Indices for stiff plates and shells (Figure E7). Aircraft and space structures make use of plates and shells. The index depends on the configuration. Here you are asked to derive the material index for

a circular plate of radius a carrying a central load W with a prescribed stiffness S =W / and of minimum mass, and

a hemispherical shell of radius a carrying a central load W with a prescribed stiffness S =W / and of minimum mass, as shown in the figures.

Use the two results listed below for the mid-point deflection of a plate or spherical shell under a load W applied over a small central, circular area.

Circular plate: = ^{3 W a2} (1² ₎ ³⁺

4 E t3 1+

Hemispherical shell = A ^{W a} (1² )

2 E t

in which A0.35 is a constant. Here E is Youngs modulus, t is the thickness of the plate or shell and is Poissons ratio.

Poissons ratio is almost the same for all structural materials and can be treated as a constant. The table summarizes the requirements.

Function		Stiff circular plate, or
		Stiff hemispherical shell
Constraints		Stiffness S under central load W specified
		Radius a of plate or shell specified
Objective		Minimize the mass of the plate or shell
Free variables		Plate or shell thickness, t
		Choice of material