Deriving euler's equation

Derivation of euler's equation

Start with the differential equation giving the deflected shape of an elastic member subjected to bending.

Set equal to zero.

Divide everything by $EI$ .

Set the variable, $\alpha ^2$

then, plug that in to get:

Since this is a second order, linear, ordinary differential equation with constant coefficients, it solves to:

Take the boundary condition that $x=0$ and $y=0$ to solve for $B$

Now, take the boundary conditions $x=L$ and $y=0$ .

Since $A$ cannot equal zero:

Take the sine inverse of both sides, and $\alpha L$ can be 0, $\pi$ , $2\pi$ , etc. So...

Solve for $\alpha ^2$

Set the two $\alpha ^2$ 's equal and solve for $P$ .

Assume that $n=1$

Now we can solve for $F_{cr}$ using this equation.

where: