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A global procedure for evaluating stability of three-dimensional slopes

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Abstract

In the setting of the three-dimensional limit equilibrium method, a procedure for bounding the factor of safety is proposed under the constraint of the reasonable conditions on the system of forces suggested by Morgenstern and Price. No assumption is made regarding the internal forces but the system of forces resulted from the procedure satisfies the reasonable conditions as well as all the equilibrium conditions. The optimization problem proposed has weak nonlinearity but no numerical problems inherent in the methods of columns and can be solved using those conventional optimization techniques. Through some typical examples, it is illustrated that even for complicated failure surfaces the proposed procedure gives rise to a very narrow interval of the factor of safety with a low degree of computational complexity.

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Abbreviations

a :

(a 1, …, a N )T

a i :

δ(x i , y i )

c :

Vector (∈R 6) [Eq. 15c]

c e :

Effective cohesion along the failure surface S

c w :

c e − f eu

d :

Vector (∈R 6) [Eq. 15d]

δ :

Remedy stress on the failure surface S

F :

The factor of safety

f e :

Effective friction factor along the failure surface S

Δr C :

r − r C

f ext :

Resultant external force vector (∈R 3) acting in the sliding mass Ω

f m :

Vector (∈R 6) [Eq. 8]

l :

(l 1, …, l N )T

l i :

Shape function of node-i of mesh T xy

m ext :

Moment vector (∈R 3) of f ext with respect to r C

N :

The number of nodes in mesh T xy

n :

Unit normal vector (∈R 3) of S pointing to the inside of Ω

n′:

Vector (∈R 6) [Eq. 8]

Ω:

Three-dimensional domain occupied by the slip mass

Ω k :

The lower part of Ω cut by section-k

Ω xy :

Projection of Ω onto horizontal plane xy

P :

6 × N matrix [Eq. 15a]

Q :

6 × N matrix [Eq. 15b]

r :

Position vector (∈R 3) of a variable point on the failure surface S

r C :

Position vector (∈R 3) of an arbitrary reference point C in Ω

R m :

An m-dimensional Euclidean Space

S :

Failure surface

s :

Unit vector (∈R 3) opposite to the slip direction

Σ:

A closed set of space F–a in which any point (F, a) (∈R N+1) corresponds to a reasonable force system

σ :

Total normal stress acting on the failure surface S

σ 0 :

Monolith stress on the failure surface S

s′:

Vector (∈R 6) [Eq. 8]

τ :

Shear stress acting on the failure surface S

t k :

Thrust on section-k

T xy :

Triangular mesh covering Ω xy

u :

Pore pressure on the failure surface S

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Acknowledgments

This study is supported by the National Basic Research Program of China (973 Program), under the Grant No. 2011CB013505, and by National Natural Science Funds for Distinguished Young Scholar, under the Project no. 50925933.

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Authors and Affiliations

  1. State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan, 430071, China

    Guanhua Sun

  2. Key Laboratory of Geological Hazards on Three gorges Reservoir Area, Ministry of Education, Three Gorges University, Yichang, 443002, China

    Hong Zheng & Wei Jiang

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  1. Guanhua Sun
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  2. Hong Zheng
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Correspondence to Hong Zheng.

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Sun, G., Zheng, H. & Jiang, W. A global procedure for evaluating stability of three-dimensional slopes. Nat Hazards 61, 1083–1098 (2012). https://doi.org/10.1007/s11069-011-9963-9

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