Question

Consider the frame shown below:


If all the members have same length $l$ and same flexural rigidity EI, then stiffness matrix for the structure corresponding to degree of freedom 1 and 2 is

• A. $\left(\begin{array}{cc}\frac{12EI}{L^3}& -\frac{6EI}{L^2}\\ -\frac{6EI}{L^2}& \frac{7EI}{L}\end{array}\right]$
• B. $\left(\begin{array}{cc}\frac{12EI}{L^3}& \frac{6EI}{L^2}\\ \frac{6EI}{L^2}& \frac{7EI}{L}\end{array}\right]$
• C. $\left(\begin{array}{cc}\frac{12EI}{L^3}& \frac{4EI}{L^2}\\ \frac{4EI}{L^2}& \frac{7EI}{L}\end{array}\right]$
• D. $\left(\begin{array}{cc}\frac{12EI}{L^3}& \frac{6EI}{L^2}\\ \frac{6EI}{L^2}& \frac{7EI}{L^2}\end{array}\right]$

Answer 1

The correct option is B $\left(\begin{array}{cc}\frac{12EI}{L^3}& \frac{6EI}{L^2}\\ \frac{6EI}{L^2}& \frac{7EI}{L}\end{array}\right]$

For the given frame, let stiffness matrix be:
$K=\left(\begin{array}{cc}{K}_{11}& K_{12}\\ K_{21}& K_{22}\end{array}\right]$

$For{k}_{11}and{k}_{12}$

$R_C\times L=\frac{12EI}{L^2}$

$\therefore k_{11}=RC=\frac{12EI}{L^3}$

$k_{12}=k_{21}=\frac{6EI}{L^2}$

$For{k}_{22}$

$k_{22}=\frac{4EI}{L}+\frac{3EI}{L}=\frac{7EI}{L}$

$\therefore$ $K=\left(\begin{array}{cc}\frac{12EI}{L^3}& \frac{6EI}{L^2}\\ \frac{6EI}{L^2}& \frac{7EI}{L^{}}\end{array}\right]$