subuA ®nite element model for simple straight wire rope strands
Anne Nawrocki *,Michel Labros 1
Department of Mechanical Engineering,Department of Biomedical Engineering,The University of Akron,Akron,OH 44325-0302,USA
月份 英语Received 7February 1998;accepted 17July 1999
Abstract
A ®nite element model of a simple straight wire rope strand is prented,which allows for the study of all the possible interwire motions.The role of the contact conditions in pure axial loading and in axial loading combined with bending is investigated.The model proves to be reliable compared with experimental and theoretical data when available.It appears that the interwire pivoting and the interwire sliding govern the cable respon,respectively,for axial and bending loads.Moreover,pivoting can be considered as free.Finally,the in¯uence of bending on wire tension is studied.Ó2000Civil-Comp Ltd.and Elvier Science Ltd.All rights rerved.
Keywords:FE model;Simple straight strand;Interwire motions;Traction;Bending
1.Introduction
Steel cables were ®rst ud in the German mines of the Harz Mountains in 1836.Since then,they have been widely employed for many di erent applications like,in particular,bridges and prestresd structures.This is why they constitute a natural ®eld of rearch in civil engineering.In France for example,two main types of cables are ud:the spiral strands (Fig.1)which consist of veral layers of helical wires wrapped around a straight core,and the parallel wire bundles (Fig.2),which are ts of parallel strands [1].The former type is especially ud in suspension bridges and the latter is dedicated to cable-stayed bridges and prestresd structures.The basic element of all the cables is a simple straight strand,which is made of a core and one layer of helical wires.Apart from the respon of cables under loading,one criterion vital to the cable design is the fatigue behavior,which is intimately in¯uenced by the contact and friction conditions between the wires.Fretting phenomena are known to drive the initiation pha of cracks in the wires,and this pha reprents most of the cable's life [2,3].Well-conducted cyclic experiments are the usual way of obtaining reliable data about the fatigue characteristics of cables,but they take much time and their costs are prohibitive in many cas.Due to the increasing demand in predicting the cable behavior,veral theoretical models of cables have been built.Hruska's pioneering work [4±7]dates back to the early ®fties.The author work
ed out a simple theory for wire ropes in tension and torsion,considering that the wires are only subjected to pure tensile forces and neglecting the clamping conditions.As a result,he did not deal with the actual contact stress.Since then,many authors (Costello [8],and later,Utting and Jones [9,10])have followed a more fundamental approach,which bene®ts from Love±Kirchho 's theory [11].They treat each wire of cable as a helically curved rod but make di ering assumptions relative to the cable ge-ometry or the interwire contacts.Indeed,Utting and Jones'analysis includes contact deformations and friction e ects,whereas in Costello's such phenomena are neglected.The di erent theories produce results,which remain clo to
the Computers and Structures 77(2000)
345±/locate/compstruc
*Corresponding author now at:Algor,Inc.,150Beta Drive,Pittsburgh,PA 15238-Z932,USA.E-mail address: (A.Nawrocki).1Now at:Heineman Medical Rearch,Inc.,P.O.Box 35457,Charlotte,NC 28235,USA.
0045-7949/00/$-e front matter Ó2000Civil-Comp Ltd.and Elvier Science Ltd.All rights rerved.PII:S 0045-7949(00)00026-2
experimental values prented by Utting and Jones [9,10],but the question of the actual relative displacements and forces within a cable is nevertheless still open.Considering multilayered strands,Hobbs and Raoof [12]have introduced a quite di erent approach in which the characteristics of each layer,including the internal friction phenomena,are homogenized.In this interesting viewpoint,a wire layer is modeled as an orthotropic cylindrical sheet,which requires a large number of wires in each layer.This is why this model is especially appropriate for multilayered strands and is not as preci as the curved rod models to predict the behavior of simple straight strands like 7-wire strands [13].At the beginning of the venties,the ®nite element method was ud for the study of cables by Carlson and Kasper [14],who built a simpli®ed model for armored cables.Then,Cutchins et
al.[15]dealt with the study of damping isolators and Chiang [16]modeled a small length of a single strand cable for geometric optimization purpos.The models u standard volumic ®nite elements,which are not suitable for the study of all the interwire motions (rotations and displacements),and as the accurate modeling of a cable requires a large number of elements,the computational cost is expensive.In order to decrea the computational cost,Jiang et al.[17]propod a conci ®nite element model for cables using three-dimensional solid brick elements,which takes bene®t from the structural and loading symmetries.This model takes into account the combined e ects of tension,shear,bending,torsion,contact,friction and local plastic yielding in axially loaded simple straight strands but cannot be generalized to the ca of bending or more complex loadings.In order to extend the range of applications of the ®nite element models,Durville [18]designed a speci®c ®nite element for cables in large perturbations with interwire friction and deformation of the wire cross-ctions.However,to the authors'knowledge,no experimental validation of the model is available.For the particular ca of bending,the existing models are bad to a greater or lesr extent on simplifying hy-pothes.In general,axial and bending problems are decoupled,except for the fact that interwire pressure is determined by the axial loading.In most of the studies [8,19,20],and although such an assumption should be limited to the bending of a cable over a sheave,a circular bending is considered for reasons of simplicity.Moreo
ver,the whole cable bending sti ness is often taken as the sum of the bending sti ness of each constitutive wire [8,21].Although veral types of loadings are studied throughout the literature,ranging from end moments [8]to uniform hydrostatic pressure [22],the global and local behavior prediction is still an open problem [13].It is to be deplored that very few experimental values related to the testing of simple straight wire rope strands are reported in the literature,except the paper by Utting and Jones [9,10].Such data would be of great interest in the checking of numerical
三月英文programs.
Fig.2.Parallel wire bundles (from Ref.
[1]).
Fig.1.Spiral strand (from Ref.[1]).
346 A.Nawrocki,M.Labros /Computers and Structures 77(2000)345±359
In this context,we have developed a speci®c ®nite element model,which takes account of every possible interwire motion.The knowledge of the contact forces and interwire movements is indeed of obvious interest as a starting point to the study of the fatigue life of cables.As we consider the experimental validation as esntial,we have restricted the applications of our ®nite element model to the simple experiments available in the literature.Nevertheless,complicated loadings can obviously be studied using a thoroughly tested ®nite element model.The prent paper is organized as follows:At ®rst,we prent our cable modeling in which each wire is studied individually.All the possible interwire contact cas are enumerated and investigated,as sliding,rolling and pivoting can occur.We will detail how to take them into account.A variational and then a ®nite element (FE)formulation of the problem are propod.Secondly,the problem of axial loading is addresd.An analysis of the relevance of the interwire contact cas is made through numerical tests in order to predict which interwire motion really drives the strand be-havior.We then compare our results with th
eoretical and experimental data,so as to derive some conclusions about the respon of simple straight strands in axial loading.In the third part,we will study about the strand subjected to axial loading and bending.The interwire motion,which drives the cable behavior in bending will be studied.Finally,an analysis will be made concerning the in¯uence of bending on the distribution of tensile forces in the wires at termi-nations.
2.Notations
In what follows:1.vectors are either denoted by bold symbols or by fÁÁÁg ,and matrices are either denoted by bold symbols or by ÁÁÁ ;2.M T stands for transpo of M ;3.cond order tensors are written with double over-bars (e.g."e );4.the cross-product is noted by and the scalar product is noted ;5.the derivative of a function f with respect to one variable x is noted either d f a d x or f Y x .The di erent variables will be de®ned throughout the paper.
3.Strand model
3.1.Strand geometry:assumptions
The cable investigated in this paper is a simple straight wire rope strand consisting of one layer of cir
cular wires of radius R w wrapped helically over a central circular straight wire (core)of radius R c (e Fig.3).The study is bad on the following assumptions:(a)Only the static behavior of cables is addresd.(b)Displacements and strains are suppod to be small.This assumption is felt to be reasonable:Velinsky [23]has shown that the results from linear and non-linear theories are very clo in the usual practical load
range.
Fig.3.Simple straight wire rope strand.
A.Nawrocki,M.Labros /Computers and Structures 77(2000)345±359347
(c)The wires are made of a homogeneous,isotropic and linearly elastic material.
(d)For each wire,a ction initially normal to the wire centerline remains plane after deformation.This hypothesis
is true especially away from terminations.
(e)Reductions in wire diameter and contact deformations are neglected.Utting and Jones have demonstrated that
in axial loading,the cable extension increas by about2%and the rotation is nearly unchanged when the e ects of Poisson's ratio and wire¯attening are taken into account[9,10].
(f)Outside wires do not touch each other,which is often a design criterion so as to minimize the friction e ects.
Moreover,Huang[24]has shown that even if the helical wires are in contact in the undeformed state,they tend to parate while loading,provided that the core and the outside wires are made of the same material.
(g)Interwire motions occur without friction,which is an initial assumption to establish the®nite element model.
3.2.Theoretical background
Preliminary remark:The straight core can be considered as a particular ca of a helical wire.All the wires are then modeled in the same way.This is why we will only dwell on the study of the helical wires in this ction.
In what follows,the wires are considered like helical beams for which the e ects of shear deformations are taken into account.The wires are modeled with displacement-bad beam elements using a Cartesian isoparametric formulation [25,26].Before treating the FE formulation,every needed theoretical tool(displacement®eld,state of stress,constitutive law)as well as the variational formulation of the contact problem are prented.
3.2.1.Helical wire:virtual work of strains
pay phoneAs shown in Fig.4,the centerline of a helical wire forms a straight single helix of radius R h R c R w and lay angle a (a is positive when the helix is right handed).Let G be a point of the centerline.Its coordinates in the global Cartesian frame R0 0Y X Y Y Y Z are de®ned as
OG
x G
y G
z G
V
told`
X
W
a
Y
R h cos/
R h sin/
R h/
tan a
V
brassiere`
X
W
a
Y Y 1
Fig.4.Centerline of a helical wire.
348 A.Nawrocki,M.Labros/Computers and Structures77(2000)345±359
where /is the polar angle.The vectors t ,n ,b are the tangential,normal and binormal unit vectors along the helix and their components in R 0are t Àsin /sin a cos /sin a cos a V `X W a Y Y n Àcos /Àsin /0V `X W a Y Y b sin /cos a Àcos /cos a sin a
V `X W a Y X 2 Let us consider a wire ction of center G and take M a point of this ction;according to the classical theory of rods,we u the displacement ®eld of Saint Venant
u M u G H GM Y 3 where u G and H denote,respectively,the displacement vector at G and the rotation vector of the cross-ction.In the same theoretical framework,we make u of the linearized Green±Lagrange tensor of deformation de®ned as
"e "h "h T 2Y where "h grad M X 4
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We adopt an anti-plane state of stress,which implies that the stress±tensor matrix in the frame t Y n Y b is mat "r r tt r tn r tb r nt 00r bt 00P R Q S X 5 Using the assumption that the material is homogeneous and isotropic,the constitutive law can be derived from Lam e 's formulñand one obtai
ns
r tt r nt r bt V `X W a Y E 000E 2 1 m 000E 2 1 m
P R Q S e tt c nt c bt V `X W a Y Y 6 where E and m are,respectively,Young's modulus and Poisson's ratio of the cable material.We write the stress±strain law in this simpli®ed manner:
f r
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g D f e g X
7 The virtual work of strains can then be expresd as follows:d V W e d Vhypervisor
留学机构排名V d f e g T f r g d V Y 8
where W e is the strain energy density.
3.2.2.Interwire contacts We have to treat here the contact problem between the outside wires and the core.The contact line is a helix (Fig.5)of radius R c and lay angle a H where tan a H R c a R h tan a as indicated in Fig.6.Let t H Y n H Y b H denote Frenet's frame at a point Q pertaining to the h
elical line of contact.The motions u R Q and H R are,respectively,de®ned as the relative displacement vector at Q and relative rotation vector between a helical wire and the core.In an attempt to explore the in¯uence of the interwire motions at Q Y we shall investigate all the possible cas of interwire contacts,namely a combination of sliding,rolling and pivoting.Sliding is the relative displacement in the tangential plane b H Y t H ,rolling is the relative rotation along t H and pivoting is the relative rotation along n H .The no-sliding condition is u R Q À u R Q Án H n H 0Y the no-rolling condition is given by H R Át H 0and the no-pivoting condition is achieved when H R Án H 0.Let us also remark that two conditions are necessary to ensure a permanent contact between a helical wire and the core.The conditions are u R Q Án H 0and H R Áb H 0,which describe the abnce of relative normal displacement and of relative binormal rotation in a locally linear contact.As a ®rst approximation,a given motion is assumed to be either free (zero friction)or fully prevented.The real behavior is bounded by the two extreme cas.Finally,the eight contact cas,which can possibly occur in a simple straight strand are summarized in Table 1,where R stands for rolling,S for sliding and P for pivoting.A.Nawrocki,M.Labros /Computers and Structures 77(2000)345±359349
