Question

A simply supported beam of span length $L$ and flexural stiffness $EI$ has another spring support at a centre span of stiffness $K$ as shown in fig. The central deflection of the beam due to a central concentrated load of $P$ would be

• A. $\left[P{L}^3/48EI\right]+\left(P/K\right)$
• B. $P/l\left(\right(48EI/L^3)-K$
• C. $\left[P{L}^3/48EI\right]+\left(P/K\right)$
• D. $P/\left(\right(48EI/L^3)+K$)

Answer 1

The correct option is D $P/\left(\right(48EI/L^3)+K$)


From compatibility condition:
$\frac{(P-R){\ell }^3}{48EI}=\frac{R}{K}$
$\Rightarrow \frac{P{\ell }^3}{48EI}=R(\frac{1}{K}+\frac{{\ell }^3}{48EI})$
$\frac{R}{K}=\frac{P{\ell }^3}{\frac{48EI}{(\frac{1}{K}+\frac{{\ell }^3}{48EI})K}}$
$=\left(\frac{\frac{P{\ell }^3}{48EI}}{1+\frac{{\ell }^{3}K}{48EI}}\right)$
$\delta =\frac{P{\ell }^3}{48EI+K{\ell }^3}$
$\delta =\frac{P}{\frac{48EI}{{\ell }^3}+K}$