Stress Distribution In Beam-Finite Element Method-Assignment Solution, Exercises of Mathematical Methods for Numerical Analysis and Optimization

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ASSIGNMENT

PROBLEM # 1.

P.1.6 FIND THE STRESS DISTRIBUTION IN THE BEAM SHOWN IN

FIGURE 1.17 USING TWO BEAM ELEMENTS.

STEP # (01): DISCRETIZATION OF BEAM

The given beam has been discretized into two elements as shown in figure 2. The variables W 1 and W 3 are displacements at node 1 and 2 respectively and W 2 and W 4 are slope at nodes 1 and 2 respectively.

STEP #(02): SELECTION OF A PROPER INTERPOLATION OR

DISPLACEMENT MODEL

Since the displacement solution of a complex structure under any specified load conditions cannot be predicted exactly, we assume some suitable solution within an element to approximate the unknown solution. The assumed solution must be simple from a computational standpoint, but it should satisfy certain convergence requirements. In general, the solution or the interpolation model is taken in the form of a polynomial. In this problem there are four variables in each element as shown in figures above therefore assuming a solution as under.

by calculating the unknown in equation (1) we get the final equation

Figure 1.

Figure 2

Let

Then

By squiring

This can be arranged into matrix form as,

Now,

Now integrating each term one by one

First term,

Similarly, second term,

Similarly all the other terms are calculated, we get

Therefore,

Also global displacement vector,

And global load vector,

Hence since,

Therefore,

STEP # (05): SOLUTION FOR THE UNKNOWN NODAL DISPLACEMENTS

Now applying boundary conditions that are W 1 =W 2 =W 5 =0. P 3 = P, P 6 = P 4 = 0. Hence,

Eliminating the row and column corresponding to W 1 , W 2 , and W 5

Simplifying the equations

Hence we get three equations.

From these three equations we get

Also from (a),

Where

STEP# (06): COMPUTATION OF STRESS DISTRIBUTION