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Solid Sections

For rectangular beams it is usually sufficiently accurate to take the shear area for deflection as 56bd\frac{5}{6}bd where bb is the breadth and dd the depth of the section. The corresponding maximum shear stress is 32Vbd\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^{12})

For circular sections the shear area for deflections is

6πr27+ν2(1+ν)2\frac{6\pi r^{2}}{7 + \frac{\nu^{2}}{(1 + \nu)^{2}}}

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

[1.5+ν1+ν]Vπr2\left\lbrack \frac{1.5 + \nu}{1 + \nu} \right\rbrack\frac{V}{\pi r^{2}}

which varies from

1.5Vπr2forν=01.5\frac{V}{\pi r^{2}} \quad\text {for} \quad \nu = 0


1.33Vπr2forν=0.51.33\frac{V}{\pi r^{2}} \quad\text {for}\quad \nu = 0.5


1.38Vπr2forν=0.31.38\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^{12} Timoshenko, S.P. and Goodier, J.N. Theory of elasticity. 3rd edition. McGraw-Hill, 1970.