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Model updating of a 48-storey building in Vancouver using ambient vibration measurements Lord, Jean-François 2003

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M O D E L U P D A T I N G OF A 48-STOREY BUILDING IN V A N C O U V E R USING AMBIENT VIBRATION MEASUREMENTS  by  JEAN-FRANCOIS LORD B.Eng., M c G i l l University, 2000  A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T OF THE REQUIREMENTS FOR THE D E G R E E OF  M A S T E R OF APPLIED SCIENCE  in  T H E F A C U L T Y OF G R A D U A T E STUDIES (Department o f C i v i l Engineering)  W e accept this thesis as conforming to the required standard,  T H E UNIVERSITY OF BRITISH C O L U M B I A A p r i l 2003 © Jean-Francois Lord, 2003  I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by t h e head o f my department o r by h i s o r her r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n .  Department o f  C'"''-  E^tJe£.'kL\\J&  The U n i v e r s i t y o f B r i t i s h Columbia Vancouver, Canada Date  ABSTRACT This study focused on the One Wall Centre, a 48-storey high building located in downtown Vancouver, British Columbia.  It is currently the highest building in  Vancouver, and it is the only structure in the region that makes use o f tuned liquid column dampers to reduce vibrations due to wind.  The "true" dynamic response o f the  One Wall Centre is o f great interest to structural engineers because o f the unusual elliptical shape o f the concrete shear core, which could present a challenge for modelling.  Ambient vibration testing was conducted on the One Wall Centre in order to determine its modal characteristics.  Such characteristics included the natural periods o f the  building, their corresponding mode shapes and damping ratios.  The analysis o f the  ambient vibration data was performed using a state-of-the-art modal identification technique in the frequency domain and a second technique in the time domain.  The natural periods and mode shapes o f the One Wall Centre were predicted analytically using a linear-elastic finite element (FE) computer model o f the building. Only the main lateral load-resisting system components were modelled, which included the reinforced concrete shear core, the outrigger columns and outrigger beams. It was found that the F E model was more flexible than the actual structure.  In order to achieve a better match between the analytical and experimental dynamic responses, the F E model was updated using two techniques: manual model updating and automated model updating.  The main change to the F E model from the manual model  updating was the inclusion o f the stiffness contribution o f the architectural components, such as the outside windows and partition walls. The intent o f the automated technique was to determine the sensitivity o f the F E models to variations in element physical properties (Young's modulus, material mass density, moment o f inertia and thickness o f elements).  A n excellent match was achieved between the analytical and experimental  results after manual and automated updating o f the F E models.  ii  TABLE OF CONTENTS ABSTRACT  .  H  TABLE OF CONTENTS  Hi  LIST OF TABLES  vi  LIST OF FIGURES.....  vii  ACKNOWLEDGMENTS  ix  CHAPTER 1 INTRODUCTION  1  1.1  Background  1  1.2  Objectives and Scope o f Study  2  1.3  Overview  2  CHAPTER 2 BACKGROUND ON CONTROL SYSTEMS  5  2.1  Passive Control Systems  5  2.2  Active Control Systems  6  2.3  Semi-Active Control Systems  6  CHAPTER 3 DESCRIPTION OF T H E BUILDING  13  3.1  Overview  13  3.2  Structural System  15  3.3  Control System  18  CHAPTER 4 AMBIENT VIBRATION TESTING  19  4.1  Test Objective  19  4.2  Test Equipment  19  4.2.1  Instrumentation  20  4.2.2  Data Acquisition System  20  4.2.3  Software  '.  iii  21  4.3  Test Procedure  21  4.3.1  Preliminary Planning  21  4.3.2  Sensor Location  22  4.3.3  Set-up Plan  24  4.3.4  Data Acquisition Plan  26  CHAPTER 5 M O D A L IDENTIFICATION RESULTS 5.1  29  Natural Periods  29  5.1.1  Enhanced Frequency Domain Decomposition ( E F D D )  29  5.1.2  Stochastic Subspace Identification (SSI)  31  5.1.3  Comparison o f Results  33  5.2  M o d e Shapes  36  5.3  Damping Ratios  39  CHAPTER 6 COMPUTER MODELLING 6.1  41  Simplified M o d e l  41  6.1.1  Motivation  41  6.1.2  Description  42  6.1.3  Results  43  E T A B S Model#l  45  6.2 6.2.1  E T A B S Software  45  6.2.2  Description  45  6.2.3  Results  48  CHAPTER 7 FINITE E L E M E N T M O D E L UPDATING  50  7.1  Manual M o d e l Updating  50  7.2  Automated M o d e l updating  54  7.2.1  F E M t o o l s Software  55  7.2.1.1  The Platform Module  56  7.2.1.2  The Correlation Analysis Module  56  7.2.1.3  Sensitivity Analysis Module  57  iv  7.2.1.4 7.2.2  57  M o d e l # l Updating  57  7.2.2.1  F E M t o o l s Model#l  57  7.2.2.2  Model#l Before Updating  58  7.2.2.3  Model#l Sensitivity Analysis  60  7.2.2.4  M o d e l # l After Updating  66  7.2.3  7.3  M o d e l Updating Module  Model#2 Updating  74  7.2.3.1  F E M t o o l s Model#2  74  7.2.3.2  Model#2 Before M o d e l Updating  74  7.2.3.3  Model#2 Sensitivity Analysis  76  7.2.3.4  Model#2 After Updating  81  Conclusions from Automated M o d e l Updating  89  CHAPTER 8 CONCLUSIONS AND RECOMMENDATIONS  91  8.1  Summary  91  8.2  Conclusions  92  8.3  Recommendations  93  REFERENCES  95  APPENDIX A STRUCTURAL AND ARCHITECTURAL DRAWINGS  99  APPENDIX B AMBIENT VIBRATION TESTING DETAILS  112  APPENDIX C SIMPLIFIED M O D E L SOLUTIONS  116  APPENDIX D A U T O M A T E D M O D E L UPDATING THEORY  122  APPENDIX E FEMTOOLS E L E M E N T MAPS  126  v  LIST OF TABLES Table 3.1. Specified concrete strengths and reinforcing steel strength  18  Table 5.1. A R T e M I S modal analysis results from the E F D D and SSI techniques  34  Table 5.2. A R T e M I S M A C matrix from the E F D D and SSI techniques (in %)  35  Table 5.3. A R T e M I S damping ratios from the E F D D and SSI techniques  40  Table 6.1. Simplified M o d e l and experimental natural periods  44  Table 6.2. E T A B S F E model material properties  .47  Table 6.3. E T A B S Model#l and experimental natural periods  48  Table 7.1. M o d a l analysis results after manual model updating  53  Table 7.2. F E M t o o l s Model#l modal analysis results before updating  59  Table 7.3. F E M t o o l s Model#l M A C matrix before updating (in %)  60  Table 7.4. F E M t o o l s M o d e l # l F E element description  63  Table 7.5. F E M t o o l s Model#l selected parameters for updating  64  Table 7.6. F E M t o o l s Model#l modal analysis results after updating  67  Table 7.7. F E M t o o l s Model#l M A C matrix after updating (in %)  68  Table 7.8. F E M t o o l s Model#l parameter comparison before and after updating  73  Table 7.9. F E M t o o l s Model#2 modal analysis results before updating  74  Table 7.10. F E M t o o l s Model#2 M A C matrix before updating (in %)  75  Table 7.11. F E M t o o l s Model#2 element description  76  Table 7.12. FEMtools Model#2 selected parameters for updating  78  Table 7.13. F E M t o o l s Model#2 modal analysis results after updating  81  Table 7.14. F E M t o o l s Model#2 M A C matrix after updating (in %)  82  Table 7.15. F E M t o o l s Model#2 parameter comparison before and after updating  87  Table 7.16. F E M t o o l s Model#l and Model#2 updated parameter comparison  89  vi  L  I  S  T  O  F  F  I  G  U  R  E  S  Figure 2.1. Scale model experimental set-up o f T L C D (Picture courtesy o f R W D I )  8  Figure 2.2. T L C D on a S D O F system  9  Figure 2.3. Effect o f the addition o f a T L C D on the response o f the main system  11  Figure 3.1. The One W a l l Centre, looking South  13  Figure 3.2. Typical floor plan  14  Figure 3.3. Typical cross-section  16  Figure 3.4 The One W a l l Centre, looking North  17  Figure 4.1. Sensor location (from level 5 to level 31)  23  Figure 4.2. Sensor location (from level 32 to level 48)  23  Figure 4.3. Elevation view o f instrument set-up #6  25  Figure 4.4. Reference and roving sensor sample signals  27  Figure 4.5. Reference and roving sensor sample signal power spectral densities  27  Figure 5.1. Singular values o f the spectral density matrices  31  Figure 5.2. Data set #6 stabilization diagram  33  Figure 5.3. Selected modes from the SSI technique  33  Figure 5.4. A R T e M I S period pairs from the E F D D and SSI techniques  34  Figure 5.5. A R T e M I S modes 1, 2 and 3  37  Figure 5.6. A R T e M I S modes 4, 5 and 6  38  Figure 5.7. A R T e M I S modes 7 and 8  39  Figure 6.1. Simplified M o d e l 1 , 2 st  n d  and 3 transverse (NS) mode shapes rd  44  Figure 6.2. E T A B S Model#l  46  Figure 6.3. E T A B S Model#l and experimental period pairs  49  Figure 7.1. E T A B S M o d e l #2: (a) wire frame, (b) full model  52  Figure 7.2. E T A B S Model#2 and experimental period pairs  53  Figure 7.3. E T A B S M o d e l #2 mode shapes  54  Figure 7.4. FEMtools Model#l and experimental period pairs before updating  59  Figure 7.5. F E M t o o l s M o d e l # l normalized sensitivity to selected parameters  63  vii  Figure 7.6.  F E M t o o l s Model#l  normalized sensitivity to selected parameters for  updating  65  Figure 7.7. F E M t o o l s Model#l F E elements 15, 18 and 21 Figure 7.8.  F E M t o o l s Model#l  65  normalized sensitivity o f each mode to selected  parameters for updating  66  Figure 7.9. F E M t o o l s M o d e l # l and experimental period pairs after updating  67  Figure 7.10. F E M t o o l s Model#l M A C values before and after updating  68  Figure 7.11 F E M t o o l s Model#l analytical and experimental reduced mode shapes after updating  69  Figure 7.12. F E M t o o l s Model#l analytical and experimental mode shapes after updating. 70 Figure 7.13. F E M t o o l s Model#l correlation tracking  71  Figure 7.14. F E M t o o l s Model#l parameter changes after updating  73  Figure 7.15. F E M t o o l s Model#2 and experimental period pairs before updating  75  Figure 7.16. F E M t o o l s Model#2 normalized sensitivity to selected  77  Figure 7.17.  F E M t o o l s Model#2 normalized sensitivity to selected parameters for  updating  79  Figure 7.18.  F E M t o o l s Model#2 normalized sensitivity o f each mode to selected  parameters for updating  80  Figure 7.19. F E M t o o l s Model#2 and experimental period pairs after updating  81  Figure 7.20. F E M t o o l s Model#2 M A C values before and after updating  83  Figure 7.21. F E M t o o l s Model#2 experimental and analytical reduced mode shapes after updating  84  Figure 7.22. F E M t o o l s ModeI#2 experimental and analytical mode shapes after updating. 85 Figure 7.23. F E M t o o l s Model#2 correlation tracking  86  Figure 7.24. F E M t o o l s Model#2 parameter changes after updating  88  viii  A  C  K  N  O  W  L  E  D  G  M  E  N  T  S  I would like to thank my advisor, D r . Carlos Ventura. H i s guidance and support during my stay at the University o f British Columbia have made the success o f this study possible.  H e inspired me and allowed me to work on fabulous projects.  I value his  insight and I hope that our paths will cross again.  I would also like to thank Dr. K e n Elwood for his enthusiasm and his valuable comments and suggestions regarding the manuscript.  The Natural Sciences and Engineering Research Council ( N S E R C ) o f Canada provided funding for this study. Their financial help is gratefully acknowledged.  Many thanks go to M r . Robert Simpson, P. Eng., o f Glotman-Simpson Consulting Engineers in Vancouver, for coordinating access to the One Wall Centre, providing architectural and structural drawings and also sharing a finite element model o f the One Wall Centre.  I would also like to thank my friends Rozlyn Bubela, Canisius Chan, Terrence Davies, Jason Esau, Wesley Novotny, Martin Turek (all graduate students in Civil Engineering at the University o f British Columbia), and Estuardo Ventura for their valuable help during the ambient vibration testing.  I am forever in debt to Rozlyn, who helped me greatly with the editing o f my thesis. Her moral support and continuous encouragements  were invaluable and will always be  remembered.  Je tiens a remercier ma famille et tout specialement mes parents, qui ont supporte mes efforts et mes reves.  toujours  Leur soutien financier et surtout moral fut d'une  valeur inestimable tout au cours de mes etudes.  ix  Chapter 1  Introduction  I  1.1  N  C  H  A  P  T  E  R  T  R  O  D  U  C  T  1  I  O  N  BACKGROUND  This study focuses on the One Wall Centre, a 48-storey building located in downtown Vancouver, British Columbia.  The response o f this building is o f interest to structural  engineers for a number o f reasons.  The structure is currently the highest building in  Vancouver, and it is the only building in the region that makes use o f tuned liquid column dampers to reduce vibrations due to wind [1]. The main lateral load resisting system for the One Wall Centre is a reinforced concrete shear core with a unique shape, which makes the study o f the dynamic response o f the building very interesting. In addition, a good understanding o f the seismic response o f the building is important since Vancouver is located in one o f the most active seismic regions o f Canada.  The modal characteristics o f a structure can be determined in a few different ways. During the design stage o f a building, a finite element (FE) model can be constructed, using the specified building geometry, material properties and section properties. modal characteristics can then be predicted analytically.  The  After construction o f the  building, the actual response o f the structure can be measured using ambient vibration testing techniques.  The data collected at these low levels o f excitation can be used to  perform output-only modal identification to obtain the natural periods and mode shapes o f the structure. B y gaining insight into the "true" response o f the structure, on can use this information to update an existing F E model. Various model-updating techniques are available but the basic fundamental concept o f model updating is to vary selected parameters in a F E model until the dynamic response predicted by the F E model corresponds to the experimental results. A n updated model provides a better analytical representation o f the dynamic response o f the building and a calibrated tool for the prediction o f seismic response.  1  Chapter I  Introduction  The One Wall Centre was a perfect candidate for this type o f ambient vibration testing and model updating study because o f the unique characteristics o f the building and the motivations for understanding its dynamic response described above.  1.2  OBJECTIVES AND SCOPE OF STUDY  The main objectives of this study were:  •  T o identify the "true" modal characteristics o f the One Wall Centre.  This was  accomplished using ambient vibration testing techniques to measure the response o f the building.  The output-only modal identification was performed using a state-of-  the-art frequency domain calculation technique to obtain the natural periods and mode shapes o f the structure.  These results were verified using a time-domain  calculation technique.  •  r  T o update an existing F E model (used in design) o f the One Wall Centre.  This  involved the exploration o f the sensitivity o f the dynamic characteristics o f the model to variations in key parameters (elements and physical properties) in the system. It was also necessary to identify the magnitudes o f parametric changes required to match the  analytical modal characteristics o f the  experimentally.  system to  those obtained  T w o different model-updating techniques were used in the study,  and the effectiveness o f each method to converge to a solution was examined.  1.3  OVERVIEW  A general overview o f the One Wall Centre study is given and a description o f the research that was done is presented in each o f the chapters that follow.  Background information is presented on semi-active control systems ( S A C S ) as the tuned liquid column damper system found on the One Wall Centre is one type o f S A C S . 2  Chapter I  Introduction  Passive and active control systems are also discussed briefly in order to better understand how S A C S have developed from a combination o f the two previous systems.  A n overall description o f the One Wall Centre is provided, including information about the building geometry and general use and occupancy.  The structural system o f the  building is discussed in detail along with pertinent information about the material properties.  The ambient vibration test performed on the One Wall Centre is described.  Specific  details o f the test procedures, test equipment and instrumentation layout are explained. The modal identification results derived from the ambient vibration test data presented.  are  The modal characteristics o f the building, such as natural periods, mode  shapes and damping ratios were calculated using two techniques and comparison o f the results will be given.  Two different analytical modelling approaches for the One Wall Centre are discussed. One approach consists o f exploring a simplified model o f the building using classical beam theory.  The second approach consists o f using a computer program to perform a  modal analysis o f a linear-elastic finite element model o f the building. Results from the various modal analyses are presented and compared  The model updating o f the linear-elastic F E model described above is explored.  Two  model-updating techniques were used in this study, namely (i) manual updating, and (ii) automated updating.  The capabilities o f the computer program used for the automated  updating process are described. The sensitivity o f the dynamic response o f the model to various parameter changes is assessed, and discussions o f results are presented.  The  main  conclusions  that  transpire  from  this  study  are  stated  and  some  recommendations are made concerning future research and development on similar topics.  Some practical considerations are discussed concerning the modelling o f similar  structures in the future.  However, any aspects related to modelling o f the tuned liquid  3  Chapter 1  Introduction  column dampers are outside the scope o f this study. The emphasis o f this study is on the model updating o f the One Wall Centre.  4  Chapter 2  Background on Control Systems  C  B  A  C  K  G  R  O  U  N  D  H  A  O  N  P  T  C  E  O  R  N  2  T  R  O  L  S  Y  S  T  E  M  S  The roof o f the One Wall Centre is fitted with two 300 ton tuned liquid column dampers to reduce vibrations due to wind. In order to better understand how tuned liquid column dampers work and why this system was chosen over other alternatives, various types o f control systems are examined. Passive and active control systems are briefly introduced. However, the focus o f this Chapter centres around semi-active control systems (found in the One Wall Centre), which combine characteristics from both passive and active systems.  2.1  P A S S I V E  C O N T R O L  S Y S T E M S  Passive control systems (PCS) could be defined as systems that do not require an external power source in order to operate and utilize the motion o f the structure to develop the control forces.  Such control forces are developed as a function o f the response o f the  structure at the location o f the P C S [2].  P C S are widely discussed in the literature and many types o f energy-dissipating devices are being studied and developed.  A small inventory o f P C S includes viscoelastic  dampers [3], friction dampers [4], slotted bolted connections [5], cladding connections [6] and base isolators, a control system frequently used in Japan [7].  Although P C S are universally used and widely studied as a mean o f earthquake protection, they are not particularly suitable for tall reinforced concrete buildings as most o f the P C S mentioned previously apply specifically to steel structures, where the effectiveness o f P C S has been demonstrated.  The concept o f base isolation is not as  effective on tall buildings as it is on short buildings. Tall structures, such as the One  5  Chapter 2  Background on Control Systems  Wall Centre, already have long fundamental period, which leads to a much smaller spectral ordinate, reducing the demand on the structure [8].  2.2  ACTIVE CONTROL SYSTEMS  Active control systems ( A C S ) typically require a large power source in order to operate the electro-hydraulic or electro-mechanical actuators that supply control forces to the structure. Control forces are developed based on feedback from sensors that measure the response o f the structure [2].  Prof. Soong o f State University o f N e w Y o r k at Buffalo has actively researched the use and implementation o f A C S in buildings.  H i s state-of-the-art-review [9] provides a  comprehensive overview o f the usage o f A C S control in civil engineering.  In another  publication, he also addresses the challenges that must be addressed before A C S can gain general acceptance by the civil engineering profession [10]. These challenges include the high maintenance costs associated with A C S , the dependence on external power and reliability during an earthquake, and the usage o f non-traditional technology.  Although A C S are currently widely used in Japan, these systems have not yet gained popularity and acceptance in Canada, most likely for the reasons described above.  2.3  SEMI-ACTIVE CONTROL SYSTEMS  A compromise between the passive and active control systems is the semi-active control systems ( S A C S ) .  S A C S maintain the reliability o f P C S , while taking advantage o f the  adjustable parameter characteristics o f A C S .  A number o f S A C S have been developed and studied over the years. Only two types o f S A C S are presented and discussed here.  The first type o f S A C S to be described is the  tuned mass damper ( T M D ) . The second type o f S A C S is the tuned liquid column damper  6  Chapter 2  Background on Control Systems  ( T L C D ) , a close relative to the T M D . T M D and T L C D are discussed in parallel since the fundamental concept of their operation is very similar.  The use o f T M D to reduce vibrations in structures was first suggested in 1909, according to Hartog [11]. A T M D is a device consisting o f a mass, a spring and a viscous damper attached to a system to reduce undesirable vibrations [12].  The T M D falls into the  category o f S A C S because o f the capacity to adjust the mass, stiffness and damping properties to achieve an intended response.  T M D are usually installed in high-rise  buildings to reduce wind-induced vibrations.  For seismic applications, however, there  has not been general agreement on the efficiency o f T M D systems to reduce the structural response.  Discussions are still ongoing regarding the optimum T M D parameters that  would achieve the greatest reduction in the seismic response o f a structure [12].  A typical T M D consists o f a mass, m, which moves relative to the structure and is attached to it by a spring with stiffness, k, and a viscous damper with damping coefficient, c.  A T M D is characterized by its tuning, mass and damping ratios.  The  tuning ratio, / , is defined as the ratio o f the fundamental period o f the structure to that o f the T M D .  The mass ratio, u, is defined as the ratio o f the total mass o f a single-degree-  of-freedom ( S D O F ) system representing the structure to that o f the T M D .  The damping  ratio, h\, of the T M D is given (Equation 2.1) as follows:  £ = c/2m©  [2.1]  t  where c and m are the damping coefficient and mass, respectively, as defined above, and Oh is the fundamental frequency o f the T M D (cot = 2it/T , where T is the fundamental t  t  period of the T M D ) . These basic concepts will be revisited later.  In principle, a T L C D functions in the same way as a T M D . However, the physical composition o f the systems is very different.  Whereas a T M D is composed o f actual  mass, spring and damper elements, a T L C D only consists o f a volume o f liquid (usually water) in a U-shaped holding structure (Figure 2.1). Due to the simple physical concepts  7  Chapter 2  Background on Control Systems  on which the restoring force is provided in a T L C D , maintenance cost is minimized. A T L C D is a flexible system in meeting architectural demands since it can be fitted easily in a variety of ways. Furthermore, a T L C D provides an excellent reservoir of water high up in the building for fire suppression [1].  Figure 2.1. Scale model experimental set-up of T L C D (Picture courtesy of RWDI). The motion of the liquid in the columns helps to counteract the movement of the building under wind loading. The mass, m, of the system is provided by the presence of the liquid in the columns. The stiffness, k, of the system can be defined numerically (as explained below) but is never referred to explicitly in the literature since the fundamental period of the T L C D is only a function of the length of the liquid column. The damping, c, of the system is introduced as the liquid passes through a variable opening sluice gate [13]. The capability to vary the opening of the sluice gate allows for the tuning of the liquid column damper to achieve a desired system response, and makes this device part of the class of SACS.  A parallel can be made between T M D and T L C D , as both can be idealized as SDOF systems [1]. Figure 2.2 shows a T L C D in this idealized SDOF form mounted on a SDOF structure.  Ms, Ks and Cs represent the mass, stiffness and damping of the SDOF  8  Chapter 2  Background on Control Systems  structure, respectively. M L , K L and C L represent the mass, stiffness and damping of the TLCD, respectively.  The equation of motion of this SDOF/TLCD system can be expressed (Equation 2.2) as follows:  M +M S  ctM  a M  L  M  T  where X , X s  s  r  L  ix  -  +  X. and X  s  "C 0  s  0" C  L  X" X.  +  ~K  S  0  0" K  L  X" X.  "F(t)  _  —  0  [2.2]  are the acceleration, velocity and displacement of the SDOF  system, respectively; X , X and X are the acceleration, velocity and displacement of L  L  L  the TLCD, respectively, a is the length ratio and F(t) is an external force acting on the primary mass.  Figure 2.2. TLCD on a SDOF system.  9  Chapter 2 A  Background on Control Systems  series o f relationships further  describe the T L C D  system.  These relationships  (Equations 2.3 to 2.7) are as follows:  M  K  C  L  L  T  = pAL  [2.3]  = 2pAg  [2.4]  =  pAdx  L  [2.5]  2  a = B/L co L = V  K  [2.6]  L / M =V2g?L  [2.7]  L  where M L is the mass o f the liquid column, p is the liquid density, A is the cross-sectional area o f the column and L is the length o f the liquid column; K L is the "stiffness" o f the liquid column and g is the gravitational constant; C L is the damping coefficient o f the liquid column; and a is the length ratio where B is the horizontal length o f the liquid column.  The effect o f adding a T L C D on the dynamic properties o f a structure is described in detail through the use o f an illustrative example.  Consider a fictitious S D O F structure,  which has a fundamental period o f Ts. B y adding a T L C D , the response o f the system will change.  Naturally, the combined S D O F / T L C D  system has a new fundamental  period simply due to the addition o f mass into the system; however, the presence o f a T L C D has much more important effects on the system than simply adding mass.  In order to investigate this concept further, the effect o f two slightly different T L C D s on the response o f the S D O F / T L C D system are examined in Figure 2.3, where the ratio o f the amplitude o f the vibratory deformation to the static deformation due to an exciting force (wind loads) serves as an indicator o f the dynamic response o f the systems. The response o f the S D O F system alone is also shown in Figure 2.3. T w o tuning ratios were considered while the damping and mass ratios o f the system remained constant.  In the  first case, a T L C D with / = 0.9 is studied and the resulting period o f the system is shown  10  Chapter 2  Background  on Control Systems  in the frequency response function. It can also be seen that there was a decrease in the dynamic response o f the structure.  In the second case, a T L C D with / = 0.8 is  investigated. Figure 2.3 demonstrates that the response o f the S D O F structure is further decreased. It is important to note that the influence o f the tuning ratio on the response o f the system is case specific and that no generalization should be made from the results shown here.  i  i  i  — S D O F system — S D O F system with T L C D , f = 0.9 S D O F system with T L C D , f = 0.8  /I =0.9 Period (s) Figure 2.3. Effect o f the addition o f a T L C D on the response o f the main system. This example shows that the tuning o f the device could have a significant impact on the response o f the structure, both in the magnitude o f the dynamic displacements and fundamental period.  However, a lot more is involved in developing a T L C D .  For  instance, wind tunnel testing is often required to predict which mode will govern the response o f the structure under wind loading. It is important that the response o f the system be governed largely by one mode (usually the first mode) because the T L C D can only be tuned to one period. This new system (structure/TLCD) will have to be studied closely in order to ensure that the wind will not excite another mode. M o r e wind tunnel  11  Chapter 2  Background on Control Systems  testing may be required to verify the effect o f the T L C D on the dynamic response o f the structure.  A s described in this section, the addition o f a T L C D on the One Wall Centre can make the dynamic response o f the building difficult to predict.  This makes the One Wall  Centre a perfect candidate for ambient vibration testing and for model updating in order to capture the true nature o f its response.  12  Chapter 3  Description  of the Building  CHAPTER 3 DESCRIPTION OF T H E BUILDING 3.1  OVERVIEW  The One W a l l Centre is part o f a three building complex located on Burrard Street in the heart o f downtown Vancouver, British Columbia, and is the home o f the Sheraton Hotel (Figure 3.1).  The building is 48 storeys high and includes 6 additional levels o f  underground parking. The bottom two thirds o f the building are used for hotel operations and the top third is for privately owned luxury suites.  Figure 3.1. The One W a l l Centre, looking South.  13  Chapter 3  Description of the Building  A t the time o f its completion, the One W a l l Centre was the highest building in Vancouver, standing 207 m above sea level.  The building is 137 m tall, which also  makes it one o f the tallest structures in the city. The parking levels and elevator shafts extend an additional 23 m into the ground. The floor heights are typically 2.615 m. The building has a 7:1 height-to-width ratio, which makes it a very slender  structure,  susceptible to vibrations due to wind. In plan, the building is 23.4 m by 48.8 m and is shaped like an ellipse with pointed ends (Figure 3.2) [14].  48.8 m  A : Outrigger Columns B: Gravity Load Columns C: Concrete Shear Walls  D: Elevator Shafts E: Service Elevator Shafts F: Stairwells  Figure 3.2. Typical floor plan. M r . Peter W a l l o f W a l l Financing Corporation developed the conceptual design o f the building.  Busby & Associates o f Vancouver, British Columbia, were the architects  involved in the project.  Glotman-Simpson Consulting Engineers, o f Vancouver, British  Columbia, performed the structural engineering [15]. The project began in M a y 1998 and the building was completed and operational in June 2001.  14  Chapter 3  3.2  Description  of the Building  STRUCTURAL SYSTEM  The primary lateral load resisting system o f the One Wall Centre consists o f two different components. The first component is a reinforced concrete shear core, which behaves like a cantilever beam, with larger deflections at the top than near the base.  The second  component is a massive outrigger frame, composed o f outrigger columns and outrigger beams.  The outrigger frame deflects like a shear building, with larger deflections near  the base than at the top. When the two components are connected together, the outrigger frame helps to control deflections o f the shear core at the top o f the building whereas the shear core controls the behaviour o f the frame near the base, therefore reducing the overall dynamic response o f the structure.  The central reinforced concrete shear core is 22 m long and 6.4 m wide and is continuous throughout the height o f the building. The core walls are 900 mm thick at the base o f the structure and taper to 600 mm thick at the top. The core contains six elevator shafts, two service elevator shafts and two stairwells (Figure 3.2). Above level 5, ten 2.5 m by 0.5 m gravity load columns are also located around the perimeter o f the building throughout the height o f the structure.  Transfer beams, located at level 5, carry the gravity loads from  these ten columns to four larger gravity columns that originate at the base o f the structure.  The outrigger frame consists o f sets o f outrigger columns and outrigger beams connected to the shear core at various locations along the height o f the building.  There are four  outrigger columns extending throughout the height o f the building, which are typically 2 m by 1 m in size. The outrigger beams connect the outrigger columns to the central shear core, and are located at level 5, level 21, level 31 and roof level (Figure 3.3). The first set o f outrigger beams, located at level 5, are 6.4 m deep. ?  The second and third sets o f  outrigger beams, located at levels 21 and 31, respectively, are 2.1 m deep.  The base o f  the concrete water tanks for the T L C D also acts as outrigger beams at the roof o f the building.  15  Chapter 3  Description  Roof  3^  Level 31  Level 21  E  Figure 3.3. Typical cross-section.  16  Level 5  of the  Building  Chapter 3  Description of the Building  The elliptical-shaped floor plan o f the building is the same from level 5 to level 22. The floor slabs are typically 175 m m thick. The floor plan becomes irregular at level 22 where a V-shaped portion o f the slab is removed at the West end o f the ellipse (as indicated by the arrow in Figure 3.4). This V-shaped inclusion increases in size at each successive floor until level 32, where the floor plan then remains constant from level 32 to level 48.  Detailed structural drawings o f typical floor plans and elevations are  included in Appendix A .  Figure 3.4 The One W a l l Centre, looking North. The strength o f concrete used in the construction o f the One W a l l Centre varied from a specified concrete strength o f 35 M P a to 50 M P a , as shown in Table 3.1.  These  variations in material properties corresponded to differences in element type (walls, columns, beams, slabs) and element location along the height o f the building. The values  17  Chapter 3  Description of the Building  reported in Table 3.1 are the specified design values, as no experimental values were made available for this study.  Table 3.1. Specified concrete strengths and reinforcing steel strength Strength (MPa)  Description Central Core W a l l s P4 to Level 20 Level 21 to Level 31 Level 32 to Level 48 Columns (outrigger and gravity) P4 to Level 20 Level 21 to Level 31 Level 32 to Level 48 Floor Slabs Outrigger Beams Reinforcing steel  50 45 40 50 45 40 35 45 Grade 400  The reinforced concrete shear core o f One Wall Centre rests on a large spread footing 2.4 m thick. The four outrigger columns rest on 6 x 6 x 2.4 m footings and the four gravity columns rest on 4.4 x 5.5 x 1.8 m footings. A l l the footings rest on sandstone.  Sandstone is not really hard but very dense.  It can be scraped with a pick and softens  when saturated for some time. In general, sandstone is good for high bearing pressures (up to 1000 kPa) in its natural state and can support up to 2000 k P a i f loaded for a short period o f time.  3.3  CONTROL SYSTEM  A structure o f the type o f the One Wall Centre is prone to excessive deformations due to wind because o f its lightness and slenderness. Thus, failure may occur at a serviceability level long before structural failure.  In order to prevent undesirable sensations for the  occupants o f the upper floors, the roof o f the One Wall Centre was fitted with two 183m tuned liquid column dampers ( T L C D ) to reduce vibrations due to wind. 3  T L C D are given in Section 2.3.  18  Details o f  Chapter 4  Ambient Vibration Testing  CHAPTER 4 A M B I E N T VIBRATION TESTING  O f the techniques available to obtain vibration data for output-only modal identification, ambient vibration testing is the most economical non-destructive testing technique to acquire vibration data from large civil engineering structures [16]. The main advantage o f ambient vibration testing is that no "artificial" excitation has to be applied to the structure in order to determine its dynamic characteristics. The structure is continuously dynamically excited by "natural" sources such as wind, nearby traffic, micro-tremors and human activity, to name a few. With proper instrumentation and data analysis tools, one can take advantage  o f these naturally occurring "loads" to evaluate the dynamic  characteristics o f a large structure such as the One Wall Centre.  4.1  TEST OBJECTIVE  The  objective  o f the  ambient  vibration testing  was  to  determine  the  dynamic  characteristics o f the One Wall Centre. The modal parameters o f interest were the lateral and torsional natural periods as well as their corresponding mode shapes.  The goal was  to capture the first six to nine natural periods and mode shapes o f the building. It was also planned to use the test data to estimate damping ratios with a state-of-the-art modal identification technique.  4.2  TEST EQUIPMENT  A l l o f the equipment required to perform the ambient vibration testing o f the One Wall Centre is described in this section.  Specific details o f the instrumentation,  acquisition system and various software used to complete this study are discussed.  19  data  Chapter 4 4.2.1  Ambient Vibration Testing  Instrumentation  A sensor used for ambient vibration testing should have: (i) a frequency range from 0-25 H z , (ii) a range o f ±0.5 g, (iii) a sensitivity o f at least 2 V / g and (iv) a dynamic range o f at least 100 db [17]. The force-balanced accelerometer is a type o f sensor commonly used for ambient vibration testing to measure building accelerations at various locations due to low level o f excitation and satisfies all the technical requirements mentioned above. Moreover, the force-balanced accelerometers are very stable below 20 H z and the behaviour from one accelerometer to another is very similar below this frequency.  This  is an important characteristic since the relative amplitudes o f individual sensors are used to  compute  the  mode  shapes o f a structure.  T w o models o f force-balanced  accelerometers were used in this study: the F B A - 1 1 and the EpiSensor E S - T . Both models o f force-balanced accelerometers are made by Kinemetrics Inc. o f California, USA.  For more details concerning the sensors used for ambient vibration testing at The  University o f British Columbia ( U B C ) , refer to [18].  4.2.2 Data Acquisition System  A VSS3000 unit, also made by Kinemetrics Inc., was used for data acquisition in this study.  The V S S 3 0 0 0 is a fully portable unit that is designed for ambient and forced-  vibration testing field measurements.  The system consists o f a laptop computer, A / D  converter and interface panel (for connecting up to 16 transducers), housed all in one unit.  The primary component o f the data acquisition system is the IOTech D A Q -  Book/216, a 100-kHz PC-based data acquisition system with an A / D resolution o f 16 bits. It connects to the P C by means o f a parallel port connector and has a maximum data throughput o f 800 Kbytes/sec. A n IOTech D B K 1 3 card with 16 programmable input channels is used for analog signal input. This card has selected gain amplifications o f 1, 10, 100 or lOOOx, which is convenient for ambient vibration testing or forced vibration testing.  20  Chapter 4 4.2.3  Ambient Vibration Testing  Software  A number o f commercially available computer programs were used to complete different tasks during the ambient vibration testing o f the One Wall Centre.  The primary data  acquisition software used for this study was a computer program called DasyLab (Version 5.0) [19]. another  On-site quality control o f the recorded data was performed using  computer program called D I A D E M  [20].  In-house  developed Mathcad  worksheets were also used to verify the quality o f the signal and to perform preliminary analyses o f the ambient vibration data [21].  The modal identification was performed  using a computer program called A R T e M I S Extractor (Version 3.1) [22]. M o r e details regarding the modal identification techniques used in this study and the dynamic characteristics extracted using A R T e M I S are presented in Chapter 5.  4.3 TEST PROCEDURE  Planning is the key to successful ambient vibration testing. Adequate preparation before testing is crucial in order to avoid "surprises" during data collection. A description o f the site visits and preparation work that were done prior to testing is included below.  In  addition, details o f the test procedure, including a discussion o f the sensor locations, setup plan and test protocol, are also provided.  4.3.1  Preliminary Planning  The first site visit to the One Wall Centre was conducted on M a r c h 19, 2001. A team from U B C and M r . Robert Simpson, P. Eng. o f Glotman-Simpson Consulting Engineers (who arranged access to the building with M r . Hank Krahn, Project Manager on the One Wall Centre project) were part o f the group visit. The team from U B C consisted o f Dr. Carlos Ventura, the author and a graduate student, M r . Mehdi Kharrazi. During the visit, preliminary discussions centred around the sensor locations (including reference sensor positioning), and the logistics o f the cable distribution from floor to floor. testing date was also scheduled.  21  A tentative  Chapter 4  Ambient Vibration Testing  On April 6, 2001, the author and M r . Kharrazi returned to the site for a second time to ensure that it would be possible to proceed with the envisioned sensor locations and setup plan, and also to confirm the testing date.  One day before testing, on April 9, 2001, the entire crew o f personnel that would be helping to perform the ambient vibration test was briefed about the project.  The crew  consisted o f the team from U B C described above, along with six other civil engineering graduate students.  The actual ambient vibration test took place on April 10, 2001 and lasted all day. The test set-up commenced on-site at 8:00, the data collection continued from 11:30 to 17:30, and the clean-up completed at 19:00.  4.3.2  Sensor Location  In order to capture the translational modes (in the transverse (North-South (NS)) and longitudinal (East-West (EW)) directions) and torsional modes o f the building, two unidirectional accelerometers were positioned in the transverse (NS) direction, and one unidirectional accelerometer was positioned in the longitudinal ( E W ) direction. The sensor locations and orientations are indicated by the arrows in Figure 4.1 and Figure 4.2. The sensors were placed as close as possible to the outside perimeter o f the concrete core from levels 5 to 31. From levels 32 to 48, a change in the floor layout made it difficult to maintain the previous sensor placement. A s a result, the sensors were positioned inside the stairwells close to the concrete core for these upper floors. Since the lateral motion o f the building was the only motion o f interest in this study, no vertical sensors were mounted.  22  Chapter 4  Ambient Vibration Testing  Figure 4.2. Sensor location (from level 32 to level 48). In every ambient vibration test, a number o f "reference" sensors should remain at the same location throughout the duration o f testing while "roving" sensors are moved throughout the building. Ideally, the reference sensors should be located at the antinodes of the expected mode shapes so that the signals from the reference sensors are always strong. A weak signal from a reference sensor (which would be the case i f it was located  23  Chapter 4  Ambient Vibration Testing  at a node o f a mode) would make the assemblage o f the mode shapes difficult [23]. With this in mind, a tri-axial EpiSensor E S - T accelerometer was placed on the 4 5  th  floor and  served as reference location. One axis o f the tri-axial accelerometer was oriented along the transverse (NS) direction and the orthogonal axis was oriented along the longitudinal ( E W ) direction. The vertical axis o f the reference sensor was disabled and no data in this direction was recorded.  Four additional uni-axial sensors (EpiSensor) were placed on  level 34 as backup reference sensors, in case the intended reference sensors on level 45 did not function properly.  4.3.3 Set-up Plan  W i t h large structures, such as the One Wall Centre, it is nearly impossible to gather ambient vibration testing data for the entire building all at one time. This is merely due to the large size o f the overall testing area in relation to the available testing equipment. Therefore, testing is often done in stages by partitioning the structure into regions and conducting a number o f tests in these smaller testing areas. One can define a test set-up by the collection o f floors in the building that are being tested concurrently.  Many factors influence the planning o f the set-ups o f the test. In this study, one o f the most important factors was time. Only one day was allotted for testing since the hotel would be operational only a few days following testing. Thus, all set-ups would have to be completed in a few hours. Another factor affecting the set-up plan was the number o f channels available to record data at one time. A s discussed in Section 4.2.2, a 16-channel data acquisition system was used. Therefore, a maximum o f 16 measurements could be taken per set-up.  A third factor influencing the set-up plan was the location o f the data  acquisition system and the length o f the available cables, as cable length dictates the maximum distance between the sensors and the data acquisition system. It was decided that the data acquisition system would be located on level 34 because o f the space available on this floor and easy access to the stairwells for the relaying o f cables to sensors located on floors above or below. This location was also selected because there  24  Chapter 4  Ambient Vibration Testing  were cables long enough to reach from the data acquisition system to any floors planned for testing.  Taking into account all the details described above, it was determined that the governing factor in the set-up plan would be the number o f channels available i n the data acquisition system. Out o f the 16 available channels, two channels were dedicated to the reference sensors and four channels were assigned to the backup reference sensors, leaving ten channels for the roving sensors.  W i t h three sensor locations per floor, a maximum o f  three floors (nine channels) could be measured per set-up.  Set-up #6 is shown as an  example o f instrument set-up in Figure 4.3 in which the dots represent sensors oriented in the transverse (NS) direction and the arrows represent sensors oriented in the longitudinal ( E W ) direction (the four backup reference sensors are not shown).  Reference  Floor 36  CN os  CN  Floor 28  OO  Floor 14  Figure 4.3. Elevation view o f instrument set-up #6. To carry out the testing set-up plan, three groups o f two crew members were responsible for moving the roving sensors from floor to floor. The building was separated into thirds;  25  Chapter 4  Ambient Vibration Testing  one group was responsible for levels 5 to 17, another group for levels 18 to 33 and a last group for levels 34 to 48. During each set-up, a group was responsible for the set o f three roving sensors on their floor, which included ensuring that the sensors were undisturbed during the data collection and moving the sensors to their next location when the data collection was over. A table with floor assignments for each group for a given set-up is presented in Appendix B .  For the sake o f time, it was decided to measure every other floor in the building, as this would be sufficient to obtain the desired results (six to nine natural periods and their corresponding mode shapes).  Floors below level 5 were not accessible by the central  stairwells, so it was not possible to conduct measurements at these lower floor levels. Therefore, a total o f seven set-ups were needed to complete the building measurements. A summary o f the test set-up details is presented in Appendix B in the form o f a channel schedule used for testing.  The channel schedule provides the exact location o f each  sensor and the data channel corresponding to each sensor for a given set-up.  4.3.4 Data Acquisition Plan  The data was recorded for a period o f 12 minutes per set-up at a gain o f ± 125 m V and at a rate o f 2000 samples per second (sps), and digitally filtered and decimated to 250 sps. Between each set-up, before the roving sensors were moved, the data was transferred to a second computer for preliminary analyses using in-house developed signal processing tools. Each data channel was visually inspected to determine i f the signal o f a particular sensor was outside o f the instrument range (saturated signal) or behaved abnormally (unusual drift o f the signal), in which case the set-up would have to be repeated.  A  sample reference sensor signal (Set-up #6, channel 8) and roving sensor signal (Set-up #6, channel 5) are shown in Figure 4.4 and their respective power spectral density (PSD) function is plotted in Figure 4.5. The power spectral density describes how the power (or variance) o f a time series is distributed with frequency.  Mathematically, it is defined as  the Fourier Transform o f the autocorrelation sequence o f the time series.  26  Chapter 4  Ambient Vibration Testing  Channel 5 2I  "350  1  1  351  352  353  354  355  353  354  355  Time (s) Channel 8  1  2i  350  r—i  351  352 Time (s)  Figure 4.4. Reference and roving sensor sample signals.  MO  3  FL  0  ~~~  0.2  0.4  0.6  0.8 1 1.2 Frequency (Hz)  1.4  1.6  1.8  2  Channel 5 Channel 8  Figure 4.5. Reference and roving sensor sample signal power spectral densities. 27  Chapter 4  Ambient Vibration Testing  If the data looked acceptable, the movement to the next set-up was initiated. U p o n the completion o f data collection, these test results would then be used for output-only modal identification.  28  Chapter 5  Modal Identification Results  CHAPTER 5 M O D A L IDENTIFICATION R E S U L T S  The modal identification results for the One Wall Centre were determined using a computer program called A R T e M I S Extractor (Version 3.1). The natural periods o f the building and their corresponding mode shapes were evaluated using two modal identification techniques  available in A R T e M I S :  the Enhanced Frequency Domain  Decomposition ( E F D D ) technique and the Stochastic Subspace Iteration (SSI) technique. The M o d a l Assurance Criterion ( M A C ) is used to numerically compare the mode shapes returned by the two techniques by means o f a correlation coefficient. The damping ratios determined by both techniques are also reported and discussed.  5.1  NATURAL PERIODS  5. 1 Enhanced Frequency Domain Decomposition (EFDD)  The Enhanced Frequency Domain Decomposition ( E F D D ) is a state-of-the-art technique for output-only modal identification in the frequency domain.  The theory behind the  technique has been well documented by Brinker [24] and will not be presented here. The E F D D is a relatively efficient technique to obtain modal identification results.  For  example, with current computer technology, the required computational time for the One Wall Centre study was about ten minutes.  The E F D D is an extension o f the classical frequency domain approach often referred to as the Basic Frequency Domain ( B F D ) technique, or the Peak Picking technique. BDF  approach  The  is based on simple signal processing using the Discrete Fourier  Transform, and is using the fact that well separated modes can be estimated directly from the power spectral density matrix at the peak [24].  29  Chapter 5  Modal Identification Results  The B D F technique gives reasonable estimates o f natural frequencies and mode shapes i f the modes are well separated.  In the case o f close modes, it can be difficult to detect  those modes, and even i f close modes are detected, the results are heavily biased. However, the B D F technique is fast and use-friendly.  The E F D D technique removes all the disadvantages associated with the B D F technique, while  remaining  fast  and  user-friendly  [24].  B y taking  the  Singular  Value  Decomposition ( S V D ) o f the spectral matrix, the spectral matrix is decomposed into a set o f auto spectral density functions, each corresponding to a S D O F system. This result is exact in the case where the "loading" is white noise, the structure is lightly damped and when the mode shapes o f close modes are geometrically orthogonal [24].  The singular values (representing the frequency locations o f maxima) o f the spectral density matrices o f all data sets recorded are displayed in the A R T e M I S E F D D PeakPicking editor (Figure 5.1). These singular values have been normalized with respect to the area under the first singular value curve (top curve in Figure 5.1). This normalization prevents "weak" modes, which could only be present in one or few data sets, from being overlooked in the analysis. If multiple data sets were recorded (as was done in the One Wall Centre study), the normalized singular values calculated for each data set are averaged, and only these averaged curves are displayed. Optionally, the singular values of each data set can be inspected individually and their peak picking position can be edited i f necessary.  Each peak in Figure 5.1 corresponds to either a structural mode or an operational mode. It is up to the user to judge i f the peak represents one type o f mode or the other, and the graphical display o f the mode shapes in A R T e M I S can assist him/her in his/her decision. The visual interpretation o f the mode shapes in A R T e M I S is an integral step in the E F D D identification technique.  The eight peaks selected in Figure 5.1 represent the natural  periods o f modes identified as structural modes in the analysis o f the ambient vibration data from the One Wall Centre.  Other peaks were examined but discarded at the end  because the visual interpretation o f their corresponding mode shapes was inconclusive  30  Chapter 5  Modal Identification Results  and also those peaks were not identified as modes by the second modal identification technique used in this study described below.  Frequency Domain Decomposition - Peak Picking Average of the Normalized Singular Values of Spectral Density Matrices of all Data Sets.  dB | 1.0 / Hz  1~  J  kg"**-  800 m  2.4  1.8  3.2  Frequency [Hz]  Figure 5.1. Singular values o f the spectral density matrices.  5.1.2 Stochastic Subspace Identification (SSI) The Stochastic Subspace Identification (SSI) is a technique for output-only modal identification in the time domain and was the second technique used in this study. The theory behind the SSI technique has been well documented by Andersen [25] and w i l l not be presented here.  The SSI is a computationally intensive technique, as it requires the  evaluation o f the response o f the system at each time step.  For example, with current  computer technology, the required computational time for the One W a l l Centre study was about 60 minutes.  31  Chapter 5  Modal Identification Results  The SSI technique consists o f fitting a parametric model directly to the raw time series returned by the sensors.  A parametric model (or state space model) is a mathematical  model in which some parameters can be adjusted to change the way that the model matches the actual data collected.  In general, the ideal parametric model is a set o f  parameters that minimize the deviation between the predicted system response (predicted sensor signal) o f the model and measured system response (actual sensor signal) [22].  The problem o f parametric model estimation is that the true model order is unknown. The model order (or state space dimension) is defined as the number o f parameters included in the model. If the model order is too small, then the dynamic response o f the structure cannot be modelled accurately.  O n the other hand, i f the model order is too  high, then the estimated parametric model becomes over-specified and, as a result, the statistical uncertainty on the estimated parameters increases unnecessarily.  One way to overcome this problem is to create a number o f probable parametric models, with the idea that the characteristics o f the structural system will be contained in all the estimated models i f the model orders are high enough. This approach is demonstrated in the stabilization diagram for data set #6 in Figure 5.2. Each cross denotes an estimated eigenvalues and each horizontal row o f crosses corresponds to one parametric model o f a given order.  Thus, the repeated trend o f crosses along a vertical line confirms the  appearance o f the same eigenvalues in all models and demonstrates the presence o f a mode. In the background o f the stabilization diagram in Figure 5.2, the singular values o f the spectral density matrices have been plotted to illustrate that i f such a repeated trend is located at an E F D D natural period, then it is a strong indication that a structural mode has been estimated [22].  In this study, seven data sets were collected from the seven set-ups that were needed to instrument the whole structure. Hence, seven different parametric models were estimated and then "linked" together to assemble the mode shapes and evaluate the natural periods of the building. Each row in Figure 5.3 represents the selected parametric model for each data set.  32  Chapter 5  Modal Identification Results  Stabilization Diagram Data Set: Setup #6 UPC [Data Driven]  State S p a c e Dimension  + +  +  1  mm F r e q u e n c y [Hz]  Figure 5.2. Data set #6 stabilization diagram.  Data Set Number  I D  Select Modes and Link Across Data Sets UPC [Data Driven]  L_  i  i_l  ;  80Dm  1  i 1.6  • •!• '  -•  I'-' 2.4  • •<• 3.2  ••  ')  _ | 4  F r e q u e n c y [Hz]  Figure 5.3. Selected modes from the SSI technique. 5.1.3  Comparison of Results  The results from the E F D D and SSI techniques are summarized in Table 5.1. The natural periods evaluated by both techniques are in good agreement with each other, as all the  33  Chapter 5  Modal Identification Results  period pairs closely follow the 1:1 correspondence line shown in Figure 5.4. standard deviation o f each natural period and is also shown in Figure 5.4.  (OEFDD  The  and cssi) was determined by A R T e M I S  The coefficient o f variation (Cv) is generally small  enough to have good confidence in the calculated values.  Table 5.1. A R T e M I S modal analysis results from the E F D D and SSI Natural period, T (s) Mode Mode Shape EFDD CyEFDD (%) SSI °~EFDD 1 3.57 ± 0.042 3.53 1.2 r'NS 2 2.07 1 EW ± 0.002 0.1 2.08 3 1 torsion 1.46 ± 0.002 0.1 1.45 4 2 NS 0.81 ±0.001 0.1 0.81 5 0.52 2 EW ±0.001 0.2 0.52 6 2 torsion 0.49 ±0.001 0.2 0.49 7 0.36 ±0.001 3 NS 0.3 0.36 8 3 torsion 0.28 ±0.001 0.4 0.28  techniques.  {  st  st  nd  nd  nd  rd  rd  o~ssi  ±0.540 ±0.187 ± 0.066 ± 0.007 ±0.001 ± 0.002 ± 0.002 ±0.001  Cvs (%) 15.3 9.0 4.6 0.1 0.2 0.4 0.6 0.4 S1  EFDD Period (s) Figure 5.4. A R T e M I S period pairs from the E F D D and SSI techniques. The M o d a l Assurance Criterion ( M A C ) [26] can be used to compare mode shape results obtained by the two techniques.  The M A C matrix calculated by A R T e M I S is shown in  34  Chapter 5  Modal Identification Results  Table 5.2. A M A C value is reported for each pair o f mode shapes found by the E F D D technique and the SSI technique. A M A C value equal to 100% for a pair o f mode shapes represents a perfect correlation between two mode shapes. Therefore, the diagonal o f the M A C matrix should approach 100%, as the mode shape vectors returned by the two techniques should, in theory, be identical.  A M A C value between 100% and 90% is  considered to be an excellent correlation. A M A C value less than 90% can be caused by noise or non-linearities in the data or poor modal analysis o f the measured data [26]. From the experience gained from ambient vibration tests done at U B C i n the last eight years, a M A C value above 70% is considered acceptable for results obtained from ambient vibration data. The M A C values show a good correlation for mode pairs 1 and 2, and excellent correlation for mode pairs 3, 4, 5, 6 and 8. Therefore, these modes can be interpreted as structural modes with confidence since they were identified by both techniques. A s for mode 7, the mode shape vectors returned by the two techniques were different, hence the low M A C value.  The mode shape was well defined in the time  domain (SSI technique) but poorly defined in the frequency domain ( E F D D technique). Rather than using the calculated average o f the normalized singular values from each data set, the singular values o f each data set was inspected and the peak picking position was edited.  However, this trial was inconclusive and mode 7 could not be identified  successfully in the frequency domain.  Table 5.2. A R T e M I S M A C matrix from the E F D D and SSI techniques (in %). EFDD Mode Shapes 1 2 3 4 5 7 6 1 65 9 16 10 5 11 18 to 2 2 71 9 18 10 11 ' !W •5, 12 2 3 5 98 7 15 7 ST c§ 4 23 1 87 8 5 3 12 1 5 7 3 1 99 14 10 6 2 99 6 19 1 7 l'» CO 14 8 5 29 4 3 36 7 3 6 8 5 1 14 28 3  35  8 4 4 4 1 13 19 5 92  Chapter 5 5.2  Modal Identification Results  MODE SHAPES  In order to simplify the visualization o f the mode shapes in A R T e M I S , the ellipticalshaped floor plan o f the One Wall Centre was idealized as the shape o f a diamond. The motions o f the apexes o f the diamond at a given floor were inferred using sets o f equations relating the position o f the apexes to the position and motion o f the sensors, based on the assumption that the floor slab behaved as a rigid diaphragm.  The mode shapes o f the One Wall Centre are typical o f bending beam-type buildings. The fundamental mode (mode 1) corresponds to the first mode in the transverse (NS) direction; mode 2 corresponds to the first mode in the longitudinal ( E W ) direction; and mode 3 corresponds to the first torsional mode. Each o f the mode shapes in this first set are evaluated using the E F D D identification technique and are well-defined graphically as shown in Figure 5.5.  The second set o f mode shapes follows a similar order as the first. M o d e 4 corresponds to the second mode in the transverse (NS) direction and the inflection point for this mode is located at level 39. M o d e 5 corresponds to the second mode in the longitudinal ( E W ) direction and its inflection point is also at level 39. Finally, mode 6 corresponds to the second torsional mode and its inflection point is located at level 36. Each o f the mode shapes in this second set are evaluated using the E F D D identification technique and are well-defined graphically as shown in Figure 5.6.  The third set o f mode shapes is not as well-defined graphically as the first and second sets, as seen in Figure 5.7. from them.  However, some valuable information can still be extracted  M o d e 7, evaluated using the SSI technique, corresponds to the third  transverse (NS) mode and its inflection points are located approximately at levels 22 and 45. Lastly, mode 8, evaluated using the E F D D identification technique, corresponds to the third torsional mode and its inflection points are located at levels 23 and 42.  36  Chapter 5  Modal Identification Results  Chapter 5  Modal Identification Results  Chapter 5  Modal Identification Results  Figure 5.7. A R T e M I S modes 7 and 8.  5.3 DAMPING RATIOS  Brincker et al. [27] and Andersen [25] describe the methods used for estimating damping from the E F D D and SSI techniques, respectively. The damping ratios obtained from the E F D D and SSI techniques are tabulated i n Table 5.3. These values represent the amount of damping i n the building at the ambient vibration level corresponding to each natural period.  The standard deviation and the coefficient o f variation o f the damping ratios  show that the uncertainty associated with some values is relatively high as the deviation can be as large as the actual value in some cases. This deviation could be due to noise in the ambient vibration data and not due to natural variability o f the damping value. However, similar trends i n the standard deviation values were obtained in a study by Tamura et al. on a 15-storey reinforced concrete office building that was done using 39  Chapter 5  Modal Identification Results  similar techniques to estimate damping values [28]. The damping values obtained from ambient vibration data must be used correctly and should not be incorporated directly in a finite element o f the One Wall Centre destined to be used for seismic analysis for example.  Table 5.3 Mode  A R T e M I S damping ratios from the E F D D and SSI techniques. Damping ratio (in %) EFDD SSI EFDD C^EFDD (%) o~ssi ±0.52 27.4 14.2 ± 14.50 1.9 0.9 ±0.19 21.1 5.7 ±4.78 0.8 ±0.19 23.8 2.0 ± 1.29 0.6 ±0.18 30.0 0.5 ±0.35 0.7 ±0.21 30.0 0.4 ±0.21 0.6 ±0.21 35.0 0.8 ±0.71 0.6 ±0.26 43.3 1.6 ±0.88 0.4 ±0.19 47.5 ±0.71 1.1 a  1 2 3 4 5 6 7 8  Cpssi (%) 102.1 83.8 64.5 70.0 52.5 88.7 55.0 64.5  The output-only modal identification results provide a great deal o f valuable information about the dynamic response the building. The first eight natural periods o f the One Wall Centre were identified using two modal identification techniques.  The good correlation  between the mode shapes evaluated by each technique adds further confidence in the results.  These experimental results will become indispensable for future updating o f an  analytical model o f the structure.  The concept o f model updating using experimental  results will be explored in more detail later in this study.  40  Chapter 6  Computer Modelling  CHAPTER 6 COMPUTER MODELLING  Computer models are useful tools that can help predict the response o f structures.  The  models can be used to estimate the dynamic properties o f a structure or predict the response o f a building under seismic loads.  T w o different approaches were explored in  this study to predict the dynamic characteristics o f the One Wall Centre.  The first  approach used traditional beam theory to develop a model o f the building and calculate its natural periods. The second approach used a linear-elastic computer model o f the One Wall Centre to compute its natural periods and mode shapes.  Results from both  approaches are presented and discussed in the following sections.  6.1  SIMPLIFIED MODEL  This section presents a simplified model o f the One Wall Centre developed using traditional beam theory.  The motivation for this type o f simplified model is explained  and a description o f the model follows.  The simplified model results are presented and  discussed, as well as compared with the experimental modal identification results.  6.1.1 Motivation  The National Building Code o f Canada ( N B C C ) [29] describes the procedures to follow for the analysis o f live loads due to earthquakes in Clause 4.1.9.1. The N B C C suggests the use o f Equation 6.1 to predict the fundamental period, T, o f a building:  T =  0.09h n  [6.1]  41  Chapter 6  Computer Modelling  where D represents the length o f the lateral force-resisting system and h represents the s  n  height o f the building.  According to the Code equation, the fundamental period o f the One Wall Centre in the transverse (NS) direction is around 4.8 seconds. The experimental fundamental period in the transverse (NS) direction is 3.6 seconds. The difference in these results can be due to a few factors.  The Code equation takes into account the effect o f strong shaking and  mostly accurate for short period buildings, whereas the experimental results were obtained at a very low level o f vibrations. Also, the Code equation relies on the fact that the lateral force-resisting system is o f simple geometry, but the geometry o f the concrete shear core o f the One Wall Centre is quite complex and irregular.  For these reasons, it would be interesting to develop a simple model from which the fundamental period o f the structure could be estimated based on more parameters than the Code equation.  The model would also be useful to predict mode shapes, which the  Code equation does not allow.  6.1.2  Description  The One Wall Centre was idealized using a simplified model. This simplified model was composed o f two base models.  The first base model consisted o f a Timoshenko  cantilever beam [30] used to represent the reinforced concrete core o f the building. The second base model consisted o f a lumped-mass system that was used to account for the floor masses. worksheets.  The two base models were implemented and solved in Mathcad The results from both base models were combined using Dunkerley's  approximation [30] to produce a complete solution.  The core o f the building is a continuous system that does not behave purely in bending or purely in shear; it is more likely that the behaviour o f the core consists o f a combination o f the two.  Therefore, a Timoshenko cantilever beam was a good candidate for  modelling the reinforced concrete shear core in the first base model because it takes both  42  Chapter 6  Computer Modelling  o f these responses into account.  Some o f the properties in the model included the  Young's modulus for reinforced concrete (30,000 M P a ) , the approximate moment o f inertia in the transverse direction (core and outrigger columns included in calculations, 1600 m ) and the floor height (2.925 m).  The complete solution for the Timoshenko  4  Beam model is presented in Appendix C.  The floor masses in the structure are significant enough that they were modelled separately in the second base model. A 46-DOF lumped-mass system model (one D O F for each level) was developed to estimate the contribution o f the floor masses to the dynamic response o f the building.  Once again, the Young's modulus for reinforced  concrete was used (30,000 M P a ) and the floor height was set to 2.925 m. The moment o f inertia in the weak direction was 25,000 m , which compares to the actual moment o f 4  inertia o f an ellipse (with a major axis o f 48.8 m and a minor axis o f 23.4 m) o f about 30,000m . The mass o f each floor was estimated at 500,000 kg, as the mass o f the floor 4  slab alone is close to 400,000 kg. The lumped-mass system was assumed to deflect in bending and not in shear.  The complete solution for the 4 6 - D O F model is presented in  Appendix C.  The results from the two analyses were combined using Dunkerley's approximation (Equation 6.2):  [6.2]  where T is the estimated value o f the period o f interest, T i is the period o f the Timoshenko Beam and T 2 is the period o f the lumped-mass system.  6.1.3 Results  The results from the simplified model are presented in Table 6.1 and the results from the E F D D and SSI modal identification techniques are repeated for comparison.  43  Chapter 6  Computer Modelling  ;  Table 6,1. Simplified M o d e l and experimental natural periods. Natural period, Ti (s) Mode Timoshenko Lumped-mass Dunkerley EFDD I 'NS 1.76 2.94 3.43 3.57 2 NS 0.52 0.47 0.70 0.81 3 NS 0.27 0.17 0.32 0.36  SSI 3.53 0.81 0.36  S  nd  rd  The calculated fundamental period from the simplified model is relatively close to the ones determined from ambient vibration data by the computer program A R T e M I S . 2  and 3  n d  from  the  r d  transverse (NS) natural periods were estimated as well. continuous and lumped-mass base models were  The  The mode shapes  averaged  in order  to  approximate the mode shapes for the Dunkerley's approximation results (Figure 6.1). The inflection point o f the 2  n d  transverse (NS) mode in the simplified model is at level  36, whereas the inflection point found using A R T e M I S is at level 39.  Mode 1  Mode 4 '  "V a  • i  /// j ' Timoshenkc Dunkerley  f  f l  1 /  a  S  * ff § f  y -  j  *  i 1  W  •' 1 ' //  V• X* •  • '/ **/  1  0  a*  • 1/ • ' / ."i /' /  a  Mode 7  i i  i i  1  If'-'  1 \ Timoshenko  1  1  Dunkerley  Figure 6.1. Simplified M o d e l 1 , 2 st  n d  '  Timoshenko Lumped-mass * " • Dunkerley  H  \  and 3 transverse ( N S ) mode shapes. r d  These results provide a relatively good approximation o f the experimental results. It can be concluded that the simplified model could be considered to be a suitable tool for a mathematical estimation o f the natural periods and mode shapes o f a high-rise building.  44  Chapter 6  Computer Modelling  6.2 ETABS  MODEL#l  A finite element (FE) model was developed using the computer program E T A B S (version 7.24) [31] for the modal analysis o f the One Wall Centre.  A short description o f the  computer program is given, followed by a detailed description o f the F E model.  The  natural periods evaluated by E T A B S and their corresponding mode shapes are presented and compared to the experimental results.  6.2.1 ETABS Software  E T A B S is a computer program developed specifically for the design and analysis o f building systems.  The F E program can be used for linear and non-linear, static and  dynamic analysis o f a three-dimensional model o f a structure. delta effects,  Other options include P -  response spectrum analysis and linear time history analysis.  Three-  dimensional mode shapes and their corresponding periods are available modal analysis outputs in E T A B S .  6.2.2 Description  A partial model o f the structure, called Model#l herein, was used by Glotman-Simpson Consulting Engineers for the wind and seismic analyses o f the One Wall Centre. It is common practice in the industry to only model the main lateral load-resisting system in an attempt to reduce the time and effort that would be required to develop a very detailed computer model o f the structure.  The reinforced concrete shear core (light green  elements in Figure 6.2), outrigger beams, and outrigger columns (black elements in Figure 6.2) were included in Model#l as they form the main structural system.  The  water tanks o f the T L C D were modeled in an unfilled state so that a comparison between the experimental results (obtained when the T L C D was empty) and analytical results could be made.  45  Chapter 6  Computer Modelling  Figure 6.2. E T A B S Mode The computer program was used to estimate the self-mass o f the elements included in the model through geometry and material properties calculations.  However, it was also  necessary to account for the mass of the elements not included in the model, such as the reinforced  concrete  slab,  gravity columns  and  other  non-structural  components.  Therefore, an additional point mass o f 410,000 k g was added to every floor level at the geometrical centre o f the model (located inside the reinforced concrete core) in the X and Y global horizontal directions. In addition, the rotational moment o f inertia o f the floor slab had to be added to the model. A moment o f inertia o f 62x10 k g * m , calculated 6  2  from an ellipse with similar dimensions as the slab, was applied at the geometrical centre of the model at every floor. Finally, to simulate the rigidity o f the floor slabs, every point located on the same X Y plane was constrained together using a " R i g i d Diaphragm." Therefore, the in-plane motion and the rotation about the Z-axis o f every point on the same floor level was the same. The out-of-plane behaviour o f the point objects was not affected by the " R i g i d Diaphragm" constraint.  46  Chapter 6  Computer Modelling  Frame elements were used to model the outrigger columns and the outrigger beams. The full cross-section o f the frame elements was used to calculate the effective moment o f inertia (i.e. Igross)-  Rigid zones were applied at the ends o f the frame elements at the  intersection between the outrigger beams and outrigger columns. The length o f the rigid zone on the beams was equal to the column thickness, and the length o f the rigid zone on the columns was equal to the beam depth. A t the base o f the structure in the model, the ends o f the frame elements were fixed against translation and rotation for the 6-DOF. N o frame element releases were used in the model.  The columns and beams o f the  underground floor levels were not modelled.  The reinforced concrete shear core was modelled with shell elements.  The thick plate  option was selected in E T A B S to take into account the out-of-plane shear deformations of the shell elements.  The shell element also has both in-plane membrane stiffness and  out-of-plane plate bending stiffness by default.  The effective shear area for the analysis  was taken as the full concrete area o f the core (i.e. Agross). Once again, at the base o f the structure in the model, the ends o f the shell elements were fixed against translation and rotation for the 6-DOF. The shear core o f the underground floor levels was not modelled.  The material properties ( M P ) in the model were divided into three groups. The elements (columns, beams, and walls) located between levels 1 and 20 were assigned M P 1; the elements located between levels 20 to 31 were assigned M P 2 ; and the elements located between level 31 and the roof were assigned M P 3 . The groups o f material properties are tabulated in Table 6.2.  The Young's modulus o f concrete shown in the table was  calculated from the expected stiffness o f concrete.  Table 6.2. E T A B S F E model material properties.  Material Property Group  Mass Density, p (kg/m ) 3  MP1 MP2 MP3  2450 2450 2450  47  Young's Modulus, E (MPa) 36,500 35,000 33,750  Chapter 6  Computer Modelling  6.2.3 Results  The natural periods and corresponding mode shapes o f the One Wall Centre were evaluated using a modal analysis o f Model#l in E T A B S .  The results from Model#l are  summarized in Table 6.3 and the ambient vibration results are repeated for comparison [32].  One should note that Model#l predicts mode 5 to be the 2  whereas the experimental results show that mode 5 is the 2 (EW) direction.  n d  n d  torsional mode,  mode in the longitudinal  In addition, Model#l predicts mode 6 to be the 2  n d  mode in the  longitudinal ( E W ) direction, whereas the ambient vibration testing results clearly show the 2  n d  torsional mode.  Table 6.3. E T A B S Model#l and Experimental Mode Mode Shape 1 1 NS 2 1 EW 3 1 torsion 4 2 NS 5 2 EW 2 torsion 6 7 3 NS 8 3 torsion st  st  st  nd  nd  nd  rd  rd  experimental natural periods. ETABS Model T (s) Mode Shape 3.57 1 NS 2.07 1 EW 1.46 1 torsion 0.81 2 NS 0.52 2 torsion 0.49 2 EW 0.36 3 NS 0.28 3 torsion t  st  st  st  nd  nd  nd  rd  rd  #1 T (s) 4.13 2.30 1.72 1.00 0.58 0.54 0.41 0.36  %Diff.  t  15.8 10.8 18.1 23.1 N/A N/A 13.7 22.7  From these results, it can be seen that the natural periods estimated by Model#l are, on average, about 15% higher than the periods estimated experimentally.  This is shown  graphically in Figure 6.3, as all the period pairs lie below the 1:1 correspondence line.  48  Chapter 6  0  Computer Modelling  0.5  1  1.5  2  2.5  3  3.5  4  ETABS Model#1 (s) Figure 6.3. E T A B S Model#l and experimental period pairs. It is fair to say that Model#l provides a good representation o f the dynamic response o f the One Wall Centre because it compared reasonably with the experimental results. However, the discrepancies between the analytical results and the experimental results obtained at low levels o f excitation suggest that there is room for improvement in the model. A number o f questions come to mind for model improvement: •  Is there anything that can be added to Model#l in order to better match the experimental results?  •  What parts of Model#l are sensitive to changes in parameters?  •  H o w much would Model#l parameters have to change in order to match the experimental results?  A l l o f these issues will be addressed to provide answers to these questions in the following sections describing Model Updating.  49  Chapter 7  Finite Element Model Updating  CHAPTER 7 FINITE E L E M E N T M O D E L UPDATING  M o d e l updating consists o f the careful adjustment o f selected parameters in a F E model such that the analytical modal characteristics more closely match the experimental ones. Generally, the differences in analytical and experimental natural periods and modal order give insight into how the parameters in the F E model should change in order for the analytical results to better reflect the observed experimental response.  T w o F E model-updating techniques were used in this study. Manual updating was used to improve Model#l and, in the process, Model#2 was created. described in detail in Section 7.1 and some results are presented.  This technique is Automated updating  was used for both Model#l and Model#2, and the details o f this technique are described in Section 7.2. The capabilities o f the computer program used for the updating process are also described briefly. The sensitivity o f both models to various parameter changes was assessed, and these, along with some numerical results, are presented and discussed.  7.1  MANUAL MODEL UPDATING  One technique used to improve the analytical results for a F E model is manual model updating. In manual updating, the user must identify, by inspection, the differences that exist between the analytical and experimental results. Then, he/she has to select physical properties from the model (such as Young's modulus, material mass density, etc.) and vary them in a way that will improve the match between these results and the experimental ones.  This iterative process (inspection/selection/modification) is repeated  until a good match is obtained between the analytical and experimental results.  B y inspection o f the results from M o d e l # l , it was concluded that the natural periods o f the model needed to be decreased and that order o f the 2  50  n d  torsional and 2  n d  E W modes  Chapter 7  Finite Element Model Updating  was reversed, according the experimental results. From these observations, it was clear that the F E model o f the structure needed to be stiffened. It should first be noted that it is not common practice to model the non-structural components (such as the exterior windows/cladding and interior partitions) o f a building. This is due to the fact that it is time consuming to include these elements in the model, as their contribution to the response o f the system under wind and earthquake loading is generally not considered in design.  However, the results from ambient vibration testing provide a measure o f the  response o f the entire building system, not only a measure o f the response o f the lateral load resisting system.  Therefore, not only the main structural components (such as the  central concrete shear core and outrigger framing system), but also the non-structural components,  should be modelled i f a good match between the experimental and  analytical results is desired.  With this in mind, Model#2 was developed using the structural elements from Model#l and additional non-structural elements in the building (Figure 7.1).  The gravity load  columns were added and modelled with the geometry/section/material properties shown in the structural drawings. Outside cladding was also added and assigned a low stiffness value.  It is important to note that the cladding was intended to model the windows as  well as the walls and partitions that are contributing to the stiffness o f these so-called non-structural elements.  With only these adjustments,  the modal analysis was repeated for Model#2 and the  resulting modal order became the same as the experimental one.  The next step in the  manual model updating was to adjust the Young's modulus ( E = 35 GPa) and the thickness (t = 0.0125 m) o f the cladding so that the analytical natural periods o f Model#2 matched the experimental ones.  These values o f the Young's modulus and the cladding  thickness were similar to those found by Ventura et al. in a similar study [33].  51  Chapter 7  Finite Element Model Updating  Figure 7.1. E T A B S M o d e l #2: (a) wire frame, (b) full model. The results from the modal analysis for Model#2 are shown in Table 7.1, along with the results for M o d e l # l and the ambient vibration test results repeated for comparison. Figure 7.2 shows good agreement between experimental and analytical results, as the natural periods for Model#2 are now all within an acceptable 10% difference. The mode shapes from Model#2 are shown in Figure 7.3.  It can be seen that the modal order  obtained analytically now corresponds to the one obtained experimentally (Figure 5.5, Figure 5.6, and Figure 5.7).  Based on these results, Model#2 has been well calibrated  and is thought to be a good representation o f the real structure.  52  Chapter 7  Finite Element Model Updating  Table l.l.  M o d a l analysis results after manual model updating. Experimental ETABS Model#l ETABS Mode Shape T (s) Mode Shape T, (s) % Diff. Mode Shape 3.57 r'NS 4.13 16 r'NS I 'NS 1 EW 2.07 I 'EW 2.30 11 r' EW 1 torsion 1.46 1 torsion 1.72 18 1 torsion 0.81 2 NS 1.00 23 2 NS 2 NS 0.52 2 EW 2 torsion 0.58 N/A 2 EW 2 torsion 0.49 0.54 2 EW N/A 2 torsion 3 NS 0.36 0.41 3 NS 14 3 NS 3 torsion 0.28 3 torsion 0.35 23 3 torsion  Mode  t  1  S  st  2  S  st  3  st  nd  4  nd  nd  5  nd  rd  rd  8  nd  nd  rd  7  nd  nd  nd  6  st  rd  rd  rd  Mode l# 2 T (s) % Diff. 3.52 -1 1.90 -8 1.42 -3 0.89 9 0.49 -6 0.48 -2 0.39 8 0.29 2 f  0.5 0+^ 0  , 0.5  , 1  , 1.5  , 2  , 2.5  , 3  ETABS Model#2 (s)  Figure 7.2. ETABS Model#2 and experimental period pairs.  53  , 3.5  1 4  Chapter 7  Finite Element Model Updating  T i = 3.519 sec  T = 0.490 sec 5  T = 1.897 sec  T = 1.415 sec  2  3  T = 0.478 sec  T = 0.386 sec  6  7  T = 0.889 sec 4  T = 0.288 sec 8  Figure 7.3. E T A B S M o d e l #2 mode shapes.  7.2  AUTOMATED MODEL UPDATING  Automated model updating follows the same general concepts as manual model updating (inspection/selection/modification), except that the process is computer aided. 54  The  Chapter 7  Finite Element Model Updating  computer program allows the user to perform an in-depth inspection o f the F E model. The sensitivity o f the dynamic characteristics o f the F E model to a wide range o f parameters can be determined by the program, providing valuable information about which parameters should be selected for updating. series  o f iterations,  changing the  Then, the program goes through a  parameter values to  experimental natural periods and mode shapes.  match  the  analytical and  It should be understood that it is not  automatic model updating, but rather automated model updating.  The user plays an  active role in making decisions as to which parameters should be selected for updating and whether or not the results extracted by the computer program make sense.  The main advantage o f using a computer program to update a F E model is that a multitude o f parameters can be changed concurrently in order to achieve the best possible match between experimental and analytical results in terms o f both natural periods and mode shapes. In addition, the automated updating process can be made more effective i f the sensitivity o f the F E model to property changes is considered. I f the sensitivity o f the model to certain parameters is known, then property changes can be explained more rationally and targeted to a specific zone o f the model instead o f varying the properties over the whole model.  7.2.1  FEMtools Software  F E M t o o l s (version 2.2) was the software used to update M o d e l # l and Model#2. F E M t o o l s is a multi-functional computer-aided engineering ( C A E ) program that includes various tools for true integration o f finite element analysis and static or dynamic testing, automation o f C A E processes and development o f data pre- and post-processing tools [34].  F E M t o o l s is not a general-purpose finite element computer program, although it  offers several finite element analysis capabilities such as an internal standard  element  library and a solver. Likewise, it is not a software for analysis o f vibration test data, but it can import experimental models and data from most existing commercially available testing computer programs.  55  Chapter 7  Finite Element Model Updating  The main purpose o f F E M t o o l s software is to give the user a wide variety o f tools that allow him/her to work on and have a better understanding  o f his/her F E model.  F E M t o o l s can be used to inspect, validate, refine, and monitor change to F E models, etc. Standard tools include data interfacing, database management, static analysis and normal modes analysis, frequency response analysis and operational mode analysis. viewer allows for graphic displays o f meshes and analysis results.  A simple  The program can also  be customized or extended using a powerful scripting language.  F E M t o o l s is a four-part modular computer software, which includes: •  A platform module;  •  A correlation analysis module;  •  A sensitivity analysis module; and,  •  A model-updating module.  The individual modules will be described briefly below. Further information concerning each module can be found in the F E M t o o l s user's guide [34].  7.2.1.1  The Platform Module  The F E M t o o l s platform is the core module o f the computer program. It includes the data interface, database management, graphics viewers, programming language and solvers for static and dynamic analyses. This module is needed in order to use all other modules. The types o f analyses available in F E M t o o l s include mass analysis, static deformation analysis, modal analysis, frequency response analysis and harmonic response analysis.  The only type o f analysis used in this study was the modal analysis, from which the dynamic properties (natural periods and mode shapes) o f the models were extracted.  To  perform the normal modes analysis, the F E M t o o l s blocked Lanczos sparse solver was used.  7.2.1.2  The Correlation Analysis Module  The correlation analysis module contains tools that allow the user to compare two sets o f results data (experimental and analytical) quantitatively and qualitatively. It includes pre-  56  Chapter 7  Finite Element Model Updating  test analysis capabilities in order to find the optimal locations and directions for exciting and measuring a structure.  Correlation analyses are also possible, and error localization  allows for spatial correlation to identify areas o f better or poorer correlation. Lastly, a user can rapidly assess the influence o f structural changes on the modal parameters using the structural dynamics modification feature.  7.2.1.3  Sensitivity Analysis Module  The sensitivity analysis module can provide the user with insight into how the dynamic properties o f a F E model are influenced by modifications to various properties, such as spring stiffness, material stiffness, or geometry, to name a few.  A sensitivity analysis o f the dynamic properties o f the two F E models o f the One Wall Centre was performed for a multitude o f parameters.  The analysis procedures and the  results from the sensitivity analysis are presented in Section 7.2.2.3 and Section 7.2.3.3 for Model#l and Model#2, respectively.  7.2.1.4  Model Updating Module  The model-updating module allows for automated F E model updating.  Through an  iterative process, it modifies selected parameters until the correlation coefficient satisfies a given convergence criterion.  The theory behind the automated  model updating  technique is given in Appendix D and [35].  Results from the automated model updating of Model#l and Model#2 follow.  7.2.2 ModelM Updating  7.2.2.1  FEMtools Model#l  A F E model had to be generated in F E M t o o l s in order to perform the automated model updating, as a direct interface between E T A B S and F E M t o o l s did not currently exist. The F E M t o o l s model was based on the F E model already created in E T A B S .  By  manipulating the *.s2k file from E T A B S (containing all the relevant information about  57  Chapter 7  Finite Element Model Updating  the F E model), a systems neutral file ( S N F ) was created.  The S N F is a text file  containing all the information about the geometry, material properties, boundary conditions, and elements o f the F E model. F E M t o o l s uses the S N F to generate the F E model.  The F E M t o o l s Model#l looked very similar to the E T A B S M o d e l # l .  Beams and  columns were modelled as 3D beam-column elements, and shear walls were modelled as 4-node plate elements.  Because the program does not support the creation o f rigid  diaphragms, the outrigger columns were linked to the concrete shear core by flat floor slabs modelled as 4-node plate elements. outrigger columns.  This was done to avoid local modes in the  The elimination o f localized modes is necessary in order to study  only the global response o f the building in the automated updating process. The material properties for all elements were the same as in the E T A B S model (Table 6.2). In total, the model consisted o f 232 beam-column elements, 1,912 4-node plate elements, 2,443 nodes, 3 different material properties, 122 different element geometry sets, and 14,658 degrees o f freedom.  7.2.2.2  Model#l Before Updating  The updating technique used by F E M t o o l s attempts to match the natural periods from the experimental  and  analytical results,  as well  as the  corresponding mode  shapes.  Therefore, well-defined experimental mode shapes are needed to obtain quality results. A s such, only the first six experimental mode shapes were selected for updating purposes. Experimental modes 7 and 8 were not well-defined graphically and were therefore excluded from the automated updating process.  However, six experimental modes  should be an adequate database o f natural periods and mode shapes for F E M t o o l s to perform the model updating.  A s a general rule in automated updating, one should extract twice as many analytical modes as experimental mode shapes.  A greater number o f analytical mode shapes gives  the computer program a large database o f responses to choose from when updating the model. Therefore, 12 analytical mode shapes were extracted from Model#l in F E M t o o l s .  58  Chapter 7  ;  Finite Element Model Updating  The results from the modal analysis in F E M t o o l s before summarized in Table 7.2.  updating Model#l  are  The table shows the experimental and analytical natural  periods and the M o d a l Assurance Criterion ( M A C ) used to numerically compare the mode shapes returned by the two models.  A l l o f the natural periods returned by the  program are lower than the experimental values, as shown by the period pairs plotted in Figure  7.4.  Ideally, the  period pairs  should fall  correspondence with the experimental results.  along the  line denoting  1:1  The analytical modal order is also  reported in Table 7.2 and corresponds well with the experimental one. Table 7.2. F E M t o o l s Model#l modal analysis results before updating. Experimental FEMtools Model# 1 Mode Mode Shape Mode Shape T (s) 1 1 NS 3.57 1 NS 2 1 EW 2.07 1 EW 3 1 torsion 1.46 1 torsion 4 2 NS 2 NS 0.81 5 0.52 2 EW 2 EW 6 2 torsion 0.49 2 torsion 7 n/a n/a 3 NS 8 n/a 3 torsion n/a 9 n/a n/a 4 torsion 10 n/a n/a 4 NS 11 n/a n/a 4 NS/3 EW 12 n/a n/a 5 Torsion t  st  st  st  st  st  st  nd  nd  nd  nd  nd  nd  rd  rd  th  th  t h  r d  th  0 4^ 0  Tt (s) 3.19 1.59 0.97 0.76 0.39 0.37 0.31 0.26 0.20 0.19 0.18 0.16  ,  ,  ,  ,  ,  ,  ,  1  0.5  1  1.5  2  2.5  3  3.5  4  FEMtools Model#1 (s) Figure 7.4. F E M t o o l s Model#l and experimental period pairs before updating.  59  Chapter 7  Finite Element Model Updating  The full M A C matrix comparing the experimental and analytical mode shapes before Model#l updating is shown in Table 7.3.  It can be seen that the first four analytical  mode shapes are already close to the experimental ones.  However, mode 5 and 6 are  being coupled with other modes, which was not the case i n the experimental results.  Mode  Mods  Table 7.3. F E M t o o l s Model#l M A C matrix before updating (in %).  •2 o  i 1  i 2 3 4 5 6 7 8 9 10 11 12  Experimental Mode Shape 5 3 4 6 2 0  1  2  99  0  0  87  0  6 1  0  3  0  7  0  99  0  0  16  3  0  0  98  0  0  0  2  3  0  60  35  1  0  15  0  21  0  0  1  13  0  WmmWmWm 0  2  0  0  0  2  23  3  0  2  1  3  54  1  0  1  3  9  0  2  0  1  3  8  0  1  0  3  1  0  1  The natural periods and mode shapes returned by F E M t o o l s are in average 15% lower than the ones estimated by E T A B S . This difference is mainly due the different modelling assumptions made in each model. The element library i n F E M t o o l s is not as elaborate as in E T A B S , therefore rigid-end offsets cannot be modelled in F E M t o o l s and rigid diaphragms cannot be created.  Taking all o f these differences into account, it makes  sense that the two models would be comparable but not exactly the same.  7.2.2.3  ModeI#l Sensitivity Analysis  Since the sensitivity analysis plays an integral role in the automated model updating process in F E M t o o l s , the concepts o f relative sensitivity and normalized sensitivity w i l l be explained herein.  60  Chapter 7 If the  Finite Element Model Updating  model  sensitivities to  different  types o f parameters are  to  be  compared  simultaneously (as it is the case in this study), the use o f relative sensitivities is advised [34]. The relative sensitivity matrix, [S ], is obtained (Equation 7.1) as follows: r  [S ] = r  5R, 5P  •hi  [7.i]  ;  where R; represents all the selected responses, Pj represents all the selected parameters, 8Rj/ 8Pj is the differential sensitivity coefficient, and [Pjj] is a diagonal square matrix holding the parameter values. (Note that the differential sensitivity coefficient is equal to the sensitivity matrix when computed for all selected responses with respect to all selected parameters)  The relative sensitivity can either be positive or negative, as a result o f a positive or negative differential sensitivity coefficient.  A positive 6Ri/ 8Pj means that an increase o f  the parameter " j " will also result in an increase in the value o f the response " i " . Conversely, a negative 5R;/ 5Pj means that an increase o f the parameter " j " will result in a decrease in the value o f the response " i " .  The relative sensitivity can also be normalized with respect to the response value (Equation 7.2) as follows:  [S„]=[S ]*[R,r  [7.2]  r  where [S ] is the normalized relative sensitivity matrix, and [Rj] is a diagonal square n  matrix holding the response values.  The normalized sensitivity is used in this study to  compare the effect o f changing parameters on the dynamic response o f the F E model.  It is important to note some definitions that will be used in the sections that follow.  In  the following sections, an element corresponds to a set o f finite elements in the model. A property corresponds to a physical property in the model, such as the Y o u n g ' s modulus. A parameter refers to a selected property o f a given element.  61  For instance, the mass  Chapter 7  Finite Element Model Updating  density (a property) o f the shear walls o f the upper floors (an element) will constitute a  parameter.  The selected elements for the sensitivity analysis o f Model#l are described in Table 7.4. Three properties were selected for the sensitivity analysis: the Y o u n g ' s modulus (E), the material mass density ( R H O ) , and the second moment o f inertia (I).  The sensitivity o f  Model#l to a change in Young's modulus and material mass density was calculated for all elements.  The sensitivity o f Model#l to a change in second moment o f inertia (in  both the Y - and Z-directions) was calculated for the columns and beams only. In total, 72 parameters were selected for the sensitivity analysis.  The results from the sensitivity analysis o f Model#l are shown in Figure 7.5. It can be observed that Model#l is not very sensitive to a change in Young's modulus in the columns and beams (parameters 1 to 13), but is fairly sensitive i f that change takes place in the concrete shear core (parameters 14 to 24). A similar trend is true for the material mass density. The model is not too sensitive to a change in mass density in the columns and beams (parameters 25 to 36), but becomes more sensitive i f the concrete shear core (parameters 37 to 48) is affected by such a change. The model is relatively unaffected by a change in second moment o f inertia in the beams and columns (parameters 49 to 72).  62  Chapter 7  ;  Finite Element Model Updating  Table 7.4. F E M t o o l s M o d e l # l F E element description. Element 1 2 3 4 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25  Element Description Outrigger columns Outrigger columns Outrigger columns Outrigger columns Outrigger beams Outrigger beams Outrigger beams Outrigger beams Coupling beam Coupling beam Transfer beams Other beams Shear walls Shear walls Shear walls Shear walls Shear walls Shear walls Shear walls Shear walls Shear walls Shear walls Shear walls Shear walls  11  Location in Building Model Bottom third Bottom third M i d d l e third Top third Level 5 L e v e l 21 L e v e l 31 Roof L e v e l 50 Roof Level 5 Throughout Bottom third Bottom third Bottom third M i d d l e third M i d d l e third M i d d l e third Top third Top third Top third Bottom third M i d d l e third Top third  RHO  E  0.8 0.6  >  0.4  '1 0) (fi T3 0) N  re E  0.2  0  # 1 ^ ^ -  TJ-  l ~ -  n fO  0.2  O -0.4  z -0.6 -0.8  Parameter  Figure 7.5. F E M t o o l s M o d e l # l normalized sensitivity to selected parameters.  63  Chapter 7  Finite Element Model Updating  A s suggested by Dascotte [36], a large number o f parameters were used in the sensitivity analysis. However, a smaller number o f parameters must be selected for the automated model updating and thus, only the parameters to which the model was most sensitive were chosen. A s such, the Young's modulus and the material mass density for elements 14 to 25 (for which the exact location in the F E model is shown in Appendix E ) were retained for the automated updating procedure, for a total o f 24 parameters.  The  parameters used for updating Model#l are summarized in Table 7.5.  Table 7.5. F E M t o o l s Model#l selected parameters for updating. Parameter Element 1 14 2 15 16 3 4 17 5 18 6 19 7 20 8 21 9 22 10 23 11 24 12 25 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25  Property E E E E E E E E E E E E RHO RHO RHO RHO RHO RHO RHO RHO RHO RHO RHO RHO  The normalized sensitivity o f Model#l to the parameters selected for updating is shown in Figure 7.6.  It can be noted that the F E model was more sensitive to a variation in  parameters 2, 5, 8, 14, 17 and 20. These parameters correspond to elements 15, 18 and 21, shown in Figure 7.7.  These six parameters will be monitored more closely in the  sections that follow and shall be referred to as the Family o f Six herein.  64  Finite Element Model Updating  Chapter 7  RHO 0.8  0.6  C  0.2  o  GO ~o 03  N  ro  1  2  3  4  5  6  7  8  9  n - i H  1 0 1 1 1 2 1 3 U 4 ^ l i l 6  17  19  20 21 22 23 24  -0.2  E c O -0.4  z  I -0.6 -0.8 -1  Parameter  Figure 7.6. F E M t o o l s Model#l normalized sensitivity to selected parameters for updating.  15  18  21  Figure 7.7. FEMtools Model#l F E elements 15, 18 and 21. The sensitivity of each mode (1 through 6) to the parameters selected for updating is shown in Figure 7.8. It is interesting to note that each of the parameters in the Family of Six influences each of the modes in the response of the model differently. 65  For example,  Finite Element Model Updating  Chapter 7  parameter 2 (Young's modulus, element 15) affects modes 2 and 3 significantly, whereas the other modes are not affected as strongly. This can be due to the fact that the base o f the concrete shear core is mostly solicited in modes 2 and 3.  Mode 1  0.3  0.2  0.2  0.1  0.1  r~| l r^pi r v - W -  0  co  in  r—  "55 c 0) CO N  to  I  -0.1  -0.1  -0.2  -0.2  -0.3  -0.3  Mode 3  0.1  m  r-  to  o>  w  h-  o  m  N-T— o  Mode 4  0.3  0.2  0.2  -  0.1  i£ 2E3  0  TJ O  0  co  0.3  >  Mode 2  0.3  • O  1  f  -  C  U  l  f  1T.V<L"—M ^  -  O  i  • -  C  O  l  O  0  teH-* CO  5  -0.1  co  CN  in  r-.  T-  co  co  1  -0.1  flj  E o z  -0.2  -0.2  -0.3  -0.3  Mode 5  0.3  0.2  0.2  0.1  0.1  0  Mode 6  0.3  T h irfft-n CO  tf>  f-  0  CO  -0.1  -0.1  -0.2  -0.2  v  1 h co  m  n-v_' r~~  o  I  ,  i—t  T - to L ^ i n '  ...  Itif ' co  TCM  1 '  CO CM  '  -0.3  Parameter Figure 7.8. F E M t o o l s Model#l normalized sensitivity o f each mode to selected parameters for updating. 7.2.2.4  Model#l After Updating  The updated results for Model#l are very interesting and w i l l be presented in two parts. The  first part w i l l examine the comparison between the experimental and analytical  natural periods and mode shapes after the automated updating; the second part w i l l 66  Chapter 7  Finite Element Model Updating  examine which parameters changed in the updating and the magnitude o f the changes needed to achieve such a match.  Beginning with the comparison o f experimental and analytical findings, the results from the modal analysis are summarized in Table 7.6.  The match between the experimental  periods and analytical periods is near perfect, as Figure 7.9 shows that the period pairs are well aligned. The modal order remained the same after the automated updating  Table 7.6. FIEMtools Model#l modal analysis results after updating. Experimental FEMtools ModeB 1 Mode Mode Shape Mode Shape T (s) Ti(s) 1 3.57 r'NS 1 NS 3.57 2 1 EW 2.07 2.07 1 EW 3 1 torsion 1.46 1 torsion 1.46 4 0.81 2 NS 0.81 2 NS 5 2 EW 0.52 2 EW 0.52 6 2 torsion 0.49 2 torsion 0.49 t  st  st  st  st  st  nd  nd  nd  nd  nd  nd  0.5 0 4^ 0  ,  ,  ,  ,  ,  ,  ,  1  0.5  1  1.5  2  2.5  3  3.5  4  FEMtools Model#1 (s)  Figure 7.9. F E M t o o l s Model#l and experimental period pairs after updating. A s shown in Table 7.7, the M A C values have not changed significantly for each o f the first four modes after model updating. However, the M A C values for modes 5 and 6 are significantly higher, suggesting a better correlation between the experimental and analytical mode shapes, and the effects o f other modes have been almost eliminated. 67  Chapter 7  Finite Element Model Updating  This phenomenon can be observed graphically in Figure 7.10, where the M A C values from before and after model updating are plotted. (In the figure, the axis labels represent the finite element analysis (FEA) and the experimental modal analysis (EMA).) Table 7.7. FEMtools Model#l M A C matrix after updating (in %).  Mode  1 as  1 -a 1 1 a  1 2 3 4 5 6 7 8 9 10 11 12  I  2  99 0 7 7 0 0 0 1 1 0 1 1  0 88 0 0 5 0 0 0 0 0 0 1  Experimental Mode Shape 3 4  7 0 99 0 2 25 1 2 5 0 2 5  Before updating  2 0 0 97 0 0 26 1 0 2 8 1  5 0 3 0 0 76 9 0 2 10 27 3 0  After updating  IK  Figure 7.10. FEMtools Model#l M A C values before and after updating.  68  6  1 0 14 1 18 87 0 44 33 4 0 0  Chapter 7  Finite Element Model Updating  The experimental and analytical mode shapes are compared i n Figure 7.12 in their reduced form and in full in Figure 7.12, where the dots represent the experimental mode shapes and the wire mesh represents the analytical mode shapes. A s demonstrated in the figure, the analytical mode shapes are now almost identical to the experimental results.  Mode 1  Mode 2  Mode 3  Mode 4  Mode 5  Mode 6  Figure 7.11 F E M t o o l s Model#l analytical and experimental reduced mode shapes after updating.  69  Chapter 7  Finite Element Model Updating  Mode 4  Mode 5  Mode 6  Figure 7.12. F E M t o o l s Model#l analytical and experimental mode shapes after updating. M o v i n g now to the second part o f the results after model updating, the focus is turned to the parametric changes in the model during the automated updating process.  To achieve a match between the experimental and analytical results, a number o f iterations were done in the updating o f M o d e l # l .  It took about five minutes for a  Pentium I V with 1 G H z o f computing power and 512 M b o f R A M to run the analysis. 70  Chapter 7  Finite Element Model Updating  After each iteration, the criterion for convergence was updated until the difference between two successive iterations was less than 0.1%.  In total, five iterations were  needed in order to converge to a solution (Figure 7.13).  In this case, the weighted  absolute relative difference ( C C A B S ) between natural periods was the criterion for convergence that was monitored.  Each natural period is weighted with respect to the  others using the expected relative error on the response value.  The expected relative  error on the experimental natural periods was 1% (default used in the program).  30  0-1  ,  ,  1  0  1  2  3  1  1  4  5  Iteration  Figure 7.13. F E M t o o l s Model#l correlation tracking. The actual parameter  values returned  by F E M t o o l s  after  updating Model#l  are  summarized in Table 7.8. The initial property value (before updating) is compared to the actual value (after updating) by calculating the variation for each parameter.  The  variation is defined as the ratio o f the difference between the initial and actual values to the initial parameter value. The parameter variations, expressed as percentage, are also illustrated in Figure 7.14.  A n Importance Index (IS) was also developed to rank the significance o f each parameter change and is listed in Table 7.8. The importance index is defined as the absolute value of the product o f the variation and the normalized sensitivity o f the model. The higher  71  Chapter 7  Finite Element Model Updating  the IS, the more influential the required parameter change is on the model updating. For example, a large variation between an initial and actual property may not have a great effect on the response o f the model i f the normalized sensitivity o f the parameter is low. The same is true for a parameter with a high normalized sensitivity but a small variation between the initial and actual property.  This is why the IS is a useful tool for targeting  which parameters o f the original model have the greatest effect on the updated model.  The highlighted rows in Table 7.8 draw attention to the Family o f Six parameters mentioned earlier (Section 7.2.2.3).  A l l o f these parameters have a high IS, which  indicates that a change in the properties o f these parameters will have a significant effect the response  o f the model greatly.  However, some others parameters also have  significant IS such as parameters 3 and 6, and should be examined further.  Parameters 3 and 6 correspond to elements 16 and 19, respectively, which are located in heavily reinforced zones in the concrete shear core (see Appendix E for the exact location of elements 16 and 19).  Therefore, it can be concluded that the large increase in the  Young's modulus shown in Table 7.8 should be interpreted as a required increase in the overall stiffness o f these elements, not as an increase in the physical property itself, in order to obtain a better correlation between experimental and analytical results.  The changes observed in the model parameters following the automated updating o f Model#l provided valuable insight into the response o f the F E model. However, before drawing conclusions about these results, the Model#2 updating will also be examined.  72  Chapter 7  Finite Element Model  Updating  Table 7.8. FEMtools Model#l parameter comparison before and after updating. Variation Normalized Importance Sensitivity Index (IS) (%) 3.65F.-!()7 1.60EH 07 0.08 -56 5 0.55 3.65Et07 1.12E+07 38 -69 3.65E+07 6.731: K)7 84 0.53 45 3.52E+07 3.56E+07 1 0.02 0 3.52E+07 1 12E-I07 -68 0.23 16 3.52E+07 7.08E+07 0.20 21 101 3.38E+07 1.04E+07 0.09 -69 6 * 'iKLJ-07 1 3 0 1 . - i r -61 0.41 3.38E+07* 3.48E+07 3 0.18 3.65E+07 4.68E+07 0.07 28 2 3.52E+07 5.23E+07 49 0.03 2 3.38E+07 4.88E+07 45 0.06 3 2.40E+00 2.10EI00 -12 -0.02 0 2.40E-.-00 I.52F.+00 -37 -0.10 2.40E+00 1.991>00 1 -17 -0.05 2.40E+00 1.90E+00 1 -21 -0.03 2.40ES00 M i l mi -37 -0.19 2.40E+00 1.98E+00 -17 2 -6.10 2.40E+00 1.63E+00 -0.06 -32 2 2 4()h' 00 3 6211-00 -0.95 48 "2.40E+00* 2.79E+00 " 16 -0.23 4 2.40E+00 2.30E+00 -4 -0.01 0 2.40E+00 2.26E+00 -6 -0.02 0 2.40E+00 2.18E+00 -0.04 -9 0 Initial  Parameter Property Element 1 2 3 4 5 6 7  r F. E E E E E  14 15 16 17 18 19 20  PIpiilM 11IBI11 9 L  10 11 12 13 14 15 16  IMIBlBil 1H 19 20 21 22 23 24  j  I  22 23 24 25 14 15 16 17 18 19 20  E E E RHO RHO RHO RHO Kl 10 RHO RHO RHi) RHO RHO RHO RHO  Actual  (kN, m, kg) (kN, m, kg)  WBi  22 23 24 25  01%  100  80 60 •  40  20 •2 .2 «  o 1  2  3  4  5  6  7  8  9  10 11 12  y  14 15 16 17 18 19 20 21  -20 -40 -60 -80  -100  Parameter  Figure 7.14. FEMtools Model#l parameter changes after updating.  73  Chapter 7  Finite Element Model  Updating  7.2.3 Model#2 Updating 7.2.3.1  FEMtools Model#2  Model#2 was generated in FEMtools from the existing E T A B S Model#2 in the same fashion as for Model#l (described in Section 7.2.2.1). Once again, the main structural element types used in the F E model were 3D beams-columns and 4-node plates.  In  addition, every floor slab was modelled, to avoid developing local modes in the columns, using 4-node and 3-node plate elements. In total, the model consisted of 616 beam-column elements, 2,916 4-node plate elements, 66 3-node plate dements, 2,862 nodes, four different material properties, 144 different element geometry sets, and 17,172 degrees of freedom. 7.2.3.2  Model#2 Before Model Updating  For the same reasons as for Model#l (described in Section 7.2.2.2), the first six experimental mode shapes were selected for updating purposes. In addition, 12 analytical modes were extracted from Model#2 in FEMtools. The results from the modal analysis in FEMtools before updating are summarized in Table 7.9. The analytical periods for Model#2 are generally lower than the experimental ones, as shown in Figure 7.15. However, as expected, these periods are somewhat closer to the experimental results than observed for Model#l before updating, since Model#2 was previously updated and calibrated manually.  Table 7.9 also shows that the  experimental modal order is reflected in the first six analytical modes as well. Table 7.9. FEMtools Model#2 modal analysis results before updating. Mode 1 2 3 4 5 6 7 8 9 10 11 12  Experimental Mode Shape  1 NS 1 EW 1 torsion 2 NS 2 EW 2 torsion n/a n/a n/a n/a n/a n/a st  st  st  nd  nd  nd  Ti(s)  FEMtools ModelH 2 Mode Shape  T (s)  3.57 2.07 1.46 0.81 0.52 0.49 n/a n/a n/a n/a n/a n/a  r'NS 1 EW 1 torsion 2 NS 2 EW 2 torsion 3 NS 3 torsion 3" EW/4 NS 3 EW/4* NS 1 vertical 4 torsion  3.01 1.52 1.05 0.76 0.40 0.36 0.34 0.22 0.20 0.20 0.18 0.16  74  st  st  nd  nd  nd  rd  rd  !  ,h  rd  st  th  t  Chapter 7  Finite Element Model  Updating  FEMtools Model#2 (s)  Figure 7.15. F E M t o o l s Model#2 and experimental period pairs before updating. The M A C matrix comparing the experimental mode shapes to the analytical ones before updating is shown in Table 7.10. In general, the M A C values in this case are better than in M o d e l # l before updating, and modes 5 and 6 are also less influenced by the corresponding components o f other modes.  The M A C values are also shown for each  period pairs in Table 7.9.  Table 7.10. F E M t o o l s Modef#2 M A C matrix before updating (in %)  •5!  1 r\  •32  i  1 2 3 4 5 6 7 8 9 10 11 12  1  2  99  0 87 0 0 1 0 0 0 0 0 0 0  0 7 2 0 3 0 0 1 0 0 0  Experimental Mode Shape 3 4 6 0  1IJI11011B 0 6 0 0 1 0 1 5  75  3 0 0 98 0 1 9 1 4 1 2 0  5  6  0 4 0 0 88  1 0 18 0 8 86 0 0 0 0 2 12  6 0 0 3 11 4 0  Chapter 7  7.2.3.3  Finite Element Model  Updating  Model#2 Sensitivity Analysis  A s was done for M o d e l # l , a series o f elements was selected for the sensitivity analysis o f Model#2. A total o f 26 elements were selected and are presented in Table 7.11.  Table 7.11. F E M t o o l s Model#2 element description. Element Element Description 1 Outrigger columns 2 Outrigger columns 3 Outrigger columns 4 Outrigger columns 6 Outrigger beams 7 Outrigger beams 8 Outrigger beams 9 Outrigger beams 10 Coupling beam 11 Coupling beam 12 Transfer beams 13 Other beams 14 Shear walls 15 Shear walls 16 Shear walls 17 Shear walls 18 Shear walls 19 Shear walls 20 Shear walls 21 Shear walls 22 Shear walls 23 Shear walls 24 Shear walls 25 Shear walls 26 Floor slabs 27-42 Gravity load columns 43-47 Other beams Cladding 48  Location in Building Model Bottom third Bottom third Middle third Top third Level 5 Level 21 Level 31 Roof Level 50 Roof Level 5 Throughout Bottom third Bottom third Bottom third Middle third Middle third Middle third Top third Top third Top third Bottom third Middle third Top third Throughout Throughout Throughout Throughout  For Model#2, four properties were selected for the sensitivity analysis:  the Young's  modulus (E), the material mass density ( R H O ) , the second moment o f inertia (I), and the thickness o f the cladding (H). Similarly to M o d e l # l , a change in Y o u n g ' s modulus and material mass density was monitored for all elements and a change in moment o f inertia was monitored for the columns and beams only. A change in thickness was monitored for the cladding only.  Therefore, a total o f 161 parameters were selected for the  sensitivity analysis.  76  Finite Element Model Updating  Chapter 7  The normalized sensitivity o f Model#2 to the selected parameters is shown Figure 7.16. It can be observed that Model#2 is not very sensitive to a change in Y o u n g ' s modulus in the columns and beams (parameters 1 to 13, and 26 to 46). However, it is more sensitive to a change in Young's modulus in the shear core, floor slabs, or cladding (parameters 14 to 24, 25, and 47, respectively). Similarly, the model is not too sensitive to a change in the material mass density in the columns and beams (parameters 48 to 59, and 73 to 93), but is fairly sensitive i f the change occurs in the shear core, floor slabs or cladding (parameters 60 to 71, 72, and 94, respectively). The model is not sensitive to a change in the second moment o f inertia in the columns and beams (parameters 95 to 160). It is highly sensitive to a change in the thickness o f the cladding (parameter 161).  2.5  1.5  E  RHO  I  I  I  H  1  0.5  w c 0)  GO TJ <D N  E o  z  nf y W L [ L m f i , 111111111111  1 h-n  CM  CO CM  *T CO  to T  m  u  c  !  rujJU co h-  m i n i m u m  cn  hmnnlilliiiuiiiiMiiiMiiirPnrtlliinniiiiiniiiiiiiilim  o o  T T -  CM CN  co co  -q-  in in  CO  -0.5  -1  -1.5 Parameter  Figure 7.16. FEMtools Model#2 normalized sensitivity to selected. Following the same approach as for M o d e l # l , a large number o f parameters were selected for the sensitivity analysis; however, a smaller number o f parameters were needed for the automated model updating.  A s such, the following combinations o f  properties and elements were selected for the automated updating o f Model#2: the Young's modulus for the core, floor slabs, and cladding; the material mass density for the  77  Chapter 7  Finite Element Model Updating  same group o f elements; and the thickness o f the cladding. In total, 29 parameters were selected for automated updating and are summarized in Table 7.12.  Table 7.12. F E M t o o l s Model#2 selected parameters for updating.  Parameter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29  Element 14 15 16 17 18 19 20 21 22 23. 24 25 26 48 14 15 16 17 18 19 20 21 22 23 24 25 26 48 48  Property E E E E E E E E E E E E E E RHO RHO RHO RHO RHO RHO RHO RHO RHO RHO RHO RHO RHO RHO H  The normalized sensitivity o f Model#2 to the parameters selected for updating is shown in Figure 7.17. It is interesting to note that the parameters in the Family o f Six to which Model#l was identified as being highly sensitive, again show a strong influence on the response o f the model. (Note that in Model#2, the numbering o f these six parameters is: 2, 5, 8, 16, 19 and 22.) Therefore, it can be said that the F E model remains very sensitive to changes in the Young's modulus and material mass density o f elements 15, 18 and 21 even with the addition o f the gravity columns and cladding.  78  Chapter 7  Finite Element Model  Updating  1.5  |  0.5  in c CO  CO •a  o  —)  f~T~i)  Mr~!—F= in  N  CM  75  I o  -0.5  z  -1.5 Parameter Figure 7.17. FEMtools Model#2 normalized sensitivity to selected parameters for updating. Model#2 was also found to be sensitive to parameters other than those included in the Family o f Six. The mass density o f the floor slabs (parameter 27) is one parameter to which Model#2 highly sensitive.  In addition, the model is sensitive to changes i n the  Young's modulus and thickness o f the cladding (parameters 14 and 29, respectively).  The sensitivity o f each o f the modes in the model to the selected parameters is shown in Figure 7.18. A l l modes appear to have relatively high sensitivity to the Family o f Six (parameters 2, 5, 8, 18, 19 and 22). However, upon closer inspection, it can be seen that each o f the modes is affected differently by each o f the parameters, similar to the trends observed in M o d e l # l .  79  Chapter 7  Finite Element Model Updating  Mode 1  0.4 0.3  0.3  0.2  0.2  0.1  0.1  -Pi-  0  '55  -0.1  -0.2  -0.2  -0.3  -0.3  -0.4  -0.4  Mode 3 r-  c  CO TJ 0) N  « E k.  o  0.4  0.2 0.1  cr  Mode 4  0.3 0.2  -  _H  0  Mode 2  0  -0.1  0.3  >  0.4  i-  LL rn in  ™  a>  -0.1  0.1  i...  p)  r--  i-  r-  T—  - J  '  ' IO CM  CO CM  0  CM  -0.1 -0.2  -0.2  -0.3  -0.3  -0.4  -0.4  Mode 5  0.4  0.4  0.3  0.3  0.2  0.2  0.1 0  Mode 6  0.1  Tn  r-HiTh  0  -0.1  -0.1  -0.2  -0.2  -0.3  -0.3  -0.4  -0.4  CM  MS  CM  Parameter  Figure 7.18. F E M t o o l s Model#2 normalized sensitivity o f each mode to selected parameters for updating. Two parameters that strongly affect the model in every mode are parameters 22 and 27 (mass density o f element 21 and mass density o f floor slabs, respectively). It is possible that the results indicate the need for improving how the mass distribution throughout the model is modelled. The model is also very sensitive to the cladding thickness (parameter 29) in every mode. This provides insight into the importance o f the stiffness contribution of the non-structural components throughout the building.  80  Chapter 7  Finite Element Model Updating  1.2.3.4 Model#2 After Updating In a format similar to that used for M o d e l # l , the results after model updating will be presented in two parts, beginning with the modal analysis results and finishing with a closer look at the parameters changes that occurred in the updating process.  The results from the modal analysis after updating Model#2 are summarized in Table 7.13. It can be seen that the analytical periods are now identical to the experimental ones. The modal order did not change in the updating process, as it already matched the experimental results before updating was done.  The excellent match between period  pairs can be appreciated in Figure 7.19, as every pair plots directly on the 1:1 correspondence line.  Table 7.13. 1FEMtools Model#2 modal analysis resu ts after updating. Experimental FEMtools ModeB 2 Mode Mode Shape Mode Shape T (s) Ti(s) 1 3.57 r'NS 3.57 1 NS 2 1 EW 2.07 1 EW 2.07 3 1 torsion 1.46 1 torsion 1.46 4 2 NS 0.81 2 NS 0.81 5 0.52 2 EW 2 EW 0.52 6 2 torsion 0.49 2 torsion 0.49 t  st  st  st  st  st  nd  nd  nd  nd  nd  nd  0.5 0+=^ 0  ,  ,  ,  ,  ,  ,  ,  1  0.5  1  1.5  2  2.5  3  3.5  4  FEMtools Model#2 (s)  Figure 7.19. F E M t o o l s Model#2 and experimental period pairs after updating.  81  Chapter 7  Finite Element Model Updating  The M A C matrix did not change significantly after model updating (Table 7.14).  This  can be due to the fact that Model#2 was already updated and calibrated manually before undergoing the automated updating process. Figure 7.20 shows plots o f the M A C values both before and after updating, and it is very difficult to observe differences between the two. (In the figure, the axis labels represent the finite element analysis ( F E A ) and the experimental modal analysis ( E M A ) . )  Table 7.14. FEMtools Model#2 M A C matrix after updating (in %). Experimental Mode Shape 1 2 3 4 1 99 0 4 6 87 2 0 0 0 §• 99 3 7 0 0 5 0 0 4 5 0 3 0 o 6 2 0 10 1 -Si 7 1 0 0 23 0 0 0 0 8 1 9 0 0 1 10 1 0 0 8 i 11 0 0 7 3 12 1 0 6 0  1  III  1  1  5 0 5 0 0 86 8 0 0 25 1 1 0  6 1 0 19 0 10 87 0 6 0 0 18 22  The result o f superimposing the analytical mode shapes (mesh) onto the experimental mode shapes (dots) is shown in Figure 7.21 (reduced form) and in Figure 7.22 (full model). This figure reinforces the findings that were shown numerically above, as the match is nearly perfect between the experimental and analytical mode shapes.  The changes in parameters that were needed to achieve a match between the experimental and analytical models represent the second part o f the results from Model#2 updating, and w i l l be presented and discussed below.  82  Chapter 7  Finite Element Model Updating  83  Chapter 7  Finite Element Model Updating  Mode 1  Mode 2  Mode 3  Mode 4  Mode 5  Mode 6  Figure 7.21. F E M t o o l s Model#2 experimental and analytical reduced mode shapes after updating.  84  Chapter 7  Finite Element Model Updating  Mode 1  Mode 4  Mode 2  Mode 5  Mode 3  Mode 6  Figure 7.22. FEMtools Model#2 experimental and analytical mode shapes after updating. During the automated updating process, five iterations were needed to converge to a solution (Figure 7.23).  It took about seven minutes for a Pentium I V with 1 G H z o f  computing power and 512 M b o f R A M to run the analysis. It can be seen that most o f the parameter changes occurred in the first iteration while the remaining iterations provided  85  Chapter 7  Finite Element Model Updating  the fine-tuning o f the model.  Once again, the weighted absolute relative difference  ( C C A B S ) between natural periods was the criterion that was monitored for convergence.  Iteration  Figure 7.23. F E M t o o l s Model#2 correlation tracking. The actual parameter values returned by F E M t o o l s after model updating are reported in Table 7.15.  The variation and IS (absolute value o f the product o f variation and  normalized sensitivity) were calculated for each o f the updated parameters, as was done for Model#l (Section 7.2.2.4). The variation for each parameter is also shown in Figure 7.24.  The IS is examined to gain an understanding  o f which parameters have the most  significant effect on the response o f Model#2. Once again, the Family o f Six parameters (highlighted in Table 7.15) all have a high importance index, which shows that they are still strongly influencing the updating o f the model.  86  Chapter 7  Finite Element Model Updating  Table 7.15. F E M t o o l s Model#2 parameter comparison before and after updating. Parameter  Property  Element  1 2 3  Y: E E E  14 15 16 17 18 19 20 21 22 23 24 25 26 48 14 15 16 17 18 19 20  4 6 7 H 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29  • HEi l l E E E E E E E E RHO RHO RHO RHO RHO KIIO RHO RHO RHO RHO RHO RHO RHO RHO H  IIBBlli 22 23 24 25 26 48 48  Initial  Actual  Variation  (kN, m, kg)  (kN, m, kg)  (%) -11 -59 -4 -19 64 109 -42 -63 4 57 31 17 84 -16 -14 -34 -17 -15 -49 -25 9 86 23 -5 -6 4 -28 1 -42  3.65E+07 8.44E+06 3.65E+07 1.49E+07 3.65E+07 3.50E+07 3.52E+07 2.86E+07 3.52E-M17 5.76E+07 3.52E+07 7.35E+07 3.38E+07 1.97E+07 3.381 •()- 1.251 - O " 3.38E+07 3.51E+07 3.65E+07 5.75E+07 3.52E+07 4.59E+07 3.38E+07 3.97E+07 3.65E+07 6.74E+07 3.25E+07 2.74E+07 2.40E+00 2.06E+00 2 401 ("i 1.5VI <»> 2.40E+00 1.99E+00 2.40E+00 2.04E+00 2.40E+00 1.22ht00 2.40E+00 1.79E+00 2.40E+00 2.62E+00 2.401 on "4.47E+00 2.40E+00 2.96E+00 2.40E+00 2.28E+00 2.40E+00 2.26E+00 2.40E+00 2.49E+00 2.40E+00 1.72E+00 2.20E+00 2.21E+00 1.25E-02 7.31E-03  87  Normalized Sensitivity 0.09 0.51 ~~ 0.29 0.01 0 IS 0.07 0.05 0.29 "o.n *  0.07 0.01 0.03 0.36 0.34 -0.02 -0.08 -0.04 -0.02 -0.09 -0.05 -0.06 -0.69 -0.15 -0.01 -0.01 -0.03 -0.89 -0.03 0.99  Importance Index (IS) 1 30 1 0 8 2 18 0 4 0 0 31 5 0 IESIIHMMMI  1 0 i i 60 3  0 0 0 25 0 41  Chapter 7  Finite Element Model Updating 109%  100 80 60 40  5?  20  C .2 .2  o  «  -20  5  6  9  10 11  \i  13 14 15  l_JUJ  17 18 19 20 21 22 23 24 25 26 27 28 29  -40 -60 -80  -100  Parameter  Figure 7.24. FEMtools Model#2 parameter changes after updating. Parameters associated with the floor slabs also have high importance indices for both the Young's modulus (parameter 13) and material mass density (parameter 27) properties. The large increase in the value o f the Y o u n g ' s modulus should interpreted as a required increase in the overall stiffness o f these elements, not as an increase in the physical property itself, in order to obtain a better correlation between experimental and analytical results. B y recalling that the floor slabs were included in the model to simulate the effect of rigid diaphragms, it is felt that this increase can be justified. The large decrease in the mass density o f the floor slabs may indicate that it would have been advisable to modify the mass o f the model, as a whole, when additional components, such as the floor slabs and cladding, were first introduced into the model.  The thickness o f the cladding (parameter 29) has a very high importance index as well. Some variation associated with the cladding element was not unexpected, as this element was introduced to the model to represent the contribution o f all non-structural components (windows, interior walls, partitions, etc.) to the overall stiffness o f the structure. Therefore, the large decrease in the value o f the cladding thickness reflects the  88  Chapter 7  Finite Element Model Updating  fact that an accurate initial value for this parameter was difficult to predict. It can be noted, however, that the actual value o f the parameter returned b y the computer program is comparable to findings from a similar study by Ventura et al. [33].  7.3  CONCLUSIONS FROM AUTOMATED MODEL UPDATING  The actual values o f the updated parameters common to both M o d e l # l and Model#2 are summarized in Table 7.16. When the importance indices o f each model are compared, it can be seen that elements 15, 18 and 21 (in the Family o f Six) are the only ones to demonstrate high importance indices in both models. Therefore, it can be concluded that, i f only a limited number o f parameters were to be selected, one should begin by adjusting the properties o f elements 15, 18 and 21 (located in the concrete shear core) to calibrate a F E model o f the One W a l l Centre. 6. F E M t o o l s Model#l and Model#2 updated parameter comparison. FEMtools Model#l FEMtools Model#2 Property Variation Actual Actual Variation Sensitivity IS E  HHP F E  mu F; L  •III E E E E RHO RHO RHO RHO RHO RHO RHO KHO RHO RHO RHO RHO  (kN, m, kg)  (%)  1.60E+07 i I:I ir 6\73E+07 3.56E+07 1.12E 0" 7.08E+07 1.04E+07 Dbl-ir 3.48E+07 4.68E+07 5.23E+07 4.88E+07 2.10E+00 1.521 -uu 1.99E+00 1.90E+00 I.51F.+00 1.98E+00 1.63EI00 3.62F.+00 2.79F. 100 2.30E+00 2.26E+00 2.18E+00  -56 -69 84 01 -68 101 -69 -61 3 28 49 45 -12 -37 -17 -21 -37 -17 -32 51 16 -4 -6 -9  IS  (kN, m, kg)  0.08 0.55 0.53 0.02 0.23 0.20 0.09 0.41 0.18 0.07 0.03 0.06 -0.02 -0.10 -0.05 -0.03 -0.19 -0.10 -0.06 -0.95 -0.23 -0.01 -0.02 -0.04 89  5 38 45 0 16 21 6 25 1 2 2 3 0 4 1 1 7 2 2 48 4 0 0 0  8.44E+06 I.49E+07 2.86E+07 5.76E+07 7.35~E+07 1.97E+07 1 25F+07 3.51E+07 5.75E+07 4.59E+07 3.97E+07 2.06E+00 1.59K-00 1.99E+00 2.04E+00^ 1.22FUO 1.79EI00 2.62E+00 4 r i (in 2.96E 100 2.28E+00 2.26E+00 2.49E+00  -77 -59 -4 -19  0.09 0.51 6.29 0.01  109 -42 -M 4 57 31 17 -14 -34 " -17 -15 -49 -25 9 86 23 -5 -6 4  0.07 0.05 u.:o 0.11 0.07 0.01 0.03 0.09 -0.02 -0.08 -0.04 -0.02 -0.09 -0.05 -0.06 -0.69 -0.15 -0.01 -0.01  0 IK  7 ' 30 1 0 11 8 2 18 0 4 0 0 0 3' " 1 0 1 1 60 , 3 0 0 0  Chapter 7  Finite Element Model Updating  Furthermore, for the case o f the One Wall Centre study, the actual values o f the parameters found after Model#2 updating should be used for model calibration. This recommendation would apply to the calibration o f either Model#l or Model#2. Because more elements were included in the construction Of Model#2, it is a more complete F E model  and  provides  a better representation  components o f the building.  o f all structural  and  non-structural  Thus any changes made to the concrete shear core  properties during Model#2 updating would bring these properties closer to the "real" values than those found in Model#l updating.  Therefore, it is recommended that, in the calibration o f a F E model o f the One Wall Centre (or the design o f a similar structure in the future), the value o f the Young's modulus for elements 15, 18 and 21 (or for a similar set o f elements located in the concrete shear core) should be reduced by about 60%. This decrease in material property is essentially used to produce a decrease in the stiffness o f the system. Therefore, it could also be compared to using Icracked instead o f Igross for the section property o f the system, which would yield similar results.  In practice, one cannot always generate a complete model (such as Model#2) due to time and financial constraints. It is more likely that a simpler model (such as M o d e l # l ) would be used for analysis and design.  However, i f experimental data for the structure is  available, the results o f Model#l updating for the One Wall Centre have demonstrated that automated model updating using a computer program (such as F E M t o o l s ) can lead to marked improvements in the F E model and more accurate prediction o f the response o f the structure in the linear range. In the case when these resources are unavailable, it has been demonstrated that, by changing only a few key parameters, the response o f a simpler model could be improved significantly.  To conclude on the philosophy behind the automated updating process, it must be emphasized that it remains the responsibility o f the user to accept or reject the changes proposed by the computer program.  The user should be able to justify any significant  changes to the model by using past experience or sound engineering judgement.  90  Chapter 8  Conclusions and Recommendations  CHAPTER 8 CONCLUSIONS AND RECOMMENDATIONS  8.1  SUMMARY  This study focused on the One Wall Centre, a 48-storey high building located in downtown Vancouver, British Columbia.  It is currently the highest building in  Vancouver, and it is the only structure in the region that makes use o f tuned liquid column dampers to reduce vibrations due to wind.  The "true" dynamic response o f the  One Wall Centre is o f great interest to structural engineers because o f the unusual elliptical shape o f the concrete shear core, which could present a challenge in modelling.  Ambient vibration testing was conducted on the One Wall Centre in order to determine its modal characteristics.  Such characteristics included the natural periods o f the  building, their corresponding mode shapes and damping ratios.  The analysis was  performed using a state-of-the-art modal identification technique in the frequency domain and a second technique in the time domain.  The natural periods and mode shapes o f the One Wall Centre were determined analytically using two different computer modelling approaches and were later compared to the ambient vibration testing results.  The first approach consisted o f a simplified  model based on a combination o f Timoshenko Beam theory and a lumped-mass multiD O F system; the reinforced concrete shear core was modelled as a Timoshenko cantilever beam and the floor masses were modelled as a 4 6 - D O F system. The results from both o f these models were combined using Dunkerley's approximation. The first three transverse (NS) natural periods and mode shapes were identified with less than a 10% difference from the ambient vibration test results. The second approach consisted o f a linear-elastic finite element ( F E ) computer model o f the building. Only the main lateral load-resisting system components were modelled, which included the reinforced concrete  91  Chapter 8  Conclusions and Recommendations  shear core, the outrigger columns and outrigger beams. The first eight natural periods and mode shapes of the building were identified and compared to the experimental results. It was found that the FE model was more flexible than the actual structure, with an average of 15% difference between the results. Thus, the results of both modelling approaches suggested that there was room for improvement in the models.  In order to achieve a better match between the analytical and experimental dynamic responses, the FE model was updated. Two model-updating techniques were used in this study: manual model updating and automated model updating. The FE model of the One Wall Centre was first updated manually, and a good match was obtained between the analytical and experimental natural periods and mode shapes. The main change to the FE model was the inclusion of the stiffness contribution of the architectural components, such as the outside windows and partition walls.  Such stiffness was added to the FE  model by surrounding the building with cladding (modelled as shell elements), and modelling the perimeter gravity load columns.  The automated model updating was  performed using a computer program. The intent of the automated technique was to determine what elements in the FE models were sensitive to variations in physical properties (Young's modulus, material mass density, moment of inertia and thickness of elements).  An excellent match was achieved between the analytical and experimental  results after manual and automated updating of the FE models.  8.2  CONCLUSIONS  The following conclusions can be drawn from this study: •  The first eight natural periods and corresponding mode shapes of the One Wall Centre were determined from ambient vibration data using two modal identification techniques. The fundamental period of the building was found to be 3.57 s.  •  A simplified model of the building, which combined two base models using Dunkerley's approximation, can be used to estimate the first three natural periods and corresponding mode shapes of the building in the transverse (NS) direction.  92  Chapter 8 •  Conclusions and Recommendations  The stiffness contribution of the non-structural components of the building was found to be an important factor in matching the experimental and analytical natural periods and mode shapes.  •  Results from automated model updating suggested that the dynamic characteristics of the FE models (Model#l and Model#2) of the structure was sensitive to variations in:  •  o  Young's modulus (for the reinforced concrete shear core and cladding)  o  Material mass density (for the reinforced concrete shear core and cladding)  o  Element thickness (for the cladding)  Results from automated updating suggested that the dynamic response of the FE models of the structure was not sensitive to variations in:  •  o  Young's moduli (for beam and column elements)  o  Material mass densities (for beam and column elements)  o  Moments of inertia (for beam and column elements)  Results from automated updating suggested that the Young's moduli of three specific zones in the reinforced concrete shear core should be decreased by 60% in order to match the experimental results.  8.3  RECOMMENDATIONS  The following recommendations are made for further research: •  Due to the limited FE element library in FEMtools, an interface program should be developed between ETABS and FEMtools, such that ETABS can be used as the FE solver program and FEMtools can be used as the model-updating program.  •  Another series of ambient vibration tests should be performed on the One Wall Centre now that the tuned liquid column damper (TLCD) is operational. The TLCD should be explicitly modelled in the ETABS model.  This could be accomplished  using a simple SDOF system composed of a mass and non-linear link elements (spring and dampers) representing the TLCD properties. Using the above-mentioned interface program, the new FE model should be updated using FEMtools. •  Further analyses should be conducted on the calibrated FE model (including the TLCD model) of the One Wall Centre, such as modal analyses and time history 93  Chapter 8  Conclusions and Recommendations  analyses, to monitor the effect of the T L C D on the dynamic and seismic response of the structure. •  Since it was the first time that automated model updating was used with ambient vibration data for a building of this type, it is being recognized that further work needs to be done to determine whether or not the solution proposed by the automated model-updating program is unique.  •  In the long term, there is a need to develop system identification tools for non-linear systems and model updating techniques for non-linear FE models.  94  References  R E F E R E N C E S  [I]  Balendra, T., Wang, C M . and Y a n , N . (2001) Control o f wind-excited towers by active tuned liquid column damper, Engineering Structures, 23: 1054-1067.  [2]  Symans, M . D . & Constantinou, M . C . (1999) Semi-active Control Systems F o r Seismic Protection O f Structures: A State-of-the-art Review, Engineering Structures, 21: 469-487.  [3]  Chang, D . C . , Soong, T.T., L a i , M . L . , Nielsen, E . J . (1993) Viscoelastic Dampers as Energy Dissipation Devices for Seismic Applications, Earthquake Spectra, 9(3): 371-418.  [4]  Cherry, S. & Filiatrault, A . (1993) Seismic Response Control o f Buildings Using Friction Dampers, Earthquake Spectra, 9(3): 447-466.  [5]  Grigorian, C . E . , Yang, T.S., Popov, E . P . (1993) Slotted Bolted Connection Energy Dissipators, Earthquake Spectra, 9(3): 491-504.  [6]  Pinelli, J-P, Craig, J.I., Goodno, B . J . , Hsu, C . C . (1993) Passive Control o f Building Response Using Energy Dissipating Cladding Connections, Earthquake Spectra, 9(3): 529-546.  [7]  Ventura, C . E . , Finn, W . D . L . , Lord, J-F, Fujita, N (2003) Dynamic Characteristics of a Base-Isolated Building from Ambient Vibration Measurements and L o w Level Earthquake Shaking, International Journal of Soil Dynamics and Earthquake Engineering. (Accepted for publication)  [8]  Chopra, A . K . (2001) Dynamics of Structures: theory and applications to earthquake engineering. Prentice-Hall, Upper Saddle River, N J , U S A  [9]  Soong, T T . (1988) State-of-the-art-Review: Active Structural Control in Civil Engineering, Engineering Structures, 10: 74-84.  [10]  Soong, T.T., Masri, S.F., Housner, G . W . (1991) A n Overview o f Active Structural Control Under Seismic Loads, Earthquake Spectra, 7(3): 483-505.  [II] Hartog, J.P.D. (1934) Mechanical Vibrations, M c G r a w - H i l l B o o k Company, N e w York.  95  References  [12] Sadek, F., Mohraz, B . Taylor, A . W . , Chung, R . M . (1997) A Method o f Estimating the Parameters o f Tuned Mass Dampers for Seismic Applications, Earthquake Engineering and Structural Dynamics, 26:617-63 5. [13] Yalla, S.K., Kareem, K . , Kantor, J.C. (2001) Semi-active tuned liquid column dampers for vibration control o f structures, Engineering Structures, 23: 1469-1479. [14] Former, B . (2001) Buildings: Water Tanks Damp M o t i o n in Vancouver High-Rise, Civil Engineering, 7: 18. [15]  Glotman Simpson Consulting Engineers (2001) Wall Centre Phase II Structural Plans (S3-S60), Vancouver, B C , Canada.  [16] L o r d , J-F and Ventura, C E . (2002) Measured and Calculated M o d a l Characteristics o f a 48-Story Tuned Mass System Building in Vancouver, International Modal Analysis Conference-XX: A Conference on Structural Dynamics, Society for Experimental Mechanics, L o s Angeles, C A , U S A , 2: 1210-1215. [17] Felber, A . J . (1993) Development of a Hybrid Bridge Evaluation System. Ph.D. Thesis, Department o f Civil Engineering, The University o f British Columbia, Vancouver, Canada. [18] Earthquake Engineering Research Facility ( E E R F ) (2002). Performing Ambient Vibration Tests using the U B C Earthquake Engineering Research Facility ( E E R F ) Dynamic Structural Behaviour Assessment System ( D S B A S ) User's Manual, The University o f British Columbia, Vancouver, Canada. [19] D A S Y T E C H , DasyLab software, © 1994-1998, www.dasvtec.com [20] G i S , D I A D E M software, © 1995-1999, www.iotech.com [21] Mathsoft, Mathcad 2001 Professional Software, © 1986-2000, w A y w M a i h c M , c o m [22]  Structural Vibration Solutions A p S (2001) ARTeMIS Extractor, Release 3.1, User's Manual, Aalborg, Denmark, www.svibs.com  [23]  Schuster, N . D . (1994) Dynamic Characteristics of a 30 Storey Building During Construction Detected From Ambient Vibration Measurements. M . A . S c . Thesis,  96  References  Department o f Civil Engineering, The University o f British Columbia, Vancouver, Canada. [24] Brinker, R., Zhang, L . and Andersen, P. (2000) M o d a l Identification from Ambient Responses using Frequency Domain Decomposition, International Modal Analysis Conference-XVIII: A Conference on Structural Dynamics, Society for Experimental Mechanics, San Antonio, T X , U S A , 1: 625-630. [25] Andersen, P (2001) Technical Paper on the Stochastic Subspace Identification Techniques, Structural Vibration Solutions A p S , Aalborg, Denmark. [26] Ewins, D . J . (1984) Modal Testing: Theory and Practice, John Wiley & Sons Inc, New York, N Y , U S A . [27] Brincker, R., Ventura, C . E . , Andersen, P. (2001) Damping Estimation by Frequency Domain Decomposition, International Modal Analysis Conference-XIX, Society for Experimental Mechanics, Kissimmee, F L , U S A , 1: 698-703. [28] Tamura, Y , Zhang, L . , Yoshida, A . , Cho, K . , Nakata, S. and Naito, S. (2002) Ambient Vibration Testing & M o d a l Idendification o f an Office Building, International Modal Analysis Conference-XX: A Conference on Structural Dynamics, Society for Experimental Mechanics, L o s Angeles, C A , U S A , 1: 141146. [29] N R C (1990) National Building Code Of Canada, Associate Committee O f The National Building Code, National Research Council O f Canada, Ottawa. [30] Rao, S.S. (1990) Mechanical Vibrations, 2 Mass, U S A .  n d  Edition, Addison-Wesley, Reading,  [31] Computers & Structures Inc (1999) ETABS User's Manual Vol. I & 2, Berkeley, C A , U S A . www.csiberkeley.com [32] Ventura, C . E . , L o r d , J-F, and Simpson, R . D . (2002) Effective Use o f Ambient Vibration Measurements for M o d a l Updating o f a 48 Storey Building in Vancouver, Canada, International Conference on "Structural Dynamics Modeling - Test, Analysis, Correlation and Validation" Instituto de Engenharia Macanica, Madeira Island, Portugal. [33] Ventura, C . E . , Brincker, R., Dascote, E . , and Andersen, P. (2001) F E M Updating o f the Heritage Court Building Structure, International Modal Analysis  97  References  Conference-XIX: A Conference on Structural Dynamics, Mechanics, Kissimmee, F L , U S A , 1: 324-330. [34] Dynamic Design Solutions N . V . (2001) FEMtools Leuven, Belgium, vvwvv.femtools.com  for  Society for Experimental  Windows Version  2.1,  [35] Dascotte, E , Vanhonacker, P. (1989) Development O f A n Automatic Mathematical M o d e l Updating Program, International Modal Analysis Conference-VII, Society for Experimental Mechanics, Las Vegas, N V , U S A , 1: 596-602. [36] Dascotte, E . (1991) Tuning O f Large-Scale Finite Element Models, International Modal Analysis Conference-IX, Society for Experimental Mechanics, Florence, Italy, 2: 1025-1028.  98  Appendix A  Structural and Architectural Drawings  A P P E N D I X  A  S T R U C T U R A L A N D A R C H I T E C T U R A L D R A W I N G S  Table A . 1. Collection o f structural and architectural drawings o f the One Wall Centre. Drawing # Drawing Title Page A202 Floor Plans: Levels 7-20 100 A211 Floor Plans: Levels 34-47, Residential (4 suite) 101 A310 Site Section A - A (Lower) 102 A311 Site Section A - A (Upper) 103 A314 Site Section C-C (Upper) 104 S22 4 Floor Plan Meeting / Service Level 105 S25 6 to 20 Floor Plan (Typical Floor Plan) 106 S28 21 Floor Plan 107 S32 32 Floor Plan 108 S34 34 to 48 Floor Plan 109 S36 Roof Plan 110 S54 Shear Wall Schedule 111 th  th  th  st  nd  th  99  th  ntrr- m « t * tivtt  t»v-  01  nu  imi  ev  Appendix B  Ambient Vibration Testing Details  APPENDIX  B  A M B I E N T VIBRATION TESTING  DETAILS  The set-up plan and sensor locations used during the ambient vibration testing o f the One Wall Centre are presented in Table B . l . In Figure B . l and Figure B . 2 , the arrows show the positive direction used for the sensors.  The nomenclature is as follow: x x O I X and  x x O l Y where "xx" corresponds to the floor level, "01" corresponds to point "01," and " X " or " Y " corresponds to the direction of the sensor. Similarly, x x 0 2 Y corresponds to a sensor pointing in the " Y " direction, located at point "02," and on floor "xx". The data acquisition system, the reference sensor and two roving sensor configurations are shown in Figure B . 3 , Figure B.4, Figure B.5 and Figure B . 6 , respectively. The floor assigned to each group of volunteers for .each set-up is shown in Table B . 2 .  Table B . l . Set-up plan and locations of sensors during ambient vibration testing. Set-up # 1 2 3 4 5 6 7 1 501X 601X 801X 1001X 1201X 1401X 160IX 2 501Y 601Y 801Y 1001Y 1201Y 1401Y 1601Y 3 502Y 602Y 802Y 1002Y 1202Y 1402Y 1602Y 4 2001X 1801X 220IX 240IX 2601X 2801X 3001X 5 1801Y 2001Y 2201Y 2401Y 2601Y 2801Y 3001Y 6 1802Y 2002Y 2202Y 2402Y 2602Y 2802Y 3002Y =tfc 7 4501X Reference 4501Y K 8 Reference rr 9 4602Y 4302Y 4202Y 4002Y 3802Y 3602Y 3202Y 10 3401X Back-up Reference 11 3401Y Back-up Reference 12 3402Y Back-up Reference 13 460IX 4301X 4201X 4001X 3801X 3601X 3201X 14 4601Y 4301Y 4201Y 4001Y 3801Y 3601Y 3201Y 15 4502Y Back-up Reference WallclOl Wallc201 Wallc301 Wallc402 Wallc501 * asc Wallc601 Wallc701 WallclOlsc Wallc201sc Wallc301sc *.prn Wallc402sc Wallc501sc Wallc601sc Wallc701sc • • • • •  Sampling rate = 2000 sps (*.asc) Decimation rate = 250 sps (*.prn) Duration = 12 min Gain = ± 125 m V High-pass filters = no  112  Sensor FBA1 FBA2 FBA3 FBA4 FBA5 FBA6 EPI456X EPI456Y FBA7 EPI9 EPI8 EPI10 EPI123X EPI123Y EPI7  Appendix B  Ambient Vibration Testing Details  Figure B . l . Sensor location and orientation (from level 5 to level 31).  Figure B . 2 . Sensor location and orientation (from level 32 to level 48).  113  Figure B . 3 . Data Acquisition System.  Figure B.4. Reference sensor (Channels 7 and 8).  Figure B . 5 . Roving sensors (Channels 1 and 2).  Figure B.6. Roving sensor (Channel 3).  114  Appendix B  Ambient Vibration Testing Details  Table B.2. Group floor assignment for each set-up. Set-up # Team 1 1 2 3 2 1 2 3 3 1 2 3 4.1 & 4.2 1 2 3 5 1 2 3 6 1 2 3 7 1 2 3  115  Floor 5 18 46 6 20 43 8 22 42 10 24 41 12 26 38 14 28 36 16 30 32  Appendix C  Simplified Model Solutions  A P P E N D I X  C  SIMPLIFIED M O D E L  SOLUTIONS  Mathcad (Version 2001 Professional) solution o f the Timoshenko Beam base model o f the simplified model o f the One W a l l Centre (Section 6.1), developed by D r . Carlos Ventura. Calculations performed by the author. Define System Parameters (using consistent units) input values: rtseg * 46  Select number of beam segments: Section:  Select value of n Ratio {>i»E/l<G,k=shape factor): » * sco FlOOr Height  FloorHeiBht2.925-rn  Total building height = Length of Beam:  i . . FioorHeigw.Nseg  Mass per unit of length:  u »72000—  Modulus of Elasticity:  E:- 3 i o  Select largest frequency of interest (in Hz):  r . 20 H Z  Select b increment:  A > .001  Tolerance:  Tolerance:- 0.001  Normal modes:  nm - 20  i « 134.55m  m  Derived  Pa  m;  values:  Moment of Inertia:  i»!600 m  1. 1.6* i d m  4  3  E  Therefore  kAG-2.4x10  g . H.R .I  This results in:  2  :  N  ==>S.O.2  {I'm ) ftoorl — I  I* )  ==>  1 3  3  2  R » 0.02  :  14  l  kAG:  E I - 4.3x i o k $ ; m 5  4  R « 0.02  Select value of Slenderness Ratio, R=r/L: _ _ ==>  1 0  3 2 kgms  and  T  -2. .L .  = = >  2  o :  s  E.l  3 H « 9.386 x 10  116  T  0  «  4.405s  and  * >T0  points at which function Beta(N,4.8,L) is evaluated.  Appendix C  Simplified Model Solutions  Find Natural Frequencies (and periods): ib:*= 0.. N  B., ib-A lb  a) Timoshenko: -p . Bota(N .a . s)  NMod  :  b,l Bending beam:fib:»Betat,(N , A) c) Shear beam:  longth(ii)  NMod  b  | i s ; - Bota{N)  => NMod  D  -2 Tb -T .|tf> :  0  [is  fs:»  s  2  0  :  ==>NMod -3  s  T > T .|f  l*b fb »  ==> NMod - 3  > length(|»b)  NMod :« Nfcdt,  s  f:~ —  13  T  0  - \ / i  TOLTolerance  Display frequency (or period) values (in Hz (or sec)):  IS  ((Kappafll.«)))  » • 3.4C9 - 10  7  -9,5564 x 10"  4.023 x 10"  6  5  _  /-  -1  To-y f>iss  -9.279 x 10"  5  1.355x 10  &  -2.025 x 10 "  3.776' 1'  Timoshenko:  J T  =(1.756  —  -(1  (l<b(f» )-(.0.966 T  0.519  3.385  -8.006  0.269  6.538  -16.654  0.186  9.416  0.143  0.116  0.098  12.319 15.153  10.54 98.348  237569  0.084  0.074  0.066  376.595  404.607  168.097  TO = (1.253 0.2 0.071 }s T -(1  6.267  17.547)  c) Shear beam: Ts  7 1 4 8 9 0 2 3 5 6 0 1.254 0.418 0.251 0.179 0.139 0.114 0.096 0.084 0.074 0.066  0  0 1 2 1 3 5  3 4 5 6 7 8 9 7 9 11 13 15 17 19  117  0.055  17.999 20.818 23.644 26.455 29.271  b) Bending beam:  ft)  0.06  -498.663  0.05)s 32.079 34.8$  -!.73x 10 -3.! 3  Appendix C  Simplified Model Solutions  Computing Normalized Mode Shapes  i:-  Nsog  j;»  i.. if(NMod >  ran.  run, NMod)  Deflections:  jb  1..  i ( ( N M o d | ) > r i m . n m . NMcxIjj)  W t  is  1..  if(NMod > n m . nm . NMod ) s  J-  d*(| ,_ s . x . )  fb.  jb.""  i*-'"  d * ( j s , 0) s  d«i. (js.O) s  Ms. dM' (is.O) s  j b  d<|.(>, x.) s  r  Vs, Ib-i"  Display Modal Components: Displacements: Mode 1  Mode 2  Bending beam  Bonding beam  Shear b e a m  Shear b e a m team  ,x.  d ^ s { j s . x.)  ib-  b  Timoshenko  .  ib-1'^) d' "M'Vr 1  ddd,i» (,ib __ x.)  —  x  s  d ' l ' ^ . S . O )  Shears:  —  i s  <l<t>sjls. X j d.js.  Mb..  1  >l'b(i»b.,  r  i'  M.  ( ' i) * (is.l) , , , s  I'M*.*.  Moments:  s  5  S  L  1  da.  is. i  Nseg  '"H'Vi-il  sb  Slopes: (o=total, v=berxling)  0..  —  Timoshenko b e a m  118  ddd* (|ib b  .0)  I -' s  d<l' (js,0) s  Appendix C  Simplified Model Solutions  Mathcad (Version 2001 Professional) solution o f the 50 D O F system base model o f the simplified model o f the One W a l l Centre (Section 6.1), developed by Dr. Carlos Ventura. Calculations performed by the author. Mode Shapes. Periods and Participation Factors of 50 DOFs (Bending & S f t w Beams) Define System Dynamic Properties:  Number of D O F :  N =5«)  Properties:  E:=.T|0  j:-i..N  :  !:=»!.. N ]:- 25000  1 0  Typical floor height: h:»2J25 and let L : - N - h  a) Mass Matrix (top row is first level) Scaling mass factor: m := 500000  Influence Coefficients, r:  Mass >  1 1  1  2  0  1  (Same for both models)  M :« M-issm  b) Stiffness Matrix of Shear Beam (top row is first level) Scaling stiffness factor: k :=  Stiffness :  1 1  12- l-l-1  2 -1  2  2 -1  3  0  -1  4  0  0  KsbStiiTness-k  2  119  1 2  1  1  Appendix C  Simplified Model Solutions  c) Stiffness Matrix of Bending Beam (top row is first level) Compute first the flexibility coefficients of Cantilever Bending Beam (origin at free end of beam): (length of beam is L, location of load is at a, and b=L-a) yfx.a. b. I.j > i [x b. : i  i)h  Then  .V : -  a.l) -(3-L-  \ - b).{L- x)  2  (N - j)-h  and  F. ' y/x.,n.,b..l.. | '\ J i i I '•J !  Kbb:=-.(F~ 2  ) 1  1 1  50  2  2.255-10 -1.426-10 13  3  5.746-10 12  1  2 125  2  125  400  3  200  701  4  275 1-103  Kbb =  13  4 -1.540-10 12 5  4.126-10"  6 -1.105-10 11  d) Mode Shapes, Frequencies and Participation Factors: Dynamic Matrix for Shear Beam:  _l  Ds:» M  -Ksb  Participation Factor  Xs:» swtjeigeiivalsfl)*))  , <i> . Xs - :» cigenwc(Ds,Xsj)  (xs'-M-r); s  i ~  j  ;—  (Xs'-MXsh .  Frequencies and Eigenvalues:  fs := ( — J x i i )  I '* 2  Modal Periods (sec):  J  Dynamic Matrix for Bending Beam: Db:- M~'-Kbb  Xb> sor.(ei envals(Db)) S  X b : - cigenvcc(Db, Xbj) i j /  ,• 0..VV, 'i Participation Factor  >bj = :  {Xb'-M-rl — ==>  |  I  >t>j 0.204 j  (xb- M-Xb), :  l  T  J.J  0 2 7 8  j  I  0  j  [0.057 J  120  ls:Ji)  Appendix C  Simplified Model Solutions  Frequencies and Eigenvalues:  ib;«l—Jib) \2-> j  Normalized Mode Shapes: ' i . j  (') I^S " > " 7 ~ Xs , . 46, j 1 1  Modal Periods (sec):  Tb :=  I.. N + I  <*> ij/b ''  mocls:- sliKk(/..>ps)  (  121  Xb^' Xb., . 46. j  modb:= stiickCz. >|»b)  {lb J  Appendix D  Automated Model Updating Theory  APPENDIX D A U T O M A T E D M O D E L UPDATING  The  theory  behind  Vanhonacker [35].  automated  model updating  was  T H E O R Y  published  by Dascotte  The essential o f the theory is summarized below.  modular, command-line driven program. o f the program is shown in Figure D . 1 .  and  F E M t o o l s is a  A tree-diagram explaining the general scheme The analyst has to enter the finite element  analysis ( F E A ) and the experimental modal analysis ( E M A ) data (measured  natural  periods, element stiffness, mass matrices, model geometry, etc.) into the program.  It is  only then that he/she can proceed with the selection o f parameters and the updating level. Input of F E A and E M A  Selection of parameters and updating level Define confidences and amplification factor  Start iterative procedure  Calculate sensitivities  o  u  Coupling analytical and experimentally obtained mode shapes  Parameter tuning  Prepare output  Figure D . 1. F E M t o o l s general scheme.  122  Appendix D  Automated Model Updating Theory  Parameters such as Young's modulus and mass density can be proportional to the model matrices or parameters such as plate thickness can be non-proportional. It is sometimes useful to "fine-tune" the individual elements o f the model matrices. Physical parameters can be tuned globally or locally. Global tuning eliminates wrongly determined structural information that was used for the F E input data.  This error is typically systematic and  therefore has an overall effect on the structure.  Local tuning refers to the individual  tuning o f physical parameters associated to elements, such as material or geometrical properties or associated to nodes, such as lumped masses or elastic constraints.  The variation o f eigen-frequencies due to parameter variations can be expressed as a Taylor series expansion (Equation D . 1) limited to the first two terms:  5co  {AP,HS]{AP,.}  [D.l]  i  where {AGO} is a column vector o f the finite difference o f real natural frequencies, 8co/8P is the partial derivative o f a natural frequency with respect to a parameter, { A P } is a column vector o f the finite difference o f parameters and [S] is the sensitivity matrix.  The factors SoVSPj are the eigen-frequencies differential sensitivity and can be calculated as follows (Equation D.2):  [D.2]  where  j is the analytically obtained eigenvector normalized to the mass matrix, [K] is  the model stiffness matrix and [ M ] is the model mass matrix.  A n alternative way to  determine the eigen-frequency sensitivities is to calculate them from the results o f two finite element runs while the parameter o f interest is modified, as shown in Equation D . 3 :  123  Appendix D  Automated Model Updating Theory  ACQ, _ aAPj)-<*>,(!>,+AP,) AP.  AP,  1  1  In Equation D . 1 , a good agreement between finite difference and differential sensitivities is shown when no extreme low or high frequencies appear and when the eigen-modes are uncoupled.  When the initial errors on the eigen-frequencies are important, the first convergence steps can be forced to be small in order to avoid divergence caused by the neglected higher order terms.  This reduction causes a higher number o f iterations.  O n the other hand,  when convergence is slow due to the sensitivity matrix characteristics, and amplification factor increases the iteration speed. matrix.  These factors are applied on the entire sensitivity  They can be constant, relative to the mean frequency deviation or determined  after two consecutive iterations.  From Equation D . 1 , the required parameter changes to obtain correlation between the analytical and experimental modal data can be calculated using a least squares, weighted least squares or Bayesian technique.  The Bayesian technique proved to be the most versatile since it allows the use o f weighting  factors  for the  measured  eigen-frequencies  and  the  initially  estimated  parameter values. F o r an undetermined system o f equations, Equation D.4 can be used as follows:  {AP} = [CP IS] §CR ] + [S1C  P  IS]  Y  M  [D.4]  where [Cp] and [CR] represent the weighted matrix o f the parameter vector and the weighted matrix o f the response vector, respectively and express the confidence in model parameters and test data, respectively.  124  Appendix D  Automated Model Updating Theory  When a dense frequency spectrum is encountered,  experimentally and analytically  obtained eigen-modes may not appear in the same order.  Interchanging o f eigen-modes  may also occur during the updating procedure. In order to guarantee the correct pairing o f experimentally and analytically obtained eigen-frequencies,  the measured modal  displacements are used with the Modal Assurance Criterion ( M A C ) to automatically verify and perform this pairing throughout the procedure.  The parameter changes obtained in Equation D.4 are used to recalculate the element mass and stiffness matrices. Eigen-value analysis yields the new analytical natural frequencies and eigen-vectors that more closely match the experimental values.  This iterative scheme can be continued until a convergence criterion is satisfied.  This  criterion is based on a correlation coefficient that includes deviation between the eigenfrequencies and the M A C values  (CC t)  or, more simply, includes only the absolute  deviation o f the eigen-frequencies  (CC b ),  as shown in Equation D.5 and Equation D.6,  to  a  S  respectively:  [D.5]  [D.6]  where N represents the number o f measured eigen-modes and W represents the weighting value for the eigen-frequencies or eigen-vectors.  The obtained parameter changes and  the variation o f the correlation coefficients can be graphically presented using customized software and commercially available F E post-processors.  125  Appendix E  FEMtools Element Maps  APPENDIX  F E M T O O L S  E  E L E M E N T  M A P S  The element sets used in FEMtools sensitivity analysis and model-updating analyses are mapped in the following figures.  Figure E . l . FEMtools elements 14, 15, 16 and 23.  126  Appendix E  FEMtools Element Maps  Appendix E  FEMtools Element Maps  Appendix E  FEMtools Element Maps  

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