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Shear center is defined as the point about which the external load has to be applied so that it produces no twisting moment.
shear stress (since the thickness is small) over the thickness, we have,
The integration being carried out over the portion ‘ab’ of the flange.
The stress per unit length of the centre line of the section,
Therefore, it is seen that q is proportional to u.
The maximum value of 
.
At the junction of the flange and web, the distribution of the shear stress is complicated, so we may assume that the equation 
holds good for u =0 and u = b.
The average shear stress 
The longitudinal shear force in the top and bottom of the flange 
The couple about the z-axis of these shear forces 
Let us assume that the vertical shear force F acts through point ‘o’, the shear centre at a distance c from O on the centre line of the web.
The twisting of this section is avoided if
which gives the position of the shear centre.
Note: the shear centre for cross-sectional areas having one axis of symmetry, is always located on the axis of symmetry. In the case of the I-beam which is symmetrical about both the x-axis and y-axis, the shear centre coincides with the centroid of the section. The exact location of the shear centre for unsymmetrical sections are complicated and can be located by inspection.
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