Civil Engineering Reference
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where the negative sign results from the introduced sign convention. In consideration of
(7.19) and using matrix notation, Equation (7.22a) can be written as
(7.22b)
Inserting (7.22b) into (7.21) yields:
(7.23)
which represents three linear differential equations for the calculation of the displace-
ments
δ
z
. In consideration of (3.113) and (7.22b) the stresses can be calculat-
ed from the displacements as follows:
δ
x
,
δ
y
and
(7.24)
vp
} are functions of stress which are obtained from viscoplas-
tic strain rates calculated for the intact rock and the different discontinuity sets accord-
ing to (3.110) by means of an iterative analysis (Wittke 1990, Wittke 2000b).
The viscoplastic strains {
ε
After introducing the boundary conditions and specifying the computational section,
the hydromechanical equations (7.17) and (7.23) can be solved numerically using the
fi nite element method (FEM) with, for example, the program systems FEST03 and
HYD03 of WBI (Section 10.7, Fig. 10.59).
Coupled seepage fl ow and stress-strain analysis
Rodatz (1973) was the fi rst to formulate a three-dimensional hydromechanical elasto-
plastic constitutive model of jointed rock according to the homogeneous model tak-
ing into account anisotropic permeability, deformability and strength. Erichsen (1987)
then formulated a corresponding model based on elastic-viscoplastic stress-strain be-
havior. In this analysis procedure, the nonlinear stress-displacement behavior of rough
joints and the effect of dilatant displacements during shearing were implemented. This
analysis procedure was later modifi ed by replacing the nonlinear relations between nor-
mal displacement, normal stress and shear displacement on discontinuities by linear
relationships as described above.
The principle of an analysis taking into account the interaction between stress-
strain behavior and seepage fl ow is illustrated in Fig. 7.7, for the example of a grav-
ity dam founded on a rock mass with horizontal bedding-parallel discontinuities B
and vertical joints J.
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