Elastic-Plastic Connections with Limited Capacity of Negative Moment
Consider a connection in which the second inequality in Eq.(2) is ignored, i. e. it is assumed that the behaviour of the connection is such that there is only a limit on negative moment but there is not any limit on positive moment, i. e. M+m = ^. In this case if a linear analysis is done and the value of moment exceeds the capacity of the connection it is necessary to add a positive moment to the connection such that after its distribution in the structure, it reduces the moment in the connection to the capacity limitMUm. Let us assume that this unknown positive moment is X+. To maintain the equilibrium and compatibility in the structure, it is necessary that this moment be accompanied by a moment at the other end and a pair of shear forces at the two ends of connected member.
Fig(2): Application of Self-Equilibrated Positive Moment X+ at two ends of a member and its related moment and shear forces. (a) Connection at i (b) Connection at j
Fig.(2) shows the application of self equilibrated moment and its accompanied moment and shear forces to a connection. From this point forward whenever it is talked about addition of a moment, it is meant a self equilibrated set of moments and shear forces.
Due to the applied moment X + in a connection or a joint named i, a moment MJt = mjtX+ will be produced in joint j. In this equation m jt is the moment produced in joint/connection j due to unit moment in i. Considering that there are n such artificial moment X + on the structure, the resultant moment Mri in a typical connection i can be obtained from the following equation:
Mrt= Mi + X+ + £ mtjX + (4)
In Eq.(4), M is the moment obtained from linear elastic analysis. It should be pointed out that since in applying unit moments at the connection i at member i-j two external moments Mi=1 and Mj=-.5 are applied, these moments should be added to mu and mjt values. This why the X + appears in Eq. (4). From this point forward, mij that comprises Mi=1 and Mj=-.5, is replaced for mtj to include the effect of external unit moments and corresponding shear forces. Therefore Eq.(4) will be written as
Mri= Mt + J mvX ++ (5)
Provided that X + are known, Eq. (5) can be used for calculation of moment in all joints/connections. Therefore It remains to find the values of X + ’s for all nonlinear connections. To that end substitute the Mri from Eq. (5) into Eq. (2) for all nonlinear connections to have:
Mri= Mt + J mvX) > M-m, t = 1,2,…,n (6)
Eq. (6) provides a set of n inequalities that can be used for determination of n unknowns. However there is a special condition that should be observed during solution of this set of inequalities. Considering that due to applied loads the value of Mri from Eq.(5) does not exceed the moment capacity limit MUm, then it would not be necessary to apply the unknown moment X +, i. e. in this case X + =0. Otherwise the artificial moment X + > 0. should be applied to the joint/ connection to makeMi=Mir These mutual conditions which are in fact the complementary conditions for the set of inequalities (6) can be written in the following form:
X,+ (M-ri – M-m) = 0. ;t = 1,2,…,n (7)
To solve the set of inequalities (6) with the conditions in Eq. (7), a Quadratic Programming (QP) problem as follows is established in which Eqs. (7) constitute its objective function and the set of inequalities in Eq. (6) comprise its constraints. It is noted that for arbitrary values of X + > 0. that satisfies Eq. (6) the value of X+(Mri—M-in)is always positive. Therefore minimization of the objective function reduces it to zero.
Mmmze ^ Г+ (Mri – MUm i)
Subject to Mri = Mt + ^mtjX + >M
The solution of QP problem in Eq. (8) while reduces the objective function to zero provides the values of Xf ’s. Then the Eq. (5) can be used to determine the moments in every joint of the structure regardless that there are or not any artificial moment.