JP2017048583A - Ground vibration prediction method - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a ground vibration prediction method capable of calculating a ground vibration at any location around a viaduct on which a train travels by using a ground property and vibration acceleration at a pier base section.SOLUTION: A ground vibration at a target point is predicted through: a first process of performing ground structural analysis (modeling) on the basis of a result of ground investigation performed previously; a second process of performing pier modeling on the basis of a result of investigation of a form/shape of a pier and foundation to be a target; a third process of calculating an acceleration waveform propagating from each pier on the basis of acceleration of the pier by calculation using a ground model obtained in the first process and a pier model obtained in the second process; a fourth process of acquiring a sum of power from respective acceleration waveforms obtained in the three processes; and a fifth process of calculating a vibration level at the target point by using the sum of power obtained in the fourth process.SELECTED DRAWING: Figure 3

Description

  The present invention relates to a ground vibration prediction method which is effective when applied to predicting vibration of surrounding ground due to a train traveling on a viaduct.

Conventionally, various vibration isolators have been proposed as countermeasures for ground vibration generated from transportation facilities such as railways and roads and business establishments such as factories. In addition, a technique for quantitatively evaluating the vibration isolation effect of such a vibration isolator has been proposed (Patent Document 1).
However, the method disclosed in Patent Document 1 is an evaluation method in the case where there is one vibration source. On the other hand, in order to predict the vibration of the surrounding ground due to the train traveling on the viaduct by numerical analysis, the train that is the excitation source moves at high speed, so the point where the vibration is predicted (hereinafter, attention is paid) It is necessary to model a wide range of ground and structures, including the dots).
Therefore, the method disclosed in Patent Document 1 can be satisfied with accuracy in a model in which vibrations propagate from a plurality of vibration sources (piers) with a time lag as a train travels like a viaduct. Such vibration analysis results cannot be obtained.

JP 2003-96808 A

“JR EAST Technical Review-No.37”, pp. 49-54 “Development of Analytical Study Method for Ground Vibration”

  In addition, in order to analyze the vibration of the surrounding ground caused by the train traveling on the viaduct and select ground vibration countermeasures, model and analyze multiple vertical sections connecting the point of interest and the surrounding viaduct pillars. Although a method for predicting vibration is proposed (for example, Non-Patent Document 1), in addition to the vibration measurement value of the pier, a vibration measurement value near the point of interest is required for the prediction. However, most vibration measurement points are roads or private land, and it takes a lot of labor to perform vibration measurement. Therefore, the point which requires time until implementation of prediction work has been a problem.

  Furthermore, conventionally, when obtaining a vibration level by transmission of vibration waves from a plurality of bridge piers, it has been considered that the vibration level should be calculated taking into account that the vibration level becomes low due to cancellation by interference between vibration waves. However, as a result of verification by the present inventors through actual measurement, it has become clear that practically no cancellation occurs due to interference between vibration waves. This is because the actual ground is thought to be caused by the fact that it is very complex, unlike the modeled ground.

The present invention has been made in view of the above problems, and provides a ground vibration prediction method capable of obtaining a highly accurate vibration level for ground vibration at an arbitrary point around a viaduct where a train travels. With the goal.
Another object of the present invention is that the ground vibration at any point around the viaduct where the train travels is easily modeled and the analysis time is relatively short from the ground physical property value and the acceleration waveform of the vibration of the pier base. An object of the present invention is to provide a ground vibration prediction method capable of utilizing a complex response analysis by an axisymmetric model.

In order to achieve the above object, the present invention provides
A ground vibration prediction method for predicting ground vibration around a bridge supported by a plurality of piers,
A first step of conducting a structural analysis (modeling) of the ground based on the results of the ground investigation conducted in advance;
The second step of modeling the pier based on the results of a survey of the target pier and foundation type and shape;
A third step of calculating an acceleration waveform propagating from each pier by calculation using the ground model obtained in the structural analysis of the first step and the pier model obtained in the second step based on the acceleration of the pier. When,
A fourth step of obtaining a sum of power (energy) from each acceleration waveform obtained in the third step;
And a fifth step of calculating a predicted value of the vibration level of the point of interest using the sum of the power obtained in the fourth step.

More specifically, a ground vibration prediction method for predicting ground vibration around a bridge supported by a plurality of piers,
A first step of analyzing the structure of the ground based on the results of the ground survey conducted in advance;
The second step of modeling the pier based on the results of a survey of the target pier and foundation type and shape;
A third step of calculating a transfer function up to a point of interest using the ground model obtained in the structural analysis of the first step and the pier model obtained in the second step;
A fourth step of calculating an acceleration waveform at a point of interest for each pier using the transfer function obtained in the third step;
A fifth step for obtaining a sum of power from the respective acceleration waveforms obtained in the fourth step;
And a sixth step of calculating a predicted value of the vibration level of the point of interest using the sum of the power obtained in the fifth step.

  According to the method as described above, the acceleration waveform at the point of interest is calculated for each pier, and the sum (positive value) of the power at the point of interest is calculated using the acceleration waveform. Based on this, the predicted value of the vibration level of the point of interest is calculated, so that it is avoided that the vibration is canceled by the interference of vibration waves from a plurality of piers in the calculation process and a vibration level lower than the actual is calculated, A highly accurate vibration level can be obtained.

In addition, the first step of conducting a structural analysis (modeling) of the ground based on the results of the ground survey conducted in advance,
The second step of modeling the pier based on the results of a survey of the target pier and foundation type and shape;
A third step of estimating an effective mass of each of a plurality of target piers and calculating an acceleration waveform of the pier based on the effective mass;
A fourth step of correcting the effective mass based on the calculated acceleration waveform and bridge vibration information;
A fifth step of calculating a transfer function to the point of interest using the ground model obtained in the first step and the pier model obtained in the second step;
A sixth step of calculating an acceleration waveform of a point of interest for each pier using the transfer function obtained in the fifth step;
A seventh step of calculating the sum of power from the respective acceleration waveforms obtained in the sixth step;
An eighth step of calculating a predicted value of the vibration level of the point of interest using the sum of the power obtained in the seventh step;
May be included.

  According to the method as described above, it is possible to avoid a vibration level that is canceled due to the interference of vibration waves from a plurality of bridge piers in the calculation process and to calculate a vibration level lower than the actual one, and to obtain a highly accurate vibration level. Can be sought. In addition, the ground vibration at any point around the viaduct where the train runs can be modeled easily and the analysis time is relatively short from the physical properties of the ground and the acceleration waveform of the vibration at the pier base. Can be sought. Furthermore, an effect is obtained that the excitation force at the pier necessary for the analysis can be obtained by calculation.

Here, preferably,
The vibration information is a vibration level of each pier base measured by a vibration measuring device,
When the vibration level of the pier calculated by calculation is L _BC , the vibration level of the measured pier is L _BO , the effective mass of the pier assumed in the calculation is m _BO , and the effective mass of the pier corresponding to the measurement result is m _BC In the fourth step, the following formulas m _BC = βm _BO , β = 10 ^{α / 20} , α = L _BO −L _BC
To correct the effective mass for each frequency.
Thereby, the predicted value of the synthetic vibration level with higher accuracy at the point of interest can be obtained.

Furthermore, preferably, while increasing the number of piers to be calculated,
Calculate the acceleration waveform of the point of interest, calculate the sum of power from the acceleration waveform, repeat the process of calculating the vibration level of the point of interest using the sum of power,
The vibration level when the difference between the calculated vibration level and the previously calculated value is smaller than a predetermined value is output as a predicted value.
Thereby, the predicted value of the vibration level with the required accuracy can be obtained with the minimum amount of calculation.

According to the present invention, the ground vibration at any point around the viaduct where the train travels is calculated and converted to the vibration level by calculating the sum of the vibration wave power from each pier. It is possible to obtain a predicted value of the vibration level with high accuracy by avoiding that the vibration is canceled due to the interference of the vibration wave and a vibration level lower than the actual is calculated.
In addition, the ground vibration at any point around the viaduct where the train runs can be obtained by complex response analysis using an axisymmetric model that is easy to model and relatively short in analysis time, based on the physical properties of the ground and the vibration acceleration of the pier base. Can do. Furthermore, an effect is obtained that the excitation force at the pier necessary for the analysis can be obtained by calculation.
In addition, since the pier whose vibration level is equal to or lower than the predetermined threshold is excluded from the final calculation target, an effect that the predicted value of the vibration level with the required accuracy can be obtained with the minimum amount of calculation can be obtained. .

It is a conceptual diagram of a plurality of bridge piers constituting a viaduct to which the ground vibration prediction method according to the present invention is applied and the ground around the pier as a vibration transmission system. It is the figure which made the axisymmetric model of the ground around the pier and the pier. It is a flowchart which shows 1st Example of the ground vibration prediction method which concerns on this invention. It is a flowchart which shows an example of the procedure of the calculation of the complex response analysis in the acceleration calculation of step S10 of FIG. It is a flowchart which shows 2nd Example of the ground vibration prediction method which concerns on this invention. It is a flowchart which shows an example of the procedure of the calculation of the time history response analysis in the acceleration calculation of step S10 of FIG. It is a figure which shows the example of the setting method of the order of the bridge pier which calculates the acceleration in the ground vibration prediction method of an Example, and the method of taking the difference of a vibration level. It is a figure which shows the example of calculation by correction of the effective mass in step S7 of the flowchart of FIG.

  Hereinafter, an embodiment of a ground vibration prediction method according to the present invention will be described in detail with reference to the drawings. The ground vibration prediction method according to the present embodiment is a personal computer including an arithmetic device such as a CPU (microprocessor), a storage device such as a ROM or a RAM, an input device such as a keyboard, and an output device such as a display device. Needless to say, this can be realized by a program stored in the storage device. Since the hardware configuration itself of such a personal computer is self-evident, its illustration is omitted.

で定義される。
環境省からの通達「環境保全上緊急を要する新幹線鉄道振動対策について（勧告）」では，通過列車ごとの振動のピークレベルを読み取ることとしている。そこで、以下の説明では、振動レベルＬvを１列車通過時に得られる時刻歴振動レベルＬv(t_m)の最大値Ｌv,maxとする。また、以下に説明する実施形態では、加速度波形から振動レベルを求めるのに、振動レベル計を再現した数値計算ツールを使用することとした。 In the following description, the term “vibration level” is used under the following rules.
When indicating the vibration level of a measurement time t _m and _{Lv (t m), Lv (} t m) , the effective value a _e of the acceleration that is frequency corrected at that point in time _{(t m) (m / s} 2) and reference Acceleration a ₀ (= 1.0 × 10 ⁻⁵ m / s ² ), the following formula [Equation 1] according to JIS C 1510

Defined by
In the notification from the Ministry of the Environment "Measures against Shinkansen railway vibration that require urgent environmental conservation (Recommendation)", the peak level of vibration for each passing train is read. Therefore, in the following description, the vibration level Lv is assumed to be the maximum value Lv, max of the time history vibration level Lv (t _m ) obtained when one train passes. In the embodiment described below, a numerical calculation tool reproducing a vibration level meter is used to obtain the vibration level from the acceleration waveform.

FIG. 1 is a conceptual diagram of a plurality of bridge piers constituting a viaduct to which the ground vibration prediction method according to the present invention is applied, and the ground around the pier as a vibration transmission system. In FIG. The bridge pier as a source, O, is a point of interest for which vibration is to be predicted. In FIG. 1, four piers are shown, but as will be apparent from the following description, the number of piers to be analyzed is not limited to four.
FIG. 2 is a diagram showing an example of a model generated for each pier used in the ground vibration prediction method of the embodiment, and FIG. 3 is a flowchart showing a procedure of the ground vibration prediction method of the first embodiment. .

  The ground vibration prediction method according to the first embodiment is a method of analyzing the vibration of the pier using the excitation force applied to the pier as an input. As shown in FIG. 3, the ground in a range where the ground vibration is desired to be predicted by boring or the like. (Step S1), and the ground is modeled up to the point of interest for each pier (step S2). In parallel with this, the shape of the piers constituting the viaduct in the prediction target area, the form of the foundation, and the shape are investigated (step S3), and the foundation is modeled for each pier (step S4).

Further, in parallel with the investigation of the ground characteristics (step S1) and the pier and foundation type and shape investigation (step S3), the vibration measurement of the base of the target pier is performed (step S5), and the excitation force is measured. Modeling is performed (step S6). The vibration measurement in step S5 is performed by attaching a vibration measurement device such as an acceleration sensor to the base of the pier. In addition, since the excitation force has a linear relationship with the acceleration of the pier, it can be converted from the measured acceleration.
As modeling of the excitation force in step S6, it can be assumed that the excitation force is input to the center of the surface of the modeled pier foundation, for example, as indicated by reference numeral F in FIG. In FIG. 2, an arrow with x is a force in a direction parallel to the track, an arrow with y is a force in a direction perpendicular to the track, and an arrow with z is a force in the vertical direction.

  Subsequently, the effective mass of the pier is estimated, and the acceleration of the pier base is calculated by a numerical calculation method such as a finite element method using the effective mass and the excitation force with respect to the model created in steps S2, S4, and S6. (Step S7). The acceleration of the pier base by the finite element method can be calculated using, for example, the analysis software “SoilPlus” provided by ITOCHU Techno-Solutions Corporation. Since it was not possible to do so, the acceleration was first calculated by giving an estimated value. In addition, in a viaduct of a train, when attention is paid to a certain pier, vibration transmitted from other piers in the vicinity through the bridge can be considered, but in the measurement results performed by the present inventors, Since it was found that the vibration hardly affects the maximum value of the vibration level measured at the pier, in this embodiment, the vibration from the other pier was ignored in the calculation.

Thereafter, it is determined whether or not the vibration level Lc obtained from the calculated acceleration Ac matches the vibration level Lp obtained from the acceleration Ap measured in step S5 (step S8). Returning to S7, the effective mass is corrected, and the acceleration of the pier base is calculated again. Thereby, even if there is no actual measurement value of the excitation force, the appropriate effective mass of the pier base can be obtained. A specific method of correcting the effective mass based on the acceleration will be described later.
If it is determined in step S8 that the vibration level Lc obtained from the acceleration Ac calculated in step S7 matches the vibration level Lp obtained from the acceleration Ap measured in step S5 (Yes), the process proceeds to step S9, and step S2 , S4, a transfer function of ground vibration to the point of interest is obtained (step S9).
In this embodiment, as the piers 21 to 24, for example, the direct foundation (footing) shown in FIG. 1 is assumed. Each pier also has an axisymmetric model of the pier foundation as shown in FIG. 2 even when the cross section is rectangular. In FIG. 2, A is a modeling area. The surface of the pier foundation should match the surface of the surrounding ground.

For the axisymmetric model shown in FIG. 2, the transfer function is obtained by grasping the ground structure from the ground survey results obtained in step S1, setting the ground constant for each layer, and performing vibration analysis using the finite element method. Can do. The calculation of the transfer function by the finite element method is a known method as described in, for example, the above-mentioned Non-Patent Document 1 and the like, and thus detailed description is omitted.
Subsequently, an effective force is obtained by multiplying the pier base acceleration calculated in step S7 by the effective mass, and the vibration propagation from each pier is performed using the excitation force and the transfer function of the ground vibration calculated in step S9. The acceleration (hereinafter referred to as ground acceleration) of the ground (excluding structures such as paved roads) at the target point of interest is calculated. At this time, an acceleration time history waveform a _wi (t) (hereinafter referred to as an acceleration waveform) is calculated in consideration of a phase difference due to the passage of a train at each pier. A detailed method of calculating the acceleration waveform will be described in detail later.

で、算出することができる。 Thereafter, the power sum of the acceleration at the point of interest is calculated (step S11). For example, the effective value a _ei (t) of the waveform a _wci (t) _obtained by performing frequency correction defined in JIS C 1510 on the acceleration waveform a _wi (t) for each pier calculated in step S10. Find the square. Here, a _wi (t) is an acceleration waveform propagated from pier i to the point of _interest , a _wci (t) is an acceleration waveform after frequency correction propagated from pier i to the point of _interest , and a _ei (t) is a pier. This is the effective value of the acceleration waveform propagated from i to the point of interest. The square of the effective value of the acceleration waveform at a certain measurement time t _m is expressed by the following equation using the time constant τ (= 0.63 s) defined in JIS C 1510:

Can be calculated.

これにより、算出された加速度のパワー和は、各橋脚における列車の通過による時間遅れを反映したものとなる。なお、本実施例では、各橋脚から伝播される加速度波形の実効値をそれぞれ求めてから[数３]よりパワー和を求めているが、[数４]から求められる各橋脚から伝播される加速度の二乗の和ａ_w ²(t)から実効値の二乗を求め、パワー和としても結果は変わらない。

The square of acceleration is proportional to vibration energy (hereinafter referred to as acceleration power). The vibration level to be calculated is evaluated by the power ratio with the reference acceleration. Therefore, the vibration energy propagated from the plurality of bridge piers can be obtained as the sum of power.
Specifically, the power sum of acceleration at the point of interest is calculated by the following equation [Equation 3].

Thereby, the calculated power sum of acceleration reflects a time delay due to the passage of the train at each pier. In this embodiment, the effective value of the acceleration waveform propagated from each pier is obtained, and then the power sum is obtained from [Equation 3]. However, the acceleration propagated from each pier obtained from [Equation 4]. The square of the effective value is obtained from the sum of squares a _w ² (t), and the result is the same as the power sum.

続いて、算出された合成振動レベルΣＬｖの前回算出値との差分ΔΣＬｖを求め（ステップＳ１３）、該差分が予め設定されているしきい値Ｌthよりも小さいか否か判定し（ステップＳ１４）、大きい（Ｎｏ）と判定したときは、ステップＳ７へ戻って、他の橋脚からの振動（加振力）に対して上記処理を繰り返し、着目点の振動レベルを算出する。 Next, the combined vibration level (sum of vibrations from a plurality of bridge piers) ΣLv at the point of interest is calculated by the following equation [Equation 5] (step S12). Here, a _O is a reference acceleration (= 1.0 × 10 ⁻⁵ m / s ² ) defined in JIS C 1510. Max (Lv (t)) is the maximum value of Lv (t).

Subsequently, a difference ΔΣLv from the previous calculated value of the calculated combined vibration level ΣLv is obtained (step S13), and it is determined whether or not the difference is smaller than a preset threshold value Lth (step S14). When it is determined to be large (No), the process returns to step S7, and the above processing is repeated with respect to vibration (excitation force) from other piers to calculate the vibration level of the point of interest.

  On the other hand, when it is determined in step S14 that the difference ΔΣLv from the previous calculated value is smaller than the threshold value Lth (Yes), the process proceeds to step S13, and the combined vibration level calculated in step S12 is used as the predicted vibration level. (Step S15). In this embodiment, the vibration level is calculated in order from the pier close to the point of interest toward the pier far away. Specifically, as shown in FIG. 7 (A), numbers # 1, # 2, # 3, # 4,... Are added in order from the pier near the point of interest O, and ground acceleration is calculated in order from the pier with the smallest number. Then, as shown in FIG. 7B, the combined vibration level ΣLv and the difference ΔΣLv at the point of interest are obtained, the vibration level at the time when the difference ΔΣLv becomes smaller than a predetermined value is set as the predicted value, and the calculation is terminated. . As a result, the amount of calculation to be performed before obtaining a vibration level with a desired accuracy can be reduced.

In the case of the analysis using the axially symmetric model as in this embodiment, the three component excitation forces in the radial direction, the circumferential direction and the vertical direction act on the ground. The vertical component acceleration generated by the excitation force is calculated and synthesized, and converted into a vibration level.
If the vibration level is calculated after combining the vibration at the point of interest at the acceleration calculation stage as in the conventional prediction method, the vibration level is actually higher than the actual level due to cancellation of vibration waves from multiple bridge piers. May become a small value. In general, the actual ground in a range that spans multiple piers varies in structure and physical properties. For this reason, the reduction of the vibration level caused by the cancellation by the interference of vibration waves that can easily occur in an ideal uniform ground is very small. According to this embodiment, since the power sum is the square of the acceleration waveform, that is, a positive value and does not take a negative value, if vibrations from a plurality of bridge piers are evaluated with the power sum, A vibration level without cancellation due to interference can be obtained. The validity of this method is shown by measurement results and comparison.

  Furthermore, since the bridge pier whose vibration level is equal to or lower than the predetermined threshold value is excluded from the final calculation target, it is possible to obtain a high-accuracy synthesized vibration level at the point of interest while reducing the calculation amount. In addition, the determination result of step S12 is stored in association with the characteristics of the target ground to create a database, and when the vibration level is predicted for a new point, the pier to be calculated based on the characteristics of the target ground and the like May be determined from the stored information in the database.

[Calculation of acceleration at the point of interest]
Next, a specific method for calculating the acceleration at the point of interest in step S10 will be described.
In the first embodiment, since the excitation force is input, complex response analysis (frequency response analysis) is applied. Note that the analysis software “SoilPlus” described above can be used for the complex response analysis.

The complex response analysis method is a method for evaluating hysteresis attenuation by complex synthesis. This analysis method is an analysis in the frequency domain. After obtaining the transfer function T (ω) between the excitation point and the target point, the frequency of the target point is multiplied by the excitation force F (ω) converted into the frequency domain. The response A (ω) is calculated. If the response in the frequency domain is converted to the time history domain, the time history waveform a (t) of the acceleration of the target point to be obtained can be obtained.
FIG. 4 shows a calculation flow of complex response analysis (corresponding to steps S9 and S10 in FIG. 3). Assuming that the ground is a linear viscoelastic body, the transfer function T (ω) is not related to the magnitude of the excitation force. Therefore, once the transfer function is calculated, the transfer function can be used even when different excitation forces are calculated.

As shown in FIG. 4, in the calculation of the complex response analysis, Fourier transform is performed on the excitation force f (t) at the excitation point (pier base), and the frequency from the excitation force f (t) in the time history region is calculated. A region excitation force F (ω) is obtained (step S21). Next, using FEM (finite element method), the transfer function T (ω) from the excitation point (pier base) to the point of interest is
T (ω) = A (ω) / F (ω)
(Step S22). Here, A (ω) is the acceleration response (frequency domain display) of the point of interest. Next, the acceleration response (frequency domain) of the point of interest
A (ω) = T (ω) × F (ω)
(Step S23). Thereafter, an inverse Fourier transform is performed on the acceleration response A (ω) at the point of interest, and an acceleration a (t) in the time history domain is obtained from the acceleration A (ω) in the frequency domain (step S24).

[Excitation force setting]
Here, a specific method of setting the excitation force on the pier foundation will be described by taking an example in which the number of piers is four. It is assumed that all of the _four piers P _{1 to} P ₄ have the same basic shape, and the acceleration waveforms measured at the respective piers are substantially the same.
If the effective mass m _{l of the} pier is given when the above conditions are satisfied, the Fourier transform A _l (ω) of the acceleration waveform measured at the pier P _l and the effective mass m _{l of the} pier Therefore, the excitation force F _l (ω) acting on the pier P _l is given by m _l × A _l (ω). Since piers P ₂ thereafter exciting force from traveling train is loading, it occurs time delayed from pier P _1.

The time point when the excitation force is loaded on the pier P ₁ is t = 0, and the delay when the train excitation force is loaded on the pier P ₂ is t ₂ . Similarly, assuming that the time delays of the piers P ₃ and P ₄ are t ₃ and t ₄ , respectively, and the function giving the time delay in the frequency domain is R (ω, t), the excitation force F (ω) of each pier Can be expressed as:
Pier P _l : F _l (ω) = m _l × A _l (ω)
Pier P ₂ : F ₂ (ω) = F _l (ω) × R (ω, t ₂ )
Pier P ₃ : F ₃ (ω) = F _l (ω) × R (ω, t ₃ )
Pier P ₄ : F ₄ (ω) = F _l (ω) × R (ω, t ₄ )

[Calculation of acceleration propagating from each pier to point of interest O]
If the transfer functions from the piers P _l , P ₂ , P ₃ , P ₄ to the point of interest O are T _1c (ω), T _2c (ω), T _3c (ω), and T _4c (ω), respectively, By using the vibration force F _i (ω) and the transfer function T _ic (ω), the accelerations A _w1 (ω), A _w2 (ω), A _w3 (ω), A _w4 propagate from each pier to the point of _interest O. (ω) and a _w1 (t), a _w2 (t), a _w3 (t), and a _w4 (t) can be expressed as follows. Here, A _wi (ω) is the acceleration in the frequency domain, and a _wi (t) is the acceleration waveform of the time history obtained by inverse Fourier transform.
Pier P _l : A _wl (ω) = T _1c (ω) × F _l (ω) Inverse Fourier transform → a _w1 (t)
Pier P ₂ : A _w2 (ω) = T _2c (ω) × F ₂ (ω) Inverse Fourier transform → a _w2 (t)
Pier P ₃ : A _w3 (ω) = T _3c (ω) × F ₃ (ω) Inverse Fourier transform → a _w3 (t)
Pier P ₄ : A _w4 (ω) = T _4c (ω) × F ₄ (ω) Inverse Fourier transform → a _w4 (t)

  The calculation of the power sum of the point of interest O in step S11 in the flowchart of FIG. 3 is calculated by substituting the acceleration at each time into the above-described equation [Equation 3] from the acceleration waveform obtained by the above calculation and conversion. Do that. In step S12 of FIG. 3, the vibration level Lv of the point of interest O is calculated by substituting the maximum value among the power sums obtained in the calculation of step S11 into the above-described equation [Equation 4]. Will be output.

Next, the calculation method of the effective mass in step S7 of FIG. 3 will be described.
In the calculation flow of the complex response analysis of FIG. 4 using the excitation force as an input, if there is an obvious difference between the vibration level L _{BC of the} pier obtained by the calculation and the vibration level L _BO of the measurement result, The force (effective mass) is corrected and recalculation is performed. Here, a _O is the reference acceleration (= 1.0 × 10 ⁻⁵ m / s ² ), a _BC is the effective value of the calculated acceleration of the pier with frequency correction, and a _BO is the measured acceleration of the pier with frequency correction. The effective value, m _BC is the effective mass used in the calculation, and m _BO is the effective mass commensurate with the measurement results.

ここで、αは実数である。
計算結果の振動レベルＬ_BCと計測結果の振動レベルＬ_BOは、振動レベルの定義式より次式［数７］のように表わせる。 Now, it is assumed that there is a relationship of the following equation [Formula 6] between the vibration level L _BC of the calculation result and the vibration level L _BO of the measurement result.

Here, α is a real number.
The vibration level L _BC of the calculation result and the vibration level L _BO of the measurement result can be expressed by the following equation [Expression 7] from the definition equation of the vibration level.

これを変形すると、次式［数９］が得られる。

Substituting the equation of [Equation 6] into the last equation of [Equation 7] and rearranging, the following equation [Equation 8] is obtained.

When this is transformed, the following equation [Equation 9] is obtained.

従って、上記のような方法によれば、橋脚の質量が不明である場合にも計算によって橋脚の実効質量を求め、この実効質量を使用して橋脚基部の加速度や加振力を算定し、有限要素法を利用して着目点の加速度、振動レベルを予測することができる。 Here, if “10 ^{α / 20} ” in the above [Equation 9] is β, the equation of the above [Equation 9] indicates that the acceleration a _{BO of the} measurement result is β times the acceleration a _BC of the calculation result. It is shown that. This means that the excitation force used for the calculation is 1 / β times the original, that is, the effective mass used for the calculation is 1 / β times. Therefore, the effective mass m _{BO to} be used for calculation of the acceleration of the pier base and the excitation force in the first embodiment can be given by the following equation [Equation 10].

Therefore, according to the method as described above, even when the mass of the pier is unknown, the effective mass of the pier is obtained by calculation, and the acceleration and the excitation force of the pier base are calculated using this effective mass. The acceleration and vibration level of the point of interest can be predicted using the element method.

In the above-described correction of the effective mass, calculation is performed for each predetermined frequency band. As a method of setting the frequency band, for example, a frequency band in octave band analysis generally performed in the field of vibration analysis can be considered.
FIG. 8 shows a calculation example performed by the present inventors. In FIG. 8, (A) shows the error of the initial value of the vibration acceleration level. From the left, the center frequency value of 1/3 octave band, the measured value, the calculated value, the error, and the effective mass adjustment value are shown. Represents. Further, (B) shows the first correction error, and in order from the left, represents the value of the center frequency, the first calculation value, the second calculation value, and the error value. Looking at the error value in the rightmost column of FIG. 8B, the error is sufficiently small, and it has become clear that the correction calculation should be performed only once. In other words, the burden on the computer associated with the effective mass correction process can be reduced.

  Next, a second embodiment of the ground vibration prediction method according to the present invention will be described. FIG. 5 shows an example of the procedure of the ground vibration prediction method of the second embodiment. The difference from the embodiment shown in FIG. 3 is that, in the calculation of the acceleration at the point of interest in step S10, as a vibration input to the pier, in the first embodiment of FIG. In the second embodiment, the acceleration of the pier measured in step S5 is used as it is and the time history response analysis is used for calculation. Therefore, in the second embodiment, steps S6, S7, and S8 in FIG. 3 can be omitted. Since the contents of the processes in steps S1 to S4 and steps S11 to S13 are substantially the same as those in the first embodiment (FIG. 3), description thereof is omitted.

FIG. 6 shows a calculation flow for calculating the ground acceleration at the point of interest in the second embodiment (corresponding to steps S9 and S10 in FIG. 5).
As described above, the complex response analysis (frequency response analysis) is applied in the first embodiment, while the time history response analysis is applied in the second embodiment. The time history response analysis can be executed using, for example, Itasca analysis software “FLAC”.

  Time history response analysis is a method that allows you to calculate the acceleration of the target point of interest by inputting the acceleration of the pier. Compared to complex response analysis, it is easier to understand because there is no work to model the input vibration. . However, a response analysis that requires a long time for calculation each time the train organization or traveling speed is changed must be performed. However, when the ground is regarded as a linear viscous body, the magnitude of the input acceleration does not change depending on the train speed or traveling speed. Therefore, in the second embodiment, the transfer function T (ω) is obtained in the same manner as in the complex response analysis, and is used to calculate the acceleration at the point of interest for different accelerations.

As shown in FIG. 6, in the calculation of the time history response analysis, the acceleration at the excitation point (pier base) is set to a (t), and the acceleration at the point of interest is set to ao (t) (step S31). Next, Fourier transformation is performed on the acceleration a (t) at the excitation point (base of the pier) and the acceleration at the point of interest with respect to ao (t) to obtain A (ω) and Ao (ω) (step S32). After that, the transfer function T (ω) from the excitation point (base of the pier) to the point of interest is
T (ω) = Ao (ω) / A (ω)
(Step S33).

Next, the acceleration response (frequency domain) of the point of interest
Ao (ω) = T (ω) × A (ω)
(Step S34). Thereafter, an inverse Fourier transform is performed on the acceleration response Ao (ω) at the point of interest, and an acceleration ao (t) in the time history domain is obtained from the acceleration Ao (ω) in the frequency domain (step S35). When analyzing different input accelerations, as in steps S36 and S37, Fourier transformation is performed on the different input accelerations a ₁ (t) to obtain A ₁ (ω) and the calculation in step S34 is performed. .
Note that, as described above, the time history response analysis is a method in which the acceleration of the target point to be obtained can be calculated by inputting the acceleration of the pier, so step S9 for calculating the transfer function in FIG. 5 is omitted. Can do.

As mentioned above, although this invention was demonstrated based on embodiment, this invention is not limited to the said embodiment, In the range which does not deviate from the summary of invention, it can change suitably. For example, in the above-described embodiment, the case where the axially symmetric model is applied to the pier foundation modeling has been described, but other modeling may be applied. By applying an axisymmetric model, the amount of calculation can be reduced compared to other models.
Moreover, although the said embodiment demonstrated the case where it applied to the prediction of the ground vibration in the attention point distant from the viaduct by predetermined distance, it is applicable also to evaluation of countermeasure work.

21-24 Pier Pier CL Pier Center Line A Axisymmetric Model O Point of Interest F Excitation Force

Claims (5)

A ground vibration prediction method for predicting ground vibration around a bridge supported by a plurality of piers,
A first step of analyzing the structure of the ground based on the results of the ground survey conducted in advance;
The second step of modeling the pier based on the results of a survey of the target pier and foundation type and shape;
A third step of calculating an acceleration waveform propagating from each pier by calculation using the ground model obtained in the structural analysis of the first step and the pier model obtained in the second step based on the acceleration of the pier. When,
A fourth step of calculating the sum of power from the respective acceleration waveforms obtained in the third step;
A fifth step of calculating a predicted value of the vibration level of the point of interest using the sum of the power obtained in the fourth step;
A ground vibration prediction method comprising: A ground vibration prediction method for predicting ground vibration around a bridge supported by a plurality of piers,
A first step of analyzing the structure of the ground based on the results of the ground survey conducted in advance;
The second step of modeling the pier based on the results of a survey of the target pier and foundation type and shape;
A third step of calculating a transfer function up to a point of interest using the ground model obtained in the structural analysis of the first step and the pier model obtained in the second step;
A fourth step of calculating an acceleration waveform at a point of interest for each pier using the transfer function obtained in the third step;
A fifth step for obtaining a sum of power from the respective acceleration waveforms obtained in the fourth step;
A sixth step of calculating a predicted value of the vibration level of the point of interest using the sum of the power obtained in the fifth step;
A ground vibration prediction method comprising: A ground vibration prediction method for predicting ground vibration around a bridge supported by a plurality of piers,
A first step of analyzing the structure of the ground based on the results of the ground survey conducted in advance;
The second step of modeling the pier based on the results of a survey of the target pier and foundation type and shape;
A third step of estimating an effective mass of each of a plurality of target piers and calculating an acceleration waveform of the pier based on the effective mass;
A fourth step of correcting the effective mass based on the calculated acceleration waveform and bridge vibration information;
A fifth step of calculating a transfer function to the point of interest using the ground model obtained in the first step and the pier model obtained in the second step;
A sixth step of calculating an acceleration waveform of a point of interest for each pier using the transfer function obtained in the fifth step;
A seventh step of calculating the sum of power from the respective acceleration waveforms obtained in the sixth step;
An eighth step of calculating a predicted value of the vibration level of the point of interest using the sum of the power obtained in the seventh step;
A ground vibration prediction method comprising: The vibration information is a vibration level of each pier base measured by a vibration measuring device,
When the vibration level of the pier calculated by calculation is L _BC , the vibration level of the measured pier is L _BO , the effective mass of the pier assumed in the calculation is m _BO , and the effective mass of the pier corresponding to the measurement result is m _BC In the fourth step, the following formulas m _BC = βm _BO , β = 10 ^{α / 20} , α = L _BO −L _BC
The ground mass prediction method according to claim 3, wherein the effective mass is corrected for each frequency by using. While increasing the number of piers to be calculated,
Calculate the acceleration waveform of the point of interest, calculate the sum of power from the acceleration waveform, repeat the process of calculating the vibration level of the point of interest using the sum of power,
The ground vibration prediction method according to any one of claims 1 to 3, wherein a vibration level when a difference between the calculated vibration level and a previously calculated value is smaller than a predetermined value is used as a predicted value.

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