A refined calculation method for the cable system of a suspension bridge

Abstract

The present invention belongs to the technical field of bridge design and construction monitoring, and proposes a refined calculation method for the cable system of a suspension bridge. Multiple parameters such as the pre-deviation of the saddle, the pre-deviation of the dispersion saddle, the compression deformation of the tower, the change in the tangent point of the wire strand at the saddle, and the elongation of the wire strand in the saddle are placed in the same calculation model. The final state of the main cable, the state of the empty cable, and the construction state under any construction load are determined by the progressive point-by-point balance iterative main cable shape-finding calculation method, gradually improving the calculation accuracy, with fast calculation speed and easy convergence. The present invention can simulate the influence of the horizontal displacement of the saddle and the change in the tangent point of the main cable on the main cable alignment, calculate the influence of the friction between the wire strand and the saddle groove, correct the elongation of the main cable in the saddle groove, and obtain a more accurate final alignment of the main cable, the pre-deviation of the saddle, and the pre-deviation of the dispersion saddle.

Description

Technical Field

[0001] The present invention belongs to the technical field of bridge design and construction monitoring, and relates to a refined calculation method for the cable system of a suspension bridge. Background Art

[0002] In the two states of the suspension bridge, namely the state of the bare cable and the state of the completed bridge, there are significant differences in the alignment and cable force of the main cable. As the construction process progresses, the weight of the stiffening girder is gradually applied to the main cable, resulting in a significantly greater increase in the cable force of the main cable in the main span than that in the side span. This uneven growth causes the horizontal component force of the main cable in the main span to be significantly greater than that in the side span, thereby generating a large bending moment on the cable tower and threatening its safety. Generally, the saddle is pushed to adjust the relative span of the main cables in the main span and the side span, and thus balance the horizontal component forces of the main cables on both sides of the cable tower. Therefore, accurately setting the pre-offset of the saddle not only concerns the structural safety but also directly affects the final alignment of the main cable. The saddle groove is usually designed as an arc, and the main cable and the saddle form two asymmetric tangent points on both sides. During the pushing process, the saddle translates relative to the cable tower, causing these two tangent points to change accordingly, which greatly increases the complexity of the main cable alignment calculation. Currently, the methods for calculating this complex process mainly include: Fan Lichu et al. in "Research on the Fine Algorithm of Structural Erection Parameters for Long-Span Suspension Bridges" use bar elements or beam elements to simulate the saddle structure and its pushing process; Li Chuanxi in "Nonlinear Fine Calculation Theory and Its Application for Hybrid Girder Suspension Bridges" uses a multi-segment circular arc composite curve saddle to solve the offset. These methods do not comprehensively consider various factors affecting the main cable alignment, such as the pre-offset of the saddle, the pre-offset of the dispersion saddle, the compression deformation of the cable tower, the change in the tangent point of the cable strands at the saddle, and the elongation of the cable strands in the saddle during the calculation. Therefore, the error of the main cable alignment of the completed bridge obtained is relatively large; some methods use the catenary for iterative calculation, and the calculation process is relatively complex and difficult to implement when considering multiple influencing factors.

[0003] Therefore, the refined calculation method of the present invention places multiple parameters, such as the pre-offset of the saddle, the pre-offset of the dispersion saddle, the compression deformation of the cable tower, the change in the tangent point of the cable strands at the saddle, and the elongation of the cable strands in the saddle, in the same calculation model, and determines the completed bridge state, the bare cable state, and the construction state under any construction load of the main cable through the progressive point-by-point balanced iterative main cable form-finding calculation method. Summary of the Invention

[0004] The purpose of the present invention is to provide a refined calculation method for the cable system of a suspension bridge, which can simulate the influence of the horizontal displacement of the saddle and the change of the main cable tangent point on the main cable alignment, calculate the influence of the friction force between the cable strands and the saddle groove, correct the elongation of the main cable in the saddle groove, and obtain a more accurate main cable alignment of the completed bridge and the saddle offset.

[0005] During the design process of a suspension bridge, designers will first determine key parameters such as the structural alignment, member dimensions, and material types in the completed bridge state, and use the finite element method for preliminary analysis and calculation to ensure that the structure meets the relevant code requirements and determine the internal forces and preliminary alignment in the completed bridge state accordingly. Based on this, the present invention conducts more refined calculations on the cable system, aiming to accurately obtain crucial construction control parameters such as the unloaded cable alignment and saddle offset, providing more detailed and accurate guidance for the construction of the suspension bridge.

[0006] The technical solution of the present invention:

[0007] A refined calculation method for the cable system of a suspension bridge, the steps are as follows:

[0008] Given the main span length, side span length, anchor span length, tower height, anchor span height, main cable sag, the abscissa and hanger force (including the weight of the cable clamp) of the hangers, the area, elastic modulus, and unit length weight of the main cable. Determine the completed bridge state of the main cable through the progressive point-by-point balance iterative main cable form-finding calculation method, including the following three major steps:

[0009] (1) Establish a main cable calculation model with the IP points of the saddle and the dispersion saddle as the reference. The main cable is only subjected to the action of the hanger force (including the weight of the cable clamp). Using the point-by-point balance iterative method, first iterate each node of the main span, then each node of the side span, and then each node of the anchor span one by one. With the equality of the horizontal component force of the main cable as the control principle and the main span sag reaching the design value as the convergence condition, conduct the form-finding calculation of the main cable to obtain the completed bridge state C1;

[0010] (2) Calculate the self-weight of the main cable based on the length of each section of the main cable obtained in the C1 completed bridge state, and superimpose it with the hanger force as the new node load, and then repeat the calculation in step (1) to obtain the completed bridge state C2;

[0011] (3) Calculate the coordinates of the tangent points of the main cable at the saddle and the dispersion saddle respectively, and re-determine parameters such as the span of each span and the sag of the main cable based on this, establish a new main cable calculation model, use the load in step (2), and conduct the point-by-point balance iterative calculation again to obtain the completed bridge state C3.

[0012] Complete the form-finding calculation of the main cable in the construction state (including the unloaded cable state and under any construction load) through the following steps:

[0013] (1) Estimate the initial value of the saddle pre-offset d0, calculate the length of each section of the main cable in the main span in the unloaded cable state according to the strain of the main cable obtained in the C1 completed bridge state. The self-weight of the unloaded cable is calculated as a concentrated force at the node. By changing the size of the saddle pre-offset d0 through multiple cycles until the two endpoints of the main cable reach the design elevation;

[0014] (2) Calculate the lengths of each section of the main cable in the bare-cable state of the anchor span according to the strain of the main cable obtained in the completed-bridge state of C1. The self-weight of the bare cable is calculated as a concentrated force at the nodes. By changing the pre-deviation θ of the dispersion saddle through multiple cycles until the endpoints of the main cable in the anchor span reach the designed elevation;

[0015] (3) Calculate the lengths of each section of the main cable in the side span in the bare-cable state according to the strain of the main cable obtained in the completed-bridge state of C1. The self-weight of the bare cable is calculated as a concentrated force at the nodes. By changing the pre-deviation d0 of the saddle through multiple cycles until the endpoints of the main cable in the side span reach the designed elevation (considering the compression deformation of the pylon);

[0016] (4) Considering the friction between the main cable and the saddle groove in the saddle and the dispersion saddle, calculate the coordinates of the main cable tangent point after the elastic elongation of the main cable in the circular arc part of the saddle groove causes a change. After correcting the tangent point coordinates, repeat steps (1)-(3) to obtain the alignment of the main cable in the bare-cable state;

[0017] (5) During a certain construction process, the load of the stiffening girder is applied to the corresponding main cable nodes in the form of nodal forces through the suspenders. The construction load is denoted as: F i , (i = 1, 2,..., n), and the F i value of the nodes without the construction load applied is 0. Calculate according to steps (1)-(3) to obtain the preliminary alignment of the main cable in the construction state and the horizontal component force H2 of the main cable. This state is called the S1 construction state;

[0018] (6) Use the formula ε i = (H0 - H2) / EA / cosθ i-1 to calculate the strain of the main cable. Use the formula α i = α i-1 + P i / H2 for the main cable angle coefficient, and then recalculate the alignment of the main cable, the offset of the saddle, and the offset of the dispersion saddle on the premise of horizontal force balance of each span according to steps (1)-(3).

[0019] Advantages of the present invention:

[0020] (1) Adopt a progressive point-by-point balance iteration method for main cable shape finding calculation to determine parameters such as the alignment, cable force, pre-deviation of the saddle, and pre-deviation of the dispersion saddle of the main cable, gradually improve the calculation accuracy, and have a fast calculation speed and easy convergence.

[0021] (2) Place multiple parameters such as the pre-deviation of the saddle, the pre-deviation of the dispersion saddle, the compression deformation of the pylon, the change in the tangent point of the wire strand at the saddle, and the elongation of the wire strand in the saddle in the same calculation model, which can simulate the influence of the horizontal displacement of the saddle and the change in the main cable tangent point on the main cable alignment, calculate the influence of the friction between the wire strand and the saddle groove, correct the elongation of the main cable in the saddle groove, and obtain a more accurate completed-bridge alignment of the main cable and the offset of the saddle.

[0022] (3) It can not only calculate the alignment of the main cable in the completed bridge state and the bare cable state, but also calculate the construction state under any construction load, providing a new algorithm for the construction control of suspension bridges. Description of the Drawings

[0023] Figure 1 It is a structural schematic diagram of the cable system of a suspension bridge.

[0024] Figure 2 It is a detailed drawing of the key points of the saddle and the cable spreader saddle.

[0025] Figure 3 It is a calculation flow chart.

[0026] In the figure: 1 main cable; 2 saddle. Detailed Implementation Manner

[0027] The following combines the technical solution and the drawings to detail the specific implementation manner of the present invention.

[0028] A refined calculation method for the cable system of a suspension bridge is as follows:

[0029] Main design parameters: main span S m , side span S s , anchor span S a , nodes are divided on the main cable at each cable clamp position, and an appropriate number of nodes are divided on the main cable without cable clamps. The abscissas of the nodes in each span are represented as x i (i = 0, 1,..., n), where n represents the maximum number of nodes in the span, tower height c, anchor span height h, main cable sag f, and the forces of each node in the completed bridge state are P i , (i = 1, 2,..., n), mainly the hanger force (including the weight of the cable clamp. In the following text, the hanger force includes the weight of the cable clamp), and the node force is 0 where there is no hanger. The area, elastic modulus, and unit length weight of the main cable are: A, E, q. Refer to the structural schematic diagram of the cable system and the detailed drawing of the key points of the saddle and the cable spreader saddle Figure 1 、 Figure 2 , and perform progressive point-by-point balance iterative calculation on the alignment of the main cable in the completed bridge state, the bare cable state, and the construction alignment under any construction load according to the following steps.

[0030] S1. Perform the main span calculation: Substitute the hanger force of the main span and the abscissa of the position of each node in each span into formula (1) to calculate the vertical force R of the IP point of the main cable in the main span y ;

[0031]

[0032] In the formula, P i is the force of each node in the completed bridge state; S mis the main span length; x i is the abscissa of each span node, i = 0, 1, …, n, where n is the maximum number of nodes within the span; P i is the force of each node in the completed bridge state, i = 1, 2, …, n;

[0033] S2. Estimate the horizontal component force H0 of the main cable force of the main span, and substitute it into formula (2) to find the angle coefficient α0 of the main cable at the IP point of the main span main cable:

[0034] α0 = -R y / H0 (2)

[0035] S3. Use formulas (3) and (4) to calculate the y coordinates of each node of the main span main cable and the corresponding angle coefficients;

[0036] y i = y i-1 + α i-1 (x i - x i-1 ) (3)

[0037] α i = α i-1 + P i / H0 (4)

[0038] In the formula, i = 1, 2, …, n;

[0039] S4. Calculate the deviation dy of the main cable mid-span alignment from the mid-span sag f of the completed bridge state:

[0040] dy = f - |y middle |

[0041] In the formula, f is the main cable sag; y middle is the y coordinate of the main cable mid-span. If the absolute value of dy is greater than 0.5 mm, then change the magnitude of H0, and return to S2 for recalculation until |dy| < 0.5 mm is satisfied, and then exit the main span alignment iterative calculation;

[0042] S5. Conduct side span calculation: Substitute the side span hanger force and the abscissa of the location of each span node into formula (1), and replace the main span length S m in formula (1) with the side span length S s , and calculate the vertical force Rs of the main cable at the IP point of the saddlespreader; y ;

[0043] S6. Take the horizontal component force H0 of the main cable force of the main span obtained by the S1 - S4 iterative calculation as the equilibrium condition, and substitute it into formula (5) to find the angle coefficient α s0 of the main cable at the IP point of the side span:

[0044] α s0 = Rsy / H s0 (5)

[0045] S7. Similarly, using formulas (3) and (4), calculate the y-coordinates of each node of the side-span main cable and the corresponding angle coefficients;

[0046] S8. Calculate the deviation dys between the coordinates of the end point of the side-span main cable and the as-built state:

[0047] dys = c - |y s |

[0048] where c is the height of the cable tower; y s is the y-coordinate of the end point of the side-span main cable. If the absolute value of dys is greater than 0.5 mm, then change the value of α s0 and return to S7 for recalculation until |dys| < 0.5 mm is satisfied, and then exit the linear calculation of the side-span main cable;

[0049] S9. Conduct the anchor-span calculation: According to the calculation steps of S5 - S7, perform iterative calculations on the anchor span to obtain the y-coordinates of each node of the anchor-span main cable and the corresponding angle coefficients;

[0050] S10. Calculate the deviation dya between the coordinates of the end point of the anchor-span main cable and the as-built state:

[0051] dya = h - |y a |

[0052] where h is the height between the theoretical anchorage point of the main cable and the IP point of the saddle; y a is the y-coordinate of the end point of the anchor-span main cable. If the absolute value of dya is greater than 0.5 mm, then change the value of α a0 and return to S9 for recalculation until |dya| < 0.5 mm is satisfied, and then exit the linear calculation of the anchor-span main cable to obtain the as-built state of C1;

[0053] S11. Calculate the length l i of each section of the main cable and the strain ε i : Calculate the self-weight of each section of the main cable based on the calculated length of each section of the main cable, superimpose it with the hanger force according to formula (6) to obtain a new hanger force, and then repeat the calculation process of S1 - S10 to obtain the as-built state of C2;

[0054] P i = P i + 0.5q(l i + l i+1 ) (6)

[0055] where q is the weight of the main cable per unit length;

[0056] S12. Respectively substitute the coordinates (x a , y a ) and (x b , y b ) of the first nodes of the main span and side span close to the IP point of the saddle in the C2 completed bridge state obtained in S11 into formulas (7)-(12) to calculate the coordinates of the saddle center as (x c , y c ), the coordinates of the saddle arc tangent points as (xt1, yt1), (xt2, yt2), and the outside arc chord lengths as L a , L b ;

[0057]

[0058] In the formula, r is the theoretical radius of the saddle;

[0059] S13. Assume that the friction coefficient between the main cable and the saddle groove is μ, and calculate the coordinates of the first pair of nodes on both sides of the saddle and the saddle tangent points after the elastic elongation of the main cable in the inner arc part of the saddle according to formulas (13)-(15);

[0060]

[0061]

[0062] In the formula, A is the cross-sectional area of the main cable, and E is the elastic modulus;

[0063] S14. Respectively substitute the coordinates (x a1 , y a1 ) and (x b1 , y b1 ) of the first nodes of the side span and anchor span close to the IP point of the dispersion saddle in the C2 completed bridge state obtained in step S11 into formulas (7)-(12) to calculate the coordinates of the dispersion saddle center as (x c1 , y c1 ), the coordinates of the dispersion saddle arc tangent points as (xt3, yt3), (xt4, yt4), and the outside arc chord lengths L a1 , L b1 ;

[0064] S15. Adopt the method in S13 to calculate the change of the tangent points on both sides of the dispersion saddle considering the influence of friction;

[0065] S16. Correct the coordinates of the two end nodes of the main cable in the main span and the main span span according to formula (16);

[0066]

[0067] S17. In the coordinate system with the IP point of the saddle as the origin, correct the coordinates of the two end nodes of the main cable in the side span, the side span length, and the tower height according to formula (17);

[0068]

[0069] S18. Correct the coordinates of the end node of the main cable in the anchor span at the saddle, the anchor span length, and the anchor span height according to formula (18);

[0070]

[0071] S19. Adopt the corrected design parameters and recalculate the steps from S1 - S11 to obtain the C3 completed bridge state considering the saddles and the dispersion saddles;

[0072] S20. Calculate the state of the main span with the cable unloaded. Assume a pre - offset of the saddle d0, and the main span length becomes S m = S m + 2d0; The self - weight of the unloaded cable is calculated as a concentrated force at the nodes, Pc i = 0.5q(l i + l i+1 ). Estimate the initial value of the horizontal force of the unloaded cable H1, and calculate the vertical force R y1 ;

[0073] S21: Substitute H1 for H0 into formula (2) to find the angle coefficient α of the main cable at the IP point of the main span 01 ;

[0074] S22: Use formulas (19) and (20) to calculate the main cable strain and the main cable angle coefficient:

[0075] ε i = H0 / EA / cosθ i-1 (19)

[0076] α i = α i-1 + Pc i / H1 (20)

[0077] S23: Assume an initial value of α c0 and use formulas (21) and (22) to calculate the main cable coordinates:

[0078] x i+j = x i+j-1 + dx i (21)

[0079] y i+j = y i+j-1 + dy i (22)

[0080] Among them, dx i , dy i respectively represent the displacement of the main cable node and are calculated using the following formula;

[0081] dx i =(1 - ε i+j )l i+j cos(atan(α i+j-1 ))

[0082] dy i =(1 - ε i+j )l i+j sin(atan(α i+j-1 ))

[0083] Among them, i = 1, 2,..., n; j = 0, 1,..., k + 1 - i. The length and strain of each section of the main cable adopt the results obtained in the last round of S11 step;

[0084] S24. If |y n - yt1| < 0.5mm is not satisfied, modify α0 and return to S23 for recalculation until |y n - yt1| < 0.5mm, and the iteration ends;

[0085] S25. Compare the difference d = x n - S m - xt1 between the assumed saddle pre - offset d0 and the actual displacement of the main cable end point obtained by calculation. If d < 0.5mm is satisfied, end the loop. Otherwise, change the magnitude of d0 and return to S20 for iterative calculation until d < 0.5mm, and the two ends of the main cable reach the design elevation;

[0086] S26. Calculate the state of the cable in the anchor span. According to the length of each section of the main cable obtained in the last round of S11, calculate the alignment of the main cable using the method of S23. Since the horizontal component forces of the main cables in the main span, side span, and anchor span are equal, H1 is the same when calculating in each span;

[0087] S27. The rotation of the dispersion saddle by θ causes changes in the coordinates (xt3, yt3), (xt4, yt4) of the tangent points of the dispersion saddle arc, and the displacement amounts are dx3, dy3, dx4, dy4 respectively;

[0088]

[0089]

[0090] In the formula, w is the vertical height from the IP point of the dispersion saddle to the rotation center of the dispersion saddle support;

[0091] If |y n-yt4 + dy4| < 0.5 mm, update the coordinates of the tangent point of the saddle circular arc using formula (25), modify α0, and return to S23 for recalculation until |y n -yt4 + dy4| < 0.5 mm, end the iteration; obtain the displacement d1 of the IP point of the saddle under the balanced state of the anchor span;

[0092] S28. Calculate the state of the empty cable in the side span. The side span length is: S s = S s -d0 + d1. According to the lengths of each section of the main cable obtained in the last round of S11, calculate the alignment of the main cable using the method of S23. At this time, the horizontal component force of the main cable is H1;

[0093] S29. If |y n -yt2 + dy3| < 0.5 mm is not satisfied, update the coordinates of the tangent point of the saddle circular arc using formula (25), modify α s0 , and return to S23 for recalculation until |y n -yt2 + dy3| < 0.5 mm is satisfied, end the iteration, and obtain the displacement d of the IP point of the saddle under the balanced state of the side span;

[0094] S30. Compare the difference between the assumed pre - offset d0 of the saddle and the actual displacement of the main cable end point calculated: d = x n -S m -xt1. If |d| < 0.5 mm is not satisfied, change the magnitude of d0, and return to S20 for iterative calculation until |d| < 0.5 mm; end the loop, and obtain the alignment of the empty cable state, the pre - offset of the saddle, and the pre - offset of the spreader saddle;

[0095] S31. Calculate the construction state. During the construction process, the load of the stiffening girder is applied to the corresponding main cable nodes in the form of nodal forces through the suspenders. Then the construction load is F i , (i = 1, 2,..., n). The F value of the nodes where the construction load is not applied i is taken as 0; calculate from S20 to S30 to obtain the preliminary construction state alignment and the horizontal component force H2 of the main cable;

[0096] S32. Use formulas (26) and (27) to calculate the main cable strain and the main cable angle coefficient:

[0097] ε i = (H0 - H2) / EA / cosθ i-1 (26)

[0098] α i = α i-1 + F i / H2 (27)

[0099] Then return to S23 to S30 to recalculate the alignment of the main cable, as well as the offsets of the saddle and the dispersion saddle, on the premise that the horizontal component forces in each span are balanced.

[0100] Embodiment

[0101] For a suspension bridge with a main span of 300m, side spans of 100m, anchor spans of 15m, hanger spacing of 10m, dead load hanger force of 1000kN, tower height of 42m, anchor span height of 10m, main cable sag of 40m, and main cable cross-sectional area of 0.137564m 2 , elastic modulus of 200GPa, and unit length weight of 11.55kN / m.

[0102] S1: First, perform the main span calculation. Substitute the hanger force and the abscissa of the hanger position into formula (1) to calculate the vertical force R at the IP point of the main cable y = 14500kN.

[0103] S2: Estimate the horizontal component force of the main cable force H0 = 25000kN, and substitute it into formula (2) to find the angle coefficient α0 = -0.58 of the main cable at the IP point of the main span

[0104] S3: Use formulas (3) and (4) to calculate the y coordinates and corresponding angle coefficients (dimensionless) of each node of the main cable (see Table 1)

[0105] Table 1 Y coordinates and corresponding angle coefficients of each node of the main cable (unit: m)

[0106] <![CDATA[x i > 10 20 30 40 50 60 70 80 90 100 <![CDATA[y i > -5.33 -10.30 -14.90 -19.13 -22.99 -26.48 -29.61 -32.37 -34.76 -36.78 αi -0.39 -0.50 -0.46 -0.42 -0.39 -0.35 -0.31 -0.28 -0.24 -0.20 <![CDATA[x i > 110 120 130 140 150 160 170 180 190 200 <![CDATA[y i > -38.44 -39.72 -40.64 -41.20 -41.38 -41.20 -40.64 -39.72 -38.44 -36.78 αi -0.13 -0.09 -0.06 -0.02 0.02 0.06 0.09 0.13 0.17 0.20 <![CDATA[x i > 210 220 230 240 250 260 270 280 290 300 <![CDATA[y i > -34.76 -32.37 -29.61 -26.48 -22.99 -19.13 -14.90 -10.30 -5.33 0.00 αi 0.24 0.28 0.31 0.35 0.39 0.42 0.46 0.50 0.53 0.57

[0107] S4: Calculate the deviation dy of the main cable mid-span alignment from the mid-span sag f in the as-built state:

[0108] dy = 40 - |-41.38| = 1.38m

[0109] Obviously, the absolute value of dy is greater than 0.5mm, so change the magnitude of H0 and return to S2 for recalculation. After 23 iterations, when |dy| < 0.5mm is satisfied, exit the main span alignment iteration calculation.

[0110] S5: Then perform the side span calculation. Substitute the side span hanger force and the abscissa of the hanger position into formula (1). For a side span of 100m, calculate the vertical force Rs of the main cable at the IP point of the dispersion saddle y = 4500kN.

[0111] S6: Take the horizontal component force of the main cable force in the main span H0 = 28125kN obtained from the S1 - S4 iterative calculation as the equilibrium condition, and substitute it into formula (5) to find the angle coefficient α of the main cable at the IP point of the side span s0 = 0.16

[0112] S7: Calculate the y - coordinates of the main cable in the side - span and the angular coefficients of each node by using formulas (3) and (4) (see Table 2).

[0113] Table 2 The y - coordinates of each node of the main cable and the corresponding angular coefficients (unit: m)

[0114] <![CDATA[x i > 10 20 30 40 50 60 70 80 90 100 <![CDATA[y i > 1.60 3.56 5.87 8.53 11.56 14.93 18.67 22.76 27.20 32.00 αsi 0.20 0.23 0.27 0.30 0.34 0.37 0.41 0.44 0.48 0.52

[0115] S8: Calculate the deviation dsy between the endpoint coordinates of the main cable in the side - span and the completed - bridge state:

[0116] dsy = 42−|32| = 10m

[0117] If the absolute value of dsy is greater than 0.5 mm, then change the magnitude of α, return to S7 for recalculation until |dsy| < 0.5 mm. Exit the linear - shape calculation of the main cable in the side - span. s0

[0118] S9: Then perform the calculation of the anchor - span. According to the calculation steps of S5 - S7, perform iterative calculation on the anchor - span to obtain the y - coordinates of the main - cable nodes in the anchor - span and the corresponding angular coefficients.

[0119] Table 3 The y - coordinates of each node of the main cable and the corresponding angular coefficients (unit: m)

[0120] <![CDATA[x i > 0 5 10 15 <![CDATA[y i > 0 3.333 6.667 10.000 <![CDATA[α ai > 0.67 0.67 0.67 0.67

[0121] S10: Calculate the deviation day between the endpoint coordinates of the main cable in the anchor - span and the completed - bridge state:

[0122] day = 10−|10| = 0

[0123] Since |day| < 0.5 mm is satisfied, exit the linear - shape calculation of the main cable in the anchor - span to obtain the completed - bridge state C1.

[0124] Table 4 The main - span main - cable coordinates of the completed - bridge state C1 (the origin is at the IP point of the saddle)

[0125]

[0126]

[0127] Table 5 The side - span main - cable coordinates of the completed - bridge state C1 (the origin is at the IP point of the cable - spreading saddle)

[0128] <![CDATA[x i > 10 20 30 40 50 60 70 80 90 100 <![CDATA[y i > 2.600 5.556 8.867 12.533 16.556 20.933 25.667 30.756 36.200 42.000

[0129] Table 6 The anchor - span main - cable coordinates of the completed - bridge state C1 (the origin is at the anchorage point)

[0130] <![CDATA[x i > 5 10 15 <![CDATA[y i > 3.333 6.667 10.000

[0131] ​S11: Calculate the length and strain of each main cable segment: l i , ε i , calculate the self-weight of each main cable segment based on the calculated length of each main cable segment, and superimpose it with the hanger force according to formula (6) to obtain the new hanger force (Table 7). Then repeat the calculation process of S1 - S10 to obtain the C2 completed bridge state.

[0132] Table 7 Updated hanger forces of the main span (unit: kN)

[0133] <![CDATA[x i > 10 20 30 40 50 60 70 80 90 100 <![CDATA[P i > 1129 1127 1126 1124 1123 1121 1120 1119 1118 1117 <![CDATA[x i > 110 120 130 140 150 160 170 180 190 200 <![CDATA[P i > 1117 1116 1116 1116 1116 1116 1116 1116 1117 1117 <![CDATA[x i > 210 220 230 240 250 260 270 280 290 <![CDATA[P i > 1118 1119 1120 1121 1123 1124 1126 1127 1129

[0134] Table 8 Updated hanger forces of the side span (unit: kN)

[0135] <![CDATA[x i > 10 20 30 40 50 60 70 80 90 <![CDATA[P i > 1120 1121 1122 1124 1125 1127 1129 1131 1133

[0136] Table 9 Coordinates of the main cable of the main span in the C2 completed bridge state (unit: m)

[0137] <![CDATA[x i > 10 20 30 40 50 60 70 80 90 100 <![CDATA[y i > -5.169 -9.978 -14.427 -18.518 -22.252 -25.627 -28.647 -31.309 -33.616 -35.567 <![CDATA[x i > 110 120 130 140 150 160 170 180 190 200 <![CDATA[y i > -37.164 -38.405 -39.291 -39.823 -40.000 -39.823 -39.291 -38.405 -37.164 -35.567 <![CDATA[x i > 210 220 230 240 250 260 270 280 290 300 <![CDATA[y i > -33.616 -31.309 -28.647 -25.627 -22.252 -18.518 -14.427 -9.978 -5.169 0.000

[0138] Table 10 Coordinates of the main cable of the side span in the C2 completed bridge state (unit: m)

[0139] <![CDATA[x i > 10 20 30 40 50 60 70 80 90 100 <![CDATA[y i > 2.590 5.537 8.840 12.500 16.519 20.895 25.631 30.727 36.183 42.000

[0140] Table 11 Coordinates of the main cable of the anchor span in the C2 completed bridge state (origin at the anchorage point, unit: m)

[0141] <![CDATA[x i > 5 10 15 <![CDATA[y i > 3.265 6.598 10.000

[0142] S12: Substitute the coordinates of the first nodes of the main span and side span close to the IP point obtained in step S11, which are (10, -5.164) and (-10, -5.817), into formulas (7) - (12) respectively to calculate the coordinates of the saddle center as (0.142, -5.702), and the coordinates of the saddle arc tangent points are: (2.438, -1.260), (-2.372, -1.380).

[0143] S13: The friction coefficient between the main cable and the saddle groove is taken as μ = 0.25, and calculate the coordinates of the main cable on the inner arc part of the saddle after the change of the saddle tangent point caused by the elastic elongation of the main cable according to formulas (13) - (15), which are (2.436, -1.258), (-2.371, -1.378).

[0144] S14: Substitute the coordinates of the first nodes of the side span and the anchor span near the IP point of the saddles obtained in step S11, which are (10.000, 2.596) and (-5.000, -3.402) respectively, into formulas (7)-(12) to calculate the coordinates of the center of the saddles as (2.0722, -4.6278), and the tangent point coordinates of the saddle arcs are: (0.816, 0.212), (-0.701, -0.468).

[0145] S15: Using the method in step S13, calculate the coordinates after the change of the tangent points on both sides of the saddles considering the influence of friction: (0.810, 0.210), (-0.695, -0.463).

[0146] S16: Correct the coordinates of the two end nodes of the main cable in the main span and the main span length to 295.128 m according to formula (16).

[0147] S17: In the coordinate system with the IP point of the saddles as the origin, correct the coordinates of the two end nodes of the side span main cable (-0.695, -0.463) and (89.299, 41.532), the side span length 97.629 m, and the tower height 40.622 m according to formula (17).

[0148] S18: Correct the coordinates of the end node of the anchor span main cable at the saddle, the anchor span length 14.305 m, and the anchor span height 9.537 m according to formula (18).

[0149] S19: Using the corrected design parameters, recalculate steps S1 - S11. Obtain the C3 completed bridge state considering the saddles and the anchor saddles.

[0150] Table 12 Coordinates of the main cable in the main span of the C3 completed bridge state (unit: m)

[0151] <![CDATA[x i > 10 20 30 40 50 60 70 80 90 100 <![CDATA[y i > -5.164 -9.974 -14.424 -18.516 -22.249 -25.626 -28.645 -31.308 -33.615 -35.567 <![CDATA[x i > 110 120 130 140 150 160 170 180 190 200 <![CDATA[y i > -37.163 -38.404 -39.291 -39.823 -40.000 -39.823 -39.291 -38.404 -37.163 -35.567 <![CDATA[x i > 210 220 230 240 250 260 270 280 290 300 <![CDATA[y i > -33.615 -31.308 -28.645 -25.626 -22.249 -18.516 -14.424 -9.974 -5.164 0.000

[0152] Table 13 Coordinates of the main cable in the side span of the C3 completed bridge state (unit: m)

[0153] <![CDATA[x i > 10 20 30 40 50 60 70 80 90 97.628 <![CDATA[y i > 2.591 5.538 8.841 12.502 16.521 20.898 25.634 30.730 36.187 40.620

[0154] Table 14 Coordinates of the main cable in the anchor span of the C3 completed bridge state (origin at the anchorage point, unit: m)

[0155] <![CDATA[x i > 5 10 14.274 <![CDATA[y i > 3.268 6.602 9.506

[0156] S20: Calculate the state of the main span with the cable unloaded. Assume a pre - deviation of the saddle of 0.15 m, and the main span length becomes: S m = 300.3 m. Calculate the self - weight of the unloaded cable as concentrated node forces, estimate the initial value of the horizontal force of the unloaded cable as 4591.6 kN, and calculate the vertical force R at the IP point of the main cable according to formula (1). y= 2507 kN.

[0157] S21: Substitute H1 = 4591.6 kN for H0 into formula (2) to find the cable angle coefficient α0 = -0.55 at the IP point of the main cable in the main span.

[0158] S22: Use formulas (19) and (20) to calculate the main cable strain and the main cable angle coefficient:

[0159] Table 15 Main cable coordinate strain

[0160] segment 1 2 3 4 5 6 7 8 9 10 unit με 1150.1 1133.9 1118.7 1104.4 1091.2 1079 1067.9 1058 1049.2 1041.6 segment 11 12 13 14 15 16 17 18 19 20 unit με 1035.3 1030.2 1026.3 1023.7 1022.4 1022.4 1023.7 1026.3 1030.2 1035.3 segment 21 22 23 24 25 26 27 28 29 30 unit με 1041.6 1049.2 1058 1067.9 1079 1091.2 1104.4 1118.7 1133.9 1150.1

[0161] Table 16 Main cable angle coefficient

[0162] segment 1 2 3 4 5 6 7 8 9 10 <![CDATA[α i > -0.514 -0.478 -0.442 -0.407 -0.372 -0.269 -0.235 -0.202 -0.168 -0.135 segment 11 12 13 14 15 16 17 18 19 20 <![CDATA[α i > -0.102 -0.069 -0.036 -0.003 0.029 0.062 0.095 0.128 0.161 0.194 segment 21 22 23 24 25 26 27 28 29 30 <![CDATA[α i > 0.227 0.261 0.295 0.330 0.364 0.399 0.435 0.471 0.503 0.516

[0163] S23: Assume an initial value of α0 and use formulas (21) and (22) to calculate the main cable coordinates:

[0164] Table 17 Main cable coordinates (unit: m)

[0165] <![CDATA[x i > 9.899 19.757 29.622 39.492 49.367 59.248 69.323 79.386 89.435 99.472 <![CDATA[y i > -1.260 -5.334 -10.400 -15.114 -19.480 -23.501 -27.181 -29.896 -32.264 -34.291 <![CDATA[x i > 109.49 119.51 129.51 139.51 149.50 159.49 169.47 179.46 189.44 199.43 <![CDATA[y i > -35.979 -37.332 -38.353 -39.043 -39.405 -39.440 -39.148 -38.529 -37.583 -36.307 <![CDATA[x i > 209.42 219.42 229.42 239.43 249.44 259.46 269.48 279.51 289.54 297.13 <![CDATA[y i > -34.702 -32.764 -30.490 -27.878 -24.924 -21.624 -17.975 -13.972 -9.611 -4.888

[0166] S24: Obviously, the above result does not satisfy |y n - yt1| < 0.5 mm. Modify α0 and return to S23 for recalculation until |y n - yt1| < 0.5 mm, and end the iteration.

[0167] S25: Compare the difference between the assumed saddle pre - offset d0 and the actual displacement of the main cable end point obtained by calculation: d = 0.193, which does not satisfy d < 0.5 mm. Change the size of d0 and return to S20 for iterative calculation until d < 0.5 mm, and the two end points of the main cable reach the design elevation.

[0168] S26: Calculate the state of the empty cable in the anchor span. According to the lengths of each section of the main cable obtained in the last step of S11, use the method in step S23 to calculate the alignment of the main cable. Since the horizontal component forces of the main cables in the main span, side span, and anchor span are equal, H1 = 4591.6 kN is the same for each span calculation.

[0169] S27: The rotation of the dispersion saddle θ causes the coordinates of the tangent points of the dispersion saddle arc to become: (0.768, 0.199), (-0.701, -0.468), and the displacement amounts dx3, dy3, dx4, dy4 are respectively: -0.042, 0.011, -0.006, 0.005 (m). It does not satisfy |y n - yt4 + dy4| < 0.5 mm. Update the coordinates of the tangent points of the dispersion saddle arc using formula (25) and modify α a0, return to S23 for recalculation until |y n -yt4 + dy4| < 0.5 mm, and end the iteration. The displacement of the IP point of the dispersion saddle under the anchor span balance state is obtained as d1 = 0.054 m.

[0170] S28: Calculate the state of the empty cable in the side span. The side span length is: S s = 99.904 m. According to the lengths of each section of the main cable obtained in the last step S11, the alignment of the main cable is calculated using the method of step S23.

[0171] S29: If |y n -yt2 + dy3| < 0.5 mm is not satisfied, update the coordinates of the tangent point of the dispersion saddle arc using formula (25), modify α s0 , return to S23 for recalculation until |y n -yt2 + dy3| < 0.5 mm, and end the iteration. The displacement of the IP point of the dispersion saddle under the side span balance state is obtained as d0 = 0.179 m.

[0172] S30: Compare the difference between the assumed pre - offset of the saddle d0 and the actual displacement of the main cable end point calculated: d = x n -S m -xt1. If |d| < 0.5 mm is not satisfied, change the magnitude of d0, and return to S20 for re - iterative calculation until |d| < 0.5 mm. End the loop, and obtain the alignment of the empty cable state and the saddle pre - offset of 0.126 m, the dispersion saddle pre - offset of 0.046 m.

[0173] Table 18 Main span main cable empty cable coordinates (unit: m)

[0174] <![CDATA[x i > 10.019 20.035 30.055 40.076 50.097 60.116 70.132 80.145 90.153 100.16 <![CDATA[y i > -5.105 -9.853 -14.228 -18.236 -21.879 -25.161 -28.087 -30.659 -32.880 -34.755 <![CDATA[x i > 110.15 120.15 130.14 140.13 150.12 160.11 170.10 180.10 190.09 200.09 <![CDATA[y i > -36.285 -37.472 -38.319 -38.826 -38.995 -38.826 -38.319 -37.472 -36.285 -34.755 <![CDATA[x i > 210.10 220.10 230.12 240.13 250.15 260.17 270.19 280.21 290.23 297.81 <![CDATA[y i > -32.880 -30.659 -28.087 -25.161 -21.879 -18.236 -14.228 -9.853 -5.105 -1.260

[0175] Table 19 Side span main cable empty cable coordinates (unit: m)

[0176] <![CDATA[x i > 9.987 19.980 29.969 39.954 49.937 59.919 69.902 79.888 89.878 97.501 <![CDATA[y i > 2.597 5.527 8.829 12.496 16.526 20.914 25.657 30.753 36.200 40.620

[0177] Table 20 Anchor span main cable empty cable coordinates (unit: m))

[0178] <![CDATA[x i > 5 10 14.274 <![CDATA[y i > 3.268 6.602 9.506

[0179] S31: Calculate the construction state. During a certain construction process, the load of the stiffening girder is applied to the corresponding main cable nodes in the form of nodal forces through the suspenders. The loads applied to the 6th suspenders in the main span and the side span are respectively: 3000 kN. Calculate the steps from S20 to S30 to obtain the preliminary construction state alignment and the horizontal component force of the main cable H2 = 4773.86 kN.

[0180] S32: Use formulas (26) and (27) to calculate the main cable strain and the main cable angle coefficient

[0181] Then return to the steps from S23 to S30, recalculate the alignment of the main cable on the premise of the balance of the horizontal component forces in each span (see Tables 21 - 23), as well as the saddle offset of 0.306 m and the anchor saddle offset of 0.027 m.

[0182] Table 21 Coordinates of the main cable in the main span with no load (unit: m)

[0183] <![CDATA[x i > 9.974 19.922 29.866 39.806 49.742 59.674 69.778 79.863 89.929 99.977 <![CDATA[y i > -5.194 -10.082 -14.626 -18.830 -22.699 -26.237 -28.842 -31.115 -33.059 -34.678 <![CDATA[x i > 110.01 120.02 130.03 140.03 150.02 160.01 169.99 179.98 189.98 199.98 <![CDATA[y i > -35.977 -36.958 -37.623 -37.974 -38.012 -37.739 -37.153 -36.254 -35.041 -33.511 <![CDATA[x i > 209.98 220.00 230.03 240.08 250.14 260.21 270.30 280.40 290.51 298.17 <![CDATA[y i > -31.663 -29.492 -26.996 -24.169 -21.009 -17.511 -13.670 -9.480 -4.938 -1.260

[0184] Table 22 Coordinates of the main cable in the side span with no load (unit: m)

[0185] <![CDATA[x i > 9.988 20.018 30.056 40.104 50.164 60.237 70.073 79.916 89.770 97.295 <![CDATA[y i > 2.479 5.279 8.427 11.919 15.753 19.926 24.969 30.335 36.025 40.610

[0186] Table 23 Coordinates of the main cable in the anchor span with no load (unit: m))

[0187] <![CDATA[x i > 5 10 14.286 <![CDATA[y i > 3.286 6.632 9.517

Claims

1. A refined calculation method for the cable system of a suspension bridge, given the main span, side span, anchor span, tower height, anchor span height, main cable sag, the abscissa of the suspender, the suspender force, the main cable area, the elastic modulus, and the weight per unit length; characterized in that, Determine the as-built state of the main cable through a progressive point-by-point balanced iterative main cable shape-finding calculation method, including the following steps: (1) Establish a main cable calculation model based on the IP points of the saddle and the dispersion saddle. The main cable only bears the action of the hanger force. Adopt the point-by-point balanced iterative method, and iterate one by one for each node in the main span, each node in the side span, and each node in the anchor span. With the principle of equal horizontal component forces of the main cable as the control principle and the main span sag reaching the design value as the convergence condition, perform the shape-finding calculation of the main cable to obtain the C1 as-built state. The specific steps are as follows: S1. Conduct the main span calculation: Substitute the main span hanger force and the abscissa of the position of each span node into formula (1) to calculate the vertical force R at the IP point of the main span main cable. y ; Wherein, P i is the force of each node in the completed bridge state; S m is the main span; x i is the abscissa of each span node, i = 0, 1, …, n, and n is the maximum number of nodes within the span; P i is the force of each node in the completed bridge state, i = 1, 2, …, n; S2. Estimate the horizontal component force H0 of the main cable force in the main span and substitute it into formula (2) to calculate the angle coefficient α0 of the main cable at the IP point of the main span main cable; α0 = -R y / H0 (2) S3. Use formulas (3) and (4) to calculate the y coordinates and corresponding angle coefficients of each node of the main span main cable; y i = y i-1 + α i-1 (x i - x i-1 ) (3) α i = α i-1 + P i / H0 (4) where i = 1, 2,..., n; S4. Calculate the deviation dy of the main cable mid-span linear shape from the mid-span sag f of the as-built state; dy = f - |y middle | where f is the sag of the main cable; y middle is the y - coordinate at the mid - span of the main cable; if the absolute value of dy is greater than 0.5 mm, then change the magnitude of H0 and return to S2 for recalculation until |dy| < 0.5 mm, and then exit the main - span alignment iterative calculation; S5. Conduct side-span calculation: Substitute the side-span hanger force and the abscissa of the position of each span node into formula (1), and replace the main-span length S in formula (1) with the side-span length S m to calculate the vertical force Rs of the main cable at the IP point of the saddles s ; y ​ S6. Taking the horizontal component force H0 of the main cable force of the main span obtained by iterative calculation of S1 - S4 as the equilibrium condition, substitute it into formula (5) to calculate the cable angle coefficient α at the IP point of the side - span main cable s0 : α s0 = Rs y / H s0 (5) S7. Similarly, use formulas (3) and (4) to calculate the y coordinates and corresponding angle coefficients of each node of the side span main cable; S8. Calculate the deviation dys of the end point coordinates of the side span main cable from the as-built state; dys=c-|y s | Where c is the height of the pylon; y s is the y coordinate of the end point of the main cable in the side span; if the absolute value of dys is greater than 0.5 mm, then change the magnitude of α s0 and return to S7 for recalculation until |dys| < 0.5 mm is satisfied, and exit the calculation of the alignment of the main cable in the side span; S9. Perform the anchor span calculation: According to the calculation steps of S5 - S7, perform iterative calculations on the anchor span to obtain the y coordinates and corresponding angle coefficients of each node of the anchor span main cable; S10. Calculate the deviation dya of the end point coordinates of the anchor span main cable from the as-built state; dya = h - |y a | Where h is the height between the theoretical anchorage point of the main cable and the IP point of the saddle; y a is the y coordinate of the end point of the main cable in the anchor span; if the absolute value of dya is greater than 0.5 mm, then change the value of α a0 and return to S9 for recalculation until |dya| < 0.5 mm is satisfied, exit the linear calculation of the main cable in the anchor span, and obtain the completed bridge state of C1; (2) Calculate the self-weight of the main cable according to the length of each section of the main cable obtained in the as-built state C1, and superimpose it with the hanger force as the new node load, and then repeat the calculation in step (1) to obtain the C2 as-built state. Proceed as follows: S11. Calculate the length \(l\) of each main cable segment i and the strain \(\varepsilon\) i : Calculate the self-weight of each main cable segment based on the calculated length of each main cable segment, superimpose it with the hanger force according to formula (6) to obtain a new hanger force, and then repeat the calculation process of S1 - S10 to obtain the C2 as-built state; P′ i = P i + 0.5q(l i + l i+1 ) (6) where q is the weight of the main cable per unit length; (3) Calculate the coordinates of the tangent points of the main cable at the saddle and the dispersion saddle respectively, and re-determine the span of each span and the sag of the main cable based on this, establish a new main cable calculation model, adopt the new node load in step (2), and perform the point-by-point balanced iterative calculation again to obtain the C3 as-built state. The specific steps are as follows: S12. Based on the coordinates (x a , y a ) and (x b , y b ) of the first nodes of the main span and side span near the saddle IP point in the C2 completed bridge state obtained in S11, calculate that the coordinates of the saddle center are (x c , y c ), the coordinates of the saddle arc tangent points are (xt1, yt1) and (xt2, yt2), and the lengths of the outer chords of the arcs are L a and L b ; where r is the theoretical radius of the saddle; S13. Assume that the friction coefficient between the main cable and the saddle groove is μ, and calculate the coordinates of the first pair of nodes on both sides of the saddle and the tangent point of the main cable after the elastic elongation of the main cable in the inner arc part of the saddle changes; where A is the cross-sectional area of the main cable and E is the elastic modulus; S14. Respectively substitute the coordinates (x a1 , y a1 ) and (x b1 , y b1 ) of the first nodes of the side span and the anchor span in the C2 completed bridge state obtained in step S11, which are close to the IP point of the dispersion saddle, into formulas (7)-(12), and calculate that the coordinates of the center of the dispersion saddle are (x c1 , y c1 ), the coordinates of the tangent points of the dispersion saddle arc are (xt3, yt3), (xt4, yt4), and the lengths of the outer chords of the arc are L a1 and L b1 ; S15. Use the method of S13 to calculate the change of the tangent points on both sides of the dispersion saddle considering the influence of friction; S16. Correct the coordinates of the two end nodes of the main span main cable and the main span span according to formula (16); is x n the value before correction; S17. In the coordinate system with the IP point of the dispersion saddle as the origin, correct the coordinates of the two end nodes of the side span main cable, the side span span, and the tower height according to formula (17); are y respectively n and S s the values before correction S18. Correct the coordinates of the end node of the anchor span main cable located at the dispersion saddle, the anchor span span, and the anchor span height according to formula (18); h 0 are S respectively a , the values before h correction; S19. Adopt the corrected design parameters and recalculate the steps of S1 - S11 to obtain the C3 as-built state considering the saddle and the dispersion saddle.

2. The refined calculation method of the suspension bridge cable system according to claim 1, wherein Complete the shape-finding calculation of the main cable in the bare cable state and the construction state through the following steps: S20. Calculate the state of the main span's empty cable. Assume a saddle pre-offset d0, and the main span's span length becomes The self-weight of the empty cable is calculated as a concentrated force at nodes, Pc i = 0.5q(l i + l i+1 ), estimate the initial value of the horizontal force of the empty cable H1, and calculate the vertical force R at the IP point of the main cable according to formula (1) y1 ; S21: Substitute H1 for H0 into formula (2) to obtain the angle coefficient α of the main cable at the IP point of the main span of the main cable. 01 ; S22: Use formulas (19) and (20) to calculate the main cable strain and the main cable angle coefficient; ε i = H0 / EA / cosθ i-1 (19) α i = α i-1 + Pc i / H1(20) S23: Assume the initial value of α c0 and calculate the coordinates of the main cable: x i+j = x i+j-1 + dx i (21) y i+j = y i+j-1 + dy i (22) where dx i , dy i respectively represent the displacement of the main cable node and are calculated using the following formula; dx i = (1 - ε i+j )l i+j cos(atan(α i+j-1 )) dy i =(1 - ε i+j )l i+j sin(atan(α i+j-1 )) Wherein, i = 1, 2, ..., m; j = 0, 1, ..., k + 1 - i, the lengths and strains of each section of the main cable adopt the results obtained in the last round of S11 step; S24. If |y n - yt1| < 0.5 mm is not satisfied, modify α0, return to S23 for recalculation until |y n - yt1| < 0.5 mm is satisfied, and the iteration ends; S25. Compare the difference d = x between the assumed saddle pre - offset d0 and the actual displacement of the main cable end point obtained through calculation n -S m -xt1. If d < 0.5 mm is satisfied, end the loop; otherwise, change the magnitude of d0, return to S20 for re - iterative calculation until d < 0.5 mm is satisfied, indicating that the two end points of the main cable reach the design elevation S26. Calculate the state of the empty cable in the anchor span. According to the lengths of each section of the main cable obtained in the last round of S11, use the method of S23 to calculate the alignment of the main cable. Since the horizontal components of the main cable in the main span, side span, and anchor span are equal, H1 is the same when calculating in each span; S27. The rotation of the cable saddle by θ causes changes in the coordinates (xt3, yt3), (xt4, yt4) of the tangent points of the cable saddle arc, and the displacement amounts are dx3, dy3, dx4, dy4 respectively; In the formula, w is the vertical height from the IP point of the cable saddle to the rotation center of the cable saddle support; If |y n -yt4 + dy4| < 0.5 mm is not satisfied, update the coordinates of the tangent point of the arc of the saddle with formula (25), and modify α s0 , return to S23 for recalculation until |y n -yt4 + dy4| < 0.5 mm, end the iteration; obtain the displacement d1 of the IP point of the saddle in the balanced state of the anchor span; S28. Calculation of the state of the side-span aerial cable. The side-span span is: Based on the lengths of each section of the main cable obtained in the last round of S11, the alignment of the main cable is calculated using the method of S23. At this time, the horizontal component force of the main cable used is H1. S29. If |y n -yt2 + dy3| < 0.5 mm is not satisfied, update the coordinates of the tangent point of the arc of the scattered cable saddle with formula (25), and modify α s0 , return to S23 to recalculate until |y n -yt2 + dy3| < 0.5 mm is satisfied, end the iteration, and obtain the displacement d of the IP point of the saddle under the balanced state of the side span; S30. Compare the difference between the assumed saddle pre - offset d0 and the actual displacement of the main cable end point obtained by calculation: d = x n -S m -xt1. If |d| < 0.5mm is not satisfied, change the magnitude of d0, return to S20 for re - iterative calculation until |d| < 0.5mm is satisfied; end the loop, and obtain the alignment, saddle pre - offset, and spreader saddle pre - offset in the bare cable state; S31. Construction state calculation: During the construction process, the load of the stiffening girder is applied to the corresponding main cable nodes in the form of nodal forces through the suspenders, so the construction load is F i , (i = 1, 2,..., n), and the value of F for the nodes without construction load applied i is taken as 0; Calculate from S20 to S30 to obtain the preliminary construction state alignment and the horizontal component force H2 of the main cable; S32. Use formulas (26) and (27) to calculate the strain of the main cable and the angle coefficient of the main cable: ε i =(H0 - H2) / EA / cosθ i-1 (26) α i = α i-1 + F i / H2 (27) Then return to S23 to S30 to recalculate the alignment of the main cable, the offset of the cable saddle, and the offset of the cable saddle under the premise of the balance of the horizontal component forces in each span.

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