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
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
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
TABLE OF CONTENTS
LIST OF TABLES ..................................................................................................... viiii
LIST OF FIGURES ........................................................................................................ x
ABSTRACT .................................................................................................................. xi
Chapter
1.
INTRODUCTION .............................................................................................. 1
1.1
1.2
1.3
1.4
1.5
2.
Background................................................................................................ 1
Motivation for Research ............................................................................ 3
Research Objectives .................................................................................. 4
Scope of Work ........................................................................................... 6
Thesis Organization ................................................................................... 7
BACKGROUND AND LITERATURE REVIEW ............................................ 9
2.1
2.2
2.3
2.4
Introduction ............................................................................................... 9
Definition ................................................................................................... 9
Abnormal Loads ...................................................................................... 10
Methods for Collapse Mitigation............................................................. 11
2.4.1
2.4.2
2.4.3
Event Control .............................................................................. 12
Indirect Methods.......................................................................... 13
Direct Methods ............................................................................ 16
2.4.3.1
2.4.3.2
2.5
Alternate Load Path Method ........................................ 16
Specific Local Resistance Method ............................... 18
Current Progressive Collapse Design Codes in the United States .......... 19
2.5.1
2.5.2
2.5.3
ASCE 2002 “Minimum Design Loads for Buildings and
Other Structures” ......................................................................... 19
ACI “Building Code Requirements for Reinforced
Concrete” (2005) ......................................................................... 20
GSA “Progressive Collapse Analysis and Design
Guidelines” (2003) ...................................................................... 20
iv
2.5.4
2.6
Current Research ..................................................................................... 24
2.6.1
2.6.2
2.6.3
2.6.4
2.6.5
2.6.6
2.7
3.
DoD “Design of Buildings to Resist Progressive Collapse”
(2005) .......................................................................................... 22
Global vs. Local Effects .............................................................. 24
Structural Response to Blast Loading ......................................... 25
Application of Seismic Design .................................................... 26
Connections ................................................................................. 27
Dynamic Analysis ....................................................................... 29
New Design and Analysis Methods ............................................ 30
Summary.................................................................................................. 33
METHODOLOGY ........................................................................................... 35
3.1
3.2
Introduction ............................................................................................. 35
Column Analysis Procedure .................................................................... 36
3.2.1
Finite Element Details ................................................................. 37
3.2.1.1
3.2.1.2
3.2.1.3
General LS-Dyna Input Format .................................... 37
Modeling Procedure ..................................................... 38
Model Input .................................................................. 40
3.2.1.3.1
3.2.1.3.2
Geometry............................................................. 40
Materials ............................................................. 43
3.2.1.3.2.1 Johnson-Cook Strength Model ..................... 45
3.2.1.3.2.1.1 Strength Model Variable Form ......... 46
3.2.1.3.2.1.2 Strength Model Constants
Determination ......................................... 47
3.2.1.3.2.2 Johnson-Cook Failure Model ........................ 57
3.2.1.3.3
3.2.1.3.4
Boundary Conditions .......................................... 60
Loading ............................................................... 61
3.2.3.3.4.1 Axial Loading .............................................. 61
3.2.1.3.4.2 Blast Loading .............................................. 65
3.2.1.3.5
Output Control .................................................... 68
v
3.2.2
3.3
Deflection Failure Criteria........................................................... 69
Missing Girder Analysis .......................................................................... 77
3.3.1
3.3.2
Procedure ..................................................................................... 78
Live Load Determination ............................................................ 81
3.3.2.1
3.3.2.2
3.3.2.3
3.3.2.4
3.3.3
3.3.4
3.3.5
3.3.6
4.
Need for Validation Model .................................................................... 100
Lawver et al (2003) Model Description ................................................ 101
Assumptions .......................................................................................... 104
LS-Dyna Validation Model ................................................................... 108
Results ................................................................................................... 109
SENSITIVITY ANALYSIS ........................................................................... 113
5.1
5.2
5.3
5.4
Introduction ........................................................................................... 113
Sensitivity Analysis Model.................................................................... 113
Displacement Results ............................................................................ 115
Stress Results ......................................................................................... 118
5.4.1
5.4.2
5.4.3
5.5
6.
Dead Load Determination ........................................................... 88
Girder Selection ........................................................................... 91
Effective Load Factor .................................................................. 95
Bridge Models for Missing Girder Analysis ............................... 98
VALIDATION STUDY ................................................................................. 100
4.1
4.2
4.3
4.4
4.5
5.
Lane Load ..................................................................... 82
Truck Load ................................................................... 83
Tandem Load ................................................................ 85
Distribution Factors ...................................................... 86
Axial Stress................................................................................ 119
Shear Stress ............................................................................... 121
Von-Mises Stress ....................................................................... 123
Conclusion ............................................................................................. 125
RESULTS ....................................................................................................... 127
6.1
6.2
Introduction ........................................................................................... 127
Column Analysis Results ...................................................................... 127
vi
6.2.1
6.2.2
Parameters Varied in Column Analysis .................................... 128
Stand-off Distance Results ........................................................ 130
6.2.2.1
6.2.2.2
6.2.2.3
6.2.2.4
6.2.2.5
6.2.2.6
6.2.3
6.3
Blast Threat Level Representation of Missing Column
Analysis ..................................................................................... 145
Missing Girder Analysis Results ........................................................... 148
6.3.1
6.3.2
6.3.3
7.
Charge Size ................................................................. 134
Vertical Position of Charge ........................................ 135
Column Height ........................................................... 136
Column Depth ............................................................ 139
Column Stiffness ........................................................ 143
Parametric Summary .................................................. 144
Span Length of 50 ft .................................................................. 148
Span Length of 100 ft ................................................................ 150
Conclusions from Missing Girder Analysis .............................. 152
SUMMARY AND CONCLUSIONS ............................................................. 156
7.1
7.2
7.3
Summary and Conclusions of Missing Column Analysis ..................... 156
Summary and Conclusions of Missing Girder Analysis ....................... 158
Synthesis ................................................................................................ 161
REFERENCES ........................................................................................................... 164
APPENDIX ................................................................................................................ 174
A.1 Johnson-Cook Parameter Results for Remaining Data Sets.................. 174
A.2 LS-Dyna Keyword File for 14-ft W14x82 Column .............................. 178
vii
LIST OF TABLES
Table 3.1 – Calculations for Johnson-Cook Constants from Figure 3.2……………...50
Table 3.2 – Johnson-Cook Strength Model Constants Used in LS-Dyna..…………...56
Table 3.3 – Johnson-Cook Failure Model Constants Used in LS-Dyna…..……….…58
Table 3.4 – Service Loadings for Each Column………………...……………………62
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…..……..95
Table 3.14 – Effective Load Factors with and without Impact……………………….97
Table 3.15 – Service Loadings and Nominal Capacity for Each Span Length..……...99
Table 4.1 – Criteria from Lawver et al (2003) to Find Bomb Size and Location...…104
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...111
viii
Table 5.1 – Percent Relative Error of Displacement along Height of Column……...117
Table 5.2 – Percent Relative Error for Various Mesh Sizes………………………...118
Table 6.1 – Failure Stand-off Distance for Each Column (Small Charge)…….........132
Table 6.2 – Failure Stand-off Distance for Each Column (Large Charge)……..…...133
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 1………………………………………………………………174
Table A.2 – Data Set 2………………………………………………………………175
Table A.3 – Data Set 3………………………………………………………………176
Table A.4 – Data Set 4………………………………………………………………176
Table A.5 – Data Set 5………………………………………………………………177
Table A.6 – Data Set 6……………………………...........….....................…………177
Table A.7 – Johnson-Cook Parameter Summary……………………………………177
ix
LIST OF FIGURES
Figure 2.1 – Schematic of Tie Forces from DoD (2005)…………………...………...14
Figure 3.1 – Stress-Strain Curve from Quasi-Static Tensile Test……...……………..48
Figure 3.2 – Linear Fit to Log-Log Data Given in Table 3.5………...………………51
Figure 3.3 – Johnson-Cook Approximation to Experimental Data……..……………52
Figure 3.4 – Plot of Equation 3.7 for Various Strain Rates…...……..………….……55
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…..…..120
Figure 5.3 – Shear Stress Distribution along Web at Mid-Height of Column............123
Figure 5.4 – Von-Mises Stress Distribution along Web at Mid-Height of Column...124
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
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
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
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
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
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
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
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
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
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
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
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
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 2.4.3.1.
2.3
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
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
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.
2.4.1
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
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.
2.4.2
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
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
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
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.
2.4.3
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.
2.4.3.1
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
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 ft2 of collapsed area for an exterior
column and 3600 ft2 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
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.
2.4.3.2
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,” Ak;
however, it is left to the designer to quantify this load.
18
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.
2.5.1
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
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.
2.5.2
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.
2.5.3
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
“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
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.
2.5.4
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
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 ft2 for an exterior column and the smaller of 30% or 1500 ft2 for an
internal column. The alternate load path analysis must also be peer reviewed.
23
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.
2.6.1
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
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
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.
2.6.3
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
loadings (Marchand and Alfawakhiri 2005). Using seismic detailing as an approach to
progressive collapse can be beneficial as strengthening for one extreme event will help
with another, but is disputed as a long term solution because blast and seismic loads
have too many differences. Thus, connection behavior in progressive collapse
situations is an area of current research. Some of this work is provided in the next
section.
2.6.4
Connections
Connections are believed to be a vital and integral part of a structure’s resistance
to collapse and are critical to satisfying the requirements of many of the indirect
methods of design. Therefore, they are currently one of the many heavily researched
topics in the field of progressive collapse. There are many traits that a connection
should have to counter the array of phenomena that it may see in a collapse situation, as
not every situation will be the same. These traits include being ductile, continuous, and
strong enough to handle the large forces and rotations seen in a collapse situation. The
capability of modern connections to achieve these demands is a topic of several recent
studies.
One of the most effective load carrying mechanisms for a structure following the
removal of one of its columns is for the beams and connections to develop catenary
action. In catenary action, a beam or a system of beams develops plastic hinges at
locations along the beam, such that it acts like a cable, carrying its load in tension
(Khandelwal and El-Tawil 2007). This requires a connection to first develop a plastic
27
hinge and then, simultaneously, have enough capacity to handle large tensile forces.
Whether or not we have connections that can adequately provide both of these
important functions is still largely unproven (Hamburger and Whittaker 2005).
In another study the authors attempted to quantify catenary forces, and found
that they frequently surpassed the design forces given in a UK design code (Liu et al
2005). Houghton and Karns (2001) cast some doubt on the presumption of catenary
action by concluding that it is not likely a girder-to-girder connection can remain
sufficiently intact when the violent nature of the column removal is taken into account;
a factor not considered in design codes. From some numerical modeling in this study
the authors concluded that a new line of innovative connections are needed. These two
studies show that there is still much to learn about the behavior of members in catenary
situations, and that design codes’ consideration of this is currently elementary.
In a study done by Khandelwal and El-Tawil (2007), they numerically modeled
a two bay moment resisting frame subject to member removal, and observed whether
the connections were ductile enough to allow the frame to deform in a catenary mode.
They found that high beam depth and yield to ultimate strength ratios adversely effected
connection ductility. Also, stronger beam web-to-column connections, achieved by
thicker shear tabs or welding, led to more ductile connections as fracture was
concentrated in the flanges. In another numerical modeling study by Munoz-Garcia et
al (2005), in which various connections were modeled under rapid rates of loading, they
found the connections to be more brittle than expected and that bolts under high strain
rate did not strain harden, leaving a reduced tensile strength and ductility compared to
28
that which was expected. The authors recommended more robust and rigid connections
than are currently being used in the UK.
2.6.5
Dynamic Analysis
The modeling currently being used to study progressive collapse is extremely
complex. The researchers carrying out these analyses have years of training and
advanced degrees. Requiring such an analysis of engineers in the design industry would
be unreasonable given their budget and time constraints, education, and training.
Therefore, simplified methods that engineers can use have been incorporated in design
codes. What follows is a discussion on the dynamic analysis methods presented in
these codes to address alternate load path method.
The GSA (2003) and DoD (2005) specifications attempt to account for the
dynamic effects associated with the loss of a member. This is done based on either a
static analysis, where the applied loads are amplified by a dynamic amplification factor
to indirectly account for the dynamic nature of the loss of a member, or a true dynamic
analysis, which typically consists of a time history analysis. In the static amplification
method, a dynamic amplification factor equal to 2.0 is applied to the static loads in both
the GSA and DOD guidelines. The work of Kaewkulchai and Williamson (2004) and
Pretlove et al (1991) clearly demonstrate that static loads must be amplified if this
analysis type is used. However, this amplification factor of 2 is based on the difference
between the static and dynamic response of a linear elastic single degree of freedom
system subjected to an instantaneously applied load of infinite duration, which
29
represents an upper-bound of possible values (Powell 2005). Case studies aimed at
examining the validity of these amplification factors have shown these methods to be
highly conservative (Ruth et al 2006, Powell 2005).
Dynamic analysis methods directly account for dynamic effects in the analysis,
and therefore, neglect the usage of the dynamic increase factor. This leads to more
accurate results and will ultimately provide a more cost effective design. These
dynamic analyses are usually avoided, however, due to their perceived complexity and
the peer review requirements stipulated by many building codes on this type of analysis
(Marjanishvili and Agnew 2006).
2.6.6
New Design and Analysis Methods
As discussed earlier, the alternate load path method requires the analyst to
consider the behavior of a structure following the removal of an element. This can be a
highly complex problem, and an analysis method that can follow the behavior of the
structure from initial damage till the point at which damage arrests while including
dynamics effects, geometric changes, high deformations, and material non-linearities is
desired (Gross and McGuire 1983).
The work of Kaewkulchai and Williamson (2003, 2004, and 2006), who have
developed a computer program that tracks progressive failure, represents one of the
state-of-the-art procedures in modeling progressive collapse. This program analyzes the
response of planar frames from the time immediately following the initiating event until
the failure arrests. Beam-column elements with multi-linear lumped plasticity
30
constitutive properties that account for strength and stiffness degradation through a
damage parameter are employed in the program along with the Newmark-Beta
integration scheme. The program also has the ability to account for the effects of failed
members impacting surrounding elements. This program is used to demonstrate
(Kaewkulchai and Williamson 2004) that the dynamic redistribution of loads is a
significant consideration in accurately modeling disproportionate collapse. Another
model that is able to track progressive collapse behavior is Extreme Loading of
Structures, an applied element-analysis software program available from Applied
Science International, which models service, blast and debris loading on the structure
dynamically (Hoon et al 2009). One disadvantage of using this program for progressive
collapse analysis is that there are no high strain rate material models currently available.
Another method, described in Grierson (2005), uses stiffness matrix methods to account
for any changes in the structure’s material properties and local or global stability. This
is done by analyzing the structure in incremental load steps and updating the matrices at
each step according to any changes.
Advances in energy equilibrium based methods have been suggested (Loizeaux
and Osborn 2006), in part because they directly consider dynamic effects of collapse
without the use of dynamic amplification factors. One approach to assessing
progressive collapse potential has recently been suggested by Dusenbury and
Hamburger (2006). In this approach, the kinetic energy of the failing member is
calculated and compared to the energy absorption capacity of the remaining elements in
the structure. If the kinetic energy exceeds the absorption energy capacity, then
31
collapse will progress. Another energy approach which has been taken is that of Bazant
and Verdure (2007), who developed a one-dimensional continuum model of progressive
collapse for tall buildings by analyzing the motion and flow of energy in the WTC
collapse. Energy based retrofit methods for existing buildings have also been
considered, such as Zhou and Yu (2004) who proposed using metal honeycomb devices
between floors to absorb energy through compaction and thus arrest collapse.
Another method that has been proposed by Liu et al (2005), called the hybrid
design method, suggests that designers select members for their structure assuming
pinned connections only, and then during construction input semi-rigid connections.
This will inherently give each member and connection additional moment capacity, and
will consequently reduce the tie forces when catenary action forms as the loads are able
to be resisted by the combination of bending and axial effects in the beam instead of
predominately axial only. While this leads to stronger members and slightly reduced tie
forces, this “over-strengthening” approach to design has been shown to lead to brittle
failure, as connections were shown to fail suddenly before the beams reached their
plastic moment capacity (Byfield 2006).
Ellingwood (1974, 1978, 2002, 2005, and 2006) has published many papers on a
risk based approach to mitigation of progressive collapse. This approach, similar to
current design methods such as “load and resistance factor design”, is an attempt to
formulate a probabilistic equation from which the likelihood of failure can be computed
for an abnormal loading acting on a given structure. This is computed by taking the
product of the probability of the abnormal loading occurring, the probability of local
32
damage given the abnormal loading, and finally the probability of structural failure
given the local damage. Once this failure probability is found, it can be equated to a
socially acceptable value (Ellingwood 2002). The primary difficulty of applying this
method is the inability to accurately define the required probabilities. Agarwal (2003)
suggests an alternative risk based approach that uses hierarchal models.
Other research objectives are aimed at finding a common language among those
in the industry with regards to general terminology of progressive collapse. Starossek
(2007) addressed some common terminology by categorizing collapses into six types,
each of which label a different type of collapse. The types of collapse are intended to
be descriptive in which failure mode they represent, such as “pancake” and “domino”.
If these types of collapse are differentiated conceptually into their mechanisms, it’s
possible that a design approach can be applied to each separately. This would avoid the
need to develop an all encompassing approach to cover all possible collapse scenarios.
In another study Starossek (2006) proposed defining robustness as “insensitivity to local
failure”, which can be used as a metric incorporated into design codes.
2.7
Summary
In this section the methods currently used for progressive collapse mitigation
were given, along with how these methods are specifically addressed in each design
code. One key approach referred to as alternate load path analysis, or missing column
analysis, is a paramount analysis method used in three of the four major codes in the
United States. The research in this thesis is intended to test some of the assumptions
33
made in alternate load path analysis as prescribed in all of these three progressive
collapse design codes, namely DoD (2005), GSA (2003), and ASCE (2002). These
assumptions are that the effects of the initial abnormal loading are not considered, there
is no consideration for the size or spacing of the members, and that only one member is
immaculately removed from the structure without any additional damage to the
structure.
34
Chapter 3
METHODOLOGY
3.1
Introduction
There are two major objectives of this thesis, which were given in Chapter 1.
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 as a means
of assessing the conservatism and uniformity in level of safety provided for this type of
abnormal loading. To achieve this, finite element models of blast loads acting on steel
columns were created. From these the range of stand-off distances causing failure for a
given blast size could be determined through iteratively evaluating various stand-off
distances. Section 3.2 provides details on the iterative procedure used, the finite
element models, their input, and the failure criteria used for the columns. The second
objective was to determine what redundancy and system load carrying capacity exists in
simple-span steel girders bridges in the event of a failed girder as a result of the reserve
strength of the system inherent to the design process. 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 building design. Details for this analysis are
located in Section 3.3.
35
3.2
Column Analysis Procedure
Finite element analyses of blast loadings acting on “typical” steel columns are
carried out to find a threat level (combination of stand-off distance and charge size) that
corresponds to their failure. This threat level will be determined as a function of the
height of the charge center, where the bottom, mid-height, and top of the column of
interest are the three heights considered. Models of three column cross-sections having
(W14x82, W14x109, and W10x77) two different heights (14 ft and 18 ft) were created
in LS-Dyna. These were then subjected to a combination of a representative axial
service loading and blast loading using two different charge sizes (small and large). The
blast was detonated in the weak direction of the column, as opposed to the strong
direction, such that it acted on the face of the web. This is the most critical orientation
of the blast (resulting in failure at smaller stand-off distances) because the column is
weaker and less stiff in this direction, and thus secondary effects will be greater.
Once the models are created, the procedure for finding the stand-off distances
causing failure of a given column for a given charge size is a relatively straight forward
process. A given charge size is input at an arbitrary horizontal stand-off distance for
each height evaluated (bottom, middle, or top) and then the maximum horizontal
displacement of the model is then compared to the failure criterion (shown in Section
3.2.2). This begins an iterative process where: if the failure criterion are exceeded then
the stand-off distance is increased and the model is run once again; and if the failure
criterion are not exceeded then the stand-off distance is decreased and the model is run
once again. This process repeats until the farthest stand-off distance that causes failure
36
is found (to the nearest ft, measured from the centerline of the web); distances less this
position will also cause failure.
By determining a stand-off distance representing failure at the bottom, top, and
middle of the column, a failure “contour” showing the vulnerability of the column to
blast loads as a function of height of the charge center is found. A discussion of the
specific parameters for the column analysis such as finite element details, input, and
failure criterion are located in the following sections.
3.2.1
Finite Element Details
The program used to create the finite element models was the commercial
software LS-Dyna. In this section the required LS-Dyna format for the finite element
models is given along with the procedure used in this work to create the models. The
numerical input parameters are also described in this section. Many of these are simply
the default values defined in LS-Dyna as they are efficient and are applicable to these
models (LSTC 2009). Input parameters that were not equal to the default values and
thus required additional consideration are also described in the following sections.
3.2.1.1
General LS-Dyna Input Format
The finite element model data that is analyzed in LS-Dyna consists of a series of
commands in a text document, referred to as a key file. These commands establish the
model’s geometry, materials, boundary conditions, loading, and output control. The
user selects appropriate commands to address each of these topics so that the model can
run successfully. These commands are achieved by using keywords, which are words
37
in all caps following an “*” in the first column (i.e., *LOAD, *CONTROL, etc.). Each
of the keywords is followed by the required input in a block of data on the following
text lines. This block of data must be input in a proper format, which is described in the
Keyword User’s Manual (LSTC 2009), for it to be read by LS-Dyna. The input for that
keyword command will stop being read once the next keyword in the file is found.
After LS-Dyna runs the analysis on the keyword file created, the results are
reported in a post processor. From the post processor the behavior of the model can be
observed visually and numeric data can be found by selecting the element or node of
interest and the data type that is needed (stresses, strains, displacements, etc). For
example, the displacement of a node can be observed by clicking on the node and
selecting the displacement in the coordinate of interest. This would then show a plot of
the displacement for that node over the period of time the model was run. From this
plot the displacements can be found.
3.2.1.2
Modeling Procedure
The first step that was taken to create a successful key file was to review various
LS-Dyna examples that the developer has on file. These were obtained by contacting
technical support and requesting information about the specific analysis that was of
interest. These files proved to be a good starting point for creating the finite element
models in this thesis. Each of the sample key files that were obtained subjected
different types of surfaces and shapes to an array of specific blast loadings, which, in
combination with the Keyword User’s Manual (LSTC 2009), helped select the needed
38
keywords to perform such an analysis. It is also possible to search the internet for
forums where users share key files and information about their analysis.
The next step, after the general keywords needed for the analysis were known,
was to start creating the key file for each model. One of the most time-consuming steps
in this procedure was creating the nodes and elements in the structure, as there were
many of these in the model. Completing this by hand, just using the key file, would be
very tedious. Instead, a program called Patran was utilized to create the nodes,
elements, and boundary conditions. The user is able to apply each keyword by using
Patran, as it is built to be compatible with LS-Dyna. However, the entire model is not
completed in Patran as some of the keywords are more easily input to the key file after
the geometry is finalized. The program also allows the user to view the direction of the
element normals, and switch them if required, which proves to be a useful feature for
applying blast loadings. Once the objectives that are otherwise too difficult to do in the
key file are accomplished, Patran can output an LS-Dyna key file. After the key file is
created using Patran, the remaining keywords, are progressively added. When all of the
keywords are added, the analysis can be run.
If the analysis abruptly stops due to a termination error, an error report will be
given. If the analysis is carried out successfully the results can be viewed in the LSDyna post-processor. However, even though the analysis is completed successfully, it
does not ensure that the results are correct. Some sort of validation study should be
carried out, which verifies the methods used in the analysis will provide results similar
39
to another that is known to be correct. The validation study for the models created in
this thesis is shown in Chapter 4.
3.2.1.3
Model Input
The sections herein contain a summary of each of the needed keyword
commands as they apply to the model’s geometry, materials, boundary conditions,
loading, and output control. Appendix A.2 shows a sample key file, containing the
keywords discussed in this thesis. The sample provided is of the blast loads being
applied to the W14x82 column. It should be noted that SI units (cm, gram, and microseconds) were used in this key file.
3.2.1.3.1
Geometry
Each column was modeled as a mesh of quadrilateral shell elements. Because
the plates of the column are relatively slender and will be loaded in a combined state of
axial force and flexure, they are prone to buckling and shell elements have the ability to
accurately respond to this loading type and represent this behavior. Quadrilateralshaped elements where used as they are efficient and each of the rectangular plates of
the column can be conveniently meshed into these shapes. The quadrilateral elements
were also desired to be as near to square as possible (each side having near equal
lengths) to reduce computational error.
The geometry of the model, as mentioned earlier, was created using the program
Patran. To begin this process the coordinates at the corners of the plates forming the
column were input into the program as nodes. As quadrilateral shell elements were
40
used, the nodes were placed at the mid-thickness of the plates for the web and flanges
and the thickness is applied as an element property later. Once the corner nodes were
shown spatially, they were connected with lines to form the shape of the column. These
lines were then used to form surfaces, which are required to create the element mesh. In
Patran, the nodes, lines, and surfaces, are created using a graphic interface to “draw” the
structure and then the program can automatically mesh it, based on input from the user.
Each flange and web was formed into a surface such that they can be meshed
separately. A mesh consisting of quadrilateral elements can be created many ways, but
if the surfaces that the mesh are created from are also quadrilaterals, the program allows
it to be created by either (1) specifying a desired mesh size or (2) giving the number of
elements desired per side of the surface. Since each of the surfaces on the column is
quadrilateral, and square elements were desired, the mesh was created by giving a
desired mesh size. The mesh size input needs to be adequately small enough to provide
accurate results, while not increasing computation time unnecessarily. The acceptable
mesh size for this thesis was determined by a sensitivity analysis, which is summarized
in Chapter 5. The program automatically creates the mesh on the surface of interest,
adjusting the specified mesh size to fit the member dimensions, which may or may not
be precisely the spacing input.
It is important to ensure that the nodes matched at connective regions between
the flanges and webs in the columns. To ensure this occurrence the meshing for one
flange was first carried out, followed by the web, and lastly the other flange. The
41
program accounts for nodes created between meshing sequences so this procedure
ensures that the connecting elements share nodes.
The *NODE keyword lists the information for each node created through mesh
creation in Patran. This keyword gives an ID for each node and its coordinates. Each
node is defined in a single line of text, creating a list in the key file.
The elements created during meshing are given in the key file by the keyword
*ELEMENT_SHELL. This keyword gives an ID for each element, the four nodes at its
corners, and can also be related to the *PART keyword, by referencing its ID, which is
described later in this section.
The *SECTION_SHELL keyword is used to assign element formulation and
thickness to shell elements. The required input is the thickness of the shell at each of
the nodes and an applicable element formulation. The thickness of the shells remained
constant across a given surface, so the same thickness (equal to either the flange or web
thickness, depending on the element location) was input at each corner of a given
element. The element type that was used for each of the models in this thesis was
Belytscho-Lin-Tsay, a four-node element formulation which employs a one-point
quadrature rule (LSTC 1998). This is the default element formulation type in LS-Dyna
because it provides accurate results and requires a relatively small amount of
computation time. The formulation achieves these low computation times by
imbedding a coordinate system that deforms with the element, thereby making
simplifications concerning co-rotational and velocity strain formations. Because of this
42
simplification, this formulation type can provide inaccurate results if the nodes in the
elements are not co-planar, which is not the case for the models in this thesis.
Since this thesis involved finite element models of columns, there were only two
different element types to distinguish between; these are the elements on the flange and
web. The only difference between these is that they have different thicknesses. To
distinguish between these, there were two *SECTION_SHELL commands used, which
were given different IDs. The keyword *PART was used to connect a defined material
(*MAT) and shell criteria (*SECTION_SHELL) to an element. Two *PART
commands were created, each referencing the same material model (shown in the
following section) but different *SECTION_SHELL IDs. The elements in the flange
and web will reference different *PART IDs, thereby defining the appropriate element
formulation data.
3.2.1.3.2
Materials
The inelastic constitutive response of steel has been shown to be sensitive to
strain rate and temperature. In general, a faster strain rate will cause the yield point in
steel to increase while higher temperatures will soften steel, reducing its yield and
ultimate strength. Steel members subjected to blast loads will likely experience high
strain rates. Thus, including a high strain rate material model was an important factor
that was included during modeling.
One prominent method for modeling steel’s constitutive response to strain rate
effects is the Johnson-Cook constitutive material model, developed in 1983 by G.R.
43
Johnson and W. H. Cook. Johnson and Cook developed an empirical relation between
stress and strain that considers strain rate and temperature effects, which is termed the
strength model. This material model also includes a failure strain calculation, which is
termed the failure model. This material model was selected based on this model’s
accepted use in modeling steel under high strain rates. While it also accounts for
temperature effects, the influence of this variable is not considered to be influential to
the objectives of this work; while a blast is accompanied by a release of heat, elevated
temperatures are not sustained for sufficient duration (in the absence of a secondary fire
caused by the blast) to be influential.
These material models are applied using the keyword *MAT_JOHNSON_
COOK, which requires the user to specify the density for the material (490 lbs/ft3),
shear modulus (11,050 ksi), elastic modulus (29,000 ksi), Poisson’s ratio (0.30), and
additional constants specific to the Johnson-Cook models. A simplified version of this
model is also available in LS-Dyna by the keyword
*MAT_SIMPLIFIED_JOHNSON_COOK which saves computation time by not
including any thermal softening effects or damage parameters. It was not used in this
thesis as it is not recommended for use in models where strain rates vary over a large
range. The strength and failure models as well as the constants that were used in the
finite element models described in this thesis are described in the next sections.
44
3.2.1.3.2.1 Johnson-Cook Strength Model
The strength model uses three independent terms to account for strain hardening,
strain rate effects, and thermal softening through the three terms in brackets shown in
Equation 3.1
1 1 Equation 3.1
where is the flow stress, A, B, C, n, and m are material constants, εpl is the plastic
strain, ⁄ is the dimensionless plastic strain rate for = 1s-1, and is termed
the homologous temperature, which is defined in Equation 3.2.
Equation 3.2
In Equation 3.2 the terms Troom and Tmelt are the room and melting temperatures,
respectively, and T is the current temperature. Since temperature is not considered in
the analyses performed in this work, the melting temperature is input as a very high
value (106 K) to make this effect negligible in the analysis. For completeness the room
temperature was input as 300 K.
The three bracketed terms in Equation 3.1 are seen, in their functional form, to
act completely independent of one another. For example, if strain rate and temperature
effects are ignored ( = 0) the last two brackets drop from the equation leaving
only the first, as shown in Equation 3.3.
Equation 3.3
The flow stress is then represented by A, which represents the yield stress, and ,
which represents the strain hardening behavior. If or is non-zero, such that
45
Equation 3.1 does not reduce to Equation 3.3, the constants A, B, and n from the latter
will remain unchanged. In other words, A, B, and n are completely independent of
strain rate and temperature. Because of this, the constants from Equation 3.3 can be
determined from experimental data that does not include these effects, such as a quasistatic tensile testing.
3.2.1.3.2.1.1
Strength Model Variable Form
The Johnson-Cook model, while being a fairly convenient and simplistic
approach, has the shortcoming that the needed constants are not readily available in the
literature for the material of interest in this thesis, Grade 50 steel. However, it is
convenient to determine some of the constants contained in the strength model from
experimental data.
The independence of the strain hardening, dynamic, and temperature terms in
each of the separate brackets shown in Equation 3.3 allows the constants within each
bracket to be determined independently of those in the other brackets. The terms in the
first bracket can be determined from a quasi-static stress-strain curve from the material
of interest. Notice that Equation 3.3 can be rearranged to form Equation 3.4.
Equation 3.4
Taking the natural log of both sides in Equation 3.4 will yield Equation 3.5.
! !
! ! Equation 3.5
If ! is plotted against ! , it can be noticed that the plot takes the
form of a line where n is the slope and ln(B) is the y-intercept. Therefore, if a quasi-
46
static experiment is carried out on the material of interest and the stress-strain data is
plotted on a log-log scale, the constants A, B and n can be determined by a best fit linear
line. This process was used in order to determine the Johnson-Cook parameters B, and
n which were used in this thesis. This is shown in the following section.
3.2.1.3.2.1.2
Strength Model Constants Determination
Stress-strain curves from quasi-static tensile testing on Grade 50 steel is given in
Righman (2005). This data was organized into seven stress-strain data sets, containing
different plate thickness from different heats of steel. For each data set, Righman
provides tablature stress-strain data that approximates the response of all samples in that
data set. Since this same material (Grade 50) is used for the models in this thesis, the
tabular data from these stress-strain data sets was used to determine the Johnson-Cook
parameters B and n for each data set. The values were then averaged to find the final B
and n values used. The process for the first data set is shown below, while the
remaining six are shown in Appendix A.1. The first stress-strain data set available from
Righman (2005) is provided in Figure 3.1 below. For clarity the data points are
connected with lines.
47
80
70
σ (ksi)
60
50
40
30
20
10
0
0
0.1
0.2
0.3
ε (in/in)
Figure 3.1 – Stress-Strain Curve from Quasi-Static
Static Tensile Test
T
One limitation
tation to determining the Johnson-Cook
Cook parameters this way is that the
experimental samples used to define the constants will all be different. Each
experimental sample will have slightly different yield and ultimate strengths,
strength which will
give different results
esults for the Johnson
Johnson-Cook constants. For this reason, it is desirable
desir
to
obtain constants from an array of samples to observe any trends and use values that are
representative of the largest sample size reasonable
reasonable. The fore-mentioned
mentioned study by
Righman (2005)
2005) gives seven stress-strain curves, similar to Figure 3.1 to base results
on. Once the analysis is carried out for each of the seven data sets, the results will be
averaged. One of the data sets is used as an example below in order to discuss the
process more clearly.
The data must be expressed in terms of the plastic strain for consistency with the
format of Equation 3.5. From Figure 3.1 it can be found that there are three distinct
48
regions of the stress-strain response: the brief initial elastic stage (0 ≤ ε ≤ 0.00211), the
plateau stage following yielding (0.00211 ≤ ε ≤ 0.045), and the strain hardening region
following the plateau (0.045 ≤ ε ≤ 0.35). For the determination of the constants, the
plateau region is ignored such that yielding strain is input as a value just before strain
hardening begins. If the yielding strain is input precisely when the plateau region ends
(i.e. – εy = 0.045) this creates an error in the derivation as log(0) will need to be
calculated, which is undefined. To avoid this, the yielding strain was input as being
0.001 before the plateau region ends, resulting in εy = 0.044. A smaller margin could
have been used for this but would not have altered the results. The plateau region is
ignored because the results are less precise when it is included, resulting in a poor fit, or
imprecise agreement between the Johnson-Cook approximation and the actual data,
near the end of the plot. This poor fit is present at the end of the plot because the
smaller values in the plateau region are weighted during linear regression analysis, and
as they do not fit the behavior necessary for a good fit in the approximation, the slope of
the fitted curve is skewed. A better fit could only be reached by using a piece-wise
material function, which is not available with this material model. The yield strain is
selected as the point at which the plateau ends and the strain hardening portion of the
curve begins.
Using the stress-strain shown in Figure 3.1, the yield strain of 0.044, and the
yield stress of 57.547 (taken directly from the stress-strain data), the calculations of the
Johnson-Cook strength constants is carried out. The calculations are summarized in
Table 3.1, beginning with the data points from Figure 3.1 being provided in the two left
49
columns of the table. The third column in Table 3.1 shows εpl for each data point,
which are the first column values minus εy.
Table 3.1 – Calculations for Johnson-Cook Constants from Figure 3.2
εpl = ε-εy
σ–A
ln(εpl)
ln(σ – A)
60.396
0.001
2.849
-6.908
1.047
0.055
62.154
0.011
4.607
-4.510
1.528
0.07
63.891
0.026
6.344
-3.650
1.848
0.09
65.355
0.046
7.808
-3.079
2.055
0.14
66.538
0.096
8.991
-2.343
2.196
0.35
66.824
0.306
9.277
-1.184
2.228
ε
σ
0
0
0.00211
57.547
0.03842
58.048
0.045
Recalling Equation 3.5, the data must then be expressed in terms of ln(εpl) and
ln(σ – A). The fourth column, σ – A, is then the second column minus the yield stress
A. The fifth and sixth columns, ln(εpl) and ln(σ – A) respectively, are then the natural
logs of the third and fourth columns. Figure 3.2 shows a plot of the data in the fifth and
sixth columns with a linear fit applied. The linear fit was found by plotting the data
points in excel and using the trend line tool.
50
3
2.5
ln (σ - A)
2
1.5
ln(σ-A) = 0.226*ln(ε pl)+ ln(2.633)
R² = 0.951
1
0.5
0
-8
-7
-6
-5
-4
-3
-2
1
-1
0
ln (εpl)
Figure 3.2 – Linear Fit to log-log Data
ata given in Table 3.5.
The linear fit gives the equation ln(σ – A) = 0.266*ln(εpl) + ln(2.633).
ln(2.633)
Therefore, the Johnson--Cook parameters B and n then become exp(2.633) = 13.92 ksi
and 0.226 respectively. Substituting these values into Equation 33.3 gives Equation 3.6.
Equation 3.6
This equation is plotted in F
Figure 3.3; the corresponding experimental data is
also shown here for comparison. As described above, the plateau region of the
experimental data was neglected to find the Johnson
Johnson-Cook
Cook parameters (i.e. – strain
yielding was assumed to begin after the plat
plateau). However, the Johnson-Cook
Johnson
model
will use the input of A = 57.547 ksi to calculate a strain (0.001984) corresponding to
this stress using the input elastic modulus (29,000 ksi). Strains exceeding 0.001984 will
then contain a plastic strain component
component. At the proportional
al limit, it can be seen in
51
Figure 3.3 that the approximation given by Equation 3.6 will begin strain hardening,
although the experimental data will plateau. Even so, the approximation is still
considered to provide good results. The Johnson-Cook
Cook fit is seen to have the same yield
limit, very similar ultimate limit (0.8% relative error), and the strain hardening behavior
is well-replicated.
70
60
σ (ksi)
50
40
30
20
Experimental Data
10
Johnson-Cook
Cook Fit
0
0
0.1
0.2
0.3
ε (in/in)
Figure 3.3 – Johnson
Johnson-Cook
Cook Approximation to Experimental Data.
D
This
his procedure was also carried out for the remaining
ing six data sets from
Righman (2005) and the constants found for each were averaged (see Appendix A.1) to
determine the
he final values that will be used
used. This resulted in B = 21.165 ksi and n =
0.241.
Although the experimental data shown in Figure 3.1 was for Grade 50 steel,
having a minimum yield point of 50 ksi, it is seen that this sample had a slightly higher
yield point. Since the yield strengths of the samples used were greater (65.1 ksi on
52
average) than the minimum specified yield strength (of 50 ksi) and the Johnson-Cook
approximation was found using these samples, the values of B and n used reflect this
extra strength. These values are not reduced for the models in this thesis because the
models are intended to be representative of members manufactured from typical steel.
To reflect this same material over-strength effect in the yield stress, A was increased by
a factor of 1.1 (to 55 ksi), which is common factor that is used in structural design and
analysis to account for such effects. While the average yield stress could have been
used for A, the yield stress increase factor of 1.1 is much more accepted for use.
However, the actual yield stress of each data set was used in deriving the Johnson-Cook
parameters B and n from each of the data sets. Obviously, the value assigned to the
variable A influences the resulting B and n values. For example, for the data set
discussed above, if A = 55 ksi is used in the derivation of B and n, these values become
B = 15.31 ksi and n = 0.151.
It should also be noted that the stress-strain data sets obtained from Righman
(2005) are according to “engineering” stress and strain, not “true” stress and strain. The
main difference between these two measures are that engineering stress and strain does
not account for changes in the cross section of the member (only elongation and force),
and therefore does not precisely indicate the actual stress-strain behavior of the
material. There are methods that can be used to convert the results to “true” stressstrain, which in general will show the material having increased strain hardening
effects. For this reason, the Johnson-Cook parameters found herein, which is according
to “engineering” stress and strain, will under predict the strain hardening in the material
53
to some degree. As the LS-Dyna and Johnson-Cook models are both calculated
according to “true” stress the effects of this will be represented in the finite element
models. In spite of this, the parameters are still believed to be reliable and will still
more accurately represent Grade 50 steel behavior than other parameters that could be
found in literature.
The other constants remaining in the strength model are C, m, Troom, and Tmelt.
The effects of temperature in the analysis will be neglected by inputting a very high
melting point (106 K). The value m is negligible, but is input as 1.03. The room
temperature in the model was input as 300 K. The constant C is taken from a study
carried out by Kajberg and Wikman (2007), in which the authors presented a way to
estimate material parameters through the use of high strain rate experiments and inverse
modeling techniques. The basic approach of this inverse modeling was to minimize an
objective function that measures the agreement between the experimental data and the
finite element models with the parameter C. The experimental data collected was strain
and displacements from high strain rate Split-Hopkinson testing of dog bone shaped
samples. These same experiments were then modeled in LS-Dyna from which the
displacements and strains at the same locations in the experiments were then recorded.
By using least squares analysis the difference in the displacement and strain between
the experiment and finite element model are minimized by varying the value of C in the
finite element model. The authors concluded that a value of 0.0327 for C can be used
for mild steel. This value was shown to be reliable through statistical analysis.
54
Given the above constants
constants, and ignoring the effects of temperature on the model,
model
the Johnson-Cook strength model used for this thesis is shown in Equation 3.7.
Equation 3.7
Figure 3.4 shows th
this strength model plotted for various strain rates ranging
from dε/dt = 0.0057 to 1000 s-1. This range is indicative of quasi-static
static to very high
strain rate values. This is the quasi
quasi-static
static range that was used in Kajberg and Wikman
(2007). It can be seen from Figure 3.5 that the strain rate significantly
ly increases the
strength of the material.
100
80
σ (ksi)
60
40
dε/dt
dt = 0.0057
dε/dt
dt = 1
dε/dt
dt = 10
20
dε/dt
dt = 100
dε/dt
dt = 1000
0
0
0.1
0.2
ε (in/in)
0.3
0.4
0.5
Figure 3.4 – Plot of Equation 3.7 for Various Strain Rates
ates
Using the constants described above in the Johnson
Johnson-Cook
Cook model will allow for
the prediction of the material behavior of Grade 50 steel in the finite element models in
this thesis. In this section it is shown that the parameters in Table 3.2 are appropriate to
55
use in the Johnson-Cook strength model for the finite element models in this thesis.
These values, however, are admittedly not perfect and could be slightly adjusted by
extensive material testing in the future, if necessary.
Table 3.2 – Johnson-Cook Strength Model Constants Used in LS-Dyna
Parameter
Value
A (ksi)
55.000
B (ksi)
21.165
C
0.0327
n
0.241
m
1.03
A study related to this thesis was published (Brown and McConnell 2009) that
contained results from a validation study similar to the one carried out in Chapter 4 of
this thesis. In that study, the Johnson-Cook material model parameters determined by
Johnson and Cook (1985) for 4340 steel were used, except the yield stress which was
reduced to 50 ksi. The remaining constants used were B = 74 ksi, n = 0.260, C = 0.014,
and m = 1.03. Using these constants gave a noticeable difference in the stress results.
Even though a smaller A and C were used, which would decrease the allowable stress
values in the model, the increased value of B made the strain hardening stress-strain
behavior sharper which allowed the material to reach higher stresses. For example the
axial stress values found in Brown and McConnell (2009) were as high as 78 ksi, while
56
the stress using the parameters in this thesis was more near 67 ksi. Therefore, it is seen
that the choice in these parameters can have a large effect on the results and should be
chosen wisely.
3.2.1.3.2.2 Johnson-Cook Failure Model
The failure model given by Johnson-Cook (1983) is shown in Equation 3.8
"# "$ % &' ( 1 ") !| |1 "+ Equation 3.8
where εf is the failure strain; D1, D2, D3, D4, and D5 are damage constants; and σ* is the
effective stress, defined as ⁄,, where is the average of the three normal stresses
and , is the von Mises equivalent stress. The other variables are the same as those
given in Equation 3.1. As with the strength model, the terms is each bracket are not
dependent on each other. The first bracket indicates that the fracture strain decreases
along with . The second bracket indicates fracture strain decreases with increasing
strain rate and the third bracket shows that the fracture strain increases with increasing
temperature.
As with the strength model, the constants available for use in the failure model
for structural steel are not readily available. In this thesis, the constants found by
Johnson and Cook (1985) for 4340 Steel will be adopted, as they are the most
representative values for Grade 50 steel that could be found. These are shown in Table
3.3.
57
Table 3.3 - Johnson-Cook Failure Model Constants Used in LS-Dyna
Parameter
Value
D1
0.050
D2
3.440
D3
-2.120
D4
0.002
D5
0.610
Once the failure strain has been found for the current conditions, a damage ratio
D is computed by Equation 3.9, where ∆ε is the accumulated strain which occurs during
the integration cycle. If the damage ratios found for each integration cycle over the run
time of the model for an element sum to greater than 1 failure will occur.
"-
∆
Equation 3.9
The behavior represented by Equation 3.8 and the damage parameters in Table
3.3 can be observed by plotting the equation for various stress states and strain rates.
Figure 3.5 shows a plot of failure strain versus effective stress, which was obtained by
substituting the constants from Table 3.3 into Equation 3.8, for two different strain
rates. The two strain rates used are 0.0057 and 1000 s-1, which correspond to the
bounding strain rates used in Figure 3.4. Because of the high melting point temperature
input, temperature effects are ignored. From Figure 3.5 it can be seen that there is little
influence on fracture strain due to strain rate for this combination of damage parameters
58
and strain rate values. However, a significant influence of stress state on failure strain
is shown.
4
εf (in/in)
3
2
1
dε/dt = 0.0057
dε/dt = 1000
0
0
0.2
0.4
0.6
0.8
1
σ*
Figure 3.5 – Failure Strain versus E
Effective Stress for Varying
arying Strain Rates
The failure
ilure strains exhibited in Figure 3.5 are larger than those that are typical of
Grade 50 steel. These were used despite this as no other constants were found that
would fit the behavior of steel better. However, it should be noted that the strains
presentt in the models were far below the failure strain values present in this model as
well as those typical of Grade 50 steel, which is on the order of 0.4. For example, the
14-ft
ft W14x82 column subjected to the large blast had strains ranging from 0.00025 to
0.0016 across the web before it buckled. The columns that were loaded with the small
charge had more localized effects and therefore saw higher strains, such as the 14-ft
14
W14x82 column which had strains ranging from 0.00048 to 0.00314. Even these,
59
however, are far below the fracture strain typical of Grade 50 steel. The columns that
were deemed failed had surpassed an unacceptable stability-based horizontal deflection
criterion. Since the failure of the columns was more due to buckling, and not due to
fracture, the values in this model did not play a role in the results. Even if these values
were altered to define the failure strain as being more near the yielding strain the results
would not have been affected.
3.2.1.3.3
Boundary Conditions
The columns modeled in this work were assumed to have moment connections
at their ends. To accomplish the fixed connection at each end of the column all of the
nodes along the cross-section at each end were restrained from translation. This
resulted in preventing rotation, since the nodes in this plane cannot move relative to one
another, and the desired fixity being achieved. Translations in all three dimensions
were restrained at the base of the column. Since axial loading was applied to the top of
the column, it needed to be able to translate freely in the vertical direction at this
location, so only the other two translations were restrained here.
To apply these boundary conditions, the keyword *BOUNDARY_SPC_SET is
used. This requires a reference to a nodal set ID (*SET_NODE_LIST) and the
applicable degrees of freedom that should be restrained. An alternate coordinate system
(labeled as “cid” in the keyword file) can also be referenced in this command if desired.
The nodal set ID is created using the *SET_NODE_LIST, which is given the ID of each
node to be included in the set. To accomplish these boundary conditions at either end
60
of the column, two node sets were created, consisting of the applicable nodes on the
bottom and top of the column. The node set at the top of the column was then
referenced by a *BOUNDARY_SPC_SET in which the two translational degrees of
freedom in the plane of the cross-section were restrained. The node set at the bottom of
the column was referenced by a *BOUNDARY_SPC_SET in which each of the three
translational degrees of freedom were restrained.
3.2.1.3.4
Loading
There were two types of loading applied to the column models in this thesis.
These were blast loads and axial loads. Each of these is discussed in the following
sections.
3.2.3.3.4.1 Axial Loading
In the column models in this thesis, it was important to accurately portray the
loading that a steel building column would see in service. As columns are designed
using load factors, in their service state they will likely carry loadings much less than
the design loads. Because each column is not loaded to its full capacity, it should have
reserve strength to resist additional loads.
An axial load was applied to the top of each column to simulate service loading
conditions. The magnitude of this load was determined by taking the given column’s
design strength, φPn,, and back calculating what service loading would likely result in
the use of the given column. This is done by using an effective load factor. An
effective load factor is intended to capture the same effect as a load combination (where
61
different load factors are applied to different loading types) but avoids the need for
defining the specific quantities of each load type. The ASCE (2002) load combination
1.2*(dead load) + 1.6*(live load) is used to derive an effective load factor in this work
since this is the load combination usually governing the design of building columns. To
complete the derivation of the load factor, some assumption about the relative
magnitude of dead and live load must be made. Because live loads are usually greater
than dead loads in building design, a live to dead load ratio of 1.67 is assumed. This
can be substituted into the above load combination to give an effective load factor of
1.45. That is, 1.45 times the service load (of dead load plus live load) produces the
same loading as 1.2*(dead load) + 1.6*(1.67*dead load). The service loading is then
calculated by taking the design strength and dividing by the effective load factor. The
service loadings applied to each column are shown in Table 3.4.
Table 3.4 - Service Loadings for Each Column
Column
Ht (ft)
Pa (kips)
14
647.6
18
590.5
14
936.6
18
895.9
14
617.5
18
567.6
W14x82
W14x109
W10x77
62
It should be noted that the effective load factor works as a weighted average, in
that if a higher percentage of dead load is present, a lower effective load factor results.
This would, in turn, result in a column with less reserve capacity (in other words,
dividing the design strength of the column by a smaller effective load factor will give a
larger service loading). The ramifications of this, while it can affect the failure standoff distance for each column, are not believed to be critical to the findings herein. This
is because altering the relative percentages of dead and live load do not significantly
change the value of the effective load factor. If the live load to dead load ratio is varied
from 1.5 to 2, the effective load factor ranges between 1.44 and 1.46. These values are
with 1% of the values used in this thesis.
The axial loading acting on the columns was input as equal magnitude
discretized nodal loads acting on all the nodes of the cross-section at the top of the
column. Since LS-Dyna captures dynamic response, static loads were simulated by
applying the loads in a linearly increasing manner at a slow loading rate such that the
structure approximately responded statically. This was accomplished in these models
by applying the axial loading as a linearly increasing function over a period of 30 ms.
After 30 ms the load stayed constant. This loading rate was determined to be
sufficiently slow as no oscillations could be seen in the vertical displacements of the top
nodes. The determination of this was simply based on a trial and error approach. For
example, a loading duration of 15 ms was also investigated and only minor oscillations
were seen when this load rate was used.
63
To further simulate the actual loading conditions in the event of a blast, the full
axial load must be applied at a time before the blast loads were applied to the column.
To achieve this, the blast loading time of initiation was varied to ensure that the
pressure loads arrived after 30 ms. A time of 30 ms can be used to ensure the desired
load sequence is achieved, but a smaller value can save computation time. For
example, if the blast takes 25 ms to arrive, an initiation time of 5 ms can be selected for
the blast. This saves 25 ms of computation time. This approach was used for the
iterative models once the time of arrival for the blast was better understood and could
be estimated with confidence. As the time of arrival of the blast is reported by LSDyna, ensuring that the axial loading was applied before the blast pressure was an easy
check to perform once the analysis was completed. This also made it easy to alter the
blast initiation time in further iterations if needed.
The axial loading was applied in LS-Dyna by using the *LOAD_NODE_SET
keyword. This option, similar to the boundary condition keyword *BOUNDARY_
SPC_SET, references a node set and applies the loading defined therein to each node in
the referenced node set. This keyword requires the nodal set ID, the direction of the
applied loading, an applicable load curve (*DEFINE_CURVE), and a scale factor. The
node set ID can be created using the *SET_NODE_LIST keyword as was previously
discussed in Section 3.2.1.3.3. Since the loading was applied to the nodes at the top of
the column, where the boundary condition already exists, this same node set ID was
referenced. The load is applied in the downward direction in a linear fashion by
referencing the *DEFINE_CURVE keyword. This keyword essentially requires the
64
user to input data points from a load vs. time curve. The loading curve input varies
linearly from 0 to 1 in 30 ms. Therefore, only two points were needed to define it as
Ls-Dyna will linearly interpolate between values by default. If linear interpolation is
not desired, the user can input a function for the load curve, such as a polynomial. The
scale factor was used to amplify the load curve values to the desired magnitude of axial
loading for each column.
3.2.1.3.4.2 Blast Loading
The blast loading was input using the *ENHANCED_LOAD_BLAST keyword,
which is an expanded method over the older *LOAD_BLAST keyword. This option
allows the user to use one of four different types of blast loadings: air burst, surface
burst, air burst from non-spherical moving warhead, and air burst strengthened by
ground reflection. These, at a minimum, require the user to input the equivalent mass of
TNT for the blast, the coordinates of the blast’s center of mass, a surface on the finite
element mesh on which the blast loading will act, and a time of initiation of the blast.
Additional input is required for the ground reflection and moving warhead options.
As discussed previously, the steel columns were modeled with charges placed
vertically aligned with the ground, mid-height, and top of the column. For charges
situated at ground level, the surface burst option was used. This option applies a
pressure history to each element using empirical equations that models an initial
incident wave immediately reflected off the ground to create a single reflected wave
stronger than the incident wave. The location of the center of mass for these charges
65
was calculated assuming a spherical mass with a density of 1.65 g/cm3 (which is typical
of TNT, Kinney and Graham 1985), whose outer perimeter was resting on the ground.
This height is less than 10% of the member height.
For charges located in the air that will interact with the ground before striking
the target surface, the air burst strengthened by ground reflection should be used. This
will apply a pressure history that models a mach front from the interaction of incident
waves from the initial blast and reflected waves off the ground. This mach front can
greatly increase the magnitude of the pressure from the blast. The charges placed at
mid-height and top of the column were modeled using this option. The input for this
option is the same as for the surface burst option except the user is also required to input
a node on the mesh that intersects with the ground and a vector that is normal to the
ground surface. The ground was placed at the base of the column with the required
vector pointing upwards, along the column’s height. The calculation procedures to
determine the pressure loadings for this option use the same equations as the surface
burst option, however, are amplified depending on the reflection angle between the blast
pressure direction and ground orientation. More specifics on this can be found in TM 5855-1 (Department of the Army 1986).
The effects of these blasts are calculated in LS-Dyna using CONWEP equations,
which assume an exponential decay of applied pressure with time as shown in Equation
3.10 (Randers-Pehrson and Bannister 1997).
./ . 01 / 1
/ 1 2 %34 0
2
66
Equation 3.10
In Equation 3.10, P(t) is the total pressure at time t, Po is the total combination of peak
incident and reflected pressure (found from Equation 3.11), Ta is the arrival time, To is
the positive phase duration (which is assumed to be identical for incident and reflected
pressure), and A is the decay coefficient. The arrival time is amount of time that it takes
for the blast pressure to arrive to the surface of interest, while the positive phase
duration is the length of the time that a positive blast pressure is applied to the surface.
The decay coefficient will model the exponential decay of this positive pressure over
time. This pressure vs. time relationship is created for each element on the surface that
the blast is defined to act on. The parameters Pi, Pr (used in Equation 3.11), A, To, and
Ta are found by referencing empirical equations that are dependent on the charge size
and stand-off distance of the charge from the element. The total peak pressure Po is
calculated by Equation 3.11, which uses the angle of incidence (θ, the angle between the
element normal and the direction of propagation of the blast wave) to find the
combination of reflected (Pr) and incident (Pi) peak pressure.
. .5 1 6789 2678 $ 9 . 678 $ 9
Equation 3.11
It can be seen, by inspection, that when θ is 0 degrees, the total pressure is the reflected
pressure, and at 90 degrees it is the incident pressure.
The ground reflection option was originally not used for the charges placed at
mid-height and top of the column, and the air-burst option was used instead. This was
before the mach front option was available in LS-Dyna. The differences between the
two results were very great.
67
3.2.1.3.5
Output Control
The results of the analysis carried out in LS-Dyna are written to a “plot file”,
which can be viewed in an LS-Dyna post-processor. This plot file essentially contains a
series of “snap shots” of the analysis results, showing a three-dimensional view of the
mesh’s response to the loading. These snap shots can be played in succession,
essentially like a movie reel. These images exist for points in time in the analysis at
which the user specifies the results to be recorded. The results become most clear when
the data is recorded at short intervals; however, the more results recorded the larger the
file becomes. The size of the output file does not affect the run-time of the program
however, so this is only considered if storage space is an issue.
In this thesis, the keywords *DATABASE_BINARY_D3PLOT and
*DATABASE_BINARY_D3THDT were used to dictate how often the results are
recorded. The former keyword controls the time interval at which the results are plotted
to the post-processor file that can be viewed. The latter keyword dictates how often the
model time is printed to the screen while it is running, which can assist in evaluating
analysis results before the completion of the analysis and estimating the amount of
analysis time left. For example if the model is being run for 100 ms and the first 25 ms
occurs within 1 hr, the user can approximate the program running for another 3 hours,
which can be helpful for scheduling purposes. Both keywords require the input of a
time interval, DT.
It was found through trial and error that the maximum deflections (which were
the key output of interest in this work) in the analyses varied in a fairly smooth fashion.
68
Thus, DT input into both keywords did not need to be particularly small. A value of 0.1
ms was considered adequate and used for both keywords in these analyses as no jumps
in deflection between “snap shots” were observed. For example, the time step before
and after the maximum deflection would exhibit only slightly smaller deflections (less
than 1%). The file size using a DT of 0.1 ms was on the order of 300 MB for most
models but differences exist depending on the run time of the models.
The keyword *CONTROL_TERMINATION is used to determine when the
analysis should end. This requires the user to input a termination time for the model to
end computation. Although this time changed depending on the model considered, the
lowest acceptable value for this was desired as it directly affects the analysis time.
Thus, during each of the iterations for determining deflection of the column, this
minimum acceptable value was explored further and successive run times were reduced
when possible. For instance, if the results showed that the column did not fail and that
the results recorded the column’s behavior well after the peak displacement was
obtained, then on the next iteration the run time could be reduced to only a few ms past
the peak displacement in the previous trial. A termination time of 50 ms or greater was
typically used in the models.
3.2.2
Deflection Failure Criteria
In the last section, the LS-Dyna models were detailed for each of the columns.
This section will discuss the criterion used to establish failure from those models. Each
of the six columns (three cross-sections with two different heights) was created in LS-
69
Dyna separately and the maximum horizontal displacement under each blast scenario
was determined. Each column was assumed to be failed when it exceeded stabilitybased deflection criterion. Using displacement as failure criteria was convenient for use
in this thesis as equations to determine stability limits are readily available and the
maximum deflection from LS-Dyna is easily found.
Other criteria were considered for defining column failure, such as localized
fracture or yielding. Fracture could not be used as a failure criterion, however, as the
strain seen in the models was far below fracture strain typical of Grade 50 steel. For
example, the 14-ft W14x82 column had a maximum strain of 0.00314 in/in which is far
below the failure strain of steel. Yielding was also difficult to use as a failure criterion
as it would occur different places of the column, some of which are not indicative of
column failure. For example, even at larger stand-off distances, yielding of the flange
tips at the fixed connections could be observed, but this was only a function of their
being idealized fixed connections and would not represent the failure of a column
anyway. If yielding of the connections was ignored and only the midsection of the
column observed, using this as a failure criteria had more potential. However, as the
failure of the columns was in large part due to buckling, the columns could begin to fail
at stresses below yield stress and would only yield as a result of buckling, which was
seen in the columns subjected to the large blast. The columns subjected to the small
blast would exhibit some localized yielding due to more localized effects before they
reached the failure deflection but would not fail. Due to the close proximity at which
this stand-off distance for the small charge occurred for yielding, and those determined
70
using a deflection based criteria (maximum of 2 ft) it did not seem important to
differentiate between them. Additionally, using yielding as a failure criteria, which is
already shown to be imprecise, was more time intensive to check in the LS-Dyna post
processor as multiple elements would have to be checked for yielding as only one node
needs to be checked for a deflection based criterion.
The failure deflection limit for each column was found by using structural
analysis equations that predict a member’s capability to resist combined compression
and bending effects. In order to use these equations the fixity and orientation of the
columns have to be known, as these are integral factors in defining effective length and
stiffness. The columns described herein are assumed to be fixed at both supports,
unsupported along their height, and have an initial concentrated load applied to the top
of them. When the blast load acts on the face of the web (in the weak direction of the
column), the column will deflect horizontally. This horizontal deflection causes the line
of action of the axial load to be out of line with the centroid of the mid-section of the
column, creating a secondary moment as a force couple is created due to the axial force
being applied at this eccentric distance. The blast loading, which initially caused the
column to deflect, disappears after only a fraction of a second and does not contribute to
the overall stability of the column. This is because the column only begins to deflect
horizontally when the greatest blast pressure is first applied, which introduces a
dynamic effect such that when the maximum deflection occurs the blast pressure is
greatly reduced. For example, for the columns subjected to the large blast, the
percentage of the peak blast pressure still present when the column reached its
71
maximum deflection was only 3-13%, the higher percentage coming from the larger
stand-off distances. The pressure due to the small blast dissipated much more quickly,
with a maximum of only 2% of the peak pressure present at the column’s maximum
deflection. The failure of the column models, which was also observed in the LS-Dyna
post processor, also supported the use of stability based criteria as each of the columns
would deflect horizontally and then “pause” until secondary moment effects became
increasingly greater and the column quickly buckled. Also, since these failure criteria
were intended to determine the greatest stand-off distance that a charge could be placed
in order to cause failure, to include the blast pressure in the equation would have
required iteratively determining the failure deflection in addition to iteratively
determining the standoff distance causing this deflection. This additional complexity
was deemed unnecessary given the generally low magnitudes of blast pressure at the
time of maximum displacement.
It is also possible, in service, that a moment may be applied to the top of the
column as a result of the fixity between the column of interest and its adjoining
members at this location. At the bottom of the column, a reaction moment is then
created to satisfy equilibrium of the structure. It has been shown above that the service
axial loading can be found rather easily if the column is assumed to carry axial loadings
only, however, attempting to quantify the service moments are more difficult. This is
because there are infinite combinations of direction of the moment (i.e., weak direction
or strong direction) and magnitudes (i.e. – 5%, 10%, etc of capacity) that may exist.
Including these effects into the analysis would require an extensive parametric study,
72
which in combination with the other parameters varied in this thesis, was considered too
extensive. Due to the simplicity sought in this work for determining a blast threat that
is indicative of single column failure, this additional force effect was not included in
addition to the axial loading. It should be noted that including these effects into the
analysis may change the resulting failure contours, so the results should be viewed as
being specific to the case of the service loading consisting of purely axial force and no
bending force.
Therefore, the only moment considered in the column is due to secondary
effects from the axial loading. Since the axial loading is known before the blast loading
is applied, it is possible to calculate the deflection in the column that creates a
secondary moment that exceeds the section’s capacity.
To find the horizontal deflection that causes failure in the column due to axial
loading and secondary moment, equations from TM5-855-1 are used (Department of the
Army 1986). TM5-855-1 is a technical manual published by the army that gives
analysis and design recommendations for structures subjected to blast loads. Thus,
these equations were used because of their accepted use in blast resistant design.
Specifically, interaction equation 9-8 from TM5-855-1 is shown in Equation 3.12 and is
followed by a discussion on the necessary parameters to calculate it.
<
.1
? 1.0
.; =1 .1 > <
.
73
Equation 3.12
This equation accounts for the effects of axial and bending forces in a member
to compare against its capacity. This is done by the addition of two ratios that account
for the fraction of the member’s capacity utilized, one each for axial loads and
moments. If either ratio is equal to 1.0, then there is no more capacity left for the other
load effect. For the purposes of this thesis, the equation can be used to find the
allowable moment that can be applied to the column, since the axial capacity ratio and
moment capacity can be readily calculated. Expressions exist for some of the variables
in Equation 3.12, which are defined below, followed by a definition of the remaining
variables.
The parameter Pcr from Equation 3.12 is the ultimate axial capacity of the
column without bending effects and is found by Equation 3.13.
.; 1.7 B1
Equation 3.13
The parameter Pe from Equation 3.12 is the Euler buckling load for the column and is
calculated by Equation 3.14.
. C$ D
EFGIHJ
$
Equation 3.14
The parameter fa from Equation 3.13 is the maximum permitted axial stress on a
compression member in the absence of bending moment and is found by Equation 3.15.
EFGIHJ
K1 L BM
2 ; $
$
B1 Q
5 3EFGIHJ EFGIHJ
3 8 ; 8 ; Q
74
Equation 3.15
The parameter Cc from Equation 3.15 is the slenderness ratio limit and is found by
Equation 3.16.
; R
2 C$ D
BM
Equation 3.16
The remaining terms in the previous equations are listed below.
Pa = applied axial load
Cm = bending coefficient (recommended value is 0.85 from TM 5-855-1 for end
restrained members subject to transverse loading)
Mf = maximum applied moment
Mm = maximum moment that can be resisted in the absence of axial load (in the
weak direction this is given by Zy*fy)
A = cross-sectional area of column
E = elastic modulus (29,000 ksi for steel is used)
K = effective length factor (use 0.5 for fixed ends)
L = height of column
r = radius of gyration for column
fy = yield stress (55 ksi, 50 ksi with 10% overstrength factor, is used)
Zy = weak axis section modulus
The failure deflection is referred to as ∆. Since the applied moment, Mf, is taken
to be secondary effects, i.e. Pa* ∆, Equation 3.12 can be rearranged to solve for ∆.
Substituting Pa* ∆ for Mf in Equation 3.11 and solving for ∆ gives Equation 3.17.
75
Δ
1
.1
.1
01 2 01 2 <
.1 .;
.
Equation 3.17
The parameters specified in Equation 3.17 are a function of the column crosssection and length. The geometric properties of each of the columns that are used in the
calculations are shown in Table 3.5, which are obtained from the AISC Steel
Construction Manual (AISC 2005).
Table 3.5 – Geometric Properties of Columns
Column
ry (in)
A (in2)
Zy (in3)
W10x77
2.60
22.6
45.9
W14x82
2.48
24.0
44.8
W14x109
3.73
32.0
92.7
Since fy and E are constant for each of the columns, Cc will also be constant and
is found to be 102.02. The remainder of the parameters are shown in Table 3.6. It is
seen from the last column of Table 3.6 that the deflection limits for all of the columns
vary between 1.81 in. and 2.75 in. These deflections are not very large considering the
size of the columns. This is a function of the high axial loading that each column is
carrying, which is essentially the highest axial loading allowed by the design process.
The applied axial loading Pa, or service loads, for each column were discussed in
Section 3.2.1.3.4.1.
76
Table 3.6 – Calculation of Parameters for Finding the Failure Distance ∆
Column
Ht (ft.)
Pa (k)
fa (ksi)
Pe (k)
Pcr (k)
∆ (in.)
14
647.6
29.09
5988
1187
1.81
18
590.5
27.51
3622
1122
1.95
14
936.6
30.7
18060
1670
2.67
18
895.9
29.82
10925
1622
2.75
14
617.5
29.33
6197
1127
1.96
18
567.6
27.85
3749
1070
2.09
W14x82
W14x109
W10x77
3.3
Missing Girder Analysis
One objective of this thesis was to discover how much reserve capacity and
redundancy is inherently built into simple-span steel girder bridges as a result of the
design process. To do this, an analysis similar to alternate load path in buildings was
formulated and carried out on “typical” simple-span steel girder bridges. As the
alternate load path method calls for the removal of a column, this procedure carried out
on a bridge will call for the removal of a girder, and is therefore referred to as missing
girder analysis.
Two key geometric parameters of steel girder bridges are span length and girder
spacing. Designs for two span lengths (50 ft and 100 ft) and three girder spacings (6 ft,
8 ft, and 10 ft) will be evaluated using the missing girder analysis. The results are
formulated in such a way as to provide a metric that can be used to evaluated bridges
77
with any possible number of girders in the cross-section. The following sections
describe the procedure used to carry out the analysis, the calculation of loads, and the
selection of the bridge models used for the missing girder analysis.
3.3.1
Procedure
In order to carry out the missing girder analysis, bridge designs for
representative “typical” bridges were created. Each is then analyzed using the missing
girder analysis. This consists of first removing one girder from the structure. It has
been argued that the alternate load path method may be unconservative in that it only
considers the removal of a single member; therefore the missing girder analysis will
also consider the removal of two adjacent members. This leaves four distinct scenarios
for girder removal: (1) one exterior girder, (2) one interior girder, (3) two interior
girders, and (4) one exterior girder and one interior girder. However, it will be shown
in the Section 3.3.5 that only two cases ultimately need to be considered. These are the
removal of one girder or two girders as differentiating between interior or exterior
girder removal is shown to be negligible.
In the missing girder analysis, girders will be instantaneously and immaculately
removed (as with the alternate path analysis method) and the service loads that they
carried will be evenly redistributed to the remaining girders in the cross-section. While
this assumption of even redistribution may seem counter to the pattern of load
redistribution that would intuitively occur (girders closer to the failed girder may likely
carry a larger percentage of the redistributed loads than girders less proximate to the
78
removed girders), this distribution pattern is accurate if the total system capacity of the
bridge is considered (as assumed herein). To illustrate this point, the system moment
capacity must first be defined, which is assumed to be equal to the sum of the moment
capacities of each girder. Further it is assumed that the remaining girders in the system
(those that are not considered removed by an initiating event) have adequate ductility to
sustain their peak moment capacity while moments exceeding this capacity are
redistributed to adjacent members. Then considering system capacity and assuming
adequate member connectivity to achieve load redistribution (e.g., through lateral
bracing members and / or the concrete deck), if a girder were to reach its maximum
capacity due to the redistributed loads (regardless of the load redistribution pattern), any
additional moment would be successively redistributed to adjacent members. Thus,
assuming all members have equal capacity and initially carry the same loads (which is
generally satisfied) the maximum reserve capacity of the system is found when the
service loading of the removed member(s) is evenly redistributed. While there are
several assumptions made to arrive at this load redistribution pattern (that may not be
satisfied in all structures), this is believed to achieve the best balance of simplicity and
accuracy for the analysis herein.
In order to provide results for as many bridge configurations as possible, the
number of girders considered for each bridge design is not a fixed quantity. However, it
is assumed that all bridges have at least three girders in the cross-section as this is the
minimum number that would be required for a redundant bridge; if a two girder bridge
lost the capacity of one girder, it would almost certainly fail, as the structure would
79
become unstable. After the girder(s) is (are) removed from the system and the
assumptions outlined above are applied, the number of girders in the cross-section
needed for the structure to survive can be calculated. During the design process for the
bridges in this work, the number of girders in the cross-section plays only a small role,
as the girders are considered to act alone in the design process. As the loads are
developed in the following sections it can be seen that the results are independent of the
number of girders in the bridge, except for the barrier loading. This is discussed further
in Section 3.3.3.
The girder spacing and span length of the bridges considered in this work will be
varied to investigate the variability in reserve capacity and to attempt to provide results
representative of a broad range of actual structures. As was seen with the service loads
applied to the columns in Section 3.2.2, structures that carry more dead load than live
load can be expected to have lower reserve capacity as load factors used to amplify
dead loads are smaller than those used with live loads. Bridges can be expected to be
more dominated by live load effects as the spans become shorter, since the dead loads
are being reduced at a greater rate than the live loading. The two span lengths that will
be used in this analysis are 50 ft and 100 ft. The girder spacings considered are 6 ft, 8
ft, and 10 ft. This results in six different bridges from which results can be compared.
In order to design each of the bridges, the dead and live loads for the prescribed
two span lengths and three girder spacings were calculated and a steel girder was
selected based on these ultimate loads. The same girder was used for the exterior and
interior girders in the bridge, where the interior girder had higher ultimate loads and
80
governed the design. The exterior girders, which see smaller loadings, will have
capacities unnecessary to carry its service loading. The governing ultimate load was
assumed to be moment forces and the girders was selected based on strength criteria.
The inherent “over-strengthening” in the design process, which is the result of
not being able to select a girder size that perfectly matches the ultimate loads present,
will be eliminated by the use of effective load factors. The effective load factor is
calculated by finding the load factor that when multiplied by the service loading will
exactly equal the design capacity of the section. Therefore, the “over-strengthening”
involved during the design process will be eliminated. This is discussed in greater
detail in Section 3.3.5. Eliminating this effect is important so that the results presented
herein will not be influenced by how well the capacities of the selected girders match
the design loads; in other words, the conclusions regarding the reserve capacity of the
system that are obtained will be independent of the reserve capacity of individual
members to carry their design loads. What follows is the determination of the live load,
dead load, ultimate load, and effective load factors such that the bridges can be designed
and the analysis can be carried out.
3.3.2
Live Load Determination
The live load is calculated according to the American Association of State and
Highway Transportation Officials (AASHTO) “LRFD Bridge Design Specifications”
(AASHTO 2007). This design code requires the analyst to consider three types of live
loads: lane load, tandem load, and truck load. Each of these is calculated for a design
81
lane of traffic and is distributed to each girder using distribution factors, which have
units of lanes/girder. This thesis only presents the results for two or more design lanes
as the distribution factor found for multiple lanes governs over the single lane loaded
case.
The determination of the lane load, tandem load, truck load, and distribution
factors are each specified in the following sections. After each is determined, the effect
of the lane load is combined with the greater of the truck or tandem load to obtain the
total live load. An impact factor of 1.33 is also applied to the tandem and truck loads to
account for dynamic effects.
3.3.2.1
Lane Load
The lane loading prescribed in AASHTO (2007) is 0.64 k/ft. This load is to be
applied at the locations along the length of the bridge that cause the maximum load
effect. For a simple-span bridge, this is achieved by placing the lane load across the
entire span.
For a simple-span beam subjected to a uniformly distributed load, the maximum
moment occurs at mid-span. This maximum moment is given by Equation 3.18 where
ω is the distributed load and L is the span length.
<TUV
WG$
8
Equation 3.18
The maximum total moment may not occur at mid-span, however; thus the lane
load may need to be calculated at a location to correspond with the maximum effects of
other loadings. The tandem and truck loading, for example, will produce maximum
82
moments in the bridge at 1 ft and 2.33 ft from mid-span, respectively. (A discussion on
how these locations are determined for each loading type can be found in the following
sections.) The moment as a function of distance along the beam can be found by
Equation 3.19, where x is the distance along the span measured from the support.
<
W3
G 3
2
Equation 3.19
Using Equations 3.18 and Equation 3.19, the moment per lane for the lane load can be
determined at key locations for both span lengths. These results are shown below in
Table 3.7.
Table 3.7 – Maximum Moment per Lane Resulting from Lane Load
Moment (k*ft/lane)
Span Length (ft)
3.3.2.2
Mid-Span
Truck Maximum
Location
Tandem Maximum
Location
50
200.0
164.4
184.3
100
800.0
727.1
768.3
Truck Load
The design truck load prescribed in AASHTO (2007) consists of three axles
with magnitudes of 8 kips, 32 kips, and 32 kips (respectively), each spaced 14 ft apart.
This design truck is to be positioned in the longitudinal location that causes the
maximum load effect. This exact location can be determined by placing the truck at a
83
location where an axle and the resultant force of the system are placed equidistant from
the centerline of the span (Hibbeler 2006).
The resultant force for the truck load is located 4.67 ft from the center 32-kip
axle towards the other 32-kip axle. This results in the center axle being placed 2.33 ft
from midspan. Using statics to solve for the moment at this location gives Equation
3.20, where L is the span length and R1 is the reaction of the support on the same side of
the longitudinal centerline as the 8-kip axle, given by Equation 3.21.
<TUVXYZ[\] ^# GI2 2.33_ 112 `a4 B/
GI 16.33_
G 9.33_
^# 32 `a48 b
d 8 `a48 K 2
L
G
G
Equation 3.20
Equation 3.21
The reaction at the opposite end of the girder from R1 is referred to as R2, which
is given by Equation 3.22.
^$ 72 `a48 ^#
Equation 3.22
R2 is used to solve for the moment at mid-span of the bridge in order to obtain the
moment value at the location corresponding to the location of the maximum moments
due to lane load and dead load such that the combined load effects at each point of
interest can be found. This calculation is shown in Equation 3.23.
<Tf&ghUiXYZ[\] ^$ GI
2 373.33 `a4 B/
Equation 3.23
Using Equations 3.20 through 3.23, the moment per lane for each span length can be
found at each location of interest. These results are shown below in Table 3.8.
84
Table 3.8 – Maximum Moment per Lane for Truck Loading
Moment (k*ft/lane)
Span Length (ft)
3.3.2.3
Mid-Span
Truck Maximum Location
50
610.7
627.8
100
1510.7
1523.9
Tandem Load
The Tandem Load in AASHTO (2007) consists of two 25-kip axles spaced 4 ft
apart. As with the truck loading in Section 3.3.2.2, this load is positioned to obtain the
maximum load effect. For a simply supported bridge, the maximum moment from the
tandem loading occurs when the tandem is placed such that the centerline of the bridge
is one foot to the inside of one of the axles. The maximum moment will occur under
the closest axle to the centerline, located (L/2 -1’) from the support. This moment is
found by Equation 3.24, which references the reaction of the support located closest to
the maximum moment location, RA, which is found by Equation 3.25.
<TUVXYUi&jT ^U EGI2 1_J
^U 25 `a48 G 2
G
Equation 3.24
Equation 3.25
The moment at mid-span is also a point of interest, as this is where the live lane
load and dead load maximum moments will occur. This moment is found by Equation
3.26.
85
<Tf&ghUiXYUi&jT ^U G
25 ` B/
2
Equation 3.26
Using Equations 3.24 through 3.26, the tandem moments per lane can be found for each
bridge at each point of interest. These values are shown below in Table 3.9.
Table 3.9 – Maximum Moment per Lane for Tandem Loading
Moment (k*ft/lane)
Span Length (ft)
Mid-Span
Tandem Maximum
Location
50
575.0
576.0
100
1200.0
1200.5
The tandem moments are shown to be very similar at the maximum location and
mid-span since the locations corresponding to these moments are located only 1 ft apart.
From comparing the data in Tables 3.8 and 3.9, it can be seen that the moment due to
the truck loading will govern over the tandem moments.
3.3.2.4
Distribution Factors
There are two distribution factors that need to be calculated, one for exterior
girders and one for interior girders. Equations for these factors are found in AASHTO
(2007) Tables 4.6.2.2.2b-1 and 4.6.2.2.2d-1, respectively. Here distribution factors are
separately calculated considering one or multiple lanes loaded, where the higher value
is used. As the distribution factors for the multiple lanes loaded case were found to be
31% to 46% greater than those of the single lane loaded case, these calculations are only
86
provided as it is the governing case and it simplifies the analysis procedure and
discussion.
The reduced form (which neglects girder stiffness as prescribed in AASHTO
2007 Section 4.6.2.2.2) of the interior girder distribution factor equation is given in
Equation 3.27. This reduced equation is recommended for use during the design
process, when the dimensions of the girder, which are needed for the full equation, are
not yet known.
k5
m o.p m o.$
0.075 l n l n
9.5
G
Equation 3.27
Here the distribution factor, g, can be seen to be a function of the girder spacing, S, and
span length, L. Both of these quantities should be expressed in terms of ft for use in
Equation 3.27. This equation shows that the distribution factor increases as the girder
spacing increases and the span length decreases.
The exterior girder distribution factor is given by Equation 3.28.
kq % k5
Equation 3.28
The variable e is a factor used to scale the interior distribution factor, and is calculated
from Equation 3.29.
% 0.77 r
9.1
Equation 3.29
The variable de in Equation 3.29 is horizontal distance from the centerline of the
exterior girder to the curb on the roadway where positive is outwards. Using a typical
bridge barrier that is 1.6875 ft wide and assuming a 2 ft overhang on the bridge, the
variable de is 0.3125 ft. One can observe that that as de gets larger, e and gext will
87
increase. Since the exterior girder distribution factor is a function of the interior girder
distribution factor, it is also dependent on the girder spacing and span length.
The distribution factors for both the exterior and interior girders for each
combination of span length and girder spacing investigated in this work were calculated
using the above equations. The results of the calculations are shown below in Table
3.10.
Table 3.10 – Distribution Factors for Interior and Exterior Girders
Span Length
(ft)
50
100
3.3.3
Girder Spacing
(ft)
gint
gext
6
0.572
0.460
8
0.700
0.563
10
0.822
0.662
6
0.507
0.408
8
0.619
0.498
10
0.726
0.584
Dead Load Determination
The dead loads accounted for in these designs are the concrete deck,
diaphragms, haunch, stay-in-place forms (SIP), future wearing surface (FWS), concrete
barrier, and self weight of the girder. As the self weight of the girder is not known at
the onset of the design, it is determined after the other loadings have been found. The
88
process of designing for the live and dead loadings and including self weight of the
girder is shown in Section 3.3.4.
The deck weight is found by taking the tributary width of the girder, multiplying
it by the assumed deck thickness (8 in), and then multiplying this product by the unit
weight of reinforced concrete, 0.15 kips/ft3. This gives a uniform weight per length of
the bridge. The tributary width for an interior girder is equal to the girder spacing,
while the tributary width of an exterior girder is equal to half of the girder spacing plus
the overhang (2 ft).
The weights of the diaphragms, haunches and SIP forms will vary depending on
numerous factors and different designers may arrive at different satisfactory designs for
each of these attributes. Furthermore, relative to the other loadings considered, the
weight of these members is fairly small. Thus, the weight of these components applied
per girder is approximated to be 15% of the concrete deck weight that was applied per
girder. This value was obtained by referencing dead load design calculations for select
steel girder bridges. These referenced dead load calculations were from bridges of span
lengths 52 ft and 87 ft (thus similar in length to those considered herein), which showed
the percentage of deck weight for diaphragm, haunch, and SIP to be 13% and 16%
respectively. This variation in percentage of deck weight that is applied to account for
additional dead load components (1-2%) only influences the total service loads by a
maximum of 0.6%. Therefore, the influence of a slightly different value for this
percentage on the results of this thesis is considered negligible.
89
The FWS weight is accounted for by using a surface area density of 0.030
kips/ft2. This value is obtained from Section 1.7.3 of the Pennsylvania Department of
Transportation’s Design Manual 4 (PennDoT 2007). This surface area density is
multiplied by the girder tributary width, which results in a weight applied per length of
the girder.
The barrier weight applied to each bridge is found by taking the weight of the
barrier and distributing it evenly between the two closest girders (exterior and adjacent
interior). This presents three possibilities for the magnitude of barrier loading applied
to the interior girders. These are a: (1) the interior girder carries the weight of a full
barrier (in a three girder bridge), (2) the interior girder carries the weight of half a
barrier (which is the case for all interior girders in four girder bridge or the two interior
girders adjacent to the exterior girders in a bridge with five or more girders in the crosssection), and (3) the interior girder carries none of the barrier weight (for interior girders
that are not adjacent to exterior girders in a bridge with 5 or more girders in the crosssection). Given that the barrier weight assumed for this calculation is 0.52 kips/ft, the
two extreme bounds for the barrier load on each interior girder are 0 kips/ft and 0.52
kips/ft. The consideration for both of these bounds was taken into account in the final
calculations in this thesis and was shown to have a minimal effect (i.e. less than 1%
difference). Thus, for the purposes of this thesis, each interior girder will be assumed to
carry the intermediate value of 0.26 kips/ft.
90
3.3.4
Girder Selection
As previously discussed, the girder self weight can vary greatly depending on
the span length and girder spacing and cannot be precisely determined at the onset of a
design. Therefore, the self-weight of the girders was initially assumed to equal 0.1
kips/ft. A rolled girder was then selected for each bridge scenario. Once a size was
selected, the actual self-weight was substituted back into the ultimate design load
calculation to ensure that the girder still had sufficient capacity.
The rolled shapes were selected using Table 3-2 in the AISC Steel Construction
Manual (AISC 2005), which categorizes rolled sections by their section modulus. A
shape was selected for evaluation and the plastic moment capacity of the section was
then calculated assuming composite action between the deck and girder. This capacity
was found per Table D6.1-1 in AASHTO (2007). The width of the deck that was used
in this calculation is the same as the tributary width used for finding the deck weight as
discussed above except for the 10 ft spaced girders. The interior and exterior girder
effective deck widths for the 10 ft girder spacing were reduced to 8 ft and 6 ft,
respectively. This is the effective deck width resulting from the calculation prescribed
in AASHTO. If the section was not adequate a larger size was selected and the process
was repeated. If the size exhibited enough capacity then the next smallest size was
evaluated. Once the girder whose capacity most exactly exceeded the requirements was
found, its weight was used in the dead load calculations to determine the effective load
factor.
91
The girders selected for the bridges are given later in the section in Table 3.13.
As these are AISC shapes, their unit weight is given in their name. For example, the
W30x90 girder weighs 90 lbs/ft. It is possible that if plate girder sections were
designed that lighter sections might result, but with the number of possible geometries
that would then exist for each girder (varying plate dimensions for both flanges and
webs) there is not a unique answer, which leads to unnecessary complexity in the
procedure.
After the weights of each of the above components were calculated, they were
summed to find the dead load acting along the length of the girder. As with the lane
load, the maximum moment created by a distributed moment occurs at mid-span. This
moment was found by Equation 3.18 and the values thus determined are shown in Table
3.11.
Table 3.11 – Maximum Dead Load Moment per Girder
Maximum Moment (k*ft)
Span Length (ft)
50
100
Girder Spacing (ft)
Interior
Exterior
6
379.4
334.1
8
471.9
381.3
10
565.3
429.4
6
1700.0
1518.8
8
2102.5
1740.0
10
2500.0
1956.3
92
The ultimate design load was calculated by using the Strength I limit state in
AASHTO (2007), which places a load factor of 1.25 on dead loads and 1.75 on live
loads. That this is the governing strength limit state is a direct consequence of the
design loads considered herein. The maximum moments from the dead load, lane load,
and tandem or truck load were multiplied by their corresponding load factor and
summed.
By comparing the data in Tables 3.7 through 3.9, it can be seen that the
moments caused by uniform distributed loads (as opposed to vehicle loads) are the most
sensitive to the location at which the moment is determined (for the range of locations
of interest). For this reason, the governing location for the ultimate loads occurs at midspan, where the lane load and dead loads are greatest. It has also been previously
determined that the truck loading will govern over the tandem loading. After the load
factors were applied to the dead and live loads, and the impact and distribution factor
was applied to the live loading, the ultimate loading was obtained. These results are
shown below in Table 3.12.
93
Table 3.12 – Ultimate Loads per Girder
Maximum Moment (k*ft)
Span Length (ft)
50
100
Girder Spacing (ft)
Interior
Exterior
6
1487.5
1232.5
8
1829.9
1473.9
10
2162.7
1709.4
6
4617.5
3904.3
8
5671.2
4623.2
10
6694.1
5316.4
As the interior ultimate loads governed for each bridge scenario, and the bridges
were designed to have all girders of the cross-section to be identical, these loads were
used in the design of all girders. The girders that were then selected for each bridge
scenario are given in Table 3.13, along with their design capacity. The interior and
exterior girders have different capacities as their effective deck widths are different. A
resistance factor of 0.90 is multiplied by the nominal moment capacity of the section,
which is the plastic moment, to reach the design capacity.
94
Table 3.13 – Girder Sizes Used for Each Bridge Span Length and Spacing
Span Length
(ft)
50
100
3.3.5
Girder Spacing
(ft)
Girder
Selection
6
Design Capacity (k*ft)
Interior
Exterior
W24X84
1624
1577
8
W30X90
2052
1985
10
W30X99
2299
2216
6
W44X230
5324
5289
8
W44X262
6132
6047
10
W44X290
6896
6748
Effective Load Factor
As previously discussed, during the design process it is inevitable that extra
capacity is built into each member of a structure since it is extremely unlikely that a
section can be selected whose capacity exactly matches its ultimate design load. This
inherent over-strengthening can introduce error into this analysis. For example, it can
be seen that the interior girder for the 50-ft span with 8-ft spacing has a capacity of
2052 k*ft and was designed to resist an ultimate load of 1829.9 k*ft. This girder,
therefore, has been over-strengthened by 12% as a result of the design process. A
similar calculation for the 100-ft bridge with 10-ft spacing shows the interior girder is
only over-strengthened by 3%. For this reason, once a girder was selected to use in
each bridge, an effective load factor was used to apply a loading that perfectly matches
the capacity of the girder. This was done to remove any variability in the results that
95
would be caused by the differences in the amount that different designs are overstrengthened in this manner.
The effective load factor was calculated by taking the ultimate design loads in
Section 3.3.3 and dividing by the service loads. These service loads were found by
simply adding the dead and live loads without any load factors applied. There is one
major consideration for this calculation, and that is whether the impact factor of 1.33
should be regarded as part of the service load of the vehicle loadings or if this should be
treated as a load factor. In the missing column analysis carried out in buildings, there is
usually a dynamic factor of 2 that is used to represent dynamic loads statically, so it
seems fitting that the impact factor should be considered as part of the service loading
for bridges. Furthermore, the dynamic effect due to impact is a true effect amplifying
the actual magnitude of stresses occurring in a bridge, while the load factors are not.
However, to gauge the effects of both considerations of the impact factor, the effective
load factor was calculated using both. The resulting effective load factors are shown for
the interior and exterior girders of all bridges in Table 3.14.
96
Table 3.14 – Effective Load Factors with and without Impact
Span Length
(ft)
50
100
Effective Load Factor
with Impact
Effective Load Factor
w/o Impact
Interior
Exterior
Interior
Exterior
6
1.55
1.54
1.76
1.74
8
1.55
1.55
1.76
1.76
10
1.55
1.55
1.76
1.77
6
1.48
1.47
1.61
1.59
8
1.48
1.47
1.61
1.60
10
1.47
1.48
1.60
1.61
Girder
Spacing (ft)
The results in Table 3.14 show the variability in the effective load factors as a
function of girder type, spacing, span length, and the consideration of the impact factor.
The girder type resulted in negligible differences, with the exterior and interior girders
having values within 1% of one another, indicating that the ratio of ultimate design
loads to service loadings for both were very similar. Similarly the influence of girder
spacing was also seen to be negligible, with the results of the different girder spacings
having results within 1% for the bridges investigated. The span length was seen to have
a greater effect; with the 100-ft span having values 5 – 9% smaller than the 50-ft span.
The effects of including impact into the results are seen to give the most influence, as
values including impact are 9 – 14% higher than without.
97
3.3.6
Bridge Models for Missing Girder Analysis
Since the girder spacing did not have a significant effect on the effective load
factor, and ultimately the service loads applied in each case, it is not necessary to carry
out the missing girder analysis for each girder spacing, as each will produce almost
identical results. Similarly, the results show that the interior and exterior girders have
almost identical effective load factors as well as nominal capacity (within 4% as shown
in Section 3.3.4). For example, the interior girder for the 50-ft bridge (8-ft girder
spacing) has a design capacity of 2052 k*ft and an effective load factor of 1.55, which
results in a service loading of 1324 k*ft that would be applied in the missing girder
analysis. Similarly, the exterior girder for the same bridge has a design capacity of
1985 k*ft and an effective load factor of 1.55, which results in a service loading of 1281
k*ft. Thus, essentially the same service loading will be applied to both in the missing
girder analysis (1.9% difference). As a result, it is unnecessary to distinguish between
exterior and interior for the missing girder analysis.
Since distinguishing between girder type and spacing is shown to be negligible,
the cross-section of each bridge was considered to consist of uniform girders having the
same capacity and carrying the same service loads as those calculated for the interior
girder in previous sections. Thus, the results in Chapter 6 will consider two possible
scenarios (removal of one or two girders) at two span lengths (50 ft and 100 ft). Each
of these scenarios will consider service loadings with and without impact. Also, the
girder spacing of 8 ft will arbitrarily be chosen, as the effects of different girder
spacings were shown to be negligible.
98
From the effective load factors shown in Table 3.14 and the design capacities
shown in Table 3.13, the service loads that were applied to the girders in each case for
the missing girder analysis was determined. These are found by taking the design
capacity of the sections and dividing by their respective effective load factor. Table
3.15 shows the service loads applied for each span length as well as the nominal
capacity for each section. The nominal capacity for each section is equal to the design
capacity given in Table 3.13 divided by the resistance factor of 0.90. The applications
of these service loadings to obtain results of the missing girder analysis are shown in
Section 6.3.
Table 3.15 – Service Loadings and Nominal Capacity for Each Span Length
Span Length
(ft)
Girder
Service Loading (k*ft) for
Interior/Exterior
w/ Impact
w/o Impact
Nominal
Capacity (k*ft)
50
W30X90
1323.9
1166.0
2280.1
100
W44X262
4143.4
3832.7
6813.7
99
Chapter 4
VALIDATION STUDY
4.1
Need for Validation Model
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 where the stand-off distance is the farthest distance from the column that the charge
can be placed to fail the column. This blast threat was found considering two charge
sizes, small and large. The stand-off distance was then iterated for each blast size and
various column designs to find the farthest stand-off distance that caused failure in each
case.
This sort of analysis could be carried out in physical tests; however, the large
number of iterations likely required for determining the precise range of stand-off
distances causing failure leads to this being an unfeasible option. This is especially true
when additional parameters such as charge elevation and member sizes are considered.
An additional problem with carrying out this investigation experimentally is that it is
difficult to compare results from two different physical tests because the results are
sensitive to changes in environmental conditions (e.g., temperature, humidity, etc.) that
are difficult to keep constant. This coupled with slight variations between specimens
(such as dimensions and boundary condition fixity) would make direct comparison
100
between the iterations at different stand-off distances difficult. Analytical methods can
be employed using the finite element method if it is believed that the results reached
will accurately depict the corresponding physical test.
Physical models with member types and loading conditions identical to the ones
under consideration in this study have been carried by Lawver et al (2003). The results
of Lawver et al (2003) are used as a validation model for the models in this thesis.
4.2
Lawver et al (2003) Model Description
In Lawver et al (2003), seven steel columns were exposed to “typical size”
charges that were described as being representative of vehicle bombs at two distances,
called “close” and “far”. Neither the weight of the charges nor the stand-off distances
were quantified in the article. These columns were AISC steel shapes: W14x38,
W14x53, W14x82, and W14x132. Each column was 14 feet tall, bolted to a concrete
foundation with an 8-bolt base plate connection at the bottom of the column, connected
to the loading apparatus with a moment connection at the top, unbraced along its height,
and was loaded simultaneously with an axial load of 150 kips at the top and a blast load
in either the strong or weak direction. This loading is smaller than the service loading
considered for the column analyses in this thesis, however, since both are well below
the nominal capacity of the column there is not a foreseeable consequence to this
difference. The varying sizes of columns were reportedly used to get a wide range of
second-order effects. During these experiments, measurements were taken using
accelerometers and pressure transducers at quarter height intervals; high speed video
101
and deflections were also recorded. As it is of present interest to validate the columns’
response to blast loading using the analysis methods described in Chapter 3, the results
of interest from this experiment are maximum displacements along the height of the
column.
Lawver et al also duplicated these same physical tests numerically using an inhouse finite element code called FLEX, as well as SDOF models as prescribed in TM 5855-1 (Department of the Army 1986). The accuracy of the FLEX model was assessed
through comparison to the experimental results; here the accuracy of the blast loading
profile applied to the structure through the use of the Kingery-Bulmash equations was
of particular interest. The accuracy of the SDOF model was evaluated through
comparison to the FLEX and physical data.
The FLEX finite element code has nonlinear, dynamic capabilities as well as
constitutive material models for ductile materials that are strain rate sensitive, such as
steel. FLEX uses Kingery-Bulmash equations to calculate forces acting on each
element, which are the same as those used in CONWEP (Kingery and Bulmash, 1984).
The columns were modeled using single-point integration quadrilateral shell elements
with shear deformation capabilities. The bolts, forming the base plate connections at
the ends of the column, were modeled by using Timoshenko beam elements with shear
deformation capabilities. The article states that pinned and fixed end models were
created, but specific details on whether these are idealized or practical connections are
not given.
102
As opposed to the FLEX models, which apply the blast pressure directly to each
element in the model, the SDOF models find the blast pressure at a single point on the
column by CONWEP equations and assume that the pressure is uniformly distributed
across the entire member. The Lawver et al (2003) study concludes that CONWEP
pressure predictions used for both the SDOF and FLEX models were a good match to
the recorded blast loading profiles in the experiments. This is a good indication that
blast pressure will be predicted accurately for the models in this thesis, as they also use
CONWEP equations. For blasts acting from large distances, the column acts more
globally and the approximations given by a uniformly distributed load are accurate,
indicating that the SDOF and FLEX model were both accurate. As the stand-off
distance decreases, however, a uniformly distributed load fails to cause the local effects
that occur in reality and becomes less accurate; thus, the SDOF model did not
accurately depict the behavior at small stand-off distances while the FLEX models did.
Since the finite element models were shown to be a much more accurate way of
modeling response to blast loading as opposed to SDOF models, only the results of the
FLEX models, along with the physical experiment results, were considered in this
thesis.
In order to validate the finite element models used in this thesis, validation
models were created that were as similar as possible to the Lawver et al (2003) physical
test and FLEX models. These two validation models were created with two different
connection types, fixed and pinned. Since the displacement results of the Lawver et al
(2003) study were only provided for the W14x82 column physical test and FLEX model
103
subjected to the close charge, the validation models for only this member and charge
size were created. Some assumptions were required to be made before the modeling
could be completed, which are discussed in the following section.
4.3
Assumptions
To duplicate the analysis carried out in Lawver et al (2003) it was necessary to
estimate the blast loading used. Because the blast loading (charge size and stand-off
distance) is only described qualitatively, the arrival time and peak pressure from the
reported pressure histories in Lawver et al (2003) were used to solve for these values.
The estimated arrival time and peak pressure from the given pressure history plots are
shown in Table 4.1. Because there are many combinations of charge size and stand-off
distance that may produce the reported pressure history, it was assumed that the same
charge size was used for both the close and far charges. While it is not explicitly stated
by Lawver et al that this is the case, this is inferred to be a reasonable assumption based
on the authors’ description of their work.
Table 4.1 - Criteria from Lawver et al (2003) to Find Bomb Size and Location
Distance
Close
Far
Arrival Time (ms)
1.45
15.2
Peak Pressure (psi)
4000
120
CONWEP was then used as an iterative tool to match the results given in Table
4.1. This was done by assuming a charge size and stand-off distance, evaluating the
104
peak pressure for this blast as compared to Table 4.1, and increasing or decreasing the
stand-off distance accordingly. If the peak pressure was too low then the stand-off
distance was reduced. Once the peak pressure was identical, the time of arrival values
were compared. If they were dissimilar then the charge size assumed was incorrect and
needed to be adjusted. If the time of arrival was too high the charge size needed to be
increased, which will result in a larger corresponding stand-off distance. This process
was continued for the close charge until a result is reached. Once the charge size was
determined that would match the close results, then this same charge size was held
constant as the stand-off distance for the far charge was found.
Carrying out this procedure yielded the results shown in Table 4.2. As can be
seen, a 947-lb TNT charge placed at 14.83 ft and 53.72 ft gave approximately the same
pressure history results as those produced in Lawver et al (2003). This is considered a
very large blast but within reason for a size that could be used experimentally,
especially when it is considered that the authors stated intent was to represent a vehicle
bomb. The calculated charges result in an exact match of the arrival time and peak
pressure for the close charge and a 2% relative error in arrival time but exact match of
peak pressure for the far charge. Given the fact that the charge size was not adjusted in
finding the far charge stand-off distances, there is a good level of confidence that this
combination of charge size and stand-off distances is correct to a reasonable level of
certainty.
105
Table 4.2 - Equivalent Charge Size Comparisons to Lawver et al (2003) Data
Close Charge Comparison
Far Charge Comparison
(947 lbs,
14.83 ft)
Lawver et al
(2003)
(947 lbs,
53.72 ft)
Lawver et al
(2003)
Arrival Time
(ms)
1.45
1.45
14.9
15.2
Peak Pressure
(psi)
4000
4000
120
120
It was also necessary to assume the vertical position of the bomb; but, there was
no definitive method for determining this value. CONWEP, which is the program that
was used by Lawver et al (2003), has two options for predicting blast loading, the
choice of which depends on the vertical location of the charge. The first option is a
“surface” blast, where the bomb is situated on the ground and reflected pressures off the
ground act immediately in combination with incident pressure to create a hemispherical
shock wave. The second is a “free-air” blast, which assumes the blast does not interact
with any surfaces (e.g., the ground profile) before striking the object of interest, thereby
excluding any effects of reflection, which can greatly amplify the effects of the blast.
These two separate blast options, which are selected and input by the user, give
different pressure-time histories and utilizing each will therefore give a different size
bomb and stand-off distance. Since it is known that the blast occurred at or near the
ground in the experimental testing, it is a reasonable assumption that the “surface” blast
106
option was utilized. Therefore, the vertical position of the charge in the validation
model was on the ground.
Connection details provided in the Lawver et al (2003) article describe the
connections as being fixed at the top, and fixed at the bottom with an 8-bolt base plate
connection. For this validation study it was assumed that that these connections could
be satisfactorily replicated by an idealized fixed connection, which was achieved by
simply restraining the nodes at the bottom and top of the column. This reduces the
amount of modeling and computational time considerably compared to modeling an
actual connection. In the Lawver et al (2003) study it was desirable to account for
damage in the bolts of the connections and thus at least some of their models appear to
have considered more detailed representations of these connections. However, for this
thesis, the bolt damage is not of interest so ignoring these effects is appropriate.
Furthermore, since details of the actual connection are not given, modeling this would
require additional speculation on the part of the author, which may introduce error into
the analysis.
Since connections are neither fully fixed nor pinned in reality, the Lawver et al
(2003) study created fixed and pinned finite element models in FLEX to try to “bound”
their physical test results. The pinned connection was assumed to be achieved by only
restraining one node at the top and bottom of the column and allowing the other nodes
to rotate freely about it. However, the specifics of how these connections were modeled
by Lawver et al (2003) (whether they are idealized or realistic connections) is not clear.
107
The constitutive properties of the steel used for the columns are not known and
were assumed to be typical of Grade 50 steel. The Johnson-Cook model was assumed
to represent the effects of strain rate. Specific information on the material input
parameters is discussed in Section 3.2.1.3.2.
4.4
LS-Dyna Validation Model
The finite element analysis program used for the validation models was LS-
Dyna. The models were built using single-point integration quadrilateral shell elements
spaced at 1.00 cm x 0.99 cm in the flange and 1.00 cm x 1.00 cm in the web. For the
fixed end model the fixed support at the bottom was achieved by restraining each of the
bottom nodes in the x, y and z directions. By restraining each of these nodes, the
bottom is unable to translate or rotate, in effect creating a fixed support. The fixed
support at the top was achieved by restraining the top nodes from translations in the
plane of the cross-section but allowing them to displace vertically (due to the axial
loading). If the vertical direction was not allowed free displacement, axial loading
could not be introduced. The pinned connections for the pinned end model were
achieved by restraining the node in the middle of the web and allowing the other nodes
to rotate freely about it.
The axial loading of 150 kips was applied as discretized nodal loads acting at
each of the nodes at the top of the column, and was applied in a linearly increasing
fashion over a period of 30 ms to avoid dynamic effects. If this loading is applied
108
instantaneously it acts dynamically, causing the column to respond in an oscillatory
manner.
LS-Dyna, similar to FLEX, uses CONWEP algorithms to calculate the blast
loading acting on each face. The center of the charge needs to be specified in the
model. If the charge is assumed to be spherical and resting on the ground, then the
center will be one radius above ground level. For a 947 lb charge with a density of 1.65
g/cm3, the radius is equal to 15.6 in., which was input as the vertical position of the
charge center (Kinney and Graham 1985). As the deflections of the column for the
“close” stand-off distance are only provided for comparative purposes in Lawver et al,
the horizontal position of the charge was input as 14.83 ft.
The constitutive model used was the Johnson-Cook model.
Discussion on the
material input parameters used in conjunction with this model can be found in Section
3.2.3.3.2.
4.5
Results
As deflection is the only additional data that was provided by Lawver et al
(2003) besides pressure vs. time histories (which were used to determine the charge
size), these are the only results used to compare with the validation models. The
maximum deflections at one-tenth intervals along the height of the columns were found
from the LS-Dyna post-processor. These maximum horizontal deflections are shown in
Figure 4.1. The deflection data from Lawver et al (2003) was also estimated from their
plots and shown for comparison purposes.
109
14.0
Height (ft)
10.5
7.0
3.5
0.0
0
1
2
3
4
Displacement (in)
Lawver Fixed
Lawver Pinned
Fixed
Pinned
5
6
Lawver Experiment
Figure 4.1
.1 – Deflection Profile for the Validation Models
Model
The fixed end validation model’s deflection is shown to be similar to the
experimental model. While the location of the maximum deflection occurs at a higher
location in the model than in the physical specimen (7 ft compared to 5 ft), there is only
a 2.8% relative error (see Table 4.3) between the two maximums. Furthermore, in both
the fixed end validation
alidation model and the physical specimen, the deflections are of greater
magnitude in the lower half than in the upper half of the column, although this effect is
more pronounced in the physical specimen. The deflected shape of the validation
model is also
so shown to be similar in shape to the Lawver et al models, which gives
110
validation to the loading assumptions made above in that the distribution of the pressure
along the height of the column must be similar to the actual distribution in order to
produce this result.
Table 4.3 - Maximum Deflection Relative Error for Fixed End Validation Model
Model
Percent Relative Error
Lawver Experiment
2.8
Lawver Fixed
23
The pinned end validation model deflection is shown to have the largest
deflection of the models and, as opposed to the fixed end validation model, does not
have a more pronounced deflection at the bottom of the column. The Lawver et al
pinned model does show a more pronounced deflection at the bottom of the column
which indicates that perhaps the pinned connections were not exactly similar and it is
demonstrated through comparing the various analysis with differing end conditions, that
the boundary conditions do have an appreciable influence on the results. The relative
error between the maximum deflections in the pinned end models is 26%, which is
similar to the 23% relative error found between the fixed end models.
Without knowing the specific details of the Lawver et al (2003) FLEX model, it
is difficult to pinpoint what discrepancies may be causing the difference. Since this
thesis only considers columns with fixed end connections and the actual method in
which the authors achieved the pinned connection is not clear, duplicating the pinned
models from Lawver et al (2003) was not a major concern. Most importantly, the fixed
111
end validation model results were shown to mimic the fixity of the fixed end model and
experiment results that were published by Lawver et al (2003).
That the displacement results are similar in shape to the Lawver et al (2003)
models and match the experimental deflection peak magnitude very closely gives
confidence in the modeling techniques and material parameters used in the validation
model. With variability in the assumptions described in Section 4.3, it is possible that a
closer match could be found if these were altered, but no attempt was made to refine the
assumptions to target particular results. Thus, for the purposes of this thesis, these
methods are considered accurate for modeling steel members subjected to simultaneous
axial and blast loading.
A previous validation study was published by the author (Brown and McConnell
2009) using this same Lawver et al (2003) data which assumed a “free-air” blast model
and different Johnson-Cook material model parameters. After a more thorough review
of the input, the input discussed herein is believed to more closely represent the Lawver
et al (2003) study.
112
Chapter 5
SENSITIVITY ANALYSIS
5.1
Introduction
In finite element modeling there is a trade-off between accuracy and
computation time. That is, up to a certain point, the smaller the element sizes in the
model (or the finer the mesh), the more accurate the results will be. The results of the
finite element model should show convergence, in that they approach a constant value
as the element size is decreased. But smaller elements lead to a higher investment in
time, so the goal becomes finding the largest mesh possible that still gives accurate
results. Due to the iterative nature of this research and the large number of analyses
thus required, efficient use of computational resources was of special importance.
Thus, a sensitivity analysis was conducted to determine the largest acceptable mesh
sizing due to the large influence of this variable on computation time. In the previous
chapter, a relatively small mesh size was selected for use in the validation study model,
but using that for the remaining models in this thesis isn’t practical because of the large
computation time required.
5.2
Sensitivity Analysis Model
Since this thesis is focused on steel columns subjected to simultaneous axial
static loading and lateral blast loading, a similar model was used in the sensitivity
113
analysis. As with the validation model analysis performed in Chapter 4, the column
used in the sensitivity study was a 14-ft W14X82 column fixed at both ends. The
charge size, charge placement, axial loading, and boundary conditions were also
identical to that used in the validation study; the charge was again modeled as a surface
blast located on the ground. Four different finite element models were built and
analyzed using element sizes of approximately 4.0 cm, 1.5 cm, 1.0 cm, and 0.50 cm,
while all other parameters were held constant. These element sizes are only
approximate because they are not an even multiple of the given column’s flange and
web dimensions, which are 10.1 in. and 14.3 in., respectively.
Differences in displacement along the height of the column were used as one
comparative measure. Displacement results are perhaps the most important metric to
use to assess convergence because failure of the specimens in this thesis is defined by
exceeding a given displacement criteria. Also, shear stress, axial stress, and von-mises
stresses were evaluated along the web at mid-height of the column. Even though these
were not used as failure criteria in this thesis it was still important to have accurate
stress values in case they wanted to be looked at. Additionally, as stress is a secondary
variable to displacement, if they are accurate it is a good indication that displacement
results are accurate as well. Stress at mid-height of the column was used as this is the
critical location for buckling and where maximum displacement occurs. The finest
mesh of 0.50 cm will be considered to give ‘very accurate’ results as convergence was
seen to occur at this value. The results of the 0.50 cm mesh are compared with the other
114
meshes to measure accuracy. Each of these results is described in the following
sections.
5.3
Displacement Results
The differences in deflection throughout the height of the column for the
different mesh sizes are shown below in Figure 5.1. The data shown here is taken from
the time interval producing the greatest displacement for each mesh size. The
maximum deflection is shown for the tenth points along the height of the column,
linearly interpolating between two nodes if necessary. To get the maximum
displacement from each node it was selected in the LS-Dyna post processor and the
horizontal displacement as a function of time could be plotted. From the plot the post
processor can give the ordinates of the maximum displacement and time. The
maximum displacement at each of the tenth points occurred at the same time step.
Between each of the different mesh size models, the maximum displacement occurred
at the same time step as well. The results were recorded every 0.1 ms in the model.
115
14
12
Column Height (ft)
10
8
0.5 cm
6
1 cm
4
1.5 cm
4 cm
2
0
0
0.5
1
1.5
2
2.5
3
Deflection (in.)
Figure 5.1 – Deflection Profile for Varying Mesh Sizes
izes
These results in Figure 5.1 show that that there is not a great variation
variat
in results
between the 0.50, 1.0,, and 1.5 cm meshes
meshes, while the 4.0 cm mesh shows more variation
from the others. The relative error of each of the 1.0, 1.5 and 4.0 cm meshes as
compared to the 0.5 cm mesh are provided in Table 5.1. The 1.0 and 1.5 cm mesh are
seen to both be very accurat
accurate, maintaining less than 2.6%
% relative error throughout the
height of the column.
116
Table 5.1 – Percent Relative Error of Displacement along Height of Column
Column Height (ft)
1 cm Mesh
1.5 cm Mesh
4.0 cm Mesh
1.4
0.9
1.9
23.0
2.8
2.6
2.4
21.1
4.2
1.4
1.5
12.7
5.6
0.4
0.5
8.3
7
0.4
0.1
6.4
8.4
1.0
0.4
5.1
9.8
1.0
0.1
3.3
11.2
0.5
0.6
3.6
12.6
2.2
0.4
2.0
The second column in Table 5.2 shows the maximum deflection of the columns
and the percent relative error with respect to the 0.50 cm mesh. The 4.0 cm mesh is
shown to have a relative error of 6.4%, while this value for the 1.0 cm and 1.5 cm
meshes are both below 1%. Since the most important output for evaluating the results is
displacement (for the reasons described above), the 4.0 cm mesh is considered too
coarse. The results for the stresses are also shown in Table 5.2; these are discussed in
their respective sections. A conclusion is given in Section 5.5.
117
Table 5.2 – Percent Relative Error for Various Mesh Sizes
Mesh Spacing
(cm)
Displacement
at mid-height
(in.)
Maximum
Axial Stress
(ksi)
Maximum
Shear Stress
(ksi)
Minimum VonMises Stress
(ksi)
0.5
3.230
66.0
13.8
23.9
1.0
3.242 (0.4%)
65.3 (1.1 %)
13.6 (1.4%)
23.4 (2.1%)
1.5
3.228 (0.1%)
66.2 (0.03%)
12.9 (6.5%)
22.4 (6.3%)
4.0
3.024 (6.4%)
69.2 (4.9%)
15.7 (13.8%)
27.8 (16.3%)
5.4
Stress Results
As previously discussed, axial, shear, and von-mises stress were recorded at
mid-height of the column. In order to find the stress distribution along the web at midheight of the column, the values had to be manually taken from each element of the web
cross-section, as was done to find the maximum displacements. To get each of the
stress results from the LS-Dyna post processor, the element was selected and then the
specific type of stress was selected and plotted as a function of time for the entirety of
the analysis duration. From this plot, the data for a specific point in time can be found
by simply clicking on the plot at this time. A time of 5 ms after blast initiation was
selected to use for the evaluation of all output. Another time interval could have been
selected; this was chosen because a high distribution of stress in the cross-section of the
column was observed for all three stress types, meaning that stress varies substantially
across the section. This is an important feature to include into the sensitivity analysis as
it not only allows for convergence of the data to be measured, but to visualize trends in
118
the stress distribution and see what behavior is being exhibited. The results may also be
sensitive to the time step used to collect data, in that data collected at say 6 ms, for
example, may show different variations between meshes. This is also why it is also
important to collect data at time steps when high stress is present as this is likely where
the most variation is present and the results most critical. Single integration point shell
elements were used, so therefore, a stress calculation for each element is performed at
the element center. Since different size elements are used between the various models,
the more coarse meshes will have less data points than the finer ones. The results for
axial, shear, and von-mises stress are summarized in the following sections.
5.4.1
Axial Stress
The axial stress distribution for the 0.50, 1.0, 1.5, and 4.0 cm meshes are shown
in Figure 5.2. The distance along the web is measured from the intersection of the web
and flange closest to the charge. The negative and positive stresses correspond to
compressive and tensile stress, respectively.
119
60
40
Axial Stress (ksi)
20
0
0
2
4
6
8
10
12
14
-20
-40
4 cm
1.5 cm
-60
1 cm
0.5 cm
-80
Distance along Web (in)
Figure 5.2 – Axial Stress Distribution along Web at Mid
Mid-Height
Height of Column
C
Each of the stress plots in Figure 5.2 is indicative of bending
ng behavior and
shows a linear distribution of stress through the middle portion of the web with yielding
at the extremities of the web. The yielding extends through a greater portion of the web
on the side of the column in closest proximity to the blast (compared to the side of the
column away from the blast), which is a logical result. It is also shown that the
maximum compressive stress (which ranges between 66-69 ksi)) is greater than the
corresponding tensile stress (which ranges between 54
54-57 ksi). This reflects the fact
that the yield strain was of a higher magnitude on the side of the column closest to the
blast because there is a greater strain rate effect here
here. There is also asymmetry of the
120
results in the linear portion of the stress response (i.e., the point of zero stress does not
correspond with the neutral axis of the cross-section) due to the combination of bending
and axial loading.
Each of the mesh sizes are seen to give very similar results as one another for
axial stress. While there are some discrepancies in the results obtained from the 4.0 cm
mesh, the results of the other three mesh sizes are nearly indistinguishable from one
another. Table 5.2 shows the percent relative error of the maximum axial stress reached
at the 5 ms time interval in the web, which occurs in the element adjacent to the webflange intersection at 0 in. Each mesh is seen to give less than 5 % relative error; the
1.5 cm mesh matches the 0.50 cm results most closely (with only 0.3% relative error),
although the 1.0 cm results are also very close (1.1% relative error). However,
evaluating the stresses at all locations along the web, the 4.0 cm mesh show a greater
discrepancy, reaching a maximum of 22% relative error at a distance of 11.2 in.
5.4.2
Shear Stress
The shear stress along the web at mid-height of the column at a time of 5 ms
was recorded and plotted in Figure 5.3. The shear stress at this location and time were
chosen to correspond with the axial stress results. Here it is shown that while all of the
meshes produce similar trends in the shear stress distribution, the magnitudes resulting
from the 4 cm mesh vary greatly from the other meshes, giving a the shear stress 2.5 to
4.4 ksi higher than the other meshes throughout the web depth. The results for the 4.0
cm mesh are also seen to be on higher than those from 0.50 cm mesh, while the 1.0 and
121
1.5 cm meshes under-predict the shear stress relative to the 0.5 cm mesh. The
maximum shear stress values resulting from each mesh are compared in Table 5.2. The
maximum shear stress locations occur at 8.2 to 8.9 in. from the web-flange intersection
closest to the charge, depending on the mesh size. At the maximum shear stress
locations, the 4.0 cm, 1.5 cm, and 1.0 cm mesh have 13.8%, 6.5% and 1.4% relative
errors, respectively. These values are generally representative of the relative error
values throughout the cross-section. It is noted that the magnitude of the shear stress is
fairly low, so higher percentages of relative error (compared to the other metrics
evaluated herein) do not correspond to significant differences in stress magnitudes.
122
20
Shear Stress (ksi)
15
10
4 cm
5
1.5 cm
1 cm
0.5 cm
0
0
2
4
6
8
10
12
14
Distance along Web (in.)
Figure 5.3 – Shear Stress Distribution along Web at Mid
Mid-Height
Height of Column
C
5.4.3
Von-Mises
Mises Stress
Another common stress measure is Von
Von-Mises
Mises stress. These values
val
were
recorded in each element across the web at mid
mid-height
height of the column and plotted in
Figure 5.4. Similar to the shear stress results discussed in the previous section,
section the
magnitudes resulting from the 4.0 cm mesh vary appreciably compared to those
thos from
the other mesh sizes, but there is very strong agreement between the results of the other
three mesh sizes. Furthermore, while the general trend resulting from the three smaller
123
mesh sizes is the same, where a plateau in stress is shown at the extr
extremities
emities of the web,
the stresses continue to increase in this region (although only slightly) in the 4.0 cm
mesh results. The distribution shows that the highest stress occurs closest to the flanges
and the minimum stress occurs towards the center. This iiss logical considering that the
Von-Mises
Mises stresses are dominated by the axial stresses shown in Figure 5.2 and these
(absolute) values are a maximum at the extremities of the web.
70
60
Von-Mises Stress (ksi)
50
40
30
4 cm
20
1.5 cm
1 cm
10
0.5 cm
0
0
2
4
6
8
10
12
14
Distance along Web (in.)
Figure 5.4 – Von-Mises
ises Stress Distribution along Web at Mid-Height
eight of Column
For this stress distribution the minimum stress was chosen for comparison
compar
in
Table 5.2, which shows the relative error in the 44.0, 1.5, and 1.0 cm meshes compared
124
to the 0.50 cm mesh. This location was chosen as local minimum or maximums are a
good place to test convergence in a sensitivity analysis. It can be seen that the 4.0, 1.5,
and 1.0 cm meshes have a relative error of 16.3%, 6.3%, and 2.1% respectively. The
other locations in the cross-section show a similar variation as that of the minimum
stress, which can be observed in Figure 5.4.
5.5
Conclusion
Of the mesh sizes evaluated, the 0.50, 1.0 and 1.5 cm meshes are considered to
give acceptable results since general trends and magnitudes show good agreement; the
4.0 cm mesh is considered unacceptable due to differences in overall trends and
significant differences in the magnitudes of the displacement and stress results. The
results converge towards the 0.5 cm results with the 1.5 cm mesh having a maximum
relative error of 6.3% (for von-mises stress) and the 1.0 cm mesh having a maximum
relative error of 1.1% (for axial stress) compared to the 0.5 cm mesh. The
computational time required to obtain 10 ms of results, using a 3.8 GHz Intel Xeon
processor for the 4.0, 1.5, 1.0, and 0.50 cm meshes is 1.0, 5.1, 17.3, and 128.0 minutes,
respectively. The above comparison of deflection and stress results and these
computational times, result in selecting the 1.5 cm mesh for use in future modeling.
This choice is further supported by considering that the FEA output that will be used to
formulate results and conclusions will be solely based on the deflection output; the
deflection results from the 1.5 cm model match those from the 0.5 cm model within
0.1% for the case evaluated herein.
125
This mesh size of 1.5 cm will be used in all of the finite element models that
require time-intensive iterative analyses to get results. These iterative analyses can
require up to 15-20 iterations. The only analysis that does not use this mesh spacing is
the validation study, which used 0.50 cm spacing because it only needed to be analyzed
once and precision in results was valued over computation time in this case.
126
Chapter 6
RESULTS
6.1
Introduction
There were two main objectives of this thesis. 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. To reach this objective, steel columns
subjected to blast loads were modeled in finite element software. The second objective
was to determine what redundancy and system load carrying capacity exists in simplespan steel girders bridges in the event of a failed girder as a result of the reserve
strength of the system inherent to the design process. This objective is reached by
carrying out a missing girder analysis, which is similar to the alternate load path
analysis carried out for building design. The methods for carrying out each of these
objectives are discussed in Chapter 3. The results of the column analysis are shown in
Section 2 of this chapter, where the missing girder analysis results are shown in Section
3.
6.2
Column Analysis Results
To achieve the first major objective in this thesis, finite element models of blast
loads acting on steel columns were created and analyzed to find what stand-off distance
for a given charge size would cause failure. After these blast threats corresponding to a
127
missing column analysis are determined for a range of different column scenarios,
conclusions can be made on the level of safety provided against blast by using the
missing column analysis. The results for this analysis are shown in this section.
Discussion on the factors that were varied during this analysis is given in the
first section and is followed by the finite element modeling results. This is followed by
an interpretation of the results to determine what blast threat level represents missing
column analysis and the variability in the level of safety that exists for typical building
columns.
6.2.1
Parameters Varied in Column Analysis
In order to determine the effects of several variables on the failure of steel
columns and to form conclusions based on a range of scenarios, a parametric study was
carried out. The variables of column depth, column stiffness, column height, charge
size, vertical position of the charge, and stand-off distance are investigated in this
parametric study. The specific values considered for each of these parameters is
described below. The first column that was considered in the analysis was a W14x82
section, as this was used in the validation study, discussed in Chapter 4. The other
sections were determined based on the criteria below and using section property data
available in the AISC Steel Construction Manual (2005). All columns were loaded such
that the blast pressures were applied to the web, thus loading the column in weak axis
bending. This is the critical loading orientation compared to strong axis bending.
128
•
The influence of column depth, and therefore surface area exposed to the blast, was
investigated by using two cross-sections (W14x82 and W10x77) with roughly the
same axial capacity (within 5%) at both column heights considered. The depth of
these two sections varied by 35% (14.3 in. versus 10.6 in.)
•
By varying the column cross-section from a W14x82 to a W14x109 the influence of
column stiffness is evaluated. Because the blast acts in the weak direction, the
stiffness of these members about the weak axis is the relevant parameter (where the
W14x109 section is 3 times stiffer about this axis). This increase in stiffness is also
accompanied by an increase in column capacity, where the plastic moment capacity
is doubled and the axial capacity is increased by 1.4 for the 14 ft columns and 1.5
for the 18 ft columns.
•
Column heights of 14 ft and 18 ft were considered to be two reasonable upper limits
for evaluation.
•
Two different charge sizes are considered, which are referred to as “large” and
“small”. The large charge has a weight twenty times the weight of the small charge.
•
Charges in three vertical positions are evaluated. These are: resting on the ground at
the bottom of the column, centered at mid-height of the column, and centered at the
top of the column.
•
Stand-off distance is iteratively varied to determine the stand-off distance causing
failure. Four-ft intervals were used to make a preliminary assessment of failure.
129
Once the failure distance was determined within this range, 1-ft intervals were used
to refine the results.
6.2.2
Stand-off Distance Results
The stand-off distance results for each of the column scenarios are first given in
tabular form in Table 6.1 (for the small charge size) and Table 6.2 (for the large charge
size). These are shown along with the applied axial loading for the reader’s
convenience, as these are used in some of the discussion that follows. The results are
shown as a function of the vertical position of the charge, where r represents the radius
of the spherical charge (i.e., the charge is resting on the ground in these cases). Here
the “failure stand-off distance” represents the farthest distance causing failure; thus,
distances less than the reported distance will also cause column failure. For example, if
the W14x82 (14 ft) column is subjected to the small charge located on the ground, it can
be expected to fail at stand-off distances of 8 ft or less; at distances of 9 ft or greater, the
column does not fail. Therefore, the smaller the failure stand-off distance, the more
resistant the column is to the blast. The failure stand-off distances for the small charge
are seen to vary between 3 and 15 ft, while these distances for the large charge size are
seen to be substantial, becoming as large as 216 ft for the 18 ft long W14x82 column
when the blast is aligned with the top of the column.
The results are then shown in a series of plots, referred to as “failure contours”
(shown in Figures 6.1 - 6.6). Each of these plots show the stand-off distance at which
the given blast needs to be placed in order to cause failure at the three vertical charge
130
positions, creating a 2-D spatial representation of charge locations causing failure.
These plots will be used to evaluate the influence of each of the parameters varied in
this analysis. The stand-off distances determined at three heights are, for illustrative
purposes, connected by straight lines in the failure contour plots. However, it is not the
intent of the author to suggest that linear behavior actually exists between the data
points. Furthermore, it should be noted that these plots are constructed by plotting the
ground charge at a vertical position of zero rather than its exact location.
Each of the
varied parameters’ influence on these results is discussed in detail in the following
sections.
131
Table 6.1 – Failure Stand-off Distance for Each Column (Small Charge)
Column Size
W14x82
W14x82
W10x77
W10x77
W14x109
W14x109
Column
Height
(ft)
14
18
14
18
14
18
Applied Axial
Load
(kips)
647.6
590.5
617.5
567.6
936.6
895.9
132
Vertical
Position of
Charge
(ft)
Failure Standoff Distance
(ft)
r
8
7
10
14
3
r
12
9
15
18
3
r
5
7
6
14
3
r
7
9
8
18
3
r
4
7
5
14
3
r
4
9
6
18
3
Table 6.2 - Failure Stand-off Distance for Each Column (Large Charge)
Column Size
W14x82
W14x82
W10x77
W10x77
W14x109
W14x109
Column
Height
(ft)
14
18
14
18
14
18
Applied Axial
Load
(kips)
647.6
590.5
617.5
567.6
936.6
895.9
133
Vertical
Position of
Charge
(ft)
Failure Standoff Distance
(ft)
r
42
7
104
14
176
r
52
9
127
18
216
r
31
7
79
14
139
r
43
9
108
18
192
r
23
7
54
14
86
r
42
9
97
18
156
6.2.2.1
Charge Size
Comparing the results in Table 6.1 and 6.2 indicates that, as expected, the large
charge will result in significantly larger failure stand-off distances. This is because the
small charge is 20 times smaller than the larger, requiring it to be much closer to cause
damage. Comparing the differences in the relative magnitudes of the failure stand-off
distances as a result of the two charge sizes, it is seen that there is great variability in
these values. For example, it was determined that the small charge placed on the
ground would fail the 14 ft W14x82 column at a distance of 8 ft or smaller; the large
charge will cause failure in the same scenario at a distance of 42 ft or smaller, which is
5.25 times greater than the value for the small charge. Performing this same
comparison for each of the common cases shows that the large charge results in a
failure stand-off distance that is 72 (in the case of the taller W14x82 column with the
vertical position of the charge being at the top of the column) to 4 (in the case of the
same column with the charge resting on the ground) times the failure stand-off distance
resulting from the small charge.
For each column, there is less variability in the relative failure stand-off
distances at the base of the column and increasing variability with increasing height of
the vertical position of the charge. This is believed to be due to two issues regarding
reflection of the blast pressures from the ground profile. First, for the small charges,
there appears to be essentially no contribution of reflected pressures when the vertical
position of the blast is in line with the top of the column as the failure stand-off distance
is 3’ in all of these cases. Secondly, the contributions of reflected pressure significantly
134
increase the applied pressures for the large charges as the failure stand-off distance
increases with increasing vertical position of the charge in these cases. In other words
the failure stand-off distance is at a minimum at the top of the column for the small
charges, but is at a maximum at this location for the large charges, resulting in the
greatest variability at this location.
Since the small charge has to be much closer to the column in order to fail it, the
effects of the pressure distribution are much more localized in these cases. The
maximum pressure on the column from the small charge occurs at the same height from
which the blast was initiated. For example, if the small charge was placed at mid-height
of the column, the maximum pressure would be present at mid-height of the column.
Because of the much greater failure distances of the large charge, its effects on the
column are much more global, being applied as a more uniformly distributed load.
6.2.2.2
Vertical Position of Charge
The sensitivity of the blast effects to the vertical position of the charge is shown
to have differing effects, depending on the charge size. The stand-off distance is seen to
increase greatly as the large charge size is moved away from the ground, while there is
less sensitivity to this parameter and the maximum damage occurs when the charge is
aligned with the mid-height of the column in the small charge cases. The large charge
shows greater sensitivity because as the charge center moves away from the ground (for
the range of charge positions considered herein) it creates larger mach wave effects,
which are related to the reflection of the blast waves from the ground profile. However,
135
if the height from which the charge was initiated was continually increased, eventually
the mach wave effects would dissipate. The small charge does not have the same mach
wave effects present because the stand-off distance is generally so small that the
pressure wave reaches the column before it can interact with the ground.
6.2.2.3
Column Height
Columns at heights of 14 ft and 18 ft were analyzed in order to gauge the effects
of this parameter on blast resistance. The failure contours for the W14x82 column (as
an example) are shown graphically in Figures 6.1 and 6.2 to illustrate the effects of
column height. Similar trends exist for the other two columns. The vertical axis of the
figures expresses the vertical position of the charge as a fraction of the column height,
such that columns of different heights can be easily compared on the same plot.
136
Veritcal Postion/Column Height
1
0.5
0
0
5
10
15
Stand-off Distance (ft)
14' Height
18' Height
Figure 6.1 – Failure Contours for W14x82 C
Column (Both Heights, Small
S
Charge)
It can be observed from Figure 6.1 that as the column size was increased to 18 ft
from 14 ft, the column was generally less resistant to the blast loading. The same result
also occurs for the W10x77 and W14x109 columns subjected to the small charge.
Comparing the two varying heights for each small blast scenario shows that there is
either no change in failure stand-off distance (for the charges vertically aligned with the
top of the column) or there is up to a 50% increase in stand-off failure distance.
For the large charge size (see Figure 6.2), the W14x82 column was also less
resistant to the blast loading as its height was increased. This behavior also exists for
the other
her two column cross
cross-sections. The variation in failure stand-off distance as a
function of column height for the large charges is between 22 and 83%.
137
Vertical Position/Column Height
1
0.5
0
0
50
100
150
200
250
Stand-off Distance (ft)
14' Height
18' Height
Figure 6.2 – Failure Contours for W14x82 Column (Both Heights, Large Charge)
C
Even though the axial loa
loading
ding was reduced to account for the increased column
height, these results show taller columns are more vulnerable to blast loading. This is
because as the column height increases, flexural stiffness decreases and consequently
the influence of bending eff
effects
ects increase. The difference in stiffness between the two
column heights of a given column cross
cross-section
section is only a function of height. If the ratio
of horizontal displacement resulting from a uniform lateral pressure on each fixed-end
fixed
column is calculated,
ed, it can be seen that the 14
14-ft
ft column is 2.73 times stiffer than the
18 ft column (found by 184/144).
138
6.2.2.4
Column Depth
W14x82 and W10x77 columns were used because they have similar axial
capacities but different web depths (10.6 in. and 14.3 in.) an
and,
d, therefore, surface areas
exposed to blast pressure. The failure contours for the 14 ft columns exposed to the
small charge can be observed in Figure 6.3 followed by the same data for the 18 ft
columns in Figure 6.4. The failure contours found for the large charge can be seen in
Figures 6.5 and 6.6, for the 14 ft and 18 ft columns, respectively. The W14x109
column, which will be discussed in Section 6.2.2.5, is also shown in these figures for
later reference.
Column Height (ft)
14
7
0
0
5
10
15
Stand-off Distance (ft)
W10x77
W14x82
W14x109
Figure 6.3 – Failure C
Contours for 14-ft Columns
olumns (Small Charge)
C
139
Column Height (ft)
18
9
0
0
5
10
15
Stand-off Distance (ft)
W10x77
W14x82
W14x109
Figure 6.4 – Failure contours for 18-ft Columns (Small Charge)
harge)
From the results in Figures 6.3 and 6.4 it can be seen that for the small charge,
the W10x77 column is generally more resistant to blast loading than the W14x82
column. The exception to this is when the charge is placed at the top of the column, in
which case the two cross
cross-sections
sections have the same resistance. For the other locations, the
variation in failure stand
stand-off distances ranges between 60 and 88%.
140
Column Height (ft)
14
7
0
0
50
100
150
200
Stand-off Distance (ft)
W10x77
W14x82
W14x109
Figure 6.5 – Failure C
Contours for the 14-ft
ft Columns (Large Charge)
C
141
Column Height (ft)
18
9
0
0
50
100
150
200
250
Stand-off Distance (ft)
W10x77
W14x82
W14x109
Figure 6.6 – Failure C
Contours for the 18-ft
ft Columns (Large Charge)
C
The failure contours for the columns exposed to the large charge also show that
the W10x77 column is more resistant to the blast load
loadings
ings than the W14x82 column.
This conclusion is fairly intuitive as the two columns are supporting similar axial
loadings (within 5%) but the W14x82 column is exposed to a larger blast loading. This
larger blast loading occurs because the web for the W14
W14x82
x82 column is 35% deeper,
giving the pressure this same increased amount to act upon. The decreased surface area
results in the failure stand
stand-off distances of the W10x77 being 13 to 35% less than those
for the W14x82 for the large charge size. Thus, this is shown to be less influential for
the large charge size than for the small charge discussed above.
142
6.2.2.5
Column Stiffness
The W14x82 and W14x109 columns have similar areas exposed to the blast
(shown to be influential in the last section) but different lateral stiffnesses. The
W14x109 column is 3.02 times stiffer in the weak direction, as this is the ratio of their
moment of inertias about the weak axis. The failure contours for these two columns, at
both heights and for both charges, are shown above in Figures 6.3 through 6.6. From
these contours it can be seen that the W14x109 column is generally more resistant to
blast loading than the W14x82 column, with the only exception being equal failure
stand-off distances when the small charge is vertically aligned with the top of the
column.
There are several differences between the W14x82 and W14x109 columns that
affect these results. Recalling that the failure criteria is based on when the applied
moments exceed the flexural capacity of the section, and that secondary effects are
assumed to be the cause of moment, differences in stiffness, moment capacity, and
applied axial force all contribute to the differing failure stand-off differences. The
W14x109 column carries 1.30 times more axial loading and, as mentioned above, is
3.02 times stiffer. For the sake of argument, consider that both of the columns are
exposed to the same charge size and stand-off distance, and that as a result the W14x82
column deflects 1.00 distance units. If the W14x82 column is under an axial loading of
1.0 force units, the total secondary moment of the section will be 1.00*1.0 = 1.00
moment units. Similarly, the W14x109 column under the same blast loading would
deflect 0.33 distance units and under an axial loading of 1.3 force units would have a
143
secondary moment of 0.33*1.3 = 0.43. Therefore, the W14x82 column would have a
2.33 times higher secondary moment. Even if these two sections had the same moment
capacity, which they don’t, the W14x109 would inherently be more resistance due to an
increased stiffness. This example shows that the effects of the increased stiffness far
out-weigh the increased axial loading and the secondary effects are, overall, smaller.
As the W14x109 has a 40% higher nominal capacity in the weak direction, it is allowed
to deflect farther than the W14x82 before it fails; only providing more resistance.
These results show that the stiffness parameter, as well as strength parameter, has a very
direct influence on the blast resistance of columns.
6.2.2.6
Parametric Summary
The parameters varied in this analysis were column depth, column stiffness,
column height, charge size, and vertical position and stand-off distance of the charge.
The column depth was investigated by evaluating results between the W14x82 and
W10x77 column, in which it was found that the shallower column (W10x77) was more
resistant to the blast loading due to its decreased cross-section (35%). The influence of
column stiffness was found by observing results between the W14x82 and W14x109
column. It was found that the W14x109 had an increased capacity for blast resistance
due to its increase stiffness (3 times) and plastic moment capacity (1.4 to 1.5 times).
Column height was also shown to be very influential, as the 14-ft columns were more
blast resistant because they were much stiffer than the 18-ft columns (2.73 times) and
less vulnerable to second order effects. Two charge sizes (small and large) were placed
144
at three vertical positions along the column (bottom, mid-height, and top) from which a
failure stand-off distance was found to the nearest 1 ft. These formed “failure contours”
which illustrated the effects of stand-off distance and vertical position for the two
charge sizes.
6.2.3
Blast Threat Level Representation of Missing Column Analysis
One of the purposes of this analysis was to determine what blast threats are
represented by a missing column analysis as prescribed in progressive collapse design
codes. For example, if the small or large charge size was applied at a given distance to
a typical column, is this representative of the missing column analysis in that one and
only one column fails? This can be determined from the failure contours, for the
columns considered in this work. This is done by using the failure stand-off distances
in combination with the column spacing to determine the number of columns that will
be failed. For example, if two columns were spaced 30 ft apart, which is a typical
column spacing in a building, and the failure stand-off distance for the columns is equal
to or greater than 15 ft (and located between two columns) then it is likely that two or
more columns will fail due to the blast. Using this reasoning, it is suggested that onehalf the column spacing should be considered to be the maximum failure stand-off
distance that is representative of a missing column analysis. It is also noted that the
results herein show that there can be significant variations in these contours if other
column cross-sections are considered and the discussion of the results presented below
should be considered with this fact in mind.
145
If the small charge size is considered, the failure stand-off distance for the steel
columns considered in this thesis is 3-15 ft, depending on the vertical location of the
blast. Because the maximum failure stand-off distance occurred when the vertical
position of the charge was aligned with the mid-height of the column, these ranges
should capture the maximum for all possible vertical positions of the charge. Therefore,
if a column spacing of at least 30 ft is used (in combination with the columns
considered herein) the missing column analysis is considered to be representative of a
blast threat of this charge size at any stand-off distance. The missing column analysis
would also be representative of this charge size if a smaller column spacing was used in
conjunction with the 14 ft columns or the 18 ft W10x77 or W14x109 columns.
For the large charge, the stand-off distances vary greatly depending on the
vertical position of the blast. For the charges resting on the ground, the failure stand-off
distances range between 23 and 52 ft; for the other positions these distances range
between 54 and 216 ft. Using the rationale described above, this would mean that a
column spacing of 46 to 432 ft would need to be provided for a missing column analysis
to be representative of this charge size. Because this is not a practical range of column
spacings, the missing girder analysis should not be considered to be representative of
large charge sizes and would be unconservative in this application. If a large charge
size is deemed to be a design consideration, the designer should perform a threat
dependent design rather than an alternate path analysis. Furthermore, while the
deviation in the results with column geometry has been discussed, the column spacings
needed in order for a large charge to fail only one column are so large that it is not
146
believed that changes in column cross-section or length would result in a situation
where a missing column analysis would be representative of a large charge size.
Alternatively, it is suggested that the failure contours such as those given above could
be used in combination with the event control method by giving guidance on the standoff perimeter that would need to be provided if this method were utilized. However,
additional column geometries should be investigated for these to be generally used in
such an application.
In conclusion, the failure contours developed herein generally support the use of
a missing girder analysis for small charge sizes and show that this type of analysis is not
representative of the failure that would be caused by large charge sizes. However, if a
small charge were to actually occur there could be additional consequences beyond the
failure of a single column. These include the fact that adjacent members and
connections could also be subjected to the blast load and while these members may not
fail, this may reduce the capacity available for carrying the redistributed loads in the
structure’s damaged state. This is a fact ignored in both the current codes as well as the
results presented herein.
Lastly, it should be noted that the failure contours presented in this thesis vary
from those previously published in Brown and McConnell (2009). The failure contours
in this thesis are different as they include more rational material parameters in the
Johnson-Cook model, mach-wave effects from blast, and different size charges.
147
6.3
Missing Girder Analysis Results
The second objective was to determine what redundancy and system load
carrying capacity exists in simple-span steel girders bridges in the event of a failed
girder as a result of the reserve strength of the system inherent to the design process.
This was accomplished by developing and carrying out a missing girder analysis for a
simple-span steel girder bridge. To do this, two bridges having span lengths of 50 ft
and 100 ft were designed and girders were removed, evenly redistributing the service
loading they carried to the remaining girders in the bridge. The girder spacing is
assumed to be 8 ft; it was shown in Section 3.3.6 that there is a negligible effect due to
variable girder spacing. The results can be presented independent of number of girders
in the cross-section because each girder was assumed to carry the same service loading
and have the same capacity. Thus, the number of girders that must remain in the crosssection to carry the service loading from the removed members can be found. The
specifics on the formulation of the service loadings and the girder sizes used for both
bridges and the how the procedure is carried out are shown in Section 3.3 of this thesis.
The results from removing one or two girders for each span length are shown in the
following sections. This is followed by discussion on the redundancy and system load
carrying capacity that exists in simple-span steel girder bridges.
6.3.1
Span Length of 50 ft
The 50-ft span bridge consists of W30x90 girders spaced at 8 ft. Each girder
was determined to carry a service load of either 1323.9 k*ft (including impact) or
148
1166.0 k*ft (without impact), and to have a nominal capacity of 2280.1 k*ft. Therefore,
the percentage of nominal capacity of the girders that is being used by the service
loading is 58% (including impact) or 51% (without impact). The reserve capacity of the
girders can be found to be 956.0 k*ft if impact is included and 1114.1 k*ft without
impact. The reserve capacity is calculated by taking the capacity and subtracting the
service loading.
If a single girder were to be removed, it is assumed that its service loading
would be transferred to the remaining girders evenly, as discussed in Section 3.3.1. If
impact is not considered, a service loading of 1166.0 k*ft would have to be carried in
order for the bridge to remain stable. Since the reserve capacity of the girders is known
to be 1114.1 k*ft, then precisely 1.05 additional girders are needed in order for the
bridge to survive. This is found by taking the load redistributed from the failed girder
and dividing by the girder reserve capacity. Since a fraction of remaining girders
cannot be provided, the number must be rounded up (i.e. – 1.05 additional girders is 2).
If impact is considered, 1323.9 k*ft of service loading is redistributed to the remaining
girders, each of which has a reserve capacity of 956.0 k*ft. Therefore, the number of
additional girders needed in the cross-section is precisely 1.38 in order for the bridge to
survive. When impact is included into the calculation it is seen to increase the service
loading (1166.0 k*ft to 1329.9 k*ft) and reduce the reserve capacity (1114.1 to 956.0
k*ft) resulting in more additional girders being required.
The calculations considering two girders removed are performed similarly. The
only difference is now the remaining girders in the cross-section must withstand twice
149
the loading of the one girder removed situation; specifically 2332.0 k*ft (without
impact) and 2647.8 k*ft (with impact). The remaining girders needed in order for the
bridge to survive are then 2.09 (without impact) and 2.77 (with impact), which is again
precisely twice those needed for the single girder removal case. These results are
summarized in Table 6.3.
Table 6.3 – Summary of Missing Girder Analysis for 50-ft Span
Number
of
Removed
Girders
6.3.2
Distributed Service
Loading (k*ft)
Girder Reserve
Capacity (k*ft)
Additional Girders
Required
w/
Impact
w/o
Impact
w/
Impact
w/o
Impact
w/
Impact
w/o
Impact
1
1323.9
1166.0
956.2
1114.1
1.38
1.05
2
2647.9
2331.9
956.2
1114.1
2.77
2.09
Span Length of 100 ft
The same analysis that was carried out for the 50 ft span bridge was carried out
for the 100-ft span. The service loading for the removed girder(s) was divided by the
reserve capacity of a single girder to find the number of additional girders needed. The
100 ft span bridge consisted of W44x262 girders spaced at 8 ft. Each girder was
determined to carry a service load of either 4143.4 k*ft (with impact) or 3832.7 k*ft
(without impact), and to have a nominal capacity of 6813.7 k*ft. The percentage of
nominal capacity that was used by the service loading in each girder was 61% including
impact or 56% without impact. These reserve capacity percentages are less than those
150
found for the 50-ft span length as the percentage of dead loads are higher, which use
lower load factors and so therefore the design strength will more closely match the
service loading. Carrying out the analysis (in the manner described for the 50-ft span
length above) gives the results shown in Table 6.4.
Table 6.4 – Summary of Missing Girder Analysis for 100-ft span
Number
of
Removed
Girders
Distributed Service
Loading (k*ft)
Girder Reserve
Capacity (k*ft)
Additional Girders
Required
w/ Impact
w/o
Impact
w/ Impact
w/o
Impact
w/ Impact
w/o
Impact
1
4143.4
3832.7
2670.2
2981.0
1.55
1.29
2
8286.9
7665.4
2670.2
2981.0
3.10
2.57
From Table 6.4 it can be seen that 1.29 (without impact) or 1.55 (considering
impact) additional girders are required if a single girder is removed. Also, 2.57
(without impact) or 3.10 (considering impact) additional girders are required if two
girders are removed, which is precisely twice those needed for the single girder removal
case. The number of additional girders needed for the 100 ft span is seen to be greater
than the number needed for the 50 ft span. This is because the 100 ft span girders have
a smaller percentage of reserve capacity available. This is because, as previously
discussed, the ratio of dead load to live load is higher for the longer span.
151
6.3.3
Conclusions from Missing Girder Analysis
The missing girder analysis can be used as a measure of system load carrying
capacity. This load carrying capacity, if considered with the number of girders in the
bridge, can give the redundancy present in the bridge. From the results above it can be
seen that, when the bridges considered in this thesis had a single girder removed, they
required between 1.05 and 1.55 additional girders to safely handle service loads
(depending on span length and whether impact is applied to the vehicle loadings). This
indicates that a minimum number of three girders should be required in the crosssection, which happens to be the minimum amount that would be provided for a bridge
to be considered as redundant. The result of three girders is obtained by rounding up
1.05 and 1.55 girders to two girders, as fractions of girders cannot be provided, and also
including the failed girder as an original part of the cross-section.
If the case of two girders being removed is considered, it can be seen that more
additional girders are required for the bridge to survive. When the bridges in this thesis
had two girders removed, they required 2.09 to 3.10 additional girders. By counting the
two initial girders that failed and rounding up the additional girders (as fractions of
girders cannot be provided) this results in a five or six girder bridge, depending on the
span length of the bridge and whether impact is included.
The effects of including impact from the truck loading are shown to be a
significant factor in calculating the additional girders needed to mitigate collapse.
Including impact into the service loading was shown to increase the amount of girders
needed by 32% and 21% for the 50 ft and 100 ft span bridges, respectively. This
152
increase in the loading required the 100-ft span bridge with two girders removed to need
an extra girder, while the remaining cases did not need additional girders. The percent
increase was lower for the 100-ft span bridge because the live load effects from the
truck loading were proportionately smaller in this case (compared to dead load effects).
While it is unconservative to entirely ignore the effects of impact, the actual effects are
likely lower than those predicted by the AASHTO (2007) impact factor, so the results
with and without impact can be regarded as bounds to the actual conditions, realizing
that these actual conditions are most likely closer to the results with impact included.
The 100-ft span conditions were seen to require 12% (when impact is
considered) to 23% (neglecting impact) more additional girders for the removal
scenarios because the dead load accounted for a higher percentage of the total loading in
the longer span, and since the load factors associated with it are smaller, the reserve
capacity of the girders is not as high. By carrying out these calculations in this thesis, it
quantified the fact that shorter spans are likely to contain more reserve capacity against
service loads.
One assumption that was made in this analysis is that the load from the removed
girder was spread equally to the remaining girders in the system. This pattern of
redistribution represents the ultimate capacity of the structure, but it is not necessarily
the case that sufficient connectivity between girders exists for this ultimate capacity to
be achieved. Achieving the ultimate capacity will be dependent upon the capacity of
the concrete deck, the robustness of the connection between the deck and the girders,
and the strength of the lateral bracing members. Additional research is needed to
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address these issues, although the work of Bechtel (2008) provides some insight on this.
This work shows that if the deck and lateral bracing components do not have sufficient
strength to permit load redistribution, they will fail prior to girders reaching their
ultimate capacity and limit the system capacity of the structure. Thus, the results
presented herein have the limitation that, while the main members of the bridge would
have adequate capacity for the conditions given, components of decks and lateral
bracing members may not, which will prevent all girders from equally participating in
load sharing. The consequence of this is that the bridge will not withstand the abnormal
loading regardless of the number of girders in the cross-section. Thus, additional
research on the strength of the deck and cross-frame components under ultimate
loadings is an important area for future research.
Additional assumptions that were made that may affect the results presented
herein are related to the applied loads and capacity of the members. In this analysis the
girders were assumed to be carrying their service loading prescribed by AASHTO. The
loading actually seen by the girders may be different in reality, but these values are
believed to be conservative. The nominal capacity of the sections in the analysis was
also the plastic moment capacity as prescribed in AASHTO, which is believed to be an
accurate assumption.
If the missing girder analysis is used as a measure of system load carrying
capacity, common simple-span steel girders can be seen as very redundant structures in
that they contain a lot of reserve strength due to the design process. In the case of
removing one girder from the bridges considered in this thesis and redistributing its
154
service loading (with or without impact), the remaining girders in a bridge consisting of
at least three girders are seen to avoid failure. Similarly, if two girders are removed
from a bridge consisting of a minimum of six girders, it will not collapse. Thus, it is
shown that a structure does not need to contain a large number of members to achieve
redundancy.
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Chapter 7
SUMMARY AND CONCLUSIONS
7.1
Summary and Conclusions of Missing Column Analysis
In an attempt to quantify the threat level represented by missing column analysis
as prescribed in progressive collapse design codes, typical steel columns were modeled
in Ls-Dyna and were exposed to selected charge sizes to determine the range of standoff distances that caused failure in these members. Three column cross-sections were
evaluated, W14x82, W10x77, and W14x109, which were each considered at two
heights (14 ft and 18 ft). Each column was subjected to small and large charges placed
at three different heights above ground. The heights were: resting on the ground,
centered at mid-height of the column, and centered at the top of the column. By varying
each of these parameters in the analysis, the effects of column height, column stiffness,
column depth, charge size, and vertical position of the charge were each able to be
examined.
The results presented herein are insightful into the effects of stiffness, height,
and depth of a column on blast resistance. A column’s height is shown to be a critical
parameter affecting a column’s blast resistance, as taller columns are shown to be much
less resistant for a given cross-section at both charge sizes considered. Also, column
web depth is a factor that consistently affected blast resistance; considering two
columns with similar axial capacity and height, the column with the smaller depth will
156
be more resistant because it has less area for the blast pressure to act upon and thus
experiences lower loads. The results also show that if two columns have similar depth
and height, the stiffer member will provide more blast resistance. While all of these
results are congruent with intuition, this work also serves to quantify these influences
for a select set of columns. This work was previously detailed in Section 6.2.2.
While there is variability in the stand-off distance causing failure as a result of
these parameters, the stand-off distances varied most dramatically as a function of
charge size. This result was also somewhat expected given that a significant (2000%)
variation in charge size was considered. Thus, separate conclusions were suggested for
the differing charge sizes. For the small charge, the failure stand-off distances
(meaning that if the failure stand-off distance is x, any stand-off distance between 0 and
x will cause failure) found for the columns in this thesis were 3 to 15 ft. The failure
stand-off distances associated with the large charge were 23 to 216 ft.
These distances were considered relative to typical column spacings to assess
whether the missing column analysis was representative of a blast threat in that the blast
threat caused the failure of one and only one column. For this consideration the
rationale used was that if the failure stand-off distance was less than half of the column
spacing, then the results would be considered representative of a missing column
analysis in that there would be no possible charge location that would cause failure of
more than one member.
Considering a typical spacing of columns in a building of 30 ft, the small charge
is thus generally representative of a missing column analysis. This conclusion is
157
reached because in only one case is the failure stand-off distance not less than this
column spacing (and in this case it is equal to the column spacing). Therefore, the
missing column analysis, considering its relative simplicity, is considered appropriate
for designing structures for relatively small charge sizes for columns of the general
sizes considered herein. Given the variation in failure stand-off distance with height,
stiffness, and depth discussed above, these results should be conservative for shorter,
less deep, or stiffer columns and the converse is also true.
The large charge results in failure stand-off distances that would cause the
failure of multiple columns. That is, for the average failure stand-off distance of 98 ft
(for example) resulting from these analyses, if the charge is within 98 ft of a given
column and the column spacing of 30 ft is again assumed, up to three columns in a
single direction could be failed. Thus, an analysis based on the removal of only one
column is highly unconservative for designing for the possible effects of large charge
sizes. Thus, it is concluded that the missing column analysis is generally representative
of relatively small charge sizes, but not large charge sizes.
7.2
Summary and Conclusions of Missing Girder Analysis
In a separate analysis, the concept of a missing column analysis (as presently
used for buildings) was conceived and applied to bridges to determine the system load
carrying capacity built into steel simple-span girder bridges through the design process.
In this analysis the cases of removing one girder or two girders were considered. After
the girder(s) was (were) removed, the service loading that they carried was distributed
158
evenly to the remaining girders in the cross-section. By knowing the reserve capacity
of the remaining girders compared to their service loads, failure can be assessed as a
function of the number of girders in the cross-section.
Service loads were applied to each bridge by making use of effective load
factors. These factors discount the inevitable situation that extra capacity is built into
each member of a structure because it is extremely unlikely that a section can be
selected whose capacity exactly matches its ultimate design load. The result of this is
that the applied service loadings are slightly increased to exactly match the capacity of
the structure, while the original ratio between dead and live load is maintained. Section
3.3.5 provides a full discussion of these effective load factors. One consideration that
was necessary in this process is the influence of the dynamic loading induced by truck
traffic, which is represented by an impact factor in AASHTO (2007) specifications.
This factor is not a “load factor” in that conservatively amplifies loads, but represents a
real load effect. However, it is obviously an upper-bound of the real dynamic effect.
Thus results are presented for the bounding cases of no impact and the impact factor of
1.33 applied to the service truck loading to investigate this influence. However, it is
suggested that actuality is most closely represented by the cases with the impact factor
applied.
Bridges having 50 ft and 100 ft span lengths and three different girder spacings
(6 ft, 8 ft, and 10 ft), were designed. However, the results showed only a 0 to 2%
influence of girder spacing, thus detailed analysis was only performed for the
intermediate (8 ft) girder spacing. Similarly, interior and exterior girders were designed
159
for the two bridges, but after discovering their capacity and service loadings (by use of
effective load factors, described below) were almost identical, the analysis considered
these to be the same for simplification.
The results for this analysis showed that when a single girder was removed, 1.05
to 1.55 additional girders were needed to carry the service loads, depending on the span
length of the bridge and whether impact was included. When the case of two girders
removed was considered, the number of additional girders required doubled to be
between 2.09 and 3.10, which was also dependent on span length and impact. When
impact was included into the analysis, it increased the service loading, which decreased
the reserve capacity, thereby requiring additional girders in the cross-section in order
for the system to have enough capacity to withstand the girder loss(es). The 100 ft span
bridge was seen to require more additional girders because the reserve capacity of these
girders was proportionately less than those of the 50 ft span due to the difference in live
to dead load ratios inherent to bridges of different span lengths. These results are
summarized in Tables 6.3 and 6.4.
If the missing girder analysis is used to assess bridge system load carrying
capacity, common simple-span steel girder bridges are shown to contain a lot of reserve
strength due to the design process. This reserve strength, combined with the fact that
these bridges usually contain at least three members makes them very redundant
structures. In the case of removing one girder from the bridges considered in this thesis
and redistributing its service loading (with or without impact), the remaining girders in
a three girder bridge are seen to have sufficient capacity to avoid failure. Similarly, if
160
two girders are removed from a bridge consisting of a minimum of six girders, the
girders will not collapse.
The reserve strength of each of the members are a direct consequence of how
much service loading was present, which in this analysis was assumed to be those
prescribed in AASHTO without any load factors applied. It is possible that the service
loading could be higher or lower, which would change the results herein. It is difficult
to gauge how much the service loads might fluctuate; however, it can be approximated
that the dead service loads considered will not fluctuate much while the live service
load will vary much more. If it is assumed that live loading is present during an
incident which causes damage to the bridge or damage to the bridge is the result of live
loading then the results are not expected to change significantly. A more significant
assumption is that the loading from a failed member would be redistributed to all of the
remaining girders. Achieving such redistribution relies on the strength of deck and
lateral bracing components, which should be a topic of further research.
7.3
Synthesis
In this thesis, the failure of columns as a result of given charge sizes at various
spatial positions was represented by “failure contours”. It is the author’s belief that
failure contours can be helpful to designers as a tool to determine how many members
of a structure will fail. For instance, the designer can place a charge (small, medium,
large, etc) at a location in the structure and then calculate its stand-off distance to the
adjacent members in the structure. If the stand-off distance is in the failure range given
161
by the particular member’s failure contours, the designer should consider that the
member will fail if a blast occurs at that given location. If the failure contours are to be
used for such a purpose, they would need to be expanded for different types of members
(beams, columns, etc) and for different blast sizes (small, medium, large, etc) acting in
different orientations on the member. This sort of procedure could prove to be a rather
simplistic way for designers (with little blast design experience) to more accurately
model damage to a structure than does the alternate load path method.
In the missing girder analysis method it was determined that simple-span girder
bridges contain a lot of reserve strength through the design process. The bridges with
the least reserve strength of those considered was ones with long spans, for which the
percentage of dead load was higher and therefore the design load was closer to the
service load. Long span bridges with only a few number of girders present could be
considered non-redundant as a result of this analysis. This combined with the fact that
structural material and construction methods of the future are becoming much more
precise, the designs become much sleeker and have a reduced amount of reserve
capacity. Such bridges could benefit from using the missing girder analysis. This
would further ensure that the structure is safe and would not fail due to the loss of one
member. Such an analysis is currently used in building design and has been shown to
be a rather simplistic way for designers to build reserve strength into the structure. In
order for this to become practical for engineers to carry out, three-dimensional analysis
techniques for bridges (as done for building design) need to advance in order for there
162
to also be proper consideration of the connecting elements (deck and cross-frame
components) of the girders in these analyses.
In this thesis it was shown that the alternate load path method, which is
primarily used as a method for mitigating progressive collapse potential in buildings,
can be applied rather simply to different types of structures. This was shown by
applying the method to a simple-span steel girder bridge, which proved to be an
insightful analysis into discovering the reserve capacity present in the bridge. In the
analysis for both bridges and buildings, the load from the removed or missing member
are easily distributed to adjacent members in which case the remaining structure can be
designed to have adequate strength. Both bridges and buildings, in theory, are able to
distribute these loadings although, perhaps, this might be more difficult for bridges to
accomplish. This is because bridges do not have as many load paths available as a
typical building and the damaged members (girders in this case) would in all likelihood
further damage bracing elements or adjacent members as they are closer together. The
alternate load path method, in lieu of more rigorous analysis and despite faults that have
been documented in literature, is still shown to be an effective, simple procedure that
can be used by lay engineers to assess structural system strength and safety for a variety
of structural systems, not only buildings.
163
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Randers-Pehrson, Glenn and Kenneth A. Bannister. (1997). Airblast Loading Model
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Righman, Jennifer. (2005). “Rotation Compatibility Approach to Moment
Redistribution for Design and Rating of Steel I-Girders”, Doctoral Dissertation,
West Virginia University, Morgantown, WV.
Rodin, J. (1969). “Statistical Design Against Progressive Collapse”, The Consulting
Engineer, London, England.
Ruth, P.; K. Marchand; and E. Williamson. (2006). “Static Equivalency in Progressive
Collapse Alternate Path Analysis: Reducing Conservatism While Retaining
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Sasani, M. and S. Sagiroglu. (2008). “Progressvie Collapse Resistance of Hotel San
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172
Scott, D.; B. Lane; and C. Gibbons. (2002). “Fire Induced Progressive Collapse”.
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173
APPENDIX
A.1 Johnson-Cook Parameter Results for Remaining Data Sets
As discussed in Section 3.2.1.3.2, this thesis used Johnson-Cook parameters that
were calculated from data sets provided by Righman (2005). An example on finding
the parameters was shown for one data set in Section 3.2.1.3.2.1.2, while the other
remaining six data sets are shown here. These data sets are followed by a summary of
the Johnson-Cook parameters determined for each.
Table A.1 – Data Set 1
σ - σy
log(ε)
log(σ - σy)
2.473
-5.896
0.905
0.021
5.850
-3.853
1.766
69.900
0.061
7.250
-2.803
1.981
0.13312
70.350
0.092
7.700
-2.385
2.041
0.40000
70.691
0.359
8.041
-1.024
2.085
ε
σ
εp
0.00000
0.000
0.00211
62.650
-0.039
0.000
0.04103
62.755
0.000
0.105
0.04375
65.123
0.003
0.06221
68.500
0.10164
-62.650
174
Table A.2 – Data Set 2
σ - σy
log(ε)
log(σ - σy)
6.321
-6.908
1.844
0.011
10.364
-4.510
2.338
80.106
0.031
15.579
-3.474
2.746
0.08000
82.921
0.051
18.394
-2.976
2.912
0.12000
84.980
0.091
20.453
-2.397
3.018
0.15000
85.186
0.121
20.659
-2.112
3.028
0.50000
85.751
0.471
21.224
-0.753
3.055
ε
σ
εp
0.00000
0.000
-64.527
0.00000
64.527
0.000
0.02000
65.381
0.854
0.03000
68.205
3.678
0.03000
70.848
0.001
0.04000
74.891
0.06000
175
Table A.3 – Data Set 3
σ - σy
log(ε)
log(σ - σy)
8.335
-6.908
2.120
0.011
12.080
-4.510
2.492
81.137
0.021
14.839
-3.863
2.697
0.06000
83.127
0.031
16.829
-3.474
2.823
0.08000
85.043
0.051
18.745
-2.976
2.931
0.09000
86.119
0.061
19.821
-2.797
2.987
0.13000
87.052
0.101
20.754
-2.293
3.033
0.40000
87.438
0.371
21.140
-0.992
3.051
εp
σ - σy
log(ε)
log(σ - σy)
ε
σ
εp
0.00000
0.000
-66.298
0.00000
66.298
0.000
0.02000
66.556
0.258
0.02000
68.487
2.189
0.03000
74.633
0.001
0.04000
78.378
0.05000
Table A.4 – Data Set 4
ε
σ
0.00000
0.000
-71.968
0.00240
71.968
0.000
0.02500
72.900
0.932
0.03100
76.600
0.001
4.632
-6.908
1.533
0.04800
82.200
0.018
10.232
-4.017
2.326
0.07200
86.150
0.042
14.182
-3.170
2.652
0.10000
88.050
0.070
16.082
-2.659
2.778
0.13000
88.557
0.100
16.589
-2.303
2.809
0.16000
88.601
0.130
16.633
-2.040
2.811
176
Table A.5 – Data Set 5
σ - σy
log(ε)
log(σ - σy)
2.189
-6.908
0.783
0.281
8.335
-1.269
2.120
εp
σ - σy
log(ε)
log(σ - σy)
ε
σ
εp
0.00000
0.000
0.00150
66.298
-0.098
0.000
0.02000
66.556
-0.079
0.258
0.10000
68.487
0.001
0.38000
74.633
-66.298
Table A.6 – Data Set 6
ε
σ
0.00000
0.000
0.00190
66.298
-0.017
0.000
0.00690
66.556
-0.012
0.258
0.02000
68.487
0.001
2.189
-6.908
0.783
0.10000
74.633
0.081
8.335
-2.513
2.120
0.30000
78.378
0.281
12.080
-1.269
2.492
-66.298
Table A.7 – Johnson-Cook Parameter Summary
Data Set
B
n
1
13.92
0.226
2
12.88
0.250
3
31.13
0.221
4
29.49
0.177
5
31.69
0.276
6
11.26
0.237
7
17.80
0.303
Average
21.17
0.241
177
A.2 LS-Dyna Keyword File for 14-ft W14x82 Column
*KEYWORD
*TITLE
cm-g-microsecond
*CONTROL_TERMINATION
$# endtim
50000.0000
*DATABASE_BINARY_D3PLOT
$#
dt
100
$
*DATABASE_BINARY_D3THDT
$#
dt
100
$
*DEFINE_CURVE
$#
lcid
1
$#
a1
0
30000
28E-6
$
*DEFINE_VECTOR
$#
vid
xt
yt
1
0
0
$
*LOAD_BLAST_ENHANCED
$#
bid
m
xbo
1
#
18.16
$#
cfm...
0
$#
gnid
gvid
1
1
$
*LOAD_BLAST_SEGMENT
$#
bid
n1
n2
n3
1
2582
2583
2593
1
2583
2584
2594
1
2584
2585
2595
1
2585
2586
2596
1
2586
2587
2597
----ETC---$
*MAT_JOHNSON_COOK
$
$ Material : Grade 50 Steel
$
$#
mid
ro
g
1
7.849
0.7615
$#
a
b
n
3.792E-3 1.459E-3
0.241
$#
cp
pc
spall
9.73E-6
-9
$#
d5
0.61
$
o1
0
1
1
zt
0
xh
0
yh
0
zh
1
ybo
#
zbo
#
tbo
30000
iunit
4
blast
4
pr
0.300000
m
1.03
d1
0.05
tm
1.0e6
d2
3.44
tr
300
d3
-2.12
epso
1E-6
d4
0.002
n4
2592
2593
2594
2595
2596
e
1.999
c
0.0327
it
178
*NODE
$#
nid
x
y
1
0
-12.827
2
0
-11.4018
3
0
-9.97656
4
0
-8.55133
5
0
-7.12611
----ETC---$
*PART
$#heading
flange
$#
pid
secid
mid
1
1
1
$
*SECTION_SHELL
$#
secid
elform
1
2
$#
t1
t2
t3
t4
2.1717
2.1717
2.1717
2.1717
$
*ELEMENT_SHELL
$#
eid
pid
n1
n2
n3
1
1
1
2
12
2
1
2
3
13
3
1
3
4
14
4
1
4
5
15
5
1
5
6
16
----ETC---$
*PART
$#heading
web
$#
pid
secid
mid
2
2
1
$
*SECTION_SHELL
$#
secid
elform
2
2
$#
t1
t2
t3
t4
1.2954
1.2954
1.2954
1.2954
$
*ELEMENT_SHELL
$#
eid
pid
n1
n2
n3
2539
2
10
2832
2856
2540
2
2832
2833
2857
2541
2
2833
2834
2858
2542
2
2834
2835
2859
2543
2
2835
2836
2860
----ETC---$
$ This is the unit load curve
$
$ Axial Loading
$
*LOAD_NODE_SET
$#
nsid
dof
lcid
sf
1
3
1 -.004801
179
z
0
0
0
0
0
n4
11
12
13
14
15
n4
20
2856
2857
2858
2859
*SET_NODE_LIST
$#
sid
1
$#
n1
n+1
2821
2822
2829
2830
9606
9607
9614
9615
9622
9905
11886
12169
12606
12609
12633
12636
$
$ Fixed Support - Bottom
$
*BOUNDARY_SPC_SET
$#
nid
cid
2
0
*SET_NODE_LIST
$#
sid
2
$#
n1
n+1
1
2
9
10
2838
2839
2846
2847
2854
9623
11604
11887
15471
15472
17455
17738
$
$ Fixed Support - Top
$
*BOUNDARY_SPC_SET
$#
nid
cid
1
0
$
*END
etc...
2823
9600
9608
9616
10188
12588
12612
12639
dofx
1
etc...
3
2832
2840
2848
9906
15465
15473
18021
dofx
1
2824
9601
9609
9617
10471
12591
12618
12642
2825
9602
9610
9618
10754
12594
12621
dofy
1
dofz
1
4
2833
2841
2849
10189
15466
16040
18304
5
2834
2842
2850
10472
15467
16323
dofy
1
dofz
0
180
2826
9603
9611
9619
11037
12597
12624
2827
9604
9612
9620
11320
12600
12627
2828
9605
9613
9621
11603
12603
12630
6
2835
2843
2851
10755
15468
16606
7
2836
2844
2852
11038
15469
16889
8
2837
2845
2853
11321
15470
17172