116
10 Mathematical Structural Imperfections
Table 10.1 Studies for the
“Kadamjay mine”
Maximum stress, MPa
Area, horizon
y, m t, m Measured Nominal
1
2
3
4
5
Horizon 960 m 240 200 11.3
7.9
Northern gate
Horizon 930 m 330
25 17.1
14.2
Southern gate
Horizon 930 m 380
10 22.0
20.0
Mine shaft
“New”
425 700
7.8
11.1
Parameters: E = 2.2 · 10 4 MPa; ν = 0.24; H =
5800 m β = 700 m; ratio of ultimate tensile strength and
compression m = 0.25; α = π/15; y—depth from day
surface; t—depth to the fault in horizontal
10.7 Pair of Fislocations in a Plane
Let us at first consider an unlimited plane where an absolutely rigid infinite stripe
δ thick is introduced along the negative half-axis Oy of the system of rectangular
coordinates x 1 Oy. Muskhelishvili functions [14] for this problem can be obtained
using the same ring cutting method as used in the previous paragraph (p. 108) to
create a Somigliana dislocation. Making a passage to the limit at R 1 → 0 and
R 2 → ∞, we obtain the functions of Muskhelishvili 1 (z) ? 1 (z) for an infinite
plane with an absolutely rigid infinite stripe δ thick introduced along the negative
half-axis Oy:
1 (z) = − 1 (z) =
Kδi
z
, K =
Eδi
8π(1 − ν 2 )
(z = x 1 + iy).
(10.35)
Let us make a parallel transfer of coordinate axes to the point (0, −t). Let us
designate the new system of coordinate axes as xOy and introduce a stripe (−δ)
thick in the point (0, −t) of the new system along the negative Y-half-axis. By
the superposition of solutions, let us find Muskhelishvili functions for an unlimited
plane where an absolutely rigid infinite stripe δ thick and 2t long is introduced at
the section (−t < y < t). Muskhelishvili functions for this problem will be
2 (z) = −
2Kδt
z 2 + t 2 , , 2 (z) =
4Kδtz 2
(z 2 + t 2 ) 2 , (z = x + iy).
(10.36)
Let us use the known formulas (9.15) associating the stress components
σ x , σ y , τ xy with Muskhelishvili functions
10 Mathematical Structural Imperfections
Table 10.1 Studies for the
“Kadamjay mine”
Maximum stress, MPa
Area, horizon
y, m t, m Measured Nominal
1
2
3
4
5
Horizon 960 m 240 200 11.3
7.9
Northern gate
Horizon 930 m 330
25 17.1
14.2
Southern gate
Horizon 930 m 380
10 22.0
20.0
Mine shaft
“New”
425 700
7.8
11.1
Parameters: E = 2.2 · 10 4 MPa; ν = 0.24; H =
5800 m β = 700 m; ratio of ultimate tensile strength and
compression m = 0.25; α = π/15; y—depth from day
surface; t—depth to the fault in horizontal
10.7 Pair of Fislocations in a Plane
Let us at first consider an unlimited plane where an absolutely rigid infinite stripe
δ thick is introduced along the negative half-axis Oy of the system of rectangular
coordinates x 1 Oy. Muskhelishvili functions [14] for this problem can be obtained
using the same ring cutting method as used in the previous paragraph (p. 108) to
create a Somigliana dislocation. Making a passage to the limit at R 1 → 0 and
R 2 → ∞, we obtain the functions of Muskhelishvili 1 (z) ? 1 (z) for an infinite
plane with an absolutely rigid infinite stripe δ thick introduced along the negative
half-axis Oy:
1 (z) = − 1 (z) =
Kδi
z
, K =
Eδi
8π(1 − ν 2 )
(z = x 1 + iy).
(10.35)
Let us make a parallel transfer of coordinate axes to the point (0, −t). Let us
designate the new system of coordinate axes as xOy and introduce a stripe (−δ)
thick in the point (0, −t) of the new system along the negative Y-half-axis. By
the superposition of solutions, let us find Muskhelishvili functions for an unlimited
plane where an absolutely rigid infinite stripe δ thick and 2t long is introduced at
the section (−t < y < t). Muskhelishvili functions for this problem will be
2 (z) = −
2Kδt
z 2 + t 2 , , 2 (z) =
4Kδtz 2
(z 2 + t 2 ) 2 , (z = x + iy).
(10.36)
Let us use the known formulas (9.15) associating the stress components
σ x , σ y , τ xy with Muskhelishvili functions