Question

Beam GHI is supported by three pontoons as shown in the figure below. The horizontal cross-sectional area of each pontoon is $8m^2,$ the flexural rigidity of the beam is $10000kN-m^2$ and the unit weight of water is $10kN/m^3.$



When the middle pontoon is brought back to its position as shwon in the figure above, the reaction at H will be

• A. $8.6kN$
• B. $15.7kN$
• C. $19.2kN$
• D. $24.2kN$

Answer 1

The correct option is C $19.2kN$



As the beam is symmetrical, so reactions at G and I will be equal
$\sum V=0$
$\Rightarrow R+R_H+R=48$
$2R+R_H=48$ $...\left(i\right)$
Depth of immersion of pontoon at $G$ and $I=h$
$R=A\times h\times {\gamma }_w$
$\Rightarrow h=\frac{R}{A.{\gamma }_w}=\frac{R}{8\times 10}$
$=\frac{R}{80}$ $...\left(ii\right)$
Also $R_H=A(h+\delta ){\gamma }_w$
$\frac{R}{80}+\delta =\frac{R_H}{80}$ [value of $h$ from $eq.\left(ii\right)]$
$\frac{48-R_H}{2\times 80}+\delta =\frac{R_H}{80}$ [value of $h$ from $eq.\left(i\right)]$
$\delta =\frac{3R_H-48}{160}$ $...\left(iii\right)$
Elastic deformation,
$\delta =\frac{(48-R_H)L^3}{48EI}$
$\frac{3R_H-48}{160}=\frac{(48-R_H){10}^3}{48\times 10000}$ $[$From $eq.\left(iii\right)]$
$\Rightarrow R_H=19.2kN$