Frames

A frame is a structure composed of members connected by joints, where at least some members carry bending moments and shear forces in addition to axial forces. This distinguishes frames from trusses, whose members are assumed to be pin-connected and to carry only axial force. Frames are classified as rigid (moment-resisting) or braced. In a rigid frame, the joints are rigid, meaning the angle between connected members remains essentially unchanged under load, and bending moments are transferred through the joint. In a braced frame, the joints may be pinned, and lateral stability is provided by diagonal bracing or shear walls. Frames may be statically determinate or indeterminate. For a planar frame, the degree of static indeterminacy is D_s = 3m + r - 3j, where m is the number of members, r is the number of reaction components, and j is the number of joints. A portal frame with two pinned bases and a rigid beam-column connection is indeterminate to the first degree; with fixed bases it is indeterminate to the third degree. Frames are analyzed by the method of consistent deformations (force method), the slope-deflection method, the moment-distribution method (Hardy Cross), or matrix stiffness methods. The slope-deflection method relates member end moments to joint rotations and translations through the member stiffness. The moment-distribution method iteratively distributes unbalanced moments at joints according to member stiffness until equilibrium is reached. In matrix stiffness analysis, the global stiffness matrix is assembled from member stiffness matrices, boundary conditions are applied, and the resulting system K u = F is solved for joint displacements; member end forces are then recovered. Rigid frames resist lateral loads primarily through bending of columns and beams, developing double curvature in columns and contraflexure points near mid-height. The location of contraflexure points is often used as a simplifying assumption in approximate analysis of portal frames under lateral load. Pinned bases cannot resist moment, so the column base rotates freely; fixed bases develop a reaction moment and reduce lateral drift. The behavior of a frame under gravity and lateral loads depends on the relative stiffness of beams and columns, which is characterized by the beam-to-column stiffness ratio. Sway frames (unbraced) and non-sway frames (braced) are treated differently in design codes because sway frames must satisfy both strength and drift limits, and their member forces are affected by second-order (P-Δ) effects.