Question

A cantilever beam of length $l$, and flexural rigidity $EI$,stiffened by a spring of stiffness k, is loaded by transverse force $P$, as shown
The transverse deflection under the load is

• A. $\frac{P{l}^3}{3EI}\left[\frac{3EI-K{l}^3}{3EI}\right]$
• B. $\frac{P{l}^3}{3EI}\left[\frac{3EI}{3EI+K{l}^3}\right]$
• C. $\frac{P{l}^3}{3EI}\left[\frac{6EI-K{l}^3}{6EI}\right]$
• D. $\frac{P{l}^3}{3EI}\left[\frac{3EI}{3EI+2K{l}^3}\right]$

Answer 1

The correct option is B $\frac{P{l}^3}{3EI}\left[\frac{3EI}{3EI+K{l}^3}\right]$

As per given configuration diagram of beam
${\Delta }_{beam}={\Delta }_{spring}$
$\frac{(P-R)l^3}{3EI}=\frac{R}{K}$
$P-R=\frac{3EI}{K{l}^3}R$
$\therefore$ $R=\frac{P}{1+\frac{3EI}{K{l}^3}}=\frac{P{l}^{3}K}{K{l}^3+3EI}$
Deflection of beam
${\Delta }_{beam}=\frac{R}{K}=\frac{P{l}^3}{K{l}^3+3EI}$