NIST Technical Note 1907

Temperature-Dependent Material 

Modeling for Structural Steels:

Formulation and Application

Mina Seif

Joseph Main

Jonathan Weigand

Fahim Sadek

Lisa Choe

Chao Zhang

John Gross

William Luecke

David McColskey

This publication is available free of charge from:

http://dx.doi.org/10.6028/NIST.TN.1907

NIST Technical Note 1907

Temperature-Dependent Material 

Modeling for Structural Steels: 

Formulation and Application

Mina Seif, Joseph Main, Jonathan Weigand, Fahim Sadek 

Materials and Structural Systems Division 

Engineering Laboratory

Lisa Choe, Chao Zhang, John Gross

Fire Research Division 

Engineering Laboratory 

William Luecke, David McColskey

Materials Science and Engineering Division

Material Measurement Laboratory

This publication is available free of charge from:

http://dx.doi.org/10.6028/NIST.TN.1907 

April 2016

U.S. Department of Commerce 

Penny Pritzker, Secretary

National Institute of Standards and Technology 

Willie E. May, Under Secretary of Commerce for Standards and Technology and Director
The policy of the National Institute of Standards and Technology is to use metric 

units in all its published materials. Because this report is intended for the U.S. 

building construction industry, which uses inch-pound units, it is more practical and 

less confusing to use inch-pound units, in some cases, rather than metric units.

However, in most cases, units are presented in both metric and the inch-pound system.

Certain commercial entities, equipment, products, or materials are identified in this 

document in order to describe a procedure or concept adequately. Such identification 

is not intended to imply recommendation or endorsement by the National Institute of 

Standards and Technology, nor is it intended to imply that the entities, products, 

materials, or equipment are necessarily the best available for the purpose.

Another policy of the National Institute of Standards and Technology is to include 

statements of uncertainty with all NIST measurements. In this document, 

however, some measurements of authors outside of NIST are presented, 

for which uncertainties were not reported and are unknown.

National Institute of Standards and Technology Technical Note 1907 

Natl. Inst. Stand. Technol. Tech. Note 1907, 125 pages (April 2016)

CODEN: NTNOEF

This publication is available free of charge from:

http://dx.doi.org/10.6028/NIST.TN.1907
iii

ABSTRACT

This report presents the formulation and application of a newly developed temperature-dependent 

material model for structural steels. First it presents a model for computing the stress-strain 

behavior of structural steels for conditions appropriate to fire. The model is based on the behavior 

of the retained high-temperature yield strength of 42 individual structural steels, augmented by the 

post-yield behavior derived from eight of those steels. The model accounts for the change in yield 

strength with temperature, the change in the amount of post-yield strain hardening with both 

temperature and ambient-temperature yield strength, and the change in strength with increasing 

strain rate. The coefficient of variation of the normalized retained yield strength increases linearly 

with temperature. The quality of the predictions of the stress-strain model are similar to those of 

the Eurocode 3 stress-strain model. The model is extensible to other steels, such as new fireresistive structural steels, quenched-and-tempered plate steels or bolt steels, with improved or 

degraded high-temperature properties, either by substituting a new description for the retained 

yield strength, or by fitting new parameters for the entire stress-strain behavior. 

This National Institute of Standards and Technology (NIST) stress-strain model is then used for 

predicting flexural buckling of steel columns subjected to elevated temperatures. These predictions 

are intended to evaluate the applicability of the NIST model for predicting the behavior of steel 

gravity columns at elevated temperatures using the finite-element method. Accurately predicting 

the fire-performance of columns is critical to evaluating the stability of buildings subjected to fire. 

In a fire, steel gravity columns can be susceptible to inelastic buckling failure because they have 

high utilization ratios with the governing loading combination, “1.2[dead load] + 1.6[live load]”. 

The failure of interior columns can significantly influence the stability of a building because loads 

supported by these columns need to be redistributed through an alternative load path to prevent 

progressive collapse. This report presents the results from analytical studies to evaluate the effect 

of material models on predicting the strength and behavior of steel columns at elevated 

temperatures. 

Besides the stress-strain behavior, another key issue in evaluating the response of structural

systems to fire (high-temperature) effects is the modeling of fracture, which is required to capture 

failure modes such as tear-out in connection plates and bolt-shear. Fracture can be simulated in 

explicit finite element analysis using element erosion, in which elements are removed from the 

analysis when specified failure criteria are satisfied. However, the basis for determining and 

implementing material failure criteria at elevated temperatures is not well-established in the 

literature. A finite element material modeling methodology is presented for structural steels and 

structural bolts at elevated temperatures that incorporates erosion-based modeling of fracture. The 

new NIST stress-strain relationships were combined with a plastic strain-based failure criterion for 

element erosion to enable modeling of fracture in analysis of structural connections and 

assemblies. The failure criterion was calibrated against high-temperature experimental data on 
elongation of tensile coupons at fracture, and its dependence on temperature and mesh size was 

investigated.

Finally, these temperature-dependent material models for structural steel and bolts that incorporate 

erosion-based modeling of fracture were implemented in high-fidelity finite element models to 

study the performance of steel moment frame assemblies at elevated temperatures. 

The performance of steel moment frames under fire-induced heating was investigated using 

detailed finite-element modeling. Assemblies consisting of two columns and a single beam span 

were modeled, with highly refined modeling of the connection regions. Structural analyses were 

performed under gravity loads in combination with a thermal loading scenario consisting of a 

heating phase followed by a cooling phase. The influence of axial restraint on the performance of 

the moment frames was investigated by considering different support conditions for the end 

columns.

Keywords: Stress-strain relationship; fire loading; structural steel; structural bolt; retained yield;

uniform strain; elevated temperatures; steel column; finite element; inelastic buckling; thermal 

strain; fracture; erosion; steel connection
v

PREFACE

The division of responsibility for the work presented in this report has been as follows. William 

Luecke, John Gross, and J. David McColskey were primarily responsible for Chapter 2. William 

Luecke conducted the high-temperature tensile tests of the fire-resistive steels, developed the 

stress-strain model, and wrote the chapter. John Gross articulated the requirements of the model 

for application to structural steel modeling and performance-based design. J. David McColskey 

conducted most of the high-temperature tensile tests that form the basis of the stress-strain model. 

Mina Seif, Joseph Main, and Jonathan Weigand developed the methodology for modeling the onset 

of necking at the ultimate tensile strength and contributed to Sections 2.2. Lisa Choe, Chao Zhang, 

William Luecke, and John Gross were responsible for Chapter 3. Chao Zhang performed the 

modeling work presented in Section 3.2 on the critical buckling temperature of steel columns, and 

Lisa Choe performed the modeling work presented in Section 3.3 on the high-temperature 

responses of steel columns. Mina Seif, Joseph Main, Jonathan Weigand, William Luecke, and 

Fahim Sadek were responsible for Chapters 4 and 5. Mina Seif was primarily responsible for the 

finite-element analyses presented in Chapters 4 and 5, while Joseph Main, Fahim Sadek, and 

Jonathan Weigand also contributed to the development of the finite element models and the 

erosion-based approach for modeling of fracture. Mina Seif served as the primary editor for this

document, developing the overall outline and coordinating the contributions from the various coauthors.
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Table of Contents

vii

TABLE OF CONTENTS

Abstract........................................................................................................................................................iii

Preface .......................................................................................................................................................... v

Table of Contents........................................................................................................................................vii

List of Figures.............................................................................................................................................. xi

List of Tables.............................................................................................................................................. xv

Notation ....................................................................................................................................................xvii

Chapter 1 Introduction.................................................................................................................1

1.1 Background...................................................................................................................................... 1

1.2 Report Outline.................................................................................................................................. 2

Chapter 2 Formulation of Temperature-Dependent Material Models .......................................3

2.1 Introduction...................................................................................................................................... 3

2.1.1 Relation to Previously Published Work ............................................................................... 3

2.1.1.1 TN1714 Sections still valid...............................................................................................4

2.1.1.2 TN1714 Sections superseded ............................................................................................4

2.1.2 Historical Background.......................................................................................................... 4

2.1.3 Existing Stress-Strain Models.............................................................................................. 5

2.1.3.1 Eurocode 3 ........................................................................................................................5

2.1.3.2 ECCS.................................................................................................................................6

2.1.3.3 NIST..................................................................................................................................6

2.2 True Stress-Strain Relationship ....................................................................................................... 6

2.2.1 Stress-Strain Behavior.......................................................................................................... 7

2.2.1.1 Features of real stress-strain behavior for structural steels...............................................7

2.2.1.2 Elastic modulus.................................................................................................................9

2.2.1.3 Yield strength behavior...................................................................................................11

2.2.1.4 Yield strength dependence of the strain hardening .........................................................14

2.2.1.5 Temperature dependence of the strain hardening ...........................................................15

2.2.1.6 Temperature dependence of the strain-rate sensitivity....................................................15

2.2.1.7 Final form of the stress-strain behavior ..........................................................................16

2.2.1.8 Method for calculating parameters..................................................................................17

2.2.1.9 Tensile strength...............................................................................................................19
Table of Contents

2.2.2 Ultimate Tensile Strength................................................................................................... 21

2.2.3 Application and Comparison to Eurocode 3 Behavior....................................................... 24

2.2.4 Uncertainty ......................................................................................................................... 27

2.2.5 Modifying the Model for Steels with Different High-Temperature Performance.............. 32

2.2.5.1 Fire-resistive steels..........................................................................................................32

2.2.5.2 Quenched-and-tempered plates.......................................................................................36

2.2.5.3 Bolts................................................................................................................................38

2.3 Summary........................................................................................................................................ 40

Chapter 3 Use of NIST Material Model to Evaluate Stability of Wide Flange Steel 

Columns at Elevated Temperatures ......................................................................................41

3.1 Introduction.................................................................................................................................... 41

3.2 Critical Buckling Temperature of Steel Columns.......................................................................... 41

3.2.1 Test Data............................................................................................................................. 41

3.2.2 Modeling Approach............................................................................................................ 44

3.2.3 Results and Discussion....................................................................................................... 46

3.3 High-temperature Responses of Steel Columns............................................................................. 48

3.3.1 Test Data............................................................................................................................. 48

3.3.2 Finite-Element Models....................................................................................................... 51

3.3.3 Results and Discussion....................................................................................................... 54

3.4 Column Design Equations............................................................................................................. 61

3.5 Summary, Conclusions, and Limitations....................................................................................... 64

Chapter 4 Modeling of Failure for Finite element Analysis of Structural Steels at 

Elevated Temperatures...........................................................................................................67

4.1 Introduction.................................................................................................................................... 67

4.2 Material Modeling and Post-Ultimate Strength............................................................................. 69

4.2.1 Structural Steel ................................................................................................................... 69

4.2.2 Structural Bolts................................................................................................................... 73

4.2.3 Assumptions and Limitations............................................................................................. 74

4.3 Analysis of Structural Steel Tensile Coupons................................................................................ 75

4.3.1 Tensile Test Data................................................................................................................ 75

4.3.2 Finite Element Modeling Approach ................................................................................... 77

4.3.3 Calibration of Temperature-Dependent Erosion Strain...................................................... 79

4.4 Analysis of Structural Bolts........................................................................................................... 83

4.4.1 Tensile Test Data................................................................................................................ 84
Table of Contents

ix

4.4.2 Finite Element Modeling Approach ................................................................................... 84

4.4.3 Calibration of Temperature-Dependent Erosion Strain...................................................... 86

4.4.4 Performance of Bolts in Shear............................................................................................ 89

4.5 Summary, Conclusions, and Future Work..................................................................................... 92

Chapter 5 Application of Material and Failure Models to the Analysis of Steel Moment 

Connections under Fire Loading ...........................................................................................95

5.1 Introduction.................................................................................................................................... 95

5.2 Prototype Moment Frames............................................................................................................. 96

5.3 Modeling and Analysis of Moment-Frame Assemblies ................................................................ 99

5.4 Results and Discussion ................................................................................................................ 104

5.4.1 IMF Assembly.................................................................................................................. 104

5.4.1.1 Fully constrained assembly...........................................................................................105

5.4.1.2 Unconstrained assembly ...............................................................................................106

5.4.1.3 Three-span assembly.....................................................................................................107

5.4.2 SMF Assembly................................................................................................................. 108

5.4.2.1 Fully constrained assembly...........................................................................................109

5.4.2.2 Unconstrained assembly ...............................................................................................110

5.4.2.3 Unconstrained assembly, with gradient temperature ....................................................111

5.5 Concluding Remarks.................................................................................................................... 112

Chapter 6 Summary, Conclusions, and Future Work............................................................113

6.1 Formulation of Temperature-Dependent Material Model............................................................ 113

6.2 Stability of Steel Columns........................................................................................................... 113

6.3 FE Component Failure Modeling ................................................................................................ 114

6.4 Behavior of Moment-Frame Assemblies..................................................................................... 115

References ............................................................................................................................117
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xi

LIST OF FIGURES

Figure 2-1: Schematic description of the behavior of the stress-strain curve with increasing 

temperature, room-temperature yield strength, and strain rate................................................ 8

Figure 2-2: Behavior of the reported elastic modulus from tensile tests. See Table 2-1 for key. .. 9

Figure 2-3: Recommended value of the elastic modulus from various organizations, see Table 

2-1 for key ............................................................................................................................. 10

Figure 2-4: Behavior of the retained yield strength for structural steels. See Table 2-1 for key.. 12

Figure 2-5: Strain-rate sensitivities, m, for literature steels. The solid line is the fit of Eq. (2.9). 16

Figure 2-6: Behavior of the retained tensile strength for structural steels. See Table 2-1 for key.

............................................................................................................................................... 20

Figure 2-7 (a) Temperature-dependent normalized uniform strain (Eq. (2.6)) compared with 

available experimental data and (b) fit to ambient temperature uniform strain based on 

ambient temperature yield stress with 95 % confidence bounds, compared with available 

experimental data................................................................................................................... 23

Figure 2-8: Tensile strength resulting from fit to uniform strain for structural steel with Fy = 345 

MPa (50 ksi), compared with experimental data (6.895 MPa = 1 ksi).................................. 24

Figure 2-9: Comparison of the predictions of the stress-strain model, Eq. (2.10), the 

Eurocode 3 (ECS, 2005) model, and the measured stress-strain behavior of A992 steel from 

Hu et al. (2009)...................................................................................................................... 25

Figure 2-10: Comparison of the predictions of the stress-strain model, Eq. (2.10), the 

Eurocode 3 (ECS, 2005) model, and the measured stress-strain behavior of AS A186:250 

steel from Skinner (1973)...................................................................................................... 26

Figure 2-11: Comparison of the predictions of the stress-strain model, Eq. (2.10), the 

Eurocode 3 (ECS, 2005) model, and the measured stress-strain behavior of A36 steel from 

Harmathy and Stanzak (1970). .............................................................................................. 27

Figure 2-12: Coefficient of variation 

VR

for elastic modulus E, as a function of temperature. 

Numbers along the 

VR

 0

axis denote the number of different steels used in each 

determination......................................................................................................................... 29

Figure 2-13: Coefficient of variation 

VR

for yield strength Fy, as a function of temperature. 

Numbers along the 

VR 0

axis denote the number of different steels used in each 

determination......................................................................................................................... 30

Figure 2-14: Coefficient of variation 

VR

for tensile strength Fu, as a function of temperature. .. 31

Figure 2-15: Comparison of the normalized retained yield strength for ordinary (Ord.) and fireresistive (FRS) steels. ............................................................................................................ 33
List of Figures

Figure 2-16: Stress-strain data (solid lines) and models (dotted lines) for two fire-resistive steels. 

Models were computed by fitting R(T), Eq. (2.4), first and then the parameters of the stressstrain model, Eq. (2.10). ........................................................................................................ 35

Figure 2-17: Comparison of the normalized retained yield strength for quenched-and-tempered 

100 ksi steel plate and the retained strength model, Eq. (2.4)............................................... 36

Figure 2-18: Stress-strain data (solid lines) and model (dotted lines) for Fy=100 ksi plate. Model 

was computed by fitting R(T) using Eq. (2.4), and then the stress-strain data using 

Eq. (2.10). .............................................................................................................................. 37

Figure 2-19: Comparison of the normalized retained yield strength for high-strength bolt steels 

and the retained strength model, Eq. (2.4). See Table 2-1 for key to data sets. .................... 38

Figure 2-20 Degradation of normalized yield strength versus the increase in temperature for 

rolled structural steel and bolts.............................................................................................. 40

Figure 3-1 Structural models for column tests............................................................................. 43

Figure 3-2 Schematic of the column model used in the analysis................................................. 44

Figure 3-3 Predicted vs. reported failure temperature for the (a) NIST and (b) Eurocode 3 

models.................................................................................................................................... 46

Figure 3-4 Normalized difference in failure temperature as a function of slenderness ratio (λ).. 47

Figure 3-5 Column test setup and instrumentation layout. ........................................................... 48

Figure 3-6 Axial load-temperature-time histories used in the Purdue column tests.................... 50

Figure 3-7 (a) Thermal expansion model of steel, (b) Eurocode 3 stress-strain model, (c) NIST 

stress-strain model. ................................................................................................................ 53

Figure 3-8 Axial load-axial displacement response of the specimen W8×35-AMB. .................. 54

Figure 3-9 Predicted buckling shape with nodal temperature (NT11) distribution..................... 55

Figure 3-10 Axial load-end rotation responses of column specimens......................................... 56

Figure 3-11 Axial load-lateral displacement responses of column specimens. ........................... 57

Figure 3-12 Axial displacements (δa) of the column specimens. ................................................ 59

Figure 3-13 Normalized difference in critical buckling loads of the FEM models. .................... 60

Figure 3-15 Comparison of column curves. ................................................................................ 64

Figure 4-1 Detailed model of a shear tab connection, (b) tear-out failure in beam web, and 

(c) shear fracture of a bolt...................................................................................................... 68

Figure 4-2 Degradation of normalized yield strength and elastic modulus with temperature for 

structural steel........................................................................................................................ 70

Figure 4-3 (a) True stress-strain and (b) engineering stress-strain curves comparing the onset of 

necking resulting from Eq. (3) with that imposed based on experimental data for eu (400 °C; 

Fy0 = 50 ksi (345 MPa))........................................................................................................ 71
List of Figures

xiii

Figure 4-4 True stress-strain curves for A572 steel, generated using Eq. (3) until the onset of 

necking (red circular marker) with linearly projected stiffness thereafter (1 ksi = 

6.895 MPa). ........................................................................................................................... 72

Figure 4-5 True stress-strain curves for A490 bolts (1 ksi = 6.895 MPa). .................................. 74

Figure 4-6 Experimental coupons; (a) picture of the four specimens after the tensile test was 

conducted at 20 oC, 400 oC, 500 oC, and 600 oC, and (b) corresponding stress-strain curves. 

* ............................................................................................................................................. 76

Figure 4-7 Experimental values of engineering strain at fracture for structural steel. *. ............ 77

Figure 4-8 Finite element mesh of a tensile coupon (1 in = 25.4 mm)........................................ 78

Figure 4-9 FE analysis results for a tensile coupon model at 500 oC (1 ksi = 6.895 MPa). ........ 79

Figure 4-10 Influence of the erosion strain ɛer on the computed engineering strain at fracture at 

400 oC .................................................................................................................................... 80

Figure 4-11 Calibrated values of erosion strain versus temperature for structural steel. ............ 80

Figure 4-12 FEA engineering stress-strain curves for selected temperatures with calibrated 

erosion strain values for structural steel (1 ksi = 6.895 MPa). .............................................. 81

Figure 4-13 Engineering stress-strain curves computed from finite-element models of a tensile 

coupon with different levels of mesh refinement (1 ksi = 6.895 MPa)................................ 83

Figure 4-14 Experimental values of engineering strain at fracture from Kodur et al. (2012) for 

A325 and A490 bolts............................................................................................................. 84

Figure 4-15 Finite element mesh of a bolt tensile coupon........................................................... 85

Figure 4-16 FE analysis results for an A490 bolt tensile coupon at 500 oC (1 ksi = 6.895 MPa).

............................................................................................................................................... 86

Figure 4-17 Calibrated values of erosion strain versus temperature for A325 and A490 bolts... 87

Figure 4-18 Computed stress-strain curves from an FE model of an A325 bolt coupon at 

elevated temperatures (1 ksi = 6.895 MPa). .......................................................................... 88

Figure 4-19 Computed stress-strain curves from and FE model of an A490 bolt coupon at 

elevated temperatures (1 ksi = 6.895 MPa). .......................................................................... 88

Figure 4-20 Detailed model of a double-shear bolt test: (a) section view, (b) bolt after fracture.

............................................................................................................................................... 89

Figure 4-21 FE model and experimental results (Wallaert and Fisher 1965) for shear 

displacement of an A325 bolt at ambient temperature (1 ksi = 6.895 MPa)......................... 90

Figure 4-22 FE results for shear displacement of an A325 bolt at selected temperatures (1 ksi = 

6.895 MPa). ........................................................................................................................... 91

Figure 4-23 FE results for shear displacement of an A490 bolt at selected temperatures (1 ksi = 

6.895 MPa). ........................................................................................................................... 91

Figure 4-24 Normalized shear strength versus temperature for A325 bolts................................ 92

Figure 5-1 Sketch of a typical assembly. ..................................................................................... 97
List of Figures

Figure 5-2 Details of WUF-B moment connection...................................................................... 98

Figure 5-3 Details of WUF-B moment connection...................................................................... 99

Figure 5-4 Temperature loading profile..................................................................................... 100

Figure 5-5 Finite-element mesh of the WUF-B assembly: (a) connection region, (b) beam with 

bolted shear tab, (c) beam, (d) bolt, and (e) shear tab. ........................................................ 101

Figure 5-6 Detailed model of the RBS connection.................................................................... 102

Figure 5-7 Finite element mesh of the 2 in gauge tensile coupon (a) solid elements, (b) shell 

elements (1 in = 25.4 mm)................................................................................................... 103

Figure 5-8 Calibrated values of erosion strain versus temperature for structural steel using shell 

elements for the RBS connection. ....................................................................................... 103

Figure 5-9 Column restraint cases considered for the IMF assembly: (a) fully constrained, 

(b) unconstrained, and (c) three-span. ................................................................................. 104

Figure 5-10 Axial forces in the beam versus temperature for the different WUF-B assembly 

cases..................................................................................................................................... 105

Figure 5-11 Local buckling mode of the connection at the end of the heating phase of the 

analysis of the fully constrained WUF-B assembly. ........................................................... 106

Figure 5-12 Failure mode of the connection at the end of the heating phase of the analysis of the 

unconstrained WUF-B assembly. ........................................................................................ 107

Figure 5-13 Axial force in the beam versus the temperature for the different RBS assembly 

cases..................................................................................................................................... 109

Figure 5-14 RBS connection failure modes for the fully constrained SMF assembly: (a) local 

buckling of connection during heating phase; (b) partial fracture during cooling phase. (1 in 

= 25.4 mm) .......................................................................................................................... 110

Figure 5-15 Cross-sectional longitudinal forces in the beam versus the temperature for the 

unconstrained RBS assembly, heated with a gradient temperature profile through the crosssection of the beam.............................................................................................................. 111
xv

LIST OF TABLES

Table 2-1: Concordance for identifying steels in Fig. 2-2, Fig. 2-3, Fig. 2-4, Fig. 2-5, and Fig. 

2-15,....................................................................................................................................... 13

Table 2-2: Values of the parameters in the stress-strain model, Eq. (2.10), for ordinary (Ord.), 

fire-resistive (FR), quenched-and-tempered plate (plate), and bolt steels............................. 19

Table 2-3: Parameters for describing the temperature dependence of the tensile strength of 

ordinary structural steel ......................................................................................................... 21

Table 2-4: Summary of coefficient of variation calculations for E, Fy, and Fu. ........................... 31

Table 3-1 Column test data .......................................................................................................... 45

Table 3-2 Column test matrix and results.................................................................................... 49

Table 3-3 Column analysis matrix................................................................................................ 52

Table 3-5 Summary statistics for the normalized column buckling load .................................... 60

Table 4-1 Engineering and erosion strain values at fracture for structural steel. ......................... 82

Table 4-2 Engineering and erosion strain values at fracture for A325 and A490 bolts................ 87

Table 5-1 Analysis cases considered for the SMF assembly...................................................... 108
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xvii

NOTATION

VR

coefficient of variation of a parameter, defined as the standard deviation normalized 

by the mean

E elastic modulus

e

i

parameters in the Young’s modulus equation, Eq. (2.2)

K prefactor that defines the strength of the steel in the stress-strain curve, defined in 

=K

n

and often called the strength coefficient

KS

portion of the strength coefficient, K, that depends on the ambient-temperature yield 

strength, Eq. (2.6)

KT

portion of the strength coefficient, K, that depends on temperature, Eq. (2.7)

k

i

parameters in the stress-strain equation, Eq. (2.10)

n strain-hardening exponent defined in =K

n

m strain rate sensitivity defined in Eq. (2.8) and Eq. (2.9)

R retained strength, usually yield strength, S

y

, normalized to its ambient-temperature 

value, S

0

y

. Eq. (2.4)

r

i

parameters in the retained yield and tensile strength equations, Eq. (2.4) and Eq. (2.14)

S

u

measured tensile strength

S

0

u

measured tensile strength at ambient temperature

F

y0

measured yield strength as defined in ASTM E6 (2009b). Sometimes denoted YS in 

other sources. Can be qualified with the type of yield strength. For example F

y0

(0.2 % 

offset) is the traditional offset yield strength in mill test reports

S

0

y

measured yield strength at ambient temperature

T temperature, C

T

*

temperature above ambient temperature: T

*=T20 C

e engineering strain: e = (l − l0) ∕ l0, where l = current length and l0 = original length

 true strain: e = ln(l ∕ l0), where l = current length and l0 = original length

p



true plastic strain, the strain in excess of the elastic strain at yield, Eq. (2.5)



true strain rate

0



true strain rate from an individual test, or 0.005/mm/mm/min, Eq. (2.8)
Notation

 true stress:  = P ∕ A, where P = force and A = current area

 p

plastic portion of the total stress, i.e. the stress in excess of yield stress, Eq. (2.5)

 0

reference stress in computing strain rate sensitivity, Eq. (2.8).  0= 1 MPa
1

Chapter 1

INTRODUCTION

1.1 BACKGROUND

Structural fire safety has historically been accomplished through hourly fire endurance ratings 

specified by the applicable building codes, and evaluated via qualification testing on individual 

members or small assemblies. More recently, the National Institute of Standards and Technology 

(NIST) and others have recognized the need for performance-based approaches to structural fire 

safety, rather than the traditional prescriptive approach. Progressing toward performance-based 

fire specifications is essential to modernizing code-based consideration of fire hazards, and would 

place fire hazards on an equal-risk basis relative to the other hazards the structure is designed to 

resist. Similar performance-based methodologies have already proven effective in seismic design,

both as a means to design structures to achieve target performance criteria given potential hazard 

levels and as an effective tool to facilitate communication between engineers and building owners 

regarding the cost-efficiency of designing more reliable structures.

While the recently constructed National Fire Research Laboratory is intended to enable 

development of performance-based design approaches by enabling full-scale testing, the imminent 

availability of full-scale validation testing makes this a critical time for development of predictive 

modeling and analysis approaches. Finite-element or other analytical models require accurate 

constitutive models for the behavior of the steels in the beams, columns, and connections, rather 

than just the typical summary data such as yield and tensile strengths. These models should be 

based on validated data, and that data should be available for long-term review.

For structural steels in fire conditions, the majority of effort to develop constitutive laws began in 

the 1980s in Europe (Anderberg, 1983, 1988; Cooke, 1988; ECCS, 1983) and culminated in the 

stress-strain model of the ECS Eurocode 3 (2005) standard for structural fire design. Despite this 

development, complete high-temperature stress-strain data sets for structural steels used in 

buildings are much less common than summary data sets. The NIST report on the collapse of the 

World Trade Center towers, (Luecke et al., 2005) contains a data set from twelve structural steels 

recovered from the World Trade Center collapse site. The technical report (Luecke et al., 2011) 

reanalyzed a subset that included 41 stress-strain curves from eight steels and combined it with 

critically evaluated literature data to create a generic constitutive model for structural steel that 

captures the temperature dependence of the yield strength and strain hardening, the yield-strength 

dependence of the strain hardening, and the overall dependence of the stress-strain behavior on the 

strain rate.
Chapter 1

2

1.2 REPORT OUTLINE

This report presents the formulation and application of a newly developed temperature-dependent 

material model for structural steels. It summarizes the model for computing the stress-strain 

behavior of structural steels at elevated temperatures in Luecke et al. (2011), and shows how this 

model can be implemented, using both analytical and numerical modeling approaches, to predict 

the fire-performance of structures. The outline of the report is as follows:

 Chapter 2 describes the formulation of the temperature-dependent constitutive model for 

structural steels (i.e., NIST Temperature-Dependent (T-D) stress-strain model) used 

throughout the subsequent chapters.

 Chapter 3 uses the NIST T-D stress-strain model to predict flexural buckling of steel 

columns at elevated temperature, using the finite-element method, to evaluate the ability 

of the material model to more accurately predict the stability of buildings subjected to fire. 

Predictions of the critical buckling temperatures of 47 tested columns are calculated using 

the finite-element models and compared with those values reported from the experiments. 

Predictions using the NIST T-D stress-strain model are also compared to predictions 

calculated using Eurocode 3, ESC (2005), and the NIST T-D stress-strain model is shown 

to provide better predictions of the buckling response, buckling strength, and failure 

temperatures of the columns at elevated temperatures.

 Chapter 4 presents a modeling approach developed for capturing fracture, which is required 

to capture failure modes such as tear-out in connection plates and bolt shear. A finite 

element material modeling methodology is presented for structural steels and structural 

bolts at elevated temperatures that incorporates erosion-based modeling of fracture. The 

new NIST stress-strain relationships were combined with a plastic strain-based failure 

criterion for element erosion to enable modeling of fracture in analysis of structural 

connections and assemblies. 

 Chapter 5 implements the NIST T-D stress-strain model into detailed continuum finite 

element models of steel-framed assemblies, including connections, to evaluate the behavior 

of connections at elevated temperatures. This chapter also shows how connection failure 

modes, such as tear-out in connection plates and bolt shear, can be captured in explicit 

thermal-structural finite element analysis using the temperature-dependent plastic strainbased material failure criterion.

 Finally, Chapter 6 provides a summary and main conclusions of the work presented 

throughout the report.
3

Chapter 2

FORMULATION OF TEMPERATURE-DEPENDENT MATERIAL MODELS

2.1 INTRODUCTION 

This chapter presents a model for computing the stress-strain behavior of structural steels for 

conditions appropriate to fire. The model is based on the behavior of the retained high-temperature 

yield strength of 42 individual structural steels, augmented by the post-yield behavior derived from 

eight of those steels. The model accounts for the change in yield strength with temperature, the 

change in the amount of post-yield strain hardening with both temperature and ambienttemperature yield strength, and the change in strength with increasing strain rate. The coefficient 

of variation of the normalized retained yield strength increase linearly with temperature. The 

quality of the predictions of the stress-strain model are similar to those of the Eurocode 3 stressstrain model. The model is extensible to other steels, such as new fire-resistive structural steels, 

quenched-and-tempered plate steels or bolt steels, with improved or degraded high-temperature 

properties, either by substituting a new description for the retained yield strength, or by fitting new 

parameters for the entire stress-strain behavior. 

Structural fire safety has historically been accomplished through the specification of steel members 

or assemblies, including passive fire protection, that meet by way of qualification testing, building 

code requirements for a specified fire endurance expressed as an hourly rating. More recently, 

there has been a trend toward performance-based approaches to structural fire safety, rather than 

the traditional prescriptive approach. For example, the American Institute of Steel Construction 

(AISC) Specification for Structural Steel Buildings (2010) beginning in the 2005 edition, 

addresses structural design for fire by analysis. 

Modeling the response of steel structures to fire requires accurate constitutive models for the 

behavior of the steels in the beams, columns, and connections. Finite-element or other analytical 

models require the entire stress-strain curve, rather than just the typical summary data such as yield 

and tensile strengths. This chapter presents a model for computing the stress-strain behavior of 

structural steels under conditions appropriate for fire modeling. Finite-element or other analytical 

models require accurate constitutive models for the behavior of the steels in the beams, columns, 

and connections, rather than just the typical summary data such as yield and tensile strengths. 

These models should be based on validated data, and that data should be available for long-term 

review.

2.1.1 Relation to Previously Published Work

This chapter represents a correction and augmentation to a previously published NIST Technical 

Report (Luecke et al., 2011) (TN1714). This chapter supersedes some portions of that document, 
Chapter 2

4

but other sections are still valid. The model development portions of TN1714 are generally 

obsolete, but the historical summary sections are still valid. 

2.1.1.1 TN1714 Sections still valid

• 3.1 Microstructure and Chemistry 

• Appendix B Experimental methods 

• Appendix C Microstructure and Chemistry 

• Appendix D.1 Tensile data 

• Appendix D.4 Calculation and analysis of the strain rate sensitivity, m

• Appendix E.1 Literature Data 

• Appendix E.2 Tabular data for the literature structural steels 

• Appendix E.4.3 Notes on literature reports of elastic modulus measurements 

• Appendix F.1 and F.2 Eurocode stress-strain model 

• Appendix G ECCS stress-strain model 

2.1.1.2 TN1714 Sections superseded

• 4.1 Retained Yield strength; superseded by Section 2.2.1.3 

• 4.2 Modeling stress-strain curves; superseded by Section 2.2

• 4.3 Comparison with Eurocode Recommended Values 

• Appendix E3 Retained Strength; superseded by Section 2.2.2

• Appendix E4.1 Elastic Modulus recommended value; superseded by Section 2.2.1.2 

In addition, this chapter corrects errors introduced in transcribing the data from Chen et al. (2006). 

These corrections did not change the values of the individual retained yield strength model 

parameters r

i

from previous unpublished versions of this chapter. 

2.1.2 Historical Background

For structural steels in fire conditions, the majority of effort to develop full constitutive laws began 

in the 1980s in Europe (Anderberg, 1983, 1988; Cooke, 1988; ECCS, 1983) and culminated in the 

stress-strain model of the ECS Eurocode 3 (2005) standard for structural fire design. Despite this 

interest, complete high-temperature stress-strain data sets for structural steels used in buildings are 

much less common than summary data sets. The NIST report on the collapse of the World Trade 

Center (WTC) towers (Luecke et al., 2005) contains a data set comprising twelve structural steels 

recovered from the WTC collapse site. These steels were important to the modeling effort. The

technical report (Luecke et al., 2011) reanalyzed a subset of that data set that included 41 stressstrain curves from eight steels and combined it with critically evaluated literature data to create a 
Material Models

5

generic constitutive model for structural steels that captures the temperature dependence of the 

yield strength and strain hardening, the yield-strength dependence of the strain hardening, and the 

overall dependence of the stress-strain behavior on the strain rate. The technical report (Luecke 

et al., 2011) includes tables of the complete data set used to develop the model so that the model 

origins are transparent. This chapter summarizes the important aspects of that model and compares 

it to the predictions of two other stress-strain models for structural steel in fire. On-going research 

into modeling column failure necessitated changes to the choice of elastic modulus reported in that 

model (Luecke et al., 2011). That substitution changed the values of the stress-strain model 

parameters slightly. 

2.1.3 Existing Stress-Strain Models

2.1.3.1 Eurocode 3

The ECS Eurocode 3 (2005) stress-strain model is used widely in Europe. Twilt (1991) documents 

its origin and the rationales for the values chosen for some of its functional forms and parameters

as follows:

A. Maximum consistency with existing data; 

B. Direct physical background; 

C. Simple mathematical model; 

D. Clearly defined effective yield stress level; 

E. Practical application range; 

F. Official recognition, either internationally or nationally. 

Based on these criteria, the committee that developed the Eurocode 3 stress-strain model (ECS, 

2005) selected the British Steel Corp. data of Kirby and Preston (1983) for steels taken from wide

flange shapes originally supplied to BS 4360 (1990) grades 43A and 50B (Fy=37 ksi (255 MPa)

and Fy=51 ksi (355 MPa)), respectively. Interestingly, these data were not generated from standard 

high-temperature tensile tests, similar to ambient-temperature mill tests. Instead, they originated 

from so-called “transient state” tests in which the specimen is held under constant force while the 

temperature increases linearly. During this test, the specimen strain increases until it runs away at 

a critical temperature. Kirby created stress-strain curves from these transient heating tests by 

assembling the collection of strain-temperature points from the transient state tests at fixed stresses 

up to engineering strains e<0.02. Twilt (1991) indicates that the Eurocode 3 model is based on data 

from tests conducted only at heating rate dT/dt=10 C/min. 

Although Kirby and Preston (1983) originally tested seven different steels at up to three heating 

rates each, the Eurocode stress-strain model is based on only six tests on one steel and eight on the 

other, at a single heating rate. Because of the shape of strain-temperature curves in the the transientstate tests, each derived stress-strain curve contains only a few (stress, strain) points past the 

proportional limit. Twilt’s summary (1991) does not explain the details of the analysis of the 
Chapter 2

6

derived stress-strain curves that produced the values of the individual parameters. Those derived 

stress-strain curves are repeated in tabular form in the 1988 paper; they must represent curve fits, 

rather than data points, because the transient-state-test analysis only produces two or three points 

per curve. The shapes of these derived stress-strain curves are very similar to the Eurocode 3 

formulation, but they are not identical. In particular, the elastic modulus, which is important for 

column-buckling calculations, is only about 90 % of the Eurocode 3 value. Twilt does not provide 

any background on the source of the data for the elastic modulus or for the method by which its 

variation with temperature was calculated. 

2.1.3.2 ECCS

The European Convention for Constructional Steelwork (ECCS) (1983) recommended a stressstrain model based on an effective yield strength that decays smoothly with temperature. The 

model claims to include time-dependent effects in the yield strength, but the published report that 

describes it does not document the method for including those time-dependent effects into the 

stress-strain curve. Very few research groups have used the model. 

2.1.3.3 NIST

As part of its World Trade Center collapse investigation, NIST developed an elevated-temperature 

stress-strain model for structural steels, based on a combination of literature data and tests 

conducted on relevant structural steel recovered from the collapse site (Luecke et al., 2005). After 

the publication of the investigation final report, a second report (Luecke et al., 2011) refined the 

model with additional literature data and improved computation schemes. 

This chapter corrects and augments that second report, adds computations of the expected 

variability of the retained high-temperature modulus of elasticity, yield strength, and tensile 

strength, and demonstrates how the stress-strain model can be used to represent new steels, with 

examples for modern construction and fire-resistive steels. For compactness, the chapter omits 

much of the supporting data for the choices of parameters, but Luecke et al., (2005) documents the 

computation of each parameter. The original report used elastic modulus data derived from slow 

dynamic measurements on recovered steel. This paper substitutes a formulation for elastic modulus 

derived from literature data from hot-tensile tests on structural steel. This substitution changed the 

absolute values of the parameters of the model. 

2.2 TRUE STRESS-STRAIN RELATIONSHIP

The goal of this chapter is to develop a generic stress-strain model for structural steel in conditions 

appropriate to fire. As much as possible it should capture the behavior of real structural steel and 

be extensible to new grades of structural steel, such as fire-resistive steels, that offer enhanced 

mechanical properties at elevated temperatures. Furthermore, its origins should be transparent and 

well documented. 
Material Models

7

A major impediment to this effort is the scarcity of full stress-strain data at high temperatures in 

literature reports of the tensile behavior of structural steel. Before the NIST WTC investigation, 

only a few investigators had reported the full stress-strain behavior of structural steels in a format 

suitable for modeling (Skinner, 1973; Harmathy and Stanzak, 1970; Hu et al., 2009). A much 

larger body of literature exists that typically only reports the high-temperature yield and tensile 

strengths. Incorporating that summary data, however, would help ensure that the modeled behavior 

truly represents the wide variety of structural steel grades. The additional data also allows

estimation of the variability in properties. The NIST World Trade Center Collapse investigation 

characterized the elevated-temperature stress-strain behavior of more than ten recovered steels. 

The approach of this study was to use the extensive literature data to model the behavior of the 

high-temperature yield strength, and to augment that behavior with the post-yield stress-strain 

behavior derived from the WTC collapse investigation data. 

2.2.1 Stress-Strain Behavior

2.2.1.1 Features of real stress-strain behavior for structural steels

The stress-strain behavior of structural steel exhibits four trends that any model should capture. 

1. The yield strength decreases with increasing temperature. 

2. The amount of strain hardening decreases with increasing ambient-temperature yield 

strength. 

3. The strain hardening decreases with increasing temperature. 

4. The strain-rate sensitivity, and therefore the increase in strength with increasing strain rate, 

increases with increasing temperature. 

Fig. 2-1 illustrates these four behaviors. 
Chapter 2

8

Figure 2-1: Schematic description of the behavior of the stress-strain curve with increasing temperature, 

ambient-temperature yield strength, and strain rate.

Another goal is to minimize the number of parameters necessary to capture the full behavior. 

Whenever possible, phenomenological parameters should follow forms that are consistent with the 

underlying materials science that governs the deformation processes, and they should exhibit 

conservative behavior outside the limits of the data they are based on. 

A general form of a stress-strain, σ-ε, equation that can represent these four behaviors is 
Material Models

9

 

 

0

0 0

0

 for elastic behavior

 for plastic behavior

y

n

y y

y

F

E

E

F F

F K

E E

  

  

 

 

       

(2.1)

where E is the modulus of elasticity, ε is the true strain, and K and n are parameters in the powerlaw equation. The form of Eq. (2.1) joins a temperature-dependent linear-elastic line, Eε, to powerlaw strain-hardening curve for plastic strain. The two branches meet at a temperature-dependent 

yield strength, Fyo. The power-law (or Holloman) stress-strain model, σ=Kεn, is a common form 

for representing the plastic behavior of metals. 

Each of the following sections details the form of an individual component of Eq. (2.1). 

2.2.1.2 Elastic modulus

The technical literature contains a rich set of elastic modulus data measured during tensile tests. 

Accurate modulus measurement during tensile testing is difficult (Lord et al., 2005), but the slope 

of the stress-strain curve determines the yield strength, so it may be a useful representation of the 

very low-strain behavior of the stress-strain curve. 

Figure 2-2: Behavior of the reported elastic modulus from tensile tests. See Table 2-1 for key.
Chapter 2

10

Fig. 2-2 shows that data set, normalized by the reported ambient-temperature value, along with the 

recommended modulus in the Eurocode 3 stress-strain model. Two characteristics stand out. The 

spread in retained modulus at every temperature is large, and all but a few of the measurements lie 

well above the line recommended in the Eurocode 3 formulation. The large spread in the data 

probably reflect, to a large extent, the difficulty in accurately measuring the modulus using small 

specimens and extensometers designed to measure large strains. Analysis of the uncertainty in the 

modulus of elasticity data is provided in Section 2.2.4, where the coefficient of variation in the 

measured modulus is characterized as a function of temperature. The data of Ritter and McPherson 

(1970), see Table 2-1, are noteworthy as they measured very different elastic moduli for the same 

steel heat treated to produce different grain sizes. This result indicates that time-dependent elastic 

processes are probably significant at elevated temperatures. 

Many organizations have produced so-called handbook values for design with structural steel. 

Fig. 2-3 plots six of them, see Table 2-1. Four of the six are very similar, and all lie well above the 

Eurocode 3 line. None of six explain the origins of the underlying data or the methods by which 

the values of the parameters were obtained. 

Figure 2-3: Recommended value of the elastic modulus from various organizations, see Table 2-1 for key
Material Models

11

The handbook values use all manner of different smooth or piecewise-continuous functions and 

discrete data points to represent the behavior of the elastic modulus. In this study, the data were 

represented by a phenomenological exponential function: 

1 2 * *

0

3 4

1 1 ( ) exp

2 2

e e

T T E T E

e e

     

                

(2.2)

Where E0 is the elastic modulus at ambient temperature (T=20 °C). The original technical 

report (Luecke et al., 2011) used a third-order polynomial fit to a set of data developed from 1 hz 

dynamic measurements at slow strain. The main advantage of the form of Eq. (2.2) over the 

polynomial form is that it evaluates to greater than zero at all temperatures. The parameters of 

Eq. (2.2), Table 2-2, were calculated by non-linear least-squares regression of the data in Fig. 2-2

with T725 C, and e2 set to unity and not fit. 1

The resulting curve is very similar to that of Schleich (1987), and to the ASME 2004 

recommendation up to 600 C. 

2.2.1.3 Yield strength behavior

The 0.2 % offset yield strength, Fy0, obtained in a high-temperature tensile test, is the most 

commonly reported parameter. By anchoring the high-temperature stress-strain behavior to the 

yield strength, the model uses forty years of data on the high-temperature strength of structural 

steel. To compare steels with different ambient-temperature yield strengths, the yield strength is 

normalized to its ambient-temperature value, S

0

y

: 

 

0 0

0

0 20

y y

y T C y

F F

R

F S 

 

(2.3)

Fig. 2-4 plots the normalized retained yield strength, R, as a function of temperature for 

42 individual structural steels from 16 different literature sources, identified by number, see

Table 2-1. An analysis of the uncertainty in the reported yield strength data is provided in 

Section 2.2.4.

 

1 At the end of the production of this report, the authors realized that they had transcribed one data set Chen et al. (2006) 

incorrectly. Because several research groups had already begun to use the stress-strain model, the authors decided to retain the 

original set of parameters. The difference between the fits amounts to less than 580 ksi (4 GPa) below T=600 °C, and less than 

1160 ksi (8 GPa) for T<725 °C . 
Chapter 2

12

Figure 2-4: Behavior of the retained yield strength for structural steels. See Table 2-1 for key.
Material Models

13

Table 2-1: Concordance for identifying steels in Fig. 2-2, Fig. 2-3, Fig. 2-4, Fig. 2-5, and Fig. 2-19.

Data Set Author

4 ASME (2004)

11 Brockenbrough and Johnston 

(1968)

12 Chen et al. (2006)

13 Chijiiwa et al. (1993)

15 ECCS (1983)

16 ECS (2005)

18 Garofalo (1952)

19 Gonzalez and Lange (2010)

20 Gowda (1978)

22 Holt (1964)

23 Holt (unk)

25 Hu et al. (2009)

31 Kirby and Preston (1988)

32 Kirby (1995)

33 Kirby and Preston (1991)

35 Kodur et al. (2012)

36 Li et al. (2003)

37 Lie (1992)

39 Lou and Northwood (1995)

40 Luecke et al. (2011)

42 Manjoine (1944)

43 Outinen et al. (2001)

45 Poh (1998)

46 Ritter and McPherson (1970)

47 Sakumoto (1999)

48 Sakumoto et al. (1993)

49 Schleich (1987)

50 Skinner (1973)

51 Spaeder Jr. and Brown (1977)

52 Stevens et al. (1971)

55 Twilt (1991)

56 USS (1972)

These data were screened from a much larger data set on structural steels, and represent data from 

reputable testing laboratories with well described test protocols. To remove the effect of strain rate 

on the strength, those data only include tests conducted in a strain-rate range that includes the range 
Chapter 2

14

of rates allowed in ASTM E21 (2009c), ISO 783 (1999), and JIS G0567 (1998) high-temperature 

tensile test standards: 3.30×10−5 mm/mm/s < 



< 1.35×10−4 mm/mm/s.

Many functions could represent the behavior of the retained yield strength, R. The solid line in

Fig. 2-4 is one that faithfully captures the steep drop-off in the range (400<T<650) °C. Its form is 

 

* *

1 2

0

0 5 5

3 4

1 1 1 exp(

2 2

r r

y

y

F T T R r r

S r r

   

             

(2.4)

where T

*=T20, measured in °C. The use of the pre-factor, r

5

, helps capture the relatively constant 

value of the retained strength at very high temperatures. Only the parameters r

1

, r

3

, and r

4

in 

Eq. (2.4) were fit, and then only in the range (300T800) °C. Limiting the fit to this range 

enhanced the fidelity of the model to the data in the temperature region where the change in 

strength is most abrupt. The parameters r

1

and r

5

were fixed and not fit; r

1

=1, and r

5

was estimated 

from the high-temperature data. The data set is identical to the one in Luecke et al., 2011. 

Fig. 2-4 also plots the Eurocode 3 (ECS, 2005) and ECCS (1983) models as well as the general 

rule that the yield strength falls in half by 538 °C (1000 °F). The values plotted for the Eurocode 3 

represent the 0.2 % offset yield strength computed from the Eurocode 3 stress-strain curve using 

its modulus as a function of temperature, instead of the proportional limit or stress at 2 % total 

elongation that the Eurocode 3 model explicitly provides. Almost all the data lie above the 

Eurocode 3 prediction. 

2.2.1.4 Yield strength dependence of the strain hardening

For structural steels, the amount of strain hardening after yield generally decreases with increasing 

ambient-temperature yield strength. The plastic portion of the stress-strain (p−p) curve can be 

represented by a power-law model: 

 0   

n   p S y T p

 K F K T

(2.5)

where the temperature and yield-strength dependence of the strength coefficient K are separated. 

A simple form of the yield-strength dependence of the strength coefficient, KS, that can represent 

the behavior is a linear decrease with increasing ambient-temperature yield strength, S

0

y

: 

0

0 3 4 ( ) K F k k S S y y

 

(2.6)

with k

3

and k

4

constrained so that the strength coefficient, KS, cannot be negative for reasonable 

values of the ambient-temperature yield strength. 
Material Models

15

2.2.1.5 Temperature dependence of the strain hardening

One approach to modeling the temperature dependence of the strain hardening would be to make 

the parameter n in Eq. (2.5) depend on temperature. The results of this approach did not match the 

observed strain hardening particularly well. An alternate approach fits a single value of the strainhardening coefficient, n, for all temperatures and makes only the strength coefficient, KT

, depend 

on temperature: 

1

2

( ) exp

k

T

T

K T

k

   

          

(2.7)

This function smoothly degrades from the ambient-temperature value to zero at high temperature 

and effectively changes the shape of the stress-strain curve after yield. 

2.2.1.6 Temperature dependence of the strain-rate sensitivity

As the temperature increases, the sensitivity of the strength of structural steel to deformation rate 

increases (Gowda, 1978; Hu et al., 2009; Lou and Northwood, 1995; Luecke et al., 2011; 

Manjoine, 1944; Poh, 1998; Skinner, 1973; Stevens et al., 1971). By 500 C, increasing the strain 

rate by ten times can increase the measured strength by 25 %. In a constant load test, this behavior 

manifests itself as creep (Williams-Leir, 1983). Some literature reports recognize the importance 

of creep effects (Twilt, 1988; ECCS, 1983; Hu et al., 2009). In particular, the ECCS (1983) stressstrain model asserts that time-dependent strength effects are included, but provides no 

documentation on the method. 

Dozens of high-temperature strength models exist that incorporate rate effects, for 

example Johnson-Cook (1983) and Zerilli-Armstrong (1987). They range from purely 

phenomenological to sophisticated micromechanical models. One simple and long-used model 

that captures the observed behavior is a power-law representation in which the stress,  is 

proportional to the strain rate, ε̇ , raised to a power, m, called the strain-rate sensitivity. 

 

0 0

m

f

 







 

     

(2.8)

The terms 

 0

and

0



are normalizing factors. Generally 

 0

= 1 MPa and

0



= 

8.33×10−5 mm/mm/s, the rate used in most high-temperature tensile testing. Fig. 2-5 plots the 

strain-rate sensitivity, m, as a function of temperature for several structural steels, see Table 2-1

for identification. The strain-rate sensitivity, m, is nearly zero at ambient temperature and increases 

to a maximum at high temperature. Takeuchi and Argon (1976, Section 2.2.4) analyzed powerlaw creep of metals theoretically and demonstrated that the strain-rate sensitivity for engineering 

alloys should be about m=0.2 in the high-temperature creep regime. A simple function that 

captures this behavior is 
Chapter 2

16

1

0 3

2

1 exp

m

T

m m m

m

     

                   

(2.9)

Figure 2-5: Strain-rate sensitivities, m, for literature steels. The solid line is the fit of Eq. (2.9).

2.2.1.7 Final form of the stress-strain behavior

The equationsfor the elastic modulus, E, retained strength, R, post-yield strain-hardening behavior, 

and strain-rate sensitivity, m, can be combined into the final form of the stress-strain model: 

 

1

0

0

0 0 0

3 4

2 0

for , elastic behavior, 

for , plastic behavior, exp

y

k n m

y y

y y

F

E

E

F RS T

RS k k S

E k E

  



  



 

         

                              

(2.10)
Material Models

17

where the parameters are given by 

E Eq. (2.2)

R Eq. (2.4)

m Eq. (2.9)

The entire model was developed for true-stress-true-strain behavior, but it can be recast in terms 

of engineering stress and strain, (Se). 

   ln(1 )e

(2.11)

     S S e exp( ) (1 )

(2.12)

Eq. (2.10) becomes

   

1

0

0

0

0 0

3 4

2 0

for , elastic behavior, 

for , ,plastic behavior, 

1

 exp ln 1

1

y

y

k n m

y

y y

F

e E

E

F

e

E

T RS

S RS k k S e

e k E







  



          

                                  

(2.13)

where the infinitesimal difference between engineering and true strain is neglected for elastic 

behavior. 

2.2.1.8 Method for calculating parameters

Calculating the values of the parameters in Eq. (2.10) comprised four steps. 

1. The values for the elastic modulus, Eq. (2.2), came from a non-linear, least-squares 

regression of Eq. (2.2) using the data in Fig. 2-2. 

2. The temperature dependence of the retained yield strength, R, came from a non-linear 

regression of Eq. (2.4) to the data in Fig. 2-4 in the range (300T700) °C. That data set 

includes both literature data and data from the WTC investigation (Luecke et al., 2011). 

3. The behavior of the strain hardening came from a non-linear regression of the parameters 

1 2 3 4 k k k k , , , 

and n in Eq. (2.10) using 41 stress-strain curves from eight different structural 

steels taken from the NIST WTC investigation, but no other literature data. In this non-linear 

regression the values of the individual e

i

in the equation for E (Eq. (2.2)) and r

i

in the 

equation for R (Eq. (2.4)) had already been determined in the first and second steps. Since 

the stress-strain behavior for these eight steels was determined at nominal strain rates ε̇ =

ε0̇ , the strain-rate sensitivity term evaluates to unity, and drops out of the regression. The 
Chapter 2

18

regression only included data in the strain range 0.0050.150. The specified yield 

strengths of those eight steels were in the range (36F

y0

70) ksi, (250F

y0

520) MPa. 

4. The strain-rate sensitivity, m, Eq. (2.9), came from a non-linear regression of the data in

Fig. 2-5. 

Table 2-2 summarizes the values of the eighteen parameters for ordinary structural steels (denoted 

by “Ord.”) in Eq. (2.10). 
Material Models

19

Table 2-2: Values of the parameters in the stress-strain model, Eq. (2.10), for ordinary (Ord.), fire-resistive 

(FR), quenched-and-tempered plate (plate), and bolt steels.

Parameter Value Value Value Value Units Eq.

(Ord.) (FR) (plate) (bolt)

r

1

7.514 9.782 10.143 4.967 (2.4)

r

2

1.000 1.000 1.000 1.000 (2.4)

r

3

588 625 589 456 °C (2.4)

r

4

676 1334 837 2040 °C (2.4)

r

5

0.090 0 0 0 °C (2.4)

k

1

7.820 9.814 10.616 (2.10)

k

2

540 616 811 °C (2.10)

k

3

1006 5835 959 MPa (2.10)

k

4

0.759 15.846 0.766 (2.10)

n 0.503 0.456 0.349 (2.10)

m0

0.0108 (2.9)

m1

7.308 (2.9)

m2

613 °C (2.9)

m3

0.126 (2.9)

ε0̇ 8.33310

5

s

1

(2.9)

E0

206.0 GPa (2.2)

e

1

3.768 (2.2)

e

2

1.000 (2.2)

e

3

639 °C (2.2)

e

4

1650 °C (2.2)

2.2.1.9 Tensile strength

A typical mill test reports the tensile strength, S

u

, (also sometimes called the ultimate tensile 

strength) in addition to the yield strength and total elongation. Generally, the tensile strength 
Chapter 2

20

represents the maximum load-carrying capacity of the material. Fig. 2-6 plots the reported tensile 

strength for the tests also shown in Fig. 2-4. 

Figure 2-6: Behavior of the retained tensile strength for structural steels. See Table 2-1 for key.

Because structural steel creeps significantly at temperatures above 500 °C, the tensile strength, 

like the yield strength, begins to depend on time as the temperature increases. Users should 

therefore exercise care in interpreting the tensile strength for high-temperature design. The hightemperature tensile strength in Fig. 2-6 only represents the maximum engineering stress that a test 

coupon achieves in a test conducted according to standard conditions. An analysis of the 

uncertainty in the tensile strength data is provided in Section 2.2.4.
Material Models

21

The behavior of the tensile strength can be represented by the same form as used for the yield 

strength: 

 

6 7 * *

0 10 10

8 9

1 1 1 exp( )

2 2

r r

u

S u

u

S T T R r r

S r r

   

             

(2.14)

where T

*=T20, measured in °C. The solid line in Fig. 2-6 is a fit to the data in the temperature 

range (350T825) °C. Table 2-3 reports the values of the parameters in Eq. (2.14). The value of 

the minimum retained yield strength, r

10, was not fit, but instead set to the its mean value for 

temperatures T>725 °C. 

Table 2-3: Parameters for describing the temperature dependence of the tensile strength of ordinary 

structural steel

Parameter Value Units Eq.

(Ord.)

r

6

4.906 (2.14)

r

7

4.906 (2.14)

r

8

560 °C (2.14)

r

9

592 °C (2.14)

r

10 0.110 (2.14)

2.2.2 Ultimate Tensile Strength

The onset of necking in a tensile coupon occurs at the ultimate tensile strength, when the increase 

in stress due to reduction of the cross-sectional area overcomes the increase in strength due to 

strain hardening. At this point the strain, which had been uniform within the gauge length, begins 

to localize. An equation for the engineering strain at the onset of necking (i.e., the uniform strain), 

was determined based on nonlinear least-squares regression of data available in the literature 

(Fig. 2-7(a)) as (Seif et al. 2016):

1

0

3

1

( ) exp

2

q

u u

T

e T e

q

      

                 

(2.15)

where eu0 is the uniform strain at ambient temperature, q1 = 3.587, and q3 = 488 °C. The data 

included in Fig. 2-7 correspond to a subset of the data considered in Section 2.2, for which uniform 

strain values were reported. The ambient-temperature uniform strain decreases with increasing 

yield strength (i.e., low-strength steels are typically more ductile than high-strength steels), as 
Chapter 2

22

shown by the experimental data in Fig. 2-7(b), and the following approximate relationship was 

obtained from linear regression of these data: 

0 0 1 0 2 ( )u y y

e F p F p  

(2.16)

where p1 = −0.00152 and p2 = 0.252. Together, Eq. (2.15) and Eq. (2.16) provide a complete 

description of the temperature-dependent uniform strain, based solely on the ambient temperature 

yield strength, that can be applied to arbitrary strength steels. Although the stress-strain model is 

limited to steels with nominal yield strengths less than 65 ksi (450 MPa), the relationship in 

Eq. (2.16) was fit to data over a wider range of nominal yield strengths with a maximum nominal 

yield strength of 110 ksi (760 MPa). The ultimate tensile strength corresponding to a given value 

of eu can be calculated using Eq. (2.2), Eq. (2.4), and Eq. (2.10). Fig. 2-8 compares tensile strength 

values obtained from the proposed uniform strain model (Eq. (2.16)) for Fy0 = 50 ksi (345 MPa) 

with corresponding experimental tensile strength data. 
Material Models

23

 (a)

(b)

Figure 2-7 (a) Temperature-dependent normalized uniform strain (Eq. (2.6)) compared with available 

experimental data and (b) fit to ambient temperature uniform strain based on ambient temperature yield 

stress with 95 % confidence bounds, compared with available experimental data.
Chapter 2

24

Figure 2-8: Tensile strength resulting from fit to uniform strain for structural steel with Fy = 50 ksi 

(345 MPa), compared with experimental data.

2.2.3 Application and Comparison to Eurocode 3 Behavior

The plastic portion of the stress-strain model, Eq. (2.10), was developed using data on eight 

structural steels used in the NIST World Trade Center Collapse investigation. Its utility can be 

assessed by comparing its predictions to stress-strain data from outside the original set. Recently, 

Hu et al. (2009) published high-temperature stress-strain data for ASTM A992 (2011) structural 

steel. Neither the retained yield strength nor the stress-strain curves from this data set were used 

in computing the values of the parameters in Table 2-2. Fig. 2-9 plots that data for some of the 

reported temperatures. In general the agreement is excellent, given that the only parameter input 

to the stress-strain model is the ambient-temperature yield strength,

y

S



 

MPa, and that the 

model did not use the data to estimate the parameters. Fig. 2-9 also plots the prediction of the 

Eurocode 3 (ECS, 2005) model. Both models over-predict the ambient-temperature behavior, but 

predict the higher temperature behavior similarly. 
Material Models

25

Figure 2-9: Comparison of the predictions of the stress-strain model, Eq. (2.10), the Eurocode 3 (ECS, 2005) 

model, and the measured stress-strain behavior of A992 steel from Hu et al. (2009)

Fig. 2-10 compares measured stress-strain data for Australian AS A186:250 steel from the 

extensive data of Skinner (1973) to the model and Eurocode 3 predictions. The yield strength data 

for this data set was used in the calculation of the retained yield strength, R (Eq. (2.4)), but the 

post-yield plastic behavior were not used. The predictions of the model of Eq. (2.10) are closer to 

the measured data than the predictions of the Eurocode 3 model. 
Chapter 2

26

Figure 2-10: Comparison of the predictions of the stress-strain model, Eq. (2.10), the Eurocode 3 (ECS, 2005) 

model, and the measured stress-strain behavior of AS A186:250 steel from Skinner (1973).

Fig. 2-11 compares measured stress-strain data for A36 steel from Harmathy and Stanzak (1970) 

to the model and Eurocode 3 predictions. 
Material Models

27

Figure 2-11: Comparison of the predictions of the stress-strain model, Eq. (2.10), the Eurocode 3 (ECS, 2005) 

model, and the measured stress-strain behavior of A36 steel from Harmathy and Stanzak (1970).

The Harmathy and Stanzak (1970) data were not used in the development of the stress-strain 

model, because the tests used a higher, non-standard strain rate. The plotted lines in Fig. 2-11

employ the strain rate sensitivity correction, Eq. (2.9), but the agreement between the measured 

behavior and the model is poorer. 

2.2.4 Uncertainty

The coefficient of variation, COV or VR, of a mechanical property, defined as the ratio of the 

standard deviation to the mean value, is an essential parameter in calculating the resistance factor, 
Chapter 2

28

, in load-and-resistance-factor design (LRFD) (Ellingwood et al., 1980; AISC, 2010). The 

uncertainty in the functions that describe the change in elastic modulus E, yield strength Fy, and 

tensile strength Fu will be major contributors to the uncertainty in the stress-strain behavior. To 

estimate the coefficient of variation as a function of temperature, the data were aggregated by 

temperature. All data that were determined at a temperature within δT = 10 °C were aggregated 

into a nominal temperature Tnom that was the mean of the individual temperatures. The coefficient 

of variation is defined as

 

  

2

nom

nom 1

1 1

1

N

R i

i

V R R T

R T N 

 

 

(2.17)

where N is the number of individual determinations of Riat the nominal temperature.

Fig. 2-12, Fig. 2-13, and Fig. 2-14 show the coefficient of variation for the modulus of elasticity

E, yield strength Fy, and tensile strength Fu, as a function of temperature, respectively. These 

figures show that the coefficient of variation for each parameter increases with temperature. The 

dashed line in each figure is a one-parameter linear regression of

V v T R 

1  20

(2.18)
Material Models

29

Figure 2-12: Coefficient of variation VR for elastic modulus E, as a function of temperature. Numbers along 

the VR=0 axis denote the number of different steels used in each determination.

In the three figures, if a nominal temperature contained fewer than Nmin individual determinations, 

it was omitted from the data set and the regression. In each figure, the horizontal uncertainty bar 

represents the range of the nominal temperatures. Where present, numbers on the x-axis represent 

the number of different steels in the determination of the coefficient of variation. Table 2-4

summarizes the determinations and regression data, where R is the retained yield strength or 

modulus function defined in the listed equations, 1 is the slope of the regression line, RSD is the 

residual standard deviation of the regression for 1, and DoF is the number of degrees-of-freedom 

of the regression for 1.
Chapter 2

30

Figure 2-13: Coefficient of variation VR for yield strength Fy, as a function of temperature. Numbers along 

the VR=0 axis denote the number of different steels used in each determination.
Material Models

31

Figure 2-14: Coefficient of variation VR for tensile strength Fu, as a function of temperature.

Table 2-4: Summary of coefficient of variation calculations for E, Fy, and Fu.

Parameter Figure R

 1

δT Nmin RSD DoF

°C

-1

°C

E 2-2 Eq. (2.2) 0.000351±0.000031 10 7 0.0402 8

Fy 2-4 Eq. (2.4) 0.000310±0.000017 10 10 0.0260 12

Fu 2-6 Eq. (2.14) 0.000273±0.000033 10 7 0.0435 7
Chapter 2

32

2.2.5 Modifying the Model for Steels with Different High-Temperature Performance

The model for structural steel at elevated temperature was developed with data exclusively from 

hot-rolled steels. Of course, not all steels used in construction are hot-rolled sections. Other steels, 

such as the new “fire-resistive” steels, quenched-and-tempered plate steels, and bolt steels are used 

in construction as well. Because of their different processing, the mechanical properties of these 

steels will certainly differ from those of hot-rolled shapes. This section shows three examples of 

how the model can be extended to represent the behavior of these steels. 

2.2.5.1 Fire-resistive steels

So-called fire-resistive structural steels have been available for some time in Europe and Japan 

(Sakumoto, 1999). These steels retain a greater fraction of their ambient-temperature yield strength 

at higher temperature than do ordinary structural steels, but can be produced to meet accepted 

standards for construction and weldability. Typically, fire-resistive steels are specified to retain 

2/3 of their specified yield strength at 600 °Cin contrast to the commonly accepted 1/2 of the 

specified yield strength at 538 °C, for ordinary structural steels. Recently these steels have been 

standardized in the United States under the designation ASTM A1077 (2012). Fig. 2-15 compares

the retained yield strength of two fire-resistive steels tested at a nominal strain rate of 8.33 ×

10−5 mm/mm/s, to the ordinary structural steel data from Fig. 2-4. Although these steels could 

meet the new A1077 standard, they were supplied to proprietary manufacturers’ standards. 
Material Models

33

Figure 2-15: Comparison of the normalized retained yield strength for ordinary (Ord.) and fire-resistive 

(FRS) steels.

Both steels were designed to meet the R=2/3 at 600 °C requirement, though in the figure, the steels 

are normalized to their measured rather than specified ambient-temperature yield strength, which 

causes the data to dip slightly below the R=2/3 line. Fig. 2-15 shows that the retained yield strength 

of the two fire-resistive steels is greater than all but a few of the ordinary structural steels. 

Modeling the performance of these steels, for example for performance-based design (Phan et al., 

2010), requires that the model parameters reflect the enhanced performance. The Eurocode 3 

(ECS, 2005) model provides no guidance on how steels with improved (or different) hightemperature performance might be modeled, though in principle new parameters for the retained 

strength, proportional limit, and modulus could be chosen. The model in this report can use high-
Chapter 2

34

temperature tensile test data from these fire-resistive steels to generate new parameters for either 

the retained yield strength, R, Eq. (2.4), data or the entire set of parameters, Eq. (2.10). 

Developing new parameters for the retained yield strength equation is the simpler approach. The 

dash/dot line in Fig. 2-15 is the non-linear regression of Eq. (2.4) to the yield strength data for the 

two fire-resistive steels. In the regression, the value of one parameter was fixed at the value for 

ordinary structural steel and not fit: r2=1.0. Fitting four parameters produced unacceptably large 

uncertainties for the ri. Table 2-2 summarizes the values of the parameters. 

With complete stress-strain data, it is also straightforward to estimate all the parameters of the 

stress-strain model. Because the yield strength of fire-resistive steels degrades more slowly than 

ordinary structural steels, it is reasonable to assume that the strain hardening behavior will also 

differ from that of ordinary structural steels. Table 2-2 shows the values for the individual k

i

and 

n for the fire-resistive steel, obtained by non-linear regression of Eq. (2.10) in the strain range 

(0.01<<0.1). 

Fig. 2-16 compares the stress-strain data for the two fire-resistive steels and two estimates of the 

stress-strain behavior. The solid lines are the measured stress-strain curves, and the dotted lines 

are the predictions of the model, Eq. (2.10), with the parameters listed in Table 2-2. In Fig. 2-14(a)

and Fig. 2-14(b), only the retained yield strength R(T), was fit, using the yield strength data in 

Fig. 2-15. That model under-predicts the strain hardening of both steels. In Fig. 2-14(a) and

Fig. 2-14(b), in addition to the retained yield strength behavior from Fig. 2-15, the values of the 

parameters k

i

and n in Eq. (2.10) were fit. 
Material Models

35

(a) FRS1, fit r

i

, k

i

, and n

(b) FRS2, fit r

i

, k

i

, and n

Figure 2-16: Stress-strain data (solid lines) and models (dotted lines) for two fire-resistive steels. Models were 

computed by fitting R(T), Eq. (2.4), first and then the parameters of the stress-strain model, Eq. (2.10).
Chapter 2

36

2.2.5.2 Quenched-and-tempered plates

Quenched-and-tempered steel plate is uncommon in buildings, but it is used in bridges (Hopwood 

et al., 2009, for example), which are also subject to fire (Kodur et al., 2010, for example). As part 

of the WTC investigation (Luecke et al., 2005), NIST characterized the high-temperature stressstrain behavior of specimens taken from Fy=100 ksi (689 MPa) plate with t=0.25 in (6.3 mm), 

denoted as specimen C10_C1M_FL. This steel was used extensively in the perimeter columns in 

the highest floors of two towers. This steel was similar to ASTM A514 (2009a), but was supplied 

to a proprietary Japanese specification. Fig. 2-17 shows that above T=600 °C the retained yield 

strength decreases faster than for hot-rolled steel. 

Figure 2-17: Comparison of the normalized retained yield strength for quenched-and-tempered 100 ksi 

(689 MPa) steel plate and the retained strength model, Eq. (2.4).
Material Models

37

Table 2-2 shows the values of the parameters used in the retained yield strength model, Eq. (2.4), 

for quenched-and-tempered plate. As with the other models, the value of r2 was not fit. 

Fig. 2-18 compares the stress-strain data for the quenched-and-tempered plate and the estimate of 

the stress-strain behavior from the model, Eq. (2.10). The solid lines are the measured stress-strain 

curves, and the dashed lines are the predictions of the model, Eq. (2.10), with the parameters listed 

in Table 2-2. In addition, because the fit employed data from a single steel, with a unique ambienttemperature yield strength, it is not reasonable to fit the parameters k

3

and k

4

, which depend only 

on the ambient-temperature yield strength. Instead those parameters were set to the values for hotrolled steel. The general agreement is acceptable, but the model over-predicts the low-strain 

strength. 

Figure 2-18: Stress-strain data (solid lines) and model (dotted lines) for Fy=100 ksi (689 MPa) plate. Model 

was computed by fitting R(T) using Eq. (2.4), and then the stress-strain data using Eq. (2.10).
Chapter 2

38

2.2.5.3 Bolts

The same process can be used to choose appropriate parameters from tensile test data for highstrength steels that are used for heat-treated bolts, whose strength also drops off faster with 

temperature than structural steel. Fig. 2-19 compares the prediction of the retained yield strength 

expression, Eq. (2.4), to data for the 0.2 % offset yield strength for steels used for structural steel 

bolts. In fitting the parameters of Eq. (2.4) to the data, only three parameters were fit, and one was 

fixed: r

2

=1. The retained strength of the bolt steels drops off much more quickly than ordinary 

structural steel as the temperature exceeds 400 °C. The figure also shows a fit of Eq. (2.4) to the 

data for the retained strength of bolt steels. Table 2-2 summarizes the values of the parameters. 

Although Kodur et al. (2012) plots stress-strain behavior of A325 and A490 bolt steels, the test 

procedure does not document the strain rate employed. 

Figure 2-19: Comparison of the normalized retained yield strength for high-strength bolt steels and the 

retained strength model, Eq. (2.4). See Table 2-1 for key to data sets.
Material Models

39

Insufficient data were available, at present, on structural bolt material properties at elevated 

temperatures to develop a power-law stress-strain model, as was presented in Eq. (2.10) for 

structural steel. Rather, a simplified multi-linear representation of the temperature-dependent true 

stress-strain behavior is proposed in this study, as follows:

     

( ) , ( )

( )

, ( )

( ) ( )

y

y

y u y y

u y

E T T

T

F T F T F T T

T T

  

 

  

 

 



 

          





(2.18)

where ɛy(T) = Fy(T)/E(T) is the temperature-dependent yield strain. The elastic modulus E is the 

same for bolts as for structural steel and is calculated from Eq. (2.2).

For A325 and A490 high-strength bolts, the temperature-dependence of the yield strength Fy is 

calculated from Eq. (2.4), with r1 = 4.967, r2 = 1.000, r3 = 456 °C, r4 = 2040 °C, and r5 = 0.000 

(see Section 2.2). Compared to rolled steel, bolts sustain their Fy value with the increase of 

temperature until about 400 °C, after which it reduces dramatically relative to structural steel. 

Fig. 2-20 shows the degradation of the normalized yield strength with increasing temperature for 

ASTM A572 rolled steel and ASTM A325 and A490 bolts (ASTM International (2012b), and 

(2010), respectively). Note that at 400 °C, both rolled steel and bolts sustain about 80 % of their 

yield capacity. At 600 °C, rolled steel sustains about 50 % of its yield capacity, while bolts sustain 

about 20 % of their yield capacity. The ultimate tensile strength Fu is calculated using Eq. (2.4) 

with the same values of r1 through r5 as for the yield strength, but with the ambient-temperature 

yield strength Fy0 replaced by the ambient-temperature ultimate strength Fu0. The temperaturedependent uniform strain, ɛu(T), is assumed to have a value of 0.1 at 20 °C and to decrease linearly 

with temperature to a value of 0.05 at 600 °C, as summarized in Eq. (2.19). 

0.1, 20 C

20 C ( ) 0.1 0.05 , 20 C 600 C

580 C

0.05, 600 C

u

T

T

T T

T



 



   

         

  

 (2.19)

Eq. (2.19) is an approximate linear fit to the elevated-temperature uniform strain data reported by 

Kodur et al. (2012) for both A325 and A490 structural bolts. 
Chapter 2

40

Figure 2-20 Degradation of normalized yield strength versus the increase in temperature for rolled structural 

steel and bolts.

2.3 SUMMARY

This chapter presents a stress-strain model for structural steel at elevated temperatures, as an 

alternative to the stress-strain model used in the Eurocode 3 standard. It is based on a large data 

set of the yield strength of structural steels, augmented by post-yield behavior derived from eight 

steels used in the construction of the World Trade Center. It accounts for the temperature 

dependence of the yield strength, the change in strain hardening with temperature, and the effect 

of strain rate on strength. The quality of the predictions of the model for stress-strain behavior for 

new steels, such as ASTM A992, are similar to those of the Eurocode 3 stress-strain model. The 

coefficient of variation of the retained yield strength, which is the largest component of the 

variability in the model, increases with increasing temperature. Finally, the model is extensible to 

steels with improved high-temperature properties by either choosing a new representation of the 

temperature dependence of the high-temperature yield strength, or by fitting stress-strain data to 

generate new values of the parameters that describe the temperature dependence of the strain 

hardening. For structural steel materials, a temperature-dependent power-law model was used to 

represent the true stress-strain behavior up to the onset of necking at the temperature-dependent 

uniform strain. The uniform strain was determined based on a least-squares fit to the uniform strain 

data for elevated temperatures available in the literature. Due to the more limited experimental 

data on structural bolts at elevated temperatures, a simpler multi-linear material model was 

proposed to represent the true stress-strain behavior of structural bolts.

0 100 200 300 400 500 600 700 800

0

0.2

0.4

0.6

0.8

1

Temp. (oC)

F

y(T)/F

y0

Steel-A572

Bolts

0 100 200 300 400 500 600

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Temperature, oC

Engineering Strain at Fracture, 



e

n

g,f

LS-DYNA

Luecke

Hu
41

Chapter 3

USE OF NIST MATERIAL MODEL TO EVALUATE STABILITY OF WIDE 

FLANGE STEEL COLUMNS AT ELEVATED TEMPERATURES

3.1 INTRODUCTION 

The main focus of the chapter is to evaluate the applicability of the NIST model for predicting the 

behavior of steel gravity columns at elevated temperatures using the finite-element method. The 

NIST stress-strain model is relatively new and has been used for very few analytical studies on 

fire-structure interaction. In this chapter, the NIST stress-strain model is used to predict flexural 

buckling of steel columns subjected to elevated temperatures. Accurately predicting the fire 

performance of columns is critical for evaluating the stability of buildings subjected to fire. In a 

fire, steel gravity columns can be susceptible to inelastic buckling failure because they have high 

utilization ratios with the governing loading combination, “1.2[dead load] + 1.6[live load],” (see 

Agarwal and Varma (2011)). The failure of interior columns can significantly influence the 

stability of a building system because loads supported by these columns need to be redistributed 

through an alternative load path to prevent progressive collapse (see Agarwal and Varma (2014)). 

This chapter presents results from analytical studies to evaluate the effect of material models on 

predicting the strength and behavior of steel columns at elevated temperatures: (1) The critical 

buckling temperatures of 47 tested columns are calculated using the finite-element models and are 

compared with those reported in the experiments. For computational efficiency and due to the

limited amount of test data, the modeling approach used in this study assumes time-independent 

loading and heating conditions, uniform temperature distribution in columns, and temperatureindependent thermal strains. (2) The high-temperature flexural buckling strength and response of 

six full-scale steel column specimens are predicted using finite-element models with two different 

material models. These models use measured steel temperature and axial load data to simulate the 

column tests as closely as possible. The responses predicted using the NIST stress-strain models 

(presented in Section 2.2 of this report) are compared with those calculated using Eurocode 3, ESC 

(2005), stress-strain model as well as test data. (3) The new flexural buckling equations are

calibrated using the NIST model and compared with those prescribed in the 2005 and 2010 editions 

of the ANSI/AISC 360 Appendix 4 (AISC, 2005 and 2010). 

3.2 CRITICAL BUCKLING TEMPERATURE OF STEEL COLUMNS

3.2.1 Test Data

The critical temperature of columns of typical proportions involves inelastic behavior that is a 

function of both the modulus of elasticity and the yield strength of the steel. At temperatures that 
Chapter 3

42

steel experiences in building-contents fires, both the stiffness and strength are significantly 

diminished and, therefore, the buckling strength of a steel column exposed to fire can be a small 

fraction of that at ambient temperature. To evaluate the behavior of steel columns at elevated 

temperatures, many laboratories have tested axially loaded columns under a variety of boundary 

conditions and heating regimes. Tests have been conducted either by loading the column specimen 

and then heating it until failure occurs, or by heating the column to a specific temperature and then 

applying an axial load to failure. For a given column geometry, fixity, and loading, the experiment 

yields the temperature at which the column buckles as evidenced by either a decrease in the applied 

load or runaway displacement. The approach used in this section is to use experimental data from 

as many laboratories as possible in an attempt to understand representative behavior of columns 

of various slenderness and fixity conditions.

To evaluate the applicability of the NIST model in predicting the critical temperature, five data 

sets from various laboratories, chosen from a larger set compiled by Zhang et al. (2012), are 

employed. Zhang’s study analyzed the literature experimental data using the Eurocode 3 column 

equations for computing the critical temperature for column buckling. In this section, the critical 

temperatures are recalculated using the finite-element method. The five laboratories represent

sources for which the original descriptions of the test protocol were available. The steels originated 

from Europe and Asia. Four of the five laboratories used pin-end connections. All were conducted 

by applying the load and then ramping the temperature until failure. Table 3-1 summarizes the data 

for the steel column tests considered in this study.

Fig. 3-1 illustrates the structural models for five different column tests as described below. The 

data sets used in this study include 47 individual column tests. All investigators took special care 

to monitor and accurately record the column temperature at multiple locations. 

 Nine tests on bare steel columns with pinned-pinned ends and length (L) of 71 in

(1800 mm) reported by Ali et al. (1998). The loading frame could apply both axial load and 

axial restraint to the test column, through half-round graphite-lubricated bearings. The 

present study considers only cases with no axial restraint. A single-burner furnace heated 

the column and produced a temperature ramp rate dT/dt = 15 ○C/min to 400 ○C and dT/dt

= 2 ○C/min above.

 Thirteen tests on bare steel columns of different lengths with pinned-pinned ends and (50 

≤ L ≤ 138) in ((1270 ≤ L ≤ 3510) mm), reported by Franssen et al. (1996, 1998). The 

loading frame employed knife edges to achieve the end fixity. Unlike the other tests which 

used a furnace, ceramic heating mats attached to the column supplied the heat. The heating 

rate was 5 ○C /min. Some data on these tests was taken from Schleich et al. (1998). Several 

data sets from Franssen et al. (1996) were omitted because of uncertainty about the test 

conditions.
Stability of Columns

43

 Three tests on protected steel columns with fixed-fixed ends and L = 150 in (3810 mm)

reported by Lie and Macauley (1989). The heating condition followed ASTM E119 (2014)

(approximately dT/dt = 3.6 ○C /min). 

 Four tests on bare steel columns with pinned-pinned ends and L = 59 in (1500 mm)

reported by Tan et al. (2007). The loading frame could apply axial forces as well as axial 

restraints, but the present study considered only cases with no axial restraint. Initial 

imperfections such as column crookedness and load eccentricity were measured by a 

specially designed facility. An electric furnace heated the columns at dT/dt = 8 ○C /min. 

 Eighteen tests on bare steel columns with pinned-rotational restrained ends through 

pinned, half-roller cylinders and L = 146 in (3710 mm), reported by Wang and Davies 

(2003). The furnace temperature increased at dT/dt = 16.7 ○C /min. The effective lengths 

of the test columns, obtained by an approximated method (Wang and Davies, 2003), were 

used in this study.

Figure 3-1 Structural models for column tests

(a) Ali and 

Tan both 

with ka = 0 

(b) Lie (c) Wang (d) Franssen (e ≠ 0) 
Chapter 3

44

3.2.2 Modeling Approach

Fig. 3-2 shows a schematic model of a steel column with pinned-pinned ends. Each column 

specimen in Table 3-1 was modeled using ANSYS 14.0.0 (2014) three-dimensional shell element, 

SHELL181, which is suitable for analyzing thin to moderately thick shell structures. It is a four 

node element with six degrees of freedom at each node: translations in the x, y, and z directions, 

and rotations about the x, y, and z axes. SHELL181 is well suited for linear, large rotation, and 

large-strain non-linear applications. The shape of initial column crookedness is assumed to be the 

same as the first buckling mode. The initial deflection amplitude at mid-height, if not specified, 

was taken as L/1000. The effect of residual stresses was omitted. For restrained column specimens, 

the model was modified such that the corresponding degrees of freedom were restrained to 

represent the support conditions as reported in test literature. Regardless of different test protocols 

used in the tests, all column models were subjected to the axial loads equal to the reported values 

of axial load capacity in test literature. Then, while the axial load was maintained constant, the 

steel temperature was increased monotonically until the analysis failed to converge. That 

temperature is defined as the failure temperature. 

Figure 3-2 Schematic of the column model used in the analysis

δ = L/1000 + e
Stability of Columns

45

Table 3-1 Column test data

Data Test Shape Fyo L λ Pr e Ends Tr Tp (

○C)

(MPa) (mm) (kN) (mm) (

○C) Eurocode 

3

NIST

Tan RS45_1 UC152×152×37 326 1500 38 708.5 1.74 P-P 647 507 588

Tan RS55_1 UB203×133×25 357 1500 47 444.3 3.19 P-P 571 520 557

Tan RS81_1 UB152×89×16 312 1500 70 260.6 2.38 P-P 499 401 567

Tan RS97_1 UB127×76×13 320 1500 83 134 4.08 P-P 606 501 617

Ali Ali1 UC152×152×23 320 1800 47 186 0 P-P 701 643 758

Ali Ali2 UC152×152×23 320 1800 47 373 0 P-P 626 550 676

Ali Ali3 UC152×152×23 320 1800 47 559 0 P-P 557 403 594

Ali Ali4 UB178×102×19 320 1800 75 202 0 P-P 629 541 662

Ali Ali5 UB178×102×19 320 1801 75 303 0 P-P 539 395 582

Ali Ali6 UB178×102×19 320 1802 75 101 0 P-P 644 643 745

Ali Ali7 UB127×76×13 320 1803 97 50 0 P-P 717 661 730

Ali Ali8 UB127×76×13 320 1804 97 101 0 P-P 658 555 647

Ali Ali9 UB127×76×13 320 1805 97 151 0 P-P 567 438 567

Lie Lie1 W10×60 300 3810 34 1760 0 F-F 565 574 591

Lie Lie2 W10×49 300 3810 34 1424 0 F-F 586 566 591

Lie Lie3 W10×49 300 3810 34 1424 0 F-F 584 566 591

Franssen AL5 HEA100 280 2770 108 127 5 P-P 457 428 589

Franssen BL3 HEA100 287 1272 50 292 5 P-P 390 377 528

Franssen BL5 HEA100 287 2772 108 73 5 P-P 587 573 699

Franssen BL6 HEA100 287 3510 137 105 5 P-P 446 356 527

Franssen CL3 HEA100 293 1271 50 251 5 P-P 474 478 581

Franssen CL5 HEA100 293 2771 108 34 5 P-P 587 677 783

Franssen CL6 HEA100 293 3510 137 90 5 P-P 493 447 601

Franssen DL3 HEA100 283 1269 49 24 5 P-P 749 875 818

Franssen DL6 HEA100 283 3510 137 12 5 P-P 727 802 840

Franssen SL40 HEA100 280 2020 79 170 5 P-P 525 475 600

Franssen SL41 HEA100 280 2026 79 174 5 P-P 509 445 584

Franssen SL42 HEA100 280 2020 79 171 5 P-P 485 458 591

Franssen SL44 HEA100 280 2023 79 173 5 P-P 495 448 586

Wang SCRI1 UC203x203x46 301 2968 57 357.5 0 R-R 646 613 707

Wang SCRI2 UC203x203x46 301 2875 56 360.3 0 R-R 681 616 707

Wang SCRI3 UC203x203x46 301 2875 56 360 0 R-R 668 616 707

Wang SCRI4 UC203x203x46 301 3061 59 600 0 R-R 604 527 640

Wang SCRI5 UC203x203x46 301 2463 48 600 0 R-R 597 552 647

Wang SCRI6 UC203x203x46 301 2745 53 600 0 R-R 613 540 644

Wang SCRI7 UC203x203x46 301 3001 58 840 0 R-R 556 400 571

Wang SCRI8 UC203x203x46 301 2894 56 840 0 R-R 553 414 574

Wang SCRI9 UC203x203x46 301 3001 58 840 0 R-R 530 400 571

Wang SCRII1 UC203x203x46 301 2820 55 358 0 R-R 680 619 708

Wang SCRII2 UC203x203x46 301 2849 55 360 0 R-R 670 617 707

Wang SCRII3 UC203x203x46 301 2876 56 360 0 R-R 675 620 707

Wang SCRII4 UC203x203x46 301 3079 60 600 0 R-R 609 526 639

Wang SCRII5 UC203x203x46 301 2883 56 600 0 R-R 604 534 643

Wang SCRII6 UC203x203x46 301 2823 55 600 0 R-R 629 537 643

Wang SCRII7 UC203x203x46 301 2916 56 840 0 R-R 542 411 573

Wang SCRII8 UC203x203x46 301 2916 56 840 0 R-R 550 411 573

Wang SCRII9 UC203x203x46 301 2842 55 840 0 R-R 549 421 575
Chapter 3

46

3.2.3 Results and Discussion

Fig. 3-3 shows the predicted failure temperature, Tp, from the ANSYS analysis as a function of 

reported failure temperature, Tr, for each test in the five data sets for both the Eurocode 3 stressstrain model and the NIST model. The solid line has unity slope and zero intercept. The Eurocode 3 

model predicts the data of Franssen et al. (1996) well since more data points are closer to the unity 

slope line, but overpredicts the four other data sets. The NIST model underpredicts the data from 

Franssen et al. (1996), but predicts the four other data sets very well.

Figure 3-3 Predicted vs. reported failure temperature for the (a) NIST and (b) Eurocode 3 models.

Fig. 3-4 shows the normalized difference between the predicted and reported failure temperatures 

for the two stress-strain models as a function of slenderness parameter, λ =L/r, where r is the 

governing radius of gyration of the column section. For four of the five data sets, the NIST models 

more accurately predict the failure temperatures than the Eurocode 3 models in terms of the 

percentage error, 

Tp Tr /Tr

100

. The NIST model tends to significantly overestimate the data 

of Franssen et al. (1996) by 21 %, on average. The Eurocode 3 model underpredicts the failure 

temperature for most of data sets. The Eurocode 3 model reasonably predicts the data set of 

Franssen et al. (1996) well (with percentage error of 9 %, on average).

Taking the total number of tests as an indicator of the quality of the model is not ideal when the 

size of the individual data sets differs. Each data set used an individual steel, probably from the 

same heat, and the same test method, so any deviations from representative behavior will be 

replicated systematically across the data set. For each data set it shows the number of times each 

model made a better prediction of the critical temperature. As shown in Fig. 3-4, the NIST model 

300

400

500

600

700

800

900

1000

Predicted failure temperature, 

300 400 500 600 700 800 900 1000

Tp

(

○C)

Reported failure temperature, Tr(

○C)

Ali

Tan

Lie

Franssen

Wang

300

400

500

600

700

800

900

1000

300 400 500 600 700 800 900 1000

Predicted failure temperature, Tp

(

○C)

Reported failure temperature, Tr(○C)

Ali

Tan

Lie

Franssen

Wang

(a) (b)
Stability of Columns

47

makes better predictions for three of the five data sets (Ali, Tan, and Wang), and overestimations 

for the data set of Franssen. The data set of Lie could be considered a tie, since both models predict 

the failure temperature to within a few percent.

Figure 3-4 Normalized difference in failure temperature as a function of slenderness ratio (λ)

-0.3

0

0.3

0 50 100 150

(Tp−Tr) / Tr

λ

NIST

Eurocode 3

-0.3

0

0.3

0 50 100 150

(Tp−Tr) / Tr

λ

NIST

Eurocode 3

-0.3

0

0.3

0 50 100 150

(Tp−Tr) / Tr

λ

NIST

Eurocode 3

-0.3

0

0.3

0 50 100 150

(Tp−Tr) / Tr

λ

NIST

Eurocode 3

-0.3

0

0.3

0 50 100 150

(Tp−Tr) / Tr

λ

NIST

Eurocode 3

(a) Tan (b) Ali

(c) Wang (d) Franssen

(e) Lie
Chapter 3

48

3.3 HIGH-TEMPERATURE RESPONSES OF STEEL COLUMNS

3.3.1 Test Data 

Choe et al. (2011b) tested full-scale steel columns at temperatures of 300 ○C, 500 ○C, or 600 ○C. 

Fig. 3-5 shows the Purdue column test setup and instrumentation layout. The flanges at the mid 

portion of the column specimen were heated with four radiant heaters. The length of heated region 

was 6 ft (1.83 m). Fig. 3-5 also shows the instrumentation layout for measuring steel temperatures, 

axial loads, deformations, and end rotations. The relevant standard uncertainty in mechanical and 

thermal measurements was estimated as ± 1 % based on calibration of the data acquisition system 

and instruments (see Choe (2011a)). Although the temperatures over the exposed flange surfaces 

were uniform (with the maximum absolute difference of 30 ○C), thermal gradients were developed 

along the web and the portions outside of heated region toward the column ends. The column 

supports remained cool (≤ 50 ○C) throughout the tests.

Figure 3-5 Column test setup and instrumentation layout.

Heater length = 2 m

508 mm 460 mm 460 mm 508 mm

Axial 

Load (P)

A B CL D E

760 mm

Thermocouples at sections A-E

Linear displacement

transducer

Rotation

transducer

Axial displacement

Hydraulic 

Ram 

Radiant 

Heaters

Column Specimen

Radiant 

Heaters
Stability of Columns

49

Table 3-2 presents the test matrix for six steel columns of W8×35 and W14×53 sections made of 

ASTM A992 steels. All columns had pinned-pinned support conditions achieved using the halfcylindrical bearings shown in Fig. 3-5. The length (L) of the W8×35 and the W14×53 column 

specimens was 11.5 ft (3.5 m) and 11.3 ft (3.45 m), respectively, which gives slenderness ratios 

(L/ry) of 69 and 71, respectively, where ry is the radius of gyration about the weak axis. The 

measured average ambient-temperature yield stress (Fyo) of the W8×35 sections was 60 ksi 

(413 MPa) and that of the W14×53 sections was 59 ksi (406 MPa). No fire protection was used 

for the column specimens tested at elevated temperatures. All column specimens were subjected 

to concentrically applied axial loads. The test matrix (in Table 3-3) included one control specimen 

(W8×35−AMB), tested at ambient temperature, to commission the structural setup and to compare 

the behavior of columns at elevated temperature. 

Each column specimen was tested under steady heating condition but subjected to increasing axial 

loads until buckling occurred. Fig. 3-6 shows the axial load-temperature-time histories recorded 

during the tests. Each plot includes the average temperature-time (T−t) curves measured at the 

flanges (Flange Temp) and the center of the web (Web Temp) of the cross-sections B, CL, and D 

shown in Fig. 3-5. Table 3-2 summarizes the column test results, including the average steel 

temperatures of the flanges (Tflange) and the center of the web (Tweb) of the cross-sections B, CL, 

and D while the axial loads were increased, the critical buckling load (Pcr), and the axial load rate 

(dP/dt). All column specimens failed by inelastic flexural buckling about the weak axis, and no 

flange distortion failures were observed. 

Table 3-2 Column test matrix and results

Specimen Fyo (MPa) L/ry Tflange(

oC) 

(

oC) (oC)

Tweb (

oC) Pcr (kN) dP/dt

W8×35−AMB 413 69 (kN/min) - - 2000 -

W8×35−T300 413 69 320 190 1413 1.3

W8×35−T500 413 69 477 406 1134 28

W8×35−T600 413 71 609 451 800 35

W14×53−T500 406 71 504 324 1435 33

W14×53−T600 406 71 608 363 1070 26
Chapter 3

50

Figure 3-6 Axial load-temperature-time histories used in the Purdue column tests.

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0

100

200

300

400

500

600

700

0 25 50 75 100 125 150

P, kN

T, 

○C

t, min

(a) W8 35-T300

Axial load

Web Temp

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0

100

200

300

400

500

600

700

0 25 50 75 100 125 150

P, kN

T, 

○C

t, min

(b) W8 35-T500

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0

100

200

300

400

500

600

700

0 25 50 75 100 125 150

P, kN

T, 

○C

t, min

(c) W8 35-T600

Flange Temp

Web Temp

Axial load

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0

100

200

300

400

500

600

700

0 25 50 75 100 125 150

P, kN

T, 

 C

t, min

(d) W14 53-T500

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0

100

200

300

400

500

600

700

0 25 50 75 100 125 150

P, kN

T, 

 C

t, min

(e) W14 53-T600

Flange Temp
Stability of Columns

51

3.3.2 Finite-Element Models 

Three-dimensional finite-element method (3D FEM) models were developed and analyzed to 

predict the high-temperature flexural buckling behavior of the column specimens. The models 

were developed and analyzed using ABAQUS ver. 6.12-2 (2012). As mentioned earlier, detailed

finite-element models were developed and analyzed to evaluate the influence of temperaturedependent material models in predicting the behavior and strength of the column specimens. The 

model geometry, element type, and boundary conditions for all models were identical. These are 

summarized below first, followed by the development of the column analysis matrix for 

comparative studies.

Each model used the general-purpose four-node shell elements with reduced integration (S4R) 

implemented in ABAQUS. The cross-section of the column specimen was discretized into 

nineteen nodes; the flanges and web were divided into six S4R elements each. These elements had 

six degrees of freedom per node, five integration points through the thickness of a shell section, 

and one integration point to form the element stiffness.

For simulating the inelastic buckling failure of steel columns, the 3D FEM model accounted for 

both material and global geometric imperfections as recommended by Agarwal and Varma (2014). 

Residual-stress distributions, observed in a typical hot-rolled steel shape, were modeled by 

creating uneven thermal strains over the cross-section. It was assumed that the flange tips and the 

center of the web cooled down faster than the flange-to-web junctions. The maximum residual 

stress was assumed as 30 % of the ambient-temperature yield stress as specified in 

ANSI/AISC 360-10 (2010). The global geometric imperfection (sweep) was assumed equal to 

L/1500, which was the mean value of the experimentally measured initial crookedness of hotrolled wide-flange shapes reported in Bjorhovde, 1972. The bending stresses and deformations 

due to the column self-weight were also included as initial imperfections because the column 

specimen was aligned horizontally as shown in Fig. 3-5. 

The column specimens were modeled with pinned-pinned boundary conditions. Both ends were 

free to rotate about the weak axis of the cross-section, but rotations about the strong axis of the 

cross-section and about the longitudinal axis of the specimen were restrained. Similar to the test

specimen, one of the column ends was free to move in the direction of axial loading to allow 

thermal elongation while temperatures increased. The axial force and the boundary conditions 

were applied to the centroid of the cross-section at both column ends. 

Table 3-3 shows the analytical matrix of 3D FEM models for the nonlinear analysis of the column 

specimens. Each model generated the output including the load-displacement (or load-end 

rotation) behavior and the critical buckling load (PFEM) of the specimens. These analytical results 

were also compared with the Purdue test results.
Chapter 3

52

Table 3-3 Column analysis matrix

Group ID Specimen EC 3 NIST

PFEM

(kN)

A

1 W8×35−T300 x 1134

2 W8×35−T500 x 921

3 W8×35−T600 x 639

4 W14×53−T500 x 1260

5 W14×53−T600 x 865

B

6 W8×35−T300 x 1392

7 W8×35−T500 x 1141

8 W8×35−T600 x 788

9 W14×53−T500 x 1442

10 W14×53−T600 x 1091

Fig. 3-7 shows the material models used in the analyses. Both the temperature-dependent stressstrain relationship and the coefficient of thermal expansion were required for simulating the 

behavior of a column specimen simultaneously subjected to axial loading and elevated 

temperatures. Each 3D FEM model in Table 3-3 used either the Eurocode 3 (EC 3) or the NIST 

material model for temperature-dependent mechanical material properties of steel. In ABAQUS, 

the stress-strain behavior was defined using two components: isotropic elastic behavior with 

Poisson’s ratio equal to 0.3 and plastic behavior defined by the Von Mises yield surface, associated 

flow rule, and kinematic hardening. The temperature-dependent elastic moduli and plastic strains 

were calculated using the average yield stress (Fyo in Table 3-2) and elastic modulus (Eo) of 

30,500 ksi (210 GPa) measured at ambient temperature. The coefficients of thermal expansion at 

temperatures varying from 20 ○C to 730 ○C were used to compute thermal strains in the column 

specimen with increasing temperatures. Thermal creep properties were not explicitly included.
Stability of Columns

53

Figure 3-7 (a) Thermal expansion model of steel, (b) Eurocode 3 stress-strain model, (c) NIST stress-strain 

model.

For simulating the Purdue column tests, a series of nonlinear static analyses were conducted 

sequentially in accordance with the testing protocol. These analyses consisted of three steps: (1) 

applying an initial axial load of 30 kip (133 kN) at ambient temperature, (2) increasing steel 

temperatures while the initial axial load was maintained, and (3) increasing the axial load under 

relatively steady heating until buckling occurred. The Purdue test conditions were modeled using 

the actual test data, including thermal gradients developed in column specimens and actual 

variations in axial loads and temperatures during the tests. The Eurocode 3 and NIST stress-strain 

models were evaluated by comparing the numerically predicted displacements and rotations with 

those measured from the experiments.

The modified Newton-Raphson algorithm was used to predict the behavior of the column 

specimens using the measured axial loads and temperatures. The 3D FEM models were analyzed 

using the measured axial load-temperature-time (P−T−t) data as input, where temperatures were 

1.10E-05

1.20E-05

1.30E-05

1.40E-05

1.50E-05

1.60E-05

1.70E-05

100 200 300 400 500 600 700

α, 

oC-1

T, oC

(a)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.005 0.01 0.015 0.02 0.0

σ/Fyo

ε

20oC

(b) 

400 500 600 700

T, oC

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.005 0.01 0.015 0.02 0.025

σ/Fyo

ε

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.005 0.01 0.015 0.02 0.025

σ/Fyo

ε

20oC

600oC

(b) (c)
Chapter 3

54

assigned at thermocouple locations as shown in Fig. 3-5. It was assumed that temperatures at 

elements between thermocouple locations were equal to the average of temperatures measured at 

two adjacent thermocouples, and temperatures at column ends were assumed to be 20 ○C. Heat 

transfer analysis was performed to obtain the temperature-time curves for all nodes of each column 

model using the Eurocode 3 thermal properties for steel. Structural responses (strains, 

displacements, and rotations) were then computed for each load and temperature increment 

following the P−T−t histories until inelastic buckling occurred. The stability failure of the column 

specimens was indicated when the analysis terminated due to non-convergence. The last axial load 

step in which convergence was achieved was defined as the critical buckling load (PFEM). 

3.3.3 Results and Discussion 

Fig. 3-8 compares the experimentally measured load-displacement responses of the specimen 

W8x35-AMB, tested at ambient temperatures, with those analytically predicted using the 3D FEM 

models. Both the Eurocode 3 and the NIST stress-strain models predicted the axial stiffness 

reasonably with differences less than 3 % with respect to the measured axial stiffness. Both models 

also reasonably predicted the critical buckling loads (PFEM) with differences less than 10 % on the 

conservative side with respect to the measured critical buckling load (Pcr). 

Figure 3-8 Axial load-axial displacement response of the specimen W8×35-AMB.

Fig. 3-9 shows the predicted buckling shape of the column specimen W8×35−T300 and 

W14×53−T500 with scale factor of 10. The results indicated that the predicted failure mode of all 

the specimens was inelastic flexural buckling regardless of the applied temperature distributions. 

Although the columns were heated unevenly, temperature distributions in the cross-section and 

about the vertical axis at mid-span remained close to symmetric; therefore, the governing failure 

0

500

1000

1500

2000

2500

0 1 2 3 4

Axial load (kN)

Axial displacement (mm)

CASE 2

CASE 1

Test

Test: Pcr = 2000 kN

EC 3: PFEM = 1812 kN

NIST: PFEM = 1818 kN

NIST

EC 3

Test
Stability of Columns

55

mode was flexural buckling. The 3D FEM models also accurately predicted the downward 

direction of lateral displacements for all column specimens due to inelastic flexural buckling. The 

initial lateral displacement resulting from the column self-weight combined with axial loading 

induced second order moments that caused the column to sag downward. 

Figure 3-9 Predicted buckling shape with nodal temperature (NT11) distribution.

Fig. 3-10 compares the axial load-end rotation responses predicted using the finite-element models 

with those measured experimentally. As shown for the specimens subjected to a target temperature 

of 300 ○C or higher, the NIST material model was better for predicting the end rotations with 

increasing axial loads. The end rotations of the specimens W8×35−T500 and W14×53−T600 

predicted using the NIST model compare very well with the test results (within ± 1 % difference). 

Fig. 3-11 compares the axial load-lateral displacement responses predicted using the finite-element 

models with those measured in the experiments at elevated temperatures. As shown, the NIST 

model accurately computed the lateral displacements of the specimens W8×35−T600, 

W14×53−T500, and W14×53−T600. Note that Fig. 3-11 shows no test results for specimens 

W8×35-T300 and W8×35-T500 because lateral displacements were not measured for these 

specimens (see Choe et al. (2011)). 

W8×35-T300

Scale factor = 10

W14×53-T500

Scale factor = 10
Chapter 3

56

Figure 3-10 Axial load-end rotation responses of column specimens.

0

200

400

600

800

1000

1200

1400

1600

1800

2000

-0.5 0 0.5 1 1.5 2 Axial load (kN)

End rotation (deg)

CASE 2 (ID = 6)

CASE 1 (ID = 1)

Test

0

200

400

600

800

1000

1200

1400

1600

1800

2000

-0.5 0 0.5 1 1.5 2 Axial load (kN)

End rotation (deg)

CASE 2 (ID = 7)

CASE 1 (ID = 2)

Test

0

200

400

600

800

1000

1200

1400

1600

1800

2000

-0.5 0 0.5 1 1.5 2 2.5 Axial load (kN)

End rotation (deg)

CASE 2 (ID = 9)

CASE 1 (ID = 4)

Test

0

200

400

600

800

1000

1200

1400

1600

1800

2000

-0.5 0 0.5 1 1.5 2 2.5 Axial load (kN)

End rotation (deg)

CASE 2 (ID = 8)

CASE 1 (ID = 3)

Test

0

200

400

600

800

1000

1200

1400

1600

1800

2000

-0.5 0 0.5 1 1.5 2 2.5 Axial load (kN)

End rotation (deg)

CASE 2 (ID = 10)

CASE 1 (ID = 5)

Test

(a) W8×35−T300

(b) W8×35−T500

(c) W8×35−T600

(d) W14×53−T500

(e) W14×53−T600

NIST (ID = 6)

EC3 (ID = 1)

Test

NIST (ID = 7)

EC3 (ID = 2)

Test

NIST (ID = 9)

EC3 (ID = 4)

Test

NIST (ID = 8)

EC3 (ID = 3)

Test

NIST (ID = 10)

EC3 (ID = 5)

Test
Stability of Columns

57

Figure 3-11 Axial load-lateral displacement responses of column specimens.

0

200

400

600

800

1000

1200

1400

1600

1800

2000

-10 0 10 20 30 40 50 Axial load (kN)

Lateral displacement (mm)

CASE 2 (ID = 6)

CASE 1 (ID = 1)

0

200

400

600

800

1000

1200

1400

1600

1800

2000

-10 0 10 20 30 40 50 Axial load (kN)

Lateral displacement (mm)

CASE 2 (ID = 7)

CASE 1 (ID = 2)

0

200

400

600

800

1000

1200

1400

1600

1800

2000

-10 0 10 20 30 40 50 Axial load (kN)

Lateral displacement (mm)

CASE 2 (ID = 8)

CASE 1 (ID = 3)

Test

0

200

400

600

800

1000

1200

1400

1600

1800

2000

-10 0 10 20 30 40 50 Axial load (kN)

Lateral displacement (mm)

CASE 2 (ID = 9)

CASE 1 (ID = 4)

Test

0

200

400

600

800

1000

1200

1400

1600

1800

2000

-10 0 10 20 30 40 50 Axial load (kN)

Lateral displacement (mm)

CASE 2 (ID = 10)

CASE 1 (ID = 5)

Test

(a) W8×35−T300

(b) W8×35−T500

(c) W8×35−T600

(d) W14×53−T500

(e) W14×53−T600

NIST (ID = 6)

EC 3 (ID = 1)

NIST (ID = 7)

EC 3 (ID = 2)

NIST (ID = 9)

EC 3 (ID = 4)

Test

NIST (ID = 8)

EC 3 (ID = 3)

Test

NIST (ID = 10)

EC 3 (ID = 5)

Test
Chapter 3

58

It was observed that the 3D FEM results with the Eurocode 3 model indicated that the column 

stiffness decreased significantly at the earlier stage of axial loading at elevated temperatures, as 

opposed to those with the NIST model. Given that the initial geometric imperfections were equally 

applied to all models, the induced second-order moments on the deformed shape remained 

comparable. As shown in Fig. 3-7, however, the Eurocode 3 model indicates that the tangent 

modulus decreases significantly beyond the elastic region (after the proportional limit, the 

maximum stress at which the stress-strain relationship is linear). This nonlinear behavior (tangent 

modulus degradation) starts at strains less than 0.1 %, and the maximum stresses (strength) in this 

small strain region are much smaller compared to the NIST model. Therefore, the Eurocode 3 

model tends to underpredict the tangent modulus in the elastic regime, and eventually underpredict 

the overall inelastic buckling behavior of a column.

Fig. 3-12 compares the predicted axial displacement-time responses with corresponding test 

results. Each plot includes the axial displacements at both thermal loading and axial loading 

phases: the positive slope indicates the thermal elongation of the pin-ended column specimen while 

steel temperature was increased to a target value. The negative slope indicates the axial shortening 

in response to increasing axial loads under steady-heating conditions. The axial shortening of the 

specimen W8×35−T300 in Fig. 3-12(a) was not experimentally measured due to the malfunction 

of an axial displacement transducer (see Choe et al (2011)). The comparison indicates that both 

the Eurocode-3 and NIST thermal expansion (α−T) models accurately predicted the thermal 

elongations of the column specimens (within ± 1 %). The predicted displacement behavior during 

the axial loading phase also shows similar trends as compared to the test results. 

The NIST models were found to be more appropriate for predicting the time of failure (at the onset 

of buckling) as shown in Fig. 3-12. The failure time is indicated with an asterisk symbol (*) on 

each plot. The Eurocode 3 models conservatively predicted the time of failure. Given that the 

period of axial loading was shorter than 1 hour and the average temperatures through crosssections were lower than 600 ○C, the effects of thermal creep on the column experiments presented 

in this paper appeared to be negligible. 

Fig. 3-13 shows the normalized difference between the measured critical buckling loads (Pcr) and 

the predicted critical buckling loads (PFEM) using the 3D FEM models. Individual values of Pcr

and PFEM are listed in Table 3-2 and Table 3-3, respectively. Table 3-5 shows the summary 

statistics including the mean ratio (X) of PFEM / Pcr, standard deviation (S), and coefficient of 

variation (CV) defined as the ratio of S / X. Overall, the critical buckling load predicted by the 

NIST models compared favorably with the test results. The mean difference was less than 1 %. In 

contrast, the Eurocode 3 model tends to underestimate the compressive strength of all the column

specimens. The critical buckling load predicted by the Eurocode 3 models (Group A, ID = 1 

through 5) was 18 % lower than the test results on average. 
Stability of Columns

59

Figure 3-12 Axial displacements (δa) of the column specimens.

0

2

4

6

8

10

12

14

16

18

0 25 50 75 100 125 150

δa, mm

t, min

CASE 2 (ID = 6)

CASE 1 (ID = 1)

Test

0

2

4

6

8

10

12

14

16

18

0 25 50 75 100 125 150

δa, mm

t, min

CASE 2 (ID = 7)

CASE 1 (ID = 2)

Test

0

2

4

6

8

10

12

14

16

18

0 25 50 75 100 125 150

δa, mm

t, min

CASE 2 (ID = 8)

CASE 1 (ID = 3)

Test

0

2

4

6

8

10

12

14

16

18

0 25 50 75 100 125 150

δa, mm

t, min

CASE 2 (ID = 9)

CASE 1 (ID = 4)

Test

0

2

4

6

8

10

12

14

16

18

0 25 50 75 100 125 150

δa, mm

t, min

CASE 2 (ID = 10)

CASE 1 (ID = 5)

Test

Axial Loading

Axial Loading

Axial Loading

Axial Loading

Axial Loading

* *

*

*

*

*

*

*

* *

* *

*

*

(a) W8×35−T300

(b) W8×35−T500

(c) W8×35−T600

(d) W14×53−T500

(e) W14×53−T600

NIST (ID = 6)

EC 3 (ID = 1)

Test

NIST (ID = 7)

EC 3 (ID = 2)

Test

NIST (ID = 9)

EC 3 (ID = 4)

Test

NIST (ID = 8)

EC 3 (ID = 3)

Test

NIST (ID = 10)

EC 3 (ID = 5)

Test
Chapter 3

60

Table 3-4 Summary statistics for the normalized column buckling load

Group A B

Material 

Model

EC 3 NIST

X 0.82 1.00

S 0.03 0.01

CV 0.04 0.01

Figure 3-13 Normalized difference in critical buckling loads of the FEM models.
Stability of Columns

61

3.4 COLUMN DESIGN EQUATIONS 

The 2005 and 2010 editions of the ANSI/AISC-360 Appendix 4 specify to use the Eurocode 3 

temperature-dependent mechanical properties of structural steel for design of steel members at 

elevated temperatures. However, those two editions include different design equations for column

strength for flexural buckling under fire conditions. According to the 2005 edition, the design 

strength for steel column for fire conditions can be computed using the standard design equations

(i.e., in Chapter E of the ANSI/AISC-360) with the temperature-dependent values of elastic 

modulus and yield strength. The same resistance factor, φ = 0.9 at ambient temperature can be 

used. Below are the standard AISC design equations for flexural buckling of columns. Fcr(T) is 

the critical buckling stress, and Fe(T) is elastic buckling stress defined in Eq. (3.3). E(T) is the

high-temperature modulus of elasticity, Fy(T) is the yield strength at elevated temperature, and 

KL/r is the column slenderness ratio.

  0.658 ( )

( )

( )

F T Fy T

F T

F T

cr

e

y



















 for 

F (T) 0.44F (T)

e  y

(3.1)

F T 0.877 F (T) cr e 

For 

F (T) 0.44F (T)

e  y

(3.2)

 

2

2

( )

e

E T F T

KL

r



     



(3.3)

However, Takagi and Deierlein (2007) showed that the standard design equations overestimate the 

flexural buckling strength of steel members at elevated temperatures. Their study indicated that 

the approach in the 2005 edition, namely, simply reducing the values of elastic modulus and yield 

strength in the standard equations, does not reflect the nonlinear inelastic behavior of the 

Eurocode 3 stress-strain relationship. To accurately account for the temperature-dependent 

material degradation of steel, new design equation (Eq. (3.4)) was introduced with the parameters, 

a0 = 0.42 and a1 = 1⁄2. These parameters were developed based on the results from the finite-element 

models with the Eurocode 3 stress-strain relationship. These equations have been adopted in the 

2010 edition while the resistance factor remains unchanged (φ = 0.9).

 

 

1

( )/ ( )

0

( )

a Fy T Fe T

Fcr T  Fy T  a

(3.4)

Takagi and Deierlein’s equation (Eq. (3.4)) was essentially a curve fit to finite element solutions 

using the Eurocode 3 material model. The critical buckling stress does not pass through 1.0 at 
Chapter 3

62

L/r = 0. So, Eq. (3.4) was limited to temperature above 200 ○C. The parameters (a0 and a1) used in 

Eq. (3.4) need to be recalibrated if other stress-strain models are used (Agarwal and Varma 2011).

This section presents the column strength computed using the AISC design equations with the 

NIST mechanical properties (temperature-dependent elastic modulus and yield strength of steel), 

compared with the results from the finite-element models. For the finite-element simulation, the 

same variables are used as in Takagi and Deierlein (2007), which are listed in Table 3-5.

Table 3-5 Variables considered in the finite-element models

Structural Shapes 2W14×90, 3W14×22

Temperatures (○C) 20, 100, 200, 300, 400, 500, 600, 700, 800

Slenderness , L/r 20, 40, 60, 80, 100, 120, 140, 160, 180, 200

Minimum yield strength, Fy (MPa) 250 , 345

Stress-strain model NIST, EC3

The finite-element models were developed using the ABAQUS software. Each model used the 

general-purpose four-node shell elements with reduced integration (S4R). The cross-section of the 

column specimen was divided into 32 elements; the flanges and web were divided into eight S4R 

elements each. Note that Takagi and Deierlein (2007) divided the column length into 32 elements, 

which resulted in inconsistent aspect ratio of the elements for all different length of columns. In 

this study, the aspect ratio (≈1:1) of the elements remained the same for all columns. The analysis 

was conducted using the modified Newton-Raphson approach to compute the critical buckling 

load at various steel temperatures.

For comparison, the finite-element models used in this study also followed the same assumptions 

as those used by Takagi and Deierlein (2007), including (i) uniform temperature distributions in 

columns, (ii) initial column imperfections with the maximum initial displacement of L/1000 at the 

midspan, and (iii) a pin-ended boundary condition. Takagi and Deierlein assumed that the residual 

stress in hot-rolled section decreases at elevated temperatures from a fire and used the temperaturedependent residual stresses, Fr(T) equal to ky(T) ×69 MPa, where ky(T) is the Eurocode-3 retention 

factor for yield strength. Instead, this study assumed that, for the high-temperature member 

strength design, the residual stress, typically observed in hot-rolled shape, is not a function of 

applied steel temperature from a fire and the maximum value of 0.2Fy (= 10 ksi (69 MPa) for Gr 50 

steel), which is similar to the assumed value (0.3Fy) in the AISC standard equation at ambient 

temperature. 

 

2 Cross-sectional Area, A = 16,832 mm2

; Plastic Modulus, Zx = 2,525,632 mm3

3 Cross-sectional Area, A = 4,094 mm2

; Plastic Modulus, Zx = 526,850 mm3
Stability of Columns

63

Fig. 3-15 compares the nominal compressive strength normalized to the elevated temperatureyield strength, which include the results from finite-element analysis (W14×90-Gr50, W14×22-

Gr50), the AISC 2005 equations (Eq. (3.1), Eq. (3.2)) with NIST proposed Fy(T) and E(T), the 

AISC 2010 equation (Eq. (3.4)) which uses the Eurocode 3 values for Fy(T) and E(T). Each plot 

also includes the re-calibrated column curve (Eq. (3.4)) with new parameters a0 and a1, using the 

NIST stress-strain model. For calibration, the results from the finite-element analyses and test data 

in Table 3-1 were used. The values of a0 and a1 were re-estimated through non-linear, least-squares 

regression of the predicted critical buckling stress, Fcr(T), on temperature, using the equations for 

yield strength and modulus of elasticity in Chapter 2. The regression analysis shows that a0 = 0.61 

± 0.0034 and a1 = 0.86 ± 0.011 with residual standard error = 8.88 on 358 degrees of freedom.

The comparison shows that the standard AISC equations with the NIST proposed Fy(T) and E(T) 

compare very well with the results from the finite-element analysis. The difference between the 

finite-element solutions and the AISC 2010 equations is greatest for columns with slenderness 

ratios between 50 and 100 at high temperatures.
Chapter 3

64

Figure 3-14 Comparison of column curves.

3.5 SUMMARY, CONCLUSIONS, AND LIMITATIONS

This chapter compares the high-temperature flexural buckling of steel columns predicted using 

two different stress-strain models: Eurocode 3 model and the model recently proposed by the 

National Institute of Standards and Technology. 
Stability of Columns

65

First, the simplified FEM models were used to compute the critical buckling temperature of 47 

column specimens. The results indicate that the NIST stress-strain model predicts the critical 

temperature for column buckling failure more accurately than the Eurocode 3 model for four of 

five data sets considered in Table 3-1. 

Then, the FEM models were developed to analyze the behavior and strength of the column

specimens tested at Purdue University. The results show that (1) the NIST stress-strain model 

accurately predict the response, buckling strength, and failure time of columns at elevated 

temperatures. The Eurocode 3 model tends to conservatively estimate the stiffness and buckling 

strength of tested specimens. (2) Both NIST-proposed and Eurocode-3 thermal expansion models 

accurately predicted the thermal elongations of column specimens within ± 1 % difference. (3) 

Overall, the NIST models accurately predicted the critical buckling load compared with the test 

results with the maximum difference of 2 %. The Eurocode 3 models systematically underestimate 

the compressive strength of all the column specimens by 18 % on average.

Finally, the member strength computed using the AISC design equations with the NIST 

mechanical properties (temperature-dependent elastic modulus and yield strength of steel), 

compared with the results from the finite-element models. Also the parameters (a0 = 0.61 and a1

= 0.86) for Eq. (3.4) were re-calibrated using the NIST stress-strain model. The comparison shows 

that the standard AISC equations with the NIST proposed Fy(T) and E(T) compare very well with 

the results from the finite-element analysis. The difference between and the AISC 2010 equation, 

which was developed using the Eurocode 3 stress-strain model, is greatest for columns with the 

slenderness ratio from 50 to 100 at elevated temperatures.

The findings from the study are limited to the range of parameters included in the experimental 

investigations and finite-element models. This paper is not intended to draw any conclusions about 

columns with failure modes other than inelastic flexural buckling or cases where the thermal creep 

behavior is critical. Further work is recommended for evaluating predictions using the NISTproposed steel material model for other failure mechanisms (yielding or lateral-torsional buckling), 

various fire loading conditions (e.g., transient heating condition or localized fire), various 

boundary conditions (e.g., the effects of thermally induced axial and rotational restraints), and 

cases where the thermal creep behavior is critical. 
66

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67

Chapter 4

MODELING OF FAILURE FOR FINITE ELEMENT ANALYSIS OF 

STRUCTURAL STEELS AT ELEVATED TEMPERATURES

A key issue in evaluating the response of structural systems to fire effects is the representation of 

material behavior at elevated temperatures. In addition to stress-strain behavior developed in 

Chapter 2, modeling of fracture is required to capture failure modes such as tear-out in connection 

plates and bolt shear. Fracture can be simulated in explicit finite element analysis using element 

erosion, in which elements are removed from the analysis when specified failure criteria are 

satisfied. However, the basis for determining and implementing material failure criteria at elevated 

temperatures is not well-established in the literature. A finite element material modeling 

methodology is presented for structural steels and structural bolts at elevated temperatures that 

incorporates erosion-based modeling of fracture. Temperature-dependent stress-strain 

relationships (presented in Chapter 2) for structural steel and structural bolts were combined with 

a plastic strain-based failure criterion for element erosion to enable modeling of fracture in analysis 

of structural connections and assemblies. The failure criterion was calibrated against hightemperature experimental data on elongation of tensile coupons at fracture, and its dependence on 

temperature and mesh size was investigated.

4.1 INTRODUCTION

Fire effects on steel structures can produce failures of connections, including fracture of 

connection plates, shear rupture of bolts, and bolt tear-out failure of beam webs or connection 

plates. Fig. 4-1 illustrates such failure modes for a typical shear tab connection at elevated 

temperatures, based on explicit finite element analyses described by Seif et al. (2013). Whether 

such failures occur depends not only on the loads that can be sustained by the various components 

of a connection, but also on the deformations that can be accommodated prior to fracture, since 

fire-induced forces result from the restraint of thermal expansion or contraction. The ductility of 

steel components thus plays an important role in the performance of connections at elevated 

temperatures. In addition, ductility can potentially allow redistribution of loads after failure of one 

or more connection components. 

While implicit finite element methods are prone to convergence problems when local failures 

occur, explicit finite element methods are well suited for simulating successive failures and the 

subsequent redistribution of loads. Fracture for both tensile and shear failure modes can be 

simulated in explicit analyses (in an approximate sense) using element erosion, in which elements 

are automatically removed from the analysis when specified failure criteria are met. The erosion 

process may continue until a component has lost elements across its entire section, representing 

complete fracture, as illustrated in Fig. 4-1(c). The basis for determining and implementing 
Chapter 4

68

material failure criteria at elevated temperatures, however, is not well-established in the literature. 

The objective of this chapter is to demonstrate a practical material modeling approach for structural 

steel and structural bolts at elevated temperatures that incorporates erosion-based modeling of 

fracture and that can be implemented in FE analysis using currently available tensile coupon data 

at elevated temperatures. Such an approach is needed in the context of performance-based design, 

to enable evaluation of the performance structural components, assemblies, and systems under fire 

effects.

Figure 4-1 Detailed model of a shear tab connection, (b) tear-out failure in beam web, and (c) shear fracture 

of a bolt.

Failure modes such as tear-out and bolt shear rupture are ductile fractures with significant plastic 

deformation before fracture, particularly at elevated temperatures. Key factors influencing the 

initiation of ductile fracture in steel are the equivalent plastic strain and the stress triaxiality, 

defined as the ratio of the mean or hydrostatic stress to the effective or von Mises stress. 

Micromechanics-based models for predicting ductile fracture generally require calibration against 

experimental fracture data at different levels of triaxiality (e.g., Kanvinde and Deierlein (2006)). 

However, experimental data on fracture of steel at elevated temperatures are currently insufficient 

to enable calibration of such micromechanics-based models. Accordingly, researchers tend to 

model strain hardening and softening, but do not address fracture at elevated temperatures (e.g. 

Garlock and Selamet (2010), Sarraj et al. (2007), and Pakala et al. (2012)). The material model for 

structural steel at elevated temperatures in the Eurocode (2001) also does not address material 

fracture. Instead, the yield strength is reduced to zero with a linear material softening between 

15 % and 20 % strain.

The proposed material modeling approach for structural steel uses the recently developed 

empirical power-law form of stress-strain relationship presented in Chapter 2, which was fit to a 

(a) (b)

(a) (b)

(a) (b) (c)
FE Failure Modeling

69

large set of experimental data at elevated temperatures. The bi-linear stress-strain relationship 

presented in Chapter 2 is proposed to model the temperature-dependent behavior of structural 

bolts, for which experimental data at elevated temperatures are more limited. Rather than using a 

micromechanics-based model, a relatively simple plastic strain-based failure criterion is proposed 

for modeling fracture of both structural steel and structural bolts in structural system analyses. 

Similar approaches have been successfully implemented in detailed finite element analyses of 

moment connections (Sadek et al. 2013) and simple shear connections (Main and Sadek 2013) 

under column removal scenarios at ambient temperature. In this study, the erosion strain (the local 

plastic strain at which element erosion is activated) was calibrated against available hightemperature experimental data on elongation of tensile coupons at fracture, and it was found that 

a temperature-dependent value of the erosion strain was required to capture the experimental data. 

The dependence of the failure criterion on temperature and mesh size was also investigated, and 

for structural bolts, analyses of double-shear tests were performed to assess the performance of the 

material modeling approach under shear loading. 

4.2 MATERIAL MODELING AND POST-ULTIMATE STRENGTH

4.2.1 Structural Steel

The temperature-dependent material model described in Chapter 2 is an empirical model for any 

structural steel with nominal yield strength less than 65 ksi (450 MPa). The equations required for 

implementation of the material model in this chapter are extracted from Chapter 2, and repeated 

here for clarity. The temperature-dependent relationship between true stress and true strain

incorporates temperature-dependent expressions for the modulus of elasticity and the yield 

strength. The temperature-dependent modulus of elasticity E is:















































 

 













 

 

1 2

3 4

0

2

1

2

1

( ) exp

e e

e

T

e

T

E T E

(4.1)

where E0 = 29, 900 ksi (206 GPa) is the value at ambient temperature, ΔT (in °C) is the increase 

in temperature above the ambient temperature, and e1 through e4 are coefficients that depend on 

the type of steel. For rolled structural steel, e1 = 3.768, e2 = 1.000, e3 = 639 °C, and e4=1650 °C. 

Fig. 4-2 shows the degradation of the normalized elastic modulus (i.e., E(T)/E0) with increasing 

temperature calculated using Eq. (4.1) with the listed coefficients. The temperature-dependence of 

the yield strength Fy is given by:
Chapter 4

70















































 

 













 

    

1 2

3 4

0 5 5

2

1

2

1

( ) (1 ) exp

r r

y y

r

T

r

T

F T F r r

(4.2)

where Fy0 is the yield strength at ambient temperature and r1 through r5 are coefficients that depend 

on the type of steel. For rolled structural steel, r1 = 7.514, r2 = 1.000, r3 = 588 °C, r4 = 676 °C, and 

r5=0.090. The degradation of the normalized yield strength (i.e., Fy(T)/Fy0) with increasing 

temperature is also shown in Fig. 4-2, calculated using Eq. (4.2) with the listed coefficients. 

Figure 4-2 Degradation of normalized yield strength and elastic modulus with temperature for structural 

steel.

Using the temperature-dependent modulus of elasticity and yield strength, the true stress  is 

expressed as a function of true strain ɛ as:

   

1

3 4 0

2

( ) , ( )

( ) exp ( ) , ( )

y

k

n

y y y y

E T T

T

F T k k F T T

k

  



   

 





                            

(4.3)
FE Failure Modeling

71

where T is temperature (in °C), y(T) = Fy(T)/E(T) is the temperature-dependent yield strain, n = 

0.503 is the strain hardening exponent, and the coefficients k1 through k4 have the following values: 

k1 = 7.820, k2 = 540 °C, k3 = 145.9 ksi (1006 MPa), and k4 = 0.759 (see Section 2.2).

The true stress-strain relationship in Eq. (4.3) was calibrated based on the available tensile coupon 

data up to the ultimate tensile strength, which corresponds to the uniform strain eu given by 

Eq. (2.15). In finite element modeling of tensile coupons, direct application of Eq. (4.3) for strains 

exceeding eu was found to result in delayed onset of necking in comparison with the experimental 

data. This was illustrated in Fig. 2-7, in which Fig. 2-7(a) shows the true stress-strain curve at

400 °C from Eq. (4.3) (for Fy0 = 50 ksi (345 MPa)) and Fig. 2-7(b) shows the corresponding 

engineering stress-strain curve generated from finite-element analysis of a tensile coupon, using 

the approach discussed subsequently. The onset of necking corresponds to the Considère criterion 

for instability in tension (e.g., Dieter (1976)), in which the true stress is equal to the slope of the 

true stress-strain curve,  = d/d. In Fig. 2-7(a), this point corresponds to the intersection of the 

solid curve, representing () from Eq. (4.3), with the dashed curve, representing d/d. This 

intersection occurs at a true strain of  = 0.293, which corresponds to an engineering strain of e = 

0.340 according to Eq. (2.11). From Eq. (2.15) and Eq. (2.16), based on experimental data, the 

expected uniform strain at 400 °C for Fy0 = 50 ksi (345 MPa) is eu = 0.150. Direct application of 

Eq. (4.3) for strains exceeding eu thus results in onset of necking at a uniform strain approximately 

twice as large as the expected experimental value.

Figure 4-3 (a) True stress-strain and (b) engineering stress-strain curves comparing the onset of necking 

resulting from Eq. (4.3) with that imposed based on experimental data for eu (400 °C; Fy0 = 50 ksi 

(345 MPa)).

To overcome this discrepancy, the approach used in this chapter was to specify the true stressstrain relationship using Eq. (4.3) only up to the uniform strain defined by Eq. (2.15). At this point, 

necking was imposed by reducing the slope of the true stress-strain curve to equal the true stress, 

thus satisfying the Considère criterion, and projecting the true stress-strain curve linearly 

thereafter, as follows:

(a) (b)

() [Eq. (4.3)]

based on [Eq. (4.3)]

onset of necking,

based on [Eq. (4.3)]

onset of necking,
Chapter 4

72 ( )

u

u

d

d  



 

 



(4.4)

where ɛu = ln(eu+1) according to Eq. (2.11). Fig. 4-4 shows true stress-strain curves for ASTM 

A572 Grade 50 steel (ASTM International (2013) at various temperatures, generated using this 

approach. The point corresponding to the onset of necking for each temperature is indicated by the 

filled red circles. Fig. 4-4 shows that at 600 °C, necking initiates at a relatively small strain of ɛu = 

6.7 %, so that the post-ultimate necking phase constitutes a greater portion of the material 

response. 

In the proposed finite element modeling approach, the failure criterion for element erosion is based 

on the effective plastic strain, a scalar measure of plastic strain that incorporates its various tensor 

components. Element erosion is activated when the effective plastic strain in any element exceeds 

a specified critical value, called the erosion strain εer. The erosion strain depends on the 

temperature and on the element discretization in the region of fracture, as is discussed 

subsequently. As described in the following section, finite element analyses of tensile coupons 

were performed to calibrate the erosion strain values against available experimental data on 

elongation of tensile coupons at fracture.

Figure 4-4 True stress-strain curves for A572 steel, generated using Eq. (4-3) until the onset of necking (red 

circular marker) with linearly projected stiffness thereafter (1 ksi = 6.895 MPa).
FE Failure Modeling

73

4.2.2 Structural Bolts 

The bi-linear temperature-dependent true stress-strain representation presented in Chapter 2, 

Eq. (2.18) is extended to a tri-linear representation in this for implementation in high-fidelity finite 

element analyses, as follows:

     

   

( ) , ( )

( )

, ( ) ( )

( ) ( )

0.0008 ( ) ( ) , ( )

y

y

y u y y u

u y

u u u

E T T

T

F T F T F T T T

T T

F T E T T T

  

 

   

 

   

 



 

           

    

(4.5)

where ɛy(T) = Fy(T)/E(T) is the temperature-dependent yield strain. The elastic modulus E is the 

same for bolts as for structural steel and is calculated from Eq. (4.1).

For A325 and A490 high-strength bolts, the temperature-dependence of the yield strength Fy is 

calculated from Eq. (4.2), with r1 = 4.967, r2 = 1.000, r3 = 456 °C, r4 = 2040 °C, and r5=0.000. 

The ultimate tensile strength Fu is calculated using Eq. (4.2) using the same values of r1 through 

r5 as for the yield strength, but with the ambient-temperature yield strength Fy0 replaced by the 

ambient-temperature ultimate tensile strength Fu0. The temperature-dependent uniform strain, 

ɛu(T), is assumed to have a value of 0.1 at 20 °C and to decrease linearly with temperature to a 

value of 0.05 at 600 °C. 

Beyond the uniform strain, the slope of the true stress-strain relationship in Eq. (4.5) was taken as 

0.08 % of the elastic modulus E(T), which was found to produce necking and softening behavior 

that were reasonably consistent with experimental data by Kodur et al. (2012). Fig. 4-5 shows the 

stress-strain relationship of the A490 bolts at temperatures of 20 °C, 400 °C, 500 °C, and 600 °C. 
Chapter 4

74

Figure 4-5 True stress-strain curves for A490 bolts (1 ksi = 6.895 MPa).

4.2.3 Assumptions and Limitations 

For strains exceeding the temperature-dependent uniform strain (Eq. (2.16)), linear projection of 

the true stress-strain curves, based on the Considère criterion, successfully initiated necking for all 

mesh sizes considered. However, it is well known that softening behavior in continuum finite 

element analyses introduces first-order mesh dependence, and thus the values of erosion strain, 

which were calibrated against the available tensile test data (see Section 4.4), are mesh-sizedependent. In addition, the erosion strains were calibrated to material within the gauge length of 

tensile coupons (at elevated temperatures) at their instantaneous triaxiality at fracture. For cases 

in which the material triaxiality at fracture may differ significantly from that of a tension test, the 

erosion strain may need to be modified accordingly.

Despite these limitations, this chapter provides a rational approach to incorporating modelling of 

fracture in structural steels at elevated temperatures with currently available FE software and 

tensile coupon data. There does not currently exist sufficient data to support a more sophisticated 

approach that includes the effects of triaxiality at elevated temperatures.
FE Failure Modeling

75

4.3 ANALYSIS OF STRUCTURAL STEEL TENSILE COUPONS

Temperature-dependent values of erosion strain for structural steel members were calibrated 

against fracture strains from elevated-temperature coupon tests reported by Luecke et al. (2005) 

and Hu et al. (2009). Using finite-element models of tensile coupons, the erosion strains were 

calibrated so that the computed engineering strain at fracture for each temperature matched the 

average experimental value. Since the simulation of post-ultimate necking and fracture can depend 

on the mesh discretization, the mesh size used in the calibration procedure needs to be consistent 

with the mesh size to be used in modeling fracture of a structural component. The influence of 

mesh size on the computed results is further discussed in Section 4.4.3.

4.3.1 Tensile Test Data

Luecke et al. (2005) tested a set of A572 Grade 50 steel tensile coupons at elevated temperatures, 

and four of these coupons are shown in Fig. 4-6(a) to illustrate the effect of elevated temperature 

on the shape and size of the necked region. The thickness of the coupons was 0.125 in (3.2 mm), 

and the reduced section (where necking and fracture occurred) had a length of Ao = 1.25 in 

(31.8 mm) and a width of 0.24 in (6.1 mm). The extensometer used in the tests had a gauge length 

of 0.5 in (12.7 mm). The coupons were first heated to the desired temperature, and then subjected 

to displacement-controlled tensile loading until fracture. Fig. 4-6(b) shows engineering stressstrain curves corresponding to the tensile coupons shown in Fig. 4-6(a). The initial portions of the 

curves, up to the ultimate tensile strength (plotted using solid lines), correspond to the strain 

measured by the extensometer. The post-ultimate portions of the curves (plotted with dashed lines) 

do not represent measured data but are simply straight-line segments connecting the point of 

necking initiation (at the ultimate tensile strength) to the point of fracture, as explained in the next 

paragraph. 
Chapter 4

76

Figure 4-6 Experimental coupons; (a) picture of the four specimens after the tensile test was conducted at 

20 °C, 400 °C, 500 °C, and 600 °C, and (b) corresponding stress-strain curves (1 ksi = 6.895 MPa). *

* Estimated coefficient of variation in measured data is 4 %.

For consistency with the experiments of Hu et al. (2009), discussed subsequently, the engineering 

strain values at fracture in Fig. 4-6(b) correspond to a gauge length of Go = 1 in (25.4 mm). This 

gauge length included 80 % of the reduced section and was sufficient to capture the necked region 

for all specimens. Fracture strain measurements for the 1 in (25.4 mm) gauge length were obtained 

by using the measured displacement of the actuator in conjunction with the extensometer 

measurements. The uniform strain at the onset of necking was first obtained from the extensometer 

measurements. The strain increment from the onset of necking to the time of fracture was then 

computed by dividing the measured actuator displacement over this time interval by the 1 in 

(25.4 mm) gauge length. Finally, the computed strain increment was added to the uniform strain 

at the onset of necking to obtain the total engineering strain at fracture. While elastic rebound of 

the tensile coupon outside of the necking region was not included in this approach, the associated 

strains are negligibly small relative to the measured strains at fracture. The resulting values of 

engineering strain at fracture are plotted at the failure point in Fig. 4-6(b) and are also plotted as a 

function of temperature in Fig. 4-7, along with data from Hu et al. (2009). 

20oC

500oC

400oC

600oC

(a) (b)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

0

10

20

30

40

50

60

70

80

Engineering Strain, 

eng Engineering Stress, eng (ksi)

20oC

400oC

500oC

600oC

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

0

10

20

30

40

50

60

70

80

Engineering Strain, 

eng

Engineering Stress, 



e

n

g (ksi)

20oC

400oC

500oC

600oC

S

e
FE Failure Modeling

77

Figure 4-7 Experimental values of engineering strain at fracture for structural steel (Gauge length: 1 in = 

25.4 mm) *.

* Estimated coefficient of variation in measured data is 4 %.

Hu et al. (2009) presented coupon test data for ASTM A992 steel (ASTM 2012c) at elevated 

temperatures, reporting values of engineering strain at fracture for two specimens at each 

temperature with a gauge length of 1 in (25.4 mm). The data from both Luecke et al. (2005) and 

Hu et al. (2009) in Fig. 4-7 show a similar trend, with the fracture strain initially decreasing with 

temperature, reaching a minimum value around 400 °C or 500 °C, and then increasing with 

temperature above 500 °C. The fracture strain values for ASTM A992 steel from Hu et al. (2009)

are generally somewhat higher than the values for ASTM A572 Grade 50 steel from Luecke et al. 

(2005). Average values of the experimental data at temperatures of 20 oC, 400 oC, 500 oC, 

and 600 oC are plotted in Fig. 4-7, connected by a dashed line. These average values were selected 

as representative failure strains for structural steel and were used as target values in calibration of 

temperature-dependent erosion strains, as described subsequently.

4.3.2 Finite Element Modeling Approach

The tensile coupons were modeled using three-dimensional solid elements, as shown in Fig. 4-8. 

Fully integrated eight-node elements were used with a typical element size of 0.06 in (1.5 mm). 

Thus, the reduced section was meshed with four elements across the width and two elements 

through the thickness. Analyses were performed at uniform temperatures of 20 °C, 400 °C, 500 °C, 

and 600 °C. The analyses were performed using explicit time integration in LS-DYNA (Livermore
Chapter 4

78

Software Technology Corporation 2012), and a smooth functional form of tensile displacement 

vs. time was used to ensure that dynamic effects were negligible (i.e., to ensure quasi-static loading 

conditions). The coupons were subjected to tensile loading in the analysis until fracture across the 

reduced section occurred through element erosion. Element erosion in the necking zone initiated 

when the effective plastic strain in any element reached the specified erosion strain ɛer, at which 

that element was removed from the analysis, leading rapidly to erosion of successive elements 

across the reduced section. 

Figure 4-8 Finite element mesh of a tensile coupon (1 in = 25.4 mm).

Fig. 4-9 shows analysis results from uniaxial extension of a tensile coupon at 500 °C. Engineering 

strain values in the stress-strain curve of Fig. 4-9 were calculated based on the relative 

displacement of two nodes at each end of the gauge section, with an initial length of Go = 1 in 

(25.4 mm). Contours of effective plastic strain are also shown in Fig. 4-9 at different points labeled 

along the computed stress-strain curve. Point a is around the onset of necking (at the ultimate 

tensile strength) and point b is shortly after. The corresponding plastic strain contours show fairly 

uniform plastic strains along the gauge length that are comparable to the corresponding 

engineering strains. For example, point b corresponds to an engineering strain of 0.15, and 

comparable plastic strains of about 0.14 are observed. However, points c and d are after the onset 

of necking, and the contours of plastic strain clearly indicate localization of strain near the center 

of the gauge length, with plastic strains significantly exceeding the corresponding engineering 

strains. For example, point d (just prior to fracture) corresponds to an engineering strain of 0.35, 

but larger plastic strains of about 0.70 in the necked region. Because of this localization of plastic 

strain that occurs during necking, erosion strains are generally larger than the engineering strains 

at which fracture occurs. An erosion strain of er = 0.80 was used for the analysis in Fig. 4-9 to 

achieve fracture at an engineering strain of 0.35. The following section discusses determination of 

temperature-dependent erosion strains for structural steel.

Go = 1 in

Ao= 1.25 in
FE Failure Modeling

79

Figure 4-9 FE analysis results for a tensile coupon model at 500 oC (1 ksi = 6.895 MPa).

4.3.3 Calibration of Temperature-Dependent Erosion Strain

Fig. 4-10 shows the influence of the erosion strain ɛer (the local plastic strain at fracture) on the 

corresponding engineering strain at fracture obtained from the finite element model of a tensile 

coupon at 400 °C. As the value of ɛer increases, the engineering strain at fracture also increases. 

For instance, when ɛer, increases from 0.55 to 0.80, the engineering strain at fracture, ef, increases 

by about 19 % from 0.32 to 0.38. Larger values of the erosion strain produce a longer descending 

branch of the engineering stress-strain curve, which is associated with more extensive necking. To 

determine the appropriate value of erosion strain at each temperature, the erosion strain was 

adjusted until the resulting engineering strain at fracture matched the target value, which is the 

average of the experimental data shown in Fig. 4-7. 

Fig. 4-11 shows the values of erosion strain ɛer used in the FE analyses to obtain the target values 

of engineering strain at fracture shown in Fig. 4-7. For each case, the erosion strain is significantly 

greater than the engineering strain at fracture (e.g., 1.63 vs. 0.46, respectively, at 600 °C), 

indicating extensive necking and associated localization of plastic strains in the post-ultimate 

phase prior to fracture. When high values of erosion strain are required in a material model, care 

must be taken to extend the stress-strain curves to the specified erosion strain. Defining stressstrain curves only up to the engineering strain at fracture would require the software to extrapolate 

stress-strain curves, possibly yielding unreliable results. 

Contours of plastic strain at labeled points:

a)

b)

c)

d)

a b

c

d

Engineering Strain, e Engineering Stress, S (ksi)
Chapter 4

80

Figure 4-10 Influence of the erosion strain ɛer on the computed engineering strain at fracture at 400 °C 

(1 ksi = 6.895 MPa).

Figure 4-11 Calibrated values of erosion strain versus temperature for structural steel.
FE Failure Modeling

81

Fig. 4-12 shows engineering stress-strain curves obtained from FE analysis of tensile coupons with 

the erosion strain values shown in Fig. 4-11 at temperatures of 20 °C, 400 °C, 500 °C, and 600 °C. 

As a consequence of the calibration procedure described above, the engineering strain values at 

fracture in Fig. 4-12 match the target values in Fig. 4-7. Note that the stress-strain curves in 

Fig. 4-12 should not be expected to closely match the experimental stress-strain curves in 

Fig. 4-6(b) for the following reasons:

 The stress-strain curves in Fig. 4-12 are based on the power-law model in Eq. (4.3), which 

was fit to a large set of experimental data, whereas the stress-strain curves in Fig. 4-6(b) 

were obtained from individual coupon tests.

 The fracture strains in Fig. 4-12 were calibrated to match the average experimental values 

in Fig. 4-7, whereas the fracture strains in Fig. 4-6(b) were obtained from individual 

coupon tests.

Figure 4-12 FEA engineering stress-strain curves for selected temperatures with calibrated erosion strain 

values for structural steel (1 ksi = 6.895 MPa).

In spite of these differences, some general similarities can be observed between the computed and 

experimental curves, including degradation in the ultimate tensile strength with increasing 
Chapter 4

82

temperature, reduced fracture strains at temperatures of 400 °C and 500 °C, and the early onset of 

necking and extensive post-ultimate portion of the stress-strain curve at 600 °C. These similarities 

confirm that the proposed modeling approach captures key aspects of the material behavior at 

elevated temperatures.

The appropriate values of erosion strain depend on the mesh discretization. The erosion strains in 

Fig. 4-11 were calibrated for coupon models with a typical element size of 0.06 in (1.5 mm) (see 

Fig. 4-8). Using a different level of mesh refinement would require recalibration of the erosion 

strains to achieve the same engineering strains at failure. Table 4-1 shows calibrated erosion strains 

for the original mesh and for a refined mesh with a typical element size of 0.03 in (0.75 mm), along 

with the target engineering strains at fracture used in the calibration. These results show that 

reducing the element size by one-half requires using erosion strains that are 41 % to 46 % higher 

for temperatures up to 500 °C and 33 % higher for 600 °C. Fig. 4-13 shows engineering stressstrain curves computed from finite-element models of a tensile coupon for both the original mesh 

and the refined mesh. These results demonstrate that the proposed calibration procedure allows 

consistent values of engineering strain at failure to be achieved using multiple levels of mesh 

refinement. Fig. 4-13 also includes markers to indicate the premature fracture that would occur if 

erosion strain values were not recalibrated for the refined mesh. Only the post-ultimate behavior 

depends on the mesh discretization, because of the softening and strain localization that occur 

during necking. Larger erosion strains are required for the refined mesh because of the greater 

localization that occurs, which also results in somewhat reduced values of engineering stress at 

fracture, as evident in Fig. 4-13. The results in Fig. 4-13 and Table 4-1 demonstrate the importance 

of using the same element size in the calibration procedure as will be used in modeling the parts 

of the structural components in which fracture is expected. A fracture-energy based approach could 

potentially be used to account for this dependence on element size, and such an approach is being 

explored in ongoing research. 

Table 4-1 Engineering and erosion strain values at fracture for structural steel.

Temperature 

(°C)

Engineering 

Strain at Fracture

Erosion Strain at Fracture

element size: 0.06 in 

*

(solid elements)

element size: 0.03 in *

(solid elements)

element size: 0.06 in *

(shell elements)

20 0.470 1.050 1.53 0.400

400 0.380 0.800 1.14 0.395

500 0.350 0.800 1.13 0.375

600 0.460 1.630 2.16 0.565

*

1 in = 25.4 mm
FE Failure Modeling

83

Figure 4-13 Engineering stress-strain curves computed from finite-element models of a tensile coupon with 

different levels of mesh refinement (1 ksi = 6.895 MPa).

The appropriate values of erosion strain also depend on the element type and formulation. Using 

2D shell elements instead of the 3D solid elements will yield lower erosion strain values. 2D shell 

elements were used to remodel the coupons, and analyses were performed at the same 20 °C, 

400 °C, 500 °C temperatures. The fifth column of Table 4-1 shows the erosion strains ɛer used in 

the FE analyses to obtain the target engineering strains at fracture shown in Fig. 4-7 and in the 

second column of Table 4-1. 

4.4 ANALYSIS OF STRUCTURAL BOLTS

Erosion strains used to model the failure of structural bolts need to be calibrated against 

experimental data at elevated temperatures. Since the shear behavior of bolts is often of primary 

concern in connection performance, we considered the possibility of calibrating erosion strain 

values using bolt shear test data at elevated temperatures from Yu (2006). However, the measured 

data included deformations of the test plates that imposed shear forces on the bolts, so that the bolt 

shear deformations could not be isolated from the published results. Therefore, similar to the 

procedure used for structural steel, erosion strains for structural bolts were calibrated against 

tensile test data at elevated temperatures, using bolt coupon data from Kodur at al. (2012),

presented in Section 4.4.1. After using finite element models of bolt coupons (Section 4.4.2) to 

0

20

40

60

80

100

0 0.1 0.2 0.3 0.4 0.5

Engineering Stress, s (ksi)

Engineering Strain, e

20 C

400 C

Original mesh (element size = 0.06 in)

Refined mesh (element size = 0.03 in)

500 C

600 C

Fracture 

point without 

recalibration
Chapter 4

84

calibrate the erosion strains against the tensile test data (Section 4.4.3), the behavior of the 

proposed modeling approach under shear loading is considered in Section 4.4.4.

4.4.1 Tensile Test Data

Fig. 4-14 presents values of engineering strain at fracture for both A325 and A490 bolts reported 

by Kodur et al. (2012) based on tensile tests of bolt coupons at temperatures of 20 oC, 400 oC, 

500 oC, and 600 oC. The reported engineering strains at fracture correspond to a gauge length of 

2.0 in (50.8 mm). The fracture strains in Fig. 4-14 were used as target values in calibration of the 

erosion strains, as described subsequently.

Figure 4-14 Experimental values of engineering strain at fracture from Kodur et al. (2012) for A325 and 

A490 bolts.

4.4.2 Finite Element Modeling Approach

Tensile coupons machined from A325 and A490 bolts were modeled using three-dimensional solid 

elements, as illustrated in Fig. 4-15. Fully integrated eight-node elements were used with a typical 

element size of 0.06 in (1.5 mm). Analyses were performed at uniform temperatures of 20 °C, 

400 °C, 500 °C, and 600 °C. The bolts were subjected to tensile loading in the analysis until 

necking and fracture occurred in the reduced section of the bolt coupon. Fracture was modeled 

using element erosion, which initiated when the effective plastic strain in elements in the necking 

zone reached the specified erosion strain, ɛer. 
FE Failure Modeling

85

Fig. 4-16 shows analysis results from uniaxial extension of an A490 bolt coupon at 500 °C. 

Engineering strain values were calculated based on the relative displacement of nodes at each end 

of the gauge section, with an initial length of Go = 2.0 in (50.8 mm), corresponding to the tests by 

Kodur et al. (2012). Contours of effective plastic strain are also shown in Fig. 4-16 for the points 

labeled on the computed stress-strain curve. Point a is at the onset of necking, and the 

corresponding plastic strain contours show uniform plastic strains of about 0.06 along the gauge 

length that are comparable to the corresponding engineering strain value of 0.06. However, points 

b and c are after the onset of necking, and the contours of plastic strain clearly indicate localization 

of strain where necking occurs near the center of the reduced section. The plastic strains in the 

necking zone significantly exceed the corresponding engineering strains. For example, point c (just 

prior to fracture) corresponds to an engineering strain of about 0.16, but much larger plastic strains 

of about 0.39 are observed in the necking region. The following section discusses determination 

of temperature-dependent erosion strains for bolts.

Figure 4-15 Finite element mesh of a bolt tensile coupon.
Chapter 4

86

Figure 4-16 FE analysis results for an A490 bolt tensile coupon at 500 oC (1 ksi = 6.895 MPa).

4.4.3 Calibration of Temperature-Dependent Erosion Strain

To determine the appropriate erosion strain at each temperature, the erosion strain was adjusted 

until the resulting engineering strain at fracture matched the target value shown in Fig. 4-14, 

obtained from the experimental data of Kodur at al. (2012). These target engineering strains at 

fracture are also listed in Table 4-2 (columns 2 and 3), along with the calibrated erosion strains

used in the FE models (columns 3 and 4). The calibrated erosion strains are plotted against 

temperature in Fig. 4-17. Fig. 4-18 and Fig. 4-19 show engineering stress-strain curves obtained 

from FE analysis of A325 and A490 bolts, respectively, at temperatures of 20 °C, 400 °C, 500 °C, 

and 600 °C. At all temperatures, the A490 bolts are stronger and fracture at lower engineering 

strains than the A325 bolts. For both types of bolts, the engineering strain at fracture is fairly 

constant from ambient temperature up to about 400 °C, but increases above 400 °C. Above 400 °C 

the strength of bolts decreases more rapidly than the strength of rolled structural steel, as shown in 

Fig. 4-13.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

0

20

40

60

80

100

120

Engineering Strain, 

eng Engineering Stress, eng (ksi)

Contours of plastic strain at labeled points:

a

b

c

d

a)

b)

c)

d)

e

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

0

10

20

30

40

50

60

70

80

Engineering Strain, 

eng

Engineering Stress, 



e

n

g (ksi)

20oC

400oC

500oC

600oC

S

e
FE Failure Modeling

87

Table 4-2 Engineering and erosion strain values at fracture for A325 and A490 bolts.

Temperature (°C)

Engineering Strain at Fracture Erosion Strain at Fracture*

A325 A490 A325 A490

20 0.21 0.16 0.50 0.35

400 0.20 0.16 0.55 0.40

500 0.25 0.19 0.75 0.55

600 0.28 0.22 0.75 0.60

*Typical element size of 0.06 in (1.50 mm).

Figure 4-17 Calibrated values of erosion strain versus temperature for A325 and A490 bolts.

0 100 200 300 400 500 600

0

0.25

0.5

0.75

1

Temperature, oC Erosion Strain, 

er

A325

A490
Chapter 4

88

Figure 4-18 Computed stress-strain curves from an FE model of an A325 bolt coupon at elevated 

temperatures (1 ksi = 6.895 MPa).

Figure 4-19 Computed stress-strain curves from and FE model of an A490 bolt coupon at elevated 

temperatures (1 ksi = 6.895 MPa).
FE Failure Modeling

89

4.4.4 Performance of Bolts in Shear

While the material model for structural bolts and the associated erosion strains were developed 

and calibrated using tensile test data, it is important that the modeling approach is also capable of 

capturing the behavior and failure of structural bolts in shear. To verify the adequacy of the 

modeling approach in representing shear behavior, FE analyses of a series of double-shear bolt 

tests were performed. Solid-element models of bolt double-shear specimens were developed, with 

dimensions corresponding to tests conducted by Wallaert and Fisher (1965) at ambient 

temperature. Material properties for ASTM A36 steel (ASTM International 2012a) were used for 

modeling the bearing plates, which are comparable to the properties of the steel plates used in the 

tests. Due to symmetry, only half of each specimen was modeled, with boundary conditions 

reflecting the symmetry as shown in Fig. 4-20(a). The center plate was subjected to tensile loading 

in the analysis, which subjected the bolts to double shear until fracture across the bolt occurred. 

Fracture initiated when the effective plastic strain in any element reached the specified erosion 

strain, ɛer, as discussed in previous sections. Fig. 4-20(b) shows the solid-element mesh of the bolt 

after shear rupture.

Figure 4-20 Detailed model of a double-shear bolt test: (a) section view, (b) bolt after fracture.

The double-shear analysis results for an A325 bolt at ambient temperature are plotted in Fig. 4-21

with experimental results from Wallaert and Fisher (1965). Note that these experimental curves 

are best fit to data from several tests; the estimated coefficient of variation in measurements is 4 %. 

At ambient temperatures, the shear capacity of the A325 bolts was 79 ksi (545 MPa), which is 

within 1 % of the experimental results, and within 5 % of the nominal shear capacity of 75 ksi

(517 MPa) specified by the Research Council on Structural Connections (RCSC 2004). The FE 

results simulated the experimental results reasonably well, particularly for the shear stress at 

fracture. The test specimen sustained somewhat larger deformations than the FE model before 

fracture, likely due to the deformations of the plates during the experiment. The shear capacity 

obtained from the FE model for the A490 bolts at ambient temperature was 96 ksi (660 MPa), 

(a) (b)

CL
Chapter 4

90

within 2 % of the nominal shear capacity of 94 ksi (646 MPa) specified by RCSC (2004). Note 

that the nominal shear capacities reported here were obtained by dividing the nominal shear 

capacity listed in Table 5.1 of the specification (RCSC 2004) by a factor of 0.80 to eliminate the 

reduction in strength that accounts for non-uniform shear force distribution, since the shear force 

is carried by a single bolt. 

Figure 4-21 FE model and experimental results (Wallaert and Fisher 1965) for shear displacement of an 

A325 bolt at ambient temperature (1 ksi = 6.895 MPa).

Using the proposed modeling approach for structural bolts, Fig. 4-22 and Fig. 4-23 show shear 

stress vs. deformation curves for bolt shear failure from the FE analyses at 20 °C, 400 °C, 500 °C, 

and 600 °C, for the A325 and A490 bolts, respectively. Fig. 4-24 shows a comparison of the 

computed shear strength values for A325 bolts from Fig. 4-22 with experimental measurements 

reported by Yu (2006) and by Kodur et al. (2012), with all results normalized by the ambienttemperature shear strength. While there is significant scatter in the experimental data in Fig. 4-24, 

the computed values are seen to be quite consistent with the measured data, confirming that the 

proposed modeling approach can capture the degradation in the shear strength of bolts with 

increasing temperature.

0 0.05 0.1 0.15 0.2 0.25

0

10

20

30

40

50

60

70

80

90

100

Displacement, (in)

Shear Stress, (ksi)

Model

Experimental
FE Failure Modeling

91

Figure 4-22 FE results for shear displacement of an A325 bolt at selected temperatures 

(1 ksi = 6.895 MPa).

Figure 4-23 FE results for shear displacement of an A490 bolt at selected temperatures 

(1 ksi = 6.895 MPa).

0 0.05 0.1 0.15 0.2 0.25

0

10

20

30

40

50

60

70

80

90

100

Displacement, (in)

Shear Stress, (ksi)

20oC

400oC

500oC

600oC

0 0.05 0.1 0.15 0.2 0.25

0

10

20

30

40

50

60

70

80

90

100

Displacement, (in)

Shear Stress, (ksi)

20oC

400oC

500oC

600oC
Chapter 4

92

Figure 4-24 Normalized shear strength versus temperature for A325 bolts.

4.5 SUMMARY, CONCLUSIONS, AND FUTURE WORK

A practical modeling approach has been developed to represent temperature-dependent nonlinear 

behavior and failure of structural steel and structural bolts at elevated temperatures, using element 

erosion to represent material failure. This modeling approach can be used in explicit finite element 

analyses to represent the successive failures of components in bolted connections under fireinduced heating and the subsequent redistribution of loads. For structural steel materials, a 

temperature-dependent power-law model was used to represent the true stress-strain behavior up 

to the onset of necking at the temperature-dependent uniform strain. The uniform strain was 

determined based on a nonlinear least-squares fit to the uniform strain data for elevated 

temperatures available in the literature. The post-ultimate behavior was represented using a linear 

projection of the true stress-strain curve, based on the Considère criterion, to initiate necking at 

the uniform strain. Due to the more limited experimental data on structural bolts at elevated 

temperatures, a simpler tri-linear material model was proposed to represent the true stress-strain 

behavior of structural bolts.

A relatively simple plastic strain-based failure criterion was used to activate element erosion, 
FE Failure Modeling

93

noting that the currently available experimental data on fracture of structural steel and structural 

bolts at elevated temperatures are insufficient to enable calibration of micromechanics-based 

models for ductile fracture. Values of erosion strain at elevated temperatures were calibrated 

against experimental data on the engineering strain at fracture of tensile coupons, using FE models 

of tensile coupons to determine appropriate erosion strains at each temperature. For structural 

steels, average failure strains from tensile coupons of ASTM A572 Grade 50 and ASTM A992 

steel were used as target values in the calibration of erosion strain. For structural bolts, erosion 

strains were calibrated separately for ASTM A325 and A490 bolts, using failure strain data from 

tensile coupons of each type of bolt. 

Because the modeling of post-ultimate softening and strain-localization behavior is known to 

depend on the mesh discretization, FE analyses of tensile coupons were performed with two levels 

of mesh refinement, to investigate the sensitivity of the results to mesh size. The results were found 

to be particularly sensitive to mesh size at high temperatures, because of the earlier onset of 

necking and the more extensive softening behavior prior to fracture. When the element size was 

reduced by one-half in an FE analysis of a tensile coupon at 600 °C, with the erosion strain 

unchanged, the computed engineering strain at fracture was reduced by 44 %. The computed 

results at lower temperatures were less sensitive to the mesh refinement, with their computed 

engineering strains at fracture reduced by approximately 20 %. However, these results confirmed 

the importance of calibrating erosion strains using the same mesh size that will be used in analyzing 

the failure of structural components.

Because the material model and erosion strains for structural bolts were developed and calibrated 

to represent tensile behavior, FE analyses of double-shear test specimens were performed to verify 

that the model was also capable of representing the behavior of structural bolts under shear loading. 

The computed shear behavior for an A325 bolt compared well with experimental data at ambient 

temperature: the ultimate shear capacity was within 1 % of the experimental value, although the 

experimental displacement at fracture was somewhat higher than that predicted by the FE analysis, 

likely due to deformation of the plates during the experiment. Comparing to the nominal shear 

capacities, the shear capacity computed from the FE analyses at ambient temperature was found 

to be within 5 % for A325 bolts and within 2 % for A490 bolts.

The primary conclusions from this research are:

 Analysis of the available experimental data showed that the uniform strain (i.e., the 

engineering strain at the onset of necking) decreased with increasing temperature.

 Imposing a reduced stiffness onto the true stress-true strain curve for strains greater than 

the uniform strain, based on the Considère criterion, was sufficient to initiate necking at 

the uniform strain at each temperature.

 The fracture strain was dependent on the temperature.

Due to the limitations both in the presented modeling approach and in the currently available 

experimental data for material behavior and fracture at elevated temperatures, NIST researchers 
Chapter 4

94

are currently planning future work that includes carefully controlled experiments with different 

levels of triaxiality at different temperatures. These tests will be used to explore ductile fracture 

characteristics at elevated temperatures.
95

Chapter 5

APPLICATION OF MATERIAL AND FAILURE MODELS TO THE ANALYSIS 

OF STEEL MOMENT CONNECTIONS UNDER FIRE LOADING

Performance-based methodologies to evaluate the fire-performance of structures are needed to 

move beyond the prescriptive procedures presently in use. Analytical methods are needed for 

simulating the performance of structural systems, including connections, subject to realistic fire 

effects. Framing connections may be subject to large unanticipated deformations and loads during 

fire events, and connection failure may lead to other failures or local collapse. To achieve this 

purpose, the materials and failure models, developed in Chapter 2 and Chapter 4, respectively, are 

used in this chapter to analyze moment-resisting frames to demonstrate how these models can be 

used for design and evaluation purposes. The performance of steel moment frames under fireinduced heating was investigated using detailed finite-element modeling. Models of two types of 

seismically designed steel moment frames were developed, including (i) an intermediate moment 

frame with welded unreinforced flange, bolted web (WUF-B) connections, and (ii) a special 

moment frame with reduced beam section (RBS) connections. Assemblies consisting of two 

columns and a single beam span were modeled, with highly refined modeling of the connection 

regions. Structural analyses were performed under gravity loads in combination with a thermal 

loading scenario consisting of a heating phase followed by a cooling phase. The temperaturedependent material models for structural steel and bolts (presented in Chapter 2) that incorporate 

erosion-based modeling of fracture (presented in Chapter 4) were implemented in the study

presented in this chapter. The influence of axial restraint on the performance of the moment frames 

was investigated by considering different support conditions for the end columns. 

5.1 INTRODUCTION

Development of design tools for evaluating fire effects requires adequately accounting for the 

relevant behaviors and failure modes of members, connections, and systems. Detailed finiteelement (FE) analyses can play an important role in identifying and understanding the relevant 

behavior and failure modes at elevated temperatures, which can include local buckling and 

fracture. 

During exposure to fire, large axial compressive and/or tensile forces may develop in floor beams 

and their connections. A number of researchers have studied the effects of fire on connections, 

though most of the literature addresses shear connections and semi-rigid connections. Sarraj et al. 

(2007) developed detailed solid element models for shear tab connections with bolts to evaluate 

bolt shear and bearing behavior. Yu et al. (2009) performed an experimental investigation of the 

behavior of shear tab connections subjected to vertical shear and tensile forces at elevated 

temperatures and measured their moment-rotation capacity. Seif et al. (2013) discussed failure 
Chapter 5

96

modes of shear-tab connections at elevated temperatures, while Seif et al. (2014) similarly 

discussed failure modes of moment connections under similar heating conditions. For moment 

connections, Yang et al. (2009) conducted experiments on welded moment connections where the 

connections and members immediately adjacent to the connection were heated up to 650 °C, and 

then loaded to failure under an applied moment. Yielding, necking, fracture, bolt shear 

deformation, and local buckling were observed, and a reduction of member stiffness to 25 % of 

ambient values was reported. Quiel and Garlock (2010) conducted detailed finite element analyses 

of shear and moment connections for 2D and 3D building frames. Their results indicated that 

thermal gradients can produce significant changes in the deflection mechanics and the plastic limitstate behavior associated with the interaction of axial forces and bending moments.

This Chapter presents results from a study employing FE analyses with geometric and material 

nonlinearities, using solid and shell elements to model the failure modes of typical steel moment 

connections in response to elevated temperatures, using the material model developed in Chapter 2 

and the failure model developed in Chapter 4. The purpose of the analyses is to examine the 

applicability of the material and failure models into the analysis of subsystems, and to gain better 

insights into the behavior of moment connections, including failure modes, when subjected to fire 

exposure and various restraining conditions. The results presented illustrate the behavior and 

failure modes of connections and assemblies under varying load, temperature, and boundary 

conditions. 

5.2 PROTOTYPE MOMENT FRAMES

As described in Lew et al. (2013), the National Institute of Standards and Technology (NIST) 

worked with a panel of practicing structural engineers across the U.S. to develop a number of 

prototype steel-frame building designs for use in assessing the robustness of structural systems. 

The buildings were designed according to the American Society of Civil Engineers 7-02 standard 

(ASCE 2002) and its referenced material design standards, including the American Institute of 

Steel Construction (AISC) Load and Resistance Factor Design Specification for Structural Steel 

Buildings (AISC 1999) and the AISC Seismic Provisions for Structural Steel Buildings (AISC 

2002). Two types of moment frames from the prototype buildings were selected for analysis in 

this study. These included an intermediate moment frame (IMF) with welded unreinforced flange, 

bolted web (WUF-B) connections, and a special moment frame (SMF) with reduced beam section 

(RBS) connections. The assemblies considered in this study were portions of these moment frames 

consisting of a beam supported on two columns and joined using two moment connections. 

Fig. 5-1 shows a sketch of a typical assembly.
Moment Connections Behavior

97

Figure 5-1 Sketch of a typical assembly.

(1 ft = 0.305 m, 1 in = 25.4 mm)

The moment-frame assembly with WUF-B connections was taken from the second-floor level of 

a seismically designed IMF in a 10-story prototype building designed for Seismic Design Category 

C. Fig. 5-2 shows details of the WUF-B connection used in the IMF assembly considered in this 

study. The beam flanges are joined to the column flange using complete joint penetration (CJP) 

groove welds. ASTM A992 structural steel, with an ambient-temperature nominal specified yield 

strength of Fy0 = 50 ksi (345 MPa) was used in all beams and columns. ASTM A36 steel, with Fy0

= 36 ksi (248 MPa) was used for the shear tabs and continuity plates at the beam-column 

connections. ASTM A490 bolts, with Fy0 = 130 ksi (896 MPa), were used to connect the shear tab 

to the beam web.

20’-0” (6096 mm)

13’-9” (4919 mm) 13’-9” (4919 mm)
Chapter 5

98

Figure 5-2 Details of WUF-B moment connection. 

(1 in = 25.4 mm)

Similarly, the moment-frame assembly with RBS connections was taken from the second-floor 

level of a seismically designed SMF in a 10-story prototype building designed for Seismic Design 

Category D. The RBS connection is created by cutting away a portion of the top and bottom flanges 

of the beam at a distance from the beam-column interface so that yielding is concentrated in this 

reduced area. Therefore, the reduced section acts as a structural fuse to protect the connection 

against premature fracture. Fig. 5-3 shows details of the RBS connection used in the SMF 

assembly considered in this study. ASTM A992 structural steel was used for the beams and 

columns.

or

Shear tab (A36):

13 mm x 305 mm x 152 mm

Continuity plate (A36): 

19 mm (int. panels)

10 mm (ext. panels)

A490 bolts, D = 25 mm 

(Class A Faying surfaces)

or

Beam W21x73

Column W18x119 8 mm

CJP 

(Typical)

CJP
Moment Connections Behavior

99

Figure 5-3 Details of WUF-B moment connection. 

(1 in = 25.4 mm)

5.3 MODELING AND ANALYSIS OF MOMENT-FRAME ASSEMBLIES

Detailed nonlinear FE analyses were conducted to simulate the behavior and failure modes of the 

moment-frame assemblies under elevated temperatures. All analyses were performed using 

explicit time integration in LS-DYNA (LSTC 2012). In each analysis, the assembly was subjected 

to a uniform gravity load along the beam (applied along the centerline of the top flange), as well 

as a prescribed temperature time history for the beam and connections. The columns were 

unheated. The gravity load applied to the beam is from a 1.2(Dead Load) + 0.5(Live Load) 

combination of 113 psf (5.41 kPa) on the adjacent slabs (refer to Main and Sadek (2012) for a 

discussion of gravity loads on the prototype buildings). The gravity load was gradually applied 

over 0.5 s to avoid any dynamic amplification. The temperature time history, which is illustrated 

in Fig. 5-4, included both a heating phase and a cooling phase. The temperature was gradually 

ramped up from ambient temperature (20 °C) to a peak temperature, was held constant at the peak 

temperature, and then was dropped back to ambient temperature. It was assumed that the material 

properties of structural steel are similar in compression and tension (heating and cooling). Most 

analyses considered a uniform temperature for the beams and connections, with a peak temperature 

of 700 °C, while specific analyses considered the influence of a temperature gradient through the 

beam depth and of heating to a lower peak temperature. Different restraint conditions for the 

columns at beam level were also considered, and details of the specific analysis cases considered 

are presented in Section 5.4.

or CJP 

(Typical)

Doubler plates: 

14 mm (int. panels)

None (ext. panels)

Shear tab (A36):

10 mm x 464 mm x 152 mm

Continuity plate (A36): 

22 mm (int. panels)

11 mm (ext. panels)

3 A490 H.S.B.

of D = 25 mm

Beam W24x94

Column W24x131 

For 

erection 

purposes 

only

Reduced Section

CJP

CJP

or CJP 

(Typical)

Doubler plates: 

14 mm (int. panels)

None (ext. panels)

Shear tab (A36):

10 mm x 464 mm x 152 mm

Continuity plate (A36): 

22 mm (int. panels)

11 mm (ext. panels)

3 A490 H.S.B.

of D = 25 mm

Beam W24x94

Column W24x131 

For 

erection 

purposes 

only

Reduced Section

CJP

CJP
Chapter 5

100

Figure 5-4 Temperature loading profile.

The WUF-B connections in the IMF assembly (Fig. 5-2) were modeled using finely meshed threedimensional solid elements for the beam, bolts, and shear tab, as shown in Fig. 5-5. Fully integrated 

eight-node solid elements were used. A typical element size of 0.12 in (3 mm) was used for the 

beam and the shear tab. A finer mesh with a typical element size of 0.06 in (1.5 mm) was used for 

the bolts. Contact was defined between the bolts, shear tab, and beam web to model the transfer of 

forces through the bolted connection, including friction, with a value of 0.3 assumed for both the 

static and dynamic coefficients of friction. No pre-tension in the bolts was considered in the 

analyses, because experimental and computational results have shown that pre-tension slightly 

affects the initial response of a bolt in shear but does not significantly affect the ultimate behavior 

or fracture of the bolt (see Kulak et al. (1986)). Fracture was modeled using element erosion, in 

which elements were removed from the analysis when specified failure criteria were satisfied, as 

discussed in Chapter 4. Element erosion was activated only for the solid elements. Outside of the 

connection regions, the beams and columns were modeled using shell elements, and nodal 

constraints were used to enforce continuity of displacements and rotations at the interfaces between 

the solid and shell elements.

0 0.5 1 1.5 2 2.5

0

100

200

300

400

500

600

700

800

Time, s

Temperature, C
Moment Connections Behavior

101

Figure 5-5 Finite-element mesh of the WUF-B assembly: (a) connection region, (b) beam with bolted shear 

tab, (c) beam, (d) bolt, and (e) shear tab.

The RBS connections in the SMF assembly (Fig. 5-3) were modeled using finely meshed twodimensional shell elements, as shown in Fig. 5-6. The reduced section of the beam was modeled 

using a fine mesh with an average element size of about 0.50 in (12.7 mm). Away from the reduced 

section, a coarser mesh was used, with an element size of about 1 in (25.4 mm). Fully integrated 

four-node shell elements were used. Fracture was considered in the analyses using element erosion, 

and the temperature-dependent erosion strain values were calibrated against experimental data for 

the specific mesh size used in the model, as discussed below. The following section discusses the 

temperature-dependent material models used for the various components of the assemblies at 

elevated temperatures.

(b)

(c)

(d) (e)

(a)
Chapter 5

102

Figure 5-6 Detailed model of the RBS connection.

The stress-strain behavior for structural steel and bolts discussed in Chapter 2, along with the 

modeling of fracture discussed in Chapter 4 are used herein to capture failure modes such as tearout in connection plates and bolt shear rupture.

As noted in Chapter 4, the erosion strains, ɛer, are mesh dependent, and must be calibrated for the 

specific element size in the area where fracture is expected. For the solid elements used in the 

critical zones of the WUF-B connections, the mesh density was the same as that used for 

calibrating the erosion strains, ɛer, in Section 4.3.3, thus those values(values are given in Table 4-1, 

and presented on Fig. 4-11) were used for the analysis of the IMF assemblies. However, a similar 

approach was needed to calibrate the required erosion strains for the shell elements used in the 

SMF assemblies herein. Thus, to calibrate the required strains, solid elements with the same mesh 

density used for the original calibration, along with their associated erosion strain values were used 

to model a 2 in gauge coupon. The engineering fracture strains, ɛf, at which these coupons fractured

were used as the target fracture strains for the shell elements. The 2 in gauge coupons were then 

modeled using shell elements with a mesh density as that used for the modeling of the RBS 

connections (shown in Fig. 5-6), and the required erosion strains were calibrated to match the 

target fracture strains. Fig. 5-7 shows the 2 in-gauge coupon models used for the calibration 

analysis, and Fig. 5-8 shows the calibrated erosion strains at different temperatures.
Moment Connections Behavior

103

Figure 5-7 Finite element mesh of the 2 in-gauge tensile coupon (a) solid elements, (b) shell elements (1 in = 

25.4 mm).

Figure 5-8 Calibrated values of erosion strain versus temperature for structural steel using shell elements for 

the RBS connection.
Chapter 5

104

5.4 RESULTS AND DISCUSSION

The discussion in this section focuses on the behavior and failure modes of the moment-frame 

assemblies.

5.4.1 IMF Assembly

As illustrated in Fig. 5-9, three cases are considered for the IMF assembly with WUF-B 

connections: (a) an assembly with fully constrained beam ends (assuming perfect rigidity of 

adjacent spans), (b) an assembly with unconstrained beam ends (assuming no adjacent spans), and 

(c) a three-span assembly, with the outer bays modeled using beam elements and reduced-order 

connection models and kept at ambient temperature (see Sadek et al. (2013) for a description of 

these reduced-order models). Fig. 5-10 shows the axial force in the beam plotted against the 

prescribed temperature for each of these three analysis cases. The results for each case are 

discussed in the following sections.

Figure 5-9 Column restraint cases considered for the IMF assembly: (a) fully constrained, (b) unconstrained, 

and (c) three-span.

(a) (b)

(c)
Moment Connections Behavior

105

Figure 5-10 Axial forces in the beam versus temperature for the different WUF-B assembly cases (1 kip = 

4.448 kN).

5.4.1.1 Fully constrained assembly

For the case of the fully constrained assembly, the beam was not allowed to expand beyond its 

original length. This led to high axial compressive forces in the beam and connections at relatively 

low temperatures in the heating phase. As shown in Fig. 5-10, the compressive axial force in the 

beam reached a peak value of about 944 kip (4200 kN), at 225 °C. At this point, the connection 

failed in a local buckling mode, relieving the axial compression in the beam. As the beam 

continued to heat, the buckled connection deformations increased and the forces in the beam 

continued to decrease. When the temperature reached about 610 °C, the bolts fractured in shear, 

and the load continued to decrease until it reached about 10 % of its peak value at 700 °C. Fig. 5-11 

shows the local buckling mode of the connection at the end of the heating phase of the analysis for 

the fully constrained WUF-B assembly.

x

x

x

x

Local buckling

Bolts fracture
Chapter 5

106

In the cooling phase, the axial compression in the beam reversed into high tensile forces, partially 

reversing the buckled connection deformations. The axial tension in the beam reached a peak of 

about 1080 kip (4804 kN) as the beam cooled to about 45 °C, triggering partial fracture of both 

the upper and lower beam flanges at the welds to the column face, as the strains exceeded the 

specified values for the erosion strains, ɛer. However, the extent of fracture was small, and the 

connection was able to sustain the applied gravity loads throughout the analysis without collapsing.

Figure 5-11 Local buckling mode of the connection at the end of the heating phase of the analysis of the fully 

constrained WUF-B assembly.

5.4.1.2 Unconstrained assembly

For the unconstrained assembly case, the beams were allowed to expand as they were only 

constrained by the stiffness of the columns (as a one-span assembly, with no beam continuity). 

Compared to the fully constrained assembly, this case led to much lower compressive forces in the 

beam and connections, at relatively higher temperatures in the heating phase. As seen in Fig. 5-10, 

the compressive axial force in the beam reached a peak value of 316 kip (1406 kN) at 610 °C, 

roughly one third of the force at nearly 400 °C higher temperature compared to the fully 

constrained case. At this point, the bolts in the connection failed due to shear fracture, in a manner 

similar to what was observed in Chapter 4, relieving the axial compression in the beam, and the 

Pcrit=944 kip (4200 kN)

Tcrit=225 °C
Moment Connections Behavior

107

beam’s flanges deformed under gravity loads, as shown in Fig. 5-12. In the cooling phase, the axial 

compression in the beam reversed into tensile forces, partially reversing the flange deformations. 

The axial tension in the beam reached a peak of about 209 kip (930 kN) as the beam cooled (about 

20 % of the peak tensile force reached in the fully constrained case). The strains due to the tensile 

forces did not exceed ɛer, and as a result no fracture was observed in the analysis.

Figure 5-12 Failure mode of the connection at the end of the heating phase of the analysis of the 

unconstrained WUF-B assembly.

5.4.1.3 Three-span assembly

A three-span assembly, as illustrated in Fig. 5-9(c), was studied as an intermediate case between 

the fully constrained and the unconstrained assemblies. Gravity loads and prescribed temperature 

were only applied to the center span. The outer spans were included to more realistically capture 

the influence of adjoining framing in restraining thermal expansion. The three-span assembly 

exhibited the same failure mode as the unconstrained assembly, with the bolts fracturing in shear 

and the flanges deforming under gravity loads (see Fig. 5-12). However, as expected, higher 

compressive forces were generated in the beam and connections at slightly lower temperatures in 

the heating phase. In the cooling phase, the axial compression again reversed into tensile forces, 

Pcrit=316 kip (1406 kN)

Tcrit=610 °C
Chapter 5

108

reaching a peak of about 432 kip (1922 kN) as the beam cooled (more than double the peak tensile 

force in the unconstrained case).

5.4.2 SMF Assembly

As listed in Table 5-1, four cases were considered for the SMF assembly with RBS connections: 

(a) an assembly with fully constrained beam ends (assuming rigid beams on adjacent spans, similar 

to the sketch in Fig. 9(a), (b) an assembly with unconstrained beam ends (assuming no continuity 

of beam, similar to the sketch in Fig. 9(b), and (c) an unconstrained assembly heated with a 

temperature gradient across the cross-section. Fig. 5-13 shows the axial force in the beam plotted 

against the prescribed temperature for all cases except case (c). Each case is discussed in detail in 

the following sections. It is noted that the gravity load was sustained in all cases without collapse.

Table 5-1 Analysis cases considered for the SMF assembly.

Case Beam end support Thermal loading

(a) Fully constrained Heated to 700 °C

(b) Unconstrained Heated to 700 °C

(c) Unconstrained Heated to 700 °C (with gradient)
Moment Connections Behavior

109

Figure 5-13 Axial force in the beam versus the temperature for the different RBS assembly cases.

(1 kip = 4.448 kN)

5.4.2.1 Fully constrained assembly

As noted previously for the IMF assembly, in the fully constrained case, the beam in the SMF 

assembly was not allowed to expand beyond its original length. This led to higher compressive 

forces in the beam and connections at lower temperatures in the heating phase. As shown in 

Fig. 5-13, the axial compression in the beam reached a peak value of 1226 kip (5454 kN) at 

175 °C. At this point, the RBS connection failed in a local buckling mode, relieving the axial 

compression in the beam. As the beam continued to heat, buckling deformations increased, and 

the axial force in the beam continued to decrease until it reached about 35 % of its peak load at 

700 °C. Fig. 5-14(a) shows the buckling failure mode of the RBS connection towards the end of 

the heating phase in the analysis of the fully constrained assembly.

In the cooling phase, the axial compression reversed into high tensile forces, partially reversing 

the buckled connection deformations. The axial tension reached a peak value of 1050 kip 

(4670 kN) as the beam cooled to ambient temperature. As the beam cooled, at about 380 °C, 

fracture initiated in the top flange. The beam was under a tensile force of about 920 kip (4092 kN) 

x

Local buckling

x

x

Flange fracture
Chapter 5

110

when fracture initiated. As the beam cooled further, the fracture started to open and propagate, 

leading to a near-complete fracture through the top flange and ultimately to partial fracture in the 

web, as shown in Fig. 5-14(b). However, the intact portion of the web, and the intact lower flange,

were able to sustain the applied gravity loads throughout without collapsing.

Figure 5-14 RBS connection failure modes for the fully constrained SMF assembly: (a) local buckling of 

connection during heating phase; (b) partial fracture during cooling phase. (1 in = 25.4 mm)

5.4.2.2 Unconstrained assembly

The unconstrained SMF assembly exhibited a similar buckling failure mode to the fully 

constrained assembly during the heated phase, but with smaller buckling deformations. As shown 

on Fig. 5-13, much lower compressive forces were generated in the beam and connections in the 

heating phase, relative to the fully constrained assembly. Again, in the cooling phase, the axial 

compression reversed into relatively low tensile forces. The axial tension reached a peak value of 

about 75 kip (334 kN) at the end of the cooling phase, roughly one third of the peak axial tension 

generated in the fully constrained case. The strains developed in the cooling phase did not exceed

ɛer, and therefore, no fracture was observed.

(a)

(b)

Tcrit = 175 °C

Pcrit = 1226 kip
Moment Connections Behavior

111

5.4.2.3 Unconstrained assembly, with gradient temperature

The unconstrained SMF assembly was analyzed with a temperature gradient through the beam 

depth, which may be a more realistic temperature distribution when heat conduction can occur 

from the top flange into the cooler floor slab above. The same time history shown in Fig. 5-4 was 

used. However, the temperature reached 700 °C only at the bottom flange. An approximately linear 

profile through the depth was assumed, reaching a peak temperature of 200 °C at the top flange. 

Fig. 5-15 shows the axial force in the beam plotted against the bottom-flange temperature for this 

case. Similar to the unconstrained SMF assembly with uniform temperature distribution 

[Fig. 5-14(a)], the assembly with the gradient distribution failed through local buckling of the RBS 

connection. However, in the gradient-temperature case, the peak axial compression developed in 

the beam during the heating phase was 250 kip (1112 kN) at 575 °C, less than half of the 

compressive force developed in the uniform-temperature case. The axial compressive forces were 

smaller because the cooler upper portion of the beam had less thermal expansion than in the 

uniform-temperature case. In the cooling phase, the axial compression again reversed into tensile 

forces. The peak axial tension in the cooling phase was about 114 kip (507 kN).

Figure 5-15 Cross-sectional longitudinal forces in the beam versus the temperature for the unconstrained 

RBS assembly, heated with a gradient temperature profile through the cross-section of the beam.

Local buckling

x
Chapter 5

112

5.5 CONCLUDING REMARKS

The material and failure models, developed in Chapter 2 and Chapter 4, respectively, were used in 

this chapter to analyze moment-resisting frames to demonstrate how these models can be used to 

gain insight into the behavior and failure modes of these frames. This chapter presented detailed 

finite-element modeling of seismically designed steel moment frames subjected to elevated 

temperatures. Models were developed for two moment-frame assemblies, which included an 

intermediate moment frame with welded unreinforced flange, bolted web (WUF-B) connections 

and a special moment frame with reduced beam section (RBS) connections. Both models 

incorporated temperature-dependent material models for structural steel and structural bolts

(Chapter 2), with an erosion-based failure criterion to simulate fracture (Chapter 4).

Each moment-frame assembly model was subjected to a uniform gravity load along the beam in 

conjunction with a prescribed temperature time-history. The prescribed temperature-history 

included both a heating phase and a cooling phase, in which the temperature was increased from 

ambient temperature (20 °C) to a peak temperature (700 °C in most cases), was held constant at 

the peak temperature, and was then decreased back to ambient temperature. 

These analyses have demonstrated the effectiveness of the developed material and failure models 

in estimating the responses of these frames, and provided useful insight into their behavior and 

failure modes.
113

Chapter 6

SUMMARY, CONCLUSIONS, AND FUTURE WORK

This report presents the formulation and applications of a newly developed temperature-dependent 

material model for structural steels. 

6.1 FORMULATION OF TEMPERATURE-DEPENDENT MATERIAL MODEL

This report first presents a stress-strain model for structural steel at elevated temperatures, as an 

alternative to the stress-strain model used in the Eurocode 3 standard. It is based on a large dataset of the yield strength of structural steels, augmented by post-yield behavior derived from eight 

steels used in the construction of the World Trade Center towers. It accounts for the temperature 

dependence of the yield strength, the change in strain hardening with temperature, and the effect 

of strain rate on strength. The quality of the predictions of the model for stress-strain behavior for 

steels, such as ASTM A992, are similar to those of the Eurocode 3 stress strain model. The 

coefficient of variation of the modulus of elasticity, retained yield strength, and ultimate tensile 

strength, which are the largest components of the variability in the model, increase linearly with 

increasing temperature. Finally, the model is extensible to steels with improved high-temperature 

properties by either choosing a new representation of the temperature dependence of the hightemperature yield strength, or by fitting stress-strain data to generate new values of the parameters 

that describe the temperature dependence of the strain hardening. For structural steel materials, a 

temperature-dependent power-law model was used to represent the true stress-strain behavior up 

to the onset of necking at the temperature-dependent uniform strain. The uniform strain was 

determined based on a nonlinear least-squares fit to the uniform strain data for elevated 

temperatures available in the literature. The post-ultimate behavior was represented using a linear 

projection of the true stress-strain curve, based on the Considère criterion, to initiate necking at 

the uniform strain. Due to the more limited experimental data on structural bolts at elevated 

temperatures, a simpler tri-linear material model was proposed to represent the true stress-strain 

behavior of structural bolts.

6.2 STABILITY OF STEEL COLUMNS

The newly developed temperature-dependent material model for structural steels was then used to 

study the flexural buckling of steel columns to (i) evaluate the applicability of, and (ii) validate,

the NIST model for predicting the behavior of steel gravity columns at elevated temperatures using 

the finite-element method. The study compares the high-temperature flexural buckling of steel 

columns predicted using two different stress-strain models: Eurocode 3 model and the NIST 

model. Phase I used the simplified FEM models to compute the critical buckling temperature of 

47 column specimens. The models assume that the effects of temperature distribution and test 
Chapter 6

114

protocols are negligible to calculate the critical temperatures of columns. The results indicate that 

the NIST stress-strain model predicts the critical temperature for column buckling failure more 

accurately than does the Eurocode 3 model. Thirty FEM models were developed to analyze the 

behavior of the columns tested at Purdue University. The variables in the FEM model include the 

temperature distributions (uniform versus non-uniform), material model (Eurocode 3 versus 

NIST), and two different numerical analysis approaches (modified Newton-Raphson iteration 

approach and modified Riks (arc-length) approach). The results show that (1) the NIST stressstrain model accurately predicts the response, buckling strength, and failure time of columns at 

elevated temperatures. The Eurocode 3 model tends to conservatively estimate the stiffness and 

buckling strength of tested specimens regardless of temperature distributions and simulation 

methods. (2) Both the NIST-proposed and Eurocode-3 thermal expansion models accurately 

predicted the thermal elongations of column specimens with thermal gradients. (3) The critical 

buckling loads predicted using both numerical analysis approaches (modified Newton-Raphson

scheme and modified Riks method) compared favorably with each other when the assumed 

temperature distribution was similar to that in the tests. (4) The behavior of columns can be 

significantly influenced by the assumed temperature distribution while the effect of axial load 

history appears to be negligible on inelastic column buckling. In Phase III, the new column curve 

is calibrated using the NIST stress-strain model. The same FEM approach for the current AISC 

2010 curve and showed that the nominal compressive strength for flexural buckling could be 

updated by substituting the behavior of the yield strength and elastic modulus from the NIST 

stress-strain model, two of the equation parameters. The findings from the study are limited to the 

range of parameters included in the experimental investigations and finite-element models. This 

paper is not intended to draw any conclusions about columns with failure modes other than 

inelastic flexural buckling or cases where the thermal creep behavior is critical. Further work is 

recommended for evaluating predictions using the NIST-proposed steel material model for other 

failure mechanisms (yielding or lateral-torsional buckling), various fire loading conditions (e.g., 

transient heating condition or localized fire), various boundary conditions (e.g., the effects of 

thermally induced axial and rotational restraints), and cases where the thermal creep behavior is 

critical. 

6.3 FE COMPONENT FAILURE MODELING

The newly developed temperature-dependent material model was then used to develop a practical 

modeling approach to represent temperature-dependent nonlinear behavior and failure of structural 

steel and structural bolts at elevated temperatures, using element erosion to represent material 

failure. This modeling approach can be used in explicit finite element analyses to represent the 

successive failures of components in bolted connections under fire-induced heating and the 

consequent redistribution of loads. A plastic strain-based failure criterion was used to activate 

element erosion, noting that the currently available experimental data on fracture of structural steel 

and structural bolts at elevated temperatures are insufficient to enable calibration of 

micromechanics-based models for ductile fracture. Values of erosion strain at elevated 
Summary

115

temperatures were calibrated against experimental data on the engineering strain at fracture of 

tensile coupons, using FE models of tensile coupons to determine appropriate values of erosion 

strain at each temperature. For structural steel, average failure strain values from tensile coupons 

of ASTM A572 Grade 50 and ASTM A992 steel were used as target values in the calibration of 

erosion strain. For structural bolts, erosion strain values were calibrated separately for ASTM 

A325 and A490 bolts, using failure strain data from tensile coupons of each type of bolt. Because 

the modeling of post-ultimate softening and strain-localization behavior is known to depend on 

the mesh discretization, FE analyses of tensile coupons were performed with two levels of mesh

refinement, to investigate the sensitivity of the results to mesh size. The results were found to be 

particularly sensitive to mesh size at high temperatures, because of the earlier onset of necking and 

the more extensive softening behavior prior to fracture. When the element size was reduced by 

one-half to 0.03 in (0.75 mm) in an FE analysis of a tensile coupon at 600 °C, with the erosion 

strain unchanged, the computed engineering strain at fracture was reduced by 44 %. The computed 

results at lower temperatures were less sensitive to the mesh refinement, with their computed 

engineering strains at fracture reduced by approximately 20 %. However, these results confirmed 

the importance of calibrating erosion strain values using the same mesh size that will be used in 

analyzing the failure of structural components. Because the material model and erosion strain 

values for structural bolts were developed and calibrated to represent tensile behavior, FE analyses 

of double-shear test specimens were performed to verify that the model was also capable of 

representing the behavior of structural bolts under shear loading. The computed shear behavior for 

an A325 bolt compared well with experimental data at ambient temperature: the ultimate shear 

capacity was within 1 % of the experimental value, although the experimental displacement at 

fracture was somewhat higher than that predicted by the FE analysis, likely due to deformation of 

the plates during the experiment. Comparing to the nominal shear capacities, the shear capacity 

computed from the FE analyses at ambient temperature was found to be within 5 % for A325 bolts 

and within 2 % for A490 bolts. This work concluded that (i) analysis of the available experimental 

data showed that the uniform strain (i.e., the engineering strain at the onset of necking) decreased 

with increasing temperature, (ii) imposing a reduced stiffness onto the true stress-true strain curve 

for strains greater than the uniform strain, based on the Considère criterion, was sufficient to 

initiate necking at the uniform strain at each temperature, and (iii) the fracture strain was dependent 

on the temperature. Due to the limitations both in the presented modeling approach and in the 

currently available experimental data for material behavior and fracture at elevated temperatures, 

NIST researchers are currently planning future work including carefully controlled experiments 

with different levels of triaxiality at different temperatures. These tests will be used to explore 

ductile fracture characteristics at elevated temperatures.

6.4 BEHAVIOR OF MOMENT-FRAME ASSEMBLIES

Finally, the newly developed temperature-dependent material model, along with the erosion based 

failure criteria was utilized to study the behavior of steel connections under fire loading. The study

presented a detailed finite-element approach developed to analyze the behavior and failure modes 
Chapter 6

116

of seismically designed steel moment frames subjected to elevated temperatures. Finite-element 

models were developed of two moment-frame assemblies, which included an intermediate moment 

frame with welded unreinforced flange, bolted web (WUF-B) connections and a special moment 

frame with reduced beam section (RBS) connections. Both models incorporated temperaturedependent material models for structural steel and structural bolts. The model of the WUF-B 

assembly also incorporated erosion-based failure criteria to simulate fracture. Each moment-frame 

assembly model was subjected to a uniform gravity load along the beam in conjunction with a 

prescribed temperature time history. The prescribed temperature history included both a heating 

phase and a cooling phase, in which the temperature was ramped up from ambient temperature 

(20 °C) to a peak temperature (700 °C in most cases), was held constant at the peak temperature, 

and was then dropped back to ambient temperature. The effects of elevated temperature on the 

behavior and failure modes of WUF-B and RBS moment-frame assemblies were presented in this 

report. The analyses demonstrated the effectiveness of the developed material and failure models 

in estimating the response of these frames, providing useful insight into their behavior and failure 

modes.
117

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