Alfano Et Al - Automatic Analysis of Multicell Thin-walled Sections

7/22/2019 Alfano Et Al - Automatic Analysis of Multicell Thin-walled Sections

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0045-7949(95)00302-9 Compulers & SIrucrures Vol. 59, No. 4, pp. 64-655, 1996Copyright 0 1996 Elsevier Science LtdPrinted m Great Britain. All rights reserved

0045.7949196 I 5.00 + 0.00

AUTOMATIC ANALYSIS OF MULTICELL THIN-WALLEDSECTIONSG. Alfanot, F. Marotti de Sciarra and L. Rosatit

TDipartimento di Scienza delle Costruzioni, Facolta di Ingegneria, Universita di Napoli Federico II,80125 Piazzale Tecchio, Napoli, ItaliaSIstituto di Ingegneria Civile ed Energetica, Facolta di Ingegneria, Universita di Reggio Calabria,

Via Cuzzocrea 48, Reggio Calabria, Italia(Recei ved 6 October 1994)

Abstract-An automatic procedure is outlined for the determination of the shear centre and the evaluationof the overall state of stress in multicell thin-walled sections subject to axial force, bending moment,shearing force and torque. Graph theory is shown to be the rationale to establish a topological modelof the section which is preliminary to a computer implementation of the shear stress analysis. Specifically,we exploit the main features of the Depth-First-Search graph algorithm in order to automaticallydetermine a number of independent circuits equal to the degree of connection m of the section. Thealgorithm also localize the m slits which make the section open, a preliminary step for the analysis ofmulticell sections subject to a shearing force. Further, the evaluation of the first elastic area moment atany point of the open section is addressed by means of the Open-Section-Cut algorithm elaborated in thispaper. The outlined procedure entails a considerable simplification of the analysis, since the geometricaldata which need to be assigned are only the strictly necessary ones, namely the coordinates of the vertices,the branches connecting them and their thickness. A numerical example, carried out for a ships hull bymeans of a computer program written in MATHEMATICA, shows the effectiveness of the proposed approach.

1. INTRODUCTION

The linear elastic analysis of thin-walled multicellsections subject to shearing force and torque is ofconsiderable interest in structural engineering. Actu-ally such an issue is common to the analysis of manysignificant structural schemes such as box girders,airplane fuselages or wings and ship hulls.

Although elastic analysis of thin-walled sectionsis based upon well established methods [l-7], it isundoubted that its computer implementation turnsout to be rather involved for multicell sections.Actually it is necessary to exploit an algorithmthat allows us to operate on the section in a waywhich is independent of its particular form, thenumeration of its vertices and upon its degree ofconnection.

For instance, to accomplish the analysis of amulticell thin-walled section subject to a shearingforce, the designer has first to define the cells of thesection and to introduce the unknown redundantshear flows at the slits which make the section open.

Such operations, besides being tedious and timeconsuming, can be performed in a completely subjec-tive way and hence they seem to be inadequate forimplementation in a computer code.

Moreover, particularly in the early stages of design,even the topology of the section can be subject tosuccessive phases of optimization in such a way thatthe input data need to be updated continuously.

For these reasons an automatic procedure has been

implemented for the calculus of the stress field and ofthe shear centre for multicell sections.

It is shown that the geometrical input data ofthe section can be reduced to the strictly necessaryones: nodes and their co-ordinates in a given refer-ence frame, and equations of the branches and theirthickness. All information concerning the topologyof the section such as the assignment of the cells canbe avoided, thus enhancing the features of the pro-cedure contributed by Maceri [8] for the automaticanalysis of thin-walled multicell sections subject totorque.The approach proposed in Ref. [8] is further gener-alized in our paper so as to determine, within thesame computational framework, the shear stressesdue to a shearing force and the shear centre of thesection.

To this end some basic concepts of graph theory [9]are introduced, since a multicell section, eventuallysupplied with open branches, can be conceived as anon-oriented graph defined by vertices and edgeswhich are coincident, respectively, with the nodes andthe branches of the section.

All geometric information necessary to evaluatethe shear field can be automatically obtained byexploiting the main features of two graph algorithms:the Depth-First-Search algorithm (DFS) due toTarjan [lo] and the Open-Section-Cut algorithm(OSC) specifically developed in this paper for theanalysis of multicell sections subject to a shearingforce.

641


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642 G. Alfano et al.Essentially the DFS allows one to successfully

address the two main issues relevant to the automaticshear stress analysis of multicell thin-walled sections:first to determine a number of independent circuitsequal to the degree of connection m of the section;second to find out an automatic procedure able tointroduce exactly m fictitious slits in the section so asto make it open.

viewed as a connected graph. Accordingly, nodes andbranches of the section can be referred to as verticesand edges when considering the relevant descriptionaccording to the graph theory.

The first result is achieved since the DFS partitionsthe edges of the graph in two disjoint sets, termedp lms and fronds and a one-to-one correspondence isestablished between the set of fronds and the set ofindependent circuits. Second, the DFS defines at theend points of each frond the fictitious slits whichmake the section open.

Hence, in order to make the paper reasonablyself-contained, we briefly introduce some standarddefinitions of graph theory [9] which will be referredto in the description of the DFS algorithm employedfor the automatic stress analysis. The reader moreinterested solely in the structural applications of theDFS algorithm can then skip this section.2.1. Basic deji nit ions of graph theory

In addition the DFS allows us to automaticallyassemble the topological matrix of the section [S]entering the linear system whose solution providesthe shear flows in the circuits.

With reference to the shear stress analysis due to ashearing force, an additional problem has to besolved for the open section obtained by the DFS.Actually, it is necessary to evaluate the first elasticarea moment, with respect to the elastic centroid, ofone of the two complementary portions of the sectionresulting from a cut at an arbitrary point.

Such a problem is solved by means of the OSCwhich determines the edges belonging to one of thetwo complementary portions of the section.

Finally, to evaluate the equivalent stress at anypoint of the section, formulas for the normal stressdistribution due to an axial force and a bendingmoment are also provided.

A graph Q = (V, E) consists of a set of elementsv = {L?, v2, } called vertices and a set of objectsE = {e, , es, } called edges such that an unorderedpair {v,, L>} s associated with each edge ek. These setscan in general be not finite but we will consider onlyfinite ones.

The vertices u, and v, associated with the edge ek arecalled the end vertices of ek When v, is an end vertexof an edge ep, v, and e, are said to be incident witheach other. Two vertices incident with the same edgeare said to be adjacent. Two edges connecting thesame end vertices are said parallel.

A graph 9 is said to be a subgraph of a graph Qif all the vertices and all the edges of 9 belong to 9,and each edge of Y has the same end vertices asin 9.

The outlined procedure has been implemented in acomputer program written in MATHEMATICA. For agiven section subject to an axial force, a bendingmoment, a torque and a shearing force the programautomatically determines the stress distributions, thevalue of the maximum equivalent stress and where itis attained. A numerical example is finally carried outfor the hull of the transatlantic ship France [7].

The easiest way to represent a graph is to use pointsfor the vertices and lines (not necessarily straight) forthe edges. Hence, an edge ekr joining two vertices v,and v,, will be represented by a line that connects thepoints associated with the vertices (Fig. la).

A graph 3 is said to be directed of an ordered pair(u,, v,) is associated with each edge. In this case theedge is said to be oriented from ~1, o v, which arecalled, respectively, the first and the second end vertexof the edge. An example of the graph and of a relateddirected one is sketched in Fig. la,b.

2. APPLICATIONS OF GRAPH THEORY TO MULTICELLTHIN-WALLEDSECTIONSThe main motivation for this section arises from

the fact that a multicell thin-walled section can be

In the sequel, an edge ek incident with the verticesv, and a, will be denoted by (ek]a,. v,) if it is notoriented, and by (ek]v,+C,) if it is oriented from u,to 2;.

A walk is defined as a finite sequence of orientededges such that the first end vertex of each edgecoincides with the second one of the previous edge,

(a)Fig. 1. (a) Undirected graph, (b) a related directed graph.


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Automatic analysis of multicell thin-walled sections 643

((bll- 2),(e 12-4) A 4-l) I l-3)) {(c 14-l),(aI 1-3),(d I3- 4),(f 14-2),(e 12-4))is an open walk is a closed walk(4 @>Fig. 2. (a) Open walk, (b) closed walk in a graph.

if there is any, and no edge appears more than once(Fig. 2).

If the second end vertex of the last edge in thewalk coincides with the first end vertex of the firstedge, the walk is said to be closed, otherwise it issaid to be open.It is assumed as positive direction of the walk theone which is consistent with the orientation of eachedge, i.e. the one for which the first end vertex of eachedge precedes the second one.

A path is an open walk in which no vertexappears more than twice in the array denoting thewalk. A circuit is a closed walk which is also a path(Fig. 3).

A graph C4 is said to be connected if there is atleast a path between every pair of vertices of S. Thedegree of connection of a graph is the maximumnumber of edges that can be eliminated withoutmaking the graph disconnected. If the degree of

b

a((a l3-l),(c 11-4),(e 14-2)) ((al3-1),(cI l-4),(d14-3))

is a path is a circuit(a) @I

connection is equal to 1 the graph is said to be simplyconnected.

A tree is a connected graph without any circuits.

2.2. Depth-First-Search (DFS) algorithmIn order to perform the automatic analysis of

the shear stress distribution in multicell thin-walledsections subject to a shearing force or a torque, twomain issues need to be addressed. First, a number ofindependent circuits equal to the degree of connectionm of the section must be determined; secondly, anautomatic procedure is needed to introduce exactlym slits in order to make the section open. It willbe shown that a solution to both problems is fullyprovided by the Depth-First-Search algorithm (DFS)contributed by Tarjan [lo].

Essentially the DFS is a powerful technique ofsystematically traversing the edges of a given graph

Fig. 3. (a) Path, (b) circuit in a graph.


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644 G. Alfano et al.such that every edge is traversed and each vertex is processed so as to determine the set of the edgesvisited at least once. incident with each vertex.

It is evident that for answering almost any non-trivial question about a given graph ~9 we mustexamine every edge and every vertex of B at leastonce. For example, before declaring a graph 9 to bedisconnected we must have looked at every edge in 9;for otherwise it might happen that the one edge wehad decided to ignore could have made the graphconnected.

Step I-Set palms = 8, fronds = 8, stack =( ), i.e. an empty array, m = 0, j = 1. Allthe vertices are initialized as unvisited andall the edges as untraversed. Set r = x andgo to next step.

The basic idea of the DFS is to traverse the graphby moving from a vertex 13 o an adjacent one assoon as possible through an untraversed edge, even ifthere are still edges incident with ~1 that remainuntraversed for the time being. When we arrive at avertex which has been visited before, a circuit isdetermined in the graph. We then have to come back,through the walk just traced, to a vertex with untra-versed edges incident with it and to restart the search.The procedure stops when there are no more verticeswith untraversed edges incident with them.

Step 2-Label v as visited and go to nextstep.

Step 3-Look for an untraversed edgeincident with v.

(a) If there is no such edge, i.e. ifevery edge incident with t hasalready been traversed, go to step 6,otherwise:

Notice that. everytime we move from a vertex 2:to another one $1 long an edge e,, we automaticallyorient ek from o to w. Hence, at the end of the process,we have transformed the given undirected graph intoa directed one.

(b) Pick an untraversed edge inci-dent with v, say (alv,w), and tra-verse it. Orient the edge from t to Mgetting (alv --+vv) and label it as tra-versed. Now you are at vertex w. Goto next step.

It is worth noting that the DFS by itself does notreveal properties of a given graph 9, except whetheror not 9 is connected. What it does, however, is topartition the edges in two disjoints sets, palms andfronds, determined as follows: at the beginning of thealgorithm palms and fronds are two empty sets; everytime we traverse an edge we add it to palms if wearrive at a not already visited vertex, otherwise weadd the edge to fronds. Hence, a frond detects acircuit since we have reached an already visitedvertex.

Step 4-(a) If w is unvisited, add the edge(alv +w) to the set pulms and at thelast position of the array stuck. Setv = w and go to step 2.(b) If u is visited, add edge (a 1 +w)to the set fronds and go to step 5.

In this way, after performing the DFS, the setpalms defines a tree which contains all vertices of thegraph and a one-to-one correspondence is establishedbetween the circuits and the fronds, their numberbeing equal to the degree of connection of the graph.Further the section can be made open by introducinga slit at the end point of each frond. A completeanswer is thus provided to the two main issues of theautomatic shear stress analysis.

We now give an accurate description of the algor-ithm employing the following terminology. The de-gree of connection of the graph, originally unknown,is denoted by m, while circuits are m arrays deter-mined by the algorithm. We shall refer to an auxiliaryarray of variable dimension as stack. The elements ofboth stack and circuits will be oriented edges, that isof the type (ala-+w).

Step S-Set circuit j = (a Iv +w), m = jand k = 0. Denote with s the length ofstuck, i.e. the number of edges that formthe array stack.

(a) Add the edge at the position(s -k) of stack to the first positionof circuit j and go to (b).(b) If this edge begins at w then setj = j + 1 and go to step 3, otherwiseset k = k + 1 and go back to (a).

Step 6-(a) If stack 8 remove its last edge,say (blu+u) which has been tra-versed to arrive at v. Move back tou, set v = u and go to step 3.(b) If stack = 8, stop (you are backat root x, having traversed everyvertex connected to x).

Let Q be a given connected and undirectedgraph assigned by its vertices and its edges. Lett be the number of edges of B and x be thechosen vertex, referred to as root, from whichthe search is to begin. The input data are further

For future convenience we further assemble anm x t matrix S, referred to as topological matrix, byassigning to the element S, the value 1 if the jthedge belongs to the ith circuit and the value 0 in


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Automatic analysis of multicell thin-walled sections 645

(a) (b)- palmsC.-W--.. fro s

Fig. 4. (a) A graph, (b) the relevant partitioning in palms and fronds.

the opposite case. Hence, the ith row of S definesthe edges which belong to the ith circuit, while thejth column defines the circuits which the jth edgebelongs to.

2.3. Appli cati on of DFS al gori thm t o mult icellthi n-w ail ed sections

We now illustrate an example of application ofDFS to the multicell section of Fig. 4a where, forthe sake of clearness, edges have been denoted byletters.

The given section, conceived as an undirectedgraph, is transformed by the DFS in the directedgraph of Fig. 4b, where the sets of palms and frondsare also marked.

We explicitly remark that the circuits determinedby the algorithm do not coincide with the physicalcells of the section; further the edges constitutingeach circuit clearly depend on the numeration ofvertices and edges adopted to input the given section.For instance, with reference to the numerationadopted in Fig. 4a, the circuits determined by DFSare (Fig. 4b):

(1) {(all-2), (bl2+3), (~13-4) (dl4-*1)}(2) {(ail-+2), @P-+3), cl3-+4), (el4+l)j(3) bl2~3), ~13~4), fl4~2)}(4) al1~2), bl2-r3), gl3-tl)}

-(4

and the corresponding topological matrix is:abcdefg

circuit 1 1111000circuit 2 1110100S= circuit 3 0110010circuit 4 1100001

Note that the columns corresponding to the frondscontain only one element different from 0 and that aone-to-one correspondence does exist between the setof the fronds and the set of the circuits. Furtherthe circulation sense along each circuit is given, bydefinition, by the positive direction of the relevantedges.2.4. Open-Section-Cut algorithm

The shear stress analysis of a multicell thin-walledsection subject to a shearing force requires one to dealwith an open section obtained from the given one byintroducing exactly m slits, m being the degree ofconnection of the section.

Such slits are introduced by the DFS at thepoints of the fronds immediately before their secondend vertex, i.e. the points which connect each frondwith the first vertex of the univocally associatedcircuit. Accordingly the multicell section can be rep-resented as in Fig. 5a where the slits are explicitlyillustrated.

(b)Fig. 5. (a) Open graph corresponding to fronds in Fig. 4b, (b) the determination of the edges preceding(c1344).


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Automatic analysis of multicell thin-walled sections 647where In the first approach the unknowns are the values

of the flows in the t branches and the rate of twist 0.A,=E, s nE(r) dA (4)

According to the relation (1) the unknowns can beA determined by writing m -t n equations. They areprovided by: (a) the n - 1 equilibrium equations

is the elastic area of the section and A is its geometricarea.The flexural curvature xr due to the bending

moment M, is expressed in the form [l l]or = Cr(k x Mr), (5) where f k s the flow in the k th branch and YP denotesthe set of indices corresponding to the branches

where k is the unit normal to the section and the incident with the nodep; (b) the equilibrium equationsymbol x denotes the axial product. Further Cr de- for the entire section about an arbitrary point Q:notes the deformability tensor of the section, i.e. theinverse of the second elastic area moment J hds=M,, (10)

J = E,, nE(r) r@r dA where h is the distance from Q of the tangent to thegeneric point of the centre line and MT is the applied

rtorque; (c) the m equations of compatibility= E,, G(S) r(s)@+) 6(s) ds, (6)

where @ denotes the tensor product and, for sim-plicity, we have dropped the index of the genericbranch. The same assumption will be made in thesequel when no confusion can arise.

An explicit expression of J for a polygonal thin-walled section is provided in the appendix. Note thatJ turns out to be positive definite unless the centreline is a unique straight segment and hence the tensorCr= J- can be easily computed.

The axial deformation corresponding to xf turnsout to be

tz(r) = xr. r = Cf(k x M,) r, (7)where the dot denotes the scalar product.

Finally the normal stress at the point r is given by

;+C,(kxM,).r 1 8)3.2. Torsion

Let us now address the analysis of a multicellthin-walled section subject to uniform torsion, Wefirst consider a section without appendices, i.e.branches not belonging to any cell.

Different solutions to this classical problem havebeen presented in the literature [l, 2,7, 121. In orderto point out the advantages of the procedure pro-posed in this paper, we briefly recall two solutiontechniques which can be found in the textbooks of

f(s)~=2G,tYA, i=l,2 ,..., m, (11)Gwhere c, denotes the centre line that defines the ith celland A, is the area enclosed by ci.

Each term on the left-hand side of eqn (10) ispositive or negative whether or not the positivedirection assumed for the flow f k etermines a mo-ment with respect to Q consistent with the oneassumed for MT.

The linear problem governed by eqns (9)


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648 G. Alfano et al.and eqn (11) becomes

,L fJ, = 2G00Ai i = 1,2,. . , m, (14)

where J%~ collects the indices of the branchesbelonging to the ith cell.We then have to solve a system of m + 1 equations

which, for the section with a high degree ofconnection, is by far less then t + 1.

The procedure developed in Ref. [8] requires inaddition to input a topological matrix S, whosegeneric entry S, is defined as follows: if the jth edgedoes not belong to the ith cell S, is set 0; otherwiseS, is set + 1 or - 1 whether or not the positivedirection of the jth flow coincides with the positivedirection of the cell (clockwise or counterclockwise).

For instance, assuming as positive direction forthe flows the one defined by the arrows of Fig. 6and a counterclockwise orientation for the cells, therelevant topological matrix S, would be

a bcde fgcell 1 0 1100 10cell 2 -1 0 0 0 -1 -1 0S,= cell 3 0 001 100cell 4 1 -10 0 0 0 1

It is then apparent that a considerable amount ofdata needs to be provided in addition to ones whichare strictly required to univocally define the topologyof the section.

It is needless to say that the burden can be substan-tial for sections with a high degree of connection,typical of aeronautical or naval applications, andwhen the topology of the section is subject to succes-sive optimizations as in the early stages of design. Adifferent approach is thus followed in our paper,while retaining the conceptual significance of thetopological matrix and its use in the determination ofthe shear stress field.

Actually, we assume as unknowns the flows q,pertinent to the circuits determined by the DFS.Recall that the circuits do not necessarily coincidewith the cells of the section and that their circulationsense is defined by the positive direction of therelevant edges. Such a direction is automaticallydetermined by the DFS so that the topological matrix

does not contain negative entries since the generic S,,simply records if the jth edge belongs to the ithcircuit.

Further the topological matrix needs no more to beassigned by the designer, but it is automatically builtup by the algorithm.Once the flows qi have been determined, the flow ineach branch is obtained by summing up the flows ofthe circuits which contain the given branch, withouttaking care of any sign.

The flows qt can be determined following thesame procedure outlined in Ref. [8]. To this end weintroduce an m x m symmetric matrix P whoseentries are

[ L,gA 1, i=j

If= C lk i j(15)

( k d, ,

where 4, and N, collect, respectively, the indices ofthe branches belonging to the ith circuit and theindices of the branches belonging both to the ith andto the jth circuit.

Accordingly, eqn (14) can be rewritten as follows:

i Yy,q, = 2GoBA , i=l,2 ,..., m. (16),=IThe consistency of the flow qi with the applied

torque is recorded through the area A, which isassumed positive or negative whether or not thecirculation sense of the ith circuit (clockwise orcounterclockwise) is the same as the positive directionassumed for the torque. A useful formula for theautomatic determination of A , and of the relevantsign is reported in the appendix.

In order to express the equilibrium eqn (10) interms of qi w e write

(17)

where Yk denotes the set of the indices of the circuitscontaining the kth branch. Accordingly the left-handside of eqn (10) can be rewritten as follows:

Grouping the terms pertaining to the same circuit,we immediately verify that the expression above canalso be written as

h ds = 2 qi 1 s h ds = $ q, h ds,i=l ke.X ck i=lFig. 6. Positive direction of flows in cells. (18)


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Automatic analysis of multicell thin-walled sections 649where ci denotes the centre line of the ith circuit.Noting that

h ds = 2A,, (19)

the equilibrium eqn (10) can be written in the form

2 f q,A,= MT,r=, (20)which represents a generaIized$rst Bredt formuIa.

The evaluation of 0 and of the flows qi can beaccomplished independently by defining the flows 4,

so that eqn (16) becomes

(21)

(22)The coefficient matrix P of the linear system

above can be given by a more compact expressionby introducing a t x t diagonal matrix A whosegeneric entry Akk is equal to 1,. To this end wenote that

(SA syo= i 1, s, S,)k l

(23)

where S is the topological matrix defined in theDFS and the superscript T denotes transposition.Actually the generic element S,S, of the sum aboveis given by

s, s, = 1 ifkE.&Y,0 ifk$Ai (24)

and

s, s, = 1 ifkEJIrU0 ifk$Mii for i j. (25)

Hence, recalling definition (15), we can writeY = SAS. (26)

Introducing the vectors 4 and A defined by

4= I927)

we finally get the linear system of equations

The matrix P is invertible by virtue of:Proposition. The matrix P = SAST is symmetric andpositive definite.Proof. The proof of the symmetry is trivial. In orderto prove the positive definiteness of P we observe that

A being positive definite, the relation above doeshold if and only if ij does not belong to the kernelof S[13].

Recall that a one-to-one correspondence does existbetween the set of the fronds and the set of circuits.Notice also that the vector STij collects the flows inthe branches of the section so that, if iZj 0, ST4 isdifferent from 0 since we get at least a non zero flowin one of the fronds.

After solving eqn (28) the rate of twist 8 isprovided, by means of eqns (20) and (21), by thefollowing formula:

,,=MT.4Goij - A (30)The torsional stiffness K, of the section is thus

given by

(31)

which stands as a generalized second Bredt formula.w Section with appendices-the analysis of a multi-

cell section supplied with appendices, that is branchesnot belonging to any cell, does not present particulardifficulties.

Let us denote by S, the portion of the sectioncollecting all the appendices and by S, the closedsection resulting from the elimination of S,. More-over let K, be the torsional stiffness of S, given byeqn (31) and KZ the one of S, provided by thewell-known formula

n,6ds, (32)

where d collects the indices of the branchesbelonging to S,.

If MT is the torque acting on the given section, theshares acting on S, and on S, are

MT, = K,-MT MT2= K2K, + KZ -MT. (33)K, + K2


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650 G. Alfano et al.Thus, by means of eqns (3lH33) it is possible to

find the flows in S, and then the shear stresses. Themaximum shear stress in a branch of S, at theco-ordinate s is given by

z(s) = g$p(s).I 23.3. Shearing force

The prominence of the approach exploited in theprevious section is further substantiated by the possi-bility of casting, within the same computationalframework, the shear stress analysis due to a shearingforce. In this respect we briefly recall the formulasproviding the shear stress due to a shearing force inthe case of an open section.

3.3.1. Shear i ng st resses or open secti ons. We shallemploy the usual assumption according to which theshear stress is uniform over the thickness of eachbranch and directed along d s)

z(s) = cd s),S (35)

where q,, denotes the shear flow.The relation between the shearing force t and the

shear flow qO(s) is provided by the equilibriumequation [141

40(s) = -v so(s), (36)where Cr is the deformability tensor defined inSubsection 3.1;

so(s) = s E(r)rdA = s E(s) r s) 6 s) ds (37)SI iis the first elastic area moment, with respect to theelastic centroid, of the portion 9; of the open sectionon one side of the point at the co-ordinate s in thedirection of decreasing s, and c denotes the relevantcentre line. In other terms, a fictitious cut introducedat the point (k, s) of the centre line disconnects thesection in two portions; S; is the one which containsthe point (k, 0). We thus get the following expressionfor so(s):

sob) = %(k s)skd= E, n&k 5) r(k, 5) 6% 5) d5 f%,(k),(k.0)(38)

where B,,(k) is the first elastic area moment, withrespect to the elastic centroid, of the branches for

which both end vertices belong to 9;. Collecting suchbranches in the set & we thus have:

Go k) =Eo 1 sh, s) r(h, s) W, s) ds,bed Chwhere c,, denotes the centre line of the h th branch.The automatic determination of d is performedby means of the OSC (see Subsection 2.4) withreference to the kth branch of the section.

3.3.2. Sheari ng str esses or mul t i cel l secti ons. In thecase of multicell sections, the equilibrium eqn (36) isno longer sufficient to determine univocally the shearstress in the section since the problem is staticallyindeterminate.The analysis is thus accomplished by defining mcircuits in the section and by making it open throughthe introduction of longitudinal slits. The continuityof the circuits is restored by writing m compatibilityconditions involving m unknown flows qi, one foreach circuit.

It is evident that the position at which the slitsare introduced can be completely arbitrary, providedthat the connectivity of the section is retained. Suchan arbitrariness, which is the typical drawback ofany computer implementation of the force method,can be very difficult to cope with for automaticcalculations.

Once again the DFS provides a solution to thisproblem since the algorithm determines not only thecircuits of the section, but also it automaticallylocates m slits at the terminal points of the fronds ofthe circuits.

The flow in the k h branch is given by

q(k s) = qo k s) +fkr (39)where fk is the flow in the k h branch due to thecircuit flows q, and qO(k, ) is the flow evaluated, bymeans of eqn (36), on the section made open.

Hence the compatibility conditions, by virtue ofthe relation above, can be rewritten as follows:

s,&=o

i=l,2 ,,.., m, (40)

where AZ collects the indices of the branchesbelonging to the ith circuit, and

s40% s)

cp n,(k, sP(k, ~1ds. (41)


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Automatic analysis of multicell thin-walled sections 651Equation (40) can be re-written as follows:

k~JJi, = 4 i=l,2 ,..., m, (42)where Lk is defined by eqn (12) and b is given by

bi= - (43)

The left-hand side of eqn (42) coincides with theleft-hand side of eqn (14). Hence, introducing thevectors Q and b defined by

(4)

we finally get the linear systemYq=b, (45)

where the coefficient matrix P can be expressed in theform of the eqn (26).Once the vector q has been determined, the shearflow at a given point (6) of a branch will be given by

the algebraic sum of the flows acting in the circuitswhich the branch belongs to plus the quantity qo(S)which is the value of the flow at S resulting from theopen section analysis. Clearly the shear flow in theappendices of the section is given solely by qo(s).

3.3.3. Shear centre. The shear centre may bedefined as the point of the cross-section throughwhich the shearing force should be applied in ordernot to produce twist. The position of the shear centrecan thus be determined by looking for the pointwith respect to which the moment of the shearstress distribution due to an arbitrary shearing forcevanishes.

The equation which determines the position vectorrc of the shear centre C with respect to the elasticcentroid is thus given bysr,-r)xdq*ds=O, (46)cwhere q* denotes an artibrary shear flow distributionover the cross-section.Denoting by k the unit vector orthogonal to thesection, we can also writesrc-r) x q*d*kds =O, (47)c

or equivalently, n = d x k beingnormal to the centre line, we have

the unit vector

s*r, q*ds = s n*rq*ds. (48)f cAs shown in the previous paragraph the flow q* ata generic point of the section is given by the sum ofthe flow qo+ valuated on the section made open byintroducing m slits plus the flow q: resulting from them circuital flows qy which restore compatibility.

Accordingly, eqn (48) becomesI *r,q,*ds + s n-r, qr dsc c= I *rq$ds + s n *r qr ds. (49)c cBy virtue of eqn (36) the first term on the left-handside of eqn (49) can be written as

I -r, qtds= - s (n - rc)(Crt* . so) dsc c= -c,t* . [~c%C3nds]rc,SO)

where t* deontes the shearing force in equilibriumwith q*.Analogously the first term on the right-hand side

becomess= -Cft*.

f(so 0 n) r ds. (51)c

The second term on the left-hand side of eqn (49)can be expressed ass*r,qrds= i f $c k=lwhere f $ ' s the flow in the kth branch, relevant to thedistribution q*, that is the sum of the flows q3 in thecircuits containing the kth branch, and _%k ollectsthe indices of the circuits containing the kth branch.Grouping the terms pertaining to the same circuit,eqn (52) becomess


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652 G. Alfano et al.Ai being the set of branches belonging to the ith Settingcircuit, we can also write

n *rc qf ds = $ q:i= ISince we get

s;=-f@s,=I

P q:=cg* .si.nds=O i=l,2 ,..., m, (55)CC Hence, eqn (57) can be written as(59)

(60)

we have finally

J* s*rqrds= -Cft*. f si P nerdsn * cc qf ds = 0. (56) ,=I c,cBy means of an analogous procedure, the second = -cC,t* .2 2 A,s,, (61)i=lterm on right-hand side of eqn (49) turns out to bes.rqrds= i f i j where use has been made of the relation (seen-rds Appendix)c k =l s

= , q : $ , n * r d s . (57) n*rds =2A,. (62)

Recalling eqns (45) and (36) and denoting by Comparing eqn (49) with eqns (50) (51) (56) and@=Y- we have (61) we get

= Crt* 5 Gii,=I

-C,t*.[l @nds]r,=

(58) - C,t* s(s, 0 n) r ds - Cft* .2 f A+,, (63)c i= 1

x16 I I17 218 3

33.20mFig. 7. Main section of the transatlantic ship France.


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Automatic analysis of multicell thin-walled sections 653or equivalently, since t* is arbitrary Table 2.

(s,,@n)rds +2f A+,. (64)i=,The solution of the linear system above providesthe position rc of the shear centre.Notice that the coefficient matrix in square brack-

ets is singular if and only if the centre line c of thesection is a straight line [151, an occurrence which isruled out for multicell sections.

4. A NUMERICAL EXAMPLEOn the basis of the outlined procedure a computer

program in MATHEMATICA symbolic language hasbeen developed. In this section we present the resultsconcerning a section with a high degree of connectionfor which hand calculations would be prohibitive.

We consider the main section of the transatlanticship France, see Fig. 7, whose data can be found inRef. [7]. The real section has been approximated bya polygonal one since the program works for suchsections with constant thickness and elastic modulifor each segment.

The actions on the section are the following:axial force N = 1.2. 10 (N), bending moment M,=3.0, 1.3) . lo9 (N m), shearing force T = 2.0,2.8)10 (N), torque M, = - 1 l . lo9 (N m), where a

Table 1.Vertex x m) y m)

1 16.002 16.103 16.204 16.355 16.506 16.607 16.008 12.809 6.2510 14.5011 14.0012 12.30

13 6.2014 8.7015 8.7016 - 16.0017 - 16.1018 - 16.2019 - 16.3520 - 16.5021 - 16.6022 - 16.0023 - 12.8024 -6.2525 - 14.5026 - 14.0027 - 12.3028 -6.2029 -8.7030 -8.7031 0.00

0 00-2.75- 5.50-8.75-11.35-14.10- 19.57- 22.95- 24.60- 14.10- 18.97-21.60-22.80-11.35-14.100.00-2.75-5.50-8.75-11.35- 14.10- 19.57- 22.95-24.60- 14.10- 18.97-21.60-22.80-11.35- 14.10-25.00

Segment Vertices 6 m)1 1 16 0.0282 2 17 0.0233 3 18 0.0114 4 19 0.0085 14 29 0.0086 1 2 0.0357 2 3 0.0258 3 4 0.0259 4 5 0.02510 5 6 0.02511 6 7 0.02612 7 8 0.02813 8 9 0.02814 9 31 0.02815 5 14 0.00816 14 15 0.01017 10 15 0.008

18 6 10 0.00819 10 11 0.01220 7 11 0.01221 11 12 0.01822 12 13 0.01823 13 28 0.01824 16 17 0.03525 17 18 0.02526 18 19 0.02527 19 20 0.02528 20 21 0.02529 21 22 0.02630 22 23 0.02831 23 24 0.02832 24 31 0.02833 20 29 0.00834 29 30 0.01035 25 30 0.00836 21 25 0.00837 25 26 0.01238 22 26 0.01239 26 27 0.01840 27 28 0.01841 9 13 0.01842 24 28 0.018

counterclockwise positive direction has been assumedfor the torque.The elastic moduli, constant for the whole sec-

tion, are: E,, = 2.1 10 (N mm2), G, = 1.05. 10(N mm2).

The geometry of the section is defined by thevertices (Table 1) and the segments (Table 2). Theprogram automatically detects in the set of the ver-tices the ones which are nodes, i.e. points where morethan two segments are incident. Accordingly eachbranch is a set of the segments connecting two nodes.

The diagrams of the shear and normal stresses arereported in Fig. 8. The shear stresses are reportednormally to the branches of the section and thenormal stresses are shown by the classical lineardiagram.

The program also calculates the maximum equival-ent stress according to the von Mises yield criterion;it is attained at the point marked with the star, seeFig. 8.


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654 G. Alfano et al

* or= 1.42-10 Nme2x shear centre

Fig. 8. Normal and shear stresses diagrams; shear centre, maximum equivalent stress.

Acknow l edgements-The financial support of the ItalianMinistry for Scientific and Technological Research is grate-fully acknowledged.

REFERENCES

1. F. M. Baron, Torsion of multiconnected thin-walledcylinders. J. appl . M ech. 9, (1942).

2. S. P. Timoshenko and J. N. Goodier, Theory ofElast icity, 2nd Edn. McGraw-Hill, New York (1951).

3. V. Z. Vlasov, Thin-Walled Elastic Beams, 2nd Edn.Israel Program for Scientific Translations, Jerusalem,Israel (1961).

4. J. T. Oden, Mechanics of El asti c Snuctures. McGrawHill, New York (1967).

5. M. Capurso, Sul calcolo delle travi di parete sottile inpresenza di forze e distorsioni, I-V. Ric. Sci. (II-A),(1971).

6. A. Gjelsvik, The Theory of Thin-Walled Bars. Wiley,New York (1981).

7. V. Franciosi, Fondamenti di Scienza dell e Costr uzioni ,Vol. 2. Liguori, Napoli (1985).8. F. Maceri, Contributo al calcolo a torsione delle sezionisottili pluriconnesse. Report of Istituto di Scienza delleCostruzioni, Faculty of Engineering, University ofNaples, 228 (1967).

9. N. Deo, Graph Theory w it h Appl icati ons to Engineeri ngand Computer Science. Prentice-Hall, London (1974).

10. R. Tarjan, Depth-first search and linear graph algor-ithms. SIAM . J. Compur. 1 146-160 (1972).11. G. Roman0 and M. Romano, Sulla deformabilita ataglio di travi in parete sottile. Report of Istituto diScienza delle Costruzioni, Faculty of Engineering,University of Naples, 312 (1983).

12. F. Maceri and R. Sparacio, Sul calcolo a torsione deliesezioni sottili pluriconnesse. Report of Act. di ScienzeFisiche e Mathematiche della Sot. Naz. di. SC. Lett. edArti Napoli, 4, XXX11 (1965).

13. P. R. Halmos, Finite Dimensional Vector Spaces, 2ndEdn. Van Nostrand Reinhold, New York (1958).

14. G. Roman0 and L. Rosati, Sul calcolo delle travi inparete sottile deformabili a taglio. Nel cinquanfenariodell a Facolt d di Archit ett ura di Napoli -Franc0 Jossa e lasua opera, Napoli, Febbraio (1988).15. G. Romano, L. Rosati, G. Ferro, Shear deformabilityof thin-walled beams with arbitrary cross-sections. Int.J. numer. M eth. Engng 35, 283-306 (1992).

APPENDIX

Let us consider a polygonal multicell thin-walled section,that is a thin-walled section with a polygonal centre line,having constant thickness and elastic moduli along each linesegment.The interest for such sections is apparent since any sectionwith curvilinear branches can be suitably replaced by apolygonal one.The parametric equation of the qth segment of thepolygon is

r=r,(A)=(l -l,)r,c,,+ir,c,, 0Gi.S 1, (Al)

where a(q) and b(q) are the indices of the end points of thesegment q and rocqj and rbcqI denote the correspondingposition vectors with respect to the elastic centroid.


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Automatic analysis of multicell thin-walled sections 655Expression of the area of a closed plane polygonal section

Let A be the area of a closed plane curve c, and n theoutward unit normal at a generic point of C. The divergencetheorem yields

where use has been made of the relation div r = 2.obtain

642)We thus

(A3)If c is polygonal, let us denote by i and i + 1 the end

vertices of its ith segment and by r, and r,+, the relevantposition vectors; we assume that i precedes i + 1 accordingto the positive circulation sense assumed for c. Accordingly,introducing the orthogonal tensor

0 1R= [ 11 0the unit normal n, to the ith segment is given by thefollowing formula:

n, = R r,+ - 4Ilr,, , - r, /Iand turns out to be outward or inward whether the positivecirculation sense is counterclockwise or clockwise.

Being the scalar product r en constant along eachsegment, we can write

r *n, = r, Rr,,, -Rr, r, Rr,, ,llr,+ I - 5 II Ilr, + - 5 II WIThe area A is thus provided by the formula

A, tl r;Rr,+,2,=, llr,+,-r,II (A7)

where I, denotes the length of the segment i and n, is thenumber of segments forming the polygonal centre line. Notethat the area A turns out to be positive if a counterclockwisecirculation sense is assumed for the centre line.

Calculus of the second elastic area tensor JLet us denote by cs, I,, 6, and E,, respectively, the centre

line, the length, the thickness and the Young modulus of theqth segment. The second elastic area tensor J, of the qthsegment, with respect to the elastic centroid of the section,is given by

J, = E,s r@rads=A, I r(li) 63 r(n) dl, (A8)CI II

where A, = E$,/, is the elastic area of the qth segment.SettingR ?=r @rI e7) J(4) (A9)

the relation (A8) can be written as follows [15]:

J, = A, I [(l-I)R~+1*R$0+ A(1 - 1)(R$ + Rg)] dlZ. (AlO)

Accordingly, the second elastic area tensor of the wholesection is given by

J= f A[Rz+symR$)+R$)](=I 3 (All)where n, is the number of segments forming the section.Calculus of the first area moment

We now give an explicit formula to calculate the firstarea moment So, with respect to the elastic centroid, ofthe portion of the segment on one side of an imaginarycut at s.

Assuming for the q th segment a co-ordinate system {0, s}with the origin coincident with an end vertex, we get theformula

sS

r&) = Ey r(5 )a, dr = A, [rtiqj + I (rMI, - r+J dl0 Al3

Alfano Et Al - Automatic Analysis of Multicell Thin-walled Sections

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