Engineering Rock Mechanics

Questions and answers: strain and the theory of elasticity 63

• Circled values are
  equal in transversely
  isotropic rock

symmetric E is a Young’s modulus,

v is a Poisson’s ratio, and

L G is a shear modulus.

Q5.5 (a) How can the strain in a particular direction be found from
the strain matrix components and hence how can a strain gauge
rosette be used to estimate the state of strain at a point, and hence
the state of stress at a point?
(b) Assume that strains measured by a strain gauge rosette are
ep = 43.0 x EQ = 7.8 x and ER = 17.0 x and that
the gauges make the following angles to the x-direction: Op =209
8~ e 80° and OR =140°. Determine the principal strains and their
orientations and then, using values for the elastic constants of
E =150 GPa and u = 0.30, determine the principal stresses and their
orientations.

A5.5 (a) In 2-D, the strain in a particular direction can be found from the
strain matrix components, E,~, E,,,,, e,,,, using the Mohr’s circle approach
or using the strain transformation equations. A strain gauge is a device
for measuring the normal strains in three directions in a plane, as indic-
ated by the sketch to the right. Foil electrical resistance strain gauges can
be glued to the rock surface, or wire or rod extensometers can be used.
The rosette device provides three normal strains at known orientations,
from which the Mohr’s circle can be constructed, the principal values
found, and the normal strains in any other required direction evaluated.
Knowing the elastic properties of the rock and making assumptions
about the type of rock anisotropy, the stress state can be established
through the generalized Hooke’s Law equations.
(b) In order to use the strain transformation equations to determine the
2-D state of strain from measurements made with strain gauges, we firstly
determine the angle each gauge makes to the x-axis: say, for gauges P, Q
and R, these are ep, ee and OR. The strains measured by the gauges are FP,
EQ and ER. Thus, the strain transformation equation linking each of the
measured strains E~, EQ and ER to the strains e,, ey and yxy are
ep = ex COS ep + cy sin2 ep + yXy sin ep cos ep
= E, cos2 ee + ey sin2 ea + yxy sin ea cos

R *p 12

2

FR = E, COS^2 OR + cy sin2 eR + yxy sin cos OR

or, in matrix form,

cos2 ep sin2 oP sin 0p cos ep

ER cos2 oR sin2 OR sin OR cos OR

[:,I = [ cos2ea sin2ea sinea

[.