xxmxxx is the rate of the kinetic energy at time t. xxCxx is the rate of the energy dissipated by damping at time t xkxx α is the rate of the elastic energy at time t xz.
Zagazig University Faculty of Engineering Structural Engineering Department
POUNDING CONTROL OF BASE ISOLATED STRUCTURES A Thesis Submitted to the Faculty of Engineering Zagazig University for The Degree of M.Sc. in Structural Engineering Presented by
Eng. Ayman Abdo Mohamed (B. Sc., civil engineering 2006) Supervisors
Prof. Dr. Sayed Abd El-Salam Prof. of Structural Engineering Zagazig University – Faculty of Engineering.
Assist. Prof., Atef Eraky Bakry Assist. Prof. Structural Engineering Dept. Zagazig University – Faculty of Eng.
Dr. H. E. Abd-El-Mottaleb Lecturer, Structural Engineering Dept. Zagazig University – Faculty of Eng.
2011
To whom I owe my life, my parents. To whom I always owe gratefulness, Noha, my lovely wife. To all my brothers.
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ACKNOWLEDGEMENT
First of all, the most gratitude to ALLAH who has guided me to complete this work.
I would like to express my sincere appreciation to those who have contributed towards the successful completion of this thesis. My supervisor Prof. Dr. Sayed Saad Abd-El-Salam, Professor of Structural Engineering Department, Faculty of Engineering, Zagazig University, for his valuable suggestions, guidance, and sincere supervisor which have greatly contributed in achieving this research.
I would like to express my thanks and gratitude to Associate Prof. Atef Eraky Bakry, Associate Professor, Structural Engineering Department, Faculty of Engineering, Zagazig University, for suggesting the project, and for his guidance, support, patient and stimulating directions throughout the work. I am grateful to him for the references he provided me with the cut out time and the time spent in refining the manuscripts and discussing the subject with me.
I would like to acknowledge Dr. hanaa El- sayed Deif, Lecture of Structure Engineering Department, Faculty of Engineering, Zagazig University, for his great support, scientific instructions, important remarks and indispensable help during the preparation of this thesis. His constant guidance, generous help and precious advice have truly helped in bringing this work to a successful end and valuable appearance.
(i)
ABSTRACT The most important property of the base-isolation system is to make the base more flexible than the elements of the superstructure, and it must still be stiff enough to resist typical pounding forces, wind loadings, and similar lowamplitude horizontal forces. However, the potential consequences of earthquake-induced poundings on seismically isolated buildings can be much more substantial, and, thus, should be assessed. This thesis investigates, through numerical simulation, firstly, the effect of the potential pounding incidences on the building response and a comparison between two fixed base buildings two story and two seismically base isolated buildings two story under this phenomenon. A parametric study is conducted on this part to investigate the effect of the important parameters of the adjacent structures on the pounding response. These parameters are the isolator shear capacity, the post-pre stiffness ratio, the ratio between the masses of the colliding structures, separation distance between the colliding structures and the superstructure and base damping ratios. Also, the study investigates the effect of the presence of the viscous dampers at the points of the collision on the reduction of the pounding force. A specialized program has been made in order to efficiently perform numerical simulation and parametric studies on the control system. The effects of certain parameters have been investigated using the developed software such as damper yield force, damper stiffness and postpre stiffness ratio to find the optimum parameter of the viscous damper. The simulation have revealed that, even if a sufficient gap is provided, with which poundings with the surrounding adjacent buildings at the base of the building could be avoided, this does not ensure that the building will not eventually collide with neighboring buildings due to the deformations of their superstructures. Also, Pounding increases the response of the lighter floors of the adjacent base isolated buildings while decreases the response of the heavier floors, but pounding increases the response of the floors of the adjacent fixed base buildings. Finally, the presence of viscous dampers at the floor levels at the points of contact increase the dispersal of energy generated by the collision.
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CONTENTS Page ●
ACKNOWLEDGMENTS ………………………………….
i
●
ABSTRACT ………………………………………………..
ii
●
TABLE OF CONTENT ……………………………...……
iii
●
LIST OF TABLES ……………………..………………….
VII
●
LIST OF FIGURES ……………………………..………...
IX
●
LIST OF SYMBOLS………... …………………………….
X
CHAPTER (1)
INTRODUCTION
1.1 Background ..…………………………………………..
1
1.2 Study Objectives …………….…………………………
5
1.3 Outline of Chapters ..…………………………………...
5
CHAPTER (2)
LITERATURE REVIEW
2.1 General …………………………………………………
6
2.2 Review Base Isolation……………………..................
7
2.2.1 Seismic Isolation Systems ….…………………....
8
2.2.1.1 Elastomeric Bearings …………..…………..
9
2.2.1.2. Sliding Isolation Systems ……………….…
15
2.2.1.3. Hybrid Bearings ……………………….…..
20
2.2.1.4. Yielding Bearings …………………………
21
2.2.1.5. Viscous Dampers (VDs) …………………..
22
2.3 A Review on Pounding Effect on Buildings ………
23
2.3.1 Observed Pounding Damage in Past Earthquakes……
24
TABLE OF CONTENTS
2.3.2 Analytical Models for Simulation of Structural Pounding ………………………………………..........
26
2.3.3 Required Seismic Separation Distance to Avoid Pounding ……………………………………………..
33
2.4. Control of Pounding Force ……………………….
37
CHAPTER (3) MODELLING & PROGRMMING 3.1 General …………………………………………………
41
3.2. Fixed Base Buildings………..…………………………
41
3.2.1
Equation Of Motion For Two Fixed Base
42
Buildings 2DOF ……………….………………... Pounding Force Numerical Models……………...
43
3.2.2.1 Classical Theory of Impact……………....
43
3.2.2.2 Contact Element Approach ……………
45
3.2.2.2.1 Elastic Spring Element ……….
46
3.2.2.2.2 Visco Elastic Element ………...
47
3.2.2.2.3 Non-linear Elastic Model …….
48
3.2.2.2.4 Non-linear Viscoelastic Model..
49
3.3 Base Isolated Techneique ……….……………………
50
3.2.2
3.3.1
Equation of Motion of Base Isolated Building …..
50
3.3.2
Hysteretic models Base Isolation Material ………
53
3.4 Energy Transmittion Equations …………………...
56
3.5 Modeling Of The 2-Mdof System With Connected Damper…………………………………………………
58
3.6. Validation Of The Numerical Model: Fixed Base…
59
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TABLE OF CONTENTS
CHAPTER (4)
EFFECT OF POUNDING ON ADJACENT STRUCTURES
4.1 General …………………………………………………
61
4.2 Analytical
Reference Building……
62
4.3. Behavior of Buildings under Sinusoidal Ground Excitation
65
4.3.1 Two Fixed Base Adjacent Buildings ……...….
66
4.3.2 Base Isolated Adjacent Buildings ...…………
69
4.3.3 Fixed base versus Base Isolated Structures ….
73
Model
for
4.4 Parameters Affecting Pounding Response………………
77
4.4.1
Effect of Mass Ratio () ……………………....
77
4.4.2
Effect of Isolator Plastic Shear Capacity (Qp)…..
81
4.4.3
Effect of post-pre yielding stiffness ratio ……...
93
4.4.4
Effect of Separation Distance (gap) ………….….
102
4.5 Behavior of Buildings under Earthquake Excitation.…….
105
4.5.1
Effect of Mass Ratio () …………………….…..
112
4.5.2
Effect of Isolator Plastic Shear Capacity (Qp)…..
115
4.5.3
Effect of Isolator Post-Pre Stiffness ………...…..
117
4.5.4
Effect of Separation Distance (Gap)……………..
119
4.5.5
Effect of Base Mass Ratio Mb/Mt (b) ……………
122
CHAPTER (5) CONTROL OF BASE ISOLATED BUILDINGS 5.1 General …..……………………………………………..
124
5.2. Numerical Model……..………………………………..
125
5.3.Parameters Affecting the Effectiveness of Viscous Damper...
129
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TABLE OF CONTENTS
5.3.1 Effect of Excitation Frequency and Damper Yield Displacement.....................................................................
129
5.3.2 Effect of Damper Stiffness and Damper Yield Force...
131
5.3.3 Effect of Post-Pre Stiffness Ratio and Stiffness Ratio of the Damper……………………………………………….
133
5.3.4 Effect of Damper Post-Pre Stiffness Ratio and Damper Yield Displacement…………………………...
135
5.3.5 Effect of Natural Frequency of the Building and the Stiffness Ratio of the Damper………………………………
136
5.3.6 Effect of Structure Damping Ratio and Yield Displacement of the Damper…………………………..
138
5.3.7 Effect of Structure Damping Ratio and the Stiffness
Ratio of the Damper………………………………………
139
5.4 Seismic Analysis of Adjacent Controlled Buildings.
140
5.4.1 Five-Story Adjacent Buildings…………………...
140
5.4.2 Ten-Story Adjacent Buildings……………………
140
5.4.3 Other Adjacent Buildings……………………………..
141
CHAPTER (6) SUMMARY AND CONCLUSIONS 6.1 Summary ….……………………………………………
146
6.2 Conclusions …………………………………………….
147
6.3 Recommendations for Future Work……..
148
APPENDIX…………………………………….…………..
150
REFERENCES.……………………………………………
163
ARABIC SUMMARY…………………………………….
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LIST OF TABLES Table (4.1) (4.2) (4.3) (4.4)
Title Page The Parameter Used for the Reference Building A and the Excitation Characteristics………………………..... 64 The Parameters of the Second Building B…………….. 64 Maximum Displacement of Building A………………. 73 Maximum Displacement of Building B………………. 73
(4.5)
The Parts of Energy Transmitted to Building A………
76
(4.6)
The Parts of Energy Transmitted to Building B………
76
(4.7)
The details of the accelerogram for the four earthquakes
105
(4.8)
The peak disp. of first and second floor of building A and B
108
(5.1) (5.2)
Parameter Used in the Adjacent Structures………….. Maximum Displacement of the Two Cases for the Adjacent Buildings……………………………………. Maximum Response of the Two Buildings 2DOF Coupled Structures A and B Without Control and With a Passive Control System…………………………….. Maximum Response of the Two Buildings 2DOF and 5DOF Coupled Structures A and B Without Control and With a Passive Control System………………………. Maximum Response of the Two Buildings 5DOF Coupled Structures A and B Without Control and With a Passive Control System……………………………… Maximum Response of the Two Buildings 5DOF and 10DOF Coupled Structures A and B Without Control and With a Passive Control System…………………… Maximum Response of the Two Buildings 10DOF Coupled Structures A and B Without Control and With a Passive Control System…………………………….
127
(5.3)
(5.4)
(5.5)
(5.6)
(5.7)
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128
145
145
146
146
147
LIST OF FIGURES Fig. No. (1-1)
(1-2)
(2-1) (2-2)
Page
Title Damage to suspended walkway in the California State University building during the 1994 Northridge earthquake: damage location (left) and close - up (right)………………... Severe damage caused by pounding between adjacent buildings with same height ( top -left ), different height ( top -right ) in the 1999 Kocaeli (Turkey) earthquake……… Elastomeric isolation bearings………………………………
3 4
Idealized force-displacement loop of isolation bearing…….
10 13
(2-3)
Sliding type isolation systems……………………………….
17
(2-4)
A common type of Hybrid bearing consisting of a laminated rubber bearing and a sliding bearing………………………...
20
(2-5) (2-6) (2-7) (2-8) (2-9)
(2-10)
(3-1) (3-2) (3-3) (3-4) (3-5) (3-6) (3-7)
Hysteretic loop of a hybrid bearing comprised of elastomer and sliding surface…………………………………………... Flexural yielding bearings…………………………………... Idealized (perfectly bilinear) hysteresis loop of a yielding Device……………………………………………………….. Longitudinal cross section of a fluid damper (a) damper with an accumulator (b) damper with a run – through rod………. Pounding damage in bridges: (a) barrier rail damage during the 1994 Northridge earthquake; (b) connector collapse during the 1994 Northridge earthquake…………………….. Pounding damage in buildings: (a) loss of column from impact during 1999 Kocaeli earthquake; (b) wall collapse during the 1989 Loma Prieta earthquake…………………… Schematic diagram of the two adjacent 2-DOF fixed base systems consider pounding………………………………….. Stereomechanical impact (a) pre-impact state (b) Post impact state
Estimation of coefficient of restitution……………………. Schematic diagram for a general contact element…………. Linear Elastic Spring Gap Element………………………… Kelvin Voigt Element………………………………………. Schematic diagram of the two building 2-DOF base isolated systems……………………………………………………….
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21 21 22 23
25
25 42 44 45 46 47 47 51
(3-8) (3-9) (3-10) (3-11) (3-12)
(4-1) (4-2) (4-3) (4-4) (4-5) (4-6)
(4-7) (4-8) (4-9) (4-10) (4-11) (4-12) (4-13) (4-14)
Elasto-plastic hysteresis model……………………………… Bilinear hysteresis model……………………………………. Structural model of two adjacent structures linked by fluid damper……………………………………………………….. Sketches of the theoretical and experimental models for modeling pounding between two adjacent structures………. The velocity time history of Tower 1 under the 1940 El Centro earthquake, (a) the numerical find from this thesis, (b) and (c) the numerical prediction and experimental result The schematic of the two adjacent buildings A & B.…….... Force-Displacement relationship of a high-damping rubber bearing ……………………………………………………… Disp. time history of the first floor of two buildings in case of FB…………………………………………………………
Impact force in case of FBP (a) Disp time history of the first floor of two buildings, (b) Impact force………….. Disp. time history of the second floor of two buildings in case of FB…………………………………………………… Results in case of FBP (a) disp. time history of the second story of two buildings, (b) impact force, (c) base shear for the two buildings…………………………………………….. The sliding displacement of the base of the two buildings in case of BI……………………………………………………. Time history of base of the two buildings in case of BIP,(a) the sliding displacement, (b) impact force............................... Force-Displacement relationship for sliding base for building A and B, (a) , (b) in case of BI, (c) and(d) in case of BIP… The total disp. time history of the second story of two buildings in case of BI………………………………………. Results for case of BIP for the two buildings (a) the disp. time history of the second story, and (b) base shear……. The disp. time history of the second story of building A in case of FB & BI……………………………………………... The disp. time history of the second story of building B in case of FB & BI……………………………………………... The disp. time history of the second story of building A in case of FBP & BIP…………………………………………...
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54 55 58 60
60 63 63 67 67 68
69 70 70 71 72 72 75 75 75
(4.15) (4-16)
(4-17)
(4-18)
(4-19)
(4-20) (4-21) (4-22) (4-23) (4-24) (4-25) (4-26) (4-27) (4-28) (4-29) (4-30) (4-31)
The disp. time history of the second story of building B in case of FBP & BIP…………………………………………... Maximum displacement for the case of FBP (a)&( b) first and second story of building A, (c)&(d) first and second story of building B. …………………………………………. Maximum relative displacement in case of BIP (a)&( b) first and second story of building A, (c)&(d) first and second story of building B………………………………………….. Total energy of the buildings under excitation frequencies (a),( b) & (c) show the SE, DE and KE in case of FBP, (d), (e)&(f) show the show the SE, DE and KE in case of BIP Maximum sliding displacement in the base of building A in case of BI under different values of plastic shear capacity of the isolator…………………………………………………… Maximum relative displacement of the first story of building A in case of BI under plastic shear capacity………………… Maximum relative displacement of the second story of building A in case BI under different plastic shear capacity. Isolator Force- Displacement relationship for building A at wex=1.6 rad.…………………………………………………. Isolator Force- Displacement relationship for building A at wex=4.0 rad.…………………………………………………. Maximum sliding displacement in the base of B in case of BI under different plastic shear capacity of the isolator…… Maximum relative displacement of the first story of building B in case of BI under different values of plastic shear……. Maximum relative displacement of the second story of B in case of BI under different plastic shear capacity…………. The maximum displacement of the base of A in case of BIP under different plastic shear capacity of the isolator……… The maximum displacement of first story of A in case of BIP under different plastic shear capacity of the isolator………… The max. disp. of the second story of A under in case of BIP under different plastic shear capacity of the isolator………… Maximum displacement of the base of building B in case of BIP under different plastic shear capacity of the isolator…… Maximum displacement of the first story of B in case of BIPunder different plastic shear capacity of the isolator…..
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76
79
80
81
82 83 84 84 85 86 86 87 88 88 89 90 90
(4-32) (4-33) (4-34) (4-35) (4-36) (4-37) (4-38) (4-39) (4-40) (4-41) (4-42) (4-43) (4-44) (4-45) (4-46) (4-47) (4-48) (4-49) (4-50) (4-51)
Maximum displacement of the second story of building B in case of BIP under different values of plastic shear………… Total energy under variable ground excitation frequencies case of BI (a) S. E, (b) D. E, (c) K. E………………………... Total energy under variable values of ground excitation frequencies in case of BIP (a) S. E, (b) D. E, (c) K. E………. Maximum sliding displacement in the base of building A in case of BI under various values of post to pre stiffness……. Maximum relative displacement of the first story of building A in case of BI under various post to pre stiffness…………. Maximum relative displacement of the second story of building A in case of BI under various post to pre stiffness… Energy dissipation through the base of building A under two different value of post to pre stiffness ratio ......... Maximum sliding displacement in the base of building B in case of BI under various post to pre stiffness……………… Maximum relative displacement of the first story of building B in case of BI under various post to pre stiffness………… Maximum relative displacement of the second story of building B in case of BI under various post to pre stiffness… Maximum sliding displacement in the base of building A in case of BIP under various post to pre stiffness…………… Maximum relative displacement of the first story of building A in case of BIP under various post to pre stiffness………… Maximum relative displacement of the second story of building A in case of BIP under various post to pre stiffness Maximum sliding displacement in the base of building B in case of BIP under various post to pre stiffness……………. Maximum relative displacement of the first story of building B in case of BIP under various post to pre stiffness………… Maximum relative displacement of the second story of building B in case of BIP under various post to pre stiffness Total S. E, D. E and K. E in case of BI under various post to pre stiffness…………………………………………………. Total SE, DE, and KE in case of BIP under various post to pre stiffness………………………………………………….. The total no of impacts In case of FBP & BIP……………… The max disp. of the two buildings in case of FBP………..
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90 91 92 94 95 95 95 96 97 97 98 99 99 100 100 101 101 102 104 104
(4-52) (4-53) (4-54) (4-55) (4-56)
(4-57) (4-58) (4-59) (4.60) (4-61) (4-62) (4-63) (4-64) (4-65) (4-66) (4-67) (4-68) (4-69) (4-70) (4-71) (4-72) (5-1)
The max rel disp. of the building A in case of BIP……….. The max rel disp. of The building B in case of BIP……….. Time history of four earthquakes (a) Elcentro (b) Northridge, (c) Loma (d) Kobe …………………………………………... Fourier Spectrum for the four earthquakes (a) Elcentro,(b) Northridge, (c) Loma earthquake and (d) Kobe earthquake Upper floor displacement against Northridge earthquake (a, b) case of FB and FBP for building A and B respectively (c, d) case BI and BIP for building A and B……………………. Base shear in case of fixed base for (a) building A, and (b) building B……………………………………………………. Base shear in case of base isolated for (a)building A, and (b)building B………………………………………………… Total impact force in case of FBP and BIP………………….. Max. disp case of FB, (a) top floor of A, (b) top floor of B Max. disp case of FBP, (a) top story of A, (b) top story of B Max. disp case of BI, (a) top story of A, (b) top story of B…. Max. disp. in case of BIP, for (a) top story of A, (b) top story of B…………………………………………………………... Max. relative disp. with plastic shear capacity in case of BI, (a) top floor of building A, (b) top floor of building B……. Max. relative disp. with plastic shear capacity in case of BIP, (a) top floor of building A, (b) top floor of B………………. Max. relative disp. with post-pre stiffness ratio in case of BI, (a) top floor of building A, (b) top floor of building B……… Max. relative disp. with post-pre stiffness ratio in case of BIP, (a) top floor of building A, (b) top floor of B………….. No of impacts under variable gap distance (a) Case of FBP, (b) case of BIP……………………………………………….. Max. disp. in case of FBP, (a) top disp of A, (b) top disp of B Max. rel. disp. in case of BIP, (a) top of A, (b) top of B......... Max. relative disp. with respect to base mass ratio in case of BI, (a) top floor of A, (b) top floor of B……………………... Max. relative disp. with respect to base mass ratio in case of BIP, (a) top floor of A, (b) top floor of B……………………. (a) schematic of the building connected with the viscous damper link (b) Elastoplastic material of the connector……
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104 105 106 107
109 110 111 111 113 113 114 114 116 116 118 118 120 121 121 123 123 126
(a)
(5-2)
(5-3)
(5-4) (5-5) (5-6) (5-7) (5-8)
(5-9) (5-10) (5-11) (5-12) (5-13) (5-14)
(5-15)
Uncontrolled vs. controlled floor displacements (a, b, c) building A, (d, e, f) building B, for the base, First and second story, respectively…………………………………………….
The excitation frequency and the yield disp. versus the total energy (a) surface area, (b) contour line, and (c) the energy ratio (TEc /TE) at wex=2.0……………………. The total energy versusyield force and damper stiffness for (a) ωo=5.0 rad, and (b) ωo=10.0 rad……………… The total energy versusyield force and damper stiffness for (a) ωo=15.0 rad, and (b) ωo=20.0 rad…………… The total energy versusyield force and damper stiffness for (a) ωo=25.0 rad, and (b) 3o=10.0 rad…………… The total energy versusyield force and damper stiffness for (a) ωo=35.0 rad, and (b) ωo=40.0 rad………….. Effect of the variation of the natural frequency of the adjacent structures, yield displacement, and stiffness of the connected damper on the total energy……………………….
Total energy versus the elastic stiffness ratio and post stiffness ratio, (a) surface area and (b) contour line….. Total energy versus yield displacement and post stiffness ratio (a) surface area, and (b) contour line……………… Total energy against natural frequency and the yield disp. of the damper (a) surface area, and (b) contour line……. Total energy against damping ratio and the yield disp. of the damper (a) surface area, and (b) contour line……….. Total energy versus damping ratio and stiffness of the damper (a) surface area, and (b) contour line…………. The max. disp. Of the controlled and uncontrolled bounded base isolated buildings (a) the base, (b) the first floor and (c) the second floor………………………………………… Five models used in the discussion of the effect of the connected damper on the total energy and absolute displacement under seismic excitation………………..
-xiii-
128
130 131 132 132 133
133 134 136 137 138 139
141
144
LIST OF SYMBOL
a
Acceleration amplitudes of base excitation along
Kb
Translational stiffness of base
Ks
Translational stiffness of superstructure
Mb
Base mass
Ms
Superstructure mass
Qb
plastic shear capacity
ωex
Frequency of base excitation
ωn
Natural frequency
b
Base damping coefficient
s
Structure damping coefficient
G
Shear modulus of rubber
Critical damping
keff
Effective stiffness
τpy
Shear yield strength of lead
δ
The frequency factor
α
Post- pre yield stiffness ratio
b
The base mass factor
The mass factor
-xiv-
INTRODUCTION
CHAPTER (1) INTRODUCTION 1.1 General: Over the years, the idea of base isolation attracted more engineers and many inventive devices have been proposed to achieve this goal, the goal of base isolation is to reduce the energy that is transmitted from the ground motion to the structure by buffering it with a bearing layer at the foundation which has relatively low stiffness. The bearing level has a longer period than the superstructure, which reduces the force and displacement demands on the superstructure, allowing it to remain elastic and generally undamaged. One of the important properties of a baseisolation system is that although it is designed to be significantly more flexible than the elements of the superstructure, it must still be stiff enough to resist typical pounding forces, wind loadings , and similar lowamplitude horizontal forces. Therefore, the bearings may have a relatively high initial stiffness but will quickly reach yield, at which point the bearings have a greatly reduced stiffness, extending the natural period of the structure, so the pounding occur. Over the past two decades, many examples of serious structural and non-structural damage due to pounding between adjacent structures during major earthquakes have been documented in the literature. The buildings are subjected to short lateral impact force in which are not accounted in the design buildings. These impact forces produce highamplitudes, short durations and local accelerations which damages the
CHAPTER (1)
Introduction
buildings. For example, pounding contributed to damage in more than one hundred structures in the San Francisco Bay Area during the 1989 Loma Prieta earthquake
(1)
. In the 1985 Mexico City earthquake,
pounding was present in over 40% of the 330 collapsed or seriously damaged buildings. Pounding occurs if the space between adjacent structures is insufficient to allow them to vibrate freely. During earthquakes, adjacent structures
with
inadequate
clear
spacing
between
them
suffer
considerable structural and non-structural damage as a result of collision. During collision, the structures are subject to high magnitude impact forces that in many cases induce energy that is higher than the capacity of the lateral resisting systems. This has led to catastrophic collapse in some cases. These impact forces are accompanied with high amplitude acceleration which can also cause severe damage to the non-structural components and contents of the buildings. Observation of previous earthquakes show certain characteristics related to pounding. Buildings of similar heights and with similar structural systems tend to suffer less damage than buildings of different height and with different structural systems. This is due to the fact that buildings with the same height will have similar natural frequencies and will tend to move in phase relative to one another. On the contrary, buildings of different height or with different structural systems will have different natural frequencies and will tend to sway out of phase with respect to each other; this may lead to more serious damage. This failure mode is caused by insufficient spacing between adjacent buildings, which should accommodate the relative displacements under earthquake ground motions. Buildings with different configurations and different materials of construction may be either in phase or in
-2-
CHAPTER (1)
Introduction
opposition of phase when oscillating. If they are sufficiently close to each other, the frames impact and may suffer significant structural damage. Adequate separation gaps should be used to prevent these failure modes. Historical data from past earthquakes show that pounding of adjacent buildings has caused enormous losses. This is a typical mechanism of failure that happens frequently in city centers, where due to the high price of land, buildings are constructed close to each other. Figures 1.1 and 1.2 provide some examples of pounding in RC buildings of similar and different heights. Major damage is frequently found in adjacent structures of different heights; taller buildings are, in fact, more flexible than their lower counterparts.
Figure 1.1: Damage to suspended walkway in the California State University building during the 1994 Northridge earthquake: damage location (left) and close - up (right) (2).
-3-
CHAPTER (1)
Introduction
Figure 1.2 Severe damage caused by pounding between adjacent buildings with same height (top - left), different height (top - right) in the 1999 Kocaeli (Turkey) earthquake (2).
-4-
CHAPTER (1)
1.2
Introduction
Study Objectives:
Assess the effect of gap width in pounding response. Assess the effect of earthquake inputs with different PGA levels and frequency content on the pounding response. Study the pounding for buildings with different characteristics. Establish a reliable analytical model to simulate building pounding. Assess the benefit of retrofitting buildings subjected to pounding by using supplemental damping devices to reduce impact forces, floor acceleration, and story drift.
1.3
Outline of Chapters: Following to this introduction, chapter (2) presents the review to the
previous works that deal with base isolation technique, the pounding effect on the structures and how to control it. Chapter (3) contains the modeling of the problem, and the derivation of the equation of motion is presented. Also the solution of the dynamic equations of motion in the domain by using different integration algorithms is illustrated. Finally, the verification for the introduced program in the appendix has been investigated and then general representations are illustrated. Chapter (4) presents the results of the pounding effect on the two cases of buildings and the parametric study using the developed computer program and the analysis of these results are presented. Chapter (5) introduces the pounding control strategies and discusses the passive method, also introduces most efficient parameters that used. In chapter (6), summary of this thesis is made and the most important conclusions are given. Also, recommendations for the future work are proposed.
-5-
LITERATURE REVIEW
CHAPTER (2) LITERATURE REVIEW 2.1
General: Seismic isolation introduces flexibility at the isolation level
of relatively stiff buildings to avoid resonance with the typical predominant frequencies of earthquakes, in order to reduce the shear forces, interstory drift, and floor accelerations of a building, and, consequently, prevent damage of its structural and non-structural elements, as well as damage of its contents. However, since the size of the seismic gap, which is provided around a seismically isolated building to facilitate the large relative displacements at the isolation level, is usually finite due to practical limitations, pounding in adjacent structures may occur during strong earthquakes. Therefore, it is important to be aware about the potential pounding of seismically isolated buildings with adjacent structures, which may be stronger than the expected effect on the fixed base, and may affect the effectiveness of seismic isolation (3). In this chapter a review on the seismic response which induce pounding in structures are presented and the models that present the structural pounding forces affected are considered. This chapter will be divided into three sections; the first section will contain introduction of base isolation technique and its different types. The second section contains pounding effect in structures; also the modeling of pounding force simulation is contained in this section. The final section is about the control of pounding effect.
CHAPTER (2 )
Literature Review
2.2 Base Isolation Review: Introduction of seismic isolation as a practical tool has provided a rich source of experimental and theoretical work both in the dynamics of the isolated structural systems and in the mechanics of the
isolators
themselves.
This section presents a summary of the
previous studies that address the modeling and analysis aspects of structures seismic. The basics and historical development of seismic isolation is outlined. After the sever earthquake in 1908 in the Italian region, a commission was setup. The commission studied two aspects for protection of buildings from earthquake, (i) to separate a building from its foundation by a layer of sand or by use of rollers at the base of building and (ii) structure should rest on a fixed foundation. The commission finally decided to follow the latter proposal with enhanced capacity of building in lateral direction to support 8% of its weight horizontally (4). Seismic isolation is an old design idea, proposing the decoupling of a structure or part of it, or even of equipment placed in the structure, from the damaging effects of ground accelerations. One of the goals of the seismic isolation is to shift the fundamental frequency of a structure away from the dominant frequencies of earthquake ground motion and fundamental frequency of the fixed base superstructure. The other purpose of an isolation system is to provide an additional means of energy dissipation, thereby reducing the transmitted acceleration into the superstructure. This innovative design approach aims mainly at the isolation of a structure from the supporting ground, generally in the
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horizontal direction, in order to reduce the transmission of the earthquake motion to the structure (5). A medical doctor named J.A. Calanterients, in 1909, proposed the first seismic base isolation method. His isolation system was totally based on sliding. Mr. Calanterients claimed that if a structure is built on a fine material such as sand, mica, or talc, this fine soil would let the superstructure to slide during an earthquake. Hence, the horizontal force transmitted to the building would be reduced and the structure would survive the event. Although the isolation system that Mr. Calanterients proposed was a primitive earthquake resistant design, the basic idea behind his method is same with the philosophy of seismic base isolation today (6).
2.2.1 Seismic Isolation Systems: There are two basic types of isolation systems ie. elastomeric bearings and sliding bearings. The elastomeric bearings with low horizontal stiffness shift fundamental time period of the structure to avoid resonance with the excitations. The sliding isolation system is based on the concept of sliding friction. An isolation system should be able to support a structure while providing additional horizontal flexibility and energy dissipation. The three functions could be concentrated into a single device or could be provided by means of different components. Various parameters to be considered in the choice of an isolation system, apart from its general ability of shifting the vibration period and adding damping to the structure are: (i) deformability under frequent quasi-static load (i.e. initial stiffness), (ii) yielding force and displacement, (iii) capacity of self-centring after deformation, and (iv) the vertical stiffness.
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2.2.1.1 Elastomeric Bearings: The laminated rubber bearing (LRB) is most commonly used base isolation system. The basic components of LRB system are steel and rubber plates built in the alternate layers as shown in Figure 2.1(a). The dominant features of LRB system are the parallel action of linear spring and damping. Generally, the LRB system exhibits high-damping capacity, horizontal flexibility and high vertical stiffness. The damping constant of the system varies considerably with the strain level of the bearing (generally of the order of 10 percent). The system operates by decoupling the structure from the horizontal components of earthquake ground motion by interposing a layer of low horizontal stiffness between structure and foundation. The isolation effects in this type of system are produced not by absorbing the earthquake energy but by deflecting through the dynamics of the system. These devices can be manufactured easily and are quite resistant to environmental effects.
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(a) Lead rubber bearing (LRB System).
(b) Lead-core-rubber bearing. Figure 2.1: Elastomeric isolation bearings (7). Skinner et al.,
(8)
has led to number of isolation concepts that has
been applied to high way bridges and in buildings. One of the buildings in Wellington, uses, as isolators, laminated rubber bearings each of which has a cylindrical plug of lead in central hole. The lead plug produces a substantial increase in damping and also increases resistance to wind. The four stories building, a reinforced concrete framed structure, has allowed to incorporate some architectural features that have not been possible in a conventionally designed structure, considering the high seismicity (4).
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A laminated rubber bearing may be approximated as a vertical shear beam, since the steel laminations severely inhibit flexural deformations while providing no impediment to shear deformations. It can be assumed that pure shear deformations occur in the rubber only. The lateral stiffness of a laminated rubber bearing kb can be approximated as:
Kb
GA h
( 2.1)
where A, is section area of the bearing G, is shear modulus of rubber h, total height of rubber Some reduction in lateral stiffness at large displacements could be experienced, partly due to flexural beam action and partly due to increased compression of the reduced overlap area. This effect is negligible for shape factor S in the order of 10 to 20 (7). Rubber is a viscoelastic material. Damping in rubber, which is predominantly velocity dependent, is provided by deforming rubber layers. The high damping rubber was initially developed by Tun Abdul Razak Research Centre (TARRC), UK. Rubber bearings are made from two major types of rubber, low and high damping rubbers. Low and high damping rubber bearing can provide ~10% and in excess of 15% damping respectively
(7)
. This property is modified by tweaking the
amount of silica and carbon black used in rubber compound. Although damping property of the rubber can be improved by adjusting the two components some other useful properties can be deteriorated too. For
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instance, carbon blacking makes the rubber stiffer and reduces its elongation-to-break capacity. The second category of elastomeric bearings is lead-rubber bearings as shown in figure 2.1b. This system provides the combined features of vertical load support, horizontal flexibility, restoring force and damping in a single unit. These bearings are similar to the laminated rubber bearing but a central lead core is used to provide an additional means of energy dissipation. These bearings are widely used in New Zealand and also referred as N-Z system. The energy absorbing capacity by the lead core reduces the lateral displacements of the isolator. Generally, the lead yields at a relatively low stress of about 10 MPa in shear and behaves approximately as an elasto-plastic solid. The interrelated simultaneous process of recovery, recrystallization and grain growth is continuously restoring the mechanical properties of the lead. The lead has good fatigue properties during cyclic loading at plastic strains and is also readily available at high purity of 99.9 percent required for its predictable mechanical properties. The lead-rubber bearings behave essentially as hysteretic damper device and widely studied in the past (8, 9). The simplest form of a lead core bearing is comprised of (1) lead that serves as energy dissipating component (2) rubber, the component that provides restoring force to bring the structure back to the point of static equilibrium. (3) Steel shims that serve as reinforcing elements for rubber to improve stability and compressive load capacity. The initial elastic stiffness, ku, is defined as sum rubber and lead contribution, (Figure 2-2): 1 K u (G p Ap Gr Ar ) h
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CHAPTER (2 )
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where: Ap = Total area of the lead plug Gp = Shear modulus of the lead plug = 5.6 GPa Ar = Area of the rubber Gr = Shear modulus of the rubber = 0.8 MPa h = Total height of the bearing
Figure 2.2: Idealized force-displacement loop of isolation bearing (7). The post elastic stiffness, kd , is mainly dominated by contribution rubber and can be approximated as: kd
Gr Ar f c h
(2.3)
where, fc is a coefficient that accounts for lead at post-yield. For current types of LRB’s the elastic stiffness ku ranges between 6.5 to 10 times post-yield stiffness, kd. The enclosed area of a hysteresis loop is the measure of energy dissipated per cycle of motion EDC. In the idealized system Figure 2.2 the resisting or damping force of the system is replaced by an equivalent viscous damping. This equivalent damping is determined in such a
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manner as to produce the same dissipation per cycle as that produced by the actual damping force. The relationship is defined as:
EDC 2k eff 2max
(2.4)
where: = Critical damping.
EDC = Energy dissipated per Cycle. keff = Effective stiffness . Δ max=Maximum bearing displacement. The yield force can be defined as the force required for yielding of the lead plus the elastic force carried by the rubber at the corresponding yield displacement: GA Fy py Ap 1 r r G p Ap
(2.5)
where: τpy =shear yield strength of lead =10.5 MPa The characteristic strength represents the shear force at zero displacement and is generally associated to the yield strength of the lead core: Q py Ap
(2.6)
The actual behavior exhibited by an elastomeric isolator depends on the interaction between shear and axial forces. Taking the shear–axial interaction into account, Koh and Kelly proposed a simple mechanical model (two-spring model) for elastomeric bearings
(5)
. The two-spring
model is a combination of a shear spring and a rotational spring and a rigid column connected above the springs. The results of seismic
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Literature Review
response analyses with the two- spring model show that neglecting P– Δ effects can lead to considerable errors for bearings with low buckling safety factors. An improvement of the two-spring model was refined to give the so-called Koh–Kelly model, which has rotational springs at the top and bottom and the rotational springs are connected by rigid columns with a shear spring at mid-height. Since the Koh–Kelly model consists of linear springs, it is not suitable for the analysis of elastomeric bearings that exhibit nonlinear behavior at large shear displacements. In order to better predict the large-displacement behavior of elastomeric bearings, Iizuka introduced finite deformation and nonlinear hysteresis into the two-spring model to produce the so-called ‘macroscopic model’. The macroscopic model successfully predicts skeleton curves enveloping the shear force–displacement relationships obtained from cyclic loading tests. Ryan et al. examined the effects of shear strain level and applied vertical load on the properties of elastomeric isolation bearings using response data from characteristic tests of three different types of bearings and proposed an analytical model for lead–rubber bearings by the modifying the Koh–Kelly model with a bilinear hysteretic relationship and an empirical equation for bearing yield strength. The authors have also proposed an analytical model for elastomeric isolation bearings under large shear deformations, through the introduction of nonlinear hysteretic relationships to the Koh–Kelly model. 2.2.1.2 Sliding Isolation Systems:
Sliding bearings include all devices that accommodate large displacements by sliding rather than stretching, shearing or compressing.
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Literature Review
In this type of bearings, energy is dissipated by friction caused by sliding of the surfaces. The simplest sliding isolation system is the pure friction (P-F) system. In this system a sliding joint separates the superstructure and the substructure. It has been developed for low rise housing in China (10). The use of layer of sand or roller in the foundation of the building is the example of P-F base isolator. The P-F type base isolator is essentially based on the mechanism of sliding friction. The horizontal frictional force offers resistance to motion and dissipates energy. Under normal conditions of ambient vibrations and small magnitude earthquakes, the system acts like a fixed base system due to the static frictional force. For large earthquake the static value of frictional force is overcome and sliding occurs thereby reducing the accelerations. There has been a significant amount of research work on the performance of P-F system in the past (5). Mostaghel et al., (11) proposed the resilient-friction base isolation (R-FBI)
system as shown in Figure 2.3(a). This base isolator consists of concentric layers of Teflon-coated plates that are in friction contact with each other and contains a central core of rubber. It combines the beneficial effect of friction damping with that of resiliency of rubber. The rubber core distributes the sliding displacement and velocity along the height of the R-FBI bearing. They do not carry any vertical loads and are vulcanised to the sliding ring. The system provides isolation through the parallel action of friction, damping and restoring force. The concept of sliding bearings is also combined with the concept of a pendulum type response, obtaining a conceptually interesting seismic isolation system known as a friction pendulum system (FPS) as shown in
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Literature Review
figure 2.3(b). In FPS, the isolation is achieved by means of an articulated slider on spherical, concave chrome surface. The slider is faced with a bearing material which when in contact with the polished chrome surface, results in a maximum sliding friction coefficient of the order of 0.1 or less at high velocity of sliding and a minimum friction coefficient of the order of 0.05 or less for very low velocities of sliding.
(a) R-FBI system
(b) FPS System Figure 2.3: Sliding type isolation systems (11).
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The dependency of coefficient of friction on velocity is a characteristic of Teflon-type materials (12). The system acts like a fuse that is activated only when the earthquake forces overcome the static value of friction. Once set in motion, the bearing develops a lateral force equal to the combination of the mobilised frictional force and the restoring force that develops as a result of the induced rising of the structure along the spherical surface. If the friction is neglected, the equation of motion of the system is similar to the equation of motion of a pendulum, with equal mass and length equal to the radius of curvature of the spherical surface. The seismic isolation is achieved by shifting the natural period of the structure. The natural period is controlled by selection of the radius of curvature of the concave surface. The enclosing cylinder of the isolator provides a lateral displacement restraint and protects the interior components from environmental contamination. The displacement restraint provided by the cylinder provides a safety measure in case of lateral forces exceeding the design values. Lin, et al.,
(13)
has proposed a new system of free circular rolling
rods located between the base and the foundation. The most attractive feature of this type of isolator is their low value of rolling friction coefficient, which allows a very low earthquake force to be transmitted to the superstructure. However, such a system suffers from re-entering capability, resulting in large peak and residual displacements. To overcome this proposed that the shape of rolling rods should be elliptical rather then circular. The low value of the rolling friction coefficient ensures the transmission of a limited earthquake force into the superstructure and the eccentricity of the elliptical rolling rods provides a
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Literature Review
restoring force that reduces peak base displacements and brings the structure back to its original position. An important friction type base isolator is a system developed under the auspices of “Electric de France” (EDF) Gueraud et al. (1985). This system is standardized for nuclear power plants in region of high seismicity. The base raft of the power plant is supported by the isolators that are in turn supported by a foundation raft built directly on the ground. The main isolator of the EDF consists of laminated (steel reinforced) neoprene pad topped by lead-bronze plate that is in friction contact with steel plate anchored to the base raft of the structure. The friction surfaces are designed to have a coefficient of friction of 0.2 during the service life of the base isolation system. The EDF base isolator essentially uses elastomeric bearing and friction plate in series. An attractive feature of EDF isolator is that for lower amplitude ground excitation the lateral flexibility of neoprene pad provides base isolation and at high level of excitation sliding will occur which provides additional protection. This dual isolation technique was intended for small earthquakes where the deformations are concentrated only in the bearings. However, for larger earthquakes the bronze and steel plates are used to slide and dissipate seismic energy. The slip plates have been designed with a friction coefficient equal to 0.2 and to maintain this for the lifetime of the plant. Ahmadi et al., (14) proposed the design of the sliding resilient-friction (S-
RF) base isolator. This isolator combines the desirable features of the EDF and the R-FBI systems. It was suggested to replace the elastomeric bearings of the EDF base isolation by the R-FBI units. It means that the friction plate replaces the upper surface of the R-FBI system in the modified design. As a result, the structure can slide on its foundation in a
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CHAPTER (2 )
Literature Review
manner similar to that of EDF base isolation system. For low level of seismic excitation the system behaves as R-FBI system. The sliding at the top friction plate occurs only for a high level of ground acceleration that provides additional safety for unexpected severe ground motion.
2.2.1.3 Hybrid Bearings: In principle hybrid bearing incorporate the functions of multiple types of bearings in one device. One example, presented in Figure (2.4) combines a laminated rubber bearing and a sliding bearing. Application of rubber and having a flat friction surface reduces the jerking of the motion caused by debris or wrinkled surface. The sliding effect can be designed to provide the required energy dissipation (EDC). The possibility of high force due to static coefficient of function as well as possible effect due to the static-slip phenomenon could be mitigated by inserting an additional service of flexibility in the system, like a laminated rubber bearing. Figure (2.5) shows an experimental forcedisplacement curve for the bearing of Figure 2.4 (5).
Figure 2.4: A common type of Hybrid bearing consisting of a laminated rubber bearing and a sliding bearing (7).
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Literature Review
Figure 2.5: Hysteretic loop of a hybrid bearing comprised of elastomer and sliding surface (7).
2.2.1.4 Yielding Bearings: Yielding dampers take advantage of the hysteretic behavior of metals when deformed into the post-elastic range. A large variety of different types of devices have been developed that utilize flexural, shear or axial deformation modes into the plastic range, Figure 2.6. In Figure 2.7, is presented the idealized force-displacement curve of a yielding device. The introduced amount of damping is related to the area bounded by the loop.
Fig 2.6: Flexural yielding bearings (7).
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Figure 2.7: Idealized (perfectly bilinear) hysteresis 0 loop of a yielding Device (7).
2.2.1.5 Viscous Dampers (VDs) The large majority of viscous dampers utilize a silicon fluid viscous material. In viscous fluids, dissipation refers to a process in which the viscous fluid absorbs energy from the imposed motion (kinetic energy) and, transforms it into internal thermal energy of the fluid. In other word viscous dissipation is the work done by the velocity against the viscous stresses. The energy transferred to heat will raise the temperature of the fluid. Viscous dissipation is an irreversible process. The viscous damper device takes advantage of this principle to consume the seismic energy. That generates effective displacement between the device ends. Viscous dampers not only provides damping of up to 40 % but also used for reduction of structural drift (15).
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Figure 2.8: longitudinal cross section of a fluid damper (a) damper with an accumulator (b) damper with a run – through rod (15).
2.3 Review of Pounding Effect on Buildings: It is interesting to note that both damage due to seismic pounding and the need to provide for its prevention or mitigation have been recognize since early in the development of earthquake engineering. One of the first books on earthquake resistance design of building structures
(ford
1926)
already
describes
seismic
pounding
of
nonstructural components against structural elements and recommends provision for sufficient separation gaps (16).
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2.3.1 Observed Pounding Damage in Past Earthquakes; More than 200 cases of seismic pounding involving more than 500 building structures were observed in the 1989 Loma Prieta earthquake (17). While the intensity of pounding damage was slight in many cases, some instances of severe pounding damage were nevertheless observed. The attention for pounding effect have increased After the 1985 Mexico City earthquake, pounding damage was reported in over 40% of the collapsed or severely damaged buildings. In at least 15% of the damaged buildings, pounding were the primary cause of collapse reported (18)
.
Based on the observations from past earthquakes, closely spaced buildings can experience infill wall damage, column shear failure and possible column collapse due to pounding. Pounding in bridges can lead to local crushing and spalling of concrete, damage to column bents, abutments, shear keys, bearing pads and restrainers and possible deck collapse. Figures 2.1 and 2.2 illustrate some instances of pounding damage, ranging from the superficial to complete collapse (19). Pounding damage has also been observed in the 1976 friuli (Italy) earthquake. In many cases, such as at the gemona hospital, the intensity of pounding damage was small (20). Instances of severe damage, however, were also observed. For example, the roof slab of a small, 1-story rigid annex was shorn off and pushed over because of pounding against the adjacent side wall panels of a storage shed. The wall panel also suffered some damage.
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Figure 2.9: pounding damage in bridges: (a) barrier rail damage during the 1994 Northridge earthquake; (b) connector collapse during the 1994 Northridge earthquake (19).
Figure 2.10: pounding damage in buildings: (a) loss of column from impact during 1999 Kocaeli earthquake; (b) wall collapse during the 1989 Loma Prieta earthquake (19).
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2.3.2 Analytical Models for Simulation of Structural Pounding: There are generally two different approaches of structural pounding the first one applies the classical theory of impact, called stereomechanics, which is based on the laws of conservation of energy and momentum and does not consider transient stresses and deformations in the impacting bodies. The second approach to modeling of pounding is to simulate directly the pounding force during impact. The experimental results shown that pounding force history depends substantially on a number of factors, such as masses of colliding structures, their relative velocity before impact, structural material properties, contact surface geometry and previous impact history (21). Kun YE, Li Li
(10)
studied Impact analytical models for
earthquake-induced pounding Simulation; a brief review on the commonly used impact analytical models is conducted. Based on this review, the formula used to determine the damping constant related to the impact spring stiffness, coefficient of restitution, and relative approaching velocity in the Hertz model with nonlinear damping is found to be incorrect. To correct this error, a more accurate approximating formula for the damping constant is theoretically derived and numerically verified. At the same time, a modified Kelvin impact model, which can reasonably account for the physical nature of pounding and conveniently implemented in the earthquake-induced pounding simulation of structural engineering, is proposed. Robert J.,
(22)
makes a Comparison of Numerical Models of
Impact Force for Simulation of Earthquake-Induced Structural Pounding, The aim of the present paper is to check the accuracy of three pounding force numerical models, such as: the linear viscoelastic model, the nonlinear elastic model following the Hertz law of contact and the non-linear
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viscoelastic model. Finally, the results of the study indicate that the nonlinear viscoelastic model is the most precise one in simulating the pounding force time history during impact. Shehata E.
(23)
studied Seismic Pounding between Adjacent
Building Structures. A parametric study on buildings pounding response as well as proper seismic hazard mitigation practice for adjacent buildings is carried out. Three categories of recorded earthquake excitation are used for input. The effect of impact is studied using linear and nonlinear contact force model for different separation distances and compared with nominal model without pounding consideration. Pounding produces acceleration and shear at various story levels that are greater than those obtained from the no pounding case, while the peak drift depends on the input excitation characteristics. Also, increasing gap width is likely to be effective when the separation is sufficiently wide practically to eliminate contact. Muthukumar and DesRoches
(24)
compared various impact
models in modeling the seismic pounding response of adjacent structures, including 261 the stereomechanical model, linear spring model, Kelvin model, Hertz model, and Hertz model with a nonlinear damper (referred to as the Hertz damp model by the authors). The results suggest that the Hertz model provides adequate results at low PGA (0.1-0.3g) levels of ground shaking, and the Hertz damp model performs better at moderate (0.4-0.6g) and high (0.7-0.9g) PGA levels, for energy loss during impact being more significant at higher levels of PGA. Robert J.,
(21)
used also a Non-linear viscoelastic modeling of
earthquake-induced structural pounding. Past severe earthquakes indicate that structural pounding may cause considerable damage or even lead to collapse of colliding structures if the separation distance between them is not sufficient. Because of its complexity, modeling of impact is an
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extremely difficult task; however, the precise numerical model of pounding is essential if an accurate structural response is to be simulated. The aim of this paper is to analyze a non-linear viscoelastic model of collisions which allows more precise simulation of the structural pounding during earthquakes. The effectiveness of the model is verified by comparing the results of numerical analyses with the results of experiments conducted on pounding between different types of structures. The results of the study indicate that, compared to other models, the proposed non-linear viscoelastic model is the most precise one in simulating the pounding-involved structural response. Robert J.,
earthquake
(25)
studied a Pounding force response spectrum under
excitation.
Earthquake-induced
pounding
between
inadequately separated structures may cause considerable damage or even lead to a structure’s total collapse. The assessment of the damage magnitude as well as the design of some pounding mitigation method requires the knowledge of the maximum impact force value expected during the time of earthquake. The aim of the present paper is to propose the idea of impact force response spectrum for two adjacent structures, which shows the plot of the peak value of pounding force as a function of the natural structural vibration periods. In the analysis, both structures have been modeled by single-degree-of-freedom systems and pounding has been simulated by the non- linear viscoelastic model. The analysis has been conducted for elastic and inelastic (Elastoplastic) structures under different ground motions. The examples of response spectra show that the selection of the structural parameters, such as the gap size between structures, their natural vibration periods, damping, mass and ductility as well as the time lag of input ground motion records, might have a substantial influence on the peak pounding force value. The results
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of the study indicate that impact force response spectra might serve as a very useful tool for the design purposes of closely-spaced structures on seismic areas. Robert J.,
(26)
make a Experimental study on earthquake-induced
pounding between structural elements made of different building materials, The aim of this paper is to show the results of two experiments concerning interactions between elements made of different building materials, such as steel, concrete, timber and ceramic. The first experiment was conducted by dropping balls from different height levels on to a rigid surface, whereas the second one was focused on poundinginvolved response of two tower models excited on a shaking table. The results of the impact experiment show that the value of the coefficient of restitution depends substantially on the prior-impact velocity as well as on the material used. Based on these results, the appropriate formulations have been suggested to be applied in the numerical simulations. The results of the shaking table tests show a considerable influence of the material used for colliding elements on the behavior of structures during earthquakes. The effect of pounding damage during Loma Prieta earthquake have been studied, the surveys on damage during past earthquakes show that interactions between insufficiently separated buildings or bridge segments may result in substantial damage or even contribute to total structural collapse caused by pounding during the 1989 Loma Prieta earthquake.
Pounding was present over a wide
geographical
area
including the cities of San Francisco, Oakland, Santa Cruz and Watsonville. The survey database contains more than 200 pounding occurrences involving more than 500 building structures. The paper discusses the distribution of pounding damage in the specific areas, types
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of pounding damage, and examples of pounding damage involving major multistory buildings. Some significant pounding occurred at sites with large epicentral distances indicating the possible catastrophic damage that may occur during future earthquakes having closer epicenters (17). Robert et al.,
(27)
studied the Reduction of pounding effects in
elevated bridges during earthquakes. Which can result in significant structural damage, several methods of reduction of the negative effects of collisions induced by the seismic wave propagation effect been analyzed in this paper. The analysis is conducted on a detailed three-dimensional structural component model of an isolated highway bridge. The results show that the influence of pounding on the structural response is significant in the longitudinal direction of the bridge and significantly depends on the gap size between superstructure segments. The smallest response can be obtained for very small gap sizes and for gap sizes large enough to prevent pounding. Further analysis indicates that the bridge behavior can be effectively improved by placing hard rubber bumpers between segments and by stiff linking the segments one with another. The experimental results show that, for the practical application of such connectors, shock transmission units can be used. Chris et al., (28) studied an Earthquake-induced interaction between
adjacent reinforced concrete structures with non-equal heights, the influence of the structural pounding on the ductility requirements and the seismic behavior of reinforced concrete structures designed to EC2 and EC8 with non-equal heights is investigated. Special purpose elements of distributed plasticity are employed for the study of the columns. Two distinct types of the problem are identified: Type A, where collisions may occur only between storey masses; and Type B, where the slabs of the first structure hit the columns of the other (72 Type A and 36 Type B
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pounding cases are examined). Type A cases yielded critical ductility requirements for the columns in the pounding area mainly for the cases where the structures were in contact from the beginning of the excitation. In both pounding types the ductility requirements of the columns of the taller building are substantially increased for the floors above the highest contact storey level probably due to whiplash behavior. The most important issue in the pounding type B is the local response of the column of the tall structure that suffers the hit of the upper floor slab of the adjacent shorter structure. In all the examined cases this column was in a critical condition due to shear action and in the cases where the structures were in contact from the beginning of the excitation, this column was also critical due to high ductility demands. It can be summarized that in situations of potential pounding, neglecting its possible effects leads to non-conservative building design or evaluation that may become critical in some cases. Abdullah et al.
(29)
studied the Using of a shared tuned mass
damper (STMD) to reduce vibration and pounding in adjacent structures, The aim of the present paper is to check the accuracy of three pounding force numerical models, such as: the linear viscoelastic model, the nonlinear elastic model following the Hertz law of contact and the non-linear viscoelastic model. Finally, the results of the study indicate that the nonlinear viscoelastic model is the most precise one in simulating the pounding force time history during impact. Kun et al.
(30)
used a modified Kelvin impact model for pounding
simulation of base-isolated building with adjacent structures; relevant parameters in the modified Kelvin model are theoretically derived and numerically verified through a simple pounding case. At the same time, inelasticity of the isolated superstructure is introduced in order to accurately evaluate the potential damage to the superstructure caused by
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the pounding of the BIB with adjacent structures. It is shown that pounding can substantially increase floor accelerations, especially at the ground floor where impacts occur. Higher modes of vibration are excited during poundings, increasing the inter-story drifts instead of keeping a nearly rigid-body motion of the superstructure. Furthermore, higher ductility demands can be imposed on lower floors of the superstructure. Moreover, impact stiffness seems to play a significant role in the acceleration response at the isolation level and the inter-story drifts of lower floors of the superstructure. Finally, the numerical results show that excessive flexibility of the isolation system used to minimize the floor acceleration may cause the BIB to be more susceptible to pounding under limited seismic gap. Panayiotis et al.
(31)
studied the Earthquake-induced poundings of
a seismically isolated building with adjacent structures, and used The effects of certain parameters, such as the size of the separation distance, the characteristics of the adjacent structures and the earthquake characteristics, have been investigated using the developed software. The simulations have revealed that even if a sufficient gap is provided, with which poundings with the surrounding moat wall at the base of the building could be avoided, this does not ensure that the building will not eventually collide with neighboring buildings due to the deformations of their superstructures. Anagnostopoulos S. et al.,
(32)
studied the earthquake induced
pounding between adjacent buildings. They idealized the building as lumped-mass, shear beam type, multi-degree-of-freedom (MDOF) systems with bilinear force- deformation characteristics and with bases supported on translational and rocking spring- dashpots. Collisions between adjacent masses can occur at any level and are simulated by means of viscoelastic impact elements. They used five real earthquake
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CHAPTER (2 )
Literature Review
motions to study the effects of the following factors: building configuration and relative size, seismic separation distance and impact element properties. It was found that pounding can cause high overstresses, mainly when the colliding buildings have significantly different heights, periods or masses. They suggest a possibility for introducing a set of conditions into the codes, combined with some special measures, as an alternative to the seismic separation requirement. Viviane Warnotte summarized basic concepts on which the seismic pounding effect occurs between adjacent buildings. He identified the conditions under which the seismic pounding will occur between buildings and adequate information and, perhaps more importantly, pounding situation analyzed. From his research it was found that an elastic model cannot predict correctly the behaviors of the structure due to seismic pounding. Therefore non-elastic analysis is to be done to predict the required seismic gap between buildings.
2.3.3 Required Seismic Separation Distance to Avoid Pounding: Seismic pounding occurs when the separation distance between adjacent buildings is not large enough to accommodate the relative motion during earthquake events. Depending on the characteristics of the colliding buildings (Anagnostopoulos), pounding might cause severe structural damage in some cases and even collapse is possible in some extreme situations
(18)
. Further, even in those cases where it does not
result in significant structural damage, pounding always induces higher floor accelerations in the form of short duration spikes, which in turn cause greater damage to building contents. For these reasons, it has been widely accepted that pounding is an undesirable phenomenon that should
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CHAPTER (2 )
Literature Review
be prevented or mitigated. This is recognized in seismic design codes and regulations worldwide, which typically specify minimum separation distances to be provided between adjacent buildings. For instance, according to the 2000 edition of the International Building Code
(32)
minimum separation distances are given by: U (t) =Ub (t) − Ua (t)
(buildings separated by a property line)
U (t ) U a2 (t ) U b2 (t )
(buildings located by a property)
2.7
2. 8
where U = separation distance and Ua, Ub = peak displacement response of adjacent structures “A” and “B”, respectively. Equations (2.7) and (2.8) are usually referred to as The ABS (ABSolute sum) and SRSS (Square Root of Sum of Squares), respectively, and are implemented in many seismic design codes and regulations worldwide. Previous studies (Jeng, Lopez Garcia), however, have shown that they give poor estimates of U, especially when the natural periods of the adjacent structures are close to each other. In these cases, the ABS and SRSS rules give excessively conservative separation distances, which are very difficult to effectively implement because of maximization of land usage. A more rational approach was presented by Jeng and can be summarized as follows: Let Ua (t) and Ub (t) be stationary random processes over a finite duration, representing the displacement response of two linear SDOF systems to a Gaussian, zero-mean stationary stochastic excitation Ug (t), and let UREL (t) = UA (t) - UB (t) be the random process representing the displacement response of the systems relative to one another. It can be shown that UA (t), UB (t) and UREL (t) are also zero-mean Gaussian processes (Soong). Assuming that the ratio of the mean peak value to the standard deviation is a constant (a
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Literature Review
reasonable approximation in the case of stationary Gaussian processes), then the critical separation distance, which is obviously equal to the peak relative displacement response, is given by: where UA, UB, UREL = mean peak values of UA (t), UB (t) and UREL (t), respectively, and is given by:
U U REL U A2 U B2 2 U AU B
EU A . U B
2.9 2.10
E U A2 E U B2
where E {} is the expectation operator. Equation (4) indicates that
is the correlation coefficient for processes UA (t) and UB (t). If it is assumed further that the excitation Ug (t) is a white noise, then correlation coefficient is given by Kiureghian et al. (33).
8 a b ( b aTb / Ta )(Tb / Ta ) 3 / 2 [1 (Tb / Ta ) 2 ]2 4 a b [1 (Tb / Ta ) 2 ](Tb / Ta ) 4( a2 b2 )(Tb / Ta ) 2
where TA,
A
and TB,
B
2.11
are natural periods and damping ratios
of systems “A” and “B”, respectively. Equation (2.10), together with Equation (2.11), is usually referred to as the Double Difference Combination (DDC) rule. In the case of more realistic nonstationary wide-band excitations, it has been shown
(34, 35)
. That despite the various
simplifying assumptions under which it was derived, the DDC rule is much more accurate than the ABS and SRSS rules, although it gives somewhat UN conservative results when TA and TB are well separated. A correction for these latter cases has been proposed.
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CHAPTER (2 )
Literature Review
In actual case scenarios, a great majority of building structures responds nonlinearly when subjected to strong ground motions. If it is assumed that Equation (2.11) is still valid for nonlinear systems (an assumption whose basis is weaker than that for linear systems), then all that is needed to extend the applicability of the DDC rule to nonlinear systems is a suitable expression for , for which no closed-form solution valid for hysteretic systems exists. This problem has been the subject of a number of studies carried out by several authors Filiatrault, A.,
(36, 49)
who have proposed different methods. Komodromos, et al.
(37)
use a numerical simulations and
parametric studies to investigate the influence of potential poundings of seismically
isolated
buildings
with
adjacent
structures
on
the
effectiveness of seismic isolation. Poundings are assumed to occur at the isolation level between the seismically isolated building and the surrounding moat wall. After assessing some common force-based impact models, a variation of the linear viscoelastic impact model is proposed to avoid tensile impact forces during detachment, while enabling the consideration of permanent plastic deformations at the vicinity of the impact. The numerical simulations demonstrate that poundings may substantially increase floor accelerations, especially at the base floor where impacts occur. Finally, the results indicate that providing excessive flexibility at the isolation system to minimize the floor accelerations may lead to a building vulnerable to poundings, if the available seismic gap is limited.
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CHAPTER (2 )
Literature Review
2.4 Control of Pounding Force: The most natural way to prevent structural pounding is to provide sufficiently large spacing between adjacent structures or structural members. However, due to the land shortage and high land prices in many cities located on seismic areas, this solution is usually very difficult to be accepted by the land owners. Moreover, there are many examples of old buildings with different dynamic characteristics, which have been constructed in contact with each other, as this was not prohibited by the old earthquake- resistant design codes. Westermo suggested, for example, linking buildings by beams, which can transmit the forces between the structures and thus eliminating collisions. The connections between adjacent structures can also have some energy dissipating properties and impacts can be partly absorbed. The idea of filling the separation gap by an energy absorbing material or providing strong collision walls protecting part of the structure has been studied.
Jankowski et al. considered the use of
bumpers, crushable devices and shock transmission units to suppress the blows of impacts in bridges. The effectiveness of variable dampers as well as restrainers has also been analyzed. For the design purposes of pounding-prone structures, the magnitude of the impact force, which can be expected during the time of earthquake, needs to be known in order to assess the potential damage due to collisions. Also the design of pounding reduction methods, such as collision walls or bumpers, for example, requires the knowledge of the peak collision forces. Nawawi, et al.,
(38)
studied the reduction of pounding responses of
bridges girders with soil-structure interaction effects to spatial nearsource ground motions. They address a possible measure for reducing the
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CHAPTER (2 )
Literature Review
pounding potential between two identical adjacent bridge girders. The using of spring and viscous damper in the investigation the influence of the spatially varying ground excitation and the soil-structure interaction are considered. The ground motions are simulated based on the Newmark-Hall response spectrum. The investigation shows that especially for structures with similar dynamic properties an assumption of uniform ground excitation and fixed base can clearly underestimate the pounding potential of the adjacent structures. They find that the reduction measure increases significantly the damping of the bridge structures, and consequently reduces the girder pounding occurrences. Orlando, et al.,
(39)
studied a numerical investigation of a variable
damping semiactive device for the mitigation of the seismic response of adjacent structures. And find that there is one way to overcome the effect of pounding is to couple the structures through elastic or damping elements. This article examines the use of a new variable damping device as a coupling element. The system, which is termed a variable damping semiactive (VDSA) device, consists of two dampers with constant parameters whose lower ends are attached to a common vertical rod whereas the upper ends are attached to the two structures. As the structures vibrate due to the ground motion, the lower end is moved up and down by means of an actuator. By changing the orientation of the dampers, the effective damping in the two structures can be changed in time in an appropriate manner to minimize the response. Anew control law is used to calculate the optimal position of the dampers. The algorithm, referred to as Qv, is a variation of the Instantaneous Optimal Control and it is based on the minimization of a performance index J quadratic in the state vector, the control force vector, and a vector of absolute velocities measured at selected points. The algorithm includes a
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CHAPTER (2 )
Literature Review
generalized LQR scheme where penalties are imposed on the state vector, on the control vector, and on the absolute velocity vector through three predefined matrices. A numerical simulation is used to verify the performance of the proposed protective system in reducing the seismic response to a series of historic earthquakes. The results show that the proposed device is able not only to eliminate the pounding effects but also to significantly reduce the response of the individual adjacent structures. Measures for reducing the pounding effects have been investigated by many researchers. For example, Abdullah et al. introduced shared tuned mass damper (STMD) that links two adjacent buildings. Their study showed that buildings with STMD could perform better than buildings with individual TMD. Westermo linked the neighboring buildings with springs and viscous dampers at possible pounding locations. Yang et al. connected two neighboring buildings with fluid dampers at the floor levels in their investigation. Kawashima et al. and Ruangrassamee et al. studied the effectiveness of different reduction measures installed between the adjacent bridge girders (39). Zhong, L.,
(30)
present a broad study on analysis and control for
seismic pounding responses of urban elevated bridges. Based on Hertz contact theory, a method to determine the spring stiffness of Kelvin impact model for seismic pounding analysis is proposed firstly. And Influence of different parameters on the seismic pounding responses is investigated. The influence of spatial variation of ground motion on the seismic pounding responses is further studied. A method to calculate critical separation gap was founded based on random vibration theory. The performance of using viscous damper to mitigate the seismic pounding response is analyzed, and its parametric design method is also
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CHAPTER (2 )
Literature Review
proposed. The semi active control on the seismic pounding responses between adjacent segments of the isolated urban elevated bridges is performed using the MRF-04K magnetor heological damper. So he find that (1) The pounding of bridges is caused by the coactions of many factors, which makes the seismic responses of the bridges more complex and non-linear. (2) The spring stiffness of pounding in the equivalent Kelvin impact model is influenced by the parameters including the Hertz contact stiffness, the relative velocity of motion of the adjacent girders and the ratio of lengths of the short girder to the long girder. (3) Considering the non-linearity of pier and the pile-soil interaction, the pounding responses may be increased, especially for the multi-span continuous girder bridge. (4) The spatial variation of ground motion has significant influence on the seismic pounding responses of the bridges. (5) The viscous damper can diminish the relative displacement of adjacent spans and the pounding effect is greatly suppressed without increasing ductility demand in the piers. (6) Installing MR dampers between superstructure and piers will get much better performance than installing them between adjacent segments, and the relative displacement and bearing deformation decrease remarkably by semi-active control.
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MODELING & PROGRAMMING
CHAPTER (3) MODELING AND PROGRAMMING
3.1 General: The theoretical formulation for modeling adjacent fixed base buildings and adjacent base isolated buildings is presented in this chapter. This chapter has been divided into four sections. In the first section, two adjacent 2DOF fixed base buildings and the discussion of the equations of pounding are presented. In the second section, a formulation that introduces two adjacent 2DOF base isolated buildings take into consideration the effecting of pounding. The third one is the energy input and transmitted to the buildings will be discussed. The last section is the validation of the numerical formulation and the verification on it.
3.2 Fixed Base Buildings:
In this part two adjacent buildings 2DOF with clearance distance make the pounding effect occur are analyzed, in order to study the pounding effect on the behavior of the adjacent buildings under this phenomenon. The pounding assumed occur at all levels of the adjacent buildings. The floors height of the corresponding adjacent buildings are the same, i.e. the pounding occur between the slabs.
CHAPTER (3)
Mathematical Development
3.2.1 Equation of Motion for Two Fixed Base Buildings 2DOF:
gap
m 12
m 22
c12
c22
k12
k22 m 21
m 11 c11
k11
c21
building A
k21
building B
Fig 3.1: Schematic diagram of the two adjacent 2-DOF fixed base systems. Figure 3.1 shows the considered two buildings; both of them are modeled as two degree of freedom systems and fixed. For each building two equations are coupled and can be written in matrix form. The response of fixed base building A is governed by the equation;
m11 0
0 x11 c11 c12 m12 x12 c12 k11 k12 k 12
c12 x 11 c12 x 12
k12 x11 m11 0 k12 x12
0 1 xg m12 1
(3.1)
The response of fixed base building B is governed by the equation;
m 21 0
0 x 21 c 21 c22 m22 x 22 c22 k 21 k 22 k 22
c22 x 21 c22 x 22
k 22 x 21 m 21 0 k 22 x 22
- 42 -
0 1 xg m22 1
(3.2)
CHAPTER (3)
Mathematical Development
Equations 3.1 and 3.2 are also coupled together leading to the overall system equation of motion; m11 0 0 0
0 m12
0 0
0 0
m21 0
0 x11 c11 c12 0 x12 c12 0 x 21 0 m22 x 22 0
k11 k12 k 12 0 0
k12 k12
0 0
0 0
k 21 k 22 k 22 m11 0 0 0
c12 c12
0 0
0 0
c 21 c22 c22
0 x 11 0 x 12 c22 x 21 c22 x 22
0 x11 0 x12 k 22 x 21 k 22 x 22 0 0 0 1 m12 0 0 1 xg 0 m21 0 1 0 0 m22 1
(3.3)
where mij is the mass of the jth floor at the ith building, kij and cij are the stiffness and damping of the jth story at the ith building respectively, and x, x , x are the absolute displacement, velocity and acceleration,
respectively.
3.2.2 Pounding Force Numerical Models: 3.2.2.1 Classical Theory of Impact: There are generally two different approaches of structural pounding. The first one applies the classical theory of impact, called stereomechanics, which is based on the laws of conservation of energy and momentum but this method does not consider transient stresses and deformations in the impacting bodies. The another method is the theory focuses on determination of post-impact velocities of colliding bodies
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Mathematical Development
based on the approaching velocities prior to contact and a coefficient of restitution which account for the energy dissipation during impact incorporating response non-linearities. The formulae for the post-impact velocities v1' and v2' of two non-rotating bodies with masses m1 and m2 in the case of the central impact are given by Jankowski, R., (21). v1'
v1
(1 e )
m2 v1 m2 v2 m1 m2
(3.4)
v2'
v2
(1 e )
m1 v1 m1 v2 m1 m2
(3.5)
where v1 and v2 are approaching velocities and e is a coefficient of restitution which can be obtained from the equation e
(a)
v2' v1' v1 v2
(3.6)
(b)
Figure 3.2: stereomechanical impact (a) pre-impact state; (b) Post impact state (20). A value of e=1 deals with the case of a fully elastic collision, for fully plastic one the restitution will be e=0 a, but for the real case the value of e will be in between. For concrete collision the value of e= 0.5 to 0.75 (e=0.65). The value of the coefficient of restitution can be determined experimentally by dropping a sphere on a massive plane plate of the same material from a height h and observing the rebound height h*.
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CHAPTER (3)
Mathematical Development
By measuring the falling distance h and the rebound distance h*, the coefficient of restitution can be determined from the following formula (20). h* e h 2
(3.7)
Figure 3.3: Estimation of coefficient of restitution (20). It has been assessed that the coefficient of restitution used to simulate real collision between structures ranges usually from 0.5 to 0.75. Based on the experimental results, suggested that e=0.65 should be used for typical concrete structures. In fact, this value has been used by a number of researchers in the analysis of pounding between different types of structures. Some of the studies indicate, however, that collisions between structural members can be more plastic in some cases (40).
3.2.2.2 Contact Element Approach: The following approach in modeling pounding between structures is to simulate the impact force during impact by the means of a finite element formulation using what is called the impact or contact element. Consequently, the contact element consists of a spring connected to a gap
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CHAPTER (3)
Mathematical Development
as shown schematically in figure 3.4. The element cannot develop a force until the opening gap is closed and the element can only develop a negative compression force.
open
Gap
k
Figure 3.4: schematic diagram for a general contact element (40).
* Elastic Spring Element: Based on the concept discussed in the last section, many contact elements have been developed over the years to model structure pounding. The simplest contact element shown in the figure 3.5 consists of a linear spring. The main disadvantage of this element is that it does not count for any energy losses during the impact. The numerical value of the spring stiffness is also a major source of uncertainty. Researchers who worked with this element suggest using large numerical value and should be taken to be equal to the in plane axial capacity of the colliding structures, or 20 times the stiffness of the stiffer structure subjected to impact. The large numerical value of the spring's stiffness often leads to a conversion problem during the solutions of equations of motion, and often the element produces unrealistic impact force.
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CHAPTER (3)
Mathematical Development
Figure 3.5: Linear Elastic Spring Gap Element (41).
* Visco Elastic Element: The Kelvin Volget element shown in figure 3.6 represents further enhancement on modeling impact between structures. The pounding force during impact Fc is counted by using the spring stiffness similar to the linear elastic element. The dashpot that is connected in parallel with the spring counts for energy losses during the impact.
Figure 3.6: Kelvin Voigt Element (41).
This linear viscoelastic model is the most frequently used one for simulation of structural pounding under earthquake excitation. The pounding force during impact, F(t), for this model is expressed as;
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CHAPTER (3)
Mathematical Development
F (t ) k (t ) c (t )
(3.8)
where (t ) describes the deformation of colliding structural members,
(t ) denotes the relative velocity between them, k is the impact element’s stiffness simulating the local stiffness at the contact point and c is the impact element’s damping, which can be obtained from the formula; c 2
k
m1 m2 m1 m2
(3.9)
where m1 , m2 are masses of structural members and is the damping ratio related to a coefficient of restitution, e, which accounts for the energy dissipation during impact. The relation between
and e in the
linear viscoelastic model is given by the formula.
ln e
2 ( ln e) 2
(3.10)
* Non-linear Elastic Model: In order to model the pounding force-deformation relation more realistically, a non-linear elastic model following the Hertz law of contact has been adopted by a number of researchers. The pounding force, F (t), for this model is expressed by the formula;
F (t )
3 2
(t )
- 48 -
(3.11)
CHAPTER (3)
Mathematical Development
where is the impact stiffness parameter that depends on material properties and geometry of colliding bodies. The disadvantage of the Hertz contact law model is that it is fully elastic and does not account for the energy dissipation during contact due to plastic deformations, local crushing, etc.
* Non-linear Viscoelastic Model: For the purposes of a more precise simulation of an impact phenomenon, a non-linear viscoelastic model has been proposed
(40)
. The
pounding force, F (t), for this model is expressed by the formula;
F (t ) F (t )
3 2 3 2
(t ) c (t ) (t )
for (t ) 0 (approach period )
(3.12)
(t )
for (t ) 0 (restitution period )
(3.13)
where is the impact stiffness parameter and c (t ) is the impact element’s damping, which at any instant of time can be obtained from the formula; c (t ) 2
(t )
m1m2 m1 m2
(3.14)
where denotes the damping ratio related to a coefficient of restitution, e. The approximate relation between
and e in the non-linear
viscoelastic model is expressed by the formula; 9 5 2
1 e2 e ( e ( 9 16 ) 16 )
- 49 -
(3.15)
CHAPTER (3)
Mathematical Development
3.3 Base Isolated Techniques: Base isolation system in a building system decreases the inertia force acting on the superstructure and hence the relative deflections and shear forces (Arya 1984).
A sliding base isolation system can be
provided by using Teflon (TFE) sliding bearings between the superstructure and its foundation and consists of Teflon-steel interfaces. In actual building a sliding base isolation system also consists of a centering device or restoring force (Mokha et al 1990) so as to avoid the residual displacement of the structure, however in the present study this consideration is neglected. For the sliding isolation system it will be assumed that a constant coefficient of friction is adequate. In an actual device the static coefficient of friction is different from the kinetic value and both of these vary as a function of bearing pressure and sliding velocity (42).
3.3.1 Equation of Motion of Base Isolated Building: The governing equation of motion will be developed for the response of two multi degree of freedom base isolated systems. Consider the two multi degree of freedom base isolated system illustrated in figure 3.7. Here the base of the building is separated from the foundation by the sliding system. The upper part of the structural system is allowed to slide with respect to the foundation, which has the same motion as the ground during an earthquake. The structure and its base slide on the lower part as one piece.
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CHAPTER (3)
Mathematical Development
second story
building A
gap
u
building B
u
first story
base ground
earthquake foundation
foundation
u
2
1
b
ug
Figure 3.7: Schematic diagram of the two building 2-DOF base isolated systems
The two degree of freedom analysis of the simple linear model developed earlier can be applied to the case of a multistoried building. The representation of the structural system of this case by mass matrix M, damping matrix C, and stiffness matrix K. for a conventionally base structure, the relative displacement u of each degree of freedom with respect to the ground is given by Kelly and Naeim (43).
M x C x K x M R xg
(3.16)
where R is a vector that couples each degree of freedom to the ground motion. And X , X , X are the displacement, velocity and acceleration vectors of the upper stories relative to the base slab; and xb is the relative
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CHAPTER (3)
Mathematical Development
acceleration of the base with respect to the ground. Also, the overall equation of motion of the combined building and base can be written as; M X C X K X M R (ug xb )
(3.17)
where X , X , X are the displacement, velocity and acceleration vectors of the upper stories relative to the base slab, Fig 3.4 shows the idealized mathematical model of the 2-story, base-isolated, building structure considered in the present study. For the system under consideration, the governing equations of motion are obtained by considering the equilibrium of forces at the location of each degree of freedom. For a fixed-base building (without any isolation system), these can be written as (43). n n RT MX mi mb xb cb xb kb xb mi mb ug i 1 i 1
(3.18)
where n is the number of stories of the building; mb, kb and cb are the mass, stiffness and damping of the base, and mi is the mass of the building’s ith floor. By combining Eqs 3.17 and 3.18 the general equation of motion for the combination of the seismically isolated building structure and the base slab can be expressed in matrix form as; M * X * C * X * K * X * M * R * ug
(3.19)
In which M mb mi
1 R* 0
k K b 0
c C* b 0
n
i 1
*
0 K
u X* b X 0 C
- 52 -
M RT M M M MR *
CHAPTER (3)
Mathematical Development
where 0 is a zero matrix. In order to solve Eq. (1), it is rewritten in statespace as Z A1Z B1P
(3.20)
where Z is the state vector, A1 the state matrix, and B1 is the input matrix. These are given as U Z , U
0 A1 1 M K
P Rug (t )
I M 1C
0 B1 1
where I is an identity matrix. The same procedure is used to solve Eq.
3.3.2 Hysteretic models Base Isolation Material: Several hysteresis models have been developed to capture the nonlinear dynamic response of base material subjected to base excitation. These range from relatively simplistic models such as the elasto-plastic and bilinear models.
* Elasto-plastic model: This is a simple model defined by three rules. The backbone curve is defined by an elastic stiffness (k) which represents cracked-section behavior and a post-yield portion with zero stiffness, as shown in Figure 3.8. The unloading stiffness is taken to be the same as the elastic loading stiffness. This model is a very poor representation of the hysteretic
- 53 -
CHAPTER (3)
Mathematical Development
behavior of concrete as it does not represent stiffness deterioration with increasing displacement amplitude reversals.
However, it has been
extensively used because of its simplicity in modeling. Lead-rubber bearings
Figure 3.8: Elasto-plastic hysteresis model.
* Bilinear model: This is very similar to the elasto-plastic model, but it also accounts for the strain hardening effect in steel using non-zero post yield stiffness, as shown in figure 3.9. Stiffness and strength degradation effects cannot be represented. Both the elasto-plastic and bilinear models do not consider hysteretic energy dissipation for small displacements. Many studies evaluating the effects of pounding have used bilinear models to represent the behavior of adjacent structures (44).
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CHAPTER (3)
Mathematical Development
Fig 3.9: Bilinear hysteresis model (45). The effective stiffness of the bearings can be calculated as; K eff
GA h
(3.21)
where A is the area of the elastomeric bearing, G is the shear modulus of the elastomer, taken as 100 psi, and h is the height of the elastomer. The effective stiffness can be related to other parameters, as shown below K eff K 2
Q D
(3.22)
where D is the maximum design deformation in the bearing, typically taken equal to the height of the elastomer. The yield displacement can be expressed in terms of the primary parameters as
Dy
Q K1 K 2
(3.23)
The yield displacement is typically taken to be one-tenth the maximum deformation (D). Thus, all the primary parameters can be calculated from Equations 3.21 and 3.23 given the bearing dimensions. In
- 55 -
CHAPTER (3)
Mathematical Development
this study, elastomeric bearings are modeled at the intermediate hinge and abutment locations.
3.4 Equations of Energy Transmitting: Strong earthquake ground motions cause severe damage in structures, and even their collapse. Nowadays, the seismic peak acceleration is believed to be the most important parameter in the seismic design in order to characterize the severity of the strong motion that damages structures. On the other hand, the evaluation of the structural performance towards seismic actions was established through the displacement ductility, defined as the ratio of the maximum to the yield displacement. This displacement ductility has then been used as a measure to establish inelastic design response spectra in the resistant design methods, ignoring the cumulative damage phenomena that occur from numerous inelastic excursions (46). Instead, it is normally accepted that the seismic structure performance is principally a problem of cumulative damage, related to the energy dissipated by plastic excursions; therefore, the energy should be considered first of all in the seismic design. The importance of energy in seismic design was first introduced by Housner in his research he explained that the seismic ground motion corresponds to the introduction into the structure of some energy. A fraction of this energy is dissipated through the viscous and hysteretic (for inelastic structures) properties of the system; the other fraction of the energy is represented by the kinetic and strain terms (47). Since the work of Housner, attention in seismic design has been focused on the energy-based methods. These are based on the theory that
- 56 -
CHAPTER (3)
Mathematical Development
the energy introduced during an earthquake into a structure (named energy demand) can be predicted and that the energy supply of the structure can be also established. The safety condition for the structure is consequently guaranteed when the energy supply is larger than the energy demand. In the following the approach to evaluating the energy terms for a non-linear system under a seismic motion is explained. First, let us write the power balance for the 1-dof system by multiplying each term of the motion equation 3.24 for the velocity term x ; mxx Cxx kxx k (1 ) zx Mxg x
(3.24)
where; mxx
is the rate of the kinetic energy at time t
Cxx
is the rate of the energy dissipated by damping at time t
kxx
is the rate of the elastic energy at time t
k (1 ) zx is the rate of the hysteretic dissipated energy at time t Mxg x
is the rate of the input energy at time t
The power balance expressed through the relation 3.24 is called the ‘relative power equation’ because it supplies the ‘relative energy equation’ obtained by integrating Equation 7 with respect to the time t;
t
0
mxx d t
t
0
c xx dt
t
0
t
kxx dt 0 k (1 ) zx dt t
0 xg x du
(3.25)
Hence, the ‘relative energy balance’ at time t can be written as; Ek (t) + Ed (t)+ Es (t)+ EH (t)= Ei (t)
- 57 -
(3.26)
CHAPTER (3)
Mathematical Development
where; t
1 M ( X t ) 2 2 C XX dt X T C X MXX d t
Ek
0
Ed
0
Es
t 0 K XX
EH Ei
t
t
0 t
0
1 T X K X 2 K (1 ) ZX dt dt
M X g X dt X T M r
3.5 Modeling of 2-Mdof System with Connected Damper: Figure 3.11 displays a structure-damper system composed of two adjacent 2DOF structures respectively, connected by dampers at some stories.
Figure 3.10: Structural model of two adjacent structures linked by fluid damper.
* Equations of Motion The equations of motion of the coupled structural system subjected to seismic excitation can be expressed as;
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CHAPTER (3)
Mathematical Development
M A . X A (t ) C A . X A (t ) K A . X A (t ) f (t ) M A . I xg (t )
(3.27 a )
M B . X B (t ) C B . X B (t ) K B . X B (t ) f (t ) M B . I xg (t )
(3.27b)
f (t ) k d . xi (t )
(3.27c)
xi (t ) xBi (t ) x Ai (t )
(3.27 d )
where; MA, KA and CA denote the mass, stiffness and damping coefficient of the building A, respectively; XA(t) is the horizontal relative displacement of the building A with respect to the ground; MB; KB and CB denote the mass, stiffness and damping coefficient of the building B, respectively; XB (t) is the horizontal relative displacement of the building B with respect to the ground; and xg (t ) is the horizontal ground acceleration.
3.6 Validation of the Numerical Model; Fixed Base: A pounding of two buildings under El Centro ground excitation has been assumed for all calculations by Chau and Wei. Therefore, in this section pounding induced by the acceleration time history of the 1940 El Centro earthquake with duration of 53.73 sec and with a peak of 0.348 g (48)
. In figure 3.11 the dynamic characteristics of building 1 and 2 are (m1
=204.0 kg, f1 =2.31 Hz, 1 =1.4%) and (m2 =146.5 kg; f2 =2.9 Hz, 2 =1.6%) with a stand-off distance of 11 mm. Figure 3.12 shows that, the numerical results from this paper for the velocity time history of building 1 agree roughly with the numerical result from Chao.
- 59 -
CHAPTER (3)
Mathematical Development
Figure 3.11: Sketches of the theoretical and experimental models for
Velocity (m/sec2)
modeling pounding between two adjacent structures.
Time
velocity (m/s)
0.9 numerical results
0.6 0.3 0 -0.3 -0.6 -0.9
0
10
20
30
40 time (s)
50
60
70
80
Figure 3.12: The velocity time history of building 1, (a) the numerical result find from Chao, (b) the numerical result in this study.
- 60 -
EFFECT OF POUNDING ON ADJACENT STRUCTURES
CHAPTER (4) EFFECT OF POUNDING ON ADJACENT STRUCTURES 4.1 General:
Past earthquakes have revealed detrimental effects of pounding on the seismic performance of conventional fixed-supported buildings, ranging from light local damage to more severe structural failure. However, the potential consequences of earthquake-induced poundings on seismically isolated buildings can be much more substantial, because seismic isolation introduces flexibility at the isolation level of relatively stiff buildings and, thus, should be assessed. Through numerical simulations, the effects of potential pounding incidences on the seismic response of typical seismically isolated buildings will be discussed. Such impact events may occur between adjacent buildings that stand at a very close distance. To perform this study, two adjacent buildings each of them two stories are studied. These buildings are studied in four cases; the first when the two buildings are fixed in base and the separation distance between them are insufficient to prevent pounding and it indicates as FB. The second case when the two buildings are very close, then the pounding is taken into account and it is indicated as FBP. The third case when the two buildings are base isolated and have a big clearance distance between them, the pounding is not taken into account and it is called as BI. The last case when the pounding between the two adjacent base isolated buildings is considered and it is called as BIP.
Chapter (4)
Effect of pounding on adjacent structures
This chapter has been divided into three sections. In the first section, the response of the four cases of the two buildings will be analyzed when subjected to harmonic excitation. In the second section, the several parameters which include the dynamic properties of the adjacent structures such as separation distance, yield force of the isolator, post to pre stiffness ratio and mass ratio are studied to show its effect on the response and the energy transmitted to the superstructure of the buildings under the harmonic excitation with various frequencies. In the last section; studying the effect of the changing of the previous parameters on the buildings when subjected to four famous earthquakes motion will be investigated.
4.2 Analytical Model for Reference Building: Two adjacent buildings shown in figure 4.1 have been used for the analysis. Building A is the reference building and its properties are described in table 4.1 and Building B has a varying mass with respect to the reference building as shown in the table 4.2, then time periods and frequencies of building B have been depicted. The damping of 5% in the first mode is kept same as that for the reference building by reconstructing the damping matrix as its mass is varied. The clearance distance between the adjacent buildings will be taken in all calculations equal to 20 mm. Figure 4.2 shows the Force-Displacement relationship of a highdamping rubber bearing of the base in case of base isolated buildings.
- 62 -
Chapter (4)
Effect of pounding on adjacent structures
second story
building A
gap
building B
first story
base material ground
earthquake foundation
foundation
Fig 4.1: The schematic of the two adjacent buildings A & B.
Fig 4.2: Force-Displacement relationship of a highDamping rubber bearing [54].
- 63 -
Chapter (4)
Effect of pounding on adjacent structures
Table 4.1: The parameter used for the reference building A and the excitation characteristics. Other parameters used in numerical modeling The mass ratio
b= ( mb / mt )
0.6
The frequency ratio
δ = ( ωb / ωt )
0.6
Post- pre yield stiffness ratio α = ( Kbp/Kbe)
0.2
The yield force for the isolator (ton)
(fy)
100
Structural damping ratio
(s)
5%
base damping ratio
(b)
8%
Coefficient of restitution
(e)
0.65
Nonlinear viscoelastic parameter Impact stiffness parameter (N/m)
( )
9.9 *1010
Impact damping ratio
( )
0.35
Harmonic excitation properties Frequency of the excitation The amplitude
(wex)
4.0 rad/sec
(a)
1.0 m/sec2
where: mb is the base mass and mt is the total mass of the superstructure. ωb is the natural frequency of the base. ωt is the fundamental natural frequency of the building. Kbp and Kbe are the plastic and elastic stiffness of the isolator.
Table 4.2: The parameters of the second building B. Properties of the second building B properties
stiff
Flexible
Mass with respect to reference building
half
Twice
First mode time period (sec)
0.7189
1.4377
Second mode time period (sec)
0.2746
0.5492
First mode frequency (rad/sec)
8.7403
4.3702
Second mode frequency (rad/sec)
22.8825
11.4412
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Chapter (4)
Effect of pounding on adjacent structures
The mass and stiffness matrices for the lumped mass model of the building are specified as
100 0 MA t .sec 2 / m 0 100 20000 10000 KA t /m 10000 10000 The mode shapes and natural frequencies of the system obtained using the above mass and stiffness matrices. - 0.0526 - 0.0851
A
6.1803
A 16.1803
- 0.0851 0.0526
rad / sec
1.0166 TA 2 / 0.3883
sec
The damping matrix is obtained by assuming that it is proportional to the stiffness matrix using the orthogonally and decoupling of mass and stiffness: 134.1641 - 44.7214 CA - 44.7214 89.4427
4.3 Behaviour of Buildings under Sinusoidal Ground Excitation: In this section the studying of the change of building behaviour for the four cases previously explained will be investigated under harmonic ground motion. All parameters will be taken in all calculations as shown later for reference building A; also building B will take the stiff parameters.
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Chapter (4)
Effect of pounding on adjacent structures
4.3.1 Two Fixed Base Adjacent Buildings: Two adjacent fixed base buildings will be studied in the process of studying building pounding phenomenon. Responses will be calculated for the reference building A and the neighbor building B which lies at 20mm from the reference building. The buildings were modeled as 2DOF systems and equations derived assuming pounding can take place at any floor levels. Figure 4.3 shows the displacement time history of the first floor of the two buildings in case of FB when subjected to sinusoidal excitation with frequency equals 4.0 rad. It is shown that each building has a peak displacement different each other, due to the change in natural frequency of each building. The maximum displacement for building A occurs at time 1.7 sec and with value equals 4.8 cm and for building B the maximum displacement equals 1.75 cm. it is shown from figure that the floors are interacted at same times due to neglecting the pounding. When pounding effect taken into consideration as shown in figure 4.4, the maximum displacement changes for the same floor because of the change in the physical manner of the buildings under pounding effect. It is clear that the maximum displacement in the reference building A decreases to 3.4 cm with decreasing ratio 31% and the peak displacement of building B increases to 3.2 cm with increasing ratio 82%. It is noticed that the interacted areas are relevant. As shown in figure 4.4(a), a part of the energy acquired by the flexible building A from the harmonic excitation loses at collision and the stiff building B takes it which increase the velocity of the building B. Figure 4.4(b) shows the impact force at impact
- 66 -
Chapter (4)
Effect of pounding on adjacent structures
points in case of FBP at the first story, the impact force equals 1120 ton at this point. 0.08
building A building B
gap=2cm
0.06 displacement(m)
0.04 0.02 0 -0.02 -0.04 -0.06 -0.08
0
2
4
6
8
10 time(sec)
12
14
16
18
20
Fig 4.3: Disp. time history of the first floor of two buildings in case of FB.
displacement(m)
(a)
0.08
building A
gap=2cm
0.06
building B
0.04 0.02 0 -0.02 -0.04 -0.06 -0.08
(b)
0
2
4
6
8
10 time(sec)
0
2
4
6
8
10 time(sec)
12
14
16
18
20
1200
impact force (ton)
1000 800 600 400 200 0
12
14
16
18
20
Fig 4.4: Impact force in case of FBP (a) Disp time history of the first floor of two buildings, (b) Impact force .
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Chapter (4)
Effect of pounding on adjacent structures
Figures 4.5 and 4.6(a) show the displacement time history of the second floor of the same cases. These figures show that the pounding decreases the peak displacement in flexible building A from 6.4cm to 5.8cm with decreasing ratio 10% and increases the peak displacement in stiff building B from 2.2 cm to 4.1 cm with increasing ratio 86.3%. Figures 4.6(b) and 4.6(c) show the impact force at the second floor of the two buildings and the base shear, respectively. From these figures, it is shown that the base shear decreases with the decrease in the impacts. From these figures, it is shown that the second stories in the two adjacent buildings make many times impact forces is higher than that of the first stories, as a result of the higher displacement and velocity of the upper stories. 0.08
building A building B
displacement(m)
0.06 0.04 0.02 0 -0.02 -0.04 -0.06 gap=2cm 0 2
4
6
8
10 12 time(sec)
14
16
Fig 4.5: Disp. Time history of second floor of The two buildings in case of FB.
- 68 -
18
20
Chapter (4)
(a)
Effect of pounding on adjacent structures
0.08 building A
displacement(m)
0.06
building B
0.04 0.02 0 -0.02 -0.04 -0.06 0
gap=2cm 2
4
6
8
(b)
10 time(sec)
12
14
16
18
20
impact force (ton)
5000 4000 3000 2000 1000 0
(c)
0
2
4
6
8
10 time(sec)
12
14
16
20
1000 building A
Base shear (t)
18
building B
500 0 -500 -1000
0
2
4
6
8
10 12 time(sec)
14
16
18
20
Fig 4.6: Results in case of FBP (a) disp. time history of the second floor of two buildings, (b) impact force, (c) base shear for the two buildings.
4.3.2 Base Isolated Adjacent Buildings: The numerical simulations of base isolated buildings are used to clarify the behaviour of the isolated buildings under pounding effect. It is assumed that pounding can take place at any of the floor levels and also at the base. Figures 4.7 and 4.8(a) show the displacement time history of the base of the two buildings for the cases base isolated without and with
- 69 -
Chapter (4)
Effect of pounding on adjacent structures
considering pounding, respectively. In these figures, it can be shown that the peak total sliding displacement of the flexible building A decreases from 16 cm to 15 cm by decreasing ratio equals 7% and increases in stiff building B from 12 cm to 15 cm with increasing ratio equals 25%. This sliding displacement increases the total displacement which required clearance distance more than in fixed base case.
displacement(m)
0.2 building A building B
0.1
0
-0.1
-0.2
0
2
4
6
8
10 time(sec)
12
14
16
18
20
Fig 4.7: The sliding displacement of the base of The two buildings in case of BI. (a)
0.2 0.15
displacement(m)
0.1 0.05 0 -0.05 -0.1 building A building B
-0.15 -0.2
0
2
4
6
8
10 time(sec)
12
14
16
8
10 12 time(sec)
14
16
(b)
18
20
impact force (ton)
8000 6000 4000 2000 0
0
2
4
6
18
Fig 4.8: Time history of base of the two buildings in case of BIP, (a) the sliding displacement, (b) impact force. - 70 -
20
Chapter (4)
Effect of pounding on adjacent structures
200
(b)
(a)
base force(ton)
base force(ton)
200
0
-200
(c) 0
-200
0 -100 -200 -0.2 -0.1 0 0.1 0.2 displacement at the base(m) (B) case of (BIP) 400
base force(ton)
base force(ton)
-400 -0.2 -0.1 0 0.1 0.2 displacement at the base(m) (A) case of (BIP) 200
100
-400 -0.2 -0.1 0 0.1 0.2 displacement at the base(m)
(d) 200
0
-200 -0.2 -0.1 0 0.1 0.2 displacement at the base(m)
Fig 4.9: Force-Displacement relationship for sliding base for building A and B, (a), (b) for case of BI, (c) and (d) for case of BIP. The same indications are shown in figures 4.10 and 4.11(a). These figures show the displacement time history of the second floor of the two buildings in case of BI and BIP, respectively. It is shown that the pounding effect changes in the peak displacement at time and values; this clearly shown in these figures where the peak total displacement at the second floor of building A changes from 17.0 cm to 18.5 with increasing ratio equals 9.4%, also in building B changes from 18.0 cm to 20 cm with increasing ratio equals 11%. Tables 4.3 and 4.4 show that the pounding effect increases the relative displacement of the stiff building B with 43% and 28% for the first and second stories respectively and increases the displacement of building A with 24% and 21 % for the two stories. From these results, it is shown that the increase in the displacement in case of base isolated buildings is more than that of the fixed base buildings. Therefore it is important to study the phenomena of pounding in order to
- 71 -
Chapter (4)
Effect of pounding on adjacent structures
evaluate existing building configurations and to provide guidance in this type of buildings. This because the base isolation increases the flexibility of the buildings with sliding displacement which make the total displacement increases.
displacement(m)
0.2 0.1 0 -0.1 -0.2
building A building B 0
2
4
6
8
10 time(sec)
12
14
16
18
20
(a)
0.2
displacement(m)
Fig 4.10: The total disp. time history of the second floor Of two buildings in case of BI.
0.1
0
-0.1 building A -0.2
(b)
0
2
4
6
8
10 12 time(sec)
14
building B 16
20
1000 building A
Base shear (t)
18
building B
500 0 -500 -1000
0
2
4
6
8
10 12 time(sec)
14
16
18
Fig 4.11: Results for case of BIP for the two buildings (a) the disp. time history of the second story, and (b) base shear.
- 72 -
20
Chapter (4)
Effect of pounding on adjacent structures
Table 4.3: Maximum Relative Displacements of Building A. Floor Max relative disp. of first floor (m) Max relative disp of second floor(m)
Fixed Base
Base Isolated
PRFB*
PRBI**
0.046 0.036 0.0145 0.018
0.77
1.24
0.072 0.057 0.0244 0.029
0.79
1.21
FB
FBP
BI
BIP
Where: PRFB* is the pounding ratio for the fixed base buildings. PRBI** is the pounding ratio for the base isolated buildings. Table 4.4: Maximum Relative Displacements of Building B. Floor Max relative disp. of First Floor(m) Max relative disp. of Second Floor
Fixed Base FB
FBP
Base Isolated BI
BIP
PRFB PRBI
0.0156 0.0324 0.0148 0.0211
2.08
1.43
0.0240 0.0422 0.0231 0.0296
1.76
1.28
4.3.3 Fixed Base versus Base Isolated Structures: Base isolation provides more flexibility for the overall structure which permits more displacement at the sliding base level. This sliding displacement dissipates the energy which reduces the transmitted energy to the superstructure. Figure 4.12 shows the relative displacement of the second floor of reference building A for the two cases FB and BI. It is shown that the
- 73 -
Chapter (4)
Effect of pounding on adjacent structures
reduction ratio in the maximum response between the two cases is equal to 198% where the displacement decreases from 7.29 cm to 2.44 cm. The displacement reduction ratio for the second floor of building B equals 4% as shown in figure 4.13 where decreases from 2.4 cm to 2.3 cm. The changes in the displacement occur when the pounding is taken into consideration. Figure 4.14 shows the displacement time history of the top floor of the building A for the two cases FBP & BIP. From this figure, the base isolation increases the displacement of the second floor from 5.7 cm to 18.5 cm with increasing ratio 224%. This increasing in the displacement occurs due to the sliding displacement. Also, figure 4.15 shows the upper floor displacement of the building B, the maximum displacement increases in this floor due to the isolation from 4.22 to 16.5 cm with increasing ratio 290%. Also by observing the results of the energy, the total energy transmitted to the building A in case of BIP equals 158,600 t. m and equals 116,500 t. m in case of FBP. These results show that the isolation makes the energy increased by 36%. In building B, the energy increased from 6.75 e+ 4t.m to 10.95 e+4 t.m when the isolation system used. Table 4.5 and 4.6 show the three parts of energy transmitted to the superstructure part of the four models used in the study for building A and B respectively. It is shown that the base isolation decrease the relative displacement which decrease the strain and damping energy, but in case of kinetic energy it is seen that the total displacement may increase in case of base isolated structures due to the increase in the sliding displacement.
- 74 -
Chapter (4)
Effect of pounding on adjacent structures
FB BI
displacement(m)
0.05
0
-0.05
0
2
4
6
8
10 12 time(sec)
14
16
18
20
Fig 4.12: The disp. time history of the second floor of Building A in case of FB & BI. FB BI
displacement(m)
0.05
0
-0.05
0
2
4
6
8
10 12 time(sec)
14
16
18
20
Fig 4.13: The disp. time history of the second floor of Building B in case of FB & BI.
displacement(m)
0.2 FBP BIP
0.1 0 -0.1 -0.2
0
2
4
6
8
10 12 time(sec)
14
16
18
Fig 4.14: The disp. time history of the second floor of building A in case of FBP & BIP.
- 75 -
20
Chapter (4)
Effect of pounding on adjacent structures
displacement(m)
0.2 0.1 0 -0.1 -0.2
FBP BIP 0
2
4
6
8
10 12 time(sec)
14
16
18
Fig 4.15: The disp. time history of the second floor of building B in case of FBP & BIP. Table 4.5: The Parts of Energy Transmitted to Building A. Energy t. m SE* DE** KE***
Fixed base FB FBP
Base isolated BI BIP
7.46 e+04 1.55 e+04
7.16 e+04 1.46 e+04
1.12 e+04 2.88 e+03
1.26 e+04 3.025 e+03
3.27 e+04
3.03 e+04
1.35 e+05
1.43 e+05
PRFBE
PRBIE
0.95
1.13
0.94
1.05
0.92
1.06
* Strain energy ** Damping energy *** Kinetic energy Table 4.6: The Parts of Energy Transmitted to Building B. Energy t. m SE DE KE
Fixed base FB 1.0 e+04 1.48 e+04 2.2 e+04
FBP 1.384 e+04 2.39 e+04 2.98 e+04
Base isolated BI 1.10 e+04 3.03 e+03 9.74 e+04
- 76 -
BIP 1.29 e+04 3.636 e+03 9.30 e+04
PRFB
PRBI
1.348
1.17
1.62
1.2
1.35
0.95
20
Chapter (4)
Effect of pounding on adjacent structures
4.4 Parameters Affecting the Building Pounding Response: In order to mitigate pounding damage in buildings, it is important to determine the factors affecting the pounding response. A comprehensive study is performed in this section to better understand the parameters affecting pounding, and to investigate the effects of building behaviour . The parameters will be studied in this work are the relative mass ratio of impacting structures, the separation distance between the to colloids buildings, the mass ratio between the sub and super structures, the damping ratio of the colliding buildings and the effect of the isolator base material such as post-pre stiffness ratio and plastic shear capacity.
4.4.1 Effect of Mass Ratio (): The effect of the mass ratio on the building behaviour under pounding phenomenon will be studied in this section. The mass ratio is the ratio between the mass of building B and the mass of building A. The mass of the colliding structures is an especially important structural parameter, which has a direct influence on structural response and on pounding force during impact. The numerical analysis has been conducted for two cases of , first the stiff case equals 0.5 and the other is more flexible with equals 2.0, as shown in the table 4.2. In both cases, the modes are assigned 5% critical damping and the buildings gap of 20mm and when subjected to harmonic excitation with different frequencies. Figure 4.16 shows the maximum displacement of the building A and B for various values of λ. It is shown that the maximum displacement
- 77 -
Chapter (4)
Effect of pounding on adjacent structures
of the first and second floor of building A with fixed base reduces from 23cm and 40cm to 20.5cm and 37 cm respectively, and for building B the response increases from 12cm and 20cm to 38cm and 62cm for first and second floor respectively with the increase in the mass ratio, because the increase in the mass ratio make the flexibility of building B increased which pound the stiff building A (the stiffer in this case) and reduce its response.
It is shown also that the peaks of the maximum floor
displacement for different mass ratio occur at different excitation frequencies, because the pounding force shift the peak of the interstory displacement. When the maximum responses of the superstructure floors are compared with that of the corresponding in the fixed base structure un pounding case, it is noticed that the peak response increase with the pounding force taken into consideration. Figure 4.17 presents the maximum relative floor displacement for the two base isolated buildings when pounding into consideration for the two buildings for the same values of mass ratios. It shows that the peak displacement of building A increases from 6cm and 10cm to 16cm and 23cm for the first and second storey respectively, and increases from 12cm and 15cm to 15cm and 21cm for building B because the base isolation increase in the flexibility of the superstructure part and also, the increase in the mass of building B make the pounding effect increases on the two buildings. The energy dissipation through the base decreases, this make the transmitted energy to the upper stories increase as shown in figure 4.18. This figure shows that the total energy increases with the increase of the flexibility of the buildings, because the increase in the mass ratio increase the time periods which increase the displacement and velocity which directly affect on the total energy.
- 78 -
Chapter (4)
Effect of pounding on adjacent structures
=.5 =2.0
Max disp(m)
0.6 (a)
unpound(FB) 0.4
0.2
0
0
1
2
3
1
2
3
4
5
6
7
8
9
10
4
5
6
7
8
9
10
Max disp(m)
0.6 0.5
(b)
0.4 0.3 0.2 0.1 0
0
ex (rad)
=.5 =2.0
Max disp(m)
0.6 (c)
unpound =.5
0.4
unpound =2
0.2
0
0
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5 ex (rad)
6
7
8
9
10
Max disp(m)
0.6 (d) 0.4
0.2
0
0
Fig 4.16: Max. disp. for the case of FBP (a)&( b) first and second floor of building A, (c)&(d) first and second floor of building B.
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Chapter (4)
Effect of pounding on adjacent structures
0.25 Max rel disp(m)
(a)
=.5 =2.0
0.2
unpounding(BI)
0.15 0.1 0.05 0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0.5
1
1.5
2
2.5 ex (rad)
3
3.5
4
4.5
5
0.25 Max rel disp(m)
(b) 0.2 0.15 0.1 0.05 0
0
0.25
=.5 =2.0
Max rel disp(m)
(c) 0.2
unpound =.5
0.15
unpound =2
0.1 0.05 0
0 0.25
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0.5
1
1.5
2
2.5 ex (rad)
3
3.5
4
4.5
5
Max rel disp(m)
(d) 0.2 0.15 0.1 0.05 0
0
Fig 4.17: Maximum disp. in case of BIP for different mass ratios (a),( b) first and second floor of A, (c),(d) first and second floor of B.
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Chapter (4)
Effect of pounding on adjacent structures
=.5
=2.0
unpound =.5
6
S.E(t.m)
10
10 (a)
(d)
4
4
10
10
2
10
5
2
0
D.E(t.m)
5
10
10
0
2
4
6
2
4
6
2
4
6
(e)
(a) 0
0
10
10
-10
-5
10
10 0
5
0
10
10
10
10
10 K.E(t.m)
10 10
10
(a)
(f) 5
0
10
-10
10
10
10
unpound =2
6
0
0
5 ex (rad)
10
0
ex (rad)
Fig 4.18: Total energy (a), (b) & (c) show the SE, DE and KE in case of FBP, (d), (e) & (f) show the show the SE, DE and KE in case of BIP.
4.4.2 Effect of Isolator Plastic Shear Capacity: Structures with different isolator plastic shear capacity (Qp) equals to 100, 150, 200 and 250 ton are analyzed when subjected to harmonic excitation with different frequencies. Figure 4.19 shows the maximum sliding displacement of the base of building A for various values of Qp. It is shown that the maximum displacement of the base is increased when the isolator plastic shear capacity is decreased, because the lower isolator plastic shear capacity means that the isolator enters the plastic zone quickly and still long time in plastic zone which has small stiffness. It is shown also that the peaks
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Chapter (4)
Effect of pounding on adjacent structures
of the maximum base displacement for different isolator plastic shear capacity occur at different excitation frequencies, because the nonlinear behaviour of the isolator leads to different participation between the elastic and plastic stages. For higher values of isolator plastic shear capacity, the isolator remains elastic most of the time, i.e. the isolator is stiffer which lead to higher resonant frequency. While for smaller values of isolator plastic shear capacity, the isolator remains plastic most of the time, i.e. the isolator is more flexible which lead to smaller resonant frequency. 1.5
Max disp(m)
Qp=250 ton Qp=200 ton
1
Qp=150 ton Qp=100 ton
0.5
0
0
1
2
3
4
5
6
7
8
9
10
ex (rad)
Fig 4.19: Maximum sliding displacement in the base of A in case of BI for different isolator plastic shear capacities. Figures 4.20 and 4.21 show the maximum relative displacement of the first and second floors of the building A respectively for different isolator plastic shear capacity.
It is noted that the two figures can be
divided into two parts. The first part is over the excitation frequency equals two, where the decrease in the isolator plastic shear capacity decreases the maximum floor displacement due to the higher energy dissipated in the isolator at this lower plastic shear capacity. The second part is under the excitation frequency equals two, where the decrease in the isolator plastic shear capacity increases the maximum floor
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Chapter (4)
Effect of pounding on adjacent structures
displacement due to increased resonant energy transmitted to the superstructure and the very small isolator plastic shear capacity decrease the dissipated energy where it behave as flexible linear system. This last conclusion is shown clearly in figure 4.22 which shows the hysteric force-displacement relationship, where the very high value of isolator plastic shear capacity (Qp = 500) remains the isolator linear which means no dissipated energy. For very small isolator plastic shear capacity (Qp = 50), the isolator behaves plastic in the most of time which means flexible linear system rather than dissipating energy. But for moderate value of isolator plastic shear capacity (Qp = 250), the isolator participates between the elastic and plastic zones which dissipates more energy. But apart the resonance case, the decrease in the isolator plastic shear capacity, increases the isolator participation between the elastic and plastic zones as shown in figure 4.23. When the maximum responses of the superstructure floors are compared with that of the corresponding in the fixed base structure, it is noticed that the peak response of the isolated buildings in its resonance case still smaller than that in the fixed base building in its resonance case. 0.2 FB Qp=250 ton
Max disp (m)
0.15
Qp=200 ton Qp=150 ton
0.1
Qp=100 ton
0.05
0
0
1
2
3
4
5
6
7
8
9
ex (rad)
Fig 4.20: Maximum relative displacement of the first floor of A in case of BI under different isolator plastic shear capacities.
- 83 -
10
Chapter (4)
Effect of pounding on adjacent structures
0.2 FB Qp=250 ton Qp=200 ton Qp=150 ton 0.1
Qp=100 ton
0.05
0
0
1
2
3
4
5
6
7
8
9
ex (rad)
Fig 4.21: Maximum relative displacement of the second floor of A in case of BI under different isolator plastic shear capacities.
1000 Qp=250
Qp=50
fo rc e (t)
fo rc e (t)
2000
0
-2000 -2
-1 0 1 base displacement (m)
2
0
-1000 -0.5
0 0.5 base displacement (m)
500 fo rc e (t)
Max disp (m)
0.15
0 -500 -1000 -0.2
Qp=500 -0.1 0 0.1 0.2 base displacement (m)
Fig 4.22: Isolator Force- Displacement relationship for building A at wex=1.6 rad.
- 84 -
10
Chapter (4)
Effect of pounding on adjacent structures
500
Qp=50 fo rc e (t)
fo rc e (t)
200
0
-200 -0.2
-0.1 0 0.1 base displacement (m)
0.2
0
-500 -0.2
Qp=250 -0.1 0 0.1 0.2 base displacement (m)
fo rc e (t)
1000
0
-1000 -0.2
Qp=500 -0.1 0 0.1 base displacement (m)
0.2
Fig 4.23 Isolator Force- Displacement relationship for building A at wex=4.0 rad.
When the building B that is stiffer than building A is analyzed separately when its base is isolated, the same results are obtained but with shifting in the excitation frequency that separates the resonant zone and the other zone. Figure 4.24 shows the maximum displacement of the base of building B when subjected to harmonic excitation with different frequencies. It is shown that the base maximum displacement of building B is smaller than that of building A (Figure 4.19), because the base frequency ratios of the two buildings are the same, i.e. the isolator of buildings B - that is stiffer than building A – is stiffer than the isolator of building A, then a smaller drift is occurred. Figures 4.25 and 4.26 show the maximum displacement of the first and second floors of the isolated building B for different isolator plastic shear capacity when the building subjected to harmonic excitation with different frequencies. It is found that the effect of the isolator plastic - 85 -
Chapter (4)
Effect of pounding on adjacent structures
shear capacity on the maximum displacement is changed at excitation frequency three as that occurs in building A at excitation frequency two. This shifting in excitation frequency is due to the extra stiffness of the building B. It is shown that reduction of the peaks of the maximum displacement when compared with that of the fixed base building is higher that reduction in building A, because the base isolation is more useful in stiff buildings than flexible buildings.
0.4 Qp=250 t Qp=200 t
Max disp(m)
0.3
Qp=150 t Qp=100 t
0.2
0.1
0
0
1
2
3
4
5
6
7
8
9
10
ex (rad)
Fig 4.24: Maximum sliding displacement in the base of B in case of BI under different isolator plastic shear capacities. 0.1
FB Qp=250 t
Max disp (m)
0.08
Qp=200 t
0.06
Qp=150 t Qp=100 t
0.04 0.02 0
0
1
2
3
4
ex (rad)
5
6
7
8
9
Fig 4.25: Maximum relative displacement of the first floor of B in case of BI under different isolator plastic shear capacities.
- 86 -
10
Chapter (4)
Effect of pounding on adjacent structures
0.2 FB Qp=250 t
Max disp (m)
0.15
Qp=200 t Qp=150 t
0.1
Qp=100 t 0.05
0
0
1
2
3
4
ex (rad)
5
6
7
8
9
10
Fig 4.26: Maximum relative displacement of the second floor of B in case of BI under different isolator plastic shear capacities. When the pounding between the two buildings is taken into consideration, the maximum displacements are increased because the impact between the two buildings creates excessive energy that increases the buildings responses.
Figure 4.27 shows the maximum base
displacement of buildings A when pounding is taken into consideration. The maximum base displacement has a peak value 152 cm against 122 cm when the pounding is not taken into consideration. Figures 4.28 and 4.29 show the maximum displacement of the first and second floors of the pounded base isolated building A. These figures show also the maximum displacement of the fixed base building responses when pounding is and is not taken into consideration. It is found that the maximum displacement of the first and second bounded floors have peak values 11 and 16 cm respectively against 7 and 11 cm for the unbounded floors. The two figures show that this increase in peak response is also for the fixed base building when pounding is taken into consideration where the peak maximum displacements for the first and second bounded floors are 23 and 39 cm respectively against 18 and 29
- 87 -
Chapter (4)
Effect of pounding on adjacent structures
cm for the unbounded floors. Although, the pounding increases the maximum displacement of the base isolated building, this increase does not overcome the reduction of response due to base isolation where the maximum response is still less than that of the fixed base unbounded building.
2 Qp=250 t Qp=200 t
Max disp(m)
1.5
Qp=150 t 1
Qp=100 t
0.5
0
0
1
2
3
4
5 ex (rad)
6
7
8
9
10
Fig 4.27: The maximum displacement of the base of A in case of BIP under different isolator plastic shear capacities. 0.25
FB FBP
Max rel disp (m)
0.2
Qp=250 t
0.15
Qp=200 t
0.1
Qp=100 t
Qp=150 t
0.05 0
0
1
2
3
4
5
6
7
8
9
10
ex (rad)
Fig 4.28: The maximum rel. disp. of first floor of A in case of BIP under different isolator plastic shear capacities.
- 88 -
Chapter (4)
Effect of pounding on adjacent structures
Max rel disp (m)
0.4
FB FBP Qp=250 t
0.3
Qp=200 t Qp=150 t
0.2
Qp=100 t
0.1
0
0
1
2
3
4
5 ex (rad)
6
7
8
9
10
Fig 4.29: The maximum rel. disp. of the second floor of A under in case of BIP under different isolator plastic shear capacities. When bounded building B is compared with that of the unbounded building, the same conclusion is found. Figure 4.30 shows the maximum displacement of the base of building B.
It has a peak maximum
displacement 100 cm against 36 cm for the unbounded one.
This
increase is due to the impact energy. Figures 4.31 and 4.32 show the maximum relative displacements of the first and second floor of the bounded building B. It is founded that the maximum displacements have a peak values 12 and 17 cm for the first and second floors respectively against 9 and 15 cm for the unbounded floors. It is shown that the peak of the maximum relative displacement of the two floors exceeds in some times the peak of the fixed unbounded building. This means that the increase in response due to impact energy overcomes the reduction in response due to base isolation, because the floors of building B is lighter than that of building A, then affected seriously than building A. It is shown that the floors maximum displacements of the bounded fixed base building B have many peaks due to the impact.
- 89 -
Chapter (4)
Effect of pounding on adjacent structures
1 Qp=250 t
Max disp(m)
0.8
Qp=200 t
0.6
Qp=150 t
0.4
Qp=100 t
0.2 0
0
1
2
3
4
5 ex (rad)
6
7
8
9
10
Fig 4.30: Maximum displacement of the base of B in case of BIP for different values of isolator plastic shear capacity 0.2
FB
Max rel disp (m)
FBP Qp=250 t
0.15
Qp=200 t Qp=150 t
0.1
Qp=100 t
0.05 0
0
1
2
3
4
5
6
7
8
9
10
ex (rad)
Fig 4.31: Maximum relative displacement of the first floor of B in case of BIP for different values of isolator plastic shear capacity 0.25
FB FBP
Max rel disp (m)
0.2
Q =250 t p
Q =200 t p
0.15
Q =150 t p
Q =100 t p
0.1 0.05 0
0
1
2
3
4
5
ex (rad)
6
7
8
9
10
Fig 4.32: Maximum relative displacement of the second floor of B in case of BIP for different values of isolator plastic shear capacity
- 90 -
Chapter (4)
Effect of pounding on adjacent structures
Figure 4.33 shows the transmitted strain, damping and kinetic energies to the base isolated adjacent buildings with different plastic shear capacity when subjected to sinusoidal ground motion with different frequencies. It is shown that at the first value of wex, the maximum transmitted energy reduces with the increase of the plastic shear because the sliding displacement of the base of the two building decreases that makes the total displacement decreases which increases the kinetic energy, and relative displacement and velocity of the building decrease also, and that make the damping and strain energy decrease. And it is shown in the stage of the increase of plastic shear the energy dissipation decreases which previously discussed which transmitted energy to the upper stories. (a)
5
S.E (T.m)
10
Qp=250 t
Qp=200 t
Qp=150 t
Qp=100 t
4
10
3
10
0
1
2
3
4
5
6
4
5
6
4
5
6
(b)
4
D.E (T.m)
10
2
10
0
10
0
1
2
0
1
2
6
3 (c)
K.E (T.m)
10
4
10
2
10
3 wex (rad) ω ex (rad)
Fig 4.33: Total energy in case of BI against ωex (a) strain energy, (b) damping energy, (c) kinetic energy. - 91 -
Chapter (4)
Effect of pounding on adjacent structures
Figure 4.34 shows the total strain, damping and kinetic energies of the two adjacent base isolated buildings effect with different plastic shear capacity when subjected to sinusoidal ground motion with different frequencies. This show that the maximum energy increases when pounding increases with respect to the case when pounding not occur this because the increase in the relative interstory drift. (a)
5
S.E (t.m)
10
4
10
3
10
0
1
2
3
4
5
6
4
5
6
(b)
5
D.E (t.m)
10
0
10
0
1
2
3 (c)
6
K.E (t.m)
10
4
10
Qp=250 t
Qp=200 t
Qp=150 t
Qp=100 t
2
10
0
1
2
3 wex (rad) ω (rad)
4
5
6
ex
Fig 4.34: Total energy for the two buildings in case of BIP (a) strain energy, (b) damping energy, (c) kinetic energy.
- 92 -
Chapter (4)
Effect of pounding on adjacent structures
4.4.3 Effect of Post-Pre Stiffness Ratio: Structures with different isolator post-pre stiffness ratio (α) equals to 0.1, 0.2, 0.3, 0.4 and 0.5 are analyzed when subjected to harmonic excitation with different frequencies. Figure 4.35 shows the maximum sliding displacement of the base of building A for various values of α. It is shown that the maximum displacement of the base is increased when the post stiffness ratio is decreased, because the lower post stiffness ratio means that the isolator plastic stiffness will be increased which lead to decrease the peak displacements. It is shown also that the peaks of the maximum base displacement for different post-pre stiffness ratio occur at different excitation frequencies, because the nonlinear behaviour of the isolator leads to different participation between the elastic and plastic stages. For higher values of α, the isolator plastic stiffness slope will be increased that reduces the hysteric force-displacement relationship, i.e. the isolator is stiffer which lead to higher resonant frequency. While for smaller values of α, the isolator plastic stiffness slope will be decreased that increases the hysteric force-displacement relationship, i.e. the isolator is more flexible which lead to lower resonant frequency.
- 93 -
Chapter (4)
Effect of pounding on adjacent structures
Max disp(m)
1.5
= 0.1 = 0.2 = 0.3 = 0.4 = 0.5
1
0.5
0
0
1
2
3
4
5 ex (rad)
6
7
8
9
10
Fig 4.35 Maximum sliding displacement in the base of A in case of BI under various post- pre stiffness ratio. Figures 4.36 and 4.37 show the maximum relative displacement of the first and second floors of the building A respectively for different post-pre stiffness ratio. The increase in the post stiffness ratio increases the maximum floor displacement due to the lower energy dissipated in the isolator at this higher post stiffness ratio. This last conclusion is shown clearly in figure 4.38 which shows the hysteric force-displacement relationship, where the very high value of post stiffness ratio (α = 0.5) increases the slope of the plastic stiffness of the isolator which means decreases in the dissipated energy. For very small isolator post stiffness ratio (α = 0.1), decreases the slope of the plastic stiffness of the isolator which means flexible nonlinear system which means increases in the dissipated energy which lead to decrease in the transmitted energy to the super structure part and decrease the interstory floor displacement. When the maximum responses of the superstructure floors are compared with that of the corresponding in the fixed base structure, it is noticed that the peak response of the isolated buildings in its resonance case still smaller than that in the fixed base building in its resonance case.
- 94 -
Chapter (4)
Effect of pounding on adjacent structures
M ax rel dis p(m )
0.2
FB
= 0.1 = 0.2 = 0.3 = 0.4 = 0.5
0.15 0.1 0.05 0
0
1
2
3
4
5 ex (rad)
6
7
8
9
10
Fig 4.36: Maximum rel. disp. of the first floor of A in case of BI under various post to pre stiffness. FB
= 0.1 = 0.2 = 0.3 = 0.4 = 0.5
0.3 0.2 0.1 0
0
1
2
3
4
5 ex (rad)
6
7
8
9
Fig 4.37: Maximum rel. disp. of the second floor of A in case of BI under various post to pre stiffness. 500
=0.1 =0.5
400 300 200 force (ton)
Max rel disp(m)
0.4
100 0 -100 -200 -300 -400 -0.2
-0.15
-0.1
-0.05 0 0.05 displacement (m)
0.1
0.15
Fig 4.38: Energy dissipation through the base of A under two different post to pre stiffness ratio
- 95 -
0.2
10
Chapter (4)
Effect of pounding on adjacent structures
When the building B that is stiffer than building A is analyzed separately when its base is isolated, the same results are obtained but with shifting in the excitation frequency that separates the resonant zone and the other zone. Figure 4.39 shows the maximum displacement of the base of building B when subjected to harmonic excitation with different frequencies. It is shown that the base maximum displacement of building B is smaller than that of building A (Figure 4.35), because the base frequency ratios of the two buildings are the same, i.e. the isolator of buildings B - that is stiffer than building A – is stiffer than the isolator of building A, then a smaller drift is occurred. Figures 4.40 and 4.41 show the maximum displacement of the first and second floors of the isolated building B for different post-pre stiffness ratios when the building subjected to harmonic excitation with different frequencies. It is found that the effect of the α on the maximum displacement is changed with respect the same floors in the flexible building A. This shifting in excitation frequency is due to the extra stiffness of the building B. It is shown that reduction of the peaks of the maximum displacement when compared with that of the fixed base building is higher that that reduction in building A, because the base isolation is more useful in stiff buildings than flexible buildings.
Max disp(m)
0.4
= 0.1 = 0.2 = 0.3 = 0.4 = 0.5
0.3 0.2 0.1 0
0
1
2
3
4
5 ex (rad)
6
7
8
Fig 4.39: Maximum sliding disp. in the base of B in case of BI under various post to pre stiffness. - 96 -
9
10
Chapter (4)
Effect of pounding on adjacent structures
Max rel disp(m)
= 0.1
FB
0.1
= 0.2
= 0.3
= 0.4
= 0.5
0.08 0.06 0.04 0.02 0
0
1
2
3
4
5
6
ex (rad)
7
8
9
10
Fig 4.40: Maximum rel. disp. of the first floor of B in case of BI under various post to pre stiffness.
M ax rel dis p(m )
= 0.1
FB
0.2
= 0.2
= 0.3
= 0.4
= 0.5
0.15 0.1 0.05 0
0
1
2
3
4
5
ex (rad)
6
7
8
9
10
Fig 4.41: Maximum rel. disp. of the second floor of B in case of BI under various post to pre stiffness. When the pounding between the two buildings is taken into consideration, the maximum displacements are increased in the stiff building B because the impact between the two buildings creates excessive energy in the opposite direction that increases the buildings responses.
Figure 4.42 shows the maximum base displacement of
buildings A when pounding is taken into consideration. The maximum base displacement has a peak value 175 cm against 140 cm when the pounding is not taken into consideration.
- 97 -
Chapter (4)
Effect of pounding on adjacent structures
Figures 4.43 and 4.44 show the maximum displacement of the first and second floors of the pounded base isolated building A. These figures show also the maximum displacement of the fixed base building responses when pounding is and is not taken into consideration. It is found that the maximum displacement of the first and second bounded floors have peak values 12 and 19 cm respectively against 13 and 20 cm for the unbounded floors. The two figures show that this increase in peak response is also for the fixed base building when pounding is taken into consideration where the peak maximum displacements for the first and second bounded floors are 23 and 39 cm respectively against 18 and 29 cm for the unbounded floors.
Although, the pounding increase the
maximum displacement of the base isolated building, this increase does not overcome the reduction of response due to base isolation where the maximum response is still less than that of the fixed base unbounded building. 2
= 0.1 = 0.2 = 0.3 = 0.4 = 0.5
Max disp(m)
1.5
1
0.5
0
0
1
2
3
4
5
6
7
8
ex (rad)
Fig 4.42: Maximum sliding disp. in the base of A in case of BIP under various post to pre stiffness.
- 98 -
9
10
Chapter (4)
Effect of pounding on adjacent structures
0.25
FB FBP
Max rel disp(m)
0.2
= 0.1 = 0.2 = 0.3 = 0.4 = 0.5
0.15 0.1 0.05 0
0
1
2
3
4
5
6
7
8
9
10
ex (rad)
Fig 4.43: Maximum rel. disp. of the first floor of A in case of BIP under various post to pre stiffness.
Max rel disp(m)
0.4
FB FBP
0.3
= 0.1 = 0.2 = 0.3 = 0.4 = 0.5
0.2
0.1
0
0
1
2
3
4
5
6
7
8
9
10
ex (rad)
Fig 4.44: Maximum rel. disp. of the second floor of A In case of BIP under various post to pre stiffness. When bounded building B is compared with that of the unbounded building, the same conclusion is found. Figure 4.45 shows the maximum displacement of the base of building B. It has a peak maximum displacement 125 cm against 38 cm for the unbounded one. This increase is due to the impact energy. Figures 4.46 and 4.47 show the maximum relative displacements of the first and second floor of the bounded building B. It is founded that the maximum displacements have a peak values 12 and 20 cm for the first and second floors respectively
- 99 -
Chapter (4)
Effect of pounding on adjacent structures
against 9 and 15 cm for the unbounded floors. It is shown that the peak of the maximum relative displacement of the two floors exceeds in some times the peak of the fixed unbounded building. This means that the increase in response due to impact energy overcomes the reduction in response due to base isolation, because the floors of building B is lighter than that of building A, then affected seriously than building A. It is shown that the floors maximum displacements of the bounded fixed base building B have many peaks due to the impact.
Max disp(m)
1.5
= 0.1 = 0.2 = 0.3 = 0.4 = 0.5
1
0.5
0
0
1
2
3
4
5 wex (rad)
6
7
8
9
10
Fig 4.45: Maximum sliding displacement in the base of B in case of BIP under various post to pre stiffness. 0.3
FB FBP
Max rel disp(m)
0.25
= 0.1 = 0.2 = 0.3 = 0.4 = 0.5
0.2 0.15 0.1 0.05 0
0
1
2
3
4
5 ex (rad)
6
7
8
9
Fig 4.46: Maximum relative displacement of the first floor of B in case of BIP under various post to pre stiffness.
- 100 -
10
Chapter (4)
Effect of pounding on adjacent structures
0.3
FB FBP
Max rel disp(m)
0.25
= 0.1 = 0.2 = 0.3 = 0.4 = 0.5
0.2 0.15 0.1 0.05 0
0
1
2
3
4
5
6
7
8
9
10
ex (rad)
Fig 4.47: Maximum relative displacement of the second floor of B in case of BIP under various post to pre stiffness. Figure 4.53 shows the total strain, damping and kinetic energies of the two base isolated buildings with different post to pre stiffness ratio when subjected to sinusoidal ground motion with different frequencies. It is shown that the maximum energy increases with the increase in the post stiffness because the relative displacement of the building increases because of the decrease in the dissipation energy in the base. (a)
5
S.E (t.m)
10
4
10
3
10
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
(b)
4
D.E (t.m)
10
2
10
=
0.1
=
0.2
=
0.3
= 0.4
= 0.5
0
10
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
3
3.5
4
4.5
5
(c)
6
K.E (t.m)
10
4
10
2
10
0
0.5
1
1.5
2
2.5 wex (rad)
ω ex (rad)
Fig 4.48: Total energy for the two buildings in case of BI (a) strain energy, (b) damping energy, (c) kinetic energy.
- 101 -
Chapter (4)
Effect of pounding on adjacent structures
Figure 4.54 shows the total strain, damping and kinetic energies increased when pounding effect taking into considering. And it is shown that the energies increased with the increase of the post stiffness because the relative displacement of the building increases due to the decrease in the dissipation energy in the base.
(a)
6
S.E (t.m)
10
4
10
= 0.1
= 0.2
= 0.3
= 0.4
= 0.5
2
10
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
3
3.5
4
4.5
5
3
3.5
4
4.5
5
(b)
5
D.E (t.m)
10
0
10
0
0.5
1
1.5
2
2.5 (c)
6
K.E (t.m)
10
4
10
2
10
0
0.5
1
1.5
2
2.5 wex (rad)
ω ex (rad)
Fig 4.49: Total energy for the two buildings in case of BIP (a) strain energy, (b) damping energy, (c) kinetic energy.
4.4.4 Effect of Separation Distance (Gap): The in-between gap size is a very important configuration parameter, which describes the spatial arrangement of neighboring structures. The easiest way to prevent structural pounding is to provide
- 102 -
Chapter (4)
Effect of pounding on adjacent structures
large spacing between adjacent structures. On the other hand, however, high land prices often generate pressure on engineers to construct as closely as possible. Separation distances between the adjacent buildings studied are ranged from zero to the value at which pounding vanishes. Also, no changes happen in the other properties of the buildings. Figure 4.50 shows the effect of the change of the separation distance on the number of impacts of building of the two cases fixed base and base isolated. It is shown that the increase in the gap reduces the number of impacts in the two cases. It is shown that the effect of pounding in case of fixed base decreases sharply and then vanish at gap equals 7cm but in case of base isolated structures the effect of the gap continue for gap more than the fixed case and vanish at 13 cm. Figure 4.51 shows the maximum displacement under different values of separation distance for case of fixed base. It is clear that the increase in the separation distance reduce the maximum displacement in the flexible building A and the opposite in the stiff building B as shown in the previous figure which turns to be constant at the gap distance which previously cleared, because the decrease in the gap decrease the chance of pounding and makes the building vibrated freely. Figures 4.52 and 4.53 show the maximum displacement of case base isolated under the variable clearance distance. It is shown that the peak of the maximum displacement decrease in the two building with the increase in the separation distance. This show that the effect of pounding is more harmful in base isolated which increase the displacement in the two buildings at small separation distance.
- 103 -
Chapter (4)
Effect of pounding on adjacent structures
50 case FBP case BIP
no of impacts
40
30
20
10
0
0
5
10
15
20
25
gap (cm)
Fig 4.50: The number of impacts for the two cases (FBP & BIP). 0.08
Max rel disp (m)
0.07
first story of A second story of A first story of B second story of B
0.06 0.05 0.04 0.03 0.02 0.01
0
5
10
15
20
25
gap (cm)
Fig 4.51: The max. disp. of the two buildings in case of FBP. 0.18 0.16
Max rel disp (m)
0.14 base of A first story of A second story of A
0.12 0.1 0.08 0.06 0.04 0.02 0
0
5
10
15
20
25
gap (cm)
Fig 4.52: The max. relative disp. of the building A in case of BIP.
- 104 -
Chapter (4)
Effect of pounding on adjacent structures
0.18 base of B first story of B second story of B
0.16
Max rel disp (m)
0.14 0.12 0.1 0.08 0.06 0.04 0.02 0
0
5
10
15
20
25
gap (cm)
Fig 4.53: The max. relative disp. of the building B in case of BIP.
4.5 Behaviour of the Adjacent Buildings under Earthquake Excitations: In this section, the behaviour s of the numerical experiments are studied under four ground motion recorded which detailed in table 4.7. These earthquakes are Elcentro and Northridge, Kobe, and Loma Prieta (Source, Earthquake Engineering Research (PEER)). Table (4.7): The details of the accelerogram for the four earthquakes: Earthquake Elcentro Kobe Loma Prita Northridge
Station S00 –North-South component at May, 1940. About 40 miles from Imperial Fault, in California. north-south component of the January 1995, (unscaled magnitude 6.9). Recorded at the Kobe Japanese Meteorological Agency (KJMA). LOMAP/CLS000 component of the October 1989, Earthquake. Recorded at 57007 Corralitos. east-west component at January 1994, Northridge, CA. USA. Tarzana (CDMG station 24436).
- 105 -
PGA 0.385 0.485 0.725 0.875
Chapter (4)
Effect of pounding on adjacent structures
The ground accelerations versus time are shown in figure 4.54. The ground motions are digitized at intervals of 0.001 seconds.
Frequency Content of Ground Motions: The Fourier transform is a frequency domain analysis technique that is used to determine dominant frequency. Figure 4.54 shows the frequency content of ground motion for the Elcentro, Kobe, Northridge, and Loma Prieta accelerograms, respectively. Each plot is annotated with the dominant frequency and corresponding period of ground shaking. (b) 1
0.5
0.5 a/g
a/g
(a) 1
0 -0.5
-0.5
0
10
20 30 time (sec) (c)
-1
40
1
1
0.5
0.5 a/g
a/g
-1
0
0 -0.5 -1
0
10
20 30 time (sec) (d)
40
0
10
20 30 time (sec)
40
0 -0.5
0
10
20 30 time (sec)
40
-1
Fig 4.54: Time history of four earthquakes (a) Elcentro earthquake (b) Northridge earthquake, (c) Loma earthquake (d) Kobe earthquake.
- 106 -
Chapter (4)
Effect of pounding on adjacent structures
(b)
15
frequency content 2) Spec. accel. (cm/sec
frequency content 2) Spec. accel. (cm/sec
(a)
10 5 0
0
2
4 time frequency (c)
6
40 30 20 10 0
0
2
4 time frequency (c) (d)
6
0
2
4 time
6
2 Spec. accel. (cm/sec frequency content )
frequency content 2) Spec. accel. (cm/sec
40 40 30 20 10 0
0
2
4 time frequency
6
30 20 10 0
frequency
Fig 4.55: Fourier Spectrum for the four earthquakes (a) Elcentro, (b) Northridge, (c) Loma earthquake and (d) Kobe earthquake. Adjacent buildings shown in figure 4.3 have been used for analysis. Building A is the reference building described in section 4.1 and Building B has a varying stiffness with respect to the reference building. A stiffer or a more flexible Building B is considered which have different mass from stiff to flexible case with respect to the reference building. Time periods and frequencies of Building B have been depicted in table 4.2. The damping of 5% in the first mode is kept same as that for the reference building by reconstructing the damping matrix as its mass is varied. Gap between the buildings have also been varied. Figure 4.56 shows the time history of the upper floor of the two adjacent buildings in all cases of study under Northridge ground excitation. In case of fixed base it is shown that the maximum displacement in the reference building A decreases from
25.24 cm to 25.18 cm as shown in
figure 4.55(a). Also, the maximum displacement decreases in building B
- 107 -
Chapter (4)
Effect of pounding on adjacent structures
from 19.46 cm to 15.44 cm as shown in figure 4.56(b). On the other hand, in case of base isolated it is shown that the effect of the pounding on the two buildings increases the maximum absolute displacement in the building A from 49.87 cm to 67.16 cm and for building B from 52.2 cm to 80 cm as shown in figure 4.56(c) and (d), respectively. The total energy transmitted to superstructure part gives an important expression for the different behaviour of the two models under pounding effect. In case of fixed base the total energy increases from 6.1940 *105 t.m to 7.6440*105 t.m with increasing factor in the total energy with 23.4%, and in case of base isolated model the total energy increased from 4.9687*105 t.m to 7.3795 *105 t.m with increasing 48.5% in the total energy. This shows that the effect of the base isolation on the pounding increase with 25% in the total energy more than fixed buildings. This because the isolation of the buildings increase the flexibility of the which affect directly on the total energy. Table 4.8 shows the peak displacement of the first and second story for the building A and B in case of fixed base buildings when pounding is and is not taking into consideration. It is shown that the pounding decrease the peak displacement of the flexible building A and increase it for the stiff building B as previously discussed. Table 4.8: The peak disp. of first and second floor of building A and B. FB earthquake Elcentro Northridge Kobe Loma
Floor First Second First Second First Second First Second
Building A 0.0799 0.1323 0.1522 0.2574 0.1539 0.2641 0.0825 0.1495
Building B 0.0567 0.0926 0.0742 0.1925 0.0882 0.1514 0.1248 0.2041
- 108 -
FBP Building Building A B 0.0639 0.0717 0.1060 0.1213 0.1127 0.1214 0.2555 0.1550 0.0755 0.1245 0.1237 0.1950 0.0998 0.1128 0.1690 0.1878
Chapter (4)
Effect of pounding on adjacent structures
0.4
FB
Abs Disp(m)
(a)
FBP
0.2
0
max(FBP)=25.18cm
-0.2
max(FB) = 25.24cm -0.4
0
5
10
15
20
25
30
35
40
ti me(sec) 0.4
FB
Abs Disp(m)
(b)
FBP
0.2
0
-0.2
-0.4
max (FB)=19.46 cm max (FBP)=15.44cm 0
5
10
15
20
25
30
ti me(sec) BI
0.6
BIP
(c)
Abs Disp(m)
0.4 0.2 0 -0.2
max(BI)=49.78 cm
-0.4 -0.6 -0.8
max(BIP)=67.16 cm 0
5
10
15
20
25
30
35
40
ti me(sec)
(d)
0.6
BI
BIP
Abs Disp(m)
0.4 0.2 0 -0.2 -0.4
max(BI)=52.2cm
-0.6
max(BIP)=80 cm
-0.8
0
5
10
15
20
25
30
35
40
ti me(sec)
Fig 4.56: Upper floor disp. for Northridge earthquake (a, b) case of FB and FBP for A and B respectively (c, d) case BI and BIP for A and B.
- 109 -
Chapter (4)
Effect of pounding on adjacent structures
Figures 4.57 and 4.58 show the base shears of the fixed base and base isolated buildings at the base of the two buildings under Northridge ground excitation respectively. By observing these figures, these figures show that the base shear in case of fixed base buildings is not changed in building A and decreases in building B from 1980 ton to 1600 ton. But, in case of base isolated buildings the base shear increased in building A from 2150 ton to 3200 ton and for building B from 2750 ton to 4150 ton. These results show the effect of pounding in case of base isolated is more than the case of fixed base. 3000
base shear (ton)
FB
(a)
2000
FBP
1000 0 -1000 -2000 -3000
0
5
10
15
20
25
30
35
40
20
25
30
35
40
2000
(b) base shear (ton)
1000
0
-1000
-2000
0
5
10
15
time(sec)
Fig 4.57: Base shear in case of fixed base for (a) building A, and (b) building B.
- 110 -
Chapter (4)
Effect of pounding on adjacent structures
3000
(a)
base shear (ton)
2000
BI
BIP
1000 0
-1000 -2000 -3000 -4000
0
5
10
15
5000
20
25
30
35
40
20
25
30
35
40
(b)
4000
base shear(ton)
3000 2000 1000 0
-1000 -2000 -3000
0
5
10
15
time(sec)
Fig 4.58: Base shear for BI case for (a) building A, and (b) building B. Figure 4.59 shows the impact force of the two adjacent buildings for the two cases of base isolated and fixed base structures under Northridge earthquake. It is shown that the impact force in case of base isolated buildings equals 28000 ton but in case of fixed base buildings equals 26000 ton this give 7.7% increase in base isolation system in impact force. This reflects that the base isolation gives sliding displacement in the base makes the total displacement in this case more than fixed base. 4
3
x 10
case FBP
case BIP
impact force(ton)
2.5
2
1.5
1
0.5
0 0
5
10
15
20
25
30
35
ti me (sec)
Fig 4.59: Total impact force in case of FBP and BIP. - 111 -
40
Chapter (4)
Effect of pounding on adjacent structures
4.5.1 Effect of Mass Ratio (): The two adjacent buildings A and B are analyzed when subjected to the four mentioned earthquakes when the buildings with different mass ratio for the two cases previously discussed. The two buildings are analyzed when they have large gap to prevent pounding and when the gap between them are very close, then the pounding is taken into consideration. The values of the mass ratio will be taken from 0.4 to 2.0 in the study. Figures 4.60 and 4.61 show the maximum response of the upper floor displacements of the two fixed base adjacent buildings against the variable values of the mass ratio when pounding is not and is considered respectively. It is shown that the increase in the mass ratio increases the peak maximum displacement in the two buildings, because the increase in the mass the superstructure part increases the flexibility of the building which makes the total displacement increases. When the pounding between the two buildings is taken into consideration, the maximum displacements are increased because the impact between the two buildings creates excessive energy that increases the buildings responses. When the response of the base isolated building is compared with that of the fixed base which stated in table 4.8, it is clear that the base isolation decreases the relative displacement in case of earthquakes. In order to demonstrate the upper stories displacement of two base isolated adjacent buildings with variable mass ratio, figure 4.62 and 4.63 are studied. it is shown that the maximum response obey the same explain that stated in the same parameter using in the previous part.
- 112 -
Chapter (4)
Effect of pounding on adjacent structures
0.5 (a)
elcentro
kobe
loma
north
max disp (m)
0.4 0.3 0.2 0.1 0 0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0.5 (b) max disp (m)
0.4 0.3 0.2 0.1 0 0.4
Fig 4.60: Maximum disp case of FB, (a) top floor of building A, (b) top floor of building B. 0.5
max disp (m)
0.4
elcentro
(a)
kobe
loma
north
0.3 0.2 0.1 0 0.4
0.6
0.8
1
0.6
0.8
1
1.2
1.4
1.6
1.8
2
1.2
1.4
1.6
1.8
2
0.5 (b) max disp (m)
0.4 0.3 0.2 0.1 0 0.4
Fig 4.61: Maximum disp in case of FBP, (a) top floor of building A, (b) top floor of building B.
- 113 -
Chapter (4)
Effect of pounding on adjacent structures
0.1 elcentro max disp (m)
0.08
kobe
loma
north
(a)
0.06 0.04 0.02 0 0.4
0.6
0.8
1
0.6
0.8
1
1.2
1.4
1.6
1.8
2
1.2
1.4
1.6
1.8
2
0.1
max disp (m)
0.08
(b)
0.06 0.04 0.02 0 0.4
Fig 4.62: Maximum relative displacement in case of BI, for (a) top floor of building A, (b) top floor of building B. 0.1
max disp (m)
0.08
elcentro
kobe
loma
north
(a)
0.06 0.04 0.02 0 0.4
0.6
0.8
1
0.6
0.8
1
1.2
1.4
1.6
1.8
2
1.2
1.4
1.6
1.8
2
0.1
max disp (m)
0.08 (b) 0.06 0.04 0.02 0 0.4
Fig 4.63: Maximum relative displacement in case of BIP, for (a) top floor of building A, (b) top floor of building B.
- 114 -
Chapter (4)
Effect of pounding on adjacent structures
4.5.2 Effect of Isolator Plastic Shear Capacity: The two adjacent isolated buildings A and B are analyzed when subjected to the four mentioned earthquakes when the bases of these buildings have varied isolator plastic shear capacity. The two buildings are analyzed when they have large gap to prevent pounding and when the gap between them are very close, then the pounding is taken into consideration. Figure 4.64 shows the maximum displacement of each case when the gap between the two buildings is large. It is expected that the decrease of the isolator plastic shear capacity increases the dissipated energy, then decrease the response. But it is shown that there is no definite manner for the behaviour of the maximum top floor drifts of the two buildings. As stated previously in the discussion of figures 4.20 and 4.21, there are two regions, at the first the decrease of the isolator plastic shear capacity increases the response, and at the second region the decrease of the isolator plastic shear capacity decreases the response. When looking in the earthquakes frequency content, there is a clear explain for the response. The response due the El-Centro earthquake is the small response because the power of the frequency content is weak. The response due Loma earthquake is larger because the energy in this earthquake is high but this energy is concentrated at definite region of frequency content. The response due Kobe earthquake is larger because the energy in this earthquake is high and distributed over wider range of frequencies.
The
response due Northridge earthquake is larger because the energy in this earthquake is high and this energy is wide distributed over the frequency content. When the response of the base isolated building is compared with that of the fixed base which stated in table 4.8, it is clear that the benefit of the base isolation also in case of earthquakes, which decrease the relative displacement between the floors of the two buildings. Figure 4.65 shows the maximum displacement of the top floor of the buildings when the gap between them is very close and pounding is taken into
- 115 -
Chapter (4)
Effect of pounding on adjacent structures
consideration. It is shown that the maximum response of the two buildings is larger than that of the unbounded buildings. And the maximum response obey the same explain that stated in the previous subsection. 0.1
elcentro
kobe
loma
north
max disp (m)
(a) 0.08 0.06 0.04 0.02 50
100
150
100
150
200
250
300
350
200
250
300
350
0.08
max disp (m)
(b) 0.06 0.04 0.02 0 50
Qp (ton)
Fig 4.64: Max. relative disp. with plastic shear capacity in case of BI, (a) top floor of building A, (b) top floor of building B. 0.1
elcentro
max disp (m)
(a)
kobe
loma
north
0.08 0.06 0.04 0.02 50
100
150
100
150
200
250
300
350
200
250
300
350
0.2
max disp (m)
(b) 0.15 0.1 0.05 0 50
Qp (ton)
Fig 4.65: Max. relative disp. with various plastic shear capacity in case of BIP, (a) top floor of building A, (b) top floor of building B. - 116 -
Chapter (4)
Effect of pounding on adjacent structures
4.5.3 Effect of Isolator Post-Pre Stiffness: The two adjacent isolated buildings A and B are analyzed when subjected to the four mentioned earthquakes when the bases of these buildings have varied post stiffness ratio. The two buildings are analyzed when they have large gap to prevent pounding and when the gap between them are very close, then the pounding is taken into consideration. Figure 4.66 shows the effect of the change in the post stiffness on the maximum displacement of the two adjacent buildings in case of base isolated (BI). It is shows that the maximum displacement increases with the increase in the post stiffness, because the increases in the post stiffness increase the rigidity of the base which makes the energy transmitted increases to the upper stories. Figure 4.67 shows the maximum displacement of the top floor of the base isolated buildings when pounding is taken into consideration. It is shown that the peak displacement increases with the increase in the post stiffness ratio, as the same explain in the previous subsection When the response of the base isolated building is compared with that of the fixed base which stated in table 4.8, it is clear that the base isolation decreases the relative displacement in case of earthquakes.
- 117 -
Chapter (4)
Effect of pounding on adjacent structures
0.08
elcentro
max rel disp (m)
(a)
kobe
loma
north
0.06 0.04 0.02 0 0.1
0.2
0.3
0.4
0.5
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.6
0.7
0.8
0.9
1
0.1 max rel disp (m)
(b) 0.08 0.06 0.04 0.02 0 0.1
Fig 4.66: Max. relative disp. with post-pre stiffness ratio in case of BI, (a) top floor of building A, (b) top floor of building B. 0.1
max disp (m)
0.08
elcentro
(a)
kobe
loma
north
0.06 0.04 0.02 0 0.1
0.2
0.3
0.4
0.5
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.6
0.7
0.8
0.9
1
0.12 (b) max disp (m)
0.1 0.08 0.06 0.04 0.02 0 0.1
Fig 4.67: Max. relative disp. with post-pre stiffness ratio in case of BIP, (a) top floor of building A, (b) top floor of building B.
- 118 -
Chapter (4)
Effect of pounding on adjacent structures
4.5.4 Effect of Separation Distance (Gap): Due to the importance of the separation distance on the pounding, the behaviour of the considered buildings with separation distance will be discussed under the effect of the four earthquakes in this part. The separation distances studied are ranged from zero to the value at which pounding vanishes at 27 cm. Figure 4.68 shows the variation in number of impacts with the increase in the gap between the buildings for the fixed and isolated case. It can be shown simply from this figure that the number of impacts decreases with the increase in gap. Also, the number of impacts decreases in case base isolated structure quickly more than the case of fixed base. But it has greater effect in some earthquakes such as Northridge earthquake, which takes clearance distance in base isolation case greater than fixed base one. This because the isolated structure lies in the Northridge frequency content range which increase the displacements allowing more impacts between the two buildings. This make the prediction of the effect of pounding on the base isolated buildings is more difficult than on the fixed case. Also in order to demonstrate the maximum displacement in case of fixed base adjacent buildings under the variation of the different clearance distance figure 4.69 demonstrates that there is difficulty in predicting the behaviour of the building under the effect of the pounding. Because the displacement increases in some cases and decreases in the other. But the total energy gives a clear prediction in this case. The total energy decreased with the increase in the clearance distance. Due to
- 119 -
Chapter (4)
Effect of pounding on adjacent structures
alimentation of the energy generated from the impact when the gap is large enough to avoid pounding. Figure 4.70 shows the maximum displacement in the upper stories of the two adjacent buildings in case of BIP. The total energy is found to be decreased with the increase in the separation distance in the four earthquakes until the impact of the collision finished and the total energy remain constant. This shows the effect of the pounding on the increase of the energy transmitted to the superstructure of the buildings especially in case of base isolated buildings because of the total displacement of the floors in this case more than fixed case. This makes the chance of the pounding increases.
no of impacts
15
elcentro
kobe
north
loma
10
(a) 5
0
5
10
15
20
25
no of impacts
15
(b)
10
5
0
5
10
15
gap (cm)
20
Fig 4.68: No of impacts under variable gap distance (a) Case of FBP, (b) case of BIP.
- 120 -
25
max disp (m)
Chapter (4)
Effect of pounding on adjacent structures
0.2
0.1 elcentro
(a) 0
kobe
north
loma
5
10
15
20
25
5
10
15 gap (cm)
20
25
0.25 (b) max disp (m)
0.2 0.15 0.1 0.05 0
Fig 4.69: max. disp. in case of FBP, (a) top disp of A, (b) top disp of B.
max rel disp (m)
0.1 0.08
elcentro
(a)
kobe
north
loma
0.06 0.04 0.02 0
5
10
15
20
25
5
10
15 gap (cm)
20
25
max rel disp (m)
0.1 0.08 (b) 0.06 0.04 0.02 0
Fig 4.70: Max. rel. disp. in case of BIP, (a) top disp of A, (b) top disp of B.
- 121 -
Chapter (4)
Effect of pounding on adjacent structures
4.5.5 Effect of Base Mass Ratio (b): The two adjacent isolated buildings A and B are analyzed when subjected to the four mentioned earthquakes when these buildings have varied base mass ratio. The two buildings are analyzed when they have large gap to prevent pounding and when the gap between them are very close, then the pounding is taken into consideration. Figure 4.71 shows the maximum relative displacement for the isolated building A and B under variable values of base mass ratio for the four earthquakes excitation when pounding do not take into consideration. It is shown that the peak displacement of the buildings decreased for the most earthquakes with the increase in the base mass, because the increase of the base mass increases the flexibility of the buildings this means that the sliding displacement increases which decreases the energy transmitted to the superstructure. There are some exceptions for this when the natural frequency obey with the frequency content of the earthquake (figure 4.55). When the response of the base isolated building is compared with that of the fixed base which stated in table 4.8, it is clear that the base isolation decreases the relative displacement in case of earthquakes. But the problem when the in between separation distance between buildings is close, the chance of pounding increased due to the increase in the sliding displacement. Figure 4.72 shows the maximum displacement of the top floor of the buildings when the gap between them is very close and pounding is taken into consideration. It is shown that the maximum response of the two buildings is larger than that of the unbounded buildings. And the maximum response obey the same explain that stated in the previous subsection.
- 122 -
Chapter (4)
Effect of pounding on adjacent structures
0.08 max rel disp (m)
elcentro
kobe
loma
north
(a) 0.06
0.04
0.02 0.2
0.3
0.4
0.5
0.6
0.7
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
max rel disp (m)
0.07 (b)
0.06 0.05 0.04 0.03 0.02 0.2
0.8
0.9
1
b
Fig 4.71: Max. relative disp. with respect to base mass ratio in case of BI, (a) top floor of A, (b) top floor of B. 0.1 elcentro
max rel disp (m)
(a)
kobe
loma
north
0.08 0.06 0.04 0.02 0.2
0.3
0.4
0.5
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.6
0.7
0.8
0.9
1
max rel disp (m)
0.12 0.1 (b) 0.08 0.06 0.04 0.02 0.2
b
Fig 4.72: Max. relative disp. with respect to base mass ratio in case of BIP, (a) top floor of building A, (b) top floor of building B.
- 123 -
CONTROL OF BASE ISOLATED BUILDINGS POUNDING
CHAPTER (5)
CONTROL OF BASE ISOLATED BUILDINGS POUNDING 5.1 General:
When two structures in close proximity and with different properties (heights, structural systems and materials) are subjected to a strong seismic ground motion there is the possibility that pounding between them may occur. Many researchers are continually striving to better understand how to reduce the effect of this phenomenon. The large impact loads induced by this phenomenon usually lead to catastrophic results. The most direct way to prevent or avoid pounding is to provide an adequate clearance distance between the two adjacent base isolated buildings. However, this measure cannot be applied for existing buildings. In this case, connected link between the buildings have been used. Link members are used to couple adjacent buildings as a proposed technique to mitigate pounding problem. Link of the structures was first suggested by Klein et al., (1972). From that time, they used a link for mitigating the response of structures due to wind loading. The characteristic of the link element between the structures plays an important role on the dynamic properties of the structures. If the structures are linked with elastic spring, their natural frequencies will
CHAPTER (5)
Control of Base Isolated Buildings Pounding
change depending on the stiffness of the spring. This can shift the natural frequencies of the structures away from the dominant frequency of the ground motion. If the natural frequencies are already on the safe side, viscous damper should be used as a link, because viscous damper changes the structural natural frequencies only slightly. Additionally, viscous damper can absorb part of the energy during the vibration of the structures. A number of researches were published on viscous damper as a link. Most of these studies only deal with uniformly distributed viscous dampers between the structures. One of the main aims of this study is to investigate the problem of earthquakes induced pounding between two buildings and how to reduce the effect of this phenomenon.
5.2 Numerical Model:
Two adjacent base isolated buildings 2DOF are idealized as lumped mass with viscous damper connected the two buildings at all levels at the clearance distance between them are analyzed, in order to study the pounding control and the behavior of the two adjacent base isolated buildings with changing the parameters of these connected damper. Figure 5.1(a) provides a schematic diagram of two base isolated adjacent buildings A and B that include the connected damper. Also, figure 5.1(b) shows the Elastoplastic material of the connector which used in all levels of structures. The basic parameters of the two base
- 125 -
CHAPTER (5)
Control of Base Isolated Buildings Pounding
isolated buildings, connected dampers and the applying excitation frequency are shown in table 5.1. x12
(a)
F2
m12 C12
x22 F2 m22
x11 k12 m11
x21 F1
F1
C22 k22
m21
C11
C21 k11
F1
k21
F1 .. Xg (t)
(b)
F Fy
Kp = Ke
Ke
X
Figure 5.1: (a) Schematic diagram of the adjacent base isolated buildings connected with the viscous dampers (b) Elastoplastic connector material.
- 126 -
CHAPTER (5)
Control of Base Isolated Buildings Pounding
Table 5.1 Parameters Used for Adjacent Structures. Building mass of each story [sec2.t/m] stiffness of each story [t/m] The mass ratio b= [ mb / mt ] The frequency ratio δ = [ ωb / ωt ] Post- pre stiffness α = [Kbp/Kbe] The isolator yield force (t) [fy] Structural damping ratio [s] base damping ratio [b]
Left
Right
100 10000 0.6 0.6 0.2 100 5% 8%
50 10000 0.6 0.6 0.2 100 5% 8%
Connected dampers Ke* Fy** α
4000 t/m 10 ton 0.0 Sinusoidal wave
ωex tmax * Elastic stiffness of the damper.
3.0 rad 20 sec ** Yield force.
Figure 5.2 shows the displacement time history of the controlled and uncontrolled base isolated structures when subjected to harmonic excitation with frequency equal to 3.0 rad for time equal 20 sec. It is shown that the peak displacement decreases with the laying of the connected damper between the two buildings. The peak absolute displacement of the top story of the reference building A reduces from 37.05 cm to 28.86 cm with reduction factor equals 22%. Also, the peak displacement of the upper story of the building B reduces from 37.66 cm to 21.1cm with reduction factor equals 44%. Also on the other hand, the total energy decreases from 5.63*105 t. m to 4.83*105 t. m with reduction factor 14.2% as shown in table 5.2. This reduction in the total energy occurs due to the absorption of the pounding force in the linked damper, and this can be attributed to the large stiffness of the connecting link in all cases which make the buildings behaves as one building. Also, this shifts the natural frequencies of the structures away from the dominant frequency of the ground motion.
- 127 -
CHAPTER (5)
displacement(m)
(a)
uncontrol
control
0.4
0.6
0.2
0.4
0
0.2
-0.2
0
-0.4
(b)
Control of Base Isolated Buildings Pounding
0
5
10
15
20
-0.2
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
(d)
0
5
10
15
20
5
10
15
20
5
10 time (sec)
15
20
displacement(m)
(e)
-0.4
displacement(m)
(c)
0
5
10
15
20
-0.4
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
0
(f)
-0.4
0
5
10 time (sec)
15
-0.4 20 0
Figure 5.2: Uncontrolled vs. controlled floor displacements (a, b, c) building A, (d, e, and f) building B, for the base, First and second story. Table 5.2: Maximum Displacement and Total Energy for Control and Uncontrolled Systems. Building
Left building
Right building
controlled uncontrolled controlled first story absolute displacement (cm) uncontrolled controlled second story absolute displacement (cm) uncontrolled Total energy (uncontrolled)*10e5 (t.m) Total energy (controlled) *10e5 (t.m)
26.26 32.95 27.99 35.56 28.86 37.05
18.34 33.66 20.07 36.35 21.10 37.66
Base story sliding displacement (cm)
- 128 -
5.6330 4.8562
CHAPTER (5)
Control of Base Isolated Buildings Pounding
5.3 Parameters Affecting the Effectiveness of Viscous Damper: To determine the optimum parameter of the dampers that used to link the two adjacent base isolated buildings, the principle of minimizing the total energy of the overall system is used. Each structure is modeled as two story base isolated structure and connected to each other through viscous dampers at each level and they are subjected to harmonic excitations with variable frequency. Many parameters are studied to obtain the optimum parameters of the viscous damper such as yield displacement, post-pre stiffness ratio and elastic stiffness. The damper elastic stiffness is expressed as a ratio from the stiffness of the columns of the first building. The properties of the adjacent base isolated buildings are changed also to show the effect of these properties on the optimum values of the damper.
5.3.1Effect of Excitation Frequency and Damper Yield Displacement: Figures 5.3a, b show the total energy of the overall system versus the excitation frequency and the damper yield displacement xy as a surface and contour lines respectively. These figures show that the total energy of the overall system have a maximum value at the excitation frequency 2.0 rad, which indicates the resonance of the overall system. The variation of the total energy of the controlled base isolated system is studied separately at resonant frequency (ωex=2.0 rad) and compared with the total energy of the uncontrolled pounded base isolated system as a ratio and versus the damper yield displacement or the damper yield force as shown in figure 5.3c. This figure shows that the energy ratio is decreased with increase of the damper yield force until specific point. After this point the increase in the damper yield force increases the energy ratio. Because at very low damper yield force, the dampers enter the plastic zone quickly, then it behaves as elastic damper with small stiffness (plastic stiffness). But at the optimum point (the lowest energy ratio), the damper obeys elasto-plastic behavior, and then the
- 129 -
CHAPTER (5)
Control of Base Isolated Buildings Pounding
dampers dissipate the energy of the system. After this point, the damper tends to be elastic with small plasticity, and then small energy dissipated. 10 T.E *105 (t.m)
(a)
5
0 300 200 100
fy (cm)
1.6712
8.3558 7.5202 6.6847 5.8491 5.0135
8.355 8 7.5202 6.6847 5.8491 5.0135 4.1779 3.3423 2.5067
0.83558 1.6712 2.5067 3.3423
fy (cm)
ex (rad)
0.83558
100
1.67 12 2.50 67 3.3 42 39 4.1 77 5.0 13 5 5. 84 91 6. 68 47 7.5202
1.6712 2.506 7 3.342 3 4.177 9 5.0135 91 5.84 7 84 6.6 20 2 7.5 58 8.35
50
0
(c)
1.6712
7.5202 6.6847 5.8491 5.0135 4.1779
150
2.5067 3.3423 4.1779
200
0
4
0.83558
250
8.3558 7.5202 6.6847 5.8491 5.0135 4.1779 3.3423 2.5067 1.6712 0.83558
(b)
0
2
10
8
6
1
2
3
4
5 6 (rad) ex
7
8
9
10
0.94 0.92
E.R (t/m)
0.9 0.88 0.86 0.84 0.82
0
50
100
150
200
250
fy (ton)
Figure 5.3: The excitation frequency and the yield disp. versus the total energy (a) surface area, (b) contour line, and (c) the energy ratio (TEc /TE) at wex=2.0.
- 130 -
CHAPTER (5)
Control of Base Isolated Buildings Pounding
5.3.2 Effect of Damper Stiffness Ratio and Damper Yield Force: Figures 5.4 – 5.8 introduce the effect of the variation of the natural frequency of the two base isolated adjacent buildings, yield displacement xy and the elastic stiffness of the connected damper kR on the total energy of the overall system. These figures show that by the increase in the natural frequency of the adjacent structures, the relative displacement and velocity in all floor levels increase which increases the force in the damper. This increase the required dissipation energy smaller value of yield force that makes the damper enter in the plastic stage rabidly, as shown in figure 5.8. Also from these figures, the decrease in the total energy transmitted in the direction of increasing of the damper stiffness at the yield force that minimize the total energy, decreased slowly after certain value that almost equal 0.6 from the reference building stiffness A. This make the increase in the stiffness of the damper after this point will not be cost- effective. (a)
(b) 9 T.E *104(t.m)
4 1 x 10 Kd (t/m) KR
50 0
0
20
40
fy (t)
8.652
60
80
0
0
fy (t)
x 10
1.5
8.837 8 9.02 3 9.2 5 09 2
8.4663
8.28 06
Kd (t/m) KR
R
4
2
0.5
50
1
6.6786 6.8286
1
100
1 Kd (t/m) K
x 10
8.2806
1.5
100 4 x 10
fy (t)
4
2
6 2
0.5
100
20
6 6 828 40
fy (t)
6.9786
8 2
7
6.8286
9
8
7.1 28 7. 5 27 7.4 85 28 5
10
w0=10 rad
6.67 86
w0=5 rad
9.395 9.58 07 Kd (t/m) KR
T.E *104(t.m)
11
60
84 57 . 7 80 100
Figure 5.4: The energy ratio versus yield force and damper stiffness ratio for (a) ωo=5.0 rad, and (b) ωo=10.0 rad.
- 131 -
CHAPTER (5)
Control of Base Isolated Buildings Pounding
(a)
(b) 8
w0=15 rad
T.E *104(t.m)
7 6 2
4 2
0
0
x 10
Kd (t/m) KR
fy (t)
Kd (t/m) KR
1.5
6. 83 4 7 6.700 5
6. 56 6 6 2 .432
5 6.16 3
KR Kd (t/m)
20
0
fy (t)
6. 297 7 40 60 fy (t)
1 0.5
69 6.9
80
x 10
5.908 2 6.0368 6.1653
6.432
0.5
2
6.163 5 6.2977
1
0
2
x 10
1.5
50
4
4
2
100
1
4
50
6.1 65 6. 3 29 3 6.4 9 6. 22 55 4 1
Kd (t/m) KR
6
100
1
4
x 10
w0=20 rad
5.908
T.E *104(t.m)
8
20
100
6.036 8 40 60 fy (t)
5 79 6.6 80 100
Figure 5.5: The energy ratio versus yield force and damper stiffness ratio for (a) ωo=15.0 rad, and (b) ωo=20.0 rad. (a)
(b) 7
w0=25 rad
T.E *104(t.m)
6.5 6 5.5 2
100
x 10
KR
20
8 5.8856.008 7 40 60 fy (t)
0
fy (t)
KR
6.2545 Kd (t/m)
6.1316
1.5 1
0.5 4 7700 3 3 . 66.5 80 100
20
33 5. 89 40 60 fy (t)
2 5677 1 2 . 6 6.3 80 100
Figure 5.6: The energy ratio versus yield force and damper stiffness ratio for (a) ωo=25.0 rad, and (b) ωo=30.0 rad.
- 132 -
6.1352
5
5.8858
9 5.76 2
6.1316 6.0087
)
KR
0.5
0
x 10
5.6515
1
2
5.7724 5.8933 6.0143
5.7629
(
x 10
5.8858
1.5
100 50
4
4
2
1 Kd (t/m)
4
50 fy (t)
0
5.5 2
6.0 14 3
0
6
5.65 1
4 1 x 10 KR Kd (t/m)
w0=30 rad
6.5
5.7 72 4
T.E *104(t.m)
7
CHAPTER (5)
Control of Base Isolated Buildings Pounding
(a)
(b) 7 T.E *104(t.m)
w0=35 rad 6 5 2
100 0
0
4
3 5.771
5.826
5 84 06 65 6.1 5.75.826 6.0 40 60 80 fy (t)
1 0.5
5
100
5.5353
Kd (t/m) KR
KR
1.5
5.6533
20
0
0
fy (t)
x 10
5.8893
5.826
0.5
100 50
4
2
5.587 5.7065
5.9455
Kd (t/m)
5 2
x 10
1.5 1
6
4 1 x 10 Kd (t/m)KR
50 fy (t)
5.5 87
2
w0=40 rad
5.535 3
4 1 x 10 Kd (t/m) KR
5. 94 55
T.E *104(t.m)
7
20
3 3 89 007 3 8 . 5 3 . 53 3 1 3 5 6.1.2243 66 5.6 5.77 40 60 80 100 fy (t)
Figure 5.7: The energy ratio versus yield force and damper stiffness ratio for (a) ωo=35.0 rad, and (b) ωo=40.0 rad. 46 44 42
y
f (ton)
40 38 36 34 32 30 28
5
10
15
20
25
30
35
40
45
50
o (rad)
Figure 5.8: Effect of the variation of the natural frequency of the adjacent structures and yield displacement at minimum values the total energy. 5.3.3 Effect of Post-Pre Stiffness Ratio and Stiffness Ratio of the Damper: Figure 5.9 shows the total energy of the overall system versus the elastic stiffness ratio kR and the post-pre stiffness ratio of the damper α of the viscous dampers. The values of the post-pre stiffness ratio will be taken from 0.0 to 1.0 and for elastic stiffness ratio from 0.0 to 2.0. This
- 133 -
CHAPTER (5)
Control of Base Isolated Buildings Pounding
figure shows that the total energy decreases with the increase in the damper stiffness until a specific value at which the dampers will be more rigid that make the two buildings behave as one, then the increase in the damper stiffness will not be cost effective. Also it is shown that the total energy decreases with the decrease in the post stiffness ratio of the damper which minimized at post-pre stiffness ratio equals zero, because the decrease in the post stiffness increases the dissipated energy in the force-displacement curve. When the post-pre stiffness ratio equals 1.0 this means that the plastic the dampers still elastic stiffness all the time which make the dissipated energy equal zero and maximize the value of the total energy of the overall system.
9.5 (a) 9
5
T.E *10 (t.m)
8.5 8 7.5 7 6.5 2 1.5
1
x 10
0.6 0.4
0.5 Kd
1.4
0
0
(b)
7.9768 8.4569
1.6
7.4966
7.0164
x 10
1.8
0.2
KR
4
2
0.8
1
4
7.0164
0.8 0.6 0.4 0.2 0
66 49 7.
7.0164
d
K
1
569 8.4 7.9768 7.4966
KR
1.2
0.1
7.9 76 8 0.2
8.9371
8. 9371
8.456 9 0.3 0.4
0.5
0.6
0.7
0.8
0.9
Figure 5.9: Total energy versus the elastic stiffness ratio and post stiffness ratio, (a) surface area and (b) contour line.
- 134 -
1
CHAPTER (5)
Control of Base Isolated Buildings Pounding
5.3.4 Effect of Damper Post-Pre Stiffness Ratio and Damper Yield Displacement: Figure 5.10 shows the effect of yield displacement xy and the postpre stiffness ratio α of the connected damper on the total energy of the overall system. The values of post stiffness ratio will be taken as the same in the last figure and for the yield displacement from 0.0 to 3.0 cm. It is shown that the total energy of the overall system decreases with the decrease in post stiffness of the connected damper which obey the same explain that stated in the previous subsection. Also, this figure shows that the total energy is decreased with the increase of the damper yield displacement until specific point. After this point the increase in the damper displacement force increases the energy. Because at very low damper yield force, the dampers enter the plastic zone quickly, then it behaves as elastic damper with small stiffness (plastic stiffness). But at the optimum point (the lowest energy ratio), the damper obeys elasto-plastic behavior, and then
the dampers dissipate the energy of the system. After this point, the damper tends to be elastic with small plasticity, and then small energy dissipated, which obey the same explain that stated in figure 5.3c.
- 135 -
CHAPTER (5)
Control of Base Isolated Buildings Pounding
9.5 (a) 9 T.E *10 (t.m)
8.5
5
8 7.5 7 6.5 3 1
2
0.8 0.6
1
0.4 0.2 0
x y (cm)
0
8.7307
8.5004
8.2701
(b)
8. 50 04
7.5791 7.8094 8.0397 8.270 1
y
7.3487 7.1184 6.8881
x (cm)
1.5
1
7 8.039
4 7.809 91 7.57
2
8.7307
2.5
7.1184 3487 7.57. 79 1 7.8 094 8.27081 .0 8..500 397 73 4 07
0.5
0
0.1
8.7307
8. 7307
8. 73 07 8.961 1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 5.10: Total energy versus yield displacement and post stiffness ratio (a) surface area, and (b) contour line. 5.3.5 Effect of Natural Frequency of the Building and the Stiffness Ratio of the Damper: The effect of the natural frequency of the building A, and the stiffness ratio of the connected damper KR on the total energy of the overall system will be study in this section. The values of the stiffness ratio will be taken from 0.0 to 2.0 and for natural frequency of the building from 10.0 to 20.0.
- 136 -
CHAPTER (5)
Control of Base Isolated Buildings Pounding
Figure 5.11 shows the total energy of the overall system versus the natural frequency and the connected damper stiffness ratio, when the ratio between the natural frequencies of the two adjacent buildings remains constant. It is shown that the total energy of the overall system decreases with the increase in the damper stiffness ratio due to the increase in the stiffness make the damper rigid enough to make the two buildings behaves as one as previously discussed. Also, it is shown that the increase in the natural frequency i.e., increase in the stiffness of the building decreases the total energy decreases the inter story drift which decreases the total energy of the overall system.
1.35 (a) 1.25
5
T.E *10 (t.m)
1.3
1.2 1.15 1.1 1.05
0.5 1 1.5
1.1304
1.2638
1.2971
K
1.0804
1.0971
1.0804
1.1 71 4 1.16348 14 15 n (rad)
71 1.09
1.18 0
38 11 1.
1.21.2 1.1 1.3 1 38 971 13 .280 1.32041 47 1 4 8 11 12 13
304 1.1
10
8 3 26 1. 71 29 1.
0.2
1.0638
1.1471
1.1638
1.1804
1.1971
1.3304
0.4
1.2471
0.6
1.2804
1.3138
0.8
16
17
10
n (rad)
1.0804
R
1.2 1
12
1.1471
1.1638
1.1804
1.2138
1.1971
1.2304
1.2138
1.2304
1.4
1.2471 1.2638 1.2804 1.2971 1.3138 1.3304
1.6
1.1138
1.1304
2 (b) 1.8
16
18
20
14
1.0971
2
1.1138
KR
1.0638
Kd
18
1.0 63 8 19
20
Figure 5.11: Total energy against natural frequency and the yield displacement of the damper (a) surface area, and (b) contour line.
- 137 -
CHAPTER (5)
Control of Base Isolated Buildings Pounding
5.3.6 Effect of Structure Damping Ratio and Yield Displacement of the Damper: The total energy of the overall system at various yield displacement of the connected damper and damping ratio of the adjacent structures will be studied in this section. The values of damping ratio will be taken from 0% to 20% and the yield displacement from 0 to 2.5 cm. Figure 5.12 shows that the total energy decreases with the increase in the yield displacement until the optimum point and then return to increase that obey the discussion in figure 5.3c. Also, the figure shows that the control effectiveness will be improved by increasing the damping of the two adjacent buildings which decreases the overall energy.
7.6
5
T.E *10 (t.m)
7.4 7.2 7 6.8
(a)
6.6 2.5 2
0.2 1.5
x y (cm)
0.15 1
2.5
7.6809
6.680 6
y
6. 6806
6.6806
6.791 8 6.902 9
0.02
0.04
0.06
6.9029 6.7918
6.7918
6. 6806
0.5
7.2363 7.1252 7.014
7.2363 7.1252 7.014 6.9029
6.9029
6 80 6.6
7.347 5
7.3475
6.7918
1
7.458 6
7.4586
7.3475 7.2363 7.1252 7.014
1.5
7.5698
7.5698
7. 5698 7.458 6
2
s
0.05
7. 6809
7.680 9
(b)
x (cm)
0.1 0.5
6. 6806
6. 7918
6.9029 0.08 0.1
s
0.12
6.902 9 0.14 0.16
6. 791 8
0.18
0.2
Figure 5.12: Total energy against damping ratio and the yield displacement of the damper (a) surface area, and (b) contour line.
- 138 -
CHAPTER (5)
Control of Base Isolated Buildings Pounding
5.3.7 Effect of Structure Damping Ratio and the Stiffness Ratio of the Damper: Figure 5.13 shows the effect of damping ratio of the adjacent base isolated buildings and the stiffness ratio of the connected dampers on the total energy of the overall system. It is clear that the total energy decreases with the increase in the stiffness of the damper and with the increase in the damping coefficient of the structure as the same in the previous figures.
7
(a)
5
T.E *10 (t.m)
6.9 6.8 6.7 6.6 6.5 0.2 0.15
0.5
s
0.1
1 1.5
0.05
K dKR
2 (b) 2
K
R
6.5 24 1
1.2 6.58 1
1
1
6.6 09 6
0.6
6. 58 1
0.2 6.95 155 76.98 008 0.02 0.04
6.6 38 1 6. 666 6 6.695 1 6.723 6 6.752 6.7805 6.80 9 6.83 75 6.866 6.8945 6.923 0.06
6.5 52 6
1
6. 60 9
0.4
6.5 24 1
26 6. 55
6. 5811
6
6. 60 9
66 951 985 0.08
0.1
6. 63 81 6.6666 6. 69 51 6.6.75 7236 2 6.78 05 6.809 6.8375 6.866 6.8945 6 923 0.12
41 6.52
0.8
6. 49 56
6.55 26
1.4
6.4 95 6
6. 52 41
6 52 6.5
1.6
26 55 6.
1.8
6.552 6
6
6.581 1
6 9515 0.14 0.16
6. 6381 6.6 66 6 6.69 51 6.7236 6.752 6.7805 6.809 6.8375 66.866 68945 923 0.18
0.2
s
Figure 5.13: Total energy versus damping ratio and stiffness of the damper (a) surface area, and (b) contour line.
- 139 -
CHAPTER (5)
Control of Base Isolated Buildings Pounding
5.3.8 The effect of optimum parameters on the reduction of the displacement: In order to check the feasibility of the parametric study that conducted to obtain the optimum parameters that reduce the system energy, these optimum parameters such as, Kd =20000 t/m, α = 0.0, and the yield force = 33.0 ton are used to check the reduction of the system response. Figure 5.14 shows the maximum displacement of the controlled and uncontrolled bounded base isolated buildings for the sliding, first and second story respectively versus a different harmonic excitation frequency. Figure 5.14a shows that the maximum displacement of the base of the base of building A (the flexible building) decreases from 100.0 m to 87.0 cm with decreasing ratio equals 13% and decreases in the base of building B (the stiff building) from 80.0cm to 37.0 cm with decreasing ratio equals 53%. Also in figures 5.14b, c it is shown that the viscous damper decrease the relative displacement of the two story of the two buildings. This results show that the effect of the viscous damper on the reduction of the pounding response take place on the different values of harmonic excitation frequency.
- 140 -
CHAPTER (5)
(a)
Control of Base Isolated Buildings Pounding
1 (base (base (base (base
Max disp.(m)
0.8
of of of of
A) A)c B) B)c
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
ex (rad/sec)
(b) 0.12
(first (first (first (first
Max rel. disp.(m)
0.1 0.08
of of of of
A) A)c B) B)c
0.06 0.04 0.02 0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
ex (rad/sec)
(c)
Max rel. disp.(m)
0.2 (second (second (second (second
0.15
of of of of
A) A)c B) B)c
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
ex (rad/sec)
Figure 5.14: The max. disp. Of the controlled and uncontrolled bounded base isolated buildings (a) the base, (b) the first floor and (c) the second floor.
- 141 -
CHAPTER (5)
Control of Base Isolated Buildings Pounding
5.4 Seismic Analysis of Adjacent Controlled Buildings: This section aims to study the effect of laying connected dampers on the behavior of the adjacent base isolated structures under earthquake excitations. Five models subjected to Elcentro, Loma Prita, Kobe and Northridge earthquakes have been made in order to study the passive control on the buildings shown in figure 5.14. The studies are performed for the adjacent base isolated buildings consisting stories building of the same floor elevations but different building heights (model 2 and 4), as shown in figure 5.14. Table 5.3 shows the peak displacements of the upper stories of the two adjacent models and the total energy of the overall system for the four earthquakes.
5.4.1 Five-Story Adjacent Buildings: Effects of damper stiffness on the dynamic response mitigation as well as the optimum value of damper stiffness are sought for the 5-story adjacent buildings with the same levels (model 3). The structural characteristics of the two buildings remain the same as discussed in the previous section. Table 5.4 presents the variations of the top floor peak displacements and the total energy, respectively. It is shown that when the dampers exists both displacements and forces are significantly reduced. Also shown that both the top floor displacement response for building A, building B and the total energy for Elcentro earthquake reduced by 39%, 16.7% and 23.5%, respectively.
5.4.2 Ten-Story Adjacent Buildings: It is seen that when the number of story levels increases, building pounding effect will be more important. So, as the previous part, 10-story two adjacent buildings will be discussed (model 5). The structural
- 142 -
CHAPTER (5)
Control of Base Isolated Buildings Pounding
characteristics of the two buildings remain the same as discussed in the previous section. Presented in table 5.5 are the variations of the top floor displacement response and the total energy transfer, respectively. It is seen that when the dampers exist, both displacement and force responses are significantly reduced. It is also seen that both the top floor displacement response for building A, B and the total energy for Elcentro earthquake reduced by 12.27%, 11.22% and 22.88%, respectively. The effects of other earthquakes are observed in table 5.5.
5.4.3 Other Adjacent Buildings: This section aims to check the effectiveness of joint dampers for the adjacent base isolated buildings of different heights (model 2, 4). The structural properties of the two 10-story buildings discussed in previous sections, which include the mass and the external damping coefficient, remain unchanged. Figure 5.14 shows the variations of top floor displacement response with damper for the two buildings having the different floor elevation. Tables 5.6 and 5.7 show that the efficiency of the passive damper increased when the two adjacent buildings have equal number of freedom where the reduced in total energy in case of model 2 which equals 10.3% and equals 12.1% in case of model 4. Finally, by observing in these tables it is indicate that by increasing the number of stories, the reduction ratio of the relative displacement and total energy increases. Also, when the two buildings have the same height, the reduction ratio of the total energy will be more efficient, because the lining of the dampers will be in all levels which increase the dissipated energy.
- 143 -
CHAPTER (5)
foundation
foundation
Model (1)
foundation
Model (2)
foundation
Control Of Base Isolated Buildings Pounding
foundation
foundation
Model (3)
foundation
foundation
Model (4)
foundation
Model (5)
Figure (5.15) Five models used in the discussion of the effect of the connected damper on the total energy and absolute displacement under seismic excitation.
- 144 -
foundation
CHAPTER (5)
Control Of Base Isolated Buildings Pounding
Table (5.3) Maximum response of the two buildings 2DOF coupled structures A and B without control and with a passive control system. Left building Right building Total energy * 10e3 (t. m) Earthquake
foundation
foundation
Controlled
uncontrolled
C.R**
controlled
uncontrolled
C.R**
Dips*. (m)
Dips*. (m)
%
Dips*. (m)
Dips*. (m)
%
- 4.00 + 13.3 +23.3 + 22.9
0.1104 0.1637 0.2309 0.4693
0.1234 0.1466 0.2680 0.6678
+ 10.5 -11.66 +13.84 +29.72
0.1349 0.1291 Elcentro 0.1733 0.2000 Loma Prita 0.2302 0.3002 Kobe 0.4701 0.6099 Northridge * Top absolute displacement.
controlled
uncontrolled
77.267 89.211 123.19 384.21
97.679 106.01 166.62 544.63
C.R** % +20.90 +15.46 +26.00 +29.50
** Reduction control
Table (5.4) Maximum response of the two buildings 5DOF coupled structures A and B without control and with a passive control system. Left building Right building Total energy * 10e3 (t. m) Earthquake
foundation
foundation
Controlled
uncontrolled
C.R**
controlled
uncontrolled
C.R**
Dips*. (m)
Dips*. (m)
%
Dips*. (m)
Dips*. (m)
%
+39.00 +11.40 +15.61 -6.700
0.1935 0.2109 0.2561 0.5462
0.2325 0.1871 0.2535 0.6084
+16.70 -12.70 -1.000 +10.22
0.1999 0.3279 Elcentro 0.2648 0.2990 Loma Prita 0.2665 0.3158 Kobe 0.6574 0.6162 Northridge * Top absolute displacement.
** Reduction control
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controlled
uncontrolled
231.23 224.01 306.69 779.94
302.18 256.22 344.00 864.49
C.R** % +23.5 +12.5 +11.0 +9.00
CHAPTER (5)
Control Of Base Isolated Buildings Pounding
Table (5.5) Maximum response of the two buildings 10DOF coupled structures A and B without control and with a passive control system. Left building Right building Total energy * 10e3 (t. m) Earthquake
foundation
Controlled
uncontrolled
C.R**
controlled
uncontrolled
C.R**
Dips*. (m)
Dips*. (m)
%
Dips*. (m)
Dips*. (m)
%
+12.27 -2.600 +8.400 -5.320
0.1954 0.2578 0.2915 0.6926
0.2201 0.2692 0.3169 0.7614
+11.22 +4.230 +8.070 +9.030
controlled
uncontrolled
434.63 566.76 520.46 127.23
563.59 601.77 539.01 138.43
C.R** %
foundation
0.2373 0.2705 Elcentro 0.4107 0.4002 Loma Prita 0.3166 0.3456 Kobe 0.7610 0.7225 Northridge * Top absolute displacement.
+22.88 +5.810 +3.500 +8.100
** Reduction control
Table (5.6) Maximum response of the two buildings 2DOF and 5DOF coupled structures A and B without control and with a passive control system. Left building Right building Total energy * 10e3 (t. m) Earthquake
foundation
foundation
Controlled
uncontrolled
C.R**
controlled
uncontrolled
C.R**
Dips*. (m)
Dips*. (m)
%
Dips*. (m)
Dips*. (m)
%
+8.110 +17.42 +34.61 +10.10
0.1822 0.2476 0.2913 0.5366
0.1840 0.2297 0.2351 0.6430
+1.00 -7.79 -23.9 +16.54
0.1393 0.1516 Elcentro 0.1754 0.2124 Loma Prita 0.2289 0.3501 Kobe 0.4720 0.5250 Northridge * Top absolute displacement.
** Reduction control
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controlled
uncontrolled
151.16 167.32 207.40 596.07
168.49 172.46 213.52 666.69
C.R** % +10.3 +3.00 +3.00 +10.6
CHAPTER (5)
Control Of Base Isolated Buildings Pounding
Table (5.7) Maximum response of the two buildings 5DOF and 10DOF coupled structures A and B without control and with a passive control system. Earthquake Left building Right building Total energy * 10e3 (t. m)
fo u n d a tio n
fo u n d a tio n
Controlled
uncontrolled
C.R**
controlled
uncontrolled
C.R**
Dips*. (m)
Dips*. (m)
%
Dips*. (m)
Dips*. (m)
%
+10.24 +18.29 +19.90 -5.720
0.2212 0.2601 0.3132 0.7216
0.2372 0.2424 0.4107 0.7753
+6.74 -7.30 +23.7 +6.90
0.1832 0.2041 Elcentro 0.2184 0.2673 Loma Prita 0.2544 0.3178 Kobe 0.6459 0.6109 Northridge * Top absolute displacement.
** Reduction control
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controlled
uncontrolled
343.76 325.99 399.90 927.30
391.33 348.92 426.75 964.38
C.R** % +12.1 +6.57 +6.29 +3.80
SUMMARY AND CONCLUSIONS
CHAPTER (6) SUMMARY AND CONCLUSIONS 6.1 Summary: This study concerns the study of the pounding problem in the base isolation technique in which the base of the structure is isolated against the ground motion by an isolator to reduce the energy transferred to the superstructure through sliding displacement in the base, that dissipate a part from the input energy from the earthquake. In this research, a MATLAB program is developed to solve the pounding problems in the fixed and base isolated structures which modeled as multi degree of freedom systems…etc, due to different dynamic load. The introduced program constructed to study the behavior of the buildings under dynamic loads when the clearance distances between the adjacent buildings not enough to prevent pounding. Building pounding can influence the response of the colliding buildings and can cause local damage at the locations where impact takes place. For base isolated building systems, pounding behavior can be favorably or unfavorably modified depending upon the nature of ground motion and sliding friction. Using the developed program, a parametric study is performed to investigate the effect of the important parameters on the pounding between the adjacent buildings. The parameters studied are the clearance distance, mass ratio between the two adjacent buildings, the post stiffness ratio in the isolator
CHAPTER 6
Summary and Conclusions
material, the isolator plastic shear capacity and the damping ratio for the base and the superstructures. Finally the effect of the passive control system developed in a new virgin of the program to study in the reduction of pounding problems. And studying the best parameters in the damper material is investigated, which minimized the three parts of energy transfer to the superstructures and happened from the pounding problems.
6.2 CONCLUSIONS: From the previous analysis, the most important conclusions can be summarized as the follow: 1.
Pounding increases the response of the lighter floors of the adjacent Base isolated buildings while decreases the response of the heavier floors.
2.
Pounding increases the response of the floors of the adjacent fixed base buildings.
3.
For the isolated buildings at the resonance region, the very small and very high isolator plastic shear capacity increases the response of the bounded and unbounded floors, while away the resonance region, the decrease of the isolator plastic shear capacity decrease the response of the adjacent bounded and unbounded floors.
4.
While small post-pre stiffness ratio decrease obviously the floors responses of the unbounded adjacent base isolated building, it makes small trend for bounded base isolated buildings especially for the lighter floors.
5.
When sufficient gap is allowed between the adjacent base isolated
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CHAPTER 6
Summary and Conclusions
buildings, the effect of the pounding in increasing their responses decreases. 6.
The response of the base isolated bounded and unbounded adjacent buildings when subjected to ground motion, depends mainly on the power of the earthquake and frequency content.
7.
When the floors of the two adjacent base isolated buildings are connected with viscous dampers, these damper have optimum yield force that minimize the overall energy of the system.
8.
The optimum yield force for the connected dampers increases for the stiff buildings rather than the flexible ones.
9.
Connected dampers with small post-pre stiffness ratio are more efficient than that with large values for all cases of dampers.
10.
The change of the system damping doesn’t change the values of the optimum yield force.
11.
After a specific value of connected dampers stiffness, the increase of the stiffness reduce system response.
6.3 RECOMMENDATIONS FOR THE FUTURE WORK: The present study could be complemented with additional research in the following areas: Experimental shake-table testing of scaled buildings models to study the effects of pounding. Dynamic testing will help in identifying the values of impact spring stiffness and coefficient of restitution to be used in analysis. The developed program may be extended to investigate other cases of pounding such that which occurred in the buildings with different story levels.
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APPENDIX
APPENDIX (A) % this program used to study the response of the adjacent base isolated buildings with considering pounding function [xx1, xx2, Yk1, Yk2, vv1, vv2,aa1,aa2 ,TEBIP, noofimpactsBIP]=PEBIP (n1, n2, mm1, mm2, kk1, kk2, zeta1, zeta2, mb1, mb2, kbe1, kbe2, kbp1, kbp2, fy1, fy2,zetabase1, zeta base2 , dt, tmax, wex, Ampl, e, gapsize) n3=min([n1 n2])+1; m1=diag(mm1); % STRUCTURAL MASS m2=diag(mm2); for i=1:n1 if (i1) k1(i,i-1)=-kk1(i); end if (i=xupper2 && stage2==1 stage2=2; kx2=kbp2; end if xx2(1,i)=xupper2 && stage2==3 stage2=2; kx2=kbp2; end if xx2(1,i)= gapsize; v1new(j) = (Mmod1(j,j)+Mmod2(j,j))^-1*(Mmod1(j,j)e*Mmod2(j,j))*vvv1(j) +(Mmod1(j,j)+Mmod2(j,j))^-1* ((1+e)*Mmod2(j,j))*vvv2(j); v2new(j) = (Mmod1(j,j)+Mmod2(j,j))^-1*(Mmod2(j,j)e*Mmod1(j,j))*vvv2(j) +(Mmod1(j,j)+Mmod2(j,j))^-1* ((1+e)*Mmod1(j,j))*vvv1(j); h=1; if j==1 v1(j)= v1new(j); v2(j)= v2new(j); else v1(j)= v1new(j)-v1(1); v2(j)= v2new(j)-v2(1); end x1(j)=(x2(j)-gapsize+x1(j))/2; x2(j)=x1(j)+gapsize; end end noofimpactsBIP=noofimpactsBIP+h; xx1=[xx1,x1]; xx2=[xx2,x2]; vv1=[vv1,v1]; vv2=[vv2,v2]; aa1=[aa1,a1]; aa2=[aa2,a2]; stage1D=[stage1D;stage1]; stage2D=[stage2D;stage2]; kkk1=[kkk1;kx1]; kkk2=[kkk2;kx2]; Yk1=[Yk1,Yk1(end)+kx1*(xx1(1,i+1)-xx1(1,i))]; Yk2=[Yk2,Yk2(end)+kx2*(xx2(1,i+1)-xx2(1,i))]; end kE1=[kbe1,zeros(1,n1);zeros(n1,1),diag(kk1)]; kE2=[kbe2,zeros(1,n2);zeros(n2,1),diag(kk2)]; cE11=[diag(c1)]; cE21=diag(c2); cE1=[cb1,zeros(1,n1);zeros(n1,1),diag(cE11)]; cE2=[cb1,zeros(1,n2);zeros(n2,1),diag(cE21)]; mE1=[mb1,zeros(1,n1);zeros(n1,1),diag(mm1)]; mE2=[mb2,zeros(1,n2);zeros(n2,1),diag(mm2)]; [TEBIP]=energycallBIP(xx1,xx2,vv1, vv2 ,kE1, kE2, mE1, mE2, cE1, cE2,dt);
- 154 -
APPENDIX
APPENDIX (B) % this program used to study the response of the adjacent base isolated buildings with considering the viscous damper lining. function [xx1, xx2, Yk1, Yk2,Yk, vv1, vv2,aa1,aa2, fcon,TSEBIP, TDEBIP, TKEBIP]=PEBIPcontrol (n1, n2, mm1, mm2, kk1, kk2, zeta1, zeta2, mb1, mb2, kbe1, kbe2, kbp1, kbp2, fy1, fy2,zetabase1, zetabase2 , dt, tmax, wex, Ampl,fcy,kce,kcp ) n3=min([n1 n2])+1; m1=diag(mm1); % STRUCTURAL MASS m2=diag(mm2); impactforce=zeros(tmax/dt,1); for i=1:n1 if (i1) k1(i,i-1)=-kk1(i); end if (i=xupper2 && stage2==1 stage2=2; kx2=kbp2; end if xx2(1,i)=xupper2 && stage2==3 stage2=2; kx2=kbp2; end if xx2(1,i)=xcup(j) && stage3(j)==1 stage3(j)=2; kc(j)=kcp(j); end if xr(j,i+1)=xcup(j) && stage3(j)==3 stage3(j)=2; kc(j)=kcp(j); end if xr(j,i+1)
