The above is a past exam question from an introductory structural analysis course, one in which although we have studied the Euler buckling load equation, we have just been given parameters for the equation based on the standard end support conditions (fixed/fixed, fixed/pinned, etc.). I don't think the above example fits any of the standard end conditions and am therefore finding it difficult to derive an equation for $P_{cr}$ I have tried to use the bending moment equation by including torque (that opposes bending) from B and the net load (due to spring C) as follows: $$ T_B = \beta_r \theta $$ Taking a small deflection x in the vertical direction and y in the horizontal: $$ tan\theta = \frac{dy}{dx} \approx \theta $$ $$ \therefore T_B = \beta_r \frac{dy}{dx} $$ Using the bending moment equation: $$ M=-EI\frac{d^2y}{dx^2} $$ $$ \therefore \sum M=-EI\frac{d^2y}{dx^2} + \beta_r \frac{dy}{dx} = Qy $$ The net load $Q$ can be expressed in term of the force P and the reaction force due to the spring at C as: $$ Q = P - \beta dx $$ Leaving the final second order equation: $$ EI\frac{d^2y}{dx^2} - \beta_r \frac{dy}{dx} + (P-\beta dx) y = 0 $$
This is beyond the scope of our course, so I have no idea if:
a) The above equation is correct or solvable (and if so how one goes about solving it)
b) There is a different approach using more basic techniques (which is more than likely the case)
