Cable structures are commonly studied with simplified analytical equations. The evaluation of the accuracy of these equations, in terms of equilibrium geometry configuration and stress distribution was performed for standard cables examples. A three-dimensional finite element analysis (hereafter FEA) procedure based on geometry-dependent stiffness coefficients was developed. The FEA follows a classical procedure in finite element programs, which uses an iterative algorithm, in terms of displacements. The theory is based on a total Lagrange formulation using Green-Lagrange strain. Pure Newton-Raphson procedure was employed to solve the non-linear equations. The results show that the rigid character of the catenary’s analytical equation, introduce errors when compared with the FEA

Cardoso, R. J. S.

Varum, H.

English

Repositório Institucional da Universidade de Aveiro

  International Conference on Textile Composites and Inflatable Structures STRUCTURAL MEMBRANES 2005 E. Onãte, and B. Kroplin, (Eds)© CIMNE, Sttugart, 2005     A GEOMETRICAL NON-LINEAR MODEL FOR CABLE  SYSTEMS ANALYSIS H. Varum*,1, R.J.S. Cardoso*,2*Department of Civil Engineering University of Aveiro  3810-193, Aveiro, Portugal E-mail: 1hvarum@civil.ua.pt, 2rcardoso@civil.ua.pt  Key words: Cables, Non-linear FEA, Green strain, Catenary shape.   Abstract. Cable structures are commonly studied with simplified analytical equations. The evaluation of the accuracy of these equations, in terms of equilibrium geometry configuration and stress distribution was performed for standard cables examples. A three-dimensional finite element analysis (hereafter FEA) procedure based on geometry-dependent stiffness coefficients was developed. The FEA follows a classical procedure in finite element programs, which uses an iterative algorithm, in terms of displacements. The theory is based on a total Lagrange formulation using Green-Lagrange strain. Pure Newton-Raphson procedure was employed to solve the non-linear equations. The results show that the rigid character of the catenary’s analytical equation, introduce errors when compared with the FEA. 1 INTRODUCTION Cables are widely used in suspended bridges, transmission lines or membrane panels [5]. These structural elements are used to transmit tensile forces along a specified curve. Cables can be divided in three categories, in accordance with the acting force field: a) Concentrated loads acting on the cable, Fig. 1-a). b) Uniformly distributed load acting along a horizontal line, Fig. 1-b). c) Uniformly distributed load acting along the cable (corresponding to its self-weight, per example), Fig. 1-c). 1   H. Varum, R.J.S. Cardoso  For the design of this type of structural elements two approaches are commonly used. The first one corresponds to the use of analytical equations, based on force equilibrium conditions. The second approach consists in the use on numerical finite element models, based on the field displacement of the nodes. The main issue of this study was to compare results of a non-linear FEA and the analytical equations. In fact, usually when the major action is the self-weight of the cable, catenary’s equations are used. The first and second categories of cables were not analyzed in this study.    Fig. 1- Cable structures. 2 ANALYTICAL EQUATIONS As already stated, the determination of the axial force field and geometry configuration of cable structures are usually performed using analytical equations [5]. Catenary’s equation is used when load is its self-weight. In the following, the analytical equations of the catenary are introduced (see Fig. 2).     Fig. 2- Catenary’s equilibrium configuration.  By the equilibrium conditions, the following equations can be found in [5]: cxsenhcs  =                                                                                                                               (1) 2   H. Varum, R.J.S. Cardoso  cwcT =0x                                                                                                                              (2)                                                                                          (3) w ecoshcy  =with,  and  wyT =                       s  hcoordinate along the cable,c  re, x, y  coordinates, wcurve constant parameter, T cable self-weight, per u it’s lenn gth, ce at catenary’s vertice, m Eqs. (1) and (3), the deformed configuration and axial force  ded of a onnected to one other by hinged connection. The structure is discretized in elements and 0  horizontal axial forT  axial force for a given y value, l  span length.  For a given value of s and frodistribution can be calculated. 3 FEA FORMULATION 3.1 Finite element mesh discretization In order to analyze an  compare the results given by the analytical equations, a series of on-linear FEA were performed.  The cable structure adopted in this study is conceivncontinuous series of discrete elements cn 1+n  nodes, as shown in Fig. 3 [4, 8, 9, 11].   Fig. 3- Cable discretization (nodes and elements).   3   H. Varum, R.J.S. Cardoso  3.2 Basic element description The basic element used to describe a cable structure acted by nodal forces submit to large displacements is represented in Fig. 4. The element has two extremity nodes and three independent orthogonal displacements at each node [1, 2, 8, 9, 11].   4- Basic element. Fig. denotes the initial e the initial configuration. The initial  Length element length, which defin0length is calculated with the nodal coordinates ( ) ( ) ( )2012012010 zzyyxxl −+−+−=                                                                                (4) where 111000  and ,,,, zyxzyx  are the nodal coordinates. }6l  The nodal displacements associated with the element { 321 ,,,, dddde ?=d , represented in ig. 4, define the displacement vector per element, and its components are the three independent displacements in each node. The displacement vector of inition of the deformed cable and the calculation of the Fthe structure, with the ( ) ( ) (initial configuration allows the deflength of each element after deformation. he length l  denotes the length of the element in a deformed configuration, T )26d−+  .                             (5) ,l0312502124011 zdzdydydxdxl −++−−+−−+= Expressions (6), defines the direction of each element in a deformed configuration,  ldxdx x=− lldydy −−+llllldzdz z=−−+= 60313α  where zyx lll   and   ,  define the projections of the length l on the three orthogonal axes. cos−+= 40111cosα y== 50212cosα ,                                         (6) 4   H. Varum, R.J.S. Cardoso  3.3 Equilibrium conditions  Eq. (7) establishes the equilibrium conditions along the three orthogonal directions at each node of the cablcalculated. Since the occurrence of large displacements was considered, the geometry is non-t. Thus, stiffness coefficient and internal forces become geometry dependent, and are fun ioK ∆                                                                                            (7) where trix obtained assembling the stiffness contribution of all the elements, r tive scheme. When the convergence is reached, the  and the nodal forces are equilibrated. The3.3.1 Ce structure. The incremental displacement vector is the variable to be  constanct n of a deformed configuration. K  int ffd −= ext ,                                               tangential stiffness mad  ∆  incremental displacement vector, extf  external forces vector (constant), intf  internal forces vector.  The p oblem is solved, in an iteradeformed configuration is calculated, expressions of the tangential stiffness matrix and internal force vector can be found in Varum and Cardoso [8, 9]. onstitutive law and strain εσ E=The constitutive law assumed in the implemented model is linear elastic (Hooke’s law), expressed by the following expression [3, 7].                                                                                                                                      (8) Since the prothe theory [10], the stiffness coefficients and internal forces were calculated using Green-blem involves large displacements, a total Lagrange formulation is employed in Lagrange strain definition [1, 6, 11]. 2220l=ε .                                04 ITERATIVE PROCEDURE      4.1 Newton-Raphson method 21 ll −                                                                                            (9) n iterative scheme [3, 7]. To start with the computation, initial stiffness and internal rces are required. Thus initial values are needed for the displacement vector, which is accomplished defininAssuming static conditions and linear elasticity, the following equation yields for all the nodes. The iterative procedure implemented to solve Eq. (10) is based on the Pure Newton-Raphsofog an initial geometry and a deformed geometry. 5   H. Varum, R.J.S. Cardoso     ffdK −=∆                                                                iextii int,, id the incremental displacement vector d∆  is computed. A new                                               (10) Eq. (10) is solved andisplacement vector is obtained with the following equation iii ddd  1 ∆+=+                                                                                                                       (11) This new displacement vector together with the initial constant geometry vector, defines the mthod [2, 3, 7, 11] the better is the estimation of the dispomputations are started given an initial configuration, an initial ector increment. The initial conbe usedTheant radius arc defines the itia . c with a slightly larger radius. re computed. 4.defor ed configuration used in the next step.   As usual, in Pure Newton-Raphson melacement vector, the faster will be the convergence to the final solution. 4.2 Computational implementation As already mentioned, the computational implementation of the numerical tool described before follows a classical pattern in finite element programs and use an iterative description of the finite element method. Cstrain vector and a set of parameters, in order to compute a strain vfiguration vector together with the new strain vector defines a deformed configuration to  in the next iteration.  computation procedure is summarised in the following steps: 1. An initial cable’s configuration is imposed, namely a constnodes position and therefore in l length’s cabled∆2. An initial strain vector is defined, considering an ar3. Initial tangential stiffness and internal forces a Eq. (7) is evaluated and vector. The new strain vector is com6. Step 2 is repeated until converge is achieved.  is computed. 5 puted using Eq. (11). nning between two rigid end supports that dist 00 m along an horizontal line, the cable initial geometry is a constant radius arc, Fig 5. A longitudinal elastic modulus of E=200 GPa, was considered. In both examples, the structures were discretized in 20 elements and 21 nodes.  5 NUMERICAL EXAMPLES The numerical examples selected for this study were established in order to compare the results of a geometric non-linear FEA and the analytical equation. In all the analyses, it was adopted a cable spa26   H. Varum, R.J.S. Cardoso   Fig. 5- Cable geometry adopted in the numerical examples. 5.1 Catenary’s equation versus FEA In the set of numerical analyses, the cable in the conditions described in Fig. 5, is subjected to the load corresponding to the self-weight of w= 0.1 kN/m. The area of the cable is constant and equal to = 3.14 cm2. Figs. 6, 7, 8 and 9 show the different cable configurations and stress distributions along the cable, obtained in the FEA and with catenary’s equation, for four different length’s cable.  Ω -6,591842 m-5,0079 m-8,00-6,00-4,00-2,000,002,004,000,00 20,00 40,00 60,00 80,00 100,00 120,00 140,00 160,00 180,00 200,00SPAN (m)241,927 MPa318,24 MPaCatenary´s lenght =200,334 mX 100 MPa319.67 MPa243,821 MPas = 200.3340 m. -10,67025 m-10,1029 m-12,00-10,00-8,00-6,00-4,00-2,000,002,004,009,8132 MPa158,15 MPa 100 MPa  s = 201.3545 m. 20756 MPa748 MPa45,551 Ms = 250.8777 m.  21,798 M a 45,174s = 279.6157 m. 0,00 20,00 40,00 60,00 80,00 100,00 120,00 140,00 160,00 180,00 200,00SPAN (m)y COORDINATE (m) FEACatenary´s eq.14Catenary´s length =201,3545 mXF.E.A length =201,50634 m152,865 MPa161,05 MPa Fig. 7- Cable stress distribution and deformed configuration, y COORDINATE (m) FEACatenary´s eqFEA lenght =200,5768 m  Fig. 6- Cable stress distribution and deformed configuration,-67,12956 m-67,0261 m-80,00-70,00-60,00-50,00-40,00-30,00-20,00y COORDINATE (m)FEACatenary´s eq. Catenary´s length=250,8777 mFEA length=250,9212 m-87,43087 m-87,3206 m-100,00-80,00-60,00-40,00y COORDINATE (m)-20,000,0020,000,00 20,00 40,00 60,00 80,00 100,00 120,00 140,00 160,00 180,00 200,00SPAN (m)FEACatenary´s eq. Catenary´s length=279,6157 mF.E.A length=279,6611 m-10,000,0010,000,00 ,00 40,00 60,00 80,00 100,00 120,00 140,00 160,00 180,00 200,00SPAN (m)26,26,x 10 MPa45,36 MPaPaFig. 8- Cable stress distribution and deformed configuration, P  MPax 10 MPa Fig. 9- Cable stress distribution and deformed configuration,7   H. Varum, R.J.S. Cardoso   From Figs. 6, 7, 8 and 9, it can be observed that the smaller is the initial length of the cable, higher is the stress level. The differences in the deformed configuration and stress level obtain by FEA and Catenary’s equation also depends on the cable length or stress level. Figs. 10, 11 and 12 illustrate the differences obtained in maximum stress, minimum stress and maximum y coordinate, using the FEA and Catenary’s equation. 0,0050,00100,00150,00200,00250,00300,00350,00200,334 201,3545 250,8777 279,6157Cable length (m)Stress (MPa)FEACatenaryFig. 10- Maximum cable stress in the support. 0,0050,00100,00150,00200,00250,00300,00350,00200,334 201,3545 250,8777 279,6157Cable length (m)Stress (MPa)FEACatenary Fig. 11- Minimum cable stress at the vertice. -100-90-80-70y -60-50-40-30-20-100200,334 201,3545 250,8777 279,6157Cable length (m) coordinate (m) FEA Catenaryble geometry a function of it. sponsible for a subsequent elastic deformation. Results from the FEA and analytical expressions are approximately equal, when  Fig. 12- Maximum y coordinate at mid-span. 6 CONCLUDING REMARKS In this research, the theoretical formulation, finite element implementation and numerical validation of cable structure are presented to compare FEA and analytical equations results. The results obtained from the analyses, shows that the solution of a non-linear finite element program are quite different from those given by the analytical expressions. The analytical expressions are based on a rigid deformed configuration, being the stress distribution and caWhen comparing the FEA results with catenary´s equation, it was concluded that the length of the cable controls the stress level. The stress level is re8   H. Varum, R.J.S. Cardoso  the maximum stress in the cable is less than 50 MPa, being the error less than 1%. For f specially for medium/high stress levels. TheRE[1] YRIS, [et al.], ‘Geometric Nonlinearity and the Finite Element Displacement  Rheological Models and Numerical Integration ty of Coimbra (in Portuguese), (2001). [8] CARDOSO ‘Finite element simulation of cables behaviour versus [9] DOSO ‘Modelo não linear geométrico para a análise de estrutras de cabos’ submitted to Revista Portuguesa de Engenharia de Estruturas, Portugal, (2005). [10] W. ZHANG, J.L. LEONARD, M.L. ACCORSI, ‘Analysis of geometrically nonlinear anisotropic membranes: theory and verification’, Finite Elements in Analysis and Design 41,963-988, (2005). [11] O.C. ZIENKIEWICZ, R.L. Taylor, The Finite Element Method, MacGraw-Hill, Vol. 2, Fourth Edition, (1989).  maximum stress levels of 150 MPa, or higher, the error is approximately 6% and easily reach 30 % (300 MPa). Finally, it is remarked that geometric non-linear analysis becomes necessary in the studies ocable structures, eAKNOWLEDGMENTS   authors would like to thanks the precious help of Professor Victor Dias da Silva, from University of Coimbra, in Portugal, for the theoretical developments included in this article.  FERENCES   J.H. ARGMethod’, in ISD Lectures on Numerical Methods in Linear and Nonlinear Mechanics, Jablona 22-28 September 1974. ISD Report N. º 174 Stuttgart, (1974). [2] J.H. ARGYRIS, I. DOLTSINIS, V.D. SILVA, ‘Constitutive Modelling and Computation of Non-Linear Viscoelastic Solids – Part I:Tecnhiques’, in Computers Methods in Applied Mechanics and Engineering, Vol. 88, Nº 2, (1991). [3] R.J.S. CARDOSO, ‘Regularização Visco-elástica de Problemas Elasto-plásticos com Amaciamento’, M.Sc Thesis, FCTUC, Universi[4] M.A. CRISFIELD, Non-linear Finite Element Analysis of Solids and Structures. Volume 1, Essentials, John Wiley & Sons, (1991). [5] D.L. SCHODEK, Structures, Fourth Edition, Prentice Hall, Inc., Upper Saddle River, New Jersey (2001). [6]  V.D. SILVA, Mecânica e Resistência dos Materiais, 3ª edição, 452 pags., edição do autor, Coimbra ISBN: 972-98155-1-8, (2004). [7]  H. VARUM, ‘Modelo Numérico para a Análise Sísmica de Pórticos Planos de Betão Armado,’ M.Sc Thesis, FEUP, University of Porto  (in Portuguese), (1995). H. VARUM, R.J.S. analytical equations’ submitted to Finite Element Analysis and Design, (2005). H. VARUM, R.J.S. CAR9  

A geometrical non-linear model for cable systems analysis

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