JP7103046B2 - Ground analysis method based on multidimensional consolidation theory - Google Patents

Description

The present invention relates to a method for analyzing the ground based on the multidimensional consolidation theory.

In various civil engineering works mainly targeting sandy land, seepage fracture may occur, and it is important to avoid this for safe construction. Preventing boiling around the water blocking wall in excavation work, deadlines for seas, rivers, lakes and marshes, and water replacement work is a typical example. Furthermore, even in the deep underground non-excavation work, which has been increasing in recent years, avoidance of infiltration failure in tunnel excavation and widening work has become a technical issue.

For the purpose of preventing such infiltration destruction of the ground, the ground is analyzed and modeled as a finite element, and the combined finite element analysis of the soil skeleton and water is performed in consideration of soil consolidation (Non-Patent Document 1). As a result, it has been conventionally practiced to grasp the behavior of the ground (Non-Patent Documents 2 and 3). The soil skeleton and water coupled finite element analysis will also be described below as soil / water coupled finite element analysis.

M.A.Biot, "General theory of three-dimensional consolidation", Jounal of applied physics, Vol. 12, February 1941 Iizuka A. and Ohta H., "A determination procedure of input parameter in elasto-viscoplastic finite element analysis", Soils and Foundation, Vol.27, No3, pp. 71-87,1987 Shigehiko Sugie: Analysis theory of 3D ground / groundwater coupled analysis program "GRASP-3D" and application to dynamic behavior analysis of clay, Obayashi Technical Research Institute Bulletin No. 51, 1995

In the conventional finite element analysis, when considering the change of the water level above the ground surface, it was necessary to input the influence of the water level change as the increase / decrease of the nodal load on the ground surface at the time of analysis. In this case, when the ground surface has a simple shape such as a flat surface, it is relatively easy to input the nodal load. However, when the shape of the ground surface is complicated, it is very difficult to input the nodal load.

In order to solve the above problems, in one aspect of the present invention, a vector {ΔWp} showing a change in ground stress due to a change in water pressure and a vector {ΔSf} showing a change in physical force due to a change in a hydraulic gradient are used. , Provide an analysis method for obtaining a numerical solution of the behavior of the ground element by giving it to the ground element for each step of the time history and applying the soil / water coupled finite element analysis based on the multidimensional consolidation theory.

It is preferable to model the ground in three dimensions by arranging a plurality of the ground elements in three dimensions and obtain the distribution of the effective stress in the ground as a numerical solution.

It is preferable to predict the location of boiling in the ground by acquiring the ground element that has lost the effective stress.

It is preferable that the above analysis method is a program implemented on a computer.

According to the present invention, since it is not necessary to input a nodal load, soil / water coupled finite element analysis based on the multidimensional consolidation theory can be easily performed without being affected by the ground surface shape, and the behavior of the ground can be grasped. can do.

It is a figure which shows the governing equation and the boundary condition of the consolidation theory of Biot in the analysis method which concerns on embodiment. It is a figure which shows the discretized matrix equation for the finite element analysis of the consolidation theory of Biot. It is a figure which shows (a) the drop of the water level on the ground and the change of the head accompanying it, and (b) the calculation result of the total head and effective stress. It is a figure which shows the photograph which took the state of the 2D model experiment just before the occurrence of boiling from the side. It is a figure which shows the distribution map at the time of boiling occurrence of the vertical effective stress ratio obtained by the reproduction analysis of a two-dimensional model experiment. It is a figure which shows the situation of the 3D model experiment just before the occurrence of boiling, (a) the photograph taken from the upper surface and (b) the photograph taken from the side surface. It is a figure which shows (a) the top surface distribution map at the time of boiling occurrence, and (b) the distribution map in the cross section of FIG. 7bb of the vertical effective stress ratio obtained by the reproduction analysis of the three-dimensional model experiment. Nodal loads to be input in conventional soil / water coupled finite element analysis, (a) an example of nodal loads to be input to flat ground, and (b) input to ground with slopes and step cuts. It is a figure which shows an example of the node load.

The soil / water coupled finite element analysis according to the embodiment of the present invention will be described below with reference to FIGS. 1 to 8.

<Boundary conditions, governing equation>
The soil / water coupled finite element analysis of the present invention is based on Biot's multidimensional consolidation theory (Non-Patent Document 1). The governing equation inside the ground (region: V, boundary: S) based on this theory is shown in FIG. The structure formed by the overlapping of soil particles in the ground is called the soil skeleton or the soil skeleton, and the water that fills the gaps in the soil skeleton is called the interstitial water.

The equations that mainly control the deformation of the earthen skeleton are the equilibrium equation and the constitutive equation (Equations (1) and (2) in FIG. 1). The equations governing the flow of pore water are the continuous conditional equation (Equation (5)) and the Darky law (Equation (6)). The relationship between strain and displacement (Equation (3)) and Terzaghi's principle of effective stress (Equation (4)) are equations that are involved in both the deformation of the soil skeleton and the flow of pore water and correlate with each other. ..

Further, as initial conditions, it is necessary to give the effective stress equation (11) and the total head (12) equation before loading (t = 0). Under these boundaries and initial conditions, the displacement / stress of the ground, the distribution of the total head / flow velocity of groundwater, and their temporal changes can be obtained by solving the governing equations in a series while following the time steps.

<Discretization>
The governing equation was discretized for use in finite element analysis. The simultaneous matrix equations obtained as a result are shown in Eqs. 2 and (13). The same method as in Non-Patent Documents 2 and 3 was used for discretizing the governing equation. More specifically, the weak form was performed by integrating the equilibrium equation and the continuous conditional equation (Equations (1) and (5)) in the region V and deriving the virtual work equation. In the process of weak formalization, the boundary conditional equations (Equations (7) and (9)) and the relationship between strain and displacement (Equation (3)) and the principle of effective stress (Equation (4)) are balanced. Substituted in the equation and the continuous conditional equation.

Furthermore, for the weak form, the constitutive equation ((2) equation) and the Darky law ((6) equation) are given to express the material properties, and then the temporal change of the total head is expressed by the receding difference. The simultaneous matrix equations of synthetic analysis (Equations 2 and 13) were derived.

By taking into consideration not only the governing equation of the soil skeleton but also the continuous conditional equation ((5) equation) and the Darky law ((6) equation) of the interstitial water as described above, the ground behavior that is greatly influenced by the groundwater The peculiarity is expressed.

<Addition of vectors>
In this embodiment, the vectors {ΔWp} (equation (14)) and { ΔSf} (Equation (15)) is added to the above-mentioned matrix equation (Equations 2 and 13).

{ΔWp} is for adjusting the total stress of the ground when the water level fluctuates on the ground surface, and is equal to ΔPw according to the amount of water pressure change ΔPw in the ground caused by the water level drop (or rise). It has the function of keeping the effective stress constant by releasing the amount of stress from the ground element (or loading it on the ground element).

{ΔSf} has a function of giving water permeability (body force) to each element when a change ΔI of the hydraulic gradient occurs in the ground. Biot's consolidation theory uses the Darky law and the principle of effective stress, and the effect of hydraulic conductivity has already been considered. If only {ΔSf} is added to this, the water permeability will be given in duplicate. However, in the present embodiment, by using {ΔSf} and {ΔWp} in combination, the effects of the water permeability can be avoided from overlapping, and the fluctuation of the ground water level (change from A to B) as shown in FIG. 3A can be avoided. ) And the process of the head change in the ground (change from B to C or D) can be calculated continuously. In the figures after FIG. 3 and the following description, the compression side is shown as positive and the decompression side is shown as negative according to the definition usually used in geotechnical engineering.

The calculation results of the effective stress under these hydraulic conditions are shown in FIG. 3 (b). In the figure, the calculation results corresponding to B, C, and D in FIG. 3A are shown in each graph. As can be seen from Graph B, the situation where the effective stress is kept constant when the ground water level fluctuates, and as can be seen from Graph C or D, the upward (or downward) hydraulic gradient due to the head change at the bottom of the ground. The situation of decrease (or increase) of effective stress due to generation is obtained.

<Comparison between 2D model experiment and reproduction analysis>
[Two-dimensional model experiment]
In order to verify the reproducibility of the above analysis method, a two-dimensional model experiment assuming deadline / water change work and its reproducibility analysis were performed, and the results were compared. Details are shown below.

FIG. 4 shows the situation immediately before the occurrence of boiling in the two-dimensional model experiment. In the model experiment, No. 6 silica sand was dropped in the air into a rectangular cuboid-shaped water tank 1 to provide a sand ground 2 having a thickness of 15 cm. Further, the water blocking wall 3 was installed in the central portion of the water tank 1 with a depth of rooting 5 cm with respect to the sand ground 2. After that, the water tank 1 was filled with water at a slow speed so as not to disturb the sand ground 2, and the initial state of the experiment was set. Based on the weight of the silica sand put in and the volume of the sand ground 2, the porosity ratio is expected to be 0.55, and the unit volume weight in water is expected to be about 1.06 g / cm ³ .

In the experiment, the water was drained from the drain hole provided on the wall surface of the water tank 1, the water level on the right side (inside) of the water stop wall 3 in FIG. 4 was lowered, and the water level on the left side (outside) of the water stop wall 3 was kept constant. Held in. In this way, a head difference ΔH was generated on the left and right sides of the water blocking wall 3, and this was continued until boiling occurred. The rate of decrease in water level on the right side of the water stop wall 3, that is, the rate of increase in head difference was set to a little less than 1 cm per minute. Boiling occurred with a head difference of ΔH = 15.3 cm.

[Reproduction analysis of 2D model experiment]
The two-dimensional model experiment was reproduced using the above analysis method. For the ground element, an isoparametric element of the Serendipity group having a first-order interpolation function was used. Gauss's quadrature method was used for numerical integration. In addition, the ground element was an elasto-plastic model, and the fracture standard of Drucker Prager was used to determine the fracture of the ground element. The ground constant value was set based on the results of the triaxial CD test. For the elements that were sheared, the rigidity was reduced according to the related flow rule. For the ground element in which the effective stress disappeared, the deformation coefficient was reduced to 1/10000. By giving the head difference ΔH to the analysis model as a time history, the behavior of the ground until the occurrence of boiling was analyzed.

The contour of the local safety factor (σ _v ′ / σ _vo ′) at the time of occurrence of boiling (head difference ΔH = 15 cm) obtained by the reproduction analysis is shown in FIG. Here, σ _v ′ is the value of the vertical effective stress after the change, and σ _v ′ is the initial value of the same stress. The time when the liquefaction area where the effective stress disappeared and the local safety factor became a negative value reached the ground surface was defined as the occurrence of boiling. As shown in FIG. 5, the local safety factor is reduced due to the upward water flow (hydraulic gradient) at each position on the right side of the water blocking wall 3.

At ΔH = 15 cm, the liquefaction region extended to the surface of the earth, and the timing and situation of boiling in the experiment matched well.

<Comparison between 3D model experiment and reproduction analysis>
In addition, a three-dimensional model experiment assuming deadline and water change work and its reproduction analysis were also performed and compared. Details are shown below.

[3D model experiment]
As shown in FIG. 6A, a water tank 4 was prepared in the three-dimensional model experiment. No. 6 silica sand was dropped in the air inside the water tank 4 to provide a sand ground 5 having a thickness of 15 cm. Further, a water blocking wall 6 bent at a right angle when viewed from above was installed to surround the corner of the water tank 4. The penetration depth of the water blocking wall 6 was set to 5 cm. After that, the water tank 4 was slowly filled with water to bring it into the initial state of the experiment. In the following description, the rectangular region in the top view, which is surrounded by the water blocking wall 6 and the wall surface of the water tank 4, is referred to as the inside, and the other regions are referred to as the outside. Based on the weight of the silica sand put in and the volume of the sand ground 5, the porosity ratio is expected to be 0.55, and the unit volume weight in water is expected to be about 1.06 g / cm ³ .

In the experiment, drainage was performed from a drain hole provided on the wall surface of the water tank 4, the water level in the inner region was lowered, and the water level in the outer region was kept constant. In this way, a head difference ΔH was generated inside and outside the water blocking wall 4, and this was continued until boiling occurred. The rate of decrease in water level, that is, the rate of increase in head difference ΔH was set to a little less than 1 cm per minute. The situation in which boiling occurred is shown in FIG. 6 (b). Boiling occurred with a head difference of ΔH = 12.5 cm. Compared with the results of the two-dimensional model experiment, boiling occurred with a small head difference.

[Reproduction analysis of 3D model experiment]
The 3D model experiment was reproduced using the above analysis method. The ground was analyzed and modeled using three-dimensional elements. For the ground element, an isoparametric element of the Serendipity group having a first-order interpolation function was used. Gauss's quadrature method was used for numerical integration. In addition, the ground element was an elasto-plastic model, and the fracture standard of Drucker Prager was used to determine the fracture of the ground element. The ground constant value was set based on the results of the triaxial CD test. For the elements that were sheared, the rigidity was reduced according to the related flow rule. For the ground element in which the effective stress disappeared, the deformation coefficient was reduced to 1/10000. Similar to the experiment, the behavior of the ground until the occurrence of boiling was analyzed by giving the time history of the head difference ΔH to the analysis model.

FIG. 7 shows the distribution of the local safety factor when boiling occurs in the reproduction analysis. In the calculation step where the water level difference ΔH = 11 cm, the portion where the effective stress disappeared and the local safety factor became a negative value, that is, the liquefaction region reached the ground surface and boiling occurred. Similar to the experiment, the occurrence of boiling is concentrated at the corners of the water blocking wall 6.

In the above two-dimensional model experiment and reproduction analysis, the head difference ΔH = 15 cm at the time of boiling is obtained. On the other hand, the head difference ΔH = 12.5 cm at the time of boiling in the three-dimensional field model experiment is smaller than this. In each reproduction analysis, as in the experiment, the three-dimensional analysis tended to have a smaller head difference when boiling occurred than the two-dimensional analysis.

Since the plane shape of the deadline / water change work is generally rectangular, it is necessary to pay attention to the distribution and magnitude of the hydrodynamic gradient that occurs in the three-dimensional field when designing and planning anti-boiling measures. It can be seen from the result of the analysis.

<Comparison and effect with conventional products>
As shown in FIG. 8 (a), in the finite element analysis method according to the prior art, in order to step-calculate the changes in the dynamics and hydraulic state of the ground during the drainage process, the nodes of the ground surface of the construction area are set for each step. It was necessary to enter the hydraulic boundary condition (total head value).

In addition, in order to reduce the total stress of the ground due to the decrease in water level, it was necessary to input stress boundary conditions (load values) at the nodes of the ground surface of the construction area for each step. Specifically, a node load equivalent to the fluctuation of water pressure was applied to the surface node. This input process does not require much labor when the ground is flat and the geometric conditions are simple.

However, in the case of a three-dimensional analysis model that simulates an actual construction site, the situation is different. Since the load is a vector quantity, as shown in FIG. 8B, especially in the case of ground having a slope or stepped ground, the applied nodal load has a three-dimensional directional component, and the load It took a lot of time and effort to calculate and input the conditions, and it was difficult to quickly apply the soil / water coupled finite element analysis to practice.

In the present invention, since the vectors {ΔSf} and {ΔWp} are added to the board elements, it is not necessary to calculate the nodal load while considering the normal direction of the ground surface as in the conventional case. Therefore, it is possible to easily perform a three-dimensional soil / water coupled finite element analysis, and it is possible to obtain a numerical solution even for a ground having a complicated shape assuming an actual construction project.

The vector {ΔWp} is represented by using the vector {ΔPw} of the water pressure fluctuation as shown in the equation (14) of FIG. Further, the vector {ΔSf} is represented by using the vector {ΔI} for the fluctuation of the hydraulic gradient. As described above, the vector corresponding to the head change is derived from the water pressure and the hydraulic gradient, and can be easily calculated. Therefore, the analysis method in the present embodiment can be executed quickly.

In the analysis method of the present invention, it is possible to obtain a numerical solution not only for a two-dimensional shape but also for a three-dimensional geometric shape. Therefore, it is possible to give an accurate numerical solution even under conditions that are difficult to predict accurately by two-dimensional modeling, such as the occurrence of boiling at the corner of the enclosure due to the water blocking wall. That is, the analysis method according to the present invention can quickly provide highly accurate predictions for a large number of construction projects that require consideration of complicated geometric shapes.

<Modification example>
The selection of the elements used in the analysis model is not limited to the above embodiment, and can be arbitrarily set according to the analysis conditions. For example, an arbitrary order can be applied to the interpolation function, and the arrangement of integration points and the method of numerical integration can be appropriately selected.

In the above embodiment, the backward difference is used for the time difference approximation, but the present invention is not limited to this. An arbitrary method such as a forward difference can be adopted instead of the backward difference.

The destruction criteria used in the analysis are not limited to the Drucker Prager, and can be arbitrarily set according to the purpose and conditions such as Mohr-Coulomb and von Mises.

If the analysis method of the present invention is configured as a program that can function on a computer and is made to function on the computer, it will be automatically, easily, and quickly executed.

Although the embodiment for carrying out the present invention has been described above, the above-described embodiment is for facilitating the understanding of the present invention, and is not for limiting the interpretation of the present invention. Further, the present invention can be modified or improved without departing from the spirit thereof, and the present invention also includes an equivalent thereof.

Aquarium 1, 4
Sand ground 2, 5
Water stop walls 3, 6

Claims (4)

Vector {ΔWp} showing changes in ground stress due to fluctuations in water pressure,
A vector {ΔSf} indicating the change in body force due to the change in the hydraulic gradient is given to the ground element at each step of the time history.
This is an analysis method for obtaining a numerical solution of the behavior of the ground element by applying a soil / water coupled finite element analysis based on the multidimensional consolidation theory.
γ _w : Unit volume weight of water {ΔI}: Fluctuation vector of hydraulic gradient {ΔP w}: Fluctuation vector of water pressure
N: Shape function
B: Strain-Displacement Matrix
V: Area of ground element
m: As individual ground elements,
The {ΔWp} and the {ΔSf} are

An analysis method characterized by being expressed as. By arranging a plurality of the ground elements in three dimensions, the ground is modeled in three dimensions.
The analysis method according to claim 1, wherein the distribution of effective stress in the ground is obtained as a numerical solution. The analysis method according to claim 1 or 2, wherein the location where boiling occurs in the ground is predicted by acquiring the ground element from which the effective stress has disappeared. A program for causing the analysis method according to any one of claims 1 to 3 to function on a computer.

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