EVALUATION OF MISSING MEMBER ANALYSES FOR PROGRESSIVE COLLAPSE DESIGN OF STEEL BUILDINGS AND GIRDER BRIDGES. Houston A. Brown

Transcription

1 EVALUATION OF MISSING MEMBER ANALYSES FOR PROGRESSIVE COLLAPSE DESIGN OF STEEL BUILDINGS AND GIRDER BRIDGES by Houston A. Brown A thesis submitted to the Faculty of the University of Delaware in partial fulfillment of the requirements for the degree of Master of Civil Engineering Winter 2010 Copyright 2010 Houston A. Brown All Rights Reserved

2 EVALUATION OF MISSING MEMBER ANALYSES FOR PROGRESSIVE COLLAPSE DESIGN OF STEEL BUILDINGS AND GIRDER BRIDGES By Houston A. Brown Approved: Jennifer Righman McConnell, Ph.D. Professor in charge of thesis on behalf of the Advisory Committee Approved: Harry W. Shenton III, Ph.D. Chair of the Department of Department Name Approved: Michael J. Chajes, Ph.D. Dean of the College of College Name Approved: Debra Hess Norris, M.S. Vice Provost for Graduate and Professional Education

3 ACKNOWLEDGMENTS There have been many people on numerous occasions who have given me a lot of support in order to finish this thesis. Fitting each of these onto a page of acknowledgements isn t possible so some may inevitably be left out. First I would like to thank my advisor, Dr. McConnell, who helped me in my academic career and introduced me to the field of progressive collapse which, while being a frustratingly difficult subject at times, is both rewarding and very interesting. She offered many contributions including helping me to form objectives, suggesting methods to reach these objectives, and providing many edits on this thesis to account for my, for lack of better words, terrible writing. I am thankful to the many friends who have kept me company and kept my spirits high while working on this thesis, including Pat, Mikey, Jim-Jim, Ludi, Jonas, Higgens, Dougy, and Peter. I am also extremely thankful to my parents and big brother Preston who have always continued to be a great example to me and remind me to pay attention to the important things in life and leave nothing behind. Most importantly, I would like to thank my wife, Heather, who has given me countless love and support and took care the little ones (Annelyse, Reagan, Cash, and Lennon) while I put in the many hours needed to finish this. For that I am forever grateful. Lastly, I would like to thank the University of Delaware University Transportation Center for partial funding provided to carry out this research. iii

4 TABLE OF CONTENTS LIST OF TABLES... viiii LIST OF FIGURES... x ABSTRACT... xi Chapter 1. INTRODUCTION Background Motivation for Research Research Objectives Scope of Work Thesis Organization BACKGROUND AND LITERATURE REVIEW Introduction Definition Abnormal Loads Methods for Collapse Mitigation Event Control Indirect Methods Direct Methods Alternate Load Path Method Specific Local Resistance Method Current Progressive Collapse Design Codes in the United States ASCE 2002 Minimum Design Loads for Buildings and Other Structures ACI Building Code Requirements for Reinforced Concrete (2005) GSA Progressive Collapse Analysis and Design Guidelines (2003) iv

5 2.5.4 DoD Design of Buildings to Resist Progressive Collapse (2005) Current Research Global vs. Local Effects Structural Response to Blast Loading Application of Seismic Design Connections Dynamic Analysis New Design and Analysis Methods Summary METHODOLOGY Introduction Column Analysis Procedure Finite Element Details General LS-Dyna Input Format Modeling Procedure Model Input Geometry Materials Johnson-Cook Strength Model Strength Model Variable Form Strength Model Constants Determination Johnson-Cook Failure Model Boundary Conditions Loading Axial Loading Blast Loading Output Control v

6 3.2.2 Deflection Failure Criteria Missing Girder Analysis Procedure Live Load Determination Lane Load Truck Load Tandem Load Distribution Factors Dead Load Determination Girder Selection Effective Load Factor Bridge Models for Missing Girder Analysis VALIDATION STUDY Need for Validation Model Lawver et al (2003) Model Description Assumptions LS-Dyna Validation Model Results SENSITIVITY ANALYSIS Introduction Sensitivity Analysis Model Displacement Results Stress Results Axial Stress Shear Stress Von-Mises Stress Conclusion RESULTS Introduction Column Analysis Results vi

7 6.2.1 Parameters Varied in Column Analysis Stand-off Distance Results Charge Size Vertical Position of Charge Column Height Column Depth Column Stiffness Parametric Summary Blast Threat Level Representation of Missing Column Analysis Missing Girder Analysis Results Span Length of 50 ft Span Length of 100 ft Conclusions from Missing Girder Analysis SUMMARY AND CONCLUSIONS Summary and Conclusions of Missing Column Analysis Summary and Conclusions of Missing Girder Analysis Synthesis REFERENCES APPENDIX A.1 Johnson-Cook Parameter Results for Remaining Data Sets A.2 LS-Dyna Keyword File for 14-ft W14x82 Column vii

8 LIST OF TABLES Table 3.1 Calculations for Johnson-Cook Constants from Figure Table 3.2 Johnson-Cook Strength Model Constants Used in LS-Dyna Table 3.3 Johnson-Cook Failure Model Constants Used in LS-Dyna Table 3.4 Service Loadings for Each Column Table 3.5 Geometric Properties of Columns..76 Table 3.6 Calculation of Parameters for Finding the Failure Distance..77 Table 3.7 Maximum Moment per Lane Resulting from Lane Load..83 Table 3.8 Maximum Moment per Lane for Truck Loading...85 Table 3.9 Maximum Moment per Lane for Tandem Loading...86 Table 3.10 Distribution Factors for Interior and Exterior Girders.88 Table 3.11 Maximum Dead Load Moment per Girder..92 Table 3.12 Ultimate Loads per Girder...94 Table 3.13 Girder Sizes Used for Each Bridge Span Length and Spacing Table 3.14 Effective Load Factors with and without Impact.97 Table 3.15 Service Loadings and Nominal Capacity for Each Span Length Table 4.1 Criteria from Lawver et al (2003) to Find Bomb Size and Location Table 4.2 Equivalent Charge Size Comparisons to Lawver et al (2003) Data 106 Table 4.3 Maximum Deflection Relative Error for Fixed End Validation Model viii

9 Table 5.1 Percent Relative Error of Displacement along Height of Column Table 5.2 Percent Relative Error for Various Mesh Sizes Table 6.1 Failure Stand-off Distance for Each Column (Small Charge) Table 6.2 Failure Stand-off Distance for Each Column (Large Charge) Table 6.3 Summary of Missing Girder Analysis for 50-ft Span..150 Table 6.4 Summary of Missing Girder Analysis for 100-ft Span 151 Table A.1 Data Set Table A.2 Data Set Table A.3 Data Set Table A.4 Data Set Table A.5 Data Set Table A.6 Data Set Table A.7 Johnson-Cook Parameter Summary 177 ix

10 LIST OF FIGURES Figure 2.1 Schematic of Tie Forces from DoD (2005) Figure 3.1 Stress-Strain Curve from Quasi-Static Tensile Test Figure 3.2 Linear Fit to Log-Log Data Given in Table Figure 3.3 Johnson-Cook Approximation to Experimental Data.. 52 Figure 3.4 Plot of Equation 3.7 for Various Strain Rates Figure 3.5 Failure Strain versus Effective Stress for Varying Strain Rates.. 59 Figure 4.1 Deflection Profile for the Validation Models.110 Figure 5.1 Deflection Profile for Varying Mesh Sizes 116 Figure 5.2 Axial Stress Distribution along Web at Mid-Height of Column Figure 5.3 Shear Stress Distribution along Web at Mid-Height of Column Figure 5.4 Von-Mises Stress Distribution along Web at Mid-Height of Column Figure 6.1 Failure Contours for W14x82 Column (Both Heights, Small Charge)..137 Figure 6.2 Failure Contours for W14x82 Column (Both Heights, Large Charge)..138 Figure 6.3 Failure Contours for 14-ft Columns (Small Charge)..139 Figure 6.4 Failure Contours for 18-ft Columns (Small Charge)..140 Figure 6.5 Failure Contours for 14-ft Columns (Large Charge)..141 Figure 6.6 Failure Contours for 18-ft Columns (Large Charge)..142 x

11 ABSTRACT The most common analysis method prescribed in progressive collapse specifications is the alternate load path analysis, which is a means for ensuring adequate structural integrity and load redistribution capability in a building. This approach is generally viewed as threat-independent in that it assumes the removal of individual columns (and is therefore termed a missing column analysis in this work), but does not consider the initiating event leading to the failure of these members. However, for reasons described herein, it is of interest to determine the blast loading that corresponds to column failure, which is an underlying threat implicitly assumed by these provisions. This is carried out through dynamic finite element modeling using the commercial software LS-Dyna. The modeling methods are validated using experimental results available in archival literature and sensitivity studies are performed to assess meshing requirements. As a result, the charge size and stand-off distance producing failure of an individual member is determined. The alternate load path analysis has been shown to be a relatively simple, yet effective method for progressive collapse mitigation in buildings, as they are made up of many members in a three-dimensional frame, resulting in redundant structures with high capacities for load redistribution. To investigate if a similar analysis method may be used for girder bridges, which are also susceptible to progressive collapse, their xi

12 load redistribution capabilities are of interest. This is explored by removing the load carrying capacity of a single girder, redistributing this load, and evaluating whether adjacent girders have enough strength to withstand the new service loading. By analyzing the results of these analyses, insight into the potential use of this method as a means of evaluating girder bridge redundancy and collapse resistance can be obtained. xii

13 Chapter 1 INTRODUCTION 1.1 Background Progressive collapse refers to the total or partial collapse of a structure stemming from a localized failure. Since nearly all failures are progressive in nature and it is not the intent to make structures perfectly resistant to all conceivable loading events, progressive collapse is distinguished as a case where the resulting damage is disproportionate to the original cause. Because of this reason, disproportionate collapse is often a term substituted for progressive collapse. Study of this began after the partial collapse of the Ronan Point Apartment Building in 1968, caused by a small explosion in an upper story kitchen. The localized failure that triggers the collapse is typically due to a loading event not considered in design; thus, it is often referred to as an abnormal load. These abnormal loads can include explosions, vehicular impact, extreme weather events, fire, and construction errors. In more recent times the consideration for these abnormal loads has expanded to include bombs intentionally placed by terrorists, such as the incident in Oklahoma City that left a large portion of the Murrah Federal Building in ruins. Such loads are very difficult to plan for as their magnitude and location are unknown. 1

14 There are three main ways to mitigate the risk of progressive collapse referred to in codes of practice. These are event control, indirect design, and direct design. Event control is intended to avoid the abnormal load from occurring. Indirect methods are intended to put more inherent strength and resiliency into the structure with the intent that after damage occurs the remaining structure will have enough strength to remain stable. Direct design specifically addresses progressive collapse by either designing for the abnormal loading or for its aftermath by assuming a predetermined level of damage has already occurred. For the structural engineer, event control is of little concern as it is largely reliant on security measures. The other two methods have been incorporated into several design codes. One prevalent direct design approach is the alternate load path analysis method. In this analysis, a member is assumed to experience some abnormal loading event and fail; this failure is modeled in the analysis by removing the affected member from the structure. The structure, minus this removed member, is then analyzed to see if it is stable or if failure propagates. This method is quite popular as it is easily incorporated into design codes, engineers are familiar with analysis methods accompanying it, and consideration of the initial loading event is not needed. The type of member that is typically removed from the structure, because it will generally govern design, is a column. For this reason the method is often referred to missing column analysis or more generically as missing member analysis. While there are some faults in this method, it is still a good method for giving the structure an increased level of ductility and redundancy. 2

15 1.2 Motivation for Research While alternate load path analyses are currently one of the most efficient options available for performing a progressive collapse analysis, this method is not without shortcomings. As there is no consideration of the initial loading event, the amount of safety that the alternate load path method provides for a given abnormal loading is unknown. In other words, the ability of the structure to pass the alternate load path method analysis gives no indication of that structures ability to withstand an abnormal loading. For this reason it has been criticized as an imprecise method to design for abnormal loadings that lead to progressive collapse, such as blast loading. In the alternate load path method, the designer is also not required to consider the size or spacing of the members in a structure when members are removed. Since the size and spacing change from structure to structure, the significance of one column removal varies. For example, removing a large column from a structure with large bay sizes is much more significant than the removal of a smaller column from a structure with smaller spacing. This is a consequence of not considering the abnormal loading effects in the analysis. Although the alternate load path method does not specifically model the effects of abnormal loadings, it does assume that some loading event has occurred and considers the behavior of the structure afterwards. In the case of missing column analysis, the structure is assumed to lose only a single column to an abnormal loading, which may or may not be an accurate portrayal. For example, an abnormal loading could cause the loss of multiple members or perhaps only the partial loss of one. 3

16 Therefore, the conservatism of modeling the structure in this damaged state (i.e. one member loss) is unknown. At this time there is a lot of research aimed toward preventing the collapse of buildings, while the collapse vulnerability of bridges is receiving little attention. Recently the collapse of the I-35 Bridge in Minneapolis showed that a bridge can also suffer from collapse in a disproportionate fashion (Cho 2008). With the loss of one connection, the entire structure collapsed. In an instant an important staple to the community was lost as well as multiple lives. 1.3 Research Objectives In the previous section it was established that criticisms of the alternate load path method arise because it does not consider the effects of the abnormal loading event. It is, therefore, desirable to know what corresponding abnormal loading event could cause such a damaged state. This will allow some conclusions on the level of conservatism that the alternate load path method provides and the uniformity of this safety between different structures. As blast loading is one of the abnormal loading events typically motivating an alternate path analysis, this load type is selected for evaluation in the present work. The first objective of this thesis was to determine what blast threat is representative of the missing column analysis prescribed in progressive collapse design codes. The blast threat was quantified by a failure stand-off distance for a given charge size (expressed in weight of TNT), where the failure stand-off distance is defined as the 4

17 farthest distance from the column that the charge can be placed to fail the column. Once this blast threat was found, an assessment of the conservatism of the alternate load path was made. Through finding the blast threat causing failure in different column sizes, the uniformity of safety provided by the alternate path method was also evaluated. As girder bridges are the most common type of bridge, understanding their susceptibility to progressive collapse is of interest. Since bridge design codes do not address progressive collapse, a bridge s only defense against such events is generally through the reserve capacity and redundancy included through design. Reserve capacity is the amount of strength that the bridge has relative to the service loads that it carries. Redundancy in the bridge is dependent on the number of girders in the system and the ability of the bridge to distribute loading between them. If this reserve capacity and redundancy are known, then an assessment on how resilient the bridge is against progressive collapse can be made. An approach similar to the alternate load path method used with buildings can be used as a measure to determine a bridge s reserve capacity. This can be achieved by assuming some type of damage to the bridge, and calculating the effects of this damage on the remaining structure or by determining what reserve capacity and redundancy are needed for the bridge to survive. The second objective was to determine what reserve capacity exists in simplespan steel girders bridges and what level of redundancy is needed in the event of a failed girder. This was accomplished by formulating a procedure for and carrying out a missing girder analysis, which is similar to the alternate load path method used in 5

18 building design. This allowed for some judgment on the resiliency of bridges to progressive collapse that inherently results from the design process to be made. 1.4 Scope of Work In order to find the blast threat that was representative of the missing column analysis method, finite element models of columns being subjected to blast loads were created in the program LS-Dyna. Three steel column sections (W14x82, W14x109, and W10x77) at two different heights (14 ft and 18 ft) were considered. These represent typical columns and comparisons of the results will lead to conclusions about the factors governing a column s resistance to blast. Two blast sizes were considered (small and large) and were placed at three different vertical positions for each column (bottom, mid-height, and top). The stand-off distance causing failure for each blast size at each of the three vertical positions for each column was found. As this stand-off distance was repeated for two blast sizes at three vertical positions, three different column sizes, and two different column heights, this resulted in the determination of 36 different failure stand-off distances to draw conclusions from. In order to determine the system load-carrying capacity that exists in steel simple-span girder bridges, a missing girder analysis was completed. This was done by considering the loss of one or two girders and redistributing their service loads to the remaining girders in the cross-section. By knowing the reserve capacity of a single girder in the cross-section, the number of remaining girders required in the cross-section given the failure of the girder(s) was found. This led to conclusions about the system 6

19 load-carrying capacity of the bridge as well as redundancy requirements, which is a function of the number of girders. 1.5 Thesis Organization Background information and a literature review pertaining to progressive collapse in buildings is provided in Chapter 2. This includes a definition of key terms used in this field, description of the methods used to provide progressive collapse resistance to buildings and how they are applied in current codes. Also research that has been completed to further understanding of and improve approaches to mitigate progressive collapse is summarized. Chapter 3 contains the methodology used to achieve the objectives of this thesis. In Section 3.2, a discussion on the procedure for finding the blast threat to cause failure in the columns that are considered is given. Details for the finite element model input and development of the failure criteria used to apply conclusions to the finite element output are given here. Section 3.3 contains details on the procedure for the missing girder analysis. The process for determining the service loads and key assumptions in the development of the analysis are described. In Chapter 4, a validation study is provided to show that the LS-Dyna modeling techniques used for the column analysis will simulate real life behavior. In Chapter 5, a sensitivity analysis is performed to find an acceptable mesh size for the LS-Dyna models. 7

20 Chapter 6 shows the results for the analyses carried out to reach both objectives in this thesis. Section 6.2 gives the failure stand-off distances that were found for each of the blast and column sizes considered. Discussion on the variability in the failure stand-off distance as a function of the various column parameters is also included here. This section concludes with a discussion on the level of conservativeness provided by the alternate load path method. Section 6.3 provides the results for the missing girder analysis. Discussion on the system load carrying capacity and the resiliency of simplespan girder bridges to progressive collapse is provided. In Chapter 7, a summary of and conclusions based on the results are given. Two appendixes show supporting calculations for the material model used in this thesis and a sample input file for the finite element models. 8

21 Chapter 2 BACKGROUND AND LITERATURE REVIEW 2.1 Introduction Since the partial collapse of the Ronan Point apartment building in England due to a small explosion in an upstairs kitchen, the sudden collapse of structures due to unforeseen events has been a very important subject in the field of structural engineering. Initial research was aimed at answering what design methods could be used to avoid collapse after losing a key structural element and whether the designs for such events were needed, since they occur so infrequently. Research has led to the adoption of design methods and continues to refine them so that they are more efficient. Often subsiding and coming back to the forefront because of disastrous events, the field has progressed successfully thus far in determining possible solutions but admittedly, it is a complex area and not perfect. The following is an overview on the current state of practice and research with respect to progressive collapse and some of the shortcomings that are pertinent to this thesis. 2.2 Definition Progressive collapse, also termed disproportionate collapse, refers to the total or partial collapse of a structure stemming from a localized failure. The currently accepted definition of progressive collapse also includes the notion that the total area or volume 9

22 of the structure that collapses is disproportionate to the area or volume of the structure destroyed by the initiating event (Nair 2006). The critical ratio between these two quantities is still a source of debate, although all definitions include the notion of disproportionality (Ellingwood 2006). Some codes quantify this to some degree, such as allowing no more than the surrounding bays of a removed column to be lost or a certain area or percentage of floor loss, depending on the design code. Specific information on these values are given in Section Abnormal Loads In modern structural design each member is designed according to probabilistic theory, viz. a log-normal probability distribution function, such that the failure probability can be defined to an acceptable level. This presents a problem for the loading events that initiate progressive collapse because they occur so infrequently that data to compile accurate probability curves is not readily available. Even extreme bounds are difficult to place on a loading event because, for example, terrorists can always use larger bombs and collisions can always occur with more mass or velocity. This necessitates a departure from the contemporary probabilistic theory of design in order to design for progressive collapse; the current progressive collapse design methods are described in the following section. The initiating events that cause damage to the structure are considered abnormal loads, as they are extremely rare and too difficult to consider in design (Breen and Siess 1979, McGuire 1974). These events can include explosions, vehicle 10

23 impact, construction and fabrication errors, and fire. Although these events are rare in occurrence, the consequences are devastating. Some studies have tried to quantify abnormal loadings and progressive collapse such as a study done by Allen and Schriever (1973), which found 495 incidents involving progressive collapse over a 10 year period in Canada. This data suggests that progressive collapse incidents are fairly common and that recording their occurrence is possible. 2.4 Methods for Collapse Mitigation Since the potential for structures to be damaged by abnormal loads is present, procedures have been developed and implemented in design codes to make existing and new structures more resistant to them. Since defining the abnormal loading is so difficult, the analysis procedures developed do not always consider loadings from these initial events. There are, in general, three ways to mitigate the risk of disproportionate collapse, being (1) event control, (2) direct methods, and (3) indirect methods (Ellingwood et al 1978, Krauthammer et al 2002). Event control is an attempt to eliminate the initial loading event that propagates collapse. Indirect methods make general recommendations with respect to continuity, redundancy, and ductility for the intended purpose of increasing the structure s ability to redistribute forces without considering the abnormal loading itself. Direct methods, on the other hand, are aimed at giving the structure enough capacity to absorb the abnormal loading locally without 11

24 further collapse. Each approach and its usage in design codes are discussed in more detail in the following sections. Several publications in the United States currently give design guidelines aimed at preventing disproportionate collapse such as Department of Defense (DoD) Design of Buildings to Resist Progressive Collapse (DoD 2005), General Service Administration (GSA) Progressive Collapse Analysis and Design Guidelines (GSA 2003), American Concrete Institute (ACI) Building Code Requirements for Structural Concrete (ACI 2005), and American Society of Civil Engineers (ASCE) Minimum Design Loads for Buildings and Other Structures (ASCE 2002). The design provisions contained in these guidelines utilize two of the three categories: direct methods and indirect methods Event Control Event control methods are intended to prevent the occurrence of the abnormal load. Methods employed include preventing the storage of explosives and high quantities of gas to lower risk of explosion and placing fenders around columns to eliminate vehicle impact (Taylor 1975, Dragosovic 1973). For buildings at risk of terrorist bombings, it is a common practice to provide a stand-off perimeter that will prevent large bombs from getting close enough to do serious damage. Another use of event control is a bollard, which is a rigid post or barrier that guards vulnerable areas from vehicles, such as a pedestrian walkway or exposed column. While this method can be a very inexpensive way to lower the probability of the initiating event and 12

25 therefore progressive collapse, it still does not ensure the risk is entirely gone and is often impractical. As this method does not include structural details, it is out of the realm of the structural engineer Indirect Methods Indirect methods are aimed at providing a structure with general integrity traits without consideration for abnormal loads. These methods involve providing continuous connections across joints and adding more ductility and redundancy to the system. These methods are easily introduced into code, which has raised their popularity. One popular approach is requiring that the structure is adequately held together by tie forces. This procedure is intended to give members adequate tensile capacity to develop catenary action after damage. This is incorporated by a network of continuous connections horizontally and vertically through the joints in the structure as well as tensile capacity requirements on select members. A schematic showing the types of tie forces considered is shown in Figure 2.1, which is taken directly from DoD (2005). 13

26 Figure 2.1 Schematic of Tie Forces from DoD (2005) Indirect methods are specified in DoD (2005), ACI (2005), ASCE (2002), and GSA (2003). These codes make general recommendations with respect to continuity, redundancy, and ductility of the structure for the intended purpose of increasing the structure s ability to redistribute forces. In the case of the ASCE disproportionate collapse provisions, it is specified that sufficient continuity, redundancy, or ductility shall be provided. However, no guidance is given on what is sufficient. ACI (2005) gives more specific requirements in the form of suggested reinforcing details to improve redundancy, continuity, and integrity, such as use of mechanical splices, 14

27 continuation of bottom reinforcement, and use of moment resisting frames. DoD (2005) recommends the use of tie forces to produce a catenary response of the structure. This is the recommended method to provide the minimal level of resistance for a building that is not believed to be a likely target for terrorist activity in this code. GSA (2003) requires the use of beam-to-beam continuity across a column, and overall redundancy and resiliency of the connections, which is to be achieved through symmetric reinforcement, welds of appropriate type and orientation, and increased torsional and minor axis bending strength. Criticisms of these methods are that they do not consider any specific loading event, so the level of safety provided against abnormal loadings is unknown. Additionally, the methods do not require analysis, so the engineer has no intuition for how the structure actually behaves. Furthermore, the tie force design method has been criticized because the minimum levels of reinforcement do not have a clear basis (Mohamed 2006). DoD (2005) is the only code that provides direct design requirements for determining the required tie forces, although the development of this required amount is still unknown. For structures requiring a high level of protection these indirect methods are often combined with direct methods. This suggests indirect methods are not considered to be the best safety measure that can be taken. The tie force method can be considered to be opposite to compartmentalization, where the structure is segmented to localize collapse. There is some fear that if the structure cannot resist collapse and is strongly tied together, it will 15

28 completely collapse into itself. This was seen in World Trade Center 7, where an interior column failed due to fire and pulled the entire structure in on itself Direct Methods In the direct design approach, the engineer specifically addresses the event of progressive collapse. This is done by either designing for the abnormal loading or by assuming a specific localized failure due to its occurrence and designing for this failure. Two approaches exist to accomplish these goals, the local resistance method and the alternate load path method Alternate Load Path Method The method of direct design for progressive collapse that is the focus of the design codes is called the alternate load path method. In this analysis, critical structural members are individually assumed to have failed and are removed from the system prior to analysis. The loads that this member carried are redistributed to adjacent members and a structural analysis is carried out to determine if any additional members in the structure fail. If additional members fail in this analysis, they are removed from the analysis, their loads are redistributed, and a new analysis is performed. This process is iteratively repeated until either the structure fails (based on the failure criteria set forth in the applicable code) or no additional failures occur. The most critical member to remove is nearly always a column and columns at different locations throughout the building are removed. Structures with abnormal geometry may have other members that will govern the analysis. 16

29 This method of analysis is implemented in GSA (2003) and DoD (2005) and suggested in the commentary of ASCE (2002). The GSA (2003) and DoD (2005) codes specify the acceptable limits of collapsed area for the floor directly above the removed column to be the greater of the adjacent bays or a given percentage or total area. This area changes depending on the code and whether an external or interior column is removed. The GSA (2003) code allows 1800 ft 2 of collapsed area for an exterior column and 3600 ft 2 for an interior column. The DoD (2005) code, similarly, allows 15% and 30% collapsed area for exterior and interior columns, respectively. If these limits are exceeded, the structure is considered to have failed. Crushing force has been shown to the leading cause of death in structural failures; therefore collapsed area is an important measure to minimize (Hayes et al 2005). It is important to note that the alternate load path method assumes that the entirety of the structure is in a perfect condition, except that one member is simply missing. Although a member is removed from the structure to simulate it being in a damaged state, the design codes do not suggest the member was removed by any specific loading. For this reason it could be argued that this method is another indirect design method, as it is only intended to test a structure s redistribution capability and not design for a specific abnormal loading. Since the removal of one element does not have any scientific bearing nor is it necessarily the result of any abnormal loading event, the basis of the exercise is arbitrary. An abnormal loading resulting in the immaculate removal of one column has been critiqued as an unconservative assumption as well as unrealistic because of the 17

30 potential for large geometric and material nonlinearities in members that survive the initiating event (Krauthammer 2005). The possibility that multiple members may be taken out of service from the initiating event is also significant, but ignored in the procedure (Mohamed 2006). The removal of any member after an abnormal loading event would in reality be a violent event, making it likely that some amount of damage would extend into the structural elements around it, such as the connections, which are important for force redistribution. For example, an examination of the Murrah Federal Building determined that the truck explosive failed one column and portions of the second, third, and fourth-floor slabs above it, leading to the failure of three more columns and consequently a significant portion of the structure (Osteraas 2006). Furthermore, Nair (2006) points out that the alternate load path method encourages the use of a larger number of smaller members, which are more vulnerable, as opposed to fewer larger ones. This is a direct consequence of the threat-independent nature of the provisions Specific Local Resistance Method A second method of direct design for preventing disproportionate collapse, contained in ASCE (2002), is called the specific local resistance approach. The purpose of this method is to prevent the loss of a critical load-carrying member under a specific extreme event such that progressive collapse cannot initiate. ASCE attempts to accomplish this by providing load combinations incorporating an accidental load, A k ; however, it is left to the designer to quantify this load. 18

31 Although the response of the structure for that specific threat level will be known, its response to other different threats is still not known (Ellingwood 2005). Another disadvantage to this method is that it gives exactly what loads are needed to exceed the design capacity of the structure (Ellingwood 2002). This information makes the structure vulnerable to attack, as the precise loading needed to cause its failure can be found by consulting the design calculations. While this method is often used as a means of retrofitting existing structures for progressive collapse (Mohamed 2006), it is likely not economical to locally strengthen every location in a structure. 2.5 Current Progressive Collapse Design Codes in the United States As mentioned above, there are currently four primary United States codes (ASCE 2002, ACI 2005, GSA 2003, DoD 2005) addressing progressive collapse in the design of buildings. The approaches utilized in these codes were summarized in Section 2.4 of this thesis. In this section more specific information on the applications of these methods in each code are given ASCE 2002 Minimum Design Loads for Buildings and Other Structures This code provides requirements to increase a building s general integrity. The requirement specific to progressive collapse, given in Section 1.4 of ASCE 2002, says that buildings should be designed to sustain local damage and not be damaged to an extent disproportionate to the original local damage. While no specific requirements are given, the code does suggest some design methods in the commentary section. These include the alternate load path method as well as the specific local resistance 19

32 method. Some specific charge sizes (for different threat levels, in weight of TNT) are recommended is association with the specific local resistance method. These charge sizes do not have corresponding stand-off distances defined, so the actual blast loading is not specified. Therefore, some judgment is needed to apply them ACI Building Code Requirements for Reinforced Concrete (2005) This code utilizes the indirect design approach to address progressive collapse. This is done by requiring specific reinforcement details to increase the overall integrity and stability of the structure. These include horizontal and vertical ties throughout the structure, continuous reinforcement in perimeter elements, a specified amount of splicing, and connections that do not rely on gravity. Since the basis of the requirements is not clear, there is no certainty that they adequately prevent progressive collapse GSA Progressive Collapse Analysis and Design Guidelines (2003) The GSA (2003) code contains a threat independent procedure meant to be used in the design and analysis of new federal office buildings. This code is largely dependent on the analyst using the alternate load path method to simulate the loss of structural members in different scenarios. The designer must first inspect the building s structural layout and find each of the members that will need to be removed. These are, at a minimum, a column at the center of the long side, a column at the center of the short side, and a corner column. Any other areas in the building that are abnormal will have members that will have to be removed. Also if the building has 20

33 uncontrolled public floor areas, an interior column must be removed. If the interior of the building is considered secured, an interior column is not removed, which contradicts the threat-independent philosophy of the code. The analysis procedures used can either be a linear-static procedure or a nonlinear-dynamic analysis. When using the linear-static procedure the code requires the analyst to double the design loading through the use of a dynamic increase factor of 2, which is meant to make the dynamic loading phenomenon be applied artificially in a static fashion. The use of the dynamic increase factor can lead to an overly conservative design, so analysts are encouraged to use the more rigorous analysis methods if a more economic design is desired (Ruth et al (2006). If the non-linear dynamic procedure is used, the analyst does not use a dynamic increase factor. In order for a building to be considered safe, the collapsed area must not exceed the greater of the bays directly connected to the removed column or a given floor area (15% for exterior columns or 30% for interior columns). A member is considered failed once its allowable demand to capacity ratio (DCR) is exceeded. To calculate the DCR, the internal capacity for each force type (shear, axial, moment) in every member is calculated by formulas provided in the code and compared to the internal forces (demands) found from the analysis. These are then compared to the allowable DCR values given in the code, which change depending on the force type, member type, and material. If the axial or shear DCR is exceeded, the member is considered failed, and removed from the structure for a consecutive analysis. If a moment DCR is exceeded, a plastic hinge is input into the structure and the analysis is restarted. If at the end of an 21

34 analysis the DCR for each member is under the allowable value and the structural failure criteria are not met, the analysis can stop and the structure is considered to meet the requirements. Conversely, if the structural failure criteria are met, the structure is considered inadequate DoD Design of Buildings to Resist Progressive Collapse (2005) This code also approaches progressive collapse without considering a specific threat. The code, as with the other codes, is not intended to eliminate the initial local damage, but reduce the casualties after its occurrence. This is done by utilizing two methods previously discussed: the alternate load path method and the tie force method. The prescribed usage of these two methods in the design process is dependent on the level of protection (LOP) desired for the building by the Project Planning Team. The LOP are broken into four categories: very low (VLLOP), low (LLOP), medium (MLOP), and high (HLOP) with the lowest two intended to cover the majority of buildings. For VLLOP and LLOP buildings, the tie force method is used. VLLOP structures are required to have a minimum level of horizontal ties and the LLOP structures must have a minimum level of horizontal and vertical ties. If an element does not pass the requirements, then it must be redesigned. In the case of the LLOP structure, if a vertical tie requirement is not met, the alternate load path method can be used to verify the structure s capacity. 22

35 In the case of the MLOP and HLOP structures, they must meet the same (vertical and horizontal) tie force requirements as a LLOP structure. Also an alternate load path analysis must be performed for critical members in the structures at each floor level including, at a minimum, a: column in the center of the short side, column in the center of the long side, and corner column. Other structural members should be included if the building has an abnormal configuration. The last requirement involves ductility requirements in the perimeter vertical load carrying members at the ground floor, specifically requiring that the lateral uniform load which defines the shear capacity is greater than the load associated with the flexural capacity. As with the GSA (2003) criteria, there are different levels of analysis that can be used to perform the alternate load path analysis. These include linear static, nonlinear static, and nonlinear dynamic analysis methods. The alternate load path method prescribed uses a Load and Resistance Factor Design approach (LRFD) to dictate which of the members in the structure have failed. If the failure of a member is flexure based, a hinge is input into the structure at that location and the analysis is restarted. If a shear or axial failure occurs, the member is removed completely from the analysis. The structure is considered to be safe if the total collapsed area of the floor above the removed column is less than prescribed limits. These limits are the smaller of 15% of the floor area or 750 ft 2 for an exterior column and the smaller of 30% or 1500 ft 2 for an internal column. The alternate load path analysis must also be peer reviewed. 23

36 2.6 Current Research The following sections review current topics of research in the field of progressive collapse. These research topics are aimed at examining current analysis approaches and providing new, innovative methods to be considered in the design and analysis process to lower the occurrence of progressive collapse Global vs. Local Effects Our design concepts today are largely based on the idea of local element failure and stability, while the influence of these failures on the global system response or stability of the structure is neglected (Starossek 2006). None of the current progressive collapse guidelines considers changes in the stability of the structure resulting from the progressive failure of members. However, Ettouney et al (2006) present a method to account for this effect through the use of alternative (larger) effective length factors, reflecting the change in boundary conditions for columns in a damaged structure. The authors also account for an increase in column axial force due to the loss of the target column. The ratios of the critical buckling load of a given column to the axial force in the column in the damaged and undamaged states are computed and compared to calculate what the authors define as the stability exceedance factor, which represents the change in the stability of the column. The authors then suggest steps to be taken for various ranges of stability exceedance factors, while noting that these are preliminary suggestions and that additional studies are needed for verification. 24

37 2.6.2 Structural Response to Blast Loading One abnormal loading that can cause severe localized effects on structures is blast loads. There are many different types of explosive devices that a structure could be subjected to, ranging from military explosives to homemade bombs such as used on the Murrah Federal Building to accidental explosions such as a gas leakage igniting in the Ronan Point Apartment Building. The vehicle bomb is, perhaps, the most common attack device of terrorists (Byfield 2006). Despite the type of bomb, its effects on the structure are a highly dynamic event, sending immense shock waves that end within a tenth of a second. Because of the rapidity at which the loads are applied to the structure, high strain rate effects are induced, which should be considered (Marchand and Alfawakhiri 2005). The blast loading has little effect on the lateral resisting system of the building as the blast duration is only a tiny fraction of the building s natural period. Therefore, the effects are far more localized on individual elements whose natural period more closely matches the blast duration (Marchand and Alfawakhiri 2005). If these individual elements near the blast fail, then the failure can propagate. Homemade bombs, which have approximately half the TNT equivalence as military bombs, detonate much slower and can impart more energy into the structures components (Byfield 2006). Many agree that blast load design is a not well understood by practicing engineers (NIST 2001, Cagley 2002). The effects of blast loading on the more intricate parts of the structure, such as connections, are complex and hard to predict or quantify. For example, the negative pressure phase of a bomb blast is known to cause failure in 25

38 steel connections which would have otherwise survived (Byfield 2006). In a study by Krauthammer (2005) it was concluded that improved design approaches for steel connections in blast resistant buildings was extremely urgent. The application of seismic design as a way of addressing progressive collapse and blast loading has been considered in the past. This concept is discussed in the following section Application of Seismic Design Some engineers (Hayes et al 2005, Carino and Lew 2001, Khandelwal and El- Tawil 2007, Corley 2002) feel that designing a new structure or rehabilitating an existing structure for a high level earthquake may be an efficient method in addressing progressive collapse, as the design for earthquakes leads to buildings with general structural integrity, ductility, and greater ability to deform in catenary modes. While this can provide a more robust structure, abnormal loadings such as blast loads are very different from earthquake loads. One difference is that their duration is orders of magnitude shorter, causing strain rate effects to become critical and necessary to account for. They also differ in that a structure responds globally to seismic forces, while its response to abnormal loads is generally much more local. Therefore, a structure subjected to seismic forces will use the inelastic response of many members to mitigate global building failure where blast loads or other localized abnormal loadings will be most severe at only a few members of a structure. Also, seismic connection details developed after the Northridge earthquake, which could possibly help in progressive collapse mitigation, were not developed to consider effects unique to blast 26