Introduction to Elasticity/Beam bending example 1

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Example 1[edit | edit source]

Given:

A long rectangular beam with cross section

Find:

A solution for the displacement and stress fields, using strong boundary conditions on the edges and .

[Hint : Assume that the displacement can be expressed as a second degree polynomial (using the Pascal's triangle to determine the terms) ]

Solution[edit | edit source]

Step 1: Boundary conditions

Step 2: Assume a solution

Let us assume antiplane strain

Step 3: Calculate the stresses

The stresses are given by , and . Therefore,

Step 4: Satisfy stress BCs

Thus we have,

Since and can be arbitrary, .

Hence, which gives us

Assume that the body force is zero. Then the equilibrium condition is . Therefore,

Therefore, the stresses are given by

Step 5: Satisfy displacement BCs

The displacement is given by

If we substitute , we cannot determine the constant uniquely.

Hence the displacement boundary conditions have to be applied in a weak sense,

Therefore,