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The problems discussed so far have involved concentrated forces and moments applied to nodes only. But real structures are subjected to distributed or concentrated loading between nodes, as shown in Figure 2.79. Loading may range from a few concentrated loads to an infinite variety of uniform or nonuniformly distributed loads. The solution method of matrix analysis must be modified to account for such load cases.

One way to treat such loads in the matrix analysis is to insert artificial nodes, such as p and q as shown in Figure 2.79. The degrees of freedom corresponding to the additional nodes are added to the total structure and the necessary additional equations are written by considering the requirements of equilibrium at these nodes. The internal member forces on each side of nodes p and q must equilibrate the external loads applied at these points. In the case of distributed loads, suitable nodes, such as I, m, n shown in Figure 2.79, are selected arbitrarily and the distributed loads are lumped as concentrated loads at these nodes. The degrees of freedom corresponding to the arbitrary and real nodes are treated as unknowns of the problem. There are different ways of obtaining equivalence between the lumped and the distributed loading. In all cases the lumped loads must be statically equivalent to the distributed loads they replace.

The method of introducing arbitrary nodes is not a very elegant procedure because the number of unknown degrees of freedom makes the solution procedure laborious. The approach that is of most general use with the displacement method is one employing the related concepts of artificial joint restraint, fixed-end forces, and equivalent nodal loads.

### 2.9.6 Semirigid End Connection

A rigid connection holds unchanged the original angles between interesting members; a simple connection allows the member end to rotate freely; a semirigid connection possesses a moment resistance intermediate between the simple and the rigid. A simplified linear relationship between the moment M acting on the connection and the resulting connection rotation C in the direction of M is assumed giving

where EI and L are the flexural rigidity and length of the member, respectively. The nondimensional quantity R, which is a measure of the degree of rigidity of the connection, is called the rigidity index. For a simple connection, R is zero and for a rigid connection, R is infinity. Considering the semirigidity of joints, the member flexibility matrix for flexure is derived as fi f 2 -

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