Indirect estimation of the ultimate bearing capacity of shallow foundations resting on rock masses
Introduction
Shallow foundations support structures at a shallow depth below the ground surface and transmit applied loads to the underlying materials such as soils, rocks or intermediate geo-materials. In general, any foundation design must satisfy at least two important criteria 1., 2., 3.: (1) obtaining sufficient bearing capacity of underlying layer against ultimate failure, and (2) achieving acceptable total or differential settlements under working loads. Although, the design of foundations resting on or in rock masses is commonly controlled by the settlement criterion, the bearing capacity of rock mass must be estimated to evaluate the stability 4. Therefore, in order to provide a precise and efficient design of a foundation, it is crucial to account for the bearing capacity of the rock mass beneath it. According to the rock mass properties and the beneath layer of it, the failure may occur in several mechanisms 5. Bearing capacity failure in an overloaded rock foundation is one of them. The mode of bearing capacity failure mainly depends on the ratio of space between joints (S) to foundation width (B), joint condition (open or closed) and direction, rock type as summarized in Table 1and schematically represented in Fig. 15., 6., 7..
The most usually utilized approaches to determine the bearing capacity (qult) of foundations on rocks can be classified into four groups: (1) codes, (2) analytical methods, (3) semi-empirical methods, and (4) in-situ and full-scaled testing methods 4. Codes often propose conservative values for estimating the allowable bearing pressure or ultimate bearing capacity 8., 9., 10.. These presumed values are derived from local experience and geology from a particular site, however, the engineer should ensure that they are applicable to the particular conditions relevant to the considered site 4. On the one hand, analytical methods are based on bearing capacity theories, including limit equilibrium methods, using initial assumptions and relate qult to footing geometry and rock properties such as those equations provided by 6., 11., 12.. On the other hand, semi-empirical and empirical methods are often obtained by the correlation between qult and rock mass properties, based on empirical observations and experimental test results such as equations made by 13., 14., 15.. General forms of mostly utilized and traditional equations proposed by various researchers in the literature are summarized in Table 24., 7., 8., 9..
As represented in Table 2, analytical methods include terms of physical and mechanical properties of rock mass and geometry of the foundation but not include terms of rock type, classification and qualitative parameters of rock mass Also, semi-empirical and empirical methods often relate qult to quantitative and qualitative of rock mass and are not prepared for the geometry of foundations or space between joints (Table 2).
The bearing capacity of a shallow foundation on a jointed rock mass mostly depends on geometry of foundation, the ratio of joint spacing to foundation breadth or loading width, as well as rock mass qualities such as joint conditions (open or closed), rock type and rock mass strength 5., 12., 15., 16., 17., 18.. In regarding to the equations in the literature, there is not a comprehensive model including simultaneously both quantitative and qualitative parameters, such as foundation geometry and RMR. Thus, the complexity of analysis of bearing capacity behavior and accounting for the influences of different parameters on the bearing capacity factor implies that there is the necessity for a more comprehensive model.
By progressing in computational software and hardware systems, several computer-aided modeling and soft computing techniques such as artificial neural networks (ANNs), adaptive neuro-fuzzy system (ANFIS), fuzzy inference system (FIS), support vector machine (SVM) and genetic programming (GP) have been realized by various researchers in several civil engineering domains. Such computing techniques have a lot of features that have made them attractive choice for predicting different problems. The first feature is they are data-driven self-adaptive methods. That means they do not require many prior assumptions about the models of the problem under study. They automatically learn from data to determine the structure of a prediction model. These techniques become more attractive because of their capability of information processing, such as non-linearity, high parallelism, robustness, fault and failure tolerance and their ability to generalize. Besides, these techniques have been successfully employed to solve problems in civil engineering field 19., 20., 21., 22., 23., 24., 25., 26., 27., 28., 29..
The aim of this paper is to utilize a powerful branch of genetic programming (GP), namely linear genetic programming (LGP), to derive a more comprehensive predictive model for the ultimate bearing capacity of shallow foundations resting in/on jointed rock masses. A comprehensive and reliable set of data including 102 rock socket, centrifuge rock socket, plate load and large-scaled footing load test results are collected to develop the model. In order to verify the robustness of the obtained model several validation and supplementary study phases are conducted.
Section snippets
Genetic programming
Genetic programming, as a subset of evolutionary computational intelligence approaches, considers the synthesis of Darwinian ideas of genetic inheritance, natural variation and selection to solve complicated problems. In general, in genetic programming (GP), inputs and corresponding output data samples are known and the main goal is to generate predictive models relating them without any prior assumptions 30., 31.. Typically in GP, a population of individuals initialized and members of the
Numerical simulation of bearing capacity
In order to reach reliable estimations of the bearing capacity of shallow foundations on rock masses, the influence of several parameters should be incorporated into the model development. Regarding to the general forms of the existing equation for indirect estimation of qult of shallow foundations on rock masses, mainly depends on embedment or excavation depth (D), angle of internal friction for the rock mass (φ), effective unit weight of the rock mass (γ), breadth or width of foundation (B),
Development of the LGP-based model
Several runs are performed to provide LGP-based models for the prediction of the ultimate bearing capacity (qult) of shallow foundation resting on/in jointed (non-fractured) rock masses. The LGP parameters are changed for each run. The parameters are selected based on previously suggested values 27., 38., 47., 48.. The developmental processes were controlled by considering evolutionary parameters such as population size, probability of crossover, probability of mutation and selecting arithmetic
Performance analysis and validation of LGP-based model
According to Smith's recommendation 51, if a model gives a correlation coefficient (R)>0.8, there is a strong correlation between the predicted and measured values. In addition, the error values should be considered in all cases. In this study, to evaluate the performance of the models correlation coefficient (R), root mean squared error (RMSE) and mean absolute error (MAE) as suggested by several researchers. These parameters were calculated using the following equations:
Conclusion
This paper aimed at developing a new formulation for indirect estimation of the bearing capacity of shallow foundations resting on/in rock masses, using a novel and powerful evolutionary computational technique, namely linear genetic programming. Unlike other intelligent methods such as ANN, FIS and ANFIS, the LGP method provides simplified equations that can be readily used for the design purposes via hand calculating. A comprehensive and reliable database, collected from the literature, is
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