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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.
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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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