A curved beam subtending 120° has a trapezoidal cross section: inner (concave) width…

A curved beam subtending 120° has a trapezoidal cross section: inner (concave) width b_i = 28 mm, outer (convex) width b_o = 52 mm, radial depth h = 60 mm, inner-fiber radius a = 70 mm, outer-fiber radius b = 130 mm. The width varies linearly with radius. A tensile load P = 45.0 kN acts along a line that passes 40 mm inside the inner fiber (measured along the radius of the critical section, which is the plane of symmetry of the 120° arc). Using the Winkler–Bach formula, determine the centroidal radius, the radius of the neutral surface, and the normal stresses at the inner fiber, the outer fiber, and the mid-depth fiber. Then compute the eccentricity of the load relative to the centroid and show that the inner-fiber stress exceeds the straight-beam prediction by more than 20 percent. The member is steel; E is not required for the stress field.

Because \(b_i/a=b_o/b\), the width is proportional to \(r\), which forces \(r_n=(a+b)/2\) exactly; \(\bar{r}\) is still the first moment of \(b(r)\,\mathrm{d}r\).