Problem $5[30$ points$]$: Figures $5($a$)$ and $5($b$)$ show the odd $($symmetric$)$ and even $($anti$-$symmetric$)$ buckling mode shapes for a clamped$-$clamped column. For example the dark thick curve in Fig. $5($a$)$ is the $5$th mode and the one in Fig. $5($b$)$ is the $6$th mode. Write the governing equations and obtain the transcendental equations for the symmetric modes

Note: in the class we found the transcendental equation for the buckling of a pinnedpinned column as: $sinkL=0$ for $k^2=\frac{P}{E}I$

$($a$)$

$($b$)$

Figure $5$