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20 kips angle applied loads arch axial force axial load axis bar forces beam element beam of Fig bending buckling load cable calculated components concentrated load cross-section deformations degrees of freedom diagram differential equation direct stiffness method displacement and rotation displacement vectors element stiffness matrices end moments equal to zero equations of statics equilibrium equations equivalent joint Example factor FIGURE Find the deflection fixed-end forces fixed-end moments force-displacement relations frames free body free body diagram global coordinates hinge indeterminate influence line internal forces joint displacements kip/ft left end left-hand end length load vector maximum member-end forces mid-span normal force problems redundant forces segment shear forces shear stress shown in Fig sidesway sign convention slope solution span statically determinate stiffness method stress structure stiffness matrix torque torsion truss unit load values vertical deflection virtual virtual displacement