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PRINCIPLE OF INITIAL STRESS ESTIMATION

Initial stress estimation is based on a principle that the induced strains on the surface of a borehole by overcoring are proportional to stresses released by the overcoring. When the initial strains are set to be zero on the surface of the borehole drilled in rock mass in a certain state of unknown initial stresses, strains develop in the process of stress relief by overcoring. If the initial stresses are completely released, we obtain strains developed by the initial stresses. Of course, the estimated initial stresses are equal to released stresses but with opposite signs.

If rock mass is in a state of initial stresses,

 
$\displaystyle \left\{ \sigma \right\}^T=\left\{ \sigma_{11},\sigma_{22},\sigma_...
...t\{ \sigma_{1},\sigma_{2},\sigma_{3},\sigma_{4},\sigma_{5},\sigma_{6} \right\},$     (1)

where $\sigma_{ij}$ imply stress components in rectangular Cartesian coordinates o-x1x2x3 with x1-axis being coincide with the borehole axis directing to the end, and strains developed on the several points of the surface by overcoring may be expressed by
$\displaystyle \left\{ \varepsilon \right\}^T=\left\{ \varepsilon_{11},\varepsilon_{12},\ldots,\varepsilon_{M1},\varepsilon_{M2} \right\},$     (2)

where the first subscript stands for the stain measuring position (points) and the second subscript 1 and 2 express the components in the directions of the generator and in perpendicular to it (circumferential direction), respectively , see Fig.3.

With an assumption of isotropic linear elasticity, we have

$\displaystyle \left\{ \varepsilon \right\}=\frac{1}{E}[A]\left\{ \sigma \right\}$     (3)

where E stands for Young's modulus of the material. The coefficient matrix [A] (strain matrix:2M$\times$6) is accurately determined by use of the FMBEM.

It is easy to construct an observation equation for determining initial stresses from the measured strains when the matrix [A] is known. If we choose more than 6 measured strains and make use of the least-squares method, for instance, we have

$\displaystyle [A]^T[A]\left\{ \sigma \right\}=E[A]^T\left\{ \varepsilon \right\},$     (4)

Therefore, the initial stresses are obtained as
 
$\displaystyle \left\{ \sigma \right\}=E[D]\left\{ \varepsilon \right\},$     (5)

where
$\displaystyle [D]=\left([A]^T[A]\right)^{-1}[A]^T$     (6)

which is called the strain-to-stress conversion matrix (6$\times$2M).

It is easily understood that the key of the stress-relief method is the strain-to-stress conversion matrix [D], and thus evaluation of the strain matrix [A] is of fundamental importance.


next up previous
Next: FORMULATION OF FMBEM FOR Up: 無題 Previous: INTRODUCTION
Toru Takahashi 平成11年10月13日