# Progressive Landslides Analysis Applications of a Finite Difference Method by Dr. Stig Bernander Case Study of the North Spur at Muskrat Falls, Labrador, Canada

(1)

## Progressive Landslides Analysis

### Civil Engineering, master's level 2017

Luleå University of Technology

Department of Civil, Environmental and Natural Resources Engineering

(2)

(3)

(4)

q

w

d2

2

(5)

(6)

q

w

d2

2

(7)

q

w

d2

2

(8)

(9)

(10)

0

𝑒𝑙

𝑝

𝑠(𝐼)

𝑐𝑟

𝑞

𝑠(𝐼𝐼)

𝑝

𝑥𝑖→𝑥𝑖+1

0

𝑝

𝑐𝑟

𝑐𝑟

𝑐𝑟

𝑞

𝑢

𝑅

𝑠

2

2

𝑒𝑙

𝑓

𝑐𝑟

N

𝜏

s

𝑥𝑖→𝑥𝑖+1

𝑥

3

𝑒𝑙

0

(11)

(12)

(13)

.

(14)

𝑟

### remains. The term ‘deformation-softening’ refers to the loss of shear resistance with increasing shear strains and displacements in the developing failure zone, the material is getting softer (less stiff) with increasing strains and deformations.

𝜏 ( )

𝑐𝑟 1 𝜏𝑒𝑙 𝑐

𝛾𝑒𝑙 , 𝛾 ( ) 𝛿𝑟 , 𝛿 ( )

𝑟

(15)

𝜏 ( )

𝛿 ( ) 𝑐𝑟

𝑐 1

𝑐𝑟 𝑐 ,

𝑐𝑟 𝑐 ,

r

r

r

-

-

𝑐𝑟𝑖𝑡

(16)

𝑟

0

o

𝑞

0



( ) , °

1 1

δ

𝑞

𝑞

0

0

0

0

0

2

2

3

0

(17)

𝑞

0

( )

1 1

( )

0

1

0

𝑞

0

q

b

1 1

( )

8 ,

1

( ) 0 18 /

0

( )

1 1

0+ ( )

8 , 0 16 /

0

𝑞

𝑞

𝑐𝑟𝑖𝑡

𝑐𝑟𝑖𝑡

𝑐𝑟𝑖𝑡

(18)

1 1

( ) 1

( ) 1 /

4, 0

( )

1 1

0+ ( )

4, 0 0+

d

r

( )

1

1

( ) N= 1 /

0

1 ,

( )

1 1

0+ ( )

, 0

instab

instab

ns b

instab

1

( )

1

( )

1 ( )

1 ,6

1 1

0+ ( )

1 ,60

𝑝

(19)

( )

1

( )

1 ( )

0

1

1

0+ ( / )

0 16 /

4

4

0

0=

0

0

p,R nk ne

0

p,R nk ne

### The deformation at the point of load application is illustrated by the following Figure 2-11

, ,4

,1 , ,

Phase 1 Phase 2 Phase 3 Phase 4 and 5

a

b

c d

e f

0

1

1 16

0+ ( )

18

1 /

δ ( )

𝑠(𝐼)

𝑐𝑟𝑖𝑡

𝑞

𝑐𝑟𝑖𝑡

𝑞

𝑠(𝐼)

(20)

𝑐𝑟𝑖𝑡

q

𝑠(𝐼𝐼)

𝑝

0

𝑚𝑎𝑥

𝑝

0

𝑚𝑎𝑥

𝑠(𝐼𝐼)

𝑝

0 𝜌𝑔𝐻2

2

202

2

0

0

0

𝑚𝑎𝑥

𝑝

𝑠(𝐼𝐼) 2800

1731

th

𝑐𝑟𝑖𝑡

𝑐𝑟𝑖𝑡

𝑐𝑟𝑖𝑡

(21)

𝑖𝑛𝑠𝑡𝑎𝑏

𝑖𝑛𝑠𝑡𝑎𝑏

,

𝑒𝑙

nd

𝑠

𝑠𝑙𝑖𝑝

### Figure 2-12.Stress-strain/deformation used by Bernander to carry out his downhill progressive failure analysis

𝜏 ( )

𝑔 𝑔

𝑐𝑟 𝜏𝑒𝑙 𝑐

𝛾𝑒𝑙 𝛾 ( ) 𝛿𝑟 𝛿 ( )

𝛿𝑠𝑙𝑖𝑝

(22)

1

3

𝐻

3

0

0

𝑞

0

𝑁

0

0

0

0

𝐻(𝑥)

0

𝑥0→𝑥1

1

0

𝑥0→𝑥1

1

1

### and vertical coordinate z. The shear deformation 𝛾(𝑥, 𝑧) is defined with an equation in diagram in Figure 2-13 and given in full in Appendix A.

𝜏0≤ 𝜏𝑒𝑙

 i n

𝜏 + ∆𝜏 > 𝜏𝑒𝑙 𝜏 + ∆𝜏 ≤ 𝜏𝑒𝑙

 i n

 i n 𝜏0> 𝜏𝑒𝑙

g 𝜏 + ∆𝜏 ≤ 𝑐

(23)

1

𝑥0→𝑥1

𝜏

1

1

𝑥0→𝑥1

𝑁

1

1

𝑥0→𝑥1

0

1

0

0

0

1

0→1

𝑁

1

0

1

0→1

𝑒𝑙

𝑥0→𝑥1

0→1

𝑥0→𝑥1

𝑥1 𝑁

𝑖

𝑥0

𝜏

1

𝑁

𝑥𝑖→𝑥𝑖+1

𝑖

(24)

0

𝑒𝑙

𝑅

𝑅

0

𝑒𝑙

𝑅

𝑅

𝑅

𝑁

𝑖

𝑥𝑛 𝑥0

𝜏

𝑛

𝑠

𝑛

𝑠

𝑠

𝑐𝑟

𝑟

𝑐𝑟

𝑟

0

𝑐𝑟𝑖𝑡

𝑐𝑟𝑖𝑡

𝑐𝑟𝑖𝑡

𝑟

𝑖𝑛𝑠𝑡𝑎𝑏

(25)

### here is applied to the right and not to the left as in Figures 2-2 to 2-10.

𝑞

𝑧 (𝑚)

𝐻

𝑥

𝛾 1

𝐸 𝛾

𝜏 , 𝜏1 𝜏

𝑥 1 ∆𝑥1−2 𝑥

∆𝜏1−2

τ τ ,

𝐻1

𝜏1

x δx

𝑥 𝐿

x 𝑁𝐿 𝑁𝑞 τ τ1

x 1

1 ∆𝜏1 ∆𝑥1

∆𝛿 1 (𝑁1+𝑁 )∆𝑥1 𝐻1 𝐸 𝛿 1= 𝛿𝜏

1

𝜏 τ τ1+ ∆𝜏1

x 1+∆ 1

𝛿 = 𝛿𝑁

1 + ∆𝛿 1 = 𝛿𝜏 𝛿𝜏

### ∑

𝛾 ∆𝑧 Lower boundary

condition

Upper boundary condition

(26)

(27)

### The North Spur on which the concrete dam is embanked is a post glacial deposit of marine and estuarine sediments which provide a partial closure of the Churchill River Valley at the Muskrat Falls site. It is about 1 km long between the Rock Knoll in the south and the Kettle Lakes in the north which represent natural boundaries. It has the following section, see Figure 3-3:

0 50 100 150 200 250 300 350 400 450 500 550 600

50

25

0

-25

Cut-off wall

(m) +39

+6 Sand

Sand

Sand Silty Clay

Silty Clay

Lower Clay +17

(28)

(29)

(30)

(31)

(32)

𝑐𝑟𝑖𝑡

𝑐𝑟𝑖𝑡

𝑐𝑟𝑖𝑡

𝑖𝑛𝑠𝑡𝑎𝑏

𝑖𝑛𝑠𝑡𝑎𝑏



0

( ) δ

x ( )

( )

Updating...