A general theory is presented for finding the deflected mode shape of a statically loaded coplanar filament as a function of the undeflected shape and the imposed loads. The solution is in terms of normal and tangential displacements from points on the original undeflected curve. The analysis results in five nonlinear differential equations. Three are equilibrium equations and two relate displacements to axial strain and to change in curvature. Analog computer solutions are presented for horizontal and circular cantilever fibers subjected to several different loadings. For a linear deflection analysis, the equations will reduce to those associated with the elastic stability of thin bars.