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## Q. 2.7

Determine the reaction in the prop of the propped cantilever shown in Figure 2.7 (a) when the prop is firm and rigid, (b) when the prop is rigid and settles an amount y, and (c) when the prop is elastic.

## Verified Solution

(a) The structure is one degree redundant, and the reaction in the prop is selected as the redundant and removed as shown at (i). The deflection of the free end of the cantilever in the line of action of V is:

\begin{aligned}\delta_{2}^{\prime} &=-w l^{3} / 6 \times 3 l / 4 E I \\&=-w l^{4} / 8 E I\end{aligned}

To the cut-back structure, the redundant V is applied as shown at (ii). The deflection of the free end of the cantilever in the line of action of V is:

\begin{aligned}\delta_{2}^{\prime \prime} &=V l^{2} / 2 \times 2 l / 3 E I \\&=V l^{3} / 3 E I\end{aligned}

The total deflection of 2 in the original structure is:

\begin{aligned}\delta_{2} &=\delta_{2}^{\prime}+\delta_{2}^{\prime \prime} \\&=0\end{aligned}

Thus:

$-w l^{4} / 8 E I+V l^{3} / 3 E I=0$

and

V = 3wl/8

(b) The total deflection of 2 in the original structure is:

\begin{aligned}\delta_{2} &=\delta_{2}^{\prime}+\delta_{2}^{\prime \prime} \\&=-y\end{aligned}

Thus:

$-w l^{4} / 8 E I+V l^{3} / 3 E I=-y$

and

$V = 3 w l / 8-3 E I y / l^{3}$

(c) The total deflection of 2 in the original structure is:

\begin{aligned}\delta_{2} &=\delta_{2}^{\prime}+\delta_{2}^{\prime \prime} \\&=-V L / A E\end{aligned}

where L, A, and E are the length, cross-section, and modulus of elasticity of the prop.

Thus:

$-w l^{4} / 8 E I+V l^{3} / 3 E I=-V L / A E$

and

$V=w l^{4} / 8 E I\left(l^{3} / 3 E I+L / A E\right)$