By William R. Spillers

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5) by selecting the elements of F to be zero except F7 = 1. If some form other than Eq. 1) is used for the node method, it is simply necessary to use some form other than Eq. 3) for the energy. 3 THE DECOMPOSITION With numerical computation in mind, it is worthwhile to pursue the details of the system matrix, NKN, a little further. The most striking feature of the matrices N and K is that they are sparse; this sparseness discourages their direct formation in the computer. In this section a scheme is discussed which allows the formation of NKN from the elements of N and K without explicitly constructing the matrices them­ selves.

3. The reduction of the moment capacity of a beam due to the presence of axial load is neglected. 4. e. P = λΡ(), where P is the usual joint load matrix, P0 is a matrix which represents the fixed ratios of the loads, and λ > 0 is a scalar. 5. The members are uniform between nodes. For this problem (see P. Hodge, Plastic Analysis of Structures. McGrawHill, 1959), one possible formulation is through linear programming: Maximize λ Subject to: NF = XP0 |w/+l ^ μ, |m,~| ^ μ, (equilibrium) (the bending moment diagram must be safe) where μ, — capacity of member / In this formulation the collapse load is the largest for which it is possible to satisfy equilibrium and not exceed the capacity of any member.

This procedure is shown schematically in Fig. 2 for a plane frame with members loads. , can be handled in a similar fashion as indicated for the case of a truss subjected to tem­ perature change in Fig. 3. 51 Automated Structural Analysis Fig. 1. Member loads. i) + nn rfn (all joint disp. zero) rm Secondary problem rfn Primary problem Fig. 2. Member loads. -AT (all joint disp. zero) Secondary problem Fig. 3. Temperature effects in a truss. 4 indicates a very simple member load problem which, while not quite in the spirit of this book, (it includes a support which is not completely fixed), illustrates the decomposition nicely.

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