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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 77
Edited by: B.H.V. Topping
Paper 104

The Computational Efficiency of Two Rigid Block Analysis Formulations for Application to Masonry Structures

H.M. Ahmed and M. Gilbert

Department of Civil and Structural Engineering, University of Sheffield, United Kingdom

Full Bibliographic Reference for this paper
H.M. Ahmed, M. Gilbert, "The Computational Efficiency of Two Rigid Block Analysis Formulations for Application to Masonry Structures", in B.H.V. Topping, (Editor), "Proceedings of the Ninth International Conference on Civil and Structural Engineering Computing", Civil-Comp Press, Stirlingshire, UK, Paper 104, 2003. doi:10.4203/ccp.77.104
Keywords: limit analysis, masonry arch bridges, rigid block, linear programming, joint equilibrium, redundant forces.

Rigid block analysis is a computational limit analysis method which is now widely applied to the analysis of masonry gravity structures such as arch bridges. Various formulations of the rigid block analysis method have been proposed, with linear programming (LP) generally being used in the solution process. However, there appears to be little information in the literature on the relative computational efficiencies of the various formulations (when used either with traditional simplex or newer interior point LP solvers). This paper aims to at least partially address this, by considering two alternative formulations.

According to the classification of LP formulations presented by Tam and Jennings for plastic frame design [1], Livesley's original rigid block analysis formulation [2] may be classified as a `joint equilibrium' approach. In this approach the problem variables comprise both the interface forces and the resultant forces acting on each and every block. Conversely, Gilbert and Melbourne's formulation for single and multi-span arch bridges [3] may be classified as a `redundant forces' approach (though the dual form of this was actually presented). In this approach the problem variables are both the interface forces and (only) the redundant forces.

In general the redundant forces method will produce LP tableaux containing fewer variables and constraints than the joint equilibrium method. However, in the case of multi-ring masonry arch analysis problems, a problem with the redundant forces method is that the presence of inter-ring contacts leads to numerous redundant forces (i.e. additional forces at every contact point on the interface between rings). Hence the resulting tableaux, although compact, will be very heavily populated with non-zero elements; this can be quite computationally expensive to solve. A recent reappraisal of the situation has also indicated that modern LP methods (i.e. interior-point based LP methods) may in fact be well suited to the solving the type of large sparse tableaux produced by the joint equilibrium method. This is investigated in the paper.

Thus `redundant forces' and `joint equilibrium' formulations are described and then tested by applying them to a number of case study problems (these comprise: multi-ring arch, multi-span arch and block wall problems). It is found that whilst the redundant forces formulation is the most computationally efficient formulation for simple single and multi-span arch problems, the joint equilibrium formulation is the most computationally efficient formulation when applied to more complex geometrical arrangements of blocks, such as those found in multi-ring arches. This is because of the low proportion of non-zero elements present in the albeit large LP tableau generated when using the joint equilibrium formulation. This sparseness greatly speeds the solution process when using most modern linear programming solvers, particularly those based on interior point methods. Furthermore, using the joint equilibrium formulation it is also relatively easy to automatically set up and then solve problems involving arbitrary two and three dimensional assemblages of rigid blocks.

However, it is well known that adoption of the standard LP limit analysis assumption of normality in the presence of sliding at frictional interfaces [i.e. using an associated flow rule] can potentially lead to unsafe estimates of the load factor. Whilst in many cases the over-estimate of the computed load factor when friction is modelled in this way will be small, unfortunately this cannot be guaranteed, with the magnitude of the over-estimate generally being highly problem dependent. Thus when certain geometrical configurations of blocks are considered the over-estimate may be quite high. For example, in the case of a wall problem documented in the paper, a load factor was obtained which was some 20 percent higher than that obtained when using a non-associative constitutive model [4]. Unfortunately modelling non-associative friction in rigid block analysis is difficult and the methods proposed to date have tended to rely on the use of specialised solution algorithms (not LP), with only relatively small problems being tractable at the present time. The authors are therefore currently working on the development of more efficient solution methods for such problems.

T. Tam, A. Jenning, "Classification and comparison of LP formulations for the Plastic Design of Frames", Engineering Structures, 11, 163-178, 1989. doi:10.1016/0141-0296(89)90004-7
R.K. Livesley, "Limit analysis of structures formed from rigid blocks", International Journal for Numerical Methods in Engineering, 12, 1853-1871, 1978. doi:10.1002/nme.1620121207
M. Gilbert, C. Melbourne, "Rigid-block analysis of masonry structures", The Structural Engineer, 72, 356-361, 1994.
M.C. Ferris, F. Tin-Loi, "Limit analysis of frictional block assemblies as a mathematical program with complementarity constraints", International Journal of Mechanical Science, 43, 209-224, 2001. doi:10.1016/S0020-7403(99)00111-3

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