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Prediction of resistance to pile driving using evolutionary neural network

  • Abdussamad Ismail
Original Paper
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Abstract

Pile driveability assessment is immensely useful prior to installation as it serves as a basis for selecting the suitable pile and driving hammer for a given ground condition. This is particularly valuable for constructions in difficult environments such as the installation of foundations for offshore and underwater facilities. In this paper, a neural network model is proposed to estimate the blow counts of an impact driving given the information regarding the hammer characteristics, the pile properties, and the SPT profile of the soil supporting the pile. The network is optimised using a self-evolution algorithm to minimise the network complexity and to avoid over-fitting. From the results obtained, the proposed model was found to reasonably estimate the hammer blow counts under a variety of sub-surface conditions. It was also found to be superior in terms of prediction accuracy compared to the network models based on convention BPN-training algorithm. The proposed model also benefits from fewer network parameters in comparison with the BPN networks.

Keywords

Pile driveability SPT test Neural network 

1 Introduction

Pile driving is one of the oldest and most widely used techniques of pile installation. The technique is usually considered, where the ground is soft enough to be penetrated by the pile without causing structural damage to the latter. One of the key parameters in the design of pile foundations is the depth of pile embedment. In the case of bored piles, the depth can be predetermined prior to installation. In the case of driven piles, however, the depth of penetration is difficult to estimate with certainty due to the influence of the operating conditions on pile driveability during installation. Although some information about the factors influencing the refusal depth of driven piles such as hammer characteristics, pile properties, and nature of the sub-surface can be obtained, the complex interaction between the variables is difficult to predict. For decades, the wave equation for pile-driving analysis, implemented numerically by Smith (1960), has been the most popular technique used in assessing pile driveability for a given sub-surface condition, pile and hammer. However, due to the simplistic assumptions about soil properties and some difficulty in accurately accounting for the effect of mechanical characteristics of the driving hammer, a huge disparity between the predictions of the wave equation model and the observed response is often obtained.

In this work, a neural network is used to develop an empirical model for estimating the driving resistance in terms of number of hammer blows required to drive a pile further into the ground (Blows/m). Blow count serves as a basis for assessing pile driveability and determining the refusal depth of impact driven piles. The parameters required by the model include the pile stiffness and geometry, current penetration depth, hammer characteristics as well as the standard penetration test (SPT) N values along the depth of penetration. Pile-driving data are used to train and test the proposed neural network model. To enhance generalisation and minimise the network size, self-evolution algorithm is used to optimise the parameters and topology. A previous study has shown that self-evolution algorithm produces less complicated networks without compromising accuracy in predictions (Jeng et al. 2013). In the next section, existing methods of pile-driving analysis are briefly discussed.

2 Wave equation analysis of pile driving

The idea of using wave theory in the analysis of pile-driving dynamics, which was first thought of in the beginning of the 1930s (Isaacs 1931), came about in recognition of the failure of the widely used pile-driving formulae to accurately account for the complex relationship between the hammer impact and the response of the soil–pile system due to a flawed assumption that a pile subject to hammer blows behaves in accordance with the principles of rigid body mechanics. In his insightful work, Isaacs (1931) analysed the pile response to driving based on the one-dimensional equation governing the wave propagation along an elastic rod credited to St. Venant (Timoshenko and Goodier 1951). In his work, he attempted to come up with a closed form solution to the wave equation by making excessively simplistic assumptions about the shaft and base resistance, which severely compromised the practical value of his solution. Attempts subsequently made to use the wave equation in a more meaningful way are based on modifying the equation to take into account the effects of soil resistance, which takes the following form:
$$\begin{aligned} \frac{\partial ^2 w }{\partial t^2}=\left( \frac{E}{\rho }\right) \frac{\partial ^2 w }{\partial x^2}\pm R, \end{aligned}$$
(1)
where R is the soil resistance. A valid solution of Eq. 1 must take into consideration the soil variability, which can only be achieved through a numerical approach. The earliest and most convenient numerical solution is the finite difference scheme proposed by Smith (1960). This solution is based on discretising the pile into a series of lumped masses interconnected by pile springs (with stiffness equal to the stiffness of the pile segment) and idealising the soil resistance along the shaft and below the pile tip as a series of slider springs and dash pots based on a simplified rheological model. The hammer ram and the pile cap are represented by discrete masses, with the interfaces between them being modelled as compression-only springs. A pile cushion is also modelled in this manner. The pile–soil–ram arrangement according to Smith’s solution is shown in Fig. 1.
Fig. 1

Smith’s finite difference representation of soil–pile–hammer system

The soil springs in this method are assumed to deform in a linearly elastic manner up to a limiting deformation (termed the quake, Q) then fail plastically at the ultimate resistance R. The dash pots represent the viscous component of soil resistance, characterized by the coefficients of viscous damping, J. The instantaneous soil resistance according to Smith (1960) is given by
$$\begin{aligned} R_\mathrm{t}=\mathrm{Min}\left( 1,\frac{ws}{Q}\right) (1+Jv)R_\mathrm{s}, \end{aligned}$$
(2)
where \(R_\mathrm{s}\) is the limiting soil resistance; w is the local pile displacement; and v is the local pile velocity. The quake parameter controls the point at which the soil moves over from the elastic zone to the plastic zone, while the damping coefficient (J) weighs the dynamic component of soil resistance, the greater the value of damping coefficient, the greater the resistance. Typical values for quake (Q) and damping constant (J) recommended by Smith (1960) are 2–3 mm, and 0.1–0.2 s/m, respectively. The analysis is carried out incrementally over a number of time intervals small enough to ensure convergence of the step-by-step computation. The initial condition is defined by the ram velocity at the moment it impacts on the pile cap. In the first time step, it is assumed that the ram travels with the initial velocity for a certain small distance and compresses the spring underneath by the same amount. The resulting force in the spring acts to decelerate the downward motion of the ram and accelerate the mass just below it. The impact force is transmitted in the same manner through the remaining parts of the soil–pile system. Based on the computed accelerations during this time interval, the velocities and net displacements of the masses in the second time step are computed. This process is repeated, with velocities and displacements of all masses being evaluated in a similar manner at the end of each time increment. The analysis is terminated when the velocity of the pile base is lost. One of the disadvantages of this method is the use of quake and damping parameters, which are unconventional soil parameters and are difficult to measure (Chow et al. 1988). In addition, the soil-resistance model used tends to oversimplify the actual soil response to such complex loading (Randolph 2003). Attempts to improve the Smith soil model dated back to a few years after its introduction. One such modification includes the TTI model (Lowery et al. 1969), in which the damping resistance is regarded as a non-linear function of pile velocity, expressed in the following form:
$$\begin{aligned} R_\mathrm{t}=\mathrm{Min}\left( 1,\frac{ws}{Q}\right) (1+Jv^N)R_\mathrm{s}, \end{aligned}$$
(3)
where N is a non-linearity exponent and the rest of parameters are the same as that of Smith’s model. Coyle and Gibson (1970) reported that the value of N is 0.18 and 0.2 for clays and sands, respectively. A roughly similar value of exponent (0.2) was also suggested by Heerema (1979) and Litkouhi and Poskitt (1980) for both sands and clays.

The common problem associated with the aforementioned soil models is that they are based on parameters that are, as mentioned earlier, difficult to determine using conventional laboratory equipments. Back analysis is often used to estimate the parameters from the results of pile-loading tests, or pile-driving analyzer (PDA) measurements and case pile wave analysis program (CAPWAP) analysis. However, the study conducted by Thendean et al. (1996) indicated that the Smith soil parameters back-figured from CAPWAP analysis varied significantly within the same type of soil and even the mean values showed very little correlation with soil type.

An attempt has also been made to develop a correlation between in-situ tests and damping parameters (McVay and Kuo 1999), but was not successful due to the extremely high scatter in the data used for developing the correlations.

An improved soil-resistance model based on elasto-dynamic theory was proposed by Simons and Randolph (1985), where the radiation damping and spring stiffness are decoupled in order more rationally to account for the damping effect. This model has the advantage of being based on known soil parameters in contrast to the previously described models. It is, however, still dependent on some coefficients, whose values have to be determined empirically. Another important parameter that profoundly affects the accuracy of wave equation analysis is the amount of energy transferred from the hammer to the pile in the event of impact. Due to the chaotic nature of the pile-driving process, an efficiency factor based on the rule of thumb is used.

3 Neural network modelling

Considering the shortcomings of the currently used method of pile driveability analysis, it is worthwhile to study the viability of a neural network as an alternative method of analysis. In this work, a self-evolving neural network is trained to predict the blow counts during pile driving given the pile properties, the hammer characteristics, as well as the properties of the sub-surface into which the pile is driven. The aim is to develop a relatively simple but reasonably accurate means of evaluating pile driveability based on easily obtainable information regarding the properties of the pile, the supporting soil and the hammer.

3.1 Self-evolving neural network (SEANN)

The network developed in this study is trained using a self-evolution algorithm, which allows the network topology to be optimised alongside the network parameters. The key advantage of this training algorithm is that it aims at minimising the network size, which tends to bolster the robustness of the network predictions.

The method adopts a bottom-to-top strategy, where the initial network topology is chosen to be a simple architecture with a single hidden node, then gradually evolve in complexity as the training progresses. The network synaptic weights are optimised during the training using a combining particle-swarm optimiser and back-propagation algorithm, while the network topology is updated alongside the synaptic weights using JPSO, a discrete version of particle-swarm optimization technique. In the following section, jumping particle is briefly described.

3.1.1 Jumping particle optimization (JPSO)

JPSO is a bio-inspired combinatorial optimisation technique proposed by Matinez-Garcia and Moreno-Perez (2008). The method is based on the social interaction between members of a swarm of particles that traverse the search space by jumping from one discrete point to another. Based on the model of frog jumping from one lily pad to another in a pool, either in search for food or to avoid an adversary, a particle updates its current position by jumping to a new position under the influence of particles’ experience, global best position as well as their drive to explore the unknown regions in the discrete space. In the end, the particle jumps to the new co-ordinate only if it results in a more favourable position over the current one. The technique is implemented based on the following equation:
$$\begin{aligned} \mathbf{x}_{t}=\lambda _1\otimes \mathbf{x}_{t}\oplus \lambda _2\otimes \mathbf{b} \oplus \lambda _3\otimes \mathbf{g}, \end{aligned}$$
(4)
where \(\mathbf{x}_{t}\) and \(\mathbf{x}_{t+1}\) are the vectors of current and future particle co-ordinates in the search space. The parameters \(\lambda _{1}\), \(\lambda _{2}\), and \(\lambda _{3}\) are probabilities of jumping randomly, towards the best particle position and to the best swarm position, respectively. \(\mathbf{b}\) and \(\mathbf{g}\) are, respectively, the particle best and global best positions. The particle position updating is carried out as follows:
$$\begin{aligned} x_{i,t+1} = {\left\{ \begin{array}{ll}x_{i,t}*\rho &{}\,\,P_{x_{i,t}\rightarrow \rho }=\lambda _1\\ x_{i,t}*\mathbf{b}&{}\,\,P_{x_{i,t}\rightarrow \mathbf{b}}=\lambda _2\\ x_{i,t}*\mathbf{g}&{}\,\,P_{x_{i,t}\rightarrow \mathbf{g}}=\lambda _3 \end{array}\right. } \end{aligned}$$
\(\lambda _1+\lambda _2+\lambda _3 = 1\) in which \(\rho \) is a random binary number and \(P_x\) refers to the probability that the particle jumps towards a position. The * operation is executed by stochastically modifying the features of the current particle with some features of its attractor in a situation, where it is attracted by either \(\mathbf{b}\) or \(\mathbf{g}\). If the particle, by chance, chooses to undertake an unguided jump, its topology will be modified by a randomly generated topology.

JPSO was successfully applied in some combinatorial optimisation problems such as the p-median problem (Matinez-Garcia and Moreno-Perez 2008), minimum labelling Steiner tree problem (Consoli et al. 2010), and gear train benchmark problem (Seren 2011). Both studies by Matinez-Garcia and Moreno-Perez (2008) and Seren (2011) compared JPSO and discrete version of PSO (Kennedy and Eberhart 1997), reporting in both cases that JPSO returned better results compared with DPSO results. It is, however, noteworthy that all the above-cited works focused on discrete problems. None of them considered problems to do with a mixture of discrete and continuous variables.

3.1.2 Self-evolution algorithm

Optimal training of neural networks is a challenging task entailing not only fine-tuning the weights of the synapses, but also simultaneously optimising the architecture. To tackle such a complex mixed-integer optimisation problem, the training algorithm used in this study involves carrying out the topology optimisation based on the JPSO technique and synaptic weight fine-tuning based on a combination of particle-swarm optimization and the BP algorithms. In a previous study, Jeng et al. (2013) used the same algorithm to optimise a network for predicting the stability factor for untrenched pipes. Figure 2 shows the self-evolving network architecture with synaptic connection options denoted by binary parameters \(C_i\). For a live connection, the \(C_i\) parameter assumes a value of 1 and 0 otherwise.
Fig. 2

SEANN topology

The self-evolution algorithm is summarised in the following steps:
  1. 1.

    Initialize a particle-swarm population of size N, with each particle representing neural networks with a single hidden node and randomly generated set of synaptic weights and connection parameters.

     
  2. 2.

    Evaluate the fitness of each particle and update the best particle and global positions.

     
  3. 3.
    Use the PSO and JPSO techniques to update particle co-ordinates for certain number of iterations in the following sub-steps:
    1. (a)

      Use PSO to update the weight vector of each particle.

       
    2. (b)

      Use JPSO to update the connection parameters of each particle.

       
    3. (c)

      Update the particle best position and the best swarm position.

       
     
  4. 4.

    If convergence is sufficient then go to 9. Else continue.

     
  5. 5.

    Reset randomly the binary and continuous parameters of duplicate particles. In addition, reset in the same manner, the binary parameters of certain fraction of the swarm with poor fitness.

     
  6. 6.

    Select best particles and update their continuous parameters using some steps of BP. If the training is satisfactory go to 9. Else continue.

     
  7. 7.

    If number of iterations is less than the maximum number then go back to step 3. Else continue.

     
  8. 8.

    Generate N particles with a number of nodes which is one greater than the current number of nodes. Replace all current particles with the newly generated particles while retaining the current particle best positions (topology and synaptic weight). Then, go back to step 3.

     
  9. 9.

    Terminate algorithm and return result.

     

3.2 Activation function

The complexity and performance of neural networks are dependent significantly on the type of activation function chosen. Depending on the characteristics of the knowledge base, some functions yield better accuracy with simpler network sizes than others. In this paper, an adaptive activation function is used to attempt to reduce the network complexity. The proposed activation function is a combination of linear and non-linear functions given by the following equation:
$$\begin{aligned} f(\mathbf{x})=\sum _{1}^{n_\mathrm{f}}k_i\alpha _i\phi _i\left( {\mathbf{w_i}}^T{\mathbf{x}}\right) \end{aligned}$$
(5)
where \(n_\mathrm{f}\) is the number of sub-functions, \(\phi _i\) refers to a sub-function, \(\alpha _i\) is an adaptive coefficient and \(k_i\) is a binary number. \(\mathbf{x}\) is the vector of inputs to the node, while \(\mathbf{w}_i\) is the vector of the weights of synapses connecting input parameters to the sub-function i. The function of the binary number is to let the associated sub-function be part of the activation function by selecting a value of 1 and excluding the associated sub-function by selecting a value of zero. The coefficient \(\alpha _i\) throws some weight behind the sub-functions making up the activation function in accordance with their relative importance to the output of the neuron. The coefficient is adjusted in the same manner as the synaptic weights are, while the binary parameter, \(k_i\), is optimised alongside the connectivity parameter \(C_i\). Two sets of functions are used for activation function in this study. They are as follows:
First set (Type I):
$$\begin{aligned}&\mathrm{Linear}{:} \phi _1 (\mathbf{x})=\mathbf{w}^T \mathbf{x} +b ,\end{aligned}$$
(6a)
$$\begin{aligned}&\mathrm{Sinusoid}{:} \phi _2 (\mathbf{x}) =\sin \left( \mathbf{w}^T \mathbf{x} +b \right) ,\end{aligned}$$
(6b)
$$\begin{aligned}&\mathrm{Sigmoid:} \phi _3(\mathbf{x}) = \frac{1}{1+\mathrm{e}^{\mathbf{w}^T \mathbf{x} +b}} ,\end{aligned}$$
(6c)
$$\begin{aligned}&\mathrm{Wavelet}{:} \phi _4(\mathbf{x})=\left( \mathbf{w}^T \mathbf{x} +b \right) \mathrm{e}^{-\frac{{\left( \mathbf{w}^T \mathbf{x} +b\right) }^2}{2}}. \end{aligned}$$
(6d)
Second set (Type II):
$$\begin{aligned}&\mathrm{Linear}{:} \phi _1 (\mathbf{x})=\mathbf{w}^T \mathbf{x} +b ,\end{aligned}$$
(7a)
$$\begin{aligned}&\mathrm{Product}\hbox {-}\mathrm{unit}{:} \phi _2 (\mathbf{x})= \prod _{1}^{n} {x_i}^{{w_2}^i}, \end{aligned}$$
(7b)

where n is the number of inputs.

3.3 Pile-driving records

The data used in training and testing the model consist of 122 pile-driving records obtained from over 60 sites, with each record consisting of number of hammer blows versus penetration, making a total of 2824 data points. The pile data are sourced from the http://www.webgeotech.com database. The same data are also available at the federal highway administration (FHWA) deep foundations database. The data consist of cylindrical and square concrete piles, H-steel piles, and closed ended steel pipe piles. Only piles installed by impact driving are considered in this study. Various types of hammer used in installing the piles are grouped broadly into open-ended single acting, closed-ended double acting and internal combustion hammers for the sake of making the inputs to the network as simple as possible and their number as small as possible. The soil test results in the database consist of uncorrected SPT-N values as well as standard soil classification. The soil classification is used to give additional information to help characterize the sub-surface more accurately. The soil types in the database range from coarse soils (sand) to fine soils (silt and clay) with varying degree of plasticity. Note that the co-ordinates of bore holes with reference to pile locations are not available in the database. Therefore, it is possible that the soil profiles are, in some cases (which are not known), not be representative of the soil properties at the point of driving. The summary of the database is given in Table 1. It can be seen from the table that steel pipe records have the smallest number. Their inclusion is mainly aimed at enriching the database with a variety of pile types as well as benefitting from other useful data contained in the records.
Table 1

Number and type of piles in the database

Type of pile material

Number of piles

Concrete piles

79

H-Steel piles

26

Pipe piles

12

3.4 Input parameters

The resistance to pile penetration, which is proportional to hammer blow counts, is a function of hammer properties such as rated energy, ram and hammer type, pile stiffness and geometry as well as the behaviour of soil along the pile shaft and around the base. To provide sufficient information about variations in soil behaviour along the pile shaft, the pile is divided into five segments. Each segment is represented by a neural net as shown in Fig. 3. The base component of soil resistance is also represented by a neural network, as shown in the figure. Inputs to the networks representing the shaft and the base appear on the left side of the networks shown in Figs. 4, 5, which represent, respectively, shaft and base components of resistance. It can be seen from the figures that the networks representing shaft segments and the pile tip are restricted to receiving information regarding the behaviour of the corresponding soils.
Fig. 3

Representation of shaft and base resistance with neural networks

Fig. 4

NN configuration (shaft)

Fig. 5

NN configuration(base)

\(\overline{N}_{\mathrm{{s}}\_i}\) and \(N_\mathrm{b}\) are the average N value along the shaft segment i, and around the base, respectively. \(A_{\mathrm{{s}}\_i}\) and \(A_\mathrm{b}\) are, respectively, the surface area of shaft segment i and the area of base. \(f_\mathrm{C}\) and \(f_\mathrm{m}\) are the fractions of silt and clay, respectively; \(f_\mathrm{P}\) is the plasticity parameter (0 for low or no plasticity and 1 for high plasticity); E is the hammer energy; \(\lambda _\mathrm{OED}\), \(\lambda _\mathrm{CED}\), and \(\lambda _\mathrm{ECH}\) are binary parameters representing open-ended, closed-ended, and internal combustion hammers. \(\lambda _\mathrm{OED}\) assumes the value of 1.0 when the hammer type is OED and 0.0; otherwise; the same applies to the other types. The parameters \(\lambda _\mathrm{M1}\) and \(\lambda _\mathrm{M2}\) inform the network about the type of pile. Both parameters are set at zero when the pile is made up of concrete, and setting \(\lambda _\mathrm{M1}\) at 1.00 when the pile is H-steel. The steel pipe is represented by setting \(\lambda _\mathrm{M2}\) to a value of 1.0. E is the elastic modulus of the pile and \(A_\mathrm{c}\) represents the cross-sectional area of the pile shaft. The impact hammer blow count (number of blows per m) is expressed by the following equation:
$$\begin{aligned} r_\mathrm{p} =\sum _{i=1}^{i=5}O_{\mathrm{{s}}\_i} +O_\mathrm{b}, \end{aligned}$$
(8)
in which \(O_{\mathrm{{s}}\_i}\) and \(O_\mathrm{b}\) are the outputs of the networks representing the shaft and base, respectively. As the training data do not contain target outputs for individual neural networks representing shaft and base components contributing to the total resistance to driving, the predicted blow count is represented by a super net (with no hidden layer) which receives input from the shaft and base networks, as shown in Fig. 6. Note that the synaptic weights are invariant (and are all equal to 1.0), thus, only the internal parameters of individual feeder networks are updated during the training process. This approach resembles the combination of experts model, a divide and conquer approach based on the concept of a committee machine, which uses parallel but simple networks to solve a complex task by putting together their output in a certain fashion (Haykin 1999). The important difference between the approach used here and the committee machines is that the input vectors associated with the networks representing different pile segments differ from each other as the characteristics of the sub-surface vary from one pile segment to another. Moreover, the training is carried out simultaneously in the case of the former as opposed to the latter’s case, where the networks are trained separately. The use of neural networks in this study to represent the soil response along the pile length and its tip is motivated by the ability of neural networks to explore relationships without having to make mechanistic assumptions apriori.
Fig. 6

Two-layered net representing the soil–pile resistance to impact driving

3.5 Model limitations

Given the complex interaction between the blow count of pile driving and the many influencing factors such as type of pile material, pile length and size, sub-surface condition, and methods of driving, it is very difficult to develop a model that is capable of accounting for all factors. Therefore, the ANN model proposed in this work can only predict the driving blow count based on the information, as described in Sect. 3.4.

It is also noteworthy that the model is based on pre-installation sub-surface condition. It, therefore, lacks the ability to account for the effect of interaction between previously driven piles and the subsequent ones, which significantly impacts on the driveability of the latter.

3.6 Network training

To commence the network training, the database of pile-driving records needs to be partitioned into training and testing data sets. Due to lack of consensus on how to divide a database into training and testing sets, the ratio of training versus testing data adopted in this case study is chosen to be around 70:30, which is reasonable considering the suggestions given in the previous studies on the subject (Swingler 1996; Looney 1996; Nelson and Illingworth 1990). The composition of training and testing sets was arrived at by conducting a series of trials in which the network training is carried out using various data combinations. The selected composition of the two data groups at the end of the trials of consists of 1845 data points and 979 points for training and testing, respectively.
Table 2

Summary of training and testing results (pile-driving analysis)

Type of activation function

Number of input links

RMSE

 
  

Training

Testing

SEANN model

   Type I

110

0.04639

0.0532

   Type II

177

0.04832

0.0677

BPN network

   Sigmoid

284

0.07656

0.08204

   Sinusoid

284

0.08001

0.09213

   Wavelet

213

0.06850

0.07656

Considering the number of networks that need to be developed simultaneously, it is desirable that network configurations be as simple as possible be developed to minimise the overall number of network parameters. To achieve this, a self-evolution algorithm described in Sect. 3.1 is used to optimise the networks. The ability of the algorithm to minimise the network complexity without a compromise in accuracy makes it ideally suitable for this problem. By its nature, the algorithm allows the networks to assume a single hidden node at the beginning, then gradually evolve into more complex configurations as training progresses, thus achieving a faster convergence than other methods of self-evolution.

Two types of SEANN networks were developed in this study: one based on a combination of sigmoid, sinusoid, and wavelet functions (Type I network) and the other based on a combination of linear and product-unit function (Type II network). BPN networks were also developed alongside the SEANN networks for the purpose of comparison.
Table 3

Convergence comparisons

Parameter

SEANN model

BPN network

 

Type I

Type II

Sigmoid

Sinusoid

Wavelet

Duration (s)

78,859.98

92,970.2

4361.198

4687.917

3756.851

No. of evaluations

540

810

1760

1910

1650

3.7 Training and testing results

The training and testing results of the networks considered in this study are summarised in Table 2. The results showed that the proposed Type I SEANN model (SEANN-I) produced the best estimate of the resistance to hammer driving, not only because it returned the least value of root mean square error with regard to testing data, but also because it surpassed the rest of the models in accuracy despite having the least number of network parameters. The SEANN model based on the combination of linear and product-unit functions (SEANN-II) is not far behind the former in terms of the prediction accuracy, but is based on a lot more constants compared with the former. The BPN network based on sigmoid function produced the worst result with respect to both training and testing data, although it has the highest number of parameters. The wavelet BPN performs much better that those based on sigmid and sinusoid, but is still far behind the SEAN-I model in terms of both the prediction error and the number of constants. Overall, The results underscore the advantages of selecting suitable activation function as well as evolving the network topology alongside the synaptic weights during the neural network training.

Speed of convergence is another important parameter to consider in assessing the performance of neural networks. In Table 3, the time to convergence of various network types is compared. It is clear from the table that two versions of the SEANN take more than twice the time required for the wavelet network to converge. This can be attributed to the computational effort needed in evaluating various combinatorial possibilities in the search for optimum topology, which is not unusual for optimisation problems dealing with a combination of discrete and continuous parameters (Stalph et al. 2012). This disadvantage is a temporal one, as the final product, by virtue of having fewer number of parameters, is computationally more convenient to operate at the prediction stage.
Fig. 7

Comparison of actual versus predicted hammer blow counts loads. a SEANN model versus actual impact hammer blow counts (training data). b SEANN model versus actual values testing data

To further assess the prediction quality of the SEAN model, the predicted values of driving resistance are compared with training and testing data using scatter-grams, as shown Fig. 7a, b. The figures indicated that a reasonably good agreement exists between the network approximations and the duo of training and testing data (\(R^2\) = 0.8642 for training and \(R^2\) = 0.8432 for testing), thus highlighting the ability of the network to respond well to the supervised learning and to make reasonably accurate predictions when compared with the data unseen by the network. It can also be seen from Fig. 8 that the network predictions, when compared with the testing data, are mostly within ± 20% of the measured blow counts; only few data points lie outside the range. This is indicative of the good agreement between the proposed model and the observed data.

Despite the large \(R^2\) values associated with the plots in Fig. 7, it is important to note that the scatter is significant in both cases. This is not unexpected considering the chaotic nature of the pile-driving process as well as the uncertainties regarding soil characterisation. It is quite possible that the soil profile obtained from borehole records are wildly different from the soil profile encountered at the location of driving. Recall that the database compiled for this study does not contain information about how far apart are the co-ordinates of the borehole and the driven pile, which could have been quite useful in fishing out those data sets with un-matching soil profile.
Fig. 8

Plot of SEANN predictions alongside the observed values ± 20%

Fig. 9

Residual plot

Fig. 10

Sorted training and testing results of SEANN network

To have a closer look at the pattern of scatter, the residuals (predicted values–measured values) were plotted against the testing data, as shown in Fig. 9. The absence of a clear trend in the scatter of the residuals suggests that the network has shown a good generalisation and that the observed prediction error is more to do with noise in the data and less to do with the network model.

The results so far presented gave a general outlook of the network simulation regardless of the type of pile. To examine the ability of the network in identifying the types of pile available in the database, results of training and testing are sorted based on the three types of pile: concrete, H-steel and pipe piles, and network performance is determined in each case. From the chart in Fig. 10, it is evident that the quality of network adaptation to the training data and the level of agreement between its prediction and testing data are low in the case of concrete piles compared to the results of H-steel and pipe piles. The network performance with regard to concrete piles does not seem to improve even when the network is trained using an isolated data, as the results shown in Table 4. This suggests that the relatively large prediction error in the case of concrete piles is more likely to be as a result of noise in the data than a difficulty in identifying the pattern of response of various types of pile.
Table 4

Training and testing results two SEANN networks (concrete piles)

Network training data

\(R^2\)

 

Training

Testing

Result extracted from lumped data

0.6801

0.7923

Concrete pile data alone

0.6950

0.7432

Fig. 11

Square concrete pile: perimeter = 1.87 m; base area = 0.208 m\(^2\); length = 13.71 m; elastic modulus = 2.8 GPa; Hammer energy = 13,350 Nm; Hammer type = ECH

Fig. 12

H-steel pile (HP360 \(\times \) 132/HP 14 \(\times \) 89); perimeter = 0.01690 m; base area m\(^2\) = 0.01690 m\(^2\) length = 35.05 m; elastic modulus =200 GPa; Hammer energy = 69,878 Nm; Hammer type= OED

Fig. 13

Steel pipe (closed ended): perimeter = 1.01787 m; base area = 0.08245 m\(^2\); length = 21.48 m; elastic modulus = 200 GPa; Hammer energy = 81,015 Nm; Hammer type = OED

As a means of demonstrating the ability of the network to simulate the pile-driving resistance profile, the network estimates are compared with three sets of field measurements which reflect, to some extent, the diversity of the database; the types pile material and the driving hammer, as well as the sub-surface characteristics. For the sake of comparison, results of BPN predictions are also plotted alongside the SEANN simulations. Figure 11 shows the results of SEANN-II predictions alongside the actual response of a concrete pile subject to impact driving, while in Figs. 12, 13, respectively, the results of H-steel and pipe piles are presented. It can be seen in the case of concrete pile that the model tends to over-estimate the resistance up to a depth of around 7 m, beyond which it under-predicts the driving resistance compared with the measured response. The reason for this significant departure from the observed behaviour can be attributed to the relatively low SPT N values (on which the model depends) around the same depth and beyond. The marked increase in the measured resistance at this depth and further downward seems to not be commensurate with the corresponding SPT values, which suggests that the soil encountered during the driving is possibly at variance with the soil profile obtained from logging data.

The predictions of SEANN model in the case of H-steel and pipe piles seem to be in a better agreement with the actual driving profile. However, the driving resistance values seem to be over-predicted by the model beyond a depth of 15 m in the case of pipe pile. While the estimated values are commensurate with the increased SPT-N counts beyond that depth, the resistance determined from the measured driving data tends to be in conflict with the soil information, possibly due to the effect of spatial variation in sub-surface characteristics as discussed previously. The figures also showed that SEANN predictions are consistently superior to that of BPN, thus corroborating the results presented in Table 2.

3.8 Optimised network parameters

The following set of equations represent the optimum networks obtained at a number of points along the shaft and at the base.

First (top most)pile shaft segment

\(O_{\mathrm{s}_1}=C_1+C_2+C_3({\mathbf{W}}^T\mathbf{X}+C_4)\mathrm{e}^{-\frac{({{W}}^T\mathbf{X}+C_5)^2}{2}}\)

\(C_1=\dfrac{11.06409}{1+\mathrm{e}^{-(0.0356 NA_\mathrm{s} + 1.8070 f_\mathrm{P} -5.1116 \lambda _\mathrm{ECH} 11.4149 \mathrm{OED} -6.3847 z/l -5.745505)}} \)

\(C_2=\dfrac{105.3469}{1+\mathrm{e}^{-(-0.0438 EA + 1.3334 \lambda _\mathrm{M1} -0.97563)}} \)

\(\mathbf{W}\)

\(\mathbf{X}\)

  

− 0.9608

\( f_\mathrm{m}\)

\(C_3\)

0.4137

− 2.0311

E

\( C_4 \)

8.6608

0.0732

\( \lambda _\mathrm{ECH} \)

\( C_5 \)

− 0.6157

1.1434

\( \lambda _\mathrm{OED} \)

  

− 2.1204

\( \lambda _\mathrm{CED} \)

  

5.9857

z / l

  

0.0410

\( \mathrm{EA} \)

  

− 0.4430

\( \lambda _\mathrm{M1} \)

  

Second segment

\(O_{\mathrm{s}_2}=C_1+C_2+C_3({\mathbf{W}}^T\mathbf{X}+C_4)\mathrm{e}^{-\frac{({\mathbf{W}}^T\mathbf{X}+C_5)^2}{2}}\)

\(C_1=28.4099 \sin ( -2.9876 \lambda _\mathrm{M1} + 1.6289 \lambda _\mathrm{M2} + 3.0416 )-0.0005 f_C -5.2687 \)

\(C_2=\dfrac{-28.7856}{1+\mathrm{e}^{-(-3.6033 f_P 0.7316 \lambda _\mathrm{ECH} 0.8780 \lambda _\mathrm{CED} 3.7713 z/l -2.9625)}}\)

\(\mathbf{W}_1\)

\(\mathbf{X}_1\)

  

12.1885

\( f_\mathrm{m} \)

\( C_1 \)

1.3845

4.3613

\( f_\mathrm{C} \)

\( C_2 \)

− 9.6998

− 0.9234

\( f_\mathrm{P} \)

\( C_3 \)

− 0.2619

− 3.4748

\( \lambda _\mathrm{OED} \)

  

5.0074

\( \lambda _\mathrm{CED} \)

  

9.5662

z / l

  

− 0.1211

\( \lambda _\mathrm{M1} \)

  

− 4.1360

\( \lambda _\mathrm{M2} \)

  

Third segment

\(O_{\mathrm{s}_3}=C_1+C_2+C_3({\mathbf{W}_1}^T\mathbf{X}_1+C_4)\mathrm{e}^{-\frac{({\mathbf{W}_1}^T\mathbf{X}_1+C_5)^2}{2}}\)

\(\,\,\,\,\,\,\,\,\,\,\,\,\,+C_6({\mathbf{W}_2}^T\mathbf{X}_2+C_7)\mathrm{e}^{-\frac{({\mathbf{W}_2}^T\mathbf{X}_2+C_8)^2}{2}}\)

\( C_1= -70.0470 \mathrm{sin}( -0.5585 \lambda _\mathrm{ECH} 1.8197 z/l + 0.5481) \lambda _\mathrm{M2}+ 2.7261 \)

\(C_2=\dfrac{-113.6115}{1+\mathrm{e}^{-(-1.2780\lambda _\mathrm{CED} + 8.172023)}}\)

\(\mathbf{W}_1\)

\(\mathbf{X}_1\)

\(\mathbf{W}_2\)

\(\mathbf{X}_2\)

  

− 0.0004

\( NA_\mathrm{s} \)

− 0.0244

\( NA_s \)

\( C_3 \)

− 39.2486

5.4456

\( f_\mathrm{m} \)

− 3.0671

\( f_\mathrm{m} \)

\( C_4 \)

5.9556

− 0.8733

\( f_\mathrm{C} \)

− 0.6816

\( f_\mathrm{C} \)

\( C_5 \)

0.3272

− 1.3399

\( f_\mathrm{P} \)

− 2.4263

\( f_\mathrm{P} \)

\( C_6 \)

-13.7852

5.4674

E

9.3707

E

\( C_7 \)

0.9976

− 6.3538

\( \lambda _\mathrm{OED} \)

1.3965

\( \lambda _\mathrm{ECH} \)

\( C_8 \)

7.7541

− 2.4244

z / l

− 5.5887

\( \lambda _\mathrm{OED} \)

  

− 0.0058

EA

− 1.5934

\( \lambda _\mathrm{CED} \)

  

− 2.0855

\( \lambda _\mathrm{M2} \)

− 0.3761

z / l

  
  

− 0.0145

EA

  
  

7.8406

\( \lambda _\mathrm{M2} \)

  

Fourth segment

\(O_{s_4}=C_1({\mathbf{W}_1}^T\mathbf{X}_1+C_2)\mathrm{e}^{-\frac{({\mathbf{W}_1}^T\mathbf{X}_1+C_3)^2}{2}}+C_4({\mathbf{W}_2}^T\mathbf{X}_2+C_5)\mathrm{e}^{-\frac{({\mathbf{W}_2}^T\mathbf{X}_2+C_6)^2}{2}}\)

\(\mathbf{W}_1\)

\(\mathbf{X}_1\)

\(\mathbf{W}_2\)

\(\mathbf{X}_2\)

  

− 1.0501

\( f_\mathrm{m} \)

− 0.0002

\( NA_\mathrm{s} \)

\( C_1 \)

8.3052

− 1.0433

\( f_\mathrm{C} \)

− 0.9923

\( f_\mathrm{P} \)

\( C_2 \)

8.9672

3.6018

\( f_\mathrm{P} \)

2.2763

E

\( C_3 \)

10.5251

− 6.0914

E

1.0196

\( \lambda _\mathrm{OED} \)

\( C_4 \)

− 0.3588

3.8386

\( \lambda _\mathrm{ECH} \)

8.1525

\( \lambda _\mathrm{CED} \)

\( C_5 \)

2.5160

2.1704

\( \lambda _\mathrm{OED} \)

− 19.8753

z / l

\( C_6 \)

− 4.3929

− 3.9504

\( \lambda _\mathrm{CED} \)

0.9121

\( \lambda _\mathrm{M1} \)

  

− 14.5595

z / l

    

0.0002

EA

    

2.4472

\( \lambda _\mathrm{M2} \)

    

Fifth segment

\(O_{s_5}=C_1({\mathbf{W}_1}^T\mathbf{X}_1+C_2)\mathrm{e}^{-\frac{({\mathbf{W}_1}^T\mathbf{X}_1+C_3)^2}{2}}+C_4({\mathbf{W}_2}^T\mathbf{X}_2+C_5)\mathrm{e}^{-\frac{({\mathbf{W}_2}^T\mathbf{X}_2+C_6)^2}{2}}\)

\(\mathbf{W}_1\)

\(\mathbf{X}_1\)

\(\mathbf{W}_2\)

\(\mathbf{X}_2\)

  

− 0.0008

\( NA_\mathrm{s} \)

2.2164

\( f_\mathrm{m} \)

\( C_1 \)

-4.9599

1.3951

\( f_\mathrm{C} \)

− 0.1838

\( f_\mathrm{C} \)

\( C_2 \)

9.0728

− 1.7797

\( \lambda _\mathrm{ECH} \)

3.2162

\( f_\mathrm{P} \)

\( C_3 \)

10.6308

0.6484

\( \lambda _\mathrm{OED} \)

1.2929

\( \lambda _{ECH} \)

\( C_4 \)

2.7971

− 2.9763

\( \lambda _\mathrm{CED} \)

2.2251

\( \lambda _\mathrm{OED} \)

\( C_5 \)

2.5156

− 0.0682

EA

− 1.6769

\( \lambda _\mathrm{CED} \)

\( C_6 \)

− 4.3933

3.3960

\( \lambda _\mathrm{M1} \)

− 3.0594

z / l

  

3.6285

\( \lambda _\mathrm{M2} \)

0.0248

EA

  

1.2629

\( \lambda _\mathrm{M2}\)

    

Pile base

\(O_\mathrm{b}=C_1+C_2({\mathbf{W}_1}^T\mathbf{X}_1+C_3)\mathrm{e}^{-\frac{({\mathbf{W}_1}^T\mathbf{X}_1+C_4)^2}{2}}+C_5({\mathbf{W}_2}^T\mathbf{X}_2+C_6)\mathrm{e}^{-\frac{({\mathbf{W}_2}^T\mathbf{X}_2+C_7)^2}{2}} \)

\(\qquad \quad + \dfrac{C_8}{1+\mathrm{e}^{-{\mathbf{W}_3}^T \mathbf{X}_3-C_9}}\)

\(C_1=\dfrac{-113.6115}{1+\mathrm{e}^{-(-1.6320 NA_b -4.219122 f_P + 8.518855 \lambda _\mathrm{OED} + 6.794539 )}}\)

\(\mathbf{W}_1\)

\(\mathbf{X}_1\)

\(\mathbf{W}_2\)

\(\mathbf{X}_2\)

\(\mathbf{W}_3\)

\(\mathbf{X}_3\)

  

− 0.0062

\( NA_\mathrm{b} \)

2.5940

\( NA_\mathrm{b} \)

− 6.9479

\( NA_\mathrm{b} \)

\( C_2 \)

− 32.6313

0.0160

\( f_\mathrm{m} \)

0.9087

\( f_\mathrm{m} \)

− 17.3699

\( f_\mathrm{m} \)

\( C_3 \)

4.6631

1.0300

\( f_\mathrm{P} \)

3.0781

\( f_P \)

− 12.3107

E

\( C_4 \)

− 0.5777

− 0.3132

\( \lambda _\mathrm{ECH} \)

2.4272

E

− 8.6232

\( \lambda _\mathrm{ECH} \)

\( C_5 \)

− 2.7889

− 0.0587

\( \lambda _\mathrm{OED} \)

− 6.0816

\( \lambda _\mathrm{ECH} \)

8.9026

\( \lambda _\mathrm{OED} \)

\( C_6 \)

− 0.4116

0.0417

\( \lambda _\mathrm{CED} \)

− 9.1764

\( \lambda _\mathrm{OED} \)

11.1659

z / l

\( C_7 \)

− 1.3502

1.4960

z / l

− 8.0379

\( \lambda _\mathrm{CED} \)

  

\( C_8 \)

6.2476

  

− 7.8970

z / l

  

\( C_9 \)

− 11.6089

  

7.0549

\( \lambda _\mathrm{M1} \)

    
  

− 0.1530

\( \lambda _\mathrm{M2} \)

    

4 Conclusion

The ability to predict the rate of advancement of a driven pile into the supporting sub-surface as well as the maximum depth of pile penetration during driving is quite important to both contractor and design engineer alike. In this paper, a neural network model self-evolution algorithm is proposed to estimate the blow counts of an impact hammer, the reciprocal of which gives an idea about how deep a pile penetrates when subjected to a blow of a certain type of hammer. The rationale for using self-evolution algorithm for the network training is to minimise network complexity and improve the quality of predictions. The information required by the model includes the type of hammer, the pile type, and the sub-surface characteristics of the soil. The proposed model was found surpass the conventional BPN networks as it produced more accurate results and fewer network parameters. The model was also found to agree well with independent test data set which was not included in the training.

Notes

Compliance with ethical standards

Conflict of interest

The author hereby declares that there is no conflict of interest in this work

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Civil EngineeringBayero UniversityKanoNigeria

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