In this tutorial we investigate **shear stresses** induced in
a beam by bending.

Analysis of shear bending stress is more complex than normal bending stress and this tutorial provides a basic introduction valid for beams of rectangular cross-section with height significantly greater than breadth.

The diagram below illustrates the origin of shear stress in bending.

The diagram shows a beam of rectangular cross-section with height h and width b.

A small element of the beam height y_{1} above the neutral axis
x-x with vertical faces A and B is subject to shearing forces V.
The bending
moment M varies along the beam and from our previous study we know:

Also from our study of normal bending stress we know σ is directly proportional to M.

It follows that the mean normal stress on face B must be greater
than on face A and thus the net the force **F**_{B} on face B must be
greater than the net force **F**_{A} on face A.

For the element to be in equilibrium the sum of horizontal forces
must equal zero. Thus a horizontal force **F**_{C} is exerted on the
bottom face of the element resulting in a shear stress **τ
**on this
face.

Note that I_{z} is the second moment of area of the entire
cross-section about the neutral z axis.

We know that the shear stress on a plane in a biaxial element is accompanied by shear stresses of equal magnitude on the other three orthogonal planes with directions such that the element is in equilibrium. Hence a shear stress of the same magnitude exists on the other faces of the element as shown below.

In this model it was assumed that V has the
same value on both vertical faces of the element, which is true for
beams with point loading. For distributed loads, V is not
constant but when the length of the element → 0 the expression above
for **τ **is valid.

We now examine the distribution of shear stress on the y-z plane in the case of the rectangular section below.

Distribution of **τ** across the section for values
of y_{1} from -c to +c is a quadratic function as shown
below. **τ _{max} **occurs at the neutral
axis where

Compare this value with the assumption of uniformly distributed shear stress over the cross-section for shearing force V giving:

i.e. **τ _{max} **is greater by a factor
of 3/2. Similar factors for solid circular and hollow circular
sections are 4/3 and 2 respectively.

I welcome feedback at:

alistair@alistairstutorials.co.uk