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This assignment solution was submitted to Amar Sharma for Finite Element Method course at Aligarh Muslim University. It includes: Stress, Distribution, Beam, Elements, Discretization, Structure, Variation, Field, Variable, Interpolation
Typology: Exercises
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Find stress distribution in beam shown in figure (below) by using two beam elements.
In this step, we discretize the given structure so that we may apply interpolation model on each of the part to get the variation of field variable within the domain.
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Desretized beam is shown below,
In this step, we choose a suitable model which represents the behavior of solution in the given domain. Let us assume a solution of the following type,
From the displacement model given above, we derive stiffness matrix and load vector. For this,
Potential energy of beam is given by,
---------------------------------------- (1)
Where,
Where π is the strain energy given by,
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Then equation (3) becomes,
Now, simplify the above equation, we get
Now, writing the above equation in matrix form, as under
Where,
Now, we have to calculate all the entries of the above matrix,
5
Similarly second term,
Similarly, all other terms are calculated below,
Now, put all these values in equation (6), we get
And equation (5) then becomes,
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And global load vector,
Put all above values in equation (7), we get
From figure, we can see the following boundary conditions;
W 1 =W 2 =W 5 = W 6 =0. V 1 = V 3 =0, M 1 =M 2 = M 3 =0. Therefore,
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So the above matrix can be reduced to,
Now replace and and in above equation, we get
Simplify it, we get two equations
Solving above two equations simultaneously, we get
For bending stress induced at distance x from node 1and y from neutral axis is given by
