Introduction to Elasticity/Energy methods example 4

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Example 4 : Bending of a cantilevered beam[edit | edit source]

Bending of a cantilevered beam

Application of the Principle of Virtual Work[edit | edit source]

The virtual work done by the external applied forces in moving through the virtual displacement is given by

The work done by the internal forces are,

From beam theory, the displacement field at a point in the beam is given by

The strains are, neglecting Poisson effects,

and the corresponding stresses are

If we also neglect the shear strains and stresses, we get

Therefore, from the principle of virtual work,

Integrating by parts and after some manipulation, we get,

where is the Dirac delta function,

The Euler equation for the beam is, therefore,

and the boundary conditions are


Application of the Hellinger-Prange-Reissner variational principle[edit | edit source]

The governing equations of the cantilever beam can be written as

Kinematics[edit | edit source]

Constitutive Equation[edit | edit source]

Equilibrium (kinetics)[edit | edit source]

Recall that the Hellinger-Prange-Reissner functional is given by

If we apply the strain-displacement constraints using the Lagrange multipliers and the displacement boundary conditions using the Lagrange multipliers , we get a modified functional

For the cantilevered beam, the above functional becomes

Taking the first variation of the functional, we can easily derive the Euler equations and the associated BCs.

and

The same process can be used to derive Euler equations using the Hu-Washizu variational principle.