Method for determining structural parameters of rock anchor cables of single-tower ground anchor suspension bridges

Abstract

This invention discloses a method for determining the structural parameters of rock anchor cables in a single-tower ground-anchored suspension bridge, comprising the following steps: Step 1, establishing an expression for the stress-free length of the main cable; Step 2, establishing an expression for the deflection of the stiffening girder; Step 3, establishing an expression for the girder end rotation angle; Step 4, establishing an analytical model of the rock anchor cable structural parameters; Step 5, solving for the unknown basic quantities in the analytical model of the rock anchor cable structural parameters; Step 6, optimizing the structural parameters of the rock anchor cable: based on the analytical model of the rock anchor cable structural parameters, establishing a multi-objective optimization function; and solving it using the multi-objective Golden Eagle optimization algorithm to obtain the optimized rock anchor cable structural parameters. This invention is applicable to calculating the deformation of suspension bridges with large differences in deformation between the main cable and the stiffening girder under live load, especially single-tower ground-anchored suspension bridges where the projected length of the main cable differs from that of the stiffening girder. This invention can not only quickly find the most unfavorable load condition for verifying the indicators, but also optimize the structural parameters of the rock anchor cables, thereby improving the safety of the suspension bridge.

Description

Technical Field

[0001] This invention relates to the field of bridge analysis theory, and in particular to a method for determining the structural parameters of rock anchor cables in a single-tower ground-anchored suspension bridge. Background Technology

[0002] Suspension bridges are favored by designers due to their superior spanning capacity and efficient material utilization. As suspension bridges continue to expand in span, their structural forms are also constantly evolving. Examples include multi-tower suspension bridges with greater spanning capacity, three-cable suspension bridges with higher traffic volume, and cable-stayed suspension bridges with better wind resistance. In addition, a new type of suspension bridge has emerged to adapt to terrain, fully utilizing the landform and integrating with nature to significantly improve economic efficiency while ensuring safety. For example, the Aizhai Bridge shortened the stiffening girder span and adopted a separated tower-girder structure to better connect the mountain tunnels on both sides of the bridge. The Tiger Leaping Gorge Jinsha River Bridge is a single-tower ground-anchored suspension bridge that uses composite cable saddles to reduce the number of bridge towers and side spans.

[0003] However, the stiffening girder length of a single-tower ground-anchored suspension bridge is much shorter than the horizontal projection length of the main cable, resulting in a long cable-free section. This is detrimental to restraining the deformation of the main cable in the cable-free section under live load, and further leads to excessive stiffening girder rotation and significant fatigue of the cables at the girder ends. Therefore, rock anchor cables are usually designed in the cable-free section to improve the structural response of the suspension bridge under live load. Since rock anchor cables differ from ordinary cables, their initial tension, cross-sectional area, and cable spacing do not depend on the self-weight of the stiffening girder. Therefore, the parameters of rock anchor cables have greater selectivity and adjustability. Existing engineering applications show that rationally designed rock anchor cables can reduce the stiffening girder rotation and stress amplitude of the cables in single-tower ground-anchored suspension bridges; however, a method for quickly and accurately evaluating the rationality of the structural parameters of rock anchor cables is lacking.

[0004] The current design process for rock anchor cables is as follows: First, the number of rock anchor cables is usually determined based on the terrain and the span of a single-tower ground-anchored suspension bridge. Second, initial parameters such as the initial tension, cross-sectional area, and cable spacing of the rock anchor cables are initially assumed. Next, the main cable of the suspension bridge is shaped to obtain structural parameters such as the stress-free length of the main cable suspenders, and a finite element model is established. Based on this, the maximum value of the verification index and the corresponding most unfavorable load condition are solved using a combination of the influence line method and the trial-and-error method. If the verification index does not meet the requirements, the rock anchor cable parameters are modified, and the above steps are repeated until the verification index meets the design requirements. It can be seen that adjusting the structural parameters of the rock anchor cables not only complicates the main cable shaping and finite element modeling of a single-tower ground-anchored suspension bridge, but also, due to the significant geometric nonlinearity of suspension bridges, the solution for the most unfavorable load condition requires a combination of the influence line method and the trial-and-error method, further increasing the computational difficulty. The difficulty in determining the appropriate structural parameters for rock anchor cables not only leads to time-consuming and labor-intensive design processes for single-tower ground-anchored suspension bridges, but also lacks a suitable model to elucidate the mechanism by which rock anchor cables act on single-tower ground-anchored suspension bridges. This seriously hinders the development of single-tower ground-anchored suspension bridges and makes it difficult for designers to understand the mechanism of rock anchor cables. Summary of the Invention

[0005] The technical problem to be solved by this invention is to address the shortcomings of the prior art by providing a method for determining the structural parameters of rock anchor cables in a single-tower ground-anchored suspension bridge. This method can quickly evaluate the analytical model of the deformation of a single-tower ground-anchored suspension bridge under arbitrary live loads; and, combined with the Golden Eagle optimization algorithm, calculate the extreme values ​​of stiffening girder deflection, beam end rotation angle, and incremental force of the suspension cables, as well as the corresponding most unfavorable live load arrangement; and obtain the Pareto front of the rock anchor cable structural parameters through multi-objective optimization.

[0006] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is as follows:

[0007] A method for determining the structural parameters of rock anchor cables in a single-tower ground-anchored suspension bridge includes the following steps.

[0008] Step 1: Establish the expression for the stress-free length of the main cable: The left side of the main cable is anchored in the rock of the mountain after passing through the composite saddle, forming the mountain anchorage end; the right side of the main cable is anchored to the ground after passing through the bridge tower, forming the ground anchorage end; the main cable located between the composite saddle and the bridge tower is called the main span main cable, the main cable located between the rock anchorage end and the composite saddle is called the left span main cable, and the main cable located between the bridge tower and the ground anchorage end is called the right span main cable; the main span main cable is connected to the stiffening beam below through n suspenders, and the main span main cable located to the left of the stiffening beam is connected to the rock in the mountain below through j rock anchors; the j rock anchors and n suspenders together constitute n+j main span suspenders, so the main span main cable is divided into the 1st segment, the 2nd segment, the 3rd segment, ..., the n+jth segment and the n+j+1th segment from left to right by the n+j main span suspenders.

[0009] The structural parameters of the j rock anchor cables are all optimized design parameters, including the initial tension P of the rock anchor cables. ri Rock anchor cable cross-sectional area A rh and the cable spacing l1, l2, l3, ..., l of the rock anchor cables. j .

[0010] Based on the forces acting on the main cable under constant load and the main cable equilibrium equation, the stress-free length S of the main cable under constant load on the left span is obtained. l The expression for the stress-free length S of the main cable under constant load in the i-th segment of the main span. i The expression for the dead load stress-free length S of the right span main cable r The expression.

[0011] Based on the force and equilibrium equations of the main cable under live load, the stress-free length S of the main cable under live load on the left span is obtained. ql The expression for the stress-free length S of the main cable under live load in the i-th main span. qi The expression for the live load stress-free length S of the right span main cable qr The expression; where 1≤i≤n+j+1; S l S r S ql S qr S i and S qi There are a total of 3(n+j)+7 basic quantities to be solved; S i The optimized design includes the cable spacing l1, l2, l3, ..., l of the rock anchor cables. j .

[0012] Step 2: Establish the expression for the stiffening girder deflection: Based on the bridge tower equilibrium equation, obtain the expression for the stiffening girder deflection w(x); where w(x) is the change in the support reaction force F at point h of the stiffening girder. h F is a function of h, where 0 ≤ h ≤ n; when h = 0, F0 represents the support reaction force at the left end of the stiffened beam under live load, which is the fundamental quantity to be solved; when 1 ≤ h ≤ n, F h This represents the support reaction force of the stiffening beam located at the lower suspension point of the h-th cable under live load.

[0013] Step 3: Establish the expression for the beam end rotation angle: Differentiate the stiffening beam deflection w(x) in Step 2 to obtain the expression for the beam end rotation angle w′(x).

[0014] Step 4: Establish an analytical model of the rock anchor cable structure parameters, including 3(n+j)+10 governing equations. The specific method for establishing the model is as follows:

[0015] Step 4-1: Based on the principle of conservation of stress-free length of each main cable segment under dead load and live load, establish n+j+3 governing equations concerning the stress-free length of the main cable.

[0016] Step 4-2: Based on the deformation coordination of the rock anchor cable and the suspender cable, establish j rock anchor cable deformation control equations and n suspender cable deformation control equations; among them, the j rock anchor cable deformation control equations include the optimized design quantity, the initial tension P of the rock anchor cable. ri The cross-sectional area A of the rock anchor cable rh The n control equations for cable deformation include the longitudinal rigid body displacement ΔU of the stiffening beam, which is the fundamental quantity to be solved.

[0017] Step 4-3: Based on the coordinates of the upper and lower suspension points of the rock anchor cable and the suspender cable under live load, establish j control equations for the rock anchor cable inclination angle and n control equations for the suspender cable inclination angle.

[0018] Step 4-4: Establish four main cable closure control equations based on the closure of the horizontal projection length of the main cable in the main span, and the closure of the height difference between the main cables in the left span, main span, and right span.

[0019] Steps 4-5: Based on the horizontal force, vertical force, and bending moment of the stiffening beam, establish three governing equations; among them, the three governing equations include the support reaction force F at the right end of the stiffening beam under live load. n+1 , where is the fundamental quantity to be solved.

[0020] Step 5: Solve for unknown fundamental quantities: the 3(n+j)+7 fundamental quantities related to the stress-free length in Step 1, F0 in Step 2, ΔU in Step 4-2, and F in Step 4-5. n+1 A total of 3(n+j)+10 unknown basic quantities are formed. The initial amplitude of the structural parameters of j rock anchor cables is determined. The 3(n+j)+10 control equations established in step 4 are used to solve for the values ​​of the 3(n+j)+10 unknown basic quantities, thus obtaining the analytical model of the rock anchor cable structural parameters with definite basic quantities.

[0021] Step 6: Optimization of structural parameters of j rock anchor cables: For the analytical model of rock anchor cable structural parameters with definite basic quantities obtained in Step 5, a multi-objective optimization function, Minimize F, is established; by solving Minimize F, the optimized rock anchor cable structural parameters are obtained; where the expression for Minimize F is:

[0022] Minimize F={f1(x),f2(x),f4(x)}

[0023] in:

[0024] f1(x) = max(|w(x)|)

[0025] f2(x)=|w′(0)|

[0026] f4(x)=[(max(Prqi -P ri )-min(P rqi -P ri )) / (2A rh ),(max(P qi -P i )-min(P qi -P i )) / (2A h )]

[0027] In the formula, f1(x) is the maximum deflection function of the stiffened beam.

[0028] f2(x) is the rotation angle function of the left end of the stiffening beam.

[0029] f4(x) is a function of the maximum stress amplitude in the suspender cable and the rock anchor cable.

[0030] P rqi and P ri These represent the tension of the rock anchor cable under live load and dead load, respectively; P ri Also known as the initial tension of rock anchor cables, optimized design quantity.

[0031] P qi and P i These represent the tension of the sling under live load and dead load, respectively.

[0032] A rh and A h These are the cross-sectional areas of the rock anchor cable and the suspension cable, respectively.

[0033] In step 1, S l S r S ql and S qr The expression is:

[0034]

[0035]

[0036]

[0037]

[0038] in:

[0039]

[0040] ΔD=(H q(n+j+1) -H qr ) / K D

[0041] In the formula, c l and cr These are the constant load catenary parameters for the left and right spans of the main cable, respectively.

[0042] H l and H r The horizontal forces under constant load on the left and right spans of the main cable are known quantities.

[0043] a l and a r The parameters for the live load catenary of the left and right main cables are respectively, and are all known design quantities.

[0044] c ql and c qr These are the live load catenary parameters for the left and right spans of the main cable, respectively; q c The weight of the main cable is known. H ql and H qr The horizontal forces under live load on the left and right spans of the main cable are respectively, and both are fundamental quantities to be solved.

[0045] l l and l r These are the horizontal projected lengths of the left and right spans of the main cable, respectively, and are known design quantities.

[0046] ΔB is the horizontal displacement of the composite saddle under live load, and is the fundamental quantity to be solved.

[0047] ΔD is the horizontal displacement of the bridge tower under live load; K D Let be the horizontal stiffness of the bridge tower, a known quantity.

[0048] a ql and a qr The parameters for the live load catenary on the left and right spans of the main cable are respectively, and both are basic quantities to be solved.

[0049] E c and A c The elastic model and cross-sectional area of ​​the main cable are given, respectively, along with known quantities.

[0050] H q(n+j+1) Let be the horizontal force of the main cable in the (n+j+1)th main span, and be the basic quantity to be solved.

[0051] In step 1, S i and S qi The expression is:

[0052]

[0053]

[0054] in:

[0055]

[0056]

[0057]

[0058] In the formula, c i Let be the constant load catenary parameters of the main cable in the i-th main span;

[0059] c q(i-1) and c qi These are the live load catenary parameters for the main span cable of the (i-1)th and i-th segments, respectively.

[0060] l i Let l be the horizontal projection length of the main cable under dead load in the i-th main span, which is a known design quantity; when 1≤i≤j, l i The distance between the i-th rock anchor cable and the (i-1)-th rock anchor cable or composite saddle is the optimized design quantity.

[0061] l qi and l q(i-1) , , are the horizontal projected lengths of the main cable under live load in the i-th and (i-1)-th main spans, respectively, and are the basic quantities to be solved.

[0062] a i Let be the parameters of the main cable under constant load in the i-th main span, and be the known design parameters.

[0063] a qi and a q(i-1) These are the live load catenary parameters of the main cable in the i-th and (i-1)-th segments, respectively; where a q1 These are the fundamental quantities to be solved.

[0064] H i Let be the horizontal force under the constant load of the main cable in the i-th main span, which is a known quantity.

[0065] H qi and H q(i-1) These represent the horizontal forces under live load on the main cable of the i-th and (i-1)-th main spans, respectively; H q1 These are the fundamental quantities to be solved.

[0066] P rq(i-1) Let be the sling force of the (i-1)th main span suspender under live load, and let be the basic quantity to be solved.

[0067] P q(i-1) Let be the sling force of the (i-1)th main span suspender under live load, and let be the basic quantity to be solved.

[0068] β (i-1) Let be the inclination angle of the (i-1)th main span hanger under live load, and be the basic quantity to be solved.

[0069] In step 2, the stiffening beam is simplified to a simply supported beam. Let the horizontal length of the live load q directly above the stiffening beam be γ2, and the distance from the left starting end of the live load q to the left end of the stiffening beam be γ1. The vertical components of the main span hangers located on both sides of the left starting end of the live load q from left to right are F z and F z+1 And F z and F z+1 The horizontal coordinates are X Gz and X G(z+1) The distance from the right end of the live load q to the right end of the stiffening girder is γ3. The vertical components of the main span hangers located on both sides of the right end of the live load q, from left to right, are F... m and F m+1 And F m and F m+1 The horizontal coordinates are X Gm and X G(m+1) Additionally, let the horizontal coordinate of the left end of the stiffening beam be X. G0 =0, the horizontal coordinate of the right end of the stiffening beam is X G(n+1) Then the expression for the stiffened beam deflection w(x) is:

[0070]

[0071]

[0072]

[0073]

[0074]

[0075]

[0076]

[0077] In the formula, X Gh Let h be the horizontal coordinate of the stiffening beam at any point h.

[0078] w a (x) represents the values ​​at F0 and F... z The deflection of the stiffened beam from point a-1 to point a in segment a.

[0079] X G(a-1) and X Ga These represent the horizontal coordinates of point a-1 and point a, respectively.

[0080] w z+1 (x) is F z The stiffening beam deflection up to the horizontal coordinate γ1 segment.

[0081] wz+2 (x) represents the horizontal coordinates from γ1 to F. z+1 Deflection of the stiffened beam segment.

[0082] w a+1 (x) is F z+1 To F m The deflection of the stiffened beam from point a-1 to point a in segment a.

[0083] w m+2 (x) is F m The stiffening beam deflection up to the horizontal coordinate segment γ1+γ2.

[0084] w m+3 (x) represents the horizontal coordinate γ1+γ2 to F. m+1 Deflection of the stiffened beam segment.

[0085] w a+2 (x) is F m To F n+1 The deflection of the stiffened beam from point a-1 to point a in segment a.

[0086] C a,1 C a,2 C z+1,1 C z+1,2 C z+2,1 C z+2,2 C a+1,1 C a+1,2 C m+2,1 C m+2,2 C m+3，1 C m+3,2 C a+2,1 and C a+2,2 These are all deflection equation coefficients, specifically obtained by establishing a corresponding set of quantitative equations based on the boundary conditions of the simply supported beam and the continuity of the stiffened beam's displacement and rotation, and then solving them.

[0087] In step 2, F h The expression is:

[0088] F h =2P q(j+h) cosβ (j+h)

[0089] In the formula, P q(j+h) Let be the sling force of the i=j+h main span suspender under live load, and let be the basic quantity to be solved.

[0090] β (j+h) Let be the cable inclination angle of the i=j+h main span suspender under live load, and let be the basic quantity to be solved.

[0091] In step 4, there are 3(n+j)+10 governing equations, specifically:

[0092] Step 4-1 and the control equations for the stress-free length of the main cable (n+j+3) are as follows:

[0093] S l =S ql

[0094] S i =S qi

[0095] S r =S qr

[0096] Step 4-2, the j rock anchor cable deformation control equations and the n suspension cable deformation control equations are as follows:

[0097]

[0098]

[0099] in:

[0100]

[0101]

[0102]

[0103]

[0104]

[0105] In the formula, l rhqi and l rhi These represent the lengths of the rock anchor cable under live load and dead load, respectively.

[0106] l hqi and l hi These represent the lengths of the slings under live load and dead load, respectively.

[0107] E rh and A rh These are the elastic modulus and cross-sectional area of ​​the rock anchor cable, respectively.

[0108] E h and A h These are the elastic modulus and cross-sectional area of ​​the sling, respectively.

[0109] and These are the horizontal and vertical coordinates of the suspension point on the i-th main span rod under live load.

[0110] and These are the horizontal and vertical coordinates of the ground anchorage point of the i-th rock anchor cable, respectively.

[0111] and X represents the horizontal and vertical coordinates of the lower suspension point of the i-th main span hanger under live load. B and Y B These are the horizontal and vertical coordinates of the IP point of the composite cable saddle under constant load.

[0112] l qk Let be the horizontal projected length of the main cable under live load in the k-th main span, and be the basic quantity to be solved.

[0113] Δh qk Let be the vertical height difference of the main cable in the k-th main span.

[0114] X Gi Let be the horizontal coordinate of the lower suspension point of the i-th main span suspender;

[0115] w(X Gi ) is X Gi Deflection of the stiffening beam at point [point].

[0116] Step 4-3, the control equations for the j rock anchor cable inclination angles and the control equations for the n suspension cable inclination angles are as follows:

[0117]

[0118]

[0119] Step 4-4, the four main cable closure control equations are as follows:

[0120]

[0121] Δh ql =Y A -Y B

[0122]

[0123] Δh qr =Y D -Y E

[0124] In the formula, L C The total horizontal projection length of the main span cable under dead load; Y D Y represents the vertical coordinate of the IP point of the bridge tower saddle. A The vertical coordinate of the anchorage end of the main cable on the left side of the mountain; Y E The vertical coordinate of the ground anchorage end on the right side of the main cable; Δh ql The vertical elevation difference of the left span main cable; Δh qiLet Δh be the vertical elevation difference of the main cable in the i-th main span. qr This represents the vertical elevation difference of the right span of the main cable.

[0125] Steps 4-5, the three governing equations are as follows:

[0126]

[0127]

[0128]

[0129] In the formula, P qxh and P qyh These are the horizontal and vertical components of the force on the h-th sling, respectively.

[0130] Let h be the horizontal coordinate of the h-th sling.

[0131] L G This refers to the length of the stiffening beam.

[0132] In step 6, the solution method for the multi-objective optimization function Minimize F is the multi-objective Golden Eagle Optimization Algorithm.

[0133] In step 5, the values ​​of 3(n+j)+10 unknown basic quantities are obtained by using the least squares method.

[0134] In step 6, before optimizing the structural parameters of the j rock anchor cables, the stiffening beam deflection, beam end rotation angle, maximum stress amplitude in the suspenders and rock anchor cables, and the corresponding most unfavorable live load conditions are calculated. The distance γ1 from the left starting end of the live load to the left end of the stiffening beam and the horizontal arrangement length of the live load are obtained as γ2. Then, the structural parameters of the j rock anchor cables are optimized based on the calculated γ1 and γ2.

[0135] This invention offers the following advantages: By establishing an analytical model of the rock anchor cable structure parameters and combining it with the Golden Eagle optimization algorithm to determine the peak value and most unfavorable live load condition of the verification index, this invention further optimizes the rock anchor cable structure parameters through multi-objective optimization to obtain its Pareto front. This clarifies the working principle of the rock anchor cable, making its physical meaning clearer, facilitating the evaluation of the contribution of the rock anchor cable to a single-tower ground-anchored suspension bridge, and enabling its application in the preliminary design of the rock anchor cable structure parameters for single-tower ground-anchored suspension bridges. Attached Figure Description

[0136] Figure 1 A simplified model of a single-tower ground-anchored suspension bridge.

[0137] Figure 2 This is a schematic diagram of the deformation of the main cable.

[0138] Figure 3 This is a schematic diagram of the force balance at the point where the sling is attached.

[0139] Figure 4 This is a schematic diagram of the stress on the stiffening beam at the lower suspension cable point.

[0140] Figure 5 This is a schematic diagram of the forces acting on a simply supported beam. Detailed Implementation

[0141] The present invention will now be described in further detail with reference to the accompanying drawings and specific preferred embodiments.

[0142] like Figure 1 As shown, the left side of the main cable is anchored in the rock of the mountain after passing through the composite saddle, forming the mountain anchorage end; the right side of the main cable is anchored to the ground surface after passing through the bridge tower, forming the ground anchorage end. The main cable located between the composite saddle and the bridge tower is called the main span main cable, the main cable located between the rock anchorage end and the composite saddle is called the left span main cable, and the main cable located between the bridge tower and the ground anchorage end is called the right span main cable. The main span main cable is connected to the stiffening beam below via n suspenders, and the main span main cable located to the left of the stiffening beam is connected to the rock in the mountain below via j rock anchor cables. In this embodiment, j = 3 is preferred.

[0143] j rock anchor cables and n suspension cables together form n+j main span suspenders. Therefore, the main span cable is divided into the 1st segment, the 2nd segment, the 3rd segment, ..., the n+jth segment and the n+j+1th segment from left to right by the n+j main span suspenders.

[0144] The structural parameters of the j rock anchor cables are all optimized design parameters, including the initial tension P of the rock anchor cables. ri (1≤i≤j), cross-sectional area A of rock anchor cable rh and the cable spacing l1, l2, l3, ..., l of the rock anchor cables. j .

[0145] A method for determining the structural parameters of rock anchor cables in a single-tower ground-anchored suspension bridge includes the following steps.

[0146] Step 1: Establish the expression for the stress-free length of the main cable

[0147] Based on the forces acting on the main cable under constant load and the main cable equilibrium equation, the stress-free length S of the main cable under constant load on the left span is obtained. l The expression for the stress-free length S of the main cable under constant load in the i-th segment of the main span. i The expression for the dead load stress-free length S of the right span main cable r The expression is given by , where 1 ≤ i ≤ n + j + 1.

[0148] Based on the force and equilibrium equations of the main cable under live load, the stress-free length S of the main cable under live load on the left span is obtained. ql The expression for the stress-free length S of the main cable under live load in the i-th main span. qiThe expression for the live load stress-free length S of the right span main cable qr The expression.

[0149] (1) The above S i and S qi The establishment method, such as Figures 2 to 4 As shown, the preferred method includes the following steps.

[0150] Step 1-1a: Based on the location of the suspenders, the main span cable is considered as multiple catenary segments, and its main span cable profile y qi for:

[0151]

[0152] In the formula, c qi Let be the live load catenary parameters of the main cable in the i-th main span, and H qi q represents the horizontal force under live load on the main cable of the i-th main span. c Represents the self-weight of the main cable; a qi Let be the parameters of the live load catenary of the main cable in the i-th segment of the main span.

[0153] Step 1-2a: Determine the vertical height difference between adjacent main spans and main cables based on the vertical coordinates of the left and right ends of the main cable.

[0154]

[0155] In the formula, Δh qi Let l be the vertical height difference of the main cable in the i-th main span; qi Let be the horizontal projection length of the main cable in the i-th main span, and let be the basic quantities to be solved, totaling n+j+1.

[0156] Step 1-3a, establish H qi With H q1 recurrence relation

[0157] Based on the equilibrium equation of the suspension points on the main span cable and main span suspenders, the following relationship can be obtained:

[0158]

[0159] In the formula, H q(i-1) It represents the horizontal force under the live load of the main cable in the i-1th main span.

[0160] P rq(i-1) Let be the cable force of the (i-1)th main span suspender (rock anchor cable) under live load, and be the basic quantity to be solved.

[0161] P q(i-1) Let be the sling force of the (i-1)th main span suspender (sling) under live load, and let be the basic quantity to be solved.

[0162] β(i-1) Let be the inclination angle of the (i-1)th main span hanger under live load, and be the basic quantity to be solved.

[0163] H q1 For the fundamental quantity to be solved, when H q1 When determined, H can be obtained recursively from the above formula. qi (i≥2).

[0164] Steps 1-4a: Establish a qi With a q1 recurrence relation

[0165] Differentiating the equilibrium equations for the suspension points on the main span cable and main span suspenders, we obtain a qi The expression is:

[0166]

[0167] In the formula, c q(i-1) S and S are the live load catenary parameters of the main cable of the i-1th main span.

[0168] l q(i-1) Let be the horizontal projected length of the main cable under live load in the (i-1)th segment of the main span, and be the basic quantity to be solved.

[0169] a q(i-1) Here are the parameters for the live load catenary of the main cable in the (i-1)th main span; where a q1 For the fundamental quantity to be solved, when a q1 When determined, a can be obtained recursively using the above formula. qi (i≥2).

[0170] Steps 1-5a, stress-free length S of the main cable under live load in each main span qi The expression is:

[0171]

[0172] In the formula, E c and A c These represent the elastic model and cross-sectional area of ​​the main cable, respectively.

[0173] Similarly, the stress-free length S of the main cable under constant load in each main span is obtained. i The expression is:

[0174]

[0175] in:

[0176] In the formula, c i Let be the constant load catenary parameters of the main cable in the i-th main span.

[0177] li Let l be the horizontal projection length of the main cable under dead load in the i-th main span, which is a known design quantity; when 1≤i≤j, l i The distance between the i-th rock anchor cable and the (i-1)-th rock anchor cable or composite saddle is the optimized design quantity.

[0178] a i Let be the parameters of the main cable under constant load in the i-th main span, and be the known design parameters.

[0179] H i Let be the horizontal force under the constant load of the main cable in the i-th main span, which is a known quantity.

[0180] (2) Stress-free length S of the left and right span main cables under live load ql and S qr The process of creating the expression preferably includes the following steps.

[0181] Step 1-1b: Establish the left and right cross-main cable alignment y ql and y qr Specifically:

[0182]

[0183]

[0184] in:

[0185] In the formula, c ql and c qr These are the live load catenary parameters for the left and right spans of the main cable, respectively.

[0186] H ql and H qr The horizontal forces under live load on the left and right spans of the main cable are respectively, and both are fundamental quantities to be solved.

[0187] a ql and a qr The parameters for the live load catenary on the left and right spans of the main cable are respectively, and both are basic quantities to be solved.

[0188] Step 1-2b: Establish the vertical elevation difference Δh of the left span main cable. ql Vertical height difference Δh between the right span and the main cable qr The expression, specifically:

[0189]

[0190]

[0191] In the formula, l l and l r These are the horizontal projected lengths of the left and right spans of the main cable, respectively, and are known design quantities.

[0192] ΔB is the horizontal displacement of the composite saddle under live load, and is the fundamental quantity to be solved.

[0193] ΔD is the horizontal displacement of the bridge tower under live load, caused by the unbalanced horizontal forces on the main cables on both sides of the tower. According to the bridge tower equilibrium equation, then:

[0194] ΔD=(H q(n+j+1) -H qr ) / K D

[0195] Among them, H q(n+j+1) Let be the horizontal force of the main cable in the (n+j+1)th main span, and be the basic quantity to be solved.

[0196] K D Let be the horizontal stiffness of the bridge tower, a known quantity.

[0197] Step 1-3b: Establish the stress-free length S of the left and right spans of the main cable under live load. ql and S qr The expression is as follows:

[0198]

[0199]

[0200] Similarly, the stress-free lengths S of the left and right span main cables under constant load are obtained. l and S r The expression is as follows:

[0201]

[0202]

[0203] in:

[0204] In the formula, c l and c r These are the constant load catenary parameters for the left and right spans of the main cable, respectively.

[0205] H l and H r The horizontal forces under constant load on the left and right spans of the main cable are known quantities.

[0206] a l and a r The parameters for the live load catenary of the left and right main cables are respectively, and are all known design quantities.

[0207] Step 2: Establish the deflection expression for the stiffened beam.

[0208] Based on the bridge tower equilibrium equations, the expression for the stiffening girder deflection w(x) is obtained; where w(x) is the change in the stiffening girder's support reaction force F at point h. h F is a function of h, where 0 ≤ h ≤ n; when h = 0, F0 represents the support reaction force at the left end of the stiffened beam under live load, which is the fundamental quantity to be solved; when 1 ≤ h ≤ n, F h This represents the support reaction force of the stiffening beam located at the lower suspension point of the h-th cable under live load.

[0209] In this embodiment, as Figure 5 As shown, the stiffening girder is simplified to a simply supported beam. Let the horizontal length of the live load q directly above the stiffening girder be γ2, and the distance from the left starting end of the live load q to the left end of the stiffening girder be γ1. The vertical components of the main span hangers located on both sides of the left starting end of the live load q, from left to right, are F... z and F z+1 And F z and F z+1 The horizontal coordinates are X Gz and X G(z+1) The distance from the right end of the live load q to the right end of the stiffening girder is γ3. The vertical components of the main span hangers located on both sides of the right end of the live load q, from left to right, are F... m and F m+1 And F m and F m+1 The horizontal coordinates are X Gm and X G(m+1) Additionally, let the horizontal coordinate of the left end of the stiffening beam be X. G0 =0, the horizontal coordinate of the right end of the stiffening beam is X G(n+1) Then the expression for the stiffened beam deflection w(x) is:

[0210]

[0211]

[0212]

[0213]

[0214]

[0215]

[0216]

[0217] In the formula, X Gh Let h be the horizontal coordinate of the stiffening beam at any point h.

[0218] w a (x) represents the values ​​at F0 and F...z The deflection of the stiffened beam from point a-1 to point a in segment a.

[0219] X G(a-1) and X Ga These represent the horizontal coordinates of point a-1 and point a, respectively.

[0220] w z+1 (x) is F z The stiffening beam deflection up to the horizontal coordinate γ1 segment.

[0221] w z+2 (x) represents the horizontal coordinates from γ1 to F. z+1 Deflection of the stiffened beam segment.

[0222] w a+1 (x) is F z+1 To F m The deflection of the stiffened beam from point a-1 to point a in segment a.

[0223] w m+2 (x) is F m The stiffening beam deflection up to the horizontal coordinate segment γ1+γ2.

[0224] w m+3 (x) represents the horizontal coordinate γ1+γ2 to F. m+1 Deflection of the stiffened beam segment.

[0225] w a+2 (x) is F m To F n+1 The deflection of the stiffened beam from point a-1 to point a in segment a.

[0226] E g and I g These are the elastic modulus and moment of inertia of the stiffening beam, respectively.

[0227] F h The vertical component of the main span hanger at point h of the stiffening girder is preferably expressed as follows:

[0228] F h =2P q(j+h) cosβ (j+h)

[0229] In the formula, P q(j+h) Let be the sling force of the i=j+h main span suspender under live load, and let be the basic quantity to be solved.

[0230] β (j+h) Let be the cable inclination angle of the i=j+h main span suspender under live load, and let be the basic quantity to be solved.

[0231] The above C a,1 C a,2 C z+1,1C z+1,2 C z+2,1 C z+2,2 C a+1,1 C a+1,2 C m+2,1 C m+2,2 C m+3，1 C m+3,2 C a+2,1 and C a+2,2 These are all deflection equation coefficients. Specifically, based on the boundary conditions of the simply supported beam (such as the deflection at both the left and right ends of the stiffened beam being equal to 0) and the continuity of the stiffened beam's displacement and rotation (that is, the displacement and rotation of two adjacent segment nodes being equal), a corresponding set of quantitative equations is established and solved.

[0232] Step 3: Establish the expression for the beam end rotation angle: Differentiate the stiffening beam deflection w(x) from Step 2 to obtain the expression for the beam end rotation angle w′(x), specifically:

[0233]

[0234]

[0235]

[0236]

[0237]

[0238]

[0239]

[0240] In this embodiment, there are a total of 2(n+3) deflection coefficients in step 2 and rotation coefficients related to beam end rotation in step 3. These are obtained by solving a system of 2(n+3) equations, specifically:

[0241] The stiffened beam is equivalently simplified to a simply supported beam, therefore the boundary conditions are:

[0242] w1(0)=w n+3 (L G ) = 0

[0243] In the formula, L G This refers to the length of the stiffening beam.

[0244] Since the stiffened beam's displacement and rotation are continuous, the following equation can be obtained:

[0245]

[0246]

[0247]

[0248]

[0249]

[0250] The system of equations contains 2(n+3) unknowns and can be solved by the above 2(n+3) system of equations.

[0251] Step 4: Establish an analytical model of the rock anchor cable structure parameters, including 3(n+j)+10 governing equations, including the following steps.

[0252] Step 4-1: Based on the principle of conservation of stress-free length of each main cable segment under dead and live loads, establish n+j+3 governing equations regarding the stress-free length of the main cable, specifically:

[0253] S l =S ql

[0254] S i =S qi

[0255] S r =S qr

[0256] Step 4-2: Based on the deformation coordination of the rock anchor cable and the suspension cable, establish j deformation control equations for the rock anchor cable and n deformation control equations for the suspension cable, as follows:

[0257]

[0258]

[0259] in:

[0260]

[0261]

[0262]

[0263]

[0264]

[0265] In the formula, l rhqi and l rhi These represent the lengths of the rock anchor cable under live load and dead load, respectively.

[0266] l hqi and l hiThese represent the lengths of the slings under live load and dead load, respectively.

[0267] E rh and A rh These are the elastic modulus and cross-sectional area of ​​the rock anchor cable, respectively.

[0268] E h and A h These are the elastic modulus and cross-sectional area of ​​the sling, respectively.

[0269] and These are the horizontal and vertical coordinates of the suspension point on the i-th main span rod under live load.

[0270] and These are the horizontal and vertical coordinates of the ground anchorage point of the i-th rock anchor cable, respectively.

[0271] and These are the horizontal and vertical coordinates of the lower suspension point of the i-th main span hanger under live load.

[0272] X B and Y B These are the horizontal and vertical coordinates of the IP point of the composite cable saddle under constant load.

[0273] l qk Let be the horizontal projected length of the main cable under live load in the k-th main span, and be the basic quantity to be solved.

[0274] Δh qk Let be the vertical height difference of the main cable in the k-th main span.

[0275] X Gi Let be the horizontal coordinate of the lower suspension point of the i-th main span suspender;

[0276] w(X Gi ) is X Gi Deflection of the stiffening beam at point [point].

[0277] Step 4-3: Based on the coordinates of the upper and lower suspension points of the rock anchor cable and the suspender cable under live load, establish j control equations for the rock anchor cable inclination angle and n control equations for the suspender cable inclination angle, as follows:

[0278]

[0279]

[0280] Step 4-4: Based on the closure of the horizontal projection length of the main cable in the main span and the closure of the elevation difference between the main cables in the left span, main span, and right span, establish four main cable closure control equations, which are as follows:

[0281]

[0282] Δh ql =Y A -Y B

[0283]

[0284] Δh qr =Y D -Y E

[0285] In the formula, L C The total horizontal projection length of the main span cable under dead load; Y D Y represents the vertical coordinate of the IP point of the bridge tower saddle. A The vertical coordinate of the anchorage end of the main cable on the left side of the mountain; Y E The vertical coordinate of the ground anchorage end on the right side of the main cable; Δh ql The vertical elevation difference of the left span main cable; Δh qi Let Δh be the vertical elevation difference of the main cable in the i-th main span. qr This represents the vertical elevation difference of the right span of the main cable.

[0286] Steps 4-5: Based on the horizontal force, vertical force, and bending moment of the stiffening beam, establish three mechanical governing equations, as follows:

[0287]

[0288]

[0289]

[0290] In the formula, P qxh and P qyh These are the horizontal and vertical components of the force on the h-th sling, respectively.

[0291] Let h be the horizontal coordinate of the h-th sling.

[0292] L G This refers to the length of the stiffening beam.

[0293] Step 5: Solve for unknown fundamental quantities: 3(n+j)+7 fundamental quantities (l) related to the stress-free length in Step 1. q1 ~l q(n+j+1) a q1 a ql a qr H q1 H qr P rq1 ~P rqj ,P q(j+1) ~Pq(j+n) ,β1~β n+j ), F0 in step 2, ΔU in step 4-2, and F in step 4-5 n+1 A total of 3(n+j)+10 unknown basic quantities are formed. The initial amplitude of the structural parameters of j rock anchor cables is determined. The 3(n+j)+10 control equations established in step 4 are used. The least squares method is preferred to solve for the values ​​of the 3(n+j)+10 unknown basic quantities, thus obtaining the analytical model of the rock anchor cable structural parameters with definite basic quantities.

[0294] Step 6a, Optimize γ1 and γ2: Optimization is achieved by calculating the stiffening beam deflection, beam end rotation angle, and the maximum stress amplitude and corresponding most unfavorable live load condition in the suspenders and rock anchors. The specific steps include the following.

[0295] Step 6a-1: Treat the load arrangement parameters γ1 and γ2 as design variables, and let X = [γ1, γ2];

[0296] Step 6a-2: Treat the extreme value of stiffening beam deflection, beam end rotation angle and maximum stress amplitude in suspension cables and rock anchor cables as an optimization problem;

[0297] f1(x) = max(|w(x)|)

[0298] f2(x)=|w′(0)|

[0299] f3(x)=|w′(L G )|

[0300] f4(x)=[(max(P rqi -P ri )-min(P rqi -P ri )) / (2A rh ),(max(P qi -P i )-min(P qi -P i )) / (2A h )]

[0301] In the formula, f1(x) is the maximum deflection function of the stiffened beam.

[0302] f2(x) is the rotation angle function of the left end of the stiffening beam.

[0303] f3(x) is the rotation angle function of the right end of the stiffening beam.

[0304] f6(x) is a function of the maximum stress amplitude in the suspender cable and the rock anchor cable.

[0305] P rqi and P riThese represent the tension of the rock anchor cable under live load and dead load, respectively; P ri Also known as the initial tension of a rock anchor cable.

[0306] P qi and P i These represent the tension of the sling under live load and dead load, respectively.

[0307] A rh and A h These are the cross-sectional areas of the rock anchor cable and the suspension cable, respectively.

[0308] Step 6a-3: Initialize the Golden Eagle particle swarm X = [γ1, γ2], and update the load arrangement parameters using the Golden Eagle foraging strategy;

[0309] The direction in which the Golden Eagle chases its prey is...

[0310] V i =X f -X i

[0311] In the formula, V i X represents the direction in which the i-th golden eagle tracks its prey. f and X i These are the best positions for the eagle and the i-th eagle so far.

[0312] The cruising direction is calculated based on the hunting direction; therefore, the mathematical equation for the golden eagle's cruising motion is defined as follows:

[0313]

[0314]

[0315] In the formula, e k For target point E i The k-th element; o j The direction of the piling for the j-th element; H = [h1, h2, ..., h n [x1, x2, ..., x] is the normal vector; X = [x1, x2, ..., x] n [] is a vector of variables.

[0316] To improve search efficiency, the utilization rate of hunting and cruising is adjusted by setting and modifying parameters linearly during the search process.

[0317]

[0318]

[0319]

[0320] In the formula, r1 and r2 are random numbers in [0,1]; t is the current iteration number; T is the maximum iteration number; and These are the initial and final values ​​of the hunting process; and These are the initial and final values ​​of the cruise trend, respectively.

[0321] Step 6a-4: Substitute the load arrangement parameters γ1 and γ2 into the analytical model to obtain the corresponding stiffening beam deflection, beam end rotation angle and cable force increment.

[0322] Step 6a-5: Set the maximum number of iterations and determine whether convergence has occurred. If convergence has occurred, end the optimization and obtain the peak value of the verification index and the corresponding worst live load condition; otherwise, return to step 6a-3 and update the particle swarm.

[0323] Step 6: Optimization of structural parameters for j-root rock anchor cables

[0324] Step 6-1: Select the initial tension, cross-sectional area, and cable spacing of the j-th rock anchor cable as design variables, and let X = [P ri A rhi ,l i ], i = 1, ... j.

[0325] Step 6-2: Establish the multi-objective function Minimize F

[0326] For the analytical model of rock anchor cable structure parameters with definite basic quantities obtained in step 5, a multi-objective optimization function Minimize F is established, the specific expression of which is:

[0327] Minimize F={f1(x),f2(x),f4(x)}

[0328] Step 6-3: Initialize the Golden Eagle Particle Swarm X = [P ri A rhi ,l i ], i=1,…j, the golden eagle foraging strategy is used to update the anchor cable structure parameters;

[0329] Step 6-4: Based on step 6a, solve for the most unfavorable load condition parameters γ1 and γ2, and the corresponding multi-objective function Minimize F.

[0330] Step 6-5: The multi-objective golden eagle foraging strategy is used for updating. The design variable update formula is the same as in step 6, except that the Pareto front needs to be stored. The multi-objective golden eagle optimization algorithm is a combination of the above single-objective optimization and three additional concepts: (1) external archive, (2) prey selection, and (3) multi-objective prey selection. The basic idea is to store promising non-dominant solutions in the external archive and update them during the optimization algorithm to eventually reach the optimal Pareto front.

[0331] Step 6-6: Set the maximum number of iterations and determine if convergence has occurred. If it has, end the optimization and obtain the optimal rock anchor cable structure parameters; otherwise, return to step 6-4 and update the particle swarm.

[0332] This invention is applicable to calculating the deformation of suspension bridges under live loads where the deformation of the main cable and stiffening girder differs significantly, particularly for single-tower ground-anchored suspension bridges where the projected length of the main cable differs from that of the stiffening girder. This invention not only quickly identifies the most unfavorable load condition for verifying parameters but also optimizes the structural parameters of the rock anchor cables, thereby improving the safety of suspension bridges. The method has high computational efficiency and provides accurate and reliable results, making it suitable for the preliminary design of the rock anchor cable structural parameters of single-tower ground-anchored suspension bridges.

[0333] The preferred embodiments of the present invention have been described in detail above. However, the present invention is not limited to the specific details in the above embodiments. Within the scope of the technical concept of the present invention, various equivalent transformations can be made to the technical solutions of the present invention, and these equivalent transformations all fall within the protection scope of the present invention.

Claims

1. A method for determining the structural parameters of rock anchor cables in a single-tower ground-anchored suspension bridge, characterized in that: Includes the following steps: Step 1: Establish the expression for the stress-free length of the main cable: The left side of the main cable is anchored in the rock of the mountain after passing through the composite saddle, forming the mountain anchorage end; the right side of the main cable is anchored to the ground after passing through the bridge tower, forming the ground anchorage end; among them, the main cable located between the composite saddle and the bridge tower is called the main span main cable, the main cable located between the rock anchorage end and the composite saddle is called the left span main cable, and the main cable located between the bridge tower and the ground anchorage end is called the right span main cable; the main span main cable is connected to the stiffening beam below through n suspenders, and the main span main cable located to the left of the stiffening beam is connected to the rock in the mountain below through j rock anchors; the j rock anchors and n suspenders together constitute n+j main span suspenders, so the main span main cable is divided into the 1st segment, the 2nd segment, the 3rd segment, ..., the n+jth segment and the n+j+1th segment from left to right by the n+j main span suspenders; The structural parameters of the j rock anchor cables are all optimized design parameters, including the initial tension P of the rock anchor cables. ri Rock anchor cable cross-sectional area A rh and the spacing of the rock anchor cables. ; Based on the forces acting on the main cable under constant load and the main cable equilibrium equations, the stress-free length of the main cable under constant load on the left span is obtained. The expression for the stress-free length of the main cable under constant load in the i-th segment of the main span. The expression and the stress-free length of the right span main cable under dead load The expression; Based on the forces acting on the main cable under live load and the main cable equilibrium equations, the stress-free length of the main cable under live load on the left span is obtained. The expression for the stress-free length of the main cable under live load in the i-th main span. The expression and the stress-free length of the right span main cable under live load The expression; where 1≤i≤n+j+1; , , , , and There are a total of 3 (n+j) + 7 basic quantities to be solved; Includes the optimized design of the rock anchor cable spacing. ; Step 2: Establish the expression for the stiffening girder deflection: Based on the bridge tower equilibrium equation, the deflection of the stiffening girder is obtained. The expression; where, For the change in support reaction force of the stiffened beam at point h The function, 0≤h≤n; when h =0, Let represent the support reaction force at the left end of the stiffened beam under live load, which is a fundamental quantity to be solved; when 1≤h≤n, This represents the support reaction force of the stiffening beam located at the lower suspension point of the h-th cable under live load; Step 3: Establish the expression for beam end rotation angle: For the stiffening beam deflection in Step 2... By taking the derivative, the beam end rotation angle is obtained. The expression; Step 4: Establish an analytical model of the rock anchor cable structure parameters, including 3(n+j) + 10 governing equations. The specific method for establishing the model is as follows: Step 4-1: Based on the principle of conservation of stress-free length of each main cable segment under dead load and live load, establish n+j+3 control equations concerning the stress-free length of the main cable. Step 4-2: Based on the deformation coordination of the rock anchor cable and the suspender cable, establish j rock anchor cable deformation control equations and n suspender cable deformation control equations; among them, the j rock anchor cable deformation control equations include the optimized design quantity, the initial tension P of the rock anchor cable. ri The cross-sectional area A of the rock anchor cable rh The n control equations for cable deformation include the longitudinal rigid body displacement ∆U of the stiffening beam, which is the basic quantity to be solved. Step 4-3: Based on the coordinates of the upper and lower suspension points of the rock anchor cable and the suspender cable under live load, establish j control equations for the rock anchor cable inclination angle and n control equations for the suspender cable inclination angle; Step 4-4: Establish four main cable closure control equations based on the closure of the horizontal projection length of the main cable in the main span, and the closure of the height difference between the main cables in the left span, main span, and right span. Steps 4-5: Based on the horizontal force, vertical force, and bending moment of the stiffening beam, establish three governing equations; among them, the three governing equations include the support reaction force F at the right end of the stiffening beam under live load. n+1 , where is the fundamental quantity to be solved; Step 5, Solve for unknown fundamental quantities: the 3(n+j)+7 fundamental quantities related to the stress-free length in Step 1, and the quantities in Step 2... ∆U in step 4-2 and F in step 4-5 n+1 A total of 3(n+j)+10 unknown basic quantities are formed; the structural parameters of j rock anchor cables are initially assigned, and the 3(n+j)+10 control equations established in step 4 are used to solve for the values ​​of the 3(n+j)+10 unknown basic quantities, thus obtaining the analytical model of rock anchor cable structural parameters with definite basic quantities. Step 6: Structural parameter optimization of j rock anchor cables: For the analytical model of rock anchor cable structural parameters with definite basic quantities obtained in Step 5, a multi-objective optimization function is established. By solving Thus, the optimized rock anchor cable structure parameters are obtained; among them, The expression is: ； in: ； ； ； In the formula, This is the maximum deflection function of the stiffened beam; The rotation angle function of the left end of the stiffening beam; It is a function of the maximum stress amplitude in the suspenders and rock anchors; and These represent the tensile forces of the rock anchor cable under live load and dead load, respectively. Also known as the initial tension of rock anchor cables, optimized design quantity; and These represent the tension of the sling under live load and dead load, respectively. and These are the cross-sectional areas of the rock anchor cable and the suspension cable, respectively.

2. The method for determining the structural parameters of a rock anchor cable in a single-tower ground-anchored suspension bridge according to claim 1, characterized in that: In step 1, , , and The expression is: ； ； ； ； in: ， ， ， ； ； In the formula, and These are the constant load catenary parameters for the left and right spans of the main cable, respectively. and The horizontal forces under dead load on the left and right spans of the main cable are known quantities. and The parameters for the live load catenary of the left and right main cables are respectively, and both are known design quantities. and These are the live load catenary parameters for the left and right spans of the main cable, respectively. The weight of the main cable is known. and These are the horizontal forces under live loads on the left and right spans of the main cable, respectively, and are both fundamental quantities to be solved. and These are the horizontal projected lengths of the left and right spans of the main cable, respectively, and the design quantities are known. Let be the horizontal displacement of the composite saddle under live load, and let be the basic quantity to be solved. This represents the horizontal displacement of the bridge tower under live load. The horizontal stiffness of the bridge tower is a known quantity; and These are the parameters of the live load catenary on the left and right spans of the main cable, respectively, and are all basic quantities to be solved. and The elastic model and cross-sectional area of ​​the main cable are given, along with known quantities. Let be the horizontal force of the main cable in the (n+j+1)th main span, and be the basic quantity to be solved.

3. The method for determining the structural parameters of the rock anchor cable of a single-tower ground-anchored suspension bridge according to claim 2, characterized in that: In step 1, and The expression is: ； ； in: ， ； ； ； In the formula, Let be the constant load catenary parameters of the main cable in the i-th main span; and These are the live load catenary parameters for the main span of the i-1 segment and the i-th segment, respectively; Let be the horizontal projected length of the main cable under dead load in the i-th main span, which is a known design quantity; when 1≤i≤j The distance between the i-th rock anchor cable and the (i-1)-th rock anchor cable or composite saddle is the optimized design amount; and ... Let be the parameters of the main cable under constant load in the i-th main span, and be the known design parameters. and These are the live load catenary parameters for the main cable of the i-th segment and the (i-1)-th segment, respectively; where, These are the fundamental quantities to be solved; Let be the horizontal force under the dead load of the main cable in the i-th main span, which is a known quantity; and These are the horizontal forces under live load on the main cable of the main span in segments i and i-1, respectively; where... These are the fundamental quantities to be solved; Let be the sling force of the (i-1)th main span suspender under live load, and let be the basic quantity to be solved. Let be the sling force of the (i-1)th main span suspender under live load, and let be the basic quantity to be solved. Let be the inclination angle of the (i-1)th main span hanger under live load, and be the basic quantity to be solved.

4. The method for determining the structural parameters of a rock anchor cable in a single-tower ground-anchored suspension bridge according to claim 1, characterized in that: In step 2, the stiffened beam is simplified to a simply supported beam, and a live load is assumed. The horizontal arrangement length directly above the stiffening girder is... Live load The distance from the left starting end to the left end of the stiffening girder is Located in live load The vertical components of the main span suspension rods on both sides of the starting end on the left are as follows, from left to right: and ,and and The horizontal coordinates are respectively and Live load The distance from the right end of the beam to the right end of the stiffening beam is Located in live load The vertical components of the main span suspension rods on both sides of the right end are, from left to right, respectively. and ,and and The horizontal coordinates are respectively and Additionally, let the horizontal coordinate of the left end of the stiffening beam be... =0, the horizontal coordinate of the right end of the stiffening beam is Then the stiffening beam deflection The expression is: ； ； ； ； ； ； ； In the formula, For stiffening beams at any point Horizontal coordinates; In order to be in and The deflection of the stiffened beam from point a-1 to point a in segment a; and Let a-1 and a represent the horizontal coordinates of points a-1 and a, respectively. for To horizontal coordinate The deflection of the stiffened beam segment; horizontal coordinates to The deflection of the stiffened beam segment; for to The deflection of the stiffened beam from point a-1 to point a in segment a; for To horizontal coordinate The deflection of the stiffened beam segment; horizontal coordinates to The deflection of the stiffened beam segment; for to The deflection of the stiffened beam from point a-1 to point a in segment a; , , , , , , , , , , , , and These are all deflection equation coefficients, specifically obtained by establishing a corresponding set of quantitative equations based on the boundary conditions of the simply supported beam and the continuity of the stiffened beam's displacement and rotation, and then solving them.

5. The method for determining the structural parameters of a rock anchor cable in a single-tower ground-anchored suspension bridge according to claim 4, characterized in that: In step 2, The expression is: ； In the formula, Let be the sling force of the i=j+h main span suspender under live load, and be the basic quantity to be solved; Let be the cable inclination angle of the i=j+h main span suspender under live load, and let be the basic quantity to be solved.

6. The method for determining the structural parameters of the rock anchor cable structure of a single-tower ground-anchored suspension bridge according to claim 1, characterized in that: In step 4, there are 3(n+j) + 10 governing equations, specifically: Step 4-1 and the control equations for the stress-free length of the main cable (n+j+3) are as follows: ； ； ； Step 4-2, the j rock anchor cable deformation control equations and the n suspension cable deformation control equations are as follows: ； ； in: ； ； ； ； ； In the formula, and These represent the lengths of the rock anchor cable under live load and dead load, respectively. and These are the lengths of the slings under live load and dead load, respectively. and These are the elastic modulus and cross-sectional area of ​​the rock anchor cable, respectively. and These are the elastic modulus and cross-sectional area of ​​the sling, respectively. These are the horizontal and vertical coordinates of the suspension point on the i-th main span hanger under live load, respectively; These are the horizontal and vertical coordinates of the ground anchorage point of the i-th rock anchor cable, respectively. and These are the horizontal and vertical coordinates of the lower suspension point of the i-th main span hanger under live load, respectively; and These are the horizontal and vertical coordinates of the IP point of the composite cable saddle under constant load, respectively. Let be the horizontal projected length of the main cable under live load in the k-th main span, and be the basic quantity to be solved. Let be the vertical height difference of the main cable in the k-th main span; Let be the horizontal coordinate of the lower suspension point of the i-th main span suspender; for Deflection of the stiffening beam at the point; Step 4-3, the control equations for the j rock anchor cable inclination angles and the control equations for the n suspension cable inclination angles are as follows: ； ； Step 4-4, the four main cable closure control equations are as follows: ； ； ； ； In the formula, The total horizontal projection length of the main span cable under constant load; The vertical coordinates of the IP point of the bridge tower saddle; The vertical coordinates of the anchorage end of the main cable on the left side of the mountain. The vertical coordinates of the ground anchorage end on the right side of the main cable; The vertical height difference of the left span main cable; Let be the vertical height difference of the main cable in the i-th main span; The vertical elevation difference of the right span main cable; Steps 4-5, the three governing equations are as follows: ； ； ； In the formula, and These are the horizontal and vertical components of the force on the h-th sling, respectively. Let h be the horizontal coordinate of the h-th sling; This refers to the length of the stiffening beam.

7. The method for determining the structural parameters of a rock anchor cable in a single-tower ground-anchored suspension bridge according to claim 1, characterized in that: In step 6, the multi-objective optimization function The solution method is the multi-objective Golden Eagle optimization algorithm.

8. The method for determining the structural parameters of a rock anchor cable in a single-tower ground-anchored suspension bridge according to claim 1, characterized in that: In step 5, the values ​​of 3(n+j)+10 unknown basic quantities are obtained by using the least squares method.

9. The method for determining the structural parameters of a rock anchor cable in a single-tower ground-anchored suspension bridge according to claim 4, characterized in that: In step 6, before optimizing the structural parameters of the j-th rock anchor cable, the distance from the left end of the live load to the left end of the stiffening beam is obtained by calculating the stiffening beam deflection, beam end rotation angle, maximum stress amplitude in the suspenders and rock anchor cables, and the corresponding most unfavorable live load condition. and the length of the live load horizontal arrangement is Then, based on the solution and Optimize the structural parameters of j rock anchor cables.

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