Structural Analysis

Count members, joints and reactions, plug them into D_s = 3m − r for frames (or use the truss form), and classify: negative means a mechanism, zero means determinate, positive means indeterminate to that degree.

Draw the influence line for the required reaction, shear or moment, then position moving loads over the maximum positive ordinates and sum load × ordinate to get the peak effect.

Write the slope-deflection equation for each member end in terms of the unknown joint rotations (and sway), apply joint moment equilibrium to form simultaneous equations, and solve for θ before back-substituting for end moments.

Compute distribution factors from member stiffnesses, lock all joints to get fixed-end moments, then release and balance each joint in turn, carrying over half the balanced moment to the far end until the unbalanced moments are negligible.

Form each element stiffness matrix in local coordinates, transform to global axes, assemble into the structure stiffness matrix by degrees of freedom, apply supports, then solve {u} = [K]⁻¹{F} for the unknown displacements.