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SOFiSTiK | 2018
VERiFiCATiON MANUAL
BE38: Calculation of Slope Stability by Phi-C Reduction
SOFiSTiK AG
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Front Cover
Project: New SOFiSTiK Office, Nuremberg | Contractor: WOLFF & MLLER, Stuttgart | Architecture: WABE-PLAN ARCHITEKTUR, Stuttgart |
Structural Engineer: Boll und Partner. Beratende Ingenieure VBI, Stuttgart | MEP: GM Planen + Beraten, Griesheim | Lead Architect: Gerhard P.
Wirth gpwirtharchitekten, Nuremberg | Vizualisation: Armin Dariz, BiMOTiON GmbH
Calculation of Slope Stability by Phi-C Reduction
Overview
1 Problem Description
In this benchmark the stability of an embankment, as shown in Fig. 1, is calculated by means of a ph− c
reduction. The factor of safety and its corresponding slip surface are verified.
αslope
h2
h1
l1 lslope l2
2 Reference Solution
The classical problem of slope stability analysis involves the investigation of the equilibrium of a mass of
soil bounded below by an assumed potential slip surface and above by the surface of the slope. Forces
and moments, tending to cause instability of the mass, are compared to those tending to resist instabil-
ity. Most procedures assume a two-dimensional cross-section and plane strain conditions for analysis.
Successive assumptions are made regarding the potential slip surface until the most critical surface, i.e.
lowest factor of safety, is found. If the shear resistance of the soil along the slip surface exceeds that
necessary to provide equilibrium, the mass is stable. If the shear resistance is insufficient, the mass
is unstable. The stability of the mass depends on its weight, the external forces acting on it, the shear
strengths and pore water pressures along the slip surface, and the strength of any internal reinforcement
crossing potential slip surfaces. The factor of safety is defined with respect to the shear strength of the
soil as the ratio of the available shear strength to the shear strength required for equilibrium [1]:
The safety definition according to F ELLENIUS is based on the investigation of the material’s shear
strength in the limit state of the system, i.e. the shear strength that leads to failure of the system.
Following this notion, in SOFiSTiK, the safety factors according to ph − c reduction are defined as the
ratio between available shear strength and the mobilized shear strength in the limit state of the system
[2]:
tn ϕnp
ηϕ = (2)
tn ϕm
cnp
ηc = (3)
cm
where c is the cohesion and ϕ the friction angle. The ph − c reduction stability analysis is based on an
incremental reduction of the shear strength adopting a synchronized increase of the safety factors η =
ηph = ηc . The reached safety η at system failure represents the computational safety against stability
failure.
The reference solution [3] is based on the finite element formulation of the upper- and lower-bound
theorems of plasticity. Thus, the finite-element limit analysis (FELA) provides a good reference for the
strength reduction method as it establishes upper and lower-bound estimates for the true stability limit.
γ = 19 kN/ m3 h2 = 10.0 m
243.4
237.3
231.2
225.1
219.0
213.0
206.9
200.8
194.7
188.6
182.5
176.4
170.4
164.3
158.2
152.1
146.0
139.9
133.9
127.8
121.7
109.5
103.4
115.6
97.3
91.3
85.2
79.1
73.0
66.9
60.8
54.8
48.7
42.6
36.5
30.4
24.3
18.3
12.2
6.1
0.0
-15.00
SOFiSTiK AG *of
Calculation Bruckmannring 38 by
Slope Stability * 85764
Phi-COberschleißheim
Reduction Page 2
SOFiSTiK 2018 WINGRAF - GRAPHICS FOR FINITE ELEMENTS (V 19.00) 2017-07-19
System
-10.00
-5.00
SOFiSTiK AG - www.sofistik.de
0.00
5.00
10.00
122.0
109.8
106.7
103.7
100.6
118.9
115.9
112.8
97.6
94.5
91.5
88.4
85.4
82.3
79.3
76.2
73.2
70.1
67.1
64.0
61.0
57.9
54.9
51.8
48.8
45.7
42.7
39.6
36.6
33.5
30.5
27.4
24.4
21.3
18.3
15.2
12.2
9.1
6.1
3.0
0.0
-15.00
Figure 2: Nodal displacements for the factor of saftey obtained with the ph − c reduction analysis
-15.00 -10.00 -5.00 0.00 5.00 10.00 15.00 20.00 25.00 30.00 m
Deformed Structure from LC 253 phi-c reduction η= 2.00 Enlarged by 20.0 M 1 : 202
-10.00
Z X
Nodal displacement vector in Node, nonlinear Loadcase 253 phi-c reduction η= 2.00 , from 0 to 243.4 step 6.08 mm
Y
-5.00
SOFiSTiK AG - www.sofistik.de
Figure 3: Deviatoric strain for the factor of saftey obtained with the ph − c reduction analysis 0.00
5.00
10.00
-15.00 -10.00 -5.00 0.00 5.00 10.00 15.00 20.00 25.00 30.00 m
Plastic deviatoric strain , nonlinear Loadcase 253 phi-c reduction η= 2.00, Material law Mohr-Coulomb , QUAD Gauss points in Node M 1 : 202
Z X
Y
Figure
o/oo, from2
0 topresents
122.0 step 3.05 the nodal displacement as a vector distribution for the factor of safety obtained with
the ph − c reduction analysis. Furthermore, the corresponding plastic deviatoric strain is shown in
Figure 3. The calculated factor of safety is compared with the reference solution [3] in Table 2, i.e. with
the results from the lower-bound and upper-bound finite element limit analysis (FELA). Additionally, the
calculated factor of safety from ph − c reduction analysis is plotted in Figure 4 as a function of the nodal
displacement in x direction for the node at the top of the embankment slope.
1.8
1.6
1.2
0.8
0.6
0.4
0.2
0
−110 −100 −90 −80 −70 −60 −50 −40 −30 −20 −10 0
Figure 4: Factor of safety as a function of displacement in x direction for the node at the top of the
embankment slope
4 Conclusion
This example verifies the stability of a soil mass and the determination of the factor of safety. The
calculated factor of safety, which is obtained with the ph − c reduction method, is compared to the finite
element limit analysis results and it is shown that the behavior of the model is captured accurately.
5 Literature
[1] USACE Engineering and Design: Slope Stability. USACE. 2003.
[2] TALPA Manual: 2D Finite Elements in Geotechnical Engineering. 2018-0. SOFiSTiK AG. Ober-
schleißheim, Germany, 2017.
[3] F. Tschuchnigg et al. “Comparison of finite-element limit analysis and strength reduction tech-
niques”. In: Geotechnique 65(4) (2015), pp. 249–257.