Frames with semi-rigid connections are investigated. Relationships between forces and displacements at the ends of a member with elastically restrained ends are derived. The force components, in terms of the end displacements, are arranged in matrix form. Stiffness matrices of a member with elastic restraint at the ends are presented, in the form of the stiffness matrices for members with rigid connections, modified by a correction matrix whose elements are functions of parameters, designated as fixity factors of the member. A formula for the force components at the ends of all the members of a frame is established. The matrix operations involved are conveniently performed by any digital computer, and problems ranging from rigid frames to pin-connected trusses can be solved by the same systematic approach. Several numerical examples are solved.