Analytical Calculation Method for Anti-Slip of Main Cables in Three-Tower Suspension Bridges with Spatial Cable Systems

Abstract

To investigate the anti-slip characteristics of the main cables in a three-tower suspension bridge with spatial cable systems, this paper proposes an analytical calculation method for the anti-slip safety factor of the main cables and establishes an equivalent mechanical analysis model for multi-tower suspension bridges with spatial cable systems. Based on the deformation of the towers and cables under live load, as well as the equilibrium relationship of the main cable forces in loaded and unloaded spans, analytical formulas for the anti-slip safety factor of the main cables at the middle tower saddle are derived. A finite element model is developed to validate the formulas. The influence of parameters such as the spatial cable inclination angle, tower-to-cable stiffness ratio, dead-to-live load ratio, sag-to-span ratio, span length, and friction coefficient between the main cable and saddle on the anti-slip safety factor is analyzed. The results indicate that the formula proposed in this paper provides a highly accurate estimation of the slip resistance safety factor for main cables in spatial cable multi-tower suspension bridges. The adoption of spatial main cable configuration enhances the stability of the slip resistance safety factor at the intermediate tower saddle. The slip resistance safety factor of the main cable decreases with the increase in the tower-to-cable stiffness ratio, while it increases with the rise in the sag-to-span ratio. Moreover, the influence of the sag-to-span ratio on the slip resistance stability of the main cable becomes more pronounced with higher tower stiffness. The slip resistance safety factor of the main cable exhibits an approximately linear increase with the rise in the dead-to-live load ratio and the coefficient of friction. Furthermore, the slip resistance safety factor increases with the span length, and this rate of increase becomes more pronounced with smaller sag-to-span ratios. The research findings presented in this paper provide a theoretical basis for the design of spatial cable multi-tower suspension bridges.

1. Introduction

Multi-tower suspension bridges are considered an ideal solution for spanning wide water bodies due to their exceptional spanning capacity [1,2]. However, as the span length increases, their lateral stiffness and dynamic stability progressively decrease. Compared to suspension bridges with planar cable systems, suspension bridges utilizing spatial cable systems exhibit superior lateral stiffness, torsional stiffness, and enhanced dynamic stability. The integration of spatial cable systems with three-tower suspension bridges can achieve enhanced structural performance [3,4,5,6]. The Xunjiang Grand Bridge, currently under construction in China (Figure 1), with a total length of 1.688 km, is the world’s first three-tower spatial cable ground-anchored suspension bridge exceeding one kilometer in total length. It is also the world’s first three-tower spatial cable ground-anchored suspension bridge with a main span surpassing 500 m, demonstrating distinctive technical features.

In multi-tower suspension bridges, the intermediate tower exhibits relatively large structural deformations under live load due to the lack of effective restraint from the side-span main cables. To reduce structural deformations, the intermediate tower of a multi-tower suspension bridge must possess adequate thrust resistance stiffness. However, if the stiffness of the intermediate tower is excessively high, it will result in a significant difference in the internal forces of the main cables on both sides of the tower under live load, potentially leading to cable slippage. Therefore, the slip resistance stability of the main cables at the intermediate tower saddle under live load determines the upper limit of the tower’s stiffness, making it one of the key design criteria for multi-tower suspension bridges [7,8]. Cheng et al. established a calculable model for the ultimate frictional resistance between cable and saddle equipped with friction plates and experimentally verified the anti-slip safety assessment method for long-span suspension bridges [9]. Wang et al. proposed a strand-element numerical framework that captures the full process from elastic slip to ultimate slip of the main cable in the saddle zone [10]. Guo et al. performed post-fire reloading tests to measure the residual slip of cable clamps, offering data for evaluating the anti-slip performance of suspension bridge cables after fire exposure [11]. Zhang et al. developed a rapid analytical algorithm to determine the strand tensions in the anchor span of a completed suspension bridge, supplying boundary conditions for cable-saddle slip checks [12].

Existing research has primarily focused on multi-tower suspension bridges with planar cable systems [13,14,15,16,17,18], while theoretical investigations into the slip resistance characteristics of main cables in multi-tower suspension bridges employing spatial cable systems are currently lacking. Research on the slip resistance characteristics of main cables at the intermediate tower saddle can be primarily categorized into two types: the first focuses on the coefficient of friction between the main cable and the saddle, while the second investigates the influence of key structural components and design load parameters on the slip resistance characteristics of the main cables. In practice, after determining the coefficient of friction between the main cable and the saddle, the primary concern for designers is how to design a suspension bridge that meets the slip resistance requirements of the main cables. This paper employs theoretical analysis methods to investigate the slip resistance characteristics of main cables in multi-tower suspension bridges with spatial cable systems.

This paper establishes a mechanical analysis model for multi-tower suspension bridges with spatial cable systems. Based on the deformation of the tower and cables under live load, as well as the equilibrium relationship of the main cable forces in the loaded and unloaded spans, an analytical formula for calculating the slip resistance safety factor of the main cables in spatial cable multi-tower suspension bridges is derived. By referencing the design data of the Xunjiang Grand Bridge, a finite element model of a three-tower, two-span suspension bridge with spatial cables is established to validate the effectiveness of the analytical formula proposed in this paper. Furthermore, the influence of key design parameters on the slip resistance safety factor of the main cables is investigated, revealing the slip resistance characteristics of spatial cables in multi-tower suspension bridges.

2. Theoretical Analysis

In this study, the following fundamental assumptions are adopted for the analysis.

(1)

The geometric shape of the main cable under dead load is parabolic.

(2)

The main cable and the hangers lie within the same inclined plane.

(3)

The self-weight of the cable clamps, hangers, and stiffening girders is equivalently distributed as a uniform load along the bridge length.

(4)

Considering the strong restraint from the side-span cable, the displacement of the side tower under live loads is assumed to be equal to zero.

(5)

A fully floating system is adopted, with no longitudinal restraints provided between the tower and the girder.

2.1. Calculation of the Slip Resistance Safety Factor for Spatial Main Cables

The calculation diagram for the slip resistance safety factor K of spatial main cables under live load is shown in Figure 2, and its calculation formula is given by [8]

$$K={\displaystyle \frac{\mu a_{\mathrm{s}}}{\ln (F_{ct}/F_{cl})}}$$

(1)

In the equation: μ represents the coefficient of friction between the main cable and the groove bottom or partition, and it is typically taken as μ = 0.15 or determined through experimental testing; α_s denotes the wrap angle of the main cable around the saddle groove; F_ct is the tensile force of the main cable in the loaded span at the tower top; F_cl is the tensile force of the main cable in the unloaded span at the tower top. The subscripts cl and ct represent the unloaded span and loaded span, respectively.

From Equation (1), it can be observed that after determining the coefficient of friction μ between the main cable and the groove bottom or partition, calculating the slip resistance safety factor K for spatial main cables requires first solving for α_s, F_cl, and F_ct.

2.1.1. Calculation of the Wrap Angle α_s

As shown in Figure 2, to determine the wrap angle α_s of the spatial main cable around the saddle, the tangent angle θ at the point where the main cable contacts the saddle can first be calculated. The wrap angle α_s of the spatial main cable around the saddle groove is given by:

$${\alpha }_s=\pi -\theta$$

(2)

By translating the main cable forces F_ct and F_cl to the tangent intersection point O, a spatial Cartesian coordinate system is established with O as the origin (as shown in Figure 3). The three components of the unloaded span main cable force F_cl in the spatial coordinate system are F_h,cl, F_v,cl, and F_z,cl, while the three components of the loaded span main cable force F_ct are F_h,ct, F_v,ct, and F_z,ct. Here, the subscripts ℎ, v, and z represent the longitudinal, vertical, and transverse bridge directions, respectively.

F_ct and F_cl can be expressed as follows:

$$\left|F_{ct}\right|=\sqrt{{\left|-F_{h,ct}\right|}^2+{\left|F_{\mathrm{v},ct}\right|}^2+{\left|F_{z,ct}\right|}^2}$$

(3)

$$\left|F_{cl}\right|=\sqrt{{\left|F_{h,cl}\right|}^2+{\left|F_{v,cl}\right|}^2+{\left|F_{z,cl}\right|}^2}$$

(4)

Based on the spatial vector angle calculation formula, θ can be expressed as follows:

$$\theta =\arccos \left({\displaystyle \frac{F_{ct}\cdot F_{cl}}{\left|F_{ct}\right|\left|F_{cl}\right|}}\right)=\arccos \left({\displaystyle \frac{-F_{h,cl}{F}_{h,ct}+F_{v,cl}{F}_{v,ct}+F_{z,cl}{F}_{z,ct}}{\sqrt{{(F_{h,cl})}^2+F_{v,cl}^2+F_{z,cl}^2}\times \sqrt{{\left(-F_{h,ct}\right)}^2+F_{v,ct}^2+F_{z,ct}^2}}}\right)$$

(5)

By substituting Equation (5) into Equation (2), the wrap angle αs between the main cable and the saddle groove can be determined.

From Equations (1)–(5), it can be observed that calculating the slip resistance safety factor K of the spatial main cable requires determining the main cable force components F_h,cl, F_v,cl, F_z,cl, F_h,ct, F_v,ct, and F_z,ct. The following section derives the main cable forces at the intermediate tower saddle under live load.

2.1.2. Calculation of Main Cable Forces at the Intermediate Tower Saddle Under Live Load

The projections of the spatial main cable in the vertical and horizontal planes are shown in Figure 4. In the figure, β represents the angle between the inclined plane of the main cable and the vertical plane, n denotes the sag-to-span ratio, f is the sag of the spatial main cable, and L is the span length of a single main span. f_v represents the sag of the main cable profile projected onto the vertical plane, given by f_v = f cosβ, while f_z represents the sag of the main cable profile projected onto the horizontal plane, given by f_z = f sinβ.

Let g denote the weight per unit length of the main cable, and let q represent the weight per unit length of the bridge deck system under dead load. Based on the assumptions in this paper, the profile of the main cable under dead load is parabolic, and the equation of the main cable profile is given by $y(x)={\displaystyle \frac{4f}{L^2}}x(L-x)$. Let G denote half of the total weight of the two main cables in a single main span, then:

$$G=g{\displaystyle {\int }_0^{L/2}\sqrt{1+y{\prime }^2}}dx={\displaystyle \frac{gL}{2}}\left(1+{\displaystyle \frac{8}{3}}{n}^2-{\displaystyle \frac{32}{5}}{n}^4\right)$$

(6)

The spatial main cable projects as parabolas in both the vertical and horizontal planes, with profile equations $y(x)={\displaystyle \frac{4f_v}{L^2}}x(L-x)$ for the vertical plane and $z(x)={\displaystyle \frac{4f_z}{L^2}}x(L-x)$ for the horizontal plane. As shown in Figure 5, a spatial Cartesian coordinate system is established at the saddle. The vertical components of the hanger forces are denoted as v₁, v₂, v₃, …, v_i, …, v_m, and the horizontal components as z₁, z₂, z₃, …, z_i, …, z_m.

Under dead load, the vertical component F_v, transverse component F_z, and longitudinal component F_h of a single main cable at the intermediate tower saddle are given by:

$$F_v={\displaystyle \frac{G}{2}}+{\displaystyle \frac{1}{4}}{\displaystyle \sum _{i=1}^m{v}_i}={\displaystyle \frac{G}{2}}+{\displaystyle \frac{qL}{4}}$$

(7)

$$F_z={\displaystyle \frac{1}{4}}{\displaystyle \sum _{i=1}^m{z}_i}={\displaystyle \frac{1}{4}}{\displaystyle \sum _{i=1}^m{v}_i\tan \beta }={\displaystyle \frac{qL\tan \beta }{4}}$$

(8)

$$F_h={\displaystyle \frac{\left(\frac{g+q}{2\cos \beta }\right)L^2}{\frac{8f_v}{\cos \beta }}}={\displaystyle \frac{(g+q)L^2}{16f_v}}$$

(9)

In Formula (9), both g and q refer to the load per unit bridge length, which is jointly borne by the two main cables.

The most critical loading condition for a three-tower suspension bridge occurs when one span is fully loaded while the adjacent span remains unloaded. As shown in Figure 6, when live load is applied to one main span, the intermediate tower top experiences a longitudinal displacement δL. The sag of the main cable in the loaded span increases by δf_v,ct, while the sag in the unloaded span decreases by δf_v,cl. Assuming the inclination angle β of the spatial main cable remains unchanged under live load, the changes in the sag of the main cable in the vertical and horizontal planes are shown in Figure 6, where δf_v,ct and δf_v,cl represent the changes in the sag in the vertical plane, and δf_z,ct and δf_z,cl represent the changes in the sag in the horizontal plane.

From Equations (7)–(9), it can be concluded that the main cable forces in the loaded span at the intermediate tower saddle, F_h,ct, F_v,ct, and F_z,ct, are given by:

$$F_{v,ct}={\displaystyle \frac{G}{2}}+{\displaystyle \frac{1}{4}}{\displaystyle \sum _{i=1}^m{v}_i}={\displaystyle \frac{G}{2}}+{\displaystyle \frac{(q+p)L}{4}}$$

(10)

$$F_{z,ct}={\displaystyle \frac{1}{4}}{\displaystyle \sum _{i=1}^m{z}_i}={\displaystyle \frac{1}{4}}{\displaystyle \sum _{i=1}^m{v}_i\tan \beta }={\displaystyle \frac{(q+p)L\tan \beta }{4}}$$

(11)

$$F_{h,ct}={\displaystyle \frac{\left(\frac{g+q+p}{2\cos \beta }\right){(L-\delta L)}^2}{8\left(\frac{f_v+\delta f_{v,ct}}{\cos \beta }\right)}}={\displaystyle \frac{\left(g+q+p\right){(L-\delta L)}^2}{16\left(f_v+\delta f_{v,ct}\right)}}$$

(12)

The main cable forces in the unloaded span, F_h,cl, F_v,cl, and F_z,cl, are given by:

$$F_{v,cl}={\displaystyle \frac{G}{2}}+{\displaystyle \frac{1}{4}}{\displaystyle \sum _{i=1}^m{v}_i}={\displaystyle \frac{G}{2}}+{\displaystyle \frac{qL}{4}}$$

(13)

$$F_{z,cl}={\displaystyle \frac{1}{4}}{\displaystyle \sum _{i=1}^m{z}_i}={\displaystyle \frac{1}{4}}{\displaystyle \sum _{i=1}^m{v}_i\tan \beta }={\displaystyle \frac{qL\tan \beta }{4}}$$

(14)

$$F_{h,cl}={\displaystyle \frac{\left(\frac{q+g}{2\cos \beta }\right){(L+\delta L)}^2}{8\left(\frac{f_v-\delta f_{v,cl}}{\cos \beta }\right)}}={\displaystyle \frac{\left(q+g\right){(L+\delta L)}^2}{16\left(f_v-\delta f_{v,cl}\right)}}$$

(15)

From Equations (10)–(15), it can be observed that the main cable forces under live load are related to the main cable profile. To solve for the main cable forces F_h,cl, F_v,cl, F_z,cl, F_h,ct, F_v,ct, and F_z,ct, it is necessary to first determine the changes in the sag of the spatial cable in the vertical plane, δf_v,ct and δf_v,cl, as well as the displacement of the intermediate tower top δL.

The following section derives the values of δf_v,ct, δf_v,c, and δL.

2.1.3. Calculation of Main Cable Profile Changes Under Live Load

Under live load, the projection of the spatial main cable in the vertical plane is shown in Figure 6.

The incremental horizontal force δH_ct,p of the main cable in the loaded span under live load is given by:

$$\delta H_{\mathrm{ct},p}={\displaystyle \frac{\left(\frac{w+p}{\cos \beta }\right){(L-\delta L)}^2}{8\left(\frac{f_v+\delta f_{\mathrm{v},\mathrm{ct}}}{\cos \beta }\right)}}-{\displaystyle \frac{\frac{w}{\cos \beta }{L}^2}{\frac{8f_v}{\cos \beta }}}={\displaystyle \frac{(w+p){(L-\delta L)}^2}{8(f_v+\delta f_{\mathrm{v},\mathrm{ct}})}}-{\displaystyle \frac{w{L}^2}{8f_v}}$$

(16)

Based on the horizontal force equilibrium between the loaded and unloaded spans, δH_ct,p can also be expressed as:

$$\delta H_{\mathrm{ct},p}=\left(K_{\mathrm{t}}+K_{\mathrm{c}}\right)\delta L$$

(17)

In the equation, K_t represents the longitudinal thrust resistance stiffness of the bridge tower, K_c denotes the longitudinal restraint stiffness of the main cable, where $K_{\mathrm{c}}={\displaystyle \frac{3w}{128n^3\cos \beta }}+{\displaystyle \frac{w}{4n\cos \beta }}$ [19]. w is the dead load per unit bridge length, given by w = q + g, and n is the sag-to-span ratio of the main cable, where $n=f/L$.

From Equations (16) and (17), the following is obtained:

$${\displaystyle \frac{(w+p){(L-\delta L)}^2}{8(f_v+\delta f_{\mathrm{v},\mathrm{ct}})}}-{\displaystyle \frac{w{L}^2}{8f_v}}=\left(K_{\mathrm{t}}+K_c\right)\delta L$$

(18)

In Equation (18), the change in the sag of the main cable in the loaded span δf_v,ct consists of two components: one is the change in the sag δf_t,p caused by the displacement of the tower top δL, and the other is the change in the sag δf_e,p resulting from the elongation of the main cable under live load. Therefore, δf_v,ct can be expressed as:

$$\delta f_{\mathrm{v},\mathrm{ct}}=\delta f_{\mathrm{t},p}+\delta f_{\mathrm{e},p}$$

(19)

First, based on the relationship between the tower top displacement and the change in the main cable sag, it can be derived that [20]

$$\delta f_{\mathrm{t},p}={\displaystyle \frac{\left(3-8n^2\right)\cos \beta }{16n}}\delta L$$

(20)

Formula (20) demonstrates that the variation in the main cable sag depends on the change in displacement at the tower top, the sag-to-span ratio n of the main cable, and its spatial inclination angle.

Then solve for δf_e,p.

Under a uniform load, the overall profile of the main cable is parabolic. With the coordinate system selected as shown in Figure 7, the profile equation of the main cable can be expressed as:

$$y={\displaystyle \frac{4f}{L^2}}x(L-x)$$

(21)

Under the action of a uniform load p, the elastic elongation of the main cable in the loaded span δS_ct,p, caused by the increase in horizontal force, is given by:

$$\delta S_{\mathrm{ct},p}={\displaystyle \frac{\delta H_{\mathrm{ct},p}}{E_{\mathrm{c}}{A}_{\mathrm{c}}}}{\displaystyle {\int }_0^L(1+{y^{\prime }}^2)dx} ={\displaystyle \frac{\delta H_{\mathrm{ct},p}L}{E_{\mathrm{c}}{A}_{\mathrm{c}}}}(1+{\displaystyle \frac{16}{3}}{n}^2)$$

(22)

In the formula, E_c represents the elastic modulus of the main cable, and A_c is the sum of the cross-sectional areas of the two main cables in a single span.

Substituting Equation (17) into Equation (22) yields:

$$\delta S_{\mathrm{ct},p}={\displaystyle \frac{\left(K_{\mathrm{c}}+K_{\mathrm{t}}\right)L}{E_{\mathrm{c}}{A}_{\mathrm{c}}}}\left(1+{\displaystyle \frac{16n^2}{3}}\right)\delta L$$

(23)

The alteration in the sag of the main cable δf_e,p, induced by the elastic elongation of the main cable δS_ct,p, is [19]:

$$\delta f_{\mathrm{e},p}={\displaystyle \frac{3L\cos \beta }{16f}}\delta S_{\mathrm{ct},p}$$

(24)

Formula (23) shows that the variation in the main cable sag is also influenced by the elongation rate of the main cable, its sag-to-span ratio, and its spatial inclination angle.

Substituting Equation (23) into Equation (24) yields:

$$\delta f_{\mathrm{e},p}={\displaystyle \frac{3L^2}{16f}}{\displaystyle \frac{\left(K_{\mathrm{c}}+K_{\mathrm{t}}\right)}{E_{\mathrm{c}}{A}_{\mathrm{c}}}}\left(1+{\displaystyle \frac{16n^2}{3}}\right)\cos \beta \delta L$$

(25)

Substituting Equations (20) and (25) into Equation (19) yields:

$$\delta f_{\mathrm{v},\mathrm{ct}}=\left[{\displaystyle \frac{\left(3-8n^2\right)\cos \beta }{16n}}+{\displaystyle \frac{wL\left(3+16n^2\right)}{16n{E}_{\mathrm{c}}{A}_{\mathrm{c}}}}\left(1+{\displaystyle \frac{K_{\mathrm{t}}}{K_{\mathrm{c}}}}\right)\left({\displaystyle \frac{3+32n^2}{128n^3}}\right)\right]\delta L$$

(26)

Let

$${\alpha }_{\mathrm{ct},p}={\displaystyle \frac{\left(3-8n^2\right)\cos \beta }{16n}}+{\displaystyle \frac{wL\left(3+16n^2\right)}{16n{E}_{\mathrm{c}}{A}_{\mathrm{c}}}}\left(1+{\displaystyle \frac{K_{\mathrm{t}}}{K_{\mathrm{c}}}}\right)\left({\displaystyle \frac{3+32n^2}{128n^3}}\right)$$

(27)

Equation (26) can be simplified as:

$$\delta f_{\mathrm{v},\mathrm{ct}}={\alpha }_{\mathrm{ct},p}\delta L$$

(28)

Substituting Equation (28) into Equation (18) yields:

$$w\left[\left(1+{\displaystyle \frac{p}{w}}\right)-{\displaystyle \frac{8{\alpha }_{\mathrm{ct},p}\left(3+32n^2\right)}{128n^3\cos \beta }}\left(1+{\displaystyle \frac{K_{\mathrm{t}}}{K_{\mathrm{c}}}}\right)\right]\delta L^2-wL\left[2\left(1+{\displaystyle \frac{p}{w}}\right)+{\displaystyle \frac{8n_v\left(3+32n^2\right)}{128n^3\cos \beta }}\left(1+{\displaystyle \frac{K_{\mathrm{t}}}{K_{\mathrm{c}}}}\right)+{\displaystyle \frac{{\alpha }_{\mathrm{ct},p}}{n_v}}\right]\delta L+p{L}^2=0$$

(29)

Equation (29) is a quadratic equation in terms of δL. Solving Equation (29) and discarding the negative value yields the expression for δL.

$$\delta L={\displaystyle \frac{-B-\sqrt{B^2-4AC}}{2A}}$$

(30)

In the formula

$$A=w\left[\left(1+{\displaystyle \frac{p}{w}}\right)-{\displaystyle \frac{8{\alpha }_{\mathrm{ct},p}\left(3+32n^2\right)}{128n^3\cos \beta }}\left(1+{\displaystyle \frac{K_{\mathrm{t}}}{K_{\mathrm{c}}}}\right)\right]$$

(31)

$$B=-wL\left[2\left(1+{\displaystyle \frac{p}{w}}\right)+{\displaystyle \frac{8n_v\left(3+32n^2\right)}{128n^3\cos \beta }}\left(1+{\displaystyle \frac{K_{\mathrm{t}}}{K_{\mathrm{c}}}}\right)+{\displaystyle \frac{{\alpha }_{\mathrm{ct},p}}{n_v}}\right]$$

(32)

$$C=p{L}^2$$

(33)

After δL is determined, substituting δL into Equation (28) allows the calculation of the change in sag δf_v,ct of the main cable in the loaded span.

The change in sag δf_v,cl of the main cable in the unloaded span is primarily caused by the displacement of the tower top. Based on the relationship between the tower top displacement and the change in sag of the main cable, the following is obtained:

$$\delta f_{\mathrm{v},\mathrm{cl}}={\displaystyle \frac{\left(3-8n^2\right)\cos \beta }{16n}}\delta L={\displaystyle \frac{\left(3-8n^2\right)\left(-B-\sqrt{B^2-4AC}\right)\cos \beta }{32An}}$$

(34)

In Equation (34), the coefficients A, B, and C can be determined from Equations (31), (32), and (33), respectively.

By substituting the obtained displacement of the mid-tower top δL and the changes in main cable sag δf_v,ct and δf_v,cl into Equations (10)–(15), the three-component forces of the main cable at the mid-tower saddle (i.e., F_h,cl, F_v,cl, F_z,cl, F_h,ct, F_v,ct, and F_z,ct) can be determined. Subsequently, by incorporating these three-component forces into Equations (1)–(5), the anti-slip safety factor K of the main cable at the mid-tower saddle for a multi-tower suspension bridge with spatial cables can be calculated.

2.2. Calculation Procedure

The calculation procedure for the anti-slip safety factor of the main cable at the saddle of the middle tower in a multi-tower suspension bridge with spatial cables is illustrated in Figure 7. The calculation flow in Figure 7 does not require iteration.

2.3. Case Study and Analysis

To verify the validity of the proposed formulas in this study, the anti-slip safety factor K of the spatial main cable at the intermediate tower was calculated using both the derived formulas and the finite element method (FEM). Based on the design parameters of the Xunjiang Bridge—a three-tower suspension bridge with spatial cables currently under construction—a finite element model was established using the structural analysis software Midas/Civil 2021 (see Figure 8). In the finite element model, the main cable is simulated using cable elements, the hangers are modeled with truss elements, and the towers and girders are modeled using beam elements. Geometric nonlinear effects are considered in the analysis. The main spans of the bridge are each 520 m in length, with a sag-to-span ratio of 1/9 for the main cables and a standard longitudinal spacing of 16 m for the hangers. The girder adopts a divided steel box configuration, comprising two steel box girders connected by transverse linking boxes. The entire bridge employs a full-floating system, with tower heights of 104.49 m. Other critical model parameters are summarized in

Table 1. Structural Properties.

Item	Symbol	Value
Length of main span (m)	L	520
Sag-to-span ratio	nv	1/9
Area of the cable (m2)	Ac	2 × 0.133
Unit weight of the deck (kN·m−1)	q	185
Unit weight of the cable (kN·m−1)	g	20.86
Unit weight of the dead load (kN·m−1)	w	205.86
Elastic modulus of cable (GPa)	Ec	195
Unit volume weight of cable and stiffening girder (kN·m−3)	γ	78.5
Live load (kN·m−1)	p	20; 25; 30; 35; 40
The angle between the inclined plane of the main cable and the vertical plane/rad	β	0.245
Longitudinal thrust resistance stiffness of the bridge tower (kN·m−1)	Kt	71,429
Friction coefficient	μ	0.35

, and the applied loading conditions are illustrated in Figure 9.

Based on the aforementioned structural parameters, the equivalent spring stiffness K_c of the main cable was calculated. Following the computational flowchart illustrated in Figure 7, the wrapping angle of the main cable at the saddle groove and the internal forces of the main cables on both sides at the tower top were determined, respectively. Finally, the anti-slip safety coefficient K of the main cable was obtained.

Based on the finite element analysis results, the internal forces of the main cables on both sides at the top of the middle tower under live load conditions, as well as the displacement at the top of the middle tower, were extracted. The wrap angle α_s between the main cables and the saddle was then calculated. Subsequently, by substituting α_s and the internal forces of the main cables on both sides at the top of the middle tower into Equation (1), the anti-slip safety factor K of the main cables was determined, which serves as the finite element analysis result.

As can be seen from Figure 10, since the structural deformation induced by the uniformly distributed load is not significant, the structural nonlinear effects are not pronounced. Consequently, the internal forces of the main cable and the magnitude of the uniformly distributed load exhibit an approximately linear relationship. The theoretical values of the wrap angle at the saddle–cable interface and the main cable tension demonstrate minimal discrepancies compared with finite element results, with computational errors within 2%. Similarly, the anti-slip safety factor exhibits calculation deviations below 4%. The analytical method proposed in this study can be effectively applied for estimating the anti-slip safety coefficient of spatial main cables during preliminary design stages.

3. Analysis of Influencing Factors

From the formula for calculating the anti-slip safety factor of the main cable (Equation (1)), it can be seen that the anti-slip safety factor depends on the friction coefficient between the main cable and the saddle, the ratio of the internal forces of the main cable on both sides of the tower top, and the wrap angle of the main cable at the saddle. Based on the analytical formulas derived in this study, the internal forces of the main cable and the wrap angle at the saddle are determined by parameters such as the span length L, the spatial main cable inclination angle β, the dead load w, the live load p, and the changes in the main cable geometry caused by the live load (tower top displacement δL, changes in the main cable sag δf_v,ct, and δf_v,cl). Furthermore, the changes in the main cable geometry are influenced by the stiffness ratio between the tower and the cable (K_t/K_c), the ratio of dead load to live load (w/p), and the sag-to-span ratio of the main cable in the vertical plane (n_v).

In summary, the main factors affecting the anti-slip stability of the spatial main cable include the friction coefficient between the main cable and the saddle μ, the span length L, the spatial main cable inclination angle β, the sag-to-span ratio of the main cable in the vertical plane n_v, the ratio of dead load to live load w/p, and the stiffness ratio between the tower and the cable K_t/K_c. Below, based on the three-tower suspension bridge model and loading conditions shown in Figure 9, the influence of these parameters on the anti-slip stability of the spatial main cable is investigated.

3.1. Friction Coefficient Between the Main Cable and the Saddle μ

The friction coefficient between the main cable and the saddle μ was assigned values ranging from 0.15 to 0.40, the live load p was set to 30 kN/m, and the sag-to-span ratio n_v was varied as 1/8, 1/9, 1/10, 1/11, and 1/12. All other parameters remained unchanged. The calculated results of the anti-slip safety factor of the main cable are shown in Figure 11.

From Figure 11, it can be observed that the anti-slip safety factor of the main cable increases linearly with the friction coefficient between the main cable and the saddle.

3.2. Spatial Cable Inclination Angle β

The spatial cable inclination angle β was assigned values ranging from 0 to 30; the sag-to-span ratio n_v was varied as 1/8, 1/9, 1/10, 1/11, and 1/12; and the main span lengths were set to 520 m (the main span length of the Xunjiang Grand Bridge currently under construction in China), 700 m, and 1000 m. The live load p was set to 30 kN/m, and the friction coefficient between the main cable and the saddle μ was set to 0.3. All other parameters remained unchanged. The calculated results are shown in Figure 12.

From Figure 12, it can be observed that the anti-slip safety factor of the main cable continuously increases with the increase in the spatial main cable inclination angle β. When the inclination angle β < 20, the influence of the inclination angle on the anti-slip safety factor K is relatively small. As the inclination angle increases from 0 to 10, the anti-slip safety factor increases by approximately 1%; from 0 to 15, it increases by approximately 2.5%; and when the inclination angle reaches 20, the anti-slip safety factor increases by approximately 5%. When the inclination angle β > 20, the effect of the inclination angle on the anti-slip safety factor becomes more pronounced as the angle increases, and when the inclination angle reaches 30, the anti-slip safety factor increases by approximately 10%.

In conclusion, the use of spatial main cables in multi-tower suspension bridges is beneficial for improving the anti-slip stability of the main cable at the middle tower saddle, as it enhances the anti-slip safety factor.

3.3. Tower–Cable Stiffness Ratio and Sag-to-Span Ratio

Let the sag-to-span ratio n_v take values of 1/8, 1/9, 1/10, 1/11, and 1/12; let the main span length L take values of 520 m and 1000 m; let the live load p be 30 kN/m; and let the friction coefficient μ between the main cable and saddle be 0.3. By varying the longitudinal resisting stiffness of the tower, the tower–cable stiffness ratio is adjusted within a range of 0.5–20, while other parameters remain unchanged. The calculation results are shown in Figure 13.

As can be seen from Figure 13:

(1)

The anti-slip safety factor of main cables exhibits an inverse correlation with the tower-to-cable stiffness ratio. When K_t/K_c < 3, the safety factor K decreases rapidly with increasing stiffness ratio. Within the range of 3 < K_t/K_c < 12, the influence of stiffness ratio variation on K diminishes significantly, demonstrating a rate of change below 5%. For stiffness ratios exceeding K_t/K_c > 12, further enhancement of the tower’s longitudinal stiffness yields negligible impact, with the alteration rate of K remaining less than 1%.

(2)

The anti-slip safety factor of the main cable exhibits a positive correlation with the sag–span ratio under a fixed tower-to-cable stiffness ratio (K_t/K_c). When K_t/K_c > 3, the sag–span ratio demonstrates a pronounced linear influence on the anti-slip safety factor. Specifically, an increase in the sag–span ratio from 1/9 to 1/8 elevates the safety factor by approximately 8%. Expanding the sag–span ratio from 1/12 to 1/8 results in a 30% enhancement of anti-slip performance. Increasing the main cable’s sag-to-span ratio alters the structural stiffness and increases the vertical deformation of the deck under asymmetric loads. Therefore, the selection of the sag-to-span ratio must also consider its effect on structural stiffness.

3.4. Ratio of Dead Load to Live Load

Variations in dead load modify the longitudinal restraint stiffness of the main cable, thereby altering the tower–cable stiffness ratio. To isolate the influence of the tower–cable stiffness ratio, the dead load was held constant, and the dead-to-live load ratio was adjusted by varying the live load. Three tower stiffness scenarios were investigated for sag-to-span ratios n_v of 1/8, 1/9, 1/10, 1/11, and 1/12: (1) constant tower stiffness: K_t = 71,429 kN/m, with other parameters unchanged; (2) reduced tower stiffness: 0.5 K_t = 35,714 kN/m, with other parameters unchanged; (3) further reduced tower stiffness: 0.3K_t = 21,428 kN/m, with other parameters unchanged. The friction coefficient μ between the main cable and saddle was set to 0.3. The computational results are presented in Figure 14.

As illustrated in Figure 14, the anti-slip safety factor of the main cable exhibits an approximately linear increase with the dead-to-live load ratio. When the tower stiffness is high, the sag-to-span ratio demonstrates a significant influence on the anti-slip safety factor. However, as the tower stiffness (represented by the tower–cable stiffness ratio Kt/Kc) decreases, the sensitivity of the anti-slip safety factor to variations in the sag-to-span ratio is reduced.

3.5. Span Length

To investigate the relationship between the anti-slip safety factor of the main cable and the span length, the span length L was varied from 500 to 2500 m, while the sag-to-span ratio n_v was set to 1/8, 1/9, 1/10, 1/11, and 1/12. The live load p was fixed at 30 kN/m, and the friction coefficient μ between the main cable and saddle was set to 0.3. When adjusting the span length and sag-to-span ratio, the cross-sectional area of the main cable was also modified to maintain a constant stress in the main cable under dead load. The computational results are presented in Figure 15.

As can be observed from Figure 15, within the main span range of 500–2500 m, the anti-slip safety factor of main cables demonstrates an increasing trend with the enlargement of span length. For smaller spans (500–1200 m), cables with larger sag-to-span ratios exhibit higher anti-slip safety factors. With the increase in span length, the anti-slip safety coefficients of main cables with different sag-to-span ratios exhibit distinct variation patterns. Main cables with smaller sag-to-span ratios demonstrate more rapid degradation in anti-slip safety performance as the span expands. This phenomenon can be attributed to the fact that reduced sag-to-span ratios induce higher cable tension forces, consequently accelerating the growth rate of required cable cross-sectional area and self-weight.

4. Conclusions

Major conclusions of the study can be summarized as follows:

(1)

An analytical method for calculating the anti-slip safety factor of main cables in multi-tower suspension bridges with spatial cable systems is proposed in this study. The derived formula for the anti-slip safety factor of spatial cables is validated through comparison with finite element results, demonstrating high computational accuracy. The proposed method is applicable to the preliminary design of multi-tower suspension bridges with spatial cable systems.

(2)

The adoption of spatial cables improves the anti-slip stability of the main cable, and the anti-slip safety factor increases with the spatial inclination angle of the main cable. The anti-slip safety factor decreases as the tower–cable stiffness ratio increases. When the tower–cable stiffness ratio Kt/Kc < 3, an increase in the stiffness ratio results in a rapid decline in the anti-slip safety factor K. When the tower–cable stiffness ratio exceeds 12, further increases in the stiffness ratio have a minimal impact on the anti-slip safety factor.

(3)

The anti-slip stability of spatial main cables increases with a larger sag-to-span ratio. When the tower–cable stiffness ratio K_t/K_c is in the range of 0.5 to 3, variations in the sag-to-span ratio have a limited effect on the anti-slip safety factor. When K_t/K_c > 3, the sag-to-span ratio significantly influences the anti-slip safety factor. Increasing the sag-to-span ratio from 1/12 to 1/8 results in an approximately 30% increase in the anti-slip safety factor. From the perspective of enhancing the anti-slip stability of the main cable, a larger sag-to-span ratio is recommended in the design.

(4)

The anti-slip safety coefficient of the main cable increases approximately linearly with the growth of the friction coefficient between the main cable and saddle, as well as the dead-to-live load ratio. The anti-slip safety coefficient of spatial main cables increases with span length, and cables with smaller sag-to-span ratios exhibit a more pronounced acceleration in anti-slip performance enhancement as the span expands.