Analytical assessment of the full-bridge response to the vertical live load: An algorithm for hybrid cable-stayed suspension bridges

Abstract

Hybrid cable-stayed suspension bridges combine the advantages of cable-stayed and suspension bridges, mitigating their deficiencies. Due to its super spanning ability, this novel form of bridge enjoys a bright application prospect as a large-span structure. This study introduces an analytical algorithm to estimate the full-bridge response of a hybrid cable-stayed suspension bridge with a vertical uniformly distributed load applied to the main beam. This method is based on the assumption that all parameters, such as the geometric configuration, internal force distribution, and material properties, of the whole bridge under a dead load are known; then, it analyzes the full-bridge response under a live load. First, the minimum and independent basic unknown parameters of the full-bridge response are determined. Next, the governing equations are derived and solved based on the conservation of unstressed length of each section of cable, closure of span lengths and elevation difference in different spans, and conditions for stress balance of the main beam. Thus, the values of basic unknown parameters are obtained. Finally, they are substituted into the governing equations to estimate the full-bridge response, including each bridge component’s internal forces and deflections. The proposed analytical method involves no iterative procedures when dealing with nonlinear problems but only focuses on the bridge’s state with and without loading. The results are obtained directly by solving the equations, which have the advantages of high efficiency, simplicity, and clear physical meaning. Finally, the feasibility and effectiveness of the proposed method are verified by a finite-element-based calculation of an exemplary asymmetric cable-stayed suspension cooperation system bridge with a main span of 1400 m.

Keywords

full-bridge response hybrid cable-stayed suspension bridge vertical loading analytical method basic unknown quantities governing equations programming solution

Introduction

Cable-stayed and suspension bridges originated in the 18th century (Starossek, 1996). After hundreds of years of development, the design theory and construction technology have matured for both bridge forms. Cable-stayed and suspension bridges of the kilometer class have become common (Mao and Hu, 2020). The Russky Island Bridge in Russia (Pipinato, 2015) is the cable-stayed bridge with the longest span in the world (1104 m) and a total length of 3100 m. The 1915 Canakkale Bridge in Turkey is the longest suspension bridge today (Mersin, 2020), with a main span of 2023 m. Recently, the construction of sea-spanning bridges on a larger scale has been put on the schedule in different parts of the world. For example, Eurasia and Africa are separated by the Strait of Gibraltar (13 km in width), the Bosporus Strait (30 km in width) separates Europe from Asia, the Tsugaru Strait (23.3 km in width) separates Honshu and Hokkaido islands of Japan. However, cable-stayed or suspension bridges can hardly suffice to cover such long distances. Each bridge type has some intrinsic deficiencies that limit its applications in more complex scenarios.

The following factors restrict the increase in the span of cable-stayed bridges: (1) As the span length increases, the stay cable sag will reduce its usage efficiency (Kim et al., 2016). (2) As the span length increases, the substantial axial force in the main beam itself will cause a significant P-Δ effect (Fleming, 1979; Freire et al., 2006). (3) Since cable-stayed bridges are often constructed using the cantilever construction method, an excessive span may cause nonnegligible bridge instability (Yoo and Choi, 2009).

The primary reasons considered to have restricted the increase in the span of the suspension bridge include the following: (1) The full-bridge is more flexible, leading to poor overall wind-resistant stability (Zhang et al., 2011). (2) The use of giant anchors during the construction of sea-spanning bridges under unfavorable conditions, such as deep water and soft soil foundations, will dramatically increase the total cost of the bridge, resulting in poor economic efficiency (Zhang et al., 2013).

Hybrid cable-stayed suspension (HCSS) bridge can combine the advantages of two bridge forms and overcome the defects of each (Zhang, 2007). The hybrid bridge enjoys bright application prospects as a large-spanning structure (Konstantakopoulos and Michaltsos, 2010), especially on soft soil foundations and in typhoon-affected areas and sea-crossing projects.

The first HCSS bridges (coined as the “Roebling” system) were designed and built in the US by an American civil engineer John A. Roebling in 1844–1867 as a cable bridge modification (Gimsing, 1997). The suspension bridge is the main structure in this bridge type, while stay cables are only arranged near the bridge towers as auxiliary components. A typical engineering case of this bridge type is the Brooklyn Bridge in the United States. It is also the first modern suspension bridge (Gandhi, 2011). In the 20th century, the German civil and structural engineer Franz Dischinger created a hybrid bridge in the modern sense, where stay cables support the main beam near the bridge towers, while hangers support the main beam near the midspan. This hybrid system was coined as the “Dischinger” system (Gimsing and Georgakis, 1983). Despite the long history of hybrid bridges, only a few such bridges have been completed worldwide. Instead, most of them stayed at the scheme design stage, including the Hamburg Elbe Bridge, Strait of Gibraltar Bridge, and Stonecutters’ Bridge. Among the several that have been completed, the Yavuz Sultan Selim Bridge is the best known, having a main span of 1408 m (Guesdon et al., 2020). Noteworthy is that China is currently building several HCSS bridges, such as the Xihoumen Highway and Railway Bridge, Tongling Rail-cum-Road Yangtze River Bridge, Libu Highway and Railway Yangtze River Bridge. All these HCSS bridges have a main span of thousands of meters.

Although HCSS bridges are developing rapidly, they face some problems that require urgent solution. For example, determining the reasonable completed bridge state, designing the full-bridge’s construction plan, solving the full-bridge’s live load response, and fatigue problem of hangers where both the hangers and the stay cables are used. As the bridge’s span length increases, the full-bridge’s live load stiffness becomes the primary concern for bridge designers. The response to live loads is closely related to safety, comfort, and economic efficiency. The main beam will be deflected downward when a vertical load is applied, which further impairs the comfort of traffic, especially for high-speed trains (Xia et al., 2000). All these factors should be considered during bridge design; otherwise, the regular use of the bridge will be affected (Jensen, 2014). It has been observed in the completed hybrid bridges that under the live load, the hanger on the outermost side of the region, where both stay cables and hangers are used, has the largest stress amplitude and therefore is susceptible to fatigue. Determining the full-bridge response under live loads lays the basis for research on the fatigue problem of hangers.

In cable-supported bridges, the full-bridge response to live loads can be analyzed by two methods: the analytical method and the finite element method (FEM). The latter usually involves the complex computational model elaboration. In addition, the computation results may be hard to verify, and it is not conducive to bridge engineers to grasp design information intuitively (Karoumi, 1999). In contrast, the analytical method can lead to universal equations with clear physical meanings. At the preliminary design stage, the estimations from the analytical method can inform the designers on the internal force and deformation of the bridge under loading. At the refined analysis stage of the full-bridge response, the analytical results can be used to verify the finite element model accuracy (Zhou and Xiao, 2013).

Zhang et al. (2008) derived differential equations for a self-anchored HCSS bridge based on the generalized variational principle for large displacement zoning and considering the bending coupling effect of the main beam and bridge towers. The equations were then used to predict the static effect of the bridge under the vertical load. Nevertheless, their method treated the forces in the stay cables and hangers as continuous film forces, and the calculation results were less precise. Xia and Zhang (2011) derived the analytical formula for the gravity stiffness and the cable deflection of the HCSS bridge. Cheng and Li (2015) derived the governing equations for the hybrid system using the potential energy method. The deflection of the main beam was described using a Fourier series and estimated iteratively. However, their approach neglected the main beam’s hanger extension and internal force under the dead load. This method did not apply to situations where the bridge tower displacements were smaller. Instead, it was only suitable for a preliminary analysis under specific conditions. Zhao et al. (2020) established a 1:20-scaled model of the Longgang Bridge, a complex continuous hybrid structure composed of two cable-stayed self-anchored suspension parts and one single-pylon cable-stayed part. The nonlinear behavior of the self-anchored suspension and cable-stayed hybrid bridge during the structural transformation of the construction process was systematically and comprehensively studied. Wang et al. (2021) presented a two-layer form-finding framework for the target configuration under dead load (TCUD) of a novel type of spatial self-anchored HCSS bridge. By integrating finite element analysis (FEA) and analytical formulas with optimization algorithms, they formed a self-regulated interactive analysis for the two related cable subsystems in an iterative manner, in which the starting control point served as an intermediate, flexibly variable connection, to couple the subsystems in a multi-nonlinear environment. Feng et al. (2022) applied the Hellinger-Reissner variational principle to study the live load response of a HCSS bridge. The internal force and deformation were estimated by solving the matrix equation. They further showed that appropriately increasing the rise-span ratio would improve the accuracy of the results. Their method also treated the forces in the hangers and stayed cables as uniformly distributed, which was contrary to reality. In addition, the actual load effect was estimated by the linear superposition of the uniformly distributed and concentrated loads. Zhang et al. (2022) proposed an analytical approach to determine the reasonable completed bridge state of HCSS bridges. For regions where both stay cables and hangers were used, the ratio of hanger force to the vertical force in the stay cable was assigned manually. Besides, the horizontal component force exerted by the main cables on both sides of the bridge tower and each pair of stay cables on the bridge tower were assumed to be equal. It can be foreseen that the stress state of the bridge tower will change under live load. Xiao et al. (2023) used the wind-vehicle-bridge coupling vibration analysis method to investigate the bridge stiffness problem of a large-span cable-stayed-suspension cooperative system. The results showed that the vertical stiffness difference between the suspension cable area and the stay cable area of the cable-stayed cooperative system bridge was obvious. The vertical acceleration response increased obviously when the vehicle travelled to the transition section of the stay cable. We urgently need a calculation method for the live load effect of the HCSS bridge that is highly efficient, accurate, and conforms to engineering practice.

With the vertical load acting on the main beam, the main beam is subjected to the synergistic constraints imposed by the towers, piers, and cables. Depending on the position, the main beam will be deflected downward or arched upward. At different elevations, the towers undergo varying degrees of lateral displacement. The hangers are no longer erect and will be elongated or inclined. The span length and elevation difference between the stay cables will also vary. The main cable will be deflected downward and undergo changes in internal force. Different components of the full-bridge are coupled, resulting in complex deformation and internal force variation patterns. An analytical algorithm proposed by this paper comprehensively considers various factors, including the lateral displacement of towers, hanger extension, and rigid body displacement of the main beam. The algorithm method is better adapted to actual bridge situations and more universally applicable.

Computational model and general principles

Computational model

Figure 1 shows the computation model of the HCSS bridge in the initial state. The full-bridge has three spans, the length of which, from left to right, are L₁′, L₂′, and L₃′. The main cable is anchored at points A′ and D′, intersecting with the towers at points B′ and C′, respectively. The main cables in the left- and right-side spans are single catenaries. There are n hangers on the main cable of the main span, dividing the main cable into n+1 segments. There are regions where both hangers and stay cables are used, and the apportioned ratio between the hanger force and the vertical force in the stay cable is manually specified. 2n₁ stay cables are anchored altogether to the left tower, and 2n₂ stay cables are anchored to the right tower. The number of stay cables is symmetrically arranged on both sides of the tower. A single stay cable has also been deemed a catenary. The vertical displacement of the main beam is constrained at the piers, auxiliary piers, and towers. The full-bridge system is a floating system in the bridge’s longitudinal direction. Due to the action of stay cables, the main beam is subjected to axial compression. The towers are kept vertical under the dead load; the tower tops have no lateral displacement; the bottom of the bridge towers is constrained in all degrees of freedom. The elevations of the left and right tower tops are H_B′ and H_C′, respectively. The elevations of the main beam at the supports are H_E′ and H_F′, respectively.

Figure 1.

Computational model of the hybrid cable-stayed suspension bridge.

Basic assumptions

The analytical approach proposed in this paper is based on the following assumptions:

(1) The stress-strain relationship of the material always obeys Hooke’s law.

(2) The main cables are ideal cables, which are only subjected to tension but not to the bending moment or pressure. In addition, the decrease in the cross-sectional area of the main cables caused by elongation is neglected.

(3) The bridge towers undergo no axial deformation under live loads. That is, the elevation of each point in the bridge tower remains constant.

(4) The main beam undergoes no axial deformation under the live load.

(5) The hangers have no dead weight.

General principles

After the main beam is subjected to the vertical live load P(x), the full-bridge response is as follows: The main beam will undergo a certain degree of downward deflection in regions subjected to loading. As a result, the internal forces in the nearby hangers, stay cables, and main cable increase, affecting the adjacent side spans via the towers. The main beam may also undergo rigid body displacement in the longitudinal direction of the bridge. In that case, the stay cables, main cables, and hangers will undergo vertical and longitudinal displacements. The internal forces and deformations of bridge components are coupled, which is a rather complex phenomenon to characterize.

To accurately solve the full-bridge response under the vertical live load, in this paper, the main cable, stay cable, bridge tower, hanger and main beam are calculated analytically separately, and then combined together through the internal force relationship. Specifically, it is assumed that the bridge’s geometric configuration and internal forces are known quantities under the dead load. Under the live load p(x), the basic unknown quantities are the critical geometric parameters and internal forces representing the full-bridge response. Then, the corresponding governing equations are written based on geometric compatibility, conservation of unstressed lengths, and mechanical equilibrium. Finally, the system of equations is solved to obtain the full-bridge response. The analytical algorithm only deals with two states of the structure: with and without the live load p(x). No intermediate process is involved. Thus, there is no need to perform geometrically nonlinear analysis, which can be very cumbersome. Therefore, the proposed analytical approach has the advantages of clear conceptualization and definite physical meanings.

Full-bridge response theory

Consider the main beam of the HCSS bridge subjected to a vertical uniformly distributed load. Besides the main beam, deformations and internal forces generated in the main cables, stay cables, and hangers as responses to the vertical loading also need to be analyzed.

Response of the main beam

The deformation of the main cables in the main span under the live load is shown in Figure 2. Both the left and right towers undergo lateral displacement, moving from points B′ and C′ in the initial state to points B and C, respectively. The lateral displacements are Δ_B and Δ_C, respectively. The lengths of the left- and right-side spans and the main span are L₁, L₃, and L₂, respectively. Correspondingly, the hanging points on the main cable move from O′₁, O′₂..., O′_n to O₁, O₂..., O_n. l_i is the horizontal projected length of the ith catenary for the main cable in the main span. The horizontal projected length of each catenary for the main cable varies, and the upper hanging points undergo varying degrees of downward deflection. The direction of hanger forces also deflects from the initial vertical direction.

Figure 2.

Main cables in the main spans under the joint action of dead and live loads.

In order to formulate the catenary equations, the overall coordinate system is established with the torsion center of the cross-section at the left end of the main beam in the initial state as the origin, with the X-axis extending along the bridge and the Z-axis pointing vertically upward. A local coordinate system is built at the left endpoint of each catenary for the deformed main cable. The X-axis extends along the bridge, and the Z-axis points vertically downward. The local coordinate system is set for the convenience of solving the catenary of each section, and the final line shapes of each component of the whole bridge is expressed by the coordinates under the overall coordinate system.

In the local coordinate system, the equation of the ith catenary in the main span was derived as follows (Zhang et al., 2019):

z (x) = c_{i} [\cosh (\frac{x}{c_{i}} + a_{i}) - \cosh a_{i}]

(1)

where c_i=-H_i/q; H_i is the horizontal component of force in the ith catenary for the main cable; q is the self-weight per linear meter of the main cable; a_i is the parameter of the equation of the ith catenary in the main span.

The elevation difference between the two endpoints of the ith catenary is given by:

Δ z_{i} = z (l_{i}) - z (0) = c_{i} [\cosh (\frac{l_{i}}{c_{i}} + a_{i}) - \cosh a_{i}]

(2)

where l_i is the horizontal projected length of the ith catenary; z(0) represents the elevation of the first end of the catenary in the local coordinate system.

The unstressed length of the ith main cable is given by (Zhang et al., 2019):

S_{i} = c_{i} [\sinh (\frac{l_{i}}{c_{i}} + a_{i}) - \sinh a_{i}] - \frac{H_{i}}{2 E A} {l_{i} + \frac{c_{i}}{2} [\sinh 2 (\frac{l_{i}}{c_{i}} + a_{i}) - \sinh 2 a_{i}]}

(3)

where E and A are the elastic modulus and the cross-sectional area of the main cable, respectively.

Parameter a_i and H_i of the equation of the ith catenary and parameter a_i+1 and H_i+1 of the equation of the i+1^-th catenary have the following recurrence relation (Zhang et al., 2018):

a_{i + 1} = asinh \frac{H_{i} \sinh (\frac{l_{i}}{c_{i}} + a_{i}) - P_{i} \cos θ_{i}}{(H_{i} - P_{i} \sin θ_{i})}

(4a)

H_{i + 1} = H_{i} - P_{i} \sin θ_{i}

(4b)

where P_i is the hanger force of the ith hanger under the live load, and θ_i is the included angle between the hanger force P_i and the z-axis, known as the vertical inclination angle, as shown in Figure 3.

Figure 3.

Schematic diagram of hanger force decomposition.

The main cables in the left- and right-side spans are only subjected to their gravitational forces. Each is a single catenary, as shown in Figure 4.

Figure 4.

Main cables in the side spans under the joint action of dead and live loads.

The catenary equations for the main cables in the left- and right-side spans have the following form:

z_{L} (x) = c_{L} [\cosh (\frac{x}{c_{L}} + a_{L}) - \cosh a_{L}]

(5a)

z_{R} (x) = c_{R} [\cosh (\frac{x}{c_{R}} + a_{R}) - \cosh a_{R}]

(5b)

where c_L=−H_L/q and c_R=−H_R/q. H_L and H_R are the horizontal components of force in the main cables in the left and right side spans under the joint action of dead and live loads, respectively; a_L and a_R are parameters of the catenary equations in the left and right side spans, respectively.

The elevation difference between the two endpoints of the catenary in the left- and right-side spans is given below:

Δ z_{L} = z (L_{1}) - z (0) = c_{L} [\cosh (\frac{L_{1}}{c_{L}} + a_{L}) - \cosh a_{L}]

(6a)

Δ z_{R} = z (L_{3}) - z (0) = c_{R} [\cosh (\frac{L_{3}}{c_{R}} + a_{R}) - \cosh a_{R}]

(6b)

where L₁ and L₃ are the horizontal projected lengths of the catenary in the left- and right-side spans under the joint action of dead and live loads, respectively.

The unstressed lengths of the main cable in the left- and right-side spans, respectively, are given as follows:

S_{L} = c_{L} [\sinh (\frac{L_{1}}{c_{L}} + a_{L}) - \sinh a_{L}] - \frac{H_{L}}{2 E A} {L_{1} + \frac{c_{L}}{2} [\sinh 2 (\frac{L_{1}}{c_{L}} + a_{L}) - \sinh 2 a_{L}]}

(7a)

S_{R} = c_{R} [\sinh (\frac{L_{3}}{c_{R}} + a_{R}) - \sinh a_{R}] - \frac{H_{R}}{2 E A} {L_{3} + \frac{c_{R}}{2} [\sinh 2 (\frac{L_{3}}{c_{R}} + a_{R}) - \sinh 2 a_{R}]}

(7b)

As shown above, the internal forces and line shapes of the main cables can be solved in the entire bridge if the following parameters are known: catenary parameters in the side spans, including H_L, a_L, L₁, H_R, a_R, and L₁; hanger force p_i and vertical inclination angle θ_i of each hanger in the main span; parameter a₁ of the equation of the first catenary for the main cable in the main span; and the horizontal component of force _H1 in the main cable in the main span.

Response of stay cables

Under the live load, a single stay cable is also a catenary due to its self-weight. Since both the left and right towers are inclined, the anchor points of the stay cables on the main beam also have longitudinal and vertical deflections. Therefore, the span length and elevation difference for each stay cable have changed, as shown in Figure 5. As above, a local coordinate system at the left endpoint of each stay cable as the origin is built.

Figure 5.

Stay cables under the joint action of dead and live loads.

In the local coordinate system, the catenary equation of the ith stay cable has the following form (Zhang et al., 2019):

{z_{c,}}_{i} (x) = {c_{c,}}_{i} [\cosh (\frac{x}{{c_{c,}}_{i}} + {a_{c,}}_{i}) - \cosh {a_{c,}}_{i}]

(8)

where c_c,i=−H_c,i/q_c, and H_c,i is the horizontal component of force in the ith stay cable under the joint action of live and dead loads; q_c is the self-weight per linear meter of the stay cable, kN/m; and a_c,i is the parameter of the catenary equation for this stay cable.

The elevation difference between the two endpoints of the catenary for the ith stay cable is given by:

Δ z_{c, i} = z (l_{c, i}) - z (0) = c_{c, i} [\cosh (\frac{l_{c, i}}{c_{c, i}} + a_{c, i}) - \cosh a_{c, i}]

(9)

where l_c,i is the horizontal projected length of the ith stay cable.

The unstressed length of the catenary for the ith stay cable is given by:

S_{c, i} = c_{c, i} [\sinh (\frac{l_{c, i}}{c_{c, i}} + a_{c, i}) - \sinh a_{c, i}] - \frac{H_{c, i}}{2 E_{c} A_{c}} {l_{c, i} + \frac{c_{c, i}}{2} [\sinh 2 (\frac{l_{c, i}}{c_{c, i}} + a_{1, i}) - \sinh 2 a_{c, i}]}

(10)

where E_c and A_c are the elastic modulus and the cross-sectional area of the stay cable, respectively.

The vertical force imposed by the ith stay cable on the main beam is given below:

F_{z, i} = H_{c, i} \sinh a_{c, i}

(11a)

F_{z, i} = H_{c, i} \sinh (\frac{l_{c, i}}{c_{c, i}} + a_{c, i})

(11b)

Formulas (11a) and Formulas (11b) respectively apply to situations where the horizontal component of force in the stay cable attached to the main beam has the same and the opposite direction as the X-axis in the overall coordinate system.

As shown above, the internal forces and line shapes of stay cables can be estimated if the horizontal component of force H_c,i in each stay cable, parameter a_c,i of the catenary equation, and horizontal projected length L_c,i of the stay cable are known.

Response of the towers

Since the towers have a considerable axial stiffness, the axial deformation of the towers is neglected. That is, the elevation of each tower point remains constant, and only displacement in the longitudinal direction of the bridge under the live load is considered. Below is an illustration of the lateral displacement calculation for the left tower. The left tower is subjected to the action of the horizontal components of forces in n₁ pairs of stay cables and one pair of main cables. This can be simplified as a situation where a cantilever beam is subjected to the action of n₁+1 horizontal components of forces. For convenience, from top to bottom, these n₁+1 horizontal components of force are numbered ΔH₁, ΔH_2…ΔH_n1+1 successively, as shown in Figure 6.

Figure 6.

Left bridge tower under the joint action of dead and live loads.

The horizontal component of force at the ith node is given by:

Δ H_{i} = H_{R, i} - H_{L, i}

(12)

where H_L,i and H_R,i are the horizontal components of the force imposed by the main cable or stay cable on the left and right sides of the ith node to the tower under the joint action of the live and dead loads, respectively.

The tower is subdivided into n₁+1 segments, and the bending moment in the mth segment of the tower (from the mth to the m+1^-th component of force) is given below (the direction of tension on one side of the side span is positive):

M_{m} (x) = \sum_{i = 1}^{m} Δ H_{i} (x - x_{i}), x_{m} \leq x < x_{m + 1}

(13)

where x_i is the distance between the tower top and the site where the ith horizontal component of force acts.

The relationship between the bending moment and rotation angle is as follows:

θ_{m} (x) = - \frac{\int M_{m} (x)}{E_{t} I_{t}}

(14)

where E_t is the elastic modulus of the tower, and I_t is the longitudinal anti-bending moment of inertia in the tower.

The relationship between the bending moment and deflection is as follows:

w_{m} (x) = - \frac{\int \int M_{m} (x)}{E_{t} I_{t}}

(15)

By substituting formula (13) into (14) and (15), the expressions of the rotation angle and deflection can be obtained:

θ_{m} (x) = - \frac{1}{E_{t} I_{t}} [\sum_{i = 1}^{m} (\frac{1}{2} x^{2} - x_{i} x) Δ H_{i} + C_{m, 1}], x_{m} \leq x < x_{m + 1}

(16)

w_{m} (x) = - \frac{1}{E_{t} I_{t}} [\sum_{i = 1}^{m} (\frac{1}{6} x^{3} - \frac{1}{2} x_{i} x^{2}) Δ H_{i} + C_{m, 1} x + C_{m, 2}], x_{m} \leq x < x_{m + 1}

(17)

where C_m,1 and C_m,2 are the constant terms after integration of the bending moment of the tower.

Hence, at the position subjected to the m+1^-th horizontal component of force, the rotation angles of the tower sections on the upper and lower sides can be expressed as:

θ_{m} (x_{m + 1}^{-}) = - \frac{1}{E_{t} I_{t}} [\sum_{i = 1}^{m} (\frac{1}{2} x_{m + 1}^{2} - x_{i} x_{m + 1}) Δ H_{i} + C_{m, 1}]

(18a)

θ_{m} (x_{m + 1}^{+}) = - \frac{1}{E_{t} I_{t}} [\sum_{i = 1}^{m + 1} (\frac{1}{2} x_{m + 1}^{2} - x_{i} x_{m + 1}) Δ H_{i} + C_{m + 1,1}]

(18b)

The deflection of the upper and lower sections at the m+1^-th horizontal force can be expressed as:

w_{m} (x_{m}^{-}) = - \frac{1}{E_{t} I_{t}} [\sum_{i = 1}^{m} (\frac{1}{6} x_{m + 1}^{3} - \frac{1}{2} x_{i} x_{m + 1}^{2}) Δ H_{i} + C_{m, 1} x_{m + 1} + C_{m, 2}]

(19a)

w_{m} (x_{m + 1}^{+}) = - \frac{1}{E_{t} I_{t}} [\sum_{i = 1}^{m + 1} (\frac{1}{6} x_{m + 1}^{3} - \frac{1}{2} x_{i} x_{m + 1}^{2}) Δ H_{i} + C_{m + 1,1} x_{m + 1} + C_{m + 1,2}]

(19b)

Since both the deflection and rotational angle of the cross-sectional area of the tower are continuous, there are:

w (x_{m}^{-}) = w (x_{m}^{+})

(20a)

θ (x_{m}^{-}) = θ (x_{m}^{+})

(20b)

Introducing formula (20) into (18) and (19) yields:

C_{m + 1,1} = C_{m, 1} + \frac{1}{2} x_{m + 1}^{2} Δ H_{m + 1}

(21a)

C_{m + 1,2} = C_{m, 2} - \frac{1}{6} x_{m + 1}^{3} Δ H_{m + 1}

(21b)

Since the bottom of the bridge tower is consolidated, the deflection and rotation angle of the tower bottom are both zero.

w (H_{B}) = 0

(22a)

θ (H_{B}) = 0

(22b)

Introducing formula (22) into the above formula gives:

C_{n_{1} + 1,1} = - \sum_{i = 1}^{n_{1} + 1} (\frac{1}{2} H_{B}^{2} - H_{B} x_{i}) Δ H_{i}

(23a)

C_{n_{1} + 1,2} = - \sum_{i = 1}^{n_{1} + 1} (\frac{1}{6} H_{B}^{3} - \frac{1}{2} H_{B}^{2} x_{i}) Δ H_{i} - C_{n_{1} + 1,1} H_{B}

(23b)

The constants of integration for each segment are calculated via formulas (21) and (23). Then, the deflection expression is derived for each segment. In other words, if the horizontal component of force ΔH_i imposed by the main cable and the stay cable to the tower is determined, we can estimate the lateral displacement at any tower position. ΔH_i can be expressed in terms of H_L, H₁, and H_c,i mentioned above.

Response of the main beam

Under the vertical live load, the main beam will be deflected downward or arched upward, depending on the position. Apart from the uniformly distributed live load p(x) in the vertical direction, the main beam is also subjected to vertical force increment ΔF_i in the stay cables and hangers. In addition, the vertical displacements of the main beam are constrained at the sites of the side piers, auxiliary piers, and towers. The theoretical computation model is shown in Figure 7. The vertical uniformly distributed live load is p(x), and its range of action of the force is l(x). E and F are the locations of the left and right pylons respectively.

Figure 7.

Schematic diagram of the computational model of the main beam under the joint action of dead and live loads.

The vertical deflection of the main beam is solved using the displacement method. There are four unknowns: rotational angles at points H, E, F, and I, numbered 1, 2, 3, and 4, respectively. The canonical equations involved in the displacement method are given below:

\begin{array}{l} r_{11} Z_{1} + r_{12} Z_{2} + r_{13} Z_{3} + r_{14} Z_{4} + R_{1 P} = 0 \\ r_{21} Z_{1} + r_{22} Z_{2} + r_{23} Z_{3} + r_{24} Z_{4} + R_{2 P} = 0 \\ r_{31} Z_{1} + r_{32} Z_{2} + r_{33} Z_{3} + r_{34} Z_{4} + R_{3 P} = 0 \\ r_{41} Z_{1} + r_{42} Z_{2} + r_{43} Z_{3} + r_{44} Z_{4} + R_{4 P} = 0 \end{array}

(24)

where Z₁, Z₂, Z₃, and Z₄ are the rotational angles at points H, E, F, and I in the man beam, respectively; r_ij is the reaction force exerted on position i upon the unit displacement of the position with the jth rotational angle; and R_ip is the reaction force exerted by the external load on the position with the ith rotational angle, which is related to the position and size of loading.

The r_ij values are given below (those not included are zero):

\begin{array}{l} r_{11} = 3 i_{GH} + 4 i_{HE} \\ r_{12} = r_{21} = 2 i_{HE} \\ r_{22} = 4 i_{HE} + 4 i_{EF} \\ r_{32} = r_{23} = 2 i_{EF} \\ r_{33} = 4 i_{EF} + 4 i_{FI} \\ r_{34} = r_{34} = 2 i_{FI} \\ r_{44} = 3 i_{IJ} + 4 i_{FI} \end{array}

(25)

where i_ij=E_bI_b/L_ij. E_b and I_b are the elastic modulus and the vertical moment of inertia of the main beam, respectively; L_ij is the length of this beam segment.

After the coefficients of the equations are obtained, the equations are solved in the form of determinants so that:

D = | \begin{array}{l} r_{11} & r_{12} & r_{13} & r_{14} \\ r_{21} & r_{22} & r_{23} & r_{24} \\ r_{31} & r_{32} & r_{33} & r_{34} \\ r_{41} & r_{42} & r_{43} & r_{44} \end{array} |

(26)

D_{1} = | \begin{array}{l} R_{1 P} & r_{12} & r_{13} & r_{14} \\ R_{2 P} & r_{22} & r_{23} & r_{24} \\ R_{3 P} & r_{32} & r_{33} & r_{34} \\ R_{4 P} & r_{42} & r_{43} & r_{44} \end{array} |

(27)

Then,

Z_{i} = \frac{D_{i}}{D}

(28)

Thus far, we have solved the rotational angles Z₁, Z₂, Z₃, and Z₄ in the canonical equation. Below, we will solve the bending moment of the beam segment, also in the case of beam segment GH.

M_{GH} = 3 i_{A B} Z_{1} + M_{GH, P}

(29)

where M_GH,P is the bending moment at the end of beam segment GH under the live load.

For the remaining nodes, the bending moment can be solved similarly. On this basis, we can solve each reaction force from the bending moment.

Knowing the bending moment of the main beam and the reaction force at each position of the main beam, one can solve the vertical deflection at any position of the main beam. Below, we solve the vertical deflection for segment GH, as above. For convenience, the concentrated forces (including the reaction forces) are, from left to right, designated ΔF₁, ΔF₂…ΔF_n, successively.

The vertical bending moment for the mth beam segment is given by

M_{m} (x) = \sum_{i = 1}^{m} Δ F_{i} (x - x_{i}), x_{m} \leq x < x_{m + 1}

(30)

Using formulas (16) and (17) and introducing the expression of the bending moment into the above formula, the expressions of the rotational angle and deflection can be obtained:

θ_{m} (x) = - \frac{1}{E_{b} I_{b}} [\sum_{i = 1}^{m} Δ F_{i} (\frac{1}{2} x^{2} - x_{i} x) + C_{m, 1}], x_{m} \leq x < x_{m + 1}

(31)

w_{m} (x) = - \frac{1}{E_{b} I_{b}} [\sum_{i = 1}^{m} Δ F_{i} (\frac{1}{6} x^{3} - \frac{1}{2} x_{i} x^{2}) + C_{m, 1} x + C_{m, 2}], x_{m} \leq x < x_{m + 1}

(32)

where C_m,1 and C_m,2 are the constant terms after integration of the bending moment of the main beam.

The rotational angles of the cross-sectional area on the left and right sides of the mth concentrated force are, respectively given by

θ_{z} (x_{m}^{-}) = - \frac{1}{E_{b} I_{b}} [\sum_{i = 1}^{m - 1} Δ F_{i} (\frac{x_{m}^{2}}{2} - x_{i} x_{m}) + {C_{m,}}_{1}]

(33a)

θ_{z} (x_{m}^{+}) = - \frac{1}{E_{b} I_{b}} [\sum_{i = 1}^{m} Δ F_{i} (\frac{x_{m}^{2}}{2} - x_{i} x_{m}) + {C_{m + 1,}}_{1}]

(33b)

The deflections of the cross-sectional area on the left and right sides of the mth concentrated force are given by:

w_{z} (x_{m}^{-}) = - \frac{1}{E_{b} I_{b}} [\sum_{i = 1}^{m - 1} Δ F_{i} (\frac{x_{m}^{3}}{6} - \frac{x_{i} {x_{m}}^{2}}{2}) + {C_{m,}}_{1} \cdot x_{m} + C_{m, 2}]

(34a)

w_{z} (x_{m}^{+}) = - \frac{1}{E_{b} I_{b}} [\sum_{i = 1}^{m} Δ F_{i} (\frac{x_{m}^{3}}{6} - \frac{x_{i} x_{m}^{2}}{2}) + {C_{m + 1,}}_{1} \cdot x_{m} + C_{m + 1,2}]

(34b)

Since the rotational angles and deflections of the cross-sectional area on the left and right sides of the mth concentrated force should be kept continuous, there are:

w (x_{m}^{-}) = w (x_{m}^{+})

(35a)

θ (x_{m}^{-}) = θ (x_{m}^{+})

(35b)

By introducing formula (35) into (33) and (34), we have:

C_{m + 1,1} = C_{m, 1} + \frac{1}{2} x_{m + 1}^{2} Δ F_{m + 1}

(36a)

C_{m + 1,2} = C_{m, 2} - \frac{1}{6} x_{m + 1}^{3} Δ F_{m + 1}

(36b)

The deflection at the two ends of the main beam is zero, that is,

w (0) = w (L_{GH}) = 0

(37)

Substituting formula (37) into the above formulas, we obtain

C_{1,2} = 0

(38a)

C_{n - 1,1} = - \frac{1}{L_{GH}} [c_{n - 1,2} + \sum_{i = 1}^{n - 1} (\frac{1}{6} L_{GH}^{3} - \frac{1}{2} x_{i} L_{GH}^{2}) Δ F_{i}]

(38b)

Thus far, we have derived the deflection at any position of the beam segment GH. The same method can be applied to the remaining beam segments to estimate the deflection. The only difference is that the shear force and bending moment cumulating in the segment on the left of the current segment should be counted. For example, for segment HE, the shear force and bending moment accumulating in segment GH should be considered. Thus, the deflection of all beam segments can be derived as long as the value of ΔF_i is known.

Response of the hangers

Hangers transmit the load from the main beam to the main cable. Since the hanger has a limited length and a small self-weight, the sag effect of the hanger itself is neglected. The direction of the hanger force is aligned with the line connecting the upper and lower hanging points. In addition, the forces exerted by the hanger on the main cable and the main beam are of equal magnitude. The coordinates of the hangers in the global coordinate system are written as follows:

X_{i} = Δ_{Β} + \sum_{j = 1}^{i} l_{j}

(39a)

X_{b, i} = x_{i} + v

(39b)

Z_{i} = H_{B} - \sum_{j = 1}^{i} Δ z_{j}

(40a)

Z_{b, i} = w_{i}

(40b)

where X_i and Z_i are the X- and Z-coordinates of the ith upper hanging point under the live load in the global coordinate system, respectively; X_b,i and Z_b,i are the X- and Z-coordinates of the ith lower hanging point under the live load in the global coordinate system, respectively; v is the rigid body displacement of the main beam in the longitudinal direction; x_i is the X-coordinate of the lower hanging point in the initial state in the global coordinate system; and w_i is the deflection of the main beam at the lower hanging point under the live load in the global coordinate system.

The hangers are no longer erect under the live load but are inclined, resulting in a vertical inclination angle θ_i. In that case, the axial force in the hanger is P_i, and the hanger will exert the component of force on the main cable and the main beam in the longitudinal direction. Here, θ_i can be expressed by the coordinates of the upper and lower hanger points in the global coordinate system:

θ_{i} = \arctan \frac{X_{b, i} - X_{i}}{Z_{i} - Z_{b, i}}

(41)

Constructing and solving the system of governing equations

Determining the basic unknown quantities

According to the full-bridge response calculation theory above, we should determine at least several key parameters to estimate the deflection and internal force of the entire bridge. Now, these parameters are defined as basic unknowns, which are then assessed by solving a system of equations. In the analytical algorithm proposed in this paper, one needs to solve 2n+6n₁+6n₂+10 unknown quantities, as shown in Table 1.

Table 1.

Summary of unknown quantities.

Unknown parameters	Left-side span	Main span	Right-side span	Number
Parameters of the catenary equation of the main cable	a _L	a ₁	a _R	3
Horizontal projected length of the catenary of the main cable	L ₁	l ₁ -l _n+1	L ₃	n+3
Horizontal component of the main cable force	H _L	H ₁	H _R	3
Hanger force increment	/	ΔP₁-ΔP_n	/	n
Horizontal projected length of the stay cable		l _c,i		2n₁+2n₂
Catenary parameters of the stay cable		a _c,i		2n₁+2n₂
Horizontal component of force in the stay cable		H _c,i		2n₁+2n₂
Horizontal rigid body displacement of the main beam in the longitudinal direction		ν		1
Total				2n+6n₁+6n₂+10

Constructing the system of governing equations

With the basic unknown quantities determined, governing equations need to be established for these unknown quantities, and the number of governing equations is equal to that of the unknown parameters. These governing equations are roughly subdivided into three types: (i) equations representing the conservation of unstressed lengths of the cables, (ii) equations describing the closure of span lengths and elevation differences in different spans, and (iii) equations representing the force balance in the main beam.

Closure of span lengths and elevation differences in different spans

According to the geometric relationship of each span with and without loading, the following equations can be constructed:

L_{1} - Δ_{B} = L_{1}^{'}

(42)

Δ_{B} + Δ_{C} + \sum_{i = 1}^{n + 1} l_{i} = L_{2}^{'}

(43)

L_{3} - Δ_{C} = L_{3}^{'}

(44)

The elevation differences between the two endpoints of the main cable in each span remain constant. That is,

Δ z_{L} = Z_{B} - Z_{A}

(45)

\sum_{i = 1}^{n + 1} Δ z_{i} = Z_{B} - Z_{C}

(46)

Δ z_{R} = Z_{C} - Z_{D}

(47)

Six governing equations can be formulated.

Conservation of unstressed lengths of the main cables

The unstressed length of each catenary of the main cable remains unchanged with and without loads acting on the main beam, that is,

S_{L} = S_{L}^{'}

(48)

S_{i} = S_{i}^{'}, 1 \leq i \leq n + 1

(49)

S_{R} = S_{R}^{'}

(50)

where

S_{L}^{'}

and

S_{R}^{'}

are the unstressed lengths of the main cable in the left and right side spans in the initial state, respectively;

S_{i}^{'}

is the unstressed length of the ith catenary of the main cable in the main span in the initial state.

Here, the n+3 governing equations can be listed.

Closure of span lengths and elevation differences between stay cables

Based on the closure of span lengths for stay cables with and without the live load, we have

l_{c, i} = | X_{c, t, i} - X_{c, b, i} |

(51)

where X_c,t,i and X_c,b,i are the X-coordinates of the anchor points of the ith catenary in the tower and the main beam, respectively.

Based on the closure of elevation differences for stay cables with and without the live load, we have

Δ z_{c, i} = Z_{c, t, i} - Z_{c, b, i}

(52)

where Z_c,t,i and Z_c,b,i are the Z-coordinates of the anchor points of this catenary in the tower and the main beam, respectively.

Then, we can write 4n₁+4n₂ governing equations.

Conservation of unstressed lengths of stay cables

Given that the unstressed lengths of stay cables remain constant with and without the live load, we have

S_{c, i} = S_{c, i}^{'}

(53)

where

S_{c, i}^{'}

is the unstressed length of the stay cable in the initial state.

Here, we can write 2n₁+2n₂ governing equations.

Stress balance of the main beam

There is a force balance in the main beam subjected to the live load acting along the x-axis:

\sum_{i = 1}^{n} (H_{c, i} + Δ P_{i} \sin θ) = 0

(54)

One governing equation can be listed here.

Conservation of unstressed lengths of hangers

The unstressed length of the hanger under the live load is represented as follows in the global coordinate system:

S_{h, i} = \frac{\sqrt{{(X_{i} - X_{b, i})}^{2} + {(Z_{i} - Z_{b, i})}^{2}}}{1 + \frac{P_{i}}{E_{h} A_{h}}}

(55)

The unstressed lengths of each hanger with and without the live load remain constant. That is,

S_{h, i} = S_{h, i}^{'}

(56)

where

S_{h, i}^{'}

is the unstressed length of the hanger in the initial state, while E_h and A_h are the hanger’s elastic modulus and cross-sectional area, respectively.

Here, we can write n governing equations.

Solving the equations

Thus far, we have written 2n+6n₁+6n₂+10 independent governing equations to solve 2n+6n₁+6n₂+10 basic unknown quantities. First, the terms on the right side of the equation are moved to the left side. Next, both sides of the equation are squared and written in the cumulative form:

\sum_{i = 1}^{2 n + 6 n_{1} + 6 n_{2} + 10} f_{i}^{2} = 0

(57)

Finally, the basic unknown quantities are solved by programming (e.g., via the nonlinear generalized reduced gradient algorithm).

A calculation example

An overview of the bridge under study

To verify the accuracy of the proposed analytical algorithm, we apply this method to solve the live load response of a real HCSS bridge. Then, the internal forces and line shapes obtained are compared with the FEM results. The bridge has a main span of 1400 m, with the left- and right-side spans in an asymmetric arrangement. The vertical layout of the full-bridge is shown in Figure 8. The known parameters under the dead load are listed in Table 2. The main span is subjected to a vertical uniformly distributed load p(x)=25 kN/m, which is from the standard named as “General Specifications for Design of Highway Bridges and Culverts (JTG D60-2015)” of China.

Figure 8.

Vertical view of the hybrid cable-stayed suspension bridge in the calculation example.

Table 2.

Known parameters of the full-bridge.

Parameter		Symbol	Unit	Value
Span length	Left side span	L₁′	m	452
	Main span	L₂′	m	1400
	Right side span	L₃′	m	464
Main cable	Elastic modulus	E	GPa	205
	Cross-sectional area	A	m²	0.24
	Weight per unit length	q	kN/m	18.85
	Elevation of the left anchor point	Z _A	m	−33.000
	Elevation of the right anchor point	Z _D	m	−33.000
	Elevation of the midpoint of the main span	Z _M	m	11.000
Stay cable	Elastic modulus	E _c	GPa	206
	Cross-sectional area	A _c	m²	0.008
	Weight per unit length	q _c	kN/m	0.628
	Elastic modulus	E _h	GPa	206
	Cross-sectional area	A _h	m²	0.0065
Hanger	Number	n	/	33
	Spacing	l₂∼l₃₃	m	16
	The horizontal distance between the left tower’s centerline and the first hanger	l ₁	m	444
	The horizontal distance between the last hanger and the right tower’s centerline	l ₃₄	m	444
Tower	Elastic modulus	E _t	GPa	36
	Cross-sectional area	A _t	m²	50
	Density	ρ _t	kg/m³	8005
	Moment of inertia	I _t	m⁴	1000
	Top elevation of the left tower	Z _B	m	236
	Top elevation of the right tower	Z _C	m	240
Stiffening girder	Elastic modulus	E _b	GPa	206
	Cross-sectional area	A _b	m²	1.8
	Weight per unit length	q _b	kN/m	138.559
	Moment of inertia	I _b	m⁴	5.824

Under the dead load (in the initial state), the vertical force ratio in the stay cable-to-hanger force is 1:1, where both stay cables and hangers are used. The initial hanger forces are shown in Figure 9, and the unstressed lengths of the hangers are demonstrated in Figure 10. The coordinates of the upper and lower hanging points in the hangers are shown in Figure 11.

Figure 9.

Initial hanger force.

Figure 10.

Unstressed length of the hangers.

Figure 11.

Initial coordinates of the hangers.

The two ends of the stay cable are anchored to the tower and the main beam. The unstressed lengths of the stay cables are shown in Figure 12. The horizontal components of the force in the stay cables are depicted in Figure 13. The stay cables are sequentially numbered based on the positions of their anchor points in the main beam from left to right.

Figure 12.

Unstressed lengths of the stay cables.

Figure 13.

Horizontal components of forces in the stay cables in the initial state.

The unstressed lengths of the main cable in the left- and right-side spans are estimated to be 587.78 and 600.31 m, respectively. The unstressed length of the main cable between the leftmost hanger and the left tower is 462.21 m. The unstressed length of the main cable between the rightmost hanger and the right tower is 479.23 m. The unstressed lengths of the main cable between the hangers are shown in Figure 14.

Figure 14.

Unstressed lengths of the main cables.

Solving the full-bridge response

The proposed analytical algorithm is then applied to assess the full-bridge response. We need to construct 400 independent governing equations and derive 400 unknown parameters. To verify the accuracy of our method, we build a finite element model in the ANSYS software (17.0), as shown in Figure 15. A vertical uniformly distributed load p(x)=25 kN/m is imposed, and the full-bridge response is estimated. Below is a comparison of results via the proposed analytical approach and the FEM.

Figure 15.

Finite element model.

The vertical deflections of the main beam estimated by the analytical algorithm and FEM are shown in Figure 16. The maximum vertical deflection is 1.267 m, occurring on the right side of the midspan of the main span. The maximum absolute error between the analytical algorithm and FEM is 0.05, and the maximum relative error is 41%. They are observed between the tower in the right side span and the auxiliary pier.

Figure 16.

Comparison of the vertical defection of the main beam estimated by the analytical algorithm and FEM.

The hanger forces, P_i, estimated by the FEM and the analytical approach are depicted in Figure 17. The maximum absolute error between the hanger forces estimated by the two methods is 1.67 kN, and the maximum relative error is 0.13%. The hanger force distribution where only hangers are installed still uniform compared with the initial state. In Figure 17, ΔP_i is the difference between the hanger force obtained by the analytical method after loading and the initial value. Under the live load, the force in the outermost hanger is significantly increased where the hangers and stay cables are installed. The variation of the hanger forces is still relatively uniform in the deck range where hangers are installed alone.

Figure 17.

Hanger forces estimated by the analytical algorithm and FEM.

Figure 18 shows the deflections of the upper hanging point of the hanger calculated by the analytical algorithm and FEM. The maximum absolute and relative errors between the two methods (0.03 m and 2.6%, respectively) occur near the midspan.

Figure 18.

Comparison of deflection of the upper hanging point of the hanger estimated by the analytical algorithm and FEM.

The horizontal components of the forces in the stay cables, H_c,i, estimated by the analytical algorithm and FEM are shown in Figure 19. The maximum absolute error of the horizontal component of force in the stay cable attached to the left tower is 8.84 kN, and the maximum relative error is 0.28%. The maximum absolute error of the horizontal force component in the stay cable attached to the right tower is 10.69 kN, and the maximum relative error is 0.41%. In Figure 19, ΔH_c,i is the difference between the horizontal component of the force obtained by analytical method after loading and the initial value in the stay cable. For the increment of the horizontal component of force in the stay cable, stay cables with larger internal forces exhibit larger variations of internal forces.

Figure 19.

Horizontal component of force in the stay cable.

The comparison of lateral displacements of the left and right towers toward the midspan estimated by the two methods is presented in Figure 20. The maximum absolute error of deflection of the left tower is 0.31 cm, and the maximum relative error is 1.45%. The maximum absolute error of deflection of the right tower is 0.46 cm, and the maximum relative error is 2.89%. The estimation of the inclination angle θ_i of the hanger is shown in Figure 21.

Figure 20.

Lateral displacement of the towers toward the midspan.

Figure 21.

Inclination angle variations in hangers.

Conclusions

This paper proposed an analytical algorithm for the full-bridge response of the HCSS bridge under a vertical uniformly distributed live load. We apply the proposed method to a HCSS bridge with a span layout of 452 m + 1400 m + 464 m to verify the effectiveness and accuracy of the proposed method. The FEM and analytical method are respectively used to calculate the full bridge response. By comparing the results of the two methods, the following conclusions can be drawn:

(1) The proposed method is based on the assumption that all parameters in the initial state are known. The basic unknown parameters representing the full-bridge response are first determined under the live load. Next, the governing equations are constructed based on geometric compatibility, conservation of unstressed lengths, and mechanical equilibrium. The number of governing equations is equal to that of the unknown quantities. Finally, the system of equations is solved to obtain the full-bridge response. The analytical algorithm only deals with two states: with and without live load. It does not involve the intermediate process that requires nonlinear analyses. Therefore, the proposed method has the advantages of high efficiency and simplicity.

(2) In this example, the main beam in the main span is subjected to a uniformly distributed load in the calculation example. A downward deflection of approximately 1.3 m occurs at the midspan, decreasing gradually toward the two towers. The left- and right-side spans produce varying degrees of the upward arch. This is because with the load acting in the main beam, the cable force in the main span increases significantly. As displacement is generated in the towers, forces in the stay cables in the side spans also increase. As a result, the main beam in the side spans arches upward. Varying degrees of downward deflection are also observed in the upper hanging points of the hangers. In addition, maximum downward deflection occurs near the midspan, reaching approximately 1.2 m.

(3) The vertical live load acts on the main beam in the examplary bridge. The horizontal components of the forces in the stay cables and the main cable in the main span increase significantly. Therefore, the left and right towers undergo varying lateral displacements toward the midspan. Within the elevation of the anchor points of the stay cables, the lateral displacement of the towers changes uniformly (with slight variation in slope). It shows that the horizontal components of the forces exerted by the cables on the towers are relatively uniform.

(4) The main beam of the bridge is subjected to a vertical uniformly distributed load. The hanger forces grow by varying degrees. The hanger forces are still uniform, where only hangers are installed. However, the hanger forces vary significantly with and without the load where both hangers and stay cables are installed. The hanger force increment increases most considerably in the outermost hanger where both the hangers and stay cables are installed. The hanger force increment is relatively uniform in the middle, which explains the fact that the outermost hanger under the live load is the most vulnerable to fatigue.

(5) The horizontal force component in the stay cable under the live load does not change monotonically. The force in the stay cable is smaller if both stay cables and hangers are used. For this reason, the force in the stay cable is also smaller in the corresponding side span. Among the remaining stay cables, the largest horizontal component of force occurs in the fifth stay cable toward the towers. Then, the closer the bridge tower, the lower the cable force. The shortest of the two stay cables has the smallest horizontal force component. The increase in the horizontal force component in the stay cable also shows a similar variation pattern above.

(6) The vertical inclination angle of the hanger also decreases toward the midspan. This is because the side spans are not symmetrically arranged, and the inclination angle of the hanger is not the smallest at the midspan.

(7) A vertical uniformly distributed live load is imposed on the main span. However, the proposed method was derived by assuming that the load acts in any position of the main beam of any length. In addition, the proposed method is also applicable to the case where the towers, stay cables, and lengths of side spans are not symmetrically arranged. As shown above, the proposed analytical algorithm has universal applicability.

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: This work was supported by the National Key R&D Program of China (No. 2022YFB3706703), the National Natural Science Foundation of China under Grant Nos. 52078134 and 52378138, and the Research and Development Project of China Communications Construction under Grant No. YSZX-02-2021-01-B.

ORCID iD

Wen-ming Zhang

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