Materials for Lightweight Table Legs
The selection methodology used in CES Materials can be encapsulated by developing a case study. Here, we will use the design of a simple table to illustrate the development of some selection criteria, we will apply them and plot them on some selection stages by using CES.The Design Problem
Luigi Tavolino, furniture designer, conceives of a lightweight table of daring simplicity: a flat sheet of toughened glass supported on slender, unbraced, cylindrical legs. The legs must be solid (to make them thin) and as light as possible (to make the table easier to move). They must support the table top and whatever is placed on it without buckling. What materials could one recommend?Design Requirements
We must first identify the Function, Objective and Constraints of our problem.
| FUNCTION | Column (support compressive loads) |
| OBJECTIVE | Minimise mass |
| CONSTRAINTS | Must not buckle |
The Model
Figure 1 A lightweight table with slender cylindrical legs
We now form the Performance Index appropriate to our design requirements.
Let us consider minimising the weight.
The leg is a slender column of density r and modulus E. Its length, l, and the maximum load, P, it must carry are determined by the design: they are fixed. The radius R of a leg is a free variable.
We wish to minimise the mass m of the leg, given by the objective function
subject to the constraint that it supports a load P without buckling. The elastic buckling load Pcrit of a column of length l and radius R is
where I is the second moment of area of the column. The load P must not exceed Pcrit. Solving for the free variable, R, and substituting it into the equation for m gives
The material properties are grouped together in the last set of brackets. The weight is minimised by selecting the subset of materials with the greatest value of the performance index
This is our performance index.
The Selection
We can now plot the material properties of our Performance Index using the CES software. In order to identify which materials maximise the performance index, we need to plot a line representing it on the graph. We use logarithmic axes on the graph and note that a simple performance index typically has the form:M = P1/P2n
Taking logs of this equation gives:
log P1 = n log P2 + log M
So, if P1 and P2 are plotted on logarithmic scales, the equation describes a line of slope n on the plot, with its position determined by the value of M. We are seeking to maximise the value of M, so our selection is optimised by moving the line to the highest value of M which still leaves a viable subset of materials exposed above the line.
For the table, we are seeking the subset of materials which have high values of E1/2 / r, so we plot a line of slope 2 on our graph.
Figure 2 shows the appropriate chart: Young's Modulus plotted against the density. The guideline is displaced upwards (retaining the slope) until a reasonably small subset of materials is isolated above it; it is shown in the position M1 = 6 GPa1/2/(Mg/m3). Materials above this line have higher values of M1. They are identified on the figure.
Figure 2 Materials for light slender legs
Woods meet the criteria and so do composites such as CFRP. Certain of the engineering ceramics also meet the stated design goals. However, ceramics, we know, are brittle—they lack fracture toughness. Table legs are exposed to abuse—they get knocked and kicked; common sense suggests that an additional constraint is required—that of adequate fracture toughness. A Selection stage that takes this into account is shown in Figure 3.
Figure 3 A Protective Selection Stage to eliminate brittle and expensive materials