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a2 cos a3 a2 sin a3 analysis applied approximate solutions assumed cantilever Castigliano's Castigliano's method changes in length Chapter complementary energy complementary-energy method components of deflection conditions of equilibrium conservation of energy conservative systems considered cos2 cosa2 cosai derived dFDv dTBD due to unit energy methods equation of virtual equations of equilibrium example flexibility coefficients follows force F Fu F2 Fv and FH geometrical Hooke's law law of conservation line of action linear systems load Fv load-deflection characteristics obtained pin-jointed position of equilibrium potential energy potential-energy method principle of virtual reciprocal theorem redundant member redundant system respectively self-straining shown in fig simultaneous equations sin2 sina2 sinat small displacement spring statically statically-determinate system stationary stiffness coefficients strain energy strain-energy method structure system of forces system shown tions variables vertical vertical deflection virtual displacements equal