Engineering ComputationsImplementing slack cables in the force density methodPetros Christou Antonis Michael Miltiades Elliotis

Article information:To cite this document:Petros Christou Antonis Michael Miltiades Elliotis , (2014),"Implementing slack cables in the force densitymethod", Engineering Computations, Vol. 31 Iss 5 pp. 1011 - 1030Permanent link to this document:http://dx.doi.org/10.1108/EC-03-2012-0054

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Implementing slack cables in theforce density method

Petros Christou and Antonis MichaelDepartment of Civil Engineering, Frederick University, Nicosia, Cyprus, and

Miltiades ElliotisDepartment of Mathematics and Statistics, University of Cyprus,

Nicosia, Cyprus

Abstract

Purpose The purpose of this paper is to present a solution strategy for the analysis of cablenetworks which includes an extension to the force density method (FDM) in an attempt to supportcable elements when they become slack. The ability to handle slack cable elements in the analysis isparticularly important especially in cases where the original cable lengths are predefined, i.e. the cablestructure has already been constructed, and there is a need for further analysis to account foradditional loading such as wind. The solution strategy is implemented in a software application.Design/methodology/approach The development of the software required the implementation ofthe FDM for the analysis of cable networks and its extension to handle constraints. The implementedconstraints included the ability to preserve the length in the stressed or the unstressed state ofpredefined cable elements. In addition, cable statics are incorporated with the development of the cableequation and its modification to be able to be handled by the FDM .Findings The implementation of the solution strategy is presented through examples using thesoftware which has been developed for these purposes.Originality/value The results suggest that for cable networks spanning large distances or cableelements with considerable self-weight the neglect of the cable slackening effects is not always conservative.

Keywords Analytical solutions, Cables, Large deflection, Nonlinear analysis, Slack cables

Paper type Research paper

The current issue and full text archive of this journal is available atwww.emeraldinsight.com/0264-4401.htm

Received 20 March 2012Revised 2 August 2013

Accepted 7 August 2013

Engineering Computations:International Journal for

Computer-Aided Engineering andSoftware

Vol. 31 No. 5, 2014pp. 1011-1030

r Emerald Group Publishing Limited0264-4401

DOI 10.1108/EC-03-2012-0054

Nomenclaturexr, yr, zr Vectors that contain the

coordinates of the releasednodes

xf, yf, zf Vectors that contain thecoordinates of the fixed nodes

Dx;Dy;Dzor

u; v;w

9=; Vectors that contain the

projections of the branches onthe coordinate axes

U,V,W Diagonal matrices whichcontain the vectors u, v, w,respectively

L1 Diagonal matrix whichcontains the inverse of thelengths of the branches

t Vector which contains thetensions in the branches

px, py, pz Vectors with the applied nodalloads

D Force Density MatrixD* Fixed Force Density Matrix

External constraint functionGT Jacobian matrixQ Diagonal matrix which contains the

FD of the cable elementsqn 1 Force density of the next iterationqn Force density of current iterationDq Incremental force densitylu Vector that contains the required

unstrained lengths of the constraintbranches

m Vector that contains the currentiteration unstrained lengths of theconstraint branches

Lu Diagonal matrix. Its entries are thevalues of lu

M Diagonal matrix. Its entries are thevalues of m

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1. IntroductionCables distinguish themselves in that they can be packaged in a compact volumeand yet retain the same structural properties as rod elements when extended. Theyfunction as tension truss members when in a stretched state but cannot support anycompressive forces once they become slack. The applications which incorporate cableelements are therefore very interesting and knowledge of their analysis and behavior isa challenge to the structural engineer. This paper is concerned with the analysis ofcable networks which contain cables that can become slack, either during the iterationprocedure of a nonlinear scheme for the determination of the initial equilibrium state,or because of the application of additional loading such as wind. The presented workincludes an extension to the Force Density Method (FDM) in order to handle cableswhich may become slack under compressive loads.

2. BackgroundThe analysis of cable networks is not always easy since traditional methods ofstructural analysis such as the direct stiffness method cannot be applied without anynonlinear modification. The difficulties in the analysis arise from both the material andgeometric nonlinearities of cable networks. The material nonlinearity is due to cableslackening, while the geometric nonlinearity stems from the large displacements andgeometric stiffness due to the applied loads (Mitsugi, 1994). A cable network is differentfrom a conventional structure, such as a space truss, in that its initial shape isstabilized by the prestress in the cables. Analysis methods of such structures includeshape finding and loading analysis. Shape finding of a cable network is a process ofobtaining the initial equilibrium configuration or initial shape, given a set ofrequirements. These requirements may consist such aspects as the connectivitypattern of the structure, its approximate overall dimensions, the support reactions andthe initial stresses in the cables. Loading analysis of a cable network is the process offinding the member stresses and structural deflections under external loads withrespect to the initial shape.

Cable networks can be analyzed using any method developed for other structuresincluding the finite element method as presented by Varum and Cardoso (2005).However, the material and geometric nonlinearities create numerical difficulties insolving the equations created from the analysis of such structures. Several methodswere developed to overcome the aforementioned difficulties and solve the requiredequations including numerical algorithms for computer programming in order toevaluate simpler three dimensional structures (e.g. Hangai and Wu, 1999; Volokh et al.,2003). These methods can be divided in two major categories based on the initial

U ;V ;W Diagonal matrices which containthe vectors u; v; w respectivelywhich, in turn, are vectors thatcontain Dx, Dy, Dz, respectively,for each constraint branch

B Matrix that contains the inverse ofthe branch constants hi (AE)i foreach branch i. The dimensions ofB equal the number of constrainedbranches by the number of allbranches

mni unstrained length for each branch i, ineach iteration n

ln

i Length in the current iteration n of theconstraint branch i

Tn

i Force in the current iteration n of theconstraint branch i

Ai Cross-sectional area of branch iEi Youngs Modulus of elasticity of

branch ihi Constant (AE)i of the constraint

branch i

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assumptions of the analysis. The first category of analysis methods, considers the initialstress-free state as a reference. Methods in this category include various iterative finiteelement methods, beam analogy of the cables, as well as the use of nonlinear constitutiveequations to mathematically denote slackening. In order to obtain the solution Buchholdtand McMillan (1971) and Buchholdt (1985) presented several minimization techniquessuch as minimization of the total potential energy. Mitsugi and Yasaga (1991) presented asimple scheme for cable analysis in which the exact axial strain and a nonlinearconstitutive equation of cable elements were used. The scheme turned out to be quiteuseful because the initial cable length can be specified exactly for the actual hardwarefabrications. This method requires the cable lengths and tensions in the initial stress-freestate as input for the analyses. Similar methods are employed by Yamamoto (1990) andby Miura and Miyazaki (1990). These methods, however, show some problems whendetailed and accurate analyses are required. Mitsugi (1994) presents a nonlinear staticcable analysis method that avoids all the above problems. In his paper the equilibriumequation which virtually provides the unbalanced force at the designated variables andthe stiffness matrix at the current configuration are derived for the Newton iteration.Cables under compressive strain are canceled at the integration for the global equilibriumto account for the effect of slackening. Krishna (1978) has used the Newton-Raphsonmethod and its variations, such as reducing load increments, to solve the nonlinearstiffness equations. Herbert and Bachtell (1986) introduced the Green-Langrange strain ofa cable and solved for the tension state in a cable network in conjunction with thedisplacement fields using Newtons method of iteration. The method successfullyeliminates rigid body motions of the cable elements but the effects of cable slackening arenot explicitly mentioned. Ren (1999) accounted for the cable slackening by considering anequivalent straight chord member with an equivalent modulus of elasticity.

When the final geometry and tension in the equilibrium state in a cable network isknown, it is difficult to back-calculate the initial lengths and tensions in the cables at theloading free state. The presence of large deformations, even if the strains are small, adds tothe complication of the calculations. When the cables constitute a complicated network,which is a typical case, the initial values are even more difficult to be prescribed. Modelinginitial conditions, e.g. initial slack ratio and pre-tension, of a cable is important in nonlinearstructural analysis of complex cable networks, which may have significant effect on theaccuracy in the numerical prediction of a final shape of cable networks subjected toexternal loading. Schek (1974) addressed this modeling challenge by using optimization ofthe initial configurations of the cable network. The difference between the proposed shapeby an architectural viewpoint, and the equilibrium shape under the loading condition, andthe differences between states of real tensions and prescribed tensions by a designer forsome of the elements, are optimized. Optimization is achieved by treating the forcedensities (FD), defined by internal tension of each element divided by its length, asindependent variables. The coordinates of the free nodes are eliminated from theoptimization problem by introducing the FD in the equilibrium equations. Theformulation is efficient when determination of a shape close to the proposed shape underthe major loading is the sole target of the optimization problem.

3. FDMThe implementation of the FDM as described by Schek (1974) is presented in thefollowing paragraphs. The linear FDM is presented first, in 3D Cartesian coordinates,together with a two dimensional (2D) example of a cable network, followed by thenonlinear extension to it to allow for external constraints. This presentation is

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considered necessary to familiarize the reader with the FDM and to aid to theunderstanding of the implementation of the slack cable element which is the mainconcept used in this study.

Force equilibrium equations in 3D Cartesian coordinatesSchek (1974) managed to transform the analysis of general networks into a linearproblem. He only considered that the final network shape should be in an equilibriumstate. The linearization is accomplished by assigning force-to-length ratios to eachmember of the network. These force-to-length ratios proved to be good parameters forthe description of the network and they are referred to as the FD. Any equilibrium statecan be obtained by solving a linear set of equations which are expressed as functions ofthe FD and the external loading. The solution that is provided by the method consistsof the final coordinates of the free nodes and a set of tensions in the cables that arerequired to maintain the cable network in equilibrium.

The FDM essentially solves similar equations for each cable network underconsideration. Obviously the number of equations increases for large cable networkswhich makes the use of the computer necessary. The FDM can be classified as a matrixmethod since the use of matrices for the solution of large systems of equations isessential. The following sections provide a description for the basic matrices that areused as well as the equilibrium equations which are expressed in terms of these matrices.

Branch node matrix [C,C*]. The first step in the FDM is to define the networkconnectivity. For this purpose the branch node matrix, as defined by J.H. Argyris (1964)and by Fenves and Brannin (1963), is incorporated. The branch node matrix consists ofm-rows, equal to the number of branches (cable elements) in the network and n-columns, equal to the number of nodes. The matrix is subdivided into two regions. TheC part of the matrix contains the columns which represent the free nodes while the C*

part contains the columns which represent the fixed nodes. The branch node matrix isa graph of the cable network denoting which branches are connected to which nodes.

Therefore, for branch i, spanning between nodes j and k, the entries in the branchnode matrix are as follows:

ci; j or k 1 when initial node of cable i j1 when end node of cable i k

0 for other nodes not belonging on cable i

* +1

where C(i,j or k) is the coefficients in the branch node matrix; i indicates the row number(cable number); j or k indicates the column number (node number). Once the branchnode matrix is defined then it can be used to calculate the projections of the brancheson the coordinate axes (Schek, 1974):

Dx u Cxr CxfDy v Cyr CyfDz w Czr Czf

2

whereDx;Dy;Dz

oru; v;w

9=; is the vectors that contain the projections of the branches on the

coordinate axes; xr, yr, zr the vectors that contain the coordinates of the released nodes;and xf, yf, zf the vectors that contain the coordinates of the fixed nodes.

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According to Schek (1974) the equilibrium equations at each joint can be expressedin matrix form as shown in the following equation:

CTUL1t pxCTVL1t pyCTWL1t pz

3

where U,V,W is the diagonal matrices which contain the vectors u, v, w, respectively;L1 the diagonal matrix which contains the inverse of the lengths of the branches; t thevector which contains the tensions in the cable elements; and px, py, pz the vectors withthe applied nodal loads.

Substituting with L1t q, Uq Qu, Vq Qv, Wq Qw and the elementprojections (u, v, w) on the axes in the above equations, the equilibrium equations canbe further modified to take their final form which is shown in the following equation(Schek, 1974):

CTQCxr CTQCxf pxCTQCyr CTQCyf pyCTQCzr CTQCzf pz

4

where Q is the diagonal matrix which contains the FD of the branches.Force density matrix (D). The product CTQC is a square matrix which is defined as the

force density matrix, D. The force density matrix has number of rows and columns equalto the number of free nodes. It turns out that the D matrix can be assembled directlyinstead of multiplying the individual matrices. Its entries consist of the FD of the branchesconnected to the free nodes and its form depends on the branch node matrix which isdefined above. The entryD(j,j) on the diagonal is equal to the sum of the FD of the branchesthat are framing into node k; the off-diagonal term D(k,j) is equal to the negative value of theFD of the branch connecting nodes k and j. The D matrix in the FDM is analogous tothe stiffness matrix in the direct stiffness method. In fact, it is actually equivalent to thelinear geometric stiffness matrix for the given structure. Therefore, it is not surprising thatthe two matrices share the same properties; they are both symmetric and positive definite.In the same manner the two matrices can be assembled in a similar way.

Fixed force density matrix (D*). The product CTQC* is defined as the fixed force densitymatrix, D*. The D* matrix has number of rows equal to the number of free nodes andnumber of columns equal to the number of fixed nodes. Its entries consist of the negativevalues of the FD of the branches that connect the free node in the corresponding row to thefixed node in the corresponding column. For example, the term D*(k,j) equals the negativevalue of the FD of the branch that connects the free node k, to the fixed node j .

Once D* is obtained the form of the structure (fixed and free nodes) is defined. Thenan equilibrium shape exists for a given set of FD in each branch and a given set ofpoint loads applied to the released (free) nodes. The equilibrium equations are solvedfor the unknown vectors xr, yr, zr as shown in the following equation (Schek, 1974):

xr D1px Dxf yr D1py Dyf zr D1pz Dzf

5

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The final lengths and forces in each branch are calculated from the final shape of the

cable network using li Dx2i Dy2i Dz2i

q. The element forces can be calculated

using the set of original FD and the equation tLq. Once the equilibrium state of thenetwork is obtained, the original unstrained length of each member can be determinedby using the principles of elasticity theory. These are the original lengths of thebranches to be used during the construction period.

A 2D example of a cable networkConsider the simple cable network of Figure 1. The network consists of three cableelements, two free nodes (node A, node B) and two fixed nodes (node C, node D). Theloading is applied on the free nodes as shown in the Figure 1.

The equations below are necessary to balance the external loading with the internaltensions and keep the structure in equilibrium. Note that the sign in the equilibriumequations is taken care from the differences in the cable end coordinates (Dx, Dy):

Dx1l1

T1 Dx2l2

T2 PxA 6

Dy1l1

T1 Dy2l2

T2 PyA 7

Dx2l2

T2 Dx3l3

T3 PxB 8

Dy2l2

T2 Dy3l3

T3 PyB 9

where Dx1, Dx2, Dx3 is the difference in X-coordinates for each cable; Dy1, Dy2, Dy3 thedifference in Y-coordinates for each cable; l1, l2, l3 the cable length; T1, T2, T3 the cabletension; PxA, PxB the applied load in X-direction at nodes A and B; and PyA, PyB theapplied load in Y-direction at nodes A and B.

In order to keep the two-dimensional structure of Figure 1 in equilibrium, the aboveequations have to be satisfied. The system of equations consists of four unknownswhich are the coordinates (x, y) of the free nodes (Note: the FD ratios Ti/li are assignedfor each cable). The system of equations is linear and since there are four equations itcan be solved for the unknown four coordinates of the free nodes (xA, yA, xB, yB).

For this example matrix C is of dimensions 3 2 and matrix Q is of dimensions3 3. Thus the D matrix of the cable network in Figure 1, is of dimensions 2 2 andhas the following expression:

D q1 q2 q2q2 q2 q3

10

B

DC

PxA

PyA

PyB

PxBA

1

2

3Figure 1.Simple cable network

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The fixed force density matrix D* of the cable network in Figure 1 is as follows:

D q1 00 q3

11

3.3 Nonlinear (extended) FDMMultiple equilibrium shapes can be obtained using the FDM in the way it is presentedso far. This is obtained by assigning different initial values to the FD. However,there are cases where an arbitrary equilibrium shape is not satisfactory becauseit cannot serve its intended purpose. In addition, there are cases where a certainshape is predefined. Therefore, it is very important for one to be able to force theFDM to allow for external constraints, so that a suitable equilibrium shape wouldbe obtained. Schek (1974) presents, The Extended Force Density Method.This method is an extension of the linear FDM and allows for external constraints.The external constraints could be anything that can be expressed as functions of theFD. In addition to that, since the coordinates can be expressed as functions of the FD,any external condition which is expressed in terms of the coordinates, can essentiallybe expressed as a function of the FD. The presence of additional conditions, however,complicate the problem. The additional conditions call for an iterative procedure.The Extended FDM is nonlinear and the degree of nonlinearity is equal to the numberof additional constraints. The Extended FDM is implemented the same way asany other nonlinear method. The set of linear equations is solved and the externalconstraints are checked whether they are satisfied. If they are not satisfied then theFD are updated and the next iteration is executed. The FD are updated usingthe following equation:

qn1 qn Dq 12

where qn 1 is the force density of the next iteration; qn the force density of currentiteration; and Dq the incremental force density.

External constraint function ( ). The external constraint function , has the form:

gx; y; z; q 0 13

and it represents additional external conditions (constraints). As aforementioned theconstraints are expressed as functions of the coordinates and essentially as functions ofthe FD. Thus, the external constraint function takes the form:

gxq; yq; zq; q 0 14

Jacobian matrix (GT). In order to express the coordinates as functions of the FD, theJacobian matrix is utilized which is obtained after applying the chain rule (Schek,1974). Therefore, the Jacobian matrix can be viewed as a characteristic description ofeach additional constraint and it can be obtained as follows:

GT qqq

qgqx

qxqq qgqy

qyqq qgqz

qzqq qgqq

15

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where:

q xq q

D1CTU

q yq q

D1CTV

q zq q

D1CTW

In the present work no damped conditions are considered. By using the definitionsof the external constraints and the Jacobian matrix, given in the previous sections,Dq can be calculated from the following expression which is also described bySchek (1974) in his classical paper:

GTDq r 16where r is the vector that holds the unbalanced external constraints.

Obviously, in many cases the number of the external constraints is less than thenumber of branches in the network. That is, not all the branches have to be constrained.In such cases the system of equations is over determinate and a unique solution does notexist. In order to obtain the best solution for the system, the least squares method and itsvariations are implemented and according to Schek (1974) the following procedure isfollowed to calculate the incremental FD, Dq:

r q0T GTGx T1rDq Gx

17

Once the incremental FD is obtained, the FD are updated and the next iteration of thenonlinear procedure is performed, with the updated values of the FD substituted inthe set of linear equations. It is obvious that when the external constraint function ,and the Jacobian matrix GT are defined, the implementation of an external constraint isquite simple.

Unstrained length constraintThe implementation of the unstrained length constraint is presented here. As it can beseen in a subsequent section the particular constraint is required for the implementationof the slack cable in the FDM. The equilibrium shape that is obtained from the FDM isbased on the equilibrium of forces at each node. The original lengths of the branches canbe back calculated based on the principles of elasticity. Thus, the material properties ofeach branch come into play only after the final shape is obtained. In order to introduce thematerial properties of each element in the analysis, the unstrained length constraint isrequired. This constraint accounts for the strains in the members. The external constraintfunction which is shown in Equation (14) takes the form below:

lu m 0 18

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Differentiating the external constraint function, substituting in Equation (15) andrearranging terms, the Jacobian matrix takes the form of Equation (19) (Schek, 1974)in which all matrices and vectors with the overbar correspond to the constraintbranches of the network:

GT M 2BM 2L3uU CD1CTU V CD1CTV W CD1CTW 19

where lu is the vector that contains the required unstrained lengths of the constraintbranches; m the vector that contains the current iteration unstrained lengths ofthe constraint branches; Lu the diagonal matrix. Its entries are the values of lu ; M thediagonal matrix. Its entries are the values of m; U ;V ;W the diagonal matrices whichcontain the vectors u; v; w respectively, which, in turn, are vectors that contain Dx, Dy,Dz, respectively, for each branch that is constraint; B the matrix that contains theinverse of the branch constants hi (AE)i for each branch i. The dimensions of B equalthe number of constrained branches by the number of all branches; and C the matrixthat contains rows of the branch node matrix which correspond to the constrainedbranches. The dimensions of C equal the number of constrained branches by thenumber of free nodes.

In each iteration a new set of coordinates is obtained which results in new lengthsfor each branch. Based on these lengths, the current branch tensions and the branchmaterial properties, the current unstrained length m for each constraint branch, iscalculated. The required unstrained lengths lui, have known values. This constraintensures that the required unstrained length and the constrained length from the finalsolution are equal (or within certain tolerance). This means that the strain in theelement is consistent with its tension. The unstrained length mni , for each branch i, ineach iteration, n, is calculated as follows:

mni hi

hi Tniln

i 20

where ln

i is the length in the current iteration n of the constraint branch i; Tn

i the forcein the current iteration n of the constraint branch i; Ai the cross-sectional area of branchi; Ei the Youngs Modulus of elasticity of branch i; and hi Constant (AE)i of theconstraint branch i.

4. Cable statics slack cablesCable structures respond in a nonlinear fashion to both pre-stressing and in-serviceforces. Pre-stressing forces are those which exist in a static equilibrium configurationof the structure subjected to static load only. They stabilize the structure and providestiffness against further load. In-service forces on the other hand are those variable liveloads which the structure is expected to encounter during its service life. During theapplication of the FDM the different branches are modeled as linear (straight) elementsand the analysis is obtained from the solution of the equilibrium equations. The resultof the FDM consists of the equilibrium shape of the network and the set of tensionsrequired to maintain the network in equilibrium.

The use of linear (one-dimensional) elements which are used in the FDM impliesthat the self-weight along the cable arc is neglected. Therefore, the linear elementsdo not explicitly model the true cable behavior. Rather, they model truss behavior.

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Depending on the nature of the load (e.g. wind load), it is possible that some of the elementsmay be in compression. Trusses can support compression; however, this is not true forcable elements. In reality, cable elements under compression become slack. This raises thequestion: What happens when a cable becomes slack? If the cable element is consideredto be weightless then the element force is zero. However, this is not true. Leonard (1988)states: The deflected geometry of a cable however, is sensitive to load patterns andmagnitudes. It is therefore, essential to consider the distributed load along the cable arc.

The linear element, which is used in the FDM in this work, is modified to modelcable behavior. This modified element is not able to support any compressive force.The tensile force is a function of the cable element weight which is uniformlydistributed along the cable arc, and the cable span.

The classic catenary solution for the analysis of a cable element subjected touniform load along the cable arc is presented. The tensile force in the cable element dueto the cable weight is obtained in this manner. When a cable supports a uniform loadper unit length, such as its own weight, it takes the form of a catenary. This sectionpresents the classic catenary solution of a cable element subjected to uniform loadalong the cable arc (Leonard, 1988). The different cable parameters, i.e. tension, etc.are expressed in terms of the horizontal component of the cable tension, Ho. Thus, allthe cable related parameters can be calculated when the horizontal component of thetension is obtained. The material is considered to be linearly elastic. The cable itselfis assumed to be perfectly flexible so that the bending moment at any point of the cablemust be zero.

The typical cable (Figure 2) catenary solution is presented by Leonard (1988) and isshown below. The cable element is supported at the cable ends which do not necessarilyhave to be on the same level. The horizontal span is Lc. The load on the cable consists ofits own weight which is uniformly distributed along the cable arc. Also shown on Figure 2is the local coordinate system of the cable and the cable related parameters.

Since the applied loads are vertical, the horizontal component of the cable tensionmust be constant throughout the cable element. Therefore, for the differential length dxin Figure 2 (Leonard, 1988):

dHc

dx 0 ! Hc Ho 21

Also, from the equilibrium of the vertical forces on the same differential length(Leonard, 1988):

dVc

dx wc ds

dx22

where wc is the uniform load along the cable arc.The cable tension must be directed along the tangent of the arc which results to the

following relation between the horizontal and the vertical forces:

Vc Ho dzdx

23The next step is to obtain the catenary equation of the cable profile. The derivative ofthe equation above yields:

dVc

dx d

dxHo

dz

dx

Ho d

2z

dx224

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Also from the differential length in Figure 2, ds dx2 dz2p which yields(Leonard, 1988):

ds 1 dz

dx

2s0@1Adx 25

Finally, the catenary equation is obtained by substituting Equations (24 and 25) inEquation (22). The catenary equation for the deflecting cable profile is shown below:

d2z

dx2 wcHo

1 dz

dx

2s 0 26

The solution of the above equation can be obtained by integrating it twice andapplying the boundary conditions z 0 at x 0 and zLc tan y at xLc. The solutionis shown below (Leonard, 1988):

zx Howc

coshg b cosh g b 1 2 xLc

27

Vc

Hc

wc

Vc +dVcdx dx

Hc +dHcdx dxds

dx

dy

Lc

wc = Load Intensity

L tan

Vo

HoV1

Ho

z

x

dx

ds

d

z

Source: Leonard (1988)

Figure 2.Typical cable elementsubjected to uniformweight along its arc

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where:

b wcLc2Ho

g sinh1 tan y bsinh b

y is the tangent angle at the left end of the element.Once the catenary equation is obtained, all the cable related parameters can be

calculated by direct substitution. The cable tension can be calculated as follows(Leonard, 1988):

T Ho1 dz

dx

2s Ho cosh g b 1 2 x

Lc

28

The sag ratio at the midspan is (Leonard, 1988):

fm 12b

cosh g b cosh g 12tan y 29

Also the vertical component of the force is obtained from (Leonard, 1988):

V Ho sinh g b 1 2 xLc

30

The cable stretched length can be calculated from (Leonard, 1988):

S Vo V1wc

31

where S is the cable stretched length; Vo the vertical force at x 0; andV1 the verticalforce at xLc.

Finally, the unstretched cable length, So, is determined from the differential equation(Leonard, 1988):

dso

dx T=Ho

1 T=AE T

Ho

1 Ho

AE

T

Ho

32

where AE is the cable constant.The solution of the equation above can be obtained by integration. The boundary

conditions so 0 at x 0 and so So at xLc are applied:

So

Lc Vo V1

wcLc wcL

2AE

Ho

wcLc Vo To V1 T1

wcLc 2" #

33

As previously stated, all the cable related parameters are expressed in terms of thehorizontal component of the cable tension, Ho. Based on the current circumstances,Ho can be determined in one of two possible ways. If the sag ratio at the cable mid-spanis specified, the nonlinear Equation (29) can be solved for b and subsequently for Ho.

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On the other hand, if the total unstretched length So is specified, then Equation (33) issolved numerically for Ho.

5. Implementation of the FDM for a slack cableThe classic catenary solution was presented in the previous section. The next step is tomodify the above solution so that it can be applied in the FDM. This section presentsthe modified cable equations as implemented in the FDM (Christou et al., 2010). TheFDM requires that the tension in each cable element is constant along the elementlength. The cable tension that is used in this work is taken as the tension at the startingnode of the cable, i.e. the tension at x 0. Therefore, from Equation (28):

T Ho coshg b 34

Also known from the discussion above the initial tension, T is:

T q S 35

where S is the cable stretched length; and q the force density.Furthermore, if both sides of Equation (31) are multiplied by q then a relation can be

obtained between the tension in the cable (which is assumed to be constant along theelement length) and the FD. Therefore, Equation (31) becomes:

q S q Vo V1wc

36

The relation between the tensions and the FD can be obtained from Equations (35and 36) and it is as follows:

T q Vo V1wc

37

It should be noted that Equation (37) is also expressed in terms of the yet unknown Howhich is the horizontal component of the cable tension. In order to calculate Ho, thetension from Equation (34) is substituted in Equation (37). Equations (37) is rearrangedto yield the equation below:

qVo V1

wc

Ho cosh g b 0 38

Equation (38) can be solved numerically for Ho. When Ho is calculated, then all of thecable parameters can be obtained by direct substitution in the equations that arepresented in the previous section. If the unstrained cable length, as an externalconstraint, is implemented, the unstrained cable length So which is the unstrainedlength that corresponds to the particular FD, is calculated from Equation (33). Also thecurrent tension in the cable can be calculated from Equation (34).

The cable element as shown above is 2D with respect to its local coordinate system.The implemented cable element in the FDM is also 2D even if the cable network thatthe element belongs to is in 3D. Therefore, the local coordinate system as shown in

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Figure 2, is obtained for each cable element and all the cable parameters are expressedin this system.

6. Case studyThe following examples are provided to demonstrate the results of the implementedmethod. The computer program SHAPE was developed by the authors to implementthe technique that is described in this paper. SHAPE provides a number of optionsfor the analysis of such structures. Of interest is the option (option 1) which considersthe network elements as links (trusses) thus allowing them to develop compressiveforces. The second option (option 2) is to treat the network elements as weightlessand thus they are only capable to support tension. If they become slack then the axialload equals zero. The third option (option 3) is to consider the cable weight and thuseven if the cables become slack they still develop tension caused by their ownweight. The implementation and use of these options allow for the study of thedifferences between the solutions. In the following examples the analysis with option 1is used to demonstrate that some elements would become slack (i.e. they areshown to develop compression). Then the same examples are analyzed again usingoption 3 to incorporate slackness. It should be noted that there is no need for themultiple analyses. In actuality option 3 can be used at all times accounting forthe weight of the cables and possible slackness. The reason that the analyseswith option 1 are presented is to identify the slack cables and therefore demonstrate thedifference when option 3 is used. Otherwise the slack cables would not be easy to identifyas the results will only provide elements with tensile forces. In particular example 1 showsa trivial solution which is used to demonstrate in simple terms the presented technique.The same network was analyzed using the commercial software SAP. Example 2 shows amore complex structure which is again analyzed using SHAPE and also the commercialsoftware SAP. The comparison of the results in both examples shows that SHAPEprovides reasonable results at a favorable computational time.

6.1 Example 1A simple cable network which consists of two cables is shown in Figure 3. Each cablehas a prescribed unstrained length of 2 m (6.55 ft). The area (A) is equal to 6.45 cm2

(1 in2) and the Youngs modulus (E) is equal to 200,000 MPa (29,000 ksi). The cableweight per unit length equals to 0.04 KN/m (0.225E-03 kips/in). The structure is loadedat node 1 as shown on Figure 3. The initial geometry of the structure is also shownon Figure 3. If the cable elements were to be modeled as links which can supportcompression (truss elements) the following results would be obtained.

Element forces:Element No. 1 1.000 KN (0.225 kips)Element No. 21.000 KN (0.225 kips)

2.0m (79 in) 2.0m (79 in)

1 23

1 2

P=2.0KN (0.45 kips)

Figure 3.Initial configuration of atwo span cable structure

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The results show that if true cables are used then element 1 is in tension whereaselement 2 is in compression satisfying the condition of equilibrium at node # 2.However, if element 2 was a cable element then it would become slack.

SHAPE is then used with the option to account for the cable weight as well as cableslackening (option 3). The results together with the required analysis time are providedbelow in Table I.

It is worth noting that there is a considerable force in the cables that were supposedto become slack. This force is a function of the cable self-weight which cannot beignored especially if the cable spans are large or if the cable weight is high (as in thecase of this example) or both.

6.2 Example 2Example 2 is of increased complexity compared to example 1. The cable networkconsists of eight cables. The material properties for each cable are the same as thoseof example 1. This network represents a structure which is initially in equilibriumunder the applied vertical loads (Figure 4). SHAPE and SAP were used to obtainthe equilibrium state due to the vertical loads. The coordinates are shown in Table II.Table III shows the internal forces obtained from the SHAPE and SAP analyses.

Once the equilibrium state was obtained then the network was subjected to anadditional horizontal (live) load as shown in Figure 5. As in example 1 the network

Element SHAPE KN (kips) CPU time (SHAPE) sec SAP KN (kips) CPU time (SAP) sec

1 4.529 (1.018) 3.1 102 4.53 (1.018) 2.192 2.529 (0.569) 2.53 (0.569)

Table I.Analysis results fornetwork in Figure 3

2

1

4

5

1kN

1.1k

N

1.1k

N

1kN1kN

1kN8

6

7

3

1.1kN

1.1kN

0.33kN

0.33k

N

0.33kN 0.33

kN

Figure 4.Equilibrium shape

under vertical loading

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elements were modeled as links to demonstrate that some elements are in compressionand if cables would actually become slack. They are also provided here to demonstratethe differences between the results. The coordinates of the equilibrium position areshown in Table IV and the element forces in Table V. The SAP solution was notobtained for this configuration as it was not considered necessary.

Node X (m) Y (m) Z (m)

1 0.0000 0.0000 0.00002 6.0000 0.0000 0.00003 0.0000 6.0000 0.00004 6.0000 6.0000 0.00005 1.5000 1.5000 4.52776 4.5000 1.5000 4.52777 1.5000 4.5000 4.52778 4.5000 4.5000 4.5277

Table II.Coordinates (fromSHAPE) for thenetwork in Figure 4

Element SHAPE KN (kips) CPU time (SHAPE) sec SAP KN (kips) CPU time (SAP), sec

1 1.104 (0.248) 3.1 102 1.106 (0.249) 2.192 1.104 (0.248) 1.106 (0.249)3 1.104 (0.248) 1.106 (0.249)4 1.104 (0.248) 1.106 (0.249)5 0.331 (0.074) 0.334 (0.075)6 0.331 (0.074) 0.334 (0.075)7 0.331 (0.074) 0.334 (0.075)8 0.331 (0.074) 0.334 (0.075)

Table III.Element forces for thenetwork in Figure 4

1

2

1.3kN

1.3kN

3.45kN

3.45kN

0.01

kN

1.3kN

1.3kN

1.65kN

1.65kN

1.03

kN

1kN

1kN6

7

8 1kN

1kN

5

4

3

0.02kN

0.02kN

Figure 5.Equilibrium stage ofnetwork including trusselements (allowingcompression)

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Figure 6 shows the geometry and the forces after the implementation of the slackelement using SHAPE. Figure 7 shows the element forces obtained from the analysesof SHAPE and SAP. The horizontal axis refers to the element number whereas thevertical axis shows the internal forces in KN.

The results obtained from SHAPE for example 2 exemplify the observations ofexample 1. If links (truss elements) are considered in the analysis then a numberof elements would normally become slack (elements under compression) under theapplication of the load. However, the implementation of the slack cable results toall elements being in tension. It is also worth mentioning the difference in thecomputational time between the analyses of SHAPE and SAP.

7. ConclusionsIn this paper a solution strategy for the analysis of cable structures is presented. Cableelements cannot support any compressive forces but rather become slack. In this paperthe implementation of a slack cable in the FDM was presented. The implementation ofthis particular element is necessary to completely model the true behavior of the cableelements which may become slack depending on the nature of the applied loads.The solution strategy is iterative and therefore highly computational which makes theuse of a digital computer necessary.

Two representative examples were also presented comparing the results of theanalysis using links (truss elements) and the implementation of slack elements.

Based on the results presented the following conclusions can be drawn:

(1) The results from the examples presented verify the original assumption thatwhen a cable becomes slack it can still influence the final equilibrium shapeand element forces.

Node X (m) Y (m) Z (m)

1 0.0000 0.0000 0.00002 6.0000 0.0000 0.00003 0.0000 6.0000 0.00004 6.0000 6.0000 0.00005 1.5000 3.7703 2.92176 4.5000 3.7703 2.92177 1.5000 6.1351 4.76788 4.5000 6.1351 4.7678

Table IV.Coordinates (from

SHAPE) for thenetwork in Figure 5

Element SHAPE KN (kips) CPU time (SHAPE), sec

1 3.449 (0.775) 6.25 1022 3.449 (0.775)3 0.016 (0.003)4 0.016 (0.003)5 1.035 (0.233)6 1.650 (0.371)7 1.650 (0.371)8 0.005 (0.001)

Table V.Element forces for the

network in Figure 5

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21.3kN

1.3kN

3.4kN

3.4kN

1kN

1kN

1kN

1kN

0.95k

N

0.09k

N

0.05kN

5

4

3

1

0.05kN

8

7

1.3kN

1.61kN

1.61kN

1.3kN

6

Figure 6.Equilibrium stage ofnetwork includingcable elements(allow cable slackening)

4

3.5

3

2.5

2

1.5

1

0.5

0

SHAP

ESA

P

SHAPE

SHAPE AND SAP RESULTS FOR EXAMPLE 2

SAP

1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8Figure 7.Element vs internal force(KN) from the SHAPEand SAP analyses

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(2) In order to ensure that the analysis of cable networks has a meaningfulpractical value the implementation of an external constraint in the FDM(the unstrained length constraint) was used. The fact that the FDM can beforced to obtain a solution which conforms to the original specifications(original cable lengths, etc.) makes the application of the FDM very attractive.This is especially true in the presence of the additional loading which isapplied on the structure after it is constructed. This method allows for theoriginal cable element lengths to be predefined.

(3) Based on the comparisons from the implementation of the proposed technique(SHAPE) and the commercial software SAP the obtained results compare wellbetween the two programs. The differences in member forces are well below 1percent. In addition, the proposed technique is computationally efficient.

References

Argyris, J.H. (1964), Recent Advances in Matrix Methods of Structural Analysis, Progress inAeronautical Sciences, Pergamon Press, Oxford, London, New York, NY and Paris.

Buchholdt, H.A. (1985), Introduction to Cable Roof Structures, Cambridge University Press,Cambridge.

Buchholdt, H.A. and McMillan, B.R. (1971), Iterative methods for the solution of prestress cablestructures and pinjointed assemblies having significant geometrical displacements,Proceedings of International Association of Shell and Spatial structures, Pacific SymposiumPart II on Tension Structures and Space Frames, Tokyo and Kyoto, pp. 305-316.

Christou, P., Michael, A. and Anastasiou, C. (2010), Accounting for cable slackening andmaterial non-linearity in the analysis of structures with cables, Proceedings of the 9thHSTAM International Congress on Mechanics, pp. 469-475.

Fenves, S.J. and Brannin, F.H. (1963), Network topological formulation of structural analysis,Proceedings of the ASCE, Journal of Structural Division, Vol. 89 No. ST4, pp. 483-514.

Hangai, Y. and Wu, M. (1999), Analytical method of structural behaviours of a hybrid structureconsisting of cables and rigid structures, Engineering Structures, Vol. 21 No. 8, pp. 726-736.

Herbert, J.J. and Bachtell, E.E. (1986), Comparison of tension stabilized structures for large spaceantenna reflectors, Proceedings of 27th AIAA/ASME/ASCE/AHS Structures, SructuralDynamics and Materials Conference, pp. 72-756.

Krishna, P. (1978), Cable Suspended Roofs, McGraw-Hill Inc., New York, NY.

Leonard, J.M. (1988), Tension Structures, McGraw-Hill, New York, NY.

Mitsugi, J. (1994), Static analysis of cable networks and their supporting structures, Computersand Structures, Vol. 51 No. 1, pp. 47-56.

Mitsugi, J. and Yasaga, T. (1991), Nonlinear static and dynamic analysis method for cablestructures, AIAA Journal, Vol. 29 No. 8, pp. 150-152.

Miura, K. and Miyazaki, Y. (1990), Concept of the tension truss antenna, AIAA Journal, Vol. 28No. 6, pp. 1098-1104.

Ren, W.X. (1999), Ultimate behavior of long-span cable-stayed bridges, Journal of BridgeEngineering, Vol. 4 No. 1, pp. 30-37.

Schek, H.J. (1974), The force density method for form finding and computation ofgeneral networks, Computer Methods in Applied Mechanics and Engineering, Vol. 3No. 1, pp. 115-134.

Varum, H. and Cardoso, R.J.S. (2005), A geometrical non-linear model for cable systems analysis,International Conference on Textile Composites and Inflatable Structures, Stuttgart, October 2-4.

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Volokh, K.Y., Vilnay, O. and Averbuh, I. (2003), Dynamics of cable structures, Journal ofEngineering Mechanics, Vol. 129 No. 2, pp. 175-180.

Yamamoto, C. (1990), Shape control of cable net structures, Theoretical and Applied Mechanics,Vol. 39 No. 9, pp. 203-208.

About the authors

Dr Petros Christou is an Assistant Professor in the Civil Engineering Department at the FrederickUniversity in Cyprus. His research interests include the analysis and design of frame and cablesupported systems, computer aided structural analysis, structural modeling and assessment ofstructural integrity, and structural analysis software development. He is also interested in soilstructure interaction and bridge analysis. Besides the research engagement Dr Christou is aRegistered Professional Engineer practicing structural engineering as a Consultant. Dr PetrosChristou is the corresponding author and can be contacted at: p.christou@frederick.ac.cy

Dr Antonis Michael is a Lecturer in the Civil Engineering Department at the FrederickUniversity in Cyprus. His research interests include repair and strengthening of wood andconcrete structures with FRP composites, material characterization, and confinement of concretewith FRP composite grids and load testing of bridges. He is also interested in the experimentaldetermination of forces in cable systems. Besides the research engagement Dr Michael is aRegistered Professional Engineer practicing structural engineering as a Consultant.

Dr Miltiades Elliotis is a Scientific Associate in the Department of Mathematics and Statistics,University of Cyprus. His research interests include the finite element method, boundarymethod, structural analysis, numerical methods and applied mathematics. Besides the researchengagement Dr Elliotis is a Registered Professional Engineer practicing structural engineering.

To purchase reprints of this article please e-mail: reprints@emeraldinsight.comOr visit our web site for further details: www.emeraldinsight.com/reprints

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