Keywords: limit state analysis, non-linear optimization, interior point methods, yield line, slip line.

Limit State analysis has been used in design for decades e.g. the yield line theory for concrete slabs or slip line solutions in geotechnics. In engineering practice manual methods have been dominating but in recent years the interest in numerical methods has been increasing. In this respect it is mandatory to formulate the methods using the well-known finite element concept in order to interface with other types of analysis.

Limit State analysis can either be formulated by the upper-bound method or the lower-bound method. In the upper-bound method the idea is to find the most critical geometrical possible collapse mechanisms. In the lower-bound method the idea is to find the stress state which does not violate the yield criteria in any point and gives the maximum load carrying capacity. Both methods can be implemented in a Finite Element type of scheme and both methods result in a non-linear optimization problem. In this work the formulation is solemnly based on the lower-bound methods, which gives great advantages for more complex yield criteria and in optimization of material layout almost is compulsory.

The lower-bound formulation results in a non-linear convex optimization problem. The variables will be the stress state in the elements and either a load parameter or design parameters such as volume of reinforcement. The object function will in this context be the load carrying capacity but could also be a weighted sum of some design parameters. The restrictions will be linear equilibrium equations and non-linear convex yield criteria. The non-linear yield criteria can be linearized resulting in a number of linear inequalities. In this way the problem can be characterized as a Linear Programming problem which can be solved by different methods. The most effective methods are based on variants of Karmarkar's interior point method.

In order to have a more efficient implementation two remedies can be used. The first is to eliminate the equilibrium constraints a priori. This gives a considerably reduction in the number of variables at the cost of a little more complex formulation of the yield criteria. The method has in previous studies shown its capability. The second is to deal with the non linear yield criteria directly and in this respect avoiding the large number of linear inequalities. This approach has also been used in previous studies and shown its validity. In the paper a new method is formulated which combines both of the remedies. The cost is a more complex programming but neither the efficiency or the generality is lost.

The mathematical theory of optimization tells that any problem has a dual counterpart. For the lower-bound formulation the dual problem is the equivalent upper-bound formulation. From the solution procedure the dual solution can be established directly. For engineering applications this is an important feature as the collapse mode can be interpreted in a sound physical manner whereas the optimal stress state gives no direct indication of the physical problem.

In the paper the method is illustrated by examples primarily taken from geotechnical applications but the method is fully general and can be used for all types of limit state problems.

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