Curved Beam: Method

Curved Beam Stresses

Problem Statement

In a beam whose axis is curved in the plane of bending, such as a crane hook, a chain link, a press frame or a ring, the fibres on the inner side of the curve are shorter than those on the outer side. A plane section that stays plane while the beam bends therefore strains the inner fibres more than the straight-beam formula assumes, and the circumferential stress is not linear across the depth but hyperbolic in the radius r: it is larger than the flexure value at the inner fibre and smaller at the outer fibre, and the neutral axis of bending moves from the centroid towards the centre of curvature. The difference matters when the centroid radius R is less than a few times the depth h; above R/h of about 8 the flexure formula is within a few percent, and above about 20 within 1 percent. The curved-beam theory of Winkler (1858), also called the Winkler-Bach theory, gives the stress in closed form once three section integrals are known. For a section drawn on the screen the tool evaluates those integrals exactly by Green's theorem, in the way the Section Properties tool computes the moments of area, and reports the circumferential and the radial stress across the depth.

Drawing the Section

The section is drawn, or loaded from the saved sections, with the same editor as the Section Properties tool. Its y axis is the radial direction: the lowest point of the drawing is the inner fibre and is placed at the entered inner radius r_i from the centre of curvature O, every other point sits at r = r_i + (y − y_min), and the axis of bending is parallel to the drawing's x axis. A section wider at the inner fibre than at the outer one, the usual shape of a hook, is drawn with its wide side down. Holes are allowed; the integrals treat them with negative area, as the moments of area do.

The Section Constants

Besides the area A and the centroid radius R, the theory needs the modified area

A_m = ∫ dA / r

which has the dimension of a length. The neutral axis of pure bending lies at r_n = A / A_m, always inside the centroid, and the shift e = R − r_n is the small difference on which the stress depends; handbooks warn that it must be computed to many digits. For a polygon the integral has a closed form: by Green's theorem the area integral of 1 / r equals the contour integral of x / r dr around the boundary, and along a straight edge x is linear in r, so each edge contributes x₁ ln(r₂ / r₁) + Δx [1 − ln(1 + u) / u] with u = Δr / r₁. The sum is exact in double precision, so the shift e is as accurate as the drawing. The same sum over the part of the section between the inner fibre and a radius r, found by clipping the polygon at r, gives the partial area A′(r) and the partial modified area A′_m(r) needed for the radial stress. For a rectangle of width b the closed form A_m = b ln(r_o / r_i) is reproduced exactly; a circle drawn as a polygon is reproduced to the accuracy of the polygon.

Circumferential Stress

With the normal force N at the centroid, positive in tension, and the bending moment M about the axis of curvature, positive when it decreases the curvature, that is opens the beam, which stretches the inner fibre and puts it in tension, the circumferential stress at radius r is

σ_θθ = N / A + M (A − r A_m) / [A r (R A_m − A)]

The bending term vanishes at r = r_n, is positive inside it and negative outside it for positive M, and varies as 1 / r plus a constant. The tool reports it at the inner and outer fibres and at an optional evaluation radius, together with the straight-beam value N / A + M (R − r) / I_xx and the ratio of the two at each fibre; Figure 2 on the results page draws both across the depth. For a crane hook loaded by a force P on its inner side the section at the horizontal centre line carries N = P and M = P R, both positive.

The theory assumes plane sections, a uniform distribution of the normal force, and neglects the radial stress in the stress-strain relation. Timoshenko compared it with the exact elasticity solution for a rectangular section in pure bending and found the inner-fibre stress within about 1 percent for r_o / r_i up to 2 and within 3 percent at r_o / r_i = 3, so the formula is used down to very sharp curvatures.

Radial Stress

Because the circumferential stresses on the two faces of a wedge of the beam are inclined to each other, they have a resultant towards the centre of curvature, and the material between the inner fibre and a radius r is held by a radial stress on the cylindrical surface at r. Equilibrium of that part of a wedge of angle dθ gives σ_rr t r = ∫ σ_θθ dA over the part, where t is the width of the section at r. Substituting the stress formula,

σ_rr = [ N A′ / A + M (A A′_m − A_m A′) / (A (R A_m − A)) ] / (t r)

The radial stress is zero at the inner fibre, has the sign of the bending moment (tensile for a positive M) and, for a compact section, peaks near the neutral axis at a small fraction of the circumferential stress. It becomes important for I, T and box sections, where the thin web must carry the radial force collected by the whole inner flange: t drops at the web-flange junction and σ_rr jumps up there, sometimes to a value comparable with the circumferential stress, and the junction is the usual location to check. The tool evaluates the formula at 300 stations and on both sides of every vertex radius, refines the maximum, and reports its location and value, Figure 3 drawing the whole distribution. The formula assumes the radial stress uniform across the width t; at a web-flange junction that is an average over the web, and the local stress at the fillet is higher.

At the outer fibre the bending part of σ_rr vanishes identically, but the normal-force part leaves N / (t r_o) where the free surface requires zero. This is a known inconsistency of the theory: a uniform N / A is not an exact stress state in a curved bar, so the radial stress from N is only approximate. The tool prints that residual next to the maximum so that the share of the bending effect can be judged.

Shear Stress

A transverse shear force V on the section produces a shear stress τ_rθ that follows from the circumferential equilibrium of the same partial wedge. The circumferential force T on the part between the inner fibre and r changes along the beam because M and N do (between loads dM/dθ = R V and dN/dθ = −V), and the partial shear force V′ on the two faces of the wedge, inclined to each other by dθ, adds a circumferential component of its own. That second term is what makes the result consistent: without it the shear stress would not vanish at the outer fibre, the same defect the normal-force part of the radial stress shows. Keeping it turns the balance into a first-order equation in r, whose solution with Q′ the first moment of the part about the centre of curvature is

V′(r) = (V / r) [ (r A′ − Q′) / A + R (A (r A′_m − A′) − A_m (r A′ − Q′)) / (A (R A_m − A)) ]

τ_rθ(r) = (V / (t r)) [ A′ / A + R (A A′_m − A_m A′) / (A (R A_m − A)) ] − V′(r) / (t r)

Two identities hold for any section: the shear stress is zero at both fibres, and V′ at the outer fibre equals V, which is the statement that the shear stress integrates to the applied force. Both follow from the integrals of A′ and of the bending kernel across the depth, which the tool's polygon integrals satisfy exactly, and both were checked numerically for a rectangle and an I-section. As R grows the expression reduces to the straight-beam V Q / (I t), which the results page draws alongside; at R/h = 50 the two agree to four digits, at R/h = 1 the curved-beam maximum of a rectangle is about 13 percent higher and sits inside the centroid. For a flanged section the maximum is at the web side of the inner web-flange junction. The shear stress carries the sign of V, is an average across the width t like the straight-beam value, and rests on the same plane-section assumption as the rest of the theory and on M varying as R V along the beam, which holds between concentrated loads. The derivation is the one the tool uses for the radial stress carried into the circumferential direction; it is not reproduced from a textbook, and the identities above are its check.

Thin-Flanged Sections

In a curved I or T beam the radial force on the flanges bends them towards the web, so the flange tips carry less circumferential stress than the theory assumes and the stress at the web-flange junction is higher. Bleich (1933) accounted for it by replacing each flange by a reduced effective width that depends on the flange projection, its thickness and its radius, and Boresi and Schmidt reproduce his table. The tool does not apply that correction: it computes the constants for the section as drawn. For a flanged section with a wide projection the circumferential stress at the junction may therefore be underestimated. Use the Curved Beam, Thin-Walled Section tool, which takes the plate dimensions of an I, T or channel section and applies the correction, or draw the reduced width by hand.

Limitations

The results are elastic stresses from the Winkler theory for a beam curved in one plane, loaded in that plane by a normal force at the centroid, a bending moment and a transverse shear force. The stress at a fillet and the deflection are not computed. The section must lie entirely at positive radius, which the inner radius input enforces. For R / h above about 20 the curved-beam and straight-beam results coincide within a percent and either may be used.