# Formulation of the Solution Procedure

For simplicity of formulation of the problem, first the elastic-plastic model of nonlinear behaviour is considered and formulated. Then the formulation is extended to more complicated behaviours. In general the elastic-plastic behaviour of a semi-rigid connection is defined as in Fig.(1) in which moment in the connection is proportional to rotation up to certain limits. Beyond these limits, despite increase in rotation, moment remains constant. In the elastic-plastic semi-rigid connections the Moment-Rotation relation in each direction can be defined by two parameters Mlim and R0 . R0 is the stiffness of the connection and for rigid connections it tends to be infinity. It is zero for hinge connections and accordingly in this case Mlim is zero. To formulate the solution procedure, it is noted that the elastic-plastic behaviour of the connection can be mathematically written as :

Mm < M < M+m (1)

in which MUm and M+m are limits of negative and positive moments in the connection respectively. This is equal to the two following relations:

MI <M-

lim – (2)

M + < M+m

Noting that the Moment in the connection is either positive (M +) or negative (M-), the above relation can be written as follows:

Mm < M +-M -< M+m

To formulate the new method, we first focus on a condition that one of the inequalities in Eq.(2) is governing. Later on the formulation will be extended for the case that both are concerned.