Question

Derive the flexibility matrix for the simple beam of span L and flexural rigidity EI shown in figure subjected to end moments $M_{1}and{M}_2$

• A. $\frac{L}{EI}\left[\begin{array}{cc}\frac{1}{3}& \frac{-1}{6}\\ \frac{-1}{6}& \frac{1}{3}\end{array}\right]$
• B. $\frac{L}{EI}\left[\begin{array}{cc}\frac{1}{3}& \frac{1}{3}\\ \frac{1}{3}& \frac{1}{6}\end{array}\right]$
• C. $\frac{L}{EI}\left[\begin{array}{cc}\frac{1}{6}& \frac{-1}{3}\\ \frac{-1}{3}& \frac{1}{6}\end{array}\right]$
• D. $\frac{L}{EI}\left[\begin{array}{cc}\frac{1}{3}& \frac{1}{6}\\ \frac{1}{6}& \frac{1}{3}\end{array}\right]$

Answer 1

The correct option is A $\frac{L}{EI}\left[\begin{array}{cc}\frac{1}{3}& \frac{-1}{6}\\ \frac{-1}{6}& \frac{1}{3}\end{array}\right]$

Apply unit moment at 'A'


${\delta }_{11}=\frac{L}{EI}$

${\delta }_{21}=\frac{-L}{6EI}$



${\delta }_{21}=-\frac{L}{6EI}$

${\delta }_{22}=\frac{L}{3EI}$

$\delta =\left[\begin{array}{cc}\frac{1}{3EI}& \frac{-1}{6EI}\\ \frac{-1}{6EI}& \frac{1}{3EI}\end{array}\right]$