Empirical modelling of shear strength of RC deep beams by genetic programming
Introduction
Reinforced concrete (RC) deep beams are characterised as being relatively short and deep, having a thickness that is small relative to their span or depth, and being primarily loaded in their own plane. They are sometimes used for load distribution, for example as transfer girders, pile caps, folded plates and foundation walls. The transition from RC shallow beam behaviour to that of deep beams is imprecise. For example, while the ACI code [1], CEB-FIP model code [5] and CIRIA Guide 2 [6] use the span/depth ratio limit to define RC deep beams, the Canadian code [7] employs the concept of shear span/depth ratio. ACI defines beams with clear span to effective depth ratios less than 5 as deep beams, whereas CEB-FIP model code treats simply supported and continuous beams of span/depth ratios less than 2 and 2.5, respectively, as deep beams.
Several possible modes of failure of deep beams have been identified from physical tests but due to their geometrical dimensions shear strength appears to control their design. Despite of the large amount of research carried out over the last century, there is no agreed rational procedure to predict the shear strength of RC deep beams [12], [22]. This is mainly because of the very complex mechanism associated with the shear failure of RC beams.
The design of RC deep beams has not yet been covered by BS8110 [4] that explicitly states, “for the design of deep beams, reference should be made to specialist literature”. Comparisons between test results and predictions from other codes, such as ACI and CIRIA Guide 2, show poor agreement [28], [30].
In this paper, the genetic programming (GP) method is used to build an empirical model to estimate the shear strength of RC deep beams subjected to two point loads. The GP model will directly evolve from a set of experimental results available in the literature. A parametric study is conducted to examine the validity of the GP model predictions.
Section snippets
Overview of the genetic programming methodology
GP [13] is a branch of genetic algorithms (GAs). Its basis is the same Darwinian concept of survival of the fittest. While a GA uses a string of numbers to represent the solution, the GP creates a population of computer programs with a tree structure. In this application, a program represents an empirical model to be used for approximation of the shear strength of RC deep beams. A typical program, representing the expression (x1/x2+x3)2, is shown in Fig. 1.
These randomly generated programs are
Genetic operators
Model structures evolve through the action of three basic genetic operators: reproduction, crossover and mutation. In the reproduction stage, a strategy must be adopted as to which programs should die. In this implementation, a small percentage of the trees with the worst fitness are killed. The population is then filled with the surviving trees according to accepted selection mechanisms, as explained below. Crossover swaps randomly selected parts of two trees to combine good information from
Model tuning
The empirical model is characterised not only by its structure (to be found by the GP) but also by a set of tuning parameters to be found by the model tuning, i.e. the least square fitting of the model into the set of experimental values:To solve this optimization problem, a combination of a GA to find an initial guess followed by a gradient-based optimization method [15] is used.
Parameters affecting shear strength of deep beams
Fig. 4 shows the geometrical dimensions and reinforcement of a typical RC deep beam tested under two point loads. The main parameters influencing the shear strength of RC deep beams are the concrete compressive strength, main longitudinal top and bottom steel reinforcement, horizontal and vertical web steel reinforcement, beam width and depth, shear span and beam span [3], [10], [24]. Those parameters can be expressed in normalised form as follows:
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Normalised shear strength λ=P/bhfc′, where P
Empirical model obtained by GP
There is a large number of test results of RC deep beams referred to in the literature. Test results of 141 deep beams reported in [10], [11], [17], [20], [21], [23], [24], [25], [27], [29] are used to create the GP response. The training dataset covers a wide range of each parameter as given in Table 1. All selected beams were tested under two point loads; other load arrangements have been excluded.
The mathematical operators addition, multiplication, division, square and negation and a
Parametric study
The influence of different parameters on the shear strength λ is studied next. In this study, expression (6) has been used to predict the shear strength outside the range where the model was constructed. Parameter x1 is predicted up to the value 2.5 and parameter x5 starts from the value 0 (see Table 1 for the initial range of the parameters of the training dataset).
Fig. 6 gives the relationship between the shear span to depth ratio x1 against the shear strength λ for different values of the
Conclusions
An empirical model to predict the shear strength of RC deep beams has been obtained by GP. Experimental results are used to build and validate the model. Good agreement between the model predictions and experiments has been achieved. As more experimental results and knowledge of the shear behaviour of deep beams become available, the GP prediction could be improved.
The GP model predicts the following behaviour between the shear strength and the influencing parameters:
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The shear span to depth and
References (31)
- ACI Committee 318. Building code requirements for structural concrete (ACI 318-99) and commentary (ACI 318R-99)....
- et al.
Experimental evaluation of design procedures for shear strength of deep reinforced concrete beams
ACI Struct. J.
(2002) Shear capacity of reinforced concrete deep beams
Struct. Eng. J. ASCE
(2000)- British Standards Institution (BSI). Structural use of concrete. Milton Keynes: BSI; BS8110: Part 1,...
- CEB-FIP. Model code 1990 for concrete structures. Thomas Telford Services, Ltd, London, Comité Euro-International du...
- CIRIA Guide 2. The design of deep beams in reinforced concrete. London: Over Arup and Partners, and Construction...
- CSA Technical Committee on Reinforced Concrete Design, A23.3-94. Design of concrete structures. Ontario, Canada:...
- et al.
Dimensionally aware genetic programming
- et al.
Genetic programming within a framework of computer-aided discovery of scientific knowledge
- et al.
Web reinforcement effects on deep beams
ACI J.
(1970)