investigation into the structural behaviour of …
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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
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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
Chantal Rudman University of Stellenbosch
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.
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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.
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Chantal Rudman University of Stellenbosch
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
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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
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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
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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
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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
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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
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Chantal Rudman University of Stellenbosch
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
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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
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Chantal Rudman University of Stellenbosch
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
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Chantal Rudman University of Stellenbosch
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
Chantal Rudman University of Stellenbosch
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
Chantal Rudman University of Stellenbosch
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
Chantal Rudman University of Stellenbosch
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
State of the art in elastic instability 2.2
Chantal Rudman University of Stellenbosch
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
State of the art in elastic instability 2.3
Chantal Rudman University of Stellenbosch
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
State of the art in elastic instability 2.4
Chantal Rudman University of Stellenbosch
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.
State of the art in elastic instability 2.5
Chantal Rudman University of Stellenbosch
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
State of the art in elastic instability 2.6
Chantal Rudman University of Stellenbosch
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]
State of the art in elastic instability 2.7
Chantal Rudman University of Stellenbosch
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
State of the art in elastic instability 2.8
Chantal Rudman University of Stellenbosch
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
State of the art in elastic instability 2.9
Chantal Rudman University of Stellenbosch
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
State of the art in elastic instability 2.10
Chantal Rudman University of Stellenbosch
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
Chantal Rudman University of Stellenbosch
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.
Investigative analysis – elastic instability 3.2
Chantal Rudman University of Stellenbosch
Figure 3.1 Various examples in ANGELINE
Figure 3.2 Graphical Model - Columns
Investigative analysis – elastic instability 3.3
Chantal Rudman University of Stellenbosch
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.
Investigative analysis – elastic instability 3.4
Chantal Rudman University of Stellenbosch
Figure 3.3 Various Editors in ANGELINE
In-plane structural behaviour of portal frames 3.5
Chantal Rudman University of Stellenbosch
(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
In-plane structural behaviour of portal frames 3.6
Chantal Rudman University of Stellenbosch
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.
In-plane structural behaviour of portal frames 3.7
Chantal Rudman University of Stellenbosch
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.
In-plane structural behaviour of portal frames 3.8
Chantal Rudman University of Stellenbosch
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
In-plane structural behaviour of portal frames 3.9
Chantal Rudman University of Stellenbosch
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
In-plane structural behaviour of portal frames 3.10
Chantal Rudman University of Stellenbosch
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
In-plane structural behaviour of portal frames 3.11
Chantal Rudman University of Stellenbosch
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
−=(%)
In-plane structural behaviour of portal frames 3.12
Chantal Rudman University of Stellenbosch
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.
In-plane structural behaviour of portal frames 3.13
Chantal Rudman University of Stellenbosch
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.
In-plane structural behaviour of portal frames 3.14
Chantal Rudman University of Stellenbosch
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.
In-plane structural behaviour of portal frames 3.15
Chantal Rudman University of Stellenbosch
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
In-plane structural behaviour of portal frames 3.16
Chantal Rudman University of Stellenbosch
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.
In-plane structural behaviour of portal frames 3.17
Chantal Rudman University of Stellenbosch
(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.
In-plane structural behaviour of portal frames 3.18
Chantal Rudman University of Stellenbosch
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.
In-plane structural behaviour of portal frames 3.19
Chantal Rudman University of Stellenbosch
(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
In-plane structural behaviour of portal frames 3.20
Chantal Rudman University of Stellenbosch
(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.
In-plane structural behaviour of portal frames 4.1
Chantal Rudman University of Stellenbosch
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.
In-plane structural behaviour of portal frames 4.2
Chantal Rudman University of Stellenbosch
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.
In-plane structural behaviour of portal frames 4.3
Chantal Rudman University of Stellenbosch
• 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
Chantal Rudman University of Stellenbosch
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.
Modelling considerations for portal frames 5.2
Chantal Rudman University of Stellenbosch
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.
Modelling considerations for portal frames 5.3
Chantal Rudman University of Stellenbosch
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,
Modelling considerations for portal frames 5.4
Chantal Rudman University of Stellenbosch
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.
Modelling considerations for portal frames 5.5
Chantal Rudman University of Stellenbosch
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.
Modelling considerations for portal frames 5.6
Chantal Rudman University of Stellenbosch
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.
Modelling considerations for portal frames 5.7
Chantal Rudman University of Stellenbosch
(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.
Modelling considerations for portal frames 5.8
Chantal Rudman University of Stellenbosch
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
Modelling considerations for portal frames 5.9
Chantal Rudman University of Stellenbosch
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.
Modelling considerations for portal frames 5.10
Chantal Rudman University of Stellenbosch
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.
Modelling considerations for portal frames 5.11
Chantal Rudman University of Stellenbosch
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].
Modelling considerations for portal frames 5.12
Chantal Rudman University of Stellenbosch
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).
Modelling considerations for portal frames 5.13
Chantal Rudman University of Stellenbosch
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.
Modelling considerations for portal frames 5.14
Chantal Rudman University of Stellenbosch
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
Modelling considerations for portal frames 5.15
Chantal Rudman University of Stellenbosch
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
Modelling considerations for portal frames 5.16
Chantal Rudman University of Stellenbosch
(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
Modelling considerations for portal frames 5.17
Chantal Rudman University of Stellenbosch
(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.
Modelling considerations for portal frames 5.18
Chantal Rudman University of Stellenbosch
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
Chantal Rudman University of Stellenbosch
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
Design of portal frames according to DRAFT SANS 10160-1 &2 and SANS 10162-1:2005 6.2
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• 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
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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”
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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 ,, .. +
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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=
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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:
Design of portal frames according to DRAFT SANS 10160-1 &2 and SANS 10162-1:2005 6.7
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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
Design of portal frames according to DRAFT SANS 10160-1 &2 and SANS 10162-1:2005 6.8
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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
Design of portal frames according to DRAFT SANS 10160-1 &2 and SANS 10162-1:2005 6.9
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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=
Design of portal frames according to DRAFT SANS 10160-1 &2 and SANS 10162-1:2005 6.10
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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=
Design of portal frames according to DRAFT SANS 10160-1 &2 and SANS 10162-1:2005 6.11
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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.
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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.
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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
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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
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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.
Design of portal frames for parameter study 7.2
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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
Design of portal frames for parameter study 7.3
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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
Design of portal frames for parameter study 7.4
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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
Design of portal frames for parameter study 7.5
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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)
Design of portal frames for parameter study 7.6
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• 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.
Design of portal frames for parameter study 7.7
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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.
Design of portal frames for parameter study 7.8
Chantal Rudman University of Stellenbosch
(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.
Design of portal frames for parameter study 7.9
Chantal Rudman University of Stellenbosch
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.
Design of portal frames for parameter study 7.10
Chantal Rudman University of Stellenbosch
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.
Design of portal frames for parameter study 8.1
Chantal Rudman University of Stellenbosch
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.
Design of portal frames for parameter study 8.2
Chantal Rudman University of Stellenbosch
• 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
Design of portal frames for parameter study 8.3
Chantal Rudman University of Stellenbosch
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
Design of portal frames for parameter study 8.4
Chantal Rudman University of Stellenbosch
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
Design of portal frames for parameter study 8.5
Chantal Rudman University of Stellenbosch
(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 ≤+.
Design of portal frames for parameter study 8.6
Chantal Rudman University of Stellenbosch
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
Design of portal frames for parameter study 8.7
Chantal Rudman University of Stellenbosch
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
Design of portal frames for parameter study 8.8
Chantal Rudman University of Stellenbosch
(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.
Design of portal frames for parameter study 8.9
Chantal Rudman University of Stellenbosch
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
Deflection vs Roof Slope
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
Deflection vs Roof Slope
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)
Design of portal frames for parameter study 8.10
Chantal Rudman University of Stellenbosch
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
Chantal Rudman University of Stellenbosch
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.
Analyses results and discussion for the parameter study 9.2
Chantal Rudman University of Stellenbosch
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.
Analyses results and discussion for the parameter study 9.3
Chantal Rudman University of Stellenbosch
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.
Analyses results and discussion for the parameter study 9.4
Chantal Rudman University of Stellenbosch
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
(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.
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
6 9 0.7869 Full 0.7974 Full 0.8933 0.8933 Maximum 2.0m to left
6 12 0.8080 Full 0.8292 Full 0.9047 0.9047 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
Analyses results and discussion for the parameter study 9.5
Chantal Rudman University of Stellenbosch
(iii) Column Length - 14.0m
Table 9.1(c) Yielding values for frames – span 24.0m - pinned supports – column length 14.0m
(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.
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
(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.
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
24 10 12 0.6741 Full 0.6741 Full 0.8000 0.8071 Increasing Top
24 14 6 0.6673 Full 0.6802 Full 0.7538 0.7599 Increasing Top
24 14 12 0.7545 Full 0.7648 Full 0.8001 0.8071 Increasing Top
Analyses results and discussion for the parameter study 9.6
Chantal Rudman University of Stellenbosch
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
(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.
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
32 14 6 0.7085 Full 0.7085 Partial 0.7449 0.7554 Decreasing 1.0m to
right
32 14 12 0.7239 Full 0.7327 Partial 0.7364 0.7703 Decreasing 1.0m to
right
Analyses results and discussion for the parameter study 9.7
Chantal Rudman University of Stellenbosch
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
Analyses results and discussion for the parameter study 9.8
Chantal Rudman University of Stellenbosch
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
Analyses results and discussion for the parameter study 9.9
Chantal Rudman University of Stellenbosch
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.
Analyses results and discussion for the parameter study 9.10
Chantal Rudman University of Stellenbosch
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.
Analyses results and discussion for the parameter study 9.11
Chantal Rudman University of Stellenbosch
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.
Analyses results and discussion for the parameter study 9.12
Chantal Rudman University of Stellenbosch
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
(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.
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.
No asymptotic behaviour of the deflection is observed and hence, no singular points are found
for these configurations of frames.
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.
Analyses results and discussion for the parameter study 9.13
Chantal Rudman University of Stellenbosch
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
(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 9.6(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
Analyses results and discussion for the parameter study 9.14
Chantal Rudman University of Stellenbosch
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.
No asymptotic behaviour of the deflection is observed and hence, no singular points are found
for these configurations of frames.
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
Analyses results and discussion for the parameter study 9.15
Chantal Rudman University of Stellenbosch
Table 9.7(b) Load factor at serviceability of respective portal frames – fixed 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 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
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
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.
Analyses results and discussion for the parameter study 9.16
Chantal Rudman University of Stellenbosch
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
−=
Analyses results and discussion for the parameter study 9.17
Chantal Rudman University of Stellenbosch
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
Analyses results and discussion for the parameter study 9.18
Chantal Rudman University of Stellenbosch
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
Analyses results and discussion for the parameter study 9.19
Chantal Rudman University of Stellenbosch
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.
Analyses results and discussion for the parameter study 9.20
Chantal Rudman University of Stellenbosch
(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.
Analyses results and discussion for the parameter study 9.21
Chantal Rudman University of Stellenbosch
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
Analyses results and discussion for the parameter study 9.22
Chantal Rudman University of Stellenbosch
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.
Analyses results and discussion for the parameter study 9.23
Chantal Rudman University of Stellenbosch
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
Chantal Rudman University of Stellenbosch
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.
Reference: Chapter 7 and Chapter 9.
Conclusions and recommendations 10.2
Chantal Rudman University of Stellenbosch
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
Conclusions and recommendations 10.3
Chantal Rudman University of Stellenbosch
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.
Reference: Chapter 6 and Chapter 9.
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
Chantal Rudman University of Stellenbosch
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
Chantal Rudman University of Stellenbosch
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
Chantal Rudman University of Stellenbosch
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
Chantal Rudman University of Stellenbosch
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
Chantal Rudman University of Stellenbosch
APPENDIX A: ELASTIC STABILITY OF COLUMNS
Appendix A: Elastic stability of columns A.1
Chantal Rudman University of Stellenbosch
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
(1)Applied as percentage of applied load.(2)Theoretical value as calculated using Euler theory.
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
Chantal Rudman University of Stellenbosch
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
DescriptionPerturbation
Load Application
Displacement Mode of Column
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)
Figure 1. Displacement at Mid Node
-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)
Figure 2. Displacement at Top Node
-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
(1)Applied as percentage of applied load.(2)Theoretical value as calculated using Euler theory.
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
(1)Applied as percentage of applied load.(2)Theoretical value as calculated using Euler theory.
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
Chantal Rudman University of Stellenbosch
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
DescriptionPerturbation
Load Application
Displacement Mode of Column
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)
Figure 1. Displacement at Mid Node
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)
Figure 2. Displacement at Top Node
-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
(1)Applied as percentage of applied load.(2)Theoretical value as calculated using Euler theory.
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
(1)Applied as percentage of applied load.(2)Theoretical value as calculated using Euler theory.
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
Chantal Rudman University of Stellenbosch
APPENDIX B: ELASTIC STABILITY OF PORTAL FRAMES
Appendix B: Elastic stability of portal frames B.1.1
Chantal Rudman University of Stellenbosch
APPENDIX B: ELASTIC STABILITY OF PORTAL FRAMES
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
0.9 12.4657 terminated
0.8 12.4653 terminated
0.6 12.4631 terminated
0.5 12.4667 terminated Table B2. Results of configuration C2:
Initial load
factor
Load Factor Comments in command
prompt
1.0 12.4231 terminated
0.9 12.4288 terminated
0.8 12.4271 terminated
0.7 12.4269 terminated Table B3. Results of configuration C3:
Initial load
factor
Load Factor Comments in command
prompt
0.9 11.5892 singular point
0.8 12.0055 singular point
0.73 78.230
singular point
0.7 78.2303 terminated
Appendix B: Elastic stability of portal frames B.1.2
Chantal Rudman University of Stellenbosch
Table B4. Results of configuration C4:
Initial load
factor
Load Factor Comments in command
prompt
0.9 11.1873 singular point
0.8 11.9516 singular point
0.79 78.0550 terminated
0.78 78.01221 singular point
0.77 78.04542 terminated
0.76 78.0212 terminated
0.75 78.0559 terminated
Table B5. Results of configuration C5:
Initial load
factor
Load Factor Comments in command
prompt
1.1 20.8429 terminated
1.0 20.84597 terminated
0.9 20.8417 terminated
0.8 20.8380 terminated
0.7 20.8354 terminated
0.6 20.8322 terminated
. Table B6. Results of configuration C6:
Initial load
factor
Load Factor Comments in command
prompt
1.2 104.739 singular point
1.15 104.7601 terminated
1.1 104.7584 terminated
1.0 104.7569 terminated
Appendix B2: Elastic stability of portal frames B.2.1
Chantal Rudman University of Stellenbosch
B.2. RESULTS FOR ANALYSES INCLUDING AN INITIAL LOAD FACTOR OF 0.1
Table B7. Results of configuration C1:
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
Table B8. Results of configuration C2:
Configuration C2: pinned, full vertical load, 0.5% horizontal perturbation load.
state u1 at ridge u2 at ridge load factor u2 / load factor
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
Appendix B2: Elastic stability of portal frames B.2.2
Chantal Rudman University of Stellenbosch
Table B9. Results of configuration C3:
Configuration C3: fixed, full vertical load, no horizontal perturbation load.
state u1 at ridge u2 at ridge load factor u2 / load factor
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
Table B10. Results of configuration C4:
Configuration C4: fixed, full vertical load, 0.5% horizontal perturbation load.
state u1 at ridge u2 at ridge load factor u2 / load factor
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
Appendix B2: Elastic stability of portal frames B.2.3
Chantal Rudman University of Stellenbosch
Table B11. Results of configuration C5:
Configuration C5: pinned, half vertical load, no horizontal perturbation load.
state u1 at ridge u2 at ridge load factor u2 / load factor
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
Table B12. Results of configuration C6:
Configuration C6: fixed, half vertical load, no horizontal perturbation load.
state u1 at ridge u2 at ridge load factor u2 / load factor
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
Appendix B3: Elastic stability of portal frames B.3.1
Chantal Rudman University of Stellenbosch
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
Appendix B3: Elastic stability of portal frames B.3.2
Chantal Rudman University of Stellenbosch
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
Chantal Rudman University of Stellenbosch
APPENDIX C: NUMBER OF ELEMENTS
Appendix C: Number of elements C.1.1
Chantal Rudman University of Stellenbosch
APPENDIX C: NUMBER OF ELEMENTS
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.
Appendix C: Number of elements C.1.2
Chantal Rudman University of Stellenbosch
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
C.2.1 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. 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.
Appendix C: Number of elements C.1.3
Chantal Rudman University of Stellenbosch
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
Displacement y1 (mm) 26.16 34.49 ‐
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
Displacement y1 (mm) 26.14 34.37 ‐
24 elemen
ts
Displacement y2 (mm) ‐ ‐ 291.33
C.3 ABAQUS
C.3.1 Basis of comparison
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.
Appendix C: Number of elements C.1.4
Chantal Rudman University of Stellenbosch
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
Chantal Rudman University of Stellenbosch
APPENDIX D: NOTIONAL HORIZONTAL LOAD
Appendix D: Notional horizontal load D.1
Chantal Rudman University of Stellenbosch
APPENDIX D: NOTIONAL HORIZONTAL LOAD
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.
Appendix D: Notional horizontal load D.2
Chantal Rudman University of Stellenbosch
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
=
Appendix D: Notional horizontal load D.3
Chantal Rudman University of Stellenbosch
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.
Appendix D: Notional horizontal load D.4
Chantal Rudman University of Stellenbosch
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
Chantal Rudman University of Stellenbosch
APPENDIX E: PORTAL FRAME DESIGN
Appendix E: Portal frame design E.1
Chantal Rudman University of Stellenbosch
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
Appendix E: Portal frame design E.2
Chantal Rudman University of Stellenbosch
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
Appendix E: Portal frame design E.3
Chantal Rudman University of Stellenbosch
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
limiting width‐to‐thickness ratio class 2
9.09
limiting width‐to‐thickness ratio class 3
10.69
class of section according to flange Class 1
width‐to‐thickness ratio of web
36.59
limiting width‐to‐thickness ratio class 1
58.20
limiting width‐to‐thickness ratio class 2
89.41
limiting width‐to‐thickness ratio class 3
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
Appendix E: Portal frame design E.4
Chantal Rudman University of Stellenbosch
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<
Appendix E: Portal frame design E.5
Chantal Rudman University of Stellenbosch
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
⎟⎟⎠
⎞⎜⎜⎝
⎛+
=
Appendix E: Portal frame design E.6
Chantal Rudman University of Stellenbosch
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
ultimate shear stress option 2
342.60 MPa
ultimate shear stress option 3
342.60 MPa
ultimate shear stress option 4
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 +=
Appendix E: Portal frame design E.7
Chantal Rudman University of Stellenbosch
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
Chantal Rudman University of Stellenbosch
APPENDIX F: DESIGN RESULTS
Appendix F: Design results F.1
Chantal Rudman University of Stellenbosch
APPENDIX F: DESIGN RESULTS
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
Chantal Rudman University of Stellenbosch
APPENDIX G: LOAD‐DISPLACEMENT HISTORY ‐ ABAQUS
Appendix G: Load‐displacement history‐ ABAQUS G.1
Chantal Rudman University of Stellenbosch
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
Appendix G: Load‐displacement history‐ ABAQUS G.2
Chantal Rudman University of Stellenbosch
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
‐0.25 ‐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 6.0m ‐ Slope 12 degrees
Span 24.0m(F) ‐ Column 10.0m ‐ Slope 6 degees
Span 24.0m(F) ‐ Column 10.0m ‐ Slope 12 degrees
Span 24.0m(F) ‐ Column 14.0m ‐ Slope 6 degrees
Span 24.0m(F) ‐ 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
‐0.80 ‐0.60 ‐0.40 ‐0.20 0.00Displacement (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 0.05 0.10 0.15 0.20 0.25Displacement (m)
Load
Factor
G.2 LOAD DISPLACEMENT HISTORY – FIXED SUPPORTS‐ ABAQUS
Appendix G: Load‐displacement history‐ ABAQUS G.3
Chantal Rudman University of Stellenbosch
G.3 LOAD DISPLACEMENT HISTORY ‐ PINNED SUPPORTS‐ VARYING SPANS ‐ ABAQUS
Right Column ‐ δh vs Load Factor
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
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 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
Ridge ‐ δv vs Load Factor
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
Chantal Rudman University of Stellenbosch
APPENDIX H: LOAD‐DISPLACEMENT HISTORY ‐ ANGELINE
Appendix H : Load-displacement history- ANGELINE (Pinned supports) H1.1
Chantal Rudman University of Stellenbosch
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
Appendix H : Load-displacement history- ANGELINE (Pinned supports) H1.2
Chantal Rudman University of Stellenbosch
P3
PORTAL FRAME COMPARISSON – (COLUMN 6.0M, ROOF SLOPE 120)
Load Factor Value Description
P10.645
Code Des ign
Ul timate Load
P20.474
Code Des ign
Serviceabi l i ty Load
P30.334
Serviceabi l i ty
(Vertica l )
P40.183
Serviceabi l i ty
(Horizonta l)
P4
P3
P4
Figure 1.1 Displacement at top node of left column
Figure 1.2 Displacement of node at ridge
P1
P2
P1
P2
Appendix H : Load-displacement history- ANGELINE (Pinned supports) H1.3
Chantal Rudman University of Stellenbosch
P3
P4
PORTAL FRAME COMPARISSON – (COLUMN 14.0M, ROOF SLOPE 60)
Load Factor Value Description
P10.627
Code Design
Ultimate Load
P20.458
Code Design
Serviceabi l i ty Load
P30.143
Serviceabi l i ty
(Verti cal )
P40.151
Serviceabi l i ty
(Hori zonta l )
P3
P4
Figure 1.1 Displacement at top node of left column
Figure 1.2 Displacement of node at ridge
P1
P2
P1
P2
Appendix H : Load-displacement history- ANGELINE (Pinned supports) H1.4
Chantal Rudman University of Stellenbosch
Load Factor Value Description
P10.630
Code Des ign
Ultimate Load
P20.461
Code Des ign
Serviceabi l i ty Load
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
Chantal Rudman University of Stellenbosch
P1P2
P3
PORTAL FRAME COMPARISON – COLUMN 6.0M, ROOF SLOPE 60
- FIXED
Load Factor Value Description
P10.641
Code Design
Ul tima te Load
P20.471
Code Design
Serviceabi l i ty Load
P30.379
Servicea bi l i ty
(Verti cal )
P40.437
Servicea bi l i ty
(Hori zonta l)
P4
P3
P4
Figure 1.1 Displacement at top node of left column
Figure 1.2 Displacement of node at ridge
P1P2
Appendix H:Load-displacement history- ANGELINE (Fixed supports) H2.2
Chantal Rudman University of Stellenbosch
Load Factor Value Description
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 )
Appendix H:Load-displacement history- ANGELINE (Fixed supports) H2.3
Chantal Rudman University of Stellenbosch
P3
PORTAL FRAME COMPARISON – COLUMN 14.0M, ROOF SLOPE 60
- FIXED
Load Factor Value Description
P10.627
Code Des ign
Ultima te Loa d
P20.458
Code Des ign
Serviceabi l i ty Load
P30.175
Servicea bi l i ty
(Verti ca l )
P40.261
Servicea bi l i ty
(Horizontal )
P4
P4P3
Figure 1.1 Displacement at top node of left column
Figure 1.2 Displacement of node at ridge
P1
P2
P1P2
Appendix H:Load-displacement history- ANGELINE (Fixed supports) H2.4
Chantal Rudman University of Stellenbosch
Load Factor Value Description
P10.630
Code Design
Ultimate Load
P20.460
Code Design
Serviceabi l i ty Load
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
Chantal Rudman University of Stellenbosch
Load Factor Value Description
P10.651
Code Design
Ultimate Load
P20.479
Code Design
Serviceabi l i ty Load
P30.314
Serviceabi l i ty
(Vertical )
P40.293
Serviceabi l i ty
(Hori zonta l )
Appendix H:Load-displacement history- ANGELINE (Varying spans) H3.2
Chantal Rudman University of Stellenbosch
Load Factor Value Description
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 )
Appendix H:Load-displacement history- ANGELINE (Varying spans) H3.3
Chantal Rudman University of Stellenbosch
Load Factor Value Description
P10.655
Code Design
Ultimate Load
P20.484
Code Design
Serviceabi l i ty Load
P30.147
Serviceabi l i ty
(Vertical )
P40.128
Serviceabi l i ty
(Hori zonta l )
Appendix H:Load-displacement history- ANGELINE (Varying spans) H3.4
Chantal Rudman University of Stellenbosch
PORTAL FRAME COMPARISON – COLUMN 14.0M, ROOF SLOPE 120
- SPAN 28.0M
Load Factor Value Description
P10.659
Code Des ign
Ultimate Loa d
P20.487
Code Des ign
Servicea bi l i ty Load
P30.157
Serviceabi l i ty
(Verti ca l)
P40.123
Serviceabi l i ty
(Hori zontal )
P3P4
Figure 1.1 Displacement at top node of left column
Figure 1.2 Displacement of node at ridge
P1
P2
P3
P4
P1
P2
Appendix H:Load-displacement history- ANGELINE (Varying spans) H3.5
Chantal Rudman University of Stellenbosch
Load Factor Value Description
P10.681
Code Design
Ultima te Loa d
P20.507
Code Design
Servicea bi l i ty Loa d
P30.313
Servicea bi l i ty
(Verti cal )
P40.214
Servicea bi l i ty
(Hori zonta l )
Appendix H:Load-displacement history- ANGELINE (Varying spans) H3.6
Chantal Rudman University of Stellenbosch
PORTAL FRAME COMPARISON – COLUMN 6.0M, ROOF SLOPE 120
- SPAN 32.0M
P3
Load Factor Value Description
P10.686
Code Des ign
Ul timate Loa d
P20.511
Code Des ign
Servicea bi l i ty Loa d
P30.210
Serviceabi l i ty
(Verti ca l)
P40.157
Serviceabi l i ty
(Hori zontal )
P4
Figure 1.1 Displacement at top node of left column
Figure 1.2 Displacement of node at ridge
P1
P2
P3
P4
P1
P2
Appendix H:Load-displacement history- ANGELINE (Varying spans) H3.7
Chantal Rudman University of Stellenbosch
PORTAL FRAME COMPARISON – COLUMN 14.0M, ROOF SLOPE 60
- SPAN 32.0M
P1
P2
P3
Load Factor Load Factor Description
P1
0.66
Code Design
Ul tima te Load
P20.488
Code Design
Serviceabi l i ty Load
P30.192
Servicea bi l i ty
(Verti cal )
P40.195
Servicea bi l i ty
(Hori zonta l)
PORTAL FRAME COMPARISON – COLUMN 14.0M, ROOF SLOPE 60
- SPAN 32.0M
P4
P4
P3
Figure 1.1 Displacement at top node of left column
Figure 1.2 Displacement of node at ridge
P1
P2
Appendix H:Load-displacement history- ANGELINE (Varying spans) H3.8
Chantal Rudman University of Stellenbosch
Load Factor Value Description
P10.667
Code Des ign
Ultimate Loa d
P20.494
Code Des ign
Servicea bi l i ty Load
P30.401
Serviceabi l i ty
(Verti ca l)
P40.196
Serviceabi l i ty
(Hori zontal )