# Euler buckling formula,triangular structure

So I tried equating the sum of forces in the x,y and torque about $C$ to zero and found $B_x$ and $C_x$.

But how to find $B_y$ and $C_y$? I have 3 equations and 4 unknowns?

Creating an FBD in joint A:

We can derive that member AC is under compression. Note: you don't need to solve for the reactions since we can answer directly using method of joints

Euler's buckling formula states that:

Therefore, if you check your column there, the critical buckling load is equal to 37.285 kN (Compression). Note: use K = 1.0 since both ends are pin-connected

So to answer the first question: NO, it is not strong enough

the second question reverses the function mentioned above, given capacity of AC = 37.285 kN, we can get AB = 74.570 kN, therefore, the point load capacity at joint A is equal to 64.580 kN.

Bars AB and AC are pin-jointed and can only carry axial forces.

Therefore looking at joint C we can say that $R_{cy}=0$ since the axial force transferred from bar AC will only be in the x direction. This leaves only three reactions left in the two pin-joints which can be found using three equations of statics: $\sum F_x=0, \sum F_y=0, \sum M=0$.

These will find the three reactions which will lead to the axial forces in the bars which can then be used in the axial stress and euler-buckling formulas respectively. Bending stress doesn't need to be checked since no bending occurs in the pin-jointed bars due to the first statement.

On closer inspection, you can take sum of forces in X and Y at A which would give two equations to solve the two unknowns in the bars at that point $F_{ac} , F_{ab}.$ This is a quicker method to get the information you need but less information on the whole structure.

This problem has nothing to do with Euler buckling. It's a Mechanics of Materials problem. The extra relationship equation you are looking for will come from the internal stress/strain in the members of the structure. Note that all members have pinned connections which cannot carry moments. You are therefore dealing with a truss problem, where all members will only carry axial forces, stresses, and strains. The equations for stress and strain in axial members will give you the final expression you need.

• the critical load on AC may well be compression buckling, (why else do they give the area moment?). You are right though, you need to solve the truss problem first before considering that. I'm sure the OP has turned in his homework by now in any case. – agentp Aug 4 '17 at 19:22