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Applied Sciences
, Volume 8 (8) – Aug 13, 2018

/lp/multidisciplinary-digital-publishing-institute/an-improved-analytical-algorithm-on-main-cable-system-of-suspension-GEGoeuY0w8

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- Multidisciplinary Digital Publishing Institute
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- © 1996-2019 MDPI (Basel, Switzerland) unless otherwise stated
- ISSN
- 2076-3417
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- 10.3390/app8081358
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applied sciences Article An Improved Analytical Algorithm on Main Cable System of Suspension Bridge 1 1 , 2 1 1 1 Chuanxi Li , Jun He *, Zhe Zhang , Yang Liu , Hongjun Ke , Chuangwen Dong and Hongli Li School of Civil Engineering, Changsha University of Science and Technology, Changsha 410114, China; lichx@csust.edu.cn (C.L.); liuyangbridge@163.com (Y.L.); khj_77@csust.edu.cn (H.K.); dd@csust.edu.cn (C.D.); hl@csust.edu.cn (H.L.) School of Civil Engineering, Dalian University of Technology, Dalian 116024, China; zhangzhe@dlut.edu.cn * Correspondence: hejun@csust.edu.cn or frankhejun@gmail.com; Tel.: +86-180-0846-6632; Fax: +86-0731-8525-6006 Received: 11 July 2018; Accepted: 7 August 2018; Published: 13 August 2018 Featured Application: An improved analytical algorithm has been successfully applied in shape ﬁnding during design and conﬁguration control during construction of main cable system for suspension bridges. Abstract: This paper develops an improved analytical algorithm on the main cable system of suspension bridge. A catenary cable element is presented for the nonlinear analysis on main cable system that is subjected to static loadings. The tangent stiffness matrix and internal force vector of the element are derived explicitly based on the exact analytical expressions of elastic catenary. Self-weight of the cables can be directly considered without any approximations. The effect of pre-tension of cable is also included in the element formulation. A search algorithm with the penalty factor is introduced to identify the initial components for convergence with high precision and fast speed. Numerical examples are presented and discussed to illustrate the accuracy and efﬁciency of the proposed analytical algorithm. Keywords: suspension bridge; main cable system; catenary cable element; search algorithm; penalty factor 1. Introduction Cable-supported structures, such as suspension bridges, have been recognized as the most appealing structures due to their aesthetic appearance as well as the structural advantages of cables [1–4]. It is well known that cables cannot behave as structural members until large tensioning forces are induced, such as pre-stressed cable in structures [5]. Therefore, in order to design a cable-supported structure economically and efficiently, it is extremely important to determine the optimized initial cable tensions or unstrained lengths. Generally, designers cannot determine the initial shape arbitrarily when the cable structures are considered. The initial shape is determined while satisfying the equilibrium condition between dead loads and internal member forces, including cable tensions in the preliminary design stage because cable members display strongly geometric nonlinear behavior as well as the conﬁguration of a cable system cannot be deﬁned in stress-free state. The process determining the initial state of cable structures is referred to as “shape ﬁnding ”, “form ﬁnding ”, or“ Initial shape or initial conﬁguration” [6–11]. Until now, nonlinear analysis procedures have been developed for shape ﬁnding problems of cable bridges: the trial-and-error method [12], the initial force method [10,13], the analytical and Appl. Sci. 2018, 8, 1358 ; doi:10.3390/app8081358 www.mdpi.com/journal/applsci Appl. Sci. 2018, 8, 1358 2 of 15 iteration method [14,15], the target conﬁguration under dead loads (TCUD) related methods [9], the optimization method [16,17], and the combined method [18]. Above mentioned various form-ﬁnding approaches are generally into three categories: (1) the simpliﬁed approach; (2) the Finite Element (FE)-based approach; and (3) the analytical method. The simpliﬁed method assumes that the load acts uniformly along the span of the main cable, which follows a parabolic shape [2,17,19]. To account for a cable’s sag effect, Ernst proposed the equivalent modulus of elasticity for a parabolic cable [20]. The simplicity of Ernst’s formula has made it widely used not only in the research ﬁeld, but also for the practical designs of suspension bridges. Owing to its simplicity, this approach has been adopted by several investigators [21–23], and has been proved to be sufﬁcient for some cases. Namely, when a cable has relatively high stress and small length, the Ernst equivalent modulus approach could achieve a good result. However, the parabolic approximation becomes inaccurate for cables with a large sag-to-span ratio (>1/8), which experience self-weight along the length of the cable and concentrated forces from the hangers. To improve the accuracy and facilitate nonlinear analyses of suspension bridges, various FE-based approaches have been developed. In these approaches, most of the ﬁnite element packages are still lack of suitable cable elements. A sagging cable is often simulated as two-node element, multi-node element, and curved element with rotational degrees of freedom [24–26]. The two-node element is only suitable for modeling the cables with high pretension and small length [27,28], and equivalent modulus are used to account for the sag effect. For cables with large sag, a series of straight elements is used to model the curved geometry of cables. The multi-node element is based on the higher order polynomials for the interpolation functions [29,30]. The tangent stiffness matrix and nodal force vector are obtained while using the iso-parametric formulation. These elements give accurate results for cables with small sag. For cable element with large sag, it is necessary to use a large number of elements to model the curved geometry of cable. Therefore, it causes computational costs. These FE-based approaches identify the target conﬁguration of main cable via updating nodal positions and internal tension of cable elements based on nonlinear structural analysis. However, these FE-based approaches elevate the computational effort, and their convergence depends to a large extent on the assumed initial cable conﬁguration and forces. The alternative approach is based on exact analytical expressions for the elastic catenary, since the equilibrium conﬁguration of a hanging cable is a catenary in nature. This method was originally proposed by O’Brien and Francis [31] and was later extensively developed [32–36]. In particular, there are various catenary-type analytical elements available, which can be used to model large sag cables in suspension bridges: (1) Inextensible catenary elements: The cable elements adopted are inﬁnitely stiff in the axial direction and cannot experience any increment of length. In practice, computer applications that are based on this type of element encounter severe difﬁculties, solving procedures tend to experience large numerical instability, causing a very difﬁcult or even impossible convergence. (2) Elastic catenary elements: An elastic catenary curve is deﬁned as the curve formed by a perfectly elastic cable, which obeys Hooke’s law and has negligible resistance to bending, when being suspended from its ends and subjected to gravity. It should be noted that the conventional formulations are based on the hypothesis of small deformations, meaning that the forces are integrated with respect to the initial conﬁguration of the catenary. Hence, the weight per unit length does not vary consistently with the elongation of the catenary. This may result in an inaccurate equilibrium of forces in the deformed conﬁguration. The main advantages of the catenary-type cable elements are the reduction of degrees of freedom, the simplicity of ﬁnding the dead load geometry of the cable system, the exact treatment of cable sag, the exact treatment of cable weight as it is included in the equations used for element formulation, and the simplicity of including the effect of pre-tension of the cable by simply giving the unstressed cable length. However, the cable segment equation is unsolvable when the initial three components are not set properly because of the so-called initial value sensitivity. Appl. Sci. 2018, 8, 1358 3 of 15 Appl. Sci. 2018, 8, x FOR PEER REVIEW 3 of 15 The purpose of this paper is to develop a catenary cable element for the nonlinear analysis of The purpose of this paper is to develop a catenary cable element for the nonlinear analysis of cable structures that are subjected to static loadings. Firstly, the tangent stiffness matrix and internal cable structures that are subjected to static loadings. Firstly, the tangent stiffness matrix and internal force vector of the element are derived explicitly based on the exact analytical expressions of elastic force vector of the element are derived explicitly based on the exact analytical expressions of elastic catenary. Self-weight of the cables can be directly considered without any approximations. The effect catenary. Self-weight of the cables can be directly considered without any approximations. The effect of pre-tension of cable is also included in the element formulation. Then, a search algorithm with the of pre-tension of cable is also included in the element formulation. Then, a search algorithm with the penalty factor is introduced to satisfy the convergence requirement with high precision and fast speed. penalty factor is introduced to satisfy the convergence requirement with high precision and fast Finally, numerical examples are presented and discussed to illustrate the accuracy and efﬁciency of the speed. Finally, numerical examples are presented and discussed to illustrate the accuracy and proposed analytical algorithm. efficiency of the proposed analytical algorithm. 2. Segmental Catenary Theory of Main Cable 2. Segmental Catenary Theory of Main Cable To accurately simulate the realistic behavior of main cables, the catenary element exactly To accurately simulate the realistic behavior of main cables, the catenary element exactly considering the effects of cable sags, cable self-weight, and cable pretension is used. considering the effects of cable sags, cable self-weight, and cable pretension is used. 2.1. Basic Equations 2.1. Basic Equations An elastic catenary cable element has been derived from the exact solution of the elastic catenary cable equation, deformed due to its self-weight [32,33]. It can be formulated in three dimensional An elastic catenary cable element has been derived from the exact solution of the elastic catenary coordinates, but only two-dimensional formulation is described in this study. cable equation, deformed due to its self-weight [32,33]. It can be formulated in three dimensional Consider a cable segment suspended between points i(x , y ) and j(x , y ), as shown in Figure 1. coordinates, but only two-dimensional formulation is described in this study. i i j j It is assumed that the cable: Consider a cable segment suspended between points i(xi, yi) and j(xj, yj), as shown in Figure 1. It is assumed that the cable: (1) is perfectly ﬂexible and can sustain only tensile forces; (1) is perfectly flexible and can sustain only tensile forces; (2) is composed of a homogeneous material which is linearly elastic; (2) is composed of a homogeneous material which is linearly elastic; (3) is subjected to a uniform distributed load q along the cable length; and, (3) is subjected to a uniform distributed load q along the cable length; and, (4) the tensile stiffness of the cable is calculated using the cross-section before deformation. (4) the tensile stiffness of the cable is calculated using the cross-section before deformation. Figure 1. An elastic catenary cable segment. Figure 1. An elastic catenary cable segment. The relative distances between two nodes (i, j) along the global x, y axis, are denoted as l (l = xj − The relative distances between two nodes (i, j) along the global x, y axis, are denoted as l (l = x xi) and h (h = yj − yi), respectively, in Figure 1, which can be expressed as a function of the global nodal x ) and h (h = y y ), respectively, in Figure 1, which can be expressed as a function of the global i j i force Hi and Vi at the node i as: nodal force H and V at the node i as: i i HS ⋅ H ii 0 q 22 2 2 l=− − ln V + H +V − ln VS−⋅q+ H + (VS−⋅q) H S H (1) i 0 i { (ii i ) (i00 i i )} 2 2 2 2 l = EA lnq V + H + V ln V S q + H + (V S q) (1) i i 0 i 0 i i i EA q qS⋅− 2V⋅S 1 00 i 22 q2 2 2 hH =− +V−H+()V−S⋅q qS 2VS 1 (2) 0 ii i i 0 i 2 0 2 2 2 2EA q h = H + V H + (V S q) (2) i 0 i i i 2EA q The force equilibriums of the elastic catenary cable require that: Appl. Sci. 2018, 8, 1358 4 of 15 The force equilibriums of the elastic catenary cable require that: H = H = H i j < V = V + S q j i 0 (3) 2 2 T = H + V i i > q 2 2 T = H + V j j Equations (1) and (2) are deﬁned as the basic equations for segmental catenary cable, showing the relation between the segmental forces and geometric parameters. Generally, the main cable is divided into several segments (number N), each segment establishes two basic equations, in total 2 times of N equations are obtained for the whole main cable system. In Equations (1)–(3), E is the elastic modulus; A is the cross sectional area, q is the self-weight of the unstressed main cable; l represents the span length of the cable segment, h represents the elevation difference of two ends, and S represents the unstressed length of cable segment; T , T are the cable tension at the left (i) and right (j) ends of the i j cable segment, respectively; H and H are the horizontal component of cable tension at the left (i) and i j right (j) ends of the cable segment, respectively; and, V and V are the vertical component of cable i j tension of the left(i) and right (j) ends of the cable segment, respectively. From Equations (1) and (2), it can be found that for a cable segment with determined S , H, and V , the length l, and high difference h can be easily obtained; similarly, for a cable segment with determined S , l, and h, the internal forces H and V can be easily solved. Thus, only three independent variables exist in these ﬁve variables (S , H, V , l and h). 0 i 2.2. Stiffness Formulation Following describe the procedure of stiffness formulation of the elastic catenary cable element. Considering q, S , EA as constants, partial differentiation of both sides of Equations (1) and (2) yield the following incremental relationships between the relative nodal displacements and nodal forces. ( ) ( ) dH dx = K (4) [ ] dV dy " # K K 11 12 1 [K] = = [B] (5) K K 11 22 2 3 i i " # b b å å i 11 12 6 7 m=1 m=1 6 7 [B] = b = (6) 4 i i 5 m=1 å b å b 21 22 m=1 m=1 (k) (k1) dH = dH = dH = dH i i (7) (k) (k1) dV = dV = dV = dV i i " # T + V ¶l S 1 H 1 1 j j b = = + ln (8) ¶H EA q T V q T (T V ) T T + V i i i i i i j j j " # H 1 1 b = b = (9) 12 21 q T T j i " # S 1 j V b = + (10) EA q T T j i where: [K] is the stiffness matrix due to cable shape change from end point (e.g., left end) to segment point i; if the segment point i become the other end point (e.g., right end), [K] is the stiffness matrix of Appl. Sci. 2018, 8, 1358 5 of 15 the main cable for the whole span; dx, dy are the cumulative amount of change in span and elevation (k ) (k ) respectively from end point to segment point i; and, dH , dV are the increment horizontal and i i vertical component of cable force at segment i, respectively. 2.3. General Solution Procedure The tangent stiffness matrix and internal force vector of cable element are determined while using an iterative procedure. This procedure requires the initial values of end forces (H, V). The iterative procedure for obtaining tangent stiffness matrix and internal force vector of cable element is brieﬂy presented, as follows: (1) input q, E, A, S , nodes I (x , y ) and J (x , y ); 0 i i j j (2) calculate l = x x , h = y y ; 0 j i 0 j i (3) initialize end forces (H, V); (4) update (l, h) using Equations (1) and (2); (5) calculate incompatibility vector of relative distances ds = {dl dh} ; (6) if ds is smaller than the permissible tolerances, calculate [K] using Equation (5) and internal forces using Equation (3), otherwise continue to next step; (7) calculate the correction vector of end forces {dH, dV}using Equation (4); (8) update the end forces H = H + dH, V = V + dV and go to Step (4). i +1 i i +1 i 2.4. No Solution Cases for Cable Segment Equation The solution to the governing equation requires the Newton-Raphson type iteration while using initial trials of the force vector of the left node in the ﬁrst cable element. However, the convergence of the gradient-based Newton-Raphson approach strongly depends on the initial value, and the estimation of initial value remains a challenge. Generally, there are two states for numerical analysis of main cable system: one is the main cable system at ﬁnished state for the whole bridge; the other is at construction state, only the main cable installation is ﬁnished [37]. The tension force at one end need to be assumed (or determined) for the main cable system calculation, the coordinates are iterated with convergence conditions. At the ﬁnished state, the tension force at one end and the horizontal distance between two ends are given, the unstressed cable length and the elevation between two end points can be solved, that is, l, H , V are i i known, to solve S , h. If the end tension force is assumed unreasonably, then there will be no solution for Equations (1) and (2). To solve unstressed length S , Equation (1) is rewritten as: q q H S H i i 2 2 2 2 f (S ) = ln(V + H + V ) ln(V S q + H + (V S q) l (11) 0 i i 0 i 0 i i i EA q Suppose that l, H , EA are constants, and EA > 0, q > 0, 0 < S < 5000 m (the length of main cable for i 0 single-span suspension bridge is currently less than 5000 m), there will be no solution for Equation (1) in the following three conditions: Condition 1. When V is positive and the absolute value of V is large enough, l and H have the same sign, i i i there will be no solution for Equation (1). It can be proved, as follows: 1+ +1 H S H H S H H S 0 i 0 0 i i i i i f (S ) = ln l ln 1 l = l 6= 0 (12) EA q EA q EA S q (V S q) 0 i i 0 1 + + V 2 2 i V V i i Appl. Sci. 2018, 8, 1358 6 of 15 Condition 2. When V is negative and the absolute value of V is large enough, l and H have the same sign, i i i there will be no solution for Equation (1). It can be proved, as follows: 1 +1 H S H i 0 i f (S ) = ln l EA S q V S q 0 i 0 i 1 1+ V V 2 i i (V S q) i (13) 1 i 1 1+ 2 2 H S H H S i 0 i i i 0 ln l l 6= 0 EA q EA S q S q 0 0 1 i 1 1 1+ V V 2 2 i i (V S q) i 0 Condition 3. When the absolute value of V is small enough, l and H have the same sign, Equation (14) is i i obtained from Equation (11), there will be no solution for Equation (1), It can be proved, as follows: 0 1 S 1 jH j S 0 0 @ A f S = H + ln q l < H l 6= 0 (14) ( ) 0 i i EA q 2 EA H + (S q) S q 0 0 3. Improved Numerical Analysis Method In order to solve the problem that no solution for basic equations since tension force at one end of the cable was assumed unreasonable, an improved numerical analysis method is proposed though searching the reasonable initial tension force at one end of the cable. Main cable system calculation in the main span and side span can be divided into two cases: one is that the theoretical vertex position is known; the other is that the saddle position is known. In the ﬁrst case, the tangent point position between saddle and main cable need not to be corrected, while in the second case, the tangent point position between saddle and main cable need to be corrected. The ﬁrst case is the special case of the second case [37]. When theoretical vertex position and saddle position are known, the calculation of main cable system in main span can adopt two iterative methods: one is the speciﬁed point elevation (or un-stressed cable length) iterates step by step, the other is the speciﬁed point elevation (or un-stressed cable length) iterates once [31]. However, the calculation of main cable system in side span generally adopts the method that the un-stressed cable length is iterated once. 3.1. The Main Cable System Calculation in Main Span at Finished State The stiffness due to cable shape change, as mentioned in Section 2.2, should be determined ﬁrst, when the iterative method was used to calculate the designated point elevation (or un-stressed cable length) for main cable system in main span. 3.1.1. Determination of Cable Force Adjustment at Start Point Equation (4) is obtained while ignoring the higher order terms of the Taylor series, which is an approximate expression. Due to strong nonlinear of suspension cable, the iterative methods for determining horizontal and vertical component of cable force adjustment at start point by Equation (4) sometimes fail to converge. Therefore, we need to revise the adjustment amount as the following Equation (15): H = H + adH L L0 (15) V = V + adV L L0 where, H , V are the initial value of horizontal and vertical component of cable force at left start L 0 L 0 end, respectively; a is called penalty factor (or Newton-Downhill factor) in the range from 0 to 1. Appl. Sci. 2018, 8, 1358 7 of 15 Obviously, if the horizontal and vertical component of initial cable force adjustment amount are much less than the value before adjustment, which means that the non-linear property of the cable force adjustment process is not strong, then a = 1; otherwise, a should be chosen between 0.1 and 1. Thus, the iteration can converge with high efﬁciency. The value of a is determined based on the above principle, and calculated, as follows: 8 p 2 2 H +V L0 L0 a = 0.1 ; i f dH > 0.1 H or dV > 0.1 V j j j j j j j j L0 L0 2 2 (dH) +(dV) (16) a = 1; i f jdHj 0.1jH j and jdVj 0.1jV j L0 L0 3.1.2. Improved Numerical Analysis Method and Its Iteration Steps The main cable system calculation in main span under the condition that theoretical vertex position is known using step by step iteration method is illustrated as an example, and the iteration steps are shown in the following: Step 1 All vertical loads in main span were simpliﬁed as uniform distributed load along the span, and the internal forces at both support ends H (1:2), V (1:2) were calculated using traditional 1 1 parabola theory (actually only the internal force at start point is needed). Step 2 The start end forces H (1:2), V (1:2) were regarded as the reference value H(1:2), V(1:2) of 1 1 initial iterated internal forces. Step 3 Input H(1:2), V(1:2) into iterative equations, and determine whether the iterative equations were solvable or not, and set initial value to sign IBZ, IBZ( 1 (Note: IBZ = 1, solvable; IBZ = 0, unsolvable). Step 4 J2 ( 1 (J2 is the modiﬁcation times of iterated initial internal force when there is no solution for iterative equations) Step 5 If IBZ = 0, obtain correction factor according to J2, modify the overall level of initial iterated internal forces (this algorithm called search algorithm, which searching a suitable internal force at start point by changing J2 to make the iterative equation solvable), namely: J2 ( J2 + 1; J2 J2 + 1 [J2int( )2+1] C3 ( 1 + (1) int 0.05; H(1 : 2) ( C3 H1(1 : 2); V(1 : 2) ( C3 V1(1 : 2); If IBZ = 1, go to Step 6. Step 6 On the basis of Step 5 or Step 2, we determine the initial iteration horizontal force multiplier (KK) at start point, and get the elevation error at different points. Then, ﬁnd suitable initial horizontal force iteration multiplier, and obtain the internal force and deformation in the main span by secant method, go to Step 7. If there is no solution, then IBZ = 0, go to Step 5 (i.e., correcting overall level of initial iterated internal forces). The details of step 6 are shown in the following: Step 6.1 Set initial iteration horizontal force, vertical force at start point: H0(1) ( KK H(1),V 0(1) ( V (1) or H0(2) ( KK H(2), V 0(2) ( V (2) Step 6.2 J3 ( 1 (set initial value of iterative times J3 based on Step 6.1). Step 6.3 Calculate internal forces from left to right point (or right to left point) in the follows: Appl. Sci. 2018, 8, 1358 8 of 15 (1) According to the internal forces (horizontal and vertical components) at one end and the horizontal distance between two ends of a cable segment k (k = 1, 2, . . . , n), the unstressed cable length S (k) and the elevation difference Dh between two ends were calculated by Equations (1) and (2). If there is no solution, then IBZ ( 0, go to Step 5. Otherwise, IBZ ( 1, calculate the (k) (k) coordinates y , horizontal and vertical component H , V of the cable k j j segment at right point k, respectively. (k + 1) (k + 1) (2) Calculate the internal force (H , V ) at left point (i) of cable segment i i k + 1 using equilibrium condition. Step 6.4 J3 ( J3 + 1. Step 6.5 Based on the elevation Y at start point and the elevation difference Dh of each cable segment (k = 1, 2, . . . , n 1), the elevation Y at end point and the elevation error D ( Y Y (Y is the actual elevation at end point) were determined. n n R R 4 6 7 9 Step 6.6 If |D | # (# = 10 m~10 m or 10 ~10 times of the main span), go to Step 6.10; if |D | > #, go to Step 6.7. Step 6.7 If the iteration time J3 > 60 (this value can be taken as 100, etc.), it was considered non-convergence, IBZ ( 0, go to Step 5; if J3 60, then go to Step 6.8. Step 6.8 Formulate stiffness matrix [K] by Equation (5), and calculate dH and dV. Step 6.9 Correction H0(1), V0(1) or H0(2), V0(2), namely: ( ( H0(1) ( H0(1) + adH H0(2) ( H0(2) + adH or V0(1) ( V0(1) + adV V0(2) ( V0(2) + adV where, a is obtained by Equation (16), H0(1), V0(1) or H0(2), V0(2) are H , V in L 0 L 0 Equation (16); then, go to Step 6.3. Step 6.10 IBZ ( 1. Step 7 End. The results of each variable at the last step are what we want. From the above calculation steps, the solution will not enter into endless loop and the elevation of key points reach a predetermined value through changing the overall level of initial iteration horizontal force multiplier. 3.2. The Main Cable System Calculation in Side Span at Finished State The main cable system calculation in side span at ﬁnished state under the condition that the horizontal component of cable at one end in side span is known. The iterative process was conducted though the proposed concept and formula of stiffness due to a vertical deformation change of the main cable. 3.2.1. Stiffness Due to Vertical Deformation Change of Main Cable Both sides of Equations (1) and (2) were differentiated, when considering side-span adjustment dl = 0 (horizontal projection length of each cable segment at ﬁnished state is known) and the horizontal component of each cable segment in side span dH = 0, the following equations were obtained: " ! !# ¶h ¶h ¶l ¶l S 1 V qS V 1 1 0 i 0 i dh = / dV = + + dV (17) i i ¶V ¶S ¶S ¶V EA q T T q T T i 0 0 i j i j i Appl. Sci. 2018, 8, 1358 9 of 15 Given dh = D dV 11 i " ! !# S 1 V qS V 1 1 0 i 0 i D = + + (18) EA q T T q T T j i j i The reciprocal of D (1/D ) is deﬁned as stiffness due to vertical deformation change of 11 11 a cable segment. It can be seen found that 1/D represents the vertical component variation of start end (or terminal end) due to unit elevation change between two points of each segment under the conditions that the horizontal force component of cable segment was unchanged, the horizontal distance between two ends of cable segment was constant, while the unstressed cable length can be varied. When considering that the change of vertical component for each segment is equal, i.e., dV = dV, the accumulated elevation difference of cable from support point to segment i can be obtained, as follows: i i i dY = dh = (D dV ) =dV D (19) 11 i 11 å å å m=1 m=1 m=1 dV = dY (20) m=1 where, is deﬁned as the stiffness due to vertical deformation of main cable at side span. å 11 m=1 3.2.2. Improved Numerical Analysis Method for Side Span and Its Iteration Steps The main cable system calculation at side span under the condition that saddle position is known, using the method that un-stressed cable length iterated once, is illustrated as an example, and the iteration steps are shown, as follows: Step 1 Set initial value of horizontal angle b (1), b (2) for tangent line of suspension cable at saddle q q point of support ends. Step 2 Set the initial value of K and reference value of K of tangent slope of suspension cable at q q 1 saddle point of start support: K ( tan b (1) , K ( K q1 q1 q1 Step 3 Determine whether the iterative equations are solvable or not, and set initial value to sign IBZ, IBZ ( 1 (Note: IBZ = 1: solvable, IBZ = 0: unsolvable). Step 4 J2 ( 1 (J2 is the modiﬁcation time of initial slope or vertical component of initial iterated internal forces, when there is no solution for iterative equations) Step 5 If IBZ = 0, obtain correction factor according to J2 to modify the vertical component of initial iterated internal forces. J2 ( J2 + 1 J2 J2 + 1 [J2int( )2+1] C3 ( 1 + (1) int 0.05 Assign initial value of the slope at start point: K ( C3 K , then go to Step 6. If IBZ = 1, go q q 1 to Step 6. Step 6 Calculate the vertical component and horizontal inclination at start point: V(1) ( K H(1); b (1) = ATAN(K ) (21) q q1 q Appl. Sci. 2018, 8, 1358 10 of 15 where, H(1) is horizontal component at start point which is determined according to the condition that the horizontal forces at both side of the saddle are equal. Step 7 Calculate tangent point coordinate (X , Y ) between the line with horizontal angle b (2) and q 2 q 2 q the end point of cable at saddle. Step 8 Calculate tangent point coordinate (X , Y ) between the line with horizontal angle b (1) and q 1 q 1 q the start point of cable at saddle. Step 9 Calculate coordinates and internal forces of cable segments from left to right point (or right to left point): (1) According to the internal forces (horizontal and vertical components) at one end and the horizontal distance between two ends of a cable segment k (k = 1, 2, . . . , n), calculate the unstressed cable length S (k) and the elevation difference Dh between two ends by 0 k Equations (1) and (2). If there is no solution, then IBZ( 0, go to Step 5. Otherwise, IBZ ( 1, and calculates the coordinates y , horizontal, vertical component H (k), V (k) of k j j the cable segment at right point k, respectively. (k +1) (k +1) (2) Calculate the internal force (H ,V ) at left end (i) of cable segment k + 1 while i i using equilibrium condition. Step 10 Calculate the elevation Y at end point and the elevation error D ( Y Y . n n n q2 4 6 Step 11 If |D | # (set # = 10 m~10 m), go to Step 16; otherwise, go to Step 12. Step 12 Calculate the stiffness due to vertical deformation of main cable at side span by Equation (18), and compute dV: dV = DY n N m=1 q q 2 2 2 2 Step 13 determine a: If dV > V(1) + H(1) , a ( 0.1 V(1) + H(1) / dV Otherwise, a ( 1. j j j j Step 14 modify V(1): V(1) ( V(1) + adV. h i V(1) Step 15 Calculate new inclination angle b (1) of cable at start point: b (1) = ATAN , then go to q q1 H(1) Step 8. Step 16 According to horizontal and vertical forces of cable at end point, calculate the error of horizontal angle Db , and set a new value of horizontal angle b (2) at end point: n q 2 3 2 3 (n) (n) V V j j 4 5 4 5 Db = b (2) ATAN ; b (2) ( ATAN n q q (n) (n) H H j j 3 5 Step 17 If |Db | # (let # = 10 ~10 ), go to Step 18; otherwise, go to Step 7. n 1 1 Step 18 End. The results of each variable at the last step are what we want. 4. Numerical Examples The numerical analysis program for calculating the main cable system of suspension bridge was developed based on the segmental catenary theory and the improved iteration method proposed in this paper. The accuracy and effectiveness of proposed numerical analysis method have been veriﬁed by a commercial ﬁnite element software ANSYS, also this method has been successfully applied to monitor the construction of some suspension bridges in China, such as Pingsheng Bridge [38], Jiangdong Bridge [39], and Taohuayu Bridge [40]. 4.1. Example 1 To illustrate the advantages of this method, a three-span suspension bridge with a main span of 400m is chosen as an example, the coordinates of two theoretical vertex positions are (200, 45) and (200, 45), the coordinate at center of main span is (0, 0), while the coordinates at both ends of side Appl. Sci. 2018, 8, 1358 11 of 15 span are (250, 10) and (250, 10), respectively. The area of cable cross section is 0.5 m , and the elastic 5 3 modulus is 2.0 10 MPa, the equivalent density is 77 kN/m . Calculate the unstressed cable length, internal forces, and other coordinates of the main cable system under two load cases, as shown in Appl. Sci. 2018, 8, x FOR PEER REVIEW 11 of 15 Figure 2: Load case 1: P = 3000 kN, P = 3500 kN, P = 3000 kN; 1 5 2 3 Load case 2: P1 = 2.0 × 10 kN, P2 = 0 kN, P3 = 0 kN. Load case 2: P = 2.0 10 kN, P = 0 kN, P = 0 kN. 1 2 3 7(200,45) 3(-200,45) 6(195,Y 6) 4(-195,Y 4) 2(-225,Y 2) 8(225,Y 8) 2 5 P1 4 P3 5(0,0) 1(-250,10) 9(-250,10) P2 Figure 2. A three-span suspension cable system. Figure 2. A three-span suspension cable system. The main cable system is calculated while using the traditional (without introduction of search The main cable system is calculated while using the traditional (without introduction of search algorithms) and the improved numerical analysis method. The calculation results are almost the same under algoritload hms) a case nd t 1, has e impro shown ved numer in Tables i1 ca and l an2 a;lys however is meth , od. The c under load alcul case ation re 2, ther su elt was s are a nolmost solution the sam using e under load case 1, as shown in Tables 1 and 2; however, under load case 2, there was no solution traditional numerical analysis method, the calculation results by improved numerical analysis method wer usin eg t shown raditio in nal Tables nume 1ric and al an 3. alysis method, the calculation results by improved numerical analysis method were shown in Tables 1 and 3. Table 1. Results of y-coordinate under two load cases (unit: m). Table 1. Results of y-coordinate under two load cases (unit: m). Item Load Case 1 Load Case 2 Item Load Case 1 Load Case 2 Node No. 2 4 6 8 2 4 6 8 Node No. 2 4 6 8 2 4 6 8 x 225.0000 195.0000 195.0000 225.0000 225.0000 195.0000 195.0000 225.0000 x −225.0000 −195.0000 195.0000 225.0000 −225.0000 −195.0000 195.0000 225.0000 y 26.9209 42.5396 42.5396 26.9209 26.9795 9.1986 43.1848 26.9795 y 26.9209 42.5396 42.5396 26.9209 26.9795 9.1986 43.1848 26.9795 Table 2. Results of the length and tension force of each cable element under load case 1. Table 2. Results of the length and tension force of each cable element under load case 1. Force at Left End/kN Force at Right End/kN Unstressed Cable Shape Force at Left End/kN Force at Right End/kN Element No. Element Unstressed Shape Horizontal Vertical Horizontal Vertical Length/m Length/m Horizontal Vertical Horizontal Vertical Component Component Component Component No. Cable Length/m Length/m Component Component Component Component 5 5 5 5 1 30.1798 30.1893 0.2585 10 0.1690 10 0.2585 10 0.1809 10 5 5 5 5 30.1798 30.1893 −0.2585 × 10 5 −0.1690 × 10 5 0.2585 × 10 5 0.1809 × 10 5 2 30.8435 30.8527 0.2585 10 0.1809 10 0.2585 10 0.1930 10 5 5 5 5 5 5 5 5 ② 3 30.8435 30.8527 −0.2585 × 10 −0.1809 × 10 0.2585 × 10 0.1930 × 10 5.5709 5.5720 0.2585 10 0.1283 10 0.2585 10 0.1261 10 5 4 5 4 5 5 5 5 4 200.2295 200.2827 0.2585 10 0.9609 10 0.2585 10 0.1750 10 ③ 5.5709 5.5720 −0.2585 × 10 0.1283 × 10 0.2585 × 10 −0.1261 × 10 5 4 5 4 5 200.2295 200.2827 0.2585 10 0.1750 10 0.2585 10 0.9609 10 5 4 5 4 ④ 200.2295 200.2827 −0.2585 × 10 0.9609 × 10 0.2585 × 10 −0.1750 × 10 5 5 5 5 5.5709 5.5720 0.2585 10 0.1261 10 0.2585 10 0.1283 10 5 4 5 4 ⑤ 200.2295 200.2827 −0.2585 × 10 −0.1750 × 10 0.2585 × 10 0.9609 × 10 5 5 5 5 7 30.8435 30.8527 0.2585 10 0.1930 10 0.2585 10 0.1809 10 5 5 55 5 5 5 5 ⑥ 5.5709 5.5720 −0.2585 × 10 −0.1261 × 10 0.2585 × 10 0.1283 × 10 8 30.1798 30.1893 0.2585 10 0.1809 10 0.2585 10 0.1690 10 5 5 5 5 ⑦ 30.8435 30.8527 −0.2585 × 10 0.1930 × 10 0.2585 × 10 −0.1809 × 10 5 5 5 5 30.1798 30.1893 −0.2585 × 10 0.1809 × 10 0.2585 × 10 −0.1690 × 10 Table 3. Results of the length and tension force of each cable element under load case 2. Table 3. Results of the length and tension for Force ce of at each Left End/kN cable element unde Force r load at Right case 2 End/kN . Unstressed Cable Shape Element No. Horizontal Vertical Horizontal Vertical Length/m Length/m Force at Left End/kN Force at Right End/kN Component Component Component Component Element Unstressed Shape Horizontal Vertical Horizontal Vertical 5 5 5 5 1 30.2114 30.2219 0.2876 10 0.1894 10 0.2876 10 0.2013 10 No. Cable Length/m Length/m Component 5 Component 5 Component 5 Component 5 2 30.8072 30.8171 0.2876 10 0.2013 10 0.2876 10 0.2134 10 5 6 5 6 5 5 5 5 ① 30.2114 36.0732 30.2219 36.1472 −0.2876 × 0.2876 10 10 −0.1894 × 0.2066 10 10 0.2876 0.2876 × 10 10 0.2052 0.2013 × 10 10 5 4 5 4 4 195.7354 195.7919 0.2876 10 0.5206 10 0.2876 10 0.2477 10 5 5 5 5 ② 30.8072 30.8171 −0.2876 × 10 −0.2013 × 10 0.2876 × 10 0.2134 × 10 5 4 5 5 5 200.2276 200.2867 0.2876 10 0.2477 10 0.2876 10 0.1034 10 5 6 5 6 ③ 36.0732 36.1472 −0.2876 × 10 0.2066 × 10 0.2876 × 10 −0.2052 × 10 5 5 5 5 5.3176 5.3188 0.2876 10 0.1034 10 0.2876 10 0.1054 10 5 4 5 4 ④ 5 5 5 5 195.7354 195.7919 −0.2876 × 10 0.5206 × 10 0.2876 × 10 0.2477 × 10 7 30.8078 30.8182 0.2876 10 0.2134 10 0.2876 10 0.2013 10 5 5 5 5 5 4 5 5 8 30.2114 30.2219 0.2876 10 0.2013 10 0.2876 10 0.1894 10 ⑤ 200.2276 200.2867 −0.2876 × 10 −0.2477 × 10 0.2876 × 10 0.1034 × 10 5 5 5 5 ⑥ 5.3176 5.3188 −0.2876 × 10 −0.1034 × 10 0.2876 × 10 0.1054 × 10 5 5 5 5 ⑦ 30.8078 30.8182 −0.2876 × 10 0.2134× 10 0.2876 × 10 −0.2013 × 10 5 5 5 5 ⑧ 30.2114 30.2219 −0.2876 × 10 0.2013 × 10 0.2876 × 10 −0.1894 × 10 Appl. Sci. 2018, 8, x FOR PEER REVIEW 12 of 15 Appl. Sci. 2018, 8, 1358 12 of 15 4.2. Example 2 4.2. Example 2 A single span flexible cable that is fixed at both ends subjected to multiple concentrated loads is adopted as an example, as shown in Figure 3, to compare the analytical results from the proposed A single span ﬂexible cable that is ﬁxed at both ends subjected to multiple concentrated loads algorithm with that from other methods. The node coordinates and unstressed lengths of cable is adopted as an example, as shown in Figure 3, to compare the analytical results from the proposed segments at initial state are shown in Tables 4 and 5, respectively. In addition, the cross-sectional area algorithm with that from other methods. The node coordinates and unstressed lengths of cable −4 2 of cable is 5.48386 × 10 m , and the elastic modulus is 13,1473.43 MPa, the weight of unit length cable segments at initial state are shown in Tables 4 and 5, respectively. In addition, the cross-sectional area 4 2 is 47.02594 kN/m. of cable is 5.48386 10 m , and the elastic modulus is 13,1473.43 MPa, the weight of unit length cable is 47.02594 kN/m. 2 9 3 8 4 4 5 6 7 1.48kN 1.48kN 1.45kN 1.45kN 1.43kN 1.43kN 37.13kN 1.41kN 1.41kN Figure 3. A single span cable subjected to multiple concentrated loads. Figure 3. A single span cable subjected to multiple concentrated loads. Table 4. Node coordinates of cable segments at initial state (unit: m). Table 4. Node coordinates of cable segments at initial state (unit: m). Node 1 2 3 4 5 6 7 8 9 10 11 Node 1 2 3 4 5 6 7 8 9 10 11 x 0.0 30.48 60.96 91.44 121.92 152.40 182.88 213.36 243.84 274.32 304.80 x 0.0 30.48 60.96 91.44 121.92 152.40 182.88 213.36 243.84 274.32 304.80 y 0.0 11.0642 19.5986 25.6565 29.2760 30.4800 29.2760 25.6565 19.5986 11.0642 0.0 y 0.0 −11.0642 −19.5986 −25.6565 −29.2760 −30.4800 −29.2760 −25.6565 −19.5986 −11.0642 0.0 Table 5. Table 5. No Node de coor coordinates dinates of of cable cable segments segments at ini at initial tial state state ( (unit: unit: m m). ). Element ① ② ③ ④ ⑤ ⑥ ⑦ ⑧ ⑨ ⑩ Element 1 2 3 4 5 6 7 8 9 10 Unstressed 32.4175 31.6441 31.0683 30.6865 30.4962 30.4962 30.6865 31.0683 31.6441 32.4175 Unstressed Length 32.4175 31.6441 31.0683 30.6865 30.4962 30.4962 30.6865 31.0683 31.6441 32.4175 Length The configuration and tension force of cable at the equilibrium state under applied load were The conﬁguration and tension force of cable at the equilibrium state under applied load were calculated by different methods, including method 1: improved analytical algorithm in present study; calculated by different methods, including method 1: improved analytical algorithm in present study; method 2: finite element method in Ref. [41]; and, method 3: traditional analytical method in Ref. [42]. method 2: ﬁnite element method in Ref. [41]; and, method 3: traditional analytical method in Ref. [42]. The segmental catenary theory is adopted for both method 1 and 3, however, method 1 uses the The segmental catenary theory is adopted for both method 1 and 3, however, method 1 uses the search search algorithm with penalty factor, while method 3 uses traditional Newton-Raphson iteration algorithm with penalty factor, while method 3 uses traditional Newton-Raphson iteration algorithm. algorithm. The comparison of cable configuration and internal forces are shown in Tables 6 and 7, The comparison of cable conﬁguration and internal forces are shown in Tables 6 and 7, respectively. respectively. It can be found that the note coordinates and tension force of cable under applied load It can be found that the note coordinates and tension force of cable under applied load calculated from calculated from improved analytical algorithm agree well with that from method 2 and 3, the improved analytical algorithm agree well with that from method 2 and 3, the maximum difference of maximum difference of node coordinate between method 1 and 2 is 6 mm (y of node 3) with relative node coordinate between method 1 and 2 is 6 mm (y of node 3) with relative error of 0.03%, and 1 mm error of 0.03%, and 1 mm (y of node 2) between method 1 and 3 with relative error of 0.01%; the (y of node 2) between method 1 and 3 with relative error of 0.01%; the maximum difference of the cable maximum difference of the cable tension force between method 1 and 2 is 0.05 kN (element 5) with a tension force between method 1 and 2 is 0.05 kN (element 5) with a relative error of 0.06%, and 0.12 kN relative error of 0.06%, and 0.12 kN (element 3) between method 1 and 3 with relative error of 0.1%. (element 3) between method 1 and 3 with relative error of 0.1%. In compression of method 2, the initial In compression of method 2, the initial node coordinates of cable segments are not necessary for the node coordinates of cable segments are not necessary for the proposed algorithm to calculate the proposed algorithm to calculate the configuration and tension force of cable at equilibrium state. In conﬁguration and tension force of cable at equilibrium state. In comparison to method 3, the initial comparison to method 3, the initial value is not sensitive to solve cable segment equations for the value is not sensitive to solve cable segment equations for the proposed algorithm especially under the proposed algorithm especially under the conditions of asymmetric and uneven loads, and also the conditions of asymmetric and uneven loads, and also the number of iterations is signiﬁcantly reduced, number of iterations is significantly reduced, resulting in faster convergence speed. resulting in faster convergence speed. Appl. Sci. 2018, 8, 1358 13 of 15 Table 6. Node coordinates of cable segments under applied load (unit: m). x y Node Method 1 Method 2 Method 3 Method 1 Method 2 Method 3 1 0.000 0.000 0.000 0.000 0.000 0.000 2 30.995 30.996 30.995 9.641 9.638 9.640 3 61.389 61.391 61.389 18.595 18.589 18.595 4 91.357 91.356 91.357 26.942 26.945 26.942 5 121.075 121.075 121.075 34.748 34.748 34.748 6 151.276 151.276 151.276 30.242 30.243 30.242 7 181.404 181.404 181.404 25.275 25.277 25.275 8 211.641 211.641 211.641 19.818 19.817 19.818 9 242.166 242.166 242.166 13.825 13.824 13.825 10 273.159 273.159 273.159 7.241 7.240 7.241 11 304.800 304.800 304.800 0.000 0.000 0.000 Table 7. Tension force of cable segments under applied load (unit: kN). Tension Force Element Method 1 Method 2 Method 3 94.41 94.40 94.40 93.98 94.00 94.00 93.58 93.60 93.70 93.21 93.20 93.30 91.15 91.20 91.20 91.37 91.40 91.40 91.61 91.60 91.70 91.87 91.90 91.90 9 92.16 92.20 92.20 10 92.48 92.50 92.50 5. Conclusions (1) It is theoretically proved that there is no solution for calculating the main cable system in main span or side span under certain loading conditions. (2) By introducing the search algorithm and penalty factor, a numerical analysis method was improved to overcome the problem of no solution under certain loading conditions, and to develop the segmental catenary theory. (3) The necessity and effectiveness of the improved analytical method were described by the theoretical calculation results and numerical examples. The program using proposed method has been successfully applied in shape ﬁnding during design and conﬁguration control during construction of main cable system for suspension bridges in China. Author Contributions: Conceptualization, methodology, software and writing, C.L., J.H. and H.K.; Investigation and validation, C.D. and H.L.; Resources, supervision and revision, C.L., Z.Z. and Y.L. Funding: This research was funded by [National Natural Science Foundation of China] grant number [51778069 and 51308070], [National Basic Research Program of China] grant number [973 Program, No. 2015CB057702], [Key Discipline Fund Project of Civil Engineering of Changsha University of Sciences and Technology] grant number [13ZDXK04, 13KA04]. Acknowledgments: The authors acknowledge funding from the National Natural Science Foundation of China (Nos. 51778069, 51308070), National Basic Research Program of China (973 Program, No. 2015CB057702), Key Discipline Fund Project of Civil Engineering of Changsha University of Sciences and Technology (13ZDXK04, 13KA04). We thank the reviewers and the editor for the valuable comments and suggestions that helped us improve the manuscript. Conﬂicts of Interest: The authors declare no conﬂicts of interest. Appl. Sci. 2018, 8, 1358 14 of 15 References 1. Pugsley, A. The Theory of Suspension Bridges, 2nd ed.; Arnold Ltd.: London, UK, 1968. 2. 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