calculation of slope stability by phi-c reduction
TRANSCRIPT
Benchmark Example No. 38
Calculation of Slope Stability by Phi-C Reduction
SOFiSTiK | 2022
VERiFiCATiONBE38 Calculation of Slope Stability by Phi-C Reduction
VERiFiCATiON Manual, Service Pack 2022-2 Build 48
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Front Cover
Arnulfsteg, Munich Photo: Hans Gossing
Calculation of Slope Stability by Phi-C Reduction
Overview
Element Type(s): C2D
Analysis Type(s): STAT, MNL
Procedure(s): LSTP, PHIC
Topic(s): SOIL
Module(s): TALPA
Input file(s): slope stability.dat
1 Problem Description
In this benchmark the stability of an embankment, as shown in Fig. 1, is calculated by means of a ph−creduction. The factor of safety and its corresponding slip surface are verified.
αslope
h 1
h 2
l1 l2lslope
Figure 1: Problem Description
2 Reference Solution
The classical problem of slope stability analysis involves the investigation of the equilibrium of a mass ofsoil bounded below by an assumed potential slip surface and above by the surface of the slope. Forcesand 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 thatnecessary to provide equilibrium, the mass is stable. If the shear resistance is insufficient, the massis unstable. The stability of the mass depends on its weight, the external forces acting on it, the shearstrengths and pore water pressures along the slip surface, and the strength of any internal reinforcementcrossing potential slip surfaces. The factor of safety is defined with respect to the shear strength of thesoil as the ratio of the available shear strength to the shear strength required for equilibrium [1]:
FS =be sher strength
eqbrm sher stress(1)
The safety definition according to FELLENIUS is based on the investigation of the material’s shearstrength 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
SOFiSTiK 2022 | Benchmark No. 38 3
Calculation of Slope Stability by Phi-C Reduction
ratio between available shear strength and the mobilized shear strength in the limit state of the system[2]:
ηϕ =tn ϕnp
tn ϕm(2)
ηc =cnp
cm(3)
where c is the cohesion and ϕ the friction angle. The ph− c reduction stability analysis is based on anincremental 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 stabilityfailure.
The reference solution [3] is based on the finite element formulation of the upper- and lower-boundtheorems of plasticity. Thus, the finite-element limit analysis (FELA) provides a good reference for thestrength reduction method as it establishes upper and lower-bound estimates for the true stability limit.
3 Model and Results
The properties of the model [3] are presented in Table 1. The embankment has a slope height of 10 mand a slope angle of 30◦. The initial stresses are generated using gravity loading. Then the embankmentis subjected to the ph − c reduction. Plane strain conditions are assumed.
Table 1: Model Properties
Material Properties Geometric Properties
E = 20000 kN/m2, ν = 0.3 h1 = 20.0m
γ = 19 kN/m3 h2 = 10.0m
ϕ = 25◦, ψ = 25◦ 1 = 2 = 15.0m
c = 20 kN/m2 αsope = 30◦ , sope = 17.321m
4 Benchmark No. 38 | SOFiSTiK 2022
Calculation of Slope Stability by Phi-C Reduction
SOFiSTiK AG * Bruckmannring 38 * 85764 OberschleißheimSOFiSTiK 2018 WINGRAF - GRAPHICS FOR FINITE ELEMENTS (V 19.00)
Page 12017-07-19
System
SO
FiS
TiK
AG
- w
ww.
sofis
tik.d
e
M 1 : 202X
YZ
Deformed Structure from LC 253 phi-c reduction η= 2.00 Enlarged by 20.0Nodal displacement vector in Node, nonlinear Loadcase 253 phi-c reduction η= 2.00 , from 0 to 243.4 step 6.08 mm
0.0
6.1
12.2
18.3
24.3
30.4
36.5
42.6
48.7
54.8
60.8
66.9
73.0
79.1
85.2
91.3
97.3
103.
4
109.
5
115.
6
121.
7
127.
8
133.
9
139.
9
146.
0
152.
1
158.
2
164.
3
170.
4
176.
4
182.
5
188.
6
194.
7
200.
8
206.
9
213.
0
219.
0
225.
1
231.
2
237.
324
3.4
m-15.00 -10.00 -5.00 0.00 5.00 10.00 15.00 20.00 25.00 30.00
10.0
05.
000.
00-5
.00
-10.
00-1
5.00
Figure 2: Nodal displacements for the factor of saftey obtained with the ph − c reduction analysis
SOFiSTiK AG * Bruckmannring 38 * 85764 OberschleißheimSOFiSTiK 2018 WINGRAF - GRAPHICS FOR FINITE ELEMENTS (V 19.00)
Page 22017-07-19
System
SO
FiS
TiK
AG
- w
ww.
sofis
tik.d
e
M 1 : 202X
YZ
Plastic deviatoric strain , nonlinear Loadcase 253 phi-c reduction η= 2.00, Material law Mohr-Coulomb , QUAD Gauss points in Nodeo/oo, from 0 to 122.0 step 3.05
0.0
3.0
6.1
9.1
12.2
15.2
18.3
21.3
24.4
27.4
30.5
33.5
36.6
39.6
42.7
45.7
48.8
51.8
54.9
57.9
61.0
64.0
67.1
70.1
73.2
76.2
79.3
82.3
85.4
88.4
91.5
94.5
97.6
100.
6
103.
7
106.
7
109.
8
112.
8
115.
9
118.
912
2.0
m-15.00 -10.00 -5.00 0.00 5.00 10.00 15.00 20.00 25.00 30.00
10.0
05.
000.
00-5
.00
-10.
00-1
5.00
Figure 3: Deviatoric strain for the factor of saftey obtained with the ph − c reduction analysis
Figure 2 presents the nodal displacement as a vector distribution for the factor of safety obtained withthe ph − c reduction analysis. Furthermore, the corresponding plastic deviatoric strain is shown inFigure 3. The calculated factor of safety is compared with the reference solution [3] in Table 2, i.e. withthe results from the lower-bound and upper-bound finite element limit analysis (FELA). Additionally, thecalculated factor of safety from ph− c reduction analysis is plotted in Figure 4 as a function of the nodaldisplacement in x direction for the node at the top of the embankment slope.
Table 2: Factor of saftey - calculated and reference values according to [3]
SOFiSTiK FEM FELAoer bond FELApper bond
2.00 1.97 2.01
SOFiSTiK 2022 | Benchmark No. 38 5
Calculation of Slope Stability by Phi-C Reduction
0−10−20−30−40−50−60−70−80−90−100−1100
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Nod dspcement n drecton[mm]
Fctoroƒsƒtey,
η[−]
Figure 4: Factor of safety as a function of displacement in x direction for the node at the top of theembankment slope
4 Conclusion
This example verifies the stability of a soil mass and the determination of the factor of safety. Thecalculated factor of safety, which is obtained with the ph− c reduction method, is compared to the finiteelement 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.
6 Benchmark No. 38 | SOFiSTiK 2022