Example 1

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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

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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,