Stress Measurements - Rock Mechanics Based on an Anisotropic Jointed Rock Model - page 514

as on the elastic constants of the rock mass and the inclusion material. For isotropic
rock mass, A ei can be expressed by the following function:

(16.2)

where E' and ν ' are the elastic constants of the inclusion material and E, ν are the elastic

constants of the rock mass.
As shown by Berry & Fairhurst (1966), Amadei & Goodman (1982) and Gonano &
Sharp (1983), anisotropic (i.e. transversely isotropic), elastic stress-strain behavior of
the rock mass needs to be accounted for when evaluating triaxial cell measurements.
Otherwise, a misinterpretation of strain readings with respect to the calculated stresses
is inevitable. Anisotropy can be detected from the results of biaxial tests and uniaxial
compression tests. For transversely isotropic rock mass with elastic constants E 1 , E 2 ,
G 2 ,

ν 1 and ν 2 the coeffi cients A ei take on the following form:

(16.3)

Explicit expressions of A ei for isotropic and anisotropic rock masses using triaxial cells
of all three types are presented by Duncan Fama & Pender (1980) and Amadei (1983),
respectively, on the basis of closed-form analytic solutions for this problem.
Normally more than the six strain components, which are required for determining the
complete three-dimensional stress state, are measured, for example, nine strain compo-
nents as specifi ed in Figs. 16.3 and 16.5. Thus, the measured strains lead to an over-de-
termined system of linear equations which can be solved by means of the least square
method or another data reduction procedure. On this basis, statements on the magni-
tude of errors are also possible. An example of such an analysis for elastic isotropic rock
mass is presented in Wittke (1990).
Once the stress components with respect to a selected coordinate system (x,y,z) are de-
termined, the magnitudes of the three principal normal stresses

σ 1 , σ 2 and σ 3 are calcu-

lated from the following equation:

(16.4)

where [ ] and [E] are the stress tensor determined from measured data and the unit ten-

sor, respectively. The roots

σ

of this cubic equation, also referred to as “eigenvalues”,
ordered according to their magnitude are the principal normal stresses. Their orienta-
tions can be determined by solving, for each principal stress

σ

σ i , the following system of

three linear equations for the eigenvector {vi}: i }: