Deriving and using indices: materials for light, strong shells

The four ecocars pictured on the title page of this chapter all have casings that are thin, doubly curved sheets, or shells. The designers wanted a cas­ing that was adequately stiff and strong and as light as possible. The double curvature of the shell helps with this: a shell, when loaded in bending, is stiffer and stronger than a flat or singly curved sheet of the same thickness, because any attempt to bend it creates membrane stresses—tensile or com­pressive stress in the plane of the sheet. Sheets support tension or compres­sion much better than they support bending.

So what is the best material for an adequately stiff and strong shell that is as light as possible? To answer that we need material indices for shells.

Modeling: indices for shells. Figure 9.6 shows a hemispherical shell of radius R and thickness t carrying a distributed load F. The load induces a deflection ё and a maximum membrane stress a. Define the stiffness S as F/6. The stiffness constraint then becomes S > S*, where S* is the desired stiffness. The strength constraint is simply a < ay, where ay is the yield strength of the material. Table 9.4 summarizes the requirements.

The mass of the hemisphere (the objective function) is

m = 2nR2 tp (9.3)

where p is the density of its material. The deflection and membrane stress a created by a distributed load like that in the figure are standard results.1 Using them, the stiffness S is


Table 9.4

Design requirements for the light stiff, strong shell


Doubly curved shell


Stiffness S* specified

} (functional constraints)

Failure load F specified Radius R specified

(geometric constraints)

Load distribution specified


Minimize mass

Free variables

Thickness of shell wall, t Choice of material

1See the compilation by Young listed under Further Reading.

and the maximum membrane stress a is

a = B-F – < ay (9.5)

where E is Young’s modulus, v is Poisson’s ratio, and A and B are constants that depend weakly on how the load is distributed on the surface of the hemisphere. Poisson’s ratio is almost the same for all structural materials and can be treated as a constant. Solving each of these for t and substitut­ing the result into the objective function gives


Everything in these two equations is specified except for the material properties in square brackets, so the two indices are


The mass of the shell is proportional to the value of the index.

The Selection. Materials for shells can be compared by evaluating the indi­ces or by plotting them onto appropriate charts. Take the index for adequate strength and minimum weight, M2, as an example. Taking logs of Equation

9.9 and rearranging gives

Log ay = 2 log p – 2 log M2

This is the equation of a family of contours of slope 2 on the ay – p chart of Chapter 8. Figure 9.7 shows this chart with the selection line of slope 2 positioned to leave the three materials with the lowest values of M2 exposed in the search area. They are CFRP and two grades of rigid polymer foam; certain ceramics come close. A similar selection line for M1, plotted on the modulus-density chart of Figure 8.11, gives the same result. Ceramics are ruled out by their brittleness. Foams are eliminated for a different reason: they are very light, but to achieve the necessary stiffness and strength, a foam shell has to be thick, increasing the frontal area and drag of the car. That leaves CFRP as the unambiguous best choice.

Postscript. CFRP is what the mileage-marathon cars use. But there is more to it than that. The lowest mass is achieved by a combination of material

A chart-based selection for a shell of prescribed strength and minimum weight. Carbon fiber reinforced polymer (CFRP) is the best choice.

and shape. CFRP offers exceptional stiffness and strength per unit weight; making it into a doubly curved shell adds shape-stiffness, further enhanc­ing performance.