Consider the elastic column shown below $($with length $L$ and flexura rigidity EI$),$ which

is fixed at its base and restrained against lateral translation at the top.

Column Configuration and Notation

a$)$ Using the general solution to the $4^{th}-$order ordinary differential equation for

flexural buckling of a column, apply the appropriate boundary conditions and

determine the characteristic equation of buckling. The coordinate $z$ is measured

from the bottom of the column along the length and the lateral displacement in the

$y$ direction is $v(z).$

The $4^{th}-$order ordinary differential equation for flexural buckling of a column is:

$v^{iv}(z)+{}^2{v}^{\text{'}\text{'}}(z)=0$

${}^2=\frac{P}{EI}$

The general solution to this $4^{th}-$order equation takes the form:

$v(z)=A+Bz+$Csin$(z)+$Dcos$(z)$

$*$Note that this is an eigenvalue problem, so you should not solve for the constants

$($A$,$ B$,$ C and D$)$ and substitute them into the general solution. Instead, use the

boundary conditions to form a coefficient matrix as shown below, take the

determinant of this matrix and set it equal to zero to obtain the characteristic

equation of buckling.

$\left[\begin{array}{c}4x4\\ \text{C}\text{o}\text{e}\text{f}\text{f}\text{i}\text{c}\text{i}\text{e}\text{n}\text{t}\\ \text{M}\text{a}\text{t}\text{r}\text{i}\text{x}\end{array}\right]\left[\begin{array}{c}A\\ B\\ C\\ D\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\right]$

b$)$ Based on the characteristic equation from a$),$ determine the critical elastic

buckling load. $P_{cr}$ for this column. PLEASE SOLVE PART A AND $B$