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DoITPoMS Teaching & Learning Packages Brittle Fracture Estimate of the strain energy in a bent beam

# Estimate of the strain energy in a bent beam

In pure bending, where the moment, M, is constant along a beam of length L, the curvature of the beam, κ, and the angle, θ, through which the beam is bent is given by

$$\kappa = {M \over {EI}}{\rm{, so }}\theta = {M \over {EI}} \cdot L$$

The strain energy, Ustrain, is given by

$${U_{{\rm{strain}}}} = {1 \over 2}M\theta = {{{M^2}L} \over {2EI}}$$

Here the moment therefore varies along the length so that Ustrain becomes

$${U_{{\rm{strain}}}} = \int\limits_0^L {{{{M^2}} \over {2EI}}} {\rm{ d}}x$$

where x is the distance along the beam from the point of application of force at the free end. The bending moment is given at any point x by M = Fx  so,

$${U_{{\rm{strain}}}} = {{{F^2}{L^3}} \over {6EI}}$$

The work done, Wforce, by the applied force when the beam is bent is

$${W_{{\rm{force}}}} = {1 \over 2}F\delta$$

This is equal to the strain energy in the beam, Ustrain, so that

$$F = {{3EI\delta } \over {{L^3}}}$$

Substituting for F in the expression for Ustrain and remembering that I = wd3/12 for a beam of thickness d and width w  gives

$${U_{{\rm{strain}}}} = \left[ {{{Ew{d^3}{\delta ^2}} \over {8{L^3}}}} \right]$$

For our peeling layer of unit width, i.e. w = 1, then L = c and d = h, so the strain energy in the peeling ligament, UE, is

$${U_{\rm{E}}} = {{E{d^3}{h^2}} \over {8{c^3}}}$$