Solid Sections

For rectangular beams it is usually sufficiently accurate to take the shear area for deflection as $\frac{5}{6}bd$ where $b$ is the breadth and $d$ the depth of the section. The corresponding maximum shear stress is $\frac{3}{2}\frac{V}{bd}$. It should be noted however that for wide beams the maximum shear stress is underestimated by this formula: for a beam with an aspect ratio of 1 the maximum stress is 12.6% higher. (For a beam with an aspect ratio of 50 (for example a slab) the maximum stress is about 2000% higher but, as this is a Poisson’s ratio effect, it is difficult to believe that this has any practical significance!$^{12}$)

For circular sections the shear area for deflections is

$$\frac{6\pi r^{2}}{7 + \frac{\nu^{2}}{(1 + \nu)^{2}}}$$

where $\nu$ is Poisson’s ratio and $r$ the radius. The expression is very insensitive to the value of $\nu$. The maximum shear stress is given by

$$\left\lbrack \frac{1.5 + \nu}{1 + \nu} \right\rbrack\frac{V}{\pi r^{2}}$$

which varies from

$$1.5\frac{V}{\pi r^{2}} \quad\text {for} \quad \nu = 0$$

to

$$1.33\frac{V}{\pi r^{2}} \quad\text {for}\quad \nu = 0.5$$

with

$$1.38\frac{V}{\pi r^{2}} \quad\text {for}\quad \nu = 0.3$$

This item was written by John Blanchard and Ian Feltham for Feedback Notes [an Ove Arup & Partners internal publication] (1992 NST/21) originally published in October 1992. Incorporates 1996NST/10 and is reproduced here with permission.

$^{12}$ Timoshenko, S.P. and Goodier, J.N. Theory of elasticity. 3rd edition. McGraw-Hill, 1970.