Applied Science, Faculty of
Civil Engineering, Department of
DSpace
UBCV
Matthews, Timothy William
Master of Applied Science - MASc
University of British Columbia
Guidelines have been developed in the United States to assist in the design of buildings resistant to progressive collapse. These guidelines outline several methods for considering the effects of dynamic amplification on load redistribution following sudden member failure in a structure. Previous analytical studies have been performed to evaluate the consistency of these guidelines, but very little testing has been conducted to verify the guidelines and analytical work performed to date. This study investigates the dynamic load redistribution occurring in a field test on a two-storey reinforced concrete frame specimen. The test consisted of removing a column with explosives and recording the deformations of the remaining columns in the structure. The explosives removed the load-carrying capacity of the column but also exerted an upward force on the structure. Observations from the test suggest the building frame remained linear-elastic. A linear analytical model of the test specimen was developed in ETABS. This model effectively captured the response of the structure during testing. The model was used to examine the response of the test specimen due to gravity load redistribution alone, as well as the total response of the structure, including the effects of the upward force from the explosion. A dynamic amplification factor of 1.89 was observed for the columns where the majority of the gravity load redistributed during in the experimental test. This value supports the maximum dynamic amplification of 2.0 proposed for linear structures by previous research and progressive collapse assessment guidelines. The response at these columns was dominated by one mode of vibration. At columns where less steady-state load was redistributed, dynamic amplification factors much greater than 2.0 were observed, although the absolute peak axial force increases were relatively small. Higher modes influenced by cantilevers along one side of the structure contributed significantly to the peak demands at these locations. The upward explosive force magnified the peak demands on the structure from gravity load redistribution by as much as 68%. The upward explosive force was also observed to act unsymmetrically on the structure. These effects must be considered when considering column failures from explosives installed inside the member.
EXPLOSIVE TESTING TO ASSESS DYNAMIC LOAD REDISTRIBUTION IN A REINFORCED CONCRETE FRAME BUILDING by T I M O T H Y W I L L I A M M A T T H E W S B.Sc. C i v i l Engineering, University of Alberta, 2002 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 T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R OF A P P L I E D S C I E N C E in T H E F A C U L T Y OF G R A D U A T E S T U D I E S (Civ i l Engineering) T H E U N I V E R S I T Y OF B R I T I S H C O L U M B I A October 2007 © Timothy Wi l l i am Matthews, 2007 ABSTRACT Guidelines have been developed in the United States to assist in the design of buildings resistant to progressive collapse. These guidelines outline several methods for considering the effects of dynamic amplification on load redistribution following sudden member failure in a structure. Previous analytical studies have been performed to evaluate the consistency of these guidelines, but very little testing has been conducted to verify the guidelines and analytical work performed to date. This study investigates the dynamic load redistribution occurring in a field test on a two-storey reinforced concrete frame specimen. The test consisted of removing a column with explosives and recording the deformations of the remaining columns in the structure. The explosives removed the load-carrying capacity of the column but also exerted an upward force on the structure. Observations from the test suggest the building frame remained linear-elastic. A linear analytical model of the test specimen was developed in E T A B S . This model effectively captured the response of the structure during testing. The model was used to examine the response of the test specimen due to gravity load redistribution alone, as well as the total response of the structure, including the effects of the upward force from the explosion. A dynamic amplification factor of 1.89 was observed for the columns where the majority of the gravity load redistributed during in the experimental test. This value supports the maximum dynamic amplification of 2.0 proposed for linear structures by previous research and progressive collapse assessment guidelines. The response at these columns was dominated by one mode o f vibration. At columns where less steady-state load was redistributed, dynamic amplification factors much greater than 2.0 were observed, although the absolute peak axial force increases were relatively small. Higher modes influenced by cantilevers along one side of the structure contributed significantly to the peak demands at these locations. The upward explosive force magnified the peak demands on the structure from gravity load redistribution by as much as 68%. The upward explosive force was also observed to act unsymmetrically on the structure. These effects must be considered when considering column failures from explosives installed inside the member. i i T A B L E OF CONTENTS ABSTRACT ii TABLE OF CONTENTS iii LIST OF TABLES vi LIST OF FIGURES vii ACKNOWLEDGEMENTS x 1 INTRODUCTION 1 1.1 Progressive Collapse - Definition 1 1.2 Progressive Collapse - How It's Considered in Design 2 1.3 How Dynamic Amplification Relates to Progressive Collapse 3 2 GUIDELINES, PREVIOUS RESEARCH & RESEARCH AIMS 8 2.1 Progressive Collapse Guidelines 8 2.1.1 Progressive Collapse in Codes, Standards and Guidelines 8 2.1.2 Approaches to Progressive Collapse-Resistant Design 9 2.1.3 The GSA Guideline 11 2.1.3.1 Indirect Design Approach in the GSA Guideline 12 2.1.3.2 Direct Design Approach in the GSA Guideline 12 2.1.3.3 Analysis Types in the GSA Guideline 13 2.1.4 The U F C Guideline 15 2.1.4.1 Indirect Design Approach in the U F C Guideline 16 2.1.4.2 Direct Design Approach in the U F C Guideline 16 2.1.4.3 Analysis Types in the UFC Guideline 17 i i i 2.1.5 Comparison of the G S A and U F C Guidelines 17 2.1.6 Dynamic Amplification in the GSA and U F C Guidelines 20 2.2 Previous Research on Dynamic Amplification 20 2.3 Research Program Objectives, Scope and Applicability 24 3 EXPERIMENTAL PROGRAM 26 3.1 Overview 26 3.2 Test Specimen 26 3.3 Material Properties 34 3.4 Preliminary Analytical Model 35 3.5 In-Situ Column Stiffness Testing 37 3.6 Instrumentation 41 3.6.1 Challenges, Simplifying Factors and Redundancy 41 3.6.2 Instrumentation Equipment 43 3.6.3 Instrumentation Layout 48 3.7 Test Preparation - Explosives 51 4 EXPERIMENTAL RESULTS 54 4.1 Qualitative Results 54 4.2 Quantitative Results 55 4.2.1 Recorded Data - Anomalies and Corrections 56 4.2.2 Steady-State Deformations 58 4.2.3 Transient Deformations 61 4.2.3.1 General Observations 62 4.2.3.2 Discrepancies Between Linear Potentiometer and PI Transducer Response 64 4.2.3.3 Comparison of Response at Different Columns in Test Specimen 66 iv 5 DATA ANALYSIS AND MODELLING 69 5.1 Overview 69 5.2 Frequency Analysis 70 5.3 Investigation of Accelerometer Data 78 5.4 Simulation of Specimen Response - S D O F Model . 83 5.5 Simulation of Specimen Response - 3D E T A B S Model 87 5.5.1 E T A B S Model Properties, Assumptions and Development Process 87 5.5.2 Comparison of E T A B S Model to Recorded Data - Steady-State Response 91 5.5.3 Comparison of E T A B S Model to Recorded Data - Transient Response 94 5.5.4 E T A B S Model - Effects of Damping, Triangular Pulse and Primary Mode 97 5.6 Dynamic Amplification in Test Specimen from 3D ETABS Model 101 6 SUMMARY, CONCLUSIONS AND RECOMMENDATIONS 112 6.1 Summary of Research Program and Conclusions 112 6.2 Recommendations for Future Testing 116 REFERENCES 119 APPENDIX A. PRELIMINARY SAP MODEL 123 APPENDIX B. IN-SITU COLUMN STIFFNESS TESTING DATA 126 APPENDIX C. INSTALLED INSTRUMENTATION LOCATIONS 147 APPENDIX D. RECORDED EXPERIMENTAL DATA 149 APPENDIX E. ADJUSTEMENT OF DATA AT COLUMN D-2 (S) 166 APPENDIX F. DETAILED ETABS MODEL 170 v LIST OF TABLES Table 4.1 Steady-State Deformations 59 Table 5.1 Comparison of E T A B S Model and Recorded Steady-State Deformations 92 Table 5.2 Comparison of Deformations at Instrument Locations and Column Centers 93 Table 5.3 Summary of A x i a l Force Changes in E T A B S Model 103 LIST OF FIGURES Figure 1.1 Peak vs. Steady-State Redistributed A x i a l Force From a Column Failure 4 Figure 1.2 Load Redistribution and Dynamic Amplification in a Spring System 5 Figure 3.1 Elevations of the Explosive Test Specimen 27 Figure 3.2 Test Specimen Following Masonry Wal l Removal - East Column Line 28 Figure 3.3 First Floor Level of Test Specimen Following Grouting 29 Figure 3.4 First Floor Structural Plan of Test Specimen 29 Figure 3.5 Column and Roof Beam Cross-Sections in the Test Specimen 30 Figure 3.6 Tapers in the Roof Beams of the Test Specimen 31 Figure 3.7 Second Floor Beam Cross-Sections in the Test Specimen 32 Figure 3.8 W B 2 Beam on Grid 2 Above Brick at Second Floor in Test Specimen 33 Figure 3.9 Slab and Flooring at the Second Floor of the Test Specimen 33 Figure 3.10 Preliminary Analytical Model of Test Specimen 35 Figure 3.11 In-Situ A x i a l Stiffness Testing on a Column in the Test Specimen 39 Figure 3.12 A x i a l Force-Deformation Plot of Column B-2 from In-Situ Stiffness Testing 40 Figure 3.13 Linear Potentiometers and Dia l Gauges Used on the Explosive Test 44 Figure 3.14 Instrument Stand and Multiple Instrument Mounting 45 Figure 3.15 PI Transducer and Accelerometer Used on the Explosive Test 47 Figure 3.16 Levels of Instrumentation on the Test Specimen - Plan V i e w 49 Figure 3.17 Elevation Showing Instrumentation Installed Along Gr id 1 50 Figure 3.18 Elevation Showing Instrumentation Installed Along Gr id 2 51 Figure 3.19 Preparation of Column C-2 for Explosive Installation 52 Figure 3.20 Explosive Shield and Safety Frame Used on the Explosive Test 53 Figure 4.1 Explosive Shield and Safety Frame Used on the Explosive Test 54 Figure 4.2 Displacement Response History - Linear Potentiometer at Column B-2 (N) 55 Figure 4.3 Anomalies and Corrections in Instrument Data 57 Figure 4.4 Deformation Recorded from 0 - 1 s by Linear Potentiometer at Column B-2 (S).... 62 Figure 4.5 Response of Dynamic Data Collection Instruments at Column B-2 (S) 65 Figure 4.6 Transient Response - Linear Potentiometers at Columns B-2, C - l & D-2 67 Figure 5.1 Fourier Spectrum of Linear Potentiometer Data from 0.13 s to 4 s: 0 to 150 H z 72 Figure 5.2 Fourier Spectrum of Linear Potentiometer Data from 0.13 s to 4 s: 0 to 20 H z 73 Figure 5.3 Fourier Spectrum of PI Transducer Data from 0.13 s to 4 s: 0 to 150 H z 73 Figure 5.4 Fourier Spectrum of PI Transducer Data from 0.13 s to 4 s: 0 to 20 H z 74 Figure 5.5 Response Recorded at Column A - 2 : 0 to 0.5 s 76 Figure 5.6 Response of Accelerometer Above Removed Column at C-2: 0 to 0.25 s 79 Figure 5.7 Delay in Response Due to Location in Specimen 80 Figure 5.8 Loading Function Shapes for Upward Explosive Force 82 Figure 5.9 Loading Functions Applied on the S D O F Model 86 Figure 5.10 Comparison of S D O F Model Response and Recorded Response 86 Figure 5.11 E T A B S Model of Test Specimen - Slab Not Shown For Clarity 88 Figure 5.12 Comparison of E T A B S Model Response and Recorded Data at B-2 (S) 95 Figure 5.13 E T A B S Model Response and Recorded Data at D-2 (N) and C - l 96 Figure 5.14 Impact of Pulse Magnitude on Model Response at B-2 (S): 3.5% Damping 99 Figure 5.15 Impact of Damping on Model Response at B-2 (S) 100 Figure 5.16 Response of Primary Mode of Vibration in the E T A B S Model at B-2 (S) 100 Figure 5.17 Components of Triangular Pulse and Step Force in Total Model Response 102 Figure 5.18 A x i a l Force Response in E T A B S Model at Column C - l - Gravity Loads 106 Figure 5.19 Participation of Modes in A x i a l Force Response at Column C - l 107 Figure 5.20 Comparison of Response from Explosive Pulse and Gravity Load at B-2 110 ix A C K N O W L E D G E M E N T S First and foremost, I would like to thank my research supervisor, Dr. K e n Elwood for all his assistance, guidance and encouragement on this often challenging research endeavour. Thanks also to Dr. Terje Haukaas for filling second reviewer duties inside a very tight time frame, and Scott Jackson for all his advice on instrumentation setup and signal analysis. Without the generous support of the National Centre for Research on Earthquake Engineering in Taipei, Taiwan, the experimental testing in this research program would never have taken place. M y thanks go out to Dr. Shyh-Jiann Hwang for spearheading the project, and Dr. Lap-Loi Chung, Wu-Wei Kuo and Chiun-Lin W u for their insight and input into the experimental test design. Special thanks go to Tsung-Chih Chiou, Fu-Pei Hsiao and Wu-Wei Kuo for all their assistance and willingness to help out despite their fantastically busy schedules. Lastly, I would like to thank the boys of Rusty Hut 129 for making the prospect of coming into the office appealing even when work on the thesis was not. Aaron, Arnaud and Soheil, it was a pleasure spending the last two years in your presence. I already look back on our adventures fondly. x 1 INTRODUCTION 1.1 Progressive Collapse - Definition The topic of dynamic amplification during gravity load redistribution in reinforced concrete frames is a subset of the much broader subject area of progressive collapse. Thus, before defining the concept of dynamic amplification, the main focus of this research study, it is necessary to first discuss progressive collapse. The exact definition of progressive collapse is still being debated. One definition of progressive collapse is a cascading sequence o f member failures resulting from the initial localized failure o f one or more elements which results in the collapse of the entire structure or a disproportionately large part of it [Ruth, 2004]. What constitutes a "disproportionately large" area of failure is also the subject of much debate and is not consistent between the various documents pertaining to progressive collapse. In more general terms, progressive collapse is a chain reaction in which a local member failure causes failures in adjacent elements, which in turn cause failure of other elements, the net result of which is structural damage far in excess of what is expected due to the initial member failure. There are two types of progressive collapse failures: inadequate load path failures and debris-induced pancake failures [Ruth, 2004]. In an inadequate load path failure, the loads originally carried by the first element to fail, or "initiator element", are not able to find another load path through neighbouring elements to the ground. Without an alternative load path, the part of the structure originally supported by the initiator element collapses, potentially inducing failure in adjacent sections of the structure. In a debris-induced pancake failure, a load path exists after the initial failure but the debris resulting from a local collapse higher up in a structure overloads the supporting structure, causing the structure at that level to collapse, generate more debris, and "pancake" onto the level below. Lower levels are often not able to support the increasing amount of debris resulting from collapse of higher levels and the structure in such cases collapses all the way to the ground. 1 In most cases, inadequate load path failures occur in a structure above the initiator element, whereas debris-induced pancake failures occur in a structure below the initiator element. The two types of progressive collapse failures can be linked: an inadequate load path failure may generate a great deal of debris and cause the structure below the initial failed member to undergo debris-induced pancake failure. 1.2 Progressive Collapse - How It's Considered in Design H o w progressive collapse is considered in the design of structures depends on the methodology used to perform the design. Three cases wi l l be discussed: structures designed according to codes, structures designed using a performance-based approach, and structures with a greater risk of experiencing abnormal loading. Despite the rise in popularity of performance-based design, a large majority of structures are still designed according to the simplified procedures in building codes. Building codes in North America do not explicitly consider progressive collapse [Ruth et al., 2006]. Many codes contain provisions that promote continuity, ductility and structural integrity, features that improve resistance to progressive collapse. Commentary C of the 1995 Edition of the National Building Code o f Canada ( N B C C ) requires that structures be designed such that they have sufficient structural integrity to ensure local failure does not lead to widespread collapse [ N R C C , 1995]. While the N B C C and other North American codes do encourage continuity, ductility and integrity, they do not currently require design engineers to consider the effect that the failure o f individual components w i l l have on the overall stability of a structure. Although progressive collapse is not directly considered in code designs, structures designed according to codes have generally performed quite well when subjected to significant localized member failures [Ruth, 2004]. For instance, when a B-25 bomber accidentally crashed into the side of the Empire State Building, a code-designed structure, in 1945, the structural damage was limited to two stories, despite the fact that the plane created an 18 foot by 20 foot hole in the side of the building [Ruth, 2004]. The overstrength inherent in code designs, combined with continuity and ductility requirements, seems to result in structures with good resistance to progressive collapse. Past history, then, would seem to suggest that the extra effort o f explicitly considering progressive collapse may be unjustified for code-designed structures. 2 In recent years, performance-based design approaches and advanced analysis techniques have been employed in the structural design of buildings. These advances have enabled designers to better analyze the forces on structures and optimize structural members to achieve a certain level of performance. In many cases, the goal of performance-based designs has been to reduce costs, whether they be construction costs or life-cycle costs. While buildings designed using performance-based approaches are more optimized, as a consequence of optimization, the amount of overstrength inherent in the structure is lower than in code-designed structures. With less overstrength, it is uncertain whether structures with performance-based designs w i l l have the same good resistance to progressive collapse characteristic of code-designed structures. Thus, it would be prudent to directly consider progressive collapse in the performance-based design of structures. Since codes do not explicitly take into account progressive collapse, they do not consider the severe and abnormal loadings (aircraft collisions, gas explosions, terrorist attacks) that cause the localized member failures that may initiate progressive collapse. A s the geopolitical situation has changed in recent years, certain structures, such as U S Government buildings, are believed to be at greater risk of experiencing abnormal loadings from terrorist attacks. To mitigate this elevated risk, the U S General Services Administration (GSA) and Department of Defense (DoD) now require progressive collapse to be directly considered in the design of their facilities [GSA, 2003; D o D , 2005]. For such facilities, progressive collapse must be explicitly considered regardless of whether the design is performance-based or completed according to a code. The additional effort of designing against progressive collapse increases the margin of safety for structures at a higher risk of terrorist attack, thereby offsetting the higher levels of building damage that terrorist attacks can cause i f they induce progressive collapse [Breen and Seiss, 1979]. 1.3 How Dynamic Amplification Relates to Progressive Collapse Progressive collapse is initiated by the localized failure of one or a few members due to a severe and abnormal loading. The abnormal loading induces forces in a locally-affected member well in excess of those anticipated in design and beyond the load-carrying capacity of the member, thereby causing localized member failure. 3 Two types of loading that can lead to progressive collapse are blast loading, from events such as natural gas or bomb explosions, and earthquake loading. Member failures, in particular those caused by blast and earthquake loads, can occur very suddenly with the result that the forces originally carried by the failed member are transferred to the remaining structure over a very short time interval. Since the time of load application is very small, the response of the remaining structure as it redistributes the force in the failed member is influenced significantly by dynamic effects. For the case of a sudden column failure, the member failure most commonly examined in progressive collapse analysis, the dynamic nature of the load redistribution results in the peak axial force increase in neighbouring columns being considerably larger than the permanent, or steady-state, axial force increase. This phenomenon is illustrated in Figure 1.1. The term dynamic amplification is used to describe the relationship between the peak and steady-state forces resulting from load redistribution. Dynamic amplification is defined simply as the ratio of the peak force increase to the steady-state force increase. 4> U o Is < U Peak Columln Force After Adjacent Column Failure Steady-State Force Alfter Adjacent ™Cdl'um"irFailure~ Column Force Adjacent Colu Column' Before mn Fl lailure Time Figure 1.1 Peak vs. Steady-State Redistributed Axia l Force F r o m a Column Failure The concepts of dynamic amplification and load redistribution are explained further by the simplified spring model in Figure 1.2. Figure 1.2a shows the spring system in initial equilibrium. The weight of the block (W) is carried partly by the upper spring (F/) and partly by 4 the lower spring (Fi) in proportion to the relative stiffness of each; in this example, the upper spring is slightly stiffer and as a result Fi is greater than F2. Referring to Figure 1.2d, prior to lower spring failure the inertial force of the block (ma) is zero because the system is at rest, and the sum of the forces in the springs (Fi + Fi) equals the weight of the block, which is represented in Figure 1.2d as a horizontal line of magnitude -W (this term is negative because W is in the opposite direction of Fi and Fi). /////// /////// ma '///// F 1 = -W ma = 0 /////// ///////////// (a) Initial Equilibrium (b) Equilibrium at Spring (c) Steady-State Equilibrium Failure F ^ / - T i m e of lower s p r i n g failure P e a k F o r c e n c r e a s e (d) Upper Spring, Lower Spring and Inertial Forces During Load Redistribution Figure 1.2 Load Redistribution and Dynamic Amplification in a Spring System When the lower spring fails instantaneously, the situation drawn in Figure 1.2b, the force in the lower spring (Fi) goes to zero, the force in the upper spring (Ff) stays the same, and an inertial force (ma) equal to the force originally carried by the lower spring starts to act on the block, which is about to start moving. A s the block begins to move and oscillate about the final equilibrium position, the upper spring force and inertial force vary; due to equilibrium, the sum of these forces is always equal to the weight of the block, assuming the damping forces on the block are negligible. Eventually the motion o f the block w i l l damp out, however, causing the acceleration, and hence the inertial force (ma), on the block to become zero, as shown in Figure 1.2c. A t this point, all the force originally carried by the lower spring is redistributed to the upper spring, making the force in the upper spring equal to —W. A s mentioned above, the magnitude of the damping force on the block while it is in motion has been assumed to be negligible, and is therefore omitted from Figure 1.2b. This assumption was made for simplicity and to make Figure 1.2 easier to comprehend. A damping force is present in the system, however, and is essential for bringing the system back to rest. A s shown in Figure 1.2d, after the lower spring fails, the force in the upper spring (Fi) and the inertial force on the block (ma) oscillate about the steady-state force values that each w i l l obtain after the motion of the block damps out: the weight of the block (-W) for Fj and zero for ma. During this transient period of load redistribution, the peak force in the upper spring is considerably greater than the weight of the block. For the system to continue to be able to support the block, the upper spring must be able to resist the peak force increase during redistribution, not just the steady-state force increase, which is equal to the original value of F2. A s noted previously, the dynamic amplification that occurs in the system during load redistribution is the ratio of the peak force increase to the steady-state force increase. The steady-state redistributed loads resulting from a spring or column failure can be determined with relative certainty using simple static analyses. Accurately estimating the peak redistributed loads is much more difficult, often requiring complex and time consuming dynamic analyses. If, however, one is designing a structure so that it w i l l remain stable after the failure of one of its columns (a strategy used to make structures resistant to progressive collapse [GSA, 2003; D o D , 2005]), it is the peak redistributed forces that are of concern, not the steady-state forces. If static analyses are to be used to design structures resistant to progressive collapse, an effective means of estimating peak redistributed forces from steady-state forces is required. Dynamic 6 amplification relates peak force increases to steady-state force increases. Thus, developing accurate dynamic amplification values is essential for design against progressive collapse using static analyses. It must be noted that in dynamic loading situations, not only are the peak forces in a structure higher, but the capacities of structural members are also observed to be higher than when loads are static in nature. For many materials (including steel and concrete), when force is applied over a short duration, thereby causing the material to deform and strain over a small time interval, the ultimate strength is greater than for cases where load is applied more slowly. This strength increase, which is often referred to using the more general term strain rate effects, is sometimes assumed to cancel out the dynamic amplification of applied forces. Dynamic amplification and strain rate effects are not explicitly coupled, however. In some situations, a loading may result in dynamic amplification but negligible strength increase due to strain rate effects. To simply assume that strain rate effects w i l l compensate for the dynamic amplification of applied forces in such a situation would be unconservative. Strain rate effects and dynamic amplification are independent phenomena and must therefore be assessed separately. For an accurate assessment of a structure's susceptibility to progressive collapse, both of these behaviours should be considered. For the purposes of conciseness, however, this study wi l l focus only on dynamic amplification. Practicing engineers in general are more familiar and more confident with using static methods to perform structural design than dynamic methods. Accordingly, i f static procedures are available for assessing the susceptibility of structures to progressive collapse, it makes progressive collapse analysis accessible to a wider range of engineers. To effectively design against progressive collapse using static methods, however, accurate dynamic amplification factors are required. Dynamic amplification factors must be large enough to conservatively estimate the peak loads on a structure, but not so large as to result in structural elements that are excessively large and expensive to construct. This study w i l l investigate appropriate values for dynamic amplification factors by examining progressive collapse prevention guidelines, consulting previous research on dynamic amplification, and performing an experimental test to assess the dynamic amplification following column failure in a full-scale two-story reinforced concrete frame building. 7 2 GUIDELINES, PREVIOUS R E S E A R C H & R E S E A R C H AIMS 2.1 Progressive Collapse Guidelines In Section 1.2, it was stated that it is desirable to explicitly consider progressive collapse when performing a performance-based design or designing structures subject to greater likelihood of abnormal loading. Unt i l quite recently, however, guidance on how to design against progressive collapse was rather limited in North American codes and guidelines. 2.1.1 Progressive Collapse in Codes, Standards and Guidelines North American codes do not explicitly consider progressive collapse, so it is not surprising that they lack a detailed procedure for progressive collapse-resistant design. The 2003 International Building Code (IBC) and 1997 Unified Building Code ( U B C ) contain no mention of requirements for progressive collapse prevention [Ruth et al., 2006]. The National Building Code of Canada ( N B C C ) [ N R C C , 1995] and American Society of C i v i l Engineers ( A S C E ) Standard 7-05 [ A S C E , 2005] acknowledge the need to prevent local member failures from propagating and suggest some general approaches to prevent this from happening. Prescriptive methodologies for progressive collapse-resistant design are not provided, however. The N B C C and A S C E 7-05 address the issue of progressive collapse primarily by emphasizing the need for structural integrity. These standards require structures to have sufficient continuity, redundancy and ductility to ensure that local failures do not lead to widespread collapse. Material standards such as American Concrete Institute (ACI) 318 [ACI , 2005] and Canadian Standards Association (CSA) A23.3 [CSA, 2004] specify detailing provisions that promote structural integrity and help achieve the requirements of the N B C C and A S C E 7-05. This indirect approach to progressive collapse prevention by promoting structural integrity is also incorporated in the IBC and U B C , although the link between structural integrity and progressive collapse resistance is not mentioned. In the United Kingdom, quantifiable and prescriptive criteria for progressive collapse-resistant design have been present in the British Standards (BS) since the early 1970's [NIST, 2007]. 8 These standards developed in response to the Ronan Point apartment building collapse of 1968, widely considered to be the watershed event initiating interest in the topic of progressive collapse [Ruth, 2004]. B S 8110-1 and other British Standards contain prescriptive procedures for design against progressive collapse (the nature of these procedures w i l l be outlined in Section 2.1.2), which are much more useful for performance-based design than the suggestions and indirect structural integrity provisions in the North American codes and standards [BSI, 1997]. Application of the British Standards for design of progressive collapse resistance in North America has been limited, however. The difficulties resulting from trying to combine standards from different countries (for instance, the inconsistencies that arise when forces computed using a British analysis code are used to design beams according to a concrete design standard from the US) have likely played a factor in this limited usage. Recognizing that a detailed U S procedure for progressive collapse-resistant design did not exist, and that such a procedure would be necessary i f their facilities were to be explicitly designed against progressive collapse, the G S A and D o D have developed guidelines for the design of their structures. The GSA Progressive Collapse and Design Guidelines 2003 [GSA, 2003] and D o D Unified Facilities Criteria (UFC) Design of Buildings to Resist Progressive Collapse 2004 [DoD, 2005] provide comprehensive instructions on how to design structures resistant to progressive collapse. These guidelines wi l l be described in detail in Sections 2.1.3 and 2.1.4. In addition to use on U S Government facilities, the G S A and U F C guidelines can be employed in the private sector when explicit consideration of progressive collapse is deemed prudent, as is the case on some performance-based designs. Guidance on how the G S A Guideline, U F C , and international standards can be used to mitigate the risk of progressive collapse in private sector structures is given in the National Institute of Standards and Technology's (NIST) Best Practices for Reducing the Potential for Progressive Collapse in Buildings [NIST, 2007]. This document is the first step towards incorporating progressive collapse mitigation procedures into North American codes and standards. 2.1.2 Approaches to Progressive Collapse-Resistant Design Before focusing in on the G S A and U F C Guidelines, it is useful to first look at the different approaches to progressive collapse-resistant design utilized by current codes, standards and guidelines. These approaches can be broken down into direct design procedures and indirect 9 design procedures. Generally speaking, progressive collapse design aids utilize a combination of direct and indirect approaches. Notable exceptions are the North American codes and standards discussed above, which lack the prescriptive design procedures needed for direct consideration of progressive collapse. Direct and indirect progressive collapse-resistant design approaches differ in the role that structural analysis plays in the design procedure. For direct design approaches, additional structural analyses must be performed for loading conditions not considered in typical building design, including situations where members are removed from the structural model. If necessary, members are strengthened to ensure the structure is adequate under the extra progressive collapse loading conditions. For indirect design approaches, additional structural analyses beyond those considered in typical building design are not required, and the focus is on ensuring that a structure contains certain characteristics that are known to improve resistance to progressive collapse. It is possible to further subdivide the categories of direct and indirect design approaches into smaller classifications. With respect to indirect design procedures, the methodology for ensuring adequate progressive collapse resistance can be either qualitative or quantitative in nature. Qualitative indirect design approaches recommend that several overall characteristics known to improve progressive collapse behaviour, such as redundancy and ductility, be incorporated into the structure; a qualitative approach is utilized by the G S A Guideline and many of the codes and standards discussed in Section 2.1.1. Quantitative indirect design approaches seek to obtain the same overall characteristics recommended by qualitative approaches, but do so through a systematic procedure complete with calculations that consider loading and geometry. The U F C Guideline and British Standards use quantitative indirect design approaches. There are two direct approaches for designing a structure to be resistant to progressive collapse: the Specific Load Resistance (SLR) method and the Alternative Path (AP) method [Smilowitz, 2002]. In the S L R method, a structure is designed to prevent collapse due to a specific abnormal load. Key structural members are hardened or reinforced to ensure progressive collapse does not occur under the abnormal loading that is considered [Ruth, 2004]. Because the S L R method considers just one abnormal loading, it is "threat-dependent" and only explicitly valid for the loading considered. Considering also that it is often difficult to determine exactly what abnormal loading should be used for design, the S L R method is not particularly well suited for 10 designing structures against the general threat of progressive collapse; ie. cases where the potential abnormal loading is not well defined. The SLR method is not used by either the GSA or UFC Guidelines but is included in the British Standards as an alternative to the AP method for some members and can be useful for retrofitting existing buildings [Smilowitz, 2002]. In the Alternative Path method, the structure is designed such that i f one member or group of members fails due to an abnormal load, the remaining structure is able to bridge across the failed element(s) and redistribute the forces originally carried by the failed element(s) [DoD, 2005]. The AP method basically consists of removing a member (or members) from the analysis model of a structure, checking that the surrounding structure can resist the forces that result from removal of the failed element(s), and revising the design until the structure is able to withstand the loss of the failed member(s). This process is repeated, re-inserting the previous member(s) and removing a new one (or new ones) until it can be concluded, with the help of symmetry and engineering judgement, that the failure of any one member (or group of members) in the structure will not bring about progressive collapse. The AP method does not concern itself with the abnormal loading causing local failure, beyond the assumption that only one member or group of members is destroyed by the loading. As a result, the AP method is a "threat-independent" procedure that can be used to assess progressive collapse resistance to any type of abnormal loading resulting in the damage pattern assumed in analysis [Smilowitz, 2002]. The GSA Guideline, UFC Guideline and British Standards all use the Alternative Path method for direct design of progressive collapse resistance. 2.1.3 The GSA Guideline The GSA Guideline specifies progressive collapse-resistant design procedures for reinforced concrete and structural steel buildings. The guideline uses a combination of direct and indirect design approaches to reduce the risk of progressive collapse in these types of buildings. For its direct progressive collapse design approach, the GSA Guideline provides procedures for four different analysis types: linear static analysis, nonlinear static analysis, linear dynamic analysis and nonlinear dynamic analysis [GSA, 2003]. The GSA Guideline has a detailed exemption process for determining i f the risk of a structure experiencing progressive collapse is low enough that a detailed progressive collapse assessment is not required. This exemption process considers many criteria, including the Level of 11 Protection (LOP) that a facility is deemed to require. Structures that are determined to be exempt are not analyzed further for susceptibility to progressive collapse. Non-exempt structures are subjected to the rigorous progressive collapse resistance assessment detailed in the guideline [GSA, 2003]. 2.1.3.1 Indirect Design Approach in the GSA Guideline The indirect design approach utilized by the G S A Guideline to improve the progressive collapse resistance of a structure is a qualitative method. Similar to many building codes and standards, the G S A Guideline recommends that redundancy, continuity and ductility be incorporated into the structure. In addition, the guideline suggests that load reversals from local member failures be considered in design and detailing, and that structural members be designed to fail by ductile flexural mechanisms rather than brittle shear mechanisms. The G S A Guideline also requires perimeter columns at the first and second levels of a building be designed for twice their unsupported length to account for the possibility of failure of the elements laterally supporting the columns at the second floor level. The above recommendations are intended to improve the robustness and structural integrity of a structure, thereby improving its resistance to progressive collapse [GSA 2003; Ruth, 2004]. 2.1.3.2 Direct Design Approach in the GSA Guideline The Alternative Path procedure in the G S A Guideline assumes that only a single member fails due to the initiating abnormal loading, rather than considering failure of a group of members. The guideline identifies critical exposure locations (e.g. along the exterior o f buildings at the ground floor level; on the interior of uncontrolled public spaces in buildings, such as parkades) where the A P method must be performed, and provides suggestions on how to recognize other critical member locations that must be checked. The G S A Guideline puts a lot of emphasis on columns at the first floor, but stresses that sound engineering judgement must be used to ensure all potentially critical member failures are identified [GSA, 2003]. A s mentioned above, the G S A Guideline assumes in its Alternative Path procedure that abnormal loadings wi l l only cause one member to fail. In reality, it is quite possible that an abnormal event could destroy several members at one location. This being the case, the G S A Guideline acknowledges that the A P method it specifies is not a rigorous procedure for 12 preventing progressive collapse from all possible abnormal loading cases. Rather, the goal of the A P method in the Guideline is to generate a consistent and systematic measure for assessing the progressive collapse resistance and structural integrity of a building. Although the A P procedure does not consider every possible scenario that may occur in reality, it does employ a prescriptive approach and incorporates the detailed characteristics of the structure. A s a result, the A P method in the G S A Guideline should provide a more consistent and dependable level of progressive-collapse resistance than that achieved by non-prescriptive and less detailed indirect design methods [Ruth, 2004]. A s wi l l be discussed in Section 2.1.4.2, the U F C Guideline also considers only the case where progressive collapse is initiated by the failure of a single element; thus, the discussion above also applies to the U F C Guideline. 2.1.3.3 Analysis Types in the GSA Guideline The direct design approach in the G S A Guideline specifies procedures for assessing progressive collapse resistance using four different analysis types: linear static analysis, nonlinear static analysis, linear dynamic analysis and nonlinear dynamic analysis. Nonlinear dynamic analysis is considered to be the most accurate form of analysis for progressive collapse design, but it is complex, time-consuming, and requires skil l and experience to get valid results. Simplified static analyses were thus included in the G S A Guideline to allow designers not wil l ing or able to conduct nonlinear dynamic analyses for all projects to still perform progressive collapse-resistant design. Although a linear dynamic analysis is included in the guideline, this type of analysis is not recommended and w i l l not be discussed further [GSA, 2003; Ruth, 2004]. A s discussed in Section 1.3, the local member failures that initiate progressive collapse can occur very suddenly and therefore induce a dynamic response on the structure. In addition, in order to redistribute the loads originally carried by the failed element, the structural members around a failed element must often resist forces very close to their capacities. Since these remaining elements are heavily loaded, it is likely that some of these members w i l l undergo nonlinear inelastic deformations. Thus, to accurately capture the response of structures suffering a local member failure, the analysis techniques used to assess progressive collapse resistance must consider the dynamic and nonlinear nature of this response. Since it inherently considers both dynamic and nonlinear behaviour, nonlinear dynamic analysis, the most complex analysis type in the G S A Guideline, is ideally suited for assessing the 13 resistance of structures to progressive collapse. The nonlinear dynamic analysis procedure in the G S A Guideline calls for a certain load combination to be applied to the analysis model of the intact structure, and then for one member to be removed to simulate local member failure. The load combination applied to the model represents the expected loads on the structure before the abnormal loading event, which the G S A Guideline defines as 100% of the maximum Dead Load plus 25% of the maximum Live Load. The nonlinear dynamic analysis automatically incorporates the dynamic effects resulting from local member failure and returns member force and deformation demands which the designer can use to determine the adequacy of the remaining elements in the structure. To account for nonlinear inelastic material behaviour, the guideline calls for plastic hinges to be inserted in the analysis model at likely locations of inelastic behaviour. The properties of these plastic hinges are prescribed in the G S A Guideline for the different building material types it considers [GSA, 2003]. The procedure in the G S A Guideline for the second most complex analysis type, static nonlinear analysis, incorporates nonlinear material behaviour in the same way as its nonlinear dynamic analysis procedure: through the use of plastic hinges. Dynamic effects, however, must be considered differently than in the nonlinear dynamic analysis procedure for the G S A Guideline, because a static analysis can not consider these effects explicitly. The G S A Guideline accounts for dynamic effects by multiplying the expected loads on the structure (the load combination stated in the guideline) at all levels and locations by a dynamic amplification factor o f 2.0. A factor of 2.0 was likely specified by the guideline because it represents the maximum dynamic amplification possible according to linear elastic theory [Chopra, 2000]. To assess the progressive collapse resistance of a structure, the dynamically amplified loads are statically applied to an analysis model with the failed local member already removed, and the response from the analysis is compared to the capacities of the members and plastic hinges [GSA, 2003]. The simplest analysis form in the G S A Guideline is a linear static analysis. In the linear static analysis procedure specified by the G S A Guideline, the expected structure loads are dynamically amplified by a factor of 2.0 and applied statically to an analysis model with the failed member missing. This is exactly the same methodology used for the nonlinear static analysis procedure. Unlike the nonlinear static procedure, however, plastic hinging of the members cannot be considered. Thus, the G S A Guideline must utilize a different strategy to incorporate nonlinear material behaviour in its linear static analysis procedure. The strategy the guideline uses is to 14 artificially increase the strength of the members in the analysis model by a Demand-Capacity Ratio (DCR) to approximate the additional member capacity due to inelastic behaviour. D C R ' s are dependent on member material, geometry, axial load and connection type [GSA, 2003]. The use of D C R ' s is somewhat analogous to reducing the base shear by an R-factor in seismic design to consider the beneficial effect of inelastic structural deformation under earthquake loads [Ruth, 2004]. In tandem with the analysis o f a structure by the three methods discussed above, the capacities of the members in a structure must also be calculated. In determining member capacities, the G S A Guideline stipulates that nominal material strengths be used. This means that the material strength reduction factors used in strength calculations for non-progressive collapse load cases are set to 1.0 for load cases checking for progressive collapse resistance. The overstrength inherent in typical structural members is also considered by the G S A Guideline. For each material type, the guideline specifies overstrength factors that are applied to member capacities to compensate for the extra static strength characteristic of members of that particular material type; it does not appear that effects of dynamic loading on material strength, such as strain rate effects, are considered in the overstrength factor [GSA, 2003]. 2.1.4 T h e U F C G u i d e l i n e The U F C Guideline is quite similar to the G S A Guideline. The U F C Guideline prescribes design procedures for mitigating progressive collapse in reinforced concrete and structural steel buildings, as well as masonry, wood and cold-formed steel structures. L ike the G S A Guideline, the U F C Guideline utilizes both direct and indirect design approaches and specifies direct design procedures using nonlinear dynamic, nonlinear static and linear static analyses [DoD, 2005]. Buildings less than three storeys tall are exempt from progressive collapse assessment by the U F C Guideline. For all other structures, susceptibility to progressive collapse must be determined. The type o f progressive collapse that must be performed is dependent on the Level of Protection (LOP) that the facility is deemed to require. For buildings requiring a Medium or High L O P , the structure must be determined to have sufficient progressive collapse resistance by both a direct and indirect design approach. For buildings deemed to need only a Very L o w or L o w L O P , adequate resistance to progressive collapse need only be proven by indirect methods; if, however, a design is found to be inadequate by the indirect approach, direct design methods 15 may be used in some circumstances to justify satisfactory progressive collapse resistance as an alternative to automatically increasing the capacity of the structure [DoD, 2005]. 2.1.4.1 Indirect Design Approach in the UFC Guideline Rather than simply specifying general provisions to improve structural integrity like the G S A Guideline, the U F C Guideline prescribes a quantitative indirect design procedure to ensure continuity, ductility and adequate overall resistance to progressive collapse. This procedure is called the Tie Force approach and it relies on the ability of structural members to act in tension, or catenary action, to resist progressive collapse following a local failure. Rather than trying to directly model the catenary response of the entire structure, which is quite difficult to do accurately, the Tie Force method uses an indirect, simplified approach in which the catenary response of the structure is approximated by evaluating the capacity of discrete tension ties. The U F C specifies where horizontal and vertical tension ties are required, and provides equations and provisions for determining the tensile forces the ties must resist; required tie locations and tensile force equations depend on the building material type. The role of tension tie can normally be accomplished by the gravity-load carrying members in a structure (columns, walls, beams, and slabs), although these members may have to be modified to increase their tensile capacity. The Tie Force method is considered to be an indirect approach since the forces in the tension ties are generated from equations rather than a static or dynamic analysis. The tie force equations do, however, consider the loads on the structure, member spans and number of stories. A s a result, the Tie Force indirect design approach generates detailing requirements that are reflective o f the characteristics of the structure [DoD, 2005; Ruth, 2004]. 2.1.4.2 Direct Design Approach in the UFC Guideline Like the G S A Guideline, the U F C Guideline uses the "threat-independent" Alternative Path (AP) approach as its direct design method for ensuring adequate progressive collapse resistance. The A P procedure in the U F C Guideline assumes that local failure consists of the destruction of a single structural element, and identifies many of the same critical exposure locations specified in the G S A Guideline. The U F C Guideline recognizes the need to perform the A P method at the first floor and at higher floors when the footprint of the building changes, and highlights the 16 important role engineering judgement plays in ensuring that all relevant member failures are considered in the progressive collapse assessment of a structure [DoD, 2005; Ruth, 2004]. 2.1.4.3 Analysis Types in the UFC Guideline The procedures for the three analysis types prescribed in the U F C Guideline (nonlinear dynamic analysis, nonlinear static analysis and linear static analysis) are the same as those specified in the G S A Guideline with a few notable exceptions. In the U F C Guideline, the load combination applied on the structure is 120% of the maximum Dead Load plus 50% of the maximum Live Load plus 20% of the maximum Wind Load. In the static analysis procedures in the U F C Guideline, a dynamic amplification factor of 2.0 is applied only to the loads in the bays adjacent to the failed member at the level of failure and all levels above, rather than everywhere in the structure. For its linear static analysis procedure, the U F C Guideline conservatively neglects nonlinear material behaviour and considers all elements in the analysis model to be elastic; it does not estimate nonlinear material behaviour with a D C R like the G S A Guideline [DoD, 2005]. Calculation of member capacities is also done slightly differently in the U F C Guideline than in its G S A counterpart. The U F C Guideline calls for factored member capacities to be used when checking the adequacy o f members; the material strength reduction factors used to calculate factored member capacities match the phi-factors stated in the codes for the different building materials. With respect to member overstrength, however, the G S A and U F C Guidelines agree on how this phenomenon should be considered (by increasing member capacities by a material-specific overstrength factor) and on the amount of overstrength inherent in members of different material types (the overstrength factors in the two guidelines are very similar for most materials). 2.1.5 Comparison of the GSA and UFC Guidelines A s discussed in Section 2.1.4, the G S A and U F C Guidelines are alike in many ways. Both guidelines utilize direct and indirect design approaches to reduce the risk of progressive collapse and prescribe the Alternative Path method for direct design of progressive collapse resistance. In the A P procedures they specify, the guidelines both assume that only one member fails as a result of abnormal loading on the structure, and give similar guidance on critical local member failure locations and the role of engineering judgement in ensuring all potentially governing 17 member failures are analyzed. Direct design methods using nonlinear dynamic analysis, nonlinear static analysis and linear static analysis are provided by both the G S A and U F C Guidelines, and the details of these methods are quite similar in many cases. Finally, the factors used to compensate for the overstrength in structural members are similar in both guidelines. Despite the many similarities between the G S A and U F C Guidelines, there are some areas where the guidelines differ. For the most part, the guidelines rely on the same underlying concepts. The specific details of the procedures embodying these concepts do vary somewhat between the guidelines, however. With respect to the types of building materials covered by the guidelines, the U F C Guideline provides progressive collapse-resistant design procedures for reinforced concrete, structural steel, masonry, wood and cold-formed steel structures. The G S A Guideline is less extensive, focusing primarily on reinforced concrete and structural steel buildings. The guidelines differ in some ways on the parameters considered in analysis. The G S A Guideline considers nonlinear material behaviour for linear static analysis while the U F C Guideline does not. The U F C Guideline requires the use of material strength reduction factors and factored member capacities, whereas the G S A Guideline specifies that nominal member capacities (which are not modified by material strength reduction factors) be used. The load combinations prescribed by the guidelines differ, with the load combination in the U F C Guideline incorporating greater proportions of the estimated maximum dead and live loads, and also including part of the expected wind load on the structure. Lastly, for static analyses, the portions of the building for which loads are dynamically amplified are different: in the G S A Guidelines, the loads are dynamically amplified everywhere, whereas for the U F C , the specified load combination is only dynamically amplified in the bays adjacent to the failed member at the level of failure and all levels above. The largest discrepancy between the guidelines is the indirect design method specified by each. Rather than just the details being dissimilar, the general approaches to indirect design taken by the guidelines are completely different. The G S A Guideline adopts a qualitative indirect design approach. The guideline suggests a number of measures to improve the structural integrity o f a building, thereby decreasing its susceptibility to progressive collapse. Due to the fact that the recommendations are general and qualitative in nature, though, it is difficult to assess how much 18 each provision improves progressive collapse resistance, and i f the measures taken to reduce the risk of progressive collapse are sufficient, inadequate or excessive for a particular structure [GSA 2003; Ruth, 2004]. The U F C Guideline, on the other hand, utilizes a quantitative indirect design procedure called the Tie Force approach. The Tie Force method approximates the tensile forces that must be resisted for a structure to prevent progressive collapse by catenary action. The members in a structure must act as horizontal and vertical tie elements to resist the calculated tensile forces. Although an indirect approach (no structural analysis is performed), the Tie Force method does consider building loads and geometry, and the structural integrity detailing requirements it generates are therefore related to the characteristics of the structure. Qualitative indirect design approaches do not Consider the make-up of a structure to the same extent as the quantitative Tie Force method, suggesting that the detailing provisions of the latter w i l l provide a more consistent level of progressive collapse resistance from building design to building design [Ruth, 2004]. A second conceptual discrepancy between the G S A and U F C Guidelines deals with how the guidelines evaluate which structures should receive which type of progressive collapse assessment. In the U F C Guideline, buildings two stories or less are exempt from analysis of progressive collapse resistance. For all other structures, the type of assessment is dependent on the Level of Protection (LOP) that must be achieved. Buildings with a Medium or High L O P must be checked by both direct and indirect design methods; buildings with a Very L o w or L o w L O P need only be designed by indirect approaches. In the G S A Guideline, structures either require a rigorous progressive collapse assessment using the Alternative Path direct design method, or are not analyzed for progressive collapse at all . The guideline has a detailed exemption process that determines i f the progressive collapse resistance of a structure must be explicitly checked. There is no less-extensive assessment process in the G S A Guideline for structures with lower L O P ' s . Thus, the indirect design approach in the G S A Guideline is used in tandem with the Alternative Path method to improve progressive collapse resistance in non-exempt buildings, and does not apply for buildings determined to be exempt from progressive collapse analysis. 19 2.1.6 Dynamic Amplification in the GSA and UFC Guidelines In Sections 2.1.3, 2.1.4 and 2.1.5, the G S A and U F C Guidelines were summarized and compared to identify their similarities and differences. The entireties of these guidelines were examined to give some context on how these documents are used to reduce the risk of progressive collapse in structures. This research study is not focused on all the material in the guidelines, but rather on just one aspect of progressive collapse design: the dynamic amplification factor used to estimate dynamic effects in linear and nonlinear static progressive collapse analyses. Both the G S A and U F C Guidelines assume a dynamic amplification factor of 2.0 for static analyses. The remainder of this report w i l l compare this assumption to analytical research conducted on the subject and an experimental test conducted to investigate dynamic amplification during gravity load redistribution in reinforced concrete frame structures. 2.2 Previous Research on Dynamic Amplification Numerous research studies have examined the impact o f dynamic effects on the load redistribution resulting from the failure of one member in a structure. Pretlove et al. [1991] performed analytical modeling, numerical simulation and experimental testing to investigate the progressive "collapse" behaviour o f a tension spoke wheel. A tension spoke wheel, which in this study consisted of a central mass supported by 12 tensioned piano wire spokes connected to a stiff steel outer ring, was selected because it is simple (only two degrees of freedom) and representative of guy-wire and cable-net structures. Kaewkulchai and Williamson [2002] demonstrated the significant impact of dynamic effects on progressive collapse in frame structures through an analytical study of a simple frame model. A finding common to both these studies is that due to dynamic amplification of redistributed loads following sudden local member failure, static analysis may underestimate the peak forces in structural members and therefore over-predict the progressive collapse resistance of structures. Ruth [2004] conducted a comprehensive analytical study on the dynamic amplification values specified in the G S A and U F C Guidelines. This study was conducted in two stages. The first stage investigated the discrepancies in the designs produced by the different analysis procedures (linear static, nonlinear static and nonlinear dynamic) detailed in the G S A and U F C Guidelines. The second stage looked at how the dynamic amplification factor for the nonlinear static analysis 20 procedures could be modified to bring the results of these procedures in line with those of the nonlinear dynamic analysis methods in the guidelines. In the first stage, a structural steel moment frame building was designed using typical techniques that do not consider progressive collapse. The building was then re-designed using the linear static, nonlinear static, and nonlinear dynamic procedures in the G S A and U F C Guidelines. In all design cases, moment capacity of pin-ended connections, catenary action in the floor structure, and lateral bracing of beams by slabs were neglected. In addition, no member failures were permitted in the structure except for the element removed to initiate the progressive collapse analysis. Analysis of the designs resulting from the different procedures in the G S A and U F C Guidelines revealed some interesting findings. First, the minimum adequate member sizes determined by the static linear procedure in the G S A Guideline were sometimes smaller than those determined by the guideline's nonlinear dynamic procedure. Assuming nonlinear dynamic analysis is the most accurate analysis type, this finding implies that the G S A Guideline linear static procedure may be unconservative in some cases, and that the Demand Capacity Ratios (DCR's ) used to approximate inelastic material behaviour in this procedure may be too large for some members [Ruth, 2004]. The second interesting finding from the first stage of the Ruth [2004] study is that the structural members determined using the G S A Guideline's nonlinear static design procedure were the same size or larger (sometimes significantly larger) than those from the nonlinear dynamic procedure in all cases. Assuming once again that nonlinear dynamic analysis is the most accurate analysis type, the above result implies that the dynamic amplification factor of 2 used in the nonlinear static procedure may be excessively conservative for this analysis type. The third significant finding identified by Ruth [2004] in the first stage of this study was that plastic hinges in the nonlinear dynamic procedure did not form at the same locations or in the same order as in the nonlinear static procedure. Design using the nonlinear dynamic procedure tended to result in a greater spreading out of plastic hinges from the location of local member failure. The second stage of the Ruth [2004] study focused on identifying a less conservative dynamic amplification factor that would bring designs completed using nonlinear static procedures in line 21 with those developed using nonlinear dynamic procedures. Eight two-dimensional and three three-dimensional computer models were analyzed in this stage of the study. The models consisted of structural steel moment frames of varying building geometries. The member sizes in the models were determined using the nonlinear dynamic procedure in the G S A Guideline. The design assumptions from Stage One of the study were implemented in Stage Two, with the additional constraint that all members must remain elastic before removal of the failed element but the structure must approach its inelastic progressive collapse capacity after removal of the failed element. For each of the computer models, the nonlinear static analysis procedure from the G S A Guideline was performed for a number of dynamic amplification factors between 1 and 2. The results of these analyses showed the nonlinear static analysis procedure to conservatively approximate the behaviour of the nonlinear dynamic procedure with a dynamic amplification factor of 1.5. None of the parameters investigated (number of bays and storeys, bay and storey dimensions, foundation constraints, strain hardening of plastic hinges) was observed to significantly affect the dynamic amplification factor at which the nonlinear static and dynamic analyses produced the same results [Ruth, 2004]. Nonlinear static analyses utilizing a dynamic amplification of 1.5 did not show as great a spreading of plastic hinges as corresponding nonlinear dynamic analyses. The additional plastic hinges that formed in the dynamic models were not, however, the critical hinges that would fail first under additional loading and bring about progressive collapse failure. Thus, although the nonlinear static models underestimated the number of plastic hinges (and therefore, the extent of structural damage), the models conservatively assessed the peak demands on the structure determined by nonlinear dynamic analyses [Ruth, 2004]. The Ruth [2004] study recommends that the dynamic amplification factor used in the nonlinear static procedures in the G S A and U F C Guidelines be reduced from 2 to 1.5 for steel moment frame structures, and structural steel buildings in general. Implicit in this recommendation is the assumption that nonlinear dynamic analysis provides the most accurate, yet still conservative, representation of the actual behaviour of a structure. The recommendation is also contingent on the structure deforming significantly into the plastic range after localized failure of one element. 22 If a structure does not exhibit substantial nonlinear behaviour, then a higher dynamic amplification factor is necessary to match static nonlinear analysis results to those from nonlinear dynamic analysis; the Ruth [2004] study implies that a dynamic amplification factor of 2.0 may be appropriate for structures behaving elastically. For structures with the ability to deform inelastically, however, the fact that the nonlinear static procedure with dynamic amplification equal to 1.5 underestimates the peak forces on the structure (since the behaviour is not significantly nonlinear) is irrelevant because the structure has reserve inelastic capacity to resist the underestimated peak forces. B y the time that the forces on the structure are large enough to mobilize most of this inelastic reserve, a dynamic amplification factor of 1.5 overestimates the peak forces on the structure, ensuring the nonlinear static procedure is conservative. If, however, a structure is brittle and not able to deform significantly in an inelastic manner, then reducing the dynamic amplification factor to 1.5 may not be conservative since this reduction relies upon inelastic deformation of the structure. Ductile steel structures can deform inelastically, however, so this latter point is rendered mute for the type of structures that Ruth [2004] proposes a dynamic amplification factor reduction for. Although much of the analysis in the Ruth [2004] study was performed using the G S A Guideline, further work on this topic [Ruth et al., 2006] has shown that analysis using the U F C Guideline generates essentially the same results as the G S A Guideline. This additional work also suggests that a reduction in the dynamic amplification factor to at least 1.5 may be conservative for reinforced concrete frame structures [Ruth et al., 2006]. Further analytical work on reinforced concrete and other types of structures is necessary, however, to confirm i f reducing the dynamic amplification for these structures is advisable. Some of the findings in the Ruth [2004] study have been confirmed by other research. Marjanishvili and Agnew [2006] assessed the progressive collapse susceptibility of a sample building by the performing each of the analysis procedures in the G S A Guideline. This study found the linear static procedure to by unconservative in some instances and the nonlinear static procedure to be overly conservative when using a dynamic amplification factor o f 2; these findings match those discerned in the Ruth [2004] study. Tsai et al. [2007] also found the nonlinear static procedure in the G S A Guideline to be very conservative compared to the other analysis methods in the guideline. Harris [2007] investigated dynamic amplification in tall buildings and concluded that dynamic amplification is different for displacements, axial forces 23 and moments in tall buildings, and is variable over the height of the building for moments and forces. The Harris [2007] study found the nonlinear static G S A procedure to generally be conservative, and recommends dynamic amplification factors less than 2 in many locations. A s discussed above, a reasonable amount of analytic research has been conducted on dynamic amplification in frame structures. Experimental testing focused on dynamic amplification is relatively rare, however. Pretlove et al. [1991] performed experimental testing as part of their investigation of the effects of dynamic amplification on the progressive "collapse" of a tension spoke wheel. Sasani and Kropelnicki [2007] conducted a static pull-down test on a concrete beam to determine the behaviour of the beam under large midspan deflections and assess its ability to resist load through catenary action; an analytical model was then calibrated with the experimental data collected and used to investigate dynamic load redistribution. A s of August 2007, however, no published work could be found describing a testing program that experimentally measured the dynamic amplification occurring during gravity load redistribution in a reinforced concrete frame test specimen. 2.3 Research Program Objectives, Scope and Applicability The overall aim of this research program is to address the current lack of experimental testing on dynamic amplification during gravity load redistribution in frame structures, and use the results collected from experimental testing to further evaluate dynamic amplification and propose future test programs on this topic. To achieve this overall aim, three objectives were identified at the outset of this research study. The first objective of the research study is to generate data on the dynamic amplification that occurs during gravity load redistribution after a column axial load failure in a frame structure. The second objective of the research study is to utilize the data collected from experimental testing to confirm the accuracy and consistency of the dynamic amplification factors in the G S A and U F C progressive collapse guidelines, and the analytical studies performed on this topic [Kaewkulchai and Williamson, 2002; Ruth, 2004; Marjanishvili and Agnew, 2006; Ruth et al., 2006]. The third objective of the research study is to evaluate a test procedure for investigating dynamic load redistribution. 24 The experimental scope of this research study consists of a single test on a two-storey reinforced concrete frame structure. In this test, a first-floor column is removed with explosives and the resulting transient and steady-state changes in the axial loads of the surrounding columns are measured. The goal of the test is to generate a set of data that can be used to evaluate the dynamic amplification in the axial force increases of the columns surrounding the removed column. This data w i l l also be used to develop an analytical model of the test specimen that w i l l be used to further evaluate the behaviour of the specimen as a whole and investigate the properties of the structure and test setup that significantly influence the redistribution of gravity loads in the frame structure. The experimental testing conducted in this research program is most directly applicable to the analysis of structures for blast loads. This area of research is also relevant, however, for the progressive collapse of structures subjected to earthquake excitation. Member failures due to earthquakes do not occur as rapidly as blast failures. Consequently, the dynamic effects resulting from member losses under seismic loading are not as severe as those resulting from blast loads. Because of this fact, dynamic amplification relationships developed from blast testing can be used conservatively to generate an upper bound for the dynamic amplification of redistributed gravity loads resulting from column failures during earthquakes. The application of the results of this research program as an upper bound or worst case could prove useful in assessing the susceptibility of structures to progressive collapse under seismic loading. 25 3 EXPERIMENTAL P R O G R A M 3.1 Overview The experimental component of this research study consisted of a single field test on a two-storey reinforced concrete frame structure. The experimental design for the test program consisted of three steps: 1) Determine the axial stiffness of the columns in the existing building selected as the test specimen; 2) Instrument the columns in the test specimen to record dynamic changes in axial deformation; 3) Explosively remove one column and measure the resulting column deformations. Central to the above experimental design is the assumption that the stiffness and deformation data collected in the test can be combined to generate axial force response histories for the instrumented columns. This axial force data can then be analyzed to determine the gravity load redistribution pattern and degree of dynamic amplification resulting from the explosive column removal. However, the assumption that axial stiffness multiplied by axial deformation equals axial force is only valid i f the columns in the test specimen behave in a linear elastic manner. A s w i l l be demonstrated, computer modelling and in-situ column stiffness testing was performed prior to the explosive test to ensure the columns would respond in the linear-elastic range; experimental observations and post-test modelling wi l l be presented that further support the assumption of linear-elastic column behaviour. 3.2 Test Specimen The test specimen for this experiment was an elementary school building scheduled for demolition in Taoyuan, Taiwan. The National Centre for Research on Earthquake Engineering ( N C R E E ) obtained permission to conduct a series of field tests on the school before it was completely demolished and removed. The majority of the building was utilized for three pushover tests [Hwang et al., 2006]. Two classrooms were separated from the rest of the 26 structure for the aforementioned explosive test. Figure 3.1 shows the two classrooms used as explosive test specimen. (b) Back (East Elevation) of the Explosive Test Specimen Figure 3.1 Elevations of the Explosive Test Specimen The school structure was initially a single storey building with masonry walls as the vertical-load-supporting elements. The school was later expanded to two storeys. To achieve this expansion, the masonry wall was removed at certain locations and reinforced concrete (RC) columns were cast against the masonry. Similarly at the tops of the walls, RC beams were cast with their soffits resting on masonry. Where there was no conflict with the original masonry, the RC structure was cast conventionally. Infill masonry walls were installed to partition the second floor level. In preparation for the test, the building was gutted so that only the RC structure remained. This was done to isolate the behaviour of the RC frame structure. Figure 3.2 shows photos of the interior of the test specimen after removal of the masonry walls. Note the interlock between the concrete structure and masonry at the first floor level (Figure 3.2a) where masonry walls were present. In contrast to the conventionally cast concrete members elsewhere at the first and on the second floor level, the interlocked masonry-concrete members have highly variable cross-sections. To make the cross-sectional areas of these members more uniform, cementitious grout was applied to the jagged surfaces of these members, as shown in Figure 3.3. Grouting the members was also deemed necessary to help prevent large chunks from dislodging during the explosive test and to make identification of cracking from test deformations easier. (a) First Floor Level (b) Second Floor Level Figure 3.2 Test Specimen Fol lowing Masonry W a l l Removal - East C o l u m n L ine 28 F i g u r e 3.3 F i r s t F l o o r L e v e l o f Tes t S p e c i m e n F o l l o w i n g G r o u t i n g CD-W B 2 ©-pa CQ W B 2 cc CQ W B 2 o oo o m |163J 314 308 | 309 313 220 Units are in cm F i g u r e 3.4 F i r s t F l o o r S t r u c t u r a l P l a n o f Tes t S p e c i m e n Figure 3.4 shows a structural plan o f the test specimen. The dimensions in Figure 3.4 were taken from measurements made on site. One-way slabs spanned between B l and R B I Beams; the loading on the W B 2 Beams was limited. The B l and R B I Beams were supported by the columns on Grids 1 and 2, and cantilevered out to the west from the columns on Gr id 2. The 29 column at Grid Intersection C-2 at the first floor level was the member selected for explosive removal. The structural plan of the second floor and roof are very similar. Not shown in Figure 3.4, but illustrated in the photos in Figure 3.1, are concrete upstand walls along the tips of the front cantilevers at the roof and second floor, and concrete canopy slabs at the roof level along the rear of the specimen. There were two column types in the test specimen; the cross-sections of these column types are illustrated in Figure 3.5; a north arrow has been included in this figure to express that the long dimensions of the columns were oriented in the east-west direction. CI columns were located at all grid intersections and were continuous from the ground floor to the roof. Floor to floor dimensions were site measured as 3750 mm ground to second floor and 3650 mm second floor to roof. Site measurements also confirmed that the actual cross-sections of the CI columns approximately matched the original design dimensions. At the first floor level, the faces of the CI Column 8-20M Long. 10M Ties (2) 25 N «r O ? — * • 0 • • • • 35 RBI - Roof 4-15M Long. 10M Ties @ 25 24 Units are in cm C2 Column 6-20M Long. 10M T i e s ® 25 B l - Roof 8-22M, 4-20M Long. 10M T i e s ® 10, 25 WB2 - Roof 4-20M Long. 10M T i e s ® 25 B1/RB1 Cant.-Roof 6-22M, 2-20M Long. 10M T i e s ® 10, 25 y, 03 • • v« * 35 Figure 3.5 C o l u m n and Roof Beam Cross-Sections in the Test Specimen 30 columns were grouted to obtain uniform cross-sections close to the design dimensions of 350 mm x 400 mm. The test specimen contained only one C2 column, located half between Grids 1 and 2 on Grid C. This column was removed with the masonry wall at the first floor level but left intact at the second floor level. The beam cross-sections at the roof level are also illustrated in Figure 3.5. The dimensions of the roof beams in the test specimen were observed to match the information on the original design drawings. The beam reinforcing shown in Figure 3.5 is indicative of that called up on the design drawings; the test specimen was only cut in places where W B 2 Beams were present, however, so it was only possible to confirm the reinforcing in this type of beam. The stirrups in the B l Beams and B1/RB1 Cantilever Beams are assumed to be at the closer spacing listed in Figure 3.5 for distances of 2000 mm and 1000 mm from the column supports, respectively. It was noted in the field that ends of the B l Beams were tapered. These tapers were approximately 100 mm deep by 900 mm long. A s well , the B1/RB1 Cantilever Beams at the front of the building were observed to taper from approximately 450 mm deep at the column to 350 mm deep at the tip of the cantilever; the part of the beam extending below the slab was measured as 325 mm at the column and 225mm at the tip o f the cantilever. Figure 3.6 shows photos of the types of tapering beams in the roof structure. (a) Taper at End of B l Beam (b) Tapering B1/RB1 Cantilever Beam F i g u r e 3.6 T a p e r s i n the R o o f B e a m s o f the Tes t S p e c i m e n Whereas the roof structure of the test specimen matched the original design drawings quite closely, many o f the beams in the second floor structure deviated considerably from the design drawings and each other. Site observations suggest that the second floor B l Beams and B1/RB1 Cantilever Beams are the same as at the roof level and in accordance with the design drawings. The observed geometry of the other beams in the second floor structure does not agree with the 31 design drawings. The cross-sections for these beams were determined from the available information and are shown in Figure 3.7. R B I 4-15M Long. 10M T i e s ® 25 W B 2 - G L 1 W B 2 - G L 2 N o Brick 4 - 1 5 M L o n g . 4 - 1 5 M L o n g . 1OM Ties @ 25 1OM Ties @ 25 W B 2 - G L 2 Brick 4-15M Long. 10M T i e s ® 25 LO o ro LO ro LO ro LO 27 35 24 Units are in cm o ro 40 F i g u r e 3.7 S e c o n d F l o o r B e a m C r o s s - S e c t i o n s i n the Tes t S p e c i m e n The original design drawings call for the R B I Beam at the second floor to be 240 mm x 350 mm, the same size as the roof R B I Beam. Observations following demolition suggest a minimum cross-section similar to the specified size and a rebar cage appropriate for that size of beam. After this beam was grouted, however, the cross-section was measured as 270 mm x 450 mm. Whether the entire cross-section of the beam is effective w i l l be discussed later in this report when modelling of the test specimen is addressed. Along Grid 2 (refer to Figure 3.4 for the structural plan of the test specimen), the size of the W B 2 Beam at the second floor depended on whether a masonry wall was present below the beam. Between Grids A and B , and Grids D and E , the exterior cladding was not brick masonry and the W B 2 Beam was measured to have a cross-section 240 mm wide x 350 mm high. The size of these beams suggests their rebar cages are similar to that assumed in the R B I Beam. Elsewhere along Gr id 2, masonry walls were originally present beneath the W B 2 Beams; the beams were measured to have a cross-section 400 mm wide x 450 mm high. A s shown in Figure 3.8 and in the cross section of Figure 3.7, however, only the upper right part of these beams was reinforced (the tube in the left part of the cross-section is an electrical conduit). Similarly, observation of the W B 2 Beams along Grid 1 (all of which topped masonry walls) revealed that the longitudinal reinforcing in these beams was also concentrated in the upper right part of the cross-section. In both cases, the rebar cage was found to be similar to that typical of a 240 mm x 350 mm beam. The effectiveness of these unconventional cross-sections w i l l be discussed, 32 along with that of the second floor R B I Beam, when modelling of the test specimen is discussed later in this report. F i g u r e 3.8 W B 2 B e a m on G r i d 2 A b o v e B r i c k at S e c o n d F l o o r i n Tes t S p e c i m e n The suspended slabs at the second floor and roof of the test specimen were very similar. These slabs were observed to be 125 mm thick and reinforced with 10M reinforcing bars spaced at approximately 250 mm. The slab in the cantilever portion of the second floor increases from a thickness of approximately 150 mm at Grid 2 to 100 mm at the tip of the cantilever; elsewhere the slab thickness is a uniform 125 mm. The slab at the second floor is covered with 25 mm of F i g u r e 3.9 S l a b a n d F l o o r i n g at the S e c o n d F l o o r o f the Tes t S p e c i m e n 33 ceramic tiling and concrete topping. A photograph of the cross-section of the second floor slab and flooring is shown in Figure 3.9. The ground floor of the test specimen consisted of concrete slab-on-grade approximately 125mm in thickness. This slab-on-grade was not sampled to ascertain its level of reinforcing. The masonry walls in the test specimen were supported on reinforced concrete strip footings and the concrete columns were supported on reinforced concrete spread footings. The depth of these footings and the degree of fixity in the column-to-footing connections were not explicitly determined on site. Observations of other pushover tests done on the field site suggest that significant moment restraint was provided at the column to foundation interface. 3.3 M a t e r i a l P r o p e r t i e s A preliminary assessment report was completed for the 2006 N C R E E Field Testing Program [Chen, 2004], and as part of this assessment, material tests were performed on concrete and reinforcing steel from the school building. In the preliminary assessment report, concrete compressive strength,/!', was found to be 17 M P a at the first level and 22 M P a at the second level; the mean yield strength of the reinforcing steel was 274 M P a . These material strengths were used to construct a preliminary model of the test specimen prior to testing, as w i l l be discussed in Section 3.4. Material testing of samples of concrete and reinforcing steel from the test building was also performed as part of the 2006 Field Testing Program [Huang, 2007]. In this round of testing, the average concrete compressive strength was determined to be 14.3 M P a , whereas the average reinforcing steel yield strength was found to be 326 M P a for longitudinal steel and 434 M P a for transverse steel. The material strengths obtained as part of the Field Testing Program were used in post-modelling of the test specimen, as w i l l be discussed in Section 5.5. The soil beneath the school building consisted of a 6 m layer of stiff, red-brown clay situated on top o f hard, clayey gravel [Chen, 2005]. The foundations o f the test specimen were founded in the stiff clay layer. 34 3.4 Preliminary Analytical Model Util iz ing the material property information in the preliminary assessment report and the geometric information from the original design drawings and site reconnaissance, a preliminary analytical model of the test specimen was constructed in S A P 2000 (Educational Version 9.0.5) [CSI, 2004]. This model is shown in Figure 3.10; further details on the preliminary model can be found in Appendix A . In this model, the structural action of the slabs was neglected and the beams in the specimen were assumed to crack, reducing their stiffness. A linear static analysis was performed on this model with the exploded column intact to estimate the member loads before the explosive test. A linear dynamic analysis, considering the instantaneous removal of the exploded column, was then conducted on the model to simulate the changes in member forces due to removal of the designated column in the test specimen. Figure 3.10 Pre l iminary Ana ly t ica l M o d e l of Test Specimen The preliminary model of the test specimen was utilized for a number of purposes. First, the peak axial compressive stresses in the columns of the test specimen were evaluated and compared to the stress limit for linear behaviour in concrete columns of 0.6 fc'. The peak axial compressive stress obtained in the preliminary model was 0.18 fc', considerably less than the 35 linear limit. This result supported the hypothesis that the columns in the specimen would behave linearly during the test, and that collected axial deformation data could be combined with the axial stiffness calculated for members to generate axial force response histories. The preliminary model of the test specimen was also used to assess the type of instrumentation that would be required to effectively collect axial column deformation data during testing. Deformations from the model were used to determine the precision and range required of the instruments. The periods corresponding to the predominant modes of vibration in the model were used to identify required instrument reporting speeds and set data acquisition filters for removing high frequency noise. The instrumentation of the test specimen w i l l be discussed more thoroughly in Section 3.6. A third use of the preliminary model was in designing the procedure for determining in-situ the stiffness o f the columns o f the test specimen. A s wi l l be described in more detail in Section 3.5, an attempt was made to verify the stiffness of each of the columns in the specimen by reducing the compressive load on each column (i.e. applying an upward force) and measuring the resulting deformation. The estimated compressive forces on the columns from the preliminary model were used to set an upper bound on the magnitude of upward force applied during column stiffness testing, with the intent of preventing the columns from going into tension. The estimated axial stiffnesses from the preliminary model were also used to assess the precision of the instruments that must be used to collect deformation data during column stiffness testing. The last task completed with the help of the preliminary model was to assess the ability of. the beams surrounding the exploded column to redistribute the load carried by the column before its removal. The peak unfactored demands on the structure were estimated from a linear dynamic analysis results generated by the preliminary model. Unfactored beam capacities were then calculated according to the Canadian Concrete Design Standard [CSA, 2004] for the R B I Beams at Grid C and the W B 2 Beams along Grid 2 between Grids B and D at both the second floor and roof. The peak dynamic demands computed by the preliminary S A P model, which did not consider slab participation as a shell or membrane element, were found to be considerably larger than the capacities of the beams in question; in fact, even the steady-state demands following column removal were larger than the beam capacities. Further modelling in E T A B S [CSI, 2005], considering the slab as a mesh of shell elements, yielded lower demands than the preliminary S A P model and suggested that the structure would at a minimum be able to support 36 the steady-state forces on the system after column removal. It was felt that inelastic beam behaviour, something not considered in the preliminary analysis models, would be sufficient in helping the structure dissipate the peak dynamic demands on the test specimen. Since the preliminary model predicted the beams connecting to the removed column would respond in the nonlinear range, it was expected that these elements would undergo significant cracking during the explosive test. The preliminary model of the test specimen was not intended to perfectly match the behaviour of the structure during the explosive test. Its primary purpose was to assist with the design of the explosive test. Because of time constraints and uncertainties, the preliminary model was conservative and simplistic. Thus, after the test, the preliminary model was abandoned. Post-test modelling efforts were instead focused on the construction of models with software better suited for the dynamic loading and response of the test specimen. Post-test modelling wi l l be discussed further in Section 5.5. 3.5 I n - S i t u C o l u m n S t i f f n e s s T e s t i n g For a typical column of uniform cross-section, axial force can be related to axial deformation by the equation: P = •A = k-A (3.1) V L j Where: P - A x i a l load on column E = Elastic modulus of column material A = Cross-sectional area of column L = Length of column A = Displacement over the column length E-A k = = A x i a l stiffness of column A s described in Section 3.2, however, the columns at the first level of the test specimen were cast directly into voids chipped into the original masonry walls, and the cross-sections of these columns are variable over their heights. Consequently, it was decided that an attempt be made to 37 measure the stiffness of the columns in-situ and confirm that the EA assumed for the columns was correct. The procedure used to determine in-situ column stiffness consisted of applying an upward force to columns at the underside of the second floor and measuring the column deformations resulting from this force. Upward force (P) was applied on the columns in increments, and the deformation (A) at each increment was recorded. The above procedure enabled force versus deformation (P-A) plots to be generated for each column. The purpose of generating P-A plots was twofold: first, to identify i f the columns deformed in a linear-elastic manner (as would be evidenced by a linear P-A relationship); and second, i f linear-elastic relationships were observed, to calculate the axial stiffness of the columns (the slope of the P-A plot). The upward force for in-situ column stiffness testing was generated using hand jacks located immediately adjacent to the column. The jacks were stationed on top of load cells so that the amount of force applied to a column could be accurately monitored. Column deformations were measured using high precision linear potentiometers. The upward force applied to the columns was kept less than the downward force being supported by the columns, thereby ensuring the columns were always in compression. A s mentioned in Section 3.4, the preliminary analytical model of the test specimen was used to estimate the dead load compressive forces in the columns (establishing the maximum upward force that could be applied) and the deformations corresponding to the applied upward forces (indicating the level of precision required by the deformation measurement instruments). Figure 3.11 shows a photo of the setup utilized for in-situ column stiffness testing. The results of the in-situ column stiffness testing are presented in Appendix B . Unfortunately, several problems were encountered during the course of this testing. First, it was difficult to maintain a constant upward load on the columns with the hand jacks utilized. When jacking was stopped to take force and deformation readings, the force and deformation values on the data acquisition systems would start to drop quickly. This introduced some uncertainty in the data. A second problem encountered during column stiffness testing was the high stiffness of the columns themselves. Because the columns were so stiff, it took a great deal of force to deform the columns by just 0.01 mm, the precision of linear potentiometers used. Accordingly, errors in 38 F i g u r e 3.11 I n - S i t u A x i a l St i f fness T e s t i n g on a C o l u m n in the Tes t S p e c i m e n reading the deformation measurements, as well as errors generated by the linear potentiometers themselves, had the potential to greatly impact the accuracy of the data recorded. A third problem experienced during in-situ testing of column stiffness was the presence of discrepancies between deformation values at zero force before and after testing. In many cases, the deformation reading after the upward force on a column had been released did not match the deformation reading before testing was commenced. Considering that the dead load forces on the columns were never exceeded during testing, the fact that deformation values at zero force did not "close" calls into question the accuracy of the data collected. The fourth problem arising from the in-situ column stiffness testing program is a result of the test design itself. The test procedure utilized assumed that the deformation of the beams between the instrument location or point of loading and the face of the column (approximately 150 mm) could be ignored and 100% of the upward force from the jacks was transferred to the closest column. Further analytical modelling of the test specimen suggests that neither of these assumptions is explicitly true and the error in these assumptions may be significant. Furthermore, since beams framed into the columns along Gr id 1 on only three sides, linear potentiometers were not placed on opposite sides of these columns (since there was no beam to mount an instrument under on the exterior side of the column). Consequently, any rotation of the beam-column joints along this gridline was not captured, with the result that the column 39 deformations along Grid 1 appear to have been underestimated (referring to Appendix B , the in-situ column stiffnesses calculated along Grid 1 are consistently higher than those along Grid 2). Because of the problems described above, the results obtained from in-situ stiffness testing of the columns in the test specimen were deemed to be unreliable. However, the results were still useful in qualitatively evaluating the general assumptions made to estimate the column stiffnesses for the analytical models. In general, the force-deformation plots generated from the data collected during stiffness testing suggest linear behaviour (refer to Figure 3.12, the P-A plot for Column B-2). Despite the similarity in the shapes of the P-A plots, the stiffness values calculated from the test data vary greatly, from a minimum of 341 kN /mm to a maximum of 617 kN /mm; considering the columns are supposed to all be identical, this wide range of measured stiffness values is suspect. Following stiffness testing, it was thus concluded that assuming linear column axial deformation behaviour was justified, but analytical means for estimating column stiffness (k = EAIL) should be used in lieu of measured values of column stiffness. 0 • 0 0.1 0.4 0.5 0,2 0.3 Deformation (mm) Figure 3.12 Axial Force-Deformation Plot of Column B-2 from In-Situ Stiffness Testing 40 3.6 Instrumentation 3.6.1 Challenges, Simplifying Factors and Redundancy For the explosive test program to succeed, it would be necessary to effectively record the small and dynamic changes in axial deformation resulting from removal of the designated column in the test specimen. Equipment availability and the nature of the explosive test presented a number of challenges to collecting a useful set of data. Thankfully, the layout of the test specimen and typical characteristics of load redistribution helped simplify the instrument layout for the explosive test. The dynamic data acquisition system designated for use on the explosive test could record information from a maximum of 16 channels. Thus, a maximum of 16 instruments could be used to collect dynamic information on the behaviour of the test specimen during the explosive test. Due to this channel limitation on the dynamic data acquisition system, it was not possible to record the dynamic movements o f the test specimen by simply putting instruments at every location of interest. A second instrumentation challenge in the explosive test was also encountered during in-situ stiffness testing. Because of the large axial stiffness of the columns in the test specimen, the column deformations expected during the explosive test were quite small. From preliminary modelling, it was deemed necessary for the instruments used in the explosive test to be able to accurately record column deformations of 0.01 mm. The level of precision necessary for the explosive test is uncharacteristically high and would require careful selection of instruments and extensive calibration efforts. The third instrumentation challenge on the explosive test was the fact that dynamic data must be collected, requiring instruments to have sufficient reporting speed and the ability to record oscillations at the frequencies of the test specimen. Some instruments are not able to collect dynamic data because of insufficient reporting speed or because the natural frequency of the instrument is in the range of the oscillations that must be recorded, thereby causing resonance problems. Simplifying the instrumentation o f the test specimen were two factors: the symmetry o f the specimen about Grid C (refer to Figure 3.4) and the tendency of load redistribution from column 41 failures to concentrate in the columns immediately adjacent to the failed column. Because the test specimen was essentially symmetric about Grid C (the column to be explosively removed being at Grid Intersection C-2), it was assumed that forces redistributed to elements on one side of this line of symmetry would match their mirror elements. This assumption reduced the number of dynamic data collection instruments that were needed by concentrating them on one side of the line of symmetry. Symmetry was used most extensively at locations where load redistribution was expected to be minimal and data collection was therefore less critical. To verify that load distribution did in fact occur symmetrically, a number of measures were taken; these measures wi l l be explained later in this section. Research done on load redistribution [Sucuoglu et al., 1994] suggests that failure of a column element primarily affects the two-dimensional frames intersecting at the column of interest, in particular, the columns directly next to the failed element. The preliminary modelling performed on the test specimen confirms this finding. The preliminary model predicted that the vast majority o f the force originally carried by the column to be removed (Column C-2 on Figure 3.4) would redistribute to the columns at B-2 and D-2 (the columns closest to Column C-2) and to a lesser extent to Column C - l (this column is adjacent to Column C-2 but further away than Columns B-2 and D-2, and would be the only column remaining in this frame, further reducing the amount of force redistributed to Column C - l ) . Knowing that most of the load redistribution in the specimen would be concentrated at Columns B-2 and D-2, it was decided that dynamic data collection instrumentation would be concentrated at these locations and lower levels of such instrumentation installed at other locations according to how much load redistribution was anticipated. In addition to the high precision and dynamic data collection challenges discussed above, there was only the opportunity for one explosive test as part of this experimental program. Thus, it was imperative that redundancy be incorporated into the instrumentation scheme for the explosive test to ensure confidence in the data collected. Since this type of test had not been conducted before, it was also felt that different types of data collection instruments should be used, in case one type of instrument turned out to perform poorly during the explosive test. To achieve added redundancy without using up valuable dynamic data acquisition channels, a separate static data acquisition system was added to run in tandem with the dynamic data acquisition. The static system would not be capable of verifying the deformation changes 42 recorded by the dynamic instruments during the transient stage following column removal, but would be well suited for validating steady-state column deformations. Correlation between dynamic and static instruments in the steady-state would thus improve confidence that the entire response history recorded by the dynamic instruments is accurate. In addition to increasing redundancy, static instruments were used to help confirm the assumption that the test specimen would respond symmetrically to removal of the column at C-2. Scarcity of channels was not a problem with the static data acquisition system, so it was possible to mount static instruments on columns without dynamic instruments located symmetrically opposite to columns with dynamic instruments. Comparing the steady-state readings from symmetric columns would validate whether the long-term deformations of the columns match, and i f so, improve confidence that the transient behaviour of the columns correspond. 3.6.2 Instrumentation Equipment Three types of displacement gauges were used on the explosive test: linear potentiometers, PI displacement transducers and dial gauges. The linear potentiometers were used as the primary source of dynamic data collection. The PI displacement transducers were installed to collect secondary dynamic data, while the dial gauges were used in a static data collection capacity to verify the steady-state column deformations recorded by the linear potentiometers. A n accelerometer was also installed on the test specimen above Column C-2 to provide some insight into the inertial forces at this location after explosive removal of the column. The linear potentiometers used in the explosive test were Novotechnik T R Series models [Novotechnik, 2006]. Models with 30 mm, 55 mm and 105 mm of range were used for the explosive test, with higher range instruments being located where larger displacements were expected. Specifications for these instruments indicate that they have a minimum accurate reporting speed of 10-18 Hz , which is greater than the 5.09 H z fundamental vertical mode of the test specimen from the preliminary analytical model. The linear potentiometers are likely capable of accurate reporting speeds well in excess of the value specified, but even at their lower bound reporting speed have the ability to capture the predicted fundamental vertical oscillations of the test specimen. Calibration of the linear potentiometers indicated that the instruments have a precision of better than 0.01 mm. It was deemed necessary that the instrumentation for the explosive test be 43 capable of accurately measuring column deformations of 0.01 mm. On the explosive test, the potentiometers were attached to 3 m high instrument stands and mounted at the underside of the second floor beams, thereby making the gauge length of the instruments the entire unsupported height of the column at the first level. Thus, the displacements recorded by the linear potentiometers were the deformations of the columns in the test specimen at the first level. This being the case, the linear potentiometers, with their better than 0.01 mm precision, were suitable for recording the column deformations during the explosive test to the desired level of accuracy. One of the linear potentiometers installed on the explosive test is shown in Figure 3.13a. (a) Linear Potentiometer with Angle Base (b) Dial Gauge with Magnetic Base Figure 3.13 L inea r Potentiometers and D i a l Gauges Used on the Explosive Test The dial gauges used in the explosive test were D D P - 5 0 A displacement transducers by Tokyo Sokki Kenkyujo Co., Ltd. [ T M L , 2006a]. One of these dial gauges is shown in Figure 3.13b. The range of the dial gauges used for the test is 50 mm. The maximum reporting speed for the dial gauges is 1 Hz , less than the predicted fundamental vertical mode of the test specimen of 5.09 Hz , and thus, the dial gauges were only used to measure static, steady-state changes in column deformation. The dial gauges are capable of a precision of 0.01 mm and were mounted at the underside of the second floor beams, similar to the linear potentiometers. Accordingly, the displacements measured by the dial gauges were column deformations and the instruments had the precision necessary to meet the required level of accuracy for the explosive test. 44 The linear potentiometers and dial gauges used on the explosive test were suspended from galvanized steel instrument stands constructed from C60x30xl Ox 1.6mm cold-formed channel sections. A photo of one of the support frames is shown in Figure 3.14a. The instrument stands were levelled with wooden shims and fastened snugly to the concrete slab-on-grade with expansion anchors. The linear potentiometers and dial gauges were attached to the instrument stands either using steel angles and C-clamps (as shown in Figure 3.13a) or magnetic bases (as shown in Figure 3.13b); the angle and magnetic bases were duct taped to the instrument stands to prevent the instruments from falling if the instruments stands were disturbed before or after testing. At locations where both linear potentiometers and dial gauges were installed, which was most locations, each instrument had its own angle/magnetic base and both bases were attached to the same instrument stand, as shown in Figure 3.14b. (a) Steel Instrument Stand (b) Linear Potentiometer and Dial Gauge at Same Location F i g u r e 3.14 I n s t r u m e n t S t a n d a n d M u l t i p l e I n s t r u m e n t M o u n t i n g 45 The model of PI displacement transducers used for the explosive test was the PI-5-200 by Tokyo Sokki Kenkyujo Co., Ltd. [ T M L , 2006b]. Figure 3.15a shows one of the PI transducers installed on the test specimen. The range of the PI-5-200 is 5 mm. Since the PI transducer is a strain gauge-based instrument and the response speed of strain gauges is very high, the response speed of PI transducers is also quite high and limited practically by the data acquisition system, rather than the instrument itself. Thus, reporting speed was not a concern for the PI transducers. O f more concern from a dynamic data collection viewpoint was the physical arrangement and mounting of the PI transducers and the possibility that the natural frequencies of the instruments might coincide with the some of the dominant vertical vibrational frequencies of the test specimen, negatively influencing the data collection abilities of the PI transducers. Testing was performed as part of the calibration of the PI transducers to identify the natural frequencies of the instruments and check for resonance with the predicted vertical vibrational modes of the test specimen. This testing suggested that resonance issues between the test specimen and PI transducers could be a problem, but in the absence of more accurate analytical modelling (which would provide better confidence as to what the significant vibrational modes o f the test specimen were), it was still considered appropriate to install the PI transducers as backup dynamic data collection instruments. Unlike the other displacement measuring instruments, the PI displacement transducers were not mounted on instruments stands, but rather connected to threaded rods embedded in the columns with epoxy adhesive at mid-height. The rods were installed to give the PI-5-200 transducers a 200 mm gauge length, much less than the full column height gauge length (approximately 3.15 m) of the linear potentiometers and dial gauges. To compare the data collected by the PI transducers to that of the other instruments, the data of the former would need to be adjusted during analysis by the of ratio the dial gauge/linear potentiometer gauge length to that of the PI transducers (it should be noted that implicit to this data manipulation is the assumption that the deformation in the 200 mm gauge length of the PI transducer is representative of the average deformation over the entire height of the column). Because of the need to later manipulate the data from the PI transducer and maintain deformation data accurate to 0.01 mm, the PI transducers had to have a precision of 0.0006 mm. Uti l iz ing an 11-point smoothing/averaging scheme to reduce random signal noise, the PI transducers were found during calibration testing to be capable of achieving this level of precision, thereby making them suitable for use on the 46 explosive test. The fact that averaging was needed to achieve the required level of precision, however, suggested that some form of post-test data processing or filtering (perhaps, something as simple as the 11 -point smoothing scheme) would likely be needed for the PI transducer data to be useful. (a) PI Displacement Transducer (b) Accelerometer F i g u r e 3.15 P I T r a n s d u c e r a n d A c c e l e r o m e t e r U s e d o n the E x p l o s i v e Tes t A Model 3028 Accelerometer by M S I Sensors was used in the explosive test [MSI Sensors, 2006]. This accelerometer is shown in Figure 3.15b. The 3028 Accelerometer chosen has a range of ± 10 g (the preliminary analytical model predicted a maximum acceleration of 3.41 g) and can record frequencies between 0 and 400 Hz , encompassing all the vertical modal frequencies for the test specimen. The dynamic data acquisition system used for the explosive test was manufactured by National Instruments (NI) [NI, 2006]. The system consisted of a N I SCXI-1000 chassis, N I SCXI-1600 digitizer, two N I SCXI-1520 modules (8 channels in each), and a Del l Inspiron 6000 notebook, which served as the data logger. The sampling rate for the dynamic data acquisition system was 1000 H z for all channels. Analog filters were set at 100 Hz for displacement instruments and 47 1000 H z for the accelerometer to reduce the amount of high frequency noise in the signals received from the various instruments. The static data acquisition system used to record the readings of the dial gauges during the explosive test was a TDS-302 Data Logger by Tokyo Sokki Kenkyujo Co. , Ltd. [ T M L , 2006c]. The sampling rate of the static data acquisition system was set at 1 Hz . The dial gauge displacements recorded during the transient motion of the test specimen were disregarded. Only the data collected by the static acquisition system before column removal and after transient vibration of the test specimen dissipated out was useful for comparison with that recorded by the dynamic data collection instruments. Two real-time digital video recorders were set up outside the front (west face) of the test specimen. One video recorder was zoomed out to capture the overall movement of the test specimen during the explosive test. The second video recorder was zoomed in to focus on the top of Column C-2, with the intention of capturing the movement of the structure directly above the removed column. 3 .6 .3 I n s t r u m e n t a t i o n L a y o u t To capture the behaviour of the test specimen during the explosive test, five different levels of instrumentation were installed on the structure. Figure 3.16 shows a plan of where the different levels of instrumentation were installed. The devices making up the various levels of instrumentation wi l l be described below this figure. Instrumentation Level 1 was the highest level of instrumentation and consisted of two linear potentiometers, two PI displacement transducers and two dial gauges. One of each type of instrument was mounted on the north face of the column and the other on the south face to account for any bending in the column or rotation of the beam-column joint. The columns at B-2 and D-2 received Instrumentation Level 1 because these columns were predicted to receive the majority of the load redistributed due to the failure of Column C-2. Instrumentation Level 2 was the next highest level of instrumentation and consisted of one linear potentiometer, one PI displacement transducer and one dial gauge. The locations that received this level of instrumentation were expected to be affected less by the failure of Column C-2 than Columns B-2 and D-2, but still to a degree significant enough to warrant two dynamic data 48 N-0-(A) (B) (C) (5) (E) — —I f 3 2 4 2 • 5 1 4 F i g u r e 3.16 L e v e l s o f I n s t r u m e n t a t i o n o n the Tes t S p e c i m e n - P l a n V i e w co l lec t ion devices. A t C o l u m n A - 2 , the instruments were mounted on the east face o f the c o l u m n , the side thought to be effected least by any bend ing or jo in t rotat ion dur ing load redistr ibut ion. O n C o l u m n C - l , the instruments were mounted on the west face o f the c o l u m n because it was not possib le to instal l the instrument stand in such a way as to permit dev ice instal lat ion on any o f the other sides o f the co lumn . Instrumentation L e v e l 3 was the same as Instrumentation L e v e l 2 except no PI d isp lacement transducer was inc luded. Th is leve l o f instrumentat ion was insta l led on C o l u m n B - l . C o l u m n B - l was expected to receive very l i tt le redistr ibuted load , but it was deemed prudent to inc lude a l inear potentiometer at this locat ion to con f i rm that the c o l u m n was not s ign i f icant ly af fected dur ing the dynamic/ t ransient phase o f load redistr ibut ion. L i k e C o l u m n C - l , the instrumentat ion on C o l u m n B - l was mounted on the west face o f the c o l u m n because o f l imi tat ions on where the instrument stand cou ld be located. Instrumentation L e v e l 4 was the lowest leve l o f instrumentat ion and consisted o f just one d ia l gauge. Th i s leve l o f instrumentat ion was instal led on the west faces o f C o l u m n D-1 and C o l u m n E - 2 ; for explanat ions on w h y the instruments were instal led on the west c o l u m n face, refer to the d iscuss ion for C o l u m n s C - l and A - 2 , respect ively. C o l u m n s D-1 and E -2 were not expected to part icipate s ign i f icant ly i n load redistr ibut ion. D i a l gauges were instal led at these locat ions to con f i rm that the steady-state deformat ions at these co lumns matched those at C o l u m n s B - l and 49 A - 2 , thereby adding confidence that the test specimen responded symmetrically to the removal of the column at C-2. N o instruments were installed at Columns A - l and E - l because preliminary modeling suggested that these columns would be relatively unaffected by the removal of Column C-2. Instrumentation Level 5 refers to the devices installed near the column to be removed by explosives at C-2. A s mentioned previously, an accelerometer was installed on top of the second floor slab above Column C-2 to help determine the inertial forces at this location. Linear potentiometers were installed at the underside of the second floor slab east and west of Column C-2 to estimate the displacement of the floor structure after removal of the column. The potentiometers were installed approximately 1 m away from Column C-2 behind the explosive shielding around this column (explosive shielding w i l l be discussed in Section 3.7). A summary of the instrumentation layout for the test specimen is provided in the elevations of Figure 3.17 and Figure 3.18. Figure 3.17 shows the instrumentation along Gr id 1, while Figure 3.18 shows the instrumentation installed on Grid 2. Refer to Appendix C for measurements of where each instrument was installed relative to the center of the nearest column. ( A ) (5) ( C ) (O) (H) 1 : : 1 ; 1 1 1 1 1 1 2 1 1 1 4 1 i i i i i | 1 1 A I 1 0 1 1 <£Z2 Dial Gauge<$£Sl Linear Potentiometer & Dial Gauge <™33 PI Displacement Transducer Figure 3.17 Elevation Showing Instrumentation Installed Along Grid 1 50 (A) (B) (c) (B) © 1 • • • | I 2 i 1 1 j m ! 4 1 1 1 1 1 1 1 1 r f r 1 i j • 'I . Column 11 /f Being | • 1 Removed 1 a [l 0 1 1 =1 Linear Potentiometer <^Z2 Dial Gauge <3§2S1 Linear Potentiometer & Dial Gauge <J33 PI Displacement Transducer <0-O Accelerometer Figure 3.18 Elevation Showing Instrumentation Installed Along Grid 2 3.7 Test Preparation - Explosives There were a number of steps involved in preparing the specimen to receive the explosives that would be used to suddenly remove Column C-2. To ensure the steel reinforcing in Column C-2 would not remain continuous after the detonation of explosives and potentially support some axial load, this reinforcing was cut prior to performing the explosive test. Four sawcuts approximately 100 mm deep were made on the column, one on each column face. These sawcuts are shown in Figure 3.19a. Sawcutting effectively sliced the bars in the column while only very slightly affecting the stiffness of the column. It was important to not alter the stiffness of Column C-2 significantly so as to ensure negligible redistribution of load away from the column prior to the explosive test. Also shown in Figure 3.19a are the six small bore holes drilled into Column C-2 for placement of explosives. These holes were drilled to the mid-depth of the column and vertically spaced at approximately 300 mm. The size of the bore holes were minimized to ensure the impact on column stiffness was very slight. Into each of the bore holes was installed 20 g of C4 explosive; 51 (a) Column C-2 Sawcuts and Boreholes (b) Installation of Explosives and Detonation Lines F i g u r e 3.19 P r e p a r a t i o n o f C o l u m n C - 2 f o r E x p l o s i v e I ns ta l l a t i on 120 g of explosive was installed in the column in total. The 20 g C4 explosive charges installed in the column were not all detonated at the same time. The firing sequence used to destroy the column saw the explosives in the outer 2 bore holes detonated first, followed by the charges adjacent to the outer bore holes, and finally the charges in the center bore holes. The time between detonation of the charges was very short, certainly less than 0.1 s. Figure 3.19b shows the installation of the explosives into the bore holes in the columns and the connection of the explosives to their respective detonation lines. To protect the surrounding structure and instrumentation from the explosion and resulting debris, and explosive shield was constructed around Column C-2. This explosive shield was constructed of wood and steel sheet panels. In addition, foam padding was installed on the second layer of the explosive shield to reduce the sound emitted by the explosion. A photograph of the explosive shield is shown in Figure 3.20a. 52 (a) Explosive Shield Around Column C-2 (b) Steel Safety Frame Figure 3.20 Explosive Shield and Safety Frame Used on the Explosive Test To protect against the complete collapse of the test specimen in the event that the structure around Column C-2 was not capable of resisting the loads redistributing from the column following its removal, structural steel safety frames were installed north and south of Column C-2. These frames are visible in Figure 3.20a; a close-up of one of the frames is pictured in Figure 3.20b. It was deemed necessary to install safety frames, thereby preventing the chance of progressive collapse, because of safety concerns and to protect the instrumentation installed on the test specimen from damage. The tops of the safety frames were 100 mm below the underside of the structure and did not come into contact with the specimen during the explosive test. 53 4 EXPERIMENTAL RESULTS 4.1 Qualitative Results In the explosive test, the load-carrying capacity of Column C-2 was effectively eliminated when the explosives installed in the column were detonated. A s shown in Figure 4.1a, the explosion disintegrated the concrete in the bottom half of Column C-2. Figure 4.1b shows the effectiveness o f the sawcuts made on the column in breaking the continuity of the longitudinal reinforcement. (a) Column C-2 After the Explosive Test (b) Close-Up of Post-Test Column C-2 Figure 4.1 Explosive Shield and Safety Frame Used on the Explosive Test 54 The response of the test specimen during the explosive test was considerably stiffer than anticipated. Very little cracking was observed in the structure surrounding Column C-2 after the test, suggesting that the test specimen behaved linearly during load redistribution. The lack of significant cracks and inelastic behaviour was surprising considering that preliminary modelling predicted extensive cracking and nonlinear deformations in the beams connected to Column C-2. The poor predictive capabilities of the preliminary model are an indication of the crudeness of this model. 4.2 Quantitative Results During the explosive test, measurements were recorded from the linear potentiometers and PI displacement transducers for 26.337 s. The first 4.310 s of this data is from before the detonation of explosives in Column C-2. The time at which initial detonation occurred (time zero) was defined as the time when the accelerometer above Column C-2 first deviated from its pre-test steady-state. Figure 4.2 shows the complete displacement response history recorded by the linear potentiometer just north of Column B-2. Note the negligible displacement before time zero and after 10 s, and the concentration of oscillatory displacement in the first 4 s after initial detonation. Response history plots for all the instruments are included in Appendix D . 0.3 T 0.2 ---0.5 -I 1 1 1 i 1 -5 0 5 10 15 20 Time (s) Figure 4.2 Displacement Response History - Linear Potentiometer at Column B-2 (N) Note: Upward displacement is positive 55 4.2.1 Recorded Data - Anomalies and Corrections Review of the video of the explosive test revealed that the instrument stand supporting the linear potentiometer east of the Column C-2 was disturbed during the course of the test. Consequently, the data collected by this instrument wi l l be neglected. Anomalies were also noted in the data recorded by the linear potentiometers on both sides of Column D-2, the PI transducer on the north face of Column D-2, and the accelerometer above Column C-2. H o w these anomalies were addressed varies from instrument to instrument, and wi l l be addressed below. For the accelerometer above Column C-2, the data acquisition system was setup to record accelerations over a range of ± 10 g, the published range of the accelerometer. From approximately 0.11 s to 0.16 s after initial detonation, the accelerometer experienced accelerations greater than ± 10 g. Due to the limits set in the data acquisition system, however, the maximum recorded readings during this time period were ± 10 g, not the actual accelerations at the location of the accelerometer. Thus, the magnitudes of the accelerations recorded from 0.11 s to 0.16 s w i l l be not be considered during analysis of the acceleration data. For the linear potentiometer and PI transducer on the north face of Column D-2, the anomalies in the data appear to be caused by electrical surges rather than physical displacements of the test specimen. Plots showing the anomalies in the data from these instruments are shown in Figure 4.3a for the linear potentiometer and Figure 4.3b for the PI transducer. In both cases, the response recorded by the instruments starts out as a regular oscillatory pattern, then the pattern is interrupted by a spike or series of spikes, and then the response returns to the regular pattern. The irregularity of the spikes and the fact that the spikes don't appear to influence the recorded response that follow them make it unlikely that the spikes represent real deformations in the test specimen; momentary malfunctions in the signals from the instruments is a much more plausible explanation for the spikes in the response data. For the linear potentiometer on the north face of Column D-2, the spike in the data only lasts from about 0.72 - 0.75 s. Thus, it is possible to approximate with reasonable accuracy the actual deformation of the test specimen during the spike using linear interpolation. Figure 4.3a shows what the data from the linear potentiometer looks like with the spike replaced by a linearly interpolated section. 56 1 T T i m e (s) (a) Data Recorded by the Linear Potentiometer on the North Face of Column D-2 0.04 T n i l -0.06 -i 1 1 1 I J , - H - ' 1 ' r 1 1 1 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 T i m e (s) (b) Data Recorded by the PI Transducer on the North Face of Column D-2 F i g u r e 4.3 A n o m a l i e s a n d C o r r e c t i o n s i n I n s t r u m e n t D a t a Note: Upward displacement is positive For the PI transducer on the north face of Column D-2, the spikes in the data last from approximately 0.35 - 0.47 s. This is too long a time period to attempt to interpolate over, since it spans more than one principal oscillation. A s a result, the data in the spiked region for the PI transducer (shown in Figure 4.3b) w i l l be neglected during data analysis. 57 The anomaly in the data for the linear potentiometer on the south face of Column D-2 is considerably more complex than the inconsistencies discussed above. Significant analysis of the transient and steady-state data for a number of instruments was required to postulate a possible reason for the peculiar response history of the linear potentiometer south of Column D-2. The procedure followed to correct the linear potentiometer data at Column C-2 (S) is detailed in Appendix E . Essentially, it was concluded that some of the plaster against which the linear potentiometer was installed flaked off during the explosive test causing an artificial, one-time jump in the recorded deformation. In Appendix E , the data after the artificial jump is shifted back down by the estimated magnitude of the jump to yield an approximate response history at the instrument location. 4.2.2 Steady-State Deformat ions The first step taken to evaluate the accuracy and consistency of the data collected during the explosive test was to calculate and compare the steady-state deformations recorded by each of the instruments. Determining steady-state deformations for the dial gauges was a simple matter of subtracting the displacement reading before initial detonation from the displacement reading several seconds after initial detonation when the post-test steady-state had been achieved. From 7 s after initial detonation onward, very little fluctuation, was observed in the dial gauge data (which was recorded to 0.01 mm), suggesting not only that a steady-state had been achieved at this point, but that the signal noise for the dial gauges was not significant enough to affect the minimum precision of 0.01 mm required for the explosive test. The procedure used to determine steady-state deformations for the linear potentiometers and PI transducers was similar to that used for the dial gauges, but slightly more complex due to the fact that dynamic data was recorded rather than the static variety. The pre-test steady-state readings for the linear potentiometers and PI transducers were determined by taking the average of the 4 s of data prior to the time of initial detonation. The maximum deviation from the average value was also calculated to get a measure of the signal noise before column removal. The maximum deviation from average in the data before the explosive test was less than 0.005 mm for the linear potentiometers and 0.0003 mm or less for the PI transducers, after applying to the latter the 11-point smoothing scheme mentioned in Section 3.6.2. Doubling these maximum deviations, the 58 instruments appear to have been operat ing at prec is ions o f at least 0.01 m m and 0.0006 m m respect ively (the m i n i m u m precis ions required) pr ior to the exp los ive test. The post-test steady-state measurements for the dynamic instruments were calcu lated by tak ing the average o f the readings between 12 s and 16 s after in i t ia l detonat ion, w h e n most o f the mot ion o f the test spec imen had dissipated out (refer to F igure 4.2). Dev ia t ions f rom average were also calculated for this t ime per iod. S ince the v ibra t ion o f the test spec imen had not complete ly damped out 16 s after the exp los ive test, the deviat ions were greater than before the test ( typ ica l ly 0.003 m m for the l inear potentiometers and 0.0004 m m for the PI transducers), but sma l l enough to suggest the prec is ion o f the instruments was suff ic ient. A f te r establ ishing pre-test and post-test steady-state measurements, ca lcu la t ing steady-state deformat ions for the l inear potentiometers was mere ly a matter o f subtract ing pre-test d isplacement f rom post-test displacement. The process for the PI transducers was the same w i t h one added step. A s was a l luded to i n Sect ion 3.6.2, to compare the d isp lacement readings f r om the PI transducers w i th those f rom the other instruments, the PI transducer d isplacements had to be mu l t ip l ied by the ratio o f the clear height o f the c o l u m n to the PI transducer gauge length. These extrapolated PI transducer d isplacements, a long w i t h the steady-state deformat ions recorded by the l inear potentiometers and d ia l gauges, are l is ted in Tab le 4 .1 . T a b l e 4.1 S teady -S ta te D e f o r m a t i o n s I n s t r u m e n t T y p e -^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ I n s t r u m e n t L o c a t i o n - G r i d In te rsec t ion (S ide o f C o l u m n ) A - 2 B - l B-2 (N) B-2 (S) C 1 C - 2 ( E ) D i a l Gauge 0.030 -0 .020* -0.080 -0.280 -0.100 -L inear Potent iometer 0.038 0 .037* -0.071 -0.299 -0.080 -5 .643* PI Transducer 0.041 - -0.090 -0.546 -0.026 -D-1 ; I)-2 (S) D i a l Gauge - 0.010 -0.390 -0.080 0.010 L inea r Potent iometer -4.205 - -0.348 0 .088* -PI Transducer - - -0.280 -0.141 -Note: All deformations in mm, upward deformations positive; * indicates deformation of concern 59 A s displayed in Table 4.1, the steady-state deformations measured during the explosive test by the linear potentiometers and dial gauges agree quite well in most instances; the deformations of concern in this table have been marked with an asterisk. Due to the fact that the instrument stand for linear potentiometer east of Column C-2 was disturbed during testing (as mentioned at the start of Section 4.2), the steady-state deformation reported by this instrument was not considered reliable. A t Column B - l , the dial gauge reported an upward deformation of 0.02 mm while the linear potentiometer measured a downward deformation of 0.037 mm. It was difficult to immediately assess which of the instruments at Column B - l provided erroneous readings; this issue w i l l be revisited in Section 5.5 during modelling of the test specimen. On the south face of Column D-2, the recorded deformations were 0.08 mm downward for the dial gauge and 0.088 mm upward for the linear potentiometer. Comparing these deformations with those recorded by the other instruments on Column D-2 and the instruments on the symmetrical column at B-2 leaves little doubt that the linear potentiometer should have recorded a downward steady-state deformation and was therefore likely disturbed during testing. Potential explanations as to what disturbed the linear potentiometer at D-2 (S) and how the data from this instrument could be modified and used in analysis of the test specimen w i l l be discussed in Appendix E . The agreement between the steady-state deformations recorded by the PI transducers and the other instruments is not as good as that between the linear potentiometers and dial gauges. Due to the fact that the PI transducers were mounted on the face of the columns, not slightly off the face of the columns like the other instruments, it was expected that the PI transducer readings would differ slightly from the readings of the linear potentiometers and dial gauges, since the former were not affected by rotation of beam-column joint. In some cases, the recorded data agrees with this general trend. Referring to Table 4.1, at B-2 (N) and D-2 (S), the PI transducer deformations are larger than those of the other instruments, while at C - l and D-2 (N), the opposite holds true; these relationships reflect the observed steady-state rotation at the top of each of the columns following the explosive test. A t other locations, however, such as A - 2 and B-2 (S), the steady-state readings of the PI transducers relative to the other instruments do not agree with the after-test beam-column rotations at these locations. A s well , even at the locations mentioned in the previous paragraph where the PI transducer deformations follow the observed beam-column rotations, the 60 proportions of the PI transducer readings to those of the other instruments seem suspect in some cases. For instance, at D-2 (N) the difference between the PI transducer and dial gauge readings is 0.11 mm, nearly double the 0.06 difference at D-2 (S). Since the instruments were placed symmetrically around the column, the dial gauge-PI transducer differences on each face of the column should be the same. This lack of proportionality calls into question the validity of the PI transducer data, at the very least in the feasibility of comparing this data to the measurements of the other instruments; the results suggest that the principal assumption used to extrapolate the PI transducer data (i.e., that the strain in the columns is constant over the entire story height) may not be applicable. The usefulness of the PI transducer data w i l l be further evaluated when examining transient data in Section 4.2.3. Before moving on to discuss the measurements recorded during the transient vibration o f the test specimen, the relative deformation of the columns in the specimen w i l l be briefly examined. The greatest amounts of steady-state column deformation were recorded at Columns B-2 and D -2; taking the average of the dial gauges on opposite faces of these columns, the center of the columns deformed approximately -0.18 mm and -0.235 mm, respectively. Looking at the instrument data in Table 4.1, significant deformations also appear to have occurred at Columns A - 2 , B - l and C - l . A s w i l l be demonstrated in the modelling of the test specimen in Section 5.5, however, the deformations at the center of Columns B - l and C - l were likely much lower than the deformations recorded by the instruments, which were located off the face of the columns. The instrument deformations in Table 4.1 imply a majority of the load redistributed during the explosive test went to Columns B-2 and D-2. Modell ing of the test specimen wi l l show that these columns received an even larger proportion of the redistributed load than suggested by the instrument data. 4.2.3 T r a n s i e n t Defo rma t ions After looking at the steady-state deformations recorded by the various instruments installed on the test specimen, the dynamic data collected by the linear potentiometers and PI transducers during the transient motion of the specimen were analyzed and compared. When comparing readings from the PI transducers to those of the linear potentiometers, the former were modified by the ratio of PI transducer to linear potentiometer gauge length. A s already discussed in Section 4.2.2, the assumption underlying this data manipulation may have been flawed. 61 4.2 .3 .1 G e n e r a l O b s e r v a t i o n s The data f rom the dynamic displacement co l lec t ion instruments instal led on the test spec imen fo l l ows the same general trend. Th i s general trend is exempl i f i ed i n F igure 4.4, a plot o f the first 1 s o f deformat ions recorded by the l inear potent iometer on the south face o f C o l u m n B - 2 . O v e r the f irst approx imate ly 0.13 s f o l l o w i n g in i t ia l detonat ion, the readings o f the l inear potentiometers and PI transducers osci l late at h igh f requency about a l ine ramp ing i n the opposite d i rect ion to the di rect ion o f the post-test steady-state deformat ion. A f te r 0.13 s, the data co l lected b y the l inear potentiometers and P I transducers is dominated by osc i l la t ions about the post-test steady-state d isplacement at a per iod o f approx imate ly 0.13 s. Phase 2 Period of Oscillations ~ 0.13 s Post-Test Steady-State Displacement 0.4 0.5 T i m e (s) F i g u r e 4.4 D e f o r m a t i o n R e c o r d e d f r o m 0 - 1 s b y L i n e a r P o t e n t i o m e t e r at C o l u m n B - 2 (S) Note: Upward displacement is positive In addi t ion, the peak deformat ions recorded after 0.13 s are greater than tw ice the post-test steady-state d isplacement, an interesting result cons ider ing that tw ice the steady-state d isplacement is the m a x i m u m peak deformat ion for a step load (the load ing cond i t ion for remova l o f a co lumn) predicted by l inear elastic theory [Chopra , 2000] . I f the structure deformed into the nonl inear range, there is a poss ib i l i t y that the peak deformat ions dur ing load redistr ibut ion cou ld be more than twice the steady-state d isplacement. H o w e v e r , the lack o f 62 cracking observed after the test and the stiff response of the structure (i.e. the very small deformations recorded by the instruments, in particular the linear potentiometers near Column C-2) suggest the structure behaved linearly during load redistribution. Thus, some factor other than nonlinear behaviour must be responsible for the peaks in the deformation data being greater than twice the steady-state displacement. The general trend of the transient data collected by the linear potentiometers and PI transducers suggests two distinct phases in the response of the specimen during the explosive test. During the first phase from 0 - 0.13 s, the recorded response is characterized by high frequency oscillations; whether these oscillations purely represent the response of the test specimen or are influenced by instrument resonance wi l l be examined in Section 5.2. Also of note during Phase 1 is the fact that the oscillations during this time period are centered about a line ramping in the direction opposite the post-test steady-state. This implies not only that the columns surrounding Column C-2 are not receiving the downward force originally carried by the removed column during Phase 1, but also that an upward force is acting at Column C-2 during this phase. The pressure pulse given off by the detonation of explosives in Column C-2 was forceful enough to disturb the instrument east of Column C-2; hence, it is conceivable that this pressure pulse may have also have travelled vertically and imposed an upward force on the test specimen. The possibility that an explosive pressure pulse may have acted upward on Column C-2 from 0 -0.13 s and caused net deformations opposite in direction to the post-test steady state w i l l be investigated further in Section 5. The second phase in the response data recorded by the linear potentiometers and PI transducers took place from 0.13 s onwards and was characterized by a dominant oscillation of period 0.13 s centered about the post-test steady-state reading for the instrument, and peak deformations greater than twice the steady-state displacement. The fact that oscillations in Phase 2 are centered about the post-test steady-state suggests that structure is redistributing the forces originally carried by Column C-2 during this time period. The fact that the response in Phase 2 is dominated by an oscillation with a period of 0.13 s implies that the remaining structure has a vertical vibrational mode of 0.13 s and that the test specimen was primarily excited in this mode by the removal of Column C-2. Lastly, the fact that the peak deformations during the explosive test were greater than twice the steady-state displacement suggests that in addition to the redistribution of the gravity forces in Column C-2, there was another force acting on the system 63 that magnified the response due to gravity load redistribution. The upward explosive pressure pulse at Column C-2 proposed in the previous paragraph to explain the deformations opposite in direction to the post-test steady state may also be responsible for the surprisingly high peak deformations in the structure. The above assertions wi l l be investigated further during analysis and modelling of the test specimen in Section 5. 4.2.3.2 Discrepancies Between Linear Potentiometer and PI Transducer Response Although the transient data collected by the linear potentiometers and PI transducers follow the same general trend, there are some significant differences in the response histories recorded by the different types of instruments. Figure 4.5a shows the response recorded by the linear potentiometer and PI transducer on the south face o f Column B-2 ; in this plot the PI transducer data has been extrapolated to approximate the deformation of the column over the entire Level 1 storey height. Looking at Figure 4.5a, three significant differences between the linear potentiometer and PI transducer data are evident. First, the PI transducer recorded a larger downward steady-state deformation than the linear potentiometer and consequently Phase 2 of the PI transducer response oscillates about a larger negative displacement than the linear potentiometer response. This issue was introduced in Section 4.2.2 and wi l l be discussed no further there. The second difference in the response of the instruments shown in Figure 4.5a is the magnitude of the high frequency oscillations at the beginning of the response histories. For the linear potentiometer data, the region where the influence of high frequency oscillations is significant is limited to Phase 1 of the response, and even in this region, the magnitudes of these oscillations are relatively low. On the other hand, for the PI transducer data, the magnitudes of the high frequency oscillations are quite high in Phase 1 of the response and these high frequency oscillations are significant well into Phase 2. One possibility for the presence of more predominant high frequency oscillations in the PI transducer data is the greater impact of random signal noise noted during calibration of these instruments (refer to Section 3.6.2). However, after extrapolating the PI transducer data to represent the deformation of the entire column, pre-test calibration would have predicted the deviations due to signal noise to be only on the order of 0.02 mm or 0.03 mm; referring to Figure 4.5a, the magnitudes of high frequency PI transducer oscillations are much greater than 0.03mm. 64 T i m e (s) (a) Response of Linear Potentiometer and PI Transducer - N o Averaging T i m e (s) (b) Response of Linear Potentiometer and PI Transducer with 11-Point Averaging F i g u r e 4.5 R e s p o n s e o f D y n a m i c D a t a C o l l e c t i o n I n s t r u m e n t s at C o l u m n B - 2 (S) Note: Upward displacement is positive The fact that random signal noise can not explain the more significant high frequency oscillations in the PI transducer data suggests that one of the higher vibrational modes of the test 65 specimen may have coincided with a natural frequency of the PI transducer, causing the PI transducer to resonate and amplify the recorded response at this frequency. The third major difference between the linear potentiometer and PI transducer data is the influence of higher mode vibrations during Phase 2 of the response of the test specimen. Observation of the response histories recorded by the linear potentiometers suggests that there is for the most part very little participation from vibrational modes other than the primary mode with a period of 0.13 s. A s mentioned in the previous paragraph, however, for the PI transducer data, higher mode vibrations are significant at the beginning of Phase 2 of the response and detectable for a considerable number of oscillations of the period 0.13 s primary mode. The continued presence of high frequency oscillations in Phase 2 of the PI transducer data is further evidence that these instruments may have resonated with one of the modes of the structure during the explosive test. In an attempt to remove the possible effects of signal noise on the PI transducer data, and identify i f the high frequency oscillations represent the actual response of the test specimen or instrument resonance, the 11-point averaging scheme described in Section 3.6.2 was applied to the PI transducer data. The averaged PI transducer data is plotted alongside the linear potentiometer data in Figure 4.5b. Referring to Figure 4.5b, the magnitude of the high frequency oscillations in the PI transducer data are somewhat muted but are still quite significant in comparison to the linear potentiometer data. The issue of resonance in the PI transducer data w i l l be examined further in Section 5.2. However, even i f it is possible to remove the apparent resonant effects in the PI transducer data, the usefulness of the data in evaluating the actual response of the test specimen, especially the response at the resonant frequency, may be limited. 4.2.3.3 Comparison of Response at Different Columns in Test Specimen Moving from a comparison of instruments at the same location to a comparison of instruments situated at different locations, a number of other interesting trends are apparent. Figure 4.6 shows the transient deformation response measured by the linear potentiometers at Columns B-2 , C - l and D-2. To estimate the center-of-column deformations at B-2 and D-2, the data from the linear potentiometers on the north and south faces were averaged (note that adjusted data was used for the linear potentiometer at Column D-2(S); the procedure and rationale for adjusting this data w i l l be discussed in Appendix E). Since only one linear potentiometer was installed at 66 Column C - l , the center-of-column deformations at this location were assumed to be the same as the readings of the one instrument installed. A s discussed in Section 4.2.2, this assumption is not explicitly correct, but the deformation trends at the instrument and column center are similar, so for the purposes of comparing the shape of the deformation response at different locations, the invalidity of the assumption was deemed to be not that significant. -0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 , T i m e (s) F i g u r e 4.6 T r a n s i e n t R e s p o n s e - L i n e a r Po ten t i ome te r s at C o l u m n s B - 2 , C - l & D-2 Note: Upward displacement is positive The most striking feature of Figure 4.6 is that the peaks and valleys of the oscillations in Phase 2 of the response for all three columns are approximately in-phase. Further comparison with the transient response recorded at Columns B - l and A - 2 (not shown here) shows that the deformations at B - l also tend to be in-phase with Columns B-2, C - l and D-2, while the oscillations at Column A - 2 are approximately 180 degrees out-of-phase with the other columns instrumented. The in-phase/out-of-phase behaviour recorded by the instruments provides insight into the primary vertical vibrational mode of the test specimen, which w i l l be examined further during modelling of the test specimen in Section 5.5. Another interesting feature of Figure 4.6 is the fact that the transition from Phase 1 to Phase 2 of the response occurs at essentially the same spot for all three columns, although the upward spike in Column C - l deformation at approximately 0.14 s makes it more difficult to identify exactly where this transition is from Phase 1 to Phase 2 for that column. That being said, it appears that 67 the delay in the redistribution of load from Column C -2 is about the same for all three columns adjacent to the removed column. This topic w i l l be explored further in Section 5. The three significant spikes in the first 0.2 s of deformation data at Column C - l are also of interest. These spikes are also present in the PI transducer response history at C - l and the linear potentiometer data at Column B - l . This suggests that the spikes represent actual deformation in the structure and are not simply instrument errors. This phenomenon w i l l be investigated further in Sections 5.5 and 5.6. 68 5 DATA ANALYSIS AND M O D E L L I N G 5.1 Overview One of the main goals of this research program was to experimentally ascertain the dynamic amplification during gravity load redistribution initiated by sudden column failure in a simple structure. As discussed in Section 4.2.3, however, the data recorded during the explosive test for the research program suggests that the specimen was acted upon by an unanticipated upward force during column removal, possibly a pressure pulse from the explosion itself, which caused the structure to deform in the direction opposite to the post-test steady-state during the first 0.13 s after initial detonation and the peak deformations in the structure to be greater than twice the post steady-state deflections. In order to achieve the goal of determining the dynamic amplification during redistribution from just the gravity loads originally carried by Column C-2 and not the aggregate of these gravity loads and the unexpected force at C-2, the response due to gravity load redistribution must be isolated. A n essential part of isolating the response from gravity load redistribution is identifying the unanticipated force on the structure. It is necessary to determine the response of the test specimen to just gravity load redistribution to make the results of this research more widely applicable. If indeed the unanticipated force on the structure can be attributed to the blast that removed Column C-2, the raw data collected during the explosive test is only representative of structures similar to the test specimen that experience a column failure due to the detonation of the equivalent of 120 g of C4 explosives embedded in a column. The raw data from the test is not useful for furthering understanding on the behaviour of structures that undergo column failures from earthquakes or the more common blast scenario in which the explosion does not originate in the column itself. In these failure scenarios, the gravity load in the failed column is the only major vertical force redistributed. Thus, for this research program to be relevant for a wide range of column failures, the dynamic amplification due to just gravity load redistribution must be evaluated. The majority of this section of the report w i l l focus on the steps taken to isolate the response o f the test specimen due to just gravity load distribution. The data recorded by the accelerometer 69 installed at C-2 w i l l be consulted for evidence supporting the hypothesis that an explosive pressure pulse acted upward on the test specimen and what the shape of the loading function representing this pulse might look like. Then, a simple, single-degree-of-freedom (SDOF) analytical model w i l l be constructed to compare the approximate deformation response for potential loading functions representing the unanticipated upward force and loss of gravity load support at Column C-2 to the data recorded during the test. After plausible loading functions are identified, a detailed, three-dimensional (3D) analytical model of the test specimen w i l l be constructed and calibrated to the recorded data, and variations of the plausible loading functions w i l l be applied to the model until a reasonable match is obtained with the recorded data. Finally, this 3-D analytical model w i l l be used to investigate the response of the structure due solely to gravity load redistribution and determine the dynamic amplification of the axial load increases in the columns of the test specimen from redistribution of gravity load. Before this section focuses in on how to isolate the response of the test specimen due to gravity load redistribution, however, the results of a frequency analysis of the test data w i l l be presented. This frequency analysis w i l l identify the major contributing frequencies in the test data, which w i l l provide valuable information about the structure for modelling, offer insight into whether the data recorded by some instruments was influenced by resonant effects, and help determine i f the high frequency oscillations recorded are actual structural deformations or instrument error. 5.2 Frequency Analysis To get a better idea of the periods of the vibrational modes contributing to the overall response recorded during the explosive test, a frequency analysis was performed on the collected deformation data. This analysis consisted of transforming the recorded data from the time domain to the frequency domain using the Fast Fourier Transform (FFT) algorithm to generate Fourier Spectra.. In a Fourier Spectrum, the relative magnitude of the individual frequencies composing the total response recorded by an instrument is calculated. This information on the major frequencies present in each data set was then used to identify the significant vibrational modes of the test specimen for later use in modelling, determine i f the linear potentiometers or PI transducers were affected significantly by resonance, and assess whether the high frequency oscillations recorded in the first 0.13 s after initial detonation represent actual deformations in the structure or are instrumentation anomalies. 70 For the purposes of frequency analysis, the recorded response was split into data sets, one from 0 s to 0.13 s, and the other from 0.13 s to 4 s. It was necessary to split the response because the data must be baseline-corrected about its average before it is transformed to the frequency domain, and the average of the first 0.13 s of the recorded data (which oscillates about a line ramping in the direction opposite the post-test steady-state deformation) is considerably different than the average o f the response after 0.13 s (which oscillates about the post-test steady-state deformation). However, splitting the data resulted in the length of the first data set being too short to generate a useful Fourier Spectrum. Thus, only the data after 0.13 s was zeroed and imported into M A T L A B [Mathworks, 2004], where it was transformed into the frequency domain using the F F T function in M A T L A B . It should be noted that a Fourier Spectrum was not generated for the data recorded by the PI transducer at D-2 (S) due to the discontinuity in the data noted in Section 4.2, or the linear potentiometer at C-2 (E) because this instrument was disturbed during testing. When a force is applied on structure over a very short duration, as was the case for the test specimen in the explosive test, the structure responds dynamically by oscillating in its modes of vibration. The applied force does not affect the frequencies that make up the response; the frequencies in the response are characteristic of the structure itself and are dependent on the stiffness and mass distribution of the structure. Thus, by looking at a Fourier Spectrum of the data recorded during the explosive test, it is possible to identify the vibrational modes of the test specimen excited during the test for later use in analytical modelling. If a mode of the test specimen is excited by the loading, the magnitude of that modal frequency wi l l be large in the Fourier Spectrum of the recorded data. That being said, it is important to keep in mind that the response reported by a deformation instrument is influenced by both the vibrations of the structure and the vibrations of the instrument itself. In most cases, instrument vibrations are small and at high frequencies, and do not contaminate the recorded response. If the instrument is excited in one of its modal frequencies, however, the magnitude of the instrument vibrations can be magnified considerably and wi l l show up noticeably in the recorded response. Thus, when looking at a Fourier Spectrum of response data, it is necessary to identify which frequencies with high magnitudes are modes of the structure and which are modes of the instrument. 71 Figure 5.1 shows a plot of the Fourier Spectrum of the 0.13 s to 4 s data set recorded by the linear potentiometers. For each instrument, the magnitudes of the individual frequencies are normalized to the maximum magnitude for that instrument; normalizing makes it possible to compare between instruments the relative importance of the individual frequencies that make up the total recorded response for each individual instrument. e WD ce Spike In All Linear Potentiometer Data At 7.82 Hz 25 50 75 Frequency (Hz) 100 125 150 Figure 5.1 Fourier Spectrum of Linear Potentiometer Data from 0.13 s to 4 s: 0 to 150 Hz Looking at Figure 5.1, the most striking feature of the plot is that the magnitude at the frequency of 7.82 H z is very high for all of the linear potentiometers. A frequency of 7.82 H z corresponds to a period of .128 s, which is very close to the dominant period of 0.13 s noted in the linear potentiometer data from 0.13 s after initial detonation onward (refer to Figure 4.4 and Figure 4.5a). A frequency of 7.82 H z is the only frequency at which there is a peak in magnitude for all the linear potentiometer data. From Figure 5.1, it is obvious that there are no frequencies larger than 25 H z that significantly contribute to the linear potentiometer data. Figure 5.2 shows a closer look at the data below 20 Hz . Some of the instruments show a spike in magnitude at frequencies of 0.244 H z and 1.95 Hz , and the data for linear potentiometer at B - l contains some significant frequencies around 14 Hz . These spikes in magnitude are not shared by all the instruments, however, and are not readily apparent in the time domain response (refer to Figure 4.5a). 72 0 2 4 6 8 10 12 14 16 18 20 Frequency (Hz) Figure 5.2 Fourier Spectrum of Linear Potentiometer Data from 0.13 s to 4 s: 0 to 20 Hz Figure 5.3 is a plot of the Fourier Spectrum of the PI transducer data from 0.13 s to 4 s. This figure shows that like the linear potentiometer data, there is a spike in magnitude at 7.82 Hz for the PI transducer data. Unlike the linear potentiometer data, the data for all the PI transducers 0 25 50 75 100 125 150 Frequency (Hz) Figure 5.3 Fourier Spectrum of PI Transducer Data from 0.13 s to 4 s: 0 to 150 Hz 73 also appears to be significantly influenced by frequencies around 117 Hz . The period associated with a frequency of 117 H z is 0.0085 s, which is approximately the period of the high frequency oscillations after 0.13 s in the PI transducer data shown in Figure 4.5a. The data recorded by two of the PI transducers have spikes in magnitude at around 60 H z but there are no frequencies other 7.82 H z and 117 H z that make up a significant component of the response of all the PI transducers. Figure 5.4, a closer look at the magnitudes of the frequencies in the PI transducer data from 0 to 20 H z , illustrates this fact more clearly for the lower frequencies. A s was the case for the linear potentiometer data, there are spikes in some of the PI transducer data at 0.244 H z and 1.95 Hz . 0 2 4 6 8 10 12 14 16 18 20 Frequency (Hz) Figure 5.4 Fourier Spectrum of PI Transducer Data from 0.13 s to 4 s: 0 to 20 Hz When considering the high magnitudes at 117 H z in the PI transducer data, it 's important to note that the analog filter in the dynamic data acquisition system was set at a cut-off frequency 100 Hz . For frequencies less than 40% of the filter cut-off frequency, the low-pass 4-pole Butterworth filter used by the data acquisition system does not affect the magnitude recorded by the instrument. However, for a frequency of 1.17 times the cut-off frequency, the magnitude recorded is reduced by 6.56 dB [Frequency Devices Inc., 1999], which is equivalent to multiplying the actual magnitude by a factor of 0.47. Thus, the actual magnitude of the component of the response at 117 H z measured by the PI transducers is approximately twice that 74 recorded by the data acquisition system. In effect then, the magnitudes of the response at 117 H z measured by the PI transducers were twice as large as shown in Figure 5.3; similarly, the magnitudes of the high frequency oscillations in the PI transducer data in Figure 4.5a were also measured twice as large as shown. Thus, the contribution of high frequencies to the response measured by the PI transducers is even higher than the recorded data suggests. The fact that the Fourier Spectra o f the linear potentiometer and PI transducer data both have spikes in magnitude at 7.82 H z suggests that the deformations at this frequency recorded by the instruments is due to vibration in the structure, not instrument vibrations. In addition, the fact that this frequency is so much more prevalent than any other implies that this particular mode was excited more than any other during testing, which should provide some insight into the shape of this mode during modelling of the test specimen in Section 5.5. The presence of spikes in magnitude at 0.244 H z and 1.95 H z in some of the recorded data suggests that the test specimen may also have modes at these frequencies, although the evidence for this hypothesis is less certain. In particular, a frequency of 0.244 H z is very low for a structural vibration, suggesting that the spike at this frequency may be a function of the F F T procedure rather an actual vibration in the recorded response. During modelling of the test specimen in Section 5.5, the model w i l l be checked to see that its primary vibrational mode during the test is around 7.82 H z or about 0.13 s, and also i f there are modes at 0.244 H z and 1.95 Hz . Considering now the spikes in magnitude at 117 H z in the PI transducer data, the fact that similar spikes are not present in the linear potentiometer data suggests the high frequency deformations recorded by the PI transducers after 0.13 s may be influenced by vibrations in the instruments and do not represent the magnitude of actual structural vibrations. If one of the vibrational modes of the PI transducers is at 117 Hz , a low magnitude vibration in the structure at this frequency would excite the instruments and cause them to resonate, i.e. oscillate at increasingly greater amplitudes. If this resonance were to take place, the component of the actual structural response at 117 H z would be greatly magnified in the data reported by the PI transducers. Resonance in the PI transducers due to a low magnitude structural vibration would explain why there is a spike in magnitude at 117 H z in the Fourier Spectrum of the PI transducer data and not in the Fourier Spectrum of the linear potentiometer data. Figure 5.5 suggests with even more certainty that the PI transducers were influenced by resonance during the course of the explosive test. 75 0.4 T -0.4 -I 1 1 1 1 1 0 0.1 ' 0.2 0.3 0.4 0.5 T i m e (s) Figure 5.5 Response Recorded at C o l u m n A - 2 : 0 to 0.5 s Figure 5.5 is a plot of the response in the time domain recorded by the linear potentiometer and PI transducer installed at Column A - 2 for the first 0.5 s after initial detonation. The steady-state deformation reported by the linear potentiometer and the equivalent steady-state deformation for the PI transducer (the deformation calculated by extrapolating the PI transducer data from its 200 mm gauge length to the full column height) are essentially the same: 0.04 mm upwards. The peak deformation in the linear potentiometer data is approximately 3.5 times the steady-state deformation. This peak deformation is larger than that predicted by linear elastic theory (2.0 times the steady-state deformation), but considering the magnifying effect of upward explosive force at Column C-2, it is still reasonable. Looking now at the PI transducer data, the peak deformation is 10 times the steady-state deformation, largely due to the high frequency component of the response. This very large peak deformation can not be attributed to the upward explosive force at Column C-2. Resonance in the PI transducer would explain the high magnitudes of the high frequency oscillations and the unreasonably high peak deformation. Considering the fact that analog filtering partially attenuated the magnitude of the high frequency oscillations, the peak deformation reported by the PI transducer was even larger than 10 times the steady-state deformation, making the proposition that these oscillations reflect actual vibration in the structure even less likely. 76 In order to use the PI transducer data for further analysis of the structure, it would have been necessary to digitally filter the data to remove its high frequency components, in particular, the 117 H z oscillations. B y further filtering the PI transducer data, however, more of the true response of the structure would be inadvertently lost, reducing the usefulness of the data. Considering the drawbacks of data filtering along with the problems stated in Section 4.2.2 regarding the steady-state deformations recorded by the PI transducers and the apparent invalidity of the assumption that PI transducer displacements can be extrapolated to represent column displacements, it was decided that the PI transducer data would not used for any further analysis of the behaviour of the test specimen. The linear potentiometer and dial gauge data was relied upon for all modelling of the test specimen. Thus far, all discussion on frequency analysis has focused on the data set from 0.13 s to 4 s. However, one of the reasons for conducting a frequency analysis was to identify the makeup of the high frequency oscillations in the first 0.13 s of the collected data and determine i f these oscillations represented structural deformations or instrumentation vibrations. A s mentioned earlier, it was not possible to get a useful Fourier Spectrum from the short data set from 0 s to 0.13 s. While Fourier Spectrum Analysis was not an option for the first 0.13 s of the linear potentiometer data, it was still possible to reason whether the high frequency oscillations in this data were 'real ' by looking at the data before and after 0.13 s in the time domain. For the PI transducer data, the resonant oscillations started in the first 0.13 s after initial detonation and carried forward past 0.13 s (refer to Figure 5.5). However, for the linear potentiometer data, the high frequency response is prevalent in the first 0.13 s and then for the most part disappears after 0.13 s (refer to Figure 4.4 or Figure 5.5). If the high frequency oscillations in the first 0.13 s of the linear potentiometer data were due to resonance, these vibrations would have continued past 0.13 s like in the PI transducer data. However, this high frequency response did not continue past 0.13 s, suggesting that the structure was actually vibrating at a higher frequency for the first 0.13 s. In Section 4.2.3, it was noted that from the trend of the recorded data, it does not appear that the gravity load supported by Column C-2 was redistributed to the rest of the structure for the first 0.13 s following initial detonation. If the forces in Column C-2 were not being redistributed, it is conceivable that this column was effectively still intact during this time. The structure with Column C-2 in place would have a much stiffer resistance to vertical forces applied at C-2, such 77 as a vertical upward force from the explosives at that location. Stiffer structures vibrate at higher frequencies, which is consistent with the higher frequency response recorded by the linear potentiometers. Thus, both the lack of load redistribution and the higher frequency response during the first 0.13 s following initial detonation of explosives suggest that Column C-2 was still effectively supporting load during this time interval. In other words, it appears that it may have taken a finite amount of time (0.13 s) for the force released by the explosives to obliterate the concrete in the column, causing it to lose its load-carrying capacity. This hypothesis, that Column C-2 remained effectively intact for the first 0.13 s wi l l be explored further in the coming sections. 5.3 Investigation of Accelerometer Data In Section 4.2.3, the idea that the explosion removing Column C-2 applied an upward force on the test specimen was proposed as a possible explanation for the net deformation of the structure in the direction opposite the post-test steady-state in the first 0.13 s after detonation and the unusually high peak deformations recorded during the course of the test. In Section 5.2, it was proposed that the high frequency response of the test specimen prior to 0.13 s was due to the fact that Column C-2 remaining effectively intact during this time period. To help assess the validity of these claims, it was thought prudent to consult the data recorded by the accelerometer installed at C-2 for evidence that an explosive pressure pulse acted upward on the test specimen and that Column C-2 retained load-carrying ability for 0.13 s following initial detonation. If the accelerometer data supported these claims, it was thought that this data might also be useful in developing loading functions to represent the forces applied on the test specimen during the explosive test. Figure 5.6 shows the data collected by the accelerometer at C-2 from 0 s to 0.25 s. This data has a couple of interesting characteristics. First, it appears that at 0 s, 0.05 s and 0.10 s something acts on the structure at C-2, causing the magnitudes of accelerations measured to increase or surge. Then at 0.11 s, the accelerometer records a massive increase in acceleration magnitude. These accelerations are very large, extending beyond the ± 10 g range set for the accelerometer (the recorded data is clipped at -9 g and 11 g) and last at this high intensity for around 0.05 s. 78 -12 -I 1 1 1 1 1 0 0.05 0.1 0.15 0.2 0.25 T i m e (s) Figure 5.6 Response of Accelerometer Above Removed Column at C-2: 0 to 0.25 s A possible explanation of the accelerometer data collected is as follows. When the pairs o f explosive charges installed in the column were detonated, it seems likely that these explosions exerted upward pressure pulse forces on the structure at C-2. Such upward pressure pulses would have registered as surges in the response history of the accelerometer. If the detonation delay between the pairs of explosive charges was around 0.05 s, (which is less than the upper bound delay of 0.10 s mentioned in Section 3.7), this would explain the acceleration surges in Figure 5.6. If, now, Column C-2 initially retained its load-carrying ability during detonation of the explosives, but then lost its structural resistance at 0.11 s, this would account for the massive increase in accelerations at this time. Considering the above, the accelerometer data tends to support the assertions made in Sections 4.2.3 and 5.2 that the explosions exerted an upward force on the structure and Column C-2 remained effectively intact for the first 0.13 s. Before moving on to how the loading condition on the test specimen during the explosive test can be represented, it is worth investigating what appears to be a discrepancy between data recorded by the accelerometer and the data of the linear potentiometers. For the linear potentiometers, Phase 1 of the response (the portion of the response where it is being proposed that Column C-2 was still intact) has been defined as lasting until 0.13 s. However, Phase 1 of the response for the accelerometer appears to end at 0.11 s. This apparent discrepancy can be explained by considering that it takes a finite amount of time for deformation at one point in a 79 building to affect points at other parts in the building due to the support initially provided by inertial forces. -1 A 1 1 1 1 1 0 0.1 0.2 0.3 0.4 0.5 Time (s) (a) Response of Linear Potentiometer at B-2 (S): 0 to 0.5 s 0.15 T -0.1 -| ! 1 1 1 1 0 0.1 0.2 0.3 0.4 0.5 Time (s) (b) Response of Linear Potentiometer at A - 2 : 0 to 0.5 s Figure 5.7 Delay in Response Due to Location in Specimen Figure 5.7a shows the first 0.5 s of the response recorded by the linear potentiometer at B-2 (S), while Figure 5.7b shows the response at A - 2 . A t B-2 (S), the response is flat for 0.01 s after 80 initial detonation and Phase 1 of the response ends (i.e. when the large magnitude, lower frequency oscillations begin) at 0.13 s; for A - 2 , these points come 0.01 s later (0.02 s for the end of the flat response and 0.14 s for the end of Phase 1). Thus, it appears to take 0.01 s for the behaviour affecting Column B-2 to make its way to Column A - 2 . Extending this idea to Column C-2, which is the same distance from B-2 as B-2 is from A - 2 , one would expect there to be 0.01 s - 0.01 s = 0 s of flat region at the start of the response, and Phase 1 of the response to end at 0.13 s - 0.01 s = 0.012 s. Sure enough, there is no flat region at the beginning of the accelerometer response at C-2, although the end of the Phase 1 response end is at 0.11 s, not 0.12 s. Nevertheless, the explanation that the physical distance between instruments is the reason for the delay between their responses appears to hold for the most part. For simplicity and consistency with the discussion presented thus far, 0.13 s w i l l be designated as the end of Phase 1 of the response of the test specimen. Returning now to the loading on the test specimen during the explosive test, discussions in Section 4.2.3, Section 5.2 and this section have suggested that during the first 0.13 s after initial detonation, the explosions in Column C-2 generated upward forces on the structure and Column C-2 retained its load-carrying capacity. In order to test these propositions further, it is necessary to simulate the response of the test specimen through analytical modelling. To so, however, the forces acting on the test specimen must be modelled as loading functions. Coming up with a loading function for the failure of Column C-2 is relatively straightforward. For modelling purposes, a system with Column C-2 intact and a system with Column C-2 replaced by the reactions at the top of the column are equivalent. Considering this second system^ the effect of Column C-2 failing instantaneously is equivalent to the reactions applied in place of this column disappearing. Thus, the change in loading experienced by the system from the failure of Column C-2 is the negative of the reaction forces applied on the system in place of Column C-2. For initial modelling, it is reasonable to assume Column C-2 supports primarily axial load and the other forces in the column can be neglected; during more sophisticated modelling, all the reactions in Column C-2 w i l l be considered (refer to Section 5.5). Including this assumption, the failure of Column C-2 can be modelled as downward axial force applied at C-2 with a magnitude equal to the axial load originally carried by the column. This downward force is not present on the system until 0.13 s; then, the full magnitude of the force is applied. The function representing this type of loading is a step function. Thus, a step force applied at 81 0.13 s after initial detonation wi l l be tried as the loading function representing the failure of Column C-2 in modelling of the test specimen. Selecting a reasonable load function for the upward explosive force on the structure is not as obvious as picking a load function to represent the failure of Column C-2. A s discussed earlier in this section, the surges in acceleration at 0 s, 0.05 s and 0.10 s may have been caused by the detonation of pairs of charges in Column C-2 at intervals of 0.05 s. One way to represent these explosions would be with rectangular pulses. If each explosive pulse was assumed to apply a force of magnitude P on the structure and last until the column failed at 0.13 s, the cumulative pressure pulse on the structure would be as shown in Figure 5.8. Modell ing the upward explosive force on. the structure with three rectangular pulses is a bit cumbersome, however. A s shown in Figure 5.8, a single triangular pulse can produce a shape very similar to the cumulative of the rectangular pressure pulses. Thus, the loading function that w i l l be used to try and represent the upward force given off by the explosive removal of Column C-2 is a triangular pressure pulse increasing from 0 s to 0.13 s. Force Cumulat ive of Rectangular Pressure Pu l ses Triangular Pressure Pu lse Individual Rectangular Pressure Pu l ses • Time (s) 0 0.05 0.10 0.13 Figure 5.8 Loading Function Shapes for Upward Explosive Force In this section, the data recorded at the accelerometer installed above the column at C-2 was examined to see i f it supports the hypotheses developed in previous sections proposing that an upward force was exerted on the structure by the explosives installed in Column C-2 and that Column C-2 remained effectively intact until 0.13 s. It was determined that the accelerometer data does indeed provide evidence to support these hypotheses. The trends in the acceleration 82 and deformation data previously identified were then used to come up with loading functions for the proposed upward force on the structure and the failure of Column C-2 at 0.13 s. These loading functions w i l l be incorporated into a simple analytical model in the next section, and evaluated by how well the deformation response from the model compares to the recorded data. 5.4 Simulation of Specimen Response - SDOF Model In the previous section, some general loading functions were proposed for the forces applied on the test specimen during the explosive test. To assess i f the shapes of these loading functions are reasonable, and i f so, determine appropriate parameters for the loading functions, an analytical model of the test specimen is required to simulate the structural response generated by the loading functions so it can be compared to the recorded data. In order to effectively capture the behaviour of the entire test specimen during the explosive test, a detailed 3D analytical model is required. Before constructing and calibrating such a model, however, it was deemed wise to determine i f there are in fact loading functions that are capable of inducing a response similar to the recorded data. With this in mind, before commencing with sophisticated 3D modelling of the test specimen, a simplified analytical model was developed to evaluate potential loading functions. This simplified model was only capable of generating the approximate response caused by the loading functions, but nonetheless proved invaluable in ascertaining that there were loading functions that could closely estimate the recorded response and what the shapes and parameters of these loading functions were. The simplified analytical model used to establish plausible loading functions was a linear elastic single-degree-of-freedom (SDOF) model. A S D O F model was chosen because it is relatively simple to calculate the response of S D O F systems to a variety of loading functions. In addition, the fact that the recorded deformation response is dominated by one frequency of vibration suggests that modelling the test specimen as a single-degree-of-freedom may actually provide a pretty good approximation to the actual behaviour of the structure. The simplified analytical model was assumed to be linear-elastic since as discussed in Section 4, the structure did not appear to deform in the nonlinear range during testing (little cracking, stiff response). Assuming the structure to be linear-elastic permitted the use of the Principle of Superposition, which allowed the response due to the triangular pulse (upward explosive force) and the response due 83 to the step force (failure of Column C-2) to be determined independently and then combined for comparison to the recorded data. The properties of the simplified analytical model were derived from the collected test data. The period of the model was chosen as 0.13 s, the dominant period in the response recorded by the deformation instruments during Phase 2. A number of different damping values were tried for the model. A damping ratio of 3.5% was found to generate the response with the best fit to the recorded data; this level of damping is within the typical range of damping values for bare, reinforced concrete frames. The recorded response used for comparison against the simplified model results was the average of the data collected by the linear potentiometers on the north and south faces of Column B-2. The average of the response at B-2 (N) and B-2 (S) was used because the quality of data collected by both these instruments appears to be quite good, and the results of the S D O F model were used to estimate dynamic amplification in the test specimen for a conference presentation [Matthews et al., 2007]. In order to calculate the dynamic amplification at Column B-2, the dynamic changes in the axial load of the column were required, which necessitated averaging the B-2 (N) and B-2 (S) data to get an estimate of the deformations at the center of Column B-2. The response of a linear elastic S D O F system to an increasing triangular pulse and a step load applied at the end of the triangular pulse is described by the following equations [Chopra, 2002]: k 2C 2C - 1 ' cos(<w/) + * , ^ s i n ( ^ f ) 1P2 (') = upi(td)+<Za>nupi(td) 0<t<t„ (5.1) s\n(a>d(t-tdj) t>td (5.2) l-e cos(ad (t -td))+ JL— sin(fl>rf (t - td)) u](t) = up](t)+o; o<t td where, (5.3) (5.4) (5.5) 84 w (t) s deformation response of Column B-2 due to the triangular pulse for 0<t <td', t= time; P = maximum magnitude of the triangular pulse; k = stiffness of Column B-2; td = duration of the triangular pulse; = damping ratio of the model; con - In IT = natural circular frequency of the model; T = period of the model; cod = con -]\-C,2 = damped circular frequency of the model; u 2 (t) = deformation response of Column B-2 due to the triangular pulse for t>td\ up] (td) = deformation response due to the triangular pulse at t - td', upX (td ) = first derivative of the deformation response due to the triangular pulse at t - td\ us it) = deformation response of Column B-2 due to step load for t >td; Pso = magnitude of the step load; w, (t), u2 (t) = total deformation response of Column B-2 during the indicated time interval The parameters defining the loading functions in the above equations are Pp0, td and Pso. The triangular pulse duration suggested in previous sections, 0.13 s, was selected for td. The value of the step load, Pso, entered into the model was -0.185 mm * k, which is the post-test steady-state deformation change in Column B-2 multiplied by the stiffness of this column. Several different values were tried for the maximum magnitude of the triangular pulse, Ppo, but the value resulting in the best response was 0.185 mm * k. A plot of the loading functions applied on the S D O F model is shown in Figure 5.9. It should be noted that the resultant loading condition resulting from the individual loading functions shown in Figure 5.9 is not unique to these loading functions. The same resultant loading, and consequently, the same model response, can be achieved from other combinations of individual loading functions. The individual loading functions chosen, though, are not merely mathematical conveniences. There is physical evidence in the test observations and recorded data, as described in Section 5.3, that supports the shape of the selected loading functions. It is this physical basis, combined with the fact that the response generated by the resultant of the loading functions is similar to recorded data, that suggests that the triangular pulse and step load shown in Figure 5.9 are reasonable representations of the actual loading on the test specimen during the explosive test. 85 0.2 T0.185 0.15 <U CZ3 s o U 0.1 + 0.05 -f 0 mm 0 0.05 -0.1 + u £ -0.15 -0.2 1 0.1 -0.185 mm Triangular Pulse 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Step Load Time (s) Figure 5.9 Loading Functions Applied on the SDOF Model Figure 5.10 compares the response of the S D O F model due to the loading functions shown in Figure 5.9 with the recorded response at Column B-2. Since the S D O F model can only capture a single period of vibration, the model response fails to capture the high frequency oscillations Response from S D O F M o d e l E B a E a Q "es Figure 5.10 Comparison of SDOF Model Response and Recorded Response 86 before 0.13 s. In addition, the model response misses the magnitude of the recorded data at some of the response extremes. On the whole, however, the S D O F model does an excellent job of approximating the recorded response. The simplified analytical model and loading functions described in this section were successful in approximating the recorded response at one location in the test specimen. However, a more detailed 3D analytical model is required to simulate the response of the entire structure and incorporate more complex behaviour such as the contribution of higher modes. In the next section, the plausible loading functions developed wi l l be applied to a sophisticated 3D model, and the model results compared to the recorded data at multiple locations. 5.5 Simulation of Specimen Response - 3D ETABS Model Having identified plausible loading functions for the forces applied on the test specimen, the next step in the modelling process was to construct a detailed, 3D analytical model. The structural modelling software E T A B S [CSI, 2005] was used for 3D modelling purposes. E T A B S is capable of simulating the dynamic behaviour o f 3D linear structures, such as the test specimen, efficiently and accurately. It features a user-friendly graphical interface and is set up to allow parameters in the structure to be changed quickly and easily. There are other programs available which enable structures to be modelled in a more detailed fashion, an asset when deformations are significantly nonlinear, but these programs tend to be more complicated and labour intensive. Considering the lack of certainty in the properties of the test specimen and the fact that the structure appeared to behave linearly during testing (refer to Section 4.1), the extra detail offered by other programs was deemed unnecessary, leaving E T A B S as the most appropriate modelling software for this research program. 5.5.1 ETABS Model Properties, Assumptions and Development Process A complete description of the construction of the E T A B S model of the test specimen is located in Appendix F. The main aspects of the E T A B S model are summarized below. The E T A B S model o f the test specimen incorporated the measured properties o f the structure (geometry, cross-sections and material properties) described in Section 3. Line elements drawn at the centroid of the section were used to represent the beams and columns in the structure. Shell elements were used to model the slabs; the shell elements were located at the mid-depth of 87 the slabs. It is important to note that the beam elements in the model represent just the portion of the beam from the underside of the slab to the underside of the beam. Modell ing the beams in this way eliminated any overlap between the beam and slab elements, but necessitated the insertion o f stiff line elements to connect the slab elements to the beam elements. Stiff line elements were also inserted at beam-column joints to consider the higher rigidity within the physical dimensions of these joints. Figure 5.11 shows a sketch of E T A B S model; for clarity the shell elements modelling the slab are not shown. Figure 5.11 E T A B S Model of Test Specimen - Slab Not Shown For Clarity A s mentioned in Section 3.2, observations from pushover tests conducted on the field test site suggested fixed connections at the base of the columns. A s a result, fixed column supports were assumed in the model. A version of the model with pinned column bases was also analyzed. This pinned-based model was only slightly more flexible than the fixed-base version, and the difference in the response of the models during removal of column C-2 was not significant. Linear elastic material was assumed for all the elements in the E T A B S model due to the linear behaviour in the test specimen observed during testing. A s part of this assumption, E T A B S assumes that member stiffness does not change (i.e. members do not crack further or degrade, reducing their stiffness) during the course of analysis. The fact that cracking in the structure was 88 not observed during or after the explosive test supports the assumption that member stiffness remained constant. Construction of the E T A B S model proceeded in a number of stages. First, a version o f the model with Column C-2 intact was developed. The loads applied on this model consisted of the self-weight of the members calculated by E T A B S and an area load representing the ceramic tile and topping on the second floor slab. A linear static analysis o f the model was performed to determine the forces in this model at the top of the portion of Column C-2 removed by the explosive blast. A second version of the E T A B S model was then created by replacing the part o f Column C-2 destroyed during the explosive test (approximately the bottom 1.65 m) with the forces at this location determined in the first version of the model. The analysis results of the two models were then compared to confirm that the models were equivalent systems. The next stage in development of the E T A B S model was to calibrate it to the recorded data. The calibration process consisted of applying on the structure the opposite of the forces originally carried by Column C-2 and no other forces (this isolated the forces and deformations in the structure due to the loss of Column C-2), comparing the results of a linear static analysis of the model with the recorded data, revising the parameters of the model to address the discrepancies with the recorded data, and then repeating the calibration process. The calibration process was repeated until the steady-state deformations in the model matched closely with the recorded data at linear potentiometer locations and the period of the primary mode of vibration excited by the applied loading was 0.13 s. The parameters that were varied to calibrate the model were the elastic modulus E of the concrete for the entire model, the effective cross-section of the second floor beams cast on top o f masonry walls, and the stiffness o f the columns at B-2 and D-2. It should be noted that when the second floor beams and the columns at B-2 and D-2 were modified, the model with Column C-2 intact was revised accordingly and re-analyzed to update the equivalent force at C-2 (this equivalent force was not affected by changes in E). For the purposes of modelling, gross section properties were assumed and then E was reduced to account for the reduced stiffness in the beams and slabs of the test specimen due to the minimal cracking in these members before the explosive test. This reduced E also compensated for the non-uniform cross-sections of the columns, including the less-stiff grout and brick intermingled with the concrete in the columns, and the softened response these elements displayed. In the first 89 E T A B S model constructed, E for the concrete was calculated as 4500^/7 = 17016 M P a using the value offc'= 14.3 M P a determined from the testing o f cores taken from the buildings on the field test site (refer to Section 3.3). The above equation for E is from the Canadian Concrete Design Standard [ C S A , 2004]. The E calculated from this equation resulted in an E T A B S model with a considerably stiffer response than the recorded data. To get a response more similar to the recorded data, E was reduced in the calibrated E T A B S model to 12075 M P a . For the beams and slabs in the test specimen, this works out to an equivalent effective stiffness of 71% of the gross stiffness with E = 17016 M P a . A n E T A B S model matching the test results could not be obtained by simply revising E. It was noted that in the model at this stage, the rotations of the beam-column joints at B-2 and D-2 were too large and the displacement o f the beam west o f C-2 was too small. B y reducing the effective cross-section of the beams cast on top of masonry walls at the second floor (Grid 1, Gr id 2 except between Grids A & B and D & E , Grid C from Grid 1 to 2), it was found that the response of the E T A B S model matched much more closely the recorded data. Considering that the bottom parts of these beams, and in some cases the sides of the beams, consisted of a mixture of brick, grout and filler concrete (refer to Figure 3.7), it was deemed justified to reduce the effective size of these cross-sections to be closer to their design dimensions. Refer to Appendix F for the effective sizes used for these beams in the calibrated E T A B S model. Although the geometry at the north and south ends of the test specimen was somewhat different, the test specimen was for the most part symmetrical about Gr id C . Despite this apparent symmetry, the deformations recorded at Column D-2 were considerably larger than those at Column B-2 , its symmetrical counterpart. There are two explanations for this discrepancy: (1) the columns had about the same stiffness, but the floor structure around Column D-2 was considerably stiffer and redistributed more load to this column than Column B-2; and (2) the amount of load distributed to both columns was approximately the same, but the stiffness of Column D-2 was less than that of Column B-2, resulting in larger deformations at the former. Referring to Table 4.1 and the modified steady-state deformation at D-2 (S) discussed in Appendix E , the differences in the steady-state displacements of the instruments on opposite faces of Columns B-2 and D-2 are, respectively, 0.299 mm - 0.071 mm = 0.228 mm and 0.348 mm - 0.122 mm = 0.226 mm. The above differences in displacement are a measure of the 90 rotations of the respective beam-column joints. The fact that the rotations of the beam-column joints are so similar for the two columns suggests that it is differences in the stiffnesses of the columns that are the cause of the discrepancies in the overall displacements at the two columns, not differences in the stiffnesses o f the surrounding floor structure. Modell ing in E T A B S confirmed that unreasonable changes in floor structure stiffness were required to achieve the recorded deformations, further supporting the conclusion that differential column stiffness was the cause of the differences in the recorded deformations at Columns B-2 and D-2. Based on the rationale above, the effective dimensions of Columns B-2 and D-2 were adjusted in the E T A B S model to modify the stiffness of these members. The effective size of Column B-2 was increased to 365 mm x 500 mm and the effective size of Column D-2 was reduced to 325 mm x 435 mm; both columns were originally entered in to E T A B S at the design column dimensions of 350 mm x 400 mm. Considering the large amount of variability in the cross-sections of the columns in the test specimen, and high degree of uncertainty in column stiffness identified by the testing described in Section 3.5, modifying the size of Column B-2 and D-2 was decided to be justified. Once the E T A B S model itself was calibrated to the recorded data, it was necessary to apply the loading functions developed in Section 5.4 to the model, perform a linear dynamic analysis, and revise these functions and the damping of the model as necessary to achieve a reasonable match with transient response recorded during the explosive test. After numerous iterations it was found that the best model results were obtained by using a damping ratio o f 3.0%, a step load o f the forces originally carried by Column C-2 applied at 0.13 s, and an upward triangular pulse from 0 s to 0.13 s increasing from 0 to Vi the axial force originally carried by Column C-2. 5.5.2 Comparison of ETABS Model to Recorded Data - Steady-State Response Table 5.1 compares the steady-state results generated by the calibrated E T A B S model with the steady-state deformations recorded during the explosive test. The recorded deformations in Table 5.1 are from the linear potentiometer data collected (including the modified linear potentiometer data at D-2 (S)) except at the locations noted, where only a dial gauge was installed on the test specimen. Linear potentiometer data was used in favour of dial gauge readings when possible because the degree of confidence in the linear potentiometer data is 91 higher than that of the dial gauges. The deformations at C-2 (E) are not included in Table 5.1 since this instrument was disturbed during testing. Table 5.1 Comparison of ETABS Model and Recorded Steady-State Deformations Instrument Location - Grid Intersection (Side of Column): Source A-2 B-l B-2 (N) B-2 (S) C-l ; "*C-2 (W) D-1 D-2 (N) D-2 (S) E-2 Recorded 0.038 0.037 -0.071 -0.299 -0.080 -4.205 0.010* -0.348 -0.122 0.010* Model 0.042 -0.017 -0.085 -0.284 -0.057 -4.393 -0.017 -0.346 -0.122 0.027 Note: All deformations in mm, upward deformations positive; * indicates dial gauge reading Referring to Table 5.1, the steady-state deformations from the model agree quite well with the recorded data in most cases. The recorded and model deformations are within 0.0025 mm at all locations except B - l , D-1 , and C-2 (W). This is about the same degree of correlation as was demonstrated between the linear potentiometer and dial gauge data in Table 4.1, indicating that the discrepancies Table 5.1 may be attributable to instrument error. A t C-2 (W), the overall displacement is quite large and the percentage difference between the deformations amounts to less than 5%; thus, this discrepancy is not likely that significant. A t B - l and D-1 , however, the overall displacement is quite small, suggesting the differences between the recorded and model deformations may be significant. Referring back to Table 4.1, there was a considerable difference between the dial gauge and linear potentiometer readings at B - l . The dial gauge recorded a steady-state displacement of 0.01 mm downwards, a measurement which agrees quite closely with the model value of-0.017 mm. Thus, it appears that the linear potentiometer data at B - l may be in error and wi l l therefore be used with caution for the remainder o f this study. A s for the deformation at D-1 , since there was only a dial gauge installed at this location, it is difficult to verify the accuracy of the recorded data. Based on the ability of the computer model to accurately estimate the steady-state deformations elsewhere in the structure, it seems likely that the dial gauge at D-1 may have been off by 0.02 or 0.03 mm. A s wi l l demonstrated next, however, the possible instrument errors at B - l and D-1 may not be that relevant. In Section 4.2.2, the steady-state deformations recorded by the instruments were examined in a preliminary attempt to determine the spatial redistribution of the gravity load originally 92 supported by Column C-2. The largest deformations were recorded in the instruments flanking Columns B-2 and D-2, implying a significant amount o f gravity load redistributed to these columns. Reasonably large instrument deformations were also recorded at A - 2 , B - l and C - l , but it was alluded to in Section 4.2.2 that later modelling would show that these instrument deformations did not necessarily translate into significant deformations at the column center. Table 5.2 compares steady-state deformations at instrument locations relative to deformations at column centers. In the first two rows of Table 5.2, deformations from the E T A B S model are shown. Comparing these two sets of the deformations, it is apparent that the deformation at the center of a column, in particular the columns at B - l , C - l and D-1 , can be significantly less than the deformation at an instrument located approximately 100 mm from the face of that column (refer to Appendix C for the exact locations of the instruments relative to the nearest column center). In the last row of Table 5.2, column center deformations are estimated for the recorded steady-state deformations. The estimates of recorded column center deformations were calculated assuming the relationship between deformation at the instrument location and the column center for the E T A B S model is representative o f this relationship in the actual test specimen. Comparing the estimated column center deformations for the recorded data with Table 5.2 Comparison of Deformations at Instrument Locations and Column Centers type of Steady-State Deformation Instrument Location - Grid Intersection (Side of Column) A-2 B-1 B-2 (N)* B-2 (S)* C-l D-1 D-2 (N)* D-2 (S)* Instrument -Model 0.042 -0.017 -0.085 -0.284 -0.057 -0.017 -0.346 -0.122 0.027 Column Center - Model 0.035 -0.001 -0.175 -0.175 -0.012 -0.001 -0.225 -0.225 0.034 Instrument -Recorded 0.038 0.037 -0.071 -0.299 -0.080 0.010 -0.348 -0.122 0.010 Column Center - Recorded** 0.032 0.002 -0.175 -0.175 -0.017 0.001 -0.225 -0.225 0.013 Note: All deformations in mm, upward deformations positive * indicates average of instruments on each side of column compared to column center displacement ** estimated deformation at column center from recorded data assuming joint rotations from ETABS model 93 those from the E T A B S model, discrepancies in deformation at B - l and D-1 that looked significant at the instrument locations, appear irrelevant when the values at the column centers are considered. Another interesting insight from looking at the column center deformations is the magnitude of the deformations at Columns B-2 and D-2 relative to the other columns. Considering that center of column deformation is directly related to change in axial load, the much larger column center deformations at B-2 and D-2 suggest that a very large proportion of the gravity load originally supported by Column C-2 redistributed to these two columns. This point wi l l be discussed further in Section 5.6. The period o f the primary mode o f vibration in the E T A B S model was 0.130 s. This period matches very closely with the spike in the Fourier Spectra of the recorded data (refer to Figure 5.2) and the period of the dominant mode of oscillation noted in the recorded data (refer to Figure 4.4, Figure 4.5 and Figure 4.6). The model was checked for modes close to 0.244 H z and 1.95 Hz , the other more minor frequencies identified in Fourier Spectra o f the recorded data (refer to Section 5.2). There were no modes in the model close to 0.244 Hz , and the only mode near 1.95 H z (at 1.91 Hz) is a lateral mode with virtually no vertical component. Thus, it appears that these minor spikes in the Fourier Spectra of the recorded data may be a side effect of the F F T algorithm rather components of the response of the test specimen Table 5.1 and Table 5.2 demonstrate the ability of the E T A B S model to approximate the steady-state response of the test specimen. It must still be shown, however, that the model can effectively capture the transient response of the structure during explosive testing. 5.5 .3 Comparison of ETABS Model to Recorded Data - Transient Response Figure 5.12 compares the transient response of the E T A B S model to the data recorded by the linear potentiometer at B-2 (S). Similar to the S D O F model, the E T A B S model response is not able to capture the high frequency oscillations in the recorded data before 0.13 s or some of the extremes in the recorded response, particularly the first two downward deformation peaks. In general, however, the E T A B S model response does quite a good job of capturing both the period and magnitude of recorded data. That being said, the damping of the system and maximum magnitude of the triangular explosive pulse were chosen to get the best fit to the data at B-2 (S). To determine the overall adequacy of the E T A B S model, the model and recorded response must be compared at a number of other locations. 94 0 0.2 0.4 0.6 0.8 1 T i m e (s) Figure 5.12 Compar ison of E T A B S M o d e l Response and Recorded Data at B-2 (S) Figure 5.13a compares the E T A B S model response and recorded data at D-2 (N), while Figure 5.13b compares the model and recorded response at the instrument location near Column C - l . Not surprisingly, the match of the model response to the recorded data at D-2 (N) and C - l is not as good as it is at B-2 (S). That being said, the model response at D-2 (S) and C - l still captures the period of the recorded data and the general trend of the recorded response. A s was the case at B-2 (S), the model response at D-2 (N) and C - l is missing the high frequency oscillations present in the recorded response, in particular the large downward spike in the C - l recorded data at 0.06 s, and fails to estimate the magnitudes of the peaks in the recorded response at some locations. A t D-2 (N), the model response significantly overshoots the recorded data at the first downward deformation peak and the first two upward deformation peaks. Inspecting closely the recorded response at this location, though, the magnitude at the bottom of the first downward peak is less extreme than the second downward peak (the same behaviour is also present at the second upward peak), and the top of the first upward peak is suspiciously flat for an extended time period. These observations go against the expected behaviour of the structure and suggest that the instrument readings at these extremes may be inexact or some unorthodox behaviour was taking place during the test at this location. If the 95 T i m e (s) (a) Comparison of Response at D-2 (N) 0.3 T T i m e (s) (b) Comparison of Response at C - l Figure 5.13 E T A B S M o d e l Response and Recorded Data at D-2 (N) and C - l recorded data looked more as would be expected, i.e. the first downward peak continued down past the magnitude of the second downward peak, the first upward peak continued up as opposed 96 to levelling off, and the second upward peak continued up past the magnitude of the third upward peak, the match with the model response would be considerably better. At C - l , the model response consistently overestimates the recorded data, although at one location there is a flat spot in the recorded response similar to D-2 (N), suggesting possible instrument error at this location. Despite overestimating the peaks in the recorded response, the model response does capture the spikes at 0.16 s and 0.19 s quite well . In Section 4.2.3, it was suggested that despite their unconventional shape, these spikes might represent the actual behaviour of the structure due to the fact that the spikes were present in both the linear potentiometer and PI transducer data. The fact that the E T A B S model also exudes this spike phenomenon further supports the assertion that the specimen actually deformed in this manner. This matter w i l l be explored further when the transient deformations and axial force changes at the column centers are examined in Section 0. Although it is by no means perfect, the E T A B S model does a reasonably good job o f approximating the transient and steady-state response of the test specimen recorded during the explosive test. The discrepancies between the model and recorded response are quite significant in some instances, but the general trends in the data agree quite well . Errors in the model, errors measuring the deformation of the structure, and the possibility for some limited nonlinear behaviour are all possible contributors to the discrepancies noted between the model response and recorded data. Although it is difficult to pinpoint the exact cause of any given discrepancy, the different potential sources of error w i l l be considered when the model is used to assess dynamic amplification in the test specimen in Section 0. Before concentrating on dynamic amplification, however, some of the significant parameters in the E T A B S model w i l l be examined for their effects on the generated response. 5.5.4 ETABS Model - Effects of Damping, Triangular Pulse and Primary Mode The damping ratio and maximum magnitude of the upward triangular pulse giving the best results for the E T A B S model were somewhat different than those used in the S D O F model. For the E T A B S model, a damping ratio of 3.0% was used and the maximum magnitude of the triangular pulse was set at 0.5 times the axial force originally carried by Column C-2. For the S D O F model, the damping ratio was 3.5% and the maximum magnitude of the triangular pulse was equivalent to 1.0 times the axial force in Column C-2. However, the S D O F model only 97 considered the primary mode of oscillation in the response (the oscillation with a period of 0.13 s) whereas the E T A B S model took into account all the modes of the structure. To try and reconcile the differences in the parameters input into S D O F and E T A B S models, the effects of damping and maximum triangular pulse magnitude on the E T A B S model response were examined, as was the response in the primary mode of vibration of the structure. Initially, the loading functions and damping from the S D O F model were tried for the E T A B S model. The E T A B S model response at B-2 (S) for 3.5 % damping and a maximum triangular pulse magnitude of 1.0 times the original axial force in Column C-2 is shown in Figure 5.14. These parameters result in pretty a good approximation o f the recorded response at B-2 (S), although the model response for this case does miss the trend of the data over the first 0.13 s and the magnitude of the first few deformation peaks, similar to the S D O F response in Figure 5.10. A t locations other than B-2 (S), however, the model response of these initial parameters grossly overestimates the peaks in the recorded data. Referring back to Figure 5.13, which shows the model response at D-2 (N) and C - l for a maximum pulse magnitude of 0.5 times the axial force in Column C-2, it is easy to see that a maximum triangular pulse magnitude double that for the model response in the figure is not going to provide a very good match with the recorded response. The model response in Figure 5.13 already overestimates the magnitude of some of the peaks in the recorded data; increasing the maximum magnitude of the triangular pulse only aggravates this problem. Because of the poor match elsewhere in the test specimen for the S D O F model parameters, a maximum triangular pulse magnitude of 0.5 times the original axial force in Column C-2 was tried; the response for this version o f the model is also shown in Figure 5.14. Dividing the maximum triangular pulse magnitude by two reduces the magnitudes of the peaks of the model response, especially in the first 0.5 s. The result was a model response that still approximates the recorded data at B-2 (S) reasonably well , but does a much better job of simulating the recorded behaviour in other parts of the structure. It appears that the main reason that the best maximum triangular pulse magnitude for the S D O F model was not the best magnitude for the E T A B S model was that the S D O F model was only calibrated using the recorded data at one location. The triangular pulse magnitude from the S D O F model actually generated a response with a slightly better match to the B-2 (S) recorded 98 0 0.2 0.4 0.6 0.8 1 Time (s) Figure 5.14 Impact of Pulse Magnitude on Model Response at B-2 (S ) : 3.5% Damping data at than the pulse magnitude that was eventually decided upon. Elsewhere, though, the E T A B S model results generated using the S D O F model pulse magnitude yielded a poor match with the recorded response. Thus, i f the S D O F model with a maximum triangular pulse magnitude of 1.0 times the original axial force in Column C-2 were applied at other locations in the test specimen, it is likely that the S D O F model response would not have matched the recorded data well . Since the purpose o f the S D O F model was merely to confirm the feasibility of the general shape of the proposed load functions, applying the model at multiple locations was determined to be unnecessary. Having determined the maximum triangular pulse magnitude for the E T A B S model, the effects of damping on the model response were then investigated. Figure 5.15 shows the response at B -2 (S) resulting from three different damping ratios. Consistent with linear elastic theory, the effects of higher damping ratios on the model response are very small at the start of transient motion and become more and more pronounced with increasing time. For all practical purposes, however, the effect of damping on the model response is quite limited. Nonetheless, a value of damping had to be chosen for the model. In studying the response at B-2 (S) and a number of other locations in the test specimen, it was determined that a damping ratio of 3% results in the best overall match with the recorded response. 99 2 % Damping 4% Damping -0.8 0.2 0.4 0.6 T i m e (s) 0.8 Figure 5.15 Impact of Damping on Model Response at B-2 (S) To explain the ability of the S D O F model to approximate the recorded data quite well despite being only able to consider a single mode of vibration, the response of the E T A B S model in just its primary mode was isolated. Figure 5.16 shows the E T A B S model response at B-2 (S) due to E T A B S Mode l Response - A l l Modes E T A B S Model Response - Primary Mode Only -0.8 4 0 0.2 0.4 0.6 T i m e (s) 0.8 Figure 5.16 Response of Primary Mode of Vibration in the E T A B S Model at B-2 (S) 100 just this primary mode for 3% damping and a maximum triangular pulse magnitude of 0.5 times the original axial force in Column C-2. Referring to Figure 5.16, the vast majority o f the response of the E T A B S model is in its primary mode of vibration. Before 0.35 s, higher mode effects amplify the response in the primary mode but from 0.5 s to 0.7 s, these higher mode effects actually reduce the magnitude of the total response; beyond 0.7s, the impact of higher modes is relatively small. The predominance of the primary mode in the E T A B S model response explains why the S D O F model was able to approximate the recorded data so well despite only considering one mode of vibration. 5.6 Dynamic Amplification in Test Specimen from 3D ETABS Model In Section 5.5, a 3D E T A B S model of the test specimen was described and the response from this model was compared to the recorded response at a number of locations. It was determined that the E T A B S model does a reasonably good job of approximating the steady-state and transient deformations of the test specimen recorded during the explosive test. That being the case, this section w i l l describe how the E T A B S model was used to approximate the axial force changes in the test specimen and assess the dynamic amplification in the structure due to just gravity load redistribution as well as the total loading on the test specimen. To get an idea of the individual contributions to the total response of the E T A B S model made by the triangular pulse representing the upward explosive force and the step load representing the gravity load originally supported by Column C-2, the deformation response at B-2 (S) due to each of these loading functions is plotted in Figure 5.17 along with the total model response. Referring to Figure 5.17, the total model response and step force response both oscillate about the same steady-state displacement. Since it is a temporary rather than permanent load on the structure, the triangular pulse oscillates about a displacement of zero. A s discussed in Section 4.2.3, the peak theoretical displacement due to a step load in a linear elastic S D O F system is twice the steady-state displacement. A s was the case for the recorded data (refer to Figure 4.4), the peak displacement of the total model response exceeds the theoretical linear elastic limit. However, the peak displacement for the step load response is around 1.75 times the steady-state displacement and therefore obeys linear elastic theory. Noting that the peak demands in the model due to the step load, i.e. gravity redistribution, were less than twice the steady-state 101 demands at one instrument location, the more critical case of peak demand at column centers wi l l be examined next. -0.7 0 0.2 0.4 0.6 0.8 1 Time (s) Figure 5.17 Components of Triangular Pulse and Step Force in Total Model Response Thus far in this report, discussion of the structural response of the test specimen during the explosive test has focused on deformations, and for the most part, deformations at instrument locations. The goals of this research program, however, all address dynamic amplification, which is a phenomenon concerned with the forces in a structure during load redistribution. Specifically, this research program set out to assess the dynamic amplification of the axial force changes in the remaining columns of the test specimen following removal of Column C-2. Thus, the focus in this section wi l l be on demands at column centers not instrument locations, and forces rather than deformations. Table 5.3 summarizes the axial force changes in the columns of the E T A B S model for a number of different loading conditions. The first row of the table displays the steady-state change in the compressive force of the columns in the model. The second row illustrates the percentage of the 206.5 k N compressive force originally carried by Column C-2 in the E T A B S model that is redistributed to each of the remaining columns in the steady-state. Consistent with the past research [Sucuoglu et al., 1994], a huge majority of the load in Column C-2 redistributed to nearest columns, Columns B-2 and D-2. Note, however, that since the steady-state compressive 102 load in some of the columns decreased as a result of the removal of Column C-2 (e.g. Columns A - 2 and E-2), the steady-state, gravity load redistributed to Columns B - 2 and D-2 is actually greater than the total gravity load originally carried by Column C-2, an interesting result. The primarily 2D nature of the load distribution and the considerably lower stiffness of the exterior columns of the frame along Grid 2 contributed to the surprisingly large steady-state axial force increases at Columns B - 2 and D-2. T a b l e 5.3 S u m m a r y o f A x i a l F o r c e C h a n g e s i n E T A B S M o d e l Type of A x i a l Force Measure G r i d In te rsec t ion o f C o l u m n / A - r A - 2 * B - l • B-2 * C - l D-1 D-2 E - l . E-2 Steady-State Increase (kN) -3.3 -18.7 0.3 122.5 5.9 0.4 120.5 -3.1 -18.0 % of Redistributed Gravity Load -1.6 -9.0 0.1 59.3 2.9 0.2 58.3 -1.5 -8.7 Peak Increase -Gravity Load (kN) -13.4 -40.4 35.6 231.9 70.7 38.1 227.4 -13.6 -39.9 Dynamic Amplification 4.13 2.17 143 1.89 11.9 97.7 1.89 4.37 2.21 Peak Increase -Total Load (kN) -18.0 -51.3 53.4 287.9 102.0 57.0 282.3 -18.5 -50.8 Amplification of Peak Gravity Load 1.34 1.27 1.50 1.24 1.44 1.50 1.24 1.36 1.27 Maximum Axial Stress (MPa) 1.07 1.83 1.32 3.81 1.53 1.35 3.74 1.07 1.84 Note: compressive forces are positive The third row in Table 5.3 gives the peak axial force increase due to gravity load redistribution in the direction of the steady-state. A t the columns where the steady-state compressive force went down due to the removal of Column C-2 (Columns A - l , A - 2 , E - l and E-2), the compressive force in these columns did increase at certain times during the transient response of the model but the magnitudes of these increases were quite small; the largest transient compressive force increase for these columns, at Column E - l , was a mere 9.9 k N . The fourth row in the table, dynamic amplification, is the most important. Dynamic amplification was calculated by dividing the peak axial force increase in the direction of the steady-state by the steady-state change in axial force. At first glance, the dynamic amplifications 103 in the model are quite startling; only at Columns B-2 and D-2 is the dynamic amplification less than 2.0, the upper limit based on linear elastic theory. Further investigation of the large dynamic amplifications at many of the columns, however, reveals that these surprisingly high dynamic amplifications are not as critical as they first appear. Referring to the third row in Table 5.3, the peak increases in axial force due to gravity load redistribution at Columns B - l , C - l and D-1 are not large, at most 31% of the peak gravity load increase at Column B-2 or D-2. Since the steady-state increases in axial load at these locations are so small, however, the dynamic amplification calculated is deceptively large, especially at Columns B - l and D-1. Thus, it would appear that for columns where the steady-state change in axial force is small, dynamic amplification is not a good measure of peak demand. A t columns where a substantial amount of gravity load redistributes, Columns B-2 and D-2 for the test specimen, doubling the steady-state axial force increase appears to conservatively estimate the peak force increase from redistribution of gravity load. For columns where the steady-state axial force increase is small, however, doubling this steady-state demand increase grossly underestimates the peak redistributed forces. If the column is not heavily loaded before load redistribution, underestimating the peak forces probably doesn't matter. However, i f a column near a failed element is heavily loaded before column failure and does not receive a large steady-state increase in axial load from load redistribution, the fact that doubling the steady-state axial force increase to account for dynamic effects underestimates the peak forces on the column would be of concern. A more appropriate procedure for such cases would be to check the heavily-loaded column for twice the maximum steady-state axial force increase in the structure or perform a dynamic analysis to directly determine the peak loads in the structure. Dynamic amplification for the columns in the model where steady-state compressive force decreased due to load redistribution (Columns A - l , A - 2 , E - l and E-2) is also greater than 2.0. Similar to Columns B - l , C - l and D-1 , however, the peak gravity force redistributed to Columns A - l , A - 2 , E - l and E-2 is quite small. More importantly, the steady-state axial force changes and peak increases in the direction of the steady-state reduced the demands on the columns, a beneficial effect considering that the peak decreases in axial load were nowhere near great enough to put the columns into tension. A s discussed earlier in this section, the compressive force in Columns A - l , A - 2 , E - l and E-2 did increase at certain times during the transient response but the magnitudes of these increases were quite small. In the more general case, it 104 would appear that i f a column experiences a steady-state decrease in compressive force due to load redistribution, the actual dynamic effects on the column w i l l not be represented well by a dynamic amplification factor of 2.0. This being said, unless the column is heavily loaded in compression or tension, the dynamic effects on the column w i l l not be significant enough to affect the load-carrying ability of the member, making the inaccuracy in the dynamic amplification factor irrelevant. Focusing now on the critical columns receiving the highest amount of redistributed load, Columns B-2 and D-2, the E T A B S model calculates at dynamic amplification factor of 1.89 for these critical columns. In Section 2.1.6, it was noted that both the G S A and U F C progressive collapse resistance guidelines stipulate a dynamic amplification factor of 2.0 to consider the peak demands on a structure during load redistribution. Similarly, analytical research by Ruth [2004] and others (refer to Section 2.2) also proposes a dynamic amplification factor of 2.0 for linear structures. In all these sources, the dynamic amplification of 2.0 is based on linear elastic theory. The results of the E T A B S model in this study suggest that the dynamic amplification factor of 2.0 suggested by current guidelines and analytical research is appropriate for members receiving substantial amounts of redistributed load in the steady-state. A s discussed in previous paragraphs, however, dynamic effects must be approximated differently for heavily-loaded structural members receiving low amounts of redistributed load in the steady-state. The fifth row in Table 5.3 displays the peak axial force increase in the columns of the E T A B S model due application o f both the gravity step load and explosive triangular pulse on the structure. In all cases the explosive force amplified the peak force redistributed to the columns. H o w much the explosive force amplified the peak axial force due to gravity load redistribution is shown in the sixth row of Table 5.3. The upward force from the explosion increased the peak demands in the model from gravity load redistribution by 24% at critical column locations (B-2 and D-2) and by as much as 50% at columns where less gravity load was redistributed. The triangular pulse also magnified the maximum compressive force increase at Columns A - l , A - 2 , E - l and E-2, although this overall increase is still quite small (a maximum of 16.8 k N at E - l ) . The model results above suggest that the upward force applied by the explosion at Column C-2 significantly amplified the peak demands on the test specimen. For design cases where explosions from within structural members are considered, the conclusion is that amplification from the explosion itself must not be neglected. 105 The sixth and final row of Table 5.3 shows the maximum axial stress in the columns of the E T A B S model due to peak force from all the loads on the structure: the step force representing the failure of Column C-2, the upward triangular pulse from the explosion and the self-weight of the structure itself. These maximum stress values approximate the largest stress experienced in the columns of the test specimen during the explosive test and were calculated to confirm that the columns deformed within the linear range during testing. Typically, concrete columns deform linearly up to an axial stress of 0.6 fc'. Considering the concrete strength o f the test specimen to be 14.3 M P a , the strength of the cores taken from the field site (refer to Section 3.3), the columns in the structure would be expected to deform linearly up to a stress of 8.58 M P a , a value considerably larger that the maximum axial stress of 3.81 M P a in Table 5.3. A s noted previously, there is some uncertainty in the actual strength and stiffness of the concrete in the test specimen. However, for a concrete strength as low as 6.35 M P a , the deformations of the columns would still be expected to linear. Thus, the assumption that the columns of the test specimen behaved linearly during testing appears to be valid. In Sections 4.2.3 and 5.5.3, large spikes were noted in both the recorded and model deformation response at the instrument location near C - l . A s shown in Figure 5.18, spikes are also present in the axial force response due to gravity load redistribution at the center of Column C-1 . Largely 0.2 0.4 0.6 T i m e (s) 0.8 Figure 5.18 A x i a l Force Response in E T A B S M o d e l at C o l u m n C - l - Grav i ty Loads 106 as a result of these spikes, the dynamic amplification at Column C - l is around 12 times the steady-data axial force increase. Also o f note is the fact that even after the large spikes in the response, the maximum magnitudes of the oscillations in axial force are still greater than twice the steady-state axial force increase. Earlier in this section, it was discussed that since the steady-state change in axial force at C - l is so small, the large value for dynamic amplification at this location is likely not that significant. Nonetheless, the shape of the response at C - l is quite peculiar and warrants further discussion. Referring to Figure 5.18, the effect of higher modes in the response at C - l , especially between 0.15 s and 0.6 s, appears to be greater than in the response at B-2 or D-2 (refer to Figure 5.16 and Figure 5.13, respectively). To determine what modes are contributing to the response at C - l and what the shapes of these modes are, the response for each individual mode was isolated in the E T A B S model. Most of the modes of the model do not contribute to the response at C - l . In fact, the response of all the modes of the model can be approximated fairly closely by just four of the modes of the structure. Figure 5.19 compares the total model response at C - l from gravity load redistribution with the aggregate response o f the four most significant modes and the individual response of the primary mode of the model. 0 0.2 0.4 0.6 0.8 1 T i m e (s) Figure 5.19 Participation of Modes in Axial Force Response at Column C-l 107 Referring to Figure 5.19, the combined response of the four significant modes effectively matches the total response from the first large upward spike in the total response onward. The main contribution of the rest of the modes in the model is to attenuate the first upward spike in the four-mode combination and amplify the first downward spike in the four-mode combination. The primary mode at C - l does not approximate the total response well until at least 0.6 s. The primary mode completely misses the big downward spike in the total response at 0.17 s and reaches less than half of the magnitude of the first two upward spikes in the total response. Contrast this with the case at B-2 (refer to Figure 5.16), where the primary mode traces the total model response quite closely for the entire response history. A t B-2 , higher mode effects are relatively negligible; at C - l , the influence of higher modes is significant, particularly with respect to the peak demands. The shapes of the four significant modes in the C - l response give an explanation as to why higher mode effects are important at C - l and not at B-2 . The vibrational shapes of the three higher modes that contribute significantly to the C - l response are all characterized by considerable movement of the cantilever on Gr id C extending out the front of the test specimen west of Grid 2. Vibrations in all three of the higher modes induce deformation at C - l , but essentially pivot about a point at C-2, explaining why the higher modes do not influence the response at B-2 significantly. In contrast, for the primary mode, the structure at C-2 moves essentially moves in-phase with the cantilever, i.e. there is not pivot at C -2, so the response in this mode registers both at B-2 and C - l . Further investigation in E T A B S revealed that higher modes, characterized usually by considerable vibration of the cantilevers on the west side of the test specimen, contributed greatly to the peak demands at all the columns along Grid 1, but did not generally have a large affect on the response at the columns along Grid 2. Hence, the higher dynamic amplification factors calculated at non-critical columns (mostly along Gr id 1) in Table 5.3 appear to be the result of higher mode effects influenced by the presence of significant cantilevers on the west side of the structure. The influence of higher mode effects and cantilevers on dynamic load redistribution has been neglected for the most part in past research on this topic. Future study looking at how different cantilever parameters affect the peak force increases during load distribution would be worthwhile and wi l l be identified as a topic requiring future research in Section 6.2. 108 In Table 5.3, the percentage by which the upward explosive force on the model magnified the peak axial force from gravity load redistribution was calculated. This table showed that the increase in peak axial force caused by the explosive force was significant at all columns. At most of the instrument locations in the test specimen, the total deformation response generated by the E T A B S model matches or slightly overestimates the magnitude o f the peaks in the recorded data, implying that the axial forces in the model conservatively approximate the actual axial force changes in the structure at these locations. For the instruments at A - 2 , B-2 (N) and B-2 (S), however, the total deformation response in the E T A B S model underestimates the peaks in the recorded data, in some cases significantly. The severity and consequences of this underestimation of peak demands by the E T A B S model wi l l now be discussed. For Column A - 2 , the maximum deformation response in the recorded data at the instrument location is approximately 17% higher than the maximum model response from both the step force and triangular pulse. One could magnify by 17% the peak axial force change at Column A - 2 due to the step load and triangular pulse on the system to compensate for the fact that the model underestimates the recorded data at the nearby instrument location. However, at Column A - 2 , the peak axial force change reduces the demand on the column and is therefore beneficial (the decrease in compressive load is not sufficient to put the column into tension). Thus, this is a somewhat redundant exercise and w i l l not be explored further. A t Column B-2, the redistribution of loads in the structure most definitely increases the demands on the column. Thus, the fact that the E T A B S model underestimates the recorded data at this location is very much of concern. To get an estimate of the changes in axial force at Column B -2 from the recorded data, the deformation response recorded at B-2 (N) and B-2 (S) was averaged and multiplied by the effective axial stiffness of the Column B-2 in the E T A B S model. This "recorded" axial force response is plotted in Figure 5.20 along with the model response at Column B-2 due to both the step load and triangular pulse. Referring to Figure 5.20, the steady-state axial force changes are the same for both responses and represent the 122.5 k N of gravity load redistributed to Column B-2. The peak axial force increase in the model response is 287.9 k N which is 2.35 times the steady-state axial force increase, and from Table 5.3, a 24% amplification of the peak axial force increase from gravity 109 -200 ± T i m e (s) Figure 5.20 Compar ison of Response from Explosive Pulse and Grav i ty L o a d at B-2 load redistribution. The peak axial force increase for the "recorded" response is 389.3 k N , a 3.18 amplification of the steady-state increase and 68% larger than the peak increase from gravity loads. The peak axial force increase for the "recorded" response at Column B-2 is considerably greater than the model response and represents a substantial increase in demand at the location in the structure already experiencing the highest levels of steady-state and peak axial force increase. In order to assess the significance of the high peak demands in the recorded response noted in Figure 5.20, it is first necessary to determine i f these high peak demands are the result of instrument error, abnormally high dynamic amplification of gravity loads or a concentrated increase in the response generated by the explosive force. If the cause is instrument error, then the peaks are actually not as large as suspected and likely no different than elsewhere in the structure; neither the data at B-2 (S) or B-2 (N) shows evidence of being in error, however. If the cause o f the high peak demands is abnormally large dynamic amplification o f gravity loads, this is very significant because it represents a departure from current research and guidelines, and has applications for all types of column failures; the fact that abnormally large dynamic amplifications are not found elsewhere in the model decrease the likelihood of this explanation. This leaves the explanation that the high peak demands in the recorded data are due to the explosive force inducing a greater response at B-2 than elsewhere in the structure. Why the 110 explosive force may have generated a greater response at B-2 is difficult to discern from the data, but one possibility is that the explosive force did not act on the structure symmetrically (i.e., it was not a force directed vertically upward at C-2) but rather in some manner that transferred more force to B-2 than elsewhere. In any case, the fact that an explosion inside a column appears to magnify peak forces during load redistribution differently at locations in a structure that appear symmetric is something that should be considered when column failures of this type are being analyzed. Regardless of what explanation above seems most likely for the unusually high peak axial force changes at Column B-2 , with the results from only one explosive test, it is difficult to identify with any certainty which of the above factors are actually contributing to the very large recorded response. This instance exemplifies the need for additional experimental testing on dynamic load redistribution, a topic that w i l l discussed in great detail in the next section. I l l 6 SUMMARY, CONCLUSIONS AND RECOMMENDATIONS 6.1 Summary of Research Program and Conclusions This research program set out to achieve three objectives: generate experimental data on the dynamic amplification that occurs during gravity load redistribution following a sudden column failure; verify the dynamic amplification factors in guidelines and analytical research on progressive collapse-resistant design; and assess an experimental test procedure for investigating dynamic load redistribution. The research program attempted to achieve the above objectives by performing a single experimental test on a reinforced concrete frame field specimen. This test consisted of removing a column with explosives and recording the deformations of the remaining columns in the structure. Observations from the test suggest the test specimen remained linear-elastic. The data collected was used to calibrate an analytical model of the structure that was in turn used to assess the dynamic amplification that occurred in the specimen during redistribution of the gravity loads originally supported by the removed column. The direction and shape of the research program were profoundly affected by the use of explosives for column removal. The explosives, detonated from within the column itself, effectively eliminated the ability of the column to support load but also exerted an unanticipated upward force on the structure. The response from this upward force mixed with and magnified the response from the redistribution of gravity load initiated by the failure of the removed column. The interference from the response of the unanticipated upward explosive force complicated the attainment of the first two objectives of the research program, which are focused on the response due to just gravity load redistribution, but also provided the opportunity to investigate, the impact of close-proximity blasts on the demands of a system, a topic not considered at the outset of the project. With respect to the first objective of the research program, the response recorded during the explosive test was the aggregate of the responses from the upward explosive force and failure of the removed column, rather than just the response due to gravity load redistribution. To isolate the response due to just gravity load redistribution, a linear analytical model of the test specimen 112 was developed, in which the failure of the removed column and upward explosive force were incorporated as loading functions. Considering both loading functions, the analytical model was able to effectively to capture the modes of response, steady-state deformations and peak transient demands on the test specimen recorded during the explosive test. Removing the loading function for the upward explosive force from the model, a reasonable approximate of the response of the test specimen due to just gravity load redistribution was generated, fulfilling the first objective o f the research program. The analytical model was then used to investigate several interesting aspects of the response of the test specimen, some focused on gravity load redistribution and others considering the effects of the upward explosive force on the structure. From the results of the analytical model, it was determined that the steady-state redistribution of load in the test specimen was concentrated overwhelmingly in the two columns closest to the removed element, one on either side of the removed column on the longitudinal frame line across the front of the test specimen. In fact, the sum of the steady-state axial force increases in these two columns was actually greater than the axial load originally carried by the removed member. Accordingly, the axial force in the exterior columns on this frame line decreased. The surprisingly high proportion of steady-state force redistributed to the columns next to the removed member is due to the primarily two-dimensional nature of the load distribution and the relative stiffnesses of the elements in frame along the front of the structure. The fact that the steady-state redistribution o f load in the test specimen concentrated in the columns closest to the removed element is consistent with previous research on this topic. The results of the analytical model were also used to achieve the second objective of the research program, verification of the dynamic amplification factors in progressive collapse-resistance design guidelines and previous research with the results from the experimental test. The analytical model estimated the dynamic amplification from gravity load redistribution of the steady-state axial load increases in the test specimen to be 1.89 in the critical columns directly adjacent to the removed column. This dynamic amplification factor agrees well with the maximum dynamic amplification of 2.0 proposed for linear structures (the test specimen responded linearly) by previous research, the G S A and U F C Guidelines, and linear elastic theory for S D O F systems. A t less critical columns in the test specimen, i.e. columns receiving a small proportion of the redistributed load in the steady-state, the analytical model calculated dynamic amplification 113 factors much larger than 2.0. For the test specimen, it was determined that these large dynamic amplification factors were not significant because the absolute peak force increases were not that great and the pre-test loads on the columns were much less than their axial capacities. However, for structures with columns heavily loaded prior to load redistribution, it was suggested that special measures such as dynamic analysis be used to determine peak axial force increases during load redistribution. For such cases, using a dynamic amplification factor of 2.0 in a static analysis underestimates the peak demands at columns receiving small amounts of steady-state redistributed load; for a heavily loaded column, using this underestimated peak demand from static analysis could lead to the conclusion that the column is adequate when the actual peak demand exceeds the capacity of the member. The larger dynamic amplification factors at less critical locations appear to be the result o f higher mode effects on the structure. Along the rear of the test specimen where the dynamic amplification factors are particularly large, higher modes involving the cantilevers along the front of the building contribute significantly to the peak demands at these columns. The potential for larger dynamic amplification factors at less critical locations, the overall effects of higher modes on dynamic amplification, and the influence of cantilevers on dynamic load redistribution are all topics not addressed by current research and guidelines. A s such, these topics w i l l be discussed further in Section 6.2. The results of the analytical model confirmed that the upward explosive force exerted on the test specimen significantly increased the peaks demands on the structure during load redistribution. In the analytical model, the response from the loading function representing the upward explosive force magnified the peak axial force increases due to gravity load redistribution by 24% at critical column locations and by as much as 50% at less critical columns. Similar to the case for dynamic amplification, higher mode vibrations in the cantilevers at the front of the structure appear to contribute to greater relative magnification of the column forces along the back o f the test specimen. A t one of the critical columns in the test specimen, the analytical model did not capture the full magnitude of the peaks in the deformation data. The recorded data suggests that the upward explosive force magnified the peak demand from gravity load redistribution at this column by 68%, rather than the 24% magnification proposed by the model. The greater influence of the 114 explosive force at this one location implies that the distribution of the explosive force on the structure was not symmetrical. The case of column failure from explosives embedded in the column is a less common failure condition than failure due to seismic forces or blasts originating outside the cross-section of a member. Nonetheless, it may be appropriate to consider this less common failure scenario in some situations. The results of this research program illustrate that the upward force given off by an explosion inside a column can significantly magnify the peak demands on a structure during load redistribution. In addition, the response induced by an upward explosive force may not be symmetrical. For design cases where explosions from within structural members are considered, the amplification and asymmetrical nature of the response resulting from the explosion must not be neglected. The third objective of the research program was to evaluate a testing method for investigating dynamic amplification. Some aspects of the test procedure worked well and others did not. The primary data collection system, consisting of linear potentiometers to record dynamic response data and dial gauges to verify steady-state deformations, worked quite well . The secondary data collection system, made up of PI transducers, experienced resonance problems and was generally unsatisfactory. Only a single experimental test was performed in the research program; this limited the extent to which the accuracy of the collected data, the analytical model and the observed patterns of behaviour in the structure could be verified. Many of the members in the test specimen selected for the explosive test had non-uniform cross-sections; choosing a specimen with non-uniform members was a poor decision. The non-uniform makeup of the columns in the test specimen contributed to the poor performance of the PI transducers, which relied upon the columns deforming uniformly to compare their data with the measurements of other instruments. The presence of non-uniform cross-sections also made calibration of the analytical model considerably more difficult and introduced uncertainty into the modelling. These modelling difficulties were exacerbated by incomplete information on the properties of the structure, in particular, the composition of several beam and column cross-sections. Finally, for the purposes of studying dynamic amplification due to the redistribution of gravity load, using explosives for column removal was ill-advised. A s discussed earlier in this section, 115 the explosives applied an upward force on the structure in addition to removing the load-carrying capacity of the column. Extensive analytical modelling was required to remove the response o f the upward explosive force, isolate the response of the test specimen due to just gravity load redistribution, and achieve the primary objectives of the research program. Using explosives did allow the effects of close-proximity blasts on dynamic load redistribution to be studied. However, the main purpose of this research program was to examine dynamic amplification from gravity load redistribution; for this purpose, column removal by some means other than explosives would have been more appropriate. 6.2 Recommendations for Future Testing One of the major motivations for conducting this research program was the lack of experimental testing focused on dynamic amplification during gravity load redistribution following a column failure. In this research program, one experimental test was performed on a reinforced concrete frame structure. Clearly, more experimental testing is required to improve understanding of dynamic load redistribution and verify i f the recommendations of previous analytical research and current progressive collapse-resistant design guidelines are appropriate. During such testing, special attention should be paid to the significant findings of this research program not addressed by previous studies: large dynamic amplification factors at less critical locations, the effects of higher modes on dynamic amplification and the influence of cantilevers on dynamic load redistribution. Since experimental testing on dynamic load redistribution is a relatively new research area, there are many different parameters that need to be investigated in order to achieve a full understanding on how this process works. To start with, it would likely be prudent to concentrate testing on two-dimensional (2D) specimens and establish a thorough understanding of dynamic amplification in 2D before expanding testing to more complex 3D specimens. Future experimental testing should examine different plan layouts (including cantilevers) and number of stories, as well as different types of structural systems (flat plates, flat slabs) and building materials (structural steel, timber). Instrumentation in future tests should be expanded to measure not only the axial force changes in the columns of a test specimen, but also the shear, bending and axial forces in the slabs and beams, and the shear and bending in columns. Instrumenting a specimen in this way would 116 generate a much clearer picture of the response of all the elements in the structure, slabs, beams and columns. Strain gauges installed on concrete and reinforcing steel at critical sections could achieve this purpose. Further testing is still required on the dynamic amplification in structures that behave linearly during load redistribution, like the specimen in this research program, but the focus of experimental testing on this topic should be on structures that deform into the nonlinear range, since this is the type of behaviour that would be expected in most real-life structures suffering a column failure. It may be possible to combine testing on dynamic amplification in linear and nonlinear structures, however, by designing test specimens to deform linearly after the removal of one column and then nonlinearly after the removal of a second column. For the most part, it would make sense for future testing on dynamic load redistribution to focus on redistribution of gravity loads, since for the column failures most commonly considered in progressive collapse assessment (failures from earthquakes or blasts not originating from within the column), gravity load from the failed column is the only major vertical force redistributed. It would also be worthwhile, however, to conduct additional testing for the less common failure case of column failure from explosives detonated from inside a column. Such testing would further increase understanding of the effects on a structure from this failure mode and verify the conclusions o f this study that the explosive force significantly increases the peaks demands on the system and can affect the structure asymmetrically. Regardless o f the focus or parameters o f future experimental testing, there are some lessons from this research program that can be applied to help ensure satisfactory results. With respect to instrumentation, high-resolution linear potentiometers are recommended for the measurement of dynamic column deformations; static dial gauges mounted beside the linear potentiometers provide a good check on steady-state displacements. It is important to install these instruments on all faces of the column i f possible for redundancy and at the undersides of beams as close as possible to the column face to mitigate the effects of beam deformations. Test specimens that are fully documented and have uniform member cross-sections are recommended; these features make post-test modelling considerably easier. Experimental test programs should include a series of tests on similar specimens to verify the accuracy o f the data collected and the applicability of analytical modelling to different sets of data. Multiple tests and 117 uniform, well-documented specimens may not be possible on field testing projects; thus, experimental testing on dynamic load redistribution may be more suited to a controlled laboratory environment. Lastly and most importantly, column removal on future testing aimed at dynamic amplification due to gravity load redistribution should be achieved by some means other than the detonation of explosives embedded in the column itself. For field specimens, an alternate procedure would be to shore a column, slice out a partial length of the column, insert a removable shim, remove the shores and then rapidly remove the shim at test time. For laboratory specimens, a removable shim could be cast into the column designated for failure during the test. Another testing option would be to use oi l jacks to shore the structure around a column, cut the column free and then release the oi l pressure in the jacks suddenly. 118 REFERENCES American Concrete Institute (ACI) , 2005, "Building Code Requirements for Structural Concrete", ACI 318-05, A C I , U S A . American Society of C i v i l Engineers ( A S C E ) , 2005, "Min imum Design Loads for Buildings and Other Structures", ASCE 7-05/ANSIA58, A S C E , U S A . Breen, J .E . and Seiss, C P . , 1979, "Progressive Collapse - Symposium Summary", American Concrete Institute (ACI) Journal, V o l . 76, No . 9, pp. 997-1004. British Standards Institute (BSI), 1997, "Structural Use of Concrete, Part 1: Code of Practice for Design and Construction", BS8110-1:1997, BSI , U K . Canadian Standard Association (CSA) , 2004, "Design of Concrete Structures", CSA-A23.3-04, C S A , Canada. Chen, G . X . , 2005, "Geological Draining Work Report - Renovation Project for Old Schoolhouse of Ruei-Pu State Elementary School, Yangmei Town, Taoyuan County [Translated from Chinese]", Site Soils Report, Lianshin Engineering Consulting Company Ltd, Taiwan. Chen, J . H . , 2004, "Safety Verification Report for Old School Structure of Ruei-Pu State Elementary School, Yangmei Town, Taoyuan County [Translated from Chinese]", Building Assessment Report, Taiwan Professional C i v i l Engineers Association, Taiwan. Chopra, A . R . , 2000, "Dynamics of Structures: Theory and Applications to Earthquake Engineering, Second Edition", Prentice Hal l . Comite Europeen de Normalization 250 (CEN) , 1994, "Eurocode 1-Actions on Structures, Part 1-Basis of Design", European Prestandard ENV1991-1, C E N , Belgium. Computers and Structures Inc. (CSI), 2004, "SAP2000 Basic Analysis Reference Manual, Version 9.0", Software User Manual, Computers and Structures Inc., U S A . 119 Computers and Structures Inc. (CSI), 2005, " E T A B S User's Guide, Version 9.0", Software User Manual, Computers and Structures Inc., U S A . Frequency Devices Inc., 1999, "Appendix A - Low-Pass 4-Pole Butterworth", Online Specification Sheet, http://www.freqdev.eom/a/7.html, Frequency Devices Inc., U S A . Harris, J., 2007, "Recommendations for Dynamic Amplification for Disproportionate Collapse Analysis o f Tall Buildings", First International Conference on Construction Materials: Performance, Protection and Strengthening of Structures Under Extreme Loading, PROTECT2007 , Canada. Huang, Y . C . , 2007, "Retrofit of School Buildings by Adding Composite Column to Partition Brick Walls: In-Situ Experiment [Translated from Chinese]", Master's Thesis, National Taiwan University, Taiwan. Hwang, S.J., Chung, L . L . and Weng, Y . T . , 2006, "In-Situ Pushover Test of Reui-Pu Elementary School Buildings in Taiwan", National Centre for Research on Earthquake Engineering (NCREE) Newsletter, V o l . 1, N o . 4, pp. 2. Kaewkulchai, G and Williamson, E . B . , 2002, "Dynamic Progressive Collapse of Frame Structures", 15th Engineering Mechanics Division Conference, A S C E , U S A . Mathworks Inc., 2004, "Programming, Computing and Graphics with M A T L A B " , Software User Manual, The Mathworks Inc., U S A . Matthews, T., Elwood, K . J . and Hwang, S.J., 2007, "Explosive Testing to Evaluate Dynamic Amplification During Gravity Load Redistribution for Reinforced Concrete Frames", 2007 ASCE Structures Congress, Long Beach, U S A . M S I Sensors, 2006, "Model 3028 Accelerometer", Online Specification Sheet, http://www.meas-spec.cn/manage/sensortypes/M_3028.pdf, M S I Sensors, U S A . National Institute of Science and Technology (NIST), 2007, "Best Practices for Reducing the Potential of Progressive Collapse in Buildings", NISTIR 7396, NIST, U S A . National Instruments (NI), 2006, " S C X I - U S B Data Acquisition and Switching System", Online Specification Sheet, http://zone.ni.eom/devzone/cda/tut/p/id/4021, NI , U S A . 120 National Research Council of Canada ( N R C C ) , 1995, "National Building Code of Canada ( N B C C ) , Part 4 and Commentary C , 1995 Edition", NBCC 1995, N R C C , Canada. Novotechnik, 2006, "Potentiometric Products", Online Specification Sheet, http://www.novotechnik.com/linear.html, Novotechnik, U S A . Pretlove, A . J . , Ramsden, M . and Atkins, A . G . , 1991, "Dynamic Effects in Progressive Failure of Structures", International Journal of Impact Engineering, V o l . 11, N o . 4, pp. 539-546. Ruth, P., 2004, "Dynamic Considerations in Progressive Collapse Guidelines", Master's Thesis, University o f Texas at Austin, U S A . Ruth, P., Marchand, K . A . and Williamson, E . B . , 2006, "Static Equivalency in Progressive Collapse Alternative Path Analysis; Reducing Conservatism While Maintaining Structural Integrity", Journal of Performance of Constructed Facilities, V o l . 20, No . 4, pp. 349-364. Sasani, M . and Kropelnicki, J., 2007, "Progressive Collapse Analysis of an R C Structure", The Structural Design of Tall and Special Buildings, http://www3.interscience.wiley.com/cgi-bin/ fu l l text /114297665/PDFSTART?CRETRY=1&SRETRY=0, John Wiley & Sons, Ltd. Smilowitz, R., 2002, "Analytical Tools for Progressive Collapse Analysis", Multihazard Mitigation Council of the National Institute of Building Sciences (NIBS) Workshop for Prevention of Progressive Collapse, N I B S , U S A . Sucuoglu, H . , Citipitioglu, E . and Al t in , S., 1994, "Resistance Mechanisms in R C Building Frames Subjected to Column Failure", Journal of Structural Engineering, V o l . 120, N o . 3, pp. 765-782. Tokyo Sokki Kenkyujo Co., Ltd. ( T M L ) , 2006a, "Displacement Transducers - D D P - A Displacement Transducer", Online Catalogue, http://www.tml.Jp/e/product/transducer/catalog_PDF/DDP-A_DP-D.pdf, T M L , Japan. Tokyo Sokki Kenkyujo Co. , Ltd. ( T M L ) , 2006b, "Displacement Transducers - PI Displacement Transducers", Online Catalogue, http://www.tml.Jp/e/product/transducer/catalog_PDF/PI.pdf, T M L , Japan. 121 Tokyo Sokki Kenkyujo Co. , Ltd. ( T M L ) , 2006c, "Displacement Transducers - PI Displacement Transducers", Online Catalogue, http://vvww.tml.Jp/e/product/instrument/pdf/tds303.pdf, T M L , Japan. Tsai, M . H . , L u , J .K . and L i n , B . H . , 2007, "Progressive Collapse Analyses o f a Seismically Designed R C Building in Taiwan", First International Conference on Construction Materials: Performance, Protection and Strengthening of Structures Under Extreme Loading, PROTECT2007 , Canada. United States Department of Defense (DoD), 2005, "Design of Buildings to Resist Progressive Collapse", Unified Facilities Criteria (UFC) 04-023-03, D o D , U S A . United States General Services Administration (GSA) , 2003, "Progressive Collapse Analysis and Design Guidelines for N e w Federal Office Buildings and Major Modernization Projects", G S A , U S A . 122 APPENDIX A. PRELIMINARY SAP M O D E L The following information and assumptions were used to create the preliminary S A P model. Geometry - From Preliminary Plans Sent From N C R E E - Long plan direction: 3.1 m bays with 0.825 m cantilevers - Short plan direction: 8 m bay with 3.1 m cantilever - Storey heights: 3.75 m first storey; 3.65 m second storey - Beams drawn at storey levels, not centroids; slabs neglected - Beam-column joints assumed rigid Materials - From Preliminary Information Sent From N C R E E - Concre te : / /= 17 M P a ; E = 4500^/7 = 18554 M P a (from C S A A23.3-04); y= 24 k N / m 3 - Rebar: fy = 21A M P a ; E = 200000 M P a - Linear-elastic material behaviour assumed Cross-Sections - From Preliminary N C R E E Info; Refer to Figure 3.4 for Location of Beams - Slabs: t = 125 mm thick; use as thickness for all beam flanges - C I (at all grid intersections): 350 mm (in long direction of building) x 400 mm - C2 (mid-bay on Grid C at second storey): 240 mm x 400 mm - B l : h = 600 mm, bw = 350 mm, b/= 2350 mm - BICant: h = 450 mm, bw = 350 mm, bf= 2350 mm - R B I : / 2 = 350 mm, 6,„ = 240 mm, 6/= 2240 mm - W B 2 - R - B : h = 450 mm, bw = 240 mm, bf= 860 mm - W B 2 - R - F : h = 450 mm, bw = 240 mm, bf= 1480 mm - W B 2 - S F - B : h = 380 mm, bw = 400 mm, bf= 1020 mm - W B 2 - S F - F : h = 380 mm, bw = 400 mm, bf= 1640 mm 123 - Flange widths based on A S C E 41:115 for all W B 2 ; 8 * t for other beams - Effective stiffness of all beams assumed to be 0.35 * Ig Foundations - Pinned foundations assumed from preliminary information sent from N C R E E Loads - 25 mm of concrete topping assumed at second floor - Weight of slabs entered as line loads on beams; tributary areas for beam based on 45° rule - Self-weight of beams and columns calculated automatically by E T A B S - Upward line load applied at beams to compensate for including beam flange weight twice - Upstand walls at front o f building and canopy slabs at rear neglected Procedure - Perform static analysis of model with Column C-2 in - Determine loading at top of Column C-2 - Replace Column C-2 with above loading - Perform static analysis of equivalent model to confirm same results - Perform linear dynamic analysis on model with Column C-2 removed. For dynamic analysis case, consider only a step load of equivalent C-2 column load with signs reversed. Analyzing this load case gives change in deformations and forces due to failure of Column C-2. To get peak forces, add peak forces from dynamic analysis to static analysis results with Column C-2 in. For steady-state forces after column removal, use static analysis results from dead load with Column C-2 out. - For modal analysis use 100 Ritz vectors; gives same results as eigenvectors with less computation time and number of vectors required - For linear analysis, use modal combination option Results - Note Structure Symmetrical About Grid C - Peak axial stress (B-2): [247.3 k N (Column In) + 189.5 k N (Peak during dynamic)] / 0.14 m 2 = 3.12 M P a = 0 . 1 8 / c ' 124 - Steady-state changes in deformation: 0.00 mm ( A - l , B - l ) , 0.02 mm (A-2), -0.01 mm ( C - l ) , -0.17 mm (B-2), -5.32 mm (C-2) - Peak changes in deformation: 0.01 mm ( A - l ) , 0.03 mm ( B - l ) , 0.04 mm (A-2), -0.04 mm (C-1), -0.27 mm (B-2),-8.18 mm (C-2) - Peak acceleration at C-2: 3.41 g - First vertical mode of vibration: 5.09 H z - C I column stiffness: 787.1 kN/mm - Min imum axial force in a column before column removal: 113 k N - Steady-state shears in critical beams: 42.2 k N (RBI ) , 60.2 k N (WB2-R) , 63.8 k N (WB2-SF) - Steady-state moments in critical beams: +35.0/-57.1 k N * m (RBI ) , +77.0/-67.7 k N * m (WB2-R), +75.3A72.8 k N * m (WB2-SF) - Peak shears in critical beams: 66.6 k N (RBI) , 88.4 k N (WB2-R) , 91.6 k N (WB2-SF) - Peak moments in critical beams: +74.8/-104.6 k N * m (RBI ) , +110.6/-103.0 k N * m (WB2-R) , +106.7/-108.8 k N * m (WB2-SF) Critical Beam Capacities - According to C S A A23.3-04 with preliminary N C R E E info - R B 1: Reinforced with 2-15M top and bottom and 10M stirrups at 250 mm, d = 310 mm Vr= 137 k N , Mr = 33.8 kN*m, Mr~ = 32.2 kN*m, - W B 2 - R : Reinforced with 2-20M top and bottom and 10M stirrups at 250 mm, d = 310 mm Vr = 181 k N , Mr = 66.8 kN*m, Mr" = 63.4 kN*m, - W B 2 - S F : Reinforced with 2 -15M top and bottom and 10M stirrups at 250 mm, d = 340 mm Vr = 187 k N , M , + = 37.0 kN*m, Mr~ = 36.2 kN*m, 125 APPENDIX B. IN-SITU C O L U M N STIFFNESS TESTING DATA The data collected during in-situ stiffness testing of the columns in the test specimen is included in the following pages. For each column, a table showing the data recorded during stiffness testing is provided followed by a plot of the data and the best-fit linear line to the data, which represents the column stiffness. The contents of the table of stiffness testing data are described below. The data collected for the load cells includes the gross readings of upward applied force taken at the different intervals in the jacking up of the columns, the net force applied by the jack at each interval (gross reading minus initial reading for each load cell) and the total amount o f force applied by the jacks at each interval (sum of net force from each jack). Deformation readings were taken simultaneously with the load cell readings. Gross deformation readings, net deformations at each instrument (gross reading minus initial reading for each instrument) and average deformation (average of net deformations) were recorded at each jacking interval. Lastly a linear regression of the total force and average deformation at each interval was performed to get the slope and y-intercept of the data. The slope of the line of best fit is an estimate of the linear stiffness of the column. 126 Column A-1 Load Cell Units South - Gross kN 0 10.08 19.05 North - Gross kN 0 9.8 20.25 South - Net kN 0 10.08 19.05 North - Net kN 0 9.8 20.25 Total kN 0 19.88 39.3 Linear Pot East - Gross mm -0.3 -0.28 -0.25 West - Gross mm -0.05 -0.01 0.02 East - Net mm 0 0.02 0.05 West - Net mm 0 0.04 0.07 Average mm 0 0.03 0.06 Stiffness By Linear Regression Slope = 617.1 kN/mm Y-lntercept = 4.12 kN 30.24 39.49 49.29 59.19 68.09 78.89 0 29.13 39.42 49.75 59.74 70.29 80.18 0 30.24 39.49 49.29 59.19 68.09 78.89 0 29.13 39.42 49.75 59.74 70.29 80.18 0 59.37 78.91 99.04 118.93 138.38 159.07 0 -0.22 -0.19 -0.16 -0.12 -0.09 -0.05 -0.29 0.04 0.07 0.1 0.13 0.17 0.23 0.01 0.08 0.11 0.14 0.18 0.21 0.25 0.01 0.09 0.12 0.15 0.18 0.22 0.28 0.06 0.085 0.115 0.145 0.18 0.215 0.265 0.035 180 160 1 1 4 0 £ 120 < © 100 80 2 -a i. e- 20 60 40 0 to oo T 0.2 0.3 Deformation (mm) Column A-2 Load Cell Units South - Gross kN 0 9.34 19.05 North - Gross kN 0 8.5 19.42 South - Net kN 0 9.34 19.05 North - Net kN 0 8.5 19.42 Total kN 0 17.84 38.47 Linear Pot East - Gross mm 0.12 0.16 0.18 West - Gross mm -0.16 -0.13 -0.1 East - Net mm 0 0.04 0.06 West - Net mm 0 0.03 0.06 Average mm 0 0.035 0.06 Stiffness By Linear Regression Slope = 504.3 kN/mm Y-lntercept = 3.74 kN 29.59 39.03 49.38 59.37 70.02 79.35 0 28.48 40.04 52.9 59 70.47 78.06 0 29.59 39.03 49.38 59.37 70.02 79.35 0 28.48 40.04 52.9 59 70.47 78.06 0 58.07 79.07 102.28 118.37 140.49 157.41 0 0.23 0.26 0.31 0.35 0.38 0.42 0.14 -0.06 -0.02 0.04 0.07 0.12 0.16 -0.16 0.11 0.14 0.19 0.23 0.26 0.3 0.02 0.1 0.14 0.2 0.23 0.28 0.32 0 0.105 0.14 0.195 0.23 0.27 0.31 0.01 COLUMN A-2 Deformation (mm) o Column B-1 Load Cell Units South - Gross kN 0 1 0 . 3 5 1 9 . 6 North - Gross kN 0 1 0 . 0 8 1 9 . 7 South - Net kN o • 1 0 . 3 5 1 9 . 6 North - Net kN 0 1 0 . 0 8 1 9 . 7 Total kN 0 2 0 . 4 3 3 9 . 3 Linear Pot East - Gross mm - 0 . 6 5 - 0 . 6 2 - 0 . 6 West - Gross mm 0 . 0 6 0 . 1 0 . 1 2 East - Net mm 0 0 . 0 3 0 . 0 5 West - Net mm 0 0 . 0 4 0 . 0 6 Average mm 0 0 . 0 3 5 0 . 0 5 5 Stiffness By Linear Regression Slope = 6 1 7 . 1 kN/mm Y-lntercept = 2 . 7 7 kN 2 9 . 6 8 3 9 . 8 6 4 9 . 2 5 9 . 5 6 6 9 . 1 8 7 9 . 3 5 0 3 0 . 0 5 4 0 . 4 1 5 0 . 7 7 6 0 . 4 8 7 0 . 1 9 8 0 . 6 5 0 2 9 . 6 8 3 9 . 8 6 4 9 . 2 5 9 . 5 6 6 9 . 1 8 7 9 . 3 5 0 3 0 . 0 5 4 0 . 4 1 5 0 . 7 7 6 0 . 4 8 7 0 . 1 9 8 0 . 6 5 0 5 9 . 7 3 8 0 . 2 7 9 9 . 9 7 1 2 0 . 0 4 1 3 9 . 3 7 1 6 0 0 - 0 . 5 7 - 0 . 5 3 - 0 . 5 1 - 0 . 4 5 - 0 . 4 3 - 0 . 4 - 0 . 6 2 0 . 1 5 0 . 1 9 0 . 2 2 0 . 2 5 0 . 2 9 0 . 3 2 0 . 0 9 0 . 0 8 0 . 1 2 0 . 1 4 0 . 2 0 . 2 2 0 . 2 5 0 . 0 3 0 . 0 9 0 . 1 3 0 . 1 6 0 . 1 9 0 . 2 3 0 . 2 6 0 . 0 3 0 . 0 8 5 0 . 1 2 5 0 . 1 5 0 . 1 9 5 0 . 2 2 5 0 . 2 5 5 0 . 0 3 COLUMN B-l o o . i 0.2 0.3 Deformation (mm) 0.4 0.5 to Column B-2 Load Cell Units South - Gross kN 0 9.98 20.44 North - Gross kN 0 10.08 19.7 South - Net kN 0 9.98 20.44 North - Net kN 0 10.08 19.7 Total kN 0 20.06 40.14 Linear Pot East - Gross mm -0.03 0 0.05 West - Gross mm -0.37 -0.33 -0.28 East - Net mm 0 0.03 0.08 West - Net mm 0 0.04 0.09 Average mm 0 0.035 0.085 Stiffness By Linear Regression Slope = 343.7 kN/mm Y-lntercept = 8.08 kN 29.87 39.58 49.29 59.84 69.64 79.63 0 30.05 40.41 50.77 60.48 70.19 80.65 0 29.87 39.58 49.29 59.84 69.64 79.63 0 30.05 40.41 50.77 60.48 70.19 80.65 0 59.92 79.99 100.06 120.32 139.83 160.28 0 0.1 0.16 0.22 0.27 0.34 0.41 0.04 -0.22 -0.16 -0.1 -0.03 0.04 0.1 -0.32 0.13 0.19 0.25 0.3 0.37 0.44 0.07 0.15 0.21 0.27 0.34 0.41 0.47 0.05 0.14 0.2 0.26 0.32 0.39 0.455 0.06 COLUMN B-2 180 0.1 0.2 0.3 Deformation (mm) 0.4 0.5 4± Column C-1 Load Cell Units South - Gross kN 0 10.63 20.07 North - Gross kN 0 10.26 20.16 South - Net kN 0 10.63 20.07 North - Net kN 0 10.26 20.16 Total kN 0 20.89 40.23 Linear Pot East - Gross mm 0.19 0.22 0.25 West - Gross mm -0.14 -0.13 -0.1 East - Net mm 0 0.03 0.06 West -Net mm 0 0.01 0.04 Average mm 0 0.02 0.05 Stiffness By Linear Regression Slope = 523.7 kN/mm Y-lntercept = 11.07 kN 30.33 40.41 49.11 59.19 68.9 81.02 0 30.05 40.32 50.59 59.47 69.73 79.98 0 30.33 40.41 49.11 59.19 68.9 81.02 0 30.05 40.32 50.59 59.47 69.73 79.98 0 60.38 80.73 99.7 118.66 138.63 161 0 0.28 0.32 0.36 0.39 0.44 0.48 0.24 -0.07 -0.02 0.02 0.06 0.11 0.16 -0.1 0.09 0.13 0.17 0.2 0.25 0.29 0.05 0.07 0.12 0.16 0.2 0.25 0.3 0.04 0.08 0.125 0.165 0.2 0.25 0.295 0.045 180 160 1 1 4 0 ft 120 < © ta 100 80 2 << ft 20 60 40 0 1 1— 0.2 0.3 Deformation (mm) Column C-2 Load Cel l Units South - G ross kN 0 10.08 20.81 North - G r o s s kN 0 9.8 20.07 South - Net kN 0 10.08 20.81 North - Net kN 0 9.8 20.07 Total kN 0 19.88 40.88 Linear Pot East - G r o s s mm -0.02 0.01 0.05 W e s t - G ross mm -0.24 -0.23 -0.19 East - Net mm 0 0.03 0.07 Wes t - Net mm 0 0.01 0.05 Average mm 0 0.02 0.06 Stiffness By Linear Regress ion Slope = 464.6 kN/mm Y-lntercept = 9.55 kN 29.5 40.41 50.03 60.02 70.56 79.54 0 30.15 40.6 49.57 59.56 69.36 80.12 0 29.5 40.41 50.03 60.02 70.56 79.54 0 30.15 40.6 49.57 59.56 69.36 80.12 0 59.65 81.01 99.6 119.58 139.92 159.66 0 0.09 0.13 0.16 0.19 0.24 0.29 0 -0.14 -0.1 -0.05 0.01 0.07 0.12 -0.21 0.11 0.15 0.18 0.21 0.26 0.31 0.02 0.1 0.14 0.19 0.25 0.31 0.36 0.03 0.105 0.145 0.185 0.23 0.285 0.335 0.025 180 160 .1 1 4 0 a 120 3 IOO © "2 La ft 20 P 0 60 40 1 1— 0.2 0.3 Deformation (mm) Column D-1 Load Cell Units South - Gross kN 0 9.8 20.25 North - Gross kN 0 10.26 19.88 South - Net kN 0 9.8 20.25 North - Net kN 0 10.26 19.88 Total kN 0 20.06 40.13 Linear Pot East - Gross mm 0.55 0.58 0.61 West - Gross mm -0.44 -0.41 -0.38 East - Net mm 0 0.03 0.06 West - Net mm 0 0.03 0.06 Average mm 0 0.03 0.06 Stiffness By Linear Regression Slope = 523.2 kN/mm Y-lntercept = 6.26 kN 29.59 39.49 49.29 59.65 70.01 80.18 0 29.31 39.67 50.59 60.11 69.73 80.46 0 29.59 39.49 49.29 59.65 70.01 80.18 0 29.31 39.67 50.59 60.11 69.73 80.46 0 58.9 79.16 99.88 119.76 139.74 160.64 0 0.64 0.67 0.71 0.75 0.79 0.85 0.61 -0.34 -0.3 -0.25 -0.21 -0.17 -0.13 -0.39 0.09 0.12 0.16 0.2 0.24 0.3 0.06 0.1 0.14 0.19 0.23 0.27 0.31 0.05 0.095 0.13 0.175 0.215 0.255 0.305 0.055 COLUMN D-1 o > o 0.1 0.2 0.3 Deformation (mm) 0.4 0.5 o Column D-2 Load Cell Units South - Gross kN 0 10.26 20.25 North - Gross kN 0 10.26 19.97 South - Net kN 0 10.26 20.25 North - Net kN 0 10.26 19.97 Total kN 0 20.52 40.22 Linear Pot East - Gross mm -0.05 -0.01 0.03 West - Gross mm -0.63 -0.6 -0.58 East - Net mm 0 0.04 0.08 West - Net mm 0 0.03 0.05 Average mm 0 0.035 0.065 Stiffness By Linear Regression Slope = 461.1 kN/mm Y-lntercept = 6.44 kN 30.33 40.51 51.23 60.3 70.01 80.65 0 30.05 . 39.58 49.66 60.11 69.18 79.54 0 30.33 40.51 . 51.23 60.3 70.01 80.65 0 30.05 39.58 49.66 60.11 69.18 79.54 0 60.38 80.09 100.89 120.41 139.19 160.19 0 0.08 0.12 0.17 0.21 0.27 0.33 -0.01 -0.54 -0.49 -0.45 -0.41 -0.37 -0.32 -0.58 0.13 0.17 0.22 0.26 0.32 0.38 0.04 0.09 0.14 0.18 0.22 0.26 0.31 0.05 0.11 0.155 0.2 0.24 0.29 0.345 0.045 0 * 0 4^ to 0.2 0.3 0.4 0.5 Deformation (mm) Column E-1 Load Cell Units South - Gross kN 0 9.24 19.33 North - Gross kN 0 10.08 20.34 South - Net kN 0 9.24 19.33 North - Net kN 0 10.08 20.34 Total kN 0 19.32 39.67 Linear Pot East - Gross mm 0.62 0.66 0.7 West - Gross mm 0.07 0.1 0.13 East - Net mm 0 0.04 0.08 West - Net mm 0 0.03 0.06 Average mm 0 0.035 0.07 Stiffness By Linear Regression Slope = 453.7 kN/mm Y-lntercept = 6.11 kN 31.53 39.58 48.92 58.17 69.64 80.28 0 30.61 39.58 50.77 58.45 69.92 78.61 0 31.53 39.58 48.92 58.17 69.64 80.28 0 30.61 39.58 50.77 58.45 69.92 78.61 0 62.14 79.16 99.69 116.62 139.56 158.89 0 0.75 0.79 0.84 0.88 0.94 1.01 0.68 0.17 0.21 0.25 0.28, 0.33 0.39 0.11 0.13 0.17 0.22 0.26 0.32 0.39 0.06 0.1 0.14 0.18 0.21 0.26 0.32 0.04 0.115 0.155 0.2 0.235 0.29 0.355 0.05 COLUMN E - l o i o 0.1 0.2 0.3 Deformation (mm) 0:4 0.5 4^ Column E-2 Load Cell Units South - Gross kN 0 9.71 19.33 North - Gross kN 0 8.97 18.68 South - Net kN 0 9.71 19.33 North - Net kN 0 8.97 18.68 Total kN 0 18.68 38.01 Linear Pot East - Gross mm 0.02 0.06 0.1 West - Gross mm 1.09 1.13 1.18 East - Net mm 0 0.04 0.08 West - Net mm 0 0.04 0.09 Average mm 0 0.04 0.085 Stiffness By Linear Regression Slope = 362.5 kN/mm Y-lntercept = 6.57 kN 29.48 38.27 47.33 60.39 70.12 75.43 0 29.13 38.19 48.64 57.43 66.22 76.3 0 29.48 38.27 47.33 60.39 70.12 75.43 0 29.13 38.19 48.64 57.43 66.22 76.3 0 58.61 76.46 95.97 117.82 136.34 151.73 0 0.14 0.19 0.22 0.28 0.33 0.39 0.07 1.24 1.29 1.34 1.42 1.51 1.56 1.14 0.12 0.17 0.2 0.26 0.31 0.37 0.05 0.15 0.2 0.25 0.33 0.42 0.47 0.05 0.135 0.185 0.225 0.295 0.365 0.42 0.05 COLUMN E-2 o . i 0.2 0.3 Deformation (mm) 0.4 0.5 4^ ON APPENDIX C. INSTALLED INSTRUMENTATION LOCATIONS The exact locations of the instrumentation installed on the test specimen were measured and recorded. The dimensions of the installed locations of the dial gauges, linear potentiometers and PI transducers from representative points (center of column, centerline of column, top of main floor slab) are listed below. Note that all the dial gauges and linear potentiometers are mounted at the underside of the second floor beam at their particular location. A - 2 - Dia l Gauge: 18 mm north and 270 mm east Of column center - Linear Potentiometer: 48 mm south and 260 mm east of column center - PI Transducer: east face, 5 mm south of column centerline, bottom embedded rod 1527 mm above main floor slab B - l - Dia l Gauge: 10 mm south and 298 mm west of column center - Linear Potentiometer: 55 mm south and 302 mm east of column center B-2 (N) - Dia l Gauge: 241 mm north and 5 mm east of column center - Linear Potentiometer: 240 mm north and 46 mm west of column center - PI Transducer: north face, 11 mm west of column centerline, bottom embedded rod 1510 mm above main floor slab B-2 (S) - Dia l Gauge: 235 mm south and 12 mm east of column center - Linear Potentiometer: 242 mm south and 57 mm east of column center - PI Transducer: south face, 23 mm west of column centerline, bottom embedded rod 1522 mm above main floor slab 147 C - l - Dia l Gauge: 8 mm north and 317 mm west of column center - Linear Potentiometer: 35 mm south and 322 mm west of column center - PI Transducer: west face, 5 mm north of column centerline, bottom embedded rod 1510 mm above main floor slab C-2 (E) - Linear Potentiometer: 3 mm north and 1060 mm east of column center c-2 cm - Linear Potentiometer: 12 mm south and 1238 mm west of column center D-1 - Dia l Gauge: 97 mm north and 297 mm west of column center D-2 (NI - Dia l Gauge: 239 mm north and 18 mm west of column center - Linear Potentiometer: 241 mm north and 59 mm west of column center - PI Transducer: north face, 14 mm east of column centerline, bottom embedded rod 1458 mm above main floor slab D-2 (S) - Dia l Gauge: 235 mm south and 12 mm east o f column center - Linear Potentiometer: 242 mm south and 57 mm east of column center - PI Transducer: south face, 12 mm west of column centerline, bottom embedded rod 1530 mm above main floor slab E-2 - Dia l Gauge: 5 mm north and 263 mm west of column center 148 APPENDIX D. RECORDED EXPERIMENTAL DATA Table D . l summarizes the dynamic data collection instruments installed on the test specimen. This table lists the instrument type, location of installation and channel name (for matching the instruments to their corresponding data set in the data log). Figure D . l shows the structural plan of the test specimen for use with Table D . l . Plots of the first 1 s of data recorded by each of the dynamic data collection instruments are included on the following pages. Table D.l Summary of Dynamic Data Collection Instruments Channel Name' Instrument T\pc -Location:* Channel Name. Instrument Type Location CustVoltageO Accelerometer C-2 CustVoltage8 Linear Potentiometer A-2 CustVoltagel Linear Potentiometer C-2 (W) CustVoltage9 Linear Potentiometer C-2 (E) CustVoltage2 Linear Potentiometer B- l CustVoltagelO PI Transducer A-2 CustVoltage3 Linear Potentiometer B-2 (N) CustVoltagel 1 PI Transducer B-2 (N) CustVoltage4 Linear Potentiometer B-2 (S) 11 CustVoltagel 2 PI Transducer B-2 (S) CustVoltage5 Linear Potentiometer C- l CustVoltagel 3 PI Transducer C - l CustVoltage6 Linear Potentiometer D-2 (N) CustVoltagel 4 PI Transducer D-2 (N) CustVoltage? Linear Potentiometer D-2 (S) CustVoltagel 5 PI Transducer D-2 (S) N -©-• ( A ) (B) ( C ) (D) ( E ) i W _ , CQ CO WB2 CO CQ WB2 |163 314 308 | 309 | 313 1220j r Units are in cm Figure D.l Structural Plan of Test Specimen 149 ( U I U I ) juaiuaoBidsiQ B - l L i n e a r P o t e n t i o m e t e r 155 en c s ! ; o ^ ( N r o o o o o o o I I I ( U l U l ) JU3lU33B |dsiQ 157 A-2 L i n e a r P o t e n t i o m e t e r C-2 (E) Linear Potentiometer 0.2 0.4 0.6 Time (s) A-2 PI Transducer 0.025 -. 0.02 --0.02 - I -0.025 -\ ; : 1 . 1 1 ' 0 0.2 0.4 0.6 0.8 1 Time (s) ON O B-2 (N) PI Transducer 0.03 i -0.04 H 1 1 r -0 0.2 0.4 0.6 Time (s) ON oo © a a H CM © o <N © O — i — p O O O p p O O © —<' (uiui) luauiaoBjdsiQ D-2 (S) PI Transducer 0.04 -. 0.03 -0.02 -Time (s) APPENDIX E. ADJUSTEMENT OF DATA AT C O L U M N D-2 (S) A s previously mentioned in Section 4.2.2, the steady-state deformations recorded at Columns B -2 and D-2 suggest that the linear potentiometer at D-2 (S) was disturbed during the explosive test. This instrument should have reported a downward deformation of approximately 0.10 mm, but instead registered a steady-state deformation reading of 0.088 mm upwards. The unmodified transient data for the linear potentiometer at D-2 (S) is shown in Figure E . l . This data starts out similar to other linear potentiometers but then records an unusually large upward deformation at 0.21 s. After this point, the response oscillates up and down similar to other instruments, but is centered about a deformation that is upward rather than downward. Recorded Post-Test Steady-State 0.1 0.2 0.3 0.4 0.5 0.6 Time (s) 0.7 0.8 0.9 Figure E.l Unmodified Data for Linear Potentiometer at Column D-2 (S) Note: Upward displacement is positive The shape of the response history for the linear potentiometer on the south face of Column D-2 suggests that an artificial, one-time increase in deformation took place at around 0.21 s. Inspection of the location where the instrument was mounted revealed that the underside of the beam at this location was plastered over to create a flat surface for the linear potentiometer to be 166 mounted against. A photo of this plastering is shown in Figure E.2. Indentation and spalling of this plaster was noted after testing, and it's possible that these occurrences could have caused the jump in deformation implied by the response data. Figure E.2 Plastering and Instrumentation at Column D-2 (S) Due to its location at one of the columns receiving the greatest amount of redistributed load, the linear potentiometer at Column D-2 (S) was one of the most important instruments installed on the test specimen, and the data collected by this instrument would be crucial for assessing the behaviour of the specimen during the explosive test. Thus, it was deemed necessary to estimate the magnitude of the artificial, one-time increase in deformation and adjust the data after this jump so as to approximate the actual deformation of the south face of Column D-2. The fact that the adjusted response at Column D-2 (S) is only approximate would then be expressly considered whenever this data was used for analysis or modelling purposes. The steady-state deformation data from the instruments at Column B-2 was used to help estimate the quantity by which the data after the artificial jump would be adjusted. Referring back to the data in Table 4.1, the average of the steady-state deformations recorded by the dial gauges on the north and south faces of Column B-2 can be calculated at -0.18 mm. Taking the average of the steady-state linear potentiometer readings at Column B-2 also yields -0.18 mm. Averaging the 167 deformations recorded by instruments installed symmetrically about a column, as the instruments at Column B-2 and D-2 were, provides an estimate o f center-of-column deformation. A t Column B-2, the steady-state center-of-column deformation estimates are the same for both types of instruments. To determine what the steady-state deformation should have been for the linear potentiometer at Column D-2 (S), it was assumed that average of the dial gauges and the average of the linear potentiometers at Column D-2 were also the same. For the averages of the dial gauges and linear potentiometers at Column D-2 to equal each other, the steady-state deformation of linear potentiometer at Column D-2 (S) would need to have been -0.122 mm. Assuming the actual deformation at the Column D-2 (S) linear potentiometer is -0.122 mm, to correct the data after the artificial jump, this data would have to be adjusted by the difference between the actual steady-state deformation (-0.122 mm) and the recorded steady-state deformation (0.088 mm). Having not yet determined exactly at what time the artificial jump in deformation took place, the entire transient response of the Column D-2 (S) linear potentiometer was adjusted by -0.122 mm - 0.088 mm = -0.210 mm. To identify when the artificial jump in deformation took place, the adjusted and unadjusted transient data for the linear potentiometer at Column D-2 (S) were plotted together. These plots are shown in Figure E.3a. Looking at Figure E.3a, it appears that the artificial jump in data occurs at around 0.21 s. The two sets of data were spliced at this point with a linear transition as shown in Figure E.3a. The spliced set of data was then plotted together with the transient data from the linear potentiometer at Column D-2 (N) to check i f the trend of the spliced data was reasonable. Figure E.3b illustrates how the spliced data and data from Column D-2 (N) compare. The trends of the two sets of data compare quite well : the shapes of the oscillations of the two data sets correlate quite well and difference in the magnitudes of the data sets stays relative constant with time. 168 a a a = U _ « "a 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 Unadjusted Data 0 0.1 0.2 0.3 Time (s) 0.4 0.5 (a) Splicing of Adjusted and Unadjusted Data for Linear Potentiometer at D-2 (S) 0 2 T ' Spliced D-2 (S) Transient Response Spliced D-2 (S) Post-Test Steady-State e B B w "a in D-2 (N) Post-Test Steady-State D-2 (N) Transient Response 0 0.1 0.2 0.3 0.4 0.5 Time (s) 0.6 0.7 0.8 0.9 1 (b) Comparison of Spliced D-2 (S) and D-2 (N) Linear Potentiometer Figure E.3 Modification of Column D-2 (S) Linear Potentiometer Data Note: Upward displacement is positive 169 APPENDIX F. DETAILED ETABS M O D E L The information, assumptions and procedure used to develop and calibrate the E T A B S model are detailed below. Geometry and Cross-Sections - Refer to Figure 3.4 for plan layout and Section 3.2 for storey heights - Refer to Figure 3.5 and Figure 3.7 for cross-sections and Section 3.2 for slab properties - C I columns located at all grid intersections - C2 column located 3.85 m west of Grid 1 in second storey - For locations of different types of beams refer to Figure 3.4 and Section 3.2 - Column, beam and slab elements are drawn at their centroid as per Section 5.5.1 Slab meshed to achieve elements approximately 500 mm x 500 mm - Beams subdivided to match slab mesh and create nodes at instrument locations - Stiff links connecting beam and slab elements are 350 mm x 350 mm - Stiff links inserted in place of beams inside beam-column joints - Inside beam-column joints, E of columns increased by a factor of 100 - N o elements have rigid end zones - N o reductions in stiffness (J) of beams and slabs Materials - Refer to Section 3.3 - Concrete:/, ' = 14.3 M P a ; E = 45007/7 = 17017 M P a (from C S A A23.3-04); y= 24 k N / m 3 - Rebar: fy = 326 M P a ; E = 200000 M P a - Stiff link material: strength =fc'\ E = 1000 * E for concrete; y=- 0 - Linear-elastic material behaviour assumed 170 - A s mentioned in Section 5.5.1, E for concrete (and consequently, E for stiff link material) reduced to consider cracking in floor structure and non-uniformity of column cross-sections Foundations - A s per Section 5.5.1, fixed foundations assumed; version of final model tried with pinned foundations: main vertical period and steady-state deformations only 0.01% larger Loads - 25 mm of concrete topping assumed at second floor - Self-weight of all elements calculated automatically by E T A B S - Upstand walls at front of building and canopy slabs at rear neglected Procedure - Develop Column C-2 Out model equivalent to Column C-2 In model as per Section 5.5.1 - Calibrate Column C-2 Out model to recorded data as explained in Section 5.5.1; note that for beams and columns where effective member dimensions are altered, self-weight is as per the original, site-measured dimensions; effective depth of beam along Gr id 1 at second floor reduced by 50 mm to consider grout and bricks at bottom of beam as ineffective; effective dimensions of beam stems on top of masonry walls at second floor along Grids 2 and C reduced to design dimensions (240 mm wide x 225 mm high) to render brick, grout and filler concrete ineffective; line elements representing beams with revised effective dimensions relocated to centroid of effective cross-section - Calibrate load functions and damping to recorded data as per Sections 5.5.1 and 5.5.4 by performing linear dynamic analysis on Column C-2 Out model with both loading functions applied; for all dynamic analyses, use an output step size of 0.001 s, neglect P -D effects (does not affect results significantly) and Ritz vector analysis considering all load cases as load vectors and 100 modes (more efficient than eigenvector analysis and modes work out the same); step load is all 6 reactions at underside of remaining part of Column C-2 with signs reversed; triangular pulse also applied at underside of remaining part of Column C-2 but consists only of axial load directed upward - For steady-state changes, use results of static analysis of step load; for peak changes due to gravity load redistribution, use results from dynamic analysis of step load; for peak changes due to total loading, use results from dynamic analysis of model with step load and triangular 171 pulse applied; for peak forces in members, add peak demand from dynamic analysis for step load and triangular pulse to static analysis with Column C-2 in. To isolate response due to one mode, damp all other modes 99% and the mode of interest by the usual amount (3%) and perform a dynamic analysis; to find the combined response of two modes, find the individual response for each mode, find the response for all modes damped 99%, subtract the damped response from the individual mode responses to get the deviations from the damped response, sum the deviations from the damped response and add back in the damped response 172
Thesis/Dissertation
eng
Civil Engineering
Vancouver : University of British Columbia Library
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.
Graduate
Explosive testing to assess dynamic load redistribution in a reinforced concrete frame building
Text