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”'3 LIBRARY
Michigan State
University

 

 

 

This is to certify that the
dissertation entitled

CAPACITY REDUCTION AND FIRE LOAD FACTORS FOR
LRFD OF STEEL MEMBERS EXPOSED TO FIRE

presented by

SHAHID IQBAL

has been accepted towards fulfillment
of the requirements for the

Ph.D degree in Civil Engineering

 

 

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CAPACITY REDUCTION AND FIRE LOAD FACTORS FOR LRFD OF
STEEL MEMBERS EXPOSED TO FIRE

By

Shahid Iqbal

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY

Civil Engineering

2010

ABSTRACT

CAPACITY REDUCTION AND FIRE LOAD FACTORS FOR LRFD OF '
STEEL MEMBERS EXPOSED TO FIRE

By
Shahid Iqbal
Until recently, structural steel members exposed to fire were designed using prescriptive
approaches that do not account for actual loading conditions and real ﬁre scenarios. The
last decade has seen the promotion of performance-based codes which allow more
rational engineering approaches for the design of steel members exposed to ﬁre. For
example, Appendix 4 of the 2005 AISC Speciﬁcations (referred to hereafter as “AISC
Specifications”) now allows steel members to be designed against ﬁre using room
temperature design speciﬁcations and reduced material properties. The AISC
Speciﬁcations allow using the same capacity reduction factors for fire design as those
used for room temperature design. For example, a capacity reduction factor of 0.9 is
suggested for steel beams and columns. Most other codes suggest that a capacity
reduction factor of 1.0 be used. This recommendation is based on arguments that the
probability of fire occurrence and the strength falling below the design value
simultaneously is very small, and that fire design is based on the most likely expected
strength. The Commentary to the AISC Speciﬁcations also states that the ﬁre load may
be reduced by up to 60 percent if a sprinkler system is installed in the building.
Automatic sprinklers reduce the probability of occurrence of a severe fire. The reduction
in ﬁre load should be based on proper reliability analysis that includes the effect of

sprinklers on the occurrence of a severe fire, and correspondingly on the probability of

failure of structural steel members. However, limited work has been done to develop
capacity reduction and ﬁre load factors based on reliability analysis.

In this study, a general reliability-based methodology is proposed for developing capacity
reduction and fire load factors for design of steel members exposed to fire. The effect of
active ﬁre protection systems (e.g., sprinklers, fire brigade, etc.) in reducing the
probability of occurrence of a severe ﬁre is included. The design parameters that
significantly affect the fire design of steel members such as fire load, opening factor,
thermal absorptivity of compartment boundaries, thickness, density and thermal
conductivity of insulation are taken as random variables. Raw experimental data
published in the literature was analyzed to obtain the statistics of parameters for which no
statistical information was available in the literature. Model errors associated with the
thermal analysis models are also characterized based on experimental data. It is found
that uncertainty associated with the fire design parameters is significantly higher than that
of room temperature design parameters.

To illustrate the proposed methodology, capacity reduction and fire load factors are
developed for simply supported steel beams and axially loaded steel columns in US.
ofﬁce buildings. It is found that the capacity reduction and fire load factors should not be
constant for all design situations as suggested in current design specifications, but should
vary depending on the presence of active ﬁre protection systems in a building.

In addition, a simpliﬁed method is proposed for computing inelastic deflections of simply
supported steel beams exposed to ﬁre, and it is shown that the fire design of most beams

is goverened by the strength limit state.

DEDICATION

I dedicate this dissertation to my mother and father. Without their support and
encouragement, I would have never been in graduate school pursuing a PhD. degree. My
late father worked hard and sacrificed days and nights of his life to support my education
throughout my academic life. I owe this degree to him. I also dedicate this dissertation to
my sons (Hassan and Ans) who did not get the time and attention they needed from me in
the last 4-5 years because of my studies, and had to participate in many events, activities

and celebrations without me.

iv

ACKNOWLEDGMENTS

I express my most sincere gratitude to my advisor Dr. Ronald S. Harichandran, Professor
and Chair of the Department of Civil and Environmental Engineering, Michigan State
University, for the encouragement and guidance received through the course of this
study. In my academic and professional life, I have worked with many people, but, no
one ever influenced my life more than Professor Harichandran. I will always remember
his humbleness, care, and moral and emotional support he provided in these 4.5 years. It
is impossible to appreciate, in few words, his patience and perseverance which made my
graduate studies a great learning experience.

I would like to thank Professor Venkatesh Kodur, Assisstant Professor Lajnef Nizar and
Associate Professor Sasha Kravchenko for serving on my Ph.D committee, and for the
valuable advice and discussions we had throughout my stay at Michigan State University.
I would also like to thank all other faculty members who taught me, especially Dr.
Rigoberto Burgueﬁo who helped me in attaining the requisites skills and training for
pursuing a Ph.D. degree. The material I learned in the courses I took with Dr. Rigoberto
Burgueﬁo, and my one-to-one academic discussions with him were very fruitful, useful,
and really helped me during my Ph.D. work.

For many reasons that I cannot count, I am thankful to my friends Monther Dwaikat,
Mahmud Dwaikat, and Mahmoodul Haq. They are great friends, who were always

available and willing to help whenever I needed it. I would like to thank the many other

students in the civil and environmental engineering department at Michigan State
University for their help and support during my Ph.D. study.

I appreciate the help provided by Laura Taylor, Mary Mroz, Margaret Conner, Joseph
Nguyen, Brooke H.M. Stokdyk, and Mary M. Gebbia-Portice, in solving my
administrative problems, and visa and passport issues.

I am extremely grateful to my brother, my wife, my uncles and aunts, and my cousins
who are always happy for me and for my successes.

I would also take this opportunity to say thanks to my friends Sher Malik, Abdul—ur-
Rehman, Arif Aalam, Nabeel Anwar Cheema, Naeem Olakh, Arif Mehmood, Attiq-ur—
Rehman, Adnan Qadir, Khurram, and Naveed, for being happy on my selection to pursue
my Ph.D. in the US, and for their prayers. I am extremely grateful to Brigadier Gulfam
and Major General Mansha for their support for pursuing my PhD studies.

I am thankful to my teachers Mr. Durez Sahib, Mr. Mehar Khan Sahib, Mr. Akseer
Sahib, Mr. Muhammad Yousaf Sahib, Mr. Malik Nawaz Sahib, Mr. Roshan Ali Sahib,
Mr. Hasnain Shah Sahib, Mr. Tariq Mehmood Sahib, Mr. Bashir Sahib, Mr. Mushtaq
Sahib, Mr. Noor Khan Sahib, Mr. Asim, Brigadier Khaleeq Kayani, Brigadier Gulfam,
Brigadier Jamil, and Brigadier Abeer, for making my academic career a success.

Finally, I thank the E-in-C Branch, Pakistan Army, and Ministry of Defence for
ﬁnancially supporting my Masters and Ph.D. studies, and I am grateful to the staff

members working at these organizations for making this Ph.D. program a success.

vi

TABLE OF CONTENTS

 

 

 

 

 

 

LIST OF TABLES xi
LIST OF FIGURES xii
CHAPTER 1 1
INTRODUCTION 1
1.1 BACKGROUND ............................................................................................................ l
1.2 DESIGN FOR FIRE HAZARD ......................................................................................... 3
1.3 FIRE AND STEEL MEMBERS ........................................................................................ 5
1.3.] F ire-Resistant Design of Steel Members ............................................................ 5
1.3.2 Capacity Reduction Factor for Fire Design ...................................................... 8
1.3.3 Reduction in Fire load ....................................................................................... 8
1.3.4 Uncertainties in Fire Design of Steel Members ................................................. 9
1.3.5 Deﬂections of Steel Beams Exposed to Fire .................................................... 10

1.4 RESEARCH OBJECTTVES ............................................................................................ 1 1
1.5 SCOPE ...................................................................................................................... 12
1.6 ORGANIZATION OF THE DISSERTATION .................................................................... 13
CHAPTER 2 15
STATE-OF-THE-ART REVIEW 15
2.1 INTRODUCTION ......................................................................................................... 15
2.2 APPROACHES FOR EVALUATING FIRE RESISTANCE OF STEEL MEMBERS ................. 16
2.2.1 Prescriptive Approach ..................................................................................... 1 7
2.2.1.1 Determination of Fire Resistance Rating .......................................... 17
2.2.1.2 Standard Fire Test ................................................................................ 18

2.2.2 Performance-Based Approach ......................................................................... 20
2.2.3 Detailed Calculation Methods ......................................................................... 21
2.2.3.1 Fire Temperature .................................................................................. 23
2.2.3.2 Temperature of Structural Member .................................................... 23
2.2.3.3 Strength Calculations ........................................................................... 24
2.3.3.4 Critical Factors in Fire Resistance Calculations .............................. 24

2.2.4 Fire Scenarios .................................................................................................. 25
2.2.4.1 Room Fires ............................................................................................. 25
2.2.4.2 Standard Fire ......................................................................................... 27
2.2.4.3 Design (or Real) Fires .......................................................................... 29

2.2.5 Loading Conditions .......................................................................................... 31
2.2.6 Failure Criteria ................................................................................................ 32

2.3 USE OF RELIABILITY THEORY IN THE DEVELOPMENT OF DESIGN SPECIFICATIONS .. 34
2.3.1 Background ...................................................................................................... 38
2.3.2 F irst—Order Second-Moment ( F OSM ) Analysis ............................................... 41
2.3.3 Hasofer-Lind Reliability Index ........................................................................ 45
2.3.4 Approximate Methods for Including Information on Distributions ................. 49

Vii

2.3.6 Selection of Load and Capacity Reduction Factors ........................................ 54
2.4 STUDIES RELATED TO RELIABILITY ANALYSES OF STRUCTURAL MEMBERS EXPOSED

 

 

 

To FIRE .......................................................................................................................... 61
2.4.1 ECSC Study, 2001 ............................................................................................ 62
2.4.2 Magnusson and Pettersson, I981 .................................................................... 63
2.4.3 Beck, 1985 ........................................................................................................ 70
2.4.4 Li and Fitzgerald, 1996 .................................................................................... 72
2.4.5 Ellingwood, 2005 ............................................................................................. 73
2.4.6 CIB W14, 1983 and 1986 ................................................................................. 74
2.4. 7 Ramachandran, 1995a ..................................................................................... 75
2.4.8 Ramachandran, 1995b ..................................................................................... 76
2.4.9 Harmathy and Mehaffey, 1985 ......................................................................... 77
2.4.10 Fellinger, 2000 ............................................................................................... 77
2.4.11 Janes, 1995 .................................................................................................... 78

2.5 SUMMARY ................................................................................................................ 78

CHAPTER 3 81
RELIABILITY-BASED METHODOLOGY FOR DEVELOPING CAPACITY
REDUCTION AND FIRE LOAD FACTORS 81

3.1 GENERAL ................................................................................................................. 81

3.2 ENGINEERING APPROACH FOR DESIGNING STEEL MEMBERS EXPOSED To FIRE ...... 82

3.3 FIRE TEMPERATURE IN THE COMPARTMENT ............................................................ 84

3.4 CALCULATING THE TEMPERATURE OF STEEL MEMBERS .......................................... 86

3.5 METHODOLOGY FOR DEVELOPING CAPACITY REDUCTION AND FIRE LOAD FACTORS

....................................................................................................................................... 89
3.5.1 Performance Function for Reliability Analysis ............................................... 89

3.5.1.1 Applied Loads ........................................................................................ 89
3.5.1.2 Capacity of Steel Members ................................................................. 90
3.5.1.3 Limit State Equation ............................................................................. 92
3.5.2 Statistics of Random Parameters ..................................................................... 92
3.5.3 Probability of Failure and Target Reliability Index ........................................ 92
3.5.4 Reliability Analysis .......................................................................................... 97
3.5.5 Determination of Partial Safety Factors .......................................................... 97
3.5.6 Combination of Partial Safety Factors into Single Capacity Reduction Factor
................................................................................................................................... 98
3.5.7 Optimal Capacity Reduction and Fire Load Factor ...................................... 100
3.6 SUMMARY .............................................................................................................. 101
CHAPTER 4 103
STATISTICS OF RANDOM PARAMETERS AND MODEL ERRORS .............. 103

4.1 STATISTICS OF RANDOM PARAMETERS .................................................................. 104
4.1.1 Fire Load Density .......................................................................................... 106
4.1.2 Ratio of Floor Area to Total Surface Area of the Compartment ................... 111
4.1.3 Opening Factor .............................................................................................. 112
4.1.4 Thermal Absorptivity of Compartment Enclosure ......................................... 113
4.1.5 Thickness of Insulation or Fire Protection Materials .................................... 117

viii

4.1.6 Thermal Conductivity and Density of Insulation ........................................... 1 18

 

 

4.2 MODEL ERRORS FOR THERMAL MODELS ............................................................... 120
4.2.1 Model Error for Steel and Fire Temperatures ............................................... 120
4.2.2 Model Error for Reduction Factors ............................................................... 122

4.3 SUMMARY .............................................................................................................. 124

CHAPTER 5 126
CAPACITY REDUCTION AND FIRE LOAD FACTORS FOR STEEL BEAMS
EXPOSED TO FIRE 126

5.1 DERIVATION 0F CAPACITY REDUCTION AND FIRE LOAD FACTORS ........................ 127

5.1.1 Performance Functions for Reliability Analysis ............................................ 127
5.1.1.1 Applied Moment under Fire ............................................................... 127
5.1.1.2 Moment Capacities of Beams under Fire ........................................ 128
5.1.1.3 Performnace Function for Reliability Analysis ................................ 133

5.1.2 Model Error (Professional Factor) for Moment Capacity Equations ........... 133

5.1.3 Probability of Failure and Target Reliability Index ...................................... 134

5.1.4 Reliability Analyses ........................................................................................ 134
5.1.4.1 Reliability Analyses of Laterally Restrained Beams ...................... 138
5.1.4.2 Reliability Analyses of Laterally Unrestrained Beams .................. 139

5.2 RESULTS ................................................................................................................ 139
5.2.1 Capacity Reduction Factor ............................................................................ I39
5.2.2Fire Load Factor ............................................................................................ 141
5.2.3 Validity of Capacity Reduction and Fire Load Factors for Multiple Fire
Scenarios ................................................................................................................. 142

5.2.4 Comparison of F ire Load Factors with those Based on the ECSC Method.. 147
5.2.5 Insulation Thicknesses from Performance-Based and Prescriptive Design

 

 

Approaches ............................................................................................................. 149
5.2.6 Reliability Inherent in AISC Fire Design Provisions ..................................... 151
5.3 DESIGN EXAMPLES ................................................................................................ 151
5.3.1 Design Example (Laterally Restrained Beam) ............................................... 152
5.3.2 Design Example (Laterally Unrestrained Beam) .......................................... 155
5.4 SUMMARY ........................................................................................... . ................... 160
CHAPTER 6 162
CAPACITY REDUCTION AND FIRE LOAD FACTORS FOR STEEL
COLUMNS EXPOSED TO FIRE 162
6.1 DERIVATION OF CAPACITY REDUCTION AND FIRE LOAD FACTORS ........................ 162
6.1.1 Performance Functions for Reliability Analysis ............................................ 162
6.1.1.1 Applied Axial Loads under Fire ......................................................... 162
6.1.1.2 Axial Capacity of Columns under Fire ............................................. 163
6.1.2 Professional Factor (Model Error) for Axial Capacity of Columns .............. 165
6.1.3 Probability of Failure and Target Reliability Index ...................................... 166
6.1.4 Reliability Analysis ........................................................................................ 166
6.2 RESULTS ................................................................................................................ 171
6.2.] Capacity Reduction Factor ............................................................................ 171
6.2.2 Fire Load Factor ............................................................................................ I 72

6.2.3 Validity of Capacity Reduction and Fire Load Factors for Diﬁ’erent

 

 

 

 

Slendemess Ratios .................................................................................................. 1 72
6.2.4 Validity of Capacity Reduction and Fire Load Factors for Different Live to
Dead Load Ratios ................................................................................................... 173
6.2.5 Validity of Capacity Reduction and Fire Load F actors for Multiple Fire
Scenarios ................................................................................................................. I 75
6.3 DESIGN EXAMPLE .................................................................................................. 177
6.4 SUMMARY .............................................................................................................. 182
CHAPTER 7 183
DEFLECTIONS OF SIMPLY SUPPORTED STEEL BEAMS EXPOSED TO FIRE
183
7.1 BACKGROUND ........................................................................................................ 183
7.2 DEVELOPENT OF SIMPLIFIED METHOD FOR ESTIMATING DEFLECTIONS OF SIMPLY
SUPPORTED BEAMS ...................................................................................................... 184
7. 2. 1 Approach ........................................................................................................ 184
7.2.1.1 Equivalent Flexural Rigidity ............................................................... 186
7.2.1.2 Finite Element Model .......................................................................... 189
7.2.1.3 Parametric Analysis ............... . ............................................................ 194
7.2.2 Validation of Simpliﬁed Method .................................................................... 198
7.2.2.1 Effect of Heating Rate of Steel ......................................................... 200
7.2.2.2 Effect of Load Ratio ............................................................................ 200
7.2.2.3 Effect of Sectional Geometry and Span Length ............................. 202
7.2.2.4 Effect of Load Configuration .............................................................. 202
7.2.3 Comparison with Test Results ........................................................................ 203
7.3 DEFLECTIONS OF SIMPLY SUPPORTED BEAMS ....................................................... 205
7.4 SUMMARY .............................................................................................................. 209
CHAPTER 8 210
CONCLUSIONS AND RECOMMENDATIONS 210
8.1 SUMMARY .............................................................................................................. 210
8.2 CONCLUSIONS ........................................................................................................ 212
8.3 RECOMMENDATIONS FOR FUTURE RESEARCH ........................................................ 214
REFERENCES 217

 

LIST OF TABLES

Table 3.1 - Range of target reliability index values based on the the effect of active fire
protection systems in reducing the probability of occurrence of a severe fire ................. 96

Table 4.1 - Mean, COV and distributions of design parameters related to loads ........... 104
Table 4.2 - Mean, COV, distributions and nominal values of fire design parameters... 105

Table 4.3 - Comparison of statistics of fire load density (MJ/m2 of floor area) in ofﬁces

......................................................................................................................................... 109
Table 4.4 - Comparison of ﬁre load density in offices (Yii , 2000) ............................... 109
Table 5.1 — Mean and COV of room temperature design parameters ............................. 129
Table 5.2 — Properties of laterally restrained simply supported beams used for reliability
analyses ........................................................................................................................... 136
Table 5.3 — Properties of laterally unrestrained simply supported beams used for
reliability analyses .......................................................................................................... 137
Table 5.4 - Combinations of b and F v used for nine fire scenarios ................................. 144

Table 5.5 — Values of b, F v and qt used for nine ﬁre scenarios corresponding to ,8, =1.5
......................................................................................................................................... 145

Table 5.6 - Comparison of insulation thickness .............................................................. 150
Table 6.1 - Properties of axially loaded steel columns used for reliability analyses ..... 168
Table 6.2 - Properties of columns used for validation .................................................... 175

Table 7.1 - Comparison of ﬁre resistance time predicted by the simpliﬁed method with
test results at the limiting deﬂection value of L/30 ......................................................... 205

Table 7.2 — Comparison of deﬂections of laterally restrained simply supported with the
limiting values of U30 or U20 ...................................................................................... 207

Table 7.3 — Comparison of deflections of laterally unrestrained simply supported with the
limiting values of U30 or U20 ....................................................................................... 207

xi

LIST OF FIGURES

Figure 1.1 — Comparison of a design (or real) and a standard fire ...................................... 6

Figure 2.1 - Time-temperature curve for ASTM E119 standard fire and two design ﬁre
exposures ........................................................................................................................... 19

Figure 2.2 — Degradation of strength of steel beam exposed to fire ................................. 21

Figure 2.3 — Flowchart for calculating the fire resistance of a structural system

(Reporduced from Kodur 2007) ........................................................................................ 22
Figure 2.4 - Various stages of ﬁre ................................................................ . ................... 27
Figure 2.5 - Fundamentals of risk-based design evaluations ........................................... 35
Figure 2.6 - Probability of failure, pp (Reproduced from Harichandran 2005) ................ 39
Figure 2.7 - Interpretation of the reliability index ,8 (Reproduced from Harichandran
2005) ................................................................................................................................. 41
Figure 2.8 - Linear approximation to non-linear function (Reproduced from Harichandran
2005) ................................................................................................................................. 43
Figure 2.9 - Illustration of failure boundary and design point in x- and z-spaces for two
variables problem (Reproduced from Harichandran 2005) .............................................. 47
Figure 2.10- Illustration of failure boundary adjustment to achieve target reliability index
(Reproduced from Harichandran 2005) ............................................................................ 56
Figure 4.1 - Comparison of yield strength and modulus of elasticity reduction models
with test data ................................................................................................................... 123
Figure 5.1 - Capacity reduction and fire load factors for steel beams ............................ 140
Figure 5.2 - Nine fire scenarios for target reliability index value of 1.5 ........................ 145
Figure 5.3 - Comparison of computed and target reliability index values ...................... 146

Figure 5.4 - Comparison of fire load factors with those obtained by the ECSC method 149

Figure 5.5 - Fire and steel temperatures VS. time used in computing ﬁre resistance for
beam in example 2 .......................................................................................................... 160

Figure 6.1 - Ratio of test capacity to nominal capacity of columns for different
Slendemess ratios ............................................................................................................ 166

xii

Figure 6.2 - Capacity reduction and fire load factors vs. target reliability index for
columns ........................................................................................................................... 171

Figure 6.3 Computed and target reliability index values for different Slendemess ratios
......................................................................................................................................... 173

Figure 6.4 - Computed and target reliability index values for different live to dead load
ratios ................................................................................................................................ 174

Figure 6.5 - Computed and target reliability index values for columns for different ﬁre
scenarios .......................................................................................................................... 176

Figure 6.6 - Computed and target reliability index values for columns with reduced

- capacity reduction factors ............................................................................................... 177
Figure 7.1 - Typical moment—curvature diagram for a steel section ............................... 186
Figure 7.2 - Variation of ultimate tensile strength of steel ............................................. 188
Figure 7.3 — Mechanical properties for structural steel at elevated temperature ............. 191
Figure 7.4 - Validation of high-temperature creep model .............................................. 192

Figure 7.5 - Comparison of deflections of Steel beams predicted by ANSYS with the test
data for (a) protected beam, and (b) an unprotected beam ............................................. 193

Figure 7.6 - Variation of normalized flexural rigidity with normalized inelastic moment
for different load and Slendemess ratios ......................................................................... 196

Figure 7.7 - Deﬂections predicted by ANSYS (broken lines) and simplified method (solid
lines) for different heating rates. Load ratio = 0.5, load type = UDL ............................. 201

Figure 7.8 - Deflections predicted by AN SYS (broken lines) and simpliﬁed method (solid
lines) for different load ratios. HR = 45°F/min, load ratio = 0.5, load type = UDL ....... 201

Figure 7.9 - Deflections predicted by ANSYS (broken lines) and simpliﬁed method (solid
lines) for different sections and span lengths. Load ratio = 0.5, HR = 45 °F/min, load type
= UDL ............................................................................................................................. 202

Figure 7.10 - Deﬂections predicted by ANSYS (broken lines) and simplified method
(solid lines) for different load conﬁgurations. Load ratio=0.5, HR: 45 °F/min ............. 203

xiii

Chapter 1

Introduction

1. 1 Background

Steel is widely used as a construction material and finds a wide range of applications in
buildings, bridges, and other infrastructure. It competes well with other construction
materials because of its light weight, high strength, reliability and speed of construction.
Steel framing is lighter and thinner compared to its counterparts such as concrete and
wood and thus results in saving of space and cost.

Although steel is non-combustible and adds no fuel in a fire, it rapidly loses its strength
and stiffness at elevated temperatures, which is a cause of concern for fire engineers.
Over the past few decades, there has been signiﬁcant research on steel structures exposed
to ﬁre. Today, the properties and behavior of steel structures exposed to ﬁre is better

known compared to many other construction materials. A wide variety of ﬁre protection

materials are used to enhance the performance of steel under ﬁre conditions. In recent
years, ﬁre-resistant steel is being used extensively in countries like Japan. However, in
the US, most steel buildings are still designed against ﬁre using ﬁre protection
materials.

Fire represents one of the severe hazards to which a structure might be subjected during
its lifetime and the performance of steel buildings under ﬁre is often questionable. Thus,
the provision of appropriate fire safety measures is an important aspect of building design
for ensuring the safety of occupants, and to protect property and the environment. Fire
safety measures have two components: active measures such as automatic sprinklers to
control and suppress the fire, and passive measures such as the provision of adequate fire
resistance to the structural and non-structural components of the building so that they do
not collapse during the ﬁre duration specified in codes. Active systems include automatic
sprinklers, smoke detectors, heat detectors, fire alarms and ﬁre brigades. Passive systems
control the ﬁre or ﬁre effects by the systems that are built into the structure or fabric of
the building, for example, structural and non-structural components like beams, columns,
and walls of the building. These building components help control the spread of ﬁre and
also ensure the structural integrity of the building, if provided with adequate ﬁre
resistance. Passive ﬁre protection can be achieved by protecting structural members in a
variety of ways, e.g., by applying spray applied materials (sprayed mineral ﬁber,
vermiculate plaster etc.), using intumescent coating, or using board materials (e.g.,
gypsum board) as insulation.

Fire resistance is the time duration during which a structural member or system exhibits

resistance with respect to structural integrity, stability and temperature transmission

(Buchanan 2001). Although, there are different ways to provide fire resistance to building
components, for Structural Steel members, ﬁre resistance is generally provided using fire
protection materials. The significance of the fire resistance can be attributed to the fact
that when other measures such as active fire protection systems fail to control the fire, the
fire resistance of building elements is the last line of defense for the safety of people,
property and the environment. Fire resistance can play a crucial role on the performance
of a structure in the event of ﬁre as seen in the collapse of the WTC Twin Towers

(FEMA 2002).

1.2 Design for Fire Hazard

Fires cause significant loss of life and property damage throughout the world. In 2007,
ﬁres in the US. caused about 3430 deaths, 97,800 injuries and property damage worth
$14 billion. The total cost of ﬁre (including direct and indirect losses) was estimated to be
around $70 billion in 2007 (Karter 2008).

Design for ﬁre safety has been improved significantly in the last century. This
improvement focused mainly on reducing the loss of life by introducing active fire safety
systems (which are self-triggered in the event of ﬁre) such as sprinklers and ﬁre
detections systems. In 1913, the death rate due to ﬁre in the United States was 9.1% per
100,000 people. However, the use of fire resistant material and improvements in design
(including smoke detectors) reduced that rate to 2% per 100,000 people in 1988
(Williams 2004). Generally, the major cause of fatalities and injuries is toxic smoke
evolving during a ﬁre event. However, some of these deaths and injuries, particularly
among ﬁrst responders, were due to structural collapse/damage under fire conditions

(FEMA 2008).

The following realistic examples illustrate the catastrophic consequences that may result

from ﬁre incidents:

On Oct. 26, 1986, a serious fire occurred in the lS-story Alexis Nihon Hotel in
Montreal, Canada (Inser 1988). This was a steel framed building with a
composite steel beams and deck floor. The fire continued for 14 hours and
resulted in the partial collapse of the 11th ﬂoor. All walls, decks and beams on
the four fire ﬂoors had to be replaced and the repair cost was $80 million.

On Feb. 23, 1991, a serious fire occurred in the 38-story Meridian Plaza office
building in Philadelphia, U.S. (Routley et al. 1991). This was a steel framed
building with composite steel beams and deck floors. The fire burned for 18
hours causing significant structural damage to 9 floors. Three fire fighters died.
On May 4, 1988, a fire occurred on the 12th floor of the First Interstate Bank in
Los Angles, U.S (Klem 1988). This was a 62 story office building having steel
framing with composite beams and deck floors. The ﬁre lasted for 3.5 hours and
caused major damage to four floors.

On Sep. 11, 2001, the 47-Story WTC 7 office building totally collapsed due to

ﬁre.

The above examples clearly illustrate the need for structural fire safety for minimizing

the catastrophic collapse of buildings and the built-infrastructure. To minimize the

possibility of structural failure during ﬁre, building codes such as the International

Building Code, specify ﬁre resistance requirements for structural systems in buildings.

The required ﬁre resistance varies based on the importance of occupancy, number of

stories and floor area of the building, type of the structural members (such as Slab, beam

or column), and also on the availability of active fire protection systems (e. g., sprinklers)
in the building. As an illustration, the fire resistance requirement for a steel beam in a

typical building can vary from 1 to 3 hours.

1.3 Fire and Steel Members

1.3.1 Fire-Resistant Design of Steel Members

The fire resistance requirement for structural members (i.e., the time during which a
structural member should perform its intended function) is speciﬁed in building codes
based on factors such as the type of occupancy, height of the building, and importance of
the structural member. Until now, the required fire resistance for Steel members (e.g.,
beams, columns, etc.) was provided using prescriptive approaches, and the ﬁre resistance
rating of a large number of steel sections is included in ﬁre resistance directories such as
the Underwriter Laboratory Fire Resistance Directory (UL 2004). The designer selects
the required thickness of the ﬁre protection material from the listed steel sections by
ensuring that the selected thickness of ﬁre protection material yields a ﬁre resistance
more than that required by the building codes. The ﬁre resistance rating of listed steel
sections is determined from tests conducted in a furnace under a standard ﬁre. If the
section being designed is not included in the ﬁre resistance directory, the thickness of the
ﬁre protection material is calculated using empirically derived correlations (Ruddy et a1.
2003).

The prescriptive approach described above has yielded adequate performance, but is
considered too conservative and simplistic because it does not account for actual loading

conditions and real ﬁre scenarios. The temperature of a typical structural steel member

exposed to a real fire (or design fire) and the standard fire used in the prescriptive
approach is compared in Figure 1.1. The two ﬁres are quite different and are likely to
yield different performances for structural members.

Over the last decade, performance-based codes which allow more rational engineering
approaches for the fire design of steel members are being promoted. For example,
Appendix 4 of the 2005 AISC Specifications (referred to hereafter as “AISC
Speciﬁcations”) now allows steel members to be designed against ﬁre using room
temperature design speciﬁcations and reduced material properties. Similar provisions
were developed by the European Convention for Constructional Steel work (ECCS 2001)
and are included in the Eurocode 3 (EN 2005). Using this engineering approach, the
veriﬁcation of design for strength during fire requires that the resistance (capacity) is

greater than the load effects. This leads to satisfying the design equation

 

 

¢fRn,f 2Qu,f (1'1)
1200 . standard fire
typical real ﬁre (or design fire)
rooo — /
a /
Q, 800 .
g -
o- 600 -r
E
8
E 400 1
'1’ steel temperature, real ﬁre
200 ~
Steel temperature, standard fire fire
0 f r I l r r
O 20 40 60 80 100 120

Time (minutes)

Figure 1.1 - Comparison of a design (or real) and a standard ﬁre

where Quf is the load effect at the time of ﬁre, Rnf is the nominal capacity at the time of

ﬁre, and ¢f is the capacity reduction factor. The AISC Specifications (AISC 2005a)

allow using the same capacity reduction factors for fire design as those used for room

temperature design. For example, ¢f = 0.9 is suggested for steel beams and columns.
Most other codes suggest that a capacity reduction factor of 1.0 be used (e.g., in the

Eurocode 3 (EN 2005), the partial safety factor TM is 1.0 for fire design).

When a steel member is exposed to fire. the temperature of the steel increases and the
strength and stiffness of the steel are reduced, leading to significant loss of capacity. The
temperature attained by a steel member depends onthe area of steel exposed to ﬁre, the
amount of protection applied, and on the duration and severity of a design fire. The AISC
Speciﬁcations (AISC 2005a) require that while calculating the steel and fire temperatures
for ﬁre design, due consideration should be given to the effectiveness of all active fire
protection systems (sprinklers, smoke and heat detectors, etc.). The commentary to the
AISC Speciﬁcations (AISC 2005b) states that the ﬁre load may be reduced by up to 60
percent if a sprinkler system is installed in the building.

The ﬁre design time, during which Equation 1-1 should be satisﬁed, is not precisely
deﬁned. Design speciﬁcations in various countries use this time in different ways for
different occupancies. For example, single Storey buildings may be designed to protect
the escape routes and to remain standing only long enough for occupants to escape the
building. On the other hand, tall and high rise buildings where people cannot escape are

designed to prevent the spread of fire, or collapse of the building.

1.3.2 Capacity Reduction Factor for Fire Design

The capacity reduction factor, ¢ f, accounts for the uncertainties in the material

properties, variations in dimensional and sectional properties, and, approximations and

assumptions used in the design and analysis procedures. As mentioned earlier, for fire

design of steel members, ¢f= 0.9 is suggested in the AISC Specifications (AISC
2005a), but most other codes suggest that a capacity reduction factor of 1.0 be used (e.g.,

in the Eurocode 3 (EN 2005), the partial safety factor 1’ M is 1.0 for fire design). This

recommendation is based on arguments that the probability of fire occurrence and the
strength falling below the design value simultaneously is very small, and that fire design
is based on the most likely expected strength (Buchanan 2001). Also, it is expected that
live loads under fire conditions are likely to be smaller than those at room temperature
conditions and hence there will be enough reserve strength available, especially for
members that have been designed for load combinations such as wind, snow or
earthquakes, or for members proportioned for deﬂection control or for architectural
reasons (Buchanan 2001). However, limited work has been done to develop capacity

reduction factors based on reliability analysis (Magnusson and Pettersson 1981), and

neither ¢ f = 0.9, nor ¢f = 1.0 are based on reliability considerations.

1.3.3 Reduction in Fire load

As mentioned previously, the AISC Specifications (AISC 2005a) suggest that while
describing the design ﬁre, due consideration should be given to the effectiveness of all

active ﬁre protection systems (sprinklers, smoke and heat detectors, etc.). The

Commentary to the 2005 AISC Specifications (AISC 2005b) states that while describing
the design ﬁre, the ﬁre load may be reduced by up to 60 percent if a sprinkler system is
installed in the building.

Automatic sprinklers reduce the probability of occurrence of a severe ﬁre. The reduction
in fire load should be based on proper reliability analysis that includes the effect of
sprinklers on the occurrence of a severe ﬁre, and correspondingly on the probability of
failure of structural steel members. Recently, a study was conducted in Europe through a
research project of the European Coal and Steel Community (ECSC) (hereinafter referred
to as the ECSC study) to develop ﬁre load factors by taking into account the variability of
the ﬁre load and the effect of active fire protection systems (ECSC 2001). However, the
fire load factors were obtained using simplified assumptions instead of rigorous
reliability analysis. It is not apparent whether rigorous reliability analysis would yield

results similar to those of the ECSC Study.

1.3.4 Uncertainties in Fire Design of Steel Members

The design of structural members exposed to fire is quite complex because both thermal
and structural analyses must be performed. Compared to room temperature design, there
are many more parameters involved in ﬁre design (e.g., ﬁre load, ventilation or opening
factor, thermal properties of compartment enclosures, thickness and thermal properties of
insulation or fire protection materials, and thermal properties of steel). The assumptions
and approximations used in thermal analyses add further uncertainty to the design
process. In the ECSC study, the variability of ﬁre load was accounted for while
developing the ﬁre load factors, but the uncertainty associated with other ﬁre design

parameters was not explicitly included. As was done for room temperature design, the

capacity reduction and fire load factors for fire design should be based on rigorous
reliability analysis that includes the uncertainty associated with all design parameters.

In CIB design guides (CIB 1983 and CIB 1986), a procedure is given for deveIOping the
load and resistance factors for ﬁre design. Ellingwood (2005) has recommended load
factors and load combinations for the ﬁre design and these recommendations were
incorporated in the ASCE Speciﬁcations (ASCE 07 2005) and Appendix 4 of the AISC
Speciﬁcations (AISC 2005a). However, not much work has been done to develop

capacity reduction and ﬁre load factors based on rigorous reliability analysis.

1.3.5 Deflections of Steel Beams Exposed to Fire

Large deformations are expected under severe fire conditions, so deflections are not
normally computed unless they affect the structural performance (Buchanan 2001). The
AISC Specifications (AISC 2005a) specifies that for structural elements, e.g., beams and
columns, the governing limit state is loss of load-bearing capacity. However, the AISC
Speciﬁcations also specify that excessive deformations are not acceptable if these
damage the integrity of the ﬁre compartment. If not limited, beam deformations may
damage the slabs and walls of the compartment to the extent that these components
cannot control the horizontal and vertical spread of fire even though the beam may still
have sufﬁcient load bearing capacity. Additionally, deformations need to be limited
because long beams may fail due to an abrupt increase of their deflection despite the fact
that they may have not reached their ultimate load bearing capacity (Skowronski 1990).
Furthermore, beams designed with a large margin of safety against collapse in fire, can
lose that margin due to the abrupt growth of strains caused by creep at elevated

temperatures even when stresses are not critical (Skowronski 1990).

10

Since design under fire may be governed either by the strength or deﬂection limit states,
both should be considered. Design speciﬁcations such as the AISC Specifications (AISC
2005a) and Eurocode 3 (EN 2005) provide simple design equations for the strength limit
state of steel beams which can be used for estimating the fire resistance. Researchers such
as Yin and Wang (2004), Skowronski (1990), Burgess, El-Rimawi and Plank (1990), and
Wang and Yin (2006) provide detailed method for tracing the time-deﬂection behavior of
steel beams, but, these approaches are quite complex. Thus no simple deﬂection
equations which can be quickly implemented in a design speciﬁcation are available for

computing the ﬁre resistance of steel beams.

1.4 Research Objectives

From the above discussion, it is clear that no substantial work has been done on

developing capacity reduction factors and ﬁre load factors for steel members subjected to

ﬁre conditions. ¢f = 0.9, nor ¢f = 1.0 are based on subjective judgment and not on

reliability analysis. The effect of active fire protection systems such as automatic
sprinklers in reducing the probability of occurrence of a severe fire was not explicitly
accounted for while specifying a reduction in fire load in the Commentary to the AISC
Speciﬁcations. In the ECSC study, the variability of the ﬁre load and effects of active ﬁre
protection systems were accounted for while developing the ﬁre load factors. However,
the fire load factors were obtained using simplified assumptions and the study also did
not account for variability in other ﬁre design parameters. Therefore, there is a need to
develop capacity reduction and ﬁre load factors based on reliability analyses for use in

the ﬁre design of steel members instead of using the current ones that are primarily based

ll

on experience and subjective judgment. In addition, although it is recognized that

excessive deﬂections of steel members need to be limited, no simplified method is given

in the AISC Speciﬁcations for estimating the inelastic deﬂections of steel members

exposed to ﬁre.

The objectives of this research are:

0 Propose a general reliability-based methodology for deriving capacity reduction
and ﬁre load factors for the LRFD of steel members exposed to ﬁre.

0 Characterize uncertainties associated with the ﬁre design parameters for deriving
capacity reduction and ﬁre load factors.

0 Develop capacity reduction and fire load factors for axially unrestrained steel
beams exposed to ﬁre.

0 Develop capacity reduction and fire load factors for axially loaded steel columns
exposed to ﬁre.

0 Suggest appropriate design/nominal values of the fire design parameters for the
performance-based design of steel members exposed to fire.

0 Develop a simplified method for predicting the deﬂections (or alternatively fire
resistance time) of simply supported steel beams exposed to fire, and check the
deﬂections of beams designed per strength limit state.

1.5 Scope

The ﬁre load for different types of occupancies varies from one country to another and

even varies signiﬁcantly for different types of occupancies within the same country (CIB

W14 1986). The probability of ﬁre occurrence is also not the same for all types of

occupancies (ECSC 2001). Therefore, it is not possible to develop one set of capacity

12

reduction and fire load factors for all types of occupancies. Accordingly, the CIB W14
Design Guide (1986) suggests that one set of factors may be developed for any type of
occupancy (say ofﬁce buildings), and then differentiated factors may be developed to
modify these basic factors for other type of occupancies. The fire load factors developed
in this study will be applicable to US. office buildings.

Capacity reduction factors developed herein are speciﬁc to Simply supported steel beams
and to axially loaded steel columns for typical fire compartments in US. office buildings.
However, the methodology presented in this study is general and can be used for deriving
capacity reduction factors for other types of structural members (e.g., restrained beams)
by using appropriate performance functions (or design equations), corresponding to the

ﬁre load factors developed in this study.

1.6 Organization of the Dissertation

This dissertation has eight chapters as outlined below.

Chapter 1: Provides background information leading to this research and also presents the
objectives and scope.

Chapter 2: Provides a state-of—the-art review of the methods used to evaluate the fire
resistance of steel members exposed to ﬁre. The chapter also includes a brief presentation
about the use of reliability theory in the development of design specifications. Based on
the literature review, previous studies related to the reliability analysis of steel Structures
exposed to ﬁre are presented and their drawbacks are brieﬂy discussed.

Chapter 3: Presents a general reliability-based methodology for developing capacity

reduction and ﬁre load factors for steel members exposed to ﬁre.

13

Chapter 4: Characterizes the fire design parameters that significantly affect the design of
steel members. Statistics obtained from the literature and those developed from the
experimental data are presented. Model errors associated with the thermal analysis are
also described in this chapter.

Chapter 5: Presents the development of capacity reduction and fire load factors for
laterally restrained and unrestrained Simply supported steel beams. Example design
problems also are given.

Chapter 6: Presents the development of capacity reduction and fire load factors for
axially loaded steel columns. Example design problems also are given.

Chapter 7: Proposes a simplified method for predicting deﬂections of simply supported
steel beams exposed to fire conditions. Also compares the deﬂections of the beams
against the limiting values of deﬂections for the beams designed per Strength limit state.
Chapter 8: Summarizes the main findings of the current study and provides

recommendations for further research.

14

Chapter 2

State-of-the-Art Review

2. 1 Introduction

In the literature, no explicit work has been reported on the derivation of capacity
reduction factors for fire design of steel members except for one case (Magnusson and
Pettersson 1981). Similarly, only one study has been reported in which fire load factors
were developed using simpliﬁed assumptions instead of rigorous reliability analysis
(ECSC 2001). However, during the last few decades, reliability analyses were performed
for estimating the probability of failure and computing reliability indices for steel
members exposed to ﬁre. In some other studies, methodologies were suggested to
develop reliability-based ﬁre safety codes. Some of the important studies are summarized

later in this chapter. Most of the studies related to reliability analysis were performed for

15

Steel members that were designed using the prescriptive design approaches instead Of the
performance-based design approaches being promoted now. Therefore, for better
understanding of the reliability Studies, these design approaches (prescriptive and
performance-based) will be reviewed first. Some important information (e.g., fire
scenarios, loading conditions, failure criterion, etc.) related to the fire design of steel
members is also reviewed in this chapter.

In addition, the use of reliability theory in the development of design speciﬁcations is

also brieﬂy described in this chapter.

2.2 Approaches for Evaluating Fire Resistance of Steel Members

The fire resistance rating requirements for structural members are specified in building
codes (e.g., IBC 2003) based on factors such as the type of occupancy, height of the
building, and importance of the structural member. The ﬁre resistance ratings are given
as a Speciﬁed amount of time the building’s structural members are required to withstand
exposure to a standard ﬁre. For example, in high rise building, a steel beam and a steel
column may be required to have a ﬁre resistance rating of 3 and 4 hours, respectively. In
the US, IBC (2003) allows both prescriptive and performance-based ﬁre-resistant
designs, although its current emphasis is clearly on the former (Ruddy et a1. 2003).

In the US, structural steel members (e.g., beams, columns, etc.) are designed in
accordance with the AISC Speciﬁcations (2005). Before 2005, AISC had a separate
design guide (Ruddy et al. 2003) for the fire-resistant design of steel members which was
based on the applications of prescriptive provisions of International Building Code.
However, presently, Appendix 4 of the AISC Speciﬁcations (2005) allows steel members

to be designed using either engineering analysis (performance—based approach) or by

16

qualification testing (prescriptive approach). According to the engineering approach,
individual members can be designed against ﬁre using room temperature design
provisions and reduced material properties. This design approach allows using one-
dimensional heat transfer analysis and a uniform temperature distribution across the steel
section. In the method of qualification testing, the AISC Specifications require that
structural members and components be qualiﬁed for the rating period in conformance '
with ASTM E 119 (2008).

Eurocode 3 (EN 2005) is one of the most widely accepted design document for fire
design. This code gives a choice of advanced or simplified calculation methods for
designing steel members for fire. As an alternative to designing using calculations, the
code allows that the ﬁre design may be based on the results of ﬁre tests, or on ﬁre tests in
combination with calculations. The simplified method in Eurocode 3 is mainly based on a
sectional analysis approach. This method may give a more economical design and/or high
ﬁre resistance periods for some ﬁre scenarios. Eurocode 3 also allows advanced methods
where detailed thermal and mechanical analyses can be performed for designing

members for ﬁre.

2.2.1 Prescriptive Approach

2.2.1.1 Determination of Fire Resistance Rating

Until now, the required fire resistance for steel members is provided using prescriptive
approaches, and the ﬁre resistance rating of a large number of steel sections is included in
building codes, standards, test reports and special directories of testing laboratories (IBC

2000, IBC 2003, ASCE/SFPE 29, NFPA 2002, UL 2003). Most Of the information about

17

fire resistance ratings is given in the form of charts and tables. The designer selects the
required thickness of the insulation for any steel section from a chart or a table by
ensuring that the selected thickness of the insulation yields a fire resistance rating more
than that required by the building codes. If the section being designed is not included in
the charts and tables, the thickness of the insulation can be determined using empirically
derived correlations (Ruddy et al. 2003).

The ﬁre resistance rating mentioned in these charts and tables is either taken directly
from experiments conducted on steel members in a furnace or extended to untested
members using empirically derived correlations. Most members are tested under a
standard fire which is described in the next sub-section. Some members are tested under
loading, while in other cases the failure criterion is the achievement of a temperature at

which steel is believed to lose most of its structural strength.

2.2.1.2 Standard Fire Test

The ﬁre resistance rating of steel members is generally evaluated by subjecting the
structural member to ﬁre in a specially constructed furnace. The purpose of the standard
ﬁre test is to determine the failure time at which the structure loses the ability to
withstand ﬁre exposure while maintaining function as a load-bearing element and as a
barrier to the spread of ﬁre. The standard ﬁre test is a comparative test that does not
reﬂect the actual performance of the member.

Standard ﬁre resistance tests are generally carried out on building elements such as walls,
ﬂoors, or columns in accordance with national standards such as ASTM E119a (2008)
and ISO 834 (1975). The standards require test specimens be constructed in a similar

manner as the building elements in practice. ASTM E119a specifies the dimensions of

18

the test specimen and the size of the furnace to be used for the standard fire test. Further,
the furnace chamber is heated by liquid fuel or gas such that the furnace follows the
speciﬁed standard ASTM E119 fire time-temperature curve shown in Figure 2.1.

During the ﬁre test, load bearing members, such as steel beams and columns, are
generally loaded at service load levels (dead load + live load) that are about 50% of the

room temperature capacities of the tested members.

 

 

 

 

 

 

 

 

 

 

1500
1200 3
900 I _
‘ —ASTME119ﬁre
, + Severe ﬁre
600 T —a— Moderate ﬁre
300 i '
O I I I I ----------- - ﬂ ] I F
O 30 60 90 120 150 180 210 240

Time (min)

Figure 2.1 - Time-temperature curve for ASTM E119 standard ﬁre and two design
ﬁre exposures
The test continues until a prescribed failure criterion is exceeded. The fire resistance of
the assembly is then recorded as the duration of ﬁre exposure until this failure. The
limiting criterion used for evaluating ﬁre resistance depends on the structural member in

question. As an illustration, for a steel column, the ﬁre resistance rating is the time at

19

which the average temperature at any level increases above 538°C, or the temperature at
any one measuring point increases above 649°C (Ruddy et a1. 2003), or the column fails
to sustain the applied load (ASTM E119a 2008). Different failure limit states are
specified for columns, beams and slabs.

The prescriptive approach described above has yielded adequate performance, but is
considered too conservative and simplistic because it does not account for actual loading

conditions and real ﬁre scenarios.

2.2.2 Performance-Based Approach

Over the last decade, performance-based codes which allow more rational engineering
approaches for the ﬁre design of steel members are being promoted. For example,
Appendix 4 of the 2005 AISC Specifications now allows steel members to be designed
against ﬁre using room temperature design specifications and reduced material
properties. Using this engineering approach, the verification of design for strength during
ﬁre requires that the resistance of the structure is greater than the load effects. This leads

to satisfying the design equation
¢mef Z me (2-1)

where Quf is the load effect at the time of fire, Rnf is the nominal capacity at the time of

ﬁre, and ¢f is the capacity reduction factor. The AISC Specifications (AISC 2005a)

state that capacity reduction factors are the same as those used for room temperature
design. Most other codes suggest that a capacity reduction factor of 1.0 be used.
At elevated temperatures, the strength and stiffness of Steel reduces significantly, and if

unprotected, steel members fail within a short time. The degradation of the strength of a

20

simply supported steel beam is illustrated in Figure 2.2. The ﬁre resistance time of the
beam should be greater than the required ﬁre resistance time speciﬁed in the building
codes. To reduce the degradation of strength and stiffness, members are generally
protected by insulation to slow down the rise of the steel temperature. The required

thickness of the insulation can be determined from Equation 2—1 using an iterative

 

 

 

procedure.
i r. 1
Load (b) Simply Supported Beam
(momeqt)
Strength of
the beam

  
 
    
 

Failure
point

 

/

Applied load

Fire resistance

.-IL---------

 

 

Time
(a) Strength Degradation of Beam
Figure 2.2 — Degradation of strength of steel beam exposed to fire
2.2.3 Detailed Calculation Methods

Whereas fire resistance provisions in the past were primarily based on full-scale tests, the

recent trend is moving toward calculation methods. This is mainly because standard ﬁre

21

tests are very expensive and time consuming. Advances in numerical models are
facilitating the application of calculation methodologies for evaluating ﬁre resistance.
These calculation methods are used both in the prescriptive and performance-based
design approaches.

In recent years, several mathematical models to calculate the ﬁre resistance of structural
members in buildings have been developed. The ﬂowchart in Figure 2.3 illustrates the

general calculation procedure employed in such methods (Kodur 2007, Buchanan 2001).

Increment time

1

Fire temperature:
Calculation of fire temperature

1

Thermal analysis:
Calculation of member temperature

 

 

 

 

 

 

 

 

 

   

 

  
   

Thermal
properties

 

 

 

 

I

 

 

 

 

Mechanical Strength Analysis:
'1. Calculations of strains, stresses,
prope '83 strength and deflection

 

 

 
   
 

 

Check failure

Figure 2.3 - Flowchart for calculating the fire resistance of a structural system
(Reporduced from Kodur 2007)

The fire resistance calculation is performed in three steps:
0 Calculation of the fire temperature.

0 Calculation of the temperatures in the fire-exposed structural member.

22

0 Calculation of the strength of the member during exposure to fire, including an

analysis of the stress and strain distributions.

2.2.3.1 Fire Temperature

For a performance—based design approach, the real or design fire temperature can be
computed using any suitable mathematical model from the literature (EN 2002. SFPE
2000 and 2004, and Buchanan 2001, etc.). In the prescriptive design approach, the fire
temperature is assumed to follow the standard time-temperature curve specified in ASTM

E119a (2008).

2.2.3.2 Temperature of Structural Member

The next step in the analysis is the calculation of the temperatures of the fire-exposed
member. These temperatures are generally calculated using a finite difference or finite
element method (Buchanan 2001, Kodur 2007). In these methods, the cross-section Of the
member is divided into a number of elemental regions, which may have various shapes
such as squares, triangles or layers, depending on the geometry of the member (Kodur
and Harmathy 2008). For each element or layer, a heat balance equation can be derived.
By solving the heat balance equations for each element or layer, the temperature history
of the member can be calculated.

The Steel temperature can be calculated using any advanced finite element software (e.g.,
ANSYS 2007, SAFIR 2003, etc.). However, most design speciﬁcations such as the AISC
Speciﬁcations (AISC 2005a) and Eurocode 3 (EN 2005) allow the steel temperature to be
calculated using simple thermal analysis methods such as the lumped heat capacity

method. The lumped heat capacity method assumes that the steel section is a lumped

23

mass at uniform temperature. The heat balance differential equations for protected and
unprotected steel members are provided in the design specifications and literature (AISC

2005a, EN 2005, Buchanan 2001, etc.).

2.2.3.3 Strength Calculations

In the third step, a Stress-Strain analysis is conducted to determine the strength of the
member during exposure to fire. This strength decreases with increasing temperature and
duration Of fire exposure. The fire resistance can be derived by determining the time at

which the strength of the member reaches the load to which the member is subjected.

2.3.3.4 Critical Factors in Fire Resistance Calculations

To achieve a reliable fire resistance estimate through detailed calculations, the following

factors must be accounted for (Kodur et al. 2009):

0 High temperature material properties: The temperature-dependent thermal and
mechanical properties that are important for establishing the fire response of steel
structures should be used. These properties vary as a function of temperature, and are
crucial for modeling the fire response of steel structural members.

0 Strain components: In addition to mechanical and thermal strains, creep strains must
be considered in computing steel strains at elevated temperatures. Creep strain which
are often ignored, may have a significant inﬂuence on the ﬁre response of steel
structural members.

0 Fire scenarios: In most fire scenarios, there always exist a growth phase followed by
a decay (cooling) phase. Further, the fire scenario is a function of compartment

characteristics such as fire load density, ventilation conditions and thermal

24

absorptivity of compartment boundaries. Therefore, it is essential, in modeling the
ﬁre response of steel structural members, to account for realistic ﬁre scenario
including the decay phase.

0 Restraint effects: Fire induced restraint is believed to inﬂuence the fire response of
structural Steel members. Thus, restraint effects must be accounted for in tracing the
response of structural steel members.

0 Geometric nonlinearity: At elevated temperature, the structural members generally
experience large deformations because of the deterioration in strength and stiffness of
the members. It is important to account for geometric nonlinearity in tracing the fire
response of structural steel members.

0 Failure limit states: Currently, the failure of structural steel members under fire
conditions is generally evaluated based on thermal and strength failure limit states.
Other failure criteria such as deﬂection and rate of deﬂection may be critical under
some conditions.

For evaluating the ﬁre resistance of steel members, especially through a performance-

based approach, three main factors must be considered, namely: (1) fire scenario, (2)

loading conditions, and (3) failure criteria. These three factors are elaborated in the

following sections.

2.2.4 Fire Scenarios

2.2.4.1 Room Fires

The ﬁre growth inside a typical room, often called a fire compartment, depends on a

number of factors including:

25

0 Fire load density that is related to the quantity of combustible materials.

0 Opening factor that is related to the degree of ventilation.

0 Dimensions of the compartment.

0 Thermal properties of the lining materials.

0 Thermal properties of compartment boundaries.

0 The amount of volatile combustibles released within the room per unit time.

0 The nature Of the fuel (of cellulose or thermoplastic nature). Cellulosic materials
are generally charting materials where a layer of char is created at the fire-
material interface. The char layer protects the inner core of the material and slows
down the burning process. Thermoplastic materials do not generally experience
charring when they are subjected to ﬁre.

Figure 2.4 Shows a typical time-temperature curve for the complete process of fire
development inside a ﬁre compartment, assuming no ﬁre suppression by sprinklers and
ﬁre ﬁghters. Most fires spread slowly at ﬁrst and then more rapidly as the ﬁre grows.
When the temperature of the upper layer reaches about 600°C, the ﬂashover stage occurs
where the burning rate increases, ﬁre engulfs most of the fuel items, and temperatures
become very high. The severity and duration of a ﬁre depends on the amount of
combustibles present in the room, the ventilation conditions, and the type of bounding
surfaces of the room. After some time, either the fuel burns out or there is a lack of
oxygen and the temperature drops during the decay phase (Buchanan 2001).

After the ﬂashover stage, the ﬁre is known as a post-ﬂashover fire which is considered
structurally significant and is used for the fire-resistant design of steel members.

Characterization of an appropriate post-ﬂashover ﬁre is difficult because of numerous

26

uncertainties arising from the multiplicity of fire scenarios and the complexity of room
fire behavior. For ease of calculation, most standard use a standard time temperature
curve to represent a room fire scenario (ASTM E119a 2008, ASTM 1529 1993, EN 2002,

ISO 834 1975).

 

 

 

 

A

Intense burning .

g Growth Decay

E ‘ '

8

.5.

I'-

Ignition Flashover Time

Figure 2.4 — Various stages of fire

2.2.4.2 Standard Fire

As mentioned earlier, at present most countries rely on ﬁre resistance tests to assess the
performance of structural steel elements. The time-temperature curve used in these fire
resistance tests is called “a standard fire.” The most widely used test speciﬁcations for a
standard ﬁre are the ASTM E119 and ISO 834. .In these standard test specifications, the
ﬁre temperature, Tf, in °C is calculated as:

ISO-834 specification:

Tf = 34510g(8t "l' 1) '1' T0 (2-2)

ASTM E119 speciﬁcation:

27

Tf = 7500— exp(—3.79553\/1;))+ 170.41); + To (2-3)

where t is the time in minutes, 1;, is the time in hours and T0 is ambient temperature in °C.

The fire temperature obtained from these two specifications is independent Of the amount
of ﬁre load, ventilation conditions and properties of the compartment boundaries. In
reality, however, these parameters have a significant effect on the severity of the fire
temperatures, and also on the duration of the ﬁre in any compartment.

Figure 2.1 illustrates the time-temperature curves for a standard and two real or design
ﬁre scenarios. In the standard ﬁre (ASTM E119a 2008), the ﬁre size is the same
(irrespective of compartment characteristics), temperature increases with time throughout
the ﬁre duration, and there is no decay (or cooling) phase. In real fires, the fire size is a
function of compartment characteristics, and there is a decay phase as shown in Figure
2.1 (severe and moderate design fires). Although, the ASTM E119 and ISO 834 standard
time-temperature curves may roughly predict the growth and burning stages of room
ﬁres, they do not consider the decay phase which always exists for real room fires, and
which has a significant inﬂuence on the ﬁre resistance of structural members. In addition,
these standard time-temperature curves represent a limited range of possible scenarios
involving ﬁre load, ventilation conditions and the bounding surfaces (Bwalya et al.
2003).

As mentioned earlier, such standard fires are used to define the temperature profile that is
generally used in standard fire tests. While standard ﬁre resistance tests are useful
benchmarks to establish the relative performance of different structural members under
the standard ﬁre condition, they should not be relied upon to determine the survival time

of structural members under realistic fire scenarios. Nor does the standard heating

28

condition bear any relation to the Often less severe heating environments encountered in
real ﬁres.

Due to drawbacks of the standard time-temperature curves, most design specifications
now require use of real or design ﬁres for the design of structural members. For example,
Appendix 4 of the 2005 AISC Specifications suggests that the design of load bearing
structural members be based on a design (or real) fire when using an engineering

/performance-based approach.

2.2.4.3 Design (or Real) Fires

The ﬁre temperature, Tf, can be estimated using a suitable mathematical model from the
literature (e.g., SFPE 2000 and 2004, EN 2002, Buchanan 2001, etc.). In most of these
models, the temperature of the design ﬁre is a function of the opening factor, F v, fire load
density, (1,, and thermal absorptivity, b.

In this study, the Eurocode parametric fire model modified by Feasey and Buchanan
(2002) is used to estimate the ﬁre temperature under design ﬁre scenarios, and this model

is brieﬂy described below.

For the burning phase, the Eurocode equation for temperature Tf (°C) is given as

* * *
Tf =1325(1— 0.324e—0‘2t - 0.204e‘1-7t — 0.472e—19" )+ T0 (2-4)
where t* is a fictious time in hours given as t = Ft ; tis the time in hours;

l.._(Fv/Fref)2
(b/bref)2

 

(2-5)

29

Fv = 1‘1,”le / A, is the ventilation or Opening factor; Av is the area of ventilation

openings; HV is the height of ventilation openings; A, is the total area of the bounding

surfaces of the fire compartment; 1) = kpcp (WSO'S/mZK) is the square root of the

thermal inertia of the bounding surfaces of the fire compartment; k, ,0 and CI) are the

thermal conductivity, density and specific heat of the bounding surfaces of the fire

. . . . 0.5 .
compartment; F ref IS the reference value of the ventilation factor given as 0.04 (m ) In

Eurocode 1; and bref is the reference value given as 1160 WsO'S/mzK in Eurocode 1.

Feasey and Buchanan (2000) have shown that that the temperatures given by the

Eurocode 1 (EN 2002) formula are often too low, and recommended that a value of bref=

1900 WSO'SlmzK be used instead of 1160 ‘WsO'S/mZK. Using bref= 1900 (WsO‘S/mZK)

and F v =0 .04 mO'S, the burning phase of the design fire can be obtained from Equation

 

2-4 by using
2
F = (F, / 0.04)2 (2_ 6)
(b/1900)
The duration of the burning period td in hours is then given as
1 — 0 00013 ‘1'
d _ ° F.— (2-7)

V

where q, is the fire load density (MJ/m2 ) of the bounding surfaces.

30

For the cooling phase, Eurocode 1 gives a reference decay rate (dT/dt)ref = 625 °C/hr for

ﬁres having a burning period of less than half an hour, decreasing to 250 °C/hr for fires
with a burning period greater than 2 hours. For incorporating the effect of thermal
insulation and opening factor, Feasey and Buchanan (2002) suggested that the reference
decay rate given by the Eurocode be modified to

dT _[dT] .lF,/0.04

—— — —— 2-8
dt dt ,6, 719/1900 ( )

Two design ﬁres shown in Figure 2.1 were obtained through Equations 2-4 to 2-8.

2.2.5 Loading Conditions

The prescriptive provisions in current codes of practice for evaluating fire resistance
through standard ﬁre tests are generally based on a load ratio of about 50%. The load
ratio is deﬁned as the ratio of the applied load on the structural member under ﬁre
conditions to the resistance of the member at room temperature. The load ratio depends
on many factors including the type of occupancy of the building, the dead load to live
load ratio, and the safety factors (load and resistance factors) used both for room
temperature and ﬁre condition designs.

However, the AISC Specifications require that while designing steel members using the
performance-based approach (engineering analysis), the applied load effects (e.g.,
bending moment, shear force, axial force, etc.) should be determined from the gravity

load combination given as:

U = 1.21),, + 0.5L" + 0.25,, + T (2—9)

31

where D", Ln and Sn are nominal dead, live and snow loads, respectively, and T includes

loads induced by the fire itself, especially, due to restraint from the surrounding
structures preventing thermal expansions, or due to a ﬂexural member becoming a
tension member after large deformations occur.

ASCE- 07 (2005) also provides the load combination similar to Equation 2-9 for the
design of structural members subjected to extra-ordinary events (e.g., ﬁre). Based on
ASCE- 07 (2005) and AISC Specifications (AISC 2005a), and for typical dead-load-to—
live-load ratios (in the range of 2 to 3), the load ratio ranges from 40% to 70%.

The load level has a significant inﬂuence on the fire resistance of structural steel
members. Thus, for innovative, realistic and cost effective fire safety design, it is
important to evaluate the ﬁre resistance of structural steel members based on actual load

levels instead of the typical load ratio of 50% often used in the prescriptive approach.

2.2.6 Failure Criteria

The conventional approach of evaluating fire resistance (defining failure under fire
conditions) is based on thermal and strength failure criteria as specified in ASTM E119a
(2008). Accordingly, thermal failure of steel members is said to occur when their
temperature exceeds the critical temperature limit of 538°C, and strength failure is said to
occur when the steel member is unable to resist the applied load during the fire design
time.

Although, most codes, such as Eurocode 3 (EN 2005), allow the fire resistance of steel
members to be evaluated using the critical temperature limit state, load bearing structural

steel members may not fail even when their temperatures reach the limiting value of

32

538°C. This is especially true for members that have low load ratios, because they were
likely designed for load combinations such as earthquake or wind, or for deﬂection
limits. Such members will have considerable reserve strength because of low load ratios
and will not fail even when their temperature reaches the limiting value of 538°C.
Therefore, it is more rational to design steel members using the strength limit state, and
now most codes recognize and allow this.

The AISC Specifications (2005) specify that for structural elements, e.g., beams and
columns, the governing limit state is loss of load-bearing capacity. However, they also
specify that excessive deformations are not acceptable if these damage the integrity of the
fire compartment. If not limited, deformations may damage the slabs and walls of the
compartment to the extent that these components cannot control the horizontal and
vertical spread of fire even though the beam may still have sufficient load bearing
capacity. Additionally, deformations need to be limited because long beams may fail due
to an abrupt increase of their deﬂection despite the fact that they may have not reached
their ultimate load bearing capacity (Skowronski 1990). Furthermore, beams designed
with a large margin of safety against collapse in fire, can lose that margin due to the
abrupt growth of strains caused by creep at elevated temperatures even when stresses are
not critical (Skowronski 1990).

The deﬂection and rate of deﬂection can play a crucial role on the fire response of
structural steel members exposed to ﬁre. In fact, a deﬂection limit is Often applied to
check the status of steel beams at ambient conditions (serviceability limit state). This
criterion should be considered to determine failure under ﬁre conditions. The resulting

deﬂections in ﬁre scenarios are generally higher than those at room temperature due to

33

deterioration of member stiffness and also due to temperature-induced creep. The British
Standard (BS 476 1987) contains deﬂection and rate of deﬂection criteria for defining
failure of a steel beam tested in a furnace. Although these deﬂection limit states might
have been set to limit damage to the furnace during fire tests, deﬂection and rate of
deﬂection can be important under certain fire conditions. This is because the integrity of
the structural member cannot be guaranteed with excessive deformations. Moreover, ﬁre
resistance based on limiting deﬂection will help to facilitate the safety of ﬁre fighters and
also to safely evacuate occupants prior to structural collapse (Kodur and Dwaikat 2008).
According to BS 476 (1987), failure is assumed to occur when:

0 The maximum deﬂection of the beam exceeds L/20 at any fire exposure time, or

0 The rate of deﬂection exceeds

L2

9000d

 

(mm/min) (2- 10)

where L = span length of the beam (mm), and d = effective depth of the beam (mm).
The other failure limit state that is relevant for ﬁre resistance evaluation is the loss of

Shear strength, which should also be considered.

2.3 Use of Reliability Theory in the Development of Design
Specifications

In general, engineering design of structures consists of proportioning the elements of a
structural system so that they satisfy various criteria of performance, safety,

serviceability, and durability under various demands. For example, a structure should be

designed so that its capacity or resistance is greater than the effects of applied loads.

34

However, there are numerous sources of uncertainty in parameters that contribute to load

and capacity.

A

11:20)

10(0)

ka Qn- Rn mR R,Q

 

Figure 2.5 - Fundamentals of risk-based design evaluations

In the presence of uncertainty, it is not simple to satisfy the basic design requirements.
Figure 2.5 shows a simple case considering two variables i.e., load Q on the structure and

capacity R of the structure. Both Q and R are random variables, and their randomness is

characterized by their means mg and mR, standard deviations (IQ and OR, and
corresponding probability density functions fR(r) and fQ(q), respectively, as shown in the
Figure 2.5. The nominal values of these parameters, Q" and R", used in the conventional

safety factor based design approach, are also shown in Figure 2.5. The concept of risk-

based design was introduced by Freudenthal (1956). Design guidelines using the LRFD

35

concept are essentially based on the risk based design format. The concept of risk-based
design is described by Haldar and Mahadevan (2000) and is summarized below.

Referring to Figure 2.5, in deterministic design, design safety is achieved by requiring

that R" be greater than Q" with a nominal safety factor given by

The nominal capacity Rn is usually a conservative value that is one or more standard

deviations below the mean value. The nominal load Q" is also a conservative value

typically several standard deviations above the mean value. The safety factor accounts for
many factors such as the uncertainty in the load and resistance and how conservatively
the nominal load and resistance values are selected. However, the nominal safety factor
may not be an indicator of the actual margin of safety in the design.

In a deterministic design, the nominal safety factor can be applied to the resistance, to the
load, or to both. In allowable stress design, the safety factor is used to compute the
allowable stresses in the member from the ultimate stress, and a satisfactory design

assumes that the stresses caused by the nominal values of the loads do not exceed the

allowable stresses; i.e., Q" < Rn/SF. In the ultimate strength design method, the loads are

multiplied by load factors to determine the ultimate loads; i.e., referring to Figure 2.5,

Rn>an, where y is load factor. In steel design using the load and resistance factor

design (LRFD) concept, the resistance factor (generally less than 1.0) and load factors

(generally greater than 1.0) are used to achieve the same objective, and the safety factors

36

are used to modify both the resistance and the loads; i.e., ¢Rn > an, where (b is the

resistance factor.

The intent of these conventional approaches can be explained by considering the area of
overlap between the two curves (the Shaded region in Figure 2.5), which provides a
qualitative measure of the probability of failure. The area of overlap depends on three
factors:

(1) The relative position of the two curves: As the distance between the two curves
increases the overlapped area reduces and the probability of failure decreases. The
positions of the curves may be represented by the means (mQ and rim) of the two
variables.

(2) The dispersion of the two curves: If the two curves are narrow, then the area of
overlap and the probability of failure are small. The dispersion may be
characterized by the standard deviations (0Q and OR) of the two variables.

(3) The shape of the two curves: The Shapes are represented by the probability
density functions fR(r) and fQ(q) and affect the area of overlap and the probability
of failure.

In deterministic design procedures safety is achieved by selecting the design variables in
such a way that the overlap between the two curves is small. This is commonly
accomplished by shifting the positions of the curves through the use of safety factors. A
more rational approach would be to compute the risk by accounting for all three overlap

factors, and selecting the design variables so that an acceptable risk of failure is achieved.

This is the foundation of the risk-based concept. This approach, however, requires

37

knowledge of the probability density functions of the resistance and loads (as in Figure
2.5), which is usually difficult to obtain. Often engineers must formulate an acceptable
design methodology using only information on means and standard deviations.

Probability theory and structural reliability methods allow safety factors and design
variables that are consistent with a desired level of performance (acceptably low
probability of failure) to be selected by using information on the means and standard
deviations of the resistance and load parameters. The probability density functions of the
resistance and loads variables can also be used, if available. Reliability-based design
approaches are able to yield more uniform safety in structures and in some cases where
design are conservative may also yield a reduction in costs. Structural reliability theory is

brieﬂy described in this section.

2.3.1 Background

In strength design, the resistance, R (e.g., bending, shear or axial strength) of a member
or connection is chosen to withstand the load effect, Q (e.g., the maximum bending
moment, shear force or axial force caused by external loads). Resistances and load effects
are random variables because of the various uncertainties associated with each.

Safety is often assessed in terms of the safety margin, R — Q, or the logarithm of the

factor of safety, In (R/Q). The probability of failure in terms of these measures is

PF = P[(R - Q) < 01 = Plln (R/Q) < 0] (33-12)
To obtain pp the probability density functions (PDF’S) of R and Q must be known. The
PDF’S of R— Q or In (R/Q) can then be derived from these, and the areas under these

PDF’S corresponding to pp are shown in Figure 2.6.

38

Two common probability distributions used for R and Q are normal and log'norrnal
distributions. If R and Q are normally distributed, then R — Q is also normally distributed.
If R and Q are lognonnally distributed, then ln (R/Q) = In R —- ln Q is normally
distributed. For these distributions, it is useful to transform the random variable R — Q or
In (R/Q) into a standard normal variate, U, with zero mean and unit standard deviation:

U : (R-Q)—mR_Q

aR—Q

 

(2-13)

U _ 111(R/Q) — Inln(R/Q)
0'In(R/Q)

 

(2-14)

in which m and 0 denote the mean and standard deviation of the subscripted quantity.

 

 

Hence,
m _ m _
pp=P[(R—Q)<O]=PU<— R Q =<1>— R Q (2-15)
UR-Q aR-Q
f(R_. QIA fin (RlQ)?

 

 

PF Pi

 

 

R IQ I In (R70)

Figure 2.6 - Probability of failure, pp (Reproduced from Harichandran 2005)

39

m m
[7,: = P[1n(R/Q)<0]=P U «M =<b ——ﬂ’3—’Q (2-16)
01n(R/Q) Uln(R/Q)

where <I>( ) is the standard normal cumulative distribution function (CDF).

A reliability (or safety) index ,6 is defined for independent normally distributed R and Q
as

mR_ mR —m
5N = Q = 2 Q2 (2-17)
(IR—Q 0R +UQ

 

 

while for independent lognorrnally distributed R and Q with COV less than about 30% it
is

mln(R/Q) ~ ln(mR /mQ)

 

 

pm 2 2 2 (2-18)
alum/Q) 1le + VQ
In terms of the reliability index, the probability of failure is
pF = P[U < —,6’J = <1>(—,6’) (2-19)

The meaning of the reliability indices are illustrated in Figure 2.7. They measure the
distance of the critical point R — Q = 0 or In (R/Q) = 0 (i.e., the origin) from mR _ Q or mm
(mg) in units of GR _ Q or O'jn (Is/Q).

Equation 2—17 was the basis of an early recommendation for a probability-based
structural design code (Cornell 1969), while Equation 2-18 was the basis for the
development of the load and resistance factor design criteria for steel structures (Ravindra

and Galambos 1978).

40

11(le

 

 

 

%
Tﬂ mu=0 U

 

 

 

 

     

 

 

 

 

 

Standard normal CDF Standard normal PDF
f (R - Q) A f It.
In (R/ Q) ﬂL

N 0' I R/Q

I I E n( )
| I
I I
I l
I l
' I
PF 5 :
ﬂN 0' R-Q: E
I l

_llllllllllll;§+- y ' a
mR- Q R-Q 0 min (R/Q) In (R/Q)
PDF of (R-O) PDF of ln(R/O)

 

 

Figure 2.7 - Interpretation of the reliability index ,6
(Reproduced from Harichandran 2005)

2.3.2 First-Order Second-Moment (FOSM) Analysis

The capacity R and load effect Q are usually related to several basic random variables

that characterize the problem. Thus the means and variances of R and Q must be

41

determined from the statistics of these other random variables. When the relationship
between R and Q and the basic random variables is non-linear, it is common to estimate
the means and variances of R and Q from those of the basic random variables by
linearizing the non-linear relation.

Consider a random variable Y derived from another random variable X through Y = g(X).
The mean and variance (ﬁrst two moments) of Y can be obtained exactly only if g( ) is
linear or if the PDF of X is known. If g( ) is non-linear and only the mean and variance of

X are known, then the mean and variance of Y can be estimated using a first order

approximation based on linearizing g(x) about some point x*. Figure 2.8 illustrates this

approximation.

The equation of the linear function is

dg
y g( )

 

x* (x " 35*) (2-20)

451
where, dx

I
— *
x* _ g (x )= slope of the straight line. Equation 2—20 represents a

first—order Taylor series expansion of g( ) at x*.

For the multi-dimensional case where the random variable Y is a non-linear function of

several basic random variables, Y = g(X 1, . . . , X“), the linearization takes the form

42

   
   

/ / -
Linear
approximation

 

 

/
/ /
:1: >
x x
Figure 2.8 - Linear approximation to non-linear function (Reproduced from
Harichandran 2005)

n a n
y = g(x1*, ..... ,xn*)+ Zilxﬁxi —x,-*) = a0 +Zai(xi —x,-*) (2-21)
i=1 '=

The mean of Y based on Equation 2-21 is

my 2: a0 + Zai(mxi —.7Cl'*)

. (222)
1:1

and the variance is

n
~Ziai2 O'X +ZZal-aj Cov(X,-,Xj ) (23)
i1: i=1j=1

43

If the Xi are uncorrelated with each other then the covariances COV (X i, Xj) = 0. Equations
2-22 and 2-23 are exact for the special case when g( ) is a linear function of the X ,2

Often the linearization point (x.*,..., xn*) is chosen as the mean

point (le ’ ' ° ' ’ an ) . In this case the method is referred to as a mean-value ﬁrst-

order second-moment method, for which Equation 2-22 simpliﬁes
mY 2:00 :g(mX1,...,an) (2-24)

and for uncorrelated random variables Xi’s, the Equation 2-23 simpliﬁes to

1/2
2

BY 02
8X,- Xi (2-25)

0T” Z:

mXi

The extent to which Equation 2-24 and 2-25 are accurate depends on the effect of

neglecting higher order terms in Equation 2-21 and the magnitudes of the coefﬁcient of

variations (COV’S) in Xi. If g( ) is linear and the variables are uncorrelated, Equations 2-

24 and 2-25 are exact.

The reliability index, ,6 (in some studies ,8 is termed the safety index), is deﬁned by
ﬂ=mﬂmr coo
which is the reciprocal of the estimated COV of Y. B is the distance from my to the

origin in standard deviation units. As such, ,0 is a measure of the probability that g ( ) will
be less than zero. This is illustrated in Figure 2.6 which shows the densities of Y for two

alternate representations of the simple two- variable problem Y = g(R, Q) = 0 discussed in

44

the previous section. Figure 2.6a shows the probability density function (generally

unknown) for Y = R - Q. The shaded area to the left of zero is equal to the probability of
failure. Observe that if 0R_Q remains constant, a positive shift in mR_Q will move the
density to the right, reducing the failure probability. Thus an increase in ,6 leads to an

increase in reliability (lower pp).

2.3.3 Hasofer-Lind Reliability Index

When R and Q are related non—linearly to the basic random variables X ,- characterizing the

problem, use of the reliability indices deﬁned in Equations 2-17 and 2-18, with means
and variances of R and Q computed by mean-value FOSM analysis, has the following
undesirable features in reliability analysis: Signiﬁcant errors may be introduced at
increasing distances from the linearization point by neglecting hi gher-order terms in the
Taylor series expansion.

The reliability index is not invariant to different but mechanically equivalent

formulations of the same problem (e.g., whether ZxFy — M < 0 or ZxFy - M/Z,C < 0 are

used to describe failure in the ﬂexural design of a continuously laterally supported beam).
In effect, this means that )6 depends on how the failure criterion is formulated.
The invariant reliability index proposed by Hasofer and Lind (1974) is the basis of

advanced reliability methods. Failure is expressed in terms of the failure criterion Y:

g(xl, xn) < 0, in which x,- is the value taken by the basic random variable X). The

function g(xl, xn) is often called the response surface and may be taken to be any one

45

of R — Q, In (R/Q), R/Q — 1, etc. Linearization of g( ) is performed at a special point (x1*,

xn*) on the failure surface g(xl, xn) = 0, often called the design (or checking)

point. The mean and variance of Y are then computed using Equations 2—22 and 2-23. The
reliability index is then deﬁned as the distance of my from the critical value of zero (at
which failure occurs) in units of O'y, i.e.,

ﬂHL = my/Gy (2-27)
Computation of the Hasofer-Lind reliability index is performed using the following steps:

(1) The basic random variables X) are transformed to uncorrelated standard random

variables Zi which have zero mean and unit variance.

 

Z,- = ' (2—28)

This may be Viewed as a coordinate transformation from x-space to z-Space.

(2) The failure surface g(xl, ..., xn) =0 in x-space is transformed to the

corresponding failure surface h(z 1, zn) = 0 in z-space as shown in Figure 2.9

with failure occurring when It ( ) < 0.

(3) The reliability index ,6 is deﬁned as the shortest distance between the surface 120

= 0 and the origin. ,BHL is the distance to (21*, zn*) from the origin.

46

Failure boundary _
obtained by linearizing x2 Fallure boundary

F '1 b d 7'2
at the design point A, g(x,,x2)=0 a1 ure oun ary A

 

 

 

 

 

 

 

,.’ h(z,,zz) = 0
/ . Design point /
Transformation
* I
x2 9: _ __ _ _ ZZ 9:
Inverse l ,3
Transformation l
I ' a: T’
x; Z1 Z1
x-space z-space

Figure 2.9 - Illustration of failure boundary and design point in x- and z-spaces
for two variables problem (Reproduced from Harichandran 2005)

(4) The design point in z-space, (31*, zn*), is the point on the failure surface

h(z 1, Zn): 0 that is closest to the origin. A numerical optimization algorithm

is usually required to ﬁnd this point. This point must be determined by solving

the system of equations (NBS 577 1980)

 

a _ Bh/az, (2 29)
Z," — : ,2 _
[ﬂak/azazl
Zi* = —0!Zl. ,6 (2—30)
h(Z1*, Z2 *, ..... , Zn*) = 0 (2-31)

searching for the drrectron cosrnes Z,- whrch minimize ,6. The derivatives are

evaluated at the point (zl*, zn*). Note this procedure is equivalent to

47

linearizing the limit state equation in reduced variables at the point (z1*,

211*), and computing the reliability associated with the linearized rather than the

original limit state. This is identical to locating the design point (x1*, xn*)in

x-space using the inverse of the transformation in step 1, and using FOSM

analysis with linearization ofg(x1, xn) at (x1*, xn*).

(5) In the original variable space , the design point variables are given by

X,*=mxi(l-a’ziﬂin) (2-32)
g (X,*,X2*, . . .,X,,*) = 0 (2-33)
Details of this procedure can be found in Madsen, et al. (1986). Note that if g(xl, ..., xn)
is linear, the Hasofer-Lind reliability index is identical to that in Equation 2-17.
If necessary, load and capacity reduction factors yi for design corresponding to a
prescribed reliability index ,6 may then be determined through (Ellingwood et a1. 1980)

71' = Xi*/Xn,i (2-34)

In which Xn,i is the nominal or design value of the load and capacity parameter speciﬁed

in the building standard. This may be the load corresponding to a mean recurrence
interval of N years, the mean maximum load during the reference period of T years, or
any one of a number of other formulations. Thus the load and resistance factors depends

on the way the nominal load and resistances are speciﬁed.

48

2.3.4 Approximate Methods for Including Information on Distributions

If the basic random variables X,- are normally distributed and the response surface g(xl,

xn) is linear, the probability of failure is <D(—,6’) (see Equation 2-19), in which ,6 is the

Hasofer-Lind reliability index or that given in Equation 2-17. If R and Q are lognormally
distributed and their means and variances can be determined from those of the Xi’s, then

the probability of failure is again <1>(—,6’), with ,6 given by Equation 2-18.

If the X,- are normally distributed but the response surface is non-linear, then pg: 2

<1>(—,6HL). Many structural problems involve random variables which are clearly non-

norrnal. As examples, instantaneous live loads appear to be modeled more appropriately
by gamma distributions, at least for relatively small loaded areas (Corotis and Doshi
1977). Studies of extreme wind load (Simiu and Filliben 1975) have Shown that that the

annual extreme wind speed due to extra tropical storms has an Extreme Value Type I
distribution. It seems appropriate that this information be used in determining ,BHL in
order to have szq)(-ﬂHL). The so-called Rackwitz-Fiessler method is often used to in-

clude information on distributions of non-normal random variables in the computation of

the reliability index (Madsen, et a1. 1986). Non—normal random variables are transformed
to equivalent normal random variables prior to computing ﬂHL-

The basic idea is to transform the non-normal variables into equivalent normal variables

prior to the solution of Equations 2-29 to 2-31. This transformation may be accomplished

by approximating the true distribution of variable Xi by a normal distribution at the value

49

X i* corresponding to a point on the failure surface. The justiﬁcation for this is that if the

normalization takes place at the point close to that where failure is most likely, (i.e., min
,6), the estimate of the failure probability obtained by the approximate procedure should
approximate the true (but unknown) failure probability quite closely.

The mean and standard deviation of the equivalent normal variable are determined such
that at the value Xi*, the cumulative probability and probability density of actual and

approximating normal variable are equal (Rackwitz and Fiessler 1976). Thus,

0N = ¢I<I>“IE<X£">J)
l fi(x:)

 

(2-35)

N _ * —ll * JUN
mX, ‘Xi ”(D FAXI) 1' (2-36)
In which Pi and fi = non-normal distribution and density functions of Xi, and ¢( ) is the

N
density function for the standard normal variate. Having determined 0,: and mN f

X ,- °
the equivalent normal random variable, the solution proceeds exactly as described in

Equations 2-9 to 2-31. In as much as the design point variable Xi* changes with each

iteration, the parameters O'iN and m? must be recomputed during each iteration cycle.
1

However, since all calculations are performed by computer, this does not materially add
to the complexity of the reliability analysis described earlier.
The following summarizes the procedure which is used to compute the reliability index ,6

associated with the particular design or, conversely, a design parameter (such as section

50

modulus) for a prescribed ,6, probability distributions, and set of means and standard
distributions (or COV) (Ellingwood et al. 1980):
(1) Deﬁne the appropriate performance function.

(2) Make an initial guess at the reliability index ,6 (or design parameter).
(3) Set the initial design point values X i* = m Xi for all i.

(4) Compute the mean and standard deviations of the equivalent normal distribution

for those variables that are non-normal according to Equations 2-35 and 2-36.

(5) Compute partial derivatives ah / 0X ,- evaluated at the design point Xi*.

(6) Compute the direction cosines a,- as

 

ai = f ah/az, (2-37)
[Z(Bh/8z,~)2l/2
zi* = -az, ,3 (2-38)
h(z1*, Z2 *, ..... , zn*) = 0 (2-39)
(7) Compute new value of X,* using
Xi* = mil, " Ctr-ﬂail, (2-40)

and repeat steps 4 through 7 until the estimate of a, stabilizes.
(8) Compute the value of 6 necessary for,

g (X1*,X2*, - - ., Xn*) = 0 2-41)

51

and repeat steps 4 through 8 until the values of 6 in successive iterations differ by some
small tolerance (say 0.05). Normally convergence is obtained within 5 cycles or less,

depending on the non-linearity of the limit state function.

2.3.5 LRFD Design Criteria

The basic design requirement is that the reliability index 6 associated with the failure
criterion R — Q < 0 or In (R/Q) < 0 should equal some assigned target value 6). Although
different design formats are in use in various countries to accomplish this, the structural
design format that is used in the US. is the load and resistance factor design (LRFD)

format given by

¢Rn Z 2 10°an (2242)
i

in which (0 is called a capacity reduction factor and the y,- are called load factors and Rn
and Qn,i are the nominal capacity and load effects. The (25 and y,- faetors account for the

variability of R and the Qi’s. For example, for dead load (Dn), lifetime maximum wind

load (Wu) and arbitrary point-in-time live load (Lmapt), the format is
¢Rn 2 70D" + llL,apth,apt + I/an (2‘43)
while for dead load, lifetime maximum live load (Ln) and arbitrary point-in-time wind

load (WWW), the format is

¢Rn Z yDDn + yLLn + yW,apth,apt (2'44)

52

For ease of use, instead of using two different live and wind loads, Equations 2-43 and 2-

44 are written as
¢Rn 2 yD D" + ”L" + yWWn (2-45)

and, 0R" 2 yDDn + yLLn + yW/Wn (2-46)

in which 7L’ = 7L,apth,apr/Ln and 7W’ 2 mﬂptwnﬂpt/Wn. Thus, while load factors
such as yL, 7L.apt’ etc. are expected to be greater than unity, yLI and yWI are less than

unity due to the scaling factors Ln.apt/Ln and Wmapt/Wn, respectively.

While several levels of sophistication for reliability—based design exist, two of the
common levels are referred to as Level I and Level II methods. Level II methods are

based on the computation of a reliability index 6 as described in Sections 2.3.3 and 2.3.4,
but 6 must be computed for each design situation (i.e., for each Ln/Dn, Wn/Dn, etc.).

Level 1 methods involve the selection of one set of load factors to be applied to all design

situations, regardless of Ln/Dn, Wn/Dn, etc., and a resistance factor which depends on the

material and limit state. Levels 1 and II can be made equivalent if the load and resistance
factors in the Level I format are allowed to vary for each design situation.

For operational convenience, practical design criteria in the US. will be of the Level I
type in the foreseeable future, with constant load and resistance factors for all design
situations. This necessarily means that 6 will be slightly different for each design

situation.

53

2.3.6 Selection of Load and Capacity Reduction Factors

In order to have a constant 6 for all design situations in LRFD, the factors o and )4 must
depend on the particular load combination, strength, and on the mean, variance and
distribution of all variables in the limit state equation. If a constant set of ¢ and Vs are
prescribed, the associated 6 will deviate from the target value for certain design
situations. However, it is possible to select one set of load factors (ys) that minimize the
extent of this deviation when considered over all likely combinations of load. While the
resistance factors (¢’s) will depend on the material and limit state of interest, the load
factors will be independent of these considerations.

For any given design situation and load combination, the resistance factor «>11 and load

factors y,” that achieve the target reliability 60 are determined using Level II reliability

analysis as follows:

(1) The failure criterion is deﬁned. e. g.,

g(R,Qi. . . .. on) = R - 01-. . .- Qn =R -ZQ,- <2-47)
i

(2) The reliability index and design point associated with Equation 2—47 are

determined as described in Sections 2.3.3 and 2.3.4. The basic variables R and Q)

(in x-space) are transformed to normalized variables zR and ZQI (in z-space). The

values of the normalized variables at the desrgn pornt are zR* and ZQi and the

reliability index is 6.

54

(3) The failure criterion is adjusted by uniformly scaling it in the uncorrelated z-space

as shown in Figure 2.10, such that the reliability index corresponding to the

adjusted criterion is 60. For uncorrelated, normally distributed R and Qi, the
factors a” and y," which achieve the target reliability index 60 by matching

Equation 2-42 with the adjusted failure criterion are:

 

11 RI mR
=——=—1+a V (248
(15 Rn Rn ( RﬂO R) )
I m .
and, y,” = Q = Q’ (1+aQi’BOVQi) (2—49)
Qn.i Qn,i

in which 01R and CZQ, are direction cosines associated with the design point in z-

space and are

 

 

 

* z *
(1R — ZR and CIQ. = Q1 (2-50)
A ' A
respectively, with
A = \/(zR*)2 + 2kg, "‘)2 (231)
i

being the distance from the origin to the design point. In most cases (when mean and

nominal values are close) OLR would be negative yielding (1)” < 1, while aQi would be

positive yielding n" > 1. Note that R’ and Q,’ are the coordinates of the design point

associated with the adjusted failure criterion in x-space as shown in Figure 2.10.

55

Design point

 

 

 

 

Design point Failure boundary , ,
associated Z Q adjusted to achieve target 3515,03?th .wcrith target Q
with target A eliability index re ‘3 ‘ "Y 1“ 6" +
reliability h ﬂu Z ’30 Z 0 Failleure boundary ’
index ’ — m— = , = 0
/ / / / ﬂ .5 Q g( Q) / / /
Inverse
/ r
4‘ Transformation Q
/ Z :1:
— " — 9:
l/ '30 Q Failure boundary Q
, l )9 >9 h(zR,zQ) = 0
L ' > r
I ZR * ZR R

 

Figure 2.10- Illustration of failure boundary adjustment to achieve target
reliability index (Reproduced from Harichandran 2005)

The required nominal capacity based on the Level II reliability analysis is therefore

1
R: = W Z 71119,...- (232)

II 11
where 1/ i and ¢i are the load and capacity reduction factors for each design

situation obtained from the reliability analysis. R: will vary for each design situation
(i.e., for each Ln/Dn, Wn/Dn, etc.). On the other hand, a Level I design format which

prescribes constant load and capacity reduction factors for all load ratios leads to a

nominal resistance

1
R5 = 32 ViQn,z (2-53)
I

The constant ¢ and yi are then selected in order to minimize an error criterion such as

56

7

60>) = 21le ,- — RI, ,1- p,- (254)

overall design situations indexed by j, and pj = weight assigned to the jth design situation
(e.g., jth set of load ratios Ln/Dn and Wn/Dn for a load combination involving D, L and W

loads). The weights pj are chosen to stress more likely load ratios over less likely ones,

and vary for different construction materials. For example, concrete structures typically

have more dead load than steel structures, so the Ln/Dn ratios prevalent for concrete

structures are smaller than those prevalent for Steel structures. Hence the weight pi

assigned for a speciﬁc Ln/Dn ratio for a concrete beam would be different from the

weight assigned for the same Ln/Dn ratio for a steel beam. Typical weights that have

been used to derive the load and resistance factors in the ANSI A58 Standard are given in
Ellingwood et al. (1980).

In developing the ANSI A58 load and resistance factors (Ellingwood et a1. 1980),
subjective judgment was used in setting some of the load factors, and the selection
process was performed in stages. For example, for gravity loads involving D + L and D +

S load combinations, the optimum value of 70 was about 1.10 for a target reliability of 6)

= 3.0. However, this is lower than the dead load factor used in all previous design

speciﬁcations and it was felt that the profession would not accept such a low value. The

minimization of Equation 2-54 was then repeated with m = 1.2 to select the optimum (ti,

)1 and ys values under this constraint. Since the n and )3 values resulting from this

57

analysis were similar, the process was repeated yet again, this time taking )1 = )4; in
addition to n; = 1.2 in order to simplify the ﬁnal load criteria. The ﬁnal gravity load case
was speciﬁed as

U = 1.2 0,, + 1.6 (Ln or 3,) (2—55)

The resistance factors resulting from the analysis ranged from 0.78—0.79 for steel and
0.81—0.86 for reinforced concrete. The ANSI A58 speciﬁcations, however, were limited
to specifying load factors, and the speciﬁcation of resistance factors were left to the

different material speciﬁcation writing groups.

2.3.7 Capacity Reduction Factors Compatible with Selected Load

Factors
Once load factors are speciﬁed independent of the construction material, the material

speciﬁcation writing bodies are expected to decide on their own target reliabilities 60,
and arrive at suitable resistance factors by minimizing Equation 2-54 with respect to ¢,

with the )7 held ﬁxed.

2.3.8 Design Criteria with Partial Safety Factors

Design criteria in other countries sometimes differ from the LRFD format used in the
US. A general but complex format is one that uses partial safety factors for each random
variable in the problem. Rather than formulating the problem in terms of a single
resistance variable R and several load effect variables Qi as in Equation 2-47, let the

overall resistance be given by

58

R =fR(r1,r3, . . ., rm) (2-56)

in which the r,- are subvariables that affect the overall resistance, and let the overall load

effect be given by

Q = ZfQ, <41, 42, ,---,q./,.) (2-57)
1

in which fQ, is the load effect due to the ith load type (e.g., dead, live, etc.), and the

qki are subvariables that affect each load effect. A design criterion based on partial

safety factors is

fR (¢lrn,l r¢2rn,2’ ° ° " ¢mrm,n) Z ZfQi (I’ll-$1.1,- ’ I’2iqn,2i ’ ' ° '9 yni qn,l,-) (2'58)
i

in which the (t),- are partial factors related to the resistance subvariables, the yki are

partial factors related to the load effect subvariables, and r", j and qui are nominal

resistance and load values used in the design.
The partial safety factors are computed using a Level II reliability analysis with the rj and

qki as basic random variables (in x-space). The basic variables are transformed to

normalized variables er and qu- (in z-space) and their values at the design point (i.e.,
I

z: and Zjlk- ) are determined. For normally distributed basic variables, the partial safety
1

factors required to obtain a target reliability index 6) can then be computed through

m

 

_ ’1'

"J

59

m

 

qk-
and, yk, = ' (1+a,,k_ 60qu_) (2-60)
qn,ki I I
in which 05,]. and aqk. are direction cosines associated with the design point in z—
1

space and are

 

 

Zr. * zqk.
or, = J and “ku =———'- (2-61)
respectively, with
2 2 2
6=\/Z(z,j*) +:;(zqki*) +....+§(zqk1*) (2-62)
J

being the distance from the origin to the design point. Again in most cases (when mean

and nominal values are close), the a . would be negative yieldin - < l, and a
r} g I (Iki

would be positive yielding 7k,- > 1.0.

For non-norrnal basic random variables, the mean and COV of the random variables
cannot be used directly in Equations 2-59 and 2-60. The “normal tail approximation”
must be performed at the design point to obtain the mean and COV of the “equivalent
normal distribution,” and these must be used in Equations 2-59 and 2-60. The equivalent
normal distribution is one whose CDF and PDF match the CDF and PDF of the basic

random variable, Xi, i.e.,

. *— . . * —- .
FX. (x,*) = (bf—hi] and fx. (x,*) = —1-¢a[§-'—ﬂ'-] (2-63)
I 0- l 0". i

,- 0'
where m,- and o,- are the mean and standard deviation of the equivalent normal

distribution. The solutions of the above two equations are:

60

a. = ¢(¢“‘<Fx, (x.*>)>
I fX. (xi*)

 

811d m,- : Xi * T0i¢—1(FX,- (xi*)) (2-64)

m,- and V,- = mi/oi are used in Equation 2—59 and 2-60 to obtain the partial safety factors.

As with the LRFD format, the partial safety factors in Equation 2-58 would change with

each design situation (i.e., for each Ln/Dn, Wn/Dn, etc.), and an optimization procedure

similar to that described earlier is necessary to arrive at constant partial safety factors for
a Level 1 design speciﬁcation.

If desired, Equation 2-58 can be converted to the LRFD format in Equation 2-42 by

computing a resistance factor 4’ and load factors y,- associated with the overall resistance

and load effects, respectively:

: fR(¢1r,,,1,¢2rn‘2, ........... ¢mrmm)

 

 

(4 (2-65)
fR(r,,’1, 01,29 .............. rm")
and
y. _ fQi (ylianli’yzianI’ """""" I’miani) (2-66)
' fQi (9n,1,-,61n,2,-, .............. an, )

2.4 Studies Related to Reliability Analyses of Structural
Members Exposed to Fire

As mentioned at the beginning of this chapter, only one study has been reported in which
ﬁre load factors were developed using simpliﬁed assumptions instead of rigorous

reliability analysis (ECSC 2001). Similarly, no explicit work has been reported on the

61

derivation of resistance factors for ﬁre design of steel members except in one case
(Magnusson and Pettersson 1981). However, during the last few decades, reliability
analyses were performed for estimating the probability of failure and computing
reliability indices for steel members exposed to ﬁre. Some of the important studies are

summarized in this section.

2.4.1 ECSC Study, 2001

In the ECSC study (ECSC 2001), ﬁre load factors were developed based on the
variability of the ﬁre load and by taking into account the effect of active ﬁre protection
systems, in reducing the probability of occurrence of a severe ﬁre. Initially, base ﬁre load
factors were derived based on the presence of some basic active ﬁre protection systems
such as a public ﬁre brigade, and the effect of occupants in reducing the ﬁre. Thereafter,
differentiated factors were derived to modify the base ﬁre load factors depending on the
presence of other advanced active ﬁre protection systems such as automatic sprinklers,
smoke and heat detectors and special ﬁre brigades, etc. The ﬁre load was assumed to be
described by the Gumbel distribution, and the ﬁre load factors were calculated through

I-ﬁvq [0.577 + ln(—ln cp(0.9p,))]}

71'

 

yq =1.05{ (2-67)

{I—qu[0.577+1n(—1n(p))]}

where Vq = COV of the ﬁre load, (D = cumulative standard normal distribution function,

and p = percentile used for obtaining the characteristic or nominal ﬁre load. If the

nominal value is taken as the 80th percentile, then p = 0.8. The value of the target

62

reliability index, 6,, depends on the effectiveness of active ﬁre protection systems present

in a building.

Fire load factors were obtained by using simpliﬁed assumptions instead of detailed
reliability analysis (ECSC 2001). For example, the value of 0.9 in the numerator in
Equation 2-67 was assumed for the direction cosine of the ﬁre load corresponding to the
failure surface instead of obtaining this from reliability analysis (see Equations 2-28 to 2-
34). In addition to the variability of the ﬁre load, there is signiﬁcant uncertainty
associated with other ﬁre design parameters (such as the thickness of insulation, thermal
properties (density, speciﬁc heat, and thermal conductivity) of insulation, cross-sectional
(e.g., plastic modulus, cross-sectional area, etc.), mechanical (yield strength, elastic
modulus, etc.) and thermal (density, speciﬁc heat, and thermal conductivity) properties
of steel, thermal properties (density, speciﬁc heat, and thermal conductivity) of
compartment enclosures, and opening factors that was not accounted for in the ECSC

study.

2.4.2 Magnusson and Pettersson, 1981

Thus far, Magnusson and Pettersson (1981) are the only one who explicitly focused on
the derivation of capacity reduction factors for the design of structural members exposed
to ﬁre.

Magnusson and Pettersson (1981) derived load and capacity reduction factors for an
insulated simply supported steel beam that was assumed to be used in an ofﬁce building.

They considered following parameters as random variables:

(1) Fire load density, qt

63

(2)

(3)

(4)

(5)

(6)

Thickness of insulation, (1)

Thermal conductivity of insulation, k,-

Insulation parameter, x = (Ai/V) x ((1)/kg). A,/ V is the ratio of exposed surface

area of insulation to the volume of steel and was considered a deterministic
parameter in the reliability study. The mean and COV of K were obtained

using the ﬁrst order Taylor series approximation from the means and COV’s
of di and kg.
Maximum steel temperature, Ts,max, given as

Ts,max = n+Tl+T2+T3 (2’68)
where,

Tn = the deterministic value of maximum steel temperature given by design

curves (graphs) for the nominal values of ﬁre load density, opening factor,

(Ai/V) and (di/kl').

T1 = the uncertainty due to variation in insulation parameter, rc.

T2 = the uncertainty reﬂecting the prediction error in the theory of
compartment ﬁres and heat ﬂow analysis.

T3 = the correction term reﬂecting the difference between a natural ﬁre in the
lab and a real ﬁre in a room.

The true capacity, Rf

(7)

Rf: (Rn, fI-R 1+R2)Mm (2-69)
where

Raf: nominal value of the capacity calculated according to creep deﬂection

theory as a function of T".

R1 = uncertainty measured by a comparison between the theoretical value of
Rn,f and laboratory tests.

R2 = uncertainty due to difference between laboratory tests and in site ﬁre
exposure.
Mm = random factor, expressing uncertainty in material strength.

Expressing the load effects as

Wf= EL(DL+LL) where EL is the random variable expressing the dispersion in

I load effects prediction. DL and LL describe the basic variability of the live and

dead loads, respectively.

The room temperature design of the beam was performed according to allowable stress
design methods given in the Swedish Codes. The load and capacity reduction factors
were derived for an arbitrarily chosen target reliability index of 2.16 and for a dead to

live load ratio of 1.0. The resulting factors were:

0 Capacity reduction factor = 0.77
0 Live load factor = 1.88

0 Dead load factor = 1.14

65

Load factors were to be applied to the mean values of dead and live loads. The value of
the capacity reduction factor found from this study is less than 1.0 suggested in most of
the ﬁre design speciﬁcations such as Eurocode 3 (EN 2005).

In Magnusson and Pettersson’s work, the statistics of thickness and thermal conductivity
of insulation were assumed due to lack of data. In the US. the test data related to
thickness and thermal conductivity of insulating materials is available which should be

characterized and incorporated in the reliability studies. Statistics of some model errors

(e. g., T3 and R2, etc.) associated with the steel temperature and capacity of the beam were

assumed in this study.

The ﬁre temperature used in Magnusson and Pettersson’s work were obtained from the
curves (graphs) given in the Fire Engineering Design of Steel Structures (Pettersson et al.
1976). These curves give time vs. ﬁre temperatures corresponding to certain values of
ﬁre load density and opening factor for a compartment that is assumed to have

boundaries made of a material having relatively high values of density and thermal

conductivity (brick, concrete). The nominal steel temperature, T", can also be calculated

from the graphs given in the Fire Engineering Design of Steel Structures (Pettersson et al.
1976) for a given value of ﬁre load density, opening factor, di/ki and Ai/V. If the
compartment boundaries are made of materials other than brick or concrete for which
graphs are given, then the manual gives a translation factor, kf, which should be

multiplied to the nominal values of ﬁre load density and Opening factor to obtain their
effective values. These effective values of ﬁre load density and opening factor should

then be used to obtain nominal values of steel temperature from the graphs. In

66

Magnusson and Pettersson’s work, they recognized that kf should be treated as a random

variable, but assumed it was deterministic stating that the inaccuracies introduced by this
assumption were negligible. Buchanan (2001) showed that thermal properties of
bounding surfaces have a signiﬁcant effect on the ﬁre temperature. The thermal
properties of different bounding surface materials differ signiﬁcantly and their variability
should be modeled. The opening factor was also considered to be deterministic in
Magnusson and Pettersson’s study.

The curves used by Magnusson and Pettersson (1981) for estimating the ﬁre and steel
temperatures differ from the new mathematical models (EN 2002, Feasey and Buchanan
2002) being suggested for use in the performance-based approaches. These models
enable ﬁre temperature to be estimated through equations for any value of ﬁre load
density, opening factor and thermal inertia of compartment boundaries. For reliability
analysis, it is convenient and more rational to use the steel and ﬁre temperatures in the
performance functions in terms of basic random variables such as ﬁre load density,
opening factor, and thermal inertia of compartment boundaries.

Magnusson and Pettersson performed the reliability analysis using both Monte Carlo
simulations and the ﬁrst order reliability method based on the Taylor series
approximation. Monte Carlo simulation was considered the preferred choice but they
stated that it was difﬁcult to check the accuracy of the method (Magnusson and

Pettersson 1981). For the ﬁrst order reliability methods, they developed a linear

expressions for the maximum steel temperature, Tsmax, in terms of the basic variables

(ﬁre load density, thickness and thermal conductivity of insulation), although the

relationship between temperature and the basic variables is non-linear. The capacity of

67

the beam was nonlinearly related to the maximum steel temperature, Twmvc. Magnusson

and Pettersson ﬁrst obtained the mean and COV of TS, max using the ﬁrst order Taylor

series approximation. They then used the ﬁrst order Taylor series approximation again to

obtain the mean and COV of the capacity this time using the mean and coefﬁcient of

T5,,nax. This may result in signiﬁcant errors because of the nonlinear relationship

between the steel temperature and the capacity of the beam (Ellingwood et al. 1980).
New mathematical models for estimating the ﬁre and steel temperatures and reliability
software afford an opportunity to use the steel temperature in the performance function in
terms of basic random variables for more accurate reliability analysis.

In Magnusson and Pettersson’s study, the effect of active ﬁre protection systems in
reducing the probability of occurrence of a severe ﬁre was not included explicitly.
Instead, resistance and load factors were developed for an arbitrarily chosen target
reliability index of 2.16. Recent studies (Ellingwood and Corotis 1991, ECSC 2001)
provide a framework for selecting a target reliability index by incorporating the effect of
active ﬁre protection systems in reducing the probability of occurrence of severe ﬁres and
correspondingly in reducing the probability of failure under ﬁre. These studies enable the
selection of more rational target reliability indices which are based on the effectiveness of
active ﬁre protection systems present in a building. The effectiveness and reliability of
active ﬁre protection systems is also now better known than three decades ago and can be
used for selecting more rational target reliability indices.

In Magnusson and Pettersson’s work, the information on the distribution of random

variables was not used when the ﬁrst order analyses were performed. Most of the ﬁre

68

design parameters are non—normal (e.g., the ﬁre load density is best described by the
Gumbel distribution (ECSC 2001)), and it is appropriate that information on distributions
be incorporated into the reliability analysis.

The statistics of the ﬁre loads used by Magnusson and Pettersson were speciﬁc to typical
Swedish ofﬁce buildings and differ from those found from the survey of US. ofﬁce
buildings performed by Culver (1976). The statistics of other parameters such as
thickness, density, thermal conductivity of insulation and bounding surfaces also differ
from country to country, and these parameters need to be characterized for conditions
speciﬁc to the US. wherever possible. The engineering approach used for the design of
steel beams and corresponding design equations used for reliability analysis were also
different than those suggested in the AISC Speciﬁcations (AISC 2005a). The load factors
of 1.0 for dead load and 1.4 for the sustained part of the live load that were used to

estimate the applied load effects for ﬁre design are different from the gravity load

combination of 1.20,, + 0.5L" suggested in the AISC Speciﬁcations for the design of

steel members exposed to ﬁre. These different loads will result in different resistance
factors. In addition, in the US, the load factors for ﬁre design have been proposed by
Ellingwood (2005), and have already been incorporated into the AISC Speciﬁcations, and
there is a need to develop resistance factors corresponding to these speciﬁc dead and live
factors.

The ﬁre load is a major parameter in ﬁre design, and uncertainty associated with it has a
signiﬁcant effect on the safety of the design. As mentioned in the introduction, the AISC
Speciﬁcations recommend that the ﬁre load be reduced by 60% if a reliable sprinkler

system is installed. The ﬁre load can be reduced appropriately if a separate safety factor

69

is used for the ﬁre load. In Magnusson and Pettersson’s work, a combined resistance
factor was developed for all the resistance parameters. Because Steel temperature was not
included in the performance function in terms of basic variables, it was not easy to

develop a separate safety factor for the ﬁre load.

2.4.3 Beck, 1985

Beck (1985) performed reliability analyses of the ﬂexural strength of Simply supported
steel beams that were protected by lightweight insulation material. The insulation
thicknesses for beams were obtained using the prescriptive approach instead of
engineering approaches currently being promoted. Fire load, opening or ventilation
factor, thickness and thermal conductivity of insulation, exposed area of insulation per
unit length, volume of steel per unit length, steel temperature, yield strength of steel, and,
dead and live loads were taken as random variables. Statistical models were developed
for all the random parameters. The probability of failures and corresponding reliability
indices were computed for various values of opening factors and for different ﬁre
resistance times. Beck found that values of the reliability index varied from zero to 5.0,
and concluded that there is considerable variation in the probabilities of failure and the
corresponding reliability indices for beams designed according to prescriptive approaches
and there is a need to adopt a more rational design approach for ensuring uniform safety
in ﬁre design.

Magnusson and Pettersson (1974) presented graphical results for the maximum steel
temperature, and using these graphical results Beck derived an equation for predicting the

maximum steel temperature, Ts,max, which is:

70

Tsmax = 25 + 0.6237", JP — 1500(F, — 0.08)(1 — [0:028]! ) (270)

where q, = ﬁre load density, F v = opening factor, and K = insulation parameter = (Al-l V) x
(di/ki). Ai/ V is the ratio of exposed surface area of insulation to volume of steel, d, is the

thickness of insulation, and k, is the thermal conductivity of the insulation. Beck used the

ﬁre load density applicable to a compartment, having an area of 120 to 400m2, from the

survey results of US. ofﬁce buildings performed by Culver (1976). The author ﬁrst

developed the statistical models for basic variables qt, F v and K. The variation of most of

the basic parameters was assumed based on judgment and experience due to lack of data.
However, now it is possible to characterize the variation based on information available
in the literature. The mean and COV of the maximum steel temperature were then
obtained through Equation 2-70 by ﬁrst order approximation (see Equations 2-22 and 2-
23) using the means and COV’S of the basic random variables. This approximation may
result in signiﬁcant errors as the relation between the basic variables and the maximum
steel temperature is nonlinear and most of the basic variables are non-normal (e.g., ﬁre
load is best described by the Gumbel distribution (ECSC 2001)).

The steel and ﬁre temperature models used in Beck’s work were different then the
advanced mathematical models being suggested for use in the performance-based
approaches. As per new ﬁre models (EN 2002, Feasey and Buchanan 2002), the ﬁre
temperature signiﬁcantly depends on the type of compartment boundaries (Buchanan

2001), but in Beck’s work, the thermal properties (density, speciﬁc heat, thermal

71

conductivity) of compartment boundaries were not explicitly included as random
variables.

In Beck’s work, the information on the distribution of random variables was not used.
Most of the ﬁre design parameters are non-normal, and it is more appropriate that the
information on distributions be incorporated in the reliability analysis.

Beck found the probability of failure and corresponding reliability indices but did not

develop resistance factors that incorporated the effect of active ﬁre protection systems.

2.4.4 Li and Fitzgerald, 1996

Li and Fitzgerald investigated the relative safety of a structural steel beam protected by
gypsum board insulation. The beam was assumed to be protected by two 5/8 inch layers
of gypsum board that would normally provide a two hours ﬁre resistance rating according
to the prescriptive design approach. The beam was assumed to be used in a typical U.S.
ofﬁce buildings’ compartment with dimensions of 5 m x 5 m x 3 m. Only the ﬁre load
and the opening factor were considered as random variables, and the remaining
parameters were assumed to be deterministic. The mean and standard deviation of the ﬁre

load and the opening factor were taken from a survey of US. ofﬁce buildings (Culver

1976). The inﬂuence of the opening factor (ranging in values from 0.01m“2 to 0.15m1/2)

on the reliability indices was studied for a simply supported beam and it was concluded

. 1/2 . .
that an opening factor of 0.025m rs most severe for the assumed compartment srze.

Reliability indices were also obtained for various values of ﬁre loads using the adverse

values of thermal inertia and opening factors by assuming different support conditions. A

72

beam ﬁxed at both ends gave higher values of reliability indices compared to a simply
supported beam.

This study was limited in scope and considered just two random variables. It was not
reported how the reliability analyses were performed (e.g., what was the performance
function), however, the structural analysis was based on the procedures published in Fire
Engineering Design of Steel Structures (Pettersson et al. 1976). Although the study was

limited in scope, it did help to demonstrate that a smaller value of the opening factor

(0.025m1/2 in this case) is likely to yield a lower reliability index and thus a higher

probability of failure.

2.4.5 Ellingwood, 2005

This work is related to the loads and load combinations to be considered in ﬁre design.

Ellingwood showed showed that the probability of coincidence of a ﬁre with maximum

values of live load, Ln, roof live load, Lr, snow load, Sn, wind, or earthquake loads at or

near the magnitudes given in ASCE Standard 7—02 (2005) is very small, and a structure is
likely to be loaded to only a fraction of the design loads when the ﬁre occurs. Therefore,

these load combinations need not be considered for ﬁre design. The companion action of

1.2Dn, 0.5L", 0.5L,, and 0.25,, represent the most probable values of load on the- structure

at the time of ﬁre. These recommendations were included in the AISC Speciﬁcations and
in the ASCE Standard 7-02 (2005). Ellingwood suggested that the capacity reduction

factors should be developed corresponding to these values of loads.

73

2.4.6 CIB W14, 1983 and 1986

The CIB design guides (CIB W14 1983 and CIB W14 1986) recognized the need for a
rational design approach for ﬁre design and provided a complete methodology for the
development of load and resistance factors for structural ﬁre design. Some of the
recommendations were:

0 The load and resistance factors can be derived in the time domain, temperature
domain and strength domain. However, it was suggested that for load bearing
members it is more appropriate to develop the safety factors in the strength
domain using real ﬁre scenarios.

0 Since ﬁre scenarios vary according to the type of occupancy, height of the
building, etc., it was suggested that separate factors be derived for different types
and categories of occupancies.

0 Partial safety factors depend on the chosen target reliability index, which in turn
reﬂects the target probability of failure. CIB W 14 (1986) suggests that while
specifying safety factors for ﬁre design, the rare occurrence of a severe ﬁre (a ﬁre
that can cause structural damage) should be, accounted for. Thus the target
reliability index is usually smaller than the index used for deriving safety factors
at room temperature. The target reliability index should vary depending on the
presence of active ﬁre protection systems such as automatic sprinklers, ﬁre
brigade, etc., because these measures affect the occurrence of a structurally
signiﬁcant ﬁre. If the target reliability index is varied, the resulting load and

resistance factors also will vary. These guidelines suggests that basic safety

74

factors such as resistance factor may be modiﬁed using differentiated factors

based on the presence of active ﬁre safety measures.

2.4.7 Ramachandran, 1995a

Ramachandran (1995) suggested that the performance of a building and installed ﬁre
protection systems may be assessed in terms of the probable area of damage for property
protection and probability of death or fatality rate per ﬁre for life safety. Acceptable
minimum levels speciﬁed for area damage and fatality rate will provide design values for
escape facilities, size of compartments, and ﬁre resistance for structural and non-
structural members. Subject to these levels, the design of a building can be altered,
depending on the presence of active ﬁre protection systems such as sprinklers, detectors,
and smoke control systems.

Occurrence of the ﬂashover stage during a ﬁre is undesirable as this would adversely
affect the stability of structural elements. Ramachandran performed reliability analysis to
calculate the probable area of damage and probability of ﬂashover in a sprinklered and
unsprinklered building and concluded that for the occupancy type considered, sprinklers
reduce the probability of ﬂashover by a factor of 3.0. He suggested that this factor may be
used to deterrninine the reduction in the ﬁre resistance requirements for a sprinklered
compartment. This study is not related to the derivation of resistance factors for steel
members exposed to ﬁre, but, does suggest that sprinklers have a signiﬁcant inﬂuence in
reducing the probability of occurrence of a severe ﬁre, and this effect should be

considered in evaluating the ﬁre resistance of steel members exposed to ﬁre.

75

2.4.8 Ramachandran, 1995b

Ramachandran, Elms and Buchanan pursed the approach proposed in the CIB Design
Guides (1983 and 1986) for the development of probability-based designs for ﬁre safety.
The authors suggested comparing the ﬁre resistance, RF, and ﬁre severity, SF, in the time

domain. The ﬁre severity is the time to ﬁre exposure, and the ﬁre resistance is the

duration of time a building element can resist ﬁre. Fire resistance, Rp, and ﬁre severity,
Sp, were assumed to follow the normal distribution. The authors suggested that if data is

not available to determine the standard deviations, O'R F and O'SF , of Rp and Sp,

respectively, a coefﬁcient of variation, r, may be assigned for both Rp and Sp so that

OR = rmR and O'SF = 177151, ,where me and mSF , are the means of Rp and

F F

Sp, respectively. Reliability indices were estimated for different m R F /m S F values by

assuming r = 0.15. The authors suggested reducing the requirement of ﬁre resistance in
sprinklered buildings according to the sprinkler factor. A procedure for calculating the
sprinkler factor was described in the paper. The authors suggested that the actual
distribution of Rp and Sp may be close to lognormal but they used the normal distribution
for convenience.

This study was again based on the prescriptive design approach, and the ﬁre resistance
was compared in the time domain, contrary to the suggestion in the ﬁre literature that the
'ﬁre resistance of load bearing members should be compared in the strength domain.

However, it shows that sprinklers have a signiﬁcant effect in reducing the probability of

76

occurrence of ﬁre, and this effect should be accounted for while determining the ﬁre

resistance of structural members.

2.4.9 Harmathy and Mehaffey, 1985

In this study, two techniques were suggested for calculating the ﬁre resistance of
buildings in such a way that the failure probability remains the same in all cases. Fire
load and experimentally determined ﬁre resistance (ﬁre resistance determined through the
prescriptive approach) were considered as random variables and the remaining variables
were taken to be deterministic. It was proposed that a deterministic value of Opening
factor that would result in worst case ﬁre scenarios be used and a method was suggested
to determine that value of the opening factor. This was done on the basis that no
information was available about the variation of the opening factor. Based on a previous
study, the COV of 0.10 to 0.15 was suggested for experimentally determined ﬁre
resistance.

This study is based on the prescriptive approach that has been frequently used in the past.
However, in recent years the use of performance-based approaches for the ﬁre design of
steel members has been recommended, which has a different format than that used in this

study.

2.4.10 Fellinger, 2000

In this study, a reliability analysis of a concrete beam and slab was done and it was found
that comparing the ﬁre resistance in the time domain results in different levels of safety
for different types of structures. It was suggested that the ﬁre resistance be determined

using the strength domain.

77

2.4.11 Janes, 1995

In this work, a statistical analysis was performed to develop an adoption factor to be used
in the analytical models proposed in Eurocode 3 (EN 2005) for determining the limiting
temperature of structural members so that the models give the same failure temperature
as found in ﬁre tests. In the statistical analysis of steel beams and columns, a target
reliability index of 2.0 was used instead of 3.0 used in developing reliability-based codes
for room temperature design of steel members. The study was done for a different
purpose but suggests that use of a reduced target reliability index of 2.0 is appropriate in
the ﬁre design, which is consistent with the recommendations of other studies

(Ellingwood and Corotis 1991, ECSC 2001).

2.5 Summary

This chapter presented an overview of the: (1) ﬁre resistance evaluation methods for
structural steel members, (2) use of reliability theory in developing design speciﬁcations,
and (3) reliability studies related to steel members exposed to ﬁre. Until now, most steel
members were designed using prescriptive design approaches and consequently most of
the reliability studies were also performed on steel members designed by prescriptive
approaches. However, in the new design speciﬁcations, there is a shift toward the use of
performance-based design approaches, which are essentially based on the risk-based
design format, wherein load and resistance factors are used for minimizing the risk of
structural failure. Thus there is a need to develop these safety factors through reliability
analysis.

There is very limited work that was done explicitly for deriving the capacity reduction

and ﬁre load factors for the design of steel members exposed to ﬁre. Only work related to

78

the derivation of ﬁre load factors was performed using simpliﬁed assumptions instead of
rigorous reliability analysis, and it is not clear whether detailed reliability analysis will
yield similar results. In studies related to the derivation of capacity reduction factors, the
variation of many parameters was assumed due to lack of data, and also the parameters
were mostly speciﬁc to European conditions. In the ﬁrst order reliability methods used in
those studies, the means and COV’S of the response variables were determined from the
basic variables using the ﬁrst order Taylor series approximation, which may result in
signiﬁcant errors due to the nonlinear relationship between the response and basic
variables. Some of the parameters (e.g., thermal absorptivity or thermal inertia of
bounding surfaces which has a signiﬁcant inﬂuence on the ﬁre temperature) were not
explicitly treated as random variables in these studies. The ﬁre and steel temperature
models used in the earlier reliability studies were different than the current models used
by many researchers. These new ﬁre models and reliability-based software afford an
opportunity to perform more accurate reliability analysis of steel structures exposed to
ﬁre. Over the last few decades, ﬁre research has advanced signiﬁcantly, and more tests
data is available that can be used to better characterize the random parameters.

The effectiveness of active ﬁre protection systems in reducing the probability of
occurrence of severe ﬁres has been well recognized in the ﬁre literature, and has been
better characterized in the last few decades. In earlier studies related to the derivation of
resistance factors of steel members, the effect of these active ﬁre protection systems was
not explicitly included.

There is no reported work related to the derivation of capacity reduction and ﬁre load

factors speciﬁc to US. conditions for the LRFD of steel members exposed to ﬁre. The

79

2005 AISC Speciﬁcations allow the use of the LRFD format for the design of steel
members exposed to ﬁre. The load factors for the LRFD of steel members have been
proposed in one study, and are included in the AISC Speciﬁcations. At present, the AISC
Speciﬁcations and the commentary to the AISC speciﬁcations Specify a capacity
reduction factor and a ﬁre load factor speciﬁc to one kind of active ﬁre protection system
(sprinklers), which are based primarily on subjective judgment. There is a need to
develop capacity reduction factors and ﬁre load factors that can account for the effect of

all ﬁre protection systems using the US. data wherever possible.

80

Chapter 3

Reliability-Based Methodology for Developing Capacity

Reduction and Fire Load Factors

3. 1 General

In this chapter, a general reliability-based methodology is presented for developing
capacity reduction and ﬁre load factors for the LRFD of steel members exposed to ﬁre.
As discussed in Chapter 2, in previous studies, the effect of active ﬁre protection systems
(e.g., sprinklers, smoke and heat detectors, ﬁre brigade, etc.) in reducing the probability
of occurrence of a severe ﬁre was not explicitly considered. In this chapter, based on the
recommendations of some earlier works (ECSC 2001, Ellingwood and Corotis 1991), an
approach is suggested for developing capacity reduction and ﬁre load factors

corresponding to a preselected target reliability index that accounts for the effect of active

81

ﬁre protection systems (e.g., sprinklers, smoke and heat detectors, etc.) in reducing the
probability of occurrence of a severe ﬁre. In the previous studies, the mean and variation
of steel temperature was obtained using ﬁrst order Tylor series approximations, which
may result in signiﬁcant errors. In the methodology proposed in this chapter, the steel
tempaerture is included in the performance function in terms of the basic variables such
as ﬁre load, thermal absorpitivity, opening factor, etc. For incorporation of the steel
temperature into the performance function in terms of its basic variables, an analytical
solution of the heat balance differential equation of steel tempaerture was developed.

The methodology developed in this chapter is then used in Chapters 5 and 6 to develop
capacity reduction and ﬁre load factors for simply supported steel beams and columns.
To better understand the performance functions used in the reliability, the engineering

approach for designing steel members subjected to ﬁre conditions is described ﬁrst.

3.2 Engineering Approach for Designing Steel Members

Exposed to Fire

In the engineering approach, the nominal capacity of steel members exposed to ﬁre, Rnf,

is a function of fabrication parameters, F i, and reduced material properties, kj(Ts)Mj, and

may be expressed as

R,,,=fR(F,, ....... F,,k,(TS)M1, ....... kk(Ts)Mk) (3-1)

where the F i are dimensional and sectional properties (e.g., depth of section, cross—

sectional area, etc.), and M j are the material properties at room temperature (e. g., yield

82

strength, etc.). kj(TS) are factors that account for reduction in strength and stiffness of

steel at elevated temperature, and their values at different values of steel temperature, TS ,

are speciﬁed in the AISC Speciﬁcations (AISC 2005a) and Eurocode 3 (EN 2005).

According to the AISC Speciﬁcations, the ultimate design action (applied axial force,

bending moment, or shear force, etc.), Quf, is determined from the load combination
given by

U = 1.21),, + 0.51., + 0.25,, + T (32)

where D", Ln and 5,, are nominal dead, live and snow loads, respectively, and T includes

loads or loads effects induced by the ﬁre itself (such as additional bending moment
induced due to thermal expansions being restrained by the surrounding structure). The

magnitude of the term T in Equation 3-2 will depend both on the type of restraint and on

the steel temperature, TS. For a simply supported beam, the term T in Equation 3-2 will

be zero because thermal expansion is free to occur.

The governing design equation for ﬁre design of steel members can then be expressed as
(if R... f 2 QM <3-3)
Under ﬁre conditions, both the nominal capacity, Rnf, estimated through Equation 3-1

and the applied load or load effect, T, in Equation 3-2, depend on the steel temperature,

Ts, which in turn depends on the design ﬁre (or time-ﬁre temperature curve). The design

ﬁre depends on many factors such as ventilation conditions, thermal properties of the

boundaries and the ﬁre load (representative of combustible materials present), etc. As

83

mentioned earlier, the Commentary to the AISC Speciﬁcations (AISC 2005b) states that
while describing the design ﬁre, the ﬁre load may be reduced by up to 60 percent if a
sprinkler system is installed in the building. In a similar vein, Eurocode 1 (EN 2002)
suggests a reduction in the ﬁre load, while the ECSC study recommends either a

reduction or increase in the ﬁre load depending on the intended reliability. This reduction

or increase (called ﬁre load factor, yq, in this study) is to be applied to the ﬁre load used

in describing the design ﬁre, and will affect the nominal capacity of all members and the
applied load effect, T, in Equation 3-2 for restrained members.

At elevated temperatures, the strength and stiffness of steel reduces signiﬁcantly, and if
unprotected, steel members fail within a short time. Therefore, steel members are
generally protected by insulation or ﬁre protection material to slow down the rise of the
steel temperature. The required thickness of insulation can be determined from Equation
3-3 using an iterative procedure, and the ﬁre temperature in the compartment and the
steel temperature of the member required for this procedure can be estimated as described

in the next two sections.

3.3 Fire Temperature in the Compartment

The ﬁre temperature, Tf, can be estimated using a suitable mathematical model from the

literature (e.g., SFPE 2000 and 2004, EN 2002, and, Feasey and Buchanan 2002, etc.). In
this study, the Eurocode parametric ﬁre model modiﬁed by Feasey and Buchanan (2002)
was used to estimate the ﬁre temperature under real ﬁre scenarios. The modiﬁed
Eurocode ﬁre model is brieﬂy described below.

The ﬁre temperature in °C during the burning phase is given by

84

T, =1325(1—0.324e_0'2’ —0.204e“-7t -0.472e‘1°t )+20 (3-4)

where r‘ is a ﬁctious time in hours related to the real time t through ti = Ft ,

__(F,,/0.04)2

r 2
(b/1900)

(3-5)

Fv = opening factor (ml/2), and b = thermal absorptivity of the bounding surfaces of the

compartment (WSO'S/mzK). The duration of the burning phase, tab in hours is given by

td = 000013-18;— (3-6)

V

where q, = ﬁre load (MJ/m2 of the total surface area of bounding surfaces).

For the cooling phase, the Eurocode gives a reference decay rate (dT/ dtlref = 625°C/hr

for ﬁres having a burning period of less than half an hour, decreasing to 250°C/hr for
ﬁres with a burning period greater than 2 hours. To incorporate the effect of thermal
insulation and the opening factor, Feasey and Buchanan (2002) suggested that the
Eurocode reference decay rate be modiﬁed to

dT _(dT] .lF,/0.04

'37 7:— ,ef ,lb/1900 (3‘7)

 

The ﬁre temperature, Tf, is thus found to be dependent upon:
(1) the amount of ventilation (represented by the opening factor, F v);

(2) the amount of combustibles in the compartment (represented by the ﬁre load, (It)?

and

85

(3) the thermal absorptivity, b, of the compartment boundaries (representing the

amount of heat absorbed by the compartment boundaries).

3.4 Calculating the Temperature of Steel Members

Once the ﬁre temperature variation with time is known, the temperature of steel elements
can be estimated through thermal analysis. The steel temperature can be calculated using
any advanced ﬁnite element software such as SAFIR (Franssen et al. 2004), ANSYS
(2007), etc. However, most design speciﬁcations such as the AISC Speciﬁcations (AISC
2005a) and Eurocode 3 (EN 2005), allow the steel temperature to be calculated using
simple thermal analysis methods such as the lumped heat capacity method.

The lumped heat capacity method assumes that the steel section is a lumped mass at
uniform temperature. The heat balance differential equation for steel members protected

by insulation can then be written as (Buchanan 2001)

dT F kl pscs
— = [—j —— (Tr ‘ Ts) (3-8)
dt V dipscs pscs +0.5(F/V)di,0,-Ci

where dT/dt = rate of change of steel temperature, F = surface area of unit length of the

 

member (m2) ,V = volume of steel per unit length of the member (m3), p5 = density of
steel (kg/m3), Cs = specific heat of steel (J/kg.K), p,- = density of the insulation (kg/m3), c,’
= speciﬁc heat of insulation (J/kg.K), cl, = thickness of insulation (m), k; = thermal

conductivity of insulation (W/m.K), T, = steel temperature (°C), and Tf= ﬁre temperature

(°C).

86

Equation 3-8 can be written in ﬁnite difference form and the steel temperature can then
be calculated at any time using a ﬁnite difference method that can be implemented in a
spreadsheet. However, for incorporation into performance functions used in reliability

analysis, a closed-form expression for calculating the maximum steel temperature,

Tmmx, is convenient. Therefore, Equation 3-8 was solved analytically to obtain the

 

 

solution
Ts,max : Tf.max — ’16“: _1) + ’15”! + C4e-aI (3-9)
where
a, _ (£)[ ki ][ pscs ]
- (3-10)
V dipscs pscs +0.5(F/V)d,-,0,-c,-
625 for td < 0.5 hours
xi. = 625 — 250(td — 0.5) for 0.5 S ta, S 2 hours (3_11)
250 for rd > 2 hours

15
z=—-l-In[-:C—§), C4 = emd (stUd) “Tf.max "3). (3-12)

a 4
5 ,lF, /0.04
= 3-13
Jib/1900 ( l

The steel temperature at the end of the burning phase of the ﬁre is

 

st (1d) = 1345+ CI(€_wd - e—O'an ) + C2(e_wd - fund )+ C3(e'wd — e—lgnd ) -1325e'wd

(3-14)

87

__ 429.36! _ 27030, C _ 625.405

h , C- , .——-—, 3—
were 1 a_0.2[‘ r a—1.7F 05—191“ (3-15)

The maximum temperature of the ﬁre, Tﬁ max, can be calculated through Equation 3-4

using I = Id. The temperature attained by a steel section protected by insulation, Ts, is
thus found to be dependent upon:
(1) cl, = thickness, ki = thermal conductivity, Ci 2 speciﬁc heat, and p,- = density of the
ﬁre protection material or insulation;

(2) p5: density of steel , and cs = speciﬁc heat of steel; and

(3) F/V = section factor (ratio of exposed surface area of insulation to volume of
steel per unit length of the member).

Equation 3-8 is used to estimate the temperature of steel members protected by
insulation. The temperature of unprotected steel members can be estimated through a
similar equation (Buchanan 2001). The heat balance differential equation for unprotected
steel members is not presented since in the US. steel columns are always protected and
steel beams are almost always protected.
The codes allow using the lumped mass method, but caution that this method may be
overly conservative for certain situations such as for a composite steel beam with a
concrete slab on top in which a signiﬁcant thermal gradient can occur through the depth.
In this study, the lumped mass method is used because it is convenient within a
reliability-based framework. The error arising from this method because of the
assumption of a uniform temperature distribution is accounted for through a model error

(or professional factor) that is presented later in this dissertation.

88

3.5 Methodology for Developing Capacity Reduction and Fire
Load Factors

Capacity reduction and ﬁre load factors can be developed by performing the following
steps:

(1) Select an appropriate performance function (design equation) for the reliability
analyses of a structural member, and identify the random design parameters.

(2) Characterize model errors (i.e., the professional factor) to account for the
differences in the capacity calculated from the design equation and that measured
in ﬁre tests.

(3) Obtain statistical parameters such as the mean, coefﬁcient of variation (COV) and
distributions of random design parameters (e.g., yield strength of steel, cross-

sectional area of steel, ﬁre load, opening factor, etc.).

(4) Select a target reliability index, 6,, which reﬂects the target probability of failure

and is a relative measure of safety.
(5) Calculate the capacity reduction and ﬁre load factor through reliability analysis.
In the succeeding paragraphs, the above steps are elaborated in the context of steel

members exposed to ﬁre.
3.5.1 Performance Function for Reliability Analysis

3.5.1.1 Applied Loads

Ellingwood (2005) showed that the probability of coincidence of a ﬁre with maximum

values of live load, roof live load, snow, wind, or earthquake loads is negligible, and a

89

structure is likely to be loaded to only a fraction of the design load when a ﬁre occurs.
Therefore, it is appropriate to use the combination of dead and arbitrary-point-in-time

live load for reliability analysis under ﬁre conditions. This is consistent with Beck’s

(1985) recommendation. Therefore, the load effect, Wf, for reliability analysis may be
calculated as
Wf= E(cDAD + CLBLapt) (3-16)

where cD and cL = deterministic inﬂuence coefﬁcients that transform the load intensities

to load effects (e.g., moment, shear, and axial force), A and B = random variables

reﬂecting the uncertainties in the transformation of loads into load effects, E = a random

variable representing the uncertainties in structural analysis, and D and Lap, = random

variables representing dead and arbitrary-point-in-time live load. The statistics of random

variables A, B, E, D and Lap, are presented in the next chapter.

3.5.1 .2. Capacity of Steel Members

The actual capacity of steel members under ﬁre can be obtained by modifying the
nominal capacity given by Equation 3-1 to

R, = P.fR(f,F,, ....... f,F,,k,(t,T,)m,M,, ....... k, (t,T,)m,M,,) (3-17)
where P, f), mj, and ts are the following non-dimensional random variables:

P= “Professional factor” reﬂecting uncertainties in the assumptions used to
determine the capacity from design equations. These uncertainties may

result from using approximations in place of exact theoretical formulas, and

90

from assumptions such as perfect elasto-plastic behavior and a uniform

temperature across the section.

f,- = Random variable that characterizes the uncertainties in “fabrication.”
m,- = Random variable that characterizes uncertainties in “material properties.”

ts: Random correction factor that accounts for differences between the steel

temperature obtained from models and that measured in actual tests.

It is assumed that the random variables fi and m,- are the same as those used for

developing LRFD speciﬁcations for ambient temperature conditions and their statistics
are available in the literature. The statistics of P are speciﬁc to each design equation,
cannot be generalized, and can be obtained from a comparison between the predicted
capacity and test results. The statistics of P speciﬁc to design equations used in this study
for simply supported steel beams and axially loaded steel columns are presented in

Chapters 5 and 6, respectively. The statistics of ts are characterized in Chapter 4.

The steel temperature, TS, in Equation 3-17 is a function of many basic parameters (e.g.,

ﬁre load, opening factor, thermal absorptivity, thickness of insulation, thermal properties
(density, speciﬁc heat, thermal conductivity) of insulation, and thermal properties
(density, speciﬁc heat, thermal conductivity) of steel, etc.). The statistics of these basic

parameters which signiﬁcantly affect the steel temperature are developed in the next

chapter. The steel tempaerture, TS, can be estimated as described in the previous section.

91

In this study, the steel tempaerture, Ts, is incorporated in Equation 3-17 in terms of basic

variables using Equations 3-4 to 3-15.

3.5.1.3 Limit State Equation

Using Equations 3-16 and 3-17, the limit state equation for reliability analysis under ﬁre

conditions may be written as
g(X)=Rf- Wf (3-18)

where X denotes a vector containing all the random variables. The probability of failure,

pp, of a steel element under ﬁre is

Pr = P1800 < 0] (3-19)

3.5.2 Statistics of Random Parameters

The design parameters that signiﬁcantly affect the ﬁre design of steel members were
identiﬁed in the previous section (Section 3.5.1), and their means, coefﬁcients of

variation (COV), and distribution types are characterized in the next chapter.

3.5.3 Probability of Failure and Target Reliability Index

Partial safety factors depend on the selected target reliability index, which in turn reﬂects
the target probability of failure. CIB W 14 (1986) suggests that the rare occurrence of a
severe ﬁre should be taken into account and therefore the partial safety factors are
generally lower than those used at room temperature. In a similar vein, the target
reliability index should vary depending on the presence of active ﬁre safety measures
such as automatic sprinklers, ﬁre brigade, etc., because these measures affect the

probability of occurrence of a severe ﬁre. Correspondingly, the target reliability index is

92

usually smaller than that used for deriving safety factors at room temperature. The
probability of failure under ﬁre conditions can be written as (AISC 2005a, ECSC 2001,

and CIB W 14 1986)

PF = PF,ﬁ Psf (3-20)
where

pp = target probability of failure (which is the same as that used for room

temperature LRFD)

pp, ﬁ = probability of failure under ﬁre

The probability of occurrence of a severe ﬁre can be expressed as (ECSC 2001, CIB W

14 1986)
p5; = 1911921031944; (3-21)
where
p1 = probability of ﬁre occurrence including the effect of occupants and standard

ﬁre brigade (per m2 of ﬂoor area and per year)

p2 = reduction factor depending on ﬁre brigade type and on time between alarm

and ﬁremen intervention (p2 is also the probability Of failure of the ﬁre

brigade in stopping the ﬁre)

p3: reduction factor if automatic ﬁre detection (by smoke or heat) and! or

automatic transmission of the alarm are present
p4 = reduction factor if sprinkler system is present (p4 is also the probability of

failure of the sprinkler in stopping the ﬁre)

93

Af= ﬂoor area of the ﬁre compartment

By rearranging Equation 3-20, the probability of failure under ﬁre becomes

PF
pF,r:_ _
f pg, (3 22)

The reliability index, 6, is related to the probability of failure through
pp = (IX-6) (323)

Therefore, for ﬁre conditions, the target reliability index can be written as

_ —1 PF
:6: — 1” 1‘— (3-24)
psf

Since the probability of occurrence of a severe ﬁre varies depending on the presence of
active ﬁre protection systems, the target reliability index varies for different design
situations. Consequently, the capacity reduction factor and ﬁre load factor will vary for
each design situation. The framework of determining the target reliability index from

Equations 3-20 to 3-24 was used to develop safety factors on the ﬁre load (ECSC 2001).

Using the values for p1, p2, p3 and p4 suggested in the ECSC study, the values of the

target reliability index were calculated using Equation 3-24 for typical ﬁre compartments

of an ofﬁce building and are shown in Table 3.1. Under normal conditions, the Eurocode

requires a maximum target probability of failure of 7.23x10-5 for building life, which
corresponds to a target reliability index of 3.8 (ECSC 2001). We used this value to
calculate the target reliability index values shown in Table 3.1.

Lower target reliability index values do not mean that the target probability of failure has

increased, but rather reﬂects the presence of active ﬁre protection systems that reduce the

94

probability of occurrence of a severe ﬁre and accordingly reduce the probability of failure

under ﬁre.

Th e AISC Speciﬁcations allow a reduction in ﬁre load for sprinkler systems only. If we

consider only the effect of sprinklers in reducing the probability of occurrence of a severe
ﬁre, then the target reliability index will depend on the effectiveness/reliability of

sprinklers and on the size of the compartment. In the ECSC study, sprinklers were
considered highly effective with effectiveness ranging from 95-99.5%. Bukowski et al.
(1999) performed a statistical analysis of the Operational reliability of sprinklers and
concluded that the reliability of sprinklers lies within 88 and 98%. Based on the
combined operational effectiveness and performance effectiveness data published by
NFPA, William (2005) assessed the overall reliability of automatic sprinkler systems to
be 91%. Column 4 of Table 3.1 shows the target reliability index values calculated from.
Equation 3-24 assuming a conservative value of 70% for the reliability/effectiveness of
sprinklers. The maximum value of target reliability index is around 2.0. Since sprinklers
are installed in most US. ofﬁce buildings, it is reasonable to use target reliability index
values ranging from zero to 2.0 for deriving capacity reduction and ﬁre load factors. .
To account for the reduced probability of occurrence of a severe ﬁre, a similar approach
to the one presented in the ECSC study was suggested by Ellingwood and Corotis (1991)

for ﬁre resistant structural design. They suggested a probability of failure for ﬁre

situations that corresponds to 6, of about 1.5, which falls within the range of zero to 2.0

found from the ECSC study.
In this study, the capacity reduction and ﬁre load factors are developed for a range of

reliability index values (zero to 2.0), both for simply supported steel beams and columns.

95

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96

3.5.4 Reliability Analysis

When the performance function (Equation 3-18) is chosen, all the random parameters and

model errors (e.g., ts and P in Equation 3-17) are characterized (i.e., their means, COV’s,

and distribution types have been established), and a target reliability index is calculated
through Equation 3—24, reliability analyses can be performed as detailed in Section 2.2.

In this study, the maximum steel temperature obtained from Equation 3—9 is incorporated
in the performance function (Equation 3-17) in terms of the basic variables and the
reliability analysis is then performed as described in Sections 2.3.2 and 2.3.3. The
information about the distribution of the basic random variables is included as described
in Section 2.3.4.

Any structural reliability analysis software can be used to perform the reliability analyses.
In this study, the FERUM (Finite Element Reliability Using Matlab) software (Der
Kiureghian 2006) is used to perform the reliability analyses. FERUM is a general purpose
structural reliability software written using Matlab. It can be used to perform reliability
analysis using different methods, including the ﬁrst order reliability method (FORM).
The output of FORM analysis for a particular design situation includes the reliability
index, the probability of failure, the values of all design parameters at the failure/design

point, and the direction cosines of the design point for each design parameter.

3.5.5 Determination of Partial Safety Factors

When reliability analysis has been performed, the partial safety factors for each of the

basic random parameter can be obtained using the direction cosine of the design point for

97

each design parameter. The detailed discussion about the partial safety factors is provided
in Sections 2.3.6 and 2.3.8.
For a normally distributed random design parameter, X, the partial safety factor resulting

from the ﬁrst-order reliability method (FORM) is given by

mx

¢X=X

 

(1+ ax ﬂ,Vx ) (3—25)

’1

where ax , mx, Vx and X" are the direction cosine of the “design point,” mean, COV and

nominal value of X, respectively, and B, is the target reliability index. For non-normal

random parameters, an explicit equation can be derived for the partial safety factor if the
PDF and CDF are simple functions (e. g., a Gumbel distribution). The safety factor can
be determined from Equation 3-25 using the mean and standard deviation of the
equivalent normal variable at the design point if the PDF and CDF are complex functions
(e.g., a lognormal distribution). The procedure for estimating the mean and standard
deviation of the equivalent normal variable at the design point is explained in Sections

2.3.4 and 2.3.8.

3.5.6 Combination of Partial Safety Factors into Single Capacity

Reduction Factor

For convenience in design, the variability of all design parameters except for the ﬁre load
was accounted for through a combined capacity reduction factor instead of using a
separate partial safety factor for each design parameter. The detailed discussion about the
combination of partial safety factors into a single capacity reduction factor is provided in
Section 2.3.8.

98

The partial safety factors, q); obtained through Equation 3-25 for each individual design

parameter except the ﬁre load can be combined into a single capacity reduction factor, (gr,

through

_ fR(Xn,q9¢an,1a¢2Xn.2, ........... ¢an,m)
_ 3-26
fR(Xn,q9Xn,l,Xn,2, .............. Xan) ( )

 

¢f

where, d,- and Xn,i are the partial safety factors and nominal values for all parameters

related to the capacity of steel members except the ﬁre load, respectively, and qu is the

nominal value of the ﬁre load.

Fire load is a major parameter in ﬁre design, and uncertainty associated with the ﬁre load
has a signiﬁcant effect on the safety of the design. Therefore, the variability of the ﬁre
load on overall safety is accounted for through the speciﬁc partial safety factor on the ﬁre
load. The use of a separate safety factor for the ﬁre load also enables the ﬁre load to be
reduced for certain target reliability index values as demonstrated later in Chapters 5 and
6. As mentioned in the introduction, the Commentary to the AISC Speciﬁcations (AISC
2005b) states that the ﬁre load be reduced by 60% if a reliable sprinkler system is
installed. In a similar vein, Eurocode 1 (EN 2002) suggests a reduction in the ﬁre load,
while the ECSC study recommends either a reduction or an increase in the ﬁre load
depending on the intended reliability. These recommendations also motivated use of a
separate safety factor for ﬁre load. Using this approach, the effect of active ﬁre protection
systems (e.g., automatic sprinklers) in reducing the probability of occurrence of a severe

ﬁre is accounted for through reliability analysis and often results in a reduction in the ﬁre

99

load as demonstrated later in Chapters 5 and 6. This reduction or increase (called the ﬁre

load factor, yq, in this study) is to be applied to the ﬁre load used in describing the design

ﬁre.

3.5.7 Optimal Capacity Reduction and Fire Load Factor

The capacity reduction factor obtained through Equation 3-26 and the ﬁre load factor
obtained through Equation 3-25 will vary for each design situation. For ease of design, it
is desirable to have a single optimal capacity reduction factor applicable to all design
situations. In addition, for ﬁre design of steel members, the AISC Speciﬁcations
recommend using dead and live load factors of 1.2 and 0.5, respectively. Therefore, the
optimization procedure described in Section 2.3.6, and summarized below is used to
develop optimal capacity reduction and ﬁre load factors corresponding to dead and live
load factors of 1.2 and 0.5, respectively.

The required nominal capacity (e. g., moment, shear, or axial capacity) based on the ﬁrst

order reliability method (FORM) is

1
Rig] = W Z YiHQn,i (3-27)

II 1’ . . . . .
where 7,- and ¢i are the load and capacrty reduction factors for each desrgn srtuatron

obtained from Equations 3-25 and 3—26, respectively. The nominal capacity

corresponding to constant dead and live load factors is

1 1
Rn =—ZYiQn.i (3-28)
¢f i

100

where the 7,- are 1.2 and 0.5 for dead and live load, respectively. The value of gt in

Equation 3-28 can be selected by minimizing the objective function

8(¢f ) = ZlRﬁj - R1. ,- 12 P j (329)

where 171' is the weight assigned to the jth design situation. In this study, each design

situation was assigned the same weight.

3.6 Summary

In this chapter, a general reliability-based methodology is proposed for developing
capacity reduction and ﬁre load factors for the design of steel members exposed to ﬁre.
As discussed in Chapter 2, active ﬁre protection systems, especially sprinklers, have a
signiﬁcant effect on the occurrence of a severe ﬁre, and this effect should be accolmted
for while calaculating the ﬁre resistance of steel members. In the methodology proposed
in this chapter, the capacity reduction and ﬁre load factors correspond to a preselected
target reliability index that accounts for the effect of active ﬁre protection systems (e.g.,
sprinklers, smoke and heat detectors, etc.) in reducing the probability of occurrence of a
severe ﬁre. Contrary to previous studies (Magnusson and Pettersson 1981, and Beck
1985), instead of approximating the mean and variance of the steel temperature, the steel
temperature was expressed in terms of the basic variables in the performance function,
and the information on the distributions of random parameters was used. In the ECSC
study, ﬁre load factors were obtained using simpliﬁed assumptions, whereas in this work

the ﬁre load factors were derived through rigorous reliability analysis.

101

To illustrate the proposed methodology, in Chapters 5 and 6, capacity reduction and ﬁre
load factors are derived for simply supported steel beams and columns in US. ofﬁce

buildings exposed to ﬁre.

102

Chapter 4

Statistics of Random Parameters and Model Errors

In Chapter 3, the design parameters that affect the ﬁre design of steel members were
identiﬁed. In this chapter, the design parameters that signiﬁcantly affect the ﬁre design of
steel members are chosen as random variables, and their statistics are obtained. In
Chapter 2, it was concluded that there is not much information available about the
varation of random ﬁre design parameters based on US. test data. In this chapter, the
information related to US. conditions is used wherever possible.

Chapter 3 descibed two types of models that are used for the ﬁre design of steel
members: (1) the thermal models used to estimate the temperature of steel member, and
(2) the structural models (or design equations). The thermal model for estimating the steel
temperature was presented in Section 3.4. The model error associated with this model is
characterized in this chapter based on experimental data. The errors associated with the

structural models are speciﬁc to design equations and cannot be generalized and are

103

characterized for simply supported steel beams and columns in Chapters 5 and 6,

respectively.

4. 1 Statistics of Random Parameters

The design parameters that signiﬁcantly affect the ﬁre design of steel members were
chosen as random variables, and their means, coefﬁcients of variation (COV), and
distribution types are summarized in Tables 4.1 and 4.2. The statistics of the arbitrary-
point—in-time live load, ﬁre load and ratio of ﬂoor area to total surface area of the ﬁre
compartment are speciﬁc to US. ofﬁce buildings. The remaining parameters are general
and apply to steel buildings of all use categories. The statistics of the dead and arbitrary-
point-in-time live loads in Table 4.1 were reported by Ellingwood (2005) and Ravindra
and Galambos (1978). The statistics of the remaining parameters in Table 4.1 were
reported by Ravindra and Galambos (1978), and were assumed to have normal
distributions.

Table 4.1 - Mean, COV and distributions of design parameters related to loads

 

Variable Characterizes variation in Mean COV Distribution
A the transformation of dead loads 1.00 004 normal
into load effects

the transformation of live loads

 

 

 

 

 

 

 

 

 

 

 

B into load effects 1'00 0'20 normal
E structural analysis 1.00 0.05 normal
D dead loads l.05*nominal 0. 10 normal
L apt arbitrary-point-in-time live load 0.24*nominal variable Gamma

 

Note: The COV of the arbitrary-point-in-time live load depends on the tributary area
(Ravindra and Galambos 1978) and is given as:
0.82[1-0.00113(A7—56)] for 56 _<_ ATS 336 square feet
0.56[l-0.0001865(A7—336)] for AT> 336 square feet

104

Table 4.2 - Mean, COV, distributions and nominal values of fire design parameters

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

No. of Dis tri- Nominal
Variable Data Mean COV .
, button
Pomts
. * 2 90th
Frre load, q f 564 MJ/m 0.62 Gumbel percentile
Ratio of floor area to total area, 1610
0.192 0.23 lognormal mean
Af/AI
Opening factor, Fv # 1*nominal 0.05 normal mean
Thermal conductivity of normal
weight concrete (NWC), k 18 1.747 W/m.K 0.171 normal mean
Speciﬁc heat of WC’ cp 15 856 J/kg.K 0.062 normal mean
Density of NWC, p 12 2258 kg/m3 0.069 normal mean
Thermal conductivity of 105
lightweight concrete (LWC), k 0.372 W/m.K 0.199 Gumbel mean
Speciﬁc heat of LWC, cp ‘3 826 J/kg.K 0.062 Gumbel mean
Density of LWC, p 22 1344 kg/m3 0.069 normal mean
Thermal absorptivity of NWC, @ 0 5 2
1830 Ws ”/m K 0.094 normal mean
b/vwc
Thermal absorptivity of LWC, @ 0 5 2
640 Ws ' /m K 0. 107 normal mean
wac
Thermal absorptivity of gypsum 16 0 5 2
423 Ws "/m K 0.09 normal mean
board, bg
Thermal absorptivity of a
compartment having a 50/50 mix $ 0 5 2
of NWC and gypsum board as 1127 WS .. /m K 0-10 normal mean
boundaries, bmix
Thickness of ﬁre protection
materials (1' & nominal+lll6 0.20 lognormal
’ l . _ # inch 0.05 normal mean
( l) spray applied materials nominal mean
(2) gypsum board systems
Density of ﬁre protection 3
materials, pi 10 307 kg/m 0.29 normal
, , 1 8 3 0.07 lognormal mean
(1) spray applied materials 745 kg/m mean
(2) gypsum board systems
Thermal conductivity of ﬁre
P'O‘w'on Tig‘236ocki' at 24 0.187 W/m. K 0.24 lognormal
temPe'a‘U’e ° ,' . 40 0.159 W/m. K 0.28 lognormal mean
(1) spray applied materials mean
(2) gypsum board systems
Note: * based on the survey of 23 ofﬁce buildings in US. and # assumed
@ derived from the statistics of density (p), thermal conductivity (k) and speciﬁc heat (cp)
for respective types of concretes
$ derived using bNWC and bg (see Equation 4-8)
& Obtained from Carino et al. (2005)

 

 

105

Raw experimental data was analyzed as discussed below to obtain the statistics of all
parameters in Table 4.2 except for the ﬁre load. To determine if data is reasonably ﬁtted
by a particular distribution goodness-of-ﬁtness test was performed using Kolmogorov-
Smimov (K-S) test. A signiﬁcance level of 5% was used while performing K-S test.
Normal, lognormal, Extreme Values and Gamma distributions were considered and the

the one which best ﬁtted the data per K-S test was selected.

4.1.1 Fire Load Density

The ﬁre load is based on the quantity of combustible materials present in a ﬁre
compartment, and is, a measure of the total energy released in a ﬁre. For calculating the

ﬁre temperature, the ﬁre load is generally expressed as the ﬁre load density (ECSC 2001)
l ,
CIf=-A—Z(ll/ixmiXHc,iXMi) (4-1)
f
where qf = ﬁre load density (MJ/m2 of the ﬂoor area), Af = ﬂoor area of the ﬁre
compartment (m2), 111,- = de—rating factor for assessing the protected amount of ﬁre load
of the material i, mi = factor between 0 and l describing the combustion behavior of
material i, He, i = net caloriﬁc value of material 1' (MJ/kg), and M,- = mass of material i

(kg). HQ,- x M,- represents the total amount of energy released from material i assuming

complete combustion. Complete combustion of some materials does not occur in a ﬁre
and this is accounted for by using a factor m. However, because there is a lack of

experimentally substantiated and veriﬁed values, the factor m may be neglected

106

(Pettersson et al. 1976). De-rating factors are applied to account for the quantity of
enclosed combustibles (such as paper in metal ﬁling cabinets) that will not burn in a ﬁre,
and the resultant ﬁre load is known as the de-rated ﬁre load.

Fire loads have been historically established by surveys of typical buildings in various
use categories (e.g., ofﬁce buildings, residential buildings). Culver (1976) reported
statistics of the total and de-rated ﬁre load for 23 typical U.S. ofﬁce buildings. Culver

reported the ﬁre load as lb/ft2 of floor area that was converted into ﬁre load density using

HG, i = 17.5 MJ/kg (caloriﬁc value of wood) and m = 1.0. The obtained statistics of ﬁre
load density are:
(1) Total ﬁre load density
0 Mean = 624 MJ/m2 of ﬂoor area
0 COV = 0.60
(2) De-rated ﬁre load density
0 Mean = 564 MJ/m2 of ﬂoor area
0 COV = 0.62
Another survey of two ofﬁce buildings was done in the US. in 1995 (Caro and Milke

1995). Both, compartmented and open types of ofﬁces, were surveyed and the de-rated

ﬁre loads were reported as lb/ft2 of floor area. These were converted into ﬁre load

density using Hc,i = 17.5 MJ/kg (caloriﬁc value of wood) and m = 1.0. The obtained

statistics of ﬁre load density are:

(1) Fire load density-open plan ofﬁces

107

0 Mean = 1243.4 MJ/m2 of floor area

0 COV = 0.39

(2) Fire load density-compartmented ofﬁces

0 Mean = 1135.7 MJ/m2 of ﬂoor area

0 COV = 0.11

The statistics of the ﬁre load density from the two surveys mentioned above (Culver
1976, Caro and Milky 1995) are compared in Table 4.3 with ﬁre load densities reported
in the literature. Table 4.4 shows the ﬁre load densities obtained from another reference
(Yii 2000). Tables 4.3 and 4.4 show a signiﬁcant scatter in the statistics of ﬁre load
densities for ofﬁce occupancies obtained from surveys carried out in different countries.
The large scatter can be attributed primarily to the different furnishings used in different
countries, different surveying techniques, and use of different values of combustion and
de-rating factors.
Even within the US, the three ﬁre load surveys reported completely different values.
The mean ﬁre load reported by Culver (1976) is much smaller than that reported by Caro
and Milke (1995). On the contrary, the COV of the ﬁre load reported by Culver is much
higher than that reported by Caro and Milke.
For this study, statistics of the de—rated ﬁre loads reported by Culver (1976) were chosen
instead of those given in the more recent survey of Caro and Milke (1995) for the
following reasons:

0 In the 1995 survey, the sample size was very small (2 buildings) compared

to the 1976 survey (23 buildings).

108

Table 4.3 - Comparison of statistics of ﬁre load density (MJ/m2 of floor area) in
ofﬁces

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fire load density
Reference Country/year 2 Remarks
Mean ( MJ/m ) COV
NCES (2001) 420 0.30
Harmothy (1985) 434 0.35
Switzerland 800
Bukowski (1986) US. 932
(l942&l957)
624 0.60 total ﬁre load
0‘1"” (1976) ”‘8' (1976) 563 0.62 de-rated ﬁre load
, l 136 0.1 1 compartmented ofﬁces
lk 9 . . 1995
Caro and M1 e (l 95) U S ( ) 1243 0.39 open plan ofﬁces
JCSS (2001) 600 0.30
Kumar (1997) India (1997) 328 0.75
Japan (1964) 1085 0.17
Netherland
.. (1963) 263
Yn (2000) 385
Newzeland 664 0.33
998

 

 

 

 

 

 

Table 4.4 - Comparison of ﬁre load density in ofﬁces (Yii , 2000)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Reference Fire Load Density
Mean Standard Deviation COV
(MJ/mz) (MJ/mz)

Mabin.( 1994) 21 10 1149 0.54
Barnett( 1984) 224 -
Narayanan (1994) 476 233 0.49
F.E.D.G (1994)** 800 , -
Petterson (l976)** 600 -
B.S.F.S.E. (1993) 420 -
Swedish data** 41 1 334 0.81
US. data* 555 625 1.13
European data* 420 370 0.88
Swiss data* 750 -
Duch data* 410 330 0.85
French data* 330 400 1.21
Notes: *CIB W14(l983),Table A 1.3.4, ** Swiss data

 

109

The COV of 0.62 reported by Culver is signiﬁcantly higher than the COV of
0.39 (open plan ofﬁces) and 0.11 (compartmented ofﬁces) reported by Caro
and Milke. The larger COV is statistically signiﬁcant and will cater for the

large variation in ﬁre loads.

In a recent study in Europe (ECSC 2001), a mean of 420 MJ/m2 and a COV

of 0.3 for ﬁre load density were used for developing the safety factor on ﬁre
load. The combustion factor m = 0.8 used in the ECSC study (ECSC 2001)
for reducing the ﬁre load is not used in this study. This will yield
conservatism and will account for the higher values of mean ﬁre load
reported by Caro and Milke.

One of the authors involved in the 1995 survey was contacted to ascertain
the reasons for the signiﬁcantly higher values of mean ﬁre load compared to
those reported by Culver. The author was of the opinion that the survey
carried out in 1995 had a very small sample size and suggested that it is

more reasonable to use the values reported by Culver for reliability analyses.

Fire engineering guideline documents, including the [PEG (International Fire
Engineering Guideline 2005) caution that mean values should not be used for design
purposes since half of the buildings would be expected to exceed this value and
recommended use of the 90th or 95th percentile values for design purposes (Bukowski
2006). Buchanan (2001) also recommended that the design ﬁre load should be that which
has less than 10% probability of being exceeded in the 50 year life of a building. For the
present study, the 90th percentile of the ﬁre load was chosen as the nominal value. The

ﬁre load was assumed to follow the Gumbel distribution (ECSC 2001).

110

4.1.2 Ratio of Floor Area to Total Surface Area of the Compartment

Culver (1976) reported the ﬁre load per unit ﬂoor area of the compartment. For
calculating the ﬁre temperature, the ﬁre load needs to be converted to per unit area of the

total surfaces of the compartment. This conversion can be done using

Ar
qt : q f E— (4‘2)

where q, = ﬁre load per unit total surface area of the compartment, (1f: ﬁre load per unit

floor area of the compartment, A f: floor area of the compartment, and A, = total surface

area of the compartment.

The ratio Af/A, varies for each compartment and should therefore be treated as a random

variable in the reliability analysis. No statistical infonnation is available in the literature
about this ratio. Culver (1976) reported the range of ﬂoor areas for 23 ofﬁce buildings in

the US, but did not explicitly report the height of the rooms. Therefore, the height of the

rooms was assumed to be 12 feet to establish the statistical parameters of the ratio Af/At.

Additionally, structural and architectural drawings of three representative ofﬁce buildings

in Detroit were examined to establish the mean, COV and distribution of the ratio Af/At.

The statistics of the ratio Af /At obtained for the three ofﬁce buildings were combined

with those obtained from the data reported by Culver (1976). The combined mean, COV

and distribution of the ratio Af/A, are:

0 Mean 2 0.192

111

0 COV = 0.23

0 Distribution = lognormal

4.1.3 Opening Factor

The opening factor represents the ventilation conditions present in a ﬁre compartment.
The duration and severity of the ﬁre depends on the value of the opening factor, which in
turn depends on the sizes of windows and doors in a compartment, and can be calculated

through
F. = Av\/Hv / A. <4-3)

where, Av = area of the openings and Hv = height of the openings.

A building and its structural components are ﬁrst” designed for room temperature
conditions and then for ﬁre. The values of the opening factor for a ﬁre compartment can
be accurately estimated from the architectural drawings of a building and is not likely to
be signiﬁcantly different from the design or nominal values. Therefore, it is reasonable to
treat the opening factor similar to the dead load in reliability analysis. For the opening
factor we assumed the nominal values to be the mean values, a COV of 0.05, and a
normal distribution.

Typical low, medium and high values of opening factors in actual building compartments

are 0.04 mm, 0.08 ml/2 and 0.12 ml/2 (Beck 1985). To determine the range of nominal

values of the opening factor the architectural drawings of three typical ofﬁce buildings in

Detroit were analyzed. In general, the opening factor was between 0.02 ml,2 and 0.3

1/2
m .

112

4.1.4 Thermal Absorptivity of Compartment Enclosure

The thermal absorptivity, b, of the compartment boundaries is a measure of the amount of

heat absorbed by the compartment boundaries and may be calculated through

b=\/k—ﬁb—p <4-4)

where, k, p and cp are thermal conductivity, density and speciﬁc heat of the bounding

material, respectively. The thermal absorptivity is a function of temperature, but the
Eurocode 3 allows room temperature properties to be used for design. We performed a
detailed analysis to study the effect of thetemperature variation of thermal absorptivity
on the steel temperature, and the results indicated that it is reasonable to use room
temperature values of thermal absorptivity.

There is no information available in the literature about the variability of the thermal
absorptivity of bounding materials. However, some researchers have reported thermal
properties of some commonly used bounding materials such as normal and lightweight
concretes and gypsum board. These reported room temperature thermal properties were
used to characterize the variability of thermal absorptivity of normal and lightweight
concretes and gypsum board as described below.

Thermal properties (density, thermal conductivity and speciﬁc heat) of gypsum boards
reported by different researchers (Carino et al. 2005, Manzello et a1. 2003, Mehaffey et

al. 1994, Thomas 2002, and Wullschleger and Wakili 2008) were ﬁrst used to obtain the
thermal absorptivity, bg, through Equation 4-4. The statistics of bg based on these

calculated values are:

. Mean of bg = 423.5 WSO'SlmZK

113

0 COV of bg = 0.23

0 Distribution of bg = lognormal

The mean value is close to the value of 410 WsO.5/m2K reported by Buchanan (2001) for
gypsum board.
In case of normal and lightweight concretes, all three corresponding thermal properties

were not available for a particular tested specimen. Therefore, ﬁrst the statistics of

density (p), thermal conductivity (k) and speciﬁc heat (cp) for both types of concretes

were obtained using test data. Thermal properties were reported by Harmathy and Allen
(1973), Lie and Kodur (1996), Shin et al. (2002), Schneider et al. (1981), and Whiting et
al. (1978) for normal weight concrete, and Harmathy and Allen (1973), Stukes et al.
(1986), and Whiting et al. (1978) for lightweight concrete. For normal weight concrete,
Schneider et al. (1981) included test data obtained in six other studies which was also
used for characterizing thermal properties of normal weight concrete. The statistics
derived from these test data are also shown in Table 4-2. The mean and variance of
thermal absorptivity for both concretes were then estimated analytically as shown below

assuming that p, k and cp are independent:

Elbl = EL/kpc, l= Tﬁf. (krdk Tﬁfprmdp 10.]? f.,, (c, >ch

= Eli; l>< Elx/El>< EL]; l (4-5)
Elf]: ENE)? = E[k]>< E[p]x Elcp] (4-6)

114

2
equaled—(150]) (.7)
where E [.] and Var L] are the mean and variance. The statistics of thermal absorptivity

obtained for normal and lightweight concretes are:

(1) Thermal absorptivity of normal weight concrete, bNWC

0 Mean of bNWC = 1830 WsO'SlmZK

. Cov of wac = 0.094
0 Distribution of wac = normal

(2) Thermal absorptivity of normal weight concrete, wac
. Mean of wac = 640 WSO'S/mZK
0 COV of bLWC = 0.107
0 Distribution of wac = normal

The mean value of wac = 1830 WsO'S/mzK compares well with the value of 1900
0.5 2 0.5 2
Ws /m K reported by Buchanan (2001). The mean value of wac = 640 Ws /m K

compares well with the value of 660.6 Wso°5/m2K reported by Kirby et al. (1994) for

lightweight concrete blocks. The distribution of thermal absorptivity for both types of
concrete was obtained through Monte Carlo simulations.

Buchanan (2001) studied the effect of two types of bounding materials (normal weight

concrete having b = 1900 WsO'SlmzK and gypsum board having b = 410 WsO'S/mzK) on

the ﬁre temperature. A typical commercial ofﬁce building constructed from a mixture of

these materials on the walls and ceiling would give values of ﬁre temperature inbetween

115

those obtained by using either of the individual materials (Buchanan 2001). Therefore,
statistics of thermal absorptivity, b, were also obtained for a compartment assuming that
50% of total surface area was constructed of normal weight concrete and the other 50%

of gypsum board. The total thermal absorptivity of this compartment can be expressed as

0.5Ab +0.5A,bNWC
173+ch = t g (4-8)

A,

 

The mean and variance of bg+ch for this mixed compartment were estimated as

E[bg+ ~wc] = 0.5E[bg 1+ 0.5E[bNWC] (4-9)

Var[bg+ we] = 0.52(Var[bg D2 + 0.52(Var[bNWC])2 (4-10)
and yielded

. Mean of 1.8,ch = 1127 wSO'S/mZK

. cov of 1.8,ch = 0.10

0 Distribution of bg+NwC = normal

To study the ﬁre and steel temperatures likely to occur in real ﬁre scenarios, Kirby et al.
( 1994) conducted 9 ﬁre tests using different materials (such as lightweight concrete
blocks, autoclaved aerated concrete slabs, ﬂuid sand, ceramic ﬁber, and ﬁreline

plasterboard) as walls, roof and ﬂoor of the compartment. The combined value of thermal

absorptivity in all tests ranged from 350-755 WsO'SlmzK. The statistics of thermal

absorptivity developed in this study effectively cover the values of thermal absorptivity

used by Kirby et a1. (1994).

116

4.1.5 Thickness of Insulation or Fire Protection Materials

Steel members may be protected either using spray applied ﬁre protection materials or
board systems.

Carino et al. (2005) studied the variation of thickness of spray applied ﬁre protection
materials used in the World Trade Center (WTC). They observed that the average
thickness is generally higher than the speciﬁed thickness and that the thickness is
distributed lognormally. Their results were used for the COV and distribution type for
insulation thickness. Because the thicknesses of ﬁre protection materials used in the
WTC were determined using prescriptive approaches, the mean of the insulation
thickness was not taken from this study. Instead, based on the analysis in the report and
conversation with a ﬁre prooﬁng expert (Ferguson 2008), the mean was taken to be 1/16-
inch higher than the thickness required using performance—based design.

There is no information available on the variability of thickness of board materials, but
since they are produced under controlled conditions, the nominal thickness was taken as
the mean value and the COV was assumed to be 0.05.

Insulation may come off when a member is exposed to ﬁre. The probability of insulation
getting damaged or falling off under ﬁre conditions may be determined by performing the
tests on protected steel members. The events of insulation falling off during different tests
may then be used for estimating the probability of insulation getting damaged or falling
off during the ﬁre conditions which could then be included in structural reliability study.
At present, there is no information about the probability of falling off for insulation,

therefore, in this study; it is assumed that insulation is not damaged under ﬁre conditions.

117

4.1.6 Thermal Conductivity and Density of Insulation

Bruls et al. (1988) studied the variation of thermal conductivity at different temperatures.
Although, thermal conductivity varies with temperature, they concluded that since the
failure of structural steel elements generally occurs at a temperature of 400 to 600°C, the
v thermal conductivity corresponding to a critical temperature of 500°C can be used in
design.

Statistical analysis of the thermal conductivity in the temperature range of 400-600°C for
eight representative materials used in the US. (ﬁve reported by Bentz and Prasad 2007,
two reported by Carino et al. 2005, and one tested at Michigan State University) was
performed. The mean, COV and distribution type established from this statistical analysis
are:'

0 Mean of thermal conductivity of spray applied ﬁre protection materials

0. 187 W/m. K

0 COV of thermal conductivity of Spray applied ﬁre protection materials

0.10
0 Distribution of thermal conductivity of spray applied ﬁre protection

materials = lognormal
Room temperature values of density reported in the literature for different ﬁre protection
materials (Bentz and Prasad 2007, and Carino et al. 2005) were used to obtain its
statistics. The mean and COV of the density of spray applied ﬁre protection materials are
given in Table 4.2. Due to insufﬁcient data, it was not possible to estimate a distribution
and a normal distribution was assumed. To reiterate:

0 Mean of density of spray applied ﬁre protection materials = 0.187 W/m. K

118

0 COV of density of spray applied ﬁre protection materials = 0.10
0 Distribution of density of spray applied ﬁre protection materials =
lognormal

Different types of board materials can be used as ﬁre protection materials (e.g., ﬁber-
silicate or ﬁber calcium silicate boards, and gypsum plaster (Buchanan 2001). Thermal
properties of all of these boards are generally not easily available because of their
proprietary nature. However, thermal properties of ﬁre rated type X gypsum boards have
‘ been reported by various researchers and were used to obtain the statistics of thermal
conductivity and density. Statistics of thermal conductivity of gypsum board materials in
the temperature range of 400-600°C were obtained using test data (Bentz and Prasad
2007, Carino et al. 2005, Manzello et al. 2003, Mehaffey et al. 1994, Sultan 1996, and
Thomas 2002) and are:

0 Mean of thermal conductivity of board materials = 0.187 W/m. K

o COV of thermal conductivity of board materials = 0.10

0 Distribution of thermal conductivity of board materials = lognormal
Room temperature values of the density of ﬁre rated type X gypsum board reported by
different researchers (Carino et al. 2005, Mehaffey et al. 1994, Samuel et al.2003,
Thomas 2002,Tsantaridis et al. 1999, and Wullschleger and Wakili 2008) were used to
obtain the statistics that are shown in Table 4.2.

Buchanan (2001) reported that typical values of thermal conductivity are 0.15 W/m.K

and 0.20 W/m.K, respectively, and typical values of densities are 600 kg/m3 and

800kg/m3, respectively for two types of board materials (i.e., ﬁber-silicate or ﬁber

119

calcium silicate boards, and gypsum plaster). The mean density of 745 kg/m3, and mean

thermal conductivity of 0.16 W/m.K, fall within the range of reported values. Therefore,
although the statistics of density and thermal conductivity were obtained using test data
of gypsum boards only, they should adequately represent other types of board materials

as well.
4.2 Model Errors for Thermal Models

4.2.1 Model Error for Steel and Fire Temperatures

The maximum temperature of steel sections estimated using Equations 3-4 to 3-8 differs
from that measured in actual ﬁre tests. Reasons for the differences include:
o The actual heat absorbed by the compartment boundaries may be different than
that used in the model through use of the thermal absorptivity, b.
0 The actual ventilation conditions may be different than those accounted for
through use of the opening factor, F v.
0 Differences in the actual duration of burning of the ﬁre from that predicted
using Equation 3-6.
0 The approximation and assumptions used in the models for estimating ﬁre and
steel temperatures.
To account for the differences in calculated and measured steel temperatures, the model
error was characterized as described below, both for steel beams (three sided exposures)
and steel columns (four sided exposure).
The experimental temperature of steel elements has been reported by many researchers

but most of these tests were carried out under standard ﬁres instead of real ﬁres, and thus

120

cannot be used to estimate the error arising from the ﬁre models. Kirby et a1. (1994)
carried out a series of nine real ﬁre tests and recorded the temperature of protected and

unprotected steel elements. The tests were performed for a range of ﬁre loads (380 — 760
2 . . . . 1/2
MJ/m of floor area), for different opening condrtrons (F v = 0.0029 — 0.062 m ), and

various types of materials were used as compartment boundaries in order to represent all

possible real ﬁre scenarios. Foster et al. (2006) reported the temperature of four protected

steel columns. In this test, the ﬁre load was 720 MJ/m2 of the floor area, and the opening

[/2
factor was 0.043 m .

The model error for the temperature of steel beams, tsb, was characterized using the test

data reported by Kirby et al. (1994), and the model error for the temperature of steel

columns, tsc. was characterized using the test data reported by Kirby et al. (1994) and
Foster et al. (2006). tsb has a mean of 0.98 and COV of 0.11, and tsc has a mean of 1.05

and COV of 0.13. Both, tsb and tsc were best described by the Gumbel distribution.

In the last decade, many real ﬁre tests were carried out all over the world, especially in
the UK. In most of these tests the steel beams were unprotected, and therefore, reported
steel temperatures cannot be used for characterizing the model error. In almost all of
these tests, steel columns were protected but various parameters (e. g., type, thickness and
properties of ﬁre protection material, type, size and thermal properties of bounding

materials) required as input data for estimating the temperature of columns were not

121

explicitly reported. Therefore, the experimental temperatures recorded in these tests could

not be used.

4.2.2 Model Error for Reduction Factors

The strength and stiffness of steel reduces signiﬁcantly at elevated temperatures. Most

design speciﬁcations give values of the reduction factors for strength and stiffness at

different steel temperatures. Values of the yield strength reduction factor, ky, and the
modulus of elasticity reduction factor, kE, given in the AISC Speciﬁcations (AISC

2005a) at different steel temperatures, T5, are plotted and compared with test data

reported by different researchers (Chen et al. 2006, Gayle et al. 2005, Kelly and Sha
1999, Kirby and Preston 1988, Li et al. 2003, Makelainen et al. 1998, and Outinen and

Makelainen 2004) in Figure 4.1.

There is some scatter in the reported values of ky and k5, even though the steel material is

generally very similar. This may have more to do with different deﬁnitions of the yield
strength and different techniques of testing than differences in the material. However, to
account for the scatter in the data, model errors were established for the reduction factors
by comparing the values given in the AISC Speciﬁcations with the test data. Only values

at temperatures between 400°C and 600°C were used for characterizing the model errors

for the reasons described in Section 4.1.6. The model error for k6,), has a mean of 1.03

and a COV of 0.12 and is best described by the Gumbel distribution. The model error for

ke, E has a mean of 1.13 and a COV of 0.15 and is normally distributed.

122

 

 

    

 

 

 

 

 

 

 

1
0.8 -
Ill
8 0.6 ~
0.4 ~
0.2 k5: EscrsyEs
o l I I f X l.
0 200 400 600 800 1000 0 200 400 600 800 1000
Temperature (°C) Temperature (°C)
(a) Yield strength reduction factor (ky) (b) MOdUIUS 0f elasticity reduction

factor (k5)

Figure 4.1 - Comparison of yield strength and modulus of elasticity reduction models
with test data

ke,y and ke,E are correction factors which account for the differences between the

nominal reduction factors (e.g., yield strength reduction factor, k =Fy(Ts)/Fy, or modulus
of elasticity reduction factor, kE=ES(TS)/ES) speciﬁed in the codes and those measured in

tests. In ﬁre tests, actual ratios of Fy(TS)/Fy and ES(TS)/ES are used and the resulting

measured capacity is based on these. When measured capacity is compared with the

design capacity based on the nominal values of F),(TS)/Fy and ES(TS)/Es, the resulting

difference includes the effect of both the uncertainties associated with the design equation

and that arising from the difference of actual and nominal ratios of Fy(Ts)/Fy and

E S( Ts)/E 3. Therefore, in such situations the uncertainty associated with reduction factors

is implicitly included in the professional factor P. Herein, the uncertainty associated with

123

reduction factors was characterized separately. Through probability concepts, the
uncertainty associated with the reduction factors and that associated with the design
equation can be separated if the design equation is simple (e.g., for laterally restrained
beams). However, separating the two model errors is difﬁcult when the design equation is
complex as for laterally unrestrained beams. Therefore, the uncertainty associated with
the professional factor, P, for both types of beams (laterally restrained and unrestrained)
and steel columns was characterized as described later in Chapters 5 and 6, respectively,
that implicitly includes the uncertainty associated with reduction factors. Accordingly,

errors associated with reduction factors are not used as a separate factor in this study.

4.3 Summary

This chapter presents the statistics of random parameters obtained from the literature and
analysis of raw experimental data. Information applicable to US. conditions is used

wherever possible. All random parameters characterized in this chapter except for the ﬁre

load, ratio Af/At, and arbitrary-point-in-time live load, are applicable for the reliability

analysis of buildings of all use categories. The statistics of the ﬁre load and arbitrary-

point-in-time load for different types of occupancies are reported in the literature.

However, not much information is available about the ratio Af/At, and these should be

obtained from architectural drawings of buildings.

Model errors associated with thermal models were also characterized based on the
experimental data. This chapter presents more complete information for the variables and
model errors compared to previous studies.The statistics of the random variables and

model errors derived in this chapter form part of the input data for deriving capacity

124

reduction and ﬁre load factors for simply supported beams and columns in Chapters 5

and 6.

125

Chapter 5

Capacity Reduction and Fire Load Factors for Steel
Beams Exposed to Fire

In this chapter, capacity reduction and ﬁre load factors are developed for simply
supported steel beams for the ﬂexural limit state using the methodology described in
Chapter 3 and using the statistics of random parameters and model errors presented in
Chapter 4.

The bending capacity equations for laterally restrained simply supported beams given in
the AISC Speciﬁcations (AISC 2005a) and those for laterally unrestrained simply
supported beams given by Takagi and Deierlein (2007) are used. Predictions of structural
capacity under ﬁre are still relatively new and evolving. With improved understanding of

structural behavior under ﬁre, performance equations may change and future design

126

reﬁnements may be necessary. However, the methodology for deriving capacity
reduction and ﬁre load factors described in Chapter 3 and the random parameters and
model errors characterized in Chapter 4 are general and may be used for deriving

capacity reduction and ﬁre load factors using improved design equations.
5. 1 Derivation of Capacity Reduction and Fire Load Factors

5.1.1 Performance Functions for Reliability Analysis

5.1.1.1 Applied Moment under Fire
According to the AISC Speciﬁcations, the applied moment, Muf, is determined from the
load combination given by

Mufz 1.2MD + 0.5ML + 0.2M5 + T (5-1)

where, MD, ML and M5 are nominal dead, live and snow load moments, respectively, and

T includes moments induced by the ﬁre itself (such as additional bending moment
induced due to thermal expansions being restrained by the surrounding structure).
In Section 3.5.1.1, it was concluded that it is appropriate to use the combination of dead

and arbitrary-point-in-time live load for reliability analysis under ﬁre conditions. Thus,

for reliability analyses, the applied moment, Maj, under ﬁre conditions may be expressed

in terms of basic variables as

Maj: E(cDAD + cLBLapt) (5-2)

127

where CD and cL = deterministic inﬂuence coefﬁcients that transform the load intensities

to moments, A and B = random variables reflecting the uncertainties in the transformation

of loads into load effects, E = a random variable representing the uncertainties in

structural analysis, and D and Lap, = random variables representing dead and arbitrary-

point-in-time live load. The statistics of A, B, E, D and Lapt are given in Table 4-1.

5.1.1.2 Moment Capacities of Beams under Fire

Moment Capacity of Laterally Restrained Beams
The nominal moment capacity of a simply supported, laterally restrained steel beam

exposed to ﬁre can be expressed as

where Z).C = plastic section modulus, F v = yield strength of steel at room temperature, and

ky(TS) = yield strength reduction factor that depends on the temperature, T3, of the steel

member.

The actual moment capacity can be obtained by modifying Equation 5-3 to

M f = P b lflzxky(tsts)mlF y (5'4)
where f1 and m1 are non-dimensional random variables that characterize uncertainty in ZyC
and Fy, respectively, and their statistics are given in Table 5.1. tsb is the model error for

steel temperature that was characterized in Section 4.2.1. Steel temperature, Ts is function

128

of many parameters (see Equations 3-4 to 3-8) and theirs statistics are given in Table 4.2.

Pbl is the professional factor (model error) that is characterized in the next section.

Table 5.1 - Mean and COV of room temperature design parameters

 

 

 

 

 

 

 

 

 

Variable Characterizes variation in Mean COV
m1 Yield strength, F y 1.03 0.063
m2 Modulus of elasticity, E 5 1.04 0.045
f 1 Plastic section modulus, Zx 1.03 0.034
f2 Elastic section modulus, SI 1.02 0.035
f3 Radius of gyration, ry 1.00 0.016
f4 Moment of inertia, IV 1.02 0.050
f5 Cross-sectional area , A s 1.03 0.031

 

 

 

 

 

Moment capacity of laterally unrestrained beams

Takagi and Deierlein (2007) compared the design provisions of AISC with ﬁnite element
simulations for laterally unrestrained beams exposed to ﬁre and reported that the AISC
speciﬁcations predict strengths 80—100% higher than the simulated results. Therefore, the
use of current AISC ﬁre design provisions for reliability analysis was considered
inappropriate, and the alternative equations proposed by Takagi and Deierlein (2007)
which have a format similar to that of the AISC design provisions were used. Takagi and
Deierlein proposed equation for inelastic lateral torsional buckling only, and maintained
the current AISC equation for elastic lateral torsional buckling of beams. Using these
proposed equations, the nominal moment capacity of laterally restrained steel beams

exposed to ﬁre can be expressed as

129

For Il 34,01):

 

 

 

 

 

 

 

 

 

A CX (Ts)
M", = M,(TS)+{M,,(TS)—M,(Ts) 1— (5-5)
MT.)
For A > /l,(Ts):
at? (T ) 2
7r
Mmf = :1— ES(TS)IyG(TS)J + Iwa[ 2 S J (5-6)
ry ry
where
2
2, (T5) = l \[E5(TS)G(TS)JA 1 1+ 1+ 4 Cw S, F21 (Ts) (5-7)
SJ, 2 FL(TS) 1y G(TS)J
. M p (T3) = Z xF y (Ts ) = reduced plastic moment
M, (T5) = SxFL (TS ) = elastic moment at the onset of yielding
F L (TS ) = F p (T, ) - F, (Ts ) = temperature dependent initial yield stress
F p (T3) = k p (T3 )F y = temperature dependent proportional limit
F, (T5) = ky (TS)F,
F r = residual stress at ambient temperature which is speciﬁed in the AISC
Speciﬁcations as 69 MPa,
TS <
CX (TS) = 0.6 + 250 _ 3.0 (5-8)

130

,1 = I/ry = Slendemess ratio, L = span length, ry = radius of gyration about minor axis, Sx

= elastic section modulus, Z = plastic section modulus, Iy = moment of inertia about

minor axis, J = torsional constant, Cw = warping constant, A = cross-sectional area,

E S(TS) = kEEs 2 reduced elastic modulus at temperature TS, E s = elastic modulus of steel,

G(TS) 2 reduced shear modulus. ky, kp and k]; are the yield strength, proportional limit

and elastic modulus reduction factors, respectively, and their values for different steel
temperatures are given in design speciﬁcations (AISC 2005a and EN 2005).

The strength of laterally unrestrained beams predicted by Takagi and Deierlein’s
equations is quite similar to the Eurocode 3 design provisions. However, Dharma and
Tan (2007) reported that the Eurocode 3 design provisions are too conservative,
especially for temperatures less than 500°C. The Eurocode 3 strength design equations
for beams were proposed by Vila Real et al. (2003) based on both numerical simulations
and experimental investigations of beam strength. Later, Vila Real et a1. (2004) also
found that their equations were overly conservative for loading other than uniform
bending, and suggested using a moment distribution adjustment factor similar to the one
used in Eurocode l for the room temperature design of laterally unrestrained beams.

Although similar to Eurocode 1, the AISC provisions for room temperature design

include a lateral torsional buckling modiﬁcation factor, Cb, for non-uniform moment

diagrams, but Takagi and Deierlein did not explicitly report whether or not the C], factor

should be used with their equations. Beam strengths predicted by Takagi and Deierlein’s

131

equations using Cb were compared with experimental data (Mesquita et al. 2005) and
were found to be conservative even if the Cb factors are used. Therefore, Takagi and

Deierlein’s equations were used with the C b factors.

The actual moment capacity of laterally unrestrained beams can be obtained by

modifying Equations 5-5 to 5-8 to

For xi. S 2., (tSst):

21 Cx (tsts)
Mf : Pb2 Mr(t.sts)+{Mp(tsts) _Mr(tsts){l—m] (5'9)
r Sb 3

For Ii. > 1,.(lsts);

 

7f
MfZPbZ

 

 

llf3ry

2
72m E t T
r m2Es(tsts)f4IyG(tsts)J+f4Iwa[ 2 S( Sb 3)] (5‘10)
3 y

where

 

 

 

 

 

2
A, ('3'st) = 7r JmZES (tSbTS)G(TS )JfSAS 1 1+ 1+4 CW ( fZSx ] F2L(tsts) (5'11)
fZSx 2 FL(Ts) f4ly C(Ts)‘,

M p (tst5) = le xml F y (2‘5st ) 2 reduced plastic moment

M ,(tst5) = fszFLUSbTS ) = elastic moment at the onset of yielding

F L (tst5) = F p (tsts) — F, (tst5) = temperature dependent initial yield stress
F p (t Sst) = k1,,(tstS)m1Fv 2 temperature dependent proportional limit

Fr(tsts) = ky(tsts)Fr

132

F r = residual stress at ambient temperature which is speciﬁed in the AISC

Speciﬁcations as 69 MPa
C(tsts) = m1 Es (tsts ) /(1 + U) , where v is the poission’s ratio, and

t T
C tsTs =0.6+—-S-9—S-S3.0 5-12
x( b ) 250 ( )

where fi and mi are non-dimensional random variables that characterize uncertainty in

fabrication and material parameters as given in Table 5.1. PM is the professional factor

(model error) that is characterized in the next section.

5.1.1.3 Performnace Function for Reliability Analysis
The performance function for reliability analysis of a beam can be expressed as

g(X) = M;— Ma,f ‘ (543)
where X denotes a vector containing all the random design parameters. The probability of

failure, pp , of a steel beam under ﬁre is

PF = P1800 < 01 (5-14)

5.1.2 Model Error (Professional Factor) for Moment Capacity
Equaﬁons
To account for the difference in the measured capacity of a laterally restrained beam in a

laboratory and that predicted by Equation 5-3, the model error, Pblr was characterized

using the test results presented by Kruppa (1979) and Wainman (1992). Kruppa (1979)

reported results for ten tests conducted in Germany and France and six tests conducted in

133

Japan. Wainman (1992) reported the applied loads and failure temperature of two beams.

Pbl has a mean of 0.99 and a COV of 0.11, and is best described by the lognormal

distribution.

To account for the difference in the measured capacity of laterally unrestrained beams in

a laboratory and that predicted by Equation 5—5 or 5-6, the model error, sz, was

characterized using the test results presented by Mesquita et al. (2005). They tested 5
beams of varying span lengths subjected to different applied loads. Each beam was

replicated three times yielding 15 test results. They reported both the failure moments and

failure temperatures which were used for characterizing, PM. PM has a mean of 1.67

and a COV of 0.19, and is best described by the Gumbel distribution. sz'has a

signiﬁcantly high mean because the model for calculating the nominal capacity of

laterally unrestrained beams is very conservative.

5.1.3 Probability of Failure and Target Reliability Index

In Section 3.5.3, it was shown that for typical U.S. ofﬁce buildings the target reliability
index is likely to vary from zero to 2.0 depending on the size of the ﬁre compartment and
on the effectiveness/reliability of active ﬁre protection systems. Therefore, in the next
section, capacity reduction and ﬁre load factors are developed for this range of target

reliability index values.

5.1.4 Reliability Analyses

Twenty ﬁve steel beams (10 laterally restrained and 15 laterally unrestrained) ranging in

length from 3 m (10 ft) to 13.7 m (45 ft) with live loads ranging from 2.4 kPa (50 psf) to

134

4.8 kPa (100 psf) were selected for the reliability study. The AISC design speciﬁcations

were used to ﬁrst design the beams for ambient temperature conditions. The same beams

were then designed for ﬁre exposure (b = 640 Wso'5/m2K and F v = 0.02 mm) using the

engineering approach described in Section 3.2, and the required thickness of insulation to
withstand the design ﬁre was determined from Equations 5-1, 5-3, 5-5 and 5-6 using an
iterative procedure. As suggested in most codes, a capacity reduction factor of 1.0 was
used for the initial design for ﬁre. The beams were assumed to be protected by spray
applied ﬁre protection materials, which is generally the case in the US. The spans of the
beams, dead and live loads, insulation thicknesses, and the section factors for laterally
restrained and laterally unrestrained beams are given in Tables 5.2 and 5.3, respectively.

The rate of temperature rise of a protected or unprotected structural steel member
exposed to ﬁre depends on the section factor, or massivity factor, which is a measure of
the ratio of the heated perimeter to the area or mass of the section (Buchanan 2001). The
section factor is an important parameter because the rate of heat input is directly
proportional to the area exposed to the ﬁre, and the subsequent rate of temperature
increase is inversely proportional to the heat capacity of the member (equal to the product
of the speciﬁc heat and mass of the steel segment). Generally a section with a lower
section factor will experience lower temperatures. Therefore, in Tables 5.2 and 5.3,

sections with lower section factors have smaller insulation thicknesses.

135

 

 

 

 

 

 

 

 

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137

5.1.4.1 Reliability Analyses of Laterally Restrained Beams

FORM analysis was performed for each design situation (each of the 10 laterally
restrained beams) using the performance function given in Equation 5-13. Using the
direction cosines obtained from the reliability analysis, the partial safety factors for each
design parameter were obtained through Equation 3-25 for each design situation. These
individual partial safety factors, except for the ﬁre load, were then combined into a single
capacity reduction factor using the procedure described in Sections 2.3.8 and 3.5.6:

.2 ($1211),((zrhszmgFy
Mn.f

 

¢f (545)

where ¢,- are the partial safety factors for each design parameter obtained through
Equation 3-25, and Mnf is the nominal bending capacity of a laterally restrained beam

calculated through Equation 5-3. Thus 10 different capacity reduction factors (one for

each beam) were obtained. Thereafter, a single optimized capacity reduction factor, (bf,

corresponding to dead and live load factors of 1.2 and 0.5, respectively, was obtained
using the optimization procedure described in Section 3.5.7. Euqal weights were used for
all beams. The dead and live load factors of 1.2 and 0.5, respectively, are proposed in

AISC Speciﬁcations (AISC 2005a) (see Euqation 5-1). Since the capacity reduction

factors were obtained for ,8, values ranging from O to 2, this procedure was repeated for
each [3, value. A similar procedure was used to obtain the ﬁre load factors corresponding

to each value of ,6,.

138

5.1.4.2 Reliability Analyses of Laterally Unrestrained Beams

A procedure similar to that used for the reliability analyses of laterally restrained beams

was followed for the unrestrained beams, and the capacity reduction factors were derived

for ﬁt ranging from O to 2.

As mentioned earlier, the equations proposed by Takagi and Deierlein (2007) for
calculating the moment capacity of laterally unrestrained beams are very conservative.
The mean of the experimentally obtained capacity was found to be 1.67 times the
nominal capacity. Since the capacity reduction factor depends signiﬁcantly on the
conservatism inherited in the design equation, it is generally not possible to have the
same capacity reduction factor for laterally restrained and unrestrained beams. However,
for ease of use, the same capacity reduction factors as those for laterally restrained beams
were maintained for unrestrained beams, and to cater for the conservatism in the design

equations, an additional factor, referred to as the capacity adjustment factor is proposed.

' A constant capacity adjustment factor, (1)0 = 1.67 yielded uniform reliability for all beams.

In other words, the design capacity given by Equations 5-5 and 5-6 is to be multiplied

both by grand ¢a.

5.2 Results

5.2.1 Capacity Reduction Factor

The plot of the capacity reduction factor, gr, vs. the target reliability index is shown in
Figure 5.1. The value of @c for a given target reliability index, ,6,, is

139

1.0 for ,6, s 1.25
f = (5-16)

1.5 — 0.4,6, for 1.25 S ,6, S 2.0

1.6 -

1.2 Q

 

>3 0.8a
"B
§
é 04< 7q
0 l l 7 l l

 

Figure 5.1 - Capacity reduction and ﬁre load factors for steel beams

According to the AISC Speciﬁcations, the capacity reduction factor for ﬁre design is 0.9
and is the same as that for room temperature design. However, most other codes suggest

that a capacity reduction factor of 1.0 be used (e.g., in the Eurocode 3, the partial safety
factor m is 1.0 for ﬁre design). Results obtained in this study indicate that the nominal

strength need not be reduced if the target reliability index is less than 1.25, which in turns
depends on the effectiveness of active ﬁre protection systems in reducing the probability
of occurrence of a severe ﬁre. Since most ofﬁce buildings in the US. are required to have
automatic sprinklers, the target reliability index is not likely to exceed 1.25 (see Table 3.1
and Section 3.5.3) even if automatic sprinklers are not as effective as commonly
assumed. Therefore, using a capacity reduction factor of 1.0 is considered reasonable for

most design situations.

140

5.2.2 Fire Load Factor

The plot of the ﬁre load factor, yq, vs. the target reliability index is shown in Figure 5.1.
The nominal value of the ﬁre load was taken as the 90th percentile (Buchanan 2001,

Bukowski 2006). The value of yq for a given target reliability index, ,6,, is

O.4+0.4,B, for ,6, £1.25
yq = (5-17)

0.15 + 0.6,6, for 1.25 S ,6, S 2.0

The yq is to be applied to the ﬁre load which is then used in describing the design ﬁre

(time-ﬁre temperature curve) using Equations 3-4 to 3-7.

When the target reliability index is less than 1.42, the ﬁre load factor given by Equation
5-17 is less than 1.0 indicating that the ﬁre load can be reduced as suggested in the ECSC
study and Eurocode 1 (EN 2002). The commentary to the AISC Speciﬁcations states that
the'ﬁre load may be reduced by up to 60% if a sprinkler system is installed. The
maximUm reduction should be considered only when the automatic sprinkler system is
considered to be of the highest reliability, i.e., having reliable and adequate water supply,
supervision of control valves, and regular schedule for maintenance in accordance with
NFPA recommendations (NFPA 2002). The reduction in ﬁre load speciﬁed in Figure 5.1
depends on the target reliability index, which in turn depends on the effectiveness of
active ﬁre protection systems in reducing the probability of occurrence of a severe ﬁre.
The proposed approach is more general and enables the reduction in ﬁre load to be
speciﬁed for sprinkler systems of all categories i.e., having low, high or medium

reliability, as well as for other active ﬁre protection systems.

141

In the ECSC study, sprinklers were given very high effectiveness ranging from 95-
99.5%. Bukowski et a1. (1999) carried out statistical analysis of the operational reliability
of sprinklers and concluded that it lies between 88% and 98% with a mean value of 93%.
Based on the combined operational effectiveness and performance effectiveness data
published by NFPA, William (2005) assessed the overall reliability of automatic sprinkler
systems to be 91%. Based on these studies, sprinklers of low and medium reliability may
be assigned reliability/effectiveness values ranging from 88% to 93%, and then the target
reliability index can be obtained using the methodology described in Section 3.5.3. Using
these target reliability index values, the ﬁre load factors can be obtained from Equation 5-
17 (or from Figure 5.1). Thus the proposed approach enables the reduction in ﬁre load to
be speciﬁed for sprinkler systems having lower effectiveness in reducing the probability
of occurrence of a severe ﬁre. For example, for a ﬁre compartment size of 200 m2, the

following ﬁre load factors are obtained for sprinklers with varying effectiveness:

0 Sprinklers of high (98%) effectiveness: ,8, = 0, n, = 0.4
0 Sprinklers of medium (93%) effectiveness: ,8, = 0.73, yq = 0.69

o Sprinklers of low (88%) effectiveness: ﬁt 2 1.09, yq = 0.84

5.2.3 Validity of Capacity Reduction and Fire Load Factors for

Multiple Fire Scenarios

The capacity reduction and ﬁre load factors shown in Figure 5.1 were developed for a ﬁre

compartment assumed to be constructed of lightweight concrete blocks having b = 640

0.5 2 . . . . . . .
Ws /m K and havrng relatively poor ventrlatron conditions represented by an openrng

142

1/2 . . .
factor, F v = 0.02 m . In reality, the compartments may be constructed usrng drfferent

bounding materials such as gypsum board, lightweight concrete blocks, a combination of
gypsum board and normal weight concrete, or by using other types of lining materials on
walls and roofs. Depending on the type of bounding materials of the compartment, the
thermal absorptivity, b, may have different values. Kirby et al. (1994) used different
types and combinations of lining materials in nine natural ﬁre tests. The tests were

carried out primarily for studying equivalency between standard ﬁre and realistic ﬁre
. . . . 0.5 2

scenarios. The value of thermal absorptrvrty in all tests ranged from 350-755 Ws /m K.

The statistics of thermal absorptivity derived in this study (values given in Table 4.2)

effectively cover the range of thermal absorptivity values used by Kirby et al. (1994), and

are used in the succeeding analyses. Similarly, the ventilation conditions in different

compartments may vary considerably. Opening factors of 0.04 mm, 0.08 ml,2 and 0.12

m“2 are typical low, medium and high values in actual building compartments (Beck

1985). Based on the architectural drawings of three typical ofﬁce buildings in Detroit, it

was found that in general the opening factor may have values between 0.02 m”2 and 0.3

mm. Therefore, opening factor values of 0.02 mm, 0.08 ml,2 and 0.12 ml,2 were used

in the reliability analyses. As shown later, higher opening factor values generally yield
higher reliability index values, so opening factor values higher than 0.12 m1/2 were not
considered.

After establishing the range of thermal absorptivity, b, and opening factor, F v, to be used

for reliability analysis, nine ﬁre scenarios obtained from different combinations of

143

opening factors and thermal absorptivity (see Table 5.4) were selected to validate the
capacity reduction and ﬁre load factors derived above. For these nine ﬁre scenarios, six
beams, 2 laterally restrained (B2 and B8 in Table 5.2) and 4 laterally unrestrained (B3,
B7, B13 and B15 in Table 5.3), were designed for ﬁre conditions using the capacity
reduction and ﬁre load factors shown in Figure 5.1. For laterally unrestrained beams, the
capacity adjustment factor of 1.67 was also used. Each beam was designed for target
reliability indices of 0, 0.5, 1.0, 1.5 and 2.0 for all nine ﬁre scenarios. The 9 ﬁre scenarios
for the target reliability index of 1.5 are shown in Figure 5.2, and the values of ﬁre load,
opening factor, and thermal absorpitivity used for describing each of these ﬁres are given
in Table 5.5. Thus, for each target reliability index value, there were 54 design situations
(9 ﬁre scenarios x 6 beams), and a total of 270 design situations for the ﬁve target
reliability index values. Reliability analysis was then performed and the computed
reliability index values for the six beams are compared with the target reliability index
values in Figure 5.3.

Table 5.4 - Combinations of b and Fv used for nine ﬁre scenarios

 

 

 

 

 

 

b (WsogmrK)
423 640 1127
F 0.02 x x x
132 0.08 x x x
(m 0. 12 x x x

 

 

 

 

 

144

 

1 400

1200 .

a 1000

2.

2 800

3

E

o 600

D.

.5.

,_ 400
200
o

50

100

Time (mins)

 

150

-<>- Fire 1
—°— Fire 2
+ Fire 3
+ Fire 4
+ Fire 5
—°— Fire 6
-+— Fire 7
—— Fire 8
-- Fire 9

200

Figure 5.2 - Nine fire scenarios for target reliability index value of 1.5

250

Table 5.5 - Values of b, Fv and qt used for nine ﬁre scenarios corresponding to ,6;

 

 

 

 

=l.5
Parameter Unit Fire Fire Fire Fire Fire Fire Fire Fire Fire
1 2 3 4 5 6 7 8 9

0.5 2
b WS [m K 423 423 423 640 640 640 1127 1127 1127
F ml/2 0.02 0.08 0.12 0.02 0.08 0.12 0.02 0.08 0.12

v

2

Qt MJ/m 206 206 206 206 206 206 206 206 206

 

 

 

 

 

 

 

 

 

 

 

 

145

 

 

3.0

2.5 —

1.04 I

0.5 1|

 

 

0.0 l l I l l

O 0.5 1 1.5 2 2.5 3
ﬁt

 

Figure 5.3 - Comparison of computed and target reliability index values

The computed reliability index values compare very well with the target values, ,6,,

indicating that the derived capacity reduction and ﬁre load factors work for all design
situations considered. The reliability index values are conservative for target reliability
index values less than about 1.25, especially for the higher opening factors. This
. conservatism is believed to be primarily due to three reasons:

(1) For target reliability indices of less than 1.25 (see Figure 5.1), the capacity

reduction factor, 9c, found from reliability analysis is greater than 1.0, and the
nominal strength can be increased. However, since Wis generally always less than

or equal to 1.0 in LRFD speciﬁcations, we restrained the gcfor ﬁre design to also

not exceed 1.0. Designed beams will therefore have reserve strength. This can be
compensated for by lowering the ﬁre load factor. In Figure 5.1, we lowered the

146

ﬁre load factor in a conservative manner so that some reserve strength is still
present. Because of this inherent conservatism, the reliability index values are
higher than the target reliability index.

(2) For reliability analyses, the mean value of insulation thickness is taken to be 1/ 16
inch higher than the nominal value. For the lower target reliability index values,
especially for higher opening factors, the nominal insulation thicknesses are very
small and the addition of 1/16 inch to the nominal values represents a large
percentage increase and yields higher reliability indices.

(3) The norrrinal insulation thicknesses are very small for the lower target reliability
index values, and thus the steel temperature is quite sensitive to variation in the

random parameters during the reliability analysis.

5.2.4 Comparison of Fire Load Factors with those Based on the ECSC

Method

As mentioned in Section 2.4.1, a study was conducted in Europe to develop ﬁre load

factors (ECSC 2001). The ﬁre load factor in the ECSC study was obtained using

simpliﬁed assumptions instead of rigorous reliability calculations, and was speciﬁed for

any ,8, value through

 

{1 — [73% [0.577 + ln(— ln (13(0916, D1}
= 1.05
J—

yq 6 (5-18)
{I — 7V q [0.577 + ln(— 1n( pm}

147

where Vq = COV of the ﬁre load, (D = cumulative standard normal distribution function,

and p = percentile used for obtaining the characteristic or nominal ﬁre load. If the

nominal value is taken as the 90th percentile, then p = 0.9.

In Figure 5.4, the yq obtained in this study is compared with that obtained using Equation

5-18 for US. ﬁre load statistics, taking the nominal value of ﬁre load to be the 90th

percentile.

yq obtained from the ECSC method is greater than that derived in the this study for ,6,
values smaller than about 1.5, and is almost the same for ,6, values greater than 1.5. As
shown earlier, yq derived in this study yields conservative ,6 values for ﬁt values less than

about 1.5, and hence the yq obtained according to the ECSC approach will yield even
more conservative results. For ,6, values greater than 1.5, the beams designed as proposed

herein yield the intended safety level or higher (see Figure 5.2) because yq is used in

combination with a q that is less than 1.0 (see Figure 5.1).

148

  

 

 

1.2 -.
0.8 ~
w
>~ O 4 3! q (present study)
- y q (ECSC method)
0 l l l l l
0 0.5 1 1.5 2 2.5
ﬁt

Figure 5.4 - Comparison of fire load factors with those obtained by the ECSC
method

5.2.5 Insulation Thicknesses from Performance-Based and

Prescriptive Design Approaches

In Table 5.6, the thicknesses of insulation computed according to the engineering
approach using the capacity reduction and ﬁre load factors developed in this study for
target reliability index values of 0.5, 1.0 and 1.5 are compared with those obtained from
the prescriptive approach for two beams (B2 and B8 in Table 5.2). Insulation thicknesses
from the prescriptive approach were obtained using Design No. N708 given in
Underwriters Laboratories (UL 2004). The nine ﬁre scenarios described in Table 5.4
were considered. The thicknesses obtained from the performance-based approach are

normalized with respect to the corresponding prescriptive thickness for a 2-hr ﬁre rating.

149

Table 5.6 - Comparison of insulation thickness

 

 

 

 

Thickness Normalized thickness from performance-based approach
from
- - b=423 b=640 b=1127
Target * ”gags: 0.5 2 05 2 0.5 2
reliability g Pg“ m) (Ws /m K) (Ws /m K) (Ws /m K)
"“1“ a Fire Rating 1/2 1/2 1/2
(hrs) Fv (m ) Fv (m ) Fv (m )
2 3 0.02 0.08 0.12 0.02 0.08 0.12 0.02 0.08 0.12

 

Bl 29 42 1.46 0.79 0.64 1.25 0.74 0.64 0.91 0.61 0.55
I?! = 0'5 82 24 35 1.41 0.72 0.58 1.19 0.68 0.59 0.84 0.55 0.49

B] 29 42 1.85 0.90 0.71 1.61 0.88 0.74 1.23 0.74 0.66
18!: 1'0 BIZ 24 35 1.83 0.84 0.65 1.57 0.82 0.68 1.17 0.68 0.60
Bl 29 42 2.41 1.08 0.85 2.15 1.08 0.89 1.72 0.94 0.82
,6! = 1-5 B2 24 35 2.45 1.02 0.78 2.15 1.02 0.82 1.68 0.88 0.76

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

In Table 5.6, the shaded thicknesses indicate situations where the performance-based
approach yields thicknesses higher than the prescriptive approach. The performance-
based approach generally yields lower insulation thicknesses, especially for lower target
reliability index values, which in turn depend on the effectiveness of active ﬁre protection
systems in reducing the probability of occurrence of a severe ﬁre. It should be noted that
a direct comparison between the thicknesses of insulation computed from the
performance-based approach with that obtained from the prescriptive approach isnot
rational. In the performance-based approach, the required thickness of insulation depends
on the real ﬁre scenario in a compartment which in turn depends on the amount of ﬁre
load present in the compartment, ventilation conditions and on the type of the bounding
materials. On the contrary, the thickness of insulation obtained from the prescriptive
approach is based on ﬁre tests conducted in a furnace under a standard ﬁre which does
not represent the real room conditions. The purpose of the comparison shown in Table

5.6 is to highlight that although the ﬁre load factors are smaller than 1.0 for target

150

reliability index values less than 1.25, the performance-based approach still yields

thicknesses that are comparable to those used in the prescriptive approach.

5.2.6 Reliability Inherent in AISC Fire Design Provisions

It is of interest to determine what ﬂ value is inherent in the AISC approach. The

insulation thicknesses for the six beams and the 9 different ﬁre scenarios described earlier

(see Table 5.4) were determined using the AISC approach for two cases: (1) using 41%:

0.9 and by reducing the 90th percentile of the ﬁre load by 60% (for sprinklers of the

highest reliability) as suggested in the Commentary to the AISC Speciﬁcations; and (2)

using ¢f = 0.9 and using the 90th percentile of the ﬁre load assuming that there are no

reliable sprinklers in the building. The reliability analyses were performed using these
insulation thicknesses, and it was found that the reliability index varied from 0.2 to 0.5

for Case 1 and from 1.45 to 1.60 for Case 2.

5.3 Design Examples

To illustrate the applicability of the proposed approach in a design situation, the method
is applied to evaluate the ﬁre resistance of two simply supported beams (one laterally

restrained and the other laterally unrestrained).

In this example, the value of the target reliability index, ,6,, is obtained using the

procedure described in Section 3.5.3. The effectiveness of active ﬁre protection systems

in reducing the probability of occurrence of a severe ﬁre were obtained from the ECSC

study (ECSC 2001). In design speciﬁcations, the speciﬁcations authority may specify ,Bt

151

values depending on the effects of active ﬁre protection systems, and the designer may

not need to calculate it.

5.3.1 Design Example (Laterally Restrained Beam)
Problem statement:

A ﬁre compartment in a lO-story open—plan ofﬁce building has a floor area of 200 m2. A

9.14 m (30 ft) long steel beam is to carry a service dead load of 11.27 kN/m (0.77 kips/ft)
and service live load of 11.68 kN/m (0.80 kips/ft). A room temperature design of the

beam was performed using the AISC design speciﬁcations, and a W18x35 section was

selected. The beam has a plastic section modulus = 1089.74 cm3 (66.5 in3). The yield

strength of the steel = 34473.8 N/cm2 (50 ksi).

The ﬁre compartment is 20 m square and 3.65 m high, and is made of lightweight

concrete with density = 2000 kg/m3, speciﬁc heat = 840 J/kg.K and thermal conductivity

= 0.8 W/mK. The compartment has four windows each 4.0 m wide and 2.0 m high. The
ﬁre load is 196 MJ/m2 per unit surface area of the compartment. The compartment is
equipped with automatic sprinklers, and smoke and heat detectors. The automatic

sprinklers are assumed to be 70% effective in reducing the probability of occurrence of a

severe ﬁre. The effect of other active ﬁre protection systems may be neglected. The beam
is to be protected using a light weight insulating material with density = 300 kg/m3,
speciﬁc heat = 1100 J/kg.K and thermal conductivity = 0.2 W/mK.

As per the building code, the beam is required to have a 2-hours ﬁre resistance rating.

Calculate the required thickness of the insulation.

152

Solution:
1. Applied moment under ﬁre

Using Equation 5-1
Muf = (1.20 + 0.5L)(span2/8)= (1.2(1 1.27) + 0.5(1 1.68))(9. 142/8) 2 202.2 kN-m

2. Capacity of the beam

Selection of target reliability index:

Probability of ﬁre occurrence in an ofﬁce building 2 4 x 10'7(per m2, per year) (ECSC
2001); probability of ﬁre occurrence for 200 m2 compartment assuming a 55-year

building life, p. = 200 x 55 x 4 x 10.7 = 0.0044 (ECSC 2001); probability of failure of

automatic sprinklers, p2 = 0.30; probability of occurrence of a severe ﬁre, psf = p1.p2 -

0.00132 (Equation 3-21) ; target probability of failure for life of the building, pp

7.23x10-5 (ECSC 2001).

Using Equation 3-24 and the normal density table, the target reliability index,

 

_ P
ﬂz=¢l 1’ f =l.6
psf

Selection of appropriate capacity reduction and ﬁre load factors:

Using Equations 5-16 and 5-17 with ,8, = 1.6:
Fire load factor, yq = 0.15 + 0.6 ,8; = 1.11

Capacity reduction factor, (13f = 1.5 - 0.4 ﬂ, = 0.86

153

Fire parameters:

Nominal ﬁre load = 196 MJ/mz; total surface area of the compartment, A, = 1092 m2;
height of windows, Hv = 2.0 m; total area of windows, Av = 32 m2; opening factor, Fv =

Fv = AM] H v / A, = 0.041 mm; density of compartment boundaries, p = 2000 kg/m3;

speciﬁc heat of compartment boundaries, c = 840 J/kg.K; thermal conductivity of

compartment boundaries, k = 0.8 W/mK; thermal absorptivity of compartment
boundaries, 193/1906,, = 1160 Wso'slmzK.

Nominal capacity of beam:

As described in Section 3.2, the required thickness of insulation by the engineering
approach can be computed using an iterative procedure.

Iteration I :

Assume thickness of insulation 2 25 mm.

Fire load = yq x nominal value of ﬁre load = 1.11 x 196 = 217.56 MJ/m2

Using this ﬁre load and the other parameters given above, the ﬁre temperature can be
computed using Equations 3-4 to 3-7 (Section 3.3), and the maximum steel temperature

can be computed using Equation 3-8 or 3-9 (Section 3.4).

Maximum steel temperature, Tamax = 630.2°C occurs after 1.25 hours

Yield strength reduction factor, ky(Ts) = 0.373

154

(The value of ky(Ts)was obtained from the AISC Speciﬁcations (AISC 2005a)

corresponding to TS, max).
Using Equation 5-3 with ¢f= 0.86, the capacity of the beam under ﬁre is:

be: ¢anf = qzxkymwy = (0.86)(1089740)(0.373)(345) = 120600545 N-mm
= 120.6 kN—m

Since be < Muf, the insulation thickness is inadequate. Perform next iteration by

assuming a higher insulation thickness.

After 4 iterations, the required thickness of insulation was obtained as 32.1 mm with a

maximum steel temperature of TS, max = 547°C that occurs after 1.35 hours. This implies

that the beam not only has required 2 hours ﬁre resistance but it will not fail because steel
temperature starts decreasing after 1.35 hours.

The iterative solution procedure can be easily implemented in an Excel spreadsheet.

Note: In this example, the required thickness of insulation was determined using an
iterative procedure. Using the same approach, the failure temperature of the steel beam
and the corresponding ﬁre resistance time can also be computed for a given value of

insulation thickness.

5.3.2 Design Example (Laterally Unrestrained Beam)

Problem statement:

155

A ﬁre compartment in a 4-story open-plan ofﬁce building has a floor area of 100 m2. A

9.14 m (30 ft) long steel beam is to carry a service dead load of 11.27 kN/m (0.77 kips/ft)
and service live load of 11.68 kN/m (0.80 kips/ft). A room temperature design of the
beam was performed using the AISC design speciﬁcations, and a W10x54 section was

selected. The properties of the beam are:

2 2
0 Cross sectional area, AS = 102 cm (15.8 in )
. . 3 . 3
0 Elastrc sectron modulus, S x = 983.2 cm (60m )
. . 3 . 3
0 Plastic sectron modulus, Zx = 1091.4 cm (66.6 in )
. . . . 4 . 4
- Moment of inertra about minor axrs, Iy = 4287.2 cm (103 in )
0 Radius of gyration about minor axis, ry = 6.5 cm (2.56 in)
. 6 . 6
o Warping constant, Cw = 623003 cm (23201n )
. 4 . 4
0 Torsronal constant, J = 75.75 cm (1.82 in )
2 .
0 Yield strength of the steel, F y = 34473.8 N/ cm (50 ksi)

0 Modulus of elasticity, E s = 19994804 N/ cm2 (29000 ksi)

The ﬁre compartment is 10 m x 10 m x 3.65 m, and is constructed from concrete with
density = 2300 kg/m3, speciﬁc heat = 980 J/kg.K and thermal conductivity = 1.6 W/mK.

The concrete walls and ceiling are covered with 12 mm thick gypsum plaster board with

3
density = 700 kg/m , speciﬁc heat = 1700 J/kg.K, and thermal conductivity = 0.2 W/mK.

156

The compartment has one window which is 3.0 m wide and 1.5 m high. The ﬁre load is

2
196 MJ/m per unit surface area of the compartment. The building is not eqipped with

any active ﬁre protection systems. However, public ﬁre brigade services are available.

The beam is provided with 44.5 mm (1.75 in.) thick lightweight insulating material with

3
density = 300 kg/m , speciﬁc heat = 1100 J/kg.K and thermal conductivity = 0.2 W/mK.

As per the building code, the beam is required to have a 2-hours ﬁre resistance rating.
Check if the provided insulation is adequate.

Solution:

1. Applied moment under ﬁre

Using Equation 5-1
Muf = (1.2D + 0.5L)(span2/8)= (1.2(11.27) + 0.5(11.68))(9.142/8) = 202.2 kN-m

2. Capacity of the beam

Selection of target reliability index:

Probability of ﬁre occurrence in an ofﬁce building = 4 x 10-7(per m2, per year) (ECSC

2
2001); probability of ﬁre occurrence for 100 m compartment assuming a 55-year

building life, p1 = 100 x 55 x 4 x 10.7 = 0.0022 (ECSC 2001); probability of occurrence

of a severe ﬁre, psf = p1 = 0.0022 (Equation 3-21) ; target probability of failure for life of

the building, p, = 7.231110”5 (ECSC 2001).

157

Using Equation 3-24 and the normal density table, the target reliability index,

.3: =¢_1[1--le]= 1.84
Sf

Selection of appropriate capacity reduction and ﬁre load factors:

Using Equations 5-16 and 5-17 with ﬂ, = 1.84:
Fire load factor, )4, = 0.15 + 0.6 ﬂ, = 1.254

Capacity reduction factor, (13c = 1.5 - 0.4 = 0.76
Fire parameters:

2 2
Nominal ﬁre load = 196 MJ/m ; total surface area of the compartment, A, = 346 m ;

. . . 2 .
height of wrndows, HV 2 1.5 m; total area of wrndows, Av = 3.75 m ; openrng factor, Fv

= F, = A,,/H,, IA, =0.015 mm.

1 .5 2
Thermal absorptivity of concrete boundaries, b = kﬂ‘p = 1900 Ws0 /m K

Thermal absorptivity of gypsum plaster board, b = 1 lkpcp = 488 WsO'SImZK

. . 0.5 2
Resultant thermal absorptrvrty of the compartment, b = 626 Ws /m K

(The procedure for calculating b is given by Buchanan (2001)).

Nominal capacity of beam:

2
Fire load = yq x nominal value of ﬁre load = 1.254 x 196 = 245.8 MJ/m

158

Using this ﬁre load and the other parameters given above, the ﬁre temperature was
computed using Euqtaions 3-4 to 3-7 (Section 3.3), and the steel temperature was
computed using Equation 3-8 (Section 3.4) using a spreadsheet. The resulting time—ﬁre
temperature and time-steel temperature curves are shown in Figure 5.5.

Steel temperature after 2 hours = 554°C

Maximum steel temperature, Ts’max = 718°C occurs after 3.54 hours
Yield strength reduction factor, ky(TS) = 0.613

Elastic modulus reduction factor, kE(TS) = 0.44

Proportional limit reduction factor, kp(Ts) = 0.26

(The values of ky(TS), kE(TS)and kp(TS) were obtained from the AISC Speciﬁcations

(AISC 20053) and Eurocode 3 (EN 2005) corresponding to Ts = 554°C).

Using Equation 5-5 with (I? = 0.76 and the capacity adjustment factor, ¢a = 1.67 (See

Section 5.1.4.2), the capacity of the beam under ﬁre is
be: @147me f (with Mn,f given by Equation 5-5)
= 214.2 kN-m

Since be > Muf , the insulation thickness is adequate for a 2- hours ﬁre resistnce

rating.

159

1200 1

 

 

Tr
g K
O
800
9 1 Ts
15
3 X
a.
E
0
'- 400 -
Ts =5 54°C
0 I it I l l l l
0 60 1 20 180 240 300 360 420

Time (mins.)

Figure 5.5 - Fire and steel temperatures vs. time used in computing ﬁre resistance
for beam in example 2

5.4 Summary

To illustrate the proposed methodology presented in Chapter 3, capacity reduction
and ﬁre load factors are developed for simply supported steel beams in US. ofﬁce
buildings exposed to ﬁre. Fire load, ratio of floor area to the total area of ﬁre
compartment, opening factor, thermal absorptivity of compartment boundaries,
thickness, density and thermal conductivity of ﬁre protection material, dead load, and
live load are taken as random variables. Mechanical and sectional properties of steel
(e.g., yield strength, section modulus, etc.) are also considered to be random
variables. The effect of active ﬁre protection systems (e. g., sprinklers, smoke and heat
detectors, ﬁre brigade, etc.) in reducing the probability of occurrence of a severe ﬁre
is included. From detailed reliability analyses, it is found that the capacity reduction

and ﬁre load factors should not be constant for all design situations as suggested in

160

design speciﬁcations, and should vary depending on the presence of active ﬁre

protection systems in a building.
The variation of the ﬁre load factor based on the presence of active ﬁre protection

systems is in agreement with the Commentary to the AISC Speciﬁcations, the Eurocode

1, and the ECSC study.

For most ofﬁce building compartments in the US. equipped with sprinklers, use of (1%:

1.0 is reasonable, and yq is likely to lie between 0.4 and 1.0.

161

Chapter 6

Capacity Reduction and Fire Load Factors for Steel

Columns Exposed to Fire

In this paper, capacity reduction and ﬁre load factors are developed for the design of

axially loaded steel columns exposed to ﬁre.

6. 1 Derivation of Capacity Reduction and Fire Load Factors

6.1.1 Performance Functions for Reliability Analysis

6.1.1.1 Applied Axial Loads under Fire
According to the AISC Speciﬁcations, the ultimate applied axial load, Puf, is determined

from the gravity load combination given by

162

Puf: 1.2PD + 0.5PL + 0.2135 + T (6-1)

where, PD, PL and P5 are nominal dead, live and snow loads, respectively, and T includes

load effects induced by the ﬁre itself.
In Section 3.5.1.1 it was concluded that it is appropriate to use the combination of dead

and arbitrary-point-in-time live load for reliability analysis under ﬁre conditions. Thus,

for reliability analysis, the applied axial load, Paf, under ﬁre conditions may be

expressed in terms of basic variables as

where A and B = random variables reflecting the uncertainties in the transformation of

loads into load effects, E = a random variable representing the uncertainties in structural

analysis, and D and Lap, = random variables representing dead and arbitrary-point-in-

time live load. The statistics of A, B, E, D and Lap, are given in Table 4.1.

6.1.1.2 Axial Capacity of Columns under Fire

Takagi and Deierlein (2007) compared the AISC and Eurocode 3 design speciﬁcations
with ﬁnite element simulations for columns exposed to ﬁre. They reported that the AISC
Speciﬁcations (AISC 20053) are unconservative at elevated temperatures, particularly for
Slendemess ratios between 40 and 100 and temperatures above 500°C. For instance, at
500°C the nominal capacities predicted by the AISC Speciﬁcations are up to 60% larger
than capacities predicted by simulations. On the other hand, the Eurocode 3 column

strength equations were within 20% of the simulated results. The equations proposed by

163

Takagi and Deierlein (2007) were used in this study, which have a format similar to that
of the AISC Speciﬁcations and predict strengths similar to the Eurocode 3 (EN 2005)

columns strength equations (see Figure 6 in Takagi and Deierlein (2007)):

' r
1ky(Ts)Fy
PM 210.42 53”“ iAsky(Ts)Fy

L ' J

(6-3)

 

 

rrzkE(TS)ES
(KL/ r)2

 

Fears) = (6—4)

Pnf = nominal capacity of column under ﬁre, A S = cross-sectional area, KL = effective
length, r = radius of gyration about the buckling axis, ES 2 elastic modulus. kE(TS) is the

elastic modulus reduction factor that depends on the temperature, TS. The actual capacity

of steel columns under ﬁre can be obtained by modifying the nominal capacity given in

Equations 6-3 and 6-7 to

 

 

 

 

 

r Jky(’scTs)mle l
Pf = PC1042 ' Fem) ffSAskyUscTslmlF y (6-5)
2
F (T )= 71' kE(lSCTS)m2ES
f3r

where fi and m,- are non-dimensional random variables that characterize the uncertainty

in the fabrication and material parameters, and their statistics are given in Table 5.1.

164

15c is the model error for steel temperature which was characterized in Section 4.2. PC is

the professional factor (model error) that is characterized in the next subsection. The

performance function for reliability analysis of column can then be expressed as
801’) = Pf- Paf (6-7)

where X denotes a vector containing all the random design parameters. The probability of

failure, pp , of a steel column under ﬁre is

PF = P[g(X) < 0] (6-8)

6.1.2 Professional Factor (Model Error) for Axial Capacity of Columns

To account for differences between axial capacities of columns measured in the

laboratory and that predicted by Equation 6—3, the professional factor, PC, was

characterized using the test results presented by Janss and Minne (1981) and Franssen et
al. (1998). Janss and Minne (1981) reported results for eighteen columns with Slendemess
ratio between 25 and 102 for which the yield strength was measured. Franssen et al.
(1998) reported test results for twenty one ﬁre tests with Slendemess ratio between 20 and
140. The nominal capacity of these tested columns was calculated through Equation 6-3.

Pc (ratio of measured axial capacity to nominal capacity) is plotted against Slendemess

ratios in Figure 6.1. PC has a mean of 1.10 and a COV of 0.18, and is best described by

the normal distribution.

165

1.5 4

 

 

 

o ’ .
O.
:0: o: . . O O . .
0 o e 2’ 3
:2 1 a O O .0 ‘
76 o
C
'9 e
U) 0
is 0.5
O
L
Q.
0 i l l
0 50 100 150

Slendemess ratio

Figure 6.1 - Ratio of test capacity to nominal capacity of columns for different
slenderness ratios

6.1.3 Probability of Failure and Target Reliability Index

In Section 3.5.3 it was shown that for typical U.S. ofﬁce buildings the target reliability
index is likely to vary from zero to 2.0, depending on the size of the ﬁre compartment
and on the effectiveness/reliability of active ﬁre protection systems. Therefore, in the
next section, capacity reduction and ﬁre load factors were developed for target reliability

index values ranging from zero to 2.0.

6.1.4 Reliability Analysis

Twenty one steel columns with slenderness ratios ranging from 25 to 200 and axial load
capacities ranging from 133 kN (30 kips) to 10,675 kN (2400 kips) were selected for the
reliability study. Columns with smaller capacity are representative of those in upper
stories and those with higher capacity are representative of those in lower stories of

typical ofﬁce buildings. Live to dead load ratios ranging from 0.5 to 5.0 were considered.

166

The AISC design speciﬁcations were used to ﬁrst design the columns for ambient

temperature conditions. The same columns were then designed for ﬁre exposure (b = 640

0. 2
Ws 5/m K and F v = 0.02 mm) using the engineering approach described in Section 3.2

and the required thickness of insulation to withstand the design ﬁre was determined from
Equations 6-1 and 6-3 using an iterative procedure. As suggested in most codes, a
capacity reduction factor of 1.0 was used for the initial design for ﬁre. The columns were
assumed to be protected by gypsum board insulation, which is generally the case in the
US. Load ratios (ratio of applied load under ﬁre to room temperature nominal capacity)
ranging from 0.35 to 0.66 were considered. The applied loads, live to dead load ratios,
slenderness ratios, section factors, load ratios, and the insulation thicknesses for the

columns used in reliability analyses are given in Table 6.1.

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Using FERUM, FORM analysis was performed for each design situation (each of the 20
columns) using the performance function given in Equation 6-7. Using the direction
cosines obtained from the reliability analysis, the partial safety factors for each design
parameter were obtained through Equation 3-25 for each design situation.

These individual partial safety factors, except for the ﬁreload, were then combined into a

single capacity reduction factor through

. ‘/ky(¢lTs)02Fy l
40.42 PM)

 

 

 

 

 

 

>¢3Asky(¢lTs)¢2Fy
¢f : L J (6—9)
me
2
_ 7T kE(¢lTs )¢4Es
where Fe (Ts) — KL 7- (6'10)
¢5rl

Q are the partial safety factors for each design parameter obtained through Equation 3-
24, and Pnfis the nominal capacity of the column calculated through Equation 6-3. Thus
20 different capacity reduction factors (one for each column) were obtained. Thereafter, a

single optimized capacity reduction factor, gr, corresponding to dead and live load factors

of 1.2 and 0.5, respectively, was obtained using the optimization procedure described in

Section 3.5.7. Equal weights were used for all columns. Since the capacity reduction

factors were obtained for ,6, ranging from 0 to 2, this procedure was repeated for each ,8,.

The ﬁre load factors were constrained to be the same as those obtained for steel beams

since it is not desirable to have different ﬁre load factors for beams and columns.

170

6.2 Results

6.2.1 Capacity Reduction Factor

The plot of the optimized capacity reduction factor, ¢f vs. ,6, is shown in Figure 6.2. The
value of (bf is given by

1.0 for ,6, S 1.25
f : (6-11)
1.25—0.25, for 1.25 3 ﬂ, s. 2.0

Again, as was the case for steel beams, the results obtained for the columns indicate that

the nominal capacity need not be reduced (i.e., (1}c = 1.0) if ,6, is less than 1.25, which in

turn depends on the effectiveness of active ﬁre protection systems in reducing the

probability of occurrence of a severe ﬁre. Since most ofﬁce buildings in the US. are

required to have automatic sprinklers, ,6, is not likely to exceed 1.25. Therefore, using q

= 1.0 is reasonable for most design situations.

1.6 —

1.21 y

 

 

3” 0.84
'3
3
‘s‘ 04* Ki
0 l I l l l
o 05 1 1.5 2 25

Figure 6.2 - Capacity reduction and ﬁre load factors vs. target reliability index for
columns

171

6.2.2 Fire Load Factor

As mentioned earlier, the ﬁre load factor, yq was constrained to be the same as that

derived for steel beams and is plotted against the target reliability index in Figure 6.2.

The value of yq for a given target reliability index, ,6,, is

0.4+0.4,6, for ,6, S 1.25
7., = (6-12)

0.15 +0.6,6, for 1.25 S ,8, S 2.0

The yq is to be applied to the ﬁre load which is then used in describing the design ﬁre '

(time-ﬁre temperature curve) using Equations 3-4 to 3-7.

6.2.3 Validity of Capacity Reduction and Fire Load Factors for

Different Slendemess Ratios

Two steel columns (a light W10x19 section and a heavy W14x90 section, having radii of
gyration of 22 mm and 94 mm, respectively) were chosen to study the effect of
slenderness ratio on the reliability index. Slendemess ratios ranging from 25 to 200 were
used for reliability analysis. As shown in the next section, a live to dead load ratio of 2.0
yields lower values of the reliability index, and therefore, in order to be conservative, this
value was used. Both columns were designed using the ﬁre load and capacity reduction

factors given in Equations 6-11 and 6-12. The reliability index values computed from the

reliability analysis, ,8, are compared with ,8, values in Figure 6.3.

The ,8 values compare quite well with the ,8, values, indicating that the derived capacity
reduction and ﬁre load factors work well for all slenderness ratios. The comparison

172

shown in Fig. 3 is for two columns, and the ,8 values coincide for both columns for each

slenderness ratio and for each ,6,.

 

 

 

 

 

 

 

3'5 — 0 ﬂt=o.o ' ﬂt=0.5 A ﬂt=1.o "

3.0 - .....
a: 25 _ x ﬂt=1.5 .
1: ° ’
E 2.0 . ’ °
32‘ 1.5 x x x x X
:3 1.0 A a A A A
7) . I I u I
a: 0.5

0.0 9 ‘3 ‘3 ‘3 ‘3

0 50 100 150 200

Selenderness ratio

Figure 6.3 Computed and target reliability index values
for different slenderness ratios

6.2.4 Validity of Capacity Reduction and Fire Load Factors for

Different Live to Dead Load Ratios

Two steel columns, a W16x57 section and a Wl4x90 section, having radii of gyration of
40 mm and 94 mm, respectively, were chosen to study the effect of live to dead load ratio
on the reliability index. Live to dead load ratios ranging from 1.0 to 5.0 were considered,
and both columns were designed using the ﬁre load and capacity reduction factors shown
in Figure 6.2. Slendemess ratios of 25 and 75 were used for the Wl4x90 and W16x57
columns, respectively. These slenderness ratios yield relatively low and high reliability
index values (see Figure 6.3), and span the most likely range of slenderness ratios for

steel columns. The reliability index values computed from reliability analysis, ,6, are

173

compared with ,8, values in Figure 6.4. The ,8 values compare quite well with the ,6,

values, indicating that the derived capacity reduction and ﬁre load factors may be used.

for all live to dead load ratios considered herein. The ,6 values are conservative (i.e.,

higher than the ﬁt) for lower live to dead load ratios. As described earlier, the applied

load effects under ﬁre are computed from the load combination 1.2Dn + 0.5L", and the

dead load factor is the same for ﬁre design as for room temperature, while the live load
factor for ﬁre design is smaller than that for room temperature gravity design (0.5 instead
of 1.6). Thus under ﬁre situations, when the design is dominated by the dead load (which
is the case for smaller live to dead load ratios), higher insulation thicknesses are required.
When the insulation thickness is large, the column is thermally more stable, the variation
of other random parameters do not have a signiﬁcant effect on the reliability of the

column, and the column design generally yields a higher ,8 value.

 

 

 

 

 

 

 

 

4.0 _. o [31:04) - ﬂt=o.5 A ﬂt=1.0
3.5 -

x 3 O _ X ﬂtzls . ﬂt=200

0 .

'g 2.5 ’ .

' E. 2.0 3‘ . 3 it ‘
g 1.5 a 35. it ’3“ ’i
g 1.0 a 3 *3 A A
a? 05 5 ' ' ' '

0.0 9 Q E? 9 9
o 1 2 3 4 5
W Ratio

Figure 6.4 - Computed and target reliability index values
for different live to dead load ratios

174

6.2.5 Validity of Capacity Reduction and Fire Load Factors for

Multiple Fire Scenarios

The capacity reduction and ﬁre load factors shown in Figure 6.2 were developed for a ﬁre

compartment assumed to be constructed of lightweight concrete blocks having a thermal

absorptivity b = 640 WsO'SImZK, and having an opening factor F v = 0.02 mm.

In reality, the compartments may be constructed using different bounding materials and
may have different ventilation conditions. Therefore, the capacity reduction and ﬁre load
factors were validated by considering 9 ﬁre scenarios described in Section 5.2.3 and
Table 5.4. For these nine ﬁre scenarios, ﬁve columnswere designed for ﬁre conditions
using the capacity reduction and ﬁre load factors shown in Figure 6.2. The steel sections
used for these columns, and their room and elevated temperature capacities are given in
Table 6.2.

Table 6.2 - Properties of columns used for validation

 

 

 

 

 

 

 

 

 

 

 

Parameter W10xl9 W10x30 W18x65 Wl4x90 Wl2xl90
H (m) 3.66 3.66 3.66 3.66 3.66
231 876 2497 5276 10751
I"n (KN)
104 394 1124 2 7 48 7
Pnf (KN) 3 5 3
Load ratio 0.45 0.45 0.45 0.45 0.45

 

 

For all columns, a live to dead load ratio of 2.0 was assumed. Each column was designed

for ,8, values of 0, 0.5, 1.0, 1.5 and 2.0 for all nine ﬁre scenarios. Thus, for each ,6, value

there were 45 design situations yielding a total of 225 design situations. Reliability

analysis was then performed and the computed reliability index values, ,6, for all ﬁve

columns are compared with the ,6, values in Figure 6.5.

175

 

    

 

 

 

2.5 .
2.0 ~
K
1.5 - 13% below [3,
R '\
1-0 ' 24% below ,3,
0.5
0.0 . . . . . 4
o o 5 1 1.5 2 2 5
ﬁt

Figure 6.5 - Computed and target reliability index values for columns for different
fire scenarios

The ,6 values compare quite well with the ,6, values, indicating that the derived capacity
reduction and ﬁre load factors work well for all design situations considered. The ,6
values are conservative for ,8, values less than about 1.5. For ,8, values of less than 1.5
(see Figure 6.2), the ¢f found from reliability analysis was greater than 1.0, and the
nominal capacity could be increased. However, since Qvis generally always taken to be

less than or equal to 1.0 in LRFD speciﬁcations, we restrained the (If for ﬁre design to
also not exceed 1.0. Because of this inherent conservatism, the ,6 values are higher than

the ,8, values.

In Figure 6.5, 24% and 13% of the [3 values fall just below the target value for ,6, values

of 1.5 and 2.0, respectively. To ensure that the ,8 values are always higher than the target

176

value, the derived capacity reduction factor was reduced from 0.95 to 0.90 and from 0.85

to 0.80 for ,6, values of 1.5 and 2.0, respectively. Five columns were redesigned using

these reduced capacity reduction factors and yielded the ,6 values shown in Figure 6.6,
which are always higher than the target values.
If greater conservatism in design is desired, the reduced values of capacity reduction

factors given by

¢ _ 1.0 for ,B,Sl.0
f _ 1.2—02,6, for 103,490 (6'13)

can be used instead of those given by Equation 6-11 (Figure 6.2).

3.0
2.5 -
2.0 -
1.5 -
1.0 -
0.5

0.0 l l f .l l l
o 0.5 1 1.5 2 2.5 3

ﬁt

 

 

 

 

Figure 6.6 - Computed and target reliability index values for columns with reduced
capacity reduction factors

6.3 Design Example

To illustrate the applicability of the proposed approach in a design situation, the

procedure is applied to evaluate the ﬁre resistance of an axially loaded steel column.

177

In this example, the value of target reliability index, ,6,, is obatined using the procedure

described in Section 3.5.3. The effectiveness values of active ﬁre protection systems in

reducing the probability of occurrence of a severe ﬁre are obtained from the ECSC study

(ECSC 2001). In design speciﬁcations, the speciﬁcations authority may specify ,6, values

depending on the effects of active ﬁre protection systems, and designer may not need to
calculate it.

Problem statement:

2
A ﬁre compartment in an 8-story open-plan ofﬁce building has a ﬂoor area of 100 m . A

3.65 m (12 ft.) long interior steel column is to carry a service dead load of 178 kN (40
kips) and service live load of 360.3 kN (81 kips). A room temperature design of the

column was performed using the AISC design speciﬁcations, and a W10x30 section was

selected. The column has a cross-sectional area = 5703 mm , and radius of gyration =
2
34.8 mm. The yield strength of the steel = 345 N/mm and elastic modulus = 200,000

2
N/mm .

The ﬁre compartment is 10 m square and 3.65 m high, and is made of lightweight

3 . .
concrete with density = 2000 kg/m , speciﬁc heat = 840 J/kg.K and thermal conductrvrty

= 0.8 W/mK. The compartment has two windows each 3.0 m wide and 1.5 m high. The

2 .
ﬁre load is 196 MJ/m per unit surface area of the compartment. The compartment is

equipped with automatic sprinklers, and smoke and heat detectors. The automatic

178

sprinklers are assumed to be 80% effective in reducing the probability of occurrence of a

severe ﬁre. The effect of other active ﬁre protection systems may be neglected. The

3
column is to be protected using gypsum board with density = 745 kg/m , speciﬁc heat =

1200 J/kg.K and thermal conductivity = 0.16 W/mK. . ,
Calculate the required thickness of the gypsum board insulation.
Solution:

1. Ultimate applied axial load under ﬁre

Using Equation 6-1:
Puf = 1.2PD + 0.5PL= 1.2(178) + 0.5(360.3) = 393.75 kN

2. Capacity of the column
Selection of target reliability index: The target reliability index will be determined using

the procedure given in Section 3.5.3.

— 2
Probability of ﬁre occurrence in an ofﬁce building = 4 x 10 7(per m , per year) (ECSC

2 .
2001); probability of ﬁre occurrence for 100 m compartment assuming a 55-year

building life, p1 = 100 x 55 x 4 x 10‘7 = 0.0022 (ECSC 2001); probability of failure of

automatic sprinklers, p2 = 0.20; probability of occurrence of a severe ﬁre, psf= p1.p2 -

0.00044 (Equation 3-21) ; target probability of failure for life of the building, pt =

7.23x10-5 (ECSC 2001).

179

Using Equation 3-24 and the normal density table, the target reliability index,

a =¢‘l[l- pf]=1.0

 

psf
Selection of appropriate capacity reduction and ﬁre load factors:

Using Equations 6-12 and 6-13 with ,6, = 1.0
Fire load factor, yq = 0.4 + 0.4 ,8, = 0.8

Capacity reduction factor, (13c: 10
Fire parameters:

2
Nominal ﬁre load = 196 MJ/m ; total surface area of the compartment, At = 346 m2;

. . 2 .
height of wrndows, Hv = 1.5 m; total area of wrndows, Av = 9 m ; openrng factor, Fv =

1/2
Fv = AM/Hv / A, = 0.032 m ; density of compartment boundaries, p = 2000 kg/m3;

speciﬁc heat of compartment boundaries, c = 840 J/kg.K; thermal conductivity of

compartment boundaries, k = 0.8 W/mK; thermal absorptivity of compartment
boundaries, b 2' kpcp =1160Wso'5/m2K

Nominal capacity of column:

As described in Section 3.2, the required thickness of insulation by the engineering
approach can be computed using an iterative procedure.

Iteration I :

Assume thickness of insulation (gypsum board) = 15 mm.

180

Fire load = yq x nominal value of ﬁre load = 0.80 x 196 = 156.8 MJ/m2

Using this ﬁre load and the other parameters given above, the ﬁre temperature can be
computed using Euqtaions 3-4 to 3-7 (Section 3.3), and the maximum steel temperature

can be computed using Equation 3-8 or 3-9 (Section 3.4).

Maximum steel temperature, Twmu = 606.5°C

Yield strength reduction factor, ky(TS) = 0.454

Elastic modulus reduction factor, kE(TS) = 0.298

(Values of ky(TS) and kE(Ts) were obtained from AISC Speciﬁcations (AISC 20053)
corresponding to Tam“).

Using Equations 6-3 and 6-4 with (13c: 1.0, the capacity of the beam under ﬁre is

Paf: (bmef (with Pnf given by Equations 6-3 and 6-4)
= 201.6 kN

Since Pc,f < Puf , the insulation thickness is inadequate. Perform next iteration by

assuming a higher insulation thickness.

After 4 iterations, the required thickness of insulation was obtained as 21.5 mm.

The iterative solution procedure can be easily implemented in an Excel spreadsheet.

Note: In this example, the required thickness of insulation was determined using an

iterative procedure. Using the same approach, the failure temperature of the steel column

181

and the corresponding ﬁre resistance time can also be computed for a given value of

insulation thickness.

6.4 Summary

Capacity reduction and ﬁre load factors are developed for steel columns in US. ofﬁce
buildings exposed to ﬁre. The effect of active ﬁre protection systems (e.g., sprinklers,
smoke and heat detectors, ﬁre brigade, etc.) in reducing the probability of occurrence of a
severe ﬁre is included by adjusting the target reliability index appropriately.

From detailed reliability analyses, it is found that the capacity reduction and ﬁre load
factors should not be constant for all design situations as suggested in design
speciﬁcations, but should vary depending on the presence of active ﬁre protection
systems in a building and the compartment size.

As with the AISC and Eurocode provisions, the ﬁre load factor should be reduced for

typical ﬁre compartment sizes when active ﬁre protection systems are present.

182

Chapter 7

Deﬂections of Simply Supported Steel Beams Exposed

to Fire

7. 1 Background

The AISC Speciﬁcations (AISC 2005a) stipulate that for structural elements, e.g., beams
and columns, the governing limit state is loss of load-bearing capacity. However, they
also specify that excessive deformations are not acceptable if these damage the integrity
of the ﬁre compartment. If not limited, deformations may damage the slabs and walls of
the compartment to the extent that these components cannot control the horizontal and
vertical spread of ﬁre, even though the beam may still have sufﬁcient load bearing
capacity.

Since design under ﬁre may be governed either by the strength or deﬂection limit states,
both should be considered. Therefore, the required thickness of insulation for beams

based on the strength limit state can be ﬁrst determined using the capacity reduction and

183

ﬁre load factors developed in Chapter 5, and then the deflections of the beams could be
checked.

Design speciﬁcations such as the AISC Speciﬁcations (2005) and Eurocode 3 (EN, 2005)
provide simple design equations for the strength limit state of steel beams which can be
used for estimating the ﬁre resistance. Researchers such as Yin and Wang (2004),
Skowronski (1990), Burgess, El-Rimawi and Plank (1990), and Wang and Yin (2006)
provide detailed methods for tracing the time-deﬂection behavior of steel beams, but
these approaches are quite complex. No simple deflection equations which can be quickly
implemented in a design speciﬁcation are available for computing the deflections of steel
beams. Therfore, a simpliﬁed method which can be quickly implemented in a design
speciﬁcation for computing the deflection of simply supported steel beams is developed
in this chapter. Subsequently, the simpliﬁed method is used to compute the deﬂections of

steel beams designed based on the strength limit state.

7.2 De velopent of Simplified Method for Estimating Deflections

of Simply Supported Beams

7.2.1 Approach

Design speciﬁcations such as the AISC Speciﬁcations (2005) allow steel beams exposed
to ﬁre to be designed using room temperature design speciﬁcations with reduced material
properties. According to this concept, under ﬁre conditions, the elastic deflections, Ae, of
steel beams can be calculated using room temperature deﬂection equations and reduced
material properties. For example, the elastic deﬂections of a uniformly loaded, simply

supported beam can be calculated as:

184

__ 5wL4
‘9 384ES(TS)I

 

(7-1)

where L = length of the beam, w = load per unit length, I = moment of inertia of the

cross-section, ES(TS) = reduced elastic modulus calculated as ES(TS) = kEES , Es =

elastic modulus used at room temperature, k1; = stiffness reduction factor that accounts

for reduction in stiffness of steel at elevated temperature.
At elevated temperature, the stiffness of the steel beams degrade rapidly, the deﬂections
are likely to be beyond the elastic range, and Equation 7-1 cannot be used in most cases.

However, in the inelastic range, Equation 7-1 may be used for calculating the plastic

deﬂection, AP’ of steel beams by using an equivalent ﬂexural rigidity, [E 3(TS)I]eq, instead

of [ES(TS)I]. This concept is similar to the one used to calculate the deﬂections of

reinforced concrete beams by using an equivalent moment of inertia that accounts for
cracking. Therefore, it is proposed that in the inelastic range the deﬂection of a uniformly
loaded simply supported beam be calculated as:

_ 5wL4
P 384w, (2)11“, (7‘2)

 

where [ES(TS)I]eq is the equivalent ﬂexural rigidity at the steel temperature, T5. In the

next section, a simpliﬁed equation is developed for calculating the equivalent ﬂexural

rigidity.

185

 

7.2.1.1 Equivalent Flexural Rigidity

The proposed method for estimating the ﬁre-induced deﬂection of simply supported

beams is based on specifying an equivalent ﬂexural rigidity, [ES(TS)I]eq of the steel
section as a function of steel temperature. [E 5(Ts)l]eq is interpolated between the elastic

ﬂexural rigidity, [ES(TS)I]1 and the ﬂexural rigidity at ultimate failure, [ES(TS)I]2. A
typical moment curvature relationship of a steel section at any steel temperature, Ts, is
shown in Figure 7.1. The curvature in the section increases linearly with increase in the
applied bending moment, M, until yielding occurs at M = My} , where My} is the yield

moment capacity at elevated temperature. After yielding, the increase in bending moment
leads to a rapid increase in curvature, thereby resulting in larger deﬂections. The beam,
however, continues to carry the applied bending moment until ultimate failure occurs

when the maximum bending moment in the beam reaches the ultimate bending capacity

A
M A[E5(TS)I]1=kEEsI

      

 

i"

 

Curvature, K K"

Figure 7.1 - Typical moment-curvature diagram for a steel section

186

of the section M = Maj. Figure 7.1 shows the elastic and ultimate ﬂexural rigidities of

the section, [ES(TS)I]1 and [ES(TS)I]2, respectively. The ultimate secant ﬂexural rigidity is

deﬁned as:

 

. Mu T
[E5 (T5 )I]2 : K" (7‘3)

where Ku = ultimate curvature.

The ultimate curvature in Equation 7-3 is assumed to occur when the maximum strain in
the section is equal to 0.2. Further, in order to account for the effect of strain hardening,

the ultimate moment capacity is calculated using the ultimate strength instead of the yield

strength. The ratio of ultimate strength to yield strength, F u/Fy, of steel is based on

material prOperty tests (Wainman and Kirby 1987, and Luecke et al. 2005), and is plotted

in Figure 7.2. Based on the experimental data, the F u/F y ratio is generally signiﬁcantly

higher than 1.25. However, to be conservative, the ultimate strength of steel was assumed

to be 1.25 times its yield strength as suggested by the Eurocode 3 (EN 2005), i.e., F =
1.25Fy. Based on these assumptions, Ku = 0.2/(d/2), Myj = SxFy(Ts) = yield moment

capacity, and Muf = F “(TS)Zx = 1.25 Fy(TS)Zx = ultimate moment capacity. Zx and Sx

are the plastic and elastic section modulus, respectively, d is the depth of the section, and

Fy(Ts) = kyFy 2 yield strength at steel temperature, T. Fy = nominal yield strength at

187

room temperature, and kv = yield strength reduction factor which accounts for reduction

in steel strength at elevated temperatures.

 

 

 

 

 

 

 

2 1
OUKtest_Flange OUK test_Web ANIST_WTC
1.7 1
5 ‘ o . o o
O O O
u? o . 2
‘5 1.5 ~ ‘
ll. . A
‘ A A O ‘
0 Q ‘
: o 8 A
1.25 0 A ‘
A
A
‘ A
1 l I T l
0 5 1O 15 20
No of Tests

Figure 7.2 - Variation of ultimate tensile strength of steel

Prior to yielding, the elastic deﬂection can be obtained using the elastic ﬂexural rigidity

of the beam [ES(TS)I]1 in Equation 7-1. However, when the beam enters the inelastic

range, the equivalent ﬂexural rigidity [ES(TS)I]eq is used to compute the inelastic
deﬂection of the beam as per Equation 7-2. In order to obtain the equivalent ﬂexural
rigidity of a deﬂected beam in the inelastic range (M > Myf), the inelastic deﬂection of

the beam is assumed to be equal to the elastic deﬂection, but with equivalent properties

used instead of the elastic properties. For example, for a uniformly loaded, simply

supported beam, using Equation 7-2, [E S(TS)I]eq can be written as

188

5wL4

[ES(TS)I]eq 2—3—8-4—A—
P

(7-4)

Using Equation 7-4, the temperature-dependent relationship between [E 3(TS)I]eq and the

applied moment M can be obtained for different values of applied load, w, provided the

inelastic deﬂection, AP, of the beam is known. For developing this relationship, the

inelastic deﬂection, AP, of beams were obtained using the ANSYS software (ANSYS,

2007) by perfomring nonlinear ﬁnite element analysis of a series of beams under ﬁre
exposure. The ANSYS software is capable of handling geometric and material

nonlinearities as well as thermal analysis.

7.2.1.2 Finite Element Model

Finite element analysis was used to generate the temperature-deﬂection history of a series
of simply supported beams. The ﬁnite element model was created in ANSYS and
comprised two sub-models; namely, the thermal model and the structural model. The
thermal model provided the temperature distribution in the steel member which was
applied as a temperature-body-load in the structural model of the steel beam. For thermal
analyses, two types of elements, namely PLANE55 and SURF151, were used. The cross-
section of the steel beam was meshed with PLANESS elements. Heat transferred through
convection and radiation was applied on the exposed boundaries of the section using the
SURF151 element. For structural analysis, the beam was discretized using 90 BEAM189

elements which account for material and geometrical nonlinearities.

189

Stress-strain curves given in ASCE (1992) and shown in Figure 7.3 were used in the
ANSYS ﬁnite element model. These stress-strain curves account for strain hardening. To
account for primary and secondary effects of creep in structural steel, the “implicit creep

model 11” built into ANSYS was used. This creep model is given by:

Atecr = e

_ 62 1 c3 -511. 66 .51
primary +“Elsecondary —Clos at €Xp( T )+C5t0.s 6Xp( T ) (7‘5)

S S

where c1 = 5><10-6 per minute, c2 = 6.95, (‘3 = — 0.4, C4 =16500°C, c5 = 20le3 per

. —5 —3. . .
minute, C6 = 6x10 , C7 = 5x10 C, t = time (mm), as = steel stress (MPa), T5 = steel

temperature (°C), A80 = increment of creep strain, eprimary = primary creep strain, and

eprimary = secondary creep strain. This creep model has been calibrated with tests as
shown in Figure 7.4. Details on the calibration and validation of the ANSYS high
temperature creep model can be found elsewhere (Kodur and Dwaikat 2009).

Test data reported by Thor (1973) and Wainman and Kirby (1987) were used to validate
the structural ﬁnite element model in ANSYS. The tested beams were a protected
HEZZOB section and an unprotected UB356x17lx67 section which were simply
supported and uniformly loaded. The protected beam was loaded to 62% of its room
temperature yield capacity and the unprotected beam was loaded to 35% of its room
temperature ultimate capacity. The mid-span deﬂections predicted by AN SYS are

compared with test data in Figure 7.5, and the predictions are very good.

190

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
   
 

 

 

-- 80
500 — T: 20°C
T= 200°C
400 — “ 60
T: 400°C
,3 ._
g 3°° T= 500°C 40 3 E
- m -'
(D «n
3 20° T= 600°C 3;
b —- 20 (D
U) 100 T: 700°C
T= 900°C I
0 fr: 1 l l I I l l O L
o 2.5 5 7.5 10 12.5 15 17.5 20 ."
Strain, % I
Temperature (°F)
32 392 752 1112 . 1472 1832
E‘ 1.0 -
3. EfTs YES
9 0.8 '1
°' 0.6 -
3 J
,5 0.4
'2 a. /
‘5 . Fun-S Wu
2 0.0 . . r . .
o 200 400 600 800 1000
Temperature (°C)

Figure 7 .3 - Mechanical properties for structural steel at elevated temperature

191

  
 
  
  
  
 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

"° A36 Steel 6 i
- ANSYS creep model ' i

0‘3 1 - -o- - - Creep test data (10°Clmin. ‘
o\° .
g 0.6 O = 50 MPa
‘5 o = 100 MPa
1:.
g 0.4“ 0:150 MP8
0

0.2 ~

0.0 1 . r

0 150 300 450 600 750
Steel temperature, °C
2.0 1
,’ o = 122.6 MPa
/ , " Gradeso Steel

1'5 i .I ANSYS creep model
32 .' 'o - - 4 - — Creep test data (600'C)
519.: o, ' \a=98MPa
3:: ' a = 73.5 MPa 0 = 49 MPa
gas .' \ ,, _, -

0.0 l —‘ - _ ’ —J¢ r

 

 

0 50 100 150 200 250 300 350
Time, mln

Figure 7.4 - Validation of high-temperature creep model

192

 

l'

Load = 62% of R.T. yield capacity

 

 

 

600 ~ Temp. of bottom flange -- 120 A
A mmnmmmmmmmmnn _ E
g 500 - .L. l . .I —- 100 z
e 400 ‘ L23200 mm . __ 80 ,g
3 - o
E 300 _ Temp. of ‘ -L 60 £3
a top ﬂange .. 3
g 200 - K ._ 40 r:
l- . - Test 3‘,
"' 100 - . uJ" -- 20 .
,3 - -/'/ Model .9
‘0 o - . ‘ . . . . . o 5

 

0 2O 40 60 80 100 1 20 140
Time (minutes)
(a)

 

 

 

 

800 . Load = 35% of R.T. ultimate capacity g 160

llllllllllllllllllllllllllllllllll __ 140 A
A .A. .I. - E
9 600 r L=4500 mm ‘” 120 E
I: T TGSt :
h emp. of bottom\A , -_ 100 o
3 flange 11':-
2 400 - ~- 80 §
8. Temp. of u .5
E top flange _- 50 D
a ./ c
'— ﬂ
_ 200 - .4/\ -- 4o 9.,
.3 - e
a) / - Model 2- 20 E

/‘
O . l I T l f 0
0 5 10 15 20 25 30

Time (minutes)
(b)

Figure 7.5 - Comparison of deflections of steel beams predicted by ANSYS with the
test data for (a) protected beam, and (b) an unprotected beam

193

7.2.1.3 Parametric Analysis

In order to obtain the equivalent ﬂexural rigidity through Equation 7-4, parametric

analysis was carried out using the ﬁnite element model validated above. The parameters

that were varied in the analysis included different slenderness ratios (Ury = 123, 185 and

246), and different load ratios (30%, 50% and 70% of room temperature moment
capacity). The analysis was carried out by applying the load gradually on the simply
supported beam until the room—temperature response was obtained, and then the steel
temperature was increased, simulating ﬁre exposure, until failure occurred. Failure was
deﬁned to occur at the last time step for which the iterative analysis converged.

Using the temperature-deﬂection history obtained from the nonlinear analysis described
above, Equation 7-4 was used to compute the equivalent ﬂexural rigidity of the beam as a
function of steel temperature. Both the equivalent ﬂexural rigidity and the bending
moment capacity of the beam signiﬁcantly reduce with increase in steel temperature.
Since the rate of reduction of these quantities depends on many factors, such as load and
geometric characteristics, both ﬂexural rigidity and the bending moment ratio were

normalized as follows:

_ [Es (T5)I]eq —[Es (T5)I]2

 

 

- 7-6
5 1E, (T,)I]1 —[E_, (Tsﬂlz ( )
and
M - M
y,T
w = (7-7)
Mu,T —My,T

where f = normalized ﬂexural rigidity, 1,11 = normalized inelastic moment and M =

applied moment.

194

Curves of normalized ﬂexural rigidity and normalized inelastic moment are shown in
Figure 7.6 for different load and slenderness ratios. The values of w from 1.0 to 1.25
correspond to the strain-hardening range. As the steel temperature increases, the bending
moment capacity of the beam reduces, and this leads to an increase in the normalized
inelastic moment, t//. Also, the deterioration of steel properties and the spread of plasticity
in the steel beam cause further deﬂection in the beam, and thus reduce the normalized
ﬂexural rigidity, f, of the beam. Figure 7.6 also shows that the inﬂuence of initial load
ratio on the normalized ﬂexural rigidity is nonlinear with the worst case of reduction in 5
occurring at a load ratio of 50%. This can be explained by the fact that the normalized
ﬂexural rigidity is a measure of the change in deﬂection with respect to a change in
moment in the beam. Both ﬂexural rigidity and moment ratio (initial applied moment to
moment capacity) vary with steel temperature. Generally, for beams with higher initial
moment ratios prior to ﬁre (higher load ratios), the spread of plasticity along the beam
occurs sooner than for beams with a lower initial moment ratio (lower load ratios).
Therefore, the rate of deﬂection for beams with higher load ratios is generally more
gradual as compared to beams with lower load ratios. The nonlinear effect of load ratios
on the normalized ﬂexural rigidity also can be attributed to geometric nonlinearities at
such high inelastic deﬂection rates and magnitudes.

Both geometrical and material nonlinearities are responsible for the nonlinear degradation
of the ﬂexural rigidity of the beam. In order to account for this degradation, an
approximate lower bound curve was ﬁtted to the curves shown in Figure 7.6. This curve

(model) is given by

195

1.00

0.80

0.60

0.40

0.20

Normalized rigidity, §

0.00

(7-8)

 
 
 
  
 
  

— Load ratio = 30%
— - Load ratio = 50%
---- Load ratio = 70%
Fitted model (Equation 7-8)
L/ry = 123
0.00 0.25 0.50 0.75 1.00 1.25

Normalized moment, at

Figure 7.6 - Variation of normalized ﬂexural rigidity with normalized inelastic

moment for different load and slenderness ratios

Equation 7-8 is also plotted in Figure 7.6. From Equations 7-6, 7-7 and 7-8, the

equivalent ﬂexural rigidity may be expressed as:

[Es (Ts

[Es(Ts)1]1 -[Es (T0112
1+ 1511/25

 

The equivalent ﬂexural rigidity computed by Equation 7-9 can be used to calculate the

ﬁre induced inelastic deﬂection of simply supported steel beams. Equation 7-9 is based

on a conservativ

ely ﬁtted curve to results from nonlinear ﬁnite element analysis. At M =

196

Myj the value of [ES(TS)I]eq = [ES(TS)I]1 as desired. However, when M = Mu], the

value of [ES(TS)[]eq = (15[ES(TS)I]2—[ES(TS)I]1)/ 16 which is not equal to [ES(TS)I]2 as

desired. Near ultimate strength failure (M = MN), the mid—span deﬂection of the beam

is quite signiﬁcant that it alters the horizontal projection of the beam length between the

supports. This is reﬂected by a stiffened response of the beam that leads to values of

[E 5(TS)I]eq that are slightly higher than the idealized [ES(TS)I]2 near failure.

In Figure 7.6, the curves for normalized ﬂexural rigidity seem to be dependent on the
load ratios. However, for simplicity, an approximate lower bound curve was ﬁtted
(Equation 7-8). This was done for two reasons: (1) Most buildings have a load ratio of 0.5
or less (Buchanan, 2001) and the ﬁtted curve matches the predicted curves well at this
load ratio; (2) As shown later, for smaller load ratios (e.g. 0.35), the simpliﬁed method
will yield conservative results but the conservatism is less than 10%. It is also shown in
the next section that Equation 78 is proposed in such a way that it takes into
consideration the inﬂuence of high temperature creep on the ﬁre-induced deﬂection of
beams.

Creep is insigniﬁcant in structural steel at room temperature, but it becomes very
signiﬁcant at temperatures above 400°C and at higher loads. Despite the signiﬁcant effect
of creep deformations on steel structures exposed to ﬁre, creep is usually not explicitly
included in the design process under ﬁre because of the lack of data and difﬁculty of
calculations. The usual assumption made is that the stress-strain relationships used for

ﬁre design are ‘effective’ relationships which implicitly include creep deformations

197

(Buchanan 2001). Other researchers (Anderberg 1986 and Poh 1995) have shown how
creep deformations can be explicitly included in computer models. Since the main focus
of this study was to develop a simple expression for calculating deﬂections, instead of
including creep deformations through a separate factor that is likely to be complex, the
effect of creep deformations on the equivalent ﬂexural rigidity is accounted for through
the ﬁtted equation (Equation 7-8) that is an approximate lower bound of the curves
shown in Figure 7.6. In the next section, while validating the simpliﬁed method, creep
deformations are explicitly included in the ANSYS analysis using the creep model given
by Equation 7-5 and it is shown that the equivalent ﬂexural rigidity expression obtained

from Equation 7-9 gives very reasonable results.

7.2.2 Validation of Simplified Method

For simplicity, most design speciﬁcations such as the AISC Speciﬁcations (2005) and
Eurocode 3, allow the use of a uniform temperature distribution across the steel section.
The uniform temperature can be calculated by simple thermal analysis methods (e.g.,
lumped heat capacity method) instead of sophisticated computer software. Therefore, the
simpliﬁed method developed herein is based on the assumption of a uniform temperature
distribution. Therefore, results from ANSYS when the temperature distribution is kept
uniform are used to validate the proposed simpliﬁed method. In actual ﬁre tests, the
temperature distribution across the section is not uniform and there is a signiﬁcant
temperature gradient. Therefore, in the next section, the ﬁre resistance computed from the
simpliﬁed method is compared with that measured in actual ﬁre tests, wherein, an'
average temperature of the steel section is used and the deﬂections resulting due to the

temperature gradient are accounted for through a separate term.

198

In performance based approaches currently being promoted, there is likely to be a limit
on deﬂections to ensure the integrity of the ﬁre compartment and to provide safe
conditions for ﬁre ﬁghters. Until now, this deﬂection limit is not well deﬁned and agreed
upon. The British Standard (BS 476-20 1987) suggests a deﬂection limit of [/20 that has
been adopted by most researchers. However, a deflection limit of U30 has also been used
by many researchers. In validating the simpliﬁed method, it is therefore appropriate to
compare the ﬁre resistance time given by the simpliﬁed method with that predicted by
ANSYS when the mid-span deﬂection of the beam reaches the limiting values of [/30
and [/20 In the succeeding sections, although the entire deﬂection curves are provided,
the discussion focuses on the comparison of the ﬁre resistance time between the limiting
deﬂection values of U30 and U20, because the intent of the simpliﬁed method is to
predict accurate deﬂections at these limits (U30 and U20).

It is desirable that the proposed method be applicable to: all types of loading
conﬁgurations; different ﬁre scenarios (real ﬁres and standard ﬁre); varying heating rates
of steel (slow or fast); all load ratios; and different sectional geometry and span lengths.
This is yet another reason why ANSYS simulations rather than experimental data were
used to initially validate the simpliﬁed method, because many parameters are not
explicitly reported for experiments.

Creep deformations are signiﬁcant at slower heating rates and higher loads. Most
buildings have a load ratio of 0.5 or less (Buchanan 2001). A load ratio (LR) of 0.5 is
used for all comparisons except when different load ratios are explicitly mentioned. A
relatively slow heating rate (HR) of 45°F/min (7°C/min) is used for most of the

comparisons. However, design ﬁres producing heating rates lower and higher than

199

45°F/min (7°C/min) are also used to validate the simpliﬁed method for different heating
rates. In reality, the heating rate of steel varies with time, but in this study the heating rate
refers to the average heating rate obtained by dividing the maximum steel temperature by

the time taken to achieve that temperature.

7.2.2.1 Effect of Heating Rate of Steel

Creep deformations form a signiﬁcant part of the total deﬂection of beams at elevated
temperatures and depend on the rate of heating and the load ratio. To determine whether
the simpliﬁed method works for all heating rates, four ﬁre scenarios producing different
heating rates of steel were selected. Slow and fast heating rates of steel sections represent
well insulated and poorly insulated (or unprotected) steel beams, respectively. The
deﬂectionspredicted by the simpliﬁed method and ANSYS are shown in Figure 7.7. The
simpliﬁed method gives deﬂections very close to those given by ANSYS and the
maximum difference in the ﬁre resistance time (i.e., the time when the deﬂection reaches

the limits of U30 and U20) from. the two methods is less than 2%.

7.2.2.2 Effect of Load Ratio

Creep deformations are strongly dependent on the load ratio and increase for higher load
ratios. The deﬂections predicted by the simpliﬁed method for ﬁve load ratios are
compared with the ANSYS predictions in Figure 7.8. The two methods give very similar
deﬂections and the maximum difference in the ﬁre resistance times is less than 7% at the
deﬂection limits of U30 and U20. The simpliﬁed method gives a relatively conservative
estimate of the deﬂections for low load ratios (e.g., 35%) because the ﬁtted curve for

normalized ﬂexural rigidity (see Equation 7 -8 and Figure 7.6) is lower than that estimated

200

from ﬁnite element simulations. As mentioned earlier, most structural members will have

a load ratio of about 0.5 under ﬁre (Buchanan 2001) and the simpliﬁed method gives very

accurate results for this case.

 

 

 

  
  
   

 

 

 

 

 

 

 

 

 

. 5 2°
450 ﬁg
Deflection limit = Ll20 5 16
375 i Deflection limit = U30 c-
E 300 " 12 2'
5’ HR = 85'Flmin. ; .3
.3 225 - e : HR = GO'Flmin. -' a g
g 150 _ .4— HR = 50'Flmin. . 8
3 - HR=40°FImin._. ' 4 .
. . o
60 80 100 120

 

 

Time, min

Figure 7.7 - Deflections predicted by ANSYS (broken lines) and simplified method
(solid lines) for different heating rates. Load ratio = 0.5, load type = UDL

 

 

 

 

 

 

 
 
 
 
 

 

 

  

 

 

I g . .1 20
450 K ' 5
375 4 i
a 243° . .5-
E 300 3 '
:- LR = 0.80 .1 12 §
.2 225 - LR = 0.70 , ‘6
i-l ' a)
3 LR = 0.60 , .- 8 g
g 150 — LR = 0.50 ,' o
0 LR = 0.35 ,' - 4
75 —
0 ‘ t ' 0

 

0 20 40 60 80 100 120 140
Time,min.

Figure 7.8 - Deﬂections predicted by ANSYS (broken lines) and simplified method
(solid lines) for different load ratios. HR = 45°F/min, load ratio = 0.5, load type =
UDL

201

7.2.2.3 Effect of Sectional Geometry and Span Length

The deﬂections predicted by the simpliﬁed method are compared with the ANSYS
predictions for three steel sections (light, heavy and intermediate) and for three span
lengths (20 ft, 30 ft and 40 ft) in Figure,7.9. The maximum difference in the ﬁre
resistance times at the deﬂection limits of U30 and U20 predicted by the two methods is
less than 1%. Since the three beams are of different lengths, the deﬂection limits of U30

and U20 will be different for each beam. In Figure 10, only the lowest U30 and highest

 

 

 

 

 

 

 

 

 

 

U20 are shown.
- 24
60° 3 H' tht L120 t: :
I e
525 ~ 9 I. a: 20
I
e 450 - ,0' - 16 -
E. 375 ~ w12x1s———> . ‘5-
C
,9 30° _ W18X35 ,r' l 12 g
E 225 _ waxes? *‘7‘ i 3 §
0) ' d)
D 150 ‘ Lowest U30 I 5" _ 4 o
75 ‘ .0...'
0 r - ' °
0 25 50 75 100 125
Time,min.

Figure 7.9 - Deflections predicted by ANSYS (broken lines) and simpliﬁed method
(solid lines) for different sections and span lengths. Load ratio = 0.5, HR = 45
°F/min, load type = UDL

7.2.2.4 Effect of Load Configuration

In all of the analyses and comparisons hitherto, the beam was subjected to a uniformly
distributed load. It is now demonstrated that the simpliﬁed method also gives good results
for other load conﬁgurations. In Figure 7.10, deﬂection proﬁles are compared for two

loading conﬁgurations: a beam subjected to a single point load at mid-span and a beam

202

subjected to two-point loads at one-third spans. The mid-span deﬂections for these cases

are PL3/ (48[ES(TS)I]eq) and 23PL3/(648[ES(TS)I]eq).The deﬂections predicted by the

simpliﬁed method agree very well with the ANSYS simulations.

 

 

 

 

 
 
   

 

 

500 "l i t E 20
- . . . /‘ i 'l' g
E 400 iDeflectronltmrt=L/20 :0 : 16 .E
g- Deflection Ilmlt = L/30\ 2': I :1 é
"3 3°°l :' .' ‘2 .8.
g 3 p t 1 ”5
.3 Two point load t '0 1:
c 200 : . 8 5
(U i ' . a
CL 1 ;: 00
'3 100 5 4 352
E ’ One i
1 point load I
o :’ : . *T’" . i J: O
O 25 50 75 100 125 150

Time. min. .

Figure 7.10 - Deflections predicted by ANSYS (broken lines) and simpliﬁed method
(solid lines) for different load conﬁgurations. Load ratio=0.5, HR: 45 °F/min

At the limiting deﬂection value of U30 and U20, the maximum difference in ﬁre

resistance time predicted by the two methods is less than 10% for the case of a single

point load.

7.2.3 Comparison with Test Results

As mentioned earlier, the temperature distribution through the depth of the steel section is
not uniform under ﬁre conditions and a signiﬁcant temperature gradient exists, especially
for steel beams supporting a concrete slab on the top ﬂange. Since deﬂections of beams at
elevated temperatures occur primarily due to a reduction in stiffness of the section and

the temperature gradient across the section, both of these effects need to be considered to

203

predict deﬂections accurately. The contribution to deﬂection due to stiffness reduction is
effectively accounted for by the use of an equivalent ﬂexural rigidity. However, a

separate term needs to be introduced to account for the contribution of the temperature
gradient to deﬂections. If the thermal gradient is assumed to be linear through the beam

depth, the additional elastic deﬂection due to the temperature gradient is

L2 AT
Ap,TG = a??? (7-10)

where, a = coefﬁcient of thermal expansion of steel, AT = the difference between the

temperature of the top and bottom ﬂanges, d = the section total depth, L = beam span
length.
The total inelastic deﬂection of the beam, including that due to temperature gradient, is

then

Ap,total = Ap + Ap,TG (7-11)

In Table 7.1, the ﬁre resistance time at the limiting deﬂection value of [/30 predicted by
the simpliﬁed method is compared with test results for 13 beams. Beams 1 to 12 were
tested in the UK. (W ainman and Kirby 1987 and Wainman 1992), and the data for beam
no. 13 was reported by Thor (1973). Beams 1 to 11. were unprotected, and beams 12 and
13 were protected by ﬁre protection material. These tests were terminated when the mid-
span deﬂection of beams reached a limiting value of [/30. For each beam, the mid-span

deﬂection and the mean temperatures of the bottom ﬂange, top ﬂange and web were

reported. In the simpliﬁed method, AP is computed using the average of the top ﬂange,

204

 

bottom ﬂange, and web temperatures of the section in Equation 7-9, and by using the

temperature difference between the top and bottom ﬂanges in Equation 7- 10.

Table 7.1 - Comparison of ﬁre resistance time predicted by the simplified method
with test results at the limiting deflection value of L/30

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Load Fire Resistance Time (min)
Test # Beam Section Length (ft) Ratio (%)

Test Model Difference
1 U8254x146x43 15.00 46 23 22.4 -0.6
2 U8254x146x43 15.00 53 22 21.3 -0.7
3 U8356x171x67 14.75 56 27 24.6 -2.4
4 UB406x178x60 14.75 35 23 2.1 -0.9
5 UB305x165x54 14.75 34 22.5 21.3 -1.2
6 U8254x146x43 15.00 34 30 30.8 0.8
7 U8356x171x67 14.75 35 24.5 24.5 0.0
8 U8254x146x43 14.75 44 21 21.6 0.6
9 U8254x146x43 14.75 45 22 21.1 -0.9
10 U8356x1 71 x67 14.75 51 29 24.6 -4.4
11 UB254x146x43 14.75 36 27 24.7 -2.3
12 UB254x146x44 14.75 67 93 91.7 -1.3
13 HE220 B 10.50 55 130 124.2 -5.8

 

 

 

 

 

 

 

 

 

7.3 Deflections of Simply Supported Beams

10 laterally restrained and 14 laterally unrestrained beams were designed for ﬁre using
the performance—based approach. The required thickness of insulation was determined
through Equations 5-1, 5-3, 5-5 and 5-6 using an iterative procedure. The capacity

reduction and ﬁre load factors given in Figure 5.1 were used. The deﬂection of the beams

were computed through Equation 7-1 if Mu,f< My], and through Equation 7-2 if Mu,f>
MyJ. E 5(Ts)l]eq was calculated through Euqation 7-9. Muf was as
Muf=MD+Mlﬂw (7-12)

205

I-nl‘muﬂ ’
..
s

where MD = applied moment due to dead load and Mum, = applied moment due to
arbitrary—point—in-time live load. Arbitrary-point-in-time live load, [apt was assumed to
be 0.24Ln (Ellingwood 2005).

The computed deﬂections are compared with limiting deﬂection values of U30 and U20
in Tables 7.2 and 7.3. It was found that deﬂections are generally smaller than the limiting

values of U30 or U20, and the strength limit state is critical in case of simply supported

beams.

 

206

 

 

 

 

 

 

 

 

 

 

 

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208

7.4 Summary

In this chapter, a simpliﬁed method was developed for computing deﬂectios of simply
supported steel beams exposed to ﬁre. In this method, deﬂections are predicted by the
same expressions used for predicting the elastic deﬂection of beams at room temperature
but using an equivalent ﬂexural rigidity instead of the elastic ﬂexural rigidity. At the
deﬂection limit of U30 to U20, the simpliﬁed method predicts ﬁre resistance times very
close to those predicted by the ANSYS ﬁnite element program which accounts for
thermal and creep effects. The ﬁre resistance time predictions by the simpliﬁed method
also compare very well with test data at the deﬂection limit of U30.

The proposed method is very simple and can be included in design speciﬁcations for the
ﬁre—resistant design of simply supported steel beams.

The deﬂections of simply supported beams designed for the strength limit states were
also computed using the simpliﬁed method, and it was found that the deﬂections were

smaller than the limiting values of U30 or U20.

209

 

l‘ﬁg

Chapter 8

Conclusions and Recommendations

8. 1 Summary

A general reliability-based methodology is proposed for developing capacity reduction
and ﬁre load factors for LRFD of steel members exposed to ﬁre. Statistics (mean, COV,
and distribution) of a variety of parameters such as ﬁre load, opening factor, thermal
absorptivity of compartment boundaries, thickness of insulation, thermal conductivity
and density of insulation were obtained from experimental data reported in the literature.
Model errors associated with the thermal models were also characterized based on
experimental data. Both 3-sided and 4-sided ﬁre esposure were considered while
developing thermal model errors. Experimental data applicable to US. conditions was
used whenever possible. In the proposed methodology, the effect of active ﬁre protection
systems (e.g., sprinklers, smoke and heat detectors, etc.) in reducing the probability of
occurrence of a severe ﬁre is accounted for by using a reduced target reliability index for

developing capacity reduction and ﬁre load factors for steel members exposed to ﬁre.

210

 

Based on the information available about the effectiveness of active ﬁre protection
systems in reducing the proability of occurrence of a severe ﬁre, a range of target
reliability index value was established for US. ofﬁce compartment ranging in ﬂoor area
from 25m2 to 500m2. Both compartmented and open plan ofﬁce buildings were
considered to establish the range of ﬂoor areas.

To illustrate the proposed methodology, capacity reduction and ﬁre load factors were
derived for simply supported steel beams and axially loaded steel columns in US. ofﬁce
buildings exposed to ﬁre. Capacity reduction and ﬁre load factors were developed for a
range of target reliability index values (0-2). Fire design provisions given in the AISC
Speciﬁcations and the literature were used for steel beams and columns, respectively. The
capacity reduction and ﬁre load factors developed in this study can be used for the ﬁre
design of simply supported steel beams and columns in US. ofﬁce buildings instead of
the current factors which are primarily based on subjective judgement. For ease of use,
linear relationships are developed for computing capacity reduction and ﬁre load factors
for a given value of target reliability index.

Capacity reduction and ﬁre load factors for other types of structural members (e.g.,
restrained beams) and for other type of occupancies may be developed using the
proposed methodology, model errors, and the statistics of ﬁre design parameters.

In addition, a simpliﬁed method is proposed for computing inelastic deﬂections of simply
supported steel beams exposed to ﬁre, and it is shown that the ﬁre design of most beams
is goverened by the strength limit state. The proposed method is very simple and can be
included in design speciﬁcations for the ﬁre-resistant design of simply supported steel

beams.

211

 

1F

8.2 Conclusions

The main conclusions resulting from this study are as follows:

The present capacity reduction and ﬁre load factors in the AISC ﬁre design
speciﬁcations are primarily based on experience and subjective judgement, and do
not consider the statistical variations of the various parameters involved in the
design.

There was only one study done in Sweden to to develop capacity reduction factors
for the design of steel members exposed to ﬁre.

No study has been done so far to develop capacity reduction and ﬁre load factors
for LRFD of steel members in the US.

The effect of active ﬁre protection systems, especially sprinklers, in reducing the.
probability of occurrence of a severe ﬁre is well recognized in the literature, but,
has not been explicitly included in developing capacity reduction factors for
design of steel members exposed to ﬁre.

From the statistics of ﬁre design parameters developed in this study, it is found
that uncertainty associated with the ﬁre design parameters is much higher than
that associated with room temperature design parameters. The errors associated
with thermal models are also much higher than those associated with structural
models used for room temperature design.

The capacity reduction and ﬁre load factors correspond to a preselected target
reliability index that accounts for the effect of active ﬁre protection systems (e.g.,

sprinklers, smoke and heat detectors, etc.) in reducing the probability of

212

 

 

occurrence of a severe ﬁre. Therfore, for uniform safety, these need to be
developed for a range of target reliability index values.

0 Based on the effectiveness of active ﬁre protection systems reported in the
literature and the typical ﬁre compartment sizes in US. ofﬁce buildings, it is
resonable to develop capacity reduction and ﬁre load factors corresponding to
target reliability index values ranging from zero to 2.0.

0 It is found that the ﬁre load factor should vary depending on the presence of
active ﬁre protection systems, and the ﬁre load may be reduced in buildings
equipped with very effective and reliabile active ﬁre protection systems,
especially, sprinklers. This is in agreement with the Commentary to the AISC
Speciﬁcations, the Eurocode l, and the ECSC study.

0 Capacity reduction factors for simply supported beams and axially loaded steel
columns in US. ofﬁce buildings should vary based on the presence of active ﬁre
protection systems in the buildings, and should not be constant as suggested in
present design speciﬁcations.

0 For most ofﬁce building compartments in the US. equipped with sprinklers, using
capacity reduction factor value of 1.0 is reasonable, and ﬁre load factor is likely to
lie between 0.4 and 1.0.

Current structural ﬁre design provisions are still relatively new and evolving, with
various remaining uncertainties and information gaps. The methodology proposed herein
is an initial attempt to characterize uncertainties in current ﬁre design provisions. It is
expected that as the research in this ﬁeld yields improved understanding of structural

behavior under ﬁre, future design reﬁnements will be necessary.

213

 

8.3 Recommendations for Future Research

While this study has advanced the state—of—the-art with respect to the development of

capacity reduction and ﬁre load factors for LRFD of steel members exposed to ﬁre,

further research is required to develop these factors for all types of structural steel

members used in different types of occupancies. The following are recommendations for

further research in this area:

Capacity reduction factors developed herein are speciﬁc to simply supported
beams for typical ﬁre compartments in US. ofﬁce buildings. However, the
methodology presented in this study and the random parameters and model errors
characterized herein are general and can be used for deriving capacity reduction
and ﬁre load factors for other types of structural members such as restrained
beams and beam - columns.

As stated in the Design Guide for Structural Fire Safety (CIB W14, 1986), the
ﬁre load varies considerably from one type of occupancy to another. Therefore, it
is not possible to develop one set of capacity reduction and ﬁre load factors for
all types of occupancies. Accordingly, one set of factors may be developed for
any type of occupancy (say ofﬁce buildings), and then differentiated factors can
be developed to modify these basic factors for other types of occupancies. All
random parameters characterized in this study except for the ﬁre load, ratio Af/At,
and arbitrary-point-in-time live load, are applicable for the reliability analysis of
buildings of all use categories. The statistics of the ﬁre load and arbitrary-point-
in-time load for different types of occupancies are reported in the literature.

However, not much information is available about the ratio Af/At, and these

214

 

should be obtained from architectural drawings of buildings. Subsequently,
differentiated factors for buildings of other use categories (e.g., residential
buildings, industrial buildings, etc.) can be developed using the methodology
described in this study.

0 The statistics of ﬁre design parameters can be further improved by using more

tests data as it becomes available.

215

 

Appendix A

Journal and conference papers resulting from the work are:

(l) Iqbal, S., and Harichandran, RS. (2009). “Capacity reduction and ﬁre load
factors for design of steel members exposed to ﬁre.” ASCE Journal of
Structural Engineering. (paper re-submitted after addressing reviewers’
comments)

(2) Iqbal, S., and Harichandran, RS. (2009). “Capacity reduction and ﬁre load
factors for LRFD of steel columns exposed to ﬁre.” Fire Safety Journal.
(paper re-submitted after addressing reviewers’ comments)

(3) Iqbal, S., Dwaikat, M.S., and Harichandran, RS. (2008). “A Simpliﬁed
Method for Determining Fire Resistance of Steel Beams Exposed to Fire.”
AISC Engineering Journal. (submitted)

 

(4) Iqbal, S., Harichandran, RS, and Kodur, V., (2008). “Capacity reduction
factor for ﬂexural strength of steel beams exposed to ﬁre.” Proceedings,
Inaugural International Conference of the Engineering Mechanics Institute,
Minnesota.

(5) Iqbal, S., and Harichandran, R.S. (2008). “Reliability-based design
speciﬁcations for simply supported steel beams exposed to ﬁre.” Proceedings,
10th International Conference on Structural Safety and Reliability, Osaka,
Japan.

(6) Iqbal, S., Dwaikat, M.S., and Harichandran, RS. (2009). “A simple approach
for calculating inelastic deﬂections of simply supported steel beams under
ﬁre.” Proceedings, ASCE Structure Congress 2009, Austin, Texas.

(7) Iqbal, S., and Harichandran, RS. (2010). “Methodology for reliability-based
design of steel members exposed to ﬁre.” Proceedings, International
Colloquium: Stability and Ductility of Steel Structures, Rio de J aneiro, Brazil.

(8) Iqbal, S., and Harichandran, RS. (2010). “Capacity reduction and ﬁre load
factors for steel columns exposed to ﬁre.” Proceedings, International
Colloquium: Stability and Ductility of Steel Structures, Rio de J aneiro, Brazil.

(9) Iqbal, S., and Harichandran, RS. (2010). “Reliability-Based capacity
reduction and ﬁre load factors for design of steel members exposed to ﬁre.”
Proceedings, 6th International Conference on Structures in Fire, East
Lansing, United States.

216

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