research article determination of the strain-free...

Research ArticleDetermination of the Strain-Free Configuration ofMultispan Cable

Chuancai Zhang1 Qiang Guo12 and Xinhua Zhang1

1State Key Lab for Mechanical Structural Strength and Vibration Xirsquoan Jiaotong University Xirsquoan Shannxi 710049 China2Shanghai Electric Power Generation Equipment Co Ltd Shanghai Turbine Plant Shanghai 200240 China

Correspondence should be addressed to Xinhua Zhang xhzhangmailxjtueducn

Received 16 March 2015 Revised 25 May 2015 Accepted 26 May 2015

Academic Editor Tai Thai

Copyright copy 2015 Chuancai Zhang et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

For building a reasonable finite element geometric model a method is proposed to determine the strain-free configuration ofthe multispan cable The geometric conditions (the end conditions and the unstretched length conditions) as constraints for theconfiguration of multispan cable are given Additionally asymptotic static equilibrium conditions are given for determining theasymptotic shape of the multispan cable By solving these constraint equations a set of parameters are determined and then thestrain-free configuration of multispan cable is determined The method reported in this paper provides a technique for buildingreasonable FEA geometric model of multispan cables Finally a three-span cable is taken as example to illustrate the effectivenessof the method and the computed results are validated via the software ADINA

1 Introduction

The cable structures are widely used in the engineering suchas bridge building and power transmission system [1 2] In[1] a new forcemethod is proposed for analysing the dynamicbehaviour of flat-sag cable structures The accepted dynamicmodel of such cables reduces to a partial differential equationand an integral equation The support reaction forces areconsidered as excitations allowing DrsquoAlembertrsquos solution tobe used In this way single or multispan cables have beendeveloped in the form of a single-degree-of-freedom systemin terms of the additional dynamic tension Finally twoexamples are presented to illustrate the accuracy of the pro-posed force method for single and multispan cable systemssubjected to harmonic forces In [2] a three-dimensionalmodeling procedure is proposed for cable-stayed bridgeswith rubber steel and lead energy dissipation devices Inthis study the cable structure can be simplified as a four-node isoparametric cable element The multispan cables andtowers are the basic structures in the power transmissionsystem Under the wind load the multispan cable is apt to

generate large displacements because of its high flexibility andlarge span So themultispan cable exhibits strong geometricalnonlinearity Usually the dynamic behavior of such cablestructures is analyzed on the platform of the finite elementanalysis (FEA) software To build a reasonable FEA modeltwo alternative configurations of the cable structures needto be determined in the modeling process one is the initialstatic equilibrium configuration and the other is the assumedstrain-free configuration As a prestressed state the staticequilibrium configuration of a cable is the starting pointof the subsequent dynamic analysis whereas the strain-freeconfiguration of a cable is only an assumed state which doesnot exist in the real world due to the ubiquitous gravity andit is the starting point for static and dynamic analyses If theFEA geometric model is built based upon the static equi-librium configuration of the cable the prestrain or prestressin each element caused by gravity should be specified oneelement by one element this is always cumbersome Thegravity load is included in the equilibrium configurationTheFEA geometric model contains two gravity loads in this wayso that the model does not match the actual cable structure

Hindawi Publishing CorporationShock and VibrationVolume 2015 Article ID 890474 6 pageshttpdxdoiorg1011552015890474

2 Shock and Vibration

The gravity load is not included in the strain-free configura-tion Alternately if the geometric model is built based on thestrain-free configuration the prestrain or prestress in eachelement needs not to be specified in advance and they areset up naturally in the nonlinear static analysis via the FEAAfter that the gravity load applied in the model the systemachieves equilibrium Therefore the assumed strain-freeconfiguration ismore preferable to the static equilibriumcon-figuration for building the FEAgeometricmodel ofmultispancable

For the static and dynamic analysis of cable structuresIrvine [3] systematically summarized the classical achieve-ments reached up to 1981 and the static equilibrium shape ofa cable is described as an elastic catenary For the modelingof cable structures in [4] a method for modeling cablesupported bridges for nonlinear finite element analysis ispresented in this paper A two-node catenary cable elementderived using exact analytical expressions for the elasticcatenary is proposed for the modeling of cables Based uponthe elastic catenary Such et al [5] proposed a method fordetermining the static equilibrium configuration of arbi-trary three-dimensional cable structures subject to gravityand point loads Srinil et al [6 7] formulated a systemof nonlinear partial differential equations describing thelarge amplitude three-dimensional free vibrations of inclinedsagged elastic cables Ai and Imai [8] proposed an iterativescheme to find the static equilibrium shape of beam-cablemixed system In [9] based on exact analytical expressionsof elastic catenary Thai and Kim presented a catenary cableelement for the nonlinear static and dynamic analysis ofcable structures In [10] Vu et al presented a spatial catenarycable element for the nonlinear analysis of cable-supportedstructures and proposed an algorithm for form-finding ofcable-supported structures Rienstra [11] derived the partialdifferential equations and boundary conditions satisfied bythe static equilibriumconfiguration ofmultiple spans coupledvia suspension strings In [12] Impollonia et al obtained thedeformed shape of the elastic cable in closed form for thecases of uniformly distributed load andmultiple-point forcesGreco and Cuomo [13] proposed a method for obtaining anexact configuration of slack cable nets by using the exactexpressions of the equilibrium derived from the equation ofthe catenary

All the above papers focused on the determinationof static equilibrium configuration of the cable structuresHowever the present paper concentrates on determiningthe strain-free or unstretched configuration of multispancable Firstly based upon the elastic catenary theory aset of geometric boundary conditions cable length con-dition and static equilibrium conditions are derived fordetermining the configuration of the multispan cableswhen they are approaching asymptotically to the strain-free state These constraint conditions constitute a sys-tem of nonlinear algebraic or transcendental functionequations Secondly these nonlinear equations are solvedand one finds a set of parameters which determine theunstretched configuration of the multispan cable Finallyan example of three-span cable is given to illustrate themethod

2 The Governing Equations ofMultispan Cable

First we assume that (1) the bending stiffness of cable isnegligible (2) the material of cable is of Hookian type thatis its constitutive relation is linear elastic and (3) only theself-weight as the static force is applied to the cable structureUnder these assumptions the governing equation for thecable in its static equilibrium state is as follows [6]

( 1198641198601205760radic1 + 119910101584020 )1015840

= 0

( 119864119860120576011991010158400radic1 + 119910101584020 )1015840

+ 120596119888radic1 + 119910101584020(1 + 1205760) = 0

(1)

Here the ldquo 1015840rdquo denotes the derivative with respect to 119909 1205760denotes the tensional strain in the cable when it is in staticequilibrium state 119864 is Youngrsquos modulus 119860 is the sectionalarea of the cable and assumed to be constant 1199100 representsthe static equilibrium shape of the cable and 120596

119888is the cable

weight per unit length in the unstretched stateThe tensional strain 1205760 is related to the horizontal ten-

sional force 119879119867as follows

1205760 = 119879119867119864119860radic1 + 119910101584020 (2)

According to (1) the final static equilibrium governingequation of the cable can be rewritten in the following form

(1+ 119879119867119864119860radic1 + 119910101584020 )119910101584010158400 + 120596

119888119879119867

radic1 + 119910101584020 = 0 (3)

This is the so-called elastic catenary equation Being differentfrom the traditional catenary equation it takes account ofthe extensibility of the cable Under the dynamic wind loadthe multispan cable usually vibrates nonlinearly due to itslarge displacement so the finite element analysis is preferableto investigate the static and dynamic behavior To build theFEA model firstly the geometric model of multispan cableis needed Therefore one of the initial configurations thestrain-free state or the static equilibrium state should bedetermined As stated in the Introduction in the followingwe only discuss how to determine the strain-free configura-tion of the multispan cable

3 Determination of the Strain-FreeConfiguration

Determining the strain-free configuration of the cable basedon the data of static equilibrium configuration can be viewedas an inverse problem in nonlinear structural mechanicsThe unstretched state of cable is an assumed state andit can be viewed as an asymptotic state that correspondsto 119864119860 rarr infin (which means the cable is inextensible)

Shock and Vibration 3

Unstretched state

Equilibrium state

y

H1

A

O

la

L1

ha hbH2

r1

L2 L3

r2

H3

CBlb

D

x

H4

P2P1

P9984002P998400

1

Figure 1 The configuration of three-span cable

and 120596119888119879119867

rarr minus119886 (note that 120596119888is negative) so the equation

for cable in unstretched state is as follows

11991010158401015840 minus 119886radic1 + 11991010158402 = 0 (4)

This is the rigid catenary equation Its solution can be foundas

119910 (119909) = 1119886 cosh 119886 (119909 + 1198881) + 1198882 (5)

For the three-span cable as illustrated in Figure 1 theshape of the left span cable in its unstretched state can beexpressed as

1199101 (119909) = 11198861 cosh 1198861 (119909 + 11988811) + 11988812 (6)

Similarly the shape of themiddle span cable can be expressedas

1199102 (119909) = 11198862 cosh 1198862 (119909 + 11988821) + 11988822 (7)

and the shape of the right span cable is expressed as

1199103 (119909) = 11198863 cosh 1198863 (119909 + 11988831) + 11988832 (8)

There are 9 unknown parameters appeared in the aboveexpressions Determination of the strain-free configurationof the three-span cable is equivalent to determining theseunknown parameters so the following constraint equationscan be built

For the left span cable

End conditions1199101 (119909)1003816100381610038161003816119909=0 = 11986711199101 (119909)1003816100381610038161003816119909=119897

119886

= ℎ119886(1198711 minus 119897119886)2 + (1198672 + 1199031 minus ℎ119886)2 = 11990321

(9)

Here the meaning of the first and second equations of (9)is obvious In engineering the rod 1198601198751 (insulator) is verystiff so it can be reasonably assumed as rigid therefore thethird equation of (9) holds too Figure 1 can be referred to forthe meaning of the symbols 1198711 1198671 1198672 119897119886 ℎ119886 and 1199031 in theabove equations In addition the following condition aboutthe length of the original unstretched cable can be built

Length conditionint1198971198860radic1 + 119910101584021 119889119909 = 11990401 (10)

Here 11990401 represents the length of the left span cable in itsunstretched state and can be expressed via the data of thestatic equilibrium configuration

It should be pointed out that in some cases (eg in theexample given later) the original length 11990401 of the cable in itsunstretched state may be less than the straight distance 1198601198751so in order to build a reasonable FEA geometric model theposition of the hanging point 11987510158401 must be determined

Note that the function 1199101(119909) is in the form of cosh theabove integral actually can be given explicitly To find thelength 11990401 the following equations can be constructed (see[3] page 18)119879

11986711990401119864119860 + 119879

119867119904011198821

sinhminus1 ( 1198811119879119867

)minus sinhminus1 (1198811 minus1198821119879

119867

) = 1198711119882111990401119864119860 ( 11988111198821

minus 12)+ 119879119867119904011198821

[1+( 1198811119879119867

)212

minus1+(1198811 minus1198821119879119867

)212] = 1198671 minus1198672

(11)

where1198821(= 12059611988811990401) is the self-weight of the left span cable and119879

119867and1198811 are the horizontal and vertical reaction forces at the

support point 119860 respectively From the above two equationsthe unknown 1198811 and 11990401 can be solved

Being similar to (9) for the middle and right span cablesthe following constraint equations can also be constructedFor the middle span cable

End conditions1199102 (119909)1003816100381610038161003816119909=119897

119886

= ℎ1198861199102 (119909)1003816100381610038161003816119909=1198711+1198712+1198713minus119897119887 = ℎ

119887(1198713 minus 119897119887)2 + (1198673 + 1199032 minus ℎ119887)2 = 11990322

Length conditionint1198711+1198712+1198713minus119897119887119897119886

radic1 + 119910101584022 119889119909 = 11990402(12)

For the right span cable

End conditions1199103 (119909)1003816100381610038161003816119909=1198711+1198712+1198713minus119897119887 = ℎ

1198871199103 (119909)1003816100381610038161003816119909=1198711+1198712+1198713 = 1198674 (13)

Length conditionint1198711+1198712+11987131198711+1198712+1198713minus119897119887

radic1 + 119910101584023 119889119909 = 11990403 (14)


Again Figure 1 can be referred to for the meaning of thesymbols11986731198674 1198712 1198713 119897119887 ℎ119887 and 1199032 in the above equations11990402 and 11990403 represent respectively the lengths of middle andright span cables in their unstretched state

Similar to (11) the determining equations for 11990402 are asfollows119879

11986711990402119864119860 + 119879

119867119904021198822

sinhminus1 ( 1198812119879119867


119867

) = 1198712119882211990402119864119860 ( 11988121198822

minus 12)+ 119879119867119904021198822

[1+( 1198812119879119867

)212

minus1+(1198812 minus1198822119879119867

)212] = 1198673 minus1198672

(15)

and the equations for 11990403 are as follows11987911986711990403119864119860 + 119879

119867119904031198823

sinhminus1 ( 1198813119879119867


119867

) = 1198713119882311990403119864119860 ( 11988131198823

minus 12)+ 119879119867119904031198823

[1+( 1198813119879119867

)212

minus1+(1198813 minus1198823119879119867

)212] = 1198673 minus1198674

(16)

Besides (9) (10) and (12)ndash(14) two other equations describ-ing the equilibrium conditions of forces in the limiting statefrom 119875

1rarr 11987510158401and 119875

2rarr 11987510158402are given as follows

At point 119875 (T1 +T2) sdot 120591 = 0 (17)

Referring to Figure 2 the unnormalized directional vector ncan be written in its coordinate-component form as

n = (119897119886minus1198711 ℎ119886 minus (1198672 + 1198971)) (18)

and the unnormalized directional vector 120591 can also be givenin its coordinate-component form as

120591 = (ℎ119886minus (1198672 + 1198971) 1198711 minus 119897119886) (19)

The tensional forces T1 and T2 in the left span and middlespan cables can be written in their component form as

T1= minus 1003817100381710038171003817T11003817100381710038171003817 ( 1radic1 + 11991010158402

1

11991010158401radic1 + 119910101584021

)10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119909=1198831198751015840

1

T2= minus 1003817100381710038171003817T21003817100381710038171003817 ( 1radic1 + 11991010158402

2

11991010158402radic1 + 119910101584022

)10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119909=1198831198751015840

1

(20)

T1

T2

Tn

P

B

120591

n

Figure 2 The forces at the suspension point

On the other hand in the limit case (120596119888rarr 0) the following

relations hold 1205961198881003817100381710038171003817T11003817100381710038171003817 radic1 + 119910101584021 = minus 119886

1

1205961198881003817100381710038171003817T21003817100381710038171003817 radic1 + 119910101584022 = minus 119886

2 (21)

Substituting these relations into (13) one finally gets

( 11198862 minus 11198861) (1198672 + 1198971 minus ℎ119886)+ (119897119886minus1198711) (119910101584021198862 minus 119910101584011198861 )

100381610038161003816100381610038161003816100381610038161003816119909=119897119886

= 0 (22)

Similar to (22) at point 119862 the following relation holds

( 11198863 minus 11198862) (1198673 + 1198972 minus ℎ119887)+ (1198713 minus 119897119887) (119910101584031198863 minus 119910101584021198862 )

100381610038161003816100381610038161003816100381610038161003816119909=1198711+1198712+1198713minus119897119887 = 0 (23)

Theoretically the strain-free configurations of cable struc-tures with any number of spans can be determined in thesame way as described above On the other hand there areinfinity of many strain-free configurations for the multispancables each of them satisfies the geometric constraints thatis the end conditions and the length conditions By imposingthe force equilibrium conditions we obtain actually theasymptotic strain-free configuration of the cable In a sensesuch an asymptotic configuration (from the static equilibriumstate to the strain-free state) is optimal because it is mostsimilar to the static equilibrium shape of the cable so ifone performs the FEA starting from this configuration theconvergence is expected to be most rapid

For determining the configuration of the three-spancable in its unstretched state totally there are 13 unknownsincluding 119886

119894 119888119894119895(119894 = 1 sdot sdot sdot 3 119895 = 1 2) and ℎ

119886 ℎ119887 119897119886 and 119897

119887


Table 1 The parameters of three-span cable

Parameter Value1198671

401198672

2061198673

2101198674

451198711

5801198712

19101198713

570119879119867

91378119864 63119890 + 10119860 66655119890 minus 4120588 45241120596119888

290668

Table 2 The solution of three-span cable


02591119890 minus 31198862

02791119890 minus 31198863

02590119890 minus 311988811

802425911988812

minus3903925711988821

minus1527592811988822

minus3502726711988831

3879889111988832

minus39035631ℎ119886

2060680ℎ119887

2100671119897119886

5788362119897119887

19123203

and 13 equations including (9)-(10) (12)ndash(14) and (22)-(23)Because of the complex nonlinearity these equations can onlybe solved numerically In the next section an engineeringexample is given as illustration

4 An Example

To illustrate the proposedmethod for determining the strain-free configuration of multispan cable in this section a realthree-span transmission line system is taken as example itsdata are listed as shown in Table 1

Substituting these data into the above equations and thenusing the Newton-Raphson method one can find a set ofsolutions as shown in Table 2

The unstretched lengths of the three span cables arefound to be 11990401 = 6027090463 11990402 = 1935110206 and11990403 = 5928032497 respectively Knowing the values of theseparameters the asymptotic configuration of the three-spancable in its strain-free state can be described by (6)ndash(8)

Via the above computed results the original unstretchedconfiguration of the three-span cable can be easily plottedas shown in Figure 3 Starting from this configuration the

X

Y

Z

Unstretched configurationStatic equilibrium configuration

ADINA

Figure 3The unstretched and the static equilibrium configurationsof the three-span cable

Figure 4 The magnification of local configuration near the leftinsulator

geometric model for FEA can be easily built and the cor-responding static equilibrium configuration can be furtherdetermined by using FEA software for example the ADINAThe static equilibrium configuration via the ADINA is alsoshown in the same figure The largest difference in heightbetween the unstretched and the static equilibrium config-urations is about 1775m

On the other hand to see clearly the difference betweenthe original unstretched and the static equilibrium configura-tions a zoom-in of the local configurations of the three-spancable near the left insulator is shown in Figure 4

5 Conclusions

In this work a method is presented for determining theasymptotic strain-free configuration of the multispan cableThe constraints of the geometric conditions and the asymp-totic static equilibrium conditions are derived and the deter-mining equations for the asymptotic strain-free configurationare obtained A set of parameters which describe the shape ofthe multispan cable are determined The proposed methodprovides a way to build reasonable FEA geometric model ofmultispan cables At last a three-span cable system is takenas example to illustrate the effectiveness of the method theresults are validated via the ADINA

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper


References

[1] X Ma and J W Butterworth ldquoA force method model fordynamic analysis of flat-sag cable structuresrdquo Shock and Vibra-tion vol 16 no 6 pp 623ndash635 2009

[2] H M Ali and A M Abdel-Ghaffar ldquoSeismic passive control ofcable-stayed bridgesrdquo Shock andVibration vol 2 no 4 pp 259ndash272 1995

[3] H M Irvine Cable Structure MIT Press Boston Mass USA1981

[4] R Karoumi ldquoSome modeling aspects in the nonlinear finiteelement analysis of cable supported bridgesrdquo Computers andStructures vol 71 no 4 pp 397ndash412 1999

[5] M Such J R Jimenez-Octavio A Carnicero and O Lopez-Garcia ldquoAn approach based on the catenary equation to dealwith static analysis of three dimensional cable structuresrdquoEngineering Structures vol 31 no 9 pp 2162ndash2170 2009

[6] N Srinil G Rega and S Chucheepsakul ldquoLarge amplitudethree-dimensional free vibrations of inclined sagged elasticcablesrdquo Nonlinear Dynamics vol 33 no 2 pp 129ndash154 2003

[7] N Srinil G Rega and S Chucheepsakul ldquoThree-dimensionalnon-linear coupling and dynamic tension in the large-amplitude free vibrations of arbitrarily sagged cablesrdquo Journalof Sound and Vibration vol 269 no 3ndash5 pp 823ndash852 2004

[8] M Ai and H Imai ldquoA shape-finding analysis of suspendedstructures on the displacement-method equilibriumrdquo StructuralEngineeringEarthquake Engineering vol 21 no 1 pp 27ndash412004

[9] H-T Thai and S-E Kim ldquoNonlinear static and dynamicanalysis of cable structuresrdquo Finite Elements in Analysis andDesign vol 47 no 3 pp 237ndash246 2011

[10] T-V Vu H-E Lee and Q-T Bui ldquoNonlinear analysis of cable-supported structures with a spatial catenary cable elementrdquoStructural Engineering and Mechanics vol 43 no 5 pp 583ndash605 2012

[11] W Rienstra ldquoNonlinear free vibrations of coupled spans of sus-pended cablesrdquo inProceedings of the 3rd EuropeanConference onMathematics in Industry J Manley S McKee and D R OwensEds pp 133ndash144 1988

[12] N Impollonia G Ricciardi and F Saitta ldquoStatics of elasticcables under 3Dpoint forcesrdquo International Journal of Solids andStructures vol 48 no 9 pp 1268ndash1276 2011

[13] L Greco andM Cuomo ldquoOn the force densitymethod for slackcable netsrdquo International Journal of Solids and Structures vol 49no 13 pp 1526ndash1540 2012

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The gravity load is not included in the strain-free configura-tion Alternately if the geometric model is built based on thestrain-free configuration the prestrain or prestress in eachelement needs not to be specified in advance and they areset up naturally in the nonlinear static analysis via the FEAAfter that the gravity load applied in the model the systemachieves equilibrium Therefore the assumed strain-freeconfiguration ismore preferable to the static equilibriumcon-figuration for building the FEAgeometricmodel ofmultispancable

For the static and dynamic analysis of cable structuresIrvine [3] systematically summarized the classical achieve-ments reached up to 1981 and the static equilibrium shape ofa cable is described as an elastic catenary For the modelingof cable structures in [4] a method for modeling cablesupported bridges for nonlinear finite element analysis ispresented in this paper A two-node catenary cable elementderived using exact analytical expressions for the elasticcatenary is proposed for the modeling of cables Based uponthe elastic catenary Such et al [5] proposed a method fordetermining the static equilibrium configuration of arbi-trary three-dimensional cable structures subject to gravityand point loads Srinil et al [6 7] formulated a systemof nonlinear partial differential equations describing thelarge amplitude three-dimensional free vibrations of inclinedsagged elastic cables Ai and Imai [8] proposed an iterativescheme to find the static equilibrium shape of beam-cablemixed system In [9] based on exact analytical expressionsof elastic catenary Thai and Kim presented a catenary cableelement for the nonlinear static and dynamic analysis ofcable structures In [10] Vu et al presented a spatial catenarycable element for the nonlinear analysis of cable-supportedstructures and proposed an algorithm for form-finding ofcable-supported structures Rienstra [11] derived the partialdifferential equations and boundary conditions satisfied bythe static equilibriumconfiguration ofmultiple spans coupledvia suspension strings In [12] Impollonia et al obtained thedeformed shape of the elastic cable in closed form for thecases of uniformly distributed load andmultiple-point forcesGreco and Cuomo [13] proposed a method for obtaining anexact configuration of slack cable nets by using the exactexpressions of the equilibrium derived from the equation ofthe catenary

All the above papers focused on the determinationof static equilibrium configuration of the cable structuresHowever the present paper concentrates on determiningthe strain-free or unstretched configuration of multispancable Firstly based upon the elastic catenary theory aset of geometric boundary conditions cable length con-dition and static equilibrium conditions are derived fordetermining the configuration of the multispan cableswhen they are approaching asymptotically to the strain-free state These constraint conditions constitute a sys-tem of nonlinear algebraic or transcendental functionequations Secondly these nonlinear equations are solvedand one finds a set of parameters which determine theunstretched configuration of the multispan cable Finallyan example of three-span cable is given to illustrate themethod

2 The Governing Equations ofMultispan Cable

First we assume that (1) the bending stiffness of cable isnegligible (2) the material of cable is of Hookian type thatis its constitutive relation is linear elastic and (3) only theself-weight as the static force is applied to the cable structureUnder these assumptions the governing equation for thecable in its static equilibrium state is as follows [6]

( 1198641198601205760radic1 + 119910101584020 )1015840

= 0

( 119864119860120576011991010158400radic1 + 119910101584020 )1015840

+ 120596119888radic1 + 119910101584020(1 + 1205760) = 0

(1)

Here the ldquo 1015840rdquo denotes the derivative with respect to 119909 1205760denotes the tensional strain in the cable when it is in staticequilibrium state 119864 is Youngrsquos modulus 119860 is the sectionalarea of the cable and assumed to be constant 1199100 representsthe static equilibrium shape of the cable and 120596

119888is the cable

weight per unit length in the unstretched stateThe tensional strain 1205760 is related to the horizontal ten-

sional force 119879119867as follows

1205760 = 119879119867119864119860radic1 + 119910101584020 (2)

According to (1) the final static equilibrium governingequation of the cable can be rewritten in the following form

(1+ 119879119867119864119860radic1 + 119910101584020 )119910101584010158400 + 120596

119888119879119867

radic1 + 119910101584020 = 0 (3)

This is the so-called elastic catenary equation Being differentfrom the traditional catenary equation it takes account ofthe extensibility of the cable Under the dynamic wind loadthe multispan cable usually vibrates nonlinearly due to itslarge displacement so the finite element analysis is preferableto investigate the static and dynamic behavior To build theFEA model firstly the geometric model of multispan cableis needed Therefore one of the initial configurations thestrain-free state or the static equilibrium state should bedetermined As stated in the Introduction in the followingwe only discuss how to determine the strain-free configura-tion of the multispan cable

3 Determination of the Strain-FreeConfiguration

Determining the strain-free configuration of the cable basedon the data of static equilibrium configuration can be viewedas an inverse problem in nonlinear structural mechanicsThe unstretched state of cable is an assumed state andit can be viewed as an asymptotic state that correspondsto 119864119860 rarr infin (which means the cable is inextensible)


Unstretched state

Equilibrium state

y

H1

A

O

la

L1

ha hbH2

r1

L2 L3

r2

H3

CBlb

D

x

H4

P2P1

P9984002P998400

1


and 120596119888119879119867



11991010158401015840 minus 119886radic1 + 11991010158402 = 0 (4)


119910 (119909) = 1119886 cosh 119886 (119909 + 1198881) + 1198882 (5)


1199101 (119909) = 11198861 cosh 1198861 (119909 + 11988811) + 11988812 (6)


1199102 (119909) = 11198862 cosh 1198862 (119909 + 11988821) + 11988822 (7)


1199103 (119909) = 11198863 cosh 1198863 (119909 + 11988831) + 11988832 (8)



End conditions1199101 (119909)1003816100381610038161003816119909=0 = 11986711199101 (119909)1003816100381610038161003816119909=119897

119886

= ℎ119886(1198711 minus 119897119886)2 + (1198672 + 1199031 minus ℎ119886)2 = 11990321

(9)






11986711990401119864119860 + 119879

119867119904011198821

sinhminus1 ( 1198811119879119867


119867

) = 1198711119882111990401119864119860 ( 11988111198821

minus 12)+ 119879119867119904011198821

[1+( 1198811119879119867

)212

minus1+(1198811 minus1198821119879119867

)212] = 1198671 minus1198672

(11)





End conditions1199102 (119909)1003816100381610038161003816119909=119897

119886

= ℎ1198861199102 (119909)1003816100381610038161003816119909=1198711+1198712+1198713minus119897119887 = ℎ

119887(1198713 minus 119897119887)2 + (1198673 + 1199032 minus ℎ119887)2 = 11990322


radic1 + 119910101584022 119889119909 = 11990402(12)



1198871199103 (119909)1003816100381610038161003816119909=1198711+1198712+1198713 = 1198674 (13)


radic1 + 119910101584023 119889119909 = 11990403 (14)




11986711990402119864119860 + 119879

119867119904021198822

sinhminus1 ( 1198812119879119867


119867

) = 1198712119882211990402119864119860 ( 11988121198822

minus 12)+ 119879119867119904021198822

[1+( 1198812119879119867

)212

minus1+(1198812 minus1198822119879119867

)212] = 1198673 minus1198672

(15)


119867119904031198823

sinhminus1 ( 1198813119879119867


119867

) = 1198713119882311990403119864119860 ( 11988131198823

minus 12)+ 119879119867119904031198823

[1+( 1198813119879119867

)212

minus1+(1198813 minus1198823119879119867

)212] = 1198673 minus1198674

(16)


1rarr 11987510158401and 119875


At point 119875 (T1 +T2) sdot 120591 = 0 (17)


n = (119897119886minus1198711 ℎ119886 minus (1198672 + 1198971)) (18)


120591 = (ℎ119886minus (1198672 + 1198971) 1198711 minus 119897119886) (19)


T1= minus 1003817100381710038171003817T11003817100381710038171003817 ( 1radic1 + 11991010158402

1

11991010158401radic1 + 119910101584021

)10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119909=1198831198751015840

1

T2= minus 1003817100381710038171003817T21003817100381710038171003817 ( 1radic1 + 11991010158402

2

11991010158402radic1 + 119910101584022

)10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119909=1198831198751015840

1

(20)

T1

T2

Tn

P

B

120591

n




1

1205961198881003817100381710038171003817T21003817100381710038171003817 radic1 + 119910101584022 = minus 119886

2 (21)



100381610038161003816100381610038161003816100381610038161003816119909=119897119886

= 0 (22)



100381610038161003816100381610038161003816100381610038161003816119909=1198711+1198712+1198713minus119897119887 = 0 (23)




119886 ℎ119887 119897119886 and 119897

119887




401198672

2061198673

2101198674

451198711

5801198712

19101198713

570119879119867

91378119864 63119890 + 10119860 66655119890 minus 4120588 45241120596119888

290668



02591119890 minus 31198862

02791119890 minus 31198863

02590119890 minus 311988811

802425911988812

minus3903925711988821

minus1527592811988822

minus3502726711988831

3879889111988832

minus39035631ℎ119886

2060680ℎ119887

2100671119897119886

5788362119897119887

19123203


4 An Example





X

Y

Z


ADINA





5 Conclusions





References
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Unstretched state

Equilibrium state

y

H1

A

O

la

L1

ha hbH2

r1

L2 L3

r2

H3

CBlb

D

x

H4

P2P1

P9984002P998400

1


and 120596119888119879119867



11991010158401015840 minus 119886radic1 + 11991010158402 = 0 (4)


119910 (119909) = 1119886 cosh 119886 (119909 + 1198881) + 1198882 (5)


1199101 (119909) = 11198861 cosh 1198861 (119909 + 11988811) + 11988812 (6)


1199102 (119909) = 11198862 cosh 1198862 (119909 + 11988821) + 11988822 (7)


1199103 (119909) = 11198863 cosh 1198863 (119909 + 11988831) + 11988832 (8)



End conditions1199101 (119909)1003816100381610038161003816119909=0 = 11986711199101 (119909)1003816100381610038161003816119909=119897

119886

= ℎ119886(1198711 minus 119897119886)2 + (1198672 + 1199031 minus ℎ119886)2 = 11990321

(9)






11986711990401119864119860 + 119879

119867119904011198821

sinhminus1 ( 1198811119879119867


119867

) = 1198711119882111990401119864119860 ( 11988111198821

minus 12)+ 119879119867119904011198821

[1+( 1198811119879119867

)212

minus1+(1198811 minus1198821119879119867

)212] = 1198671 minus1198672

(11)





End conditions1199102 (119909)1003816100381610038161003816119909=119897

119886

= ℎ1198861199102 (119909)1003816100381610038161003816119909=1198711+1198712+1198713minus119897119887 = ℎ

119887(1198713 minus 119897119887)2 + (1198673 + 1199032 minus ℎ119887)2 = 11990322


radic1 + 119910101584022 119889119909 = 11990402(12)



1198871199103 (119909)1003816100381610038161003816119909=1198711+1198712+1198713 = 1198674 (13)


radic1 + 119910101584023 119889119909 = 11990403 (14)




11986711990402119864119860 + 119879

119867119904021198822

sinhminus1 ( 1198812119879119867


119867

) = 1198712119882211990402119864119860 ( 11988121198822

minus 12)+ 119879119867119904021198822

[1+( 1198812119879119867

)212

minus1+(1198812 minus1198822119879119867

)212] = 1198673 minus1198672

(15)


119867119904031198823

sinhminus1 ( 1198813119879119867


119867

) = 1198713119882311990403119864119860 ( 11988131198823

minus 12)+ 119879119867119904031198823

[1+( 1198813119879119867

)212

minus1+(1198813 minus1198823119879119867

)212] = 1198673 minus1198674

(16)


1rarr 11987510158401and 119875


At point 119875 (T1 +T2) sdot 120591 = 0 (17)


n = (119897119886minus1198711 ℎ119886 minus (1198672 + 1198971)) (18)


120591 = (ℎ119886minus (1198672 + 1198971) 1198711 minus 119897119886) (19)


T1= minus 1003817100381710038171003817T11003817100381710038171003817 ( 1radic1 + 11991010158402

1

11991010158401radic1 + 119910101584021

)10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119909=1198831198751015840

1

T2= minus 1003817100381710038171003817T21003817100381710038171003817 ( 1radic1 + 11991010158402

2

11991010158402radic1 + 119910101584022

)10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119909=1198831198751015840

1

(20)

T1

T2

Tn

P

B

120591

n




1

1205961198881003817100381710038171003817T21003817100381710038171003817 radic1 + 119910101584022 = minus 119886

2 (21)



100381610038161003816100381610038161003816100381610038161003816119909=119897119886

= 0 (22)



100381610038161003816100381610038161003816100381610038161003816119909=1198711+1198712+1198713minus119897119887 = 0 (23)




119886 ℎ119887 119897119886 and 119897

119887




401198672

2061198673

2101198674

451198711

5801198712

19101198713

570119879119867

91378119864 63119890 + 10119860 66655119890 minus 4120588 45241120596119888

290668



02591119890 minus 31198862

02791119890 minus 31198863

02590119890 minus 311988811

802425911988812

minus3903925711988821

minus1527592811988822

minus3502726711988831

3879889111988832

minus39035631ℎ119886

2060680ℎ119887

2100671119897119886

5788362119897119887

19123203


4 An Example





X

Y

Z


ADINA





5 Conclusions





References
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












11986711990402119864119860 + 119879

119867119904021198822

sinhminus1 ( 1198812119879119867


119867

) = 1198712119882211990402119864119860 ( 11988121198822

minus 12)+ 119879119867119904021198822

[1+( 1198812119879119867

)212

minus1+(1198812 minus1198822119879119867

)212] = 1198673 minus1198672

(15)


119867119904031198823

sinhminus1 ( 1198813119879119867


119867

) = 1198713119882311990403119864119860 ( 11988131198823

minus 12)+ 119879119867119904031198823

[1+( 1198813119879119867

)212

minus1+(1198813 minus1198823119879119867

)212] = 1198673 minus1198674

(16)


1rarr 11987510158401and 119875


At point 119875 (T1 +T2) sdot 120591 = 0 (17)


n = (119897119886minus1198711 ℎ119886 minus (1198672 + 1198971)) (18)


120591 = (ℎ119886minus (1198672 + 1198971) 1198711 minus 119897119886) (19)


T1= minus 1003817100381710038171003817T11003817100381710038171003817 ( 1radic1 + 11991010158402

1

11991010158401radic1 + 119910101584021

)10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119909=1198831198751015840

1

T2= minus 1003817100381710038171003817T21003817100381710038171003817 ( 1radic1 + 11991010158402

2

11991010158402radic1 + 119910101584022

)10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119909=1198831198751015840

1

(20)

T1

T2

Tn

P

B

120591

n




1

1205961198881003817100381710038171003817T21003817100381710038171003817 radic1 + 119910101584022 = minus 119886

2 (21)



100381610038161003816100381610038161003816100381610038161003816119909=119897119886

= 0 (22)



100381610038161003816100381610038161003816100381610038161003816119909=1198711+1198712+1198713minus119897119887 = 0 (23)




119886 ℎ119887 119897119886 and 119897

119887




401198672

2061198673

2101198674

451198711

5801198712

19101198713

570119879119867

91378119864 63119890 + 10119860 66655119890 minus 4120588 45241120596119888

290668



02591119890 minus 31198862

02791119890 minus 31198863

02590119890 minus 311988811

802425911988812

minus3903925711988821

minus1527592811988822

minus3502726711988831

3879889111988832

minus39035631ℎ119886

2060680ℎ119887

2100671119897119886

5788362119897119887

19123203


4 An Example





X

Y

Z


ADINA





5 Conclusions





References
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












401198672

2061198673

2101198674

451198711

5801198712

19101198713

570119879119867

91378119864 63119890 + 10119860 66655119890 minus 4120588 45241120596119888

290668



02591119890 minus 31198862

02791119890 minus 31198863

02590119890 minus 311988811

802425911988812

minus3903925711988821

minus1527592811988822

minus3502726711988831

3879889111988832

minus39035631ℎ119886

2060680ℎ119887

2100671119897119886

5788362119897119887

19123203


4 An Example





X

Y

Z


ADINA





5 Conclusions





References
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










References
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article determination of the strain-free...

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