**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}