ANN-based empirical modelling of pile behaviour under static compressive loading

Abstract

Artificial neural networks have been widely used over the past two decades to successfully develop empirical models for a variety of geotechnical problems. In this paper, an empirical model based on the product-unit neural network (PUNN) is developed to predict the load-deformation behaviour of piles based SPT values of the supporting soil. Other parameters used as inputs include particle grading, pile geometry, method of installation as well as the elastic modulus of the pile material. The model is trained using full-scale pile loading tests data retrieved from FHWA deep foundations database. From the results obtained, it is observed that the proposed model gives a better simulation of pile load-deformation curves compared to the Fleming’s hyperbolic model and t-z approach.

Keywords

piles in compression / load-deformation behaviour / product-unit neural network

Introduction

Piles have been widely used as foundations for various important structures such as tall buildings, bridges and wind farms etc. One of the major functions of a pile foundation is to transmit axial loading fromthe superstructure to the subsurface through shaft friction and end bearing. Prior to construction, load-deformation analysis is normally carried out to determine whether or not a pile settles beyond the serviceability limit under a working load. Considering the nature of inter relationship between pile movement and that of the supporting soil during the load transfer process, accurate estimation of pile settlement is quite a challenging task. The complex soil behaviour, as well as the uncertainty with regards to parameter estimation of in-situ soil due to spatial variability and effects of pile installation make the soil-pile response even more difficult to accurately predict. This particularly undermines the worthiness of rigorous computational models such as non-linear Finite Element analysis, which require more detailed material information in order to predict the pile deformation behaviour. While suitably accurate information about the soil and soil-pile interface for FEM analysis may be difficult to determine due to sampling and pile installation related problems as Poulos [¹] rightly observed, much less rigorous methods such as the elastic solution, which can rely on correlations with in-situ tests for estimating the relevant soil parameters used in the analysis, are overly simplistic in their the assumption of soil behaviour [²], and their usage is limited to very low strain levels where the soil can be safely regarded as such [³]. Hyperbolic models proposed by Chin [⁴] and Fleming [⁵] do not have the rigour of computationally intensive FE analysis approach but are capable of making a more rational assumption about soil behaviour than elastic solutions. The limitation of hyperbolic models is that, despite being in agreement with general consensus that soil is a non-linear geo-material, the constants of a single hyperbolic equation may be too few to adequately represent the complex behaviour of load transfer along the pile shaft for various types of soil [⁶].

A full scale pile loading test provides the most accurate information about the axial pile behaviour, but the budget constraint makes it infeasible to carry out in every piling project, and its use beyond the site it was carried out at is severely limited by the variability in site conditions. The use of archived Pile loading test results remains of limited value due to the difficulty faced by conventional empirical modelling tools in producing a correlation between pre-installation soil information and post-installation behaviour in the pile-soil system.

In this paper, an attempt is made to utilise the available pile loading test data to develop an em piricalmodel of load- settlement behaviour of an axially loaded pile based on a product-unit neural network. This work is inspired by the highly successful applications of neural networks in such a wide range of engineering problems as materials and structural modelling [⁷,⁸], geotechnical modelling [⁹], hydraulic and hydrologic analyses [¹⁰–¹²], etc. Other modelling tools based on soft-computing such as the moving least square method (MLS) have also been successfully used in simulating the behaviour of complex materials [¹³,¹⁴]. The proposed model is intended to predict the relationship between the pile head deformation and the magnitude of axial loading exerted on the pile given the pile geometry, the elastic modulus of pile material, the properties of the foun dation soil, as well as some information about pile installation procedure. The depth un-corrected SPT number of blows is used to represent the sub-surface properties, with soil classification infor mation as additional information to make-up for the limitation of SPT in reflecting the variation in soil types. The aim is to provide a simple but reliable means of analysing the behaviour of axially loaded piles by developing a correlation between full scale pile loading tests and simple in-situ soil tests such as the SPT.

The neuronet is trained using a hybrid optimisation method combining PSO and BP algorithms in order to benefit from the advantage of the former with respect to its local search capability and the advantage of the latter with respects to its ability to escape from local optima. Based on prediction accuracy with regards to unseen (testing) data, the performance of the hybrid training algorithm adopted in this work will be compared with BP, PSO and existing PSO-BP hybrid algorithms. To further evaluate the ability of the ANN-based empirical model in simulating the load-deformation curve, comparisons will be made with predictions based on hyperbolic and nonlinear load transfer (t-z) methods.

Neural network modelling

Neural networks provide a unique alternative for modelling the behaviour of pile foundations. Their ability to map complex relationship between variables makes them quite suitable for such complex problems as load deformation analysis of axially loaded piles. According to recent studies, ANN gives a better estimate of axial load-settlement curves compared with some of the most widely used methods [¹⁵,¹⁶]. In Ref. [¹⁵], a 4-hidden layer BPN with a sigmoid activation was used to estimate the pile head settlement, but with synaptic connections around 2000, the number of network parameters tends to be on the high side. Ismail and Jeng [¹⁶] attempted to address the size issue by proposing a network model based on product-units (PUNN), with a considerably smaller size, to simulate the load deformation curves. The common feature in both 63 studies is that the networks were trained using the back-propagation (BP) algorithm- which is often criticised for its tendency to converge towards a sub-optimal solution. In this paper, the PUNN network is be used to develop an empirical relationship describing the load-deformation behaviour of pile subject,. to axial loading with the aim of keeping the resulting formula as simple as possible. For the sake of enhancing the search for optimumnetwork parameters during training, the BP algorithmis coupled with particle swarm optimisation algorithm (PSO), a meta-heuristic and population-based search technique [¹⁷], to develop a robust hybrid training algorithm with both local and global search capabilities. The idea of putting together the two methods is based on the fact that neither gradient free nor gradient based algorithms are efficient under all circumstances.

Hybrid BP-PSO network training algorithm

The proposed training approach involves running PSO and BP routines intermittently in the course of ANN training, unlike other hybrid PSO-BP algorithms in which PSO is used essentially for weight initialisation such as Zhang et al.’s model [¹⁸]. The aim is to more efficiently utilise the search capabilities of both algorithms towards locating a best solution and reducing the risk of settling at a sub-optimal solution. The algorithm is executed by first generating a set of neural networks with randomly generated synaptic parameters, followed by a series of optimisation cycles; each of which entails updating the network parameters using BP and PSO algorithms. The cycles are repeated until a sufficiently accurate result is obtained. The flow chart in Figure 1 describes the steps involved in the parameter updating cycles during the network training.

As shown in the flow chart, the Levenberg-Marquardt (L-M) algorithm is used to carry out the BP phase of the updating cycle. L-M, a variant of gradient based methods, is chosen for its superior local search power compared to other methods such as gradient descent and conjugate gradient methods. In the case of PSO, the equations based on the modifications proposed by Clerc and Kennedy [¹⁹] are used in updating the network parameters. This is to avoid either a pre-mature convergence or solution divergence. Note that ‘particles’ referred to in the flowchart mean the set of neural networks generated at the beginning of the process.

To implement the proposed algorithm, the two constituent algorithms (BP and PSO) require a number of parameters to be defined by the user. The L-M BP algorithm requires the learning parameter and the adjustment parameter to be defined at the beginning of training. In this work, proposed by Fun and Hagan [²⁰] are adopted. A fairly large initial value of learning parameter, \eta, is selected on the premise that the solution likely to be far away from the local optimum at the beginning of the process.

As for the PSO algorithm, the swarm population size, the constriction parameter and acceler ation coefficients have to be set at certain values by the user. In this work, swarm (population) size is set at 60, while the optimised values of constriction parameter and acceleration coefficients suggested in Ref. [¹⁹] adopted. With regards to the number of steps which PSO is used to update the network parameters in each cycle, 100 iterations is suggested in this paper. This is not far away from 80 iterations proposed by Yu et al. [²¹]. The increased number of iterations in this case is an attempt to create more space between successive BP cycles in order to help reduce the overall computing time. At the end of PSO turn and at the point of BP takeover, the duplicate particles are removed as indicated in the flowchart (see Figure 1). Particles are regarded as duplicate when the difference between their training errors is less than 5%. This is based on the observation that particles within this range of error difference are likely to produce similar results when subjected to steps of local search based on BP algorithm. The search around best particles is carried out at the beginning by arranging the particles in the order of increasing error after the resetting the duplicate particles, then positions of the best 10% of the particles are updated using the BP algorithm.

Product-Unit processing function

Product-Unit neurons perform neural computation by processing the products of input they receive, instead of adding together the weighted input signals fed into them. In the case of commonly used feed-forward neural networks (i.e. neural nets based on sigmoid and radial basis functions), summation units are used to compute the processing signals, which often lead to large network sizes when dealing with highly non-linear functions involving higher order combinations of input variables. In such cases, Product-unit neurons, by virtue of their graeter information capacity, could reduce significantly reduce the required number of processing functions [²²]. The mathematical model of a product-unit neuron is:

(1){{\displaystyle O}}_j={\displaystyle \prod _{i=\mathrm{1}}^{i=n}{{{\displaystyle x}}_i}^{{{\displaystyle p}}_{i,j}}}

where O_j , x and p are the output of the product-unit neuron, the input signal and the synaptic weight respectively. The basic structure of product-unit network, as proposed by Durbin and Rumelhart [²²], consists of three hidden layers; the input layer, the product unit layer and the summation layer.

Pile loading test database

The data used in training and testing the model consists of results of static load tests carried out on 115 piles with 1285 data points. The pile data is sourced from the FHWA deep foundations load test database. The data is also available online at www.webgeotech.com. The data base consists of various types of pile, which include cylindrical and square concrete piles, H-steel piles and closed end steel pipe piles. The methods of installation include impact driving in the case of driven piles and wet and dry excavations in the case of bored piles. The number and type of piles in the database is given in Table 1, while the range of parameter values in the database is given in Table 2. The soil test results in the database consist of uncorrected SPT N values and soil classification. The soil classification is used to give additional information about the foundation soil. The soil types in the database range from coarse soils (sand) to fine soils (silt and clay). Only sites whose sub-surface consists of fairly uniform soil are considered for this study. This is to provide a rational basis for computing the average SPT values along the pile shaft. The important information that is not included in the database is the co-ordinate of the borehole logs relative to the point of pile installation. This is due to the unavailability of the information in the majority of the sites. If this information were available, it would have been helpful in judging whether the soil profile can be confidently regarded as representative of the soil encountered at the installation point.

Input variables

As a data-based model, the ability of a neural network to make an accurate prediction is largely dependent on the choice of input parameters. For a reliable network to be developed, a sound understanding of the factors controlling the system under study is therefore necessary. Considering the behaviour of single piles embedded into a soil medium and subjected to an axial load, the factors that control its deformation include parameters such as pile stiffness, mobilised shear resistance of the soil around the pile shaft and the bearing resistance of the soil at the pile base. The pile stiffness is dependent on the pile geometry and elastic modulus of the pile material. The soil resistance depends on mechanical properties of the soil around the pile and the characteristics of soil-pile interface. In this study, depth uncorrected SPT blow counts (N60) are chosen to represent the soil resistance. SPT is the most widely used in-situ test due to its lack of sophistication and low cost. Although less accurate and lacking in standardisation compared to other in-situ tests, it still gives some idea about strength and deformability of geo-materials [²³]. Numerous studies have been carried out with the aim of correlating SPT N and mechanical properties of soil, results of which were reported in the literature [²⁴–²⁶].

The magnitude of axial compressive load exerted on a pile head can be expressed as a non-linear function of related variables as:

(2)P=f(s,{{\displaystyle k}}_s,{{\displaystyle E}}_p,D,L)

where s, k_s, E_p, D and L are pile head settlement, soil resistance, pile modulus, pile diameter and pile length, respectively. The unknown function f() relates the parameters in the bracket and the pile head load.

For convenience sake, the soil resistance around the pile can be split into shaft resistance k_shaft and base resistance k_base. The shaft and base components of soil stiffness are mainly represented in this work by SPT N values around the shaft and at the base respectively. The SPT value around the shaft is determined by computing the weighted average value of N along the pile shaft:

(3){{\displaystyle \bar{N}}}_s={\displaystyle \frac{{\displaystyle \sum _{\mathrm{1}}^n{{\displaystyle N}}_i{{\displaystyle l}}_i}}{L}}

n is the number of layers along the shaft of pile segment; the thickness of pile segment i ; N_i the average value of SPT-N over the pile segment and L is the pile length. Other factors affecting the soil stiffness are the contact area (i.e., the shaft area in the case of k_shaft and base area in the case of k_base), the soil and pile types as well as the method of installation. By replacing the soil stiffness parameter in Eq. (2) with a number of related variables and splitting shaft and base components, the following expressions result:

(4a)P=Q_s+Q_b

(4b)Q_b={{\displaystyle f}}_{\mathrm{1}}\left(s,{{\displaystyle \overline{N}}}_s{{\displaystyle A}}_s,{\displaystyle \frac{{{\displaystyle E}}_p{{\displaystyle A}}_p}{L}},{{\displaystyle f}}_C,{{\displaystyle f}}_m,{{\displaystyle f}}_P,{{\displaystyle \lambda }}_{T\mathrm{1}},{{\displaystyle \lambda }}_{T\mathrm{2}},{{\displaystyle \lambda }}_{M\mathrm{1}},{{\displaystyle \lambda }}_{M\mathrm{2}}\right)

(4c)Q_s={{\displaystyle f}}_{\mathrm{2}}(s,{{\displaystyle N}}_s{{\displaystyle A}}_b,{{\displaystyle f}}_C,{{\displaystyle f}}_m,{{\displaystyle f}}_P,{{\displaystyle \lambda }}_{T\mathrm{1}},{{\displaystyle \lambda }}_{T\mathrm{2}},{{\displaystyle \lambda }}_{M\mathrm{1}},{{\displaystyle \lambda }}_{M\mathrm{2}})

It can be noted that pile head settlement is used in both Eqs. (4b) and (4c) instead of shaft or base deformations as the case may be. This decision is taken out of the desire to develop a simpler empirical relationship. The functions f₁ and f₂ in the expression, are representing the contributions of shaft and base resistance to the pile response respectively. Both f₁ and f₂ are unknown functions that will be replaced with a neural network. The term {\displaystyle \frac{{{\displaystyle E}}_p{{\displaystyle A}}_p}{L}} stands for the pile stiffness. A_s and A_b are the shaft and base areas respectively. A_p is the pile shaft cross-sectional area. The parameters f_C and f_m are the fractions of silt and clay respectively. Further information about the fine fraction is also provided with the help of f_P parameter. For high plasticity silts and clays, the parameter f_P is set at 1.0 and switched to 0.0 in the case of low or no plasticity. In this study, both percentage of fines and plasticity are approximately deduced from soil classification.

The binary variables {{\displaystyle \lambda }}_{T\mathrm{1}} and {{\displaystyle \lambda }}_{T\mathrm{2}} provide information about the method of pile installation. {{\displaystyle \lambda }}_{T\mathrm{1}} assumes a value of 1.0 when the pile is installed by driving, and 0 when the pile is bored. {{\displaystyle \lambda }}_{T\mathrm{2}} helps the network differentiate bored piles constructed using dry excavation and those installed using wet excavation. It is assigned a value of 0 when the excavation is dry and switched to 1.0 otherwise. The parameters {{\displaystyle \lambda }}_{M\mathrm{1}} and {{\displaystyle \lambda }}_{M\mathrm{2}}, which inform the network about the type of pile, operate in a similar manner. Both nodes are deactivated when the pile is made up of concrete. {{\displaystyle \lambda }}_{M\mathrm{1}} is switched on to 1 in the case of H-steel piles, while {{\displaystyle \lambda }}_{M\mathrm{2}} assumes 1.0 when the pile type is a steel pipe. The network structures representing the shaft and base components are shown in Figures 2-3.

To determine the shaft area in the case of H-steel piles, the dimensions 189 of an equivalent cylindrical pile is used. The radius of the equivalent pile is estimated based of the following suggestion by Fleming et al. [²⁷]:

(5){{\displaystyle r}}_{\mathrm{0}}={{\displaystyle \left({\displaystyle \frac{DH}{\mathrm{\pi }}}\right)}}^{\mathrm{0.5}}

where r₀ is the equivalent cross-sectional radius of the H-pile; Dand Hare the overall dimensions of H-section. It is important, however, to note that when the cross-sectional area or base area are required, only the cross-section of the H-steel is relevant, with the exception of the case when stiffening plates are welded onto the pile around the base.

Networks training and validation

The data was split into training and testing sets. The training data was used to teach the network while the latter was used to test the performance of the model. Data division is a crucial stage in developing a neural network model, as it significantly affects the network’s quality of prediction. While testing a network with as many sets of data as possible makes the validation more robust, provision of sufficiently large and diverse set of training data is necessary if the network is to reasonably learn the behaviour of the underlying function on which the data is based, given the sole dependence of the former on the data with which it is trained. There are no well defined rules on dividing the data into training and testing sets, but some rules of thumb have been suggested in the literature. For instance, Swingler [²⁸] and Looney [²⁹] proposed 20% and 25% of the data, respectively, to be used as testing set, while Nelson and Illingworth [³⁰] recommended between 20% and 30% of data for testing. Methods such as cross-validation and bootstrapping can also be used in data division in order to improve the network’s ability to generalise [³¹,³²]. In this work, the training data is kept around 70% of the database, while the remaining portion of the data was used for testing, the ratio of which constitutes the lower end of the spectrum of training/testing ratios suggested by Nelson and Illingworth [³⁰]. In order to avoid the detrimental effect of bad data points on the prediction quality of the network, the data groups (load-settlement curves) that are associated with a large network training error and whose exclusion from training significantly reduces the training error are regarded as an outliers and removed from both training and testing subsets. The network was trained using the proposed PSO-BP hybrid training algorithm described in section 2.1 (which is referred to as BP-PSO (II) henceforth). For the sake of comparison, a number of network models were also trained using the BP algorithm, the PSO technique and the BP-PSO algorithm proposed by Zhang et al. [¹⁸] (which is referred to as BP-PSO (I) henceforth).

The data division into training and testing subsets was carried out randomly, while at the same time ensuring that the training subset contains, as much as possible, the widest variations in input and output patterns in the database. This is to enable the network to capture the widest available range of the pile behaviour.

The accuracy of the model prediction with respect to both training and testing are assessed based on the non-dimensional root mean square error (R-RMSE) and the index of model performance mance (I_d). The R-RMSE is defined using the Eq. (6).

(6)R-RMSE={\displaystyle \frac{\sqrt{{\displaystyle \frac{\mathrm{1}}{N}}{{\displaystyle \Sigma }}_{i=\mathrm{1}}^{i=n}{({{\displaystyle O}}_i-{{\displaystyle y}}_i)}^{\mathrm{2}}}}{{{\displaystyle O}}_{rms}}}

where O_i, y_i and O_rms are, respectively, the experimental value, the predicted value and the root mean square value of the observed data. The index of model performance, I_d is defined as:

(7){{\displaystyle I}}_d=\mathrm{1}-{\displaystyle \frac{{{\displaystyle \Sigma }}_{i=\mathrm{1}}^{i=n}\left|{{\displaystyle O}}_i-{{\displaystyle y}}_i\right|}{{{\displaystyle \Sigma }}_{i=\mathrm{1}}^{i=n}\left(\left|{{\displaystyle O}}_i-\bar{O}\right|+\left|{{\displaystyle y}}_i-\bar{O}\right|\right)}}

where \bar{O} is the mean of the observed values. The index of performance has the advantage over the coefficient of determination (R²) for its greater sensitivity to lower values in the database, as well as its ability to capture systematic errors.

Based on the selected training and testing data, the values of root mean square error and the performance index for the networks are compared in Table 3, with the proposed algorithm producing the most accurate model, followed by the BP-PSO procedure suggested by Zhang et al. [¹⁸], the least performing being the stand alone PSO algorithm. From the standpoint of convergence speed, the proposed algorithm takes relatively shorter time to converge, as the results in Table 4 showed. Overall, the results summarised in Tables 3 and 4 seem to be supportive of the hypothesis that the combination of BP-PSO algorithm improves the ability to search for the best solution and quicker convergence compared to the stand-alone BP or PSO approaches.

The following empirical equations can be derived from the optimised PUNN model.

(8a)\begin{array}{l}{Q}_s={{\displaystyle C}}_{\mathrm{1}}{{\displaystyle \left({{\displaystyle \overline{N}}}_s{{\displaystyle A}}_s\right)}}^{\mathrm{0.5997}}{s}^{\mathrm{1.1231}}{{\displaystyle \left({\displaystyle \frac{{{\displaystyle E}}_{p}A}{L}}\right)}}^{-\mathrm{0.0662}}+{{\displaystyle C}}_{\mathrm{2}}{{\displaystyle \left({{\displaystyle \overline{N}}}_s{{\displaystyle A}}_s\right)}}^{\mathrm{0.7228}}{s}^{\mathrm{0.6550}}{{\displaystyle \left({\displaystyle \frac{{{\displaystyle E}}_{p}A}{L}}\right)}}^{\mathrm{0.1438}}\\ +{{\displaystyle C}}_{\mathrm{3}}{{\displaystyle \left({{\displaystyle \overline{N}}}_s{{\displaystyle A}}_s\right)}}^{-\mathrm{0.0719}}{s}^{\mathrm{0.5866}}{{\displaystyle \left({\displaystyle \frac{{{\displaystyle E}}_{p}A}{L}}\right)}}^{\mathrm{0.0650}}\end{array}

(8b)Q_b={{\displaystyle C}}_{\mathrm{4}}{({{\displaystyle N}}_b{{\displaystyle A}}_s)}^{\mathrm{0.8607}}{s}^{\mathrm{0.6034}}+{{\displaystyle C}}_{\mathrm{5}}{({{\displaystyle N}}_b{{\displaystyle A}}_s)}^{\mathrm{0.0141}}{s}^{\mathrm{0.5827}}+{{\displaystyle C}}_{\mathrm{6}}{({{\displaystyle N}}_b{{\displaystyle A}}_s)}^{\mathrm{2.1046}}{s}^{\mathrm{0.6106}}

where the elastic modulus E_p is in GPa and s is mm. The unit of both A_s and A_b is m². The parameters C₁-C₆ referred to the equations depend on the type of pile material, method of installation, as well as type of soil. In the case of piles in non-cohesive deposits, the parameters are denoted by {{\displaystyle C}}_{\mathrm{1}}*-{{\displaystyle C}}_{\mathrm{6}}*, the values of which are given in Table 5 below.

For piles in silty or clayey soils, the C- parameters are defined by the relations given in Table 6 The parameters \alpha and \beta are constants, the values of which depend on the method of installation. The values of the parameters are given in Table 7.

The performance of the proposed hybrid algorithms is further assessed by plotting their results against the training and testing data on scatter-grams. Figures 4(a) and 4(b) show the prediction results of PUNN based on the proposed algorithm plotted alongside the training and testing data respectively. The low scatter level observed in Figure 4(a) is indicative of the positive response of the model to the learning process it has gone through, while the good agreement between the testing data and the model’s estimate, as Figure 4(a) showed, is a demonstration of the ability of the model to make a reasonable predictions.

As the database consist of a range of pile types, it is important to examine the performance of various network types for each type of pile. To this end, both the training and testing data are sorted according to types of pile and the approximation accuracy of the networks is evaluated in each case. In Figures 5-8 the networks’ performance with respect to various pile types, expressed in terms of I_d value, is compared using bar charts. It is apparent from the figures that the network based on the proposed hybrid algorithm performed consistently well across the board, indicating that the superior performance it recorded with respect to the lumped data (Figure 4) is in tune with its performance with respect to various types of piles when isolated and considered separately. The result also suggest that the adoption of the switch-like input variables to represent parameters of qualitative nature such as the method of installation and the type of pile material provided quite a useful information to help the network identify the piles types and methods of installation.

It can also be seen from the result that the performance of BP-PSO (I) is comparable to that of the BP-PSO (II) in most of the cases. It however struggled to learn and predict the behaviour of H-steel piles (see Figure 7), although it is by no means the worst predictor for that case. The performance of the stand-alone BP algorithm, although generally lower than the hybrid algorithms, looks very impressive in the case of steel pipe pile, with an I_d value comparable to the hybrids. This is indicative of the ability of BP to focus much more effectively on the best solution PSO algorithm.

To further assess the ability of the proposed model to predict the load-deformation curves, comparisons are made between the model estimates and the Fleming’s hyperbolic model [⁵] and the t-z approach alongside the measured values for the different piles (see Figures 9-12). The load-deformation curves based on Fleming’s (hyperbolic)model are generated using the following equations.

(9){{\displaystyle P}}_t={\displaystyle \frac{a{{\displaystyle \Delta }}_t}{c+{{\displaystyle \Delta }}_t}}+{\displaystyle \frac{b{{\displaystyle \Delta }}_t}{d+e{{\displaystyle \Delta }}_t}}

where P_t and {{\displaystyle \Delta }}_t represent the pile load and pile head settlements respectively, while a= U_s, b= D_BE_BU_B, c= M_sD_B, d= 0.6 U_Band e= D_BE_B. U_s, U_B, D_B, E_B and M_s are the ultimate shaft resistance, ultimate base resistance, pile base diameter, soil deformation soil modulus at the pile base and flexibility factor respectively. Note that the above equation excludes the pile compression, which is separately estimated using the following expression,

(10){{\displaystyle \Delta }}_E={\displaystyle \frac{\mathrm{4}{{\displaystyle P}}_t({{\displaystyle L}}_{\mathrm{0}}+{{\displaystyle K}}_E{{\displaystyle L}}_F)}{{{{\displaystyle D}}_s}^{\mathrm{2}}{{\displaystyle E}}_P}}

where {{\displaystyle \Delta }}_E is the elastic pile shortening, L₀ is the upper length of pile carrying no load or small loads by friction, L_F is the length of pile transferring load to the soil by friction, K_E is ratio of effective column length of shaft transferring friction to L_F and D_s and E_P are the diameter of pile shaft and young’s modulus of pile material respectively.

The t-z method used here is based on the equation proposed by McVay et al. [³³], which is expressed as follows:

(11a)Side\hspace{0.17em}spring:z={\displaystyle \frac{{{\displaystyle \tau }}_0{{\displaystyle r}}_0}{{{\displaystyle G}}_{ib}}}\left[ln{\displaystyle \frac{{{\displaystyle r}}_m-\omega }{{{\displaystyle r}}_o-\omega }}+{\displaystyle \frac{\omega ({{\displaystyle r}}_m-{{\displaystyle r}}_0)}{({{\displaystyle r}}_0-\omega )({{\displaystyle r}}_m-\omega )}}\right]

(11b)Base\hspace{0.17em}spring:{{\displaystyle z}}_b={\displaystyle \frac{{{\displaystyle q}}_b{{\displaystyle A}}_b(\mathrm{1}-\nu )}{\mathrm{4}{{\displaystyle r}}_0{{\displaystyle G}}_{ib}{{\displaystyle \left[\mathrm{1}-{{\displaystyle R}}_f{\displaystyle \frac{{{\displaystyle q}}_b}{{{\displaystyle q}}_{max}}}\right]}}^{\mathrm{2}}}}

where z and z_b are pile shaft and base movements, respectively; G_ib is shear modulus of soil layer at low strain; r₀ and r_m are the pile radius and radius of influence of pile, respectively. {{\displaystyle \tau }}_{\mathrm{0}} is the shear stress at the soil-pile interface, while R_f is the stress-strain curve fitting constant. {{\displaystyle \tau }}_{\mathrm{m}\mathrm{a}\mathrm{x}} and q_max are the limiting shaft friction and pile end bearing pressure respectively. \omega is defined as expressed as \omega ={\displaystyle \frac{{{\displaystyle \tau }}_{\mathrm{0}}{{\displaystyle r}}_{\mathrm{0}}{{\displaystyle R}}_f}{{{\displaystyle \tau }}_{max}}}.

From the above equations, it is apparent that both models are based on the limiting values of shaft and base resistance, as well the elastic modulus. In this study, the limiting values of limiting shaft and base resistance for non-cohesive soils are estimated based on suggestions given by Meyerhof [³⁴]. For cohesive soils, both components of resistance around the pile are expressed as factored undrained shear strength, with the coefficient for shaft adhesion being 0.45 as proposed in [³⁵], while the coefficient for base resistance is taken as 9.0. With the SPT as the only available information about soil mechanical properties in the ground investigation result, the shear strength values are estimated as equal to 12 N; with N being the SPT blow-count. The elastic modulus, another parameter required by the Fleming’s and t-z methods, is estimated using the correlations reported in [²⁶].

As the space may not be sufficient to plot all curves in the database, an attempt is made in the selection of piles used for this demonstration to reflect the diversity in the database, which consist of a number of pile material types, different methods of installation and a range of subsurface characteristics. To this end, at least one test set is chosen from each of tests on bored and driven concrete piles, H-steel, as well as steel pipe piles. The selected tests also reflect a variety of soil profiles available in the database. In Figure 9 the result of a pile loading test carried out on a concrete pile driven into a non-cohesive soil deposit is shown. Also displayed in the figure is the closest available sub-surface information. The PUNN model can be seen to give the closest approximation to field measurements as the figures showed. Both hyperbolic and t-zmodels produced very conservative estimates. Better approximation of load-settlement behaviour on the part of the PUNN model results is also noted in another case involving a bored pile installed in clay (Figure 10). In this case the hyperbolic model tends to under estimate the settlement up to 4% of pile diameter, while the t-z approximation is slightly on the conservative side.

Two more cases involving H-steel and concrete filled pipe piles are shown in figures 11 and 12 respectively. In both cases, the PUNN managed to produce a good approximation, while both the hyperbolic and t-zmodels grossly underestimated the settlement in the case 318 shown in Figure 11 and overestimated the pile head deformation in the other case (Figure 12). So far, the general trend is that the proposedmodel gives the best estimate of load-settlement relationship. However, this is not the case with the pile test result shown in Figure 13. As shown in the figure, the PUNN predictions are relatively poor in comparison with field measurements. The estimates of the hyperbolic model are remarkably good in this case.

Overall, the quality of load-deformation curves predicted using Fleming’s method and load transfer approach have been much lower than PUNN approximations. However, the factors responsible for this inferior performance may not necessarily be connected to the nature of the models. The empirical relations used in estimating the limiting values of the shaft and base resistance, as well as the elastic modulus are lacking in reliability, as mentioned earlier, and may well be the culprits behind the poor predictions seen.

Parametric study

The results of a parametric study carried out to assess the generalisation ability of PUNNmodel are presented in Figures 14-20. The study is conducted by adopting a commonly used approach [³⁶,³⁷], which entails keeping all input variables, except the subject parameter, to values around the mean of training values, while the subject parameter is allowed to vary. In this manner, various controlling parameters are investigated in this study. The response of the model to the variation of a given parameter is examined in the light of the current knowledge about the behaviour of piles under axial loading. To provide a suitable reference point on the load- deformation curve, the values of computed pile head load are based on the settlement magnitude of 10% pile diameter, which is regarded as the load corresponding to the ultimate state [³⁸]. The sub-surface is assumed to be a uniform soil, with N-value increasing linearly with depth, starting from zero at the ground level.

In Figure 14 results of PUNN predictions of the behaviour of piles installed in a non-cohesive soil are presented. Note that the value of N at the pile base (N_b) is used throughout the parametric study considering the fact that the average N-value along the shaft is a function of N_b for the sub-surface model chosen. Note also that, as aforementioned, the average values of pile diameter (D=0.6 m) and length (11.0 m) obtained from the training database are used whenever they are not subjects to variation. As the figure showed, the pile head load at a settlement of 10% pile diameter tends to increase with the number of SPT blow counts. This is quite a rational response as the stiffness of the soil, which increases with increasing SPT-N values, is expected to vary proportionally with pile load capacity. Further, it is noticeable from the figure that, for a given SPT count, the model returns the highest value of head load for a driven pile. This is in harmony with the general understanding that the granular soil around the pile gets densified in the process of pile driving. The lower value of pile head load returned by the model in the case of bored piles installed in a wet hole compared to the one which is constructed in a dry hole is also a normal response. The wet excavation method results in weaker soil resistance compared to the dry method especially with regards to the resistance along the pile shaft [³⁹].

In order to examine the model’s response to variation in the type of soil, separate curves, each representing one of sand, clay and silt are compared in Figures 15-17. Apart from the general trend of proportional variation of pile head load at 10% diameter with SPT-N regardless of the type of soil and method of installation, it can be seen from figure 15 that sand is associated with highest value of pile head load corresponding to 10% D settlement for the same value of N. This is suggestive of a possible effect of soil densification as a result of pile driving in the case of sand. Conversely, in the case of both types of bored pile (Figures 16 and 17), it is the granular soil that yields the least value of head load, possibly due to loosening during the excavation process.

The effect of type of pile (material) on PUNN prediction is also studied. In this case, only driven piles are chosen because of the different types of pile available under this category. The results displayed in Figure 18 showed that that the model is able to make a distinction between different types of pile without ignoring the general trend of increasing ultimate head load with soil stiffness. It can also be observed from the curves in the figure that the concrete pile seems to return the lowest value of ultimate load for the same values of SPT-N and pile dimensions, as a result of the lower stiffness on the part of the same compared to steel.

The effect of variation in slenderness ratio on the model’s simulation is also examined. In Figure 19, the predicted load is plotted against the slenderness ratio, where the former appears to vary proportionally with slenderness ratio (with diameter varied while length is kept constant). The consistent increase in the predicted pile head load (at 10% Dsettlement) with increase in pile stiffness in Figure 20 is indicative of the good agreement between the PUNN model predictions and actual pile behaviour, where the resistance to compression is proportional to pile stiffness for a given soil strength.

Conclusion

Due to the complexity of soil-pile interaction and the difficulty of taking into account the effects of pile construction on the pile behaviour, accurate prediction of pile load-deformation relationship is a very challenging task. In this paper, an empirical model based on product unit network is proposed to simulate the pile load-deformation curve using easily obtainable input data. The model is based on input parameters including the average value of SPT along the pile shaft, the SPT value at the pile base, the pile stiffness and the area of the shaft and base. Other parameters include soil type and installation method. The proposed formula predicts the pile head load for a given head settlement and the other parameters mentioned above. The model is trained using a robust raining algorithm based on the L-M algorithm and particle swarm optimiser.

From the results obtained, the model optimised using the proposed hybrid algorithm gives a more accurate simulation of load-deformation curve than the ANN models based on conventional learning algorithms. Furthermore, the predictions of the optimised model are in a better agreement with pile loading tests compared with both hyperbolic and t-z methods. The results are also in3 dicative of the ability of PUNN to use relatively simple data such as SPT, soil classification, pile type and method of installation to predict the axial load deformation behaviour of piles with a reasonable accuracy.

Also, based on the parametric study carried out, the PUNN model was found to respond reasonably well to various input parameters in a manner consistent with the anticipated behaviour of an axially loaded pile.

Also, based on the parametric study carried out, the PUNN model was found to respond reasonably well to various input parameters in a manner consistent with the anticipated behaviour of an axially loaded pile.

References

[1] Poulos H G. Settlement of pile foundations. Numerical methods in Geotechnical Engineerng, McGraw-Hill, New York, 1977, 326–363

[2] Desai C S. Deep foundations. Numerical methods in Geotechnical Engineerng. McGraw-Hill, New York, 1977, 235–271

[3] Tomlinson M J. Pile Design and Construction Practice. Longman, 6th edition, 1995

[4] Chin F K. Estimation of the ultimate load of piles from tests not carried to failure. In: Proc. 2nd SE Asian Conf. Soil Engrg, Singapore, 1970, 81–92

[5] Fleming W. A new method of single pile settlement and analysis. Geotechnique, 1992, 42(3): 411–425

[6] Poskitt T J. A new method for single pile settlement prediction and analysis. Geotechnique, 1993, 43(4): 615–619

[7] Hamdia K M, Lahmer T, Nguyen-Thoi T, Rabczuk T. Predicting the fracture toughness of pncs: A stochastic approach based on ann and anfis. Computational Materials Science, 2015, 102: 304–313

[8] Rafiq M Y, Bugmann G, Easterbrook D J. Neural network design for engineering applications. Computers & Structures, 2001, 79(17): 1541–1552

[9] Ghaboussi J, Sidarta D E. New nested adaptive neural networks (nann) for constitutive modelling. Computers and Geotechnics, 1998, 22(1): 29–52

[10] Lee T L, Jeng D S. Artificial neural networks for tide forecasting. Ocean Engineering, 2002, 29(9): 1003–1022

[11] Kabiri-Samani A R, Aghaee-Tarazjani J, Borghei S M, Jeng D S. Application of neural networks and fuzzy logic models to long-shore sediment transport. Applied Soft Computing, 2011, 11(2): 2880–2887

[12] Ismail A, Jeng D S, Zhang L L, Zhang J S. Predictions of bridge scour: Application of a feed-forward neural network with an adaptive activation function. Engineering Applications of Artificial Intelligence, 2013, 26(5-6): 1540–1549

[13] Vu-Bac N, Lahmer T, Keitel H, Zhao J, Zhuang X, Rabczuk T. Stochastic predictions of bulk properties of amorphous polyethylene based on molecular dynamics simulations. Mechanics of Materials, 2014, 68: 70–84

[14] Vu-Bac N, Lahmer T, Zhang Y, Zhuang X, Rabczuk T. Stochastic predictions of interfacial characteristic of polymeric nanocomposites (pncs). Composites. Part B, Engineering, 2014, 59: 80–95

[15] Pooya Nejad F, Jaksa M B, Kakhi M, McCabe B A. Prediction of pile settlement using artificial neural networks based on standard penetration test data. Computers and Geotechnics, 2009, 36(7): 1125–1133

[16] Ismail A, Jeng D S. Modelling load settlement behaviour of piles using high-order neural network (hon-pile model). Eng. Appl. of AI., 2011, 24(5): 813–821

[17] Eberhart R C, Kennedy J. A new optimizer using particle swarm theory. Proceedings of the Sixth International Symposium on Micro Machine and Human Science, Nagoya, Japan, 1995, 39–43

[18] Zhang J R, Zhang J, Lok T M, Lyu M R. A hybrid particle swarm optimizationback propagation algorithm for feedforward neural network training. Applied Mathematics and Computation, 2007, 185(2): 1026–1037

[19] Clerc M, Kennedy J. The particle swarm-explosion, stability and convergence in a multidimensional complex space. IEEE Transactions on Evolutionary Computation, 2002, 6(1): 58–73

[20] Fun M H, Hagan M T. Levenberg-marquardt training for modular networks. In IEEE International Conference on Neural Networks, 1996, 468–473

[21] Yu J, Wang S, Xi L. Evolving artificial neural networks using an improved pso and dpso. Neurocomputing, 2008, 71(4-6): 1054–1060

[22] Durbin R, Rumelhart R. Product units: A computationally powerful and biologically plausible extension to backpropagation networks. Neural Computation, 1989, 1(1): 133–142

[23] Prevost J H, Popescu R. Constitutive relations for soil materials. Electronic Journal of Geotechnical Engineering, 1996, 1

[24] Poulos H G, Davis E H. Pile foundation analysis and design. Wiley, New Jersey, 1980

[25] Kulhawy F H, Mayne P W. Manual on estimating soil properties fro foundation design. Technical Report 2-38, Electric power Res. Inst. EL-6800; Palo Alto Carlifonia, 1990

[26] Bowles J E. Foundation analysis and design. McGraw-Hill, New York, 1997

[27] Fleming W G K, Weltman A J, Randolph M F, Elson W K. Piling Engineering. Taylor and Francis, Oxford, 2009

[28] Swingler K. Applying neural networks: a practical guide. Academic Press, New York, 1996

[29] Looney C G. Advances in feed-forward neural networks: demystifying knowledge acquiring black boxes. IEEE Transactions on Knowledge and Data Engineering, 1996, 8(2): 211–226

[30] Nelson M, Illingworth W T. A practical guide to neural nets. Addisin-Wesley, Reading MA, 1990

[31] Vu-Bac N, Silani M, Lahmer T, Zhuang X, Rabczuk T. A unified framework for stochastic predictions ofmechanical properties of polymeric nanocomposites. Computational Materials Science, 2015, 96: 520–535

[32] Vu-Bac N, Lahmer T, Zhuang X, Nguyen-Thoi T, Rabczuk T. A software framework for probabilistic sensitivity analysis for computationally expensive models. Advances in Engineering Software, 2016, 100: 19–31

[33] McVay M C, Towsend F C, Bloomquist D G, O’Brien M O, Caliendo J A. Numerical analysis of vertically loaded pile groups. Proceedings of the Foundation Engineering Congress, North Western University, Illinois, 1989, 675–690

[34] Meyerhof G G. Bearing capacity and settlement of pile foundations. Journal of Geotechnical Engineering, 1976, GT3(102): 197–228

[35] Skempton A W. Cast in situ bored piles in london clay. Geotechnique, 1959, 3(4): 153–173

[36] Goh A T C. Back-propagation neural networks for modeling complex systems. Artificial Intelligence in Engineering, 1995, 9(9): 143–151

[37] Rahman M S, Wang J, Deng W, Carter J P. A neural network model of the uplift capacity of suction caissons. Computers and Geotechnics, 2001, 28(4): 269–287

[38] Reese L C, O’Neill M W. Drilled shafts: Construction and design. Technical Report HI-88-042, FHWA, 1988

[39] Balakrishnan E G, Balasubramaniam A S, Phien-Wej N. Load deformation analysis of bored piles in residual weathered formation. Journal of Geotechnical and Geoenvironmental Engineering, 1999, 125(2): 122–131

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