This can be rewritten as

 

E I v’’ + P v = 0

 

or by dividing out the E I, we get

 

v’’ + [ P / (E I)] v = 0

 

So the buckling problem is governed by a second order differential equation.  The solution to this equation can be written in the following form.

 

v = A cos( k x ) + B sin( k x )               where k² = P / (E I)

 

We then apply our boundary conditions to try to find the constants of integration.

 

v(x = 0) = 0 = A (1) + B (0) = A

 

v(x = L) = 0 = B sin( k L)

 

This second equation can be solved in two ways.  Either B = 0 or sin( k L ) = 0.  If B = 0, the entire solution is zero and there is no buckling.  This is called the trivial solution.  If sin( k L ) = 0, then ( k L ) = n p where n is an integer.  We want to find the first value of P which creates this situation so we can set n = 1.  Solving for P, we find

 

P_cr = ( p² E I ) / L²

 

This is called Euler’s buckling formula.  Note that this equation assumes that the material is linear and elastic.

 

If we have a beam with different boundary conditions, the critical load will be different.

 

Let’s consider the case of a cantilever.

 

 

 

 

SM = 0:          -M + P ( v(L) - v ) = 0 

                       

                        M = P v(L) - P v

 

 

 

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