Page 184 - Rock Mechanics For Underground Mining

METHODS OF STRESS ANALYSIS

quantitative experimental determination of the internal state of stress in a body sub-
ject to applied load was the photoelastic method. The principle exploited in the method
is that, in two dimensions, and for isotropic elasticity, the stress distribution is inde-
pendent of the elastic properties of the material, and is the same for plane stress
and plane strain. In its original application, a two-dimensional model of a structure
was prepared from a transparent material such as glass or plastic, and mounted in
a beam of monochromatic, polarised light. Application of loads to the model, and
passage of the light beam through an analyser onto a screen, produced a series of
bands, or fringes, of light extinction and enhancement. Generation of the fringes is
due to dependence of the propagation velocity of light through the medium on the
local principal stress components. A fringe, also called an isochromatic, represents
a contour line of constant principal stress difference. Thus a fringe pattern produced
by a photoelastic model represents a mapping of contours of maximum shear stress
throughout the medium. Calibration of the system allows the shear stress magnitude
of any contour level to be determined. For excavation design in rock, it is necessary to
establish the distribution of principal stresses throughout the medium. Thus in addi-
tion to the maximum shear stress distribution, it is necessary to establish contour plots
of the first stress invariant. Since, as is shown later, this quantity satisfies the Laplace
equation, various analogues can be used to define its spatial variation in terms of a set
of isopachs, or contour plots of ( 1 + 3 ). Taken together with the photoelastic data,
these plots allow the development of contour plots of the principal stresses throughout
the problem domain.
It is clear from this brief discussion that the photoelastic method of stress analysis
is a rather tedious way of predicting the stress distribution in a mine structure. It is
therefore rarely used in design practice. However, the method is a useful research
technique, for examining such problems as blocky media (Gaziev and Erlikmann,
1971) and three-dimensional structures (Timoshenko and Goodier, 1970) using the
frozen-stress method.
A major detraction from the use of physical models of any sort for prediction of the
rock mass response to mining is their high cost in time and effort. Since many mine
design exercises involve parameter studies to identify an optimum mining strategy,
construction and testing of models is inherently unsuited to the demands of the design
process. Their use can be justified only for a single, confirmatory study of a proposed
extraction strategy, to verify key aspects of the mine structural design.

6.2 Principles of classical stress analysis

A comprehensive description of the fundamentals of stress analysis is beyond the
scope of this book. Texts such as those by Timoshenko and Goodier (1970) and
Prager (1959) may be consulted as general discourses on engineering elasticity and
plasticity, and related methods of the analysis of stress. The intention here is to
identify key elements in the analytical determination of the stress and displacement
fields in a body under applied load. The particular concern is to ensure that the
conditions to be satisfied in any closed-form solution for the stress distribution in a
body are appreciated. Techniques can then be established to verify the accuracy of any
solution to a particular problem, such as the stress distribution around an underground
excavation with a defined shape. This procedure is important, since there exist many
166