analysis of industrial steel portal frames under fire conditions · 2014. 11. 27. · analysis of...

The University of Sheffield

Department of Civil and Structural Engineering

Analysis of Industrial Steel Portal

Frames under Fire Conditions

by:

Yuanyuan Song

A thesis submitted in partial fulfilment of the requirements for the degree of

Doctor of Philosophy

November 2008

ABSTRACT

A new quasi-static solver has been developed and incorporated in to Vulcan for

dealing with temporary instabilities, such as the snap-through of the pitched roof of a

portal frame, which occurs in static analysis at elevated temperatures. A new

dynamic model is designed to work following a quasi-static solution procedure, so

that divergence of the numerical analysis is avoided in situations where the stiffness

matrix becomes singular, when part of the structure loses stability in fire, and the

structural behaviour after that can be traced. The capabilities and accuracy of this

new solver have been validated, using several benchmark problems, against the

results of reduced-scale fire tests and numerical analysis performed using

commercial software.

Using this new solver, the final failure mechanism of single-span pitched-roof portal

frames, at much higher deflections than that at which the frame initially loses stability

due to snap-through, is shown in the numerical modelling. Factors which may affect

the failure mechanisms of industrial portal frames in fire have been investigated by a

series of parametric studies. The effects of semi-rigid column bases, load ratio,

haunch length, rafter section and different heating profiles have been tested using

two-dimensional models. The restraint provided by secondary members has also

been investigated using three-dimensional models. Two distinct phases of failure,

one of which involves a mechanism which is different from that assumed in the

widely used UK design guide, have been shown from both the fire tests and the

numerical studies.

A new design method based on these two phases of failure has been developed to

determine the final failure mechanisms and critical temperatures of portal frames in

fire. This new method has been compared with the predictions of the current design

guide and numerical tests.

Acknowledgments

This work would not have been possible without the help of my supervisors,

Professor Ian Burgess, Dr Zhaohui Huang and Professor Roger Plank. They have

provided assistance in numerous ways, including encouragement and heuristic

discussion during the whole process of this research, and the sound advice and

patience in reviewing this thesis. The financial support provided by The University of

Sheffield and Vulcan Solutions is also greatly appreciated.

I would like to express my deep and sincere gratitude to my family, their love gave

me strength to accomplish this work. Also thanks to colleagues in the Fire

Research group who provided helpful discussions on my research topic and also

great opportunities to acquire knowledge in different fields.

Special thanks goes to SAFE Ltd for their support during writing this thesis.

Declaration

Except where specific reference has been made to the work of others, this thesis is

the result of my own work. No part of it has been submitted to any organization for a

degree or qualification.

Yuanyuan Song

i

CONTENTS

List of Figures iv

List of Tables xiv

Notation xv

1. INTRODUCTION 1

1.1. Design of Portal Frame at Ambient Temperature ................................. 1

1.2. Fire Concept ........................................................................................ 3

1.3. Steel Properties at Elevated Temperatures ......................................... 5

1.4. Current Design Method for Portal Frames In Fire ...............................12

1.5. Behaviour of Portal Frames In Fire .....................................................14

1.5.1. Fire Test ......................................................................................15

1.5.2. Behaviour of Single-Storey Portal Frames in Fire ........................18

1.5.3. Overall Behaviour of Industrial Frames in Fire .............................24

1.6. Numerical Modelling ...........................................................................32

1.6.1. Vulcan .........................................................................................32

1.7. Scopes of The Research and Layout of The Thesis ...........................33

2. NEW QUASI-STATIC PROCEDURE 35

2.1. Introduction .........................................................................................35

2.2. Dynamic Methods ...............................................................................41

2.2.1. Direct Integration Method ............................................................42

2.2.2. Mode Base Method .....................................................................45

2.3. Developed Dynamic Model .................................................................47

2.3.1. Predictor ......................................................................................47

2.3.2. Corrector .....................................................................................48

2.3.3. Lumped Mass Assumption ..........................................................50

2.3.4. Damping Matrix ...........................................................................54

ii

2.3.5. Transfer Matrix for Three-Noded Beam Elements .......................58

2.4. Quasi-Static Solution Procedure .........................................................63

2.5. Conclusion..........................................................................................64

3. VALIDATIONS 67

3.1. Validation of the Dynamic Model ........................................................67

3.1.1. Hinge Supported Beam ...............................................................67

3.1.2. Half Roof Frame ..........................................................................71

3.1.3. Pitched Portal Frame ...................................................................74

3.2. Validation of the Quasi-Static Solution Procedure ...............................76

3.2.1. Twin-Bay Single-Storey Steel Portal Frame .................................77

3.3. Validation Using Fire Tests .................................................................81

3.3.1. Wong’s Test ................................................................................81

3.3.2. Numerical Model .........................................................................85

3.4. Comparisons Against Static Analysis ..................................................91

3.5. Conclusion..........................................................................................95

4. BEHAVIOUR OF PITCHED PORTAL FRAMES IN FIRE 96

4.1. Introduction .........................................................................................96

4.2. Behaviour of Portal Frames with Semi-Rigid Bases ............................97

4.2.1. Frames without Fire Protection ....................................................98

4.2.2. Frames with Columns Protected ................................................ 101

4.3. Parametric Studies ........................................................................... 103

4.3.1. Numerical Model ....................................................................... 104

4.3.2. The Effect of Base Capacity ...................................................... 111

4.3.3. The Effect of Base Stiffness ...................................................... 117

4.3.4. The Effect of Loading Ratio ....................................................... 121

4.3.5. The Effect of Haunch Length ..................................................... 124

4.3.6. The Effect of the Rafter Section ................................................. 127

4.3.7. The Effect of Column Heating .................................................... 129

iii

4.3.8. The Effect of Asymmetrical Localised Fire ................................. 134

4.3.9. The Effect of Purlins .................................................................. 140

4.4. Conclusion........................................................................................ 148

5. A NEW DESIGN METHOD FOR INDUSTRIAL FRAMES IN FIRE 151

5.1. Discussion on Current Design Method .............................................. 152

5.1.1. Current Design Model ................................................................ 152

5.1.2. Comparison with Numerical Results .......................................... 156

5.2. Development of the Simple Design Method ...................................... 161

5.2.1. First Phase Failure Mechanism ................................................. 161

5.2.2. Second Phase Failure Mechanism ............................................ 165

5.3. Calibration of the New Design Method .............................................. 172

5.3.1. Cases with Different Base Strength ........................................... 172

5.3.2. Cases with Different Haunches ................................................. 175

5.3.3. Cases with Different Rafters ...................................................... 176

5.3.4. Cases with Different Loading Ratio ............................................ 177

5.4. Conclusions ...................................................................................... 180

6. CONCLUSION 182

6.1. New Dynamic Model and Quasi-Static Procedure ............................ 182

6.2. Paramatric Studies ........................................................................... 183

6.3. New Simple Design Method ............................................................. 186

6.4. Recommendations for Further Work ................................................. 188

APPENDIX 190

REFERENCES 193

iv

List of Figures

Figure Page

1.1 Cross section showing the portal frame and its restraints 1

1.2 A typical nominally pinned base 2

1.3 Development of a natural fire 4

1.4 Heat release rate for a fire in an industrial building 5

1.5 Stress-strain curves for steel tested at 550°C 6

1.6 Stress-strain curves of steel tested at the medium strain-rate 7

1.7 The comparison of the stress-strain curves between the

Ramberg-Osgood and Eurocode model up to 2% strain 9

1.8 The comparison of the stress-strain curves between the

Ramberg-Osgood and Eurocode models up to 20% strain 9

1.9 Strength reduction factors from the stress-strain relationships

of steel at elevated temperatures according to the Eurocode

and British Standard 10

1.10 Thermal elongation of carbon steel as a function of the

temperature 11

1.11 Failure mechanism used in current design method 13

1.12 Frame dimensions used in calculation of overturing moment 13

1.13 The first test: heating of the whole rafte 16

1.14 The second test: edge fire 16

1.15 The deformed portal frame after the third fire test 17

1.16 Variation of overturning moment with the deformation of the

portal frame in fire 19

1.17 Heating situation considered in O’Meagher et al.’s model 20

1.18 Failure temperatures of portal frames 21

v

1.19 Comparison of failure temperatures with pinned and

semi-rigid bases 22

1.20 Variation of deflected shape showing the out-of-plane

collapse of frames heated by the ISO834 fire 23

1.21 Overall building behaviour 25

1.22 Deflected shape for the model with purlins at the moment of

failure, with an amplification factor of 10 25

1.23 Deflected shape at 660°C of the entire roof heated case 26

1.24 Vertical displacement at apex -2D vs 3D analysis 26

1.25 Vertical displacements at apex of the central frame 27

1.26 The analytical model in Bong’s study 28

1.27 Deflected shapes immediately before and after the rapid

sagging of roof of the pinned-base case with purlin axial

restraint 28

1.28 Sidesway collapse of the pinned-base case without purlin

axial restraint 28

1.29 Inward collapse of the fixed-base case without purlin axial

restraint 29

1.30 Finally deflected shape of the pinned-base case without

purlin axial restraint and with full concrete encasement of

columns 29

1.31 Lateral displacements at the tops of columns in Phase 1 31

1.32 Lateral displacements at the tops of columns in Phase 2 31

2.1 Illustration of Newton-Raphson iterative procedure 36

2.2 Illustration of the ill-conditioning of stiffness matrix in static

analysis 36

2.3 Illustration of the mechanism of snap-through for portal frame

37

vi

2.4 Load-deflection relationships for the pitched roof frame 39

2.5 Effect of damping in the snap-through process 40

2.6 Calculation of mass based on the segments of the beam

sections 51

2.7 Illustration of lumped mass assumption for a three-noded

beam element 52

2.8 The relationship between damping ratio and frequency 55

2.9 Calculation of direction cosines of beam element 60

2.10 Illustration of the quasi-static solution procedure 64

3.1 Illustration of the loading on the pinned connected beam 68

3.2 Comparison of the mid-span deflections of the elastic beam

with geometrically linear dynamic modelling by VULCAN and

ABAQUS 68

3.3 Comparison of the mid-span deflections of the elastic beam

with geometrically linear dynamic modelling by VULCAN and

ABAQUS (from t = 4 to t = 6 second) 69

3.4 Predicted mid-span displacements of the elastic beam with

geometrically non-linear behaviour using different time steps 70

3.5 Predicted mid-span displacements of the elastic beam with

geometrically non-linear behaviour using different time steps

(from 4.7 to 5.4 second) 70

3.6 Single element model to e snap-through behaviour 71

3.7 Deflection curves at the moving end of the element (ξ =1.0) 72

3.8 Detailed deflection curves at the moving end of the element

(ξ =1.0) 72

3.9 Effect of damping during snap-through of the frame 74

3.10 Single pitched roof frame with loading P on apex 74

3.11 Deflections of the apex of the pitched frame 75

vii

3.12 Deflections of the mid-point of the left rafter of the pitched

frame 75

3.13 Deflections of the mid-point of the right rafter of the pitched

frame 76

3.14 Initial configuration and loading arrangement of Franssen’s

frame 77

3.15 Vertical displacement at node A 78

3.16 Horizontal displacement at node A 78

3.17 Vertical displacement at node B 79

3.18 Horizontal displacement at node B 79

3.19 Vertical displacement at node C 80

3.20 Horizontal displacement at node C 80

3.21 Horizontal displacement at node D 81

3.22 Deflected shape of the twin-bay steel portal frame 81

3.23 General layout of the structure in the fire test 82

3.24 The loading system for the portal frame 82

3.25 Lateral support system 83

3.26 Dimensions of base connection 83

3.27 Recorded displacements from the third test 84

3.28 The deformation of the rafter and purlins after the fire test 84

3.29 The details of the base connection in the third test 85

3.30 Comparison between numerical modelling and test results 86

3.31 Deflected shape predicted by Vulcan in two dimensional view

with pinned-base assumption 87

3.32 Vertical displacements of the apex from different numerical

models performed by Vulcan quasi-static analysis 88

3.33 Z-Purlin section modelled by the equivalent I section 88

viii

3.34 Deflected shape predicted by Vulcan in three-dimensional

view 90

3.35 Comparison between the test result and the two dimensional

numerical models with nominally pinned bases 91

3.36 Initial configuration of the pitched portal frame for the

comparison between static and quasi-static analysis 92

3.37 Vertical apex displacements of the frames with different

columns 92

3.38 Horizontal displacements of the left eaves with different

columns 93

3.39 Horizontal displacements of the right eaves of the frames

with different columns 93

3.40 Failure modes of the pitched portal frames with different

column sections 94

4.1 Vertical displacement of the apex of the frame 98

4.2 Horizontal displacement of the left eaves of the frame 99

4.3 Horizontal displacement of the right eaves of the frame 99

4.4 Failure progress of portal frame with nominally rigid bases

when only rafters are heated 100

4.5 Vertical deflection of the apex 102

4.6 Horizontal deflection of the left eaves 102

4.7 Horizontal deflection of the right eaves 103

4.8 Failure progress of portal frame with nominally rigid bases

with only rafters heated case 103

4.9 Layout of the pitched portal frame for parametrical studies 105

4.10 Details of haunch models 106

4.11 Vertical displacements at the apex for different haunch

models 107

ix

4.12 Horizontal displacements at the eaves for different haunch

models 107

4.13 The comparison between the simple assumption and test

data for the initial stiffness of the steel column bases 109

4.14 The comparison of the ultimate resistance of the bases

obtained in different tests 109

4.15 Illustration of the column base model 110

4.16 Effect of base capacity: vertical displacements at apex 113

4.17 Effect of base capacity: horizontal displacements at left

eaves 113

4.18 Effect of base capacity: horizontal displacements at right

eaves 114

4.19 Effect of base capacity: failure progress of the frame with

pinned base 115


semi-rigid base (a=0.5, k=0.1) 115





4.23 Effect of base stiffness: vertical displacements at apex 119

4.24 Effect of base stiffness: horizontal displacements at left

eaves 119

4.25 Effect of base stiffness: horizontal displacements at right

eaves 120

4.26 Effect of base capacity: failure sequence of the frame with a

semi-rigid base ( a =4.0, k =0.5) 120

4.27 Effect of load ratio: vertical displacement at apex 122

x

4.28 Effect of load ratio: horizontal displacement at left eaves 123

4.29 Effect of load ratio: horizontal displacement at right eaves 123

4.30 Effect of haunch length: vertical displacement at apex 125

4.31 Effect of haunch length: horizontal displacement at left eaves

126

4.32 Effect of haunch length: horizontal displacement at right

eaves 127

4.33 Effect of rafter section: vertical displacement at apex 128

4.34 Effect of rafter section: horizontal displacement at left eaves 128

4.35 Effect of rafter section: horizontal displacement at right eaves

129

4.36 Effect of column heating: vertical displacement at apex 131

4.37 Effect of column heating: horizontal displacement at left

eaves .. 131

4.38 Effect of column heating: horizontal displacement at right

eaves 132

4.39 Effect of column heating: failure sequence of the frame 133

4.40 Effect of column heating: failure sequence of the frame 133

4.41 Heating profiles of the frame modelled 134

4.42 Effect of different localised fire: vertical displacements at

apex (part 1) 136

4.43 Effect of different localised fires: failure sequence of frame

under heating Profiles 1 and 7 137

4.44 Effect of different localised fires: failure sequence of frame

under heating Profiles 2 and 8 138

4.45 Effect of different localised fires: vertical displacements at

apex(part 2) 139

xi

4.46 Effects of different localised fire: failure progress of frame

under heating Profile 3 139

4.47 Comparison of the bending moment diagrams given by

heating Profiles 5 and 6 140

4.48 Effects of purlins: a part of the portal frame in a

three-dimensional view 141

4.49 Effects of purlins: the model of the half frame 142

4.50 Heating profiles assumed for the three dimensional model 143

4.51 Comparison of vertical displacements of the apex of the

central frame 144

4.52 Vertical displacement of the apex of the internal frame 145

4.53 Vertical displacement of the apex of the external frame 145

4.54 Deflected shape of the frames at 1015°C in Test 1 146

4.55 Deflected shape of the frame at 1015°C in Test 2 146

4.56 Deflected shape of the frames in Test 3 147

4.57 Deflected shape of the frames in Test 4 147

5.1 Forces and moments acting on the frame 152

5.2 Base overturning moment of the column changed with

column inclination and rafter temperatures 155

5.3 Vertical displacements of the apex of the frame with different

column’s base strength 156

5.4 Comparisons: current design model against the numerical

model 157

5.5 Vertical displacements of the apex of the frame with different

column’s base strength 158

5.6 Comparisons: the current design method against the

numerical test results 159

5.7 The boundary wall of the portal frame after fire 160

xii

5.8 The model of Wong’s simple design method 163

5.9 The haunch included model of Wong’s method 164

5.10 Illustration of the second phase failure mechanism 166

5.11 The iterative procedure for estimating the critical temperature

of the second phase failure mechanism of the frame 167

5.12 Model for the calculation of the re-stabilised position of the

frame 168

5.13 Haunch included model for the calculation of the re-stabilised

position of the frame 170

5.14 Haunch included model for the second phase failure

mechanism of the frame 172

5.15 Comparison of predicted critical temperatures between new

design method and numerical analysis results for the frames

without haunch 174

5.16 Comparison of predicted critical temperatures between new

design method and numerical analysis results for the frames

with haunch 175

5.17 Comparison of predicted critical temperatures for the first

phase mechanism between new design method and

numerical analysis 176






numerical analysis, together with current design method 178

xiii

5.20 Comparison of predicted critical temperatures for the second



xiv

LIST OF TABLES

Table Page

1 Mass-proportional damping factor and referential damping

ratio values for single element model 71

2 Load combinations 97

3 List of the numerical studies conducted 143

4 Recommended horizontal deflection limits at eaves level and

the corresponding lean limit of columns in a pitched portal

frame 154

xv

NOTATION

x,y,z global axis

r,s,t elemental axis

t time increment

F internal force vector

U displacement vector

U velocity vector

U acceleration vector

K stiffness matrix

M mass matrix

R external load vector

C damping matrix

zyx FFF ,, force in direction x, y, z

zyx MMM ,, moment about axis x, y, z

wvu ,, translational displacement in directions x, y, z

zyx ,, rotational displacement about axis x, y, z

density of an element.

L length of an element

J rotational inertia

Chapter 1 Introduction

1

1. INTRODUCTION

1.1. DESIGN OF PORTAL FRAME AT AMBIENT TEMPERATURE

Over half of the usage of the steel in UK is on the primary framework of the industrial

portal frames. For its significant advantages in simple assemblage and cost

efficiency, the portal frame has become the most common construction form for

single-storey industrial buildings in the UK. A typical single-span symmetrical portal

frame, as shown in Figure 1.1 could have a span up to 50m, eaves height up to 10m

and a roof pitch between 5° and 10°. Because the bending resistance of the

connections at the eavess and apex determines the capacity of the frame, rigid

connections between each column and the deep rafter section are required. In the

most common situation, haunches are adopted at the eavess and apex to reduce the

depth of the rafters. For ultimate limit state design at ambient temperature, such

frames are usually designed on the assumption that the column bases act as

frictionless hinges for convenience in foundation construction.

Figure 1.1 Cross-section showing a portal frame and its restraints. (Salter et al.,

2004)

A dado masonry wall up to a height of 2.5m is normally provided along the edges of

the building. It is required to be designed in compliance with the deflection criteria

which are specified for masonry construction. In order to provide stability for the


2

building, both during erection and in the completed building under full loading, the

resistance of the frames to wind load in the longitudinal direction is needed.

Therefore, an adequate anchorage is required for the purlins and sheeting rails to

ensure their function of restraining the rafters and columns. Also, bracing is required

both in the plane of the rafters and vertically in the side walls. Purlins and side rails,

to support the roof and side wall sheeting, are designed to be continuous over the

rafters and also provide restraint to the rafters or columns. (Salter et al., 2004)

Nominally pinned bases are normally chosen for the columns due to the difficulty and

expense of providing moment-resisting bases. These nominally pinned bases are

normally designed to provide resistance to the vertical loading on the roof and to

horizontal loading caused by the wind. It is formed by a base plate, holding down

bolts and a concrete foundation block as shown in Figure 1.2. If a moment-resisting

base is required, a bigger lever arm for bolts, a stiffer base plate and additional

gusset plates should be designed, as well as a much larger concrete foundation.

Holding down bolts

Base

plate

Bedding

space

Anchor

plates

Location

tube

Top of

concrete

foundation

Holding down bolts

Base

plate

Bedding

space

Anchor

plates

Location

tube

Top of

concrete

foundation

Figure 1.2 A typical nominally pinned base.


3

1.2. FIRE CONCEPT

The development of a natural fire is described by Purkiss (1996) as consisting of

three periods: pre-flashover, post-flashover and decay, as shown in Figure 1.3.

During the pre-flashover stage, the fire is still localized within the compartment, so

the rise in the overall temperature is small. The highly non-uniform temperature

distribution within the compartment, the hazard posed by the toxic gases, and the

“back-draught” caused by sudden ventilation are also stressed by Wong (2002). At

this stage, the enclosure is filled by hot air and smoke, with combustible products

near the ceiling and cool air at the floor. When the temperature of the hot layer is high

enough it can ignite all the rest of the unburned combustible materials within the

room, and the fire develops to the flashover stage. Purkiss (1996) defined flashover

as the moment when the local fire spreads to all the combustible material within the

compartment, so the temperature in the compartment increases dramatically after

flashover, because all the available fuel is burning at this stage. The fire reaches its

highest temperature at this stage, and the duration of the post-flashover period

largely depends on the exposed surface of the fuel and the ventilation conditions of

the fire compartment. When the amount of combustible material begins to reduce

after a period of steady burning, the heating rate decreases and the compartment

begins to cool. Normally, it takes a longer time to achieve flashover in a large

enclosure.


4

Figure 1.3 Development of a natural fire.

For fires in large compartments, the concept of progressive burning is discribed by

Buchanan (2001), on the basis of fire tests and numerical modelling. It is claimed that

the development of the fire at different positions within the compartment largely

depends on the distance from the fire source. The maximum temperature is

developed progressively at different positions due to the delayed burning of the fuel

at more remote positions.

The industrial portal frame is a special case in fire. When a fire is ignited in a

single-storey industrial frame, cool air is drawn into the space, and hot air and smoke

rise to the roof due to the buoyancy caused by the fire plume. The hot gas layer can

then spread across the ceiling and either escape at the edge of the ceiling or decend

towards the floor, provided that the ceiling is closed. However, in most portal frames,

the hot air can escape either through skylights in the roof, or through openings

caused by the burning out of flammable cladding, so that flashover is unlikely to

happen in portal frames. The developing fire concept was introduced by O’Meagher

(1992) into numerical modeling to investigate the behaviour of industrial steel frames

at the pre-flashover stage. A possible heat release rate senario for industrial portal

frame is presented by Buchanan (2001) as the sequence shown in Figure 1.4.

Time

Temperature

Flashover

Pre-flashover Post-

flashover Decay


5

Figure 1.4 Heat release rate for a fire in an industrial building (Buchanan, 2001).

To assess the performance of building materials and structural elements, a standard

heating curve is defined for full-scale fire tests. This is the so-called “Standard

Fire”(Buchanan, 2001). In the UK, the widely-used Standard Fire temperature-time

( tT ) curve is defined in ISO834 (ISO, 1975) which is defined by Equation (1.1) as

0100 8log345 TtTT (1.1)

where T (0C ) is the furnace temperature at time t, 0T (0C ) is the initial furnace

temperature , and t is the time in minutes. This is also adopted as the design fire in

Eurocode 1 Part 1.2 (CEN, 2002) for wood-based, or cellulosic, fires.

1.3. STEEL PROPERTIES AT ELEVATED TEMPERATURES

For steel at ambient temperature, plastic yield can be observed as a sudden change

in the stress-strain curve. However, the change of steel strain at elevated

temperatures is determined as a combination of thermal strain, creep strain and

stress-related strain, so the mechanical properties of steel at elevated temperatures

Heat release rate

Ventilation

Control

Fuel Control

Time

Growth

Period

Skylights melt

Roof collapse

Decay period


6

are quite different from those at ambient temperature. Some typical stress-strain

curves for steel at elevated temperatures presented by Harmathy (1993) show that

the elastic tangent modulus decreases gradually before the steel reaches plastic

yield at elevated temperatures, and that the ultimate strength of steel is higher than

that of cold steel when it is heated to a temperature range between 200°C and

320°C.

In 1988, a series of tensile capacity tests on hot-rolled steel was performed by Kirby

and Preston (1988). Strength reduction factors, which are the ratios of the steel

strength at elevated temperature to its strength at ambient temperature, was

obtained from these tests. The definitions of the stress-strain curves for steel at

elevated temperatures in BS5950 Part 8 (BSI, 2003), Eurocode 3 Part 1.2 (CEN,

2005a) and Eurocode 4 Part 1.2 (CEN, 2005c) are based on this test data.

0

50

100

150

200

250

300

350

400

450

500

0 0.1 0.2 0.3 0.4 0.5 0.6

Fast speed (0.7mm/min)

Middle speed (3.1mm/min)

Slow speed (6mm/min)

EC-Curve

Stress (N/mm2)

Strain

0

50

100

150

200

250

300

350

400

450

500

0 0.1 0.2 0.3 0.4 0.5 0.60

50

100

150

200

250

300

350

400

450

500

0 0.1 0.2 0.3 0.4 0.5 0.6




EC-Curve




EC-Curve

Stress (N/mm2)

Strain

Figure 1.5 Stress-strain curves for steel tested at 550°C (Renner, 2005).


7

A group of tensile tests on S275 steel aimed to study the effect of strain-rate was

performed by Renner (2005) at Sheffield University. The tests were focused on the

steel properties between 400°C and 700°C, and three displacement speeds:

0.7mm/min, 3.1mm/min and 6mm/min were applied. The results presented in Figure

1.5 prove that the ultimate strength of the heated steel changes with the loading

speed, as a result of the re-arrangement of microstructure during the deformation. It

is also observed that the true stress at 15% strain is higher than the assumption

made in Eurocode 3 (CEN, 2005a). Results shown in Figure 1.6 are an example

taken from these tests.

0

50

100

150

200

250

300

350

400

450

500

0 0.1 0.2 0.3 0.4 0.5 0.6

Strain

Stress (N/mm2)

ambient°C Test

ambient°C EC

400°C Test

400°C EC

450°C Test

450°C EC

500°C Test

500°C EC

550°C Test

550°C EC

600°C Test

600°C EC

700°C Test

700°C EC

0

50

100

150

200

250

300

350

400

450

500

0 0.1 0.2 0.3 0.4 0.5 0.6

Strain

0

50

100

150

200

250

300

350

400

450

500

0 0.1 0.2 0.3 0.4 0.5 0.6

Strain

Stress (N/mm2)

ambient°C Test

ambient°C EC

400°C Test

400°C EC

450°C Test

450°C EC

500°C Test

500°C EC

550°C Test

550°C EC

600°C Test

600°C EC

700°C Test

700°C EC

ambient°C Test

ambient°C EC

400°C Test

400°C EC

450°C Test

450°C EC

500°C Test

500°C EC

550°C Test

550°C EC

600°C Test

600°C EC

700°C Test

700°C EC

Figure 1.6 Stress-strain curves of steel tested at the medium strain-rate (Renner,

2005).

The stress-strain curves at elevated temperatures have been defined by a

Ramberg-Osgood (Ramberg & Osgood,1943) type of equation:


8

TN

T

T

T

TT

bBaA

01.0 (1.2)

where the value of TA , TB and TN are temperature-dependent.. The terms T

and T are the strain and stress at the temperature T , respectively.

In Eurocode 3 Part 1.2 (CEN, 2005a), the stress-strain curves at elevated

temperature are represented by one elliptical and three linear equations. In the first

stage, the stress increases with strain following the slope ,aE until the stress

exceeds the proportional limit ,pf and a perfect plastic stress plateau is defined

from 2% strain to 15% strain at a constant stress which equals the effective yield

stress ,yf . An elliptical curve is used to connect the linear elastic and plastic

curves. The stress decays to zero linearly from 15% to 20% strain. The detailed

formulation of the curve is given in Figure 3.1 of Eurocode 3 Part 1.2 and values of

,aE , ,pf and ,yf at elevated temperatures are givien in Table 3.1 of Eurocode 3

Part 1.2.

The stress-strain curves for S275 steel calculated using the Ramberg-Osgood

(Ramberg & Osgood, 1943) model (BSI, 1985) and the Eurocode model are

presented in Figures 1.7 and 1.8. It is shown that Ramberg-Osgood is very close to

the Eurocode model when the strain is less than 2%. After this, the Eurocode model

gives much lower effective stress than the Ramberg-Osgood model.


9

0

50

100

150

200

250

300

0 0.005 0.01 0.015 0.02

Strain

20°C

100°C

200°C

300°C

400°C

500°C

600°C

700°C

800°C

Stress (N/mm2)

0

50

100

150

200

250

300

0 0.005 0.01 0.015 0.02

Strain

20°C

100°C

200°C

300°C

400°C

500°C

600°C

700°C

800°C

Stress (N/mm2)

Figure 1.7 The comparison of the stress-strain curves between the

Ramberg-Osgood (dashed line) and Eurocode model (solid line) up to 2% strain.

0

50

100

150

200

250

300

350

400

450

500

0 0.05 0.1 0.15 0.2

Strain

20°C

100°C

200°C

300°C

400°C

500°C

600°C

700°C

800°C

Stress (N/mm2)

0

50

100

150

200

250

300

350

400

450

500

0 0.05 0.1 0.15 0.2

Strain

20°C

100°C

200°C

300°C

400°C

500°C

600°C

700°C

800°C

Stress (N/mm2)

Figure 1.8 The comparison of the stress-strain curves between the

Ramberg-Osgood (dashed line) and Eurocode models (solid line) up to 20% strain.


10

Figure 1.9 Strength reduction factors from the stress-strain relationships of steel at

elevated temperatures according to the Eurocode and British Standard.

For design proposes, the strength retention factors for steel strains of 0.5%, 1.5%

and 2% are given in BS5950 Part 8 (BSI, 2003). The strain limit required for a

composite member is 2% and for a non-composite member is 1.5%. 0.5% strain is

required for all other members. In Eurocode 3 Part 1.2 (CEN, 2005a), 2% strain is

used in almost all calculations. Figure 1.9 shows that the strength reduction factors

obtained from BS5950 and Eurocode are similar; this is not surprising since they are

all based on the result from the same test programme (Kirby & Preston,1988).

Thermal elongation of steel is critical to structural behaviour at elevated

temperatures. It can lead to the thermal bowing due to temperature gradients, or

buckling when the beam or slab is restrained at the boundary and an extra internal

compression is generated on the cross-section. Because the internal forces through

out the structure are affected by the thermal strains in the structural members, the

load distribution within the structure become complex under high temperatures. Also,

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800 1000

0.5% strain BS5950 Part 8 0.5% strain Eurocode 3&4

1.5% strain BS5950 Part 8 1.5% strain Eurocode 3&4

2% strain BS5950 Part 8 2% strain Eurocode 3&4

Strength reduction factor

Temperature (°C)


11

for a long span structure, such as the single-span pitched portal frame, the

elongation of the members can be significant and can not be ignored in design.

0

4

8

12

16

20

0 200 400 600 800 1000 1200

Elongation 310/ ll

Temperature (°C)

0

4

8

12

16

20

0 200 400 600 800 1000 1200

Elongation 310/ llElongation 310/ ll

Temperature (°C)

Figure 1.10 Thermal elongation of carbon steel as a function of the temperature

(CEN, 2005a)

The thermal elongation of steel is defined as a function of temperature in BS5950

Part 8 (BSI, 2003) and Eurocode 3 Part 1.2 (CEN, 2005a). As shown in Figure 1.10,

the steel elongates quite linearly with increasing temperature until 750°C, and then

stops from 750°C to 860°C due to a phase-change of the microstructure of steel. The

steel then continues to expand linearly when the temperature is beyond 860°C.

The basic steel properties at 20°C, assumed for the numerical studies, are listed as

follows:

Mass density: 3/7850 mkg ;

Young’ modulus: MPaE 210000 ;

Yield strength for S275 Steel: MPaf y 275 .

The thermal properties of steel adopted in this study are basically those according to

Eurocode 3 Part 1.2 (CEN, 2005a). Considering that the decay of stress beyond 15%


12

strain could be much slower than the assumption in Eurocode, in this study the steel

stress remains constant as the ultimate stress when the strain exceeds 15%.

1.4. CURRENT DESIGN METHOD FOR PORTAL FRAMES IN FIRE

In UK, the fire safety of the buildings has to comply with the Building Regulations

2000, Approved Document B (DETR, 2006). To prevent the collapse of buildings in

fire, a degree of fire protection of the structural elements is required for multi-storey

buildings, but the main concern for single-storey buildings is stopping fire spread

between buildings. In order to meet this requirement, when a single-storey building is

designed to satisfy fire safety requirements, the distance from its external wall to the

site boundary must be at least 15m, or fire protection of the external wall is

necessary. Moreover, all structural elements which are used to support the external

wall must have the same fire resistance as the boundary wall itself. A portal frame is

considered as a single structural element, due to the rigid connection between rafters

and columns, so if any part of it needs to be protected against fire, the whole frame

has to be protected. As a result, the whole portal frame has to be fully protected

against fire to guarantee the longitudinal stability of the boundary wall when the

frame is in a fire boundary condition. Fire protection to the whole portal frame is very

expensive compared with the cost of the portal frame itself. To avoid this, a simple

design method (Simms & Newman, 2002), which allows no fire protection for portal

frames, was published by the Steel Construction Institute. This method is widely

used in the UK.

In order to guarantee the longitudinal stability of the boundary wall, the inclination of

the columns in the boundary frame should be limited if they are not protected against

fire. When the roof collapses below the eavess level, due to the mechanism shown in

Figure 1.11, the catenary force generated on the rafters becomes dominant in pulling

the columns inward, and causes an overturning moment (OTM ) at the column


13

bases. The idea of Newman’s method is to specify a very strong base connection to

resist the OTM generated at the column bases.

Figure 1.11 Failure mechanism used in current design method (Newman, 1990).

Referring to Figure 1.12, the vertical reaction RV , the horizontal reaction RH and

OTM at the column bases can be calculated by Equations (1.3) to (1.5).

wallofweightdeadSLWV fR 2

1 (1.3)

Y

M

G

CMSGAWKH C

p

fR10

(1.4)

10065.0 Cpf

M

G

CYM

Y

BASGAWKOTM (1.5)

G

L

Y

RV

RHOTM

G

L

Y

RV

RHOTM

Figure 1.12 Frame dimensions used in calculation of overturing moment

(Newman, 1990).

Fire hinges


14

where

G

GLB

8

22 (1.6)

and

fW = load at time of collapse;

S = distance between frame centers;

G = distance between ends of haunches;

Y = vertical height of end of haunch;

L = span;

pM = plastic moment of resistance of rafter

cM = plastic moment of resistance of column

K = 1 for single bay frames or is taken from Table 2 of SCI Publication 087

(Newman, 1990);

A and C are taken from Table 1 of SCI Publication 087 (Newman, 1990).

In the latest design guide, SCI Publication P313 (Simms&Newman, 2002), for single

storey steel frame buildings in fire boundary conditions, this simple method is still

recommended for the design of single- or multi-span symmetrical pitched portal

frames, either with or without haunches. Instead of using the single value of A for

frames with various span-to-height ratios, the new design guide gives modified

values of A for frames with low span-to-height ratio. In addition, the derivation of

the simple calculation method for symmetrically pitched portal frames is given based

on the same assumptions used in the old design guide (Newman, 1990).

1.5. BEHAVIOUR OF PORTAL FRAMES IN FIRE

Many studies(De Souza Jonior et al., 2002, Bong, 2005, Vassart, et al., 2005, Moss

et al., 2006) have been carried out on the thermal and structural performance of


15

single-storey industrial buildings. Only a limited number of fire tests (Wong, 1999,

2001) have been carried out on the portal frames in the past 20 years. Thanks to the

development of advanced finite element software, the behaviour of industrial steel

frames at elevated temperatures can now be modelled in two- and three-dimensions.

The basic failure mechanism of industrial steel frames and the effect of the

secondary elements under different heating scenarios have been investigated using

such models. Although the fire safety regulations differ in different countries, with

different design goals, the resistance of fire-walls is a common concern for industrial

buildings under fire conditions. Column bases may remain at a relatively low

temperature, but the effect of their strength and flexibility can be significant on the

global behaviour when the rafters are very hot and the catenary forces on the column

tops are significant. Previous works related to the behaviour of industrial portal

frames under fire conditions are briefly summarised in the next section.

1.5.1. FIRE TESTS

In order to investigate the behaviour of industrial frames in fire, three fire tests on a

scaled pitched-roof portal frame were performed by Wong (Wong et al., 1999) at the

University of Sheffield. In the first experiment (see Figure 1.13), the rafter of one

frame was heated by a line fire under sufficient ventilation. The fire was produced by

liquid heptane contained by a metal tray hanging below the frame, supported by a

separate frame, and lasted for about 10 minutes; some parts of the steel were heated

to just over 900°C. A certain amount of deflection was recorded, but the frame did not

totally collapse. The second experiment (See Figure 1.14) was carried out on

another internal bay of the same frame. In order to obtain a longer-lasting fire, the

liquid fuel was replaced by timber cribs. This test lasted for well over 40 minutes,

before the fire was manually extinguished. The compartment temperature achieved a

state of equilibrium after about 10 minutes, and this continued for at least 30 minutes.


16

The maximum steel temperature recorded in the test was only about 800°C, and no

collapse mechanism of the portal frame was observed during the test.

Figure 1.13 The first test: heating of the whole rafter (Wong, 2001).

Figure 1.14 The second test: edge fire (Wong, 2001).

In order to produce a structural collapse, the overall rafter-heated test was repeated

on the rebuilt internal bay, with some improvements to the test arrangement.

Insulation was applied to the roof so that the temperature of the rafter could be

maintained for longer, and the entire rafter section could be heated evenly by the fire.

It had been shown in the first test that the use of liquid heptane as fuel was efficient in

heating, but that the heating time was limited by the amount of fuel, so a controllable

fuel supply system, which could provide sufficient fuel for burning until the collapse of

the frame, was devised. For convenience in the numerical modelling, the uncertainty


17

of the base rotational stiffness was minimized by artificially creating pinned bases for

the columns.

In the test the fire lasted for approximately 500 seconds, and the average gas

temperature was around 900°C. The column sections were not exposed to much

radiation, and remained relatively cool. According to the test data, the apex was seen

to start to deflect downward after about 280 seconds, and then underwent a vertical

dive as both raters were seen to collapse rapidly. This phenomenon is believed to be

a snap-through failure of the rafters. Both eavess were initially deflected outward due

to the thermal expansion of the rafters, until the snap-through took place, and they

were then pulled inward by the catenary force generated in the rafters which was in

highly curvature. The different horizontal displacements at the eavess indicated that

the runaway deflection of the portal frame was not induced by a ‘beam mechanism’

but more likely by a ‘combined mechanism’. The post-test inspection showed that

three plastic hinges (marked in Figure 1.15), generated near each eaves and to the

right of the apex could have led to the snap-though collapse observed in the test.

Figure 1.15 The deformed portal frame after the third fire test

Plastic Hinges


18

1.5.2. BEHAVIOUR OF SINGLE-STOREY PORTAL FRAMES IN FIRE

In industrial warehouses, the fire-wall is normally made of masonry or concrete, and

can either be built embedded between the flanges of the steel column, or stand as a

cantilever wall. If the fire-wall is supported laterally by the steel columns of the portal

frames, its stability would mainly be based on the amount of lateral movement of the

column in fire.

As early as 1980, the behaviour of steel portal frames in accidental fires was

described as the result of a study (CONSTRADO, 1979) of fires in a number of portal

frames in the UK. At the early stage of a fire, the frame begins to heat up and

expand, so the apex deforms upward while the eavess deform outward.

Subsequently, due to the moments caused by the increasing thermal expansion of

the rafters, plastic hinges tend to form at the maximum moment positions on the

rafters, which are at the ends of the haunches and near to the apex. Because these

hinges are generated due to the degradation of steel at elevated temperature, they

are described as “fire hinges”. The formation of two or three fire hinges completes the

mechanism in the rafter which leads to the collapse of the pitched roof. When the

rafters collapse to near eavess level and the pitched roof deforms to a roughly level

shape, as shown in Figure 1.16, the rafter is still laterally restrained by the purlins,

and is capable of supporting itself, and the columns are still upright. When the rafter

falls below eavess level, most of its stiffness is lost and it tends to deform to a

catenary, and the pull-in of the tops of the columns becomes significant. It is found

that columns with fixed bases stand when the rafters are collapsing, while in other

cases they can be pulled inward by the collapse of the rafters when the bases are

pinned or partially fixed. Therefore, the design of the bases becomes important when

the rafters collapse below eavess level.

A typical variation of the overturning moment at the column base with time, after a

fire is ignited in a pitched portal frame is described by CONSTRADO (1980) as


19

shown in Figure 1.16. At an early stage, the thermal expansion of the heated roof

pushes the columns outward, so that the base rotates outwards. The moment at the

base then reduces to zero when the rafter deforms to roughly eavess level, with the

columns standing upright. When the rafter collapses below eavess level, the

catenary force developed in the rafters becomes dominant which pulls the columns

inwards, so that the bases tend to rotate in the opposite direction and the moment at

the base increases again. It is believed that the stability of portal frames in fire is

mainly determined by the resistance provided the column base connections. In other

words, inclination of the column should be stopped provided that the moment

resistance at the column base is bigger than the maximum overturning moment

generated during the collapse of the rafters.

OTM

Time

Maximum

outward

inclination

No

inclination

Inward

inclination

OTM

Time

OTM

Time

Maximum

outward

inclination

Maximum

outward

inclination

No

inclination

No

inclination

Inward

inclination

Inward

inclination

Figure 1.16 Variation of overturning moment with the deformation of the portal frame

in fire.

In order to compare the behaviour of single-storey industrial frames against the

Building Code of Australia (BCA, 1990), two-dimensional numerical modelling of


20

portal frames under seven critical heating profiles (see Figure 1.17) in a developing

fire situation was performed by O’Meagher et al. (1992) using ABAQUS. Pitched

portal frames with and without haunches were modelled two-dimensionally, and

nominal rotational stiffness was applied at the column bases. The effects of column

fire protection, different heated regions of the rafters, and the lateral restraint

provided by adjacent cooler parts of the roof were considered. The results showed

that single-span pitched portal frames, tested with nominal moment resistance at the

column bases, always collapsed inward so that they would not injure people outside

the building, or damage adjacent buildings, as required by the Building Code of

Australia.

L

L/4

L

L/2

Situation A Situation B

L

L/4

L

L/4

L

L/2

Situation A Situation B

L

L/4

L

L/2

Situation C Situation D

L

L/4

L

L/2

Situation C Situation D

L

3L/4

L

Situation E Situation F

L

3L/4

L

Situation E Situation F

L

Situation G

Temperature = T

Temperature = 3T/4

CoolL

Situation G

Temperature = T

Temperature = 3T/4

Cool

Temperature = TTemperature = T

Temperature = 3T/4Temperature = 3T/4

Cool

Figure 1.17 Heating situation considered in O’Meagher et al. (1992)’s model.


21

The effects of the load ratio, span and column height, heating profiles, horizontal load

and rotational stiffness at the bases were investigated by Wong (2001) using Vulcan

(Huang, et al. 2003a, 2003b & 2004). Portal frames with different spans, column

heights and different loadings were modelled two-dimensionally. The results

presented in Figure 1.18 show that the load ratio has the greatest effect on the failure

temperature of the frame. Frames heated by different heating profiles, including both

symmetrical and localised fire scenarios, initially collapsed in a combined

mechanism, and the heated column showed no significant influence on the limiting

temperatures. The semi-rigid base represented by a rotational spring element, which

gives a certain amount of rotational stiffness, showed a beneficial effect on the

collapse temperature of the rafters compared with the pinned base as shown in

Figure 1.19.

400

450

500

550

600

650

700

750

0 2 4 6 8 10 12

Span/Height Ratio

Failure Temperature (°C)

Load Ratio =0.2

Load Ratio =0.4

Load Ratio =0.7

Load Ratio =0.5

0.2 span 300.2 ht 70.2 ht 40.2 span 500.4 ht 40.4 span 500.5 span 300.5 ht 70.5 ht 40.5 span 500.7 span 300.7 ht 70.7 ht 40.7 span 50

Load Ratio Dim (m)

400

450

500

550

600

650

700

750

0 2 4 6 8 10 12

Span/Height Ratio

Failure Temperature (°C)

Load Ratio =0.2

Load Ratio =0.4

Load Ratio =0.7

Load Ratio =0.5

0.2 span 300.2 ht 70.2 ht 40.2 span 500.4 ht 40.4 span 500.5 span 300.5 ht 70.5 ht 40.5 span 500.7 span 300.7 ht 70.7 ht 40.7 span 50

Load Ratio Dim (m)

Figure 1.18 Failure temperatures of portal frames (Wong, 2001) (ht – height of

column in metres).


22

0

200

400

600

800

1000

1200

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Load Ratio

Temperature (°C)

0

5

10

15

20

25Percentage %

Failure Temp for pinned bases (°C)

Failure Temp for Semi-Rigid Bases (°C)Actual Difference in °C

Difference in %

0

200

400

600

800

1000

1200

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Load Ratio

Temperature (°C)

0

5

10

15

20

25Percentage %

Failure Temp for pinned bases (°C)

Failure Temp for Semi-Rigid Bases (°C)Actual Difference in °C

Difference in %

Figure 1.19 Comparison of failure temperatures with pinned and semi-rigid bases

(Wong, 2001)

The comparisons between two- and three-dimensional analyses on a laterally

unrestrained single-span portal frame performed by de Souza Junior et al.(2002)

using SAFIR (2003) shows that the simulation of the structural behaviour of a

single-storey industrial portal frame by two-dimensional modelling in which lateral

instability of the members is not taken into account is not realistic. The collapse

progress of a single portal frame in fire due to out-of-plane deformation was also

observed in three-dimensional analysis by Bong (2005). Because no lateral restraint

was applied, in three-dimensional analysis the fames collapsed at almost the same

load level regardless of whether the column bases were pinned or fixed, as shown in

Figure 1.20.


23

Pinned Support Frame Fixed Support Frame

Time = 16 second

Time = 19 second

Time = 17 second

Time = 20 second

Time = 18 second

Time = 21 second

Figure 1.20 Variation of deflected shape showing the out-of-plane collapse of frames

heated by the ISO834 fire (UDL=1.269kN/m) (Bong, 2005).

Another solution to provide stability to the boundary fire-wall is to build it seperate

from the boundary frame. A clearance between the frames and the fire-wall is

essential in this case, to prevent the damage caused by excessive deformation of the

steel frames in fire. It is observed from a series of numerical simulations on

single-storey steel frames in the USA (Hosam et al., 2004) that the later deformation

of the frame could be caused either by the expansion of the steel roof, or by outward

buckling of the columns if a localized fire was ignited close to the column nearest to

the fire wall. If the fire is assumed to occupy both spans in a double-span frame, the

collapse of the frame is likely to be caused by inward collapse of the heated column


24

which is pulled in by the catenary force generated in the beam. The minimum

clearance between frames and fire-walls is mainly determined by the spatial extent of

the fire, and it can be reduced if stiffer or shorter columns, which provide more

restraint to the frame, are adopted. The simulations also found that higher loads on

the roof could reduce both the time to collapse and the maximum lateral expansion of

the frames, which means that the response of the frame in fire will depend on the

load remaining on the roof.

1.5.3. OVERALL BEHAVIOUR OF INDUSTRIAL FRAMES IN FIRE

Industrial sheds are normally formed by several parallel frames. They are connected

in the out-of-plane direction by secondary members such as purlins, and the sheeting

anchored on the purlin finishes, the roof and part of the external wall. The purlins are

relatively slender, so they are heated quickly and fail earlier in a real fire. As

mentioned in Section 1.2, fires rarely occupy the whole industrial building

immediately, so the frames near to the fire source can be very hot while the rest of

the frames remain relatively cold. The collapse of hot frames could be suppressed by

the restraint from secondary members connected to the relatively cold frames, and

therefore the overall behaviour of whole industrial buildings may be different from

that of individual frames.

It is postulated by O’Meagher et al. (1992) that the collapsed frame could act as

“anchors” to the rest of the building as shown in Figure 1.21, provided that the force

developed in the purlins is very small and that they have sufficient capacity at high

temperatures. Therefore, the nearby frames will deform in a similar mode as the

collapsed frame, so the whole building will collapse in an inward mode which is

acceptable under the Building Code of Australia (AUBRCC, 1990).


25

Figure 1.21 Overall building behaviour (O’Meagher et al., 1992).

It is observed in the three-dimensional modelling by de Souza Junior et al. (2002)

that, although the stability of the portal frame was not endangered by the buckling of

the purlins, after this failure the purlins tend to deform in a catenary shape which may

contribute to the lateral stability of the portal frame at a later stage, as shown in

Figure 1.22.

Figure 1.22 Deflected shape for the model with purlins at the moment of failure,

with an amplification factor of 10 ( de Souza Junior et al., 2002).

Wong (2001) recognized that the analysis performed by the preliminary static Vulcan

model including portal frames and purlins, could not easily be extended to high

temperatures because of the buckling failure of purlins at an early stage. This

problem was overcome by adding imaginary members (see Figure 1.23) that connect

the purlins together to simulate the effect of the restraint from the roof cladding.

These imaginary members were not heated in the analysis, which implies that the


26

cladding is infinitely strong and will not detach in the fire condition. This assumption

makes the roof stronger than in reality. It was found from the analysis that the purlins

lose stability very quickly in comparison with the portal frame when the entire roof

was heated, so the two- and three-dimensional analyses show very similar results

when different load ratios were tested (Figure 1.24).

Portal frames

Purlins

Imaginary membersHeated members

Unheated members

Portal frames

Purlins

Imaginary membersHeated members

Unheated members

Figure 1.23 Deflected shape of the entire-roof-heated case at 660°C steel

temperature. (Wong, 2001).

-1400

-1200

-1000

-800

-600

-400

-200

0

200

0 100 200 300 400 500 600 700 800

Temperature (°C)

Vertical Displacement (mm)

Load Ratio 0.2 - 2D Analysis




-1400

-1200

-1000

-800

-600

-400

-200

0

200

0 100 200 300 400 500 600 700 800

Temperature (°C)










Figure 1.24 Vertical displacement at apex. 2D vs 3D analysis (Wong, 2001).


27

The effect of the different fire senarios shown in Figure 1.25 as a three-dimensional

model was tested by Wong (2001). The tests showed that the major deformation

always occurs to the rafters, especially with low load ratios and long-span portal

frames. The comparison of vertical deflections at the apex for frames under different

fire senarios shows that the heating of columns has little effect on the overall

behaviour of the portal frame, whether the purlins are heated fully or partially.


Steel Temperature (°C)

(a)

(b)

(c)

(d)

Entire roof

heated


Steel Temperature (°C)

(a)

(b)

(c)

(d)

Entire roof

heated

Figure 1.25 Vertical displacements at apex of the central frame (Wong, 2001)

A comprehensive study (Bong, 2005) on the overall behaviour of a full-sized

industrial frame shown in Figure 1.26, was carried out at the University of Canterbury

in 2005. In this study, a three-dimensional numerical model was set up using the

advanced finite element analysis software SAFIR (2003). The effects of different

support conditions at the column bases, the out-of-plane restraint to columns and the

axial restraint of purlins were investigated. Three heating profiles which simulate fully

developed and localized fires near the centre or the end of the building were tested

on frames with two extreme base support conditions: fixed and pinned. A recovery of

the frame’s stability could be observed if the axial restraint from the purlins was


28

applied when the frame was heated by the ISO standard fire (ISO, 1975) up to 1

hour. This was also observed in pin-supported frames with restraint from purlins

heated by the External Fire curve (BSI, 1999), in which a well-ventilated fire is

simulated and the maximum temperature is taken as 660°C. It was concluded from

an inspection of the deformed shape that the degree of the axial restraint from purlins

is much less important to the failure mode than the flexural fixity of the bases of the

columns.

Figure 1.26 The analytical model in Bong’s study (Bong, 2005).

Time=14.14 minutes Time=19.6 minutes

Figure 1.27 Deflected shapes

immediately before and after the rapid

sagging of roof of the pinned-base case

with purlin axial restraint (Peter et al.,

2006)

Figure 1.28 Sidesway collapse of the

pinned-base case without purlin axial

restraint (Peter et al., 2006)


29

Time =14.92 minutes

Figure 1.29 Inward collapse of the

fixed-base case without purlin axial

restraint (Peter et al., 2006)

Time =15.9 minute

Figure 1.30 Finally deflected shape of the

pinned-base case without purlin axial

restraint and with full concrete

encasement of columns (Peter et al.,

2006)

The passive fire protection achieved by concrete encasement of either the full height

or two-thirds of the height of the column was also simulated by Bong (2002). Frames

with partially fixed bases, which were simulated by halving the stiffness and strength

of the elements adjacent to the fixed bases, were also tested when passive fire

protection to columns was considered. Protection of columns with pinned bases

shows no improvement in structural performance. For frames with partially rigid and

rigid bases, because the strength and stiffness of the concrete-encased parts of the

steel columns are largely unaffected, protected columns remain relatively straight

during the fire.

The structural fire performance of steel portal frame buildings was also studied by

Peter et al. (2006) on the basis of Bong’s work. Four modes of deformation:

Catenary, Sway, Inward and Upright were observed in Bong’s tests on frames in a

fully developed ISO fire (ISO, 1975). When rigid bases and purlin restraint were

assumed for a frame, its deformation tended to be a Catenary mode, which means

that the frame deformed almost symmetrically and the roof structure deformed into a

catenary shape, as shown in Figure 1.27. For a frame with pinned bases, the roof

lost its stiffness due to the thermal degradation of the steel and sagged relatively

quickly, so that the frame could lose stability in a Sway mechanism, as shown in


30

Figure 1.28. When the restraint from the purlins was released, the heated frame

could collapse into the building in an Inward mode of failure as described in Figure

1.29. When columns with pinned bases were protected in fire, they did not collapse

inwards together with the collapse of the rafters, and stood upright after the fire, as

shown in Figure 1.30. It is suggested by Peter et al. (2006) that, in practical design,

some level of rotational restraint should be provided to the frame base connections to

prevent sidesway of frames and outward collapse of walls. Passive fire protection to

the column legs was also recommended, to ensure the stability of the columns and

walls in fire.

The simple assumption for partial fixed bases made by Bong (2002) can produce

stiffness and strength between the fully rigid and pinned conditions for the base

connections. However it could introduce extra horizontal movement to the column

when the column element adjacent to the base begins to plastically yield, which is

different from the real behaviour of the frame. It could also influence the overall

behaviour of the frame if the artificial base element is not short enough. The required

design capacity of the column base can not be indicated by the model with this

simple assumption. Although the cooling phase of the fire was considered in this

study, a fully developed fire was assumed, which means that all the frames in the

building begin to cool down at the same time. It is very possible in a real fire that,

when the temperature of some frames begins to decrease the other frames are still

hot. The interaction between the cool and hot frames can be extreme, and so some

localized fire including cooling phase, should be considered in a three-dimensional

analysis of industrial portal frames.

The structural behaviour of multi-span portal frames was analysed by Vassart et al.

(2005). The collapse of a frame was divided into two successive phases, according

to the inclination of columns. In the first phase, the lateral displacements at the tops

of columns increases progressively due to the thermal expansion of heated

members, so that the cold frames are pushed outwards, as shown in Figure 3.30.


31

The second phase is defined by collapse of the heated roof. Tensile forces then

become dominant in the heated members, so the unheated frames are pulled inward

in this stage. A simple method was developed for frames with more than one span, to

estimate the outward deflection at the tops of columns in the expansion phase.

Figure 1.31 Lateral displacements at the tops of columns in Phase 1 (Vassart et al.

2005).

Figure 1.32 Lateral displacements at the tops of columns in Phase 2. (Vassart et al.

2005).

A dynamic analysis, performed by ABAQUS (2004), was adopted to model the failure

progression of the multi-span portal frame. Semi-rigid bases were simulated by

spring elements defined using a bi-linear M - curve. Because this study concerns

the French fire safety requirements for industrial buildings, which specify no

progressive collapse between different fire compartments and no collapse outwards,

an inward collapse of the frame was not important. Inward collapse normally leads to

a much higher inclination of the columns comparing to the outward inclination in the

initial expansion stage, which may be more harmful to the stability of a boundary wall,

and so the results from this study are not very helpful for portal frames designed in

the UK.


32

1.6. NUMERICAL MODELLING

Structural behaviour in fire involves highly geometric and material nonlinearity which

can not be represented by a closed form solution in most cases. In order to solve this

problem the finite element method and advanced solution procedures have been

applied to simulate the behaviour of structures in fire. Compared to laboratory tests,

numerical analysis has a significant advantage in cost efficiency, and is capable of

providing results with acceptable accuracy.

1.6.1. VULCAN

Many finite element analysis programs, such as ABAQUS (2004), ANSYS (1992),

have been developed for the numerical modelling of structural behaviours under

different circumstances. They are normally designed to provide a comprehensive tool

for modelling of structures at ambient temperature. It may sometimes be difficult to

use them for structural analysis under fire conditions. In recent years a specialised

finite element analysis software Vulcan has been developed at the University of

Sheffield for three-dimensional modelling of steel, composite and reinforced concrete

buildings under fire conditions. In this program buildings are modelled as an

assembly of a finite beam-column, slab and connection elements, and both material

and geometric non-linearities are considered. Currently, a Newton-Raphson iterative

method is adopted to solve for the incremental displacements at each node, on the

basis of static force equilibrium. This program has been well validated against fire

tests (Huang et al. 2003b). The academic version of Vulcan, coded in FORTRAN,

has mainly been developed for research purposes, which allows incorporation of new

developments, such as new elements or solvers, into the original program.


33

1.7. SCOPE OF THE RESEARCH AND LAYOUT OF THE THESIS

As mentioned above, in the last decade several studies have been carried out to

investigate the response of steel portal frames to fire conditions. However, for

numerical modelling, most of them are based on static analysis. Hence, the

behaviour of the pitched-roof portal frame is not represented fully, especially when

temporary instability is encountered during the collapse of the frame. Of course a

fully dynamic procedure can be employed to deal with the instability encountered in

the analysis, but the computational efficiency is very low under such conditions

because the analysis is mainly of stable equilibrium, apart from the period of the

dynamic snap-through.

A fundamental feature of the collapse of pitched-roof portal frames in fire is the loss

of stability at the point where the snap-through mechanism happens, which may

re-stabilise when the roof is inverted. Hence, one of the main objectives of this

research is to investagate the behaviour of the portal frame after re-stabilisation, and

to identify the eventual failure point. Hence, the first step of the research is to develop

a robust quasi-static procedure to effectively model the snap-through mechanism

of portal frames under fire conditions. The new procedure is validated using both test

data and fully dynamic application of other software, such as ABAQUS (2004) and

ANSYS (1992).

In previous researches numerical modelling based on the fully dynamic procedure

has been used to trace the behaviour of portal frames in fire up to very big

deformations. However, the failure mechanisms of portal frames under different

conditions have not been fully investigated. The current design guide used in the UK

is based on several arbitrary assumptions, and it is still doubtful whether designs are

reasonable and conservative. Hence, the newly developed procedure in Stage 1 is

used to conduct a series of numerical studies on typical industrial frames under


34

different conditions. The research has revealed the common failure mechanism of

the steel portal frames at elevated temperatures.

As required by the current design method, specification of a strong moment-resisting

column base is the only way to guarantee the fire safety of a portal frame in a fire

boundary condition. However, this solution is complex and expensive from the

practical point of view. Therefore, based on the data generated at Stage 2 of the

research, a new simplified design method is proposed for fire safety design of portal

frames in boundary conditions.

An overview of the remaining chapters of this thesis is given below:

Chapter 2: The detailed theory and formulation of the robust quasi-static procedure

are presented.

Chapter 3: A detailed validation of the newly-developed procedure is provided.

Chapter 4: A series of numerical studies on typical industrial portal frames are

conducted.

Chapter 5: The current design method is validated by the results from the

numerical tests. A new design method is developed.

Chapter 6: The main findings of this research are reiterated and the scope for

possible future work on this topic is discussed.

Chapter 2 New Quasi-static Procedure

35

2. NEW QUASI-STATIC PROCEDURE

2.1. INTRODUCTION

At present, most numerical analysis codes developed for modelling the behaviour of

structures subject to fire are based on static formulations, which assume that the

variation of loading and temperature does not introduce inertial effects to the

structure and that the loading process is infinitely slow. The current static procedures

with the geometrical and material nonlinearity have been able to capture the major

aspects of the behaviour of steel and composite structures under fire conditions,

such as catenary action of beams and membrane action of composite slabs.

In static analysis the external forces at nodes are balanced by the internal forces to

achieve equilibrium, which can be expressed as

tt ExtInt FF (2.1)

where tIntF is the internal nodal force vector, and tExtF is the external nodal

force vector. In geometrically and materially nonlinear problems, the stiffness of the

structure is not constant but changes with the variation of the strain in the elements,

so Newton-Raphson iterative solvers are commonly used. This procedure can be

represented mathmatically as Equation (2.2) for a load-controlled analysis.

1,,1

,

,1,

nktt

Int

kttk

Int

nttInt

ktt

Int

ktt UKFUU

FFF (2.2)

where Int

ktt ,F is the unbalanced force at the k th iteration of step tt . Equilibrium

between external and internal forces at time tt is achieved when the unbalance

force ktt ,g has a very small value, say StaticTOL , after the k th iteration. That is,

Static

Int

ktt

Ext

ttktt TOL ,, FFg (2.3)


36

The incremental displacement U from time t to time tt can be calculated

by summing the incremental displacements from each iteration, as shown in Figure

2.1. That is,

kUUUUU 321 (2.4)

Figure 2.1 Illustration of Newton-Raphson iterative procedure.

Figure 2.2 Illustration of the ill-conditioning of stiffness matrix in static analysis

δU1 δU2

tU ttU

Load

Displacement

Ext

ttF

Int

t

Ext

t FF

Ext

tt 0,g

δU3

Divergence

Negative stiffness

Limiting point

δU1 δU2

tU ttU

Load

Displacement

Int

ttF 2,

Ext

ttF

Int

ttF 1,

Int

t

Ext

t FF

Ext

tt 0,g

Ext

tt 1,g

Ext

tt 2,g

δU3 δU4 … δUn


37

This iterative solver only converges for incremental displacements on a continuously

ascending part of the load-displacement curve, as shown in Figure 2.1. However,

when the stiffness of the structure becomes negative, as shown in Figure 2.2,

equilibrium cannot be achieved in the opposite direction, so the analysis becomes

divergent after the limit point.

In previous studies on industrial frames under fire conditions, the numerical failure

caused by ill-conditioning of the stiffness matrix has been observed by O’Meagher et

al (1992), Wong (2001) and De Souza Junior (2002). Unless special solution

processes, such as displacement control, are used the analysis cannot be continued

beyond any point in the fire at which the structural stiffness matrix is not

positive-definite. The appearance of negative stiffness in a static analysis may be

caused by either true structural failure, or numerical failure. This can easily happen to

pitched portal frames in fire, because of the snap-through behaviour of the frame due

to load on the roof and the degradation of steel at elevated temperatures.

P

a

b

c

yh

S

PP

a

b

c

yh

S

Figure 2.3 Illustration of the mechanism of snap-through for portal frame.

The mechanism of snap-through which happens to a pitched portal frame roof can be

idealised by the two-strut model shown in Figure 2.3. The theory has been explained

by Crisfield (1991) using a bar and spring model. When a downward force acts on

the apex and pinned bases are assumed, the deformation of the struts follows the

sequence from shape a to shape b, and then from shape b to shape c. The internal


38

force in the vertical direction, VN , at the apex can be simply calculated by Equation

(2.5) if the shallow roof pitch assumption is adopted, so that the change of the length

of the truss can be ignored in the calculations.

L

yhNNNV sin (2.5)

where is the modified pitch as shown in Figure 2.3, and N is the axial force

developed in the truss element which can be calculated using Equation (2.6)

EAN (2.6)

The strain can be estimated by Equation (2.7).

2

222

3

2

3

222

3

2

1

S

y

S

hy

hS

SyhhS (2.7)

where 3S is the strut length when the both struts deform to the horizontal positions,

and for the shallow roof assumption 3S can be represented by the original strut

length S , h represents the apex height, and y is the displacement of the apex

from the original position as shown in Figure 2.3. The relationship between the

vertical reaction VN and the apex deflection y can be obtained by subs

analysis of industrial steel portal frames under fire conditions · 2014. 11. 27. · analysis of...

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