$1.$

E$4\mathrm{.}15$ Indices for stiff plates and shell.

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$)$ a circular plate of radius a carrying a central load $W$ with a prescribed stiffness $S=\frac{W}{}$ and of minimum mass, and

$($b$)$ a hemispherical shell of radius a carrying a central load $W$ with a prescribed stiffness $S=\frac{W}{}$ and of minimum mass, as shown in the figure.

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: $,=\frac{3}{4}\frac{W{a}^2}{E{t}^3}(1-v^2)(\frac{3+v}{1+v})$

Hemispherical shell $,=A\frac{Wa}{E{t}^2}(1-v^2)$

in which $A$~~$0\mathrm{.}35$ is a constant. Here $E$ is Young's modulus, $t$ is the thickness of the plate or shell and $v$ is Poisson's ratio. Poisson's ratio is almost the same for all structural materials and can be treated as a constant. The table summarizes the requirements.

$\backslash$table$[[$Function$,\backslash$table$[[-$ Stiff circular plate, or$],[-$ Stiff hemispherical shell$]]],[$Constraints$,\backslash$table$[[-$ 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,$\backslash$table$[[-$ Plate or shell thickness, $t$