Consider a tapered shape elastic bar which is subjected to an applied tensile of force P at the bottom end and it is attached to a fixed support at the other end (which is shown in below figure 1). The area of cross-section A varies linearly from area A0 at the fixed support end at the distance x = 0 to area A0/2 at the distance x = L.
Here we need to calculate the displacement of the end of the bar
From figure a and b, the area of the cross-section for a single element is 3A0/4 and the element “spring constant k” is
and element equation are:
Now, applying the constraint condition U1 = 0, and we find for U2 as the displacement at x = L.
From figure c, there are two elements of same length L/2 with the related nodal displacements.
For element 1, area is A1 = 7A0/8 and so
And for the element 2, we have the area is A1 = 5A0/8 and the stiffness k2 will be written as
As no load is applied at the midpoint of the bar, the equations of equilibrium for the system of the two elements is written as
After applying the constraint condition, U1 = 0, results in
By adding the two equations, it gives
By substituting this U2 into equation 1, the result is
By comparing the displacement U for three solution at the length x = L is
a) U1 = 1.333 PL/A0E
b) U3 = 1.371 PL/A0E
c) The resultant exact solution is 1.386 PL/A0E.
Hence, the above derived expression is example for the truss element and its solution.
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