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IX Internatinal Conference on Computational Plasticity

COMPLAS IX

E. O˜nate and D.R.J. Owen (Eds)

c

°CIMNE, Barcelona, 2007

DISCONTINUITY LAYOUT OPTIMIZATION: A NEW

NUMERICAL PROCEDURE FOR UPPER BOUND LIMIT

ANALYSIS

Matthew Gilbert∗and Colin C. Smith†

Computational Limit Analysis & Design Unit

Department of Civil & Structural Engineering

University of Sheﬃeld, Mappin Street, Sheﬃeld S1 3JD, UK

e-mail: ∗m.gilbert@sheﬃeld.ac.uk †c.c.smith@sheﬃeld.ac.uk

web page: http://www.cladu.shef.ac.uk/

Key words: Computational Plasticity, Limit Analysis

Summary. Discontinuity Layout Optimization (DLO) is a recently developed and extremely

promising alternative to more well established numerical limit analysis procedures (e.g. ﬁnite

element limit analysis). With DLO a comparatively simple discontinuum problem formulation

is considered: a given planar body is discretised using a suitably large number of nodes laid out

on a grid and the failure mechanism is deemed to comprise the most critical sub-set of potential

discontinuities inter-connecting these nodes, identiﬁed using mathematical programming tech-

niques. Here the procedure is outlined and its promise is demonstrated by applying it to various

standard limit analysis problems.

1 INTRODUCTION

When an estimate of the ultimate load carrying capacity of a body or component is required,

computational limit analysis tools have the potential to play an invaluable role in bridging

the gap between existing hand-type calculations and highly complex non-linear modelling tech-

niques. One such tool which has been investigated by academic researchers over a period of

several decades is ﬁnite element limit analysis. However, despite its promise, for a variety of

reasons this method has not yet found its way into general engineering practice.

Discontinuity Layout Optimization (DLO) is a recently developed alternative to ﬁnite element

limit analysis which shows signiﬁcant promise. Though there are certain similarities between

DLO and previously proposed numerical formulations (e.g. as with the formulation of Munro

& da Fonseca1, the zones lying between potential discontinuities are simply assumed to be

rigid), a key diﬀerentiator is that the DLO problem is formulated entirely in terms of potential

discontinuities interconnecting nodes, rather than in terms of (solid) elements. This means that

discontinuities can be allowed to freely cross-over one another, considerably increasing the search

space and hence the ability of the procedure to identify complex failure mechanisms. Beneﬁts of

the procedure are that singularities are identiﬁed without diﬃculty and, as failure mechanisms

are explicitly identiﬁed, output is easy to interpret (e.g. see Fig. 1).

1

Matthew Gilbert and Colin C. Smith

(a)

(b) (c) (d)

Figure 1: Sample DLO output, showing discontinuities at failure: (a) bearing capacity problem (weight-

less sand with φ= 25oand uniform surcharge, also showing nodal discretization); (b) max. height of

vertical cut in clay problem (translational mechanism); (c) horizontal anchor in clay pullout problem; (d)

uniformly loaded square concrete slab problem (ﬁxed supports)

2 BRIEF OUTLINE OF THE DLO PROCEDURE

The Discontinuity Layout Optimization (DLO) procedure was devised following (re)discovery

by the authors of the similarity in the forms of optimum trusses†and critical arrangements of

lines of failure (i.e. discontinuities) in limit analysis problems (e.g. see Fig. 2). In fact the

analogy was formally identiﬁed almost half a century ago3,4,5, but, apart from leading to a

realization that established analytical solution methods used in limit analysis could be applied

to the then emerging ﬁeld of truss layout optimization, little use appears to have been made of

the analogy in the intervening years. However, with DLO well developed approximate-discretized

truss layout optimization techniques are eﬀectively being transferred back to the ﬁeld of limit

analysis. Consider for example a plane strain problem involving a Tresca (cohesive) material:

slip displacements along discontinuities correspond to bar forces (Fig. 2), and the objectives of

ﬁnding the minimum (slip) energy dissipation and minimum (truss) volume solutions are also

found to be analogous. Full details are provided in a forthcoming publication6.

†An optimum truss (after Michell2) comprises an arrangement of bars of minimum total weight to support

speciﬁed applied loads without yielding.

2

Matthew Gilbert and Colin C. Smith

3

3

(a) Truss layout problem: loading & support con-

ditions; domain, nodal discretisation & ‘ground

structure’; optimal bar layout & pre-existing bars

along top of domain are highlighted

3

−3−3

√18

√20 √20

√18

−√2

−4−√2

line of zero cohesion

(b) Discontinuity layout problem: imposed dis-

placement conditions; domain, nodal discretisation

& optimal slip-line layout (i.e. as (a)); labelled

slip line displacements (corresponding to truss bar

forces)

Figure 2: Analogy between (a) truss layout and (b) discontinuity layout optimization problems

The use of Linear Programming (LP) guarantees that a globally optimal solution is obtained

for a given nodal discretization. Increased accuracy can be obtained by increasing the nodal

density and by making use of an adaptive nodal connection procedure originally developed for

truss optimization7, allowing problems with up to 109potential discontinuities to be treated.

Although the simple truss analogy exists only for in-plane problems involving weightless cohesive

media, DLO can be applied to problems involving various constitutive relations and loading

regimes (e.g. a bearing capacity on sand problem with ﬁne nodal reﬁnement is shown on Fig.

1(a); a problem involving body forces is shown on Fig. 1(b); a concrete slab problem subjected

to distributed out-of-plane live loading is shown on Fig. 1(d)).

A selection of DLO results for standard problems are tabulated in Table 1. In all cases nodes

were laid out uniformly over the problem domains (i.e. no manual, problem speciﬁc, reﬁnement

was undertaken). The close agreement with known exact analytical solutions using the basic

DLO procedure demonstrates its promise (solutions are within 1% in all but one case). For two

problems exact solutions were not available: (i) for the vertical cut problem a solution reasonably

close to that previously obtained using a carefully tailored ﬁnite element mesh geometry8was

obtained‡; (ii) for the horizontal anchor problem the DLO procedure produced a signiﬁcantly

better upper bound solution than that previously obtained using ﬁnite element limit analysis9.

3 CONCLUDING REMARKS

Discontinuity Layout Optimization (DLO) is a powerful new numerical procedure which

shows considerable promise. In contrast with ﬁnite element limit analysis, no mesh is required

and the underlying formulation is simpler. Furthermore, stress / displacement singularities can

‡For the quoted DLO solution only translational displacements were permitted but the solution may have been

better still had rotations also been allowed.

3

Matthew Gilbert and Colin C. Smith

Literature DLO

Problem solution solution % diﬀ.

Square concrete slab (simple supports) 24 24 0.00

Square concrete slab (ﬁxed supports) - Fig. 1(d) 42.851 42.869 0.04

Compressed metal block (width / height = 3.64) 3.334 3.335 0.03

Smooth footing on sand (Nqwith φ= 25o) - Fig. 1(a) 10.662 10.684 0.21

Smooth footing on sand (Nγwith φ= 25o) 3.461 3.563 2.94

Maximum height of a vertical cut in clay - Fig. 1(b) 3.782†3.797‡0.4

Horizontal anchor pullout in clay (embedment ratio = 8) - Fig. 1(c) 7.6†7.247 (-4.6)

†Upper bound ﬁnite-element limit analysis solution

‡Assuming translational failure mechanism

Table 1: Comparison of literature and DLO solutions for various well known problems (solutions are

expressed in the form commonly used for each problem type, e.g. as a dimensionless load factor)

be handled automatically (i.e. without the need for tailored meshes) and failure mechanisms

are explicitly identiﬁed so output can be interpreted easily. For the same reason the method can

also be used as a tool for users wishing to identify new analytical solutions.

4 ACKNOWLEDGEMENTS

The ﬁrst author acknowledges the support provided by EPSRC (grant ref: GR/S53329/01).

REFERENCES

[1] J. Munro and A. Da Fonseca. Yield line method by ﬁnite elements and linear programming.

The Structural Engineer,56B(2), 37-44, (1978).

[2] A.G.M. Michell. The limits of economy of material in frame-structures, Phil. Mag.,8, 589-

597, (1904)

[3] W.S. Hemp. Rep. 115: Theory of structural design, College Aeronautics, Cranﬁeld, (1958).

[4] M.P. Neilson. Plasticitetsteorien for jernbetonplader (Theory of plasticity for reinforced con-

crete slabs): Licentiatafhandling, Danmarks Tekniske Hojskole, Copenhagen, (1962).

[5] W. Prager. On a problem of optimal design, Proc. IUTAM Symposium on non-homogeneity

in elasticity and plasticity, Warsaw, London: Pergamon, 125-132, (1959).

[6] C.C. Smith and M. Gilbert. Application of discontinuity layout optimization to plane plas-

ticity problems, accepted for publication in Proc. R. Soc. A, (2007)

[7] M. Gilbert and A. Tyas. Layout optimization of large-scale pin-jointed frames, Engineering

Computations,20 (8), 1044-1064, (2003).

[8] J. Pastor, T.H. Thai, and P. Fracescato. Interior point optimization and limit analysis: an

application, Commun. Numer. Meth. Engng.,19, 779-785, (2003).

[9] R. Meriﬁeld, S.W. Sloan and H. Yu, Stability of plate anchors in undrained clay, Geotech-

nique,51(2), 141-153, (2001).

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