A cantilever beam of uniform EI has a span equal to ‘L’. An upward force W acts at the midpoint of the beam and a downward force P acts at the free end. In order that the deflection at the free end is zero, the relation between P and W should bea) W =3P/2b) W = 2Pc) W =16P/5d) W = 5PCorrect answer is option 'C'. Can you explain this answer?

Question

Answer

Solution:

Given, a cantilever beam of uniform EI has a span equal to ‘L’. An upward force W acts at the midpoint of the beam and a downward force P acts at the free end.

To find: The relation between P and W should be in order that the deflection at the free end is zero.

Let the deflection at the free end be zero, then the equation of the deflection curve is given by;

∂^2y/∂x^2 = -P/(EI) (x-L)

where, y = deflection of the beam at any point x

Now, integrating this equation twice, we get;

y = (P/(2EI))(x-L)^2 + C1x + C2

Applying the boundary conditions, at x = 0, y = 0 and at x = L, y = 0

0 = C2

0 = (P/(2EI))(L^2) + C1L

C1 = -PL/(2EI)

Thus, substituting the value of C1 in the deflection equation, we get;

y = (P/(2EI))(x-L)^2 - (PL^2/(2EI))x

Now, at x = L/2, the deflection is maximum and it is given by;

δmax = (PL^3)/(192EI)

For the deflection at the free end to be zero, we have;

δ(L) = 0

(P/(2EI))(L^3/4) - (PL^3/(2EI)) = 0

P = (16/5)W

Hence, the relation between P and W should be W = 16P/5 in order that the deflection at the free end is zero.

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