Shearing deformations in beams

Abstract

Euler-Bernoulli beam theory is developed in chapter 5 based on the purely kinematic assumptions discussed in section 5.1. In particular, the cross-section of the beam is assumed to remain plane after deformation, and furthermore, this plane is assumed to remain normal to the deformed axis of the beam. This second assumption implies the vanishing of the transverse shear strains, γ 12 = 0, and leads to the following result for a beam made from a linearly elastic, homogeneous and isotropic material V 2 = A τ 12 dA = A Gγ 12 dA = 0, (15.1) where the second integral is the result of using the constitutive law relating shear stresses to shear strains, τ 12 = Gγ 12. On the other hand, equilibrium conditions require a non-vanishing transverse shear force, V 2 , to equilibrate the distributed transverse load, p 2 (x 1), applied to the beam, see eq. (5.38). This apparent contradiction with eq. (15.1) can be resolved through the following reasoning: as required by equilibrium, the shear stress, τ 12 , does not vanish, but the corresponding shear strain is vanishingly small. This implies a very large shearing modulus, G → ∞, so that a vanishing shear strain γ 12 → 0, results in a product, Gγ 12 = τ 12 , that becomes a finite, non-vanishing quantity. In view of this reasoning, the assumption "plane sections remain normal to the deformed axis of the beam," which implies the vanishing of the transverse shear strains, could be replaced by "the beam is made of a material with an infinite shear modulus." Because such a constitutive law is awkward, the transverse shear force (the stress resultant associated with the shear stress), is not evaluated from this constitutive law but from equilibrium considerations instead. In fact, the shear force is altogether eliminated from Euler-Bernoulli beam theory using equilibrium considerations, see eq. (5.39), and can be recovered from the bending moment as V 2 = −dM c 3 /dx 1 , eq. (5.38). In reality, the shear modulus is of the order of Young's modulus and for isotropic materials, G = E/(2(1 + ν). To investigate the effects of the additional flexibility