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This is to certify that the
thesis entitled

ENERGY BASED EQUIVALENT APPROACH FOR
EVALUATING FIRE RESISTANCE OF REINFORCED
CONCRETE BEAMS

presented by

PURUSHOTHAM PAKALA

has been accepted towards fulfillment
of the requirements for the

MS. degree in CIVIL ENGINEERING

 

 

MW

Mafior Professor’s Signature

9th December 2009

 

Date

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ENERGY BASED EQUIVALENT APPROACH FOR EVALUATING FIRE
RESISTANCE OF REINFORCED CONCRETE BEAMS

By

Purushotham Pakala

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE
Civil Engineering

2009

ABSTRACT

ENERGY BASED EQUIVALENT APPROACH FOR EVALUATING FIRE
RESISTANCE OF REINFORCED CONCRETE BEAMS

By

Purushotham Pakala

Provisions of fire resistance is one of the key considerations in building design since fire
represents most severe hazard experienced by built infrastructure during their life time.
Current fire resistance provisions do not account for critical factors in evaluating fire
resistance of reinforced concrete (RC) members and hence are not applicable for realistic
fire performance assessment of RC members. Rational design approaches for evaluating
realistic fire resistance of RC members can be achieved through the use of time
equivalency. Existing time equivalency methods have a number of limitations and are
derived for protected steel members and may not be applicable for RC members.

This thesis presents the development of an energy based approach for evaluating time
equivalent of RC members under realistic fire scenarios. To generate large amount of
data required for development of energy based method, a macroscopic finite element
model has been extended to cover beams with T and I cross-sections. The validated
numerical model is used to undertake a set of parametric studies on RC beams. Data from
the parametric studies is used for validating the proposed energy based approach.
Regression analysis has been applied to the data set and a correlation between the ratio of
time equivalent predicted by the finite element model and equal energy method has been
established. Based on the results, it is shown that the proposed energy based approach

provides a better estimate of time equivalent for RC beams than current approaches.

Dedicated to
my family and friends
who have always been with me

during various phases of my life.

iii

ACKNOWLEDGEMENTS

I would like to thank a number of people who have contributed to the successful
completion of this thesis. Many thanks to Dr. Venkatesh Kodur, Professor of Civil
Engineering, Michigan State University, for giving me the wonderful opportunity of
doing thesis under his guidance and for his constant encouragement and support. I would
like to thank Dr. Ronald Harichandran and Dr. Neeraj Buch for joining my Master’s
Thesis committee.

I am highly grateful to Dr. Monther Dwaikat for his help and constant guidance
throughtout the course of this study without which this thesis would not have been
completed.

The success of this study was due to the support of faculty, staff and students at Michigan
State University. My special thanks to Laura Taylor, Margaret Conner and Mary Mroz
for all the help they provided.

I would like to thank Aqeel Ahmed, Mahmud Dwaikat , Nickolas Hattinger, Nikhil Raut,
Rustin Fike for the valuable advice, guidance and troubleshooting they provided in
developing the numerical model. Thanks to my co-students Kavita Kamat, Kishore
Balsubramaniam, Lensir Gu, Megan Vivian, Nikhil Choudhary, Sonali Kand, Wasim
Khaliq who have contributed directly or indirectly to this thesis.

My extended thanks to Indumathy Jayamani, Janardhan Madala, Kashinath Telsang and
Siva Rama Krishna Chalasani for making my stay at Michigan State University fun and

interesting.

iv

Finally I would like to thank my family members for my academic success. Without their
moral support and sacrifices, it would not have been possible for me to pursue this

graduate degree.

TABLE OF CONTENTS

 

 

 

 

 

LIST OF TABLES _- _ ...... ix
LIST OF FIGURES - - - .................... - xi
1. INTRODUCTION - - -- 1
1.1 GENERAL ................................................................................................................... 1
1.2 FIRE RESISTANCE OF STRUCTURAL MEMBERS ............................................................. 2
1.3 FIRE PERFORMANCE OF RC BEAMS UNDER DESIGN FIRES ........................................... 5
1.4 OBJECTIVES ............................................................................................................... 6
1.5 OUTLINE OF THESIS .................................................................................................... 7

2. LITERATURE REVIEW - -- - -- - - - 11
2.1 GENERAL ................................................................................................................. 1 l
2.2 RESPONSE OF RC BEAMS UNDER FIRE ...................................................................... 12
2.3 STATE-OF-THE-ART REVIEW — FIRE RESISTANCE OF RC BEAMS ............................... 14
2.3.1 Experimental studies ........................................................................................ 14

2. 3.2 Analytical studies ............................................................................................. 18
2.3.3 Code provisions ............................................................................................... 22
2.3.4 Summary ................................................. 23

2.4 TIME EQUIVALENT METHODS ................................................................................... 24

2. 4. 1 Equal area method ........................................................................................... 25
2.4.2 Maximum temperature method ........................................................................ 26

2. 4. 3 Minimum load capacity method ....................................................................... 27

2. 4. 4 Maximum deflection method ............................................................................ 28
2.4.5 Empirical formulae .......................................................................................... 29
2.4.5.1 CIB formula .............................................................................................. 30
2.4.5.2 Law formula: ............................................................................................. 31

2.4.5.3 Eurocode formula: .................................................................................... 31

2.5 COMPARISON OF VARIOUS TIME EQUIVALENT METHODS .......................................... 33
2.6 SUMMARY ................................................................................................................ 34

3. NUMERICAL MODEL 44
3.1 GENERAL ................................................................................................................. 44
3.2 MACROSCOPIC FINITE ELEMENT MODEL ................................................................... 45
3.3 EXTENSION OF MACROSCOPIC FINITE ELEMENT MODEL FOR T/I BEAMS ................... 48

3. 3. 1 Fire temperatures ............................................................................................. 48
3.3.2 Thermal analysis .............................................................................................. 50
3.3.3 Strength analysis .............................................................................................. 55
3.3.3.]. General ..................................................................................................... 55
3.3.3.2 Generation of M-K relationships ............................................................... 61

3.3.3.3 Computation of deflection ........................................................................ 62

3.4 COMPUTER PROGRAM .............................................................................................. 63
3.4.1 General ............................................................................................................ 63

Vi

3.4.2 Beam discretization .......................................................................................... 63

 

 

 

3.4.3 Material properties .......................................................................................... 63

3. 4. 4 Program input .................................................................................................. 64
3.4.5 Program output ................................................................................................ 64
3.5 VALIDATION ............................................................................................................ 65
3.5.1 Rectangular beam B1 ....................................................................................... 65
3.5.2 Tbeam B2 ........................................................................................................ 67
3.5.3 1 beam B3 ......................................................................................................... 70
3.6 SUMMARY ................................................................................................................ 72
4. PARAMETRIC STUDIES--- . - - -- - - - - - 90
4.1 GENERAL ................................................................................................................. 90
4.2 CRITICAL FACTORS INFLUENCING FIRE RESISTANCE ................................................. 91
4.3 NUMERICAL STUDIES ON T AND I CROSS-SECTIONS .................................................. 93
4.3.1 Beam characteristics ........................................................................................ 93
4.3.2 Analysis variables ............................................................................................ 94
4.4 RESULTS OF PARAMETRIC STUDIES ........................................................................... 95
4.4.1 Effect of load ratio ........................................................................................... 96

4. 4. 2 Effect of aggregate type ................................................................................... 96
4.4.3 Effect offire scenario ............... 97

4. 4. 4 Effect of concrete strength ............................................................................... 98
4.4.5 Effect of failure criteria ................................................................................... 98
4.5 SUMMARY ................................................................................................................ 99
5. ENERGY BASED EQUIVALENT APPROACH - -- -- 111
5.1 GENERAL ............................................................................................................... 111
5.2 ENERGY BASED TIME EQUIVALENT APPROACH ....................................................... 112
5.2.1 General approach .......................................................................................... 112
5.2.2 Evaluating fire energy transferred to beam ................................................... 113

5. 2. 3 Computation of time equivalent ..................................................................... 116
5.3 NUMERICAL STUDIES ............................................................................................. 1 17
5.3.1 Design parameters ......................................................................................... 117
5.3.2 Fire exposure scenarios ................................................................................. 118

5. 3. 3 Determining time equivalent .......................................................................... 119

5. 3. 4 Evaluating time equivalent fiom current methods ......................................... 120
5.4 CALIBRATION ......................................................................................................... 122
5.5 VALIDATION .......................................................................................................... 124
5.6 NUMERICAL EXAMPLE ............................................................................................ 125
5.7 SUMMARY .............................................................................................................. 126
6. CONCLUSIONS AND RECOMMENDATIONS - - - - 137
6.1 GENERAL ............................................................................................................... 137
6.2 CONCLUSIONS ........................................................................................................ 138
6.3 RECOMMENDATIONS .............................................................................................. 139
APPENDICES“ - - - - - - - - . - - - 140

 

Vii

REFERENCES ........... -- -- -- - ...... - ..... 161

 

Viii

LIST OF TABLES

Table 2.1 - ACI minimum width and cover thickness requirements for achieving fire
resistance in RC beams [AC1216.1 2007] ............................................................ 35

Table 2.2 - Eurocode 2 Specifications for fire resistance rating of Simply supported RC
beams [Eurocode 2 2004] ..................................................................................... 35

Table 2.3 - Values of kb and kc for computing time equivalency using CIB and Eurocode
empirical formulae ................................................................................................ 35

Table 2.4 - Time equivalent (in minutes) of an RC beam as obtained from existing

methods ................................................................................................................. 36
Table 3.1 - Summary of properties of Beam Bl ............................................................... 73
Table 3.2 - Summary of fire resistance values predicted for the analyzed beams using

various failure criterion ......................................................................................... 73
Table 3.3 - Summary of various properties used in the analysis of Beam B2 .................. 74
Table 3.4 - Summary of various properties used in the analysis of Beam B3 .................. 74
Table 4.1 — Summary of fire resistance values for the rectangular beams ..................... 100

Table 4.2 — Properties for concrete cross-sections used in the analysis of rectangular

beams .................................................................................................................. 102
Table 4.3 — Compartment characteristics used for developing different design fire
scenarios .............................................................................................................. 102
Table 4.4 — Summary of fire resistance values for the beams analyzed ......................... 103

Table 5.1 — Compartment characteristics used for arriving at different design fire

scenarios .............................................................................................................. 128
Table 5.2 — Summary of computed time equivalent values from various methods ........ 129
Table 5.3 — Cross-sectional details and properties of beams used in the analysis .......... 130

Table 8.1 — Constitutive relationships for high temperature properties of concrete ...... 143

Table 3.2 - Values for the main parameters of the stress—strain relationships of normal
strength concrete at elevated temperatures [Eurocode 2] ................................... 147

Table 8.3 — Constitutive relationships for high temperature properties Of reinforcing steel
............................................................................................................................. 148

ix

Table 8.4 — Values for the main parameters of the stress-strain relationships of
reinforcing steel at elevated temperatures [Eurocode 2] ..................................... 150

Table D.1 — Step-by-step calculations for evaluating time equivalent of fire scenariOS
(F SS) by equal energy method ............................................................................ 158

LIST OF FIGURES

Figure 1.1 - Time-temperature curves of standard and representative design fires ............ 9

Figure 1.2 - Comparison of loading and boundary conditions of structural members
encountered in standard fire testing and an actual building .................................. 10

Figure 2.1 - Variation of strength capacity of a typical RC beam under fire conditions. 37
Figure 2.2 - Layout of beams tested by Lin and Ellingwood [1987] ................................ 38
Figure 2.3 - Typical layout of beam tested by Dwaikat [2009] ........................................ 39

Figure 2.4 - Layout of RC beams used by Kodur and Dwaikat[2008b] in parametric
studies ................................................................................................................... 40

Figure 2.5 - Illustration of equivalent fire severity calculation using equal area method 41

Figure 2.6 - Illustration of equivalent fire severity computation using maximum
temperature method .............................................................................................. 41

Figure 2.7 - Illustration of equivalent fire severity calculation using minimum load

capacity method .................................................................................................... 42
Figure 2.8 - Illustration of equivalent fire severity calculation using maximum deflection
method ................................................................................................................... 43
Figure 2.9 — Representative design fire scenario used in comparison Of various time
equivalent methods ............................................................................................... 43
Figure 3.1 - Layout of typical RC beam and its idealization for analysis ........................ 75
Figure 3.2 — Typical moment-curvature relationships for an RC beam ........................... 76

Figure 3.3 — Flow chart showing the numerical procedure associated with analysis of RC
beam exposed to fire ............................................................................................. 77

Figure 3.4 — Cross-sectional discretization of T-beam for analysis .................................. 79

Figure 3.5 -— Variation of strain, stress and internal forces in a typical beam cross-section

exposed to fire ....................................................................................................... 80
Figure 3.6 -— Step-by-step approach for evaluating the mid-Span deflection of the beam
using M-Ic relationship .......................................................................................... 81
Figure 3.7 — Elevation and cross section of RC beam Bl ................................................ 82

xi

Figure 3.8 — Comparison of measured and predicted average rebar and concrete
temperature as a function of time for beam Bl ..................................................... 83

Figure 3.9 — Comparison of measured and predicted mid-span deflection as a function of

time for beam Bl .................................................................................................. 83
Figure 3.10 — Elevation and cross section of RC Beam B2 .............................................. 84
Figure 3.11 — Variation of rebar temperature for Beam B2 as obtained from proposed
model and SAFIR ................................................................................................. 85
Figure 3.12 — Variation of temperature at three different concrete locations in beam BZ
predicted by the proposed model and SAP IR ....................................................... 85
Figure 3.13 — Variation of mid-span deflection for Beam B2 as predicted by the proposed
model and SAFIR ................................................................................................. 86
Figure 3.14 — Representative stress-strain curves of concrete with ultimate strain and
strain corresponding to peak stress value .............................................................. 86
Figure 3.15 — Elevation and cross section of RC I beam B3 ............................................ 87
Figure 3.16 — Variation of rebar temperature for Beam B3 as obtained from proposed
model and SAFIR ................................................................................................. 88
Figure 3.17 — Variation of temperature at three different concrete locations in Beam B3
predicted by the proposed model and SAP IR ....................................................... 88
Figure 3.18 — Variation of mid-Span deflection for beam B3 (1 beam) as predicted by the
proposed model and SAF IR .................................................................................. 89
Figure 4.1 — Elevation and cross-sectional details of RC beams used in parametric studies
............................................................................................................................. 104
Figure 4.2 - Time-temperature curves for design and standard fire exposures used in the
analysis ................................................................................................................ 105
Figure 4.3 — Cross-section and segmental discretization of the analyzed beams ........... 106

Figure 4.4 — Influence of load ratio on the mid-span deflection of simply supported T-
beam exposed to fire ........................................................................................... 107

Figure 4.5 — Influence of load ratio on the mid-Span deflection of Simply supported I-
beam exposed to fire ........................................................................................... 107

Figure 4.6 — Effect of aggregate type on the mid-span deflection of Simply supported T-
beam exposed to fire ........................................................................................... 108

xii

Figure 4.7 — Effect of aggregate type on the mid-span deflection of Simply supported I-
bearn exposed to fire ........................................................................................... 108

Figure 4.8 — Effect of fire scenario on the deflection of a simply supported T beam
exposed to six design fires and a standard fire ................................................... 109

Figure 4.9 -— Effect of fire scenario on the deflection of Simply supported I beam exposed
to six design fires and a standard fire .................................................................. 109

Figure 4.10 — Effect of concrete strength on the deflection of Simply supported T beam

exposed to fire ..................................................................................................... 110
Figure 4.11 — Effect of concrete strength on the deflection of simply supported I beam
exposed to fire ..................................................................................................... 110
Figure 5.1 — Equivalent energy concept for standard and design fire ............................ 131
Figure 5.2 — Cross section and elevation of RC beam used in the analysis .................... 132

Figure 5.3 — Time temperature curves for design and ASTM E119 standard fire exposure

............................................................................................................................. 133
Figure 5.4 — Comparison of time equivalent computed based on FE analysis with that of
other methods ................. 133
te(FE)
Figure 5.5 — Variation of — with maximum fire temperature ....................... 134
te(Energy)

Figure 5.6 - Comparison of time equivalent from equal energy method with predictions
from other methods for a rectangular beam ........................................................ 134

Figure 5.7 — Comparison of time equivalent from equal energy method with predictions
from other methods for a T beam ....................................................................... 135

Figure 5.8 — Comparison of time equivalent from equal energy method with predictions
from other methods for an I beam ....................................................................... 135

Figure 5.9 — Fire scenarios used in the analysis of RC beams ........................................ 136

Figure 5.10 — Comparison of time equivalents from the proposed equation and the finite
element analysis .................................................................................................. 136

Figure C.1 -— Cross-section, Elevation, Bending moment diagram and Shear force
diagram from T and I beams ............................................................................... 154

Figure D.1 - Illustration of computation of area under heat flux curves using Trapezoidal
rule ...................................................................................................................... 160

xiii

CHAPTER 1

1. INTRODUCTION

1.1 General

Reinforced Concrete (RC) structural systems are frequently used in high rise buildings
due to several advantages they provide over other construction materials. Provision of
appropriate fire safety measures is one of the key considerations in buildings since fire
represents one of the severe hazards to which buildings may be subjected to in their
design life time. One of the main advantages of RC construction is its high fire resistance
properties. This leads to achieving required fire resistance ratings without the use of any
external fire protection measures.

The main objective of fire safety provisions is to limit the probability of death, injury,
and property loss, during a fire [Buchanan, 2002]. These fire safety objectives are
achieved by providing active and passive fire protection systems. Active fire protection
systems, such as Sprinklers, are those which become functional in the event of fire and

control the fire through an external device. Passive fire protection systems, such as fire

resistance, are inherent within the structural system and do not require any device to get
activated. Thus, active and passive fire protective systems can be used independently or
jointly to achieve fire safety objectives in buildings. The most important component of
passive fire protection is fire resistance which is defined as the duration during which a
structural element exhibits resistance with respect to structural integrity, stability, and
temperature transmission, when exposed to fire. Fire resistance depends on type of fire
exposure, structural member type, applied load and characteristics of the constituent

materials of the structural member.

1.2 Fire resistance of structural members

Fire resistance provisions for RC members in codes and standards are prescriptive in
nature and are derived based on results from standard fire resistance tests or through
empirical calculation methods. Standard fire resistance tests are conducted on structural
components such as beams, walls, floors or columns according to national standards
[ASTM E119a or ISO 834]. The standards require the test specimens be constructed in a
Similar manner as in real building. Some standards such as ASTM E119 specify the
dimensions of the Specimen to be tested along with the size of the fumace being used for
the standard fire test. The test specimens are often loaded with service loads (generally
50% of the room temperature capacity) and are subjected to standard fire exposure which
follows a predefined time-temperature relationship as shown in Figure 1.1. The test is
continued until a prescribed failure criterion is met. The time to reach failure point is

termed as the fire resistance of the member. This time is rounded off to nearest half hour

(30 or 90 minutes) or hour ( 1, 2, 3 or 4) and this value represents the fire resistance
rating.

The fire resistance of structural component depends on a number of factors including
type of fire scenario, applied load, material properties and structural configuration
(restraint). Figure 1.1 also shows time-temperature curves of two representative design
fires. Design fire (Fire 1) is Short hot fire whereas the other design fire (Fire 2) is
moderate fire. It can be seen from Figure 1.1 that the temperature in standard fire
increases continuously without a decay (cooling) phase, whereas in design fires there is a
well defined decay phase. The presence of decay phase in design fires Significantly
influences the fire response of structural members. Though standard fire resistance tests
are very helpful in assessing the comparative performance of structural members, they do
not account for important factors such as realistic fire scenario, loading and failure
criterion. Also, the standard fire resistance tests are often carried out on individual
elements without any consideration to structural interactions and restraint conditions at
supports. These interactions provide considerable amount Of redistribution of moments
that will have significant influence on the fire resistance of RC members as Shown in a
number Of studies [Dwaikat 2009]. Figure 1.2 illustrates the loading and support
conditions encountered by structural members during standard fire resistance test and in a
real building. The presence of end restrains, member continuity, and structural members’
interaction at joints in a building will modify the fire response of RC members compared
to that under a standard fire resistance test.

Fire response of RC members is highly influenced by the material properties of

constituent materials, namely concrete and reinforcing steel. The high temperature

material properties that are important are thermal, mechanical and deformation
properties. Thermal properties influence the amount of heat transferred to a structural
member whereas mechanical properties determine the extent of loss in strength and
stiffness of the member. Deformation properties influence the extent of deformations in
the structural components. All these properties change with increasing temperature and
also depend on the characteristics of concrete and steel.

Structural configuration refers to geometric and support conditions of the structural
member. The support conditions define the level of restraint at the ends which in turn
influences the fire resistance of RC beams. Recent studies showed that axial restraint can
improve the fire resistance of RC beam by arch action. However, the axial restraint under
large deflections in the span may lead to buckling and premature failure of the beam
[Dwaikat 2009].

Designing a structural component through prescriptive based methods involves adherence
to predetermined requirements specified in codes and does not give the designer the
freedom to implement innovative and cost-effective engineering approaches to fire
safety. Also prescriptive based approaches provide same level of fire safety to RC
members irrespective of the fire severity, magnitude of loading, compartment
characteristics and member dimensions. In addition, these ratings are derived based on
standard fire exposure without due consideration to important factors such as fire
exposure, load level and support conditions. Thus the current provisions in codes and
standards may not provide a realistic assessment of the fire performance of RC beams.

In contrast to prescriptive based codes, performance based fire design comprises of

evaluating performance of structural elements under realistic fire, loading and restraint

scenarios. In a performance based evaluation, the structural components are designed
taking into account realistic fire exposure, support conditions, loading and failure
criterion. Performance based approach gives the designers the opportunity to use
innovative strategies for design of structural components, provided that equivalent fire
safety can be demonstrated. Generally performance based design requires detailed
calculations and numerical simulations.

Of the various factors, fire scenario has the greatest influence on fire resistance and is not
properly accounted for in prescriptive based approaches. In order to evaluate or design a
structural component for fire resistance, the worst case fire scenario that the structural
component might be subjected to during its design life time should be known. In other
words, the designer Should be aware of the destructive potential of a fire to which the
structure might be subjected to. The destructive potential of fire iS defined as the fire
severity and it depends on a number of factors such as the fuel load (occupancy type),
compartment Size and location of openings, type of lining materials and area of
compartment. Knowing the fire severity, a designer could design the structure by making
sure that the structural fire resistance is greater than the severity of fire. In order to ensure
that the above criteria are met for performance based design, severity of realistic fire
exposure should be related to that of standard fire exposure. This correspondence is often

established by a simplified approach termed as equivalent fire severity.

1.3 Fire performance of RC beams under design fires
AS illustrated above, fire severity is one of the main factors that influence the

performance of a beam under fire. If the fire severity under design fire is linked to the

performance of an RC beam under a standard fire scenario then an equivalency can be
established. Therefore, a designer can evaluate the performance of RC beam exposed to a
design fire using the currently available data from standard fire tests, without the need of
conducting fire tests or detailed finite element analysis on the beams under design fire
exposure. The equivalency for beam response under design and standard fire exposures is
generally expressed using the term “time equivalent”, which can be defined as the time of
exposure to standard fire that would result in the same fire severity as that under design
fire exposure. A review of the literature Shows that there are very few methods and
empirical formulae for evaluating time equivalent of RC members [CIB, Eurocode,
Ingberg, Law, Pettersson]. Further, there are a number of drawbacks in these methods
and empirical formulae since they do not account for all the critical factors governing fire
response. Also, these methods are mainly derived for protected steel members and are not
validated over the full range of fire scenarios for RC members. Furthermore, there is
significant variation among various time equivalent approaches. Hence, there is a need
for an appropriate time equivalent method for establishing equivalency between standard

and design fire exposure.

1.4 Objectives

Currently, fire resistance of RC beams is established based on standard fire tests which
have numerous drawbacks. To overcome these limitations, the current Study is focused on
developing a simplified engineering approach for evaluating fire resistance of RC beams

under realistic fire, loading and restraint conditions. Establishing such simplified

approaches require large set of data which can be generated through numerical studies
utilizing finite element based computer models.

Recently Dwaikat and Kodur [2008] have developed a macroscopic finite element based
numerical model for fire resistance analysis of RC beams. However this model is for RC
beams with rectangular sections only, but in real life situations RC beams with T and I
cross-sections are often used. Hence the current study aims at extending the macroscopic
finite element model,to evaluate the fire performance of RC beams with rectangular, T
and l cross-section. Using the updated numerical model, parametric studies will be
conducted to quantify the influence of various parameters on the fire response of RC
beams. Data generated from parametric studies will be used to develop a time equivalent
approach for evaluating equivalent fire resistance of RC beams under design fire

exposure.

1.5 Outline of thesis

This thesis is divided into six chapters and four appendices. Chapter 2 provides a review
of literature relevant to the behavior of RC beams under fire conditionSPrevious
experimental and analytical studies on fire response of RC beams are reviewed and
currently available time equivalent methods for RC beams are discussed.

Chapter 3 presents the extension of macroscopic FE model for tracing the fire response of
RC beams with T and I cross sections. Predictions from the model are compared with
those obtained from test data and other numerical models to demonstrate the validity of

the model.

Results from a set of parametric studies on the fire response of RC beams are presented in
Chapter 4. Factors influencing the fire resistance of RC beams are discussed.

Chapter 5 presents the development of semi-empirical energy based approach for
evaluating the time equivalent of RC beams under realistic fire, loading and support
conditions. Chapter 6 presents the key conclusions derived from the research and main

recommendations for further research.

Figures

 

 

 

 

 

 

 

 

1400 ,
1200§ _
610004: ,
9.4 3 . +Standardfire
o , .
:5: 300 i / ‘ “ +Design firel
“ ’ +D ' f 2
1.. , eSIgn Ire
8. 600 1 I -
E .
“ I
F" 400 ,
200 I
0....,....,....,....T..f......
0 50 100 150 200 250 300

Time (minutes)

Figure 1.1 - Time-temperature curves of standard and representative design fires

 

 

 

 

 

 

 

 

 

w
IV V V V V V V]
. . _A_ _O_
FIre FIre
Fire
1 i
L
'/////////1 Beam

Colum

(a) Fire resistance evaluation through standard fire test

 

Beam

 

Column /

\ A/

’///////////////////7////////////.

 

 

 

 

 

(b) Actual conditions in a building

Figure 1.2 - Comparison of loading and boundary conditions of structural members
encountered in standard fire testing and an actual building

10

CHAPTER 2

2. LITERATURE REVIEW

2.1 General

Current methods of evaluating fire resistance of RC beams utilize prescriptive based
approaches which are developed using data from standard fire resistance tests. The
standard fire resistance tests are conducted under a standard fire exposure and do not take
into consideration realistic fire scenarios that depend on compartment characteristics such
as fuel load and ventilation properties. Thus the prescriptive based methods for
evaluating fire resistance Of RC beams may not be applicable for evaluating fire
resistance under performance based codes wherein the fire resistance of the member is to
be evaluated under realistic conditions. Evaluation of fire resistance under realistic
conditions requires detailed finite element analysis. However, currently there are only
very limited finite element based computer models for modeling the fire response of RC

members.

11

This chapter presents a brief overview of the behavior of RC beams under fire conditions.
Both experimental and numerical studies that are reported in the literature to characterize
the fire behavior of RC beams are reviewed. In addition, different time equivalent
methodologies for evaluating the equivalent fire resistance of structural members are

reviewed.

2.2 Response of RC beams under fire

The response of RC beams under fire conditions is different from that of room
temperature. This is due to the fact that under fire conditions the loads on the beam
remain the same or may even reduce (due to evacuation of people), but the strength and
stiffness of the beam deteriorates with fire exposure time. The temperatures in the beam
increase with time due to heat transfer from the fire through the exposed surface of the
beam. The increasing temperatures results in loss of strength and stiffness of the
constituent materials (namely concrete and steel) which leads to reduction in the moment
(capacity) as well as increase in the deflection of the beam. This process will continue
until the capacity of the beam deteriorates to the level of applied moment (load), at which
point the beam is considered to have failed. The duration at which the capacity of beam
equals the applied load is termed as the fire resistance of the beam.

Behavior of an RC beam subject to a standard fire exposure is illustrated in Figure 2.1. It
can be seen from the figure the capacity of the beam decreases with time under increasing
fire temperatures. The decrease in the capacity of the beam is not significant in the initial
stages of fire exposure but as the time progresses the capacity of the beam decreases

gradually due to increase in fire temperature. The degradation in the capacity of the beam

12

continues until it reaches a value where it can no longer sustain the applied moment
(load) resulting in failure of the beam. The fire exposure 'time at which the failure of the
beam occurs is defined as the fire resistance of the beam.

The fire resistance of RC beams is mostly evaluated through full-scale fire-resistance
tests or prescriptive based methodologies. In the fire test, the building element is
subjected to a standard fire exposure specified in ASTM E119a [2008] or 180834 [1975]
and the fire exposure iS continued until failure of the element or required fire resistance
rating is attained. The failure of a structural element in standard fire resistance test iS
based on stability, integrity and insulation failure criterion. To meet the stability failure
criterion, the structural element Should be able to perform its load-bearing capacity for
the duration of the test without collapse. Integrity failure criterion requires that the
structural element does not develop any cracks/openings that allow the fire/smoke to pass
through it. The insulation failure criterion is said to be met when the average increase in
unexposed Side temperature of the assembly is below 140°C and a maximum temperature
increase does not exceed 180°C at a single point.

In lieu of standard fire tests or complex set of calculations, codes and standard provisions
often relate fire resistance of RC members to minimum cross-section dimensions and the
concrete cover to the reinforcement. These prescriptive based approaches, derived based
on standard fire resistance tests have numerous drawbacks and do not lead to realistic

assessment of fire resistance.

13

2.3 State-of-the-art review - Fire resistance of RC beams

A review of literature shows that there has been limited number of experimental and
numerical studies on the behavior of RC flexural members (beams) exposed to fire. An
overview of previous experimental and numerical (analytical) fire resistance studies on

RC beams iS presented here.

2.3.1 Experimental studies

Experimental studies aimed at characterizing the behavior of RC beams under fire have
been undertaken by Dotreppe and Franssen [1985], Lin et al. [1981], Lin and Ellingwood
[1987], Shi et a1. [2004] and Dwaikat [2009]. Dotreppe and Franssen [1985] tested an RC
beam Of rectangular cross-section under ISO834 standard fire exposure to evaluate its fire
resistance rating. The beam had a Span length of 6.5 m with 200 mm x 600 mm cross-
sectional dimensions. The beam was reinforced with 2012 mm bars as compression
reinforcement and 3022 mm bars as tension reinforcement. The beam was made up of
siliceous aggregate concrete having a compressive strength of 15 MPa. Two concentrated
loads were applied on the beam, each at a distance of 1.625 m from the support. The
study concluded that the presence of steep fire induced thermal gradients in the cross-
section produces large deflections in the beam even in early stages of fire exposure.

Lin et a1. [1981] tested eleven rectangular RC beams subjected to ASTM E119 standard
fire exposure. All the test specimens had a cross-sectional dimension of 305 mm x 355
mm with a total length of 9.76 m.The beams were reinforced with #6 (019 mm) and #8
(025 mm) bars of ASTM A615 designation. The measured average yield strength of #6

bars was 435.8 MPa while that of #8 bars was 434.4 MPa. Normal weight concrete was

14

used for fabrication of ten beams while sanded lightweight concrete was used to fabricate
Single beam. The compressive strength of concrete at 28 days ranged between 27-33
MPa. The study investigated the effect of aggregate type, beam continuity and moment
redistribution on the fire behavior of RC beams. Lin et al. [1981] observed that the loads
required on overhang of the beam to accomplish the continuity effect in beams increased
during the first 15 minutes of fire test, reached maximum value after 30 to 45 minutes
into the test and then remained constant for the remaining duration. Also mid-span
deflection of specimens tested as continuous beams increased at a constant rate for the
first two-third duration of fire test and then increased Sharply for the remaining duration.
The study concluded that continuous beams as compared to Simply supported beams
undergo Significant redistribution of moments during fire exposure and this helps in
enhancing their fire resistance.

Lin and Ellingwood [1987] tested Six full-scale concrete beams to study the behavior of
RC beams exposed to fire. Five of the Six test Specimens had a cross-sectional dimension
of 533 mm x 229 mm, while the Sixth one was 610 mm deep and 254 mm wide. All the
six beams were 8.2 m in total span with a fire exposed span of 6.1 m and an unexposed
cantilever loaded in order to provide continuity over one support. Cross-sectional details
as well as the elevation of typical beams are shown in Figure 2.2. All the six beams were
fabricated with normal-weight concrete having a compressive strength of 2.76 MPa. Four
beams were exposed to ASTM E119 standard fire, while the other two beams were
subjected to short duration high-intensity (SDHI) design fire. During the test furnace
atmosphere, concrete and steel temperatures, deformations in beam such as deflection,

expansion were measured for each beam. Though shear cracks developed during the early

15

stages of fire tests, all the beams failed by flexure rather than by Shear. The fire resistance
of the beams(failure times) ranged from 3 hr 26 min to 4 hr 8 min. Major conclusion of
this study is that the shear failure in RC beams is unlikely under fire exposure conditions.
Other notable conclusion was that the temperature history in the reinforcement iS the
most important factor that affects fire response of RC beams.

Shi et a1. [2004] tested Six RC rectangular beams with varying concrete cover thickness
by exposing them to time-temperature curve in a specially designed electric furnace. All
the six beams were of 1.3 m long with a cross-section width Of 100 mm. The section
depth of the cross-section varied between 180 mm to 200 mm. The beams were
reinforced with 2010 mm bars in compression and 2010 mm bars in tension. The test
Specimens were made with crushed limestone aggregate concrete having 28 day cube
strength of 39 MPa and rebars having yield strength of 270 MPa. The main objective of
the study was to study the effect of bottom and lateral concrete cover thickness on the fire
response of RC beams. All beams were Simply supported at both ends. One of the beams
was tested in order to Obtain the ultimate moment capacity at room temperature while
another beam was heated without any applied load. Remaining four beams were
subjected to load level of about 50% of ultimate capacity at room temperature in the form
of two concentrated loads.

Shi et al. [2004] observed that the temperature distribution across the section depth is
nonuniform and the gradients change continuously with an increase in the furnace
temperature. The authors also Observed that the temperature distribution in the top zone is
similar for all test Specimens irrespective of the amount of bottom concrete cover

thickness and whether the specimen is loaded first before subjecting to heating. They

l6

concluded that an increase in bottom concrete cover thickness up to a certain threshold
increases the fire resistance of RC beams. Further increase in bottom concrete cover
thickness, beyond the threshold value, does not improve the behavior (fire resistance) due
to the widening of concrete cracks in the tension zone of concrete. The study also found
that any increase in lateral concrete cover thickness beyond that of bottom concrete cover
thickness, has negligible effect on the fire resistance of RC beams.

Dwaikat [2009] tested six RC beams to study the influence of critical factors such as
concrete strength, load level, fire scenario, fire induced spalling, and axial restraint on the
fire resistance of RC beams. Two of the beams were tested under ASTM E119 standard
fire exposure [ASTM E119a 2008] while the remaining four beams were tested under
typical design fire scenarios. Typical layout of tested beams is illustrated in Figure 2.3
along with the cross-sectional dimensions and material properties. Two of the six beams
were fabricated with normal strength concrete (NSC), while the other four beams were
fabricated with high strength concrete (HSC). Coarse aggregate used in both type of
concretes were of carbonate aggregate. The average compressive cylinder strength at 28
days for NSC and HSC beams was 52.2 MPa and 93.3 MPa respectively. These tests
were carried out to investigate the effect of variables such as concrete strength, axial
restraint, type of fire exposure and load level on the fire behavior of RC beams.
Dwaikat[2009] observed that the measured temperatures in HSC and NSC beams were
close to each other. Measured mid-span deflection and spalling of HSC beam was higher
than that of NSC beams. In addition, the axial restraint force in axially restrained beams
increased with fire exposure time. Major conclusions from the study include that the fire

resistance of HSC beams is lower than that of NSC beams due to faster degradation of

17

concrete strength and fire induced spalling in HSC. Fire resistance of RC beams exposed
to design fire was higher as compared to standard fire exposure, due to the presence of
the decay phase which leads to recovery of strength and stiffness.

The above discussed experimental studies were aimed at characterizing the behavior of
RC beams under fire conditions. Effect of various parameters such as continuity, moment
redistribution, concrete cover, concrete strength, load ration, fire scenario, fire induced
spalling and axial restraint force on behavior of RC beams was studied. In most cases
standard fire exposure was used and the results Obtained from such tests might not
represent actual performance under realistic fire conditions. Further, most of the fire
resistance experiments were conducted on beams with rectangular crOSS section, but T

and I cross sections were not considered.

2.3.2 Analytical studies

A review of literature indicates that a number of numerical studies were carried out by
researchers ( Dotreppe and Franssen [1985], Ellingwood and Lin [1991], Poh et al.
[1995], Kang and Hong [2004], and Kodur and Dwaikat [2008, 2008b] ) on the fire
behavior of RC beams. Dotreppe and Franssen [1985] applied a finite element approach
for the fire resistance analysis of RC beams. In this approach, the structure is divided into
beam elements and sectional approach is used to evaluate strength at various time steps.
As the study of mechanical behavior of concrete at high temperatures is complicated, the
model employed a step-by-step analysis taking into account material and geometrical
non-linearities. The model used Newton-Raphson procedure to perform iterative

nonlinear structural analysis.

18

Ellingwood and Lin [1991] developed mathematical models for predicting thermal and
structural response of RC beams subjected to standard and design fire exposures. The
model utilizes sectional analysis approach for evaluating fire resistance of RC beams and
accounts for change in thermal and mechanical properties of constituent materials, high
temperature creep and shrinkage. Based on the results, the authors concluded that the
most important factor that affects the behavior of RC beams is the temperature history in
the reinforcement. Also the authors indicated that compressive strength and stiffness of
concrete have less influence on fire resistance of RC beam.

Poh et al. [1995] developed a general numerical model to calculate the nonlinear
behavior of load-bearing members (beams and coltunns) exposed to fire. According to
this model, the structural member is discretized into a series of segments of appropriate
lengths. Each segment is assumed to be represented by a cross section at its mid-length.
The cross section is further divided into a number of small elements (subareas). The
method uses a three tiered approach involving the elemental analysis, cross-section
analysis and the overall global analysis of the structure. The structural behavior of each
tier is characterized by appropriate action (internal or external) and deformations
(including strains and curvatures) associated with the element. The governing equations
are expressed in a succinct matrix format in terms of action-deformation relationships.
These governing equations are solved by an iterative technique using two separate
iterative loops namely, first loop which determines the behavior of the cross section and
the other determines the behavior of the member. The first loop takes into account
material nonlinearity while the second loop iterates on geometric nonlinearity. The

material nonlinearity in the first loop is treated as a combination of linear component and

19

a correction component that accounts for material nonlinearity. Thus, the behavior of
cross section is determined by a linear equation, and it is progressively corrected for
material nonlinearity effects to be used in subsequent iterations. The method takes into
account various factors such as axial forces, biaxial bending, restraints, material and
geometric nonlinearity, unloading and reloading, residual or initial stresses, and initial
out-of-straightness of the member. The numerical model is validated by comparing the
results Obtained from the model with those obtained from a series of restrained and
unrestrained columns tested in laboratory by Poh and Bennetts [1995].

Kang and Hong [2004] proposed an analytical method for the thermal behavior of RC
beams under fire conditions. The analysis was performed in two different levels; cross-
sectional analysis and member solution. Sectional analysis is performed by the use of
segmentation scheme which divides the length of the beam into segments of appropriate
length. Mechanical changes at segmented section such as strain changes due to increase
in temperature are integrated into the member behavior in the member solution. The
model is based on the analytical formulation suggested by Poh et al. [1995] which uses
action-deformation relationships, and numerical method proposed by Lie et al. [1993].
The method accounts for material deterioration, material nonlinearity and nonlinear strain
changes of concrete with increasing temperature.

Apart from the above mentioned numerical models, Special purpose finite element based
computer programs such as SAFIR, developed at University of Liege in Belgium
[Franssen et al. 2004] , can be applied to evaluate the performance of RC beams
subjected to fire. A separate or combined (coupled or uncoupled) thermal and structural

analysis can be performed in SAFIR to model the behavior of RC beams exposed to fire.

20

Fire resistance analysis of structural members can also be done using various commercial
microscopic finite element based packages such as ANSYS, ABAQUS. However the
analysis involves significant amount of complexity and effort. Results obtained from such
programs are difficult to interpret and such complex level Of analysis might not be
necessary for the analysis of conventional RC beams. These factors limit the use of such
microscopic finite element based models in practical situations.

To overcome the complexities of microscopic finite element models, Kodur and Dwaikat
[2008] developed a macroscopic finite element based numerical model for tracing the
response of RC members (beams and columns) exposed to fire. The model uses moment-
curvature relationships to predict the behavior Of an RC beam in the entire range of
loading up to collapse under fire. The beam is divided into number of longitudinal
segments and the mid-section of each segment is assumed to represent the behavior of the
whole segment. The fire resistance analysis is carried out at various time steps, till failure
occurs in the beam. The model accounts for creep and transient strain components, high
temperature material properties, geometric and material nonlinearity and material
softening. Based on the detailed validation the authors concluded that the model is
capable of undertaking fire resistance analysis of RC beams for any value Of Span length,
aggregate type, fire scenario, loading and sectional dimensions.

Kodur and Dwaikat [2008b] applied above developed finite element model to study the
effect of parameters such as fire scenario, load level, concrete cover thickness, failure
criteria, aggregate type and span length on the flexural response of RC beams exposed to
fire. Sectional dimensions and properties of beam are illustrated in Figure 2.4. Based on

these parametric studies, Kodur and Dwaikat [2008b] concluded that the type of failure

21

criterion, load level, fire scenario, concrete cover thickness and aggregate type have
Significant influence on fire resistance of RC beams while the Span length has a minor

effect.

2.3.3 Code provisions

Provisions for evaluating fire resistance of RC beams are generally specified in building
codes and standards. These provisions, derived based on results of standard fire resistance
tests, are generally related to factors such as minimum member dimensions and concrete
cover thickness.

ACI 216.1 standard [2007] gives specifications for fire design of concrete and masonry
structures. The standard Specifies minimum beam width and concrete cover thickness to
achieve a required fire resistance rating in RC beam as presented in Table 2.1. These
requirements differ for restrained and unrestrained support conditions, but the definition
of support conditions is not clearly addressed in the code. Hence, it is unclear whether
restrained case refers to rotational restraint, or axial restraint or both rotational and axial
restraints. Canadian provisions for fire resistance design, which are available in NBCC
[2005], are Similar to that of ACI 216.1.

Eurocode 2 [2004] provides tabulated data for evaluating fire resistance of RC beams.
The required fire resistance of RC beams is related to minimum width and nominal axis
distance, which is defined as the distance measured from the center of main reinforcing
bar to the fire exposed surface of the member. Table 2.2 shows the minimum dimensions

and axis distances, as specified in Eurocode 2, required for achieving fire resistance

22

rating in simply supported beams. These tables also provide possible combinations of
minimum width and axis distance for achieving a required fire resistance rating.

In addition, Eurocode 2 [2004] also provides Simplified calculation method, advanced
calculation methods and tabulated data for determining fire resistance of RC beams.
Simplified calculation method is based on cross-section analysis. Advanced calculation
methods provide a realistic fire resistance assessment of a structure. These methods
generally involve a detailed thermal and structural analysis model and require the use of
advanced computer packages.

The above review of fire resistance provisions for RC beams in standards and codes
Shows that the fire resistance of a RC beam is mostly related to the cross-section
dimensions and the concrete cover thickness without consideration to important facts
such as fire scenario, material strength, amount of ventilation, loading and failure criteria.
Thus these prescriptive based approaches may not yield realistic performance of beam

under practical design Situations.

2.3.4 Summary

Most of the previous experimental and analytical studies on RC beams were conducted
under standard fire exposure. These standard fire resistance tests though helpful in
assessing comparative performance of structural members, they do not take into account
important factors such as realistic fire scenario, loading, restraint and failure criteria,
which influence the fire behavior of RC beams. Thus, current prescriptive based fire
resistance approaches may not be applicable for evaluating fire resistance under recently

introduced performance based codes.

23

2.4 Time equivalent methods

AS illustrated in Section 2.3, the main drawback in current prescriptive based fire
resistance approaches is that they are derived for standard fire exposure without any
consideration to realistic fire scenario. The fire scenario experienced by a beam varies
from one compartment to other and is a function of fuel load, ventilation and Size of
compartment. Also, the absence of decay phase in the standard fire tests is a serious
misrepresentation of fire scenario and this can have significant influence on fire
resistance of RC beams.

Recently, there is a move towards performance based fire safety design Since this
facilitates realistic and rational assessment of fire safety under actual conditions present
in a building. Such performance based fire safety design can be carried out through the
use of calculation methods. However, undertaking performance based approach can
present design challenges since numerous fire scenarios have to be considered. Further,
tests or detailed finite element analysis are quite expensive and it may not be possible to
undertake fire resistance tests or simulations for all possible fire scenarios. One way of
overcoming such problem is by establishing an equivalency between the severity of a
design fire and a standard fire exposure. Such an approach will facilitate the
establishment of fire resistance information under any possible design fire scenario,
provided the fire resistance is known for a standard fire exposure. Therefore, existing
data from standard fire resistance tests or detailed finite element analysis can be utilized
for evaluating performance under design fire scenarios. In order to ensure that the above
criteria are met for performance based design, the severity of design fires should be

related to standard fire exposures and can be done using equivalent fire severity.

24

Equivalent fire severity, commonly referred to as time equivalent, can be used to gauge
the fire resistance of a structural member by comparing the severity of a design fire to
that of a standard fire exposure. A review of literature indicates that there are a number of
methods and empirical formulae for evaluating the equivalent fire severity. These
methods include equal area method, maximum temperature method, minimum load
capacity method, maximum deflection method and other empirical formulae such as CIB,
Law and Eurocode methods. More details on these methods are presented in the

following sub sections.

2.4.1 Equal area method

The equal area method proposed by Ingberg [1928], is based on the concept that two fires
have equal fire severity if the areas under each of the time-temperature curves are equal,
beyond a certain reference temperature (about 20°C — room temperature). The concept of
equal area method is illustrated in Figure 2.5 where the fire temperatures from standard
and design fire exposures are plotted as a function of time. According to this method, the
area under the design fire (time-temperature) curve is computed first (area B in Figure
2.5). Then the area bound under the standard fire (time-temperature) curve, which is a

function of time, is calculated at various time steps (area A in Figure 2.5). The time

equivalency (te) is defined as the time at which the area under the standard time

temperature curve (area A in Figure 2.5) is equal to that of design fire curve (area B in
Figure 2.5).
Though the equal area concept was one of the earlier advancements in the area Of fire

engineering, it has many technical insufficiencies. This method is purely empirical

25

because the units of areas under comparison are not meaningful [Buchanan 2002]. The
equal area concept can give an incorrect fire resistance assessment in case of a design fire
that has Significant variations from standard time-temperature curve. This is because the
heat transfer from a fire to the structural element is mainly by radiation which iS not
directly proportional to the temperature difference, but it is proportional to the fourth
power of the temperature difference. This method of comparing standard fire and a
realistic fire underestimates the heat transfer in a short hot fire and overestimate the heat
transfer in long cold fire, though both of them have equal areas under time-temperature
curves [Nyman 2002]. Despite these technical inadequacies, equal area concept can be
used to correct the results from a standard fire-resistance test if the time-temperature

curve measured in a furnace is not followed within the limits described in the standard

[ASTM E1 19a 2008].

2.4.2 Maximum temperature method

To overcome some of the deficiencies in equal area concept, Law [1971], Pettersson et al.
[1976] and others developed maximum temperature concept for predicting equivalency
between standard and design fire exposure in protected steel members. According to this
method, equivalent fire severity is the time of exposure to standard fire that would result
in the same maximum temperature in a protected steel member as would occur in a
complete burnout of the fire compartment [Buchanan 2002]. Maximum temperature
concept is illustrated in Figure 2.6, which Shows the time-temperature curves for a design
fire and a standard fire and also the corresponding protected steel temperatures. To

determine the time equivalency, first the peak protected steel temperature resulting from

26

the design fire exposure (temperature at point B in Figure 2.6) iS computed. The second
step is to evaluate the steel temperature resulting from the standard fire exposure at
various time steps. Time equivalency is then defined as the time at which the protected
steel temperature under standard fire exposure (temperature at point A in Figure 2.6) is
equal to the maximum steel temperature computed for the design fire exposure
(temperature at point B in Figure 2.6).

This method is mainly derived for protected steel structural members exposed to fire and
may not be applicable for RC members. In addition, this method may not be accurate if
the maximum temperatures, used for computing the time equivalent, are much greater or
lower than those which would cause failure in a particular building [Buchanan 2002].

In this method, rebar temperatures (instead of protected steel temperatures) can be used to
evaluate time equivalency in the case of RC beams. However, computing rebar
temperatures requires thermal analysis of the beam and Significant computational effort,
which limits the usefulness of this method in practical applications. Further, the method

is not validated for RC beams.

2.4.3 Minimum load capacity method

In lieu of equal area and maximum temperature methods, minimum load carrying
capacity method can be employed to establish equivalency between standard fire and
design fire exposure. The concept of minimum load capacity is illustrated in Figure 2.7
where the variation of load carrying capacity is plotted as a function of time for an RC
beam under both Standard and design fire exposures. It can be seen from the figure that

the capacity of RC beam decreases with fire exposure time due to gradual increase in

27

cross sectional temperature. However, for an RC beam exposed to design (realistic) fire,
the strength decreases initially during the growth phase of fire, followed by an increase as
the temperatures decrease in the decay phase as shown in Figure 2.7.

To calculate time equivalency in this method, the first step is to compute the minimum
load capacity of the RC beam under design fire (load capacity at point B in Figure 2.7).
The second step is to compute the load capacity of the beam under standard fire exposure
as a function of fire exposure time. Time equivalency is the time at which the load
capacity at point A in Figure 2.7 is equal the load capacity at point B in Figure 2.7.

This method utilizes various governing factors, such as fire scenario, beam
characteristics, high temperature material properties, load level and support conditions, to
establish time equivalency. Thus, the method generally results in a better estimate of the
time equivalency as compared to other methods. However this method bases the failure
of an RC beam only on strength capacity and does not account for deflection limit state
which may be the governing failure criterion under some scenarios. In addition, the

method requires detailed finite element analysis which limits its use in practice.

2.4.4 Maximum deflection method

In addition to the above discussed methods, time equivalency can be computed using
maximum deflection method. The concept of computing time equivalency using
maximum deflection method is illustrated in Figure 2.8, where the variation of deflection
is plotted as a function of time for an RC beam subjected to standard and design fire
exposures. To compute the time equivalency, first the maximum deflection of the beam

subjected to design fire exposure is calculated (deflection at point B in Figure 2.8). Next

28

the deflection of the beam under standard fire exposure is computed as a function of fire
exposure time. Time equivalent is defined as the time point when deflection at point A is
equal to that at point B (Figure 2.8).

The maximtun deflection method takes into account the deflection failure criteria, which
makes it more reliable for estimating time equivalency for RC beams. A deflection failure
criterion is important because failure in an RC beam can occur if the integrity cannot be
maintained under large deflections. Large deflections can lead to the development of
wider cracks that leave the reinforcement directly exposed to fire and cause failure of the
beam.

With the exception of the equal area method, all three methods namely maximum
temperature method, minimum load capacity method and maximum deflection method
require detailed finite element analysis and thus can not be easily applied in design

Situations.

2.4.5 Empirical formulae

In addition to above methods, several empirical time equivalent formulae have been
developed for evaluating time equivalency between standard and design fire exposure.
These formulae are mainly derived based on the maximum temperature of protected steel
members exposed to design fires. Widely used time equivalent formulae include the CIB

formula, Law formula and Eurocode formula.

29

2.4.5.1 CIB formula: CIB formula [CIB 1986], derived by Pettersson [1973] is one of
the most widely used time equivalent formula for evaluating fire resistance and
establishes equivalency based on the fuel load (amount of combustible materials present)
and ventilation parameters of the compartment. According to this formula, the equivalent
time of design fire exposure to a standard fire exposure as per ISO 834 [1974] test is
expressed as:

te =kcwef (2.1)
where
te = time equivalency in minutes,

ef = fuel load (Ml/m2 floor area),

kc = parameter to account for different compartment linings,

. . -0.25
ventilation factor ( m ),

Wf

= Af (2.2)

Ami;

A f = floor area of the compartment ( m2 ),

 

Av = total area of the openings in the walls ( m2 ),
. . 2
At = total area of the Internal bounding surfaces of the compartment ( m ) and

Hv = height of the windows (m).

CIB formula is valid only for compartments with vertical openings in walls and is not

valid for compartments having horizontal openings in the roof [Buchanan 2002].

30

2.4.5.2 Law formula: Another expression for calculating the time [equivalency was
developed by Law[197l] based on a series of tests conducted on small scale and larger-

scale compartments. The time equivalent is given by:

 

A
= fef (2.3)
AHm/AVM, — A,)

 

e

where

te = time equivalency in minutes, and

AHC = calorific value of the fuel (MJ /kg).

Similar to CIB formula, Law’s formula is valid only for compartments with vertical
openings in the walls and is not applicable for rooms with horizontal Openings in the roof.
In general, time equivalent predicted by Law formula is Slightly higher than that time

equivalent predicted by CIB formula [Buchanan 2002].

2.4.5.3 Eurocode formula: Eurocode [1994] provides an empirical formula for
calculating the time equivalency and is given by:

te = kbwef (2.4)
where
te = time equivalency in minutes,
kg, = parameter to account for compartment linings,

w = ventilation factor,

0.3 4 '
.4 -
= Q 0.62 + 90(0 av) > 0.5 (2.5)
Hr 1+ bvah

 

31

a, = Av/Af, 0.025 s a, s 0.25 (2.6)

9:
2:-
ll

Ah/Af , (IV S 0.2 (2.7)
2

by =12.5(1 +10 av—av ), (2.8)

H, = ceiling height (m) and

. . . 2
A h = area of horizontal openings In roof ( m ).

Eurocode formula is Similar to CIB formula, however, the compartment lining parameter

(kc) is replaced by kb and the ventilation factor (w) is also altered to take into account the

effect of horizontal Opening that might be present in the roof.

Values for the compartment lining factors kc and kg, for thermal conductivity (k), Specific

heat (cp) and density (p) are shown in Table 2.3. The value of b iS defined as follows

 

b = \/ Thermal Inertia = , / kpc p (2.9)

where
k = thermal conductivity,
p = density and

cp = specific heat

Unlike CIB formula, time equivalency predicted by Eurocode formula is independent of
the height of opening, but depends on the height of compartment ceiling. Both CIB and
Eurocode formulae give identical results for small compartments with tall windows, but
the Eurocode formula gives lower fire severities for large compartments having tall

ceilings and low window heights [Buchanan 2002].

32

2.5 Comparison of various time equivalent methods

Though the above empirical formulae are useful in computing equivalency, the results
from these formulae are not consistent and the time equivalent computed by these
formulae shows a Significant variation even for Similar fire exposure. Further, these
formulae are mainly derived for protected steel members and may not be fully applicable
for RC members.

To illustrate the variation in time equivalency predicted by various methods and
empirical formulae, the above formulae was applied for evaluating the equivalent fire
resistance of an RC beam. The RC beam used in the case study is a simply supported
beam of 6 m Span length and is made of concrete with a compressive strength of 30 MPa
and reinforced with steel rebars having yield strength of 400 MPa. This beam is subjected
to a representative design fire Shown in Figure 2.9 and the time equivalency is calculated
based on equal area method and empirical formulae discussed above. CIB formula gives
a time equivalent of 174 minutes whereas Law and Eurocode formulae predict a time
equivalent to be 132 and 155 minutes respectively. Time equivalent predicted by equal
area method iS about 232 minutes. It can be clearly seen that there is a wide variation in
time equivalent predicted by equal area method and empirical formulae. This can be
attributed to the fact that the above empirical formulae have been derived based on the
maximum temperature in protected steel members and may not be fully applicable to RC
members. Detailed computation of time equivalency by equal area method and formulae

is illustrated in Appendix A and the results are summarized in Table 2.4.

33

2.6 Summary

The response of RC beams under fire conditions iS different from that of room
temperature due to deterioration in strength and stiffness of the beam with temperatures.
The fire response depends on various parameters such as fire scenario, loading condition
and failure criterion. Most of the previous experimental and analytical studies on RC
beams were conducted under standard fire exposure and the fire resistance analysis was
carried out using sectional analysis procedure. The current provisions in codes and
standards base the fire resistance on sectional dimensions and concrete cover thickness
without any consideration to fire exposure, load and concrete strength.

The use of time equivalent approach provides an attractive proposition for evaluating
equivalent fire resistance under design fire scenarios. However, most of the existing time
equivalent methods are derived for protected steel members and may not be applicable
for RC beams. Thus there is a clear need for a time equivalent approach for evaluating

fire resistance of RC beam under design fire scenarios.

34

Tables

Table 2.1 - ACI minimum width and cover thickness requirements for achieving fire
resistance in RC beams [ACI 216.1 2007]

 

 

 

 

 

 

 

Restraint Beam Cover for corresponding fire-resistance rating, mm
condition Iii/"111?; 1 hour 1-1/2 hour 2 hours 3 hours 4 hours

125 20 20 20 25 30
Restrained 175 20 20 20 20 20
Z 250 20 20 20 20 20

125 20 25 30 Not permitted Not permitted
Unrestrained 175 20 20 20 45 75
Z 250 20 20 20 25 45

 

 

 

 

 

 

 

 

 

 

Table 2.2 - Eurocode 2 Specifications for fire resistance rating of Simply supported RC
beams [Eurocode 2 2004]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Beam dimensions (mm)/ 30 60 90 120 180 2 40
Fire resistance (min)
383‘“ 80 120 150 200 240 280
WIdth
,AXiS 25 40 55 65 80 90
distance
383’“ 120 160 200 240 300 350
Possible w'dth
combInatIons .AXIS 20 35 45 6O 70 80
of beam distance
width and Beam
axis distance width 160 200 300 300 400 500
(mm) .
5"“ 15 30 40 55 65 75
dIStance
38“” 200 300 400 500 600 700
WIdth _
.AXiS I5 25 35 50 60 70
dIstance

 

 

Table 2.3 - Values of kb and kc for computing time equivalency using CIB and Eurocode
empirical formulae

 

 

 

 

 

 

 

Compartment = \/——‘
Formula lining b kpcp General
parameter High (>2500) Medium Low
(720-2500) (<720)
ClB k 0.05 0.07 0.09 0.10
c
Eurocode kb 0.04 0.055 0.07 0.07

 

 

 

 

35

 

Table 2.4 - Time equivalent (in minutes) of an RC beam as obtained from existing
methods

 

CIB formula Law formula Eurocode formula Equal area method

 

 

 

 

 

 

174 132 155 232

 

36

Figures

 

 

 

 

 

 

 

 

w o ._
v V + v + i +
l l d
_A_ _Q_ ...

 

h

(a) Elevation and cross section of Simply Supported Beam

 

 

 

 

 

 

3* Capacity of the
a beam
CG 0.)
U S
*6 / *6
g Failure g1
2 J g
T ‘\
Applied l
moment ;
‘ Fire resistance r: e
. > Time
Time
(b) Variation of capacity of beam (c) Time-temperature curve for a
with fire exposure time standard fire exposure

Figure 2.1 - Variation of strength capacity of a typical RC beam under fire conditions

37

229 mm

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

254 mm
' I < >
A 1‘
// 5#8*//
4 # 8 Ar 0 O O C
bars bars
533 mm 6‘0 mm
# 3 r/
stirrups # 3 r/
4 # 7 stirrups
O O ,/' bars O O /”
O__O Q_Q_O
v
v
Beams 1 to 5 Beam 6
(a) Cross-sectional details
P P P P P P 508 mm P-
508
mm 1016mm 1016 mm 1016mm 1016 mm 1016 mm 1524 mm

7; 'v‘ r ‘ v
14.14 .12 .1
1"1‘ 7" 7'

304.8 mm 6096 mm 1828.8 mm

(b) Elevation

Figure 2.2 - Layout of beams tested by Lin and Ellingwood [1987]

38

38 mm

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-i __
V f ’c = 58.2 MPa
//
= 420 MPa
2 d) 13 mm fy
3 6 19 mm 406 mm
K
\ \
m \ ‘1’ 6 mm
38 mm i __ stirrups
1 I
I 1
254 mm
(a) Cross Section
50 kN 50 kN
V i
1‘ >1 F a‘
1.4 m 1.4 m
T
3.96 m

(b) Elevation

Figure 2.3 — Typical layout of beam tested by Dwaikat [2009]

39

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

fir: f ’c = 30 MPa
2014mm fv= 400MPa
3 ¢ 20 mm 500 mm
1K .
x
\: \
\ \ ¢ 10 mm
40 mm : _ stirrups
1 1
300 mm
(a) Cross Section
20 kN/m
V V V V V V V
l l
l 6 m l
(b) Elevation

Figure 2.4 - Layout of RC beams used by Kodur and Dwaikat[2008b] in parametric

studies

40

 

 

 

 

1400 ,
1200 5
_" + Standard fire
9L: 1000 ? +Design fire
2 _ Area A
E
E
S.
E Area B
[-
iliilllllllllliilllll..
O 100 200 L 300 400 500 600
Time (minutes)

Figure 2.5 - Illustration of equivalent fire severity calculation using equal area method

 

1000
900
800
700
600
500
400

Temperature (°C )

 

 

 
  
 
 

 

+ Standard fire

—I— Desi 11 fire

+ Stee tempéStandard fire)
—)(— Steel temp Design fire)

 

 

 

 

OVI'IITVVY‘II'vav-
0 100 200 te/300

Time (minutes)

I I y u v l l x

 

400 500 600

Figure 2.6 - Illustration of equivalent fire severity computation using maximum

temperature method

41

 

1.0

 

Real fire

Load Capacity

Standard fire

 

 

 

V

Time

Figure 2.7 - Illustration of equivalent fire severity calculation using minimum load
capacity method

42

Time (minutes).

0 100 200 te 300 400 500

 

 

--"

    
 

 

ASTM E119
.......... Design Fire

Deflection(mm)

 

 

 

 

 

 

Figure 2.8 - Illustration of equivalent fire severity calculation using maximum deflection
method

1 000

 

800 3

  
 

 

' +Design fire

 

 

 

l I

400

Temperature(°c )

200 j

 

 

.1
0 I I I I l I I Y I I I I T I r W I I I I

O 100 200 300 400 500
Time (minutes)

 

Figure 2.9 - Representative design fire scenario used in comparison of various time
equivalent methods

43

CHAPTER 3

3. NUMERICAL MODEL

3.1 General

Fire resistance of RC beams can be evaluated by conducting detailed analysis using
microscopic finite element models. In this approach an RC beam is discretized into a two
or three dimensional mesh, and the fire performance iS evaluated by undertaking coupled
or uncoupled thermal and structural analysis at various time increments. This analysis is
quiet complex and generally requires Significant amount of computational time and
effort.

In lieu of microscopic finite element models, a macroscopic finite element analysis can
be applied to evaluate fire resistance, wherein a RC beam is divided into a number of
segments along the span length. For each segment, moment-curvature (M- K) relations are
generated at various time steps. These M- K relationships are utilized to trace the response

of the beam at different time steps till failure occurs. Strain components such as creep and

44

transient strain, high temperature material properties, geometric, material nonlinearity
and material softening can be accounted for in this analysis. The results obtained from
such analysis (models) are easy to interpret and can be used in practical design situations.
This type of macroscopic models has been developed for evaluating fire resistance of RC
beams with rectangular cross section. In the current study the approach is extended for

predicting fire response of RC beams with T, I and inverted T sections.

3.2 Macroscopic finite element model
In the macroscopic finite method, the structural member (beam or column) is divided into
a number of segments along its length, and the mid-section of each segment is assumed to
represent the behavior of the whole segment. The cross section of the segment is further
divided into elements forming a two-dimensional mesh. The fire resistance analysis is
carried out at various time steps, until failure occurs in a beam. At each time step, the
analysis is carried out by
0 Evaluating fire temperature resulting from fire exposure,
0 Conducting heat transfer analysis in each segment to predict cross sectional
temperature,
0 Performing strength and deflection analysis by:
o Generating moment-curvature (M-K) relationships in various segments,
0 Conduct nonlinear structural analysis to compute strength and deflections.
The fire temperature is evaluated for a given fire scenario, i.e., standard or design, which
is generally represented by temperature-time relations. Then, heat transfer analysis is

performed by applying the finite element approach and temperature distribution across

45

the mid-section of each segment is developed. Figure 3.1 shows layout of a typical RC
beam and it’s idealization for analysis. The beam is idealized as a set of segments along
its Span (Figure 3.1(c)). The mid-section of each segment is firrther discretized into
elements as Shown in Figure 3.1(d).

The computed cross-sectional temperature distribution is utilized to undertake strength
analysis of various segments of the beam. As part of this strength analysis moment-
curvature relationships are generated for various segments. These relationships are
generated as a function of time for each segment, based on the high temperature
properties of constituent materials. Thus material nonlinearity is accounted for in the
analysis. Strain components namely thermal, mechanical and creep Strains are included in
generation of M-K relationships for both concrete and reinforcing steel. Additionally,
transient strain for concrete is also accounted for in computing total strain. The model
accounts for geometrical nonlinearity, material nonlinearity and material softening. The
generated moment-curvature relationships are used to undertake a nonlinear structural
analysis to predict the fire response of the RC beam. Typical moment-curvature
relationships of an RC beam, generated by the model are shown in Figure 3.2. It can be
seen from figure 3.2 that the moment capacity of the beam decreases with fire exposure
time. This can be attributed to reduction in material strength and stiffness as a result of
increased temperatures in concrete and steel. In addition, the ultimate curvature
(curvature at collapse) increases with time of fire exposure. This is due to deterioration of
stiffness as well as the increase in creep strain which becomes Significant prior to failure.
The model is capable of estimating fire resistance of beams exposed to any given fire

time temperature curve. The computer model also generates temperatures, stresses,

46

strains, moments and deflections at each time step. These output parameters are used to
check the failure of the beam. At every time step, each beam segment is checked against
four pre-determined sets of failure criteria, which include prescriptive thermal criteria,
and performance-based strength and deflection considerations. The analysis terminates
when failure of the beam occurs under any specified limiting criteria. The four sets of
failure criteria that can be applied to define failure Of an RC beam are:
1) When the temperature in steel rebars (tension reinforcement) exceeds the critical
temperature which is 593 °C for reinforcing steel.
2) When the beam is unable to resist the applied load.
3) When the maximum deflection in the beam exceeds L/20 at any fire exposure
time, where L is span length.

4) When the rate of deflection exceeds the limit given by the following expression:

L2
9000d

 

(mm/min)

where: L = Span length of the beam (mm), and d = effective depth of the beam
(mm).
The above discussed model is capable of predicting the fire performance of beam with
rectangular cross section only. However, in practice T and I cross section beams are often
used in buildings and bridge applications. To cater for wide range of design Situations,
the macroscopic finite element model has been extended to cover T, I and inverted T

Shaped cross-sections.

47

3.3 Extension of macroscopic finite element model for T/I beams

In the extended model, the analysis starts by dividing the beam into longitudinal
segments and each segment cross-section is further discretized into a number of elements.
Then sectional analysis is carried out at various cross sections along the length of the
beam. The sectional response generated at different cross-sections is used to predict the
fire response of RC member. A flow chart Showing the numerical procedure for tracing
the fire behavior of RC beams is shown in Figure 3.3 and the detailed description of the

procedure associated with the analysis is presented in the following sections.

3.3.1 Fire temperatures
The fire temperatures are calculated from the Specified time-temperature relationships for
a given type of fire exposure i.e., standard or design. The time—temperature relationship

for ASTM E119a [2008] standard fire exposure is approximated by the following

equation:
T = 7500— exp(—3.79553,/7,,_))+170.41,/Z + To (3.1)
where

t}, = time (hours),

To = initial temperature (°C) and

T = fire temperature (°C)
The temperatures for design fire exposure are calculated based on parametric fire time-
temperature equations given in Eurocodel [1994] or SFPE [2004]. These equations can

be used to predict the temperature of design fires for any combination of fuel load,

48

ventilation openings and lining materials. According to Eurocodel specifications the

approximate time-temperature relationship for the burning period of fire is given as:

T =1325(1— 0.324e-0'2t* — 0.204e—1'7t * — 0.472e‘19’*)+ To (3.2)
where
To = initial temperature (°C),

T= fire temperature (°C), and
t* = fictitious time (hours) = Ft

t = time (hours), and

= (Fv/Fref)2
(b/bref)2

Fv = ventilation factor (mo's) = Av 1/Hv/A, ,
F ref: 0.04,
. . 2
Av = area ofw1ndow openmg (m ),
Hv = height of window opening (m),
A , = total internal surface area of room (m2),

6 = .lkpcp (WSO'S/mzK),

k,ocp = thermal inertia,
k = thermal conductivity (W/mK),

p = density (kg/m3),

49

cp = specific heat (J/kgK), and

bref: reference value of thermal inertia = 1900.

During the decay phase of design fire, the fire temperatures are computed using the

modified decay rate specified by Feasey and Buchanan [2000] and is given as follows:

11(511] ./F,/0.04
dt dt ref 1/b/1900

(3.3)

Eurocodel gives a reference decay rate (Cali—i] of 625°C per hour for fire with duration

ref
of burning less than half an hour and 250°C per hour for fires with over two hours of

burning period.

3.3.2 Thermal analysis

For the thermal analysis, the beam is divided into a number of segments along its length
and the mid-section of each segment is assumed to represent the behavior of the whole
segment. The mid-section is further divided into elements forming a two-dimensional
mesh. Temperature distribution in this mid-section is computed using finite element
method based heat transfer analysis. Steel reinforcement is not specifically considered in
the thermal analysis because it does not Significantly influence the temperature
distribution in the beam cross section [Lie and Irwin 1993]. The beam is assumed to be
exposed to fire from three Sides and maintained at ambient conditions on the fourth side
as shown in Figure 3.4(d).

The governing heat transfer equation used for analysis of beam cross-section is given as:

50

6T
kV2T+Q=pc—a—t— (3.4)

where
k = thermal conductivity,
,oc = heat capacity,
T = temperature,
= time, and
Q = heat source.
The mechanism for heat transfer on the boundary of RC beam is by convection and

radiation. Therefore, the heat flow per unit area (heat flux) to RC beam can be written as:

qcon :hcon (T—Tf) and (3-5)
qrad = 08(T4 _Tf4)

= 05(T2 + Tf2)(T + Tf)(T — Tf) =hmd (T — Tf)

(3.6)

where

. 2
qcon = convect1ve heat flux(W/m ),
. . 2
qmd = radIatlve heat flux(W/m ),
. . 2
ham = convect1ve heat transfer coeffic1ent(W/m °K ),
. . . 2
hmd = radiatlve heat transfer coeffiCIent(W/m °K ),

0' = Stefan- Boltzmann constant = 5.67x10-8(W/m2°K4),

E = emissivity,

51

T = temperature on the boundary of RC beam, and

Tf = temperature of the fire.

The total heat flux transferred to the exposed surface of an RC beam through its

boundaries is:
q : qcon +qrad : (hcon + hrad )(T — Tf) (3'7)

Therefore, the governing heat transfer equation on the boundary of RC beam is obtained

by balancing the energy transfer across the boundary and can be written as:

6T 6T
k[Ey—ny+-é—Z—nz]=—h(T—Tf) (3.8)
where

ny and nz = components of the vector normal to the boundary in the plane of the

cross section, and
h = hcon + hrad-
AS the cross-section of the beam is exposed to fire from three sides and ambient
conditions prevail on the fourth Side, two types of boundary conditions exists, namely:

0 Fire exposed boundaries where the heat flux is governed by the following

equation:

 

6T 6T
Z = — —— 3.9
k[—ny+ 62 n ] hf(T Tf) ( )

o Unexposed boundary where the heat flux equation is given by:

 

6T 6T
__ z =— c — 0 3.10
k[éyny+azn] 11(T T) ( )

where

52

hf= heat transfer coefficient on the fire Side,
C = heat transfer coefficient on the cold Side,
Tf= fire side temperature, and

T0 = cold Side temperature.

Eq. (3.4) is solved through Galerkin finite element formulation using four node

quadrilateral elements (Q4). In this approach, element stiffness matrix (K e ), mass matrix

(M e ), and nodal load vectors (F e) are formulated as follows:

 

T T
Kg: ka—NaN “ca—NEIL dA+[NaNTds (3.11)
6x 6x 6y By
A F
Me: [chNTdA (3.12)
A
Fe: [NQdA+ INades (3.13)
A F
where

N = vector of shape function,

01 = he or hfdepending on the boundary F,

T00 = fire or ambient temperature depending on the boundary F, and

s = distance along the boundary.
After computing the elemental matrices, they are assembled into global stiffness and

mass matrices and formulated into a system of differential equations as follows:

MT+KT=F(t) (3.14)

53

where
K = global stiffness matrix, '
M = global mass matrix,

F = equivalent nodal heat flux, and

T = temperature derivative with respect to time.
Eq. (3.14) is solved using finite difference algorithm of trapezoidal family (0 algorithm)
in the time domain. This algorithm computes the temperature distribution at any time step
using the information available at previous time step, and can be written as [Williams

1990]:

T...=T.+h<6in+.+(1—6)Tn) (3.15)
Multiplying both sides of Eq. (3.15) by M and using Eq. (3.14) at the beginning and the
end of the time interval (tn, tn+ 1), the following equation can be obtained:

(M + h6K)T,,+1=(M — h(l —6)K)Tn +h(6F,,+1 + (1 — 6)F,,) (3.16)

where

h = time step,

Tn and Tn+1 = temperature at the beginning and the end of time step,

F n and Fn+1 = equivalent nodal heat flux at the beginning and the end of time

step, and

6 = a constant between 0 and 1.
Knowing the temperature at ambient conditions (zeroth time step), Eq. (3.16) begins
marching in time to give temperatures at the following time step (first time step), and this

procedure is repeated for subsequent time steps. At each time step, Eq. (3.16) Should be

54

solved using an iterative process due to nonlinearity of both material properties and
boundary conditions. Furthermore, for unconditional stability of the numerical

calculations, the value of 6 Should be more than or equal to 0.5 [Williams 1990].

3.3.3 Strength analysis

3.3.3.]. General

The computation of temperature distribution is followed by strength analysis in which the
moment-curvature relationships are generated for each segment represented by the mid-
section. The basic assumptions made in the strength analysis are:

0 Plane sections remains plane before and after bending.

o Tensile strength of concrete, at elevated temperatures, is taken into account in the ‘

model based on the reduction factors given in Eurocode 2 [2004].

o No bond-Slip exists between concrete and steel reinforcement.
The strength analysis is carried out in two stages. First, the moment-curvature (M-K)
relationships for each segment are generated. Then the deflection of the beam is
computed at each time step using the generated moment—curvature relationships.
Strength calculations are carried out using the same discretized cross-section as for
thermal analysis as shown in Figure 3.4(c). It is assumed that temperatures, strains,
stresses, deformations in each element are represented by those at the centre of the
element. The temperature in each element is computing by taking an average Of the
temperatures at the nodes of that element. Similar assumption is used for steel
reinforcement, where in the values of temperature, stress, strain in each bar is represented

by those at the centre of the bar. Temperature at the center of steel reinforcement is

55

approximated by the temperature in the element at the location of the center of the rebar

cross-section (in the concrete).
The total strain in concrete at any time step comprises of four components namely

thermal, mechanical (or stress related strain), creep strain, and transient strain. Thus the

total strain can be written as:
e, = 5,}, + eme + cc, + 6,, (3.18)
where

e, = total strain in concrete element,

8,), = thermal strain in concrete element,

em = mechanical strain (or stress related strain) in concrete element,
cc, = creep strain in concrete element, and

8,, = transient strain in concrete element.

The thermal strain (8”,) is a function of temperature and can be obtained by knowing the

temperature and thermal expansion Of type of concrete. In general, the thermal strain can

be written as follows:
gthzaTc (3.19)
where

a = coefficient of thermal expansion, and

TC = concrete temperature.

56

The mechanical strain (or stress related strain) (eme) is a function of both the applied

stress and the temperature. It includes the elastic and plastic components of strain

resulting from applied stress.
The creep strain (ccr) is a function of time, temperature and the applied stress level. This

creep strain can be computed based on Harmathy’s [1993] approach, which can be

written as:

cc, = p, +1.6 edV—z”) (3.20)
fc,T

where

m = 6.28x10’6 3'05,

d= 2.6583(10'3 K",
T = concrete temperature (°K) at time I (see),
fa T = concrete strength at temperature T, and
O'= stress level in the concrete at the current temperature.
The transient strain (81,-) is a function of applied stress and the temperature [Anderberg

and Thelandersson 1976]. This transient strain occurs only during the first time heating of
concrete under load to approximately 600°C, and can be computed using the relationship
proposed by Anderberg and Thelandersson [1976]. Accordingly, the transient strain can

be written as follows:

 

A5,, = k2 A5,}, (3.21)

0,20

where

57

k2 = a constant ranges between 1.8 and 2.35 (a value of 2 will be used in the

analysis),

A8,}, = change in thermal strain,
A5,, = change in transient strain, and.

fc, 20 = strength of concrete at room temperature.

Similar to concrete, the total strain in steel reinforcement at any time step comprises of
three components namely thermal, mechanical (or stress related strain) and creep strain.

Thus the total strain in steel reinforcement can be written aS follows:
51s = £ths + gmes + gcrs (3'22)

where

8,5 = total strain in steel reinforcement,
ems = thermal strain in steel reinforcement,
emes = mechanical strain in steel reinforcement, and

8ch = creep strain in steel reinforcement.

The thermal strain can be computed by knowing the values of coefficient of thermal
expansion and temperature of the reinforcing steel. Eurocode 3 [1995] provides a linear
coefficient of thermal expansion for use in the design equations. According to Eurocode
3, the thermal strain of steel can be approximated aS follows:

8m, =14 x10‘6 (T — 20) (3.23)

where

58

T = temperature in °C.
The creep strain in steel is computed based on Dorn’s theory and the model proposed by
Harmathy [1967] with some modifications to account for different values of yield

strength of steel. Accordingly, the creep strain in steel can be written as:

l 1

gm =(3Ze,20)/3 o 3 +20 (3.24)
where

’ 4.7 ‘

6.755x10‘°[—i'—] i s 153-

Z =< fy fy >,
1.23x10'6(e'°'8(°/fy)) 3— > —5—
f, 12

 

 

e = Je‘AH/RTdt,

% = 38900°K,

t = time (hours),

1.75
5,0 =0.016 —°— ,
fy

O = stress in the steel, and
j} = yield strength of steel.

The distribution of strain, stress, and internal forces for the beam cross-section at any fire
exposure time is shown in Figure 3.5. At each time step, for an assumed value of
curvature of the beam, the total strain in any element of concrete or rebar can be written

as:

81:80 +Ky (3.25)

59

where

so = total strain at the geometrical centroid of beam cross-section,

K' = curvature of the beam, and
y = the distance measured from geometrical centroid of beam cross-section.

Thus, the total strain can be computed for a segment at any given fire exposure time

using Eqs. (3.18)-(3.25) . At each time step, a value of 80 and x are assumed and the total

strain is computed using Eq.(3.25) and an iterative process is applied till equilibrium and

compatibility conditions are satisfied. The analysis at each time starts with the converged

value of strain (80) at previous time step. The individual strain components like thermal,

transient, creep strain for concrete are evaluated using Eqs.(3.19)-(3.21). The thermal,
creep strain for reinforcing steel are also evaluated using Eqs. (3.23)— (3.24). Using the
value of total strain and individual strain components, the mechanical strain of each
element can be computed by subtracting the various strain components from the total
strain (using Eqs. (3.18) and (3.22)) as follows:

For concrete,

gme =5t ‘8th ’gcr “Etr (3'26)
For steel,
gmes = gts — °ths _ Ecrs (3'27)

Having known the value of mechanical strain, the stress in the element can be computed

using the material Specific stress-strain relationships.

60

3.3.3.2 Generation of M-K relationships

The M-Ic relationships are generated using the same rectangular network of elements used

for thermal analysis (Figure 3.4(c)). The M-K relationships are established by iterating the

central total strain (80) and the curvature (K). At the beginning of each time step,

curvature and central total strain in concrete is assumed. Then, the total strain in concrete
and rebar elements are computed from the assumed values of strain and curvature. These
strain values are used to compute the stresses in rebars and concrete elements by using
the constitutive laws of respective materials. Once the stresses are known, the forces are
computed in the concrete elements and rebars as a product of stress and the area of
element. The computed force in each element is subsequently used to verify the
equilibrium condition (total compressive force to be equal to total tensile force) for the
assumed value of curvature and strain. If the equilibrium is not satisfied the curvature is
incremented and the iteration process is continued until the computed forces satisfy
compatibility, equilibrium and convergence criteria. When these conditions are satisfied
the corresponding moment and curvature are computed. Thus at the end of iterative
procedure, a point on the moment-curvature curve is obtained.

The value of the central total strain is incremented to generate subsequent points on the
moment curvature curve. This process is repeated again for each time step, to compute
various points on the moment-curvature curve at that given time step. Thus, various
points on the moment-curvature curve are generated at a given time step and also at

various time steps.

61

3.3.3.3 Computation 'of deflection

The mid-span deflection of the beam at each time step is computed using the generated
moment-curvature relationships. For any given time step, the curvatures along the
member is integrated twice numerically to estimate the mid-Span deflection of the beam
using the following equation

B
6AB=[0Ade (3.28)

A

where, 6 A B iS the rotation between any two points A and B of the member, which can be

calculated as:
B

9A3 = [$43095 (3.29)
A

where, (p A B is the curvature between any two points A and B of the member.

The step-by-step procedure involved in the numerical integration is illustrated in Figure
3.6. In this approach the beam is divided into segments and using the moment-curvature
relationship for each segment, the curvature for the corresponding moment distribution at
any location along the length of the beam is determined. The curvature is then integrated
twice along the member using Eq.(3.28) and Eq.(3.29), to compute the mid-Span
deflection of the beam.

Therefore this model can be used to compute the temperatures, moment and curvature
values, deflection of the beam for any specified duration of fire exposure. These output
parameters namely temperature, strength (capacity) of the beam, deflection can be
subsequently used to evaluate the failure of the beam using the failure conditions

described earlier.

62

3.4 Computer program

3.4.1 General

The numerical procedure described in Section 3.2 through Section 3.3 requires significant
computational effort at each time step as iterative process is needed to satisfy
equilibritun, compatibility and convergence criterion. To facilitate the vast amount of
calculations, the numerical procedure has been incorporated into a computer program

written in FORTRAN language.

3.4.2 Beam discretization

The given RC beam is divided into several longitudinal segments and the central cross
section at mid-length of the segment is assumed to represent the behavior of each
segment. The cross section is further divided into elements. The number of longitudinal
segments and the number of elements in the cross section along with the dimensions of
the elements in each direction has to be specified in the input file. The program allows
the user to specify uniform and non-uniform grid Size in the cross sectional mesh

generation.

3.4.3 Material properties

Two sets of material properties for both concrete and the reinforcing steel as specified in
Eurocode 2 [2004] and ASCE Manual [Lie 1992], and reproduced in Appendix B, are
incorporated into the computer program. Appropriate formulas for both thermal and
mechanical properties of concrete and steel as a function of temperature are built into the

program. The user has the Option of selecting the desired material properties and Should

63

Specify the choice of relevant material model codes in the input file. Further, the user can
select either siliceous aggregate concrete or carbonate aggregate concrete. The user has to
specify the modulus of elasticity of steel and concrete, the 28-day compressive strength
of concrete, yield strength of steel, initial moisture content of concrete, material model

code and type of aggregate in the concrete as input in the input file.

3.4.4 Program input

The input for the program is divided into two categories namely input for thermal
analysis and input for structural analysis. Input for the thermal analysis includes the
number of horizontal and vertical division of the beam cross-section, grid Size along
vertical and horizontal directions, aggregate type, moisture content in concrete, number
of reinforcing bars, location of center of rebars in the cross section, type of fire exposure,
and duration of fire exposure time.

Input for structural analysis requires the user to Specify the material properties (concrete
and reinforcing steel), structural geometry, boundary conditions, loading conditions and

the number of longitudinal segments required along the span of the beam.

3.4.5 Program output

Similar to the input for the model, the output from the model comprises of thermal output
file and structural output file. At each time increment, the thermal analysis generates the
temperature at each node. This temperature distribution serves as one of the input for
structural analysis for generating moment curvature curves. Additionally, the deflection
of the beam and rate of deflection of the beam is written in the output file at each time

step.

64

3.5 Validation

In order to validate the numerical model, presented above, predictions from the model are
compared with data generated from tests and finite element analysis for three RC beams
namely Bl, B2 and B3. Beam B1 is a RC beam with a rectangular cross-section tested by
Dwaikat [2009], while beams BZ and B3 are typical RC beams with T, I cross-sections
respectively. Since there is no test data available on the fire performance of RC beams
with T and I cross-section, results obtained for beams B2 and B3 from the model are

compared with the results obtained from SAP 1R [Franssen et al. 2004].

3.5.1 Rectangular beam Bl

Predictions from the computer program are compared with test data generated on a
rectangular RC beam. Beam Bl, shown in Figure 3.7, was tested by Dwaikat [2009]
under standard fire conditions. The Simply supported rectangular beam of 254 mm x 406
mm had a span length of 3960 mm. The beam was made of carbonate aggregate concrete
with a 28 day compressive strength of 52.2 MPa. The beam was reinforced with 2013
mm bars as compressive reinforcement and 3019 mm bars as tension reinforcement.
Shear reinforcement consisted of 06 mm stirrups spaces at 150 mm over the length of the
beam and bent at 135° into the concrete core. The main reinforcing bars (tensile and
compressive rebars) and Shear reinforcement were having yield strength of 420 MPa and
280 MPa respectively. The beam had a clear concrete cover thickness of 38 mm to the
surface of stirrups. The beam was tested after 16 months of casting. The measured
compressive strength of concrete on the day of testing was 58.2 MPa. The beam was

tested by applying two point loads of 50 kN/m at a distance of 1.4 m from the end

65

supports and this represented an equivalent load ratio of 54%. Load ratio is defined as the
ratio of applied load to the capacity at room temperature. Details about the beam
properties are presented in Table 3.1, and the Sketch Of the beam is shown in Figure 3.7.
This beam was analyzed using the macroscopic finite element model by Simulating
ASTM E119 fire conditions. Figure 3.8 shows the comparison Of measured and predicted
temperatures for rebar and two concrete locatiOns (at a depth of 200 mm and 300 mm
from the surface of beam). It can be seen that there is a good agreement between the
measured and predicted temperatures. Initially there is a temperature plateau at about
100°C for rebar and concrete temperatures and this can be attributed to the evaporation of
water in concrete which consumes Significant amount of energy [Dwaikat 2009]. The
figure also shows that the measured temperatures in concrete decrease with increasing
depth from the bottom surface. This is mainly due to low thermal conductivity and high
thermal capacity of concrete which reduces heat penetration into the inner layers of
concrete. The slight discrepancies in the measured and predicted temperatures can be
attributed to variation in the high temperature material properties reported. The same
discrepancies between the temperatures measured from the test and those predicted by the
model were observed by Dwaikat[2009].

The measured and predicted mid-span deflection is presented in Figure 3.9. It can be
seen from the figure that the mid-span deflection predicted by the current model vary
significantly from those measured in the test. This difference can be attributed to
variability in the high temperature properties of concrete and rebar Specified in codes.
High temperature material properties specified in codes are developed using data

obtained from limited material property tests. However these material properties Show

66

considerable variations due to differences in test methods, conditions and procedures
used during test. Kodur et. al. [2008] reviewed high-temperature concrete constitutive
relationships and illustrated the considerable variations in the high-temperature properties
of concrete.

Based on the results obtained from the model, fire resistance of the beam is evaluated
according to the four failure criterion discussed in Section 3.2 and is presented in Table
3.2 along with the measured fire resistance from test. According to deflection and the rate
of deflection failure criteria, the model predicts a fire resistance value of 125 and 120
minutes respectively, which is lower than that predicted for temperature and strength
criteria. The fire resistance predicted by the model using strength criterion is 145 minutes
which is lower than that in the test (180 minutes). The difference can be attributed to
variation in the high temperature material properties used in the analysis, as discussed

above.

3.5.2 T beam B2

The second beam (B2) selected to validate the proposed model is of T cross-section.
Since there is no tested beams in literature a representative T beam was designed as per
ACI 318 Specifications [2008]. Layout of this beam is Shown in Figure 3.10 and design
details are presented in Appendix C.

The simply supported beam has a Span length of 7.62 m (25 feet). The flange is of 1905
mm wide x 152 mm deep while the web is of 317 mm wide x 550 mm deep. The beam is
assumed to be made up of siliceous aggregate concrete having a compressive strength of

28 MPa. The beam has 3020 mm bars as tensile reinforcement and 2020 mm bars as

67

compressive reinforcement. The Shear reinforcement consists of #4 (012.7mm) stirrups
spaced at 317.5 mm. The yield strength of the reinforcing steel is assumed to be equal to
413 MPa. A concrete cover thickness of 61 mm (clear cover of 38 mm) was provided to
the center of tension reinforcement from both Sides while that to the center of
compression reinforcement was 61 mm from the top of beam. The room temperature
capacity of the beam was calculated according to AC1 318 specification [2008] and was
found to be 260 kN—m. The beam was analyzed using the current model and SAFIR
computer program. The applied load on the beam was 18.1 kN/m, which corresponds to a
load ratio of 50%. For the analysis of the beam, the thermal and mechanical properties
Specified in Eurocode 2 and incorporated into SAFIR were used. The analysis was carried
out by exposing the beam from three sides (bottom and two sides) to ASTM E119
standard fire exposure while the top surface was maintained at ambient conditions.
Layout of the beam along with the cross-sectional details is Shown in Figure 3.10 and the
parameters used in the analysis are shown in Table 3.3.

Figure 3.11 and Figure 3.12 shows the variation of temperatures at bottom central rebar
and at three different concrete locations respectively as predicted by the current model
and SAFIR. The three different concrete locations used to compare the temperatures
correspond to quarter depth (175.5 mm), half depth (351 mm) and three quarter depth
(526.5 mm) of the beam B2. It can be seen from the figures, that the temperatures
predicted by the current model and SAFIR are in good agreement with each other in the
entire range of fire exposure time. The temperature at various locations in concrete, as
well as the rebar, increases with fire exposure time. Also from Figure 3.12, it can be

observed that the temperatures decrease with increasing distance from fire exposed

68

bottom surface of the beam. This can be attributed to low thermal conductivity and high
thermal capacity of concrete that slows down heat transfer to the inner layers of concrete.
Variation of mid-span deflection predicted by the current model and SAP IR are plotted in
Figure 3.13 as a function of fire exposure time. The predicted fire resistance from the
current macroscopic finite element model is Slightly higher and also there is large
increase in deflection of the beam prior to failure. The higher fire resistance predicted by
the proposed model can be attributed to the fact that the current model accounts for
softening of the concrete and SAP IR does not take this into consideration [Franssen et al.
2004, Kodur and Dwaikat 2008]. Rapid increase in deflection prior to failure can be
attributed to yielding of steel and creep strains which become significant during the later
stages of fire exposure. SAFIR does not explicitly account for high temperature creep in
concrete and steel.

Results from current model was used to evaluate the fire resistance for beam BZ using the
four sets of failure criterion and is compared with fire resistance obtained from SAFIR
analysis in Table 3.2. The current model predicts a fire resistance of 205 minutes (based
on strength criteria) while SAF IR predicts a value of 187 minutes. The difference can be
attributed to the fact that the current model accounts for softening effect in concrete
(descending branch of stress-strain curve). However SAFIR does not take this into
consideration. Therefore failure in the current model corresponds to concrete attaining
ultimate strain while in SAP IR it corresponds to concrete reaching peak strain. A
representative stress-strain curve for concrete is Shown in Figure 3.14 along with the

ultimate strain and the strain corresponding to peak stress.

69

As can be seen from Table 3.2, the fire resistances predicted by rebar temperature and
strength failure criterion is higher than that of deflection and rate of deflection failure
criterion. Though actual failure in a beam occurs when the strength limit state is reached,
deflection and rate of deflection may be important in various applications in order to
maintain the integrity of the structure. Therefore under some fire exposures, deflection
and rate of deflection failure criterion might govern the failure of RC beam and hence
governs its fire resistance. It should be noted that the current provisions in ASTM E119

does not specify deflection and rate of deflection limit states.

3.5.3 I beam B3

The third beam selected for validation is of I cross-section (B3) and the beam was
designed as per ACI 318 Specifications [2008]. The room temperature capacity of this I
beam, calculated according to AC1 318 specification [2008], is 260 kN-m, which is same
as that of a T beam BZ. Design details of beam B3 are presented in Appendix C.
Sectional dimensions and other characteristics of the beam are illustrated in Figure 3.15.
The beam is assumed to have same material properties as that of Beam BZ and analyzed
using the current model and SAFIR. The analysis is carried out with an applied load of
18.1 kN/m by exposing it to ASTM E119 standard fire exposure from three sides. Layout
of the beam along with the cross-sectional details is Shown in Figure 3.15 and the various
parameters used for the analysis are presented in Table 3.4.

Variation of temperatures at bottom central rebar and at three different concrete locations
as predicted by the current model and SAFIR are presented in Figures 3.16 and 3.17

respectively. It can be seen from Figs 3.16 and 3.17 that the temperatures predicted by the

70

proposed model and SAFIR are in good agreement with each other in the entire range Of
fire exposure time. As expected, the predicted temperatures increase with increasing
depth from the top surface of the beam due to low thermal conductivity and high thermal
capacity of concrete.

Figure 3.18 compares mid-Span deflections of beam B3 as predicted by the proposed
model and SAFIR. It can be seen in the figure that the mid-span deflection increases
continuously throughout the fire exposure time. This can be attributed to the gradual
decrease in stiffness of the beam with increasing fire temperature. It can also be seen
from the figure that the deflections predicted by the proposed model and SAFIR are in
close agreement with each other in the entire duration of fire exposure. The current model
predicts a large increase in deflection prior to failure as compared to SAFIR. This is on
the expected lines as the current model accounts for high temperature creep and transient
strains which play a Significant role during the later stages of fire exposure.

Table 3.2 shows the fire resistance values for beam B3 computed using the four sets of
failure criterion discussed in Section 3.2 along with the fire resistance obtained from
SAFIR analysis. The current model predicts a fire resistance of 270 minutes (based on
strength criteria) while SAFIR analysis gives a values of 230 minutes. The difference in
the fire resistance values can be attributed to the fact that the current model uses ultimate
strain to predict the failure in the beam whereas SAF IR predicts failure Of the beam at a
strain corresponding to maximum stress, as explained in previous section. It can be seen
from Table 3.2 that the fire resistance values predicted by deflection and rate of
deflection are lower than those predicted by rebar temperature and strength failure

criterion. Thus the deflection or rate Of deflection failure criteria might play a crucial role

71

in some applications where the integrity of the structure has to be maintained and hence

govern fire resistance Of beams.

3.6 Summary

This Chapter presents the development of a macroscopic finite element based numerical
model for tracing the fire response of RC beams with rectangular, T, I or inverted T
cross-section. The main steps associated with the numerical procedure namely:
calculating fire temperature, thermal analysis, and structural analysis of the beam are
described in detail. The high temperature material properties of concrete and steel used in
the model are also described in this chapter. The validity of the computer model is
established by comparing the predictions from the model with that obtained from test
data and other numerical models such as SAFIR. Based on the results obtained in the
study, it is concluded that the proposed model is capable of predicting the fire response of
RC beams under realistic conditions. The numerical model described here will be used to
undertake a set of parametric studies for quantifying the influence of various factors on

fire resistance of RC beams.

72

Tables

Table 3.1 - Summary of properties of Beam Bl

 

Summary of various properties used in the analysis of Beam Bl

 

 

 

 

 

 

 

 

 

 

 

 

 

Concrete fc' = 58.2 MPa
Strength
Steel fy = 420 MPa
Aggregate type Carbonate
. Concrete ASCE manual
Thermal properties
Steel ASCE manual
. . Concrete ASCE manual
Mechamcal propertres
Steel ASCE manual
Number of longitudinal segments 20
As 3019 mm bars
As. 2013 mm bars

 

 

Table 3.2 - Summary of fire resistance values predicted for the analyzed beams using

various failure criterion

 

Fire resistance(min)

 

 

 

 

 

 

 

 

 

 

B
eam Rebar Strength Deflection Rate (if Test/SAFIR
deflectIon
R°°t°"f§:‘;' °°°m 220 145 125 120 180
T beam (82) 245 205 195 190 187
I beam (B3) 280 270 260 255 230

 

73

 

Table 3.3 - Summary of various properties used in the analysis of Beam B2

 

Summary of various properties used in the analysis of Beam BZ

 

 

 

 

 

 

 

 

 

 

 

 

 

Concrete fc' = 28 MPa
Strength
Steel fy = 413 MPa
Aggregate type Siliceous
, Concrete Eurocode
Thermal propertIes
Steel Eurocode
M h _ l rt' Concrete Eurocode
ec anIca prope Ies Steel Eurocode
Number of longitudinal segments 20
As 3(D20 mm bars
' 2<D20 mm bars
As

 

Table 3.4 — Summary of various properties used in the analysis of Beam B3

 

SummarLof various properties used in the analysis of Beam B3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Concrete f(3 = 28 MPa
Strength
Steel fy = 413 MPa
Aggregate type Siliceous
Thermal ro erties Concrete Eurocode
p p Steel Eurocode
M h , l rt' Concrete Eurocode
ec amca prope Ies Steel Eurocode
Number of longitudinal segments 20
A5 3¢20 mm bars
' 2¢20 mm bars
As

 

74

 

 

Figures

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

bf
w H AdI
, 41’ hfI \
V I I As ~\ 3
d bw AS d
’ A
c I s As k
L J V: 1w V
1‘ '1 c I

. H

(a) SImply supported beam c

Rectangular T

(b) Cross section
*Segment number

 

 

1‘2 3 45 6 7 8 91011121314

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

(c) Idealized along span Rectangular T

(d) Idealized along cross section (segment)

Figure 3.1 - Layout of typical RC beam and its idealization for analysis

75

Moment Capacity (kN-m)

 

 

 

 

+ t=0min
-I— t=30min
—0— t=60min
—a-— t=90min
+ t=120min
+ t=180min

 

 

I j r I 7 I I I I I T T T 1’ Y I 1'

0 100 200 300
Curvature (l/Km)

Figure 3.2 — Typical moment-curvature relationships for an RC beam

76

I I Y

400

500

 

600

 

 

/ ‘25:: /

 

 

Computation of fire

 

V

 

 

temperature

 

 

I
Thermal Calculation of

 

 

 

properties ____> cross-sectional
temperature

 

 

 

 

 

 

Initial total strain at the

 

tOp most fiber of
concrete

 

 

 

 

 

Initial curvature

 

 

 

 

Stress-strain

 

 

 

 

 

 

relations .
and ‘ Calculate strams and
deformation stresses
properties

 

        
   

Compressive force

Tensile force

Yes

No

 

 

Increment
curvature

 

_

 

 

 

 

 

Figure 3.3 — Flow chart Showing the numerical procedure associated with analysis of RC

beam exposed to fire

77

 

 

Write curvature and internal moment
(a point on moment curvature curve)

 

 

 

 

     

Ultimate
curvature"

 

Increment
strain

 

 

   

 

 

 

 

Segment = n**

   

 

Segment =
i+1

 

 

 

 

 

 

 

Nonlinear stiffness analysis of beam

 

 

 

 

 

Calculate deflection and rate of
deflections

 

 

 

*either crushing of concrete or
rupture of steel

"n = total number of segments
In the beam

 

 

Failure limit
states*

 

Increment

 

 

Time step

 

   

 

 

 

 

C E... )

Figure 3.3 (continued) — Flow chart showing the numerical procedure associated with
analysis of RC beam exposed to fire

78

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

IV V V V V V VI
_A_ _Q_
l i
L
Ambient
conditions
(a) Simply supported T beam
b
A
hf I J I
1" ’ O
bw AS d .
Fire CLCLC Fire
AS *‘~
000 v
°I O
H
c
Fire
(b) Cross-sectional details (c) Discretization (d) Boundary conditions
for

Heat transfer analysis

Figure 3.4 — Cross-sectional discretization of T-beam for analysis

79

Elements

 

 

 

 

 

 

 

 
  
 
 

 

 

 

 

 

 

 

 

 

 

in beam Geometrical .
centrOIdal axIs
Element
area gS’C
\ 81 o Ct= CS+ Cc
S.C Cq
0’1 3 | Neutral ‘— ‘ I
1111111111111 .
.—...__. ._..._._.._. ......_._ ._. 3x15 «C
y , c
--_ ....... V ..... .Z "AXIS szerc
/ Concrete m mechanical Tc
8 Tensron strain Tt= TS+TC
O ES,T _’
0.0.0 ‘——* °S,T 9T5
Force
Cross-section Total strain Stress Internal equllrbnum
diagram diagram forces Tt + Ct = 0

Figure 3.5 — Variation of strain, stress and internal forces in a typical beam cross-section
exposed to fire

80

 

A B

a) Bending moment diagram

V

 

AF \ b) Moment-curvature relationship

A

é B

c) Curvature along the Span of the beam

 

 

 

 

 

 

 

 

 

 

 

 

d) Rotation along the Span of the beam

 

e) Deflection along the Span of the beam

Figure 3.6 — Step-by-step approach for evaluating the mid-Span deflection of the beam
using M-Ic relationship

81

 

 

 

 

 

 

 

 

 

 

50 kN 50 kN
V V
1‘ VI i< *1
1.4 m 1.4 m
I J
I 1
3.96 m
(a) Elevation for Simply Supported Beam
38 mm
—+ __
2 d) 13 mm
3 d) 19 mm 406 mm
§ \
‘-\_‘\-_'9 \ 1’ 6 mm
3 3 mm i __ stirrups

 

 

 

—_

(b) Cross Section

254 mm

 

Figure 3.7 — Elevation and cross section of RC beam B1

82

 

 

 

 

 

 

 

 

 

 

 

 

 

~ 1M
1000 . —+—ASTM E119
r-s 1 ——I— Rebar-Model
g) « + Concrete(200mm from top) - Model
V 800 j -—)(—— Concrete(300nm from top) - Model
2 . - - °X- - - Rebar- Test
3 1 - - -O- - - ConcreteéZOOrrm from top; -Test
2 600 j 200 X 300 - - -+- - . Concrete 300mm'fi'om top - Test
a : mm mm
E 400 j X
200j + ”WP” __..-o--o"
‘ .X g—-.“O ‘5- +¢ v H. .
0 ‘ ' ' 5 ' ' ' l T T I 1 I T I T T
0 30 60 90 120 150 180 210
Time(min)

Figure 3.8 — Comparison of measured and predicted average rebar and concrete
temperature as a fimction of time for beam Bl

Time(min)
O 30 60 90 120 150 180 210

 

1 1 L A 1 1 I I l
1 I

 

 

 

 

 

 

 

A
E
E
S
'5
u
é 250 : —O—Deflection- Model
0 - .
G i - - -I- - - Deflection-Test
-300 5
-350 €
-400 ,

 

Figure 3.9 — Comparison of measured and predicted mid-Span deflection as a function of
time for beam BI

83

18.1 kN/m

 

V V V

 

 

 

 

 

7.62 m

(a) Elevation for Simply Supported T Beam

 

 

 

 

 

 

 

 

 

1905 mm
L 41
I‘ '1
152 mm m
r
\\
\‘ 2020mm
550 mm
3020 mm
K~~
[1.0.0
61 mm I _

 

 

 

 

 

 

61 mm

317mm

(b) Cross Section

 

——()—

I 61mm

635 mm

Figure 3.10 — Elevation and cross section of RC Beam B2

84

 

 

  
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

600 4 .0’
G 500 j
0" .
2 4003
B :
a .
a .
E: 200 2 ---o---Rebar-Model
——-l— Rebar -SA FIR
100 r
0,....,....,....,....,....,....
0 50 100 150 200 250 300
Time(min)
Figure 3.11 — Variation of rebar temperature for Beam B2 as obtained from
proposed model and SAP IR
1200 ,
1000 : —O—ASTM E119
—-— 1/4 depth-Model
g: 8007 +l/2depth-Model
e A ——)(—- 3/4 depth-Model
g 600 --x--- 1/4depth-SAFIR
g. , ------o 1/2depth- SAFIR
E : ~-+— -- 3/4 depth-SAFIR
*9 400 -
H -
j . l a . . . .
O‘—- I"I'II'IIIT'I'IIIIIIIII'
0 50 100 150 200 250 300
Time(min)

Figure 3.12 — Variation of temperature at three different concrete locations in beam BZ
predicted by the proposed model and SAF IR

85

200 250

 

 

01
-200
E ‘ ,
E ‘------o SAFIR 9..
'§ -400 — ._i—l—‘Model 2..
a '0.
5 a.
- O
-600 e
-800

 

 

 

Figure 3.13 — Variation of mid-Span deflection for Beam BZ as predicted by the proposed
model and SAP IR

Peak stress

 

 

 

0
3 j " \ +20°C I
25 5 ' ' +4°°°C1
: : : +600°C1
: : I H Ultimate
20 3 . i ..... strain
. I I .. _.
' l
' I

..v-——q::,

Strain corresponding...
1 to peak stress

Stress (MPa)
3

   

 

l
l
l
l
l
l
'5
I:
l5
I;
I;

v-..,

 

 

I
1
I
!
I
I
I
I
I
l
1
1
I
1

0____-______a

 

0W ._
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

<1
4

Strain

Figure 3.14 — Representative stress-strain curves of concrete with ultimate strain and
strain corresponding to peak stress value

86

18.1 kN/m

 

 

 

 

 

 

1 1 1 i 1 1 l
i i
7.62 m
(3) Elevation for Simply Supported Beam
1905 mm
IA
1‘ 'l
7
I52 mm C7—C3 I 6] mm
\\
\‘ 2o20mm
635 mm
398 mm
3020 mm
152 mm \
I 61 mm b_o_c .1
H
61 mm
I<——+I
317 mm

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(b) Cross Section

87

 

Figure 3.15 — Elevation and cross section of RC I beam B3

 

600 .

l l

500

b

O

O
J

    

 

Temperature(°C)
L»)
8

—+— Rebar-Model

 

 

 

 

 

 

100% 4 777777 , . -~I---Rebar-SAFIR
0..T.......................r....
O 50 100 150 200 250 300
Time(min)

Figure 3.16 — Variation of rebar temperature for Beam B3 as obtained from proposed
model and SAP IR

 

   
 
   
   

 

—-0— ASTM E119

—l— 1/4 depth-Model
+ 1/2 depth-Model
—)(—— 3/4 depth-Model
- - -x- - - 1/4 depth-SAFIR
- - -o- - - l/2depth-SAFIR
- - -+- - - 3/4 depth-SAFIR

 

 

 

Temperature(°C]

iLLuiiiil

200

1 l

 

 

 

300

Figure 3.17 — Variation of temperature at three different concrete locations in
Beam B3 predicted by the proposed model and SAFIR

88

Time(min)
0 50 100 150 200 250 300

 

 

 

 

 

-400 :

 

Deflection(mm)

-500 5
-600 E

-1

-700 “

 

 

 

Figure 3.18 — Variation of mid-span deflection for beam B3 (I beam) as predicted by the
proposed model and SAP IR

89

CHAPTER 4

4. PARAMETRIC STUDIES

4.1 General

The review of experimental and analytical studies on RC beams, presented in Chapter 2,
indicate that most of the previous studies were carried out under standard conditions and
did not take into consideration critical factors such as realistic fire scenario, loading and
failure criteria, which influence the fire response of RC beams. Therefor to investigate
the effect of various parameters on the fire performance of RC beams, a parametric study
was conducted on RC beams using the macroscopic finite element model presented in
Chapter 3. The parameters that are studied include load level, aggregate type, fire
scenario and concrete strength. Results from the analysis were used to quantify the
influence of various parameters on the fire response of RC beams. Details of the

parametric study are presented in this chapter.

90

4.2 Critical factors influencing fire resistance
A review of literature presented in Chapter 2 indicates that fire response of RC beams is
influenced by a number of factors. Recent numerical studies (conducted by Kodur and
Dwaikat [2008b] and Dwaikat and Kodur [2008a]) on RC beams with rectangular cross-
section have shown that the most important factors that influence the fire response of RC
beams are:

0 Load ratio,

0 Aggregate type,

0 Fire scenario,

0 Concrete strength,

0 Failure criteria,

0 Concrete cover thickness,

0 Location of axial restraint,

0 Span-to-depth ratio, and

0 Axial and rotational restraint
Dwaikat [2009] conducted a parametric study to investigatie the effect of section

characteristics, load ratio (LR) , concrete type, aggregate type, span-to-depth ratio, degree
of axial restraint (kr), location of axial (Y/H) restraint, rotational restraint, fire scenario,
concrete strength, spalling and failure criteria on the fire performance of rectangular RC
beams. Results of the parametric study obtained for rectangular RC beams are reproduced

in Table 4.1. Different cross-sections used for the analysis are presented in Table 4.2. All

the analyzed beams were designated by five characters from left to right as follows:

0 Concrete type (N for NSC, H for HSC),

91

0 Aggregate type( S for siliceous and C for carbonate aggregate),

o Cross-section number (1 ,2,3,4 as described in Table 4.2),

0 Support conditions (S for simply supported, A for axially restrained, R for

rotationally restrained and AR for axially and rotationally restrained beams),
0 Span-to-depth ratio (S for small span-to-depth ratio of 8, M for medium span-to-
depth ratio of 13 and L for large span-to-depth ratio of 18)

For some beams and additional sixth character (number) is used for designation of
different beams whose first five characters are same.
Results from the parametric study indicate that an increase in beam width and concrete
cover thickness enhances the fire resistance of the beam by reducing the heat
transmission to the cross-section and subsequently to the rebars. Increasing load ratio
increases deflection in the beam due to yielding of steel reinforcement and increase in
plastic and creep strains. As seen from Table 4.1, beams made with carbonate aggregate
have lower deflections than those made with siliceous aggregate due to high thermal
capacity and low thermal conductivity of carbonate aggregate. Axial restraint of the
beams has significant effect on the fire resistance of the beams and it depends on span-to-
depth ratio of the beam. Rotationally restrained beams have higher fire resistance due to
the presence of moment redistribution in the beam. Beams subjected to design fire
exposure have higher fire resistance than those subjected to standard fire exposure. This
can be attributed to the presence of decay phase in design fires during which the beam
recovers part of its strength and stiffness.
In summary, the author concluded that fire scenario, load ratio, span-to-depth ratio,

location of axial and rotational restraint have significant influence while sectional

92

dimensions, aggregate, concrete strength, spalling and failure criteria have moderate
influence on fire response of rectangular beams.

However the effect of these parameters on RC beams with T and I cross-sections have
not been studies. Therefore a set of numerical studies have been conducted to quantify
the influence of the above mentioned factors on the fire resistance of T and 1 cross-

section beams.

4.3 Numerical studies on T and I cross-sections

4.3.1 Beam characteristics

Reinforced concrete beams with two different cross-sectional shapes namely T cross-
section and I cross-sectional shapes were selected for the analysis. The T beam and I
beam used for the parametric studies are the same as those described in Section 3.5.2 and
Section 3.5.3 respectively. These beams are designed as under reinforced sections as per
the specifications described in AC1 318 [2008]. Both beams are fabricated with concrete
having a compressive strength of 28 MPa and the yield strength of steel reinforcement is
assumed to be 413 MPa. Cross-sectional dimensions and other characteristics of T and 1
beams are illustrated in Figures 4.1. For concrete and steel, high temperature property
relationships specified in ASCE manual are used as input data for the analysis. All the
beams are assumed to be made up of carbonate aggregate as coarse aggregate except

those used to study the effect of aggregate type.

93

._ -. __—--—
‘ 5"

99-5.“ Emma

 

4.3.2 Analysis variables

The RC beams were analyzed using three load ratios (30%, 50% and 60%), two
aggregate types (siliceous and carbonate aggregate), four values of compressive strength
for concrete (30 MPa, 40 MPa, 50 MPa and 60 MPa). The effect of fire scenarios is
investigated by subjecting the beams to a standard fire (ASTM E119) exposure and six
design fire scenarios namely: F83, F86, F87, F8] 1, F16 and F817. Figure 4.2 shows the
time-temperature curves for the six design fire scenarios and the ASTM E119a standard
fire scenario. It can be seen from Figure 4.2 that the standard fire scenario does not have
a decay phase. However, the six design fire scenarios have a well defined decay phase.
The parametric fire time temperature curve [Magnusson SE, Thelandersson S 1970]
proposed in Eurocode 1 [1994] and the recent modifications suggested by Feasey and
Buchanan [2002] are implemented to arrive at different design fire scenarios. According
to Eurocode l, a design fire consists of a growth phase and a decay phase. Feasey and
Buchanan [2002] showed that both the growth and decay phases of the fire are influenced
by compartment properties such as the fuel load, ventilation opening and wall linings.
The design fires are selected to cover wide range of compartment characteristics and fuel
loads that are encountered in different types of occupancies (buildings). The design fires

are assumed to occur in a room of dimension 6 m x 4 m x 3 m. Values of fuel loads

ranging from 400 MJ/m2 to 1200 MJ/m2 of floor area are used. Opening dimensions are

. . . 0.5
assumed such that the ventilatlon factor 15 between 0.02—0.04 m . In order to account

for the realistic nature of lining material such as gypsum board, concrete and composite

construction material, values b (given by Eq. (2.9)) are assumed to vary from 488

94

 

WsO'S/mZK to 1900 WsO'S/mzK. The different compartment characteristics utilized to

establish the design fire scenarios are presented in Table 4.3.

All the beams were analyzed as having simply supported end condition and each beam is
discretized into twenty longitudinal segments. The cross-section of each segment was
idealized into elements having different sizes in horizontal and vertical directions. Since
the dimensions of the flange in T beam and flanges in I beam are relatively larger
compared to that of the web and also due to the presence of thermal gradients, the cross-
section is idealized into elements having sizes ranging from 2m to 10mm in the vertical
direction and 2.5mm to 25mm in the horizontal direction. Segmental and cross-sectional
discretization of the analyzed beams is illustrated in Figure 4.3.The analysis was carried
out using the macroscopic finite element model, presented in Chapter 3, in 5 minute time
increments for a total duration of four hours or till failure occurred in the beam. Fire
resistance of the beams is evaluated based on the thermal, strength, deflection and rate of
deflection failure criterion (described in Section 3.2). Results from the parametric studies

are used to quantify the influence of various parameters on the overall fire performance

of RC beams.

4.4 Results of parametric studies
Results from the parametric studies are presented in Table 4.4 and Figures 4.4 to 4.11.

These results are used to discuss the influence of various parameters on the fire response

of RC beams.

95

4.4.1 Effect of load ratio

In order to evaluate the effect of load ratio on the fire response, the beams were analyzed
by subjecting them to ASTM E119 standard fire exposure under three load ratios of 30%,
50% and 60%. The load ratio is defined as the ratio of applied load under fire conditions
to the capacity of the beam at room temperature.

The influence of load ratio on the fire performance of beams is illustrated in figures 4.4-
4.5 and Table 4.4. Figures 4.4 and 4.5 show the variation of mid-span deflection of the
beam with fire exposure time and it can be seen that the load ratio has significant
influence on the response of the beam. The deflection as well as rate of deflection, of the
beam increases with increasing load ratio and this can be attributed to early yielding of
the steel reinforcement under higher loads and also increasing high temperature creep
effects under large loads. The early yielding leads to decrease in the stiffness of the beam
and increases the deflection.

Fire resistance values evaluated based on different failure criterion are presented in Table
4.4. As can be seen from the table an increase in load ratio significantly reduces the fire
resistance of RC beams. However, the current prescriptive based approaches evaluate the
fire resistance of the beam under an assumed load ratio of 50%. Thus the results from the
current approaches may not represent the realistic assessment of RC beams if a different

load ratio is used.

4.4.2 Effect of aggregate type

The effect of aggregate type on the fire resistance of RC beams is illustrated in figures

4.6 and 4.7 and Table 4.4. The two figures show the variation of deflection with fire

96

exposure time for the simply supported T and I beams respectively. It can be seen from
both figures that the deflection of the beam with carbonate (calcareous) aggregate type is
lower that of the siliceous aggregate RC beam. The difference in deflection can be
attributed to the fact that the carbonate aggregate has a high thermal capacity resulting
from an endothermic reaction that occurs due to dissociation of dolomite. Also the
thermal conductivity of carbonate aggregate concrete is generally lower that that of
siliceous aggregate concrete [Dwaikat 2009]. High thermal capacity and low thermal
conductivity of carbonate aggregate results in lower temperature increases which in turn
produces lower deflection. Hence, the deflection of beam made with carbonate aggregate
is lower that that of siliceous aggregate RC beam. A summary of fire resistance values

computed based on various failure criteria is presented in Table 4.4.

4.4.3 Effect of fire scenario

The effect of fire scenario on the fire response of RC T and I beams was evaluated by
subjecting them to one standard fire scenario namely ASTM E119a [2008], and six
design fire scenarios.

The variation of mid-span deflection with fire exposure time for T and I beams are shown
in Figures 4.8 - 4.9 respectively. It can be seen from the figures that the deflection and the
rate of deflection of the beam exposed to fire increases during the early stages of fire
exposure. However, beams exposed to design fires show decrease in deflection towards
the later stages of fire exposure. This is due to the presence of decay phase in design fires

during which the beam starts cooling and regains part of its strength and stiffness.

97

Results of fire resistance values computed based on different failure criteria (see Table
4.4) show that the fire exposure has significant influence on the fire resistance of RC
beams. It can be seen that the beams exposed to ASTM E119 standard fire exposure
attained failure while those subjected to design fire exposure did not fail. Thus the results
show that the fire resistance value of beam computed using standard fire exposure will

give conservative results if the same beam is exposed to design fire scenario.

4.4.4 Effect of concrete strength

The effect of concrete strength on the fire resistance of T and I beams is investigated by
analyzing each of the beams with four different concrete strengths namely: 30, 40, 50 and
60 MPa. Figures 4.10 - 4.11 shows the effect of concrete strength on the mid-span
deflection of simply supported T and I beams respectively. It can be seen from both the
figures that the concrete strength does not have a significant effect on the fire response of
beams. This is due to the fact that the moment capacity of simply supported beams
depends mostly on the tension rebar temperature.

From Table 4.4 it can be seen that the fire resistance values computed for different
concrete strengths are almost similar implying that the concrete strength has a minor

influence on the fire performance of RC beams.

4.4.5 Effect of failure criteria
The fire resistance of T and I beams analyzed in the study was computed according to
strength, deflection and rate of deflection failure criterion as shown in Table 4.4. Rebar

temperature failure criterion is not considered as it does not represent the actual failure of

98

the beam [Dwaikat 2009]. It can be seen from Table 4.1 that the deflection and rate of
deflection failure criteria predicts lower fire resistance values as compared to strength
failure criteria for all the beams. Deflection and rate of deflection failure criterion are
important because the integrity of the beam cannot be maintained if it experiences large
deflections. The increased deflections lead to cracking in the bottom portion of the beam
which exposes the tension rebars to fire. Exposing the tension rebars to fire results in
rapid reduction of rebar strength and hence the beam fails early. However, the current

provisions in ASTM E119 do not specify deflection or rate of deflection failure criteria.

4.5 Summary

This chapter presents results of parametric studies conducted to quantify the effect of
load ratio, aggregate type, fire scenario, concrete strength and failure criterion on the fire
response of RC beams. Results from the parametric study show that load ratio, fire
scenario and failure limit states have significant influence on the fire resistance of RC
beams, while aggregate type and failure criteria have moderate influence. However, the
variation in concrete strength has negligible influence on the fire response of RC beams.
Further, current codes and standards do not account for critical factors such as realistic
fire exposure, deflection and rate of deflection failure criterion, which play an important
role in determining the realistic fire resistance of RC beams. Results obtained from the
parametric studies are used for developing an energy based time equivalent approach for

evaluating fire resistance of RC beams under design fire scenarios.

99

 

Tables

Table 4.1 — Summary of fire resistance values for the rectangular beams

 

Fire resistance based on failure

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Studied Beam LR k criterion (minutes)
parameter designation (%) Y/H (kN/rrnm) Rate of
Strength Deflection .
DeflectIon

NSlSM 50 0.5 O 135 123 115
NS2SM 50 0.5 0 215 192 189
NS3SM 50 0.5 O 183 165 166
Section NS4SM 50 0.5 0 233 217 218
characteristics NC 1 AM 50 0.5 50 143 NF 13 8
NC2AM 50 0.5 50 228 227 223
NC3AM 50 0.5 50 195 NF 192
NC3AM 50 0.5 50 253 NF 249

NSlSSl 30 0.5 0 168 160 143

NSlSSZ 50 0.5 0 135 128 115

Load ratio NSlSS3 70 0.5 0 108 105 93
NC3AM1 30 0.5 50 258 NF 254
NC3AM 50 0.5 50 195 NF 192

NC3AM2 70 0.5 50 145 NF NF
NS2SM 50 0.5 0 215 192 189
Aggregate NCZSM 50 0.5 0 283 245 244
type NS4AM 50 0.5 50 198 NF 194
NC4AM 50 0.5 50 253 NF 249
Span-to- depth NS 1 AS 50 0.5 50 200 NF 194
ratio NS 1 AM 50 0.5 50 110 NF 106
NSIAL 50 0.5 50 75 74 70

NSlSSZ 50 0.5 0 135 128 115
NSlAS 50 0.5 50 200 NF 194
Degree of NSlAS] 50 0.5 100 210 NF 206
axial restraint NSlSLl 50 0.5 0 135 117 115
NS [AL 50 0.5 50 75 74 70

NSlAL3 50 0.5 100 63 61 56

NSlAS] 50 0.3 50 123 NF 115
NSlASZ 50 0.4 50 143 NF 137
NSlAS 50 0.5 50 200 NF 194

NSlAS4 50 0.6 50 378 NF 373
Location of NSlASS 50 0.7 50 588 NF 582
axial restraint NSIAL] 50 0.3 50 98 96 91
NSlAL2 50 0.4 50 90 88 83

NSIAL 50 0.5 50 75 74 7O

NSlAL4 50 0.6 50 80 NF 77
NSIALS 50 0.7 50 110 NF 106
NSZSM 50 0.5 O 215 192 189

Rotational NSZRM 50 0.5 0 363 NF NF
restraint NS4AM 50 0.5 50 198 NF 194
NS4ARM 50 0.5 50 498 NF NF

 

 

100

 

“"fir -—a -

Table 4.1 (Continued) - Summary of fire resistance values for the rectangular beams

 

Fire resistance based on failure

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Studied Beam LR k criterion (minutes)
. . Y/H r
parameter deSIgnatIon (%) (kN/mm) . Rate of
Strength Deflection D H .
e ectIon
NClSLl 50 0.5 0 170 146 149
NClSL2 50 0.5 0 133 113 113
NClSL3 50 0.5 0 NF 105 105
NClSL4 50 0.5 0 NF NF NF
Fire NClSLS 50 0.5 0 NF NF NF
scenario NClASl 30 0.5 200 428 NF 425
NClAS2 30 0.5 200 425 NF 423
NClAS3 30 0.5 200 NF NF NF
NClAS4 30 0.5 200 NF NF NF
NClASS 30 0.5 200 NF NF NF
NSlSM 50 0.5 0 135 123 113
Concrete HSISM 50 0.5 0 130 118 105
type and NSlAMl 50 0.5 200 98 NF 95
spalling HSlAM 50 0.5 200 80 NF 75
NSlRM 50 0.5 0 318 307 298
HSIRM 50 0.5 0 120 NF NF

 

N/H — NSC/HSC S/C - Siliceous/Carbonate aggregate
1/2/3/4 — Cross-sectional size (refer to Table 4.2 for cross-sectional sizes)
S/A/R/AR - simply supported/axially restrained/rotationally restrained/axially and rotationally restrained
SIM/L — small/medium/ large span-to-depth ratio
NF — No failure

101

 

 

Table 4.2 — Properties for concrete cross-sections used in the analysis of rectangular
beams

 

 

 

 

 

 

 

 

 

 

 

 

Cross-section #
Pmpeny I 2 3 4
C'OSS'SeCt‘O" 300 x 900 400 x 800 700 x 400 600 x 600
(mm x mm)
R'f 2(Dl4mmtop 2<Dl4mm 3<Dl4mm 3d>14mmtop
em orcement bars top bars top bars bars
for SImply
Supported a?“ 5 o 20 mm 3 (D 35 mm 6 (D 20 mm 8 a) 20 mm
aXIally r estramed bottom bars bottom bars bottom bars bottom bars
' 5<D20mmtop 3¢35mm 6<D20mm 8<l>20mmtop
Reinforcement bars top bars top bars bars
for rotationally
restrained beams 4 (D 20 mm 3 (I) 30 mm 4 (D 20 mm 5 (I) 20 mm
bottom bars bottom bars bottom bars bottom bars
fc' (MP3) (for 40 6O 30 50
NSC beams)
1
f6 (MP3) (for 100 100 100 100
HSC beams)
fy(Mpa) 413 413 413 413
Concrete cover
thickness (mm) 40 60 4O 50

 

 

 

 

 

 

Table 4.3 — Compartment characteristics used for developing different design fire
scenarios

 

 

 

 

 

 

 

 

 

 

 

 

 

F' Fuel Lozzld Ventilation b
Ire factor 0 5 2
Scenario (MJ/ m ()5 W - / °K
floor area) (m ) ( S m )
FS 3 1200 0.04 488
FS 6 800 0.04 1900
F3 7 400 0.026 1900
FS 1 l 1000 0.02 1900
FS 16 l 100 0.04 800
FS 17 1000 0.02 1200
102

T T 1.." ‘ ._L dlmafiw—I

Table 4.4 — Summary of fire resistance values for the beams analyzed

 

Fire resistance (min)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

{‘3 E Beam Concrete Fire Load
:3 2‘ Aggregate strength . ratio _ Rate
a 2 type (MPa) scenano ((y Strength Deflection of
°‘ N 0) deflection
T C 28 E119 30 NF 430 NF
'9 T C 28 E119 50 300 240 275
E T C 28 E119 60 190 160 170
g 1 C 28 E119 30 NF 430 470
"l I C 28 E119 50 305 245 270
I C 28 E119 60 210 185 190
3 T C 28 E119 50 300 230 275
g0 a T S 28 E119 50 240 195 210
53 b I C 28 E119 50 375 290 360
< I s 23 E119 50 305 245 275
T C 30 E119 50 300 230 275
a T C 40 E119 50 305 235 275
5 T C 50 E119 50 310 240 280
‘3 T C 60 E119 50 310 240 280
0
g 1 C 30 E119 50 305 245 280
‘g’ 1 c 40 E119 50 305 250 280
U 1 C 50 E119 50 305 250 280
I C 60 E119 50 305 250 285
T C 28 FS3 50 NF NF NF
T C 28 FS6 50 NF NF NF
T C 28 FS7 50 NF NF NF
T C 28 F811 50 NF NF NF
T C 28 F816 50 NF NF NF
g T C 28 FS17 50 NF NF NF
5 T C 28 E119 50 300 240 275
O
3 I C 28 F53 50 NF NF NF
'5 l C 28 FS6 50 NF NF NF
1 C 28 FS7 50 NF NF NF
I C 28 F811 50 NF NF NF
I C 28 F816 50 NF NF NF
1 C 28 F317 50 NF NF NF
1 C 28 E119 50 305 245 270

 

*NF — No failure; C/S - Carbonate/Siliceous aggregate;

103

 

Figures

 

I
L
(a) Elevation of simply supported beam
1905 mm

 

 

 

 

 

\ 2020
550 mm mm

 

 

N‘

O_Q_O

H
| 61' mm

317mm

 

 

 

 

 

(b) Beam with T cross-section

1905 mm

A

 

 

152mm 0 C I 61mm
\

 

 

 

2020mm 63 5 mm
393 mm 3020 mm

 

 

 

 

 

 

\
152mm 61mm 0 Q g

 

 

 

H
61 mm

H—fi
317mm

(c) Beam with I cross-section

Figure 4.1 — Elevation and cross-sectional details of RC beams used in parametric studies

104

 

 

 

—ir-FS3
-—0—-FS6
-—+—-FS7
+FSII 1'
—I—FS16
~“.~ -—-—-FS17

‘ ——ASTM Ell9

Temperature(°C )

 

 

 

 

 

 

T r I I I I I I I I I I T I L

200 300 400 500

 

Time (minutes)

Figure 4.2 - Time-temperature curves for design and standard fire exposures used in the
analysis

105

 

 

(a) Elevation of simply supported beam
Segment number

 

1 | 2| [19 Lzoj

. I
L

(b) Segmental discretization of the beam

2.5 mm to 25 mm elements

L .1
1‘ '1

 

l

 

 

 

 

bps

 

 

L 2 mm to 10 mm elements A

 

 

 

 

 

(C) Cross-section details ((1) Cross-sectional discretization

Figure 4.3 — Cross-section and segmental discretization of the analyzed beams

106

 

Time (min)

 

 

 

 

 

 

 

 

0 100 200 300 400 500 600
O i i n J r 1 i J ‘ L ‘
-200 {
3-400
E
g 1
z: -600 f
8 1
E . + Load ratio = 30%
G -800 j I -I— Load ratio= 50%
I + Load ratio = 60%
-1000 j
-1200 1

Figure 4.4 — Influence of load ratio on the mid-span deflection of simply supported T—
beam exposed to fire
Time(min)
0 100 200 300 400 500 600

IIIIIIIIIII

 

 

—+— Load ratio = 30%
—l— Load ratio = 50%

 

 

+ Load ratio = 60%

 

 

 

 

 

Figure 4.5 — Influence of load ratio on the mid-span deflection of simply supported I-
beam exposed to fire

107

Time(min)
0 50 100 150 200 250 300 350

-200 ,

  

 

is
o
o

+ Carbonate Agg

 

 

 

—l— Siliceous Agg

 

Deflection(mm)
c'»
O
O

§

1 1 1 J I

 

—1000 ,

 

 

 

-1200 4
Figure 4.6 -— Effect of aggregate type on the mid-span deflection of simply supported T-
beam exposed to fire

Time(min)
0 50 100 150 200 250 300 350 400

   

1 1 l L 1 L I 1 l I I 1 1 I I I L
1 I T I

   

A 1 l l l l 1 1 l J.
l 7

     

 

 

  

+ Carbonate Aggr
+ Siliceous Aggr

 

 

 

 

 

 

Figure 4.7 — Effect of aggregate type on the mid—span deflection of simply supported I-
beam exposed to fire

108

 

   
   

 

 

 

0 50 100 150 200 250
0 1 .....
-100 5
’a‘ :
3-200 -
fl 1 l
.3 I—I—FSB
O , .
o >—i——FS7
c: -300 - . .
a l—x—Fs11 l
l—x—FS16 l
400 _ I—o—FS17 l
‘ —I—ASTME119l
-500 _

Figure 4.8 — Effect of fire scenario on the deflection of a simply supported T beam
exposed to six design fires and a standard fire

 

 

    
 

,
—+—FS1
+ F82
+ F83

—x——FS4

Deflection(mm)
('6
O
C

-—)I(—- F55

+ FS6
—I—ASTM E119

 

 

 

-400

 

Figure 4.9 — Effect of fire scenario on the deflection of simply supported I beam exposed
to six design fires and a standard fire

109

   
   

0 _

-200 {

 

-400 5
- +fc'=30MPa

-600§ +fc'=40MPa
+fc'=50MPa ‘

-800 f
* —-x—fc'=60MPa

 

 

 

Deflection(mm)

-1000 3
-1200 f

-1400 f

 

 

 

-l600 1

Figure 4.10 — Effect of concrete strength on the deflection of simply supported T beam
exposed to fire

Time(min)
0 50 100 150 200 250 300 350

 

L l l l l I l L L l l L l l A I L I 1 l I I
I T 1
q
-<

l

-200 q

is

c:

c:
I

 

Deflection(mm)
65
E3

I
00
O
O
1

 

 

 

111411

 

-1000

 

 

 

-1200 1

Figure 4.11 — Effect of concrete strength on the deflection of simply supported I beam
exposed to fire

110

 

J i It'll l

CHAPTER 5

5. ENERGY BASED EQUIVALENT APPROACH

5.1 General

The current approach of evaluating fire resistance of RC beams is based on prescriptive
based approaches which are derived from standard fire tests. In these standard fire tests,
the fire resistance of RC beams is evaluated under a standard fire exposure which follows
a predefined time-temperature relationship. This time-temperature relationship does not
provide a realistic representation of typical compartmentation fire conditions since it does
not consider the actual fuel and ventilation conditions present in buildings. Hence the
standard fire tests do not provide a rational and realistic fire resistance assessment of RC
beams. Thus the current prescriptive based methods may not be fiilly applicable for
undertaking performance based fire design of structural members. For realistic
assessment of fire resistance of the beam, performance of the beam should be evaluated

under realistic fire exposure, loading and failure criteria. However, conducting full scale

111

 

fire tests under realistic fire exposure may not be practical or economical since numerous
types of fire exposure conditions have to be considered. Alternatively “time equivalency”
concept which relates the severity of a design fire to that of a standard fire exposure can
be applied to evaluate fire resistance under design fire. However, there is lack of reliable
time equivalency methods for evaluating fire resistance of RC members. In this chapter,
an energy based time equivalent approach is developed for evaluating the fire resistance

of RC beams under any given design fire scenarios.

5.2 Energy based time equivalent approach
5.2.1 General approach
A review of literature presented in Chapter 2 indicates that the current methods for
evaluating time equivalent of structural members are not consistent. The time equivalent
(fire resistance) predicted by various methods shows a significant variation even for the
same fire exposure. In addition, most of the empirical formulae are derived for protected
steel members and hence they may not be applicable for RC members. To overcome
some of the drawbacks, an energy based time equivalent approach has been developed for
establishing equivalency between design and standard fire exposure.
The proposed approach utilizes energy equivalence to compute the time equivalent of
design fires with respect to standard fire exposure. According to this method, time
equivalent can be computed through the following steps:

0 Evaluate fire temperatures (both for design fire and standard fire exposures) in the

entire duration of fire exposure time,

112

 

0 Using the above calculated fire temperatures, compute the total amount of energy
transferred to RC beam from design fire exposure,

0 Similarly using the standard fire temperatures, the amount of energy transferred to
RC beam from standard fire exposure is calculated at various time steps,

0 Identify a time on the standard fire energy curve where the total amount of energy
transferred by design fire exposure is equal to the cumulative energy transferred
by standard fire exposure up to that time. The time point at which the energies are
equal is defined as the time equivalent.

The fire temperatures for the design fire are computed, as a function of time, based on the
compartment characteristics such as dimensions of the compartment, size of the openings
and the amount of combustible fuel load present. These fire temperatures are evaluated
using the parametric fire time-temperature relationships provided in Eurocode 1 [1994].
For the standard fire exposure, the temperatures are computed using the predefined time-
temperature relationships specified in codes. Following this energy transferred to the RC
beam under standard and design fire exposures is computed. By equating two energies

under standard and real fire exposure, time equivalency is established for an RC beam.

5.2.2 Evaluating fire energy transferred to beam

The proposed approach computes time equivalent based on amount of energy transferred
to the beam (energy equivalence) from two fires under comparison. The approach is
based on the principle that a structural member is said to experience same level of fire
severity if same amount of energy is transferred to the member from two different fire

scenarios. The amount of energy transferred is mainly by the heat flux transferred from

113

_ _i-

 

the fire exposed surfaces to the beam through convection and radiation heat transfer
mechanisms. The convection and radiation heat flux on the boundary of an RC beam

exposed to fire can be expressed by the following two formulae respectively [Buchanan

2002]
qc=hc(Tf—Tc) (5.1)
q, = our; - T64) (52)
‘1
|
where I
. 2 «.l
qc = convect1ve heat flux (W/m ), f i

q, = radiative heat flux (W/mz),

hc = convective heat transfer coefficient (W/m2°K),
Tf= temperature of fire (°C or °K),

Tc = temperature on the surface of boundary (°C or °K),

o = Stefan-Boltzmann constant (5.67x10-8 W/m2°K4) and
a = emissivity.

Generally the temperature on the surface of boundary (TC) is close the temperature of fire

(Tf) itself. Thus, the radiative heat flux can be approximated as follows:

114

 

q, = 019(Tf4 - T64)
2 2 2 2
=a£(Tf + Tc )(Tf —-TC)
:04sz + T} )(Tf + TC)(Tf — TC) (5.3)
z 05(sz + T} )(Tf + Tf)(Tf — TC)
z 4ang3(Tf — Tc)
Hence, the total heat flux transferred to an RC beam can be written as:
q = qc + q, z hc(Tf — Tc) + 4ang3(Tf — Tc) (5.4)

Further if we assume Tf— TC = an, (where a is a constant), Eq.(5.4) can be rewritten as:

q z a(40'£Tf4 + thf) (5.5)

Then the total amount of energy transferred to an RC beam for a particular fire exposure

(design or standard) can be approximated by the following formula:
E: qudz z j A 01(40an + thf)dt (5.6)

where
A = area of boundary exposed to fire and
E = total energy.

As both A and a are constants, Eq (5.6) can be rearranged as follows

E = out [(40504 + thf)dt (5.7)

or

E = aA x ( Area under heat flux curve (_q_ ) ) (5.8)
a

This ‘E’ is the total energy bound by the time-temperature curve for a given fire

exposure.

115

m

f!'”' ‘

 

 

5.2.3 Computation of time equivalent

The third step in the energy based approach involves the computation of time equivalent.

The amount of energy transferred to RC beam from the design fire exposure (Ed) is

computed as described in the previous section. Similarly the amount of energy transferred

to RC beam from standard fire exposure (Es) is also computed. Thus using this approach,

a design fire will have the same severity as that of the standard fire if

E, = Ed (5.9)

E S = total energy under the heat flux (1) curve of the standard fire, and
a

Ed = total energy under the heat flux (—q—) curve of the design fire
0:

Consequently, the equivalent time can be computed by equating the total area under the
heat flux (1) curve for the design fire with the area under the heat flux (—q-) curve for
a a

the standard fire as shown in Figure 5.1 (for the standard and design fires). To arrive at
equivalency, first the total area under the heat flux curve for the design fire (area B in
Figure 5.1) is computed. The area under the heat flux of a standard fire (area A in Figure
5.1) is computed at various time steps. The time at which area A (which varies as a

function of time) equals area B is defined as the time equivalent of the design fire.

116

 

5.3 Numerical studies

The development of above discussed energy based time equivalent approach requires
large set of fire resistance data on RC beams under standard and design fire exposures.
Such data can be generated through numerical simulations on RC beams under various
fire scenarios. For the analysis, the macroscopic finite element (FE) based computer
program developed in Chapter 3 was selected. The advantage of utilizing the FE program
is that various fire, and support conditions can be accounted for in evaluating the fire
response of RC beams.

Rectangular RC beams were analyzed under 18 fire scenarios and four different support
conditions resulting in 72 beam-fire combinations. Results from FE analysis was used to
establish fire resistance time equivalent of each beam under a given design fire exposure.
This predicted time equivalent is considered to be reliable since finite element analysis
accounts for various factors such as support conditions, high temperature material
properties and fire scenario that influence fire resistance of RC beams. Fire resistance
data generated through these numerical simulations is used to verify the energy based
approach under different scenarios. Details on the computer program, the analysis and the

analyzed beams are discussed in the following section.

5.3.1 Design parameters

For generating time equivalent data, four rectangular RC beams (each with different
support conditions) were analyzed. The four beams had a simply supported, axially
restrained, rotationally restrained and both axially and rotationally restrained end

conditions. Details of the beam and boundary conditions are illustrated in Figure 5.2. The

117

Inimi—

1
.. -

boundaries restraining the beam are idealized as a spring of stiffness (k) as shown in
Figure 5.2. Two values of axial restraint stiffness, namely 0 and 20 kN/mm, are selected
for the analyzed beams. An axial restraint stiffness value of zero represents a simply
supported condition and stiffness of 20 kN/mm represents a restrained condition.

All the beams are of rectangular cross section (300 mm x 500 mm) having a span of 6 m
as shown in Figure 5.2. The beams are assumed to be made of concrete with a
compressive strength of 30 MPa and reinforced with steel rebars having yield strength of
400 MPa. The applied load on simply supported and axially restrained beams was 14
kN/m (equivalent to a load ratio of 40%), while the corresponding applied load on
rotationally restrained, rotational and axially restrained beams was 30 kN/m (equivalent
to a load ratio of 40%). Load ratio is defined as the ratio of expected loads on the beam
during a fire to the loads that would cause collapse of beam at room temperatures. The
fire resistance analysis was carried out in 2.5 minute time increments for a maximum fire
exposure time of 8 hours. Data from fire resistance analysis was used to derive time

equivalent for the seventeen design fires with respect to that of standard fire exposure.

5.3.2 Fire exposure scenarios

In order to generate data for applying equivalent fire severity concept, the beams were
analyzed under one standard fire [ASTM E119a, 2008] and seventeen design fire
scenarios namely FSl through FS17. ASTM E119 fire represents the fire scenario used in
standard fire resistance tests and is similar to other standard fire scenarios specified in
standards such as ISO 834 [1974]. Figure 5.3 shows the time-temperature curves for the

standard and various design fires (FSl through F 817) used in the analysis. The design

118

 

fires are selected to cover wide range of compartment characteristics and fuel loads that
are encountered in different types of occupancies (buildings). The parametric fire time
temperature curve [Magnusson SE, Thelandersson S 1970] proposed in Eurocode 1
[1994] and the recent modifications suggested by Feasey and Buchanan [2002] are
implemented to arrive at different design fire scenarios. According to Eurocode 1, a
design fire consists of a growth phase and a decay phase. Feasey and Buchanan [2002]
showed that both the growth and decay phases of the fire are influenced by compartment
properties such as the fuel load, ventilation opening and wall linings. These design fires

are assumed to occur in a room of dimension 6 m x 4 m x 3 m. Values of fuel loads

ranging from 400 MJ/m2 to 1600 MJ/m2 of floor area are used. Opening dimensions are

. . . 0.5
assumed such that the ventilation factor 1S between 0.02—0.04 m . In order to account

for the realistic nature of lining material such as gypsum board, concrete and composite

construction material, values of b (given by Eq. (2.9)) are assumed to vary from 488

WsO'S/mZK to 1900 WSO'S/mZK. The different compartment characteristics utilized to

establish the design fire scenarios are presented in Table 5.1.

5.3.3 Determining time equivalent

The time equivalent for the analyzed beams was evaluated by the maximum deflection
method using the results obtained from FE analysis described in previous chapters. The
maximum deflection method is selected (over minimum load capacity method) since the
failure of an RC beam under fire exposure is generally governed by deflection failure

criteria. This was the case for most of the beam analyzed in Chapters 3 and 4. It should be

119

 

 

 

noted that the fire resistance based on deflection criteria is very close to that obtained

from strength criterion.

5.3.4 Evaluating time equivalent from current methods

The analysis of four types of rectangular RC beams under one standard fire and seventeen
design fires produced seventy two time-deflection curves. The time-deflection curve of a
beam under design fire is compared with standard fire exposure to establish the time
equivalent for each beam-fire combination using the maximum deflection method. The
time equivalent for each combination is also evaluated using the currently available
empirical formulae and the equal area method. The comparative study did not include the
minimum load capacity and the maximum temperature methods since these methods may
not provide realistic fire resistance values and also because these methods require
detailed finite element analysis which limits their use for design purposes, as discussed in
Section 2.4.4. The time equivalency predicted by maximum deflection method is
considered to be the most reliable value for time equivalency. Maximum deflection
method is considered to be reliable because the integrity of the beam cannot be
maintained under large deflections. Further increasing deflection results in cracking of
the beam, which exposed the rebars directly to the fire, causing premature failure of the
beam. A comparison of estimated time equivalent values based on equal area method and
empirical formulae (CIB, Law and Eurocode) with that predicted by the maximum
deflection method for the 72 beam-fire combinations are shown in Figure 5.4. The

conservative and unconservative regions for the data points are also shown in Figure 5.4.

120

 

An observation of time equivalent values presented in Figure 5.4 show that Law formula

predicts lower fire resistance for all the fire scenarios.

It can be seen from Figure 5.4, that there is significant variation in the time equivalent (te)

values predicted by empirical formulae and equal area method. Almost all the time

equivalent values predicted by CIB formula are unconservative and the variation in the

predicted te values increases as the severity of the design fire increases (as the time

equivalent value increase). Similar to the CIB formula, the time equivalent values
predicted by the Law formula and Eurocode formula are also on the unconservative side
and are highly scattered for increasing values of time equivalent (i.e., for severe design
fire exposures). In general, the time equivalent values predicted by empirical formulae
are unconservative and the accuracy of prediction decreases as the fire severity increases.
The unconservative nature of time equivalent values predicted by empirical formulae can
be attributed to the fact that these formulae were derived using the maximum temperature
concept for protected steel members under design fire scenarios. Thus they may not be
fully applicable for predicting the time equivalent of RC members.

As can be seen in Figure 5.4, the time equivalent values predicted by the equal area
method have less variation than those predicted by the empirical formulae. However,
similar to different empirical formulae, the variation in the time equivalent values
predicted by equal area method becomes significant for severe design fire scenarios.
Also, equal area method gives unconservative predictions for almost half of the time
equivalent values computed in this study. This can be related to the fact that the equal
area method predicts the same time equivalent value for different shaped time-

temperature curves provided that they have equal areas under time-temperature curves.

121

.7

w:-

m lent“ I‘m!

 

I-a

 

This clearly indicates that the time equivalent computed based on the equal area method
may not be conservative under many fire scenarios.

In summary, the time equivalent values predicted by equal area method and empirical
formulae are generally unconservative and have significant variation. Thus, the current
time equivalent methods do not yield reliable fire resistance predictions for RC beams.
To overcome these drawbacks, a semi-empirical energy based approach for establishing

time equivalency of RC beams is proposed.

5.4 Calibration
To improve the accuracy of predictions from the proposed method, estimated time
equivalency values from energy based approach have been calibrated against time

equivalent values obtained from finite element analysis. An analysis of results indicate

that there exists a correlation between the ratio (te(FE)/ te(energy)) of the two time

equivalent values (predicted by the FE method and the equal energy method) and the
maximum temperature of design fire as shown in Figure 5.5. It can be seen from Figure
5.5 that the ratio of time equivalent predicted by the FE method to that predicted by equal
energy method decreases with increase in the maximum temperature of design fire. This
can be attributed to the fact that the extent of fire damage in the beam depends not only
on energy transferred from fire but also on other factors such as temperature distribution
and resulting thermal gradients across the beam. These thermal gradients are large in case
of fires with high maximum temperatures.

Data generated from FE analysis was randomly divided into two sets. The first set was

used for the calibration of the method and the second set was used to validate the method.

122

 

 

Almost half of the beam-fire combinations have been selected at random and the ratio of
time equivalents obtained by FE and energy method are plotted against the maximum
temperature reached in respective fire scenarios as can be seen from Figure 5.5. Least
sum of square of error analysis is carried out to obtain a best fit and a conservative line

for that correlation and is shown in Figure 5.5. Accordingly, the equation of conservative

 

line for predicting the ratio between the two time equivalents (te(FE)/ te(energy)) is given h
as
’e(FE) -=
—— =1.6—0.00042 *T (5.10) “
t max
e(energy)
where

tearE) = time equivalent computed from maximum deflection method (or FE

analysis),

te(energy) = time equivalent computed from equivalent energy method and

Tmax = maximum temperature of design fire

Thus, the actual time equivalent of a design fire can be estimated by the following

equation

=(1.6—0.00042*T )*z (5“)
max e(energy)

te(FE)
A 95% confidence interval for the variation of temperature was determined for the above

data set used for calibration. Accordingly, the lower and upper confidence interval limits

are found to be equal to 1008°C and 1168°C.

123

5.5 Validation

To illustrate the validity of the proposed method, time-equivalent values computed based
on the energy method are compared with those obtained from FE analysis and equal area
method. Three cross-sectional types of RC beams namely rectangular, T and I beams are
used for validation. The rectangular beams are same as that described in Section 5.3.1
and shown in Figure 5.2, while T and I beams are those described in Section 3.7.2 and
Section 3.7.3 respectively. As mentioned earlier (Section 5.4) data generated from FE
analysis for rectangular beams was divided into two sets. First set of data was used for
calibration of the proposed method while the second set of data was used for validation.
In order to generate data for T and I beams, both beams are subjected to six randomly
selected design fire scenarios (namely FS3, FS6, FS7, F 81 1, F816 and FS17) and time

equivalent values are computed through finite element analysis using the macroscopic

finite element model. A summary of computed te values for rectangular RC beams is

presented in Table 5.2.

Figures 5.6 -5.8 shows variation of time equivalent values predicted by equal area
method, empirical formulae, FE analysis and proposed equal energy method along with
conservative and unconservative regions for rectangular, T and I beams respectively. It
can be seen from the figures that almost all the time equivalent values predicted by the
equal energy method are on the conservative side throughout the range of fire scenarios
considered. Time equivalents predicted by equal area method shows less scatter as
compared to empirical formulae. The figures also show that the time equivalent

computed based on equal energy method has less variation as compared to other methods.

124

 

Thus, equal energy method can be considered as a reliable method for estimating time
equivalent of design fires.

In order to illustrate the applicability of the proposed method (Eq. 5.11) to an
independent set of data, four RC beams were analyzed under six design fire scenarios.
Cross-sectional details and properties of the beams used for this analysis are presented in
Table 5.3 and selected fire scenarios are shown in Figure 5.9. A comparison of time
equivalent obtained from the proposed method with those obtained from macroscopic
finite element model (described in Chapter 3) is shown in Figure 5.10. It can be seen
from Figure 5.10 that the predictions from the proposed method are in good agreement
with finite element results, and are conservative for most of the beams. The coefficient of
determination was found to be equal to 0.94 indicating that the proposed equation is
sufficiently accurate in estimating time equivalent in comparison with detailed finite
element analysis. A comparison of figures 5.6 and 5.10 show that time equivalent
computed from the proposed method has less variation than other methods. Thus the
proposed method can be used in predicting conservative and relatively better estimate of

time equivalent under design fires.

5.6 Numerical example

In order to illustrate the applicability of the proposed energy based time equivalent
method to practical design situations, fire resistance of an RC beam is evaluated by
applying the proposed energy based approach. A simply supported rectangular beam of 6
m span length and made of concrete with a compressive strength of 30 MPa and

reinforced with steel rebars having yield strength of 400 MPa is selected. The beam is

125

 

assumed to be exposed to design fire F85 (as shown in Figure 5.3) and the time
equivalent is evaluated. Detail step-by-step procedure involved in computing the time
equivalent is presented in Appendix D.

The total amount of energy transferred to RC beam from F85 is calculated to be equal to
39814995 Joules. Next, the amount of energy transferred to RC beam from ASTM E119
standard fire is evaluated at various time intervals. Subsequently the difference in total
energy of design fire and cumulative energy of standard fire is computed at each time
step to find a minimum value. The minimum difference in energy occurs at 150 minutes.
Hence according to the equal energy method, design fire F85 has a time equivalency of
150 minutes. The same beam is analyzed (using the numerical model described in
Chapter 3) under ASTM E119 standard fire exposure and fire resistance of the beam is
found to be equal to 272 minutes. Since the computed time equivalent is less than the fire
resistance (failure time) of the beam, the beam can survive complete burnout under
design fire F85 (without failure). In summary, failure occurs in the beam if time

equivalent is greater than the fire resistance of the beam.

5.7 Summary

Currently available empirical formulae for computing the time equivalent are mainly
derived for protected steel members and may not be fully applicable for RC members.
Further, the time equivalent values predicted by current methods and empirical formulae
show significant variation and the accuracy of prediction decreases with increasing fire
severity. To overcome these drawbacks, a reliable semi-empirical energy based approach

for evaluating the time equivalent of RC beams is presented in this chapter. The proposed

126

 

 

 

approach is validated by comparing the predicted time equivalent values form the
approach with those obtained from existing time equivalent methods and detailed finite
element analysis. Based on the results obtained in this study, it is concluded that the
proposed approach gives a reliable estimate of time equivalent for RC beams compared
to existing methods. In order to facilitate the use of the proposed approach in practical
design situations, a simple linear design equation has been developed and validated.
Using this equation, time equivalent of RC beams for any design fire exposure with
respect to standard fire exposure can be estimated with sufficient accuracy without the

need for expensive full-scale fire resistance tests.

127

 

Tables

Table 5.1 — Compartment characteristics used for arriving at different design fire

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

scenarios
. ' Fuel Load (M J /m2 Ventilatign5 Factor 0 5b 2
FIre ScenarIo floor area) (m - ) (Ws - /m °K)

F81 1600 0.02 488
F8 2 1200 0.02 488
FS 3 1200 0.04 488
FS 4 800 0.04 488
F8 5 1600 0.02 1900
F 8 6 800 0.04 1900
F S 7 400 0.026 1900
F8 8 1 100 0.04 488
F8 9 1300 0.03 1900
F8 10 900 0.04 488
F8 11 1000 0.02 1900
F8 12 700 0.035 1900
F8 13 1100 0.02 488
FS 14 800 0.04 1200
F8 15 800 0.04 1000
F8 16 1100 0.04 800
FS 17 1000 0.02 1200

 

 

 

 

 

 

128

 

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Table 5.3 — Cross-sectional details and properties of beams used in the analysis

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Property Beam I Beam 11 Beam III Beam 1V
Simulated by Tested by D'Ol‘terztedebgind Tested by
Kodur and Lin et at “3:556" Dwaikat
DwaIkat [2007] [1981] [1985] [2009]
Width (mm) 300 305 200 255
Depth (mm) 500 355 600 405
Length (m) 6 6.1 6.5 3.96
Tension reinforcement 3 (1)20 mm 4 (1)19 mm 3 (1)22 mm 3:11:19
Compressmn 2<I>14 mm 2<I>l9 mm 2<D12 mm 24)”
reinforcement mm
fc' (Mp3) 30 3o 15 58.2
Load ratio 0.5 0.42 0.263 0.576
Applied load (kN) 120 80 65 141.5
25 (bottom)
Concrete cover (mm) 40 38 (si de) 40 38
Aggregate type Carbonate Carbonate Siliceous Carbonate

 

 

130

 

 

Figures

180
160
140

120

 

100

M +Design fire

80 —I— Standard fire

Heat flux

 

 

 

Area A

20

   

0 50 100 l 50 200

Time(minutes)

Figure 5.1 — Equivalent energy concept for standard and design fire

131

 

40 mm 150 mm

 

 

 

 

 

 

 

 

 

 

 

 

+ +——+
'1'
$2 ; fc = 30 MPa
/
// fy = 400 MPa
6 ¢ 20 mm 500 mm
\ x
\\ \
d) 10 mm
40 mmi 2_ stirrups
If 1
300 mm
(a) Cross section
14 kN/m
F + v v v v t k I
_A_ _O_
l i
6 m

(b) Elevation for simply supported restrained beam

30 kN/m
v v v v v

 

 

   

6m

(c) Elevation for fixed ended restrained beam

Figure 5.2 — Cross section and elevation of RC beam used in the analysis

132

 

 

 

1400 .
1200

1000 '
800

600

Temperature(°C )

400

200

 

 

Time (minutes)

Figure 5.3 — Time temperature curves for design and ASTM E119 standard fire exposure

 

450 l , , ‘7‘. _-7_--,. *—

A400 !
350 3‘ UnconservatiVe / ‘
300 :1 /

3 I ‘ ‘ / Conservative

 

te from FE analysis (minutes)

 

 

250 1 g i 1
200 i: I ‘ . Tcg 777 %
150 " ' MW 1.
100 % lg/{i A Eurocode “1
f ‘ X EqualArea '
5° . 133211992? 803
0 if 'fi'fi ‘fi ’fi ' '_'_ fi" 7 —T‘ ‘—fi "—T"f ‘7" 2,, ' " v i

0 100 200 300 400 500

Estimated te using various methods (minutes)

Figure 5.4 — Comparison of time equivalent computed based on FE analysis with that of
other methods

133

 

 

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._a
N
U]
. rr.r+Li_i_i_l_L._r_i_r_l_er.l .L_L_A 11 r r
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Ratio of b(FE)/te(Energy)
J

 

 

w; ,, _ * I
0.95 jl 0 Ratio of . oz.
. i te(FE)/teéenergy_) o ‘
035 _j‘ -—l—-Propose equation
ii - - - -Bestfitline
0.75 “‘“fi;~ x . . . , . , . Ts r ,# ‘ ‘fifi, fl fly

 

 

600 700 800 900 1000 1100 1200 1300 1400

 

Maximum Fire Temperature (°C)

 

 

 

 

 

 

 

t
. . . e FE . .
Figure 5.5 — Vanation of 4 With maxrmum fire temperature
te(Energy)
450 :
’5? 400 Q
*3 Unconservative
,5 350 a
S E
g 300 g I r s a
M
>. 250 r ' '
T" j ! § g A § Conservative
E'- 2
°‘ 200 : I A o x
H 1 AL
‘2 150 r: I. ‘§ C o CIB l
< X I La 1
E 100 5 A Eurocode ‘
“-I : X EqualArea
4’ 50 L + Safe Line Boundary
3 . 0 Energy method points used for calibration
0 . y . y . ‘ . v V ‘ T 11 A Energy method pomts used for validation ;
0 100 200 300 400 500

Estimated te using various methods (minutes)

Figure 5.6 — Comparison of time equivalent from equal energy method with predictions
from other methods for a rectangular beam

134

te from FE anal

 

  
 
 
   

ysis (mimit’es)

 

 

  

 

 

 

 

 

250
~ Unconservative
00 —*
Conservative
150 r I. ‘A. ox
I O
100 ~ ' 0 cm
i I Law
A Eurocode
* x EqualArea E
50 —)tt— Safe line boundary
0 Equalenergy
0 T T ‘ V Y 1 r T

 

 

0 50 100 150 200 250
Estimate te using various methods (minutes)

Figure 5.7 — Comparison of time equivalent from equal energy method with
predictions from other methods for a T beam

 

   
   
 

 

 

te from FE analysis (minutes)

 

 

 

 

 

250
Unconservative
200 -
. Conservative
150 _
.{
l00 + CIB
Law
Eurocode
: Equal Area
50 . " +Safe line boundary
4 O Equalenergy
0

 

T l’ T 1' T I T T T T I Y T

0 50 100 150 200 250
Estimate te using various methods (minutes)

 

Figure 5.8 — Comparison of time equivalent from equal energy method with
predictions from other methods for an I beam

135

Temperature(°C )

te from FE method (min)

 

 

 

 

0 100 200 300 400 500

Time (minutes)

Figure 5.9 — Fire scenarios used in the analysis of RC beams

 

 

240 T— ~#7‘
* l—“* ‘__7 :
J l—Best fit line] 1
180 i
i l
120 1 ‘
w l
60 l i
1
0 i 7 fi—Vifi—i Y—fi fi—— Viuhww fii V fifi
0 50 100 150 200

te from proposed equation (min)

Figure 5.10 — Comparison of time equivalents from the proposed equation and the

finite element analysis

136

CHAPTER 6

6. CONCLUSIONS AND RECOMMENDATIONS

6.1 General

The fire response of RC beams exposed to fire is presented in this thesis. A numerical
model was deve10ped for tracing the fire response of RC beams under realistic fire and
loading conditions. The model is based on macroscopic finite element approach and uses
moment-curvature relationships to predict the response of rectangular, T and I cross-
section RC beams.‘ The model accounts for high temperature constitutive material
properties, various strain components and different fire exposure conditions. The validity
of the model is established by comparing the predictions from the model with those
obtained from tests and other numerical programs such as SAFIR. The proposed
numerical model was used to undertake a set of parametric studies to quantify the
influence of various parameters on the fire response of RC beams. Data generated from
the parametric studies was used to develop a semi-empirical energy based approach for

evaluating time equivalent of RC beams under design fire scenario. The proposed

137

 

approach, which equates energy between a standard and a design fire exposure, can be

applied to evaluate fire resistance of RC beams under design fires thus facilitating

performance based design.

6.2 Conclusions

Based on the information presented in the thesis, the following conclusions are made:

There is limited information on response of RC beams under realistic fire and
loading conditions, specifically for beams with T and I cross-sections. The current
fire resistance provisions in codes and standards are based on prescriptive
approaches and may not be applicable for performance based fire design.

The macroscopic finite element based numerical model developed in this thesis is
capable of tracing the fire response of RC beams, with rectangular, T and I corss-
sections, under realistic loading and fire scenarios. The model account for critical
factors, - such as high temperature material properties and various strain
components that govern the fire response of RC beams.

Results from the parametric study show that load ratio and fire scenario have
significant influence on the fire resistance of RC beams while aggregate type and
failure criteria have moderate influence.

The proposed energy based approach accounts for critical compartmentation
factors such as fuel load and ventilation parameters in establishing time
equivalency for design fire exposures. Thus it is capable of providing a realistic

assessment of fire resistance of RC beams.

138

 

0 The energy based approach provides better estimate of time equivalent (fire

resistance) of RC beams than current approaches.
6.3 Recommendations
Though the data presented in this study provides valuable information in understanding
the response of RC beams under realistic fire conditions, further studies are needed to
provide a comprehensive understanding of the fire behavior of RC beams. The following
are some of the topics that need to be explored for developing such a comprehensive
understanding:

0 The macroscopic finite element model proposed here is applicable for reinforced
concrete beams only. The model can be extended to cover other cases such as
prestressed and FRP strengthened concrete beams. This can be done by modifying
the thermal and structural models to account for temperature dependent material
properties for FRP, insulation and prestressing steel.

0 The numerical model can be further enhanced to model the effects of spalling in
concrete. Previous studies have indicated that explosive spalling, which generally
occurs in new concrete types such as high strength concrete, has a significant
influence on the fire response of concrete structural systems.

0 The macroscopic finite element model, presented in this thesis, can model the
behavior of simply supported RC beams (for T, I beams) only. The model can be
extended for RC beams with different support conditions such as axial restraint,
rotational restraint, axial and rotational restraint by incorporating stiffness based
formulation to the structural analysis (following the generation of M-K

relationships).

139

APPENDICES
APPENDIX A
To illustrate the applicability of time equivalence formula for an RC beam per existing
methods and empirical formulae, equivalent time subjected to a design fire is computed.
The characteristics of design fire are shown in Figure 2.5 and presented in Table 5.1. The
RC beam is simply supported and the properties of this beam are explained in section 2.5.

Time equivalency computation from different methods is presented below.

 

CIB formula

te = kcwef
A

w = f

 

AVA! V Hv
0.5 2 0.5 2
b = 1900 Ws /m K, Fv = 0.02 m , ef= 1600 MJ/m (From Table 5.1)

kc =0.07 ( From Table 2.1)
A, =2"‘(6"‘4+6*3+3"‘4)=108m2 , Af =6*4=24m2, Av =2.2"‘1=2.2m2

A,./H,, = (2.2*1)*fi = 2.2m2'5

w=——2—4—-=1.56

VlO8*2.2

te = 0.07 * 1.56 *1600 = 174 minutes

Law formula

 

_ A/e/
_ AHCJAV(AI _ Av)

 

 

e

ef= 1600 MJ/m2 (From Table 5.1)

140

 

AHC=19MJ/kg
A, =2*(6*4+6*3+3*4)=108m2, Af =6*4=24m2, A, =2.2*l=2.2m2

_ 24 *1600
19J2.2* (108 - 2.2)

 

= 132 minutes

 

e

Eurocode formula
(e = kb WBf
kg, = 0.055 (from Table 2.1)

ef= 1600 MJ/m2 (From Table 5.1)

0.3 4
w = (:0) [0.62 + 90(0'4 av) ]> 0.5

 

 

r 1+bvah
av=fi1=£=00917 0.02550:v $0.25
Af 24
A
ah=——h—=—9—=0 avSO.2
Af 24

b, =12.5(1+10a, -a,,2)=125*(1+10*0.0917—0.09172) = 23.86

 

 

0.3 _ 4 0.3 _ 4
w = g). 0.62 + 90(0'4 a") = (99-) 0.62 + 90(0'4 0'09”) = 1.76 > 0.5
H 1+bvah 3.0 1+23.86*O

,
1, = kbwef = 0.055 *1.76 *1600 =155 minutes.

Equal area method

Cumulative area under time-temperature curve for FSS = 3740.3 minutes centigrade.

141

 

Time at which area under ASTM E119 time-temperature curve is 3740.3 minutes
centigrade is equal to 232 minutes (Refer to Figure.2.1). These areas are computed using

the procedure discussed in Chapter 2.

Therefore, (e = 232 minutes.

 

142

 

13.] Material Properties

Table B.1 — Constitutive relationships for high temperature properties of concrete

APPENDIX B

 

ASCE Manual 1992 (NSC)

 

Stress-strain relationships

0.
0
||

 

 

: 5-8max,T
fc,T1-[

5 max, T

p—

 

 

2
7 8max,T ‘5
fc,T 1_[ J ’ 5>8max,T

3“"max, T

_

T—2O

21
J 9 gsgmax,T

i l

 

 

- J

,20°C 5 T 3 450°C ‘

' =4 ' 2.011-2353 —— ,450°C<TSS74°C

 

0

k

,874°C<T

em,” = 0.0025 + (6.0T + 0.04T2)x10‘6

 

 

Thermal capacity

Siliceous Aggregate Concrete

 

 

 

 

 

‘

 

 

“0.005T +1.7 20°C 3 T 5 200°C
2.7 200 °C < T 3 400°C
pc=<0.0l3T—2.5 400°C<T3500°Cl
10.5—0.013T 500°C <T $600°C
12.7 600°C < T
Carbonate Aggregate Concrete.
r2.566 20°C _<_ T s 400 °C ‘
0.1765T—68.034 400°C <Ts410°C
25.00671 — 0.05043T 410°C < T 3 445°C
_ 2.566 445°C < T _<_ 500°C
pc —<0.01603T—5.44881 500°C <TS635°C
0.16635T —100.90225 635°C < T 3 715°C
17607343 — 0.22103T 715°C < T 3 785°C
£2566 785°C < T

 

143

 

 

Table 8.] (Continued) — Constitutive relationships for high temperature properties of
concrete

 

 

 

 

 

ASCE Manual 1992 (NSC)
Siliceous Aggregate Concrete.
3? — 0.000625T +1.5 20 °C 3 T s 800 °C
> =
'3 c 1.0 800°C < T
:3
“O
E Carbonate Aggregate Concrete.
f5; _ 1.355 20°CsTsz93°C
g C _ —0.001241T+1.7162 293°C<T
E
I:
g All types :
7° 2 —6
g a”, = [0.004(T — 400)+ 6(T — 20)Jx 10
d.)
.1:
E—

 

 

 

144

 

Table 8.] (Continued) — Constitutive relationships for high temperature properties of

concrete

 

Normal strength and high strength concrete — Eurocode 2[2004]

 

Stress-strain relationships

38fa,T

3
8
ECLT 2+[g ]
cl,T

For a C 1(7) < e S 8 “11(7) , the Eurocode permits the use of linear as well as

 

ac : ’5 S gcul,T

 

nonlinear descending branch in the numerical analysis.

For the parameters in this equation refer to Table A2

 

 

 

Thermal capacity

 

Specific heat (J/kg C)
C: 900, for 20°C 5 Ts 100°C
c = 900 + (T- 100), for 100°C < TS 200°C
c = 1000 + (T - 200)/2, for 200°C < TS 400°C
c = 1100, for 400°C < T_<_ [200°C

Density change (kg/m3)
p = p(20°C) = Reference density
for 20°C 5 TS 115°C
,0 =p(20°C) (1— 0.02(T- 115)/85)
' for 115°C< T5200°C
p= p(20°C) (0.98 — 0.03(T— 200)/200)
for 200°C < T 5 400°C
,0: p(20°C) (0.95 — 0.07(T- 400)/800)
for 400°C < T 5 [200°C

Thermal Capacity = p X c

 

145

 

 

Table 8.1 (Continued) — Constitutive relationships for high temperature properties of

concrete

 

Normal strength and high strength concrete —- Eurocode 2[2004]

 

Thermal conductivity

All types :

Upper limit:
kc = 2 — 0.2451 (T/100) + 0.0107 (T/100)2
for 20°C 5 T 51200°C

Lower limit:

kc = 1.36 — 0.136 (T/ 100) + 0.0057 (T/100)2
for 20°C 5 T s 1200°C

 

 

Thermal strain

 

Siliceous aggregates:

a”, = -1.8><10'4+9x10'6T+2.3 x 10 ””73
for 20°C _<_ T_<_ 700°C
a”, = 14 x 10'3
for 700°C < T51200°C

C alcareous aggregates:

a”, = -1.2><10‘4+6><10'6T+1.4x10 “73
for 20°C 5 T 5 805°C
a”, = 12 x 10'3
Afar 805°C < T s 1200°C

 

 

146

 

 

Table B.2 — Values for the main parameters of the stress-strain relationships of normal
strength concrete at elevated temperatures [Eurocode 2]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Normal Strength Concrete
Temp. Siliceous Aggre ate Calcareous Aggre ate
C fc,T 8 8 fc,T 8 8
f;(20°C) c1,T cu1,T f;(20°C) cI,T cu1,T
20 1 0.0025 0.02 1 0.0025 0.02
100 1 0.004 0.0225 1 0.004 0.023
200 0.95 0.0055 0.025 0.97 0.0055 0.025
300 0.85 0.007 0.0275 0.91 0.007 0.028
400 0.75 0.01 0.03 0.85 0.01 0.03
500 0.6 0.015 0.0325 0.74 0.015 0.033
600 0.45 0.025 0.035 0.6 0.025 0.035
700 0.3 0.025 0.0375 0.43 0.025 0.038
800 0.15 0.025 0.04 0.27 0.025 0.04
900 0.08 0.025 0.0425 0.15 0.025 0.043
1000 0.04 0.025 0.045 0.06 0.025 0.045
1100 0.01 0.025 0.0475 0.02 0.025 0.048
1200 0 - - 0 - -
147

 

 

Table 8.3 -— Constitutive relationships for high temperature properties of reinforcing steel

 

ASCE Manual [1992]

 

 

Stress-strain relationships

f(T,0.001)
as
0.001

0S =
f(T,0.001)

f(T, x) = 6.9(50 - 0.04T)[1— exp((—30 + 0.03T)\/§]

—6
8p=4><10 fy,20

8335!,

where: as and 83 = stress(MPa) and strain in steel reinforcement, respectively,

and f y,20 is the yield strength of reinforcing steel(MPa) at room temperature.

 

Thermal strain

 

am, =[0.004(T2 — 400) + 6(T — 20)]x10“6 T <10000C

 

148

 

 

 

Table B.3 (Continued) - Constitutive relationships for high temperature properties of
reinforcing steel

 

Eurocode 2[2004]

 

Stress-strain relationships

r \

gsEsJ 5s S‘c:sp,T

.5
2 2
fsp,T —c+(b/a)(a _(£sy,T —£S) )0 emf <85 S as)”

 

 

 

as =1fSy,T Esy,T < 5s -<- est,T i
5s _gst,T
fsy,T 1‘ £st,T <83 SEsu,T
5su,T —€st,T
K0 (is > 8514,77 J
Parameters
fs T
_ p, _ _ _
55p,T ._.—ET— Esyfl‘ —0.02 851,7‘ — 0.15 gqu‘ —0.2

s,T

Functions

 

2 C
a = (55y,T —Esp,T) 5sy,T _5sp,T + E
s,T

2 2
b = C(85)),T -8Sp,T)Es,T +C

= (fsy,T ‘fsp,T)2
(Esy,T “gap,T)Es,T —(fsy,T —fsp,T)

 

C

Values of f m7- , f 5),; and E s,T can be obtained from Table B.4

 

 

i Thermal strain

 

W

r1.2x10—5T+0.4x10—8T2 —2.416x10‘4 200C s T s 7500C
am, =11.1x10"2 7500C<Tss600C
2x10"5T-6.2x10‘3 8600C<T51200°C

 

 

 

149

 

 

Table B.4 — Values for the main parameters of the stress-strain relationships of
reinforcing steel at elevated temperatures [Eurocode 2]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Steel temperature T(°C) [1: fig—’0 EST
fy fy Es
20 1 1 1
100 1 1 1
200 1 0.807 0.9
300 1 0.613 0.8
400 1 0.42 0.7
500 0.78 0.36 0.6
600 0.47 0.18 0.31
700 0.23 0.075 0.13
800 0.11 0.05 0.09
900 0.06 0.0375 0.0675
1000 0.04 0.025 0.045
1100 0.02 0.0125 0.0225
1200 0 0 0

 

 

* f y and E s are yield strength and modulus of elasticity at room temperature

150

 

 

APPENDIX C
This appendix presents a detailed step-by—step calculations involved in the design of T
and I beams. Both the beams are designed as under reinforced section using ACI 318
[2008] provisions. Theoretically design of I beams is exactly similar to that of T beams.
Hence the detailed design procedure for T beams is presented here and it is also used for
the design of I beams. Both the beams are assumed to be located in a building having a
span of 7.62m (25 feet). The effective depth of the section is assumed to be equal to one-
twelfth of the span length. The width of the top flange for T beam and top and bottom
flanges for I beam is assumed to be equal to one-fourth of the span length while the width
of the web is equal to one-half of the depth of the section. The height of the top flange for
T beam and top, bottom flanges for I beam is assumed to be equal to 152mm (6 inches).
Each beam is assumed to be made of siliceous aggregate concrete having a compressive
strength of 28 MPa. The beams were provided with 3020 mm bars as tensile
reinforcement and 2020 mm bars as compressive reinforcement. The yield strength of
the reinforcing steel is 413 MPa. A clear concrete cover thickness of 51mm was provided
to the center of tension reinforcement from both sides while that to the top of
compression reinforcement was 51mm from the top of beam. The cross-sectional details
along with bending moment diagram and shear force diagram are shown in Figure C.1

while the design calculations are as follows:

fcl = 28 MPa (4 ksi) fy = 413 MPa (60 ksi)
Modulus of elasticity of steel E5 = 2068428 MPa (30000 ksi)

Yield strain of steel 8 - -—41—3—~ 2 *1 ’3

i- ..
y E, 2068428

151

 

 

Length of the beam (L) = 7.62 m = 7620 mm

Length _ 7620

 

 

Effective depth of the section ((1) = 12 — ~1—2— = 635 mm
Width of the flange (bf) = Length = 764g = 1905 mm
Width ofthe web (bw) = Effecm depth = 933 = 317 mm

 

2 2

Thickness of the flanges = 152 mm
Clear concrete cover to top and bottom reinforcement = 51 mm
Diameter of tension reinforcement = 20 mm

Diameter of compressive reinforcement = 20 mm

Area of tension reinforcement A s = 3 * Z— * 202 =942.5 mm2

Area of compressive reinforcement A S = 2 1‘14“" 202 =628.3 mm2

Tensile force in bottom steel T = AS * fy = 942.5 * 413: 3892525 N

Assume that the neutral axis is within the top flange and the top steel reinforcement has
also yielded. Let the depth of neutral axis be c from the surface of top flange.

Compressive force in concrete

CC =0.85*f; *bf *(fllc)=0.85*28*1905*0.85*c
=38538.15c

Force in top steel C; = A; * fy =628.3 * 413=259487.9 N
For equilibrium

T+C§=Cc

152

 

“I

 

 

3892525 + 2594879 = 38538.15c
= 16.83 mm
a = file = 0.85 *16.83 = 14.3 mm
The strains in the tensile and compressive steel satisfy the assumptions made:

8s 8c

 

d — c _—c—
83 _0.003
635—16.83 16.83

 

gs=OJl>£y

 

a; _0.003
61—1683 16.83

 

a; =7.87*10‘3>e 0k

y

Thus, the factored moment resistance of the section is:

a . . a
Mn=T.(d——2')+CS.(d —-§)

Mn =389252.5.(635 — 1352) + 259487.9.(61— 1:23)
2258.4 kN — m

Mn =258.4 kN — m

153

 

 

Figures

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1905 :< 1905 mm 7'
IA I
T 61
152 , I 61...... 152 \ mm
mm W \N mm N\
‘A 2020
2020 635 393 3020 mm :13":
550 3020 mm mm mm mm
mm mm *‘N 152 5.9.0 _V.
0.00 mm 61
JL 61 mm]: mm H
H 6] mm
l 31 mm
317 mm 317 mm

(a) Cross-sectional details of T and I beam

W

[111111 1
_A_ _Q_
l
l

 

7.62

(b) Elevation of the beam

 

(c) Bending moment diagram

((1) Shear force diagram

Figure C.1 — Cross-section, Elevation, Bending moment diagram and Shear force
diagram from T and I beams

154

APPENDIX D

Step-by-step procedure
Time equivalent of an RC beam exposed to design fire can be evaluated by applying the
proposed energy based approach. Various calculations involved in computing the time
equivalent can be performed using a spreadsheet program. The three main steps
associated with the approach are:

0 Computing the total energy of design fire.

0 Computing cumulative energy of standard fire and.

0 Finding the time equivalent.
The properties of RC beam are as discussed in Section 5.6. The beam is assumed to be
exposed to design fire FSS with a total duration of 425 minutes, as shown in Figure DI,
and the time equivalent is evaluated as follows. Detailed calculations for computing the
time equivalent are illustrated in Table D. 1.
Computing the total energy of design fire can be divided into the following sub-steps:

o Dividing the total duration of fire (425 minutes), computed according to Eurocode

equations, into half minute (8.333’110-3 hour) time increments (At).

0 Calculation of fire temperatures at each time increment using the time-
temperature relationships specified in standards (Eq. 3.2).

o The fire temperature computed above is used to compute the heal flux (q/a) using
Eq.(5.5).

0 At each time step, area under the heat flux curve (energy) is calculated by

integrating the area using Trapezoidal rule as shown in Figure D]. For example,

155

at any time step, the area is computed as a product of the time increment (At) and
average value of heat flux during the same time increment.

The computed values of energy at each time step are summed up to give the total
energy of design fire. Thus a total energy transferred to the beam under design

fire exposure is of 39814995 Joules.

Computing the cumulative energy of standard fire can be done through the following sub-

steps:

0

where

The standard fire used in this case study is ASTME119 with a maximum duration

of six hours (480 minutes).

The total duration of 480 minutes is divided into half minute (8.333’110.3 hour)
time increments.

At each time step, the temperature of standard fire is computed using the

approximate time-temperature relationship provided by Lie[1995] which is

expressed below:

T = 7500- exp(—3.79553\/;)) + 170.4171: + T0 (12.1)

th = time (hours),

To = initial temperature (°C) and

T = fire temperature (°C)
Using this value of fire temperature, heat flux at each time step is computed using
Eq.(5.5) followed by the computation of area under heat flux curve (energy),

using Trapezoidal rule. (similar procedure as that for design fire)

156

0 At each time step the cumulative area under heat flux curve (energy) of the
standard fire exposure, which is equal to the energy at the current time step plus
the sum of energies till the previous time step is computed.

0 At each time step, the difference between the total energy of design fire and
cumulative energy of standard fire is computed.

The difference between two energy values is minimum (approximately zero) at a time of
150 minutes. Hence this time value is defined as the time equivalent by the current
method.

In summary, the proposed method predicts a time equivalent value of 150 minutes with
respect to ASTM E119 standard fire exposure for the RC beam described in the case

study subjected to design fire scenario F SS.

157

 

 

 

 

 

 

 

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Figures

 

4005 , --+--FSS
‘ —I—ASTMEll9

/ Average heat flux

100 7 v "m,
— 9 ‘0 .
‘o 9‘

'Ow-.
..$Q*©Q0i
0,,,1,rr.rr,,,,r..,,,,.

Heat flux (Joules)

 

 

 

O 100 200 300 400 500

Time (min)

Figure D.1 — Illustration of computation of area under heat flux curves using Trapezoidal
rule

160

10.

11

12.

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15.

16.

17.

18.

19.

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22.

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24.

25.

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Franssen J.M., Kodur V.K.R., and Mason J.(2004), User Manual for SAFIR,
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Harmathy T.Z.(1967), “ A comprehensive Creep Model”, Journal of Basic
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Ingberg, SH. (1928), Tests of the severity of building fires. National Fire
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ISO 834-1975 (1975), Fire resistance tests -— elements of building construction.
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