Investigation into the structural behaviour of portal frames

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(1)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.

(2) Investigation into the structural behaviour of portal frames. i. 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. Chantal Rudman. University of Stellenbosch.

(3) Investigation into the structural behaviour of portal frames. ii. SYNOPSIS The current trend of the building industry by which stronger but more slender elements are designed, due to economical considerations, contributes to the serious consideration of the stability of structures. The Southern African Institute of Steel Construction (SAISC) has expressed its concerns about the stability of steel structures with specific interest to the elastic instability of portal frames.. The research will focus on the in-plane purely elastic stability of portal frames. In this investigation a distinction is made between the prediction of instability by means of evaluating the nonlinear load-path and instability without prior warning. The analyses done in this research uses a software programme ANGELINE which addresses both of these aspects. This software programme is especially developed for the academic research into geometric nonlinear behaviour of slender structures.. The structural analyses reveal that elastic instability is not a concern for portal frames with practical dimensions. Further investigation includes determining what the limiting in-plane behaviour is. This is done by evaluating a benchmark portal frame and it is shown that plastic deformation in the frame is the limiting criterion. This is done using the commercial software programme, ABAQUS.. The research is concluded by evaluating a selection of portal frames, with practical dimensions, in order to substantiate the conclusions above.. This is done by designing the selection of. portal frames according to the DRAFT SANS 10160-1 & 2:2008, and SANS 10162-1:2005. Subsequently, these frames are analysed using ANGELINE (including geometric nonlinearity) and ABAQUS (second-order elastic perfectly plastic analysis).. Although it is shown that the limiting in-plane behaviour of portal frames is governed by the plastic deformation of the members it becomes clear that the design of the selection of portal frames in this research is governed by the serviceability limit state requirements.. Chantal Rudman. University of Stellenbosch.

(4) Investigation into the structural behaviour of portal frames. iii. SAMEVATTING Die huidige neiging in die konstruksie industrie om sterker strukture met ‘n hoër slankheid te ontwerp deur gebruik te maak van hoër sterkte materiale het aanleiding gegee tot die ernstige oorweging van die stabiliteit van hierdie strukture. Die Suider-Afrikaanse Instituut vir Staal Konstruksie het besorgdheid uitgespreek oor die stabiliteit van staalstrukture met spesifieke fokus op die elastiese onstabiliteit van portaalrame. Hierdie navorsing sal fokus op die suiwer elastiese in-vlak stabiliteit van portaalrame. In hierdie ondersoek word ‘n onderskeiding gemaak tussen die voorspelling van onstabiliteit deur die nie-lineêre belasting-roete te evalueer asook onstabiliteit sonder enige vooraf waarskuwing. Die analises wat uitgevoer is in hierdie ondersoek gebruik ‘n sagteware paket ANGELINE wat beide hierdie aspekte aanspreek. Hierdie sagteware is spesifiek vir akademiese navorsing in geometriese nie-lineêre gedrag ontwikkel. Die strukturele analises toon dat elastiese onstabiliteit nie van groot belang is vir portaalrame met praktiese afmetings nie. Verdere ondersoek sluit die bepaling van die beperkende in-vlak gedrag in. Dit is uitgevoer deur ‘n voorbeeld portaalraam te evalueer en daar word getoon dat plastiese vervorming van die raam die beperkende maatstaf is. Die kommersiële sagteware paket ABAQUS is vir hierdie doel gebruik. Die ondersoek is afgesluit deur ‘n reeks portaalrame met praktiese afmetings te evalueer ten einde die bogenoemde gevolgtrekkings te staaf. Dit is gedoen deur die reeks portaalrame te ontwerp volgens die konsep kode SANS 10160-1 & 2:2008 en die ontwerpkode SANS 101621:2005. Hierna is analises op die rame uitgevoer deur van ANGELINE (wat geometriese nielineêriteit 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.. Chantal Rudman. University of Stellenbosch.

(5) Investigation into the structural behaviour of portal frames. iv. ACKNOWLEDGEMENTS. The author of this thesis would like to express her gratitude to the following people:. -. Professor PE Dunaiski for his patience and guidance and teaching me that an elephant should be eaten one bite at a time.. -. Professor PJ Pahl for his expert knowledge and time.. -. My classmates who made the last two years an experience of a life time.. -. And last but not least: my mother, father, brother and fiancé. Without them I would never have seen the light at the end of the tunnel.. Chantal Rudman. University of Stellenbosch.

(6) Investigation into the structural behaviour of portal frames. v. TABLE OF CONTENTS. DECLARATION……………………………………………………………………………………………i SYNOPSIS…………………………………………………………………………………………….…..ii SAMEVATTING…………………………………………………………..………………..…………..iii ACKNOWLEDGEMENTS…….……………………..……………………….…….……….…………iv TABLE OF CONTENTS…………………………………………….…………….………………….….v LIST OF APPENDICES.................................................................................. viii LIST OF FIGURES ......................................................................................... ix LIST OF TABLES........................................................................................... xi LIST OF SYMBOLS ..................................................................................... .xii LIST OF ABBREVIATIONS .............................................................................xiv. 1. INTRODUCTION ..........................................................................1.1 1.1 1.2 1.3. 2. THE PROBLEM........................................................................................................................... 1.1 OBJECTIVES ............................................................................................................................... 1.1 FLOW CHART FOR PART 1 ....................................................................................................... 1.2. STATE OF THE ART IN ELASTIC INSTABILITY .................................2.1 2.1 2.2 2.3 2.4 2.5. 3. THE REAL BEHAVIOUR OF STRUCTURES ............................................................................... 2.1 ELASTIC INSTABILITY IN PITCHED ROOF STEEL FRAMES..................................................... 2.6 DETERMINING THE POINT OF ELASTIC INSTABILITY – INCLUDING GEOMETRIC NONLINEARITY.......................................................................................................................... 2.7 ANGELINE .................................................................................................................................. 2.8 SUMMARY ............................................................................................................................... 2.10. INVESTIGATIVE ANALYSIS – ELASTIC INSTABILITY........................3.1 3.1 3.2 3.3 3.4 3.5. 4. ANGELINE .................................................................................................................................. 3.1 COLUMN INVESTIGATION ....................................................................................................... 3.9 INVESTIGATIVE ANALYSES: PORTAL FRAMES .................................................................... 3.14 CONCLUSIONS ........................................................................................................................ 3.18 SUMMARY ............................................................................................................................... 3.20. IN-PLANE STRUCTURAL BEHAVIOUR OF PORTAL FRAMES...........4.1 4.1 4.2. INTRODUCTION ........................................................................................................................ 4.1 OBJECTIVES ............................................................................................................................... 4.2. Chantal Rudman. University of Stellenbosch.

(7) Investigation into the structural behaviour of portal frames 4.3. 5. vi. METHOD OF APPROACH ......................................................................................................... 4.2. MODELLING CONSIDERATIONS FOR PORTAL FRAMES.................5.1 5.1 5.2 5.3 5.4 5.5 5.5 5.6. 6. IDENTIFICATION OF A TYPICAL PORTAL FRAME AND LOAD PATTERN............................. 5.1 TYPES OF ELEMENTS TO BE USED IN MODELLING .............................................................. 5.3 IMPERFECTIONS ....................................................................................................................... 5.5 MODELLING OF HAUNCHES.................................................................................................... 5.8 PLASTIC DEFORMATION OF STRUCTURAL MEMBERS ........................................................ 5.9 COMPATIBILITY OF SOFTWARE PACKAGES ........................................................................ 5.17 SUMMARY ............................................................................................................................... 5.18. DESIGN OF PORTAL FRAMES ACCORDING TO DRAFT SANS 10160-1, & 2 : 2008 AND SANS 10162-1:2005. ..........................................6.1 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8. 7. INTRODUCTION ........................................................................................................................ 6.1 LIMIT STATE DESIGN ................................................................................................................ 6.1 DESIGN OF A PORTAL FRAME ACCORDING TO DRAFT SANS 10160-1 & 2 : 2008 AND SANS 10162-1:2005 ................................................................................................................. 6.2 LOAD COMBINATIONS............................................................................................................. 6.3 CAPACITY OF MEMBERS – ULTIMATE LIMIT STATE ............................................................ 6.5 SERVICEABILITY LIMIT STATE ................................................................................................ 6.11 DESIGNING THE BENCHMARK EXAMPLE ............................................................................ 6.11 SUMMARY ............................................................................................................................... 6.14. ANALYSIS OF BENCHMARK PORTAL FRAME ................................7.1 7.1 7.2 7.3. 8. ANALYSIS OF BENCHMARK PORTAL FRAME ........................................................................ 7.1 CONCLUSIONS ........................................................................................................................ 7.10 SUMMARY ............................................................................................................................... 7.10. DESIGN OF PORTAL FRAMES FOR PARAMETER STUDY ................8.1 8.1 8.2 8.3 8.4. 9. DEFINITION OF PORTAL FRAMES ........................................................................................... 8.1 DESIGN OF PORTAL FRAMES FOR THE PARAMETER STUDY .............................................. 8.4 CONCLUSIONS ........................................................................................................................ 8.10 SUMMARY ............................................................................................................................... 8.10. ANALYSES RESULTS AND DISCUSSION - PARAMETER STUDY .......9.1 9.1 9.2 9.3 9.4. 10. RESULTS ..................................................................................................................................... 9.2 DISCUSSION ON RESULTS...................................................................................................... 9.16 CONCLUSIONS ........................................................................................................................ 9.21 SUMMARY ............................................................................................................................... 9.23. CONCLUSIONS AND RECOMMENDATIONS ................................10.1. 10.1 INTRODUCTION ...................................................................................................................... 10.1 10.2 CONCLUSIONS AND RECOMMENDATIONS ........................................................................ 10.1. Chantal Rudman. University of Stellenbosch.

(8) Investigation into the structural behaviour of portal frames. 11 11.1 11.2 11.3 11.4 11.5. vii. REFERENCES..............................................................................11.1 BOOKS ...................................................................................................................................... 11.1 PUBLICATIONS ........................................................................................................................ 11.2 DESIGN CODES ........................................................................................................................ 11.2 INTERVIEWS ............................................................................................................................ 11.3 ELECTRONIC REFERENCES ..................................................................................................... 11.3. Chantal Rudman. University of Stellenbosch.

(9) Investigation into the structural behaviour of portal frames. viii. LIST OF APPENDICES APPENDIX A: APPENDIX B: APPENDIX C: APPENDIX D: APPENDIX E: APPENDIX F: APPENDIX G: APPENDIX H:. ELASTIC STABILITY OF COLUMNS ELASTIC STABILITY OF PORTAL FRAMES NUMBER OF ELEMENTS NOTIONAL HORIZONTAL LOAD PORTAL FRAME DESIGN DESIGN RESULTS LOAD-DISPLACEMENT HISTORY – ABAQUS LOAD-DISPLACEMENT HISTORY - ANGELINE. Chantal Rudman. University of Stellenbosch.

(10) Investigation into the structural behaviour of portal frames. ix. LIST OF FIGURES Figure 1.1 Flow chart for Part 1 ................................................................................................. 1.2 Figure 2.1 Nonlinear behaviour of structures............................................................................ 2.2 Figure 2.2 Frame second-order effects: (a) P- effects and (b) P- effects .............................. 2.3 Figure 2.3 Load deflection paths of a structure ........................................................................ 2.5 Figure 2.4 Snap-through behaviour .......................................................................................... 2.5 Figure 2.5 Elastic instability of portal frames............................................................................ 2.6 Figure 3.1 Various examples in ANGELINE................................................................................ 3.2 Figure 3.2 Graphical Model - Columns...................................................................................... 3.2 Figure 3.3 Various Editors in ANGELINE.................................................................................... 3.4 Figure 3.4 Session.java ............................................................................................................. 3.5 Figure 3.5 Portal frame default model..................................................................................... 3.6 Figure 3.6 Graphical model of portal frame............................................................................. 3.7 Figure 3.7 Generator.java......................................................................................................... 3.8 Figure 3.8 Profile.java............................................................................................................... 3.8 Figure 3.9 K-values for different end restraints ..................................................................... 3.10 Figure 3.10 Flow diagram illustrating the analysis procedure and selection of columns ...... 3.10 Figure 3.11. Selection of portal frames................................................................................... 3.14 Figure 3.12 Vertical deflection u2 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. Chantal Rudman. University of Stellenbosch.

(11) Investigation into the structural behaviour of portal frames. x. Figure 7.12 Load-Axial force history.......................................................................................... 7.9 Figure 7.13 Load-Bending moment history............................................................................... 7.9 Figure 8.1 Sequence of analyses for each frame ...................................................................... 8.1 Figure 8.2 Portal frames with pinned supports with varying column length and roof slope ... 8.2 Figure 8.3 Portal frames with fixed supports with varying column length and roof slope....... 8.3 Figure 8.4 Portal frames with varying spans, column length and roof slope........................... 8.3 Figure 8.5 Distribution of forces - illustrating maximum forces ............................................... 8.4 Figure 8.6 Design values used ................................................................................................... 8.6 Figure 8.7 Maximum vertical and horizontal deflection........................................................... 8.9 Figure 9.1 Flow chart of procedure........................................................................................... 9.1 Figure 9.2 Material model......................................................................................................... 9.3 Figure 9.3 Comparison of percentage difference -right hand column and max load factor .. 9.17 Figure 9.4 Behaviour compared to ABAQUS results ............................................................... 9.18 Figure 9.5 Comparison of load factor...................................................................................... 9.21 Figure 10.1 Portal frame with tapered members .................................................................... 10.2. Chantal Rudman. University of Stellenbosch.

(12) Investigation into the structural behaviour of portal frames. xi. LIST OF TABLES Table 3.1 Values obtained for column analyses...................................................................... 3.11 Table 3.2. Example of effect of axial shortening..................................................................... 3.13 Table 5.1 Forces at allocated elements – various software programmes ............................... 5.18 Table 5.2 Percentage differences in forces............................................................................. 5.18 Table 6.1 Classification of sections in axial compression.......................................................... 6.5 Table 6.2 Classification of flanges – flexural ............................................................................. 6.7 Table 6.3 Classification of webs– flexural ................................................................................. 6.7 Table 6.4 Example for calculation of dead weight of the structure........................................ 6.12 Table 6.5 Example for calculation of imposed loads of the structure .................................... 6.12 Table 6.6 Column resistances – I-section 254 x 146 x 37........................................................ 6.13 Table 6.7 Rafter resistances – I-section 254 x 146 x 37 .......................................................... 6.14 Table 8.1 Designated sections – span 24.0m, pinned supports............................................... 8.6 Table 8.2 Designated sections – span 24.0m, fixed supports ................................................... 8.7 Table 8.3 Designated sections – varying span lengths.............................................................. 8.8 Table 9.1(a) Yielding values for frames – span 24.0m - pinned supports – 6.0m ................... 9.4 Table 9.1(b) Yielding values for frames – span 24.0m - pinned supports – 10.0m ................. 9.4 Table 9.1(c) Yielding values for frames – span 24.0m - pinned supports – 14.0m ................. 9.4 Table 9.2 Yielding values for frames – span 24.0m – fixed supports........................................ 9.5 Table 9.3 Yielding values for frames – varying length spans .................................................... 9.6 Table 9.4(a) Deflection at selected nodes – pinned supports ................................................ 9.10 Table 9.4(b) Deflection at selected nodes – pinned supports-ridge....................................... 9.10 Table 9.5 Deflection at selected nodes –fixed supports ......................................................... 9.12 Table 9.6(a) Deflection at selected nodes – varying spans ..................................................... 9.13 Table 9.6(b) Deflection at selected nodes – varying spans - ridge.......................................... 9.13 Table 9.7(a) Load factor at serviceability of portal frames – pinned supports – span 24.0m . 9.14 Table 9.7(b) Load factor at serviceability of portal frames – fixed supports – span 24.0m .... 9.14 Table 9.7(c) Load factor at serviceability of portal frames – varying spans ............................ 9.14. Chantal Rudman. University of Stellenbosch.

(13) Investigation into the structural behaviour of portal frames. xii. LIST OF SYMBOLS A. cross-sectional area. Ad. design value of accidental action. Av. shear area. Cr. critical axial compressive force. Cu. Ultimate compressive force in member. Cy. axial compressive force in member at yield stress. E. elastic modulus of steel. G. shear modulus of steel. Gk,j. characteristic value of permanent action j, self weight. I. moment of inertia. K. effective length factor. L. gross length of member. Mr. Factored moment resistance of member. Mu. Ultimate bending moment in member. P. relevant representative value of prestressing action. Qk,1. characteristic value of leading variable action, imposed load. Qk,i. characteristic value of accompanying variable action i. Tr. Factored tensile resistance of member. Tu. Ultimate tensile force in member. U1. factor to account for moment gradient and for second-order effects of axial force acting on the deformed member. Vr. Factored shear resistance of member. W. width to thickness ratio. Wlim Limit of width to thickness ratio Ze. elastic section modulus of steel section. Zpl. plastic section modulus of steel section. B. half of width of flange of column. F. calculated compressive stress in element. fe. elastic critical buckling stress in axial compression. fs. Ultimate shear stress. fy. Yield stress. H. height of section. hw. Clear depth of web between flanges. Chantal Rudman. University of Stellenbosch.

(14) Investigation into the structural behaviour of portal frames. xiii. kv. shear buckling coefficient. N. material regression factor. R. radius of gyration. S. centre-to-centre distance between transverse web stiffeners. tf. thickness of flange. tw. thickness of web. Σ. combined effect. Φ. resistance factor for structural steel. γG,j. partial factor for permanent action j. γQ,1. partial factor for leading variable action. γQ,i. partial factor for accompanying variable action i. Λ. non-dimensional slenderness ratio. ψi. action combination factor corresponding to accompanying variable action i. Chantal Rudman. University of Stellenbosch.

(15) Investigation into the structural behaviour of portal frames. xiv. LIST OF ABBREVIATIONS ANGELINE. Analysis of geometrically nonlinear structures. SAISC. Southern African Institute of Steel Construction. SANS. South African National Standards. TUB. Technical University Berlin. LL. Live Load. DL. Dead Load. Chantal Rudman. University of Stellenbosch.

(16) Introduction. 1. 1.1. INTRODUCTION. Due to the ever increasing complexity of structures being designed, it has become an absolute necessity that the behaviour of structures related to the overall and member stability is understood. A recent article published in the Journal of Engineering Mechanics [14] states the following:. “As far as structural engineering is concerned, scientific and technological advances are often fostered by the occurrence of collapses involving a more or less relevant amount of damage and in the most unfortunate cases, also the loss of human lives”. This statement was made due to tragic collapse of the World Trade Centre twin towers, on the September 11, 2001, which highlights the importance of the understanding of behaviour of real structures.. 1.1. THE PROBLEM. The current trend of the building industry by which stronger but more slender elements are designed, due to economical considerations, contributes to the serious consideration of the stability of structures. Portal frames are widely used in the industrial sector in South Africa and the possible elastic instability of these frames has raised concerns at the Southern African Institute of Steel Construction (SAISC).. 1.2. OBJECTIVES. This research is subdivided into two parts. The first part and main focus of the research will include the investigation into the in-plane stability of pitched roof steel frames. This means that only strong-axis bending is considered and it is assumed that the portal frame is sufficiently laterally restrained.. The question that must be answered is the following:. Is purely geometric elastic instability a problem in portal frames?. In the first part it becomes clear that elastic instability is not a problem in portal frames. The second part shifts the focus of the research towards the inclusion of material nonlinearity.. Chantal Rudman. University of Stellenbosch.

(17) Introduction. 1.2. The objective in this part of the research is to determine:. The limiting in-plane behaviour of portal frames by including plastic deformation.. A detailed approach and flow chart for the second part of the research project is included in Chapter 4. The flow chart for the first part of the thesis is shown below.. 1.3. FLOW CHART FOR PART 1. In Chapter 2 the elastic behaviour and stability of structures are discussed with reference to portal frames. The software programme that is used for this investigation is explained.. This is followed by an investigative analysis in Chapter 3, which entails the behaviour of the frames by determining the elastic instability of selected portal frames. This is done by means of verifying the behaviour in columns and the influence of the perturbation load. Subsequently, selected portal frames are investigated and their elastic stability evaluated.. A flow chart for Part 1 of this research is shown.. Figure 1.1 Flow chart for Part 1. Chantal Rudman. University of Stellenbosch.

(18) State of the art in elastic instability. 2. 2.1. 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. Chantal Rudman. University of Stellenbosch.

(19) State of the art in elastic instability. 2.2. 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.. Displacement u. 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.. Chantal Rudman. University of Stellenbosch.

(20) State of the art in elastic instability. 2.1.2. 2.3. Types of nonlinearity. Two types of nonlinearities are distinguished [4]:. (a). •. Geometric nonlinearity and. •. Material nonlinearity. 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).. (a). (b). P H B B’ A. P C D. C’. C. B A. D. Figure 2.2 Frame second-order effects: (a) P-∆ effects and (b) P-δ effects. Chantal Rudman. University of Stellenbosch.

(21) State of the art in elastic instability. 2.4. 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 crosssectional 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.. Chantal Rudman. University of Stellenbosch.

(22) State of the art in elastic instability. 2.5. 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.. th le pa Stab. e nc ue eq h g S pat din le loa tab Un Uns. Load Factor. Limit state. Displacement. 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. Chantal Rudman. University of Stellenbosch.

(23) State of the art in elastic instability. 2.6. 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]. Chantal Rudman. University of Stellenbosch.

(24) State of the art in elastic instability. 2.3. 2.7. 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 snapthrough, 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. Chantal Rudman. University of Stellenbosch.

(25) State of the art in elastic instability. 2.8. 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 inplane 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. Chantal Rudman. University of Stellenbosch.

(26) State of the art in elastic instability. 2.9. 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 Chantal Rudman. University of Stellenbosch.

(27) State of the art in elastic instability. 2.10. 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.. Chantal Rudman. University of Stellenbosch.

(28) Investigative analysis – elastic instability. 3. 3.1. 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.. Chantal Rudman. University of Stellenbosch.

(29) Investigative analysis – elastic instability. 3.2. Figure 3.1 Various examples in ANGELINE. Figure 3.2 Graphical Model - Columns. Chantal Rudman. University of Stellenbosch.

(30) Investigative analysis – elastic instability. 3.3. 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.. Chantal Rudman. University of Stellenbosch.

(31) Investigative analysis – elastic instability. 3.4. Figure 3.3 Various Editors in ANGELINE. Chantal Rudman. University of Stellenbosch.

(32) In-plane structural behaviour of portal frames. (c). 3.5. 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. Chantal Rudman. University of Stellenbosch.

(33) In-plane structural behaviour of portal frames. 3.1.3. Part 2:. (a). Graphical User Interface. 3.6. Portal frame analyses. 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.. Chantal Rudman. University of Stellenbosch.

(34) In-plane structural behaviour of portal frames. 3.7. 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.. Chantal Rudman. University of Stellenbosch.

(35) In-plane structural behaviour of portal frames. 3.8. 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. Chantal Rudman. University of Stellenbosch.

(36) In-plane structural behaviour of portal frames. 3.2. 3.9. 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:. PE =. π 2EI (KL )2. 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. Chantal Rudman. University of Stellenbosch.

(37) In-plane structural behaviour of portal frames. 3.10. 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. Chantal Rudman. University of Stellenbosch.

(38) In-plane structural behaviour of portal frames. 3.11. 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. 203x133x25. 457x191x75. 533x210x122. Support Fixity. Column Length (m). ANGELINE Result. Euler Value. % Difference. Pinned Pinned Pinned Fixed Fixed Fixed Pinned Pinned Pinned Fixed Fixed Fixed Pinned Pinned Pinned Fixed Fixed Fixed. 6 8 10 6 8 10 6 8 10 6 8 10 6 8 10 6 8 10. 0.1290 0.0726 0.0465 0.5184 0.2916 0.1866 1.8340 1.0316 0.6602 7.3673 4.1441 2.6523 4.1841 2.3536 1.5063 16.8082 9.5461 6.0510. 0.1289 0.0725 0.0464 0.5154 0.2899 0.1856 1.8314 1.0301 0.6593 7.3254 4.1206 2.6372 4.1781 2.3502 1.5041 16.7125 9.5008 6.0165. 0.1424 0.1419 0.1356 0.5697 0.5690 0.5686 0.1426 0.1427 0.1427 0.5687 0.5692 0.5691 0.1427 0.1426 0.1425 0.5692 0.4749 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:. Difference (%) =. Chantal Rudman. ANGELINE Value − Euler Value x 100 ANGELINE Value. University of Stellenbosch.

(39) In-plane structural behaviour of portal frames. 3.12. 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.. Chantal Rudman. University of Stellenbosch.

(40) In-plane structural behaviour of portal frames. 3.13. 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.. Chantal Rudman. University of Stellenbosch.

(41) In-plane structural behaviour of portal frames. 3.14. 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.. Chantal Rudman. University of Stellenbosch.

(42) In-plane structural behaviour of portal frames. 3.3.2. 3.15. 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.. 8.0. C3 C4. Displacement u2 (m). 6.0 C6. 4.0. C2 C5. C1. 2.0. 0.0 0.0. 10.0. 20.0. 30.0. 40.0 Load factor. Figure 3.12 Vertical deflection u2 of the ridge of portal frames as a function of the load factor Chantal Rudman. University of Stellenbosch.

(43) In-plane structural behaviour of portal frames. 3.16. 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.. Chantal Rudman. University of Stellenbosch.

(44) In-plane structural behaviour of portal frames (ii). 3.17. 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.. Chantal Rudman. University of Stellenbosch.

(45) In-plane structural behaviour of portal frames. 3.18. 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.. Chantal Rudman. University of Stellenbosch.

(46) In-plane structural behaviour of portal frames (b). 3.19. 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. Chantal Rudman. University of Stellenbosch.

(47) In-plane structural behaviour of portal frames. 3.20. (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.. Chantal Rudman. University of Stellenbosch.

(48) In-plane structural behaviour of portal frames. 4.1. 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. Chantal Rudman. University of Stellenbosch.

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