Numerical modeling of concrete strength under multiaxial confinement pressures using linear genetic programming
Introduction
High strength concrete, high performance concrete and ultra-high performance concrete are nowadays widely used in civil engineering tasks. This implies the necessity of having a better understanding of the relationship between the mechanical properties of these materials. One of the major technical interests for different concrete types is the failure mechanisms under uniaxial and triaxial loading. On the other hand, as known, there is a wide variety of investigations which have focused on the development and strengthening of structural elements under confinement pressures. Steel fibers, fiber-reinforced polymer (FRP) composites and many other types of composites are the devices which improve the loading capabilities of structural concrete elements under confinement pressures [1], [2], [3], [4]. However, according to literature, there is a huge gap between the experimental and the numerical knowledge of failure mechanisms for the triaxially compressed concrete and composite structures. Hence, it is necessary to develop some equations correlating the confining pressures, uniaxial compressive strength, peak triaxial strengths and other mechanical characteristics for an efficient design of the triaxially compressed concrete structures. In this context, Cordon and Gillespie [5] performed triaxial tests on the plain concrete under multiaxial loading conditions to investigate the effect of different parameters on the strength of concrete. Lahlou et al. [6] presented the preliminary results of an experimental investigation on the confined high strength concrete. In this manner, Chern et al. [7] studied the response of fiber-reinforced concrete under multiaxial loading conditions. Xie et al. [8] performed a series of triaxial tests on the high strength concrete specimens. Also, Attard and Setunge [9] and Setunge et al. [10] performed a series of triaxial tests on larger cylindrical samples involving concretes with strengths as high as 109 MPa. Bohwan et al. [11] presented the preliminary results of an experimental investigation on confined concrete. Imran and Pantazopoulou [12] studied the behavior of plain concrete under triaxial stress states and the damage characteristics of concrete under dry and saturated conditions. In a similar manner, Nielsen [13] prepared an experimental study on the assessment of the triaxial behavior of ultra-high strength mortar. Some researchers performed a series of triaxial tests on the high strength concrete specimens involving concretes with strengths as high as 103 MPa [14], [15]. Candappa et al. [16], [17] performed a series of triaxial tests under low confining pressures on the four high-strength concretes with uniaxial compressive strengths of 41.9 to 103.3 MPa. Moreover, Mei et al. [18] performed an experimental investigation on the stress–strain characteristics of steel sleeve confined high strength concrete. Sfer et al. [19] conducted an experimental study of the confined compression behavior of concrete subjected to hydrostatic pressure for triaxial tests. The behavior of high strength concrete and steel fiber reinforced high strength concrete (1% steel fiber volume fraction) under triaxial compression was investigated by other researchers [20], [21], [22]. Recently, Farnam et al. [23] assessed multiaxial compressive behaviors of high strength concrete, high performance fiber reinforced concrete (HPFRC) and slurry infiltrated fiber concrete (SIFCON) under different confining pressures.
Prediction of the properties of concrete using statistical empirical relationship and numerical methods is a difficult task due to the extreme complexity involved [24], [25], [26], [27]. Besides, the current in-use empirical and numerical models are based on the theories of strength of material that in some cases cannot release the exact and precise results. This is caused due to the lack of data and insufficient knowledge of the micro and macro properties of materials. Such models are derived on the basis of several assumptions, approximations, and simplifications. An efficient idea to cope with this problem is to use machine learning methods. These systems learn adaptively from experience and extract various discriminators. They rely on the real data and natural rules in the nature. In most cases, they can release more exact, precise and reliable results than conventional modeling approaches [26]. Thus, they can be used as valuable tools for facilitating the computer-aided design of various civil engineering projects. Artificial neural networks (ANNs) are the most widely used machine learning methods. ANNs and other machine learning methods have been applied to assess different concrete properties [24], [25], [28], [29], [30]. Despite the acceptable performance of ANNs, they usually do not give a certain function to calculate the outcome using the input values.
Genetic programming (GP) [31] is a new machine learning approach for behavioral modeling of structural engineering problems. The main advantage of the GP-based methods is their ability to generate prediction equations without assuming prior form of the existing relationship. Recently, a robust variant of GP, namely gene expression programming was utilized to predict the ultimate strength of concrete under triaxial compression loading [32]. The triaxial strength of concrete was formulated in terms of mix design parameters such as triaxial confining pressure, ratio of water and superplasticizer summation to binder, ratio of coarse aggregate content to fine aggregate content, fiber index, and age of specimens [32]. Linear genetic programming (LGP) [33], [34] is a new subset of GP with a linear structure similar to the DNA molecule in biological genomes. Unlike classic GP and other soft computing tools like neural networks, the LGP applications to solve problems in civil engineering are restricted to fewer areas [35], [36].
The machine learning methods use the construction information database to automate and computerize the design stages without any need to perform the experimental and numerical tests. The main contribution of this paper to the domain knowledge is to propose and validate the novel machine learning technique, LGP, to automate concrete design for civil infrastructure and building construction. More specifically, the LGP technique is utilized to build predictive models for the strength of concrete under triaxial compression. The concrete strength is formulated in terms of uniaxial compressive strength and confining pressure. The predictions made by models are further compared with those provided by several models found in the literature [8], [9], [10], [11], [14], [15], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50].
Section snippets
Genetic programming
GP is a developing subarea of evolutionary algorithms inspired from Darwin's evolution theory. It may generally be defined as a supervised machine learning technique that searches a program space instead of a data space [31], [51]. GP is an extension of genetic algorithms (GAs). Most of the genetic operators used in GA can also be implemented in GP with minor changes. The main difference between GP and GA is the representation of the solution. The GP solutions are computer programs that are
Failure models
Table 1 presents the empirical strength criteria proposed for high strength concretes obtained from the literature [32].
Experimental database
The model is developed based on a comprehensive database obtained from the literature. The gathered database contains 370 test results for the strength capacity of concrete under triaxial compression [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23]. Table 2 shows the geometrical and mechanical properties of the samples
Model validity
Smith [61] states that if a model gives R > 0.8, there is a strong correlation between the predicted and measured values. The model can therefore be judged as very good. Based on the results, the proposed LGP models with low RMSE and MAE, and high R values are able to predict the target values to an acceptable degree of accuracy. Frank and Todeschini [62] argue that the minimum ratio of the number of objects over the number of selected variables for model acceptability is 3. They also suggest
Conclusion
A robust variant of GP, namely LGP, was utilized to formulate the strength capacity of the concrete under confinement conditions. Accurate empirical models were derived for the prediction of the concrete ultimate triaxial strength. A reliable database from the previously published triaxial test results was used for developing the models. The proposed LGP-based models are capable of predicting the ultimate strength of concrete under triaxial compression with high accuracy. The validity of the
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