Background Reading on Selection of a Material

2.5 Shape Factors

2.5.1 Elastic Shape Factors

The bending stiffness of a beam and the resistance of a column to elastic buckling depend on the properties of the material and on the second moment of area of its section, I. A beam with a cross-section shaped to give a large I is stiffer and more resistant to buckling than one with a less efficient shape and lower I, even though both have the same cross-section area A. Thin-walled tubes, I-sections and box-sections are efficient in this sense whereas a beam with a solid circular cross section is not.

The second moment of area I depends on scale and has dimensions of (length)4. The effect of shape can be separated from the effect of scale by defining the elastic shape factor for bending elastic shape factor for bending (the 'e' means elastic), such that:

equation

(13)

Defined in this way, elastic shape factor for bending is a dimensionless number with a value of unity for a solid square section. Other solid sections such as circular and hexagonal all have shape factors near 1. Efficient shapes have larger shape factors: thin-walled tubes have shape factors for bending in the range 10–30; standard I-sections have shape factors as high as 60. Efficient shapes are stiffer in bending for a given cross-sectional area and hence stiffer for a given weight.

There is an upper limit to the shape factor for a given material. The limit is usually set by the onset of local buckling, and thus by the properties of the material. Therefore the upper limit on elastic shape factor for bending can be thought of as a material property. For steel it is about 60, for aluminum alloys about 40 and for polymers it is between 5 and 10. Occasionally it is set by manufacturing constraints. Wood, and most composites, fall into this category: current technology for shaping them does not yet allow the local buckling limit to be reached.

2.5.2 Failure Shape Factors

Plasticity starts when the stress at some location first reaches the yield strength (elastic limit). Fracture starts when the stress first exceeds the fracture strength. Either one of these constitutes failure. The symbol σf is used here to denote the failure stress which is the local stress that will cause yielding, or fracture. One shape factor covers both.

In bending, the stress σ is largest at the point ym on the surface of the beam which lies furthest from the neutral axis. It is given by the simple bending stress formula:

equation

(14)

where M is the bending moment and Z is called the section modulus. In problems of failure in beams, shape enters through the section modulus, Z. The shape factor for failure in bending, shape factor for failure in bending, is defined by

equation

(15)

Like the elastic shape factor elastic shape factor for bending, it is dimensionless, and so independent of scale, and its value for a solid square section is 1. References [1, 3] tabulate values of shape factor for failure in bending for various common shapes.

A complete description of the stiffness and strength of a component requires four shape factors. This is because additional moments of the section are needed to describe torsional stiffness and torsional strength. They need not concern us here: examples will be based on elastic bending only.

2.5.3 Performance Index for a light, stiff beam with efficient cross-section

Reconsider the selection of a light, stiff beam with section-shape as an additional variable, as shown in figure 2b. The mass of the beam is still given by equation (6), and the stiffness by equation (7). Using the definition of elastic shape factor for bending given in equation (13) to eliminate I in equation (7) gives:

equation

(16)

Now eliminating A between this and equation (6), gives

equation

(17)

The form of equation (17) is the same as equations (2) and (9): function in the first bracket, geometry in the second and material properties in the third. The first two are fixed by the design. Performance is maximized by seeking the material with the greatest value of

equation

(18)

The selection could be made on a property chart by choosing materials with a high value of M1 and then modifying the value of the performance indices, multiplying each by (elastic shape factor for bending)1/2. However, it would be much better to plot a Materials Selection Chart with elastic shape factor for bendingE as one axis and the shape factor ρ as the other, then use a selection line of slope 2. This can be done automatically by CES.


References

  1. Ashby MF, "Materials Selection in Mechanical Design", 3rd edition, Elsevier Butterworth Heinemann, Oxford, UK, 2005.

  2. Ashby MF, 'On the engineering properties of materials.' Acta Metall., 37 pp 1273–1293, 1989.

  3. Ashby MF, 'Materials and shape.' Acta Metall., 39 pp 1025–1039, 1991.