Solid Mechanics & Structural Design
Winkler-Bach Curved-Beam Neutral-Axis Shift and Stress Formula
Calculate curved-beam neutral-axis radius R_n = h / LN(r_o / r_i), eccentricity e = R_c - R_n, and hyperbolic inner-fiber stress sigma_i = M*(R_n - r_i) / (A * e * r_i) for crane hooks and C-clamps.
Catch the 29% unconservative error of applying straight-beam M*c/I to a curved crane hook!
In a sharply curved crane hook or C-clamp of inner radius r_i and outer radius r_o (depth h = r_o - r_i), inner arc fibers are shorter than outer arc fibers and strain much more per unit rotation! Winkler-Bach curved-beam theory proves that for a rectangular section of width b, the neutral axis shifts inward from centroidal radius R_c = (r_i + r_o) / 2 to R_n = h / LN(r_o / r_i), creating eccentricity e = R_c - R_n > 0 and hyperbolic inner-fiber bending stress sigma_inner = (M * (R_n - r_i)) / (A * e * r_i).
Audit a rectangular steel hook (r_i = 40 mm in B2, r_o = 80 mm in B3 -> h = 40 mm, R_c = 60 mm; width b = 20 mm in B4 -> A = 800 mm^2; opening moment M = 1,600,000 N*mm in B5): naive straight-beam 6*M / (b*h^2) predicts 300.0000 MPa. Exact Winkler-Bach computes R_n = 40 / LN(2) = 57.7078 mm, inward shift e = 2.2922 mm, and true inner-fiber tensile stress sigma_inner = 386.2624 MPa — a +28.75% stress concentration factor K_i = 1.2875!
| Curved-Beam Quantity (r_i=40, r_o=80, b=20) | Formula | Verified Output |
|---|---|---|
| Naive Straight-Beam Stress | 6 * M / (b * h^2) | 300.0000 MPa (Unconservative!) |
| Neutral-Axis Radius (R_n) | (r_o - r_i) / LN(r_o / r_i) | 57.7078 mm (< R_c = 60 mm) |
| Neutral-Axis Eccentricity (e) | R_c - R_n | 2.2922 mm |
| Winkler-Bach Inner-Fiber Stress | M * (R_n - r_i) / (A * e * r_i) | 386.2624 MPa (K_i = 1.2875x) |
Rectangular Curved-Beam Neutral-Axis Radius R_n = (r_o - r_i) / LN(r_o / r_i) Formula
Because R_c - R_n can be small (millimeters or fractions of a millimeter), NEVER pre-round R_n before computing e = R_c - R_n inside the stress denominator — keep full IEEE-754 precision inside LET!
=LET(r_in, B2, r_out, B3, IF(OR(r_in <= 0, r_out <= r_in), "Invalid Hook Geometry", ROUND((r_out - r_in) / LN(r_out / r_in), 4)))Exact Winkler-Bach Hyperbolic Inner-Fiber Stress sigma_inner (MPa) Formula
Compute R_c, R_n, e = R_c - R_n, and A = b * (r_o - r_i) unrounded inside LET to avoid subtractive cancellation even at mild curvature ratios.
=LET(r_in, B2, r_out, B3, b_wd, B4, m_ben, B5, IF(OR(r_in <= 0, r_out <= r_in, b_wd <= 0, m_ben < 0), "Invalid Hook Geometry", LET(h_dp, r_out - r_in, r_cen, 0.5 * (r_in + r_out), r_neu, h_dp / LN(r_out / r_in), e_ecc, r_cen - r_neu, area_s, b_wd * h_dp, ROUND((m_ben * (r_neu - r_in)) / (area_s * e_ecc * r_in), 4))))Try the example
Use the original values or change them to test your understanding.