The Basic Differential Equations of Self-Anchored Cable-Stayed Suspension Bridge

Abstract

The static behavior of self-anchored cable-stayed suspension bridge under vertical load is described with the continuum method. Based on the partition generalized variation principle, considering the compression-bending coupling effect of the main girder and the tower, the large displacement incomplete generalized potential energy functional of three-span self-anchored cable-stayed suspension bridge is established. Then, the basic differential equations of self-anchored cable-stayed suspension bridge are derived through constraint variation. Taking a self-anchored cable-stayed suspension bridge with main span 100 m, for example, the results by the proposed analytic method agree with that of numerical analysis. Therefore, the basic differential equations proposed in this paper could be applied to the preliminary analysis of self-anchored cable-stayed suspension bridge. The equations also provide a theoretical basis to describe the static behavior of this type of bridge.

1. Introduction

Sea-crossing bridge projects in the 21st century are facing adverse natural conditions such as deep water foundations, strong winds, and so on. Either suspension bridge or cable-stayed bridge for large span bridge has shown shortcomings and inadequacies [1]. Therefore, the cable-stayed suspension bridge, which combines the advantages of cable-stayed bridge and suspension bridge to make up for their shortcomings, provides a suitable option for long-span bridge, particularly those sea-crossing bridge. There are some cable-stayed suspension bridges in the world, such as Albert Bridge in London, Brooklyn Bridge in New York, and so on [2]. Wujiang Bridge in China, with spans of 66+288+66m is the longest modern cable-stayed suspension bridge. Nagisa Bridge, a single span footbridge in Japan, is a hybrid structure of cable-stayed prestressed concrete bridge and steel suspension bridge. And the schemes of cable-stayed suspension bridge have been proposed many times, such as Turkey Izmit Bridge, Gibraltar Strait Bridge, Lingdingyang Bridge, and so on. However, due to lack of systematic study on such bridge type, the proposals are not implemented. At present, the schemes of cable-stayed suspension bridge just stay in the program design phase, and the proposed schemes all adopt the earth-anchored system with large anchor. Using a self-anchored system could not only save the anchor, but also possess the advantages of cable-stayed suspension bridge. The Jianshe Bridge in Dalian is the first self-anchored cable-stayed suspension bridge in the world [3]. And self-anchored cable-stayed suspension bridge proposals have been brought up for Dalian Bay Bridge, Dalian Jinzhou Bay Bridge, and Eastern Hubei Yangtze River Bridge.

The bridge structural behavior under static load is an important basis for its design. Ichiro Konishi gave a detailed elaboration on the static and dynamic basic differential equations of earth-anchored suspension bridge [4]. Based on the large-displacement partition generalized variation principle, the incomplete generalized potential energy functional of a three-span self-anchored cable-stayed suspension bridge is established with the continuum method. The differential equations of self-anchored cable-stayed suspension bridge are derived through constraint variation. The differential equations could analyze the structural behavior of self-anchored cable-stayed suspension bridge under vertical load and could provide a theoretical basis for static analysis of such type bridge.

2. The Establishment of the Large Displacement Partition Generalized Potential Energy Functional

Figure 1 shows a self-anchored cable-stayed suspension bridge. Based on the structural characteristics, the following items are assumed [4].(1) All the materials conform with Hooke's Law.(2) In operational bridge status, the dead load is distributed uniformly along the span, and the main cables are of parabolic shape.(3) The main girder is a continuous beam with constant cross section, and vertical supports exist at the towers. Ignore the vertical curve and camber of the girder.(4) The stay cables and the hangers are dense and could be compared to a homogeneous membrane with only axial resistance.

Figure 1 

Illustration of a self-anchored cable-stayed suspension bridge.

𝑞(𝑥) is the dead load on the girder, while 𝑝(𝑥) the live load. Define the displacement and strain of various parts as follows: 𝑢_𝑐,𝑣_𝑐 are the longitudinal and vertical displacements of the main cable respectively, 𝑢_𝑏,𝑣_𝑏 are that of the girder, and 𝑢_𝑡,𝑣_𝑡 are that of the tower. And 𝜀_𝑐,𝜀_ℎ,𝜀_𝑙 are the strain increments of the main cable, the hanger, and the stay cable caused by the live load, respectively. In Figure 1, [𝑥₁,𝑥₂]∪[𝑥₃,𝑥₄] is the cable-stayed interval, while [𝑥₂,𝑥₃] the suspension interval. Due to different boundary conditions between the cable-stayed interval and the suspension interval, derivation of each will be undertaken separately.

2.1. Establishment of Strain Energy of Components within the Cable-Stayed Interval

2.1.1. Establishment of Strain Energy of the Main Cable

Based on the geometric relationship shown in Figure 2, the following exists for the main cable𝑑𝑠cos𝜃=𝑑𝑥,1+𝜀_𝑐𝑑𝑠cos𝜃+𝜑=𝑑𝑠cos𝜃+𝑑𝑢_𝑐,1+𝜀_𝑐𝑑𝑠sin𝜃+𝜑=𝑑𝑠sin𝜃+𝑑𝑣_𝑐, (2.1) where 𝜃+𝜑 is the inclination angle of the main cable after deformation and 𝜑 the change of the main cable inclination. Elimination of 𝜑 and element length 𝑑𝑠 leads to the cable strain increment given as 𝜀_𝑐=cos𝜃+𝑢^′_𝑐cos𝜃²+sin𝜃+𝑣^′_𝑐cos𝜃²−1. (2.2) The strain energy of the main cable is [5]𝑈_𝑐1=^𝑥2_𝑥1^{𝜀𝑐0+𝜀𝑐}_𝜀𝑐0𝑇sec𝜃𝑑𝜀_𝑐𝑑𝑥, (2.3) where 𝜀_𝑐0 is the initial strain of the main cable and 𝑇 is the cable tension.

Figure 2 

Deformation of the main cable element.

2.1.2. Establishment of Strain Energy of the Stay Cables

According to the assumptions, the strain energy of the stay cables is [6] 𝑈_𝑙=^𝑥2_𝑥1^{𝜀𝑙0+𝜀𝑙}_𝜀𝑙0𝑙(𝑥)𝑐(𝑥)𝑑𝜀_𝑙𝑑𝑥, (2.4) where 𝜀_𝑙0 is the initial strain of the stay cables, 𝑙(𝑥) the stay cable length, and 𝑐(𝑥) the tension of stay cables under dead and live loads.

2.1.3. Establishment of Strain Energy of the Stay Cables

Due to the action of the horizontal component force of the main cable, the girder is under compression and bending. The assumption that the structure is in an elastic stable state and shear deformation is neglected will lead to the expression of the axial strain [7]𝜀_𝑥=𝜕𝑢_𝑏𝜕𝜕𝑥−𝑦²𝑣_𝑏𝜕𝑥²+12𝜕𝑣_𝑏𝜕𝑥². (2.5) Then, the strain energy of the girder gives [8, 9]𝑈_𝑏1=12^𝑥2_𝑥1𝑁_𝑥𝜕𝑢_𝑏𝜕𝑥−𝑀_𝑧𝜕²𝑣_𝑏𝜕𝑥²+𝑁_𝑥𝜕𝑣_𝑏𝜕𝑥²𝑑𝑥, (2.6) where 𝑁_𝑥 is the girder axial force and 𝑀_𝑧 is the girder bending moment. In (2.6), the first item is the strain energy caused by compression, the second that caused by bending, and the third that due to compression-bending coupling and could be neglected in linear structural analysis.

2.1.4. Establishment of Strain Energy of the Stay Cables

Based on the above derivation, strain energy of the tower gives [10, 11] 𝑈_𝑡=12^𝐻₀𝑁_𝑡𝜕𝑣_𝑡𝜕𝑦−𝑀_𝑡𝜕²𝑢_𝑡𝜕𝑦²+𝑁_𝑡𝜕𝑢_𝑡𝜕𝑦²𝑑𝑦, (2.7) where 𝑁_𝑡 is the axial force of the tower and 𝑀_𝑡 is the bending moment. And due to the symmetry of the structure, the strain energy of the main cable, the tower, and the girder within the interval [𝑥₃,𝑥₄] equals that within the interval [𝑥₁,𝑥₂].

2.2. Establishment of Strain Energy of Components within the Suspension Interval

According to (2.3)–(2.7), the strain energy of the main cable, the girder, and the hanger is given as follows.

The strain energy of the main cable is𝑈_𝑐2=^𝑥3_𝑥2^{𝜀𝑐0+𝜀𝑐}_𝜀𝑐0𝑇𝑑𝜀_𝑐𝑑𝑠=^𝑥3_𝑥2^{𝜀𝑐0+𝜀𝑐}_𝜀𝑐0𝑇sec𝜃𝑑𝜀_𝑐𝑑𝑥. (2.8)

The strain energy of the girder is𝑈_𝑏2=12^𝑥3_𝑥2𝑁_𝑥𝜕𝑢_𝑏𝜕𝑥−𝑀_𝑧𝜕²𝑣_𝑏𝜕𝑥²+𝑁_𝑥𝜕𝑣_𝑏𝜕𝑥²𝑑𝑥. (2.9)

The strain energy of the hanger is𝑈_ℎ=^𝑥3_𝑥2^{𝜀ℎ0+𝜀ℎ}_𝜀ℎ0ℎ(𝑥)𝑠(𝑥)𝑑𝜀_ℎ𝑑𝑥, (2.10) where 𝜀_ℎ0 is the initial strain of the hanger, ℎ(𝑥) is the length of hanger and 𝑠(𝑥) is the tension of hanger under dead and live loads.

2.3. Establishment of the Potential Energy Functional

Based on the partition generalized variation principle, the potential energy functional of the system may be given as [12]=2×𝑈_𝑐1+𝑈_𝑙+𝑈_𝑏1+𝑈_𝑡+𝑈_𝑐2+𝑈_𝑏2+𝑈_ℎ−^𝑥4_𝑥1^𝑣𝑏₀𝑞𝑑𝑣_𝑏𝑑𝑥−^𝑥4_𝑥1^𝑣𝑏₀𝑝𝑑𝑣_𝑏𝑑𝑥. (2.11)

And the constraint conditions are as follows.

2.3.1. The Cable-Stayed Interval

(1) The deformation compatibility condition of the main cable is 𝐺_𝑐1=cos𝜃+𝑢^′_𝑐cos𝜃²+sin𝜃+𝑣^′_𝑐cos𝜃²−1−𝜀_𝑐=0. (2.12)(2) The deformation compatibility condition of the stay cable is 𝐺_𝑙=1𝑙(𝑥)𝑙(𝑥)cos𝛼+𝑢_𝑏−𝑢_𝑡²+𝑙(𝑥)sin𝛼+𝑣_𝑏−𝑣_𝑡²−1−𝜀_𝑙=0. (2.13)

2.3.2. The Suspension Interval

(1) The deformation compatibility condition of the main cable is the same as (2.12) 𝐺_𝑐2=cos𝜃+𝑢^′_𝑐cos𝜃²+sin𝜃+𝑣^′_𝑐cos𝜃²−1−𝜀_𝑐=0. (2.14)(2) The deformation compatibility condition of the hanger is 𝐺_ℎ=1ℎ(𝑥)ℎ(𝑥)+𝑣_𝑏−𝑣_𝑐²+𝑢_𝑏−𝑢_𝑐²−1−𝜀_ℎ=0. (2.15)

Then, the large-displacement partition incomplete generalized potential energy functional is ^∗=+^𝑥2_𝑥1𝜆_𝑐1𝐺_𝑐1𝑑𝑥+^𝑥2_𝑥1𝜆_𝑙𝐺_𝑙𝑑𝑥+^𝑥3_𝑥2𝜆_𝑐2𝐺_𝑐2𝑑𝑥+^𝑥3_𝑥2𝜆_ℎ𝐺_ℎ𝑑𝑥+^𝑥4_𝑥3𝜆_𝑐1𝐺_𝑐1𝑑𝑥+^𝑥4_𝑥3𝜆_𝑙𝐺_𝑙𝑑𝑥+2^𝐻₀𝜆_𝑙𝐺_𝑙𝑑𝑥=+2×^𝑥2_𝑥1𝜆_𝑐1𝐺_𝑐1𝑑𝑥+^𝑥2_𝑥1𝜆_𝑙𝐺_𝑙𝑑𝑥+^𝐻₀𝜆_𝑙𝐺_𝑙𝑑𝑥+^𝑥3_𝑥2𝜆_𝑐2𝐺_𝑐2𝑑𝑥+^𝑥3_𝑥2𝜆_ℎ𝐺_ℎ𝑑𝑥. (2.16)

3. Establishment of the Basic Differential Equations of Self-Anchored Cable-Stayed Suspension Bridge

Perform variation of the generalized functional ∏^∗ on 𝑢_𝑐,𝑣_𝑐,𝑢_𝑏,𝑣_𝑏,𝑢_𝑡,𝑣_𝑡,𝜀_𝑐,𝜀_𝑙,𝜀_ℎ,𝜆_𝑐1,𝜆_𝑐2,𝜆_𝑙, and 𝜆_ℎ, respectively, and correspondent Euler equations could be achieved, that is, 𝛿∏^∗=(𝜕∏^∗/𝜕𝜙_𝑖)𝛿𝜙_𝑖=0 [13], where 𝜙_𝑖=𝑢_𝑐,𝑣_𝑐,𝑢_𝑏,𝑣_𝑏,𝑢_𝑡,𝑣_𝑡,𝜀_𝑐,𝜀_𝑙,𝜀_ℎ,𝜆_𝑐1,𝜆_𝑐2,𝜆_𝑙,𝜆_ℎ.

3.1. Basic Differential Equations of the Cable-Stayed Interval

For the cable-stayed interval 𝑥∈[𝑥₁,𝑥₂]∪[𝑥₃,𝑥₄], the balance equations are,−𝑑𝑇=𝐶𝐶representstheconstant𝑐cos𝛼+𝛽𝑁𝑑𝑥^𝑥+12𝐸𝐴_𝑏𝑣_𝑏^′2=0, (3.1) where (𝛼+𝛽) is the angle between the stay cable and the 𝑥-axis positive direction after deformation𝑑²𝑑𝑥²𝑀_𝑧+𝑑𝑁𝑑𝑥^𝑥𝑣^′_𝑏𝑑−𝑐sin𝛼+𝛽+𝑝+𝑞=0,₂𝑑𝑦²𝑀_𝑡+𝑑𝑁𝑑𝑦^𝑡𝜕𝑢_𝑡𝑑𝜕𝑦+𝑐cos𝛼+𝛽=0,𝑁𝑑𝑦^𝑡+12𝐸𝐴_𝑡𝜕𝑢_𝑡𝜕𝑦²+𝑐sin𝛼+𝛽=0. (3.2)

3.2. Basic Differential Equations of the Suspension Interval

For the suspension interval 𝑥∈[𝑥₂,𝑥₃], the balance equations are𝑑(𝑥)𝑢𝑑𝑥𝑇cos𝜃+𝜑+𝑠^𝑏−𝑢_𝑐ℎ(𝑥)+𝑣_𝑏−𝑣_𝑐²+𝑢_𝑏−𝑢_𝑐²𝑑=0,(𝑥)ℎ(𝑥)𝑑𝑥𝑇sin𝜃+𝜑+𝑠+𝑣_𝑏−𝑣_𝑐ℎ(𝑥)+𝑣_𝑏−𝑣_𝑐²+𝑢_𝑏−𝑢_𝑐²𝑑=0,𝑁𝑑𝑥^𝑥+12𝐸𝐴_𝑏𝑣_𝑏^′2(𝑥)𝑢−𝑠^𝑏−𝑢_𝑐ℎ(𝑥)+𝑣_𝑏−𝑣_𝑐²+𝑢_𝑏−𝑢_𝑐²𝑑=0,₂𝑑𝑥²𝑀_𝑧+𝑑𝑁𝑑𝑥^𝑥𝑣^′_𝑏(𝑥)ℎ(𝑥)+𝑝+𝑞−𝑠+𝑣_𝑏−𝑣_𝑐ℎ(𝑥)+𝑣_𝑏−𝑣_𝑐²+𝑢_𝑏−𝑢_𝑐²=0. (3.3)(3.4)(3.5)(3.6)

The differential equations of either the cable-stayed interval or the suspension interval are quite complicated, and their solutions are difficult to reach. A convenient way to solve the equations is setting a family function of displacement for each parameter that meets the boundary conditions and approaches an approximate solution by utilizing the principle of stationary potential energy [14], but this is not the purpose of this paper. This paper intends to indicate that a relatively precise description of the static behavior of self-anchored cable-stayed suspension bridge under vertical load can be achieved through the continuum approach. And further assumptions are introduced to simplify the above equations.

4. Simplification of the Basic Differential Equations of Self-Anchored Cable-Stayed Suspension Bridge

4.1. Simplification of the Basic Differential Equations of the Cable-Stayed Interval

Based on the above-mentioned assumptions and excluding the impact of compression-bending coupling, the strain energy of the main girder and the tower become 𝑈_𝑏1=12^𝑥2_𝑥1𝑁_𝑥𝜕𝑢_𝑏𝜕𝑥−𝑀_𝑧𝜕²𝑣_𝑏𝜕𝑥²𝑈𝑑𝑥,_𝑡=12^𝐻₀𝑁_𝑡𝜕𝑣_𝑡𝜕𝑦−𝑀_𝑡𝜕²𝑢_𝑡𝜕𝑦²𝑑𝑦. (4.1)

Then, the basic differential equations of the cable-stayed interval could be simplified into,𝑑𝑇=𝐶𝐶representstheconstant𝑁𝑑𝑥^𝑥,𝑑=𝑐cos𝛼+𝛽₂𝑑𝑥²𝑀_𝑧𝑑−𝑐sin𝛼+𝛽+𝑝+𝑞=0,₂𝑑𝑦²𝑀_𝑡𝑑+𝑐cos𝛼+𝛽=0,𝑁𝑑𝑦^𝑡+𝑐sin𝛼+𝛽=0. (4.2)

It can be seen from the above equations that the action of stay cables on the main girder and the tower is equivalent to the action of distributed load, and the basic differential equations of the main girder and the tower accord with the equation of a classic beam.

4.2. Simplification of the Basic Differential Equations of the Suspension Interval

On the basis of the above assumptions, introduce some more as follows: excluding the elongation of the hangers and assuming the vertical displacement of the main girder equals that of the main cable.

Then, (3.5) can be simplified into 𝑁_𝑥+(1/2)𝐸𝐴_𝑏𝑣_𝑏^′2=𝐶 (𝐶 represents the constant). Omitting the higher-order terms leads to 𝑁_𝑥=𝐶. (4.3)

Equation (4.3) indicates that without considering the compression-bending coupling effect, the axial force in the main girder is constant and its value equals that of the horizontal component force of the main cable. And (3.6) can be simplified into𝑑²𝑑𝑥²𝑀_𝑧+𝑑𝑁𝑑𝑥^𝑥𝑣^′_𝑏+𝑇sin𝜃+𝜑+𝑝+𝑞=0. (4.4)

Because the horizontal component force of the main cable accords 𝐻=𝑇cos(𝜃+𝜑) and the axial force of the main girder 𝑁_𝑥=−𝐻, the above equation could be transformed into𝑀^′′𝑑+𝑝+𝑞+𝑑𝑥𝐻tan𝜃+𝜑−𝐻𝑣^′_𝑏=0. (4.5)

Introduction of (2.1) can transform (4.5) into𝑀^′′+𝑝+𝑞+𝐻(𝑑/𝑑𝑥)(𝑦^′+𝑣^′_𝑐−𝑣^′_𝑐)=0. Considering that 𝑞=−𝐻_𝑞𝑦^′′ and 𝐻=𝐻_𝑞+𝐻_𝑝, where 𝐻_𝑝 represents the increment of the horizontal component force of the main cable caused by live load, then the above equation could be further transformed into𝑀^′′+𝑝+𝐻_𝑝𝑦^′′=0, integration of which leads to𝑀=𝑀⁰_𝑝−𝐻_𝑝𝑦, (4.6) where 𝑀⁰_𝑝 is the bending moment of a simply supported beam with the same span under live load 𝑝(𝑥). It can be seen from (4.6) that the horizontal component force of the main cable caused by live load can reduce the live load bending moment of the main girder. Equation (4.6) is just the equilibrium differential equation of the deflection theory of self-anchored suspension bridge [15], and it is same as that of the elasticity theory of earth-anchored suspension bridge.

Equation (4.6) is an approximate equivalent of the linear equation of the elasticity theory. The additional bending moment of the girder caused by the enormous axial force offsets that caused by the live-load deflection when considering the compression-bending coupling effect. Therefore, the main girder of self-anchored cable-stayed suspension bridge can be analyzed utilizing the elasticity theory.

An equation of the compatibility condition is required since two unknowns 𝐻_𝑝,𝑣 exist in (4.6), and according to the compatibility equation in the deflection theory of self-anchored suspension bridge, the specific form of the compatibility equation for the suspension interval is given as follows:𝐻_𝑝𝐸_𝑐𝐴_𝑐^𝑥4_𝑥1𝑑𝑥cos³𝜃±𝛼𝑡^𝑥4_𝑥1𝑑𝑥cos³𝜃−^𝑥4_𝑥1𝑦^′𝑣^′𝐻𝑑𝑥=^𝑝𝐿𝐸_𝑏𝐴_𝑏±𝛼𝑡𝐿−2^𝑥2_𝑥1𝑁_𝑥𝐸_𝑏𝐴_𝑏𝑑𝑥, (4.7) where cos𝜃=𝑑𝑥/𝑑𝑠,𝐸_𝑐𝐴_𝑐, and 𝐸_𝑏𝐴_𝑏, represent the axial stiffness of the main cable and the main girder, respectively, 𝛼 is the temperature linear expansion coefficient, 𝑡, is the temperature variations, 𝐿, is the length of the main girder, and 𝑁_𝑥 is the axial force of the main girder within the cable-stayed interval. And the above equation indicates that the horizontal displacement at the anchor end of the main cable equals that at both ends of the main girder.

5. Example of Verification

Below is a simple example of verification of the above-derived differential equations. As shown in Figure 3, there is a self-anchored cable-stayed suspension bridge with span of 34.5 m + 100 m + 34.5 m. The spacing between both the stay cables and the hangers is 3.1 m, the inclination angles of the stay cables are all 30^∘, the sag of the main cable is 17 m, and a uniformly distributed load of 𝑞=20kN/m is imposed on the main girder. The material and sectional properties of the main components are shown in Table 1.

Table 1 

Material and section properties of the structure components.

Figure 3 

Example of a self-anchored cable-stayed suspension bridge (Unit: m).

Note the continuity of the internal force and the displacement at the junction between the cable-stayed interval and the suspension one, solve the above equations referencing the solution approach [16], and compare the results with that of the nonlinear numerical solution. The comparison result is given as shown in Table 2.

Table 2 

Analytic solution and numerical solution.

It can be seen from Table 2 that the analytic solution is relatively quite close to the nonlinear numerical solution. The error is mainly due to the neglecting of the strain energy of hangers and the compression-bending coupling effect, and in particular, the error of the middle-span bending moment in side span reaches 12%. The result indicates that the analytic approach proposed in this paper can be applied to the preliminary analysis of simple self-anchored cable-stayed suspension bridge, but should not be used in design stage.

6. Conclusion

Based on the large displacement partition, incomplete generalized variation principle, and considering the compression-bending coupling effect of the main girder and the tower, the potential energy functional of a self-anchored cable-stayed suspension bridge is established with the continuum method. Then, the differential equations of that are derived, which describe the static behavior of self-anchored cable-stayed suspension bridge under vertical load. And the following could be seen from the basic differential equations.(1)The axial force along the main cable within the cable-stayed interval is constant under vertical load. In addition, the action imposed on the main girder and the tower by the stay cables is equivalent to that by the distributed load, and the basic differential equations of the main girder and the tower accord with that of a classic beam.(2) The additional bending moment caused by the enormous axial force imposed on the main girder by the main cable offsets the moment caused by the live load deflection when considering the compression-bending coupling effect. Therefore, the main girder of a self-anchored cable-stayed suspension bridge can be analyzed utilizing the elasticity theory.(3) The analytic method proposed in this paper is mainly to provide a theoretical basis to describe the static behavior of self-anchored cable-stayed suspension bridge; the analytic solution is relatively close to the nonlinear numerical solution; therefore, the analytic approach can be applied to the preliminary analysis of simple self-anchored cable-stayed suspension bridge.

Nomenclature

𝑞(𝑥):	Dead load on the girder
𝑝(𝑥):	Live load
𝑢𝑐:	Longitudinal displacements of the main cable
𝑣𝑐:	Vertical displacements of the main cable
𝑢𝑏:	Longitudinal displacements of the girder
𝑣𝑏:	Vertical displacements of the girder
𝑢𝑡:	Longitudinal displacements of the tower
𝑣𝑡:	Vertical displacements of the tower
𝜀𝑐:	Strain increments of the main cable
𝜀ℎ:	Strain increments of the hanger
𝜀𝑙:	Strain increments stay cable
[𝑥1,𝑥2]∪[𝑥3,𝑥4]:	Cable-stayed interval
[𝑥2,𝑥3]:	Suspension interval
𝜑:	Change of the main cable inclination
𝜀𝑐0:	Initial strain of the main cable
𝑇:	Cable tension
𝜀𝑙0:	Initial strain of the stay cables
𝑙(𝑥):	Stay cable length
𝑐(𝑥):	Tension of stay cables
𝑁𝑥:	Axial force of the girder
𝑀𝑧:	Bending moment of the girder
𝑁𝑡:	Axial force of the tower
𝑀𝑡:	Bending moment of the tower
𝜀ℎ0:	Initial strain of the hanger
ℎ(𝑥):	Length of hanger
𝑠(𝑥):	Tension of hanger
𝑈𝑐:	Strain energy of the main cable
𝑈𝑏:	Strain energy of the girder
𝑈𝑙:	Strain energy of the stay cables
𝑈𝑡:	Strain energy of the tower
𝑈ℎ:	Strain energy of the hanger.

Acknowledgments

The authors wish to thank the anonymous referees for their very careful reading of the paper and fruitful comments and suggestions. This work was supported by National Natural Science Foundation of China (no. 50708012) and West Transportation Construction Projects Foundation of Ministry of Communications, China (no. 2006 318 823 50).