An X, Gosling PD, Zhou X. Analytical structural reliability analysis of a suspended cable. Structural Safety 2016, 58,

Transcription

1 An X, Gosing PD, Zhou X. Anaytica structura reiabiity anaysis o a suspended cabe. Structura Saety 2016, 58, Copyright: 2015 Esevier. This manuscript version is made avaiabe under the CC-BY-NC-ND 4.0 icense DOI ink to artice: Date deposited: 16/09/2015 Embargo reease date: 10 September 2016 This work is icensed under a Creative Commons Attribution-NonCommercia-NoDerivatives 4.0 Internationa icence Newcaste University eprints - eprint.nc.ac.uk

2 Anaytica structura reiabiity anaysis o a suspended cabe Dr. Xuwen An, Associate Proessor, Wuhan University, P. R. China, Emai: anxw@whu.edu.cn, Pro. P.D. Gosing*, Proessor, Newcaste University, UK, Emai: p.d.gosing@nc.ac.uk (* corresponding author) Dr. Xiaoyi Zhou, Researcher Associate, Newcaste University, UK, Emai: xiaoyi.zhou@nc.ac.uk Abstract: Suspended cabes, incuding transmission ines, suspension bridge cabes, and edge cabes o roo structures, eature in high proie and arge span projects or architectura reasons, or their unctiona eiciency, and or ease o construction, particuary over arge spans. A suspended cabe predominanty reacts externa oads by means o axia tension, and is, thereore, abe to make u use o the materia strength. Because o the senderness o the suspended cabe, the structura response is noninear, even i the materia property is within the eastic range. From a mechanics perspective, thereore, these types o structure exhibit high eves o geometric non-inearity. For this reason, the noninear reationships between tension orce, norma dispacement, and the externa oads need to be considered. In aiming to determine the structura saety o a suspended cabe, and to understand which uncertainty eatures have the greatest inuence, these reationships are written within a probabiistic ramework. This artice briey sets the anaysis o suspended cabes within the context o geometricay noninear eastic structures and corresponding inite eement anaysis methodoogies. Anaytica soutions to the tension and norma dispacement o a suspended cabe subjected to externa oading are presented. Noninear perormance unctions, in the orm o either the cabe tension or norma dispacement are stated. Anaytica expressions or the required gradients o the perormance unction o a suspended cabe with respect to the basic variabes under static oads are deveoped. The structura reiabiity o a suspended cabe is studied using a irst-order reiabiity method (FOSM) and veriied by comparison with Monte Caro simuation (MCS) and Monte Caro simuation based optimization principes (MCOP) or a number o exampes. Load cases incuding, wind, snow, and temperature variation are incuded. Key words: suspended cabe, anaytica soution, reiabiity anaysis, FOSM, Monte Caro simuation. 1. Introduction Suspended cabes, incuding transmission ines, suspension bridges and roos, are widey used as ong-span engineering soutions. Unike conventiona structures, a suspended cabe is a exibe structura system, the shape o which cannot be prescribed, but must take a orm-ound coniguration determined by equiibrium and geometric constraints on the basis o a predeined initia stress state [1-3]. Because o the senderness and exibiity o a suspended cabe, the structura responses are noninear even i the materia property is within the eastic range. For this reason, geometric noninearity shoud be considered in the anaysis o a suspended cabe. The deterministic geometricay noninear anaysis o a suspended cabe is expained in reerences [4-6] and others. In reaity, the variabes aecting the saety o structure are random because these parameters contain uncertainties introduced in the orm o epistemic uncertainties in the design process, and aeatoric uncertainties in the orm o materia characteristics, construction toerances, and service conditions 1

3 incuding oading. An accurate prediction o the perormance o an anayticay described suspended cabe in the presence o uncertainty is presented in this paper. Previousy pubished work on the reiabiity anaysis o cabe structures has mainy ocused on evauating the reiabiity o geometricay noninear structures through the use o the inite eement method. For instance, P.-L. Liu, A.D. Kiureghian [7] introduced the inite eement-based reiabiity method, ormuated using FOSM and SORM principes, or geometricay noninear eastic structures, and created a genera purpose reiabiity anaysis code to evauate structura reiabiities. Xinpei Zhang [8], proposed an agorithm to cacuate the saety index with imit states based on eement strength and in service perormance o a cabe structure using a noda dispacement agorithm and the checking point (JC) method o reiabiity. K.Imai, D.M.Frangopo [9-10] and D. M. Frangopo, K. Imai [11] considered the noninear reationships between strains and dispacements and investigated the system reiabiity evauation o suspension bridges by a probabiistic inite eement anaysis approach. The above methods, combining probabiistic theory with the traditiona inite eement anaysis, are one o the more practicay eective methods to anayse the reiabiity o compex structures in the broadest sense. However, itte work has been done on the reiabiity anaysis o a suspended cabe based on cassica anaytica soutions at present, which, once avaiabe, negates the need to consider urther numerica modeing, and oers both computationa eiciency and cear deinitions o assumptions that may contribute to otherwise unknown or uncear epistemic uncertainty. The aim o this paper is to ormuate and quantiy the reiabiity o a suspended cabe on the basis o a cassica anaytica structura mechanics soution. In the oowing section, the theory o suspended cabe is described and the anaytica soution or the tension o a suspended cabe is provided. In section 3, the imit state unction o suspended cabe is estabished and the gradients o the imit state unction with respect to the basic random variabes are deduced. In section 4, the computationa accuracy o FOSM used in suspended cabe is demonstrated by comparison with Monte Caro simuation (MCS) and Monte Caro simuation based optimization principe (MCOP) by using an exampe o transmission ine (suspended cabe). Concusions rom the present study are drawn in the ast section. 2. Anaytic soution o a suspended cabe 2.1 Seection o cabe equation Foowing the theoretica basis o Irvine [1], we consider here a proie adopted by a uniorm cabe suspended between two rigid supports that are at the same eve and subjected to a uniorm distributed oad mg aong cabe ength, as shown in ig.1. It is assumed that the cabe: (1) is perecty exibe and devoid o exura rigidity; (2) can sustain ony tensie orces; (2) is composed o a homogeneous materia which is ineary eastic. A z ds mg ds H V mgds dz H+dH dx V+dV Fig.1 a cabe under a distributed oad aong the cabe ength and equiirium o an eement o cabe B x 2

4 Considering the sketches in ig.1, the vertica and horizonta equiibrium o the isoated eement o the cabe ocated at (x, z) require that, dv + mgds = 0 (1) dh = 0, (2) where, H is the horizonta component o cabe tension. H is constant everywhere no ongitudina oads act on the cabe, or Equation (2) may be directy integrated; V is the vertica component o cabe tension and can be written as V = Htanθ = H dz dx. (3) Dierentiating equation (3) with respect to x and substituting it into equation (1), and noting that the oowing geometric constraint that must be satisied, the governing dierentia equation o the cabe is obtained as, ( dx ds ) 2 + ( dz ds ) 2 = 1, (4) H d2 z dx 2 + mg 1 + ( dz dx ) 2 = 0. (5) Soving the dierentia equation (5), the proie unction o the cabe that satisies the boundary conditions in Fig.1 can be obtained as, z = H q [cosh mg 2H cosh (mgx H mg )] (6) 2H This is a catenary equation o a suspended cabe, uy determined by the coordinate x at any point, (or exampe, the sag at mid-span). I the sag at mid-span o cabe is, namey, x = /2, z =, the horizonta component o cabe tension H can be cacuated rom equation (6), as in, = H mg (cosh 1). (7) mg 2H Because the catenary equation o the suspended cabe and the equations derived rom it a invove hyperboic unctions and, thereore, transcendenta equations as unctions o the probem deining variabes, the soution o the system o dierentia equations is overy compicated. It is, however, possibe to derive some reativey simpe soutions or speciic oading and boundary conditions, such as or a proie under a distributed oad mg aong the cabe span, or exampe. In the context o a transmission ine ideaisation, we may consider the proie o a uniorm cabe spanning between two supports at the same eve, generated by a uniormy distributed vertica oad 3

5 mg, as shown in ig.2. The horizonta equiibrium o the isoated eement o cabe is the same as deined in equation (2). The vertica equiibrium o an eement requires that, dv + mgds = 0 (8) A mg B x x dx z V H mgdx H+dH dx V+dV Fig.2 a cabe under a distributed oad and equiirium o an ement o the cabe. Dierentiating equation (3) with respect to x and substituting it into equation (8), then, H d2 z = mg (9) dx2 I the proie is reativey at, so that the ratio o sag span is 1:8 or ess, the dierentia equation governing the vertica equiibrium o an eement is accuratey speciied by equation (9) [1]. Integrating equation (9) twice and appying the boundary conditions identiied in Fig.2, the soution or z as a unction o the system characteristic vaues and the ongitudina co-ordinate, x, is, z = mg x( x). (10) 2H Given that the soution to the sag at mid-span o the cabe is, that is, at x = /2, z =, and substituting this condition into equation (10), the horizonta component o cabe tension H can be obtained as, H = mg2 8. (11) Substituting equation (11) into equation (10), we can have the proie unction o cabe as, z = 4x( x) 2. (12) Equation 12 describes a paraboa that is uy determined by the sag at the mid-span o cabe. Comparing the deormed geometry o the catenary computed using iterative method using (6) and (7) with the paraboa in equation (12), when the sag at the mid-span o two proie unctions o the cabe are equa, the maximum dierences o the deormed geometry are summarised in Tabe 1 and iustrated in Fig.3 (or a sag to span ratio o approximatey 0.15). The dierences in the predicted 4

6 horizonta component o the cabe orce or the catenary and paraboa, H = H c H p (H c and H p are the deormed geometries o the catenary and paraboa soutions, respectivey) as a percentage o the appied norma oad, are isted in Tabe 1. x z d /2 paroboic catenary Fig. 3 Iustrative (exaggerated) dierence H o the deormed geometries o the catenary (equations (6) & (7)) and the paraboa (equation (12)) Tabe 1 the maximum dierence o the deormation and H o catenary and paraboic / δ/ 0.08% 0.32% 0.70% 1.19% 1.75% 2.37% H/mg 0.83% 1.63% 2.39% 3.09% 3.72% 4.28% It is cear that the dierences in the predicted deormed geometries and the horizonta components o cabe tension, are very sma, and as the sag tends to zero, the two soutions tend to converge. With the cacuation o the catenary equation overy compex and invoving an iterative soution approach, we may note that when mg/h is sma such that the cabe ength is ony ractionay onger than the span, the substitution o a power series approximation or a hyperboic unction yieds the properties associated with a paraboa, which is then the imiting orm o the catenary as the proie attens. In a practica engineering context, given that the sag to span ratios o suspended cabes are requenty reativey sma, and that i the actua oad distribution can be approximated as a uniormy distributed, then the errors arising rom the paraboic approximation are acceptabe [1,2], and thereore orms the theoretica basis or the remainder o the paper. 2.2 Paraboic proie and response to a uniormy distributed oad It is an essentia requirement o cabe structura theory that the eects o geometric noninearity shoud be incuded in the anaysis o suspended cabe structures. In comparison with the initia sag o a suspended cabe, the vertica deormation generated under a oad increment suppementary to the se-weight o the cabe may be substantia, especiay in cases where the initia sag is sma. The equiibrium equation o a suspended cabe is not set by its initia state, but rather, it is deveoped by considering the change o the cabe proie produced with the change o externa oads and the preceding deormed state. We consider a simpy supported cabe, with the two supports at the same eve, subjected to a uniormy distributed oad o intensity mg + q per unit span on. The proie o the initia state o the cabe under the action o its se-weight mg is deined by equation (12) and the horizonta component o cabe tension H is cacuated using equation (11). w denotes the additiona vertica deection o the cabe and h is the increment in the horizonta component o cabe tension, arising rom the action o the suppementary (to the se-weight mg) uniormy distributed oad q. The 5

7 deormed proie o the cabe under the combined oading mg + q is obtained by augmenting equation (10), i.e., (H + h)(z + w) = (mg + q)x ( x) (13) 2 Expanding the et-hand side o equation (10) and making use o the coniguration o the cabe under se-weight (e.g. equation (10), we obtain, w = where w = w/(q 2 /H), h = h/h, x = x/ and q = q/mg. 1 (1 h/q)x(1 x) (14) 2(1 + h) To compete the soution, h must now be cacuated. The equiibrium equation or the cabe (equation (14)) simpy provides the reationship between the externa oad q, the vertica deection w, and the horizonta component h o cabe tension under the current coniguration. The unknown increments in horizonta component o cabe tension, and additiona sag or deection, h and w, are not independent. The current coniguration represents an intermediate state in deining the ina equiibrated orm that cannot be deined by the singe equiibrium equation (14). Given the appied oad parameter q a priori, equation (14) is insuicient to mathematicay sove or the unknown parameters w and h. The deormation o the suspended cabe needs to be considered during the transition process rom the initia state to the ina state so as to both estabish the equation o equiibrium and ensure compatibiity and continuity aong the cabe ength. As constitutive modeing or cabes is not within the scope o this paper, Hooke s aw is used here to reate the changes in cabe tension to changes in the cabe geometry when the cabe is dispaced rom its origina se-weight equiibrium proie. Thereore, a change in ength o a component ength o the cabe may be reated to a corresponding change in axia tension according to, t EA = ds ds (15) ds where t is the increment in tension exerted on the eement, t = hds/dx; E is Young s moduus; A is the cross-section area o the cabe (assumed uniorm); ds is the origina ength o the eement; and ds is its ina ength,ds = (dx + du) 2 + (dz + dw) 2 ; and u is the ongitudina components o the dispacement. ds may be expanded as a Tayor series, remaining suicient to the second order o sma quantities or a suspended cabe with a shaow sag (i.e. sag to span ratio o 1:8 or ess). Meanwhie, considering the increment o horizonta tension, h, to be constant given the absence o ongitudina oads are acting on the cabe, the oowing equation may be derived i u and w are both considered to vanish at the supports; hl e EA = dzdw dx 2 dx (dw dx ) dx, (16) 0 0 6

8 where L e is a cacuating quantity ony a itte greater than the span itse, L e [1 + 8(/) 2 ]. Substituting the deinition or w rom equation (14) into (16), and using the convenient dimensioness orms o the variabes to compete the integration, we obtain a cubic unction o h o the orm, h 3 + (2 + λ2 24 ) h2 + (1 + λ2 12 ) h λ2 12 q (1 + 1 q) = 0 (17) 2 λ 2 = (mg/h)2 HL e /EA (18) The independent parameter λ 2, accounting or geometric and eastic eects, is o undamenta importance in the static response o suspended cabes. Equation (17) characterises the inherent geometric non-inearity o a suspended cabe. With the deinitions, a = 2 + λ 2 /24 { b = 1 + λ 2 /12 (19) c = λ 2 q(1 + q/2)/12 equation (17) may be reduced to the orm, h 3 + ah 2 + bh c = 0 (21) where the coeicients a, b and c are rea constants. The positive rea root o equation (17) is obtained as; deining, { B = b a2 /3 D = 2a 3 /27 ab/3 + c, (22) and = (D/2) 2 + (B/3) 3 0, then, 1 h = a/3 + ( D/2 + (D/2) 2 + (B/3) 3 ) 3 + ( D/2 (D/2) 2 + (B/3) 3 ) 1 3 (23) or = (D/2) 2 + (B/3) 3 < 0, h = a/3 + 2( B/3) 1 D/2 2cos(θ/3), and θ = arccos [ ( B/3) 3/2], (24) such that the method appicabe to the cacuation h depends on the parameter, i.e. the reative magnitudes o B and D, which are determined by the parameter λ 2 and the oads q appied on cabe. This competes the undamenta anaytica approach in deining the soution (e.g. cabe tension and dispacement) to the probem o a suspended cabe subjected to se-weight (to deine the origina coniguration) and a uniormy distributed oad. 7

9 Variations in the ambient temperature can change the ength o a suspended cabe, with correspondingy non-inear impacts on the cabe sag and tension. Equation (17) may then be extended to; h 3 + (2 + λ EA αδtl HL t ) h 2 + (1 + λ2 e EA αδtl HL t ) h λ2 e 12 q (1 + 1 EA q) + αδtl 2 HL t = 0 e (25) where α is the coeicient o expansion, t describes a uniorm temperature change, and L t is deined as L t = [1 + 16(/) 2 /3]. Writing, a = 2 + λ 2 /24 + EAαΔtL t /HL e { b = 1 + λ 2 /12 + 2EAαΔtL t /HL e, (26) c = λ 2 q(1 + q/2)/12 + EAαΔtL t /HL e the required soution or h (equation (25)) may be cacuated rom equation (23) or (24) according to the magnitude o. In practice, a cabe is subjected to the action o vertica oads such as se-weight mg, ice accumuation, or other, simiary vertica, ive oads, q i ; simiary, a suspended cabe may aso be subjected to the action o horizonta ive oads, q w, arising rom wind pressure, or exampe, and acting norma to the projected surace o the cabe. Other secondary structura oads rom other attached cabes or structura membranes, or exampe, where the actions may be appied to the cabe at arbitrary directions, may be resoved into vertica, q i, and horizonta, q w, components. The tota imposed oad on the cabe may be deined by the vector sum o vertica and horizonta oads. The tota oad acting on the cabe per unit ength is, q = (mg + q i ) 2 + q w 2. (27) Under the combined orces q i and q w, the cabe rotates to an equiibrated pane at an ange β 0, known as the windage yaw, to the vertica, where, q w β 0 = actan ( ) (28) mg + q i The components o se-weight mg, ice oad q i and wind oad q w in the oca co-ordinate direction deined by the windage yaw is, mg = mgcosβ 0 { q i = q i cosβ 0 q w = q w sinβ 0 (29) and q = mg + q = mg + q i + q w, and the reationships between q and the basic oads mg, q i and q w is showed as in Fig.4 (b). 8

10 q w b 0 b 0 mg' mg q w q q' w q i q i +mg q (a) Tota oad acting on a suspended cabe per unit ength q' i (b) The reationship between q and mg, q i and q w Fig.4 Tota oad acting on a suspended cabe under vertica and horizonta oads and the reationships between and the basic oads and mg. 3. Reiabiity cacuation o a suspended cabe 3.1 Structura reiabiity anaysis approach The irst-order (FOSM) and second-order (SORM) reiabiity methods perorm anaytica probabiity integration to provide nomina saety indices or probabiities o aiure [10, 12-14]. FOSM is considered to be one o the most reiabe computationa methods or expanding the imit state unction as a irst-order Tayor series expansion about the checking point, approximating the imit state unction by a tangent (hyper-) pane. Simiary, SORM expands the imit state unction as a second-order Tayor series expansion, approximating the imit state unction by a parabooid. The reiabiity index in FOSM or SORM, representing the shortest distance rom the origin in standardized norma space to the hyper-pane or parabooid, can be cacuated by soving an optimization probem [13]. I the imit state unction is noninear near the checking point, SORM may provide more accurate resuts, but it is an approach that is inherenty more compex. However, i the imit state unction is neary inear in the vicinity o the checking point, the reiabiity indices provided by both methods may be cosey equivaent [10]. From equation (23), (24) and (30), (31), it is cear that in principe the imit state unction the anayticay derived description o a suspended cabe is noninear. Liu and Der. Kiureghian [7,11] investigated the dispacement reiabiity o a two-dimensiona geometricay noninear eastic structure in the orm o a stochastic pate under random static oads, and concuded that, in spite o strong noninearity o the structura response, the resuts provided by FOSM and SORM were simiar, with FOSM providing a good measure o the structura reiabiity. Initiay based on these resuts, FOSM has been seected to estimate the reiabiity o a suspended cabe described in the preceding section. Concurrenty, in order to assess the vaidity o adopting FOSM, Monte Caro simuation is aso used to estimate the imit state probabiities. 3.2 Deining the imit state unction The tension, T, at any point, x, aong the ength o a cabe with supports that are at the same eve, is, T = [(H + h) 2 + (mg + q)(/2 x)] 1/2, (30) and at the maximum sag point, x = /2, the cabe tension is T = H + h. 9

11 The imit state unction, Z, at any point aong the cabe may then be written as, Z = R S = y A T (31) where, R is the resistance o cabe in tension; y is yied strength o materia used or cabe; A is area o crossed section o cabe; S is oad eect under dierent oad; H, cacuated by equation (11), and h, cacuated by equation (23) or (24), respectivey, are the horizonta component o cabe tension under se-weight mg and externa oad q. 3.3 Anaytica gradients temperature independent case The basic random variabes aecting the reiabiity o a suspended cabe, without considering the eects o a change o temperature, incude materia properties o the cabe (yied (or working) strength o the materia y, Young s moduus E), oads or actions (se-weight mg, horizonta ive oad q w, vertica ive oad q i ), and geometric parameters (cross-sectiona area A, cabe sag, and span ength ). The dimensioness horizonta component o tension orce h o the cabe in the imit state unction Z is cacuated using equation (23) I Δ 0, and (24) i Δ < 0. The reationships between the imit state unction and the basic random variabes, without considering the temperature change, is shown in Fig. 5 and the reationship between q, in the windage yaw pane arising rom combined horizonta and vertica oads, and mg, q i and q w is shown in Fig.4(b). Z h y A H B D mg' b a c 2 q E A mg' q' Z h y A H q mg' B D b a c 2 q E A mg' q' (a) For Δ 0 (b) For Δ < 0 Fig. 5 Reationships between the imit state unction and basic variabes excuding temperature eects. The gradients o the imit state unction Z o a geometricay noninear eastic cabe with respect to the basic random variabes are derived rom the compex reationships shown schematicay in Fig.5 and Fig.4(b), using the chain rue o dierentiation or the cases that Δ 0 and Δ < 0. I Δ 0, the parameter h in the imit state unction Z is expressed by equation (23) and the ogica reationships between Z and basic variabes are shown in Fig.5 (a). The gradients are computed by taking the derivative o Z with respect to the random variabes as; (mg) = [ η 1 EA(8/) 3 (mgcosβ 0 ) 2 2 e 8 (1 + h)] (sin2 β 0 cosβ 0 + cosβ 0 ) Hλ2 ξ 1 72mg (1 + q) (tan2 β 0 + q w mg tanβ 0 + q i mg ) (32a) 10

12 = η 1 4EA(8/) 3 [1 + 4(/) 2 ] mg e 2 mg (1 + h) 4 (32b) = η 1 8EA(8/) 2 [3 + 8(/) 2 ] mg e 2 + mg2 (1 + h) (32c) 82 = λ2 Hξ 1 q w 36 (1 + q) tanβ 0 mg = λ2 Hξ 1 q i 72 (1 + q) (1 tan2 β 0 ) mg E = η A(8/) 3 1 mg e A = E(8/) 3 y + η 1 mg e (32d) (32e) (32) (32g) where, y = A η 1 = H 72 H 1 B [( D ) 3 D ( 2 ) 2 3 ] (1 a (32h) 3 ) Hξ 1 72 [a2 9 b 6 a 3 q (1 + q 2 )] (32J) ξ 1 = ( D ) 3 D ( ) ( D 2 2 ) 3 D (1 + 2 ) (32k) I Δ < 0, the parameter h in the imit state unction Z is expressed by equation (24) and the ogica reationships between the imit state and the basic variabes are shown in Fig.5 (b). Considering the imit state unctions in the cases o Δ 0 and < 0, the ony dierence is in the expression or h (e.g., c.. equation (23) with (24)). The dierence in the gradients between the two cases is in the partia derivatives o h with respect to a, B and D, which can be cacuated rom equation (24). The remaining gradients are unchanged rom the case 0. Foowing the same anaytica computationa procedure, the gradients in the case o < 0, with the exception o the gradients o the imit state unction with respect to the oads mg, q w and q i, are quite dierent rom equation (32a), (32d) and (32e). In the case < 0, the gradient / y is identica to equation (32h) where 0, whist the remaining gradients o the imit state unction with respect to the parameters,, E and A (iustrated in equation (33a) (33c)), require repacing η 1 in the corresponding equations with η 2 as deined in equation (33d). 11

13 (mg) = [ η EA(8/) 3 2 (mg) 2 2 e 8 (1 + h)] (sin2 β 0 cosβ 0 + cosβ 0 ) + λ2 Hξ 2 18mg (1 + q) (tan2 β 0 + q w mg tanβ 0 + q i mg ) (33a) = λ2 Hξ 2 q w 9 (1 + q) tanβ 0 mg = λ2 Hξ 2 q i 18 (1 + q) (1 tan2 β 0 ) mg (33b) (33c) η 2 = H 72 + H 36 [( B 1/2 3 ) cos ( θ 3 ) BDξ 2 3 ] (1 a 3 ) + Hξ 2 18 [a2 9 b 6 a 3 q (1 + q )] (33d) 2 ξ 2 = ( B/3) 1 1 2sin(θ/3) 2 ( B/3) 3 (D/2) 2 (33e) 3.4 Anaytica gradients temperature dependent case I the eects o a uniorm temperature rise t need to be incorporated, a coeicient o therma expansion α t or the cabe materia and eects o a temperature change t are introduced in addition to the eight basic variabes incuded in the case o without considering temperature change. The expressions o the imit state unction and a parameters among the cacuation proceeding are the same as that without considering temperature, with the exception o the expressions o a, b, c, in which a new item is added due to a temperature rise (see equation (26)). To obtain the partia derivatives it is convenient to introduce an intermediate variabe, h t, taking into account the eect o temperature in (26), is deined as h t = EAα t ΔtL t /HL e (34) The reationships between the imit state unction and the basic random variabes, considering temperature eect, are shown in Fig.6, and the reationships between q in the windage yaw pane and mg, q i and q w are the same as without considering the temperature change, as shown in Fig.4(b). In comparison with the temperature independent case, it is required to add the gradients o the imit state unction with respect to α t and Δt, aong with the partia derivatives o parameters a, b, c with respect to the basic random variabes where there is dependency on temperature variation. 12

14 Z A h y H B D mg' b a c h t 2 q a t Dt E A mg' q' Z h y A H q mg' B D b a c h t 2 q a t Dt E A mg' q' (a) For Δ 0 (b) For Δ < 0 Fig. 6 Reationships between the imit state unction and basic variabes incuding temperature eects. The gradients considering the eect o temperature are obtained as; For Δ 0 α t = η t1 8EAΔt t mg 2 e t = η 8EAα t t t1 mg 2 e (35a) (35b) (mg) = [ η EA(8/) 3 1 (mgcosβ 0 ) 2 2 e 8 (1 + h) η 8EAα t t t t1 (mgcosβ 0 ) 2 2 ] (sin 2 β 0 cosβ 0 + cosβ 0 ) e Hλ2 ξ 1 72mg (1 + q) (tan2 β 0 + q w mg tanβ 0 + q i mg ) (35c) = η 1 = η 1 4EA(8/) 3 [1 + 4(/) 2 ] mg e 2 mg 4 (1 + h) η t1eaα t Δt 8 (4/) 4 /3 + (8/) 2 /3 + 2 mg2 e (35d) 8EA(8/) 2 [3 + 8(/) 2 ] mg2 + mg2 e 8 2 (1 + h) + η t1eaα t Δt 4(4/)4 /3 + (8/) mg2 e E = η A(8/) 3 8Aα t Δt t 1 + η mg t1 e mg 2 e (35e) (35) A = E(8/) 3 8Eα t Δt t y + η 1 + η mg t1 e mg 2 e (35g) η t1 = H1 3 H B 2 27 [( D ) 3 D ( 2 ) 2 3 ] (1 a 3 ) + ξ 1 6 (a2 9 b 6 a ) (35h) 13

15 Comparing the corresponding case in Fig.5 and in Fig.6, the reationships between the imit state unction Z and the ive oads q w and q i, are independent o whether the eect o temperature is considered or not. Hence, the gradients o the imit state unction Z with respect to the ive oads q w and q i are same as equation (32d) and (32e), in which the eect o temperature is not considered and Δ 0, respectivey. The gradient o the imit state unction Z with respect to y is the same as equation (32h) in the case o 0 without considering the eect o temperature. I Δ < 0 and considering the eect o temperature, the dimensioness horizonta component o the cabe tension orce h in the imit state unction Z is expressed by equation (24) and the reationships between the imit state and basic variabes are shown in ig.6 (b). The gradients o the imit state unction with respect to the oads mg remain unchanged with the incusion o temperature eects. For < 0, equation (35c) is repaced by equation (36a). The gradient / y remains as equation (32h) in the case o 0 without considering the eect o temperature. The outstanding gradients o the imit state unction with respect to the parameters α t, t,,, E and A are simiar with the corresponding equations in the case o 0, requiring repacement o η 1 and η t1 with η 2 expressed in (33d) and η t2 in (33d), respectivey. The gradients o the imit state unction Z with respect to the ive oads q w and q i are same as equation (33b) and (33c), in which the eect o temperature is not considered and Δ < 0, respectivey. (mg) = [ η EA(8/) 3 2 (mgcosβ 0 ) 2 2 e 8 (1 + h) η 8EAα t t t t2 (mgcosβ 0 ) 2 2 ] (sin 2 β 0 cosβ 0 + cosβ 0 ) e + λ2 Hξ 2 18mg (1 + q) (tan2 β 0 + q w mg tanβ 0 + q i mg ) (36a) η t2 = H 3 + 2H 3 [( B/3) 1 2cos(θ/3) ξ 2 BD/3] (1 a 3 ) + 4Hξ 2 3 (a2 9 b 6 a ) (36b) Comparing the corresponding gradients o the imit state unction with respect to the basic random variabes in section 3.3 and section 3.4, the eect o incuding temperature changes are that new items, denoted η t1 or η t2, are generated in a gradients except or / y, aong with two new gradients that are concerned with the temperature change and the coeicients o therma expansion. 4. Cabe reiabiity anaysis: exampes and discussion Eight basic random variabes are deined in the absence o temperature eects, which are increased to ten when considering changes due to temperature, a o which aect the reiabiity o the cabe to varying degrees. More generay, there may be many basic random variabes X i (i = 1,2,, n) describing the structura reiabiity probem at the imit state unction G(X) = 0. The undamenta FOSM agorithm described by Mechers is adopted in this work [25]. A conductor or wire, widey used in overhead transmission ines in eectrica engineering, is typica a suspended cabe. An overhead transmission ine is seected as an exampe to demonstrate the impementation o the preceding ormuation. The structura saety and sensitivities o suspension cabes are cacuated. An overhead transmission ine may be subjected to externa uncertainties incuding environmenta conditions (wind and ice), secondary environmenta actors, such as temperature 14

16 variations, or exampe, and intrinsic manuacturing uncertainties incuding axia stiness o the cabe, and se-weight. In design practice, these uncertainties are normay accounted or by the use o a actor (requenty described as a actor o saety) to give a permissibe stress vaue (the utimate tensie strength divided by a minimum actor saety) or the maximum permissibe tension when scaed by a measure o the cross-sectiona area. Any ack o conidence in the resuts o the anaysis, in the orm o epistemic uncertainty, may aso be reected in this singe saety actor. Current practice in the design o the transmission ine conductors adopts the catenary equation (6) [18] or the paraboic equation (12) [19] and appies a oads to the conductor as a singe oad case to cacuate the sag, whie the tension at the owest point (i.e. the horizonta tension) is set in advance to be equa to the maximum permissibe tension. In this case, however, it is not convenient to estabish the imit state unction and compute the reiabiity o the conductor in tension. Hence, the imit state unction (31), considering the geometricay non-inear behaviour o suspension cabe, is used to compute the reiabiity o the conductor in tension, and the sag is assumed to be known priori in terms o engineering experience. The minimum actor o saety in respect o conductor tension is 2.5 at the owest point on ine and 2.25 at the suspension point [18]. Exampe 1: A 330kV overhead transmission ine is suspended between two equa height supports. The transmission ine is a round wire concentric ayup overhead eectrica stranded conductor, comprising auminium cad stee wires (designation JL/G1A-240/30). It has a mass per unit ength m=0.9207kg/m. The equivaent diameter o the conductor ine, d, is 21.6mm, and the cross-sectiona area, A, is mm 2 [15]. The span o the transmission ine is 300m and the height o wire hanging on the structure is h s = 45m, the cacuating height or the wind oad o wire is h c = 40m computed by the equation in reerence [17], as shown in Fig.7. Conductor 5 hs=45 Poe or Tower =300 Poe or Tower hc=40 (Note: A dimensions are in meters) Fig.7 The diagrammatic drawing o the suspended conductor A singe basic oad case combination is considered: extreme radia ice thickness δ max = 20mm with a concurrent wind speed v = 10m/s which is perpendicuar to the ine and a temperature variation o t = 5. The wind oads and ice cover oad acting on the ine in this case o oads combination can be cacuated according to the corresponding rues as oows [16, 17] : I the density o ice is taken as 900kg/m 3, the ice cover oad on ine per meter is: q i = 0.9 πδ(δ + d)g 10 3 N/m (37) In genera, the density o air is taken as 1.25kg/m 3 at a temperature o 15 and an atmospheric pressure o kpa at sea eve, and the wind oad on ine per meter is: q w = 0.625v 2 (d + 2δ)α μ z μ sc 10 3 N/m (38) 15

17 where, g represents the acceeration o gravity, g = N/mm 2 ; α represents the wind pressure coeicient o non-uniormity, shown in Tabe 2; μ sc is the shape coeicient o conductor ine, shown in Tabe 3; δ is the icing thick on ines, and δ=0 without ice cover; and μ z represents the wind pressure exposure coeicients, varying with the height o wires and shown in Tabe 4. Tabe 2 Wind pressure coeicient α [18] wind speeds v(m/s) v <20 20 v <27 27 v <31.5 v 31.5 α Tabe 3 Conductor shape coeicient μ sc [18] conductor surace condition without ice cover with ice cover conductor diameter d(mm) d < 17 d 17 μ sc Tabe 4 Wind pressure exposure coeicients (excerpts) [18] height o the conductor type o the ground roughness above ground or sea eves (m) A B C D Notes: or the type o ground roughness, Type A represents the paces near the sea, isands, seashores, akeshores or desert areas; Type B represents the ieds, countryside, hiy areas, or towns and city suburbs where the houses is sparse; Type C represents urban districts with the crowded buidings; Type D represents urban districts with the crowded and high buidings. Supposing that the transmission ine is ocated within an environment o ground roughness B, the oowing oad case combination is considered: the extreme ice cover with the concurrent wind and the temperature, mg=9.03n/m; q w =7.21 N/m; q i =23.07 N/m; t=-5. For this exampe, a random variabes are assumed to be normay distributed, and the coeicients o variation o variabes mg, q w and q i are, respectivey, 0.07, and or a 50 year reerence period [20] ; that o the tensie strength y and the Young s moduus E o conductor are assumed 0.10 and 0.05, respectivey, with reerence to [11, 21]; that o the geometrica variabes is assumed 0.1 because the minor change o cabe ength may generate a remarkabe variation o [1,2] ; and that o the other parameters are assumed as in Tabe 5. The statistica characteristics o the basic variabes are isted in Tabe 5. Tabe 5 Statistica characteristics o the basic variabes Basic Variabe Statistica standard coeicient mean vaue distribution deviation o variation Se-weight mg (N/m) Norma Length o span (m) Assumed Norma Sag (m) Assumed norma Wind oad q w (N/m) Assumed Norma Icing oad q i (N/m) Assumed Norma

18 Young s moduus E (N/mm 2 ) Norma Section areas A (mm 2 ) Assumed Norma Tensie strength y (N/mm 2 ) Norma Coeicient o therma expansion α t (1/ ) Assumed Norma temperature change t Assumed Norma Based on the statistica characteristics o the basic random variabes isted in Tabe 5, the preceding anaytica FOSM-based agorithm is used to cacuate the reiabiity o the conductors. The resuting vaues o β and p or the dierent cases are isted in tabe 6. To demonstrate the vaidity and rationaity o the resuts computed by the FOSM-based agorithm, the direct Monte-Caro simuation (MCS) and the Monte-Caro simuation based optimization (MCOP) methods are aso used to evauate the aiure probabiity and/or the corresponding reiabiity index o the same exampe. The we known direct Monte-Caro simuation method determines the probabiity o aiure by means o a arge number o simpe repeated samping; it is adaptabe to a genera cass o probems, incuding non-inear imit state unctions. Convergence is reated to the aiure probabiity or the reiabiity index and is independent o the dimensionaity o the probem, and may be sow and computationay expensive i not prohibitive. I a conidence vaue is seected as 95% to ensure the samping error and the reative error is 0.2, the samping numbers required is o the order o 100/p to obtain reiabe estimates or p or the corresponding reiabiity index [14]. In the ight o the resuts o aiure probabiity p cacuated by FOSM in Tabe 6, the required samping number or direct Monte-Caro simuation is at east need 10 7 to obtain reiabe p estimates in this exampe. To reduce the samping number and assure simuation precision, MCOP is introduced or the sma aiure probabiity probem. The principe o the Monte-Caro simuation based optimization method is to deine the random variabes with known distributions by means o samping on the imit state unction and cacuating the distance rom the samping points to the origin. The shortest distance is then taken to be the reiabiity index β. The speciic cacuation methodoogy initiay invoves transorming the constrained optimization probem into a non-constrained probem by combining the imit state unction with n stochastic variabes; that is, one o variabes y i in the imit state unction Z = g(y 1,, y i 1, y i+1, y n ) is expressed by the others, i.e., y i = (y 1,, y i 1, y i+1, y n ), or exampe, y in equation (31); secondy, in order to deine a combination o samping points that are ocated on the imit state surace, n 1 variabes are samped with the exception o y i that is then cacuated rom y i = (y 1,, y i 1, y i+1, y n ); inay, compute the distance rom the samping points to the origin and ind the reiabiity index β [22]. Cacuated vaues o β using the MCS and MCOP approaches as a unction o the number o simuations are aso isted in Tabe 6. Tabe 6 Cacuated vaues o reiabiity index β and aiure probabiity p Cacuating method FOSM MCS MCOP Numbers o simuation or iteration Probabiity o aiure p ( 10-5 ) Reiabiity indices β

19 Dierence D(%) Sensitivity coeicient α i cacuated using FOSM α L =0.308; α =-0.247; α A =-0.226; α qw =0.050; α mg =0.100; α qi =0.372; α y =-0.800; α E =0.055; α αt = ; α t = The reiabiity indices or exampe 1, isted in Tabe 6, and computed using the FOSM, is used as the basis or estimating the reative dierences between vaues obtained rom the MCS and MCOP agorithms, shown in ine 5 in Tabe 6, and vaidating the FOSM approach. The oowing observations are made: (1) The aiure probabiity and the reevant reiabiity index o the conductor in this exampe computed using FOSM is very cose to the resuts computed rom MCS, with the dierence in the reiabiity index being -0.22%. Estimates based on the MCOP agorithm are aso isted in Tabe 6 or comparison with the FOSM. Again, the dierences in vaues are sma, in the range 2.02%~0.95%, and reduce with increasing numbers o MCOP simuations. The resuts vaidate the anaytica reiabiity anaysis o a suspended cabe using the FOSM ormuation presented in this paper. (2) Anaysing the reiabiity estimates computed by the FOSM and that by MCS or MCOP, two main reasons may be identiied or their dierences: irsty, the predictions rom the MCS and MCOP are dependent on the number o simuations, athough the reiabiity graduay converges with increased numbers o simuations but with randomness remaining; the second reason is that the imit state unction is compex or such a strongy geometrica non-inear probem, such that empoying ony irst order terms at the expansion o the checking point in the FOSM wi introduce either an overestimate or an underestimate o the aiure content. It is or this reason that the reiabiity anaysis o a suspended cabe using SORM may be beneicia, athough it is noted that the FOSM predictions are suicient or design. (3) As expected, it is demonstrated that not a o the random variabes have the same eve o eect on the reiabiity or a speciic case. The inuence o the random variabes on the reiabiity prediction can be, o course, identiied by the magnitudes o the sensitivity coeicients α i computed as part o the FOSM. In exampe 1, materia strength y have high sensitivity coeicients, meaning that this random variabe pay signiicant roes in the reiabiity o the conductor. The icing oad q i and the geometrica variabes such as span ength L, sag and cross-section area A o conductor have intermediate eves o sensitivity. The reiabiity is not sensitive to the other variabes incuding se-weight mg, eastic moduus E and wind oadq w, especiay to temperature change t and coeicient o therma expansion α t. I the temperature change t and the coeicient o therma expansion α t is assumed to be deterministic, the reiabiity index o the conductor in Exampe 1 is β = and the corresponding aiure probabiity p = Comparing with the resuts o deaing variabes t and α t with random variabes isted in Tabe 6, the dierence is so sma that they can be assumed to be deterministic. Exampe 2: The overhead transmission ine is same as that in Exampe 1 with the exception that two basic oad case combinations are considered: (1) a maximum wind speed v max = 40 m/s perpendicuar to the ine and a coincident temperature variation t = 10 with respect to the reerence temperature at instaation o the conductor. The ine is taken to be not subjected to ice accretion, such that the radia ice thickness is δ = 0 mm; (2) extreme radia ice thickness δ max = 18

20 15mm with a concurrent wind speed v = 10m/s perpendicuar to the ine and a temperature variation o t = 5. In this exampe, the FOSM is just used to compute the reiabiity o the conductor. Simiary, supposing that the transmission ine is ocated within an environment o ground roughness B, the wind oads and ice cover oad acting on the ine in two cases o oads combination can be cacuated by equations (37) and (38) as oows: In case 1: the extreme wind speed and the concurrent temperature, q w =25.95 N/m; t=10. The standard deviation o wind oad is cacuated in ine with the coeicient o variation in Tabe 5. In case 2: the extreme ice cover with the concurrent wind and the temperature, q w =6.04 N/m; q i =15.22 N/m; t=-5. The standard deviations o wind oad and icing oad are and 2.275, respectivey, cacuated in ine with the coeicient o variation in Tabe 5. The probabiity distribution types o these random variabes are assumed to be normay distributed. The statistic characteristics o the other variabes are same as Exampe 1, as shown in tabe 5, and the variabes t and α t are assumed to be deterministic. Based on the statistica characteristics o the basic random variabes shown above and in Tabe 5, the resuting vaues o β and p at the owest point are cacuated by FOSM or the dierent cases are isted in Tabe 7(a), meanwhie, the actor o saety or this wire are cacuated as K = y A/(H + h) and isted in Tabe 7(a). Tabe 7(a) Reiabiity index β and aiure probabiity p cacuated using FOSM Cases o oads Case 1: extreme wind speed Case 2: extreme ice cover with combination and concurrent temperature concurrent wind and the temperature Sag (m) Factor o saety K Reiabiity indices β Probabiity o aiure p Sensitivity coeicient α i Tabe 7(b) α L =0.212; α =-0.127; α A =-0.173; α L =0.292; α =-0.269; α A =-0.208; α qw =0.698; α mg =0.019; α qw =0.056; α mg =0.117; α qi =0.282; α y =-0.647; α E =0.048; α y =-0.838; α E =0.044 Reiabiity index β and aiure probabiity p cacuated using FOSM Cases o oads Case 1: extreme wind speed Case 2: extreme ice cover with combination and concurrent temperature concurrent wind and the temperature Sag (m) Factor o saety K Reiabiity indices β Probabiity o aiure p Sensitivity coeicient α i α L =0.214; α =-0.132; α A =-0.173; α L =0.290; α =-0.271; α A =-0.206; α qw =0.697; α mg =0.018; α qw =0.056; α mg =0.114; α qi =0.282; α y = α E =0.047; α y =-0.838; α E =0.043 The Eurocode - Basis o structura design (EN 1990:2002) [23] divides the Reiabiity Consequence (RC) into RC1, RC2 and RC3 associated with the three consequences casses. The recommended minimum reiabiity index or utimate imit states associated with dierent RC is 3.3, 19

21 3.8 and 4.3, respectivey, or a 50 year reerence period. The Uniied standard or reiabiity design o engineering structures (GB ) [24] in China aso divides the saety eve o structura members into eve 1, 2 and 3 according to the importance o structure. The target reiabiity index o a structura member subject to a ductie aiure is speciied or each eve as 3.7, 3.2 and 2.7, respectivey. The resuts o exampe 2, isted in Tabe 7, and computed using the FOSM, show that the reiabiity indices in both cases are achieved above 5.0, ar exceeding the recommended eve o RC3 speciied in EN 1990:2002 and eve 1 speciied in GB The actor o saety K in case 2, however, is not satisied with the minimum vaue at the owest point on ine speciied by reerence [18]. For the geometric noninearity o the cabe, the tension orce o cabe wi decrease with the increment o sag i a the other parameters remain the same, and the actor o saety o cabe K wi increase. For exampe, the sag o the conductor is increased to =7.76m to et the actor o saety K just meet the demands o reerence [18], the reiabiity index β is 5.140, as isted in Tabe 7(b). The inuence o the random variabes on the reiabiity prediction in dierent case o oads combination is very dierent. In case 1 o the extreme wind with the concurrent temperature, materia strength y and wind oad q w have high sensitivity coeicients, meaning that these random variabes pay signiicant roes in the reiabiity o the conductor. The geometrica variabes such as span ength L, sag and crosssection area A o conductor have intermediate eves o sensitivity. The reiabiity is not sensitive to the other variabes incuding se-weight mg, and eastic moduus E. In case 2 comprising extreme ice and the concurrent wind and temperature eects, the materia strength y remains a signiicant inuence on the reiabiity o conductor. The sensitivity o the saety o the suspended cabe to the wind oad q w is reduced, consistent with the reduction in magnitude o this appied oad. The sensitivity o ice oad q i is just above that o the geometrica variabes, with the inuence o the cabe span becoming more prominent. 5. Concusions (1) Based on the cassica paraboic anaytica soution o a suspension cabe, the reiabiity o suspended cabe is expored using the FOSM. The necessary incusion o geometric noninearity creates a imit state unction that is compex. The cacuation o the structura reiabiity o a suspended cabe is a compicated probem with a arge number o basic random variabes. Nevertheess, it has been demonstrated that it is easibe to compute the reiabiity o a suspension cabe using the FOSM. (2) The irst-order derivatives o the imit state unction with respect to the basic random variabes can be derived by means o the successive appication o the chain rue o dierentiation. This methodoogy provides a soid oundation or computing the reiabiity o suspension cabe using FOSM. (3) To veriy the rationaity and correctness o the resuts o FOSM, the Monte Caro Simuation method (MCS) and the Monte Caro Simuation Based Optimization principe (MCOP) have been impemented. The reiabiity or the aiure probabiity evauated using MCS and MCOP are very cose to that by obtained rom the FOSM. These outcomes impy that the structura reiabiity soutions or a suspended cabe estimated by the impemented anaytica FOSM are rationa and correct. 20

22 (4) The stress-based structura reiabiity anaysis o a suspended cabe, with an assumed paraboic proie, with the same height supports, and subjected to horizonta and vertica oads, using the FOSM has been demonstrated in this paper. The structura reiabiity o a suspended cabe with ewer restrictions on geometric orm and boundary conditions and incuding deormation imit state unctions wi be studied urther in a separate manuscript. (5) I the saety actor K o the conductor is satisied with a minimum vaue o 2.5 at the owest points o suspended cabe, the reiabiity indices β in dierent cases are ar higher than the eve o RC3 in EN 1990:2002 and that in GB A saety index in excess o 5.1 woud appear to be required to achieve an equivaent saety actor o 2.5. Acknowedgements Financia support provided to Dr. Xuwen An (Fie No ) by China Schoarship Counci (CAS) to pursue research in the United Kingdom as a visiting schoar is grateuy acknowedged. Reerences [1] H.M. Irvine, Cabe Structures. The Massachusetts Institute o Technoogy Press, 1981 [2] Shizhao Shen, Chunbao Xu, Chen Zhao, Design o cabe structure. China Architecture & Buiding Press, 1997 [3] P.D. Gosing, B.N. Bridgens, L. Zhang. Adoption o a reiabiity approach or membrane structure anaysis. Structura Saety 2013; 40:39-50 [4] Crisied MA. Non-inear inite eement anaysis o soid and structures. Chichester, UK: John Wiey; [5] Jianmin Tang, Yin Zhao. Non-inear inite eement method or equiibrium and natura requencies o cabe structures. Journa o Hohai University 1999; 27(1): [6] V Gattuia, L Martinei, F Perotti, F Vestron. Noninear osciations o cabes under harmonic oading using anaytica and inite eement modes. Comput. Methods App. Mech. Engrg 2004; 193: [7] P.-L. Liu, A.D. Kiureghian. Finite Eement Reiabiity o Geometricay Noninear Uncertain Structures. Journa o Engineering Mechanics, ASCE 1991; 117(8): [8] Zhang Xinpei, The Reiabiity Anaysis o Cabe-suspended structures. Journa o UEST o China 1991; 20 (6): [9] K. Imai, D.M. Frangopo. Geometricay noninear inite eement reiabiity anaysis o structura systems. I: theory. Computers and Structures 2000; 77: [10] K.Imai, D.M.Frangopo. System reiabiity o suspension bridges. Structura saety 2002; 24: [11] D. M. Frangopo, K. Imai. Geometricay noninear inite eement reiabiity anaysis o structura systems. II: appications. Computers and Structures 2000; 77: [12] Y.-G. Zhao, T. One. Moment methods or structura reiabiity. Structura saety 2001; 23: [13] Robert E. Mechers. Structura reiabiity anaysis and Prediction (Second Edition). John Wiey & Sons, 1999 [14] Guoan Zhao, Weiiang Jin, Jinxin Gong. Theory o Structura Reiabiity. China Architecture & Buiding Press,