diagonal braces. Therefore the diagonal cross-braces were chosen with the largest possible cross-sectional area within the limits imposed by architecture, cost and availability of material. As a first trial an HSS406x406x15.9 section and an HSS304X304X15.9 section were selected among the largest possible sections according the AISC –provisions. The cross-braces were used along the six stories of the building. A spreadsheet in MathCAD was used to calculate the optimum activation loads at each damper for the proposed cross sections and for the different ground motions. Calculations corresponding to the ground motion LA2 and section HSS406x406x15.9 are shown in the following page. The idea is to get a felling on which are the best braces configuration and member cross section to take into account for the optimum activation load study to be carried out in the intermediate design.
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(Total seismic weight of the structure)
(Number of floors)
(Fundamental period of the braced structures)
(Fundamental period of the unbraced structure)
(Predominant period of the design ground motion LA2)
(Design Peak Ground Acceleration)
(Unknown single valued function)
(Optimum activation shear)
(Story base shear, uniformly distributed)
W 14475 kN
Nf 6
Tb 0.615 s
Tu 1.304 s
Tg 0.683 s
Tb
Tu0.472
Tg
Tu0.524
ag 0.676 g
QTg
Tu1.24 Nf 0.31( )
Tb
Tu 1.04 Nf 0.43
0Tg
Tu 1if
Tb
Tu0.01 Nf 0.02( )
Tg
Tu 1.25 Nf 0.32
Tg
Tu0.002 0.002 Nf( ) 1.04 Nf 0.42
Tg
Tu1if
Q 1.579
Vo Qag
g W
Vo 15451.9 kN
i 1 Nf 56.3
180 46.2
180
Vsi
1
2
Vo
Nf
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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.
(Optimum activation load for each damper)
Vs
1287.7
1287.7
1287.7
1287.7
1287.7
1287.7
kN
i 2 Nf
Fa1
Vs1
2 cos ( ) Fa
i
Vsi
2 cos ( )
Fa1
1160.4kN
Fa
1160.4
930.2
930.2
930.2
930.2
930.2
kN
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Figure 48: Preliminary design
The proposed brace size member from the previous Figure seems to work fine for ground motions LA2 and LA7, with an approximate increase on performance of about 30% with respect of the existing structure. Nonetheless for ground motion LA16 not significant improvement was found. This fact can be justified by arguing that for LA16 the predominant period of the ground motion is high producing high activation forces that will eventually prevent the damper to activate and contribute to the energy dissipation. In the intermediate design based on a comparative study aiming to improve the performance of our building, the optimum activation loads and brace sections will be found.
5 Intermediate design
In the preliminary design the members size and the activation forces were intuitively chosen based on the recommendation of Tb/Tu = 0.4 and the design procedures provided by Christopoulos and Filiatrault (2006). In order to optimize the design we proceed to perform multiple analyses in RUAMOKO but this time considering different cross sections and braces configurations. Regarding the members size we tried to get closer to the recommended ratio of Tb/Tu = 0.4 by proposing a composite section capable of increase the brace stiffness without violating the design specifications that states that hollow shape section are to be considered in the design. Details on this composite section are presented in Appendix B. For this approach the activation load corresponding to each configuration were still calculated based on the procedure suggested by Christopoulos and Filiatrault (2006). Figure 49 shows a sketch of the proposed cross sections to be considered in the analyses.
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Figure 49: Sections
A second approach was followed by proposing an alternative configuration which presents frame braces in the three bays of the first floor (See Figure 50). For this particular configuration the way the optimum activation shear was redistributed in height was also studied.
Figure 50: Alternative retrofit scheme considered in the analyses (Configuration-C2)
A total of 4 brace cross sections were used in the two different brace configurations just shown in Figure 45 and Figure 50). The HSS406x406x15.9 and HSS304x304x15.9 section were used as described in the preliminary design but in this part were compared with the composite sections and evaluated in the two different configurations (C1 and C2). A total of 21 analyses were performed since each alternative had to be evaluated for each of the three design ground motions specified. (See Table 22 for reference)
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It is important to mention that when pursuing an optimum design, performance indices were used to make a comparison between the different alternatives. These indices were presented in Chapter 3. Excel macros were used to get the relevant values used to compute the performance indices and accelerate the design process. Batch files were also created in RUAMOKO to efficiently get the relevant results regarding our performances indices. Appendix A shows a detailed explanation on how the macro works.
Table 22: Parameters
The highlighted values in the previous table were inputted in the RUAMOKO files when defining the elasto-plastic hysteretic loop shown in Figure 46. A plot summarizing the performances indices obtained for each of the proposed configurations and the relevant member sizes are shown in the figure below.
Figure 51: Optimum size study
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It can be seen in the previous figure that in overall all the proposed alternatives reached performances indices higher than the existing building. The higher PI corresponds to the composite section HSS406&304 in the brace configuration C2. The same results are presented in the figure below but this time in terms of Tb/Tu. It can be inferred from this graph that the closer we get to 0.4 the higher the PI is.
Figure 52: Optimum size study
The previous results give us an idea on which sections and brace configurations should be considered for the final design. Nonetheless a study on the optimum activations loads will be performed in order to optimize the design. This optimum activation study will be considered among the brace sizes and configuration that performed better in the previous comparison. The selected configurations to be studied are the HSS406-C1, HSS406&304-C1 and HSS406&304-C2. (See Figure 45, Figure 46 and Figure 50 for details on the configurations and cross sections selected). For this study the optimum activation load were evaluated in the range of 200kN to 2000kN based on previous calculations (See Table 22). Analyses in RUAMOKO were performed for each configuration under study for activation load increments of 200kN. In total 30 analysis were run per proposed configuration and the optimum activation load corresponds to the maximum PI value for the most critical earthquake (In this case LA16).
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Figure 53: Optimum activation load study for HSS406-C1
Figure 54: Optimum activation load study for HSS304-C1
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Figure 55: Optimum activation load study for HSS406&304-C1
The following plot summarizes the three previous analyses, and shows the higher PI obtained for each configuration. It is important to state that these performances indices correspond to different activation loads as shown in the previous plots.
Figure 56: Optimum Design
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Even though the HSS406&304-C2 configuration reached the higher performance, we assumed for this case a uniform load redistribution along the height of the building, even though the second configuration present higher stiffness in the first floor with respect to the other floors. In order to deal with this uncertainty the way the forces were redistributed was studied. In each of the plots presented below the ratio shown in brackets corresponds to the ratio of the first floor activation load to the other floors activation loads. A triangular distribution of the activation loads based on the first mode was also considered.
Figure 57: Optimum activation load study for HSS406&304C2 (1/1)
Figure 58: Optimum activation load study for HSS406&304C2 (2/1)
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Figure 59: Optimum activation load study for HSS406&304C2 (3/1)
Figure 60: Optimum activation load study for HSS406&304C2 (4/1)
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Figure 61: Optimum activation load study for HSS406&304C2 (1st Mode proportional)
Figure 62: Optimum Design
6 Final design
Based on the parametric study performed in the preliminary and intermediate design the HSS406&304 composite sections in the bracing configuration C2 was proved to be the more optimum in terms of performance.
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6.1 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.
Kinetic EnergyViscous DampingStrain EnergyTotal EnergyInput Energy
Time (sec.)
Ene
rgy
(kN
-mm
)
Time History Energy ComponentsLA - 07
0 10 20 30 40 50 60 70 80 90 1000
2E+5
4E+5
6E+5
8E+5
1E+6
1.2E+6
Kinetic EnergyViscous DampingStrain EnergyTotal EnergyInput Energy
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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.
Time (sec.)
Ene
rgy
(kN
-mm
)
Time History Energy ComponentsLA - 16
0 5 10 15 20 25 300
6E+5
1.2E+6
1.8E+6
2.4E+6
3E+6
Kinetic EnergyViscous DampingStrain EnergyTotal EnergyInput Energy
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Table 23: Maximum plastic rotations for LA-02 ground motion.
Hinge Member Prop. Type Lp
(mm) ϕp
Ductility μ
Lp(Int) 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.
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.
7 Flow Chart for Hysteretic dampers optimum design
A flow chart summarizing the procedure to obtain the optimum design is shown below. As is described in Appendix A a macro were used to get the relevant information used to computed the performance indices which allowed us to perform a parametric study in terms of optimum size, optimum activation load and optimum distribution of the dampers.
Figure 78: Flow Chart for hysteretic dampers optimum design
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CHAPTER 5 - VISCOUS DAMPERS
1 Description
The retrofit strategy for the structure consists on introducing chevron braced frame in the middle bay of each moment resisting frame and installing viscous-type energy dissipating devices at one end of the bracing members, as shown in Figure 79. The bracing members must be designed to sustain the maximum load developed by the viscous damper. Brace forces induced by gravity loads will be ignored in the design of the bracing and viscous energy dissipating systems, as the braces would be installed to the existing building and that live loads will have a negligible effect on the bracing members.
Figure 79: Location of added bracing and viscous dampers
The retrofit system considered incorporates at one end of the bracing members, viscous damper connections with an axial force linearly proportional to the relative velocity between ends. This system exhibits the elliptical hysteretic behavior shown in Figure 80. The behavior of the damper element will be proven when referring to the DAMPER element in RUAMOKO and the validation process of this element presented at the end of the chapter.
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Figure 80: Hysteretic Behavior of Viscous Dampers
2 Procedures to calculate the damping coefficients Viscous dampers were installed at all floors. The retrofit procedure included the calculation of the damping constants of the viscous dampers as well as their distribution along the height of the building. The first approach used to determine the target viscous damping constants was achieved by providing damping constant for each floor level proportional to the lateral inter-story stiffness of the story at which the damper is to be placed. By imposing the damping constants to be proportional to the inter-story lateral stiffness of the structure, this ensures that classical normal modes will be maintained (Christopoulos and Filiatrault, 2006).
2
ˆ10Tk
CL
Where k0 are the spring constants, T1 the fundamental period of the building and CL are the damping coefficients. The drawback of this procedure is that the dampers will be different at each floor level, which would cause a construction issue. Secondly, a preliminary analysis for the most significant earthquake for the building showed that the structure would have a highly nonlinear behavior due to the large amount of plastic hinges occurring in the structure and the achievement of a linear response under the specific earthquake would not be possible. A second approach was implemented by using the same damping coefficient for all the floor of the building with the assumption that the building will behave mostly in the first mode of vibration. For this method, the fundamental natural period of the existing building and the target damping ratio(s) of the building retrofitted with viscous dampers, and the maximum inter-story drifts are needed. With these three parameters known, the calculation of the damping constant can be determined for a given time history and damping level desired using the equation below.
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21 1
1
2 2
1
cos
fN
i ii
L Nd
i ii
T kC
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.
iL i LC C
Where d is the inter-story drift of the normalized first mode of vibration. In the above equation it is considered as non-dimensional.
2 2 2
1 1
2 ( ) cosdNi L i i
vdi
CE
T
2
1
1
2
fN
es i ii
E k
1 4vd
es
E
E
21 1
1
3 2
1
cos
fN
i ii
L Nd
i ii
T kC
For the complete design protocol used the different approaches, refer to the MathCAD calculations on the preliminary design.
3 Modeling of dampers
Before modeling the damper and in order to obtain the corresponding trial value of the fundamental period, fictitious spring elements were modeled in RUAMOKO as brace elements.
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The spring constants were determined as shown in Table 27 and the trial period needed to correct the stiffness was computed. This procedure was followed in the first approach only. Having calculated the damping constants to be used in the model for the three approaches we proceeded to model the dampers. For the modeling of the dampers phantom nodes were placed directly on the node at mid-span of the beam in the middle bay of the structure. The phantom nodes were located at the same coordinates than the existing nodes, but have different degrees of freedom. The reason for using phantom nodes is to eliminate the effect of gravity dead loads on the damper as this is a retrofit of an existing building and these loads are already supported by the existing structure. These nodes are locked to the horizontal component of the node it is connected to but have different y-axis displacements and z-axis rotations. Then the damping constants were assigned to these damper elements. For the first approach were the damping constants varies along the height of the building multiple properties were defined. In Figure 81 history displacements for the viscous damping using the different methods were plotted in conjunction with the Rayleigh damping. A good correlation was found among the plots, proving a good estimate of the damping coefficient and validating the behavior of the dampers.
Figure 81: Plot showing comparison among viscous damping and Rayleigh damping
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.
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.
Time (s)
Dis
plac
emen
t (m
m)
Lateral diplacement
0 3 6 9 12 15 18 21 24 27 30-80
-60
-40
-20
0
20
40
60
80
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Figure 84: Spring and viscous damper forces
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.
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).
Figure 86: Spectra accelerations for LA2 under different damping ratios
Period
Res
pons
e A
cele
ratio
n (
g)
0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.20
0.25
0.5
0.75
1
1.25
1.5
1.75
2
0.46g
0.27g0.28g
LA2 Response Spectra5%10%20%30%35%
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Figure 87: Spectra accelerations for LA7 under different damping ratios
Figure 88: Spectra accelerations for LA16 under different damping ratios
Period
Re
spon
se A
cce
lera
tion
(g
)
0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.20
0.15
0.3
0.45
0.6
0.75
0.9
1.05
1.2
0.39g
0.26g0.25g
LA7 Response Spectra5%10%20%30%35%
Period
Re
spon
se A
cce
lera
tion
(g
)
0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.20
0.25
0.5
0.75
1
1.25
1.5
1.75
2
0.99g
0.60g 0.59g
LA16 Response Spectra5%10%20%30%35%
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Figure 89: Spectra displacements for LA2 under different damping ratios
Figure 90: Spectra displacements for LA7 under different damping ratios
Period
Res
pon
se d
isp
lace
men
t (c
m)
0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.20
8
16
24
32
40
48
56
64
19.4cm
9.1cm9.9cm
LA2 Response Spectra5%10%20%30%35%
Period
Res
pon
se d
isp
lace
men
t (c
m)
0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.20
10
20
30
40
50
60
70
80
16.0cm
9.0cm
8.3cm
LA7 Response Spectra5%10%20%30%35%
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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.
11 arg
2 1t et
TT
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
Period
Res
pon
se d
isp
lace
men
t (c
m)
0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.20
8
16
24
32
40
48
56
64
41.0cm
20.0cm
18.5cm
LA16 Response Spectra5%10%20%30%35%
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5.1 Stiffness proportional approach
(Fundamental period of un-braced structure)
(Assumed damping ratio)
(Target fundamental period of the structured braced with thefictitious springs)
(Fictitious spring constants at proposed locations, fromRuamoko and proportional to drifts)
(Trial value of the fundamental period of the fictitious braced structure obtained with first trial spring constants,from Ruamoko)
(Final spring constants)
(Damping coefficient of each viscous damper)
T1 1.304 s
1 0.30
T1targetT1
2 1 1
T1target 1.031s
ko
70.72
85.67
123.25
137.47
164.2
95.04
kN
mm
T1tr 0.950 s
kfinalko
1T1target
2T1tr
2
T1target2
T12
kfinal
57
68
98
110
131
76
kN
mm
CLkfinal T1
2
CL
12
14
20
23
27
16
kN s
mm
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5.2 Constant damping approach
(Number of dampers)
(Number of floors)
(Fundamental period of umbraced structure)
(Assumed damping ratio)
(Fictitious spring constants at proposed locations, fromRuamoko and proportional to drifts)
(Inclination angle of the dampers)
(Inter-story drift at the storey where the damper is located )
Nd 6
Nf 6
T1 1.304s
1 0.30
k
97.95
118.65
170.71
190.40
227.43
105.44
kN
mm
0.806
0.806
0.806
0.806
0.806
0.983
0.09
0.14
0.15
0.18
0.18
0.27
CL
1 T1
1
Nf
i
kii 2
2
1
Nd
i
i 2 cos
i 2
CL 22.628skN
mm
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5.3 First mode proportional damping
(Number of dampers)
(Number of floors)
(Fundamental period of umbraced structure)
(Assumed damping ratio)
(Fictitious spring constants at proposed locations, from Ruamoko and proportional to drifts)
(Inclination angle of the damper is defined on the left andInter-story drift at the storey where the damper is located isdefined on the right)
Nd 6
Nf 6
T1 1.304s
1 0.30
k
97.95
118.65
170.71
190.40
227.43
105.44
kN
mm
0.806
0.806
0.806
0.806
0.806
0.983
0.09
0.14
0.15
0.18
0.18
0.27
CL
1 T1
1
Nf
i
kii 2
2
1
Nd
i
i 3 cos
i 2
CL 117.556skN
mm
CLfm CL
CLfm
10.6
16.5
17.6
21.2
21.2
31.7
skN
mm
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6 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 30: First Mode proportional Approach damping coefficients
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Based on the figures presented below we can infer that there is not significant improvement in terms of performance for damping ratios higher than 30%. It is for this reason that for the optimum design just performance indices corresponding to 30% damping will be compared as seen in Figure 95.
It is clear that the first mode proportional method reaches the higher performance of the building. Therefore this method is chosen for the optimum design.
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7 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.
Kinetic EnergyViscous DampingStrain EnergyTotal EnergyInput Energy
Time (sec.)
Ene
rgy
(kN
-mm
)
Time History Energy ComponentsLA - 07
0 10 20 30 40 50 60 70 80 90 1000
2E+5
4E+5
6E+5
8E+5
1E+6
1.2E+6
Kinetic EnergyViscous DampingStrain EnergyTotal EnergyInput Energy
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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.
Time (sec.)
Ene
rgy
(kN
-mm
)
Time History Energy ComponentsLA - 16
0 5 10 15 20 25 300
6E+5
1.2E+6
1.8E+6
2.4E+6
3E+6
Kinetic EnergyViscous DampingStrain EnergyTotal EnergyInput Energy
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Table 31: Maximum plastic rotations for LA-02 ground motion.
Hinge Member Prop. Type Lp
(mm) ϕp
Ductility μ
Lp(Int) θp
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
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
Ductilityμ
Lp(Int) θp
1 57 23 Beam 678 3.84E-05 1.061 71.936 0.0028
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Table 33: Maximum plastic rotations for LA-16 ground motion
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.
A flow chart describing the procedure that was followed to achieve the optimum design is presented below.
Figure 112: Flow chart for optimum design for viscous dampers
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CHAPTER 6 - BASE ISOLATION
1 Description
The retrofit strategy for the structure consists on introducing lead-rubber bearings at the base of the structure as shown in Figure 113. For this purpose, it will be assumed that a large foundation mat supports the building and that retrofit work will be required to introduce a link-frame between the columns and this mat. The isolators will be installed between this link-frame and the top surface of the mat. For modeling purposes, it will be assumed that all bearings operate in parallel and the complete base isolation system will be modeled as a single horizontal bilinear spring at the base of the structure.
Figure 113: Modelling of Building Structure with Lead-Rubber Base-Isolation System
The base isolation system used for this retrofit strategy is lead-rubber bearings. These isolation elements are comprised of two distinct components; a laminated rubber bearing and a lead core as seen in Figure 114.
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Figure 114: Components of Lead-Rubber base isolation
The first component of this type of isolation system is the laminated rubber bearing which is the primary mechanism of the isolation system and consists of thin layers of rubber and steel shim plates laminated together in an alternating pattern as shown in the figure above. The physics behind the use of laminated rubber bearings for base isolation is that the lateral stiffness of the bearings is significantly less than that of the supported structure. Consequently, the objective of the use of a base isolation system is to provide a shift of the structure’s fundamental natural period out of the frequency range at which most buildings are more vulnerable to damage due to the affects of ground motion during a seismic event. The second component of a lead-rubber bearing isolator is the lead core plug. The stiffness of the laminated rubber bearing is low, providing little damping by itself and as a result is susceptible to large lateral displacements. The lead core element is introduced to compensate for this by providing an element to increase damping as well as to dissipate hysteretic energy. To model the Lead-rubber bearings in our RUAMOKO model we used a non-linear spring with a bi-linear hysteretic model. The bi-linear hysteretic model is shown in Figure 115.
Figure 115: Lead-Rubber Bi-Linear Model
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If the mechanical properties based on experimental tests are not known, the determination of the mechanical properties requires an iterative approach for the preliminary bearing design. The three parameters that define the bi-linear model are k1, k2 and Fy. Where k1 is the combined elastic stiffness of the laminated rubber and the lead core assemblage, k2 is the post-yield stiffness equal to the stiffness of only the laminated rubber, and Fy is the yield force at which the lead core starts to yield. For modeling of the base isolators in RUAMOKO, a fixed node was introduced at the ground level. The horizontal degrees of freedom at the base of all the column nodes at the ground level were released. A non-linear spring element was connected to the base node of one of the exterior bay columns. Next, the base nodes at the remainder of the ground level columns were slaved to horizontal degree of freedom of the aforementioned column. This is shown schematically in Figure 113. The determination of the hysteretic model of the isolators is an iterative approach. The preliminary approach was used to determine the bi-linear properties using a MathCAD worksheet developed linking the assumptions listed below.
Overlap factor was fixed to 0.6. '
0.6r
A
A
The diameter of the bearing was fixed based on the following recommendation:
'0.8(1 )
bDb A
Ar
x
The thickness of the rubber was computed based on the following recommendation:
4
br
Dt
S
The shape factor was considered in the following interval
10 20S
The total height of the rubber layers was set to remain in the following range:
2
3 3
b bD Dhr
b isoD h
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2r st t
The number of isolator was fixed to 6 to ensure enough redundancy.
6isolatorsn
The plug diameter was contained in the following range
1 1
3 6b p bD D D
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.
2r r
r
G Ak
h
1 210k k
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.
(1 )r r
py pp p
G AFy A
G A
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).
b Dx S
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2 Preliminary Design
(Rubber Shear Modulus)
(Lead shear Modulus)
(Rubber compression modulus)
(Lead Shear Yield Strength)
(Maximum seismic displacement of lead-rubber bearings)
(Maximum overlap factor for individual bearings, A1/A)
(Shape factor for individual bearings)
(Total Weight of structure)
(Short-term failure strain of the rubber)
(Allowable shear strain under gravity load
(Diameter of the rubber bearing)
(Rounded rubber bearing)
(Rubber cross area)
(Rubber thickness per layer)
(Rounded rubber thickness)
(Overlap area)
(Maximum allowable vertical load)
(Number of isolators)
(Rounded number of isolators)
Gr 1 MPa
Gp 150 MPa
kr 2000MPa
py 10 MPa
xb 300 mm
OverlapFactor 0.6
S 12
BldgWtTotal 30950kN
v 4.5
w 0.4 v 1.8
Dbearingxb
0.8 1 OverlapFactor( )
937.5 mm
Dbearing 940 mm
Ar Dbearing
2
2
693978mm2
trDbearing
4 S
19.583mm
tr 20 mm
A1 OverlapFactor Ar 416387 mm2
Wmax A1 Gr S w 8994kN
nisolatorsBldgWtTotal
Wmax
3.441
nisolators 6
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(Number of rubber layers)
(Lead plug diameter, taken between 1/3 and 1/6 of Dbearing)
(Rounded lead plug diameter)
(Area of the lead plug)
(Lead rubber Post-yield stiffness)
(Lead rubber approximation of the lateral elastic stiffness)
(Yield force of the bearing)
(Total elastic stiffness of the system)
(Total yield force of the system)
(Equivalent Stiffness of the system)
(Equivalent period of the system)
(Equivalent damping of the system)
(Total rubber height)
nlayers 20
hr nlayers tr 400 mm
DplugDbearing
3
313.333mm
Dplug 315 mm
Ap Dplug
3
2
34636mm2
k2Gr Ar
hr1.735
kN
mm
k1 10 k2 17.3494kN
mm
Fy py Ap 1Gr Ar
Gp Ap
392.626kN
Tk1nisolators k1
252.048
kN
mm
TFynisolators Fy
21177.877kN
keffFy
xbk2 1
Fy
k1 xb
nisolators
keff 17.477kN
mm
teff 2 BldgWtTotal
keff g 2.67s
dyFy
k122.63 mm
beff2 TFy xb dy( )
keff xb2
0.132
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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:
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.
Period (s)
Spe
ctra
l Dis
pla
cem
ent
(mm
)
0 0.5 1 1.5 2 2.5 3 3.5 40
80
160
240
320
400
480
560
640
300mm
LA713.2%
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3 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.
2
3 3
b bD Dhr
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
3 6b p bD D D
Therefore the upper and lower bound of Fy can also be established. This range of application was found to vary from 750kN to 2500kN. Performances indices were calculated for increments of 250kN in the range of 750kN to 2500kN for each of the three assumed values of k1.
Table 36: Summary of parameters to be studied in intermediate design
In Table 36 the different values of k1 and the range of application of Fy is shown. Figure 117, Figure 118 and Figure 119 shows the performance indices obtained for each run.
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Figure 117: Optimum Fy study for k1=30kN/mm
Figure 118: Optimum Fy study for k1=45kN/mm
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Figure 119: Optimum Fy study for k1=65kN/mm
From Figure 120 we can observe that the highest performance is obtained when k1 equals 30kN/mm and from Figure 117 it is evident that this happens when Fy is 750 kN. This optimum configuration reached a performance index of 81%.
Figure 120: Optimum Design
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MathCAD calculations to obtain the optimum parameters just presented are shown below.
(Rubber Shear Modulus)
(Lead shear Modulus)
(Rubber compression modulus)
(Lead Shear Yield Strength)
(Maximum seismic displacement of lead-rubber bearings)
(Maximum overlap factor for individual bearings, A1/A)
(Shape factor for individual bearings)
(Total Weight of structure)
(Short-term failure strain of the rubber)
(Allowable shear strain under gravity load)
(Diameter of the rubber bearing)
(Rounded rubber bearing)
(Rubber cross area)
(Rubber thickness per layer)
(Rounded rubber thickness)
(Overlap area)
(Maximum allowable vertical load)
(Number of isolators)
(Rounded number of isolators)
(Number of rubber layers)
Gr 1 MPa
Gp 150 MPa
kr 2000MPa
py 10 MPa
xb 300 mm
OverlapFactor 0.6
S 13
BldgWtTotal 30950kN
v 4.5
w 0.4 v 1.8
Dbearingxb
0.8 1 OverlapFactor( )
937.5 mm
Dbearing 940 mm
Ar Dbearing
2
2
693978mm2
trDbearing
4 S
18.077mm
tr 20 mm
A1 OverlapFactor Ar 416387 mm2
Wmax A1 Gr S w 9743.4kN
nisolatorsBldgWtTotal
Wmax
3.176
nisolators 6
nlayers 34
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A summary of the optimum design is shown in Table 37.
Table 37: Summary of Design Parameters
Parameter Value Diameter of the bearing 940 mm
Diameter of the plug 160 mm Rubber thickness / number of layers 20 mm / 34
Shape factor 13 Number of isolators 6
(Lead plug diameter, taken with 1/3 and 1/6 of Dbearing)
(Rounded lead plug diameter)
(Area of the lead plug)
(Lead rubber Post-yield stiffness)
(Lead rubber approximation of the lateral elastic stiffness)
(Yield force of the bearing)
(Total elastic stiffness of the system)
(Total yield force of the system)
(Total rubber height)
DplugDbearing
6
156.667mm
Dplug 161 mm
Ap Dplug
2
2
20358mm2
k2Gr Ar
hr1.021
kN
mm
k1 10 k2 10.2056kN
mm
Fy py Ap 1Gr Ar
Gp Ap
249.848kN
Tk1nisolators k1
230.6
kN
mm
TFynisolators Fy
2750 kN
hr nlayers tr 680 mm
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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.
Figure 121: Energy components time history for LA-02.
Figure 122: Energy components time history for LA-07.
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.
A flow chart describing the procedure that was followed to achieve the optimum base isolation design is presented below.
Figure 142: Flow chart for optimum design of base isolation systems
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CHAPTER 7 - OPTIMUM DESIGN AND NEAR-FAULT GROUND MOTION PERFORMANCE
1 Optimum retrofit strategy
The base isolation retrofit option was shown to achieve the highest performance of the building as indicated in the table below. In this table all the retrofit schemes are compared.
The performance of all retrofit options compared against the performance levels in the PI scale is summarized in the next chart. Using the base isolation system we can reach the immediate occupancy level and this option will be chosen to be the optimum design for this project and considered for the near fault event study.
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Table 40: Performance level category
CP LS IO
Existing Structure X (36%)
Hysteretic Damping X (63%)
Viscous Damping X (65%)
Base Isolation X (81%)
2 Performance under near-fault ground motion
The performance of the existing and the optimally retrofitted structure in the case that the construction site would be located at proximity (less than 10 km) of an active fault will be studied.
2.1 Near-Fault Ground Motion
The optimally retrofitted structure will be analyzed under a particular historically derived near-fault ground motion. This ground motion, called NF13, has been derived from one horizontal component of the ground motion recorded at the Rinaldi station (distance = 7.5 km) during the 1994 Northridge earthquake (Moment Magnitude = 6.7). The ground motion has a PGA of 0.89g at 2.69 sec.
Figure 143: Near fault ground motion horizontal component
Period
Res
pons
e A
cele
ratio
n (g
)
Northridge,17 Jan 94,04:31PST; Rinaldi Receiving Station FF
0 2 4 6 8 10 12 14 16-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
Ground motion NF13
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2.2 Assessment of the existing structure under near fault ground motion
As expected the structure experienced significant damage. The energy plots show a significant contribution of strain energy due to plastic hinge formation in the structure as seen in Figure 145.
Figure 144: Energy components time history for Near Fault Ground motion.
Figure 145: Distribution of plastic hinges for Existing Structure.
Time (sec.)
Ene
rgy
(kN
-mm
)
Energy Components Time History - Existing StructureNear Fault G.M.
0 3 6 9 12 15 18 210
6E+5
1.2E+6
1.8E+6
2.4E+6
3E+6
3.6E+6
4.2E+6
4.8E+6
Kinetic EnergyViscous DampingStrain EnergyTotal EnergyInput Energy
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Table 41: Maximum plastic rotations for Near Fault ground motion in existing structure.
Acceleration Time History - Existing StructureNear Fault G.M.
0 3 6 9 12 15 18 21-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
1st floor peak -0.582
2nd floor peak -0.4923rd floor peak -0.463
4th floor peak -0.509
5th floor peak -0.462
Roof peak -0.913
1st floor2nd floor3rd floor4th floor5th floorRoof
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The performance evaluation of the existing building is presented in the table below:
Table 42: Performance Indexes of existing structure for near fault ground motion.
Ground Motion
μ μ ∆ (%) ∆ (%) a(g) PI
N.F. 9.743 10.9 3.33 1.97 0.913 19.60%
2.3 Retrofitted building performance under near fault ground motion
The input energy as seen in the figure below was significantly reduced compared with the existing building due to the strain energy and viscous energy provided by the base isolators.
Figure 148: Energy components time history for Retrofitted structure.
Time (sec.)
Ene
rgy
(kN
-mm
)
Energy Components Time History - Retrofit with Bae IsolationNear Fault G. M.
0 3 6 9 12 15 18 210
6E+5
1.2E+6
1.8E+6
2.4E+6
3E+6
Kinetic EnergyViscous DampingStrain EnergyTotal EnergyInput Energy
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Figure 149: Inter storey - drifts time history for retrofitted structure.
Figure 150: Displacement time history for Bearings in Base isolation system.
Time (s)
Inte
r-S
tore
y dr
ift (
mm
)
Inter-Storey drift time history - Retrofit with Base IsolationNear Fault G.M.
Displacement history - Lead Rubbers Base IsolationNear Fault G.M.
0 3 6 9 12 15 18 21-600
-450
-300
-150
0
150
Peak Displacement -483
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Figure 151: Acceleration time history for retrofitted structure.
Figure 152: Acceleration time history for bearings in base isolation.
Time (s)
Tot
al A
ccel
erat
ion
(g)
Acceleration Time History - Retrofit with Base Isolation Near Fault G.M.
0 3 6 9 12 15 18 21-0.5
-0.25
0
0.25
0.5
0.75
1st floor peak -0.257
2nd floor peak -0.231
3rd floor peak 0.1984th floor peak 0.229
5th floor peak 0.265
Roof peak -0.489
1st floor2nd floor3rd floor4th floor5th floorRoof
Time (s)
Tot
al A
ccel
erat
ion
(g)
Acceleration History - Lead Rubbers Base IsolationNear Fault G.M.
0 2.5 5 7.5 10 12.5 15 17.5 20 22.5-0.45
-0.35
-0.25
-0.15
-0.05
0.05
0.15
0.25
0.35
Base Isolation peak -0.436
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Figure 153: Comparison of peak inter-storey drifts.
Figure 154: Comparison of residual inter-storey drifts.
Figure 155: Comparison of peak accelerations.
Displacement (mm.)
Hei
ght (
m.)
Peak Inter-storey Drifts
-200 -100 0 100 2000
5
10
15
20
25
-200 -100 0 100 2000
5
10
15
20
25
Inter Storey Drifts2.5% Drift (LS)0.7% Drift (IO)Existing StructureRetrof. Base isolation
Displacement (mm)
He
igh
t (m
)
Residual Inter-Storey Drifts
-120 -60 0 60 1200
5
10
15
20
25
Residual Inter-Storey Drifts1% Drift (LS)Existing StructureRetrof. Base Isolation
Acceleration (g)
Hei
gh
t (m
)
Peak Accelerations
-1 -0.5 0 0.5 10
5
10
15
20
25
-1 -0.5 0 0.5 10
5
10
15
20
25
Peak AccelerationExisting StructureRetrof. Base Islation
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The retrofitted structure performed well under the near-fault effect. Nonetheless the allowable bearing displacement was exceeded by 180 mm. Other than that the super structure performs well in the elastic range with a performance index of 81.75% and overall reduction of all the parameters shown in Table 43. The increase in performance compared to the original structure is shown in Figure 156.
Table 43: Performance of existing and retrofitted structure for the near fault ground motion
Existing μ μ ∆ (%) ∆ (%) a(g) PI
N.F. 9.743 10.9 3.33 1.97 0.913 19.60%
Retrofitted μ μ ∆ (%) ∆ (%) a(g) PI
N.F. <1.00 <1.00 0.46 0.05 0.489 81.75%
Figure 156: Performance of the existing building compared to the optimum retrofit strategy
In summary the chosen optimum design is proved to perform well for the design ground motions with improvement from the collapse prevention to immediate occupancy. The big margin gained in the optimum retrofitted performance allows the structure to achieve a good performance under unexpected ground motion uncertainties as the near-fault phenomena.
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APPENDIX A – RESULTS ANALYSIS WITH VBA SCRIPT
In order to facilitate the laborious task of analyzing output results from several runs done for each retrofitting strategy, a VBA Script was coded to do this task. This code reads RUAMOKO output files and synthesizes the information into a single Excel file per run. Then, it chooses the necessary values from these tables to compute the Performance Index for that run. Finally another VBA Script was coded to set all the Performance Indexes determined in each of the runs in one graph. Moreover in order to assess the overall performance of the set of runs done, it extracts the corresponding values to generate comparison graphs for acceleration, drifts, ductility in beam and columns.
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APPENDIX B – COMPOSITE SECTION
Sketch of composite sections used in the hysteretic damper scheme are detailed below. Two HSS tubes are intended to work as a composite section. The inside HSS with the stiffeners is slided into the bigger section and the cover plate is applied and welded to both HSS sections. There is no direct welding between the two HSS sections. The stiffener in the small HSS is introduced to enable the composite behavior of the two HSS if there is bending.
Figure 157: Details for the composite section
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APPENDIX C: PEER REVIEW LETTERS
From: CIE 626 Peer Review Panel To: Team #4 Date: 04/17/2011 Subject: "Peer Review Panel's Suggestions/Recommendations to Team #4 of CIE 626 Structural Control Project - First Meeting" On April 15, 2011, the members of Team #4, namely, (1) Nguyen Nam Hoai, (2) Gonzalez Sanchez Efrain, (3) Roberts Cervantes Gonzalo and (4) Rosas Espinoza Jorge, met with the Peer Review Panel to discuss the progress of the CIE 626 class project. Team #4 presented the progress of phase 4 (hysteretic dampers retrofit) of the project. The suggestions/commendations listed below are provided by the Peer Review Panel (PRP) based on the 4/15/2011 progress presentation:
- For the retrofit scheme using hysteretic dampers, preliminary case studies were presented assuming a uniform activation load distribution. It is recommended to investigate different activation load distributions of along the height of the building.
- Three different performance indexes were presented. It is recommended to justify the use
of three performance indexes. It is suggested to combine the three performance indexes in one to better identify the performance level of the structure.
Shall you have questions regarding the above suggestions/recommendations; do not hesitate to contact us. Peer Review Panel Members Maria Koliou Maikol Del Carpio Ramos Response:
- An activation load study was carried out and presented for the second review. - Three indices were combined in one single index as suggested and presented in the
second review.
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From: CIE 626 Peer Review Panel To: Team #4 Date: 04/25/2011 Subject: "Peer Review Panel's Suggestions/Recommendations to Team #4 of CIE 626 Structural Control Project – Second Meeting" On April 25, 2011, the members of Team #4, namely, (1) Nguyen Nam Hoai, (2) Gonzalez Sanchez Efrain, (3) Roberts Cervantes Gonzalo and (4) Rosas Espinoza Jorge, met with the Peer Review Panel to discuss the progress of the CIE 626 class project. Team #4 presented results of phase 4 (hysteretic dampers), phase 5 (viscous dampers), and phase 6 (base isolation) of the project. The suggestions/commendations listed below are provided by the Peer Review Panel (PRP) based on the 4/25/2011 progress presentation:
- Plots should include normalized responses (i.e., inter-story drift ratios (%), peak accelerations (g), etc).
- Envelopes of responses of the three ground motions should be used to compare different
retrofit cases.
Shall you have questions regarding the above suggestions/recommendations, please do not hesitate to contact us. Peer Review Panel Members Maria Koliou Maikol Del Carpio Ramos Response:
- Plots include the inter-story drift ratios (%) as well as the absolute value. - Performance indices were calculated for each ground motion therefore they were
presented separately.
STRUCTURAL CONTROL PROJECT TEAM 4
101
REFERENCES
Christopoulos C. and Filiatrault, A. (2006) “ Principles of Passive Supplemental Damping and Seismic Isolation”, IUSS Press, Italy. Romero, M.L. and Martinez-Rodrigo, M., (2003) “An Optimum Retrofit Strategy for Moment Resisting Frames with Nonlinear Viscous Dampers for Seismic Applications” Engineering Structures 25, p913-925. Federal Emergency Management Agency – ASCE (1997) “FEMA 273 - NEHRP Guidelines for the Seismic Rehabilitation of Buildings” Washington, DC. Federal Emergency Management Agency – ASCE (1997) “FEMA 274 Commentary on the NEHRP Guidelines for the Seismic Rehabilitation of Buildings” Washington, DC
