Load–Settlement Modeling of Micropiled Rafts in Cohesive Soils Using an Artificial Intelligence Technique

Abstract

The traditional design of foundations in soft clay often relies on large-diameter piles, which, although effective, are costly and impractical for low- to medium-rise buildings. Micropiles have emerged as a cost-effective alternative, offering an efficient solution to these challenges. To advance the adoption of micropiles in geotechnical practice, this study employs a multi-objective genetic algorithm-based evolutionary polynomial regression (EPR-MOGA), a hybrid artificial intelligence method, to develop a robust and straightforward model for predicting the load–settlement response of micropiled rafts in cohesive soils under vertical loads. The model was created using an extensive database comprising 458 data points derived from field tests, centrifuge experiments, laboratory studies, and numerical simulations reported in the literature. This comprehensive database covers a wide range of scenarios by varying key parameters of micropiles within a group, including their length, diameter, number, spacing, construction method, and raft thickness. The proposed EPR model could deliver accurate predictions, providing a practical approach for geotechnical applications. In addition, the predictions of the model could support the conclusion that pressure-grouted micropiles are more efficient than gravity-grouted ones in enhancing the performance of micropiled rafts.

1. Introduction

1.1. Background of Micropiles

A micropiled raft is a deep foundation system that has recently gained significant popularity in civil engineering. They are widely applied in projects involving restricted access, complex soil conditions, or the need to strengthen or stabilize existing structures [1,2]. However, the load-bearing behavior of micropiled rafts is complex and is influenced by such factors as soil–structure interaction, group effects, load transfer mechanisms, and construction methods [3]. Therefore, predicting the response of micropiled foundations remains a notable challenge, particularly given the misconception that micropiles cannot support substantial loads due to their small diameter. This underscores the pressing need for reliable tools to accurately and promptly predict the performance of micropiled rafts, equipping engineers with the resources required to design cost-effective and efficient foundations.

Micropiles are slender reinforced concrete or steel piles installed using drilling and grouting techniques. These special piles feature several advantages, including the high bearing capacity of the pressure-grouted types, adaptability to diverse soil conditions, a speedy installation process, and reduced disruption during construction (e.g., low noise and vibration levels) [4,5]. The reinforcement of concrete micropiles may incorporate casings in addition to a central rebar [6,7]. The central rebar is surrounded by a cement-based grout, poured either through gravity or under high pressure [8,9,10]. Although micropiles were originally employed to enhance the bearing capacity of existing shallow foundations [11], researchers have recently promoted their use in under-construction footings [12,13].

The load-carrying capacity of an individual micropile can be assessed based on the mobilized interface strength along its length, as expressed in the following equation:

$$Q_u={\alpha }_b \pi d L$$

(1)

where $Q_u$ is the ultimate load capacity of the micropile; ${\alpha }_b$ is the grout-to-soil bond strength; d is the drill-hole diameter; and L is the bond length. The base resistance is often neglected due to the small diameter of micropiles. According to FHWA [3], the ultimate bond strengths for soft-to-medium plastic silt and clay with some sand ranges from 35–70 kPa for gravity-grouted micropiles and 35–95 kPa for pressure-grouted micropiles. Notably, the FHWA permits a bond strength for pressure-grouted micropiles that is approximately 1.36 times that of gravity-grouted ones. However, given that the present study focuses on soft clay devoid of sand, ${\alpha }_b$ can be conservatively taken as equal to the undrained shear strength of the clay. This assumption is based on the recommendation of the CFEM [14], which suggests adopting an adhesion factor of unity for small-diameter piles. Based on these considerations, the axial load capacity of a single micropile in soft clay typically ranges from 70 kN to 700 kN, depending on ground conditions, micropile diameter, bond length, and installation method. Similar capacities have been reported in previous studies [15,16,17,18].

Conventional pile–raft systems in some scenarios, such as low- and medium-rise buildings, may result in uneconomical designs. In terms of building height classification, low-rise buildings typically consist of one to three stories, with a maximum height of 9 m, whereas medium-rise buildings range from four to ten stories, with a maximum height of 30 m [19]. The strategic implementation of micropile–raft systems in these structures can provide cost-effective and efficient alternative solutions. However, utilizing micropiles as floating piles in under-construction footings positioned on soft clay is an innovative approach that requires detailed evaluation. For high-rise buildings founded on deep soft clay deposits, larger-diameter piles extending to greater depths are typically required [20].

Extensive experimental and numerical studies have been conducted to investigate various aspects of micropiles, including their bearing capacity and settlement behavior [16,17,21,22,23,24,25], lateral performance [26,27,28], seismic response [29,30,31], and application in foundation retrofitting [32,33,34,35,36]. Notably, Alnuaim et al. [13,15] advanced the use of micropiles in primary foundations on soft clay through comprehensive centrifugal and numerical testing. Moreover, Borthakur and Dey [37] contributed by examining the vertical load-carrying capacity of micropile groups in highly plastic soft clay with cohesion values between 18 and 20 kPa. Their experimental program, conducted in a large-scale pit measuring 2.0 m × 4.0 m × 3.0 m, should have minimized the scaling effects typically associated with experimental tests. In a recent study, El Sawwaf et al. [38] demonstrated the effectiveness of floating pile groups in soft clay by highlighting their ability to reduce settlement and enhance ultimate load-carrying capacity.

1.2. Artificial Intelligence (AI) Applications

With advancements in technology, artificial intelligence (AI) has found widespread application among several researchers in predicting the axial performance of piles. For instance, Nejad and Jaksa [39] developed an artificial neural network (ANN) model to predict the load–settlement response of single piles using cone penetration test (CPT) data, demonstrating superior accuracy over traditional methods and providing ANN-based design charts for practical use. Similarly, Huynh et al. [20] applied ANN modeling to analyze the base resistance of super-large and long piles in soft soil, using a dataset from 37 real projects. Their study highlighted the critical role of displacement in governing the pile base resistance and introduced a training approach with multiple learning rates. Besides, few studies have been conducted to compare between different machine learning algorithms and evaluate their performance in predicting the pile capacity [40,41,42].

With regard to micropiles, Borthakur and Dey [43] used support vector machines (SVMs) to develop an empirical equation for predicting the capacity of free-standing micropile groups and micropiled rafts in soft clay, based on the results of 54 large-scale load tests. Similarly, Borthakur and Das [44] employed an artificial neural network approach (ANN), while Mukherjee and Borthakur [45] explored five different machine learning algorithms to construct predictive models for the settlement of micropiled rafts.

A detailed review of these AI-driven studies [43,44,45] reveals that they were primarily based on the comprehensive field load tests conducted by Borthakur and Dey [37]. However, there is substantial scope to enlarge this database with more data points and extend the AI-based predictions to encompass a broader range of scenarios. First, the database used for training the machine learning models in [43,44,45] was constrained to a maximum of 16 micropiles, whereas real-world applications may involve a larger number of micropiles. Second, the database was limited to gravity-grouted micropiles only. Incorporating other cases involving pressure-grouted micropiles could provide valuable insights into this alternative construction method. Thirdly, the AI-driven models in [43,44,45] did not account for the raft thickness. Since Alnuaim et al. [13] observed an 8% increase in the capacity when the thickness of micropiled rafts was increased from 0.3 m to 1.2 m, incorporating the raft thickness as a feature in the database would be beneficial. Finally, the multi-objective genetic algorithm-based evolutionary polynomial regression (EPR-MOGA), which excels in providing direct equations between input features and output parameters, has not yet been applied to predict the capacity of micropiled rafts in cohesive soils.

In light of the aforementioned points, the current study evaluates the performance of micropiled rafts composed of either gravity-grouted or pressure-grouted micropiles in cohesive soils. To achieve this, a multi-objective genetic algorithm-based evolutionary polynomial regression (EPR-MOGA), a hybrid artificial intelligence approach, was employed to develop a robust yet straightforward predictive model for the load–settlement behavior of micropiled rafts under vertical loads. The model was constructed using an extensive database comprising 458 data points sourced from a variety of field tests, centrifuge experiments, laboratory investigations, and numerical simulations reported in the literature. This comprehensive database covers a wide range of scenarios by varying key geometric parameters of micropiles within a group, including their length, diameter, number, spacing, and construction method. The developed equation was rigorously validated using a subset of the data that was not utilized during the training phase of the model, ensuring its reliability and generalization capabilities. Additionally, a sensitivity analysis was conducted to assess the relative influence of input variables on the model’s predictions, offering valuable insights into the factors that govern the performance of micropiled rafts in cohesive soils.

2. Data Used in the Modeling

A total of 458 data points were compiled for this study from the existing literature sources, as outlined in the following subsections. The dataset includes seven distinct input features that describe various physical characteristics of the micropiled raft and one output variable representing the bearing capacity. All micropile-related features were normalized using the raft width (b).

Table 1. The features selected for the study and their definitions.

Feature Name	Definition
d/b	The ratio of the diameter of a micropile to the width of a micropiled raft
L/b	The ratio of the length of a micropile to the width of a micropiled raft
n	Number of micropiles in the group
s/b	The ratio of the micropile spacing to the width of a micropiled raft
${\mathrm{K}}_{\mathrm{s}}$	Coefficient of lateral earth pressure on the micropile shaft
t/b	The ratio of the thickness to the width of a micropiled raft
${\mathrm{s}}_{\mathrm{e}}$/b	The ratio of the induced settlement to the width of a micropiled raft
$\mathrm{q}/c_u$	The ratio of the bearing pressure to the undrained shear strength

provides detailed definitions of all input variables and the output feature. The database comprises square-shaped micropile rafts constructed with varying numbers of micropiles (n), including configurations of 4, 9, 16, 25, 36, 81, 121, and 289 micropiles. Figure 1 shows a micropiled raft with nine numbers (3 × 3) of micropiles.

An innovative input feature was introduced to the database: the coefficient of the lateral earth pressure (${\mathrm{K}}_{\mathrm{s}}$), which accounts for the confining pressure exerted by the soil on the micropile shaft. In pressure-grouted micropiles, ${\mathrm{K}}_{\mathrm{s}}$ tends to be higher than the at-rest coefficient of lateral earth pressure (${\mathrm{K}}_0$) [4,6,9,13,46], serving as a key differentiating factor between gravity-grouted and pressure-grouted micropiles in the database. Accurate determination of ${\mathrm{K}}_{\mathrm{s}}$ was a critical aspect, and the methodology used to establish its value will be demonstrated. Another innovative input feature introduced to account for the raft’s flexibility is the raft thickness (t), normalized by the raft width ($\mathrm{b})$. The output variable was formulated as a dimensionless parameter, defined as the ratio of the pressure applied (q) to the micropile group and the undrained shear strength of the clay (${\mathrm{c}}_{\mathrm{u}}$). The pressure (q) was calculated by dividing the applied load for a particular settlement by the area of the associated raft size.

Table 2. Statistics of the features of the database used in the model development.

Statistical Indicator	d/b	L/b	n	s/b	${\mathbf{K}}_{\mathbf{s}}$	t/b	${\mathbf{s}}_{\mathbf{e}}$/b	$\mathbf{q}/{\mathbf{c}}_{\mathbf{u}}$
Minimum	0.0119	0.4762	4	0.0595	0.561	0.0143	0.0002	0.2258
Maximum	0.2	6.6667	289	0.8571	1.2	0.4833	0.3513	17.0124
Mean	0.0871	2.6088	15.345	0.3836	0.71	0.1553	0.0645	4.5674
Standard deviation	0.0476	1.5563	35.3726	0.1674	0.2107	0.1147	0.0738	2.8398

presents the statistical parameters of the database, including maximum, minimum, mean, and standard deviation values of each feature.

2.1. Centrifuge Tests and Numerical Simulations by Alnuaim et al. [13,15]

The dataset incorporated data points derived from one centrifuge test [15] and 26 numerical simulations [13] conducted on micropiled rafts in clay soils (see Appendix A,

Table A1. Data points derived from the work performed by Alnuaim et al. [13,15].

d/b	L/b	n	s/b	Ks	t/b	se/b	q/cu
0.0286	1.9048	4	0.2286	1.2	0.1143	0.0003	0.2594
0.0286	1.9048	4	0.2286	1.2	0.1143	0.0006	0.5594
0.0286	1.9048	4	0.2286	1.2	0.1143	0.0009	0.8918
0.0286	1.9048	4	0.2286	1.2	0.1143	0.0014	1.3134
0.0286	1.9048	4	0.2286	1.2	0.1143	0.0018	1.6134
0.0286	1.9048	4	0.2286	1.2	0.1143	0.0023	1.9135
0.0286	1.9048	4	0.2286	1.2	0.1143	0.0028	2.2055
0.0286	1.9048	4	0.2286	1.2	0.1143	0.0035	2.4246
0.0286	1.9048	4	0.2286	1.2	0.1143	0.0042	2.6600
0.0286	1.9048	4	0.2286	1.2	0.1143	0.0051	2.8549
0.0286	1.9048	4	0.2286	1.2	0.1143	0.0059	3.0094
0.0286	1.9048	4	0.2286	1.2	0.1143	0.0070	3.1477
0.0286	1.9048	4	0.2286	1.2	0.1143	0.0079	3.2778
0.0286	1.9048	4	0.2286	1.2	0.1143	0.0090	3.4081
0.0286	1.9048	4	0.2286	1.2	0.1143	0.0102	3.5383
0.0286	1.9048	4	0.2286	1.2	0.1143	0.0115	3.7011
0.0286	1.9048	4	0.2286	1.2	0.1143	0.0124	3.7988
0.0286	1.9048	4	0.2286	1.2	0.1143	0.0133	3.8803
0.0119	0.4762	289	0.0595	1.2	0.0286	0.0036	6.0000
0.0119	0.4762	121	0.0952	1.2	0.0286	0.0036	5.0000
0.0119	0.4762	81	0.119	1.2	0.0286	0.0036	4.6667
0.0119	0.4762	36	0.1905	1.2	0.0286	0.0036	4.0000
0.0119	0.4762	25	0.2381	1.2	0.0286	0.0036	3.6667
0.0119	0.4762	289	0.0595	1.2	0.0286	0.0071	8.3330
0.0119	0.4762	121	0.0952	1.2	0.0286	0.0071	6.6667
0.0119	0.4762	81	0.119	1.2	0.0286	0.0071	6.0000
0.0119	0.4762	36	0.1905	1.2	0.0286	0.0071	5.0000
0.0119	0.4762	25	0.2381	1.2	0.0286	0.0071	4.3333
0.0119	0.4762	289	0.0595	1.2	0.0143	0.0036	6.0000
0.0119	0.4762	121	0.0952	1.2	0.0143	0.0036	4.6667
0.0119	0.4762	81	0.119	1.2	0.0143	0.0036	4.3333
0.0119	0.4762	36	0.1905	1.2	0.0143	0.0036	3.6667
0.0119	0.4762	25	0.2381	1.2	0.0143	0.0036	3.3333
0.0119	0.4762	289	0.0595	1.2	0.0143	0.0071	8.0000
0.0119	0.4762	121	0.0952	1.2	0.0143	0.0071	6.3330
0.0119	0.4762	81	0.119	1.2	0.0143	0.0071	5.6667
0.0119	0.4762	36	0.1905	1.2	0.0143	0.0071	4.6667
0.0119	0.4762	25	0.2381	1.2	0.0143	0.0071	4.0000
0.0119	0.4762	289	0.0595	1.2	0.0571	0.0036	6.6667
0.0119	0.4762	121	0.0952	1.2	0.0571	0.0036	5.3333
0.0119	0.4762	81	0.119	1.2	0.0571	0.0036	5.3333
0.0119	0.4762	36	0.1905	1.2	0.0571	0.0036	4.3333
0.0119	0.4762	25	0.2381	1.2	0.0571	0.0036	4.0000
0.0119	0.4762	289	0.0595	1.2	0.0571	0.0071	8.6667
0.0119	0.4762	121	0.0952	1.2	0.0571	0.0071	7.0000
0.0119	0.4762	81	0.119	1.2	0.0571	0.0071	6.0000
0.0119	0.4762	36	0.1905	1.2	0.0571	0.0071	5.0000
0.0119	0.4762	25	0.2381	1.2	0.0571	0.0071	4.6667

). The centrifuge test involved a micropiled raft with a prototype-scale raft thickness of 0.6 m. The raft was supported by four micropiles, each with a diameter of 150 mm, a length of 10 m, and a spacing of 8d (where d is the micropile diameter). The soil bed consisted of a kaolin-silt mixture with an average undrained shear strength of 30 kPa. To replicate the behavior of pressure-grouted micropiles, the micropiles were jacked into the clay.

Additionally, the numerical simulations analyzed 26 cases of large-scale micropiled rafts of 21 m × 21 m using a finite element (FE) model calibrated using the centrifuge test results. The number of micropiles was varied from 25 to 289, and the bearing pressure was recorded at a settlement level of 75 mm and 150 mm. Alnuaim et al. [13] determined that the effective friction angle (${\mathrm{\varphi }}^{\prime }$) of the soil was 25°, with an over-consolidation ratio (OCR) ranging from 2 to 7.5, resulting in an estimated $K_0$ of approximately 1.0 using the following formula suggested by Mayne and Kulhawy [47]:

$$K_0=\left(1-\sin {\mathrm{\varphi }}^{\prime }\right){\left(OCR\right)}^{\sin {\mathrm{\varphi }}^{\prime }}$$

(2)

To evaluate the installation effect, the impact of increased confining pressure on the lateral earth pressure coefficient (${\mathrm{K}}_{\mathrm{s}}$) was assessed by Alnuaim et al. [13] using an advanced approach detailed in [4,48]. This method allows for the expansion of the volume elements representing the micropiles, simulating the effects of pressurized grouting. Based on the resulting increase in confining stress, the average post-expansion ${\mathrm{K}}_{\mathrm{s}}$ was found to be approximately 1.2.

2.2. Large-Scale Tests by Borthakur and Dey [37]

The dataset incorporated data points derived from 28 large-scale tests [37] conducted on micropiled rafts in soft clay with ${\mathrm{c}}_{\mathrm{u}}$ of 18–20 kPa (see Appendix A,

Table A2. Data points derived from the work performed by Borthakur and Dey [37].

d/b	L/b	n	s/b	Ks	t/b	se/b	q/cu
0.1695	4.0678	4	0.5085	0.5610	0.2458	0.0003	0.6384
0.1695	4.0678	4	0.5085	0.5610	0.2458	0.0013	1.4364
0.1695	4.0678	4	0.5085	0.5610	0.2458	0.0034	2.0428
0.1695	4.0678	4	0.5085	0.5610	0.2458	0.0067	2.5855
0.1695	4.0678	4	0.5085	0.5610	0.2458	0.0102	3.3196
0.1695	4.0678	4	0.5085	0.5610	0.2458	0.0218	3.9516
0.1695	4.0678	4	0.5085	0.5610	0.2458	0.0407	4.4049
0.1695	4.0678	4	0.5085	0.5610	0.2458	0.0678	4.8517
0.1695	4.0678	4	0.5085	0.5610	0.2458	0.0925	5.1901
0.1695	4.0678	4	0.5085	0.5610	0.2458	0.1153	5.4454
0.1695	4.0678	4	0.5085	0.5610	0.2458	0.1373	5.6178
0.1695	4.0678	4	0.5085	0.5610	0.2458	0.1684	5.8476
0.125	3	9	0.375	0.5610	0.1813	0.001	1.0417
0.125	3	9	0.375	0.5610	0.1813	0.0027	2.1938
0.125	3	9	0.375	0.5610	0.1813	0.0086	3.2181
0.125	3	9	0.375	0.5610	0.1813	0.0275	4.5139
0.125	3	9	0.375	0.5610	0.1813	0.0469	5.301
0.125	3	9	0.375	0.5610	0.1813	0.0763	5.9028
0.125	3	9	0.375	0.5610	0.1813	0.11	6.4236
0.125	3	9	0.375	0.5610	0.1813	0.1456	6.9444
0.0893	2.1429	16	0.2679	0.5610	0.1295	0.0002	1.0346
0.0893	2.1429	16	0.2679	0.5610	0.1295	0.0014	2.2003
0.0893	2.1429	16	0.2679	0.5610	0.1295	0.0063	3.2348
0.0893	2.1429	16	0.2679	0.5610	0.1295	0.0179	3.7202
0.0893	2.1429	16	0.2679	0.5610	0.1295	0.0309	4.0745
0.0893	2.1429	16	0.2679	0.5610	0.1295	0.0443	4.4483
0.0893	2.1429	16	0.2679	0.5610	0.1295	0.0571	4.7832
0.0893	2.1429	16	0.2679	0.5610	0.1295	0.0732	5.1375
0.0893	2.1429	16	0.2679	0.5610	0.1295	0.0864	5.4439
0.1667	4	4	0.75	0.5610	0.2417	0.0003	0.7407
0.1667	4	4	0.75	0.5610	0.2417	0.0008	1.7037
0.1667	4	4	0.75	0.5610	0.2417	0.0038	2.7845
0.1667	4	4	0.75	0.5610	0.2417	0.0159	3.9898
0.1667	4	4	0.75	0.5610	0.2417	0.037	4.7115
0.1667	4	4	0.75	0.5610	0.2417	0.0667	5.3086
0.1667	4	4	0.75	0.5610	0.2417	0.094	5.7428
0.1667	4	4	0.75	0.5610	0.2417	0.128	6.1728
0.1667	4	4	0.75	0.5610	0.2417	0.16	6.5432
0.1667	4	4	0.75	0.5610	0.2417	0.2049	7.047
0.0893	2.1429	9	0.4018	0.5610	0.1295	0.0009	0.713
0.0893	2.1429	9	0.4018	0.5610	0.1295	0.0036	1.3287
0.0893	2.1429	9	0.4018	0.5610	0.1295	0.0068	1.729
0.0893	2.1429	9	0.4018	0.5610	0.1295	0.0107	2.2144
0.0893	2.1429	9	0.4018	0.5610	0.1295	0.0197	2.6272
0.0893	2.1429	9	0.4018	0.5610	0.1295	0.0357	3.0116
0.0893	2.1429	9	0.4018	0.5610	0.1295	0.0623	3.4315
0.0893	2.1429	9	0.4018	0.5610	0.1295	0.0911	3.7202
0.0893	2.1429	9	0.4018	0.5610	0.1295	0.1235	3.9718
0.0658	1.5789	16	0.2961	0.5610	0.0954	0.0006	0.6189
0.0658	1.5789	16	0.2961	0.5610	0.0954	0.0016	1.2037
0.0658	1.5789	16	0.2961	0.5610	0.0954	0.0042	1.826
0.0658	1.5789	16	0.2961	0.5610	0.0954	0.0114	2.3615
0.0658	1.5789	16	0.2961	0.5610	0.0954	0.0201	2.8597
0.0658	1.5789	16	0.2961	0.5610	0.0954	0.0283	3.1741
0.0658	1.5789	16	0.2961	0.5610	0.0954	0.0397	3.5595
0.0658	1.5789	16	0.2961	0.5610	0.0954	0.05	3.8582
0.1429	3.4286	4	0.8571	0.5610	0.2071	0.0004	0.5442
0.1429	3.4286	4	0.8571	0.5610	0.2071	0.0012	1.3902
0.1429	3.4286	4	0.8571	0.5610	0.2071	0.0029	2.309
0.1429	3.4286	4	0.8571	0.5610	0.2071	0.0075	3.3596
0.1429	3.4286	4	0.8571	0.5610	0.2071	0.0262	4.3681
0.1429	3.4286	4	0.8571	0.5610	0.2071	0.0571	4.9887
0.1429	3.4286	4	0.8571	0.5610	0.2071	0.0815	5.3446
0.1429	3.4286	4	0.8571	0.5610	0.2071	0.1011	5.6236
0.1429	3.4286	4	0.8571	0.5610	0.2071	0.1371	6.0771
0.1429	3.4286	4	0.8571	0.5610	0.2071	0.1602	6.3581
0.0667	1.6	9	0.4	0.5610	0.0967	0.0004	0.5378
0.0667	1.6	9	0.4	0.5610	0.0967	0.0014	1.044
0.0667	1.6	9	0.4	0.5610	0.0967	0.0056	1.6258
0.0667	1.6	9	0.4	0.5610	0.0967	0.0114	2.1944
0.0667	1.6	9	0.4	0.5610	0.0967	0.0177	2.6168
0.0667	1.6	9	0.4	0.5610	0.0967	0.0278	3.12
0.0667	1.6	9	0.4	0.5610	0.0967	0.0377	3.4568
0.0667	1.6	9	0.4	0.5610	0.0967	0.0473	3.7824
0.0495	1.1881	16	0.297	0.5610	0.0718	0.0004	0.2723
0.0495	1.1881	16	0.297	0.5610	0.0718	0.0009	0.5462
0.0495	1.1881	16	0.297	0.5610	0.0718	0.0013	0.6808
0.0495	1.1881	16	0.297	0.5610	0.0718	0.002	1.0072
0.0495	1.1881	16	0.297	0.5610	0.0718	0.0026	1.15
0.0495	1.1881	16	0.297	0.5610	0.0718	0.0041	1.4944
0.0495	1.1881	16	0.297	0.5610	0.0718	0.0061	1.7486
0.0495	1.1881	16	0.297	0.5610	0.0718	0.0098	2.0751
0.0495	1.1881	16	0.297	0.5610	0.0718	0.0131	2.2874
0.0495	1.1881	16	0.297	0.5610	0.0718	0.0166	2.4914
0.0495	1.1881	16	0.297	0.5610	0.0718	0.0198	2.6141
0.0495	1.1881	16	0.297	0.5610	0.0718	0.0234	2.7128
0.1667	5.3333	4	0.5	0.5610	0.2417	0.0008	1.2346
0.1667	5.3333	4	0.5	0.5610	0.2417	0.0021	2.6694
0.1667	5.3333	4	0.5	0.5610	0.2417	0.0027	3.7037
0.1667	5.3333	4	0.5	0.5610	0.2417	0.0046	4.9333
0.1667	5.3333	4	0.5	0.5610	0.2417	0.0123	6.1728
0.1667	5.3333	4	0.5	0.5610	0.2417	0.0332	7.1014
0.1667	5.3333	4	0.5	0.5610	0.2417	0.0767	8.0247
0.1667	5.3333	4	0.5	0.5610	0.2417	0.1143	8.642
0.1667	5.3333	4	0.5	0.5610	0.2417	0.15	9.2593
0.1667	5.3333	4	0.5	0.5610	0.2417	0.1891	9.8778
0.125	4	9	0.375	0.5610	0.1813	0.0002	2.7778
0.125	4	9	0.375	0.5610	0.1813	0.0007	4.1014
0.125	4	9	0.375	0.5610	0.1813	0.0027	5.454
0.125	4	9	0.375	0.5610	0.1813	0.0139	7.2396
0.125	4	9	0.375	0.5610	0.1813	0.0263	8.1597
0.125	4	9	0.375	0.5610	0.1813	0.0417	8.9788
0.125	4	9	0.375	0.5610	0.1813	0.0575	9.5486
0.125	4	9	0.375	0.5610	0.1813	0.0733	10.069
0.125	4	9	0.375	0.5610	0.1813	0.0896	10.576
0.1563	5	4	0.7031	0.5610	0.2266	0.0003	0.651
0.1563	5	4	0.7031	0.5610	0.2266	0.0007	1.5516
0.1563	5	4	0.7031	0.5610	0.2266	0.0019	2.7669
0.1563	5	4	0.7031	0.5610	0.2266	0.0025	3.3637
0.1563	5	4	0.7031	0.5610	0.2266	0.0041	4.1829
0.1563	5	4	0.7031	0.5610	0.2266	0.0141	4.8828
0.1563	5	4	0.7031	0.5610	0.2266	0.0236	5.3993
0.1563	5	4	0.7031	0.5610	0.2266	0.0431	6.1051
0.1563	5	4	0.7031	0.5610	0.2266	0.0765	6.8229
0.1563	5	4	0.7031	0.5610	0.2266	0.0938	7.053
0.1563	5	4	0.7031	0.5610	0.2266	0.1141	7.27
0.1563	5	4	0.7031	0.5610	0.2266	0.1425	7.5684
0.0909	2.9091	9	0.4091	0.5610	0.1318	0.0009	0.9183
0.0909	2.9091	9	0.4091	0.5610	0.1318	0.0025	1.844
0.0909	2.9091	9	0.4091	0.5610	0.1318	0.0058	2.7548
0.0909	2.9091	9	0.4091	0.5610	0.1318	0.011	3.5152
0.0909	2.9091	9	0.4091	0.5610	0.1318	0.0182	4.1322
0.0909	2.9091	9	0.4091	0.5610	0.1318	0.0255	4.5914
0.0909	2.9091	9	0.4091	0.5610	0.1318	0.0387	5.1978
0.0909	2.9091	9	0.4091	0.5610	0.1318	0.0551	5.6933
0.0909	2.9091	9	0.4091	0.5610	0.1318	0.0727	6.1524
0.0909	2.9091	9	0.4091	0.5610	0.1318	0.0949	6.6412
0.125	4	4	0.75	0.5610	0.1813	0.0004	1.0417
0.125	4	4	0.75	0.5610	0.1813	0.0009	1.8392
0.125	4	4	0.75	0.5610	0.1813	0.0045	2.7778
0.125	4	4	0.75	0.5610	0.1813	0.013	3.5888
0.125	4	4	0.75	0.5610	0.1813	0.0345	4.5139
0.125	4	4	0.75	0.5610	0.1813	0.0611	5.2365
0.125	4	4	0.75	0.5610	0.1813	0.09	5.7292
0.125	4	4	0.75	0.5610	0.1813	0.125	6.25
0.125	4	4	0.75	0.5610	0.1813	0.1591	6.7275
0.0714	2.2857	9	0.4286	0.5610	0.1036	0.0009	0.5669
0.0714	2.2857	9	0.4286	0.5610	0.1036	0.0019	1.01633
0.0714	2.2857	9	0.4286	0.5610	0.1036	0.0029	1.5873
0.0714	2.2857	9	0.4286	0.5610	0.1036	0.0063	2.20321
0.0714	2.2857	9	0.4286	0.5610	0.1036	0.0157	2.83447
0.0714	2.2857	9	0.4286	0.5610	0.1036	0.0304	3.40332
0.0714	2.2857	9	0.4286	0.5610	0.1036	0.0429	3.68481
0.0714	2.2857	9	0.4286	0.5610	0.1036	0.06	3.96825
0.0714	2.2857	9	0.4286	0.5610	0.1036	0.0855	4.37501
0.1667	6.6667	4	0.5	0.5610	0.2417	0.0003	1.8179
0.1667	6.6667	4	0.5	0.5610	0.2417	0.0021	3.95679
0.1667	6.6667	4	0.5	0.5610	0.2417	0.0083	5.55556
0.1667	6.6667	4	0.5	0.5610	0.2417	0.0192	6.83642
0.1667	6.6667	4	0.5	0.5610	0.2417	0.0367	8.02469
0.1667	6.6667	4	0.5	0.5610	0.2417	0.06	9.25926
0.1667	6.6667	4	0.5	0.5610	0.2417	0.0799	10.0877
0.1667	6.6667	4	0.5	0.5610	0.2417	0.1067	10.8025
0.1667	6.6667	4	0.5	0.5610	0.2417	0.1333	11.4198
0.1667	6.6667	4	0.5	0.5610	0.2417	0.164	12.0889
0.1111	4.4444	9	0.3333	0.5610	0.1611	0.0004	1.48486
0.1111	4.4444	9	0.3333	0.5610	0.1611	0.0019	2.63336
0.1111	4.4444	9	0.3333	0.5610	0.1611	0.0067	4.049
0.1111	4.4444	9	0.3333	0.5610	0.1611	0.0117	5.2594
0.1111	4.4444	9	0.3333	0.5610	0.1611	0.0222	6.39232
0.1111	4.4444	9	0.3333	0.5610	0.1611	0.0462	7.22908
0.1111	4.4444	9	0.3333	0.5610	0.1611	0.0756	7.54458
0.1111	4.4444	9	0.3333	0.5610	0.1611	0.1156	7.81893
0.1111	4.4444	9	0.3333	0.5610	0.1611	0.1444	7.9561
0.1111	4.4444	9	0.3333	0.5610	0.1611	0.1903	8.19479
0.1429	5.7143	4	0.6429	0.5610	0.2071	0.0017	1.36054
0.1429	5.7143	4	0.6429	0.5610	0.2071	0.0046	3.02222
0.1429	5.7143	4	0.6429	0.5610	0.2071	0.0086	4.30839
0.1429	5.7143	4	0.6429	0.5610	0.2071	0.0267	5.73787
0.1429	5.7143	4	0.6429	0.5610	0.2071	0.0543	6.80272
0.1429	5.7143	4	0.6429	0.5610	0.2071	0.0886	7.93651
0.1429	5.7143	4	0.6429	0.5610	0.2071	0.1419	9.4381
0.1429	5.7143	4	0.6429	0.5610	0.2071	0.1771	10.2041
0.1429	5.7143	4	0.6429	0.5610	0.2071	0.2171	10.8844
0.1429	5.7143	4	0.6429	0.5610	0.2071	0.2543	11.4522
0.0926	3.7037	9	0.4167	0.5610	0.1343	0.0013	1.11797
0.0926	3.7037	9	0.4167	0.5610	0.1343	0.0041	2.4257
0.0926	3.7037	9	0.4167	0.5610	0.1343	0.0081	3.46174
0.0926	3.7037	9	0.4167	0.5610	0.1343	0.013	4.28669
0.0926	3.7037	9	0.4167	0.5610	0.1343	0.0179	4.83482
0.0926	3.7037	9	0.4167	0.5610	0.1343	0.0265	5.52507
0.0926	3.7037	9	0.4167	0.5610	0.1343	0.0396	6.14426
0.0926	3.7037	9	0.4167	0.5610	0.1343	0.0556	6.57293
0.0926	3.7037	9	0.4167	0.5610	0.1343	0.0722	6.95397
0.0926	3.7037	9	0.4167	0.5610	0.1343	0.0881	7.27176
0.2	5	4	0.6	0.5610	0.4833	0.0027	1.7284
0.2	5	4	0.6	0.5610	0.4833	0.008	3.7284
0.2	5	4	0.6	0.5610	0.4833	0.0167	5.92593
0.2	5	4	0.6	0.5610	0.4833	0.0233	7.90124
0.2	5	4	0.6	0.5610	0.4833	0.044	9.98025
0.2	5	4	0.6	0.5610	0.4833	0.09	11.8519
0.2	5	4	0.6	0.5610	0.4833	0.156	13.4568
0.2	5	4	0.6	0.5610	0.4833	0.2067	14.5679
0.2	5	4	0.6	0.5610	0.4833	0.2753	16.0494
0.2	5	4	0.6	0.5610	0.4833	0.3249	17.0124
0.12	3	9	0.36	0.5610	0.2900	0.0024	1.51111
0.12	3	9	0.36	0.5610	0.2900	0.004	2.84444
0.12	3	9	0.36	0.5610	0.2900	0.0102	4.26133
0.12	3	9	0.36	0.5610	0.2900	0.02	6.04444
0.12	3	9	0.36	0.5610	0.2900	0.0432	7.50133
0.12	3	9	0.36	0.5610	0.2900	0.088	8.88889
0.12	3	9	0.36	0.5610	0.2900	0.124	9.77778
0.12	3	9	0.36	0.5610	0.2900	0.1787	11.0711
0.0857	2.1429	16	0.2571	0.5610	0.2071	0.0009	0.90703
0.0857	2.1429	16	0.2571	0.5610	0.2071	0.0024	2.04082
0.0857	2.1429	16	0.2571	0.5610	0.2071	0.0054	3.1746
0.0857	2.1429	16	0.2571	0.5610	0.2071	0.0092	3.99683
0.0857	2.1429	16	0.2571	0.5610	0.2071	0.0137	4.98866
0.0857	2.1429	16	0.2571	0.5610	0.2071	0.02	6.12245
0.0857	2.1429	16	0.2571	0.5610	0.2071	0.0333	7.161
0.0857	2.1429	16	0.2571	0.5610	0.2071	0.0486	7.93651
0.0857	2.1429	16	0.2571	0.5610	0.2071	0.0671	8.61678
0.0857	2.1429	16	0.2571	0.5610	0.2071	0.0929	9.52381
0.0857	2.1429	16	0.2571	0.5610	0.2071	0.1172	10.3444
0.13	3.75	4	0.6	0.5610	0.3625	0.0003	0.83333
0.13	3.75	4	0.6	0.5610	0.3625	0.0006	1.70556
0.13	3.75	4	0.6	0.5610	0.3625	0.0078	2.89139
0.13	3.75	4	0.6	0.5610	0.3625	0.015	3.88889
0.13	3.75	4	0.6	0.5610	0.3625	0.0245	4.78236
0.13	3.75	4	0.6	0.5610	0.3625	0.04	5.55556
0.13	3.75	4	0.6	0.5610	0.3625	0.0634	6.34847
0.13	3.75	4	0.6	0.5610	0.3625	0.0975	7.22222
0.13	3.75	4	0.6	0.5610	0.3625	0.1309	7.97667
0.13	3.75	4	0.6	0.5610	0.3625	0.161	8.61111
0.13	3.75	4	0.6	0.5610	0.3625	0.1939	9.23157
0.0867	2.5	9	0.4	0.5610	0.2417	0.0007	0.61728
0.0867	2.5	9	0.4	0.5610	0.2417	0.0023	1.46895
0.0867	2.5	9	0.4	0.5610	0.2417	0.0083	2.09877
0.0867	2.5	9	0.4	0.5610	0.2417	0.0159	2.68765
0.0867	2.5	9	0.4	0.5610	0.2417	0.0267	3.45679
0.0867	2.5	9	0.4	0.5610	0.2417	0.0366	3.89704
0.0867	2.5	9	0.4	0.5610	0.2417	0.0607	4.69136
0.0867	2.5	9	0.4	0.5610	0.2417	0.0869	5.29605
0.0867	2.5	9	0.4	0.5610	0.2417	0.1093	5.67901
0.0867	2.5	9	0.4	0.5610	0.2417	0.1433	6.17284
0.0867	2.5	9	0.4	0.5610	0.2417	0.1802	6.69614
0.0605	1.7442	16	0.2791	0.5610	0.1686	0.0002	0.48074
0.0605	1.7442	16	0.2791	0.5610	0.1686	0.0009	0.90581
0.0605	1.7442	16	0.2791	0.5610	0.1686	0.0039	1.69966
0.0605	1.7442	16	0.2791	0.5610	0.1686	0.0111	2.3946
0.0605	1.7442	16	0.2791	0.5610	0.1686	0.0225	3.28195
0.0605	1.7442	16	0.2791	0.5610	0.1686	0.0337	3.72574
0.0605	1.7442	16	0.2791	0.5610	0.1686	0.0499	4.1877
0.0605	1.7442	16	0.2791	0.5610	0.1686	0.0721	4.32666
0.0605	1.7442	16	0.2791	0.5610	0.1686	0.0898	4.93225
0.0605	1.7442	16	0.2791	0.5610	0.1686	0.1128	5.28814
0.0605	1.7442	16	0.2791	0.5610	0.1686	0.1379	5.6487
0.1333	5	4	0.4	0.5610	0.4833	0.002	1.23457
0.1333	5	4	0.4	0.5610	0.4833	0.0056	2.59259
0.1333	5	4	0.4	0.5610	0.4833	0.0133	4.19753
0.1333	5	4	0.4	0.5610	0.4833	0.0283	5.34321
0.1333	5	4	0.4	0.5610	0.4833	0.06	7.03704
0.1333	5	4	0.4	0.5610	0.4833	0.1078	8.29877
0.1333	5	4	0.4	0.5610	0.4833	0.2033	9.87654
0.1333	5	4	0.4	0.5610	0.4833	0.2933	11.1111
0.1333	5	4	0.4	0.5610	0.4833	0.3513	11.9309
0.1	3.75	9	0.3	0.5610	0.3625	0.0015	1.11111
0.1	3.75	9	0.3	0.5610	0.3625	0.0032	2.36111
0.1	3.75	9	0.3	0.5610	0.3625	0.005	3.33333
0.1	3.75	9	0.3	0.5610	0.3625	0.0112	4.30556
0.1	3.75	9	0.3	0.5610	0.3625	0.025	5.41667
0.1	3.75	9	0.3	0.5610	0.3625	0.0405	6.26111
0.1	3.75	9	0.3	0.5610	0.3625	0.07	7.36111
0.1	3.75	9	0.3	0.5610	0.3625	0.1076	8.37222
0.1	3.75	9	0.3	0.5610	0.3625	0.16	9.58333
0.1	3.75	9	0.3	0.5610	0.3625	0.209	10.5556
0.1	3.75	9	0.3	0.5610	0.3625	0.26	11.6667
0.1	3.75	9	0.3	0.5610	0.3625	0.3095	12.6333
0.0741	2.7778	16	0.2222	0.5610	0.2685	0.0021	1.29553
0.0741	2.7778	16	0.2222	0.5610	0.2685	0.0037	2.74348
0.0741	2.7778	16	0.2222	0.5610	0.2685	0.0075	3.82716
0.0741	2.7778	16	0.2222	0.5610	0.2685	0.0148	4.87731
0.0741	2.7778	16	0.2222	0.5610	0.2685	0.0315	6.25194
0.0741	2.7778	16	0.2222	0.5610	0.2685	0.0537	7.31596
0.0741	2.7778	16	0.2222	0.5610	0.2685	0.0815	8.38287
0.0741	2.7778	16	0.2222	0.5610	0.2685	0.0985	8.97318
0.0741	2.7778	16	0.2222	0.5610	0.2685	0.1296	9.90703
0.0741	2.7778	16	0.2222	0.5610	0.2685	0.1556	10.6691
0.0741	2.7778	16	0.2222	0.5610	0.2685	0.1889	11.5836
0.0741	2.7778	16	0.2222	0.5610	0.2685	0.2292	12.7206

). The experiments were conducted in a test pit of size 2.0 m × 4.0 m × 3.0 m. Studied variables included micropile diameter (25 mm and 50 mm), length (24d, 32d, and 40d, where d is the diameter), number of micropiles in a group (2, 4, 9, and 16), and spacing between micropiles (3d, 4.5d, and 6d). The micropiles were constructed within the test pit as cast-in situ model piles (gravity-grouted). Since the effective friction angle (${\mathrm{\varphi }}^{\prime }$) of the soil was not reported in [37], ${\mathrm{K}}_0$ was estimated to be 0.561 using the formula proposed by Massarsch [49], based on the reported plasticity index (PI) of 28.9% [37].

$$K_0=0.44+0.42\left[{\displaystyle \frac{PI\left(\%\right)}{100}}\right]$$

(3)

2.3. Physical Model Tests by El Sawwaf et al. [38]

The dataset incorporated data points derived from nine physical model tests [38] conducted on piled rafts in soft clay with ${\mathrm{c}}_{\mathrm{u}}$ of 16 kPa (see Appendix A,

Table A3. Data points derived from the work performed by El Sawwaf et al. [38].

d/b	L/b	n	s/b	Ks	t/b	se/b	q/cu
0.05	0.75	9	0.3	0.85	0.05	0.0018	0.2258
0.05	0.75	9	0.3	0.85	0.05	0.0083	0.5130
0.05	0.75	9	0.3	0.85	0.05	0.0166	0.8926
0.05	0.75	9	0.3	0.85	0.05	0.0250	1.1695
0.05	0.75	9	0.3	0.85	0.05	0.0350	1.4566
0.05	0.75	9	0.3	0.85	0.05	0.0486	1.7846
0.05	0.75	9	0.3	0.85	0.05	0.0648	2.1434
0.05	0.75	9	0.3	0.85	0.05	0.0801	2.4714
0.05	0.75	9	0.3	0.85	0.05	0.0985	2.8198
0.05	0.75	9	0.3	0.85	0.05	0.1161	3.2196
0.05	0.75	9	0.3	0.85	0.05	0.1345	3.5783
0.05	0.75	9	0.3	0.85	0.05	0.1542	3.9266
0.05	0.75	9	0.3	0.85	0.05	0.1818	4.1413
0.05	0.75	9	0.3	0.85	0.05	0.2063	4.2020
0.05	0.75	9	0.3	0.85	0.05	0.2308	4.2525
0.05	0.75	9	0.3	0.85	0.05	0.2500	4.3031
0.05	1.25	9	0.3	0.85	0.05	0.0061	0.6260
0.05	1.25	9	0.3	0.85	0.05	0.0131	1.2212
0.05	1.25	9	0.3	0.85	0.05	0.0263	1.8367
0.05	1.25	9	0.3	0.85	0.05	0.0372	2.2573
0.05	1.25	9	0.3	0.85	0.05	0.0530	2.7392
0.05	1.25	9	0.3	0.85	0.05	0.0718	3.2314
0.05	1.25	9	0.3	0.85	0.05	0.0880	3.5799
0.05	1.25	9	0.3	0.85	0.05	0.1099	3.9898
0.05	1.25	9	0.3	0.85	0.05	0.1345	4.4201
0.05	1.25	9	0.3	0.85	0.05	0.1612	4.8196
0.05	1.25	9	0.3	0.85	0.05	0.1918	5.0033
0.05	1.25	9	0.3	0.85	0.05	0.2212	5.0536
0.05	1.25	9	0.3	0.85	0.05	0.2500	5.1038
0.05	1.75	9	0.3	0.85	0.05	0.0031	0.6877
0.05	1.75	9	0.3	0.85	0.05	0.0092	1.4267
0.05	1.75	9	0.3	0.85	0.05	0.0188	1.9602
0.05	1.75	9	0.3	0.85	0.05	0.0298	2.4834
0.05	1.75	9	0.3	0.85	0.05	0.0451	3.0783
0.05	1.75	9	0.3	0.85	0.05	0.0609	3.5808
0.05	1.75	9	0.3	0.85	0.05	0.0823	4.1139
0.05	1.75	9	0.3	0.85	0.05	0.1038	4.6573
0.05	1.75	9	0.3	0.85	0.05	0.1331	5.2928
0.05	1.75	9	0.3	0.85	0.05	0.1564	5.7847
0.05	1.75	9	0.3	0.85	0.05	0.1804	6.1330
0.05	1.75	9	0.3	0.85	0.05	0.2120	6.3577
0.05	1.75	9	0.3	0.85	0.05	0.2339	6.4391
0.05	1.75	9	0.3	0.85	0.05	0.2500	6.5206
0.05	1.25	4	0.3	0.85	0.05	0.0070	0.6616
0.05	1.25	4	0.3	0.85	0.05	0.0163	1.1110
0.05	1.25	4	0.3	0.85	0.05	0.0317	1.5230
0.05	1.25	4	0.3	0.85	0.05	0.0510	1.9724
0.05	1.25	4	0.3	0.85	0.05	0.0765	2.4218
0.05	1.25	4	0.3	0.85	0.05	0.1012	2.8088
0.05	1.25	4	0.3	0.85	0.05	0.1311	3.2582
0.05	1.25	4	0.3	0.85	0.05	0.1610	3.5328
0.05	1.25	4	0.3	0.85	0.05	0.1887	3.7076
0.05	1.25	4	0.3	0.85	0.05	0.2182	3.7949
0.05	1.25	4	0.3	0.85	0.05	0.2500	3.8823
0.05	1.25	16	0.3	0.85	0.05	0.0013	0.7490
0.05	1.25	16	0.3	0.85	0.05	0.0048	1.4855
0.05	1.25	16	0.3	0.85	0.05	0.0097	2.1471
0.05	1.25	16	0.3	0.85	0.05	0.0150	2.6590
0.05	1.25	16	0.3	0.85	0.05	0.0229	3.1084
0.05	1.25	16	0.3	0.85	0.05	0.0321	3.5827
0.05	1.25	16	0.3	0.85	0.05	0.0457	4.1195
0.05	1.25	16	0.3	0.85	0.05	0.0576	4.5564
0.05	1.25	16	0.3	0.85	0.05	0.0717	4.9809
0.05	1.25	16	0.3	0.85	0.05	0.0831	5.3429
0.05	1.25	16	0.3	0.85	0.05	0.1003	5.8172
0.05	1.25	16	0.3	0.85	0.05	0.1157	6.1793
0.05	1.25	16	0.3	0.85	0.05	0.1315	6.4913
0.05	1.25	16	0.3	0.85	0.05	0.1452	6.8658
0.05	1.25	16	0.3	0.85	0.05	0.1623	7.1030
0.05	1.25	16	0.3	0.85	0.05	0.1856	7.3277
0.05	1.25	16	0.3	0.85	0.05	0.2094	7.4526
0.05	1.25	16	0.3	0.85	0.05	0.2301	7.5025
0.05	1.25	16	0.3	0.85	0.05	0.2500	7.5399
0.06	1.5	9	0.36	0.85	0.05	0.0049	0.7169
0.06	1.5	9	0.36	0.85	0.05	0.0132	1.4340
0.06	1.5	9	0.36	0.85	0.05	0.0256	2.0590
0.06	1.5	9	0.36	0.85	0.05	0.0441	2.7354
0.06	1.5	9	0.36	0.85	0.05	0.0613	3.2377
0.06	1.5	9	0.36	0.85	0.05	0.0794	3.6888
0.06	1.5	9	0.36	0.85	0.05	0.0962	4.0478
0.06	1.5	9	0.36	0.85	0.05	0.1156	4.4068
0.06	1.5	9	0.36	0.85	0.05	0.1416	4.8480
0.06	1.5	9	0.36	0.85	0.05	0.1707	5.2278
0.06	1.5	9	0.36	0.85	0.05	0.1937	5.4539
0.06	1.5	9	0.36	0.85	0.05	0.2237	5.6187
0.06	1.5	9	0.36	0.85	0.05	0.2500	5.7017
0.07	1.75	9	0.42	0.85	0.05	0.0026	0.7988
0.07	1.75	9	0.42	0.85	0.05	0.0071	1.2699
0.07	1.75	9	0.42	0.85	0.05	0.0163	1.8129
0.07	1.75	9	0.42	0.85	0.05	0.0304	2.4380
0.07	1.75	9	0.42	0.85	0.05	0.0454	3.0221
0.07	1.75	9	0.42	0.85	0.05	0.0653	3.6269
0.07	1.75	9	0.42	0.85	0.05	0.0865	4.2420
0.07	1.75	9	0.42	0.85	0.05	0.1041	4.6931
0.07	1.75	9	0.42	0.85	0.05	0.1231	5.1238
0.07	1.75	9	0.42	0.85	0.05	0.1434	5.5136
0.07	1.75	9	0.42	0.85	0.05	0.1685	5.8728
0.07	1.75	9	0.42	0.85	0.05	0.1954	6.2014
0.07	1.75	9	0.42	0.85	0.05	0.2246	6.4584
0.07	1.75	9	0.42	0.85	0.05	0.2500	6.5822
0.05	1.25	9	0.15	0.85	0.05	0.0107	0.6512
0.05	1.25	9	0.15	0.85	0.05	0.0192	1.0197
0.05	1.25	9	0.15	0.85	0.05	0.0317	1.3794
0.05	1.25	9	0.15	0.85	0.05	0.0465	1.7219
0.05	1.25	9	0.15	0.85	0.05	0.0608	2.0387
0.05	1.25	9	0.15	0.85	0.05	0.0770	2.3812
0.05	1.25	9	0.15	0.85	0.05	0.0949	2.7750
0.05	1.25	9	0.15	0.85	0.05	0.1146	3.1689
0.05	1.25	9	0.15	0.85	0.05	0.1316	3.5027
0.05	1.25	9	0.15	0.85	0.05	0.1544	3.8793
0.05	1.25	9	0.15	0.85	0.05	0.1755	4.1102
0.05	1.25	9	0.15	0.85	0.05	0.2019	4.2209
0.05	1.25	9	0.15	0.85	0.05	0.2212	4.2461
0.05	1.25	9	0.15	0.85	0.05	0.2500	4.2794
0.05	1.25	9	0.4	0.85	0.05	0.0026	0.6343
0.05	1.25	9	0.4	0.85	0.05	0.0075	1.1743
0.05	1.25	9	0.4	0.85	0.05	0.0138	1.5857
0.05	1.25	9	0.4	0.85	0.05	0.0218	1.9969
0.05	1.25	9	0.4	0.85	0.05	0.0303	2.3482
0.05	1.25	9	0.4	0.85	0.05	0.0411	2.6823
0.05	1.25	9	0.4	0.85	0.05	0.0536	3.0506
0.05	1.25	9	0.4	0.85	0.05	0.0688	3.4359
0.05	1.25	9	0.4	0.85	0.05	0.0836	3.7870
0.05	1.25	9	0.4	0.85	0.05	0.0993	4.1124
0.05	1.25	9	0.4	0.85	0.05	0.1163	4.4377
0.05	1.25	9	0.4	0.85	0.05	0.1391	4.7542
0.05	1.25	9	0.4	0.85	0.05	0.1629	5.0022
0.05	1.25	9	0.4	0.85	0.05	0.1884	5.2244
0.05	1.25	9	0.4	0.85	0.05	0.2202	5.4121
0.05	1.25	9	0.4	0.85	0.05	0.2500	5.5140

). The model raft was a square steel plate (200 mm × 200 mm) with a thickness of 10 mm, while the model piles were steel closed-end pipes with diameters of 10 mm, 12 mm, and 14 mm and lengths of 15d, 25d, and 35d. The tests were conducted in a steel tank with dimensions of 600 mm × 600 mm × 600 mm. The piled raft was jacked vertically (driven) into the ground. Since the effective friction angle (${\mathrm{\varphi }}^{\prime }$) of the soil was not reported in [38], ${\mathrm{K}}_0$ was estimated to be 0.503 using Equation (3), based on the reported plasticity index (PI) of 16% [38]. To evaluate the installation effect, a thorough review of the literature revealed that ${\mathrm{K}}_{\mathrm{s}}$ for driven piles in clay can vary within the following ranges: up to 1.5${\mathrm{K}}_0$ [49,50], between 1.4–1.8${\mathrm{K}}_0$ [51], up to about 1.75${\mathrm{K}}_0$ [52], and 2${\mathrm{K}}_0$ [14]. In the current study, an average ${\mathrm{K}}_{\mathrm{s}}$ value of 0.85 was taken for the data points from [38].

3. The EPR Predictive Model

Evolutionary polynomial regression using a multi-objective genetic algorithm (EPR-MOGA) is an advanced regression technique that autonomously identifies the optimal polynomial structure to represent the relationship between input and output variables [53,54,55]. By integrating the flexibility of genetic algorithms with the interpretability of polynomial regression, EPR-MOGA dynamically discovers explicit mathematical models that accurately capture complex relationships within the data. Unlike many machine learning methods that operate as black-box models, EPR-MOGA generates transparent equations, enabling physical interpretation and making it particularly suitable for engineering applications. Additionally, it simultaneously optimizes multiple objectives, balancing model accuracy and complexity to ensure that the resulting expressions are both precise and straightforward. In contrast to conventional regression techniques, which require predefined functional forms and produce fixed regression coefficients for user-selected models (e.g., linear or polynomial), EPR-MOGA autonomously determines the most suitable model structure, reducing the risk of overfitting while improving predictive performance.

3.1. Methodology

The development of candidate relationships is influenced by several factors, including the size and quality of the dataset used in the analysis, the predefined form of the relationship between input and output variables, the predefined range of exponents for the polynomial terms, and the predefined number of terms for the resulting relationship. The general formulation of the EPR model is expressed as follows:

$$Y=a_0+{\sum }_1^m(a_j.{\left(X_i\right)}^{ES\left(j,1\right)}\dots {\left(X_k\right)}^{ES\left(j,k\right)}.f\left[{\left(X_i\right)}^{ES\left(j,k+1\right)}\dots {\left(X_k\right)}^{ES\left(j,2k\right)}\right])$$

(4)

where Y is the output value, m is the user-defined maximum number of terms in the produced equation, ${\mathrm{X}}_{\mathrm{i}}$ represents the input features, ES is the matrix of exponents selected via the genetic algorithm, f is the user-defined internal function, $a_j$ is a coefficient, and $a_0$ is the bias.

The least squares fitting technique is employed to solve the equation. Additionally, the EPR methodology integrates genetic algorithms (GAs) to identify the optimal mathematical equation based on user-defined exponents [54]. This approach enhances the search for the most suitable relationship between input and output variables. The GA operates based on Darwinian evolutionary principles, beginning with the random generation of an initial population of solutions. Each parameter set within this population represents the chromosomes of individual solutions. A fitness value is assigned to each individual, reflecting its performance within the given environment. Subsequently, the next generation is formed through crossover and mutation operations [53], which introduce variability and improve the population’s diversity. These iterations are repeated until the desired number of terms in the developed model is achieved, ensuring a robust and well-fitted solution to the problem.

EPR-MOGA integrates the flexibility of genetic algorithms with the interpretability of polynomial regression, allowing for the automatic discovery of explicit mathematical models that capture complex relationships within the data. Unlike black-box models, such as artificial neural networks (ANNs) and support vector machines (SVMs), MOGA-EPR provides transparent equations that facilitate physical interpretation, making it highly suitable for engineering applications.

Additionally, MOGA-EPR optimizes multiple objectives simultaneously, balancing model accuracy and complexity. This ensures that the resulting expressions are not only precise but also generalizable. Compared to conventional regression techniques, which often require predefined functional forms, MOGA-EPR autonomously determines the optimal model structure, reducing the risk of overfitting.

3.2. Model Development

To conduct the EPR analysis, the MATLAB-based software (EPR-MOGA-XL 1.0) was utilized to develop a relationship that describes the bearing capacity of micropiled rafts in cohesive clays at a predefined settlement level. The initial step involved categorizing the 458 data points into 2 subsets, with 366 data points used for training (80%) and 92 used for testing (20%). The training subset was employed to train the machine learning algorithm, while the testing subset was used to validate the model’s predictive accuracy. Data selection was conducted randomly to ensure that both subsets reflected the overall data distribution, thereby minimizing biases. Moreover, the training and testing subsets were designed to have consistent statistical parameters, including maximum, minimum, mean, and standard deviation values, as shown in

Table 3. Statistics of the features of the training data.

Statistical Indicator	d/b	L/b	n	s/b	${\mathit{K}}_{\mathit{s}}$	t/b	${\mathit{s}}_{\mathit{e}}$/b	$\mathit{q}/{\mathit{c}}_{\mathit{u}}$
Minimum	0.0119	0.4762	4	0.0595	0.5610	0.0143	0.0002	0.2258
Maximum	0.2	6.6667	289	0.8571	1.2	0.4833	0.3513	17.0124
Mean	0.0865	2.6012	16.7896	0.3778	0.7128	0.1533	0.0615	4.5724
Standard deviation	0.0485	1.6067	39.1803	0.1634	0.2116	0.1166	0.0715	2.8842

and

Table 4. Statistics of the features of the testing data.

Statistical Indicator	d/b	L/b	n	s/b	${\mathit{K}}_{\mathit{s}}$	t/b	${\mathit{s}}_{\mathit{e}}$/b	$\mathit{q}/{\mathit{c}}_{\mathit{u}}$
Minimum	0.0119	0.4762	4	0.119	0.561	0.0286	0.0002	0.4807
Maximum	0.2	5.7143	81	0.8571	1.2	0.4833	0.3095	12.7206
Mean	0.0894	2.6387	9.5978	0.4069	0.6989	0.1631	0.0766	4.5476
Standard deviation	0.044	1.3445	9.1963	0.1817	0.2078	0.1071	0.0817	2.6709

. Such consistency ensures that the model generalizes effectively to unseen data and prevents issues related to data shifts.

The model was developed following a comprehensive iterative analysis, considering various numbers of terms, exponents, and internal function types to ensure optimal performance. For the predefined internal function in Equation (1), options included in the EPR-MOGA-XL are “no function”, logarithmic, exponential, tangent, or hyperbolic secant functions. The “no function” option was selected to allow EPR-MOGA to autonomously determine the most suitable functional form, thereby avoiding the imposition of an incorrect predefined structure. The candidate exponents ranged from −2 to 2 in 0.5 increments, while the number of terms was set to 6, with the optional inclusion of a bias term. Model optimization was achieved by minimizing the sum of squared errors (SSE) while simultaneously simplifying the equation terms. The performance of each iteration was evaluated using statistical indicators, such as the coefficient of determination ($R^2$), root mean squared error (RMSE), and mean absolute error (MAE). The $R^2$ (Equation (5)) measures the closeness of predictions to the zero-error line, where the predictions perfectly match observed values. A higher $R^2$ percentage reflects greater accuracy, with 100% indicating a perfect fit. RMSE (Equation (6)) and MAE (Equation (7)) evaluate the overall prediction error, where lower values denote more accurate models. A value of 0 indicates an exact match between observed and predicted data. Equations (5)–(7) are as follows:

$$R^2=1-{\displaystyle \frac{{\sum }_{i=1}^n{(X_i-X)}^2}{{\sum }_{i=1}^n{(X_i-X_m)}^2}}$$

(5)

$$RMSE=\sqrt{{\displaystyle \frac{{\sum }_{i=1}^n{(X_i-X)}^2}{n}}}$$

(6)

$$MAE={\displaystyle \frac{{\sum }_{i=1}^n\left|X_i-X\right|}{n}}$$

(7)

where n is the number of observations, $X_i$ is the value of the ith observation, X is the value of the calculated data, and $X_m$ is the mean value of the observed data.

The final model for the bearing pressure of micropiled rafts is as follows:

$$\begin{gathered} {\displaystyle \frac{q}{c_u}}=0.48125\times {\displaystyle \frac{K_s^2\times {\left(\frac{s_e}{b}\right)}^{0.5}}{\frac{d}{b}}}+0.80541\times {\left({\displaystyle \frac{L}{b}}\right)}^{0.5}\times \mathit{ln}\left({\displaystyle \frac{L}{b}}\times n^{1.5}\times {\displaystyle \frac{s}{b}}\times {\left({\displaystyle \frac{s_e}{b}}\right)}^{0.5}\right)+ \\ 0.17359\times {\displaystyle \frac{{\left(\frac{L}{b}\right)}^2\times K_s^2}{\frac{s}{b}}}\times \mathit{ln}\left({\left({\displaystyle \frac{L}{b}}\right)}^{0.5}\times K_s\right)+1.9272\times {\left({\displaystyle \frac{L}{b}}\right)}^2\times {\displaystyle \frac{s}{b}}\times {\displaystyle \frac{t}{b}}\times {\left({\displaystyle \frac{s_e}{b}}\right)}^{0.5}+3.0206\times {\displaystyle \frac{{\left(\frac{d}{b}\right)}^{0.5}}{K_s^{0.5}}}+ \\ 337.9512\times {\left({\displaystyle \frac{d}{b}}\right)}^2\times n\times K_s^{1.5}\times {\left({\displaystyle \frac{t}{b}}\right)}^{0.5}\times {\left({\displaystyle \frac{s_e}{b}}\right)}^{0.5}\times \mathit{ln}\left({\displaystyle \frac{d}{b}}\times n^2\times K_s\times {\left({\displaystyle \frac{t}{b}}\right)}^{0.5}\right) \end{gathered}$$

(8)

Figure 2 and Figure 3 illustrate the comparison between the predicted and measured values of the EPR-MOGA model against the hypothetical no-error line for the training and testing datasets, respectively. The visual analysis of both figures indicates that the majority of the predicted values closely align with the no-error line, which demonstrates the model’s high predictive accuracy. The figures also show the statistical performance indicators of the proposed EPR model (Equation (8)). For the training data, the mean absolute error (MAE) and root mean squared error (RMSE) in the predicted q/$c_u$ were calculated as 0.45 and 0.60, respectively. For the testing data, the corresponding values were 0.47 and 0.67, respectively. The model’s coefficient of determination ($R^2$) was 0.96 for the training data and 0.93 for the testing data. Based on the performance indicators, it can be concluded that the developed model is robust and can be utilized for quick and straightforward modeling of the load–settlement response of micropiled rafts in cohesive clays.

4. Model Performance and Sensitivity Analyses

The performance of the EPR model in modeling the load–settlement response of a micropiled raft under vertical loads was evaluated through graphical analysis, as shown in Figure 4a–f. The figure includes representative simulation results from experimental tests performed by Borthakur and Dey [37], a centrifuge test by Alnuaim et al. [15], and a model test by El Sawwaf et al. [38]. It can be seen that there is a strong correlation between the measured response and the predictions made by the EPR model. The model effectively captured the nonlinear load–settlement behavior, demonstrating its robustness in accurately simulating the load–settlement response of micropiled rafts in cohesive soils.

Furthermore, to assess the generalization ability of the EPR model, sensitivity analyses were performed to evaluate how varying L/b, n, or t/b affect predictions of the dimensionless bearing capacity (q/$c_u$). Figure 5 presents the variation of the load–settlement response by changing the L/b ratio. The micropile length was allowed to change, while all other input parameters were set to constant values. The figure effectively demonstrates how micropile geometry influences the bearing capacity of micropiled rafts in cohesive soils. It can be seen that q/$c_u$ increases and improves as L/b grows, indicating that longer micropiles contribute to a greater bearing capacity. Figure 6 demonstrates the influence of varying the number of micropiles (n) on the load–settlement response, with n ranging from 25 to 289. Correspondingly, the associated s/b ratios decrease from 0.2381 to 0.06. The results show that the q/${\mathrm{c}}_{\mathrm{u}}$ ratio increases as the number of micropiles (n) rises, signifying that a greater number of micropiles enhances the overall bearing capacity of the micropiled raft. Additionally, Figure 7 illustrates the impact of the normalized raft thickness (t/b) on the load–settlement response. The figure shows that a greater raft thickness enhances the bearing capacity of the micropiled raft. The results of these sensitivity analyses show that the EPR model’s predictions align well with the expected geotechnical behavior and previously published experimental and numerical findings. The model could capture the influence of key factors on micropiled raft behavior, highlighting its robustness and reliability for practical applications.

5. Further Validation Using a Case Study

Field test data from a case study of a micropiled raft in Shanghai, reported by Han and Ye [25] and not included in formulating the empirical equation, were used for further validation and comparison purposes. A surface footing measuring 1.5 m × 1.5 m was initially preloaded to a stress level of 89 kPa. Following this, it was underpinned with four micropiles and subsequently subjected to additional loading. While the test does not represent a primary micropiled raft constructed with micropiles from the outset, it shares similar field conditions and falls within the normalization range of the current dataset, making it the most relevant reference for comparison. Furthermore, the relevance of this case study lies in its ability to validate the model’s predictions for pressure-grouted micropiles, a key focus of this research. The soil profile in the site was composed of a crust layer near the ground surface, then clay layers with an undrained shear strength ranging from 25 to 35 kPa [25]. A weighted average of 29 kPa was considered in the present analyses [25]. Additionally, the micropiles were constructed by injecting the grout under a pressure of 0.2 to 0.5 MPa [17] and were classified as Type B per FHWA standards [3]. The input data are as follows: undrained cohesion of clay: ${\mathrm{c}}_{\mathrm{u}}$ = 30, micropile diameter: d = 0.15, micropile length: L = 8 m, number of micropiles = 4, spacing between micropiles = 0.75 m, raft thickness = 0.55 m, and width of raft = 1.5 m.

Gu et al. [56] suggested that the $K_0$ value of Shanghai clay typically lies in the range of 0.40–0.66 and proposed the following empirical equation to estimate $K_0$ using the initial voids ratio ($e_0)$ and the plasticity index ($PI$):

$$K_0=0.049e_0+0.02PI(\%)+0.139$$

(9)

Based on the physical properties reported by Han and Ye [25], $K_0$ was estimated to be 0.513. However, it is anticipated that the actual ${\mathrm{K}}_{\mathrm{s}}$ would be in the order of 1.5${\mathrm{K}}_{\mathrm{s}}$ to account for the Type B pressure grouting process [49,50,51].

The EPR model was used to predict the load–settlement response of the case study, while considering three ${\mathrm{K}}_{\mathrm{s}}$ values of 0.513, 0.641, and 0.77 (see Figure 8). As anticipated, the field results significantly exceeded the predicted response for ${\mathrm{K}}_{\mathrm{s}}$ = 0.513, as this ${\mathrm{K}}_{\mathrm{s}}$ value is specifically intended for gravity-grouted micropiles. In contrast, the field results closely align with the predictions for ${\mathrm{K}}_{\mathrm{s}}$ = 0.77, indicating that this higher lateral earth pressure coefficient is more representative of Type B micropiles constructed using a grouting pressure of 0.2 to 0.5 MPa. The EPR model’s predictions for varying ${\mathrm{K}}_{\mathrm{s}}$ values further support the conclusion that pressure-grouted micropiles are more effective than gravity-grouted micropiles in improving the performance of micropiled raft foundations.

6. Strengths and Limitations

This study presents an accurate and transparent predictive tool, integrating key design parameters across a diverse dataset, making it valuable for preliminary micropiled raft design. Notably, while the model was specifically developed for micropiles, it is also applicable to slender pile–raft foundations, provided the input variables fall within the training data range, as it performs more reliably in interpolation than in extrapolation.

The model was developed using a dataset that primarily covers the pre-failure range of settlement (see Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8). As a result, the ability of the EPR model to capture the post-failure behavior, where the load capacity declines or plateaus, is limited. Moreover, the EPR model is not designed to handle highly complex or exceptional site conditions that may yield unexpected or difficult-to-interpret results. Instead, it is well-suited for common site conditions involving single or multilayer cohesive soils. The coefficient of lateral earth pressure (${\mathrm{K}}_{\mathrm{s}}$) plays a crucial role in the model predictions, and uncertainties in its measurement could introduce errors. If field data on ${\mathrm{K}}_0$ is unavailable, it is recommended to estimate it using the correlations in Equations (2), (3) and (9). To account for pressure grouting effects, ${\mathrm{K}}_{\mathrm{s}}$ can be reasonably assumed within the range of 1.2 to 1.7 times ${\mathrm{K}}_0$.

7. Conclusions

A reliable predictive model for the load–settlement behavior of micropiled rafts in cohesive soils was developed using the multi-objective genetic algorithm-based evolutionary polynomial regression (EPR-MOGA) approach. The model was trained and validated using an extensive database of 458 data points from field tests, centrifuge experiments, laboratory studies, and numerical simulations. The following key conclusions can be drawn from this study:

• The model demonstrated excellent performance, with a coefficient of determination (R²) of 0.96 for the training data and 0.93 for the testing data. It effectively captured the nonlinear load–settlement behavior of micropiled rafts.
• The sensitivity analyses further validated that the EPR model’s predictions align well with the expected geotechnical behavior. The predicted response of micropiled rafts improves by increasing the micropile length, number, or raft thickness.
• Pressure-grouted micropiles typically exhibit higher values of ${\mathrm{K}}_{\mathrm{s}}$ compared to gravity-grouted ones. This difference reflects the fact that pressure grouting creates greater lateral confinement on the micropile shaft.
• The EPR model was successfully trained to differentiate between the gravity and pressure grouting construction methods, by incorporating the coefficient of lateral earth pressure on the micropile shaft (${\mathrm{K}}_{\mathrm{s}}$). A higher bearing capacity could be predicted by using elevated Ks values.
• The model’s sensitivity to ${\mathrm{K}}_{\mathrm{s}}$ was evident in the case study validation, where pressure-grouted micropiles with a higher ${\mathrm{K}}_{\mathrm{s}}$ value (1.5${\mathrm{K}}_0$) provided a closer match to field measurements compared to ${\mathrm{K}}_{\mathrm{s}}={\mathrm{K}}_0$.