SUMMARYThe estimation of the parameters ('fictitious densities') which control the convergence and numerical stability of a non-linear Dynamic Relaxation solution is described. The optimal values of these parameters vary during the iterative solution and they are predicted from the Gerschgorin bounds, that is rowsums of the stiffness matrix, which are divided into constant and variable parts for computational convenience. The procedure is illustrated by reference to the analysis of an axially loaded beam on a non-uniform elastic foundation.
SUMMARYA non-linear 9-node stress resultant shell finite element with six degrees of freedom per node is formulated.The material non-linearity is based on an implicit integration scheme using the von Mises yield criterion and linear isotropic hardening. The small strain geometric non-linearity is formulated using the polar decomposition theorem of continuum mechanics via a corotational updated Lagrangian method, which represents finite rotations with accuracy. Reduced integration is used to remove locking and calculate the stresses at their optimal stress accuracy points. A practical procedure is employed to stabilize the troublesome spurious zero energy modes. A number of tests covering the non-linear material and geometry ranges and buckling show the good performance of the new element.
The development of the flexibility method of analysis of skeletal structures has been hindered by the difficulty of determining a suitable statical basis on which to form the flexibility matrix. A combinatorial approach reduces the difficulty to one of selecting a minimal basis of the cycle vector space. After an introduction to flexibility analysis and a brief review of earlier work using combinatorics, the paper presents a procedure to construct a finite sub-set of the cycle vector space containing the elements of all minimal bases. This makes the generation of the required basis feasible by a finite procedure, such as Welsh’s generalization of the Kruskal algorithm. It is thus possible to have an automatic method for the analysis of skeletal structures which uses an optimal combinatorial approach.
SUMMARYTWO methods are presented for the automatic selection of a cycle basis leading to a sparse flexibility matrix for the analysis of rigid-jointed skeletal structures.The first method having a local approach forms a maximal set of admissible miimal cycles, while the second having a global approach constructs admissible minimal cycles on the ordered chords of a shortest route tree. A cycle ordering algorithm is also given to reduce the band width of the corresponding flexibility matrix. DEFINITIONSIn order to describe the methods in a self-contained manner, a number of definitions are given, which in the main follow those of Tutte.'A graph S consists of a set N(S) of elements called nodes (vertices) and a set M(S) of elements called members (edges), together with a-relation of incidence, which associates with each member two nodes called its ends. Two nodes of S are adjacent if these nodes are the ends of a member. A member is called incident with a node if this node is an end of the member. The ualency of a node of S is the number of members incident with that node. The intersection of two subgraphs Sk and S,+ , is similarly defined using the intersections of the node sets and member sets of the two subgraphs.A path P, in S is a finite sequence P, = (no, mi , . . . , m p , np) whose terms are alternately nodes n, and members mi of S if for 1 < i < p , n,-1 and ni are the two ends of mi. A simple path in S is a path in which no member and node of S appears more than once. If the end nodes no and np of a simple path correspond to the end nodes of a member mf then Pi U mf forms a simple cycle c, of length L(c,) = p + 1. The length of 4.. L(Pi) = p, is taken as the number of its members. Pi is called a shortest path between the two nodes no and np if for any other path, P,, between these nodes L(PJ < L(P,). For convenience, in this paper, simple cycles will be referred to as cycles.The member mf is called the generator of ci. A cycle generated an a member mf is called a minimal cycle if PI is a shortest path other than the generator.A tree, T, is a connected subgraph of S which contains all the nodes of S and has no cycle. The complement of the members of T i n S are the chords of this tree. The number of the chords in Tin a connected S is given by r, the cycle rank (first Betti number) of S. r = M -N + 1 = b,(S) where M and N are the numbers of members and nodes of S respectively. A set of tree cycles (fundamental cycles or circuits) of S for Tare r cycles formed by each chord and its unique simple path in T.
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