Three‐dimensional Mohr–Coulomb limit analysis using semidefinite programming

Krabbenhøft, K.; Lyamin, A. V.; Sloan, Scott W.

doi:10.1002/cnm.1018

Cited by 126 publications

(95 citation statements)

References 26 publications

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“…Inspired by the progress in general convex programming, these linear programming formulations have recently been replaced by more general nonlinear formulations avoiding the need to linearize [8,13]. The most recent development on this front has been the applications of the so-called conic programming algorithms to solve typical limit analysis problems such as the ones considered here as well other plasticity problems [10,11]. These algorithms are particularly suited for dealing with nonsmooth strength domains such as those typically characterizing the strength of cohesive, frictional materials (Drucker-Prager, Coulomb, etc.…”

Section: Introductionmentioning

confidence: 98%

Failure in accretionary wedges with the maximum strength theorem: numerical algorithm and 2D validation

Souloumiac¹,

Krabbenhøft²,

Leroy³

et al. 2010

Comput Geosci

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International audienceThe objective is to capture the 3D spatial variation in the failure mode occurring in accretionary wedges and their analog experiments in the laboratory from the sole knowledge of the material strength and the structure geometry. The proposed methodology relies on the maximum strength theorem which is in- herited from the kinematic approach of the classical limit analysis. It selects the optimum virtual velocity field which minimizes the tectonic force. These velocity fields are constructed by interpolation thanks to the spatial discretization conducted with ten-noded tetra- hedra in 3D and six-noded triangles in 2D. The re- sulting, discrete optimization problem is first presented emphasizing the dual formalism found most appropri- ate in the presence of nonlinear strength criteria, such as the Drucker-Prager criterion used in all reported examples. The numerical scheme is first applied to a perfectly triangular 2D wedge. It is known that fail- ure occurs to the back for topographic slope smaller than and to the front for slope larger than a critical slope, defining subcritical and supercritical slope stabil- ity conditions, respectively. The failure mode is char- acterized by the activation of a ramp, its conjugate back thrust, and the partial or complete activation of the décollement. It is shown that the critical slope is captured precisely by the proposed numerical scheme, the ramp, and the back thrust corresponding to regions of localized virtual strain. The influence of the back- wall friction on this critical slope is explored. It is found that the failure mechanism reduces to a thrust rooting at the base of the back wall and the absence of back thrust, for small enough values of the friction angle. This influence is well explained by the Mohr construction and further validated with experimental results with sand, considered as an analog material. 3D applications of the same methodology are presented in a companion paper

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Section: Introductionmentioning

confidence: 98%

Failure in accretionary wedges with the maximum strength theorem: numerical algorithm and 2D validation

Souloumiac¹,

Krabbenhøft²,

Leroy³

et al. 2010

Comput Geosci

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“…1459 the variation nor non-variation of the yield strength plays any role in Equations (9), (11) and (12) of Reference [2]. Finally, they end up with a form containing the actual yield strength.…”

Section: Properties Of Conic Yield Restrictionsmentioning

confidence: 98%

“…The yield restriction can be considered as the intersection of two semi-definite cones [11,12,16] kI 3 +aI 3 t aux −r 0 r−I 3 t aux 0 (28)…”

Section: Mohr-coulomb In 3dmentioning

confidence: 99%

“…Numerical implementations of conic optimization are not included, as this was the subject of previous work, e.g. [3,[5][6][7][8][9][10] for second-order cone programming and [11,12] for semi-definite programming (SDP). From a historical point of view, Psomiadis [13] applied SDP in shakedown analysis in 2005.…”

Section: Introduction-issues On the Application Of The Upper Bound Thmentioning

confidence: 99%

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Remarks on some properties of conic yield restrictions in limit analysis

Makrodimopoulos

2010

Numer Methods Biomed Eng

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SUMMARYA major difficulty when applying the kinematic theorem in limit analysis is the derivation of expressions of the dissipation functions and the set of plastically admissible strains. At present, no standard methodology exists. Here, it is shown that they can be readily obtained, provided that the yield restriction can be rewritten as an intersection of cones, and that the expression defining the dual cones is available. This is always possible for the case of self-dual cones and some other classes, and covers many of the well-known criteria. Therefore, a difficult obstacle with respect to the use of the kinematic theorem in conjunction with any numerical method can be overcome. The methodology is illustrated by giving the expressions of the dissipation functions for various conic yield restrictions. A special emphasis is given on upper bound finite element limit analysis. Taking advantage of duality in conic programming, we can obtain the dual problem, where knowledge of the dual cone is not necessary. Therefore, this formulation is feasible for any cone. Finally, it is interesting that the form of the dual problem, for varying yield strength within the finite element, differs from that presented in other papers.

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“…In this framework, the first applications of the finite element method have been proposed around 40 years ago [6,7], with a linearization of the classical strength criteria. The recent development of powerful numerical tools has brought new opportunities for analyzing numerically three-dimensional geotechnical problems with real strength criteria, such as von Mises and Drucker-Prager [8,9] or Tresca and Mohr-Coulomb [10,11]. An interested reader would find a comprehensive literature review dedicated to homogeneous geotechnical problems in [12].…”

Section: Introductionmentioning

confidence: 99%

Stability analysis of homogenized stone column reinforced foundations using a numerical yield design approach

Gueguin

Hassen²,

Buhan³

2015

Computers and Geotechnics

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This paper deals with the ultimate bearing capacity of soft clayey soils, reinforced by stone columns, analyzed in the framework of the yield design theory. Since such geotechnical structures are almost impossible to analyze directly due to the strong heterogeneity of the reinforced soil, an alternative homogenization approach is advocated here. First, numerical lower and upper bound estimates for the macroscopic strength criterion of the stone column reinforced soil are approximated in a rigorous way with convex ellipsoidal sets, which makes the approximated criteria much easier to handle than the initial ones. Then, both static and kinematic approaches are carried out numerically on the homogenized problem using the above approximated macroscopic strength domains in an adapted finite element method. The whole numerical procedure is applied on one classical geotechnical problem: the ultimate bearing capacity of stone column reinforced foundations. The strength capacity of the structure is rigorously framed and the efficiency of the proposed numerical method is highlighted in terms of accuracy and calculation time.

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Three‐dimensional Mohr–Coulomb limit analysis using semidefinite programming

Cited by 126 publications

References 26 publications

Failure in accretionary wedges with the maximum strength theorem: numerical algorithm and 2D validation

Failure in accretionary wedges with the maximum strength theorem: numerical algorithm and 2D validation

Remarks on some properties of conic yield restrictions in limit analysis

Stability analysis of homogenized stone column reinforced foundations using a numerical yield design approach

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