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To analytically investigate this scenario, consider a column under eccentric loading. A section PR of the column is selected and its free-body diagram is drawn, which helps visualize the forces and moments acting on it. Choosing an appropriate coordinate system allows us to determine the couple moment at a given point (e.g., point R), which is integral to the subsequent mathematical modeling of the column's behavior. The next step incorporates this couple moment into the differential equation governing the elastic curve of the column. The solution to this differential equation gives the equation of the elastic curve, which describes how the column bends under the applied load. By applying boundary value conditions, the solution's coefficients can be determined, further refining the model's accuracy.

A critical aspect of this analysis is identifying the maximum deflection of the column, which typically occurs at its midpoint. This maximum deflection is pivotal for assessing the column's stability, as it indicates how much the column bends under the applied load. The equation for this deflection points towards an intriguing phenomenon: it suggests that the deflection approaches infinity as the secant term within the equation becomes infinite. This condition marks the threshold beyond which the column loses stability and undergoes buckling. The critical loading condition, derived from the infinite deflection criterion, is fundamental for engineers to ensure columns are designed within safe operational limits. By substituting the critical loading condition into the expression for maximum deflection, we can derive an equation expressing maximum deflection in terms of critical loading. This relationship is pivotal for designing columns that can withstand eccentric loads without excessive deformation or failure.

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