The soils found in the coastline and offshore region are soft clay type, and usually have low shear strength. The methods used for the foundation of the different structures in such soil conditions are pile foundation. Increasing the need of construction of structures like bridges, transmission towers, tall chimneys, and offshore structures like jetty, quay and harbor require pile foundations with lateral and axial load (Varghese, 2005). Lateral loads act on piles are in the form of wind forces, ship swing, sway and impact, earthquake, water forces, and lateral earth pressures. The resistance to the lateral load of pile foundations is important in the design of structures under static and dynamic loading. For structures to be safe and serviceable, two criteria must be satisfied, viz., a pile should be safe against ultimate failure and normal deflection at working loads should be well within the permissible limit. The analysis of behavior of the pile under lateral load is required for the calculation of maximum bending moment along with the depth of fixity and surface deflection under different soil conditions (Kavitha, Beena, & Narayanan, 2016). To predict the behavior of a pile structure, due consideration of the properties of pile structure as well as properties of surrounding soil is necessary, especially when the structure is to built in a weak soil stratum. Depending on load transfer mechanism, axially loaded piles and laterally loaded piles are the two classes of pile foundation. Frictional resistance between pile and surrounding soil and end stratum bearing capacity defines load carrying capacity of axially loaded pile. In case of laterally loaded pile,stiffness and depth of fixity forms the basis of load carrying capacity. This depth of fixity of pile depends on soil interaction with pile material. In most of the soils, depth of fixity is observed to be less than six times the diameter of pile (Poulos & Davis, 1980).

The different approaches are used for the analysis of the laterally loaded pile like an experimental method using a lab scale prototype model or full length pile. Many researchers have studied on experimental approach. Beena, Kavitha, and Narayanan (2018) have studied an effect of sloping bed on the deflection and bending moment of the pile under lateral load. A single pile with different slenderness ratio was considered for analysis under homogeneous soil characteristics and loading conditions. PLAXIS-3D, the commercial software, was used for the numerical analysis and results were validated with the experimental results found. The authors concluded that, the maximum bending moment increases with depth, diameter and bed slop of the pile. The depth of the fixity moves down with increased bed slope. Abdrabbo and Gaaver (2012) have been presented a simplified approach to analyze a laterally loaded pile group with p-multiplier factors in a horizontal modulus of subgrade reaction. They proved that, laterally loaded pile can be analyzed within a working load by assuming a linear relationship between a lateral displacement and load. The method was used in preliminary design stage. Apart from this method, a pile groups can be analyzed using elastic continuum approach and group equivalent pile procedure. Matlock (1970) have developed the p-y relationship for analysis of single pile and soil interaction. The method is then modified by introducing p-multiplier for the analysis of pile in group. Gupta and Basu (2017) developed a continuum based method for the analysis of pile under lateral load in multilayered and heterogeneous soil. The modulus of the layered soil varies linearly or non- linearly with the depth. A finite difference method and Ritz method, were used for solving the differential equations for pile and soil displacements. They calculated the maximum bending moment, deflection and rotation of the pile and compared the results with 3D finite element analysis software. An analysis of pile under lateral load using commercial software ANSYS was presented by George and Lovely (2015). A linear elastic Winkler model was used for the analysis of single pile with soil structure. They investigated the pile behavior in uniform clay soils and different soil layer in clay and results were compared with theoretical results. Johny, Binu, and Issac, (2014) did a study on the basis of finite element analysis using PLAXIS 3D software. The different lateral loads applied on cohesionless soil and calculated the maximum bending moment and deflection. Kalavathi and Muralidhar (2015) presented the subgrade reaction method for piles in cohesionless soils subjected to lateral loads to calculate the deflection, slope, bending moment and shear force which then validated using ANSYS software. Bahrami and Nikraz (2017) performed a mathematical analysis on the basis of governing electrodynamics equations for Winkler model used for the analysis of laterally loaded pile. The solution from the model leads to generalized Winkler model. Beena et al. (2018) conducted a model study to investigate the behavior of pile under lateral load in sloping bed. The deflection and the bending moment of the pile were calculated using analytical formulations and results are validated with numerical software PLAXIS 3D. Deendayal, Muthukkumaran, and Sitharam, (2016) carried an experimental analysis of single pile on the sloping bed carrying soft clay. A single aluminum pile was considered with different length to diameter ratio (L/D= 20, 25, 30) with different sloping beds. The experimental results had been compared with those obtained by FEA analysis using PLAXIS 3D. The correlation between the subgrade reaction and the soil stiffness properties for laterally loaded pile was presented by Medjitna and Bouzid (2019). They found correlations for the soil stiffness properties and the subgrade reaction coefficient for different soil conditions. The Winkler model had been modified with these studies. Fatahi, Basack, Ryan, Zhou, and Khabbaz (2014) found that, the lateral loads on piles in soft clay can initiate the gap formation, clay surge and tensile cracks at the surface of clay-pile interface, whereas semi-elliptical narrow dip can develop in medium dense sandy soil. Kanakeswararao and Ganesh (2017) have carried out the analytical and numerical investigation on pile in cohesive and cohesionless soils, whereas Uddin and Islam (2010) have carried out an experimental and numerical investigation on pile in sandy soil under lateral loads to study the effectiveness and influence of pilesoil parameters.

From the above literature, it is clear that a limited research work has been carried out on the investigation of pile under lateral load in homogeneous layered soil. Also, the study of pile behavior in different subgrade coefficient of soil was found less. In the present work, the analysis of a pile under lateral load with different soils in their homogeneous layers is carried out numerically. A finite element based mathematical code has been developed in commercial software MATLAB to find out the maximum deflection and bending moment of pile under different operating conditions. Heavy duty PVC pipe material is considered for the model pile while, different sand and clay conditions are used with different subgrade modulus present within the range. A single hollow pile of 24 mm outside diameter with thickness of 2.3 mm is considered for the analysis. The length of the pile varies between 720 mm to 912 mm with an eccentricity of 61 mm from the ground level. The numerical results are validated with the experimental results found in lab scale setup and also the results found in literature.

Laterally loaded pile analysis is carried on homogeneous soils. Solution to this problem is studied by application of static lateral load on pile through model, analytical, and the numerical investigation. To understand the present scenario of the problem under consideration, an extensive literature review has been conducted. An analysis is carried on a single pile for static load at pile top. A schematic representation of the single pile setup and the actual photographs are shown in Figure 1. In present study, model investigations have given preference than to have investigations on prototype study because prototype study is tedious and expensive in nature. Therefore, pile model testing setup was established in the laboratory to investigate the variations in pile top deflection under lateral loads. An extensive experimental work has been carried out in the laboratory to obtain load and corresponding deflection relations. The study on structural behavior of laterally loaded pile is carried out in homogeneous soil bed with the change of pile lengths. An analytical solution of the problem has been derived from the governing equation suggested by Reese, Isenhower, and Wang (2006). A MATLAB code was developed to get the solutions of finite difference equations on account of tedious manual calculations. The analytical solution is validated by an onsite pile load test results. Model tests are also simulated using analytical formulations and the results are compared. A detailed parametric study of the laterally loaded pile is conducted by varying the pile and soil parameters. The influence of change in soil and pile parameters in the structural behavior of pile is studied.

Figure 1. (a) Experimental setup Photograph (b) Schematic Diagram of the Test Setup

In this research study, a simple mathematical model is developed to study the effect of soil structure interaction in laterally loaded pile. A MATLAB R2010a, computer program has been developed for the analysis of the laterally loaded pile using a classical theories proposed by the Reese et al. (2016). 3.1 Analytical Formulations A finite difference scheme is used to develop a mathematical code for the calculation of the pile deflection at the top, bending moment and depth of fixity along the length. Figure 2 shows the discretization method used for the analysis of the pile in finite difference method. Several classical methods are used for calculating the pile deflection under lateral load, viz., modulus of subgrade reaction method and the elastic approach method, in which, modulus of subgrade reaction approach ignored continuous nature of soil medium and considered the pile reaction at a point is related with the deflection. Modulus of subgrade reaction method is simple and incorporates factors such as nonlinearity, soil layers and variation in subgrade modulus with depth. The following equation from the beam theory represents the equilibrium equation for the laterally loaded pile,

(1)

Figure 2. Finite Element Discretization Considered for the Pile Deflection Analysis

Where, E_py is the soil reaction and varying uniformly along the pile depth and valid for over consolidated clays. However, E_py is assumed to be linearly varying with the depth of pile in case of cohesionless soil. Hence, the linear variation of the soil reaction, (E_py = k_pyx) is assumed for the present analysis. A finite difference method with central difference discretization scheme is employed in present study for the analysis of the pile behavior. The node '1' at the pile top is assumed to be free headed and node 'n' at the tip of the pile as floating tip and their boundary conditions are also described. By employing the central difference scheme,

(2)

By adding the values of equation (2) into equation (1),

(3)

By rearranging,

(4)

where,

(5)

n' is the number of element along the pile, ℎ = L/n is the segmented length, E_py is the modulus of subgrade reaction, k_py at n. E_pI_p is the bending stiffness and y is lateral deflection at point x along the length of the pile. By applying different boundary conditions and from equations (4), the n-1 and remaining unknown can be obtained using numerical methods. The following assumptions are involved in the formulations,

• A straight and uniform cross section of the pile is considered.
• The pile material is considered as homogenous and isotropic.
• The pile has longitudinal plane of symmetry, loads, and reflections lie in the plane.
• The modulus of elasticity is same in tension and compression.
• Transverse deflection of the pile is neglected.
• The pile is not subjected to dynamic loading.

3.2 Validation of the Mathematical Model

A finite difference based mathematical model developed for the analysis of the pile deflection is validated with the available solutions in the literature Reese et al. (2016). A long pile of length 20 m and the outside diameter 380 mm having material steel with thickness 25 mm is considered for the analysis. A lateral load of 300 kN is applied at the top of the pile to check the failure of the pile. A steel pile having an elastic modulus of E = 2.0 x 10⁸ kPa and calculated Moment of Inertia (MI) is I = 4.414 x 10^-4 m⁴ is considered. At ground level, the deflection of the pile is critical. Over consolidated clay is assumed and the effect of axial load is neglected. Two boundary conditions each at top and bottom of the pile are applied. A displacement and bending moment are set to zero at the bottom of the pile. A point load is applied at the top of the pile. The bending moment and deflection of the pile are plotted. It can be seen that, there exists a good agreement between the modulus of subgrade reaction of the model and the solution available in at Reese et al. (2016). It is found in literature that, the maximum deflection at the top of the pile is 47.6 mm, while the bending moment is 410 kN-m (Figure 3). Similarly, the corresponding results are calculated using mathematical code and shown in Figure 4 (a) and (b). From the results, the deflection observed is 48.8 mm and that of bending moment 413.56 kN-m. The maximum error found in deflection is 2.5% and for BM it is 0.86%.

Figure 3. Deflection and Bending Moment Results at Reese et al. (2006)

Figure 4. Results obtained for validation of (a) Deflection of the Pile (b) Bending Moment of the Pile in MATLB Code

3.3 Validation of the Mathematical Model with Experimental Results

A MATLAB code developed and validated with solution available in Reese et al. (2016) is used for the analysis of the experimental results in present study. The experimental investigation includes model study on 24 mm external and 19.4 mm internal diameter with 2.3 mm thickness, as the model pile. The length of the pile is varied in the range 720 mm and 912 mm to have a flexible behavior of pile. The modulus of elasticity of pile material is adopted to be 30.45 GPa and the moment of inertia of the model pile is 4 calculated to be 9328.21 mm⁴. Figure 5 (a) to (d) show the load vs. deflection curves for sand and BC soil conditions with different slenderness ratio. The experimental results are plotted and compared with the results produced from the MATLAB code. A single pile with 720 mm and 912 mm length and 24 mm diameter is considered for the experiment as well as numerical purpose. The numerical results are showing a good agreement with experimental data. A subgrade reaction of 25 MN/m³ is considered for the sand and 12 MN/m³ is selected for BC soil. The experimental results of deflection of the pile with load is found good in agreement with the numerical results captured in MATLAB within a minimum error.

Figure 5. (a) to (d) Comparison of the Numerical Results with Experimental Results with Different Slenderness Ratio and Different Soil and Sand Conditions (a) Sand SL=30 (b) Sand SL=38 (c) BC Soil SL=30 (d) BC Soil SL=38

4.1 Parametric Study for the Analysis of Prototype Pile with MATLAB

A parametric study is done to examine the effect of length, diameter of the pile and subgrade reaction of the sand and BC soil conditions on pile under lateral load. A deflection and bending moment plots are plotted by changing different parameters.

4.2 Effect of Pile Length

The effect of change in length of the pile on the maximum deflection and the bending moment is illustrated in Figure 6 (a) and (b). The effect of the change in length of the pile at constant diameter and subgrade modulus are listed in Tables 1 and 2. The lateral load of 2500 kN is applied at the top of the pile for two different clay conditions, sand and soil. Table 1 shows the results for the sand with subgrade modulus of 20 MN/m³ while the diameter of the pile is set to constant at 1000 mm. It can be seen from the results that, as the length of the pile increases, the deflection of the pile reduces up to a certain length. It is observed that the reducing deflection difference of the pile is going towards the stabilization with increase in length. The maximum deflection (224.13 mm) is found in pile of length 14 m, while, piles of lengths 20 m and 22 m respectively, showed very less deflection difference in them. As seen in Table 2, the BC soil with subgrade modulus 5 MN/m³ showed more deflection compared to sand. Therefore, it can be concluded that, the subgrade modulus affects the deflection as well as bending moment of the pile. It can be seen from both the cases that, for increase in pile length, there is decrease in deflection and increase in bending moment. It can also be observed that, these values are going towards stabilization. From this it can be inferred that, the pile is losing its significance beyond a particular length i.e., 22 m in view of deflection and bending moment. This indicates the flexibility behavior of pile. The deflection and bending moment observed in pile of length 22 m can be seen in Table 2. Figure 6 (a) and (b), shows the plots of deflection and bending moment for the effect of change in length of the pile with soil (5 MN/m³).

Table 1. Effect of Length of Pile and Loading Conditions of 2500 kN in Sand

Table 2. Effect of Length of Pile and Loading Conditions of 2500 kN in Soil

Figure 6. Effect of Length of Pile in Soil on the (a) Deflection and (b) Bending Moment of the Pile

4.3 Effect of Diameter of the Pile

Table 3 shows the effect of diameter of the pile on the deflection and bending moment. The 18 m length pile is considered with change in different diameter. The subgrade modulus is kept constant so as to capture the effect of diameter of the pile on the deflection and bending moment of the pile. The diameters considered for the analysis are 800 mm to 1400 mm and subgrade modulus of 20 MN/m³. A constant 2500 kN lateral load is applied on the top of the pile. Table 3 shows the results for the change in diameter. It can be seen, from the values that, the deflection of pile reduces with the increasing pile diameter. The maximum deflection is found with lowest diameter (800 mm), while the bending moment is increased with increase in diameter of the pile. The change in deflection and the bending moment for different diameters of the pile is seen in Figure 7 (a) and (b). It can be concluded from the figure that, the deflection of the pile reduces with increase in diameter of the pile. The maximum deflection of the pile found 293.08 mm for the 800 mm pile diameter while the minimum deflection 151.88 mm is found for 1400 mm diameter. From Figure 7, it is found that, the bending moment is maximum for the maximum diameter of the pile.

Table 3. Effect of Diameter of Pile and Loading Condition of 2500 kN

Figure 7. Effect of Diameter on the Deflection and Bending Moment of the Pile

4.4 Effect of Subgrade Modulus

The effect of subgrade modulus has been analyzed for the two different soils with different range of subgrade modulus. Tables 4 and 5 shows the effect of subgrade modulus on the deflection and bending moment of the pile for sand and BC subgrade modulus of 20 to 50 MN/m³ for sand and 5 to 15 MN/m³ for BC soil. Figure 8 (a) and (b) shows that the deflection and bending moment for the different subgrade modulus of range between 20 to 50 MN/m³ of soil is considered for the analysis. It can be concluded from the table that, as the subgrade modulus reduces the deflection increases. The soil stiffness is less with the lower subgrade modulus hence the deflection found maximum with lower subgrade value. The maximum deflection of pile is found 223.90 mm³ in sand for 20 MN/m³ while in soil 467.83 mm³ for 5 MN/m for soil. The change in deflection and the bending moments of 3 the soil with 5 to 15 MN/m subgrade modulus depicts in Figure 9 (a) and (b). Figure 9 shows the graphs of deflection and bending moment for the sand and soil with different subgrade modulus.

Table 4. Effect of Subgrade Reaction and Loading Conditions of 2500 kN

Table 5. Effect of Subgrade Reaction and Loading Conditions of 2500 kN

Figure 8. Effect of Subgrade Reaction of Sand on the Deflection and Bending Moment of the Pile

Figure 9. Effect of Subgrade Reaction of Soil on the Deflection and Bending Moment of the Pile

An experimental and numerical investigation of pile under lateral load is carried out. A lab scale setup has been developed to perform a set of experiments on a single pile under different soil conditions for different loads. A finite element analysis based numerical study is done to investigate the effect of different parameters on the deflection and bending moment of the pile. A numerical model validation is done with experimental results found in lab scale setup and the results available in literature. The following conclusions can be drawn from this study,

• From the experimental investigation, it is revealed that the maximum deflection decreases as pile embedment depth increases.
• More deflection was observed in homogeneous BC soil than homogeneous sand soil.
• There is marginal decrease of 6.99% in the maximum deflection of the pile when its embedment depth increases from 720 mm to 912 mm in homogeneous BC soil bed, whereas the decrease of 4.8% in maximum deflection for same increase in pile embedment depth in homogeneous sand soil bed.
• The percentage variation in experimental deflection results and deflection results obtained in MATLAB are in the range of 3.19% to 19.77% for sand soil and for BC soil, they are in the range of 5% to 16% which can be seen well within the acceptable limit.
• Based on the parametric study in a MATLAB code developed for laboratory model pile system, it reveals that,

(i) There is decrease in pile deflection and increase in bending moment with increase in pile length. There is marginal change in deflection and bending moment beyond certain pile depth.

(ii) There is decrease in pile deflection and increase in the bending moment with increase in pile diameter.

(iii) There is a significant effect of subgrade reaction of the soil on the deflection and bending moment of the pile. A reverse effect is found in deflection and bending moment with subgrade reaction.