JP6543863B2 - Structural performance survey method of railway bridge - Google Patents

Description

  The present invention relates to a method of investigating the structural performance of a railway bridge.

  There is a need for a method for efficiently and effectively maintaining and managing a large existing railway bridge. For example, development of a method for measuring a structural performance of a bridge as simply and quantitatively as possible as possible by installing sensors etc. Is a very important issue.

  On the other hand, with the progress of price reduction of sensors and simplification of measurement, a method of estimating the structural performance by measuring the acceleration of a bridge during train travel is widely adopted as a method of obtaining quantitative data.

  For example, modal characteristics such as the natural frequency of the bridge and the modal damping ratio are estimated from the measured acceleration. Since the mode characteristics such as the natural frequency and mode damping ratio of this bridge indirectly indicate the structural performance, by continuously measuring the acceleration of the bridge with a sensor, the natural frequency and mode damping ratio etc. It is tried to grasp the condition of the abnormality of the bridge (structure) and the aged deterioration by accumulating the data of the modal characteristics and the structural performance of the bridge and making the relative comparison and capturing the aged change.

  Moreover, the impact coefficient is used as an important index for evaluating the dynamic response at the time of train travel. The impact coefficient is also an important indicator in maintenance as it is closely related to deterioration phenomena such as cracks in railway bridges. And when evaluating the impact coefficient of a railway bridge, using a ring displacement meter, video measurement, U-Doppler (non-contact vibration measurement system for a structure incorporating a laser Doppler velocimeter), etc. The vertical displacement is measured (deflection measurement) on the bridge side (ground side), and the impact coefficient is determined using this measurement result (see, for example, Patent Document 1).

JP, 2012-233758, A

  Here, in Japan's railway technology field, in order to secure and maintain the railway bridge and hence the safety and usability of the railway bridge, a great deal of knowledge about the design of the railway bridge and the like has already been accumulated. Most of the structural performance used in the design of railway bridges based on such vast knowledge is based on displacement, and in addition to static displacement at low speed travel, an impact coefficient representing response amplification at high speed travel is representative It is considered as a structural performance index.

  However, measurement of the displacement of the bridge during train operation is an absolute requirement that the fixed points be always accurately and precisely set and installed to ensure measurement accuracy, and also the environment at the time of measurement. In addition to severe environmental constraints, the measurement load is large compared to acceleration measurement. For this reason, it is practically difficult to use as a method for investigating the structural performance of a huge existing railway bridge and, consequently, a method for maintaining and managing a huge existing railway bridge.

  On the other hand, many estimation methods of displacement response based on acceleration response during train running have been proposed, but while problems remain in terms of estimation accuracy and calculation load, indicators that require extrapolation calculations such as impact coefficient can not be obtained Therefore, the application in practice is limited at present. For this reason, development of a method capable of accurately estimating the displacement response and vibration characteristics of a bridge based on the acceleration response during train travel has been strongly desired.

  In view of the above circumstances, the present invention provides a method of surveying structural performance of a railway bridge that enables surveying and evaluating structural performance of the bridge to be suitably performed using observation data of acceleration response of the bridge during train travel. The purpose is

  In order to achieve the above object, the present invention provides the following means.

  The structural performance investigation method of a railway bridge according to the present invention uses a train as a moving load train and uses the bridge as a simple beam to formulate a theoretical analysis model of the dynamic response of the railway bridge at the time of train traveling as Equation (1) below. The present invention is characterized in that the acceleration of a bridge at the time of train travel is measured, and an unknown parameter of the theoretical analysis model is estimated by the inverse analysis method from the acceleration data.

here,
x is the distance (position), t is the time when the time when the first axle of the leading vehicle enters the bridge is 0, ν (t) is a step function of 0 at t <0, 1 at 0 ≦ t, τ _{k, i} is the point at which the k-th vehicle passes the i-th axle.

  u (above...) (x, t) is an acceleration response, and is expressed by the following equation (2), equation (3) to equation (10).

here,
u (above) (x, t) is a velocity response, and u (x, t) is a displacement response.
m (above-) is the bridge mass per unit length (per unit length mass), c is a damping coefficient, and EI is bending stiffness. L _b is the span length of the bridge, n is the mode order, T is the bridge transit time, ω _n is the nth natural angular frequency, ξ _n is the nth mode damping ratio, ω _n (-above) is one axle The n-order component of each frequency passing through the axle with half cycle from the bridge approach to the exit, _{n n} (x) is the n-order vibration mode type, and u ₀ is the static deflection amount when the axial weight acts on the middle of the span It is.

Further, in the method of investigating the structural performance of a railway bridge according to the present invention, an error term is introduced to the theoretical analysis model of the above equation (1) to define a probability model of the following equation (11) to give an unknown parameter θ. And the co-occurrence probability (likelihood function) L (u ₀ | θ) of the following equation (12) in which the acceleration data u ₀ (······) is generated from the equation (11) The prior probability density function π (θ) of the unknown parameter of the following equation (13) is substituted for the following equation (14) obtained by the Bayes theorem and the acceleration data is given as the unknown parameter It is desirable to determine the simultaneous a posteriori probability density function π (θ | u ₀ ) and to evaluate the structural performance of the railway bridge reflecting the estimated parameters and the uncertainty of the parameters.

here,
N _omega mean mu _omega, normal distribution of variance σ _ω ^2, GJ (ligature) _xi] shows the gamma distribution shape factor alpha _xi], scale factor gamma _xi]. Θ is the parameter space.

  Furthermore, in the method for investigating the structural performance of a railway bridge according to the present invention, an MCMC method for obtaining a probability density function that approximates the simultaneous a posteriori probability density function by numerical calculation is used to sequentially obtain plural unknown parameters. It is more desirable to repeat the sampling from the conditional a posteriori probability density function to obtain the joint a posteriori probability density function.

  Further, in the method of investigating the structural performance of a railway bridge according to the present invention, the sample of the sampled parameter reaches the steady state of the Markov chain, and the expected value and the confidence interval are obtained from the samples of the parameters after the sampling number reaching the steady state. It is further desirable to calculate the parameter estimates and evaluate the reliability of the parameter estimates and the parameter estimates.

Furthermore, in the method for investigating the structural performance of a railway bridge according to the present invention, unknown parameters θ are ω (natural period), m (above-) (bridge mass per unit length), ξ (mode attenuation ratio), σ (probability error term variance), and π (ω | u ₀ , m (above-), ξ, σ) of the conditional posterior probability density function is a random wall MH method, π (m above ) | U ₀ , ω, ξ, σ) is the ratio of the maximum amplitude, π (ξ | u ₀ , ω, m (above-), σ) is the independent MH method, π (σ | u ₀ , ω, m It is desirable to use the acceleration data and the standard deviation of the theoretical analysis model to perform sampling from the conditional posterior probability density function (above-), ξ).

  Further, in the method of investigating the structural performance of a railway bridge according to the present invention, it is desirable to obtain a sample of displacement response and impact coefficient according to the simultaneous a posteriori probability density function by theoretical analysis of MCMC method using samples of sampled parameters. .

Furthermore, in the method for investigating the structural performance of a railway bridge according to the present invention, response prediction analysis during train travel is incorporated in the m-th sampling process using the arbitrary train speed v _s represented by the following equation (15) as a parameter By obtaining the maximum value at any given speed and incorporating the response prediction process into the calculation flow of MCMC method, it is possible to obtain the sample of the maximum displacement according to the simultaneous posterior probability density function simultaneously with the estimation calculation of unknown parameters. More desirable.

  In the method of investigating the structural performance of a railway bridge according to the present invention, it is possible to accurately obtain displacement response during train travel necessary for estimation of bridge condition, travel safety and comfort evaluation from acceleration response at one point of the bridge. It will be possible.

  Further, among the estimation parameters based on the actual acceleration of the bridge during train travel, the natural frequency is related to the bridge stiffness, so the natural frequency can be accurately estimated, so that the structure performance can be accurately estimated. .

  Furthermore, it is possible to predict responses at other train speeds from the measured acceleration of the bridge during train travel.

  In addition, since the uncertainty of the estimation parameter can be quantified, it is possible to evaluate the risk of changing according to the estimation accuracy, such as adoption of the 95% confidence interval lower limit value.

  Furthermore, based on the uncertainty of the estimation parameter that changes according to the estimation accuracy, it is also possible to evaluate the value of more accurate measurement from the aspect of estimation accuracy.

  In addition, calculating the margin with respect to the maintenance management reference value may make it possible to determine the maintenance management priority and to extract weak structures.

  Therefore, according to the method for investigating the structural performance of a railway bridge of the present invention, it is possible to appropriately investigate and evaluate the structural performance of the bridge using observation data of acceleration response of the bridge during train traveling.

It is a figure which shows the simple beam model at the time of train travel. It is a figure which shows a vehicle model (interaction analysis). It is a figure showing a structure model (interaction analysis and load sequence analysis). It is a figure which shows the displacement response computed by each analysis method of a load sequence analysis method, a dynamic interaction analysis method, and a simple analysis method (structure performance investigation method of the railway bridge concerning one embodiment of the present invention). It is a figure which shows the impact coefficient calculated by each analysis method of a load-train analysis method, a dynamic interaction analysis method, and a simple analysis method (structure performance investigation method of the railway bridge which concerns on this invention). It is a flowchart which shows the structure performance investigation method (detailed reverse analysis method) of the railway bridge which concerns on one Embodiment of this invention. It is a flowchart which shows the structure performance investigation method (inverse analysis and response prediction by MCMC method) of the railway bridge which concerns on one Embodiment of this invention. It is a flowchart which shows the simple reverse analysis method. It is a flowchart which shows the simple reverse analysis method. It is a figure which shows an example of acceleration time-series data. It is a figure which shows the example of convergence of the structure performance investigation method (detailed reverse analysis method) of the railway bridge which concerns on one Embodiment of this invention. It is the figure which compared the estimation accuracy of the structure performance investigation method (detailed reverse analysis method) of the railway bridge which concerns on one Embodiment of this invention, and the existing method. It is a figure which shows the reverse analysis result (estimated value (expected value of posterior distribution) and its confidence interval) based on an acceleration response. It is a figure which shows the estimation result of a displacement response. It is a figure which shows the estimation result of the largest displacement. It is the figure which compared the correct value and the estimated value of the largest displacement of C bridge. It is a figure which shows the track | orbit displacement introduce | transduced into interaction analysis. It is a figure which shows the influence which noise, the interaction with a vehicle, and track | orbit displacement have on detailed reverse analysis in 10 m of span lengths. It is a figure which shows the influence which a noise, interaction with a vehicle, and track | orbit displacement have on detailed reverse analysis in 45 m of span lengths. It is a figure which shows the relationship between the confidence interval of the natural frequency estimated value in 10 m of span lengths, and an impact coefficient. It is a figure which shows the relationship between the confidence interval of the natural frequency estimated value in 45 m of span lengths, and an impact coefficient. It is a figure which shows the estimation precision of the largest displacement by detailed reverse analysis in 10 m of span lengths. It is a figure which shows the estimation precision of the largest displacement by detailed reverse analysis in 45 m of span lengths. It is a figure which shows an example of the application target bridge of the structure performance investigation method of the railway bridge which concerns on one Embodiment of this invention. It is a figure which shows an example of measurement acceleration. It is the figure which compared the acceleration before and behind detailed reverse analysis. It is a figure which shows the parameter estimation result by inverse analysis in 25 m of span lengths. It is a figure which shows the parameter estimation result by inverse analysis in 35 m of span lengths. It is a figure which shows the displacement estimation result based on the measurement acceleration in 25 m of span lengths. It is a figure which shows the displacement estimation result based on the measurement acceleration in 35 m of span lengths. It is a figure which shows the relationship between the largest displacement and train speed based on a reverse analysis result.

  Hereinafter, with reference to FIG. 1 to FIG. 31, a method for investigating a structural performance of a railway bridge according to an embodiment of the present invention will be described.

  Here, the present embodiment relates to a method for estimating the displacement response and impact coefficient of the bridge by the inverse analysis method based on the measured acceleration of the bridge at the time of train traveling, and investigating and evaluating the structural performance of the railway bridge. is there.

  In the present embodiment, first, it is shown that the formulated theoretical analysis model can reproduce the dynamic response at the time of train traveling with constant accuracy, and then, based on the formulated theoretical analysis model, estimation based on uncertainty is A possible detailed inverse analysis method (the structural performance investigation method of a railway bridge according to the present invention) and a simple inverse analysis method with a small calculation load will be described.

  In addition, the parameters, displacement response, and impact coefficient estimation accuracy obtained by the method of investigating the structural performance of a railway bridge according to the present invention are verified by twin experiments, and accuracy evaluation is performed for various influential factors. The result which applied to a response and confirmed the effectiveness of this invention is demonstrated.

  In the method for investigating the structural performance of a railway bridge according to this embodiment, a dynamic response theory analysis method at the time of train traveling on a simple girder is formulated as follows.

(The response of the bridge when running the train based on the theory)
FIG. 1 shows a simple beam model when running a train.
Here, the traveling train is regarded as a moving load train, and the bridge is modeled as a simple beam having no change in cross section in the track direction.

Note that FIG. 1 shows that a train 1 formed by vehicles n _v with an axle spacing a, a bogie spacing b, and a vehicle length L _v is a bridge 2 with a span L _b at a constant speed v from left to right in the figure. It shows the state of running. Further, time t is 0 when the first axle of the leading vehicle enters the bridge 2.

  And, assuming that the surface is a sufficiently smooth Bernoulli-Eiler beam, the equation of motion at any point and time on the bridge can be expressed by equation (16).

Here, m (above-) is a bridge mass (per unit length mass) per unit length, c is a damping coefficient, and EI is bending stiffness. Further, J is a moving load sequence expressed by equation (17), and δ represents a Delta function of Dirac. Further, τ _{k, i} is the point at which the k-th vehicle passes by the i-th axle according to equation (18), and P _{k, i} is the axial weight of the same axle.

  For each moving load of equation (16), if the bridge can be assumed to be a linear system, it can be decomposed as equation (19). It can be expressed by superposition.

Further, the solution of the equation (21) of the differential equation relating to the bridge response when the k-th vehicle passes the i-th axle under the boundary condition of FIG. 1 is the equation (22). Where n is the mode order, T is the bridge transit time, ω _n is the nth natural angular frequency, ξ _n is the nth mode damping ratio, and ω _n (-above) is from one bridge entry to exit The n-order component of each frequency passing through the axle with half cycle, _{n n} (x) is the n-order vibration mode type, and u ₀ is the static deflection amount when the axial weight acts on the middle of the span, ) To equation (27).

The natural frequency f _n is f _n = ω _n / 2π. Further, α _n , β and 中 in the formula are respectively the formula (28), the formula (29) and the formula (30). Furthermore, assuming that the attenuation is sufficiently smaller than 1 for the derivation of the equation (22), an approximation of ω _n √ (1−ξ _n ² ) ≒ ω _n was used.

Next, the velocity response u _{k, j} (above ·) and acceleration response u _{k, j} (above · · ·) when the k-th vehicle passes through the i-th axle are respectively given by the equation (31), It can be expressed as equation (32).

Next, a theoretical analysis model will be described.
First, the maximum mode amplitude depends on the static deflection (Equation (27)) and the mode shape Φ and mode orders n and β as well known. Based on the past research, when the bridge natural period is approximated by ω ₁ = 2π · 105 L _b ⁻¹ , equation (33) is derived from equations (24) and (29). Here, vkm / h = v / 3.6 [km / h].

Next, even if the current maximum operating speed of the railway is assumed, since β ² << 1, the equation (34) can be simply considered.

From this equation (34), with the increase of the mode order n, the maximum amplitude of each mode is rapidly reduced in inverse proportion to the fourth power of the order. Furthermore, considering that the displacement is maximum at the middle of the span for most simple girders, the secondary mode amplitude at the middle of the span is _{S S} (L _b / 2) = 0, which is the second largest amplitude after the primary mode The third mode is the third mode. On the other hand, from the equation (34), the maximum mode amplitude of the third order mode is 1/34 ≒ 0.011 for the first order mode, which is about 1%. Thus, in the present embodiment, only the amplitude of the primary mode is considered. Moreover, at this time, Formula (22) becomes Formula (35). The subscripts for the mode order are omitted.

  Further, by superposing equation (22) which is a solution of the equation of motion under single moving load action, a dynamic response theoretical analysis model during train traveling can be formulated as equation (36). Note that ν (t) is a step function of 0 at t <0 and 1 at 0 ≦ t.

  At this time, the velocity response u (····) (x, t) and the acceleration response u (··· · · (x, t) are expressions (37) and (38).

  And since these equations (36) to (38) are only superposition of trigonometric functions, there is an error due to the omission of higher order modes, but there is no error mixing through numerical integration, and comparison with the finite element method The computational load is extremely small. This makes it possible to calculate the response even with common spreadsheet software.

Next, the validity of the theoretical analysis model will be described.
In order to verify the validity of the theoretical analysis model (theoretical analysis method / simplified inverse analysis method) derived based on multiple hypotheses, high-speed railway cars of 10 cars consisting of 10m, 35m, 45m simple girders shown in Table 1 In the case where the vehicle traveled, we compared with two types of numerical analysis methods: load sequence analysis method (load sequence analysis) and dynamic interaction analysis method (interaction analysis).

  Here, in the load train analysis method (load train analysis), a vehicle is used as a load train of a non-vibration system, and a structure is modeled as a finite element model. A general purpose structural analysis program DIARIST (Dynamic and Impact Analysis for Railway Structure) of the track structure was used for the calculation. In DIARIST, the equation of motion in the modal coordinate system is solved by Newmark's average acceleration method in time increments Δt. The Δt was 0.0005 seconds, and the mode order was the 50th.

  In the dynamic interaction analysis method (interaction analysis), a traveling vehicle is modeled as a multi-body model, and a structure is modeled as a finite element model. The calculation was carried out using the dynamic interaction analysis program DIASTARS III between the Shinkansen vehicle and the railway structure.

Further, an analysis model of the vehicle (train) is as shown in FIG.
The vehicle has a three-dimensional model in which each component of the car body, the bogie frame, and the wheelset is a rigid body, and these are coupled by a spring and a damper. A model with 31 degrees of freedom per vehicle was linked in 10 cars and modeled. The validity of the analysis model is confirmed by verification experiments using an actual vehicle and a vehicle test stand.

  In addition, the specifications of the vehicle were set based on the recent high-speed railway vehicle. The interaction force between the wheel and the rail was calculated using the vertical relative displacement and the horizontal relative displacement of the two. Specifically, the normal direction of both contact surfaces considered the contact spring of Hertz, and the tangential direction considered the creep force.

  The numerical calculation was solved using Newmark's average acceleration method as in the case of load sequence analysis, but since the equation of motion is non-linear, iterative calculation is performed within Δt until the unbalance force is sufficiently reduced at each time step. did. The Δt was 0.0005 seconds, and the mode order was up to the 50th.

Next, the structure model will be described.
FIG. 3 shows a finite element model of a simple girder used for load sequence analysis and interaction analysis. Bridges and rails were used as beam elements, and track pads were used as spring elements, and three bridges with span lengths of 10m, 35m and 45m were modeled respectively.

  The track pad and the bridge were connected by a rigid beam element. The cross-sectional specifications were determined as shown in Table 1 with reference to the measurement results. The mode attenuation ratios were all 2%. In the theoretical analysis, the same natural frequency, unit length mass, and modal damping ratio as in the structure model of Table 1 were input.

  FIG. 4 shows the vertical deflection response at the center of the span calculated respectively by theoretical analysis (simple inverse analysis method), load sequence analysis and interaction analysis. In the case of a train, it is assumed that a high-speed train of 10 cars runs at the speed shown in the figure. And from this figure, regardless of the traveling speed and the span length, all analysis results agree very well, and it is confirmed by the theoretical analysis according to the present invention that the displacement response during train traveling can be reproduced with high accuracy. The

Next, the dynamic response component due to the speed effect of the railway bridge is also considered in the design according to the impact coefficient i _a of equation (39). Here, S _d is the maximum value of dynamic deflection, and S _s is the maximum value of static deflection.

And FIG. 5 has shown the impact coefficient calculated by each analysis method of a load sequence analysis method (load sequence analysis) and a dynamic interaction analysis method (interaction analysis).
As for the vehicle specifications, as in the above, the train speed was 100 km / h to 420 km / h, and the maximum deflection at 100 km / h was taken as the static deflection.

  From this figure, it is confirmed that the impact coefficient differs slightly between the theoretical analysis (simple inverse analysis method) and the interaction analysis at the resonance peak, but good agreement is obtained at other speed bands. In addition, since the impact coefficients of the theoretical analysis and the load sequence analysis match well, it is presumed that the difference between the resonance peaks is due to the vibration system (mass or damping) of the vehicle.

  Based on the above results, although the theoretical analysis method is limited to simple digits, the evaluation of time series response and impact coefficient is at least as accurate as load series analysis with much less freedom of the model than finite element method. It became clear that it was possible.

  Based on these results, in this embodiment, the detailed inverse analysis method (the structural performance investigation method of the railway bridge of the present invention) and the simple inverse analysis method for estimating the parameters required for the theoretical analysis model from the actual acceleration Is shown below.

(Detailed reverse analysis method: Structural performance investigation method of railway bridge of the present invention)
First, the detailed inverse analysis method will be described with reference to FIG.
The detailed inverse analysis method in the present embodiment is a detailed inverse analysis method of unknown parameters and displacement response by MCMC (Markov chain Monte Carlo) method, and the degree of freedom is obtained by using the above-described formulated theoretical analysis method as a basic model. It is possible to avoid the underdetermination problem of parameter estimation process caused by excessiveness, and to realize the stability and accuracy improvement of estimation.

  Specifically, in the detailed inverse analysis method in the present embodiment, first, Bayesian estimation is used to evaluate parameters and their uncertainties with high accuracy.

  The general Bayesian estimation method estimates the posterior distribution of unknown parameters based on the parameter prior distribution and the likelihood function defined by observation data.

The likelihood function is expressed as L (u ₀ (·····) | θ). Note that θ represents an unknown parameter vector, and u ₀ (above, ···) represents observed data of acceleration response during train traveling. Further, it is assumed that the random variable θ follows the prior probability density π (θ).

Then, when the observation data u ₀ (above) is a given condition, the simultaneous a posteriori probability density function π (θ | u ₀ (above)) of the unknown parameter vector θ is an equation (from the Bayesian theorem) 40) However, Θ is a parameter space. The denominator of Equation (40) is a constant, and the simultaneous a posteriori probability density function π (θ | u ₀ (······)) is Equation (41).

Thus, the simultaneous a posteriori probability density function π (θ | u ₀ (·····)) is nothing but a parameter distribution with the observation data known, that is, model parameters estimated from actual values.

Next, the formulation of the likelihood function and the prior distribution will be described.
Here, an error term represented by ε _t is introduced into the theoretical analysis model, and probability models represented by the equations (42) and (43) are defined. N (0, σ ² ) is a normal distribution with an average of 0 and a variance of σ ² .

Let natural frequency (eigen period) ω, unit length mass m (-above), mode damping ratio ξ, random error term dispersion σ be unknown parameter θ = (ω, m (-above), ξ, σ), Assuming these to be given, the co-occurrence probability (likelihood function) L (u ₀ (above · · · | θ) that the observation data u ₀ (· · · above) is generated from the model equation (42) is Expression (44) is obtained. Here, the prior probability density function π (θ) of the unknown parameter θ = (ω, m (-above), ξ, σ) is set as shown in equation (45).

Incidentally, showing N _omega mean mu _omega, normal distribution of variance sigma _omega, GJ (ligature) _xi] is the shape factor alpha _xi], the gamma distribution scale factor gamma _xi], respectively. Further, from the equation (27), among the unknown parameters θ = (ω, m (-above), ξ, σ), the unit length mass m (-above) is in inverse proportion to the static deflection amount.

That is, if the natural frequency f and the mode damping ratio ξ are obtained, for example, the unit length mass m (− on the top) can be calculated backward from the ratio of the maximum amplitude to the observation data. Similarly, the variance σ ² of the random error term can be uniquely calculated from the observation data and the reproduction data if other parameters are determined. As a result, it is decided that Bayesian estimation is not performed on the unit length mass m (above-) and the variance σ ^{2 of the} error term.

Then, by substituting the likelihood of equation (44) and the a priori probability density function of equation (45) into equation (40), the simultaneous a posteriori probability density function π (θ | u ₀ (above) is defined. However, it is very difficult to solve this analytically.

  Therefore, in the method of investigating the structural performance of a railway bridge according to the present embodiment, a numerical solution method by MCMC method is used.

(Parameter estimation based on MCMC method)
a) Estimation of Simultaneous Posterior Probability Density Function FIG. 7 shows a calculation flow including the estimation of the joint posterior probability density function by the MCMC method of the present embodiment.

The MCMC method is a method of obtaining a probability density function that approximates the simultaneous a posteriori probability density function π (θ | u ₀ (above) by numerical calculation). In this embodiment, the Gibbs method is one of the MCMC methods. To adopt.

  Then, using Gibbs's method, the simultaneous posterior probability density function is obtained by repeating sampling from the conditional a posteriori probability density function in order for each element of the unknown parameter θ = (ω, m (-above), σ, σ). calculate.

  The parameter samples θ (m) and θ (m−1) sampled at arbitrary m-th and m−1-th times in the Gibbs method have Markov chains whose transition kernel is the equation (46).

By sampling a sufficient number, specimen theta (m) together with reaching steady state Markov chain, the number of sampling has reached a steady state _{m s} subsequent samples _{θ (m = m s + 1} , m s +2, ···, m _e ) is equal to the sample from the joint a posteriori probability density function π (θ | u ₀ (····)).

Thus, the sample _{θ (m = m s + 1} , m s +2, ···, m e) by calculating the expected value and confidence intervals from each makes it possible to evaluate a parameter estimates and their reliability .

b) Sampling from the conditional a posteriori probability density function Conditional a posteriori probability density function π (α | u ₀ , m (above-), ξ, σ) is the random walk MH method, π (m above) | u ₀ , ω, ξ, σ) is the ratio of the maximum amplitude, π (π | u ₀ , ω, m (-above), σ) is the independent MH method, π (σ | u ₀ , ω, m ( Above-), ξ) use the standard deviations of the observed data and theoretical analysis model respectively.

  Then, in the present embodiment, the natural frequency ω is based on the random walk MH method, and the m-th sample candidate ω ′ is sampled by the equation (47). Note that ω (m−1) is the m−1-th sample, and N (0 | ν) is a normal distribution with an average of 0 and a variance ν.

The candidate sample ω ′ sampled according to equation (47) is accepted as a specimen in establishing equation (48). In the actual numerical calculation, a random number u _α is generated from the uniform distribution defined in the interval [0, 1], and determined according to equation (49).

  GJ (ligature) is based on the independent MH method, and sample candidate '' of the m-th sample is sampled from prior distribution by Formula (50).

  The sampled sample candidate '' is accepted as a sample with the probability of equation (51). In the actual numerical calculation, acceptance / rejection is determined in the same manner as equation (49).

The unit long mass m ^(m) (above-) is based on the sampled m-th sample ω ^(m) , ξ ^(m) , and the m-1 -th sample m ^(m-1) (-above) Formula (52) can be calculated based on the ratio between the maximum value of theoretical analysis and the maximum value of observation data.
Equation (52) means that the MCMC method incorporates the constraint that the maximum amplitude always matches the observed value. Thereby, it is possible to suppress the convergence to the parameter value such that the time series is always zero.

The variance σ ^{2 of the} error term can be calculated by equation (53).

(Response prediction analysis of MCMC method)
Next, response prediction analysis of the MCMC method will be described.

Parameters specimens _{θ (m = m s + 1} , m s +2, ···, m e) by theoretical analysis using, joint posterior probability density function [pi | displacement response according to (θ _{u 0} (·· above)) And samples of impact coefficient can be obtained.

Then, in the present embodiment, response prediction analysis at the time of running a train using the arbitrary train speed v _s expressed by the equation (54) as a parameter is incorporated into the m-th sampling process, and the maximum value (Eq. 55) Record.

The train speed v _s is, for example, 10 km / h to 400 km / h in 5 km / h steps.

In the present embodiment, by incorporating this way the response prediction process to calculate flow MCMC method, the unknown parameters _{θ (m = m s + 1} , m s +2, ···, m e) and estimation calculations at the same time, To obtain a sample u _max (v _s , m = m _s +1, m _s +2,..., M _e ) of maximum displacement according to the simultaneous a posteriori probability density function π (θ | u ₀ (····)) it can.

  Detailed inverse analysis and response prediction were implemented on MATRAB. The calculation time for one acceleration waveform (200 Hz, 20 seconds, 32 kBytes) is about 330 seconds (five and a half minutes) including inverse analysis (3000 samplings) and response prediction with a PC of CPU 3.5 GHz and RAM 4.00 GB is there. According to the dynamic FEM and estimation method based on data assimilation shown in the previous literature, it takes about 83,080 seconds (one day) for the acceleration waveform under the same conditions, which is about 1/250 of the calculation time. .

(Simplified inverse analysis method)
Here, although the above-mentioned detailed inverse analysis method (the structural performance investigation method of a railway bridge according to the present invention) aims to finely evaluate unknown parameters of the theoretical analysis model and the uncertainty thereof, MCMC method is repeated A return calculation is required, and some time and computational load can not be avoided. On the other hand, reverse calculation with less calculation load while securing a certain degree of accuracy, such as on-site measurement and survey with severe time constraints, programs embedded in smart sensors etc., batch processing of mass measurement data accumulated so far It is often desirable to be able to analyze.

  Therefore, in the present embodiment, a simplified inverse analysis method which satisfies such a demand will be described below.

  8 and 9 show the flow of the simple inverse analysis method according to the present embodiment.

  The unknown parameters of the bridge to be estimated from the measured data are the natural frequency f (= 2πω), the modal damping ratio 長, and the unit length mass m (-above). Among them, the estimation of the natural frequency f and the mode damping ratio 同 定 can be identified by the vibration characteristic identification method which has already been proposed, by regarding the residual waveform after passing the train as the free vibration waveform. Also, the unit length mass m (above-) can be uniquely calculated by determining other parameters as described above.

  Thereby, parameters necessary for theoretical analysis can be obtained based on observation data by using the existing vibration characteristic identification method, and displacement response and impact coefficient can be calculated by theoretical analysis based on the obtained parameters.

(Method of identifying vibration characteristics)
Here, the results of examining the applicability of the following three methods as a method of identifying the natural frequency and the mode damping ratio based on the residual waveform at the time of train passing (hereinafter, waveform after train passing) will be described.
1) FFT method and half power (HP) method 2) autocorrelation (AR) model 3) Eigensystem Realization Algorithm (ERA) method

1) The FFT method and the half power (HP) method are methods for identifying the natural frequency based on the peak frequency of the FFT spectrum and the modal damping ratio based on the HP method.
2) The AR model is a method of identifying the natural frequency and the modal damping ratio based on the estimated AR coefficient.
3) The ERA method is an identification method of natural frequency and modal damping ratio based on the minimum realization algorithm in linear time invariant system.
There have already been application results for many bridges in any of the identification methods, and in this embodiment, based on the accuracy verification results, an identification method that can accurately evaluate the natural frequency and modal damping ratio from the waveform after train passage is selected , Used in the simple inverse analysis method.

(Method for identifying unit long mass)
As described above, the unit length mass m (above-) controls the mode amplitude of the primary mode, and can be uniquely determined if other parameters are given.

In the simplified inverse analysis method of the present embodiment, theoretical analysis is performed using the natural frequency f identified based on the above identification method, the modal damping ratio ξ, and the initial unit length mass m ₀ (− above) = 1. 32)), calculate the ratio of the measured value to the maximum amplitude, and identify the unit length mass m (-on the top) with formula (56).

Then, by substituting the obtained parameters into theoretical analysis, it becomes possible to estimate the displacement response and the impact coefficient during train traveling.
The simplified inverse analysis time including the estimation of the impact coefficient is about 1.6 seconds under the same conditions as the above-mentioned detailed inverse analysis, which is about 1/200 of the detailed inverse analysis.

(Verification of accuracy by twin experiments)
Next, twin experiments will be performed to explain the verification results of the accuracy with respect to the detailed inverse analysis method (the structural performance survey method for a railway bridge according to the present invention) and the simple inverse analysis method.

  First, in the case of an underdetermined problem due to the excess degree of model freedom or the correlation between unknown parameters, the inverse analysis result does not necessarily converge to the correct value.

  The twin experiment is a basic method of verifying the validity of parameter estimation in such inverse analysis. First, time series data is created in advance by numerical calculation, and noise is added to create simulated observation data. Next, a parameter (prior distribution) different from time series data creation is set as an initial value, and inverse estimation based on simulated observation data is performed. 4 Verify the validity of parameter estimation based on whether the estimation results converge to the correct value.

  In this embodiment, based on the acceleration response when the train travels at speeds of 100 km / h, 200 km / h, and 300 km / h on the bridges of 10 m, 35 m, and 45 m with A, B, and C shown in FIG. A twin experiment was performed.

Table 2 shows parameters set at the time of time series data creation.
Moreover, FIG. 10 is an example of the simulation observation data which added the time series data and noise which were created by the theoretical analysis model, and the train passed the bridge (A bridge) of case A with a span of 10 m at speed 100km / h. The case is shown.
In addition, the noise which generate | occur | produced from the normal distribution which makes 5% of a largest displacement a standard deviation was added to the acceleration waveform with reference to the existing measurement result.

(Result of parameter estimation)
FIG. 11 shows the posterior distribution obtained by the MCMC method.
In this MCMC method, 3000 samplings were performed, the first 1000 samplings were removed as a convergence process to a steady distribution, and the remaining 2000 sampling results were used as samples from the posterior distribution.
Then, it was confirmed from FIG. 11 that the posterior distribution is updated near the correct value regardless of the speed of any parameter.

FIG. 12 shows the estimation results of the natural frequency, the modal damping ratio and the unit length mass.
It was confirmed that the natural frequency, modal damping ratio and unit length mass based on the detailed inverse analysis all converge to the correct value. Moreover, it was confirmed that the detailed inverse analysis method can estimate the parameter with high accuracy compared with the existing identification method shown together in FIG.

  Thereby, in the detailed inverse analysis method of the present embodiment (the structural performance survey method of a railway bridge according to the present invention), the problem of underdetermination can be avoided, and the parameters can be accurately determined even if the prior information is somewhat different from the correct value. It was confirmed that it could be estimated.

  Further, FIG. 12 shows the natural frequency identified by the existing method, the modal damping ratio, and the unit length mass calculated by the equation (56) in order to select the identification method used for the simple inverse analysis method.

  In addition, in the FFT, 0 vector is added after time series to secure a frequency resolution of 0.2 Hz. In addition, the Berg's method was used to estimate the AR coefficient, and the model order of the AIC minimum was adopted. In the ERA method, a degree of freedom was 10, and a mode satisfying an observability MAC value of 0.99 or more and a mode attenuation ratio [0.001 to 0.05] was identified as a mode of a railway bridge.

  And from FIG. 12, it was confirmed that a comparatively big error arises with train speed in the natural frequency based on ERA method, the mode damping ratio based on AR model, and unit length mass. As for the identification method used for the simple inverse analysis method, the simple and basic method is practically convenient if there is no significant difference in accuracy.

  On the other hand, the results based on the FFT method and the HP method have errors of up to 2% for natural frequency, up to about 45% for modal damping ratio, and up to 25% for unit length mass. Among the methods, it was the most satisfying method.

  Thus, in the present embodiment, in the simple inverse analysis method, the FFT method is used to identify the natural frequency, and the HP method is used to identify the modal damping ratio.

(Reliability of parameter estimates)
Next, we examined the reliability of the parameter estimates.

  FIG. 13 shows parameter estimates by the detailed inverse analysis method and 90% confidence intervals. In FIG. 13, the vertical axis of each parameter is different. Also, the vertical axis shows the correct value as 1. Furthermore, FIG. 13 also shows the 90% confidence interval of the estimated value.

  As shown in FIG. 13, it was confirmed that the confidence interval has the smallest natural frequency and the largest modal damping ratio. However, the size of the 90% confidence interval was about + 5% to -5% even at the largest mode attenuation ratio. In addition, the reliability increased at a train speed of 200 km / h in the case C bridge (C bridge) with a span length of 45 m due to the increase of natural vibration accompanying the occurrence of resonance.

  From this, it was inferred that the magnitude of the dynamic response affects the estimation accuracy.

(Estimated result of displacement response)
Next, FIG. 14 shows displacement responses estimated based on the simple and detailed inverse analysis method.
Further, FIG. 14 shows the result of the C bridge (100 km / h, 300 km / h) with a span length of 45 m with the largest error, which is implemented by the FFT method and the HP method.

  As shown in FIG. 14, it was confirmed that the displacement response estimated from the acceleration by the detailed inverse analysis method matches the correct value with high accuracy. Moreover, the displacement estimated by the simple inverse analysis method has a value close to the boundary of the 90% confidence interval in the detailed inverse analysis method, and it is confirmed that the displacement estimation accuracy is slightly reduced compared to the detailed inverse analysis method The However, the estimation error was at most about 10% of the maximum displacement.

(Estimated result of maximum displacement)
Next, FIG. 15 shows estimated values of the maximum displacement other than the estimated velocity calculated based on the detailed inverse analysis method and the simple inverse analysis method.

  Here, although an impact coefficient is practically used, an error is mixed in for each of the static displacement and the dynamic response magnification, and the impact coefficient is an impact in the subsequent analysis for the purpose of grasping the accuracy in more detail. Use the maximum displacement instead of the factor.

  As shown in FIG. 15, as a result of the detailed inverse analysis of the solid line, the maximum displacement can be evaluated accurately even at train speeds other than the estimated speed. Moreover, in the detailed inverse analysis method, when based on the acceleration of the train speed 100 km / h in the C bridge (45 m), the error was about 3% at the maximum.

On the other hand, in the simplified inverse analysis method indicated by the dotted line, it was confirmed that the estimation error tends to be larger as the bridge is longer. When the span length is long, the natural frequency is low, so it can be considered that the error due to the frequency step is relatively large.
Moreover, when based on the acceleration of the train speed 100 km / h in the C bridge, the error with respect to the correct value was 10% on average and 18% at maximum at resonance.

  As a result, in the simple inverse analysis method, the identification accuracy of the eigenfrequency and modal damping ratio greatly affects the displacement estimation. Therefore, particularly when a bridge with a low eigenfrequency is targeted, it is possible to measure the fine movement or multiple times. Therefore, it is desirable to average those results and improve the estimation accuracy.

Next, FIG. 16 shows the details of C bridge and the maximum value estimation result based on the simplified inverse analysis method.
Although the simple inverse analysis has the above-mentioned error, it has been confirmed that the characteristic of response amplification by resonance can be consistently evaluated.

(Influencing factor analysis by numerical analysis)
Next, analysis results of influence factors by numerical analysis will be described.

  The influence of the train speed etc. on the inverse analysis method of this embodiment is analyzed in more detail. Table 3 shows a list of analysis cases.

  Taking train speed, noise, interaction with vehicles, and track displacement as influence factors, we decided to analyze the influence of each parameter on the inverse analysis accuracy.

  The theoretical analysis result of train speed 100 to 400 km / h (in 10 km / h increments) with noise of 1% of maximum amplitude for the bridge with 10 m span and 45 m span with the highest estimation accuracy The detailed inverse analysis was performed based on this.

  Case I is a basic case to compare the effects of train speed and other cases.

  Case II is a case in which noise with a standard deviation of 5% of the maximum acceleration is added as observation data, and the effect of noise on estimation accuracy is clarified by comparison with case I.

  Case III is a case where the detailed inverse analysis method based on the analysis of interaction with DIASTARS was carried out for bridges with a span of 10 m and 45 m in order to examine the influence of the dynamic interaction between the vehicle and the bridge on the inverse analysis accuracy.

  Case IV is an inverse analysis based on the interaction analysis with the orbital displacement to verify the influence of the orbital displacement which is one of the main influencing factors to the dynamic response.

  The track | orbit displacement introduced in FIG. 17 is shown. Trajectory displacement is described in "Sogabe Masamichi, Matsumoto Nobuyuki, Fujino Yozo, Sasai Hajime, Kanamori Makoto, Miyamoto Masaaki: Dynamic Design of Concrete Railway Bridge in Resonant Region. Journal of the Japan Society of Civil Engineers, Vol. 724, pp. 83-102 , 2003. "was used. The structure / vehicle model is as shown in FIG. In addition, the number of repetitions of the MCMC method was 3000, and the first 1000 were removed as a convergence process, and the remaining 2000 were used as samples from the posterior distribution. Furthermore, by making the prior distribution uniform (without prior information), the influence of prior information was eliminated.

(Influence on parameter estimation)
FIG. 18 and FIG. 19 show the results of detailed reverse analysis of each parameter at a train speed of 100 to 400 km / h for a bridge with a span of 10 m and 45 m.
FIG. 20 and FIG. 21 show 90% confidence intervals of the natural frequency as an example of the estimated 90% confidence intervals.

  In each case, the correct value is 1 and represented by a non-dimensionalized value. Also, the impact coefficient is shown together for analysis of the relationship with the dynamic response component. Furthermore, in the simple inverse analysis method, the variation of the method itself is large, and no clear change was observed in each case, so only the result of the detailed inverse analysis method is shown. In the simple inverse analysis method, the maximum error with respect to the correct value for any span length and case is about ± 7% for natural frequency, about −80% to + 100% for mode damping ratio, and about ± 30% for unit length mass there were.

(Basic case)
From FIG. 18 and FIG. 19, the estimation result when the load string analysis result is an observed value accurately matches the correct value with any span length, parameter, and train speed, and the method of this embodiment is an underdetermined problem. It was confirmed that it could be avoided.

  From FIGS. 20 and 21, it has been confirmed that the 90% confidence section of the natural frequency shows a particularly small (high accuracy) tendency particularly near the peak of the impact coefficient in a bridge with a span of 10 m. It is considered that this is because the natural vibration component is increased by the resonance. On the contrary, it was confirmed that the 90% confidence interval tends to show a large value near the velocity of the shock coefficient minimum where the natural frequency component of the bridge is hard to be excited.

  Although the reliability of the estimated value fluctuates depending on the impact coefficient representing the magnitude of the dynamic response component, a bridge whose 90% confidence interval of the natural frequency estimated by load string analysis is 10m in span The maximum is about 0.1%, and the maximum length is about 0.05% for a 45m long bridge. Both can be said to be sufficiently high as the reliability of the estimated value.

(Effect of noise)
Next, the influence of noise will be described.

  As shown in FIGS. 18 and 19, in the time series estimation result with 5% noise added, errors of up to about 5% in mode attenuation ratio and up to about 1% in unit length mass are recognized. It can be estimated accurately. From this, it was confirmed that the influence of the noise on the estimation accuracy is limited, and in particular, it hardly affects the estimation of the natural frequency.

  Moreover, as shown in FIG. 20 and FIG. 21, it was confirmed that the 90% confidence section of the natural frequency shows the same tendency as the result of the load sequence analysis. However, the value is increased by the addition of noise, and the maximum value is about 0.4% for a 10m long bridge and about 0.1% for a 45m long bridge. From these results, it was confirmed that the noise contained in the observation data tends to reduce the reliability of the estimated value.

(Influence of interaction with vehicle)
Next, the influence of the interaction with the vehicle will be described.

  As shown in FIG. 18 and FIG. 19, in the estimation results when analysis results by DIASTARS (interaction analysis with vehicle) are given as observation data, the deviation from 1 which is the correct value is confirmed for any estimation parameter. It was done.

  In addition, regarding the natural frequency, it was confirmed that the 10m span was estimated to be lower by about 5% from the correct value, and the 45m bridge was reduced by about 3% at maximum. This is considered to be due to the apparent natural frequency decreasing due to the passing train mass. That is, it is estimated that it is an error due to modeling of the variable axial load caused by the influence of the vehicle vibration system with a time-invariant moving load.

  In addition, with regard to the modal damping ratio, it was also found that the maximum deviation from the correct value by 50% and the minimum deviation by about 60%. This error is also considered to be the additional damping that the damping mechanism of the vehicle exerts on the bridge through the coupling of the bridge primary mode and the vehicle vibration mode. On the other hand, it is also seen that the attenuation is lower than the correct value. Both occur at a speed slightly shifted to the high speed side and the low speed side from the resonance speed at which the impact coefficient is maximized, which is considered to be the effect of the beating phenomenon.

  Since the theoretical model can not evaluate the decrease in apparent natural frequency during train travel in interaction analysis, a difference also occurs in the beat phenomenon during train passing that occurs around the resonance speed. It is inferred that the mode attenuation ratio has apparently decreased to compensate for this difference.

  The unit length mass tended to be higher than the correct value for any bridge and contrary to the estimated natural frequency. This is because the natural frequency is apparently lowered by the train mass in order to match the static displacement (acceleration) at the time of train traveling determined by the natural frequency and the unit length mass to the time series waveform, the unit length mass is incidentally Is considered to have increased.

  As shown in FIG. 20 and FIG. 21, the 90% confidence section of the natural frequency is larger by one digit or more especially for a bridge with a span of 10 m compared to the result of load string analysis (only in FIG. Scale is different). It has been confirmed that the confidence interval is increased at the antiresonance speed at which the impact coefficient is minimized at any of the bridges.

(Influence of orbital displacement)
Next, the influence of track displacement will be described.

  As shown in FIG. 18 and FIG. 19, the estimated value in the case of giving DIASTARS (interaction analysis with a vehicle) in consideration of the track displacement as observation data was almost the same as the interaction analysis in any of the bridges. From this, it was confirmed that the influence of the track displacement on the estimation accuracy is small compared to the influence of the interaction with the vehicle.

  As shown in FIGS. 20 and 21, the 90% confidence interval of the natural frequency has a tendency to increase in particular in the velocity zone where the impact coefficient is minimized, when the trajectory displacement is considered.

  As described above, as the influence of various factors on the estimation accuracy of the inverse analysis method of the present embodiment, the interaction with the vehicle is the largest, and the value is about ± 5% with respect to the natural frequency and the mode damping ratio On the other hand, it is confirmed that the maximum is -60% to + 50%, and it is about + 10% with respect to the unit length mass.

  In particular, it was confirmed that the natural frequency tends to be more influential when the span length is short. In addition, it has become clear that the noise and track deviation affect the confidence interval of the estimated value but the influence is small compared to the interaction with the vehicle.

(Influence on estimation of maximum displacement)
Next, the influence on the estimation of the maximum displacement is explained.

22 and 23 show the estimation results of the maximum displacement for each case.
In parameter estimation, the influence of interaction with the vehicle was seen, but as shown in Fig. 22 and Fig. 23, the maximum displacement was accurately estimated, and the influence of the interaction with the vehicle on the maximum displacement estimation was small Was confirmed.

  From this, it is considered that the difference from the correct value found in the natural frequency and the mode damping ratio is the result of equivalently evaluating the effects of the additional mass and the additional attenuation caused by the traveling vehicle using a simple model.

  As described above, it has become clear that the maximum displacement can be accurately estimated by the inverse analysis method of the present embodiment, and the influence of noise, interaction with a vehicle, and track displacement on the estimated value of the maximum displacement is small.

(Inverse analysis method of this embodiment based on actual measurement response / accuracy verification of structural performance investigation method of railway bridge)
Next, the result of verifying the accuracy of the method of this embodiment using the actual measurement response will be described.

First, specifications and appearances of application targets are shown in Table 4 and FIG.
The target bridges are an open floor RCT girder bridge with a span of 25 m and an open floor PC box girder with a span of 35 m. In addition, the acceleration response during train travel measured in the center of the span of the bridge in the past was used as observation data. Moreover, the displacement measured at the same place was used for verification. The acceleration and displacement are measured by a piezoelectric accelerometer and a ring displacement meter, respectively. The method is applied to the acceleration at the time of passing each of two bridges on a limited express train (vehicle length 21.3 m, 6-car train formation) passing through the girder on the sensor installation side.

Pre-processing was performed to use the measured acceleration for the inverse analysis.
First, since only the first order mode is considered in the theoretical analysis model, high frequency components are removed by low pass filter processing. In the present embodiment, low-pass filter processing of 20 Hz is performed. Next, the train speed was identified. In addition, each axle weight of the train was a capacity ride. Finally, time synchronization was performed using cross correlation between theoretical analysis data and observation data in the detailed inverse analysis method.

  FIG. 25 shows an example of a measured acceleration waveform when the train passes a bridge with a span of 35 m at 136.5 km / h. In the filtered waveform after filtering, it was confirmed that the amplitude increased as the train passed.

(Result of parameter estimation)
Next, the estimation results of the parameters will be described.

FIG. 26 shows acceleration by theoretical analysis before and after estimation by MCMC method. Further, FIG. 27 and FIG. 28 show parameter estimation results by the simple and detailed inverse analysis method.
27 and 28 also show the average values of the two cases of the simplified inverse analysis method results. In the detailed inverse analysis method, the MCMC method was repeated 3000 times, the first 1000 times were removed, and the remaining 2000 times were used as samples from the posterior distribution. The initial value is the design value shown in Table 4, and the modal damping ratio is 2%.

  It was confirmed from FIG. 26 that the parameters were corrected in accordance with the actual measurement value by the detailed inverse analysis method in any of the bridges, and the reproducibility of the measured acceleration was significantly improved.

  Moreover, it was confirmed from FIGS. 27 and 28 that the natural frequency and unit length mass according to the simplified inverse analysis method tend to have a larger variation than the natural frequency and unit length mass based on the detailed inverse analysis method. Furthermore, it was confirmed that the modal damping ratio estimated by the simplified inverse analysis method is 30 to 50% lower than that of the detailed inverse analysis method. The simple inverse analysis method is for the waveform after passing the train without interaction with the train, but the detailed inverse analysis method is for the waveform at the time of passing the train and is generated by the interaction with the vehicle It is considered that the additional damping is equivalently evaluated as the bridge mode damping ratio.

  Further, FIGS. 27 and 28 show 90% confidence intervals of the estimated values by error bars. In any case, the range was small, and it was confirmed that the underdecision problem could be avoided even when the measured acceleration was used.

(Estimated result of displacement response)
Next, estimation results of displacement response will be described.

  The displacement response estimated by the detailed inverse analysis method and the simple inverse analysis method is shown in FIG. 29 and FIG. These figures also show the measured displacements (measured values). Also, the enlarged view shows the response near the maximum displacement. In addition, the value of measurement displacement is correct | amended by the correction coefficient according to the theoretical analysis model which does not consider the torsion component. The correction factor was calculated from the maximum displacement when the train passed the measuring side and the non-measuring side at almost the same speed, and was 0.59 for a span of 25 m and 0.68 for a span of 35 m.

  As shown in Fig. 29 and Fig. 30, the displacement estimated by the detailed inverse analysis tends to evaluate the upper amplitude slightly larger for a bridge with a span of 35 m compared to the measured displacement, but the average at the time of train passing In each case, it was confirmed that the typical deflection and the minimum value were in good agreement with the measured displacement. In addition, the displacement based on the simple inverse analysis method agrees well with the measured displacement in the case of a train speed of 138.8 km / h with a span length of 25 m, but it was confirmed that there is a clear error in other cases. . On the other hand, when the average value of the parameter estimated by the simple inverse analysis method was used in each bridge, it was confirmed that the accuracy was improved by approaching the measured displacement which is the correct value.

(Estimated result of maximum displacement)
Next, the estimation results of the maximum displacement will be described.

Table 5 shows values obtained by arranging the estimation error of the displacement response by the maximum displacement.
As shown in Table 5, it was confirmed that the detailed inverse analysis can estimate the maximum displacement within the range of -3% to -6% of the correct value. On the other hand, it was confirmed that the simple inverse analysis has a relatively large error of -9% to + 23% with respect to the correct value. However, it was also confirmed that the error with respect to the correct value can be suppressed from -8% to + 5% by using the average of the parameter estimation values of two cases even in the simplified inverse analysis method.

  FIG. 31 shows the relationship between the train speed and the maximum displacement calculated by the detailed inverse analysis method and the simple inverse analysis method. Peaks were observed at almost the same speed for all target bridges by the simple inverse analysis method and the detailed inverse analysis method. In addition, it was confirmed that the average value of the parameter estimation value by the simple inverse analysis method is used for any of the bridges to approach the estimation value of the detailed inverse analysis method other than the measurement speed.

  Here, since the uncertainty is quantified in the detailed inverse analysis method, the upper limit of the 90% confidence interval can also be used as an evaluation on the safety side. For example, in the case of speed improvement up to 300 km / h for a bridge with a span of 35 m, the maximum displacement is about 1.3 mm at the expected value at 145 km / h traveling and about 1 at the upper limit of 90% It can be predicted to be .4 mm.

(Evaluation of estimated displacement)
Next, evaluation of estimated displacement will be described.

  Based on the theoretical model estimated by the inverse analysis, we checked the deflection that is determined from the driving safety and riding comfort that always reflect the current situation. Regarding maintenance and management of existing structures, railway structures maintenance management standards and explanations (hereinafter referred to as maintenance management standards) stipulate the performance check by equation (57).

Here, K _m is a coefficient for maintenance management index J, γ _i is a structure coefficient, I _Rm and I _Lm are a maintenance management response value and a maintenance management limit value, respectively.

With regard to the checking of traveling safety and ride comfort, in the railway structure design standard and description (displacement limitation), K _m = 1 and γ _i = 1. The calculation of response values and limit values at the time of survey requires the application of more realistic values, and this is the case with theoretical analysis models estimated as described above.

  Therefore, by using the theoretical analysis model obtained by the method of investigating the structural performance of a railway bridge according to this embodiment and the train load for calculating the response value shown below, seamless verification based on the maintenance management standard becomes possible.

  Here, based on the operation status of the relevant route, a maximum speed of 140 km / h, a six-car limited express vehicle (vehicle length 21.3 m) is assumed. As the condition of the traveling vehicle, single ride on capacity ride is adopted for ride comfort and double carriage on 350% ride for traveling safety ・ Simplified vehicle fluctuation can not be considered in theoretical analysis model, so the design impact coefficient of equation (58) Based on the impact factor.

Here, _ic is an impact coefficient of vehicle sway, and in the target bridge, it is respectively set to 0.11 (25 m) and 0.10 (35 m) according to the existing literature. Moreover, the reduction coefficient (beta) _{i with} respect to the impact coefficient at the time of double wire loading was 0.875 (25 m) and 0.825 (35 m). The impact coefficient i _α of the velocity effect was calculated by the theoretical analysis in which the moving load was determined as described above.

  Table 6 summarizes the driving safety and ride comfort limit values, response values, and maintenance management indexes at all times. Response values were calculated using the expected value in the detailed inverse analysis method, the upper limit of 90%, and the average value in the simplified inverse analysis method.

  By estimating the displacement based on the acceleration by the method of investigating the structural performance of the railway bridge of this embodiment, the response value of the two bridges targeted is sufficiently smaller than the limit value, and the deflection limit determined from the traveling safety and the riding comfort is obtained. It can verify quantitatively that it fills. In addition, it was confirmed that by using the average value of the simple inverse analysis method, it is possible to evaluate the running safety and the maintenance management index J for the ride with almost the same accuracy as the detailed inverse analysis method.

  As described above, in this embodiment, in order to estimate the displacement response and the impact coefficient based on the acceleration during train traveling, the theoretical analysis model is formulated, and the detailed inverse analysis method based on MCMC method and the FFT method and HP method are used. While explaining the simple inverse analysis method, verification based on numerical analysis and actual measurement was conducted. Then, the following findings and effects were obtained.

(1) Formulating a theoretical analysis model during train operation that can evaluate displacement and impact coefficient with the same accuracy as moving load analysis based on the finite element method, and simple reverse using the detailed inverse analysis method by MCMC method and the existing identification method I came up with an analysis method.

(2) Although this method is limited to simple digit application, it can be calculated in about 1/250 of the existing inverse analysis method in the detailed inverse analysis method and in about 1/200 of the detailed inverse analysis method in the simple inverse analysis method. is there.

(3) The parameter estimation error of the detailed inverse analysis method based on the twin experiment can be estimated with high accuracy compared with the existing identification method with -2% to + 1%. In addition, it becomes possible for the FFT method and the HP method to identify the natural frequency and the modal damping ratio with relatively high accuracy for the waveform after passing the train.

(4) As a result of studying the influence of train speed, noise, interaction with vehicles and track displacement on inverse analysis by numerical analysis, the modeling error for interaction with vehicles is the largest, and for natural frequency There is a variation of about -50% to -50% to the modal damping ratio, but any significant change can not be seen in the displacement estimation, and accurate estimation can be made.

(5) As a result of performing the simple and detailed inverse analysis method on the acceleration measurement results of two types of actual bridges, the simplified inverse analysis method has particular variation of natural frequency and unit length mass compared with the detailed inverse analysis method Although the maximum displacement is -3% to + 6% in the detailed inversion method and -9% to + 23% in the simple inversion method, the average value of a plurality of measurement results is also used in the simple inversion method. The maximum displacement can be estimated with an accuracy of -8% to + 5%.

  Therefore, in the method for investigating the structural performance of a railway bridge according to the present embodiment, displacement response during train travel necessary for estimation of the bridge condition, travel safety and comfort evaluation is accurately obtained from the acceleration response of one point of the bridge It will be possible to

  Further, among the estimation parameters based on the actual acceleration of the bridge during train travel, the natural frequency is related to the bridge stiffness, so the natural frequency can be accurately estimated, so that the structure performance can be accurately estimated. .

  Furthermore, it is possible to predict responses at other train speeds from the measured acceleration of the bridge during train travel.

  In addition, since the uncertainty of the estimation parameter can be quantified, it is possible to evaluate the risk of changing according to the estimation accuracy, such as adoption of the 95% confidence interval lower limit value.

  Furthermore, based on the uncertainty of the estimation parameter that changes according to the estimation accuracy, it is also possible to evaluate the value of more accurate measurement from the aspect of estimation accuracy.

  In addition, calculating the margin with respect to the maintenance management reference value may make it possible to determine the maintenance management priority and to extract weak structures.

  Therefore, according to the method for investigating the structural performance of a railway bridge of the present embodiment, it is possible to appropriately investigate and evaluate the structural performance of the bridge using observation data of the acceleration response of the bridge during train travel.

  As mentioned above, although one embodiment of the structural performance investigation method of the railway bridge of this embodiment concerning the present invention was described, the present invention is not limited to the one above-mentioned embodiment, In the range which does not deviate from the meaning, suitably It can be changed.

1 train (vehicle)
2 Bridge

Claims (7)

The theoretical analysis model of the dynamic response of the railway bridge during train operation with the train as the moving load train and the bridge as the simple beam is formulated as the following equation (1),
A method of investigating the structural performance of a railway bridge characterized in that the acceleration of a bridge at the time of train travel is measured, and the unknown parameters of the theoretical analysis model are estimated by the inverse analysis method from the acceleration data.

here,
x is the distance (position), t is the time when the time when the first axle of the leading vehicle enters the bridge is 0, ν (t) is a step function of 0 at t <0, 1 at 0 ≦ t, τ _{k, i} is the point at which the k-th vehicle passes the i-th axle.
u (above...) (x, t) is an acceleration response, and is expressed by the following equation (2), equation (3) to equation (10).

here,
u (above) (x, t) is a velocity response, and u (x, t) is a displacement response.
m (above-) is the bridge mass per unit length (per unit length mass), c is a damping coefficient, and EI is bending stiffness. L _b is the span length of the bridge, n is the mode order, T is the bridge transit time, ω _n is the nth natural angular frequency, ξ _n is the nth mode damping ratio, ω _n (-above) is one axle The n-order component of each frequency passing through the axle with half cycle from the bridge approach to the exit, _{n n} (x) is the n-order vibration mode type, and u ₀ is the static deflection amount when the axial weight acts on the middle of the span It is. In the method for investigating the structural performance of a railway bridge according to claim 1,
An error term is introduced into the theoretical analysis model of the above equation (1) to define a probability model of the following equation (11),
The simultaneous occurrence probability (likelihood function) L (u) of the following equation (12) in which the acceleration data u ₀ (······) is generated from the equation (11) when an unknown parameter θ is given as a condition ₀ ) and the a priori probability density function π (θ) of the unknown parameter of the following equation (13) into the following equation (14) obtained by the Bayes theorem, Railway bridge characterized by determining the simultaneous a posteriori probability density function π (θ | u ₀ ) of unknown parameters at the same time and reflecting the estimated parameters and the uncertainty of the parameters to evaluate the structural performance of the railway bridge Structural performance survey method.

here,
N _omega mean mu _omega, normal distribution of variance σ _ω ^2, GJ (ligature) _xi] shows the gamma distribution shape factor alpha _xi], scale factor gamma _xi]. Θ is the parameter space. In the method for investigating the structural performance of a railway bridge according to claim 2,
Using the MCMC method for obtaining a probability density function that approximates the simultaneous a posteriori probability density function by numerical calculation, sampling from the conditional a posteriori probability density function is repeated in order for each of a plurality of elements that are unknown parameters A method of investigating structural performance of a railway bridge characterized by determining a posterior probability density function. In the method for investigating structural performance of a railway bridge according to claim 3,
The sample of the sampled parameter reaches the steady state of the Markov chain, and the expected value and the confidence interval are calculated from the sample of the parameters after the sampling number which reached the steady state, and the reliability of the parameter estimated value and the parameter estimated value A method of examining structural performance of a railway bridge characterized in that it is evaluated. In the method for investigating the structural performance of a railway bridge according to claim 3 or claim 4,
The unknown parameters θ are ω (natural period), m (-above) (bridge mass per unit length), ξ (mode attenuation ratio), σ (probability error term dispersion),
The conditional a posteriori probability density function π (ω | u ₀ , m (above −), ξ, σ) is a random wall MH method, π (m (above −) | u ₀ , ω, ξ, σ ) Is the ratio of the maximum amplitude, π (ξ | u ₀ , ω, m (above-), σ) is the independent MH method, π (σ | u ₀ , ω, m (above-), ξ) is the acceleration A method of investigating the structural performance of a railway bridge characterized by performing sampling from the conditional a posteriori probability density function using data and standard deviation of a theoretical analysis model respectively. In the method of investigating the performance of a railway bridge according to any one of claims 3 to 5,
A method of investigating the structural performance of a railway bridge characterized by obtaining a sample of displacement response and impact coefficient according to the simultaneous a posteriori probability density function by theoretical analysis of MCMC method using samples of sampled parameters. In the method of investigating the structural performance of a railway bridge according to claim 6,
Response prediction analysis at the time of train running with the arbitrary train speed v _s expressed by the following equation (15) as a parameter is incorporated into the m-th sampling process to obtain the maximum value at each arbitrary speed, and MCMC method A method of structural performance investigation of a railway bridge characterized in that a response prediction process is incorporated into the calculation flow of, so that a sample of the maximum displacement according to the simultaneous a posteriori probability density function is obtained simultaneously with the estimation calculation of unknown parameters.

Images