An improved method of substructure analysis

Coapufers & Structures, Vol. 8, pp. 147-152.

Pergamon Press 1978.

Printed in Great Britain

AN IMPROVED METHOD OF SUBSTRUCTURE ANALYSIS C. S. GURUJEE? and V. L. DESHPANDES Indian Institute of Technology, Bombay 400 076, India (Received 20 August 1976) Abstract-Substructure method is an established way of overcoming the difficulty of large dimensionality in analysing structures. An improved substructure analysis method suitable for structures like multi-storied buildings and towers is presented in this paper. The method is based on peculiar geometry of these structures which could be used for numbering the boundary joints for a substructure either in the beginning or towards the end. Extra advantage could also be taken of the substructures identical in terms of geometry and loading both or in terms of geometry alone. Special static condensation and substitutions routines are developed. The methodis shown to be more efficient than any of the existing substructure analysis methods.

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NOTATION the number of the first internal displacement of substructure I (this is called m, in the text of the paper) the number of the last internal displacement of substructure 1 (this is called m, in the text of the paper) the half band width of a stiffness matrix of Ith substructure (called m in the text) the total number of internal displacements the stiffness matrix of a substructure the load vector of a substructure the total number of displacements plus one the total number of loading cases the number of the last member appearing in a substructure the modified half band width this denotes whether the element to be modified lies in the region to the left of the column which is being eliminated. INTRODUCTION

Major part of time in analysing any structure is spent in finding the solution of the system of simultaneous equations

where K is the structure stiffness matrix, S is the displacement matrix and F is the load matrix. As is known K is a sparsely populated, symmetric, semidefinite matrix. The efficiency of solution of this system of equations depends on how the trivial arithmatical operations are eliminated. Whenever a given problem can be accommodated in core memory, ordinary half band width solution with Gaussian Elimination as basis is one of the preferred methods. The more sophisticated and specialised techniques like substructure method, wavefront technique or special sparse matrix techniques are used only when ordinary half band solution is ruled out. A number of methods [ 141 have appeared recently for reducing the bandwidth of K and also for eliminating all trivial operations involved due to sparsity. Williams[5] has shown that there is a complete equivalence between the sparse matrix methods and the substructure method tlecturer, Civil Engineering Department. *Formerly graduate Student.

when the rows are not interchanged. Further it has been indicated in the same reference that, by selecting the substructures suitably, substructure method can be made more efficient than the sparse matrix techniques. He has further shown that the number of arithmatic operations in sparse matrix techniques can never be less than those in substructure method, however if one can identify identical substructures (loading conditions may or may not be identical) a large amount of saving in computational effort can be achieved by following a substructure method. The main advantage of sparse matrix techniques is all trivial arithmatic is got rid off. This advantage is obviously shared by substructure method. In this paper a substructure analysis method has been presented which is specifically developed for structures having large expense in one direction. Multistoried buildings, transmission or communication towers are some of the examples of this type. The technique used here makes use of the peculiar structure configuration and substructure node numbering is done in such a way that the boundary nodes are numbered either in the beginning or towards the end. All the interior nodes are accommodated in the middle. Special static condensation and substitution routines are developed for obtaining substructure stiffness matrix in terms of boundary displacements. The repetition of calculations in this phase can be avoided, if identical substructures can be identified. Such a repetition of substructures is obviously not present in multi-storied buildings or towers but is likely to be present in some continuum problems. The loading on identical substructures may or may not be identical. The proposed method is shown to be more efficient than any of the existing methods. AVAILABLE SUBSTRUCTURE METHODS

The pivotal idea in all substructure analysis methods is reducing the stiffness matrix of a substructure only in terms of boundary displacements and then using such reduced stiffness matrices to form the overall structure stiffness matrix. Broadly there are two methods available for reducing a substructure stiffness matrix. The first method is proposed originally by Livesley and Charlton[6]. This was modified by Rubinstein[7, 81 to avoid matrix inversion. This method will hence be termed as Rubinstein’s approach in this paper. In this method node numbering is done such that all the internal 147

C. S.

148

GURUJEE

and V. L.

DESHPANDE

nodes are numbered first and boundary nodes are numThe equations can be expanded to get, bered later. Because of such a numbering system the stiffness matrix equation can be written in a convenient K s,e,6s,fKs,,,S~+Ke,,srssz= FB, (2) form for reduction. The second method is called unit Km, a,, + Ku& t K,,szSm = Fr (3) displacement method which is used in some computer programs without giving it a status of a different method. K BZ.B1 6 B2 + Km,, S, + Km&h2 = Fm. (4) As against numbering all the internal displacements in the beginning and boundary displacements towards the From (3) end as required by Rubinstein’s[7] approach, the displacement numbering can be done in order to get mini(5) V,] = [Km-’ IFI -Km, 88,- K132bi. mum band width. When numbered this way the boundary displacements are distributed throughout the sub- Using this in (2) and (4) structure equilibrium equations. To derive the substructure stiffness matrix from these equations, all the boundary stiffnesses are given a very high value, say 103’ Ku-’ Km1 hn} = IFB11 - [KwK~,;‘l {SI (6) and on the right hand side fictitious load conditions, one corresponding to each boundary displacement are added. - Km&,’ In each of these fictitious load vectors one element [Kms, - KB,IK,;’ Km,1&,I+ t&2.82 corresponding to one boundary displacement is given the &s2I @,2} = I&I - t&2,1 K,,,-‘I VJ. (7) same high value 103’. The solution of this system of equations will give displacements at all the joints when Equations (6) and (7) written in matrix form give, all the boundaries are restrained and also when each of them is given unit displacement. Knowing these displacements substructure stiffness matrix is then formulated knowing the forces in the members meeting at the boundaries. In addition to these approaches Rubinstein[9] has suggested one more variation which is the coefficients I? and P in eqn (8) are the required cobased on matrix inversion and tri-diagonalisation. efficients for the substructure stiffness matrix and the substructure force vector in terms of the boundary disTHE PROPOSEDMETHOD placements. If there are two or more substructures with As mentioned earlier the method is developed spe- identical stiffness properties, the load vectors of all these cially for structures such as multi-storey frames and could be written one after the other in eqn (1) and eqn (8) towers. A typical example of such a structure is shown in will then be replaced by Fig. 1. Considering any one of the substructures it is readily recognised that it has two boundaries. The substructure node numbering therefore is started from one end of a boundary. After finishing all the nodes on this boundary internal nodes are numbered so as to get the least band width. With all the internal nodes numbered where substructures 1 to p are identical in stiffness second boundary nodes are numbered last. When num- properties. bering is done in this manner equilibrium equations for a In the proposed method inversion is completely avoided and the substructure stiffness matrix is obtained substructure can be written as by static condensation scheme described below. The procedure is described with help of a typical example. K S,,B, K,,,r K,,,B, %z, FB, (1) The description given here is for the stiffness matrix K I&1 &,I Kr.82 ] { 8i ]={Fi ) written in original form through while making use of the I KB&31 I&r KBz.Bz S,, Fsz method the stiffness matrix is written in a half band where K denotes stiffness matrices, S denotes the dis- form. The appropriate equivalents in a general case have placement vectors and F denotes the load vectors. The been mentioned in the brackets. Figure 2(a) shows the subscripts Bl and B2 stand for the first and second stiffness matrix of a typical substructure. The number of total displacements here is 10(n), the half band width is boundaries respectively and I stands for the internal 5(m). The first internal displacement is numbered 4th(m,) nodes. and the last internal displacement is numbered 7th(m,) and therefore the total number of internal displacements is 4(mz- m, t 1). The static condensation scheme is stari-----l B”“N~ARY ’ -/ SUB STRUCTURE ted from the first internal displacement, i.e. from the 4th row. The operations involved for this are as follows: BOUNDARY

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149

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A new quantity called modified half band width is = 2m - 2 whichever is defined as rn, =mi-m,-2orm, minimum subjected to an overall minimum of m. This is the actual band width needed for the condensation process. The rows that are affected by the elimination of the first internal displacement, i.e. 4th in the present case are from l(N1) to S(N2). Nl and N2 in a general case are defined as follows: N, = ml--m-k 1 subjected to the minimum of unity and N2 = m, t m - 1 subjected to the maximum of n. Further rows are not affected by this condensation. The column numbers which satisfy the equation j - i z m + m, - 1 also remain unaffected. The affected region for the elimination of the 4th variable has been shown hatched in Fig. 2(a). The 4th column is now reduced to zero as a result of the application of eqn (10) except for &, which is unity and the c~~cients of the 4th variable do not appear in any other equation except the 4th. The 4th row therefore can be transferred as the last row and can be used at substitution stage. The co~esponding operations are also performed on force vector. After affecting the horizontal and diagonal shifts shown in Figs. 2(b) and 2(c), next internal displacement now occupies the fourth row. The operations described above are performed on this internal displacement. The end product of these successive modifications is shown in Fig. 2(d). After obtaining boundary displacement matrix for each substructure, it is synthesised with the global boundary displacement matrix for the entire structure. Boundary

force vector for each substructure is also synthesised with global boundary force vector. At this stage usual half band solution procedure will yield bound~y displacements for all the substructures. Once the boundary displacements of the substructure are obtained internal displacements can be obtained by recalling the pivotal equations stored in the lower regions of the substructure stiffness matrix which are kept in an auxiIiary storage. OPERATION COUNTS

The number of arithmatic operations involved in the proposed method have been given in Table 1. Since in the proposed method static condensation and substitution parts are separate, operation counts have been given separately. The expressions given are for individual substructures and not for the entire solution process. EXAMPLES FORCOhQ’AREON

It is intended here to make comparisons of different substructure methods namely Rubinstein’s and the unit displacement method with the proposed method on the basis of number of arithmatic operations involved, and the memory space used. Amount of book-keeping, tape operations etc. will be about the same for all the methods. So for the comparison, c~culations involved in actual solution process, that is condensation and substitution phase, only have been compared. Two typical examples (Fig. 3) are considered for the comparison and the different methods are compared in Table 2.

150

C. S. GURUIEE

and V. L. DESHPANDE

Table 1. Operation counts Operation

Condensation

Reciprocals

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Table 2. Comparison of substructure methods Operations

Suggested method

Rubinstein’s

Unit displacement

Half band

Maximum band width Number of r.h.s. Reciprocals Multiplications Additions Memory space

Problem 1, Fig. 3(a) 9 10 4 4 40 40 2510 2703 6009 3289 156 168

6 12 40 6892 6218 216

6 4 40 2370 2025 240

Maximum band width Number of r.h.s. Reciprocals Multiplications Additions Memory space

Problem 2, Fig. 3(b) 20 21 4 4 174 174 42488 54698 112106 84102 1286 1348

12 22 174 108236 102542 1836

12 4 174 31360 29002 2784

Table 3. Comparison for identical substructures

Operation

Without identical substructures

With identical substructures With identical Without identical loading loading

Reciprocals Multiplications Additions

Problem 1, Fig. 3(a) 40 32 2510 1566 6009 4349

32 1246 3949

Reciprocals Multiplications Additions

Problem 2, Fig. 3(b) I74 102 42488 25889 112106 60425

102 19860 52505

Note. In all calculations above number of loading cases is assumed to be 4. For the proposed method it has also been shown that the number of calculations saved is quite large if identical substructures can be identified with or without identical loading conditions. Table 3 shows such a comparison for solution of the same examples. Both the problems chosen can be accomodated in core memory and hence as can be seen from Table 2, ordinary half band solution is the most efficient. It must be mentioned here that the two problems chosen are only to bring out the comparison. DISCUSSIONS As can be seen from Table 2 the band width requirement of the unit displacement method is the least while

that required for Rubinstein’s method, it is the largest. The band width requirement of the proposed method is always in between these two extremes. The unit displacement method though requiring the least band width for the stiffness matrix needs extra locations on the right hand side, so whatever it gains in saving in the space required for the stiffness matrix it loses in the storage required on the right hand side. If the modified half band width required in the proposed method is comparable with the half band width, with all the internal nodes are numbered first, as required in Rubinstein’s approach, Rubinstein’s approach would obviously be more efficient than the proposed method. But such a case is very unlikely to occur in practice. The suggested method

151

An improved method of substructure analysis

respect it is much more efficient than the sparse matrix techniques such as wave front. The special condensation subroutine developed for this method is given in the Appendix which can be used for any structural analysis. The other parts of a structural analysis program are standard and are not given here. Appropriate changes should be made in the backward substitution part while making use of this subroutine. CONCLUSIONS

An efficient substructure analysis method has been presented which can be useful for the structures such as multistorey frames and towers. The method could be applied in conjunction with the finite element method for continuum problems which have larger expanse in one direction. REFERENCES

(a) Fig. 3. Examples for comparison. (a). Example 1 (plane struss); (b). Example 2 (plane frame).

therefore requires least storage as compared to both the unit displacement method and Rubinstein’s approach for

the practical problems. As regards the operations the suggested method is better than any of the other methods as can be seen from Table 2, the only exception being Rubinstein’s approach when the band width requirements are comparable. As can be seen from Table 3 if identical substructures can be identified there is large amount of saving in the number of computations. This advantage can be made use of only through the substructure method and in this

SUBROUTiNE

MOBiNo

1. W. R. Spillers, Techniques for analysis of large structures. J. Swucf. Div. AXE. 94, 2521-2534(1%8). 2. R. J. Melosh and R. M. Bamford, Efficient solution of load deflection equations. J. Struct. Div. AXE. 95.661-676 (1969). 3. H. G. Jensen and G. A. Parks, Efficient solution of linear matrix equations. J. Strut. Dio. ASCE. %, 49-64 (1970). 4. B. M. Irons, A frontal solution program for finite element analysis. ht. J. Numer. Meth. Engng 2, 5-32 (1970). 5. F. W. Williams, Comparison between sparse stiffness matrix and substructure methods. Int. L Numer. Meth. Engng. 5, 383-394 (1973). 6. R. K. Livesley and T. M. Charlton, Analysis of rigid jointed plane frameworks. Engng. 177, 239-241 (1954). 7. M. F. Rubinstein, Matrix Computer Analysis of Structures. Prentice Hall, Englewood Cliff, New Jersey (1966). 8. R. Rosen and M. F. Rubinstein, Structural analysis by matrix decomposition. J. Struct. Div. ASCE. %, 663-671 (1970). 9. M. F. Rubinstein, Combined analysis by substructures and recursion. L Struct. Div. ASCE. 93, 231-235(1%7).

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