THREE-DIMENSIONAL NUMERICAL MODELING OF SKIRTED ANCHOR AND RETICULATED MICROPILE FOUNDATION SYSTEMS

The development of novel foundation systems can render projects ranging from infrastructure rehabilitation to sustainable energy applications more cost-effective. A reticulated micropile system is one such foundation and is studied herein. A reticulated micropile network is a three-dimensional lattice structure that involves the creation of a laterally confined soil-pile composite structure. A numerical analysis using FLAC3D was performed to better understand the mechanics of the ‘knot’ effect in a reticulated micropile system. FLAC (Fast Lagrangian Analysis of Continua) is a numerical modeling software used for advanced geotechnical analyses of soil, rock, constructs, and ground support. The ‘knot’ effect is the formation of an improved soil mass within the reticulated geometry that has been credited with improved group performance. An increase in confining pressure in the soil mass due to shear and dilation of the soil surrounding the pile is the anticipated cause of the ‘knot’ effect. The formation of this soil block, coinciding with an increase in confining pressure and a decrease in shear stress, was observed in both reticulated and plumb pile geometries under vertical compression loading, but it did not improve the group capacity. Improved group performance was observed, however, in groups of both reticulated and vertical piles when laterally loaded.


Preface
This dissertation is organized into three manuscripts contained in three separate chapters.
It is the intention of the author that these manuscripts will be submitted for publication in appropriate peer-reviewed journals. The first two chapters deal with the numerical analysis of a proprietary anchor system for floating offshore wind turbines, while the third chapter investigates the mechanics of a reticulated micropile system used for foundation stabilization and/or rehabilitation. Chapters 1 and 2 contain confidential information and have been removed from the dissertation. Chapter 3 investigates the group efficiency and horizontal stress response of a reticulated micropile system when compared to a similar micropile group consisting of vertical piles for conditions of vertical compression and lateral loading.
The three manuscripts utilize the software FLAC 3D 6.0 to perform numerical analyses. FLAC 3D is a finite difference program used to study the mechanical behavior of a continuous three-dimensional medium as it reaches equilibrium or steady plastic flow. It uses an explicit Lagrangian calculation scheme and a mixed discretization zoning technique to ensure that elastic behavior and plastic failure and flow are modeled accurately (Itasca, 2017c.). FLAC 3D solves Netwon's second law at each gridpoint and the stress-strain compatibility for each zone, based on the constitutive equations for the zone, by analyzing a series of timesteps as the model is brought to equilibrium. The model is considered to be in equilibrium when the maximum out-of-balance force is small compared to the total applied v forces within the model. A large number of small timesteps are required, to ensure stability, as stresses and strains propagate though the numerical grid with each timestep as the model is solved. A detailed description of the calculation scheme used by FLAC 3D can be found in Appendix A.
The Mohr-Coulomb (Chapters 1 and 2) and plastic hardening (Chapter 3) constitutive models were implemented. The Mohr-Coulomb constitutive model is the most common model in the context of defining the behavior of soil for general engineering studies, representing the medium as a linear-elastic perfectly-plastic material with a failure envelope corresponding to a Mohr-Coulomb criterion. The plastic hardening constitutive model is a shear and volumetric hardening model used for the simulation of soil behavior.
It features a hyperbolic stress-strain relationship and stress-dependent elastic stiffness, obtained using a power law. The plastic hardening model also uses a Mohr-Coulomb failure criterion. A detailed description of the Mohr-Coulomb and plastic hardening constitutive models can be found in Appendix B.
vi    (Durot and Plumelle, 1996;Gagneux and Plumelle, 1997    An increase in group efficiency has been observed in the literature for a reticulated micropile system when compared to a similar vertical micropile group geometry. This increase in efficiency has been credited to the improvement of the soil mass contained within the reticulated system originally described by Lizzi as the 'knot effect'. This paper examines the knot effect, which is anticipated to be caused by an increase in effective confining pressure in the soil mass due to shear and dilation of the soil surrounding the pile. This is achieved through a 3D numerical analysis of a full-scale reticulated system consisting of 18 micropiles. The plastic hardening constitutive model was employed to more accurately model the pre-failure stress-strain behavior of the soil. To ensure the constitutive model would capture the pre-failure effects of dilation, it was calibrated using consolidated-drained triaxial tests and validated using constant normal stiffness tests. The numerical model used to simulate the reticulated system was validated using a series of four full-scale field tests, with good agreement being observed between the numerical simulations and field tests. A parametric study was performed to investigate the effects of soil strength and stiffness on the lateral stress response and group efficiency of the system. The results of the study indicate up to a 6-fold increase in the horizontal effective stresses in the soil mass in both reticulated and vertical micropile groups under vertical loading, but did not result in positive group efficiency (i.e. >1). However, under lateral loading positive group efficiencies between 1.5 and 2 were observed for both a reticulated system and vertical micropile group due to an increase in effective stress within the soil mass.

Introduction
The need for cost-effective foundation solutions for offshore energy systems or infrastructure rehabilitation has never been more pressing. As offshore wind farms look to expand to deeper waters, the cost of the foundation system becomes increasingly cost prohibitive. The cost of a foundation placed at water depths between 40-50 m is 1.9 times greater than a foundation placed between 10-20 m (Oh et al., 2018). In their Rhode Works Ten-Year Transportation Improvement Program (2015), the Rhode Island Department of Transportation (RIDOT) stresses the importance of investing more money in bridge preservation in an effort to arrest the downward trend of bridge deterioration. The RIDOT estimates bridge replacement is 6 times more expensive than bridge preservation.
"Root piles', or micropiles, were developed by Fernando Lizzi in post-World War II Italy as a means to rehabilitate the Neapolitan seaport (Bares, 2007). Soon after introducing the concept of the micropile, Lizzi experimented with networks of such piles he called a "reticulated root pile" system (reticulated micropile structure or reticulated micropile system, henceforth) (Lizzi, 1981). A reticulated micropile system is a three-dimensional lattice structure that involves the creation of a laterally confined soil/pile composite structure that can work for underpinning, stabilization, and earth retention as seen in Figure   3.1 (Bruce et. al., 1997).
Lizzi performed a series 1g model tests in sand comparing the group efficiency of a reticulated system of 18 micropiles to that of both a vertical group of 18 micropiles and a single micropile (Lizzi, 1978), group efficiency being defined as: where ηg = group efficiency, N = number of piles in group, Qg = capacity of group, and Qs = capacity of a single pile not within a group. Lizzi's results indicated that the reticulated system had an efficiency of 2.22 when compared to the single pile and 1.32 compared to the vertical group; the vertical group had an efficiency of 1.68 when compared to the single pile. Lizzi concluded that the capacity of the reticulated and vertical micropiles was supplied by a reinforced mass of soil within the group (Lizzi, 1978).
The French National Project on Micropiles (FOREVER, 2008) was a major research initiative aimed at gaining a better understanding of the behavior of micropile networks.
The studies included in this project included both small-scale 1g testing as well as centrifuge testing, on relatively loose sand (relative density between 45 and 55%). Vertical load tests performed on full-scale networks consisting of 4 micropiles (Durot and Plumelle, 1996;Gagneux and Plumelle, 1997) rendered group efficiencies between 0.81 and 0.85.
Experimental tests by Rault and Noblet (2000) and Haza et al. (2001Haza et al. ( & 2002  The use of reticulated micropile structures in slope stabilization has been described by several investigators including Lizzi (1978), Dash (1987), andCantoni et. al. (1989). Lizzi's (1978) design envisioned the structure as a retaining wall, whereas the piles act as the "lines of force" that allow the whole mass to support compression, tension, and shear.
In reviewing Lizzi's design, Dash (1987) commented that there is no guarantee that the reinforced soil mass behaves like a composite material. Dash proposed a design chart to aid in the selection of pile spacing and diameter. Similar to Lizzi (1978), the design methodology presented by Cantoni et. al. (1989) assumes the formation of a composite structure. The formation of this composite structure assumes that the reticulated micropile structure acts as a unit whereas the stresses acting on the whole structure are fully distributed to the ground and the piles, a phenomenon Lizzi called the 'knot' effect. The 'knot' effect assumes stresses acting on a pile are partially transferred to nearby piles due to the interaction of the piles and the ground. Lizzi attributed this to the high bond between the individual micropiles and the surrounding soil (Lizzi, 1978). Lizzi (1978) attributed the capacity achieved through reticulation to the interaction of the micropiles and the ground. Masarucci et al. (2014) noted that pile response to axial loading is mainly governed by the shear band surrounding the pile. The shear band thickness, ts, depends mainly on an average grain size (D50), and typically approaches 10 times the D50 of the soil (Balachowski, 2006). Wernick (1972) postulated that the dilation of the sand in this thin shear band is the reason shaft resistance noticeably increases with decreasing pile diameter, D. The relatively large size of sand grains in relation to a pile modeled in a centrifuge test renders a disproportionately thick shear band resulting in scale effects that inflate the shaft resistance (Lehane et al., 2005). Additionally, higher dilative effects occur at lower confining stresses, leading to soils with a higher strength in small-scale 1g tests (e.g., Giampa et al., 2019).
The observed group efficiencies of a micropile networks have been wide-ranging. Lizzi (1978) and Plumelle (1984) observed 'positive' group effects, while experiments included as part of the FOREVER project saw efficiencies ranging from 0.51 to 2.93. Soil density being a key parameter in the behavior of a reticulated system, was not investigated as part of the FOREVER project, and has not been investigated in detail in the literature. Many of the experiments included in the FOREVER project were either small-scale 1g load tests or were performed in a centrifuge, both of which introduce scale effects that could result in an exaggerated group efficiency. If the 'knot' effect has the potential to result in an increased group efficiency, an investigation to better understand the mechanics of its formation is necessary.
The objectives of this study were therefore to investigate the 'knot' effect numerically, more specifically to see if the effect is due to an increase in effective confining pressure between the micropiles due to dilation of the soil at the pile interface. The objectives of this study were achieved through numerical analysis using the software FLAC 3D . Full-scale numerical models were simulated to avoid scale effects. Simulations were performed in loose, medium, and dense sand to investigate the effects of soil strength and stiffness on both the capacity and horizontal stress response for vertical and lateral loading conditions.
The remaining sections of this paper detail the choice of constitutive model, the development and validation of the numerical model, the numerical study, and the analysis of the numerical results.

Constitutive Model
Capturing the dilative phenomena along the pile-soil interface is crucial to determining whether or not the 'knot' effect is caused by an increase in confining pressure due to dilation. The plastic hardening (PH) constitutive model extends the hyperbolic Duncan-Chang non-linear elastic model (Duncan & Chang 1970) and is based on the work of Schanz et al. (1999). Because the PH model utilizes different stiffness values for primary loading and unloading/reloading, it provides a better pre-failure stress-strain relationship than the more commonly used Mohr-Coulomb model which uses a constant stiffness for both loading and unloading (Cheng & Lucarelli, 2016). To ensure the PH constitutive model would capture the dilative behavior of the soil along the interface, constant normal stiffness (CNS) tests were modeled using soil parameters calibrated from the results of three CD triaxial tests performed on medium dense Monterey sands having void ratios ranging from 0.65 to 0.66 with relative densities ranging from 68% to 71%. CNS tests were selected because they most closely represent the interface between the pile and sand, and also the stiffness of the surrounding soil constraining the dilation (Lehane et al., 2005).
The PH constitutive model was calibrated following the procedure outlined by Cheng and Lucarelli (2016). Using the calibrated material parameters presented in Table 3.1, the CD triaxial tests were modeled using the PH model in FLAC 3D (after Itasca, 2017a.). The deviatoric stress -axial strain data, presented in Figure 3.2a, showed good agreement between the laboratory tests and the numerical models. The volumetric strain -axial strain data presented in Figure 3.2b. displayed the dilative behavior of the soil samples, and was in relatively good agreement with the laboratory tests.
To ensure the PH constitutive model can capture the pre-failure effects of dilation and changes in effective stress under CNS boundary conditions, CNS tests were modeled in FLAC 3D and compared to the results of six laboratory tests, three each having initial normal stress of 100 and 200 kPa.
The geometry of the numerical model representing the soil mass within the shear box used for the CNS tests is shown in Figure 3.3. The diameter of the bottom half of the shear "box" was made larger than the top half to maintain a soil-to-soil interface throughout the model run. The soil input parameters for the plastic hardening constitutive model obtained from CD triaxial testing (Table 3. Pile structural elements, two-noded, straight finite elements consisting of uniform, bisymmetric cross-sectional properties, are available in FLAC 3D and are ideal for complex pile group configurations (Itasca, 2017b.). When initially validating the numerical model of a single micropile in dilative sand, a structural element was used in the simulation. It was observed, however, that the capacity achieved in a dilative sand was better simulated by representing the micropile as a zone group with an interface between the micropile and surrounding soil. To create the mesh required to represent the micropile networks, Griddle 2.0 was utilized. Griddle is a plug-in for the computer-aided design (CAD) software Rhinoceros 3D that allows for the creation of unstructured mesh for complex geometries that can be imported into FLAC 3D . Once imported into FLAC 3D , zone groups consisting of the pile cap, micropiles, and soil were created in addition to the interface between the micropiles and soil.
The three networks (Durot and Plumelle, 1996;Gagneux and Plumelle, 1997)  ( 3) where patm = atmospheric pressure (100 kPa), p' = mean effective stress, and C = site specific soil constant that accounts for the effect of the void ratio, e, on Gmax. As Fontainebleau sand having relative density values between 53% and 62% was used in the Saint-Rémy-lès-Chevreuse tests (FOREVER, 2008), C values were estimated using the equations derived by Richart et al. (1970) for Gmax as a function e. Using the corresponding void ratios of Fontainebleau sand (Latini and Zania, 2016), a C value of 985 was used for the numerical simulations.
Using elastic theory, the small strain Young's modulus, Emax, was then obtained: where R is the reduction factor. A commonly used value of R is 3, values between 1.5 and 3 were used for the validation study which yielded average values between 1.5e5 and 3.0e5 kPa, reasonable values for a loose to medium loose sand (Itasca, 2017b. Latini andZania, 2016). The unit weight of the soil was obtained from the literature (FOREVER, 2008), and the friction angle, φ', was varied between 34 o and 37 o , a representative range for loose to medium loose Fontainebleau sand (Latini and Zania, 2016). The dilation angle was estimated using the following relationship (Bolton, 1968): where φ' = peak friction angle and φ'cs = critical state friction angle. A value of φ'cs = 33 o was used for Fontainebleau sand (Yang et al., 2010). The remaining soil input parameters for the numerical validation can be found in Table 3 The micropiles and reinforced concrete cap were modeled using the isotropic elastic constitutive model defined by an equivalent bulk, K, and shear, G, modulus. Gravitational stresses were initiated and therefore densities were also defined. Table 3.3 displays the material properties for the micropiles and cap.
The pile-soil interface was modeled using the Coulomb friction model. The interface had the properties of friction, cohesion, dilation, and normal and shear stiffness. A FISH function was written so that the interface properties were a function of the surrounding soil.
The normal and shear stiffness of the interface were calculated as: (7) where zavg is the average dimension of an adjoining zone in the normal direction (Lucarelli, 2018 1 and 2, Figures 3.8a. and 3.8b., used a range of soil strength and stiffness parameters consistent with a loose to medium loose Fontainebleau sand to form an envelope of results. In both cases, the results of the Saint-Rémy-lès-Chevreuse full-scale tests were within the range of the numerical simulations and found best agreement using a reduction factor of 2, yielding an average of 2.0e5 kPa, and friction and dilation angles of 36 o and 4 o , respectively. The load test simulation performed on Network 3, Figure 3.8c., used the soil parameters obtained from the simulations performed on Networks 1 and 2, validating the plausibility of the parameters used to define a loose to medium loose sand.
An additional model validation was performed using the results of a series of load tests on single full-scale micropile . The load tests were also performed at the experimental site in Saint-Rémy-lès-Chevreuse on micropiles of identical geometry placed in identical soil to the networks previously simulated. The mesh for the single micropile was created in FLAC 3D using the extrusion pane, rendering a refined mesh of zones consisting of eight nodes and six faces. The symmetric nature of the problem allowed for one-half of the pile to be considered for modeling. A single vertical pile was used to further validate the numerical model to ensure the model was applicable to both vertical and battered micropiles created using either unrefined or refined mesh. Figure 3.9 plots the load-displacement curve comparisons of the numerical simulation and the field tests for the single micropiles. Using R = 2, φ' = 37 o , and ψ = 5 o , the ultimate capacity obtained through numerical simulation falls within the envelope of field test results, again validating the numerical model and the parameters used to define a loose to medium loose sand.

Numerical Load Test Simulations
Group effects were investigated by performing numerical load test simulations on Lizzi's micropile groups that showed positive group efficiencies under compressive loading in loose sand (Lizzi, 1978). Modeling was performed on three group configurations as shown in Figure 3. Load test simulations were performed for three different soil relative densities. The 'loose' sand had a unit weight of 15 kN/m3 with φ'=35 o , the 'medium' sand was 16 kN/m3 with φ'=40 o , and the 'dense' sand was 17 kN/m3 with φ'=45 o . The dilation angle was calculated using Equation (6), assuming φ'cs = 33 o . The soil stiffness was calculated using Equations (3) through (5) for C values of 500, 1000, and 1500 representing loose, medium dense, and dense sand, respectively. The soil properties used in the parametric study are found in Table 3.4.
Groups were displaced approximately 2.5 cm and capacity was defined at a displacement of 10% of the single micropile diameter, 1.8 cm. Micropile and pile cap properties were identical to those found in Table 3.3.

Vertical Loading
Figure 3.12a. presents the normalized change in horizontal stress at the center of the reticulated system and plumb pile group, defined as the ratio of the increase in horizontal stress to the initial horizontal stress before loading (i.e., Ko conditions). A 5.5-fold increase in horizontal stress was observed in dense sand near the surface, with increases of 4.5-fold and 3-fold seen in medium and loose sand, respectively. Further examination of the horizontal stress response upon loading at different points within the reticulated system and plumb pile group in medium sand is depicted in Figures 3.12b. and 3.12c., respectively.
It is interesting to note that for both geometries, starting at a depth of 4 m, the greatest increase in horizontal stress is seen at the center of the system, suggesting that regardless of geometry, there is an increase in effective stress at the center that is caused by shear of the soil surrounding piles.
When observing the horizontal stress response with depth at the center of the geometries, the greatest increases, noted above, were seen near the surface, decreasing with depth and merging to a similar ratio for each geometry and density at approximately 13 m, Figure   3.12a. A similar response was observed at different points within each geometry in medium sand, Figures 3.12b. and 3.12c. This response is consistent with higher dilative effects being observed in lower confining stresses (e.g., Giampa et al., 2019). The depth of the horizontal stress increase was directly connected to the formation of a 'block' within each geometry. For the purposes of this discussion, a 'block' is soil that has been strengthened by an increase in horizontal stress and is constrained by the geometry of the surrounding micropiles. causing an area of low shear stress due to low relative soil movement within the 'block'.
The numerical results were used to calculate group efficiencies for each case as defined in Equation (1) and summarized in Table 3.5. Although the formation of a 'block' was observed, there was no benefit in terms of group efficiency as each simulation resulted in an efficiency less than unity. The efficiency for the plumb pile group decreased with increasing sand density, having efficiencies of 0.75, 0.66, and 0.65 in loose, medium, and dense sand, respectively. The efficiencies observed for the reticulated system were similar to those seen in the plumb pile group for loose and dense sand, 0.74 and 0.63, respectively but was 12% higher, 0.74, in medium sand. It is reasonable to assume an interaction between the piles due to high shear stress negatively affects group efficiency, as this effect is most evident in the dense sand (Figure 3.13f.), where efficiencies are lowest. An area of high shear stress over the bottom 2 m of the geometry was observed in the plumb pile group relative to the reticulated system in the medium sand, suggesting this area of high shear stress may account for the lower efficiency observed in the plumb pile group. The numerical results were used to calculate the group efficiencies for lateral loading, again using Equation (1) and shown in Table 3.5. The efficiencies for all cases were greater than unity. The high group efficiencies for lateral loading are likely a function of the (i) low lateral capacity of a single micropile combined with (ii) the increase in effective stress observed within both geometries that strengthened the soil to increase lateral resistance.

Lateral Loading
Group efficiencies were identical for both geometries in dense sand, but the reticulated system achieved efficiencies 8% and 15% greater than the plumb pile group in loose and medium sand, respectively, Table 3.5. It is interesting to note that the soil is displaced to greater depths in the reticulated system in the loose and medium sand simulations, Figures 3.15a. and 3.15b., suggesting a greater percentage of the reticulated system is being activated to resist the lateral load. This behavior is reflected by the reticulated system having an increase in horizontal stress approximately 1.2x greater than the plumb pile group in medium sand to a depth of 9 m, Figures 3.14b. and 3.14c. The displacement profile of both geometries is similar in dense sand, Figure 3.15c, coinciding with identical efficiencies.

Conclusions
The objectives of this study were to investigate the 'knot' effect numerically, specifically to see if this effect is due to an increase in effective confining pressure between the micropiles due to dilation of the soil surrounding the pile. The plastic hardening constitutive model was implemented to more accurately capture the pre-failure stress-strain behavior of the system. The constitutive model was calibrated using consolidated drained triaxial tests, the properties of which were then used to simulate a constant normal stiffness test to ensure the constitutive model would capture the dilative effects under CNS boundary conditions. Existing full-scale field tests were used to validate the numerical model, which was then used to perform a numerical study.
The results of the numerical study under the condition of pure vertical compression loading indicated an increase of horizontal stress of up to nearly 6-fold near the surface at the center of the geometries. The formation of an improved soil 'block' was evident in both geometries, though the group efficiencies were less than unity for all cases. While the formation of a soil 'block' was observed, the increase in shear stress in the vicinity of the micropiles negatively affected group efficiency.
Under the condition of pure lateral loading, an increase of horizontal stress near the surface of up to 3-fold was observed at the center of the geometries and up to nearly 5-fold within the outermost concentric circle in the direction of loading. The smaller increase in horizontal stress at the center of the laterally loaded geometries suggests the greater increases observed in the vertically loaded study were due to the shear and dilation of the soil surrounding the micropiles. The formation of a 'block' was not observed. Efficiencies for all of the lateral loading simulations were greater than unity, as the increase in effective stress observed within both geometries strengthened the soil, thereby increasing lateral resistance.
The nodal mass is calculated only once, before cycling, in the small strain mode. Newton's law at the nodes can then be expressed as: where nn is the total number of nodes in the medium being represented, d[v]/dt is the material derivative of the velocity, and l F   is the out-of-balance force given by: where ρ is the mass per unit volume of the medium, [b] is the body force per unit mass, and P <l> are the contributions of applied loads and concentrated forces. The symbol [[.]] <l> is used to represent the sum of the contributions at global node l of all tetrahedra meeting at that node. The out-of-balance forces are monitored to check whether the system has reached a state of equilibrium or steady flow (Itasca, 2017c.).
Using the equations of motion, new nodal velocities are derived from the known out-ofbalance forces: