JP2012037305A - Sequential nonlinear earthquake response analysis method for foundation and storage medium with analysis program stored thereon - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a method for accurately making a behavior of the foundation by an earthquake motion into a nonlinear model through numerical simulation from a fine strain area to a large strain area together with shear rigidity and attenuation, and applying the nonlinear model to sequential nonlinear earthquake response analysis.SOLUTION: A skeleton curve, a shear rigidity ratio/strain relationship and an attenuation curve obtained by a soil test are simulated in all strain areas by using a mixed model (an H-R model) obtained by mixing a hyperbolic curve model and an R-O model, and a parameter of the H-R model is automatically set according to approximation. Namely, the H-R model used for the present invention is obtained by correcting and mixing the hyperbolic curve model and the R-O model, thereby compensating for an application range of the hyperbolic curve model and the R-O model, and suppressing a tendency to evaluate large the shear rigidity in a large strain area in the R-O model or a tendency to evaluate small the shear rigidity in a large strain area in the hyperbolic curve model.

Description

  The present invention relates to a mathematical model for forming a hysteresis curve used when performing a sequential nonlinear seismic response analysis of the ground, and a sequential nonlinear using a nonlinear model that can simulate the strain dependence of the shear stiffness ratio and damping constant of the ground. The present invention relates to an earthquake response analysis method and a storage medium storing the analysis program.

  Earthquake motion propagates while being gradually amplified from the hard ground deep in the ground, which can be considered as an elastic body, and reaches the ground surface through the soft ground of the surface layer. The soft ground of the surface layer exhibits non-linearity (a phenomenon in which the rigidity decreases and the damping increases) even when the strain is small, and when the ground motion is large, it behaves as an elasto-plastic body. It is necessary to adopt an appropriate nonlinear model that takes into account the nonlinearity of the stress-strain relationship that changes every moment in the time history. The non-linear characteristics of the ground at that time include indoor soil tests (vibration triaxial test and hollow torsion test). From the stress-strain relationship obtained, the shear stiffness ratio and damping constant of the ground are determined. It is shown in a form dependent on.

  The non-linear model of the ground is represented by three curves of a stress-strain relationship (skeleton curve), a stress-damping relationship (damping curve), and a hysteresis curve that can be applied to repeated loading-unloading by combining them. Conventionally, as a typical nonlinear model, a hyperbola model (also referred to as a hyperbola model here, which is also referred to as an HD model) and a Ramberg-Osgood (RO) model are known.

  Moreover, as a method of applying to a sequential nonlinear seismic response analysis using the stress-strain model by a RO model or a hyperbola model, what was described in the following patent document 1 is known.

JP 2006-266940 A

  However, the hyperbolic model of the conventional nonlinear model expresses the skeletal curve with two clear parameters, the standard strain and the shear rigidity at the time of micro strain, and although it is simple, the attenuation curve and the hysteresis curve are defined. Therefore, it cannot be applied to sequential nonlinear seismic response analysis. Moreover, it is difficult to accurately represent the skeletal curve of the hyperbolic model from the small strain region to the large strain region. There is also a method of defining a hysteresis curve by applying the Masing rule to this hyperbolic model (modified HD model). However, in the modified HD model, the attenuation constant does not take into account the maximum attenuation. There is a problem that the attenuation constant tends to be overestimated.

  In addition, among the conventional nonlinear models, the RO model is composed of four parameters, so it has a variety of applications and high applicability. However, the shear stiffness G is underestimated in the small strain region, and in the large strain region. There is a tendency that evaluation tends to be excessive, and the accuracy of attenuation is not high due to the influence.

  The present invention has been made in view of the above points, and its technical problem is to accurately analyze the behavior of the ground due to earthquake motion by numerical simulation from a small strain to a large strain region together with shear stiffness and damping. And to provide a method for applying this nonlinear model to sequential nonlinear earthquake response analysis.

ここに、τは応力、γはせん断ひずみ、Ｇ_０は初期せん断剛性、γ_0.5は基準ひずみ、γ_ｍは微小ひずみ、λはパラメータ。

ここに、α、βはパラメータ。 As a means for effectively solving the above-mentioned technical problem, the invention of claim 1 is directed to a skeleton obtained by soil tests in modeling of non-linear characteristics of the ground used when performing sequential non-linear seismic response analysis of the ground. The curve, the shear stiffness ratio G / G ₀ -strain γ relationship, and the damping curve are a mixed model (hereinafter referred to as H) in which the modified hyperbola model represented by Equation (3) and the R—O model represented by Equation (4) are mixed. -R model) is used to simulate the entire strain region, and the parameters of the HR model are automatically set by an approximation method.

Where τ is stress, γ is shear strain, G ₀ is initial shear stiffness, γ _0.5 is reference strain, γ _m is micro strain, and λ is a parameter.

Where α and β are parameters.

  That is, since the HR model used in the present invention is a modified and mixed hyperbola model and RO model, it supplements the applicable range of the hyperbola model and the RO model, and the RO model is large in the RO model. It is possible to suppress a tendency to largely evaluate the shear rigidity in the strain region and a tendency to evaluate the shear rigidity in the large strain region to be small in the hyperbolic model.

  The method of sequential nonlinear earthquake response analysis of ground according to the invention of claim 2 is characterized in that, in the method of claim 1, the approximation method for setting the parameter λ is based on the least square method. is there. That is, since the least square method is used as a method for simulating the shear rigidity ratio-strain relationship and the damping constant-strain relationship of the soil test results, approximation with high reproducibility can be performed with high accuracy.

The ground sequential nonlinear seismic response analysis method according to claim 3 is the method according to claim 1, wherein the modified hyperbola model represented by equation (3) is a skeleton curve of the hyperbola model represented by equation (2). And a parameter λ and a small strain γ _m are added. By this method, the skeleton curve can be accurately simulated in the entire strain region.

The ground sequential nonlinear seismic response analysis method according to the invention of claim 4 is the method of claim 1, wherein the value of the parameter α of the RO model represented by the equation (4) is set to the right side of the equation (9). The strain dependency is imparted by substituting the values obtained from the soil test into G ₀ , γ, τ, τ ₀ , and β. By this method, the skeleton curve and the attenuation curve can be accurately simulated in the entire strain region.

The ground sequential nonlinear seismic response analysis method according to the invention of claim 5 adds the parameter α value of the RO model represented by the equation (4) and the parameter to the hyperbolic model in the method of claim 1. The strain dependence is imparted by obtaining the equation (10) using the skeleton curve display equation (3) of the modified hyperbolic model.

The sequential nonlinear seismic response analysis method for ground according to the invention of claim 6 is the method according to claim 1, wherein the parameter α deviates from the skeletal curve by changing from unloading to loading or vice versa. In addition, the value of the parameter α _a when reaching the past maximum or minimum strain according to the following equation (11) is set as a constant. In other words, if the response stress-strain is on the skeletal curve in the history curve when loading-unloading is repeated as in an earthquake response, the parameter α changes momentarily depending on the strain, but it is unloaded from loading. Alternatively, when the parameter α deviates from the skeletal curve by changing from unloading to loading, the value of the parameter α _a when reaching the past maximum or minimum strain is set as a constant, so that the application of the Masing law It becomes possible.

ここに、ｈ_maxは最大減衰比。 In the method of sequential non-linear seismic response analysis of ground according to the invention of claim 7, in the method of claim 1, the attenuation is underestimated because the attenuation constant h becomes 0 at the time of minute strain in the RO model. In order to avoid this, in the modification of the R-O model, the attenuation constant h is modified to have the minimum attenuation constant h _min by the following equation (12). In this way, the attenuation constant is not underestimated in the small strain region, and an excessive response value can be prevented.

Here, h _max is the maximum attenuation ratio.

  A storage medium according to an eighth aspect of the invention stores a program for executing the ground sequential nonlinear seismic response analysis method according to any one of the first to seventh aspects.

  According to the ground sequential nonlinear seismic response analysis method according to the present invention, it is possible to accurately simulate the skeletal curve and the attenuation curve of the ground survey result, by extreme rigidity reduction, rigidity overestimation or attenuation constant underestimation. It is possible to obtain a nonlinear model in which unstable behavior is unlikely to occur. Approximation with high accuracy and high reproducibility is possible.

It is schematic structure explanatory drawing which shows the analyzer used in order to implement the sequential nonlinear earthquake response analysis method of the ground which concerns on this invention. It is a hysteresis diagram which shows the skeleton curve and hysteresis curve of the stress-strain relationship by the sequential nonlinear earthquake response analysis method of the ground which concerns on this invention. It is a flowchart which shows the flow of the rough process by the analyzer shown by FIG. It is a flowchart which shows the flow of the analysis in the sequential nonlinear earthquake response analysis method of the ground which concerns on this invention. H-R model according to the invention the soil test results, the relationship between the conventional R-O model and strain and shear stiffness ratio G / G ₀ which simulates hyperbolic model gamma, relationship damping constant h and strain gamma, and skeletal curve These are diagrams shown in comparison with the ground survey results. It is a diagram which shows the skeletal curve of the HR model by this invention, and the skeleton model of the conventional RO model compared with a ground investigation result as a soil test result. It is a diagram which shows a skeleton curve and a hysteresis curve by comparing the HR model according to the present invention with the conventional RO model and hyperbola model. It is a diagram which compares and shows the acceleration waveform of a sequential nonlinear earthquake response analysis result with the HR model which concerns on this invention, the conventional RO model, and a hyperbola model. The acceleration response spectrum (h = 5%) of the sequential nonlinear seismic response analysis result and the pseudo velocity response spectrum (h = 5%) of the sequential nonlinear seismic response analysis result are used as the HR model according to the present invention and the conventional RO model. It is a diagram shown by comparing with a hyperbola model. The maximum absolute acceleration, maximum relative velocity, maximum relative displacement, maximum strain, shear wave velocity, and vertical distribution of the damping constant of the results of the sequential nonlinear seismic response analysis are shown in the HR model according to the present invention and the conventional RO model. It is a diagram shown in comparison with a hyperbola model.

  The ground sequential nonlinear earthquake response analysis method according to the present invention will be described in detail below with reference to the drawings. First, FIG. 1 is a schematic configuration explanatory view showing an analysis apparatus used for a ground sequential nonlinear seismic response analysis according to the present invention.

  This analysis apparatus is a general-purpose personal computer 100 including a CPU (Central Processing Unit) 101, an input unit 102 such as a keyboard and a mouse connected to the CPU 101, an interface 103, a memory 104, a hard disk 105, a display 106, and the like. The server 111 and the printer 112 are connected via a network. Further, measurement data from a test apparatus (not shown) (vibration triaxial test apparatus, hollow torsion test apparatus, etc.) can be input to the CPU 101 via the interface 103.

  The hard disk 105 stores a calculation program necessary for sequential nonlinear earthquake response analysis using a ground nonlinear model according to the present invention. This program is recorded in advance on an appropriate storage medium 107 such as a CD-ROM or a USB memory, read from the storage medium 107 and installed on the hard disk 105.

  The CPU 101 executes sequential nonlinear seismic response analysis by processing steps to be described later in accordance with a calculation program stored in the hard disk 105, whereby the personal computer 100 can perform the ground nonlinear sequential earthquake response analysis method according to the present invention. It functions as an analysis device that realizes

  FIG. 2 is a hysteresis diagram showing a skeleton curve and a history curve of a stress-strain relationship by a ground sequential nonlinear seismic response analysis method according to the present invention.

  In this diagram, reference numeral 1 is a virgin loading skeleton curve. The virgin loading skeleton curve 1 shows the relationship between stress τ and strain γ when stress and strain both start from 0 and there is no repeated loading (virgin loading). The maximum amplitude of the stress and strain history curve is substituted. Further, the suitability between the test data and the model calculation value is determined by the virgin loading skeleton curve 1.

Reference numeral 2 is a history curve. The hysteresis curve 2, virgin loading when it is unloaded at an strain gamma _a in relation to the stress τ and strain gamma from unloading point P on the virgin loading skeleton curve 1 when it is unloaded and strain-gamma _a The relationship between the stress τ and the strain γ from the negative point P ′ on the skeleton curve 1 is shown.

In the present invention, the Masing rule that connects the skeleton curve 1 and the history curve 2 is such that the model parameter changes when the strain γ exceeds the past maximum strain γ _a (or −γ _a ) on the skeleton curve 1. Set to (irreversible). For this reason, the history curve 3 of the small strain when the strain γ is unloaded at the point γ _b exceeding the past maximum strain γ _a is not the same as the history curve before the large strain is generated, and experiences a large strain. Compared with the prior art in which the history curves are the same before and after, approximation with high reproducibility becomes possible.

  FIG. 3 is a flowchart showing a schematic processing flow by the analysis apparatus shown in FIG.

  First, when the operator starts up the personal computer 100 and activates the program, the CPU 101 reads basic data necessary for the nonlinear earthquake response analysis sequentially from the hard disk 105 (step S1).

  Next, regarding the ground material collected by boring the ground to be analyzed, various stresses, strains, shear wave velocity, shear stiffness, damping constant, etc. obtained by indoor soil tests (vibration triaxial test and hollow torsion test) The measurement data of the ground physical property value is read into the CPU 101 from the test apparatus (not shown) via the interface 103 (step S2). These ground data can also be input by an operator by operating the input unit 102 such as a keyboard.

  Next, from the basic data acquired in step S1 and the ground data acquired in step S2, a mass matrix M, a damping matrix C, and a stiffness matrix K indicating characteristics necessary for performing ground response analysis are created (step S3). ).

Next, the seismic motion data input to the lower part of the surface ground (engineering base) is read (step S4) and substituted into the right side of the equation of motion represented by the following equation (1) (step S5).

  Next, the CPU 101 calculates the response acceleration, the response displacement, the response speed, the response maximum value distribution, and the like that change every moment of the ground to be analyzed by the direct integration method (linear acceleration method) using the equation (1). Perform time history response analysis.

In the time history response analysis using only the direct integration method described above, it is difficult to accurately analyze the ground behavior in consideration of the influence of nonlinear strain dependence. Of the ground shear stiffness ratio G / G ₀ (G: shear stiffness, G ₀ : shear stiffness at the time of micro strain at the initial strain) and damping constant h are determined in a form depending on the shear strain γ of the ground (step S6).

  FIG. 4 is a flowchart showing a flow of analysis by a ground nonlinear earthquake response analysis method according to the present invention corresponding to step S6.

First, a minute strain γ _m is set (step S61), and a reference strain γ _0.5 is obtained from the measurement data of the physical property values of the ground obtained by the indoor soil test described above (step S62). The reference strain γ _0.5 is a strain where G / G ₀ = 0.5.

The following equation (2) is a model equation representing a conventional hyperbolic model. In this equation (2), τ is stress, γ is shear strain, G ₀ is initial shear stiffness, and γ _0.5 is reference strain.

In the present invention, the parameter λ is obtained by applying the least square method to the measurement data of the physical property values of the ground obtained by the indoor soil test described above (step S63), and the hyperbolic model represented by the above equation (2) is obtained. A modified hyperbola model represented by the following equation (3) in which a parameter λ and a small strain γ _m are added to the skeleton curve is set. Therefore, this equation (3) is equivalent to equation (2) if γ _m = 0 and λ = 1, that is, is equivalent to a hyperbolic model.

FIG. 5 shows the relationship between the shear stiffness ratio G / G ₀ and the strain γ, and the relationship between the damping constant h and the strain γ, in which the soil test results are simulated by the HR model according to the present invention, the conventional RO model and the hyperbolic model. It is a diagram which shows a skeleton curve in comparison with a ground survey result, respectively. From these simulation results, it can be seen that the HR model according to the present invention corresponds well to the ground survey results as compared with the conventional RO model and hyperbolic model.

Moreover, following Formula (4) is a model formula showing a well-known RO model. In this equation, τ ₀ is a fracture stress (τ ₀ = G ₀ · γ _0.5 ), and α and β are parameters.

When the expression (4) of the RO model is expressed by the ratio G / G ₀ of the shear rigidity G and the initial shear rigidity G ₀ , the following expression (5) is obtained.

Further, the attenuation constant h in the R—O model can be obtained by the following equation (6) from the history area when the Masing rule is applied to the above equation (4) with α and β as constants.

また、ｈ_maxはβの関数であるため、βは次式（８）に式（７）で求められたｈ_maxの値を代入することによって求められる（ステップＳ６５）。

Here, h _max in Equation (6) is the maximum attenuation ratio, and is obtained by the least square method using the attenuation constant h and soil test data as G / G ₀ (step S64).

Since h _max is a function of β, β is obtained by substituting the value of h _max obtained by equation (7) into the following equation (8) (step S65).

Next, how to obtain α is expressed by the following equation (9) by solving equation (4) for α.

  In the HR model according to the present invention, the parameter α in the equation (9) is treated as strain dependency, and the following two methods are adopted for obtaining the parameter α.

One is a method of obtaining directly from the test data by substituting the values obtained from the soil test into G ₀ , γ, τ, τ ₀ , β on the right side of Equation (9). FIG. 6 shows the HR according to the present invention, which is simulated by substituting the values obtained from the soil test into G ₀ , γ, τ, τ ₀ , β on the right side of the formula (9) in this way. It is a diagram which shows the skeleton curve of a model and the conventional RO model compared with a ground investigation result. As can be seen from FIG. 6, in the RO model, the value of the parameter α is constant even if the value of the strain γ changes, that is, α has no strain dependence, which is the case of the RO model. This was one of the causes of the decrease in accuracy. On the other hand, in the HR model according to the present invention, α is not a constant but a strain-dependent function, and corresponds well with the ground survey result.

  However, in this method, when the distortion of the seismic response result exceeds the range of the test data, it is necessary to extrapolate the value of α in the exceeded range, so the accuracy depends on the extrapolation method.

The other method is a method using the skeleton curve of the modified hyperbolic model expressed by Equation (3). That is, if τ in the modified hyperbolic model equation (3) is substituted into the equation (9), the parameter α is obtained by the following equation (10).

Here, when equation (10) is used as it is in the sequential nonlinear seismic response analysis, the transition from loading to unloading or vice versa in the hysteresis diagram shown in FIG. 2 shifts from skeleton curve 1 to hysteresis curve. Even in this case, since α changes every moment according to the response strain γ, the history curve cannot be expressed by the Masing rule. Therefore, in the present invention, when changing from unloading to unloading or conversely from unloading to loading and shifting from the skeleton curve 1 to the history curve, α at the time when the past maximum strain γ _a or minimum strain −γ _a is reached. By using the value as a constant, it is possible to analyze the hysteresis curve 3 by applying the Masing rule. With this rule, the parameter α becomes _a function of the past maximum strain γ _a and has irreversible properties. This corresponds to the following formula by replacing the gamma of the formula (10) to gamma _a (11) (step S66).

  In the sequential nonlinear seismic response analysis method according to the present invention, a skeletal curve is obtained by applying an ever-changing response strain γ and αa of equation (11) to equation (4), and applying to equation (5). Thus, the shear rigidity G is obtained every moment, and the nonlinear earthquake response analysis can be sequentially performed (step S67).

In Equation (6), when γ → 0, that is, G / G ₀ → 1, the attenuation constant h becomes 0, and the soil test result is underestimated. Therefore, Equation (6) is changed to the minimum attenuation constant h. It corrects like following Formula (12) which has _min .

  The result of sequential nonlinear earthquake response analysis by the above-described analysis steps is output from the CPU 101 as response acceleration, response speed, response displacement, response maximum value distribution, etc. according to time history (step S7 in FIG. 3). The output analysis data is displayed on the display 106, can be stored in the memory 104 or the hard disk 105, and can also be output to the printer 112 via the network.

  FIG. 7 is a diagram showing a skeleton curve and a hysteresis curve based on the stress τ-strain γ relationship obtained from the soil test by comparing the HR model according to the present invention with the conventional RO model and hyperbola model. is there. As shown in FIG. 5A described above, the RO model has a tendency to largely evaluate the shear stiffness in the large strain region, and the hyperbolic model has a tendency to evaluate the shear stiffness to be small in the large strain region. Therefore, the RO model has a skeletal curve with a τ-γ relationship as shown in FIG. 7B, and the hyperbolic model has a τ-γ relationship as shown in FIG. 7C. While the skeletal curve is lying on its side due to a sudden decrease in rigidity in the large strain region, that is, in the large strain region, the result is that the strain greatly increases with a slight increase in stress, whereas the HR model according to the present invention is An intermediate analysis result between the two is obtained, that is, as shown in FIG. 7A, it can be said that the application range of the RO model and the hyperbola model is supplemented.

  FIG. 8 is a diagram showing the acceleration waveform of the result of sequential nonlinear seismic response analysis by comparing the HR model according to the present invention with the conventional RO model and hyperbolic model, and FIG. 9 is the result of sequential nonlinear seismic response analysis. The acceleration response spectrum (h = 5%) and the pseudo-velocity response spectrum (h = 5%) of the results of sequential nonlinear seismic response analysis were compared between the HR model according to the present invention, the conventional RO model and the hyperbolic model. FIG. 10 shows the vertical distribution of the maximum absolute acceleration, maximum relative velocity, maximum relative displacement, maximum strain, shear wave velocity, and damping constant of the sequential nonlinear earthquake response analysis results according to the present invention. FIG. 6 is a diagram showing a model in comparison with a conventional RO model and a hyperbola model. From these diagrams, it can be seen that the HR model according to the present invention can obtain an intermediate analysis result between the RO model and the hyperbolic model. In other words, it can be said that the model is unlikely to cause unstable behavior due to extreme rigidity reduction, overestimation of rigidity, or underestimation of damping constant.

Further, according to the method of the present invention, since the least square method is used as a method for simulating the shear stiffness ratio G / G ₀ -strain γ relationship and the damping constant h-strain γ relationship of the indoor soil test results, the reproducibility is high with accuracy. A high approximation is possible.

Moreover, the HR model used in the present invention can express the RO model if the parameter α is a constant value, while the hyperbolic model can be expressed if the parameters λ = 1 and γ _m = 0. The -R model can be said to be a highly versatile nonlinear model.

101 CPU
102 Input unit 104 Memory 105 Hard disk 106 Display 107 Storage medium

Claims (8)

In modeling the non-linear characteristics of the ground used when performing sequential nonlinear seismic response analysis of the ground, the skeletal curve, the shear stiffness ratio G / G ₀ -strain γ relationship, and the attenuation curve obtained by the soil test are expressed by the equation (3) ) And a mixed model obtained by mixing the modified hyperbolic model represented by formula (4) and the RO model represented by formula (4), and the parameters of the mixed model are automatically set by an approximation method. Sequential nonlinear seismic response analysis method of ground characterized by

Here, τ is stress, γ is shear strain, G ₀ is initial shear stiffness, γ _0.5 is reference strain, γ _m is initial micro strain, and λ is a parameter.

Where α and β are parameters.   2. The method for analyzing a sequential nonlinear seismic response of a ground according to claim 1, wherein the approximation method for setting the parameter [lambda] is a least square method. 2. The ground according to claim 1, wherein the modified hyperbola model represented by formula (3) is obtained by adding a parameter λ and a small strain γ _m to the skeleton curve of the hyperbola model represented by formula (2). The sequential nonlinear seismic response analysis method.

The value of the parameter α of the RO model represented by the equation (4) is obtained by substituting the values obtained from the soil test into G ₀ , γ, τ, τ ₀ , β on the right side of the equation (9). The method according to claim 1, further comprising imparting strain dependence.

The value of the parameter α of the RO model represented by the equation (4) is strain-dependent by obtaining the value by the equation (10) using the display equation (3) of the skeleton curve of the modified hyperbola model in which the parameter is added to the hyperbola model. The method for analyzing a sequential nonlinear seismic response of a ground according to claim 1, wherein the property is imparted.

When the parameter α deviates from the skeleton curve by changing from unloading to unloading or conversely from unloading to loading, the value of the parameter α _a when reaching the past maximum or minimum strain according to the following equation (11) The ground nonlinear sequential earthquake response analysis method according to claim 1, wherein a constant is used.

In order to avoid an underestimation of the attenuation due to the attenuation constant h being 0 at a minute strain in the RO model, in the modification of the RO model, the attenuation constant h is minimized by the following equation (12). The ground nonlinear sequential earthquake response analysis method according to claim 1, wherein the ground is corrected so as to have an attenuation constant h _min .

Here, h _max is the maximum attenuation ratio.   A program for executing the method of ground sequential nonlinear seismic response analysis according to any one of claims 1 to 7 is stored. Storage medium.

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