A method for determining equivalent pre-tension of cables

Abstract

The present invention discloses a method for determining the equivalent pretension of cables. The method is based on nonlinear implicit equation construction and the Steffensen iteration method. Under the condition that the structural load and support state are known, the cable force value under the load state is targeted. Considering geometric nonlinear effects, the problem of determining the equivalent pretension of cables is transformed into the problem of solving multiple relatively independent nonlinear implicit equations. An iterative formula for the equivalent pretension is constructed based on the principle of the Steffensen iteration method. A set of equivalent pretensions of cables with a cable force error within an allowable error range is obtained through cyclic iterative solutions. This method has a small number of iterations, a short solution time, and has a convergence efficiency of second order or higher, high applicability, outstanding computational efficiency, and stability. It is widely applicable to the problem of determining the equivalent pretension of cables in various prestressed structures.

Description

Technical Field

[0001] The present invention relates to the technical field of prestressed structures, and more particularly to a method for determining equivalent pretension of a cable. Background Art

[0002] The equivalent pretension of cables in prestressed structures is an unbalanced force applied to the cable unit in the form of initial strain or equivalent temperature difference. It is primarily used in simulation analysis of the entire prestressed cable construction process. Determining the equivalent pretension of cables involves determining the equivalent pretension that should be applied to the cables, given the actual cable force under a specific load state. This is a key issue in the theory of prestressed structural morphology analysis and forms the basis for simulation analysis of prestressed structure construction.

[0003] Currently, the ratio method and compensation method are commonly used to determine the equivalent pretension of cables in prestressed structures. When the cable forces in a group of cables significantly influence each other, the incremental ratio method and the quantitative ratio method are more aggressive and have unstable convergence. Furthermore, the incremental ratio method requires that the cable forces obtained from two iterations are unequal, while the quantitative ratio method requires that the cable forces remain non-zero and unchanged in sign during the iteration process. The convergence speed of the compensation method depends on the compensation factor λ: a smaller λ results in slower convergence, while a larger λ leads to divergence and unstable iterations.

[0004] Therefore, it is urgent to propose a method for determining the equivalent pre-tension of cables that is highly applicable, computationally stable and efficient. Summary of the Invention

[0005] In response to the shortcomings of the existing technology, the present invention provides a method for determining the equivalent pretension of cables. In view of the shortcomings of the current methods for determining the equivalent pretension of prestressed structures, such as large limitations, poor stability, and low calculation efficiency, the present method has strong applicability, stable calculation, and high calculation efficiency.

[0006] To achieve the above-mentioned purpose, the present invention provides the following technical solutions: a method for determining the equivalent pre-tension of cables, which is a method for determining the equivalent pre-tension of cables based on the construction of nonlinear implicit equations and the Steffensen iteration method. Under the condition that the structural load and support state are known, the cable force value under the load state is taken as the target, and the geometric nonlinear effect is considered. The problem of determining the equivalent pre-tension of cables is converted into a problem of solving multiple relatively independent nonlinear implicit equations. Based on the principle of the Steffensen iteration method, an iterative formula of equivalent pre-tension is constructed, and a set of equivalent pre-tensions of cables with cable force errors within the allowable error range are obtained by cyclic iterative solution.

[0007] Furthermore, the method for determining the equivalent pre-tension of the cable specifically comprises the following steps:

[0008] (1) Clarify the load and support state of the structure, establish a structural model, group the cables, and set the cable force value F for each group under the load state. i As the target, set the allowable error δ of the cable force;

[0009] (2) Considering the geometric nonlinear effect, the problem of determining the equivalent pre-tension of the cable is transformed into the problem of solving multiple relatively independent nonlinear implicit equations. The problem of determining the equivalent pre-tension of the cable is transformed into the solution of the following equations:

[0010]

[0011] Among them, F i is the target cable force value of the i-th group of cables under load state, f (i) (P i ) is the P applied to the i-th group of cables i The cable force value after equivalent pre-tension is an implicit nonlinear function;

[0012] (3) Based on the principle of Steffensen iteration method, an iterative formula for equivalent pre-tension is constructed, and f i (P i,n )=F i -f (i) (P i,n ), then the iterative formula of equivalent pretension can be constructed as:

[0013]

[0014] Among them, F i is the target cable force value of the i-th group of cables under load state, P i,n is the equivalent pre-tension value of the i-th group of cables in the n-th iteration, f (i) (P i,n ) is the P applied to the i-th group of cables in the n-th iteration i,n Cable force after equivalent pre-tension, f i (P i,n ) is the P applied to the i-th group of cables in the n-th iteration i,n The difference between the cable force value after equivalent pre-tension and the target cable force value, P i,n+1 is the equivalent pre-tension value of the i-th group of cables in the n+1-th iteration;

[0015] (4) Assign the iterative initial value P of the equivalent pre-tension of each group of cables i,1 ;

[0016] (5) Perform structural finite element analysis to obtain the actual cable force values ​​of each group of cables;

[0017] (6) Determine whether the errors between the actual cable forces and the target cable forces of each group of cables are within the allowable range δ; if so, output the equivalent pretension of the cables at this time; if not, calculate the equivalent pretension of each group of cables in the next iteration according to the above iterative formula, and return to step (5).

[0018] In summary, compared with the existing technology, the patent of the present invention has the following beneficial effects: First, compared with the commonly used ratio method, it greatly improves the iterative stability and has stronger applicability; second, compared with the commonly used compensation method, it has a convergence efficiency of the second order or above, especially in high-precision calculations, it still has a high computational efficiency; third, the number of iterations is small and the solution time is short, and it is widely applicable to the problem of determining the equivalent pretension of cables in various prestressed structures. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] Figure 1 This is a flow chart of a method for determining equivalent pre-tension of a cable according to the present invention;

[0020] Figure 2 1 is a diagram of a prestressed structure in an embodiment of the present invention;

[0021] Figure 3 The efficiency comparison of each method under the condition of 5% allowable error of cable force in the embodiment of the present invention is shown;

[0022] Figure 4 The efficiency comparison of each method under the condition of 1% allowable error of cable force in the embodiment of the present invention is shown;

[0023] Figure 5 The efficiency comparison of various methods is shown in the embodiment of the present invention when the cable tension tolerance is 1% after the initial value of the iteration is modified. DETAILED DESCRIPTION

[0024] Reference Figures 1 to 5 The specific implementation of a method for determining equivalent pre-tension of a cable of the present invention is further described by comparison. The methods of the present invention include but are not limited to the following embodiments.

[0025] The prestressed structure involved in the embodiment of the present invention is specifically a suspend-dome structure with a span of 117 meters. The upper part is a rib-ring single-layer grid shell, and the lower part is arranged with a total of five circles of levy cable systems. The roof suspend-dome is supported by an octagonal ring truss structure, and the exterior wall adopts a single-layer grid shell structure system.

[0026] By applying the method for determining equivalent pretension of a cable described in the present invention, the problem of the equivalent pretension that should be applied to the cable is solved under the premise that the actual cable force value of the cable under a certain load state is known.

[0027] The method for determining the equivalent pre-tension of the cable is a method for determining the equivalent pre-tension of the cable based on the construction of nonlinear implicit equations and the Steffensen iteration method. Under the condition that the structural load and support state are known, this method takes the cable force value under the load state as the target, considers the geometric nonlinear effect, and transforms the problem of determining the equivalent pre-tension of the cable into a problem of solving multiple relatively independent nonlinear implicit equations. Based on the principle of the Steffensen iteration method, an iterative formula of equivalent pre-tension is constructed, and a set of equivalent pre-tensions of the cable with cable force errors within the allowable error range is obtained through cyclic iterative solution.

[0028] Furthermore, the method for determining the equivalent pre-tension of a cable comprises the following steps (eg Figure 1 ):

[0029] (1) Clarify the load and support status of the structure and establish a structural model (such as Figure 2 ), group the cables, each circle of circumferential cable as a group, a total of 5 groups of cables, set the cable force value F of each group under load state i As a target, the allowable error δ of the cable force is set to 5%, as shown in Table 1;

[0030] Table 1 Target cable force values ​​and cable force tolerance ratios for each group of hoop cables

[0031]

[0032] (2) Considering the geometric nonlinear effect, the problem of determining the equivalent pre-tension of the cable is transformed into the problem of solving n relatively independent nonlinear implicit equations. The problem of determining the equivalent pre-tension of the cable is transformed into the solution of the following equation:

[0033]

[0034] Among them, F i is the target cable force value of the i-th group of cables under load state; f (i) (P i ) is the P applied to the i-th group of cables i The cable force value after equivalent pre-tension is an implicit nonlinear function;

[0035] (3) Based on the principle of Steffensen iteration method, an iterative formula of equivalent pretension is constructed. Let f i (P i,n )=F i -f (i) (P i,n ), then the iterative formula of equivalent pretension can be constructed as:

[0036]

[0037] Among them, F iis the target cable force value of the i-th group of cables under load state; P i,n is the equivalent pre-tension value of the i-th group of cables in the n-th iteration; f (i) (P i,n ) is the P applied to the i-th group of cables in the n-th iteration i,n Cable force value after equivalent pre-tension; f i (P i,n ) is the P applied to the i-th group of cables in the n-th iteration i,n The difference between the cable force value after equivalent pre-tension and the target cable force value; P i,n+1 is the equivalent pre-tension value of the i-th group of cables in the n+1-th iteration.

[0038] (4) Assign the iterative initial value P of the equivalent pre-tension of each group of cables i,1 ;

[0039] (5) Perform structural finite element analysis to obtain the actual cable force values ​​of each group of cables;

[0040] (6) Determine whether the errors between the actual cable forces and the target cable forces of each group of cables are within the allowable range δ; if so, output the equivalent pretension of the cables at this time; if not, calculate the equivalent pretension of each group of cables in the next iteration according to the above iterative formula, and return to step (5).

[0041] For the two cases where the allowable error ratio δ of the cable force is 5% and 1%, the target cable force value is used as the initial value of the equivalent pretension iteration. The cable equivalent pretension determination method proposed in this invention, the commonly used ratio method, and the compensation method are used to determine the cable equivalent pretension. The difference between the cable force value after applying the equivalent pretension and the target cable force changes with the number of iterations. Figure 3 and Figure 4 shown.

[0042] from Figure 3 and Figure 4 It can be seen that the method for determining the equivalent pre-tension of the cable proposed in the present invention has the highest computational efficiency, and the cable tension error is reduced to 0.4% after only three iterations; for the ratio method, when the cable tension allowable error is 5%, 5 iterations are required, and when the cable tension allowable error is 1%, 6 iterations are required, and the cable tension error fluctuates during the iteration process, the calculation process is relatively unstable, and the calculation speed is relatively fast; for the compensation method, when the cable tension allowable error is 5%, 4 iterations are required, and when the cable tension allowable error is 1%, 8 iterations are required. It can be seen that when the calculation accuracy requirements are high, the calculation speed of the compensation method is relatively slow.

[0043] In order to further verify the stability of the method of the present invention, the iterative initial value of the equivalent pre-tension is modified. The iterative initial values ​​of the equivalent pre-tension of the 1st, 2nd, 3rd, 4th and 5th groups of cables are modified to 100, 300, 1000, 2000 and 6000 kN respectively. The allowable error δ of the cable force is 1%. The equivalent pre-tension of the cables is determined by the method for determining the equivalent pre-tension of the cables proposed in the present invention, the commonly used ratio method and the compensation method respectively.

[0044] The difference between the cable force after applying equivalent pre-tension and the target cable force changes with the number of iterations for the three methods as shown in the following figure: Figure 5 As shown in the figure, the proposed method for determining the equivalent pretension of the cable still has the highest computational efficiency, reducing the cable tension error to 0.3% after only three iterations. For the ratio method, the error reached 260% in the second iteration, and was ultimately reduced to 0.6% after nine iterations, making the computational process relatively unstable. For the compensation method, the error was reduced to 0.9% after eight iterations, making the computational process relatively stable but slow.

[0045] Therefore, compared with the ratio method and the compensation method, the cable equivalent pretension determination method proposed in the present invention has outstanding computational efficiency and stability.

[0046] The above description is merely a preferred embodiment of the present invention. The scope of protection of the present invention is not limited to the above embodiment. All technical solutions based on the concept of the present invention are within the scope of protection of the present invention. It should be noted that for those skilled in the art, various improvements and modifications that do not depart from the principles of the present invention should also be considered within the scope of protection of the present invention.

Claims

1. A method for determining equivalent pretension of a cable, characterized by: The method for determining the equivalent pre-tension of cables is based on the construction of nonlinear implicit equations and the Steffensen iteration method. Under the condition that the structural load and support state are known, the cable force value under the load state is taken as the target. Considering the geometric nonlinear effect, the problem of determining the equivalent pre-tension of cables is transformed into a problem of solving multiple relatively independent nonlinear implicit equations. Based on the principle of the Steffensen iteration method, an iterative formula for equivalent pre-tension is constructed. A set of equivalent pre-tensions of cables with cable force errors within the allowable error range are obtained through cyclic iterative solution. The specific steps include the following: (1) Clarify the load and support state of the structure, establish a structural model, group the cables, and set the cable force value F for each group under the load state. i As the target, set the allowable error δ of the cable force; (2) Considering the geometric nonlinear effect, the problem of determining the equivalent pre-tension of the cable is transformed into the problem of solving multiple relatively independent nonlinear implicit equations. The problem of determining the equivalent pre-tension of the cable is transformed into the solution of the following equations: Among them, F i is the target cable force value of the i-th group of cables under load state, f (i) (P i ) is the P applied to the i-th group of cables i The cable force value after equivalent pre-tension is an implicit nonlinear function; (3) Based on the principle of Steffensen iteration method, an iterative formula for equivalent pre-tension is constructed, and f i (P i,n )=F i -f (i) (P i,n ), then the iterative formula of equivalent pretension can be constructed as: Among them, F i is the target cable force value of the i-th group of cables under load state, P i,n is the equivalent pre-tension value of the i-th group of cables in the n-th iteration, f (i) (P i,n ) is the P applied to the i-th group of cables in the n-th iteration i,n Cable force after equivalent pre-tension, f i (P i,n ) is the P applied to the i-th group of cables in the n-th iteration i,n The difference between the cable force value after equivalent pre-tension and the target cable force value, P i,n+1 is the equivalent pre-tension value of the i-th group of cables in the n+1-th iteration; (4) Assign the iterative initial value P of the equivalent pre-tension of each group of cables i,1 ; (5) Perform structural finite element analysis to obtain the actual cable force values ​​of each group of cables; (6) Determine whether the errors between the actual cable forces and the target cable forces of each group of cables are within the allowable range δ. If so, output the equivalent pretension of the cables at this time; if not, calculate the equivalent pretension of each group of cables in the next iteration according to the above iterative formula, and return to step (5).

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