Curved-beam theory is used to determine stresses in members whose longitudinal axis is initially curved, including crane hooks, rings, chain links and other curved structural or machine components.
For a straight beam, or a beam whose cross-sectional depth is small compared with its radius of curvature, ordinary beam theory can provide a good approximation. As the initial curvature becomes more significant, however, the stress distribution differs increasingly from the linear distribution predicted by straight-beam theory.
An important feature of curved-beam theory is that the neutral axis does not generally coincide with the centroidal axis of the cross-section.
| Symbol | Description | SI Unit |
|---|---|---|
| A | Cross-sectional area | m2 |
| cc | Distance from centroidal axis to inner surface | m |
| ci | Distance from neutral axis to inner surface | m |
| co | Distance from neutral axis to outer surface | m |
| E | Young's modulus | Pa (N/m2) |
| e | Eccentricity between centroidal and neutral axes, rc - rn | m |
| M | Applied bending moment | N m |
| r | Radius to the point under consideration | m |
| rc | Radius of the centroidal axis | m |
| rn | Radius of the neutral axis | m |
| ri | Inner radius of curved member | m |
| ro | Outer radius of curved member | m |
| y | Distance from neutral axis | m |
| ε | Normal strain | dimensionless |
| σ | Normal stress | Pa (N/m2) |
| dφ | Change in angular position caused by bending | rad |
The sketch below shows an initially curved member subjected to a bending moment M. The centroidal axis has radius rc, while the neutral axis has radius rn.
Unlike the corresponding straight-beam case, these two axes do not generally coincide.
The strain at a radius r is:
The strain is zero at the neutral axis, where r = rn. Unlike the linear strain variation across a straight beam when expressed against radial position, the curved geometry leads to the characteristic curved-beam stress distribution.
For a linear-elastic material, Hooke's law gives the corresponding normal stress:
For pure bending, there is no resultant axial force on the section. Consequently, integration of the normal stress over the complete cross-sectional area must give zero:
This relationship determines the radius rn of the neutral axis from the geometry of the cross-section.
The eccentricity between the centroidal and neutral axes is:
For a rectangular curved-beam section, the neutral-axis position can be obtained by integrating the general neutral-axis relationship over the section depth.
The following diagrams give neutral-axis relationships for a range of common curved-beam cross-sections. The appropriate relationship depends on the variation of section width with radius.
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The stress resulting from an applied bending moment is obtained by requiring the internal stresses over the complete cross-section to produce a resisting moment equal to the applied moment.
For an elemental area dA, the elemental force is:
The resisting moment is obtained by integrating the product of this force and its moment arm over the complete cross-section.
The original derivation is retained below:
For the usual curved-beam formulation, the bending stress varies hyperbolically with radius rather than linearly through the section depth. Using the eccentricity e = rc - rn, the bending stress may be written:
The sign of the stress depends upon the adopted moment and stress convention. The important feature is that the stress magnitude is not symmetrical about the centroidal axis.
At the inner surface, r = ri:
At the outer surface, r = ro:
As the radius of curvature becomes large relative to the section depth, curved-beam results approach those obtained from ordinary straight-beam flexure theory.
For rectangular sections, the original RoyMech comparison indicates that the curved-beam flexure formula and straight-beam formula are in reasonably close agreement when the ratio rc/h exceeds approximately 5. At a ratio of about 8, the difference in calculated maximum stress is of the order of 5%. These values should be regarded as useful engineering guidance rather than universal limits, since the difference also depends upon section geometry.
The equations above describe pure bending. Many practical curved members, including crane hooks, are subjected to an applied force whose line of action does not pass through the section centroid.
The force then produces both a direct normal stress and a bending moment. The bending moment is determined about the centroidal axis of the section, and the direct and bending stress components must be combined to obtain the resultant stress.
The direct normal stress is:
The sign of each component must be retained when calculating the resultant stress at the inner and outer surfaces.
The hook shown below carries a load of 25,000 N.
The section geometry is first used to determine the centroidal radius, neutral-axis radius and eccentricity. The applied load then produces both direct stress and curved-beam bending stress.
The calculated normal stress plotted against radius is shown below.
For this example, the calculated tensile stress at the inner surface is:
and the calculated compressive stress at the outer surface is:
The magnitude of the inner-surface tensile stress is therefore approximately three times the outer-surface compressive stress. This illustrates the strongly non-symmetrical stress distribution that can occur in a curved member.
For this loading direction, a section having more material toward the inner radius and less toward the outer radius can provide a more favourable balance between the maximum tensile and compressive stresses. Actual hook design must, of course, also satisfy the applicable material, fatigue, manufacturing and design-code requirements.