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Shear center is a point on the beam-section where the application of loads does not cause its twisting. The shear center position is dependent on the cross-section of the beam. For instance, shear center and center of gravity are the same in a symmetrical section, but it may not coincide with the centroid in case of an unsymmetrical cross-section.
So, in unsymmetrical cross-sections, the external forces shall pass through the shear center rather than the center of gravity of the section otherwise the produced bending moment would be accompanied by twisting.
How to Compute Location of Shear Center?
Consider a channel section as shown in fig. 1. Now we shall find the position of the plane through which the vertical loads must act so as to produce simple bending, with the x-axis as the neutral axis.
It may be assumed that the vertical shearing force F at the section is taken up by the web alone. In the flanges, there will be horizontal shear stresses which are denoted by q.
Let us consider an element ‘abcd’ cut from the lower flange by two adjacent cross-sections (delta z) apart, and by a vertical plane parallel to the web and at distance ’u’ (which is variable) from the free end of the lower flange.
The difference in tensile forces T and T+Delta z must be equal to the shear force on the side ‘ad’ of the element. Assuming a uniform distribution of shear stress (since the thickness is small) over the thickness, we have:
The integration is carried out over the portion ‘ab’ of the flange. The stress per unit length of the center line of the section:
Therefore, it is seen that q is proportional to u. The maximum value of q :
At the junction of the flange and web, the distribution of the shear stress is complicated, so we may assume that the equation 4 holds good for u = 0 and u = b.
Let us assume that the vertical shear force F acts through point ‘o’, the shear center at a distance c from O on the center line of the web. The twisting of this section is avoided if:
which gives the position of the shear center.
Note: The shear center 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 center coincides with the centroid of the section. The exact location of the shear center for unsymmetrical sections are complicated and can be located by inspection.
Locate the shear center of the unsymmetrical I-beam cross-section as shown in the figure below :
Taking moment about the point D: