Curved Beams

Why it matters

Initial curvature changes the normal-stress distribution in hooks and curved members. The neutral radius generally differs from the centroidal radius, so straight-beam flexure is inaccurate when curvature is significant.

Curved-beam model

For homogeneous linear-elastic curved-beam bending with the section remaining plane, define cross-sectional area A, centroidal radius R_c, and neutral radius R_n = A/∫(dA/r). Let e = R_c − R_n. Radius r is measured from the centre of curvature.

For pure bending, choose positive M to produce inner-fibre tension. The circumferential stress is:

σ_θ(r) = [M/(Ae)](R_n/r − 1).

It vanishes at r = R_n and varies nonlinearly through the radial depth. Reversing M reverses the stress signs. The dimensions are N·m/m³ = Pa. See NPTEL, Bending of Curved Beams.

Rectangular section

For constant width b and radial boundaries r_i and r_o, A = b(r_o − r_i), R_c = (r_i + r_o)/2, and R_n = (r_o − r_i)/ln(r_o/r_i). The logarithm is essential; ∫dA/r cannot be replaced by the area.

Worked example

A rectangular curved beam has width 100 mm, radial depth 200 mm, and centroidal radius 500 mm. It carries a pure bending moment of 5 kN·m, with inner-fibre tension. Find inner and outer stresses.

In metres, b = 0.1, r_i = 0.4, r_o = 0.6, and A = 0.02 m².

R_n = 0.2/ln(0.6/0.4) = 0.49326069 m.

e = 0.5 − 0.49326069 = 0.00673931 m.

At the inner surface: σ_i = [5000/(0.02 × 0.00673931)](0.49326069/0.4 − 1) = 8.649 MPa tension.

At the outer surface: σ_o = [5000/(0.02 × 0.00673931)](0.49326069/0.6 − 1) = −6.599 MPa, or 6.599 MPa compression.

Model limits and common mistakes

A real hook load may produce axial force and shear as well as bending; the example is pure bending only. Local contact stresses and end effects require separate analysis. Distinguish e from the distance between a selected fibre and the neutral surface. State whether a supplied radius refers to the inner surface, centroid, or neutral surface. As curvature becomes small, compare with the appropriate straight-beam limit rather than assuming that increasing radius always reduces stress for every loading arrangement.

Quick check

1. Where is the bending normal stress zero? At R_n.
2. Is the stress variation linear in r? No; it contains 1/r.
3. Does the section centroid coincide with the neutral radius? Generally no for an initially curved beam.