A beam structure is forced by an axial load $P$ (Figure P12.2). When $P$ is increased to its critical value, the beam will buckle. Prove that the critical force $P$ to cause buckling can be expressed by a function
Figure Can't Copy
$$
P=\frac{n^2 \pi^2 E I}{L}
$$
where
$E I=$ stiffness of the beam
$L=$ span length of the beam
$n=$ number of sine waves the beam shape takes when it buckles (assume it to be continuous).
If $0 \leq n \leq 2$, assume that $n$ is constrained by the fuzzy member function
$$
\mu_c(n)=\left\{\begin{array}{cc}
1-n, & 1 \leq n \leq 2 \\
0, & n<1
\end{array}\right.
$$