Derive the expression for the shape-efficiency factor \(\phi_{B}^{e}\) for stiffness-limited design for a circular tube with outer radius \(5 t\) and wall thickness \(t\), loaded in bending (Fig. E10.2). Do not assume that the thin-wall approximation is valid.

Data From Fig, E10.2

YA 10t Outer radius FIGURE E10.2 r = 5t