Euler's formula is used in structural engineering to determine the buckling load of columns under various conditions. However, when dealing with systems that incorporate both rigid elements and elastic components, such as springs, the analysis requires a finer approach to determine the critical load. The problem described involves two rigid bars connected at a pivot point with a spring attached and a vertical load applied at one end.

The system comprises two vertical rigid bars, AB and BC, of equal length, connected at pivot point B, similar to a knee joint. A tension spring of constant k is attached to B, acting horizontally. When a vertical load F is applied downwards at point A, the bars rotate around the pivot point B. The spring acts against this force, pulling point B back to its original location and resulting in an elongation X in the spring.

First, to analyze this system, a free-body diagram for each bar is used, and the principles of static equilibrium are applied to analyze the forces acting on the bars. The axial load F causes the system to pivot at point B, which results in both a horizontal and vertical force in the rods.

An equilibrium equation for the horizontal forces acting on the system is set up, which includes the spring force due to the elongation X. The spring force counteracts the sideways movement of point B. The system's pivoting at point B also induces a bending moment in both bars. The moment equilibrium at B for Bar AB allows the determination of the reaction force at point A. This involves calculating the moment generated by the applied load F and equating it to the moment due to the reaction force. Similarly, examining the moment of forces around point B for the bar BC yields the reaction force at point C. This step also includes the spring force and its moment arm relative to point B. We can calculate the total spring force by substituting the reactions at points A and C into the horizontal force equilibrium equation. This force counterbalances the moments generated by the reaction forces at points A and C.

The critical load for the system is determined by recognizing that the elongation X of the spring directly relates to the applied load F. The system's stability is compromised when the spring elongation (and thus the spring force) reaches a value that cannot be balanced by the structural integrity of the assembly, leading to buckling. The exact mathematical expression for the critical load can be derived from the equilibrium equations, incorporating the spring constant k, the length of the bars, and the geometry of the system at the onset of buckling.

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