Progressive Landslide Analysis with Bernander Finite Difference Method

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Progressive Landslide Analysis with Bernander Finite Difference Method

R. Dury 1 , S. Bernander 1 , A. Kullingsjö 2 , J. Laue 1 , S. Knutsson 1 , R. Pusch 1 and L. Elfgren 1

1 Luleå University of Technology, Luleå, Sweden, 2 Skanska Teknik, Göteborg, Sweden

Robin DURY

Luleå Tekniska Universitet

Email: robdur-6@ltu.student.se Phone: + 33 6 80 38 87 19

Contact

Bernander, S. and Olofsson, I. (1981). On formation of progressive failure in slopes. In Proceedings of the 10

th

, ICSMFE, Stockholm 1981. Vol 3, 11/6, pp 357-362.

Bernander, S. 2011. Progressive landslides in long natural slopes. Formation, potential extension and configuration of finished slides in strain- softening soils. Doctoral thesis, Luleå University of Technology, , 3rd rev. version, April 2012. ISBN 978-91-7439- 283-8.

Bernander, S., Kullingsjö, A., Gylland, A. S., Bengtsson, P. E., Knutsson, S., Pusch, R., Olofsson, J., & Elfgren, L. (2016). Downhill progressive landslides in long natural slopes: triggering agents and landslide phases modelled with a finite difference method. Canadian Geotechnical Journal, Vol. 53, No. 10, pp. 1565-1582,

Dury, Robin (2017). Progressive Landslide Analysis. MSc Thesis, Luleå University of Technology, Luleå, Sweden. To be published at http://ltu.diva- portal.org/

References

• A spreadsheet is presented of the Finite Difference

Method (FDM) by Stig Bernander et al. (1981, 2011, and 2016) and Dury (2017)

• Contrary to the classic Limit Equilibrium Method (LEM) the softening of the soil is taken into consideration.

• The method is illustrated in Fig. 1-3, Dury (2017).

Introduction

• An easy to use spreadsheet has been developed, Dury (2017)

• Different geometries and material properties can be studied

• The method may readily be applied by Consultant Engineers

Discussion

Finite Differential Method Material Properties

Figure 1. Stress-strain deformation relationship of a typical ‘deformation softening’

clay from southwestern Sweden. The letters a to f refer to Fig. 3 for L=0.

z (m)

∆γ

E 0 H

x

∆x

∆τ

τ

∆z

E x δ N = δ τ

• The mean deformation δ in each element caused by normal forces N is maintained compatible with the deformation generated by the shear stresses τ.

Figure 2. Illustration of the Finite Differential Method

γ

el

= 3,75 % γ (%) δ

r

= 0,3 m δ (m)

τ (kPa)

c

r

= 15 kPa c = 30 kPa

τ

el

= 20 kPa

Progressive Failure Process

Figure 3. Illustration of the progressive failure process

a

b

c at τ

0

=20,8 kPa

d → e, f

Phase 1, Moment a: In-situ conditions. No load q or N

q

and in situ stress τ

0

= 20,8 kPa

Phase 2, Moment b: A load q is applied giving τ = c = 30 kPa. The shear stresses can be integrated to the force N

q

= 189 kN for an influence length L

b

= 85,5 m.

End of Phase 2, start of Phase 3, Moment c: The shear stress has now decreased to τ

0

= 20,8 kPa at the point of the application of q and the load has reached its maximum value N

crit

= 231 kN for an influence length L

crit

= 94,3 m.

Phase 3, Moment d: Now an unstable dynamic phase starts and the load that can be taken is reduced to N = 215 kN for an influence length of L

d

= 99,7 m. The shear stress is reduced to its minimum value c

r

= 15kPa

End of Phase 3, Moment e: The negative shear forces balance the positive forces so that N = 0 at the point of load application. The maximum shear force 231 kN has travelled downslope for a total influence length of L

e

= 139,6 m

Phase 4 (and 5), Moment f: The in situ shear stress 𝛕 𝟎 decreases from L = 150 m where the slope turns horizontal. The pressure N is caused by the weight of the sliding mass, N = L ∗ H ∗ ρ ∗ g ∗ sin(β). The residual shear stress 𝐜 𝐫 is reduced due to dynamic action. The pressure is "permanently" or "temporarily"

balanced by passive resistance if (E 0 + N) max < E p,Rankine . The failure plane develops far into the unsloping ground before equilibrium is reached. If (E 0 + N) max > E p,Rankine a final collapse will occur in Phase 5.

Downhill progressive failure triggered by an additional load q can be divided in 5 different phases :

q

N

q

H

L (m)

0

τ

β = 3,727°

50 100 150 200

β = 0

E

z

δ

L (m)

0 100 150

τ (kPa)

τ

0

= 20,8 kPa 30

50 200

20

10 c

r

= 15 kPa

0 50 100 150 200

τ (kPa) 30

20 10

L (m)

N

crit

= 231 kN/m

L

crit

= 94,3

τ

0

L (m)

0 50 100 150

E = E

0

+ N (kN/m)

L = 94,3 E

0

E

max

= E

0

+ N

crit

τ (kPa)

0 50 150 200

30 20 10

L (m)

N=215 kN/m

τ

0

c

r

100

L = 99,7

L (m)

0 50 100 150

E = E

0

+ N (kN/m)

L = 97,7 E

0

N

crit

30 20 10

L (m)

0 50 150 200

τ (kPa)

100 L (m)

N = 0

c

r

L = 139,6

0 50 100 150

E = E

0

+ N (kN/m)

L = 139,6 E

0

N

crit

0 0,1 0,2 0,3 0,4 0,5

Phase 1 Phase 2 Phase 3 Phase 4 and 5

a

b

c d

e

f E

0

1900

1700 1600

E = E

0

+ N (kN/m) at L=0 1800

N

crit

= 231 kN/m

Deformation δ at L=0

δ (m)

0 50 100 150 200

τ (kPa)

L = 85,5

30 20 10

L (m)

N = τ − τ

0

dL = 189 kN/m

τ

0

L (m)

0 50 100 150

E = E

0

+ N (kN/m)

L = 85,5 E

0

= 1600 kN/m

Smaller length scale L (m)

30 20 10

L (m)

2000 c

r

0 100

τ (kPa)

200

τ

0

3000

1000

0 100 200

E = E

0

+ N (kN/m)

E

0

= 1600 kN/m τ

0

300 400

300 400

N

max

Smaller length scale τ

0

τ

0

=c

r

Figur

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