Keywords: configurational forces, energy minimisation, pre-stressed concrete structures, tendon layouts, finite element method.

The concept of configurational forces (also known as material forces) has been explored in recent years as a tool for optimisation based on sound mechanical principles of energy minimisation. Whereas the usual physical forces (or spatial forces) can be defined as the derivative of the energy with respect to the displacements, configurational forces are the derivative of the energy with respect to the initial coordinates of the material. In order words, physical forces are conjugated to the movement of material through space, while configurational forces are conjugated to the movement of a spatial configuration through the material. Solving for a zero residual of physical forces is equivalent to requiring equilibrium to be fulfilled. Arguing along similar lines, solving for a zero residual of configurational forces implies that no energy can be released by changing the material positions, which indicates an initial configuration that is optimal in terms of energy.

Configurational forces have been used successfully in the optimisation of trusses and finite element meshes, but also in shape optimisation problems [1,2]. The advantage of using configurational forces is that they are firmly rooted in continuum mechanics, by which the same principles and solution methods can be used for physical forces and configurational forces. Furthermore, many applications of optimisation in biological systems are based on energy minimisation.

In this contribution, the method of configurational forces will be applied to the layout optimisation of tendons in pre-stressed concrete structures. The dual set of equilibrium equations (in terms of physical and configurational forces) has been derived and implemented in a finite element code. A load balancing method is adopted as an analysing method for obtaining the equivalent loads in pre-stressed concrete structures [3]. The accuracy of the analytic procedure can be evaluated using the method of minimum energy. The convergence rate is also employed so as to clarify the precision of the proposed scheme [4]. Some benchmark examples have been studied to analyse the potential of the method.

1

Bargmann, S., Kuhl, E., Askes, H., Steinmann, P., "Application of the Material Force Method to Structural Optimisation", XXI ICTAM, 15-21 August, Warsaw, Poland, 2004.

2

Askes, H., Barmann, S., Kuhl, E., Steinmann, P., "Structural Optimisation by Simultaneous Equilibration of Spatial and Material Forces", Commun. Numer. Meth. Engng, 21, 433-442, 2005. doi:10.1002/cnm.758

3

Aalami, B.O., "Structural Modeling of Positioned Members", Journal of Structural Engineering. 126(2), 157-162, 2000. doi:10.1061/(ASCE)0733-9445(2000)126:2(157)

4

Zienkiewicz, O.C., The finite element method. - Vol.1: The basis, 5th ed, Oxford: Butterworth-Heinemann, 2000.

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