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CivilComp Proceedings
ISSN 17593433 CCP: 81
PROCEEDINGS OF THE TENTH INTERNATIONAL CONFERENCE ON CIVIL, STRUCTURAL AND ENVIRONMENTAL ENGINEERING COMPUTING Edited by: B.H.V. Topping
Paper 97
An Efficient Implementation of NonLinear Limit State Analysis based on LowerBound Solutions L. Damkilde and L. Juhl Schmidt
Esbjerg Department of Engineering, Aalborg University Esbjerg, Denmark L. Damkilde, L. Juhl Schmidt, "An Efficient Implementation of NonLinear Limit State Analysis based on LowerBound Solutions", in B.H.V. Topping, (Editor), "Proceedings of the Tenth International Conference on Civil, Structural and Environmental Engineering Computing", CivilComp Press, Stirlingshire, UK, Paper 97, 2005. doi:10.4203/ccp.81.97
Keywords: limit state analysis, nonlinear optimization, interior point methods, yield line, slip line.
Summary
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 wellknown finite
element concept in order to interface with other types of
analysis.
Limit State analysis can either be formulated by the upperbound method or the lowerbound method. In the upperbound method the idea is to find the most critical geometrical possible collapse mechanisms. In the lowerbound 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 nonlinear optimization problem. In this work the formulation is solemnly based on the lowerbound methods, which gives great advantages for more complex yield criteria and in optimization of material layout almost is compulsory. The lowerbound formulation results in a nonlinear 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 nonlinear convex yield criteria. The nonlinear 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 lowerbound formulation the dual problem is the equivalent upperbound 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. purchase the fulltext of this paper (price £20)
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