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 ds 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.
