STATIC ANALYSIS OF CABLE STRUCTURE

Applied Mathematics and Mechanics (English Edition), 2006, 27(10):1425-1430CEditorial Committee of Appl. Math. Mech., ISSN 0253 4827STATIC ANALYSIS OF CABLE STRUCTURE *HUANG Yan (黄炎)"，LAN Wei-ren (兰伟仁)2(1. College of Aerospace and Material Engineering, National University of Defense Technology,Changsha 410073, P. R. China;2. Head General Staff Hydrometeorological Center, Bejng 100081, P. R. China)(Communicated by HE Fu bao, Original Member of Editorial Comittee, AMM)Abstract: Based on the nonlinear geometric relation between strain and displacementfor flexible cable, the equilibrium equation under self- weight and infuence of tempera-ture was established and an analytical solution of displacement and tension distributiondefined in Eulerian coordinate system was accurately obtained. The nonlinear algebraicequations caused by cable structure were solved directly using the modified Powell hy-brid algorithm with high precision routine DNEQNE of Fortran. For example, a cablestructure consisting of three cables jointly supported by a vertical spring and all the otherends fixed was calculated and compared with various methods by other scholars.Key words: cable structure; nonlinear deformation; analytical method; displacement;Chinese Library Classifcation: 0343.52000 Mathematics Subject Classification: 74B20; 74K10; 34A34Digital Object Identifier(DOI): 10.1007/s 10483-006- 1015-yIntroductionCable structure is a kind of special engineering structures. Many analytical methods havebeen presented, such as nonlinear finite elementl1- 4] and elastic catenery methods5-9). Allof them are defined in Largrangian coordinate system and realized using Newton-Raphsoniterative algorithm. The process of calculation is troublesome. In this paper, based on nonlineardeformation theory for fexible catenery cable defined in Eularian coordinate systeml10, ananalytical solution is obtained. The nonlinear algebraic equations can be solved directly byroutine DNEQNF1I] in IMSL of Visual Fortran. Routine NEQNF is based on the MINPACKsubroutine HYBRID1, which uses a modification of M.J.D. Powell's hybrid algorithm. As avariation of Newton's method, this algorithm uses a finite-diference approximation to computeJacobian Matrix and takes precautions to avoid large step sizes or error accumulation. Sincea finite diference method is used to estimate the Jacobian, for single precision calculation, theJacobian may be so incorrect that the algorithm terminates far from tree roots. In such cases,high precision arithmetic DNEQNF is recommended. So long as given the appropriate firstguess, then the high precision solution will be obtained, and it will be showed at the followingnumerical examination.1 Basic EquationsConsider an elastic cable element stretched in the中国煤化Ieplaneoftheelement as in Fig.1. The cable is horizontal before:YHCNMHG:line coincides* Received Dec.16, 2005; Revised Apr.27, 2006Project supported by the National Natural Science Foundation of China (No.19872076)Corresponding author LAN Wei-ren, E mail: lan _wr@163.com1426HUANG Yan and LAN Wei-ren1F+dF.dr←i↓qx 卡u>HKX32号(C.D .1400Side viewP23001|4\3Top vicwFig.2 Example of cable structureAccording to the continuous condition of displacement and equilibrium conditions of forceat point A' and assuming that points B, C, D are the origins in local system, it must beun(ln)= Ln-ln，wn(ln)= Wn + wo,n= 1,2,3,(16)-2 Xx(h)cosyhn=0， P-2 Xn(hn)sinyn =Q =Knn-570)=0， (17)中国煤化工Ln=γU呢+V，cosψn=Unsinψn =VTYHCNMHGLn'LnStatic Analysis of Cable Structure1429U1=uo+400，Uz= Uz= uo- 400,V1=v， Vz=v-300， Vs=w + 300, .W1=400，W2= W3= 0,where uo, VO, Wo are the displacement components of A to A' in global system (x, y, z). un(ln),wn(ln) are the displacements at the end Tn = In in local system. From Eqs.(10) and (14) weobtainXn(ln)=Cn，Zn(n)= 9nln + Dn,1+aT√CH+ (gnIn+ Dn2 +9nln+ Dnun(n)=Cn log.In√CH+D+ Dn(最-1),wr(n)= 1+aT[(+(anh+D.jp. -√C+叫+hoht2D-.9n2EAFrom these equations we can obtain the values Cn, Dn, 40, Vo and wo. The displacementcomponents of point A to A' (40 = Xx, Vo = Xy, Wo = Xz), the tension components ofindividual cable in global system (x,y,z) at joint A' and the sag are shown in Tables 1, 2 and3, respectively, and compared with other references.Table 1 Displacement components from A to A'Table 2 Sag of the cable and its positionuo(Xx)vo(Xy)wo(Xz)Cable3This paper26.471541.1385-2.8745fn104.1639.31Ref.[5]26.47341.135- 2.874Ref.[2]26.0237.58- - 2.94n257.07259.94Table 3 Tension components at joint A'Test1(J)2(1)3(I)Cn cos t如n1686.44-437.58- 1284.86Ref.[5}1686.0-437.6- 1284.5Cn sin 4ψn162.68-303.251140.57yRef.[5|162.6- -303.31140.29nln+Dn.1868.55505.86500.11 .Ref.5]1868.1505.9500.1From Table 3 we can conclude that the results of this paper satisfy exactly the equilibriumequation (17) and very near that of Ref.[5].4 ConclusionA completely general analytical method is developed accurately for the static analysis ofcable structures. A general analytical solution for fAexible cable under self-weight and tempera-ture changes is used. The integral constants may be solved hv end mnditions nf the cable. Foreach cable, there are four integral constants. There a中国煤化工is for each end,namely displacement component u or w, and force CfYCNMH Gd z-diretions.At the node points, there are continuous conditions oI alsplacemnent ana equprium conditionsof force. The theoretical analysis is simple. The procedure in calculation is easy and availablein practical engineering.1430HUANG Yan and LAN Wei-renReferences1] Yuan Xingfei, Dong Shilin. A two-node curved cable element for nonlinear analyis[]. EngineeringMechanics, 1999, 16(4):59-64 (in Chinese).2] Hu Song, He Yanli, Wang Zhaomin. Nonlinear analysis of fexible cable structures using the finiteelement method[J]. Engineering Mechanics, 2000, 17(2):36-43 (in Chinese).3] Yang Menggang, Chen Zhengqing. 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