Curved Beams Notes

Last reviewed 16 Sept 2026 · 4 min read

When the straight-beam formula fails

The flexure formula $σ = M y / I$ assumes a beam that is straight before loading. Many machine and structural parts are initially curved: crane hooks, chain links, rings, C-clamps, piston rings and the frames of presses.

If the radius of curvature is large compared with the depth (roughly $R / h > 10$ , as in most arches and bent girders), the straight-beam formula is accurate enough. If it is small ( $R / h$ of about 5 or less), a curved-beam analysis is needed.

What changes in a curved beam

Take a curved beam with centre of curvature O. Fibres on the inner side are shorter than those on the outer side before loading. When bending changes the angle of an element by the same amount at every radius:

the deformation of a fibre is proportional to its distance from the neutral axis (plane sections remain plane), but
the strain is that deformation divided by the fibre's own original length, which is smaller for inner fibres.

So strain — and stress — varies hyperbolically, not linearly, across the depth.

DefinitionConsequences

The neutral axis does not pass through the centroid; it shifts towards the centre of curvature.
The stress at the inner fibre is larger than a straight-beam calculation predicts; the stress at the outer fibre is smaller.
The distance between the centroidal axis and the neutral axis is called $e$ .

Winkler–Bach theory

Assumptions: material elastic and homogeneous; plane sections remain plane; the section is symmetric about the plane of bending; no radial stress (a reasonable approximation for solid sections).

Let $r_{i}$ = inner radius, $r_{o}$ = outer radius, $R$ = radius of the centroidal axis, $r_{n}$ = radius of the neutral axis, $A$ = area.

FormulaNeutral axis and stresses

r_{n} = \frac{A}{\int \frac{d A}{r}}, e = R - r_{n}

Stress at a fibre at radius $r$ (distance $y = r_{n} - r$ from the neutral axis, positive towards the centre):

σ = \frac{M y}{A e r}

Extreme fibres, with $c_{i} = r_{n} - r_{i}$ and $c_{o} = r_{o} - r_{n}$ :

σ_{i} = \frac{M c _{i}}{A e r _{i}}, σ_{o} = \frac{M c _{o}}{A e r _{o}}

The sign is set by inspection: a moment that decreases the curvature (straightens the beam) produces tension on the inner fibre; one that increases curvature produces compression there.

Radius of the neutral axis for common sections

Section	$r_{n}$
Rectangle, depth $h$ ( $r_{o} - r_{i} = h$ )	$\frac{h}{ln ( r _{o} / r _{i} )}$
Solid circle, section radius $c$ , centroid radius $R$	$\frac{R + R ^{2} - c ^{2}}{2}$
Triangle and trapezium	from $\int d A / r$ for the shape (tabulated in handbooks)

Because $e$ is a small difference between two nearly equal numbers, compute $r_{n}$ to enough significant figures before subtracting.

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Curved Beams

When the straight-beam formula fails

What changes in a curved beam

Winkler–Bach theory

Radius of the neutral axis for common sections

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