I have a stepped cantilever and I want to derive deflection formula to calculate the deflection at any point along this cantilever !
I've tried Castigliano's theorem but it gave me the tip only.
The cross sections are hollow rectangles with different hights and widths.
Thanks in advance All.
Dear Ahmed
It is a pity that the FEM has over run other techniques in structural mechanics. Consider a cantilever with an overall length of L and with the step occurring at a distance s = a from the free end. You did not define the load on the cantilever. But it may not be vital to the solution. You should use the method of 'unit virtual load' as devised by the Russian Vereschagin. Draw the BMD for the real loading or obtain the equation(s) of the BMD. E.g. if the Cantilever is subject to a udl q and a tip point load of P the equation of the BMD is M = P(L-s) + .5q(L-s)2 . Apply a unit load at any point x from the clamped end. The BMD has the equation m1 = x-s if s < x and m1 = 0 if s > x; where s is an auxiliary parameter measured from the clamped end. The deflection of the cantilever at any point x is then given by v = Integral 0a(Mm1ds)/EI1 + Integral aL (Mm1ds)/EI2 where EI1 is the flexural rigidity from s = 0 to s =a and EI2 is the flexural rigidity from s = a to s = L. Recall the step occurs at s = a from the clamped end. Refer to any old text book on Structures for this concept. You need only worry about the integration limits. If x < a only the 1st integration need be carried out from s = 0 to s = x. If x > a the 1st integration will be carried out and the limits of the 2nd integration will be s = a to s = x. Using this method and the assumed loading the deflection of the beam at x from clamped end when x < = a is given by
v(x) = Px2(3L-x)/(6EI1) + qx2(6L2- 4Lx + x2)/(24EI1) which can be shown to be correct when the beam is uniform and x = L when it reduces to
v(L) = PL3/(3EI) + qL4/(8EI). When x > a we obtain the lengthy solution
v(x) = Pa2(3L-a)/(6EI1) + qa2(6L2-4La+a2)/(24EI1) + P(x-a)2(3L-2a+x)/(6EI2) + q(x-a)2[6(L-a)2 - 4(L-a)(x-a) +(x-a)2]/(24EI2)
Sincerely
Amaechi. J. Anyaaegbunam
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