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Assignment
FINITE ELEMENT
METHODS
Q.1.7 Find the stress distribution in the beam shown in figure 1.18 using two beam elements.
STEP (I): DISCRETIZATION OF THE STRUCTURE
The first step in the finite element method is to divide the structure or solution region into subdivisions or elements. Hence, the structure is to be modeled with suitable finite elements. The number, type, size, and arrangement of the elements are to be decided. For given problem free body diagram is shown below.
Structure has been discretized into two elements which are two node and linear. The variables W 1 and W 3 are displacements at node 1 and 2 respectively and W 2 and W 4 are rotations at nodes 1 and 2 respectively.
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STEP (II): 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.
And
Now putting 1st^ and 2nd^ boundary conditions
Putting in (1) and (2) respectively gives,
Also
Putting in (3) and (4) gives two equations.
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Potential energy of beam is given by,
Where π is the strain energy given by,
For 2-node bar element,
Where E(1)^ = E(2)^ = E(e)^ = E. For beams,
Where I(1)^ = I(2)^ = I(e)=I. Thus
Since by definition,
For beams moment is related to displacement as (note that derivative is exact since y(x) is a function of only x and not time in this case)
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But here y is denoted by w(x) thus,
Thus,
Simplifying,
If E and I are independent of x as in this case then,
Now since,
First derivative is,
Second derivative is,
Let us consider that,
Where,
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First term,
Similarly, second term,
Similarly all the other terms are calculated to get,
Therefore,
Where,
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STEP (IV): ASSEMBLAGE OF ELEMENT EQUATIONS TO OBTAIN THE OVERALL EQUILIBRIUM
EQUATIONS
Since the structure is composed of two elements, the individual element stiffness matrices and load vectors are to be assembled in a suitable manner and the overall equilibrium equations have to be formulated.
For element no. 1,
For element no. 2
Since there are three nodes each having two degrees of freedom, therefore global stiffness matrix will be of 6×6 order. Also l(1)=l(2)=l.
Since length of both elements is same therefore,
Also global displacement vector,
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From (b),
From (c),
Also from (a),
Realizing that,
STEP (VI): COMPUTATION OF REACTIONS AND END MOMENTS
We know that
Three equations can be written as,
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Therefore,
Now since, therefore,
And
Hence completes the solution.
