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An efficient flexibility analysis of structures
Compurm & Snucrure~ Vol. 22. No. 6. pp. 973-977. Pnnted in Great Bntain.
co45-794986 53.00 l .oo S 1986 Pcrgamon Press Ltd.
1986
AN EFFICIENT FLEXIBILITY ANALYSIS OF STRUCTURES A.
KAVEHf
Iran University of Science and Technology, Narmak, Tehran, Iran (Received
23 January 1985)
Abstract-An algorithm is developed for generating suboptimal cycle bases corresponding to localized statical bases, leading to highly sparse flexibility matrices. For this purpose, an expansion process is used, in each step of which an independent optimal cycle having the least possible overlap with the neighbouring cycles is generated. The elements of the statical basis are then formed on the selected
cycles. These cycles are also ordered to reduce the bandwidth of the corresponding flexibility matrix. This is achieved by ordering the nodes of the interchange graph, defined for the selected cycle basis.
Many improved methods have recently been deIn the theory of structural analysis there is a dis- veloped for generating cycle bases corresponding parity in the development of the flexibility method to sparse C, using different criterion for selection: and the stiffness approach despite the dual nature see Kavehll-41 and Cassell et a1.[5, 61. Although of the two methods. As a result, some of the ad- the generation of such bases has automated the flexvantages of flexibility analysis have been neglected. ibility method, however, it is not yet comparable It is interesting to explore the problems involved in with the simple formation of cocycle bases for stiffthe flexibility method for the advantages it offers ness analysis. On the other .hand. maximizing the in cases such as optimization or nonlinear analysis sparsity of C does not necessarily lead to D with and for the sake of the completeness of structural maximal sparsity. In this paper a new method is presented which theory. leads to the formation of highly sparse D, hence G. The stiffness matrix K of a skeletal structure is pattern equivalent to the cut-set-basis adjacency matrices for flexibility analysis of skeletal strucmatrix N = LL’ of its graph model S, where L is tures. A two-step approach is also presented for the cut-set-edge incidence matrix of the corre- ordering the cycles of the selected bases to reduce sponding basis. Similarly the flexibility matrix G is the bandwidth of the corresponding flexibility mapattern equivalent to the cycle basis adjacency ma- trices. This method is a combinatorial approach; trix D = CC, where C is the cycle-edge incidence however, there are also algebraic methods for generating sparse flexibility matrices, a complete rematrix of the selected basis. For an efficient solution of a structure by the view of which may be found in Kaneko er ~1.171. stiffness method or flexibility approach, the sparsity of K or G should be maximized. This can be FLEXIBILITY ANALYSIS achieved by maximizing the sparsity of N and D, respectively. The stress resultant distribution due to a given The simplicity of the stiffness method is due to loading on a skeletal structure S. which has been the natural existence of a special cut-set basis, obtained by means of a linear flexibility analysis, known as a cocycle basis, consisting of the cuts can be given as around the nodes of S, except the ground node. This basis corresponds to a highly sparse N matrix, alr = BoP + B,X. though the sparsity is not maximal for all the structures. However, no such simple and efficient cycle where BoP is any particular solution satisfying equibasis can be generated for the flexibility analysis of librium with the imposed loads, and B,X is a comthe structures. plementary solution formed from a maximum set of A simple but inefficient cycle basis, the so-called independent self-equilibrating stress distributions tree (fundamental) cycle basis, has been used in net- known as a statical basis. The number of indepenwork analysis for a long time. Although the for- dent self-equilibrating stress distributions is equal mation of such a basis is simple, in general, how- to the degree of statical indeterminacy of the strucever, it does not correspond to a sparse D matrix. ture, denoted by y(S). The particular solution may be obtained by considering cuts or releases to make the structure statically determinate. Gaps existing at the imaginary t Guest professor at lnstitut fur Allgemeine Mechanik, Technische Universitiit Wien, A-1040, Wien, Austria. cuts may be removed by applying a set of pairs of INTRODUCTION
973
datum node Fig. I. (a) A plane frame. (b) Graph model of the frame.
equal and opposite forces and moments X = {X,X2*.+X,,s,} at the cuts or releases. The compatibility eqns will then yield:
where F,, defines the flexibility of the individual disconnected members, and the matrix 8, consists of y(S) independent columns of self-equilibrating stress dist~butions, the ith column being associated with a unit value of Xi. The ith column of the matrix BO represents the stress resultant in the members of S due to a unit value of Pi. The stress resultant r is then obtained from r = [B. - B,(B~F,,B,)-’
B’,F,Eo]P,
where G = B:F,,Bx is the flexibility matrix of the structure. Allied to a statical basis, there is another set associated with the mathematical model S of a structure, This set consists of independent cycles of S. On each cycle three or six self-equilibrating stress dist~butions can be formed, depending on whether S is a planar or a space structure. respectively. This set forms a cycie basis of S. For an efficient flexibility method, G should be sparse. narrowly banded and well conditioned. The
sparseness and bandedness of G are handled as a totally combinatorial problem, while the conditioning of G is partially combinatorial. In this paper only the former properties are being studied. The graph-theoretical definitions used in this paper are similar to those of Ref. [
The mathematical model S of a structure is considered to be a fioite graph without loops or multiple edges (members), The structure is assumed to be rigid-jointed, and releases such as hinges and slides are omitted from the present argument. This, however, does not impose a significant restriction on the problem. since the effect of releases and various boundary conditions can be easily included for each special case. The graph model S is constructed by representing each element of the structure by a member and each joint by a node in a one-to-one correspondence, except for support joints. lvhich are in a many-to-one correspondence. For example. the graph model of a plane frame in Fig. Ital is shown in Fig. I(b). and the model of a space structure in Fig. 2(a) is given in Fig. 2(b).
datum node
Fig. 1. (a) A space frame. fb) Graph model of the frame.
Flexibility
analysisof structures
The dimension of a cycle basis b,(S) of S is related to that of a statical basis of the structure y(S) by the following: y(S) = 17[h,(S~l = N [N - N -t boWI, where M, N and hu(S) are the numbers of members, nodes and components of S, respectively. In this relation n = 6 or 3. according to whether the SWUCture is a space structure with a general loading or a planar structure with an inplane loading.
975
it can be shown that x(D) = hlfS) + 2
hr(St- I x ai( i= I
where uifC) of row i for C is the number of rowj such that: j E {i + I, i i- 2, . . . , b,(S)):
Ci 17Cj f 8’; i.e. there is at least one k such that both columns k of cycle Ci and Cj (row i and j) contain nonzero entries. AN ADWSSIBLE CYCLE The numerical value of b,(S) for a given graph S is THEOREM. Let S be the graph model of a structure+ constant, and in order to minimize x(D). the value andletS= UY~,&.where&(i= l,....q)are 0f x;it;!s,- ’ ai should be minimized: i.e. cycles subgraphs of S. Define Sk = Ux I S.andAk = Sk-’ of minimal overlaps should be chosen as the cycle n Sk. Then it can be proved[lO] th& basis of S. A cycle basis corresponding to the minimum [b,(S) - bo(S)l value of x(D) is called an optimal cycle basis. If a cycle basis corresponds to the minimum value of = i: ibr(Sii) - &(&)I - 5 fb,fAi) - bo(Ai)l, x(C) = x/Z(:) L(C;), it is called a rn~tz~nzai cycle i-? i=I basis of S. For planar graphs, the boundaries of the regions where b, and b. denote the first and zero Betti num- obtained by embedding S in a plane, except the exbers, respectively. terior cycle, form a cycle basis known as a mesh As already defined, S’ = U?=, Si. Let the basis of S. subgraphs Si fi = 1, . , . , k) be simple cycles Ci (i If a cycle basis corresponds to a near minimum I * . * . , k); then C”’ = Uf_ 1 Ci and Ak,l = C” value of x(D) or x(C), it is called a suboptimal or EC. I ,., . A cycle Ck_, is called an admissible cycle a subminimal cycle basis, respectively. if bl(C’+‘) = bl(Ck U C*+f) = b!(P) + 1. Considering the above theorem, it is easy to show that such an admissibility condition is satisfied in each CYCLE BASIS SELECTION ALGORITHM of the following cases: 1. An+ I = 8, where B is an empty intersection; Step. 1. Delete every node of degree 2 and the 2. b,(Ak+gf = r - s, where r and s are the num- corresponding members and replace them by a sinbers of components of Ck*’ and C’, respec- gle member, Repeat this pi-ocess also for those tively; nodes of degree 2 which contain the newly formed 3. &(A,,,) = 0 when C” and C”” are con- members. In this process, each path containing all nected (r = sl. the nodes of degree 2 will be effectively replaced In the above relations zI(Al) = ;i?i - Ri + 1, where by one member. Hence, S will be contracted to S’. mii and zi are the numbers of members and nodes Step 2. For each member of S’, find the number of Ai, respectively. of shortest cycles which can be formed on that member and call it the i~cide~ee ~ttrn~er of that member. If S’ is a space graph and the incidence CRITERION FOR CYCLE BASIS SELECTION number of a member is I, then find the number of Let C = {C,, Cz. . . , , Cbl(sf} be a cycle basis the next shortest cycles, considering the sum as the of S. The cycle basis incidence matrix C is a b,(S) incidence number of that member. Step 3. Assign the length of the shortest cycle x M matrix whose entries cij = 1 or 0 according to whether Cicontains or does not contain the mem- generated on a member as the cycle /en& number ber mj. The cycle adjacency matrix D = CC’ is a of that member. Again, for space graphs, the mean b!(S) X b,(S) matrix whose typical entry dii is non- value should be taken as the cycle length number zero when Ci and Ci have at least one common of a member, if its incidence number is I and next member. shortest cycles are being considered as well. Step 4. Start with a member of the least cycle The matrix B, is pattern equivalent to C’, and B:F,,,B, is pattern equivalent to CIC’ or CC’. This length number and generate the least weight cycte correspondence transforms certain structural prob- on this member. The sum of the incidence numbers lems associated with the characterization of of the members contained in a cycle is used as the B:FmBx into combinatorial problems dealing with measure of the cycle weight. D= CC’. If the spar&y coefficient x of a matrix Step 5. On the next unused member of the least is taken as the number of its nonzero entries, then cycle length number, generate an admissible min-
976
A. KAVEH
imal weight cycle. Continue this process as far as the length of the selected cycle is the same as the cycle length number of the generator. If no such a cycle can be found, the least weight admissjb~e cycle should be chosen on the member with the least cycle length number. After a member is used as many times as its incidence number. before each extra usage increase the incidence number of such a member by 1. Sfep 6. Repeat the process of selecting the least weight admissible cycles using the up-to-date incidence numbers, until b,(S) = bt(S’) admissible minimal weight and least weight cycles are being generated. Srep 7. A reverse process to that of Step I transforms the selected cycles of S’ to those of S. This algorithm leads to the formation of a suboptimal cycle basis, and for many practical models the selected bases have been optimal.
EXAMPLES
In this section, simple examples are studied. The cycle bases seiected by the present algorithm are compared to those generated by one of the author’s previous methods. The examples chosen are simple in order to give a clear ilIustration of the algorithmic process. ~bviousiy for multjmember strrmtures with higher nodal valences, the improvement obtained by using the present method becomes more evident.
support joints are considered as a single ground node [Fig. 3(b)]. A minimal basis containjng the following cycles is selected by using the authors previous algorithm~2~. ct: I, 2,3+
C-1: 4, 5, 8. 9.
c:: I, 2, 5,
Cs : 6, 7, 8, 9.
CJ: I, 3, 4,
Cs : 7, 8. il. 12.
C.$: I,S,4,
Clo: 6. 7, IO, 1f.
cs: 2. 3, 6, 7,
c,,: 9, 8, 12. 13.
cij: 3, 4, 7, 8,
ct2: IO, 11. 12, 13,
which corresponds to x(C) = 4 x 3 + 8 x 4 = 44 and x(D) = 12 +- 2 x 23 = 58. Using the algorithm of this paper leads to the formation of a similar basis with the difference C8: 6, 7, 3, 9 being replaced by Cl,: 6, 9, 10, 13, which leads to x(C) = 4 X 3 + 8 x 4 = 44, corresponding to x(D) = 12 + 2 x 20 = 52. ~.~ff~p~e2 S is the graph model of a space frame, for which blfS) = 7 (Fig. 4). Using the previous algorithm of the author&?] leads to the following minimal cyefe basis: c1: I, 2,3,
Cs: 1, 4, 3, 6,
Cz: 4, S, 6,
cg: 4, 5, 7, 8,
cj: 7, 8, 9,
62: 4, 7, 6, 9,
cj: 1, 2, 4, 5, In this example, S is the graph model of a space frame with bt(Sf = 12 [Fig. 3(a)]. Hence, 12 in- which has x(C) = 3 x 3 + 4 x 4 = 25, corredependent cycles are being selected as a basis. The sponding to x(r>> = 7 + 2 x 12 = 31. @,4
12
Fig. 3. (a) Sand the incidence numbers and cycle length numbers of its members. (br The graph model of s.
Flexibility
2
02 ,4
02
977
analysis of structures
,4
5
0’4
6
~314
8
Fig. 4. The space frame S.
The algorithm of the present paper. however, leads to the following optimal cycle basis: c,: 1. 2, 3,
Cg: 4, 7, 6. 9,
Cz: I, 2.4, 5,
Cg: 5, 6, 8, 9,
C>: 1, 4, 3, 6,
C,: 7, 8, 9.
c1: 5, 4,l. 8, for which x(C) = 5 x 4 + 2 x 3 = 26, corresponding to x(D) = 7 + 2 x 11 = 29. Notice that in this example only one three-sided cycle has been generated on members like f I, 3). Hence, higher length cycles are being considered (four-sided cycle), and a mean value of 3.5 is used. For a typical member (4. 6), this mean value is equal to 3.67. B..#.YDWIDTHREDUCTION OF THE FLEXIBfLITY
MATRIX
In order to obtain a banded G, the bandwidth of D is reduced. For this purpose the cycles of C are reordered by means of an interchange graph i(C) of the selected cycle basis C of S. This graph has its nodes in one-to-one correspondence with cycles of C, and two of its nodes are being connected if the co~esponding cycles have a common overlap (at least one member). The nodes of the interchange graph are renumbered by a “bandwidth reducing algorithm” (see, for example, Kaveh[ 1l- 131).This numbering of nodes leads to ordering the cycles of S corresponding to a small bandwidth of D, hence that of G.
,
Y
Y
Fig. 5. Graph model S and I(c) of the selected basis C.
Exumnple 3 S is the graph model of a planar structure (Fig. 5). For this graph b,(S) = 30, and the selected optimal cycle basis consists of the boundaries of the three-sided cells. This basis is also the mesh basis of the graph. The interchange graph I(C) of S is formed as shown, in broken lines, in Fig. 5. The bandwidth reducing algorithm is applied to Z(C), and its nodes are numbered as shown. Thus the order of the cycles is obtained, which corresponds to a banded D matrix, leading to a banded flexibility matrix G of the structure.
REFERENCES I. A. Kaveh, Application of topology and matroid theory to the analysis of structures. Ph. D. Thesis. Imperial College, London (1974). 2. A. Kaveh, improved cycle bases for the flexibility analysis of structures. ~~mpt~f. Mefh. Appl. Me&. Engng 9, 267-272(1976). 3. A. Kaveh, A combinatorial optimization problem: Op timal generalized cycle bases. Compur. Merh. Appl. Mech. Engng 20, 39-52 (19791. 4. A. Kaveh, An efficient program for generating cycle bases for the flexibility analysis of structures. Comms. Appf. Namer. Mefhs. 1 (1986). 5. A. C. Cassell, J. C. de C. Henderson and A. Kaveh, Cycle bases for the flexibility analysis of structures. Inf. J. Num. Meth. Engng 8, 521-528 (19741. 6. A. C. Cassell, J. C. de C. Henderson and K. Ramachandran. Cycle bases of minimal measure for the analysis of skeletal structures by the flexibility method. Proc, Roy. Sot. London A 350,61-70 (1976). 7. I. Kaneko, M. Lawo and 0. Thierauf, On computational procedures for the force method. Inr. 1. Nttm. Mefh. Engng 18, 1460-1495 (1982). 8. C. Berge, Graphs and Hypergruphs. North-Holland, Amsterdam (1975). 9. F. Harary. Graph Theory. Addison-Wesley, Reading, Mass. (1969). IO. A. Kaveh, Static and kinematic indeterminacy of skeieta1 structures. Iranian J. Sri. Tech. 7, 37-45 f 1978). II. A. Kaveh, A note on a two-step approach for finite element orderinn. Inf. J. Nrtm. Mefh. Enana ._ . 20. 15531554 (1984). 12. A. Kaveh and A. M. Behzadi, An associate graph for finite element orderina. Pruc. l5fh Irctnirtn Mu//t. Con&, Shiraz (19841. 13. A. Kaveh and K. Ramachandran. Graph theoretical approach for bandwidth and frontwidth reductions. Proc. 3rd hf. Conf, on Spnce Sfrttcfures (Edited by H. Nooshin), pp. 245-249 (1984).
co45-794986 53.00 l .oo S 1986 Pcrgamon Press Ltd.
1986
AN EFFICIENT FLEXIBILITY ANALYSIS OF STRUCTURES A.
KAVEHf
Iran University of Science and Technology, Narmak, Tehran, Iran (Received
23 January 1985)
Abstract-An algorithm is developed for generating suboptimal cycle bases corresponding to localized statical bases, leading to highly sparse flexibility matrices. For this purpose, an expansion process is used, in each step of which an independent optimal cycle having the least possible overlap with the neighbouring cycles is generated. The elements of the statical basis are then formed on the selected
cycles. These cycles are also ordered to reduce the bandwidth of the corresponding flexibility matrix. This is achieved by ordering the nodes of the interchange graph, defined for the selected cycle basis.
Many improved methods have recently been deIn the theory of structural analysis there is a dis- veloped for generating cycle bases corresponding parity in the development of the flexibility method to sparse C, using different criterion for selection: and the stiffness approach despite the dual nature see Kavehll-41 and Cassell et a1.[5, 61. Although of the two methods. As a result, some of the ad- the generation of such bases has automated the flexvantages of flexibility analysis have been neglected. ibility method, however, it is not yet comparable It is interesting to explore the problems involved in with the simple formation of cocycle bases for stiffthe flexibility method for the advantages it offers ness analysis. On the other .hand. maximizing the in cases such as optimization or nonlinear analysis sparsity of C does not necessarily lead to D with and for the sake of the completeness of structural maximal sparsity. In this paper a new method is presented which theory. leads to the formation of highly sparse D, hence G. The stiffness matrix K of a skeletal structure is pattern equivalent to the cut-set-basis adjacency matrices for flexibility analysis of skeletal strucmatrix N = LL’ of its graph model S, where L is tures. A two-step approach is also presented for the cut-set-edge incidence matrix of the corre- ordering the cycles of the selected bases to reduce sponding basis. Similarly the flexibility matrix G is the bandwidth of the corresponding flexibility mapattern equivalent to the cycle basis adjacency ma- trices. This method is a combinatorial approach; trix D = CC, where C is the cycle-edge incidence however, there are also algebraic methods for generating sparse flexibility matrices, a complete rematrix of the selected basis. For an efficient solution of a structure by the view of which may be found in Kaneko er ~1.171. stiffness method or flexibility approach, the sparsity of K or G should be maximized. This can be FLEXIBILITY ANALYSIS achieved by maximizing the sparsity of N and D, respectively. The stress resultant distribution due to a given The simplicity of the stiffness method is due to loading on a skeletal structure S. which has been the natural existence of a special cut-set basis, obtained by means of a linear flexibility analysis, known as a cocycle basis, consisting of the cuts can be given as around the nodes of S, except the ground node. This basis corresponds to a highly sparse N matrix, alr = BoP + B,X. though the sparsity is not maximal for all the structures. However, no such simple and efficient cycle where BoP is any particular solution satisfying equibasis can be generated for the flexibility analysis of librium with the imposed loads, and B,X is a comthe structures. plementary solution formed from a maximum set of A simple but inefficient cycle basis, the so-called independent self-equilibrating stress distributions tree (fundamental) cycle basis, has been used in net- known as a statical basis. The number of indepenwork analysis for a long time. Although the for- dent self-equilibrating stress distributions is equal mation of such a basis is simple, in general, how- to the degree of statical indeterminacy of the strucever, it does not correspond to a sparse D matrix. ture, denoted by y(S). The particular solution may be obtained by considering cuts or releases to make the structure statically determinate. Gaps existing at the imaginary t Guest professor at lnstitut fur Allgemeine Mechanik, Technische Universitiit Wien, A-1040, Wien, Austria. cuts may be removed by applying a set of pairs of INTRODUCTION
973
datum node Fig. I. (a) A plane frame. (b) Graph model of the frame.
equal and opposite forces and moments X = {X,X2*.+X,,s,} at the cuts or releases. The compatibility eqns will then yield:
where F,, defines the flexibility of the individual disconnected members, and the matrix 8, consists of y(S) independent columns of self-equilibrating stress dist~butions, the ith column being associated with a unit value of Xi. The ith column of the matrix BO represents the stress resultant in the members of S due to a unit value of Pi. The stress resultant r is then obtained from r = [B. - B,(B~F,,B,)-’
B’,F,Eo]P,
where G = B:F,,Bx is the flexibility matrix of the structure. Allied to a statical basis, there is another set associated with the mathematical model S of a structure, This set consists of independent cycles of S. On each cycle three or six self-equilibrating stress dist~butions can be formed, depending on whether S is a planar or a space structure. respectively. This set forms a cycie basis of S. For an efficient flexibility method, G should be sparse. narrowly banded and well conditioned. The
sparseness and bandedness of G are handled as a totally combinatorial problem, while the conditioning of G is partially combinatorial. In this paper only the former properties are being studied. The graph-theoretical definitions used in this paper are similar to those of Ref. [
The mathematical model S of a structure is considered to be a fioite graph without loops or multiple edges (members), The structure is assumed to be rigid-jointed, and releases such as hinges and slides are omitted from the present argument. This, however, does not impose a significant restriction on the problem. since the effect of releases and various boundary conditions can be easily included for each special case. The graph model S is constructed by representing each element of the structure by a member and each joint by a node in a one-to-one correspondence, except for support joints. lvhich are in a many-to-one correspondence. For example. the graph model of a plane frame in Fig. Ital is shown in Fig. I(b). and the model of a space structure in Fig. 2(a) is given in Fig. 2(b).
datum node
Fig. 1. (a) A space frame. fb) Graph model of the frame.
Flexibility
analysisof structures
The dimension of a cycle basis b,(S) of S is related to that of a statical basis of the structure y(S) by the following: y(S) = 17[h,(S~l = N [N - N -t boWI, where M, N and hu(S) are the numbers of members, nodes and components of S, respectively. In this relation n = 6 or 3. according to whether the SWUCture is a space structure with a general loading or a planar structure with an inplane loading.
975
it can be shown that x(D) = hlfS) + 2
hr(St- I x ai( i= I
where uifC) of row i for C is the number of rowj such that: j E {i + I, i i- 2, . . . , b,(S)):
Ci 17Cj f 8’; i.e. there is at least one k such that both columns k of cycle Ci and Cj (row i and j) contain nonzero entries. AN ADWSSIBLE CYCLE The numerical value of b,(S) for a given graph S is THEOREM. Let S be the graph model of a structure+ constant, and in order to minimize x(D). the value andletS= UY~,&.where&(i= l,....q)are 0f x;it;!s,- ’ ai should be minimized: i.e. cycles subgraphs of S. Define Sk = Ux I S.andAk = Sk-’ of minimal overlaps should be chosen as the cycle n Sk. Then it can be proved[lO] th& basis of S. A cycle basis corresponding to the minimum [b,(S) - bo(S)l value of x(D) is called an optimal cycle basis. If a cycle basis corresponds to the minimum value of = i: ibr(Sii) - &(&)I - 5 fb,fAi) - bo(Ai)l, x(C) = x/Z(:) L(C;), it is called a rn~tz~nzai cycle i-? i=I basis of S. For planar graphs, the boundaries of the regions where b, and b. denote the first and zero Betti num- obtained by embedding S in a plane, except the exbers, respectively. terior cycle, form a cycle basis known as a mesh As already defined, S’ = U?=, Si. Let the basis of S. subgraphs Si fi = 1, . , . , k) be simple cycles Ci (i If a cycle basis corresponds to a near minimum I * . * . , k); then C”’ = Uf_ 1 Ci and Ak,l = C” value of x(D) or x(C), it is called a suboptimal or EC. I ,., . A cycle Ck_, is called an admissible cycle a subminimal cycle basis, respectively. if bl(C’+‘) = bl(Ck U C*+f) = b!(P) + 1. Considering the above theorem, it is easy to show that such an admissibility condition is satisfied in each CYCLE BASIS SELECTION ALGORITHM of the following cases: 1. An+ I = 8, where B is an empty intersection; Step. 1. Delete every node of degree 2 and the 2. b,(Ak+gf = r - s, where r and s are the num- corresponding members and replace them by a sinbers of components of Ck*’ and C’, respec- gle member, Repeat this pi-ocess also for those tively; nodes of degree 2 which contain the newly formed 3. &(A,,,) = 0 when C” and C”” are con- members. In this process, each path containing all nected (r = sl. the nodes of degree 2 will be effectively replaced In the above relations zI(Al) = ;i?i - Ri + 1, where by one member. Hence, S will be contracted to S’. mii and zi are the numbers of members and nodes Step 2. For each member of S’, find the number of Ai, respectively. of shortest cycles which can be formed on that member and call it the i~cide~ee ~ttrn~er of that member. If S’ is a space graph and the incidence CRITERION FOR CYCLE BASIS SELECTION number of a member is I, then find the number of Let C = {C,, Cz. . . , , Cbl(sf} be a cycle basis the next shortest cycles, considering the sum as the of S. The cycle basis incidence matrix C is a b,(S) incidence number of that member. Step 3. Assign the length of the shortest cycle x M matrix whose entries cij = 1 or 0 according to whether Cicontains or does not contain the mem- generated on a member as the cycle /en& number ber mj. The cycle adjacency matrix D = CC’ is a of that member. Again, for space graphs, the mean b!(S) X b,(S) matrix whose typical entry dii is non- value should be taken as the cycle length number zero when Ci and Ci have at least one common of a member, if its incidence number is I and next member. shortest cycles are being considered as well. Step 4. Start with a member of the least cycle The matrix B, is pattern equivalent to C’, and B:F,,,B, is pattern equivalent to CIC’ or CC’. This length number and generate the least weight cycte correspondence transforms certain structural prob- on this member. The sum of the incidence numbers lems associated with the characterization of of the members contained in a cycle is used as the B:FmBx into combinatorial problems dealing with measure of the cycle weight. D= CC’. If the spar&y coefficient x of a matrix Step 5. On the next unused member of the least is taken as the number of its nonzero entries, then cycle length number, generate an admissible min-
976
A. KAVEH
imal weight cycle. Continue this process as far as the length of the selected cycle is the same as the cycle length number of the generator. If no such a cycle can be found, the least weight admissjb~e cycle should be chosen on the member with the least cycle length number. After a member is used as many times as its incidence number. before each extra usage increase the incidence number of such a member by 1. Sfep 6. Repeat the process of selecting the least weight admissible cycles using the up-to-date incidence numbers, until b,(S) = bt(S’) admissible minimal weight and least weight cycles are being generated. Srep 7. A reverse process to that of Step I transforms the selected cycles of S’ to those of S. This algorithm leads to the formation of a suboptimal cycle basis, and for many practical models the selected bases have been optimal.
EXAMPLES
In this section, simple examples are studied. The cycle bases seiected by the present algorithm are compared to those generated by one of the author’s previous methods. The examples chosen are simple in order to give a clear ilIustration of the algorithmic process. ~bviousiy for multjmember strrmtures with higher nodal valences, the improvement obtained by using the present method becomes more evident.
support joints are considered as a single ground node [Fig. 3(b)]. A minimal basis containjng the following cycles is selected by using the authors previous algorithm~2~. ct: I, 2,3+
C-1: 4, 5, 8. 9.
c:: I, 2, 5,
Cs : 6, 7, 8, 9.
CJ: I, 3, 4,
Cs : 7, 8. il. 12.
C.$: I,S,4,
Clo: 6. 7, IO, 1f.
cs: 2. 3, 6, 7,
c,,: 9, 8, 12. 13.
cij: 3, 4, 7, 8,
ct2: IO, 11. 12, 13,
which corresponds to x(C) = 4 x 3 + 8 x 4 = 44 and x(D) = 12 +- 2 x 23 = 58. Using the algorithm of this paper leads to the formation of a similar basis with the difference C8: 6, 7, 3, 9 being replaced by Cl,: 6, 9, 10, 13, which leads to x(C) = 4 X 3 + 8 x 4 = 44, corresponding to x(D) = 12 + 2 x 20 = 52. ~.~ff~p~e2 S is the graph model of a space frame, for which blfS) = 7 (Fig. 4). Using the previous algorithm of the author&?] leads to the following minimal cyefe basis: c1: I, 2,3,
Cs: 1, 4, 3, 6,
Cz: 4, S, 6,
cg: 4, 5, 7, 8,
cj: 7, 8, 9,
62: 4, 7, 6, 9,
cj: 1, 2, 4, 5, In this example, S is the graph model of a space frame with bt(Sf = 12 [Fig. 3(a)]. Hence, 12 in- which has x(C) = 3 x 3 + 4 x 4 = 25, corredependent cycles are being selected as a basis. The sponding to x(r>> = 7 + 2 x 12 = 31. @,4
12
Fig. 3. (a) Sand the incidence numbers and cycle length numbers of its members. (br The graph model of s.
Flexibility
2
02 ,4
02
977
analysis of structures
,4
5
0’4
6
~314
8
Fig. 4. The space frame S.
The algorithm of the present paper. however, leads to the following optimal cycle basis: c,: 1. 2, 3,
Cg: 4, 7, 6. 9,
Cz: I, 2.4, 5,
Cg: 5, 6, 8, 9,
C>: 1, 4, 3, 6,
C,: 7, 8, 9.
c1: 5, 4,l. 8, for which x(C) = 5 x 4 + 2 x 3 = 26, corresponding to x(D) = 7 + 2 x 11 = 29. Notice that in this example only one three-sided cycle has been generated on members like f I, 3). Hence, higher length cycles are being considered (four-sided cycle), and a mean value of 3.5 is used. For a typical member (4. 6), this mean value is equal to 3.67. B..#.YDWIDTHREDUCTION OF THE FLEXIBfLITY
MATRIX
In order to obtain a banded G, the bandwidth of D is reduced. For this purpose the cycles of C are reordered by means of an interchange graph i(C) of the selected cycle basis C of S. This graph has its nodes in one-to-one correspondence with cycles of C, and two of its nodes are being connected if the co~esponding cycles have a common overlap (at least one member). The nodes of the interchange graph are renumbered by a “bandwidth reducing algorithm” (see, for example, Kaveh[ 1l- 131).This numbering of nodes leads to ordering the cycles of S corresponding to a small bandwidth of D, hence that of G.
,
Y
Y
Fig. 5. Graph model S and I(c) of the selected basis C.
Exumnple 3 S is the graph model of a planar structure (Fig. 5). For this graph b,(S) = 30, and the selected optimal cycle basis consists of the boundaries of the three-sided cells. This basis is also the mesh basis of the graph. The interchange graph I(C) of S is formed as shown, in broken lines, in Fig. 5. The bandwidth reducing algorithm is applied to Z(C), and its nodes are numbered as shown. Thus the order of the cycles is obtained, which corresponds to a banded D matrix, leading to a banded flexibility matrix G of the structure.
REFERENCES I. A. Kaveh, Application of topology and matroid theory to the analysis of structures. Ph. D. Thesis. Imperial College, London (1974). 2. A. Kaveh, improved cycle bases for the flexibility analysis of structures. ~~mpt~f. Mefh. Appl. Me&. Engng 9, 267-272(1976). 3. A. Kaveh, A combinatorial optimization problem: Op timal generalized cycle bases. Compur. Merh. Appl. Mech. Engng 20, 39-52 (19791. 4. A. Kaveh, An efficient program for generating cycle bases for the flexibility analysis of structures. Comms. Appf. Namer. Mefhs. 1 (1986). 5. A. C. Cassell, J. C. de C. Henderson and A. Kaveh, Cycle bases for the flexibility analysis of structures. Inf. J. Num. Meth. Engng 8, 521-528 (19741. 6. A. C. Cassell, J. C. de C. Henderson and K. Ramachandran. Cycle bases of minimal measure for the analysis of skeletal structures by the flexibility method. Proc, Roy. Sot. London A 350,61-70 (1976). 7. I. Kaneko, M. Lawo and 0. Thierauf, On computational procedures for the force method. Inr. 1. Nttm. Mefh. Engng 18, 1460-1495 (1982). 8. C. Berge, Graphs and Hypergruphs. North-Holland, Amsterdam (1975). 9. F. Harary. Graph Theory. Addison-Wesley, Reading, Mass. (1969). IO. A. Kaveh, Static and kinematic indeterminacy of skeieta1 structures. Iranian J. Sri. Tech. 7, 37-45 f 1978). II. A. Kaveh, A note on a two-step approach for finite element orderinn. Inf. J. Nrtm. Mefh. Enana ._ . 20. 15531554 (1984). 12. A. Kaveh and A. M. Behzadi, An associate graph for finite element orderina. Pruc. l5fh Irctnirtn Mu//t. Con&, Shiraz (19841. 13. A. Kaveh and K. Ramachandran. Graph theoretical approach for bandwidth and frontwidth reductions. Proc. 3rd hf. Conf, on Spnce Sfrttcfures (Edited by H. Nooshin), pp. 245-249 (1984).