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Modelling and analysis on flutter stability of bridge section based on simulated annealing algorithm.

1. Introduction

So far, scholars at home and abroad have carried out a lot of research work on the suspension bridge system bridges mainly concentrated in the structural system, static and dynamic performance and economic performance of its wind stability research is very small and insufficient (Dai, 2013). With that becomes the bridge condition to compare, in the construction hangs pulls the combination system bridge because not yet forms the final structure system, and the structure rigidity in particular torsional rigidity weakens obviously, while under the wind action the structure will produce the bigger distortion gentle breeze to inspire the response, the structure wind resistant stability problem hastens sternly (Manavizadeh, 2013). Therefore, the research of wind resistant stability early only for bridge state of the cable-stayed suspension bridge is not enough, we should study on the construction of cable-stayed suspension bridge wind stability strengthen the key to ensure the structure safety of the construction process (Ma, 2012; Kolahan, 2012). Along with the bridge span unceasing increase as well as the light quality excel in the material application, the greatly cross bridge flutter stability problem is more and more prominent, the greatly cross bridge flutter occurs how mechanism as well as fast analyze the appraisal bridge flutter stability already to then become universal precisely the matter of concern (Matusiak, 2014). Due to the differences in different cross section shape of the girder, the aerodynamic measures for the bridge, on a bridge vibration suppression effect remarkable suppression measures for another bridge may not be effective need through the wind tunnel test study on the corresponding pneumatic suppression measures.

[FIGURE 1 OMITTED]

As shown in the figure one, the stability factors are demonstrated, goes against the wind today in the great span bridge which the stable performance can guarantee and the excitation asks the topic with serviceable and the durable correlation to be day by day prominent, how effective anti-vibration or that damps inspires into an important research topic. Starting with this goal, this paper proposes the novel and new flutter stability of the bridge section model based on the simulated annealing algorithm (Soni, 2013; Wang, 2014). The method is used to calculate the flutter of two large span bridges. The reliability of the flutter stability is verified by the analysis and verification of the calculation results. This method can quickly and efficiently evaluate the flutter stability of long span bridges the goal. In the later sections, we will introduce the proposed methodology in detail. The rest of the paper is organized as follows. In the section 2, we modified the traditional simulated annealing algorithm to make it fit with our application scenario. In the section 3, we propose the model from the perspectives of general bridge structure, section of bridge and stability modeling, respectively. In the section 4, the numerical simulation is conducted. In the section 5, we summarize the work and propose the potential optimization methods.

2. The Modified Simulated Annealing Algorithm

In many optimization problems with non-linear constraints, especially engineering optimization problems, how to find the optimal solution of this type of problem does not exist foolproof method. At present, there are analytic and some numerical methods to solve the problem of the functions optimization (Matsumoto, 2013). In order to solve the problem of function optimization, we must derive the objective function and the constraint function, and then use the extremum of the function to find the extreme point of the function and finally, the solution of the extremum as the function optimization problem (Xin, 2013). The simulated annealing algorithm is a heuristic random search algorithm based on Monte Carlo iterative algorithm, algorithm proposed by the N Metropolis et al. most as early as 1953, but it is used in the combinatorial optimization and VLSI design is in the 1983 by S Kirkpatrick and V. Cemy et al. proposed respectively. The algorithm will combine in the optimized question and statistics thermal equilibrium question analogy has in the addition warded off the combination optimization question new way (Gouder, 2015). Its starting point is in the physics annealing process, namely when carries on the annealing processing to the solid matter, usually is first it heats up causes its granule to be possible the free motion, then the temperature decrease, the granule forms the low energy state gradually the crystal, if nearby dew point under temperature enough slowly, then the solid matter can certainly form the lowest energy the ground state.

Due to the slow convergence speed of genetic algorithm, early maturity and fall into local optimum, and the simulated annealing algorithm in convergence speed and jump out of local optimal advantage is very significant, so the NSGA is still can be room for improvement (Korlin, 2012; Brusiani, 2013). Before the optimization, we firstly introduce the original version of the SAA. Simulated annealing algorithm is a kind of algorithm based on the mechanism of metal annealing. The specific steps of the simulated annealing operation are as follows. Firstly, it has an initial optimum point stochastically, and it took the current optimum point, and calculates the goal function value. Then, we set the simulated annealing initial temperature as follows.

[T.sub.initial] [right arrow] [theta] (1)

And the temperature can be then updated as the follows.

[T.sub.updated] = [T.sub.initial]/[t.sup.[beta]] (2)

Randomly change the current optimal point to produce a new optimal point, and calculate the new objective function value, and calculate the increment of objective function value. The optimal point could be then defined as the formula 3.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Although the simulation annealing algorithm can discover the objective function by the probability search technology from the probability significance and the overall situation most dot, while grasps the entire search process ability to be insufficient, it is not advantageous for the causes the search process to enter the most hopeful search region, the cause simulation annealing operation efficiency is not high. In figure 2, we show the visualized demonstration of SAA optimization procedures.

[FIGURE 2 OMITTED]

The genetic algorithm first encodes the feasible solution of the problem in some form. The coded solution is called the individual and randomly generates the initial population of the M individuals. Then the fitness of each individual is calculated according to the predetermined evaluation function, and then the next generation population is generated by genetic operation such as selection, crossover mutation and so on, and the individual adaptability is improved. NSAG can be served as the optimization of the traditional SAA, its main mentality is that in genetic algorithm foundation, before each generation of choice pair first carries on sorting according to the solution individual non-control relations, the identical rank is one, then to each level according to density sorting, then the choice, overlapping, the variation obtains the filial generation, merge father generation of and the filial generation, according to the level order, after selects the merge a half solution to take next time iterates father generation. If selects level and size cannot satisfy condition exactly the size, then finally selects to it the level according to density comparison operator sorting, selects according to sequence needs the integer solution. In evolutionary algorithm, in order to avoid falling into the local optimum, there will be a certain probability to accept the poor quality of the solution, those who are about to be replaced the best solution to save up that is the elite strategy. In order to preserve the convergence and diversity of the solutions, we use fast non- dominated sorting and density comparison operators as shown in the figure three.

[FIGURE 3 OMITTED]

The state transition of immune algorithm can be described by the Markov chain, algorithm restraining is refers after the algorithm iterates to enough many number of the times, in the community contains the overall situation best individual the probability to approach in one. This kind of definition namely for probability one restraining which usually said. The equation 4 expresses this feature.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

When the NSGA-II algorithm is improved, a multi-objective simulated annealing algorithm is combined to retain the essence of the NSGA-II algorithm that is the elites retention strategy, followed by its non-dominant sort operator and density comparison operator and can be defined as equation 5.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

3. The Proposed Flutter Stability of Bridge Section Model

3.1. The bridge structure review and analysis

In recent years, most of the important long-span bridges at home and abroad set up structural safety and the health monitoring system, and has made many gratifying achievements and progress which can be generally reflected from the listed aspects. (1) By measuring the response of the structure of the sensing device can be obtained to reflect the behavior of the structure of the various records, and bridge operation after the continuous or intermittent monitoring of structural status, and strive to obtain the bridge structure of the continuous, complete information. (2) Comprehensive monitoring content. In addition to monitoring the general state and behavior of the structure itself (stress, displacement, inclination, acceleration, dynamic characteristics, etc.), monitoring and recording of modern environmental conditions (wind, earthquake, temperature, vehicle load, etc.) is also emphasized. (3) Diversified monitoring instruments, advanced monitoring capabilities continue to improve. Many monitoring systems are fast and large capacity of information collection, communication and processing capabilities, and network sharing.

The entity degeneration unit is in senate unit in and so on three dimensional the entity foundations, through uses the revision elasticity coefficient matrix and the restraint corresponding relative displacement method, introduces the board shell directly and so on each kind of basic component simplification hypothesis obtains. Because is based on senate unit structure and so on three dimensional an entity, therefore entity senate unit and so on degeneration unit and entity have the same grid shape, the node degree of freedom and the shape function. Entity degeneration unit and the entity isoparametric compared to the biggest feature is reflected in the elastic matrix, the entity isoparametric elastic matrix of elastic mechanics space problem to reflect the stress and strain relations, while the entity degradation unit through the elastic matrix to reflect different types mechanical characteristics of the components. For the degraded series unit, through block integral technique, a unit can be divided into the plurality of regions, each partition can be of different materials, such as the steel and concrete can have different geometry and can have different mechanical characteristics, such as plate and beam the empty region as an independent partition as only the physical and the mechanical parameters of the materials are set to zero. Then, we can start modelling some parameters.

[FIGURE 4 OMITTED]

The figure 4 illustrates the modelling connections and the relationships. When the moving load is moved on the simply supported girder bridge and the stress in the structure is directly proportional to the magnitude of the load in the elastic range. The lower edge stress of i-th main girder is defined as follows.

[[sigma].sub.i] = [M.sub.i]/[W.sub.i] (6)

Assuming that the main beam section is the same and then the sum of the main beam bending moment can be defined as the equation 7. Therefore, the king post obtains sum of the strain the proportion the bending moment which receives in the beam bridge whole. Sum of the strain may obtain by the WIM system strain sensor, EW respectively is the material and the section attribute, speaking of in the service bridge that may actual obtain. Thus, the WIM system turned the strain which the vehicles load caused to be possible to know, but vehicles load actually unknown inverse. Because serves in the service bridge certain age limit as well as the construction with becomes between the bridges the error, the section EW already between the same design values has possibly had the certain change as this value should by the scene gauging result primarily.

M = [[summation].sup.m.sub.i=1][M.sub.i] = [[summation].sup.m.sub.i=1] [EM.sub.i][[epsilon].sub.i] = EM [[summation].sup.m.sub.i=1] [[epsilon].sub.i] (7)

The effective stress of the bridge structure is the effective stress after the deduction of the stress loss, and the loss of the pre-stress should be reasonably estimated in the analysis of the pre-stressing effect. In this, according to the pre-stressed muscle action mechanism, divides into the loss of pre-stress two kinds: One kind is the loss which has nothing to do with the structure distortion, like the pipeline friction and the anchorage retraction causes loss and so on; Another kind is distorts the related loss with the structure, while like the concrete elastic compression, the contraction continuous variation and so on cause loss. Regarding first kind of the loss, when computation equivalent node load according to standard its deduction as regarding the second kind of loss, uses this article the steel bar model in the structure analysis time includes automatically which can be defined as the formula 8,

[sigma] + 1/[beta] [??] = [g.sub.0] [epsilon] + [[g.sub.0] + [g.sub.1]/[beta]] [epsilon] (8)

3.2. Section of bridge modelling and analysis

In flutter analysis, the unsteady aerodynamic forces caused by the movement of the bridge girder are usually expressed by the aerodynamic derivative, which depends on the profile of the girder cross-section, the wind speed and the wind speed. At the present the hydromechanics to holds flows the mechanism not to understand, causes in the numerical method to hold with the aid of the experience flows the model also causes the value simulation existence difference and the uncertainty. Compares with the CFD numerical method, usually thought the stage model wind tunnel test obtains the bridge cross section flutter derivative to be most direct and the most effective method. CBHM algorithm is a multi-input and multi output time-domain modal parameter identification method. Compared with the ERA algorithm, the algorithm has a small amount of computation fast recognition speed, and with the algorithm has the same recognition accuracy. Without considering the lateral movement of the structure, a bridge section with two degrees of freedom of vertical bending and torsional motion is used. The equation of motion is expressed as the equation 9 and 10, respectively.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

The ultimate goal of flutter analysis is to calculate the structure of the bridge or flutter critical wind speed, in order to evaluate the flutter stability. There are many ways to calculate structure bridge flutter critical wind speed, more classic is the flutter critical wind speed of the Scanlan solving method are used to calculate two degree of freedom structure flutter critical wind speed, by solving dimensionless flutter motion equations, flutter critical wind speed can be obtained. For solving this challenge, in the following equations, we provide the solutions.

f(x) = [[infinity].summation over (n=0)] [c.sub.n] [(x - [x.sub.0]).sup.n] = [c.sub.0] + [c.sub.1] [(x - [x.sub.0]).sup.2] + [c.sub.3][(x - [x.sub.0]).sup.3] + O ([(x - [x.sub.0]).sup.4]) (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

From the above, we know that the covariance block Hankel matrix method uses the covariance of the output vector, which can effectively reduce the impact of output noise, and through the output covariance Toeplitz matrix SVD, from the process of determining the order of the system the elimination of high-order noise mode for which has a strong anti-noise ability under the following restriction condition.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

In order to simplify the numerical realization of forced vibration test, the model aerodynamic force is obtained, and the flutter analysis theory can be used to bridge flutter analysis and the specific steps are as follows.

1. Establishment of bridge cross section model. When under the guarantee computation speed premise, as far as possible the earth selection model and the solid bridge cross section proportion causes the wind field value simulation the flow field and the actual Reynold's number is close.

2. Selected model motion the whole period of time discrete number n. The selected n must be able to fully characterize the motion state of the model in a period, and to calculate the convenience, the time is equal to interval.

3. Wind field simulation. The boundary conditions of the computational domain can be obtained from the boundary conditions of the bridge section model wall surface and the outflow velocity U, and the calculation can be obtained by solving the two-dimensional incompressible Navier-Stokes equations to calculate the wind pressure field.

3.3. Stability of the bridge modelling and discussion

Based on the discussion on the 3.1 and 3.2 parts, this section proposes systematic implementation of the model. The capacity spectrum method is the ability to reflect the structure of the pushover analysis capacity spectrum curve with representative seismic demand response spectrum curve to the same coordinate system, calculate the intersection point is the performance of structure in the blurring the solving steps are as follows. Based on the static analysis method, the shear force and displacement curve of the structure are calculated as the follows.

d/r = [[6 (C/I)].sup.1/a] (15)

[N.sub.R] = 1/3 [[M x K x A x [(C/I).sub.T]].sup.2/a] (16)

By using the capacity spectrum of each order mode and the demand response spectrum of the 5% damping ratio, the maximum displacement and pseudo acceleration of each mode are determined by simplified capacity spectrum method. The essential parameters can ben then calculated as the follows.

[n.sub.s] = [n.sub.f] [1/1.6v ([W/R]/[[[E.sub.b]/[N.sub.o]]] - W[N.sub.th]/[P.sub.s]) + 1]. (17)

[P.sub.s] = W[N.sub.th]/[[W/R/[[E.sub.b]/[N.sub.0]]]] - v (1 + [I.sub.out]/[I.sub.in])([n.sub.s]/[n.sub.f] -1). (18)

The present probabilistic model and the statistical parameter that aims at plans to construct the bridge to say while regarding in the service bridge, its load had own particularity, if still according to planned to construct the structure probabilistic model and the statistical parameter to carries on the probability reliability in the service bridge to appraise, then had the unfairness. Theoretically said that, in the service bridge load may through the examination, the experiment be equal to the section obtains the enough information, thus carries on the analysis to it, definite load probability distribution and statistical parameter as follows.

[I.sub.in] = v ([n.sub.s]/[n.sub.f] - 1) [P.sub.s]/W (19)

The main purpose of the bridge in service is to compare the roughness of the bridge in order to grasp the main parameters of the bridge.

4. The Experiment on the Proposed Model

In this section, we simulate the proposed method. In the bridge earthquake resistance design, ability spectral method is the most commonly used one kind, regarding the base local oscillation main function regular bridge, may neglect the higher order to inspire, carries on the comparison with the time interval method computed result, may obtain a more precise solution, the relative error in 10% that satisfies the precision request. The figure 5 shows the result under different levels.

[FIGURE 5 OMITTED]

Through two kinds of bridge structure with diaphragm beam test data comparison and theoretical calculation analysis, found that the bridge structure of cross beam both analysis process than the diaphragm beam stability data in dynamic analysis or static load, the stress strain and displacement of the number of the values are relatively small the transverse coefficient analysis of field test data are calculated, the bridge structure of cross beam than horizontal beam should be uniform. And in the figure 6, we show the overall bridge structure stability test simulation.

[FIGURE 6 OMITTED]

5. Conclusion

This paper conducts the modelling and analysis on flutter stability of bridge section based on simulated annealing algorithm. In view of this new combination bridge structure in the design aspect material flaw question, while through establishes the ANSYS finite element model, research internode web member rigidity these design parameters to the overall bridge structure stress performance influence rule. The conclusion indicated that, along with internode web member rigidity increase, the combination bridge structure cross most larges deflection non-linearity reduces. The revised SAA model is applied into the system for the systematic implementation. Compares with the CFD numerical method, usually thought the stage model wind tunnel test obtains the bridge cross section flutter derivative to be most direct and the most effective method. CBHM algorithm is a multi-input and multi output time-domain modal parameter identification method. Referring to the numerical simulation, the performance of the proposed model is well presented. In the future, we will integrate neural network to predict the stability of bridges under different conditions and restrictions to test of the overall robustness.

Recebido/Submission: 05/06/2016

Aceitacao/Acceptance: 25/09/2016

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Cheng Kang *

Sichuan College of Architectural Technology, Deyang, Sichuan, 618000 China

* qwert296@sina.com
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Author:Kang, Cheng
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Date:Dec 1, 2016
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