a study of pounding between adjacent structures

A STUDY OF POUNDING BETWEEN ADJACENT STRUCTURES

MS by Research Thesis submitted by Chenna Rajaram (200814001)

in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE (BY RESEARCH) in Computer Aided Structural Engineering under the guidance of Dr. Ramancharla Pradeep Kumar

Earthquake Engineering Research Centre, International Institute of Information Technology, Hyderabad-500 032 April 2011

CERTIFICATE

It is certified that the work contained in this thesis, titled ”A study of Pounding between Adjacent Structures” by Mr. Chenna Rajaram, has been carried out under my supervision and is not submitted elsewhere for a degree.

Advisor: Dr.Ramancharla Pradeep Kumar Associate Professor Earthquake Engineering Research Centre International Institute of Information Technology Hyderabad-500 032 INDIA

Dedicated to my mother Bharathi and father Rajagopal who have worked hard throughout my education and gave me the opportunity to do research.

Acknowledgements

I would like to express my sincere gratitude to my thesis supervisor Associate Prof. Dr. Pradeep Kumar Ramancharla, for the continuous support of my research, for his patience, motivation, enthusiasm, and immense knowledge. His guidance helped me alot in all the time of research and writing of this thesis. I would like to thank to him for teaching me various courses and Jeevan Vidya, which will be very useful for everyone. I would also like to thanks Prof. M. Venkateswarlu for teaching me Mathematical Foundations of Solid Mechanics and Theory of Elasticity and Assistant Prof. Dr. Neelima Satyam for her support. My sincere thanks goes to Prof. B.Venkat Reddy for his encouragement to do research. I would like to express sincere thanks to all my colleagues (CASE), EERC members (Admin Staff, Project Staff, Research Staff and office boy) and M.Tech Students which contributes directly and indirectly in various ways for my research. I would like to express sincere thanks to Prof. Wijeyewikrema (Tokyo Institute of Technology, JAPAN), Prof. Maekawa (Univ. of Tokyo, JAPAN), Prof. Jankowski (Gdansk Univ. of Technology, POLAND), Prof. Anagnostopoulos (Univ. of Patras, GREECE) and Prof. Kun Ye (Huazhong Univ. of science and Technology, CHINA) for sending their journal papers which are very useful for my literature review. Special thanks to NICEE (National Information Centre of Earthquake Engineering)for sending the codes which are very useful for my study. Finally, I would like to dedicate this work to my mother Chenna Bharathi and father Chenna Rajagopal, whose continuous love and support guided me. Without their encouragement and understanding it would have been impossible for me to finish this work. My special thanks goes to my brother Chenna Sai Krishna for their loving support.

Chenna Rajaram Earthquake Engineering Research Centre IIIT-H, Hyderabad INDIA

Abstract

Pounding between adjacent structures is commonly observed phenomenon during major earthquakes which may cause both architectural and structural damages. To satisfy the functional requirements, the adjacent buildings are constructed with equal and unequal heights, which may cause great damage to structures during earthquakes. To mitigate the amount of damage from pounding, the most simplest and effective way is to provide minimum separation distance. Generally most of the existing buildings in seismically moderate regions are built without codal provisions. Past earthquakes have shown an evidence that the buildings are more vulnerable to pounding. Building codes provide a set of guidelines for the practice of structural engineering and play an important role of transferring technology from research to practice. In numerical modeling, different combination of structures are considered for doing the analysis using Applied Element Method (AEM). The separation distance between the structures is provided according to various codes from different countries and are subjected to ten different ground motions. Some codal provisions failed to satisfy the requirements. The shortcomings in codal provisions are identified and provided with proper suggestions to them. To study the behavior of structures due to structural pounding, linear and nonlinear analysis are done for different structures subjected to ground motion. The analysis considers equal and unequal height of structures. The behavior of adjacent structures is similar as linear till failure of first spring or first collision. The displacement responses for flexible structures are less compared to stiff structures when structures vibrate at dominant period and also the responses for flexible structures are more when structures vibrate at non-dominant period. Also we estimate the amount of damage for structures in terms of stiffness degradation. For unequal height of structures, the interaction is between slab and column. During this interaction, shear causes more damage to the column which leads to collapse of structure. To study the torsional effects due to pounding, buildings with different setbacks and unequal storey levels are analyzed using SAP 2000. The effect of collision is more when structures are kept at extreme levels of setback. When the structures are kept at different elevation levels (setback=0), the pounding response changes significantly as the height of structure decreases. At mid height of structure, the collision force is more compared to other height levels because of shear amplification.

List of Publications 1. Chenna Rajaram and Ramancharla Pradeep Kumar: ”Three Dimensional Pounding Analysis between Two Structures”, Journal of Structural Engineering SERC-Chennai, Vol.**, No.** (Under Review). 2. Chenna Rajaram and Ramancharla Pradeep Kumar: ”Pounding Analysis of Adjacent Structures with Equal and Unequal Heights”, Earthquake Spectra, Vol.**, No.** (Under Review). 3. Chenna Rajaram and Ramancharla Pradeep Kumar., ”Linear and Nonlinear Numerical Analysis of Pounding Between Adjacent Buildings”, 8th International conference on Earthquake Resistant Engineering Structures, Chianciano Terme, Italy (Abstract accepted). 4. Chenna Rajaram and Ramancharla Pradeep Kumar., ”Numerical Modeling of Pounding Between Adjacent Buildings: Some Corrections to Codal Provisions”, Earthquake Engineering and Engineering Vibrations, Vol.**, No.** (submitted). 5. Chenna Rajaram and Ramancharla Pradeep Kumar., ”Comparison of codal provisions on pounding between adjacent buildings”, International Journal of Earth science and Engineering (IJEE), Vol**, No**, (Under Review). 6. Chenna Rajaram, Bodige Narender, Neelima Satyam and Ramancharla Pradeep Kumar., ”Preliminary Seismic Hazard Map of Peninsular India”, Proc. 14th Symposium on Earthquake Engineering, IIT Roorkee 2010, pp 497-491. 7. Chenna Rajaram and Ramancharla Pradeep Kumar., ”Comparison of codal provisions on pounding between adjacent buildings: Some Corrections to Codal Provisions”, Indian Concrete Journal (ICJ), Vol**, No**, (Under Review).

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Contents List of Publications

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1 Introduction and Literature Review 1.1 Introduction . . . . . . . . . . . . . 1.2 Causes of pounding . . . . . . . . . 1.3 Case studies . . . . . . . . . . . . . 1.3.1 Worldwide observations . . 1.3.2 Indian observations . . . . . 1.4 Literature review . . . . . . . . . . 1.4.1 Analytical studies . . . . . . 1.4.2 Experimental studies . . . . 1.4.3 Numerical studies . . . . . . 1.5 Summary of contribution . . . . . . 1.5.1 Mitigation measures . . . . 1.5.2 Impact models . . . . . . . 1.5.3 Codal provisions . . . . . . 1.6 Objective of the study . . . . . . . 1.7 Organization of thesis . . . . . . . .

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2 Numerical Modeling of Pounding 2.1 Introduction . . . . . . . . . . . . . . . . 2.2 Selection of buildings . . . . . . . . . . . 2.2.1 Building geometry . . . . . . . . 2.2.2 Material properties . . . . . . . . 2.3 Selection of ground motions . . . . . . . 2.3.1 Characteristics of ground motions 2.4 Numerical method . . . . . . . . . . . . 2.4.1 Introduction . . . . . . . . . . . . 2.4.2 Mathematical formulation . . . . 2.4.3 Element size . . . . . . . . . . . . 2.4.4 Material model . . . . . . . . . . 2.4.5 Collision model . . . . . . . . . . 2.4.6 Failure criteria . . . . . . . . . .

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CONTENTS

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2.4.7 Limitations . . . . . . . . . . . Linear pounding analysis of structures 2.5.1 Structures with equal heights . 2.5.2 Structures with unequal heights Summary . . . . . . . . . . . . . . . .

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3 Pounding Analysis With Equal Heights 3.1 Introduction . . . . . . . . . . . . . . . . 3.2 Non-linear analysis of pounding . . . . . 3.3 Damage analysis . . . . . . . . . . . . . 3.3.1 Damage model . . . . . . . . . . 3.3.2 Damage calculation . . . . . . . . 3.3.3 Proposed damage scale . . . . . . 3.4 Summary . . . . . . . . . . . . . . . . .

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4 Pounding Analysis With Un-equal Heights 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Pounding analysis for structures with un-equal heights 4.2.1 Single-single storey structural pounding . . . . . 4.2.2 Single-two storey structural pounding . . . . . . 4.2.3 Two-two storey structural pounding . . . . . . . 4.2.4 Two-three storey structural pounding . . . . . . 4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 5 3D Analysis of Pounding 5.1 Introduction . . . . . . . . . . . . . . . 5.2 Modeling of structures in 3D . . . . . . 5.2.1 Geometry and material details . 5.2.2 Gap element model . . . . . . . 5.2.3 Non-linear analysis of pounding 5.3 Summary . . . . . . . . . . . . . . . .

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6 Conclusions 104 6.1 General Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 A Comparision of codal provisions on pounding A.1 Review on codal provisions . . . . . . . . . . . A.2 Minimum separation between buildings . . . . A.3 Case study . . . . . . . . . . . . . . . . . . . . A.4 Conclusions . . . . . . . . . . . . . . . . . . .

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B Calculation of separation distance from codes 115 B.1 Calculation of separation distance from codes . . . . . . . . . . . . . . . . 115 v

CONTENTS

Bibliography

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List of Figures 1.1 1.2

1.3 1.4 1.5 1.6 1.7 1.8

Representation of different places where pounding occurs . . . . . . . . . . 2 Pounding damage of Olive View hospital. (a) View of Olive View hospital (b)Permanent tilting of a stairway tower during San Fernando earthquake, 1971 (Courtesy: EERC, University of California, Berkeley). . . . . . . . . . 3 Pounding damage due to insufficient separation distance during 1999 ChiChi earthquake, Taiwan . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Pounding damage due to unequal slab levels during 2007 Niigata ChuetsuOki Japan earthquake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Significant pounding was observed Santa Clara River Bridge during Northridge earthquake. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Pounding damage at the intersection of approach to main jetty at Diglipur harbor during Diglipur Earthquake . . . . . . . . . . . . . . . . . . . . . . 5 Linear Spring Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Kelvin Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1 2.2 2.3 2.4 2.5 2.6 2.7

Geometry of single storey structure . . . . . . . . . . . . . . . . . . . . . . Geometry of two storey structure . . . . . . . . . . . . . . . . . . . . . . . Geometry of three storey structure . . . . . . . . . . . . . . . . . . . . . . Geometry of five storey structure . . . . . . . . . . . . . . . . . . . . . . . Reinforcement details of single and two storey structures . . . . . . . . . . Reinforcement details of three and five storey structures . . . . . . . . . . Athens ground motion record and its fourier spectrum amplitude (a) Ground motion record (b) Fourier amplitude spectrum . . . . . . . . . . . . . . . 2.8 Athens(tran) ground motion record and its fourier spectrum amplitude (a) Ground motion record (b) Fourier amplitude spectrum . . . . . . . . . . . 2.9 Ionian ground motion record and its fourier spectrum amplitude (a) Ground motion record (b) Fourier amplitude spectrum . . . . . . . . . . . . . . . 2.10 Kalamata ground motion record and its fourier spectrum amplitude (a) Ground motion record (b) Fourier amplitude spectrum . . . . . . . . . . . 2.11 Umbro ground motion record and its fourier spectrum amplitude (a) Ground motion record (b) Fourier amplitude spectrum . . . . . . . . . . . . . . . 2.12 Elcentro ground motion record and its fourier spectrum amplitude (a) Ground motion record (b) Fourier amplitude spectrum . . . . . . . . . . .

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22 23 24 25 26 26 27 27 27 28 28 29

LIST OF FIGURES

2.13 Olympia ground motion record and its fourier spectrum amplitude(a) Ground motion record (b) Fourier amplitude spectrum . . . . . . . . . . . . . . . 2.14 Parkfield ground motion record and its fourier spectrum amplitude(a) Ground motion record (b) Fourier amplitude spectrum . . . . . . . . . . . . . . . 2.15 Northridge ground motion record and its fourier spectrum amplitude(a) Ground motion record (b) Fourier amplitude spectrum . . . . . . . . . . . 2.16 Lomaprieta ground motion record and its fourier spectrum amplitude(a) Ground motion record (b) Fourier amplitude spectrum . . . . . . . . . . . 2.17 Element components for formulating stiffness matrix (SOURCE: Kimuro Meguro and Hatem, 2001) . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.18 Quarter portion of stiffness matrix . . . . . . . . . . . . . . . . . . . . . . 2.19 Uttarkasi ground motion record and its fourier spectrum amplitude(a) Ground motion record (b) Fourier amplitude spectrum . . . . . . . . . . . 2.20 Material models for concrete and steel (a) Tension and compression concrete Maekawa model (b) Bi-linear stress strain relation model for steel reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.21 Arrangement of collision springs . . . . . . . . . . . . . . . . . . . . . . . . 2.22 (a) Principal Stress determination and (b) Redistribution of spring forces at element edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.23 Linear pounding response of two single storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.0 m) . . . . . . 2.24 Pounding force between two single storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.0 m) . . . . . . . . 2.25 Linear pounding response of single-two storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.0 m) . . . . . . 2.26 Pounding force between single-two storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.0 m) . . . . . . . . 2.27 Linear pounding response of two-two storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.0 m) . . . . . . . . 2.28 Pounding force between two-two storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.0 m) . . . . . . . . . . 2.29 Linear pounding response of two-three storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.0 m) . . . . . . . . 2.30 Pounding force between two-three storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.0 m) . . . . . . . . 2.31 Geometry details of structure-B @2.75 m slab levels in the single storey structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.32 Geometry details of structure-B @3.25 m slab levels in the single storey structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.33 Geometry details of structure-B @3.5 m slab levels in the single storey structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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29 29 30 30 32 32 34

34 35 37 40 40 41 42 43 43 44 44 46 46 47

LIST OF FIGURES

2.34 Geometry details of structure-B @2.75 m slab levels in the two storey structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.35 Geometry details of structure-B @3.25 m slab levels in the two storey structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.36 Geometry details of structure-B @3.5 m slab levels in the two storey structure 2.37 Geometry details of structure-B @2.75 m slab levels in the three storey structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.38 Geometry details of structure-B @3.25 m slab levels in the three storey structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.39 Geometry details of structure-B @3.5 m slab levels in the three storey structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.40 Linear pounding response of two single storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 2.75 m) . . . . . 2.41 Pounding force between two single storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 2.75 m) . . . . . . . . 2.42 Linear pounding response of two single storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.25 m) . . . . . 2.43 Pounding force between two single storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.25 m) . . . . . . . . 2.44 Linear pounding response of two single storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.5 m) . . . . . . 2.45 Pounding force between two single storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.5 m) . . . . . . . . Pounding nonlinear response of two single storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.0 m) . . . . 3.2 Pounding force between two single storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.0 m) . . . . . . . 3.3 Pounding nonlinear response of single-two storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.0 m) . . . . 3.4 Pounding force between single-two storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.0 m) . . . . . . . 3.5 Pounding nonlinear response of two-two storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.0 m) . . . . . 3.6 Pounding force between two-two storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.0 m) . . . . . . . . . 3.7 Pounding nonlinear response of two-three storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.0 m) . . . . . 3.8 Pounding force between two-three storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.0 m) . . . . . . . 3.9 Load vs displacement curve for structure having a period of 0.127 sec . . 3.10 Load vs displacement curve for structure having a period of 0.155 sec . .

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LIST OF FIGURES

3.11 Load vs displacement curve for structure having a period of 0.253 sec . . . 67 3.12 Load vs displacement curve for structure having a period of 0.304 sec . . . 67 3.13 Load vs displacement curve for structure having a period of 0.334 sec . . . 68 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 4.15 4.16 4.17 4.18 4.19

Pounding nonlinear response of two single storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 2.75 m) . . . Pounding force between two single storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 2.75 m) . . . . . . . Pounding nonlinear response of two single storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.25 m) . . . Pounding force between two single storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.25 m) . . . . . . . Pounding nonlinear response of two single storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.5 m) . . . . Pounding force between two single storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.5 m) . . . . . . . Pounding nonlinear response of single-two storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 2.75 m) . . . Pounding force between single-two storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 2.75 m) . . . . . . . Pounding nonlinear response of single-two storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.25 m) . . . Pounding force between single-two storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.25 m) . . . . . . . Pounding nonlinear response of single-two storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.5 m) . . . . Pounding force between single-two storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.5 m) . . . . . . . Pounding nonlinear response of two-two storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 2.75 m) . . . . Pounding force between two-two storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 2.75 m) . . . . . . . . . Pounding nonlinear response of two-two storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.25 m) . . . . Pounding force between two-two storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.25 m) . . . . . . . . . Pounding nonlinear response of two-two storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.5 m) . . . . . Pounding force between two-two storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.5 m) . . . . . . . . . Pounding nonlinear response of two-three storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 2.75 m) . . . .

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. 70 . 71 . 72 . 72 . 73 . 73 . 74 . 75 . 76 . 76 . 77 . 78 . 79 . 79 . 80 . 81 . 81 . 82 . 83

LIST OF FIGURES

4.20 Pounding force between two-three storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 2.75 m) . . . . . . . 4.21 Pounding nonlinear response of two-three storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.25 m) . . . . 4.22 Pounding force between two-three storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.25 m) . . . . . . . 4.23 Pounding nonlinear response of two-three storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.5 m) . . . . . 4.24 Pounding force between two-three storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.5 m) . . . . . . . 5.1 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 5.16 5.17 5.18 5.19 5.20 5.21 5.22 5.23

Geometry details of structures . . . . . . . . . . . . . . . . . . . . . . . . Gap-joint element from SAP 2000 . . . . . . . . . . . . . . . . . . . . . . Link element internal forces and moments at the joints . . . . . . . . . . Response of structures in x-direction at location Ct with setback of 1.5 m Pounding force between structures with setback of 1.5 m . . . . . . . . . Response of structures in y-direction at location Ct with setback of 1.5 m Response of structures in x-direction at location Cb with setback of 1.5 m Response of structures in y-direction at location Cb with setback of 1.5 m Response of structures in x-direction at location Ct with setback of 3.0 m Pounding force between structures with setback of 3.0 m . . . . . . . . . Response of structures in y-direction at location Ct with setback of 3.0 m Response of structures in x-direction at location Cb with setback of 3.0 m Response of structures in x-direction at location Ct with setback of 6.0 m Pounding force between structures with setback of 6.0 m . . . . . . . . . Response of structures in y-direction at location Ct with setback of 6.0 m Response of structures in x-direction at location Ct with height of 2.25 m Pounding force between structures with height of 2.25 m . . . . . . . . . Response of structures in x-direction at location Cb with height of 2.25 m Pounding force between structures with height of 2.25 m . . . . . . . . . Response of structures in x-direction at location Ct with height of 1.5 m . Pounding force between structures with height of 1.5 m . . . . . . . . . . Response of structures in x-direction at location Cb with height of 1.5 m Pounding force between structures with height of 1.5 m . . . . . . . . . .

. 83 . 84 . 85 . 86 . 86 . . . . . . . . . . . . . . . . . . . . . . .

89 90 90 92 92 93 93 94 95 95 96 96 97 97 98 99 99 100 100 101 102 102 103

A.1 Idealized model of SDOF system . . . . . . . . . . . . . . . . . . . . . . . 107 A.2 Minimum space provided between two structures having different dynamic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

xi

List of Tables 1.1

List of codal provisions on pounding . . . . . . . . . . . . . . . . . . . . . 20

2.1 2.2 2.3

Details of ground motions . . . . . . . . . . . . . . . . . . . . . . . . . . Fundamental period of the structures . . . . . . . . . . . . . . . . . . . . Maximum displacement response of structures, pounding forces and number of collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fundamental period of the structures at different slab levels . . . . . . . Maximum displacement response of structures, pounding forces and number of collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4 2.5

3.1 3.2 4.1

. 26 . 39 . 45 . 49 . 55

Maximum nonlinear displacement response of structures, pounding forces and number of collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Stiffness degradation for different structures during pounding . . . . . . . . 65 Maximum nonlinear displacement response of structures, pounding forces and number of collisions for structures with unequal heights . . . . . . . . 87

A.1 Details of codal provisions on pounding . . . . . . . . . . . . . . . . . . . . A.2 Details about mass, stiffness and spacing provided between two structures . A.3 Lomaprieta ground motion record having amplitude of 0.22 g, duration 9.58 sec and predominant time period 0.41-1.61 sec . . . . . . . . . . . . . A.4 Elcentro ground motion (S00E) record having amplitude of 0.348 g, duration 24.44 sec and predominant time period ranges from 0.45-0.87 sec . . . A.5 Parkfield ground motion record having amplitude of 0.430 g, duration 6.76 sec and predominant time period 0.3-1.20 sec . . . . . . . . . . . . . . . . . A.6 Petrolia ground motion record having amplitude of 0.662 g, duration 48.74 sec and predominant time period 0.50-0.83 sec . . . . . . . . . . . . . . . . A.7 Northridge ground motion record having amplitude of 0.883 g, duration 8.94 sec and predominant time period ranges from 0.2-2.2 sec . . . . . . . .

107 108 110 111 112 112 113

B.1 Separation distances from codes . . . . . . . . . . . . . . . . . . . . . . . . 118 B.2 Status on separation distance from codes for single-single storey structures in group-A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 B.3 Status on separation distance from codes for single-two storey structures in group-A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 xii

LIST OF TABLES

B.4 Status on separation distance from codes for two-two storey structures in group-A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 B.5 Status on separation distance from codes for three-three storey structures in group-B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

xiii

Chapter 1 Introduction and Literature Review 1.1

Introduction

Structures are built very close to each other in metropolitan areas where the cost of land is very high. Due to closeness of the structures, they collide with each other when subjected to earthquake or any vibration. This collision of buildings or different parts of the building during any vibration is called pounding which may cause either architectural and structural damage or collapse of the whole structure. This may happen not only in buildings but also in bridges and towers which are constructed close to each other. Although some modern codes have included seismic separation requirement for adjacent structures, large areas of cities in seismically active regions were built before such requirements were introduced. Many investigations have been carried out on pounding damage caused by previous earthquakes.

1.2

Causes of pounding

Structural pounding damage in structures can arise from the following: (1) Adjacent buildings with the same heights and the same floor levels (fig 1.1a). (2) Adjacent buildings with the same floor levels but with different heights (fig 1.1b). (3) Adjacent structures with different total height and with different floor levels (fig 1.1c). (4) Structures are situated in a row (fig 1.1d). (5) Adjacent units of the same buildings which are connected by one or more bridges or through expansion joints. (6) Structures having different dynamic characteristics, which are separated by a distance small enough so that pounding can occur. (7) Pounding occurred at the unsupported part (e.g., mid-height) of column or wall resulting in severe pounding damage. (8) The majority of buildings were constructed according to the earlier code that was vague on separation distance. (9) Possible settlement and rocking of the structures located on soft soils lead to large lateral deflections which results in pounding. (10) Buildings having irregular lateral load resisting systems in plan rotate during an earthquake, and due to the torsional rotations, pounding occurs near the building periphery against the adjacent buildings (fig 1.1e).

1

CHAPTER 1. INTRODUCTION AND LITERATURE REVIEW

Figure 1.1: Representation of different places where pounding occurs

1.3

Case studies

During past earthquakes, structural pounding was noticed in buildings and bridges. Some of the cases observed on pounding are listed below:

1.3.1

Worldwide observations

In the Alaska earthquake of 1964, the tower of Anchorage westward hotel was damaged by pounding with an adjoining three storey ballroom portion of the hotel (Pantelides et al., 1998). In Sanfernando earthquake (Jankowski et al., 2009) of 1971, the second storey of the Olive View hospital struck the outside stairway; in addition, the first floor of the hospital was hit against a neighboring warehouse. The pounding of the main building against the stair way tower during the earthquake evoked considerable damage at the contact points and caused permanent tilting of the tower (fig 1.2). During Mexico City earthquake (Aguilar et al., 1989) on 19th September 1985, more than 20% of the 114 affected structures were damaged because of pounding. Among them more than 10% were due to failure of concrete frames and 3% were due to failure of concrete walls and frames. During Loma Prieta earthquake (Kasai et al., 1997) in 1989, significant pounding was observed at sites over 90 km from the epicenter. Many old buildings constructed prior to 1930 suffered. The typical floor mass of the five-storey building is about eight times that of the ten-storey building. Pounding was located at the 6th level in the ten-storey building and at the roof level in the five-storey building because of less separation distance of 1.0 to 1.5 inches was present. The ten storey building suffered structural pounding damage result large diagonal shear cracks in the masonry piers. In 1999, the chi-chi earthquake (Lin et al., 2002) in central Taiwan, structural pounding were also observed during the

2


earthquake. Having constructed at different times, the old and the new classrooms could be different in height, weight or stiffness. Thus, the two structures may possess different fundamental vibration periods. In Taiwan, a large number of old school buildings had been constructed and later expanded. The old and new classrooms may not have vibrate in-phase during the Taiwan earthquake (Chung et al., 2007). The class rooms may have pounded with each other because of lack of sufficient space between them (fig 1.3). During 2007 Niigata Chuetsu-Oki Japan Earthquake (Global risk Miyamoto, 2007) one type of observed damage in school buildings was the pounding of buildings against adjacent units. This type of damage occurred when adjacent structures had floor slabs located at different elevations and insufficient separation distance between them (fig 1.4). Pounding damages were also observed in recent massive Wenchuan earthquake on May 12, 2008.

(a)

(b)

Figure 1.2: Pounding damage of Olive View hospital. (a) View of Olive View hospital (b)Permanent tilting of a stairway tower during San Fernando earthquake, 1971 (Courtesy: EERC, University of California, Berkeley). The 4-in wide seismic joint used to separate both structures was not sufficient to accommodate the actual relative displacements that were developed during the ground motion. In 1994 Northridge earthquake (Pantelides et al., 1998) at the interstate 5 and state road 14 interchange, which was located approximately 12 km from the epicenter, significant pounding damage was observed at expansion hinges of the Santa Clara River Bridge (fig 1.5). Poundings between adjacent decks or between a deck and an abutment occurred in 1995 Hyogo-ken Nanbu earthquake (Kawashima et al., 2000). Pounding bring damage at not only expansion joints and contact faces of decks but also other elastomeric bearings and column.

1.3.2

Indian observations

The observations on pounding from the Indian earthquakes are as follows: The powerful 2001 Bhuj earthquake (Jain et al., 2001) has been the most damaging earthquake in the last five decades in India. Reinforced concrete buildings suffered the 3


Figure 1.3: Pounding damage due to insufficient separation distance during 1999 Chi-Chi earthquake, Taiwan

Figure 1.4: Pounding damage due to unequal slab levels during 2007 Niigata Chuetsu-Oki Japan earthquake heaviest damage during the earthquake because of poor design and construction practices. Pounding of adjacent structures was evident at Ayodhya apartments in Ahmedabad with significant damages. The Sikkim earthquake (Kaushik et al., 2006) on 14th February 2006 of 5.3 magnitude caused damage to a nine storey masonry infill RC frame hostel building at Sikkim Manipal Institute of Medical Sciences (SMIMS) Tadong, Gangtok which caused severe damages in walls and columns. Pounding damages were observed between two long wings in the building and corridors connecting the wings.

4


(a)

(b)

Figure 1.5: Significant pounding was observed Santa Clara River Bridge during Northridge earthquake.

The only road link between Kutch and Saurashtra areas is the road bridge at Surajbadi, which was damaged. Pre-stressed concrete girder bridge spans sustained substantial damage like pounding of the deck slab, horizontal movement of girder, and damage at the bottom of girders(Mistry et al., 2001). In Diglipur(Rai et al., 2003) harbor pounding damage was observed at the intersection of the approach segment and the main berthing structure (fig 1.6). During Sumatra earthquake (Rai et al., 2005) of 26th December 2004, pounding damage at junctions was noticed at the same top ends of piles of the approach jetty, which were covered up.

Figure 1.6: Pounding damage at the intersection of approach to main jetty at Diglipur harbor during Diglipur Earthquake From above observations, it is concluded that major pounding damages are caused due to insufficient separation distance. Hence, there is a need to do research on sepa5


ration distance between adjacent structures and pounding behavior of structures during earthquakes.

1.4

Literature review

Impact between adjacent structures during an earthquake is a phenomenon that has attracted considerable research interest in the recent past. Pounding is a non-linear problem due to its impact. Most of the numerical studies utilize single-degree-of-freedom (SDOF) and multi-degree-of-freedom (MDOF) systems in order to simplify the problem and concentrate on the non-linear aspects. The studies presented for literature review are categorized as: X Analytical studies X Experimental studies X Numerical studies

1.4.1

Analytical studies

Lin (1997) analyzed the uncertainty of the separation distance required to avoid seismic pounding of two adjacent buildings. The analytical procedures were based on random vibrations. The results indicated that the theoretical results agree well with simulated results. A larger separation distance is required for both adjacent buildings having a longer fundamental period. Lin (2002) investigated the pounding probability of buildings designed according to 1997 Taiwan building code (TBC) to gain an insight into the validity of the pounding related provisions. A total of 36 cases of adjacent buildings A and B are considered. The conditional probabilities of adjacent buildings separated by minimum code-specified separation distance under earthquakes with different peak ground acceleration (PGA) are investigated under 1000 artificial earthquakes. From the results it was revealed, the building separation specified by TBC is approximately 1.6 times that specified by Uniform building code (UBC) for the same building and soil properties. The probability of exceeding the design basis ground motion specified in UBC-94 during 50 year period is 10%. Garcia (2004) proposed a new method to calculate critical separation distances between adjacent nonlinear hysteretic structures. A pair of single degree of freedom (SDOF) systems was considered and the mean peak displacement responses were obtained through numerical simulations (nonlinear time history analysis). He examined the correlation co6


efficient ρ, which was obtained for calculating the separation distance proposed by Filiatrault, Penzien, Kasai and Valles. Results were expressed in terms of the ratio minimum separation to peak relative displacement response. Results showed that the proposed method provides consistently conservative estimates of critical separation distances, the degree of conservatism being slight in most cases. When compared to other existing methods, the proposed approach exhibited a number of convenient advantages. The main disadvantage was the fact that the proposed values of ρ are available only in charts. Ye et al., (2008) reexamined the derivation of the formula for damping constant ζ in Hertz damp contact model. They found that the derivation is based on the following assumptions: (1) The energy dissipated during impact is small compared with maximum absorbed elastic energy. (2) The penetration velocities during the compression and restitution are equal. In order to remedy the Hertz damp model valid for pounding analysis in structural engineering, the corrected expression for the damping constant ζ should be derived again. Through numerical analysis, the correctness of formula and its corresponding theoretical derivation has been verified. More reliable results of pounding simulation in structural engineering can be provided by using the Hertz damp model with the corrected formula for damping constant. It is observed from the past analytical studies on pounding, that they assure the dynamic response of a building can be well simulated by using lumped mass structural system and the excitation can be considered as a non-stationary Gaussian random process with zero mean. Torsional effects on structural responses are ignored. It is also assumed that floor elevations are same for all buildings so that pounding occurs only at those elevations where the masses are lumped. It has been observed that the adjacent buildings may be constructed with different materials and exhibit different hysteretic behavior(Jankowski, 2009). To better simulate the actual pounding probability of adjacent buildings with different materials, the use of different hysteretic loops for each building will be necessary.

1.4.2

Experimental studies

Papadrakakis et al., (1995) performed shaking table experiments on pounding between two-storey reinforced concrete buildings with zero gap separation, subject to sinusoidal excitation. The test structures were designed to remain elastic under an excitation with an acceleration design spectrum of 1.0 g. A shaking table test was conducted with a ramped sinusoidal displacement signal having a peak displacement of 0.13 cm and at resonance with the fundamental frequency(f) of the flexible structure (f=4.1 Hz). Both pounding and no-pounding cases were studied. The results indicated that pounding amplified the displacement responses of the stiffer structure and reduced the responses of the flexible structure. Increase in accelerations peaks up to six was recorded during impact but they take place within a very short-time duration. Comparison of the experimental results

7


with analytical predictions using the Lagrange multiplier method showed good agreement. Chau et al., (2003) performed shake table tests on pounding between two steel towers subject to both harmonic and El-Centro ground motions. The natural frequency, damping, the stand-off distance between the towers and the forcing frequency were varied during the experiment. Under sinusoidal excitations, impacts were either periodic (one impact within each excitation cycle or within every other excitation cycle) or chaotic. Chaotic motions dominated when there was a large difference in the natural frequencies of the two towers. It was observed that pounding amplified the response of the stiffer structure and reduced the flexible tower response. The maximum relative impact velocity was found to occur at an excitation frequency between the natural frequencies of the two towers. The experimental findings were then compared with results from an analytical model where impact was modeled using the Hertz contact law [Chau and Wei, 2000]. The region of excitation frequency within which impact occurred was well predicted by the analytical model. The estimated relative impact velocity and the maximum stand-off distance to prevent pounding agreed qualitatively with the experiments.

1.4.3

Numerical studies

The numerical studies can be categorized into SDOF and MDOF cases, Studies on SDOF structures Anagnostopoulos (1988) studied the case of several adjacent buildings in a row subjected to pounding. Pounding is simulated using linear visco-elastic impact elements that are introduced between the masses and act only when the masses are in contact. Elastic and inelastic systems have been examined using a set of five real earthquake motions and a wide variation of the problem parameters. Results are given in terms of displacement amplifications for exterior and interior structures. The results indicate that the displacement of exterior structures may be considerably amplified, while interior structures may experience amplification or deamplification, depending on the ratio of structural periods. A gap size equal to square root of sum of squares(SRSS) of the design peak displacements of the adjacent structures could be sufficient to avoid pounding. The results show that the effects of pounding diminish as the gap increases. Larger differences in the masses of two adjacent structures make the effect of pounding more pronounced for the structure with the smaller mass. Other parameters, like the stiffness of the contact element, play a minor role in the response. Impact generated accelerations which can cause damage to the contents of the building but have a little effect on the displacement response of the colliding masses. Davis (1992) has used a SDOF oscillator interacting with either a stationary or moving neighboring barrier. Impact forces are described by non-linear Hertz law of contact and results are given in the form of impact velocity spectra for harmonic excitation. These 8


spectra are characterized by a strong peak near a period equal to one half the natural period of a similar non-impacting oscillator. In effect, the impact oscillator has a natural period equal to roughly half the value it would have had if the neighboring structure had not been present. Athanassiadou et al., (1994) studied the seismic response of adjacent series of structures with similar and different dynamic characteristics subjected to five different seismic excitations. The inelastic load displacement relation for the structures followed Clough hysteresis model. From the results it was concluded that the effect of pounding from adjacent buildings on the seismic behavior of a structure is more pronounced for the end structures in a row. The effect of pounding is smaller when the adjacent structures have similar dynamic characteristics with equal heights. In case of adjacent structures with different natural periods, the most affected by pounding are rigid ones, irrespective of their relative position in the row. The most adverse case is when the rigid structures are placed at the end of the row. It was found that the seismic response of adjacent buildings is not considerably affected by their strength, coefficient of restitution and relative mass size. Pantelides et al., (1998) considered poundings between a damped SDOF structure with either elastic and inelastic structural behavior and a rigid barrier. The pounding phenomenon is modeled as a Hertz impact force. Artificial as well as actual earthquake excitations were used in numerical evaluations of the seismic response. The response of the inelastic structural system is compared to that of an elastic structure. The inelastic structure has considerably smaller accelerations as compared to the elastic structure and the maximum displacement of the inelastic structure is larger than that of the elastic structure. Moreover, the maximum pounding force and number of pounding occurrences are considerably less in the inelastic case as compared to the elastic case. The inelastic behavior of structures under pounding is less conservative than the elastic behavior assumption. This could be one of the explanations of why in general buildings experiencing pounding have shown satisfactory response in past earthquakes. Muthukumar et al., (2004) examined the effectiveness of various analytical impact models for two closely spaced SDOF adjacent structures. Only elastic responses are considered in the analysis. A suite of twenty seven ground motion records from thirteen different earthquakes was selected in parametric study. The records are chosen such that the ground motion period ratio (T2 /Tg = flexible system period over the ground motion characteristic period) is less than one. To examine the effects of energy loss during impact, two values of the coefficient of restitution(e) are chosen, e = 1.0 (no energy loss) and e = 0.6 (some energy loss). The contact force-based models predict higher accelerations due to pounding. The acceleration responses from the stereo-mechanical model are much smaller than those from the contact models. The displacement amplifications get closer to unity, as the system gets more in-phase. Neglecting energy loss during impact overestimates the stiff system displacement, when subjected to high levels of PGA. Pounding models that 9


account for energy loss during impact are best suited to simulate pounding. The Hertzdamp model appears to be an effective contact based approach, as it can model energy loss. Jankowski (2006) proposed 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. The structures had modeled as SDOF systems and pounding had simulated by a nonlinear visco elastic model. The structural natural periods T1 , T2 had ranged from 0.05s to 3s with an increment of 0.05s under different ground motion records. The results are shown as pounding force spectra. In a pounding force spectra, where the pounding force is becomes equal to zero region concerns the cases when the natural vibration periods are very small for both structures. When the damping ratios of two structures are different, pounding force for the cases of identical natural vibration periods is not always equal to zero. It is indicated that peak impact force values depends on the consideration of which of the structures is elastic and which is the inelastic one. The results indicated that impact force response spectra might serve as a very helpful tool for the design purposes of closely spaced adjacent structures. It is observed from the past analytical studies on pounding of SDOF structures, the local effects such as damage of a column being pounded by a slab of adjacent building was not considered whereas, the effects of pounding on the overall structural response was concerned. It is assumed that all systems are subjected to same input ground motion i.e. the effects of phase difference due to travelling waves are not considered. It is assumed that the various seismic waves which compose the complex motion represented by an accelerogram propagate with the same average velocity. Studies on MDOF structures Westermo (1989) examined the dynamic implications of connecting closely neighboring structures for the purpose of eliminating pounding. The structures were assumed and modeled as linear, MDOF systems where the mass is concentrated at each floor and the stiffness is provided by walls and columns and subjected to harmonic and earthquake excitation. Coupling reduces the potential for pounding by maintaining a separation distance between the structures. The contact is likely to occur at a single point (due to predominant first mode vibration), a single connection may be sufficient. However, at higher modes multiple connection points are necessary. The linkage not only reduces the relative overlap deflection of the structures at large amplitudes but also increase the base shear on the stiffer of the two structures at excitation frequencies below the fundamental frequency. Maison et al., (1990) presented a formulation and solution of the multiple degree of freedom equations of motion. The studied building undergoes pounding at a single floor level with a rigid adjacent building. A single linear spring represents the local flexibility of the buildings at their locations of contact. They found that even at the relatively large separation (90% of the sum of maximum displacements obtained without pounding) the 10


increases in drifts and shears are significant. In situations where pounding may potentially occur, neglecting its possible effects leads to unconservative building design/evaluation. The story drifts, shears, and overturning moments in the stories above the pounding elevation will be underestimated (pounding occurred at 8th level in a 15 story structure). Anagnostopoulos et al., (1992) extended their studies on a series of SDOF systems to MDOF. They investigated the linear as well as the non-linear response of several adjacent buildings in a row under conditions of pounding. They idealized the buildings as lumped mass, shear beam type and MDOF systems with bilinear force-deformation characteristics. Furthermore, the structural models include foundation compliance by means of a linear spring for translational and rotational motions. The constants for the foundation springs were determined. Collisions were simulated by visco-elastic impact elements (a Kelvin solid) and five real earthquake motions were used. The coefficient restitution used to simulate real collision in structural engineering varies in the range of 0.5-0.75. If there are large differences in the masses of the colliding buildings then pounding can cause high overstresses in the building with the smaller mass. Greater consequences for the tall building can be expected if the lower building were more massive and stronger. The effects of pounding are reduced as the separation distance increases, even if the code (Uniform Building Code and Euro No. 8) specified gap proved inadequate in some future strong shaking. Rahman et al., (2000) investigated the influence of soil flexibility on the dynamic response of structures subjected to pounding. The 12 and 6-storey reinforced concrete moment-resisting frames were subjected to ground motion and the time-history response of this system (linear soil behavior and nonlinear structural response) was evaluated by means of the structural analysis software RUAUMOKO. The soil flexibility was modeled as discrete elements and the parameters were chosen from previous investigators. The results indicated that the effect of foundation compliance is to increase of the natural periods of both structures from the fixed base condition. Consideration of the flexibility of the soil mass between the structures, even for the large separation distance, shows a slight difference from the compliant foundation case. Raheem et al., (2006) developed a methodology for the formulation of adjacent building pounding problems based on classical impact theory through parametric study to identify the most important parameters. The steel moment resisting frame building of 8 storey and 13 storey was modeled as 3D finite element model and nonlinear time history analyses were performed. Nine ground motion records [Muthu kumar 2004] are taken and grouped into three levels depending on PGA. For the purpose of evaluating the effect of torsion, a torsional unbalanced model is defined where the centre of mass lies at a distance ’e’ from centre of rigidity. Impact energy dissipation is also introduced. Bilinear truss contact model is used for the analysis. It was found that pounding in severe load condition could result in high acceleration pulses in the form of short duration spikes which in 11


turn cause greater damage. Pounding response is increased in the flexible building and reduced in stiff building at dominant frequency. The amplification in building response is a function of TA , TB , (TA /TB ) and dominant frequency of input ground motion. An increase in the damping energy absorption capacity of pounding element results in the reduction of the acceleration spikes, impact force and building global response. It is clear that an energy dissipation system installed at potential pounding level could be an effective tool to reduce the effect of impact. Polycarpou et al., (2008) investigated the influence of the impacts on the overall structural response, seismically isolated buildings in series with adjacent structures numerically. A four storey seismically isolated building and neighboring two fixed supported buildings were subjected to six strong motions. A linear visco elastic impact model with plastic deformations is employed for pounding simulation. A series of dynamic analyses conducted in order to investigate the responses for isolated building adjacent to other fixed supported buildings, compared to the case of the seismically isolated building standing alone surrounded only by the moat wall. It was found that the presence of a fixed-supported building in close proximity with the seismically isolated building may cause unexpected impact phenomena at upper floors due to the deformation of the buildings in series. Also, the number of stories of the adjacent fixed-supported buildings seems to play a significant role to the severity of the impact. Goltabar et al., (2008) studied the impact between adjacent structures of different heights using gap joint element and nonlinear time history analyses during earthquakes and their effective parameters. In this case 10 and 13 storey structures were chosen, the connection modeling was done using gap joint element. The two structures were subjected under three accelerographs. The analysis was done using SAP 2000 software. From the analysis, maximum responses (lateral displacements and storey shears) in the shorter building decreased throughout the whole building except for the impact point. Maximum responses in the taller buildings increased throughout the building. One of the ways to decrease impact effects is considering a proper distance between two structures. As a result the responses will be similar to non-impact case. By hardening the building also we can reduce the impact effects. Tande et al., (2009) investigated the optimal seismic response of two adjacent structures through passive energy dissipation devices in order to minimize the pounding effect. Two twenty storey buildings having same floor elevation with dampers connecting two neighboring floors were used. The damping ratio for both the buildings is taken as 5%. Time history analysis is carried out using Elcentro ground motion 1940 earthquake data to find out the maximum top floor displacements for three cases. The first case is that of unconnected structures, second is when the two buildings are connected at all floor levels and third case is at optimal locations. The damper stiffness coefficient is chosen such that the addition of dampers doesn’t change the modal frequencies of the individual buildings. 12


The value of 5.0x104 N/m is selected as optimal stiffness for the dampers. When the value of stiffness is increased to 1.0x109 N/m, the relative displacement between the adjacent buildings becomes nearly zero, implying that both buildings are rigidly connected. The maximum top floor displacement is lowest when the damper damping coefficient is 1.0x104 Ns/m. Beyond this value there is no significance reduction in the displacements of both the buildings. If a very high of damping coefficient is chosen, the structure becomes over-damped and hence becomes stiff. Optimal damper damping coefficient is in between 5.0x105 and 3.0x106 Ns/m. The natural frequencies for both the buildings should not change after the insertion of external damping devices. As the damping coefficient of visco-elastic damper is increased, there is no significant decrease of response beyond certain value. Hence this value is selected as optimal damper damping coefficient. It is not necessary to connect the two adjacent buildings by dampers at all floor levels. Dampers at appropriate locations can significantly reduce the earthquake response of the combined system. Mahmoud et al., (2009) studied and compared the maximum elastic with inelastic responses under three ground motions. Two adjacent four storey buildings are considered in which the building parameters are assumed to be same for all floor levels. The buildings are modeled as MDOF systems and the nonlinear visco-elastic model is used to simulate impact force during collisions. The peak responses are considerably reduced for all ground motions for different values of separation distances. For both elastic and inelastic systems, the peak responses are of the flexible building increase up to a certain value of the gap distance and with further increase in gap a decrease trend can be observed. The responses of the elastic systems are significantly different comparing to the responses of inelastic one. Especially, the flexible building easily enters into the yielding range as a result of pounding. The maximum impact forces and the number of impacts are larger in the elastic case. The normalized errors increase with the increase of levels of PGA of the earthquake records. It is observed from the past numerical studies on pounding of MDOF structures, are modeled as linear systems using link and beam elements and considered MDOF systems where the mass is concentrated at each floor level and the stiffness is provided by walls and columns. The building of interest dynamically vibrates and laterally collides with an adjacent rigid building. This implies that the building of interest is very flexible and possesses low mass relative to the adjacent building. The pounding occurs at a single floor level in the building of interest and the floor diaphragms are rigid in plane. A single linear spring represents the local flexibility of the buildings at their locations of contact. It is assumed that floor elevations are the same for all buildings so that pounding can occur only at these elevations where the masses are lumped. Local damage effects due to pounding are not addressed. All the systems are subjected to same base acceleration and so any effects due to spatial variations of the ground motion or due to soil structure interactions are neglected. The material damping of the soil was not accounted for in the 13


soil structure interaction models and only radiation damping was present.

1.5 1.5.1

Summary of contribution Mitigation measures

In the past, large metropolitan areas (eg.Mexico) were affected by major earthquakes which induced severe pounding damage. To minimize the effects of pounding, mitigation techniques are used and some of them are listed below. Mitigation techniques are used to minimize the damages from pounding as follows: 1. In link element technique, forces in links can be same order of magnitude of base shear. This link may sometimes totally alter the distribution of the forces. 2. Bumper damper elements are link elements that are activated when gap is closed. Such elements reduce energy transfer during pounding and the high frequency pulses. The damper will yield a smaller value for the coefficient of restitution. These bumper elements have already been incorporated in the Greek code and in Euro code-08 for earthquake resistant design. 3. Supplemental energy devices can be used in the structures for pounding mitigation depending on the additional damping supplied. 4. The use of shear walls that are constructed at right angles to the divided line between two buildings in contact, so that they can be used as bumper elements in the case of pounding. 5. Provide sufficient minimum distance between adjacent structures. 6. Provide sufficient seated length between the decks or provide shock absorbing devices between the decks and bearings under the extremities of the decks in bridges. 7. Buildings having simple regular geometry, uniformly distributed mass and stiffness in plan as well as in elevation, suffer very less damage than buildings with irregular configurations. From the above reasons, it is widely accepted that pounding is an undesirable phenomenon that should be prevented or mitigated. Several impact models are available to understand the pounding behavior of structures.

1.5.2

Impact models

The collisions between adjacent buildings are simulated by means of contact elements that are activated when the bodies come in contact and deactivated if they are separated. A brief summary with advantages and disadvantages of the various modeling techniques are presented below. 14


Coefficient of restitution model Stereo-mechanical model The stereo-mechanical theory of impact is the classical formulation to the problem of impacting bodies. The stereo-mechanical approach assumes instantaneous impact and uses momentum balance and the coefficient of restitution to modify velocities of the colliding bodies after impact. The original theory considered the impacting bodies as rigid; later a correction factor to account for energy losses was introduced. The theory concentrates on determining the final velocities of two impacting bodies depending on their initial velocities and a coefficient of restitution to account for plasticity during impact. The final velocities of the bodies are determined from equation 1.1 to 1.2. The disadvantage of the method is that it is no longer valid if the impact duration is large enough so that significant changes occur in the configuration of the system. This implies that the duration of impact is neglected. Traditionally the value of the coefficient of restitution was assumed to depend only on the material properties. The stereo-mechanical model is not a force-based model. Hence, there is no impact force and consequently, no amplification in the acceleration response. v10 = v1 − (1 + e)

m2 (v1 − v2 ) m1 + m2

(1.1)

v20 = v2 + (1 + e)

m1 (v1 − v2 ) m1 + m2

(1.2)

Where v10 and v20 are final velocities, v1 , v2 are initial velocities of the colliding bodies, e is co.efficient of restitution, m1 , m2 are the mass of bodies. Contact force based models Linear spring element model The simplest contact element consists of a linear elastic element (fig 1.7). The spring

Figure 1.7: Linear Spring Model is assumed to have restoring force characteristics such that only when the relative distance between the masses becomes smaller than the initial distance, the spring contracts

15


and generates forces which enable us to consider the phenomenon of pounding. This collision spring is assumed to be the axial stiffness of the floors and the beams in each storey. The force in the contact element can be expressed according to equation 1.3:  k (u − u − δ), (u − u − δ) ≥ 0 l 1 2 1 2 Fc = 0, (u1 − u2 − δ) < 0

(1.3)

Where u1 and u2 are the displacements of the impacting bodies, kl is the spring constant of the element and δ is the separation distance between the structures. However, energy loss during impact cannot be modeled. Whenever two mechanical systems collide there is an exchange of momentum and also energy is dissipated in the high stress region of contact. Kelvin-Voight element model The Kelvin-Voight element is represented by a linear spring in parallel with a damper (fig 1.8). This model has been widely used in some studies [Anagnostopoulos 1988, Jankowski

Figure 1.8: Kelvin Model 2004]. This impact model is capable of modeling energy dissipation during impact and the impact force is represented by equation 1.4.  k (u − u − δ) + c (u˙ − u˙ ), (u − u − δ) ≥ 0 k 1 2 k 1 2 1 2 Fc = (1.4) 0, (u1 − u2 − δ) < 0 Where u1 , u2 and its derivatives are the displacements and velocities of the impacting bodies, kk is the spring constant of the element and δ is the separation distance between the structures. The damping coefficient ck can be related to the coefficient of restitution (e), by equating the energy losses during impact. r ck = 2ζ

kk

m1 m2 lne ; ζ = −p m1 + m2 π 2 + (lne)2

(1.5)

Where m1 and m2 are the mass of colliding bodies and ζ is the damping co.efficient.

16


Modified Kelvin element model The disadvantage of the Kelvin model is that its viscous component is active with the same damping coefficient during the whole time of collision. The damping forces causes negative impact forces that pull the colliding bodies together, during the unloading phase, instead of pushing them apart. Ye et al., (2009) reexamined and modified the Kelvin model and theoretical derivation has been verified with numerical experiment. The corrected damping ratio value can be expressed as: ζk =

3kk (1 + e) 2e(v1 − v2 )

(1.6)

Hertz contact model In pounding, one would expect the contact area between neighboring structures to increase as the contact force grows, leading to a non-linear stiffness. In order to model highly non-linear pounding more-realistically, Hertz impact model has been adopted by various authors [Davis 1992, Jing 1990, Chau and Wei, 2001, Chau et al., 2003]. This model uses the Hertz contact law: a non-linear spring in an impact oscillator. The main restriction of their works is that only pounding of a SDOF oscillator on a stationary barrier or on a barrier moving with ’locked-to-ground-motion’ is considered. The force in the contact element can be expressed as:  k (u − u − δ) 23 , (u − u − δ) ≥ 0 h 1 2 1 2 Fc = 0, (u1 − u2 − δ) < 0

(1.7)

The coefficient kh depends on material properties and geometry of colliding bodies. The Hertz contact law, is incapable of taking into account dissipation during impact phenomenon. The value of the Hertz exponent, 3/2, may be different for real pounding, but Davis [1992] has shown that the exact value may be altered without radically changing the oscillator response. Hertz-damp contact model An improved version of the Hertz model, called Hertz-damp model, has been considered by Muthukumar and DesRoches [2004] whereby a non-linear damper is used in addition with the Hertz spring. The force in the contact element can be expressed as:  k (u − u − δ) 32 + c (u˙ − u˙ ), (u − u − δ) ≥ 0 h 1 2 h 1 2 1 2 Fc = (1.8) 0, (u1 − u2 − δ) < 0 Where ch is the damping coefficient, (u1 − u2 − δ) is the relative penetration and derivative of (u1 − u2 ) is the penetration velocity. A nonlinear damping coefficient (ch ) is proposed so that the hysteresis loop matches the expected loop due to a compressive 17


load that is applied to and removed from a body within its elastic range at a slow rate. The damping co.efficient can be expressed as, ch = ζ(u1 − u2 − δ)n

(1.9)

Where ζ is damping constant. Equating the energy loss during stereo-mechanical impact to the energy dissipated by the damper, the value of ζ can be related to the spring constant, kh , the coefficient of restitution (e), and the relative velocity of the bodies at the instant of impact, (v1 − v2 ), as shown below. ζ=

3kh (1 − e2 ) 4(v1 − v2 )

(1.10)

Hence, the force during contact can be expressed as:  k (u − u − δ) 23 + [1 + h 1 2 Fc = 0,

3(1−e2 ) (u˙1 4(v1 −v2 )

− u˙2 )], (u1 − u2 − δ) ≥ 0 (u1 − u2 − δ) < 0

(1.11)

The Hertz model with nonlinear damper shall be referred to as the Hertz-damp model.

1.5.3

Codal provisions

Most of the world regulations for seismic design do not take into account this phenomenon and some of the ones which do, do not provide specific rules that must be followed (Paz, 1994). Among the exceptions are the codes of Argentina, Australia, Canada, France, India, Indonesia, Mexico, Taiwan and USA which specify a minimum gap size between adjacent buildings. In some cases gap depends only on the maximum displacements of the each building. The rule to determine the size is nevertheless variable, being in some cases the simple sum of the displacements of each building (eg. Canada and Israel) and in other cases a small value that may be either a percentage of previous one or a quadratic combination of the maximum displacements (eg. France). In other cases the gap size is made dependent on the building height (eg.Taiwan), in some cases a combination of two rules is implemented and in others there is a even a minimum gap size which varies between 2.5 cm (eg. Argentina) and 1.5 cm (eg.Taiwan). In some cases these values depend on the type of soil and seismic action. However most of the studies assume only two dimensional behaviors i.e. only translational pounding is considered. But actually torsional pounding tends to be more common than uni-directional pounding during real ground motions. This problem is particularly 18


common in many cities located in seismically active regions. The list of various country codal provisions on pounding is shown in table 1.1 (Source: IAEE):

1.6

Objective of the study

Structural pounding is a complex phenomenon which involves local damage to structures during earthquakes. It is necessary to quantify the amount of damage due to structural pounding. In numerical modeling, the research on structural pounding has not been upgraded till quantification of damage and damage scale. The main objective of this research is to study pounding of structures using applied element method (AEM).

1.7

Organization of thesis

The body of this thesis is organised in six chapters. Chapter(2) deals about modeling of structures which are subjected to pounding. Also, linear pounding behavior of structures with equal and un-equal heights are discussed. Chapter(3) discusses about nonlinear pounding behavior of different structures with equal heights subjected to ground motion. Furthermore, a damage scale has proposed for all structures to categorize the level of damage due to pounding. Chapter(4) discusses the nonlinear pounding behavior of different structures with unequal heights. Different un-equal heights are considered in this analysis and the level of damage for all structures are estimated from the above damage scale. Chapter(5) extends the analysis to 3D. This chapter introduces nonlinear torsional pounding between structures and structures are placed in staggered manner. Finally, the general conclusions and future direction of research are introduced in chapter(6).

19


Table 1.1: List of codal provisions on pounding Country

Australia

Canada

Egypt

Ethiopia

Greece

India

Mexico

Nepal

Provision on pounding Structures over 15 m shall be separated from adjacent structures or setback from building boundary by a distance sufficient to avoid damaging contact. This clause is deemed to be satisfied if the primary seismic force-resisting elements are structural walls that extend to the base, or the setback from a boundary is more than 1% of the structure height. [clause 5.4.5] Adjacent structures shall be separated by the sum of their individual lateral deflections obtained from an elastic analysis. [clause 4.1.9.2(6)] • Each building separated from its neighbor shall have a minimum clear space from the property boundary, other than adjoining a public space, either by 2.0 times the computed deflections or 0.002 times its height whichever is larger, and in many cases, not less than 2.5 cms. • Parts of the same building or buildings on the same site which are not designed to act as an integral unit shall be separated from each other by a distance of at least 2.0 times the sum of the individual computed deflections or 0.004 times its height whichever is larger, and in many cases, not less than 5.0 cms. [clause 2.7.2] To prevent collision of buildings in an earthquake, adjacent structures shall either be separated by twice the sum of their individual deflections obtained from an elastic analysis. [clause 7.7] For buildings which are in contact with each other but there is no possibility for any columns to be rammed, the width of the respective joint, in the absence of more accurate analysis may be determined on the basis of the total number of storeys in contact above the ground as follows: • 4 cm up to and including 3 storeys in contact • 8 cm from 4-8 storeys in contact • 10 cm for more than 8 storeys in contact. For underground floors a seismic joint is not obligatory. [clause 4.1.7.2] R times the sum of the calculated storey displacements as per clause7.11.1. When floors levels of two similar adjacent units or buildings are at the same elevation levels, factor R in this requirement may be replaced by R/2.[clause7.11.10.] When using the simplified seismic analysis method the separation distance shall be neither smaller than 5 cm nor smaller than the height of the level over the ground multiplied by 0.007, 0.009 or 0.012 depending on the site in zones I, II or III respectively. [article 211] • Above ground level, each building of greater than three storeys shall have a separation from the boundary, except adjacent to a designed street or public way, of not less than the design lateral deflection or 0.002 hi or 25 mm whichever is greater. • Parts of buildings or buildings on the same site which are not designed to act as an integral unit shall be separated from each other by a distance of not less than the sum of the design lateral deflections or 0.004 hi or 50 mm whichever is greater. [clause 9.2]

20


Peru

Serbia

Taiwan

• The minimum distance shall not be less than 2/3 of the sum of the maximum displacements of the adjacent blocks, nor shall it be less than: S=3+0.004(h-500) (h and S are in cms) S>3 cm. • The building shall be set back from adjacent property lines of empty plots that can built on or from existing buildings, by distances no less than 2/3 of the maximum displacement nor less than S/2. [article 15.2] • The minimum width of the aseismic joint shall be 3.0 cm. It shall be increased by 1.0 cm for every increase of 3.0 meters of height above 5.0 mts. • In the case of buildings higher than 15 m, as well as in the case of flexible structures of lesser height such as unbraced frames, it shall not be less than the sum of the maximum deflections of adjacent parts of the building, nor shall it be less than the above. [article 47] Pounding may be presumed not to occur wherever buildings are separated by a distance greater than or equal to 0.6x1.4fy Ra times the displacement caused by the determined seismic base shear. [clause 2.5.4] *The factor 0.6 is used because of low probability that two adjacent buildings will move in the opposite directions and reach the maximum displacement simultaneously.

21

Chapter 2 Numerical Modeling of Pounding 2.1

Introduction

The main focus of this chapter is to study the linear pounding behavior of structures with equal and un-equal heights. Also discuss on separation distance from different country seismic codes for different structures subjected to different ground motions. Furthermore, provide suggestions for the codes which are underestimated on separation distance.

2.2 2.2.1

Selection of buildings Building geometry

The analysis considers single storey, two storey, three storey and five storey structures and the details of the structures are shown in fig 2.1 to 2.4.

Figure 2.1: Geometry of single storey structure The single storey structure has a total height of 3.12 m including the thickness of slab. The dimension of column is 0.24 m x 0.24 m. For two storey structure the total height is 22

CHAPTER 2. NUMERICAL MODELING OF POUNDING

Figure 2.2: Geometry of two storey structure 6.24 m and the thickness of slab is 0.12 m. The total height of three storey structure is 9.36 m having a column size of 0.3 m x 0.3 m. The total height of five storey structure is 15.6 m having a column size of 0.3 m x 0.3 m. All the above mentioned structures are designed as per IS-456:2000 and minimum reinforcement is provided. The reinforcement details for single and two storey structures are shown in fig 2.5 and for three and five storey structures are shown in fig 2.6. For all the structures the slab thickness is considered as 0.12 m. The foundation reinforcement details are not considered and assumed as fixed at ground surface level.

2.2.2

Material properties

The grade of concrete, grade of steel for reinforcement and poission’s ratio are same for all the structures. The details of material properties for the structures (ref 2.2.1) are as follows: Grade of concrete : M25 Grade of steel for reinforcement : Fe415 Poission’s ratio : 0.2 For the pounding analysis minimum two structures are considered namely structure-A and B. It is assumed that the live load acting on the structure-A is 2 kN/m2 and for structure-B is 5 kN/m2 . The above mentioned live load is applied for the structures hereafter.

23


Figure 2.3: Geometry of three storey structure

2.3

Selection of ground motions

Ten moderate ground motions are choosen for the analysis whose peak ground acceleration (PGA) ranging from 0.25-0.55 g. The PGAs and duration of ground motions ranges from low to high and the frequency content ranges from resonating to non-resonating conditions. The details of the ground motions are listed in table 2.1 and the ground motion records and its fourier amplitude spectrums are shown in fig 2.7 to 2.16. The characteristics of corresponding ground motions are described in next sub-section.

2.3.1

Characteristics of ground motions

For engineering purposes, (1) amplitude (2) frequency and (3) duration of the motion are the important characteristics (Kramer, 1996). Horizontal accelerations have commonly been used to describe the ground motions. The peak horizontal acceleration for a given component of motion is simply the largest

24


Figure 2.4: Geometry of five storey structure

25


Figure 2.5: Reinforcement details of single and two storey structures

Figure 2.6: Reinforcement details of three and five storey structures Table 2.1: Details of ground motions

S.No

Earthquake

Date

Station

Comp

1 2 3 4 5 6 7 8 9 10

Athens Athens Ionian Kalamata Umbro Elcentro Olympia Parkfield Northridge Lomaprieta

1999 1999 1973 1986 1997

Kallithea Sepolia Garage Lefkada-Ote Kalamata Nocera Umbra Imperial Valley Washington Parkfield New Hall LA Lomaprieta

N46 Tran NS N355 NS S00E N86E N85E Up 2700

1994 1989

Amplitude (g) 0.265 0.31 0.525 0.297 0.47 0.348 0.28 0.43 0.548 0.276

Duration (sec) 4.1 4.44 6.9 5.27 9.35 29 20.82 6.55 12.44 9.78

Frequency (Hz) 1.5-4.5 2.0-6.0 0.85-2.4 0.7-1.7 6.2-7.2 1.15-2.2 1.12-4.66 0.8-3.3 2.2-5.3 0.65-1.72

(absolute) value of horizontal acceleration obtained from the accelerogram of that component. The largest dynamic forces induced in certain types of structures (very stiff) are closely relate to the peak horizontal accelerations (Kramer, 1996). Earthquakes produce complicated loading with components of motion that span a broad range of frequencies. The frequency content describes how the amplitude of ground 26


(a)

(b)

Figure 2.7: Athens ground motion record and its fourier spectrum amplitude (a) Ground motion record (b) Fourier amplitude spectrum

(a)

(b)

Figure 2.8: Athens(tran) ground motion record and its fourier spectrum amplitude (a) Ground motion record (b) Fourier amplitude spectrum

(a)

(b)

Figure 2.9: Ionian ground motion record and its fourier spectrum amplitude (a) Ground motion record (b) Fourier amplitude spectrum

27


(a)

(b)

Figure 2.10: Kalamata ground motion record and its fourier spectrum amplitude (a) Ground motion record (b) Fourier amplitude spectrum

(a)

(b)

Figure 2.11: Umbro ground motion record and its fourier spectrum amplitude (a) Ground motion record (b) Fourier amplitude spectrum motion is distributed among different frequencies. The frequency content of an earthquake motion will strongly influence the motion of structure. The broad band width of the Fourier amplitude spectrum is the range of frequencies over which some level of Fourier amplitude is exceeded. Generally band width is measured at a level of √12 times of maximum Fourier amplitude. The fourier transform of an accelerogram x¨(t) is given by, 1 X(ω) = 2π

Z

∞

x¨(t)e−iωt dt

(2.1)

−∞

Where, x¨(t) is the acceleration record and ω is frequency. The duration of strong ground motion can have a strong influence on earthquake dam-

28


(a)

(b)

Figure 2.12: Elcentro ground motion record and its fourier spectrum amplitude (a) Ground motion record (b) Fourier amplitude spectrum

(a)

(b)

Figure 2.13: Olympia ground motion record and its fourier spectrum amplitude(a) Ground motion record (b) Fourier amplitude spectrum

(a)

(b)

Figure 2.14: Parkfield ground motion record and its fourier spectrum amplitude(a) Ground motion record (b) Fourier amplitude spectrum

29


(a)

(b)

Figure 2.15: Northridge ground motion record and its fourier spectrum amplitude(a) Ground motion record (b) Fourier amplitude spectrum

(a)

(b)

Figure 2.16: Lomaprieta ground motion record and its fourier spectrum amplitude(a) Ground motion record (b) Fourier amplitude spectrum age. It is related to the time required for accumulation of strain energy by rupture along the fault. There are different procedures for calculating the duration of ground motion, Brackted Duration: It is the time between the first and last exceedances of a threshold acceleration (usually 0.05 g) Trifunac and Brady Duration: It is the time interval between the points at which 5% and 95% of the total energy has been recorded. In our analysis brackted duration is used for calculating the duration of ground motion.

30


2.4 2.4.1

Numerical method Introduction

The numerical techniques can be categorize in two ways. The first case assumes that the material as continnum like finite element method (FEM). In this method, elements are connected by nodes where the degrees of freedom are defined. The displacement, stresses and strains inside the element are related to the nodal displacements. The analysis can be done in elastic and nonlinear materials, small and large deformations except collapse behavior. At failure, the location of cracks should be defined before analysis which is not possible in collapse analysis. The other catogory assumes that the material as discrete model like rigid body spring model (RBSM), extended distinct element method (EDEM) and applied element method (AEM) (Hatem, 1998). The RBSM performs only in small deformation range. EDEM overcomes all the difficulties in FEM, but the accuracy is less than FEM in small deformation range. Till now there is no method among all the available numerical techniques, in which the behaviour of the structure from zero loading to total complete collapse can be calculated with high accuracy. The modeling of AEM and formulation are as follows. Applied element method is a discrete method in which the elements are connected by pair of normal and shear springs which are distributed around the element edges. These springs represents the stresses and deformations of the studied element. The elements motion is rigid body motion and the internal deformations are taken by springs only. It is advisable to increase the number of elements than connecting springs for improving the accuracy. The general stiffness matrix components corresponding to each degree of freedom are determined by assuming unit displacement and the forces are at the centroid of each element. The element stiffness matrix size is 6x6. The stiffness matrix components diagram is shown in fig 2.17. The first quarter portion of the stiffness matrix is shown in fig 2.18. However the global stiffness matrix is generated by summing up all the local stiffness matrices for each element.

2.4.2

Mathematical formulation

In order to begin any physical system, it is necessary to formulate it in a mathematical form. The general dynamic equation for a structure is given in equation 2.2 M u¨ + C u˙ + Ku = ∆f (t) − M u¨g

(2.2)

Where [M ] is mass matrix; [C] is damping matrix; [K] is nonlinear stiffness matrix; ∆f (t) is incremental applied load vector ∆U and its derivatives are the incremental displacement, velocity and acceleration vectors respectively. The above equation is solved numerically using Newmark’s beta method.

31


Figure 2.17: Element components for formulating stiffness matrix (SOURCE: Kimuro Meguro and Hatem, 2001)

Figure 2.18: Quarter portion of stiffness matrix

For mass matrix the elemental mass and mass moment of inertia are assumed lumped at the element centroid so that it will act as continuous system. The elemental mass matrix in case of square shaped elements is given in equation 2.3     M1 D2 tρ     M2  =  D2 tρ  M3 D4 tρ/6.0

(2.3)

Where D is the element size; t is element thickness and ρ is the density of material. From the above equation it is noticed that [M1 ] and [M2 ] are the element masses and [M3 ] is the mass moment of inertia about centroid of the element. The mass matrix is a diagonal matrix. The response of the structure is very near to the continuous/distributed mass system if the element size becomes small. If the damping is present, the response of the structure will get reduced. The damping matrix is calculated from the first mode as follows: 32


C = 2ζM ωn

(2.4)

Where ζ is damping ratio and ωn is the first natural frequency of the structure. For finding out the dynamic properties such as natural frequencies of a structure requires eigen values. The general equation for free vibration without damping is: M u¨ + Ku = 0

(2.5)

For a non trival solution, the determinant of the above matrix must be equal to zero. The solution of determinant of matrix gives the natural frequencies of the structure.

2.4.3

Element size

In numerical modeling, element size of the structure is important. Using large element size decreases the displacement of the structure and finally leads to the larger failure load than actual failure load. For any kind of numerical analysis three important requirements (convergence, stability and accuracy) are necessary. 1. Convergence - As the time step decreases, the numerical solution should close to the exact/theoretical solution. 2. Stability - The numerical solution should be stable in the presence of numerical round off errors. 3. Accuracy - The numerical procedure should provide results that are close to the exact solution. The details of the structures are taken from fig 2.1 to 2.4 for fixing the element size and the structures are subjected to uttarkasi ground motion having PGA of 0.252 g shown in fig 2.19. The displacement response of the structure is calculated using Newmarks beta method. Initially the response is calculated with an element size of 0.24 m. As the size of the element decreases, the response level will get saturate. This means the response will be same with decreasing the element size further. The same analysis is done for two storey, three storey and five storey structures. Finally, the element size is fixed at 0.06 m for all the structures.

2.4.4

Material model

The material model used in this analysis is Maekawa compression model (Tagel-Din Hatem, 1998). In this model, the tangent modulus is calculated according to the strain at the spring location. After peak stresses, spring stiffness is assumed as a minimum value to avoid having a singular matrix. The difference between spring stress and stress corresponding to strain at the spring location are redistributed in each increment in reverse 33


(a)

(b)

Figure 2.19: Uttarkasi ground motion record and its fourier spectrum amplitude(a) Ground motion record (b) Fourier amplitude spectrum direction. For concrete springs are subjected to tension, spring stiffness is assumed as the initial stiffness till it reaches crack point. After cracking, stiffness of the springs subjected to tension are assumed to be zero. For reinforcement, bi-linear stress strain relationship is assumed. After yield of reinforcement, steel spring stiffness is assumed as 0.01 of initial stiffness. After reaching 10% of strain, it is assumed that the reinforcement bar is cut. The force carried by the reinforcement bar is redistributed force to the corresponding elements in reverse direction. For cracking criteria (Hatem, 1998), principal stress based on failure criteria is adopted. The models for concrete, both in compression and tension and the reinforcement bi-linear model are shown in fig 2.20 (a) and (b).

(a)

(b)

Figure 2.20: Material models for concrete and steel (a) Tension and compression concrete Maekawa model (b) Bi-linear stress strain relation model for steel reinforcement

2.4.5

Collision model

Collision springs are added between the collide elements to represent the material behavior during contact. As the collision check of irregularly shaped elements is more difficult 34


Figure 2.21: Arrangement of collision springs and time consuming, the element shape during collision is assumed as circle. This assumption is acceptable even in case of relatively large elements because the sharp corners of elements are broken due to stress concentration during collision and edge of elements become round shape. The arrangement of collision springs are as shown in fig 2.21 The normal spring stiffness is calculated as follows: Edt (2.6) D Where, ’E’ is minimum Young’s modulus of the material, ’t’ is the element thickness, ’D’ is the distance between element centroids and ’d’ is the contact distance which is assumed as 0.1 times of the element size. The shear spring stiffness is assumed as 0.01 times of normal spring stiffness. The normal contact spring is not allowed to fail, instead of that compression failure of the distributed springs connecting to the elements allowed to fail. Here the objective of the collision spring is to transmit the stress wave to other elements. The tensile force in the normal spring indicates that elements tend to separate each other. Then, the residual tension is redistributed in the next increment. kn =

Procedure for finding out the response of colliding structures is as follows: The displacement response of the structure is calculated using Newmarks β technique. Using geometric coordinate technique, contact between elements is checked and its neighbour elements instead of all elements. If collision occurs, the following steps to be followed: STEP 1: The time increment should be reduced to follow the material behavior properly during collision. This value depends on many factors like relative velocity between elements before collision and force transmitted during collision. After separation of ele-

35


ments the time increment is increased. Shorter time increment should be used because of excessive overlapping between elements if time increment value is relatively large which leads to numerical errors. STEP 2: The collision of normal and shear springs is added between two elements. The normal spring direction passes through the elements centroid, while shear spring direction is tangent to the assumed entire circles. These springs exist if the elements are in contact and removed after separation. When elements separate, residual tension is redistributed by applying the forces in reverse direction. Set collision spring stiffness equal to zero if separation occurs. STEP 3: Finally the stiffness values are added in the global stiffness matrix. STEP 4: Determine the resultant spring forces including forces from springs at location of each element centroid. The geometrical residuals are calculated as, Fg = F (t) − M u¨ − C u˙ − Fm

(2.7)

Determine the new stiffness matrix for the new configuration. STEP 5: Calculate the incremental displacement, velocity and acceleration for each element in the next increment. Based on the calculated displacement from collision, collision force is calculated as follows: The collision force is kn times to the relative displacement response at the contact point.

2.4.6

Failure criteria

To determine the principal stresses at each spring location, the following technique is used in this analysis. The shear and normal stress components (τ and σ1 ) at point A are determined from the normal and shear springs attached at the contact point location shown in figure 2.22 (a) & (b). The secondary stress σ2 from normal stresses and at point B and C can be calculated by using the equation given below: a−x x σB + σC a a The principal tension is calculated as: σ2 =

σ1 + σ2 σP = + 2

r ( 36

σ1 − σ2 2 ) + τ2 2

(2.8)

(2.9)


(a)

(b)

Figure 2.22: (a) Principal Stress determination and (b) Redistribution of spring forces at element edges The value of principal stress (σP ) is compared with the tension resistance of the studied material. When σP exceeds the critical value of tension resistance, the normal and shear spring forces are redistributed in the next increment by applying the normal and shear spring forces in the reverse direction. These redistributed forces are transferred to the element center as a force and moment, and then these redistributed forces are applied to the structure in the next increment. When the element is subjected to shear, the crack propagation is mainly on shear stress value. To represent the occurance of cracks, two techniques can be used. Among these techniques, the one which we adopt is as follows: It is assuming that failure inside the element is represented by failure of attached springs (Hatem et al., 2000). If the spring gets failed, then the force in the spring is redistributed. During this process, springs near the crack portion tend to fail easily. However, the main disadvantage of this technique is that the crack width can not be calculated accurately. In each increment, stresses and strains are calculated for reinforcement and concrete springs. In case of springs subjected to tension, the failure criterion is checked (refer failure criteria).

2.4.7

Limitations

1. When the collision takes place, virtual collision springs are developed between the collided elements. But virtual dampers were not provided in this model to describe 37


the collision behavior. These virtual collision springs and dampers together will describe the collision behavior in numerical model. 2. Buckling of reinforcement was not included in this method.

2.5

Linear pounding analysis of structures

A study has been conducted in order to investigate the minimum separation distance and structural pounding behavior between two adjacent structures using linear analysis. Study on minimum separation distance is discussed in appendix-B. To study the linear pounding behavior, structures with equal heights separated by 2 mm under Northridge ground motion are considered in this analysis. The material and geometrical details of the structures which are considered in this analysis are mentioned in previous section. The behavior of structures at different slab levels are discussed in subsequent sections.

2.5.1

Structures with equal heights

Pounding doesn’t happen for the structures having same dynamic properties, even though the separation distance is zero. When the response between two adjacent structures is in same direction, those vibrations are called inphase vibrations and when they are in opposite direction called as out-of-phase vibrations. Inphase or out-of-phase or both vibrations can occur during collision of structures. It depends on the dynamic properties and velocities of the structures. It is difficult to find out whether the structures have undergone inphase or out-of-phase during earthquake. During collision, there is sudden change in velocity (vertical drop) in phase potrait diagrams and also have a great impact on the structures. A large amount of force is generated when two adjacent structures collide. This collision/pounding force is dependent on many parameters like velocity of the structures, stiffness provided between two structures and separation distance between them. During compression phase of collision, the force is getting increased from zero force and for restitution phase, the force reduces to zero. The procedure for calculating the collision response and force are described in earlier sections. As inertia is main important criteria for accelerations, higher amplification pulses are observed in acceleration when two structures collide. These amplification pulses cause great damage to the structures. These pulses occur at the places where collision takes place and will have short duration. The present study is carried out in four cases: (1) single-single storey, (2) single-two storey, (3) two-two storey and (4) two-three storey structural pounding. For the above cases, the damping ratio is taken as 5%. The fundamental natural periods of the structure-

38


A and B are listed in table 2.2 given below:

Table 2.2: Fundamental period of the structures Fundamental period (sec) Case Structure-A Structure-B single-single 0.127 0.155 single-two 0.127 0.304 two-two 0.253 0.304 two-three 0.253 0.334

Case-I: Structures with equal slab levels (a) Single-single storey structural pounding The analysis is carried out for the structures with equal slab levels. The time history of pounding displacement responses and corresponding forces for the single-single storey structures are shown in figure 2.23 to 2.24. Since the dynamic properties of the structures are different, collision takes place between them. In practical situation, pounding between adjacent buildings may also occur when the structures have same dynamic properties due to time lag which depends mainly on the propagation of seismic waves. This effect is not considered in this study. Structural pounding is mainly dependent on the input ground motion and displacement response of the structures. From the analysis, the maximum displacement response for structure-A and B are 0.0064 and 0.0083 m respectively. Since flexible structure vibrates far from the dominant period of ground motion, the response of flexible structure is more compared to stiffer structure. The response of structures doesn’t change with their positions till they reach initial collision. It means if structure-A is replaced by structure-B and vice versa, the responses will not change till first collision. After first collision, the responses are catastrophic. But in case of nonlinear, the responses are same till first collision starts or failure of spring, whichever comes first. The calculated maximum collision force between them is 40.11x103 kN and the number of collisions are 30. At initial stage of collision, the collision force is less and its value is 15.44x103 kN. This initial collision force may become maximum depending on response of both structures. (b) Single-two storey structural pounding Since structure-B is flexible, the displacement responses are higher than in single-single storey structural pounding. The time history of pounding displacement responses and corresponding forces for the single-two storey structures are shown in figure 2.25 to 2.26. The maximum displacement response for structure-A and B are 0.0114 and 0.0112 m respectively. When any one of the structures or both vibrate near to the dominant period of the input ground motion, the response of flexible structure amplifies the response 39


Figure 2.23: Linear pounding response of two single storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.0 m)

Figure 2.24: Pounding force between two single storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.0 m)

40


of stiffer structure. If both structures vibrate far from dominant period, the response of flexible structure will be more than stiffer one. From the analysis, structure-B is vibrating near to dominant period of input ground motion, resulting high initial impact force between the structures. Because of reduction in period ratio, the first collision force increases to 18x103 kN. It means, as the period ratio approaches to zero, the first impact force will be high. The maximum collision force between two structures is 58.76x103 kN. The maximum pounding force between them is more than the force for single-single storey structural pounding, because of flexibility and less separation distance.

Figure 2.25: Linear pounding response of single-two storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.0 m)

(c) Two-two storey structural pounding The fundamental period of the structure-A is changed to 0.253 sec. The time history of pounding displacement responses and corresponding forces for the two-two storey structures are shown in figure 2.27 to 2.28. As both structures vibrate in the dominant frequency of ground motion, the responses of structure are high. The maximum displacement response for structure-A and B are 0.0330 and 0.0250 m respectively. The period ratio of structure-A to B is 0.83. Here, the period ratio is increased due to change in storey level of structure-A and fundamental period. As period ratio of structures increases, the first collision force decreases to 15x103 kN. In case-I(a)&(b), the period ratio is decreased from (a) to (b) due to structure-A. Because of decrease in period ratio, the first collision force increases. The maximum collision force between them is 147.30x103 kN. Since 41


Figure 2.26: Pounding force between single-two storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.0 m) both structures vibrate at the dominant frequency of ground motion, the displacement responses are higher than in case-I(a)&(b). Here the response of structure-A is more than structure-B. Because flexible structure amplifies the response of stiffer one. (d) Two-three storey structural pounding The fundamental period of the structure-B is changed to 0.334 sec. The time history of pounding displacement responses and corresponding forces for the two-three storey structures are shown in figure 2.29 to 2.30. The response of both structures are higher than the response in case-I(a), (b) and (c). Because they vibrate at dominant frequency. The maximum displacement response for structure-A and B are 0.0450 and 0.0200 m respectively. The response in the stiffer structure is more than the response in flexible structure, because both structures vibrate in the dominant frequency. The period ratio of the structures decreased to 0.75. Because of decrease in the period ratio, the first collision force increased to 22x103 kN. The maximum collision force between them is 147.30x103 kN. In case of nonlinear, this maximum pounding force will be much higher than linear case pounding which will be discuss in subsequent chapters. The maximum displacement responses for structure-A and B, initial and maximum collision forces and number of collisions are listed in table-2.3.

42


Figure 2.27: Linear pounding response of two-two storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.0 m)

Figure 2.28: Pounding force between two-two storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.0 m)

43


Figure 2.29: Linear pounding response of two-three storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.0 m)

Figure 2.30: Pounding force between two-three storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.0 m)

44


Table 2.3: Maximum displacement response of structures, pounding forces and number of collisions Displacement response (m) Case Structure-A Structure-B single-single 0.0064 0.0083 single-two 0.0114 0.0112 two-two 0.0330 0.0250 two-three 0.0450 0.0200

Pounding force (MN) First Maximum 15.44 40.11 18.00 58.76 15.00 147.30 22.00 175.1

No. of collisions 30 44 48 30

Discussion on structures with equal slab levels In this study, the analysis considered four different structures subjected to Northridge ground motion. From the analysis, the response of stiffer structure is more than flexible structure, when structures vibrate at dominant frequency. Also, response of flexible structure is more, when they vibrate at non-dominant frequency. As the period ratio of structures increases, the first collision force decreases. But same collision force and responses will not occur if position of structures are exchanged. It means, in case of adjacent structures with different natural periods, the most affected are the rigid structures irrespective of their position due to pounding. After first collision force, the response of structures are catastrophic. The effect of pounding is less when the adjacent structures having almost similar dynamic properties. Also, the larger the difference in the periods of adjacent structures, the effect of pounding will be more. For the design of structures against lateral loads, maximum pounding force is needed. In general, the adjacent structures are not at the same level. So it is necessary to understand the behavior of adjacent structures at unequal levels.

2.5.2

Structures with unequal heights

Pounding of adjacent structures with unequal heights is the most critical case between adjacent buildings and also a common problem in practice. To understand the pounding behavior of adjacent structures at unequal levels we consider four different structures(singlesingle, single-two, two-two and two-three) with different slab levels(2.75 m, 3.25 m and 3.5 m) subjected to Northridge ground motion and the separation distance provided between them is 2 mm. The material properties and reinforcement details are same as mentioned earlier. The geometry details of structure-B for all cases are as shown in fig 2.31 to 2.39. Also the fundamental periods of structures are tabulated in table 2.4.

45


Figure 2.31: Geometry details of structure-B @2.75 m slab levels in the single storey structure


Case-II: Structures with unequal slab levels (a) Single-single storey structural pounding XAt 2.75 m slab level In this analysis, the total height of the structure-B is 2.88 m. The time history of pounding displacement responses and corresponding forces for the single-single storey structures at 2.75 m slab level are shown in figure 2.40 to 2.41. The maximum pounding displacement response for structure-A and B are 0.0040 and 0.0070 m respectively. From the analysis, the response of flexible structure is more compared to stiff structure because, structure46



Figure 2.34: Geometry details of structure-B @2.75 m slab levels in the two storey structure B vibrates at non-dominant period of ground motion. The first and maximum collision force between them are 15x103 kN and 16.33x103 kN respectively. Here the slab level of structure-B is located just below to the slab level of structure-A. The maximum displacement responses for structure A and B are less compared with structures at equal slab level because of reduction in the period of structure-B. Now the analysis move towards to the slab level at 3.25 m. 47


Figure 2.35: Geometry details of structure-B @3.25 m slab levels in the two storey structure

Figure 2.36: Geometry details of structure-B @3.5 m slab levels in the two storey structure

Single-single storey structural pounding at 3.25 m slab level Here the slab level of structure-B is located just above to the slab level of structure-A. The time history of pounding displacement responses and corresponding forces for the 48


Figure 2.37: Geometry details of structure-B @2.75 m slab levels in the three storey structure Table 2.4: Fundamental period of the structures at different slab levels Fundamental period (sec) Structure-A Structure-B Case @ 3.0 m @ 2.75 m @ 3.25 m @ 3.50 m single-single 0.127 0.143 0.181 0.200 single-two 0.127 0.273 0.342 0.360 two-two 0.253 0.273 0.342 0.360 two-three 0.253 0.380 0.413 0.430

single-single storey structures at 3.25 m slab level are shown in figure 2.42 to 2.43. Since the structure is vibrating predominantly in the first mode, the response of the structure-B is less at 3.0 m level. The maximum displacement response for structure-A and B are 0.0063 and 0.0061 m respectively. The maximum collision force between them is 29.34x103 kN. As structure-B is flexible than structure placed at 2.75 m level, the collision force is slightly higher. Now the analysis move towards to the slab level at 3.50 m.

49


Figure 2.38: Geometry details of structure-B @3.25 m slab levels in the three storey structure

Single-single storey structural pounding at 3.50 m slab level In this analysis, the height of the structure-B changes keeping the height of structure-A constant. The time history of pounding displacement responses and corresponding forces for the single-single storey structures at 3.25 m slab level are shown in figure 2.44 to 2.45. The maximum displacement response for structure-A and B are 0.0040 and 0.0070 m respectively. The maximum collision force between them is 16.33x103 kN. Here the slab level of structure-B is located just below to the slab level of structure-A. The maximum displacement responses for structure A and B are less compared with structures at equal slab level because of reduction in the period of structure-B. The summary of all the cases is tabulated in table 2.5. In case of unequal slab levels, local damage occurs to the exterior column of the taller structure due to shorter structure.

2.6

Summary

In this analysis, structures with equal storey heights are considered which are separated from Australia, Canada, Egypt, Ethiopia, Greece, India, Mexico, Nepal, Peru, Serbia and 50


Figure 2.39: Geometry details of structure-B @3.5 m slab levels in the three storey structure Taiwan country codal provisions on separation distance. These structures are subjected to ground motions (Ref. table 2.1) having PGA ranges from 0.25-0.54 g. Using AEM, parametric study (Ref. Appendix-B) has been conducted on separation distances. These separation distances are modified with factors if the provided separation distance from codes are not sufficient for the structures. To study the linear pounding behavior, equal and unequal heights are considered under Northridge ground motion. From the analysis results, the displacement response of stiff structure is more than flexible structure, when they vibrate at dominant frequency. Also, the displacement response of flexible structure is more, when they vibrate at non-dominant frequency.

51




52




53




54


Table 2.5: Maximum displacement response of structures, pounding forces and number of collisions Case single-single @ 2.75 m @ 3.25 m @ 3.50 m single-two @ 2.75 m @ 3.25 m @ 3.50 m two-two @ 2.75 m @ 3.25 m @ 3.50 m two-three @ 2.75 m @ 3.25 m @ 3.50 m

Displacement response (m) Structure-A Structure-B

Pounding force (MN) First Maximum

No. of collisions

0.0040 0.0063 0.0082

0.0070 0.0061 0.0086

15.00 25.00 19.20

16.33 29.34 54.85

6 32 40

0.0140 0.0115 0.0108

0.0100 0.0118 0.0139

16.64 18.22 18.20

48.00 64.20 73.30

38 40 48

0.0286 0.0292 0.0350

0.0285 0.0299 0.0360

22.60 16.20 18.10

27.56 57.30 33.51

34 42 40

0.0440 0.0420 0.0440

0.0450 0.0440 0.0446

23.40 25.20 25.70

48.80 234.0 44.40

34 36 48

55

Chapter 3 Pounding Analysis With Equal Heights 3.1

Introduction

The analysis for providing minimum separation distance without pounding was discussed in Appendix-B. To study the structural pounding behavior of adjacent structures, different structures with equal roof levels are considered in this chapter. Also explained about damage analysis during pounding.

3.2

Non-linear analysis of pounding

In order to achieve a safe economical seismic structural design during seismic events, nonlinear analysis is essential. To investigate the structural pounding behavior using nonlinear analysis, a study has been conducted on the structures with equal heights separated by 2 mm under Northridge ground motion. The material and geometrical details of the structures which are considered in this analysis are mentioned in chapter-2. Also discussed about the material model and failure criteria of the material in the previous chapter. The analysis is carried out for structures(single-single, single-two, two-two and two-three) with equal heights. Case-I: Structures with equal slab levels (a) Single-single storey structural pounding In order to achieve a safe structural pounding during earthquakes, a correct seismic analysis should take into account by means of nonlinear time history analysis. The nonlinear behavior of structures will be same as linear, till failure of first spring in the structures. This failure of spring occurred, when the principle stress of spring exceeds the tensile resistance value. Also the forces in normal and shear spring are redistributed in next increment by applying the forces in opposite direction. The stiffness of the structure will reduce when the spring has failed. Because of reduction in stiffness, the frequency of 56

CHAPTER 3. POUNDING ANALYSIS WITH EQUAL HEIGHTS

structure will come down and structure vibrates with lesser frequency than fundamental frequency of structure. The pounding displacement responses and corresponding forces for structure-A & B are shown in figure 3.1 to 3.2. From the result, structure-B is vibrating with lesser frequency compared to structure-A. Also, pounding increases the response of flexible structure(structure-B) because, both structures vibrate in non-dominant frequency. Here, the first spring has failed at the column base(3.71 m, 0.06 m) of second structure. The first collision occurred after failure of spring. Since the first collision occurred after failure of springs, the first collision force is not equal to the force obtained from linear analysis. The magnitude of collision forces and accelerations are less compared to linear analysis. The maximum displacement responses for structure-A and B are 0.092 m and 0.098 m respectively. These responses are 5-10 times more than the responses obtained from linear analysis. The first and maximum collision forces are 14.57x103 and 23.22x103 kN respectively.

Figure 3.1: Pounding nonlinear response of two single storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.0 m)

(b) Single-two storey structural pounding A two storey structure with period 0.304 sec is considered in this analysis. The nonlinear responses for structure-A and B are similar as linear analysis till first collision. The pounding displacement responses and corresponding forces for structure-A & B are shown 57


Figure 3.2: Pounding force between two single storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.0 m) in figure 3.3 to 3.4 respectively. During this process, the first spring has failed at beam location(3.72 m, 3.11 m) of structure-B and the force generated between them is 18x103 kN which is equal to the first collision force in linear analysis. Because of failure of springs, both structures vibrate at lesser frequency than fundamental frequency of structures and structure-B frequency is lesser than structure-A frequency. The maximum displacement responses for structure-A and B are 0.126 and 0.127 m respectively. The response of structures are more than the responses from case-I(a), because of increase in flexibility of structure-B. Also structure-B is responding at dominant frequency. The nonlinear responses for both structures are nearly 10 times more than the responses obtained from linear analysis. The maximum collision force between them is 35x103 kN, also number of occurances of collisions are less than linear case. From figure 3.4, the duration of last collision is around 2 sec. It is because of more intact time for both structures. (c) Two-two storey structural pounding Two two storey structures having period of 0.253 sec and 0.304 sec are considered in this analysis. The pounding displacement responses and corresponding forces for structure-A & B are shown in figure 3.5 to 3.6 respectively. The maximum displacement responses for structure-A and B are 0.20 and 0.19 m. Since both structures are vibrating at dominant frequency, the response for flexible structure is lesser than stiff structure. Whereas, in the above two cases(I(a) & (b)) the response of flexible structure was more compared to

58


Figure 3.3: Pounding nonlinear response of single-two storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.0 m)

Figure 3.4: Pounding force between single-two storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.0 m)

59


stiff structure because, they vibrate at non-dominant frequency. Also, the response for both structures are similar till first failure or collision. The first collision force generated between them is same as the force in linear case and its magnitude is 15x103 kN. During initial collision, failure of spring occurred at beam location(0.24 m, 3.10 m) of structure-A. Because of this failure, both structures vibrate at lesser frequencies. But the maximum collision force depends on displacement and velocity of the structure. Since the separation distance and input ground motion are same for all the cases, the maximum collision forces are increased as the flexibility of the structures increases. The maximum collision force between them is 73.12x103 kN and also the number of collisions are less than linear case.

Figure 3.5: Pounding nonlinear response of two-two storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.0 m)

(d) Two-three storey structural pounding A two storey structure with period 0.304 sec is considered in this analysis. The pounding displacement responses and corresponding forces for structure-A & B are shown in figure 3.7 to 3.8 respectively. The maximum displacement responses for structure-A and B are 0.246 and 0.210 m respectively. Since the structures are vibrating at dominant period, the response of flexible structure is lesser than stiff structure. During collision between two structures, the first spring has failed at beam location(6.78 m, 3.11 m) of structureA. This is because of reduction in stiffness of structure. The maximum collision force is much higher than case-I(a), (b) and (c) due to less separation distance, more flexible structures and high velocities of structures during motion. The damage will also be higher 60


Figure 3.6: Pounding force between two-two storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.0 m) for structures than the damage for case-I(a), (b) and (c) structures which will be discussed in damage analysis. The maximum collision force between them is 267.5x103 kN and also the number of collisions are less than linear case. The brief overview of structures which are discussed above are summarized and tabulated below: Table 3.1: Maximum nonlinear displacement response of structures, pounding forces and number of collisions Displacement response (m) Case Structure-1 Structure-2 single-single 0.0926 0.0979 single-two 0.1260 0.1271 two-two 0.2000 0.1900 two-three 0.2460 0.2100

Pounding force (MN) First Maximum 14.57 23.22 18.00 35.00 15.00 73.12 22.00 267.5

No. of collisions 2 8 16 8

Discussion This analysis considered four different cases with different structures to study the pounding behavior. This behavior is similar as linear till failure of either first spring or first collision. The responses for flexible structures are less compared to stiff structures when structures vibrate at dominant period and also the responses for flexible structures are 61


Figure 3.7: Pounding nonlinear response of two-three storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.0 m)


62


more when structures vibrate at non-dominant period. Almost every structure vibrate at lesser frequencies because of failure of springs. The maximum pounding forces increases, as the period of structure increases with the same separation distance. The quantification of damage will be discussed in next section.

3.3

Damage analysis

From the past earthquakes such as 1991 Uttarkasi earthquake, 1993 Killari earthquake, 1997 Jabalpur earthquake, 1999 Chamoli earthquake, 2001 Bhuj earthquake and 2006 Sikkim earthquake it has been observed that the potential damage for RC structures is more due strong ground shaking. In order to implement seismic safety for RC structures, the quantitative analysis of structural damage is necessary. In this section, the quantification of damage is proposed interms of stiffness and strength degradation for structures under Northridge ground motion.

3.3.1

Damage model

In case of concrete structures, damage indices has been developed to provide a way to quantify numerically the seismic damage sustained by individual elements, storeys or complete structures. These indices are based on the results of a nonlinear dynamic analysis during an earthquake or on a comparison of a structure’s physical properties before and after an earthquake. The brief decription about Park-Ang damage model is as follows: Under seismic loading, RC structures are generally getting damaged. This damage of RC structures can be expressed as a linear combination of maximum deformation and absorbed hysteretic energy. Park et al., (1985) expressed seismic structural damage as follows: D=

EH xm +β xu Qy xu

(3.1)

Where, xm is the maximum displacement that the linear equivalent linear SDOF system would be subjected to during the base excitation, xu (=µ xy where µ is the ductility) is the ultimate displacement of the system under monotonic loading, β represents the effect of cyclic loading on structural damage, EH represents the total energy dissipation in the structure during excitaiton and Qy is the yield strength of the structure. Later it was modified by Kunnath (Ramancharla, 1997) and is as follows: D=

xm − xy EH +β xu − xy Q y xu

(3.2)

In the above equation, the first term doesn’t take an account of cumulative damage whereas the second term does. The advantage of this model is its simplicity. In this study, eq 3.2 is used for estimating the amount of damage. In 1985 Park et.al proposed regression equations for strength detoriation parameter β, interms of numerical variables 63


including shear span ratio, axial load, longitudinal and confining reinforcement ratios and material strengths. Generally in RC structures, failure occurs at beam-column joints. Hence it is necessary to find out the global damage of the structure. It is to be obtained as the summation of local damage. The global damage (Gomez et al., 1990) is estimated as follows: Pn i=1 Di Ei (3.3) Dg = P n i=1 Ei Where, Di is the local damage index at ith location, Ei is the energy dissipated at ith location and n is the number of locations at which the local damage computed.

3.3.2

Damage calculation

Damage index is one of the parameters which describes the state of structures. In this analysis, global damage of structure is calculated. The ductility (Paulay et al., 1992) values calculated for the beam and column members in a single storey structure(Ref. 2.2.1) are 1.85 and 1.84 respectively. β is a component which is obtained from test data. Mathematically, the β value can be found out from the below equation: l β = (−0.447 + 0.73 + 0.24no + 0.314Pt )0.7ρω d

(3.4)

Where, dl is shear span ratio, Pt is longitudinal steel ratio, no is axial stress and ρω is confinement ratio. Generally β ranges from -0.2 - 2.0. Park et al., (1985) established the value of β is of the order of 0.05 for reinforced concrete members. In this study the value of β is taken as 0.05. Yielding of the reinforced concrete is defined as when the extreme fibre compressive strain in the concrete exceeds 1.5 times the crushing strain. Theoretically, the yield deformation is the combination of deformation due to flexural component, bond slippage of the reinforcing bar from anchorage, inelastic and elastic shear deformation. The yield displacement obtained from the above relation is 0.01 m. The local and global damage of the single-storey structure(Ref. 2.2.1) are calculated using eq 3.2 and 3.3. The global damage of the structure is 3.0 under Northridge ground motion. It is dependent on crack length, width and location of crack such as beam column joint, base of column etc. The damage values are not same for all the structural members. Depending on location of crack, the damage value changes. A damage scale has been introduced for estimating damage interms of stiffness and strength degradation.

3.3.3

Proposed damage scale

In this analysis, a damage scale has been proposed for the structures which are mentioned earlier. The load-displacement plot has drawn for each structure till collapse. The collapse of the structure might be stiffness or strength degradation. The actual load-displacement 64


plot is smoothened using smoothing techniques. The damage state of structure can be found out from smooth curve. Now the whole curve is divided into two categories. (i) Based on stiffness degradation-the range is from zero to maximum loading and (ii) strength degradation-the range is from maximum loading to complete failure of structure. From the displacement response of each structure, the end response is considered. If the end response is at category-1, the damage of structure will be from stiffness degradation. Otherwise, it will be from strength degradation. From the figure 3.9, it has been observed that as the applied displacement on structure increases, the lateral load also increases till the displacement reaches 0.04 m. Continuous failure of springs are observed up to 0.055 m. It means that damage has taken place in the structure. Thereafter, readjustment of particles in the structure takes place. Since the analysis allows failure of the reinforcement also, it occurred in the structure at 0.218 m. Because of failure in the reinforcement at critical locations, the load drastically comes down with small increment in the displacement. Thereafter, no failure in reinforcement is observed till 0.23 m. There might be a possibility of more number of reinforcement bar failures at single increment also. From figure 3.1, the end response of structure-A is 0.075 m. The initial stiffness of structure-A is 1.161x104 kN/m. From figure 3.9, the decrease in stiffness at 0.075 m is 17.3%. Now the end response of structure-A has come under category-1. From this analysis, one can able to find out that the structure has under gone stiffness or strength degradation without much doing the analysis. The same analysis is carried out for the remaining structures and the load vs displacement curves are shown in figure 3.10 to 3.13. The stiffness degradation for all structures which are considered in this analysis are tabulated below: Table 3.2: Stiffness degradation for different structures during pounding Stiffness Degradation Case Structure-1 Structure-2 single-single 17.3 1.3 single-two 2.16 3.5 two-two 3.1 2.7 two-three 3.4 5.5

3.4

Summary

This study is carried out four(single-single, single-two, two-two and two-three storey) structures under Northridge ground motion with equal heights. To study the pounding behavior, nonlinear analysis is conducted on the structures. The behavior of the structures is similar to linear behavior till failure of either first spring or first collision. Because of springs failure, structures vibrate at lesser frequency and also reduce the pounding force. 65


Figure 3.9: Load vs displacement curve for structure having a period of 0.127 sec

Figure 3.10: Load vs displacement curve for structure having a period of 0.155 sec The damage is proposed interms of stiffness degradation for all the structures.

66




67



68

Chapter 4 Pounding Analysis With Un-equal Heights 4.1

Introduction

Most of the studies on pounding are with buildings that have equal storey levels. In general, the slabs of structure hit the columns of other structure which are very common pounding cases during earthquakes. The case of colliding structures with unequal heights has not yet been studied effectively. In this chapter, different buildings with unequal roof levels of different storey heights are considered. The analysis is carried out for the buildings to study the damage due to pounding.

4.2

Pounding analysis for structures with un-equal heights

In this case, nonlinear analysis is considered for the structures with unequal heights separated by 2 mm subjected to Northridge ground motion. The geometry details of structures are discussed in chapter-2. For nonlinear analysis the material model and failure criteria (Hatem, 1998) has already been discussed in chapter-2. The analysis considers fourteen different structures with different heights subjected to Northridge ground motion.

4.2.1

Single-single storey structural pounding

XAt 2.75 m slab level: In this analysis, structure-B is kept just below to the slab level of structure-A. The pounding displacement responses and corresponding forces for structure-A & B are shown in fig 4.1 to 4.2 respectively. From results, the maximum pounding displacement responses for structure-A and B are 0.084 m and 0.106 m respectively. The maximum response of flexible structure is more than stiff structure, because both structures vibrate at nondominant period of ground motion. From these pounding responses collision force will 69

CHAPTER 4. POUNDING ANALYSIS WITH UN-EQUAL HEIGHTS

generate. From the results of equal slab levels, it is clearly shows that the first collision forces are more in case of unequal slab levels. After first collision, the behavior is catastrophic. In case of unequal storey levels, the shear on the column will be more than the shear at equal slab levels. This shear force may change at different height levels of the same structure which depends on the pounding response of the structures. This shear amplification will cause more damage to the column member. The first collision force between them is 18x103 kN which occured after failure of first spring. It has failed at the base(3.71 m, 0.06 m) of structure-B.


XAt 3.25 m slab level: In this analysis, structure-B is kept just above to the slab level of structure-A. The pounding displacement responses and corresponding forces for structure-A & B are shown in fig 4.3 to 4.4 respectively. From results, the maximum pounding displacement responses for structure-A & B are 0.103 m and 0.107 m respectively. The maximum pounding response of flexible structure is more than stiff structure, because both structures vibrate at non-dominant period of ground motion. The first spring has failed at the base(3.71 m, 0.06 m) of structure-B. The stiffness of the structure gets reduced because of failure of spring which causes the first collision force between them and its magnitude is 19.30x103 kN. Also this collision force is less than the force obtained from linear analysis. If the first spring has failed after first collision, the collision force from nonlinear analysis would 70


Figure 4.2: Pounding force between two single storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 2.75 m) be same as linear analysis. Because of nonlinearity, both structures vibrate at lesser frequencies.

XAt 3.50 m slab level: In this analysis, the slab level of structure-B is kept at 3.50 m. The pounding displacement responses and corresponding forces for structure-A & B are shown in fig 4.5 to 4.6 respectively. From results, the maximum pounding displacement responses for structureA & B are 0.107 m and 0.104 m respectively. The maximum response of stiff structure is more than flexible structure because structure-B vibrates at dominant period of ground motion. The first collision force generated between them is 16.20x103 kN. Because of this force, the response of stiffer structure gets amplified. The collision force is less than the force obtained from linear analysis. The first spring has failed at the base(3.71 m, 0.06 m) of structure-B. Because of nonlinearity, both structures vibrate at lesser frequencies.

Discussion: In this analysis, two single storey structures are considered with unequal heights. From the results, we can conclude that the response of stiff structure is more than flexible structure at dominant period of ground motion irrespective of equal and unequal levels. If the structures vibrate at non-dominant period of ground motion then, the response of flexible structure will be more than stiff structure. In case of equal slab levels, the collision 71




72




73


between slab to slab is a rigid body motion whereas, for unequal levels the interaction is in between slab to column. During this interaction, shear causes more damage to the column which leads to collapse of structure.

4.2.2

Single-two storey structural pounding

XAt 2.75 m slab level: The same analysis is for single-two storey structures. Now the storey height of structureB is kept as 2.75 m which is just below to the slab level of structure-A. The pounding displacement responses and corresponding forces for structure-A & B are shown in fig 4.7 to 4.8 respectively. From results, the maximum pounding displacement response for structure-A & B are 0.115 m and 0.102 m respectively. Since structure-B is vibrating at dominant period of ground motion, the response of stiff structure is more than flexible structure. Whereas, in the earlier case(single-single) the response of flexible structure is more because of non-dominant period of ground motion. The initial collision force generated between them is 16.80x103 kN which is almost similar as linear analysis. This force occurred during failure of springs and the initial spring has failed at column level(3.72 m, 2.86 m) of structure-B. The maximum collision force between them is 80x103 kN. Due to reduction in stiffness, structures vibrate at lesser frequencies.


74



XAt 3.25 m slab level: The storey height of structure-B is kept at 3.25 m which is just above to the slab level of structure-A. The pounding displacement responses and corresponding forces for structureA & B are shown in fig 4.9 to 4.10 respectively. From results, the maximum pounding displacement responses for structure-A & B are 0.128 m and 0.132 m respectively. Even though structure-B is vibrating at dominant period of ground motion, the response of flexible structure is more compared to stiff structure. But the difference in the responses is less. In case of structure(single-two) with equal levels, the amplification in stiff structure is more comapred to flexible structure. The initial collision force between them is 18.20x103 kN which is almost similar to the collision from linear analysis. This force occurred during failure of springs. The initial spring has failed at the column level(3.72 m, 3.33 m) of structure-B. Due to reduction in stiffness structures vibrate at lesser frequencies. The maximum collision force between them is 54.10x103 kN.

XAt 3.50 m slab level: The storey height of structure-B is kept at 3.50 m. The pounding displacement responses and corresponding forces for structure-A & B are shown in fig 4.11 to 4.12 respectively. From results, the maximum pounding displacement responses for structure-A & B are 0.124 m and 0.123 m respectively. Since structure-B is vibrating at dominant period of ground motion, response amplification in stiff structure is more. Depending on structural 75




76


properties and response of structures, the initial collision force changes. During failure of spring at column level(6.72 m, 3.47 m), the first collision force generated between them is 18.10x103 kN. Because of reduction in stiffness, both structures vibrate at lesser frequencies. The maximum collision force between them is 19.92x103 kN. The maximum collision force between them is 54.10x103 kN.


XDiscussion: In this analysis single-two storey structures are considered with unequal heights. In most of the cases, flexible structures amplifies the response of stiff structure because of dominant period of ground motion. The number of occurrances of collisions are also decreased when compared with linear analysis. This is due to vibration of structures at lesser frequencies. In case of single-single(Ref. 4.2.1) structural pounding, initial pounding forces are different than linear analysis. This is because of reduction in stiffness before first collision.

4.2.3

Two-two storey structural pounding

XAt 2.75 m slab level: The analysis is carriedout for two-two storey structures with a storey height of 2.75 m. The pounding displacement responses and corresponding forces for structure-A & B are

77


Figure 4.12: Pounding force between single-two storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.5 m) shown in fig 4.13 to 4.14 respectively. From results, the maximum responses for structureA & B are 0.195 m and 0.185 m respectively. Because of non-dominant period of ground motion, the response of stiff structure gets amplified. The first spring has failed at the column level(0.24 m, 3.10 m) of structure-A which causes the initial collision force with magnitude of 15.50x103 kN. The initial collision force from linear analysis is different from nonlinear analysis. This is due to failure of first spring before collision. The stiffness of structure gets reduced because of failure of springs resulting structures vibrate at lesser frequencies.

XAt 3.25 m slab level: The storey height of structure-B is kept at 3.25 m which is just above to the slab level of structure-A. The pounding displacement responses and corresponding forces for structureA & B are shown in fig 4.15 to 4.16 respectively. From results, the maximum pounding displacement responses for structure-A & B are 0.218 m and 0.217 m respectively. Here, the maximum responses for both structures are same even though they vibrate at dominant period of ground motion. Whereas, in the above two cases(single-single, single-two) the maximum responses for flexible structure are more. The initial collision force between them is 15.80x103 kN. This force occurred during failure of springs. The initial spring has failed at the column level(0.24 m, 3.10 m) of structure-A. Due to reduction in stiffness structures vibrate at lesser frequencies. The maximum collision force between them is

78




79


70x103 kN.


XAt 3.50 m slab level: The storey height of structure-B is kept at 3.50 m. The pounding displacement responses and corresponding forces for structure-A & B are shown in fig 4.17 to 4.18 respectively. From results, the maximum pounding displacement responses for structure-A & B are 0.216 m and 0.215 m respectively. Here, the maximum responses for both structures are same even though they vibrate at dominant period of ground motion. Depending on structural properties and response of structures, the initial collision force changes. During failure of spring at column level(6.72 m, 3.47 m), the first collision force generated between them is 17.50x103 kN. Because of reduction in stiffness, both structures vibrate at lesser frequencies. The maximum collision force between them is 136.80x103 kN.

XDiscussion: In this analysis, two-two storey structures are considered with unequal heights. From the results, we can conclude that the response of stiff structure is more than flexible structure at dominant period of ground motion irrespective of equal and unequal levels. But the initial collision force changes at different levels. These are dependent on the response of both structures and are less than from linear analysis. The initial collision occurs during 80




81


Figure 4.18: Pounding force between two-two storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.5 m) failure of springs at 3.25 m and 2.75 m. After initial collision, the response of structures are catastrophic. Due to this, the maximum collision force would be high value.

4.2.4

Two-three storey structural pounding

XAt 2.75 m slab level: The same analysis is carriedout for two-three storey structures. Now the storey height of structure-B is kept as 2.75 m which is just below to the slab level of structure-A. The pounding displacement responses and corresponding forces for structure-A & B are shown in fig 4.19 to 4.20 respectively. From results, the maximum pounding displacement response for structure-A & B are 0.225 m and 0.190 m respectively. Since structure-B is vibrating at dominant period of ground motion, the response of stiff structure is more than flexible structure. Except single-single storey structural pounding, in all other cases, the response of stiff structures is more compared to flexible structures. This is because of dominant vibration of ground motion. The initial collision force generated between them is 25.95x103 kN. This force occurred during failure of springs and the initial spring has failed at column level(6.78 m, 2.87 m) of structure-B. The maximum collision force between them is 25.95x103 kN. This means that the maximum force and initial force are equal. Due to reduction in stiffness, structures vibrate at lesser frequencies.

82




83


XAt 3.25 m slab level: In this analysis, structure-B is kept just above to the slab level of structure-A. The pounding displacement responses and corresponding forces for structure-A & B are shown in fig 4.21 to 4.22 respectively. From results, the maximum pounding displacement responses for structure-A & B are 0.217 m and 0.216 m respectively. The maximum pounding response of stiff structure is more than flexible structure, because both structures vibrate at dominant period of ground motion. The first spring has failed at the base(6.78 m, 3.35 m) of structure-B. The stiffness of the structure gets reduced because of failure of spring which causes the first collision force between them and its magnitude is 25.40x103 kN. Also this collision force is less than the force obtained from linear analysis. If the first spring has failed after first collision, the collision force from nonlinear analysis would be same as linear analysis. Because of nonlinearity, both structures vibrate at lesser frequencies.


XAt 3.50 m slab level: In this analysis, the slab level of structure-B is kept at 3.50 m. The pounding displacement responses and corresponding forces for structure-A & B are shown in fig 4.23 to 4.24 respectively. From results, the maximum pounding displacement responses for structureA & B are 0.223 m and 0.222 m respectively. The maximum response of stiff structure is more than flexible structure because structure-B vibrates at dominant period of ground motion. The first collision force generated between them is 25.37x103 kN. Because of this 84


Figure 4.22: Pounding force between two-three storey structures separated by 2 mm subjected to Northridge ground motion (@ slab level 3.25 m) force, the response of stiff structure gets amplified. The collision force is less than the force obtained from linear analysis. The first spring has failed at the base(6.78 m, 3.59 m) of structure-B. Because of nonlinearity, both structures vibrate at lesser frequencies.

Discussion: In this analysis, two-three storey structures are considered with unequal heights. From the results, we can conclude that the response of stiff structure is more than flexible structure at dominant period of ground motion irrespective of equal and unequal levels. If the structures vibrate at non-dominant period of ground motion then, the response of flexible structure will be more than stiff structure. The summary of all cases are tabulated in table 4.1.

4.3

Summary

The analysis considered for the structures under Northridge ground motion with unequal heights. The nonlinear pounding responses are higher than linear responses because of reduction in stiffness. The pounding forces produced by two adjacent structures are higher than forces from linear case. The maximum collision forces are different due to catastrophic behavior of structures after first collision. This can be clearly seen from table 4.1. The maximum pounding responses are more for stiffer structures, when they vibrate at 85




86


Table 4.1: Maximum nonlinear displacement response of structures, pounding forces and number of collisions for structures with unequal heights

Case single-single @ 2.75 m @ 3.25 m @ 3.50 m single-two @ 2.75 m @ 3.25 m @ 3.50 m two-two @ 2.75 m @ 3.25 m @ 3.50 m two-three @ 2.75 m @ 3.25 m @ 3.50 m

Displacement response (m) Structure-1 Structure-2

Pounding force (MN) First Maximum

No. of collisions

0.084 0.103 0.107

0.106 0.107 0.104

18.00 19.30 16.20

22.83 58.05 16.23

6 6 14

0.115 0.128 0.124

0.102 0.132 0.123

16.80 18.20 18.10

80.00 54.10 19.92

8 6 10

0.195 0.218 0.216

0.185 0.217 0.215

15.50 15.80 17.50

116.30 70.00 136.80

10 6 16

0.225 0.217 0.223

0.190 0.216 0.222

25.95 25.40 25.37

25.95 26.65 64.50

8 8 6

predominant frequency of ground motion irrespective of equal or unequal slab levels.

87

Chapter 5 3D Analysis of Pounding 5.1

Introduction

Numerical simulation of contact between two adjacent structures under the action of seismic load involves many complexities. The methods for solving contact problems can be categorize into mass to mass, node to node and node to surface contact(i.e., arbitrary contact in 3D). The objective of this chapter is to study the torsional pounding behavior of adjacent structures subjected to El-centro ground motion.

5.2 5.2.1

Modeling of structures in 3D Geometry and material details

The analysis considers two single storey structures with symmetric (structure-A) and asymmetric (structure-B) configuration. Different setback levels (1.5 m, 3.0 m and 6.0 m) and structure at different height levels (at (2/4)th and (3/4)th of column height) are considered in this study. The modeling of structures has done using SAP 2000 (CSI). The total height for both structures is 3 m. The thickness of the slab is 0.12 m and the dimensions for all columns are 0.24 x 0.24 m. The slab dimensions for both structures are taken as 6 x 6 m. It is assumed that the live load acting on the structure-A is 2 kN/m2 . To study the torsional behavior of structures due to pounding an eccentricity is provided in both x and y directions for structure-B. The centre of mass (CM) and centre of stiffness (CS) for structure-B are (2.7 m, 3.3 m) and (3.0 m, 3.0 m) respectively. An additional load of 4 kN/m2 is provided at top left corner of slab portion for structure-B. The plan and elevation view of structures are shown in fig 5.1. The material properties such as grade of concrete, grade of steel reinforcement and poission’s ratio are same as mentioned in chapter-2. The details of gap element model and material model used for nonlinearity in SAP 2000 are described in the following sections.

88

CHAPTER 5. 3D ANALYSIS OF POUNDING

Figure 5.1: Geometry details of structures

5.2.2

Gap element model

Gap joint element is an element which connects two adjacent nodes to model the contact. This is activated when structures come closer and deactivate when they go far away. A collision force will be generated when they come closer. From figure 5.2, it is shown that, the gap element will activate if ’open’ is equal to zero. In SAP modeling each element is assumed to be composed of six separate springs with six deformational degree

89


of freedom (DOF) as shown in fig 5.3. Every DOF may has linear effective stiffness and damping properties. The mass contributed by the link or support element is lumped at the joints i & j and half of the mass is assigned to the three translational degrees of freedom at each of the elements joint. No inertial effects are considered within the element itself. Generally the effective stiffness value is in a range from 102 to 104 times more than the stiffness in any connected elements. Larger values of effective stiffness may leads to numerical difficulties during solution. During nonlinear analysis, the nonlinear force-deformation relationships are used at all degrees of freedom for which nonlinear properties were specified. For all other degrees of freedom, the linear effective stiffnesses are used during a nonlinear analysis. The results for linear analyses are based upon linear effective stiffness and damping properties. Only the results for nonlinear analysis cases include the nonlinear behavior. The force-deformation relationship is as follows:  k(d + open), if (d + open) < 0 (5.1) f= 0, otherwise Where, k is spring constant, open is the gap opening which must be positive or zero and d is the relative deformation across the spring.

Figure 5.2: Gap-joint element from SAP 2000

Figure 5.3: Link element internal forces and moments at the joints

90


5.2.3

Non-linear analysis of pounding

To study torsional pounding between the structures, different setbacks are considered in this analysis. The separation distance between two adjacent structures as 0.01 m. The structures which are considered in this analysis are subjected to El-centro ground motion and gap element is provided between them. The stiffness of gap element is 477.6x103 kN/m (Muthukumar et al., 2004). The fundamental natural periods of structure-A & B are 0.176 and 0.217 sec respectively which are far from predominant period of the ground motion. Case-I: Different setback (a) At setback of 1.5 m The analysis is carried out for structures with setback of 1.5 m. The structures have responses in both x and y directions, though the ground motion is restricted to x-direction. This is because of un-symmetric mass properties about X & Y axes. The pounding at top edge of structure-B is referred as location ’Ct ’ and the bottom edge of structure-A is referred as location ’Cb ’ (Ref. fig 5.1). The maximum pounding responses at location Ct for structure-A and B are 0.0351 m and 0.043 m respectively. The pounding responses would be more if structures vibrate near to the predominant period of ground motion. In this case, both structures vibrate far from predominant period of ground motion. Resulting more response in flexible structure compared to stiff structure. During vibration, the response of structure-A in y-direction is amplified due to collision between structure-A & B. The maximum pounding responses for structure-A and B in y-direction are 0.00138 m and 0.0073 m respectively. From results, it has been clearly observed that the responses for stiffer structure get amplified due to collision between them. Also, structure-B vibrates at lesser frequency because of reduction in stiffness. The maximum pounding force between them is 53.16 kN. The displacement responses for structure-A & B in x-y directions and collision forces are shown from fig 5.4 to 5.6. Due to unsymmetrical mass property of structure-B, the possible pounding may occur at Cb also. From the results, the maximum responses of structure-A & B in X and Y directions are 0.0344 m, 0.033 m, 0.00138 m and 0.0073 m respectively. It is clearly show that the maximum pounding responses at Ct and Cb are same in Y-direction. The displacement responses for structure-A & B in x-y directions and collision forces are shown from fig 5.7 to 5.8. (b) At setback of 3.0 m The similar analysis is carried out for structures with setback of 3.0 m. The maximum pounding responses at location Ct for structure-A and B are 0.0347 m and 0.0432 m respectively. The response of flexible structure is more compared to flexible structure, because of non-dominant period of ground motion. In the earlier case, more response 91


Figure 5.4: Response of structures in x-direction at location Ct with setback of 1.5 m

Figure 5.5: Pounding force between structures with setback of 1.5 m is observed at location Ct than at Cb . This is due to eccentricity in mass distribution resulting torsion induced in the system. This torsion in structure-B initiates the colli92


Figure 5.6: Response of structures in y-direction at location Ct with setback of 1.5 m

Figure 5.7: Response of structures in x-direction at location Cb with setback of 1.5 m sion to structure-A with a magnitude of 73.7 kN. The maximum pounding responses for structure-A and B in y-direction are 0.00428 m and 0.0090 m respectively. The collision 93


Figure 5.8: Response of structures in y-direction at location Cb with setback of 1.5 m force increases as the setback level increases. The displacement responses for structure-A & B in x-y directions and collision forces are shown from fig 5.9 to 5.11. The pounding occurred at location Cb also. From the results, the responses at location Cb are less compared at location Ct because of uneven distribution of mass for structureB. The responses are similar at location Ct and Cb in Y-direction (Ref. figure 5.8. For more details refer figure 5.1). The maximum pounding responses for structure-A and B in x-direction are 0.0344 m and 0.0376 m respectively. The displacement responses for structure-A & B in x direction are shown from fig 5.12. (c) At setback of 6.0 m The analysis is carried out for structures with setback of 6.0 m. In this analysis, the possible pounding location is at one place only. From the results, the maximum pounding responses for structure-A and B are 0.0344 m and 0.0432 m respectively. Because of nondominant period of ground motion, response of flexible structure is more compared to stiff structure. The maximum force generated between them is 169.4 kN. From the observation of all results, it is clearly shown that the maximum pounding forces are increasing as the setback level increases. The number of collisions are also same for all the cases. The displacement responses for structure-A & B in x-y directions and collision forces are shown from fig 5.13 to 5.15.

94



Figure 5.10: Pounding force between structures with setback of 3.0 m From results, the response for both structures increases as the setback level increases. But this increase in response is not significant. But the collision force between them 95


Figure 5.11: Response of structures in y-direction at location Ct with setback of 3.0 m

Figure 5.12: Response of structures in x-direction at location Cb with setback of 3.0 m increase significantly. The number of collision occurrences are same in all the cases. It can be concluded that the effect of collision is more when structures are kept at extreme 96



Figure 5.14: Pounding force between structures with setback of 6.0 m levels of setback.

97


Figure 5.15: Response of structures in y-direction at location Ct with setback of 6.0 m Case-II: Different height levels (a) At (3/4)th height of structure Pounding analysis is carried out for structures at different elevation levels. From the analysis, the maximum pounding response of structure-A and B in X-direction are 0.21 m and 0.120 m respectively. Due to unsymmetrical mass of structure-B, they have responses in both directions. Because of non-dominant period of ground motion, the response of flexible structure is more than stiff structure. The responses of structure-A and B in Y-direction are 0.052 m and 0.025 m respectively. The maximum collision force between them is 1244 kN. The displacement responses for structure-A & B and collision forces are shown from fig 5.16 to 5.17. The maximum pounding response of structure-A and B in X and Y direction are 0.126 m, 0.170 m, 0.016 m and 0.062 m respectively. The maximum pounding force generated between them is 777 kN. It is clearly show that, the pounding forces are more at where mass concentration is more. If we change the mass concentration to other location (top right corner of structure-B), the responses and collision forces will be change. The displacement responses for structure-A & B and collision forces are shown from fig 5.18 to 5.19.

98


Figure 5.16: Response of structures in x-direction at location Ct with height of 2.25 m

Figure 5.17: Pounding force between structures with height of 2.25 m (b) At (2/4)th height of structure The same analysis is carried out for structures at mid height of structure-A. From the analysis, the maximum pounding response of structure-A and B in X and Y direction are 99


Figure 5.18: Response of structures in x-direction at location Cb with height of 2.25 m

Figure 5.19: Pounding force between structures with height of 2.25 m 0.20 m, 0.168 m, 0.0053 m and 0.02 m respectively. The response of flexible structure is more than stiff structure due to non-dominant period of ground motion. The collision 100


force between them is 5350 kN. This force is more than the force when structures are kept at (3/4)th height. From the results it can conclude that the collision force at mid height level is more than (3/4)th height level because of more shear amplification. Depending on the response of structures, this collision force changes at different levels. The displacement responses for structure-A & B and collision forces are shown from fig 5.20 to 5.21.

Figure 5.20: Response of structures in x-direction at location Ct with height of 1.5 m The maximum pounding response of structure-A and B in X and Y direction are 0.20 m, 0.127 m, 0.005 m and 0.022 m respectively. The maximum pounding force generated between them is 3493 kN. From results, the maximum pounding force at location Ct is more than at Cb . This is because of uneven mass distribution resulting torsional effect. The displacement responses for structure-A & B and collision forces are shown from fig 5.22 to 5.23. From the results, the pounding response changes significantly as the height of structure decreases. At (2/4)th height of structure, the collision force is more compared to (3/4)th height. It can be concluded that as the height of structure increases, the collision force decreases.

5.3

Summary

This analysis considered different setback levels for a single-single storey structure. The setback levels were 1.5 m, 3.0 m and 6.0 m. The separation distance between them is 0.01 101


Figure 5.21: Pounding force between structures with height of 1.5 m

Figure 5.22: Response of structures in x-direction at location Cb with height of 1.5 m m and they subjected to El-centro ground motion.

102


Figure 5.23: Pounding force between structures with height of 1.5 m Setback Levels: From results, the response for both structures increases as the setback level increases. But this increase in response is not significant. But the collision force between them increase significantly. The number of collision occurrences are same in all the cases. It can be concluded that the effect of collision is more when structures are kept at extreme levels of setback. The response of flexible structure is more than stiff structure when they vibrate at non-dominant period of ground motion. As the setback distance increases, the collision force between them also increases. Height Levels: From the results, the pounding response changes significantly as the height of structure decreases. At (2/4)th height of structure, the collision force is more compared to (3/4)th height because of shear amplification. It can be concluded that as the height of structure increases, the collision force decreases.

103

Chapter 6 Conclusions 6.1

General Conclusions

The analysis considered linear and nonlinear pounding analysis of structures with equal and unequal heights. The damage of structures are estimated from stiffness degradation. This analysis was extended to 3D also. The main conclusions for all the cases can be summarized as follows: Separation distance from different codal provisions: The separation distances are modified with modification factor in which it is insufficient. The separation distance does not only depends on PGA but also depends on several other factors like displacement and velocity responses, material properties of the structures and characteristics of ground motion. Linear and Nonlinear analysis: From the analysis, the response of stiffer structure is more than flexible structure, when structures vibrate at dominant frequency. Also, response of flexible structure is more, when they vibrate at non-dominant frequency. But same collision force and responses will not occur if position of structures are exchanged. It means, in case of adjacent structures with different natural periods, the most affected are the rigid structures irrespective of their position due to pounding. The behavior of structures is similar as linear, till failure of either first spring or first collision. Almost every structure vibrate at lesser frequencies because of failure of springs. The maximum pounding forces increases, as the period of structure increases with the same separation distance. 3D analysis of pounding: The effect of collision is more when structures are kept at extreme levels of setback. When the structures are kept at different elevation levels (setback=0), the pounding response changes significantly as the height of structure decreases. At mid height of structure, the collision force is more compared to other height levels because of shear amplification. 104

CHAPTER 6. CONCLUSIONS

6.2

Future Work

There are some suggestions for future research work on numerical modeling of pounding between adjacent structures. 1. Extension of this work needs to consider the soil and brick parameters. 2. Modeling the structures using expansion joints such as filler or rubber material etc. 3. Damage of structure to be implemented in AEM using crack length and crack width due to pounding. 4. Relationship between setback and collision force need to be calculated. 5. Effect of separation distance on collision force with constant setback need to be studied.

105

Appendix A Comparision of codal provisions on pounding A.1

Review on codal provisions

Most of the world regulations for seismic design do not take into account the pounding phenomenon. Among of the ones who do consider it, do not provide specific rules that must be followed. A list of codal provisions on pounding is described in chapter 1 (Ref 1.5.3). These codes specify a minimum separation distance between adjacent buildings. In some cases this depends only on the maximum displacements of the each building (eg., Canada and Israel) and in other cases a small value that may be either a percentage of previous one or a quadratic combination of the maximum displacements (eg., France). In other cases the separation distance is made dependent on the building height (eg., Taiwan). According to International Building Code (IBC) all the structures shall be separated from adjoining structures. If the adjacent buildings are on the same property line, the minimum separation distance simply follows SRSS rule and if they are not located on the same property line (adjacent buildings separated by property line) simply follows the sum of maximum displacements of the structures. In 2006 version there is no such type of codal provision on building separation. Uniform Building Code (UBC-1997) also follows the same codal provisions. According to Federal Emergency Management Agency (FEMA: 273-1997) the potential effects of building pounding whenever the side of the adjacent structure is located closer to the building by less than 4% of the building height above grade at the location of potential impacts. ASCE (7-05) states that all portions of the structure shall be designed and constructed to act as an integral unit in resisting seismic forces unless separated structurally by a distance sufficient to avoid damaging contact under total deflection as determined in section 12.8.6. The details of codal provisions are listed in table below: From the observation of all codal provisions it is seen that most of the codal provisions 106

APPENDIX A. COMPARISION OF CODAL PROVISIONS ON POUNDING

Table A.1: Details of codal provisions on pounding S.No CODE

FORMULAE q

1

IBC 2003

2

UBC 1997

3

FEMA: 273-1997

4

ASCE/SEI 7-05

2 + δ2 δM = δM 1 M 2 ..... (Adjacent Buildings located on the same property line) [Clause 1620.4.5] q 2 + δ2 δM = δM 1 M 2 ..... (Adjacent Buildings located on the same property line) [Clause 1633.2.11] Data on adjacent structures should be collected to permit investigation of the potential effects of building pounding whenever the side of the adjacent structure is located closer to the building than 4% of the building height above grade at q the location of potential impacts. The value of Si (Si =

∆2i1 + ∆2i2 ) calculated by the equation need not exceed 0.04 times the height of the buildings above grade at the zone of potential impacts. [(Clause 2.7.4), (Clause 2.11.10)] δx = CdIδxe [Clause 12.12.3]

follow SRSS method only. The minimum separation distance is not only depending on the response of the structure but also various factors like importance factor, amplification factor etc. This case study deals about the collision force of first impact of the structure by using linear impact models. The response considered is in translational direction only and not consider in torsional direction.

A.2

Minimum separation between buildings

For studying pounding between adjacent structures numerically, we considered two buildings as shown in figure A.1. These buildings were idealized as two equivalent linear single

Figure A.1: Idealized model of SDOF system

107


degree of freedom (SDOF) systems. The two structures have lumped masses m1 = 11400 kg, m2 = 6410 kg, equal stiffnesses k = 45000 kN/m and equal damping ratios ζ. Let u1 (t) and u2 (t) are independent responses of structure 1 and structure 2. The governing differential equation of motion of SDOF system is expressed as follows: mi u¨i (t) + ci u˙ i (t) + ki ui (t) = −mi u¨g (t)

(A.1)

Where,’i’denotes the building under consideration. For the purpose of studying the collison between the buildings we considered SE component of El-Centro ground motion(Ref 2.3.1) whose PGA is 0.348 g. Newmark’s approach is used for finding the response of the structures. In case of pounding collision condition to be checked and the condition is as follows: u1 (t) − u2 (t) ≥ δ

(A.2)

If the above condition satisfies then collision occurs. For the purpose of finding the minimum gap between two buildings, we considered different time periods for structure 2 ie., 0.075, 0.10, 0.125, 0.15, 0.175, 0.20, 0.225 and 0.25 sec. The details are given in the below table. Table A.2: Details about mass, stiffness and spacing provided between two structures S.No 1 2 3 4 5 6 7 8

Mass(kg) 06410 11400 17810 25645 34900 45600 57700 71240

Stiffness(MN/m) 45 45 45 45 45 45 45 45

Time Period(sec) 0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.250

Space Provided(m) 0.009 0.000 0.006 0.010 0.013 0.016 0.017 0.018

The peak of relative response of adjacent buildings gives the minimum separation distance between them. The minimum separation distance between two adjacent structures is as shown in Figure A.2. From this figure it can be observed that as the time period of the structure increases, minimum distance is increasing to avoid pounding and for the two structures with same time period, there is no need to provide any separation distance because these buildings will vibrate in phase and does not collide at any point of time. However, this situation is not realistic because it is very difficult to construct two structures with same natural period. Also, it can be observed from the figure that the minimum separation distance is getting saturated when time period of building 2 is increasing say beyond 1sec.

108


Figure A.2: Minimum space provided between two structures having different dynamic properties

A.3

Case study

For the purpose of studying the impact force by providing minimum separation distance between buildings, we selected structure 1 with time period 0.1 sec time period and varied time period of other structure i.e, 0.075, 0.1, 0.15, 0.2 sec. Also for the purpose of doing time history analysis we selected five earthquake records, viz., Loma-Prieta earthquake, Elcentro earthquake, Parkfield earthquake, Petrolia earthquake and Northridge earthquake. For the purpose of analysis, Kelvin impact model (Ref 1.5.2) is choosen. For the calculation of impact force between two structures stiffness of the spring, kk is assumed as 4378x103 kN/m. The co-efficient of restitution, e = 0.6 is assumed and it is defined as the ratio of the relative velocities of the bodies after collision to the relative velocities of the bodies before collision. Lomaprieta earthquake occurred in 1989 having a magnitude of 6.9 and PGA value of 0.22 g. The duration of this ground motion is 9.58 sec according to trifunac and broady calculation. The predominant frequencies range present in the ground motion is 0.62-2.44 hz (0.41-1.61 sec). In this study structures having time period range from 0.075 sec to 0.2 sec with an interval of 0.025 sec has taken. Structure having time period 0.1 sec is kept constant and others are varying and the minimum separation distances are calculated from above codal provisions. As the structures time period increases, the response of the structure is also increases for a given ground motion and damping. According to IBC, UBC and FEMA follows SRSS rule. According to ASCE codal provisions the minimum separation distance is high compared to others. But ASCE codal provision deals importance factor also. This importance factor is based on occupancy category. The predominant time period range (0.41-1.61 sec) is not presented in this case and there is no impact of the structures. Hence the collision force is zero for all the structures. 109


Table A.3: Lomaprieta ground motion record having amplitude of 0.22 g, duration 9.58 sec and predominant time period 0.41-1.61 sec

SNo. 1 2 3 4

Code IBC2003 UBC1997 FEMA273 ASCE

T1 =0.075 sec T1 =0.075 sec T1 =0.1 sec T1 =0.15 sec T1 =0.2 sec Gap(m) Force(MN) Gap(m) Force(MN) Gap(m) Force(MN) Gap(m) Force(MN) 0.0065

0

0.0082

0

0.016

0

0.03

0

0.0065

0

0.0082

0

0.016

0

0.03

0

0.0065

0

0.0082

0

0.016

0

0.03

0

0.022 0.017 0.014

0 0 0

0.029 0.023 0.019

0 0 0

0.052 0.042 0.035

0 0 0

0.09 0.071 0.06

0 0 0

Elcentro earthquake occurred in 1940 having a magnitude of 7.1 and PGA value of 0.348 g. The duration of this ground motion is 24.44 sec according to trifunac and broady calculation. The predominant frequencies range present in the ground motion is 1.15-2.22 hz (0.45-0.87 sec). In this study structures having time period range from 0.075 sec to 0.2 sec with an interval of 0.025 sec has taken. Structure having time period 0.1 sec is kept constant and others are varying and the minimum separation distances are calculated from above codal provisions. For the structures having time period 0.1 and 0.075 sec the amount of impact is 20.58x103 kN by providing the minimum separation distance 0.012 m according to IBC, UBC and FEMA. As the minimum space between the structures decreases the amount of impact increases, but this impact occurs at the same time even the separation distance decreases. For the structures having same time period, no need to provide minimum space between them. Because both structures response is same. For the structures having time period 0.1 and 0.15 sec, the amount of impact is 26.28x103 kN by providing the minimum separation distance 0.028 m according to IBC, UBC and FEMA. For the structures having time period 0.1 and 0.2 sec, the amount of impact is 42.92x103 kN by providing the minimum separation distance 0.056 m according to IBC, UBC and FEMA. The amount of impact depends on response of the structures at particular time, minimum space between the structures and velocity of the structures. Even though the predominant time period range (0.45-0.87 sec) is not presented, there are collisions for the structures. If the predominant time period range structures are present, the collision will be more. As the time period of the structures near to the predominant time period range, the response of the structures are more and the impact, damage are more and finally may lead to collapse of the structure. Parkfield earthquake occurred in 1966 having a magnitude of 6.0 and PGA value of 110


Table A.4: Elcentro ground motion (S00E) record having amplitude of 0.348 g, duration 24.44 sec and predominant time period ranges from 0.45-0.87 sec

SNo. 1 2 3 4



20.58

0.014

0

0.028

26.28

0.056

42.92

0.012

20.58

0.014

0

0.028

26.28

0.056

42.92

0.012

20.58

0.014

0

0.028

26.28

0.056

42.92

0.042 0.034 0.028

0 0 0

0.032 0.026 0.022

0 0 0

0.091 0.073 0.066

0 0 0

0.164 0.131 0.110

0 0 0

0.43 g. The duration of this ground motion is 6.76 sec according to trifunac and broady calculation. The predominant frequencies range present in the ground motion is 0.83-3.33 hz (0.30-1.2 sec). In this study structures having time period range from 0.075 sec to 0.2 sec with an interval of 0.025 sec has taken. Structure having time period 0.1 sec is kept constant and others are varying and the minimum separation distances are calculated from above codal provisions. For the structures having time period 0.1 and 0.075 sec there is no collision and for the structures having same time period also there is no collision. For the structures having time period 0.1 and 0.15 sec the amount of impact is 38.98x103 kN by providing the minimum separation distance 0.03 m according to IBC, UBC and FEMA. In this case the impact occurs at the same time, when the distance between two structures reduced. For the structures having time period 0.1 and 0.15 sec there is collision when the provided minimum space is 0.03 m, but if the structure changed to 0.15 to 0.2 sec there is no collision according to IBC, UBC and FEMA, because the provided space is more. Petrolia earthquake occurred in 1992 having a magnitude of 7.2 and PGA value of 0.662 g. The duration of this ground motion is 48.74 sec according to trifunac and broady calculation. The predominant frequencies range present in the ground motion is 1.20-2.00 hz (0.50-0.83 sec). In this study structures having time period range from 0.075 sec to 0.2 sec with an interval of 0.025 sec has taken. Structure having time period 0.1 sec is kept constant and others are varying and the minimum separation distances are calculated from above codal provisions. For all the structures there is no impact as per the codal provisions. The predominant time period range is far away from the existing time period structures. Northridge earthquake occurred in 1994 having a magnitude of 6.70 and PGA value of 0.883 g. The duration of this ground motion is 8.94 sec according to trifunac and broady

111


Table A.5: Parkfield ground motion record having amplitude of 0.430 g, duration 6.76 sec and predominant time period 0.3-1.20 sec

SNo. 1 2 3 4



0

0.021

0

0.03

38.98

0.091

0

0.016

0

0.021

0

0.03

38.98

0.091

0

0.016

0

0.021

0

0.03

38.98

0.091

0

0.05 0.04 0.033

0 0 0

0.075 0.06 0.05

0 0 0

0.1 0.08 0.066

0 0 0

0.26 0.21 0.17

0 0 0

Table A.6: Petrolia ground motion record having amplitude of 0.662 g, duration 48.74 sec and predominant time period 0.50-0.83 sec

SNo. 1 2 3 4



0

0.0356

0

0.0630

0

0.109

0

0.0317

0

0.0356

0

0.0630

0

0.109

0

0.0317

0

0.0356

0

0.0630

0

0.109

0

0.111 0.088 0.074

0 0 0

0.126 0.100 0.084

0 0 0

0.208 0.166 0.138

0 0 0

0.328 0.262 0.218

0 0 0

calculation. The predominant frequencies range present in the ground motion is 0.45-5.00 hz (0.20-2.2 sec). In this study structures having time period range from 0.075 sec to 0.2 sec with an interval of 0.025 sec has taken. For the structures having time period 0.1 and 0.15 sec has no impact. For the structures 0.1 and 0.2 sec time period the minimum separation distance is 0.23 m according to IBC, UBC and FEMA. The amount of impact is 56.5x103 kN.

A.4

Conclusions

From the above observations, the duration of strong motion increases with an increase of magnitude of ground motion. As the PGA value increases, the minimum separation 112


Table A.7: Northridge ground motion record having amplitude of 0.883 g, duration 8.94 sec and predominant time period ranges from 0.2-2.2 sec

SNo. 1 2 3 4



0

0.053

0

0.093

0

0.23

56.5

0.041

0

0.053

0

0.093

0

0.23

56.5

0.041

0

0.053

0

0.093

0

0.23

56.5

0.134 0.100 0.09

0 0 0

0.190 0.152 0.126

0 0 0

0.307 0.240 0.204

0 0 0

0.670 0.530 0.446

0 0 0

between the structures also increases. • The separation distance between the two structures decreases, the amount of impact is increases, which is not applicable in all cases. It is only applicable when the impact time is same. It may also decreases when separation distance decreases, which leads to less impact time. • At resonance condition the response of the structure is more and may lead to collapse of the whole structure. In this case even though the predominant time period range is not present, the impact occurs, but this impact is more when the predominant time period structures present. • For Elcentro earthquake, the PGA value and duration are slightly less than Petrolia earthquake, but the collision is significant. The minimum separation distances are different in both cases and less in Elcentro earthquake. • For Parkfield earthquake, magnitude and duration are less and predominant time period structures are near to the existing structures. Hence collision happens. • For Northridge earthquake which are less magnitude and duration than Parkfield, the collision is more because of resonant frequencies. The amount of impact is not only depending on response and velocity of the structure but also magnitude and duration of earthquake. • For IBC, UBC and FEMA codal provisions pounding happens almost structures having different dynamic properties when El-Centro ground motion is given to the struc-

113


tures. This happens for moderate earthquakes. • From the all above observation, the duration of strong motion increases with an increase of magnitude of ground motion. As the PGA value increases, the minimum separation distance is also increases between the structures.

114

Appendix B Calculation of separation distance from codes B.1

Calculation of separation distance from codes

A series of separation distances are studied to estimate the minimum separation distance (MSD) between two adjacent structures which are separated with an interval of 5 mm. A set of structures (equal heights of single, two, three and five storey structures and unequal heights of single-two and three-five storey structures) are subjected to series of ground motions (Ref 2.3.1) to estimate MSD. The distance where collision ceases will give the MSD between them. The separation distances are also calculated from codes (ref 1.5.3). The calculations for single storey structures are as follows: According to Greece, the minimum separation distance should be 4 cm upto three storey level, 8 cm from four to eight storey level and 10 cm for more than eight storey levels. According to Mexico, it shall be neither smaller than 5 cm nor smaller than the height of the level over the ground multiplied by 0.007, 0.009, 0.012 depending on site of zones I, II and III respectively. For Australia, it has given 1% of structure height. From Serbia, it shall be 3 cm and increased by 1.0 cm for every increase of 3.0 meters of height above 5.0 mts. From the above codal provisions, it is not given any specific formulation for the separation of distance. Some of them gave formulations which are described below: The separation distances are calculated from codes (ref 1.5.3) for single storey structures. According to Canada codal provisions, the separation distance is the sum of their individual lateral deflections obtained from an elastic analysis. The equivalent lateral seismic force representing elastic response, Ve shall be calculated in accordance with the following: Ve = V SIF W

115

(B.1)

APPENDIX B. CALCULATION OF SEPARATION DISTANCE FROM CODES

Where ’V’ is zonal velocity ratio, ’S’ is seismic response factors, ’I’ is seismic importance factor and ’F’ is foundation factor. The time period of the structure is calculated 3/4 as Tn =0.075hn (concrete moment resisting frames). For calculations, V is taken as 1.0 and S is based on fundamental period of structure and its value is 3.0. I is taken as 1.0 for all buildings. F is taken as 1.3 for firm soils. W is weight of structure which is 100 kN. From the above calculations, the value of Ve is 390 kN. The minimum lateral seismic force is given as, Vl = (Ve /R)U

(B.2)

Where R is force modification factor and its value is 2.0 and U is 0.6. Finally the value of Vl is 117 kN. This load is applied on to the structure to get displacements. Finally the separation distance is 0.025 m. According to Egypt codal provisions, it shall be either 2.0 times the computed deflections or 0.002 times of its height whichever is larger and in many cases not less than 2.5 cms. The total horizontal seismic force is calculated as: V = ZISM RQW

(B.3)

Where, ’I’ is importance factor = 1.0, ’S’ is structural system type factor = 1.0 (for moment resisting frames), ’M’ is material factor =1.0 (for RC material), ’R’ is risk factor =1.0 (normal buildings), ’Q’ is construction quality control factor = 1.0 and ’Z’ can be calculated as Z=ACF, where ’A’ is horizontal acceleration ratio = 0.02 (for zone I and intensity VI), ’F’ is foundation soil factor = 1.3 (for fine grained soils). The value of C √ . The is calculated from time period of the structure. It can be calculated as, T= 0.09H d calculated value for C is 1.0. Finally the value of V is 2.6 kN and the separation distance is 0.012 m. The minimum value should be 0.025 m. According to Ethiopia codal provisions, it shall be twice the sum of their individual deflections obtained from an elastic analysis. The total seismic force is given as, F = αβγ(Gk + ψQk )

(B.4)

Where, ’α’ is design bedrock acceleration ratio (α=αo I). For zone II αo is taken as 0.05 and ’I’ is taken as 1.0 for buuildings and structures occupancy. β is elastic design 0.09h √ n , ’S’ is site condition response factor and its value is β=βo S ≤ 2.5. βo = T1.2 1/2 and T= d factor = 1.0 for gravels. Finally the value of β is 2.5, γ is structural system type factor = 0.5 for RC buildings. Gk is characteristic dead load, Qk is characteristic live load and ψ is live load incident factor = 0.25 for public buildings. The seismic force is 2.64 kN and the separation distance is 2 mm. According to Peru codal provisions, the shear in the base of the structure is calculated as, 116


ZU SC P (B.5) R The minimum value for C/R is taken as 0.125. Where U is occupancy factor = 1.0 for common buildings. Z is zone factor = 0.30 for zone II and S is soil parameters = 1.2 for intermediate soils. The shear value is 4.5 kN. The other formulation is S=3+0.004(h-500) or greater than 3 cms. Where h is in cms. It must be greater than above claculations. Finally the separation distance is 3 cms. V =

According to Indian codal provision, the design seismic base shear is calculated as, VB = Ah W

(B.6)

a Where Ah = ZIS , Ah is design horizontal seismic coefficient, ’Z’ is zone factor=0.24 2Rg for zone IV(assumed), ’I’ is importance factor=1.0 for general buildings, ’R’ is response reduction factor=3.0 for moment resisting frames and Sga is taken for medium soil sites. The calculated fundamental period without considering brick infill panels is T=0.075h0.75 = 0.17 sec. Where h=height of building in m. From all the calculations, the design seismic base shear is 1.7 kN and the separation distance is 0.01 m.

According to Taiwan codal provision, the seiamic base shear is calculated as, V =

1 SaD ( )m W 1.4αy Fu

(B.7)

Where αy is defined as the first yield seismic force amplification factor that is dependent on the structure types and design method which will be equal to 1.5 for RC structures 3/4 using strength design method. The fundamental period is calculated as T=0.07hn for RC moment resisting frames and its value is 0.159 sec. The value To is 1.3 sec for zone II (Source: Design response spectrum for Taipei basin).

SaD =

  SDs (0.4 +     S ,

3T ), To

T ≤ 0.2To 0.2To < T ≤ To

Ds

 SDs ( TTo ),     0.4S , Ds

To < T ≤ 2.5To

(B.8)

T > 2.5To

Where SDs is site-adjusted spectral response acceleration parameter, the design spectral response acceleration SaD for a given site can be developed directly from the design spectral response acceleration at short periods and its value is 0.6 g. The value of SaD is 0.46 g. The reduction factor Fu is given by,

117


  Ra ,     √2R − 1 + (R − √2R − 1) T −0.6To , a a a 0.4To Fu = √  2Ra − 1,    √   2R − 1 + (√2R − 1 − 1) T −0.2To , a a 0.2To

T ≥ To 0.6To ≤ T ≤ To 0.2To ≤ T ≤ 0.6To

(B.9)

T ≤ 0.2To

The value of Ra = 1 + R−1 =1.33. Finally the value of V is 18.6 kN. The calculated 1.5 separation diatance is 1 cm. The total cases can be categorized into two groups. The first group (group-A) deals with single-single, single-two and two-two storey structures. Whereas, the second group (group-B) deals with the remaining structures in the analysis.

Table B.1: Separation distances from codes Code Australia Canada Egypt Ethiopia Greece India Mexico Peru Serbia Taiwan

Single-single 3.0 2.5 2.5 0.2 4.0 1.0 5.0 3.0 3.0 1.0

Single-two 6.0 3.2 2.5 0.5 4.0 3.2 7.2 3.5 4.0 3.9

Two-two 6.0 3.5 2.5 0.7 4.0 4.0 7.2 3.5 4.0 5.0

Three-three 12.0 4.7 2.5 1.8 4.0 6.3 14.4 4.6 5.0 7.4

Three-five 20.0 6.0 3.0 2.0 8.0 7.4 24.0 7.0 7.0 8.5

Five-five 20.0 6.3 3.0 2.7 8.0 9.6 24.0 7.0 7.0 9.0

* All units are in cms

In first group, Taiwan, Indian, Egyptian and Ethiopian seismic codes doesn’t satisfy the minimum separation requirement. As per the Indian and Taiwan codal provisions, the separation distance for single-single storey structures are 1.0 cm. This separation distance doesn’t satisfy for earthquakes 2 and 5 (Table B.2), because of underestimation of separation distance between the structures. As per the Ethiopian codal provision, the separation distance is 2 mm which is insufficient. From the analysis, 15 mm will be the minimum separation distance between two single storey structures which has passed from all the considerable ground motions without collision. For single-two storey structures, Egyptian and Ethiopian codal provisions doesn’t satisfy the minimum requirement on separation distance. As per Egyptian codal provision, the separation distance is 2.5 cms for single-two storey structures which has not satisfied for earthquakes 8 and 9. As per the Ethiopian codal provision, the separation distance is 5 mm which is insufficient. From the analysis, 30 mm will be the minimum separation distance between two single storey structures which has passed from all the considerable 118


Table B.2: Status on separation distance from codes for single-single storey structures in group-A Code Australia Canada Egypt Ethiopia Greece India Mexico Peru Serbia Taiwan

1 √ √ √

2 √ √ √

3 √ √ √

4 √ √ √

5 √ √ √

6 √ √ √

7 √ √ √

8 √ √ √

9 √ √ √

10 √ √ √

x √ √ √ √ √ √

x √

x √ √ √ √ √ √

x √ √ √ √ √ √

x √

x √ √ √ √ √ √

x √ √ √ √ √ √

x √ √ √ √ √ √

x √ √ √ √ √ √

x √ √ √ √ √ √

x √ √ √ x

x √ √ √ x

* 1-Athens ground motion, 2-Athens(trans) ground motion, 3-Ionian ground motion, 4-Kalamata ground motion, 5-Umbro ground motion, 6-Elcentro ground motion, 7-Olympia ground motion, 8-Parkfield ground motion, 9-Northridge ground motion and 10-Lomaprieta ground motion √ -satisfies the separation distance from codes x-does not satisfy the separation distance from codes

Table B.3: Status on separation distance from codes for single-two storey structures in group-A Code Australia Canada Egypt Ethiopia Greece India Mexico Peru Serbia Taiwan

1 √ √ √

2 √ √ √

3 √ √ √

4 √ √ √

5 √ √ √

6 √ √ √

7 √ √ √

x √ √ √ √ √ √

x √ √ √ √ √ √

x √ √ √ √ √ √

x √ √ √ √ √ √

x √ √ √ √ √ √

x √ √ √ √ √ √

x √ √ √ √ √ √

8 √ √

9 √ √

x x √ √ √ √ √ √

x x √ √ √ √ √ √

10 √ √ √ x √ √ √ √ √ √

ground motions without collision. For two-two storey structures, the separation distance is 35 mm which has passed from all the ground motions without collision. It is necessary to change some (Taiwan, Indian, Egyptian and Ethopian) codal provisions from the observation of above structures in group A. The change on separation distance should be such that, it should pass from all the ground motions in group A. The modified formulae on separation distance with modification factor are as follows: Taiwan: The separation distance should be 1.5 times of [0.6x1.4αy Ra ]. Indian: The separation distance should be 1.5 times of [R times the sum of the 119


Table B.4: Status on separation distance from codes for two-two storey structures in group-A Code Australia Canada Egypt Ethiopia Greece India Mexico Peru Serbia Taiwan

1 √ √ √ x √ √ √ √ √ √

2 √ √

3 √ √

4 √ √

x x √ √ √ √ √ √

x x √ √ √ √ √ √

x x √ √ √ √ √ √

5 √ √ √

6 √ √ √

7 √ √ √

x √ √ √ √ √ √

x √ √ √ √ √ √

x √ √ √ √ √ √

8 √ √

9 √ √

x x √ √ √ √ √ √

x x √ √ √ √ √ √

10 √ √ √ x √ √ √ √ √ √

calculated storey displacements]. Egypt: The separation distance should be 0.8% of storey height or not less than 3.5 cms. Ethiopia: The separation distance should be 8 times of [twice the sum of their individual deflections].

Table B.5: Status on separation distance from codes for three-three storey structures in group-B Code Australia Canada Egypt Ethiopia Greece India Mexico Peru Serbia Taiwan

1 √ √ x x √ √ √ √ √ √

2 √ x x x x √ √ x x √

3 √ √ x x √ √ √ √ √ √

4 √ x x x x √ √ x x √

5 √ √ √ x √ √ √ √ √ √

6 √ √

7 √ √

x x √ √ √ √ √ √

x x √ √ √ √ √ √

8 √

9 √

x x x x x √

x x x x x √

x x √

x x x

10 √ √ √ x √ √ √ √ √ √

In second group, all the codal provisions doesn’t satisfy the minimum separation distance except Australian and Mexican codes. According to Canadian seismic codal provision, the MSD should be 4.7 cms which doesn’t satisfy when the structures (three-three storey structures) subjected to earthquakes 2, 4, 8 and 9. From the analysis, the MSD value should be 8.0 cms which passes from all the ground motions. The complete status for all the ground motions and codes are listed in table B.5. 120


It is necessary to change the codal provisions from the observation of above structures in group B. The modified formulae on separation distance are as follows: Taiwan: The separation distance should be 1.16 times of [0.6x1.4αy Ra ]. Indian: The separation distance should be 1.35 times of [R times the sum of the calculated storey displacements]. Egypt: The separation distance should be 0.8% of storey height or not less than 8.0 cms. Ethiopia: The separation distance should be 5 times of [twice the sum of their individual deflections]. Canada: The separation distance should be 1.8 times of [sum of their individual deflections]. Greece: The separation distance should be 8 cms upto three storeys. Peru: The separation distance should be S = 7+0.004(h-500)...’h’ is in cms. Serbia: The separation distance should be increased by 1.7 cms for every increase of 3.0 m height of height above 6.0 mts.

121

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a study of pounding between adjacent structures

a study of pounding between adjacent structures

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