Interpretation of the Pile Static Load Test Using Artificial Neural Networks

Abstract

This study presents a novel approach for interpreting static load tests (SLT) of piles using Artificial Neural Networks (ANNs) integrated with the Meyer and Kowalow load-settlement mathematical model. Reliable estimation of pile bearing capacity and settlement behavior is critical for safe and economical geotechnical design, particularly given the nonlinear and heterogeneous nature of soils. Traditional SLT interpretation methods, such as Chin-Kondner, Decourt, and hyperbolic fitting approaches, provide useful extrapolation of the ultimate capacity but are sensitive to test termination levels and parameter estimation uncertainties. The Meyer and Kowalow function offers a robust mathematical representation of the load-settlement curve, allowing decomposition of the total pile resistance into the shaft and base components. In this work, ANN models were trained to solve both the direct and inverse forms of the Meyer and Kowalow problem, enabling rapid identification of constitutive parameters (initial stiffness, nonlinearity coefficient, and ultimate capacity) from measured SLT data. Numerical experiments demonstrated that networks with a single hidden layer achieved accurate predictions with low RMSE for both training and test sets. The proposed ANN-based framework facilitates improved parameter identification, supports partial-load SLT interpretation, and provides a practical tool for engineers seeking the reliable prediction of pile performance under service loads.

1. Introduction

For safety and economic considerations, it is impossible to build constructions without the precise knowledge needed concerning the bearing capacity of the structure. In the field of civil engineering, there is an obvious need to analyze the work of the entire structure before it is built. Mostly, these problems are on a large scale, e.g., skyscrapers, bridges, and dams. In geotechnics, it is not only the scale of the structure that can be a source of problems. While the constitutive behaviors of construction materials used by builders are quite well known, in geotechnical designs, the soil medium has mechanical properties that are difficult to recognize. The sources of the difficulties here are the need for in situ tests of the mechanical characteristics of the soil, and the natural heterogeneity of the ground and its nonlinear behavior (except for in the simplest situations). Such tests are usually expensive, so their number is limited. In addition, in geotechnical research, the mechanical parameters of the soil are determined indirectly, by measuring the values correlated with them. Structures such as pile foundations also require knowledge of the nature of the soil-structure interaction. This is another factor that makes the geotechnical design more complicated. Precise data concerning the soil properties and the behavior of the soil-structure interface is thus crucial for the quality of the numerical model and the quality of the design. It is obvious that the reliability of the structure must always be assured, but its optimality is doubtful in cases of uncertain geotechnical surveys or untrustworthy material properties, particularly those of soils.

Modern civil engineering, thanks to the quick development of computer-aided design methods, is able to face more and more complex problems as described above. Nowadays, the invaluable development of numerical modeling tools, such as the Finite Element Method (FEM), Discrete Element Method (DEM), or Finite Difference Method (FDM), is noted. However, the use of commercial software implementing the above-mentioned methods is not regulated in any norms or guidelines. Additionally, code developers are not responsible for design errors resulting from a software defect or bugs. Moreover, the cost of specialized design software is still a barrier in some areas of the globe, and it is not used in everyday design practice.

Based on studies and guidelines [1,2], it should be noted that when designing pile foundations, it is necessary to verify the pile response to loads from the supported structure or confirm that this response is consistent with the adopted assumptions. It should not be forgotten that information about the total pile settlement is as important as knowledge of the pile’s bearing capacity [1]. Verification of pile bearing capacity, which can be performed in accordance with modern geotechnical standards, can be performed using geotechnical calculations based on pile and soil properties, or using a full-scale pile load test, which is also recommended in the above guidelines. Verification of pile bearing capacity is most often performed using a static load test (SLT). It is obvious that the purpose of such a test is to determine the pile’s bearing capacity when the load only approaches this value. However, the sources cited above and the normative documents do not directly indicate how to determine the bearing capacity itself based on data obtained from this test (the exception is the Eurocode, which specifies the method for determining bearing capacity). Publications on this topic do not clearly indicate a preferred method for calculating bearing capacity. Bearing capacities calculated using methods used by engineers and researchers may differ in the safety factor and resistance factor applied to their values [3,4]. Determining the allowable load by reducing the estimated bearing capacity with a safety factor or resistance factor does not provide much insight into how the pile will react to the actual load [5].

The most advanced methods for estimating pile bearing capacity are based on back testing, which matches a theoretical simulation of the measured settlement from the applied load to the in situ SLT results. The results allow for an assessment of how the pile will react to the applied load from the structure and enable the extrapolation of results not captured by the SLT. In this approach, the pile is treated as an axial structure divided into a series of short sections (elements) that are subjected to the soil resistance at the shaft or the soil resistance under the pile base. The pile head settlement is the sum of the pile displacement in the soil and the pile shortening (or elongation in tension). The relationship between the displacement of a pile element and the stress mobilized in its shaft is called the load transfer function, which is a mathematical expression of the load-dependent settlement relationship. The load-transfer function for shaft stress (the shear stress displacement curve, called the ‘t-z’ curve) and base stress (the reaction displacement curve, called the ‘q-z’ curve) expresses the load as a function of settlement according to a nonlinear mathematical relationship with some additional parameters controlling the shape of the curve [5]. The load transfer function is adapted to the actual test by numerical simulation, using load or stress-settlement relationships for the elements [5]. One proposition is very popular among researchers [6]. In the paper, the authors present the fundamentals of the numerical calculation of pile load-bearing capacity, based on assumptions regarding the relationship between the stress developed in the pile shaft and at the base, and between the pile and the displacement of the pile (which is not necessarily treated as an infinitely stiff structure). The method uses a formalism inspired by the finite element method. The constitutive relations proposed in these works lead to a good approximation of the pile settlement-load curve, although they are assumed based on other laboratory tests and theoretical considerations. Commercial software based on this principle demonstrates very good agreement with experience for a variety of such relationships (as in a prior study [7], for example). Despite the high efficiency of such programs, problems in modern pile design remain relevant. Despite the advancement of modern approaches, the engineering community still desires simple, straightforward methods for estimating pile ultimate bearing capacity (and ultimate settlement) from the SLT that do not require specialized software. These methods correspond to the determination of mathematical functions representing the load-settlement relationship recorded in the SLT.

The primary goal of this article is to analyze existing methods for estimating pile bearing capacity based on the analysis of data collected from the static pile load test (SLT) and to present our own algorithm for such an analysis, leveraging the capabilities of artificial neural networks. The article is organized as follows. In Section 2, we present existing methods for approximating SLT results and discuss methods for extrapolating these data to estimate ultimate bearing capacity and settlement. In Section 3, we present an approximation method using an artificial neural network—one of the forms of experimental curves describing the SLT. A Meyer and Kowalow curve was chosen to represent the SLT results. The central point of the method is the estimation of parameters describing the settlement-load curve as a solution to the inverse problem obtained by a suitably trained ANN. The network used is as simple as possible, allowing it to be used as a numerical tool by geotechnical engineers. The results of our numerical experiments are discussed in Section 4. We believe they are very promising.

2. Methods That Determine Load-Movement Relation to Ultimate Bearing Capacity from Load-Movement Records of a Static Loading Test

According to the old Polish standard [8], the ultimate load-bearing capacity of a pile is determined from the load-settlement diagram using a semi-graphical method. An auxiliary graph of dQ/ds, the derivative of movement over load versus load, is constructed. We will limit ourselves to stating that its result strongly depends on the accuracy of the constructed nomogram. Changing the scale of the nomogram results in a different ultimate capacity reading.

While an approximate value can be produced by looking at the curve, researchers agree that a mathematically strict definition is required that is independent of judgment and plotting scales [1] (p. 258). The approach that comes closest to fulfilling the described conditions is the mathematical function describing the load-settlement relationship. A literature review shows that over the years, researchers have proposed different approaches to estimating the ultimate bearing capacity of a pile from load-movement records of a static loading test.

Table 1. Compilation of methods (selected).

No.	Author	Mathematical Procedure				Reference
1	Hansen (1963)	$Q={\displaystyle \frac{\sqrt{\delta }}{C_1\delta +C_2}}$	$Q_u={\displaystyle \frac{1}{2\sqrt{C_1{C}_2}}}$		${\delta }_u={\displaystyle \frac{C_2}{C_1}}$	[9,10,11]
		$Q$$\text{—}\mathrm{any} \mathrm{applied} \mathrm{load}, \delta$$\text{—}\mathrm{the} \mathrm{movement} \mathrm{associated} \mathrm{with} \mathrm{load} \mathrm{Q}, Q_u$$\text{—}\mathrm{ultimate} \mathrm{load}, {\delta }_u$$\text{—}\mathrm{movement} \mathrm{at} \mathrm{the} \mathrm{ultimate} \mathrm{load}, C_1$$\text{—}\mathrm{slope} \mathrm{of} \mathrm{the} \mathrm{straight} \mathrm{line} \mathrm{in} \mathrm{the} \mathrm{auxiliary} \sqrt{\delta }/Q$$\mathrm{versus} \mathrm{settlement} \mathrm{graph}, C_2$$\text{—}\mathrm{y-intercept} \mathrm{of} \mathrm{the} \mathrm{straight} \mathrm{line} \mathrm{in} \mathrm{the} \mathrm{auxiliary} \sqrt{\delta }/Q$ versus settlement graph
2	Chin-Kondner (1970)	$Q={\displaystyle \frac{\delta }{C_1\delta +C_2}}$	$\delta ={\displaystyle \frac{Q{C}_2}{1-C_{1}Q}}$		$Q_u={\displaystyle \frac{1}{C_1}}$	[12,13]
		$C_1$$\text{—}\mathrm{slope} \mathrm{of} \mathrm{the} \mathrm{straight} \mathrm{line} \mathrm{in} \mathrm{the} \mathrm{auxiliary} \delta /Q$$\mathrm{versus} \mathrm{settlement} \mathrm{graph}, C_2$$\text{—}\mathrm{y-intercept} \mathrm{of} \mathrm{the} \mathrm{straight} \mathrm{line} \mathrm{in} \mathrm{the} \mathrm{auxiliary} \delta /Q$ versus settlement graph
3	Decourt (1999)	$Q={\displaystyle \frac{C_2\delta }{1-C_1\delta }}$		$Q_u={\displaystyle \frac{C_2}{C_1}}$		[14,15]
		$C_1$$\text{—}\mathrm{slope} \mathrm{of} \mathrm{the} \mathrm{straight} \mathrm{line} \mathrm{in} \mathrm{the} \mathrm{auxiliary} Q/\delta$$\mathrm{versus} \mathrm{settlement} \mathrm{graph}, C_2$$\text{—}\mathrm{y-intercept} \mathrm{of} \mathrm{the} \mathrm{straight} \mathrm{line} \mathrm{in} \mathrm{the} \mathrm{auxiliary} Q/\delta$ versus settlement graph
4	Poulos (2000)	$\delta ={\displaystyle \frac{Q}{k_{p0}\left(1-R_{fp}\frac{Q}{Q_u}\right)}}$ $k_{p0}$$\text{—}\mathrm{initial} \mathrm{stiffness} \mathrm{of} \mathrm{the} \mathrm{pile} \mathrm{determined} \mathrm{as} \mathrm{the} Q/\delta$$\mathrm{ratio} \mathrm{for} \mathrm{initial} \mathrm{settlements} \mathrm{for} \mathrm{which} \mathrm{linear} \mathrm{relation} \mathrm{is} \mathrm{observed}, R_{fp}$—constant hyperbolic coefficient				[16]
5	Meyer and Kowalow (2010)	$\delta =C_2{Q}_u{\displaystyle \frac{{\left(1-\frac{Q}{Q_u}\right)}^{{-\kappa }_2}-1}{{\kappa }_2}}$		$Q=Q_u\left(1-{\left(1+{\displaystyle \frac{{\kappa }_2\delta }{C_2{Q}_u}}\right)}^{-\left({\displaystyle \frac{1}{{\kappa }_2}}\right)}\right)$		[17]
		direct formula		reversible formula
		$C_2$$\text{—}\mathrm{inverse} \mathrm{of} \mathrm{the} \mathrm{aggregate} \mathrm{Winkler} \mathrm{constant} [\mathrm{mm}/\mathrm{kN}], {\kappa }_2$—dimensionless parameter defining the resistance distribution at the base and the shaft of the pile
6	van der Veen (1953)	$Q=Q_u\left(1-e^{-\alpha \delta }\right)$ $\alpha$—coefficient that influences the shape of the load-settlement curve > 0				[18]
7	Vijayvergiya (1977)	$Q=Q_{trg}\left(V\sqrt{{\displaystyle \frac{\delta }{{\delta }_{trg}}}}-\left(V-1\right){\displaystyle \frac{\delta }{{\delta }_{trg}}}\right)$ $Q_{trg}$$\text{—}\mathrm{target} \mathrm{load} \mathrm{or} \mathrm{resistance}, {\delta }_{trg}$$\text{—}\mathrm{target} \mathrm{movement} \mathrm{paired} \mathrm{with} Q_{trg}$$, V$—function coefficient > 0				[19]
8	Gwizdala (1996)	$Q=Q_{trg}{\left({\displaystyle \frac{\delta }{{\delta }_{trg}}}\right)}^{\theta }$ $\theta$$\text{—}\mathrm{function} \mathrm{coefficient} 0 \le \theta$ ≤ 1				[6]
9	Zhang and Zhang (2012)	$Q={\displaystyle \frac{\delta \left(a+c\delta \right)}{{\left(a+b\delta \right)}^2}}$		$Q_{trg}={\displaystyle \frac{1}{4\left(b-c\right)}}$		[20]
		${\delta }_{trg}={\displaystyle \frac{a}{b-2c}}$		$Q_{inf}={\displaystyle \frac{c}{Q_{trg}{b}^2}}$
		$Q_{trg}$$\text{—}\mathrm{target} \left(\mathrm{peak}\right) \mathrm{load} \mathrm{or} \mathrm{resistance}, {\delta }_{trg}$$\text{—}\mathrm{target} \left(\mathrm{peak}\right) \mathrm{movement} \mathrm{paired} \mathrm{with} Q_{trg}$$, Q_{inf}$$\text{—}\mathrm{resistance} \mathrm{at} \mathrm{infinitely} \mathrm{large} \mathrm{movement} (\mathrm{must} \mathrm{always} \mathrm{be} \ge 0), b$$\text{—}\mathrm{parameter} {\displaystyle \frac{1}{2Q_{trg}}}-{\displaystyle \frac{a}{{\delta }_{trg}}}$$, c$$\text{—}\mathrm{parameter} {\displaystyle \frac{1}{4Q_{trg}}}-{\displaystyle \frac{a}{{\delta }_{trg}}}$
10	Rahman (2019)	$Q=Q_{trg}{\left({\displaystyle \frac{{\delta }_{trg}{\delta }^{F-1}+{\delta }_{trg}^{F-1}\delta }{{\delta }_{trg}^F+{\delta }^F}}\right)}^{{\displaystyle \frac{1}{M}}}$ $M$$\text{—}\mathrm{function} \mathrm{coefficient} > 0, F$—function coefficient > 1.0				[5]
11	Attributed to Miad and Mahious (2021)	$Q=2Q_u{\displaystyle \frac{\delta }{{\delta }^R}}\left(1-{\displaystyle \frac{\delta }{2{\delta }^R}}\right)$		${\delta }^R=2{\displaystyle \frac{Q_u}{K_{v0}}}$		[2,21]
		${\delta }^R$—reference settlement defined as threshold of full $\mathrm{mobilization} \mathrm{of} \mathrm{the} \mathrm{ultimate} \mathrm{bearing} \mathrm{capacity} Q_u$$, K_{v0}$—initial vertical pile stiffness

presents the methods elaborated for mapping the load-movement records of a static loading test. The methods are numbered in the order in which they appear in Table 1. The authors refer to the methods by number in the text. However, the legends in the figures include the full names of the methods used. Methods 1 to 7 may be classified as interpolators to the response of a pile element and extrapolators to the ultimate bearing capacity. Methods 8 to 11 may be classified as functions describing t-z/q-z relations for short pile elements. The second group presented in this paper shows the load as governed by movement and as correlated to a selected Target Load and a Target Movement. The Target can be, but is not usually, the estimated ultimate capacity of the load-movement response [5].

The presented functions are based on different assumptions, i.e., the hyperbolical or exponential shape of the load-movement relation. Researchers stated that SLTs conducted on a simply instrumented pile cannot directly measure ultimate load capacity, especially when the load-settlement relation is recorded only for the pile head. It yields no quantitative information on the load-transfer mechanism, e.g., compression (shortening) of the pile or base resistance and the distribution of shaft resistance. Load-settlement measured at the pile head is interpreted by a design criterion (per one definition or another) to estimate pile bearing capacity. To fill the gap, conventional SLT is frequently expanded to include instrumentation (telltales and strain gages) to obtain the required information. As stated by Fellenius [1], only the load-settlement curves for pile-elements (at the gage’s location) should be interpreted. Figure 1 presents a compilation of the mathematical functions (selected) fitted to a measured pile-element load-settlement curve. The filed data from the gage locations reported in [1] (p. 280) were used. The test curve indicates a slight strain-hardening response. The initial part of all curves except number 4 shows a good fit to the test data. The best fit for the test results was obtained by functions 2, 5, and 6.

Usually, commercial SLTs for the private sector are simply instrumented at the top with dial gauges, with a usual accuracy of 0.01 mm for settlement reading. The load is usually applied in increments by a hydraulic jack and a pump assembly fitted with a pressure gauge, against a weighted platform [2]. Only the load-settlement in the pile head is recorded. This is the situation that practical engineers must face. Sometimes, the SLT is performed over the full load range until failure. Figure 2 presents a compilation of the curves fitted to a measured load-settlement curve for the pile head. The test data for the 31-10L pile are taken from a prior study [22]. The best fit for the test results was obtained by functions 2, 3, 4, 5, 7, and 10. It is worth mentioning that method 5 was a direct and reversible formula that provided exact results. As was previously mentioned, methods 2, 3, 4, and 5 are extrapolation functions and allow us to estimate the ultimate bearing capacity. For methods 2 and 3, estimation depends on auxiliary graph. For methods 4 and 5, ultimate bearing capacity is included as a function variable that needs to be estimated. The analytical functions quoted refer to the hyperbolic load-settlement characteristic, but their effectiveness depends on knowledge of the full range of characteristics until failure.

Even more frequently, commercial static load testing is performed for a partial load range, meaning it is terminated before the ultimate load capacity is reached. This has its economic justification. Furthermore, standards allow for the application of a load equal to 1.5 times the design load capacity. With incomplete knowledge of the load-settlement curve, it seems that good fits can be obtained, but the selection of parameters of individual functions is largely based on guesswork.

The results of the static load test used in the analysis were provided thanks to the courtesy of professor Kazimierz Gwizdala [6]. The results of SLT presented were part of his commercial work. The obtained materials contain the full geotechnical details, but only the SLT logs are available for readers. We used the SLT database of piles installed for the foundation of the supports of access flyovers to the MA-91 highway bridge constructed as part of the construction of the A-1 highway, section I, Nowe Marzy–Grudziądz, km 89 + 494.76 to km 98 + 400.00. A static load test was performed during winter 2009/2010 and related to twelve ϕ508/560 mm Vibro piles. In this case study, only data from SLT field tests for piles 11-41 was used.

2.1. Driven Piles Formed into Subgrade Construction Technology

Historically, displacement pile technology has included driven piles. However, for some time, it has been usual practice to use it mainly for screwed piles. Driven pile technologies include prefabricated piles as well as piles formed in the ground. Driven displacement pile technology is based on the fact that during the construction of the pile, the soil is not moved outside, but is instead displaced by the execution elements into the lateral zones around the pile. This results in the compaction and tightening of the soil medium around the pile—around the pile shaft, and most importantly, under the base of the pile. Driven piles can be used in a variety of soil conditions, but they are mainly recommended for soil substrates in which the non-cohesive bearing layers need to be compacted (e.g., medium-compacted sands or gravels). In situations where the subsoil is inherently well compacted or consolidated, the use of driven piles could disturb the soil structure and reduce its mechanical performance, and encounter technical difficulties due to high pile driving resistance. The construction of driven displacement piles formed into the ground consists of the following steps. In the first stage, the cast pipe is inserted into the ground. The cast pipe is closed from the bottom with a cone or flat steel base (Vibro, Simplex) and concrete plug (Franki). The cast pipe is then embedded in the ground using a diesel or hydraulic hammer. The steel pipe is driven into the ground by the hammer free-falling onto the plug. Knocking out the cork with the hammer causes the formation of an enlarged base (which depends on technology used) [23]. The steel base is always lost. When measurements show that the base of the pipe has reached soil of sufficient mechanical parameters, driving stops. In the second stage, the pile is formed in the cast pipe. This consists of two processes: the injection of the concrete mixture and the introduction of the reinforcing structure. Depending on the technology used, this happens in a certain order. In some solutions, reinforcement happens even before concreting, e.g., Simplex, Franki, Vibro. This avoids the problems of reinforcing the base of the pile. The next characteristic stage is the process of pulling out the cast pipe. This can occur with the simultaneous compaction of concrete, e.g., Simplex, Franki. In Vibro piles, the cast pipe is pulled out using a special frequency hammer or vibrator. Attached to the pipe, the vibrator causes compaction of the concrete mixture and ensures that the pile shaft is well connected to the ground. Reinforcing steel protects the pile from breaking when the cast pipe is lifted. The process of the installation of a Vibro pile is shown in Figure 3. In their present form, Vibro piles were implemented as early as the 1960s [24].

2.2. Geological Condition of Tested Area

In order to identify the soils of the investigated area, a number of geotechnical and geological field studies were carried out: 75 boreholes with a depth of up to 35.0 m (with a total length of 1757.5 m), 36 CPT static soundings with a depth of up to 24.1 m (with a total length of 645.1 m), and 40 DPH or DPSH dynamic soundings with a depth of up to 28.8 m (with a total length of 868.0 m). The test points were located at bridge supports and flyovers. On the discussed section of the A1 highway, there are variable geological and engineering conditions caused by geomorphological formations within the Vistula Valley. To the depth recognized by the field survey, there are Quaternary soils: Holocene and Pleistocene. In the axis of the present river valley lie Quaternary sand and gravel soils. It is dominated by medium-grained sands and gravels with admixtures of stones and pebbles in states ranging from loose, with ID = 0.35, through to medium-compacted with ID = 0.55 and compacted with ID = 0.85. The layer has a thickness of 10–20 m. Within it, there are interlayers of cohesive soils, including Tertiary soils (silt, compacted sandy clays). In the layer 15.0–20.0 m below surface level lies the bottom of a discontinuous series of Quaternary cohesive soils in the form of sandy silt, sandy clay, and fine sand, in states from plastic with IL = 0.35 to hard plastic with IL = 0.05. In the layer about 9.0–16.0 m below surface level, there is an irregular system of layers of varying thickness organic soils interlayered in some cross sections with sandy sediments. The thickness of organic soils reaches 6.0 m in some places. The stabilized groundwater table is formed at a depth of 0.5–3.0 m below surface level, and locally lies at ground level. The groundwater level depends on the water level of the Vistula River.

For support no. 11 (pile 11–41), the geological condition of the tested area is presented in Figure 4. The geological parameters are given in

Table 2. Soil geotechnical parameters.

Soil Layer	Soil Condition		Natural Moisture Content	Unit Weight	Cohesion	Friction Angle	CPT Cone Resistance	Oedometric Modulus from CPT	Oedometric Modulus from Polish Codes	Filtration Ratio
	Compaction Rate	Plasticity Rate
	ID	IL	Wn [%]	γ [kN/m3]	c [kPa]	Phi []	qc [MPa]	M0 [MPa]	M0 [MPa]	k [m/Day]
I A2	-	≥0.35	23.4 ÷ 43.9	16.6	6.3	8.0	0.3 ÷ 1.0	3.0 ÷ 12.0	12.0	-
I A4	~0.35	-	14.3 ÷ 28.7	8.2	-	24.8	1.0 ÷ 5.0	10.0 ÷ 35.0	40.0	-
I B1	0.35	-	25.0	9.5	-	29.7	2.5 ÷ 4.0	20.0 ÷ 30.0	45.0	0.1 ÷ 1
I B2	0.45	-	24.6	9.5	-	30.2	4.0 ÷ 6.0	30.0 ÷ 50.0	55.0	1 ÷ 10
I B3	0.55	-	23.5	9.6	-	30.7	6.0 ÷ 9.0	40.0 ÷ 60.0	65.0	1 ÷ 10
I B4	0.66	-	23.0	9.6	-	31.2	9.0 ÷ 14.0	55.0 ÷ 80.0	80.0	1 ÷ 10
I C2	0.45	-	22.5	10.2	-	32.7	5.0 ÷ 8.0	35.0 ÷ 60.0	85.0	10 ÷ 25
I C3	0.55	-	21.0	10.3	-	33.3	8.0 ÷ 12.0	55.0 ÷ 75.0	100.0	10 ÷ 25
I C4	0.65	-	20.0	10.4	-	33.9	12.0 ÷ 17.0	75.0 ÷ 95.0	120.0	10 ÷ 25
I C5	0.75	-	19.0	10.7	-	34.5	17.0 ÷ 25.0	95.0 ÷ 125.0	140.0	10 ÷ 25
I E1	-	0.35	19.7 ÷ 31.9	20.0	11.9	12.4	0.4 ÷ 0.8	5.0 ÷ 15.0	20.0	0.1 ÷ 1
II A2	0.65	-	23.0	9.6	-	31.2	9.0 ÷ 14.0	55.0 ÷ 80.0	80.0	1 ÷ 10
II A3	0.75	-	22.5	10.2	-	31.6	14.0 ÷ 22.0	80.0 ÷ 100.0	95.0	1 ÷ 10
II A4	0.85	-	22.0	10.2	-	32.1	>22.0	>100.0	110.0	1 ÷ 10
II B1	0.55	-	21.0	10.3	-	33.3	8.0 ÷ 12.0	55.0 ÷ 75.0	100.0	10 ÷ 25
II B3	0.75	-	19.0	10.7	-	34.5	17.0 ÷ 25.0	95.0 ÷ 125.0	140.0	10 ÷ 25
II B4	0.85	-	18.0	10.8	-	35.2	>25.0	>125.0	165.0	10 ÷ 25
Legend
Soil Layer	Soil Type (in Polish)					Soil Type (in English)
I A2	G(pi)H, (pi)H, G(pi)zH, G(pi)H//(pi), G(pi)H//P(pi)					Clayey silt, silty sand, silty clay with sand—with humous soil, silty sand
I A4	PdH, PsH, P(pi)H					Fine sand, medium sand, silty sand—with humous sand
I B1	P(pi), Pd					Silty sand, fine sand
I B2	Pd, P(pi)					Fine sand, silty sand
I B3	Pd					Fine sand
I B4	Pd					Fine sand
I C2	Ps, Pr					Medium sand, coarse sand
I C3	Ps, Pr					Medium sand, coarse sand
I C4	Ps, Pr					Medium sand, coarse sand
I C5	Ps, Pr					Medium sand, coarse sand
I E1	G(pi), Gp, Pg					Clayey silt, clayey sand, slightly clayey sand
II A2	Pd					Fine sand
II A3	Pd					Fine sand
II A4	Pd					Fine sand
II B1	Ps, Pr					Medium sand, coarse sand
II B3	Ps, Pr					Medium sand, coarse sand
II B4	Ps, Pr					Medium sand, coarse sand

. The geological configuration of the ground layers is an alternation of cohesive and (mainly) non-cohesive soils. The first layer has a thickness of around 9.0 m and consists of sands that gradually change particle size to form silty sand (siSa) on top, through to medium sand (MSa) and coarse sand (CSa). A stable groundwater level is expected at 3.5 m below the surface. The next geotechnical layer has a thickness of 2.5 m and consists of fine sand mixed with humus (FSaH) and sandy clays mixed with humus (saCl H). This layer has pressurized groundwater. The pore pressure is unknown due to a lack information from field test reports. A geotechnical layer of fine sand (FSa) with a thickness of 5.5 m is located 13.0 m below surface level, where the second geotechnical layer of fine sand (FSa) occurs. The borehole stops at 22.0 m below surface level.

2.3. Description of SLT

Test loads in accordance with the recommendations of the designer and Polish codes were carried out on piles in eleven supports. A total of 12 static compressive load tests of Vibro piles with a diameter of ϕ 508/560 mm were performed under the entire north and south flyover. Test loads were performed using test stands anchored to adjacent construction piles. The main structural element of the stand was a steel plate beam of 1100 mm height and 12.0 m or 16.0 m length. This beam was supported on working supports and anchored to adjacent construction piles using transverse steel beams and a system of tendons in the form of steel bars. An example of a workstation is shown in Figure 5.

The test load on the pile was applied using a set of three hydraulic cylinders with a total range of Q_max = 7500 kN. The force value was determined by measuring the oil pressure on a 0.6 class gauge. Settlement of the test pile was measured using four dial sensors with a reading accuracy of 0.01 mm based on an independent measurement base. In addition, the pull of the anchor piles was controlled. This control showed safe operation of the anchor piles during the entire load test run. According to the requirements of the standard, such piles can be used fully in the construction of foundations (with a pile uplift of less than 5 mm). According to the guidelines, it was assumed that conventional stabilization of the pile settlement occurred when the settlement did not increase by more than 0.05 mm within 10 min. After reaching a force close to the design load Q_d and stabilizing the settlement, the pile was completely unloaded to measure its permanent settlement. The pre-unloading force was then restored and loading continued in stages until a force of Q_max ≈ 1.5·Q_d was applied. After the final unloading, the permanent final settlement of the piles was measured.

As stated by Fellenius [1] (p. 258), it is improper to conduct unloading/reloading cycles. If unloading/reloading cycles cannot be omitted, the primary test should be completed first. Due to fact that any presented mathematical functions cannot extrapolate unloading/reloading cycles, this part will not be taken under consideration.

Figure 6 presents a compilation of the curves numbered 2, 3, 4, and 5 fitted to a measured load-settlement curve for pile head no. 11–41. All presented methods have built-in formulas to estimate ultimate bearing capacity. Firstly, estimation was conducted according to the old Polish standard [8]. The procedure is based on visual reading from an auxiliary graph. The Q_ult was estimated as 8826 kN. The main disadvantage is that the graph must be drawn in a certain scale, which is based only on practical experience. The second problem is the sufficiently large dataset. SLT must be conducted to the point where a straight line on the auxiliary graph is observed. If this is not possible, the ultimate bearing capacity cannot be determined. SLT stopped at Q_max ≈ 1.5⋅Q_d often does not indicate a part with a straight line. The Chin-Kondner and Decourt curves also allow us to estimate ultimate bearing capacity thanks to auxiliary graphs, but these methods do not consider the scale of the graph. However, the problem with SLT’s early stop remains. Poulos’ mathematical function uses the ultimate bearing capacity as an unknown inside the equation, which is determined by trial and error. One of the coefficients is $k_{p0}$—the initial stiffness of the pile determined as the Q/δ ratio for initial settlements for which linear relation is observed, which is easy to determine. The last coefficient is $R_{fp}$—the constant hyperbolic coefficient, which is very speculative. The author’s general recommendation is to adopt a coefficient of 0.75. In the above examples, the coefficients 0.75, 0.98, and 0.85 were used, respectively. The Meyer and Kowalow mathematical function is described by three variables. One of them is the ultimate bearing capacity, determined in the back-calculation process.

The Meyer and Kowalow mathematical function is described by three variables: $C_2$, ${\kappa }_2$, and Q_ult. These unknowns are determined in the back-calculation process. Classic backward calculations, i.e., those leading to the determination of mathematical function parameters, consist of minimizing the sum-of-squares differences between the measured and theoretical SLT results [25], which can be written as follows:

$$\underset{p_i}{min}\left[{\displaystyle \sum _{k=1}^m}{\left(u_k\left(p_i\right)-d_k\right)}^2\right]$$

(1)

where $p_i$ is the assumed parameter of mathematical function, $i$ is the parameter index, $m$ is the number of load increaments, $u_k\left(p_i\right)$ is the calculated theoretical settlement at the k-th load increament, and $d_k$ is the measured settlement at the k-th load increament.

Obtained in this way, the parameters of the Meyer and Kowalow mathematical function are a best-fit for the theoretical load-settlement relation to the field SLT results. The Meyer and Kowalow proposal is free from the problems of the previous functions. However, it depends on the accuracy of the optimization performed. The optimization problem mentioned above can be solved by different methods of statistical analysis. What distinguishes the Meyer and Kowalow proposal from other mathematical descriptions is that the authors state that using their function, it is possible to separate the total load-settlement relation of the pile into the shaft and under the pile base load-settlement relation. This will be disscussed in the next section.

3. Interpretation of the Static Pile Load Test Using Artificial Neural Networks

In this section, the application of artificial neural networks (ANNs) in the analysis of experimental data will be presented for the example of the interpretation of the static pile driving test. The presented approach involving ANNs and notions of direct and inverse problems is important, but it does not fully solve the issue of interpretation of the static pile bearing capacity test. The most important simplifications are assumed: limiting the analysis of the static test without unloading and omitting the rheological phenomena that are observed during the test execution. Only the simplest hypotheses regarding pile skin and soil interaction will be presented. It should be emphasized that there exist at least a few methods of performing backward calculations. The most conventional is curve fitting, which is the process of finding a mathematical function that best represents a series of data points. The most common method is least-squares fitting, which minimizes the sum of the squared differences between the data and the curve. Optimization algorithms are used for this purpose, searching for local minima, e.g., Nelder-Mead, or global minima, e.g., the differential evolution method. The application of certain optimizers is presented in [26]. However, authors and engineers using the Meyer and Kowalow (abbreviated from now on as ‘M-K’) method have raised questions about the physical meaning of deviations from this curve. These deviations may be caused, for example, by the dependence of measurement accuracy on the load level. It can also be assumed that some subsequences in the measurement sequence are less reliable than others. It is also likely that the experimental shape of the settlement-load curve may be qualitatively different from that assumed by the M-K formula due to the specific distribution of the load-bearing capacity of the pile base and its shaft. In our opinion, current methods for examining deviations between approximated and recorded values do not provide information about the physical nature of these deviations. Identification of M-K curve parameters using ANNs is based on solving an inverse problem. In this problem, additional physical parameters can be introduced into the model on which the inverse neural network is trained. This allows us to answer the question of whether these additional parameters can be identified as responsible for any qualitative deviations from the M-K curve, assumed as an “a priori” settlement-load constitutive relationship.

3.1. Meyer and Kowalow Representation of a Static Load Test as a Direct Problem

In 2010, Meyer and Kowalow presented a proposal for a full description of the Q-s curve [17] that can be used both for pile foundations [27,28,29,30,31,32,33,34] and slab-and-pile foundations [35]. The equation describing the pile settlement curve in the Meyer and Kowalow model can be written as:

$$s(N)=C_2{N}_{gr}{\displaystyle \frac{{\left(1-\frac{N}{N_{gr2}}\right)}^{-{\kappa }_2}-1}{{\kappa }_2}}$$

(2)

where s denotes the displacement of the head of the pile, and N stands for the concentrated vertical force applied at the point when settlement is measured, namely at the head of the pile. Two parameters, $C_2$ and ${\kappa }_2$, can be considered as typical constitutive descriptors of the observed physical properties of the pile-ground system. A third parameter, $N_{gr2}$, has the dimension of a concentrated force. It can be observed that the parameters of this representation have a clear physical meaning. The limit of the proposed load-settlement relation when load approaches zero reveals that the dependence (2) is linear in the region of the origin. Coefficient $C_2$ reflects the initial rigidity of the soil-structure (pile-ground) interaction. This is in accordance with elementary intuition and most observations.

$$\underset{N\to 0}{\lim } s\left(N\right)=C_{2}N$$

(3)

In the limit of the proposed load-setlement dependence, when load approaches the arbitrarily assumed parameter $N_{gr2}$, displacement of the head of the pile approaches infinity. The coefficient $N_{gr2}$ can be interpreted as the value of the force N, at which the bearing capacity of the pile is exhausted, since its displacements increase indefinitely.

$$\underset{N\to N_{\mathit{gr}2}}{\mathit{lim}} s\left(N\right)=\infty$$

(4)

A typical course of the Meyer and Kowalow curve is shown in Figure 7. Taking into account the fact that this curve is drawn between the straight tangent to it in the vicinity of zero and the asymptote drawn at the level of the $N_{gr2}$ force, it should be noted that it qualitatively corresponds to the course of the displacement of the pile head loaded with increasing vertical force. Therefore, the Meyer and Kowalow curve can be useful in describing the pile SLT.

Formulate the ‘direct’ problem related to the SLT.

Problem 1.

During the static loading of the pile, the dependence s(N) of the vertical displacement of the point on the pile head s under the monotonically increasing vertical force N is recorded. Let us assume that the interaction of the pile with the ground medium is described by two constitutive parameters, C₂ and κ₂, so that: (1) at the initial moment of the load, the displacement s depends linearly on the load N; (2) at the value N = N_gr2, the displacements increase to infinity. Find a relation, s(N), that satisfies the above requirements.

The solution to Problem 1 can be obviously assumed in the form of Formula (2), and the proof is trivial. In order to emphasize the dependence of the solution on the M-K problem given explicitly by Formula (2) on the three values of $C_2$, ${\kappa }_2$, and $N_{gr2}$, the following notation can be used:

$$\{s_i,N_i\}=\left\{F\right(C_2, {\kappa }_2, N_{gr2}), N_i\} \mathrm{for} \mathrm{any} N_i$$

(5)

For a given set of control points $N_i$, the pairs $\{s_i,N_i\}$ are a function $F$ of the three-element set ($C_2$, ${\kappa }_2$, $N_{gr2}$).

Formulate the ‘inverse’ problem related to the SLT. This ‘inverse’ problem is based on the formulation of the ‘direct’ Problem 1.

Problem 2.

During the static loading of the pile, the dependence s(N) of the vertical displacement of the point on the pile head s under the monotonically increasing vertical force N is recorded. Let us assume that the dependence s(N) is known for n control points in the form of an ordered set of pairs {N_i, s_i} i = 3…n. Find two constitutive parameters, C₂ and κ₂, and the value of N = N_gr2—such that the given pairs {N_i, s_i} belong to the curve that verifies the requirements postulated in the problem.

Inverse Problem 2 can be formulated in accordance with Equation (5). For a given set of control points $N_i$, pairs $\{s_i,N_i\}$ of the three element sets ($C_2$, ${\kappa }_2$, $N_{gr2}$) are the inverse function of the measured set of displacements:

$$\left\{C_2, {\kappa }_2, N_{gr2}\right\}=F^{-1}\left(s_i\right)$$

(6)

where $s_i$ is the same sequence of displacements as in Equation (5), and the solutions of direct Problem 1 are obtained for the same sequence of $N_i$.

3.2. Existing Methods of Finding Meyer and Kowalow Curve Coefficients

The solution to the M-K problem is given explicitly by Formula (2). This formula is a function that assigns the displacement of the pile head, $s_i$, for each given load, $N_i$, to the three values, $C_2$, ${\kappa }_2$, and $N_{gr2}$. Solving the inverse problem is no longer easy, because it will not be possible to solve the system of Equation (2) written for each of the pairs {$s_i$, $N_i$}. This leads to a nonlinear system of equations for $C_2$, ${\kappa }_2$, and $N_{gr}$. However, the solution of the inverse problem is very important, since $N_{gr2}$ is a key parameter for the engineer waiting for the results of the static pile load test. It is the pile bearing capacity, which should be determined as a result of SLT. In previous work [36], three different numerical optimization algorithms were written for this purpose. The properties of various algorithms applied for various choices of subsets of the measured pairs $\{s_i,N_i\}$ were discussed in several chapters of the paper [36]. The choice of subsets of the given data can be important, since only the three pairs $\{s_i,N_i\}$ are necessary to solve the above problem. Thus, some perturbation of the results of the minimization (3) can be interpreted to be due to possible errors in the measurements in SLT. In the following part of this section, the algorithm based on ANNs will be proposed for quick identification of the parameters, assuming that all the pairs are equally meaningful in consideration. The workflow is as follows:

• Determining the possible range of M-K curve parameters, consistent with their physical interpretation (three parameters according to Formula (8) or five parameters according to Formula (22) for the generalized version).
• Selecting the “sample” values of the triplets (or fives) using any method that provides representative “coverage” of their physical range of variability.
• Performing the calculations “directly” according to Formula (7) or (20), to obtain a database with Formula (10) or (24) for ANN training.
• Dividing the database into a subset used for network training and a subset used for testing the network’s response.
• Training the inverse networks for all M-K curve parameters according to Formula (12) or (24) until the network’s response error for the testing set starts to increase.

3.3. Identification of Meyer and Kowalow Curve Coefficients as an Inverse Problem Solved with Artificial Neural Networks

The theoretical basis for using ANNs to solve the Meyer and Kowalow problem relates to its approximation properties. ANNs can approximate an unknown function with the needed accuracy, provided we have enough number examples of pairs: {arguments of function, corresponding function values}.

The training of the ANN proceeds in such a way that the arguments of the function should be entered into the input of the ANN (containing the correct number of input neurons), and the function values will serve as the appropriate target for the output neurons of the network. The training process will end when the neural network assigns the correct function values to all arguments. If the training is carried out correctly, the neural network will calculate the correct function values also for the arguments that were not used during the training of the network. Usually, a sufficient number of examples of pairs needed to train the network is provided by solving the ‘direct’ problem. In order to train a network that solves the inverse problem, it is enough to introduce the function values into the input of the network and use the corresponding function arguments as the neural network response patterns on its output layer. Also in this case, when the ‘direct’ solutions to the problem are known, it is enough to use them at the input of the network and put the arguments corresponding to these solutions on the output of the network.

An important observation for the numerical part of this chapter is as follows. Coming back to Formula (2), it is seen that this formula is easy invertible. Instead, the normal force can be computed as a function of displacement s. This approach has been chosen in the following parts of this chapter, so the following formula replaces the classical Meyer and Kowalow Formula (2):

$$N\left(s\right)=N_{gr2}\left[1-{\left(1+{\displaystyle \frac{{\kappa }_{2}s}{C_2{N}_{gr2}}}\right)}^{-{\displaystyle \frac{1}{{\kappa }_2}}}\right]$$

(7)

We stress that the two formulae, (2) and (7), are fully equivalent, but the chosen one is easier from a numerical point of view. To assure the correct location of the pairs $\{N_i,s_i\}$, a special set of displacements has been assumed in the numerical example that follows. In practical analysis of the data from SLT, it is advisable to train the ANN with the obtained, observed displacements as independent variables.

3.3.1. Computation Steps to Be Executed

According to this overall plan, the following computation steps will be executed:

Random uniform distribution of $C_2$, ${\kappa }_2$, and $N_{gr2}$ has been assumed. According to this, from an assumed range of values, $\left[C_2\left(min\right),\dots ,C_2\left(max\right)\right]$, $\left[{\kappa }_2\left(min\right),\dots ,{\kappa }_2\left(max\right)\right]$, and $\left[N_{gr2}\left(min\right),\dots ,N_{gr2}\left(max\right)\right]$, the trial values of the triple $C_2$, ${\kappa }_2$, and $N_{gr2}$ are randomly generated. The training pairs are built: $\left\{\left\{C_{2,tr}, {\kappa }_{2,tr}, N_{gr2,tr}\right\},\left\{s,N\left(s\right)\right\}\right\}$. Since the trial independent variables are always the same, they can be omitted from the training. Thus, the training pattern looks like, for a chosen set of $s_i$:

$$\left\{\left\{C_{2,tr}, {\kappa }_{2,tr}, N_{gr2,tr}\right\},N_i\left(s_i,C_{2,tr}, {\kappa }_{2,tr}, N_{gr2,tr}\right)\right\}$$

(8)

For the direct problem, as an approximator of relation (9) (corresponding to (5) in classical notation):

$$\left\{N_i,s_i\right\}=\left\{F\left(C_2, {\kappa }_2, N_{gr2}\right),s_i\right\} \mathrm{for} \mathrm{chosen} s_i$$

(9)

The function, F, will be replaced by an artificial neural network, ANN_F. It is assumed that there are as many neurons in the output layer as the number of chosen $s_i$ values (in a practical application—the number of data points issued from SLT). The $s_i$ values are presented to the ANN_F as the target values. (1) Hidden layers (one hidden layer can be sufficient). (2) Input layer with three neurons for $C_{2,tr}$, ${\kappa }_{2,tr}$, and $N_{gr2,tr}$. This ANN acts as an approximator of F, which we denote as follows:

$$\left\{N_i\right\}=ANN\_F@\left({\left\{C_2, {\kappa }_2, N_{gr2}\right\}}_i\right)$$

(10)

(The symbol @ denotes an action of the ANN considered as an operator on the set of its arguments).

After the suitable training, we can check that the forces are accurately approximated. This step is not important in the analysis and can be omitted, but here, it is evoked just to show that the analytical computations can be replaced easily by the suitably trained ANN.

Important step—Training the ANN for unknown inverse relation, corresponding to (6):

$$\left\{C_2, {\kappa }_2, N_{gr2}\right\}=F^{-1}\left(N_i\right) \mathrm{for} \mathrm{a} \mathrm{given} \mathrm{set} \mathrm{of} s_i$$

(11)

In this case, the artificial neural network, named ANN_F, acts as an approximation of the unknown inverse operator $F^{-1}$:

$${\left\{C_2, {\kappa }_2, N_{gr2}\right\}}_i\widetilde{=}ANN\_F@\left(\left\{N_i\right\}\right) \mathrm{for} \mathrm{a} \mathrm{given} \mathrm{set} \mathrm{of} s_i$$

(12)

The function $F^{-1}$ will be approximated by an artificial neural network, ANN_F, in the following form. It is assumed that there are as many neurons in the input layer as the number of chosen $s_i$ values (in a practical application—the number of data issued from SLT). (1) Hidden layers (one hidden layer can be sufficient). (2) Output layer with three neurons for $C_2$, ${\kappa }_2$, and $N_{gr2}$. The $C_{2,tr}$, ${\kappa }_{2,tr}$, and $N_{gr2,tr}$ values are presented to the ANN_F as the target values.

In ‘recall mode’, when the input layer will be presented with the true $s_i$ issued from SLT, the ANN_F will respond with the true, corresponding values of $C_2$, ${\kappa }_2$, and $N_{gr2}$, which are the unknowns of the problem.

3.3.2. Results of the Computations Executed According the Proposed Scheme

Neural Network for Direct Problem and Testing Patterns Used

In the considered problem of Meyer and Kowalow, defined above in this section, the function to be approximated is subjective and smooth, thus the numerical complexity of approximation of both direct and inverse relations should be easy. Indeed, we show that the ANN with a unique and small hidden layer is sufficient to approximate these functions with accuracy satisfactory for engineering problems. Numerical complexity depends on two important factors. The first one is the number of samples of the direct solution that allow the needed degree of accuracy of the solution to the inverse problem. The second is the complexity of the structure of the ANN itself, which is necessary to approximate, with small tolerance, both inverse and direct problems. The well-known theorems that assure that the ANN is a universal approximator of any function are not constructive. They mean that neither the number of neurons in hidden layers nor the number of hidden layers are clearly prescribed. Also, another problem is that the sampling of the multidimensional space of the solution depends strongly on the properties of the approximated multivariable function, F.

For the first example, we constructed a trial space with 180 elements, and 70 of them were taken as the test set (35 were used only for verification of the quality of generalization during the training). The rest were used as the learning set of examples. In the symbol of the networks, a description of the structure is shown. The first number is the number of neurons in the input layers, followed by the numbers of neurons in the consecutive hidden layers, and then the number of neurons in the output layers. According to this, the first network we successfully tested, ANN_F 3_5_20, is the network that approximates the direct relation (11) with the three coefficients $C_2$, ${\kappa }_2$, and $N_{gr2}$ at the input, with 5 hidden neurons and 20 forces corresponding to the prescribed set of displacements. Similarly, the ANN denoted by ANN_F 20_5_3 is the network that approximates the inverse relation (12) with the three coefficients $C_2$, ${\kappa }_2$, and $N_{gr2}$ at the output, with 5 hidden neurons and 20 forces corresponding to the prescribed set of displacements at the input.

Range of the coefficients:

• Trial values of $N_{gr2}$ are randomly selected in the range from 4000 kN to 12,000 kN;
• Trial values of constitutive coefficients, ${\kappa }_2$, are randomly selected in the range from 0.0 to 2.0;
• Trial values of initial flexibility, $C_2$, are randomly selected in the range from 0.0000001 mm/kN to 0.001 mm/kN.

Training Description

One of the most sensitive elements in building a neural network model is dataset selection. The dataset that produces the neural network is divided into two subsets: training data and testing data. Both the stability and precision of the neural network model depend on the training phase. In all computations, the same, simplest commercial network, namely Qnet2k [37], has been used. In this paper, sigmoidal functions were used in the neural network models, both in the hidden layers and in the output layer. Despite the possibility of using other activation functions, this function was deliberately chosen. For all examples in the paper, the root mean squared error (RMSE) cost function was used. The cost function characterizes the effectiveness of the model on the training dataset. Loss functions express the discrepancy between the predictions of the trained model and the actual occurrences of the problem. If the deviation between the predicted and actual results is too large, then the loss function will have a very high value. To accelerate the training of the network, supporting algorithms—optimization functions—are also used. Optimizers are algorithms or methods used to change neural network attributes, such as weights and learning speed, to reduce losses. Gradient descent is the simplest but most commonly used optimization algorithm. In this paper, only the Qnet2k built-in backpropagation algorithm was used.

Table 3. Information on the network parameters and training results for direct artificial neural network ANN_F 3_5_20.

Network Definition		Training Controls
Network Layers	3	Max Iterations	10,000,000
Input Nodes	3	Learn Control Start	10,001
Output Nodes	20	Learn Rate	0.022258
Hidden Nodes	5	Learn Rate Max	0.30
Transfer Functions	Sigmoid	Learn Rate Min	0.001
Connections	115	Momentum	0.80
Training Patterns	180	Patterns per Update	180
Test Patterns	70	FAST-Prop	0.00
Network Size (Bytes)	56,842	Screen Update	5
AutoSave Rate	500	Tolerance	0.000
Training Results
	RMS Error	Correlation	Tol. Correct
Training Set	0.012280	0.99744	-
Test Set	0.012480	0.99744	-

presents information on the ANN_F 3_5_20 network parameters and training results, including correlations and RMSEs for training and test sets of the input patterns. All data contained in the following tables were read from the Qnet2k programme.

Figure 8, Figure 9 and Figure 10 show a comparison of the expected and actual neural network output for the test set and the training set of input patterns. The red and black graphs represent the network’s response to the input data. This is always one of 20 values of the axial pile load. The orange graph represents the expected network response. The network was trained to calculate this output value (target) for the given input values. The red graph is for the target’s value that was not used (“shown” to the network) during the training process, while the black graph is for the set of target values that was used in training. The closer these values are to the values in the orange graph, the better is the network’s training result. One can see that the correspondence between both graphs is very good; the red graph is almost obscured by the black and red graphs, which overwrite each other with almost the same values of the expected output. This description applies to all the following graphs illustrating the quality of network training.

Neural Network for Inverse Problem

Similar illustrations of the performance of the inverse approximations are given in the figures that follow.

Table 4. Information on the network parameters and training results for direct artificial neural network ANN_F 20_5_3.

Network Definition		Training Controls
Network Layers	3	Max Iterations	10,000,000
Input Nodes	20	Learn Control Start	1001
Output Nodes	3	Learn Rate	0.001008
Hidden Nodes	5	Learn Rate Max	0.30
Transfer Functions	Sigmoid	Learn Rate Min	0.001
Connections	115	Momentum	0.80
Training Patterns	180	Patterns per Update	180
Test Patterns	70	FAST-Prop	0.00
Network Size (Bytes)	39,230	Screen Update	5
AutoSave Rate	500	Tolerance	0.000
Training Results
	RMS Error	Correlation	Tol. Correct
Training Set	0.021015	0.994866	-
Test Set	0.021131	0.994343	-

presents information on the inverse network, ANN_F 20_5_3 (parameters and training results, including correlations and RMSEs for training and test sets of the input patterns). Figure 11, Figure 12, Figure 13 and Figure 14 present the comparison between the expected output and the actual output of the ANN for the test set and for the learning set of the input patterns. In these cases, at the output, we have the parameters of the Meyer and Kowalow curve, including the bearing capacity of the pile. It is seen that the agreement of the two graphs is very good, thus the approximation is correct.

3.4. Separation of the Pile Load Capacity into the Bearing Capacity of the Base and the Bearing Capacity of the Shaft by the Meyer and Kowalow Approach

It is important for the engineer to assess how much of the total bearing capacity of the pile is taken up by the skin and which part is carried by the base of the pile. Due to the fact that both of these forces can develop independently of each other, the search for traces of such an evolution in the study of the Q-s curve will be the goal of this subsection.

Usually, the decomposition of the force applied to the pile head into additive components, being the reaction under the pile foot and the force resulting from the interaction of the soil with the side surface of the pile, is possible after considering additional information or hypotheses. Additional information may come from the study of the distribution of forces in the pile [38] or from additional theoretical hypotheses. Thanks to additional theoretical hypotheses, like the research described in, e.g., the works of Meyer and Żarkiewicz [39,40,41,42], it is possible to demonstrate the relationships between the parameters of the base resistance curve and the sidewall resistance curve present in the model. Below are the formulas for the force under the base $N_1$ and the total load on the pile $N_2$, expressed in the parameters of the Meyer and Kowalow curve. In this approach, the force $T$ developing on the pile skin is the difference in the two mentioned forces and is also expressed by the parameters of the Meyer and Kowalow curve (Formula (2)). Again, as in Equation (7), the formula for the force rather than for displacement will be useful:

$$N_2\left(s\right)=N_{gr2}\left[1-{\left(1+{\displaystyle \frac{{\kappa }_{2}s}{C_2{N}_{gr2}}}\right)}^{-{\displaystyle \frac{1}{{\kappa }_2}}}\right]$$

(13)

$$N_1\left(s\right)=N_{gr1}\left[1-{\left(1+{\displaystyle \frac{{\kappa }_{1}s}{C_1{N}_{gr1}}}\right)}^{-{\displaystyle \frac{1}{{\kappa }_1}}}\right]$$

(14)

$$T\left(s\right)=N_2\left(s\right)-N_1\left(s\right)$$

(15)

All parameters of the M-K curve in the formula for total force were written with a subscript of 2, and for the force under the base, the index of 1 was used. Further studies described in the works of Meyer and Żarkiewicz made it possible to demonstrate the relationships between the parameters of the base resistance curve and the sidewall resistance curve present in the model. These relationships are as follows:

$${\displaystyle \frac{C_1}{C_2}}={({\kappa }_2+1)}^2$$

(16)

$${\kappa }_1=\ln (1+{\kappa }_2)$$

(17)

$$N_{gr1}={\displaystyle \frac{N_{gr2}}{2^{{\kappa }_2}}}$$

(18)

Therefore, it is enough to identify the parameters of the Meyer and Kowalow curve obtained from the static pile test to obtain information about the decomposition of the bearing capacity into the pile skin and footing. Such an interpretation approach does not allow obtaining, from the Meyer and Kowalow interpretation of the SLT records, information on the interaction of the pile side with the soil in the case when this interaction is qualitatively different than under the footing. In order to avoid this limitation, we will adopt (in the next section) a hypothesis assuming a qualitatively different description of the side-ground interaction, defined by additional constitutive parameters. We will show that these additional parameters affect the shape of the Meyer and Kowalow curve and that they can be read from it by solving the inverse problem using artificial neural networks. This reasoning has the advantage that in many cases, the Meyer and Kowalow curve allows us to identify different values of its parameters for the case when the data identifying this curve are taken from different fragments of the curve obtained from the static pile test (three points taken from the Q-s curve are enough to determine the parameters of the Meyer and Kowalow curve). In this work, it is assumed that on different sections of the Meyer and Kowalow curve, the measurements are burdened with different errors. In this work, we will assume that these discrepancies result from the cooperation of the sidewall with the soil, which is qualitatively different than assumed for the interaction under the pile base.

3.5. Separation of the Pile Load Capacity into the Bearing Capacity of the Base and the Bearing Capacity of the Shaft According to the Adopted Constitutive Hypothesis

Assuming that the two components of the total bearing capacity of the pile (the one bearing by the skin and the other carried by the base of the pile) can develop independently of each other, the search for traces of such an evolution in the Q-s curve will be the goal of the considerations in this subsection.

Two assumptions were made regarding the static work of the pile under the pressing force. Firstly, the component of the pile load capacity transferred by the pile base, $N_1$, is described by the formula consistent with the Meyer and Kowalow theory, and its value as a function of the displacement of the pile head is consistent with the notation (14). Secondly, the force transmitted by the pile skin $T$ is limited by a linear tangent function for the value of the displacement of the pile head close to zero, and is limited by the limit value $T_{gr}$, which is reached when the displacement of the pile head increases infinitely. The functional dependence of this force on the displacement of the head of the pile is different than the one for $N_1$ and is defined by an assumed constitutive hypothesis.

The force observed in the static load test is given by the formula:

$$N_2\left(s\right)=T\left(s\right)+N_1\left(s\right)$$

(19)

For the simplest case, this force depends on the independent variable, s, and on the set of parameters as indicated below:

$$N_2\left(s,C_T,T_{gr},C_1, {\kappa }_1, N_{gr1}\right)=T\left(s,C_T,T_{gr}\right)+N_1\left(s,C_1, {\kappa }_1, N_{gr1}\right)$$

(20)

where $C_T$ and $T_{gr}$ are parameters of the simplest constitutive hypothesis concerning the reaction on the lateral surface of the pile:

$$T\left(s\right)=\left\{\begin{array}{c}s{·C}_T if s<{\displaystyle \frac{T_{gr}}{C_T}}\\ T_{gr} if s\ge {\displaystyle \frac{T_{gr}}{C_T}}\end{array}\right.$$

(21)

Observed in the static load test graph, Q-s can be interpreted as the classical Meyer and Kowalow curve, perturbed with the qualitatively different constitutive properties on the pile sidewall. Having this simplest additional hypothesis, the driving static force can be decomposed into the two components of the bearing capacity of the pile in the more general case. Of course, the tentative hypothesis (21) can be replaced with any more general, smooth form, but the number of parameters will be higher. The preliminary computations will be performed with the formulae of the forces as specified above by Equations (20) and (21).

3.6. Trace of the Assumed Constitutive Hypothesis in the Results of the Static Test and Its Identification Using Inverse ANN and the Modified Meyer and Kowalow Curve

In this subsection, we will present an algorithm leading to the identification of the parameters of the modified Meyer and Kowalow curve using an artificial neural network approximating the inverse function. This procedure is almost identical to the one presented in Section 3.3.

3.6.1. Computational Steps to Be Executed

In the case of the modified Meyer and Kowalow curve, adopted as a representation of the static pile loading test, the ‘direct’ problem is described by Formulas (19), (21), and (14), which is an element of the original Meyer and Kowalow curve. These formulae will allow us to prepare a database of the form:

$$\left\{\left\{C_{T,tr},T_{gr,tr},C_{1,tr}, {\kappa }_{1,tr}, N_{gr1,tr}\right\}, N_2\left(C_{T,tr},T_{gr,tr},C_{1,tr}, {\kappa }_{1,tr}, N_{gr1,tr}\right)\right\}$$

(22)

Then, in order to check the representativeness of this database, a ‘direct’ artificial neural network will be trained. This ANN is approximating the relationship between the model’s coefficients and the values of forces calculated for subsequent deflections (the independent variable is the deflection $s_i$ and the five parameters of the model, and the value of the approximated function is the force Ni calculated according to Formulas (19) and (21)):

$$\left\{N_i\right\}=ANN\_mF@\left({\left\{C_T,T_{gr},C_1, {\kappa }_1, N_{gr1}\right\}}_i\right)$$

(23)

Subsequently, for a sufficiently rich database, an inverse artificial neural network will be trained. This network will approximate an unknown, inverse function, which assigns the parameters of the modified Meyer and Kowalow problem to provide the values of forces (computed for assumed displacements $s_i$). This problem serves as a physical model of the interaction of the pile with the soil, adopted for the interpretation of the SLT.

$${\left\{C_T,T_{gr},C_1, {\kappa }_1, N_{gr1}\right\}}_i\widetilde{=}ANN\_mF@\left(\left\{N_i\right\}\right) \mathrm{for} \mathrm{a} \mathrm{given} \mathrm{set} \mathrm{of} s_i$$

(24)

The artificial neural network, ANN_mF, that approximates the inverse function, will have the following form: (1) As many neurons at the input layer as the number of chosen $s_i$ values is assumed (in a practical application—the number of data issued from SLT). (2) Hidden layers. (3) Output layer with five neurons: for $C_T,T_{gr},C_1, {\kappa }_1, N_{gr1}$. The corresponding $C_{T,tr},T_{gr,tr},C_{1,tr}, {\kappa }_{1,tr}, N_{gr1,tr}$ are presented to the ANN_mF as the target values.

In ‘recall mode’ when the input layer will be presented with the true $s_i$ issued from SLT, the ANN_mF will respond with the true, corresponding values of $C_T,T_{gr},C_1, {\kappa }_1, N_{gr1}$ which are unknowns of the problem, thus the bearing capacity of the pile, we are looking for, is equal to:

$$N_{gr2}=T_{gr}+N_{gr1}$$

(25)

3.6.2. Results of the Computations Executed According to the Proposed Scheme

As in the former case, in the modified problem of Meyer and Kowalow defined above in this section, the function to be approximated is subjective and smooth, thus the numerical complexity of approximation of both direct and inverse relations should be easy. For the direct and inverse approximations, we construct a trial space with 150 elements, and a quarter of them are taken as the test set (the y are used only for verification of the quality of generalization during the training). The rest are used as the learning set of examples.

Similarly, the ANN approximating the direct problems solution is denoted by ANN_mF, and the network that approximates the inverse relation with five coefficients at the output is denoted by ANN_mF. The range of the coefficients and corresponding errors of approximations are computed according to Formulae (16)–(18):

• Trial values of $N_{gr1}$ are randomly selected in the range from 2000 kN to 8000 kN;
• Trial values of constitutive coefficients ${\kappa }_1$ are randomly selected in the range from 0.0 to 2.0;
• Trial values of initial flexibility $C_1$ are randomly selected in the range from 0.0000001 mm/kN to 0.001 mm/kN;
• Trial values of the bearing capacity of the lateral surface $T_{gr}$ are randomly selected in the range from 2000 kN to 8000 kN;
• Trial values of displacement $s_T$ at which the bearing capacity of the lateral surface is exhausted are randomly selected in the range from 0.0 mm to 80 mm.

In the numerical studies, we decided to use the values of displacement $s_T$ at which the bearing capacity of the lateral surface is exhausted, instead of $C_T$. Obviously, these two approaches are equivalent. The performance of the approximations is illustrated in the figures that follow. The first direct problem was analyzed.

Table 5. Information on the network parameters and training results for direct artificial neural network ANN_mF 5_10_10_20.

Network Definition		Training Controls
Network Layers	4	Max Iterations	10,000,000
Input Nodes	5	Learn Control Start	10,001
Output Nodes	20	Learn Rate	0.001001
Hidden Nodes	20	Learn Rate Max	0.30
Transfer Functions	Sigmoid	Learn Rate Min	0.001
Connections	350	Momentum	0.80
Training Patterns	550	Patterns per Update	550
Test Patterns	150	FAST-Prop	0.00
Network Size (Bytes)	175,514	Screen Update	5
AutoSave Rate	500	Tolerance	0.000
Training Results
	RMS Error	Correlation	Tol. Correct
Training Set	0.009909	0.998469	-
Test Set	0.010838	0.998195	-

presents information on the network ANN_mF 5_10_10_20 parameters and training results, including correlations and RMSEs for training and test sets of the input patterns.

Figure 15, Figure 16 and Figure 17 show a comparison of the expected and actual neural network results for the test and training sets of input patterns. As in Figure 11, Figure 12 and Figure 13, the red and black graphs show the network’s response to the input data. This is always one of 20 values of the pile’s axial load. The orange graph shows the expected network response. It can be seen that the agreement between the two graphs is very good, even though this time, the network is richer in data (it has five input parameters); the values in the red and black graphs almost coincide with the expected result values.

Similar illustration of the performance of the inverse approximations of the modified Meyer and Kowalow curves is given in the figures that follow. First, we train the inverse network that approximates only the bearing capacity $N_{gr1}$ at the basis of the pile, ${\kappa }_1$, and $C_1$ from the modified M-K curve. The network is denoted by the symbol ANN_mF 20_10_10_3.

Table 6. Information on the network parameters and training results for inverse artificial neural network ANN_mF 20_10_10_3.

Network Definition		Training Controls
Network Layers	4	Max Iterations	10,000,000
Input Nodes	20	Learn Control Start	10,001
Output Nodes	3	Learn Rate	0.001565
Hidden Nodes	20	Learn Rate Max	0.30
Transfer Functions	Sigmoid	Learn Rate Min	0.001
Connections	330	Momentum	0.80
Training Patterns	550	Patterns per Update	550
Test Patterns	150	FAST-Prop	0.00
Network Size (Bytes)	152,930	Screen Update	5
AutoSave Rate	500	Tolerance	0.000
Training Results
	RMS Error	Correlation	Tol. Correct
Training Set	0.024820	0.992054	-
Test Set	0.061957	0.947312	-

presents information on the inverse network ANN_mF 20_10_10_3 (parameters and training results, including correlations and RMSEs for training and test sets of the input patterns).

In Figure 18, Figure 19, Figure 20 and Figure 21, we present the comparison between the expected output and the actual output of the ANN for the test set and for the learning set of the input patterns. In this case, at the output, we have the parameters of the Meyer and Kowalow curve: the bearing capacity $N_{gr1}$ at the basis of the pile, ${\kappa }_1$, and $C_1$. It is seen that the agreement of the two graphs is very good, thus the approximation is correct.

Finally, we train the inverse network that approximates only the bearing capacity of the lateral surface of the pile T_gr and the displacement $s_T$ at which this bearing capacity is exhausted. All data are from the modified M-K curve. The network is denoted by the symbol ANN_mF 20_10_10_2.

Table 7. Information on the network parameters and training results for inverse artificial neural network ANN_mF 20_10_10_2.

Network Definition		Training Controls
Network Layers	4	Max Iterations	10,000,000
Input Nodes	20	Learn Control Start	10,001
Output Nodes	2	Learn Rate	0.001000
Hidden Nodes	20	Learn Rate Max	0.30
Transfer Functions	Sigmoid	Learn Rate Min	0.001
Connections	320	Momentum	0.80
Training Patterns	550	Patterns per Update	250
Test Patterns	150	FAST-Prop	0.00
Network Size (Bytes)	131,908	Screen Update	5
AutoSave Rate	500	Tolerance	0.000
Training Results
	RMS Error	Correlation	Tol. Correct
Training Set	0.047994	0.968974	-
Test Set	0.061935	0.949299	-

presents information on the inverse network ANN_mF 20_10_10_2 (parameters and training results, including correlations and RMSEs for training and test sets of the input patterns). Figure 22 and Figure 23 present the comparison between the expected output and the actual output of the ANN for the test set and for the learning set of the input patterns. In this case, at the output, we have the parameters of the Meyer and Kowalow curve: T_gr and $s_T$.

From the results of the numerical experiments so far executed, we conclude that the training of the ANN approximating the inverse problem is more difficult. It needs a more complicated structure of networks and the parameters of learning are worse than in the classical Meyer and Kowalow representation of SLT results.

In the last numerical example, we will train the inverse network with only one output node. In the first case, we will train it for approximation of the bearing capacity at the foot of the pile, and in the second case, for bearing capacity of the lateral surface of the pile. In the authors’ opinion, the training results are promising, based on the evaluation of training quality indicators such as RMSE and correlation. The clustering of results along a straight line on the input-output graph is also satisfactory. Analysis of individual deviations indicates that the network training process requires further refinement and improvement. This is the subject of our ongoing work on network modification and optimization. Given that direct training yields much better results than the inverse problem, it can be assumed that training errors for the inverse problem may be caused by the ambiguous dependence of the output parameter on the network’s input parameters. This is currently under analysis.

Table 8. Information on the network parameters and training results for inverse artificial neural network ANN_mF 20_10_10_1—the network that approximates the bearing capacity of the base of the pile.

Network Definition		Training Controls
Network Layers	4	Max Iterations	10,000,000
Input Nodes	20	Learn Control Start	10,001
Output Nodes	1	Learn Rate	0.001038
Hidden Nodes	20	Learn Rate Max	0.30
Transfer Functions	Sigmoid	Learn Rate Min	0.001
Connections	310	Momentum	0.80
Training Patterns	550	Patterns per Update	550
Test Patterns	150	FAST-Prop	0.00
Network Size (Bytes)	141,246	Screen Update	5
AutoSave Rate	500	Tolerance	0.000
Training Results
	RMS Error	Correlation	Tol. Correct
Training Set	0.016170	0.993935	-
Test Set	0.029605	0.980966	-

and

Table 9. Information on the network parameters and training results for inverse artificial neural network ANN_mF 20_10_5_1—the network that approximates the bearing capacity of the shaft of the pile.

Network Definition		Training Controls
Network Layers	4	Max Iterations	10,000,000
Input Nodes	20	Learn Control Start	10,001
Output Nodes	1	Learn Rate	0.001504
Hidden Nodes	20	Learn Rate Max	0.30
Transfer Functions	Sigmoid	Learn Rate Min	0.001
Connections	310	Momentum	0.80
Training Patterns	550	Patterns per Update	550
Test Patterns	150	FAST-Prop	0.00
Network Size (Bytes)	141,246	Screen Update	5
AutoSave Rate	500	Tolerance	0.000
Training Results
	RMS Error	Correlation	Tol. Correct
Training Set	0.018229	0.994051	-
Test Set	0.032264	0.981887	-

present information on the inverse network ANN_mF 20_10_10_1. The results are shown in Figure 24, Figure 25, Figure 26 and Figure 27.

4. Discussion and Conclusions

The previous section demonstrated that the use of neural networks (ANNs) in analyzing the Q-s plot of SLT results for a pile can be useful from an engineering perspective. Assuming these data are approximated by the Meyer and Kowalow curve, the presented method allows for the identification of the parameters of this curve. To do this, it is sufficient to use the simplest neural network with one or two hidden layers and train it according to the presented procedure. We find that the accuracy of this identification is very high (approximation errors are only about 2%). We also note that the training process is very fast (less than 100,000 epochs), and the representation of the sample space of variables by the input patterns can be very sparse (less than 250 random examples for direct network, about 700 examples for extended inverse nets). We note that an ANN, even for a large set of observations (20 pairs of Q-s in our example), can be rather small and easily constructed numerically. A single small hidden layer is sufficient to approximate this quantity with sufficient accuracy for engineering problems. We therefore conclude that the numerical complexity of the problem of identifying pile bearing capacity based on SLT data is low. The numerical complexity of the problem of identifying the parameters of the Meyer and Kowalow curve is also low. Furthermore, in the previous chapter, the classical Meyer and Kowalow approach was modified (generalized) by introducing two parameters enabling separate estimation of the bearing capacity of the pile shaft. It was shown that the use of artificial intelligence (ANN) in analyzing the Q-s plot of the pile LST results, under this additional hypothesis, can also be useful. Unfortunately, in the latter case, the numerical complexity of the procedure increases. However, we find that the accuracy of this identification is also satisfactory (approximate errors are around 5%). We also conclude that training is not overly tedious. Of course, the representation of the sample space of the five independent variables by the input patterns must be larger (around 750 random examples). An ANN with two hidden layers performs very well, especially when the network is trained separately for a single identified parameter. These observed properties of the presented algorithm justify the conclusion that ANN can be considered a convenient algorithm for identifying the parameters of the Meyer and Kowalow curve. We believe it can be applied in engineering practice to assess pile bearing capacity based on a static load test.

Applying the presented ANN-based method relieves the engineer of the need for work requiring high mathematical erudition and the use of sophisticated optimization algorithms. The presented method is fully automatic. Using pre-trained networks (such as those prepared in this article), it is enough to simply input the coordinates of the points measured during SLT to the inverse network’s input to obtain the required coefficients. Among them—the pile bearing capacity and, in the extended version—the pile skin bearing capacity. To ensure the obtained data are correct, it is enough to simply input it directly to the “direct” network input and compare the resulting graph with the experimental graph. The method of transmitting network data (structure, weights, and biases) to the user is, of course, a practical issue and can be implemented in many ways, which we do not discuss in this paper. We believe that the method presented in this article is automatic and easy to apply. It only requires verification that the ANN has been trained on a sufficient range of parameters. Another argument in favor of this method is the ease with which the number of identified parameters can be expanded relative to the three parameters of the M-K curve.

The presented studies are preliminary. Their goal was to verify that the ANN easily “learns” the settlement-load relationship for both direct calculations and reverse analysis. It is now necessary to investigate whether different pile types can be associated with a specific representation of the settlement-load curve, from those listed in Table 1. The network training process can be performed on sets generated by advanced pile models, in which the unloading test is numerically simulated using the Finite Element Method. As is known, such models use specific functions describing the pile-soil interactions along the pile shaft. In light of the presented method and based on a set of experimental data from load-bearing tests, the adequacy of these models can be verified.

The limitation here will always be the number of physical parameters that we intend to identify, but the good results of research to date and the extremely rapid development of networks trained with deep learning methods, which we have not used so far because classical networks have proven to be sufficiently effective, allow us to look at this perspective with optimism.

In our opinion, the presented procedure is quite general and can be used for different hypotheses concerning the constitutive interaction ground-structure on the lateral surface of the pile.