An iterative calculation method for hanger tensions and the cable shape of a suspension bridge based on the catenary theory and finite element method

Abstract

Construction of suspension bridges and their structural analysis are challenged by the presence of elements (chains or main cables) capable of large deflections leading to a geometric nonlinearity. For an accurate prediction of the main cable geometry of a suspension bridge, an innovative iterative method is proposed in this article. In the iteration process, hanger tensions and the cable shape are, in turns, used as inputs. The cable shape is analytically predicted with an account of the pylon saddle arc effect, while finite element method is employed to calculate hanger tensions with an account of the combined effects of the cable-hanger-stiffening girder. The cable static equilibrium state is expressed by three coupled nonlinear governing equations, which are solved by their transformation into a form corresponding to the unconstrained optimization problem. The numerical test results for the hanger tensions in an existing suspension bridge were obtained by the proposed iterative method and two conventional ones, namely, the weight distribution and continuous multiple-rigid-support beam methods. The latter two reference methods produced the respective deviations of 10% and 5% for the side hangers, respectively, which resulted in significant errors in the elevations of the suspension points. To obtain more accurate hanger tensile forces, especially for the side hangers, as well as the cable shape, the iterative method proposed in this article is recommended.

Keywords

cable shape hanger tensile force iterative method pylon saddle suspension bridge tangent point unconstrained optimization problem

Introduction

Suspension bridges are known to have the road deck suspended by strung cables: with no support in the span center, the deck hangs below the supports rather than resting upon them. Suspension bridges, due to their aesthetical look, high material utilization rate, and excellent static and dynamic performances, are considered very lucrative choices in the construction of long-span bridges. In recent years, the dimensions of suspension bridges erected worldwide are permanently increased due to the ongoing improvement of the design and construction technologies. Moreover, to reduce the cable weight and material usage, the cable safety factor was subsequently dropped to about 2.0 (Chen et al., 2013). This safety margin reduction puts more stringent requirements on the prediction accuracy of the optimal configuration and distribution of internal forces in the suspension bridge components (cable, hangers, suspenders, girders, etc.) at the design stage.

In contrast to cable-stayed bridges (where deck is directly connected by cables to the towers/pylons), the design configuration of suspension bridges (with main cables holding the deck via suspender/hanger cables) cannot be amended through the tracking and adjustment during the construction process (Atmaca and Ates, 2012). Once the main cables are installed, their shapes and tensions are entirely determined by the external loads. Therefore, the completed bridge state is significantly affected by the free cable parameters. Consequently, the determination of hanger tensile forces and cable shape in the completed bridge state, from which the free cable parameters can be calculated adversely, is mandatory for the construction control.

Since the hanger tensile forces of a suspension bridge are not uniformly distributed in the completed bridge state (Huang et al., 2017), it is unreasonable to equate a hanger tensile force with the dead load of the deck between two adjacent hangers, especially for multiple hangers located near pylons. Moreover, the accuracy of estimating hanger tensile forces directly affects the predicted cable shapes in both the completed state and unloaded state, including the unstrained length values of cables and hangers. As adopted in literatures (Jung et al., 2013; Kim and Lee, 2001; Thai and Choi, 2013), the continuous multiple-rigid-support beam (CMRSB) concept provides a straightforward way to derive hanger tensions, which are approximated with the CMRSB reaction forces. On this basis, the cable shape can be determined. However, the CMRSB method is not accurate enough because the deck of a suspension bridge is supported by elastic springs, whose stiffness values are determined by both the axial stiffness of hangers and the cable contribution to the vertical stiffness at the locations of hangers. At the design stage, the known parameters of the completed bridge state include the span arrangement, cable sag-to-span ratio in the main span, cable elevation at the midspan point of the main span, elevations of the cable intersection points over the saddles, hanger spacing, and deck elevation. However, the cable elevations at each suspension point are unknown, and the length of each hanger cannot be determined as a result. In the case where both the cable shape and hanger lengths are unknown, the spring stiffness of the continuous multiple-elastic-support beam cannot be determined explicitly. Therefore, an iterative solution is required between the hanger tensile forces and the cable shape. On the contrary, the final hanger forces are greatly influenced by the construction sequence. In the general construction of a suspension bridge, the hanger forces are implemented following two dead load stages. The first stage using the lifting construction technology provides simply supported beam reaction forces, whereas the second one (floor system) provides the elastic reactions in a similar form with the continuous beam reactions after the hoisted beams are connected as a whole. The sum of the two cannot be identical to the reactions of a complete CMRSB.

A full-bridge finite element model can be used to calculate the hanger tensile forces, but it is difficult to simulate the pylon saddle in the cable shape-finding analysis. No matter whether multiple rigid links or saddle-cable elements are used to simulate the pylon saddle, the cable tangent point in the pylon saddle cannot be renewed with the change of the external forces acting on the cable.

The methods for cable shape-finding of suspension bridges can be subdivided into the following two categories: (1) catenary-element-based methods and (2) analytical methods. The former techniques identify the curved shape of the target cable by updating the nodal positions and tensile forces of the cable catenary elements through the nonlinear structural analysis via the Newton iterations (Cao et al., 2017; Chen et al., 2000, 2013; Hassan, 2013; Karoumi, 1999; Kim et al., 2002; Kim and Kim, 2012; Kim and Lee, 2001; Lu et al., 2014). In contrast to the catenary-element-based methods, the analytical techniques possess a much higher computational efficiency and a faster convergence (Cao et al., 2017). Over the past decades, the analytical approaches for cable shape-finding have been significantly improved based on either the segmental parabola theory or the segmental catenary theory. Within the framework of the Newton–Raphson method, Chen et al. (2013) proposed a flexible iterative approach, in which the nonlinear governing equations were approximatively linearized using the first-order Taylor expansions. Based on the segmental parabola theory, Jung et al. (2015) proposed a simplified analytical method for the optimization of the initial shape analysis in self-anchored suspension bridges. It is noteworthy that the above studies did not consider the joint effect of the cable-hanger-stiffening girder combinations, but treated hanger tensions as known parameters in the cable shape-finding analysis, as well as neglected the impact of the pylon saddle’s arc. Insofar as the latter impact on the cable shape was found to be quite strong, the coordinates of the cable’s tangent point on the pylon saddle should be accurately determined. However, the calculation of the tangent point coordinates will increase the number of nonlinear governing equations required for the calculation of the cable shape from two to three. Moreover, the revealed coupling of the resulting three nonlinear governing equations hinders their solution.

In this article, an iterative algorithm is proposed to determine the final state of a suspension bridge: the analytical method is used to calculate the cable shape, while the full-bridge finite element model is used to calculate the hanger tensile forces. The hanger tensile forces and cable shape are used, in turn, as inputs in the iteration process. Besides, the pylon saddle effect is taken into account by determining the position of the tangent point. When the analytical method is used to calculate the cable shape, three coupled nonlinear governing equations are transformed into a convenient form, which reduces the problem to an unconstrained optimization task. Finally, the algorithm is applied to a steel truss suspension bridge with a main span of 730 m, and the feasibility and effectiveness of this algorithm are verified.

An analytical method for cable form-finding

Under the action of lump/concentrated forces transferred via hangers, the cable in the bridge’s final state consists of many catenary segments hanging between adjacent hangers. Each cable segment and hanger should meet the following underlying requirements:

The materials work in their elastic range (i.e. below their yield stress level), and their stress–strain dependences follow Hooke’s law;

The small-scale deformation conditions are assumed, which imply that the cable deformation is small enough to disregard the cross-sectional cable contraction;

The cable is ideally flexible, which excludes the generation of bending stresses in it.

The flexural stiffness of the main cable is neglected in this study for the following two reasons (Zhang et al., 2018b): (1) During the whole construction process of a suspension bridge, the flexural stiffness of the main cable is insignificant, so is the clamping effect of a cable clamp on the main cable and (2) in long-span suspension bridges, clamps occupy a low percentage (about 6%) of the total length of the main cable in the central span.

The cable shape-finding in the main span has to be conducted first. As shown in Figure 1, several coordinate systems are established, with their origin being at the left tangent point, O₀, and the suspension points, O₁–O_n, along the cable, the positive x-axis pointing right, and the positive y-axis pointing downward. The shape of an arbitrary cable segment is derived from the following catenary equation

y = c \cosh (\frac{x}{c} + a_{i}) + b_{i}

(1)

where c = –H / q, in which H is the horizontal component of the cable tension in the completed bridge state (kN) and q is the cable weight per unit length (kN/m); a_i and b_i are parameters of the catenary equation.

Figure 1.

Cable configuration in the completed bridge state.

From the boundary condition y(0) = ccosha_i + b_i = 0, we obtain b_i = –ccosha_i. By substituting this term into equation (1), it can be reduced to the following form

y = c [\cosh (\frac{x}{c} + a_{i}) - \cosh a_{i}]

(2)

Three constraint conditions are introduced: (1) the elevation difference between the left tangent point and the midspan point is closed; (2) the elevation difference between the two tangent points on the left and right saddles is closed; and (3) the length of the orthogonal projection of the rightmost catenary segment on the horizontal plane, l_n+₁, meets the relevant design requirement. The corresponding equations are as follows

\sum_{i = 1}^{m} Δ h_{i} = Δ h_{O_{0} O_{m}}

(3a)

\sum_{i = 1}^{n + 1} Δ h_{i} = Δ h_{O_{0} O_{n + 1}}

(3b)

l_{n + 1} = l_{O_{n} D'} - Δ_{2}

(3c)

where m is the number of the cable segments between the left tangent point, O₀, and the midspan point, O_m; n is the number of suspension points along the cable; Δh_i denotes the elevation difference between two endpoints of an arbitrary catenary segment of the cable; $Δ {h_{O_{0}}}_{O_{m}}$ is the elevation difference between the left tangent point, O₁, and the midspan point, O_m; $Δ h_{O_{0} O_{n + 1}}$ is the elevation difference between the left (O₀) and right (O_n+1) tangent points; $l_{O_{n} D'}$ is the designed horizontal distance between the rightmost hanger and the actual apex of the right tower saddle, D’, as shown in Figure 2(b); Δ₂ is the horizontal distance between points O_n+1 and D’, as shown in Figure 2(b).

Figure 2.

The layout of tower saddles and critical parameters: (a) left tower saddle and (b) right tower saddle.

The three unknown parameters in the above three equations are H, a₁, and l_n+₁, which refer to the horizontal cable tension, the coefficient in the catenary equation for the first catenary segment, and the length of the orthogonal projection of the rightmost catenary segment on the horizontal plane, respectively. The next step is aimed to express all other terms in the above equations as functions of these three parameters.

Thus, Δh_i can be expressed as follows

Δ h_{i} = y (l_{i}) - y (0) = c [\cosh (\frac{l_{i}}{c} + a_{i}) - \cosh a_{i}]

(4)

where l_i is the length of the orthogonal projection of an arbitrary catenary segment on the horizontal plane, as shown in Figure 1.

For a given designed elevation of the circle center, C, in the geodetic system (denoted as h_C), the elevation of the left tangent point in the geodetic system, $h_{O_{0}}$ , can be written as

h_{O_{0}} = h_{C} + R \cos θ

(5)

where R is the radius of the left tower saddle arc-shaped top, as shown in Figure 2(a); θ is the angle between the vertical segment BC and the segment-connecting point O₀ and the circle center, C.

Insofar as

\tan θ = \frac{d y}{d x} |_{x = 0} = \sinh a_{1}

(6)

we get

\cos θ = sech a_{1}

(7)

The substitution of equation (7) into equation (5) yields

h_{O_{0}} = h_{C} + R sech a_{1}

(8)

For a given design elevation of point O_m in the geodetic system (denoted as $h_{O_{m}}$ ), the elevation difference between points O₀ and O_m (denoted as $Δ {h_{O_{0}}}_{O_{m}}$ ) can be reduced to

Δ h_{O_{0} O_{m}} = h_{O_{0}} - h_{O_{m}} = h_{C} + R sech a_{1} - h_{O_{m}}

(9)

For a given design elevation of the circle center, C’, in the geodetic system, denoted h_C’, the elevation of the right tangent point in the geodetic system, $h_{O_{n + 1}}$ , can be expressed as

h_{O_{n + 1}} = h_{C'} + R' \cos θ'

(10)

where R’ is the radius of the right tower saddle arc-shaped top, as shown in Figure 2(a); θ’ is the angle between the vertical segment B’C’ and the segment-connecting point O_n+1 and the circle center, C.’

Similarly, from

\tan θ' = - \frac{d y}{d x} |_{x = l_{n + 1}} = - \sinh (\frac{l_{n + 1}}{c} + a_{n + 1})

(11)

we get

\cos θ' = sech (\frac{{l_{n}}_{+ 1}}{c} + {a_{n +}}_{1})

(12)

\sin θ' = - \tanh (\frac{{l_{n}}_{+ 1}}{c} + {a_{n +}}_{1})

(13)

The substitution of equation (12) into equation (10) yields

h_{O_{n + 1}} = h_{C'} + R' sech (\frac{{l_{n}}_{+ 1}}{c} + {a_{n +}}_{1})

(14)

Then, the elevation difference between points O₀ and O_n+1, denoted as $Δ h_{O_{0} O_{n + 1}}$ , can be expressed as follows

\begin{matrix} Δ h_{O_{0} O_{n + 1}} & = h_{O_{0}} - h_{O_{n + 1}} = h_{C} - h_{C'} + R sech a_{1} \\ - R' sech (\frac{{l_{n}}_{+ 1}}{c} + {a_{n +}}_{1}) \end{matrix}

(15)

The expression for Δ₂ is

Δ_{2} = R' \sin θ' - Δ_{3} = - R' \tanh (\frac{{l_{n}}_{+ 1}}{c} + {a_{n +}}_{1}) - Δ_{3}

(16)

where Δ₃ is the horizontal distance between points C’ and D’, as shown in Figure 2(b).

At an arbitrary suspension point on the main cable, the axial tensile force can be decomposed into a horizontal and a vertical component, as shown in Figure 3 (Zhang et al., 2018b). From the force equilibrium condition in the vertical direction, we can obtain

H \tan φ_{i} = H \tan φ_{i + 1} + P_{i}

(17)

where P_i is the hanger tensile force and φ_i and φ_i+₁ are the inclination angles for the cable segments at the left and right of the suspension point, O_i, respectively.

Figure 3.

The equilibrium between forces at a suspension point.

Upon substituting tanφ_i = sinh(l_i / c + a_i) and tanφ_i+₁ = sinha_i+1 into equation (17), we get

H \sinh (\frac{l_{i}}{c} + a_{i}) = H \sinh {a_{i +}}_{1} + P_{i}

(18)

Then

{a_{i +}}_{1} = asinh [\sinh (\frac{l_{i}}{c} + a_{i}) - \frac{P_{i}}{H}]

(19)

The horizontal distance between the tangent point on the left tower saddle and the first hanger, l₁, can be expressed as

l_{1} = {l_{D}}_{O_{1}} - (R \sin θ - Δ_{1}) = {l_{D}}_{O_{1}} - (R \tanh a_{1} - Δ_{1})

(20)

where ${l_{D}}_{O_{1}}$ is the designed horizontal distance between the left tower saddle actual apex, D, and the first hanger, and Δ₁ is the horizontal distance from the circle center, C, to the actual cable apex on the left tower saddle, D, as shown in Figure 2(a).

Substituting equations (4), (9), (15), (16), (19), and (20) into equation (3) yields three nonlinear governing equations that are coupled to each other

\begin{matrix} f_{1} (H, a_{1}) = & c \sum_{i = 1}^{m} [\cosh (\frac{l_{i}}{c} + a_{i}) - \cosh a_{i}] \\ - (h_{C} + R sech a_{1} - h_{O_{m}}) = 0 \end{matrix}

(20a)

\begin{matrix} f_{2} (H, a_{1}, {l_{n +}}_{1}) = c \sum_{i = 1}^{n + 1} [\cosh (\frac{l_{i}}{c} + a_{i}) - \cosh a_{i}] \\ - [h_{C} - h_{C'} + R sech a_{1} - R' sech (\frac{{l_{n}}_{+ 1}}{c} + {a_{n +}}_{1})] = 0 \end{matrix}

(20b)

\begin{matrix} f_{3} (H, a_{1}, {l_{n +}}_{1}) = l_{n + 1} - l_{O_{n} D'} \\ - R' \tanh (\frac{{l_{n}}_{+ 1}}{c} + {a_{n +}}_{1}) - Δ_{3} = 0 \end{matrix}

(20c)

where f₁(), f₂(), and f₃() denote functions.

For solving the resulting system of equations (20), it can be reduced by transformation to the following unconstrained optimization problem

min F (H, a_{1}, {l_{n +}}_{1})

(21)

where $F () = f_{1}^{2} () + f_{2}^{2} () + f_{3}^{2} ()$ .

Equation (21) can be solved either by the Newton–Raphson method or the conjugate technique, which are incorporated into the Microsoft Excel program, making their application more user-friendly.

By solving equation (21) via any of the above methods, one can derive the three unknown parameters, namely, H, a₁, and l_n+₁. This permits a further determination of the cable shape and internal forces in the main span, which includes the position of each tangent point, elevation of each suspending point, stresses and strains in each cable segment, and so on.

The above analysis was focused on the main span, in which the cable sag was specified at the design stage. In the side spans, the shape-finding method for the cable is the same as the main span, except for slightly different known conditions: the cable sag in the side span cannot be used as a known parameter. However, for the side span of a suspension bridge, the horizontal component of the cable tensile force is always equal to that in the main span to keep the pylons under the optimal stress state. Therefore, the shape-finding analysis for the side span is quite straightforward but, for brevity sake, the details of the cable shape calculation in the side span are omitted in this article.

Finite element method simulation of tensile forces in hangers

Hanger tensile force values were used in the cable shape calculation by the proposed analytical method. However, the particular values are unknown, and some assumptions had to be used. To justify the assumed values, the finite element method (FEM) can be used to simulate cables, hangers, and the deck in the main span, as shown in Figure 4. However, despite the effect of pylon displacement caused by its elastic deformation, it is not necessary to model pylons. During the erection of the stiffening girder and floor system, the vertical pressure acting on the pylon top via the main cable gradually increases and induces an elastic contraction, Δh, of the tower in the completed bridge state. To get the precise pylon top position in the completed bridge state, the pylon top is usually built with a pre-raised height, Δh, which vanishes in the completed bridge state (Zhang et al., 2018b). Given this, pylons can be treated as incompressible and excluded from the further FEM analysis.

Figure 4.

Sketch of the FEM simulation scheme.

Link elements are employed to model cables and hangers, while beam elements are used to simulate the deck. The link element has double linear stiffness matrices, which enable it to be a tension-only or compression-only element. If it is selected as a tension-only element, the stiffness will vanish when it is compressed. Therefore, it is suitable to model the cables and hangers. The two ends of the cable are at the cable’s tangent points on the pylon saddles, and they are hinged. The coordinates of each node along the cable are those of each suspension point, which are derived via the above analytical shape-finding process that is also used to assess the cable tensile forces by their conversion to initial strains via Hooke’s law. The node coordinates of the beam are controlled by the designed deck geometry. Moreover, according to the planned arrangement of the bearings, the corresponding degrees of freedom (DoF) at the two beam ends are constrained in the FEM model. To avoid the excessive deflection of the deck under its weight, the initial strains that can be obtained from the assumed hanger forces should be incorporated into the hangers. The hanger tensile forces of the suspension bridge under its weight can be obtained by the static problem calculations. Furthermore, iterative calculations are required because the newly calculated hanger forces may differ from the previously assumed ones.

Iterative calculation steps

Since the initial strains are first obtained based on the assumed hanger tensile forces, the iterative process, in which the hanger tensile forces are updated, is required to determine the cable shape. The flowchart of the iteration process is shown in Figure 5, and the calculation steps are as follows:

Assume the initial values for the tensile forces acting on all hangers.

Determine the cable shape and internal forces in the completed bridge state, including positions of each tangent point, elevations of each suspending point, as well as stresses and strains of each cable segment, by the proposed analytical shape-finding method.

Model the suspension bridge based on the parameters obtained in step (2), and calculate the hanger tensile forces by the FEM model.

Verify the error of the assumed hanger tensile forces using the following inequality

max_{1 \leq i \leq n} | \frac{P_{i, k} - P_{i, k - 1}}{P_{i, k - 1}} | \leq ε

(22)

where P_i,k is the hanger tensile force calculated by FEM, P_i,k–1 is the assumed hanger tensile force, and ε is the error tolerance.

Figure 5.

Flowchart of the iteration process.

If inequality (22) is not satisfied, value P_i,k is used as the assumed hanger tensile force, and the procedure returns to step (2) for the next iteration. Otherwise, the calculation is completed, and both the cable shape and hanger tensions of the suspension bridge are determined.

The weight distribution method can be alternatively used to determine the initial values of the hanger tensile forces. As shown in Figure 6, it is assumed that each pair of hangers supports half of the weight of the left and right deck segments. Therefore, the hanger tensile force can be expressed as follows

P_{0} = \frac{0.5 w (S_{l} + S_{r})}{2} = \frac{w (S_{l} + S_{r})}{4}

(23)

where w is the dead load (including the girder and floor system) per unit length, while S_l and S_r are the lengths of the left and right deck segments, respectively.

Figure 6.

The weight distribution method applied to the assessment of hanger tensile forces.

Verification study

In order to verify how well the proposed analytical method predicts the cable geometry with known hanger tensions, the Great Belt suspension bridge is selected as a verification example. This problem was first considered by Karoumi (1999), and later analyzed by Kim and Lee (2001), Kim and Kim (2012), and Cao et al. (2017). The length of the main span is 1624 m. The cable sag in the main span is 180 m. The structural model used in this study mostly follows Karoumi’s model (Karoumi 1999), in which every third hanger from the original bridge is included. The detailed geometry of the cable in the main span is described in Figure 7. The cross-sectional area (A), elastic modulus (E), and weight per unit length (w) of the cable are 0.4 m², 210 GPa, and 32.9 kN/m, respectively. The saddles’ radii can be set to be 0. The hanger tensile forces, P_i, are known and listed in Table 1.

Figure 7.

The main cable and geometric parameters of the main span (unit: m).

Table 1.

Known hanger tensile forces and calculated nodal coordinates of the main cable.

Node no.	P_i (kN)	y (m)			(3) – (1)	(3) – (2)
Node no.	P_i (kN)	(1) Kim and Kim (2012)	(2) Cao et al. (2017)	(3) Present work	(m)	(m)
0	/	180.000	180.000	180.000	0.000	0.000
1	4906.596	151.132	151.129	151.129	−0.003	0.000
2	4897.055	124.812	124.811	124.810	−0.002	−0.001
3	4899.638	101.032	101.031	101.030	−0.002	−0.001
4	4898.905	79.781	79.78	79.779	−0.002	−0.001
5	4899.079	61.050	61.050	61.049	−0.001	−0.001
6	4899.105	44.833	44.833	44.833	0.000	0.000
7	4899.045	31.122	31.123	31.122	0.000	−0.001
8	4899.042	19.912	19.913	19.912	0.000	−0.001
9	4899.115	11.198	11.198	11.198	0.000	0.000
10	4899.043	4.977	4.977	4.977	0.000	0.000
11	4899.043	1.245	1.245	1.245	0.000	0.000
12	4899.115	0.001	0.001	0.001	0.000	0.000

The nodal coordinates of the main cable calculated by the proposed analytical method are compared with above-mentioned literatures in Table 1. Table 2 summarizes the results for the horizontal tension of the main cable. It can be seen that the obtained results are almost identical with those given by other literatures.

Table 2.

Horizontal tension of the main cable.

Literature	Kim and Lee (2001)	Kim and Kim (2012)	Cao et al. (2017)	Present work
Value (×10⁶ N)	194.00	193.75	193.69	193.83

Numerical simulation example

To verify the proposed method applicability to full-scale suspension bridges, the Jindong Bridge across the Jinsha River located in Kunming, China, was used as an example. The Jindong Bridge, as depicted in Figure 8(a) and schematically presented in Figure 8(b), has a main span of 730 m, with the left and right spans of 240 and 120 m, respectively. Thus, the total length is 1090 m.

Figure 8.

(a) Photo and (b) elevation view of Jindong Bridge (units in meters).

The sag-to-span ratio of the cable in the central span is 1 / 10. The spacing between the upstream and downstream cables is 17.5 m. Prefabricated parallel wire strands are adopted for both cables. Each strand consists of 91Φ5.2 high-strength galvanized steel wires, and its cross section has a regular hexagonal shape. Each cable is made of 91 strands and, thus, contains 8281 wires in total. Cables are anchored by tunnel-type anchorages at the left (Huidong) side and gravity-type anchorages at the right (Dongchuan) one. The total number of hangers in each cable plane is 71, and their layout is shown in Figure 8(b). Each hanger is composed of 109Φ5.0 high-strength galvanized steel wires.

The Jindong Bridge structure includes a stiffening steel truss girder, which consists of the main truss, top and bottom bracings, and transverse trusses, as shown in Figure 9. The main truss is a Warren-type truss of 5 m in height, 17.5 m in width, and 5.0 m in panel length. Each panel contains a transverse truss, while the bridge floor system combines longitudinal I-beams and a concrete slab deck. The transverse spacing between longitudinal beams is 2.0 m, while their heights range from 0.4 to 0.56 m. Beams are simply supported by the top chords of the cross-members in the main truss. The bridge deck is constructed of precast concrete slabs with a length of 4.96 m, a width of 1.70 m, and a thickness of 0.16 m each. The slab deck and longitudinal beams are fastened by shear pins. The dead load is 177.12 kN/m, including the secondary dead load of 36.40 kN/m. The horizontal distance between the side hanger and the deck endpoint is 13 m.

Figure 9.

Steel truss girder and bridge floor system (Zhang et al., 2018a).

The initial values of the hanger tensile forces, P_i,0, were first determined by using the weight distribution method. This yielded P_1,0 = P_71,0 = 1018.44 kN, and P_i,0 = 885.6 kN, where i = 2–70. With these initial values of the hanger tensile forces, the cable shape and internal forces were determined by the analytical shape-finding method proposed in this article. Then the bridge was modeled by the FEM method using the commercial ANSYS software package via the scheme depicted in Figure 10. There are 2940 nodes and 5785 elements in total. The element numbers for cables, hangers, and the truss girder are 144, 142, and 5499, respectively. The vertical and lateral DoF at the two girder-ends are constrained, as shown in Figure 10(b). The input parameters required for the cable shape-finding using the analytical method and the FEM simulation are listed in Tables 3 and 4, respectively.

Figure 10.

(a) Full-bridge finite element model and (b) a detailed part of the girder.

Table 3.

Input parameters used to calculate the cable geometry by the analytical method.

Basic parameters in the completed bridge state		Symbol	Unit	Value
The number of hangers/suspension points along the cable		n	/	71
The number of cable segments between the left tangent point, O₀, and the midspan point, O_m		m	/	36
The elastic modulus of wires in the cable		E	GPa	197.03
The cross-sectional area of the cable		A	m²	0.1759
Main cable weight per unit length		q	kN/m	14.268
Hanger spacing		l₂–l₇₁	m	10
Designed elevation of point O_m in the geodetic system		$h_{O_{m}}$	m	856.000
Left tower saddle	Radius of the tower saddle arc-shaped top	R	m	5.5
	Designed elevation of point C in the geodetic system	h_C	m	923.210
	Designed horizontal distance between the actual apex of the left tower saddle, D, and the first hanger	${l_{D}}_{O_{1}}$	m	15
	Horizontal distance from point C to the cable actual apex, D	Δ₁	m	0.227
Right tower saddle	Radius of the tower saddle arc-shaped top	R’	m	5.5
	Designed elevation of point C’ in the geodetic system	h _C’	m	923.090
	Designed horizontal distance between the rightmost hanger and the actual apex of the right tower saddle, D’	$l_{O_{72} D'}$	m	15
	Horizontal distance between points C’ and D’	Δ₃	m	−0.061^a

Sign “–” indicates that point C’ is located to the left of point D’.

Table 4.

Input parameters for the FEM simulation.

Basic parameters in the completed bridge state	Symbol	Unit	Value
The elastic modulus of wires in the hanger	E’	GPa	197.03
The hanger cross-sectional area	A’	m²	2.14e–3
Hanger weight per unit length	q’	kN/m	0.1835
Elastic modulus of the steel truss	E”	GPa	206

FEM: finite element method. The significance of bold values are: They are big values.

For the error tolerance ε = 0.035%, the convergent solution for the hanger tensile forces can be attained after 51 iterations. The variation of the hanger tensile forces at the upper endpoints of hanger Nos. 1–6 versus the iteration number is depicted in Figure 11. As seen in Figure 11, the maximal number of iterations was required to attain a convergent value of the tensile force for hanger No. 1, followed by hanger No. 2, while the remaining hangers need the minimal number of iterations. The convergent values of the hanger tensile forces at the upper and lower endpoints calculated by the iterative method are listed in Table 5.

Figure 11.

Variation of hanger tensile force versus the iteration number.

Table 5.

Hanger tensions calculated by different methods.

Hanger No.	Weight distribution method (kN)	CMRSB method (kN)	Iterative method (kN)		Error (%)
	Weight distribution method (kN)	CMRSB method (kN)	Upper endpoint	Lower endpoint	Error (%)
	(A)	(B)	(C)	(D)	(A – D) / D	(B – D) / D
1	1018.4	1074.5	1133.7	1120.8	−9.14	−4.13
2	885.6	900.7	907.5	895.4	−1.09	0.59
3	885.6	881.5	886.0	874.5	1.27	0.80
4	885.6	882.9	890.1	879.3	0.72	0.41
5	885.6	884.4	893.9	883.6	0.23	0.09
6	885.6	885.3	895.1	885.4	0.02	−0.01
7	885.6	885.5	894.7	885.7	−0.01	−0.02
8–64	885.6	885.5–885.8	886.5–894.3	885.7–885.9	−0.03 to −0.01	−0.02 to −0.01
65	885.6	885.6	894.8	885.8	−0.02	−0.02
66	885.6	885.2	894.9	885.3	0.03	−0.01
67	885.6	884.6	893.8	883.6	0.23	0.11
68	885.6	882.8	889.5	878.6	0.80	0.48
69	885.6	881.6	885.0	873.5	1.39	0.93
70	885.6	900.6	906.8	894.6	−1.01	0.67
71	1018.4	1074.6	1140.7	1127.9	−9.70	−4.73

CMRSB: continuous multiple-rigid-support beam.

For the comparative analysis, hanger tensile forces were alternatively calculated via the CMRSB method. The finite element model of the Jindong Bridge for the CMRSB application was constructed by keeping the girder mesh in the previous full-bridge finite element model, and replacing the nodes of cables and hangers by rigid supports at the hanger locations, as shown in Figure 12. After a static equilibrium analysis under the dead load, the reaction forces at the newly added supports were obtained, as listed in Table 5.

Figure 12.

A part of the CMRSB model of the Jindong Bridge.

As seen in Table 5, the results on the tensile forces in side hangers (Nos. 1 and 71) by the weight distribution and CMRSB methods deviate from those calculated by the proposed iterative method by 10% and 5%, respectively. For the remaining hangers (Nos. 2–70), the three methods provide nearly identical estimates, with minimal deviations. By the way, the maximum tensile force capacity of each hanger is 3570 MPa, which is much greater than all hanger tensile forces.

To analyze the impact of deviations in the assessment of hanger tensions on the cable shape, the following three cases were considered. Hanger tensions used for the cable shape-finding were calculated by (1) the weight distribution method (case A), as listed in column A in Table 5; (2) the CMRSB method (case B), as recorded in column B in Table 5; and (3) the proposed iterative method (case C), as listed in column C in Table 5.

The tangent point positions, cable tensile forces, and suspension point elevations in the above three cases were computed by the analytical shape-finding method, as listed in Tables 6 and 7. As follows from the results in Table 6, the tangent point positions in the three cases are nearly identical, with the maximal error not exceeding 1 cm. The errors for the horizontal components of the cable tensile forces are also negligible. However, as listed in Table 7, the elevation errors for several suspension points are about 4 cm, which are not insignificant anymore.

Table 6.

Tangent point positions and cable tensile forces.

Compared items	Case A	Case B	Case C
Left tangent point
Distance from the left pylon central line (m)	1.783	1.791	1.786
Elevation (m)	928.330	928.327	928.328
Right tangent point
Distance from the right pylon central line (m)	2.071	2.079	2.074
Elevation (m)	928.210	928.207	928.209
Horizontal component of the cable tensile force (kN)	94,131.6	94,249.9	93,962.8

Table 7.

Elevations at the suspension points.

No.	Elevation (m)			Error (cm)		No.	Elevation (m)			Error (cm)
No.	Case A	Case B	Case C	A – C	B – C	No.	Case A	Case B	Case C	A – C	B – C
1	923.154	923.146	923.132	2.2	1.4	37	856.055	856.055	856.054	0.1	0.1
2	919.366	919.357	919.339	2.7	1.8	38	856.219	856.219	856.218	0.1	0.1
3	915.688	915.681	915.658	3	2.3	39	856.492	856.492	856.491	0.1	0.1
4	912.121	912.114	912.088	3.3	2.6	40	856.875	856.875	856.873	0.2	0.2
5	908.663	908.657	908.628	3.5	2.9	41	857.368	857.368	857.365	0.3	0.3
6	905.317	905.311	905.278	3.9	3.3	42	857.970	857.970	857.965	0.5	0.5
7	902.080	902.075	902.040	4	3.5	43	858.681	858.681	858.675	0.6	0.6
8	898.954	898.949	898.913	4.1	3.6	44	859.502	859.502	859.495	0.7	0.7
9	895.937	895.933	895.896	4.1	3.7	45	860.433	860.432	860.423	1	0.9
10	893.031	893.027	892.990	4.1	3.7	46	861.472	861.472	861.461	1.1	1.1
11	890.235	890.231	890.194	4.1	3.7	47	862.622	862.621	862.609	1.3	1.2
12	887.549	887.545	887.509	4	3.6	48	863.881	863.880	863.865	1.6	1.5
13	884.973	884.970	884.934	3.9	3.6	49	865.250	865.249	865.232	1.8	1.7
14	882.507	882.504	882.469	3.8	3.5	50	866.728	866.727	866.708	2	1.9
15	880.151	880.148	880.114	3.7	3.4	51	868.316	868.314	868.293	2.3	2.1
16	877.904	877.902	877.870	3.4	3.2	52	870.013	870.012	869.988	2.5	2.4
17	875.768	875.765	875.735	3.3	3	53	871.821	871.819	871.793	2.8	2.6
18	873.741	873.739	873.711	3	2.8	54	873.737	873.735	873.707	3	2.8
19	871.824	871.822	871.796	2.8	2.6	55	875.764	875.762	875.732	3.2	3
20	870.016	870.015	869.991	2.5	2.4	56	877.901	877.898	877.866	3.5	3.2
21	868.319	868.317	868.296	2.3	2.1	57	880.147	880.144	880.110	3.7	3.4
22	866.730	866.729	866.710	2	1.9	58	882.503	882.500	882.465	3.8	3.5
23	865.252	865.251	865.234	1.8	1.7	59	884.969	884.965	884.929	4	3.6
24	863.883	863.882	863.868	1.5	1.4	60	887.545	887.541	887.504	4.1	3.7
25	862.624	862.623	862.611	1.3	1.2	61	890.231	890.227	890.189	4.2	3.8
26	861.474	861.474	861.463	1.1	1.1	62	893.027	893.022	892.985	4.2	3.7
27	860.434	860.434	860.425	0.9	0.9	63	895.932	895.928	895.891	4.1	3.7
28	859.504	859.503	859.496	0.8	0.7	64	898.949	898.944	898.907	4.2	3.7
29	858.682	858.682	858.677	0.5	0.5	65	902.075	902.069	902.035	4	3.4
30	857.971	857.971	857.967	0.4	0.4	66	905.311	905.305	905.273	3.8	3.2
31	857.369	857.369	857.366	0.3	0.3	67	908.658	908.652	908.622	3.6	3
32	856.876	856.876	856.874	0.2	0.2	68	912.115	912.108	912.082	3.3	2.6
33	856.493	856.493	856.492	0.1	0.1	69	915.682	915.675	915.652	3	2.3
34	856.219	856.219	856.219	0	0	70	919.360	919.351	919.333	2.7	1.8
35	856.055	856.055	856.055	0	0	71	923.148	923.140	923.125	2.3	1.5
36	856.000	856.000	856.000	0	0

The significance of bold values are: They are big values.

The stresses of bridge components should be checked. According to the FEM result, the maximum tensile stresses of the hanger and the cable are 533 and 579 MPa, respectively, which are less than the yield stress limit, 1670 MPa. The maximum tensile stress and compressive stress of the steel truss are 142 and 95 MPa, respectively, which are less than the yield stress limit, 345 MPa.

It is noteworthy that the proposed method and the respective calculation results were successfully implemented in the design and construction of the Jindong Bridge, which ensured its improved cable geometry.

Conclusion

In this article, we proposed the iterative method to determine the hanger tensile forces and the cable geometry of a suspension bridge. The FEM analysis is used to calculate the hanger tensile forces with an account of the joint effects of the cable-hanger-stiffening girder, while the cable shape and the impact of pylon saddle arc are predicted by the analytical cable shape-finding method. Since it is hard to model the pylon saddle arc and the sliding of the cable tangent point on the pylon saddle through FE/multi-physics software, the joint application of the analytical approach and FEM analysis is found to be more instrumental. In the process of iteration, the input of hanger tensile forces and cable shape characteristics is made in turns. The feasibility and validity of the proposed method are proved by the numerical test example for the suspension bridge, namely, Jindong Bridge across the Jinsha River located in Kunming, China. The results obtained make it possible to draw the following conclusions.

In the cable shape-finding analytical method, the effect of the pylon saddle can be taken into account by determining the position of the tangent point, which increases the number of nonlinear governing equations from 2 to 3. The three coupled nonlinear governing equations can be conveniently solved by transforming them into the form corresponding to the unconstrained optimization problem.

When the convergent values of hanger tensile force are derived, the maximal number of iterations is required for the side hangers, the adjacent hangers occupy the second place in the decreasing order, while the remaining ones need a small number of iterations.

As compared to the side hanger tensile forces assessed by the proposed iterative method, those calculated by the weight distribution and CMRSB methods have significant errors, for example, 10% and 5% in the Jindong bridge, respectively. For the remaining hangers, the three methods provide nearly identical results.

In three cases of analytical cable shape-finding, where hanger tensile forces calculated by the three above methods were used as inputs, quite significant errors in the suspension point elevations were found to be induced, even though the errors of resulting tangent point positions and cable tensions are negligible.

The iterative method proposed in this study was proved to be instrumental in the assessment of hanger tensile forces, especially for the side hangers, as well as accurate cable shape prediction.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The research described in this article was financially supported by the NSFC under Grant 51678148, a project supported by the Natural Science Foundation of Jiangsu Province (BK20181277), and the National Key R&D Program of China (No. 2017YFC0806009), which are gratefully acknowledged.

ORCID iD

Wen-ming Zhang

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