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Limit Analysis: Advantages and Limitations

Introduction

The following subsections discuss some of the advantages and disadvantages of limit analysis. This is not intended to be an exhaustive list.

Simplicity

Unlike elasto-plastic FE analysis which typically requires many iterations in order to arrive at a solution, numerical limit analysis seeks out the solution directly by coupling optimization techniques and rigorous plasticity theory. Limit analysis thus has the advantage that it can directly determine a solution and normally suffers from no numerical instabilities. Formulated as a linear programming problem, it is also guaranteed to find the global optimum (for a given nodal discretization).

Limit analysis also has the advantage that it typically only requires two strength parameters for any material modelled, the cohesion () and the angle of shearing resistance (). However more complex yield surfaces may be specified.

Stress states in yielding and non-yielding zones

In the context of DLO the aim is to find a mechanism that results in collapse under the lowest load. The corollary of this is to find a network of discontinuities where the yield condition is just not violated. Solutions can thus be used to correctly represent force distributions in yielding (failing areas). Outside these areas the solver needs only to find a set of forces that does not cause yield, no other conditions being stipulated. If these are examined, then erratic distributions may be observed. This is not an error, but an inherent result of the application of plasticity theory which is only concerned with the collapse state.

Thus LimitState:GEO will generate suitable force distributions that can be used to check ultimate limit state in a structural element, but it should not be used (or expected) to generate force distributions for determining deflections in structural members that are not yielding or adjacent to yielding soil.

Strain compatibility

The inherent assumption in limit analysis that collapsing bodies mobilize the same strength everywhere at the same time. In reality strains are likely to differ within the problem domain and thus different soil strengths will be mobilized in different locations at collapse. This may be exacerbated where there are two or more materials present in the problem with very different stress-strain characteristics.

This is an issue that has always required consideration by geotechnical engineers when using any form of limit analysis solution. A conservative recommendation often made is to assume the critical state strength for all materials. At sufficiently high strains most soil materials should reach their critical state (or ultimate) strengths thus rendering the limit analysis assumption valid as long as excessive geometry change has not occurred in the process of mobilising this critical state strength everywhere. It may be reasonable to assume strengths higher than critical state but this requires the judgement of the engineer.

Non-associativity

The upper and lower bound theorems of limit analysis require that the materials modelled obey the associative flow rule. In effect this requires that all shearing resistance is modelled as dilation rather than a combination of dilation and friction as occurs with real soils. This is accurate for undrained problems where the angle of shearing resistance is zero, however for drained problems it typically leads to a small overestimate of load capacity. In extreme cases it can lead to volumetric locking and no collapse.

Experience has shown that for moderately unconstrained problems, the increase in load estimate is minor. However it is not possible to give guidance on its effect on all problems and the user is referred to the literature for specific guidance. It may be noted that limit analysis and therefore an assumption of associative flow is commonly used for typical geotechnical problem types in geotechnical textbooks and design codes. For example the formula for bearing capacity recommended by Eurocode 7 implies use of a limit analysis model.

For further information on the effects of non-associativity for specific problem types, reference may be made to the following:

Slopes

Manzari & Nour (2000), indicate that non-associative results for cohesive-frictional slope stability problems typically give values 3-10% lower than for the associated flow rule case (pure plasticity model). To put this into context, this corresponds to using an angle of shearing resistance in an associated flow model approximately 3% lower than the actual angle. This is of the order of 1.

Footings

Loukidis et al. (2008) indicate that non-associative results for the problem (i.e. a rigid rough footing on the surface of a cohesionless soil) typically give values 15-30% lower than for the associated flow rule case (pure plasticity model). To put this into context, this corresponds to using an angle of shearing resistance in an associated flow model approximately 3% lower than the actual angle. This is of the order of 1.

Kinematic Constraints

If the problem kinematics are significantly constrained, then limit analysis solutions can significantly overestimate the collapse load. This can occur in the case of a frictional/dilational soil, where geometric locking can occur.