Circular Ring — Bending Moment, Hoop & Shear Force

Thin circular ring under in-plane radial point loads — two opposing forces or n equally spaced forces

Circular Ring — Bending Moment, Hoop & Shear Force

This calculator analyses a thin circular ring of constant cross-section under in-plane radial point loads, using classical thin-ring (curved-beam/Castigliano) bending theory. The results are closed-form exact solutions — not numerical approximations. Two classic load cases are covered:

• Two equal and opposite radial forces along one diameter (the classic "ring being squeezed or expanded" problem). The reference moment M_A = WRk_2/\pi and the diameter changes \Delta D_H, \Delta D_V are given by well-known formulae.
• n equal radial forces equally spaced around the ring (n \geq 2). The half-angle between adjacent loads is \theta = \pi/n, and the closed-form solution gives the load-point moment M_A and the mid-point moment M_C as well as the radial deflections \Delta R_{load} and \Delta R_{mid}.

The optional correction constants k_1 = 1 - \alpha + \beta and k_2 = 1 - \alpha allow for hoop-stress and shear deformation in thicker rings (\alpha = I/AR^2, \beta = FEI/GAR^2). For slender rings (cross-section dimension \ll R), set \alpha = \beta = 0 (the default), giving k_1 = k_2 = 1.

The bending stress is estimated as \sigma_{max} = |M|_{max} / Z where Z = I/c_{sec} is the section modulus you supply. This is the straight-beam (outer-fibre) approximation; for rings where R/\text{depth} < 5, a curved-beam correction (Winkler–Bach) would be needed — see a dedicated curved-beam calculator for that.

Sign convention: Roark Table 9.2 notation. The angle x runs from reference point A (load point or midpoint, depending on the case). Hoop force N is positive in tension. Diameter increases are positive.

Model limits: equal cross-section, linear-elastic, in-plane loading, small deflections. Does not cover out-of-plane loading, tangential forces, couples, distributed loads, or thermal loads (see the spec §6.4 for planned extensions). Outputs are structural response quantities — no pass/fail safety check.

Covered cases of this table:

• Case 1: two equal opposing radial forces
• Case 7: n equal radial forces, equally spaced
• Not included: the other load cases of Table 9.2

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