A semi-rigid connection must have the required strength, stiffness and ductility. Some typical moment versus rotation (M-θ) curves and various analytical and mathematical models representing the M-θ relationships can be found in previous chapters. Conveniently, a joint can be considered in an analysis as dimensionless with location at the intersection of the element centre lines. Further, a rotational spring element satisfying the M-θ relationship is inserted into each end of beam element to model the connection behaviour. The joint equilibrium condition can be expressed as,
in which Me and Mi are the moments at the two ends of a connection (see Figure1). The corresponding external node is connected to the global node and the internal node is joined to the beam element.
(a) Equilibrium at a joint (b) External and internal rotations
Figure1 Modeling of Semi-Rigid Jointed Member
The stiffness of the connection, S, can be related to relative rotations at the two ends of the connection spring as,
where θe and θi are the conjugate rotations for the moments Meand Mi (see figure above). Rewriting Equation (2) in matrix form, the stiffness matrix of a connection spring can be written as,
A typical beam element bending stiffness matrix can be expressed as,
in which kij are the stiffness coefficients of a prismatic beam. Here, the imperfect PEP element proposed by Chan and Zhou (1995) is adopted and more details can be referred to the original reference. Therefore, a hybrid element can be obtained by directly adding the two ends connection stiffness to the PEP element bending stiffness matrix as,
In which the first subscript refers to node 1 or node 2. The internal degrees of freedom of the stiffness expression can be eliminated by a standard static condensation procedure. The stiffness expression of a beam element with both ends connected to a pair of springs can be finally written as,
and β is given by,
where S1 and S2 are the connection stiffness at nodes 1 and 2, respectively. When the spring stiffness Si is zero, it means that the corresponding end is pinned end; when the spring stiffness Si is infinite, it means that the corresponding end is rigid end. For semi-rigid case, the spring stiffness Si can be determined by the given M-θ function.