When a beam’s centerline is already curved, the inner and outer fibers do not carry the same stress pattern predicted by the straight-beam formula. The Winkler-Bach model shifts the neutral axis away from the centroid and produces a curved-beam stress distribution. The factors here compare the calculated inner and outer stress magnitudes with the straight-beam value Mc/I for selected rectangular-section radius-to-depth ratios.

Curved beam stress correction factors

Definitions and equations

Let r_i and r_o be the inner and outer radii measured from the center of curvature, h = r_o − r_i the radial section depth, R = (r_o + r_i)/2 the centroidal radius, and r_n the neutral-axis radius. For a rectangular section, r_n = h / ln(r_o/r_i). The eccentricity is e = R − r_n. The stress magnitude at a fiber radius r is |σ| = |M|·|r_n − r| / (A·e·r), with A the section area and M the bending moment.

The correction factors K_i and K_o are the curved inner and outer stress magnitudes divided by the straight-beam value Mc/I. The factors in the table are calculated for a rectangular section with R/h = 2, 3, 5, 10, and 20. They are dimensionless and depend on geometry, not material strength.

For a rectangular section of width b, the area is A = bh and the centroidal second moment is I = bh³/12. The same section should be used in both the curved and straight-beam calculations so the ratio isolates the effect of initial curvature. The table uses a constant depth and varies R/h; it does not change width, section shape, material, or loading between rows.

Worked example

Consider a rectangular section 20 mm wide by 40 mm deep in the radial direction, with centroidal radius R/h = 5 and a 100 N·m bending moment. Its second moment of area is I = bh³/12 = 106,667 mm⁴ and c = h/2 = 20 mm. The straight-beam estimate is Mc/I ≈ 18.75 MPa. At R/h = 5, the inner factor is about 1.0711, giving an inner stress magnitude of about 20.1 MPa; the outer factor is about 0.9370, giving about 17.6 MPa. This example assumes pure elastic bending and ideal rectangular geometry.

The neutral radius lies closer to the center of curvature than the centroidal radius for the rectangular section. Consequently, the inner side carries a larger bending-stress magnitude than Mc/I while the outer side carries a smaller magnitude in this comparison. As R/h grows, the inner and outer factors move toward 1, which is the straight-beam limit. The trend is a useful hand-check of the calculation, but it does not decide whether curved-beam theory alone is adequate.

Applicability limits

The table is not a universal notch factor. It applies to the stated rectangular geometry and the Winkler-Bach assumptions; it does not directly cover a crane hook with a nonrectangular section, direct force, contact pressure, holes, welds, or local stress concentrations. Tight curvature increases the difference between inner and outer stresses, while the factors approach 1 as the radius becomes large relative to depth.

For a real hook, clamp, chain link, or frame, model the actual section and load path. Check combined direct and bending stress, yielding, fatigue, contact, and three-dimensional effects as applicable. Do not treat this comparison with Mc/I as a design allowable or code check.

Also keep the moment arm distinct from the center of curvature: the section moment comes from the actual force line and load path. Using the radius as a force lever arm without checking the diagram can introduce a large error. Confirm units before comparing stress; with newtons, millimeters, and N·mm moments, the result is N/mm², or MPa.

Source

The rectangular neutral-axis and Winkler-Bach stress equations are presented in NPTEL’s Bending of Curved Beams lecture notes. The displayed ratios are calculations from those equations relative to Mc/I.