## Shear Deflections in Beams

Shear deformations in a beam add to the deflections due to bending discussed in Art. 3.18. Deflections due to shear are generally small, but in some cases they should be taken into account.

When a cantilever is subjected to load P (Fig. 3.51a), a portion dx of the span undergoes a shear deformation (Fig. 3.51b). For an elastic material, the angle equals the ratio of the shear stress v to the shear modulus of elasticity G. Assuming that the shear on the element is distributed uniformly, which is an approximation, the deflection of the beam ds caused by the deformation of the element is

Figure 3.52c shows the corresponding shear deformation. The total shear deformation at the free end of a cantilever is

The shear deflection given by Eq. (3.84) is usually small compared with the flexural
deflection for different materials and cross-sectional shapes. For example, the flexural deflection at the free end of a cantilever is f  PL3/3EI. For a rectangular section made of steel with G 0.4E, the ratio of shear deflection to flexural deflection is

where h  depth of the beam. Thus, for a beam of rectangular section when h/L = 0.1, the shear deflection is less than 1% of the flexural deflection.
Shear deflections can be approximated for other types of beams in a similar way. For example, the midspan shear deflection for a simply supported beam loaded with a concentrated load at the center is PL/4AG.