고체역학 (Gere) — Stresses in Beams (Advanced Topics): Problem 6.9-9

6.9-9 A U-shaped cross section of constant thickness is shown in the figure. Derive the following formula for the distance \(e\) from the center of the semicircle to the shear center \(S\): \[ e = \frac{2(2r^2 + b^2 + \pi br)}{4b + \pi r} \] Also, plot a graph showing how the distance \(e\) (expressed as the nondimensional ratio \(e/r\)) varies as a function of the ratio \(b/r\). (Let \(b/r\) range from 0 to 2.)