The formula for the critical load of a column was derived in 1757 by Leonhard Euler, the great Swiss mathematician. Euler’s analysis was based on the differential equation of the elastic curve:
\(\frac{d^2v}{dx\ }+\frac{p}{EI}v\ =0\)
Find the solution to this equation and apply the following conditions to obtain the values for the constants of integration:
\(v\left|x=0\right|\ =0\)
\(v\left|x=L\right|\ =0\)
\(v\left|x=L\right|\ =0\)
Finally explain how to get the following result :
\(P=n^{2\ }\ \frac{\pi^2EI}{L^2}\)
1° The first step is to derive
\(v=C_1\ \sin\left|x\ +C_2\right|x\)
\(v^!=\frac{dv}{dx}=C_1\left|\cos-C_2\right|\left|\sin\right|x\)
\(v^{!!}=d^{\frac{2v}{dx^2}}=C_1\left|^{2\ \sin}\right|x-C_2\left|^{2^{ }}\cos\right|x\)
The second step is to factor
\(-C\left|^{^2}\sin\right|x-C_2\left|^{^2}\cos\right|x\)
\(\left(\frac{p}{EI}\right)\left(C_1\sin\left|x+C_2\cos\right|x=0\right)\)
\(C_1\left|^{^2}\sin\right|x-C_2\left|^{^2}\cos+C_1\right|\left(\frac{P}{EI}\right)\sin\left|x+C_{2_{\ }}\right|\left(\frac{P}{EI}\right)\cos\left|x\right|=0\)
\(C_1\sin\left|x\left(\frac{P}{EI}-x^2\right)+C_2\right|x\left(\frac{P}{EI-}x^{2\ }\right)=0\)
\(\frac{P}{EI}\left|2\right|=\sqrt{\frac{P}{EI}}\)
\(v=C_1\sin\sqrt{\frac{P}{EI}}x+C_2\cos\sqrt{\frac{P}{EI}}x\)
\(v=o\left|x\right|=0\)
\(v=0\left|x\right|x=L\)
\(C_1\sin\sqrt{\frac{P}{EI}}\ \left(0\right)+C_2\cos\sqrt{\frac{P}{EI}}\ \left(0\right)=0\)
For V=0 X=L
\(v\left(x=L\right)=C_1\sin\sqrt{\frac{P}{EI}}L=0v\left(x=L\right)=C_1\sin\sqrt{\frac{P}{EI}\ L\ =0}\)
\(\sin\left(\sqrt{\frac{P}{EI}\ \ L}\right)=\ 0\ \ \ =\sqrt{\frac{P}{EI}}-L=n\ \pi\)
it will clear P
\(\frac{P}{EI}L^2=n^2\ \pi^2=P=n^2\ \frac{\pi^2EI}{L^2}\)
To calculate the critical P
\(n=1\ =\ P\ =\frac{\pi^2EI}{L^2}\)
Conclution:
En este presente documento se llevo a cabo una serie de operaciones,una de ellas fue derivar,otra factorizar y por ultimo despejar p.