Crystal Elasticity

7Beam Bending

7.1. Homogeneous beam bending without shear

Consider a homogeneous beam of length L along x, thickness B along z, modulus E and density ρ subjected to bending. With the usual notations, u is the displacement along x and w along z. According to the Euler–Bernoulli kinematic hypothesis (zero shear γ_xz), the following can be written as:

[7.1]

Since the displacement w does not depend on z, integration with respect to z yields:

[7.2]

The only existing deformation is longitudinal along x, which can be written as:

[7.3]

Denoting by F_F the bending resonance frequency, the deformation and displacement energies are:

[7.4]

[7.5]

The rotational kinetic energy is negligible compared to the translational energy according to the hypothesis of no deformation under shear force and therefore no shear. The expressions of the Lagrangian and its minimization are:

[7.6]

[7.7]

A first integration ...