Improving semi-empirical equations of ultimate bearing capacity of shallow foundations using soft computing polynomials
Highlights
► Improved weighted genetic programming (WGP) and novel soft computing polynomials (SCP) are proposed. ► The ultimate bearing capacity of shallow foundations is predicted and programmed. ► The SCP is further used to improve existed semi-empirical equations. ► Visible polynomials facilitate parameter studies. ► Sensitivity analysis for parameters and pruning techniques for polynomials are investigated.
Introduction
Soft computing approaches include neural networks (NNs), fuzzy logic, support vector machines, genetic algorithms (GAs), and genetic programming (GP). Each has unique benefits when applied to particular application categories. NNs are the most commonly used soft computing approaches for inference tasks, from which many NN derivatives have been developed and applied (Tran et al., 2007, Mehrjoo et al., 2008, Moghaddas Tafreshi and Tavakoli Mehrjardi, 2008, Behzad et al., 2009, Tsai, 2009, Tsai, 2010, Ismail and Jeng, 2011, Rezania et al., 2011). However, NNs have been characterized as “black box” models due to the extremely large number of nodes and connections within their structures. Since it was first proposed by Koza (1992), GP has garnered considerable attention due to its ability to model nonlinear relationships for input–output mappings. Baykasoglu et al. (2008) compared a promising set of GP approaches, including Multi Expression Programming (MEP), Gene Expression Programming (GEP), and Linear Genetic Programming (LGP) (Oltean and Dumitrescu, 2002, Ferreira, 2001, Bhattacharya et al., 2001). Notably, LGP was the most efficient algorithm for studied limestone strengths. Differences between these algorithms are rooted in the methodology utilized to generate a GP individual. A chromosome representation, a tree topology, and a linear string are used by MEP, GEP, and LGP, respectively. Although, some formulas generated by MEP, GEP, and LGP have coefficients, all coefficients are fixed constants (Baykasoglu et al., 2008). Several studies have utilized GP derivatives for construction industry problems. Baykasoglu et al. (2009) applied GEP to determine concrete strength, cost, and slump. Yeh and Lien (2009) developed a genetic operation tree (GOT) to investigate concrete strength. The GOT uses a tree topology (as does GEP) and optimized coefficients that differ from other GP derivatives. Coefficients do not frequently appear in formulas programmed using any of these GP models. Tsai (2011) proposed a weighted GP (WGP) to introduce weight coefficients into tree connections and generate a fully weighted formula.
Shallow foundations have a depth-to-width ratio less than four. Two fundamental performance criteria are ultimate bearing capacity and foundation settlement. Ultimate bearing capacity, which depends on soil strength, is calculated using equations developed by Terzaghi (1943), Meyerhof (1963), Vesic (1973), and so on. Available methods for determining the ultimate bearing capacity include limit equilibrium method (Silvestri, 2003), slip line method (Bolton and Lau, 1993), limit analysis method (Soubra, 1999), finite element method (Griffiths, 1982), etc. Of these, the limit equilibrium method is the most popular. However, figuring out various factors affecting on the ultimate bearing capacity is difficult. Most methods require assumptions that are inconsistent with experimental data. A basic method of determining the ultimate bearing capacity of a foundation is in situ testing. However, this method is time consuming and costly. Soft computing approaches are therefore alternatives for estimating ultimate bearing capacity based on historical datasets.
Lee and Lee (1996) successfully applied NN to predict pile bearing capacity. Padmini et al. (2007) used neurofuzzy to analyze shallow foundations on cohesionless soils. Kalinli et al. (2011) also used hybrid models with fuzzy theory and NN for shallow foundations. Although these studies agreed that soft computing approaches are more accurate compared to analytical formulas, they could not meaningfully represent ultimate bearing capacity. Although, Kalinli et al. (2011) provided formulas with ant colony optimization to calculate ultimate bearing capacity, the formula formats were formatted before the optimization. Soft computing approaches can achieve good prediction accuracy. Particularly, GP and its derivatives are potential to provide visible formulas additionally without any assumption of formula types.
Giustolisi and Savic (2006) proposed an evolutionary polynomial regression (EPR) and its applications had been validated in civil engineering (Berardi et al., 2008, Doglioni et al., 2010). Giustolisi and Savic (2006) argued that GP is not very powerful in finding constants and that it tends to produce functions that grow in length over time. This paper significantly improves GP in finding constants and controls formula length with layered tree structures and a terminal. Additionally, EPR has to set up the number of additive terms. Detailed comparisons on EPR and the proposed model will be revealed after resultant formulas are available.
The main aims of this paper are as follows:
- (1)
improve WGP;
- (2)
modeling ultimate bearing capacity of shallow foundations with good prediction accuracy and visible formulas;
- (3)
provide polynomials with a modified WGP, namely, soft computing polynomials (SCP);
- (4)
improving semi-empirical equations with the SCP;
- (5)
study parameter impact using sensitivity analysis;
- (6)
prune techniques for compacting formulas.
The remainder of this paper is organized as follows. Section 2 presents the proposed WGP, SCP methods and GAs. Section 3 characterizes details of ultimate bearing capacity of shallow foundations. Section 4 gives analytical results, comparisons, and discussions. Section 5 gives conclusions.
Section snippets
Weighted genetic programming
Following the study by Tsai (2011), this study presents a WGP method with an NL-layered tree structure (Fig. 1). The eventual layer has 2NL−1 parameter nodes and each parameter node (xiNL) selects one input (including a unit parameter “1”). When a unit parameter is selected, the value of the parameter node uses its weight (i.e., value of w is not 1) to create a coefficient.where xiNL represents nodes in the NL-th layer and i is a related node number; Pj is the j
Ultimate bearing capacity of shallow foundations
Terzaghi (1943) proposed a well known semi-empirical equation for calculating ultimate bearing capacity. In accordance with Terzaghi (1943), Meyerhof (1963) developed an equation considering shearing resistance caused by a failure surface above the bottom of the foundation. The ultimate bearing capacity can be expressed aswhere qu is the ultimate bearing capacity of shallow foundations; c is the cohesion; γ is the soil density; D is the depth of the
Predictions and visible formulas
This study utilizes NLs in the range of 2–6 to model ultimate bearing capacity (qu) of shallow foundations. Statistical results adopts 20 runs for qu. Results focus on training/testing RMSE and “Count”. The “Count” is used to count the number of activated operators. For instance, a fully linked four-layer tree has 7 operator nodes; thus, the “Count” is 7. When a “T” function is used in the third layer, the “Count” is 6. As a “T” occurs in the second layer, three operator nodes are eliminated
Conclusions
This study proposes WGP and SCP for modeling the ultimate bearing capacity of shallow foundations and improving its semi-empirical equations. Both models achieve good prediction accuracy and visible formulas. The significant findings of this study are as follows:
- 1.
Previous WGP is improved with a terminate operator, which reduces tree complexity and makes formulas compact.
- 2.
Visible formulas are bonus production of the WGP and SCP compared with black-box approaches. Unlike analytical models, the
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