JP2012032214A - Response analyzer, method and program - Google Patents

Abstract

PROBLEM TO BE SOLVED: To improve analysis precision of response analysis such that a range outside an inner area of an analysis object model shows nonlinear behavior.SOLUTION: Property values of the ground at respective analysis time points (at intervals Δt) are calculated (38) first by performing nonlinear time history response analysis of the ground in an outer area with respect to the analysis object model provided with an energy transmission border as a wave motion border model between an inner area and the outer area. A transmission border matrix and an impulse response of the energy transmission border including the ground in the outer area are calculated (40-44) at intervals Δtb (>Δt) based upon the calculated property values. Impulse responses at all the analysis time points are calculated (46) through interpolation operation. Time history response analysis of the analysis object model is carried out (52) at all the analysis time points.

Description

  The present invention relates to a response analysis apparatus, method, and program, and in particular, a response analysis apparatus that performs a response analysis of a vibration problem in a time domain with respect to an analysis target model including a wave boundary model, and a response applicable to the response analysis apparatus The present invention relates to an analysis method and a response analysis program for causing a computer to function as the response analysis apparatus.

  In order to accurately evaluate the behavior and damage of a building during an earthquake, it is necessary to perform an earthquake response analysis in consideration of the interaction effect between the building and the ground. Earthquake response analysis is broadly divided into frequency response analysis, which performs response analysis in the frequency domain, and time history response analysis, which performs response analysis in the time domain. In order to accurately evaluate the behavior of the building and the like, it is desirable to perform a time history response analysis capable of nonlinear analysis.

  In earthquake response analysis, the ground is often modeled using a finite element method (FEM) nonlinear solid element. The ground originally has a semi-infinite extent, but in FEM analysis, it is necessary to model the analysis region as finite, so that a wave boundary model is set on the side surface and bottom surface of the analysis target model. The wave excited in the building mainly propagates as a body wave downward and as a surface wave laterally, but the surface wave propagates farther, so the side surface of the model to be analyzed is the side surface. The wave boundary model set to has a greater effect on the analysis results. As side boundary models applicable to time domain analysis, for example, repetitive boundaries, viscous boundaries, and non-reflective boundaries are known. Of these, it is extremely easy to model the repetitive boundary, but the accuracy as a wave boundary is low, and it is necessary to take a large modeling area (analysis target model), so the analysis load is large. In addition, the viscous boundary has low accuracy with respect to waves from an oblique direction, and it is necessary to increase the modeling area in the same manner in order to increase the analysis accuracy. Furthermore, the non-reflective boundary has many problems in application to real problems and is not often used.

  On the other hand, an energy transmission boundary (Energy Transmitting Boundary) is known as a more accurate wave boundary model. The energy transfer boundary is installed at the outer edge of the ground that forms parallel stratification on the rigid base, and is strictly in the horizontal direction, and the vertical direction follows the element's displacement assumption (if linear primary element, the displacement changes linearly) It is a highly accurate boundary and has a characteristic of absorbing the wave from an arbitrary direction almost completely. For this reason, if the above-mentioned energy transfer boundary is used as the wave boundary model on the side of the model to be analyzed, response analysis with the same analysis accuracy can be realized with a smaller analysis load by significantly reducing the modeling area. .

  However, although the characteristics of the energy transfer boundary are expressed by the transfer boundary matrix and the correction force vector, the transfer boundary matrix is a complex number in the frequency domain and has strong frequency dependence, so it is converted to the time domain. Therefore, the response analysis using the analysis target model to which the energy transfer boundary is applied as the side boundary model is limited to the equivalent linear analysis in the frequency domain. In order to solve this, the inventor of the present application has already proposed a technique for realizing response analysis in the time domain for an analysis target model provided with an energy transfer boundary as a wave boundary model (see Patent Document 1). The inventor has already proposed a technique for converting the dynamic stiffness of the ground, which is a complex function in the frequency domain, into an impulse response represented in the time domain (see, for example, Patent Document 2 and Non-Patent Document 1). ).

JP 2009-250805 A Japanese Patent No. 3878626

Naohiro Nakamura, “Earthquake response analysis of structures embedded in stratified ground by time domain transformation of ground impedance”, Architectural Institute of Japan, May 2003, No. 567, p. 63-70

  However, the technique described in Patent Document 1 is a technique for performing linear analysis on the energy transfer boundary and the range outside the energy transfer boundary (external region) in the analysis target model, and is constructed on, for example, hard ground Analytical conditions such as the arrival of an earthquake in a built building can provide good analysis accuracy. When response analysis is performed under analysis conditions that extend to the energy transfer boundary, the accuracy of the analysis decreases because the plasticity (non-linearization) of the energy transfer boundary and the external region cannot be expressed. was there.

  The present invention has been made in consideration of the above facts, and a response analysis apparatus, a response analysis method, and a response analysis method capable of improving the analysis accuracy in a response analysis in which the range outside the internal region of the analysis target model exhibits nonlinear behavior The purpose is to obtain a response analysis program.

  In order to achieve the above object, a response analysis apparatus according to claim 1 is formed by modeling an object to be analyzed consisting of a single ground or a ground and a building, and a wave boundary between an internal region and an external region of the model. Response to analyze the behavior when a specific external force that vibrates the object to be analyzed is input to the external region of the model to be analyzed provided with an energy transfer boundary as a model at each time of the object to be analyzed By performing the analysis, the physical property value calculating means for calculating the physical property value of the object to be analyzed in the external region for each time of the analysis target, and the physical property value calculating means for calculating the physical property value for each time of the analysis target The main component of the energy transfer boundary when the specific external force is input based on the physical property value of the object to be analyzed in the external region, and the frequency region Matrix calculation means for calculating the value of a transmission boundary matrix defined as a complex number and having a strong frequency dependence for each of two or more times of each time to be analyzed, and two or more by the matrix calculation means The energy when the specific external force is input as an impulse response representing the relationship between the external force that vibrates the object and the behavior of the object in the time domain based on the value of the transmission boundary matrix calculated for each time of An impulse response calculating means for calculating the impulse response of the transmission boundary at each time to be analyzed, and the impulse response of the energy transfer boundary calculated at each time of the analysis target by the impulse response calculating means Then, the behavior when the specific external force is input to the analysis target model is solved. It is configured to include an analysis unit that performs response analysis for analyzing for each time of the target, the.

  In the first aspect of the present invention, an analysis object in which an analysis target object composed of the ground alone or the ground and a building is modeled, and an energy transmission boundary as a wave boundary model is provided between the internal region and the external region of the model. Response analysis that analyzes the behavior when a specific external force that vibrates the object to be analyzed is input at each time of the analysis target is performed on the external area of the model by the physical property value calculation means. The physical property value of the analysis target object in the region is calculated for each time of the analysis target. As the physical property value of the analysis target object, for example, the shear strain amplitude value γ, the shear wave velocity Vs, and the attenuation constant h of the analysis target object can be applied. Thereby, data (physical property value at each time of the analysis target) representing the behavior of the analysis target object in the external region when a specific external force is input is obtained.

  In addition, the matrix calculation means is based on the physical property value of the analysis target object in the external area calculated at each time of the analysis target by the physical property value calculation means, and the main energy transfer boundary when a specific external force is input. A component, which is defined as a complex number in the frequency domain and has a strong frequency dependence, calculates a value of a transmission boundary matrix for each of two or more times of each time to be analyzed. Based on the value of the transmission boundary matrix calculated for each of two or more times by the matrix calculation means, a specific external force is input as an impulse response that represents the relationship between the external force that vibrates the object and the behavior of the object in the time domain. The impulse response of the energy transfer boundary is calculated for each time to be analyzed.

  As described above, in the first aspect of the present invention, a response for analyzing the behavior when a specific external force that vibrates the analysis target object is input to the external region of the analysis target model at each time of the analysis target. By performing the analysis, the physical property value of the analysis target object in the external region when a specific external force is input is calculated for each time of the analysis target, and at least two or more times of the analysis target time Each transmission boundary matrix is calculated, and the impulse response of the energy transmission boundary is calculated for each time to be analyzed based on the transmission boundary matrix calculated for two or more times. Therefore, analysis is performed according to the input of a specific external force. Even when the outer region of the target model exhibits nonlinear behavior, an impulse response that accurately reflects the behavior of the outer region at each time to be analyzed can be obtained.

  In the first aspect of the invention, the analysis means inputs a specific external force to the analysis target model using the impulse response of the energy transmission boundary calculated at each time of the analysis target by the impulse response calculation means. Response analysis is performed to analyze the behavior at each time of the analysis target, so that it is possible to improve the analysis accuracy in the response analysis in which the range outside the internal region of the analysis target model exhibits nonlinear behavior.

In the invention described in claim 1, the transmission boundary matrix of the energy transmission boundary is, for example, as described in claim 2, the mass matrix of the internal region is [M _I ], the stiffness matrix is [K _I ], and the displacement vector {U (t)}, the boundary force vector is {F _R (t)}, the response displacement of the object to be analyzed in the outer region is {u _R ^* (t)}, and the correction force vector acting on the boundary is − [D _R (t)] {u _R ^* (t)}
[M _I ] {u "(t)} + ([K _I (t)] + [R (t)]) {u (t)}
= −y ″ (t) [M _I ] {1} + {F _R (t)} (1)
{F _R (t)} = ([R (t)] − [D _R (t)]) {u _R ^* (t)} (2)
When the equation of motion of the entire analysis target model is expressed by the above equations (1) and (2), it can be expressed by [R (t)] in the above equations (1) and (2).

Hereinafter, the above equations (1) and (2) will be described. In FIG. 1, the ground is modeled as an assembly of a large number of components of a certain size, and an energy transfer boundary is provided as a wave boundary model between the internal region and the external region of the model. ½ model on the right side formed by ½ model with inverse symmetry condition (the energy transfer boundary exists only on the right side of the model, the internal area on the left side across the energy transfer boundary, the external area on the right side (Model where free ground is also located). In this 1/2 model, the wave equation of the external region (free ground) in the two-dimensional in-plane problem is expressed by the following equation (3).
([A (t)] k ² + i [B (t)] k + [G (t)] − ω ² [M]) {u ^* (ω)} = {0} (3)
Here, k is the wave number, and {u ^* (ω)} is the displacement vector of the external region (free ground). [A (t)], [B (t)], [G (t)], and [M] are 2n × 2n (n is the number of nodes) matrix, and the sub-matrix [A ( t)] _j , [B (t)] _j , [G (t)] _j , [M] _j . Each sub-matrix is expressed by the following equations (4) to (7) using the lame constants G (t) _j and λ (t) _j of each element. h _j and ρ _j are the height and density of each element.

here,
[C] = [G (t)] − ω ² [M (t)] (8)
And solve the eigenvalue problem concerning the wave number k in the following equation (9).
| [A (t)] k2 + i [B (t)] k + [C (t)] | = {0} (9)
From the obtained 4n eigenmodes, 2n components propagating in the right direction are extracted and set as a mode matrix [V]. Thus, the transfer boundary matrix of the energy transfer boundary existing on the right side of the 1/2 model is represented by [R (t)] in the following equation (10). However, [D (t)] is represented by superposition of sub-matrix [D (t)] _j represented by the following equation (11).
[R (t)] = i · [A (t)] [V] [K] [V] ⁻¹ + [D (t)] (10)

In the above equation (10), the matrix is asymmetric, but in the following, the matrix is symmetrized by averaging the terms at the symmetrical positions. As a result, the above equation (1) is obtained as the equation of motion of the entire model to be analyzed, and the above equation (2) is obtained as the arithmetic expression of the boundary force vector {F _R (t)}. The correction force (also referred to as notch force) vector − [D _R (t)] {u _R ^* (t)} in the above equation (2) is [Q (t )] = [D (t)], otherwise [Q (t)] = ik [A (t)] + [D (t)]. Where k is the wave number. [R (t)], {u _R ^* (t)}, {F _R (t)} are defined with 2n degrees of freedom at the boundary up to equation (10). In order to superimpose, the equations (1) and (2) are used by expanding the number of degrees of freedom in the inner region.

  Further, in the invention according to claim 1 or claim 2, the physical property value calculating means, for example, as described in claim 3, has a frequency ω in the range of (n−1) · ωm to n · ωm (2n -1) is a sum of a regular component of the imaginary part showing a value of (2ω / ωm) (where n is an integer) and a singular component of an imaginary part showing a value of (2ω / ωm) regardless of the frequency ω. An imaginary part representing a value of (2n-1) in the range of (n-1) · ωm to n · ωm, and a real part corresponding to the Hilbert transform value of the regular component of the imaginary part, Using the impulse response value of the causal unit imaginary function obtained by converting only the imaginary part into the time domain, or using the impulse response value of the causal unit imaginary function obtained by converting only the imaginary part into the time domain. By performing a response analysis on the outer region, the shear strain amplitude value is used as the physical property value of the object to be analyzed in the outer region. It is preferably configured to compute the shear wave velocity Vs and damping constant h.

When a large energy and large amplitude external force is input to the analysis target object consisting of the ground alone or the ground and the building as a specific external force that vibrates the analysis target object, such as during a large earthquake, The ground has a frequency-independent characteristic (a characteristic in which the damping constant h shows a constant value regardless of the frequency ω) as a nonlinear behavior, and the hysteresis absorption energy (damping constant h) of the internal damping does not depend much on the frequency ω. In addition, a strain amplitude dependence characteristic in which the stiffness reduction rate α (= rigidity G / initial stiffness G ₀ ) and the damping constant h change according to the shear strain amplitude value γ of the object (ground) is also shown.

  On the other hand, in the invention described in claim 3, since the response analysis for the external region of the analysis target model is performed using the impulse response value of the causal unit imaginary function, the invention described in Japanese Patent Application Laid-Open No. 2008-304227. As with, the behavior of the object to be analyzed in the external region can be analyzed with high accuracy without being affected by the fact that the ground also exhibits frequency-independent characteristics and strain amplitude-dependent characteristics. As the physical property value, a physical property value that represents the behavior of the object to be analyzed in the external region with high accuracy can be obtained.

  Further, in the invention according to any one of claims 1 to 3, the impulse response calculation means includes a transmission boundary matrix calculated for each of two or more times by the matrix calculation means as described in claim 4, for example. After calculating the impulse response of the energy transfer boundary when a specific external force is input based on the value of the value at two or more times at which the values of the transfer boundary matrix are calculated, the energy transfer of each time to be analyzed It is preferable that the impulse response of the energy transfer boundary at the time when the impulse response of the boundary is not calculated is obtained by interpolation from the impulse response of the energy transfer boundary calculated for each of two or more times. This reduces the number of computations of the transmission boundary matrix with a relatively large computation load, thereby reducing the computation load caused by the computation of the transmission boundary matrix while suppressing the deterioration of the analysis accuracy in the response analysis by the analysis means. Can do.

  Further, in the invention described in claim 4, the impulse response interpolation calculation by the impulse response calculation means is the analysis target in the external region at the time when the impulse response of the energy transfer boundary is not calculated as described in claim 5, for example. Based on the ratio that the shear strain amplitude value γ of the object internally divides the shear strain amplitude value γ of the object to be analyzed in the external region at two or more times when the value of the transmission boundary matrix is calculated, the impulse of the energy transfer boundary The impulse response of the energy transfer boundary at the time when the response is not calculated can be realized by obtaining the impulse response of the energy transfer boundary calculated at two or more times by interpolation.

  In the invention according to claim 4, the impulse response interpolation calculation by the impulse response calculation means is performed, for example, as described in claim 6, the time when the impulse response of the energy transfer boundary is not calculated is the value of the transfer boundary matrix. The impulse response of the energy transfer boundary at the time when the impulse response of the energy transfer boundary is not calculated is calculated based on the ratio that internally divides the time of 2 or more when is calculated. It can also be realized by obtaining the impulse response by interpolation calculation.

Further, in the invention according to any one of claims 1 to 6, the impulse response calculation means, as described in claim 7, for example, sets the simultaneous component of the impulse response depending on the displacement of the object to k ₀ , The simultaneous component of the impulse response that depends on the velocity of the object is c ₀ , the simultaneous component of the impulse response that depends on the acceleration of the object is m ₀ , the time delay component in increments of Δt of the impulse response that depends on the displacement of the object is k _j , The time delay component in increments of Δt of the impulse response depending on the velocity is c _j (where j is a natural number t _j = Δt · j), the displacement of the object in the time domain is u (t), and the velocity is u ′ (t) , When the acceleration is u "(t), it is an equation that defines the impulse response F _B (t) as an equation including a mass term that depends on the acceleration of the object and includes at least a simultaneous component.

Using the above equation (12), the impulse response of the energy transfer boundary at the time when the transfer boundary matrix is calculated by the matrix calculation means is the transfer boundary when the vibration at the time has N types (N = n + 1) frequencies. The calculation can be performed based on the value of the matrix. Thus, as is clear from Patent Document 1, the present inventor is more accurate as the impulse response of the energy transfer boundary than when the impulse response of the energy transfer boundary is calculated according to the calculation method proposed in Non-Patent Document 1. An accurate value can be obtained, and the analysis accuracy of the response analysis in the time domain for the analysis target model can be improved.

In the seventh aspect of the invention, the impulse response calculation means, for example, as described in claim 8, provides a correction value Δm ₀ for the simultaneous component m ₀ of the impulse response depending on the acceleration of the object in the equation (12). The correction value Δk ₀ for the simultaneous component k ₀ of the impulse response depending on the displacement of the object is calculated by the following equation (13):

(However, in the above equation (13)

Re (S _B (ω _i )) in the above equation (14) is the following equation (15) reproduced from the impulse response of the energy transfer boundary calculated using the above equation (12). The value of the real part at the frequency ω _i of the transmission boundary matrix S _B (ω) represented

Re (D (ω _i )) in the above equation (14) is the data D (of the transmission boundary matrix at the frequency ω _i used in the calculation of the impulse response of the energy transmission boundary based on the equation (12). represents the value of the real part of ω _i)), the calculated correction value Delta] m _0, it is preferably configured to modify the co-component m _0, k ₀ using .DELTA.k _0. As a result, as is clear from Patent Document 1, it is possible to obtain a more accurate value as the impulse response at the energy transfer boundary as compared with the case where the above-described simultaneous components m ₀ and k ₀ are not corrected. The analysis accuracy of the response analysis in the time domain for the analysis target model can be further improved.

In the seventh aspect of the present invention, the impulse response calculating means, for example, as described in the ninth aspect of the present invention, sets the simultaneous component of the impulse response depending on the speed of the object in the equation (3) as a correction value Δc for c ₀ . _{0 is} also calculated by the following equation (16),
Δc ₀ = −E / B (16)
(However, in the above equation (16)

Im (S _B (ωi)) in the above equation (17) is a transmission represented by the above equation (15) reproduced from the impulse response of the energy transmission boundary calculated using the above equation (12). The value of the imaginary part at the frequency ω _i of the boundary matrix S _B (ω) is expressed, and Im (S _B (ω _i )) in the above equation (17) is the impulse response of the energy transfer boundary based on the equation (12). (This represents the value of the imaginary part of the transmission boundary matrix data D (ω _i ) at the frequency ω _i ) and the calculated correction value Δc ₀ to correct the simultaneous component c _0. It is preferable to configure.

Thus, as is clear from Patent Document 1, the positions of the data points used for the calculation of the impulse response, for example, on the frequency axis are not equally spaced compared to the case where the correction of the simultaneous component c ₀ is not performed. In the case where the accuracy of the imaginary part of the calculated impulse response is likely to be reduced, a more accurate value can be obtained as the impulse response of the energy transfer boundary, and the time domain for the model to be analyzed The analysis accuracy of the response analysis at can be further improved.

  The response analysis method according to the invention of claim 10 is formed by modeling an object to be analyzed consisting of a single ground or a ground and a building on a computer, and energy as a wave boundary model between an internal region and an external region of the model. A response analysis for analyzing the behavior when a specific external force that vibrates the object to be analyzed is input at each time of the object to be analyzed is performed on the external region of the model to be analyzed provided with a transmission boundary. Thus, the physical property value of the analysis target object in the outer region is calculated at each time of the analysis target, and the physical property value of the analysis target object in the outer region is calculated at each time of the analysis target. Based on this, it is the main component of the energy transfer boundary when the specific external force is input, is defined as a complex number in the frequency domain, and has a strong frequency dependence. The value of the transmission boundary matrix having the characteristic is calculated for each of two or more times among the respective times to be analyzed, and the object is vibrated based on the value of the transmission boundary matrix calculated for each of the two or more times. The impulse response representing the relationship between the external force and the behavior of the object in the time domain is calculated at each time to be analyzed by calculating the impulse response of the energy transfer boundary when the specific external force is input. Using the impulse response of the energy transfer boundary calculated at each time of the target, the behavior when the specific external force is input to the analysis target model is analyzed at each time of the analysis target Since the response analysis is performed, as in the first aspect of the invention, the response analysis in which the range outside the internal region of the analysis target model exhibits nonlinear behavior It can be realized to improve the definitive analysis accuracy.

  The response analysis program according to the invention as claimed in claim 11 is formed by modeling an object to be analyzed consisting of a single ground or a ground and a building, and energy as a wave boundary model between an internal region and an external region of the model. A response analysis is performed to analyze the behavior when a specific external force that vibrates the analysis target object is input to the external region of the analysis target model provided with a transmission boundary at each time of the analysis target. The physical property value calculating means for calculating the physical property value of the analysis object in the external region at each time of the analysis target, and the external region calculated at each time of the analysis target by the physical property value calculation means Is a main component of the energy transmission boundary when the specific external force is input based on the physical property value of the analysis target object in the frequency domain Matrix calculation means for calculating a value of a transmission boundary matrix defined as a complex number and having strong frequency dependence for each of two or more times of the time to be analyzed, two or more times by the matrix calculation means The energy transfer boundary when the specific external force is input as an impulse response that represents the relationship between the external force that vibrates the object and the behavior of the object in the time domain based on the value of the transfer boundary matrix calculated for each An impulse response calculating means for calculating the impulse response of each time to be analyzed, and the impulse response of the energy transfer boundary calculated at each time of the analysis target by the impulse response calculating means. , The behavior when the specific external force is input to the analysis target model. To function as an analysis means for performing a response analysis for analyzing for each time of the target.

  The response analysis program according to the invention described in claim 11 is a program for causing a computer to function as the physical property value calculation means, matrix calculation means, impulse response calculation means, and analysis means. By executing the response analysis program according to the present invention, the computer functions as the response analysis device according to claim 1, and, similar to the invention according to claim 1, outside the internal region of the analysis target model. It is possible to realize improvement in analysis accuracy in response analysis in which the range of non-linear behavior behaves.

  As described above, the present invention is an analysis in which an object to be analyzed consisting of the ground alone or the ground and a building is modeled, and an energy transfer boundary is provided as a wave boundary model between the internal region and the external region of the model. By performing a response analysis to analyze the behavior when a specific external force that vibrates the analysis target object is input to the external region of the target model at each time of the analysis target, the analysis target object in the external region is analyzed. Calculate the physical property value at each analysis target time, and based on the physical property value calculated at each analysis target time, the value of the transmission boundary matrix when a specific external force is input The impulse of the energy transfer boundary when a specific external force is input based on the value of the transfer boundary matrix calculated for each of the two or more times Answers are calculated at each time to be analyzed, and the impulse response of the energy transfer boundary calculated at each time to be analyzed is used to analyze the behavior when a specific external force is input to the model to be analyzed. Since the response analysis is performed at each time of the target, the analysis accuracy can be improved in the response analysis in which the range outside the internal region of the analysis target model exhibits nonlinear behavior.

It is an image figure which shows an example of a ground model (1/2 model). It is a block diagram which shows schematic structure of the computer which concerns on embodiment. It is a flowchart which shows the content of an earthquake response analysis process. It is a conceptual diagram which shows the relationship (data flow) of each step in an earthquake response analysis process. It is an image figure which shows an example of the analysis object model used by an earthquake response analysis process. It is a diagram showing an example of the shear strain amplitude value γ-stiffness reduction rate α characteristic and the shear strain amplitude value γ-damping constant h characteristic of the ground. In the causal unit imaginary function, (A) is a diagram showing the entire imaginary part, (B) is a regular component, and (C) is a singular component. It is an image figure which shows the other example of an analysis object model. In the analysis study conducted by the inventor of the present application, (A) is the maximum acceleration, (B) is the maximum strain, and (C) is the maximum when the maximum strain storage time is each value in the time history response analysis of the ground in the external region. It is a diagram which shows distribution of the response value of a shear stress, respectively. In the analysis examination which this inventor implemented, it is a diagram which shows the time change of the shear strain obtained by the time history response analysis of the ground of an external area | region for every site | part of a ground model. In the analysis examination which this inventor implemented, it is a diagram which shows the time change of the shear wave velocity Vs obtained by the time history response analysis of the ground of an external area | region, and the attenuation constant h for every site | part of a ground model. In the analysis examination which this inventor implemented, it is a diagram which shows distribution of the shear strain, the shear wave velocity Vs, and the damping constant h for every representative time obtained by the time history response analysis of the ground of an external area | region. It is a diagram which shows the response value in the time history response analysis which applied this invention in the analysis examination which this inventor implemented. In the analysis examination which this inventor implemented, it is a graph which shows the ratio with respect to the reference value of the response value of each model in the time history response analysis to which this invention is applied. In the analysis examination which this inventor implemented, it is a diagram which shows the response value in the time history response analysis to which the invention of patent document 1 is applied. 10 is a chart showing a ratio of response values of each model to a reference value in a time history response analysis to which the invention described in Patent Document 1 is applied in an analysis study performed by the inventor of the present application. In the analysis examination which this inventor implemented, it is a diagram which shows the response value in the equivalent linear response analysis with respect to the viscous boundary as a wave boundary model. In the analysis examination which this inventor implemented, it is a table | surface which shows the ratio with respect to the reference value of the response value of each model in the equivalent linear response analysis with respect to the viscous boundary as a wave boundary model.

  Hereinafter, an example of an embodiment of the present invention will be described in detail with reference to the drawings. FIG. 2 shows a computer 10 to which the present invention can be applied. The computer 10 includes a CPU 10A, a memory 10B including a ROM and a RAM, a non-volatile storage unit 10C including an HDD (Hard Disk Drive), a flash memory, and the like, a display 12 including a CRT or LCD, a keyboard 14, and a mouse. 16 are connected to each other.

  An earthquake response analysis program for performing an earthquake response analysis process to be described later is installed in the storage unit 10C of the computer 10. This earthquake response analysis program corresponds to the response analysis program according to claim 11. The computer 10 corresponds to the computer according to claims 10 and 11, and the CPU 10A functions as a response analysis apparatus according to the invention according to claim 1 or the like when the CPU 10A executes the earthquake response analysis program. . The computer 10 is preferably a personal computer (PC), but is not limited to this. For example, a workstation or a general-purpose large computer may be used.

  Next, as an operation of the present embodiment, an earthquake response analysis process realized by the CPU 10A executing the earthquake response analysis program will be described with reference to FIG. 3 (and FIG. 4). In the earthquake response analysis processing according to the present embodiment, the behavior when the ground motion is input to the object to be analyzed in which the ground and the building having the underground part embedded in the ground are integrated is expressed in increments of Δt in the time domain. The response analysis to be analyzed is modeled as an analysis target model, and as shown in FIG. 5 as an example, the ground constituting the analysis target object is modeled as an aggregate of a large number of constituent elements (also referred to as mesh regions) of a certain size. In addition, the building that constitutes the object to be analyzed is modeled as a structure model including multiple nodes, and an energy transfer boundary as a wave boundary model is provided between the internal region and the external region of the ground model. ½ model on the right side, which is modeled by ½ model with an anti-symmetric condition with respect to the center position (the energy transfer boundary exists only on the right side of the model, Inner region on the left across the boundary on the right side performed by using the outer region (the free soil also called) model is located).

  In step 30 of the seismic response analysis process, data for calculating the ground and building dynamic characteristic matrix of the internal region of the analysis target model is acquired. In the present embodiment, the dynamic characteristic matrix means a matrix representing characteristics necessary for performing response analysis of the ground or the building, and specifically includes a mass matrix, an attenuation matrix, and a stiffness matrix, As the data acquired at 30, for example, physical property values for each ground layer in the inner region (for example, shear wave velocity Vs, Poisson's ratio ν, density ρ, damping constant h, stiffness reduction rate characteristic γ-α and damping constant characteristic γ− h (an example is shown in FIG. 6) etc. (however, the shear wave velocity Vs and the damping constant h are values at the analysis start time (initial values)) and constants (mass, shear rigidity, rotational inertia, damping rate) for each floor of the building. h) and the like.

In the next step 32, based on the data acquired in step 30, the dynamic characteristic matrix of the ground and building of the inner region is calculated by performing analysis by the finite element method (FEM). As a result, the mass matrix [M _I ] of the internal region shown in the above equation (1), the rigidity matrix [K _I ] of the internal region, and the attenuation matrix [C] shown in the above equation (8) are calculated. The In the present embodiment, a causal history attenuation model is used as an attenuation model representing the frequency independence of material attenuation, and the above attenuation matrix [C] is calculated using the causal history attenuation model.

In step 34, input seismic motion data (data representing “specific external force” in the present invention) is acquired, and in the next step 36, the dynamic characteristic matrix and seismic response of the ground in the external region (infinitely distant ground) are calculated. As data, physical property values (for example, shear wave velocity V _s , Poisson's ratio ν, density ρ, damping constant h, stiffness reduction rate characteristic γ-α, damping constant characteristic γ-h, etc.) The speed Vs and the attenuation constant h are values at the analysis start time (initial values))).

In step 38, the acquired data is obtained based on the input ground motion data acquired in the previous step 34 and the data for calculating the earthquake response of the ground of the external area (infinitely distant ground) acquired in the previous step 36. Calculate and analyze the non-linear seismic response of the ground in the external region (ground at infinity) in the time domain when the input seismic motion is expressed (see also Fig. 4). The ground where large energy and large amplitude external force is input, such as during a large earthquake, has a frequency-independent characteristic in which the history absorption energy (damping constant h) of internal damping does not depend much on the frequency ω. (A characteristic indicating a constant value regardless of the frequency ω), and the stiffness reduction rate α (= rigidity G / initial stiffness G ₀ ) and the damping constant h vary according to the shear strain amplitude value γ of the object (ground). The distortion amplitude dependence characteristics are also shown.

For this reason, in this embodiment, as shown in FIG. 7 as an example, the value of the frequency ω is (2n−1) − (2ω / ωm) in the range of (n−1) · ωm to n · ωm (however, n is an integer) regular component of the imaginary part (see FIG. 7B), and imaginary part singular component of (2ω / ωm) regardless of the frequency ω (see FIG. 7C) Corresponding to the imaginary part showing the value of (2n-1) in the range of (n-1) · ωm to n · ωm, and the Hilbert transform value of the regular component of the imaginary part. An impulse of the causal unit imaginary function Z ′ (ω) by converting the causal unit imaginary function Z ′ (ω) composed of the real part and the time domain, or by converting only the imaginary part into the time domain As response values, a simultaneous component c (t ₀ ) that depends on the speed of the object, a simultaneous component k (t ₀ ) that depends on the displacement of the object, and a time delay component k (t _j ) in increments of Δt that depends on the displacement of the object ( Where j is a natural number t _j = Δt · _j) are stored in advance calculated and stored portion 10C.

Then, the response analysis in step 38 is performed as follows. That is, first, the impulse response values (c (t ₀ ), k (t ₀ ), k (t _j )) of the causal unit imaginary function Z ′ (ω) are read from the storage unit 10C, and the object (in this case, the external region) The mass matrix of the ground) is [Ms], the initial stiffness matrix of the object is [K ₀ ], the displacement vector of the object in the time domain is {u (t)}, the velocity vector is {u '(t)}, When the force vector is {F (γ, t)}, the acceleration in the time domain of the external force that vibrates the object is y ₀ "(t), and the total number of time delay components k (t _j ) is n, the storage unit The impulse response values (c (t ₀ ), k (t ₀ ), k (t _j )) of the causal unit imaginary function Z ′ (ω) read out from
[Ms] {u "(t)} + {F (γ, t)} = − y ₀ “ (t) [Ms] {1} (18)
However,

Substitute into the above formula.

  Then, data representing the external force applied to the ground in the external region at the time t (initial value is t = 0) to be analyzed is extracted from the input ground motion data acquired in step 34, and based on the external force represented by the extracted data. , The displacement vector {u (t)}, velocity vector {u ′ (t)}, acceleration vector {u ”(t)} of the ground in the external region at the analysis target time t are estimated, and the estimated displacement vector {u (t)} to calculate the shear strain amplitude value γ of the ground in the outer region at the analysis target time t, and based on the stiffness reduction rate characteristic γ-α and the damping constant property γ-h of the ground in the outer region, The stiffness reduction rate α corresponding to the amplitude value γ (where the shear strain amplitude value γ is the maximum value of the shear strain amplitude value γ from the time t to be analyzed to the past Δtm seconds (= maximum strain storage time)). And the damping constant h, respectively, and the estimated displacement vector {u (t)}, velocity vector {u ′ (t)}, acceleration Substituting the degree vector {u "(t)}, the calculated stiffness reduction rate α, and the damping constant h into the above equations (18) and (19) to calculate the behavior of the ground in the external region at the time t to be analyzed As a result of the calculation, it is determined whether or not the deviation between the external force applied to the ground of the external region at the analysis target time t and the reaction force of the ground of the external region at the analysis target time t is within the allowable range. Otherwise, the displacement vector {u (t)}, velocity vector {u '(t)}, acceleration vector {u "(t)} is corrected and the above calculation is repeated, so that the external force with respect to the external force is analyzed at the analysis target time t. The values of the displacement vector {u (t)}, velocity vector {u '(t)}, acceleration vector {u "(t)} of the ground of the external region where the deviation of the ground ground reaction force falls within the allowable range The shear strain amplitude value γ, shear wave velocity Vs, and attenuation constant of the ground in the external region at the time when the deviation of the reaction force of the ground in the external region with respect to the external force is within the allowable range. The h, is stored in the memory 10B as a physical property value of the ground in the outer region in the analysis target time t.

  The series of processes described above are processes for analyzing the behavior of the ground in the external region at a single analysis target time t when an external force represented by the input ground motion data is applied, while increasing the analysis target time t by Δt. By performing the above-described series of processes for all times to be analyzed (see also Japanese Patent Application Laid-Open No. 2008-304227), the external region at all times to be analyzed when the external force represented by the input earthquake motion data is applied. Physical property values (shear strain amplitude value γ, shear wave velocity Vs, damping constant h, etc.) representing the ground behavior are obtained. The above step 38 is an example of processing by the physical property value calculating means according to the present invention (more specifically, the physical property value calculating means according to claim 3).

In the next step 40, when the external force represented by the input ground motion data obtained by the response analysis of step 38 and the data for calculating the earthquake response of the ground in the external region obtained in the previous step 36 is applied. From the physical property values representing the behavior of the ground in the outer region at all times to be analyzed in Δt increments, the physical property values representing the ground behavior in the outer region (shear wave velocity Vs, attenuation constant h) are in increments of Δtb (where Δtb> Δt Yes, for example, values of Δtb = 1 second and Δt = 0.01 seconds are suitable), and based on physical property values at each time of the extracted Δtb increments (note that Poisson's ratio ν and density ρ are Regardless of the constant value used), the dynamic characteristic matrix of the ground in the external region at each time in Δtb increments when the external force represented by the input seismic motion data is applied is calculated (see also FIG. 4). As a result, the matrix [A (t)], [B (t)], [G (t)], [M] and the mode matrix [V] in the above equation (3) are used as the dynamic characteristic matrix of the ground in the outer region. ], The matrix [D (t)] represented by the superposition of the sub-matrix [D (t)] _j represented by the above equation (11) is calculated for each time in increments of Δtb.

  In the next step 42, the external force represented by the input earthquake motion data is obtained by substituting the dynamic characteristic matrix of the ground in the external region at each time in Δtb increments obtained by the calculation of step 40 into the above equation (10). When added, the transmission boundary matrix of the energy transmission boundary including the ground of the outer region is calculated for each time in increments of Δtb (see also FIG. 4). As a result, the transmission boundary matrix [R (t)] expressed by the above equation (10) as the transmission boundary matrix of the energy transmission boundary including the ground of the external region is expressed in Δtb increments for each time in Δtb increments. Calculated based on physical properties (shear wave velocity Vs, Poisson's ratio ν, density ρ, damping constant h: Poisson's ratio ν and density ρ are constant regardless of time) representing the behavior of the ground in the external region at each time Will be.

  In step 44, the energy transfer boundary matrix including the ground in the external region is converted into the time domain by converting the energy transfer boundary matrix including the ground in the external region calculated at each time in Δtb increments in step 42, respectively. The impulse response is calculated for each time in increments of Δtb. Specifically, the impulse response of the energy transfer boundary including the ground of the external region at a certain time can be calculated as follows.

That is, first, transmission at N types of frequencies (N types of angular frequencies ω ₁ ,..., Ω _N ) within a preset frequency range to be calculated from the data of the transmission boundary matrix obtained by the calculation at step 48. N complex data S _B (ω ₁ ),..., S _B (ω _N ) representing the values of the boundary matrix are extracted. In addition, as a frequency range of calculation object, the range of 0-20 (Hz) is applicable, for example. Next, using the previous equation (12) as the mathematical formula for defining the impulse response and the previous formula (15) as the mathematical formula for defining the transmission boundary matrix, the following derived from the formulas (12) and (15) Substituting _N complex data S (ω ₁ ),..., S (ω _N ) into the simultaneous equations of equations (20) and (21) and finding the solution of these simultaneous equations includes the ground in the external region The impulse response at the energy transfer boundary is calculated in increments of Δt set in advance.

As a result of this calculation, the impulse response of the energy transfer boundary including the ground of the external region is converted into the simultaneous component k ₀ of the impulse response that depends on the displacement of the object, the simultaneous component c ₀ of the impulse response that depends on the velocity of the object, and the acceleration of the object. The data of the simultaneous component m ₀ of the dependent impulse response is obtained, and n pieces (n = N−1) of data of the time delay component k _j of the impulse response depending on the displacement of the object are obtained in increments of Δt. N-1 pieces of data of the time delay component c _j of the impulse response depending on the speed are obtained in increments of Δt.

Subsequently, the correction values Δm ₀ and Δk ₀ are calculated according to the above equations (13) and (14), and the simultaneous components m ₀ and k ₀ of the impulse response are corrected using the calculated correction values Δm ₀ and Δk _0. . Furthermore, when the conditions for the imaginary part of the calculated impulse response are likely to decrease, such as the positions on the frequency axis of the data points used for the calculation of the impulse response are not equally spaced, The correction value Δc ₀ is also calculated according to the previous equations (16) and (17), and the simultaneous component c ₀ is corrected using the calculated correction value Δc ₀ . By the above calculation, an impulse response of the energy transfer boundary including the ground of the external region at a certain analysis target time t is obtained. Then, by performing the above calculation for each time in increments of Δtb, an impulse response of the energy transfer boundary including the ground in the outer region is obtained for each time in increments of Δtb.

In the next step 46, based on the impulse response of the energy transfer boundary calculated for each time in increments of Δtb, the impulse response of the energy transfer boundary in all the analysis target times in increments of Δt is obtained by interpolation calculation. By performing this interpolation calculation, the number of times of the calculation is compared with the case where the calculation of the transmission boundary matrix with a relatively large calculation load and the calculation of the impulse response of the energy transmission boundary from the transmission boundary matrix are performed at all times to be analyzed. The calculation load applied to the computer 10 can be reduced by performing the calculation.
This step 46 is an example of the calculation by the impulse response calculation means.

The interpolation calculation in step 46 is specifically performed at the time when the transfer boundary matrix (and the impulse response of the energy transfer boundary) has been calculated based on the shear strain amplitude value γ at the time t of the impulse response calculation target, for example. The time t1, t2 satisfying the relationship that the shear strain amplitude values γ1, γ2 at each time “γ1 <γ <γ2” are selected from among them, and the impulse response at the energy transfer boundary at the time t1 is X1, and the energy at the time t2. When the impulse response at the transmission boundary is X2, the following equation (22) that defines the impulse response X at the energy transmission boundary at the time t of the impulse response calculation target is used,
X = X1 + (X2−X1) · {γ / (γ2−γ1)} (22)
This can be realized by calculating each component (k ₀ , c ₀ , m ₀ , k _j , c _j ) constituting the impulse response of the energy transfer boundary at the time t to be calculated by the above equation (22). By performing this calculation for all times in which the transfer boundary matrix (and the impulse response of the energy transfer boundary) is not calculated among all the times to be analyzed, the impulse response of the energy transfer boundary at all the times to be analyzed in increments of Δt is obtained. can get.

  “{Γ / (γ 2 −γ 1)}” in the above equation (22) is “the shear strain amplitude of the object to be analyzed in the external region at the time when the impulse response of the energy transfer boundary is not calculated” according to claim 5. The value γ is an example of a ratio that internally divides the shear strain amplitude value γ of the object to be analyzed in the outer region at two or more times when the value of the transmission boundary matrix is calculated. The process for obtaining the impulse response value is an example of the process performed by the impulse response calculation means.

Further, the interpolation calculation in step 46 is not limited to using the expression (22). For example, the transfer boundary matrix (and the impulse response of the energy transfer boundary) has already been calculated based on the time t of the impulse response calculation target. When times t1 and t2 satisfying the relationship of “t1 <γ <t2” are selected from the times, and the impulse response at the energy transfer boundary at time t1 is X1, and the impulse response at the energy transfer boundary at time t2 is X2. Using the following equation (23) that defines the impulse response X of the energy transfer boundary at time t of the impulse response calculation target:
X = X1 + (X2-X1). {T / (t2-t1)} (23)
Each component (k ₀ , c ₀ , m ₀ , k _j , c _j ) constituting the impulse response of the energy transfer boundary at the calculation target time t may be calculated by the above equation (23). Even when this calculation is performed for all times in which the transfer boundary matrix (and the impulse response of the energy transfer boundary) is not calculated among all the times to be analyzed, the impulses of the energy transfer boundary at all the times to be analyzed in increments of Δt. A response is obtained.

  “{T / (t 2 −t 1)}” in the above equation (22) is “2 when the impulse response of the energy transfer boundary is not calculated, and the value of the transfer boundary matrix is calculated 2 according to claim 6. The above-described ratio of dividing the time internally is an example, and the process of obtaining the impulse response value by the interpolation calculation of equation (23) is an example of the process by the impulse response calculation means according to claim 6.

In the next step 48, the earthquake response of the ground in the external region obtained by the calculation of step 38, the data acquired in the previous step 36, and the impulse response of the energy transfer boundary calculated for all the times to be analyzed in step 46 are obtained. Based on this, the correction force vector of the ground in the external region in the time region is calculated for all times to be analyzed (see also FIG. 4). Thereby, the correction force vector − [D _R (t)] {u _R ^* (t)} in the above equation (1) is calculated for all the times to be analyzed as the correction force vector of the ground in the outer region. .

In step 50, the ground and building dynamic characteristics matrix obtained by the calculation in the previous step 32, the earthquake response of the ground in the outer region in the time domain obtained by the previous step 38, the previous step 40 To impulse response at the analysis target time t (initial value is the analysis target time t = 0) out of the impulse response of the energy transfer boundary including the ground of the outer region obtained by the calculation of step 46, and the calculation of the previous step 48 By combining the correction force vector at the analysis target time t (initial value is the analysis target time t = 0) among the correction force vectors of the ground of the external region obtained by the above, a time domain response analysis is performed on the analysis target model. (See also the following equation (24)) is obtained (see also FIG. 4).
[M _I ] {u ″ (t)} + [K _I ] {u (t)} + {R ₀ } = − y ″ (t) [M _I ] {1} + {F (t)} + { R _f (t)} (24)
However,

  In step 52, the seismic motion at the analysis target time t represented by the input seismic motion data acquired in the previous step 34 is substituted into the equation of motion of the equations (24) and (25), and as in step 38 described above, Repeating the calculation until the calculation result converges to obtain the behavior of the analysis target object at the analysis target time t, increasing the analysis target time t by Δt and substituting it into the equations of motion of equations (24) and (25) The time domain seismic response analysis for the model to be analyzed is repeated by switching the impulse response of the energy transfer boundary, the correction force vector of the ground in the external region and the ground motion while switching to a value corresponding to the time t to be analyzed after the increase of Δt ( Time history response analysis) is performed for all times to be analyzed. As a result, the behavior of the analysis target object when the ground motion is input to the analysis target object in which the ground in the inner region, the building, and the ground in the outer region are integrated is considered nonlinear while considering the plasticization of the ground and the building. Even if the analysis condition is such that the range showing the behavior of the above extends to the energy transfer boundary, it can be evaluated accurately without being affected.

  In the above, as an analysis target model, as shown in FIG. 5 as an example, a mode has been described in which a model that is ½ modeled by providing an inverse symmetry condition with respect to the center position of the model has been described, but the present invention is not limited to this. As an analysis target model, as shown in FIG. 8 as an example, external regions are provided on the left and right sides of the internal region, and a wave boundary model is provided between the internal region and the left and right external regions. A model in which energy transfer boundaries are provided may be used. By performing response analysis using such an analysis target model, the center position of the analysis target model is sandwiched because the ground physical property values of the left and right outer areas are different or the ground surface is inclined. Thus, even when the symmetry between the left part and the right part is not guaranteed, it is possible to prevent the accuracy of the response analysis from being lowered.

As described above, when using the analysis target model in which the energy transfer boundaries are provided on the left and right sides of the inner region, the left and right transfer boundary matrices are expressed as [L (t)], [R (t)], The left and right boundary force vectors are {F _L (t)} and {F _R (t)}, and the left and right outer ground response displacements are {u _L ^* (t)} and {u _R ^* ( t)}, and the correction force vectors acting on the left and right boundaries are − [D _L (t)] {u _L ^* (t)}, − [D _R (t)] {u _R ^* (t)} Then, the equation of motion of the entire model to be analyzed is
[M _I ] {u "(t)} + ([K _I (t)] + [L (t)] + [R (t)]) {u (t)}
= −y ″ (t) [M _I ] {1} + {F _L (t)} + {F _R (t)} (26)
{F _L (t)} = ([L (t)] − [D _L (t)]) {u _L ^* (t)} (27)
{F _R (t)} = ([R (t)] − [D _R (t)]) {u _R ^* (t)} (28)
First, instead of the expressions (1) and (2), the above expressions (26) to (28) are used.

  In the above description, the case where the analysis target object is modeled as a two-dimensional model and response analysis is performed has been described. However, the present invention is not limited to this, and the analysis target object is modeled as a three-dimensional model. It can also be applied to response analysis.

  In the above, the physical property values (shear strain amplitude value γ, shear wave velocity Vs, damping constant h, etc.) of the analysis target object at each time of the analysis target in the external region are calculated, and the impulse response of the causal unit imaginary function is calculated. Although the embodiment to be used (an embodiment corresponding to the invention described in claim 3) has been described, the present invention is not limited to this, and it is also possible to apply other calculation methods in the above calculation. Included in the range.

  Further, in the above, when the impulse response of the energy transfer boundary at the time when the impulse response of the energy transfer boundary is not calculated among the respective times to be analyzed is obtained from the impulse response of the energy transfer boundary calculated for each of two or more times. As the interpolation calculation, the shear strain amplitude value γ of the object to be analyzed in the external region at the time when the impulse response of the energy transfer boundary is not calculated is calculated in the external region at two or more times when the value of the transfer boundary matrix is calculated. The calculation method using the ratio that internally divides the shear strain amplitude value γ of the object to be analyzed, and the time when the impulse response of the energy transfer boundary is not calculated divides the two or more times when the values of the transfer boundary matrix are calculated Although the calculation method using the ratio is illustrated, the present invention is not limited to this, and other calculation methods are used for the above-described interpolation calculation. Is also within the scope of the present invention.

  Furthermore, although the aspect which the earthquake response analysis program corresponding to the response analysis program which concerns on this invention was previously memorize | stored (installed) in the memory | storage part 10C of the computer 10 was demonstrated above, the response analysis program which concerns on this invention is CD -It is also possible to provide in the form recorded on recording media, such as ROM and DVD-ROM.

  Next, in order to confirm the usefulness of the present invention, the details of the analysis study conducted by the present inventor will be described. In this analysis study, as shown in FIG. 8, an analysis target model in which an outer region is provided on each of the left and right sides of the inner region and a wave boundary model is provided between the inner region and the left and right outer regions. Using.

  As shown in Table 1 above, the ground was a two-layer ground in which a surface layer ground having a layer thickness of 40 m and a shear wave velocity Vs = 300 m / s was present on a base having a shear wave velocity Vs = 500 m / s. Of the models to be analyzed, the ground model was modeled only on the surface ground, and the characteristics of the foundation were evaluated at the bottom viscous boundary. Further, the surface layer ground has the γ-α characteristic and γ-h characteristic shown in FIG.

  In addition, the building has a planar shape of 20 m × 20 m, and as shown in Table 2 above, the building is a 6-story structure with a height of 24 m. In the analysis target model, this building is represented by a mass point shear model. The attenuation of the building was causal history attenuation with an attenuation constant h of 3%.

  The input seismic motion was defined as 2E (twice ascending waves) using El Centro 1940 NS wave (duration 10 seconds, time increment Δt = 0.01 seconds) with maximum acceleration = 500 Gal. Time integration was Newmark-β method and average acceleration method (β = 1/4).

  In this analysis study, first, under the above analysis conditions, (1) time history response analysis of the ground in the external region (analysis corresponding to step 38 in FIGS. 3 and 4), (2) calculation of the transmission boundary matrix including the external region ( Calculations corresponding to steps 40 and 42 in FIGS. 3 and 4) and (3) calculation of impulse response of the transmission boundary matrix (calculation corresponding to step 44 in FIGS. 3 and 4) were sequentially performed.

(1) Time history response analysis of the ground in the external region In the time history response analysis of the ground in the external region, a nonlinear causal history attenuation model (fmax = 10 Hz, 18-term model) was used. As described above, in this response analysis, the stiffness reduction rate α and the damping constant h at each analysis target time are based on the stiffness reduction rate characteristic γ-α and the damping constant characteristic γ-h, and are the maximum from the analysis target time. A value corresponding to the maximum value of the shear strain amplitude value γ from the time before the strain storage time Δtm seconds to the analysis target time is used, but in order to confirm the influence of the difference in the maximum strain storage time Δtm on the response value. The response analysis was performed by switching the maximum strain storage time Δtm to 0.1 seconds, 0.5 seconds, 1.0 seconds, and 2.0 seconds. The results are shown in FIG. For any of the maximum acceleration shown in FIG. 9 (A), the maximum strain shown in FIG. 9 (B), and the maximum shear stress shown in FIG. 9 (C), if the maximum strain storage time Δtm = 0.1 seconds, the response value The difference is very small. From this result, the maximum strain storage time Δtm = 1.0 seconds was adopted in the subsequent analysis studies.

  In addition, when the maximum strain storage time Δtm = 1.0 second, the temporal change of the shear strain response value and the absolute value of the shear strain obtained by the time history response analysis is shown for each position along the depth direction of the ground model. 10 shows. However, (A) is a position 0.5 m deep from the ground surface (GL) of the ground model (position near the ground surface), (B) is a position 20.5 m deep from the ground surface (position at the center in the depth direction), (C) Indicates the time variation of shear strain at a position of 39.5m depth from the ground surface (position near the bottom of the ground model). In addition, as a result of the time history response analysis of the ground in the external region in the case of the maximum strain storage time Δtm = 1.0 second, FIG. 11 shows the time variation of the shear wave velocity Vs and the attenuation constant h in the depth direction of the ground model. FIG. 12 shows the distribution of response values of shear strain, shear wave velocity Vs, and damping constant h along the depth direction of the ground model at typical times (1 second, 3 seconds, Shown every 8 seconds).

(2) Calculation of transmission boundary matrix including external region Based on the results of time history response analysis of the ground in the external region, the transmission boundary matrix of the energy transmission boundary including the ground in the external region was calculated for each Δtb. In this analysis study, Δtb = 1 second, and the analysis frequency was 0.5 Hz to 20 Hz in increments of 0.5 Hz. From the physical property values (shear wave velocity Vs and damping constant h: see FIG. 11) of the ground in the external region for each time obtained by time history response analysis of the ground in the external region, the external at each time in Δtb increments The shear wave velocity Vs and the attenuation constant h of the ground in the region were respectively extracted, and the transmission boundary matrix at each time in Δtb increments was calculated using the extracted shear wave velocity Vs and the attenuation constant h. The degree of freedom of the energy transfer boundary is 82 of 41 nodes x 2 (horizontal / vertical degrees of freedom). By the above calculation, the transfer boundary matrix of the energy transfer boundary including the ground of the external area at each time in increments of Δtb As a result, a frequency-dependent complex matrix of 82 × 82 components was calculated.

(3) Calculation of the impulse response of the transfer boundary matrix By converting the transfer boundary matrix of the energy transfer boundary calculated at every time in Δtb to the time domain, the impulse response of the energy transfer boundary is changed at every time in Δtb Calculated. The conditions for time domain conversion in this calculation are shown in Table 3 below.

  Also, the impulse response of the energy transfer boundary calculated at each time in Δtb is interpolated based on the shear strain amplitude value γ at each time to be analyzed (each time in Δt = 0.01 seconds in this analysis study). As a result, the impulse response of the energy transfer boundary at all times to be analyzed was obtained by interpolation. In addition, as the shear strain amplitude value γ used for the interpolation calculation, the value in the lowermost component of the ground model that generates the maximum strain was used.

  In this analysis study, subsequently, the impulse response of the energy transfer boundary obtained by the above series of operations (case NT described below) is used to confirm the effectiveness of the time history response analysis to which the present invention is applied. For the purpose, as shown in Table 4 below, the present invention is applied to each of a plurality of types of models in which the distance L (see FIG. 8) between the building edge and the wave boundary model in the analysis target model is different from each other. Time history response analysis (equivalent linear response analysis for energy transfer boundary as wave boundary model: case NT shown in the following Table 5), time history response analysis applying the invention described in Patent Document 1 (as wave boundary model) Equivalent linear response analysis for energy transfer boundary: Case LT shown in Table 5 below), Equivalent linear response analysis using viscous boundary as wave boundary model (Case V shown in Table 5 below) The analysis results were compared.

  In this analysis study, as shown in FIG. 8, an object to be analyzed in which the ground and a building having an underground portion embedded in the ground are integrated, and exists in the inner area and the left and right sides of the inner area. In order to clarify the effect of the difference in the wave boundary model on the response analysis, the response analysis for the external region was unified with nonlinear analysis.

  FIG. 13 shows the analysis result of the maximum response acceleration for each model of distance L = 5 m, 20 m, 40 m, and 120 m as an analysis result in the time history response analysis (case NT) to which the present invention is applied. FIG. 13B shows the analysis result of the maximum response displacement, and FIG. 13C shows the analysis result of the maximum response shear force. Of the models whose analysis results are shown in FIG. 13, the most accurate analysis result can be obtained in principle with a maximum distance L = 120 m in the internal region. As is clear from (C), in the time history response analysis (case NT) to which the present invention is applied, there is a difference in whether the response value of the model with smaller L matches the response value of the model with distance L = 120 m. It can be understood that the response analysis accuracy is relatively high even when the distance L is small.

  14A to 14C show the response values in the model with the distance L = 120 m among the above analysis results (analysis results in the time history response analysis (case NT) to which the present invention is applied). And the ratio of response values in each model when this response value is used as a reference value. Also, in each column in the figure, the column whose deviation from the reference value (response value in the model with distance L = 120 m) is ± 10% or less is white, and the deviation is larger than ± 10% but ± 20% or less Columns are displayed in gray, and columns where the deviation is greater than ± 20% are displayed in black.

  As is clear from FIG. 14, in the time history response analysis (case NT) to which the present invention is applied, in the model with the distance L = 30 m or more, all response values are within ± 10% of the reference value. Even if the model is less than distance L = 30m, the number of response values with a deviation from the reference value greater than ± 10% is one for the model with distance L = 20m, and three for the model with distance L = 15m. In the model with L = 5,10 m, there are relatively few response values, and there is no response value with a deviation from the reference value larger than ± 20%. From the above results, in the time history response analysis (case NT) to which the present invention is applied, an analysis result with good accuracy can be obtained even when the size (distance L) of the analysis target model is relatively small.

  On the other hand, the analysis result in the time history response analysis (case LT) to which the invention described in Patent Document 1 is applied is shown in FIG. As is clear from comparison of FIG. 15 with FIG. 13, in the time history response analysis (case LT) to which the invention described in Patent Document 1 is applied, the response value in the model with the distance L = 120 m Although the response value in the model with the distance L = 120 m in the applied time history response analysis (case NT) is approximately the same, the response value in the model with a smaller distance L is the same as the model with the distance L = 120 m. It can be seen that the deviation from the response value is slightly increased.

  16 (A) to 16 (C) show the above analysis results (analysis results in time history response analysis (case NT) to which the present invention is applied) in the same manner as FIGS. 14 (A) to (C). As shown in FIG. 16, the response value in the model of the distance L = 120 m in the time history response analysis (case NT) to which the present invention is applied is used as a reference value. In the time history response analysis to which the invention is applied (case LT), all response values are within ± 10% of the reference value compared to the time history response analysis to which the present invention is applied (case NT). The minimum value of the distance L in the model group is as large as the distance L = 40 m, and the number of response values whose deviation from the reference value is larger than ± 10% also increases. In the model of the distance L = 5,10 m, the reference value Since response values with deviations greater than ± 20% also appear, time history response solutions to which the present invention is applied Analysis accuracy than (case NT) it is clear that the reduced.

  In addition, FIG. 17 shows an analysis result in an equivalent linear response analysis (case V) using a viscous boundary as a wave boundary model. As is clear from comparing FIG. 17 with FIG. 13 and FIG. 15, in the equivalent linear response analysis using the viscous boundary as the wave boundary model (case V), the response value in the model with the distance L = 120 m is as follows. The time history response analysis to which the present invention is applied (case NT) and the time history response analysis to which the invention described in Patent Document 1 is applied (case LT), but the response in a model with a smaller distance L As for the values, the variation of response values is increased as compared with the time history response analysis to which the present invention is applied (case NT) and the time history response analysis to which the invention described in Patent Document 1 is applied (case LT). .

  18A to 18C show the above analysis results (analysis results in an equivalent linear response analysis (case V) using a viscous boundary as a wave boundary model) similar to FIGS. (The reference value is a response value in a model with a distance L = 120 m in time history response analysis (case NT) to which the present invention is applied). As is apparent from FIG. 18, in the equivalent linear response analysis (case V) to the viscous boundary as the wave boundary model, the distance L in the model group in which all the response values are within ± 10% of the reference value. The distance L is significantly larger than the distance L = 100 m, and the deviation from the reference value is larger than ± 20% compared to the time history response analysis (case LT) to which the invention described in Patent Document 1 is applied. The number of response values has also increased significantly. In addition, in an equivalent linear response analysis using a viscous boundary as a wave boundary model (Case V), the analysis accuracy tends not to improve as the distance L increases unless the distance L is sufficiently large. I can confirm.

  From the above results, the time history response analysis (case NT) to which the present invention is applied is equivalent to the time history response analysis (case LT) to which the invention described in Patent Document 1 is applied, and an equivalent using a viscous boundary as a wave boundary model. Compared to the linear response analysis (Case V), it was confirmed that the same analysis accuracy can be obtained with a model having a smaller size (distance L) of the inner region.

  In the above analysis study, each response analysis is performed using a computer equipped with an IBM Power5 + (registered trademark) processor (2.2 GHz), and the processing time (calculation time) in each response analysis is analyzed by time history response analysis. And the time required for pre-processing (processing such as matrix calculation of the ground in the external region and calculation of impulse response at the energy transfer boundary) were measured separately. The measurement results are shown in Table 6 below.

  As is apparent from Table 6, as a whole, the processing time (calculation time) tends to increase as the size (distance L) of the analysis target model increases. Note that, in part, for example, the processing time (calculation time) regardless of the increase in the size (distance L) of the model to be analyzed, such as the processing time of the model with the distance L = 30 m and the model with the distance L = 40 m in the case LT. ) May decrease, but this is thought to be due to the difference in the number of computations required to converge the computation results in the nonlinear response analysis for the internal region.

  Comparing the processing time (calculation time) for each case, the equivalent linear response analysis using the viscous boundary as the wave boundary model (Case V) shows that the wave boundary model is a viscous boundary. Is simple and the processing time is the shortest. On the other hand, the time history response analysis (case NT) to which the present invention is applied and the time history response analysis (case LT) to which the invention described in Patent Document 1 is applied are both because the wave boundary model is an energy transfer boundary. The time required for time history response analysis is almost the same, slightly longer than equivalent linear response analysis (case V) using a viscous boundary as a wave boundary model, and further preprocessing (matrix calculation and energy of the ground in the external region) Processing such as calculation of impulse response at the transmission boundary is also required.

  The time history response analysis (case NT) to which the present invention is applied has a longer pre-processing time than the time history response analysis (case LT) to which the invention described in Patent Document 1 is applied. In the time history response analysis (case LT) to which the invention described in Patent Document 1 is applied, the ground matrix in the external region and the impulse response at the energy transfer boundary can be calculated by one operation, whereas the present invention is applied. In the time history response analysis (case NT), the calculation of the impulse response of the ground matrix of the external region and the energy transfer boundary is repeated in increments of Δtb, and the impulse response of the energy transfer boundary is calculated by interpolation calculation for all analysis target times. This is necessary.

  As is clear from the above results, in the time history response analysis to which the present invention is applied (case NT), if a model having the same size (the same distance L) is used as an analysis target model, a viscous boundary is used as a wave boundary model. The processing time becomes longer than the equivalent linear response analysis (case V) used and the time history response analysis (case LT) to which the invention described in Patent Document 1 is applied. However, the time history response analysis (case NT) to which the present invention is applied applies the equivalent linear response analysis (case V) using the viscous boundary as the wave boundary model as described above, or the invention described in Patent Document 1. Since the analysis accuracy is higher than the time history response analysis (case LT), the same analysis accuracy can be obtained with an analysis target model having a smaller size (distance L).

  For example, according to the results shown in FIGS. 14, 16, and 18, when the deviation of the response value with respect to the reference value is within ± 10%, the present invention is applied to the minimum size of the analysis target model (minimum value of the distance L). In the time history response analysis (case NT), the distance L = 30 m, in the time history response analysis (case LT) to which the invention described in Patent Document 1 is applied, the distance L = 40 m, and an equivalent linear using a viscous boundary as a wave boundary model In the response analysis (case V), the distance L is 100 m (in Table 6, each is indicated by a gray column). Therefore, the processing time (calculation time) including the preprocessing is 17.4 minutes (= 13.8 minutes + 3.6 minutes) in the model of distance L = 30 m in the time history response analysis (case NT) to which the present invention is applied. In the time history response analysis (case LT) to which the invention described in 1 is applied, the distance L = 40m model is 16.6 minutes (= 16.3 minutes + 0.3 minutes), and the equivalent linear response analysis using the viscous boundary as the wave boundary model In (Case V), the distance L = 100m model is 17.2 minutes, and there is no significant difference.

  The response analysis described above is a two-dimensional response analysis, and a two-dimensional model is used as an analysis target model. However, in a three-dimensional response analysis using a three-dimensional analysis target model, the analysis target model is configured. Since the number of components to be processed is very large, the computation load is high and the response analysis takes a long time. On the other hand, when the analysis target model is reduced in size (distance L is reduced), the constituent elements constituting the analysis target model are accordingly accompanied. The number of computers is greatly reduced, and the computation load is also greatly reduced. Therefore, in the time history response analysis to which the present invention is applied, particularly when performing a three-dimensional response analysis, the time history response analysis to which the invention described in Patent Document 1 is applied, or an equivalent using a viscous boundary as a wave boundary model. Compared with linear response analysis, it is considered to have a great advantage in terms of processing time (calculation time).

10 Computer 10C Storage Unit 12 Display 14 Keyboard 16 Mouse

Claims (11)

An analysis target object consisting of the ground alone or the ground and a building is modeled, and an energy transfer boundary as a wave boundary model is provided between the internal region and the external region of the model. On the other hand, by performing a response analysis that analyzes the behavior when a specific external force that vibrates the analysis target object is input at each time of the analysis target, the physical property value of the analysis target object in the external region is analyzed. Physical property value calculating means for calculating each target time,
Main configuration of the energy transmission boundary when the specific external force is input based on the physical property value of the analysis target object in the external region calculated at each time of the analysis target by the physical property value calculation means Matrix calculation means for calculating values of a transmission boundary matrix that is an element and is defined as a complex number in the frequency domain and has strong frequency dependence for each of two or more times of the time to be analyzed;
Based on the value of the transmission boundary matrix calculated for each of two or more times by the matrix calculating means, the specific external force is expressed as an impulse response that represents the relationship between the external force that vibrates the object and the behavior of the object in the time domain. Impulse response calculation means for calculating the impulse response of the energy transfer boundary when input, for each time to be analyzed;
Using the impulse response of the energy transfer boundary calculated at each time of the analysis target by the impulse response calculation means, the behavior when the specific external force is input to the analysis target model is analyzed. An analysis means for performing a response analysis for analyzing each target time;
Including a response analysis device. The transmission boundary matrix of the energy transmission boundary includes a mass matrix [M _I ], a stiffness matrix [K _I ], a displacement vector {u (t)}, and a boundary force vector {F _R (t )}, The response displacement of the object to be analyzed in the external region is {u _R ^* (t)}, the correction force vector acting on the boundary is − [D _R (t)] {u _R ^* (t)},
[M _I ] {u "(t)} + ([K _I (t)] + [R (t)]) {u (t)}
= −y ″ (t) [M _I ] {1} + {F _R (t)} (1)
{F _R (t)} = ([R (t)] − [D _R (t)]) {u _R ^* (t)} (2)
The equation of motion of the entire model to be analyzed is expressed by [R (t)] in the above equations (1) and (2) when the equations of motion are expressed by the above equations (1) and (2). Response analysis device.   The physical property value calculating means is a regular imaginary part indicating a value (2n-1)-(2ω / ωm) (where n is an integer) in the range of the frequency ω from (n−1) · ωm to n · ωm. It is represented by the sum of the singular component of the imaginary part indicating the component and the value of (2ω / ωm) regardless of the frequency ω, and the frequency ω is in the range of (n−1) · ωm to n · ωm (2n -1) A causal unit imaginary function consisting of an imaginary part indicating the value of -1 and a real part corresponding to the Hilbert transform value of the regular component of the imaginary part is converted into the time domain, or only the imaginary part is timed. Using the impulse response value of the causal unit imaginary function obtained by converting to a region, by performing a response analysis on the external region of the analysis target model, the analysis target object in the external region The shear strain amplitude value γ, the shear wave velocity Vs, and the damping constant h are calculated as physical property values. Response Analysis device according.   The impulse response calculation means, based on the value of the transfer boundary matrix calculated for each of two or more times by the matrix calculation means, the impulse response of the energy transfer boundary when the specific external force is input, After calculating each of the two or more times at which the values of the transfer boundary matrix are calculated, the energy transfer boundary at the time when the impulse response of the energy transfer boundary is not calculated among the times to be analyzed. The response analysis apparatus according to any one of claims 1 to 3, wherein an impulse response is obtained by interpolation calculation from the impulse response of the energy transmission boundary calculated for each of the two or more times.   The impulse response calculation means is configured to calculate the shear strain amplitude value γ of the object to be analyzed in the external region at the time when the impulse response of the energy transfer boundary is not calculated, Based on the ratio of internally dividing the shear strain amplitude value γ of the object to be analyzed in the external region at the above time, the impulse of the energy transfer boundary at a time when the impulse response of the energy transfer boundary is not calculated. The response analysis apparatus according to claim 4, wherein a response is obtained by interpolation calculation from the impulse response of the energy transfer boundary calculated for each of the two or more times.   The impulse response calculation means is configured to calculate the energy transfer boundary based on a ratio of dividing the time at which the impulse response of the energy transfer boundary is not calculated into the two or more times at which the values of the transfer boundary matrix are calculated. 5. The response analysis according to claim 4, wherein the impulse response of the energy transfer boundary at a time when the impulse response is not calculated is obtained by interpolation from the impulse response of the energy transfer boundary calculated for each of the two or more times. apparatus. The impulse response calculation means includes k _{0 as} a simultaneous component of the impulse response depending on the displacement of the object, c _{0 as} a simultaneous component of the impulse response depending on the velocity of the object, and m as a simultaneous component of the impulse response depending on the acceleration of the object. ₀ , the time delay component of Δt in the impulse response depending on the displacement of the object is k _j , the time delay component of the impulse response in the Δt step depending on the speed of the object is c _j (where j is a natural number, t _j = Δt · j) When the displacement of the object in the time domain is u (t), the velocity is u '(t), and the acceleration is u "(t), it depends on the acceleration of the object and contains at least simultaneous components As a mathematical formula that also includes a mass term, it is a mathematical formula that defines the impulse response F _B (t).

Using the above equation (3), the impulse response of the energy transfer boundary at the time when the transfer boundary matrix is calculated by the matrix calculation means is the frequency of the N vibrations (N = n + 1) at the time. The response analysis device according to claim 1, wherein the response analysis device performs calculation based on a value of the transmission boundary matrix. The impulse response calculation means is a correction value Δm ₀ for the simultaneous component m ₀ of the impulse response depending on the acceleration of the object in the equation (3), and a correction value Δk ₀ for the simultaneous component k ₀ of the impulse response depending on the displacement of the object. Is calculated by the following equation (4),

(However, in the above equation (4),

Re (S _B (ω _i )) in the above equation (5) is the following equation (6) reproduced from the impulse response of the energy transfer boundary calculated using the above equation (3). The value of the real part at the frequency ω _i of the transmission boundary matrix S _B (ω) represented

Re (D (ω _i )) in the above equation (5) is the data D (of the transmission boundary matrix at the frequency ω _i used in the calculation of the impulse response of the energy transmission boundary based on the equation (3). ω represents the value of the real part of the _i)), the calculated correction value Delta] m _0, response analysis apparatus according to claim 7, wherein modifying the simultaneous component m _0, k ₀ using .DELTA.k _0. The impulse response calculating means calculates a correction value Δc ₀ for c _{0 with} respect to the simultaneous component of the impulse response depending on the speed of the object in the equation (3) according to the following equation (7):
Δc ₀ = −E / B (7)
(However, in the above equation (7),

In the above equation (8), Im (S _B (ωi)) is expressed by the equation (6) reproduced from the impulse response of the energy transfer boundary calculated using the equation (3). Represents the value of the imaginary part at the frequency ω _i of the transmission boundary matrix S _B (ω), and Im (S _B (ω _i )) in the above equation (8) is the energy transfer based on the equation (3). This represents the value of the imaginary part of the transmission boundary matrix data D (ω _i ) at the frequency ω _i used in the calculation of the impulse response of the boundary), and the calculated correction value Δc ₀ The response analysis apparatus according to claim 7, wherein component c _{0 is} also corrected. On the computer,
An analysis target object consisting of the ground alone or the ground and a building is modeled, and an energy transfer boundary as a wave boundary model is provided between the internal region and the external region of the model. On the other hand, by performing a response analysis that analyzes the behavior when a specific external force that vibrates the analysis target object is input at each time of the analysis target, the physical property value of the analysis target object in the external region Calculate at each time to be analyzed,
The main component of the energy transfer boundary when the specific external force is input based on the physical property value of the analysis target object in the external region calculated at each time of the analysis target, the frequency domain A transmission boundary matrix value defined as a complex number and having a strong frequency dependence is calculated for each of two or more times of the time to be analyzed,
Based on the value of the transmission boundary matrix calculated for each of two or more times, the impulse response that represents the relationship between the external force that vibrates the object and the behavior of the object in the time domain is obtained when the specific external force is input. The impulse response of the energy transfer boundary is calculated for each time to be analyzed,
Using the impulse response of the energy transfer boundary calculated at each time of the analysis target, the behavior when the specific external force is input to the analysis target model at each time of the analysis target. A response analysis method for performing a response analysis to be analyzed. Computer
An analysis target object consisting of the ground alone or the ground and a building is modeled, and an energy transfer boundary as a wave boundary model is provided between the internal region and the external region of the model. On the other hand, by performing a response analysis that analyzes the behavior when a specific external force that vibrates the analysis target object is input at each time of the analysis target, the physical property value of the analysis target object in the external region is analyzed. Physical property value calculation means for calculating each target time,
Main configuration of the energy transmission boundary when the specific external force is input based on the physical property value of the analysis target object in the external region calculated at each time of the analysis target by the physical property value calculation means Matrix calculation means for calculating values of a transmission boundary matrix that is an element and is defined as a complex number in the frequency domain and has strong frequency dependence, for each of two or more times of each time to be analyzed,
Based on the value of the transmission boundary matrix calculated for each of two or more times by the matrix calculating means, the specific external force is expressed as an impulse response that represents the relationship between the external force that vibrates the object and the behavior of the object in the time domain. Impulse response calculation means for calculating the impulse response of the energy transfer boundary when input, for each time to be analyzed,
And using the impulse response of the energy transfer boundary calculated at each time of the analysis target by the impulse response calculation means, the behavior when the specific external force is input to the analysis target model A response analysis program that functions as an analysis unit that performs a response analysis to be analyzed at each time to be analyzed.