Shear Stresses in Beams

• Review simple beam theory.

• Consider effects of bending on shear forces in beams.

Shear stresses in beams is given by:

$\tau = \dfrac{VQ}{Ib}$

where Q is the first moment of area $\int ydA$ of the surface section of the cross-section.

For the rectangular beam, one has the maximum shear stress in the cross section arises at the neutral surface: $\tau_{max} = \dfrac{3V}{2bh} = \dfrac{3V}{2A}$ and the shear stress dies away towards the upper and lower surfaces. Note that the average shear stress over the cross-section is $\dfrac{V}{A}$ and the maximum shear stress is 150% of this value.

Solved Example:

44-1-01

A symmetrical I-section is subjected to shear force. The shear stress is induced across the section is maximum at which location?

Solution:
Shear stress in beams is maximum at the neutral axis and it is zero at the extreme top and bottom layers.

Solved Example:

44-1-02

Shear stress is zero at the:

Solved Example:

44-1-03

Shear stress in the beam acting on the cross section is:

Solved Example:

44-1-04

A symmetric I-section (with width of each flange = 50 mm, thickness of each flange = 10 mm, depth of web = 100 mm, and thickness of web = 10 mm) of steel is subjected to a shear force of 100 kN. Find the magnitude of the shear stress in the web at its junction with the top flange.

Solution:
$I =\dfrac {50\times 120^{3}}{12}-\dfrac {40\times 100^{3}}{12} =3.866\times 10^{6}\ mm^{4}$ $q =\dfrac {SA\overline {y}}{Ib} =\dfrac {100\times 10^{3}\times 50\times 10\times 55}{3.866\times 10^{6}\times 10} =71.12\ \mathrm{N/mm^{2}}$