Determine the vertical deflection of the free end of the cantilever beam shown in Fig. 4.13(a).

Repeat using the Symbolic Math Toolbox in MATLAB.

Question

Determine the vertical deflection of the free end of the cantilever beam shown in Fig. 4.13(a).

Accepted Answer

Let us suppose that the actual deflection of the cantilever at B produced by the uniformly distributed load is [latex]V_{ B }[/latex] and that a vertically downward virtual unit load is applied at B before the actual deflection takes place. The external virtual work done by the unit load is, from Fig. 4.13(b), [latex]1 V_{ B }[/latex] The deflection, [latex]V_{ B }[/latex], is assumed to be caused by bending only; that is, we ignore any deflections due to shear. The internal virtual work is given by Eq. (4.21), which, since only one member is involved, becomes

[latex]W_{i, M}=\sum \int_{L} \frac{M_{ A } M_{v}}{E I} d x[/latex] (4.21)

[latex]W_{i, M}=\int_{0}^{L} \frac{M_{ A } M_{v}}{E I} d x[/latex] (i)

The virtual moments, [latex]M_{v}[/latex], are produced by a unit load, so we replace [latex]M_{v}[/latex] by [latex]M_{1}[/latex]. Then,

[latex]W_{i, M}=\int_{0}^{L} \frac{M_{ A } M_{1}}{E I} d x[/latex] (ii)

At any section of the beam a distance x from the built-in end,

[latex]M_{ A }=-\frac{w}{2}(L-x)^{2}, \quad M_{1}=-1(L-x)[/latex]

Substituting for [latex]M_{ A }[/latex] and [latex]M_{ 1 }[/latex] in Eq. (ii) and equating the external virtual work done by the unit load to the internal virtual work, we have

[latex]1 v_{ B }=\int_{0}^{L} \frac{w}{2 E I}(L-x)^{3} d x[/latex]

which gives

[latex]1 v_{ B }=-\frac{w}{2 E I}\left[\frac{1}{4}(L-x)^{4}\right]_{0}^{L}[/latex]

so that

[latex]v_{ B }=\frac{w L^{4}}{8 E I}[/latex]

Note that [latex]V_{ B }[/latex] is in fact negative, but the positive sign here indicates that it is in the same direction as the unit load.

Calculation of the vertical deflection of the free end of the cantilever shown in Fig. 4.13(a) is obtained through the following MATLAB file:

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