Keywords: verified numerical computing, interval analysis, reinforced concrete beams, optimisation, automatic design.

For the automation of the shear reinforcement design [5] a mathematical model for stepped overestimations with left-bounded step widths of the shear force function is defined. This overestimating function delivers the data for the mass of the reinforcement steel to be utilized. If the overestimation (the difference between the shear forces and the effects the designed reinforcement can absorb in fact) is low, then the structure is inexpensive.

Although several new optimization methods exist (for example in reference [2]) the problem is treated by interval analysis tools. This not only guarantees the results mathematically [4] but makes is easy to deliver global solutions for the problem. The applied methods are finite subdivision and simple branch-and-bound principle based algorithms (see for example reference [3]). The stepped overestimating functions are constructed using various heuristics, as listed in [1], assuring fast functioning and robust behaviour.

A further condition also has to be taken into account. Due to standards, at least half of the shear force has to be absorbed by stirrups. If a stepping method creating an overestimating step function is applied twice, an optimal reinforcement combining stirrups and bent-up bars complying with this condition can be constructed. Such an algorithm is introduced and presented on a numerical example. The saving on that particular beam is around 13% as compared to a traditional method using only stirrups.

1

A.E. Csallner, "Step Function Overestimators With Left-Bounded Step Widths", Book of Abstracts of the SCAN 2006, pp.137, Duisburg, Germany, September 26-29, 2006.

2

A. Csébfalvi, "Nonlinear Path-Following Method for Computing Equilibrium Curve of Structures", Annals of Operation Research 81, 15-23, 1998. doi:10.1023/A:1018944804979

3

R.B. Kearfott, Rigorous global search: continuous problems, Kluwer, Dordrecht, 1996.

4

M.C. Markót, T. Csendes, "A new verified optimization technique for the "packing circles in a unit square" problems", SIAM J. on Optimization 16, 193-219, 2005. doi:10.1137/S1052623403425617

5

MSZ EN 1992-1-1:2005 Eurocode 2: Design of concrete structures. Part 1-1: General rules and rules of buildings.

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