STABILITY OF ANGLE—SECTION MEMBERS by R.T.G. Green, B.E. (Hons.) submitted in partial fulfilment of the requirements for the degree of Master of Engineering Science, 0 UNIVERSITY OF TASMANIA HOBART. — AUGUST, 1967 — SUMMARY A simple treatment of the elastic stability of angle-piectio# members, both columns and beams, has been developed, based on the measured deformations of typical members loaded in the laboratory° Detailed mathematical models describing the torsiOnal,or local buckling modes of the members are presented° Other buckling modes have been considered and the interaction of the various modes has P been discussed. Angle.-section columns, eccentrically loaded columns, cantilevers, centrally-loaded simply-supported beams, and laterally loaded columns, have been studied in particular. I hereby declare that, except as stated herein, this thesis contains no material which has been accepted for the award of any other degree or diploma in any University, and that, to the best of my knowledge or belief the thesis contains no copy of paraphrase of material previously published or written by another person, except where due reference is made in the text of this thesis. - INDEX - INTRODUCTION 1 GEOMETRY AS A WORKING TOOL 4 Ligtenberg Moire Method Crossed Diffraction Grating Moire Method BUCKLING 11 Columns Plate Structures Energy Stability of Systems Governed by Differential Equations Stability of Systems Governed by Simultaneous Linear Equations Finite Difference Methods A System of n Degrees of Freedom TORSION 24 BUCKLING OF COLUMNS Load Applied Through a Base Plate 29 First Mathematical Model - Small Elastic Deflections Second Mathematical Model - Small Elastic Deflections Initial Shape Third Mathematical Model - Large Elastic Deflections Eccentric Loading Applied Torque Columns with Bolted End Conditions 45 First Mathematical Model Second Mathematical Model Stress Distribution Around a Bolt Torsional-Flexural Buckling 58 - INDEX- STABILITY OF ANGLE-SECTION BEAMS Stability of Angle-Section Cantilevers 65 Small Elastic Deflection Mathematical Model Large Elastic Deflection Mathematical Model Fully Plastic Mathematical Model Lateral Buckling of Cantilever 74 Simply Supported Beams 76 Central Load Uniform Bending Moment Combined Axial and Lateral Buckling 79 CONCLUSIONS Appendix A Appendix B Notation ACKNOWLEDGEMENTS This work was carried out in the Civil Engineering Department of the University of Tasmania. The author wishes to thank the members of the staff of the University. In particular the author wishes to thank Professor A.R. Oliver, Professor of Civil Engineering, and Dr. M.S. Gregory, my supervisor of research, for their help and encouragement. INTRODUCTION Angle-section members, under various loading conditions, have been found to be unstable. In this thesis the conditions under which the instability occurs are presented and mathematical models are form- ulated to describe the geometry of the deformed member and to calculate the load capacity of the member. Equal-leg angle-section members were tested as columns, eccentrically loaded columns, cantilevers, centrally loaded beams, and laterally loaded columns. The mathematical models which are developed herein describe the torsional and local buckling of the members. However, where applicable, other types of instability have been investigated; also some types of interaction which can occur between the possible modes of buckling have been considered. Only relatively recently, the torsional and local properties of structural members have become important. With the introduction of slender high-strength steel members, and materials such as aluminium and its alloys with low moduli of elasticity, the problem has been accentuated. Even today most design codes are based upon practices developed for mild steel members. In this century, considerable work has been carried out on the instability properties of columns. Two organisations which are particularly interested in the problem are the 1 Column Research Council of America and the Aluminium Research Develop- 2 ment Association of Britain. Both organisations have published results or codes which could be used by practising engineers. The German code is one of the most progressive codes. In this thesis the problem of the buckling of angle-section mem- bers has been investigated using a new approach. Large field methods of measuring geometrical shapes have been used to obtain the deformed shape of the member. The basic geometry is then described analytically and the analytic function is used as a basis for the mathematical model. The forces required to sustain the measured deformation are calculated and a differential equation is obtained by considering the statical equilibrium of the whole member. The analytic function describing the geometry can be specified to any order of approximation and consequently Ref. 1 Column Research Council Guide to Design Criteria for Metal Com- pression Members, John Wiley & Sons, Inc. Ref. 2 Series of Aluminium Research Development Association Reports. - 2- a number of successively more satisfactory mathematical models can be derived to describe the physical behaviour. For most cross-sections the member can buckle either by the whole cross-section rotating, in an undeformed state, or by part of the cross- section defoIrming. The first type of instability is referred to as a torsional instability and the second as a local instability. For an equal-leg angle-section member, with both leg's loaded identically, torsional buckling and local buckling are the same phenomena, and the two terms are interchanged freely in all the literature. Under this loading each leg acts as a simply supported plate, and there is no mom- ent acting around the corner of the cross-section. The buckling of an angle member has been treated by these various methods, each of which will be considered. The models tested were of such dimensions that the torsional mode was prominent, To emphasize torsional buckling behaviour the members tested had thinner walls, relative to width of leg, than are common in practice, but it has been indicated how the results obtained can be amended to give an understanding of the behaviour of more practical sections. This thesis does not set out to present a large quantity of results and to derive empirical formulae or relationships. Rather, it relies on the similarity of the geometry of the deformation of members of different proportions and sizes. The mathematical models developed are based upon the deformed geometry of a number of members tested in the laboratory, and the results derived are compared with those obtained from a few physical models. In the future, large-scale testing programmes for more practical members might be contemplated; it is thought that the necessary basic ideas are established in this thesis, This thesis is divided into four parts. The first part is devoted to forming a foundation upon which the author's work is built. Although no new ideas are presented therein, the understanding of the ideas is basic to the remainder of the thesis, The second section deals with the stability of a column which is axially loaded with a uniform stress distribution. Results presented - 3 - in this part have been derived in detail, because many of the ideas are used in the following sections. The work on a column under a pure axial load was presented by the author as a partial requirement for an honours bachelor's degree. In this thesis, the topic has been expanded by including the effect of non-uniform stress distribution produced by load- ing the member through a bolt. A detailed comparison with other math- ematical models is also presented. The third section deals with the instability of angle-section beams. This section leans heavily on the preceding section as it uses basically the same logic, although the mathematical models developed are more complicated. To the author's knowledge, there exist no other math- ematical models which describe this problem, although beam-columns have been treated empirically. The fourth section, a detailed comparison between the results obtained in this thesis and those obtained by other mathematical models is given. Present design codes are considered, and possible amendments are suggested in the light of the results of the work described in the thesis, and the fundamental understanding which it has encouraged or made possible. This thesis establishes a new, simple mathematical model for the elastic behaviour of a column, and original mathematical models for the torsional buckling of a cantilever and a centrally loaded simply supported beam. Although the author has presented a mathematical model for a laterally loaded column, this topic needs further investigation. The thesis also considers the mathematical models describing the torsional- flexural buckling of a column developed by other workers, and the rel- evance of the lateral buckling model of beams developed by Tunoshenko. GEOMETRY AS A WORKING TOOL In the history of engineering structural science, the basic under- standing of the geometry of the deformations of a loaded member has led to the necessary valuable simplifications on which all analysis is based, and has thus played an important part in the advancement of the science. Geometry has the advantage that it can be easily measured. From the earliest problem, that of a loaded cantilever, the geometry has been the basis for the mathematical description. The theory of bending is based upon the geometrical assumption that plane sections remain plane. However, the parameters of the geometry must be evaluated by considering the stat- ical equilibrium of the member. The early development of the theory of bending was slow, as the experimenters failed to combine the geometry and the equations of equilibrium. In fact, this lack of completeness in the model led early engineers to assume that the neutral axis of a beam in bend- ing was at the lower edge. Later, prominent men, such as Timoshenko, have made advances because they have been able to base their mathematical descriptions upon geometry. One example, which was developed during this century, is the plastic analysis of members. Plastic analysis has become important because of the simplicity of its application, which in turn depends upon a simple deformation pattern. A framed structure in a fully plastic state is described by a rigid-plastic load-deformation relationship, in which all deformations occur within local regions known as plastic hinges. Lately, more sophisticated descriptions of elastic-plastic bending have been produced in which other deformations have been included. The author will consider the torsional buckling of angle-section members by measuring the geometry of the member in its deformed or buckled state. The geometrical approach has been made possible by the develop- ment of optical methods of measuring geometry over large fields of view. Two such methods are the photo-elastic method, which measures stress in the plane of the model, and the moire fringe techniques, which measure deflections both in and normal to the plane of the model. For the work described here the moire fringe methods have been used, as they measure geometry directly. white grey fringe fringe MOIRE FRINGES FIGURE 1. AB and A'Bi are the initial and the final slopes of the plate at 0. ROI and Q0I are two rays. I is the image of both Q and R. LIGTENBERG APPARATUS FIGURE 2. 5 Moire Fringes When two sets of parallel lines are interfered, a moire fringe pattern is formed as shown in fig. 1. Where two lines intersect, a "white" fringe is formed, while a grey fringe occurs when a white and a black line intersect. In interference terms, a "white" fringe occurs when the lines are in phase, that is displaced by an integral number of line spaces. The "grey" fringes are formed when the lines are out of phase. Two of the available moire fringe methods have been used by the author in his experimental work. One method, the Ligtenberg method, measures changes in slope of a surface. The other method measures deflections in the plane of the surface. In this section of the thesis the author will only outline the experimental methods used. For complete details, such as the production of gratings, and the preparation of the 1 models, reference may be made to Ligtenberg's paper and two papers by 2 3 Middleton, Jenkins and Stephenson;' the latter workers are engaged in the development of the techniques used at the University ofTasmania. The basic ideas involved in using the two methods are described in the follow- ing sections. * * * * 1 The Ligtenberg Method The Ligtenberg method produces moire fringes which are contours of equal change in slope of a surface. The surface to be examined is made reflective by gluing a sheet of Melanex, a commercially available sheet of plastic coated with aluminium, to the surface. Kodaflat matte solution, a pressure-sensitive glue is used. A set of photographically reproduced lines is mounted on a part of a cylindrical surface and a camera is arranged so that the lens is at the centre of the screen. The lines are reflected from the model's surface and an image is produced on the camera film. The model is loaded and the second exposureis taken. Ref. 1 Ligtenberg: "The Moire Method as a new experimental method for the determination of moments in small slab models". Vol. XII, No, 2, Proc, Soc. Experimental Stress Analysis. Ref. 2 E. Middleton and C. Jenkins: 'Moire methods for Strain Analysis for Student Use", Bulletin of Mech. Eng, Education, Issue 3, Vol, 5, 1966. Ref. 3 E. Middleton and L.P. Stephenton: "A reflex Spectrographic Tech- nique for in-plane Strain Analysis". In printers hands. SESA Paper No. 1250.