Question

$\text{Flexibility matrix of the beam shown below is}\delta =\frac{1}{3EI}\left[\begin{array}{cc}2& 2\\ 2& 8\end{array}\right]$



$\text{If support B settles by}\frac{\Delta }{EI}\text{units, what is the reaction at B}$

• A. $4\Delta$
• B. $\Delta$
• C. $2\Delta$
• D. $8\Delta$

Answer 1

The correct option is C $2\Delta$

$\text{Stiffness matrix [K] is inverse of flexibility matrix}\left[\delta \right]$

$\therefore \left[K\right]=[\delta {]}^{-1}={\left(\frac{1}{3EI}\right)}^{-1}{\left[\begin{array}{cc}2& 2\\ 2& 8\end{array}\right]}^{-1}$
$=3EI\left[\begin{array}{cc}2/3& -1/6\\ -1/6& 1/6\end{array}\right]$

We know [p] = [k]$\left[\Delta \right]$
$\left[\begin{array}{c}{p}_1\\ P_2\end{array}\right]=\left[K\right]\left[\begin{array}{c}{\Delta }_1\\ {\Delta }_2\end{array}\right]$

$\left[\begin{array}{c}{p}_1\\ p_2\end{array}\right]=3EI\left[\begin{array}{cc}2/3& -1/6\\ -1/6& 1/6\end{array}\right]\left[\begin{array}{c}\frac{\Delta }{EI}\\ 0\end{array}\right]$

${\Delta }_2=0asnodisplacementat2$
$p_1=reactionat{}^{\prime }{1}^{\prime }duetosettlementat1$
$p_1=3EI\times \frac{2}{3}\times \frac{\Delta }{EI}$
$p_1=2\Delta$