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ALTERNATE PATH METHOD IN PROGRESSIVE COLLAPSE ANALYSIS:
VARIATION OF DYNAMIC AND NON-LINEAR LOAD INCREASE FACTORS
APPROVED BY SUPERVISING COMMITTEE:
________________________________________ Manuel Diaz, Ph.D., P.E Chair
________________________________________
Jose Weissmann, Ph.D.
________________________________________ Mijia Yang, Ph.D.
Accepted: _________________________________________
Dean, Graduate School
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DEDICATION
This Thesis is dedicated to God, without him in my life, nothing is possible. I also want to dedicate this to my dear Mother and Father, who have given me more than I ever needed and have always supported me in all my endeavors.
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ALTERNATE PATH METHOD IN PROGRESSIVE COLLAPSE ANALYSIS:
VARIATION OF DYNAMIC AND NON-LINEAR LOAD INCREASE FACTORS
by
ALDO E. MCKAY, B.E.
THESIS Presented to the Graduate Faculty of
The University of Texas at San Antonio In partial Fulfillment Of the Requirements
For the Degree of
MASTER OF SCIENCE IN CIVIL ENGINEERING
THE UNIVERSITY OF TEXAS AT SAN ANTONIO College of Engineering
Department of Civil and Environmental Engineering August, 2008
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1454510
1454510 2008
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ACKNOWLEDGEMENTS
I would like to thank Dr. David Stevens and Mr. Kirk Marchand from Protection Engineering
Consultants for their support and guidance during this research. The input provided by them
helped greatly in the completion of this study. I also would like to thank Dr. Eric Williamson
and Daniel Williams (PhD candidate) at the University of Texas in Austin for their contributions
to this effort.
August, 2008
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ALTERNATE PATH METHOD IN PROGRESSIVE COLLAPSE ANALYSIS:
VARIATION OF DYNAMIC AND NON-LINEAR LOAD INCREASE FACTORS
Aldo E. McKay, M.S
The University of Texas at San Antonio, 2008
Supervising Professor: Manuel Diaz, Ph.D.
As a result of the increasing number of terrorist attacks registered against American
facilities in the United States or abroad, United States government agencies continue to improve
the design of their buildings to make them safer and less vulnerable to terrorist attacks. One of
the factors typically considered in designing safer buildings and structures, is their ability to
prevent total collapse after the loss of load-carrying components (Progressive Collapse) resulting
from a terrorist attack. The consequences of not having a building capable of reducing the
potential for progressive collapse could be catastrophic, as it was the case of the Oklahoma City
bombing in 1995 where 42% of the Alfred P. Murrah Federal Building was destroyed by
progressive collapse and only 4% by the explosion or blast. This attack claimed 168 lives and
left over 800 injured.
Over the last 10 years, two United States government agencies have developed guidelines
for the design of their structures to resist progressive collapse: 1. The General Services
Administration, Progressive Collapse Analysis and Design Guidelines, (GSA Guidelines) and
2. The Department of Defense Unified Facilities Criteria 4-023-03 Design of Buildings to
Resist Progressive Collapse (UFC 4-023-03). Within both approaches, the main direct design
procedure is the Alternate Path (AP) method, in which a structure is analyzed for collapse
potential after the removal of a column or section of wall. Different analytical procedures may
be used, including Linear Static (LS), Nonlinear Static (NLS), and Nonlinear Dynamic (NLD).
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v
Typically, NLD procedures give better and more accurate results, but are more
complicated and expensive. As a result, designers often choose static procedures, which tend to
be simpler, requiring less labor. As progressive collapse is a dynamic and nonlinear event, the
load cases for the static procedures require the use of factors to account for inertial and nonlinear
effects, similar to the approach used in ASCE Standard 41 Seismic Rehabilitation of Existing
Buildings (ASCE 41).
A number of inconsistencies have been indentified in the way the existing guidelines
applied dynamic and non-linear load factors to their static approaches. As part of an existing
effort to update the existing guidelines, this study used SAP2000 to perform several AP analyses
on a variety of Reinforced Concrete and Steel Moment Frame buildings to investigate the
magnitude and variation of the dynamic and non-linear load increase factors. The study
concluded that the factors in the existing guidelines tend to yield overly conservative results,
which often translate into expensive design and retrofits. This study indentified new load
increase factors and proposes a new approach to utilize these factors when performing AP
analyses for Progressive Collapse.
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TABLE OF CONTENTS
Acknowledgements........................................................................................................................ iii
List of Tables ............................................................................................................................... viii
List of Figures ................................................................................................................................ ix
CHAPTER 1: Introduction ........................................................................................................... 11
CHAPTER 2: Progressive Collapse............................................................................................. 13
U.S. Existing Guidelines for Design against Progressive Collapse.................................. 14
Design Approaches to Resist Progressive Collapse.......................................................... 15
Overview of GSA Guidelines ........................................................................................... 15
Overview of DoD Guidelines UFC 4-023-03................................................................... 16
CHAPTER 3: Procedures for the Alternate Path Method ............................................................ 18
CHAPTER 4: Inconsistencies of Existing Factors ....................................................................... 20
CHAPTER 5: Variation of Load and Dynamic Increase Factors Research Procedure ................ 23
3- Dimensional Analytical Models ................................................................................... 25
3-Dimmensinal Building Designs..................................................................................... 33
2-Dimmensional Models................................................................................................... 35
CHAPTER 6: Analysis Results and Data Analysis ...................................................................... 38
Analysis of Data from Linear Static AP Analyses (LIF).................................................. 39
Analysis of Data from Nonlinear Static Analysis (DIF)................................................... 46
CHAPTER 7: Proposed Procedures for Using LIFs and DIFs in Static AP Analyses ................ 50
Linear Static AP Procedure............................................................................................... 50
Non-Linear Static AP Procedure ...................................................................................... 52
CHAPTER 8: Conclusions and Recommendations...................................................................... 55
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BIBLIOGRAPHY......................................................................................................................... 56
Appendix - RESULT TABLES .................................................................................................... 57
VITA
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viii
LIST OF TABLES
Table 1. Study Matrix of AP Analyses Performed ....................................................... 34
Table 2. Design Loads .................................................................................................. 34
Table 3. Steel Buildings, Material Properties ............................................................... 35
Table 4. Reinforced Concrete Buildings, Material Properties ...................................... 35
Table 5. 2-Dimmensionals Analyses ............................................................................ 37
Table 6. Steel Building, 10-Story - 25 ft Bay, Interior Column Removal .................... 38
Table 7. Reinforced Concrete Building, 3-Story-25 ft, Interior Column Removal ..... 39
Table 8 LIF Data for RC Sections ............................................................................... 45
Table 9 LIF Data for Steel Sections............................................................................. 45
Table 10. Complete Results for RC 3-Dimmensional AP Analyses............................... 60
Table 11. Complete Results, Steel Building 3-Dimmensional AP Analysis .................. 76
Table 12. Complete Results, RC Double Span Beams, SDOF Analyses ....................... 78
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ix
LIST OF FIGURES
Figure 1. Ronan Point Collapse...................................................................................... 14
Figure 2. Procedure to Determine Load Increase Factors. ............................................. 24
Figure 3. Steel Frame Building Hinge Definition (ASCE 41). ...................................... 26
Figure 4. Reinforced Concrete Hinge Definition. .......................................................... 27
Figure 5. NLD Analysis Procedure. ............................................................................... 28
Figure 6. Results of NLD Procedure. ............................................................................. 29
Figure 7. Stage Construction Setup................................................................................ 30
Figure 8. NLS Analysis Stage 1. .................................................................................... 30
Figure 9. NLS Stage 2, Load Around Loss Location..................................................... 31
Figure 10. NLS Stage 3, Nonlinear Analysis of Structure Response............................... 31
Figure 11. Typical Floor Plan........................................................................................... 33
Figure 12. 2-Dimmensional Models................................................................................. 36
Figure 13. LIF vs Plastic Rotation for RC sections.......................................................... 40
Figure 14. RC Sections Strong Dependence on Stiffness ................................................ 41
Figure 15. LIF vs. Total Rotation for Steel Sections........................................................ 41
Figure 16. Normalized LIF for RC sections..................................................................... 42
Figure 17. Normalized LIF for Steel Sections ................................................................. 43
Figure 18. DIFs for RC Buildings .................................................................................... 46
Figure 19. DIFs for Steel Buildings ................................................................................. 47
Figure 20 . Normalized DIFs for RC Buildings ................................................................ 48
Figure 21. Normalized DIFs for Steel Buildings ............................................................. 48
Figure 22. LIF for LS Analysis ........................................................................................ 51
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Figure 23. DIF for NLS Analysis..................................................................................... 53
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CHAPTER ONE: INTRODUCTION
As a result of the increasing number of terrorist attacks registered against American
facilities in the United States or abroad, United States government agencies continue to improve
the design of their buildings to make them safer and less vulnerable to terrorist attacks. One of
the factors typically considered in designing safer buildings and structures, is their ability to
prevent total collapse after the loss of load-carrying components (Progressive Collapse) resulting
from a terrorist attack. The consequences of not having a building capable of reducing the
potential for progressive collapse could be catastrophic, as it was the case of the Oklahoma City
bombing in 1995 where 42% of the Alfred P. Murrah Federal Building was destroyed by
progressive collapse and only 4% by the explosion or blast. This attack claimed 168 lives and
left over 800 injured.
Over the last 10 years, two United States government agencies have developed guidelines
for the design of their structures to resist progressive collapse: 1. The General Services
Administration, Progressive Collapse Analysis and Design Guidelines, (GSA Guidelines) and
2. The Department of Defense Unified Facilities Criteria 4-023-03 Design of Buildings to
Resist Progressive Collapse (UFC 4-023-03). Although both documents incorporate some of
the same approaches, there are notable differences in the application of these procedures. Within
both approaches, the main direct design procedure is the Alternate Path (AP) method, in which a
structure is analyzed for collapse potential after the removal of a column or section of wall.
Different analytical procedures may be used, including Linear Static (LS), Nonlinear Static
(NLS), and Nonlinear Dynamic (NLD). Typically, NLD procedures give better and more
accurate results, but are more complicated and expensive. As a result, designers often choose
static procedures, which tend to be simpler, requiring less labor. As progressive collapse is a
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12
dynamic and nonlinear event, the load cases for the static procedures require the use of factors to
account for inertial and nonlinear effects, similar to the approach used in ASCE Standard 41
Seismic Rehabilitation of Existing Buildings (ASCE 41). It is important that design
requirements for progressive collapse incorporate appropriate dynamic and nonlinear factors
such that the linear static and nonlinear static designs are more representative of the actual
nonlinear and dynamic response of the structure.
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CHAPTER 2: PROGRESSIVE COLLAPSE
Progressive collapse is defined in the commentary of the American Society of Civil
Engineers Standard 7-05 Minimum Design Loads for Buildings and Other Structures (ASCE 7-
05) as the spread of an initial local failure from element to element, eventually resulting in the
collapse of an entire structure or a disproportionately large part of it.
There have been a number of progressive collapse failures where the above definition can
be clearly observed. Probably one of the most famous progressive collapse failures was the 1968
collapse of the Ronan Point apartment building. The building was a 22-story precast concrete
bearing wall system. An explosion in a corner kitchen on the 18th floor blew out the exterior
wall panel and failure of the corner bay propagated up and down to cover almost the complete
height of the building. Figure 1 illustrates the final state of the collapse of the Ronan Point
apartment building. After this event, England was the first nation to address progressive collapse
explicitly in their building standards. Another famous case of progressive collapse was the
Oklahoma City bombing in 1995 of the Alfred P. Murrah Federal Building mentioned
previously.
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14
Figure 1. Ronan Point Collapse It is important to point out that, as stated in (Ellingwood, Smilowitz, Dusenberry,
Duthinh, Carino, 2006) there have been numerous cases of progressive collapse of buildings
during construction and data suggest that buildings under construction have a higher probability
of sustaining collapse. However, the design approaches against progressive collapse mentioned
in this paper relate only to progressive collapse of finished buildings in service.
U.S. Existing Guidelines for Design against Progressive Collapse
Existing U.S building codes do not address progressive collapse explicitly. Standards
such as ASCE 7 and ACI-318 include references to improve structural integrity but do not
provide quantifiable or enforceable requirements to resist progressive collapse. Only two U.S
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15
agencies have developed guidelines that provide quantifiable and prescriptive requirements to
reduce the potential for progressive collapse. These guidelines are: The General Services
Administration Progressive Collapse Analysis and Design Guidelines 2003 and The Unified
Facilities Criteria Design of Buildings to Resist Progressive Collapse 2005 by the Department of
Defense ( DoD).
Currently, work is being done to update the Unified Facilities Criteria Design of
Buildings to Resist Progressive Collapse. One of the improvements in the new updated
guidelines will be more realistic load factors for static analysis procedures based on the approach
used in ASCE41.
Design Approaches to Resist Progressive Collapse
Prevention or mitigation of progressive collapse can be achieved using two different
methods: indirect design and direct design. The indirect method consists of improving the
structural integrity of the building by providing redundancy of load paths and ductile detailing.
Currently, only UFC 4-023-03 allows the use of indirect methods. The direct method is divided
into two approaches: Specific Load Resistance (SLR) and Alternate Path (AP), with the latter
being the most widely used in the US. The updated guidelines will incorporate a combination of
indirect and direct methods with AP being the main direct design procedure. This research
focuses primarily on the AP method for progressive collapse.
Overview of GSA Guidelines
The purpose of the Guidelines is to: Assist in the reduction of the potential for
progressive collapse in new Federal Office Buildings, assist in the assessment of the potential for
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16
progressive collapse in existing Federal Office Buildings, and assist in the development of
potential upgrades to facilities if required.
To meet this purpose, these Guidelines provide a threat independent methodology for
minimizing the potential for progressive collapse in the design of new and upgraded buildings,
and for assessing the potential for progressive collapse in existing buildings (GSA, 2003). The
GSA guidelines only provide requirements for Reinforced Concrete and Steel structures. The
main design procedure is the AP method. In the AP method, designers and analysts are allowed
to choose between linear/non-linear, dynamic/static and 2-dimmensional/3-dimmensional
procedures and models. The load combination used for dynamic analyses is D + 0.25L and a
factor of 2 is applied for static cases to account for dynamic and inertial effects 2(D + 0.25L).
For non-linear analyses the acceptance criteria is based upon ductility and rotation limits
specified in tables for different component types. For linear analyses, the capacity of the
members is artificially enhanced using demand capacity ratios DCR (specified in the guidelines)
to account for non-linearity effects not explicitly included in the model. If the enhanced capacity
is more than the demand or acting force in the component, the member is said to be acceptable.
DCR values range from 1 to 3 based on construction type and configurations. The guidelines
recommend that non-linear procedures should be use for buildings with more than 10 stories.
Finally, vertical support removal locations are explicitly provided and each loss location must be
considered as an independent analysis.
Overview of DoD Guidelines UFC 4-023-03
The purpose of these guidelines is to provide a design to reduce the potential of
progressive collapse for new and existing DoD facilities, (UFC, 2005). The DoD guidelines use
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17
a combination of direct and indirect approaches and must be applied to all DoD building with 3
or more stories. In addition to RC and steel structures, the DoD guidelines include Masonry,
Wood and Cold-formed structural components. The requirements can be applied to provide four
different levels of protection (LOP). For Very Low LOP, only indirect design is employed by
specifying the required levels of Tie Forces. If a structural element does not provide the required
tie force, the element must be re-designed or retrofitted. For Low LOP, a combination of the
Indirect and Direct method is used. The design must incorporate both horizontal and vertical tie
forces. However, if the vertical tie forces are insufficient, the designer must upgrade the vertical
ties or perform an AP analysis to prove the structure is capable of bridging over the deficient
vertical member. For Medium and High LOP, both, Tie Forces and AP requirements are
mandatory. In the AP method, the DoD guidelines allowed the use of three procedures: Linear
Static, Non-linear Static and Non-linear Dynamic. The load combination used for the AP
analysis is 1.2D + 0.5L and as with the GSA guidelines a factor of 2 is applied for static
procedures 2(1.2D+0.5L). Response criteria are given for non-linear analyses in terms of
allowable levels of ductility and rotation presented in tables for each construction type. For
static analyses, the un-enhanced capacity of the members is compared to the demand or acting
force on the component. This differs drastically with the GSA guidelines, which use DCR values
ranging from 1 to 3. A more detail explanation of the use of these capacity increase factors is
presented in Chapter 4.
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CHAPTER 3: PROCEDURES FOR THE ALTERNATE PATH METHOD
In the AP method, the designer must show that the building is capable of bridging over a
removed structural element and that the resulting extent of damage does not exceed the damage
limits. In the updated UFC 4-023-03, an AP analysis may be performed using one of three
procedures, Nonlinear Dynamic, Nonlinear Static, or Linear Static, as described next.
Linear Static (LS): In general, this is the simplest of the three procedures to apply. A
linear static model of the structure is created and two load cases are considered: one is used to
calculate the deformation-controlled (or ductile) actions (or internal forces and moments or
demands) and the second load case is used to calculate the force-controlled (or brittle) actions.
For the analysis of the deformation-controlled actions, the applied load is enhanced by a Load
Increase Factor that approximately accounts for both dynamic and nonlinear effects. The
enhanced load is applied to the linear static model that has been modified by removal of a
column, wall section or other vertical load-bearing member. The calculated internal member
forces (actions) due to the enhanced loads are compared to the expected member capacities. For
deformation-controlled actions, the expected member capacities are increased by a capacity
increase factor (CIF, similar to the m-factor in ASCE 41) that accounts for the expected
ductility and the resulting values are compared to the deformation-controlled actions. For force-
controlled actions, the model is re-analyzed with a different Load Increase Factor that accounts
for only the inertial effects and the calculated demand is directly compared to the un-modified
member capacity.
Nonlinear Static (NLS): After the materially- and geometrically-nonlinear model is built,
the loads are magnified by a Dynamic Increase Factor that accounts only for inertia effects and
the resulting load is applied to the model with the removed vertical load-bearing element. For
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19
deformation-controlled actions, the resulting member deformations are compared to the
deformation limits based on the desired performance level; for force-controlled actions, the
member strength is not modified and is compared to the maximum actions (internal member
forces).
Nonlinear Dynamic (NLD): In this case, the un-modified load case is directly applied to
a materially- and geometrically-nonlinear model of the structure. In the first phase of the
dynamic analysis, the structure is allowed to reach equilibrium under the applied load case. In
the second phase, the column or wall section is removed almost instantaneously and the software
tool calculates the resulting motion of the structure. As with the NLS case, the resulting
maximum member deformations are compared to the deformation limits and for force-controlled
actions, the member strength is compared to the maximum internal member force. Dynamic
nonlinear analysis explicitly includes nonlinearity and inertial effects and therefore no correction
factors are needed.
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CHAPTER 4: INCONSISTENCIES OF EXISTING FACTORS
As mentioned earlier, the linear static procedure requires the use of a load increase factor
(LIF) to account for both dynamic and non-linear effects. The nonlinear static procedure
requires a dynamic increase factor (DIF) to account for just the inertial effects. For linear and
nonlinear static analysis methods, the current UFC 4-023-03 and the GSA Guidelines use the
same load multiplier of 2.0, which is applied directly to the progressive collapse load
combination. Four major issues have been identified in the static procedures.
1. The same load enhancement factor is used for Linear Static and Nonlinear Static
analyses. To approximate the actual nonlinear and dynamic response of a damaged
structure, the load on a LS model must be increased by a factor that accounts for both
effects. For a NLS model, the load must be increased by a factor that accounts only for
the dynamic effects, as the nonlinear behavior has already been addressed. The current
UFC 4-023-03 and GSA Guidelines use the same increase factor of 2.0 for both types of
analyses, which is incorrect.
2. The dynamic increase factor of 2.0 is not appropriate for the majority NLS cases. As is
well known from structural dynamics, the maximum dynamic displacement of an
instantaneously applied, constant load in a linear analysis is twice the displacement
achieved when the load is applied statically. If a structure is designed to remain elastic, a
factor of 2.0 would be appropriate. However, in extreme loading events, it is more
economical and typical to design structures to respond in the nonlinear range. Thus, as
will be shown later for the buildings that were analyzed, the dynamic increase factor
(DIF) that allows a Nonlinear Static solution to approximate a Nonlinear Dynamic
solution, is typically less than 2.
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3. Load enhancement factors do not vary with the performance level. The current
guidelines apply the same multiplier to the loads independent of the performance level
being used in the design. In other words, a structure is assigned a load enhancement
factor of 2.0 regardless of whether the designer wants to allow significant structural
damage (Collapse Prevention, as described in ASCE 41) or very little damage
(Immediate Occupancy in ASCE 41). As will be shown later, the load enhancement
factors can be defined as functions of the desired building performance level and the
building characteristics.
4. Inconsistency of Capacity Increase Factors (CIF) in LS procedures. UFC 4-023-03 uses a
CIF (m-factor) of 1.0. A CIF (m-factor) of 1.0 combined with a dynamic multiplier of
2.0, can produce overly-conservative designs as the resulting double-span condition after
the removal of a vertical load bearing element is required to carry 2 times the progressive
collapse load. GSA uses CIFs (or DCRs) between 1.0 and 3.0. As shown by Ruth 2004,
the design could be either overly conservative or un-conservative depending on the DCR
value being used. For example:
In both cases, a dynamic multiplier of 2.0 is applied to the progressive collapse load as
required in the existing GSA guidelines. For the first case, a DCR of 3.0 is applied to the
GSA LS Acceptance Equation: Dyn. Multiplier * (PC load) < (DCR) x (Capacity)
DCR = 3: 2 x (PC load) = (3) x (Capacity)
(2/3) x (PC load) = (Capacity)
DCR = 1: 2 x (PC load) = (1) x (Capacity)
(2) x (PC load) = (Capacity)
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capacity of the member, which could be a beam in flexure. If the dynamic multiplier (2.0) and
the DCR (3.0) are combined and applied to the progressive collapse load, it can be seen that the
member would be designed for 2/3 of the original progressive collapse load, which is un-
conservative. Conversely, if the structural member under consideration has a DCR of 1.0, which
could correspond to a column in flexure, the combined factor (Dynamic Multiplier / DCR) would
be 2.0. In this case, the member would be designed for a load of 2 times the progressive collapse
load, which could be overly conservative.
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CHAPTER 5: VARIATION OF LOAD AND DYNAMIC INCREASE FACTORS
RESEARCH PROCEDURE
As a result of the inconsistencies presented in the previous chapter, a study was
undertaken to investigate the factors needed to better match the LS and NLS static procedures to
the NLD procedure in AP analysis for Progressive Collapse. The variation of the enhanced load
with respect to structure deformation was investigated. As in ASCE 41, structural deformation is
considered to be the best metric for approximating structural damage.
To study the variation of load increase factors (LIFs for LS analyses) and dynamic
increase factors (DIFs for NLS analyses), a series of 3-dimensional reinforced concrete and
moment-frame steel building and 2-dimensional double span beam models were developed. The
3-dimensional models were used to perform AP analyses using SAP2000, and the 2-dimensional
double-span beam models were use to simulate column removals using a Single-Degree-of-
Freedom (SODF) software. The basic procedure to determine the LIFs and DIFs consisted of
3 steps:
1. Starting with a baseline model of a building designed using conventional design loads, a
NLD AP analysis was performed for a given column removal location (Corner,
perimeter or interior). The analysis used the ASCE 7 extreme event load case without
any enhancement; the values of plastic rotation and displacement at the column removal
location were recorded.
2. Using the exact same design and column removal location in the model from Step 1, a
NLS analysis was performed, with a trial DIF applied to the ASCE extreme event load
case. The DIF was adjusted and the model was re-run until the maximum plastic rotation
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24
matched the rotation measured in Step 1. This step yielded the DIF for the first design of
the first building configuration.
3. Using the same design and column removal location in the model from Step 1, a LS
analysis was performed. A trial LIF was applied to the ASCE extreme event load case.
The LIF was adjusted and the model was re-run until the maximum displacement
matched the displacement that corresponds to plastic rotation measured in Step 1. This
step yielded the LIF for the first design of the first building configuration.
After a value for the DIF and LIF had been determined for the initial design, the beams
and girders were re-designed to produce a new design using the same building configuration
(building height and bay spacing), and Steps 1 through 3 were repeated. This process is
illustrated in Figure 2 .
Figure 2. Procedure to Determine Load Increase Factors.
After a series of values of LIF and DIF were recorded for a given column removal
location, the procedure was repeated for a different column removal location using the same
building configuration (same building height and bay spacing). After all three column removal
locations had been analyzed for a particular building configuration, a new building (new building
height and bay spacing) would be analyzed following the same steps described above.
1 (NLD)
(1.0) PC
2 (NLS)
(DIF) PC
3 (LS)
(LIF) PC NLD NLD
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25
3- Dimensional Analytical Models
The study included reinforced concrete and steel moment frame buildings. For each
building type, different configurations of building height and bay spacing were analyzed to
determine how the variation of these parameters affected the load and dynamic increase factors.
Constant material over-strength factors were employed. The ASCE 7 extreme event load case
was used for all analyses; ignoring wind and snow loads, this load combination is 1.2D + 0.5L,
where D is the dead load and L is the live load. For each model, different factors were applied to
the load to match a given deformation level.
All 3-dimensional structures were analyzed using SAP2000, a well-know structural
software commonly used in conventional design and other applications. The lateral resisting
frames for both; RC and steel buildings were modeled using full moment connections. The
connections at the foundations were modeled as pinned connections and secondary members
were not included. For RC buildings, the analyses were performed assuming appropriate
detailing practices for progressive collapse. In other words, it was assumed that the reinforcing
steel was continuous through the supports and that it was properly anchored at the ends to
develop the full tension capacity of the bars. Appropriate detailing practices are necessary to
allow the structure to achieve large deformations typical of progressive collapse. A more detail
description of the set-up and properties for each type of analysis follows:
Nonlinear Dynamic Analysis: As mentioned earlier, the NLD procedure is the most
comprehensive and realistic method of analysis for progressive collapse. The important
modeling parameters included the damping ratio, time step, column removal time and plastic
hinge definitions. For these analyses, these parameters were taken as follows:
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26
- Damping ratio = 1%
- Column removal and time step = 1/20 of the structures natural period
- Analysis Time Step = 1/200 of the structures natural period
The natural period was determined by performing a Modal Analysis, and selecting the
Natural Period (T) of the dominating mode of vibration. The dominating mode of vibration was
selected visually based on the location of the column removal and the motion of the structure.
Non-linearity was included in the model by using Plastic hinges at both ends and mid-
point of every beam element and at both ends of the column elements. No hinge offsets were
used. The hinge definition for the steel buildings was taken from the pre-set options available in
SAP2000 corresponding to the hinge definition given in Chapter 5 (Steel Frame Structures) of
ASCE 41. A graphical representation of this hinge definition is shown next in Figure 3.
Figure 3. Steel Frame Building Hinge Definition (ASCE 41).
For reinforced concrete structures, the hinge definition (Figure 4) was designed to allow
strain hardening of 5% at the point expected to be the maximum allowed rotation (0.07 radians).
This differs from the 10% hardening at 0.025 radians used in ASCE 41. The reason for this
difference is the larger allowable rotations used in progressive collapse analyses. In other words,
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27
if the same slope used in the ASCE 41 hinge definition from points B to point C (Figure 4) was
used in this analysis, this would result in an increase in moment capacity of approximately 30%
at the point of maximum allowed displacement (0.07 radians) which is unrealistic.
Figure 4. Reinforced Concrete Hinge Definition.
As seen in Figures 3 and 4, SAP2000 does not allow the user to enter a rotation value for
the yield point (See point B in Figures 3 and 4). In other words, the hinges in SAP2000 by
default use an initial stiffness of 1.0. For this type of non-linear analysis were large deformations
are expected, the yield rotation is often negligible when measuring total displacement
particularly for RC beams which are very stiff. However, during the data analysis stage of the
study, yield rotation values were calculated using the formulas of ASCE 41 and included in the
normalization of the data.
The expected value of maximum allowable rotation for reinforced concrete structures of
0.07 radians was taken from the acceptance criteria in ASCE 41 with a factor of 3.5 applied to it.
ASCE 41 will largely be the basis for the allowable performance levels in the new UFC 4-023-
03, although some modifications are anticipated.
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28
The deformation limits for Life Safety for steel buildings were taken identical to those
values in Table 5-6 of ASCE 41. However, for reinforced concrete (RC) buildings, the Life
Safety values in Table 6-7 were increased by a factor of 3.5. This is because, within the seismic
community, the RC limits in FEMA are considered to be conservative [EERI/PEER 2006] and,
in the blast-design community, the allowable deformation criteria in ASCE 41 are much smaller
than indicated by test data from blast- and impact-loaded RC structural members. In addition, the
conservative ASCE 41 RC criteria are based on backbone curves derived from cyclic testing of
members and joints, whereas only one-half cycle is applied in a progressive collapse event.
To simulate the instantaneous removal a given column, the column was replaced with
equivalent reactions obtained from a static analysis of the building using the progressive collapse
load applied to the entire structure (1.2D + 0.5L). These loads were then removed over time to
simulate the removal of the column. This process is shown in Figure 5.
Figure 5. NLD Analysis Procedure.
1.2D +0.5L
=
1.2D +0.5L
Equivalent loads
Equivalent loads
t
1
0(1/20) T
Removal
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29
After the equivalent column loads were removed, the building was allowed to deform until it
settled and the maximum plastic rotation was recorded for all hinges formed during the analyses.
(See Figure 6)
Figure 6. Results of NLD Procedure.
Nonlinear Static Analysis: In the NLS analysis, non-linearity was modeled identically as
with the NLD model discussed above (with plastic hinges). However, to simulate the column
removal, the non-linear staged construction feature in SAP2000 was used (Figure 7). The
model was analyzed in three stages using 100 steps per stage.
Structure Settles
Plastic Hinge Rotation
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30
Figure 7. Stage Construction Setup. In the first stage, the progressive collapse load case was applied to all elements; see
Figure 8.
Figure 8. NLS Analysis Stage 1.
In the second stage, only the bays around the loss location were loaded with the
progressive collapse load, multiplied by the trial DIF, as shown in Figure 9.
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31
Figure 9. NLS Stage 2, Load Around Loss Location.
In the final stage, the column was removed and the analysis was run until the building
settled; see Figure 10. After the building had settled, the maximum plastic hinge rotations were
recorded in a similar manner to the NLD case. If the maximum plastic rotation was not equal to
the plastic rotation from the NLD analysis, the DIF was adjusted and the analysis was repeated,
until the plastic rotations from the NLD and NLS analyses matched within 2%.
Figure 10. NLS Stage 3, Nonlinear Analysis of Structure Response.
Non-linear Hinge Rotation
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32
Linear Static Analysis: The linear static procedure is simpler in that it does not require
the use of dynamic and non-linear parameters such as time step, damping ratio, plastic hinges,
etc. In these analyses, two sets of loads were applied to the building model, from which a
column has been removed: one set of loads was applied to the whole structure, and the other set
of loads, which includes the trial LIF, was applied only around the column removal locations as
directed in UFC 4-023-03.
The analysis was run using the linear elastic option in SAP2000 and the displacement was
measured at the loss location. If the displacement did not match the displacement from the NLD
procedure, the trial LIF was adjusted and the analysis was run again.
The rigidity (EI) of the steel beams was modeled implicitly in SAP2000 by defining the
size and Elastic Modulus (E) of the structural components. For concrete however, the rigidity
must be explicitly modified to account for the effects of cracking at large rotations, which tend to
reduce the effective Moment of Inertia (I). Therefore, the Rigidity of the RC linear models was
taken as 0.5 E I as indicated in Table 6.5 of ASCE 41.
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33
3-Dimmensinal Building Designs
The baseline for all 3D models (reinforced concrete and steel moment frame) was taken
from the examples in the current UFC 4-023-03. This floor plan is illustrated in next figure.
Figure 11. Typical Floor Plan.
Using the floor plan illustrated above, different building configurations were obtained by
varying the bay spacing and the number of stories. After the building designs were finalized, the
variation and magnitude of load and dynamic increase factors was investigated following the
procedure explained above. AP analyses were performed in all buildings. For each building, the
AP analysis included corner exterior and interior column removals. The following table presents
a summary of all the AP analyses included in this research. As seen in Table 1, a total of 408 AP
analyses were performed in this research study.
5 @ 25ft
4 @
25f
t
Spandrel
Spandrel-girder
Girder
Interior Beam
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34
Table 1. Study Matrix of AP Analyses Performed
Alternate Path Analysis Performed for Steel Moment Frame Buildings
Building Configuration Corner Column Removal Perimeter Column Removal Interior Column
Removal
3-Story, 25 ft bay Spacing 12 12 24 10-Story, 25 ft bay Spacing 12 9 30
Total AP Analyses for
Steel Buildings 99
Alternate Path Analysis Performed for Reinforced Concrete Buildings
Building Configuration Corner Column Removal Perimeter Column Removal Interior Column
Removal
3-Story, 20 ft bay Spacing 15 30 15 10-Story, 20 ft bay Spacing 15 30 18
* 10-Story, 20 ft bay Spacing 15 33 15 10-Story, 25 ft bay Spacing 9 9 9
** 10-Story, 25 ft bay Spacing 9 9 9 *** 10-Story, 25 ft bay Spacing 9 9 9
3-Story, 25 ft bay Spacing 0 0 21 10-Story, 30 ft bay Spacing 21 0 0
Total AP Analyses for RC
Buildings 309 * Denotes removal of column at 6th floor level ** Denotes removal of column at 5th floor level *** Denotes removal of column at 8th floor level
The loads and material properties used for the analyses are presented in the following
tables.
Table 2. Design Loads
DL 49 psf (steel)
54 psf (RC) Includes self weight of members not modeled
SDL 35 psf Includes partitions, ceiling weight and mechanical equipment
CL 15 psf Cladding load, only in the perimeter
LL 50 psf Live load
The DL values in Table 2 are abased on lightweight RC floor systems as indicated in the
examples of the current UFC 4-023-03.
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35
Table 3. Steel Buildings, Material Properties fy 52.5 ksi Includes 1.05 over-strength factor
E 29,000 ksi Modulus of elasticity
Table 4. Reinforced Concrete Buildings, Material Properties fc 6.25 ksi Includes 1.25 over-strength factor
fy 75 ksi Reinforcing steel w/ 1.25 over-strength factor
In progressive collapse, strength increase factors applied to material properties are used to
account for the average ratio of the actual static strength of materials to the nominal specified
value, and the rapid application of the load. These values are specified in the UFC 4-023-03
guidelines as 1.05 for structural steel, and 1.25 for concrete compressive strength and reinforcing
steel.
2-Dimmensional Models
Additional data needed to be generated to develop a larger baseline for comparison of
results. However, AP analysis using 3D models can be time consuming and expensive.
Therefore, additional data was generated using simple 2-dimensional double-span models
analyzed with a SDOF approach. The models consisted of a fixed-fixed double span beam. The
analysis procedure was similar to the procedure describe previously for 3D models. First, the
double span was loaded with a given design load, and the deflection of the beam was calculated
performing a non-linear dynamic analysis using time integration techniques and SDOF approach.
Next, the LIF corresponding to that design was calculated using the following equation:
wkLIF = * , where k is the beam stiffness calculated per ASCE 41 procedures, is the
calculated displacement, and w is the applied load. The DIF was calculated as follows:
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36
1)( +=R
wRDIF , where R is the ultimate flexural resistance of the beam. The ultimate
resistance R is calculated using the equations for maximum moment of single span beams and
solving for w. When divided w by tributary area, R can be expresses in units of psi. An
example follows for a simply supported beam.
2**8
LBMpR = , where w is the beam loading in lb/ft, and L is the beam span. Mp is the cross-
sectional moment capacity and B is the tributary width.
An illustration of the 2D models and their typical resistance function is presented next in
schematic form.
Figure 12. 2-Dimmensional Models
Different beam configurations were obtained by changing the beam stiffness and ultimate
resistance. The concrete compressive strength used for the 2-dimensional analyses was 5000 psi,
and the steel yield strength was 75 ksi. A total of 48 column removal simulations were
Span, LSpan, L
Load, w
k
R
Beam Resistance
Deflection
-
37
performed using 2D double-span beam models and SDOF approach. Table presented below
presents a summary of all the simulations performed.
Table 5. 2-Dimmensionals Analyses Double-Span Beam Configuration
Beam Span (ft)
Width (in)
Depth (in)
Avg. Steel Area (in2)
Stiffness k
(psi/in)
Ultimate Resistance, R
(psi)
Number of Column
Removals
20 24 20 1.45 1.20 0.58 8 20 24 20 1.70 1.22 0.68 8 20 24 20 2.40 1.29 0.95 8 20 12 30 0.98 1.98 0.58 8 20 12 30 1.30 2.05 0.76 8 20 12 30 1.70 2.13 0.98 8
Total No. of
runs 48
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38
CHAPTER 6: ANALYSIS RESULTS AND DATA ANALYSIS
A sample of the result tables generated for the 3-dimensional building analyses is
presented next. Similar tables were generated for the 2-dimensional study. A complete set of
results is provided in the Appendix.
Each table below lists the different structural designs that were evaluated for the
particular building configuration and column removal location. The first column in each table
indicates the design number. As mentioned earlier, a baseline design model was developed
using standard structural design software and then modified (Re-Design 1 through X) by
changing the beam, spandrel, girder, and spandrel-girder cross-sections to acquire different
displacements and plastic rotations. The following columns provide the section properties and
geometry of the structural elements framing into the loss location for that particular re-design.
Finally, the last three columns on the right hand side of the tables show the displacement and/or
plastic rotation measured with the NLD analysis and the values of DIF and LIF obtained from
the NLS and LS analysis of that particular re-design.
Table 6. Steel Building, 10-Story - 25 ft Bay, Interior Column Removal
Run # Frame Section Section Zx Ix Weight NLD, Disp. NLD
Plastic Rotation
NLS DIF
LS LIF
sap name in^3 in^4 lb in rad Re-Design 6 Girder w24x76 200 2100. 76 3.77 0.0021 1.82 1.80
Int. Beam w18x60 123 984. 60 Re-Design 5 Girder w21x73 172 1600. 73 4.76 0.0080 1.68 1.84
Int. Beam w16x57 105 758. 57 Re-Design 4 Girder w24x62 153 1550. 62 4.80 0.0103 1.60 1.90
Int. Beam w21x44 95.4 843. 44 Re-Design 3 Girder w24x55 134 1350. 55 6.42 0.0167 1.44 2.12
Int. Beam w18x40 78.4 612. 40 Re-Design 2 Girder w18x60 123 984. 60 8.99 0.0260 1.38 2.37
Int. Beam w16x40 72.9 518. 40 Baseline Girder w18x55 112 890. 55 11.44 0.0349 1.29 2.84
Int. Beam w18x35 66.5 510. 35 Re-Design 1 Girder w16x57 105 758. 57 16.91 0.0532 1.23 3.49
Int. Beam w14x38 61.5 385. 38
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39
Table 7. Reinforced Concrete Building, 3-Story-25 ft, Interior Column Removal
Frame Section Top. As Bot. As Steel % NLD,Plastic
Rotation. Run # sap name in^2 in^2 increment in
NLS DIF
LS LIF
Girder 4.48 2.42 --- --- Baseline Int. Beam 2.40 1.76 ---
Girder 7.84 4.24 75% 0.021 1.14 4 Re-Design 1 Int. Beam 4.20 3.08 75%
Girder 7.48 4.04 67% 0.028 1.09 5.5 Re-Design 2 Int. Beam 4.01 2.94 67%
Girder 7.17 3.87 60% 0.042 1.06 8 Re-Design 3 Int. Beam 3.84 2.82 60%
Girder 7.03 3.80 57% 0.053 1.05 10 Re-Design 4 Int. Beam 3.77 2.76 57%
Girder 6.94 3.75 55% 0.064 1.05 13 Re-Design 5 Int. Beam 3.72 2.73 55%
Girder 6.81 3.68 52% 0.086 1.04 17 Re-Design 6 Int. Beam 3.65 2.68 52%
Girder 6.76 3.65 51% 0.097 1.04 19 Re-Design 7 Int. Beam 3.62 2.66 51%
The results obtained from the 3-dimensional and 2-dimensional models presented above
were use to generate plots with proposed normalized factors for static procedures.
Analysis of Data from Linear Static AP Analyses (LIF)
The results obtained in this study demonstrated that LIFs are a function of section
properties and geometry; particularly for RC sections where stiffness can vary significantly
based on rebar placement and section aspect ratio. For steel structures, the LIFs results were
found to be less dependent on the selected section. The plot in Figure 13 shows the LIF plotted
versus plastic rotation for reinforced concrete for selected analysis cases. Figure 15 shows the
LIF plotted versus total rotation for steel sections. Plastic rotation was use to plot the LIF values
for concrete sections to be consistent with ASCE41 which treats reinforced concrete sections as
having negligible elastic rotations. Steel sections are in general more ductile and exhibit more
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40
considerable elastic rotations; hence, the LIF values for steel sections are plotted against total
rotation as in ASCE 41.
Figure 13. LIF vs Plastic Rotation for RC sections
The dispersion of data points in Figure 13 represents the LIFs strong dependence on
section properties for the concrete members mentioned earlier. In concrete members, two beams
with the same moment capacity can have different stiffness values. Therefore, when analyzed as
non-linear members, the maximum calculated deflection will be similar since, most of the
response will be plastic as concrete members exhibit small elastic rotations. However, if the
same two beams are analyzed as linear members, the stiffness (not the capacity) becomes the
controlling parameter, and the factor applied to the stiffness of the member to achieve the same
deflection calculated with the non-linear analysis could differ significantly. This is illustrated
0.0
10.0
20.0
30.0
40.0
50.0
60.0
70.0
80.0
90.0
100.0
0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090 0.100
Plastic Rotation (rad)
LIF
3-story interior column removal (20x22 girder, L=240, rho=.0037) 3-story interior column removal (30x20 girder, L=300, rho~.01)3-story interior column removal (20x22 girder, L=240, rho~.004) 3-story corner column removal (6x15 girder, L=240, rho~.03)3-Story, 4x8 bay, L=240, Mid Side Long, 1st 3-Story, 4x8 bay, L=240, Corner, 1st3-Story, 4x8 bay, L=240, Interior, 1st 3-Story, 4x8 bay, L=240, Mid Side Short, 1st3-Story, 4x8 bay, L=240, Interior Corner, 1st 3-Story, 4x8 bay, L=240, Corner Optimized, 1st10-Story, 4x8 bays, L=240, Mid Side Long 1st 10-Story, 4x8 bays, L=240, Corner 1st10-Story, 4x8 bays, L=240, Interior 1st 10-Story, 4x8 bays, L=240, Mid Side Short 1st10-Story, 4x8 bays, L=240, Interior Corner 1st 10-Story, 4x8 bays, L=240, Mid Side Long 6th10-Story, 4x8 bays, L=240, Corner 6th 10-Story, 4x8 bays, L=240, Interior 6th10-Story, 4x8 bays, L=240, Mid Side Short Edge 6th 10-Story, 4x8 bays, L=240, Interior Corner 6th10-Story, 4x8 bays, L=240, Corner 9th 10-Story, 4x8 bays, L=300, Mid Side Long 1st10-Story, 4x8 bays, L=300, Corner 1st 10-Story, 4x8 bays, L=300, Interior 1st10-Story, 4x8 bays, L=300, Mid Side Long 5th 10-Story, 4x8 bays, L=300, Corner 5th10-Story, 4x8 bays, L=300, Interior 5th 10-Story, 4x8 bays, L=300, Mid Side Long 8th10-Story, 4x8 bays, L=300, Corner 8th 10-Story, 4x8 bays, L=300, Interior 8th
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41
next. CIF, in the figure below, corresponds to the artificial increase factor applied to the stiffness
of the element to allow it to achieve the same displacement as in the plastic response.
Figure 14. RC Sections Strong Dependence on Stiffness
Figure 15. LIF vs. Total Rotation for Steel Sections
0.00
1.00
2.00
3.00
4.00
5.00
6.00
0.000 0.020 0.040 0.060 0.080 0.100 0.120 0.140
Total Rotation (rad)
LIF
3-story corner column removal3-story interior column removal3-story perimeter column removal10-story interior column removal3-story perimeter column; rev hinge
K1, MK2, M
K2
K1
RPlastic Response
K2
K1
kElastic Response
CIF 1
CIF 2
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42
Because the LIFs need to be applied consistently to different structural elements
regardless of their stiffness or shape, the data above was normalized and plotted against the ratio
of total rotation to the calculated yield rotation of the element, which corresponds to the m-
factors in ASCE 41. A more detail explanation about the use of m-factors if provided in
Chapter 7. The plots in Figure 16 and 17 show the same LIF data when normalized by yield
rotation.
Figure 16. Normalized LIF for RC sections
0.0
10.0
20.0
30.0
40.0
50.0
60.0
70.0
80.0
90.0
0.0 10.0 20.0 30.0 40.0
Norm Rotation (total rot/member yield) (m factor)
LIF
3-story interior column removal (20x22 girder, L=240, rho=.0037) 3-story interior column removal (30x20 girder, L=300, rho~.01)3-story interior column removal (20x22 girder, L=240, rho~.004) 3-story corner column removal (6x15 girder, L=240, rho~.03)3-Story, 4x8 bay, L=240, Mid Side Long, 1st 3-Story, 4x8 bay, L=240, Corner, 1st3-Story, 4x8 bay, L=240, Interior, 1st 3-Story, 4x8 bay, L=240, Mid Side Short, 1st3-Story, 4x8 bay, L=240, Interior Corner, 1st 3-Story, 4x8 bay, L=240, Corner Optimized, 1st10-Story, 4x8 bays, L=240, Mid Side Long 1st 10-Story, 4x8 bays, L=240, Corner 1st10-Story, 4x8 bays, L=240, Interior 1st 10-Story, 4x8 bays, L=240, Mid Side Short 1st10-Story, 4x8 bays, L=240, Interior Corner 1st 10-Story, 4x8 bays, L=240, Mid Side Long 6th10-Story, 4x8 bays, L=240, Corner 6th 10-Story, 4x8 bays, L=240, Interior 6th10-Story, 4x8 bays, L=240, Mid Side Short Edge 6th 10-Story, 4x8 bays, L=240, Interior Corner 6th10-Story, 4x8 bays, L=240, Corner 9th 10-Story, 4x8 bays, L=300, Mid Side Long 1st10-Story, 4x8 bays, L=300, Corner 1st 10-Story, 4x8 bays, L=300, Interior 1st10-Story, 4x8 bays, L=300, Mid Side Long 5th 10-Story, 4x8 bays, L=300, Corner 5th10-Story, 4x8 bays, L=300, Interior 5th 10-Story, 4x8 bays, L=300, Mid Side Long 8th10-Story, 4x8 bays, L=300, Corner 8th 10-Story, 4x8 bays, L=300, Interior 8thFit LIF eqnASCE41(Mod) Concrete Beams (Primary) ASCE41(Mod) Concrete Beams (Primary/NC shear steel)ASCE41(Mod) Concrete Beams (Secondary) ASCE41Concrete Columns (P/Agf'c < 0.1)ASCE41Concrete Columns (P/Agf'c > 0.4) Linear (Fit)
recommended eqnLIF = 1.2m + 0.8
Linear fit to all data except 10-story L=240: y = 1.1259x + 0.87 R2 = 0.987
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43
Figure 17. Normalized LIF for Steel Sections
Using the normalized plots of Figure 16 and 17, a linear fit of the data was performed to
support the development of equations for determination of the required LIF for LS procedures.
The linear fit for RC sections and steel sections had an R-square value of 0.99 and 0.95
respectively, which indicate a good fit. The linear fit for each plot is shown in black. However,
to provide conservatism and avoid effective multipliers (LIF/m) smaller than 1.0, the linear fits
were shifted up and manually adjusted. Therefore, an upper bound fit (the line in red) which
encloses the envelope of data is the final recommended equation. The upper bound fit provides a
conservative factor on the effective multiplier of 1.05 for RC and 1.35 for steel sections. A
larger factor was required for steel sections to keep all effective multipliers greater than 1.0.
This is demonstrated in Tables 8 and 9. Also presented on the plots are typical acceptance
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
11.0
12.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0
Norm Rotation (total rotation/member yield)) (m factor)
LIF
3-story corner column removal3-story interior column removal3-story perimeter column removal10-story interior column removal3-story perimeter column; rev hingeFitLIF FitASCE41 "Compact" Beams and Columns (P/Pcl < 2.0)ASCE41 "Non-Compact" BeamsASCE41 "Compact" Columns (P/Pcl = 0.5)ASCE41 (Mod) WUF for W18ASCE41 (Mod) RBS for W18ASCE41 "Compact" Secondary BeamsASCE41 (Mod) Shear Tab for W14Linear (Fit)
recommended eqn:LIF = 0.9m + 1.1
Linear fit to data: y = 0.6674x + 0.8027 R2 = 0.9493
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44
values for both concrete and steel components. This illustrates the general range of applicability
of the fits. The equations are:
RC Structures: LIF = 1.2 m + 0.80
Steel Structures: LIF = 0.9 m + 1.1
m is the component m-factor. The m-factor will be a direct multiplier on the expected
component strengths given in the revised UFC 4-023-03, which will correspond to the existing
values for acceptance criteria on ASCE 41. Although the LIF values from Figure 16 and 17
seem high, it should be noted that the effective multiplier on the static load case for LS
analysis is the LIF divided by the m-factor (LIF/m). This is demonstrated next.
General Equation: (LIF) x (PC load) < m x (Capacity)
LS: (LIF/m) x (PC load) < (Capacity)
The m-factors in ASCE41 need to be modified before being added to the revised UFC 4-
023-03. However, for reinforced concrete, the proposed m-factors will nominally range from 5
to 20 after adjustments are made to account for the conservatism in the existing concrete criteria
of ASCE 41. For steel, the range of m-factors will be a function of the component but will not be
significantly different from the existing ASCE41 criteria, i.e., 1.5 to 7. Hence, final effective
load multipliers for LS analysis will generally vary from 1.0 to 2.0, never below 1.0.
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45
Table 8 LIF Data for RC Sections
Typical LIF LIF/m LIF LIF/m Consv. Factor
m-factors Upp.
Bound Upp.
Bound Linear
Fit Linear Fit (Up. Bound/ Lin.
fit) 5 6.80 1.36 6.52 1.30 1.04 6 8.00 1.33 7.65 1.28 1.05 7 9.20 1.31 8.78 1.25 1.05 8 10.40 1.30 9.91 1.24 1.05 9 11.60 1.29 11.04 1.23 1.05
10 12.80 1.28 12.17 1.22 1.05 11 14.00 1.27 13.30 1.21 1.05 12 15.20 1.27 14.43 1.20 1.05 13 16.40 1.26 15.56 1.20 1.05 14 17.60 1.26 16.69 1.19 1.05 15 18.80 1.25 17.82 1.19 1.05 16 20.00 1.25 18.95 1.18 1.06 17 21.20 1.25 20.08 1.18 1.06 18 22.40 1.24 21.21 1.18 1.06 19 23.60 1.24 22.34 1.18 1.06 20 24.80 1.24 23.47 1.17 1.06
Table 9 LIF Data for Steel Sections
Typical LIF LIF/m LIF LIF/m Consv. Factor
m-factors Upp.
Bound Upp.
Bound Linear Fit Linear Fit (Up. Bound/ Lin.
fit) 1 2.00 2.00 1.47 1.47 1.36
1.5 2.45 1.63 1.81 1.20 1.36 2 2.90 1.45 2.14 1.07 1.36
2.5 3.35 1.34 2.48 0.99 1.35 3 3.80 1.27 2.81 0.94 1.35
3.5 4.25 1.21 3.15 0.90 1.35 4 4.70 1.18 3.48 0.87 1.35
4.5 5.15 1.14 3.82 0.85 1.35 5 5.60 1.12 4.15 0.83 1.35
5.5 6.05 1.10 4.49 0.82 1.35 6 6.50 1.08 4.82 0.80 1.35
6.5 6.95 1.07 5.16 0.79 1.35 7 7.40 1.06 5.49 0.78 1.35
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46
Analysis of Data from Nonlinear Static Analysis (DIF)
In NLS procedures, non-linearity is explicitly included in the model by use of plastic
hinges and the capacity of the members does not need to be adjusted using m-factors. Therefore,
the values of DIFs obtained in this study are a direct representation of the dynamic multiplier on
the load. The application of the DIFs is demonstrated below:
General Equation: (DIF) x (PC load) < (Reaction)
NLS Deformation-Controlled: (DIF) x (PC load) = measured < allowed
NLS Force-Controlled: (DIF) x (PC load) < (Capacity)
The results showed a range of variation in DIFs with respect to plastic rotation from 1.00
to 1.40 for concrete buildings and 1.20 to 1.85 for steel buildings as illustrated in Figure 18 and
19, respectively.
Figure 18. DIFs for RC Buildings
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090 0.100
Plastic Rotation
DIF
3-Story, 4x8 bay, L=240, Mid Side Long, 1st 3-Story, 4x8 bay, L=240, Corner, 1st3-Story, 4x8 bay, L=240, Interior, 1st 3-Story, 4x8 bay, L=240, Mid Side Short, 1st3-Story, 4x8 bay, L=240, Interior Corner, 1st 10-Story, 4x8 bays, L=240, Mid Side Long 1st10-Story, 4x8 bays, L=240, Corner 1st 10-Story, 4x8 bays, L=240, Interior 1st10-Story, 4x8 bays, L=240, Mid Side Short 1st 10-Story, 4x8 bays, L=240, Interior Corner 1st10-Story, 4x8 bays, L=240, Mid Side Long 6th 10-Story, 4x8 bays, L=240, Corner 6th10-Story, 4x8 bays, L=240, Interior 6th 10-Story, 4x8 bays, L=240, Mid Side Short Edge 6th10-Story, 4x8 bays, L=240, Interior Corner 6th 10-Story, 4x8 bays, L=240, Corner 9th10-Story, 4x8 bays, L=300, Mid Side Long 1st 10-Story, 4x8 bays, L=300, Corner 1st10-Story, 4x8 bays, L=300, Interior 1st 10-Story, 4x8 bays, L=300, Mid Side Long 5th10-Story, 4x8 bays, L=300, Corner 5th 10-Story, 4x8 bays, L=300, Interior 5th10-Story, 4x8 bays, L=300, Mid Side Long 8th 10-Story, 4x8 bays, L=300, Corner 8th10-Story, 4x8 bays, L=300, Interior 8th 10-Story, 4x8 bays, L=360, Corner 1st10-Story, 4x8 bays, L=360, Corner 1st Additional data)
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47
Figure 19. DIFs for Steel Buildings
DIFs needed to be expressed in terms of allowable plastic rotation since these are
values that an analyst is expected to look up in the tables provided in the revised progressive
collapse guidelines. Therefore, the DIFs were plotted as a function of the ratio of allowable
plastic rotation to member yield rotation. Figure 20 and 21 show the DIFs values from Figure
and 19 normalized using the ratio of allowable plastic rotation over the calculated yield rotation
of the (typically) beam element.
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
0.000 0.050 0.100 0.150 0.200
Plastic Rotation (rad)
DIF
3-story corner column removal
3-story interior column removal
3-story perimeter column removal
10-story interior column removal
-
48
Figure 20 . Normalized DIFs for RC Buildings
Figure 21. Normalized DIFs for Steel Buildings
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
0.0 2.0 4.0 6.0 8.0 10.0
Norm Rotation (allowable plastic rot/member yield)
DIF
3-story corner column removal
3-story perimeter column removal
3-story interior column removal
10-story interior column removal
DIF Fit
recommended eqn:DIF = 1.08+(0.76/((allow plastic rot/member yield)+0.83))
R2 = 0.83
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0
Norm Rotation (allowable plastic rot/member yield)
DIF
3-Story, 4x8 bay, L=240, Mid Side Long, 1st 3-Story, 4x8 bay, L=240, Corner, 1st3-Story, 4x8 bay, L=240, Interior, 1st 3-Story, 4x8 bay, L=240, Mid Side Short, 1st3-Story, 4x8 bay, L=240, Interior Corner, 1st 10-Story, 4x8 bays, L=240, Mid Side Long 1st10-Story, 4x8 bays, L=240, Corner 1st 10-Story, 4x8 bays, L=240, Interior 1st10-Story, 4x8 bays, L=240, Mid Side Short 1st 10-Story, 4x8 bays, L=240, Interior Corner 1st10-Story, 4x8 bays, L=240, Mid Side Long 6th 10-Story, 4x8 bays, L=240, Corner 6th10-Story, 4x8 bays, L=240, Interior 6th 10-Story, 4x8 bays, L=240, Mid Side Short Edge 6th10-Story, 4x8 bays, L=240, Interior Corner 6th 10-Story, 4x8 bays, L=240, Corner 9th10-Story, 4x8 bays, L=300, Mid Side Long 1st 10-Story, 4x8 bays, L=300, Corner 1st10-Story, 4x8 bays, L=300, Interior 1st 10-Story, 4x8 bays, L=300, Mid Side Long 5th10-Story, 4x8 bays, L=300, Corner 5th 10-Story, 4x8 bays, L=300, Interior 5th10-Story, 4x8 bays, L=300, Mid Side Long 8th 10-Story, 4x8 bays, L=300, Corner 8th10-Story, 4x8 bays, L=300, Interior 8th 10-Story, 4x8 bays, L=360, Corner 1st10-Story, 4x8 bays, L=360, Corner 1st Additional data) Fit
recommended eqn:DIF = 1.04+(0.45/((allow plastic rot/member yield)+0.48))
R2 = 0.80
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49
Similar to the LIFs for LS procedures, a mathematical fit of the data in Figure 20 and 21
was performed to develop the equations that represent the required DIF for NLS procedures.
The mathematical fit was performed manually using the general equation for a rectangular
hyperbola. The data from Figures 20 and 21 was used to adjust the equations until an acceptable
fit was determined. The R-square values calculated for the RC sections and steel sections were
0.80 and 0.83 respectively. The DIF fits determined in this study are purposely shifted towards
the upper limit of the data to add a level of conservatism. Because of this, the calculated R-
square values are lower than typical acceptable values for R-square in the range of 0.9-1.0.
These equations are presented next:
RC Structures: 48.0
45.004.1+
+=yieldall
DIF
Steel Structures: 83.0
76.008.1+
+=yieldall
DIF
In the equations above, the allowable plastic rotation (all) is taken from the nonlinear acceptance criteria tables in the revised guidelines, which will be taken from ASCE41. For RC
concrete structures, the allowable plastic rotations in ASCE 41 will be modified to account for
the extra conservatism included due to the cyclic nature of seismic events. yield, corresponds to the yield rotation of the member calculated per ASCE 41 procedures.
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50
CHAPTER 7: PROPOSED PROCEDURES FOR USING LIFS AND DIFS IN
STATIC AP ANALYSES
After developing new equations for the application of LIFs and DIFs, procedures were
developed for the application of these factors. These procedures are described next.
Linear Static AP Procedure
The LS approach will often be used in concept development of complex structural
systems required to satisfy AP requirements. The proposed procedure for using LIF values to
perform these preliminary analyses is presented next:
1. Select most restrictive structural component (smallest m-factor) from bays immediately
around loss location at all floor levels. Separate analyses will be performed for
horizontal flexural (beam element and connection) and vertical column components.
2. Select the m-factor corresponding to the element found on step 1 from tables given in the
revised UFC 4-023-03.
3. Using the m-factor from step 2, and the LIF equations developed previously; calculate the
LIF to be used in the LS analysis.
4. Divide the LIF from step 3 by m-factor from step 2 to determine the effective load
multiplier.
5. Apply effective load multiplier (LIF/m) to progressive collapse load (1.2D + 0.5L) and
perform LS analysis.
The proposed procedure presented above is inherently conservative as it uses the most
restrictive component around the loss location to select the m-factor used to calculate the
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51
effective load multiplier for the analysis. Therefore, the majority of the structural components
are designed or analyzed for a progressive collapse load larger than that required for that
particular member based on the equations developed for LIF presented in this document. This is
demonstrated next for a corner column removal on a typical steel moment frame structure. The
following example assumes that the structural elements in the corner bay are the same at all floor
levels.
Figure 22. LIF for LS Analysis
As seen in the example of Figure 22, based on the corresponding m-factors and the
equations developed for calculating LIFs, the W14x38 beam and WUF connection should be
W14x38 W
14x3
8
Loss Location W12x190 Corner Column
WUF Fully Restrained Connection
W14x38
m-factor (CP value from Table 5-5 ASCE41) = 8.0
LIF = LIF = 0.9(8) + 1.1 = 8.3
LIF/m (Effective Load Multiplier) = 8.3 / 8.0 = 1.04
General Equation: 1.04 (PC load) < (Capacity)
W12x190 Column
m-factor (LS value from Table 5-5 ASCE41, P/Pcl < 0.20) = 6.0
LIF = 0.9(6) + 1.1 = 6.5
LIF/m (Effective Load Multiplier) = 6.5 / 6.0 = 1.08
General Equation: 1.08 (PC load) < (Capacity) WUF Connection
m-factor (CP value from Table 5-5 ASCE41) = 3.9 0.043d = 3.30
LIF = 0.9(3.30) + 1.1 = 4.07
LIF/m (Effective Load Multiplier) = 4.07 / 3.34 = 1.22 (CONTROLS OVER BEAM)
General Equation: 1.22 (PC load) < (Capacity)
Typical Corner bay
W12x190 Columns
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52
designed for a load equal to 122% of the progressive collapse load. The 122% for the WUF
connection controls as it is larger than the 104% multiplier for the beam. The effective load
multiplier calculated for the W12x190 column, 108%, could be used if the designer chooses to
perform a separate LS analysis to check columns.
Comparison with existing UFC 4-023-03
The LS procedure for AP analysis of progressive collapse given in the existing UFC 4-
023-03 specifies a factor of 2.0 to be applied directly on the load without increasing the capacity
of the members, sine there are no m-factors in the existing UFC criteria. Therefore, if expressed
in terms of this study, the existing UFC criteria use an LIF of 2.0 and an m-factor of 1.0. This
results in an effective multiplier per UFC 4-023-03 of 2.0. This value is overly conservative
when compared to 1.22, the calculated value in Figure 22.
Non-Linear Static AP Procedure
In the new PC guidelines, the practitioner will select a value of DIF to be applied to the
progressive collapse load combination in NLS procedures based on the ratio of allowable plastic
rotation to yield rotation specified in the acceptance criteria. The proposed steps to select the DIF
for NLS analysis are presented next:
1. Select most restrictive structural component (smallest allowable plastic rotation) from
bays immediately around loss location at all floor levels. Separate analyses will be
performed for horizontal flexural (beam element and connection) and vertical column
components.
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53
2. Calculate the yield rotation of the flexural element or column using procedures in ASCE
41. Calculate the ratio of allowable plastic rotation to this yield rotation for the element
found on step 1.
3. Using the ratio from step 2, and the DIF equations developed previously; calculate the
DIF to be used in the NLS analysis.
4. Apply DIF to the progressive collapse load and performed NLS analysis.
An example of how to select the appropriate DIF for a NLS analysis of a RC
structure is presented next.
Figure 23. DIF for NLS Analysis
Spandrel
Typical Floors Roof
Spandrel
Gird
er
Loss Location
Roof Spandrel
Roo
f Gird
er
Loss Location
Roof Spandrel
Top. Columns
Bot. Columns
Spandrel
Typical Floors Roof
Spandrel
Gird
er
Loss Location
Roof Spandrel
Roo
f Gird
er
Loss Location
Roof Spandrel
Top. Columns
Bot. Columns
Spandrels:
b=15 in, d=24 in, As = 5.3 in2, As = 5.3 in2
allowable plastic rotation (ASCE41, table 6-11, NC) = 0.05 rad
Calculated yield rotation = 0.0096 rad
DIF = 1.04+(0.45/((0.05/0.0096)+0.48)) = 1.12
Girder:
b=10 in, d=24 in, As = 4.5 in2, As = 4.5 in2
allowable plastic rotation (ASCE41, table 6-11, C) = 0.0625 rad
Calculated yield rotation = 0.0118 rad
DIF = 1.04+(0.45/((0.0625/0.0118)+0.48)) = 1.12
Bot. Columns:
b=30 in, d=30 in, As = 15.24 in2
allowable plastic rotation (ASCE41, table 6-11, P / Ag fc > 0.4,
C) = 0.015 rad
Calculated yield rotation = 0.0007 rad
DIF = 1.04+(0.45/((0.015/0.0007)+0.48)) = 1.07
Roof Spandrels:
b=15 in, d=24 in, As=4.17 in2, As=4.17 in2
allowable plastic rotation (ASCE41, table 6-11, NC) = 0.05 rad
Calculated yield rotation = 0.0110 rad
DIF = 1.04+(0.45/((0.05/0.0110)+0.48)) = 1.13 (CONTROLS) Girder:
b=10 in, d=24 in, As=2.62 in2, As=2.62 in2
allowable plastic rotation (ASCE41, table 6-11, C) = 0.0625 rad
Calculated yield rotation = 0.0073 rad
DIF = 1.04+(0.45/((0.0625/0.0073)+0.48)) = 1.09
Top. Columns:
b=18 in, d=18 in, As=10.16 in2
allowable plastic rotation (ASCE41, table 6-11, P / Ag fc > 0.4,
C) = 0.015 rad
Calculated yield rotation = 0.0019
DIF = 1.04+(0.45/((0.015/0.0019)+0.48)) = 1.09 (CONTROLS)
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54
As seen in the example of Figure 23, based the equation developed for calculating DIFs,
the roof spandrels should be designed or analyzed for a load of 113% of the progressive collapse
load. The DIF calculated for all the other components is less than 1.13, therefore, 1.13 is the
multiplier on the load used for the progressive collapse analysis or design of this building.
Comparison with existing UFC 4-023-03
Similarly to the LS procedure, the NLS procedure for AP analysis of progressive collapse
given in the existing UFC 4-023-03 specifies a factor of 2.0 to be applied directly on the load.
Capacity increase factors are not necessary in NLS procedures. Therefore, the 2.0 value per the
UFC criteria is overly conservative when compared to 1.13, the calculated value in Figure 23.
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55
CHAPTER 8: CONCLUSIONS AND RECOMMENDATIONS
The dynamic multiplier on the load of 2.0 currently used for AP analysis can produce
overly conservative designs. This is particularly true for cases where large deformations are
allowed. The results of this study showed that for RC buildings, the dynamic multiplier on the
load (DIF) ranges from 1.05 to 1.75, which is significantly less than 2.0, particularly at
normalized rotations greater than 1. Similarly, for steel buildings analyzed, the dynamic
multiplier on the load (DIF) ranged from 1.2 to 1.8.
Additionally, the results from the analyses of LS procedures demonstrated that LIFs
depend on the total deformation as was expected, but are also strongly dependent on section
properties. As there are no LIFs in the current guidelines, there is no direct comparison that can
be made between the conservatism of the existing approach and the proposed use of the LIFs.
However, the current DoD and GSA procedures use a load multiplier of 2.0, and capacity
increase factors (DCRs only in GSA) of between 1.5 and 2.0. Therefore, while the LIFs
proposed in this work are greater than 2.0, the CIFs (m-factors) are much larger than the capacity
increase factors used in the current criteria. Hence, relative conservatism must be evaluated
based on the effective multiplier on the load. With a multiplier of 2.0 and a capacity increase
factor of 1.0, the effective multiplier in the current UFC criteria, for example, is 2.0. For an LIF
of 20 in the current research, the corresponding m-factor for a RC section would be 16, and the
effective multiplier is 1.25, significantly smaller than the current criteria. This reinforces that the
current UFC 4-023-03 could be considered overly conservative. Likewise, for the GSA criteria if
a DCR value of 2.0 is coupled with the specified load increase factor in the current criteria of
2.0. The resulting effective multiplier would be 1.0, which could be under-conservative.
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56
BIBLIOGRAPHY
ASCE/SEI 41-06 Prestandard and Commentary for the Seismic Rehabilitation of Buildings, American Society of Civil Engineers, 2007. ASCE/SEI 7-05 Minimum Design Loads for Buildings and Other Structures, American Society of Civil Engineers, Reston, VA, 2006 Biggs, John M., Introduction to Structural Dynamics A McGraw-Hill Publication: McGraw-Hill Inc., 1964 Design of Buildings to Resist Progressive Collapse, Unified Facilities Criteria (UFC) 4-023-03, Department of Defense (DoD), January, 2005. EERI/PEER, 2006, New Information on the Seismic Performance of Existing Concrete Buildings, Seminar Notes, Earthquake Engineering Research Institute, Oakland, California. Ellingwood, B., Smilowitz, R., Dusenberry, D., Duthinh, D., Carino, N., Best Practices for Reducing the Potential for Progressive Collapse, August 2006 Herrle, K., Mckay, A., Development and Application or Progressive Collapse Design Criteria for the Federal Government, ARA Technology Review, Volume 2, Number 2, June 2006 Mckay, A., Marchand, K., Stevens, D., Dynamic Increase Factors (DIF) and Load Increase Factors (LIF) for Alternate Path Procedures, A Report prepared for UFC 4-023-03 Steering Group, January, 2008 Powell, G., Progressive Collapse: Case Studies Using Nonlinear Analysis, SEAOC Annual Convention, Monterrey, 2004. Progressive Collapse Analysis and Design Guidelines for New Federal Office Buildings and Major Modernizations Projects, U.S General Services Administration (GSA), 2003. Ruth, P., Dynamic Considerations in Progressive Collapse Guidelines, MS Thesis, Department of Civil Engineering, University of Texas at Austin, 2004.
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57
APPENDIX - RESULT TABLES
Table 10, Columns Description
Column 1: Beams span (L), center to center.
Column 2: beam width (b)
Column 3: beam depth (d)
Column 4: average of top steel and bottom steel (Aavg)
Column 5: distance from edge of beam to center of reinforcing bars.
Column 6: gross moment of inertia of section (Ig) = (1/12) b h3
Column 7: ASCE 41 beam stiffness (k) = (384 EI/ B L4) for a fixed-fixed beam, EI = 0.5EIg, and
B is the tributary width
Column 8: beam ultimate resistance (R) = (8*(2M)/B*L2) for a fixed-fixed beam
Column 9: average moment (M) = Aavg fy * (d-0.5a), where a is depth of compression block
Column 10: maximum displacement calculated at loss location with SAP2000 for NLD
procedure
Column 11: maximum calculated rotation = maximum displacement / L
Column 12: plastic rotation = Maximum calculated rotation yield rotation
Column 13: yield rotation = ((R / k) / L)
Column 14: normalized rotation = plastic rotation / yield rotation
Column 15: LIF calculated with NLS procedure
Column 16: DIF calculated with LS procedure
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58
Table 11, Columns Description
Column 1: Beams span (L), center to center.
Column 2: beam Spacing (B)
Column 3: steel section
Column 4: Moment of Inertia (I)
Column 5: plastic section modulus (Z).
Column 6: ASCE 41 beam stiffness (k) = (384 EI/ B L4) for a fixed-fixed beam, EI = 0.5EIg, and
B is the tributary width
Column 7: beam ultimate resistance (R) = (8*(2M)/B*L2) for a fixed-fixed beam
Column 8: moment capacity (M) = Z* fy
Column 9: maximum displacement calculated at loss location with SAP2000 for NLD procedure
Column 10: maximum calculated rotation = maximum displacement / L
Column 11: plastic rotation = Maximum calculated rotation yield rotation
Column 12: LIF calculated with NLS procedure
Column 13: yield rotation = ((R / k) / L)
Column 14: normalized rotation = plastic rotation / yield rotation
Column 15: DIF calculated with LS procedure
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59
Table 12, Columns Description
Column 1: Beams span (L), center to center.
Column 2: beam width (b)
Column 3: beam depth (d)
Column 4: average of top steel and bottom steel (Aavg)
Column 5: distance from edge of beam to center of reinforcing bars.
Column 6: gross moment of inertia of section (Ig) = (1/12) b h3
Column 7: cracked moment of inertia (Icr)
Column 8: average moment of inertia (Iavg)
Column 9: beam stiffness (k) = (384 EIavg/ B L4) for a fixed-fixed beam, B is the tributary width
Column 10: beam ultimate resistance (R) = (8*(2M)/B*L2) for a fixed-fixed beam
Column 11: average moment (M) = Aavg fy * (d-0.5a), where a is depth of compression block
Column 12: applied load in psi
Column 13: maximum displacement calculated at loss location with SDOF approach
Column 14: maximum calculated rotation = maximum displacement / L
Column 15: plastic rotation = Maximum calculated rotation yield rotation
Column 16: LIF calculated with NLS procedure
Column 17: DIF calculated with LS procedure
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Table 10. Complete Results for RC 3-Dimmensional AP Analyses Type beam
span beam width
beam depth
avg steel area
bar ctr cover
calc beam Ig
ASCE 41 beam
stiffness
calc beam R
calc beam M
SAP calc disp
calc rotation
plastic rotation
ASCE41 beam yield
rotation
norm rotation
calc LIF calc DIF
(in) (in) (in) (in^2) (in) (in^4) (psi/in) (psi) (k-ft) (in) (rad) (rad) (rad) 3-story, 20 ft bay,
interior 240 20 22 2.21 2.5 17747 1.21 0.90 258.46 2.4 0.010 0.0071 0.0031 2.284 4.70 1.129
240 20 22 2.09 2.5 17747 1.21 0.85 244.63 4.1 0.017 0.0142 0.0029 4.837 8.00 1.072 240 20 22 1.99 2.5 17747 1.21 0.81 233.57 9.8 0.041 0.0381 0.0028 13.586 19.00 1.038 240 20 22 1.95 2.5 17747 1.21 0.80 229.27 15.8 0.066 0.0629 0.0028 22.859 30.70 1.039 240 20 22 1.93 2.5 17747 1.21 0.79 226.63 20.6 0.086 0.0831 0.0027 30.550 39.20 1.044
3-Story, 25 ft bay, Interior
Girder 300 30 20 6.04 2 20000 0.45 1.11 625.61 6.3 0.021 0.0128 0.0083 1.533 4.00 1.14Int. Beam 300 24 20 3.64 2 16000 0.36 0.68 385.14 6.3 0.021 0.0147 0.0064 2.292 4.00
Girder 300 30 20 5.76 2 20000 0.45 1.07 599.35 8.4 0.028 0.0200 0.0080 2.504 5.50 1.09Int. Beam 300 24 20 3.47 2 16000 0.36 0.66 368.60 8.4 0.028 0.0218 0.0061 3.559 5.50
Girder 300 30 20 5.52 2 20000 0.45 1.02 576.19 12.5 0.042 0.0339 0.0077 4.418 8.00 1.06Int. Beam 300 24 20 3.33 2 16000 0.36 0.63 354.04 12.5 0.042 0.0357 0.0059 6.054 8.00
Girder 300 30 20 5.42 2 20000 0.45 1.01 566.21 15.9 0.053 0.0456 0.0075 6.043 10.00 1.05Int. Beam 300 24 20 3.27 2 16000 0.36 0.62 347.78 15.9 0.053 0.0473 0.0058 8.174 10.00
Girder 300 30 20 5.35 2 20000 0.45 0.99 559.54 19.2 0.064 0.0566 0.0075 7.599 13.00 1.05Int. Beam 300 24 20 3.22 2 16000 0.36 0.61 343.59 19.2 0.064 0.0583 0.0057 10.203 13.00
Girder 300 30 20 5.24 2 20000 0.45 0.98 549.51 25.8 0.086 0.0788 0.0073 10.772 17.00 1.04Int. Beam 300 24 20 3.16 2 16000 0.36 0.60 337.31 25.8 0.086 0.0805 0.0056 14.343 17.00
Girder 300 30 20 5.21 2 20000 0.45 0.97 546.16 29.2 0.097 0.0901 0.0073 12.385 19.00 1.04Int. Beam 300 24 20 3.14 2 16000 0.36 0.60 335.21 29.2 0.097 0.0918 0.0056 16.446 19.00
3-story (over-reinforced), 20 ft bay,
corner
240