research article a parametric study of nonlinear seismic...

Research ArticleA Parametric Study of Nonlinear Seismic Response Analysis ofTransmission Line Structures

Li Tian1 Yanming Wang1 Zhenhua Yi1 and Hui Qian2

1 School of Civil and Hydraulic Engineering Shandong University Jinan 250061 China2 School of Civil Engineering Zhengzhou University Zhengzhou 450001 China

Correspondence should be addressed to Li Tian tianlisdueducn

Received 13 May 2014 Accepted 1 July 2014 Published 15 July 2014

Academic Editor Fei Kang

Copyright copy 2014 Li Tian et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A parametric study of nonlinear seismic response analysis of transmission line structures subjected to earthquake loading is studiedin this paper The transmission lines are modeled by cable element which accounts for the nonlinearity of the cable based on a realproject Nonuniform ground motions are generated using a stochastic approach based on random vibration analysis The effectsof multicomponent ground motions correlations among multicomponent ground motions wave travel coherency loss and localsite on the responses of the cables are investigated using nonlinear time history analysis method respectively The results show themulticomponent seismic excitations should be considered but the correlations among multicomponent ground motions could beneglected The wave passage effect has a significant influence on the responses of the cables The change of the degree of coherencyloss has little influence on the response of the cables but the responses of the cables are affected significantly by the effect ofcoherency loss The responses of the cables change little with the degree of the difference of site condition changing The effect ofmulticomponent ground motions wave passage coherency loss and local site should be considered for the seismic design of thetransmission line structures

1 Introduction

Transmission line structures are very important to electricengineering Most of the transmission lines cross the highlyseismic region The past earthquakes indicate that trans-mission lines are often damaged under earthquake loadingAbout 100 transmission lines of Los Angeles city weredestroyed in the 1992 Landers earthquake [1] 38 transmissionlines were broken during the 1995 Kobe earthquake [2]Manytransmission lines were pulled off in the 2008 Wenchuanearthquake [3] A lot of transmission lines were rupturedduring the 2013 Lushan earthquake To guarantee the safety oftransmission lines during earthquake the parametric study ofnonlinear responses of the transmission line structures underearthquake loading should be accurately obtained

Most of research has focused on the actions of static loadimpulsive load and equivalent static wind load There are nocalculation methods about how to consider the transmissionline structures under earthquake loading in current seismic

codes [4 5] Some research has been performed to analyzethe seismic responses of transmission tower and transmissionline under earthquake loading Li et al [6 7] have completeda number of studies on seismic effects on transmission towersand have verified that the effect of transmission lines inseismic design should not be neglected

With the development of techniques the spans of trans-mission lines have increased dramatically It is unrealisticto assume that earthquake ground motions for long spantransmission tower-line system are the same and singlecomponent Ghobarah et al [8] investigated the effects ofmultisupport excitations on the response of overhead powertransmission tower and line The results indicated that theassumption of uniform ground motions at all supports ofa transmission line does not provide the most critical casefor the response calculations Tian et al [9] studied thebehavior of power transmission tower-line system subjectedto spatially varying ground motions The effects of theincident angle of the seismic wave coherency loss and wave

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 271586 9 pageshttpdxdoiorg1011552014271586

2 The Scientific World Journal

travel on the transmission tower are investigated Li et al[10 11] investigated the response of a transmission tower-line system at a canyon site to spatially varying groundmotionsThe results showed that the effect of groundmotionspatial variations should be incorporated in seismic analysisof the transmission tower-line system In addition Tian etal [12] analyzed the effect of multicomponent multisupportexcitations on the response of the transmission tower-linesystem Multiple effect parameters were considered but theresponses of the transmission tower were obtained onlyWang et al [13] researched the progressive collapse analysisof a transmission tower-line system under earthquake Theresults indicated that the proposed procedure can providecollapsemode and vulnerable points for use in seismic perfor-mance and retrofit evaluation of structure Furthermore Tianet al [14] studied seismic responses of straight line type andbroken line type transmission tower-line systems subjected tononuniform seismic excitations The results showed that theeffect of nonuniform ground motions should be consideredin seismic design for the straight line type and broken linetype transmission lines practical engineering The previousstudy concluded that the responses of transmission towerunder nonuniform and multicomponent ground motionswere different from that of under uniform and single groundmotion A lot of studies about the response of transmissiontowers are obtained but there is little research about theparametric study of nonlinear response of transmission linestructures under earthquake loading

The parametric studies of seismic response analysis oftransmission line structures considering geometric nonlin-earity subjected to earthquake loading are carried out inthis paper The transmission lines are modeled by cableelement account for the nonlinearity of the cable based on areal project The effects of multicomponent ground motionscorrelations among multicomponent ground motions wavetravel coherency loss and local site on the responses of thecables are investigated using nonlinear time history analysismethod respectively The analysis results could providereference for the seismic design of the transmission linestructures

2 Transmission Line Structures Model

A typical three-dimensional finite element model of trans-mission line structures is established based on a real electricproject in the north of China SAP2000 finite elementprogram is used to simulate the transmission line structuresAs shown in Figure 1 the transmission line structures includethree towers and four span conductors lines and groundlines Figure 1 shows the conductor and ground lines Thelongitudinal transverse and vertical directions of the trans-mission line structures are shown in Figure 1 Conductorline and ground line properties are shown in Table 1 Thetransmission line is modeled by 40 two-node isoparametriccable elements with three translational degrees of freedomat each node The upper one layer line is ground cableand the lower three layer lines are four-bundled conductorcables The distance between adjacent transmission towers

Table 1 Conductor line and ground line properties

Type Conductor line Ground lineTransmission line LGJ-40035 LGJ-9555Outside diameter (m) 2682119864 minus 3 1600119864 minus 3

Modulus (GPa) 65 105Transversal cross-section (m2) 42524119864 minus 6 15281119864 minus 6

Mass per unit length (Kgm) 13490 06967Expansion coefficient (1∘C) 205119864 minus 005 155119864 minus 005

is 400m The connections between transmission towers andtransmission lines are hinged using insulatorsThe side spansof the transmission lines are hinged at the same height ofmiddle transmission tower

Under self-weight the cablesrsquo configuration is a catenaryBased on the coordinate system illustrated in Figure 2 (1) wasused to define the initial geometry of the cable profile [15]

119911 =119867

119902

10038161003816100381610038161003816100381610038161003816

cosh (120572) minus cosh10038161003816100381610038161003816100381610038161003816

2120573119909

119897minus 120572

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

(1)

where 120572 = sin hminus1|120573(119888119897) sin(120573)| + 120573 120573 = 1199021198972119867 119867represents the initial horizontal tension which can beobtained from a preliminary static analysis and 119902 denotes theuniformly distributed gravity loads along the conductor andground lines

3 Simulation of Nonuniform Ground Motions

An empirical coherency loss function derived from SMART-1 array is used in the paper [16] The coherency loss functionbetween two points 119894 and 119895 is

10038161003816100381610038161003816120574119894119895(120596 119889119894119895)10038161003816100381610038161003816= expminus (120573119889

119894119895) sdot expminus119886 (120596)radic119889

119894119895(120596

2120587)

2

(2)

in which 119889119894119895is the projected distance in the wave propagation

direction between points 119894 and 119895 in the wave propagationdirection120573 is a constant and 119886(120596) is a functionwith the form

119886 (120596) =

2120587119886

120596+119887120596

2120587+ 119888 0314 rads le 120596 le 6283 rads

01119886 + 10119887 + 119888 120596 ge 6283 rads

(3)

where the constants 119886 119887 and 119888 can be obtained by least-squares fitting the coherency function of recorded motionsThe constants in coherency function are 119886 = 3583 times 10

minus3119887 = minus1811 times 10

minus5 119888 = 1177 times 10minus4 and 120573 = 1019 times

10minus4 which were obtained by processing recorded motions

during Event 45 at the SMART-1 array [16] and it representshighly correlated ground motions To compare the changeof the coherency loss different degrees of coherency loss areselected based on Bi et alrsquos studies [17]

A stochastic approach based on random vibration analy-sis is used and the simulated ground motion time history isiterated to be compatible with the response spectrum defined

The Scientific World Journal 3

1st

2nd

3rd

Vertical

Longitudinal Transverse

Conductor line

Ground line

400m

400m

Figure 1 Finite element model of transmission line structures

O

i

l

c

x

j

q

dx

ds

z

Figure 2 Coordinates of a single cable under self-weight

in Code for Design of Seismic of Electrical InstallationsReference [18] gives the parameters of Clough-Penzienmodelaccording to the Code for Design of Seismic of ElectricalInstallations The transmission cable structures are assumedto locate in the mid-firm soil The peak ground motion ofthe longitudinal component is 04 g The intensities of thetransverse component and vertical component as stated inthe code are 085 and 065 times of the longitudinal com-ponent respectively The three components of the groundmotion are assumed to coincide with the principal axes Thethree components of ground motions along a set of principalaxes are uncorrelated based on Penzien and Watabersquos studies[19] Figure 3 shows acceleration time histories of threetransmission tower points in longitudinal direction on mid-form soil with apparent velocity 1000ms

4 Numerical Simulation and Discussion

The parametric studies of nonlinear seismic responses of thetransmission line structures under earthquake loading areanalyzed using nonlinear time history analysis method Thegeometric nonlinearity is taken into account due to largedeformation of the transmission lines The HHT (Hilber-Hughes-Taylor) method is applied in the numerical integra-tion The layers of cables shown in Figure 1 from upper todown are numbered 1 2 3 and 4 respectively

Table 2 Selection of seismic wave

Number Earthquake Event date Magnitude StationA Imperial Valley May 18 1940 67 El CentroB Kobe January 16 1995 69 OkaC Kern County July 21 1952 74 Taft

41 Effect of Multicomponent Ground Motions To study theeffect of multicomponent ground motions three typicalnatural seismic waves are selected which are El Centro waveOka wave and Taft wave The selection of seismic waves isshown in Table 2 Three components of the natural seismicwaves are considered in the paper The direction of themaximum acceleration component of the horizontal seismicwave is denoted by the horizontal 1 while the other directioncomponent of the horizontal seismic wave is denoted by thehorizontal 2 and the vertical component of seismic waveis denoted by vertical The maximum acceleration value ofthe ground motion is adjusted to 04 g and the other twodirections are scaled according to the proportion

Four cases are considered longitudinal excitation only(Case 1) transverse excitation only (Case 2) vertical exci-tation only (Case 3) and multicomponent excitations (Case4) Case 1 is longitudinal excitation only and the horizontal 1component of the seismic wave is inputted along longitudinaldirection of the transmission line structures model Case 2 istransverse excitation only and the horizontal 2 componentof the seismic wave is inputted along transverse directionof the transmission line structures model Case 3 is verticalexcitation only and the vertical component of the seismicwave is inputted along vertical direction of the transmissionline structures model Case 4 is multicomponent excitationsand the horizontal 1 horizontal 2 and vertical componentof seismic wave are inputted together along longitudinaltransverse and vertical direction of the transmission linestructures model respectively

The maximum value curves of the vertical displacementsof the cable under different analysis cases are shown inFigure 4 It can be seen from Figure 4 that the vertical dis-placement of the cable under longitudinal or transverse exci-tation only is larger than that of under vertical seismic exci-tation only so the longitudinal or transverse excitation has agreat influence on the response of the vertical displacementof the cable The vertical displacements of the cable under


0 10 20

Tower 1

Time (s)

4

2

0

minus2

minus4

Acce

lera

tion(m

s2)

(a) 1st tower

0 10 20

Tower 2

Time (s)

4

2

0

minus2

minus4

Acce

lera

tion(m

s2)

(b) 2nd tower

0 10 20

Tower 3

Time (s)

4

2

0

minus2

minus4

Acce

lera

tion(m

s2)

(c) 3rd tower

Figure 3 Acceleration time histories of three tower points in longitudinal direction

multicomponent excitations are significantly larger than thatof under vertical longitudinal or transverse excitation onlyThe longitudinal and transverse seismic excitations have alarge coupling with the response of the vertical displacementof the cable Therefore multicomponent seismic excitationsshould be considered for the transmission line structures

42 Effect of Correlations among Multicomponent GroundMotions To research the effect of the correlations amongmulticomponent ground motions four cases are considereduniform (Case 1) 120572 = 0

∘ (Case 2) 120572 = 18∘ (Case 3) and

120572 = 45∘ (Case 4) The correlations among multicomponent

ground motions are selected based on previous studies [12]The maximum values of the tension forces of the cables

under different degrees of the coherence are shown inTable 3It can be seen from Table 3 that the tension forces of thecables have an increasing tendency with the increasing of thedegree of the coherence Ignoring the correlations among themulticomponent ground motions the results may be smallbut the changes are very little The above analysis indicates

Table 3 Tension forces of the cables under different degrees of thecoherence (kN)

Layer Case 1 Case 2 Case 3 Case 4A 1398 2297 2321 2391B 9551 14280 14312 14379C 9552 14313 14377 14422D 9520 14301 14363 14391

that the effect of correlations amongmulticomponent groundmotions could be neglected

43 Effect of GroundMotion Spatial Variations To investigatethe effect of ground motion spatial variations four casesare considered uniform (Case 1) wave passage effect only(Case 2) coherency loss effect only (Case 3) and local siteeffect only (Case 4) Case 1 is the uniform excitation becausethe apparent velocity coherency loss and soil condition of


0 100 200 300 40000

03

06Ve

rtic

al d

ispla

cem

ent (

m)

Cable span (m)

Case 1 Case 2

Case 3 Case 4

(a) El Centro

0 100 200 300 40000

02

04

Vert

ical

disp

lace

men

t (m

)

Cable span (m)

Case 1 Case 2

Case 3 Case 4

(b) Oka

0 100 200 300 40000

04

08

Vert

ical

disp

lace

men

t (m

)

Cable span (m)

Case 1 Case 2

Case 3 Case 4

(c) Taft

Figure 4 Vertical displacements of the cable under different analysis cases

ground motion are assumed to be infinite highly correlatedand mid-firm site respectively

The maximum value curves of the vertical displacementsof the cable under different analysis cases are shown inFigure 5 It can be seen from Figure 5 that the verticaldisplacements of the cable considering wave travel effectonly coherency loss effect only or local site effect only arelarger than that of under uniform excitation The verticaldisplacements of the cable considering wave travel effectonly are larger than that of considering coherency loss effectonly or local site effect only Existing research [20] hasshown that the wave travel effect is very important to theresponses of structure when the structure is flexible andthe responses are mainly decided by dynamic response ofthe structure The coherency loss effect is very important tothe responses of structure when the structure is rigid andthe responses are mainly decided by quasistatic response ofthe structure Therefore wave travel effect of ground motion

is more obvious to the influence of the structure than theother effect for the flexible structure of the transmissionlines

44 Wave Travel Effect To study the effect of apparentvelocity ten different velocities of wave propagation areconsidered in the analysis uniform (Case 1) 200ms (Case2) 400ms (Case 3) 600ms (Case 4) 800ms (Case 5)1000ms (Case 6) 1200ms (Case 7) 1600ms (Case 8)2000ms (Case 9) and 3000ms (Case 10) to cover the rangeof practical propagation velocities in engineering In all thesecases the coherency loss and soil condition of groundmotionare assumed to be highly correlated and the mid-firm siterespectively

The maximum value curves of the vertical displace-ments of the cable under different traveling wave velocitiesare shown in Figure 6 It can be seen from Figure 6 that


0 100 200 300 4000

1

2

3

4

Vert

ical

disp

lace

men

t (m

)

Cable span (m)

Case 1 Case 2 Case 4

Case 3

Figure 5 Vertical displacements of the cable under different anal-ysis cases

the vertical displacements of the cable increase with thedecreasing wave velocity The maximum vertical displace-ments of the cable appear when the velocity is 200msThe vertical displacements of the cable decrease with theincreasing wave velocity but it is larger than that of underuniform excitation Therefore the vertical displacement ofthe cable is very sensitive to the traveling wave velocity of theseismic wave

The maximum value curves of tension forces of thecables under different traveling wave velocities are shown inFigure 7 The tension forces of the cables change very littlewhen the traveling wave velocity is less than 200ms butit is larger than that of under uniform excitation With thetraveling wave velocity increasing the tension forces of thecables decrease gradually Neglecting the wave passage effectof ground motion the maximum tension forces of the cablescould be underestimated by more than 50

Based on the variations of the displacements and tensionforces of the cables considering the change of traveling wavevelocity the wave travel effect has a significant influence onthe response of the cables The vertical displacements of thecable are amplified greatly considering the wave travel effectThe vibration of the cable is very large which would lead todischarge and short circuit The tension forces of the cablesconsidering the wave travel effect are larger than that ofunder uniform excitation Because the tension forces of thecables are too large the transmission lines would be pulledoff and the situations usually occur in the past earthquakesTherefore it is necessary to estimate the traveling wavevelocity accurately

45 Coherency Loss Effect To investigate the effect of co-herency loss uniform (Case 1) uncorrelated (Case 2) weakly(Case 3) intermediately (Case 4) highly (Case 5) and com-pletely correlated (Case 6) ground motions are consideredrespectively It should be noted that the correlation as low

0 100 200 300 4000

4

8

12

Vert

ical

disp

lace

men

t (m

)

Cable span (m)

Case 1Case 2Case 3Case 4Case 5


Figure 6 Vertical displacements of the cable under differenttraveling wave velocities

1 2 3 40

40

80

120

160

200

Tens

ion

forc

e (kN

)

Case 1 Case 6 Case 7 Case 8 Case 9 Case 10

Case 2 Case 3 Case 4 Case 5

Layer of cables

Figure 7 Tension forces of the cables under different traveling wavevelocities

as uncorrelated does not usually occur at short distancesunless there are considerable changes in the local geologyfrom one support to the other In all these cases the apparentvelocity and soil condition of ground motion are assumed tobe 1000ms and the mid-firm site respectively

The maximum value curves of the vertical displacementsof the cable under different degrees of coherency loss areshown in Figure 8 It can be seen from Figure 8 that the


0 100 200 300 4000

4

8

12

Vert

ical

disp

lace

men

t (m

)

Cable span (m)

Case 1Case 2Case 3

Case 4Case 5Case 6

Figure 8 Vertical displacements of the cable under different degreesof coherency loss

maximum vertical displacements of the cable appear whenthe coherency loss is uncorrelatedThe vertical displacementsof the cable have an increasing tendency with the decreaseof the degree of coherency loss The changes of the verticaldisplacements are very little when the coherency losses areintermediately highly and completely correlated

The maximum value curves of tension forces of thecables under different degrees of coherency loss are shownin Figure 9 It can be seen from Figure 9 that the change ofcoherency loss has little influence on the tension forces ofthe cables so the change of coherency loss can be ignoredThe tension forces of the cables considering coherency losseffect are larger than that of under uniform excitation sothe effect of coherency loss should be considered Neglectingthe coherency loss effect of ground motion the maximumtension forces of the cables could be underestimated by morethan 50

The variations of the displacement and force responses ofthe cables considering the change of coherency loss can beobtained from the above analysis The vertical displacementsof the cable have an increasing tendency with the decreaseof the degree of coherency loss The change of coherencyloss can be ignored but the effect of coherency loss must beconsidered Therefore it is very important to consider thecoherency loss effect of groundmotion for the seismic designof the transmission line structures

46 Local Site Effect To research the effect of local siteinfluence on the cable responses eight cases are consideredCase 1simCase 8 Analysis cases considering the effect oflocal site are shown in Table 4 Mid-firm mid-soft and softsites are denoted by F MF MS and S respectively In allthese cases the apparent velocity and coherency of groundmotion are assumed to be 1000ms and highly correlatedrespectively

1 2 3 40

40

80

120

160

200

Tens

ion

forc

e (kN

)

Layer of cables

Case 1Case 2Case 3

Case 4Case 5Case 6

Figure 9 Tension forces of the cables under different degrees ofcoherency loss

The maximum value curves of the vertical displacementsof the cable under different site conditions are shown inFigure 10 It can be seen from Figure 10 that the verticaldisplacements of the cable have an increasing tendency withthe site condition growing soft The vertical displacementsof the cable increase with the degree of the difference of sitecondition increasing

Themaximum value curves of tension forces of the cablesunder different site conditions are shown in Figure 11 It canbe seen from Figure 11 that the tension forces of the cableshave an increasing tendency with the site condition growingsoft and the maximum tension forces of the cables appearwhen the three transmission towers are located on soft sitesThe tension forces of the cables change very little when thesite is located in different types and it could be ignored

Based on the above analysis the variations of the displace-ment and tension force responses of the cables consideringdifferent site conditions can be summarizedThe vertical dis-placements and tension forces of the cable have an increasingtendency with the site condition growing soft The responsesof the cables change little with the degree of the differenceof site condition changing especially for the tension forces ofthe cablesTherefore the local site effect should be consideredfor the seismic design of the transmission line structures

5 Conclusion

Theparametric studies of nonlinear dynamic responses of thetransmission line structures subjected to earthquake loadingare investigated in the paper The effects of multicompo-nent ground motions correlations among multicomponentground motions ground motion spatial variation wavepassage coherency loss and local site on the transmission


Table 4 Analysis cases considering the effect of local site

Case Apparent velocity Coherency Soil condition1st tower 2nd tower 3rd tower

Case 1 Infinite Perfectly MF MF MFCase 2 1000ms Highly F F FCase 3 1000ms Highly MF MF MFCase 4 1000ms Highly MS MS MSCase 5 1000ms Highly S S SCase 6 1000ms Highly F MF FCase 7 1000ms Highly MS MF MSCase 8 1000ms Highly S MF S

0 100 200 300 4000

4

8

Vert

ical

disp

lace

men

t (m

)

Cable span (m)



Figure 10 Vertical displacements of the cable under different siteconditions

line structures are considered respectively Based on thenumerical results the following conclusions are drawn

(1) The vertical displacements of the cable under mul-ticomponent excitations are significantly larger thanthat of under vertical longitudinal or transverseexcitation only Multicomponent seismic excitationsshould be considered

(2) Ignoring the correlations among themulticomponentground motions the response of the cable may besmall but the changes are very little The correla-tions amongmulticomponent groundmotions can beneglected

(3) The responses of the cables considering the effectof ground motion spatial variations are larger thanthat of under uniform excitation Wave travel effectof ground motion is more obvious to the influenceof the structure than the other effect for the flexiblestructure of the transmission line structures

1 2 3 40

40

80

120

160

200

240

Tens

ion

forc

e (kN

)

Layer of cables



Figure 11 Tension forces of the cables under different site condi-tions

(4) The wave passage effect has a significant influenceon the responses of the cables Neglecting the wavepassage effect in analysis the cables responses wouldbe underestimated Because the tension forces of thecables are too large the transmission lines would bepulled off It is necessary to estimate the travelingwave velocity accurately

(5) The change of the degree of coherency loss has littleinfluence on the response of the cablesThe responsesof the cables are affected significantly by the effect ofcoherency loss It is very important to consider thecoherency loss effect of groundmotion for the seismicdesign of the transmission line structures

(6) The vertical displacements and tension forces of thecables have an increasing tendency with the sitecondition growing soft The responses of the cableschange little with the degree of the difference of sitecondition changing especially for the tension forces


Conflict of Interests

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

Acknowledgment

This work is supported by the National Natural ScienceFoundation of China under Grant no 51208285The supportsfor this research are greatly appreciated

References

[1] J F Hall W T Holmes and P Somers Northridge Earthquakeof January 17 1994 Earthquake Engineering Research InstituteCalifornia Calif USA 1994

[2] M Shinozuka ldquoThe Hanshin-Awaji earthquake of January17 1995 performance of lifelinesrdquo Report NCEER-95-0015NCEER 1995

[3] P Zhang G Song H Li and Y Lin ldquoSeismic control ofpower transmission tower using pounding TMDrdquo Journal ofEngineering Mechanics vol 139 no 10 pp 1395ndash1406 2013

[4] GB 50260-96 Code for Seismic Design of Electrical FacilitiesNational Standard of the Peoples Republic of China China PlanPress Beijing China 1996 (Chinese)

[5] C J Wong and M D Miller Guidelines for Electrical Trans-mission Line Structural Loading American Society of CivilEngineers New York NY USA 2009

[6] H Li S Xiao and S Wang ldquoStudy on limits of height-to-widthratio for base isolated buildings under earthquakerdquoASME PVPvol 445 no 2 pp 143ndash147 2002

[7] H Li W Shi G Wang and L Jia ldquoSimplified models andexperimental verification for coupled transmission tower-linesystem to seismic excitationsrdquo Journal of Sound and Vibrationvol 286 no 3 pp 569ndash585 2005

[8] A Ghobarah T S Aziz and M El-Attar ldquoResponse oftransmission lines to multiple support excitationrdquo EngineeringStructures vol 18 no 12 pp 936ndash946 1996

[9] L Tian H Li and G Liu ldquoSeismic response of powertransmission tower-line system subjected to spatially varyingground motionsrdquo Mathematical Problems in Engineering vol2010 Article ID 587317 20 pages 2010

[10] H Li F Bai L Tian and H Hao ldquoResponse of a transmissiontower-line system at a canyon site to spatially varying groundmotionsrdquo Journal of Zhejiang University vol 12 no 2 pp 103ndash120 2011

[11] F Bai H Hao K Bi and H Li ldquoSeismic response analysisof transmission tower-line system on a heterogeneous siteto multi-component spatial ground motionsrdquo Advances inStructural Engineering vol 14 no 3 pp 457ndash474 2011

[12] L Tian H Li and G Liu ldquoSeismic response of power transmis-sion tower-line system under multi-component multi-supportexcitationsrdquo Journal of Earthquake and Tsunami vol 6 no 4pp 1ndash21 2012

[13] WMWangHN Li andL Tian ldquoProgressive collapse analysisof transmission tower-line system under earthquakerdquoAdvancedSteel Construction vol 9 no 2 pp 161ndash174 2013

[14] L Tian RMaH Li and P Zhang ldquoSeismic response of straightline type and broken line type transmission lines subjected tonon-uniform seismic excitationsrdquo Advanced Steel Constructionvol 10 no 1 pp 85ndash98 2014

[15] S Shen C Xu and C Zhao Design of Suspension StructureChina Architecture and Building Press Beijing China 1997(Chinese)

[16] H Hao C S Oliveira and J Penzien ldquoMultiple-station groundmotion processing and simulation based on smart-1 array datardquoNuclear Engineering andDesign vol 111 no 3 pp 293ndash310 1989

[17] K Bi H Hao and N Chouw ldquoRequired separation distancebetween decks and at abutments of a bridge crossing a canyonsite to avoid seismic poundingrdquo Earthquake Engineering andStructural Dynamics vol 39 no 3 pp 303ndash323 2010

[18] L Tian and H Li ldquoParameter study on seismic random modelbased on code for design of seismic electrical installationsrdquoJournal of Disaster Prevention and Mitigation Engineering vol30 no 1 pp 17ndash22 2010 (Chinese)

[19] J Penzien and M Watabe ldquoCharacteristics of 3-dimensionalearthquake ground motionsrdquo Earthquake Engineering andStructural Dynamics vol 3 no 4 pp 365ndash373 1975

[20] H Hao ldquoArch responses to correlated multiple excitationsrdquoEarthquake Engineering and Structural Dynamics vol 22 no5 pp 389ndash404 1993

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travel on the transmission tower are investigated Li et al[10 11] investigated the response of a transmission tower-line system at a canyon site to spatially varying groundmotionsThe results showed that the effect of groundmotionspatial variations should be incorporated in seismic analysisof the transmission tower-line system In addition Tian etal [12] analyzed the effect of multicomponent multisupportexcitations on the response of the transmission tower-linesystem Multiple effect parameters were considered but theresponses of the transmission tower were obtained onlyWang et al [13] researched the progressive collapse analysisof a transmission tower-line system under earthquake Theresults indicated that the proposed procedure can providecollapsemode and vulnerable points for use in seismic perfor-mance and retrofit evaluation of structure Furthermore Tianet al [14] studied seismic responses of straight line type andbroken line type transmission tower-line systems subjected tononuniform seismic excitations The results showed that theeffect of nonuniform ground motions should be consideredin seismic design for the straight line type and broken linetype transmission lines practical engineering The previousstudy concluded that the responses of transmission towerunder nonuniform and multicomponent ground motionswere different from that of under uniform and single groundmotion A lot of studies about the response of transmissiontowers are obtained but there is little research about theparametric study of nonlinear response of transmission linestructures under earthquake loading

The parametric studies of seismic response analysis oftransmission line structures considering geometric nonlin-earity subjected to earthquake loading are carried out inthis paper The transmission lines are modeled by cableelement account for the nonlinearity of the cable based on areal project The effects of multicomponent ground motionscorrelations among multicomponent ground motions wavetravel coherency loss and local site on the responses of thecables are investigated using nonlinear time history analysismethod respectively The analysis results could providereference for the seismic design of the transmission linestructures

2 Transmission Line Structures Model

A typical three-dimensional finite element model of trans-mission line structures is established based on a real electricproject in the north of China SAP2000 finite elementprogram is used to simulate the transmission line structuresAs shown in Figure 1 the transmission line structures includethree towers and four span conductors lines and groundlines Figure 1 shows the conductor and ground lines Thelongitudinal transverse and vertical directions of the trans-mission line structures are shown in Figure 1 Conductorline and ground line properties are shown in Table 1 Thetransmission line is modeled by 40 two-node isoparametriccable elements with three translational degrees of freedomat each node The upper one layer line is ground cableand the lower three layer lines are four-bundled conductorcables The distance between adjacent transmission towers

Table 1 Conductor line and ground line properties

Type Conductor line Ground lineTransmission line LGJ-40035 LGJ-9555Outside diameter (m) 2682119864 minus 3 1600119864 minus 3

Modulus (GPa) 65 105Transversal cross-section (m2) 42524119864 minus 6 15281119864 minus 6

Mass per unit length (Kgm) 13490 06967Expansion coefficient (1∘C) 205119864 minus 005 155119864 minus 005

is 400m The connections between transmission towers andtransmission lines are hinged using insulatorsThe side spansof the transmission lines are hinged at the same height ofmiddle transmission tower

Under self-weight the cablesrsquo configuration is a catenaryBased on the coordinate system illustrated in Figure 2 (1) wasused to define the initial geometry of the cable profile [15]

119911 =119867

119902

10038161003816100381610038161003816100381610038161003816

cosh (120572) minus cosh10038161003816100381610038161003816100381610038161003816

2120573119909

119897minus 120572

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

(1)

where 120572 = sin hminus1|120573(119888119897) sin(120573)| + 120573 120573 = 1199021198972119867 119867represents the initial horizontal tension which can beobtained from a preliminary static analysis and 119902 denotes theuniformly distributed gravity loads along the conductor andground lines

3 Simulation of Nonuniform Ground Motions

An empirical coherency loss function derived from SMART-1 array is used in the paper [16] The coherency loss functionbetween two points 119894 and 119895 is

10038161003816100381610038161003816120574119894119895(120596 119889119894119895)10038161003816100381610038161003816= expminus (120573119889

119894119895) sdot expminus119886 (120596)radic119889

119894119895(120596

2120587)

2

(2)

in which 119889119894119895is the projected distance in the wave propagation

direction between points 119894 and 119895 in the wave propagationdirection120573 is a constant and 119886(120596) is a functionwith the form

119886 (120596) =

2120587119886

120596+119887120596

2120587+ 119888 0314 rads le 120596 le 6283 rads

01119886 + 10119887 + 119888 120596 ge 6283 rads

(3)

where the constants 119886 119887 and 119888 can be obtained by least-squares fitting the coherency function of recorded motionsThe constants in coherency function are 119886 = 3583 times 10

minus3119887 = minus1811 times 10

minus5 119888 = 1177 times 10minus4 and 120573 = 1019 times

10minus4 which were obtained by processing recorded motions

during Event 45 at the SMART-1 array [16] and it representshighly correlated ground motions To compare the changeof the coherency loss different degrees of coherency loss areselected based on Bi et alrsquos studies [17]

A stochastic approach based on random vibration analy-sis is used and the simulated ground motion time history isiterated to be compatible with the response spectrum defined


1st

2nd

3rd

Vertical


Conductor line

Ground line

400m

400m


O

i

l

c

x

j

q

dx

ds

z











0 10 20

Tower 1

Time (s)

4

2

0

minus2

minus4

Acce

lera

tion(m

s2)

(a) 1st tower

0 10 20

Tower 2

Time (s)

4

2

0

minus2

minus4

Acce

lera

tion(m

s2)

(b) 2nd tower

0 10 20

Tower 3

Time (s)

4

2

0

minus2

minus4

Acce

lera

tion(m

s2)

(c) 3rd tower




∘ (Case 2) 120572 = 18∘ (Case 3) and









0 100 200 300 40000

03

06Ve

rtic

al d

ispla

cem

ent (

m)

Cable span (m)

Case 1 Case 2

Case 3 Case 4

(a) El Centro

0 100 200 300 40000

02

04

Vert

ical

disp

lace

men

t (m

)

Cable span (m)

Case 1 Case 2

Case 3 Case 4

(b) Oka

0 100 200 300 40000

04

08

Vert

ical

disp

lace

men

t (m

)

Cable span (m)

Case 1 Case 2

Case 3 Case 4

(c) Taft








0 100 200 300 4000

1

2

3

4

Vert

ical

disp

lace

men

t (m

)

Cable span (m)


Case 3






0 100 200 300 4000

4

8

12

Vert

ical

disp

lace

men

t (m

)

Cable span (m)




1 2 3 40

40

80

120

160

200

Tens

ion

forc

e (kN

)



Layer of cables





0 100 200 300 4000

4

8

12

Vert

ical

disp

lace

men

t (m

)

Cable span (m)

Case 1Case 2Case 3

Case 4Case 5Case 6






1 2 3 40

40

80

120

160

200

Tens

ion

forc

e (kN

)

Layer of cables

Case 1Case 2Case 3

Case 4Case 5Case 6





5 Conclusion






0 100 200 300 4000

4

8

Vert

ical

disp

lace

men

t (m

)

Cable span (m)








1 2 3 40

40

80

120

160

200

240

Tens

ion

forc

e (kN

)

Layer of cables










Acknowledgment


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1st

2nd

3rd

Vertical


Conductor line

Ground line

400m

400m


O

i

l

c

x

j

q

dx

ds

z











0 10 20

Tower 1

Time (s)

4

2

0

minus2

minus4

Acce

lera

tion(m

s2)

(a) 1st tower

0 10 20

Tower 2

Time (s)

4

2

0

minus2

minus4

Acce

lera

tion(m

s2)

(b) 2nd tower

0 10 20

Tower 3

Time (s)

4

2

0

minus2

minus4

Acce

lera

tion(m

s2)

(c) 3rd tower




∘ (Case 2) 120572 = 18∘ (Case 3) and









0 100 200 300 40000

03

06Ve

rtic

al d

ispla

cem

ent (

m)

Cable span (m)

Case 1 Case 2

Case 3 Case 4

(a) El Centro

0 100 200 300 40000

02

04

Vert

ical

disp

lace

men

t (m

)

Cable span (m)

Case 1 Case 2

Case 3 Case 4

(b) Oka

0 100 200 300 40000

04

08

Vert

ical

disp

lace

men

t (m

)

Cable span (m)

Case 1 Case 2

Case 3 Case 4

(c) Taft








0 100 200 300 4000

1

2

3

4

Vert

ical

disp

lace

men

t (m

)

Cable span (m)


Case 3






0 100 200 300 4000

4

8

12

Vert

ical

disp

lace

men

t (m

)

Cable span (m)




1 2 3 40

40

80

120

160

200

Tens

ion

forc

e (kN

)



Layer of cables





0 100 200 300 4000

4

8

12

Vert

ical

disp

lace

men

t (m

)

Cable span (m)

Case 1Case 2Case 3

Case 4Case 5Case 6






1 2 3 40

40

80

120

160

200

Tens

ion

forc

e (kN

)

Layer of cables

Case 1Case 2Case 3

Case 4Case 5Case 6





5 Conclusion






0 100 200 300 4000

4

8

Vert

ical

disp

lace

men

t (m

)

Cable span (m)








1 2 3 40

40

80

120

160

200

240

Tens

ion

forc

e (kN

)

Layer of cables










Acknowledgment


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 10 20

Tower 1

Time (s)

4

2

0

minus2

minus4

Acce

lera

tion(m

s2)

(a) 1st tower

0 10 20

Tower 2

Time (s)

4

2

0

minus2

minus4

Acce

lera

tion(m

s2)

(b) 2nd tower

0 10 20

Tower 3

Time (s)

4

2

0

minus2

minus4

Acce

lera

tion(m

s2)

(c) 3rd tower




∘ (Case 2) 120572 = 18∘ (Case 3) and









0 100 200 300 40000

03

06Ve

rtic

al d

ispla

cem

ent (

m)

Cable span (m)

Case 1 Case 2

Case 3 Case 4

(a) El Centro

0 100 200 300 40000

02

04

Vert

ical

disp

lace

men

t (m

)

Cable span (m)

Case 1 Case 2

Case 3 Case 4

(b) Oka

0 100 200 300 40000

04

08

Vert

ical

disp

lace

men

t (m

)

Cable span (m)

Case 1 Case 2

Case 3 Case 4

(c) Taft








0 100 200 300 4000

1

2

3

4

Vert

ical

disp

lace

men

t (m

)

Cable span (m)


Case 3






0 100 200 300 4000

4

8

12

Vert

ical

disp

lace

men

t (m

)

Cable span (m)




1 2 3 40

40

80

120

160

200

Tens

ion

forc

e (kN

)



Layer of cables





0 100 200 300 4000

4

8

12

Vert

ical

disp

lace

men

t (m

)

Cable span (m)

Case 1Case 2Case 3

Case 4Case 5Case 6






1 2 3 40

40

80

120

160

200

Tens

ion

forc

e (kN

)

Layer of cables

Case 1Case 2Case 3

Case 4Case 5Case 6





5 Conclusion






0 100 200 300 4000

4

8

Vert

ical

disp

lace

men

t (m

)

Cable span (m)








1 2 3 40

40

80

120

160

200

240

Tens

ion

forc

e (kN

)

Layer of cables










Acknowledgment


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 100 200 300 40000

03

06Ve

rtic

al d

ispla

cem

ent (

m)

Cable span (m)

Case 1 Case 2

Case 3 Case 4

(a) El Centro

0 100 200 300 40000

02

04

Vert

ical

disp

lace

men

t (m

)

Cable span (m)

Case 1 Case 2

Case 3 Case 4

(b) Oka

0 100 200 300 40000

04

08

Vert

ical

disp

lace

men

t (m

)

Cable span (m)

Case 1 Case 2

Case 3 Case 4

(c) Taft








0 100 200 300 4000

1

2

3

4

Vert

ical

disp

lace

men

t (m

)

Cable span (m)


Case 3






0 100 200 300 4000

4

8

12

Vert

ical

disp

lace

men

t (m

)

Cable span (m)




1 2 3 40

40

80

120

160

200

Tens

ion

forc

e (kN

)



Layer of cables





0 100 200 300 4000

4

8

12

Vert

ical

disp

lace

men

t (m

)

Cable span (m)

Case 1Case 2Case 3

Case 4Case 5Case 6






1 2 3 40

40

80

120

160

200

Tens

ion

forc

e (kN

)

Layer of cables

Case 1Case 2Case 3

Case 4Case 5Case 6





5 Conclusion






0 100 200 300 4000

4

8

Vert

ical

disp

lace

men

t (m

)

Cable span (m)








1 2 3 40

40

80

120

160

200

240

Tens

ion

forc

e (kN

)

Layer of cables










Acknowledgment


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 100 200 300 4000

1

2

3

4

Vert

ical

disp

lace

men

t (m

)

Cable span (m)


Case 3






0 100 200 300 4000

4

8

12

Vert

ical

disp

lace

men

t (m

)

Cable span (m)




1 2 3 40

40

80

120

160

200

Tens

ion

forc

e (kN

)



Layer of cables





0 100 200 300 4000

4

8

12

Vert

ical

disp

lace

men

t (m

)

Cable span (m)

Case 1Case 2Case 3

Case 4Case 5Case 6






1 2 3 40

40

80

120

160

200

Tens

ion

forc

e (kN

)

Layer of cables

Case 1Case 2Case 3

Case 4Case 5Case 6





5 Conclusion






0 100 200 300 4000

4

8

Vert

ical

disp

lace

men

t (m

)

Cable span (m)








1 2 3 40

40

80

120

160

200

240

Tens

ion

forc

e (kN

)

Layer of cables










Acknowledgment


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 100 200 300 4000

4

8

12

Vert

ical

disp

lace

men

t (m

)

Cable span (m)

Case 1Case 2Case 3

Case 4Case 5Case 6






1 2 3 40

40

80

120

160

200

Tens

ion

forc

e (kN

)

Layer of cables

Case 1Case 2Case 3

Case 4Case 5Case 6





5 Conclusion






0 100 200 300 4000

4

8

Vert

ical

disp

lace

men

t (m

)

Cable span (m)








1 2 3 40

40

80

120

160

200

240

Tens

ion

forc

e (kN

)

Layer of cables










Acknowledgment


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













0 100 200 300 4000

4

8

Vert

ical

disp

lace

men

t (m

)

Cable span (m)








1 2 3 40

40

80

120

160

200

240

Tens

ion

forc

e (kN

)

Layer of cables










Acknowledgment


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Acknowledgment


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 a parametric study of nonlinear seismic...

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