JP2018031187A - Examination method of structure performance of railway bridge - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide an examination method of structure performance of a railway bridge capable of properly investigating and evaluating structure performance of a bridge using a piece of measurement data of acceleration response of the bridge when a train travels thereon.SOLUTION: The examination method of structure performance of a railway bridge includes the steps of: assuming a train as a moving load train and a bridge as a simple beam, formulating a theoretical analysis model of dynamic response of the railway bridge when the train runs thereon; measuring acceleration of the bridge when the train runs thereon; and estimating an unknown parameter of a theoretical analytic model from the acceleration data using an inverse analysis method.SELECTED DRAWING: Figure 6

Description

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

  There is a demand for a method for efficiently and effectively maintaining and managing a huge number of existing railway bridges. For example, the development of a method for quantitatively and continuously measuring the structural performance of a bridge as easily as possible by installing sensors, etc. Is considered a very important issue.

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

  For example, the mode characteristics such as the natural frequency of the bridge and the mode damping ratio are estimated from the measured acceleration. Mode characteristics such as the natural frequency and mode damping ratio of this bridge indirectly represent structural performance. Therefore, by continuously measuring the acceleration of the bridge with a sensor, the natural frequency, mode damping ratio, etc. Attempts have been made to ascertain abnormalities of bridges (structures) and aging deterioration by accumulating mode characteristics and structural data of bridges, comparing them relative to each other, and capturing changes over time.

  In addition, the impact coefficient is used as an important index for evaluating the dynamic response during train travel. The impact coefficient is an important indicator in maintenance management because it is closely related to deterioration phenomena such as cracks in railway bridges. When evaluating the impact coefficient of a railway bridge, a ring displacement meter, video measurement, U Doppler (non-contact vibration measurement system for structures with a built-in laser Doppler velocimeter), etc. are used. The vertical displacement is measured on the bridge side (ground side) (deflection measurement), and the impact coefficient is obtained using this measurement result (see, for example, Patent Document 1).

JP 2012-233758 A

  Here, in the field of railway technology in Japan, a great deal of knowledge about the design of railway bridges and the like has already been accumulated in order to suitably secure and maintain the safety and usability of railway bridges and thus railways. Most of the structural performance used in the design of railway bridges based on such a vast amount of knowledge is based on displacement, and in addition to static displacement at low speeds, an impact coefficient that represents response amplification at high speeds is typical. It is regarded as a structural performance index.

  However, the measurement of the displacement of the bridge during train travel is an absolute requirement that the fixed points are always set accurately and strictly in order to ensure the measurement accuracy, and the measurement environment In addition to severe restrictions, the measurement load is larger than acceleration measurement. For this reason, it is practically difficult to use as a method for investigating the structural performance of a huge number of existing railway bridges, and as a method for maintaining and managing a huge number of existing railway bridges.

  On the other hand, many methods for estimating the displacement response based on the acceleration response during train travel have been proposed, but problems remain in terms of estimation accuracy and calculation load, and indicators that require extrapolation such as impact coefficient cannot be obtained. Therefore, at present, the practical application is limited. For this reason, the development of a method that can accurately estimate the displacement response and vibration characteristics of the 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 for investigating the structural performance of a railway bridge, which makes it possible to suitably investigate and evaluate the structural performance of the bridge using observation data of the acceleration response of the bridge during train travel. For the purpose.

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

  The method for investigating the structural performance of a railway bridge according to the present invention formulates a theoretical analysis model of the dynamic response of a railway bridge when the train is run using a train as a moving load train and a bridge as a formula (1) below. The present invention is characterized in that the acceleration of the bridge when the train travels is measured and the unknown parameter of the theoretical analysis model is estimated from the acceleration data by an inverse analysis method.

here,
x is a distance (position), t is a 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 when t <0, 1 when 0 ≦ t, τ _{k, i} is the time when the k-th vehicle passes the i-th axle.

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

here,
u (above) (x, t) is a velocity response, and u (x, t) is a displacement response.
m (upward-) is the bridge mass per unit length (unit length mass), c is the damping coefficient, and EI is the bending stiffness. L _b is span length of the bridge, n represents mode order, T is a bridge transit time, omega _n is n th order natural angular frequency, xi] _n the n-th order mode damping ratio, omega _{n (above} -) is the primary axle The nth order component of each frequency passing through the axle with a half cycle from entry to exit of the bridge, Φ _n (x) is the nth order vibration mode shape, and u ₀ is the amount of static deflection when axial load acts on the center of the span It is.

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

here,
N _ω is a normal distribution with mean μ _ω and variance σ _ω ² , GJ (ligature) _ξ is a gamma distribution with shape factor α _ξ and scale factor γ _ξ . Θ is a parameter space.

  Further, in the method for investigating the structural performance of a railway bridge according to the present invention, the MCMC method is used to obtain a probability density function that approximates the simultaneous posterior probability density function by numerical calculation, and each of a plurality of elements that are unknown parameters is sequentially used. It is more desirable to obtain the simultaneous posterior probability density function by repeating sampling from the conditional posterior probability density function.

  In the method for 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 from the sample of the parameter after the number of samplings reaching the steady state. More preferably, the parameter estimation value and the reliability of the parameter estimation value are evaluated.

Furthermore, in the method for investigating the structural performance of a railway bridge according to the present invention, the unknown parameter θ is ω (natural period), m (upward-) (bridge mass per unit length), ξ (mode damping ratio), σ (probability error term variance), and π (ω | u ₀ , m (upward −), ξ, σ) of the conditional posterior probability density function is the random wall MH method and π (m (upward − ) | U ₀ , ω, ξ, σ) is the maximum amplitude ratio, π (ξ | u ₀ , ω, m (upper −), σ) is the independent MH method, and π (σ | u ₀ , ω, m). (Above-) and ξ) are preferably sampled from the conditional posterior probability density function using acceleration data and the standard deviation of the theoretical analysis model, respectively.

  In the method for 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 posterior probability density function by theoretical analysis of the MCMC method using a sample of sampled parameters. .

Furthermore, in the structural performance survey method railway bridge of the present invention, include responses predictive analysis at a train traveling that any train speed v _s represented by the following formula (15) as a parameter to the m-th sampling process In addition to obtaining the maximum value at any speed and incorporating the response prediction process in the MCMC calculation flow, it is possible to obtain a sample of the maximum displacement according to the simultaneous posterior probability density function at the same time as estimating the unknown parameter. More desirable.

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

  Of the estimated parameters based on the measured acceleration of the bridge during train travel, the natural frequency is related to the bridge stiffness, so the natural frequency can be estimated accurately, so that the structural performance can be estimated accurately. .

  Furthermore, response prediction at other train speeds can be performed based on the actual acceleration of the bridge during train travel.

  In addition, since the uncertainty of the estimation parameter can be quantified, for example, an evaluation that guarantees a risk that changes according to the estimation accuracy, such as adoption of a lower limit value of 95% confidence interval, can be performed.

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

  Also, by calculating the margin with respect to the maintenance management reference value, it can be used to determine the priority of maintenance management and extract weak point structures.

  Therefore, according to the structural performance investigation method of the railway bridge of the present invention, it is possible to suitably investigate and evaluate the structural performance of the bridge using the observation data of the acceleration response of the bridge during train travel.

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 which shows a structure model (interaction analysis and load train analysis). It is a figure which shows the displacement response computed with each analysis method of a load train analysis method, a dynamic interaction analysis method, and a simple analysis method (the structure performance investigation method of the railway bridge which concerns on one Embodiment of this invention). It is a figure which shows the impact coefficient computed by each analysis method of a load train analysis method, a dynamic interaction analysis method, and a simple analysis method (the structure performance investigation method of a railway bridge according to the present 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 a simple reverse analysis method. It is a flowchart which shows a simple reverse analysis method. It is a figure which shows an example of acceleration time series data. It is a figure which shows the convergence example 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 a maximum displacement. It is the figure which compared the correct value and estimated value of the maximum displacement of C bridge. It is a figure which shows the track | orbit displacement introduced in the interaction analysis. It is a figure which shows the influence which noise, interaction with a vehicle, and track | orbit displacement exert on the detailed reverse analysis in span length 10m. It is a figure which shows the influence which noise, interaction with a vehicle, and track | orbit displacement exert on the detailed reverse analysis in span length 45m. It is a figure which shows the relationship between the confidence interval of the natural frequency estimated value in span length 10m, and an impact coefficient. It is a figure which shows the relationship between the confidence interval of the natural frequency estimated value in span length 45m, and an impact coefficient. It is a figure which shows the estimation precision of the maximum displacement by the detailed reverse analysis in span length 10m. It is a figure which shows the estimation precision of the largest displacement by the detailed reverse analysis in span length 45m. It is a figure which shows an example of the bridge for which the structure performance investigation method of the railway bridge which concerns on one Embodiment of this invention is applied. It is a figure which shows an example of a measurement acceleration. It is the figure which compared the acceleration before and behind a detailed reverse analysis. It is a figure which shows the parameter estimation result by the reverse analysis in span length 25m. It is a figure which shows the parameter estimation result by the reverse analysis in span length 35m. It is a figure which shows the displacement estimation result based on the measurement acceleration in span length 25m. It is a figure which shows the displacement estimation result based on the measurement acceleration in span length 35m. It is a figure which shows the relationship between the maximum displacement and train speed based on a reverse analysis result.

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

  Here, the present embodiment relates to a method for investigating and evaluating the structural performance of a railway bridge by estimating the displacement response and impact coefficient of the bridge by an inverse analysis method based on the measured acceleration of the bridge during train travel. 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 travel with a constant accuracy.Next, based on the formulated theoretical analysis model, estimation based on uncertainty is performed. A possible detailed inverse analysis method (a method for investigating the structural performance 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, we verified the accuracy of estimation of parameters, displacement response, and impact coefficient obtained by the method for investigating the structural performance of railway bridges according to the present invention through twin experiments, evaluated the accuracy of various influencing factors, and measured acceleration on actual bridges. The result of confirming the effectiveness of the present invention applied to the response will be described.

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

(The response of the bridge during train travel based on theory)
FIG. 1 shows a simple beam model during train travel.
Here, the traveling train is a moving load train, and the bridge is modeled as a simple beam with no cross-sectional change in the track direction.

Incidentally, FIG. 1, the axle spacing a, bogie distance b, the vehicle length L _v train 1 which is organized in the vehicle n _v platform is from right to left in FIG bridges 2 of span length L _b with a constant velocity v It shows the state of running. Time t is set to 0 when the first axle of the leading vehicle enters the bridge 2.

  If the surface of the Bernoulli Euler beam is sufficiently smooth, the equation of motion at an arbitrary point on the bridge and at a point in time can be expressed by Equation (16).

Here, m (upward-) is the bridge mass per unit length (unit length mass), c is the damping coefficient, and EI is the bending stiffness. J is a moving load sequence expressed by the equation (17), and δ represents a Dirac delta function. Also, τ _{k, i} is the time when the k-th vehicle passes through the i-th axle shown in Equation (18), and P _{k, i} is the axle load of the same axle.

  If the bridge can be assumed to be a linear system for each moving load in equation (16), it can be decomposed as in equation (19). Therefore, the response during train travel as shown in equation (20) It can be expressed by overlapping.

Further, the solution of the differential equation (21) relating to the bridge response when the k-th vehicle passes through the i-th axle under the boundary condition of FIG. Where n is the mode order, T is the bridge transit time, ω _n is the n-order natural angular frequency, ξ _n is the n-order mode damping ratio, and ω _n (upward-) is the one-axle bridge entry to exit. The n-th order component of each frequency passing through the axle in a half cycle, Φ _n (x) is the n-th order vibration mode shape, u ₀ is the static deflection amount when the axial load acts on the center of the span, ) To Expression (27).

The natural frequency f _n is f _n = ω _n / 2π. In addition, α _n , β, and ψ in the formula are respectively the formula (28), the formula (29), and the formula (30). Furthermore, assuming that the attenuation in the derivation of formula (22) is sufficiently smaller than 1, using an approximation of _{ω n √ (1-ξ n} 2) ≒ ω n.

Next, the speed response u _{k, j} (above) and the acceleration response u _{k, j} (up) of the k-th vehicle passing through the i-th axle are represented by the formula (31), It can be expressed as equation (32).

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

Next, even assuming the current maximum operating speed of the railway, since β ² << 1, it can be simply considered as in Expression (34).

From the equation (34), as the mode order n increases, the maximum amplitude of each mode rapidly decreases in inverse proportion to the fourth power of the order. Furthermore, considering that the displacement is maximum at the center of the span in most simple girders, the secondary mode amplitude at the center of the span is Φ _S (L _b / 2) = 0, which is the next largest amplitude after the primary mode. It is the third-order mode that has. On the other hand, the maximum mode amplitude of the third-order mode is 1 / 34≈0.011 with respect to the first-order mode, which is about 1% from the equation (34). Thus, in this embodiment, only the amplitude of the primary mode is considered. At this time, Expression (22) becomes Expression (35). Note that subscripts for the mode order are omitted.

  Further, by superimposing the equation (22) which is the solution of the equation of motion when a single moving load is applied, the dynamic response theory analysis model during train travel can be formulated as the equation (36). Note that ν (t) is a step function of 0 when t <0 and 1 when 0 ≦ t.

  At this time, the speed response u (upward) (x, t) and the acceleration response u (upper...) (X, t) are the expressions (37) and (38).

  Since these equations (36) to (38) are only the superposition of trigonometric functions, there is an error due to omission of higher-order modes, but there is no error mixing through numerical integration and a comparison with the finite element method. The calculation load is extremely small. As a result, response calculation is possible even with general 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 assumptions, the 10 m, 35 m, and 45 m simple girders shown in Table 1 have a 10-car train. As an example, a comparison was made between two types of numerical analysis methods, a load train analysis method (load train analysis) and a dynamic interaction analysis method (interaction analysis).

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

  In the dynamic interaction analysis method (interaction analysis), a traveling vehicle is modeled as a multibody model and a structure as a finite element model. The calculation used the dynamic interaction analysis program DIASTARS III between the Shinkansen and the railway structure.

An analysis model of the vehicle (train) is as shown in FIG.
The vehicle is a three-dimensional model in which the components of the vehicle body, the bogie frame, and the wheel shaft are rigid bodies, which are connected by springs and dampers. 10 models of 31 degrees of freedom per vehicle were connected and modeled. The validity of the analysis model has been confirmed by a verification experiment using an actual vehicle and a vehicle test bench.

  The vehicle specifications were set with reference to recent high-speed rail vehicles. The interaction force between the wheels / rails was calculated using the vertical relative displacement and the horizontal relative displacement of both. Specifically, the normal direction of both contact surfaces considered a Hertz contact spring, and the tangential direction considered creep force.

  The numerical calculation is solved by Newmark's average acceleration method as in the load train analysis. However, since the equation of motion is nonlinear, iterative calculation is performed within Δt until the unbalance force becomes sufficiently small at each time step. did. Δt was 0.0005 seconds, and the mode order was up to 50th order.

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

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

  FIG. 4 shows the vertical deflection response at the center of the span calculated by theoretical analysis (simple inverse analysis method), load train analysis, and interaction analysis. In addition, the train was a case where a 10-car train high-speed rail car traveled at the speed shown in the figure. And from this figure, it is confirmed that all the analysis results are in good agreement regardless of the running speed and span length, and that the displacement response during train running can be accurately reproduced by the theoretical analysis according to the present invention. It was.

Then, dynamic response component due to speed effects of railway bridges are considered in the design by the impact factor i _a of formula (39). S _d is the maximum value of dynamic deflection, and S _s is the maximum value of static deflection.

FIG. 5 shows the impact coefficients calculated by the analysis methods of the load train analysis method (load train analysis) and the dynamic interaction analysis method (interaction analysis).
The vehicle specifications were the same as described above, and the maximum deflection at a train speed of 100 km / h to 420 km / h and 100 km / h was defined as static deflection.

  From this figure, it was confirmed that the impact coefficient was slightly different between the theoretical analysis (simple inverse analysis method) and the interaction analysis at the resonance peak, but well matched in the other velocity bands. Since the impact coefficients of the theoretical analysis and the load train analysis are in good agreement, it is assumed that the difference in resonance peak is due to the vibration system (mass and damping) of the vehicle.

  Based on the above results, although the theoretical analysis method is limited to simple digits, the time series response and impact coefficient are evaluated with at least the same accuracy as the load train analysis with much less model freedom than the finite element method. It became clear that we could do it.

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

(Detailed inverse analysis method: Method for investigating structural performance of railway bridges of the present invention)
First, the detailed inverse analysis method will be described with reference to FIG.
The detailed inverse analysis method in this embodiment is a detailed inverse analysis method of unknown parameters and displacement response by the MCMC (Markov chain Monte Carlo) method. It is possible to avoid the problem of inferior decision of the parameter estimation process due to the excess and to improve the stability and accuracy of the estimation.

  Specifically, in the detailed inverse analysis method in the present embodiment, first, Bayesian estimation is used in order to evaluate a parameter and its uncertainty with high accuracy.

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

The likelihood function is represented as L (u ₀ (above ..) | θ). Here, θ represents an unknown parameter vector, and u ₀ (above) represents observation data of acceleration response during train travel. 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 given, the simultaneous posterior probability density function π (θ | u ₀ (above...)) Of the unknown parameter vector θ is expressed by the equation ( 40). Where Θ is a parameter space. The denominator of Equation (40) is a constant, and the simultaneous posterior probability density function π (θ | u ₀ (above...)) Becomes Equation (41).

Thus, the simultaneous posterior probability density function π (θ | u ₀ (above...)) Is nothing but a parameter distribution in which the observation data is known, that is, a model parameter estimated from the actual measurement value.

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

The natural frequency (natural period) ω, unit length mass m (upward-), mode damping ratio ξ, probability error term variance σ are unknown parameters θ = (ω, m (upward-), ξ, σ), Given these, 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 Equation (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 the equation (45).

N _ω is a normal distribution with mean μ _ω and variance σ _ω , and GJ (ligature) _ξ is a gamma distribution with shape factor α _ξ and scale factor γ _ξ . From the equation (27), among the unknown parameters θ = (ω, m (upward--), ξ, σ), the unit length mass m (upward--) is inversely proportional to the static deflection amount.

That is, if the natural frequency f and the mode damping ratio ξ are obtained, the unit length mass m (upward) can be calculated backward from the maximum amplitude ratio with the observation data, for example. Similarly, the variance σ ² of the probability error term can be uniquely calculated from the observation data and the reproduction data if other parameters are determined. Accordingly, the unit length mass m (upward-) and the variance σ ^{2 of the} error term are not subjected to Bayesian estimation.

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

  For this reason, in the structural performance investigation method of the railway bridge of this embodiment, the numerical solution method by MCMC method was used.

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

The MCMC method is a method for obtaining a probability density function that approximates the simultaneous posterior probability density function π (θ | u ₀ (above...)) By numerical calculation. In this embodiment, the Gibbs method is one of MCMC methods. Is adopted.

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

  Samples θ (m) and θ (m−1) of parameters sampled at an arbitrary mth and m−1th times in the Gibbs method have a Markov chain with Expression (46) as a transition nucleus.

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 simultaneous posterior probability density function π (θ | u ₀ (above...)).

Therefore, by calculating the expected value and the confidence interval from the sample θ (m = m _s +1, m _s +2,..., M _e ), it is possible to evaluate the parameter estimation value and its reliability, respectively. .

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

  In this 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−1th sample, and N (0 | ν) is a normal distribution with an average of 0 and a variance ν.

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

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

  Sampled sample candidates ξ 'are accepted as samples with the probability of equation (51). In actual numerical calculation, acceptance / rejection is determined in the same manner as in equation (49).

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

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

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

Displacement response according to simultaneous posterior probability density function π (θ | u ₀ (above)) by theoretical analysis using parameter sample θ (m = m _s + 1, m _s +2,..., M _e ) And a sample of impact coefficient can be obtained.

In the present embodiment, include responses predictive analysis of the train travel times that any train speed v _s of the formula (54) as a parameter to the m-th sampling process, the maximum value at each speed (formula ( 55)) is recorded.

Train speed _{v s} is, for example, and from 10km / h to 400km / h in the 5km / h increments.

In the present embodiment, by incorporating the response prediction process into the MCMC calculation flow in this way, simultaneously with the estimation calculation of the unknown parameter θ (m = m _s +1, m _s +2,..., M _e ) It is possible to obtain a sample u _max (v _s , m = m _s +1, m _s +2,..., M _e ) of the maximum displacement according to the simultaneous posterior probability density function π (θ | u ₀ (above)). it can.

  The detailed inverse analysis method and response prediction were implemented on MATRAB. The calculation time for one acceleration waveform (200 Hz, 20 seconds, 32 kBytes) is about 330 seconds (5 and a half minutes) including back analysis (3000 samplings) and response prediction on a PC with CPU 3.5 GHz and RAM 4.00 GB. is there. In the estimation method based on dynamic FEM and data assimilation shown in the past literature, the acceleration waveform under the same condition is about 83,080 seconds (one day), which is about 1/250 of the calculation time. .

(Simple inverse analysis method)
Here, the detailed inverse analysis method (the method for investigating the structural performance of a railway bridge according to the present invention) is intended to precisely evaluate the unknown parameters of the theoretical analysis model and its uncertainty, but the MCMC method is repeated. Return calculation is necessary, and a certain amount of time and calculation load are inevitable. In contrast, on-site measurement / surveys with strict time constraints, programs embedded in smart sensors, etc., batch processing of large amounts of accumulated measurement data, etc., while maintaining a certain degree of accuracy, reverse with less computational load It is often desirable to be able to analyze.

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

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

  The unknown parameters of the bridge to be estimated from the measured data are the natural frequency f (= 2πω), the mode damping ratio ξ, and the unit length mass m (above −). Among these, the estimation of the natural frequency f and the mode damping ratio ξ can be identified by the already proposed vibration characteristic identification method by regarding the remaining waveform after passing the train as a free vibration waveform. Further, the unit length mass m (above-) can be uniquely calculated by determining other parameters as described above.

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

(Vibration characteristics identification method)
Here, the result of examining the applicability of the following three methods as the identification method of the natural frequency and the mode damping ratio based on the residual waveform at the time of passing the train (hereinafter, the waveform after passing the train) 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 identifying the mode damping ratio based on the HP method.
2) The AR model is a method for identifying the natural frequency and the mode damping ratio based on the estimated AR coefficient.
3) The ERA method is a method for identifying a natural frequency and a mode damping ratio based on a minimum realization algorithm in a linear time-invariant system.
Both identification methods have already been applied to many bridges.In this embodiment, an identification method that can accurately evaluate the natural frequency and mode damping ratio from the post-train waveform is selected based on the accuracy verification results. Used for simple inverse analysis method.

(Unit long mass identification method)
As described above, the unit length mass m (upward-) can be uniquely determined by controlling the mode amplitude of the primary mode and taking other parameters as given.

In the simple inverse analysis method of the present embodiment, a theoretical analysis (formula (equation (1)) in which the natural frequency f, the mode damping ratio ξ, and the initial unit length mass m ₀ (above −) = 1 are identified based on the above identification method. 32)) is performed, the ratio of the measured value to the maximum amplitude is calculated, and the unit length mass m (upward) is identified by the equation (56).

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

(Accuracy verification by twin experiment)
Next, a twin experiment is performed, and the detailed inverse analysis method (the method for investigating the structural performance of a railway bridge according to the present invention) and the accuracy verification results for the simple inverse analysis method will be described.

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

  The twin experiment is a basic method for verifying the validity of parameter estimation in such an inverse analysis. First, time series data is created by numerical calculation in advance, and noise is added to obtain simulated observation data. Next, parameters (prior distribution) different from the time series data creation are set to initial values, and reverse estimation based on simulated observation data is performed. The validity of the parameter estimation is verified by whether or not the four estimation results converge to the correct value.

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

Table 2 shows the parameters set when creating the time series data.
FIG. 10 is an example of time-series data created by a theoretical analysis model and simulated observation data added with noise. The train passed through the bridge of Case A with a span length of 10 m (A bridge) at a speed of 100 km / h. Shows the case.
In addition, with reference to the existing measurement results, noise generated from a normal distribution with a standard deviation of 5% of the maximum displacement was added to the acceleration waveform.

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

FIG. 12 shows the estimation results of natural frequency, mode damping ratio, and unit length mass.
It was confirmed that the natural frequency, mode damping ratio, and unit length mass based on the detailed inverse analysis method all converge to the correct value. In addition, it was confirmed that the detailed inverse analysis method can estimate the parameters very accurately as compared with the existing identification method shown in FIG.

  As a result, in the detailed inverse analysis method of the present embodiment (the method for investigating the structural performance of a railway bridge according to the present invention), an underdetermined problem can be avoided and the parameters can be accurately set even if the prior information is slightly different from the correct value. It was confirmed that estimation was possible.

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

  In FFT, 0 vector was added after time series to ensure a frequency resolution of 0.2 Hz. Also, the Burg method was used for the estimation of the AR coefficient, and the model order with the smallest AIC was adopted. The ERA method has a degree of freedom of 10, a mode satisfying the observability MAC value of 0.99 or more and the mode attenuation ratio [0.001-0.05] is identified as a railway bridge mode.

  From FIG. 12, it was confirmed that a relatively large error occurs depending on the train speed in the natural frequency based on the ERA method, the mode damping ratio based on the AR model, and the unit length mass. As long as there is no great difference in accuracy, the simple and basic method is more practical in terms of the identification method used for the simple inverse analysis method.

  On the other hand, the results based on the FFT method and the HP method have an error of 2% at maximum for the natural frequency, about 45% for the mode damping ratio, and 25% for the unit length mass. It was the method that met the most requirements.

  Thus, in the present embodiment, the simple inverse analysis method adopts the FFT method for identifying the natural frequency and the HP method for identifying the mode damping ratio.

(Reliability of parameter estimates)
Next, the reliability of parameter estimation values was examined.

  FIG. 13 shows parameter estimation values and 90% confidence intervals by the detailed inverse analysis method. In FIG. 13, the vertical axis of each parameter is different. The vertical axis indicates the correct answer value as 1. Further, FIG. 13 also shows a 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 mode 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 increase in the natural vibration accompanying the generation of resonance increased the reliability at a train speed of 200 km / h in the case C bridge (C bridge) having a span length of 45 m.

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

(Displacement response estimation result)
Next, FIG. 14 shows the displacement response estimated based on the simple and detailed inverse analysis method.
FIG. 14 shows an example of a result of a C bridge (100 km / h, 300 km / h) having a span length of 45 m with the largest error when the simple inverse analysis method is performed 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 coincided with the correct value with high accuracy. In addition, the displacement estimated by the simple inverse analysis method is close to the boundary line of the 90% confidence interval in the detailed inverse analysis method, and it was confirmed that the displacement estimation accuracy is slightly reduced compared to the detailed inverse analysis method. It was. However, the estimation error was about 10% of the maximum displacement at most.

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

  Here, the impact coefficient is practically used, but the impact coefficient contains an error with respect to each of the static displacement and the dynamic response magnification. Use maximum displacement instead of coefficient.

  As shown in FIG. 15, the result of the solid line detailed inverse analysis method can accurately evaluate the maximum displacement even at a train speed other than the estimated speed. Further, in the detailed inverse analysis method, when it was based on the acceleration of the train speed of 100 km / h on 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 in the bridge with a longer span length. When the span length is long, the natural frequency is low, so it can be considered that the error due to the frequency increment becomes relatively large.
Moreover, when based on the acceleration of the train speed of 100 km / h in the C bridge, the error with respect to the correct value was 10% on average and 18% at the maximum at resonance.

  As a result, the identification accuracy of the natural frequency and mode damping ratio greatly affects the displacement estimation in the simple inverse analysis method. Therefore, especially when a bridge with a low natural frequency is targeted, microtremor and multiple measurements are possible. For example, it can be said that it is desirable to average the results together to improve the estimation accuracy.

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

(Influencing factor analysis by numerical analysis)
Next, an analysis result of influential factors by numerical analysis will be described.

  The influence of train speed and the like on the inverse analysis method of this embodiment will be analyzed in more detail. Table 3 shows a list of analysis cases.

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

  The results of theoretical analysis of train speeds of 100 to 400 km / h (in increments of 10 km / h) for bridges with the highest span accuracy of 10 m and the shortest span length of 45 m, with 1% noise of maximum amplitude added. Based on the detailed inverse analysis method.

  Case I is a basic case for comparing the effects of train speed and other cases.

  Case II is a case in which noise having a standard deviation of 5% of 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 a detailed inverse analysis method based on an interaction analysis by DIASTARS is performed on bridges having span lengths of 10 m and 45 m in order to examine the influence of the dynamic interaction between the vehicle and the bridge on the accuracy of the inverse analysis.

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

  FIG. 17 shows the orbital displacement introduced. Orbital displacement is described by “Masayuki Sogabe, Nobuyuki Matsumoto, Yozo Fujino, Hajime Sakurai, Makoto Kanamori, Masaaki Miyamoto: A study on the dynamic design method of concrete railway bridges in the resonance region. , 2003. "was used. The structure / vehicle model is as shown in FIG. The MCMC method was repeated 3000 times, the first 1000 times were removed as a convergence process, and the remaining 2000 times were used as samples from the posterior distribution. Furthermore, the influence of prior information was eliminated by making the prior distribution uniform (no prior information).

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

  In all cases, the correct answer value is set to 1 and the value is made dimensionless. In addition, the impact coefficient is also shown for analysis of the relationship with the dynamic response component. Furthermore, since the simple inverse analysis method has a large variation in the method itself and no clear change was observed in each case, 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% in terms of natural frequency, about −80% to + 100% in mode damping ratio, and about ± 30% in unit length mass. there were.

(Basic case)
From FIG. 18 and FIG. 19, the estimation result when the load train analysis result is an observed value agrees accurately with the correct value for any span length, parameter, and train speed, and the method of the present embodiment solves the problem of underdetermination. It was confirmed that it can be avoided.

  20 and 21, it was confirmed that the 90% confidence interval of the natural frequency tends to be particularly small (high accuracy) in the vicinity of the peak of the impact coefficient particularly in a bridge with a span length of 10 m. This is presumably because the natural vibration component increases due to resonance. Conversely, it was confirmed that the 90% confidence interval tends to show a large value in the vicinity of the velocity of the impact coefficient minimum at which the natural frequency component of the bridge is difficult to be excited.

  The reliability of the estimated value varies depending on the impact coefficient indicating the magnitude of the dynamic response component, but the 90% confidence interval of the natural frequency estimated by the load train analysis has a span length of 10 m. The maximum is about 0.1%, and the maximum is about 0.05% for a bridge with a span length of 45m, which can be said to be sufficiently high in reliability of the estimated values.

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

  As shown in FIG. 18 and FIG. 19, in the time-series estimation results with 5% noise added, an error of about 5% at maximum in the mode attenuation ratio and about 1% at maximum in the unit length mass is recognized. Estimated with high accuracy. From this, it was confirmed that the influence of noise on the estimation accuracy is limited, and in particular has little influence on the estimation of the natural frequency.

  Further, as shown in FIGS. 20 and 21, it was confirmed that the 90% confidence interval of the natural frequency shows the same tendency as the result of the load train analysis. However, the value increases due to the addition of noise, and the maximum is about 0.4% for a bridge with a span length of 10 m, and the maximum is about 0.1% for a bridge with a span length of 45 m. From these results, it was confirmed that the noise included in the observation data tends to reduce the reliability of the estimated values.

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

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

  In addition, it was confirmed that the natural frequency tends to be estimated as low as about 5% at maximum from the correct value for bridges with a span length of 10m and about 3% at maximum for bridges with 45m. This is thought to be due to the apparent decrease in natural frequency due to the passing train mass. That is, it is presumed that the error is caused by modeling the fluctuating axle load due to the influence of the vehicle vibration system with a time-invariant moving load.

  Further, the mode damping ratio tended to vary from the correct value to a maximum of 50% and a minimum of about 60%. This error is also considered to be an additional attenuation that the vehicle damping mechanism exerts on the bridge through the coupling of the bridge primary mode and the vehicle vibration mode. On the other hand, there are cases where the attenuation is lower than the correct value. Both occur at a speed slightly deviating from the resonance speed at which the impact coefficient is maximized to the high speed and low speed sides, and this is considered to be the effect of the beat phenomenon.

  Since the theoretical model cannot evaluate the decrease in the apparent natural frequency when the train is running in the interaction analysis, there is a difference in the beat phenomenon when the train passes around the resonance speed. In order to fill this difference, it is presumed that the mode attenuation ratio apparently decreased.

  The unit length mass of each bridge was higher than the correct value and showed a tendency opposite to the estimated natural frequency. In order to match the static displacement (acceleration) during train travel determined by the natural frequency and unit length mass to the time-series waveform, the unit frequency mass is incidentally reduced as the natural frequency is apparently lowered by the train mass. Is thought to have increased.

  As shown in FIGS. 20 and 21, the 90% confidence interval of the natural frequency is larger by one digit or more than the result of the load train analysis, particularly in the bridge having a span length of 10 m (in FIG. 19, only the vertical axis Are different scales). It was confirmed that the confidence interval was large at the anti-resonance speed at which the impact coefficient was minimized in any bridge.

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

  As shown in FIG. 18 and FIG. 19, the estimated values when DIASTARS (analysis of interaction with the vehicle) considering the trajectory displacement was given as observation data were almost the same as the interaction analysis in any bridge. From this, it was confirmed that the influence of the orbital displacement on the estimation accuracy is small compared with the influence of the interaction with the vehicle.

  As shown in FIGS. 20 and 21, it was recognized that the 90% confidence interval of the natural frequency tends to be larger when the orbital displacement is taken into consideration particularly in the speed zone where the impact coefficient is minimized.

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

  In particular, it has been confirmed that the natural frequency tends to have a greater effect when the span length is shorter. In addition, noise and trajectory displacement have an influence on the confidence interval of the estimated value, but it has been clarified that the influence is small compared to the interaction with the vehicle.

(Effect on estimation of maximum displacement)
Next, the influence on the estimation of the maximum displacement will be described.

22 and 23 show the estimation results of the maximum displacement for each case.
In the parameter estimation, the influence of the interaction with the vehicle was observed, but as shown in FIGS. 22 and 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 can be considered that the difference from the correct values found in the natural frequency and the mode damping ratio is a result of equivalently evaluating the effects of additional mass and additional damping caused by the traveling vehicle with a simple model.

  From the above, it has been clarified that the maximum displacement can be accurately estimated by the inverse analysis method of the present embodiment, and the influence of noise, interaction with the vehicle, and orbital 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 the present embodiment using the actual measurement response will be described.

First, the specifications and appearance of the application target are shown in Table 4 and FIG.
The target bridges are an open floor type RCT girder bridge with a span length of 25 m and an open floor type PC box girder bridge with a span length of 35 m. In addition, the acceleration response during train travel, measured in the past at the center of the span of the bridge, was used as observation data. Moreover, the displacement measured in the same location was used for verification. The acceleration and displacement are measured by a piezoelectric accelerometer and a ring displacement meter, respectively. Both bridges were intended for two express trains (vehicle length 21.3 m, 6-car train) that passed through the girder on the sensor installation side, and the method of this embodiment was applied to the acceleration at the time of passage.

In order to use the measured acceleration for the inverse analysis, pre-processing was performed.
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 this embodiment, 20 Hz low-pass filter processing is performed. Next, the train speed was identified. In addition, each axle load of the train was a capacity ride. Finally, time synchronization processing was performed by 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 a train passes through a bridge having a span length of 35 m at 136.5 km / h. In the filter waveform after the filter processing, it was confirmed that the amplitude increases as the train passes.

(Parameter estimation results)
Next, parameter estimation results will be described.

FIG. 26 shows the acceleration by the theoretical analysis before and after the estimation by the MCMC method. 27 and 28 show the parameter estimation results by the simple and detailed inverse analysis method.
27 and 28 also show the average values of the results of the simple inverse analysis in two cases. 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 was the design value shown in Table 4, and the mode damping ratio was 2%.

  From FIG. 26, it was confirmed that the parameters were corrected according to the actual measurement values by the detailed inverse analysis method in any bridge, and the reproducibility of the measured acceleration was greatly improved.

  27 and 28, it was confirmed that the natural frequency and the unit length mass by the simple inverse analysis method tend to have larger variations than the natural frequency and the unit length mass based on the detailed inverse analysis method. Further, it was confirmed that the mode damping ratio estimated by the simple inverse analysis method was 30 to 50% lower than that of the detailed inverse analysis method. This is because the simplified inverse analysis method targets the waveform after passing through the train without interaction with the train, but the detailed inverse analysis method targets the waveform at the time of passing the train and is caused by the interaction with the vehicle. This is thought to be because the additional damping was equivalently evaluated as the bridge mode damping ratio.

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

(Displacement response estimation result)
Next, the estimation result of the displacement response will be described.

  29 and 30 show displacement responses estimated by the detailed inverse analysis method and the simple inverse analysis method. These figures also show the measured displacement (actual value). The enlarged view shows the response near the maximum displacement. The value of the measured displacement is corrected by a correction coefficient in accordance with a theoretical analysis model that does not consider the torsional component. The correction factor was calculated from the maximum displacement when the train passed through the measurement side and the non-measurement side at almost the same speed, and was 0.59 for the span length of 25 m and 0.68 for the span length of 35 m.

  As shown in FIG. 29 and FIG. 30, the displacement estimated by the detailed inverse analysis method tends to evaluate the upper amplitude slightly larger in the bridge with a span length of 35 m than the measured displacement. It was confirmed that the actual deflection amount and the minimum value were in good agreement with the measured displacement in any case. The displacement based on the simplified inverse analysis method is in good agreement with the measured displacement in the case of a span length of 25 m and a train speed of 138.8 km / h, but in other cases it was confirmed that there was an obvious error. . On the other hand, when using the average value of the parameters estimated by the simple inverse analysis method for each bridge, it was confirmed that the accuracy was improved by approaching the measured displacement which is the correct value.

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

Table 5 shows values obtained by arranging the estimation errors of the displacement response by the maximum displacement.
As shown in Table 5, it was confirmed that the detailed inverse analysis method can estimate the maximum displacement in the range of error −3% to −6% with respect to the correct value. On the other hand, it was confirmed that the simple inverse analysis method has a relatively large error of −9% to + 23% with respect to the correct answer value. However, it was 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 the two cases even in the simple inverse analysis method.

  FIG. 31 shows the relationship between the train speed and maximum displacement calculated by the detailed inverse analysis method and the simple inverse analysis method. Peaks were confirmed at almost the same speed for both 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 estimated value by the simple inverse analysis method was used for any bridge, and that the estimated value of the detailed inverse analysis method was approached except for the measurement speed.

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

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

  Based on the theoretical model estimated by inverse analysis, we checked the deflection determined from the normal driving safety and riding comfort reflecting the current state. Regarding the maintenance and management of existing structures, the railway structure maintenance management standard and the description (hereinafter referred to as maintenance management standard) stipulate the performance check according to equation (57).

Here, K _m is a coefficient for the maintenance index J, γ _i is a structure coefficient, and _IRm and _ILm are a maintenance response value and a maintenance limit value, respectively.

Regarding the verification of travel safety and ride comfort, K _m = 1 and γ _i = 1 in the railway structure design standard and the description (displacement restriction). For the response value and limit value calculation at the time of the survey, application of values closer to the actual conditions is required, and the theoretical analysis model estimated as described above corresponds to this.

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

  Here, a maximum speed of 140 km / h and a six-car limited express vehicle (vehicle length: 21.3 m) were assumed from the operational status of the route. As the condition of the traveling vehicle, the single-line loading of the occupant riding for the ride comfort and the double-track loading of the 350% riding for the driving safety. Since the theoretical analysis model cannot consider the vehicle fluctuation, the design impact coefficient of the equation (58) is used. Based on this, the impact coefficient was increased.

Here, _ic is an impact coefficient of vehicle shake, and is set to 0.11 (25 m) and 0.10 (35 m), respectively, in the target bridge according to the past literature. Further, the reduction coefficient β _{i with} respect to the impact coefficient at the time of double-track loading was set to 0.875 (25 m) and 0.825 (35 m). The impact coefficient i _α of the speed effect was calculated by a theoretical analysis in which the moving load was determined as described above.

  Table 6 summarizes the running safety and ride comfort limit values, response values, and maintenance management indices at all times. The response value was calculated using the expected value in the detailed inverse analysis method, the 90% upper limit value, and the average value in the simple inverse analysis method.

  By estimating the displacement based on the acceleration by the structural performance investigation method of the railway bridge of this embodiment, the response value of the two target bridges is sufficiently smaller than the limit value, and the deflection limit determined from the driving safety and riding comfort is limited. Satisfaction can be quantitatively verified. In addition, it was confirmed that the maintenance management index J for running safety and riding comfort can be evaluated with the same accuracy as the detailed inverse analysis method by using the average value of the simple inverse analysis method for the bridge.

  As described above, in this embodiment, in order to estimate the displacement response and the impact coefficient based on the acceleration at the time of running the train, the theoretical analysis model is formulated, and the detailed inverse analysis method based on the MCMC method, the FFT method, and the HP method are used. A simple inverse analysis method was explained, and verification based on numerical analysis and actual measurement was performed. And the following knowledge and effects could be obtained.

(1) Formulate a theoretical analysis model for train travel that can evaluate displacement and impact coefficient with the same accuracy as the moving load analysis based on the finite element method, and the simple inverse using the detailed inverse analysis method by the MCMC method and the existing identification method An analytical method was devised.

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

(3) The parameter estimation error of the detailed inverse analysis method based on the twin experiment is -2% to + 1%, which can be estimated with higher accuracy than the existing identification method. Further, the natural frequency and the mode damping ratio can be identified with relatively high accuracy by the FFT method and the HP method for the waveform after passing through the train.

(4) As a result of studying the effects of train speed, noise, vehicle interaction, and track displacement on the inverse analysis by numerical analysis, the modeling error for the interaction with the vehicle is the largest. Although it varies about + 5% to -60% with respect to the mode damping ratio, the displacement estimation does not show any significant effect and can be estimated with high accuracy.

(5) As a result of performing simple and detailed inverse analysis methods on the acceleration measurement results of two types of actual bridges, the simplified inverse analysis method has particular variations in natural frequency and unit length mass compared to the detailed inverse analysis method. The maximum displacement is -3% to + 6% in the detailed inverse analysis method, and -9% to + 23% in the simple inverse analysis method. However, by using the average value of the results of multiple measurements, 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 this embodiment, the displacement response during train travel necessary for estimation of the bridge state, travel safety and comfort evaluation is accurately obtained from the acceleration response of one point of the bridge. It becomes possible to do.

  Of the estimated parameters based on the measured acceleration of the bridge during train travel, the natural frequency is related to the bridge stiffness, so the natural frequency can be estimated accurately, so that the structural performance can be estimated accurately. .

  Furthermore, response prediction at other train speeds can be performed based on the actual acceleration of the bridge during train travel.

  In addition, since the uncertainty of the estimation parameter can be quantified, for example, an evaluation that guarantees a risk that changes according to the estimation accuracy, such as adoption of a lower limit value of 95% confidence interval, can be performed.

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

  Also, by calculating the margin with respect to the maintenance management reference value, it can be used to determine the priority of maintenance management and extract weak point structures.

  Therefore, according to the structural performance investigation method of the railway bridge of the present embodiment, it is possible to suitably investigate and evaluate the structural performance of the bridge using observation data of the acceleration response of the bridge when the train is running.

  As mentioned above, although one Embodiment of the structural performance investigation method of the railway bridge of this embodiment which concerns on this invention was described, this invention is not limited to said one Embodiment, In the range which does not deviate from the meaning, it is appropriate. It can be changed.

1 Train (vehicle)
2 Bridge

Claims (7)

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

here,
x is a distance (position), t is a 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 when t <0, 1 when 0 ≦ t, τ _{k, i} is the time when the k-th vehicle passes the i-th axle.
u (above...) (x, t) is an acceleration response and is expressed by the following equations (2) and (3) to (10).

here,
u (above) (x, t) is a velocity response, and u (x, t) is a displacement response.
m (upward-) is the bridge mass per unit length (unit length mass), c is the damping coefficient, and EI is the bending stiffness. L _b is span length of the bridge, n represents mode order, T is a bridge transit time, omega _n is n th order natural angular frequency, xi] _n the n-th order mode damping ratio, omega _{n (above} -) is the primary axle The nth order component of each frequency passing through the axle with a half cycle from entry to exit of the bridge, Φ _n (x) is the nth order vibration mode shape, and u ₀ is the amount of static deflection when axial load acts on the center 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 formula (1) to define a probability model of the following formula (11),
The co-occurrence probability (likelihood function) L (u) of the following equation (12) in which the acceleration data u ₀ (above) is generated from the equation (11) when the unknown parameter θ is given. ₀ | θ) and the prior probability density function π (θ) of the unknown parameter of the following equation (13) are substituted into the following equation (14) obtained by Bayes' theorem, and the acceleration data is given as A railway bridge characterized in that a simultaneous posterior probability density function π (θ | u ₀ ) of an unknown parameter is determined, and the structural performance of the railway bridge is evaluated by reflecting the estimated parameter and the uncertainty of the parameter. Structural performance investigation method.

here,
N _ω is a normal distribution with mean μ _ω and variance σ _ω ² , GJ (ligature) _ξ is a gamma distribution with shape factor α _ξ and scale factor γ _ξ . Θ is a parameter space. In the method for investigating the structural performance of a railway bridge according to claim 2,
The MCMC method for obtaining a probability density function that approximates the simultaneous posterior probability density function by numerical calculation is used, and sampling from the conditional posterior probability density function is sequentially repeated for each of a plurality of elements that are unknown parameters. A method for investigating the structural performance of railway bridges, characterized by obtaining a posteriori probability density function. In the structural performance investigation method of the railway bridge according to claim 3,
When the sample of the sampled parameter reaches the steady state of the Markov chain, the expected value and the confidence interval are calculated from the sample of the parameter after the number of samplings reaching the steady state, and the reliability of the parameter estimated value and the parameter estimated value is increased. A method for investigating the structural performance of railway bridges, characterized by being evaluated. In the structural performance investigation method for a railway bridge according to claim 3 or claim 4,
The unknown parameters θ are ω (natural period), m (upward-) (bridge mass per unit length), ξ (mode damping ratio), σ (probability error term variance),
In the conditional posterior probability density function, π (ω | u ₀ , m (upward −), ξ, σ) is the random wall MH method, π (m (upward −) | u ₀ , ω, ξ, σ ) Is the maximum amplitude ratio, π (ξ | u ₀ , ω, m (upward −), σ) is the independent MH method, π (σ | u ₀ , ω, m (upward −), ξ) is the acceleration A method for investigating the structural performance of a railway bridge, wherein sampling is performed from the conditional posterior probability density function using data and a standard deviation of a theoretical analysis model, respectively. In the structural performance investigation method for a railway bridge according to any one of claims 3 to 5,
A method for investigating the structural performance of a railway bridge, characterized in that a sample of displacement response and impact coefficient according to the simultaneous posterior probability density function is obtained by theoretical analysis of the MCMC method using a sample of sampled parameters. In the structural performance investigation method of the railway bridge according to claim 6,
It includes responses predictive analysis at a train traveling that any train speed v _s parameters represented by the formula (15) below the m-th sampling process, with obtaining the maximum value at each arbitrary speed, MCMC methods A method for investigating the structural performance of railway bridges, which incorporates a response prediction process into the calculation flow to obtain a sample of the maximum displacement according to the simultaneous posterior probability density function simultaneously with the estimation calculation of unknown parameters.

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