Material limits for shape efficiency

The efficiency of a material in a load bearing application depends on the shape of its cross-section. By "shape" we mean that the section is formed to a tube, a box-section, an I-section, or the like. By "efficient" we mean that it supports a given load safely while using as little material as possible. Steel tubes, boxes, and I-sections are examples: loaded in bending, in torsion, or axially as columns, they can be more than 60 times suffer, and carry loads safely which are more than 12 times greater than a solid circular section with the same area. Any material can be formed to efficient shapes, but the ultimate gains in stiffness and strength differ from material to material: those for aluminium alloys, for example, are about 2/3 of those for steel; those for GFRP or nylon, or wood are lower still. The upper limits for shape efficiency are important. They are central to the design of light-weight structures, and structures in which, for other reasons (cost, perhaps) the material content should be minimised. Three questions then arise. What sets the upper limit on shape efficiency? Why does the limit depend on material? And what, in a given application where efficiency is sought, is the best combination of material and shape? The paper addresses these issues from a materials perspective. Shape is characterised by a set of four shape factors which directly measure efficiency; one each for bending and for torsion with stiffness or strength as the design constraint. The useful range of shape factors extends from 1 (no gain in efficiency) to an upper limit, φ_opt, which depends strongly on the material itself. An analysis of standard sections, presented in the paper, establishes empirical values for these upper limiting efficiencies for steel, aluminium, GFRP and wood. The origins of the efficiency limits derive from the competition between failure modes; "failure" here meaning "inadequate stiffness" or "inadequate strength". Inefficient sections fail in a simple way: they yield, they fracture, or they suffer large-scale buckling. In seeking efficiency, a shape is chosen which raises the load required for the simple failure mode, but in doing so the structure is pushed nearer the load at which other modes - particularly those involving local buckling - become dominant. It is a characteristic of shapes which approach their limiting efficiency that two or more failure modes occur at almost the same load, and when they do, they interact. To analyse this, and to predict the limiting efficiency for materials in general, we draw on results from the disciplines of mechanics, and of aircraft and civil structures. Here are to be found expressions for the failure load of thin-walled sections in bending, torsion and compression, by each of a number of failure. The interaction between modes, too, is explored there, providing a basis, much of it empirical but well-tried, for estimating the consequent loss of load-bearing capacity. The approach adopted in this paper has three steps. 1. The mass of the section which will safely carry a given design load and (where necessary) meet a specified requirement for stiffness is calculated as a function of the parameter φ which characterises shape, allowing for all possible failure modes and, in an approximate way, their interaction. The results are displayed graphically as "failure-mode charts" that plot load factor against the shape factor F. They are useful in the early stages of design in determining the failure mode and ultimate efficiency of simple shapes. They also provide scope for existing section data to be superimposed that permit performance comparisons with potential made-to-order sections. 2. The value of φ which minimises this mass is sought; it generally lies at or near a boundary between two or more failure modes, one of them involving local buckling. Local buckling is defect-sensitive, and can lead to unstable load-deflection response, so the best practical choice of shape is one which is a little less efficient than that at the boundary, ensuring that failure is benign, not catastrophic. The paper surveys the failure of tubes, boxes and I-sections under bending, torsion and compression, listing expressions for the best practical shape, which, it is shown, are very similar and depend primarily on material properties. At their simplest, the best value for φ for tubes has the form scale buckling. In seeking efficiency, a shape is chosen which raises the load required for the simple failure mode, but in doing so the structure is pushed nearer the load at which other modes - particularly those involving local buckling - become dominant. It is a characteristic of shapes which approach their limiting efficiency that two or more failure modes occur at almost the same load, and when they do, they interact. To analyse this, and to predict the limiting efficiency for materials in general, we draw on results from the disciplines of mechanics, and of aircraft and civil structures. Here are to be found expressions for the failure load of thin-walled sections in bending, torsion and compression, by each of a number of failure. The interaction between modes, too, s explored there, providing a basis, much of it empirical but well-tried, for estimating the consequent loss of load-bearing capacity. φ_opt = (E/σ_y)^1/2 that for box sections can be approximated by φ_opt = 0.2(E/σ_y) and that for optimum I-sections can be written φ_opt = (E/σ_y)^1/2 These expressions (or more elaborate versions of them, given in the text) explain why optimum shape depends on material; and they allow prediction of optimum shape for a material where this is not known. 3. The final step is to substitute the optimum φ back into the expressions for the mass, giving a relationship between mass, load and material properties. The resulting minimum mass expressions share a common form. The minimum mass, represented by the mass index m/l³, is given in terms of a function of the structural loading coefficient and a group of material properties which is called the material index; the relevant material index is not only a function of the mode of loading and cross-section but also the magnitude of the load factor. Structural efficiency is maximised by selecting the material with the largest material index. The resulting selection includes not only the direct influence of material properties on the load-bearing capacity of the section but also their indirect effect in dictating the limits on shape efficiency.