A lot of the theoretical foundation for limit analysis of reinforced concrete structures came about in the 60’ties. Much of this work is due to M. P. Nielsen (Nielsen, M.P., 1999).
The first numerical formulations were presented in the late 70’ties. Since then formulations for beams, stringers, disks and plates have been presented. These formulations were all based on linear constraints and solved using Linear Programming (LP). Most of these structural components were designed without interaction capabilities but the disk element by Poulsen and Damkilde 2000, can work in combination with beams and bars in 2D.
Today more efficient optimization algorithms exist such as Second-Order Cone Programming and Semidefinite Programming. The Second-Order Cone Programming makes the formulation of disks and plates much more direct and eliminates the need for linearization of the yield surface. Niebling, Vinther and Larsen 2007, applied the Second-Order Cone Programming to reinforced concrete beams and shells with interaction in 3D. The Semidefinite Programming is very efficient as it can be applied for a 3D Mohr-Coulomb material, see e.g. Krabbenhøft et. al., 2008 and Larsen et. al. 2009, has applied the Semidefinite Programming to modeling of 3D reinforced concrete structures, see figure. This 3D modeling capability opens up for a new and much more detailed analysis of reinforced concrete structures.
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Rigid plastic 3D modelling of reinforced concrete structures.
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References
Nielsen, M.P. Limit Analysis and Concrete Plasticity. CRCPress. 1999.
Poulsen, P.N. & Damkilde, L. Limit state analysis of reinforced concrete plates subjected to in-plane forces. International Journal of Solids and Structures 2000, 37(42), 6011–6029.
Niebling, J., Vinther, A. & Larsen, K.P., Numerisk Modellering af Plastiske Betonkonstruk-tioner. Master thesis, Department of civil engineering, BYG-DTU. 2007.
Krabbenhøft, K., Lyamin, A.V. & Sloan, S.W. Three-dimensional Mohr-Coulomb limit analysis using semidefinite programming. Communications In Numerical Methods in Engineering, 24(11), 1107–1119. 2008.
Larsen, K.P., Poulsen, P.N. & Nielsen, L.O., Limit analysis of solid reinforced concrete structures. Submitted for publication, 2009.