Cable force adjustment method based on sequential quadratic programming method

Abstract

The application discloses a cable force adjusting method based on a sequence quadratic programming method, and the method comprises the following steps: S1, establishing a calculation model according to actual loads and construction steps; S2, applying boundary conditions and loads in the calculation model, and gradually increasing cable forces of each cable to obtain cable force change values Ki of the whole bridge; S3, combining all the cable force change values Ki to obtain a cable force influence matrix to obtain an influence matrix equation of cable force adjustment; S4, solving cable force adjusting amounts {T x} by applying the influence matrix equation, and judging whether the solved cable force adjusting amounts {T x} meet actual construction conditions; and S5, if the solved cable force adjusting amounts {T x} in the step S4 cannot meet the construction conditions, optimizing the cable force adjusting amounts {T x} by using the sequence quadratic programming method until the optimized cable force adjusting amounts {T x ′} meet the construction conditions. The application can solve a group of cable force adjusting amounts meeting actual conditions, and has small deviation, small error and high efficiency.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of bridge construction monitoring, and particularly relates to a cable force adjustment method based on a sequential quadratic programming method. BACKGROUND

[0002] For the adjustment and optimization of structural cable force, many scholars have conducted in-depth research. Xiao Rucheng et al. unified the objective function of the optimization of cable-stayed bridges with the cable force variable and the generalized influence matrix, and proposed the influence matrix method for cable force optimization. Fang Hong et al. applied the influence matrix method to the adjustment of the cable force of the construction stage of the tied-arch bridge. Wang Weikun et al. based on the influence matrix method, carried out the secondary cable force adjustment of the single-pylon cable-stayed bridge.

[0003] Some scholars also transformed the bridge cable force optimization problem into a mathematical optimization model, taking the structural internal force and linear shape as the objective function, and adding various constraint conditions, and transformed the bridge cable force optimization problem into a constrained nonlinear programming model. Based on the influence matrix method and genetic algorithm, Yan Song et al. carried out reasonable bridge cable force optimization calculation under multi-objective linear programming. Chen Zhijun et al. introduced an improved particle swarm algorithm to solve the bridge cable force, which can efficiently and accurately solve the cable force that meets the reasonable bridge state. Based on the response surface method and particle swarm algorithm, Zhu Yulin et al. carried out the cable force optimization of the special-shaped cable-stayed bridge. Chaolinsong et al. took the minimum total bending energy as the objective function, considered the bending moment of the main beam and the tower, the weight distribution of the side span, and the bearing reaction of the transition pier and auxiliary pier as the constraint condition, and proposed an optimization method for determining the cable force of the long-span cable-stayed bridge considering the weight distribution. Zhang Tao et al. took the minimum bending energy of the structure as the objective function, took the deflection and stress of the main beam and the cable force as the constraint, and completed the cable force optimization of the reasonable bridge state based on the zero-order and first-order algorithms in the ANSYS optimization module.

[0004] Using the above influence matrix method to adjust the cable force in the construction stage, since the order of the cable force influence matrix is the same as the number of cable roots, there is a unique cable force solution, but due to the limitation of actual construction conditions, the unique solution may not be applicable, and the result obtained by solving has large error and deviation, and the solving time is long. SUMMARY

[0005] The purpose of the present application is to overcome the shortcomings of the prior art, and to provide a cable force adjustment method based on a sequential quadratic programming method. The cable force adjustment method based on the sequential quadratic programming method can quickly solve the result with small error and deviation to meet the requirements of the construction conditions.

[0006] The purpose of the present application is achieved by the following technical solutions: the cable force adjustment method based on the sequential quadratic programming method comprises the following steps:

[0007] S1, a calculation model is established according to the actual load and construction steps;

[0008] S2, boundary conditions and loads are applied in the calculation model, and the cable force of each cable is increased successively to obtain the cable force variation value Ki of the whole bridge;

[0009] S3, all cable force variation values Ki are combined to obtain a cable force influence matrix to obtain a cable force adjustment influence matrix equation;

[0010] S4, the cable force adjustment amount {T x} is solved by applying the influence matrix equation, and it is judged whether the solved cable force adjustment amount {T x} meets the actual construction condition;

[0011] S5, if the cable force adjustment amount {T x} solved in step S4 cannot meet the construction condition, the sequential quadratic programming method is used to optimize the cable force adjustment amount {T x} until the optimized cable force adjustment vector {T x ′} meets the construction condition.

[0012] Preferably, the specific process of step S5 includes the following steps:

[0013] S51, the cable force adjustment amount is solved as an unknown quantity of optimization calculation:

[0014] {T′ x}={T′ x1 T′ x2 …T′ xn} T , wherein {T′ x} is the optimized cable force adjustment vector; T′ xi is the optimized i-th cable adjustment cable force.

[0015] Let the difference between the target cable force vectors before and after the cable force optimization be {ΔT}:

[0016] {ΔT}={ΔT1 ΔT2…ΔT n} T

[0017] ={T′ m1 -T m1 T′ m2 -T m2 …T′ mn -T mn} T

[0018] , wherein ΔT i is the difference between the i-th cable target cable force before and after optimization; T mi is the i-th cable target cable force before optimization; T′mi the target cable force of the i-th cable after optimization;

[0019] Let the standard deviation of the difference between the target cable force vectors before and after optimization be σ:

[0020]

[0021] where ΔT i is the difference between the target cable force of the i-th cable before and after optimization; μ is the mean of ΔT i ; and r is the maximum allowable dispersion value of the cable force.

[0022] S52, selecting appropriate objective functions and constraints;

[0023] S53, optimizing the cable force adjustment vector {T x} based on the sequential quadratic programming method to obtain the optimized cable force adjustment vector {T′ x};

[0024] S54, if the optimized cable force adjustment vector {T′ x} cannot meet the construction conditions, repeating steps S52 and S53 until the last optimized cable force adjustment vector {T′ x} meets the construction conditions.

[0025] Preferably, the objective functions and constraints in step S52 include:

[0026] Objective function and constraint I:

[0027]

[0028] s.t. 0 ≤ {T s} + {T′ x} ≤ {T u}

[0029] {|ΔT|} ≤ {T p}

[0030] σ ≤ r

[0031] Objective function and constraint II:

[0032]

[0033] s.t. 0 ≤ {T s} + {T′ x} ≤ {T u}

[0034] {|ΔT|} ≤ {T p}

[0035] Objective function and constraint III:

[0036]

[0037] st0≤{T s}+{T′ x}≤{T u}

[0038] {|ΔT|}≤{T p}

[0039] σ≤r

[0040] Where, |ΔT i | represents the absolute value of the difference in cable force between the i-th cable target before and after weighting; {T s} represents the actual cable force vector before cable adjustment; {T′ x} represents the optimized cable force application vector; {T u} represents the maximum allowable value of cable force; {|ΔT|} represents the vector of differences in target cable force before and after optimization; {T p} represents the maximum permissible error of the cable force; σ represents the standard deviation of the difference between the target cable force vectors before and after cable force optimization; and r represents the maximum permissible discrete value of the cable force.

[0041] Preferably, step S2 includes the following steps:

[0042] By successively increasing the force of the i-th cable, the change in force of each of the n cables in the entire bridge is obtained. The change in the total cable force of the bridge caused by a unit change in the i-th cable is expressed as:

[0043] K i =[k 1i ,k 2i ,…k ni ] T .

[0044] Preferably, step S3 includes the following steps:

[0045] The unit change in all cables of the bridge causes the change in cable force K of the entire bridge. i The combined force influence matrix [K] is obtained, i.e.:

[0046] [K] = [K1,K2,…K] n ],

[0047] The basic formula for the influence matrix method is expressed as follows:

[0048] [K]{X}={D},

[0049] Where [K] is the influence matrix; {X} is the vector exerting the modulation; and {D} is the vector being modulated.

[0050] The influence matrix equation in the cable force adjustment process is represented as:

[0051] {T m}={T s}+[K]{T x},

[0052] Wherein, {T x} is the cable force adjustment amount; {T s} is the actual cable force vector before adjustment; {T m} is the cable force target vector; and [K] is the cable force influence matrix.

[0053] The present application has the following advantages over the prior art:

[0054] 1. In view of the problem that the adjustment amount obtained by directly inverting the influence matrix cannot be implemented, the method of sequential quadratic programming optimization is introduced on the basis of the influence matrix, the cable force adjustment amount is calculated based on the influence matrix method, the sequential quadratic programming method is used to optimize the cable force adjustment amount, and a group of cable force adjustment amounts meeting the actual conditions are solved, and the error and deviation are small.

[0055] 2. The existing cable force optimization method focuses on the cable force optimization of the reasonable bridge state, and the cable force adjustment in the construction process is less studied, the method based on the sequential quadratic programming method is used to adjust the cable force of the whole bridge in the construction stage, and reliable basis is provided for the construction monitoring of the bridge, so that the construction is effectively carried out. BRIEF DESCRIPTION OF DRAWINGS

[0056] Figure 1 It is the calculation flow chart of the cable force adjustment method based on the sequential quadratic programming method of the present application.

[0057] Figure 2 It is the calculation model diagram of the present application.

[0058] Figure 3 It is the result comparison diagram of the actual cable force and the theoretical cable force.

[0059] Figure 4 It is the calculation result of the present application using the objective function and constraint condition I.

[0060] Figure 5 It is the calculation result of the present application using the objective function and constraint condition II.

[0061] Figure 6 It is the calculation result of the present application using the objective function and constraint condition III.

[0062] Figure 7 It is the cable force result comparison and analysis diagram before and after optimization. DETAILED DESCRIPTION

[0063] The application will be further described below in connection with the accompanying drawings and examples.

[0064] The cable force adjustment method based on the sequence quadratic programming method comprises the following steps:

[0065] S1, a calculation model is established according to actual loads and construction steps;

[0066] Specifically, in the calculation model, the main girder and the main arch are plate elements, the suspender is a string element, the lower foundation is a beam element, and the suspender and the arch rib and the main girder are all elastically connected. According to the design drawing and the actual construction scheme, a calculation model is established by using the MidasCivil finite element analysis software.

[0067] S2, boundary conditions and loads are applied to the calculation model, and the cable force of each cable is gradually increased to obtain the cable force change value KI of the whole bridge;

[0068] The specific process of step S2 comprises the following steps:

[0069] The cable force of the i-th cable is gradually increased, and the cable force change value of the n cables of the whole bridge is obtained. The unit change of the i-th cable leading to the cable force change value of the whole bridge is represented as:

[0070] K i =[k 1i ,k 2i ,…k ni ] T .

[0071] S3, all cable force change values KI are combined to obtain a cable force influence matrix to obtain an influence matrix equation of cable force adjustment. The specific process of step S3 comprises the following steps:

[0072] The unit change of all cables of the whole bridge leading to the cable force change value K i is combined to obtain the cable force influence matrix [K], that is:

[0073] [K]=[K1,K2,…K n ],

[0074] The basic formula of the influence matrix method is represented as:

[0075] [K]{X}={D},

[0076] Wherein, [K] is the influence matrix; {X} is the adjustment vector; {D} is the adjusted vector;

[0077] Then the influence matrix equation in the cable force adjustment process is represented as:

[0078] {T m}={T s}+[K]{T x},

[0079] wherein, {T x} is the cable force adjustment amount; {T s} is the actual cable force vector before adjustment; {T m} is the target cable force vector; and [K] is the cable force influence matrix.

[0080] S4, the cable force adjustment amount {T x} is solved by applying the influence matrix equation, and it is determined whether the solved cable force adjustment amount {T x} meets the actual construction condition.

[0081] The specific process of step S4 includes the following steps:

[0082] S41, according to the formula

[0083] {T x} = [K] -1 ({T m} - {T s})

[0084] the cable force adjustment amount {T x} is solved.

[0085] S42, it is determined whether the cable force adjustment amount {T x} meets the construction condition and has actual operability.

[0086] S5, if the solved cable force adjustment amount {T x} in step S4 cannot meet the construction condition, the sequential quadratic programming method is used to optimize the cable force adjustment amount {T x} until the optimized cable force adjustment vector {T' x} meets the construction condition.

[0087] The specific process of step S5 includes the following steps:

[0088] S51, the cable force adjustment amount is solved as the unknown quantity of optimization calculation:

[0089] {T' x} = {T' x1 T' x2 …T' xn} T wherein, {T' x} is the optimized cable force adjustment vector; and T' xi is the optimized cable force adjustment of the i th cable.

[0090] Let the difference between the target cable force vectors before and after the cable force optimization be {ΔT}:

[0091] {ΔT} = {ΔT1 ΔT2…ΔTn} T

[0092] = {T m1 -T m1 T m2 -T m2 …T mn -T mn} T

[0093] wherein: ΔT i is the difference of the target cable force of the i-th cable before and after optimization; T mi is the target cable force of the i-th cable before optimization; T mi is the target cable force of the i-th cable after optimization;

[0094] Let the standard deviation of the difference of the target cable force vectors before and after optimization be σ:

[0095]

[0096] wherein: ΔT i is the difference of the target cable force of the i-th cable before and after optimization; μ is the mean of ΔT i ; and r is the maximum allowable dispersion value of the cable force;

[0097] S52, select appropriate objective function and constraint condition;

[0098] S53, optimize the cable force adjustment vector {T x} based on the sequential quadratic programming method, to obtain the optimized cable force adjustment vector {T x};

[0099] S54, if the optimized cable force adjustment vector {T x} cannot meet the construction condition, repeat steps S52 and S53 until the last optimized cable force adjustment vector {T x} meets the construction condition.

[0100] The objective function and constraint condition in step S52 include:

[0101] Objective function and constraint condition I:

[0102]

[0103] s.t. 0 ≤ {T s} + {T x} ≤ {T u}

[0104] {|ΔT|} ≤ {T p}

[0105] σ ≤ r

[0106] Objective function and constraint condition II:

[0107]

[0108] s.t.0≤{T s}+{T′ x}≤{T u}

[0109] {|ΔT|}≤{T p}

[0110] Objective function and constraint condition III:

[0111]

[0112] s.t.0≤{T s}+{T′ x}≤{T u}

[0113] {|ΔT|}≤{T p}

[0114] σ≤r

[0115] Where |ΔT i | is the absolute value of the difference between the i-th cable force before and after weighting; {T s} is the actual cable force vector before adjustment; {T′ x} is the adjusted cable force vector after optimization; {T u} is the maximum allowable cable force; {|ΔT|} is the difference between the target cable force before and after optimization; {T p} is the maximum allowable cable force error; σ is the standard deviation of the difference between the target cable force before and after optimization; and r is the maximum allowable dispersion value of the cable force.

[0116] Taking the Guangzhou Tower Pedestrian Bridge (now known as the Haixin Bridge) as an example, the overall bridge spans the Pearl River in a north-south direction, connecting the Ersha Island and the Haizhu Island, and is located about 300 m downstream of the Guangzhou Bridge. The main bridge is a half-through steel arch bridge with a span of 198 m, and the span combination of the approach bridges on both sides is 2x40 m continuous steel box girder.

[0117] There are a total of 23 hangers in the whole bridge. The upper anchor point of the hanger is the anchoring end, which is connected to the ear plate on the arch through the insertion of ears and pins. The lower anchor point is the tensioning end, which is connected to the beam anchor plate through the use of an integral anchor head.

[0118] A full bridge model is established by using the finite element analysis software Midas Civil, the main girder and the main arch are simulated by plate elements, the suspenders are simulated by string elements, the lower foundation is simulated by beam elements, the suspenders and the arch rib main girder are simulated by elastic connection, the full bridge has 7834 nodes and 11564 elements, as shown in Figure 2 .

[0119] The bridge appeared a problem of large deviation between the calculation result and the measured result after the first overall cable, and the specific situation is as follows:

[0120] After the first overall cable tensioning is completed, the actual cable force of the full bridge is measured, and the comparison between the actual cable force and the theoretical cable force is as shown in Figure 3 .

[0121] Before the cable force adjustment, the parameters of the calculation model are identified, and it is found by analysis that there is a large difference between the expected boundary condition in the support model on both sides of the main girder and the actual boundary condition in the first tensioning process of the suspenders, which leads to a large difference between the cable force on both sides of the main girder and the theoretical cable force after the first overall bridge tensioning is completed. The boundary condition of the model is modified and adjusted, and the adjusted model is applied to the calculation of the cable force influence matrix.

[0122] In the Midas model, the unit cable force value is added to each suspender in sequence, the cable force change amount of the full bridge under the unit change of each suspender cable force is obtained, the cable force influence matrix [K] is obtained by combination, and the unique solution of the cable force adjustment vector is obtained by applying formula (4):

[0123] {T x}={886 1070 1234 1317 1472 1594 1584 1547 1559 1513 1426 16541451 1596}T, unit: kN.

[0124] The cable adjustment sequence is from short cable to long cable, and the two sides are symmetrically tensioned, and it is found that the cable force in the cable adjustment stage has the following three problems:

[0125] 1) The tensioning cable force of most suspenders exceeds the upper limit of the operation of the jack (1000kN), and cannot be implemented.

[0126] 2) The cable force in the process is much larger than the cable force in the bridge state (about 900kN), and the cable itself and the anchoring ear plate may appear stress overrun problem.

[0127] 3) The cable force of part of the suspenders appears negative value in the process, which is impossible in the actual situation, and the cable adjustment matrix is inconsistent with the actual situation.

[0128] Under the condition that the cable adjustment amount obtained by directly inverting the influence matrix cannot be implemented, the sequential quadratic programming method is used to adjust the cable force {Tx} is optimized to obtain a new cable force application vector {T′}. x} and cable force target vector {T′ m To ensure that the cable force adjustment is within a reasonable range, the target cable force error after adjustment is within the allowable range, and the cable force is uniform, different objective functions and constraints are selected. The cable force optimization is solved using the following three methods:

[0129] Objective function and constraints I:

[0130]

[0131] st0≤{T s}+{T′ x}≤{T u}

[0132] {|ΔT|}≤{T p}

[0133] σ≤r

[0134] Objective function and constraints II:

[0135]

[0136] st0≤{T s}+{T′ x}≤{T u}

[0137] {|ΔT|}≤{T p}

[0138] Objective function and constraints III:

[0139]

[0140] st0≤{T s}+{T′ x}≤{T u}

[0141] {|ΔT|}≤{T p}

[0142] σ≤r

[0143] The calculation results of each algorithm are as follows Figures 4 to 6 As shown.

[0144] As can be seen from the above, the cable force calculated by the three cable force adjustment methods based on the sequential quadratic programming method is less than the limit of the jack, which is practically operable. The deviation between the target cable force and the initial target is less than 3%, and the theoretical cable force in the adjustment stage is basically positive, which meets the actual cable adjustment requirements.

[0145] After the optimization of cable force based on the sequence quadratic programming method, compared with the three calculation methods, the target function and the target cable force of constraint condition I have the smallest deviation from the initial target cable force, the cable force deviation of constraint condition II is the most uniform, and the calculation time is the shortest, which indicates that the calculation efficiency is higher when the constraint condition is linear constraint, and the mean of the adjusted cable force of constraint condition III is the smallest. Therefore, different optimization methods can be selected according to the actual requirements on site.

[0146] After the optimization of cable force, the cable force optimized by algorithm ② is applied to the actual construction, and the comparison and analysis of the cable force results before and after the adjustment are as shown in Figure 7

[0147] According to the calculation results, after the adjustment of the cable force, the error of the measured cable force is basically controlled within 5% of the theoretical cable force, the maximum error value is 37kN, the deviation is 7%, and the overall meets the construction control requirements.

[0148] The above specific embodiments are preferred embodiments of the present application, and cannot limit the present application, and any changes or other equivalent replacement methods without deviating from the technical solutions of the present application are included in the protection scope of the present application.​

Claims

1. A method for adjusting cable force based on a sequence quadratic programming method, characterized by, Comprise the following steps: S1, according to the actual load and construction steps to establish the calculation model; S2, in the calculation model to apply boundary conditions and load, and gradually increase the cable force of each cable, get the whole bridge cable force variation value Ki; S3, all the cable force variation value Ki combination to get the cable force influence matrix, to get the cable force adjustment influence matrix equation; S4, the cable force adjusting amount {T x} is solved by using the influence matrix equation, and whether the solved cable force adjusting amount {T x} meets the actual construction condition is judged. S5, if the adjusted cable force vector {T x} in step S4 cannot meet the construction conditions, the sequential quadratic programming method is used to optimize the adjusted cable force vector {T x}, until the optimized cable force vector {T x ′} meets the construction conditions. The specific process of step S5 includes the following steps: S51, the cable force adjustment amount as the unknown quantity of optimization calculation is solved: {T′ x}={T′ x1 T′ x2 …T′ xn} T , where {T′ x} is the optimized cable force adjustment vector; T′ xi is the optimized i-th cable force adjustment cable force; Let the target cable force vector difference before and after the optimization of cable force be {ΔT}: {ΔT} = {ΔT1 ΔT2...ΔT n} T = {T m - T m1 T m2 - T m2 … T mn - T mn} T wherein: ΔT i is the difference between the target cable force of the ith cable before and after optimization; T mi is the target cable force of the ith cable before optimization; T' mi is the target cable force of the ith cable after optimization; Let the standard deviation of the target cable force vector difference before and after the optimization of cable force be σ: where ΔT i is the difference between the target cable force of the ith cable before and after optimization; μ is the mean of ΔT i ; and r is the maximum allowable dispersion of cable force. S52, select appropriate objective function and constraint condition; The objective function and constraint condition in step S52 includes: Objective function and constraint condition I: s.t. 0 ≤ {T s} + {T′ x} ≤ {T u} {|ΔT|}≤{T p} σ≤r Objective function and constraint condition II: s.t. 0≤{T s}+{T′ x}≤{T u} {|ΔT|}≤{T p} Objective function and constraint condition III: s.t. 0≤{T s}+{T′ x}≤{T u} {|ΔT|}≤{T p} σ≤r Where, |ΔT i | represents the absolute value of the difference in cable force between the i-th cable target before and after weighting; {T s } represents the actual cable force vector before adjustment; {T′ x } represents the optimized cable force application vector; {T u } represents the maximum allowable value of cable force; {|ΔT|} represents the vector of differences in target cable force before and after optimization; {T p } represents the maximum permissible error of the cable force; σ represents the standard deviation of the difference between the target cable force vectors before and after cable force optimization; r represents the maximum permissible discrete value of the cable force; S53, based on the sequence quadratic programming method, optimizing the cable force adjustment vector {T x} to obtain the optimized cable force adjustment vector {T' x} S54, if the optimized cable force adjustment vector {T' cannot satisfy the construction condition, repeat steps S52 and S53 until the last optimized cable force adjustment vector {T' satisfies the construction condition. x x} satisfies the construction condition.​ 2. The method of claim 1, wherein the adjustment of the cable force is performed by a sequential quadratic programming method. The specific process of step S2 includes the following steps: Gradually increase the cable force of the i-th cable, get the whole bridge n cable force variation value, the unit change of the i-th cable leads to the whole bridge cable force variation value is expressed as: K i = [k 1i ,k 2i ,…k ni ] T .

3. The method of claim 2, wherein the adjustment of the cable force is performed by a sequential quadratic programming method. The specific process of step S3 includes the following steps: The unit change of all bridge cables causes the cable force change value K of the full bridge cable i The combination obtains the cable force influence matrix [K], that is: [K] = [K1, K2,... K n ], The basic formula of influence matrix method is expressed as: [K]{X}={D}, Wherein, [K] is influence matrix; {X} is adjustment vector; {D} is the adjusted vector; Then the influence matrix equation in the process of cable force adjustment is expressed as: {T m} = {T s} + [K] {T x}, wherein, {T x} is the cable force adjustment amount; {T s} is the actual cable force vector before adjustment; {T m} is the cable force target vector; and [K] is the cable force influence matrix.

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