JP7702862B2 - Principal stress calculation program and principal stress calculation method - Google Patents

Description

This invention relates to technology related to the principal stress of rock masses that exhibit anisotropic deformation characteristics, and more specifically, to technology for determining the two-dimensional initial principal stress of rock masses by performing analysis based on anisotropic elastic theory on the results obtained from borehole loading tests using a constant displacement loading method.

When planning the design of a structure to be built on or within rock, it is extremely important to understand the mechanical properties of the rock. Examples of the mechanical properties of rock include the stress state and deformation coefficient of the rock. Of these, the stress state of the rock is essential information for the design of new tunnels or deep underground cavities, or for the design of reinforcing existing tunnels that are showing more deformation than expected or old tunnels that have deteriorated significantly over time.

To grasp the stress state of rock, it is common to carry out in situ testing, i.e., to actually measure stress on-site. In particular, these methods can be broadly divided into the "hydraulic fracturing method" shown in Patent Document 1 and the "stress release method" shown in Patent Documents 2 and 3. The applicant of the present application has also disclosed in Patent Document 4 a suitable invention that can obtain the initial rock stress based solely on the results obtained from a borehole jack test.

Japanese Unexamined Patent Publication No. 62-050591 Japanese Unexamined Patent Publication No. 62-220823 JP 2005-69937 A JP 2021-60295 A

The hydraulic fracturing method shown in Patent Document 1 involves plugging a specified section of a borehole with a packer, applying water pressure to this section to generate artificial cracks, and determining the stress in the rock mass from the relationship between water pressure and the cracks. Therefore, hydraulic fracturing requires the preparation of a variety of equipment, such as packers, high-pressure pumps, water tanks, and cable winches, meaning that the entire system is large-scale.

On the other hand, the stress relief method shown in Patent Documents 2 and 3 involves fixing a rock pressure detector (or a specially processed strain gauge) in a borehole with cement milk or adhesive, then overcoring, and measuring the released strain continuously for a certain period of time before and after overcoring to determine the stress in the rock. Therefore, to carry out the stress relief method, it is necessary to construct an overcoring, and it is also necessary to fix the rock pressure detector or special strain gauge in the borehole using grout or the like.

The method shown in Patent Document 2 is a stress relief method, specifically called the "Denchuken Buried Strain Method," which is considered unsuitable for cases where there is a lot of spring water, because it is difficult to fix the rock pressure detector with grout. In addition, it requires overcoring with a large hole diameter (usually about φ200 mm), which requires a lot of labor and time, and in order to determine the sensitivity coefficient of each strain gauge to be used, indoor tests must be conducted using the cores collected by overcoring. The method shown in Patent Document 3 is a stress relief method, specifically called the "cone hole bottom strain method," which is considered unsuitable for cases where there is a small amount of spring water or when the target is porous soft rock, because it is difficult to glue and fix the strain gauge, and indoor tests must also be conducted to determine the deformation coefficient and Poisson's ratio.

Incidentally, since stress distribution within rock generally varies significantly, it is considered desirable to measure at multiple points rather than just one point (one borehole). However, the method of Patent Document 1 requires a large-scale device, which requires time for advance preparation, and the method of Patent Document 2 requires a great deal of time and effort. Furthermore, the method of Patent Document 3 is subject to the above-mentioned constraints on strain gauge installation. Thus, it is not realistic to measure at many points (boreholes) using conventional methods, as it requires the construction of overcoring and grout fixing work.

In this regard, the invention disclosed in Patent Document 4 requires only a normal borehole jack test, meaning that the test can be performed easily and at low cost, making it possible to easily carry out measurements at multiple points. However, this invention is an analysis technique that assumes an isotropic rock mass, and does not produce sufficient results when applied to anisotropic rock masses.

The objective of the present invention is to solve the problems of the conventional technology, that is, to provide a principal stress calculation program that can relatively easily determine the principal stress of anisotropic rock mass from the results of in-situ tests, and a principal stress calculation method using the program.

The present invention was made by focusing on the fact that since there is a linear relationship between the initial loading pressure _p0 in an in-situ test and the principal stresses (maximum principal stress σ ₁ ^∞ and minimum principal stress σ ₂ ^∞ ), it is possible to convert the complex implicit function used to calculate the initial loading pressure _p0 into a relatively simple explicit function, and is an invention based on an unprecedented idea.

The program causes a computer to execute a function of calculating the maximum principal stress σ 1 ∞ , minimum principal stress σ 2 ∞ , and action angle α of the maximum principal stress σ 1 ∞ of the rock mass based on the first deformation coefficient E ₁ , the second deformation coefficient E ₂ , and the initial loading pressure p ₀ obtained by performing N ( _N is a natural number of ₃ ^{or more)} _loading ^tests in which the wall surface of a borehole in an anisotropic rock ^mass is pressurized by a uniform displacement loading method while changing the loading direction, and has the functions of causing the computer to execute a condition value import process, an action angle setting process, a provisional principal stress calculation process, a sensitivity coefficient calculation process, a sum of squared differences calculation process, an action angle determination process, and a principal stress determination process. Note that the first deformation coefficient E ₁ is the deformation coefficient in the first axis direction of three orthogonal axes (the first axis, the second axis, and the third axis), and the second deformation coefficient E ₂ is the deformation coefficient in the second axis direction. Of these, the condition value import process is a process that imports the test hole radius r and loading surface curvature β in the load test, the first Poisson's ratio _v12 (in a plane including the first and second axes), the second Poisson's ratio _v13 (in a plane including the first and third axes), the first deformation coefficient _E1 , the second deformation coefficient _E2 , and the initial loading pressure _p0 . The duration angle setting process is a process that sets a plurality of candidate duration angles α(i) that are candidates for the duration angle α, and the sensitivity coefficient calculation process is a process that calculates the first sensitivity coefficient _Ω1 and the second sensitivity coefficient _Ω2 given by the following equation for each candidate duration angle α(i).
p ₀ = Ω ₁ ×σ ₁ ^∞ (i) + Ω ₂ ×σ ₂ ^∞ (i)
The provisional principal stress calculation process is a process for calculating a provisional maximum principal stress σ ₁ ^∞ (i) and a provisional minimum principal stress σ ₂ ^∞ (i) for each candidate operating angle α(i). The difference squared sum calculation process is a process for calculating, for each candidate operating angle α(i), a squared difference s given by the following equation for each of N tests, and for calculating a sum of squared differences for the N tests.
s=[p ₀ -Ω ₁ ×σ ₁ ^∞ (i) + Ω ₂ ×σ ₂ ^∞ (i)] ²
The duration angle determination process selects the smallest sum squared difference from the sum squared differences obtained for each candidate duration angle α(i), and determines the candidate duration angle α(i) associated with the smallest sum squared difference as the duration angle α. The principal stress determination process determines the provisional maximum principal stress σ ₁ ^∞ (i) and provisional minimum principal stress σ ₂ ^∞ (i) associated with the duration angle α determined in the duration angle determination process as the maximum principal stress σ ₁ ^∞ and minimum principal stress σ ₂ ^∞ , respectively.

The principal stress calculation program of the present invention may also be configured so that the provisional principal stress calculation process calculates provisional maximum principal stress σ ₁ ^∞ (i) and provisional minimum principal stress σ ₂ ^∞ (i) by the least squares method using the following equations.

The principal stress calculation method of the present invention is a method for determining the maximum principal stress σ ₁ ^∞ , minimum principal stress σ ₂ ^∞ , and action angle α of the maximum principal stress σ ₁ ^∞ of an anisotropic rock mass by performing a loading test (a test in which pressure is applied to the borehole wall surface using a constant displacement loading method), and is a method including a loading test process and an analysis process. In the loading test process, N (N is a natural number of 3 or more) load tests are performed on the rock mass while changing the loading direction, and in the analysis process, the principal stress calculation program of the present invention is used to determine the maximum principal stress σ ₁ ^∞ , minimum principal stress σ ₂ ^∞ , and action angle α of the principal stress of the rock mass based on the results of the N tests.

The principal stress calculation program and the principal stress calculation method of the present invention have the following effects.
(1) The principal stress of anisotropic rock mass can be determined relatively easily based on the results of in-situ tests.
(2) By taking anisotropy into account, the variation in the measurement results of the principal stress can be reduced.
(3) When conducting numerical analysis of cavities, structures, etc. within rock mass, it is possible to adopt input values that take into account the anisotropy of the principal stress of the rock mass, enabling more accurate evaluation of cavities, structures, etc. than was previously possible.

(a) is a vertical cross-sectional view illustrating the borehole jack test, and (b) is a horizontal cross-sectional view illustrating the borehole jack test. (a) is a horizontal cross-sectional view showing the maximum and minimum principal stresses when loaded at a loading direction of 0°, and (b) is a horizontal cross-sectional view showing the maximum and minimum principal stresses when loaded at a loading direction of -45°. FIG. 2A is a perspective view showing the deformation coefficients in three axes of an anisotropic rock mass whose layer direction is a horizontal plane, and FIG. 2B is a perspective view showing the deformation coefficients in three axes of an anisotropic rock mass that is inclined with respect to the horizontal plane. Graph showing the relationship between initial loading pressure and subgrade reaction coefficient. 1 is a mathematical diagram for explaining Lekhnitskii's theory of anisotropic elasticity. A mathematical diagram to explain the theory in Kamata's paper. FIG. 11 is a formula diagram for explaining various formulas for deriving an implicit function. A mathematical diagram to explain implicit and explicit functions. FIG. 4 is a flowchart showing a main process flow of a principal stress calculation program according to the present invention. FIG. 1A is a data diagram showing the loading directions of four borehole jack tests, FIG. 1B is an analysis result diagram showing the analysis results using the principal stress calculation program of the present invention, and FIG. 1C is a graph showing the change in the sum squared difference with the change in the action angle. FIG. 11 is an analysis result diagram showing the change in the sum square difference with the change in the operating angle. FIG. 2 is a flow chart showing main steps of the deformation coefficient calculation method of the present invention.

An example of an embodiment of the principal stress calculation program and the principal stress calculation method of the present invention will be described with reference to the figures.

1. Definitions Before describing the embodiments of the principal stress calculation program and the principal stress calculation method of the present invention, the definitions of terms used herein will be given.

(Borehole jack test)
One of the technical features of the present invention is to use the results of an in-situ test to determine the principal stress, which is a mechanical property of rock. For this in-situ test, a load test in which pressure is applied to the borehole wall using a constant displacement loading method may be adopted. For convenience, this load test method will be referred to as the "borehole jack test." The method of the borehole jack test in the present invention will be described below.

Figure 1 is a diagram explaining the borehole jack test, where (a) is a vertical cross-sectional view, and (b) is a horizontal cross-sectional view of the borehole BH. As shown in this figure, when performing the borehole jack test, a loading device (hereinafter referred to as the "borehole jack JB") is placed at a predetermined position in the borehole BH that has been constructed in advance. This borehole jack JB is composed of a piston jack and a loading plate LB, and the piston jack operated by hydraulic pressure presses the loading plate LB toward the borehole wall, and a pressure gauge and a displacement gauge are built into the borehole jack JB. The hydraulic pressure is transmitted from a pressure source CP installed on the ground through a hose HS. The data logger DL records the pressure (hydraulic pressure) and displacement values through a communication cable SC connected to the borehole jack JB. As shown in Figure 1(b), the loading plate LB has a predetermined width, that is, the borehole wall can be pressurized by the width dimension of the loading plate LB. For convenience, we refer to half the central angle extending from the center of the borehole jack JB to both ends of the loading plate LB as the "loading surface curvature β," and the radius of the borehole BH (test hole) as the "test hole radius r" (Figure 1 (b)).

The borehole jack test in the present invention is performed N times (N is a natural number of 3 or more) while changing the loading direction, and the test results of N times are used for analysis. Like FIG. 1(b), FIG. 2 is a cross-sectional view of the borehole BH cut in the horizontal direction, and shows the relationship between the loading direction θ and the direction in which the maximum principal stress σ ₁ ^∞ (or the minimum principal stress σ ₂ ^∞ ) acts (hereinafter referred to as the "working angle α"). (a) shows the maximum principal stress σ ₁ ^∞ and the minimum principal stress σ ₂ ^∞ when the loading direction θ is 0°, and (b) shows the maximum principal stress σ ₁ ^∞ and the minimum principal stress σ ₂ ^∞ when the loading direction θ is -45°. Here, the loading direction θ is the direction in which the loading plate LB presses the hole wall, and is the rotation angle in which the positive direction of the X axis is 0° and the counterclockwise direction is positive when two orthogonal axes consisting of the X axis and the Y axis are set on a horizontal plane. Of course, the X-axis-Y-axis can be set arbitrarily, the 0° rotation angle direction can be set arbitrarily, for example, to the positive direction of the Y-axis, and the positive direction of the rotation angle can be set to clockwise instead of counterclockwise. Note that, although this figure shows an example of two borehole jack tests with different loading directions, as described above, in the present invention, it is necessary to perform at least three (e.g., four) borehole jack tests while changing the loading direction θ.

(First deformation coefficient and second deformation coefficient)
The present invention is to obtain the principal stress of "anisotropic rock mass" that shows anisotropy in deformation characteristics, such as sedimentary rock with well-developed bedding, metamorphic rock with well-developed schistosity, and igneous rock with well-developed joints. Figure 3 is a diagram showing the deformation coefficients in the three axial directions of this anisotropic rock mass, where (a) shows a case where the layer direction (direction of bedding, schistosity, and joints) of the anisotropic rock mass is a horizontal plane (in this case, a plane including the x-axis and z-axis), and (b) shows a case where the layer direction of the anisotropic rock mass is inclined to the horizontal plane.

As shown in Fig. 3, an anisotropic rock mass has a deformation coefficient in each of three orthogonal axial directions set along the strike and inclination of the rock mass, and for convenience, the three orthogonal axial directions set in the rock mass are referred to as the "first axis", the "second axis", and the "third axis", respectively, and the deformation coefficient in the first axis direction is referred to as the "first deformation coefficient _E1 ", the deformation coefficient in the second axis direction is referred to as the "second deformation coefficient _E2 ", and the deformation coefficient in the third axis direction is referred to as the "third deformation coefficient _E3 ". For example, in the case of Fig. 3(a), since the layer direction of the anisotropic rock mass is a horizontal plane, the first deformation coefficient _E1 can be set in the x-axis direction, the second deformation coefficient _E2 in the y-axis direction, and the third deformation coefficient _E3 in the z-axis direction. On the other hand, in the case of Figure 3 (b), since the layer direction of the anisotropic rock mass is inclined at a predetermined angle (hereinafter referred to as the "inclination angle φ") from the horizontal plane, the first deformation coefficient E1 can be set in a direction inclined at the inclination angle φ from the x-axis, the second deformation coefficient _E2 can be set in a direction inclined at the inclination angle _φ from the y-axis, and the third deformation coefficient _E3 can be set in the z-axis direction.

Usually, in anisotropic rock mass, the two deformation coefficients parallel to the plane of each layer (in this case, the first deformation coefficient _E1 and the third deformation coefficient _E3 in the z-axis direction) can be treated as being approximately equal in magnitude, but differing in direction. On the other hand, the deformation coefficient perpendicular to the plane of each layer (in this case, the second deformation coefficient _E2 ) is significantly different in magnitude from the deformation coefficients in the other two directions (in this case, the first deformation coefficient _E1 and the third deformation coefficient _E3 in the z-axis direction). Therefore, for convenience, an example using the first deformation coefficient _E1 , the second deformation coefficient _E2 , and the inclination angle φ will be described here.

(Poisson's ratio and shear modulus of elasticity)
Since anisotropic rock mass has a Poisson's ratio in each of three orthogonal planes, for convenience, the Poisson's ratio in the plane including the first axis (x-axis in FIG. 3) and the second axis (y-axis in FIG. 3) is referred to as the "first Poisson's ratio v ₁₂ ", the Poisson's ratio in the plane including the first axis (x-axis in FIG. 3) and the third axis (z-axis in FIG. 3) is referred to as the "second Poisson's ratio v ₁₃ ", and the Poisson's ratio in the plane including the second axis (y-axis in FIG. 3) and the third axis (z-axis in FIG. 3) is referred to as the "third Poisson's ratio v ₂₃ ". In addition, the shear modulus of the rock mass in the plane including the first axis and the second axis is referred to as the "first shear modulus G ₁₂ ", and the shear modulus of the rock mass in the plane including the first axis and the third axis is referred to as the "second shear modulus G ₁₃ ". For the reasons mentioned above, the shear modulus of the rock mass in the plane including the second axis and the third axis (third shear modulus G ₂₃ ) can be treated as being substantially equal to the first shear modulus G ₁₂ .

(Initial loading pressure and subgrade reaction coefficient)
When a borehole jack test is performed, the loading pressure and the amount of displacement of the borehole wall can be plotted on a graph with two orthogonal axes, and the initial loading pressure _p0 can be obtained as shown in Figure 4. Furthermore, the gradient of the straight line showing the relationship between the loading pressure and the amount of displacement (i.e., the ratio of the loading pressure increment to the displacement increment) can be obtained as the "subgrade reaction coefficient K".

2. Principal stress calculation program The principal stress calculation program of the present invention will be described in detail with reference to the drawings. Note that the principal stress calculation method of the present invention is a method for calculating the principal stress of a rock mass using the principal stress calculation program of the present invention, so the principal stress calculation program of the present invention will be described first, and then the principal stress calculation method of the present invention will be described.

The principal stress calculation program of the present invention uses "Lekhnitskii's theory of anisotropic elasticity (S.G. Lekhnitskii: Theory of Elasticity of an Anisotropic Body, Mir Publishers)" to theoretically treat anisotropic rock masses, and also derives various numerical values using "theory in Kamada's paper (Takeshi Kamada: Two-dimensional mixed boundary value problem for an anisotropic infinite plate with an elliptical hole (Transactions of the Japan Society of Mechanical Engineers))". Therefore, we will briefly explain "Lekhnitskii's theory of anisotropic elasticity" and "theory in Kamada's paper".

(Lekhnitskii's theory of anisotropic elasticity)
5 is a mathematical diagram for explaining Lekhnitskii's theory of anisotropic elasticity. Among these, mathematical formula (1) is a constitutive equation showing the relationship between strain and stress in an anisotropic elastic body. In the formula, σ _x is the normal stress in the x-axis direction, σ _y is the normal stress in the y-axis direction, σ _z is the normal stress in the z-axis direction, τ _xy is the shear stress in the plane including the x-axis and y-axis, ε _x is the normal strain in the x-axis direction, ε _y is the normal strain in the y-axis direction, and γ _xy is the shear strain in the plane including the x-axis and y-axis. In the constitutive equation, the coefficients of each stress component (b ₁₁ to b ₆₆ ) are referred to as "elastic compliance".

The elastic compliance (b ₁₁ to b ₆₆ ) can be expressed by the elastic compliance (a ₁₁ to a ₆₆ ) as shown in the formula (4) of Fig. 5. The elastic compliance (a ₁₁ to a ₆₆ ) can be expressed by the first Poisson's ratio v ₁₂ , the second Poisson's ratio v ₁₃ , the third Poisson's ratio v ₂₃ , the first shear modulus G ₁₂ , the first deformation coefficient E ₁ , the second deformation coefficient E ₂ , and the inclination angle φ as shown in the formula (2) of Fig. 5. The first shear modulus G ₁₂ can be expressed by the first Poisson's ratio v ₁₂ , the first deformation coefficient E ₁ , and the second deformation coefficient E ₂ , and the second shear modulus G ₁₃ can be expressed by the second Poisson's ratio v ₁₃ and the first deformation coefficient E ₁ . Since the elastic compliance is symmetric, a _ij =a _ji and b _ij =b _ji hold.

(Theory of Kamata's paper)
Fig. 6 is a formula diagram for explaining the theory of Kamada's paper. Among them, formula (5) is a fourth-order algebraic equation for finding characteristic values μ _j (j=1, 2) for quantitatively treating anisotropy. In the theory of Kamada's paper, characteristic values μ _j (j=1, 2) are defined as two of the four solutions of formula (5) that satisfy |μ _j |<1. Note that A ₁ to A ₄ in the formula are complex constants expressed by formula (6) in Fig. 6, the bar attached to the letters indicates the conjugate, and i indicates the imaginary unit.

When complex constants δ _j and ρ _j (j=1, 2) are defined by equation (7) in Fig. 6 and P, Q, and R are defined by equation (8) in Fig. 6, a solution λ _j to the quadratic equation shown in equation (9) in Fig. 6 is obtained. Then, a real constant κ _j (j=1, 2) and a real number κ are set so as to satisfy equation (10) in Fig. 6. Furthermore, when this real number κ is used, γ, γ _j , and γ _j ' can be defined as in equation (11) in Fig. 6.

(Implicit and Explicit Functions)
The maximum principal stress σ ₁ ^∞ , minimum principal stress σ ₂ ^∞ , and action angle α can be found from equation (12) in Fig. 7 using the various numerical values explained so far (hereinafter, for convenience, referred to as "Kamada constants") and complex constants B ₁ and B ₂ (the bars indicate conjugates). In other words, if the maximum principal stress σ ₁ ^∞ , minimum principal stress σ ₂ ^∞ , and action angle α are known, the complex constants B ₁ and B ₂ can be found from the Kamada constants and equation (12). Then, equation (16) in Fig. 7 can be derived from equations (13) to (15) in Fig. 7 using the Kamada constants and complex constants _{B 1} and B ₂ . In addition, _r0 in formula (13) is the test hole radius r, β in formulas (14) to (16) is the loading surface curvature β, θ in formula (15) and σ in formula (16) are integral constants, and Λ' in formula (16) is the loading range shown in Figure 1 (b) (4β, unit is radian).

When formula (16) is obtained, the initial loading pressure p ₀ can be expressed by formula (17) in FIG. 8. In formula (16), r ₀ is the test hole radius r, β is the loading surface curvature β, Re [ ] is the real part in the brackets, and Im [ ] is the imaginary part in the brackets. Here, the right side of the second formula of formula (17) (i.e., X) is an implicit function for the first deformation coefficient E ₁ and the second deformation coefficient E ₂ , the inclination angle φ, the action angle α, and the maximum principal stress σ ₁ ^∞ and the minimum principal stress σ ₂ ^∞ , and since the initial loading pressure p ₀ is linearly related to the initial principal stress (maximum principal stress σ ₁ ^∞ , minimum principal stress σ ₂ ^∞ ), formula (17) can be rewritten in the explicit form (hereinafter referred to as "explicit function") shown in formula (18) in FIG. 8. In addition, _Ω1 and _Ω2 in formula (18) mean the initial loading pressure _p0 when the unit maximum principal stress _σ1∞ ^{and the} unit minimum principal stress _σ2∞ act, and for convenience, _Ω1 and _Ω2 are referred to as sensitivity coefficients here. These sensitivity coefficients _Ω1 and _Ω2 are calculated using formula (19) in Figure 8 in formulas (17) and (18).

(Initial loading pressure)
When N borehole jack tests are performed, the initial principal stresses (maximum principal stress σ ₁ ^∞ , minimum principal stress σ ₂ ^∞ ) can be calculated by equation (20) in Fig. 8 using sensitivity coefficients Ω ₁ and Ω ₂ and initial loading pressure p _0. More specifically, the initial principal stresses (maximum principal stress σ ₁ ^∞ , minimum principal stress σ ₂ ^∞ ) can be calculated by solving equation (20) using the least squares method. Note that equation (18) shows the sensitivity coefficients Ω ₁ and Ω ₂ and initial loading pressure p ₀ for each borehole jack test, and for convenience, the number of tests is represented by i (i = 1 to N) shown in parentheses at the top.

The initial principal stresses (maximum principal stress σ ₁ ^∞ , minimum principal stress σ ₂ ^∞ ) obtained by formula (20) are, so to speak, analytical provisional values for each borehole jack test, and are not the initial principal stresses that are finally determined. Therefore, these analytical maximum principal stress σ ₁ ^∞ and minimum principal stress σ ₂ ^∞ are referred to as the "provisional maximum principal stress" and the "provisional minimum principal stress", respectively. In addition, once the provisional maximum principal stress and the provisional minimum principal stress are obtained, the analytical initial loading pressure p ₀ (hereinafter referred to as the "analytical initial loading pressure p _0c ") for each borehole jack test can be calculated using formula (22) in FIG. 8 . Furthermore, the square of the difference (hereinafter referred to as "square difference _S ") between the initial loading pressure _p0 actually obtained for each test in the borehole jack test (hereinafter referred to as "actual initial loading pressure p0" for convenience) and the analytical initial loading pressure _p0c can be calculated, and the sum of the square differences S for N borehole jack tests (hereinafter referred to as "sum square difference ΣS") can be calculated using equation (21) in Figure 8.

(Determination of initial principal stress)
As described above, the formula (18) in FIG. 8 is an explicit function with the duration angle α as a variable. Therefore, the values of the sensitivity coefficients Ω ₁ and Ω ₂ change depending on the duration angle α, and the values of the provisional maximum principal stress, provisional minimum principal stress, analytical initial loading pressure p _0c , and sum square difference ΣS also change. Therefore, the sum square difference ΣS is calculated while changing the value of the duration angle α, and the sum square difference ΣS showing the smallest value is selected (hereinafter, specifically referred to as the "minimum sum square difference"). For convenience, the duration angle α set at various values is specifically referred to as the "candidate duration angle". The candidate duration angle when the minimum sum square difference is obtained is then determined as the final "duty angle α", and the provisional maximum principal stress and provisional minimum principal stress associated with that duration angle α are determined as the final maximum principal stress σ ₁ ^∞ and minimum principal stress σ ₂ ^∞ .

(Processing flow)
Next, the flow of main processes of the principal stress calculation program of the present invention will be described with reference to Fig. 9. Fig. 9 is a flow diagram showing the flow of main processes of the principal stress calculation program of the present invention, in which the central column shows the processes to be performed, the left column shows information necessary for the processes, and the right column shows information resulting from the processes.

N borehole jack tests are performed on a borehole BH (FIG. 1) in an anisotropic rock mass, and when each test result is obtained, a series of processes shown in FIG. 9 is executed by a computer using the principal stress calculation program of the present invention. When the process is started, various condition values including the N test results obtained by the tests (test hole radius r, loading surface curvature β, first Poisson's ratio v ₁₂ , second Poisson's ratio v ₁₃ , first deformation coefficient E ₁ , second deformation coefficient E ₂ , inclination angle φ, measured initial loading pressure p ₀ ) are first imported (Step 101 in FIG. 9 ). Specifically, the test results are imported by reading out the stored condition values or by the operator inputting the condition values.

Once the condition values are input, the candidate operating angles are set while varying these values (Step 102 in FIG. 9). For convenience, M (a natural number equal to or greater than 2) types of candidate operating angles are set, and the jth type (j=1 to M) of candidate operating angles is expressed as "candidate operating angle α(j)." Once the candidate operating angle α(j) is set, the elastic compliances (b ₁₁ to b ₆₆ ) are calculated (Step 103 in FIG. 9), and the complex constants B ₁ (j) and B ₂ (j) are calculated (Step 104 in FIG. 9). Then, an implicit function (Equation (17) in FIG. 8 ) representing the analytical initial loading pressure p _0c is set (Step 105 in FIG. 9 ), and an explicit function (Equation (18) in FIG. 8 ) representing the analytical initial loading pressure p _0c is set by rewriting this implicit function, and two sensitivity coefficients (hereinafter referred to as the "first sensitivity coefficient Ω ₁ (i, j)" and the "second sensitivity coefficient Ω ₂ (i, j)") are obtained (Step 106 in FIG. 9 ). Here, the (j) attached to each value means that the value changes for each candidate operating angle α(j), and (i, j) means that the value changes for each candidate operating angle α(j) for each test.

Once the first sensitivity coefficient Ω ₁ (i, j) and the second sensitivity coefficient Ω ₂ (i, j) are obtained, the provisional maximum principal stress σ ₁ ^∞ (i, j) and the provisional minimum principal stress σ ₂ ^∞ (i, j) are calculated using equation (20) in Figure 8, and the analytical initial loading pressure p _0c is obtained using equation (22) in Figure 8 (Step 107 in Figure 9), and further the sum squared difference ΣS(j) is obtained using equation (21) in Figure 8 (Step 108 in Figure 9).

Once the sum squared difference ΣS(j) is obtained, a different candidate operating angle α(j) is set (Step 102 in FIG. 9 ), and a series of processes (Steps 102 to 108 in FIG. 9 ) are repeatedly executed. Then, after M iterations of the process have been executed, the sum squared difference ΣS(j) showing the smallest value is selected as the "minimum sum squared difference" (Step 109 in FIG. 9 ), and the candidate operating angle α(j) at which the smallest sum squared difference was obtained is determined as the "operating angle α." Furthermore, the provisional maximum principal stress σ ₁ ^∞ (i, j) and provisional minimum principal stress σ ₂ ^∞ (i, j) associated with that operating angle α are determined as the final maximum principal stress σ ₁ ^∞ and minimum principal stress σ ₂ ^∞ (Step 110 in FIG. 9 ).

(Verification of validity)
The inventor has verified the validity of the analysis method using the principal stress calculation program of the present invention. The details of the verification are described below.

In this verification, four borehole jack tests were conducted while changing the loading direction (θ = 90°, 45°, 0°, -45°) as shown in Figure 10 (a), and 90 types of candidate action angles α(j) (-88° to 90°) were set as shown in Figure 11, and then an analysis was performed using a principal stress calculation program. As a result, as shown in Figures 10 (c) and 11, when the candidate action angle α(j) was 90°, the sum square difference ΣS(j) showed the smallest value. Note that action angles α(j) = 0° and action angles α(j) = 90° are mechanically the same state, so either can be adopted, and since the power of -28 in double precision calculation indicates 0, the sum square difference ΣS(j) of both can be treated as the same value. Therefore, it was decided to adopt the action angle α(j) = 90° here. That is, in this case, the action angle α at which the minimum sum square difference is obtained is set to 90°, and further, the maximum principal stress σ ₁ ^∞ = 5.0 MPa and the minimum principal stress σ ₂ ^∞ = 3.0 MPa associated with this action angle α of 90° are determined. The inventors performed an isotropic analysis in this verification for comparison, and the results are shown in FIG. 10(b). According to this, the anisotropic analysis using the principal stress calculation program of the present invention obtains values equal to the actual results (set values), whereas the isotropic analysis results calculate values different from the actual results (maximum principal stress σ ₁ ^∞ is 5.0 MPa ≠ 5.5 MPa, and action angle α is 90° ≠ 106°). That is, it can be said that the anisotropic analysis using the principal stress calculation program of the present invention is appropriate.

3. Deformation Modulus Calculation Method Next, the principal stress calculation method of the present invention will be described with reference to the drawings. Note that the principal stress calculation method of the present invention is a method for calculating the principal stress of a rock mass using the principal stress calculation program of the present invention described so far, and therefore, we will avoid explanations that overlap with the contents explained in the principal stress calculation program of the present invention, and will only explain the contents unique to the principal stress calculation method of the present invention. In other words, the contents not described here are the same as those explained in "2. Principal Stress Calculation Program" and "1. Definition".

Fig. 12 is a flow diagram showing the main steps of the principal stress calculation method of the present invention. As shown in this figure, first, a boring machine or the like is used to drill (excavate) an anisotropic rock mass to construct a borehole BH (Step 201). After the borehole BH is constructed, a borehole jack test as shown in Fig. 1 is carried out (Step 202). At this time, as described above, the loading direction θ The borehole jack test is carried out repeatedly N times (a natural number of 3 or more) while changing the

After conducting the borehole jack test and obtaining the test results for N times, the maximum principal stress σ 1 ∞ and minimum principal stress σ ₂ ^∞ of the anisotropic rock mass and the principal stress action angle α are calculated by having a computer execute a series of processes shown in ^FIG . ₉ using the principal stress calculation program of the present invention (Step 203).

The principal stress calculation program and method of the present invention can be used in the design and stability study of various structures, such as the design of new pile foundations, tunnels and underground cavities, the stability study of underground cavities under construction, and the reinforcement design of deformed tunnels. According to the present invention, the principal stress of rock can be appropriately evaluated based on the premise of anisotropy, which enables more accurate design of structures, etc., and as a result, leads to higher quality of construction infrastructure in Japan. Considering this, the present invention can be said to be an invention that can be expected to not only be used industrially, but also to make a great contribution to society.

BH Borehole CP Pressure source HS Hose JB Borehole jack LB Loading plate DL Data logger SC Communication cable

Claims (3)

A program for causing a computer to execute a function of calculating the maximum principal stress σ 1 ∞ , the minimum principal stress σ 2 ∞ , and the action angle α of the maximum principal stress σ 1 ∞ of a rock mass based on the first deformation coefficient E ₁ , the second deformation coefficient E ₂ , and the initial loading pressure p ₀ obtained by performing N (N is a natural number of ₃ ^{or more)} _loading ^tests in which pressure is applied to the wall surface of a borehole in ^an _anisotropic rock mass by a uniform displacement loading method while changing the loading direction,
The first deformation coefficient _E1 is a deformation coefficient in a first axis direction among a first axis, a second axis, and a third axis which are perpendicular to each other, and the second deformation coefficient _E2 is a deformation coefficient in a second axis direction,
a condition value input process for inputting the test hole radius r and the loading surface curvature β in the loading test, the first Poisson's ratio v ₁₂ in the plane including the first axis and the second axis, the second Poisson's ratio v ₁₃ in the plane including the first axis and the third axis, the first deformation coefficient E ₁ , the second deformation coefficient E ₂ , the inclination angle φ between the horizontal plane and the first axis, and the initial loading pressure p ₀ described above;
a duration setting process for setting a plurality of candidate duration angles α(j) that are candidates for the duration angle α;
a sensitivity coefficient calculation process for calculating a first sensitivity coefficient Ω 1 (i, j) and a second sensitivity coefficient Ω ₂ (i, j) given by the following formula, with the first deformation coefficient E ₁ , the second deformation coefficient E ₂ , the inclination angle φ, and the duration angle α as variables, for each of the candidate duration angles _α (j);
p ₀ = Ω ₁ (i, j) × σ ₁ ^∞ (i, j) + Ω ₂ (i, j) × σ ₂ ^∞ (i, j)
a provisional maximum principal stress σ ₁ ^∞ (i, j) and a provisional minimum principal stress σ ₂ ^∞ (i, j) are calculated for each of the candidate operating angles α(j);
a sum-of-squared-difference calculation process for calculating a squared difference s(i,j) given by the following formula for each of N tests for each of the candidate operating angles α(j), and calculating a sum of squared differences, which is the sum of the squared differences for the N tests;
s (i, j) = [p ₀ - Ω ₁ (i, j) × σ ₁ ^∞ (i, j) + Ω ₂ (i, j) × σ ₂ ^∞ (i, j)] ²
a duration determination process for selecting a minimum sum squared difference from the sum squared differences obtained for each candidate duration angle α(j), and determining the candidate duration angle α(j) associated with the minimum sum squared difference as the duration angle α;
a principal stress determination process for determining the provisional maximum principal stress σ ₁ ^∞ (i, j) and the provisional minimum principal stress σ ₂ ^∞ (i, j) associated with the duration angle α determined in the duration angle determination process as the provisional maximum principal stress σ ₁ ^∞ and the minimum principal stress σ ₂ ^∞ ,
A principal stress calculation program characterized by: The provisional principal stress calculation process calculates the provisional maximum principal stress σ ₁ ^∞ (i, j) and the provisional minimum principal stress σ ₂ ^∞ (i, j) by a least squares method using the following equation:

2. The principal stress calculation program according to claim 1, A method for determining the maximum principal stress σ ₁ ^∞ , the minimum principal stress σ ₂ ^∞ , and the action angle α of the maximum principal stress σ ₁ ^∞ of the rock mass exhibiting anisotropy by performing a loading test in which pressure is applied to the wall surface of a borehole by a uniform displacement loading method, comprising:
A load test process in which the load test is performed N times (N is a natural number of 3 or more) on the rock mass while changing the loading direction;
and an analysis step of determining the maximum principal stress σ ₁ ^∞ , the minimum principal stress σ ₂ ^∞ , and the operating angle α of the rock mass based on the results of N tests,
In the analysis step, the maximum principal stress σ ₁ ^∞ , the minimum principal stress σ ₂ ^∞ , and the action angle α of the principal stress are obtained using the principal stress calculation program according to claim 1 or 2.
A method for calculating a principal stress.

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