What Determines the Critical Load in a Pinned-Fixed Rod with a Torsional Spring?

[me_ia]

Homework Statement

A rod of leangth L and flexural rigidity EI is pinned at one end by means of torsional spring having a constant beta, and fixed on the other end.
The rod is subjected to a compressive axial force P.
Determine the critical load for instability.(image of the problem is attached)

Homework Equations

As far as I understand, the way to solve this problem is:
M(x)=-R*x-P*v(x)+beta*v'(0) , where R is the reaction on the left end at the y direction v(x) is the deflection function and v'(x) is the angle function.
M(x)=v''(x)*IE

The Attempt at a Solution

the two equations above yields:
v''(x)+(P/EI)*v(x)= (beta*v'(0)-R*x)/EI ->

v(x)=A*sin(sqrt(P/EI)*x)+B*cos(sqrt(P/EI)*x)+beta*v'(0)-R*x

to find those constants the boundry conditions are: v(o)=0 v(L)=0 v'(L)=0

How do I represent v'(0) which is unknown ?

 

[nilesh_pat]

A:The boundary conditions need some modifications in order to solve for the unknowns. $v(0)=0$$v(L)=0$$v'(0)=\alpha$$v'(L)=0$where $\alpha$ is a constant.By solving these equations, you will find the constants $A$, $B$, and $\alpha$.