investigation into the structural behaviour of …

INVESTIGATION INTO THE STRUCTURAL BEHAVIOUR OF PORTAL FRAMES

by

Chantal Rudman

Thesis presented in partial fulfilment of the requirements for the degree of Master of Science

in Engineering at the University of Stellenbosch

Professor PE Dunaiski

Professor PJ Pahl

March 2009

Investigation into the structural behaviour of portal frames i

Chantal Rudman University of Stellenbosch

DECLARATION

By submitting this thesis electronically, I declare that the entirety of the work contained

therein is my own, original work, that I am the owner of the copyright thereof (unless to the

extent explicitly otherwise stated) and that I have not previously in its entirety or in part

submitted it for obtaining any qualification.

March 2009

Copyright © 2008 Stellenbosch University

All rights reserved

Investigation into the structural behaviour of portal frames ii


SYNOPSIS

The current trend of the building industry by which stronger but more slender elements are

designed, due to economical considerations, contributes to the serious consideration of the

stability of structures. The Southern African Institute of Steel Construction (SAISC) has

expressed its concerns about the stability of steel structures with specific interest to the elastic

instability of portal frames.

The research will focus on the in-plane purely elastic stability of portal frames. In this

investigation a distinction is made between the prediction of instability by means of evaluating

the nonlinear load-path and instability without prior warning. The analyses done in this

research uses a software programme ANGELINE which addresses both of these aspects. This

software programme is especially developed for the academic research into geometric

nonlinear behaviour of slender structures.

The structural analyses reveal that elastic instability is not a concern for portal frames with

practical dimensions. Further investigation includes determining what the limiting in-plane

behaviour is. This is done by evaluating a benchmark portal frame and it is shown that plastic

deformation in the frame is the limiting criterion. This is done using the commercial software

programme, ABAQUS.

The research is concluded by evaluating a selection of portal frames, with practical dimensions,

in order to substantiate the conclusions above. This is done by designing the selection of

portal frames according to the DRAFT SANS 10160-1 & 2:2008, and SANS 10162-1:2005.

Subsequently, these frames are analysed using ANGELINE (including geometric nonlinearity)

and ABAQUS (second-order elastic perfectly plastic analysis).

Although it is shown that the limiting in-plane behaviour of portal frames is governed by the

plastic deformation of the members it becomes clear that the design of the selection of portal

frames in this research is governed by the serviceability limit state requirements.

Investigation into the structural behaviour of portal frames iii


SAMEVATTING

Die huidige neiging in die konstruksie industrie om sterker strukture met ‘n hoër slankheid te

ontwerp deur gebruik te maak van hoër sterkte materiale het aanleiding gegee tot die ernstige

oorweging van die stabiliteit van hierdie strukture. Die Suider-Afrikaanse Instituut vir Staal

Konstruksie het besorgdheid uitgespreek oor die stabiliteit van staalstrukture met spesifieke

fokus op die elastiese onstabiliteit van portaalrame.

Hierdie navorsing sal fokus op die suiwer elastiese in-vlak stabiliteit van portaalrame. In hierdie

ondersoek word ‘n onderskeiding gemaak tussen die voorspelling van onstabiliteit deur die

nie-lineêre belasting-roete te evalueer asook onstabiliteit sonder enige vooraf waarskuwing.

Die analises wat uitgevoer is in hierdie ondersoek gebruik ‘n sagteware paket ANGELINE wat

beide hierdie aspekte aanspreek. Hierdie sagteware is spesifiek vir akademiese navorsing in

geometriese nie-lineêre gedrag ontwikkel.

Die strukturele analises toon dat elastiese onstabiliteit nie van groot belang is vir portaalrame

met praktiese afmetings nie. Verdere ondersoek sluit die bepaling van die beperkende in-vlak

gedrag in. Dit is uitgevoer deur ‘n voorbeeld portaalraam te evalueer en daar word getoon dat

plastiese vervorming van die raam die beperkende maatstaf is. Die kommersiële sagteware

paket ABAQUS is vir hierdie doel gebruik.

Die ondersoek is afgesluit deur ‘n reeks portaalrame met praktiese afmetings te evalueer ten

einde die bogenoemde gevolgtrekkings te staaf. Dit is gedoen deur die reeks portaalrame te

ontwerp volgens die konsep kode SANS 10160-1 & 2:2008 en die ontwerpkode SANS 10162-

1:2005. Hierna is analises op die rame uitgevoer deur van ANGELINE (wat geometriese nie-

lineêriteit insluit) en ABAQUS (wat ‘n tweede-orde elasties perfek plastiese analise uitvoer).

Alhoewel daar getoon is dat die beperkende in-vlak gedrag van portaalrame deur die plastiese

vervorming van elemente beheer word, is dit duidelik dat die ontwerp van die reeks portaal

rame in hierdie ondersoek beheer word deur vereistes vir die grenstoestand van

diensbaarheid.

Investigation into the structural behaviour of portal frames iv


ACKNOWLEDGEMENTS

The author of this thesis would like to express her gratitude to the following people:

- Professor PE Dunaiski for his patience and guidance and teaching me that an elephant

should be eaten one bite at a time.

- Professor PJ Pahl for his expert knowledge and time.

- My classmates who made the last two years an experience of a life time.

- And last but not least: my mother, father, brother and fiancé. Without them I would never

have seen the light at the end of the tunnel.

Investigation into the structural behaviour of portal frames v


TABLE OF CONTENTS

DECLARATION……………………………………………………………………………………………i

SYNOPSIS…………………………………………………………………………………………….…..ii

SAMEVATTING…………………………………………………………..………………..…………..iii

ACKNOWLEDGEMENTS…….……………………..……………………….…….……….…………iv

TABLE OF CONTENTS…………………………………………….…………….………………….….v

LIST OF APPENDICES.................................................................................. viii

LIST OF FIGURES ......................................................................................... ix

LIST OF TABLES........................................................................................... xi

LIST OF SYMBOLS ..................................................................................... .xii

LIST OF ABBREVIATIONS .............................................................................xiv

1 INTRODUCTION ..........................................................................1.1

1.1 THE PROBLEM........................................................................................................................... 1.1

1.2 OBJECTIVES ............................................................................................................................... 1.1

1.3 FLOW CHART FOR PART 1 ....................................................................................................... 1.2

2 STATE OF THE ART IN ELASTIC INSTABILITY .................................2.1

2.1 THE REAL BEHAVIOUR OF STRUCTURES ............................................................................... 2.1

2.2 ELASTIC INSTABILITY IN PITCHED ROOF STEEL FRAMES..................................................... 2.6

2.3 DETERMINING THE POINT OF ELASTIC INSTABILITY – INCLUDING GEOMETRIC

NONLINEARITY.......................................................................................................................... 2.7

2.4 ANGELINE .................................................................................................................................. 2.8

2.5 SUMMARY ............................................................................................................................... 2.10

3 INVESTIGATIVE ANALYSIS – ELASTIC INSTABILITY........................3.1

3.1 ANGELINE .................................................................................................................................. 3.1

3.2 COLUMN INVESTIGATION....................................................................................................... 3.9

3.3 INVESTIGATIVE ANALYSES: PORTAL FRAMES .................................................................... 3.14

3.4 CONCLUSIONS ........................................................................................................................ 3.18

3.5 SUMMARY ............................................................................................................................... 3.20

4 IN-PLANE STRUCTURAL BEHAVIOUR OF PORTAL FRAMES...........4.1

4.1 INTRODUCTION ........................................................................................................................ 4.1

4.2 OBJECTIVES ............................................................................................................................... 4.2

Investigation into the structural behaviour of portal frames vi


4.3 METHOD OF APPROACH ......................................................................................................... 4.2

5 MODELLING CONSIDERATIONS FOR PORTAL FRAMES.................5.1

5.1 IDENTIFICATION OF A TYPICAL PORTAL FRAME AND LOAD PATTERN............................. 5.1

5.2 TYPES OF ELEMENTS TO BE USED IN MODELLING .............................................................. 5.3

5.3 IMPERFECTIONS ....................................................................................................................... 5.5

5.4 MODELLING OF HAUNCHES.................................................................................................... 5.8

5.5 PLASTIC DEFORMATION OF STRUCTURAL MEMBERS ........................................................ 5.9

5.5 COMPATIBILITY OF SOFTWARE PACKAGES ........................................................................ 5.17

5.6 SUMMARY ............................................................................................................................... 5.18

6 DESIGN OF PORTAL FRAMES ACCORDING TO DRAFT SANS 10160-1,

& 2 : 2008 AND SANS 10162-1:2005. ..........................................6.1

6.1 INTRODUCTION ........................................................................................................................ 6.1

6.2 LIMIT STATE DESIGN ................................................................................................................ 6.1

6.3 DESIGN OF A PORTAL FRAME ACCORDING TO DRAFT SANS 10160-1 & 2 : 2008 AND

SANS 10162-1:2005 ................................................................................................................. 6.2

6.4 LOAD COMBINATIONS............................................................................................................. 6.3

6.5 CAPACITY OF MEMBERS – ULTIMATE LIMIT STATE ............................................................ 6.5

6.6 SERVICEABILITY LIMIT STATE................................................................................................ 6.11

6.7 DESIGNING THE BENCHMARK EXAMPLE ............................................................................ 6.11

6.8 SUMMARY ............................................................................................................................... 6.14

7 ANALYSIS OF BENCHMARK PORTAL FRAME ................................7.1

7.1 ANALYSIS OF BENCHMARK PORTAL FRAME ........................................................................ 7.1

7.2 CONCLUSIONS ........................................................................................................................ 7.10

7.3 SUMMARY ............................................................................................................................... 7.10

8 DESIGN OF PORTAL FRAMES FOR PARAMETER STUDY ................8.1

8.1 DEFINITION OF PORTAL FRAMES........................................................................................... 8.1

8.2 DESIGN OF PORTAL FRAMES FOR THE PARAMETER STUDY .............................................. 8.4

8.3 CONCLUSIONS ........................................................................................................................ 8.10

8.4 SUMMARY ............................................................................................................................... 8.10

9 ANALYSES RESULTS AND DISCUSSION - PARAMETER STUDY .......9.1

9.1 RESULTS ..................................................................................................................................... 9.2

9.2 DISCUSSION ON RESULTS...................................................................................................... 9.16

9.3 CONCLUSIONS ........................................................................................................................ 9.21

9.4 SUMMARY ............................................................................................................................... 9.23

10 CONCLUSIONS AND RECOMMENDATIONS ................................10.1

10.1 INTRODUCTION ...................................................................................................................... 10.1

10.2 CONCLUSIONS AND RECOMMENDATIONS ........................................................................ 10.1

Investigation into the structural behaviour of portal frames vii


11 REFERENCES..............................................................................11.1

11.1 BOOKS...................................................................................................................................... 11.1

11.2 PUBLICATIONS ........................................................................................................................ 11.2

11.3 DESIGN CODES........................................................................................................................ 11.2

11.4 INTERVIEWS ............................................................................................................................ 11.3

11.5 ELECTRONIC REFERENCES..................................................................................................... 11.3

Investigation into the structural behaviour of portal frames viii


LIST OF APPENDICES

APPENDIX A: ELASTIC STABILITY OF COLUMNS

APPENDIX B: ELASTIC STABILITY OF PORTAL FRAMES

APPENDIX C: NUMBER OF ELEMENTS

APPENDIX D: NOTIONAL HORIZONTAL LOAD

APPENDIX E: PORTAL FRAME DESIGN

APPENDIX F: DESIGN RESULTS

APPENDIX G: LOAD-DISPLACEMENT HISTORY – ABAQUS

APPENDIX H: LOAD-DISPLACEMENT HISTORY - ANGELINE

Investigation into the structural behaviour of portal frames ix


LIST OF FIGURES Figure 1.1 Flow chart for Part 1 ................................................................................................. 1.2

Figure 2.1 Nonlinear behaviour of structures............................................................................ 2.2

Figure 2.2 Frame second-order effects: (a) P- effects and (b) P- effects .............................. 2.3

Figure 2.3 Load deflection paths of a structure ........................................................................ 2.5

Figure 2.4 Snap-through behaviour .......................................................................................... 2.5

Figure 2.5 Elastic instability of portal frames............................................................................ 2.6

Figure 3.1 Various examples in ANGELINE................................................................................ 3.2

Figure 3.2 Graphical Model - Columns...................................................................................... 3.2

Figure 3.3 Various Editors in ANGELINE.................................................................................... 3.4

Figure 3.4 Session.java ............................................................................................................. 3.5

Figure 3.5 Portal frame default model..................................................................................... 3.6

Figure 3.6 Graphical model of portal frame............................................................................. 3.7

Figure 3.7 Generator.java......................................................................................................... 3.8

Figure 3.8 Profile.java............................................................................................................... 3.8

Figure 3.9 K-values for different end restraints ..................................................................... 3.10

Figure 3.10 Flow diagram illustrating the analysis procedure and selection of columns...... 3.10

Figure 3.11. Selection of portal frames................................................................................... 3.14

Figure 3.12 Vertical deflection 2u of the ridge as a function of the load factor .................... 3.15

Figure 3.13 Variation of the minimum diagonal coefficient (Configuration C1)..................... 3.17

Figure 4.1 Flow chart for investigation into the structural behaviour of portal frames........... 4.3

Figure 5.1 Benchmark portal frame .......................................................................................... 5.2

Figure 5.2 Load pattern across roof .......................................................................................... 5.3

Figure 5.3 Load-deflection at mid node.................................................................................... 5.8

Figure 5.4 Haunches in ANGELINE ............................................................................................ 5.9

Figure 5.5 Equivalent I-sections ................................................................................................ 5.9

Figure 5.6 Stress distribution in cross-section ........................................................................ 5.10

Figure 5.7 Idealised stress-strain curve................................................................................... 5.11

Figure 5.8 Various stages in the forming of plastic hinges in beam........................................ 5.12

Figure 5.9 Collapse modes in portal frames............................................................................ 5.13

Figure 5.10 Verification of ABAQUS ....................................................................................... 5.14

Figure 5.11(a) Load-deflection path at mid node ................................................................... 5.14

Figure 5.11(b) Stresses in beams ............................................................................................ 5.14

Figure 5.12(a) Load-deflection path at the top node and........................................................ 5.16

Figure 5.12(b) Stresses in cantilever column ........................................................................... 5.16

Figure 6.1 Numbering of nodes in PROKON – Benchmark example....................................... 6.11

Figure 6.2 Axial Force, Shear Force and Bending Moment Diagram ...................................... 6.13

Figure 7.1 Configuration of portal frame analysed in ANGELINE and ABAQUS........................ 7.1

Figure 7.2(a) .Location of highest stresses at yielding of cross-section in rafter....................... 7.2

Figure 7.2(b) .Location of highest stresses at first yielding of cross-section ............................. 7.2

Figure 7.3 Location on cross-section where ABAQUS calculates stresses ................................ 7.3

Figure 7.4 Load deflection paths of the allocated elements..................................................... 7.3

Figure 7.5(a) Location of members ........................................................................................... 7.4

Figure 7.5(b) Load-stress history of critical members............................................................... 7.4

Figure 7.6 Displacement of frame at load factor 1.736 ............................................................ 7.6

Figure 7.7 Deflection-load path of frame at top of left hand column ...................................... 7.7

Figure 7.8 Load-deflection path of portal frame at ridge ......................................................... 7.7

Figure 7.9 Axial force diagram at a load factor of 1.0............................................................... 7.8

Figure 7.10 Shear force diagram at a load factor of 1.0 ........................................................... 7.8

Figure 7.11 Bending moment diagram at a load factor of 1.0 .................................................. 7.8

Investigation into the structural behaviour of portal frames x


Figure 7.12 Load-Axial force history.......................................................................................... 7.9

Figure 7.13 Load-Bending moment history............................................................................... 7.9

Figure 8.1 Sequence of analyses for each frame ...................................................................... 8.1

Figure 8.2 Portal frames with pinned supports with varying column length and roof slope ... 8.2

Figure 8.3 Portal frames with fixed supports with varying column length and roof slope....... 8.3

Figure 8.4 Portal frames with varying spans, column length and roof slope........................... 8.3

Figure 8.5 Distribution of forces - illustrating maximum forces ............................................... 8.4

Figure 8.6 Design values used ................................................................................................... 8.6

Figure 8.7 Maximum vertical and horizontal deflection........................................................... 8.9

Figure 9.1 Flow chart of procedure........................................................................................... 9.1

Figure 9.2 Material model......................................................................................................... 9.3

Figure 9.3 Comparison of percentage difference -right hand column and max load factor .. 9.17

Figure 9.4 Behaviour compared to ABAQUS results ............................................................... 9.18

Figure 9.5 Comparison of load factor...................................................................................... 9.21

Figure 10.1 Portal frame with tapered members .................................................................... 10.2

Investigation into the structural behaviour of portal frames xi


LIST OF TABLES

Table 3.1 Values obtained for column analyses...................................................................... 3.11

Table 3.2. Example of effect of axial shortening..................................................................... 3.13

Table 5.1 Forces at allocated elements – various software programmes ............................... 5.18

Table 5.2 Percentage differences in forces............................................................................. 5.18

Table 6.1 Classification of sections in axial compression.......................................................... 6.5

Table 6.2 Classification of flanges – flexural ............................................................................. 6.7

Table 6.3 Classification of webs– flexural ................................................................................. 6.7

Table 6.4 Example for calculation of dead weight of the structure........................................ 6.12

Table 6.5 Example for calculation of imposed loads of the structure .................................... 6.12

Table 6.6 Column resistances – I-section 254 x 146 x 37........................................................ 6.13

Table 6.7 Rafter resistances – I-section 254 x 146 x 37 .......................................................... 6.14

Table 8.1 Designated sections – span 24.0m, pinned supports............................................... 8.6

Table 8.2 Designated sections – span 24.0m, fixed supports ................................................... 8.7

Table 8.3 Designated sections – varying span lengths.............................................................. 8.8

Table 9.1(a) Yielding values for frames – span 24.0m - pinned supports – 6.0m ................... 9.4

Table 9.1(b) Yielding values for frames – span 24.0m - pinned supports – 10.0m................. 9.4

Table 9.1(c) Yielding values for frames – span 24.0m - pinned supports – 14.0m ................. 9.4

Table 9.2 Yielding values for frames – span 24.0m – fixed supports........................................ 9.5

Table 9.3 Yielding values for frames – varying length spans .................................................... 9.6

Table 9.4(a) Deflection at selected nodes – pinned supports ................................................ 9.10

Table 9.4(b) Deflection at selected nodes – pinned supports-ridge....................................... 9.10

Table 9.5 Deflection at selected nodes –fixed supports ......................................................... 9.12

Table 9.6(a) Deflection at selected nodes – varying spans ..................................................... 9.13

Table 9.6(b) Deflection at selected nodes – varying spans - ridge.......................................... 9.13

Table 9.7(a) Load factor at serviceability of portal frames – pinned supports – span 24.0m . 9.14

Table 9.7(b) Load factor at serviceability of portal frames – fixed supports – span 24.0m .... 9.14

Table 9.7(c) Load factor at serviceability of portal frames – varying spans ............................ 9.14

Investigation into the structural behaviour of portal frames xii


LIST OF SYMBOLS

A cross-sectional area

Ad design value of accidental action

Av shear area

Cr critical axial compressive force

Cu Ultimate compressive force in member

Cy axial compressive force in member at yield stress

E elastic modulus of steel

G shear modulus of steel

Gk,j characteristic value of permanent action j, self weight

I moment of inertia

K effective length factor

L gross length of member

Mr Factored moment resistance of member

Mu Ultimate bending moment in member

P relevant representative value of prestressing action

Qk,1 characteristic value of leading variable action, imposed load

Qk,i characteristic value of accompanying variable action i

Tr Factored tensile resistance of member

Tu Ultimate tensile force in member

U1 factor to account for moment gradient and for second-order effects of axial force acting

on the deformed member

Vr Factored shear resistance of member

W width to thickness ratio

Wlim Limit of width to thickness ratio

Ze elastic section modulus of steel section

Zpl plastic section modulus of steel section

B half of width of flange of column

F calculated compressive stress in element

fe elastic critical buckling stress in axial compression

fs Ultimate shear stress

fy Yield stress

H height of section

hw Clear depth of web between flanges

Investigation into the structural behaviour of portal frames xiii


kv shear buckling coefficient

N material regression factor

R radius of gyration

S centre-to-centre distance between transverse web stiffeners

tf thickness of flange

tw thickness of web

Σ combined effect

Φ resistance factor for structural steel

γG,j partial factor for permanent action j

γQ,1 partial factor for leading variable action

γQ,i partial factor for accompanying variable action i

Λ non-dimensional slenderness ratio

ψi action combination factor corresponding to accompanying variable action i

Investigation into the structural behaviour of portal frames xiv


LIST OF ABBREVIATIONS

ANGELINE Analysis of geometrically nonlinear structures

SAISC Southern African Institute of Steel Construction

SANS South African National Standards

TUB Technical University Berlin

LL Live Load

DL Dead Load

Introduction 1.1


1 INTRODUCTION

Due to the ever increasing complexity of structures being designed, it has become an absolute

necessity that the behaviour of structures related to the overall and member stability is

understood. A recent article published in the Journal of Engineering Mechanics [14] states the

following:

“As far as structural engineering is concerned, scientific and technological advances are often

fostered by the occurrence of collapses involving a more or less relevant amount of damage

and in the most unfortunate cases, also the loss of human lives”. This statement was made due

to tragic collapse of the World Trade Centre twin towers, on the September 11, 2001, which

highlights the importance of the understanding of behaviour of real structures.

1.1 THE PROBLEM

The current trend of the building industry by which stronger but more slender elements are

designed, due to economical considerations, contributes to the serious consideration of the

stability of structures. Portal frames are widely used in the industrial sector in South Africa and

the possible elastic instability of these frames has raised concerns at the Southern African

Institute of Steel Construction (SAISC).

1.2 OBJECTIVES

This research is subdivided into two parts. The first part and main focus of the research will

include the investigation into the in-plane stability of pitched roof steel frames. This means

that only strong-axis bending is considered and it is assumed that the portal frame is

sufficiently laterally restrained.

The question that must be answered is the following:

Is purely geometric elastic instability a problem in portal frames?

In the first part it becomes clear that elastic instability is not a problem in portal frames. The

second part shifts the focus of the research towards the inclusion of material nonlinearity.

Introduction 1.2


The objective in this part of the research is to determine:

The limiting in-plane behaviour of portal frames by including plastic deformation.

A detailed approach and flow chart for the second part of the research project is included in

Chapter 4. The flow chart for the first part of the thesis is shown below.

1.3 FLOW CHART FOR PART 1

In Chapter 2 the elastic behaviour and stability of structures are discussed with reference to

portal frames. The software programme that is used for this investigation is explained.

This is followed by an investigative analysis in Chapter 3, which entails the behaviour of the

frames by determining the elastic instability of selected portal frames. This is done by means

of verifying the behaviour in columns and the influence of the perturbation load.

Subsequently, selected portal frames are investigated and their elastic stability evaluated.

A flow chart for Part 1 of this research is shown.

Figure 1.1 Flow chart for Part 1

State of the art in elastic instability 2.1


2 STATE OF THE ART IN ELASTIC INSTABILITY

The first part of this research includes the investigation into the stability of portal frames if

purely geometric nonlinearity is included. The discussion in this chapter will serve as an

introduction to the concept of geometric nonlinear behaviour and the difficulties arising in

determining the instability of portal frames.

Discussions in this chapter are subdivided into the following sections:

• The real behaviour of structures and the concept of nonlinearity

• The failure modes as a result of purely geometric instability

• The difficulty of determining instability in structures

• ANGELINE (Analysis of Geometrical Nonlinear Structures) is introduced and

explained

2.1 THE REAL BEHAVIOUR OF STRUCTURES

2.1.1 Nonlinear behaviour of structures

A structure that is subjected to a vertical loading and a proportional horizontal load will deflect

as a result of the load application. Engineering practice simplifies true structural behaviour by

not including the influence of the deflection of the structure as a result of the applied load on

the geometry in the equilibrium state.

This is known as first order linear theory and in some cases the influence of this deflection on

the structure is neglible [3]. However, the fundamental behaviour of a true structure includes

nonlinearities that are not included in simplified theory.

The effect of the nonlinearities can be extremely important as this change in geometry can

have weakening effects on the structure. For example, the deflection may add a significant



additional moment to the members due to the eccentricity of the normal force and thus

collapse may occur at loads below predicted failure loads [3]. All structures will exhibit

nonlinear behaviour and deviate from the straight path implied by the linear theory as shown

in Figure 2.1.

Figure 2.1 Nonlinear behaviour of structures

There are fundamental differences between linear and nonlinear theories, which necessitate

such theories and are explained as follows [21]:

(a) The relationship between the strains and the displacements of a member is

highly nonlinear and implies that even if the strains are small the translations

and rotations of the members can be large due to rigid body displacements.

This is not included in linear theory.

(b) The linear problem can be solved directly by solving a set of linear equations

based on the reference state which contains an equal number of unknowns

and equations. The nature of the solution which is obtained with linear frame

theory does not depend on the load level. The nature of the solution that is

obtained with the nonlinear theory depends strongly on the load level.

(c) Due to the nonlinearity of the governing equations, the principle of

superposition is not valid for nonlinear analysis.

Displacement

u



2.1.2 Types of nonlinearity

Two types of nonlinearities are distinguished [4]:

• Geometric nonlinearity and

• Material nonlinearity

(a) Geometric nonlinearity

Geometric nonlinearity can be as a result of many effects. These effects include the influence

of the axial force on the bending moment, the effect of relative horizontal joint displacements,

changes in member chord lengths and initial crookedness of members. Geometric nonlinearity

is also referred to as second order effects or P-delta effects. In the literature distinction is

made between two types of delta effects.

• P-∆ effects

This is the sway displacements taking place between column ends as a result

of the vertical forces applied to the structure. The additional bending moment

is obtained from the equilibrium equations taken from the frame in the

partially deformed structure.

This is shown in Figure 2.2 (a). It should be noted that the P-∆ effects only

occur in unbraced frames and not in braced frames.

• P-δ effects

The concept of P-δ effects is shown in Figure 2.2 (b).

P P

A

C’CB’B

D

H

(a) (b)

A

CB

D

Figure 2.2 Frame second-order effects: (a) P-∆ effects and (b) P-δ effects



P-δ effects are a result of the compressive axial forces acting on the various frame members

and concern the individual deformation of these members i.e the displacements that take

place between the member deformed configurations and chord positions [15].

(b) Material nonlinearity

The stress-strain relationship in a member is nonlinear due to a variety of reasons i.e residual

stresses present in members prior to loading, spread of inelastic zone in members as member

forces increase, variations in member strength due to variations in the theoretical cross-

sectional dimensions, shearing deformations, local buckling, out of plane movement of frames,

connection flexibility and strain hardening [4].

2.1.3 Types of elastic instability

(a) General concept of elastic stability

Galambos [4] states that instability is a condition wherein a compression member loses the

ability to resist increasing loads and exhibits instead a decrease in load-carrying capacity. In

other words instability occurs at the maximum point of the load –deflection curve.

However, this does not give full understanding of the concept, which can be better explained

by looking at a structure in a certain equilibrium configuration. If it is possible for that

structure to displace to another configuration without the change in loading the configuration

is said to be unstable. The following is stated by Pahl [21]:

“In some configurations of a structure, its shape can change significantly while there is little

change in the loading and the strains remain small. This type of behaviour is considered to be a

failure of the structure, even though the material does not rupture.”

Figure 2.3 shows a typical load deflection path of a structure. In the case of geometrical failure

the possibility of the structural deflection following either of the paths is possible. This

indicates two type of elastic instabilities: namely snap-through and bifurcation. If there is a

single continuation of the load path after the stiffness matrix becomes singular the instability is

called a snap-through (turning point). If there is more than one possible continuation of the

load path at a singular point, the instability is called a bifurcation. The differences in these two

instability phenomena are explained in the following sections.



Figure 2.3 Load deflection paths of a structure [3]

(b) Limit stability load

Limit state or snap-through buckling is usually a primary cause of failure when looking at

shallow arches, shallow trusses and shallow spherical domes. The load deformation path

increases until a maximum load is reached and beyond this the system becomes unstable. This

is shown in Figure 2.4.

Displacement

Load Factor

Stable path

Limit state

Unloading S

equence

Unstab

le pa

th

Figure 2.4 Snap-through behaviour

If a load is applied, the load deformation path is positive up to a point where stability is lost,

and a non-equilibrium state occurs where there is a dynamic jump-through to another



equilibrium state, where the load path once again becomes stable and follows a positive load

deflection path [22].

(c) Bifurcation buckling of the system

If the system is at a point of bifurcation and there exists another equilibrium position in a

slightly deflected configuration; and if, at this load, the system is deflected by some small

disturbance, it will not return to the straight configuration and start to buckle. If the load

exceeds the critical value, the straight position is unstable and a slight disturbance leads to

large displacements of the system and, finally, to the collapse or buckling. The critical point,

after which the deflections of the system become very large, is called the "bifurcation point" of

the system [22].

If small imperfections exist in the system, deflection starts from the beginning of the loading.

2.2 ELASTIC INSTABILITY IN PITCHED ROOF STEEL FRAMES

Silvestre et al [15] state that portal frames are governed by two modes of failure as shown in

Figure 2.5. This is the symmetric and anti-symmetric configuration of which both involve the

horizontal displacement in the columns.

This implies that elastic in-plane failure modes of pitched roof steel portal frames are

considered to be either through side sway of the frame due to the buckling of the columns or

the snap through of the roof.

Figure 2.5 Elastic instability of portal frames [15]



2.3 DETERMINING THE POINT OF ELASTIC INSTABILITY – INCLUDING

GEOMETRIC NONLINEARITY

In this section the difficulty of analysing structures which include nonlinearity is discussed.

These problems are subdivided into two parts and are discussed in Section 2.3.1 and 2.3.2. A

solution is proposed which is described in the first part of this thesis.

2.3.1 The difficulty in analysing structures which include nonlinear behaviour

The equilibrium equations of linear frame theory are formulated in the reference configuration

of the frame. The linear equations are solved by setting up governing equations, which have

the same number of equations as unknowns.

However, nonlinear theory necessitates the formulation of equilibrium equations in the instant

configuration of the frame.

The nonlinear problem cannot be solved directly as in the case of linear theory and must be

solved by iteration because the governing equations are nonlinear expressions in the

displacements. The most common approach is to treat the nonlinear behaviour as an initial

value problem [5].

To determine the nonlinear behaviour of structures and the point of bifurcation or snap-

through, special numerical methods and data structures are required [21].

2.3.2 Determining instability using commercial software programmes

Various software packages are available that employ different methods of nonlinear analysis.

The problem with these software programmes is that they are usually general software

programmes of which analysis of nonlinear behaviour is only one component.

The theory behind the nonlinear analysis is normally not sufficiently explained in the

accompanying documentation. Therefore, a full academic research cannot be achieved using

these packages because the results cannot be fully explained.

It is also not explicitly stated in most software package manuals that nonlinear structural

behaviour comprises of various stages that should be investigated:

(a) This first stage includes investigating the behaviour of the linear structure

gradually having the nonlinear behaviour affecting the load displacement



curve as the load increases. The instability of the structure is indicated by large

displacements.

(b) However, other stages of analysis exist that are not recognised by designers.

The second stage includes the necessity of understanding the difference

between the deformation (as explained in the first stage) of the structure and

that of the stability of the structure.

An example of this is the Euler column. It could be possible that buckling is preceded by small

deformations and no initial “warning” is given to the forming of elastic instability by means of

large displacements.

This means that designers cannot rely on displacements to predict collapse. The second stage

of nonlinear behaviour should include these instability phenomena. This, however, is not

automatically included in commercial software packages.

Other stages that should be included in the full understanding of nonlinear behaviour also

include the post-buckling behaviour of the structure. This is not included in the explanation as

this research study defines the point of instability at the point where a bifurcation point or a

snap-through point exists.

2.4 ANGELINE

ANGELINE is a software structural analysis programme developed through academic

collaboration between Professor P J Pahl from the Technische Universität Berlin in Germany,

Professor Vera Galishnikova from the University of Architecture and Civil Engineering in Russia

and Professor P E Dunaiski from the University of Stellenbosch.

ANGELINE includes both stages of the nonlinear behaviour in its theoretical implementation.

This software package can also be used as an academic tool as the necessary theory through

all stages of the nonlinear theory is available. ANGELINE is used for investigation into the in-

plane behaviour of columns and portal frames under various loading and support conditions.

ANGELINE’s theory is based on the fundamentals of nonlinear structural behaviour which is

developed through the Theory of Elasticity. Since the number of unknowns in the equations of

kinematics and statics exceed the number of the equations, constitutive equations are

established for different models of material behaviour. These relate the stresses to the strains



in the body. The total number of equations now equals the number of unknowns, so that the

governing equations can be solved with suitable boundary conditions for the unknown stresses

and displacements [5].

It is not possible to solve these equations analytically and numerical methods are needed,

which is implemented by finite elements into suitable software. The governing equations are

partially integrated by using the weighted residual method so that it can be used for numerical

treatment [5].

2.4.1 Algorithm implemented in ANGELINE

The equations that describe the configuration of a structure are nonlinear. Various

mathematical solution methods have been investigated to solve these nonlinear equations as

discussed in the previous section.

The algorithm used in ANGELINE is called the Constant Arc Increment method and is used for

the solution of the governing equations for the geometrically nonlinear behaviour of trusses

and frames [5].

The Constant Arc Increment method is a modification of previous mathematical methods of

solution. This includes the Direct Iteration method, Newton Raphson Iteration Method and the

Modified Newton Raphson Iteration Method.

These earlier mathematical methods are not sufficient as they do not treat the nonlinear

analysis as an initial value problem. It is possible for a load to result in very small

displacements if the structure is still in the reference state but can be quite large if the load is

applied in the deformed state of the structure. It is then beneficial to rather control the arc

increment of the load-displacement path, than the load factor.

The Basic Arc Increment method allows for this. However, some errors still occur due to the

linearization of the governing equations. This method has been modified so that the arc length

increment after each iteration is the same in all load steps of the procedure to form the

constant arc increment method [5].

2.4.2 Instability of the structure

The buckling of a structure is identified by the singularity of its tangent stiffness matrix.For

each step of the Constant Arc Increment method the first iteration includes the calculation of

the decomposed stiffness matrix, so that a trial equilibrium is found. This is done by using the



tangent stiffness matrix. However, the load path is curved and the secant stiffness matrix is

used to determine the displacement load-path more accurately. A set of iterations of the

secant matrix is done until the iteration converges. If the decomposed secant matrix which

includes the frame in equilibrium shows a singular state, instability of the frame occurs.

2.5 SUMMARY

(a) Nonlinear theory is explained.

(b) The problems associated with solving the behaviour of frames if geometric

nonlinear theory is included are discussed.

(c) ANGELINE, a software programme which include the implementation of the

theory of geometric nonlinearity is discussed.

Investigative analysis – elastic instability 3.1


3 INVESTIGATIVE ANALYSIS – ELASTIC INSTABILITY

This chapter includes an investigative analysis into the purely geometric instability of portal

frames. The investigation is divided into three sections:

• The use and implementation of ANGELINE is explained.

• An investigation into the elastic instability of a selection of columns which

will serve as verification and a preliminary study of the influence of the

perturbation load.

• An investigation into the elastic instability of portal frames.

3.1 ANGELINE

3.1.1 Using ANGELINE

The use of ANGELINE is explained to demonstrate to the reader the transparency of the

programme. ANGELINE consists of several parts in which specialised 2D models are created

for analysis. The two parts of interest for this investigation includes 2D columns and 2D portal

frames. An explanation on the use of and modelling in the software follows.

3.1.2 Part 1: Column Analysis

(a) Graphical User Interface

With the initialisation of this part of the software a grid with eight tabs at the top of the screen

appears. The Model Editor enables the user to choose various configurations of columns.

Many examples are given, ranging from columns with simple, clamped or cantilever support

conditions. The number of elements per member can be varied as well as the inclusion of a

perturbation load as shown in Figure 3.1. A simply supported column with a length of 6.0m

and 12 elements is shown in Figure 3.2. The graphical model shows the placement of the

nodes, applied load and placement of supports.



Figure 3.1 Various examples in ANGELINE

Figure 3.2 Graphical Model - Columns



Parameters can be changed by making use of the tabs at the top of the screen, showing the

various Editors. Nodes are marked alphabetically and the Node Editor is used to change the

dimensions of the column, the Element Editor is used to change section properties, and the

Load Editor is used to define new forces or change the magnitude of the defined forces. The

Format Editor is used for changes to the screen visualisation of the graphical model and the

Support Editor can be used to change the fixity of the supports between fixed or pinned.

Displacements y1 and y2 relate to the translational degrees of freedom of the support in

question. A blank space indicates that the parameter is not active. An active “fixity” is

indicated by 0. The various editors are shown in Figure 3.3.

(b) Analysis and Output

The nonlinear analysis is performed incrementally. The configuration of the column at the

beginning and at the end of a step is called a state of the column. The number of steps in the

incremental analysis is set by the user before the analysis is started in the Analysis Editor. If a

singular point is not reached within the number of steps specified, the termination of the

analysis is determined by the number of steps. The initial load factor is set in the Analysis

Editor and this value should be chosen with careful consideration. The choice of the initial

load factor is described in more detail in Section 3.3.3.

Output is obtained in the Result Editor shown in Figure 3.3. Values at the nodes can be

obtained for displacements, rotations and reaction forces. These are given in the form of a

load force history graph. Member results include displacements, axial and shear forces and

bending moments for the member chosen in the component name space. The Frame option is

used to obtain values for the overall distribution of forces and displacements of the whole

model. Visually, the user can obtain the displacement of the frame by changing the state of the

model under a particular loading condition.



Figure 3.3 Various Editors in ANGELINE

In-plane structural behaviour of portal frames 3.5


(c) Modeleditor.java and Session.java

To change parameters in the software, direct access can be obtained through the java files.

Session.java and Modeleditor.java contain information used in the default examples.

Modeleditor.java contains the names of examples and the visualisation parameters of the

Model Editor. This does not change any of the physical properties of the model. Information

needed to change these parameters is collected in Session.java. The collection of parameters

for a 2-element column is shown in Figure 3.4.

Figure 3.4 Session.java



3.1.3 Part 2: Portal frame analyses

(a) Graphical User Interface

With the initialisation of the Model Editor a default frame appears where parameters of the

portal frame can be changed. In this case configuration C1 described in Section 3.3. is used to

illustrate the parameters shown in Figure 3.5.

Figure 3.5 Portal frame default model

Parameters for support fixity and inclusion of haunches at the eaves and the ridge are also

included.

When all the parameters have been chosen the model can be initialised as shown in Figure 3.6.



Figure 3.6 Graphical model of portal frame

(b) Analysis and results

Analyses are performed and results are obtained through the Result Editor. Results are

obtained graphically, by incrementing the state of the frame or by means of history graphs.

(c) Generator.java and Profile.java

The number of elements per column and per haunch can be changed in Generator.java, see

Figure 3.7.

By changing these parameters, the section, support, load and haunch properties can be

changed so that values are given as a default in the initial model and minimal changes have to

be made in the graphical interface.



Figure 3.7 Generator.java

Sections not included in the current selection can be added in Profile.java, see Figure 3.8. The

section properties are added by defining the mass per metre of the section, the height and

width of the section, the thickness of web and flange, the cross sectional area and moment of

inertia.

Figure 3.8 Profile.java



3.2 COLUMN INVESTIGATION

The following is investigated for selected columns:

(a) Verification of ANGELINE

The verification of solutions obtained from ANGELINE is done in Section 3.2.2 by

means of evaluation of examples for which theoretical solutions are available.

The Euler buckling loads for specific columns are compared with results obtained

in ANGELINE to illustrate the accuracy of the theory and the implemented

algorithm.

(b) The influence of a perturbation load on stability of the columns

The influence of the perturbation load on the stability of columns is investigated

in this section. The significance of the perturbation load is explained in Section

5.3.

3.2.1 Euler Buckling

The theory developed by Euler in 1759 is the cornerstone of column theory. The Euler buckling

load is the critical load for an ideal elastic column [2].

The formula for the Euler buckling load is given as:

2

2

EKL

EIP

)(

π=

where ,

EI is the elastic stiffness

KL is the effective length of the column, also defined as the portion of the buckled

column between points of zero curvature.

From the definition of KL it is apparent that end restraints will have a considerable

influence on the buckling load of the column. Figure 3.9 indicates these K-values for



various end restraints. The load applied is P=10.0kN. Results are given as a factor of

this value.

Figure 3.9 K-values for different end restraints [3]

3.2.2 Verification of ANGELINE

(a) Definition of columns

Figure 3.10 Flow diagram illustrating the analysis procedure and selection of columns



The procedure of analysis includes the selection of sections that are used in practice. Columns

with varying lengths and support conditions are analysed. Column lengths between 6.0m and

10.0m are commonly used in industry. Figure 3.10 illustrates these alternatives. An initial load

factor of 0.1 is used.

(b) Results for verification

The values obtained for the various analyses in ANGELINE and the calculated Euler values are

shown in Table 3.1.

Table 3.1 Values obtained for column analyses

Section

Designation

Support

Fixity

Column

Length

(m)

ANGELINE

Result

Euler

Value

%

Difference

Pinned 6 0.1290 0.1289 0.1424

Pinned 8 0.0726 0.0725 0.1419

Pinned 10 0.0465 0.0464 0.1356

Fixed 6 0.5184 0.5154 0.5697

Fixed 8 0.2916 0.2899 0.5690

203x133x25

Fixed 10 0.1866 0.1856 0.5686

Pinned 6 1.8340 1.8314 0.1426

Pinned 8 1.0316 1.0301 0.1427

Pinned 10 0.6602 0.6593 0.1427

Fixed 6 7.3673 7.3254 0.5687

Fixed 8 4.1441 4.1206 0.5692

457x191x75

Fixed 10 2.6523 2.6372 0.5691

Pinned 6 4.1841 4.1781 0.1427

Pinned 8 2.3536 2.3502 0.1426

Pinned 10 1.5063 1.5041 0.1425

Fixed 6 16.8082 16.7125 0.5692

Fixed 8 9.5461 9.5008 0.4749

533x210x122

Fixed 10 6.0510 6.0165 0.5692

Results of ANGELINE analyses are compared to the Euler value of the frame under

consideration. The percentage difference between the two values is calculated by the

following formula:

100xValueANGELINE

ValueEulerValueANGELINEDifference

−=(%)



3.2.3 Investigation into the effect of the perturbation load on column stability

(a) Definition of perturbation load

This part of the investigation includes the application of a perturbation load of 0.25%, 0.5%

and 0.75% of the applied vertical load at the mid node of the column. Columns of 6.0m

lengths and simply supported conditions are analysed for the following I-sections:

203 x 133 x 25, 457 x 191 x 75 and 533 x 210 x 122.

(b) Results of investigation of columns with perturbation loads

Results for the selection of columns with perturbation loads are shown in Appendix A. Each

result page includes the respective column configuration and the results of the varying

perturbation load at mid node. Results include the mode of instability and the load deflection

path of the mid node and the top node of the column. The load at which instability occurs is

also shown.

3.2.4 Discussion on results for column analysis

(a) Verification of ANGELINE

Analysis results in all cases are found to be a fraction higher than the theoretical Euler buckling

loads. The difference between the analysis results and theoretical Euler buckling loads vary

between 0.13% and 0.14% for simply supported columns and 0.47% to 0.56% for columns with

fixed supports.

The reason for the difference becomes apparent when evaluating the theoretical Euler

buckling loads. The applied axial force causes an axial shortening of the column, the effect

which is not taken into account in the theory. The Euler buckling load computed using

traditional Euler formula neglects axial shortening before buckling. For example if an axial

shortening of 0.1% occurs as a result of axial strain, the length of the column reduces to 0.999

L, and a higher buckling load is obtained [21]. This is shown by means of an example in Table

3.2.



Table 3.2. Example of effect of axial shortening

The second reason for the difference is as a result of the approximate nature of the finite

element approach. The numerical nature of the solution leads to round-off errors which do not

occur in analytical solutions. The result of the finite element analysis is dependent on the finite

element net. A cubic interpolation for the displacement of a finite element is used, which is

exact if linear frame theory is considered. Euler column theory leads to a sinusoidal

displacement function, which can only be approximated by a cubic function. As the number of

elements in the column increases, the approximation is reduced and this means that the

accuracy of the results improves. The accuracy of the approximation of the displacement

increases as the stiffness of the column increases since the displacement approach is used and

not the force approach. The buckling loads computed by means of the algorithm are therefore

marginally larger than those of the sinusoidal Euler theory [21].

(b) Inclusion of the perturbation load

Column analyses terminate before a singular point is reached. However, the values obtained

at this point are very close to the singular value. It can clearly be seen that the column

displacement approaches the singular point asymptotically.

Termination of the nonlinear analysis and detection of a singular point are different events in

the analysis. In this case the accuracy limit of the computer has been reached in a normal step

of the constant arc increment method, without change of sign of any of the diagonal

coefficients.



This is an important feature of the perturbation load and shows that the singular point is

approached only after large horizontal displacement at the mid node has occurred.

3.3 INVESTIGATIVE ANALYSES: PORTAL FRAMES

3.3.1 Definition of portal frames

The selection of portal frames analysed is shown in Figure 3.11. The portal frames include

column lengths of 5.0m, a roof slope of 3o, span of 24.0m and a 457 x 191 x 82 I-section. The

load pattern, support conditions and the application of the perturbation load is varied.

Figure 3.11. Selection of portal frames

Note: Each arrow in red presents a value of P=10.0kN applied at each node as indicated in the

figure, unless otherwise specified. Results are given as a load factor of P.



3.3.2 Results – Portal frame analyses

Table B1 to B6 in Appendix B1 show preliminary analyses which were performed to obtain

suitable values for the initial load factor increment. The results for factors larger than 0.10 are

discarded because they can be unreliable.

The results of the analyses of the configurations shown in Figure 3.11 are shown in Table B7 to

B12. An initial load factor of 0.10 is used throughout this set of analyses. The absolute value of

the displacement coordinates of the ridge of the frame is shown in the tables. The ratio of the

displacement u2 to the load factor is the total stiffness of the ridge in the current state of the

frame. The displacement behaviour of configurations C1 to C6 is shown in Figure 3.12.

0.0

2.0

4.0

6.0

8.0

0.0 10.0 20.0 30.0 40.0

Load factor

C2

C1

C3

C4

C6

C5

Dis

pla

cem

en

t u

2

(m)

Figure 3.12 Vertical deflection 2u of the ridge of portal frames as a function of the load factor



3.3.3 Discussion of results – Portal frame analyses

(a) The results in Table B1 to B6 show that the initial load factor increments are too

large for reliable analysis.

The initial load factor should be chosen so that the behaviour in the first load step does not

deviate significantly from linear elastic behaviour. Ridge displacement of 0.160m is reached in

the case of an initial load factor of 1.0. The height of the ridge above the edge of the roof at

the column is only 0.629 m, and displacement in the first load step is 25% of the difference in

height between eaves and ridge. The behaviour already deviates from the linear elastic path

and makes the load factor too large for a reliable analysis.

If an initial load factor of 0.1 is chosen, this results in a displacement of 0.015m, which is a

2.4% of the difference in height between the eaves and the ridge.

If the initial load factor increment is too large, the chord length of the constant arc increment

method becomes too large. The tangent stiffness matrix at the start of the load step, which is

used in cycle 0 of the iteration, then deviates significantly from the correct secant stiffness

matrix for the step. The trial tangent matrix is then not suitable for continuation as the load

step may become so large that the iteration procedure is not able to handle the nonlinearity.

This leads to termination of the analysis with the message “too many iterations in step …”

It can also happen that a diagonal coefficient of the decomposed incremental stiffness matrix

becomes negative, even though there is no singular state of the frame in the neighbourhood.

This occurs because the trial displacement state is not an exact equilibrium state. The

algorithm then tries to find a singular point which does not exist, and fails at one of several

possible code locations [21].

(b) The results in Table B7 to B12 are obtained with a suitable initial load factor

increment. The behaviour is characterised by the following phenomena:

(i) The portal frames C1, C2 and C5 with simple supports reach a singular

configuration. Similarities are observed in the instability behaviour between

portal frames with perturbation loads and portal frames without

perturbation loads. This is discussed later in this section.



(ii) Portal frame configurations C3, C4 and C6 with fixed support conditions do

not reach a singular point over the full nonlinear path analysed as shown in

Table B9, B10 and B12.

The behaviour prior to the singular point is discussed by looking at the smallest diagonal

coefficient of the decomposed secant matrix for each state of the frame. For illustration

Configuration C1 is used. The variation of the smallest coefficient with the load factor is shown

numerically in Appendix B2 and shown graphically in Figure 3.13.

Figure 3.13 Variation of the minimum diagonal coefficient (Configuration C1)

In this figure it can be seen that a rapid decrease in the diagonal coefficients of the

decomposed stiffness matrix occurs as it approaches the singular point. The instability of the

singular point is defined by the asymptotic behaviour of the smallest diagonal coefficients. This

is an important consideration as no perturbation load is applied for configuration C1.

The second phenomenon shows that portal frames with fixed supports do not reach a singular

configuration on the computed load path. This is confirmed by the variation of the smallest

diagonal coefficient of the converged secant stiffness matrix of Configuration C3 as shown in

Appendix B2.



The results show that the smallest diagonal coefficient minD decreases by about 8.3% as the

load factor increases from 0.00 to 9.45 and the smallest diagonal coefficient increases

continuously if the load factor increases further. There is no sign that the structure is

approaching a singular point.

Portal frames with fixed supports are not investigated further than a load factor of 22.5 as the

displacement of the ridge is already below the floor of the portal frame. The axial force in the

rafter at this load level is tensile near the corners of the frame and compressive near the ridge.

The variation of the ridge displacement with the load factor shows that the snap-through

behaviour does not exist and elastic instability does not occur.

3.4 CONCLUSIONS

3.4.1 Column behaviour

(a) Method of Nonlinear Analysis -verification

Column analysis shows that for the selection of columns in this investigation

consistent results are obtained.

(b) Perturbation loads

The inclusion of the perturbation load shows that the singular point of the column

is approached asymptotically in the displacement behaviour of the frame. This

means that large displacements occur before the singular point is reached and is a

very important difference of the perturbation approach to the classical eigen value

approach towards instability.

3.4.2 Portal frame behaviour

The numerical algorithm in ANGELINE for the range of portal frames within the range of

practical design configurations is robust.

(a) Choice in load steps

The reliability and accuracy of the numerical methods for the solution of the

initial value problem depend on the size of the load steps. The demands on the

theoretical background and computational experience of the engineer significantly

exceed those for linear structural analysis, which is nearly automatic.



(b) Identification of Singular Points

Singular points are reached for portal frame configurations with pinned supports,

but at a load level that is well beyond values of engineering significance.

The fixed frames C3 with full load and C6 with partial load do not approach a

singular point which is of practical interest.

Note that it is not necessary for a portal frame to reach a singular point and very

large displacements occur without any indication of a singular point.

(c) Inclusion of Perturbation loads

Instability in the case of portal frame configurations with simply supported

conditions show an asymptotic approach in the displacement as it reaches the

singular point. This is also the case with the portal frame which does not include

the applied perturbation load.

The deformation behaviour of the portal frame in the absence of the perturbation

load is as a result of the horizontal component of the axial force in the rafter

acting as a perturbation load on the corresponding columns. This leads to the

same behaviour explained in column analysis when a perturbation load is applied.

3.4.3 Evaluation of the Nonlinear Structural Behaviour of Portal Frames

(a) Lateral Displacement of the Ridge

(i) The lateral displacements of the ridge of the fully loaded frame C2 vary from

0.008m at a load factor of 5 to 0.045m at the singular point.

(ii) In the case of the partially loaded frame C5 with pinned supports the lateral

displacements of the ridge are much larger. They vary from 0.140 m for a load

factor of 5.0 to 1.53 m at the singular point with load factor of 12.4.

(iii) The lateral displacements of the ridge of the fully loaded frame C4 with fixed

supports and a perturbation load vary from 0.001 m at a load factor of 5 to

0.005m at a load factor of 30.0



(iv) The lateral displacements of the ridge of the partially loaded frame C6 with

fixed supports are significantly less than those of the frame with pinned supports

but larger than in the case with a full load. They vary from 0.034 m for a load

factor of 5.0 to 0.190 m for a load factor of 30.0.

The 0.5% horizontal perturbation load acting at the left corner of the frames does not

influence the deflection of the frames significantly. However, the influence of the partial

loading on the deflection behaviour of the frame is more significant.

(b) Comparison between pinned and fixed supports

The analysis of frames C1 to C6 shows that the nonlinear behaviour of portal

frames with pinned supports is less favourable than that of portal frames with

fixed supports.

3.5 SUMMARY

• Portal frames do not necessarily have a singular point.

• The load factor at which a singular point occurs for the selection of frame

configurations is beyond the point where the material becomes inelastic.

• The instability of the portal frames is visible in the asymptotic behaviour in the

displacement as the singular point is approached.



4 IN-PLANE STRUCTURAL BEHAVIOUR OF PORTAL

FRAMES

4.1 INTRODUCTION

The research undertaken in Chapter 3 shows that the point of elastic instability for the

investigated portal frames, which represent portal frames with dimensions commonly used in

practice, is far beyond the loading which causes yielding in the material.

The theory implemented in ANGELINE does not include the plastic deformation of members.

To study the behaviour of portal frames that include the plastic deformation it is necessary to

employ a software programme that models the plastic deformation and subsequently the

forming of plastic hinges correctly. Many commercial software packages are available that

claim to include the correct application of a plastic deformation analysis. However, the

following problems exist:

• Manuals include only part of the theory and often do not explain

implementation of the theory from first principles.

• The implementation of the theory in the software is often seen as intellectual

property of the developers and therefore not freely available. This leaves the

researcher with not enough insight to adapt the same procedure as followed

in the first part of the research, for which the software is developed

specifically as a research tool.

This dilemma enables the user only to identify the type of behaviour but accurate insight into

the theory which is implemented and the real problem behind the symptoms might not be

truly understood.

A form of reliability can be achieved by investigating benchmark examples using the

commercial software and comparing this to theoretical values, but from these verifications it

cannot explicitly be known if the theory is correct. This necessitates the development of a

software programme from first principles. The development of such a research tool is not

within the scope of the investigation.



As a result of the problems stated previously, the second part of this investigation necessitates

a shift in the focus of research. In this part of the research an engineering view is adopted and

further structural behaviour is investigated from this point of view.

4.2 OBJECTIVES

To evaluate the factors influencing the in-plane structural behaviour of portal frames in order

to determine the following:

What behaviour governs the in-plane behaviour of 2D portal frames, including plastic

deformations.

4.3 METHOD OF APPROACH

A summary is given in Figure 4.1, which includes a detailed method of the approach.

The approach includes factors taken into account for modelling purposes, and these are

evaluated by means of a benchmark example. The benchmark and subsequently portal frames

selected for the parameter study are analysed using the following procedure:

• Portal frames are designed according to the DRAFT SANS 10160-1 (Basis of

Structural Design and Actions for Buildings and Industrial Structures - Basis of

structural design) [17], the DRAFT SANS 10160-2 (Basis of Structural Design

and Actions for Buildings and Industrial Structures - Self-weight and imposed

loads)[18] and SANS 10162-1:2005 (The structural use of steelwork) [16].

• A second order analysis is done with a commercially accepted design package

namely PROKON, widely used in South Africa. This is done so that a practical

section is chosen, of which the capacity is sufficient according to SANS 10162-

1:2005, similarly to what is done in the industry.

• A second order elastic-plastic analysis is done using ABAQUS. ABAQUS is an

commercial general finite element software package.

• The selection of portal frames is analysed using ANGELINE.



• Serviceability requirements as set out in SANS 10162-1:2005 are identified

and compared to the displacement of the frame, as calculated using

ANGELINE.

• This also gives the opportunity to show by means of a parameter study, that

elastic buckling is not a limiting criterion in the design of portal frames with

practical dimensions.

Figure 4.1 Flow chart for investigation into the structural behaviour of portal frames

Modelling considerations for portal frames 5.1


5 MODELLING CONSIDERATIONS FOR PORTAL FRAMES

This chapter describes factors to be taken into account when modelling in-plane behaviour of

2D portal frames. The following considerations are discussed:

• The benchmark portal frame with typical geometric properties and the load

pattern to be applied.

• The type of elements used for modelling.

• The inclusion of imperfections.

• Identifying a software programme that includes plastic deformation in its

analysis procedure and verifying correct implementation by means of

benchmark examples.

• Identifying compatibility of the various software packages.

5.1 IDENTIFICATION OF A TYPICAL PORTAL FRAME AND LOAD PATTERN

5.1.1 Identifying the portal frame

In general portal frames with pinned supports are preferred to portal frames with fixed

supports, as this reduces the required foundation size and simplifies the connection detail.

The disadvantage of using pinned supports is that the stiffness of the overall structure is

reduced, which results in higher deflections, subsequently larger profiles are used to satisfy

serviceability limit state requirements.



A typical pitched roof steel frame has the following configuration [6]:

• A span between 15m and 50m (25m to 35 m is the most efficient)

• An eaves height between 5m and 10 m (5m to 6 m is the most common)

• A roof pitch between 5° and 10° (6° is commonly adopted)

• A frame spacing between 5m and 8m

• Haunches are used to accommodate bolted connections

A portal frame configuration is identified using these suggestions and is shown in Figure 5.1.

6.0m

1.00

Figure 5.1 Benchmark portal frame

This figure shows the span of the frame, length of the columns and indicates the nodal point

spacing of the rafters and columns. For modeling of the haunches correctly, additional nodal

points are required, which are not indicated in Figure 5.1. Further description of the frame is

given at the right hand side of the figure. Stability is provided by rigid frame action in the form

of fixed connections, at the column-rafter connection and the ridge connection. Only in-plane

behaviour of the frame is investigated.

5.1.2 Load application

The identified load pattern is given in Figure 5.2. This also shows the applied perturbation load

which models imperfections and in the case of a symmetric structure and a symmetric load is

required to initiate horizontal deflections for the second order analysis. This perturbation load

is discussed in further detail in Section 5.3.



Of the actions on structures as set out in the DRAFT SANS 10160-2 and 3, two types of actions

are of interest in addition to the own weight of the structure, namely imposed loads and wind

actions. However, as stated previously, this study will include vertical gravitation loads only.

Figure 5.2 Load pattern across roof

5.2 TYPES OF ELEMENTS TO BE USED IN MODELLING

For clarity it is repeated that PROKON will be used to obtain results used in the design

according to SANS 10162-1:2005, ABAQUS to do a second-order elastic perfectly-plastic

analysis and ANGELINE for the purely geometric nonlinear analysis.

5.2.1 Selection of elements for the finite element analysis

Finite elements based on beam theory have a number of advantages, including the simple

geometric description and the reduced number of degrees of freedom compared to I-type

section modelling using 3D elements. The selection of the type of element to be used is based

on the fact that the primary investigation considers the global instability of the frame. For this

consideration beam theory is sufficient as this is widely used for framed structures composed

of slender members. The reason for this is that although beam theory is a one-dimensional

approximation of the three-dimensional continuum, this reduction can be made due to the

fact that the dimensions of the cross-section are small compared to typical dimensions along

the axis of the beam [24]. A beam element can undergo deformations due to axial forces,



bending moments and torsion. However, torsion is not applicable to in-plane behaviour of the

frame.

Two of the most popular beam element types that are used is the Euler Bernoulli beam and

the Timoshenko beam. The underlying assumption of the Euler Bernoulli beam is that plane

sections remain plane, i.e the plane which is perpendicular to the longitudinal axis of the beam

remains plane after bending. This is an important factor to consider if the structural elements

are subjected to large bending moments or axial tension or compression. The Euler Bernoulli

beam is used.

5.2.2 Number of elements

In linear frame analysis the exact displacement interpolation can be obtained from the

differential equations. This means that the subdivision of elements does not influence the

results of the analysis. Linear analysis is a specialised form of nonlinear analysis.

For geometric nonlinear analysis, Pahl [21] states that: “The exact solution of the nonlinear

differential equations for a beam is a highly complex series which is not suited for the

construction of finite elements. The displacement variation is therefore approximated with a

polynomial, as in the case of other types of finite elements, for example plates. The number of

elements per member therefore influences the results of a nonlinear frame analysis. “

An investigation is done into the selection of the correct number of elements in PROKON,

ABAQUS and ANGELINE. This investigation does not include the study of optimum design but

to ensure the correct number of elements for accurate results. The benchmark example is

used and the numbers of elements are varied. Element subdivision is varied between 6, 12

and 24 elements per member. In ANGELINE and PROKON the frame is evaluated on axial

force, shear force, bending moment and deflection of the frame. In ABAQUS the frame is

evaluated at the load factor at which the first plastic hinge forms. The concept of plastic

hinges is discussed in Section 5.5.

Comprehensive results of the varying number of elements are shown in Appendix C. None of

these results exceed a difference of 0.5% and according to these results six elements are

sufficient for the benchmark example. However, 12 elements are chosen.



5.3 IMPERFECTIONS

The influence of imperfections in portal frames must be included in the analyses. Principal

causes of imperfections are [4]:

• Unavoidable eccentricities in the application of loads and construction of the

frame.

• Initial curvature of the member. Flat rolled sections are fabricated to

specified tolerances.

Usually values and shapes are assumed for initial curvature with the maximum

straightness given at mid node.

• Residual stresses in the member. This is caused primarily by the uneven

cooling after the rolling of the structural steel profiles.

The magnitude and type of residual stresses depend on the cross-section,

rolling temperature, cooling conditions, straightening procedures and metal

properties.

5.3.1 Making provision for imperfections in the analysis of portal frames

(a) Methods proposed in literature

The following three approaches are proposed by Chan et Al ( 2005)[11]:

(i) Buckling mode approach

Chan refers to Kitipornchai (1987), Schafer and Pekoz, and Dubina and Ungureanu and points

out that initial imperfections for the geometrically non-linear analysis of a structure can be

selected by means of a buckling analysis.



This is done by applying an initial imperfection in the form of the lowest eigen-mode. The

general eigen-value problem is as follows:

{ [KL] + li [KG] }[fi] = 0

where,

[KL] = linear matrix

[KG] = geometric stiffness matrix or initial stress matrix

[fi] = ith eigen-value and eigen-mode

A global analysis is performed to obtain the lowest eigen-value and accompanying eigen-

mode. This analysis is usually performed with the equilibrium equations of the undeformed

shape of the structure. The scaled down mode of this eigen-mode is then used as the initial

displacement of the structure for the second order analysis. However, a disadvantage of this

method is that the real collapse mode could differ from that of the lowest eigen-mode.

(ii) Applying a notional horizontal force

This method applies a notional horizontal load. In current codes this value is prescribed as

0.5% of the gravity load. In this way the geometric imperfections of the undeformed model of

the structure are replaced by the horizontal deflections due to the notional horizontal load.

(iii) Initial geometric imperfection approach

The allowable tolerance for the out of straightness for a member as specified in SANS 2001 CS

[19] is L/1000. This bow imperfection is usually assumed to be in the form of a sine curve. By

defining the coordinates along the element the out-of straightness can be modelled in terms

of the coordinate system.

The disadvantage of this method is that determining these coordinate positions can be very

time consuming and the accuracy is directly proportional to the number of elements used. The

advantage of this method is that it yields accurate results, if compared to experimentally

determined values.



(iv) Extensive research including all three methods concludes the following of interest

to the current study:

• Initial imperfections do influence the instability limit of the structure and

cannot be ignored.

• All three methods give consistent results.

5.3.2 The inclusion of the perturbation load as prescribed by SANS 10162-

1:2005

The following is stated in SANS 10162-1: 2005 [16], Section 8.7:

“The translational load effects produced by notional lateral loads, applied at each storey, equal

to (0.005 x factored gravity loads contributed by that storey, shall be added……”

The perturbation load is applied according to the clause in this standard. Verification of the

approach by which this method is applied is verified in the next section.

5.3.3 Verification of application of perturbation load

The notional horizontal force method and the initial geometric imperfection approach are

compared by means of an example in Appendix D. ANGELINE is used for this comparison.

This study looks at an example of a column configuration with an I-section of 203 x 133 x 25

and simply supported conditions.

The investigation includes the application of the perturbation load of 0.25%, 0.5% and 0.75% at

the mid node and a column with an initial curvature.

The load-deflection path at mid node, of the column with varying perturbation loads and the

column assuming an initial curvature is shown in Figure 5.3 to illustrate how well the various

methods compare with each other.



Load Factor

0.10

0.00

-0.10

0.30

0.40

0.20

0.50

Figure 5.3 Load-deflection at mid node

This comparison verifies that the application of the notional horizontal approach compares

well with the initial geometric imperfection approach and is used for further modelling

purposes in this thesis due to the simplicity of it.

A perturbation load of 0.5% of the gravity load is applied at the top of the left hand column as

shown in Figure 5.2 in each of the frame configurations.

5.4 MODELLING OF HAUNCHES

The use of haunches in portal frames is important as it facilitates the bolted connections and

improves the overall stiffness of the portal frame for the serviceability limit state of the frame.

Portal frames are fabricated from hot rolled I-beams, which are cut to use as haunch members.

In South Africa the common trend is to design portal frames including haunches at the eaves.

The option of including haunches at the eaves and the ridge in the model is provided for

analysis with ANGELINE. The choice of using haunches is given in the Model Editor in the

graphical user interface. The length of the haunch to be modelled is subdivided into four

prismatic elements. Each of these elements has a representative stiffness of that prismatic part

of the member. The choice of the number of elements can be changed by accessing



Generator.java. The number of elements in the haunches can be seen by the number of nodes

in the user interface as shown in Figure 5.4.

2.0m

24.0m

0

2

4

11 13 15 17

27

25

23

5

6

7 8 910 19

21

Figure 5.4 Haunches in ANGELINE

In ABAQUS the modelling of the haunches is not automatically included and calculations are

done by hand and implemented with equivalent I-sections of the relevant stiffness, see Figure

5.5. The procedure undertaken is as follows: the designer firstly decides on how many

equivalent members the haunch should be subdivided into and the stiffness of each of these

subdivisions is calculated in the middle of the member. The equivalent I-section with the same

stiffness is then substituted into that part of the haunch.

Figure 5.5 Equivalent I-sections

5.5 PLASTIC DEFORMATION OF STRUCTURAL MEMBERS

In this section the development of stresses in members and how plastic hinges are formed is

explained.



5.5.1 Stresses in members

As a load in a member is increased, so the stresses in the member are increased until yielding

of the material occurs as shown in Figure 5.6 (a). In this phase the bending stress is linear along

the cross-section of the beam and the bending moment M is proportional to the curvature

2

2

zd

d υ− for a cross section of a typical I-section. Beyond this point the increase of load will

induce resistance of more inner fibers as shown in Figure 5.6(b), each in turn reaching yield

stress until ultimately the yield stress propagates to the neutral axis and the section becomes

fully plastic as shown in Figure 5.6(c). When the yield moment M = fy Zex is exceeded, the

curvature increases rapidly as yielding progresses and the stresses become nonlinear. At high

curvatures the limiting situation is reached and a full plastic moment is formed at Mp=fy Zpl

[10].

Figure 5.6 Stress distribution in cross-section

However, in the analysis of steel structures it is acceptable to assume elastic perfectly-plastic

behaviour of the beam as shown in Figure 5.7. This shows an idealised stress-strain curve for

structural steel in direct tension. Line AB represents the elastic stress in the material according

to Hooke’s law. In elastic-perfectly plastic analysis it is assumed that the cross-section remains

fully elastic until the plastic moment resistance is reached. Concentrations of plastic

deformations cause the formation of plastic hinges at critical locations in a member.



Figure 5.7 Idealised stress-strain curve

It should be noted that in real members strain hardening commences just before Mp is reached

and the real moment-curvature relationship rises above the fully plastic limit of the plastic

moment, on the other hand, shear forces could cause small reductions in the value of Mp,

principally due to the reduction in plastic bending [1].

Lim et al states [13] that this beneficial effect of strain hardening can increase the capacity to

8% above the calculated value of the plastic moment. This effect is conservatively ignored in

this research.

5.5.2 Plastic Hinges

The formation of a plastic hinge allows the parts of the member at either sides of the hinge to

rotate freely. The formation of enough hinges can cause collapse of the member.

This concept is explained by looking at a simply supported beam, as shown in Figure 5.8. Stage

(a) of Figure 5.8 shows the beam under loading and remains entirely elastic. As the load is

increased in Figure 5.8(b) the outer fibres yield at section C where the maximum stress in the

beam is located and a plastic zone develops in these parts. As the load increases the section

becomes fully plastic and a plastic hinge forms as shown in Figure 5.8(c).

This causes large deflection and ultimately the collapse of the beam. The collapse is at the

cross-section in the beam where the plastic moment in the section is formed [9].



Figure 5.8 Various stages in the forming of plastic hinges in beam

5.5.3 Collapse mode in portal frames

Portal frames require a certain number of plastic hinges to form a failure mechanism. This is

determined by the number of redundancies + 1. This implies that the number of plastic hinges

with pinned supports and fixed supports are 2 and 4 respectively.

Various failure mechanisms are possible [23]:

(a) Mode 1. Pitched portal frame mechanism

This type of mechanism forms due to a dominant vertical loading on the portal

frame.

(b) Mode 2. Sway mechanism

This type of mechanism forms due to a dominant horizontal loading on the portal

frame.

(c) Mode 3. Combined mechanism

This is due to a combination of the portal frame mechanism (Mode 1) and the side

sway mechanism (Mode 2) reducing the combined rotations such that point A

does not form a plastic hinge as shown in Figure 5.9 (c).



Figure 5.9 Collapse modes in portal frames [23]

It should be noted that this explanation does not show the combined effect of buckling and

plastic deformation.

5.5.4 Verification of correct implementation of the programme

ABAQUS is used for the identification of the collapse load of the structure through means of

formation of hinges. A second order (inclusion of geometric nonlinearity) bilinear elastic-

perfectly plastic approach is used and this is a selectable analysis option in ABAQUS.

Geometric nonlinearity is included through the RIKS method in ABAQUS.

The modified RIKS method is an algorithm that obtains nonlinear static equilibrium solutions of

unstable problems. The basic algorithm used is the Newton method.

Two examples are used to verify correct implementation of the plastic hinge theory in these

two examples is shown in Figure 5.10.

The points where plastic hinges should form are shown. The load factor of collapse is

determined through numerical analysis and theoretical calculations. The criterion that is used

is the load-displacement path at specified nodes.



I-se

ctio

n –

20

3 x

13

3 x

25

6.0

m

Figure 5.10 Verification of ABAQUS

(a) Example 1

(i) Numerical analysis (ABAQUS)

The load deflection path at mid node of the analysed beam is shown in Figure 5.11(a). The

load deflection path is linear up to a load factor of 6. After which the mid node deflects

asymptotically. This occurs at a load factor of 6.18. This relates to a force of 61.86kN.

Figure 5.11(a) Load-deflection path at mid node

Load Factor vs Displacement

0

1

2

3

4

5

6

7

0 0.1 0.2 0.3 0.4 0.5

Displacement at Mid Node (m)

Lo

ad

Fa

cto

r (x

10

kN

)

P=61.86kN



Figure 5.11(b) Stresses in beam

(ii) Theoretical results

The bending moment at which the beam will form a plastic hinge is calculated. This is

compared to the theoretical bending moment at mid node for Example 1 and subsequently the

concentrated force at the mid node which will result in this bending moment is calculated.

The plastic moment of a 203 x 133 x 25 I-section is given by:

Plastic Moment (Mp) = Zpl x fy

where,

Zpl = Plastic Section Modulus

fy = material yield point

Mp = 259 x 103 mm6 x 350 mPa = 90.65kN.m

Bending Moment at Mid Node = 4

PL

where,

P = the applied load

L = the length of the beam

Comparing the bending moment to the plastic moment, the applied value of P is calculated as:

P = 60.43kN

Stresses in beam



(b) Example 2

(i) Numerical analysis (ABAQUS)

The load deflection path the top node is shown in Figure 5.12(a). The load-deflection path is

linear up to a load of approximately 14.0. After which the top node deflects asymptotically.

This happens at a load of 15.19kN.

Figure 5.12(a) Load-deflection path at the top node and (b) stresses in cantilever column

(ii) Theoretical results

Bending Moment at Support = P x L

where,

P = the applied load

L = the length of the beam

Comparing the bending moment to the plastic moment, the applied value of P is calculated as:

P = 15.11kN

Load Factor vs Displacement

0

2

4

6

8

10

12

14

16

0 0.1 0.2 0.3 0.4 0.5Displacement at Top Node (m)

Loa

d F

acto

r (x

1.0

kN)

P=15.19kN

Stresses in Cantilever



(c) Verification

A 2.3% and 0.53% difference between the numerical and theoretical results are obtained for

example 1 and example 2, respectively.

The theory upon which the theoretical bending moment is calculated is linear. In real

behaviour the beam has undergone deflections at the point of hinge forming which is not

taken into account in the linear calculation of the bending moment.

The differences in the results are considered acceptable and correct implementation is

assumed.

5.5 COMPATIBILITY OF SOFTWARE PACKAGES

To evaluate the correct implementation of the models in the various software programmes a

comparison is made of the forces at a load of 6.41kN at the nodes as shown in Figure 5.2.

Clarification of the load selection becomes apparent in the next chapter as this is the value

applied if the benchmark portal frame is designed according to SANS 10162-1:2005 [16]. It is

also in the elastic range of the portal frame.

The results that are obtained with the different software packages are compared at the

locations in the frame where the maximum axial force, shear force and bending moment

occur. This is illustrated in Table 5.1.

The difference in the values (%) is calculated by assuming the smallest value to be the correct

one, and determining the percentage difference to that of the other software programmes.

The lowest value is not taken because it is assumed to be the correct value but because this

gives the most conservative highest value when differences are computed. These percentage

differences are shown in Table 5.2.



Table 5.1 Forces at allocated elements – various software programmes

Axial Forces (kN)

Shear Forces

(kN) Bending Moment (kN.m)

Deflections

(mm)

L

oa

d (

kN

)

To

p L

eft

Co

lum

n

To

p R

igh

t C

olu

mn

To

p L

eft

Ra

fte

r

To

p R

igh

t R

aft

er

To

p L

eft

Co

lum

n

To

p R

igh

t C

olu

mn

To

p L

eft

Co

lum

n

To

p R

igh

t C

olu

mn

To

p L

eft

Ra

fte

r

To

p R

igh

t R

aft

er

Ve

rtic

al

To

p

Ra

fte

r

Ho

rizo

nta

l T

op

Left

Co

lum

n

ANGELINE results

38.74 38.95 22.62 22.60 21.36 21.77 133.10 135.70 75.30 75.26 297.70 24.80

ABAQUS results

38.80 39.90 22.62 22.60 21.29 21.69 130.80 133.40 74.70 75.18 299.00 23.70

PROKON results

6.41

38.90 39.20 22.50 22.50 21.10 21.82 132.10 135.20 75.20 76.80 291.30 25.11

Table 5.2 Percentage differences in forces

Axial Forces Shear Forces Bending Moment Deflections

Lo

ad

(k

N)

To

p L

eft

Co

lum

n

To

p R

igh

t C

olu

mn

To

p L

eft

Ra

fte

r

To

p R

igh

t R

aft

er

To

p L

eft

Co

lum

n

To

p R

igh

t C

olu

mn

To

p L

eft

Co

lum

n

To

p R

igh

t C

olu

mn

To

p L

eft

Ra

fte

r

To

p R

igh

t R

aft

er

Ve

rtic

al

To

p

Ra

fte

r

Ho

rizo

nta

l T

op

Left

Co

lum

n

ANGELINE results

0.00 0.00 0.53 0.62 1.33 0.37 1.72 1.72 0.74 0.11 2.15 4.60

ABAQUS results

0.16 2.44 0.53 0.62 1.00 0.00 0.00 0.00 0.00 0.00 2.59 0.00

PROKON results

6.41

0.41 0.64 0.00 0.00 0.00 0.41 0.96 1.36 0.66 2.16 0.00 5.90

No significant differences are found between the various software programmes for axial and

shear forces. However, the bending moment results between the various results differ. This

influence ranges in a deviation of 2% for vertical deflection and between 4% and 5% for

horizontal deflection. One possible cause for this difference can be as a result of the different

approaches to geometrically nonlinear analysis.

5.6 SUMMARY

• A benchmark frame and load pattern is identified.

• Beam elements used for modelling and the modelling of haunches are discussed.

• ABAQUS results are verified by means of benchmark examples.

• Results in ANGELINE, ABAQUS and PROKON are compared and consistent results

are obtained.

Design of portal frames according to DRAFT SANS 10160-1 &2 and SANS 10162-1:2005 6.1


6 DESIGN OF PORTAL FRAMES ACCORDING TO DRAFT

SANS 10160-1 & 2: 2008 AND SANS 10162-1:2005.

6.1 INTRODUCTION

In this chapter the design of portal frames according to the DRAFT SANS 10160-1 (Basis of

Structural Design and Actions for Buildings and Industrial Structures - Basis of structural

design) [17], the DRAFT SANS 10160-2 (Basis of Structural Design and Actions for Buildings and

Industrial Structures - Self-weight and imposed loads) [18] and SANS 10162-1:2005 (The

structural use of steelwork) [16] is discussed.

The benchmark portal frame is used as an example to show design calculations.

6.2 LIMIT STATE DESIGN

The DRAFT SANS 10160-1 & 2 :2008 and the current SANS 10162-1:2005, employ the limit-

state design procedure in general procedures and calculation of loadings and the calculation of

member capacities. This method is based on the fact that loads are treated as random

variables. Different actions (self weight, imposed load and wind actions etc) have different

probability of occurrences and different degrees of variability.

Each variable action is taken in turn as the leading variable action with the life-time maximum

value of the variable, combined with accompanying variable actions with the arbitrary point in

time value of these variables.

Limit state design approach allows for differentiation of reliability and ensures that the

required level of reliability is achieved. This approach entails that the structure must satisfy

different limit state design requirements e.g.

• Sufficient strength capacity

• Stable against overturning

• Stable against uplift

• Serviceability requirements

• Durability



• Fire protection

• Fatigue

The current code allows for these requirements by specifying the ultimate and serviceability

limit state. The ultimate limit state design must be considered separately from the

serviceability limit state and the success of one does not necessarily mean the success of the

other. Only ultimate limit state design is considered in the first part of this research.

6.3 DESIGN OF A PORTAL FRAME ACCORDING TO DRAFT SANS 10160-1 & 2:

2008 AND SANS 10162-1:2005

6.3.1 Applied loads

Two types of loads are considered when investigating vertical gravitational loads, namely the

self weight and imposed loads.

(a) Self weight

The self weight on a structure arise from the own weight of structural and non-

structural elements. This may vary during construction but becomes permanent

after completion. Self weight can be determined by means of an iterative

procedure whereas members are identified and the weight of these members

determined until a section with sufficient resistance is obtained.

(b) Imposed loads

In the design of portal frames, imposed loads during construction and

maintenance must be taken into consideration when calculating roof loads. These

conditions are almost constantly changing and are rather more difficult to quantify

than the self weight.

Portal frame roofs are generally classified as inaccessible roofs, and treated as

such in this investigation. The guidelines for the determination of the roof loads



for inaccessible roofs are set out in DRAFT SANS 10160-2, Table 5 and are as

follows:

(i) For transient load design situation

• 0,75kN/m2 for 23m≤A

• 0,25kN/m2 for 215m≥A

• 1.0kN concentrated load over an area of 0.1m x 0.1m

(ii) For long term load design situations:

• 0,50kN/m2 for 23m≤A

• 0,25kN/m2 for 215m≥A

• 1.0kN concentrated load over an area of 0.1m x 0.1m

For areas between 3m2 and 15m

2 interpolation is allowed. The distributed loads and the

concentrated loads must not be applied simultaneously.

Imposed loads can be determined for each node by using these values. The tributary area for

the representing portal frame is more than 15m2. The imposed load will therefore be

0.25kN/m2 of the vertically projected area, applied as nodal forces at the purlin connection

points.

6.4 LOAD COMBINATIONS

6.4.1 Ultimate limit state

The combination of actions is given in Clause 7.3.2.1 of DRAFT SANS 10160-1. The combination

of actions for use in the ultimate limit state is given by the following equation:

∑≥

∑≥

+ψγ+γ++γ1j 1i

dikiiQ1k1QjkjG AQxxQxPGx """""""" ,,,,,,

where,

"" + implies “to be combined with”

∑ implies ”the combined effect of”



jG,γ the partial factors for the permanent action j

jkG , the characteristic value of permanent action j

P the relevant representative value of the prestressing action

1Q,γ the partial factors for the leading variable action

1kQ , the characteristic value of the leading variable action

iQ,γ the partial factor for the accompanying variable action i

ikQ , the characteristic value of the accompanying variable action i

iψ the action combination factor corresponding to the accompanying variable

action i

dA the design value of the accidental action

The corresponding partial factors are given in Table 2 of DRAFT SANS 10160-1.

The following load combinations are applicable:

1kjk Qx01Gx351 ,, .. +

1kjk Qx61Gx21 ,, .. +

where,

jkG , = the self weight also referred to as DL (Dead load)

1kQ , = imposed load also referred to as LL (Live Load)

6.4.2 Serviceability limit state

As previously indicated, serviceability is not included in the first part of the design. This

criterion however, will be checked after the frame has been designed according to the

ultimate limit state. The partial load factors to be used are given in Clause 8.2 of DRAFT SANS

10160-1. The following combination is applicable:

1kjk Qx01Gx11 ,, .. +



6.5 CAPACITY OF MEMBERS – ULTIMATE LIMIT STATE

6.5.1 Analysis of structure

Analysis of the frame is done using PROKON’s second order analysis.

6.5.2 Design of member of ultimate limit state

The design procedure as set out below determines the resistances of each of the individual

members. Typically, the maximum forces are obtained in the column and in the rafter.

Sections are designed according to the more critical of these two and hot rolled I-sections are

used as main structural members for the portal frame.

6.5.3 Classification of profile

Members are subjected to axial tension, bending and axial compression depending on which

load combination is under consideration. When the flange or the web is in compression and

the web or flange is too slender, local buckling occurs.

For this reason the limiting width to thickness ratios are specified so that local buckling is

eliminated. These limiting width-to-thickness ratios categorise the members into various

classes and capacity calculations must be done according to these classes.

(a) Members in axial compression

Classifications of sections in axial compression as set out in SANS 10162-1, Table 3 apply:

Table 6.1 Classification of sections in axial compression

Conditions Description General

yf

200

t

b<

for flanges of I-sections

f2t

b

t

b=

yf

670

t

b<

for webs of I-sections

wt

h

t

b=



where,

b = width of half of the flange in the first column

tf = thickness of the flange

tw = thickness of the web

fy = yield stress of material

If the t

b ratio is larger than these values, the section is classified as a class 4, and treated as a

thin-walled section.

(b) Design of class 4 members

If the web or the flange is a class 4 member it is necessary to determine the factored

compressive resistance as set out in SANS 10162-1 Clause 13.3.3.3. Using this procedure the

width of the component of the cross-section is reduced to an effective element width under

the calculated compressive stress until the requirements of class 3 are met.

f

kE6440W

iml .=

If Wlim <t

bis then

−=

f

kE

W

20801

f

kEt950b

..

where,

k = 4 for webs and (Laterally supported at both edges)

k = 0.43 for flanges (Laterally supported at one edge)

f = the calculated compressive stress in the element, using gross element properties

otherwise, no reduction in area needs to be done.

(c) Members in flexural compression

For classification of elements in flexural compression formulas as set out in SANS 10162-1,

Table 4 apply for flanges of I-sections:



Table 6.2 Classification of flanges – flexural

or webs of I-sections

Table 6.3 Classification of webs– flexural

Conditions Description

φ−≤

y

u

yw

w

C

C3901

f

1100

t

h. class 1 sections

φ−≤

y

u

yw

w

C

C6101

f

1700

t

h. class 2 sections

φ−≤

y

u

yw

w

C

C6501

f

1900

t

h.

class 3 sections

where,

fy= yield stress

Cu = ultimate compressive force in member or component

Cy = axial compressive force in member at yield stress

6.5.4 Compression capacity of element

The slenderness ratio of members under compression must be calculated and shall not exceed

the value of 200 as set out in clause 10.4.2.1 of SANS 10160-1:2005.

Conditions Description

yf

145

t

b<

class 1 section

yf

200

t

b< class 2 sections

yf

170

t

b<

class 3 sections



Based on research, including the effects of material and geometric non-linearity, the code

defines the maximum compressive strength of the column in terms of a function on the non-

dimensional slenderness ratio. The following formulas apply for in-plane behaviour:

[SANS 10162-1:2005 Clause 13.3.2(a)]

where,

Kx = the effective length factor

E = elastic modulus

Lx = the effective length for buckling about the x-axis

rx = the radius of gyration about the x-axis

φ = 0.9 the material factor in order to account for the possibility of under-strength in

materials

fy= yield stress

n =dependent on pattern of the residual stresses

Note: 2D in-plane behaviour is considered

6.5.5 Bending capacity of element

The bending capacity of the element must be checked. Bending around the strong axis is

determined by its plastic section modulus. In plane bending is considered, and thus assumed

that members is sufficiently laterally supported.

(a) Moment capacity of laterally supported members

The moment capacity of a member must be checked according to clause 13.5 of SANS 10162-

1:2005

2

x

xx

2

ex

r

LK

Ef

π=

n1n2yr 1AfC

/)( −λ+φ=

e

y

f

f=λ



yplrx fZM φ= for class 1 and 2 sections

yexr fZM φ= for class 3 sections

(b) Interaction between bending and compression

Interaction between bending and compression must be checked as the combination of the two

may be excessive and cause the member to reach its capacity. The interaction formula is as

follows, for class 1 and 2 I-shaped sections.

U1x= 1.0 for unbraced frames

(c) Tension capacity of profile

The slenderness ratio of members under tension shall not exceed the value of 300 as set out in

clause 10.4.2.2 of SANS 10160-1:2005.

Secondly, the calculation of the tension capacity of the member is done as follow:

where,

φ =0.9 =resistance factor

fy = yield stress

A =gross area of the section

6.5.6 Interaction between tension and bending

In addition to checking the tension capacity of the member it is also necessary to check the

interaction between bending and tension. The following formulas are prescribed by the code

in clause 13.9(a) and (b):

1M

MU850

C

C

rx

uxx1

r

u ≤+.

yr AfT φ=



where Tr is calculated as in paragraph 5.4.7

Mr as determined in 13.5 of SANS 10160-1:2005

for class 1 and 2 sections

where,

Tr is calculated as in paragraph 5.4.7

Mr as determined in section 13.5 or 13.6 of SANS 10162:1-2005, whichever is

applicable

6.5.7 Checking for shear capacity

The shear capacity is calculated with the following formulas:

2

w

v

h

s

4345k

+= . for s ≥ hw

if

then

and

where wv htA =

where,

kv = shear buckling coefficient ( kv = 5.34 for hot rolled sections with web stiffeners)

fs = ultimate shear stress

hw= clear depth of web between flanges

tw= web thickness

Av = shear area

Vr = factored shear resistance of member

s = spacing of web stiffeners

r

u

r

u

M

M

T

T+

AM

ZT

M

M

r

plu

r

u −

y

v

w

w

f

k440

t

h≤

ys f660f .=

svr fAV φ=



6.6 SERVICEABILITY LIMIT STATE

Portal frames need to be checked for serviceability requirements. Horizontal deflections are

limited to height-to-deflection limits and vertical deflections must adhere to span-to-deflection

limits. These are set out in the informative Appendix D of SANS 10162-1:2005 and are as

follow:

• minimum span/vertical deflection for simple span members supporting elastic roofing

= 180

• minimum height/horizontal deflection for simple span members supporting elastic

roofing =300

6.7 DESIGNING THE BENCHMARK EXAMPLE

6.7.1 Modelling of frame

The dimensions of the benchmark are explained in Section 5.1. This is modelled using PROKON

with 12 elements per column and 24 elements over the span of the roof. Node numbering is

shown in Figure 6.1 for explanation of force application.

6.0m

Figure 6.1 Numbering of nodes in PROKON – Benchmark example

Column node spacing is 0.5m, and roof node spacing is 1.0m.



6.7.2 Load application

Following an iterative process, the size of the section is identified as a 254 x 146 x37 I-section.

Self weight is determined by calculating the forces at the allocated nodes. This includes the

weight of the section, purlins which are converted from a line load to a nodal load, as well as

the services and sheeting which need to be converted from an area load to a nodal load.

Members used for purlins, isolation and sheeting is shown in Table 6.4. The self weight of

these structural and non-structural members is also indicated in the table. The total given at

the bottom of the table is the total self weight to be applied at allocated roof nodes 12 to 36

at every second node number.

Table 6.4 Example for calculation of dead weight of the structure

Member Section Length

(m)

Width

(m)

Weight

(kg/m* or

kg/m2)

Force

(kN)

Reference

(SAISC Handbook)

Purlins Lipped

Channel 5 Not applicable 5.92 kg/m -0.290 Table 2.30

Isolation Expanded 5 2.011 1.00 kg/m2 -0.099 Table 13.14

Services 5 2.011 2.50 kg/m2 -0.247

Sheeting IBR (0.6mm) 5 2.011 6.53 kg/m2 -0.644 Table 13.14

Member 254 x 146 x 37 5 Not applicable 37.00 kg/m -0.730 Table 2.1

Total - 2.010kN

The imposed loads to be applied on roof nodes are shown in Table 6.5.

Table 6.5 Example for calculation of imposed loads of the structure

Load Type Roof Type Area Load

(kN/m2)

Width

(m)

Force/node

(kN)

Reference

Imposed Inaccessible >15m2 0.25kN/m

2 2.000 -2.500

DRAFT SANS 10160-2,

Table 5

Total - 2.500 kN

Values obtained in Table 6.4 and Table 6.5 are the characteristic values, these values must be

multiplied by the appropriate partial load factors for the ultimate limit state.



6.7.3 Analysis results

The critical force combination is 1.2DL + 1.6LL. The axial force diagram, the shear force

diagram and the bending moment diagram for the ultimate limit state are shown in Figure 6.2.

Figure 6.2 Axial Force, Shear Force and Bending Moment Diagram

6.7.4 Summary of member analysis

Comprehensive resistance calculations are set out in Appendix E in the form of spreadsheet

calculations. Appendix E includes capacity calculations for forces obtained for the critical right

hand column. The summary of the calculations for the column and the rafter are shown below

in Table 6.6 and 6.7, respectively:

Table 6.6 Column resistances – I-section 254 x 146 x 37

Force type Force Resistance

Axial 38.9kN 1133.96kN

Bending 135.2kN.m 152.775kN.m

Bending - Axial Interaction As Above 78.67% utilised

Tension Not Applicable Not Applicable

Bending - Tension interaction Not Applicable Not Applicable

Shear 22.5kN 340.62kN



Table 6.7 Rafter resistances – I-section 254 x 146 x 37

Force type Force Resistance

Axial 25.80kN 540.78kN

Bending 76.8kN.m 152.775kN.m

Bending - Axial Interaction As Above 46. 5% utilised

Tension Not Applicable Not Applicable

Bending - Tension interaction Not Applicable Not Applicable

Shear 33.5kN 340.62kN

The maximum bending moment in the rafter is located at the rafter-column connection. The

haunch assists in the resistance at this point and makes this a non-critical point. The resistance

of the rafter section is determined by obtaining the maximum bending moment 2.0m away

from the rafter-column connection. At this point no haunch is present and resistance is

calculated by determining if the capacity of the I-section is sufficient at this location or at the

ridge of the roof, whichever is the critical bending moment. This is explained in better detail in

Section 8.2.1.b (i) and Figure 8.6.

6.8 SUMMARY

• Portal frame design according DRAFT SANS 10160-1 & 2:2008 and SANS 10162-

1:2005 is explained.

• The benchmark portal frame is designed according to DRAFT SANS 10160-1 & 2

:2008 and SANS 10162-1:2005.

• A 254 x 146 x 37 I-section is identified for the benchmark example.

• The design (ultimate limit state) is governed by the bending moment at the top of

the right hand column.

Design of portal frames for parameter study 7.1


7 ANALYSIS OF BENCHMARK PORTAL FRAME

In this chapter the analysis of the benchmark example using ABAQUS and ANGELINE is

described. This is done to identify the limiting behaviour of the portal frame.

7.1 ANALYSIS OF BENCHMARK PORTAL FRAME

7.1.1 Portal frame configuration

The benchmark portal frame identified and designed in the previous chapter is shown in Figure

7.1. This section describes results of the analysis of the benchmark portal frame using ABAQUS

and ANGELINE. The benchmark portal frame includes 12 elements for each rafter and column

member, respectively. The load P applied at the nodes is 10.0kN and the results of the analysis

are expressed in terms of the load factor. To determine the actual load applied to the

structure, the load P=10.0kN must be multiplied by the load factor. The loads are applied at

the connection points between purlins and rafters.

6.0m

12 elements

Figure 7.1 Configuration of portal frame analysed in ANGELINE and ABAQUS

7.1.2 Analysis using ABAQUS – Benchmark portal frame

Stresses in the various elements are observed and the location of cross-sections which reach

yielding in the portal frame, is identified by observing the sequence of behaviour as the

stresses in the elements increase.



Values of stresses at these members are obtained from the results output file.

(a) Results and discussion – ABAQUS analysis

(i) Load-displacement behaviour and stresses in critical elements

Locations of highest values of stresses are indicated in red in Figure 7.2 (a) and 7.2 (b) and

indicate yielding of the material at that point.

Figure 7.2(a) .Location of highest stresses at first yielding of cross-section

Figure 7.2(b) Location of highest stresses at yielding of cross-section in rafter

These figures indicate that the locations of the highest stresses are at the top right hand

column, top left hand column and at the ridge of the portal frame. Results of stresses in

ABAQUS are given by a maximum positive stress and maximum negative stress, at the furthest

fibre of the cross-section as shown in Figure 7.3.

The load factor at which the first cross-section in the portal frame starts to yield is determined.

This yielding is identified by the furthest fibres of this cross-section reaching the yielding stress

(350 MPa). If the analysis done for the benchmark portal frame in Section 7.1.2 (using

ANGELINE) does not reach a singular point before this load factor, the frame does not become

unstable as a result of purely geometric instability and plastic deformation is the governing

behaviour.

X

Y



XX

Stress taken at furthest fibre

Stress taken at furthest fibre

Figure 7.3 Location on cross-section where ABAQUS calculates stresses

The sequence of events include the first cross-section to yield at the element at the top of the

right hand column, and is followed by the yielding of the top left hand column as indicated in

Figure 7.2(a) The frame then starts to deflect in the positive x-direction and the left hand

column starts unloading. Subsequently the cross-section of the member at the ridge of the

portal frame starts yielding and the maximum load on the load path is reached shortly after

this. This is illustrated in Figure 7.2(b).

The right hand column reaches yield stress at a load factor of 0.7626, followed by the yielding

of the top element of the left hand column at a load factor 0.7734. The vertical displacement

of the roof increases due to the gravitational load until the elements at the ridge reach yield

stress. The maximum load factor on the load path is 0.8769.

Figure 7.4 illustrates the load-displacement paths of the top right and left hand column’s

horizontal deflection and the vertical deflection at the ridge of the roof as the load in the

portal frame is increased.

Figure 7.4 Load deflection paths of the allocated elements

Deflection vs Load Factor

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

-1 -0.5 0 0.5Displacement (m)

Loa

d F

ac

tor

Horizontal top right column

Horizontal top left column

Vertical ridge of roof

0.760.763

0.876



The graph illustrates the outward deflection of the respective columns as a result of the thrust

of the rafter on the columns.

Shortly after the top element in the right hand column reaches yielding, the notional

horizontal load causes the frame to sway. This explains the sudden deflection of both columns

to the right. This deflection causes the stresses in the left hand column to unload.

From the slope of the load deflection path after the yielding of the first cross-section,

considerable decrease in the stiffness of the frame is observed but global system failure only

occurs when the cross-sections in the right hand column and at the ridge of the portal frame

has yielded.

This predicts the same behaviour as discussed in Section 5.5 and shown in Figure 5.9, a

combined mechanism occurs. As a result, the frame is unable to take on any more loads and

the frame becomes instable.

(ii) Stresses in elements

The load-stress history of the critical elements is shown in Figure 7.5. Figure 7.5(a) shows the

location of the elements that are included in the load-stress history graph and Figure 7.5 (b)

shows the stresses associated with these members.

Figure 7.5(a) Location of members



Load Factor vs Stresses

0

50

100

150

200

250

300

350

400

0.00 0.20 0.40 0.60 0.80 1.00

Load Factor

Ma

xim

um

Str

ess

in

Ele

me

nt

(MP

a)

Left Column - Top

Left Rafter - Eaves

Left Rafter - Ridge

Right Rafter - Ridge

Right Rafter - Eaves

Right Column - Top

Figure 7.5(b) Load-stress history of critical elements

These graphs show that the yielding of the cross-sections in the right hand column and shortly

after in the left hand column. This is followed by the yielding of the members at the ridge.

Stresses in members at the eaves in the rafter do not reach yield due to the presence of the

haunches. From these stresses it can also be seen that the stresses in the rafter are much

lower than the stresses in the column throughout the load path.

However, as the yielding of the column is reached, a redistribution of forces in the frame

occurs and an increase in the stress-load factor slope is observed in the rafter member as

shown at (a) of Figure 7.5(b).

7.1.2 Analysis using ANGELINE – Benchmark example

The benchmark portal frame is analysed using ANGELINE. An initial load of 1.0kN per node is

used which represents an initial load factor of 0.1. The analysis includes the following

observations:

• The displacement behaviour of the frame.

(a)



• To confirm that purely geometric instability has not occurred in the range

before a cross-section in any of the members in the portal frame has yielded.

• To investigate the axial forces, shear forces and bending moment in critical

members.

(a) Results and discussions - ANGELINE

(i) Displacement behaviour of frame - elastic behaviour

The global load-displacement behaviour for the frame is shown in Figure 7.6 at a load factor of

1.736.

Figure 7.6 Displacement of frame at load factor 1.736

Figure 7.7 and Figure 7.8 illustrate the load-deflection path of the node at the top left hand

column and ridge of the rafter, respectively. Y1 pertains to the horizontal deflection and y2 to

the vertical deflection.

In the absence of any plastic hinges, no sway to the right is observed up to a load factor of

0.7626. At this load factor of 0.7626 the yielding of the right hand column occurs as analysed

in ABAQUS and shown in Figure 7.2 (a). The columns are pushed further outward due to the

thrust of the rafters and at a load factor of 1.70 a turning point in the load-deflection path is

observed.



Figure 7.7 Deflection-load path of frame at top of left hand column

The load factor at serviceability (1), the load factor at ultimate state (2) and the load factor at

the yielding of the column (3), is illustrated in Figure 7.7 and 7.8. The load-displacement path

of the left hand column node is almost linear up to a load factor of 0.641. Deviation from the

linear path is observed after this value up to the load factor where yielding of the right hand

column commences in Figure 7.7.

Figure 7.8 Load-deflection path of portal frame at ridge

In Figure 7.8 the vertical deflection of the rafter indicates a linear path up to a point where the

load reaches the value prescribed by SANS 10162-1: 2005 for the serviceability limit state. The

horizontal deflection at the ridge indicates that the perturbation load has very little effect on

the portal frame.



(ii) Axial force, bending moment and shear force behaviour of the benchmark

portal frame

The axial force diagram, shear force diagram and the bending moment diagram is shown in

Figure 7.9, Figure 7.10 and Figure 7.11, respectively.

Figure 7.9 Axial force diagram at a load factor of 1.0

Shear Force diagram

36.25kN 37.01kN

52.26kN

52.58kN

Figure 7.10 Shear force diagram at a load factor of 1.0

Figure 7.11 Bending moment diagram at a load factor of 1.0

A constant distribution is shown for the axial forces with the axial forces in the columns being

larger than in the roof. The shear forces increase from the ridge to the eaves and from the top

of the columns to the supports.



Figure 7.9 illustrates the same distribution for axial forces, shear forces and bending moment

as compared to Figure 6.2 in PROKON. Maximum bending moments similarly occur at the

columns-to-roof connection and at the ridge of the roof.

Figure 7.12 and 7.13 show the load-axial force history and load-bending moment history of the

critical elements.

Axial Forces in members

0

0.2

0.4

0.6

0.8

1

1.2

0 20 40 60Axial Forces (kN)

Loa

d F

ac

tor Top

Right Rafter

Right Column

Figure 7.12 Load-Axial force history

Bending Moment in members

0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150 200Bending Moment (kN.m)

Loa

d F

ac

tor Top

Right Column

Figure 7.13 Load-Bending moment history

The axial force and bending moments in the rafter and column increase linearly.



7.2 CONCLUSIONS

7.2.1 ABAQUS

(a) The yielding of the cross-section in the portal frames is consistent with the

predicted behaviour explained in Section 5.5.

(b) The stresses in the rafter, including the eaves of the rafter, are lower than the

column stresses. This indicates that the haunch assists in the resistance against

yielding of the roof and smaller members can be used for the rafter

7.2.2 ANGELINE

(a) It is clearly indicated that design is governed by serviceability requirements. The

serviceability design value for the horizontal deflection is 20mm and the deflection

at a load factor of 1.1DL + 1.0LL is reached is at a value of 25mm. In the case of

vertical deflection the prescribed deflection is at 132mm and the maximum

vertical deflection is 291.0mm located at the ridge.

To adhere to the prescribed serviceability requirements, a 305 x 165 x 46 I-section

is required as opposed to the 254 x 146 x 37 I-section if the frame is designed

according to the ultimate limit state.

(b) The axial forces in the columns and in the rafters are very small.

(c) The load-deflection path does not indicate any singularity through means of

asymptotic approach of the load-deflection path.

7.3 SUMMARY

• The benchmark portal frame is analysed using ABAQUS and ANGELINE.

• The failure of the frame is not governed by the elastic instability of the frame but

the plastic deformation of the frame.



8 DESIGN OF PORTAL FRAMES FOR PARAMETER STUDY

A selection of portal frames is analysed in Chapter 8. The sequence in the study of each portal

frame is shown in Figure 8.1.

Figure 8.1 Sequence of analyses for each frame

8.1 DEFINITION OF PORTAL FRAMES

The parameter study includes the investigation into the behaviour and stability of portal

frames.

The selection of portal frames that is investigated is shown schematically in Figure 8.2, Figure

8.3 and Figure 8.4. The selection of portal frames is subdivided into three sections namely:

• Section 1: Portal frames with pinned supports and varying column lengths

and roof slopes.

Support conditions entail the translational degrees of freedom to be fixed.



• Section 2: Portal frames with fixed supports and varying column lengths and

roof slopes.

Support conditions entail the translational and rotational degrees of freedom

to be fixed.

• Section 3: Portal frames with varying spans and column lengths and roof

slopes.

This section includes pinned support conditions but a selection of frames of

which the span is varied is included.

Figure 8.2 Portal frames with pinned supports with varying column length and roof slope



Column Length

Figure 8.3 Portal frames with fixed supports with varying column length and roof slope

pinned supportspan

pinned support

Column Length

PPPPPP0.5P

P P P P P 0.5P

0.005x12P

applied perturbation

load

Roof Slope

Span – 28.0m

Roof Slope

Column Length

6.0m

6o 12

o

14.0m

Span – 24.0m

Roof Slope

Column Length

6.0m

6o 12

o

14.0m

Span – 32.0m

Roof Slope

Column Length

6.0m

6o 12

o

14.0m

Figure 8.4 Portal frames with varying spans, column length and roof slope



8.2 DESIGN OF PORTAL FRAMES FOR THE PARAMETER STUDY

This section describes the design of portal frames according to the DRAFT SANS 10160-1 (Basis

of Structural Design and Actions for Buildings and Industrial Structures - Basis of structural

design), the DRAFT SANS 10160-2 (Basis of Structural Design and Actions for Buildings and

Industrial Structures - Self-weight and imposed loads) and SANS 10162-1:2005 (The structural

use of steelwork). The structures are analysed using a second-order analysis in PROKON. The

following is included in this chapter:

• Identification of the critical member and the location of maximum forces.

• Identification of an I-section with sufficient capacity for the respective portal

frame as determined using SANS 10162-1:2005.

8.2.1 Results

(a) Location of maximum forces

Similar distributions of the axial forces, shear forces and bending moments are obtained for

the various portal frame configurations and the values of maximum forces are obtained at the

same locations for the selection of portal frames. In Figure 8.5 the locations of the maximum

forces and moments are illustrated by means of generic force and bending moment diagrams.

Figure 8.5 Distribution of forces - illustrating maximum forces



(b) Predicted failure of frames according to DRAFT SANS 10160-1 & 2: 2008 and

SANS 10162-1:2005

(i) Column member

The critical member for all portal frame configurations is identified as the right hand column

with maximum bending and axial forces occurring at the top of this member. Forces in the

right hand column are slightly larger than the left hand column due to the perturbation load

applied in the x-direction.

The selection of the I-section for the right hand column determines the type of section to be

used for the rest of the portal frame members.

The critical capacity of the right hand column member is in all cases determined by the

bending moment and the capacity of the section is determined by the plastic moment. The

critical formula as described in SANS 10162-1:2005 is:

1M

M

r

u ≤ and not

The reason for this is that a factor of 0.85 is applicable to class 1 and 2 sections allowing for a

limited redistribution of moments for the forming of plastic hinges.

In the case of fixed supports the maximum bending moment is also located at the top of the

right hand column and not at the supports of the portal frame.

(ii) Rafter member

The maximum forces in the rafter are also checked to ensure that the capacities are not

exceeded. However, as mentioned previously these forces are never found to be critical for

the selection of portal frames analysed in the current parameter study. It should be noted that

maximum values are taken at (1) or (2) as shown in Figure 8.6 although maximum forces in

reality occur at (3). However the introduction of the haunches makes this a non-critical

member.

1M

MU850

C

C

rx

uxx1

r

u ≤+.



Figure 8.6 Design values used

(c) Section Identification

The design procedure includes the checking of the axial compression, bending moment, the

interaction between bending and axial compression and shear force so that the capacity of the

member is sufficient. The sections identified are discussed in the following sections.

(i) Section 1 : Pinned Supports

A comprehensive result sheet of forces obtained for the portal fames in the selected

parameter study is shown in Appendix F. A summary of the identified sections are given in the

following sections. Table 8.1 shows the sections identified for portal frame configurations with

pinned supports and span lengths of 24.0m.

Table 8.1 Designated sections – span 24.0m, pinned supports

Span Support

Fixity

Column

Height

(m)

Roof

Slope

(o)

Section

Designation

3 254x146x37

6 254x146x37

9 254x146x37 6

12 254x146x37

3 254x146x37

6 254x146x37

9 254x146x37 10

12 254x146x37

3 254x146x31

6 254x146x31

9 254x146x31

24

Pin

ne

d

14

12 254x146x31



The choice in the section used in the standard configuration (frames with column lengths of

6.0m) as well as frames with column lengths of 10.0m are identical i.e 254x146x37 I-section

(Mr = 152.775kN.m). It should be mentioned that a 305x102x33 I-section (Mr = 151.15kN.m) is

sufficient for this purpose and is lighter in weight.

However, in the current South African industry the 254 x 146 x 37 I-section is more economical

and used due to the wider width of the flanges to accommodate bolted connections.

A 254 x 146 x 31 I-section (Mr = 124.425kN.m) is used for all selections of frame

configurations with a column length of 14.0m as a result of the smaller bending moment at the

top of the column. Again, the 305 x 102 x 29 I-section (Mr = 128.52kN.m) is lighter in weight

but in practice the 254 x 146 x 31 selection is more economical.

(ii) Section 2 : Fixed Supports

Table 8.2 shows the sections identified for portal frame configurations with fixed supports and

span length of 24.0m.

With the exception of portal frames with column lengths of 6.0m and 10.0m with a roof slope

of 12 degrees, the selected I-sections are similar to I-sections identified for portal frames with

pinned supports.

Table 8.2 Designated sections – span 24.0m, fixed supports

Span Support

Fixity

Column

Height

(m)

Roof

Slope

(o)

Section

Designation

6 254x146x37 6

12 254x146x31

6 254x146x37 10

12 254x146x31

6 254x146x31

24

Fix

ed

14 12 254x146x31



(iii) Varying spans

Table 8.3 show sections chosen for portal frame designs with varying span lengths.

Table 8.3 Designated sections – varying span lengths

Span

(m)

Support

Fixity

Column

Height

(m)

Roof

Slope

(o)

Section

Designation

6 305x165x41 6

12 254x146x43

6 254x146x43

28

14 12 254x146x43

6 305x165x54 6

12 305x165x54

6 356x171x45

32

Pin

ne

d

14 12 356x171x45

Sections with larger moment capacities are needed for the longer spans and it is not possible

to keep the choice of sections in the 254 I-section category as bending moment values in the

critical members exceed these capacities.

(d) Deflection of frames

Figure 8.7 illustrates the maximum horizontal and vertical deflections of the various frames

under the serviceability load combination.

Figure 8.7(a), Figure 8.7(b), Figure 8.7(c), illustrate portal frame configurations with pinned

supports for spans of 24.0m, portal frames with fixed supports and portal frames with varying

span lengths, respectively.

The serviceability requirements are plotted against the deflections obtained in PROKON.



Different Spans

Deflection vs Roof Slope

0

100

200

300

400

500

600

6 12 6 12 6 12 6 12 6 12 6 12

Roof slope (o)

De

fle

cti

on

(m

m)

Span 24.0m - Pinned Supports


0

50

100

150

200

250

300

350

400

450

500

3 6 9 12 3 6 9 12 3 6 9 12

Roof slope (o)

De

fle

cti

on

(m

m)

Span 24.0m - Fixed Supports


0

50

100

150

200

250

300

350

400

6 12 6 12 6 12

Roof s lope (o)

De

fle

ctio

n (

mm

)

Calculated Horizontal Deflection

Allowable Horizontal Deflection

Calculated Vertical Deflection

Allowable Vertical Deflection

Figure 8.7 Maximum vertical and horizontal deflection

Figure 8.7 indicates that deflections of designed configurations do not comply with

serviceability requirements.

6.0m 10.0m 14.0m

Span 24.0m Span 28.0m Span 32.0m

(b)

6.0m 10.0m 14.0m

(c)

(a)



8.3 CONCLUSIONS

8.3.1 Failure modes in columns

The plastic moment determines the capacity of the critical member in the portal frame. It is

also shown that the axial compression in the members is very small in comparison with the

axial capacity. This failure mode is consistent for all portal frames analysed in the parameter

study.

8.3.2 Deflection in frames

Design is governed by the serviceability requirements of the frame if designed according to

recommended serviceability design criteria in SANS 10162-1:2005. This means that the design

does not depend on the capacity of the member but the stiffness of the portal frame. In

Section 9.1.4 comparison is made between the load factors at which the serviceability

requirements (for vertical and horizontal deflection) of the portal frames in the parameter

study are exceeded. This load factor is compared to the serviceability load if the frame is

designed according to the ultimate limit state.

8.4 SUMMARY

• The benchmark portal frame is analysed using ABAQUS and PROKON.

• The failure of the frame is not governed by the elastic instability of the frame but

the plastic deformation of the frame.

• A selection of portal frames is chosen for the parameter study.

• The selection of portal frame configurations identified in the parameter study is

designed according to SANS 10162-1:2005.

• Serviceability requirements govern the design of the benchmark portal frame.

Analyses results and discussion for the parameter study 9.1


9 ANALYSIS RESULTS AND DISCUSSION OF THE

PARAMETER STUDY

The objective in this chapter is to determine if portal frames, with dimensions commonly used

in practice, are governed by the plastic deformation of the frame as proven for the benchmark

portal frame in Chapter 7. This is done by using the selection of portal frames identified and

designed in Chapter 8.

In order to understand the path followed in this chapter, a flow chart is shown in Figure 9.1.

ANGELINE

ABAQUS

Figure 9.1 Flow chart of procedure

The flow chart is explained in more detail:

(a ) Evaluating the behaviour of the frame using a second-order elastic-perfectly

plastic analysis

The behaviour of the frames is evaluated by identifying the location and load factor at which

the first cross-section in the portal frame reaches yielding. Beyond this point behaviour cannot

be considered purely geometric and hence failure is not defined by the purely geometric

instability of the frame.



However, it is also necessary to evaluate the behaviour of the frame further to determine how

the failure compares to what is predicted in Section 5.5 or if a combination of buckling and

plastic deformation occurs.

(b) The load-displacement path for the frame is plotted at selected nodes

The load-displacement path is used as the failure criterion for the frames. The failure is

determined by the point where the maximum load is reached on the load path.

(c) Analysis of frames using ANGELINE beyond the maximum load factor obtained in

ABAQUS.

Analyses using ANGELINE will include the evaluation of the frame beyond the maximum load

factor achieved in the second-order elastic-plastic analysis.

(d) Analysis of portal frames in order to determine at which point the serviceability

requirements are exceeded.

This analysis is done to show that portal frames are not governed by the structural capacity of

the frames but the stiffness i.e. serviceability requirements of the frame.

It should once again be noted that analysis done in ABAQUS includes a second-order elastic

perfectly-plastic analysis and analysis using ANGELINE includes a geometric nonlinear analysis.

9.1 RESULTS

9.1.1 Evaluating the behaviour of the frame using ABAQUS

The portal frames selected for the parameter study in Chapter 8 are analysed by means of a

second-order elastic perfectly-plastic with the material model as shown in Figure 9.2.



Figure 9.2 Material model

Table 9.1, Table 9.2 and Table 9.3 show the load factor and location of yielding of the cross-

sections for each portal frame configuration. The maximum load factor of the portal frame is

also shown.

The discussion of these results follows and is subdivided into the selection of portal frame

configurations with pinned supports and span lengths of 24.0m (section 9.1.1.a), portal frame

configurations with fixed supports and span lengths of 24.0m (section 9.1.1.b) and portal

frame configurations with pinned supports and varying span lengths (section 9.1.1.c).

(a) Pinned Supports – Span 24.0m

(i) Column Length – 6.0m

Portal frames in this range reach yielding of the cross-section at the top of the right hand

column. Subsequently, the left hand column starts yielding but unloading of the left hand

column occurs as the load factor increases. Unloading of the left hand column is due to the

frame deflecting in the positive x-direction.

The rafter yields shortly after that and the maximum load to be carried by the system is

reached. With the exception of the portal frame configuration with a roof slope of 3 degrees,

yielding does not occur at the ridge, but 2.0m to the left of the ridge.



Table 9.1(a) Yielding values for frames – span 24.0m - pinned supports – 6.0m

(a) This indicates if the stresses reached in the column are at the top and bottom fibre or if only partial or no

yielding occurs.

(b) This indicates if (i) the yielding of the rafter happens on the increasing load-deflection path before the

maximum load factor is reached, (ii) or yielding of the rafter happens at the maximum load factor on the

load-deflection path or (iii) if the rafter yields only after the maximum load factor is reached and the load

factor on the load-deflection path shows a decrease. This is indicated in the table by (i) Increasing, (ii)

Maximum or (iii) Decreasing, respectively.

(ii) Column Length - 10.0m

Table 9.1(b) Yielding values frames – span 24.0m - pinned supports – column 10.0m


yielding occurs.






In portal frames with column lengths of 10.0m, only partial or no yielding occurs in the

respective column lengths as indicated in Table 9.1.b.

Yielding of the rafter does not occur before the maximum load factor is reached.

Column

height

(m)

Roof

slope

(o)

Right

column

yielding

(LF)

Right

column

yielding (a)

Left

column

yielding

(LF)

Left

column

yielding (a)

Rafter

yielding

(LF)

Max

load

factor

(LF)

Rafter

yielding (b)

Rafter

yielding

location

6 3 0.7320 Full 0.7550 Full 0.8540 0.8540 Maximum Top

6 6 0.7626 Full 0.7734 Full 0.8769 0.8769 Maximum 2.0m to left



Column

height

(m)

Roof

slope

(o)

Right

column

yielding

(LF)

Right

column

yielding (a)

Left

column

yielding

(LF)

Left

column

yielding (a)

Rafter

yielding

(LF)

Max

load

factor

(LF)

Rafter

yielding (b)

Rafter

yielding

location

10 3 0.7884 Full 0.7957 Full 0.83574 0.83655 Decreasing Top

10 6 0.7987 Full 0.8074 Partial 0.8346 0.8426 Decreasing Top

10 9 0.8046 Full 0.8131 Partial 0.8310 0.8497 Decreasing 2.0m to left

10 12 0.8170 Full 0.8250 Partial 0.8247 0.85673 Decreasing 2.0m to left



(iii) Column Length - 14.0m

Table 9.1(c) Yielding values for frames – span 24.0m - pinned supports – column length 14.0m


yielding occurs.






In portal frames with column lengths of 14.0m, no yielding of the left column occurs.

However, similar behaviour is observed to that of column lengths of 10.0m and only the

yielding of the right column is observed before the maximum load factor is reached.

(b) Fixed Supports – Span 24.0m

Table 9.2 shows the behaviour of portal frames with fixed supports.

Table 9.2 Yielding values for frames – span 24.0m – fixed supports


yielding occurs.






Column

height

(m)

Roof

slope

(o)

Right

column

yielding

(LF)

Left

column

yielding

(LF)

Left

column

yielding (a)

Rafter

yielding

(LF)

Max

load

factor

(LF)

Rafter

yielding (b)

Rafter

yielding

location

14 3 0.6722 NA None 0.6897 0.6933 Decreasing Top

14 6 0.6753 NA None 0.6744 0.6965 Decreasing Top

14 9 0.6759 NA None 0.6326 0.6955 Decreasing 2.0m to left

14 12 0.6780 NA None 0.6065 0.6931 Decreasing 2.0m to left

Span

(m)

Column

height

(m)

Roof

slope

(o)

Right

column

yielding

(LF)

Right

column

yielding (a)

Left

column

yielding

(LF)

Left

column

yielding (a)

Rafter

yielding

(LF)

Max

load

factor

(LF)

Rafter yielding (b)

Rafter

yielding

location

24 6 6 0.7831 Full 0.7831 Full 0.9823 0.9822 Maximum 1.0m

to left

24 6 12 0.7242 Full 0.7242 Full 0.9260 0.9260 Maximum Top

24 10 6 0.7891 Full 0.7942 Full 0.9475 0.9516 Increasing Top






Similar to portal frame configurations with pinned supports, the cross-section at the top of the

right hand column starts yielding followed by the yielding at the top of the left hand column.

Subsequently, the ridge of the rafter yields.

The location of yielding is at the top of the rafter, with the exception of portal frame

configuration with a column length of 6.0m and roof slope of 3 degrees. In this particular

frame the rafter yields one meter to the left of the ridge of the roof.

Frames configurations in this selection reach a maximum load shortly after the onset of

yielding in the rafter and a decrease in the load path is observed.

(c) Varying span lengths – 24.0m, 28.0m and 32.0m – pinned supports

Table 9.3 shows the sequence of behaviour in this category of frames.

Table 9.3 Yielding values for frames – varying length spans


yielding occurs.






Span

(m)

Column

height

(m)

Roof

slope

(o)

Right

column

yielding

(LF)

Right

column

yielding (a)

Left

column

yielding

(LF)

Left

column

yielding (a)

Rafter

yielding

(LF)

Max

load

factor

(LF)

Rafter

yielding (b)

Rafter

yielding

location

28 6 6 0.7123 Full 0.7243 Full 0.84215 0.8421 Maximum 1.0m to

right

28 6 12 0.7055 Full 0.7163 Full 0.773 0.7730 Maximum 1.0m to

right

28 14 6 0.7076 Full NA None 0.6720 0.7293 Decreasing 1.0m to

right

28 14 12 0.7149 Full NA None 0.6359 0.7325 Decreasing 1.0m to

right

32 6 6 0.7398 Full 0.7513 Full 0.8668 0.8668 Maximum Top

32 6 12 0.8182 Full 0.8182 Partial 0.9133 0.9135 Decreasing 2.0m to

right


right


right



Similar sequence of behaviour is observed compared to frames with a span length of 24.0m.

The yielding of the left hand column is also observed only for frames with column lengths of

6.0m. The maximum load factor is reached shortly after the rafter yields for frame

configurations with 6.0m column lengths and span of 28.0m.

However, in the portal frame configuration with a span of 32.0m the yielding of the rafter is

only observed in the portal fame configuration with a column length of 6.0m and roof slope of

six degrees.

Frames with column lengths of 10.0m and 14.0m with varying span length, similarly exhibit the

same absence in the yielding of the rafter on the increasing path as in the case of portal frame

configurations with span lengths of 24.0m

9.1.2 Displacement behaviour of portal frames analysed

Appendix G1, Appendix G2 and Appendix G3 show the load displacement paths of the various

portal frames analysed. δh pertains to the horizontal deflection at the node under

consideration and δv indicates the vertical deflection of the node under consideration. The

deflection of the allocated nodes is taken at the top node of the left hand column, at the ridge,

and the top node of the right hand column.

(a) Pinned Supports – span 24.0m

Appendix G1 contains the graphical representation of the displacement behaviour for portal

frames with pinned supports and span of 24.0m.

The load displacement of the left hand column for frames under consideration indicates a

deflection in the negative x-direction followed by a deflection in the positive x-direction as

soon as the structure undergoes sway. A maximum load factor is reached and the load path

decreases after this point. The absolute value of the negative deflection of the left hand

column increases as the roof slope increases in the respective portal frames. The behaviour of

the columns can be attributed to the thrust of the rafter on the column which pushes the

columns outward. The rapid change in deflection occurs shortly after the left and the right



hand columns have reached yielding point. The perturbation load applied at the top of the left

hand column then encourages side sway to the right. The rapid deflection to the right is

followed by failure of the frame as the maximum load factor is reached.

The load deflection path of these respective portal frame configurations indicate that after the

yielding of the right hand column, only a very small increase in the load path is observed and a

deflection in the positive direction of the portal frame occurs after which load path decreases.

This is most visible in column lengths of 14.0m.

(b) Fixed supports

Appendix G2 shows the load deflection path for frame configurations with fixed supports. As in

the case of the pinned supports there is a general outward deflection of the columns with the

increase in roof slope influencing the increasing outward deflection of the columns. Shortly

after, the left and right hand column yields and the load path increases and the deflection of

the left hand column increase slightly in the negative direction. A maximum load is reached

shortly after the ridge has yielded.

(c) Varying spans – pinned supports

The load-deflection graphs of the selection of portal frames which include varying span lengths

are shown in Appendix G3. These displacement graphs indicate that portal frames with column

lengths of 6.0m for varying spans show a further increase in the load path after the yielding of

the first cross-section. The slope of this increase however, is much smaller in portal frames

with 12 degrees. Portal frames with column lengths of 10.0m and 14.0m show a small

increase in the load path after the yielding of the column.

9.1.3 Evaluating the possibility of geometric instability

For the identification of the possibility of the singular point the load-displacement history is

observed at the node of the top left hand column and at the ridge of the rafter for portal

frames selected for the parameter study. The analyses include incrementing the load factor up

to a value of 1.0. An initial load factor of 0.1 is used. A selection of the load-history

displacement graphs are shown in Appendix H. This selection includes the highlighted portal

frame configurations indicated in Table 9.4, Table 9.5 and Table 9.6. Each page in Appendix H



includes the displacement at the ridge and at the top node of the left hand column for the

respective portal frame. y1 depicts the horizontal displacement and y2 the vertical

displacement of the node. The load factor at the following points is also identified on these

graphs:

• The load factor at which the portal frame is designed for i.e ultimate limit state

(denoted as P1).

• The serviceability load for that frame (denoted as P2).

• The load factors at which the serviceability requirements, which limit the

vertical and horizontal deflection, are exceeded (denoted as P3 and P4).

Table 9.4, 9.5 and 9.6 show the numerical results of the nonlinear analysis. These tables

include the numerical results of the displacements at the allocated nodes if a load factor of 1.0

is applied (See (c) and (d) in Table 9.4.

If the horizontal displacement at the top node of the left hand column shows a turning point in

the deflection behaviour i.e from a negative to a positive displacement, this is illustrated in the

tables by indicating the numerical value of the displacement and the load factor at which this

turning point occurs. See (a) and (b) of Table 9.4.

(a) Pinned supports and spans of 24.0m

In Table 9.4(a) portal frames with column lengths of 6.0m show a negative horizontal

displacement at the top node of the left hand column. The absolute value of the negative

displacement increases as the roof slope increases.

Similar negative displacement behaviour is observed in portal frame configurations which

comprise of 10.0m and 14.0m columns, but the absolute value of the outward thrust

decreases as the column length increases.

The side sway of the portal frames becomes more visible as the column length of the portal

frame increases.



Table 9.4(a) Deflection at selected nodes – pinned supports

Top Left Column

Maximum

Negative

Displacement

(mm)

(a)

Load Factor at

Maximum Negative

Displacement

(b)

At Load Factor 1

(mm)

(c)

Column

Length

(m)

Roof

Slope

(o)

Horizontal Deflection (mm)

Vertical

deflection

(mm)

3 -8.4 0.8 -7.7 -1.3

6 -32.5 No turning point -32.5 -1.5

9 -51.5 No turning point -51.5 -1.5 6.0

12 -68.0 No turning point -68.0 -1.5

3 -2.2 0.3 14.6 -4.4

6 -20.6 0.8 -19.0 -4.4

9 -49.5 No turning point -49.5 -4.5 10.0

12 -77.1 No turning point -77.0 -4.4

3 0.0 0.0 293.4 -1.5

6 -6.2 0.3 245.6 -1.5

9 -20.3 0.4 221.4 -1.5 14.0

12 -40.2 0.5 168.8 -1.5

(a) Indicates the maximum negative horizontal displacement at the top node of the left column.

(b) Indicates the load factor at which the turning point of the left column is reached and the deflection at

the top of the left column displaces in the positive direction.

If no turning point is observed in the load path up to a load factor of 1 this is indicated by “no turning

point”. This implies that the corresponding value given in column (a) is not the maximum negative value

before the turning point is reached.

(c) Indicates the values at a load factor of 1.0.

Table 9.4 (a) show the vertical displacement in the top of the left hand column for portal frame

configurations do not exceed vertical deflections of 4.5mm.

In the case of portal frames with column lengths of 6.0m the horizontal deflection of the frame

is still negative with respect to its original position.

As the column length increases the positive displacement becomes more visible and in portal

frames with column lengths of 14.0m the displacement at the top node of the left column is

positive for all roof slopes.



Table 9.4(b) Deflection at selected nodes –ridge

Ridge of portal frame (At load factor 1)

(d)

Column

Length

(m)

Roof

Slope

(o)

Horizontal

Deflection

(mm)

Vertical

Deflection

(mm)

3 5.9 528.3

6 5.9 472.6

9 6.0 427.1 6.0

12 6.1 385.3

3 25.0 685.3

6 25.4 650.3

9 27.0 620.7 10.0

12 30.6 585.3

3 281.0 1053.9

6 300 1024.9

9 323.0 1000.6 14.0

12 364.0 987.7

Table 9.4 (b) shows that the horizontal displacement at the ridge node is small for portal

frames comprising of 6.0m column lengths.

This shows that the perturbation load has very little effect. However, as the length of the

column in the portal frames increase the horizontal displacement becomes more evident.

No asymptotic behaviour of the deflection is observed and hence, no singular points are found

for these configurations of frames.

However, at a load factor of 1.0 the horizontal displacements at the top left column and the

vertical displacement at the ridge becomes very large as the column length is increased.

This results in values beyond practical design.

(b) Fixed Supports

Results in Table 9.5 include portal frames with fixed support conditions.



Table 9.5 Deflection at selected nodes –fixed supports

Top Left Column Ridge of portal frame

Horizontal

Deflection

(mm)

Vertical

Deflection

(mm)

Horizontal

Deflection

(mm)

Vertical

Deflection

(mm)

Column

Length

(m)

Roof Slope

(o)

At Load Factor 1

6 -30.9 -7.1 1.0 -377.7 6.0

12 -63.4 -1.0 1.3 -337.3

6 -37.4 -2.0 4.2 -532.2 10.0

12 -97.6 -2.9 5.5 -564.9

6 -53.7 -7.0 7.1 -843.4 14.0

12 -115.2 -6.5 15.1 -752.4


(b) Indicates the load factor at which the turning point of the left column is reached and the deflection at the

top of the left column displaces in the positive direction.





No turning point of the left hand column is observed up to a load factor of 1.0.

An increase in the vertical deflection is observed as the column length increases. However for

each category of column lengths the vertical deflection decreases as the roof slope increases.



The deflections at a load factor of 1.0 become large as the column length is increased,

especially in the case of the vertical deflections. This results in values beyond practical design.

(c) Varying span lengths and pinned supports

Table 9.6(a) show the deflection behaviour at the top of the left column and Table 9.6(b) show

the deflection at the ridge of the roof for portal frames with varying spans and pinned

supports.



Table 9.6(a) Deflection at selected nodes – varying spans

Top Left Column

Span

(m)

Maximum

Negative

Displacement

(mm)

(a)

Load Factor at

Maximum Negative

Displacement

(b)

At Load Factor 1

(mm)

(c)

Column

Length

(m)

Roof

Slope

(o)

Horizontal Deflection (mm)

Vertical

deflection

(mm)

6 -38.0 1.0 -38.0 -1.2 6.0

12 -96.0 1.0 -96.0 -2.2

6 13.1 0.4 81.5 -13.8

28.0

14.0 12 70.0 0.7 22.1 -13.3

6 -45.4 1.0 -45.4 -1.2 6.0

12 -85.1 1.0 -85.1 -1.5

6 -23.6 0.7 -12.8 -7.6

32.0

14.0 12 -103.3 1.0 -103.3 -7.7


(b) Indicates the load factor at which the turning point of the left column is reached and the deflection at the

top of the left column displaces in the positive direction.





Table 9.6(b) Deflection at selected nodes – varying spans - pinned supports-ridge

Ridge of portal frame

(d)

Column

Length

(m)

Column

Length

(m)

Roof

Slope

(o)

Horizontal

Deflection

(mm)

Vertical

Deflection

(mm)

3 5.1 -527.0 6.0

6 7.4 -555.0

9 146.3 -1181.0 28.0

14.0 12 157.9 -1105.0

3 4.7 620.1 6.0

6 4.8 -467.1

9 5.5 -989.5 32.0

14.0 12 5.7 906.7

The turning point of the left hand column is more visible in portal frame configurations with

column lengths of 14.0m for spans of 28.0m and 32.0m. The vertical deflections at the ridge

are higher in portal frames with spans of 28.0m compared to portal frames with 24.0m spans.

However, a decrease in the deflection in frame configurations of column lengths with 14.0m is



smaller than other frames. This is as a result of the different selection in section. The

horizontal deflection at the ridge is small for most portal frames in this category.



9.1.4 Analysis of the nonlinear behaviour – serviceability

Table 9.7 (a), Table 9.7 (b), Table 9.7 (c) show the results obtained for the load factor at which

serviceability is exceeded.

The tables include the limiting deflection values as set out in SANS 10162-1:2005 shown in

column 5 and 6. The accompanying load factors at which these deflections are reached for the

analysis using ANGELINE, are given for the respective portal frames in column 7 and 8. The

portal frames in the parameter study are designed for the ultimate limit state. Column 9 shows

the load factor of the load applied to the portal frame under serviceability conditions

(according to DRAFT SANS 10160-1).

Table 9.7(a) Load factor at serviceability of portal frames – pinned supports – span 24.0m

Limiting deflections

according to SANS

10162-1:2005

ANGELINE

1 2 3 4 5 6 7 8 9

Span Support Column

Height

Roof

Slope

Horizontal

deflection

(mm)

Vertical

deflection

(mm)

Load

Factor at

Horizontal

Deflection

Limit

Load

Factor at

Vertical

Deflection

Limit

Load factor

at

serviceability

24 Hinged 6 3 20.0 133.3 0.342 0.271 0.470

24 Hinged 6 6 20.0 133.3 0.339 0.297 0.471

24 Hinged 6 9 20.0 133.3 0.265 0.329 0.472

24 Hinged 6 12 20.0 133.3 0.183 0.334 0.474

24 Hinged 10 3 33.3 133.3 0.265 0.179 0.470

24 Hinged 10 6 33.3 133.3 0.219 0.220 0.471

24 Hinged 10 9 33.3 133.3 0.192 0.231 0.472

24 Hinged 10 12 33.3 133.3 0.173 0.244 0.474

24 Hinged 14 3 46.7 133.3 0.177 0.140 0.457

24 Hinged 14 6 46.7 133.3 0.151 0.143 0.458

24 Hinged 14 9 46.7 133.3 0.123 0.150 0.459

24 Hinged 14 12 46.7 133.3 0.118 0.154 0.461



Table 9.7(b) Load factor at serviceability of respective portal frames – fixed supports – span

24.0m


according to SANS

10162-1:2005

ANGELINE

1 2 3 4 5 6 7 8 9

Span Support Column

Height

Roof

Slope

Horizontal

deflection

(mm)

Vertical

deflection

(mm)

Load

Factor at

Horizontal

Deflection

Limit

Load

Factor at

Vertical

Deflection

Limit

Load factor

at

serviceability

24 Fixed 6 6 20.0 133.3 0.437 0.379 0.471

24 Fixed 6 12 20.0 133.3 0.291 0.423 0.460

24 Fixed 10 6 33.3 133.3 0.350 0.267 0.471

24 Fixed 10 12 33.3 133.3 0.233 0.285 0.460

24 Fixed 14 6 46.7 133.3 0.261 0.175 0.458

24 Fixed 14 12 46.7 133.3 0.193 0.195 0.460

Table 9.7(c) Load factor at serviceability of portal frames – pinned supports – varying spans


according to SANS

10162-1:2005

ANGELINE

1 2 3 4 5 6 7 8 9

Span Support Column

Height

Roof

Slope

Horizontal

deflection

(mm)

Vertical

deflection

(mm)

Load

Factor at

Horizontal

Deflection

Limit

Load

Factor at

Vertical

Deflection

Limit

Load factor

at

serviceability

28 Hinged 6 6 20.0 155.6 0.293 0.315 0.479

28 Hinged 6 12 20.0 155.6 0.160 0.302 0.487

28 Hinged 14 6 46.7 155.6 0.128 0.147 0.484

28 Hinged 14 12 46.7 155.6 0.123 0.157 0.487

32 Hinged 6 6 20.0 177.8 0.214 0.313 0.507

32 Hinged 6 12 20.0 177.8 0.157 0.210 0.511

32 Hinged 14 6 46.7 177.8 0.195 0.192 0.488

32 Hinged 14 12 46.7 177.8 0.196 0.401 0.494

It is evident from the results obtained in ANGELINE that the serviceability requirements is

exceeded long before the ultimate limit state of the structure is reached.



9.2 DISCUSSION ON RESULTS

The design of portal frames is governed by the serviceability requirements of the portal frame.

In the following sections the failure of the frames and the deflection behaviour of the portal

frames are discussed. Subsequently, it is shown that the serviceability requirements are the

governing design criterion.

9.2.1 Failure of frames

(a) Pinned supports

With the exception of the portal frame configurations with a span length of 32.0m and roof

slope of 12 degrees, portal frames with pinned supports (with varying span lengths) and 6.0m

column lengths exhibit the combined sway behaviour as explained in Section 5.5.

Portal frames with column lengths of 10.0m and 14.0m (pinned supports for varying spans) do

not exhibit the behaviour as predicted in Section 5.5 and after the first cross-section has

yielded in the column, the maximum load factor is reached in the portal frame before the

yielding of the rafter occurs.

The reason for this is that the buckling behaviour of the more slender columns are greatly

influenced by the effect of the plastic deformations, and the final failure of these portal frames

is a combination of the plastic deformation and the side sway due to the buckling columns of

the portal frame.

Figure 9.3 indicates the difference between the maximum load factor reached and the load

factor at yielding of the right hand column.

The percentage difference is calculated by using the maximum load factor as the reference

value. i,e:

100xFactorLoadMax

YieldingColumnRightFactorLoadFactorLoadMaxDifferencePercentage

−=



Percentage Difference

0.00

5.00

10.00

15.00

20.00

25.00

3 6 9 12 3 6 9 12 3 6 9 12 6 12 6 12 6 12 6 12 6 12 6 12 6 12

Roof Slope (o)

Pe

rcen

tag

e D

iffe

ren

ce

Figure 9.3 Comparison of percentage difference between right hand column and max load

factor

Portal frame configurations with pinned and fixed supports show a general decrease between

the difference in the maximum load factor and the yielding of the cross-section at the top of

the right column, as the column length increases.

(b) Fixed supports

For frames with fixed supports, three cross-sections in the frame yield before the ultimate

collapse of the portal frame occurs. This occurrence is as a result of the frame not failing due

to global failure, but failure and descend of the load path is defined by a localised failure as the

roof member is unable to carry the vertical load any further.

(c) Comparison between analysis – Second-order elastic-plastic versus second-order

elastic analysis

In this section the failure of the frame is compared to that of the portal frames designed

according to SANS 10162-1:2005. According to SANS 10162-1:2005 design of portal frames is

governed by the maximum bending moment at the top of the right hand column.

6.0m 10.0m 14.0m 6.0m 10.0m 14.0m 6.0m 14.0m 6.0m 14.0m

Pinned – Span 24.0m Fixed – Span 24.0m Span 28.0m

Pinned

Span 32.0m

Pinned



The design of the frame is not allowed to exceed the plastic moment of the section. The

applied load factor which causes the critical section to become plastic as calculated according

to SANS 10162-1:2005 is compared to that of the yielding in the right hand column as

predicted by analyses and multiplied by its form factor using ABAQUS.

Current analysis done in PROKON include the prescribed load according to SANS 10162-1,2 and

3, and the chosen sections are not necessarily used to its full capacity for the designed portal

frame configurations.

To compare these values the load factor applied in PROKON is increased until the plastic

moment for the respective portal frame is reached in the top of the right hand column as

calculated using SANS 10162-1:2005 (i.e. Mp = Zpl x Fy). The material factor φ is excluded in

this calculation. This is indicated by Mp in Figure 9.4.

The values of the right column yielding in ABAQUS are multiplied with its corresponding form

factor in order to obtain the load factor at which the plastic moment is reached. These values

are indicated in yellow in Figure 9.4.

Load Factors of right column

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

3 6 9 12 3 6 9 12 3 6 9 12 6 12 6 12 6 12 6 12 6 12 6 12 6 12

Roof slope(0)

Loa

d f

act

or

Mp

ABAQUS

Figure 9.4 Behaviour compared to ABAQUS results

6.0m 10.0m 14.0m 6.0m 10.0m 14.0m 6.0m 14.0m 6.0m 14.0m

Pinned Fixed Varying Span

28.0m 32.0m



These values indicate that results obtained from SANS 10162-1:2005 prove to be conservative

as the values of Mp (Plastic moment of I-section) in all cases are smaller than the

corresponding ABAQUS values that are multiplied with the form factor.

9.2.2 Deflection of portal frames elastic perfectly-plastic analysis

The deflection of frames analysed using an elastic perfectly-plastic analysis shows that none of

the portal frames reach a maximum load factor higher than 1. 0.

9.2.3 Identification of the possibility of geometric instability

(a) Deflection of portal frames elastic second-order analyses

(i) Vertical deflection at the ridge

Results show deflections of fixed supports are more favourable than pinned supports and that

the increase in the span generally shows an increase in the vertical deflection.

(ii) Horizontal deflection in the columns

A higher horizontal outward thrust is observed for frames in column lengths of 6.0m and lower

horizontal displacement in column lengths of 10.0m and 14.0m for pinned supports compared

to portal frame configurations with fixed supports.

It should be noted that the frame works as a unit and the column has an influence on the

rafter and the rafter also has an influence on the column behaviour. This effect on the outward

thrust can be answered by observing the vertical displacement of the ridge. Pinned supports

show larger deflection than those of fixed supports due to the fixed supports adding stiffness

to the frame.

However, although larger vertical deflections are observed in the case of frames with column

lengths of 10.0m and 14.0m in pinned supports, the increased slenderness of the columns

influence the behaviour of frames with pinned supports.



(b) Influence of the perturbation load

In most portal frame configurations the perturbation load has a small influence on the frames

and the displacement of the top of the column is largely governed by the downward vertical

displacement.

This can be seen by the little effect the perturbation load has on the horizontal displacement

at the ridge of the roof. The influence does become more evident with pinned supports of

column lengths of 14.0m in the case of frame configurations with pinned supports and spans

of 24.0m and 28.0m.

Less influence is observed in portal frames configurations with fixed supports and are more

favourable in terms of the perturbation load and hence imperfections.

(c) Identification of singular points

Investigation of portal frames yields no singular points in any of the portal frame

configurations. This can be seen by the absence of the asymptotic behaviour of the

displacement for selected nodes. Elastic instability is not the critical failure mode.

9.2.4 Serviceability requirements of portal frames

It is evident from this comparison that none of the frames meet serviceability requirements as

set out in SANS 10162-1:2005 and design is governed by this requirement. Figure 9.5 show the

graphical results obtained from Table 9.7.

This indicates the applied load factor to the load factor at which serviceability requirements

are exceeded.

The column length and supports conditions are shown at the top of the graph. A and B

denotes span lengths of 28.0m and 32.0m, respectively.



Load Factor at Serviceability Limit

0

0.1

0.2

0.3

0.4

0.5

0.6

3 6 9 12 3 6 9 12 3 6 9 12 6 12 6 12 6 12 12 6 12 6 12 6 12 6 12Roof Slope(

o)

Loa

d F

ac

tor

(10

kN

)

Applied load

Serviceabil ity load factor - Vertical Deflection

Serviceabilty load factor - Horizontal Deflection

Figure 9.5 Comparison of the applied load factor to load factor at serviceability

requirements

The observation is made that in roof slopes of 3 and 6 degrees the critical serviceability

requirement is the vertical deflection criteria and in higher roof slopes the critical deflection

criteria is the horizontal deflection.

9.3 CONCLUSIONS

(a) Failure of frames

The critical behaviour in the failure of the portal frame is as a result of the plastic deformation

of the cross-section. In all cases of portal frames analysed in this research the first cross-

section to yield is in the right-hand column.

Portal frame configurations with 6.0m columns and hinged supports exhibit similar behaviour

as predicted in Section 5.5. In portal frames with column lengths of 10.0m and 14.0m, the

rafter does not yield before the maximum load is reached.

This indicates a combination of buckling and plastic deformation. However, to identify this

behaviour correctly it is necessary to acquire the fundamental theory of the implementation of

this software.

6.0m 10.0m 14.0m 6.0m 10.0m 14.0m 6.0m 14.0m 6.0m 14.0m

A B



In the case of fixed supports failure is governed by local collapse of the roof in the respective

portal frames.

(b) Elastic perfectly-plastic analysis

The results of analysis of portal frames designed according to the procedure as set out in SANS

10162-1:2005, which prescribes a second-order elastic analysis, compare well with portal

results obtained in the second-order elastic plastic analysis using ABAQUS.

(c) Evaluation of the nonlinear analyses

In Chapter 3 the conclusion is made that for portal frames with practical dimensions, no

singular point in the geometric nonlinear analysis could be found (In the practical range of

studies).

It is also further shown that the near singular point is found by means of the asymptotic

behaviour of the displacement of the nodes in the portal frame.

The evaluation of the portal frames included in the parameter study and analysed using

ANGELINE confirms these conclusions.

(d) Serviceability requirements

The nonlinear analyses are evaluated by means of results obtained. Frames do not meet

serviceability requirements. It should be noted that SANS 10162-1:2005 prescribes the

checking of serviceability requirements as normative.

However the prescribed limitations are informative as set in Annex D of SANS 10162-1:2005

and not comprehensive.

This shows that the design of portal frames is not governed by the capacity of the members

but the stiffness of the portal frames.



9.4 SUMMARY

• Portal frame failure is governed by plastic deformation of the frame.

• Elastic instability does not occur in any of the selected frames.

• The selected portal frames for the parameter study do not meet serviceability

requirements as set out in SANS 10162-1:2005 and shows that design is governed

by this requirement.

Conclusions and recommendations 10.1


10 CONCLUSIONS AND RECOMMENDATIONS

10.1 INTRODUCTION

This chapter includes the conclusions derived from the current research. Recommendations

are made according to these conclusions. These conclusions are subdivided into two sections:

• Failure of portal frames

• Design considerations

The main conclusions under each of these sections are summarised and discussed.

10.2 CONCLUSIONS AND RECOMMENDATIONS

10.2.1 Structural instability

(a) Conclusion: In-plane elastic instability is not a concern in the structural

failure of portal frames with practical dimensions.

Reference: Chapter 3 and Chapter 9.

Recommendation: This is proven for in-plane behaviour of portal frames. It is

quite possible to design a portal frame so that in-plane behaviour governs by

means of sufficient lateral support.

However, this might not always be the most economical approach to design and it

is necessary to do the same evaluation for portal frames including out-of-plane

effects. A further development is proposed that includes out-of-plane effects.

(b) Conclusion: Portal frames failure is governed by the plastic deformation of

the members.




Recommendations:

The plastic deformation of portal frames must be understood as this governs the

behaviour of theses type of frames. In commercial software packages it is difficult

to obtain the necessary information as the theory of the implementation is not

available. This means that the influence of plastic deformation behaviour of the

portal frames cannot be fully understood.

It is necessary that a software programme like ANGELINE is developed which

includes the development of plastic deformation in its formulation. This enables

the researcher to have full knowledge of the implemented theory.

This will enable the researcher not only to understand the development of plastic

deformation but also to understand the influence of the plastic deformation on

the buckling behaviour of the frame.

10.2.2 Design considerations

(a) Conclusion: The economy in using materials is an ethical obligation to

future generations. The members of portal frames must be designed accordingly

and the optimum amount of steel used.

Reference: Chapter 8.

Recommendation: The design of portal frames with tapered sections is proposed.

See Figure 10.1.

Figure 10.1 Portal frame with tapered members



The larger cross-section at critical locations in the member will resist the maximum

bending moment at these points. This type of design consideration is used in

other countries but it is not a general practice approach in South Africa. However,

by means of prefabrication of the single portal frame bays of this type, labour,

material cost and time can be saved on the current design practice.

The problem exists that SANS 10162-1:2005 does not make allowance for this type

of design and it is necessary to use a performance based design approach. This

again brings the problem back to the necessity of a software programme that can

analyse the behaviour correctly as discussed in the previous section.

(b) Conclusion: The portal frames in this research are governed by

serviceability requirements.


Recommendation:

All portal frames designed in this section are governed by the serviceability

requirements as set out in Annex D of SANS 10162-1:2005.

It is recommended that research should include an investigation into more

practical guidelines with regard to serviceability requirements. Some of the factors

that should be considered are:

• One example of such a practical guideline is illustrated by the following

example: The effect of the vertical displacement on the ability for water to

run-off of the roof.

• The effect of displacements on the sheeting used for the portal frames

• The effect of displacements on the internal structures of the portal frame

It should be noted that these recommendations are made under the consideration of in-plane

behaviour of portal frames and a vertical gravitational load pattern.

References 11.1


11 REFERENCES

11.1 BOOKS

1. Baker, Heyman (1980). Plastic Design of Frames. Cambridge University Press,

Cambridge.

2. Chen, Atsuta (1956). Theory of Beam-Columns, Volume 1: In-Plane Behaviour

and Design. McGraw Hill, New York.

3. Chen, Liew, Goto (1996). Stability Design of Semi-Rigid Frames. WILEY IEEE,

England.

4. Galambos, T. V. (1988). Guide to Stability Design Criteria for Metal Structures.

Canada: John Wiley and Sons, New York.

Chapter 1: Stability Theory

Chapter 2: Centrally Loaded Column

Chapter 16: Frame Stability

5. Galishinikova, Pahl and Dunaiski. Geometrically Nonlinear Analysis of Plane

Trusses and Frames. To be published.

Chapter 1 : State of the art in nonlinear structural analysis

Chapter 2 : Nonlinear behaviour of plane trusses and frames

Chapter 4 : Plane Frames

Chapter 7 : Stability Analysis

6. Mahachi, J (2004). Design of Structural Steelwork to SANS 10162-1:2005. CSIR,

Pretoria.

7. Timoshenko, Gere (1961). Theory of Elastic Stability. McGraw-Hill, New York.

8. Trahair,N.Bradford,M (2001). The Behaviour and Design of Steel Structures to

BS5950. Taylor and Francis.

References 11.2


11.2 PUBLICATIONS

9. BBCSA Committee. The Collapse Method of design – Being the Application of the

Plastic theory to the Design of Mild Steel Beams and Rigid Frames. British

Constructional Steelwork Association, No 5, 1957.

10. Johnson, Morris, Randall, Thompson. Plastic Design. British Constructional

Steelwork Association, No 28, 1965.

11. Chan, Huang, Fang. Advanced Analysis of Imperfect Portal Frames with Semirigid

Base Connections. Journal of Engineering Mechanics, Volume 131, No 6 (633-

640), 2005.

12. Davies, J.M. Inplane stability in portal frames. The Structural Engineer, Volume

68, No 8 (141-147), 1990.

13. Lim, King, Rathbone, Davies, Edmondson. Eurocode 3 and the In-plane Stability of

Portal Frames, The Structural Engineer, Volume 83, No 21 , 2005.

14. Rasheed, Camotim. Advances in the Stability of Frame Structures. Journal of

Engineering Mechanics, Volume 131 (557-558), 2005.

15. Silvestre, Camotim. Elastic Buckling and Second-order Behaviour of Pitched-Roof

Steel Frames. Journal of Constructional Steel Research, Volume 6 (804-818), 2007.

11.3 DESIGN CODES

16. Standards South Africa (2005). SANS 10162-1:2005, The structural use of

steelwork. Standards South Africa, Pretoria.

References 11.3


17. Standards South Africa (2005). Draft SANS 10160-2.Basis of Structural Design and

Actions for Buildings and Industrial Structures - Self-weight and imposed loads.

To be published.

18. Standards South Africa (2005). Draft SANS 10160-3.Basis of Structural Design and

Actions for Buildings and Industrial Structures –Wind Actions. To be published.

19. Southern African Institute of Steel Construction (2000). South African Steelwork

Specification for Construction. Southern African Institute of Steel Construction,

Pretoria.

20. Southern African Institute of Steel Construction(2005). South African Steel

Construction Handbook. Southern African Institute of Steel Construction, Pretoria.

11.4 INTERVIEWS

21. Discussions with Prof PJ Pahl, Technische Universitat Berlin (TUB), Germany

12-18 March 2008

8 May 2008

17 July 2008

14 November 2008

18 November 2008

11.5 ELECTRONIC REFERENCES

22. Lecture 6.1: Concepts of Stable and Unstable Elastic Equilibrium.

http:/www.kuleuven.ac.be/bwk/materials/Teaching/master/wg06/l0410.htm

Date accessed: 3 March 2008.

23. Zhuge, Y. Plastic Analysis: Structural Analysis and Computer Applications.

www.unisanet.unisa.edu.au/courses/course

Date accessed: 23 September 2008

References 11.4


24. Hibbitt, Karlsson, Sorensen. ABAQUS – Documentation.

www.scientific-mputing.de/organization/aw/services/ abaqus/Documentation

Date accessed: January 2007 to August 2008

Appendix A: Elastic stability of columns


APPENDIX A: ELASTIC STABILITY OF COLUMNS

Appendix A: Elastic stability of columns A.1


lNo P

ertu

rbat

ion

Load

l0.25%

Per

turb

atio

n Lo

adl0.5

0% P

ertu

rbat

ion

Load

l0.75%

Per

turb

atio

n Lo

ad

App

licat

ion

of P

ertu

rbat

ion

Load

DescriptionPerturbation

Load Application

Displacement Mode of Column

Other Information

1287 0

0 0

0

1287

Axi

al F

orce

(kN

)

She

ar F

orce

(kN

)

Ben

ding

Mom

ent

(kN

.m)

Mid Node

Top Node

1279.3 12.2

245.2 0

468.2

1301.9

Axi

al F

orce

(kN

)

She

ar F

orce

(kN

)

Ben

ding

Mom

ent

(kN

.m)

Mid Node

Top Node

1278.4 24.1

316.5 0

612.3

1316.7

Axi

al F

orce

(kN

)

She

ar F

orce

(kN

)

Ben

ding

Mom

ent

(kN

.m)

Mid Node

Top Node

1278.1 29.6

374.5 0

719.2

1332.2

Axi

al F

orce

(kN

)

She

ar F

orce

(kN

)

Ben

ding

Mom

ent

(kN

.m)

Mid Node

Top Node

Column Section

Length

Support Conditions

203 x 133 x 25

6.0m

Pinned Pinned

I xx =

23.

5x10

6 mm

4

A =

3.2

2x10

3 mm

2 l Page 1

Buc

klin

gBu

cklin

gB

uckl

ing

Buc

klin

g

6.0m

P1=10 000kN

3.0m

P2 = 0kN

6.0m

P1=10 000kN

3.0m

P2 = 50kN

6.0m

P1=10 000kN

3.0m

P2 = 75kN

6.0m

P1=10 000kN

3.0m

P2 = 25kN

Load

Fac

tor

-0.0

4

-0.0

2

-0.0

8

0.00

0.04

0.08

Verti

cal D

ispl

acem

ent

Hor

izon

tal D

ispl

acem

ent

-0.0

6

Displacement (m)

Figure 2. Displacement at Top Node

-0.1

4

-0.1

2

0.12

-0.1

0

Load

Fac

tor

0.10

0.00

0.00

0.04

0.08

Ver

tical

Dis

plac

emen

t

Hor

izon

tal D

ispl

acem

ent

Displacement (m)

Figure 1. Displacement at Mid Node

-0.1

0

0.30

0.40

0.20

0.50

0.12

0.0025P1

TheoreticalValue

None

0.0050P1

0.0075P1

0.128304

0.128203

0.128179

0.1288532

0.129037

0.066382

0.102867

0.139522

Not Applicable

0.011986

Appl

ied

Per

turb

atio

n Lo

ad(1

)

Load

Fac

tor

Ver

tical

D

ispl

acem

ent

Top

Nod

e (m

)

Legend

(1)Applied as percentage of applied load.(2)Theoretical value as calculated using Euler theory.

0.0025P1

TheoreticalValue

None

0.0050P1

0.0075P1

0.128304

0.128203

0.128179

0.1288532

0.129037

0.00331

0.051434

0.069761

Not Applicable

0.005993

App

lied

Per

turb

atio

n Lo

ad(1

)

Load

Fac

tor

Ver

tical

D

ispl

acem

ent

Mid

Nod

e (m

)

Legend


0.362481

0.467573

0.552793

Not Applicable

0

Hor

izon

tal

Dis

plac

emen

t M

id N

ode

(m)

Appendix A: Elastic instability of columns A.2


No

Per

turb

atio

n Lo

ad0.

25%

Per

turb

atio

n Lo

ad0.

50%

Per

turb

atio

n Lo

ad0.

75%

Per

turb

atio

n Lo

ad

App

licat

ion

of P

ertu

rbat

ion

Load


Load Application


Other Information

18081.12 0

0 0

0

18081.12

Axi

al F

orce

(kN

)

She

ar F

orce

(kN

)

Ben

ding

Mom

ent

(kN

.m)

Mid Node

Top Node

18066.98 420.9

5972.3 0

11525.7

19016.37

Axi

al F

orce

(kN

)

She

ar F

orce

(kN

)

Ben

ding

Mom

ent

(kN

.m)

Mid Node

Top Node

18040.6 478.9

6484.4 0

12551.6

19163.9

Axia

l For

ce(k

N)

She

ar F

orce

(kN

)

Ben

ding

Mom

ent

(kN

.m)

Mid Node

Top Node

18012.6 528.2

6867.3 0

13328.0

19273.6

Axi

al F

orce

(kN

)

She

ar F

orce

(kN

)

Ben

ding

Mom

ent

(kN

.m)

Mid Node

Top Node

Column Section

Length

Support Conditions

457 x 191 x 75

6.0m

Pinned Pinned

I xx=

334x

106 m

m4

A =

23.

5x10

3 mm

2

Page 1

Buc

klin

gB

uckl

ing

Buc

klin

gB

uckl

ing

Load

Fac

tor

0.00

0.00

1.00

1.50

Ver

tical

Dis

plac

emen

t

Hor

izon

tal D

ispl

acem

ent

Displacement (m)


-0.1

0

0.40

0.60

0.20

0.50

Load

Fac

tor

-0.0

8

-0.0

4

-0.1

6

0.00

0.50

1.00

Ver

tical

Dis

plac

emen

t

Hor

izon

tal D

ispl

acem

ent

-0.1

2

Displacement (m)


-0.2

4

1.50

-0.2

0

6.0m

P1=10 000kN

3.0m

P2 = 0kN

6.0m

P1=10 000kN

3.0m

P2 = 25kN

6.0m

P1=10 000kN

3.0m

P2 = 50kN

6.0m

P1=10 000kN

3.0m

P2 = 75kN

0.0025P1

TheoreticalValue

None

0.0050P1

0.0075P1

1.832166

1.829427

1.826329

1.83136

1.833976

0.214154

0.242585

0.265484

Not Applicable

0.057038

App

lied

Per

turb

atio

n Lo

ad(1

)

Load

Fac

tor

Ver

tical

D

ispl

acem

ent

Top

Nod

e (m

)

Legend


0.0025P1

TheoreticalValue

None

0.0050P1

0.0075P1

1.832166

1.829427

1.826329

1.83136

1.833976

0.107077

0.121293

0.132742

Not Applicable

0.028519

App

lied

Per

turb

atio

n Lo

ad(1

)

Load

Fac

tor

Ver

tical

D

ispl

acem

ent

Mid

Nod

e (m

)

Legend


0.607697

0.722392

0.698392

Not Applicable

0

Hor

izon

tal

Dis

plac

emen

t M

id N

ode

(m)

Appendix A: Elastic instability of columns A.3


No

Per

turb

atio

n Lo

ad0.

25%

Per

turb

atio

n Lo

ad0.

50%

Per

turb

atio

n Lo

ad0.

75%

Per

turb

atio

n Lo

ad

App

licat

ion

of P

ertu

rbat

ion

Load


Load Application


Other Information

41028.2 0

0 0

0

41028.2

Axi

al F

orce

(kN

)

She

ar F

orce

(kN

)

Ben

ding

Mom

ent

(kN

.m)

Mid Node

Top Node

41008.2 1075.5

15282.8 0

29503.77

43700.9

Axi

al F

orce

(kN

)

She

ar F

orce

(kN

)

Ben

ding

Mom

ent

(kN

.m)

Mid Node

Top Node

40998.44 1223.5

16653.4 0

32243.8

44220.2

Axia

l For

ce(k

N)

She

ar F

orce

(kN

)

Ben

ding

Mom

ent

(kN

.m)

Mid Node

Top Node

40886.3 1291.2

16967.1 0

32924.0

44213.7

Axi

al F

orce

(kN

)

She

ar F

orce

(kN

)

Ben

ding

Mom

ent

(kN

.m)

Mid Node

Top Node

Column Section

Length

Support Conditions

533 x 210 x 122

6.0m

Pinned Pinned

I xx=

762x

106 m

m4

A =

15.

6x10

3 mm

2

Page 1

Buc

klin

gB

uckl

ing

Buc

klin

gB

uckl

ing

Load

Fac

tor

0.00

3.00

Ver

tical

Dis

plac

emen

t

Hor

izon

tal D

ispl

acem

ent

Displacement (m)


0.20

0.40

1.00

0.00

2.00

0.60

4.00

Load

Fac

tor

-0.4

0

-0.2

0

0.00

2.00

Ver

tical

Dis

plac

emen

t

Hor

izon

tal D

ispl

acem

ent

-0.6

0Displacement (m)


-0.0

0

3.00

1.00

4.00

0.0025P1

TheoreticalValue

None

0.0050P1

0.0075P1

4.180999

4.179893

4.168319

4.17813

4.184102

0.274834

0.311938

0.321055

Not Applicable

0.078900

App

lied

Per

turb

atio

n Lo

ad(1

)

Load

Fac

tor

Ver

tical

D

ispl

acem

ent

Top

Nod

e (m

)

Legend


0.0025P1

TheoreticalValue

None

0.0050P1

0.0075P1

4.180999

4.179893

4.168319

4.17813

4.184102

0.137417

0.155969

0.160527

Not Applicable

0.039450

App

lied

Per

turb

atio

n Lo

ad(1

)

Load

Fac

tor

Ver

tical

D

ispl

acem

ent

Mid

Nod

e (m

)

Legend


0.674628

0.734294

0.748570

Not Applicable

0

Hor

izon

tal

Dis

plac

emen

t M

id N

ode

(m)

6.0m

P1=10 000kN

3.0m

P2 = 75kN

6.0m

P1=10 000kN

3.0m

P2 = 25kN

6.0m

P1=10 000kN

3.0m

P2 = 50kN

6.0m

P1=10 000kN

3.0m

P2 = 0kN

Appendix B: Elastic stability of portal frames



Appendix B: Elastic stability of portal frames B.1.1



B.1 RESULTS FOR VARYING INITIAL LOAD FACTORS

Table B1. Results of configuration C1:

Initial load

factor

Load Factor Comments in command

prompt

1.0 12.4622 terminated




0.5 12.4667 terminated Table B2. Results of configuration C2:

Initial load

factor


prompt




0.7 12.4269 terminated Table B3. Results of configuration C3:

Initial load

factor


prompt

0.9 11.5892 singular point


0.73 78.230

singular point


Appendix B: Elastic stability of portal frames B.1.2



Initial load

factor


prompt









Initial load

factor


prompt







. Table B6. Results of configuration C6:

Initial load

factor


prompt





Appendix B2: Elastic stability of portal frames B.2.1


B.2. RESULTS FOR ANALYSES INCLUDING AN INITIAL LOAD FACTOR OF 0.1


Configuration C1: pinned, full vertical load, no horizontal perturbation load.

state u1 at ridge u2 at ridge load factor u2 / load factor

20 0 0.318 1.946 0.163

40 0 0.669 3.847 0.173

60 0 1.059 5.686 0.186

100 0 1.983 9.061 0.219

150 0 3.391 11.951 0.284

200 0 3.710 12.269 0.302

400 0 3.895 12.401 0.314

465 0 3.914 12.424 0.317


Configuration C2: pinned, full vertical load, 0.5% horizontal perturbation load.


20 0.002 0.318 1.946 0.163

40 0.006 0.669 3.846 0.174

60 0.009 1.060 5.684 0.186

100 0.020 1.985 9.055 0.219

150 0.039 3.395 11.928 0.285

200 0.042 3.703 12.233 0.303

400 0.044 3.889 12.365 0.315

465 0.045 3.927 12.387 0.317




Configuration C3: fixed, full vertical load, no horizontal perturbation load.


20 0 0.287 1.981 0.145

50 0 0.886 5.177 0.171

100 0 2.760 10.609 0.260

200 0 4.215 14.090 0.299

400 0 5.537 19.483 0.284

600 0 6.506 25.699 0.253

800 0 7.297 32.475 0.225

1000 0 7.971 39.540 0.202


Configuration C4: fixed, full vertical load, 0.5% horizontal perturbation load.


20 0.000 0.287 1.980 0.145

50 0.001 0.885 5.173 0.171

100 0.002 2.759 10.588 0.261

200 0.003 4.213 14.048 0.300

400 0.004 5.535 19.407 0.285

600 0.004 6.503 25.588 0.254

800 0.005 7.294 32.331 0.226

1000 0.005 7.968 39.369 0.202




Configuration C5: pinned, half vertical load, no horizontal perturbation load.


50 0.134 0.406 4.843 0.084

100 0.312 0.876 9.500 0.092

150 0.555 1.434 13.825 0.104

200 0.886 2.113 17.518 0.121

250 1.287 2.940 20.040 0.147

300 1.440 3.293 20.562 0.160

500 1.503 3.452 20.739 0.166

928 1.531 3.524 20.807 0.169


Configuration C6: fixed, half vertical load, no horizontal perturbation load.


20 0.013 0.137 1.971 0.070

50 0.034 0.371 4.978 0.075

100 0.074 0.901 10.395 0.087

150 0.131 2.062 17.934 0.115

200 0.163 3.312 23.553 0.141

400 0.191 4.481 30.190 0.148

600 0.220 5.139 35.648 0.144

800 0.236 5.666 41.286 0.137



B.3. RESULTS FOR DECOMPOSED STIFFNESS MATRIX

B.3.1 Diagonal Coefficients for Configuration C1

The load factor LF, the ridge displacement u and the smallest diagonal coefficient minD of the decomposed secant stiffness matrix of configuration C1 (pinned portal frame) are shown in the following table as functions of the load step number:

step LF u minD

0 0.000 0.000 13799.88

10 0.979 ‐0.155 13299.00

20 1.946 ‐0.318 12779.98

30 2.903 ‐0.489 12240.62

40 3.847 ‐0.669 11679.67

50 4.775 ‐0.859 11095.95

60 5.686 ‐1.059 10488.33

70 6.575 ‐1.271 9855.86

80 7.438 ‐1.495 9197.81

90 8.269 ‐1.732 8513.84

100 9.061 ‐1.983 7804.55

110 9.806 ‐2.250 7070.61

120 10.491 ‐2.533 6314.68

130 11.102 ‐2.831 5540.49

140 11.621 ‐3.142 4755.81

150 11.951 ‐3.391 4091.60

160 12.102 ‐3.528 3768.12

170 12.175 ‐3.602 3576.16

180 12.217 ‐3.649 3464.02

190 12.248 ‐3.684 3368.61

200 12.272 ‐3.713 3298.14

250 12.398 ‐3.890 2742.54

300 12.453 ‐3.988 2276.72

350 12.481 ‐4.045 1838.24

400 12.495 ‐4.078 1415.03

450 12.503 ‐4.098 1018.85

500 12.507 ‐4.109 689.31

550 12.510 ‐4.116 438.77

590 12.511 ‐4.120 284.49

593 12.512 ‐4.121 266.39

594 12.512 ‐4.126 234.70

595 12.512 ‐6.841 32.42



B.3.2 Diagonal Coefficients for Configuration C3

The load factor LF, the ridge displacement u and the smallest diagonal coefficient minD of the decomposed secant stiffness matrix of configuration C3 (fixed portal frame) are shown in the following table as functions of the load step number:

step LF u minD

0 0.000 0.000 88553.87

20 1.981 ‐0.287 86792.13

40 4.050 ‐0.650 84659.18

60 6.467 ‐1.202 82001.82

80 9.448 ‐2.228 81224.93

100 10.609 ‐2.760 83369.13

120 11.414 ‐3.140 85488.69

140 12.484 ‐3.616 86569.18

160 13.059 ‐3.846 87007.92

180 13.581 ‐4.039 87419.94

200 14.090 ‐4.215 87848.32

220 14.597 ‐4.377 88246.51

240 15.108 ‐4.530 88533.80

260 15.625 ‐4.674 88833.63

280 16.149 ‐4.812 89142.66

300 16.682 ‐4.944 89458.21

320 17.224 ‐5.071 89778.07

340 17.775 ‐5.193 90100.43

360 18.335 ‐5.312 90423.77

380 18.904 ‐5.426 90702.10

400 19.483 ‐5.537 90931.07

420 20.070 ‐5.645 91161.91

440 20.666 ‐5.750 91393.82

460 21.270 ‐5.853 91626.13

480 21.882 ‐5.952 91858.25

500 22.501 ‐6.050 92078.14

Appendix C: Number of elements



Appendix C: Number of elements C.1.1



C.1 ANGELINE

C.1.1 Basis of comparison

The analyses include the number of elements to be varied between 6, 12 and 24 elements per

column and rafter, respectively as shown in Figure C1.

6.0m

12 e

lem

ents

12 e

lem

ents

6.0m

6 el

emen

ts

6 el

emen

ts

6.0m

24 e

lem

ents

24 e

lem

ents

Figure C1. Varying the number of elements

Frames are analysed up to a load factor of 1.00. Forces are compared for the varying number of

elements at this load factor. Evaluation includes comparison between maximum axial, shear and

bending in the columns and the rafters as well as displacement of the top left and right hand column

and the ridge of the roof. These values are shown in Table C1 for the respective number of elements.



Table C1. Comparative values

Elements Column Rafter Force Type Top Left Top Right Left Right Axial (kN) ‐60.75 ‐61.07 ‐39.22 ‐39.3 Bending Moment (kN.m) 210.61 214.96 210.6 241.94 Shear (kN) ‐33.33 ‐34.01 ‐52.27 ‐52.58 Top Left Top Right Ridge

Displacement y1 (m) ‐0.03224 0.04498 0.006091

6 elem

ents

Displacement y2 (m) ‐0.00133 ‐0.00145 ‐0.471708 Elements Column Rafter

Force Type Top Left Top Right Left Right Axial (kN) ‐60.9 ‐61.23 ‐39.22 ‐39.3 Bending Moment (kN.m) 210.62 214.97 210.59 214.94 Shear (kN) ‐33.07 ‐33.74 ‐52.26 ‐52.58 Top Left Top Right Ridge

Displacement y1 (m) ‐0.03219 0.044431 0.006095

12elem

ents

Displacement y2 (m) ‐0.00135 ‐0.00147 ‐0.472036 Elements Column Rafter

Force Type Top Left Top Right Left Right Axial (kN) ‐60.91 ‐61.23 ‐39.22 ‐39.3 Bending Moment (kN.m) ‐210.62 214.97 210.61 214.94 Shear (kN) ‐30.03 ‐33.7 ‐52.27 ‐52.58 Top Left Top Right Ridge

Displacement y1 (m) ‐0.03217 0.044431 0.006096

24 elemen

ts

Displacement y2 (m) ‐0.00136 ‐0.00147 ‐0.472156

C.2 PROKON


The analyses include the number of elements to be varied between 6, 12 and 24 elements per

column and rafter, respectively as shown in Figure C1. Results are evaluated at a load factor of

0.641. This is the designed value for the benchmark portal frame according to SANS 10162‐1 and 2.

Evaluation includes the comparison of maximum axial, shear and bending in the columns and the

rafters as well as displacement of the top left and right hand column and the ridge of the roof. These

values are shown in Table C2.



Table C2. Comparative values

Elements Column Rafter Force Type Top Left Top Right Left Right Axial (kN) 38.93 39.18 25.86 25.88 Bending Moment (kN.m) 132.1 135.2 132.1 135.2 Shear (kN) 21.23 22.02 33.45 33.43 Top Left Top Right Ridge

Displacement y1 (mm) 26.16 34.49 ‐

6 elem

ents

Displacement y2 (mm) ‐ ‐ 291.30 Elements Column Rafter

Force Type Top Left Top Right Left Right Axial (kN) 38.93 39.18 25.86 25.88 Bending Moment (kN.m) 132.1 135.2 132.1 135.2 Shear (kN) 21.12 21.82 33.45 33.43 Top Left Top Right Ridge


12elem

ents

Displacement y2 (mm) ‐ ‐ 291.30 Elements Column Rafter

Force Type Top Left Top Right Left Right Axial (kN) 38.93 39.18 25.85 25.88 Bending Moment (kN.m) 132.1 135.2 132.1 135.2 Shear (kN) 21.23 21.71 33.46 33.43 Top Left Top Right Ridge


24 elemen

ts

Displacement y2 (mm) ‐ ‐ 291.33

C.3 ABAQUS


ABAQUS is used to determine the plastic deformation of the frame and therefore analysis included

observing the yielding of the first cross‐section in the member.

The frames are evaluated on the load factor where the first yielding of the cross‐section is identified.

The frame is evaluated for 6,12 and 24 elements as shown in Figure C1.



Stresses versus Load Factor

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

‐400 ‐300 ‐200 ‐100 0Stresses (mPa)

Load

Factor

6 elements

12 elements

24 elements

Figure C2. Stresses vs Load Factor

The load factor at which the first hinge is formed is at a load factor of 0.771, 0.7626, and 0.7620 for

6, 12 and 24 elements, respectively.

C.4 CONCLUSION The differences in results obtained in varying the number of elements are very small. These small

differences are found for all software packages. According to these results the use of 6 elements is

sufficient. However, 12 elements are used in the benchmark example.

Appendix D: Notional horizontal load



Appendix D: Notional horizontal load D.1



D.1 VERIFICATION OF APPROACH USING THE NOTIONAL HORIZONTAL LOAD

A short example is done to verify the validity of the notional horizontal approach. The software

programme ANGELINE is used. An explanation in using ANGELINE follows in the literature. A

comparison between the load‐deflection curve of a straight column with an applied perturbation

load, to the load‐deflection path of an initially curved column is done. This study looks at an example

of a column configuration with a 203 x 133 x 25 I‐section and simply supported conditions.

D.1.1 Analysis and Results –Notional horizontal approach

The first part of the investigation includes the application of the perturbation load of 0.25%, 0.5%

and 0.75% at the mid node. A certain value P is applied at the top node of the column. For this

analysis a value of 10 000kN is chosen. The implementation into ANGELINE is shown in Figure D.1.

Figure D.1 Application of compressive force and perturbation load

The results of this analysis is given in Appendix A1.



Note that the load deflection curve consists of a linear part and the curve starts to reach the Euler

value asymptotically. However, values are very close to each other. The load deflection‐curve differs

for each application of the perturbation load. The stable part of the curve is less for higher values of

perturbation loads but the deflection increases at this point. However, the slope of the curve in the

linear part of the graph is similar for all applications.

D.1.2 Analysis and Results – Initial curvature

Secondly, a column with an initial curvature is programmed in ANGELINE to represent the actual

imperfection in practice. A half sine curve is assumed for the curvature, with a maximum

displacement at mid height. The latter is usually modelled as a half sine wave.

The magnitude of the initial out‐of‐straightness is usually limited by the tolerances or specifications

given as a fraction of the length. The calculation of the curvature is shown in Table D.1. Two

graphical models are also shown in this table. These models indicate the geometric properties of the

out of plumbness of the column and the graphical presentation as shown in ANGELINE, respectively.

Table D.1. Calculations of initial curvature of column Parameters Maximum allowed tolerance

greater of 1000L

or 3mm

Choose 6mm Calculate C Length of column (L) 6000mm Mode shape number (n)

1

x 3000mm Solving for C

)(sin6000

3000xx1C6

π=

Value of C 6

Displacement formula )(sin)(6000

Lxx16xv xπ

=



The following formula is used to describe the curvature of the column to be modelled :

)L

xn(sinC)x(v π=

Where C is a constant determined by setting the maximum value at midpoint.

Figure D.2 illustrates the load‐deflection curve obtained by modelling a column with initial curvature

at the mid node.

Figure D.2: Load‐deflection curve of a column with initial curvature

D.1.3 Comparison between two methods

The load deflection‐curve of the column with an initial curvature is superimposed onto the load‐

defection curves of the straight column which include analyses with varying perturbation loads

(Figure D.3). The load‐deflection curves are very similar and the load deflection curve for a straight

column with a perturbation load of 0.5% and an initially curved column with proposed tolerances

match almost exactly.



Load

Fac

tor

0.10

0.00

-0.1

0

0.30

0.40

0.20

0.50

Figure D.3 Load‐deflection curves between various methods

D.1.4 Conclusion

The use of a perturbation load as representation of the imperfections in columns is justified and can

be used for modelling of imperfection in the study of portal frames.

Appendix E: Portal frame design


APPENDIX E: PORTAL FRAME DESIGN

Appendix E: Portal frame design E.1


APPENDIX E: PORTAL FRAME DESIGN ACCORDING TO DRAFT SANS 10160‐1 AND 2 : 2008 AND SANS 10162‐1:2005

DESIGN OF A PORTAL FRAME ACCORDING TO SANS 10162‐2005

COLUMN DESIGN

a. portal frame geometric properties

column height hc 6000 mm

roof height hr 7262 mm

roof angle a 6.004 o

span of portal frame S 24000 mm

length of portal frame lt 35000 mm

length of single bay lb 5000 mm

b. section properties ‐ column

section I 254 X 146 X 37

height of section h 256 mm

Width of flange b 146.4 mm

thickness of web tw 6.4 mm

thickness of flange tf 10.9 mm

cross sectional area A 4740 mm2

second moment of inertia about the x‐axis Ixx 55500000 mm4

second moment of inertia about the y‐axis Iyy 5710000 mm4

radius of gyration xx rxx 108 mm

radius of gyration yy ryy 34.7 mm

elastic section modulus about x‐axis Zex 433000 mm3

elastic section modulus about y‐axis Zey 78000 mm3

plastic section modulus about x‐axis Zplx 485000 mm3

plastic section modulus about y‐axis Zply 119000 mm3

St‐Venant torsional constant J 155000 mm4

warping torsional constant Cw 8.57E+10 mm6

yield stress of steel fy 350 MPa

elastic modulus E 200 GPa

shear modulus G 77 GPa

resistance factor f 0.9

column behaviour identifier n 1.34



c. serviceability limit state

horizontal deflection dh 25.07 mm

vertical deflection dv 233.89 mm

span/vertical deflection

102.00

height/horizontal deflection

240.00

minimum span/vertical deflection for simple span members supporting elastic roofing 180

102.00 < 180 NOT ACCEPTABLE

minimum height/horizontal deflection for simple span members supporting elastic roofing 300

240 < 300 NOT ACCEPTABLE

d. ultimate limit state

d.1. classification of profile

d1.1. axial compression ‐ column

width‐to‐thickness ratio of flange

6.72

limiting width‐to thickness ratio for flange

10.69

class of section according to flange Class 3

width‐to‐thickness ratio of web

36.59

limiting width‐to thickness ratio for web

35.81

class of section according to web Class 4

d.1.2. initial member forces flexural compression

classification ‐ column

maximum axial compressive force Cu 39.2 kN

Cy 1659 kN

yf

200

w

f

t

t2h −

ft2b

yf

670

v

sδ

h

chδ



Note: Cu as indicated is only used for classification of members.

d.1.3. flexural compression ‐ column

width‐to‐thickness ratio of flange

6.72

limiting width‐to‐thickness ratio class 1

7.75


9.09


10.69

class of section according to flange Class 1

width‐to‐thickness ratio of web

36.59


58.20


89.41


99.83

class of section according to web Class 1

d.2.axial compression capacity ‐ column

d.2.1. member dimensions and axial forces‐

column

effective length for axial buckling about x‐axis Lx 6000 mm

applied axial compressive force Cu 39.2 kN

d.2.2. class 4 members in compression‐column

stress due to maximum load

0.008

ft2b

yf

145

yf

200

w

f

tt2h −

⎟⎟⎠

⎞⎜⎜⎝

⎛

φ−

y

u

y C

C3901

f

1100 .

⎟⎟⎠

⎞⎜⎜⎝

⎛

φ−

y

u

y C

C6101

f

1700 .

⎟⎟⎠

⎞⎜⎜⎝

⎛

φ−

yy CCu6501

f

1900 .

A

Cf u=

yf

170



width‐to‐thickness ratio of flange W(flange)= 6.72

k(flange)= 0.43

width‐to‐thickness ratio limit of flange

flange Class 3

area effective flange Aeff 4740 mm2

width‐to‐thickness ratio of web W(web)= 36.59

k(web)= 4.00

width‐to‐thickness ratio limit of web

200.30

area effective of web Aeff 4740.00 mm2

area to be used Aeff 4740.00 mm2

d.2.3. axial compression capacity‐column

slenderness ratio x‐x

0.7398

axial compression capacity

1133.96 kN

slendernes ratio check

55.55 ACCEPTABLE

Axial capacity of section

sufficient

d.3. flexural compression capacity ‐ column

d.3.1. member bending moment forces‐column

maximum bending forces Mu 135.20 kN.m

d.3.2 flexural compression capacity

moment resistance for class 1 and 2

152.78 kN.m

choose class of section Class 1

fkE

6440W .lim =

E

f

rLK

2y

x

xx

π=λ

yplrx fZM φ=

fkE

6440W .lim =

n/1n2yr )1(AfC −λ+φ=

200rL<



determining moment resistance Mrx 152.78 kN.m

Bending capacity of section

sufficient

d.3.3 interaction‐overall member strength‐column

factor U1x 1

maximum axial compressive force* Cu 39.2 kN.m

Maximum moment* Mu 135.2 kN.m

interaction formula

0.79

Interaction acceptable

d.4. shear capacity ‐ column

height of section h 256 mm

thickness of web tw 6.4 mm

height of web hw 234.2

shear area Av 1638.4 mm2

spacing S 100000000 mm

height to web ratio

36.59

spacing to web ratio 426985.48

shear buckling coefficient for s/hw ≥1

5.34

limiting height to web ratio

54.35

aspect coefficient

2.342E‐06

rx

uxx1

r

u

M

MU850

C

C .+

w

wth

y

v

f

k440

whs

2

w

v

hs

4345k

⎟⎟⎠

⎞⎜⎜⎝

⎛+= .

2

w

a

hs

1

1k

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=



critical shear stress

342.60 MPa

elastic shear buckling resistance

717.79 MPa

tension field post‐buckling stress ‐

0.00028502 MPa

tension field post‐buckling stress ‐

0.00104596 MPa

ultimate shear stress option 1

231 MPa


342.60 MPa


342.60 MPa


717.79 Mpa

ultimate shear stress fs 231.00 MPa

ultimate shear resistance

340.62 kN

maximum shear force Vu 22.5 kN

Shear capacity of section

sufficient

d.5. tensile capacity ‐ column

axial tension force Tu 0.0 kN

axial tension resistance

1493.1 kN

Tensile capacity of section

sufficient

d.5.1 interaction tension end bending‐column

ultimate moment Mu 135.2

moment resistance Mr 152.8 kN.m

interaction

0.8850

Interaction acceptable

ys f660f .=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

w

w

vycri

t

h

kf290f

)..( criyati f8660f500kf −=

)f866.0f50.0(kf creyate −=

2

w

w

vcre

t

h

k180000f

⎟⎟⎠

⎞⎜⎜⎝

⎛=

cris ff =

tecres fff +=

svr fAV φ=

yr AfT φ=

r

u

r

u

M

M

T

T+

tecris fff +=



d.7. summary

reference force /deflection resistance /limit status

c. span/vertical deflection 180 113.3 NOT

ACCEPTABLE

c. height/horizontal deflection 300 166.3 NOT

ACCEPTABLE

d.2.3. axial compression capacity‐column 39 1133.96 ACCEPTABLE

d.3.2 flexural compression capacity 135.20 152.78 ACCEPTABLE

d.3.3 interaction‐overall member strength‐column 0.787 1 ACCEPTABLE

d.4. shear capacity ‐ column 22.5 340.62 ACCEPTABLE

d.5. tensile capacity ‐ column 0.0 1493.10 ACCEPTABLE

d.5.1 interaction tension end bending‐column 0.8850 1.00 ACCEPTABLE

Appendix F:Design results



Appendix F: Design results F.1



Column Rafter Column Rafter Deflection Column Rafter

Span Support Column Height

Roof Slope

Axial Force Top (kN)(1)

Axial Force Ridge (kN)

Axial Force Eaves (kN)

Bending Moment (kN.m) (2)

Bending Moment Ridge (kN.m)

Bending Moment Eaves (kN.m)

Section Choice (3)

Horizontal (mm)

Vertical (mm)

Critical Load (kN)(4)

Ratio (%) (5)

Bending (kN.m)

Bending Resistance (kN.m)(6)

Ratio (%) (7)

Interaction Ratio (8)

Axial Load (kN)

Critical Load (kN)

Ratio (%)

Bending (kN.m)

(9)

Bending Resistance (kN.m)

Ratio (%)

Interaction Ratio

24 Simple 6 3 39.13 23.20 24.89 139.40 85.90 69.79 254x146x37 29.98 234.26 1133.96 3.45 139.40 152.78 91.25 81.01 24.89 544.11 4.57 85.90 152.78 56.23 52.06 24 Simple 6 6 39.18 22.44 25.85 135.20 76.20 67.83 254x146x37 36.11 211.99 1133.96 3.46 135.20 152.78 88.50 78.68 25.85 540.79 4.78 76.20 152.78 49.88 46.55 24 Simple 6 9 39.22 21.77 26.80 131.10 67.40 65.90 254x146x37 41.68 192.74 1133.96 3.46 131.10 152.78 85.81 76.40 26.80 535.26 5.01 67.40 152.78 44.12 41.57 24 Simple 6 12 39.39 21.03 27.79 127.30 61.84 63.88 254x146x37 46.37 175.91 1133.96 3.47 127.30 152.78 83.33 74.30 27.79 527.53 5.27 63.88 152.78 41.81 38.39 24 Simple 10 3 39.27 12.82 14.50 129.50 102.50 58.14 254x146x37 71.75 302.47 1133.96 3.46 129.50 152.78 84.77 75.51 14.50 544.11 2.66 102.50 152.78 67.09 59.38 24 Simple 10 6 39.40 12.70 16.75 127.70 96.40 57.25 254x146x37 80.62 287.40 1133.96 3.47 127.70 152.78 83.59 74.52 16.75 540.79 3.10 96.40 152.78 63.10 55.98 24 Simple 10 9 39.41 12.64 17.76 126.60 90.67 57.93 254x146x37 89.00 273.85 1133.96 3.48 126.60 152.78 82.87 73.91 17.76 535.26 3.32 90.67 152.78 59.35 52.81 24 Simple 10 12 39.53 12.42 19.23 125.50 85.12 57.86 254x146x37 97.25 261.65 1133.96 3.49 125.50 152.78 82.15 73.31 19.23 527.53 3.65 85.12 152.78 55.72 49.71 24 Simple 14 3 38.67 8.11 9.92 118.60 111.80 48.18 254x146x31 178.67 440.67 439.10 8.81 118.60 124.43 95.32 89.83 9.92 439.10 2.26 111.80 124.43 89.85 78.22 24 Simple 14 6 38.73 8.28 11.69 118.50 109.70 48.71 254x146x31 193.06 427.68 439.10 8.82 118.50 124.43 95.24 89.77 11.69 436.37 2.68 109.70 124.43 88.17 76.84 24 Simple 14 9 38.89 8.31 13.38 118.60 106.30 49.32 254x146x31 197.40 415.22 439.10 8.86 118.60 124.43 95.32 89.88 13.38 431.83 3.10 106.30 124.43 85.43 74.54 24 Simple 14 12 38.93 8.52 15.07 118.80 101.30 51.50 254x146x31 211.78 403.36 439.10 8.87 118.80 124.43 95.48 90.02 15.07 425.48 3.54 101.30 124.43 81.41 71.20 24 Fixed 6 6 39.10 37.05 40.43 132.60 61.75 68.37 254x146x37 22.28 168.16 1133.96 3.45 132.60 152.78 86.79 77.22 40.43 540.79 7.48 68.37 152.78 44.75 41.21 24 Fixed 6 12 38.40 33.51 40.08 113.50 36.02 57.08 254x146x31 33.36 147.45 937.66 4.10 113.50 124.43 91.22 81.63 40.08 425.48 9.42 57.08 124.25 45.94 32.52 24 Fixed 10 6 39.13 21.04 24.36 130.70 82.20 62.92 254x146x37 46.75 237.20 701.29 5.58 130.70 152.78 85.55 78.30 24.36 540.79 4.50 82.20 152.78 53.80 49.62 24 Fixed 10 12 38.44 20.17 26.75 121.10 64.07 58.53 254x146x31 69.45 243.85 569.25 6.75 121.10 124.43 97.33 89.48 26.75 425.48 6.29 64.07 124.25 51.57 48.57 24 Fixed 14 6 38.31 13.97 17.27 122.40 95.60 54.39 254x146x31 97.84 356.45 344.84 11.11 122.40 124.43 98.37 94.73 17.27 436.37 3.96 95.60 124.25 76.94 68.60 24 Fixed 14 12 38.48 13.98 22.54 119.20 82.52 53.77 254x146x31 114.30 320.40 344.84 11.16 119.20 124.43 83.50 82.14 22.54 425.84 5.29 82.52 124.25 66.41 59.74 24 Simple 6 6 39.18 22.44 25.85 135.20 76.20 67.83 254x146x37 36.11 211.99 1133.96 3.97 135.20 152.78 88.50 78.68 25.85 540.79 4.78 76.20 152.78 49.88 46.55 24 Simple 6 12 39.39 21.03 27.75 127.30 61.84 63.88 254x146x37 46.37 175.91 1133.96 3.99 127.30 152.78 83.33 74.30 27.75 527.53 5.26 63.88 152.78 41.81 38.39 24 Simple 14 6 38.89 8.10 11.53 120.20 110.20 50.04 254x146x31 193.06 427.68 439.10 12.11 120.20 124.43 96.60 90.97 11.53 436.37 2.64 110.20 124.43 88.57 77.14 24 Simple 14 12 39.91 8.06 14.91 120.70 102.30 51.50 254x146x31 211.78 403.36 439.10 12.16 120.70 124.43 97.01 91.54 14.91 425.48 3.50 102.30 124.43 82.22 71.78 28 Simple 6 6 46.28 31.10 35.22 187.70 98.30 107.90 305x165x41 35.96 238.76 1354.35 3.87 187.70 197.19 95.19 84.33 35.22 609.50 5.78 107.90 197.19 54.72 47.48 28 Simple 6 12 46.89 28.80 37.10 176.80 77.03 101.60 254x146x43 62.39 253.39 1323.10 4.01 176.80 178.92 98.82 87.54 37.10 487.43 7.61 101.60 178.92 56.79 42.50 28 Simple 14 6 47.04 12.19 16.41 175.60 146.10 89.65 254x146x43 186.70 521.19 505.25 12.52 175.60 178.92 98.14 92.73 16.41 500.78 3.28 146.10 178.92 81.66 71.84 28 Simple 14 12 47.43 11.70 20.45 175.10 133.30 90.73 254x146x43 217.22 486.69 505.25 12.57 175.10 178.92 97.86 92.57 20.45 487.43 4.20 133.30 178.92 74.50 65.73 32 Simple 6 6 55.33 42.60 47.76 257.80 162.40 127.10 305x165x54 40.40 293.80 1801.80 3.45 257.80 265.55 97.08 85.59 47.76 673.02 7.10 162.40 265.55 61.16 58.31 32 Simple 6 12 55.66 42.10 49.92 259.40 163.40 128.00 305x165x54 40.69 296.08 1801.84 3.47 259.40 265.55 97.68 86.12 49.92 655.54 7.62 163.40 265.55 61.53 58.72 32 Simple 14 6 53.90 16.40 21.20 232.50 181.30 134.10 356x171x45 133.39 448.67 805.96 8.72 232.50 243.50 95.48 87.85 21.20 663.73 3.19 181.30 243.50 74.46 65.76 32 Simple 14 12 54.41 15.80 25.75 230.40 163.50 130.10 356x171x45 163.13 414.56 805.96 8.81 230.40 243.50 94.62 87.18 25.75 647.61 3.98 163.50 243.50 67.15 59.51

[1] The axial force value at the top of the critical right column. [2] The bending moment value at the top of the critical right column. [3] Section choice taken from the Southern African Steel Construction Handbook. [4] Calculated by formula as set out in Section 6.5.3.4 of the literature. [5] Axial capacity utilised by section: Axial Load / Critical Load x 100 = Ratio. [6] Bending capacity calculated as discussed in Section 6.5.5 [7] Bending capacity utilised by section: Bending Moment / Bending Capacity x 100 = Ratio. [8] Calculated as discussed in Section 6.5.5. [9] Maximum bending moment in the rafter. Blue pertains to maximum value located at the eaves. Yellow pertains to maximum value located at the ridge.

Appendix G: Load‐Displacement History‐ ABAQUS


APPENDIX G: LOAD‐DISPLACEMENT HISTORY ‐ ABAQUS

Appendix G: Load‐displacement history‐ ABAQUS G.1


G.1 LOAD DISPLACEMENT HISTORY ‐ PINNED SUPPORTS‐ ABAQUS

Left Column ‐ δh vs Load Factor

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

‐0.10 0.00 0.10 0.20 0.30 0.40 0.50 0.60Displacement (m)

Load Factor

Span 24.0m ‐ Column 6.0m ‐ Slope 3 degreesSpan 24.0m ‐ Column 6.0m ‐ Slope 6 degreesSpan 24.0m ‐ Column 6.0m ‐ Slope 9 degeesSpan 24.0m ‐ Column 6.0m ‐ Slope 12 degreesSpan 24.0m ‐ Column 10.0m ‐ Slope 3 degreesSpan 24.0m ‐ Column 10.0m ‐ Slope 6 degreesSpan 24.0m ‐ Column 10.0m ‐ Slope 9 degreesSpan 24.0m ‐ Column 10.0m ‐ Slope 12 degreesSpan 24.0m ‐ Column 14.0m ‐ Slope 3 degreesSpan 24.0m ‐ Column 14.0m ‐ Slope 6 degreesSpan 24.0m ‐ Column 14.0m ‐ Slope 9 degreesSpan 24.0m ‐ Column 14.0m ‐ Slope 12 degrees

Ridge ‐ δv vs Load Factor

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

‐1.30 ‐0.80 ‐0.30Displacement (m)

Load Factor

Right Column ‐ δh vs Load Factor

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.20 0.40 0.60Displacement (m)

Load Factor




0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

‐0.25 ‐0.15 ‐0.05 0.05 0.15 0.25 0.35 0.45 0.55Displacement (m)

Load

Factor

Span 24.0m(F) ‐ Column 6.0m ‐ Slope 6 degrees


Span 24.0m(F) ‐ Column 10.0m ‐ Slope 6 degees





0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

‐0.80 ‐0.60 ‐0.40 ‐0.20 0.00Displacement (m)

Load

Factor


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20 0.25Displacement (m)

Load

Factor

G.2 LOAD DISPLACEMENT HISTORY – FIXED SUPPORTS‐ ABAQUS



G.3 LOAD DISPLACEMENT HISTORY ‐ PINNED SUPPORTS‐ VARYING SPANS ‐ ABAQUS


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.00 0.20 0.40 0.60Displacement (m)

Load

Factor


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

‐0.10 0.00 0.10 0.20 0.30 0.40 0.50 0.60Displacement (m)

Load

Factor

Span 24.0m ‐ Column 6.0m ‐ Slope 6 degreesSpan 24.0m ‐ Column 6.0m ‐ Slope 12 degreesSpan 24.0m ‐ Column 14.0m ‐ Slope 6 degeesSpan 24.0m ‐ Column 14.0m ‐ Slope 12 degreesSpan 28.0m ‐ Column 6.0m ‐ Slope 6 degreesSpan 28.0m ‐ Column 6.0m ‐ Slope 12 degreesSpan 28.0m ‐ Column 14.0m ‐ Slope 6 degreesSpan 28.0m ‐ Column 14.0m ‐ Slope 12 degreesSpan 32.0m ‐ Column 6.0m ‐ Slope 6 degreesSpan 32.0m ‐ Column 6.0m ‐ Slope 12 degreesSpan 32.0m ‐ Column 14.0m ‐ Slope 6 degreesSpan 32.0m ‐ Column 14.0m ‐ Slope 12 degrees


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

‐1.30 ‐1.10 ‐0.90 ‐0.70 ‐0.50 ‐0.30 ‐0.10Displacement (m)

Load

Factor

Appendix H:Load‐displacement history‐ ANGELINE


APPENDIX H: LOAD‐DISPLACEMENT HISTORY ‐ ANGELINE

Appendix H : Load-displacement history- ANGELINE (Pinned supports) H1.1


Load Factor Value Description

P10.641

Code Design

Ul timate Load

P20.471

Code Design

Serviceabi l i ty Load

P30.297

Serviceabi l i ty

(Vertical )

P40.339

Serviceabi l i ty

(Horizonta l )

P3

P4

PORTAL FRAME COMPARISSON – (COLUMN 6.0M, ROOF SLOPE 60)

P3

P4

Figure 1.1 Displacement at top node of left column

Figure 1.2 Displacement of node at ridge

P1

P2

P1

P2



P3



P10.645

Code Des ign

Ul timate Load

P20.474

Code Des ign


P30.334

Serviceabi l i ty

(Vertica l )

P40.183

Serviceabi l i ty

(Horizonta l)

P4

P3

P4



P1

P2

P1

P2



P3

P4



P10.627

Code Design

Ultimate Load

P20.458

Code Design


P30.143

Serviceabi l i ty

(Verti cal )

P40.151

Serviceabi l i ty

(Hori zonta l )

P3

P4



P1

P2

P1

P2




P10.630

Code Des ign

Ultimate Load

P20.461

Code Des ign


P30.154

Serviceabi l i ty

(Vertica l )

P40.118

Serviceabi l i ty

(Horizonta l )

Appendix H:Load-displacement history- ANGELINE (Fixed supports) H2.1


P1P2

P3

PORTAL FRAME COMPARISON – COLUMN 6.0M, ROOF SLOPE 60

- FIXED


P10.641

Code Design

Ul tima te Load

P20.471

Code Design


P30.379

Servicea bi l i ty

(Verti cal )

P40.437

Servicea bi l i ty

(Hori zonta l)

P4

P3

P4



P1P2




P10.630

Code Design

Ultima te Load

P20.460

Code Design

Servicea bi l i ty Loa d

P30.423

Servicea bi l i ty

(Verti cal )

P40.291

Servicea bi l i ty

(Hori zonta l )



P3


- FIXED


P10.627

Code Des ign

Ultima te Loa d

P20.458

Code Des ign


P30.175

Servicea bi l i ty

(Verti ca l )

P40.261

Servicea bi l i ty

(Horizontal )

P4

P4P3



P1

P2

P1P2




P10.630

Code Design

Ultimate Load

P20.460

Code Design


P30.195

Serviceabi l i ty

(Vertical )

P40.193

Serviceabi l i ty

(Hori zonta l )

Appendix H:Load-displacement history- ANGELINE (Varying spans) H3.1



P10.651

Code Design

Ultimate Load

P20.479

Code Design


P30.314

Serviceabi l i ty

(Vertical )

P40.293

Serviceabi l i ty

(Hori zonta l )




P10.659

Code Des ign

Ultimate Load

P20.487

Code Des ign

Servicea bi l i ty Load

P30.302

Serviceabi l i ty

(Verti ca l)

P40.160

Serviceabi l i ty

(Hori zontal )




P10.655

Code Design

Ultimate Load

P20.484

Code Design


P30.147

Serviceabi l i ty

(Vertical )

P40.128

Serviceabi l i ty

(Hori zonta l )




- SPAN 28.0M


P10.659

Code Des ign

Ultimate Loa d

P20.487

Code Des ign


P30.157

Serviceabi l i ty

(Verti ca l)

P40.123

Serviceabi l i ty

(Hori zontal )

P3P4



P1

P2

P3

P4

P1

P2




P10.681

Code Design

Ultima te Loa d

P20.507

Code Design


P30.313

Servicea bi l i ty

(Verti cal )

P40.214

Servicea bi l i ty

(Hori zonta l )




- SPAN 32.0M

P3


P10.686

Code Des ign

Ul timate Loa d

P20.511

Code Des ign


P30.210

Serviceabi l i ty

(Verti ca l)

P40.157

Serviceabi l i ty

(Hori zontal )

P4



P1

P2

P3

P4

P1

P2




- SPAN 32.0M

P1

P2

P3

Load Factor Load Factor Description

P1

0.66

Code Design

Ul tima te Load

P20.488

Code Design


P30.192

Servicea bi l i ty

(Verti cal )

P40.195

Servicea bi l i ty

(Hori zonta l)


- SPAN 32.0M

P4

P4

P3



P1

P2




P10.667

Code Des ign

Ultimate Loa d

P20.494

Code Des ign


P30.401

Serviceabi l i ty

(Verti ca l)

P40.196

Serviceabi l i ty

(Hori zontal )

investigation into the structural behaviour of …

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