# Modeling Dynamics of Laterally Impacted Piles in Gravel Using Erosion Method

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Element Erosion Technique

## 3. Constitutive Models

#### 3.1. Granular (MASH Strong) Soil

#### 3.2. Steel Pile

_{ij}is the deviator stress tensor, and

_{p}.

_{p}is the current hardening modulus, and G is the shear modulus. The trial deviator stress state, ${\stackrel{~}{s}}_{ij}$, is scaled back as follows:

#### 3.3. Erosion Criteria

_{max}) to incorporate element erosion into the soil model. As depicted in Equation (6), the maximum damage within the soil element is associated with the peak (${\varphi}_{peak}$) and residual (${\varphi}_{res}$) friction angles of the soil. These values for element erosion were utilized as adaptivity thresholds in simulating various post-soil systems subjected to lateral impact loading.

_{f}, and volumetric strain at the initial damage threshold, ${\xi}_{0}$. The void formation parameter, representing the area under the softening region in the pressure–volumetric strain curve, is quantitatively described as

## 4. Model Development and Simulation Details

## 5. Simulating the Impact Response of Flexible Pile in Soil

#### 5.1. Comparison between Simulation and Physical Impact Tests

#### 5.2. Discussion of Results

## 6. Simulating Impact Response of Rigid Pile in Soil

#### 6.1. Comparison between Simulation and Physical Impact Tests

#### 6.2. Discussion of Results

## 7. Effect of Soil Mesh Density on Response of Laterally Impacted Pile–Soil Systems

#### 7.1. Model Geometry and Discretization

#### 7.2. Results

#### 7.3. Guidelines and Recommendations

## 8. Effect of Soil Domain Size on Response of Laterally Impacted Pile–Soil Systems

#### 8.1. Model Geometry and Discretization

#### 8.2. Results

## 9. Effect of Boundary Condition on Response of Laterally Impacted Pile–Soil Systems

#### 9.1. Results

#### 9.2. Guidelines and Recommendations

## 10. Summary, Conclusions, and Future Research

- The proposed large-deformation soil modeling method for pile–soil impact analysis based on the element erosion algorithm within the UL-FEM framework agreed well the measured pile–soil impact response. The applicability of the soil modeling method has been successfully demonstrated for both “long” or “flexible” and “short” or “rigid” pile behavior under impact loading.
- The simulation method presented in this study overcomes the inherent limitations of popular soil modeling techniques typically used for modeling piles embedded in soil under vehicular impacts, such as the lumped parameter method, subgrade reaction approach, modified subgrade reaction method, and direct method.
- This study investigated the effect that soil domain sizes and boundary conditions had on the dynamic impact response of pile–soil systems using field-scale physical impact test data. This study should help engineers and researchers better understand the influence of soil domain sizes and boundary conditions on the dynamic response of piles embedded in granular soil when subjected to lateral vehicular impacts. Furthermore, guidelines and recommendations were provided on optimum soil domain size and boundary conditions.
- Computational time studies were conducted to assess the efficiency of the various soil domain sizes and boundary conditions. This investigation demonstrated the effect that soil domain sizes and boundary conditions had on the performance of LS-DYNA pile–soil impact simulations.
- The modeling method developed in this study can be used to enhance and advance the current pile–soil system modeling methods and be extended for future research, such as modeling full-scale, soil-embedded barrier and containment systems.
- This research work will significantly contribute to the numerical modeling techniques currently used by engineers and researchers in the analysis and design of piles subjected to vehicular impact loading. The findings of this study will facilitate efficient and economically feasible pile design by reducing the required number of component crash tests of pile–soil systems.
- The research presented herein has made substantive advances in soil modeling techniques for simulating dynamic impact interactions within pile–soil systems. The erosion method introduced in this paper has undergone rigorous validation at a component scale. It is apparent, however, that there remains a pressing need to extend this approach to encompass full-scale, soil-embedded roadside safety structures. This expansion is essential to facilitate an exploration of the dynamic interplay of factors—ranging from varied soil characteristics, pile embedment depths, and terrain conditions—on the structural response and resilience of soil-embedded roadside safety structures under vehicular impacts. The advancement of this research will provide valuable insights, with potential implications for enhancing safety infrastructure design and contributing to reducing vehicular impact-related fatalities and injuries.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Pajouh, M.A.; Schmidt, J.; Bielenberg, R.W.; Reid, J.D.; Faller, R.K. Simplified Soil-Pile Interaction Modeling under Impact Loading. In Geotechnical Earthquake Engineering and Soil Dynamics V; American Society of Civil Engineers: Reston, VA, USA, 2018; pp. 269–280. [Google Scholar]
- Schmidt, J.; Reid, J.; Stolle, C.; Faller, R.; Bielenberg, R.; Asselin, N.; Rilett, L. Analysis supporting development of a new, non-proprietary ASTM F2656-15 M30 barrier. In Final Report to the Surface Deployment and Distribution Command Transportation Engineering Agency; Midwest Roadside Safety Facility, University of Nebraska-Lincoln: Lincoln, NE, USA, 2017. [Google Scholar]
- Plaxico, C.A.; Patzner, G.S.; Ray, M.H. Finite-element modeling of guardrail timber posts and the post-soil interaction. Transp. Res. Rec.
**1998**, 1647, 139–146. [Google Scholar] [CrossRef] - Patzner, G.S.; Plaxico, C.A.; Ray, M.H. Effects of post and soil strength on performance of modified eccentric loader breakaway cable terminal. Transp. Res. Rec.
**1999**, 1690, 78–83. [Google Scholar] [CrossRef] - Sassi, A. Analysis of W-beam guardrail systems subjected to lateral impact. In Department of Civil and Environmental Engineering; University of Windsor (Canada): Windsor, ON, Canada, 2011. [Google Scholar]
- Sassi, A.; Ghrib, F. Development of finite element model for the analysis of a guardrail post subjected to dynamic lateral loading. Int. J. Crashworthiness
**2014**, 19, 457–468. [Google Scholar] [CrossRef] - Tabiei, A.; Wu, J. Roadmap for crashworthiness finite element simulation of roadside safety structures. Finite Elem. Anal. Des.
**2000**, 34, 145–157. [Google Scholar] [CrossRef] - Wu, W.; Thomson, R. A study of the interaction between a guardrail post and soil during quasi-static and dynamic loading. Int. J. Impact Eng.
**2007**, 34, 883–898. [Google Scholar] [CrossRef] - Opiela, K.; Kan, S.; Marzougui, D. Development of a Finite Element Model for W-Beam Guardrails; Report No. NCAC 2007-T-002; National Crash Analysis Center, George Washington University: Washington, DC, USA, 2007. [Google Scholar]
- Bligh, R.P.; Abu-Odeh, A.Y.; Hamilton, M.E.; Seckinger, N.R. Evaluation of roadside safety devices using finite element analysis. In Sponsored by the Texas Department of Transportation in Cooperation with the U.S. Department of Transportation Federal Highway Administration; Texas Transportation Institute, Texas A&M University: College Station, TX, USA, 2004. [Google Scholar]
- Marzougui, D.; Mahadevaiah, U.; Opiela, K.S. Development of a modified MGS design for test level 2 impact conditions using crash simulation. In Working Paper, NCAC 2010-W-005; National Crash Analysis Center: Ashburn, VA, USA, 2010. [Google Scholar]
- Hendricks, B.F.; Wekezer, J.W. Finite-element modeling of G2 guardrail. Transp. Res. Rec.
**1996**, 1528, 130–137. [Google Scholar] [CrossRef] - Whitworth, H.; Bendidi, R.; Marzougui, D.; Reiss, R. Finite element modeling of the crash performance of roadside barriers. Int. J. Crashworthiness
**2004**, 9, 35–43. [Google Scholar] [CrossRef] - Kulak, R.F.; Bojanowski, C. Modeling of cone penetration test using SPH and MM-ALE approaches. In Proceedings of the 8th European LS-DYNA Users Conference, Strasbourg, France, 23–24 May 2011. [Google Scholar]
- Kulak, R.F.; Schwer, L. Effect of soil material models on SPH simulations for soil-structure interaction. In Proceedings of the 12th International LS-DYNA Users Conference, Detroit, MI, USA, 3–5 June 2012. [Google Scholar]
- Ceccato, F.; Beuth, L.; Simonini, P. Adhesive contact algorithm for MPM and its application to the simulation of cone penetration in clay. Procedia Eng.
**2017**, 175, 182–188. [Google Scholar] [CrossRef] - Ortiz, D.; Gravish, N.; Tolley, M.T. Soft robot actuation strategies for locomotion in granular substrates. IEEE Robot. Autom. Lett.
**2019**, 4, 2630–2636. [Google Scholar] [CrossRef] - Butlanska, J.; Arroyo, M.; Gens, A.; O’Sullivan, C. Multi-scale analysis of cone penetration test (CPT) in a virtual calibration chamber. Can. Geotech. J.
**2014**, 51, 51–66. [Google Scholar] [CrossRef] - Evans, T.M.; Zhang, N. Three-dimensional simulations of plate anchor pullout in granular materials. Int. J. Geomech.
**2019**, 19, 04019004. [Google Scholar] [CrossRef] - Gens, A.; Arroyo, M.; Butlanska, J.; O’Sullivan, C. Discrete simulation of cone penetration in granular materials. In Advances in Computational Plasticity: A Book in Honour of D. Roger J. Owen; Springer: Berlin/Heidelberg, Germany, 2018; pp. 95–111. [Google Scholar]
- Khosravi, A.; Martinez, A.; DeJong, J. Discrete element model (DEM) simulations of cone penetration test (CPT) measurements and soil classification. Can. Geotech. J.
**2020**, 57, 1369–1387. [Google Scholar] [CrossRef] - Liang, W.; Zhao, J.; Wu, H.; Soga, K. Multiscale modeling of anchor pullout in sand. J. Geotech. Geoenvironmental Eng.
**2021**, 147, 04021091. [Google Scholar] [CrossRef] - Hallquist, J.O. LS-DYNA Theory Manual; 10–82–10–102; Livermore Software Technology Corporation: Livermore, CA, USA, 2014. [Google Scholar]
- Hallquist, J.O. LS-DYNA Keyword User’s Manual (r:13107); 10–82–10–102; Livermore Software Technology Corporation: Livermore, CA, USA, 2020. [Google Scholar]
- Beppu, M.; Miwa, K.; Itoh, M.; Katayama, M.; Ohno, T. Damage evaluation of concrete plates by high-velocity impact. Int. J. Impact Eng.
**2008**, 35, 1419–1426. [Google Scholar] [CrossRef] - Nyström, U.; Gylltoft, K. Numerical studies of the combined effects of blast and fragment loading. Int. J. Impact Eng.
**2009**, 36, 995–1005. [Google Scholar] [CrossRef] - Riedel, W.; Kawai, N.; Kondo, K.-I. Numerical assessment for impact strength measurements in concrete materials. Int. J. Impact Eng.
**2009**, 36, 283–293. [Google Scholar] [CrossRef] - Tu, Z.; Lu, Y. Modifications of RHT material model for improved numerical simulation of dynamic response of concrete. Int. J. Impact Eng.
**2010**, 37, 1072–1082. [Google Scholar] [CrossRef] - Tu, Z.; Lu, Y. Evaluation of typical concrete material models used in hydrocodes for high dynamic response simulations. Int. J. Impact Eng.
**2009**, 36, 132–146. [Google Scholar] [CrossRef] - Farnam, Y.; Mohammadi, S.; Shekarchi, M. Experimental and numerical investigations of low velocity impact behavior of high-performance fiber-reinforced cement based composite. Int. J. Impact Eng.
**2010**, 37, 220–229. [Google Scholar] [CrossRef] - Teng, T.-L.; Chu, Y.-A.; Chang, F.-A.; Shen, B.-C.; Cheng, D.-S. Development and validation of numerical model of steel fiber reinforced concrete for high-velocity impact. Comput. Mater. Sci.
**2008**, 42, 90–99. [Google Scholar] [CrossRef] - Wang, Z.; Konietzky, H.; Huang, R. Elastic–plastic-hydrodynamic analysis of crater blasting in steel fiber reinforced concrete. Theor. Appl. Fract. Mech.
**2009**, 52, 111–116. [Google Scholar] [CrossRef] - Zhou, X.; Hao, H. Mesoscale modelling and analysis of damage and fragmentation of concrete slab under contact detonation. Int. J. Impact Eng.
**2009**, 36, 1315–1326. [Google Scholar] [CrossRef] - Coughlin, A.; Musselman, E.; Schokker, A.J.; Linzell, D. Behavior of portable fiber reinforced concrete vehicle barriers subject to blasts from contact charges. Int. J. Impact Eng.
**2010**, 37, 521–529. [Google Scholar] [CrossRef] - Luccioni, B.M.; Aráoz, G.F.; Labanda, N.A. Defining erosion limit for concrete. Int. J. Prot. Struct.
**2013**, 4, 315–340. [Google Scholar] [CrossRef] - Yosef, T.Y. Development of advanced computational methodologies and guidelines for modeling impact dynamics of post-granular soil systems. In Department of Civil and Environmental Engineering; University of Nebraska-Lincoln: Lincoln, NE, USA, 2021. [Google Scholar]
- Ross, H.; Sicking, D.; Zimmer, R.; Michie, J. Recommended procedures for the safety performance evaluation of highway features. In National Cooperative Highway Research Program (NCHRP) Report 350; Transportation Research Board: Washington, DC, USA, 2009. [Google Scholar]
- Saleh, M.; Edwards, L. Evaluation of soil and fluid structure interaction in blast modelling of the flying plate test. Comput. Struct.
**2015**, 151, 96–114. [Google Scholar] [CrossRef] - Busch CLAimone-Martin, C.T.; Tarefder, R.A. Experimental evaluation and finite-element simulations of explosive airblast tests on clay soils. Int. J. Geomech.
**2016**, 16, 04015097. [Google Scholar] [CrossRef] - Tagar, A.; Changying, J.; Adamowski, J.; Malard, J.; Qi, C.S.; Qishuo, D.; Abbasi, N. Finite element simulation of soil failure patterns under soil bin and field testing conditions. Soil Tillage Res.
**2015**, 145, 157–170. [Google Scholar] [CrossRef] - Linforth, S.; Tran, P.; Rupasinghe, M.; Nguyen, N.; Ngo, T.; Saleh, M.; Odish, R.; Shanmugam, D. Unsaturated soil blast: Flying plate experiment and numerical investigations. Int. J. Impact Eng.
**2019**, 125, 212–228. [Google Scholar] [CrossRef] - Symonds, P. Survey of methods of analysis for plastic deformation of structures under dynamic loading. In Division Engineering Report BU/NSRDC/1–67; Brown University: Providence, RI, USA, 1967. [Google Scholar]
- Schmidt, J.; Mongiardini, M.; Bielenberg, R.L.K.; Reid, J.; Faller, R. Dynamic testing of MGS W6x8.5 posts at decreased embedment. In Final Report to Nebraska Department of Roads, Transportation Research Report No. TRP-03-271-12; Midwest Roadside Safety Facility, University of Nebraska-Lincoln: Lincoln, NE, USA, 2012. [Google Scholar]
- Schrum, K.; Sicking, D.; Faller, R.; Reid, J. Predicting the Dynamic Fracture of Steel via a Non- Local Strain Energy Density Failure Criterion. In Final Report to Federal Highway Administration, MwRSF Research Report No. TRP-03-311-14; Midwest Roadside Safety Facility, University of Nebraska-Lincoln: Lincoln, NE, USA, 2014. [Google Scholar]
- Deladi, E.L. Static friction in rubber-metal contacts with application to rubber pad forming processes. In Department of Civil and Environmental Engineering; University of Twente: Enschede, The Netherlands, 2006. [Google Scholar]
- Yoshimi, Y.; Kishida, T. A ring torsion apparatus for evaluating friction between soil and metal surfaces. Geotech. Test. J.
**1981**, 4, 145–152. [Google Scholar] [CrossRef] - Uesugi, M.; Kishida, H. Frictional resistance at yield between dry sand and mild steel. Soils Found.
**1986**, 26, 139–149. [Google Scholar] [CrossRef] - Homan, D.; Thiele, J.; Faller, R.; Rosenbaugh, S.; Rohde, J.; Arens, S.; Lechtenberg, K.; Sicking, D.; Reid, J. Investigation and dynamic testing of wood and steel posts for MGS on a wire-faced mse wall. In Final Report to the Federal Highway Administration, Transportation Research; Midwest Roadside Safety Facility, University of Nebraska-Lincoln: Lincoln, NE, USA, 2012. [Google Scholar]
- Mongiardini, M.; Ray, M.; Plaxico, C.; Anghileri, M. Procedures for verification and validation of computer simulations used for roadside safety applications. In Final Report to the National Cooperative Highway Research Program, NCHRP Report No. W179, Project No. 22-24; Worcester Polytechnic Institute: Worcester, MA, USA, 2010. [Google Scholar]
- Meyer, D.; Ammon, T.; Bielenberg, R.; Stolle, C.; Holloway, C.; Faller, R. Quasi-static tensile and dynamic impact testing of guardrail components. In Draft Report to the U.S. Army Surface Deployment and Distribution Command Traffic Engineering Agency, Transportation Research Report No. TRP-03-350-17; Midwest Roadside Safety Facility, University of Nebraska-Lincoln: Lincoln, NE, USA, 2017. [Google Scholar]
- Reese, L.; Qiu, T.; Linzell, D.; O’hare, E.; Rado, Z. Field tests and numerical modeling of vehicle impacts on a boulder embedded in compacted fill. Int. J. Prot. Struct.
**2014**, 5, 435–451. [Google Scholar] [CrossRef] - Lim, S.G. Development of design guidelines for soil embedded post systems using wide-flange I-beams to contain truck impact. In Department of Civil and Environmental Engineering; Texas A&M University: College Station, TX, USA, 2011. [Google Scholar]
- Mirdamadi, A. Deterministic and probabilistic simple model for single pile behavior under lateral truck impact. In Department of Civil and Environmental Engineering; Texas A&M University: College Station, TX, USA, 2014. [Google Scholar]

**Figure 2.**Computational model geometry, set-up, and initial conditions of a laterally impacted W152 × 12.6 steel pile embedded in granular (MASH strong) soil.

**Figure 3.**(

**a**) Force vs. displacement and (

**b**) energy vs. displacement plots from simulation using erosion method and physical impact tests (i.e., test nos. MH-1 and MH-4).

**Figure 4.**Post-impact photographs of buckled W152 × 12.6 steel pile, physical impact test, and simulation using erosion method.

**Figure 5.**Comparison of time-sequential images derived from physical impact test and simulation using erosion method for 1830-mm long, W152 × 12.6 steel pile embedded in granular (MASH strong) soil. Note that 1.461 × 10

^{3}on the scale indicates 1.461 × 10

^{3}.

**Figure 6.**Von Mises stress distribution within granular (MASH strong) soil in laterally impacted “flexible” or “long” I-shaped W152 × 12.6 steel pile embedded in granular (MASH strong) soil. Note that 3 × 10

^{−3}on the scale refers to 3 × 10

^{−3}.

**Figure 7.**Comparison of: (

**a**) force vs. displacement and (

**b**) energy vs. displacement plots from simulation using erosion method and physical impact tests (i.e., test nos. BL-8 and BL-20).

**Figure 8.**Time-sequential images of dynamic impact test (test no. BL-8) and simulated test using the erosion method for an 1830-mm long W152 × 23.6 steel pile embedded in granular (MASH strong) soil. Note that 1.544 × 10

^{−3}on the scale indicates 1.544 × 10

^{−3}.

**Figure 9.**Von Mises stress distribution within granular (MASH strong) soil in laterally impacted rigid W152 × 23.6 steel pile embedded in granular (MASH strong) soil. Note that 4.000 × 10

^{−3}on the scale refers to 4.000 × 10

^{−3}.

**Figure 10.**Baseline mesh-size model: (

**a**) initial conditions, model set-up, and geometry; and (

**b**) soil mesh pattern in X–Y plane.

**Figure 11.**Uniform mesh model: (

**a**) model set-up, geometry, and initial conditions; and (

**b**) soil mesh pattern in X–Y plane.

**Figure 12.**(

**a**) Average force vs. displacement comparison between baseline and uniform mesh-size simulations and impact test data (i.e., test no. P3G-7); and (

**b**) average force percentage difference between the baseline and uniform mesh-size simulation and physical impact test (i.e., test no. P3G-7) for 125 mm through 500 mm pile displacements.

**Figure 13.**Computational model geometry, set-up, and initial conditions of laterally impacted, 2134-mm long, ASTM A500 Grade B steel tube pile embedded in 1.5 d × 1.5 d × 1.5 d granular (MASH strong) soil domain (Note: figure not drawn to scale).

**Figure 14.**Force vs. displacement and energy vs. displacement comparisons between physical impact test data (i.e., test no. P3G-7) and simulation with (

**a**) 1.5 d × 1.5 d × 1.5 d; (

**b**) 2 d × 2 d × 1.5 d; and (

**c**) 2 d × 2 d × 2 d soil domain sizes.

**Figure 15.**Force vs. displacement and energy vs. displacement comparisons between physical impact test data (i.e., test no. P3G-7) and simulation with (

**a**) 3 d × 3 d × 2 d; (

**b**) 3 d × 3 d × 3 d; and (

**c**) 4 d × 4 d × 3 d soil domain sizes.

**Figure 16.**Computation time as a function of number of CPUs for the six soil domain sizes for (

**a**) BNR boundary condition and (

**b**) SPC boundary condition.

**Figure 17.**Relationship between computation time and number of soil elements at 8, 16, 32, 64, and 124 CPUs for (

**a**) BNR boundary condition and (

**b**) SPC boundary condition.

**Table 1.**Fully calibrated granular (MASH strong) soil input parameters [36].

Item | Soil Parameter | Unit | Value |
---|---|---|---|

Soil characteristic parameters | Specific gravity, G_{S} | [-] | 2.65 |

Moisture content, w | [%] | 3.4 | |

Density of soil, ${\rho}_{soil}$ | [kg/mm^{3}] | 1.9 × 10^{−6} | |

Elasticity parameters | Shear modulus, G | [MPa] | 12.0 |

Bulk modulus, K | [MPa] | 20.0 | |

Plasticity parameters | Peak friction angle, ${\varphi}_{peak}$ | [Degrees] | 45.0 |

Cohesion, c | [kPa] | 5.0 | |

Modified MC surface coefficient, a | [kPa] | 3.7 | |

Eccentricity parameter, e | [-] | 0.7 | |

Viscoplasticity parameters | Viscoplasticity parameter, $\gamma $ | [-] | 1.0 × 10^{−3} |

Viscoplasticity parameter, n | [-] | 2.0 | |

Strain softening parameters | Volumetric strain at initial damage threshold, ${\xi}_{0}$ | [-] | 1.0 × 10^{−5} |

Void formation energy, G_{f} | [kN/mm] | 6.0 × 10^{−8} | |

Residual friction angle, ${\varphi}_{res}$ | [Degrees] | 15 |

**Table 2.**Material input parameters for a W152 × 12.6 ASTM A36 steel pile [43].

Input Parameters | Value | |||||||
---|---|---|---|---|---|---|---|---|

Density (kg/mm^{3}) | 7.86 × 10^{−6} | |||||||

Elastic modulus (GPa) | 200 | |||||||

Poisson’s ratio | 0.30 | |||||||

Effective plastic strain | ep1 | ep2 | ep3 | ep4 | ep5 | ep6 | ep7 | ep8 |

0.000 | 0.0243 | 0.0303 | 0.0368 | 0.0776 | 0.1425 | 0.1794 | 0.9050 | |

Effective stress (GPa) | es1 | es2 | es3 | es4 | es5 | es6 | es7 | es8 |

0.370 | 0.3701 | 0.4050 | 0.4236 | 0.5026 | 0.5638 | 0.5858 | 0.8731 |

**Table 3.**Material input parameters for a W152 × 23.6 ASTM A992 steel pile [44].

Input Parameters | Value | |||||||
---|---|---|---|---|---|---|---|---|

Density (kg/mm^{3}) | 7.86 × 10^{−6} | |||||||

Young’s modulus (GPa) | 200 | |||||||

Poisson’s ratio | 0.30 | |||||||

Effective plastic strain | ep1 | ep2 | ep3 | ep4 | ep5 | ep6 | ep7 | ep8 |

0.000 | 0.0160 | 0.0470 | 0.0890 | 0.1170 | 0.1410 | 0.1850 | 2.0000 | |

Effective stress (GPa) | es1 | es2 | es3 | es4 | es5 | es6 | es7 | es8 |

0.439 | 0.4730 | 0.5200 | 0.5610 | 0.5860 | 0.6010 | 0.6210 | 1.8000 |

**Table 4.**Average force comparison at pile displacements of 125 mm, 250 mm, 375 mm, and 500 mm between simulation and test nos. MH-1 and MH-4.

Item | Average Force (kN) | |||
---|---|---|---|---|

at 125 mm | at 250 mm | at 375 mm | at 500 mm | |

Test No. MH-1 | 43.19 | 43.56 | 42.55 | 39.87 |

Test No. MH-4 | 41.99 | 42.79 | 42.49 | 39.66 |

Test Average | 42.59 | 43.17 | 42.52 | 39.77 |

Simulation Test No. MH-1: Erosion Method | 45.30 | 50.78 | 45.68 | 39.70 |

% Difference: Simulation vs. Test No. MH-1 | 4.8% | 15.3% | 7.1% | 0.4% |

% Difference: Simulation vs. Test Average | 6.2% | 16.2% | 7.2% | 0.2% |

**Table 5.**Average force comparison at pile displacements of 125 mm, 250 mm, 375 mm, and 500 mm between simulation and test nos. BL-8 and BL-20.

Item | Average Force (kN) | |||
---|---|---|---|---|

at 125 mm | at 250 mm | at 375 mm | at 500 mm | |

Test No. BL-8 | 43.09 | 45.49 | 45.89 | 45.24 |

Test No. BL-20 | 47.16 | 46.03 | 43.68 | 40.47 |

Test Average | 45.13 | 45.76 | 44.78 | 42.85 |

Simulation Test No. BL-8: Erosion Method | 54.09 | 51.47 | 44.89 | 40.78 |

% Difference: Simulation vs. Test No. BL-8 | 22.6% | 12.3% | 2.2% | 10.4% |

% Difference: Simulation vs. Test Average | 18.1% | 11.8% | 0.3% | 5.0% |

**Table 6.**Average force percentage difference between simulation with BNR boundary condition and impact test data (test no. P3G-7) for six soil domain sizes.

Soil Domain Size | Average Force Percentage Difference between Simulation with BNR and Impact Test Data (Test No. P3G-7) [%] | |||
---|---|---|---|---|

At 125 mm | At 250 mm | At 375 mm | At 500 mm | |

1.5 d × 1.5 d × 1.5 d | 1.5% | 11.8% | 10.6% | 9.2% |

2 d × 2 d × 1.5 d | 3.2% | 10.5% | 10.2% | 9.4% |

2 d × 2 d × 2 d | 2.2% | 12.6% | 11.2% | 7.3% |

3 d × 3 d × 2 d | 5.1% | 4.1% | 6.2% | 4.7% |

3 d × 3 d × 3 d | 6.8% | 3.1% | 4.2% | 3.2% |

4 d × 4 d × 3 d | 9.6% | 2.3% | 4.6% | 3.2% |

**Table 7.**Average force percentage difference between simulation with SPC boundary condition of the soil domain and impact test data (test no. P3G-7) for six soil domain sizes.

Soil Domain Size | Average Force Percentage Difference between Simulation with BNR and Impact Test Data (Test No. P3G-7) [%] | |||
---|---|---|---|---|

At 125 mm | At 250 mm | At 375 mm | At 500 mm | |

1.5 d × 1.5 d × 1.5 d | 0.8% | 17.2% | 15.2% | 11.3% |

2 d × 2 d × 1.5 d | 0.3% | 17.8% | 15.1% | 12.3% |

2 d × 2 d × 2 d | 0.6% | 15.6% | 15.2% | 8.4% |

3 d × 3 d × 2 d | 3.4% | 8.5% | 9.6% | 8.1% |

3 d × 3 d × 3 d | 4.3% | 5.1% | 5.2% | 6.3% |

4 d × 4 d × 3 d | 9.6% | 2.3% | 4.6% | 3.2% |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Yosef, T.Y.; Faller, R.K.; Fang, C.; Kim, S.
Modeling Dynamics of Laterally Impacted Piles in Gravel Using Erosion Method. *Geotechnics* **2023**, *3*, 1251-1278.
https://doi.org/10.3390/geotechnics3040068

**AMA Style**

Yosef TY, Faller RK, Fang C, Kim S.
Modeling Dynamics of Laterally Impacted Piles in Gravel Using Erosion Method. *Geotechnics*. 2023; 3(4):1251-1278.
https://doi.org/10.3390/geotechnics3040068

**Chicago/Turabian Style**

Yosef, Tewodros Y., Ronald K. Faller, Chen Fang, and Seunghee Kim.
2023. "Modeling Dynamics of Laterally Impacted Piles in Gravel Using Erosion Method" *Geotechnics* 3, no. 4: 1251-1278.
https://doi.org/10.3390/geotechnics3040068