Shakedown Analysis of Unsaturated Soils Considering the Variation of Hydraulic States
Publication: International Journal of Geomechanics
Volume 20, Issue 9
Abstract
Long-term stability evaluation of unsaturated earth structures has always been one of the most important issues in geotechnical engineering. Most available methods are appropriate for calculating the maximum loads that the structures can resist when the hydraulic state is fixed, and the external loads are monotonically increased. In practical engineering, however, the hydraulic condition is ever changing, and so are the matric suction and the shearing strength of unsaturated soils. In this situation, the obtained minimized collapse loads can no longer be used as reference for the design criterion, which may greatly overestimate the stability of the structures. Classical shakedown analysis is a more appropriate method, which can only consider the variation of the loads but not the variation of hydraulic states. This paper presents a novel formulation for shakedown analysis of unsaturated soils considering both the variation of external loads and the variation of the shearing strength caused by drying–wetting cycles. By using the suction stress-based effective stress of unsaturated soils, the three-phased mixture is dealt with as a single-phased material. The suction-stress equivalent forces are established, and the variation of the hydraulic state is dealt with as the variation of equivalent forces. Numerical formulations are then developed by the combination of the finite-element method and the second-order cone programming. Some shakedown problems are solved, and the effect of hydraulic state variation on the shakedown limits of foundations, pavements, and slopes is studied in detail. It is shown that the neglection of moisture content variation would greatly overestimate the safety of earth structures, and the produced errors become more significant with the increase of the suction-stress increment and the decrease of the shearing strength. The shakedown limit reaches the smallest when both external loads and suction stress are varying. When the variation of equivalent forces is complicated, the loading domain can be approximated by a polyhedron using a uniform distribution of sampling points, and the shakedown factor of safety converges to the true value with the increase of the number of sampling points. Shakedown analysis is necessary in engineering designs when several wetting and drying cycles happen.
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Acknowledgments
The present investigation was performed with the support of the National Natural Science Foundation of China (No. 51908053) and Natural Science Basic Research Plan in Shaanxi Province of China (No. 2018JQ5098).
Notation
The following symbols are used in this paper:
- A
- area of the element;
- B
- strain matrix;
- Bd
- deviatoric strain matrix;
- Bm
- volumetric strain matrix;
- c′
- effective cohesion;
- reduced effective cohesion;
- rate of plastic dissipation;
- E
- Young's modulus;
- ered
- deviatoric plastic strain;
- modified deviatoric plastic strain;
- F
- factor of safety;
- Flim
- factor of safety obtained by limit analysis;
- Fsd
- shakedown factor of safety;
- f
- body force;
- equivalent body force;
- H
- total hydraulic head of the pore water;
- I
- identity vector;
- L
- differential operator relating the strain and the displacement;
- Le
- differential operator of the deviatoric strain;
- Lm
- differential operator of the volumetric strain;
- N
- number of load vertexes;
- Nel
- number of elements;
- Nu
- coefficient matrix consisting of shape functions of the displacement;
- Nɛ
- coefficient matrix consisting of shape functions of the strain;
- normal of the interface between soil layers;
- outward normal of the stress boundary;
- load vertex;
- q
- surface force;
- equivalent surface force;
- T
- period of a loading cycle;
- t
- time;
- equivalent surface force across the soil layers;
- velocity of the soil skeleton;
- ua
- pore air pressure;
- ue
- element displacement vector;
- uw
- pore water pressure;
- v
- Poisson's ratio;
- α, n
- parameters of the soil water characteristic curve;
- Γs
- portion of boundary where surface forces are specified;
- Γu
- portion of boundary where displacement is specified;
- γw
- unit weight of the pore water;
- Δu
- displacement increment over a loading cycle;
- Δug
- global displacement vector;
- (Δu)l
- displacement increment at the lth node;
- Δɛp
- accumulated plastic strain over a loading cycle;
- Δσs
- discontinuity of suction stress across the interface between soil layers;
- strain rate;
- ɛpe
- element strain vector;
- produced plastic strain increment by the load vertex;
- plastic strain rate;
- volumetric strain rate;
- plastic volumetric strain;
- and
- components of the plastic strain;
- θp
- plastic volumetric strain;
- λ
- load multiplier;
- λsd
- shakedown load multiplier;
- σ
- total stress;
- σ′
- effective stress;
- fictious elastic stress produced by the load vertex;
- normal effective stress on the failure surface;
- σe
- fictious elastic stress;
- σs
- suction stress;
- τf
- tangential stress on the failure surface;
- τk
- time increment of a load vertex;
- φ′
- effective angle of internal friction;
- reduced angle of internal friction; and
- Ωi
- domain of the ith element.
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Published In
International Journal of Geomechanics
Volume 20 • Issue 9 • September 2020
Copyright
© 2020 American Society of Civil Engineers.
History
Received: Nov 15, 2019
Accepted: Mar 9, 2020
Published online: Jun 22, 2020
Published in print: Sep 1, 2020
Discussion open until: Nov 22, 2020
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