Examine2D

2D stress analysis for underground excavations

Verification Manual

© 1989 - 2007 Rocscience Inc.

1 Stresses and Displacements around Circular Excavations 1.1 Problem description A circular opening with a diameter (a) of 0.5 m is considered in this example. The vertical in-situ field stress (p) is assumed to be 10 MPa. Three different values of horizontal in-situ field stress (Kp) were evaluated. They are:

- Case 1: Kp = 0 MPa

- Case 2: Kp = 10 MPa

- Case 3: Kp = 20 MPa

The Young’s Modulus (E) and Poisson’s ratio (ν) of the material around the opening is 10,000 MPa and 0.25 respectively. The stresses (σ rr & σ θθ) and the total displacements around the opening at θ = 0o are verified. Variation of σ θθ with θ on the excavation boundary is also confirmed. The model geometry is shown in Figure 1.1.

Figure 1.1 – Model Geometry

p = 10 MPa

Kp Kp

p = 10 MPa

2a = 1 m

r

θ

σ rr

τ rθ

σ θθ

uruθ

1.2 Closed Form Solution

Stresses and displacements around circular openings can be solved analytically using the Kirsch solution [1, 2]:

( ) ( ) ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−−−⎟⎟

⎠

⎞⎜⎜⎝

⎛−+= θσ 2cos341111

2 4

4

2

2

2

2

ra

raK

raKp

rr

( ) ( ) ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−+⎟⎟

⎠

⎞⎜⎜⎝

⎛++= θσθθ 2cos31111

2 4

4

2

2

raK

raKp

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−+−= θτ θ 2sin3211

2 4

4

2

2

ra

raKp

r

( ) ( ) ( ) ⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

−−−−+−= θυ 2cos14114 2

22

raKK

rGapur

( ) ( ) ⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

+−−−= θυθ 2sin21214 2

22

raK

rGapu

For plane strain and isotropic conditions, the shear modulus (G) is defined as:

( )'12'υ+

=EG

where

21'

υ−=

EE

υ

υυ−

=1

'

1.3 Results and Discussion

Figures 1.2 – 1.4 show the radial stress profiles and Figure 1.5 illustrates the total displacement profiles for all three cases. Figure 1.6 demonstrates variation of σ θθ on the excavation boundary with θ. The results from Examine2D are compared to the analytical solutions and are in good agreement.

0

5

10

15

20

25

30

35

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Distance (m)

Stre

ss (M

Pa)

Examine2D

Analytical Solution [1]

σ θθ

σ rr

Fig. 1.2 Radial and Hoop Stress Profiles for Case 1

0

5

10

15

20

25

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Distance (m)

Stre

ss (M

Pa)

σ θθ

σ rr

Examine2D

Analytical Solution [1]

Fig. 1.3 Radial and Hoop Stress Profiles for Case 2

0

2

4

6

8

10

12

14

16

18

20

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Distance (m)

Stre

ss (M

Pa)

σ θθ

σ rr

Examine2D

Analytical Solution [1]

Fig. 1.4 Radial and Hoop Stress Profiles for Case 3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Distance (m)

Tota

l Dis

plac

emen

t (m

m)

1.6

Case 1 (Kp = 0 MPa)

Case 2 (Kp = 10 MPa)

Case 3 (Kp = 20 MPa)

Examine2D

Analytical Solution [1]

Fig. 1.5 Total Displacements Profile

-20

-10

0

10

20

30

40

50

60

0 30 60 90 120 150 180 210 240 270 300 330 360

θ (degree)

σ θθ (M

Pa)

Case 1 (Kp = 0 MPa) Case 2 (Kp = 10 MPa) Case 3 (Kp = 20 MPa)

Examine2D

Analytical Solution [1]

Fig. 1.6 Variation of σ θθ with θ on the excavation boundary

1.4 References

1. B. H. G. Brady and E. T. Brown (1993), Rock Mechanics: for underground mining, 2nd Ed., London: Chapman & Hall.

2. H. G. Poulos and E. H. Davis (1974), Elastic Solutions for Soil and Rock Mechanics, New York: John Wiley & Sons.

2 Stresses and Displacements around Elliptical Excavations 2.1 Problem description An elliptical opening with a dimension (W x H) of 1 m x 0.5 m is considered in this example. The vertical in-situ field stress (p) is assumed to be 10 MPa. Three different values of horizontal in-situ field stress (Kp): 0, 10 MPa and 20 MPa, were evaluated. The vertical stresses (σyy) and horizontal stresses (σxx) along the x-axis are verified. The model geometry is shown in Figure 2.1.

Figure 2.1 – Model Geometry

p = 10 MPa

Kp Kp

p = 10 MPa

W = 1 m

H = 0.5 m x

z

2.2 Results and Discussion

Figures 2.2 to 2.4 show the vertical and horizontal stress profiles of the three cases. The results from Examine2D are compared to the analytical solutions and are in good agreement.

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Distance (m)

Stre

ss (M

Pa)

Examine2D

Analytical Solution [1]

σyy

σxx

Fig. 2.2 Vertical and Horizontal Stress Profiles for Kp = 0

0

5

10

15

20

25

30

35

40

45

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Distance (m)

Stre

ss (M

Pa)

Examine2D

Analytical Solution [1]σyy

σxx

Fig. 2.3 Vertical and Horizontal Stress Profiles for Kp = 10 MPa

0

5

10

15

20

25

30

35

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Distance (m)

Stre

ss (M

Pa)

Examine2D

Analytical Solution [1]σyy

σxx

Fig. 2.4 Vertical and Horizontal Stress Profiles for Kp = 20 MPa

2.3 References

1. B. H. G. Brady and E. T. Brown (1993), Rock Mechanics: for underground mining, 2nd Ed., London: Chapman & Hall.

2. H. G. Poulos and E. H. Davis (1974), Elastic Solutions for Soil and Rock Mechanics, New York: John Wiley & Sons.

3 Vertical Stresses and Relative Surface Displacements due to an Infinite Strip of Uniform Loading 3.1 Problem description This problem verifies the vertical stresses beneath an infinite strip footing subjected to uniform loading. The relative vertical displacements at the surface due to the strip footing were also evaluated. The model geometry is shown in Figure 3.1. The results are compared to the analytical solution [1]. The material below the strip footing has a Young’s Modulus (E) of 10,000 MPa and a Poisson’s ratio (ν) of 0.25.

Figure 3.1 – Model Geometry

3.2 Closed Form Solution

For the general case shown below:

Figure 3.2 – General Case of Uniform Loading on an Infinite Strip

q

2b

(x, z)

x

z

α δ

1 MPa

Point 1 Point 2

1 m

2b = 2 m

1 m

Point 3

Vertical stress at any point (x, z) is given by:

( )[ ]δαααπ

σ 2cossin ++=q

z

and relative vertical displacement on the surface can be solved analytically by using:

( ) ( ) ( ) ( ) ( )[ ]bbbxbxbxbxE

qxzz ln2lnln120,0,02

+++−−−−

−=−π

υρρ

3.3 Results and Discussion

Figure 3.3 shows vertical stress profiles underneath the three points given by Examine2D compared to the analytical solutions [1]. Figure 3.4 illustrates the relative vertical displacements along the surface predicted by Examine2D compared to that from [1].

0.0

1.0

2.0

3.0

4.0

5.0

0 0.2 0.4 0.6 0.8 1 1.2

Vertical Stress (MPa)

Dep

th (m

)

Examine2D

Analytical Solution [1]Point = 3

Point = 2

Point = 1

Fig. 3.3 Vertical Stress Profiles

0.0E+00

1.0E-04

2.0E-04

3.0E-04

4.0E-04

5.0E-04

6.0E-040 5 10 15 20 25 30 35 40 45 50

Distance from Center (m)

Rel

ativ

e Ve

rtic

al D

ispl

acem

ent (

m)

Examine2D

Analytical Solution [1]

Fig. 3.4 Relative Vertical Displacement Profiles

3.4 References

1. H. G. Poulos and E. H. Davis (1974), Elastic Solutions for Soil and Rock Mechanics, New York: John Wiley & Sons.