A transversely loaded beam is generally subjected to both bending moment and shear force. Classical beam flexure equations determine the longitudinal bending stress, while the transverse shear force produces an additional shear stress distribution through the cross-section.
For many slender beams the contribution of shear deformation to the total deflection is small compared with bending deformation. Shear effects can, however, become important in short or deep beams, at locations of high shear force, and in members or materials having relatively low shear stiffness.
This page develops the beam shear-stress relationship, examines the stress distribution for typical sections, and considers the contribution of shear strain to beam deflection.
| Symbol | Description | SI Unit |
|---|---|---|
| A | Total cross-sectional area | m2 |
| A' | Area of the section between the level under consideration and the outer surface | m2 |
| b | Section width at the level where the shear stress is being calculated | m |
| E | Young's modulus | Pa |
| F | Concentrated force | N |
| Fa, Fb, Fc | Longitudinal forces used in the shear-stress derivation | N |
| G | Shear modulus (modulus of rigidity) | Pa |
| I | Second moment of area about the neutral axis | m4 |
| L | Beam length | m |
| M | Bending moment | N m |
| Q | First moment of area A' about the neutral axis | m3 |
| V | Transverse shear force | N |
| x | Distance along beam | m |
| y | Distance from the neutral axis | m |
| γ | Shear strain | dimensionless |
| δ | Deflection | m |
| σ | Normal stress | Pa |
| τ | Shear stress | Pa |
Consider a short element of a beam subjected to both bending moment and transverse shear force. The following derivation initially considers a rectangular beam of constant width b.
The longitudinal bending stress at a distance y from the neutral axis is obtained from the flexure equation:
If the bending moment changes along the beam, the longitudinal bending stress also changes. The difference between the longitudinal forces acting on adjacent transverse faces must be balanced by longitudinal shear stress on the horizontal face of the element.
The longitudinal force acting over a selected part A' of the cross-section is obtained by integrating the bending stress over that area.
The change in these longitudinal forces over a beam length dx must be balanced by the shear force acting over the longitudinal surface.
The integral occurring in the derivation is the first moment of the selected area A' about the neutral axis:
The resulting shear stress at the level being considered is:
where b is the width of the cross-section at the level where the stress is required. The value of Q is the first moment, about the neutral axis, of the portion of the section on one side of that level.
For a rectangular beam, substitution of the section geometry into the general shear-stress equation produces a parabolic stress distribution through the depth.
The shear stress is zero at the upper and lower free surfaces and reaches its maximum value at the neutral axis.
For a rectangular section:
Thus the maximum shear stress is 1.5 times the average shear stress V/A.
Maximum shear stresses for several other section geometries are summarised in the original RoyMech reference diagram below.
The transverse shear stress acting on a beam section is accompanied by an equal complementary longitudinal shear stress at the same point. In the usual plane-stress notation:
This complementary shear relationship follows from rotational equilibrium of an infinitesimal stress element.
The following figure, also used in the RoyMech Mohr's Circle notes, illustrates the stress components.
The following example illustrates how transverse shear stress can influence the principal stress in a beam.
The section considered is a hollow square section measuring 100 mm square with a wall thickness of 5 mm. Corner radii are ignored for simplicity. The maximum bending stress occurs at the outer fibre, y = 50 mm, while the shear stress is evaluated at the location shown in the calculation.
The resulting normal and shear stresses at a point may be combined to obtain the principal stresses. For plane stress with σy = 0:
Applying these relationships to the example gives the principal stresses shown in the original calculation:
This example demonstrates that shear stress can affect the governing stress state even when the longitudinal bending stress is the dominant component. Its significance depends on the beam geometry, loading and the point within the section being considered.
The following treatment relates primarily to rectangular-section beams and uses strain energy to determine the additional deflection caused by transverse shear.
When an elastic material is subjected to shear stress, shear strain stores elastic strain energy in the same way that normal strain stores energy under direct loading.
Consider the elemental block shown below.
For linear-elastic shear:
The strain energy stored by the element is developed in the original derivation below.
In terms of shear stress, the strain-energy density is:
Consequently, the shear strain energy stored throughout a volume is:
Consider a cantilever of length L carrying a transverse end force. If the shear stress were uniformly distributed over the complete cross-section, the shear strain would be:
and the corresponding end displacement would be:
The actual shear stress in a rectangular section is not uniform, so the non-uniform distribution must be included in the energy calculation.
Consider a longitudinal strip of a rectangular beam having width b, depth dy, and located a distance y from the neutral axis.
The complete derivation using the parabolic shear-stress distribution is retained below:
For a rectangular section, integration of the non-uniform shear-stress distribution gives the shear strain energy:
The same result may be expressed using an energy multiplier K:
For a rectangular section in this convention:
Using Castigliano's theorem, the corresponding end deflection for a cantilever carrying a constant transverse shear force V is:
The shear-deflection expressions for several common loading arrangements are summarised in the original RoyMech diagrams below.
For the following expressions, b and d refer to the web dimensions of the beam.
For thin-webbed I-sections, most of the transverse shear is carried by the web. The suitability of a web-area approximation depends on the section geometry and the accuracy required.