KAVEII, A. : Topology. and Skeletal Structures 347

ZAMM . Z. angew. Math. Mech. 68 (1988) 8, 347 -353

KAVEH, A.

Topology and Skeletal Structures

Es werden Methoden zum titudium der topologischen Eigenschaften verschiedener Typen von Ceruststrukturen angegeben. Eine Anwendung dieser Methoden unter Einbeziehung der Eulerschen Formel ergibt einige neue Ansatze zur Bestimmung des Grades der statischen Unbestimmtheit von Strukturen. FGr Raum-Strukturen vereinfacht sich der Z6iihlprozep bei den vorgestellten Verfahren betrachtlich.

Hethods are presented for the study of the topological properties of different types of skeletal structures. A n application of these methods together with Euler's jormula, results in some new approaches for determining the degree of statical indetermi- nacy of structures. The counting process in the presented methods is considerably simplified lor space structures.

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1. Introduction

Analysis of a skeletal structure involves the study of three different properties; topological, geometrical and material. Separate study of these properties results in a considerable simplification in the analysis and leads to a clear understanding of the structural behaviour. I n this paper only the topological properties are studied.

For the stiffness analysis of a structure S, a kinematical basis leading to a set of independent equilibrium equa- tions should be formed. The dimension of such a basis, q(S), is the same as the degree of kinematical indeterminacy (total degrees of freedom) of the structure. For the flexibility analysis, a statical basis corresponding to it set of independent compatibility equations should be constructed. The dimension of such a basis, y ( S ) , is the same as the degree of statical indeterminacy of the structure.

Adopting a combinatorial approach, a kinematical (statical) basis can be formed on the substructures corre- sponding to the elements of a maximal set of independent cut-sets (generalized cycles, cf. KAVEH [l]). This set is called a cut-set (generalized cycle) basis of the topological model S of the structure.

The stiffness matrix K of a skeletal structure is pattern equivalent to the cut-set basis adjacency matrix N = LLT of its graph model S, where L is the cut-set-edge incidence matrix of the corresponding basis. Similarly theflexibility matrix G is pattern equivalent to the generalized cycle basis adjacency matrix D = CCT ,where C is the generalized cycle-edge incidence matrix of the selected basis.

For an efficient analysis of a structure by stiffness method or flexibility approach, the sparsity of K or G should he maximized. This can be achieved to a great extent by maximizing the sparsity of Nand D, respectively.

The simplicity of the stiffness method is due to the natural existence of a special cut-set basis, known as a cocycle basis, consisting of the cuts around the nodes of S, except the ground node. This basis corresponds to a highly sparse N matrix, although the sparsity is not maximal for all the structures. However, no such a simple generalized cycle basis can begenerated for the flexibility analysis of the structures. Thus in this article only the concepts rele- vant to the flexibility method will be studied.

The concept of statical indeterminacy is central to a clear understanding of the mechanics of a skeletal structure, when analyzed by means of the flexibility method. It is not surprising that most of the methods presented for an efficient flexibility analysis are based on a logical approach for obtaining formulae used for calculating the degrees of statical indeterminacies (cf. HENDERSON [2 ] , CASSELL et al. [3] and KAVEH [4, 5, 6, 71).

I n this paper topological properties for different types of skeletal structures, such as rigid-jointed frames, pin- jointed and ball-jointed trusses are studied. A general approach is presented for determining the degree of statzml indvterrninacy (DSI) of structures and it's application is extended to the study of models with some identifications. Bew methods are developed for calculating the DSI of space frames and ball-jointed trusses by drawing their models on a plane. These methods simplify the counting process and give a new insight to the problem of selecting a suitable statical basis for these types of structures. Simple examples are included to illustrate the proposed methods. The graph theoretical definitions used in this paper may be found in [l, 91.

9 . A study of the topological properties of structures

The mathematical model of a skeletal structure X is considered to be a simple graph, i.e. loops and multiple members (edges) are excluded. I n this section an efficient method is described for the study of the properties of S, using the propertiea of its substructures (subgraphs). A particular property, namely the DSI is used to illustrate the proposed approach. However, the application of the method is by no means limited to this particular problem.

Definit ion 1: Consider a linear relationship in the following form:

where M , N and y,(S) are the numbers of members, nodes and components of S, respectively. The coefficients a, b and c are integers depending on both the property which y(S) is expected to represent and the type of the correspond- ing structure. 24 *

__- 348 ZAMM - Z. angew. Math. Mech. 88 (1988) 8

Tor example y(S) = Bil - N + yo(S) represents the first Betti number (cyclomatic number) of S, i.e. b,(S) = = M - N + b,(S). The value of the coefficients representing the DSI for different types of skeletal structures are given in Table 1.

Table 1 Structural type U b C

Plane rigid-jointed frame Space rigid-jointed frame Plane pin-jointed truss Space ball-jointed truss

+3 -3 1 3 i-6 -6 +6 3.1 -2 +3 i-1 -3 i-6

Throughout this paper, the structures are assumed to have no critical forms. Def in i t i on 2: Consider S as the union of q substructures. S can be reformed by joining its substructures

S,, S,, ... , S, in q steps as follows: k

i = l S, = 81 /" S2 f ... /" Sq /* S , where St = u Xi is the union of S,, S,, ... , Sk.

Def in i t i on 3: Let S be the union of q substructuress,, X,, ... , X,, with the followingfunctions being defined:

y(S) = ulM + bN + cyO(S) , Y(A,) = + b N i + cyo(At) for i = 2,3 , ... , 4 ,

Iris) - cyo(S)I = z Ir(Sd - C Y o ( m 1 -,z [ y ( 4 - cYo(AJ1 *

y(&) = a N t + 6Ni 4 cy,,(Xt) for .i = 1, 2, ... , q ,

where As = 8;-1 n Sf, and M, and Rt are the numbers of members and nodes of At, respectively. Then q L?

i=l 1 = 2 (1)

A simple proof of (1) is given in [8, 91. Special Case: When a structure S and each of its substructures Si (for i = 1, 2, ... , q ) are non-disjoint

(connected), then equation (1) can be simplified as

2.1. Union-intersection method

Let S be the union of its repeated andlor simple pattern substructure SiJ i = 1, ... , q. Calculate the DSI of

Step 1: Join Sl to S, to form S2 = S, u S,, and calculate the DSI of their intersection A, = S, n S,, The

Step 2 : Join S3 to S2 to obtain S3 = X2 u S3, and determine the DSI of 243 = S2 n S,, Similar to Step 1, cal-

Step 3 : Subsequently join SE+I to Sk, calculating the DSI of A k t ~ = Sk n S k + 1 and evaluating the value of

each substructure using the appropriate coefficients from Table 1.

value of y(S2) can be found from (1) or (2), as appropriate.

culate y(S3).

y(Sk++1). Repeat this process until the entire structural model S = U Sr is reformed and y(S) is determined. Q

i= l 27

i=l Example 1 : Let S be the model of a space structure. This model may be considered as X = U S,. A typical Si

is shown in Fig. l a and the interfaces of Xi for i = 1, ... , q are shown in Fig. 1 b, c, in which, for the sake of clarity Borne of their members are omitted.

For this structure y(S) can be calculated as follows: 27 27 27 7 2 1 27

i = l i = 2 i = 2 i = 2 i-8 i = 2 2 y(S) = Z: y(&) - 2 ?(At) 3 where Z ?(Ad = Z ?(A',) + Z ?(A?) + Z ?(A!).

Fig. l a . A typical substructure Si - Kd u Li

KAVEH, A.: Topology and Skcletal Structures BAR

2 7 7 A, =A,uA,

27

i -1 Fig. Ib. A space structure 9 = u Si Fig. l c . Typical intersections of substructures Si, i = 1, 2, ... , p

A typica.1 S, can also be considered as Si = Kt u Li, where Ki is a triangulated cube and Li is a star connecting the middle node of the corner nodes of the cube. Thus

Similarly A: and A t can be decomposed as Y ( 8 d = y(&) + y(L0 - y(& n Ll) - AS = A t u A t and A: = A: u A t .

Now if S is viewed as a ball-jointed truss; then y ( 8 ) can be calculated as follows: y(Si) = (0) + (8 - 3 x 9 + 6) - (0 - 3 X 8 + 6) = +5.

The reason for y ( K 0 = 0 is given by Theorem 2 of Section 3. ?(A:) = M - 3N + 6 = 8 - 3 X 5 + 6 = -1 , y(Ai) = (-1) + (-3) - (2 - 3 x 3 + 6) = - 3 .

?(A:) = (-1) + (-1) - (1 - 3 X 2 + 6) = -3,

Hence

When S is taken as a space rigid-jointed frame, then y(8) = 3612 is obtained. ~ ( 8 ) = 27(5) - [6(-1) + 14(-3) + 6(-3)J = 201.

Notice that classic formulae require to count 804 members and 203 nodes, which is not an easy task.

2.2. Identification method

This approach is based on the union-intersection method of the previous section and provides an efficient means for finding the topological properties of a structural model X after identifying two subgraphs of S. For example, consider a model as shown in Fig. 2a. Identifying ab with cd as in Fig. 2b yields a cylindrical space model S2. Identification of ac and bd results in a toric structure as shown in Fig. 2c.

a1 S ' S ,

c ) S3=SZu $3 Fig. 2. Steps of the ldentifications (b) shows the plane diagram of S', c) that of Sa)

350 ZAMM - Z. angew. Math. Mech. 68 (1988) 8

The following equation can be employed in such an approach, which is the same as equation (2) of the pre- vious section :

?(Xi u Sf) = y(8; ) + y(&) - y(Si n Sf) . (3) I n this relation Xj is a substructure of Si through which the identification is made. Obviously Xi n XI contains two disjoint S,.

E x a m p l e 2: Let 8, be the graph model of a rigid-jointed frame, Fig. 2a. The first Betti number of 8 3 , the 1-skeleton of a torus, is obtained by two identifications, as shown in Figs. 2b and 2c, as follows:

I n this case ?(AS) = b,(AS') = M - N + b,(AS') and for Ohe first identification holds - - -

~,(AS') = 18, ~,(AS',) = 0, b,(X' n AS,) = M - N + 1 = 6 - 8 + 1 = -1,

b,(S1 u S,) = b,(AS'Z) = (18) + (0) - (-1) = 19 . and for the second identification

- b,(S2) = 19, b,(S,) = 1 , b,(S2 n 8,) = 6 - 6 + 1 = 1,

b , ( P u 8,) = b,(83) = (19) + (+1) - (+ l ) = 19.

When X is viewed as a ball-jointed truss, then for the first identification obtains y ( A S 1 ) = M - 3 3 N + 6 = 3 3 - 3 X 1 6 + 6 = - 9 , y ( A S ' 2 ) = 3 - 3 X 4 + 6 = - 3 , - y(S1 n 8,) = 6 - 3 x 8 + 6 = -12,

y(X2) = 0,

y(S1 u AS',) = ?(AS',) = (-9) + (-3) - (-12) = 0.

j+!P n AS',) = 6 - 3 x 3 + 6 = - 6 ,

and for the second identification

~ ( 8 , ) = 3 - 3 x 3 + 6 = 0, y(S2 u AS',) = y(S3) = (0) + (0) - (-6) = + G .

3. Special methods

I n the following, five theorems are proved which can be employed in the determillation of the DSI for different types of skeletal structures.

T h e o r e m 1: For a fu l ly triangulated planar truss (except the exterior boundary), as in Fig. 3, the DSI i s the same as i ts internal nodes, i.e. y ( S ) = N,.

Proof : Embed X on a sphere. According to Euler

(4) R - - M + N = 2 , where R = Ri + 1 is the total number of regions and Ri is the number of triangles. Hence

3Ri = 2M - M , ( M e being the number of members in the boundary of S which may be non-triangulated) and

3Rt = 2 M - N , = 221.1 - N + Ni = 3 ( M - N + 1) .

Hence M - 2N + 3 = Ni , i.e.

y ( 8 ) = Ni . ( 5 )

?(AS') = Ni.- N c .

For trusses which are not fully triangulated (Fig. 4), let M, be the number of members required for comple- tion of the triangulation, then

(6) E x a m p l e 3: For the truss shown in Fig. 3, using equation (5) yields

~ ( 8 ) = Ni = 39.

The use of the following formula from text books leads to

which requires to count 154 members and 59 nodes in comparison with counting 39 internal nodes. y(X)= M - 2 X + 3 = 1 5 4 - 2 x 59 + 3 = 39,

Fig. 3. A triangulated planar truss Fig. 4. A planar triihs and the necessary members for its Lrinngiilation

KAVEH, A. : Topology and Skeletal Structures 351

E x a m p l e 4: Let S be a planar truss as shown in Fig. 4. For the completion of the triangulation, 7 members are required, as shown in dashed lines. Thus using equation (6) yields

y(S) = Ni - ill, = 12 - 7 = 5 , T h e o r e m 2: A ball-jointed space truss embedded on a sphere is statically determinate, i f all the regions are

P r o o f : According to Euler one has triangles.

R - M + N = 2 .

Due to the full triangulation of the regions, 3 R = 2 M ,

combining the above two equations yields

M - 3N + 6 = 0 , i.e. y ( S ) = 0 .

As an example, the ball-jointed truss K , of Fig. l a is statically determinate, i.e. y(&) = 0. Theorem 3: The DSI of a planar rigid-jointed frame S is three times as the number of its internal regions, i.e.

y(S) = 3Ri. Proof: A simple proof can be found in HENDERSON and BICKELY [lo]. E x a m p l e 5: For the planar frame of Fig. 5

y(B) = 3b,(S) = 3Rj = 3 x 12 = 36.

( 7 )

Fig. 5. A plane rigid-joint.ed frame

The standard formula of the text books yields the same result: y(S) = 3(49-38 + 1) =z 36.

b

However, this requires to count 49 members and 38 nodes in comparison to counting 12 internal regions. Drawing of a s t r u c t u r a l model : A drawing Sp of a graph S is a mapping of S into a surface. The nodes

of S go into distinct nodes of Sp. A member and incidence nodes map into a homeomorphic image of the closed inter- val [0, 11 with the relevant nodes as end points and the interior, a member, containing no node. A good drawing is one in which no two members incident with a common node have a common point, and no two members have more than one point in common. A common point of two members is a crossing. An optimal drawing in a given surface is one which exhibits the least possible crossings.

In this article all the drawings are good, but not necessarily optimal. The number of crossing points of S after drawing on a plane or sphere, S p , is denoted by v(Sp). For cases when the drawing is optimal, v(SP) becomes the cross- ing number.

E x a m p l e 6: A good drawing of S is shown in Fig. Gb, where the crossing points are marked by X. For this drawing v ( S p ) = 2. An optimal drawing of X, shown in Fig. 6c, corresponds to v(XP) = 0 .

Lemma 1 : Identification of two points from two distinct members of a graph model increases its Betti number by unity.

Proof : Insert two nodes on two members of 8. Obviously the first Betti number remains unchanged, since for each additional node the number of members is increased by one. Now identify the newly introduced nodes to

Fig. 6 % A space model S Wig. 6b. A drawing of S Fig. 6c. An optimal drawing of S

352 ZAMM * Z. angew. Math. Mech. 68 (1988) 8

Fig. 7a. A space model S Fig. 7b. An arbitrary drawing S’ of S

obtain a crossing point. The model S is changed to S1 and by equation (3) bl(X1) = b, (S) + bl(a single node) - & (two disjoint nodes)

= b,(S) + (0) - (0 - 2 + 1) = bl(S) + 1 .

Example 7: A space graph model S after identification is shown in Fig. 7b. Use of equation (8) leads to b,(S1) = b,(S) + 1 = 3 + L = 4 .

Lemma 2: For S P obtained from S by p = v(SP) identifications the first Betti numbers of X p and X are related by

bl(SP) = b,(S) + v(XP) . The proof of this lemma follows from multiple application of Lemma 1. Theorem 4: Let S be a space rigid-jointed frame. Then

y(X) = 6b,(S) = 6[Rj - Y(P)],

where Ri and v(SP) are the numbers of internal regions and crossing points of SP (a good drawing of X), respectivelg.

X P we have bl(SP) = Ri. Hence y(S) = 6[Ri - v(SP)]. Proof: From Lemma 2 b,(S) = bl(SP) - ~(SP) and by Ref. [lo] y ( S ) = 6b,(S). From Theorem 3 for planar

Example 8: Let S be the graph model of a space frame as shown in Fig. 8a. For an arbitrary drawing of this frame, as shown in Fig. 8 b, 011 a plane, v(8p) = 12 crossing points exist. The

number of internal regions of this drawing is Ri = 35. Thus

y (S ) = 6b1(S) = 6[35 - 121 = 138.

Theorem 5 : Let S be a ball-jointed truss. For a good drawing XP ( p = v(SP)) of SJ the DSI of S is given by

y(’) = v(sp) - M C J (11) where M , is the number of members required for the full triangulation of #I).

Proof: With anargument similar to that of Lemma 1, for each identificat,ion of two added points of two distinct members, M and N are increased by 2. Notice that a real crossing point of two members does not alter t h e indeterminacy and must be counted aa crossing of a drawing; e.g. if Ki in Rg. l a is viewed as a space truss, one should write y(&) = v(KI) - Mc = 18 - 18 = 0. Crossing points of diagonal members of 6 faces of the cube should not be counted in calculating v (Kf ) . Thus before identification

y ( S ) = ( M + 2) - 3(N + 2) $- 6 = 1M - 3N + 6 - 4 = ~ ( 8 ) - 4 .

After identification through one point

y(X1) = y ( S ) + y (a single node) - (two disjoint nodes)

= EYP) - 41 + (3) - (0) = y(s) - 1 .

Fig. 8a. A space frame S Fig. 8 b. A good drawing of S

KAVEH, A. : Topology and Skeletal Structures 353

Fig. Qa. A space truss S Fig. 9b. An arbitrary drawing of 8 with added members for full triangulation

For v(Sp) such identifications,

But

where S, is a fully triangulated mode1 (all the regions are triangles) and y(8,) = 0 (see Theorem 2 ) . Thus y ( S ) = = y ( 8 P ) + v(XP) = -Mc + v ( 8 p ) and the proof is completed.

Example 9: Consider a space ball-jointed truss S as shown in Fig. 9a. A good drawing of S is given in Fig.9b. This drawing contains 3 crossing points as marked by X. For full triangulation Me = 2 members are added as shown in dashed lines. Thus

y ( 8 P ) = y ( 8 ) - Y(SP) . ~ ( 8 ’ ) = M - 3N + 6 = M + M , - 3N + 6 - Me = ~(8,) - M e ,

y(S) = Y(XP) - Mc = 3 - 2 = 1 , Theorems 4 and 5 can be proved directly as follows. Proof of Theorem4: For a space frameS,let M,, Nl be the numbers of members and nodes, and for its planar

drawing SP, let M , and N, be the numbers of members and nodes. Then

but

and by substitution

For the planar drawing fP results M2 - N , + 1 = Ri. Hence

748) = 6(Ml - Nl + 1) Y

M2 = N1 + 2 ~ ( 8 P ) , N , = iVl + v(SP)

y($) = 6[(HS - ZY(@)) - (N, - Y(SP)) + 11 = 6[(M2 - A72 + 1 - Y(SP)] *

y(S) = 6[R, - Y(SP)].

Proof of Theorem 5 : For space truss 8, by full triangulation of 81), drawitig S’p is obtained, for which M i = M, + Mc = MI + 248Y) + Mc , N i = N , = iVl + ~ ( 8 p ) .

The DSI of 8 is

By Theorem 2 obtains y($’P) = M i - 3Ni + 6 = 0 and hence ~(8) = MI - 3N1 + 6 = M i - 2 ~ ( 8 P ) - M , - 3Ni + 3 ~ ( 8 P ) + 6 = M i - 3Ni + 6 + ~ ( 8 p ) - 2 M C .

y ( 8 ) = v(SP) - M , .

4. Conclusions

The methods presented in this paper provide an insight to the connectivity properties of skeletal Rtructures. Thus resulting in an efficient means for generating localized self-equilibrating stress systems for their flexibility analysis. These methods simplify the evaluation of the DSI of structures, considerably.

References 1 KAVEH, A.: A combinatorial optimization problem, optimal generalized cycle bases. Comput. Neth. Appl. Mech. Engng. 20

2 HENDERSON, J. C. DE C. : Topological aspects of structural analysis. Aircr. Eng. 82 (1960), 1-6. 3 CASSELL, A. C.; HENDERSON, J. C. DE C.; KAVEH, A.: Cycle bases for the flexibility analysis of structures. Internat. J. Numer.

4 KAVEH, A. : Application of topology and matroid theory to the analysis of structures. Ph. D. thesis, Imperial College, Lond. Univ.

5 KAVEH, A.: Improved cycle bases for the flexibility analysis of structures. Comput. Meth. Appl. Mech. Engng. 9 (1976), 267--272. 6 KAVEH, A. : An efficient program for generating cycle bases for the flexibility analysis of structures. Commun. Appl. Numer.

7 KAVEH, A.; An efficient flexibility analysis of structurea. Comput. Struct. 22 (1986), 973-977. 8 KAVEH, A.: Static and kinematic indeterminacy of skeletal structures. Iranian J. Sci. Tech. 7 (1978), 37-45. 9 KAVEH, A.: Statical bases for efficient flexibility analysis of planar trusses. J. Struct. Mech. 14 (1986), 475-488.

10 HENDERSON, J. C. DE C.; BICKELY, W. G.: Statical indeterminacy of a structure. Aircr. Eng. 27 (1955), 400-402.

Received July 13, 1987

Addre8.s: Professor Dr. A. KAVEH, Department of civil Engineering, Iran University of Science and Technologg, Narmak, Teheran 16, Iran

(1979), 39-52.

Meth. Engng. 8 (1974), 521-528.

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