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Computational Science, Engineering & Technology Series
ISSN 1759-3158
Edited by: B.H.V. Topping
Chapter 9

Numerical Limit Analysis: An Engineering Tool for both Ultimate Limit Load Analysis and Optimal Material Layout

L. Damkilde

Esbjerg Institute of Technology, Aalborg University, Denmark

Full Bibliographic Reference for this chapter
L. Damkilde, "Numerical Limit Analysis: An Engineering Tool for both Ultimate Limit Load Analysis and Optimal Material Layout", in B.H.V. Topping, (Editor), "Civil Engineering Computations: Tools and Techniques", Saxe-Coburg Publications, Stirlingshire, UK, Chapter 9, pp 199-216, 2007. doi:10.4203/csets.16.9
Keywords: limit state analysis, non-linear optimization, finite element, reinforced concrete, geotechnics, material optimization.

Limit State analysis has a long history and many prominent researchers have contributed. The theoretical foundation is based on the upper- and lower-bound theorems which give a very comprehensive and elegant formulation on complicated physical problems. In the pre-computer age Limit State analysis also enabled engineers to solve practical problems within reinforced concrete, steel structures and geotechnics.

The development within non-linear Finite Element analysis has reached a very advanced level today, and complicated non-linear structural analyses are in many engineering firms every day business. Limit State analysis has not yet reached this level even though it can be a very efficient tool especially in the design and verification process.

In the paper an overview of Limit State analysis including a Finite Element formulation is given. Contrary to linear or non-linear formulations the governing equations result in an optimization problem which can be solved in different ways. The solution gives both the optimal stress distribution and the collapse mode, which are used in the engineering design analysis.

In recent years numerical Limit State analysis has attracted research which has improved element types and optimization algorithms. The paper will describe the trends within this area, and also give examples of application areas.

In engineering design one of the key problems is to find a suitable design layout. In reinforced concrete it could be the number and size of reinforcement bars. The paper presents a formulation of the Limit State analysis which for given loads allows for an optimization of the material layout. This feature has not yet been widely used, but may turn out to be very useful even in cases where the Limit State conditions of unlimited plastic strains cannot be justified. The Limit State analysis could be used for the initial design specifications, and full non-linear Finite Element analysis in the final verification.

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