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## Q. 2.5

Determine the deflection at the free end of the cantilever shown in Figure 2.5.

## Verified Solution

The origin of coordinates is taken at the free end and the functions M and m derived from (i) and (ii) as:

M = Wx

and m = x

The deflection at the free end is given by:

\begin{aligned}1 \times \delta &=\int_{0}^{l} M m \mathrm{~d} x / E I \\&=W \int_{0}^{l} x^{2} \mathrm{~d} x / E I \\&=W l^{3} / 3 E I\end{aligned}

Alternatively, the solid defined by the functions M and m is shown at (iii); its volume is:

$W l^{2} / 2 \times 2 l / 3=W l^{3} / 3$

and the deflection at the free end is given by:

$\delta=W l^{3} / 3 E I .$

Alternatively, from Table 2.3, the value of $\int$ Mm dx/EI is given by:

\begin{aligned}\delta &=l a c / 3 E I \\&=W l^{3} / 3\end{aligned}
 Table 2.3 Volume integrals M m lac lac/2 la(c + d)/2 lac/2 la(c + 4d + e)/6 lac/2 lac/3 la(2c + d)/6 lac (1 + β)/6 la(c + 2d)/6 lac/2 lac/6 la(c + 2d)/6 lac (1 +  α)/6 la (2d + e)/6 lc (a + b)/2 lc(2a + b)/6 la(2c + d)/6 + lb (c + 2d)/6 lac (1 + β)/6 +  lbc (1 + α)/6 la(c + 2d)/6 +  lb(2d + e)/6