flexural behaviour and design of the new litesteel beams · flexural behaviour and design of the...

Flexural Behaviour and Design of the New

LiteSteel Beams

By

Cyrilus Winatama Kurniawan

A thesis submitted to the School of Urban Development

Queensland University of Technology in partial fulfilment

of requirements for the degree of

Master of Engineering,

December 2007

Cyrilus Winatama Kurniawan i

Keywords

Lateral distortional buckling, lateral torsional buckling, LiteSteel beam, moment

distribution effect, moment gradient, load height effect, finite element analysis,

elastic buckling analysis, non-linear static analysis, flexural behaviour, and structural

stability.

Cyrilus Winatama Kurniawan ii

Abstract

The flexural capacity of the new hollow flange steel section known as LiteSteel

beam (LSB) is limited by lateral distortional buckling for intermediate spans, which

is characterised by simultaneous lateral deflection, twist and web distortion. Recent

research based on finite element analysis and testing has developed design rules for

the member capacity of LiteSteel beams subject to this unique lateral distortional

buckling. These design rules are limited to a uniform bending moment distribution.

However, uniform bending moment conditions rarely exist in practice despite being

considered as the worst case due to uniform yielding across the span. Loading

position or load height is also known to have significant effects on the lateral

buckling strength of beams. Therefore it is important to include the effects of these

loading conditions in the assessment of LSB member capacities.

Many steel design codes have adopted equivalent uniform moment distribution and

load height factors for this purpose. But they were derived mostly based on data for

conventional hot-rolled, doubly symmetric I-beams subject to lateral torsional

buckling. In contrast LSBs are made of high strength steel and have a unique cross-

section with specific residual stresses and geometrical imperfections along with a

unique lateral distortional buckling mode. The moment distribution and load height

effects for LSBs, and the suitability of the current steel design code methods to

accommodate these effects for LSBs are not yet known. The research study

presented in this thesis was therefore undertaken to investigate the effects of non-

uniform moment distribution and load height on the lateral buckling strength of

simply supported and cantilever LSBs.

Finite element analyses of LSBs subject to lateral buckling formed the main

component of this study. As the first step the original finite element model used to

develop the current LSB design rules for uniform moment was improved to eliminate

some of the modelling inaccuracies. The modified finite element model was

validated using the elastic buckling analysis results from well established finite strip

analysis programs. It was used to review the current LSB design curve for uniform

moment distribution, based on which appropriate recommendations were made.

Cyrilus Winatama Kurniawan iii

The modified finite element model was further modified to simulate various loading

and support configurations and used to investigate the effects of many commonly

used moment distributions and load height for both simply supported and cantilever

LSBs. The results were compared with the predictions based on the current steel

code design rules. Based on these comparisons, appropriate recommendations were

made on the suitability of the current steel code design methods. New design

recommendations were made for LSBs subjected to non-uniform moment

distributions and varying load positions. A number of LSB experiments was also

undertaken to confirm the results of finite element analysis study.

In summary the research reported in this thesis has developed an improved finite

element model that can be used to investigate the buckling behaviour of LSBs for the

purpose of developing design rules. It has increased the understanding and

knowledge of simply supported and cantilever LSBs subject to non-uniform moment

distributions and load height effects. Finally it has proposed suitable design rules for

LSBs in the form of equations and factors within the current steel code design

provisions. All of these advances have thus further enhanced the economical and

safe design of LSBs.

Cyrilus Winatama Kurniawan iv

Table of Content

1. Introduction 1-1

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

1.2. Research Problem .......................................................................................... 1-3

1.3. Research Objectives....................................................................................... 1-4

1.4. Research Methodology .................................................................................. 1-4

1.5. Thesis Outline ................................................................................................ 1-5

2. Literature Review 1-1

2.1. Cold-formed Steel Sections ........................................................................... 2-1

2.2. Material Properties of Cold-formed Steel Sections ....................................... 2-2

2.3. Flexural Behaviour of Cold-formed Steel Beams.......................................... 2-4

2.3.1. Local Buckling...................................................................................... 2-4

2.3.2. Distortional Buckling............................................................................ 2-5

2.3.3. Lateral Torsional Buckling ................................................................... 2-5

2.4. Beams Subjected to Torsion .......................................................................... 2-6

2.5. Elastic Lateral Torsional Buckling Moments ................................................ 2-8

2.5.1. Effects of Bending Moment Distribution ........................................... 2-10

2.5.2. Effects of Load Height........................................................................ 2-12

2.6. Strength of Real Beams ............................................................................... 2-14

2.7. Equivalent Uniform Moment and Load Height Factors in the Current Steel Design Standards ................................................................................ 2-16

2.7.1. Australian Standards ........................................................................... 2-17

2.7.2. American Standards ............................................................................ 2-20

2.7.3. Bristish Standards ............................................................................... 2-21

2.7.4. Limitation of Current Equivalent Uniform Moment and Load Height Factors..................................................................................... 2-22

2.7.5. Moment Distribution and Load Height Effects on Hollow Flange Beams.................................................................................................. 2-24

2.8. LiteSteel Beam Research ............................................................................. 2-26

2.8.1. Material Properties of LiteSteel Beams .............................................. 2-27

2.8.2. Flexural Behaviour of LiteSteel Beam ............................................... 2-29

2.8.3. Flexural Capacity and Design of LiteSteel Beam Sections ................ 2-31

2.8.4. Finite Element Analysis and Model for LiteSteel Beam .................... 2-33

Cyrilus Winatama Kurniawan v

2.9. Lateral Buckling Experimental Methods ..................................................... 2-35

3. Finite Element Modelling 3-1

3.1. General ........................................................................................................... 3-1

3.2. Mahaarachchi and Mahendran’s (2005c) Finite Element Models of LSB Flexural Members .................................................................................. 3-1

3.3. Validation of Mahaarachchi and Mahendran’s (2005c) Ideal FE Model ...... 3-3

3.4. Modified Ideal Finite Element Model of LSB Flexural Members................. 3-7

3.4.1. Discretization of the Finite Element Mesh............................................ 3-7

3.4.2. Material Model and Properties.............................................................. 3-8

3.4.3. Idealised Load and Boundary Conditions ............................................. 3-9

3.4.4. Geometric Imperfections..................................................................... 3-14

3.4.5. Residual Stresses................................................................................. 3-16

3.4.6. Analysis Methods................................................................................ 3-19

3.5. Typical Buckling Modes of the Modified Ideal FE Model.......................... 3-20

3.6. Validation of the Modified Ideal FE Model................................................. 3-21

3.7. Modifications for Various Loading and Support Boundary Conditions...... 3-25

3.7.1. Ideal Simply Supported LSB Model with a Moment Gradient .......... 3-26

3.7.2. Ideal Simply Supported LSB Model with a Mid-span Point Load (PL) ..................................................................................................... 3-27

3.7.3. Ideal Simply Supported LSB Model with a Uniformly Distributed Load (UDL)...................................................................... 3-30

3.7.4. Ideal Simply Supported LSB Model with Quarter Point Loads (QL)..................................................................................................... 3-31

3.7.5. Cantilever LSB Model with a Point Load (PL) at the Free End ......... 3-31

3.7.6. Cantilever LSB Model with a Uniformly Distributed Load (UDL).................................................................................................. 3-33

3.7.7. Experimental Finite Element Model used in This Research............... 3-33

4. Review of the Current Design Rules of LSBs 4-1

4.1. General ........................................................................................................... 4-1

4.2. Elastic Buckling Analysis Results using the Modified Ideal FE Model (First Version) ................................................................................................ 4-1

4.3. Non-linear Static Analysis Results using the Modified Ideal FE Model (First Version) ................................................................................................ 4-3

4.4. Review of Mahaarachchi and Mahendran’s Nonlinear Static Analysis Results ............................................................................................................ 4-5

Cyrilus Winatama Kurniawan vi

4.5. Comparison of the Non-linear Static Analysis Results using the Final Version of the Modified Ideal FE Model with the Current LSB’s Design Curve ................................................................................................. 4-7

5. Effect of Non-Uniform Bending Moment Distributions on the

Lateral Buckling Strength of LSBs 5-1

5.1. General........................................................................................................... 5-1

5.2. Simply Supported LSBs with a Moment Gradient ........................................ 5-1

5.2.1. Elastic Buckling Analysis Results and Discussions of Moment Gradient Cases ...................................................................................... 5-3

5.2.2. Non-linear Static Analysis Results and Discussions of Moment Gradient Cases .................................................................................... 5-10

5.3. Simply Supported LSBs with Transverse Loads of a Uniformly Distributed Load (UDL) and a Mid-span Point Load (PL).......................... 5-18

5.3.1. Elastic Buckling Analysis Results and Discussions of UDL and PL Cases.............................................................................................. 5-19

5.3.2. Non-linear Static Analysis Results and Discussions of UDL and PL Cases.............................................................................................. 5-25

5.4. Simply Supported LSBs with Quarter Point Loads (QL) ............................ 5-31

5.4.1. Elastic Buckling Analysis Results and Discussions of QL Cases ...... 5-31

5.4.2. Non-linear Static Analysis Results and Discussions of QL Cases ..... 5-34

5.5. Cantilever LSBs with a Uniformly Distributed Load (UDL) and a Point Load (PL) at the Free End .................................................................. 5-35

5.5.1. Elastic Buckling Analysis Results and Discussions of Cantilever LSBs (UDL and PL) ........................................................................... 5-36

5.5.2. Non-linear Static Analysis Results and Discussions of Cantilever LSBs.................................................................................................... 5-41

5.6. Design Recommendation Summary ............................................................ 5-43

6. Experimental Investigation of LSBs with Moment Gradient 6-1

6.1. General........................................................................................................... 6-1

6.2. Experimental Method..................................................................................... 6-1

6.3. Test Specimens .............................................................................................. 6-2

6.4. Test Set-up ..................................................................................................... 6-3

6.5. Support System .............................................................................................. 6-4

6.6. Loading System ............................................................................................. 6-5

6.7. Measurement System..................................................................................... 6-6

6.8. Measurement System..................................................................................... 6-8

6.9. Results and Discussions................................................................................. 6-9

Cyrilus Winatama Kurniawan vii

7. Effect of Load Height on the Lateral Buckling Strength of

LSBs 7-1

7.1. General ........................................................................................................... 7-1

7.2. Load Height Effects for Simply Supported LSBs.......................................... 7-2

7.2.1. Elastic Buckling Analysis Results ........................................................ 7-2

7.2.2. Discussions on the Load Height Effects for Simply Supported LSBs...................................................................................................... 7-3

7.3. Load Height Effects for Cantilever LSBs .................................................... 7-10

7.3.1. Elastic Buckling Analysis Results ...................................................... 7-10

7.3.2. Discussions on the Load Height Effects for Cantilever LSBs ............ 7-11

7.4. Design Recommendation Summary............................................................. 7-17

8. Conclusions and Recommendations 8-1

9. References 9-1

Appendices

Appendix A (Section Properties, Capacities and Tolerances)

Appendix B (Elastic Buckling Moments)

Appendix C (Residual Stresses Subroutine and Example Calculations)

Appendix D (Test Results)

Cyrilus Winatama Kurniawan viii

List of Figures

Figure 1.1 – LiteSteel Beam Section (Kurniawan, 2005).........................................1-1

Figure 1.2 – Lateral Buckling Modes of Beams .......................................................1-2

Figure 2.1 – Common Cold-formed Steel Shapes (Yu, 2000)..................................2-1

Figure 2.2 – Common Cold-formed Steel Shapes (Yu, 2000)..................................2-2

Figure 2.3 – Effects on strain hardening and strain ageing (Yu, 2000) ....................2-3

Figure 2.4 – Definition of Flexural and Membrane Residual Stresses (Schafer and

Pekoz, 1998) ........................................................................................2-4

Figure 2.5 – Beams Subjected to Local Buckling (Rogers et al., 1997)...................2-4

Figure 2.6 – Beams Subjected to Distortional Buckling (Rogers et al., 1997).........2-5

Figure 2.7 – Beams Subjected to Lateral Torsional Buckling (Hancock, 2002) ......2-6

Figure 2.8 – Shear Stress Distribution due to Torsions (Yuan, 2004)......................2-7

Figure 2.9 – Warping of I-sections (Kirby and Nethercot, 1985).............................2-7

Figure 2.10 – Deformations Associated with Lateral Torsional Buckling

(Nethercot, 1992) .................................................................................2-8

Figure 2.11 – Internal and External Moments Relationship (Kirby and Nethercot,

1985) ....................................................................................................2-9

Figure 2.12 – Bending Moment Distribution Examples (Suryoatmono, 2002)......2-11

Figure 2.13 – Doubly-symmetric Beams under Moment Gradient (Trahair,

1993) ..................................................................................................2-12

Figure 2.14 – Effects of Load Height (Trahair, 1993)............................................2-13

Figure 2.15 – Behaviour of Real Beams (Trahair, 1991) .......................................2-14

Figure 2.16 – Typical Design Curve of I-Beams (Trahair, 1991) ..........................2-15

Figure 2.17 – Simply Supported mono-symmetric I-beam with a Uniformly

Distributed Load using 5th Order Polynomial (Ma and Hughes,

1996) ..................................................................................................2-24


Distributed Load Using Cubic Function (Ma and Hughes, 1996) .....2-24

Figure 2.19 – Moment Distribution Effects on Hollow Flange Beams (Pi et al.,

1997) ..................................................................................................2-25

Figure 2.20 – LiteSteel Beam Section ....................................................................2-27

Cyrilus Winatama Kurniawan ix

Figure 2.21 – Typical Stress-Strain Curves from Tensile Tests (Mahaarachchi

and Mahendran, 2005e)..................................................................... 2-28

Figure 2.22 – Residual Stress Distributions in LSB Sections (Mahaarachchi and

Mahendran, 2005e) ........................................................................... 2-28

Figure 2.23 – Lateral Torsional and Lateral Distortional Buckling Modes

(Mahaarachchi and Mahendran, 2005c)............................................ 2-29

Figure 2.24 – Buckling Modes of LiteSteel Beam................................................. 2-30

Figure 2.25 – Schematic Diagrams of the Experimental and Ideal Finite element

Models used by Mahaarachchi and Mahendran (2005c) .................. 2-33

Figure 2.26 – Applied Loads and Boundary Conditions for the Experimental

Model (Mahaarachchi and Mahendran, 2005c) ................................ 2-34

Figure 2.27 – Applied Loads and Boundary Conditions for the Ideal Model

(Mahaarachchi and Mahendran, 2005c)............................................ 2-35

Figure 2.28 – Schematic Diagrams of Overhang Loading Method ....................... 2-35

Figure 2.29 – Quarter Point Loading Method used by Mahaarachchi and

Mahendran (2005a) ........................................................................... 2-36

Figure 3.1 - Schematic Diagrams of the Experimental and Ideal FE Models

Developed by Mahaarachchi and Mahendran (2005c) ....................... 3-1

Figure 3.2 – Typical Finite Element Mesh for LSB Model ..................................... 3-8

Figure 3.3 – “Idealised” Simply Supported Boundary Conditions ........................ 3-10

Figure 3.4 – Idealised Finite Element Models ....................................................... 3-10

Figure 3.5 – Load and Boundary Condition Modelling used in the Modified Ideal

Finite Element Model (First Version) ............................................... 3-11

Figure 3.6 – Load and Boundary Condition Modelling used in the Modified Ideal

Finite Element Model (Final Version) .............................................. 3-13

Figure 3.7 – Positive and Negative Initial Overall Geometric Imperfections........ 3-15

Figure 3.8 – Effects of Geometric Imperfection Direction on LSB’s Ultimate

Strength ............................................................................................. 3-16

Figure 3.9 – Residual Stress Distribution Model for LSB Sections (Mahaarachchi

and Mahendran, 2005e)..................................................................... 3-17

Figure 3.10 – Typical Residual Stress Distribution for LSB Sections................... 3-18

Figure 3.11 – Typical Buckling Modes of LSB Flexural Members ...................... 3-21

Figure 3.12 – Comparison of Elastic Buckling Moments vs. Span from Finite

Element Analysis, THINWALL and the Mod Equation for LSBs .... 3-23

Cyrilus Winatama Kurniawan x

Figure 3.13 – Bending Moment vs. Vertical Deflection Curves for 4m

LSB200x60x2.5 LSB (Mahaarachchi and Mahendran, 2005c).........3-25

Figure 3.14 – Loading and Support Types Considered in This Study....................3-26

Figure 3.15 – Transverse Loading Positions Considered in This Study.................3-26

Figure 3.16 – Ideal LSB Model with a Moment Gradient ......................................3-27

Figure 3.17 – Comparison of Lateral Distortional Buckling Mode and Yielding

(von Mises) Distribution at Failure (1st Yield) using the MPC Link

and Adopted Methods ........................................................................3-28

Figure 3.18 – Schematic View of the Adopted Method .........................................3-29

Figure 3.19 – Schematic View of the Adopted Method for Various Levels ..........3-29

Figure 3.20 – Ideal LSB Model with a Mid-span Point Load at the Shear

Centre.................................................................................................3-30

Figure 3.21 – Ideal LSB Model with a Uniformly Distributed Load at the Shear

Centre.................................................................................................3-30

Figure 3.22 – Ideal LSB Model with Quarter Point Loads at the Shear Centre .....3-31

Figure 3.23 – Cantilever LSB Model......................................................................3-32

Figure 3.24 – Cantilever LSB Model with a Point Load at the Free End (through

the Shear Centre)................................................................................3-32

Figure 3.25 – Cantilever LSB Model with a Uniformly Distributed Load (through

the Shear Centre)................................................................................3-33

Figure 3.26 – Applied Loads and Boundary Conditions for the Experimental FE

Model used in This Research.............................................................3-34

Figure 4.1 – Comparison of Elastic Buckling Moments vs. Span Curves from

Finite Element Analysis using the Modified Ideal FE Model (First

Version) and THINWALL...................................................................4-3

Figure 4.2 – Dimensionless Non-linear Finite Element Analysis Results using the

First Version of the Modified Ideal FE Model (Parsons, 2007) ..........4-4

Figure 4.3 – Typical Yielding Distribution (von Mises) at Failure (1st Yield) using

the First Version of the Modified Ideal FE model...............................4-4

Figure 4.4 – Current LSB’s Design Curve and Non-linear Finite Element Analysis

Results Used for its Development (Mahaarachchi and Mahendran,

2005d) ..................................................................................................4-5

Figure 4.5 – Mahaarachchi and Mahendran’s Non-linear Finite Element Analysis

Results Used for Developing the Current Design Curve of LSBs.......4-6

Cyrilus Winatama Kurniawan xi

Figure 4.6 – Comparison of the Non-linear Finite Element Analysis Results using

the Modified Ideal FE Model (Final Version) and the Current LSB

Design Curve....................................................................................... 4-8

Figure 5.1 – Simply Supported LSB Subjected to a Moment Gradient................... 5-2

Figure 5.2 – “Idealised” Simply Supported Boundary Condition............................ 5-2

Figure 5.3 – αm Factors for Simply Supported LSBs Subjected to Moment

Gradient Based on Elastic Buckling Analyses.................................... 5-4

Figure 5.4 – Local Buckling Mode due to Moment Gradient Action...................... 5-5


Gradient Based on Elastic Buckling Analyses (Grouped) .................. 5-6

Figure 5.6 – Typical Lateral Buckling Modes of LSBs with Moment Gradient ..... 5-8

Figure 5.7 – Comparison of αm Factors from FE Elastic Buckling Analyses and

Current Design Equations ................................................................... 5-9

Figure 5.8 – New αm Equation for LSB................................................................. 5-10

Figure 5.9 – Moment Gradient Effects (β= -0.4 & 0) for Simply Supported

LSB250x60x2.0 based on FE Non-linear Analyses.......................... 5-13

Figure 5.10 – Moment Gradient Effects (β = -0.4 & 0) for Simply Supported




Figure 5.12 – Comparison of Strength Ratios (Mult-non / Mult) and αm factors

(Equation 5.3).................................................................................... 5-15

Figure 5.13 – Typical Yielding Distribution (von Mises) at Failure (First Yield)

on the Inside Surface of the Section.................................................. 5-16

Figure 5.14 – Simply Supported LSB Subjected to Transverse Loads (UDL and

PL)..................................................................................................... 5-18

Figure 5.15 – αm Factors for the UDL Case Based on Elastic Buckling

Analyses ............................................................................................ 5-21

Figure 5.16 – αm Factors for the PL Case Based on Elastic Buckling Analyses... 5-21

Figure 5.17 – Other Critical Buckling Modes of LSBs Subjected to Transverse

Loads ................................................................................................. 5-23

Cyrilus Winatama Kurniawan xii

Figure 5.18 – Comparison of Typical Elastic Lateral Buckling Moment versus

Span curves for Transverse Loads (UDL and PL) and Uniform

Moment Cases....................................................................................5-24

Figure 5.19 – Yielding Distribution (von Mises) at Failure (First Yield) of 3m

LSB2506020 with a UDL ..................................................................5-26

Figure 5.20 – Moment Distribution Effects of UDL on Simply Supported

LSB125x45x2.0 based on Non-linear FE Analyses ..........................5-28

Figure 5.21 – Moment Distribution Effects of PL on Simply Supported

LSB250x60x2.0 based on Non-linear FE Analyses ..........................5-29

Figure 5.22 – Typical Failures other than Lateral Buckling Mode for Short to

Intermediate Span LSBs with Transverse loads (UDL and PL) ........5-30

Figure 5.23 – Simply Supported LSB Subjected to Quarter Point Loads (QL) .....5-31

Figure 5.24 – αm Factors for the QL Case Based on Elastic Buckling

Analyses.............................................................................................5-33

Figure 5.25 – Other Critical Buckling Modes (Both Shear and Local Web

Buckling) for 1.5m LSB250x60x2.0 Subjected to QL ......................5-34

Figure 5.26 – Cantilever LSB Subjected to Transverse Loads (UDL & PL) .........5-36

Figure 5.27 – αm Factors for the PL Case Based on Elastic Buckling

Analyses.............................................................................................5-38

Figure 5.28 – αm Factors for the UDL Case Based on Elastic Buckling

Analyses.............................................................................................5-38

Figure 5.29 – Typical Lateral Buckling Modes of Cantilever LSBs ......................5-39

Figure 5.30 – Other Critical Buckling Modes of Cantilever LSBs.........................5-39

Figure 6.1 – Schematic Diagram of the Overhang Loading Method Used in the

Experimental Study..............................................................................6-1

Figure 6.2 – Test Specimens.....................................................................................6-3

Figure 6.3 – Overall View of Test Rig .....................................................................6-4

Figure 6.4 – Support System (Mahaarachchi and Mahendran, 2005a).....................6-5

Figure 6.5 – Loading System....................................................................................6-6

Figure 6.6 – Measurement and Data Acquisition Systems .......................................6-7

Figure 6.7 – Test and FEA Results for 2.5m LSB125x45x2.0 .................................6-9

Figure 6.8 – Test and FEA Results for 3.5m LSB250x60x2.0 ...............................6-10

Cyrilus Winatama Kurniawan xiii

Figure 6.9 – Typical Specimen Deformation at Failure and Associated Yielding

(von Mises) Distribution based on FEA and Test............................. 6-12

Figure 7.1 – Transverse Loading Levels Considered in This Study ........................ 7-1

Figure 7.2 – Load Height Effects (Top Flange and Bottom Flange Levels) for...... 7-3

Simply Supported LSBs Based on Elastic Buckling Analyses........... 7-3

Figure 7.3 – Effect of Loading above the Shear Centre........................................... 7-4

Figure 7.4 – Load Height Effects on the Lateral Distortional Buckling Model

(2.5m LSB250x60x20 Subjected to a UDL)....................................... 7-4

Figure 7.5 – Other Critical Buckling Modes of LSBs Subjected to TF and BF

Loading ............................................................................................... 7-5

Figure 7.6 – Comparison with Current Design Methods in Predicting the Lateral

Buckling Strength of Simply Supported LSBs Subjected to

Load Height Effects (UDL Case)........................................................ 7-6


Buckling Strength of Simply Supported LSBs Subjected to Load

Height Effects (PL Case) .................................................................... 7-7

Figure 7.8 – Load Height Ratios Comparison ......................................................... 7-9

Figure 7.9 – Load Height Effects (Top Flange and Bottom Flange Levels) for

Cantilever LSBs Based on Elastic Buckling Analyses ..................... 7-11

Figure 7.10 – Load Height Effects on the Lateral Distortional Buckling Mode (1 m

Cantilever LSB125x45x20 Subjected to a PL at the free end) ......... 7-12

Figure 7.11 – Other Critical Buckling Modes of Cantilever LSBs with TF

Loading ............................................................................................. 7-13

Figure 7.12 – Comparison with Current Design Methods to Predict Cantilever

LSB’s Lateral Buckling Strength Subjected to Load Height Effect

(PL Case)........................................................................................... 7-13


Buckling Strength of Cantilever LSBs Subjected to Load Height

Effect (PL Case) (Continued)............................................................ 7-14


Buckling Strength of Cantilever LSBs Subjected to Load Height

Effect (UDL Case) ............................................................................ 7-15

Cyrilus Winatama Kurniawan xiv

List of Tables

Table 2.1 – Equivalent Uniform Moment Factors (AS4100 Tables 5.6.1 and

5.6.2) ..................................................................................................2-17


5.6.2) (Continued)..............................................................................2-18

Table 2.2 – Load Height Factors kl (AS4100 Table 5.6.3(2)) ................................2-19

Table 2.3 – Cb Factors for Simply Supported Beams with Uniformly Distributed

Loads within the Span (AS/NZS4600 Table 3.3.3.2) ........................2-19

Table 2.4 – Effective Length for Beams without Intermediate Restraint

(BS5950-1 Table 13)..........................................................................2-21

Table 2.5 – Effective Length for Cantilevers without Intermediate Restraint

(BS5950-1 Table 14)..........................................................................2-22

Table 3.1 – Nominal Dimensions of LSB Sections Used by Mahaarachchi and

Mahendran (2005c)..............................................................................3-2

Table 3.2 – Comparison of Elastic Lateral Buckling Moments from Finite Element

Analysis and THINWALL (Mahaarachchi and Mahendran, 2005c)...3-3

Table 3.3 – Nominal Centreline Dimensions of LSB Sections Used in this

Research...............................................................................................3-5

Table 3.4 – Percentage Difference in Elastic Lateral Buckling Moments obtained

from THINWALL using External and Centreline Dimensions...........3-5

Table 3.5 – Percentage Differences between Elastic Buckling Moments obtained

from Mahaarachchi and Mahendran’s FEA and THINWALL

Analysis in this Research using External Dimensions of LSBs ..........3-6

Table 3.6 – Percentage Difference in Elastic Buckling Moments obtained from

THINWALL using Centreline Dimension with and without Corner

Radii.....................................................................................................3-7

Table 3.7 – Comparison of Elastic Buckling Moments of LSBs from Finite Element

Analysis, THINWALL and the Mod Equation ...................................3-22

Table 4.1 – Comparison of Elastic Lateral Buckling Moments from FEA using

the Modified Ideal FE Model (First Version) and THINWALL .........4-2

Table 5.1 – Elastic Buckling Moments of Simply Supported LSBs Subjected to

Moment Gradient .................................................................................5-3

Cyrilus Winatama Kurniawan xv

Table 5.1 – Elastic Buckling Moments of Simply Supported LSBs Subjected to

Moment Gradient (Continued)............................................................ 5-4

Table 5.2 – Ultimate Moments of Simply Supported LSBs Subjected to Moment

Gradient............................................................................................. 5-12

Table 5.3 – Strength Ratio Comparison of Mult-non / Mult (FEA Results) and

Mb-non / Mb ......................................................................................... 5-17

Table 5.4 – Elastic Lateral Buckling Moments of Simply Supported LSBs

Subjected to a Uniformly Distributed Load (UDL) .......................... 5-19


Subjected to a Mid-span Point Load (PL)......................................... 5-20

Table 5.6 – Effects of Initial Geometric Imperfection Direction on the Ultimate

Moments of LSBs ............................................................................. 5-26

Table 5.7 – Ultimate Moments of Simply Supported LSBs Subjected to a UDL . 5-27

Table 5.8 – Ultimate Moments of Simply Supported LSBs Subjected to a PL..... 5-28


Subjected to Quarter Point Loads (QL) ............................................ 5-32

Table 5.10 – Ultimate Moments of Simply Supported LSBs Subjected to QL ..... 5-35

Table 5.11 – Elastic Lateral Buckling Moments of Cantilever LSBs Subjected to

a Point Load at the Free End (PL)..................................................... 5-37


a Uniformly Distributed Load (UDL)............................................... 5-37

Table 5.13 – Effects of Initial Geometric Imperfection Direction......................... 5-41

Table 5.14 – Ultimate Moments of Cantilever LSBs Subjected to a PL at the

Free End ............................................................................................ 5-42

Table 6.1 – Test Program......................................................................................... 6-2

Table 6.2 – Average Measured Dimensions of LSB Sections used in

Experiments ........................................................................................ 6-3

Table 6.2 – Summary of Test Results ...................................................................... 6-9

Table 6.3 – Moment Distribution Effects (Test Results and Exp FEA Results).... 6-11


Subjected to Top Flange (TF) and Bottom Flange (BF) Loading....... 7-2


Top Flange (TF) and Bottom Flange (BF) Loading.......................... 7-10

Cyrilus Winatama Kurniawan xvi

Notations

E – Young’s modulus of elasticity (MPa)

fy – Yield stress (MPa)

fu – Tensile strength (MPa)

Iw – Warping constant (mm6)

G – Shear modulus (MPa)

GJ – Torsional rigidity (Nmm2)

GJe – Effective torsional rigidity (Nmm4)

GJf – Effective torsional rigidity of flange (Nmm2)

Iw – Warping constant (mm6)

J – Torsional constant (mm2)

K – Torsion parameter

Ke – Modified torsion parameter

kl – Load height factor

L – Length (m)

Le – Effective length (m)

M – Applied moment (kNm)

Mo – Elastic lateral torsional buckling moment (kNm)

Mod – Elastic lateral distortional buckling moment (kNm)

Mod-non – Elastic lateral distortional buckling moment (non-uniform moment) (kNm)

Mcr – Critical moment (kNm)

Myz – Uniform elastic buckling moment (kNm)

Mb – Member moment capacity (kNm)

Ms – Section capacity (kNm)

My – 1st yield moment capacity (kNm)

Mutlt – Ultimate moment (kNm)

Mutlt-non – Ultimate moment for non-uniform moment case (kNm)

P – Applied force (kN)

yQ – Load height

Zf – Full section modulus (mm3)

Ze – Effective section modulus (mm3)

αm – Equivalent uniform moment factor

Cyrilus Winatama Kurniawan xvii

αs – Slenderness reduction factor

β – End moment ratio

Cb – Equivalent uniform moment factor

mLT – Equivalent uniform moment factor

λ – Beam slenderness

λd – Modified beam slenderness

ρ – Density (kg/m3)

ν – Poisson’s ratio

φb – Capacity reduction factor

Cyrilus Winatama Kurniawan xviii

Statement of Original Authorship

The work contained in this thesis has not been previously submitted to meet

requirements for an award at this or any other higher education institution. To the

best of my knowledge and belief, the thesis contains no material previously

published or written by another person except where due reference is made.


Signature: ________________

Date: …..19-12-2007…...

Cyrilus Winatama Kurniawan xix

Acknowledgement

I would like to express my gratitude to my supervisor, Professor Mahen Mahendran

for his invaluable expertise, patient guidance, continuous support and rigorous

discussions throughout the course of this research project. Many thanks to the School

of Urban Development, Queensland University of Technology (QUT) for providing

financial support through the Australian Postgraduate Award (APA) and the Faculty

of Built Environment and Engineering Postgraduate Scholarship, and for providing

testing facilities at the Steel Structure Laboratory. Special thank also to Smorgon

Steel Tube Mills for providing the test materials.

I wish to thank my fellow research students, Sivapathasunderam Jeyaragan,

Tharmarajah Anapayan, Poologanathan Keerthan for their generous assistance

during experiments, Jeremy Parson, Yasintha Bandula Heva, Prakash Kolarkar, and

Nirosha Dolamune Kankanamge for their friendship and support.

Finally, I would like to express my appreciation to my family for their love and

support during the difficult times. Without them the completion of thesis would not

have been possible.

Chapter 1 – Introduction

Cyrilus Winatama Kurniawan 1-1

1. Introduction

1.1. Background

Smorgon Steel Tube Mills has recently developed a new cold-formed, thin-walled

and high strength steel section, known as LiteSteel beam (LSB) which has a unique

shape comprising two rectangular hollow flanges and a slender web as shown in

Figure 1.1. The LSB section is manufactured using a combined process of cold roll-

forming and dual electrical resistance welding. The unique profile avoids many of

the problems commonly associated with other cold-formed sections, while

maintaining a high level of structural efficiency. One of the main advantages of LSB

is that it is 40% lighter than traditional hot-rolled steel members of equivalent

bending strength. LSB can be manually lifted thus places LSB into the same weight

category as structural timber.

Figure 1.1 – LiteSteel Beam Section (Kurniawan, 2005)

However, the unique shape and dual welding process of the new LiteSteel Beam

introduce many structural issues that may not be adequately covered in the current

steel design standards. Recent researches have identified one of the major problems

of LSBs as their lateral distortional buckling performance for intermediate spans.

The presence of two stiff hollow flanges and a slender web is the main reason for

lateral distortional buckling during which a web distortion occurs in addition to the

lateral deflection and twist that occur in the common lateral torsional buckling. This

therefore reduces its buckling resistance to be lower that based on the lateral

torsional buckling, a common buckling mode for medium to longer span steel I-

Two rectangular hollow flanges

Slender web



beams. This unique lateral distortional buckling (LDB) mode of LSB is shown in

Figure 1.2.

(a) Lateral torsional buckling (LTB) (b) Lateral distortional buckling (LDB)

Figure 1.2 – Lateral Buckling Modes of Beams

Since there were inadequate steel design rules that include the effects of LDB,

extensive researches were conducted by Mahaarachchi and Mahendran (2005a-e) at

the Queensland University of Technology to study the flexural stability of LiteSteel

beams. Following experimental and computer modelling studies, appropriate design

rules were developed to conservatively estimate the member capacity of LiteSteel

beams based on lateral distortional buckling effects. This new design rule has been

adopted in the Design Capacity Tables for LiteSteel Beam and in the new cold-

formed steel structures code, AS/NZS 4600: 2005, although no validation has been

reported on its adequacy for other sections that may exhibit lateral distortional

buckling.

Other researches on LiteSteel beams have also been conducted recently. The Centre

for Advanced Structural Engineering at the University of Sydney investigated the

bearing strength of LSB and the results have also been adopted in the Design

Capacity Table for LiteSteel beams. In 2005, Kurniawan (2005), Perren (2005) and

Hateley (2005) investigated the effects of web stiffeners in eliminating lateral

distortional buckling, the buckling strength of back to back LSBs, and the shear

capacity of LSBs, respectively. Further research on LiteSteel beams is still

progressing since various structural aspects of LiteSteel beams have not been fully



understood and current design methods can be either too conservative or un-

conservative for LSBs.

1.2. Research Problem

A uniform bending moment distribution is commonly regarded as the worst

condition due to uniform yielding across the section, but its existence is rarely found

in practice. Thus ignoring the actual bending moment distribution is very

conservative and leads to uneconomical design. Properly addressing the loading

position (load height effect) in the design is also very important as it can

significantly affect the lateral buckling strength. In the current steel design standards

(i.e. Australian, American and British), a simple modification with an equivalent

uniform moment factor (moment modification) is used to accommodate different

loading conditions, while a load height factor is used in the determination of a

modified effective length to allow for the effect of loading positions.

The current LiteSteel beam design rules developed by Mahaarachchi and Mahendran

(2005d) are also limited to uniform bending moment distributions. The equivalent

uniform moment and load height factors adopted in the current design standards are

mostly based on research data for simply supported hot-rolled double symmetric I-

beams subjected to lateral torsional buckling. No research has been conducted to

study the effect of moment distribution and load height on the lateral buckling

strength of LSB. The cold-formed LiteSteel beam has different stress-strain curves,

cross sections, residual stresses, and geometrical imperfections while being subjected

to unique lateral distortional buckling mode. This therefore raises a question about

the applicability the design rules in the current design codes to LSB sections.

Nevertheless, the current Design Capacity Tables for LSBs provide guidance for

load height effects simply by referring to AS4100 (SA, 1998), the Australian hot-

rolled steel design code, despite the lack of evidence on its applicability for LSB

sections. Therefore there is a need to study the adequacy of current equivalent

uniform moment and load height factors in the design of LiteSteel beams and to

recommend new factors if necessary.



Further, preliminary investigations of Mahaarachchi and Mahendran’s (2005c,d)

research showed that there were some inaccuracies in their finite element models

used in developing the design curve for LSB sections. Therefore this research was

also intended to study the details of their finite element models, review the current

LSB design rules, and to develop accurate finite element models of LSB flexural

members.

1.3. Research Objectives

The main objective of this research is to improve the understanding and knowledge

of the lateral buckling behaviour of the new LiteSteel beam under different bending

moment distributions and loading positions. Specific objectives of this research as

are follows:

• Investigate the accuracy of Mahaarachchi and Mahendran’s (2005c) finite

element model in simulating the lateral buckling behaviour of LSBs.

• Develop a suitable finite element model of LiteSteel beam that is also capable

of simulating various moment distribution and load height conditions.

• Investigate the suitability of the current LSB design rules.

• Investigate the effect of non-uniform bending moment distribution and load

height (loading position) using finite element analysis.

• Conduct experimental investigation for comparison with the results from

finite element analyses and to validate finite element models used in this

research.

• Developed suitable modification factors to include the effect of moment

distribution and load height, and provide design recommendations for

LiteSteel beams.

1.4. Research Methodology

Following research methods were adopted to meet the objectives above:

• Phase 1 – Preliminary studies and literature review particularly in the area

(but not limited to) of cold-formed steel beams, buckling, LiteSteeel beam,

moment distribution effect, load height effect, current steel design standards,

finite element modelling, and any associated researches.



• Phase 2 – Thorough investigation of Mahaarachchi and Mahendran’s (2005c)

finite element analysis work used in developing the current LSB design

curve. Other elastic buckling analysis techniques such as finite strip methods

were also used. The finite strip analysis program used was THINWALL

(Hancock and Papangelis, 1994).

• Phase 3 – Developing a suitable finite element model for simulating LiteSteel

beam sections under various loading conditions and positions. The developed

model was required to be capable for elastic buckling and non-linear static

analyses. The latter was for simulating the real beam behaviour which

includes the effect of material inelasticity, buckling deformations, member

instability, residual stresses, and initial geometric imperfections. The finite

element software used was ABAQUS (HKS, 2005) with MSC.PATRAN as

the interface software (pre and post processing).

• Phase 4 – Investigation using finite element analysis into the effects of non-

uniform bending moment distribution and load height on the lateral buckling

strength (both lateral distortional and lateral torsional buckling) of LiteSteel

beam sections. Various load and boundary conditions that are commonly

encountered in practice were studied. Loading away from the cross section

shear centre is not within the scope of this thesis, thus only a loading through

the shear centre was considered in all the cases to simulate an ideal condition

of zero torsion effect from loading eccentricity.

• Phase 5 – Experimental investigation of LiteSteel beam sections subjected to

a moment gradient for comparison with finite element analysis study.

• Phase 6 – Analyses and discussions of the results from finite element and

experimental studies. It included a comparison study using the method used

in the current steel design codes. The final outcome from this research was to

provide design recommendations for LiteSteel beams.

1.5. Thesis Outline

To present the research processes and integrity of the works, this thesis is presented

in accordance with the adopted methodology. There are nine chapters as listed

below:

• Chapter 1 – Introduction



• Chapter 2 – Literature Review

• Chapter 3 – Finite Element Modelling

• Chapter 4 – Review of the Current Design Rules of LSBs

• Chapter 5 – Effect of Non-Uniform Bending Moment Distributions on the

Lateral Buckling Strength of LSBs

• Chapter 6 – Experimental Investigation of LSB with Moment Gradient

• Chapter 7 – Effect of Load Height on the Lateral Buckling Strength of LSBs

• Chapter 8 – Conclusions and Recommendations

• Chapter 9 – References

Chapter 2 – Literature Review


2. Literature Review

2.1. Cold-formed Steel Sections

Structural steel sections can be classified into two categories, hot-rolled steel and

cold-formed steel sections (thin-walled). The latter is less prominent but its market

share is growing significantly in recent years. Cold-formed steel sections are

fabricated by plastically deforming thin steel sheet, strip, plate or flat bar into a

desired shape under ambient temperature using roll forming, brake pressing or

bending brake technique. The thickness of the materials most frequently used for

structural members ranges from 0.4 mm to about 6.4 mm and the common grades

used range from 250 to 550 MPa (Yu, 2000). Compared with traditional hot-rolled

steel sections, cold-formed steel sections provide great advantages of lightness,

higher capacity to weight ratio, ease of handling, erection and installation

(economical construction), flexibility for custom shape, free from creep and

shrinkage at ambient temperatures. While the available technology and level of

understanding in the past had limited the development of cold-formed steel sections,

vast improvements in material, and manufacturing technologies and knowledge have

allowed cold-formed sections into a new era of competitiveness with conventional

hot-rolled sections.

Figure 2.1 – Common Cold-formed Steel Shapes (Yu, 2000)

With the trend of growing importance in the construction industry, cold-formed steel

sections are extensively used in various applications such as:

• Roof and wall systems of industrial, domestic and commercial buildings

• Steel storage racks, structural members for plane and space trusses

• Frameless stressed-skin buildings

• Floor bearers and joists, steel decking for composite construction

• Transmission towers, storage silos

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This figure is not available online. Please consult the hardcopy thesis available from the QUT Library



2.2. Material Properties of Cold-formed Steel Sections

The typical stress-strain curve for medium strength cold-formed section is a sharp

yielding type that has a linear region followed by a distinct plateau and then the

strain hardening up to the ultimate tensile strength before reaching the failure (Figure

2.2(a)). Higher strength steels (i.e. G450) do not exhibit a yield point and plateau but

instead a gradual yielding as shown in Figure 2.2(b). The modulus of elasticity is

defined by the slope of the initial straight portion of the stress-strain curve. The yield

strength is determined by either the offset method which is usually 0.2% of offset or

the strain-under load method, in which the yield strength is the stress corresponding

to a specified elongation under load (usually 0.5% of total elongation).

(a) Sharp yielding (b) Gradual yielding

Figure 2.2 – Common Cold-formed Steel Shapes (Yu, 2000)

The mechanical properties of cold-formed sections are different from the virgin plate

due to the cold work. The effect of cold work, particularly around the bend regions,

increases the yield strength and the tensile strength and in turns decreases the

ductility, depending upon the amount of cold work. In the bend regions, the amount

of cold work is considerably higher, thus the mechanical properties also vary across

the cross section. For this reason, buckling and yielding always begin in the flat

portion due to lower yield point of the material and any additional loads applied to

the section will spread to the corners (Yu, 2000). For ductility criterion of cold-

formed sections, Clause 1.5.1.5(a) of AS/NZS 4600 states that the ratio of tensile

strength to yield stress shall be not less than 1.08 and the total elongation shall be not

less than 10% for a 50 mm gauge length.

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Figure 2.3 – Effects on strain hardening and strain ageing (Yu, 2000)

The changes of mechanical properties due to cold work are caused mainly by strain

hardening and strain ageing as shown in Figure 2.3. Ageing of steel restores or

partially restores the sharp yielding characteristic. However, some steels and highly

worked steels, as the corners of cold-formed tubes, do not return to a sharp yielding

characteristic (Hancock, 1998). Generally, the larger the effect of strain hardening,

then the larger is the ratio of the ultimate tensile strength to the yield strength. In

addition, the changes in mechanical properties are also caused by Bauschinger effect;

where difference occurs between the yield strengths in tension and compression.

Residual stresses are stresses that occur as a result of manufacturing and fabricating

processes. These stresses are often dispersed non-uniformly across the cross section.

Addressing the residual stresses is important as it causes premature yielding than is

expected and reduces the member stiffness. Hancock (1998) stated that the increased

residual stress is one of the factors that cause rapid fracture. In hot-rolled steel

sections, residual stresses do not vary markedly through the thickness, but in cold-

formed sections residual stresses are dominated by a flexural, or through thickness

variation (Schafer et al., 1998). Using past experimental works, Schafer et al. (1998)

idealised the residual stresses on cold-formed steel as a summation of two types,

flexural and membrane as shown in Figure 2.4, in which the idealisation is a

pragmatic rather than scientific choice. Flexural residual stresses are more regularly

observed, in which residual stresses can be up to 50% of the yield stress of the virgin

plate. Membrane residual stresses are more common in cold-formed steel sections

shaped with roll forming rather than brake pressing technique.

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Figure 2.4 – Definition of Flexural and Membrane Residual Stresses (Schafer

and Pekoz, 1998)

2.3. Flexural Behaviour of Cold-formed Steel Beams

Under the flexural actions, the cold-formed steel beam is prone to loss of stability

through local, distortional and lateral torsional buckling.

2.3.1. Local Buckling

Cold-formed sections often have thin plate elements with a high plate slenderness

ratio (width to thickness (b/t) ratio) and thus local buckling may occur before section

yielding. Local buckling is a mode involving plate flexure independently without

transverse deformation at the intersection of adjoining plates. It is characterised by

wavelike deformations in one or more elements of the section under compression. It

is well known that local buckling of the plate element does not mean failure. Unlike

in a pure compression member (column) in which the entire cross-section starts to

yield at the same time, only the extreme fibres yield in beams. The cold-formed

beams are able to support further load until the whole cross-section yields (post

buckling strength). Hancock (1998) stated that the plate element will continue to

carry load though the stiffness reduces to 40.8% for a square stiffened element and to

44.4% for a square unstiffened element.

Figure 2.5 – Beams Subjected to Local Buckling (Rogers et al., 1997)

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2.3.2. Distortional Buckling

Cold-formed steel beams with high plate slenderness ratio can be subjected to a

buckling mode at half-wavelength intermediate between local and flexural-torsional

buckling. This mode is commonly called distortional buckling. It is most common

for edge-stiffened sections such as C and Z purlins (Hancock, 1997). The distortional

buckling may consist of flange or web distortional buckling. The most common is

flange distortional buckling in which the lip and flange rotate about the flange/web

corner with some elastic restraint to rotation provided by the web as shown in Figure

2.6. The use of thin material has caused the flanges to have low stability, stiffness

and tendency to move in or out.

Figure 2.6 – Beams Subjected to Distortional Buckling (Rogers et al., 1997)

The web distortional buckling in the C and Z section usually occurs under point load

condition which is often called as web crippling. Experimental tests have shown that

high strength steels with narrow flanges and web slenderness ratios of up to 200 can

fail by distortional buckling (Rogers and Schuster, 1997).

2.3.3. Lateral Torsional Buckling

Under bending about the major principal axis, cold-formed steel beams may buckle

out of plane by combined twist and lateral bending at long half-wavelengths. This

mode is called lateral torsional buckling. Compared with the two buckling modes of

short and intermediate half-wavelengths (local and distortional buckling), the

common cold-formed open sections are likely to endure lateral torsional buckling

because of their low torsional rigidity (Hancock, 2002). Furthermore, the sections are

often loaded eccentrically from their shear centres and consequently they will rotate

and deflect laterally as shown in Figure 2.7. In the design of cold-formed steel

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beams, this mode is the most significant factor governing the flexural capacity, in

which collapse is initiated as a result of lateral deflection (u) and twist (φ).

Figure 2.7 – Beams Subjected to Lateral Torsional Buckling (Hancock, 2002)

2.4. Beams Subjected to Torsion

When an I-beam is subjected to twisting about its longitudinal axis, it is said to be in

a pure uniform torsion and only shearing stresses occurred as shown in Figure 2.8(a).

It is usually assumed that these shear stresses at any point act parallel to the tangent

to the midline of the cross section and the magnitudes will be proportional to the

distance from the midline of the component plate. The applied torque (T1) is given

by:

T1 = GJ dφ/dz (2.1)

where, J = torsion constant for the section

G = shear modulus of the material

dφ/dz = rate of twist

The application of a torque causes the initially straight longitudinal fibres to twist. In

the conventional I-section, the flanges will twist by an angle of φh/2L thus inducing

axial displacement on the flanges, which is often interpreted as bending of the

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flanges in opposite directions about the vertical axis. This type of deformation is

called as warping (Figure 2.9).

(a) Uniform torsion (warping) (b) Non-uniform torsion (no warping)

Figure 2.8 – Shear Stress Distribution due to Torsions (Yuan, 2004)

Figure 2.9 – Warping of I-sections (Kirby and Nethercot, 1985)

If warping is prevented by fixing the end of the member, axial stresses will develop

in the flanges, and the rate of twist will no longer be constant thus leading to non-

uniform torsion distribution. These axial stresses in the two flanges create a pair of

equal and opposite moments (Figure 2.8(b)). The couple from the pair of shear

forces (Vf) associated with the moments (Mf) provide a warping restraint torsion

(T2):

T2 = -EIw d3φ/dz3 (2.2)

where, Iw = warping constant (Iw = If h3/2 for an I section)

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E = elastic Young’s modulus

In general, the applied torque is resisted partly by the shear stresses associated with

pure torsion and the warping restraint, and hence is given as follows:

T = T1+ T2 = GJ dφ/dz - EIw d3φ/dz3 (2.3)

Although the uniform torsion is always present, the warping resistance only occurs

when a cross section is prevented from warping. For sections with negligible

warping stiffness such as narrow rectangular sections, the second part of Equation

2.3 may be ignored.

2.5. Elastic Lateral Torsional Buckling Moments

The derivation of the elastic lateral torsional buckling moment for doubly symmetric

sections such as I-beams under uniform bending moment is based on the following

idealised conditions:

• Simply supported in-plane (fixed against in-plane transverse deflection but

unrestrained against in-plane rotation) and out-of plane (fixed against out-of

plane defection and twist rotation, but are unrestrained against minor axis

rotation and warping displacements).

• The beam is initially undistorted (perfectly straight) and unstressed.

• The beam behaves elastically and linear (geometry remains unchanged).

Figure 2.10 – Deformations Associated with Lateral Torsional Buckling

(Nethercot, 1992)

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Figure 2.10 shows the buckle shape of an I-beam subjected to uniform bending

moment. It involves lateral deflection (u) and a twist (φ) about an axis parallel to z

axis. By defining a local coordinate system of 123, the internal and external force

relationships can be simplified. The internal resisting moments at any cross section

are given by:

M1(int) = EIx d2v/dz2 (2.4)

M2(int) = EIx d2u/dz2 (2.5)

M3(int) = GJ dφ/dz - EIw d3φ/dz3 (2.6)

The relationship between the applied moments with the internal moments as shown

in Figure 2.11 enable the equations above to be written as:

M1(ext) = M EIx d2v/dz2 = M (2.7)

M2(ext) = Mφ EIx d2u/dz2 = Mφ (2.8)

M3(ext) = M du/dz GJ dφ/dz - EIw d3φ/dz3 = M du/dz (2.9)

Figure 2.11 – Internal and External Moments Relationship (Kirby and

Nethercot, 1985)

It appears that Equation 2.7 is independent of the other two equations. It actually

describes the in-plane behaviour whereas the last two equations express the lateral

buckling behaviour. Combination of these two equations leads to the following:

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EIw d4φ/dz4 - GJ d2φ/dz2 + M2/EIy φ = 0 (2.10)

By solving this equation with simply supported conditions where the lateral

deflection u and twist φ at both ends are fixed:

uo,L = u"o,L = φo,L = φ"o,L = 0 (2.11)

Then the elastic critical moment of lateral torsional buckling can be obtained as:

GJLEIGJEI

LM w

ycr 2

2

1 ππ+= (2.12)

Alternatively, this equation can also be obtained by using the energy method as

presented in Trahair (1993). It appears from Equation 2.12 that when the load is

applied on the principal plane then lateral torsional buckling will not occur. Further,

the derivation of the equation above assumed that the in-plane deflection has no

effect on the lateral stability. This is insignificant provided that the minor axis

rigidity is small compared to major axis, such as narrow flange I sections (Universal

Beams). To account for this effect, an approximate solution is given in Kirby and

Nethercot (1985). Nevertheless, it is always safe to neglect this in practice because

the effect is always beneficial. For point-symmetric sections such as Z-sections that

bend only in its stiffer principal plane (rare) and mono-symmetric sections that bend

about an axis of symmetry (principal axis) such as channels and equal angles, the

elastic buckling resistance under uniform bending moment is also given by Equation

2.12 above. For other sections, more accurate closed form solutions of critical lateral

buckling moment using the energy approach are provided in Trahair (1993).

2.5.1. Effects of Bending Moment Distribution

In real situations, beams are subjected to various loading conditions (Figure 2.12)

thus in turn produce a variety of bending moment distributions (non-uniform

bending). In fact, uniform bending moment is rarely found in practice although it is

regarded as the worst condition due to uniform yielding along the length.



Appropriately addressing the actual bending pattern is important to achieve

economical designs.

Figure 2.12 – Bending Moment Distribution Examples (Suryoatmono, 2002)

Elastic lateral buckling moment solutions for various load conditions are much more

difficult to obtain. Approximate solutions of elastic buckling moments for several

conditions such as beams with a mid-span load and beams under moment gradient,

may be obtained by using hand calculations based on energy method as given in

Trahair (1993). Finite element programs can provide more accurate outcomes but

this is not feasible in normal design practices.

Nevertheless, equivalent uniform moment approach has been a popular method to

accommodate this issue in the design based on a comparison of the elastic critical

buckling moment for the actual condition with the basic condition (uniform bending

moment). The reason is because the case of uniform bending moment is always the

most severe condition and the easiest to treat analytically.

Mcr = αm Myz (2.13)

where, αm = equivalent uniform moment factor or moment modification factor

Myz = elastic buckling moment for uniform moment case

Mcr = elastic critical buckling moment for the actual condition of non-

uniform moment

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One of the early works for moment distribution effects, Salvadori (1955) has

demonstrated that the effects of moment gradient that occurs commonly in a simply

supported beam can be estimated by using an approximate factor (see Figure 2.13):

αm = 1.75 + 1.05β + 0.3β2 ≤ 2.3 (2.14)

Figure 2.13 – Doubly-symmetric Beams under Moment Gradient (Trahair,

1993)

Various common critical loading conditions as well as support conditions (i.e.

cantilever) have been studied extensively since then by many methods including the

most advanced numerical analysis, finite element analysis. Studies using inelastic

buckling moments have also been thoroughly investigated. Appropriate equivalent

uniform moment factors or equations are available in many steel design standards

which will be discussed later. In addition, a general formula that covers a wide range

of loading has also been developed. Although its accuracy may vary from case to

case, but it might become the only practical way to deal with cases that are not

common.

2.5.2. Effects of Load Height

For beams that are subjected to transverse loads, the lateral stability not only depends

on the bending moment distribution but is also significantly affected by the level of

application of the load in relation to the shear centre axis. When a transverse point

load acts at a distance (yQ-yo) below the shear centre and moves along with the beam

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during buckling as shown in Figure 2.14, it creates an additional torque of Q(yQ-yo)φ

about the shear centre axis due to an eccentricity. This additional torque resists the

twist rotation (φ) of the beam thus increases the buckling resistance of the beam.

Conversely, when the load acts above the shear centre, the additional torque

magnifies the twist rotation (destabilising effect), thus reduces the resistance to

buckling of the beam. On the other hand, no additional torque will occur when the

load acts at the shear centre.

Figure 2.14 – Effects of Load Height (Trahair, 1993)

Figure 2.14 also shows that the load height effect diminishes “linearly” with the

slenderness of the beam. The destabilising effect will also not develop if the load

cannot move freely with the beam or there is resistance to lateral movement, e.g.

loads from floors do not constitute a destabilising load. This destabilising effect may

be included in the theoretical analysis leading to the determination of the critical

buckling moment. For example the value of Mcr for a beam with a mid-span point

load applied at the top flange level may be closely approximated by using the

following equation (Trahair, 1993):

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧+

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

yyz

Qm

yyz

Qmm

yz

cr

PMy

PMy

MM

/4.0

/4.0

12

ααα (2.15)

where, αm = equivalent uniform moment factor (1.35 for mid-span point load)

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yQ = load height

Py = π2EIy/L2

The equation above reflects both the moment distribution and the level of load

application, thus showing the complexity of the problem. Further, an alternative

approximate method of “modified effective length” has been a popular approach

adopted in design practice and many steel design standards. This is done by

increasing the effective length of the beam used for determining the elastic buckling

resistance (reduced due to greater length), in which the load height factors are

generally 1.2 to 1.4 for simply supported beams and higher (i.e. 2.0) for cantilevers.

This is a convenient approach as the moment distribution effect is addressed

separately by using the αm factor as discussed previously.

2.6. Strength of Real Beams

Figure 2.15 – Behaviour of Real Beams (Trahair, 1991)

Real beam behaviour is different from the ideal beams discussed above. Real beams

are not perfectly straight, but have small imperfections of initial crookedness or

curvature and twist which cause them to deflect laterally and twist at the beginning

of loading. The effect material non-linearity (yielding) of the steel introduces an

inelastic buckling behaviour in the beam. Further the presence of residual stresses

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induced by the method of manufacture amplifies the buckling inelasticity behaviour.

Thus these factors lead a steel beam to yield before it reaches the elastic buckling

load as shown in Figure 2.15, except for a beam with high beam slenderness (long

span beam).

The current design curves are usually based on a semi-empirical approach that

relates the ultimate moment capacity with the section moment capacity by a

slenderness reduction factor that provides allowances for the effects of initial

imperfections of real beams. This approach has been adopted in the Australian

Standards which is based on the lower bound of experimental results. The member

moment capacity (Mb) of a doubly-symmetric I-beam (hot-rolled section) is given

by:

136.0

2/12

≤⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧−

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

yz

s

yz

sm

s

b

MM

MM

MM α or Mb = αm αs Ms (2.16)

where, Ms = nominal section capacity

Myz = elastic buckling moment for uniform moment condition

αs = slenderness reduction factor

Figure 2.16 – Typical Design Curve of I-Beams (Trahair, 1991)

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Generally, there are three distinct regions in the real beam design curve (see Figure

2.15):

• Beams of high slenderness, for which the moment capacity is close to the

elastic buckling.

• Beams of low slenderness or stocky beams are capable of attaining the yield

strength without out of plane buckling (however, local buckling may govern).

• Beams of intermediate slenderness provide a transition between the two

regions.

2.7. Equivalent Uniform Moment and Load Height Factors in the Current Steel

Design Standards

There are three approaches of applying αm factors in design, which are commonly

adopted in the current standards:

• The effect of equivalent non-uniform moment is applied directly to the

nominal bending moment capacity (Mb) under uniform moment conditions.

• The effect of non-uniform moment is applied to the elastic buckling moment

(Myz) under uniform moment conditions.

• Design by buckling analysis.

Generally, the design codes for hot-rolled steel beams adopt the first method while

the second is implemented in the design codes for cold-formed steel beams. The

second method is more conservative and does not take into account the important

effect of localized yielding on inelastic buckling (Trahair, 1993). The latter is a

method of using the results from an elastic buckling analysis that may allow for a

range of loading conditions, supports and restraint conditions, in which it replaces

the uniform bending buckling resistance (Myz) in the design process. Although its

application is not yet widely used, the Australian standard allows explicitly the

design by buckling analysis as discussed in the following sections.

The destabilising effect such as a top flange loading is accounted approximately in

design codes by increasing the effective length of the beam used in determining the

elastic buckling resistance, while the stabilising effect such as bottom flange loading

is simply neglected as a conservative measure. A more accurate method is also



available and permitted in the design codes by using approximate equations that

consider the effect of load height on elastic buckling resistance (Trahair, 1993).

Nevertheless, it should be noted that in no case can the modified maximum moment

capacity exceed the plastic moment (Mp).

2.7.1. Australian Standards

The AS4100 (SA, 1998) is a steel design code for hot-rolled steel sections. Clause

5.6.1.1(a) of AS4100 allows a conservative use of equivalent uniform moment factor

(αm) as 1.0 (ignoring moment distribution effect) while recommending the use of

equations and factors for different loading conditions as given in Table 2.1.

Alternatively, a simple approximation by using one equation to accommodate any

bending moment distribution is given in AS4100:

( ) ( ) ( )[ ] 5.27.12*

42*

32*

2

*

≤++

=MMM

M mmα (2.17)

where, M*m = maximum design bending moment in the segment

M*2, M*

4 = design bending moments at the quarter points of the segment

M*3 = design bending moment at the midpoint of the segment


5.6.2)




5.6.2) (Continued)

All of the αm factors above are applied directly to the member moment capacity Mb

as shown in Equation 2.16. In addition, Clause 5.6.4 of AS4100 also allows the

design by buckling analysis where the value of Moa or Myz used in the calculation of

slenderness reduction factor (αs) shall be as follows:

Moa = Mob / αm (2.18)

where, the elastic buckling bending moment (Mob) at the most critical section of the

member shall be determined by using the results of an elastic buckling analysis (shall

take proper account of the member support, restraint and loading conditions). The αm

factors shall be determined from Clause 5.6.1.1 (a) as mentioned above or from the

equation given below:



αm = Mos / Moo (2.19)

where, Mos is the elastic buckling moment for a segment, fully restrained at both

ends, which is unrestrained against lateral rotation and loaded at shear centre, and

Moo is the elastic buckling moment for uniform moment conditions as given by

Equation 5.6.1.1(3) of AS4100 with effective length of 1. The αm is then

reintroduced in the final calculation of the member moment capacity.

Clause 5.6.3 of AS4100 allows the effect of load height by increasing the effective

length with a factor (Le = kl x L) for use in calculating the elastic buckling resistance.

The maximum load height factor allowed for loading above the shear centre is

approximately 1.4 for simply supported beams and higher for cantilevers (2.0), while

for loading below the shear centre it is taken as 1.0 (see Table 2.2). A more accurate

method is permitted by using approximate equations that consider the effect of load

height on elastic buckling resistance such as Equation 2.15 (Trahair, 1993).

Table 2.2 – Load Height Factors kl (AS4100 Table 5.6.3(2))

The design code for cold-formed steel sections, AS/NZS4600 (SA, 2005) allows a

moment modification (Cb) equation that is similar to the American code (see next

section), where in design, the moment modification factor is applied conservatively

to the elastic buckling resistance moment as discussed previously.

Table 2.3 – Cb Factors for Simply Supported Beams with Uniformly Distributed

Loads within the Span (AS/NZS4600 Table 3.3.3.2)

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This table is not available online. Please consult the hardcopy thesis available from the QUT Library



AS/NZS4600 generally does not provide any provisions to account for the effect of

load height except the alternative Cb factors in Table 3.3.3.2 of AS/NZS4600 (see

Table 2.3) which implicitly considers the load height effect. However they are only

applicable for beams with uniformly distributed load and lateral torsional buckling.

2.7.2. American Standards

The ANSI/AISC 360 (ANSI, 2005), the steel design code for hot-rolled steel

sections, provides a moment modification (Cb) equation which was developed

originally by Kirby and Nethercot (1979) that applies to various shapes of moment

diagrams. As used in the Australian Code, the Cb factor is applied directly to the

member moment capacity.

27.23435.12

5.12

max

max ≤+++

= mCBA

b RMMMM

MC (2.20)

where, Mmax = maximum design bending moment in the segment

MA, MC = design bending moments at the quarter points of the segment

MB = design bending moment at the midpoint of the segment

Rm = mono-symmetric parameter (for mono-symmetric I section)

For cantilevers where the free end is unbraced, Cb is taken as 1.0. For doubly-

symmetric members without transverse load, Cb equation limit reduces to 2.27 for

the case of equal end moments of opposite signs and to 1.67 when one end moment

is equal to zero. For single angle sections, Cb equation limit reduces to 1.5. Cb is

permitted to be conservatively taken as 1.0 for all cases with the exception of some

cases involving unbraced cantilevers or members with no bracing within the span

and with significant top flange loading (above shear centre). For unbraced top flange

loading on compact I-shaped beam, the reduced critical moment may be

conservatively approximated by setting the square root expression in the critical

uniform moment equation (Equation F2-4 in ANSI/AISC 360) equal to unity.

The design code for cold-formed steel sections (AISI, 2001) provides a moment

modification (Cb) equation that is similar to the code for hot-rolled steel sections

(Equation 2.20 above), but for design purposes, it is applied conservatively to the

elastic buckling resistance moment. For cantilevers or overhangs where the free end



is unbraced, Cb factor shall be taken as unity. On the other hand, no provision on

load height effect is provided in this code.

2.7.3. Bristish Standards

The BS5950-1 (BSI, 2000), the design code for hot-rolled steel sections, provides a

moment modification (mLT) equation analogous to ANSI/AISC 360 equation

(Equation 2.20 above) that applies to various shapes of moment diagrams.

44.015.05.015.02.0max

432 ≥++

+=M

MMMmLT (2.21)

where, Mmax = maximum design bending moment in the segment

M2, M4= design bending moments at the quarter points of the segment

M3 = design bending moment at the midpoint of the segment

Table 2.4 – Effective Length for Beams without Intermediate Restraint

(BS5950-1 Table 13)

The destabilising effect of load height should be taken where a load is applied at the

top flange of a beam or a cantilever, otherwise the normal loading condition should

be assumed. Table 2.4 shows the approximate approach of BS5950-1 Table 13 that

increases the effective lengths by 20% to allow for the destabilising effect in beams.

For cantilevers without intermediate lateral restraint, the effective length factors are

given in BS5950-1 Table 14 (Table 2.5). When a bending moment is applied at the

cantilever tip, the effective length from the Table 14 of BS5950-1increased by the

greater of 30% or 0.3L.

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Table 2.5 – Effective Length for Cantilevers without Intermediate Restraint

(BS5950-1 Table 14)

The design code for cold-formed steel sections, BS5950-5 (BSI, 1998) only provides

a moment modification (Cb) equation for beams subjected to moment gradient, and

in the design it is applied conservatively to the elastic buckling resistance moment:

Cb = 1.75 + 1.05β + 0.3β2 ≤ 2.3 (2.22)

Note that this Cb equation in BS5950-5 is similar to the αm equation given in Table

5.6.1 of AS4100. Clause 5.6.4 of BS5950-5 allows the destabilizing effect of top

flange loading in a similar manner to BS5950-1, i.e. by simply increasing the

effective lengths by 20% for simply supported beams. For other conditions that are

not covered in BS5950-5 (i.e. cantilever), reference should be made to BS5950-1.

2.7.4. Limitation of Current Equivalent Uniform Moment and Load Height

Factors

Despite the fact that the current equivalent uniform moment factors are being widely

used in routine steel designs, various past investigations have indicated their

discrepancy for different material and cross sections, as their development was

mainly based on conventional hot-rolled I-beams.

Following are some of the studies reported on the current moment distribution

factors for different materials and cross sections. Suryoatmono and Ho (2002) and

Serna (2005) demonstrated that AISC’s equivalent uniform moment factors are

sometimes very conservative and at other times non-conservative for doubly-

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symmetric I-beams. Wilkerson (2005) also reported that AISC is unconservative for

doubly-symmetric I-beams with top flange loading (load height effect). Kitipornchai

et al. (1985), Helwi et al. (1997), and Lim et al. (2003) showed that AISC 1994

moment modification factors cannot predict satisfactory results for mono-symmetric

I-beams about principal axis. Nevertheless, the new AISC 2005 has included the

effect of mono-symmetry (Rm factor in Equation 2.20). On the other hand, AS4100

does not have any provisions regarding mono-symmetry effect. While Pi et al.

(1998) demonstrated that moment modification and load height factors in AS4100

are reasonably accurate (conservative) for cold-formed channel sections, Put et al.

(1999) reported that AS4100 factors are not accurate for cold-formed Z-sections.

Kitipornchai and Wang (1986) also reported that the moment modification factors

given in the current design standards are potentially unsafe for Tee beams under

moment gradient.

Ma and Hughes (1996) studied the lateral distortional buckling of mono-symmetric

I-beams under uniform bending moment, and a transverse load of uniformly

distributed load and mid-span point load (including load height effect), using the

classical energy method. For the latter, shear stresses were necessary to be included

in the energy method analysis. They reported that the inclusion of web distortion

from lateral distortional buckling reduces the critical moment for short beams under

uniform moment, but the effect is not significant. However for short beams under

transverse load, the classical method seriously overestimates the critical load as

shown in Figure 2.17, although they also noted that for short beams other modes of

failure may precede lateral distortional buckling (Ma and Hughes, 1996). The

classical method refers to the Trahair’s solution for non-uniform moment distribution

and load height effect (i.e. Equations 2.14 and 2.15).

Further, Ma and Hughes (1996) compared a cubic polynomial shape function to

represent the web out-of plate buckling (the web distortion of lateral distortional

buckling) with a formulation using 5th order polynomial as shown in Figures 2.17

and 2.18, respectively. It can be seen that the flexible web formulation using the

cubic polynomial shape function only agrees qualitatively with the results obtained

from finite element analyses (using ABAQUS), while using the 5th order polynomial

shape function greatly improves the accuracy of the results.




Distributed Load using 5th Order Polynomial (Ma and Hughes, 1996)


Distributed Load Using Cubic Function (Ma and Hughes, 1996)

2.7.5. Moment Distribution and Load Height Effects on Hollow Flange Beams

Pi et al. (1997) investigated the effects of moment distribution and load height on

Hollow Flange Beams (HFB), which are the cold-formed steel beams that exhibit a

unique lateral distortional buckling similar to the new LiteSteel Beams. Although

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AS4100 provisions on moment distribution effects are generally based on lateral

torsional buckling of conventional I-beams, it was found that its application was

adequate for HFBs except for HFBs with low modified slenderness (Figure 2.19),

which is relatively similar to Ma and Hughes’s (1996) findings. The latter is unlikely

to be used in practice because their span-to-height ratios are unrealistically small (Pi

et al., 1997). For design purposes, they recommended the application of the

equivalent uniform moment factor directly to the uniform bending moment capacity

(Mb).

For HFBs subjected to top flange loading, Pi et al. (1997) suggested that the A4100

provision by increasing the effective length by 1.4 is generally conservative. In

addition, a more accurate method can also be used by using an approximate equation

that considers the effect of load height on elastic buckling resistance as an given in

Trahair (1993) (Equation 2.15). In this equation the lateral torsional buckling

moment shall be replaced with the lateral distortional buckling moment.

Figure 2.19 – Moment Distribution Effects on Hollow Flange Beams (Pi et al.,

1997)

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However, Pi et al.’s (1997) study was entirely based on finite element analysis using

a simple beam element to simulate HFB, where a cubic polynomial function was

assumed to represent the web out-of plate buckling shape (the web distortion of

lateral distortional buckling). According to Ma and Hughes (1996), the use of cubic

polynomial is not accurate, thus may suggest an overestimation of beam strength in

Pi et al.’s (1997) study, particualry for beams with low beam slenderness. Further, it

is unclear whether Pi et al. (1997) had included the shear stresses due to transverse

loading in their analyses, which are important to obtain accurate results.

2.8. LiteSteel Beam Research

LiteSteel beam (LSB) is a new Australian cold-formed high strength steel section

developed by Smorgon Steel Tube Mills (SSTM). The grounds for the LSB’s unique

shape of mono-symmetric channel shape with two hollow flanges (Figure 2.20) are

the beneficial characteristics of high torsional rigidity from the closed flanges,

practical advantages for connection and economical fabrication processes from a

single strip steel using an advanced SSTM patented technique, and the simultaneous

Dual Electric Resistance Welding process (DERW). This product was released to the

Australian market in mid of 2005.

Smorgon Steel Tube Mills claims that the unique profile of LSB avoids many

problems commonly associated with other open cold-formed sections, while

maintaining a high level of structural efficiency.

• LSB provides an excellent light weight member, 40% lighter than the

traditional hot-rolled steel of equivalent bending strength. It has a similar

weight to structural timber that can be manually lifted without the need for a

heavy mechanical lifting device.

• Low propensity of local buckling because of no free edges on the sections.

• High torsional rigidity due to the hollow flanges and thus gives better

stability.

• Lower flat width to thickness ratios (b/t) than many other cold-formed

sections.



• High efficiency and flexibility in handling. Cutting, screwing, nailing and

joining LSB with other materials can be done easily with only using standard

power tools.

Figure 2.20 – LiteSteel Beam Section

There are thirteen available LSB section sizes ranging from 125 mm to 300 mm deep

with thicknesses ranging from 1.6 mm to 3 mm. The available maximum standard

length for smaller sections is 12 m while for deeper sections is 14.5 m. During the

combined manufacturing process, LiteSteel Beam section is also encrusted with the

environmentally friendly EnviroKote water-based primer paint protective coating

system with thickness of 18-24 microns, which is claimed to be superior than the

common primer in traditional hollow sections.

2.8.1. Material Properties of LiteSteel Beams

The LiteSteel Beam is manufactured from a base steel with yield stress (fy) of 380

MPa and tensile strength (fu) of 490 MPa. As a result of the cold-forming process,

the flanges gain higher fy and fu of 450 MPa and 500 MPa, respectively, while the

web remains at the virgin plate’s strength (DuoSteel - 380/450 Grade). A typical

tensile coupon test undertaken by Mahaarachchi and Mahendran (2005e) as shown in

Figure 2.21 demonstrates the higher level of cold-working in the flanges. In addition,

the flanges exhibit a gradual yielding stress-strain curve without a yield plateau and

this results in lower ductility compared with the web element.



Figure 2.21 – Typical Stress-Strain Curves from Tensile Tests (Mahaarachchi

and Mahendran, 2005e)

Other LSB steel mechanical properties are similar to the common steels. The

Young’s Modulus of Elasticity (E) = 200x103 MPa, Shear Modulus of Elasticity (G)

= 80x103 MPa, Density (ρ) = 7850 kg/m3, and the Poisson’s Ratio (ν) = 0.25.

Figure 2.22 – Residual Stress Distributions in LSB Sections (Mahaarachchi and

Mahendran, 2005e)

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Mahaarachchi and Mahendran (2005e) conducted measurements of LSB residual

stresses and developed an ideal residual stress distribution model to be used for any

advanced numerical analyses. Due to the manufacturing process using a combined

cold-forming and electric resistance welding process, LSB sections were found to

have membrane and flexural residual stresses as shown in Figure 2.22. The residual

stress model is expressed as a ratio of the virgin plate’s yield stress value of 380

MPa. The maximum flexural residual stress occurs in the corner of the outside flange

(1.07fy), which has a higher yield stress than the virgin plate due to the cold-forming

process. Mahaarachchi and Mahendran (2005e) also measured the initial geometric

imperfection of initial crookedness (lack of straightness) and twist along the web and

the flanges of LSBs. It was concluded that the geometric imperfections are less than

the currently accepted fabrication tolerances. The measured local plate imperfections

are within the manufacturer’s fabrication tolerance limits while the overall member

imperfections are less than the AS4100 recommended limit of span/1000 (SA, 1998).

2.8.2. Flexural Behaviour of LiteSteel Beam

The unique shape of LSB section has introduced many structural issues that are not

commonly associated with the current steel beams. One of the major issues of LSBs

is their lateral distortional buckling (LDB) performance for intermediate spans.

Figure 2.23 – Lateral Torsional and Lateral Distortional Buckling Modes

(Mahaarachchi and Mahendran, 2005c)

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Unlike the generally observed lateral torsional buckling (LTB) of steel beams, the

LDB is characterised by simultaneous lateral deflection with twist and cross-section

change as shown in Figures 2.23 and 2.24. The presence of two stiff hollow flanges

and a slender web is the main reason for this buckling mode during which a web

distortion occurs in addition to the lateral torsional mode. This LDB is relatively a

new buckling mode and hence research in this area has been limited (with an

exception to mono-symmetric hot-rolled I-beams). Hollow Flange Beam (HFB), a

LSB predecessor whose production has been ceased, is also susceptible to LDB due

to its similar feature of stiff triangular hollow flanges with a slender web. The

implication from the LDB is a significant strength reduction, below that

corresponding to lateral torsional buckling (LTB). Since the current Australian

Standards do not include the complicated LDB behaviour, extensive researches were

conducted by Mahaarachchi and Mahendran (2005a-e) using full scale testing and

finite element analysis at the Queensland University of Technology. Appropriate

design rules for LSBs have been developed to conservatively estimate the member

capacity of LiteSteel beams allowing for LDB effects. Further, the lateral distortional

buckling was found to occur only for intermediate spans while for shorter and longer

spans the buckling modes of LSB are as in common steel beams, local buckling and

lateral torsional buckling, respectively.

Figure 2.24 – Buckling Modes of LiteSteel Beam

LDB

LTB

Local buckling



2.8.3. Flexural Capacity and Design of LiteSteel Beam Sections

For bending in-plane, the section moment capacity (Ms) of LSB can be estimated in

accordance with Clauses 3.3.1 and 3.3.2.2 of AS/NZS 4600 as follows:

φb Ms = φb Ze fy (2.22)

where; Ze = effective section modulus calculated with the extreme compression

or tension fibre at yield stress fy

φb = capacity reduction factor (0.9 for bending).

Generally, LiteSteel beam sections are fully effective for bending about the major

principal x-axis. This is due to the closed nature of the section (no free edges), where

every flat element of the cross-section is a stiffened element. Mahaarachchi and

Mahendran (2005b) stated that AS/NZS 4600 (SA, 2005) is conservative and safe for

predicting the section moment capacity of LSB, as it considers only the first yield

moment. Mahaarachchi and Mahendran (2005d) developed appropriate design rules

by modifying the previous AS/NZS 4600 rules to conservatively estimate the

member capacity (Mb) of LSBs allowing for lateral distortional buckling effects.

This has been adopted in the Design Capacity Tables for LiteSteel Beam and in the

new Australian cold-formed steel design code, AS/NZS 4600 (SA, 2005), although

no validation has been reported on its adequacy for other sections that may exhibit

lateral distortional buckling. The member capacity equations of simply supported

LSB under a uniform bending moment are given as follows:

φb Mb = φb Zc fc (2.23)

where: fc = Mc / Zf

λd = (My / Mod)1/2

For; λd ≤ 0.59 Mc = My (2.24a)

0.59 < λd < 1.7 Mc = My (0.59 / λd) (2.24b)

λd ≥ 1.7 Mc = My (1 / λd2) (2.24c)

where; φb = capacity reduction factor (0.9 for bending)

Zc = effective section modulus calculated at a stress

fc = extreme compression fibre

Zf = full section modulus



λd = non-dimensional slenderness

Mc = critical moment for bending about the major principal axis

My = moment causing initial yield at the extreme compression fibre

of the full section

The elastic lateral distortional buckling moment (Mod) for simply supported beam

under uniform bending can be estimated by using Pi and Trahair‘s (1997) formula:

⎥⎦

⎤⎢⎣

⎡+= 2

2

2

2

LEI

GJLEI

M we

yod

ππ (2.25)

The approximate effective torsional rigidity (GJe) is given by:

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

12

231

2

23

91.02

91.02

dLEtGJ

dLEtGJ

GJ

f

f

e

π

π (2.26)

where; EIy = minor axis flexural rigidity

EIw = warping rigidity

JF = torsion constant for a single hollow flange

d1 = depth of the flat portion of the web

L = beam length

This Mod equation is a modification of the AS4100 equation (Mo - elastic lateral

torsional buckling moment), which was developed by Pi and Trahair (1997) to

predict the effects of web distortion on the lateral buckling of HFB, a section with a

similar characteristic and flexural behaviour to LSB as discussed earlier.

Comparisons with finite strip analyses (THINWALL) and Mahaarachchi and

Mahendran’s (2005d) finite element analyses verify that Pi and Trahair’s (1997)

equation accurately predicts the elastic distortional buckling moment of LSB.

The moment distribution and load height effects for beams subjected to lateral

distortional buckling are not provided in any of the current steel design codes and no

research has been conducted yet on LSBs subjected to different loading conditions.

At present, the Design Capacity Tables for LiteSteel Beam conservatively



recommends an αm factor of 1.0 for all cases, while allows the use of load height

factor (kl) based on AS4100 for load height effects.

2.8.4. Finite Element Analysis and Model for LiteSteel Beam

The current LSB design curve was developed from Mahaarachchi and Mahendran’s

(2005c,d) finite element analyses using ABAQUS. They developed a finite element

model of LSB that accounts for all the significant behavioural effects including

material inelasticity, local buckling and lateral distortional buckling deformations,

member instability, web distortion, residual stresses, and geometric imperfections. In

summary there were two type of models developed for LSBs (Figure 2.25) as

described next:

Figure 2.25 – Schematic Diagrams of the Experimental and Ideal Finite element

Models used by Mahaarachchi and Mahendran (2005c)

• The experimental models, developed with the objective of creating

appropriate finite element model of LSB that simulates the actual test

members’ physical geometry, loads, constraints, simply supported conditions,

material properties, residual stresses and initial geometric imperfections as

closely as possible. They were beams subject to quarter point loading (Figure

2.25(a)). The quarter point loading was used because the use of a small

overhang to simulate a uniform bending moment within the span causes

warping restraint. The loads and boundary conditions of the experimental

model are shown in Figure 2.26. Simply supported boundary conditions were

applied to the node at the shear centre, combining with a rigid body using

R3D4 elements in order to provide an ideal pinned support. The concentrated

load was applied at the shear centre at the quarter point of the span using

rigid beam elements.

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Figure 2.26 – Applied Loads and Boundary Conditions for the Experimental

Model (Mahaarachchi and Mahendran, 2005c)

• The ideal models, developed to generate member capacity curves suitable for

the design of LSB. These models incorporated “idealised” simply supported

boundary (no warping restraint) conditions with a uniform bending moment

within the span as shown in Figure 2.25(b). In the ideal model, the idealised

boundary conditions were achieved by using a system of Multiple Point

Constraints (MPC) and the required uniform bending moment distribution

was implemented using a concentrated nodal moment applied at the cross-

section shear centre as shown in Figure 2.27.

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Figure 2.27 – Applied Loads and Boundary Conditions for the Ideal Model

(Mahaarachchi and Mahendran, 2005c)

2.9. Lateral Buckling Experimental Methods

The first lateral buckling experiments were introduced simultaneously in 1899 by

Prandtl and Michell (Singer, 1998). It was a simple system of simply supported

beam with cantilevers loaded at the free end to simulate uniform bending moment

within the beam span. Thereafter, lateral buckling experiment method has developed

extensively, in particular improvements in loading, lateral restraint and end

conditions. Overhang method is one of the most commonly used methods for testing

a simply supported beam with equal end moments as shown in Figure 2.28.

However, it is also commonly recognised that this system introduces a warping

restraint to the test beam, thus leading to strength overestimation.

Figure 2.28 – Schematic Diagrams of Overhang Loading Method

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Mahaarachchi and Mahendran (2005a) investigated the warping restraint of

overhang loading method on their experiments of LiteSteel beams and reported that

it over-predicted the failure moment by about 12%. For that reason, they introduced

a new system of quarter point loading method as shown in Figure 2.29. The loading

system includes two hydraulic rams connected to a wheel system, load cell and other

components to which the load was applied vertically upward. This system is capable

of eliminating the effect of warping restraint while still providing a uniform bending

moment although only between the points of load applications.

Figure 2.29 – Quarter Point Loading Method used by Mahaarachchi and

Mahendran (2005a)

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Chapter 3 – Finite Element Modelling


3. Finite Element Modelling

3.1. General

Finite element analysis (FEA) is the main tool used in this research to investigate the

effects of moment distribution and load height or loading position on the flexural

strength of LiteSteel beam. Therefore its accuracy is critical for the development of

reliable research outcomes. Considerable amount of time was spent in developing an

appropriate finite element model for LSB flexural members under uniform moment

and ideal support conditions, which was essentially a modification of the earlier

model developed by Mahaarachchi and Mahendran (2005c). This chapter presents

the details of the modified ideal finite element model of LSB developed in this

research that is capable of simulating the significant behavioural effects of material

inelasticity, buckling deformations including local buckling and web distortion,

member instability, residual stresses, and geometric imperfections. Further

modifications were also made to the finite element model to simulate other types of

loading and support configurations.

3.2. Mahaarachchi and Mahendran’s (2005c) Finite Element Models of LSB

Flexural Members

As described in Chapter 2, the development of LSB member capacity curves by

Mahaarachchi and Mahendran (2005c,d) was based on two types of finite element

(FE) models.

Figure 3.1 - Schematic Diagrams of the Experimental and Ideal FE Models

Developed by Mahaarachchi and Mahendran (2005c)

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• The ideal FE models, with “idealised” simply supported boundary (no

warping restraint) conditions and a uniform bending moment within the span

as shown in Figure 3.1. These idealised conditions usually simulate the worst

case, and therefore they are commonly adopted in the development of design

capacity curves of steel beams including LSBs.

• The experimental FE models, used for comparison with Mahaarachchi and

Mahendran’s (2005c) experimental test results of LSBs subjected to quarter

point loads. This comparison was intended to establish the validity of the FE

model for explicit modelling of initial geometric imperfections, residual

stresses, lateral distortional and local buckling deformations, and the

associated material yielding in non-linear static analyses. Although this does

not directly verify the suitability of the ideal FE model for use in the

development of LSB’s design curve, this approach is reasonably acceptable

as the ideal conditions are simply a theoretical assumption and are difficult to

simulate in the experiments.

Table 3.1 – Nominal Dimensions of LSB Sections Used by Mahaarachchi and

Mahendran (2005c)

The cross-section geometry of their experimental FE model was based on the

measured dimensions of the test specimens (Mahaarachchi and Mahendran, 2005a),

while their ideal FE model was based on the nominal dimensions as shown in Table

3.1. The corner radius was ignored for modelling simplicity (the effect is small).

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3.3. Validation of Mahaarachchi and Mahendran’s (2005c) Ideal FE Model

Mahaarachchi and Mahendran (2005c) confirmed the accuracy of their ideal FE

model for elastic buckling analysis by conducting a series of elastic buckling

analyses. They obtained the elastic lateral buckling moments (lateral distortional and

torsional buckling), and compared them with solutions obtained from the well

established finite strip analysis program, THINWALL (Hancock and Papangelis,

1994) as shown in Table 3.2. Finite strip method has been widely used for elastic

buckling analyses with great success, especially for cold-formed steel sections.

Table 3.2 – Comparison of Elastic Lateral Buckling Moments from Finite

Element Analysis and THINWALL (Mahaarachchi and Mahendran, 2005c)

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The comparison was made for 13 LSB sections with 10 different spans ranging from

1500 mm to 10000 mm. It was reported that the results from FE buckling analyses

exceed those from THINWALL by 5% on average, suggesting that their ideal FE

model accurately predicts the elastic lateral buckling moments of all LiteSteel beam

sections for a range of member slenderness values. However, this research indicates

that their conclusion may not be accurate, when the elastic lateral buckling moments

from FEA and THINWALL given in Table 3.2 were compared, it was found that in

most cases the difference exceeds 5% and hence casts some doubt over the accuracy

of their ideal FE model for use in future research.

A detailed investigation of their finite element model showed that they have used the

nominal dimensions of LSBs as the centreline dimensions of the ideal FE model. The

nominal dimensions listed in Table 3.1 are in fact the external dimensions (see the

LSB figure in Table 3.1). Shell elements that were used in the LSB model, discretize

a body by defining the geometry at the reference surface (the centreline of the body)

and the thickness is defined through the section property definition. Hence, their FE

model using the nominal external dimensions as the centreline or reference

dimensions simulated a larger LSB section, and this resulted in strength over

prediction. The actual dimensions of practical LSB sections differ from the nominal

dimensions and hence the real difference between the external and centreline

dimensions may be small, particularly for thinner LSB sections. However, when

comparisons are made to validate FE models, it is important that appropriate

dimensions are used.

Elastic buckling analyses of LSBs were undertaken using THINWALL to compare

the effect of using external dimensions instead of centreline dimensions. Table 3.3

shows the nominal centreline dimensions used in these analyses where the corner

radius was ignored (note that this effect is relatively small). The difference in the

elastic lateral buckling moments for these two cases is shown as a percentage in

Table 3.4 (the elastic buckling moments are given in Appendix B). The results

highlight the importance of using accurate dimensions in numerical modelling,

where the models using external dimensions has over-predicted the lateral buckling

moments by 17% on average. This is because the models using external dimensions

simulate larger LSB sections (thicker and deeper), thus leading to greater buckling



capacities. This effect will be more significant for LSBs with higher beam

slenderness (long span) as indicated in Table 3.4.

Table 3.3 – Nominal Centreline Dimensions of LSB Sections Used in this

Research Depth Flange Width Flange Depth Thickness

LSB Sections d (mm) bf (mm) df (mm) t (mm)

300 x 75 x 3.0 297.0 72.0 22.0 3.0 2.5 297.5 72.5 22.5 2.5

300 x 60 x 2.0 298.0 58.0 18.0 2.0 250 x 75 x 3.0 247.0 72.0 22.0 3.0

2.5 247.5 72.5 22.5 2.5 250 x 60 x 2.0 248.0 58.0 18.0 2.0 200 x 60 x 2.5 197.5 57.5 17.5 2.5

2.0 198.0 58.0 18.0 2.0 200 x 45 x 1.5 198.4 43.4 13.4 1.6 150 x 45 x 2.0 148.0 43.0 13.0 2.0

1.5 148.4 43.4 13.4 1.6 125 x 45 x 2.0 123.0 43.0 13.0 2.0

1.5 123.4 43.4 13.4 1.6

Table 3.4 – Percentage Difference in Elastic Lateral Buckling Moments

obtained from THINWALL using External and Centreline Dimensions

LSB Sections Difference in Elastic Lateral Buckling Moments (%)

d x bf x t (mm) 1.5 m 2.0 m 2.5 m 3.0 m 4.0 m 5.0 m 6.0 m 7.0 m 8.0 m 10 m 300 x 75 x 3.0 13.8% 13.4% 13.6% 14.3% 15.9% 17.4% 18.6% 19.4% 20.0% 20.8%

2.5 11.5% 10.9% 10.8% 11.0% 12.1% 13.3% 14.2% 15.0% 15.6% 16.4% 300 x 60 x 2.0 11.3% 10.8% 10.8% 11.1% 12.3% 13.6% 14.5% 15.3% 15.9% 16.7% 250 x 75 x 3.0 13.7% 13.6% 14.2% 15.1% 17.0% 18.4% 19.3% 20.0% 20.6% 21.3%

2.5 11.3% 10.9% 11.1% 11.6% 12.9% 14.1% 15.0% 15.7% 16.2% 16.9% 250 x 60 x 2.0 11.1% 10.8% 11.0% 11.7% 13.1% 14.3% 15.2% 15.8% 16.4% 17.0% 200 x 60 x 2.5 14.2% 15.1% 16.4% 17.6% 19.5% 20.7% 21.4% 22.0% 22.4% 22.9%

2.0 10.9% 11.1% 11.7% 12.6% 14.1% 15.2% 15.9% 16.5% 16.9% 17.4% 200 x 45 x 1.5 11.6% 12.1% 13.1% 14.1% 15.8% 16.9% 17.6% 18.1% 18.4% 18.9% 150 x 45 x 2.0 16.5% 18.5% 20.1% 21.3% 22.9% 23.7% 24.2% 24.5% 24.8% 25.0%

1.5 12.1% 13.2% 14.4% 15.5% 16.9% 17.8% 18.3% 18.7% 18.9% 19.2% 125 x 45 x 2.0 17.7% 19.8% 21.3% 22.2% 23.5% 24.2% 24.6% 24.8% 25.0% 25.2%

1.5 12.8% 14.1% 15.5% 16.4% 17.6% 18.3% 18.7% 19.0% 19.2% 19.4%

An investigation of Mahaarachchi and Mahendran’s (2005c) THINWALL analyses

indicated that they might have used the centreline nominal dimensions given in the

manufacturer’s specification (Appendix A), which includes the corner radius. This

contradicts with the assumed dimensions in their finite element model, contributing

to the differences shown between their FEA and THINWALL results.



Table 3.5 – Percentage Differences between Elastic Buckling Moments obtained

from Mahaarachchi and Mahendran’s FEA and THINWALL Analysis in this

Research using External Dimensions of LSBs

LSB section Difference in Elastic Lateral Buckling Moments (%) d x bf x t (mm) 1.5 m 2.0 m 2.5 m 3.0 m 4.0 m 5.0 m 6.0 m 7.0 m 8.0 m 10 m

300 x 75 x 3.0 NA 11.2% 12.1% 12.6% 15.2% 15.4% 15.1% 14.4% 13.8% NA 2.5 NA 11.4% 11.2% 11.1% 11.9% 12.6% 12.5% 12.5% 12.0% 15.3%

300 x 60 x 2.0 NA 9.7% 14.9% 6.0% 6.7% 7.1% 7.1% 6.9% 7.4% 6.4% 250 x 75 x 3.0 NA 11.9% 2.0% 1.9% 1.9% 2.0% 2.1% 2.2% 13.6% NA

2.5 NA 12.2% 13.5% 11.6% 1.4% 1.6% 1.7% 1.9% 11.9% NA 250 x 60 x 2.0 NA NA 6.1% 6.6% 7.3% 7.4% 17.7% 12.2% 6.8% 6.3% 200 x 60 x 2.5 NA 11.6% 1.4% 1.5% 1.7% 1.9% 10.0% 8.8% 2.2% 7.6%

2.0 NA 19.7% 0.7% 0.9% 1.2% 1.5% 7.9% 7.5% 2.0% 6.5% 200 x 45 x 1.5 13.3% 7.8% 8.4% 8.8% 8.8% 8.4% 8.1% 7.6% 7.2% 6.3% 150 x 45 x 2.0 5.1% 5.8% 6.0% 5.9% 5.7% 5.2% 5.0% 4.7% 4.5% 4.2%

1.5 2.1% 3.2% 3.8% 4.2% 4.4% 4.2% 4.1% 3.9% 3.9% 3.6% 125 x 45 x 2.0 12.4% 12.7% 12.3% 11.7% 10.4% 9.4% 12.4% 7.6% 7.1% 6.4%

1.5 8.7% 9.3% 17.6% 9.3% 8.6% 7.9% 7.2% 6.7% 6.2% 5.6%

Mahaarachchi and Mahendran’s (2005c) FE analysis results were also compared

with the THINWALL analysis results from this research using the same assumption

of external dimensions in both analyses (see Table 3.5). The percentage differences

are still too large. This indicates that there are some modelling inaccuracies in

Mahaarachchi and Mahendran’s ideal FE model, assuming that the THINWALL

analyses provided the benchmark solutions.

This preliminary investigation based on elastic buckling analysis (both FEA and

THINWALL) indicates that Mahaarachchi and Mahendran’s ideal FE model may not

be accurate with one of the reasons being the use of external LSB dimensions whilst

the other being possible problems with finite element modelling related to support

conditions. Detailed investigations were therefore undertaken to review

Mahaarachchi and Mahendran’s ideal FE model. They confirmed that their ideal FE

model had inaccuracies due to the modelling of support conditions and the use of

external dimensions. This raises a question on the adequacy of the current LSB’s

design curve, which was based on Mahaarachchi and Mahendran’s finite element

analyses. A review on the current LSB design curve as well as the effect of their

modelling inaccuracies are discussed in Chapter 4. The original model was modified

to achieve an improved finite element model that is suitable for use in this research.

The following section presents the details of the modified ideal finite element model.



3.4. Modified Ideal Finite Element Model of LSB Flexural Members

Finite element analyses were carried out using ABAQUS (HKS, 2005) with

MSC/PATRAN as the modelling interface program for pre- and post-processing.

This section presents the details of the modified ideal FE model of LSBs that

accounts for the significant LSB behavioural effects of material inelasticity, buckling

deformations that include local buckling and web distortion, member instability,

residual stresses, and initial geometric imperfections.

In the modified model, the nominal centreline dimensions as shown in Table 3.3

were adopted to accurately model the LSB sections. However, the corner radius was

ignored in the model for simplicity. It was found that this simplification has

insignificant effect on the results as shown in Table 3.6. The elastic buckling

moments from THINWALL analyses using centreline dimensions with and without

the corner radii agreed closely with a difference of only 2% on average (the elastic

buckling moments are given in Appendix B).

Table 3.6 – Percentage Difference in Elastic Buckling Moments obtained from

THINWALL using Centreline Dimension with and without Corner Radii

LSB section Span (mm) d x bf x t (mm) 1500 2000 2500 3000 4000 5000 6000 7000 8000 10000

300 x 75 x 3.0 LSB 4.1% 3.5% 3.0% 2.6% 2.2% 1.9% 1.7% 1.6% 1.6% 1.5% 2.5 LSB 3.5% 3.1% 2.7% 2.3% 1.9% 1.6% 1.5% 1.4% 1.3% 1.2%

300 x 60 x 2.0 LSB 3.5% 3.1% 2.7% 2.3% 1.9% 1.6% 1.5% 1.3% 1.3% 1.2% 250 x 75 x 3.0 LSB 3.9% 3.2% 2.7% 2.4% 2.0% 1.8% 1.7% 1.6% 1.5% 1.5%

2.5 LSB 3.4% 2.9% 2.4% 2.1% 1.8% 1.5% 1.4% 1.3% 1.3% 1.2% 250 x 60 x 2.0 LSB 3.4% 2.9% 2.4% 2.1% 1.7% 1.5% 1.4% 1.3% 1.2% 1.2% 200 x 60 x 2.5 LSB 3.6% 2.9% 2.5% 2.3% 2.0% 1.8% 1.8% 1.7% 1.6% 1.6%

2.0 LSB 3.1% 2.6% 2.2% 1.9% 1.6% 1.5% 1.4% 1.3% 1.3% 1.2% 200 x 45 x 1.5 LSB 3.2% 2.7% 2.2% 2.0% 1.7% 1.5% 1.4% 1.4% 1.4% 1.3% 150 x 45 x 2.0 LSB 3.3% 2.8% 2.5% 2.2% 2.1% 2.0% 1.9% 1.9% 1.9% 1.9%

1.5 LSB 2.8% 2.4% 2.1% 1.9% 1.7% 1.6% 1.5% 1.4% 1.4% 1.4% 125 x 45 x 2.0 LSB 3.1% 2.6% 2.4% 2.3% 2.1% 2.1% 2.0% 2.0% 2.0% 1.9%

1.5 LSB 2.6% 2.3% 2.0% 1.9% 1.7% 1.6% 1.5% 1.5% 1.5% 1.4%

3.4.1. Discretization of the Finite Element Mesh

The FE models of LSBs were developed using shell elements, which are capable of

providing sufficient degrees of freedom to explicitly model buckling deformations

and spread of plasticity effects. The ABAQUS - S4R5 element type was selected for



this purpose. It is a thin, shear flexible, isoparametric quadrilateral shell with four

nodes and five degrees of freedom per node, utilizing reduced integration and

bilinear interpolation schemes.

Appropriate selection of mesh size is critical in finite element analysis for improved

accuracy of its results. A fine mesh density is desirable for greater accuracy, but it

may lead to excessive computation time and resources. Mahaarachchi and

Mahendran (2005c) have conducted convergence studies to select the suitable

number and size of S4R5 elements that provide sufficient accuracy of results without

excessive use of computer time. An element size of 5mm for both the flanges and

web of LSB was recommended for an accurate representation of the spread of

plasticity, residual stress distribution and local buckling deformation. In the

longitudinal direction, an element length of 10mm was recommended to provide

suitable accuracy. A typical finite element mesh used for the LiteSteel Beam (LSB)

model is shown in Figure 3.2. The suitability of this selection was justified by

comparing their FEA results (using experimental FE model) with corresponding

experimental results of LSBs

Figure 3.2 – Typical Finite Element Mesh for LSB Model

3.4.2. Material Model and Properties

The ABAQUS classical metal plasticity model was used to model the material non-

linearity. This model implements the von Mises yield surface to define isotropic



yielding and associated plastic flow theory, i.e. as the material yields, the inelastic

deformation rate is in the direction of the normal to the yield surface. This

assumption is generally acceptable for most calculations with metals (Yuan, 2004).

A perfect plasticity model based on a simplified bilinear stress-strain curve without

strain hardening was assumed for all the models. Isotropic hardening model that

allows strain hardening behaviour where yield stresses increase as plastic strain

occurs was not considered in this study. It may be important for modelling sections

subjected to local yielding where strain hardening can take place (Mahaarachchi and

Mahendran, 2005c). A simple, perfect plasticity model was sufficient for modelling

sections subject to a dominant failure mode of lateral buckling.

The LiteSteel beam is manufactured from a single base steel, but because of the

cold-forming process, the flanges have a higher yield stress (fy) than the web. The

nominal yield stresses given in the LSB specification were adopted, i.e. yield stresses

of 380 and 450 MPa for the web and flanges, respectively. These values are the

minimum specified for the range of LSB sections as presented by CASE (2001) and

also confirmed by the tensile coupon tests undertaken by Mahaarachchi and

Mahendran (2005e). Other LSB steel mechanical properties were taken based on the

manufacturer’s specification, i.e. the Young’s Modulus of Elasticity (E) of 200x103

MPa and the Poisson’s Ratio (ν) of 0.25.

3.4.3. Idealised Load and Boundary Conditions

An “idealised” simply supported beam with a uniform bending moment within the

span has generally been assumed as the fundamental case providng a lower bound

solution in the development of steel beam design curves including LiteSteel beams.

The “idealised” simply supported boundary conditions were assumed as the basis in

this research. The following requirements are required to satisfy the idealised

boundary conditions (Trahair, 1993).

• Simply supported in-plane; both ends fixed against in-plane vertical

deflection, but unrestrained against in-plane rotation, and one end fixed

against longitudinal horizontal displacement.

• Simply supported out-of-plane; both ends fixed against out-of-plane

horizontal deflection and twist rotation, but unrestrained against minor axis

rotation and warping displacement. An illustration is provided in Figure 3.3.



Figure 3.3 – “Idealised” Simply Supported Boundary Conditions

Mahaarachchi and Mahendran’s ideal finite element model was developed to achieve

all of the requirements described above. A system of Multiple Point Constraints

(MPC) was used to simulate the required idealised boundary condition. The

modelling details of their MPC support system are summarised in the literature

review chapter. The preliminary research described earlier has highlighted the

modelling inaccuracy of their model. One of the inaccuracies was found to be due to

the inadequacy of the MPC support system to simulate the required idealised simply

supported boundary conditions.

(a) Modified ideal FE model (first version) (b) Mahaarachchi’s ideal FE model

Figure 3.4 – Idealised Finite Element Models

Figure 3.4(b) shows that Mahaarachchi and Mahendran’s ideal FE model is

subjected to a local flange twisting due to lack of restraint, despite the full twist

restraint assumptions in the idealised boundary conditions. The local flange twisting

TY, TZ=0

TX=0

TY, TZ=0

YZ

X

Free to warp Free to warp

RX=0 RX=0

Local Flange Twisting

Buckling Buckling

Undeformed Undeformed



unfavourably affects the buckling resistance as the high torsional rigidity provided

by the two hollow flanges has an important role in the buckling strength of LSB.

This may explain the lower buckling capacity prediction of their FE analyses

compared with the THINWALL analyses using the same dimensions in this research

(see Table 3.5), where the effect can be quite significant in some cases. This suggests

that the assumption of idealised boundary condition may not yield a lower bound

solution. For conventional I-beam sections, this is not an issue because the local

flange twisting effect is small due to the negligible torsional rigidity of their flanges.

Nevertheless, the LSB design curve development has simply adopted the generally

assumed idealised boundary conditions. Since in practice LSBs are likely to be

connected to their supports via their web elements only, thus allowing local flange

twists, future research may be worthwhile to quantify accurately the effect of local

flange twisting on the LSB design rules.

Appropriate modifications were therefore made to develop a model with idealised

boundary conditions including local flange twist restraint (see Figure 3.4(a)). The

load and boundary condition modelling of this modified ideal FE model (First

version) is illustrated in Figure 3.5.

Figure 3.5 – Load and Boundary Condition Modelling used in the Modified

Ideal Finite Element Model (First Version)

Tie MPC to link flange nodes and Rigid Beam

Explicit MPC “UX”, “UY”, “UZ”, “RX”, “RZ”

Single Tie MPC

Tie MPC to link web nodes and Rigid Beam

RIGID Beam MPC

Tie MPC to link flange nodes and Rigid Beam

M =

SPC



The modified ideal FE model (First version) is similar to the original model as it

retains the use of a system of Multiple Point Constraints (MPC) but with some

improvement as described next:

• Ten “rigid beam” type Multiple Point Constraint (MPC) elements which were

connected to the shear centre of the cross section. These elements were used

to spread the concentrated moment evenly to the web and flanges at the shear

centre.

• Two “Explicit” type MPCs connecting the web and flange MPC elements. In

this case only the y-rotational degree of freedom was unlinked, so the flanges

are free to rotate independently about the minor axis (ie, warping restraint

was eliminated) without any local flange twist.

• TIE type MPC elements linking “rigid beams” to the corresponding nodes on

the edge of the section. This will allow all the degrees of freedom of the

nodes on “rigid beams” and the edges of the section to be equal. Single TIE

type MPC was used to connect the centre node in the web at the support to

the independent node at the web rigid beam, which allows the concentrated

moment to be transferred to the beam section. The concentrated moment was

applied at the shear centre of each support to provide a uniform bending

moment within the span.

• A strip of elastic elements (20mm width) was included adjacent to the MPC

support system or the pinned end to eliminate any undesirable stress

concentration. The dimensions of the elastic elements are the same as for

LSBs, but the material properties are modelled as elastic. All the nodes on the

“rigid beam” are dummy nodes located in the same plane as the beam’s end.

A single point constraint (SPC) of “1234” was applied to the shear centre

node to provide a pinned support at one end while a SPC of “234” was

applied to provide a roller support at the other end, thus this combination

simulates a simply supported condition. The notation for degrees of freedom

“123” corresponds to translations along x, y and z axes whereas “456” relate

to rotations about x, y and z axes, respectively.

The elastic buckling moments obtained using this modified ideal FE model (First

version) provide a good prediction compared with those from THINWALL analyses,



but only for intermediate to long span LSBs. For short to intermediate span LSBs,

the use of this modified model underestimated the buckling strength, and its non-

linear strength (ultimate) was also found to be inaccurate. Discussions and details of

the elastic buckling and non-linear strengths from this first version of the modified

ideal FE model are presented in Chapter 4.

Figure 3.6 – Load and Boundary Condition Modelling used in the Modified

Ideal Finite Element Model (Final Version)

It was found that the use of Multiple Point Constraint (MPC) elements for warping

free simulation was always associated with undesirable stress concentrations and

thus reduced the elastic buckling and non-linear ultimate strengths of LSBs with

short to intermediate spans. This will be clearly shown in Chapter 4. Further

modification was therefore made to the finite element model. The MPCs system was

replaced with a simpler method of directly restraining the degrees of freedom of the

nodes at beam ends, thus eliminating the need for dummy nodes. The load and

boundary condition modelling of the final version of the modified ideal FE model is

shown in Figure 3.6, and is described as follows:

• The pin support at one end was modelled by using a single point constraint

(SPC) of “1234” applied to the node at the middle of the web element and the

SPC “1234” Restrained DOF

“234” for all other nodes

Restrained DOF “234” for all nodes at the other end

End moment simulation

Beam end

Linear compressive forces at every node (max at the top flange)

Linear tensile forces at every node (max at the bottom flange) Moment



degrees of freedom “234” of the other nodes were restrained. To simulate the

roller support at the other end, all the nodes degrees of freedom “234” were

restrained. This combination simulates a simply supported condition with

warping free and local flange twist restrained. The degrees of freedom

notation “123” corresponds to translations in x, y and z axes whereas “456”

relate to rotations about x, y and z axes, respectively.

• To simulate a uniform end moment across the section, linear forces were

applied at every node of the beam end, where the upper part of the section

was subject to compressive forces while the lower part was subject to tensile

forces. The required uniform bending moment distribution within the span

was achieved by applying equal end moments using linear forces at each

support, but in opposite directions. Example calculations of nodal forces are

given in Appendix C.

• Full span beam model was used in all cases as the boundary conditions is not

symmetrical and it was also required to simulate various unsymmetrical

loading conditions within the span. Note that with this final version of the

modified ideal FE model a strip of elastic elements adjacent to the support is

no longer required due to negligible stress concentrations.

3.4.4. Geometric Imperfections

Real beams are not perfectly straight and are often associated with geometric

imperfections, which may affect their buckling behaviour and strength. Due to the

limited suitable data available for LSB’s geometry imperfections except for the LSB

section’s fabrication tolerances given in the Design Capacity Tables for LiteSteel

Beams, Mahaarachchi and Mahendran (2005e) measured the geometric

imperfections of a range of LSB sections using a specially made imperfection

measuring equipment to establish the required geometric imperfection distribution

and magnitudes for numerical modelling of LSB sections. The local plate

imperfections were reported to be within the fabrication LSB tolerance limit while

the overall member imperfections of out of straightness were less than span/1000

based on the AS4100 (SA, 1998) fabrication tolerance for compression members.

Therefore it was recommended to conservatively adopt the fabrication tolerances for

local imperfection and span/1000 for overall (global) member imperfections. The

initial imperfection shape was introduced by ABAQUS *IMPERFECTION option



with the buckling eigenvector obtained from an elastic buckling analysis. Hence the

imperfection for lateral distortional buckling will include lateral displacement, twist

rotation and cross section distortion. The node that has the maximum out-of-plane

deformation of lateral distortional buckling from elastic buckling analysis will have

the largest imperfection magnitude of span/1000.

(a) Negative direction (-) (b) Positive direction (+)

Figure 3.7 – Positive and Negative Initial Overall Geometric Imperfections

Since LSB is a mono-symmetric section, the direction of initial global geometric

imperfection may affect its out of plane bending strength. A preliminary series of

non-linear static analyses was conducted to investigate the effect of imperfection

direction. In this research, the positive direction was considered as lateral

displacement in a direction to the right hand side of the section with clockwise twist

while the negative direction was for lateral displacement to the left hand side with

anticlockwise twist as shown in Figure 3.7. The effect of geometric imperfection

direction on LSB’s ultimate strength (non-linear strength) under uniform bending

moment is shown in Figure 3.8. The beams with negative initial imperfection

displace laterally in the negative direction and twist anti-clockwise, while the

opposite happens for positive initial imperfection. The preliminary analysis indicates

that the ultimate strengths (Mult) of LSBs with positive initial imperfections are

slightly higher (less strength reduction below the elastic buckling strength) than

those of the beams with negative imperfections. Figure 3.8 also shows that the

postbuckling behaviour with positive imperfection is stiffer. Therefore negative

initial imperfection was adopted in the non-linear static analysis to obtain a lower

bound solution of LSB’s lateral buckling strength.

Maximum imperfection magnitude (span/1000) at midspan Maximum imperfection

magnitude (span/1000) at midspan



0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4Modified Beam Slenderness (λd)

Mod

/My ,

Mul

t /My

Mult/My with (-) imperfection

Mult/My with (+) imperfection

Figure 3.8 – Effects of Geometric Imperfection Direction on LSB’s Ultimate

Strength

3.4.5. Residual Stresses

The residual stress is an important parameter in the flexural strength of steel beams

as it could lead to premature yielding, and reduce the beam strength. Residual

stresses in steel beam sections vary markedly because they occur as a result of

manufacturing and fabrication processes. The dual electric resistance welding and

cold-forming technologies used to make LSBs produce a unique residual stress

distribution in LSBs. Mahaarachchi and Mahendran (2005c) conducted tests using

the well known sectioning method to determine the residual stresses of LSB sections.

Based on the test results, a simplified residual stress distribution model was

developed for LSB sections as shown in Figure 3.9, which can be used in any

advanced numerical analyses. There are two types of residual stresses in LSB

sections, namely the membrane residual stresses in the web (mainly) due to the

welding process and the flexural residual stresses in the flanges (mainly) due to the

cold-forming process. The residual stress is expressed as a ratio of the virgin plate’s

yield stress (fy) value of 380 MPa. The maximum flexural residual stress is in the

corner of the outside flange (1.07fy) while the maximum membrane residual stress is

in the web (0.60fy). The maximum residual stresses in the flange corners are larger

My = Ms

Elastic buckling

LSB250x60x2.0

LSB125x45x2.0



than the virgin plate’s yield stress of 380 MPa, but it does not mean that they are

larger than the flange yield stress (average of 516 MPa), which is higher than the

virgin plate’s yield stress due to the cold-forming process. The flexural residual

stresses also vary with the section thickness and hence the entire outside surface of

the cross-section is in tension while the inside surface of the section is in

compression.

Figure 3.9 – Residual Stress Distribution Model for LSB Sections

(Mahaarachchi and Mahendran, 2005e)

The idealised residual stress model in Figure 3.9 was included in the analysis using

the ABAQUS *INITIAL CONDITIONS option, with TYPE = STRESS. The initial

stresses were created using the SIGINI Fortran user subroutine. This subroutine

defines the local components of the initial stress as a function of the global

coordinates. Since the global coordinates were used to define the local stress

components, member imperfections had to be included in determining the residual

stresses. Equations with the member length as a variable and constant deformation

factors obtained from the buckling analysis were used to represent the imperfection

of top and bottom flanges approximately. To vary the flexural residual stresses

through the thickness, they were applied as a function of the integration point

numbers through the thickness. An example of this subroutine defining the residual

stress distribution is provided in Appendix C.

halla




Figure 3.10 – Typical Residual Stress Distribution for LSB Sections

When initial stresses are given in ABAQUS models, the initial stress state may not

be in an exact equilibrium state for the finite element model. Therefore, the initial

stresses were applied in a *STATIC step with no loading and the standard model

boundary conditions to allow equilibration of the initial stress field before starting

the response history. The contours of residual stress after equilibration are shown in

Figure 3.10. However, the equilibration of the initial stress may require additional

deformations to bring the model into equilibrium due to the unbalanced stress. In the

finite element modelling of hot-rolled I sections, Yuan (2004) applied an additional

“force field” in the *STATIC step to reverse this extra initial deformations. This

(a) Residual stresses on the inside surface of the section

(b) Residual stresses on the outside surface of the section



force field was the reaction forces obtained from a preliminary analysis with all the

FE nodes fixed in the x, y, and z translation degrees of freedom. But this technique

was found inappropriate because the force field remains in the subsequent non-linear

analysis step, providing further restraint to the section. Nevertheless, this initial

deformation effect was considered small to significantly affect the analysis. Thus no

attempt was made in this research to eliminate the deformations caused by the

residual stresses.

3.4.6. Analysis Methods

Two methods of analysis were used, elastic buckling and non-linear static analyses.

Elastic buckling analysis was undertaken first. This method of analysis is generally

used in the development of design rules for moment distribution and load height

effects. It was also used to obtain the eigenvectors required for the inclusion of

geometric imperfections in the subsequent non-linear analysis. The first eigenvector

was adopted as it provides the critical buckling mode.

Nonlinear static analysis was adopted to investigate the behaviour and strength of

LiteSteel beam sections up to failure. ABAQUS uses the Newton-Raphson method

to solve the nonlinear equilibrium equations. The RIKS method in ABAQUS was

included in the nonlinear analysis. It is generally used to predict unstable structures

and is also useful for solving ill-conditioned problems such as limit load problems or

almost unstable problems that exhibit softening. The RIKS method uses the load

magnitude as an additional unknown; it solves simultaneously for loads and

displacements. Therefore, another quantity must be used to measure the progress of

the solution; ABAQUS uses the “arc length,” along the static equilibrium path in

load-displacement space (HKS, 2005). This approach provides solutions regardless

of whether the response is stable or unstable. Large displacement theory was also

considered in the analyses. The parameters used for non-linear static analyses are as

follows:

• Typical maximum number of load increments = 100 (may vary)

• Initial increment size = 0.001

• Minimum increment size = 0.0000001

• Automatic increment reduction enabled



The finite element models of LSB were developed in MSC/Patran and submitted to

ABAQUS for analysis. The following is a summary of the non-linear analysis

procedure used in this research:

• Define geometric surfaces for the web and hollow flanges, and mesh (shell

element – S4R5) those surfaces.

• Define load and support boundary conditions (including the MPC support

system), elastic material properties, and element properties.

• Define buckling analysis parameters and run bifurcation buckling analysis

using ABAQUS.

• Obtain the critical buckling eigenvector and the required maximum

deformation factors for member imperfection equations to be included in the

residual stress input subroutine.

• Prepare the residual stress input subroutine.

• Define the non-linear material properties and non-linear static analysis

parameters.

• Using ABQUS, run a non-linear analysis which consist of two “load steps”:

a) Equilibration ”STATIC” step – with the standard boundary conditions,

initial geometric imperfection and residual stress input subroutine, b)

Ultimate capacity load factor step – with the applied moment or load.

3.5. Typical Buckling Modes of the Modified Ideal FE Model

A series of buckling analyses using the modified ideal finite element model (final

version) was conducted. The typical buckling modes of LSB are shown in Figure

3.11. The results show that LSBs are associated with three distinct buckling modes,

namely local buckling for short span LSBs, lateral distortional buckling (LDB) for

intermediate span LSBs, and lateral torsional buckling (LTB) for long span LSBs.

This therefore confirms that LSBs with intermediate spans are prone to lateral

distortional buckling. Further, it was also found that the level of web distortion in

LDB varies as a function of beam slenderness, where the increasing beam

slenderness reduces the web distortion and thus approaches lateral torsional

buckling. Figure 3.11 shows this transformation from lateral distortional buckling to

lateral torsional buckling when the beam slenderness is increased.



(a) Local buckling (b) Lateral distortional buckling

(c) Lateral distortional buckling with reduced web distortion

(d) Lateral torsional buckling (no web distortion)

Figure 3.11 – Typical Buckling Modes of LSB Flexural Members

3.6. Validation of the Modified Ideal FE Model

The elastic buckling analyses using the final version of the modified finite element

model were compared with the solutions obtained from THINWALL and the

equation for elastic lateral distortional buckling moment (Mod equation). The Mod

equation (Eq. 2.25 and 2.26 in Chapter 2) developed by Pi and Trahair‘s (1997) has

been verified and adopted in the Design Capacity Tables for LiteSteel Beam. The

purpose of the comparison was to verify the accuracy of the finite element type,

Due to unsymmetrical boundary conditions

Note: all views are exaggerated for clarity

10m LSB250x60x2.0

4m LSB250x60x2.0

2.5m LSB250x60x2.0

1m LSB250x60x2.0



mesh density, load and boundary conditions used in the modified ideal finite element

model for LSBs. The elastic buckling moments from FEA, THINWALL and Mod

equation and the percentage differences from the comparison are summarised in

Table 3.7.

Table 3.7 – Comparison of Elastic Buckling Moments of LSBs from Finite

Element Analysis, THINWALL and the Mod Equation

LSB section d x bf x t

Span Elastic Buckling Moment (kNm) Differencve in Elastic Buckling Moments

(mm) (mm) FEA THINWALL Mod Eq THINWALL Mod Eq 300 53.57* 54.73* 176.94 2.1% - 750 33.67 34.11 35.40 1.3% 4.9% 1000 24.85 25.03 25.40 0.7% 2.2% 1500 17.83 18.10 18.06 1.5% 1.3% 2500 12.02 12.25 12.14 1.8% 1.0% 4000 8.00 8.17 8.08 2.1% 1.0% 6000 5.47 5.59 5.53 2.2% 1.0%

125 x 45 x 2.0 LSB

10000 3.33 3.41 3.36 2.2% 1.0% 500 52.69* 53.24* 358.89 1.0% - 1000 52.91* 53.24* 93.32 0.6% - 1500 45.10 45.45 47.11 0.8% 4.3% 2000 31.17 31.95 32.61 2.4% 4.4% 2500 25.76 26.01 26.34 0.9% 2.2% 3000 22.37 22.61 22.80 1.1% 1.9% 4000 18.19 18.44 18.51 1.3% 1.7% 6000 13.37 13.60 13.58 1.7% 1.5%

250 x 60 x 2.0 LSB

10000 8.60 8.77 8.72 1.9% 1.4% 500 177.82* 184.20* 1215.7 3.5% - 1000 179.56* 184.20* 314.50 2.5% - 1500 144.13 145.80 152.56 1.1% 5.5% 2500 76.95 77.93 79.06 1.3% 2.7% 3000 65.87 66.43 66.98 0.8% 1.7% 4000 52.19 53.10 53.20 1.7% 1.9% 6000 37.82 38.65 38.50 2.2% 1.8%

300 x 75 x 3.0 LSB

10000 24.12 24.77 24.58 2.6% 1.9% * Subjected to local buckling mode (as the critical buckling mode that precedes lateral buckling)

For clarity, Figure 3.12 also presents the elastic buckling moments vs. span

comparison of the three prediction methods. Three LSB sections were investigated:

LSB 125x45x2.0, LSB 250x60x2.0 and LSB 300x75x3.0. Based on AS4100 rules,

the selected sections are classified as compact, non-compact and slender sections,

respectively. Among the full range of LSB sections, the smallest section is 125 mm

deep LSB while 300 mm deep LSB is the largest section. The beam lengths were



varied from short to long spans to fully observe all the buckling modes associated

with LSB sections.

0

20

40

60

80

100

120

140

160

180

200

0 2000 4000 6000 8000 10000Span (mm)

Elas

tic B

uckl

ing

Mom

ent (

kNm

)Mod Equation for LSBTHINWALL (critical buckling load)THINWALL (buckling load for lateral buckling)FEA (critical buckling load) for LSB125x45x2.0FEA (critical buckling load) for LSB250x60x2.0FEA (critical buckling load) for LSB300x75x3.0

Figure 3.12 – Comparison of Elastic Buckling Moments vs. Span from Finite

Element Analysis, THINWALL and the Mod Equation for LSBs

The comparison shows that FEA results agree well with the results from both

THINWALL and Mod equation, where the average difference is less than 2% and

3%, respectively. While the Mod equation is an approximate solution, the small

difference with THINWALL is probably due to a very fine mesh density used in the

model for FEA. As discussed in the previous section, short span LSBs will be

governed by a local buckling mode. This is evident from both FEA and THINWALL

results (Figure 3.12). They predicted local buckling as the critical buckling mode (1st

eigenmode), which precedes lateral distortional buckling for short span LSBs.

LSB250x60x2.0 and LSB300x75x3.0 appear to be subjected to local buckling when

the span is between 1.0 and 1.5m or less, while for LSB125x45x2.0 the relevant span

is between 0.3 and 0.75m or less. The local buckling loads obtained from FEA were

similar to THINWALL results, where the difference is also less than 2% on average.

The Mod equation only provides solution for lateral distortional buckling, thus its

results for short span LSBs can not be compared with the FEA results.

Lateral buckling mode (LDB or LTB)

Local buckling mode



Figure 3.12 also demonstrates that the elastic lateral distortional buckling moment

(LDB) can be obtained from THINWALL for short span LSBs, but not from FEA.

Therefore it is not possible to use FEA to investigate the moment distribution and

load height effects on lateral buckling strength for the full range of beam

slenderness. Nevertheless, these comparisons suggest that the modifications made to

the current ideal FE model are sufficient to predict the elastic buckling moments for

all the buckling modes associated with LSB sections, i.e. local buckling, lateral

distortional buckling and lateral torsional buckling modes. Therefore, the suitability

of the element type, mesh density, geometry, load and boundary conditions used in

the modified ideal FE model is considered adequate.

The suitability of the modified ideal FE model for non-linear analysis is also

important to be verified. As discussed previously, the non-linear analysis validation

made by Mahaarachchi and Mahendran (2005c) was achieved by comparing the non-

linear analysis results of the experimental finite element model with their LSB

experimental results. The experimental finite element models simulated the actual

test members’ physical geometry, loads at quarter points, simply supported

conditions, material properties, residual stresses and initial geometric imperfections.

It was reported that the results from the experimental finite element model agreed

very well with the experimental test results, where the typical comparison is shown

in Figure 3.13. The figure also implies the accuracy of Mahaarachchi and

Mahendran’s residual stress model for LSB sections. This comparison had

established the validity of the FE mesh density and the shell element for explicit

modelling of initial geometric imperfections, residual stresses, lateral distortional

deformation, and the associated material yielding. This validated experimental FE

model was taken as the basis for Mahaarachchi and Mahendran’s ideal FE model,

and only the load and boundary conditions were improved in this research. Since the

experimental comparisons had been assumed to validate their ideal FE model, it is

therefore also reasonable to assume the suitability of the modified ideal FE model for

non-linear analyses in this research. The non-linear analysis results using the

modified ideal finite element model are presented in Chapter 4 for comparison with

the current design curve for LSBs.



Figure 3.13 – Bending Moment vs. Vertical Deflection Curves for 4m

LSB200x60x2.5 LSB (Mahaarachchi and Mahendran, 2005c)

3.7. Modifications for Various Loading and Support Boundary Conditions

Figure 3.14 illustrates the various load and support conditions investigated in this

research;

• simply supported beam with moment gradient, a mid-span point load, a

uniformly distributed load and quarter point loads, and

• cantilever beam with a point load at the free end and a uniformly distributed

load.

All the considered cases assume the transverse load application at the shear centre of

the cross section to avoid any additional torsional load. The transverse load

applications at positions above and below the shear centre (SC) were also studied in

this research. The load positions considered were at the top flange (TF) and bottom

flange (BF) level as these are the common cases in practice (Figure 3.15). Therefore,

suitable modifications were made to the ideal finite element model to simulate a

variety of loading and support types as well as loading positions (i.e. loading above

and below the shear centre).

halla




Figure 3.14 – Loading and Support Types Considered in This Study

(a) Load applied at shear centre (b) Loads applied at top flange and bottom flange

Figure 3.15 – Transverse Loading Positions Considered in This Study

3.7.1. Ideal Simply Supported LSB Model with a Moment Gradient

The ideal finite element model described in the last section was simply extended to

unequal bending moments by factoring the applied end moment (simulated using

linear forces) with a moment gradient ratio (β) at one support as shown in Figure

3.16.

(a) Simply supported beam with a moment gradient

(b) Simply supported beam with a mid-span point load

(c) Simply supported beam with quarter point loads

(d) Simply supported beam with a uniformly distributed load

(e) Cantilever beam with a point load at the free end

(f) Cantilever beam with a uniformly distributed load



Figure 3.16 – Ideal LSB Model with a Moment Gradient

3.7.2. Ideal Simply Supported LSB Model with a Mid-span Point Load (PL)

This research required the transverse load application to be at the cross section shear

centre. To simulate this condition, the finite element modelling was initially

attempted by using a RIGID MPC (Multiple Point Constraint) to link a dummy node

at the cross section shear centre to the node at the centre of web. This allows the

mid-span point load applied at the dummy node to be transferred to the web of the

beam. However, this attempt was unsuccessful as it caused a localised buckling

deformation and high stress concentrations at the centre of web, resulting in

premature failures as shown in Figure 3.17(a). A simulation of loading at the shear

centre which is located away from the cross section (unlike with doubly symmetric

section) in three dimensional modelling using shell element is complex and difficult

to achieve ideally. A considerable amount of time was spent to investigate a suitable

modelling technique to simulate the required loading conditions, including

simulations for loading above and below the shear centre.

M

βM



Figure 3.17 – Comparison of Lateral Distortional Buckling Mode and Yielding

(von Mises) Distribution at Failure (1st Yield) using the MPC Link and

Adopted Methods

The preliminary study resulted in an approximate method to simulate the shear

centre loading by applying the transverse loads (P) evenly to the web elements of

LSB’s top and bottom hollow flanges and with lateral forces (P’) applied through the

nodes at the corners of the outer flange plate element and the web element of LSB

hollow flanges (both top and bottom flanges) as shown in Figure 3.18. The lateral

forces (P’) create a torque to counter the torque due to loading away from the shear

centre. It was calculated with P’ = (P x a)/d, where P is the total transverse loads, a is

the distance from the centre of web to the shear centre, and d is the depth of the

beam. Thus this approach provides an equivalent condition to the ideal shear centre

loading. The transverse loads were not applied to the LSB’s main web element to

reduce local deformation or bearing effect and they were also distributed evenly to

(a) With the MPC Link method

Colour Code for Stress Red: 450 MPa (flange fy) Dark Yellow: 380 MPa (web fy)

Stress concentration around the centre of web

A localised web distortion due to a concentrated load (at the shear centre) applied to the node at the centre of web using a MPC Link

Lateral distortional buckling mode No stress

concentration

(b) With the adopted method



the web element of LSB’s hollow flanges to reduce stress concentration (see Figure

3.17(b)).

Figure 3.18 – Schematic View of the Adopted Method

Figure 3.19 – Schematic View of the Adopted Method for Various Levels

This modelling approach was further developed to be able to simulate the load height

effects. A method of simply removing the transverse loads on the web element of

LSB’s bottom flange was adopted to simulate the top flange loading while the

removal of transverse loads on the web element of top flange was adopted to

simulate the bottom flange loading. The lateral forces remained the same to provide

the required counter torque as shown in Figure 3.19. The three dimensional view of

the ideal finite element model using this method to simulate the mid-span point

loading at the shear centre is illustrated in Figure 3.20. The critical moment from the

finite element analyses was calculated using PL / 4, where P is the total applied load

and L is the beam span.

Top flange loading (TF)

Shear centre loading (SC)

Bottom flange loading (BF)

Resisting torque due to P’

Ideal condition

Equivalent shear centre loading



Figure 3.20 – Ideal LSB Model with a Mid-span Point Load at the Shear Centre

3.7.3. Ideal Simply Supported LSB Model with a Uniformly Distributed Load

(UDL)

Figure 3.21 – Ideal LSB Model with a Uniformly Distributed Load at the Shear

Centre

The modelling for UDL was also achieved using the method shown in Figure 3.18,

where a series of point loads (P) was applied evenly to all of the nodes of the web

element of LSB’s top and bottom hollow flanges while lateral forces (P’) were

P’

P

P

Transverse loads applied at every node of the web elements of hollow flanges

P’

Lateral loads applied at the corner node

P

P’ P

Transverse loads applied at every node of the web elements of hollow flanges P’

Note: boundary conditions are not shown for clarity




applied to all the nodes at the corners of the outer flange plate element and the web

element of LSB hollow flanges (both top and bottom flanges). Details of the finite

element model for the UDL case are shown in Figure 3.21. Simple modification as

shown in Figure 3.19 can be made to allow for loading above and below the shear

centre. The critical moment from the finite element analyses was calculated using

ωL2 / 8, where ω is the uniformly distributed load and L is the beam span.

3.7.4. Ideal Simply Supported LSB Model with Quarter Point Loads (QL)

The finite element model for this case is equivalent to the mid-span point load model

except that two point loads were applied at quarter points of the span as shown in

Figure 3.22. The critical moment from the finite element analyses was calculated

using PL / 4, where P is half of the total applied load and L is the beam span.

Figure 3.22 – Ideal LSB Model with Quarter Point Loads at the Shear Centre

3.7.5. Cantilever LSB Model with a Point Load (PL) at the Free End

Cantilevers are usually considered to be flexural members, which in the plane of

loading are built-in at the support (fully fixed end) and unrestrained or free at the

other end (Trahair, 1993). The boundary conditions in the ideal finite element model

for LSB were modified to achieve this condition. To simulate a fully fixed end

condition, all the degrees of freedom “123456” at the support nodes were restrained

in all translations, rotations, and warping, while at the free end they were

P

P’P


P’

P’

P

P

P’




unrestrained, thus free to translate and rotate. Figure 3.23 presents the modified FE

model for cantilever LSBs. The degrees of freedom notation “123” correspond to

translations in x, y and z axes whereas “456” relate to rotations about x, y and z axes,

respectively.

Figure 3.23 – Cantilever LSB Model

Figure 3.24 – Cantilever LSB Model with a Point Load at the Free End

(through the Shear Centre)

Restrained DOF “123456” for all nodes at the beam end (fully rigid support)

The free end is unrestrained

Beam end

P’ P

P


P’

Lateral load applied at the corner node



The modelling methods for the mid-span point load applied at the free end and

through the shear centre, above and below the shear centre were similar to that for

simply supported LSBs (Figures 3.18 and 19). Figure 3.24 shows the modified finite

element model for cantilever LSB subjected to a mid-span point load at the free end

(shear centre loading). The critical moment from the finite analyses was calculated

using PL, where P is the total applied load and L is the cantilever span.

3.7.6. Cantilever LSB Model with a Uniformly Distributed Load (UDL)

Figure 3.25 – Cantilever LSB Model with a Uniformly Distributed Load

(through the Shear Centre)

The modelling methods for the uniformly distributed loading at the shear centre,

above and below the shear centre for cantilever LSB were similar to that for the

simply supported LSBs. Figure 3.25 shows the modified finite element model for

cantilever LSB subjected to a uniformly distributed load (shear centre loading). The

critical moment from the finite element analyses was calculated using wL2 / 2, where

w is the uniformly distributed load and L is the cantilever span.

3.7.7. Experimental Finite Element Model used in This Research

Experiments on LSBs subjected to a moment gradient were conducted in the

Queensland University of Technology Steel Structures Laboratory. An LSB flexural

P

P’

P

Transverse loads applied at every node of the web elements of hollow flanges P’

Lateral load applied at the corner node

P P



member with overhangs as shown in Figure 3.26(a) was used to create equal or

unequal end moments for simulating the moment gradient action on a beam with

simply supported conditions. Detail descriptions of the physical implementation of

the experimental loads and boundary conditions are presented in Chapter 6. The

actual physical test system (members’ physical geometry, loads, constraints, etc) was

simulated by finite element analyses for the purpose of comparison of results.

(a) Schematic diagram

(b) Plan view (overhang segment)

(c) Isometric view (overhang segment)

Figure 3.26 – Applied Loads and Boundary Conditions for the Experimental FE

Model used in This Research

The summary of the applied loads and boundary conditions for the experimental

finite element model developed in this research is summarised in Figures 3.26(b) and

Overhang OverhangTest beam

Rigid body

10mm thick shell element to simulate the clamping plate

Loading at shear centre

4 RIGID MPCs to link the loading node (at shear centre) to the beam web

Support at shear centre (SPC “1234”)

Loading at shear centre

Support at shear centre (SPC “1234”)

SPC “234” for support at the other end

SPC “234” for support at the other end



3.26(c). The test members included a rigid plate at each support to prevent distortion

and twisting of the cross-section. These stiffening plates were modelled as rigid body

using R3D4 elements. In ABAQUS (HKS, 2003) a rigid body is a collection of

nodes and elements whose motion is governed by the motion of a single node,

known as the rigid body reference node. The motion of the rigid body can be

prescribed by applying boundary conditions at the rigid body reference node. A

single point constraint (SPC) of “1234” was applied at one of the supports and a SPC

“234” was applied at the other end, thus the experimental FE model simulated the

actual simply supported boundary conditions in the inelastic buckling tests (in-plane

translations restrained, in-plane and out-of-plane rotations free, and twisting

restrained, but warping restraint is inevitable due to the overhang segment). The

degrees of freedom notation “123” correspond to translations in x, y and z axes

whereas “456” relate to rotations about x, y and z axes, respectively.

In the experimental test set-up, a concentrated load was applied at the end of each

overhang. A steel plate connected to the web using four bolts was used to position

the loading arm at the cross section shear centre to avoid the additional torsional

load. Same loading arrangement was implemented in the experimental finite element

model using a concentrated nodal load applied at the cross-section shear centre while

simulating the bolts using RIGID MPCs as shown in Figures 3.26(c). Thicker shell

element (i.e. 10 mm) with elastic material properties was used to represent the steel

plate. Other modelling details were similar to the ideal FE model such as finite

element mesh discretization (using nominal centreline dimensions), material

properties, residual stresses and initial geometric imperfections (using the actual

imperfection direction reported from the test results).

Chapter 4 – Review of the Current Design Rules of LSBs


4. Review of the Current Design Rules of LSBs

4.1. General

This chapter presents a review of the current LSB design curve and the finite element

analysis results that were used for its development (Mahaarachchi and Mahendran,

2005d). Finite element analysis results using the first and the final versions of the

modified ideal finite element model are also presented for comparison with the

current LSB design curve.

4.2. Elastic Buckling Analysis Results using the Modified Ideal FE Model (First

Version)

As discussed in the previous chapter, the MPC support system of Mahaarachchi and

Mahendran’s ideal finite element model for LSBs was modified to eliminate the

local flange twist in lateral buckling, particularly for elastic buckling analysis. The

first version of the modified ideal FE model was developed for this purpose. A series

of buckling analyses using the modified ideal finite element model (first version)

was conducted, and the results of lateral buckling moments (lateral distortional or

lateral torsional buckling) are summarised and compared with those obtained from

the elastic buckling analyses using THINWALL in Table 4.1. It covers the full range

of LSB sections (13 sections) with different spans ranging from 1500 mm to 10000

mm.

In general, this comparison may indicate that this modified model provides a good

agreement with the results from THINWALL, however not for all the cases. For

instance, FEA shows that deep LSB sections (size greater than 200 mm deep) with

span less than 2.5 m are subjected to a local buckling mode that precedes the lateral

buckling mode. This contradicts the corresponding results from THINWALL. Figure

4.1 compares the elastic buckling moments vs. span curves from FEA using the

modified ideal FE model (First Version) and THINWALL for LSB 250x60x2.0 and

LSB300x75x3.0. It clearly indicates that this modified model underestimates the

local buckling load of LSBs. Note that all the results from FEA and THINWLL in

Table 4.1 and Figure 4.1 were based on the critical buckling mode (1st eigenmode).



This observation is similar to that in the comparison of lateral buckling moments

from Mahaarachchi and Mahendran’s FEA and THINWALL analyses shown

previously in Table 3.2 of Chapter 3. This finding may suggest that the adopted MPC

system to simulate the required ideal boundary conditions both in the original model

and the first version of the modified model is not accurate for short to intermediate

span LSBs. Although the modified model (First version) which eliminates the local

flange twist at the support is accurate for intermediate to long span LSBs, it is

evident that the MPC system is not the best option for simulating the ideal boundary

conditions.

Table 4.1 – Comparison of Elastic Lateral Buckling Moments from FEA using

the Modified Ideal FE Model (First Version) and THINWALL

Elastic Lateral Buckling Moment (kNm) Designation 1500 mm 2000 mm 2500 mm 3000 mm 4000 mm

d x bf x t (mm) FEA THIN WALL FEA THIN

WALL FEA THIN WALL FEA THIN

WALL FEA THIN WALL

300 x 75 x 3.0 - 145.80 - 98.70 77.03 77.93 65.86 66.43 52.67 53.10 2.5 - 119.80 - 78.99 - 61.36 50.91 52.10 41.57 42.08

300 x 60 x 2.0 - 53.40 - 35.29 - 27.41 22.92 23.21 18.41 18.58 250 x 75 x 3.0 - 125.50 89.59 90.62 74.03 74.61 64.48 64.96 52.10 52.58

2.5 - 101.50 - 71.40 57.74 58.32 50.56 50.96 41.63 41.99 250 x 60 x 2.0 - 45.45 - 31.95 25.70 26.01 22.41 22.61 18.27 18.44 200 x 60 x 2.5 51.47 52.13 39.97 40.29 33.71 33.97 29.38 29.65 23.40 23.68

2 - 39.18 29.55 29.86 25.16 25.37 22.26 22.45 18.21 18.41 200 x 45 x 1.5 - 15.09 11.36 11.50 9.58 9.67 8.37 8.45 6.70 6.79 150 x 45 x 2.0 18.30 18.46 14.62 14.76 12.26 12.40 10.55 10.68 8.21 8.33

1.5 13.62 13.76 11.07 11.17 9.47 9.57 8.28 8.38 6.59 6.69 125 x 45 x 2.0 17.96 18.10 14.46 14.61 12.10 12.25 10.37 10.53 8.03 8.17

1.5 13.43 13.53 11.07 11.18 9.46 9.57 8.24 8.35 6.50 6.61 5000 mm 6000 mm 7000 mm 8000 mm 10000 mm

FEA THIN WALL FEA THIN

WALL FEA THIN WALL FEA THIN

WALL FEA THIN WALL

300 x 75 x 3.0 44.29 44.73 38.20 38.65 33.53 33.98 29.84 30.27 24.38 24.77 2.5 35.54 36.00 31.09 31.53 27.59 28.01 24.75 25.16 20.45 20.82

300 x 60 x 2.0 15.61 15.76 13.56 13.72 11.97 12.12 10.69 10.85 8.79 8.93 250 x 75 x 3.0 43.77 44.28 37.64 38.15 32.95 33.44 29.25 29.71 23.82 24.23

2.5 35.58 35.96 31.00 31.39 27.40 27.78 24.49 24.86 20.13 20.46 250 x 60 x 2.0 15.50 15.67 13.43 13.60 11.81 11.99 10.53 10.69 8.61 8.77 200 x 60 x 2.5 19.38 19.65 16.49 16.75 14.32 14.56 12.64 12.86 10.22 10.41

2 15.37 15.57 13.24 13.44 11.60 11.79 10.30 10.48 8.39 8.55 200 x 45 x 1.5 5.58 5.66 4.76 4.84 4.14 4.22 3.66 3.73 2.97 3.03 150 x 45 x 2.0 6.69 6.81 5.64 5.74 4.86 4.96 4.27 4.36 3.44 3.51

1.5 5.44 5.53 4.62 4.70 4.00 4.08 3.53 3.60 2.85 2.91 125 x 45 x 2.0 6.53 6.65 5.49 5.59 4.73 4.82 4.15 4.24 3.33 3.41

1.5 5.34 5.43 4.52 4.60 3.91 3.98 3.44 3.51 2.77 2.83 "-" Not available as local buckling is the critical buckling mode



0

20

40

60

80

100

120

140

160

180

200

0 2000 4000 6000 8000 10000Span (mm)

Elas

tic B

uckl

ing

Mom

ent (

kNm

)

THINWALL (critical buckling load)

THINWALL (buckling load for lateral buckling)

FEA (critical buckling load) for LSB250x60x2.0

FEA (critical buckling load) for LSB300x75x3.0

Figure 4.1 – Comparison of Elastic Buckling Moments vs. Span Curves from

Finite Element Analysis using the Modified Ideal FE Model (First Version) and

THINWALL

4.3. Non-linear Static Analysis Results using the Modified Ideal FE Model (First

Version)

Prior to the development of the final version of the modified ideal finite element

model for LSB, Parsons (2007) studied the flexural strength of LSBs under the ideal

boundary conditions with a uniform moment distribution using the first version of

the modified FE model. Elastic buckling and non-linear static analyses using finite

element analysis (ABAQUS) were conducted to obtain the elastic lateral bucking

moment (LDB or LTB) as well as the ultimate moment capacity (Mult). The elastic

buckling analysis results are similar to those shown in the last section. The non-

linear analysis results are presented in a non-dimenionalised format in Figure 4.2,

where the modified beam slenderness (λd) equals to √[My/Mod] and My is taken as

the section capacity (Ms) based on AS/NZS 4600 (SA, 2005). Parsons’s (2007) non-

linear analyses included only the positive initial geometric imperfections, thus his

results may not provide the lower bound solution.

Lateral buckling mode (LDB or LTB)

Local buckling mode



Figure 4.2 shows that the ultimate moment capacities of all the LSB sections are

reduced in the low and intermediate beam slenderness region (instead of increasing

with decreasing beam slenderness). This means that the section capacity (Ms) will

not be achieved in the low beam slenderness region. It appears that the MPC system

induces a stress concentration in the section (particularly on the web) near the

support and thus underestimates the strength (Figure 4.3).

Figure 4.2 – Dimensionless Non-linear Finite Element Analysis Results using the

First Version of the Modified Ideal FE Model (Parsons, 2007)

(a) 1.5m LSB125x45x2.0 (b) 6m LSB125x45x2.0

Figure 4.3 – Typical Yielding Distribution (von Mises) at Failure (1st Yield)

using the First Version of the Modified Ideal FE model


Web yielding occurs along the span

Cross section view

halla




The non-linear strengths for intermediate and long span LSBs also appear to be

reduced, but not significantly as the stress concentration is small for LSBs with high

beam slenderness. Thus it is evident that the MPC system is not adequate for both

elastic buckling and nonlinear analyses of short span LSBs (only suitable for

intermediate and long spans). Further modifications were made that resulted in the

final version of the modified ideal FEA model as discussed in Chapter 3. This

observation therefore indicates that Mahaarachchi and Mahendran’s ideal FE model

should have a similar issue with their MPC support system in simulating the ideal

boundary conditions. This is discussed in the following section.

4.4. Review of Mahaarachchi and Mahendran’s Nonlinear Static Analysis

Results

Figure 4.4 – Current LSB’s Design Curve and Non-linear Finite Element

Analysis Results Used for its Development (Mahaarachchi and Mahendran,

2005d)

Mahaarachchi and Mahendran’s non-linear finite element analysis results are shown

in a non-dimensionalised format in Figure 4.4. These results were used to develop

halla




the current LSB design curve as a lower bound of the results. It was categorised into

three different regions as follows (the equations are given in Chapter 2):

• λd ≤ 0.59 Plastic region

• 0.59 < λd < 1.7 Inelastic region

• λd ≥ 1.7 Elastic region

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5Modified Beam Slenderness (λd)

Mu /

My,

Mb /

My,

Mod

/My

Current LSB's Design Curve300 x 75 x 3.0 LSB 300 x 75 x 2.5 LSB300 x 60 x 2.0 LSB250 x 75 x 3.0 LSB250 x 75 x 2.5 LSB250 x 60 x 2.0 LSB200 x 60 x 2.5 LSB200 x 60 x 2.0 LSB200 x 45 x 1.6 LSB150 x 45 x 2.0 LSB150 x 45 x 1.6 LSB125 x 45 x 2.0 LSB125 x 45 x 1.6 LSB

Figure 4.5 – Mahaarachchi and Mahendran’s Non-linear Finite Element

Analysis Results Used for Developing the Current Design Curve of LSBs

The scattered data in the results may appear reasonable as they represent the effect of

section geometry. However, this may not be entirely true. The results in Figure 4.4

are replotted in Figure 4.5, where the data points are connected. This enhances the

presentation and at the same time reveals the actual variation in Mahaarachchi and

Mahendran’s non-linear FEA results. In many cases, the results in the inelastic

region were reduced, but suddenly increased (sudden jump) to the plastic or section

capacity region. If the data in the “sudden jump” and plastic region are neglected,

then they would be similar to that of using the first version of the modified ideal FE

model (Figure 4.2), i.e. they were also reduced due to the inadequate modelling

using MPC system. It was found that Mahaarachchi and Mahendran (2005d) used a

Warping restraint was applied in the Mahaarachci’s ideal finite element model

Elastic buckling



warping restraint at the support in their ideal finite element model to avoid the

problems associated with the use of the MPC system in those “sudden jump” and

plastic regions and to obtain reasonable results in the plastic and inelastic regions.

The use of warping restraint at the beam supports will increase the flexural strength

considerably when compared with the ideal warping free support case. Nevertheless,

if the moment curve is plotted in a non-dimensionalised format using the relevant

appropriate elastic lateral buckling moment of beams (i.e. effect of warping restraint

is included in the modified beam slenderness (λd)), the difference will be minimised.

However, it appears that Mahaarachchi and Mahendran (2005d) used the elastic

lateral buckling moment (Mod) with warping free conditions. Thus this may

contribute to the overestimation shown in Figure 4.5, i.e. in the “sudden jump” and

plastic regions. Further, the approach of using warping restraint for some cases only

can not be justified as it is not consistent and the basis when the warping restraint has

to be included is unclear. It should also be noted that only the positive initial

imperfection was considered in their FEA, thus ignoring the unfavourable effect of

negative imperfections.

It can be concluded from the discussions above that non-linear analyses of all the

LSB sections shall be undertaken using the final version of the modified ideal model

to determine the overall effect of the modification on the non-linear ultimate moment

capacities of LSBs and their comparison with the current LSB design curve.

4.5. Comparison of the Non-linear Static Analysis Results using the Final

Version of the Modified Ideal FE Model with the Current LSB’s Design

Curve

The non-dimesionalised non-linear analysis results using the final version of the

modified ideal finite element model are summarised in Figure 4.6. The critical

negative imperfection was adopted in these analyses. Three LSB sections were

investigated, namely LSB125x45x2.0, LSB250x60x2.0 and LSB300x75x3.0.

According to AS4100 (SA, 2005), they are categorised as compact, slender and non-

compact sections, respectively. The results demonstrate that the final version of the

modified ideal FE model eliminates all the problems associated with the original

model as described above, and thus was capable of producing a reasonable non-



dimensionalised curve for all three regions, plastic, inelastic and elastic. A variation

exists in the dimensionless curve, which indicates the effect of sections slenderness.

The dimensionless strength of compact section (i.e. LSB125x45x2.0) is substantially

greater in the inelastic region and its postbuckling behaviour is also stiffer than the

other two less compact sections. Compact section is less prone to lateral distortional

buckling (particularly the web), which may explain why its non-linear strength is not

significantly reduced below the elastic strength.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0Modified Beam Slenderness (λd)

Mod

/My ,

Mb /

My o

r M

ult /M

y

Mod/MyMb/My (current LSB's design curve)Mult/My (LSB125x45x2.0) with (-) imperfectionMult/My (LSB250x60x2.0) with (-) imperfectionMult/My (LSB300x75x3.0) with (-) imperfection

Figure 4.6 – Comparison of the Non-linear Finite Element Analysis Results

using the Modified Ideal FE Model (Final Version) and the Current LSB Design

Curve

The three dimensionless strength curves are compared with the current LSB design

curve, which demonstrates that the adopted design curve is too conservative in the

inelastic region due to the modelling issues described above, in particular for less

compact sections. Note that the LSB250x60x2.0 is the second most slender section

among the available sections thus it should provide a good indicator as the lower

bound. It is expected that the curve for the most slender section will only be slightly

lower than that of LSB250x60x2.0. Nevertheless, non-linear analyses of all the LSB

My = Ms

Elastic buckling

Current LSB’s Design Curve



sections are being undertaken using the accurate finite element model developed in

this thesis.

In summary it is evident that the current LSB design curve is still safe for use in

routine designs despite the problems with the FEA used for its development.

Producing a suitable modified ideal finite element model for LSBs is also another

major outcome from this research. Further, this modified model brings benefits to

other or future LSB research projects that will fully exploit FEA modelling.

Chapter 5 – Effect of Non-Uniform Bending Moment Distributions


5. Effect of Non-Uniform Bending Moment

Distributions on the Lateral Buckling Strength of

LSBs

5.1. General

This chapter describes a detailed study using finite element analysis (FEA) to

investigate the effects of non-uniform moment distributions on the lateral buckling

strength of LiteSteel beams (LSBs). Finite element analyses were based on both

elastic buckling and non-linear static analyses. The modified finite element model

(final version) developed in Chapter 3 was used in this study. The FEA results from

these two analyses as well as a discussion of suitable design methods for LSBs

subjected to non-uniform bending moment distributions are presented in this chapter.

This research investigated the effect of a number of commonly found non-uniform

bending moment distributions for the following load and support conditions:

1. Simply supported LSBs with a moment gradient, and a transverse loading of

mid-span point load, uniformly distributed load and quarter point loads.

2. Cantilever LSBs with a point load at the free end and a uniformly distributed

load.

For the investigation with transverse load cases, the load was applied at the shear

centre to avoid any additional torsional loading. Three LSB sections were

investigated to include the effect of section geometry. They were LSB125x45x2.0,

LSB250x60x2.0 and LSB300x75x3.0, where the height of the smallest and the

largest sections are 125 mm and 300 mm, respectively. Based on AS4100 rules, the

selected sections are classified as compact, non-compact and slender sections,

respectively. The beam spans were generally varied from 0.75 m (intermediate spans

for LSB125x45x2.0) to 10 m (long spans) to study the relationship of lateral

buckling modes (Lateral Distortional Buckling versus Lateral Torsional Buckling) to

the moment distribution effects.

5.2. Simply Supported LSBs with a Moment Gradient

Two moments M and βM applied at beam ends produce a linear bending moment

distribution within the span as shown in Figure 5.1, where β is the ratio of the two



applied end moments (moment gradient ratio or the end moment ratio). When β

equals to -1, the beam is subjected to a uniform bending moment distribution.

Figure 5.1 – Simply Supported LSB Subjected to a Moment Gradient

An “idealised” simply supported beam with a uniform bending moment within the

span usually provides the worst condition due to uniform yielding across the entire

length. This condition has generally been assumed as the fundamental case for steel

beam design curves including the LiteSteel Beams’ design curves. Although this

condition rarely exists in practice, such an assumption is used for reasons of

conservatism and simplicity required for the design curves. Other combinations are

generally considered in the design using appropriate modification factors to the basic

design curves, which are commonly provided in many steel design codes. In this

LSB research, an idealised simply supported condition was assumed as the basis for

this study into the effects of non-uniform moment distributions. The study was also

extended to include another common support condition, the cantilever end support.

The idealised simply supported boundary condition is illustrated in Figure 5.2.

Figure 5.2 – “Idealised” Simply Supported Boundary Condition

TX=0

X

TY, TZ=0 TY, TZ=0

Y

Z

Free to warp Free to warp

RX=0 RX=0

βM M

Bending moment distribution



5.2.1. Elastic Buckling Analysis Results and Discussions of Moment Gradient

Cases

Finite element analyses were conducted for LSBs with a moment gradient that varied

from β = -1.0 (uniform moment) to β = 1.0 at intervals of 0.2. The elastic buckling

moment results under moment gradient are given in Table 5.1 and plotted in Figure

5.3, where the well known equivalent uniform moment or moment distribution factor

(αm) is used to represent the moment gradient effects.

od

nonodm M

M −=α (5.1)

Where; Mod-non = elastic lateral buckling moment for non-uniform moment

Mod = elastic lateral buckling moment for uniform moment

Table 5.1 – Elastic Buckling Moments of Simply Supported LSBs

Subjected to Moment Gradient

LSB FEA Buckling Moments (Mod for β = -1 & Mod-non for other β values) d x bf x t

Span β = -1 β = -0.8 β = -0.6 β = -0.4 β = -0.2 β = 0

(mm) (mm) kNm Mode kNm Mode kNm Mode kNm Mode kNm Mode kNm Mode 750 33.67 LDB 37.31 LDB 41.51 LDB 46.24 LDB 51.28 LDB 56.15 LDB

1500 17.83 LDB 19.76 LDB 22.04 LDB 24.70 LDB 27.73 LDB 31.08 LDB 2500 12.02 LDB 13.32 LDB 14.84 LDB 16.58 LDB 18.54 LDB 20.66 LDB 4000 8.00 LDB* 8.83 LDB* 9.88 LDB* 11.06 LDB* 12.41 LDB* 13.88 LDB* 6000 5.47 LTB 6.06 LTB 6.77 LTB 7.59 LTB 8.53 LTB 9.59 LTB

125 x 45 x 2.0 LSB

10000 3.33 LTB 3.69 LTB 4.12 LTB 4.62 LTB 5.21 LTB 5.87 LTB 1500 45.10 LDB 49.90 LDB 54.41 nLB 54.55 nLB 54.16 nLB 53.23 nLB 2500 25.76 LDB 28.57 LDB 31.89 LDB 35.79 LDB 40.27 LDB 45.22 LDB 3000 22.37 LDB 24.80 LDB 27.67 LDB 31.03 LDB 34.88 LDB 39.16 LDB 4000 18.19 LDB 20.16 LDB 22.47 LDB 25.12 LDB 28.10 LDB 31.36 LDB 6000 13.37 LDB* 14.82 LDB* 16.50 LDB* 18.42 LDB* 20.57 LDB* 22.89 LDB*

250 x 60 x 2.0 LSB

10000 8.60 LTB 9.53 LTB 10.63 LTB 11.90 LTB 13.34 LTB 14.92 LTB 1500 144.1 LDB 159.3 LDB 175.6 LDB 185.3 nLB 182.2 nLB 176.1 nLB 2500 76.95 LDB 85.35 LDB 95.3 LDB 106.9 LDB 120.3 LDB 134.9 LDB 4000 52.19 LDB 57.87 LDB 64.56 LDB 72.35 LDB 81.25 LDB 91.13 LDB 6000 37.82 LDB* 41.93 LDB* 46.74 LDB* 52.30 LDB* 58.56 LDB* 65.42 LDB*

300 x 75 x 3.0 LSB

10000 24.12 LTB 26.76 LTB 29.86 LTB 33.49 LTB 37.62 LTB 42.21 LTB LDB Lateral distortional buckling mode LTB Lateral torsional buckling mode LDB* Lateral distortional buckling mode with negligible web distortion nLB Non-lateral buckling mode (local buckling near the support) that precedes LDB



Table 5.1 – Elastic Buckling Moments of Simply Supported LSBs

Subjected to Moment Gradient (Continued)

LSB FEA Buckling Moment (Mod & Mod-non) d x bf x t

Span β = 0.2 β = 0.4 β = 0.6 β = 0.8 β = 1

(mm) (mm) kNm Mode kNm Mode kNm Mode kNm Mode kNm Mode 750 56.76 nLB 57.17 nLB 57.55 nLB 53.88 nLB 49.08 nLB 1500 34.65 LDB 38.27 LDB 41.71 LDB 44.50 LDB 44.14 LDB 2500 22.90 LDB 25.17 LDB 27.40 LDB 29.35 LDB 29.37 LDB 4000 15.44 LDB* 17.02 LDB* 18.52 LDB* 19.74 LDB* 19.52 LDB* 6000 10.71 LTB 11.86 LTB 12.96 LTB 13.81 LTB 13.59 LTB

125 x 45 x 2.0 LSB

10000 6.58 LTB 7.32 LTB 8.02 LTB 8.56 LTB 8.41 LTB 1500 51.71 nLB 49.65 nLB 47.16 nLB 44.40 nLB 41.56 nLB 2500 50.36 LDB 54.01 nLB 53.45 nLB 52.69 nLB 51.69 nLB 3000 43.69 LDB 48.18 LDB 52.24 LDB 53.76 nLB 53.21 nLB 4000 34.82 LDB 38.35 LDB 41.80 LDB 44.82 LDB 44.94 LDB 6000 25.31 LDB* 27.77 LDB* 30.17 LDB* 32.29 LDB* 32.31 LDB*

250 x 60 x 2.0 LSB

10000 16.60 LTB 18.28 LTB 19.88 LTB 21.18 LTB 20.93 LTB 1500 167.3 nLB 156.5 nLB 144.7 nLB 133.1 nLB 121.1 nLB 2500 149.7 LDB 162.8 LDB 171.6 nLB 171.7 nLB 161.9 nLB 4000 101.7 LDB 112.4 LDB 122.7 LDB 131.1 LDB 130.0 LDB 6000 72.69 LDB* 80.10 LDB* 87.37 LDB* 93.67 LDB* 93.50 LDB*

300 x 75 x 3.0 LSB

10000 47.09 LTB 52.01 LTB 56.70 LTB 60.48 LTB 59.73 LTB

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1End Moment Ratio (β )

αm

Fac

tor

LSB1254520-0.75mLSB1254520-1.5mLSB1254520-2.5mLSB1254520-4mLSB1254520-6mLSB1254520-10mLSB2506020-1.5mLSB2506020-2.5mLSB2506020-3mLSB2506020-4mLSB2506020-6mLSB2506020-10mLSB3007520-1.5mLSB3007520-2.5mLSB3007520-4mLSB3007520-6mLSB3007520-10m


Gradient Based on Elastic Buckling Analyses



The results show that moment gradient increases the lateral buckling strength of

LSBs (αm factor is greater than 1.0) with increasing end moment ratio (β). However,

this moment gradient benefit is reduced in some cases (i.e LSBs with low to

intermediate beam slenderness) as indicated by the skewed αm curve in Figure 5.3.

The moment gradient action actually reduces the effective length of the beam, which

explains the increase of lateral buckling strength, but in turn it also increases the

likelihood of local buckling to be the critical mode.

As described in the previous chapter, the FEA results have confirmed that the critical

buckling mode for LSBs under uniform bending moment (basic case) is local, lateral

distortional and lateral torsional buckling for low, intermediate and high beam

slenderness cases (three distinct regions), respectively. Therefore the presence of the

skewed αm curve indicates that they have already reached the critical local buckling

mode (within the low beam slenderness region) and does not contribute to the

variation of moment gradient effects. This limiting effect is generally seen in the

case of LSBs with short to intermediate spans and those with intermediate spans and

a high positive end moment ratio (β). Nevertheless, the purpose of αm factor is

generally only for the pure lateral buckling mode, thus such results that are limited

by the local buckling mode will not be considered in this research.

Figure 5.4 – Local Buckling Mode due to Moment Gradient Action

Local buckling near the support

Note: view is exaggerated for clarity

1.5m LSB250x60x2.0 (β = 0)



The typical deformation of local buckling as the governing mode due to moment

gradient action is shown in Figure 5.4. The local buckling is concentrated near the

support because it is where the high moment region exists in the moment gradient

case.

The FEA results of αm factor are also replotted in Figure 5.5, where they are grouped

according to the lateral buckling modes, namely, lateral distortional buckling (LDB)

and lateral torsional buckling (LTB) to observe their relationship. The LDB mode is

further differentiated for cases with less web distortion, i.e. LDB deformation is

close to that of LTB. This is because the level of web distortion associated with

lateral distortional buckling was found to be a function of beam slenderness, and

diminishes with increasing beam slenderness as already discussed earlier.

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8


αm

Fac

tor

LTBLDB (Limited due to LB) LDBLDB (Less web distortion)


Gradient Based on Elastic Buckling Analyses (Grouped)

Pi and Trahair (1997) developed a closed form solution for elastic lateral distortional

buckling moment (Mod) of hollow flange beams subject to a uniform moment. This

Reduced as local buckling is the governing mode

Upper bound results (LTB mode)

Ke = 0.05

Ke = 0.48

Ke = 0.65

Ke = 1.30

Assumed lower bound results (LDB mode)



has been adopted for LSBs by modifying the torsional rigidity (GJ) in the elastic

lateral torsional buckling moment (Mo) equation with the effective torsional rigidity

(GJe) to account for the web distortion effect of LDB.

⎟⎟⎠

⎞⎜⎜⎝

⎛+= 2

2

2

2

LEI

GJLEI

M we

yod

ππ (5.2)

where GJe is given by Equation 2.26 in Chapter 2.

Alternatively it can be written as;

( )eey

od KL

GJEIM += 12

2π (5.2a)

where;

22 / LGJEIK ewe π= (5.2b)

Ke is a modified torsion parameter which expresses not only the torsion component

of lateral buckling but also the web distortion. Note that the original form of torsion

parameter (K) is used to express the torsion component of lateral torsional buckling.

Hence the use of the modified torsion parameter Ke is appropriate for this research

involving lateral distortional buckling as used in Figure 5.5. High Ke value means

low beam slenderness and vice versa.

Figure 5.5 shows that the benefits of moment gradient with high end moment ratios

(β) vary in the case of lateral buckling mode. It appears that αm factors are

unfavourably influenced by the level of web distortion during lateral buckling. At

very low Ke value where the buckling mode is almost an LTB mode, αm factors are

in the region of upper bound results, but the increase of Ke value (ie. increasing web

distortion) reduces the αm factors, thus a variation exists in the results. However, this

variation due to LDB mode can be considered insignificant (neglecting the results

associated with local buckling mode). For the moment gradient case with β = 0.8, the

maximum αm factor was approximately 2.6 (with LTB as the governing mode) and

the assumed lower bound was 2.4 (with LDB as the governing mode), which is only

7.7% difference at the most. While the variation exists for cases with higher



β values, the αm factor variation was found to be almost negligible for cases with

lower β values, i.e. β less than 0. Figure 5.6 shows the typical lateral buckling

deformations under moment gradient action.

Figure 5.6 – Typical Lateral Buckling Modes of LSBs with Moment Gradient

Figure 5.7 provides a comparison of the αm factors based on FEA results and the

current hot-rolled steel design codes AS4100 (SA, 1998), ANSI/AISC 360 (AISI,

2005) and BS5950-1 (BSI, 2000). Comparison with the cold-formed steel design

codes was not made because they adopt similar factors to the hot-rolled steel codes.

Details of the current αm equations are given in Chapter 2. The comparison shown in

Figure 5.7 appears to indicate that the current steel design codes do not provide

accurate predictions. The AS4100 equation in its Table 5.6.1 (Table 2.1 in Chapter

2) and the general equation (Eq. 2.17 in Chapter 2) are unconservative for higher β

value cases, in particular the general equation. The equations in ANSI/AISC 360 and

BS5950 may provide a better prediction, but they are still not very accurate for

LSBs. Further, the comparison shows that the equation from Table 5.6.1 of AS4100

predicted closer results with the upper bound results (associated with LTB mode),

Lateral distortional buckling

Plan view of lateral distortional buckling mode with end moment ratio (β) = 1

Lateral torsional buckling

2.5m LSB300x70x3.0 (β = 0) 6m LSB125x60x2.0 (β = -0.4)

1.5m LSB125x60x2.0



implying that it may be only suitable for LSBs subjected to lateral torsional buckling.

This observation is sensible as the αm equation was originally developed for LTB

mode. This finding highlights the significance of LDB mode on the behaviour of

LSB under moment gradient and the FEA results have showed the inaccuracy of the

current αm equation for LSBs, especially for moment gradients with high β values.

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8


αm

Fac

tor

LTB

LDB

LDB (Less web distortion)

AS4100 Table 5.6.1AS4100 (Eq 2.17)

ANSI/AISC 360-05 (Eq 2.20)

BS5950-1:2000 (Eq 2.21)

Figure 5.7 – Comparison of αm Factors from FE Elastic Buckling Analyses and

Current Design Equations

However, it will be shown later in the section on the discussion of non-linear

analysis results that accurate αm factors for design purposes are not necessarily

important, especially for LSBs subjected to high end moment ratios (β) and also for

LSBs with high Ke values. Therefore the αm factor variation in the high β cases

“considered as small” can be neglected, and the current αm factors (AS4100:1998,

ANSI/AISC 360:2005 and BS5950-1:2000) may be still appropriate for use in

design. Nevertheless, a more accurate αm equation is developed for LSBs subjected

to moment gradient based on the lower bound results from the FE elastic buckling

analyses as shown below.

Upper bound results (LTB mode)

Note: results which are governed by local bucking mode are excluded

Ke = 0.05

Ke = 0.48

Assumed lower bound results (LDB mode)



αm = 1.7 + 0.86 β + 0.16 β2 ≤ 2.25 (5.3)

This new equation is limited to a maximum αm factor of 2.25 as a conservative

measure. This value is closer to the limit of 2.27 given in the general equations of

ANSI/AISC 360 and BS5950-1. The new equation is plotted with the FE results in

Figure 5.8, which shows that it does not reflect the variation due to LDB mode, but

instead the variation for the LDB mode with less web distortion. This was done to

achieve a simple αm equation and also to provide further conservatism.

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8


αm

Fac

tor

LTB

LDB

LDB (Less web distortion)

New Equation (Eq 5.3)

Figure 5.8 – New αm Equation for LSB

5.2.2. Non-linear Static Analysis Results and Discussions of Moment Gradient

Cases

A new αm equation based on the elastic FE buckling analyses is provided in the last

section, but the effect of moment gradient loading on “real beam strength” may be

different. Therefore the application of the new αm factors as determined above to the

“real beam strength” has to be investigated. Non-linear static finite element analyses

were conducted to study the ultimate strength behaviour up to failure, which

accounted for the important parameters in a real beam such as material yielding,

Note: results which are governed by local bucking mode are excluded



residual stresses and initial geometric imperfections. Since non-linear analysis is a

time consuming process, only selected spans and moment gradient cases were

investigated. As for the basic case of uniform moment, the use of negative initial

geometric imperfection is critical for LSBs with moment gradient, and thus it was

adopted in all the finite element analyses of moment gradient cases.

Generally the moment distribution effect on the non-linear behaviour is presented as

a comparison with the basic design curve equation (uniform moment case), however

this approach was not feasible in this research because of:

• The original FE model of LSB has been modified for use in this research as

already discussed in the previous chapter. The modified model is considered

more accurate than the original model. Due to the use of the modified FE model

in this study, the non-linear analysis results are not comparable with the current

LSB design curve (LSB’s Mb equation) developed based on the original model.

• Further, the current LSB design curve was derived from the lower bound FEA

results (Mahaarachchi and Mahendran, 2005d), but the actual FE ultimate

moment capacity of LSBs varies with the section geometry, hence indicating that

a direct comparison with the current LSB design curve may not be adequate as

the current design curve was intentionally developed to be conservative for non-

slender sections, i.e. LSB300x75x3.0 and LSB125x45x2.0.

In this thesis, the comparison to demonstrate the moment gradient benefits from

nonlinear analyses was entirely based on the current FEA results, thus independent

of the current LSB design curve. Table 5.2 summarises the non-linear finite element

analysis results, where Mult and Mult-non are the FEA ultimate moment capacities for

the basic case (uniform moment) and non-uniform moment case, respectively. The

section moment capacities (Ms) obtained using AS/NZS 4600 (SA, 2005) are also

provided, which are conservative for LSBs according to Mahaarachchi and

Mahendran (2005b, d). The results are also presented in a non-dimensionalised

format in Figures 5.9, 5.10 and 5.11, where Mult/My and Mult-non/My are plotted

against the modified beam slenderness (λd=√[My/αmMod]). In these calculations, My

is taken as Ms because the LSB sections are fully effective according to the Design

Capacity Tables of LiteSteel Beams while the αm factors are obtained from the



elastic buckling analyses reported in the last section. Appendix C provides example

calculations used to plot Figures 5.9, 5.10 and 5.11. The results show that the

moment gradient action increases the lateral buckling strength of intermediate and

long span LSBs as a function of the end moment ratio (β).

Table 5.2 – Ultimate Moments of Simply Supported LSBs

Subjected to Moment Gradient

LSB Ms FEA Ultimate Moment (Mult & Mult-non) d x bf x t

Span AS4600 β = -1 β = -0.4 β = 0

(mm) (mm) (kNm) kNm Failure Mode kNm Failure

Mode kNm Failure Mode

750 10.90 11.05 VLB 11.63 INB 11.71 INB 1500 10.90 10.43 LDB 11.55 VLB 11.60 VLB 2500 10.90 9.20 LDB 11.45 VLB 11.52 VLB 4000 10.90 7.25 LDB* 9.65 LDB* 11.03 LDB* 6000 10.90 5.48 LTB 7.43 LTB 8.94 LTB

125 x 45 x 2.0 LSB

10000 10.90 3.83 LTB 5.08 LTB 6.15 LTB 1500 34.65 25.4 LDB 35.3 INB + nLB 35.5 INB + nLB 2500 34.65 19.9 LDB 26.7 LDB 31.0 LDB 4000 34.65 15.9 LDB 21.5 LDB 25.3 LDB 6000 34.65 12.5 LDB* 17.0 LDB* 20.3 LDB*

250 x 60 x 2.0 LSB

10000 34.65 8.6 LTB 11.7 LTB 14.4 LTB 1500 76.05 61.1 LDB 78.7 VLB + nLB 79.3 VLB + nLB 2500 76.05 52.8 LDB 69.5 LDB 78.3 LDB 4000 76.05 43.6 LDB 58.2 LDB 67.5 LDB 6000 76.05 34.2 LDB* 46.3 LDB* 54.9 LDB*

300 x 75 x 3.0 LSB

10000 76.05 23.5 LTB 32.3 LTB 39.6 LTB LDB* Lateral distortional buckling mode with negligible web distortion nLB Non-lateral buckling mode (local buckling near the support) VLB Very small lateral bending mode (close to in-plane bending) INB In-plane bending mode

The common method used in the hot-rolled steel structures design code to take into

account the moment gradient benefits in non-linear behaviour is by applying the αm

factor (based on elastic buckling analysis) directly to the basic design curve as

shown below:

Mb-non = αm Mb ≤ Ms (5.4)

Where, Mb-non = member moment capacity for non-uniform moment case

Mb = member moment capacity for uniform moment case (basic)

Ms = section moment capacity

αm = equivalent uniform moment factor



0.0

0.2

0.4

0.6

0.8

1.0

1.2


Mod

/My ,

Mul

t /My o

r M

ult-

non /

My

Mult/My (beta:-1)Alpha m.Mult/My (beta:-0.4)Mult-non/My (beta:-0.4)Alpha m.Mult/My (beta:0)Mult-non/My (beta:0)


LSB250x60x2.0 based on FE Non-linear Analyses

0.0

0.2

0.4

0.6

0.8

1.0

1.2


Mod

/My ,

Mul

t /My o

r M

ult-

non /

My




My = Ms

Elastic buckling

My = Ms

Elastic buckling

(-) Imperfection

(-) Imperfection



0.0

0.2

0.4

0.6

0.8

1.0

1.2


Mod

/My ,

Mul

t /My o

r M

ult-

non /

My




Equation 5.4 was used to predict the member moment capacities for non-uniform

moments (Mult-non) by multiplying the Mult values with the actual αm factors obtained

from elastic buckling analyses and the results are compared in Figures 5.9, 5.10 and

5.11. The comparison demonstrates that this method may over-predict the actual

moment gradient benefits for LSBs. Although it may not be significant for moment

gradient cases with low β values as seen in the comparison for β = -0.4 cases, it

overestimates the moment gradient benefits for higher β cases as seen in the

comparison for β = 0 cases. The overestimation using this method decreases with

increasing beam slenderness (lateral torsional buckling mode region). Note that a

non-linear analysis for the moment gradient case with positive β values was not

conducted because the αm factor would be very high, increasing the strength of short

and even intermediate span LSBs to the section capacity region. Thus the results in

this case would be often limited to long spans.

The variation can also be demonstrated by a comparison of strength ratios (moment

capacity ratios) of Mult-non / Mult with the αm factors obtained from Equation 5.3 in

My = Ms

Elastic buckling

(-) Imperfection



Figure 5.12, i.e. elastic buckling behaviour vs non-linear ultimate strength behaviour.

As some of the results of Mult-non are limited to the section capacity, they are not

presented in Figure 5.12. Moment gradient effect on the non-linear behaviour is

usually more favourable than in elastic buckling for conventional I-beams because it

ensures yielding of flanges only within a short region closer to the support while the

rest remains elastic. Therefore αm factors can be applied conservatively to the Mb

equation as specified in AS4100 (SA, 1998). Figure 5.13(c) shows the von Mises

yield distribution of 250UB37.3 (hot rolled I-beam) under a moment gradient, β = 0.

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6End Moment Ratio (β )

Stre

ngth

Rat

io (M

ult-

non

/ M

ult)LSB125x45x2.0 (4m)

LSB125x45x2.0 (6m)LSB125x45x2.0 (10m)LSB250x60x2.0 (2.5m)LSB250x60x2.0 (4m)LSB250x60x2.0 (6m)LSB250x60x2.0 (10m)LSB300x75x3.0 (2.5m)LSB300x75x3.0 (4m)LSB300x75x3.0 (6m)LSB300x75x3.0 (10m)

Figure 5.12 – Comparison of Strength Ratios (Mult-non / Mult) and αm factors

(Equation 5.3)

In contrast, the LSB subjected to a uniform moment and LDB mode is generally

governed by uniform flange yielding along the span as well as some web yielding

(on the compressive side) due to web distortion effect as shown in Figure 5.13(a).

While the moment gradient may confine the yielding in LSB sections to be closer to

the support, the FEA results as shown typically in Figure 5.13(b) indicate that it is

not in a short region near the support and the web yielding appears to be greater

(with higher end moment ratio (β)). This may result in greater strength reduction

below elastic buckling strength in comparison to I-beams. These yielding effects

αm curve based on Equation 5.3

Reduced due to non-linear behaviour



may explain why the non-linear analyses provide less moment gradient benefits

compared to the elastic buckling analyses, especially for high β cases, i.e. β = 0. For

LSBs with higher beam slenderness where the LDB mode changes towards the LTB

mode, the unfavourable yielding effects, in particular the web yielding, are

minimised, and hence increases the strength ratios closer to αm factors.

(a) 2.5m LSB250x60x2.0 (β= -1) (b) 2.5m LSB250x60x2.0 (β= 0)

(c) 6m 250UB37.3 (Hot-rolled I-beam) (β = 0)

Figure 5.13 – Typical Yielding Distribution (von Mises) at Failure (First Yield)

on the Inside Surface of the Section

Many cold-formed steel codes use another method of using αm factors in design. In

this method the elastic buckling moment is multiplied by the relevant αm factors and

used in the member capacity calculations (Mb).

Mod-non = αm Mod (5.5)

where; Mod-non = Elastic buckling moment for non-uniform moment case

Mod = Elastic buckling moment for uniform moment case

αm = Equivalent uniform moment factor

Web yielding

Less web yielding

Colour Code for Stress Red: 300 MPa (flange & web fy)

Lateral torsional buckling

Yielding on the flanges only within a short region closer to the support

Uniform flange yielding along the span

Non-uniform flange yielding

Colour Code for Stress Red: 450 MPa (flange fy)Dark Yellow: 380 MPa (web fy)



Figures 5.9, 5.10 and 5.11 clearly indicate that the use of this method is safe and may

be too conservative for design purposes. The conservatism of this method is also

demonstrated in Table 5.3 which compares the strength ratios from FEA results

(Mult-non/Mult) and the current LSB member capacities (Mb-non/Mb). The latter

calculation used Equation 5.5 to account for the moment gradient effects, where the

actual αm factors based on the elastic buckling analyses were used. The current LSB

design curve (Equation 2.23) was used to predict Mb-non and Mb based on the

modified beam slenderness (λd=√[My/αmMod]). This comparison should be still

acceptable since only the ratios (moment gradient benefits) from the current LSB

design rules are employed. Therefore, it is recommended that the design method in

the cold-formed steel code (Equation 5.5) is used in the design of LSBs to allow for

the moment gradient effects as it always provide safer solutions unlike the design

method in the hot-rolled steel structures code.

Table 5.3 – Strength Ratio Comparison of Mult-non / Mult (FEA Results) and

Mb-non / Mb

LSB A= Mult-non / Mult B = Mb-non / Mb A/B d x bf x t

Span β β β

(mm) (mm) -0.4 0.0 -0.4 0.0 -0.4 0.0 750 1.05* 1.06* 1.00 1.00 - - 1500 1.11* 1.11* 1.14 1.27 - - 2500 1.24* 1.25* 1.17 1.30 - - 4000 1.33 1.52 1.17 1.30 1.13 1.17 6000 1.36 1.63 1.17 1.30 1.15 1.25

125 x 45 x 2.0 LSB

10000 1.33 1.61 1.24 1.37 1.07 1.17 1500 1.39* 1.40* 1.12 1.24 - - 2500 1.34 1.56 1.17 1.30 1.14 1.20 4000 1.35 1.59 1.17 1.30 1.15 1.22 6000 1.36 1.63 1.17 1.30 1.16 1.25

250 x 60 x 2.0 LSB

10000 1.36 1.67 1.37 1.52 1.0 1.10 1500 1.29* 1.30* 1.11 1.14 - - 2500 1.32 1.48 1.17 1.30 1.12 1.14 4000 1.34 1.55 1.17 1.30 1.14 1.19 6000 1.35 1.60 1.17 1.30 1.15 1.23

300 x 75 x 3.0 LSB

10000 1.38 1.69 1.21 1.35 1.14 1.25 A Strength ratio from FEA Results B Strength ratio from current LSB member capacity equation (modified with Eq. 5.5) * Not relevant due to failure mode of non-lateral buckling (or limited by section capacity)

Given its conservatism, the use of Equation 5.5 indicates that accurate αm factors for

design purposes are not necessarily important. This means that the “considered



small” αm factor variation in the high β cases (Figure 5.5) as a result from LDB

mode can be neglected. Accurate αm factors are also not important for intermediate

span LSBs subjected to high end moment ratios (β) due to the limiting effect (upper

bound) from the section capacity. Therefore, it can be concluded that as long as

Equation 5.5 is adopted, the current αm factors (AS4100:1998, ANSI/AISC

360:2005 and BS5950-1:2000) as well as the more accurate αm equation (Equation

5.3) are deemed suitable for LSB design purposes. Note that the αm factors adopted

in the current cold-formed steel codes are similar to the hot-rolled steel codes (refer

to Chapter 2).

On the other hand, the contradicting observations of FEA results of elastic buckling

with non-linear analyses as discussed above may raise questions on the accuracy of

FE modelling. Therefore, limited number of experiments of LSBs subjected to

moment gradient were conducted at the Queensland University of Technology Steel

Structure Laboratory to evaluate the findings from FEA. The details of this

experimental study, results and discussions are presented in the next chapter.

5.3. Simply Supported LSBs with Transverse Loads of a Uniformly Distributed

Load (UDL) and a Mid-span Point Load (PL)

When a uniformly distributed load or a mid-span point load is applied to a simply

supported beam, the bending moment varies along the beam as shown in Figure 5.14.

Figure 5.14 – Simply Supported LSB Subjected to Transverse Loads

(UDL and PL)

q

Uniformly Distributed Load (UDL)

Bending moment

distribution

P

Mid-span Point Load (PL)



5.3.1. Elastic Buckling Analysis Results and Discussions of UDL and PL Cases

Finite element analyses were conducted for simply supported LSBs subjected to a

uniformly distributed load and a mid-span point load. Three LSB sections with spans

ranging from 750 to 10000 mm were analysed. The elastic buckling moment results

(Mod-non) and the equivalent uniform moment or moment distribution factors

calculated using Eq.5.1 are given in Tables 5.4 and 5.5. The results in Tables 5.4 and

5.5 are also plotted against the modified torsion parameter (Ke) in Figures 5.15 and

5.16, respectively. The FEA results show that higher strength benefit is provided for

the PL case because of less severe moment distribution than in the UDL case, i.e.

high moment region is concentrated at mid-span.


Subjected to a Uniformly Distributed Load (UDL)

LSB Current αm Factors d x bf x t

Span FEA Buckling Moment (Mod-non) AS4100 AISC BS5950

(mm) (mm) (kNm) Mode

αm Factor AS4100

Table 5.6.1 Eq 2.17 Eq 2.20 Eq 2.21 750 29.76 LB - 1.13 1.17 1.14 1.08

1000 25.83 LDB+ 1.039 1.13 1.17 1.14 1.08 1500 19.76 LDB 1.108 1.13 1.17 1.14 1.08 2500 13.48 LDB 1.121 1.13 1.17 1.14 1.08 4000 8.98 LDB* 1.122 1.13 1.17 1.14 1.08 6000 6.12 LTB 1.119 1.13 1.17 1.14 1.08

125 x 45 x 2.0 LSB

10000 3.74 LTB 1.122 1.13 1.17 1.14 1.08 1500 29.62 nLB - 1.13 1.17 1.14 1.08 2000 30.80 LDB+ 0.988 1.13 1.17 1.14 1.08 2500 27.34 LDB 1.061 1.13 1.17 1.14 1.08 3000 24.47 LDB 1.092 1.13 1.17 1.14 1.08 4000 20.28 LDB 1.115 1.13 1.17 1.14 1.08 6000 14.99 LDB* 1.121 1.13 1.17 1.14 1.08

250 x 60 x 2.0 LSB

10000 9.65 LTB 1.122 1.13 1.17 1.14 1.08 1500 96.53 nLB - 1.13 1.17 1.14 1.08 2500 79.91 LDB+ 1.038 1.13 1.17 1.14 1.08 3000 70.97 LDB 1.084 1.13 1.17 1.14 1.08 4000 58.08 LDB 1.113 1.13 1.17 1.14 1.08 6000 42.55 LDB* 1.125 1.13 1.17 1.14 1.08

300 x 75 x 3.0 LSB

10000 27.15 LTB 1.126 1.13 1.17 1.14 1.08 LDB* Lateral distortional buckling mode with negligible web distortion LDB+ Lateral distortional buckling mode with shear buckling near the supports (interaction) nLB Non-lateral buckling mode (shear buckling near the supports) that precedes LDB




Subjected to a Mid-span Point Load (PL)




αm Factor AS4100

Table 5.6.1 Eq 2.17 Eq 2.20 Eq 2.21 750 36.41 nLB - 1.35 1.39 1.32 1.18

1000 30.79 LDB+ 1.239 1.35 1.39 1.32 1.18 1500 23.58 LDB 1.323 1.35 1.39 1.32 1.18 2500 16.04 LDB 1.334 1.35 1.39 1.32 1.18 4000 10.69 LDB* 1.337 1.35 1.39 1.32 1.18 6000 7.34 LTB 1.341 1.35 1.39 1.32 1.18

125 x 45 x 2.0 LSB

10000 4.48 LTB 1.345 1.35 1.39 1.32 1.18 1500 34.17 nLB - 1.35 1.39 1.32 1.18 2000 36.41 LDB+ 1.168 1.35 1.39 1.32 1.18 2500 32.68 LDB 1.269 1.35 1.39 1.32 1.18 3000 29.27 LDB 1.309 1.35 1.39 1.32 1.18 4000 24.20 LDB 1.330 1.35 1.39 1.32 1.18 6000 17.81 LDB* 1.332 1.35 1.39 1.32 1.18

250 x 60 x 2.0 LSB

10000 11.49 LTB 1.336 1.35 1.39 1.32 1.18 1500 116.3 nLB - 1.35 1.39 1.32 1.18 2500 95.64 LDB+ 1.243 1.35 1.39 1.32 1.18 3000 84.96 LDB 1.297 1.35 1.39 1.32 1.18 4000 69.50 LDB 1.332 1.35 1.39 1.32 1.18 6000 50.72 LDB* 1.341 1.35 1.39 1.32 1.18

300 x 75 x 3.0 LSB

10000 32.51 LTB 1.348 1.35 1.39 1.32 1.18 LDB* Lateral distortional buckling mode with negligible web distortion LDB+ Lateral distortional buckling mode with web local buckling at mid-span (interaction) nLB Non-lateral buckling mode (web local buckling at mid-span) that precedes LDB

The FEA results also show the variation of αm factors for the UDL and PL cases,

which indicate that they are also unfavourably influenced by lateral distortional

buckling (LDB). The αm factors appear to be a function of the modified torsion

parameter (Ke). They decrease with increasing Ke or lower beam slenderness due to

the presence of higher level of web distortion during lateral buckling.

Comparison with the currently used αm factors in Figures 5.15 and 5.16 indicates

that although the presence of web distortion leads to lower αm factors, the αm factor

increases to that predicted by AS4100 Table 5.6.1 and ANSI/AISC 360-05’s general

equation as the web distortion diminishes with increasing beam slenderness

(decreasing Ke values). This supports the previous finding in the case of moment

gradient about the suitability of αm factors for beams subjected to LTB. It is also



evident that the current αm factors for the UDL case are unconservative as they are

given as constant values (1.13 for UDL and 1.35 for PL based on AS4100 Table

5.6.1), independent of beam and section slenderness. This does not reflect the

observed αm variation due to web distortion

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

0.0 0.2 0.4 0.6 0.8 1.0Modified Torsion Parameter (Ke)

αm

Fac

tor

LSB250x60x2.0 (slender)LSB300x75x3.0 (non-compact)LSB125x45x2.0 (compact)AS4100-1998 Table 5.6.1AS4100-1998 (Eq 2.17)ANSI/AISC 360-05 (Eq 2.20)BS5950-1:2000 (Eq 2.21)

Figure 5.15 – αm Factors for the UDL Case Based on Elastic Buckling Analyses

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4


αm

Fac

tor


Figure 5.16 – αm Factors for the PL Case Based on Elastic Buckling Analyses

New αm equation based on elastic buckling analysis (see Eq. 5.6)

Ke = √[π2EIw/GJeL2]

Lateral buckling (LTB to LDB) nLB Interaction

New αm equation based on elastic buckling analysis (see Eq. 5.7)





The αm factor variation also demonstrates that it is more severe for LSBs with

transverse loading compared to the moment gradient case. In the moment gradient

case, the variation only occurs in the case of high end moment ratio (β) and can be

considered small. This may be because unlike for LSBs subject to a moment

gradient, shear stresses are not negligible for the transverse loading cases, which

appear to increase the reduction of non-uniform moment benefits due to web

distortion effect of LDB mode. A study by Ma and Hughes (1996) also showed the

significant effects of uniformly distributed vertical load on the lateral distortional

buckling strength of monosymmetric I-beams in comparison to the uniform moment

case.

When the beam span is reduced (higher Ke value), stresses may become more critical

causing LSBs to be governed by other buckling modes, which do not represent

lateral buckling and also precede lateral distortional buckling (providing lower

buckling load) as noted in Tables 5.4 and 5.5. For the UDL case, this non-lateral

buckling mode is shear buckling near both supports, and for the PL case it is local

web buckling at mid-span, which can also be considered as web bearing buckling at

mid-span. Figures 5.17(a), (b) and (c) show these other buckling modes of LSBs

with high Ke value (lower beam slenderness). But before this non-lateral buckling

mode completely becomes the governing mode, there will be a transition from pure

lateral buckling mode to the non-lateral buckling mode which appears to be an

interaction of the two buckling modes. Figures 5.15 and 5.16 show that LSBs will be

subjected to an interaction buckling when the Ke value is greater than 0.8

(approximately) for both UDL and PL cases and thereafter the shear stresses will be

more dominant and thus at some stage the non-lateral buckling mode will be the

governing mode. Figures 5.17(d) and (e) show the interaction buckling of lateral

distortional buckling with non-lateral buckling mode for UDL and PL cases,

respectively. The FEA results suggest that the interaction buckling may also reduce

the buckling load to be lower than that of pure lateral distortional buckling.



(a) 1.5m LSB300x75x3.0 with a UDL

(b) 1.5m LSB300x75x3.0 with a UDL (c) 1.5m LSB250x60x2.0 with a PL

(d) 2m LSB250x60x2.0 with a UDL (e) 2m LSB250x60x2.0 with a PL

Figure 5.17 – Other Critical Buckling Modes of LSBs Subjected to

Transverse Loads

The pattern of αm factor variation seen in Figures 5.15 and 5.16 is also quite

interesting as it appears to keep decreasing with increasing Ke value. Figure 5.18

shows the moment distribution effect in terms of buckling moment, indicating that

the decreasing αm factor is as a result of differences in the slope of buckling load

curves of transverse loading case compared with uniform moment case. This means

when they intersect, the αm factor is one and is less than one beyond it (i.e. LSB’s

lateral buckling strength is below that of the basic uniform moment case). But again,

Interaction of lateral distortional buckling with shear buckling near the support

Interaction of lateral distortional buckling with local web buckling (bearing) at mid-span

Local web buckling (bearing) at mid-span

Shear buckling near the support

Front view (deformation)




in that region FEA predicted non-lateral buckling (as well as interaction buckling),

which will be most likely to be the actual critical mode. It is also close to the local

buckling region, where moment distribution effect is negligible and is normally not

considered.

0

10

20

30

40

50

60

70

0 2000 4000 6000 8000 10000Span (mm)

Elas

tic B

uckl

ing

Mom

ent (

kNm

)

THINWALL (lateral buckling load for uniform moment)THINWALL (critical buckling load for uniform moment)FEA (critical buckling load for uniform moment)FEA (critical buckling load for UDL case)FEA (critical buckling load for PL case)

Figure 5.18 – Comparison of Typical Elastic Lateral Buckling Moment versus

Span curves for Transverse Loads (UDL and PL) and Uniform Moment Cases

The interaction of bending and shear actions is a natural cause from 3D modelling

using shell elements and it is inevitable to exclusively treat one of these components.

Just like in the moment gradient case, the results with shear stresses are dominant

(including the interaction) and are best ignored. Due to this reason, the buckling

analysis was then limited. Higher eigenmodes that represent LDB mode were

searched, however, they were always associated with interaction buckling. Other

buckling analysis technique such as energy methods may have to be used in future

research to be able to derive solutions exclusively for lateral distortional buckling

(particularly in the high Ke region).

Nevertheless, it is true that the αm factor should not be less than one due to the non-

lateral buckling mode. A more accurate αm factor prediction than the current factors

LSB250x60x2.0

Expected buckling curve direction for shorter span

Non-lateral buckling mode as well as interaction



can therefore be obtained by developing an equation as a function of Ke to reflect the

level of web distortion of LDB and assuming the lower bound of αm factor as 1.0.

UDL: αm = 1.125 - 0.145 Ke2 + 0.008 Ke (1 ≤ αm ≤ 1.125) (5.6)

PL: αm = 1.34 - 0.25 Ke2 + 0.06 Ke (1 ≤ αm ≤ 1.34) (5.7)

Equations 5.6 and 5.7 were derived based on the best fit of data in Figures 5.15 and

5.16 with an average percentage error less than 1% for the UDL and PL cases,

respectively. They are more complex than that developed for the moment gradient

case (Eq. 5.3), which is treated as independent of Ke because the variation due to

web distortion effect is not significant and occurs only in the case of high β values.

This will be further confirmed in the discussion of non-linear analyses. Nevertheless,

these equations are considered simple to use as all the required parameters to

calculate the modified torsion parameter (Ke) are readily available from the Design

Capacity Tables of LSBs.

5.3.2. Non-linear Static Analysis Results and Discussions of UDL and PL Cases

Non-linear static FE analyses were conducted for UDL and PL cases to observe the

effects on LSB’s ultimate strength. Preliminary analyses were conducted prior to a

detailed analysis to study the effect of imperfection direction as summarised in Table

5.6. The comparison shows that unlike in the case of moment only, negative initial

geometric imperfection does not always yield lower capacity solutions for LSBs with

a transverse load. Based on this preliminary analysis, it can be said that when the

web distortion of LDB is quite dominant, the associated web yielding with positive

imperfection will be large due to the applied loading directly on the web. This may

result in greater strength reduction below elastic buckling strength than with negative

imperfection. Figure 5.19 compares the typical yielding distribution for LSBs with

positive and negative imperfections subjected to a UDL and LDB mode. Table 5.6

also shows that negative imperfection is always the critical case for a compact

section (LSB125x45x2.0). This may be because the web distortion is not as severe as

for slender sections (LSB250x60x2.0). Therefore the selection of imperfection

direction in the non-linear FE analysis was based on whichever provides the critical

solution (lower ultimate moment).



Table 5.6 – Effects of Initial Geometric Imperfection Direction on

the Ultimate Moments of LSBs

LSB section d x bf x t Span Ms

AS4600

FEA Ultimate Moment (kNm) for

UDL case

FEA Ultimate Moment (kNm) for

PL case

(mm) (mm)

Ke

(kNm) (-) IMP (+) IMP (-) IMP (+) IMP

Failure Mode

750 1.30 10.90 - - - - INB + nLB 1500 0.79 10.90 10.26 11.75 10.44 12.09 LDB 2500 0.42 10.90 9.44 10.31 10.95 11.00 LDB 4000 0.21 10.90 7.76 8.30 8.72 9.44 LDB*

125 x 45 x 2.0 LSB

6000 0.08 10.90 6.05 6.34 6.94 7.30 LTB 1500 1.57 34.65 - - - - INB + nLB 2500 0.65 34.65 22.81 20.47 26.63 22.50 LDB 3000 0.48 34.65 20.81 19.35 24.30 21.98 LDB 4000 0.32 34.65 17.86 17.44 20.90 20.10 LDB 6000 0.19 34.65 13.86 14.13 16.20 16.35 LDB*

250 x 60 x 2.0 LSB

10000 0.10 34.65 9.58 10.54 11.18 11.93 LTB IMP Imperfection direction LDB* Lateral distortional buckling mode with negligible web distortion nLB Non-lateral buckling mode (shear or web bearing failure) INB In-plane bending mode

(a) With positive initial imperfection (b) With negative initial imperfection

Figure 5.19 – Yielding Distribution (von Mises) at Failure (First Yield) of

3m LSB2506020 with a UDL

The non-linear FEA results for UDL and PL cases are summarised in Tables 5.7 and

5.8, respectively. The strength ratios of Mult-non / Mult are again used to represent the

moment distribution effect in the non-linear analyses. It clearly shows that a

variation exists in the results in that the non-uniform moment distribution benefits

Large web yielding




are reduced with increasing level of web distortion (decreasing beam slenderness).

This reduction is similar to the findings based on elastic buckling analyses.

Table 5.7 – Ultimate Moments of Simply Supported LSBs Subjected to a UDL

LSB Ms Critical Mult-non / Mult D x bf x t

Span AS4600 Buckling

FEA Ultimate Moment (Mult-non) Strength

(mm) (mm) (kNm) (kNm) (kNm) Mode Ratio

αm Factor

Mb-non / Mb

750 10.90 29.76 10.30 INB + nLB - - - 1500 10.90 19.76 10.26 LDB 0.98 1.108 1.03 2500 10.90 13.48 9.44 LDB 1.03 1.121 1.06 4000 10.90 8.98 7.76 LDB* 1.07 1.122 1.06

125 x 45 x 2.0 LSB

6000 10.90 6.12 6.05 LTB 1.10 1.119 1.06 1500 34.65 29.62 28.01 INB + nLB - - - 2500 34.65 27.34 20.47 LDB 1.03 1.061 1.03 3000 34.65 24.47 19.35 LDB 1.06 1.092 1.04 4000 34.65 20.28 17.44 LDB 1.10 1.115 1.06 6000 34.65 14.99 13.86 LDB* 1.11 1.121 1.06

250 x 60 x 2.0 LSB

10000 34.65 9.65 9.58 LTB 1.11 1.122 1.12 1500 76.05 96.53 75.94 INB + nLB - - - 2500 76.05 79.91 50.44 LDB+ 0.96 1.038 1.02 3000 76.05 70.97 48.47 LDB 1.00 1.084 1.04 4000 76.05 58.08 45.36 LDB 1.04 1.113 1.05

300 x 75 x 3.0 LSB

6000 76.05 42.55 37.91 LDB* 1.11 1.125 1.06 LDB* Lateral distortional buckling mode with negligible web distortion LDB+ Lateral distortional buckling mode with shear failure near the supports (interaction) nLB Non-lateral buckling mode (shear failure near the supports) INB In-plane bending mode Mb-non / Mb Strength ratio from current LSB member capacity equation (modified by using Eq. 5.5)

Typical non-dimensionalised strength curves that represent the moment distribution

effects for both transverse load cases are shown in Figures 5.20, and 5.21, where

Mu/My and Mu-non/My are plotted against the modified beam slenderness

(λd=√[My/αmMod]). In these calculations, My is taken as equal to Ms (LSB sections

are fully effective) while the αm factors are based from the elastic buckling analyses

reported in the last section. Appendix C provides example calculations used to plot

Figures 5.20 and 5.21. The figures also provide a prediction (red curves) using the

method commonly used in the hot-rolled steel code, Mult-non = αmMult (see Equation

5.4).



Table 5.8 – Ultimate Moments of Simply Supported LSBs Subjected to a PL

LSB Ms Critical Mult-non / Mult D x bf x t


FEA Ultimate Moment (Mult-non) Strength


αm Factor

Mb-non / Mb


125 x 45 x 2.0 LSB

6000 10.90 10.69 6.94 LTB 1.27 1.341 1.16 1500 34.65 34.17 20.81 INB + nLB - - - 2500 34.65 32.68 22.50 LDB 1.13 1.269 1.13 3000 34.65 29.27 21.98 LDB 1.20 1.309 1.14 4000 34.65 24.20 20.10 LDB 1.26 1.330 1.15 6000 34.65 17.81 16.35 LDB* 1.31 1.332 1.15

250 x 60 x 2.0 LSB

10000 34.65 11.49 11.93 LTB 1.39 1.336 1.34 1500 76.05 116.3 80.55 INB + nLB - - - 2500 76.05 95.64 51.68 LDB+ 0.98 1.243 1.11 3000 76.05 84.96 52.74 LDB 1.07 1.297 1.14 4000 76.05 69.50 51.24 LDB 1.18 1.332 1.15

300 x 75 x 3.0 LSB

6000 76.05 50.72 43.56 LDB* 1.27 1.341 1.16 LDB* Lateral distortional buckling mode with negligible web distortion LDB+ Lateral distortional buckling mode with a web bearing failure at mid-span (interaction) nLB Non-lateral buckling mode (web bearing failure at mid-span) INB In-plane bending mode Mb-non / Mb Strength ratio from current LSB member capacity equation (modified by using Eq. 5.5)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.4 0.6 0.8 1.0 1.2 1.4 1.6Modified Beam Slenderness (λd)

Mod

/My ,

Mul

t /My o

r M

ult-

non /

My

Mult/My (uniform moment)Alpha m.Mult/My (UDL case)Mult-non/My (UDL case)

Figure 5.20 – Moment Distribution Effects of UDL on Simply Supported

LSB125x45x2.0 based on Non-linear FE Analyses

My = Ms

Elastic buckling

UDL



0.0

0.2

0.4

0.6

0.8

1.0

1.2


Mod

/My ,

Mul

t /My o

r M

ult-

non /

My

Mult/My (uniform moment)Alpha m.Mult/My (UDL case)Mult-non/My (UDL case)

Figure 5.21 – Moment Distribution Effects of PL on Simply Supported

LSB250x60x2.0 based on Non-linear FE Analyses

Figures 5.20 and 5.21 demonstrate that the method in the hot-rolled structures code

may also over-predict the actual moment distribution benefits for LSBs subject to a

transverse load. Thus it is apparent that the other common method used in many

cold-formed steel codes of using the appropriately modified Mod using the αm factors

(Equation 5.5) in the member capacity calculation is more suitable for LSBs. Its

suitability is better shown in Tables 5.7 and 5.8, which provide a comparison of the

strength ratios from FEA (Mult-non/Mult) and the current LSB member capacities (Mb-

non/Mb). The latter calculation used Equation 5.5 to account for the moment gradient

effects with the actual αm factors based on the elastic buckling analyses reported in

the last section. The current LSB design curve (Eq. 2.23) was used to predict Mb-non

and Mb based on the modified beam slenderness (λd=√[My/αmMod]). Appendix C

provides example calculations.

The strength reduction in transverse load cases seems to be greater than in the

moment gradient case because the direct loading into the web may cause greater

yielding in the web while it distorts during LDB. Further, the non-linear FE analyses

for both transverse load cases were also limited to other failure modes (non-lateral

buckling mode), resembling the critical buckling mode based on the elastic buckling

My = Ms

Elastic buckling

Governed by web bearing failure or interaction with LDB

PL



analysis. For LSBs with higher Ke value (i.e. short to intermediate spans), the

ultimate strength will be governed by shear failure when subjected to a UDL and by

web bearing failure when subjected to a PL. In the transition zone, the FEA results

also show a failure formed by an interaction of LDB with shear or web bearing

failure. Figure 5.22 illustrates these failure modes. Note that these results are not

plotted in Figures 5.20 and 5.21. This may suggest that the design for UDL and PL

cases against bending can be ignored for LSBs with high Ke values, and instead it

shall be based on the criteria for shear strength or web bearing.

Experimental evidence may be important and useful in future research, particularly

for the interaction cases. In fact, a research project on the shear strength of LSB is

currently being undertaken at the Queensland University of Technology. In practice

it is common that a web stiffener would be used to prevent the shear and web bearing

failures, which can make the lateral buckling to be the governing mode. However, it

was not considered in this study because the inclusion of a web stiffener in the FE

model would restrain the web distortion and increase the lateral buckling strength

(changing LDB towards LTB mode) and hence would not allow the determination of

the moment distribution effect. Nevertheless, the Ke region for LSBs where shear

stresses become critical is actually close to the local buckling region as mentioned

earlier in the elastic buckling analysis discussion.

(a) 1.5m LSB300x75x3.0 with a UDL (b) 1.5m LSB250x60x2.0 with a PL

Figure 5.22 – Typical Failures other than Lateral Buckling Mode for Short to

Intermediate Span LSBs with Transverse loads (UDL and PL)

Web bearing failure at mid-span

Shear failure near the supports



Therefore, based on the elastic buckling and non-linear analyses it is recommended

that when shear stresses become important (i.e. Ke greater than 0.8), the moment

distribution effect should be ignored (αm = 1). Further, Figures 5.20 and 5.21 also

indicate that with the proposed αm equations for transverse loading cases, LSBs,

which are less prone to LDB mode (i.e. LSB125x45x2.0), will gain the moment

distribution benefits for a greater range of beam slenderness because the region

where shear stresses become important is closer to the section capacity region

(normally in the local buckling region). For slender LSBs (i.e. LSB250x60x2.0), the

range of beam slenderness will be small.

5.4. Simply Supported LSBs with Quarter Point Loads (QL)

Two point loads at quarter points of span in a simply supported beam produce a

bending moment distribution as shown in Figure 5.23. This transverse load case is

not presented together with the other two studied cases (UDL and PL) for the

purpose of better presentation.

Figure 5.23 – Simply Supported LSB Subjected to Quarter Point Loads (QL)

5.4.1. Elastic Buckling Analysis Results and Discussions of QL Cases

Finite element analyses were conducted for simply supported LSBs subjected to

quarter point loads (QL). Three LSB sections with spans ranging from 0.75 to 10 m

were analysed. The elastic buckling moment results (Mod-non) and the equivalent

uniform moment or moment distribution factors calculated using Equation 5.1 are

given in Table 5.9 and Figure 5.24.

Bending moment distribution

P P




Subjected to Quarter Point Loads (QL)




αm Factor AS4100

Table 5.6.1 Eq. 2.17 Eq. 2.20 Eq. 2.21 750 25.05 nLB - 1.09 1.00 1.00 1.00 1000 22.89 LDB+ 0.921 1.09 1.00 1.00 1.00 1500 18.13 LDB 1.017 1.09 1.00 1.00 1.00 2500 12.44 LDB 1.035 1.09 1.00 1.00 1.00 4000 8.29 LDB* 1.037 1.09 1.00 1.00 1.00 6000 5.67 LTB 1.037 1.09 1.00 1.00 1.00

125 x 45 x 2.0 LSB

10000 3.45 LTB 1.037 1.09 1.00 1.00 1.00 1500 23.31 nLB - 1.09 1.00 1.00 1.00 2000 21.78 nLB - 1.09 1.00 1.00 1.00 2500 24.61 LDB+ 0.955 1.09 1.00 1.00 1.00 3000 22.27 LDB 1.000 1.09 1.00 1.00 1.00 4000 18.62 LDB 1.024 1.09 1.00 1.00 1.00 6000 13.82 LDB* 1.034 1.09 1.00 1.00 1.00

250 x 60 x 2.0 LSB

10000 8.91 LTB 1.036 1.09 1.00 1.00 1.00 1500 71.6 nLB - 1.09 1.00 1.00 1.00 2500 71.42 LDB+ 0.928 1.09 1.00 1.00 1.00 3000 64.34 LDB+ 0.983 1.09 1.00 1.00 1.00 4000 53.31 LDB 1.022 1.09 1.00 1.00 1.00 6000 39.24 LDB* 1.038 1.09 1.00 1.00 1.00

300 x 75 x 3.0 LSB

10000 25.18 LTB 1.040 1.09 1.00 1.00 1.00 LDB* Lateral distortional buckling mode with negligible web distortion LDB+ Lateral distortional buckling mode with other buckling mode (shear, web local buckling or both) nLB Non-lateral buckling mode (shear, web local buckling or both) that precedes LDB

The results clearly show that the benefits for the QL case are very small for LSBs.

The highest αm factor was found to be 1.04 (4% improvement from the basic case of

uniform moment), which was obtained for LSBs subject to LTB. This is because the

bending moment distribution for the QL case is closer to a uniform moment

distribution, thus only a negligible benefit will result. As for the other two transverse

load cases, the level of web distortion in LDB mode was also found to be

unfavourable to the already small strength benefits for the QL case.



0.5

0.6

0.7

0.8

0.9

1.0

1.1

0.0 0.2 0.4 0.6 0.8Modified Torsion Parameter (Ke)

αm

Fac

tor


Figure 5.24 – αm Factors for the QL Case Based on Elastic Buckling Analyses.

Figure 5.24 shows that the web distortion effect may diminish the benefit due to the

non-uniform moment distribution and even reduce the lateral buckling strength

below the basic case of uniform moment. This further suggests the significance of

transverse loading in increasing the web distortion effect for LSBs. However, the

results also indicate that with increasing Ke value, the associated shear stresses are

more important where at Ke value greater than 0.6 (approximately), the buckling

analyses indicate an interaction with either shear or local web buckling (or web

bearing buckling) or both precede lateral distortional buckling. When the shear

stresses are more dominant than the bending stresses, these non-lateral buckling

modes would be the governing mode as shown in Figure 5.25. The elastic buckling

analysis for lateral buckling was again limited due to this shear and bending

interaction. As in the other two transverse load cases, it is reasonable to ignore the

moment distribution effect for LSBs subjected to non-lateral buckling modes as well

as the interaction buckling mode. However, this may still leave a short Ke region in

LSBs which has a slight unfavourable moment distribution effect as indicated in

Figure 5.24, but nevertheless this is not critical for design purposes.

nLB Lateral buckling (LTB to LDB) Interaction

αm equation based on elastic buckling analysis (see Eq 5.8)




Figure 5.25 – Other Critical Buckling Modes (Both Shear and Local Web

Buckling) for 1.5m LSB250x60x2.0 Subjected to QL

From the above discussion, it appears that completely neglecting the benefits for QL

is an ideal solution as it is not worth increasing the design complexity for a small

moment capacity improvement at the most. Comparison with the currently used αm

factors generally agree with this recommendation as they also recommend an αm

factor equal to 1.0, except for the prediction by AS4100 Table 5.6.1 (αm = 1.09). The

prediction of Table 5.6.1 in AS4100 is very unconservative and is not even close to

the FEA results of the case subjected to LTB mode. Hence its application is not

suitable for LSBs. Nevertheless, if a more accurate prediction is required, then the

following αm equation can be used to take into account the benefits from QL with an

assumption that αm shall not be less than one.

αm = 1.04 - 0.22 Ke2 + 0.025 Ke (1.0 ≤ αm ≤ 1.04) (5.8)

5.4.2. Non-linear Static Analysis Results and Discussions of QL Cases

The non-linear analysis results for the QL case and the associated strength ratios of

Mult-non/Mult are summarised in Table 5.10. The moment distribution benefits based

on non-linear analysis are small and also reduced with decreasing beam slenderness

(unfavourable web distortion effect). When the beam slenderness is low (high Ke

value) the results were also found to be limited by other failure modes (either shear,

web bearing failure or both) due to the associated high shear stresses.

Web local buckling (bearing) at loading point


Top view

Front view



Table 5.10 – Ultimate Moments of Simply Supported LSBs Subjected to QL

LSB Ms Critical Mult-non / Mult d x bf x t


FE Ultimate Moment (Mult-non) Strength


αm Factor

Mb-non / Mb


125 x 45 x 2.0 LSB

6000 10.90 8.29 5.66 LTB 1.03 1.037 1.02 1500 34.65 23.31 22.50 INB + nLB - - - 2500 34.65 24.61 18.77 LDB+ 0.95 0.96 0.98 3000 34.65 22.27 17.93 LDB 1.00 1.00 1.00 4000 34.65 18.62 16.20 LDB 1.02 1.024 1.01 6000 34.65 13.82 12.89 LDB* 1.03 1.034 1.02

250 x 60 x 2.0 LSB

10000 34.65 8.91 8.95 LTB 1.04 1.036 1.04 1500 76.05 71.6 43.00 INB + nLB - - - 2500 76.05 71.42 45.89 LDB+ 0.90 0.928 0.96 3000 76.05 64.34 44.64 LDB+ 0.98 0.983 1.00 4000 76.05 53.31 42.24 LDB 1.02 1.022 1.01

300 x 75 x 3.0 LSB

6000 76.05 39.24 35.46 LDB* 1.04 1.038 1.02 LDB* Lateral distortional buckling mode with negligible web distortion LDB+ Lateral distortional buckling mode with web bearing failure at mid-span (interaction) nLB Non-lateral buckling mode (shear, web bearing failure or both) that precedes LDB INB In-plane bending mode

Failure by interaction mode was also found in the transition zone as indicated from

the buckling analysis. These findings confirm the recommendations based on the

buckling analysis that the moment distribution effects from QL for LSBs can be

simply neglected. Nevertheless, it should be noted that the recommendations

provided in this study are based on LSBs with unstiffened web. Stiffened LSBs may

have higher shear strengths, but at the same time also improve the lateral bending

strength, affecting the moment distribution effect. Other analysis techniques such as

energy methods may have to be used in future research to be able to derive solutions

exclusively for lateral buckling.

5.5. Cantilever LSBs with a Uniformly Distributed Load (UDL) and a Point

Load (PL) at the Free End

Cantilevers are usually considered to be flexural members which in the plane of

loading are built-in at the support (fully fixed end) as shown in Figure 5.26 and

unrestrained or free at the other end (Trahair, 1993). This ideal cantilever condition



which provides a full restraint against warping and twisting at the support was

adopted in this research. Other cantilever conditions that are not fully fixed at the

support such as cantilever formed by overhang segment of continuous beam were not

considered.

Figure 5.26 – Cantilever LSB Subjected to Transverse Loads (UDL & PL)

Two transverse load cases were considered, a uniformly distributed load (UDL) and

a point load at the cantilever free end (PL). The bending moment distribution on a

cantilever subjected to these two loading types are shown in Figure 5.26.

5.5.1. Elastic Buckling Analysis Results and Discussions of Cantilever LSBs

(UDL and PL)

The elastic buckling results for cantilever LSBs subjected to a PL and a UDL are

summarised in Tables 5.11 and 5.12, and the associated αm factors are plotted

against the modified torsion parameter (Ke) in Figures 5.27 and 5.28, respectively.

Figures 5.27 and 5.28 show that a variation exists in the results, which show that the

αm factor for cantilever LSBs is significantly increased with increasing Ke value.

This significant benefit is expected because the moment is confined to a short region

near the support and the presence of the rigid support. For cases with high beam

slenderness (low Ke value), the αm factor is low because the instability effect due to

the free end condition is significant. A study by Nethercot (1971) reported that the

αm factor for cantilevers of conventional hot-rolled I-beams under shear centre

loading is also increased as a function of beam slenderness.

q P

BMD BMD Bending moment

distribution

Cantilever LSB with a UDL Cantilever LSB with a PL

Rigid support Rigid support



Table 5.11 – Elastic Lateral Buckling Moments of Cantilever LSBs

Subjected to a Point Load at the Free End (PL)


Span FEA Buckling Moment (Mod-non) BS5950


αm Factor AS4100

Table 5.6.2 AISC Le = 0.8L

750 57.35 nLB* - 1.25 1.00 1.39 1000 48.72 LDB 1.96 1.25 1.00 1.27 1500 31.10 LDB* 1.74 1.25 1.00 1.19 2500 18.01 LDB* 1.50 1.25 1.00 1.21

125 x 45 x 2.0 LSB

4000 11.18 LTB 1.40 1.25 1.00 1.24 1500 54.65 nLB - 1.25 1.00 1.42 2000 56.69 nLB - 1.25 1.00 1.31 2500 50.23 LDB 1.95 1.25 1.00 1.23 3000 41.14 LDB* 1.84 1.25 1.00 1.19 4000 30.12 LDB* 1.66 1.25 1.00 1.17

250 x 60 x 2.0 LSB

6000 19.87 LTB 1.49 1.25 1.00 1.20 1500 158.58 nLB* - 1.25 1.00 1.46 2500 149.46 LDB 1.94 1.25 1.00 1.27 3000 123.29 LDB* 1.88 1.25 1.00 1.22 4000 88.83 LDB* 1.70 1.25 1.00 1.19

300 x 75 x 3.0 LSB

6000 57.31 LTB 1.52 1.25 1.00 1.20 LDB* Lateral distortional buckling mode with negligible web distortion nLB Non-lateral buckling mode (local buckling near the support) that precedes LDB nLB* Non-lateral buckling mode (web local buckling at the load position) that precedes LDB

Table 5.12 – Elastic Lateral Buckling Moments of Cantilever LSBs

Subjected to a Uniformly Distributed Load (UDL)


Span FEA Buckling Moment (Mod-non) BS5950


αm Factor AS4100

Table 5.6.2 AISC Le = 0.8L

750 58.90 nLB - 2.25 1.00 1.39 1000 64.32 nLB - 2.25 1.00 1.27 1500 51.96 LDB* 2.91 2.25 1.00 1.19 2500 29.53 LDB* 2.46 2.25 1.00 1.19

125 x 45 x 2.0 LSB

4000 17.92 LTB 2.24 2.25 1.00 1.24 2500 54.50 nLB - 2.25 1.00 1.23 3000 55.85 nLB - 2.25 1.00 1.19 4000 50.16 LDB* 2.76 2.25 1.00 1.17

250 x 60 x 2.0 LSB

6000 32.58 LTB 2.44 2.25 1.00 1.20 2500 175.31 nLB - 2.25 1.00 1.27 3000 183.01 nLB - 2.25 1.00 1.22 4000 150.05 LDB* 2.88 2.25 1.00 1.19

300 x 75 x 3.0 LSB

6000 95.90 LTB 2.54 2.25 1.00 1.20 LDB* Lateral distortional buckling mode with negligible web distortion nLB Non-lateral buckling mode (local buckling near the support) that precedes LDB



0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0


αm

Fac

tor

LSB250x60x2.0 (slender)LSB300x75x3.0 (non-compact)LSB125x45x2.0 (compact)

Figure 5.27 – αm Factors for the PL Case Based on Elastic Buckling Analyses

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.0 0.1 0.2 0.3 0.4 0.5 0.6Modified Torsion Parameter (Ke)

αm

Fac

tor


Figure 5.28 – αm Factors for the UDL Case Based on Elastic Buckling Analyses

However, Figures 5.27 indicates that the increase of αm factor will diminish after a

certain Ke value, i.e. greater than 0.6 approximately. The αm factor may be further

reduced with higher Ke values. In these regions, the buckling analysis results indicate

that they are associated with a lateral distortional buckling (LDB) mode as shown in

AS4100 1998 Table 5.6.2


ANSI/AISC 360-05

αm equation based on elastic buckling analysis (see Eq. 5.9)

BS5950-1:2000 (Le = 0.8L)

nLB Lateral buckling (LTB to LDB)


nLB Lateral buckling (LTB to LDB)

αm equation based on elastic buckling analysis (see Eq. 5.10)

AS4100 1998 Table 5.6.2

ANSI/AISC 360-05

BS5950-1:2000 (Le = 0.8L)



Figure 5.29(b), which may indicate that the limitation in the benefits is due to the

web distortion effect. This is different compared to the case of simply supported

beams with a transverse load where the unfavourable web distortion effect generally

starts at lower Ke values of 0.3 approximately.

(a) 1.5m LSB125x45x2.0 with a PL (b) 2.5m LSB250x60x2.0 with a PL

Figure 5.29 – Typical Lateral Buckling Modes of Cantilever LSBs

The unfavourable web distortion effect in cantilever LSBs affects the results for

higher Ke values. This may be because the web distortion of LSBs with low Ke

values occurs near the support or the high moment region (Figure 5.29(a)), while the

free end (the critical region) remains in the LTB mode (noted as LDB* in Tables

5.11 and 5.2). Therefore the unfavourable web distortion effect is negligible for

cases with Ke values < 0.6 approximately.

(a) 1.5m LSB300x75x3.0 with a PL (b) 0.75m LSB125x45x2.0 with a UDL

Figure 5.30 – Other Critical Buckling Modes of Cantilever LSBs



Local web buckling (web bearing) at the load position

Negligible lateral deformation

LDB occurs near the high moment region

Lateral torsional buckling (LTB) at the free end

Lateral distortional buckling (LDB) at the free end



Further, the elastic buckling analyses for cantilever LSBs with high Ke values were

also limited to other critical buckling mode that precedes lateral distortional

buckling. The non-uniform moment benefits with high Ke values increase the

buckling load up to the region where local buckling (near the support) is the critical

mode, or a local web buckling (web bearing) at the load location which may also be

the governing mode particularly for the case with PL at the free end. Figure 5.30

shows these non-lateral buckling modes. Further, the buckling resistances of

cantilever LSBs with UDL were significantly increased (very high αm factor), which

caused cantilever LSBs to be limited by non-lateral buckling modes with Ke values

only greater than 0.45 approximately. Note that in Figures 5.27 and 5.28 the results

associated with these non-lateral buckling modes are not included. Other buckling

analysis technique such as energy methods may have to be used for future research

to be able to derive solutions exclusively for lateral distortional buckling.

Comparison with the currently used αm factors in Figures 5.27 and 5.28 indicates

that they are conservative for cantilever LSBs despite not predicting them well nor

reflecting the observed αm variation. ANSI/AISC 360-05 (αm = 1.0) provisions are

very conservative due to its αm factor of 1.0. BS5950-1 (BSI, 2000) recommends the

use of effective length method (Le = 0.8L) to allow for the moment distribution effect

for cantilever. For comparison purposes, this effective length method is presented as

in αm factor form (Appendix C shows the example calculation). For the PL case,

BS5950-1 only provides a close prediction to the results for low Ke values. This is

the same with AS4100 Table 5.6.2 except that it is also valid for the UDL case. This

indicates that the moment distribution benefits of cantilevers subjected to shear

centre loading are taken with great conservatism in the current design codes, which

do not consider the favourable variation of αm factor. It may be to provide

conservatism in the case with loading above the shear centre, i.e. top flange loading.

This will be discussed further in Chapter 7. Ignoring the favourable variation of αm

factor especially with the UDL case is actually quite reasonable because using high

αm factor may not be very important due to other strength limiting effect such as

section capacity and other buckling modes as discussed later. Therefore AS4100

Table 5.6.2 is recommended for cantilever LSBs. For very low beam slenderness, it



may be slightly unconservative as shown in the elastic buckling results, however its

practical application rarely exists.

Nevertheless, an empirical αm equation for cantilever LSBs can also be developed

based on the elastic buckling analysis results (with an average error < 2%) as given

in the following equations. However, it should be noted that these equations may not

be suitable for LSBs with high Ke values due to the limited data especially with the

UDL case. This equation is also less convenient for use in design offices than

AS4100 Table 5.6.2.

Cantilever with PL: αm = 1.2 - 1.065 Ke2 + 1.78 Ke (5.9)

Cantilever with UDL: αm = 2.0 - 0.7 Ke2 + 2.85 Ke (5.10)

5.5.2. Non-linear Static Analysis Results and Discussions of Cantilever LSBs

Non-linear static FE analyses were conducted for cantilever LSBs with PL cases to

observe the effects on LSB’s ultimate strength. Preliminary analyses were conducted

prior to a detailed analysis to study the effect of positive and negative imperfections

on the ultimate strength of cantilever LSBs as summarised in Table 5.13. It

demonstrates a similar finding to the case with simply supported beams subjected to

moment gradient, i.e. the negative imperfection will provide the critical solution.

Therefore it was then used in the non-linear FEA of cantilever LSBs.

Table 5.13 – Effects of Initial Geometric Imperfection Direction

LSB d x bf x t Span Ke

Ms AS4600

FEA Ultimate Moment (kNm) for UDL Case

(mm) (mm) (kNm) (-) IMP (+) IMP

Failure Mode

1500 0.79 10.90 12.72 12.84 INB 2500 0.42 10.90 12.16 12.38 VLB 125 x 45 x 2.0

LSB 4000 0.21 10.90 10.18 10.78 LTB 2500 0.65 34.65 35.25 38.0(INB) VLB 3000 0.48 34.65 32.40 35.70 LDB* 4000 0.32 34.65 26.80 29.20 LDB*

250 x 60 x 2.0 LSB

6000 0.19 34.65 19.20 19.86 LTB IMP Imperfection direction LDB* Lateral distortional buckling mode with negligible web distortion INB In-plane bending mode



The non-linear FEA results for cantilever LSBs subjected to a PL are summarised in

Table 5.14, where strength ratios are used to represent the moment distribution

benefits in the non-linear behaviour. The non-linear FEA results confirm the

favourable moment distribution variations as indicated in the elastic buckling

analyses. However, it is evident that most of the intermediate spans considered here

are limited by the section capacity, highlighting the significant moment distribution

benefits in increasing the strength of cantilever LSBs. Due to this reason, the non-

linear FEA study for the UDL case was not conducted as the moment distribution

benefit is greater. Therefore a recommendation based on the elastic buckling analysis

is considered adequate, which suggest to adopt AS4100 Table 5.6.2 method for both

UDL and PL cases.

Table 5.14 – Ultimate Moments of Cantilever LSBs Subjected to

a PL at the Free End

LSB Ms Critical Mult-non / Mult

d x bf x t Span

AS4600 Buckling FEA Ultimate

Moment (Mult-non) Strength


αm Factor

Mb-non / Mb

750 10.90 57.35 12.96 INB - - 1500 10.90 48.72 12.72 INB - 1.96 1.33 2500 10.90 31.10 12.16 VLB 1.32 1.74 1.32

125 x 45 x 2.0 LSB

4000 10.90 18.01 10.18 LTB 1.40 1.50 1.23 1500 34.65 54.65 37.05 INB - - 2500 34.65 50.23 35.25 VLB 1.77 1.95 1.4 3000 34.65 41.14 32.40 LDB* 1.77 1.84 1.36 4000 34.65 30.12 26.80 LDB* 1.68 1.66 1.29

250 x 60 x 2.0 LSB

6000 34.65 19.87 19.20 LTB 1.49 1.49 1.22 1500 76.05 158.58 73.62 INB - - 2500 76.05 149.46 85.20 INB - 1.94 1.39 3000 76.05 123.29 83.88 VLB 1.70 1.88 1.37 4000 76.05 88.83 72.48 LDB* 1.66 1.70 1.30

300 x 75 x 3.0 LSB

6000 76.05 57.31 54.36 LTB 1.59 1.52 1.23 LDB* Lateral distortional buckling mode with negligible web distortion VLB Very small lateral bending mode (close to in-plane bending) INB In-plane bending mode

Table 5.14 also provides the strength ratios for cases subject to lateral buckling. It

shows that the strength ratios in many cases are less than the corresponding αm

factors. This means that the design method used in many cold-formed steel codes is

more suitable and conservatice for cantilever LSBs. That is, the αm factor is used to

modify the elastic buckling moment of uniform moment case (Eq. 5.5, Mod-non = αm



Mod), and the resulting buckling moment is then used to calculate the member

moment capacity (Mb) through the use of modified slenderness (λd) in the design

moment capacity equation.

5.6. Design Recommendation Summary

The strength benefit due to non-uniform bending moment distributions for simply

supported LSBs is unfavourably influenced by lateral distortional buckling. The αm

factor reaches the upper bound with high beam slenderness (subject to lateral

torsional buckling), but it reduces with lower beam slenderness due to the increasing

level of web distortion of lateral distortional buckling, until other buckling modes

that precede lateral buckling govern (i.e. local buckling, shear buckling, etc).

For the moment gradient case, the benefit variation is insignificant and thus the

currently available αm equations in AS4100, BS5950-1 and ANSI/AISC 360 are

adequate and recommended for design purposes, although a more accurate equation

is also provided in this thesis. However, the reduction of moment distribution

benefits due to lateral distortional buckling for simply supported LSBs with

transverse loading (uniformly distributed load, mid-span point load, and quarter

point loads) is more significant, where the presence of shear stresses may contribute

to greater reduction in αm factors. It is therefore recommended to use the proposed

αm equations for simply supported LSBs with a uniformly distributed load and a

mid-span point load. The proposed αm equations suggest neglecting the moment

distribution effect when shear stresses are significant. For quarter point load cases, it

is recommended to simply neglect any moment distribution effect (αm = 1). For

cantilever LSBs subjected to transverse loading (uniformly distributed load and point

load at the free end), the unfavourable effect of lateral distortional buckling is less

than in the case of simply supported beams. The currently available αm factors were

found to be conservative for intermediate to long span cantilever LSBs (other

buckling modes govern for short spans). The AS4100 design approach is less

conservative than BS5950-1 and ANSI/AISC 360, and is recommended for

designing cantilever LSBs. More accurate αm equations are also provided in this

thesis, but are less favoured.



For the application of αm factors to determine the design moment capacities of LSBs

with any load and boundary conditions, it is recommended that the design method in

many cold-formed steel codes is used in which the elastic lateral buckling moment

Mod for the uniform moment case is modified by using the appropriate αm factor and

used in the member capacity calculation.

Chapter 6 – Experimental Investigation of LSBs with Moment Gradient


6. Experimental Investigation of LSBs with Moment

Gradient

6.1. General

A limited number of experiments was conducted on LSBs subjected to moment

gradient in order to evaluate the findings from FE analyses in the last chapter. This

chapter presents the details of this experimental study, the results and associated

discussion. Experimental LSB behaviour was also simulated using finite element

modelling and the results are compared in this chapter.

6.2. Experimental Method

It was initially proposed to build an experimental set-up with limited warping

restraint at its supports in order to compare the experimental results with those

obtained using the ideal finite element model that assumes warping free support

conditions.

Figure 6.1 – Schematic Diagram of the Overhang Loading Method Used in the

Experimental Study

However, an ideal simply supported beam with a moment gradient is impossible to

simulate in practice (non-existing case), thus an experimental arrangement of LSB

flexural members with short overhangs as shown in Figure 6.1 was used to produce

Uniform bending moment distribution (β= -1)

Overhang OverhangTest beam

Non-uniform bending moment distribution (β= 0)

OverhangTest beam Overhang

P

P P



equal or unequal end moments as part of the moment gradient action on a simply

supported beam. Although this method is commonly used for testing of beams with a

moment gradient loading, it is also commonly recognised that it introduces a partial

warping restraint due to the presence of continuing flanges in the overhangs. The

level of warping restraint with this method will depend on the overhang length, i.e.

less warping restraint with shorter overhang. However, simply using a very short

overhang may not be possible as additional problems may arise such as a local

failure in the overhang due to a very high load requirement. An appropriate overhang

length was then chosen based on a series of preliminary FE analyses. Four

experiments were conducted on 2.5 m span LSB125x45x2.0 and 3.5 m span

LSB250x60x2.0 subjected to a uniform moment (β = -1) and a non-uniform moment

with a moment gradient of β= 0, respectively. Table 6.1 summarises the test

program.

Table 6.1 – Test Program

LSB Section d x bf x t

Span Overhang Length

Total Length Test

No (mm) (mm) (mm) (mm)

Test Type

1 2500 500 3000 Uniform Moment, β = -1 2

125 x 45 x 2.0 LSB 2500 500 3000 Moment Gradient, β = 0

3 3500 750 5000 Uniform Moment, β = -1 4

250 x 60 x 2.0 LSB 3500 750 5000 Moment Gradient, β = 0

6.3. Test Specimens

Test specimens were provided by the manufacturer, Smorgon Steel Tube Mills Pty

Ltd (see Figure 6.2). The test beam was connected to the support and loading devices

using M10 (3/8” diameter) high strength bolts. Hence 10 mm holes were inserted in

the test specimens to enable the required connections. The actual test specimen

dimensions were also measured and given in Table 6.2. The nominal section

properties of LSB sections can be found in the Design Capacity Tables for LSBs.

The measured dimensions were found to be different to the nominal values, however

they were within the manufacturer’s fabrication tolerance limits (Appendix A).



Figure 6.2 – Test Specimens

Due to time constraints in this study, the initial global imperfections of the test

specimens, in particular to determine whether they were in the positive or negative

directions, were not measured. Mahaarachchi and Mahendran’s (2005e) report

includes the geometric imperfection measurements of the test specimens they used,

but further measurements are needed to confirm the type of global imperfections of

LSBs.

Table 6.2 – Average Measured Dimensions of LSB Sections used in Experiments

LSB Section Thickness (t) d x bf x t

Span Depth (d)

Flange Width (bf) Flange Web

Flange Depth (df)

(mm) (mm) (mm) (mm) (mm) (mm) (mm) 125 x 45 x 2.0 LSB 2500 125.7 45.6 2.14 1.98 14.63 250 x 60 x 2.0 LSB 3500 252.0 60.4 2.11 2.00 20.30

6.4. Test Set-up

A special full scale bending test rig for lateral distortional buckling tests was

designed, fabricated and built by Mahaarachchi and Mahendran (2005a) in the QUT

Structures Laboratory. While it was mainly used to apply quarter point loads, it

could also be used to apply overhang loads to produce a uniform bending moment

within the test beam span. The same test rig and its supporting equipment were

reassembled and rebuilt for use in this experimental study. Figure 6.3 shows the

overall view of the test rig with the overhang loading system. The test rig included a

frame of two main beams and four columns, two smaller frames providing the

LSB250x60x2.0

LSB125x45x2.0



support system, and the loading system. The smaller support frames were set up

within the main frame by fixing the top and bottom of the support frames to the main

beams and the rigid floor, respectively.

Figure 6.3 – Overall View of Test Rig

6.5. Support System

The support system was designed to ensure that the test beam was simply supported

in-plane and out-of-plane at the supports by restraining the in-plane vertical

deflections, out of plane deflections and twist rotations, but permitting major and

minor axis rotations. The support system is shown in Figure 6.4.

The in-plane vertical movements and lateral movements were prevented by the

running tracks and side guides. The box-frames with ball bearings were designed to

provide major and minor axis rotations of the test beam. The side ball bearings

allowed major axis rotation, whilst the top and bottom ball bearings allowed out of

plane rotation and differential flange rotations (about the minor axis) associated with

the warping displacement rotations. The two supports were aligned to ensure that the

vertical deflections remained in the same plane. One of the supports had horizontal

stops to prevent the movement of the side bearing along the running track. The test

beam was connected to the support system by using four M10 (3/8” diameter) bolts

Test beam Test beam

Hydraulic pump Overhang parts Support frames



and a clamping plate. This plate was used to prevent web crippling and twisting of

the section at the supports.

Figure 6.4 – Support System (Mahaarachchi and Mahendran, 2005a)

6.6. Loading System

The overhang loading system consisted of two hydraulic jacks located at the free end

of overhang parts and connected to a wheel system, a load cell, universal joints and

other components as shown in Figure 6.5. It provided a symmetric and uniform

bending moment to the test beam between the two supports (ie. within the main

beam span). The loading system was designed so that there was no restraint on

displacements or rotations in any direction from the loading device to the test beam

(overhang part) at the loading positions. The wheel system allowed the loading arm

to move in plane when the test beam (overhang part) deformed. The universal joints

ensured the load was applied eccentrically without forming a torque. The load was

also applied to the shear centre of the test beam through the loading arm and thus

halla




eliminated the load height effect. Therefore all the six degrees of freedom were

considered unrestrained at the loading positions of the test beam. Four M10 bolts

were used to connect the test beam to the loading device. In addition, the load was

applied vertically upward to avoid damage to the joints. A hydraulic pump connected

to the loading jacks was used to apply the load based on displacement control.

Figure 6.5 – Loading System

The test of LSB250x60x2.0 with moment gradient β = 0 required a very high applied

load at one end and none at the other end. A single hydraulic jack (maximum tension

capacity = 40kN) was considered inadequate to load the test beam to failure.

Therefore two hydraulic jacks were positioned side by side on one of the overhangs

to produce the required load as shown in Figure 6.5.

6.7. Measurement System

There were two important parameters to be measured in this experiment, the applied

load and deformations. Two 60 kN load cells were attached to each loading arm as

Load cell

Universal joint

Loading arm

Loading position at shear centreTest beam

Loading arm at overhang

Wheel system

Two hydraulic jacks at overhang



shown in Figures 6.6 to measure the applied load. The recorded deformations were

the in-plane and out of plane deflections of top and bottom flanges at midspan, and

the vertical deflection under each loading point of the overhang. In total five wire

potentiometer type displacement transducers (WDT) were used to measure these

displacements. Figure 6.6(a) shows the arrangement of the WDTs. A strain gauge

was not used in this test, instead the yielding distribution observation was attempted

with a lime coated to the test beam. The coating was expected to come off when

yielding commenced, but unfortunately this method did not work well.

(a) Overall view of measuring system

(b) Data logger

Figure 6.6 – Measurement and Data Acquisition Systems

WDTs

Test beam Overhang part



These measurements were recorded by using a C10DAS1402/12 Data Acquisition

Unit and a PC as shown in Figure 6.6(b). Calibration factors (i.e. unit of voltage per

mm) of the WDTs were determined and input to the recording unit before the

commencement of tests.

6.8. Measurement System

The typical test procedure used is described as follows:

1. Specimen measurements, and detailing works were conducted prior the test

set up.

2. The test rig was set for the required beam span (i.e. 2.5m or 3.5m)

3. The test beam was inserted within the two box frames (supports) and the

clamping plates were bolted to the test beam.

4. The loading devices were connected at the free end of each overhang. For the

test with moment gradient β = 0, only one overhang was loaded. The loading

jacks and arms were set and aligned to prevent any eccentricities.

5. Locations for the measurements of deflections were marked on the top

flange, bottom flange and web at midspan and at the loading points.

6. The WDTs were installed at the required positions. The WDTs for lateral

displacements were held by tripod whilst those for vertical displacements

were held by weights on the ground.

7. The load cells and WDTs were connected to the data logger. The accuracy of

WDTs was checked by comparing with manual measurement.

8. A small load was applied first to allow the loading and support system

components to settle evenly.

9. The measuring system was set to zero values. A trial load of 10% of the

expected ultimate capacity was applied and released to remove any slack in

the system and to ensure functionality. Subsequently, the load was applied

gradually until the test beam failed by lateral buckling.

The applied bending moment (M) was calculated using:

M = Applied load (P) x Lever arm (L) (6.1)



Where; 22valL Δ−=

al = Initial lever arm length

vΔ = Vertical deflection at the loading position

6.9. Results and Discussions

The test results are summarised in Table 6.2 while the moment versus vertical

deflection at mid-span curves are provided in Figures 6.7 and 6.8. Other test results

are given in Appendix D.

Table 6.2 – Summary of Test Results

Test Results LSB Section d x bf x t Span Overhang

Length Exp Mult* Test No

(mm) (mm) (mm)

Test Type

(kNm) Failure Mode Direction

2500 500 β = -1 9.6 LDB (+) 1 2

125 x 45 x 2.0 LSB 2500 500 β = 0 14.1 LDB (-)

3500 750 β = -1 21.1 LDB (-) 3 4

250 x 60 x 2.0 LSB 3500 750 β = 0 35.5 LDB (+)

* Exp Mult = Experimental maximum moment capacity

0

3

6

9

12

15

0 5 10 15 20 25 30 35 40 45

Vertical Displacement at Mid-span (mm)

Mom

ent (

kNm

)

Exp: LSB125x45x2.0 beta=-1 Exp: LSB125x45x2.0 beta=0FE w/o RS: LSB125x45x2.0-2.5 beta=-1FE with RS: LSB125x45x2.0-2.5 beta=-1FE w/o RS: LSB125x45x2.0-2.5 beta=0FE with RS: LSB125x45x2.0-2.5 beta=0

Figure 6.7 – Test and FEA Results for 2.5m LSB125x45x2.0

Due to error in measurements during final stages of test



Table 6.2 shows that the lateral displacement (lateral distortional buckling) of the

tests beams can be either in the positive or the negative direction (the positive and

negative direction descriptions are given in Chapter 3). Although the initial global

imperfections of the test beams were not measured in this experimental study, the

test results imply that the initial global imperfections could be in either direction.

This is because the lateral buckling direction is mainly determined by the initial

global imperfection. Based on these observations the parametric study based on FEA

in this research used both global imperfections.

0

5

10

15

20

25

30

35

40

0 5 10 15 20 25 30 35Vertical Displacement at Mid-span (mm)

Mom

ent (

kNm

)

Exp : LSB250x60x2.0 beta=-1 Exp : LSB250x60x2.0 beta=0FE w/o RS: LSB250x60x2.0 beta=-1FE with RS: LSB250x60x2.0 beta=-1FE w/o RS: LSB250x60x2.0 beta=0FE with RS: LSB25x60x2.0 beta=0

Figure 6.8 – Test and FEA Results for 3.5m LSB250x60x2.0

The actual test beams’ physical geometry, loads, constraints, material properties,

residual stresses and initial geometric imperfections (resembling the actual lateral

buckling direction in the test) were simulated as closely as possible using finite

element modelling (see experimental FE model in Chapter 3). The purpose of this

study was to validate the adopted FE mesh density, shell element for explicit

modelling of initial geometric imperfections, residual stresses, lateral distortional

deformation and buckling effects, and the associated material yielding. It was also

used to obtain the elastic buckling moments for uniform (Mod), non-uniform (Mod-non)

moment distribution and load height cases. A detailed description of finite element

modelling is given in Chapter 3.

Higher strength was achieved due to the over restraining effect from the use of two hydraulic jacks side by side



A comparison of experimental and FEA results in Figures 6.7 and 6.8 indicate that

they agree reasonably well, especially when residual stresses were included. The

FEA may provide slightly less prediction due to the unaccounted restraining factors

involved in the testing such as friction in the support and loading systems, partial

restraint from the loading system, measurement errors, etc. Therefore, this

comparison further confirms the validity of the adopted ideal FE model which has a

similar modelling arrangement except for the loading and boundary conditions. It

also confirms the suitability of the adopted residual stress model for use in the

numerical modelling of LSBs.

Both the test and FEA results demonstrate that LSB’s flexural strength (lateral

distortional buckling) is significantly improved under the moment gradient action of

β = 0. In Table 6.3, the moment gradient benefits are presented as strength ratios,

which are taken as the ratio of ultimate moment capacities for non-uniform and

uniform moment distribution loading (Mult-non / Mult). They are also compared with

the corresponding αm Factor based on the elastic buckling analysis results, i.e. Mod-

non / Mod. For 2.5m LSB125x45x2.0 tests, the strength ratio is not relevant because it

is limited by its section capacity. For 3.5m LSB250x60x2.0 tests, it indicates a

similar result to the study using the ideal FE model as described in Chapter 5, where

the strength ratio is less than the αm factor. This confirms that the actual moment

gradient benefit (for high β cases) on LSB subjected to LDB mode is less than that

based on elastic buckling analyses.

Table 6.3 – Moment Distribution Effects (Test Results and Exp FEA Results)

LSB Section Experimental FEA Results with RS Test Results d x bf x t

Span Strength Strength

(mm) (mm) β Mod or Mod-

non (kNm) αm

Factor Mult or Mult-

non (kNm) Ratio Mult or Mult-

non (kNm) Ratio 2500 -1 11.71 9.49 9.6 125 x 45 x 2.0

LSB 2500 0 20.87 1.78

13.67 1.44+

14.1+ 1.47+

3500 -1 24.57 20.49 21.1 250 x 60 x 2.0 LSB 3500 0 44.84

1.82 31.93

1.64 35.5

1.68*

+ Limited by its section capacity * Associated with over restraining effect (two hydraulic jacks used side by side)

Figure 6.9 shows the typical deformation of the test beams at failure subjected to a

moment gradient (β = 0) and the associated yielding distribution based on the FEA



(experimental FE model), showing good agreement between the FEA and the actual

test (in terms of moment and displacement). A method to observe the yielding

patterns using a lime coating was attempted, but unfortunately it did not work well.

(a) Test – 3.5m LSB250x60x2.0 (β = 0)

(b) FEA – 3.5m LSB250x60x2.0 (β = 0)

Figure 6.9 – Typical Specimen Deformation at Failure and Associated Yielding

(von Mises) Distribution based on FEA and Test

This yielding (von Mises) distribution is similar to that of using ideal FE model as

shown in Figure 5.12 above, where with moment gradient action the yielding is

actually not confined to a short region near the support and it is associated with


Lateral distortional buckling mode with (+) direction (at failure)

Web yielding Note: overhangs are not shown

Loaded overhang part

Loaded overhang

Unloaded overhang



greater web yielding which may explain greater strength reduction below elastic

buckling strength. Experiments for the case with LTB mode (high beam slenderness)

were not conducted. However it is expected to have less unfavourable yielding effect

or in other words, strength ratio is closer to αm factor.

Although a direct comparison of test results with FEA results using the ideal FE

model could not be made due to the partial warping restraint in the tests, this

experimental study was able to confirm the actual moment gradient benefits for

LSBs. In this study only a limited number of tests was undertaken due to time

constraints and hence more tests are recommended for future research. Effects of

other loading and support conditions associated with transverse loads and cantilever

beams should also be investigated. However, such tests will need greater care as

other factors may influence the results, i.e. an interaction with shear in the case of

low to intermediate spans of deep sections.

Chapter 7 – Effect of Load Height on the Lateral Buckling Strength of LSBs


7. Effect of Load Height on the Lateral Buckling

Strength of LSBs

7.1. General

This chapter describes a detailed study using finite element analysis (FEA) to

investigate the load height effects on the lateral buckling strength of LiteSteel beams

(LSBs). As in the moment distribution study, two types of support were investigated

in this research, simply supported LSBs and cantilever LSBs. The transverse load

types used in this study were a uniformly distributed load (UDL), and a point load

(PL) applied at mid-span for the case with simply supported beams, and a point load

(PL) applied at the free end for the cantilever beams. All the cases considered

assume a loading through the shear centre of the LSB cross section to avoid any

additional torsional load. The load heights investigated in this research were the top

flange (TF) and bottom flange (BF) loading levels. They are the most common cases

encountered in practice (Figure 7.1). The case of transverse load applied at the shear

centre (SC) has already been investigated and the results are presented in Chapter 5.

The results in Chapter 5 thus form the basis for the evaluation of load height effects.

(a) Load applied at the shear centre (b) Loads applied at above or below the shear centre

Figure 7.1 – Transverse Loading Levels Considered in This Study

Since non-linear finite element analysis is a time consuming process and the moment

distribution study in Chapter 5 has shown that an analysis based on elastic buckling

only is adequate to investigate its effects, only the elastic buckling analysis was

Bottom Flange Loading (BF)

Shear Centre Loading

Top Flange Loading (TF)



conducted in this study on load height effects. Three LSB sections, LSB 125x45x2.0,

250x60x2.0 and 300x75x3.0 were investigated to include the effect of section

geometry, while their spans were varied from intermediate to long spans (0.75 to 10

m) in order to observe the relationship of lateral buckling modes (Lateral

Distortional Buckling vs. Lateral Torsional Buckling) to the load height effects. The

results and discussions of a suitable method for design purposes are presented in this

chapter.

7.2. Load Height Effects for Simply Supported LSBs

7.2.1. Elastic Buckling Analysis Results


Subjected to Top Flange (TF) and Bottom Flange (BF) Loading

LSB d x bf x t

Span For TF case (kNm) For BF case (kNm)

(mm) (mm)

General Buckling

Mode UDL PL UDL PL 750 LB - - - - 1000 LDB 18.56(+) 19.64(+) 36.27 45.37 1500 LDB 16.16 22.67 24.17 37.82 2500 LDB 12.13 17.29 14.98 23.08 4000 LDB* 8.44 7.65 9.55 9.09 6000 LTB 5.90 8.67 6.39 9.69

125 x 45 x 2.0 LSB

10000 LTB 3.65 5.42 3.83 5.78 1500 nLB - - - - 2000 LDB 21.72(+) 20.25(nLB) 46.83(+) 55.48(+) 2500 LDB 20.44 21.87(+) 36.71 46.33 3000 LDB 19.35 21.47 30.97 38.99 4000 LDB 17.25 19.47 23.81 29.74 6000 LDB* 13.59 15.53 16.47 20.30

250 x 60 x 2.0 LSB

10000 LTB 9.13 10.63 10.19 12.38 1500 nLB - - - - 2500 LDB 57.99(+) 62.06(+) 110.95 139.76 3000 LDB 54.36 60.23 92.87 117.15 4000 LDB 48.14 54.24 70.22 88.03 6000 LDB* 37.94 43.26 47.74 59.05

300 x 75 x 3.0 LSB

10000 LTB 25.50 29.57 29.10 35.77 LDB* Lateral distortional buckling mode with negligible web distortion (+) Lateral distortional buckling mode with shear buckling or web local buckling (interaction) nLB Non-lateral buckling mode (shear buckling or web local buckling) that precedes LDB



Elastic buckling analyses of simply supported LSBs with top flange (TF) and bottom

flange (BF) loading were undertaken using the finite element models described in

Chapter 3 (Section 3.6). Table 7.1 summarises the elastic lateral bucking moments of

simply supported LSBs subjected TF and BF loading for both transverse load cases

(UDL and PL). The results are also presented in a dimensionless format in Figure

7.2. The modified torsion parameter (Ke) in this figure was defined and used in

Chapter 5 – Equation 5.2(b), and the dimensionless buckling load (DBL) is given as

follows:

For UDL: DBL = ey3 GJEI/QL (7.1a)

For PL: DBL = ey2 GJEI/QL (7.1b)

where the buckling load (Q) is obtained from the FE elastic buckling analysis and L

is the span.

0

5

10

15

20

25

30

35

40

45

50


Dim

ensi

onle

ss B

uckl

ing

Load



Simply Supported LSBs Based on Elastic Buckling Analyses

7.2.2. Discussions on the Load Height Effects for Simply Supported LSBs

Figure 7.2 demonstrates the load height effects based on the elastic buckling results

of LSBs. It shows that the destabilising effect of loading above the shear centre (TF)

PL case

UDL case

BF

BFSC

SCTF

TF




decreases the buckling resistance while the loading below the shear centre (BF)

produces the opposite effect. The mechanism is illustrated in Figure 7.3. When a

transverse load acts above the shear centre and moves with the beam during lateral

buckling, it exerts an additional torque about the shear centre axis, subjecting the

section to an additional twisting thus reducing the buckling resistance. Conversely,

for loading below the shear centre the additional torque opposes the twist rotation of

the beam, thus increasing the buckling resistance. Figure 7.4 shows the differences in

the torsion level of LDB for various load heights (shear centre, top flange and

bottom flange). This effect is more important for beams with low beam slenderness

as the torsion level is significant to create a larger additional torque than for beams

with higher beam slenderness for which its lateral component is more dominant than

the torsion components. This is evident from the results that the load height effect is

more important for LSBs with high modified torsion parameter (Ke).

Figure 7.3 – Effect of Loading above the Shear Centre

Figure 7.4 – Load Height Effects on the Lateral Distortional Buckling Model

(2.5m LSB250x60x20 Subjected to a UDL)

Creates additional torque

Shear centre loading (SC) Bottom flange loading (BF) Top flange loading (TF)

LDB with amplified torsion

LDB with reduced torsion (greater lateral component)

Lateral distortional buckling (LDB)



(a) 1.5m LSB300x75x3.0 with a UDL (BF)

(b) 1.5m LSB300x75x3.0 with a UDL (BF) (c) 1.5m LSB250x60x2.0 with a PL (TF)

(d) 2m LSB250x60x2.0 with a UDL (BF) (e) 2.5m LSB250x60x2.0 with a PL (TF)

Figure 7.5 – Other Critical Buckling Modes of LSBs Subjected to TF and BF

Loading

However for cases with high Ke values, the elastic buckling analyses were always

limited by other buckling modes, which precede lateral distortional buckling (similar

to the moment distribution study). For the UDL case, the non-lateral buckling mode

is shear buckling (due to significant shear stresses caused by the transverse load)

near the supports, and for PL case it is local web buckling or web bearing buckling at

mid-span as shown in Figure 7.5. An interaction mode of lateral distortional buckling

with non-lateral buckling was also encountered as shown in Figure 7.5, where it

Interaction of lateral distortional buckling with shear buckling near the support

Interaction of lateral distortional buckling with local web buckling (bearing) at mid-span

Local web buckling (bearing) at mid-span


Front view (deformation)




shows that LSBs are subjected to an interaction buckling when the Ke value is

greater than 0.8 (approximately). Further, the FEA results also suggest that LSBs

subjected to a mid-span point load at TF level are more prone to local web buckling

(web bearing). Therefore, the results associated with non-lateral buckling mode

(including the interaction) are not considered in this study on load height effects.

15

20

25

30

35

40

45

50

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Modified Torsion Parameter (Ke)

Dim

ensi

onle

ss B

uckl

ing

Load

LSB250x60x2.0 (slender)LSB300x75x3.0 (non-compact)LSB125x45x2.0 (compact)Prediction by Trahair's equation (Eq. 7.2)BS5950-1:2000 (kl=1.2)AS4100-1998 (kl=1.4)

Figure 7.6 – Comparison with Current Design Methods in Predicting the

Lateral Buckling Strength of Simply Supported LSBs Subjected to

Load Height Effects (UDL Case)

BS5950-1 (BSI, 2000) and AS4100 (SA, 1998) treat the destabilising effect of TF

loading by using a factor to increase the effective length (Le = L x kl) which in turn is

used to calculate the elastic buckling resistance (Mod-non for top flange loading). This

factor is referred to as the load height factor (kl) in AS4100. However for loading

below the shear centre (i.e. BF loading), both design codes conservatively ignore its

beneficial effects. BS5950-5 (BSI, 1998), British cold-formed steel design code,

adopts a similar load height factor as in the BS5950-1, British hot-rolled steel design

code. However, AS/NZ4600 (SA, 2005), ANSI/AISC 360 (ANSI, 2005) and AISI

(AISI, 2001), Australian cold-formed steel design code do not provide explicit

provisions to account for the effect of load height effect (refer to Chapter 2).

BF

SC

TF



5

10

15

20

25

30

35

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Modified Torsion Parameter (Ke)

Dim

ensi

onle

ss B

uckl

ing

Load

LSB250x60x2.0 (slender)LSB300x75x3.0 (non-compact)LSB125x45x2.0 (compact)Prediction by Trahair's equation (Eq. 7.2)BS5950-1:2000 (kl=1.4)AS4100-1998 (kl=1.2)


Lateral Buckling Strength of Simply Supported LSBs Subjected to Load Height

Effects (PL Case)

Figures 7.6 and 7.7 compare the dimensionless buckling loads from the elastic FE

buckling analysis results with the predictions using BS5950-1 and AS4100. In

calculating the Mod-non for TF loading using the design code method of effective

length, the actual αm factors based on the elastic buckling analysis in Chapter 5 was

used to include the moment distribution effects. Appendix C gives the example

calculation for Figures 7.6 and 7.7 (i.e. dimensionless buckling load of using the

design code method). In general, the comparison indicates that the design code

prediction does not represent the actual load height effect (TF) variation, i.e. too

conservative for lower Ke value (higher beam slenderness) and unconservative for

the opposite case, particularly with the BS5950 prediction. AS4100 prediction is

better than that of BS5950-1 as it is only slightly unconservative in the higher Ke

region because of its higher load height factor (kl) of 1.4 (1.2 in BS5950-1).

Trahair (1993) provides an approximate solution to predict the elastic lateral

buckling strength of a beam subjected to load height effects as given by the

following equation.

BF

SC

TF



⎪⎭

⎪⎬

⎫α

⎪⎩

⎪⎨

⎧+

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛ α+α=

yyz

Qm

2

yyz

Qmm

yz

cr

P/My4.0

P/My4.0

1MM

(7.2)

Where; Mcr = elastic lateral buckling moment including load height effect

Myz = elastic lateral buckling moment (Mod)

yQ = load height

Py = π2 E Iy / L2

The use of this more accurate equation for design purposes is permitted in AS4100

(SA, 1998). Figures 7.6 and 7.7 also plot the dimensionless buckling load calculated

using Trahair’s solution. In using Equation 7.2, actual αm factors based on the elastic

buckling analyses presented in Chapter 5 were used to include the moment

distribution effect. The comparison demonstrates that Trahair’s equation is

reasonably accurate with the elastic buckling results for both TF and BF cases.

Therefore this equation can be safely implemented to estimate LSB’s lateral

buckling moments (LDB and LTB), provided the appropriate αm factor is used. The

use of an accurate αm factor which includes the effect of web distortion in lateral

distortional buckling, allows treating the load height effects without considering the

web distortion effect. In other words, the web distortion should have no effect on the

load height effects if it is already considered in the moment distribution effects. Note

that accurate αm factors are also used in the comparisons using BS5950-1 and

AS4100 above.

The moment distribution effect study in Chapter 5 suggests modifying the elastic

buckling moment with the αm factor (Mod-non = αm Mod) for use in the member

capacity (Mb) calculation. A similar approach therefore can be applied for this load

height case, i.e. the Mod-non for load height is obtained using Trahair’s equation and

used in the Mb calculation to include the load height effect. But it should be noted

that Trahair’s equation may not be accurate for LSBs with high Ke values (i.e.

greater Ke value than shown in Figures 7.6 and 7.7) because of the use of a unity αm

factor (αm = 1) as suggested from the study in Chapter 5. However, other non-lateral

buckling will be the governing mode in this case.



Further, the use of Trahair’s equation is superior to the current design method of

effective length as it can be applied for any load heights using the yQ parameter in

Equation 7.2 (not limited to TF and BF cases). A comparison of predictions from

Trahair’s equation and AS4100 in Figures 7.6 and 7.7 indicates that for LSBs with

lower beam slenderness (higher Ke value), they are quite close although the latter is

less accurate (slightly unconservative). The slight overestimation from AS4100

method may not be very significant for design purposes as there are many other

unaccounted factors that may compensate this inaccuracy. The effective length

method is also convenient than Equation 7.2 due to its simplicity and conservatism

for cases with high beam slenderness. Hence the choice of using the AS4100 method

based on effective length may be still adequate for LSBs.

For loading below the shear centre, its benefit is usually ignored in the current steel

design codes. Its significant benefits may not be very important as its ultimate

strength is very likely to be limited to its section capacity, particularly for

intermediate spans and less. This conservative approach can be safely adopted for

LSB design. Alternatively, Trahair’s equation (Eq. 7.2) can be simply used to obtain

the Mod for loading below the shear centre of LSBs.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Modifed Torsion Parameter (Ke)

Load

Hei

ght R

atio

LSB250x60x2.0 (slender)LSB300x75x3.0 (non-compact)LSB125x45x2.0 (compact)Prediction by Trahair's equation (UDL case)Prediction by Trahair's equation (PL case)

Figure 7.8 – Load Height Ratios Comparison

TF

UDL case

PL case

BF AS4100-1998 (kl = 1.4)

BS5950-1:2000 (kl = 1.2)



The load height effects from TF and BF loading can also be expressed in a load

height ratio format, which is a ratio of lateral buckling moment with load height and

shear centre loading. Figure 7.8 presents the load height ratios from the elastic

buckling results as well as the predictions using the current design code methods and

Trahair’s equation (Eq. 7.2). It confirms the many observations made based on

Figures 7.6 and 7.7. It also demonstrates that the variation among the two transverse

load cases (PL and UDL) can be considered small, indicating that these two cases are

sufficient to represent the load height effect on LSBs. Nevertheless, further research

into other transverse load types may be useful.

7.3. Load Height Effects for Cantilever LSBs

7.3.1. Elastic Buckling Analysis Results


Top Flange (TF) and Bottom Flange (BF) Loading

LSB d x bf x t

Span for TF case (kNm) for BF case (kNm)

(mm) (mm) PL Mode UDL Mode PL Mode UDL Mode 750 21.81 LDB+ 43.75 LDB - nLB - nLB 1000 24.29 LDB 47.59 LDB 53.47 LDB - nLB 1500 24.85 LDB 40.47 LDB 32.81 LDB* 60.92 LDB* 2500 16.58 LDB* 25.96 LDB* 18.84 LDB* 32.66 LDB*

125 x 45 x 2.0 LSB

4000 10.61 LTB 16.61 LTB 11.53 LTB 19.20 LTB 1500 - nLB* - nLB - nLB - nLB 2000 25.99 LDB+ - nLB - nLB - nLB 2500 28.70 LDB 53.69 LDB 54.10 LDB - nLB 3000 29.73 LDB 51.03 LDB 43.45 LDB* - nLB 4000 26.26 LDB* 41.40 LDB* 31.52 LDB* 57.28 LDB*

250 x 60 x 2.0 LSB

6000 18.42 LTB 28.89 LTB 20.65 LTB 35.55 LTB 1500 - nLB* 123.69 LDB+ - nLB - nLB 2500 74.07 LDB 144.23 LDB 167.60 LDB - nLB 3000 77.36 LDB 139.70 LDB 132.84 LDB* - nLB 4000 72.55 LDB* 117.12 LDB* 94.18 LDB* 175.97 LDB*

300 x 75 x 3.0 LSB

6000 52.14 LTB 82.40 LTB 60.22 LTB 106.70 LTB LDB* Lateral distortional buckling mode with negligible web distortion nLB Non-lateral buckling mode (local buckling near the support) that precedes LDB nLB* Non-lateral buckling mode (local web buckling at the load position) that precedes LDB LDB+ Lateral distortional buckling mode with interaction (nLB or nLB*)



Elastic buckling analyses of cantilever LSBs with top and bottom flange loading

were undertaken using the finite element models described in Chapter 3 (Section

3.6). Table 7.2 summarises the elastic lateral bucking moments of cantilever LSBs

subjected to top flange (TF) and bottom flange (BF) loading for both transverse load

cases (UDL and PL at the free end). Non-dimesionalised results are presented in

Figure 7.9, where the modified torsion parameter (Ke), dimensionless buckling load

for UDL case, and PL at the free end case are obtained using Equations 5.2(b), 7.1(a)

and 7.1(b), respectively.

0

2

4

6

8

10

12

14

16

18

20

22

24

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4Modifed Torsion Parameter (Ke)

Dim

ensi

onle

ss B

uckl

ing

Load



Cantilever LSBs Based on Elastic Buckling Analyses

7.3.2. Discussions on the Load Height Effects for Cantilever LSBs

Figure 7.9 demonstrates that the effect of bottom flange (BF) loading increases the

buckling resistance while the top flange (TF) loading reduces the resistance

substantially, especially for cantilevers with high Ke values. It has been shown in

Chapter 5 that the elastic buckling analyses for the shear centre loading cases were

limited to sections with high Ke values, because the buckling load was increased to

the region where local buckling was the critical mode. However for cantilever LSBs

with top flange loading, the lateral distortional buckling remain the critical mode at

BF

TF

SC

TF

BF SC PL (free end)

case

UDL case

Interaction

Interaction



longer Ke region (than in SC and BF cases) as shown in Figure 7.9. The free end

condition from a cantilever offers less stability that may cause larger coupled

twisting (due to TF loading action) unlike in the case of simply supported beam, thus

resulting in significant buckling strength reduction. Figure 7.10 demonstrates the

significance of top flange loading to augment the cross section twisting on cantilever

LSBs subject to a PL at the free end.

Figure 7.10 – Load Height Effects on the Lateral Distortional Buckling Mode

(1 m Cantilever LSB125x45x20 Subjected to a PL at the free end)

Nevertheless, interaction buckling with non-lateral buckling mode was also observed

in the elastic buckling analysis for the TF loading case with high Ke values as shown

in Figure 7.9. For very high Ke values (short cantilever), the elastic buckling analysis

gave to a non-lateral buckling as the critical mode. This non-lateral buckling mode

was a local buckling near the support for the UDL case, while for the PL case it was

more prone to local web buckling (web bearing) at the load location. Figure 7.11

shows these other critical modes that precede lateral distortional buckling. These

buckling modes are not relevant to this study which is only concerned with lateral

buckling.

Shear centre loading (SC) Bottom flange loading (BF) Top flange loading (TF)

LDB with amplified torsion

LDB with reduced torsion (greater lateral component)

Lateral distortional buckling (LDB)



(a) 2m LSB250x60x2.0 with a PL (TF) (b) 2m LSB250x60x2.0 with a UDL (TF)

Figure 7.11 – Other Critical Buckling Modes of Cantilever LSBs with TF

Loading

1

2

3

4

5

6

7

8

9

10

11

12


Dim

ensi

onle

ss B

uckl

ing

Load

LSB250x60x2.0 (slender)LSB300x75x3.0 (non-compact)LSB125x45x2.0 (compact)Prediction by Trahair's equation (Eq 7.3)

(a) With αm Factor Based on the Equation Developed using FEA (Eq. 5.9)

Figure 7.12 – Comparison with Current Design Methods to Predict Cantilever

LSB’s Lateral Buckling Strength Subjected to Load Height Effect (PL Case)


Interaction with local web buckling (web bearing) at the load position (free end)

BFSC

TFAS4100-1998 (kl = 2)

BS5950-1:2000 (kl = 1.4)

Interaction



1

2

3

4

5

6

7

8

9

10

11

12


Dim

ensi

onle

ss B

uckl

ing

Load

LSB250x60x2.0 (slender)LSB300x75x3.0 (non-compact)LSB125x45x2.0 (compact)Prediction by Trahair's equation (Eq 7.3)

(b) With αm Factor Based on the Respective Design Codes


Lateral Buckling Strength of Cantilever LSBs Subjected to Load Height Effect

(PL Case) (Continued)

Due to the significance effect of top flange loading in a cantilever, British (BS5950-

1) and Australian steel design codes (AS4100) use a higher load height factor (kl) to

increase the beam’s effective length (Le = L x kl) than in the case of simply

supported beam. This effective length method is used to calculate the elastic

buckling moment (Mod-non for TF loading). For loading below the shear centre (i.e.

BF) both design codes conservatively ignore its beneficial effects.

Figure 7.12(a) compares the dimensionless buckling loads from the elastic buckling

analysis results with the predictions using BS5950-1 (BS1, 2000) and AS4100 (SA,

1998). In calculating the Mod-non for TF loading using the design code method of

effective length, αm factors obtained from the proposed Equation 5.9 (developed in

Chapter 5) were used to include the moment distribution effects. Appendix C gives

the example calculation for Figures 7.12 and 7.13 (i.e. dimensionless buckling load

of using the design code method). The comparison indicates that they are

BFSC

TF

BS5950-1:2000 (kl = 1.4) with αm = 1.0

AS4100-1998 (kl = 2) with αm = 1.25

Interaction



conservative for low Ke values (high beam slenderness) but unsafe for the opposite

case, particularly the BS5950-1 prediction.

A single αm factor of 1.25 for cantilever with a PL (free end) is suggested in AS4100

Table 5.6.2, while BS5950-1 appears to incorporate the moment distribution effect in

the effective length method, meaning that αm factor is equal to 1.0. These factors

were used in calculating the Mod for TF loading with the design code method of

effective length and compared with elastic buckling analysis results as shown in

Figure 7.12 (b). Both AS4100 and BS5950-1 show better prediction (although

become very conservative for the case with low Ke value) when the respective αm

factor is used accordingly. For very high Ke value (short cantilever), their prediction

may not be appropriate for use, however in such cases it may not be very important

as other critical buckling modes may precede the lateral distortional buckling and

also it is close to the region of section capacity.

0

5

10

15

20

25

30

35

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4Modified Torsion Parameter (Ke)

Dim

ensi

onle

ss B

uckl

ing

Load

LSB250x60x2.0 (slender)LSB300x75x3.0 (non-compact)LSB125x45x2.0 (compact)Prediction by Trahair's equation (Eq 7.3a)


Lateral Buckling Strength of Cantilever LSBs Subjected to Load Height Effect

(UDL Case)

BF

SC

TF

BS5950-1:2000 (kl = 1.4) with αm = 1.0

AS4100-1998 (kl = 2) with αm = 2.25



A comparison with the current design code method for the UDL case is also given in

Figure 7.13 (with the αm factor obtained from the respective design codes). It is

similar to the case with PL at the free end except that the BS5950-1 prediction

appears very conservative. This may be because BS5950-1 method is mainly based

on the PL case which is often considered as the worst transverse loading case for

cantilever.

Trahair (1993) also provides an approximate solution to predict the elastic lateral

buckling strength of a cantilever subjected to loading above and below the shear

centre as given by Equations 7.3(a) and 7.3(b) for the UDL and PL cases,

respectively.

For PL at the free end case:

( ) ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−+

−+−+⎟⎟

⎠

⎞⎜⎜⎝

⎛

++=

2222

2

1.02.11

)1.0(2.11)2(42.11

2.1111ε

ε

ε

εe

ey

KGJEI

QL (7.3a)

For UDL case:

( ) ( ) ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−+

−+−+⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

−+

−+=

2222 1.03.11

)1.0(3.11)2(101.04.11

)1.0(4.11272

3

ε

ε

ε

εe

ey

KGJEI

QL (7.3b)

Where, ε = ⎟⎟⎠

⎞⎜⎜⎝

⎛

e

Q

GJEI

Ly

Ke = modified torsion parameter

yQ = load height

GJe = effective torsional rigidity

Note that these equations are independent of the αm factor unlike the equation given

for the case of simply supported beams (Eq. 7.2). The effect of web distortion from

LDB mode can be incorporated in these equations by using the modified torsion

parameter (Ke) and effective torsional rigidity (GJe) as opposed to the original

equations which use the torsion parameter (K) and torsional rigidity (GJ). The

predictions using these equations are also compared with the elastic buckling

analysis results as shown in Figures 7.12 and 7.13 (non-dimensionalised format).

The comparison of top flange (TF) loading for the UDL case shows the adequacy of



this equation, but not for the PL case as it is unconservative even in the intermediate

Ke regions.

This means that AS4100 and BS5950-1 methods are more suitable than those

equations above, therefore they are recommended to allow for the load height effect

of TF loading on cantilever LSBs. As AS4100 is less conservative, it is preferred to

the latter code. In modifying the elastic bucking moment with the load height effect

(Mod-non with load height), the αm factor is suggested in accordance with the

provision in the design code. For this reason also, the moment distribution effect

study in Chapter 5 suggests to adopt a constant αm factor based on AS4100.

For the case of bottom flange (BF) loading, the comparison in Figures 7.12 and 7.13

may indicate that Equations 7.3(a) and 7.3(b) are adequate for use, provided that

other buckling mode is assumed to be the critical for the case with high Ke values.

Nevertheless, AS4100 and BS5950-1 methods that ignore the benefits of loading

below the shear centre can be conservatively used in cantilever LSBs. Its significant

benefits may not be very important as its ultimate strength is very likely to be limited

to its section capacity particularly for intermediate spans and less. The study in

Chapter 5 shows that the ultimate strength of intermediate span cantilever LSB with

shear centre loading is often controlled by its section capacity.

7.4. Design Recommendation Summary

The effect of loading above the shear centre creates an additional torque about the

shear centre axis, subjecting LSBs with an additional twisting thus reducing the

buckling resistance. The approximate AS4100 (SA, 1998) method of using a

modified effective length, using a load height factor (kl), is suitable to allow for the

destabilising effect from top flange loading on the lateral buckling resistance of

simply supported LSBs. A more accurate method using Trahair’s equation can also

be used to estimate the lateral buckling strengths (LDB and LTB) of LSBs subject to

top flange loading, provided that the appropriate αm factor is used. The accurate αm

factor is also recommended to be used in using the AS4100 method. The

destabilising effect of top flange loading on the lateral buckling resistance of

cantilever LSBs can be accounted for by using either the AS4100 or the BS5950-1

methods of effective length, however with the αm factor based on the respective



design codes (not the accurate αm factor) due to the significance of top flange

loading in cantilever LSBs. The AS4100 method is less conservative, thus it is

recommended for cantilever LSBs subject to top flange loading.

To allow for the load height effects for LSB design purposes, the method adopted in

many cold-formed steel codes can be used. The elastic buckling moment (Mod) for

load height is obtained using the recommendations above and used for the member

moment capacity (Mb) calculation. It should also be noted that the recommendations

above may not be accurate for LSBs with high Ke values, however they are likely to

be governed with other non-lateral buckling modes. The loading below the shear

centre exerts additional torque that increases the lateral buckling resistance of LSBs.

Its beneficial effect is always conservatively neglected in the current steel design

codes, and the same method can be also conservatively used for both simply

supported and cantilever LSBs. More accurate equations to predict the elastic

buckling moment are also available but less important and convenient for LSB

design.

Chapter 8 – Conclusions and Recommendations


8. Conclusions and Recommendations

This thesis has described a detailed investigation into the flexural behaviour and

design of the new LiteSteel beam sections subject to lateral distortional buckling.

This unique lateral buckling mode characterised by lateral deflection, twist and web

distortion was found to limit the flexural capacity of LiteSteel beam (LSB) for

intermediate spans. Effects of moment distribution and load height on the lateral

buckling strength of LSBs were also not known. This research used both finite

element analyses and laboratory experiments, in particular, the former method, to

improve the knowledge and understanding of the lateral distortional buckling

behaviour of LSB flexural members subject to various loading and support

conditions.

The first phase of this research investigated the suitability and accuracy of the

original ideal finite element model used by Mahaarachchi and Mahendran (2005c,d)

to develop the current design rules for the member capacity of LiteSteel beams under

uniform conditions. It was found that this ideal finite element model had some

modelling inaccuracies relating to the use of external dimensions and the modelling

of ideal support conditions required for lateral buckling. Hence, a new finite element

model was developed for simply supported LiteSteel beam sections based on a

modification of the original model developed by Mahaarachchi and Mahendran

(2005c), which accounted for all significant behavioural effects including material

inelasticity, buckling deformations (local, lateral distortional and lateral torsional

buckling), member instability, residual stresses, and initial geometric imperfections.

The new model was successfully validated using available results from

Mahaarachchi and Mahendran (2005c) and other well established numerical methods

such as finite strip method. A comparison of the lateral distortional buckling strength

results using the modified finite element model with the current LSB design rules

showed that the current design rules are over-conservative for LSB sections with

intermediate beam slenderness.



In the second phase of this research, the effects of non-uniform moment distribution

and load height on the lateral buckling strength of LiteSteel beams were investigated

using finite element analyses for both their elastic buckling and non-linear

behaviours. The new finite element model developed for LSB sections subject to

uniform moment and simple support conditions was further modified to simulate

various loading and support configurations, i.e. non-uniform moment distributions,

load heights, simply supported and cantilever LSBs. Following conclusions and

design recommendations have been drawn from the results for the moment

distribution and load height effects on LSBs:

• The finite element analysis study found that the strength benefit due to non-

uniform bending moment distributions for simply supported LSBs is

unfavourably influenced by lateral distortional buckling. The moment

distribution factor (αm) reaches the upper bound with high beam slenderness

(subject to lateral torsional buckling), but it reduces with lower beam

slenderness due to the increasing level of web distortion of lateral distortional

buckling, until other buckling modes that precede lateral buckling govern

(i.e. local buckling, shear buckling, etc). For the moment gradient case, the

benefit variation is insignificant and thus the currently available αm equations

in AS4100 (SA, 1998), BS5950-1 (BSI, 2000) and ANSI/AISC 360 (AISI,

2005) are adequate and hence recommended for the design of LSBs, while a

more accurate equation is also developed in this thesis.

• The reduction of moment distribution benefits due to lateral distortional

buckling for simply supported LSBs with transverse loading (uniformly

distributed load, mid-span point load, and quarter point loads) is more

significant, where the presence of shear stresses may contribute to greater

reduction in αm factors. It is recommended to use the new αm equations

developed in this thesis for simply supported LSBs with a uniformly

distributed load and a mid-span point load. The proposed αm equations

suggest neglecting the moment distribution effect when shear stresses are

significant. For quarter point load cases, it is recommended to neglect any

moment distribution effect (αm = 1).

• For cantilever LSBs subjected to transverse loading (uniformly distributed

load and point load at the free end), the unfavourable effect of lateral



distortional buckling is less than in the case of simply supported beams. The

currently available αm factors were found to be conservative for intermediate

to long span cantilever LSBs (other buckling modes govern for short spans).

The AS4100 design approach is less conservative than BS5950-1 and

ANSI/AISC 360, and is recommended for cantilever LSBs. More accurate αm

equations are also provided in this thesis, but are less favoured for design

purposes.

• A limited number of experiments was conducted on LSBs subjected to

moment gradient to confirm the findings from the finite element analyses.

Experimental results validated the suitability of the adopted finite element

model for LSBs and the proposed recommendation to include the moment

gradient benefits for LSB design purposes.

• The effect of loading above the shear centre creates an additional torque

about the shear centre axis, subjecting LSBs with an additional twisting thus

reducing the buckling resistance. The approximate AS4100 method of using

an effective length modified by a load height factor (kl) is suitable to allow

for the destabilising effect from top flange loading on the lateral buckling

resistance of simply supported LSBs. A more accurate method using

Trahair’s equation (1993) can also be used to estimate LSB’s lateral buckling

moments (lateral distortional and lateral torsional buckling) subject to top

flange loading, provided an appropriate αm factor is used. The accurate αm

factor is also recommended to be used in using the AS4100 design method.

• The destabilising effect of top flange loading on the lateral buckling

resistance of cantilever LSBs can be accounted for by using either AS4100 or

BS5950-1 methods of effective length, however with the αm factor based on

the respective design codes (not the accurate αm factor) due to the

significance of top flange loading in cantilever LSBs. The AS4100 method is

less conservative, and hence it is recommended for cantilever LSBs subject to

top flange loading.

• The loading below the shear centre exerts an additional torque that increases

the lateral buckling resistance of LSBs. Its beneficial effect is always

conservatively neglected in the current steel design codes, and the same

approach can also be conservatively used for both simply supported and



cantilever LSBs. More accurate equations are available to predict the elastic

buckling moment, but are less important and convenient for LSB design

purposes.

• The application of αm and kl factors suggested above for LSB design

purposes is recommended for use in the design method in many cold-formed

steel codes in which the elastic lateral distortional buckling moment Mod for

the uniform moment case is modified by using the appropriate αm and kl

factors, and used in the member capacity calculations.

• The design method used in many hot-rolled steel codes allows additional

moment distribution benefits to lateral buckling strength in the non-linear

behaviour by using an enhanced member capacity of αm times the member

capacity for uniform moment conditions. This research has shown that this

approach is not suitable for LSBs.

The study and design recommendation in this research are intended only for LSB

sections. Their application to other sections that have similar characteristics and

flexural behaviours may be still suitable; however such assumption should be taken

with care. Despite the recommendations provided in this research, further research is

required in the following areas:

• Non-linear static finite element analysis of all the available LSB sections

using the new finite element model of LSB section (the final modified

version) for a complete review of the current LSB design curve and to

investigate the effects of LSB section geometry on lateral distortional

buckling strength.

• The finite element analysis studies of the lateral buckling of LSBs under

transverse loading were often limited in this research due to the interaction of

bending and shear effects. The interaction is a result of 3D modelling using

shell elements and it is inevitable to exclusively treat one of these

components. Therefore other buckling analysis techniques such as energy

methods are recommended to derive solutions exclusively for lateral

distortional buckling (particularly in the high Ke region). Further

investigation into the interaction of bending and shear effects is also

important.



• Experimental study of LSBs with other loading and support conditions

associated with transverse loads and cantilever beams. It is recommended

that such tests are taken with greater care as other factors may influence the

results, i.e. an interaction of bending and shear in the case of intermediate

spans (and less) of deep sections.

• This research was limited to the case with loading through the cross section

shear centre (transverse loading case). However loading directly onto the

LSB section is the most likely case encountered in practice. The effect of

loading away from the shear centre on the lateral distortional buckling

behaviour and strength of LSB section is not yet known, and therefore further

research is recommended.

Chapter 9 – References


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Appendix A


Appendix A (Section Properties, Capacities and Tolerances)

Appendix A


Appendix A


Section Capacities Based on AS/NZS4600 (with Corner Radius)

Section Effective Yield Stress Section Modulus Section Modulus Flange Web Capacity LSB Section

Zx Zex fyf fyw Ms x-x d x bf x t (mm) kg/m 103 mm3 MPa MPa kNm

300 x 75 x 3.0 163.8 163.8 450 380 73.72 2.5 138.5 138.5 450 380 62.32

300 x 60 x 2.0 96.8 96.8 450 380 43.57 250 x 75 x 3.0 126.9 126.9 450 380 57.12

2.5 107.4 107.4 450 380 48.32 250 x 60 x 2.0 75.0 75.0 450 380 33.76 200 x 60 x 2.5 67.4 67.4 450 380 30.34

2.0 55.0 55.0 450 380 24.73 200 x 45 x 1.6 36.7 36.7 450 380 16.54 150 x 45 x 2.0 30.1 30.1 450 380 13.55

1.6 24.6 24.6 450 380 11.06 125 x 45 x 2.0 23.3 23.3 450 380 10.48

1.6 19.0 19.0 450 380 8.57

Section Capacities Based on AS/NZS4600 (without Corner Radius)

Section Effective Yield Stress Section Modulus Section Modulus Flange Web Capacity LSB Section

Zx Zex fyf fyw Ms x-x d x bf x t (mm) kg/m 103 mm3 MPa MPa kNm

300 x 75 x 3.0 169.0 169.0 450 380 76.05 2.5 142.1 142.1 450 380 63.95

300 x 60 x 2.0 99.3 99.3 450 380 44.66 250 x 75 x 3.0 131.1 131.1 450 380 59.01

2.5 110.3 110.3 450 380 49.65 250 x 60 x 2.0 77.0 77.0 450 380 34.65 200 x 60 x 2.5 69.8 69.8 450 380 31.40

2.0 56.5 56.5 450 380 25.42 200 x 45 x 1.6 37.8 37.8 450 380 17.01 150 x 45 x 2.0 31.3 31.3 450 380 14.07

1.6 25.3 25.3 450 380 11.40 125 x 45 x 2.0 24.2 24.2 450 380 10.90

1.6 19.6 19.6 450 380 8.84

Appendix A


Fabrication Tolerances for LSB Sections (SSTM, 2005)

halla


Appendix B


Appendix B (Elastic Buckling Moments)

Appendix B


Elastic Lateral Buckling Moments from Mod Equation (with Corner Radius)

LSB Section Elastic Lateral Buckling Moment (kNm) d x bf x t (mm) 0.2m 0.3m 0.5m 0.75m 1.0m 1.5m 2.0m 2.5m

300 x 75 x 3.0 LSB 7156.82 3181.56 1147.58 514.46 295.66 145.34 97.29 76.55 2.5 LSB 6165.72 2740.75 987.96 441.71 252.25 120.96 78.67 60.86

300 x 60 x 2.0 LSB 2710.00 1204.66 434.31 194.30 111.13 53.60 35.08 27.20 250 x 75 x 3.0 LSB 5598.66 2489.42 899.52 406.34 237.64 124.46 89.06 73.10

2.5 LSB 4824.64 2144.94 774.13 347.96 201.28 101.77 70.81 57.63 250 x 60 x 2.0 LSB 2131.25 947.54 342.06 153.90 89.22 45.41 31.72 25.78 200 x 60 x 2.5 LSB 1892.84 842.28 306.17 141.65 86.84 51.24 39.46 33.21

2.0 LSB 1579.17 702.38 254.40 116.09 69.41 38.95 29.54 25.06 200 x 45 x 1.6 LSB 580.34 258.18 93.68 43.03 26.04 14.95 11.38 9.56 150 x 45 x 2.0 LSB 471.63 210.53 78.33 39.07 26.57 18.04 14.38 12.06

1.6 LSB 394.59 175.85 64.67 31.21 20.49 13.61 11.02 9.42 125 x 45 x 2.0 LSB 367.38 164.68 63.01 33.76 24.57 17.66 14.20 11.89

1.6 LSB 307.44 137.37 51.49 26.35 18.57 13.34 10.98 9.39 LSB Section Elastic Lateral Buckling Moment (kNm)

d x bf x t (mm) 3.0m 4.0m 5.0m 6.0m 7.0m 8.0m 10m 300 x 75 x 3.0 LSB 65.15 51.97 43.69 37.70 33.10 29.46 24.08

2.5 LSB 51.59 41.59 35.53 31.07 27.57 24.73 20.44 300 x 60 x 2.0 LSB 23.02 18.41 15.59 13.54 11.95 10.68 8.77 250 x 75 x 3.0 LSB 63.54 51.31 43.13 37.12 32.50 28.87 23.52

2.5 LSB 50.28 41.35 35.36 30.83 27.26 24.38 20.05 250 x 60 x 2.0 LSB 22.40 18.23 15.46 13.39 11.79 10.51 8.60 200 x 60 x 2.5 LSB 28.94 23.06 19.11 16.26 14.13 12.48 10.10

2.0 LSB 22.14 18.12 15.30 13.19 11.56 10.27 8.37 200 x 45 x 1.6 LSB 8.34 6.68 5.55 4.74 4.13 3.65 2.96 150 x 45 x 2.0 LSB 10.37 8.07 6.58 5.55 4.79 4.21 3.39

1.6 LSB 8.23 6.55 5.41 4.59 3.98 3.51 2.84 125 x 45 x 2.0 LSB 10.19 7.89 6.42 5.40 4.65 4.09 3.28

1.6 LSB 8.17 6.45 5.30 4.49 3.88 3.42 2.76

Appendix B


Elastic Lateral Buckling Moments from THINWALL (with Corner Radius)

LSB Section Elastic Lateral Buckling Moment (kNm) d x bf x t (mm) 0.2m 0.3m 0.5m 0.75m 1.0m 1.5m 2.0m 2.5m 300 x 75 x 3.0 LSB 179.9* 179.9* 179.9* 179.9* 179.9* 140.02 95.32 75.64

2.5 LSB 106.0* 106.0* 106.0* 106.0* 106.0* 115.78 76.60 59.76 300 x 60 x 2.0 LSB 43.3* 43.3* 43.3* 43.3* 43.3* 51.60 34.23 26.70 250 x 75 x 3.0 LSB 218.4* 218.4* 218.4* 218.4* 218.4* 120.80 87.77 72.62

2.5 LSB 129.1* 129.1* 129.1* 129.1* 129.1* 98.19 69.40 56.93 250 x 60 x 2.0 LSB 52.1* 52.1* 52.1* 52.1* 52.1* 43.96 31.06 25.39 200 x 60 x 2.5 LSB 126.0* 126.0* 126.0* 126.0* 83.14 50.31 39.14 33.14

2.0 LSB 66.3* 66.3* 66.3* 66.3* 66.3* 37.99 29.12 24.83 200 x 45 x 1.6 LSB 24.9* 24.9* 24.9* 24.9* 24.9* 14.62 11.20 9.46 150 x 45 x 2.0 LSB 64.3* 64.3* 64.3* 37.42 25.92 17.87 14.36 12.10

1.6 LSB 34.0* 34.0* 34.0* 29.68 19.85 13.38 10.91 9.38 125 x 45 x 2.0 LSB 55.2* 55.2* 55.2* 32.55 24.09 17.56 14.23 11.97

1.6 LSB 28.1* 28.1* 28.1* 27.18 18.11 13.18 10.93 9.38 LSB Section Elastic Lateral Buckling Moment (kNm)


2.5 LSB 50.91 41.30 35.42 31.07 27.64 24.84 20.57 300 x 60 x 2.0 LSB 22.68 18.24 15.52 13.52 11.97 10.71 8.83 250 x 75 x 3.0 LSB 63.43 51.54 43.48 37.51 32.91 29.26 23.88

2.5 LSB 49.90 41.26 35.41 30.95 27.41 24.55 20.22 250 x 60 x 2.0 LSB 22.15 18.13 15.44 13.42 11.83 10.56 8.67 200 x 60 x 2.5 LSB 28.99 23.22 19.30 16.46 14.32 12.66 10.25

2.0 LSB 22.02 18.12 15.34 13.26 11.63 10.35 8.45 200 x 45 x 1.6 LSB 8.29 6.68 5.57 4.77 4.16 3.68 2.99 150 x 45 x 2.0 LSB 10.45 8.16 6.67 5.63 4.86 4.28 3.44

1.6 LSB 8.23 6.58 5.45 4.63 4.02 3.55 2.87 125 x 45 x 2.0 LSB 10.29 8.00 6.51 5.48 4.73 4.15 3.34

1.6 LSB 8.20 6.50 5.35 4.53 3.92 3.46 2.79 * Subjected to local buckling mode (critical buckling mode) that precedes lateral buckling)

Appendix B


Elastic Lateral Buckling Moments from THINWALL (without Corner Radius)


2.5 LSB 108.5* 108.5* 108.5* 108.5* 108.5* 119.80 78.99 61.36 300 x 60 x 2.0 LSB 44.4* 44.4* 44.4* 44.4* 44.4* 53.40 35.29 27.41 250 x 75 x 3.0 LSB 224.4* 224.4* 224.4* 224.4* 224.4* 125.50 90.62 74.61

2.5 LSB 132.0* 132.0* 132.0* 132.0* 132.0* 101.50 71.40 58.32 250 x 60 x 2.0 LSB 53.2* 53.2* 53.2* 53.2* 53.2* 45.45 31.95 26.01 200 x 60 x 2.5 LSB 129.3* 129.3* 129.3* 129.3* 86.90 52.13 40.29 33.97

2.0 LSB 67.6* 67.6* 67.6* 67.6* 67.6* 39.18 29.86 25.37 200 x 45 x 1.6 LSB 25.4* 25.4* 25.4* 25.4* 25.4* 15.09 11.50 9.67 150 x 45 x 2.0 LSB 65.7* 65.7* 65.7* 39.31 27.05 18.46 14.76 12.40

1.6 LSB 34.4* 34.4* 34.4* 30.92 20.59 13.76 11.17 9.57 125 x 45 x 2.0 LSB 54.7* 54.7* 54.7* 34.11 25.05 18.10 14.61 12.25



2.5 LSB 52.10 42.08 36.00 31.53 28.01 25.16 20.82 300 x 60 x 2.0 LSB 23.21 18.58 15.76 13.72 12.12 10.85 8.93 250 x 75 x 3.0 LSB 64.96 52.58 44.28 38.15 33.44 29.71 24.23

2.5 LSB 50.96 41.99 35.96 31.39 27.78 24.86 20.46 250 x 60 x 2.0 LSB 22.61 18.44 15.67 13.60 11.99 10.69 8.77 200 x 60 x 2.5 LSB 29.65 23.68 19.65 16.75 14.56 12.86 10.41

2.0 LSB 22.45 18.41 15.57 13.44 11.79 10.48 8.55 200 x 45 x 1.6 LSB 8.45 6.79 5.66 4.84 4.22 3.73 3.03 150 x 45 x 2.0 LSB 10.68 8.33 6.81 5.74 4.96 4.36 3.51

1.6 LSB 8.38 6.69 5.53 4.70 4.08 3.60 2.91 125 x 45 x 2.0 LSB 10.53 8.17 6.65 5.59 4.82 4.24 3.41


Appendix B


Elastic Lateral Buckling Moments from THINWALL using External Dimension

(without Corner Radius)


2.5 LSB 115.5* 115.5* 115.5* 115.5* 115.5* 133.60 87.63 67.98 300 x 60 x 2.0 LSB 46.7* 46.7* 46.7* 46.7* 46.7* 59.46 39.11 30.36 250 x 75 x 3.0 LSB 242.2* 242.2* 242.2* 242.2* 242.2* 142.70 102.90 85.21

2.5 LSB 140.6* 140.6* 140.6* 140.6* 140.6* 113.00 79.17 64.77 250 x 60 x 2.0 LSB 56.6* 56.6* 56.6* 56.6* 56.6* 50.50 35.39 28.88 200 x 60 x 2.5 LSB 140.0* 140.0* 140.0* 140.0* 99.58 59.54 46.36 39.53

2.0 LSB 72.0* 72.0* 72.0* 72.0* 72.0* 43.47 33.16 28.35 200 x 45 x 1.6 LSB 27.0* 27.0* 27.0* 27.0* 27.0* 16.84 12.89 10.93 150 x 45 x 2.0 LSB 71.6* 71.6* 71.6* 45.45 31.19 21.51 17.49 14.89

1.6 LSB 36.8* 36.8* 36.8* 34.76 23.04 15.42 12.65 10.95 125 x 45 x 2.0 LSB 53.2* 53.2* 53.2* 39.43 29.01 21.31 17.50 14.86



2.5 LSB 57.84 47.17 40.78 36.02 32.22 29.09 24.24 300 x 60 x 2.0 LSB 25.79 20.87 17.90 15.71 13.98 12.57 10.42 250 x 75 x 3.0 LSB 74.79 61.51 52.41 45.53 40.14 35.83 29.38

2.5 LSB 56.87 47.41 41.03 36.10 32.13 28.89 23.92 250 x 60 x 2.0 LSB 25.25 20.86 17.91 15.67 13.89 12.44 10.26 200 x 60 x 2.5 LSB 34.86 28.29 23.71 20.34 17.76 15.74 12.79

2.0 LSB 25.28 21.01 17.94 15.58 13.73 12.25 10.03 200 x 45 x 1.6 LSB 9.65 7.86 6.62 5.69 4.98 4.42 3.60 150 x 45 x 2.0 LSB 12.96 10.24 8.42 7.13 6.17 5.44 4.38

1.6 LSB 9.68 7.82 6.51 5.56 4.84 4.28 3.47 125 x 45 x 2.0 LSB 12.87 10.09 8.25 6.97 6.02 5.29 4.26


Appendix C


Appendix C (Residual Stresses Subroutine and Example Calculations)

Appendix D


Typical Subroutine for Residual Stresses Input:

---------------------------------------------- start -------------------------------------------------

SUBROUTINE SIGINI

(SIGMA,COORDS,NTENS,NCRDS,NOEL,NPT,LAYER,KSPT)

C

INCLUDE 'ABA_PARAM.INC'

C

REAL X,Y,Z,nipt,ipt,Fy,IMPBUF,IMPBLF,IMPTLF,IMPTUF,

sigmaout,MEMB

DIMENSION SIGMA(NTENS), COORDS(NCRDS)

C

X=COORDS(1)

Y=COORDS(2)

Z=COORDS(3)

fullspan=2500.

midspan1=1250.

midspan2=fullspan-midspan1

IMP=-2.5

TUF=1.0

TLF=0.964

BUF=0.627

BLF=0.594

BF=43.

tk=1.

nipt=9.

Fy=380.

C

IF(KSPT.EQ.1.) THEN

ipt=1.

ENDIF

IF(KSPT.EQ.2.) THEN

ipt=2.

ENDIF

Appendix D


IF(KSPT.EQ.3.) THEN

ipt=3.

ENDIF

IF(KSPT.EQ.4.) THEN

ipt=4.

ENDIF

IF(KSPT.EQ.5.) THEN

ipt=5.

ENDIF

IF(KSPT.EQ.6.) THEN

ipt=6.

ENDIF

IF(KSPT.EQ.7.) THEN

ipt=7.

ENDIF

IF(KSPT.EQ.8.) THEN

ipt=8.

ENDIF

IF(KSPT.EQ.9.) THEN

ipt=9.

ENDIF

C

IF(X.LE.(midspan1)) THEN

IMPTUF=Z-tk-(IMP*TUF)*X/midspan1

IMPTLF=Z-tk-(IMP*TLF)*X/midspan1

IMPBUF=Z-tk-(IMP*BUF)*X/midspan1

IMPBLF=Z-tk-(IMP*BLF)*X/midspan1

ELSEIF(X.GE.(midspan1)) THEN

IMPTUF=Z-tk-(IMP*TUF)*(fullspan-X)/midspan2

IMPTLF=Z-tk-(IMP*TLF)*(fullspan-X)/midspan2

IMPBUF=Z-tk-(IMP*BUF)*(fullspan-X)/midspan2

IMPBLF=Z-tk-(IMP*BLF)*(fullspan-X)/midspan2

ENDIF

C

Appendix D


C FLEXURAL RESIDUAL STRESS

IF((NOEL.GE.11751.).AND.(NOEL.LE.14000.)) THEN

sigmaout=(0.24*Fy+0.83*Fy*IMPTUF/BF)*(1.-2.*(nipt-ipt)/(nipt-1.))

ELSEIF((NOEL.GE.1.).AND.(NOEL.LE.2250.)) THEN

sigmaout=(0.24*Fy+0.83*Fy*IMPBLF/BF)*(1.-2.*(nipt-ipt)/(nipt-1.))


sigmaout=(0.38*Fy+0.42*Fy*IMPTLF/BF)*(1.-2.*(nipt-ipt)/(nipt-1.))


sigmaout=(0.38*Fy+0.42*Fy*IMPBUF/BF)*(1.-2.*(nipt-ipt)/(nipt-1.))

ELSEIF(((NOEL.GE.14001.).AND.(NOEL.LE.14750.)).OR.

& ((NOEL.GE.2251.).AND.(NOEL.LE.3000.))) THEN

sigmaout=(0.41*Fy)*(1.-2.*(nipt-ipt)/(nipt-1.))






ENDIF

C

C MEMBRANE RESIDUAL STRESS

IF((NOEL.GE.6001.).AND.(NOEL.LE.11000.)) THEN

IF((Y.GE.14).AND.(Y.LE.62.5)) THEN

MEMB=(0.0208*Y-0.7015)*Fy

ELSEIF((Y.GE.62.5).AND.(Y.LE.111)) THEN

MEMB=(-0.0208*Y+1.9015)*Fy

ENDIF


MEMB=0.11*Fy-0.08*Fy*IMPTLF/BF


MEMB=0.11*Fy-0.08*Fy*IMPBUF/BF



MEMB=0.03*Fy


Appendix D



MEMB=-0.23*Fy

ENDIF

SIGMA(1)=sigmaout+MEMB

C

SIGMA(2)=0

SIGMA(3)=0

C

RETURN

END

----------------------------------------------- end -------------------------------------------------

Note:

IMP = imperfection value (i.e. span/1000)

TUF = top-upper flange displacement factor

TLF = top-lower flange displacement factor

BUF = bottom-upper flange displacement factor

BLF = bottom-lower flange displacement factor

BF = flange width

tk = thickness

TUF

TLF

BUF

BLF

Appendix D


Example moment calculation from the finite element analysis using the ideal

finite element model (final version) which adopts linear forces at the support to

simulate the applied end moment:

For 2500mm LSB250x60x2.0 subject to uniform bending moment; Linear (-) Forces at Beam End Nodes

Lever Arm* Moment Total

Moment Moment Node Order N mm Nmm Nmm

Load Factor (Output from

FEA) kNm 1 -10000.4 248.0 2480110.4 80408576.8 0.3204 25.76 2 -10000.4 248.0 2480110.4 3 -10000.4 248.0 2480110.4 4 -10000.4 248.0 2480110.4 5 -10000.4 248.0 2480110.4 6 -10000.4 248.0 2480110.4 7 -10000.4 248.0 2480110.4 8 -10000.4 248.0 2480110.4 9 -10000.4 248.0 2480110.4

10 -10000.4 248.0 2480110.4 11 -10000.4 248.0 2480110.4 12 -10000.4 248.0 2480110.4 13 -10000.4 248.0 2480110.4 14 -9637.45 239.0 2303350.0 15 -9274.55 230.0 2133146.5 16 -8911.65 221.0 1969475.2 17 -8548.75 212.0 1812336.1 18 -8548.75 212.0 1812336.1 19 -8548.75 212.0 1812336.1 20 -8548.75 212.0 1812336.1

SPC “1234” Restrained DOF

“234” for all other nodes

Restrained DOF “234” for all nodes at the other end

End moment simulation

Beam end

Linear compressive (-) forces at every node (max at the top flange)

Linear tensile (+) forces at every node (max at the bottom flange) Moment

Node Order

Appendix D


Linear (-) Forces at Beam End Nodes

Lever Arm* Moment Total

Moment Moment Node Order N mm Nmm Nmm

Load Factor from FEA kNm

21 -8548.75 212.0 1812336.1 22 -8548.75 212.0 1812336.1 23 -8548.75 212.0 1812336.1 24 -8548.75 212.0 1812336.1 25 -8548.75 212.0 1812336.1 26 -8548.75 212.0 1812336.1 27 -8548.75 212.0 1812336.1 28 -8548.75 212.0 1812336.1 29 -8548.75 212.0 1812336.1 30 -9637.45 239.0 2303350.0 31 -9274.55 230.0 2133146.5 32 -8911.65 221.0 1969475.2 33 -8141.68 201.9 1643844.0 34 -7734.62 191.8 1483574.8 35 -7327.55 181.7 1331520.6 36 -6920.48 171.6 1187685.5 37 -6513.42 161.5 1052072.6 38 -6106.35 151.4 924675.3 39 -5699.29 141.3 805499.9 40 -5292.22 131.2 694540.5 41 -4885.16 121.1 591802.5 42 -4478.09 111.0 497280.9 43 -4071.03 101.0 410980.4 44 -3663.96 90.9 332896.7 45 -3256.9 80.8 263033.7 46 -2849.83 70.7 201387.9 47 -2442.77 60.6 147962.3 48 -2035.7 50.5 102754.3 49 -1628.6 40.4 65763.0 50 -1221.58 30.3 36996.7 51 -814.486 20.2 16444.5 52 -407.471 10.1 4113.8

* Note that the lever arm is taken double, thus the tensile forces can be neglected

Appendix D


Other example calculations:

250x60x2.0 LSB section – based on nominal dimensions and yield stresses, and

without corner radius:

d = 250 mm

d1 = 210 mm

df = 20 mm

bf = 60 mm

fyf = 450 MPa (flange)

fyw = 380 MPa (web)

Iy = 0.46 x 106 mm4

Iw = 4.46 x 109 mm6

E = 200000 MPa

(note: without corner radius) G = 80000 MPa

• Section capacity (Ms):

Ms = Ze fy (AS4600 Clause 2.22)

Zx = Zex (fully effective)

Zex = 77 x 103 mm3 (Appendix A)

Ms = 77 x 103 x 450

Ms = 34.65 kNm = My (i.e. Used in Table 5.2 of Chapter 5)

• Elastic lateral distortional buckling moment (Mod) for 2500 mm:

⎥⎦

⎤⎢⎣

⎡+= 2

2

2

2

LEI

GJLEI

M we

yod

ππ (Pi and Trahair, 1999)

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

12

231

2

23

91.02

91.02

dLEtGJ

dLEtGJ

GJ

f

f

e

π

π (Pi and Trahair, 1999)

GJf = 4589.2 x 106 Nmm2 (Appendix A)

Appendix D


⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

210x x π0.912500x x320000010 x 4489.2 x 2

210x x π0.912500x x320000010 x 4489.2 x 2

2

236

2

236

eGJ

GJe = 3360.67 x 106 Nmm2

⎥⎦

⎤⎢⎣

⎡+= 2

926

2

62

250010 x 4.46 x 200000 x 10 x 3360.67

250010 x 0.46 x 200000 x ππ

odM

Mod = 26.34 kNm (i.e. Used in Table 3.7 of Chapter 3)

Mod / My = 0.76 (i.e. Used in Figure 4.6 of Chapter 4)

• Elastic lateral distortional buckling moment with moment gradient β = 0

(Mod-non) for 2500 mm (using the currently available αm factor):

Mod-non = αm Mod

αm = equivalent uniform moment factor

For example; αm = 1.75 + 1.05β + 0.3β 2 ≤ 2.3 (AS4100 Table 5.6.1)

αm = 1.75 + 0 + 0 = 1.75 (i.e. Used in Figure 5.7 of Chapter 5)

Mod-non = 1.75 x 26.34

Mod-non = 46.10 kNm

• To obtain the αm factor from Mod-non of 2500 mm cantilever LSB obtained

using the effective length method of BS5950-1:

For cantilever; Le = 0.8 L (BS5950-1)

Le = 0.8 L = 2000 mm

Mod-non ⎥⎥⎦

⎤

⎢⎢⎣

⎡+= 2

2

2

2

e

we

e

y

LEI

GJL

EI ππ

Mod-non = 32.61 kNm

αm = Mod-non / Mod = 32.61 / 26.34

αm = 1.23 (i.e. Used in Table 5.11 of Chapter 5)

• Elastic lateral distortional buckling moment for transverse loading case with

top flange loading (Mod-non for load height) for 2500 mm:

With the effective length method, Le = L x kl

Appendix D


kl = load height factor

For example;

kl = 1.4 for beam with top flange loading (AS4100 Table 5.6.3(2))

Mod-non for load height ⎥⎥⎦

⎤

⎢⎢⎣

⎡+= 2

2

2

2

e

we

e

ym L

EIGJ

L

EI ππα

αm = 1.06 for uniformly distributed load (obtained from elastic

buckling analysis using FEA)

Mod-non for load height =

⎥⎦

⎤⎢⎣

⎡+ 2

926

2

62

1.4) x (250010 x 4.46 x 200000 x 10 x 3360.67

1.4) x (250010 x 0.46 x 200000 x 06.1 ππ

Mod-non for load height = 21.28 kNm (top flange loading)

With Trahair’s equation (1993);

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧+

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

yyz

Qm

yyz

Qmm

yz

cr

PMy

PMy

MM

/4.0

/4.0

12

ααα (Trahair, 1993)

where; 22 / LEIP yy π=

3262 10 x 0.1452500 / 10 x 0.46 x 200000 == πyP kN

yQ = -115 mm (resultant vertical load is at the middle of the web of

the flange, refer to the FE model for load height)

αm = 1.06 for uniformly distributed load (obtained from elastic

buckling analysis using FEA)

Mod-non for load height =

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+

0.145 / 26.34115- x 1.06 x 0.4

0.145 / 26.34115- x 1.06 x 0.411.06 x 26.34

2

Mod-non for load height = 20.82 kNm (top flange loading)

• Dimensionless buckling load (DBL) for transverse loading case with top

flange loading for 2500 mm:

With the effective length method, Le = L x kl

As above;

Mod-non for load height (TF) = 21.28 kNm (with αm = 1.06)

Appendix D


which is for uniformly distributed load case

DBL for uniformly distributed load = ey3 GJEI/QL

Buckling load (Q) = 8 Mod-non / L2= 8 x 21.28 / 2.52 = 27.24 kN/m

DBL = 663 10 x 3360.67 x 10 x 0.46 x 200000/2500 x 27.24

DBL = 24.18 (i.e. Used in Figure 7.6 of Chapter 7)

With Trahair’s equation (1993);

As above;

Mod-non for load height (TF) = 20.401 kNm (with αm = 1.06)

which is for uniformly distributed load case

DBL for uniformly distributed load = ey3 GJEI/QL

Buckling load (Q) = 8 Mod-non / L2= 8 x 20.82 / 2.52 = 26.65 kN/m

DBL = 663 10 x 3360.67 x 10 x 0.46 x 200000/2500 x 26.65

DBL = 23.67 (i.e. Used in Figure 7.6 of Chapter 7)

• Modified torsion parameter (Ke) for 2500 mm:

⎥⎦

⎤⎢⎣

⎡+= 2

2

2

2

LEI

GJLEI

M we

yod

ππ (Pi and Trahair, 1999)

Alternatively it can be written as;

( )eey

od KL

GJEIM += 12

2π

where;

22 / LGJEIK ewe π=

2692 2500 x 10 x 3360.67/ 10 x 4.46 x 200000x π=eK

Ke = 0.647 (i.e. Used in Figure 5.15 of Chapter 5)

Mod-non = Q L2 / 8

Q

Mod-non = Q L2 / 8

Q

Appendix D


• Member moment capacity (Mb) for 2500 mm with the current LSB design

rules:

φb Mb = φb Zc fc (AS4600 Clause 3.3.3.3(1))

where; fc = Mc / Zf (Clause 3.3.3.3(1))

λd = (My / Mod)1/2 (Clause 3.3.3.3(8))

For; λd ≤ 0.59 Mc = My (Clause 3.3.3.3(5))

0.59 < λd < 1.7 Mc = My (0.59 / λd) (Clause 3.3.3.3(6))

λd ≥ 1.7 Mc = My (1 / λd2) (Clause 3.3.3.3(7))

My = Z fy (AS4600 Clause 2.22)

Zx = 77 x 103 mm3 (Appendix A)

My = 34.65 kNm (fully effective)

λd = (34.65 / 26.34)1/2 = 1.147 (i.e. Used in Figure 4.6 of Chapter 4)

Since 0.59 < λd < 1.7

Mc = 34.65 (0.59 / 1.147) = 17.82 kNm

fc = 17.82 x 106 / 77 x 103 = 231.43 kPa

Mb = 77 x 103 x 231.43

Mb = 17.82 kNm < Ms

Mb / My = 0.52 (i.e. Used in Figure 4.6 of Chapter 4)

• Member moment capacity of non-uniform moment (Mb-non) for 2500 mm

using hot-rolled steel design codes (Chapter 5 - Equation 5.4)

Mb-non = αm Mb (AS4100 Clause 5.6.1.1(a))


αm = 1.75 + 0 + 0 = 1.75

Mb-non = 1.75 x 17.82

Mb-non = 31.19 kNm < Ms

• Member moment capacity of non-uniform moment (Mb-non) for 2500 mm

using cold-formed steel design codes (Chapter 5 - Equation 5.5)

Mo-non = Cb Mo (AS4600 Clause 3.3.3.2.1(a))

which is equivalent to;

Appendix D


Mod-non = αm Mod


αm = 1.75 + 0 + 0 = 1.75

Mod-non = 1.75 x 26.34= 46.05 kNm

λd = (34.65 / 46.05)1/2 = 0.87

Since 0.59 < λd < 1.7

Mc = 34.65 (0.59 / 0.87) = 23.5 kNm

fc = 23.5 x 106 / 77 x 103 = 305.19 kPa

Mb-non = 77 x 103 x 305.19

Mb-non = 23.5 kNm < Ms

Strength ratio = Mb-non / Mb = 1.30 (i.e. Used in Table 5.3 of Chapter 5)

• Example calculations to non-dimesionalised the ultimate moment capacities

obtained from finite element analyses (Mult, αmMult-non and Mult-non), i.e. used

in Figure 5.9 of Chapter 5 (for 2500 mm LSB):

For Mult/My vs. λd;

Mult = 19.9 kNm (Non-linear FEA result)

Mult/My= 19.9 / 34.65 = 0.573

Corresponding Mod = 25.76 kNm (Elastic buckling FEA result)

λd = (34.65 / 25.76)1/2 = 1.16

For αmMult/My vs. λd;

Mult = 19.9 kNm (Non-linear FEA result)

αm = 1.74 (Elastic buckling FEA result)

αmMult/My= 1.74 x 19.9 / 34.65 = 1.0

λd = (My / αm Mod)1/2

λd = (34.65 / 1.74 x 25.76)1/2 = 0.879

For Mult-non/My vs. λd (with moment gradient β=0):

Mult-non = 31.04 kNm (Non-linear FEA result)

Mult-non/My= 31.04 / 34.65 = 0.896

Corresponding Mod-non = 45.22 kNm (Elastic buckling FEA result)

λd = (34.65 / 45.22)1/2 = 0.875

Appendix D


Appendix D (Test Results)

Appendix D


0

2

4

6

8

10

12

14

16

0 5 10 15 20 25 30 35 40 45 50 55 60Overhang Vertical Displacement (mm)

Ove

rhan

g M

omen

t (kN

m)

Case beta=-1 (overhang 1)

Case beta=-1 (overhang 2)

Case beta=0

Overhang Moment vs. Overhang Vertical Displacement of 2.5m LSB125x45x2.0

0

2

4

6

8

10

12

14

16

-60 -40 -20 0 20 40 60 80Lateral Displacement at Mid-span (mm)

Mom

ent (

kNm

)

Case beta=-1 (top flange)Case beta=-1 (bottom flange)

Case beta=0 (top flange)Case beta=0 (bottom flange)

Moment vs. Lateral Displacement of 2.5m LSB125x45x2.0

Appendix D


0

5

10

15

20

25

0 5 10 15 20 25 30 35 40 45Overhang Vertical Displacement (mm)

Ove

rhan

g M

omen

t (kN

m)

Case beta=-1 (overhang 1)Case beta=-1 (overhang2)Case beta=0 (overhang 1a)Case beta=0 (overhang 1b)

Overhang Moment vs. Overhang Vertical Displacement of 3.5m LSB250x60x2.0

0

5

10

15

20

25

30

35

40

-120 -80 -40 0 40 80Lateral Displacement (mm)

Mom

ent (

kNm

)

Case beta=-1 (top flange)Case beta=-1 (bottom flange)Case beta=0 (top flange)Case beta=0 (bottom flange)

Moment vs. Lateral Displacement of 3.5m LSB250x60x2.0

Appendix D


Deformation after Failure of 2.5m LSB125x45x2.0 with Uniform Moment β = -1

Deformation after Failure of 2.5m LSB125x45x2.0 with Moment Gradient β = 0

Appendix D


Deformation after Failure of 3.5m LSB250x60x2.0 with Uniform Moment β = -1

Deformation after Failure of 3.5m LSB250x60x2.0 with Moment Gradient β = 0

flexural behaviour and design of the new litesteel beams · flexural behaviour and design of the...

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