# Numerical Analysis of Horizontal Impact Resistance of Rock Socketed Pile with Steel Casing

Authors: Xiaolong Zhang; Lunliang Duan; Duoyin Wang
Journal: Advances in Civil Engineering
 (2022)

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## Abstract

Based on a Wharf project in Chongqing, this paper carries out experimental research and nonlinear finite element analysis on the impact resistance of steel casing rock socketed pile of inland river frame Wharf under horizontal impact load. The results show that the impact force and its growth rate, the bending moment of concrete pile core, and the maximum lateral displacement of pile top increase significantly with the increase of impact energy, and the linear increase of the maximum displacement of pile top is better. Under the same impact energy, with the increase of impact mass, the maximum impact force decreases, the maximum bending moment of concrete pile core and its vibration center value (platform value) increase, and the maximum displacement of pile top and its vibration center value (platform value) increase. Based on the impact massm threshold, the criterion of different dynamic response forms of pile body is proposed. The impact mass threshold corresponds to the development of plastic hinge at the root of pile body. Through the parametric analysis method, it is obtained that the impact mass, impact velocity, and the maximum displacement of pile top meet the parabolic spherical relationship.


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## Body

## 1. Introduction

The building of the Three Gorges Dam has effectively promoted the development of inland river shipping and enabled the improved utilization of inland river shipping resources. However, at the same time, the improvement of the construction water level in the reservoir area has introduced new challenges to the construction of inland river terminals and addressed the construction difficulties resulting from negative factors such as the large water level difference, large flow velocity, and complex geological conditions faced by the construction of terminals in the upper reaches of the Yangtze River; therefore, Steel casing rock socketed pile structure is introduced [1]. The structure is composed of a steel casing and reinforced concrete pile core, and its mechanical properties are determined by the material constitutive relationship and structural relationship of each component. The load distribution mechanism and dynamic response of a pile body under a horizontal impact load are extremely complex, and there have been few studies to investigate this topic.Under the circumferential constraint of the steel pipe, the radial deformation of the concrete pile core is restrained, its strength and toughness are improved, and the local buckling of the steel pipe is restrained, enabling the performance of the steel to be maximized [2]. Under an impact load, the cooperative working mechanisms of the steel pipe and core concrete are particularly complex. This study focuses on the following two perspectives: the dynamic response of concrete-filled steel tubular members under a drop hammer impact load and the dynamic response of concrete-filled steel tubular members under a horizontal impact load.


### 1.1. Dynamic Response of Concrete-Filled Steel Tubular Members under a Drop Hammer Impact Load

Concrete filled steel tubular structures are generally used to bear vertical and transverse loads [3]; however, to facilitate research, scholars position concrete-filled steel tubular members horizontally; under different constraint conditions at both ends (two ends are consolidated, the ends are simply supported, and one end supports one end), from the section shape of the members (solid/hollow, round/square), the structure comprises a double hollow steel pipe, fillers (ultra-light cement composite, high-strength concrete, rigid polyurethane foam, recycled concrete), and the concrete is embedded with a steel structure (I-steel, duplex steel, round pipe), steel pipe composite members, etc.Zhu et al. [4] found that, for general concrete-filled steel tubular members, concrete columns wrapped with steel pipes can significantly improve the transverse impact resistance of members, and the deformation after impact increases significantly with an increase in the axial compression ratio. Yousuf et al. [5] found that concrete-filled square steel tubular members inhibited local buckling and improved impact resistance. Subsequently, Deng et al. [6] found that the failure of steel pipes is usually a tensile perimeter or fracture, and the concrete pile core in the impact area is usually broken under compression and cracked under tension. Wang [7] proposed that the impact speed is the most influential factor on the degree of damage to members and proposed that the residual bearing capacity should be employed as the standard for the evaluation of member damage. Hou et al. [8] reported that, with an increase in impact energy, the duration of the impact force increases, and the increase in steel content and steel yield strength can significantly improve the impact resistance, while the change in concrete strength has little effect. Based on energy and local deformation, Ci [9] obtained a correction calculation formula for the impact point deflection by performing a statistical analysis of test data. Wang et al. [10] found that when the constraint factor is 1.23, the component toughness is good, and the impact force has a platform stage; when the constraint factor is 0.44, the member exhibits a brittle mechanism, and the impact force platform stage disappears. Kang et al. [11] used a mass spring damper to simulate the dynamic contact behavior between the impact member and the concrete-filled steel tubular column, and they used a fiber-based nonlinear beam column element to simulate the behavior of concrete-filled steel tubular columns under impact loads. Then, they verified the accuracy of the simulation method.Scholars have also studied the influence of different fillers in steel pipes on their dynamic response. Wang et al. [12] studied the transverse impact behavior of an ultra-light cement composite- (ULCC-) filled steel pipe structure. The cement composite pile core effectively limited the indentation in the local area near the impact point and the deformation of the steel pipe. Based on the local indentation stage, indentation expansion stage, and load redistribution stage of the composite cement, there is the P–δ relationship. Yang et al. [13] found that filling high-strength concrete in square steel tubular has limited improvement on its impact resistance. Han et al. [14] studied high-strength concrete-filled steel tubular column members and found that the performance of members is completely different from that under static load owing to the strain rate effect of materials and the influence of inertial force. Parameters that are affected include the force state, internal force distribution, and bending capacity. The yield stress, steel content, section diameter, and impact velocity of steel are the primary parameters that affect the dynamic factors of the bending capacity. Remennikov et al. [15] and other studies analyzed steel tube filled rigid polyurethane foam (RPF) steel tube components. It was found that RPF could enhance the energy absorption capacity of square steel tubes, and its impact resistance and energy absorption were smaller than those of concrete pipes, which are larger than those of hollow steel tubes. The advantages of RPF relative to concrete are that foam filling only requires a very short time (2–3 min). Yang et al. [16] found that the impact resistance of square recycled concrete-filled steel tubular (RACFST) members is almost the same as that of ordinary concrete-filled steel tubular, and its plastic deformation is mainly concentrated at the impact position. However, at the same impact height, the impact of the axial compression load ratio on the performance of recycled aggregate concrete-filled steel tubular members is different from that of ordinary concrete-filled steel tubular members.The dynamic response of hollow Sandwich concrete-filled steel tubular (CFDST) members has also become the focus of research. Yang et al. [17] found that, for hollow double-layer concrete-filled steel tubular columns, increasing the steel ratio and reducing the slenderness ratio can improve the transverse stiffness and transverse impact resistance of the samples. Agahdmy et al. [18] proposed that the outer diameter thickness ratio, slenderness ratio, and hollow ratio have the greatest influence on the impact response of CFDST columns. Wang et al. [19] found that, compared with a hollow double-layer steel pipe, owing to the interaction between a Sandwich concrete and double-layer steel pipe, hollow Sandwich concrete-filled steel pipe members absorb higher energy and exhibit less overall and local deformation. Meanwhile, Wang et al. [20] also found that when the void ratio is less than 0.7, the dynamic response can be divided into three stages, namely, the peak, platform, and unloading stages; when the void ratio exceeds 0.7, the dynamic response will only go through the platform stage and unloading stage; when the hollow ratio reaches 0.8, the impact resistance of CFDST members decreases sharply. As the constraint factor ξ changes from 0.17 to 3.1, the dynamic increase factor (DIF) increased from 0.9 to 1.1. Wang et al. [21] reported that the hollow ratio increases, and the member changes from overall deformation to local deformation. When the hollow ratio was greater than 0.6, the brittle failure probability of the member increased significantly, and when the hollow ratio was greater than 0.729, the impact resistance of the member decreased significantly. Shi et al. [22, 23] found that the hollow ratio increases, the peak value of impact force decreases obviously, the impact duration shortens, the axial compression ratio increases, the second-order effect is more obvious, and the impact resistance decreases. When the hollow ratio is the same, the impact resistance of concrete-filled square hollow steel tubular members with outer circles is greater than that of double-layer circular steel tubular members. At the same time, for concrete-filled square circular steel tubular members, the impact height, boundary constraints, and axial compression ratio are important factors that affect the dynamic response of specimens. Increasing the impact height will increase the local depression of the members, and increasing the axial compression ratio will increase the disturbance and accelerate the impact process.With respect to the dynamic response of concrete-filled steel tubular members with different steel structures embedded in concrete, Zhang et al. [24] studied concrete-filled steel tubular structures with different sections. Appropriately reducing the core concrete within a certain range and replacing it with a hollow steel tube can improve the impact resistance. When the amounts of concrete and steel are the same, the impact resistance of solid concrete-filled steel tubular, embedded double H-shaped concrete-filled steel tubular, and hollow concrete-filled steel tubular decreases in turn. Xian et al. [25] found that the energy absorption and bending deformation resistance of duplex-shaped section steel embedded in concrete-filled steel tubular were slightly higher than those of I-shaped steel. Shi et al. [26] found that after the concrete-filled steel tubular members embedded with steel tubes are subjected to a transverse impact load, the outer steel tube is the main energy-consuming part, and the inner steel tube consumes less energy. The yield strength and steel content of the outer steel tube significantly affect the impact resistance of the members, and the change in concrete strength has little impact.With respect to the dynamic response of concrete-filled steel tubular composite members, Hu et al. [27, 28] studied CFST composite members and found that the increase in impact energy has a significant impact on the peak value of the impact force, the peak value of the mid-span disturbance, and the increase in impact duration, but it has no significant impact on the platform value of the impact force. The plastic deformation of the steel tube appeared in the middle and both ends of the span, and the crack perpendicular to the axial direction appears in the core concrete in the middle of the span, increasing the strength of concrete outside the pipe, the yield strength of the steel pipe, the ratio of longitudinal reinforcement, and the steel content of the concrete-filled steel pipe. Meanwhile, the geometric size ratio of the concrete-filled steel pipe parts and reinforced concrete parts has an obvious impact on the impact resistance of members, but the change in core concrete strength has no obvious impact. Guan et al. [29] studied the impact resistance of steel pipes strengthened with carbon fiber-reinforced polymer (CFRP) laminate outside the steel pipe. The CFRP bonding area increased from one-third to two-thirds, which had a considerable impact on the stiffness and total displacement of the column. However, regardless of the L/D ratio, the impact of full CFRP wrapping was almost the same as that of two-thirds CFRP wrapping.


### 1.2. Dynamic Response of Concrete-Filled Steel Tubular Piles under Horizontal Impact Load

In practical engineering applications, concrete-filled steel tubular piles or columns are mostly vertical components. To determine the dynamic response law of concrete-filled steel tubular structures after transverse impact loads in practical engineering applications, it is necessary to study the vertical concrete-filled steel tubular components under different constraints at the upper and lower ends. However, there is a lack of relevant research at home and abroad, and only vertical concrete-filled steel tubular piles and columns have been studied as separate components.The main research contents are as follows. Wang et al. [30–32] added a neck rib at the bottom of the steel pipe and poured a concrete foundation to form a fixed support at the bottom. Then, a horizontal impact load test was carried out under the constraints of boundary conditions such as a free or hinged upper part. Compared with the simple support at the upper end, when the upper end is a cantilever, the deformation and impact force platform values in the middle of the pile body are smaller, and the impact force peak value and residual displacement are larger. The impact velocity has a significant influence on the failure mode of concrete-filled steel tubular piles, and the impact quality has a significant influence on the duration of the impact force and residual displacement. Zhou et al. [33] carried out a comparative analysis and research on hollow and partially filled concrete rectangular steel pipe piles under fixed support at the lower end and simple support at the upper end, and they found that a concrete-filled steel pipe can significantly improve the local buckling in the impact load area, and the local buckling of steel pipe and the crushing of filled concrete often occur in areas with a relatively weak interaction and relatively large impact load. With the increase of the concrete filling height, there is a decreased influence of the impact energy on the typical displacement and strain response and overall bending deformation. Fan and Ren [34] conducted research on a concrete-filled steel tubular column with a hollow interlayer in the outer and inner circles under a horizontal impact load, and they found that the impact side of the member has local bulging, the other side has bending deformation, the local deformation of the cantilever column is small, and the energy absorption of the member is stronger when an axial force is applied to the cantilever section.


## 1.1. Dynamic Response of Concrete-Filled Steel Tubular Members under a Drop Hammer Impact Load

Concrete filled steel tubular structures are generally used to bear vertical and transverse loads [3]; however, to facilitate research, scholars position concrete-filled steel tubular members horizontally; under different constraint conditions at both ends (two ends are consolidated, the ends are simply supported, and one end supports one end), from the section shape of the members (solid/hollow, round/square), the structure comprises a double hollow steel pipe, fillers (ultra-light cement composite, high-strength concrete, rigid polyurethane foam, recycled concrete), and the concrete is embedded with a steel structure (I-steel, duplex steel, round pipe), steel pipe composite members, etc.Zhu et al. [4] found that, for general concrete-filled steel tubular members, concrete columns wrapped with steel pipes can significantly improve the transverse impact resistance of members, and the deformation after impact increases significantly with an increase in the axial compression ratio. Yousuf et al. [5] found that concrete-filled square steel tubular members inhibited local buckling and improved impact resistance. Subsequently, Deng et al. [6] found that the failure of steel pipes is usually a tensile perimeter or fracture, and the concrete pile core in the impact area is usually broken under compression and cracked under tension. Wang [7] proposed that the impact speed is the most influential factor on the degree of damage to members and proposed that the residual bearing capacity should be employed as the standard for the evaluation of member damage. Hou et al. [8] reported that, with an increase in impact energy, the duration of the impact force increases, and the increase in steel content and steel yield strength can significantly improve the impact resistance, while the change in concrete strength has little effect. Based on energy and local deformation, Ci [9] obtained a correction calculation formula for the impact point deflection by performing a statistical analysis of test data. Wang et al. [10] found that when the constraint factor is 1.23, the component toughness is good, and the impact force has a platform stage; when the constraint factor is 0.44, the member exhibits a brittle mechanism, and the impact force platform stage disappears. Kang et al. [11] used a mass spring damper to simulate the dynamic contact behavior between the impact member and the concrete-filled steel tubular column, and they used a fiber-based nonlinear beam column element to simulate the behavior of concrete-filled steel tubular columns under impact loads. Then, they verified the accuracy of the simulation method.Scholars have also studied the influence of different fillers in steel pipes on their dynamic response. Wang et al. [12] studied the transverse impact behavior of an ultra-light cement composite- (ULCC-) filled steel pipe structure. The cement composite pile core effectively limited the indentation in the local area near the impact point and the deformation of the steel pipe. Based on the local indentation stage, indentation expansion stage, and load redistribution stage of the composite cement, there is the P–δ relationship. Yang et al. [13] found that filling high-strength concrete in square steel tubular has limited improvement on its impact resistance. Han et al. [14] studied high-strength concrete-filled steel tubular column members and found that the performance of members is completely different from that under static load owing to the strain rate effect of materials and the influence of inertial force. Parameters that are affected include the force state, internal force distribution, and bending capacity. The yield stress, steel content, section diameter, and impact velocity of steel are the primary parameters that affect the dynamic factors of the bending capacity. Remennikov et al. [15] and other studies analyzed steel tube filled rigid polyurethane foam (RPF) steel tube components. It was found that RPF could enhance the energy absorption capacity of square steel tubes, and its impact resistance and energy absorption were smaller than those of concrete pipes, which are larger than those of hollow steel tubes. The advantages of RPF relative to concrete are that foam filling only requires a very short time (2–3 min). Yang et al. [16] found that the impact resistance of square recycled concrete-filled steel tubular (RACFST) members is almost the same as that of ordinary concrete-filled steel tubular, and its plastic deformation is mainly concentrated at the impact position. However, at the same impact height, the impact of the axial compression load ratio on the performance of recycled aggregate concrete-filled steel tubular members is different from that of ordinary concrete-filled steel tubular members.The dynamic response of hollow Sandwich concrete-filled steel tubular (CFDST) members has also become the focus of research. Yang et al. [17] found that, for hollow double-layer concrete-filled steel tubular columns, increasing the steel ratio and reducing the slenderness ratio can improve the transverse stiffness and transverse impact resistance of the samples. Agahdmy et al. [18] proposed that the outer diameter thickness ratio, slenderness ratio, and hollow ratio have the greatest influence on the impact response of CFDST columns. Wang et al. [19] found that, compared with a hollow double-layer steel pipe, owing to the interaction between a Sandwich concrete and double-layer steel pipe, hollow Sandwich concrete-filled steel pipe members absorb higher energy and exhibit less overall and local deformation. Meanwhile, Wang et al. [20] also found that when the void ratio is less than 0.7, the dynamic response can be divided into three stages, namely, the peak, platform, and unloading stages; when the void ratio exceeds 0.7, the dynamic response will only go through the platform stage and unloading stage; when the hollow ratio reaches 0.8, the impact resistance of CFDST members decreases sharply. As the constraint factor ξ changes from 0.17 to 3.1, the dynamic increase factor (DIF) increased from 0.9 to 1.1. Wang et al. [21] reported that the hollow ratio increases, and the member changes from overall deformation to local deformation. When the hollow ratio was greater than 0.6, the brittle failure probability of the member increased significantly, and when the hollow ratio was greater than 0.729, the impact resistance of the member decreased significantly. Shi et al. [22, 23] found that the hollow ratio increases, the peak value of impact force decreases obviously, the impact duration shortens, the axial compression ratio increases, the second-order effect is more obvious, and the impact resistance decreases. When the hollow ratio is the same, the impact resistance of concrete-filled square hollow steel tubular members with outer circles is greater than that of double-layer circular steel tubular members. At the same time, for concrete-filled square circular steel tubular members, the impact height, boundary constraints, and axial compression ratio are important factors that affect the dynamic response of specimens. Increasing the impact height will increase the local depression of the members, and increasing the axial compression ratio will increase the disturbance and accelerate the impact process.With respect to the dynamic response of concrete-filled steel tubular members with different steel structures embedded in concrete, Zhang et al. [24] studied concrete-filled steel tubular structures with different sections. Appropriately reducing the core concrete within a certain range and replacing it with a hollow steel tube can improve the impact resistance. When the amounts of concrete and steel are the same, the impact resistance of solid concrete-filled steel tubular, embedded double H-shaped concrete-filled steel tubular, and hollow concrete-filled steel tubular decreases in turn. Xian et al. [25] found that the energy absorption and bending deformation resistance of duplex-shaped section steel embedded in concrete-filled steel tubular were slightly higher than those of I-shaped steel. Shi et al. [26] found that after the concrete-filled steel tubular members embedded with steel tubes are subjected to a transverse impact load, the outer steel tube is the main energy-consuming part, and the inner steel tube consumes less energy. The yield strength and steel content of the outer steel tube significantly affect the impact resistance of the members, and the change in concrete strength has little impact.With respect to the dynamic response of concrete-filled steel tubular composite members, Hu et al. [27, 28] studied CFST composite members and found that the increase in impact energy has a significant impact on the peak value of the impact force, the peak value of the mid-span disturbance, and the increase in impact duration, but it has no significant impact on the platform value of the impact force. The plastic deformation of the steel tube appeared in the middle and both ends of the span, and the crack perpendicular to the axial direction appears in the core concrete in the middle of the span, increasing the strength of concrete outside the pipe, the yield strength of the steel pipe, the ratio of longitudinal reinforcement, and the steel content of the concrete-filled steel pipe. Meanwhile, the geometric size ratio of the concrete-filled steel pipe parts and reinforced concrete parts has an obvious impact on the impact resistance of members, but the change in core concrete strength has no obvious impact. Guan et al. [29] studied the impact resistance of steel pipes strengthened with carbon fiber-reinforced polymer (CFRP) laminate outside the steel pipe. The CFRP bonding area increased from one-third to two-thirds, which had a considerable impact on the stiffness and total displacement of the column. However, regardless of the L/D ratio, the impact of full CFRP wrapping was almost the same as that of two-thirds CFRP wrapping.


## 1.2. Dynamic Response of Concrete-Filled Steel Tubular Piles under Horizontal Impact Load

In practical engineering applications, concrete-filled steel tubular piles or columns are mostly vertical components. To determine the dynamic response law of concrete-filled steel tubular structures after transverse impact loads in practical engineering applications, it is necessary to study the vertical concrete-filled steel tubular components under different constraints at the upper and lower ends. However, there is a lack of relevant research at home and abroad, and only vertical concrete-filled steel tubular piles and columns have been studied as separate components.The main research contents are as follows. Wang et al. [30–32] added a neck rib at the bottom of the steel pipe and poured a concrete foundation to form a fixed support at the bottom. Then, a horizontal impact load test was carried out under the constraints of boundary conditions such as a free or hinged upper part. Compared with the simple support at the upper end, when the upper end is a cantilever, the deformation and impact force platform values in the middle of the pile body are smaller, and the impact force peak value and residual displacement are larger. The impact velocity has a significant influence on the failure mode of concrete-filled steel tubular piles, and the impact quality has a significant influence on the duration of the impact force and residual displacement. Zhou et al. [33] carried out a comparative analysis and research on hollow and partially filled concrete rectangular steel pipe piles under fixed support at the lower end and simple support at the upper end, and they found that a concrete-filled steel pipe can significantly improve the local buckling in the impact load area, and the local buckling of steel pipe and the crushing of filled concrete often occur in areas with a relatively weak interaction and relatively large impact load. With the increase of the concrete filling height, there is a decreased influence of the impact energy on the typical displacement and strain response and overall bending deformation. Fan and Ren [34] conducted research on a concrete-filled steel tubular column with a hollow interlayer in the outer and inner circles under a horizontal impact load, and they found that the impact side of the member has local bulging, the other side has bending deformation, the local deformation of the cantilever column is small, and the energy absorption of the member is stronger when an axial force is applied to the cantilever section.


## 2. Numerical Modeling

### 2.1. Project Overview

Overview of the prototype piles: the test takes the second-phase rock-socketed cast-in-place pile of the Chongqing orchard port as the prototype. As shown in Figure1, the steel casing was made of Q235 steel, the thickness of the protective layer of the main reinforcement was 80 mm, and the concrete material grade was C30. The rock stratum at the pile bottom is mainly composed of sandstone and mudstone, where the natural compressive strength and saturated compressive strength of sandstone are 24.6 MPa and 18.9 MPa, respectively, and the natural compressive strength and saturated compressive strength of mudstone are 7.4 MPa and 4.3 MPa, respectively.Figure 1
Schematic diagram of model pile structure (unit: mm). (a) Structural elevation; (b) physical model.
(a)(b)


### 2.2. Test Boundary Conditions

The model test boundary conditions have a significant impact on the model stress transfer and accuracy of the test data. The key problems to be considered are as follows:Pile embedment depth. Depending on the code for the design and construction of a rock-socketed pile in port engineering [35], the rock socketed depth of the rock-socketed pile should meet the requirements of the bearing axial force and horizontal force simultaneously. The rock-socketed depth of the rock-socketed pile should not be less than 1.5 times that of the pile diameter in order to ensure that the rock-socketed end can be considered as consolidation in structural design. The test shows that when the rock-socketed depth of the rock-socketed pile is approximately three times the pile diameter, the axial bearing capacity is the best. The model mainly considers the dynamic response of the pile body under lateral action, so the rock-socketed depth is three times the pile diameter; that is, the embedded depth is 0.9 m.Plane dimensions of the foundation. The proposed foundation plane size is 0.7 m × 0.7 m × 1.5 m.


### 2.3. Numerical Model

#### 2.3.1. Geometric Model

The geometric model of a steel casing reinforced concrete rock-socketed pile was established using the nonlinear finite software ABAQUS. The model components mainly include the steel casing, reinforcement cage, concrete pile core, concrete foundation, and loading pendulum.Because the impact pressure sensor has less mass and inertia than the pendulum, it has a greater stiffness than the steel pile casing rock-socketed pile body, and the plastic deformation of the pressure sensor during impact is very small, the influence of the inertia of the pressure sensor is ignored during pendulum impact. In the numerical simulation, a hemispherical cylindrical steel body with simple geometry is used to replace the sensor and pendulum. The horizontal impact load in all calculation models below adopts the hemispherical cylinder to impact the pile body. The length of the latter half of the cylinder can be varied to adjust the mass of the impact body. The structure of the impact hemispherical cylinder is shown in Figure2.Figure 2
Mesh generation of model.Calculation of model dimensions. According to the physical model test size, the diameter of the head hemisphere of the pendulum was 0.1 m, and the rear of the pendulum was a cylinder with a diameter of 0.1 m and a height of 0.1 m (adjustable). The outer diameter of the steel casing was 0.3 m, the thickness was 0.002 m, and the length was 1.85 m. The diameter of the concrete pile core was 0.298 m, and the length was 2.6 m. The foundation size was 0.7 m × 0.7 m × 1.5 m.Table 1
Performance parameters of steel materials.
DensityElasticPoisson’s ratioUltimate strain7.8 × 10−92.1 × 10−50.30.2


#### 2.3.2. Material Properties

(1)
Concrete.Concrete includes C30 concrete for pile core and C15 concrete for foundation. The concrete material adopts the plastic damage model (CDP) in abaqus material library modified by Ye [36]. The stress-strain relationship of CDP model adopts the concrete constitutive relationship given in Appendix C of code for design of concrete structures [37] (hereinafter referred to as the code). The initial elastic modulus E0 of concrete is the secant modulus of concrete during tensile cracking, and Poisson’s ratio used in the calculation is 0.2. For the stress-strain curve in plastic stage, it is taken according to the concrete code [36].(2)
SteelSteel includes reinforcement and a steel casing. The binary line model was used to simulate the constitutive relationship of steel, and the yield strength and compressive strength [38] were obtained from the measured values in the tensile test. See Table 1 for further details.


#### 2.3.3. Constraints, Contact, and Boundary Conditions

Constraints. To simulate the foundation constraint boundary conditions in the actual test, fixed constraints are made on the six degrees of freedom directions of the five surfaces of the foundation in Figure3 (including four surfaces around the foundation and one surface below the foundation), including displacement constraints in X, Y and Z directions and three angular displacement constraints, all defined as 0, that is, encastre (U1 = U3 = U3 = UR1 = UR2 = UR3 = 0).Contact. The embedded method was adopted for the contact of the reinforced cage concrete pile core (embedded region) method simulation. Other surface-to-surface contacts are used to describe the contact between the steel casing, concrete pile core, and foundation, which mainly include two types: one is the steel-concrete contact, steel casing and concrete pile core, steel casing embedded section, and foundation belonging to the first type of contact, and the side of the steel casing embedded section and ground and foundation belong to the second type of contact. Regarding the steel–concrete interface contact, contact attribute package including the tangential behavior, and tangential behavior, the tangential friction coefficient is 0.2 and is calculated by the penalty function penalty. For the concrete–concrete interface contact, the normal behavior attribute of the normal contact is hard contact. The contact between the pendulum and the steel casing surface of the pile body is with the same steel interface, and the contact attribute is a tangent behavior, where the tangential friction tangent behavior coefficient is 0.4, which is calculated by the penalty function penalty.Boundary conditions. Under the action of a horizontal impact load, because the initial speed needs to be predefined in the initial step, the initial distance between the head of the impact body and the surface of the steel casing pile body is set to 0, and the initial speed isv0.


#### 2.3.4. Grid Element

ABAQUS/explicit was used to mesh the concrete entity with a three-dimensional (3D) solid element. The pile core concrete, foundation concrete, and pendulum solid components were meshed using the element c3d8i. For the mesh generation of the reinforcement, the linear element B31 considering shear deformation was adopted, as shown in Figure2.Figure 3
Test verification impact load condition.The element size directly affects the calculation cost and accuracy. The smaller the cell size, the greater the number of grids and the number of calculation cells, and the results are generally more accurate; however, the calculation time and cost will be too high. Therefore, to consider the calculation accuracy and efficiency, it is necessary to evaluate the rationality of the grid cell division. According to the requirements of the structural calculations, different components adopt different degrees of division accuracy. The volume of the foundation was large, and the mechanical response law of the foundation was not the focus of our study. Considering the calculation cost of the numerical model, we set the approximate global size of the model seed of the foundation to 0.04.


#### 2.3.5. Loading Scheme

With respect to the design data of the prototype pile and the actual scenario, the normal impact velocity of ship berthing varies owing to different ship displacements, and under the same displacement, a river ship is faster than a sea ship. For specific conditions, please refer to the load code for port engineering [39]. According to the energy similarity scale, the following is obtained:(1)
The prototype 5000DWT ship impacts the prototype at a speed of 0.3 m/s. According to the load code for port engineering [39], from the ship impact energy formula E0 = ρ/2MVn2 (ρ = 0.8), it is concluded that the effective impact energy when the ship berths is 180 kJ according to the performance curve of the DA-A500HL1500 rubber fender. After conversion according to a similar scale, the impact of the model requires 180 J of energy, and the converted reaction force is 0.45 kN. In the model test, a pendulum of 90 kg was required to impact at 2 m/s, and the pendulum volume was 0.0115 m3.(2)
The prototype 1000DWT ship impacts the prototype at a speed of 0.3 m/s, and the effective impact energy of the ship is 36 kJ. After conversion according to a similar scale, the model impact requires a 36 J energy impact, and the converted reaction force is 0.09 kN. An 18 kg pendulum was required to impact at 2 m/s, and the pendulum volume was 0.00231 m3.(3)
The prototype 2000DWT ship impacts the prototype at a speed of 0.3 m/s, and the effective impact energy of the ship is 72 kJ. After conversion according to a similar scale, the model impact requires a 72 J energy impact, and the converted reaction force is 0.18 kN. A 36 kg pendulum is required to impact at 2 m/s, and the pendulum volume is 0.00462 m3.(4)
The prototype 3000DWT ship impacts the prototype at a speed of 0.3 m/s, and the effective impact energy of the ship is 108 kJ. After conversion according to a similar scale, the model impact requires 108 J of energy impact, and the converted reaction force is 0.27 kN. A 54 kg pendulum is required to impact at 2 m/s, and the pendulum volume is 0.00231 m3.In comprehensive consideration, the action point of pendulum impact load is on the pile body at a height of 1.2 m from the foundation, an 8 kg pendulum impact loading is proposed for test verification, and the numerical calculation loading conditions are 49 working conditions from 0.02 kJ to 1.28 kJ, as shown in Figure4.Figure 4
Displacement of pile body under impact of 0.12 m pendulum height. (a) Maximum lateral displacement; (b) time history curve of u1; (c) time history curve of u2; (d) time history curve of u3.
(a)(b)(c)(d)


### 2.4. Test Verification

To verify the numerical calculation model, the test values and numerical calculation values of the lateral displacement of the three measuring points of the pile body under impact load were compared and analyzed (Figure3). The maximum displacement and time history curve of the three measuring points u1, u2, and u3 are compared and analyzed under the impact of pendulum heights of h = 0.12 m (v0 = 1.55 m/s), h = 0.14 m (v0 = 1.67 m/s), and h = 0.16 m (v0 = 1.79 m/s) (Figures 4–6).Figure 5
Displacement of pile body under impact of 0.14 m pendulum height. (a) Maximum lateral displacement; (b) time history curve of u1; (c) time history curve of u2; (d) time history curve of u3.
(a)(b)(c)(d)Figure 6
Displacement of pile body under impact of 0.16 m pendulum height. (a) Maximum lateral displacement; (b) time history curve of u1; (c) time history curve of u2; (d) time history curve of u3.
(a)(b)(c)(d)After the pile body is impacted by the pendulum at a height of 0.12 m from the foundation surface, the comparison between the test values and numerical calculation values of the lateral displacement dynamic response of three displacement measuring points opposite the pile body is as shown in Table2.Table 2
Maximum displacement of two measuring points.
Working conditionu1max (mm)u2max (mm)u3max (mm)TestFEMTestFEMTestFEM0.12 m (1.55 m/s)0.5070.50660.3570.43280.0690.0330.14 m (1.67 m/s)0.5500.5580.3880.3750.07380.03180.16 m (1.67 m/s)0.5990.61590.4210.41350.08090.0313To maintain a higher contrast accuracy, the accuracy of the test displacement dial indicator was 0.001, and the calculated data retained one decimal place with an accuracy of 0.0001.From Figures4–6, it can be seen that, under three kinds of horizontal impact loads, the dynamic response test data of lateral displacement u1, u2, and u3 of the pile at three measuring points 0 m, 1.5 m, and 1.7 m away from the foundation are in good agreement with the model calculation data.Comparison of maximum value of first displacement (maximum displacement): when the pile body is subjected to an impact load under three working conditions, the maximum value of the first transverse displacement of the three measuring points of the pile body is as shown in Figures4(a), 5(a), and 6(a), respectively. The test values of u1max, u2max, and u3max are in good agreement with the numerical calculation values, and the u3max test data are large, but they all meet the requirements.Comparison of displacement time history curves: under the three working conditions, the displacement of the three measuring points of the pile presents an attenuated sinusoidal waveform, and the test data are in good agreement with the numerical calculation data. In particular, the curves of u1 and u2 were in good agreement. After the pile is subjected to a lateral impact load, the response extreme value in the numerical calculation results of u1 and u2 attenuates slightly in the stage of 0∼75 ms, and after 75 ms, the numerical results show that the amplitude of the displacement response decays slightly faster, but they all show good agreement. Under the three working conditions, the numerical results of the displacement time history response of u3 are slightly different from the test results, but they all meet the requirements.In conclusion, the numerical model calculation results are in good agreement with the test results, and it is possible to simulate the numerical calculation of the dynamic response of a steel casing rock-socketed pile under a horizontal impact load.


## 2.1. Project Overview

Overview of the prototype piles: the test takes the second-phase rock-socketed cast-in-place pile of the Chongqing orchard port as the prototype. As shown in Figure1, the steel casing was made of Q235 steel, the thickness of the protective layer of the main reinforcement was 80 mm, and the concrete material grade was C30. The rock stratum at the pile bottom is mainly composed of sandstone and mudstone, where the natural compressive strength and saturated compressive strength of sandstone are 24.6 MPa and 18.9 MPa, respectively, and the natural compressive strength and saturated compressive strength of mudstone are 7.4 MPa and 4.3 MPa, respectively.Figure 1
Schematic diagram of model pile structure (unit: mm). (a) Structural elevation; (b) physical model.
(a)(b)


## 2.2. Test Boundary Conditions

The model test boundary conditions have a significant impact on the model stress transfer and accuracy of the test data. The key problems to be considered are as follows:Pile embedment depth. Depending on the code for the design and construction of a rock-socketed pile in port engineering [35], the rock socketed depth of the rock-socketed pile should meet the requirements of the bearing axial force and horizontal force simultaneously. The rock-socketed depth of the rock-socketed pile should not be less than 1.5 times that of the pile diameter in order to ensure that the rock-socketed end can be considered as consolidation in structural design. The test shows that when the rock-socketed depth of the rock-socketed pile is approximately three times the pile diameter, the axial bearing capacity is the best. The model mainly considers the dynamic response of the pile body under lateral action, so the rock-socketed depth is three times the pile diameter; that is, the embedded depth is 0.9 m.Plane dimensions of the foundation. The proposed foundation plane size is 0.7 m × 0.7 m × 1.5 m.


## 2.3. Numerical Model

### 2.3.1. Geometric Model

The geometric model of a steel casing reinforced concrete rock-socketed pile was established using the nonlinear finite software ABAQUS. The model components mainly include the steel casing, reinforcement cage, concrete pile core, concrete foundation, and loading pendulum.Because the impact pressure sensor has less mass and inertia than the pendulum, it has a greater stiffness than the steel pile casing rock-socketed pile body, and the plastic deformation of the pressure sensor during impact is very small, the influence of the inertia of the pressure sensor is ignored during pendulum impact. In the numerical simulation, a hemispherical cylindrical steel body with simple geometry is used to replace the sensor and pendulum. The horizontal impact load in all calculation models below adopts the hemispherical cylinder to impact the pile body. The length of the latter half of the cylinder can be varied to adjust the mass of the impact body. The structure of the impact hemispherical cylinder is shown in Figure2.Figure 2
Mesh generation of model.Calculation of model dimensions. According to the physical model test size, the diameter of the head hemisphere of the pendulum was 0.1 m, and the rear of the pendulum was a cylinder with a diameter of 0.1 m and a height of 0.1 m (adjustable). The outer diameter of the steel casing was 0.3 m, the thickness was 0.002 m, and the length was 1.85 m. The diameter of the concrete pile core was 0.298 m, and the length was 2.6 m. The foundation size was 0.7 m × 0.7 m × 1.5 m.Table 1
Performance parameters of steel materials.
DensityElasticPoisson’s ratioUltimate strain7.8 × 10−92.1 × 10−50.30.2


### 2.3.2. Material Properties

(1)
Concrete.Concrete includes C30 concrete for pile core and C15 concrete for foundation. The concrete material adopts the plastic damage model (CDP) in abaqus material library modified by Ye [36]. The stress-strain relationship of CDP model adopts the concrete constitutive relationship given in Appendix C of code for design of concrete structures [37] (hereinafter referred to as the code). The initial elastic modulus E0 of concrete is the secant modulus of concrete during tensile cracking, and Poisson’s ratio used in the calculation is 0.2. For the stress-strain curve in plastic stage, it is taken according to the concrete code [36].(2)
SteelSteel includes reinforcement and a steel casing. The binary line model was used to simulate the constitutive relationship of steel, and the yield strength and compressive strength [38] were obtained from the measured values in the tensile test. See Table 1 for further details.


### 2.3.3. Constraints, Contact, and Boundary Conditions

Constraints. To simulate the foundation constraint boundary conditions in the actual test, fixed constraints are made on the six degrees of freedom directions of the five surfaces of the foundation in Figure3 (including four surfaces around the foundation and one surface below the foundation), including displacement constraints in X, Y and Z directions and three angular displacement constraints, all defined as 0, that is, encastre (U1 = U3 = U3 = UR1 = UR2 = UR3 = 0).Contact. The embedded method was adopted for the contact of the reinforced cage concrete pile core (embedded region) method simulation. Other surface-to-surface contacts are used to describe the contact between the steel casing, concrete pile core, and foundation, which mainly include two types: one is the steel-concrete contact, steel casing and concrete pile core, steel casing embedded section, and foundation belonging to the first type of contact, and the side of the steel casing embedded section and ground and foundation belong to the second type of contact. Regarding the steel–concrete interface contact, contact attribute package including the tangential behavior, and tangential behavior, the tangential friction coefficient is 0.2 and is calculated by the penalty function penalty. For the concrete–concrete interface contact, the normal behavior attribute of the normal contact is hard contact. The contact between the pendulum and the steel casing surface of the pile body is with the same steel interface, and the contact attribute is a tangent behavior, where the tangential friction tangent behavior coefficient is 0.4, which is calculated by the penalty function penalty.Boundary conditions. Under the action of a horizontal impact load, because the initial speed needs to be predefined in the initial step, the initial distance between the head of the impact body and the surface of the steel casing pile body is set to 0, and the initial speed isv0.


### 2.3.4. Grid Element

ABAQUS/explicit was used to mesh the concrete entity with a three-dimensional (3D) solid element. The pile core concrete, foundation concrete, and pendulum solid components were meshed using the element c3d8i. For the mesh generation of the reinforcement, the linear element B31 considering shear deformation was adopted, as shown in Figure2.Figure 3
Test verification impact load condition.The element size directly affects the calculation cost and accuracy. The smaller the cell size, the greater the number of grids and the number of calculation cells, and the results are generally more accurate; however, the calculation time and cost will be too high. Therefore, to consider the calculation accuracy and efficiency, it is necessary to evaluate the rationality of the grid cell division. According to the requirements of the structural calculations, different components adopt different degrees of division accuracy. The volume of the foundation was large, and the mechanical response law of the foundation was not the focus of our study. Considering the calculation cost of the numerical model, we set the approximate global size of the model seed of the foundation to 0.04.


### 2.3.5. Loading Scheme

With respect to the design data of the prototype pile and the actual scenario, the normal impact velocity of ship berthing varies owing to different ship displacements, and under the same displacement, a river ship is faster than a sea ship. For specific conditions, please refer to the load code for port engineering [39]. According to the energy similarity scale, the following is obtained:(1)
The prototype 5000DWT ship impacts the prototype at a speed of 0.3 m/s. According to the load code for port engineering [39], from the ship impact energy formula E0 = ρ/2MVn2 (ρ = 0.8), it is concluded that the effective impact energy when the ship berths is 180 kJ according to the performance curve of the DA-A500HL1500 rubber fender. After conversion according to a similar scale, the impact of the model requires 180 J of energy, and the converted reaction force is 0.45 kN. In the model test, a pendulum of 90 kg was required to impact at 2 m/s, and the pendulum volume was 0.0115 m3.(2)
The prototype 1000DWT ship impacts the prototype at a speed of 0.3 m/s, and the effective impact energy of the ship is 36 kJ. After conversion according to a similar scale, the model impact requires a 36 J energy impact, and the converted reaction force is 0.09 kN. An 18 kg pendulum was required to impact at 2 m/s, and the pendulum volume was 0.00231 m3.(3)
The prototype 2000DWT ship impacts the prototype at a speed of 0.3 m/s, and the effective impact energy of the ship is 72 kJ. After conversion according to a similar scale, the model impact requires a 72 J energy impact, and the converted reaction force is 0.18 kN. A 36 kg pendulum is required to impact at 2 m/s, and the pendulum volume is 0.00462 m3.(4)
The prototype 3000DWT ship impacts the prototype at a speed of 0.3 m/s, and the effective impact energy of the ship is 108 kJ. After conversion according to a similar scale, the model impact requires 108 J of energy impact, and the converted reaction force is 0.27 kN. A 54 kg pendulum is required to impact at 2 m/s, and the pendulum volume is 0.00231 m3.In comprehensive consideration, the action point of pendulum impact load is on the pile body at a height of 1.2 m from the foundation, an 8 kg pendulum impact loading is proposed for test verification, and the numerical calculation loading conditions are 49 working conditions from 0.02 kJ to 1.28 kJ, as shown in Figure4.Figure 4
Displacement of pile body under impact of 0.12 m pendulum height. (a) Maximum lateral displacement; (b) time history curve of u1; (c) time history curve of u2; (d) time history curve of u3.
(a)(b)(c)(d)


## 2.3.1. Geometric Model

The geometric model of a steel casing reinforced concrete rock-socketed pile was established using the nonlinear finite software ABAQUS. The model components mainly include the steel casing, reinforcement cage, concrete pile core, concrete foundation, and loading pendulum.Because the impact pressure sensor has less mass and inertia than the pendulum, it has a greater stiffness than the steel pile casing rock-socketed pile body, and the plastic deformation of the pressure sensor during impact is very small, the influence of the inertia of the pressure sensor is ignored during pendulum impact. In the numerical simulation, a hemispherical cylindrical steel body with simple geometry is used to replace the sensor and pendulum. The horizontal impact load in all calculation models below adopts the hemispherical cylinder to impact the pile body. The length of the latter half of the cylinder can be varied to adjust the mass of the impact body. The structure of the impact hemispherical cylinder is shown in Figure2.Figure 2
Mesh generation of model.Calculation of model dimensions. According to the physical model test size, the diameter of the head hemisphere of the pendulum was 0.1 m, and the rear of the pendulum was a cylinder with a diameter of 0.1 m and a height of 0.1 m (adjustable). The outer diameter of the steel casing was 0.3 m, the thickness was 0.002 m, and the length was 1.85 m. The diameter of the concrete pile core was 0.298 m, and the length was 2.6 m. The foundation size was 0.7 m × 0.7 m × 1.5 m.Table 1
Performance parameters of steel materials.
DensityElasticPoisson’s ratioUltimate strain7.8 × 10−92.1 × 10−50.30.2


## 2.3.2. Material Properties

(1)
Concrete.Concrete includes C30 concrete for pile core and C15 concrete for foundation. The concrete material adopts the plastic damage model (CDP) in abaqus material library modified by Ye [36]. The stress-strain relationship of CDP model adopts the concrete constitutive relationship given in Appendix C of code for design of concrete structures [37] (hereinafter referred to as the code). The initial elastic modulus E0 of concrete is the secant modulus of concrete during tensile cracking, and Poisson’s ratio used in the calculation is 0.2. For the stress-strain curve in plastic stage, it is taken according to the concrete code [36].(2)
SteelSteel includes reinforcement and a steel casing. The binary line model was used to simulate the constitutive relationship of steel, and the yield strength and compressive strength [38] were obtained from the measured values in the tensile test. See Table 1 for further details.


## 2.3.3. Constraints, Contact, and Boundary Conditions

Constraints. To simulate the foundation constraint boundary conditions in the actual test, fixed constraints are made on the six degrees of freedom directions of the five surfaces of the foundation in Figure3 (including four surfaces around the foundation and one surface below the foundation), including displacement constraints in X, Y and Z directions and three angular displacement constraints, all defined as 0, that is, encastre (U1 = U3 = U3 = UR1 = UR2 = UR3 = 0).Contact. The embedded method was adopted for the contact of the reinforced cage concrete pile core (embedded region) method simulation. Other surface-to-surface contacts are used to describe the contact between the steel casing, concrete pile core, and foundation, which mainly include two types: one is the steel-concrete contact, steel casing and concrete pile core, steel casing embedded section, and foundation belonging to the first type of contact, and the side of the steel casing embedded section and ground and foundation belong to the second type of contact. Regarding the steel–concrete interface contact, contact attribute package including the tangential behavior, and tangential behavior, the tangential friction coefficient is 0.2 and is calculated by the penalty function penalty. For the concrete–concrete interface contact, the normal behavior attribute of the normal contact is hard contact. The contact between the pendulum and the steel casing surface of the pile body is with the same steel interface, and the contact attribute is a tangent behavior, where the tangential friction tangent behavior coefficient is 0.4, which is calculated by the penalty function penalty.Boundary conditions. Under the action of a horizontal impact load, because the initial speed needs to be predefined in the initial step, the initial distance between the head of the impact body and the surface of the steel casing pile body is set to 0, and the initial speed isv0.


## 2.3.4. Grid Element

ABAQUS/explicit was used to mesh the concrete entity with a three-dimensional (3D) solid element. The pile core concrete, foundation concrete, and pendulum solid components were meshed using the element c3d8i. For the mesh generation of the reinforcement, the linear element B31 considering shear deformation was adopted, as shown in Figure2.Figure 3
Test verification impact load condition.The element size directly affects the calculation cost and accuracy. The smaller the cell size, the greater the number of grids and the number of calculation cells, and the results are generally more accurate; however, the calculation time and cost will be too high. Therefore, to consider the calculation accuracy and efficiency, it is necessary to evaluate the rationality of the grid cell division. According to the requirements of the structural calculations, different components adopt different degrees of division accuracy. The volume of the foundation was large, and the mechanical response law of the foundation was not the focus of our study. Considering the calculation cost of the numerical model, we set the approximate global size of the model seed of the foundation to 0.04.


## 2.3.5. Loading Scheme

With respect to the design data of the prototype pile and the actual scenario, the normal impact velocity of ship berthing varies owing to different ship displacements, and under the same displacement, a river ship is faster than a sea ship. For specific conditions, please refer to the load code for port engineering [39]. According to the energy similarity scale, the following is obtained:(1)
The prototype 5000DWT ship impacts the prototype at a speed of 0.3 m/s. According to the load code for port engineering [39], from the ship impact energy formula E0 = ρ/2MVn2 (ρ = 0.8), it is concluded that the effective impact energy when the ship berths is 180 kJ according to the performance curve of the DA-A500HL1500 rubber fender. After conversion according to a similar scale, the impact of the model requires 180 J of energy, and the converted reaction force is 0.45 kN. In the model test, a pendulum of 90 kg was required to impact at 2 m/s, and the pendulum volume was 0.0115 m3.(2)
The prototype 1000DWT ship impacts the prototype at a speed of 0.3 m/s, and the effective impact energy of the ship is 36 kJ. After conversion according to a similar scale, the model impact requires a 36 J energy impact, and the converted reaction force is 0.09 kN. An 18 kg pendulum was required to impact at 2 m/s, and the pendulum volume was 0.00231 m3.(3)
The prototype 2000DWT ship impacts the prototype at a speed of 0.3 m/s, and the effective impact energy of the ship is 72 kJ. After conversion according to a similar scale, the model impact requires a 72 J energy impact, and the converted reaction force is 0.18 kN. A 36 kg pendulum is required to impact at 2 m/s, and the pendulum volume is 0.00462 m3.(4)
The prototype 3000DWT ship impacts the prototype at a speed of 0.3 m/s, and the effective impact energy of the ship is 108 kJ. After conversion according to a similar scale, the model impact requires 108 J of energy impact, and the converted reaction force is 0.27 kN. A 54 kg pendulum is required to impact at 2 m/s, and the pendulum volume is 0.00231 m3.In comprehensive consideration, the action point of pendulum impact load is on the pile body at a height of 1.2 m from the foundation, an 8 kg pendulum impact loading is proposed for test verification, and the numerical calculation loading conditions are 49 working conditions from 0.02 kJ to 1.28 kJ, as shown in Figure4.Figure 4
Displacement of pile body under impact of 0.12 m pendulum height. (a) Maximum lateral displacement; (b) time history curve of u1; (c) time history curve of u2; (d) time history curve of u3.
(a)(b)(c)(d)


## 2.4. Test Verification

To verify the numerical calculation model, the test values and numerical calculation values of the lateral displacement of the three measuring points of the pile body under impact load were compared and analyzed (Figure3). The maximum displacement and time history curve of the three measuring points u1, u2, and u3 are compared and analyzed under the impact of pendulum heights of h = 0.12 m (v0 = 1.55 m/s), h = 0.14 m (v0 = 1.67 m/s), and h = 0.16 m (v0 = 1.79 m/s) (Figures 4–6).Figure 5
Displacement of pile body under impact of 0.14 m pendulum height. (a) Maximum lateral displacement; (b) time history curve of u1; (c) time history curve of u2; (d) time history curve of u3.
(a)(b)(c)(d)Figure 6
Displacement of pile body under impact of 0.16 m pendulum height. (a) Maximum lateral displacement; (b) time history curve of u1; (c) time history curve of u2; (d) time history curve of u3.
(a)(b)(c)(d)After the pile body is impacted by the pendulum at a height of 0.12 m from the foundation surface, the comparison between the test values and numerical calculation values of the lateral displacement dynamic response of three displacement measuring points opposite the pile body is as shown in Table2.Table 2
Maximum displacement of two measuring points.
Working conditionu1max (mm)u2max (mm)u3max (mm)TestFEMTestFEMTestFEM0.12 m (1.55 m/s)0.5070.50660.3570.43280.0690.0330.14 m (1.67 m/s)0.5500.5580.3880.3750.07380.03180.16 m (1.67 m/s)0.5990.61590.4210.41350.08090.0313To maintain a higher contrast accuracy, the accuracy of the test displacement dial indicator was 0.001, and the calculated data retained one decimal place with an accuracy of 0.0001.From Figures4–6, it can be seen that, under three kinds of horizontal impact loads, the dynamic response test data of lateral displacement u1, u2, and u3 of the pile at three measuring points 0 m, 1.5 m, and 1.7 m away from the foundation are in good agreement with the model calculation data.Comparison of maximum value of first displacement (maximum displacement): when the pile body is subjected to an impact load under three working conditions, the maximum value of the first transverse displacement of the three measuring points of the pile body is as shown in Figures4(a), 5(a), and 6(a), respectively. The test values of u1max, u2max, and u3max are in good agreement with the numerical calculation values, and the u3max test data are large, but they all meet the requirements.Comparison of displacement time history curves: under the three working conditions, the displacement of the three measuring points of the pile presents an attenuated sinusoidal waveform, and the test data are in good agreement with the numerical calculation data. In particular, the curves of u1 and u2 were in good agreement. After the pile is subjected to a lateral impact load, the response extreme value in the numerical calculation results of u1 and u2 attenuates slightly in the stage of 0∼75 ms, and after 75 ms, the numerical results show that the amplitude of the displacement response decays slightly faster, but they all show good agreement. Under the three working conditions, the numerical results of the displacement time history response of u3 are slightly different from the test results, but they all meet the requirements.In conclusion, the numerical model calculation results are in good agreement with the test results, and it is possible to simulate the numerical calculation of the dynamic response of a steel casing rock-socketed pile under a horizontal impact load.


## 3. Numerical Analysis

### 3.1. Influence of Different Mass and Speed on Structural Dynamic Response Based on Impact Energy

Changing the velocity and mass of the impact body has a significant impact on the dynamic response of the pile body. Increasing the impact velocity increases the amplitude of the impact force, the internal force response of the structure is faster, and the amplitude of the response time history curve is larger. When the velocity increases to a certain value, the central value of the internal force oscillation time history curve also increases markedly. By increasing the impact mass, the speed of the impact force time history curve from a positive to negative amplitude is accelerated. The displacement amplitude of the pile increases linearly with the impact mass and impact velocity. However, it is not possible to reliably evaluate the dynamic response of a steel casing rock socketed pile under a horizontal impact load only by changing the mass or speed. In practice, to evaluate the impact safety or horizontal impact resistance of a structure, people usually load a certain amount of impact energyEk (1/2 mv02) or momentum P (mv0) to analyze the dynamic response of the structure in order to evaluate the impact resistance of the structure. Because this analysis needs to consider more working conditions, the adoption of the physical model test method will require too much manpower and material resources. Because the analysis methods are the same under the same impact energy or momentum, this section proposes the numerical simulation method for use with reinforced concrete piles. The dynamic response law of steel casing rock-socketed pile is discussed by changing the mass (m) and velocity (v0) of the impact body under different impact energy values Ek (1/2 mv02).The calculation and analysis results of 49 combined working conditions of the pile body under seven different impact energies are shown in Figures7–13 and include primarily the time history curve of the impact force F, the reinforced concrete pile core section bending moment M at the horizon of H = 0 m at the root of the pile body, and the transverse displacement u1 at the top of the pile.(1)
The same impact kinetic energyEk. Under different impact energies, the impact force, concrete pile core bending moment, and pile top displacement all show the same response law. (1) The increase in the impact mass has a significant impact on the maximum impact force, and the maximum impact force gradually decreases with the increase in impact mass. (2) The increase in impact mass has an impact on the maximum, and the center of concrete bending moment vibration (platform value) has a significant impact, which gradually increases with the increase in impact mass. (3) The increase in the impact mass has an impact on the maximum displacement, and the vibration center of the pile top (platform value) also has a significant impact and gradually increases with an increase in the impact mass. The increase in the impact mass m has an obvious impact on the vibration change form of the bending moment M and pile top displacement u1, and the response law is consistent. When the impact mass m is less than the threshold, the concrete pile core bending moment M and pile top displacement vibrate around point 0, and the amplitude of the M vibration is positively correlated with m. When m is greater than the threshold, the concrete pile core bends, the moment M and the pile top displacement oscillation center are offset towards the impact direction, and the offset value is positively correlated with m. As shown in Table 3, the impact mass threshold m is negatively correlated with the impact energy. When m is greater than the impact threshold, obvious plastic hinges begin to appear at the bottom of the pile body. To prevent obvious plastic deformation at the root of the pile body after the impact load, the key is to determine the corresponding impact mass m threshold.(2)
Varying impact kinetic energyEk. With the increase in Ek, the dynamic response laws of various factors are as follows: (1) the maximum impact force becomes increasingly large, and the time to reach the maximum impact force becomes shorter and shorter because of the increase in impact speed; (2) the maximum bending moment M increases significantly. When Ek is large, the pile body reaches the maximum value after being impacted, and then the vibration center value in the stable vibration stage (platform value) becomes larger and larger, indicating that when the impact energy is large, the concrete pile core produces a plastic hinge near the foundation surface. The plastic hinge begins to appear, which is mainly dominated by the impact energy, followed by m, which has little correlation with the impact velocity v0, as shown in Figure 7(b). Even when the impact energy is smaller than 0.02 kJ, when m is 64 kg, the center value of moment oscillation still increases markedly (Figure 7); (3) the maximum value of u1 on the top of the pile becomes larger and larger. The oscillation of u1 is similar to that of M. It is worth noting that, for each impact energy, when m = 512 kg, two or three continuous peaks appear near the maximum value of displacement, which is due to the impact of the pile body many times. By performing a detailed analysis, it is found that when m = 512 kg, there are two consecutive peaks at the maximum value of u1 under the four impact energies of Ek = 0.02 kJ, 0.04 kJ, 0.08 kJ, and 0.16 kJ, and three consecutive peaks at the maximum value of u1 under the three impact energies of Ek = 0.32 kJ, 0.64 kJ, and 1.28 kJ. When m = 256 kg, Ek = 0.16 kJ, 0.32 kJ, 0.64 kJ, and 1.28 kJ, there are two consecutive peaks at the maximum value of u1. In fact, it can be seen from the F-t time history curves of seven different impact energy Ek working conditions that when m = 256 kg, there are two consecutive impacts, and when m = 512 kg, there are three consecutive impacts. However, there is no obvious secondary impact on the displacement u1 time history response curve with small impact energy. This is because of the following: (1) because the secondary impact energy is small, the impact force during the second impact is small; (2) the secondary impact time is when the pile body swings back after the positive impact reaches the maximum displacement. At this time, the secondary impact only reduces the negative displacement in the process of pile body vibration, so there are no obvious positive continuous displacement maxima on the u1 time history curve. As shown in Figure 14, it can be seen that the pile body reduces the negative vibration amplitude of u1 owing to the secondary impact. In fact, when m = 256 kg, under the four energy shocks of Ek = 0.16 kJ, 0.32 kJ, 0.64 kJ, and 1.28 kJ, two consecutive peaks at the maximum value of u1 are also caused by the secondary impact during the swing back process when u1 reaches the first positive maximum displacement. This is because the pile body begins to rebound in the negative direction, and the secondary impact energy is large, and the impact force is large. Therefore, there is an obvious second impact phenomenon on the u1 time history curve.Figure 7
Dynamic response when Ek = 0.02 kJ. (a) F-t; (b) M-t; (c) u1-t.
(a)(b)(c)Figure 8
Dynamic response when Ek = 0.04 kJ. (a) F-t; (b) M-t; (c) u1-t.
(a)(b)(c)Figure 9
Dynamic response when Ek = 0.08 kJ. (a) F-t; (b) M-t; (c) u1-t.
(a)(b)(c)Figure 10
Dynamic response when Ek = 0.16 kJ. (a) F-t; (b) M-t; (c) u1-t.
(a)(b)(c)Figure 11
Dynamic response when Ek = 0.32 kJ. (a) F-t; (b) M-t; (c) u1-t.
(a)(b)(c)Figure 12
Dynamic response when Ek = 0.64 kJ. (a) F-t; (b) M-t; (c) u1-t.
(a)(b)(c)Figure 13
Dynamic response when Ek = 1.28 kJ. (a) F-t; (b) M-t; (c) u1-t.
(a)(b)(c)Figure 14
Displacement time history curve whenEk = 0.02 kJ, M = 256 kg.Table 3
Impact mass thresholdm.
Ek (kJ)Impact mass thresholdm (kg)From the perspective of u1 (mm)From the perspective ofM (kN·m)0.0232320.0416160.0816160.16Below 8Below 80.32Below 8Below 80.64Below 8Below 81.28Below 8Below 8


### 3.2. Parametric Analysis of Horizontal Antiscour Capacity of Steel Casing Rock-Socketed Pile

After the steel casing rock-socketed pile is subjected to horizontal impact, owing to the action of inertial force and foundation constraint, the pile body swings freely with damping, and the response parameters, such as the lateral displacement and internal force of the pile body, show attenuation vibrations that are similar to sine waves. In the time domain, both sides of the pile body are in the working state of repeated and alternating tensile and compressive stresses, and the structural bearing capacity deteriorates. The maximum value (extreme value) of each response parameter of the pile body can be used as a typical value to reflect its bearing capacity or impact resistance. To clarify the horizontal impact resistance of the steel casing rock-socketed pile, this subsection intends to analyze the influence of impact parameters on the dynamic response of pile shaft under different impact loading conditions, so as to obtain the physical relationship between impact parameters (independent variables) and the maximum value of dynamic response parameters (dependent variables).Below, it is proposed to study the maximum value of the dynamic response parameters of the pile body under seven different impact energies (Ek = 20 × 10−3 kJ, 40 × 10−3 kJ, 80 × 10−3 kJ, 160 × 10−3 kJ, 320 × 10−3 kJ, 640 × 10−3 kJ, 1280 × 10−3 kJ), when the impact mass and impact speed are changed, including the pile top displacement u1max, concrete pile core bending moment Mmax at the junction with the foundation surface, and impact force Fmax on the pile body.u1max, Mmax, and Fmax increased significantly by increasing the impact energy Ek (Figure 15), and the detailed rules are as follows: (1) there is a good linear relationship between the impact mass m and the maximum displacement u1max at the pile top, and there is a positive correlation. When the impact mass m is less than 128 kg, u1max is very sensitive to the increase in M, and when the impact mass m ≥ 128 kg, the sensitivity of u1max to the increase of M decreases significantly. (2) When the impact mass m is less than 128 kg, the influence of the increase in m on Mmax is obvious, but the law is not in agreement, while Ek varies; when Ek is small, they are positively correlated, and when Ek is large, they are negatively correlated; when m ≥ 128 kg, the sensitivity of Mmax to the increase in m decreases significantly. (3) When the impact mass m < 128 kg, an increase in m can cause an obvious increase in Fmax, while when m ≥ 128 kg, the change is very small. In Figures 15(d)–15(f), when m = 128, 256, and 512 kg, the three curves are very close, which also illustrates this.Figure 15
Relationship between impact energy and response parameters. (a) u1max-m; (b) Mmax-m; (c) Fmax-m; (d) u1max-Ek; (e) Mmax-Ek; (f) Fmax-Ek.
(a)(b)(c)(d)(e)(f)Using MATLAB, the data of different impactsm, velocity v0, and transverse maximum displacement u1max of the pile under seven different impact energies in Table 4 were fitted by the linear regression method. The three parameter values were found to be located on an elliptical parabolic sphere, as shown in Figure 16.Table 4
Numerical calculation results under each impact energy.
Ek (kJ)m (kg)V0 (m/s)u1max (mm)Mmax (kN·m)Fmax (kN)0.0282.240.685−8.6122.214161.580.461−6.23135.578321.121.068−13.80250.396640.791.885−13.74670.0701280.562.316−12.57260.7612560.402.480−12.97855.8865120.282.376−12.43545.3790.0483.160.914−12.51080.864162.240.733−9.23115.142321.582.112−14.13281.674641.122.852−13.44169.8371280.793.589−11.22279.6812560.563.793−10.67766.1565120.403.918−10.95660.2860.0884.471.373−11.3242.889163.160.915−11.93043.803322.242.710−12.457112.317641.584.493−10.17486.9271281.125.409−12.790118.4712560.796.062−11.21187.4045120.566.309−11.66478.8850.1686.321.905−13.91196.989164.471.311−10.99195.448323.163.909−14.468147.183642.246.383−12.139148.1491281.588.296−13.197127.2812561.129.195−14.039134.9885120.7910.050−13.905108.5800.3288.943.168−13.741179.245166.322.019−14.79842.950324.476.667−13.419114.412643.169.860−14.796157.7111282.2412.314−15.508167.9012561.5813.729−15.970144.3645121.1215.419−15.786147.6670.64812.655.192−10.732129.509168.942.816−13.432102.544326.3210.444−15.171136.888644.4716.733−16.862186.5361283.1619.667−17.829188.7362562.2421.785−18.044170.8805121.5823.064−18.471174.0561.28817.898.173−14.332142.0151612.654.831−12.15451.102328.9416.786−18.182271.797646.3224.807−21.549138.8811284.4730.163−20.545227.4412563.1633.677−18.791185.0485122.2438.090−19.081231.157Figure 16
Relationship between (m) v0, and u1max.Under different impact energies, the formula fitted by the linear regression method has a common form of (1), namely,(1)u1max=a+bm+c2+dv0+e2,where m is the mass of the impactor, and v0 is the impact velocity; a, b, c, d, and e are real constants.The real constant values corresponding to different impact energies are listed in Table5.Table 5
Different values ofEk and their corresponding real constants.
Ek (kJ)abcde0.0214.77−5.203e − 0752492.02−1.7470.0420.47−5.439e − 0760271.328−2.5380.089.141−3.823e − 0747740.4087−3.580.1628.28−6.44e − 0764780.6562−5.160.32−6.6644.92e − 05−428.30.5283−7.5380.64−18.311.367 e − 04−422.10.6751−10.641.28−18.481.388 e − 04−420.40.3384−15.05


## 3.1. Influence of Different Mass and Speed on Structural Dynamic Response Based on Impact Energy

Changing the velocity and mass of the impact body has a significant impact on the dynamic response of the pile body. Increasing the impact velocity increases the amplitude of the impact force, the internal force response of the structure is faster, and the amplitude of the response time history curve is larger. When the velocity increases to a certain value, the central value of the internal force oscillation time history curve also increases markedly. By increasing the impact mass, the speed of the impact force time history curve from a positive to negative amplitude is accelerated. The displacement amplitude of the pile increases linearly with the impact mass and impact velocity. However, it is not possible to reliably evaluate the dynamic response of a steel casing rock socketed pile under a horizontal impact load only by changing the mass or speed. In practice, to evaluate the impact safety or horizontal impact resistance of a structure, people usually load a certain amount of impact energyEk (1/2 mv02) or momentum P (mv0) to analyze the dynamic response of the structure in order to evaluate the impact resistance of the structure. Because this analysis needs to consider more working conditions, the adoption of the physical model test method will require too much manpower and material resources. Because the analysis methods are the same under the same impact energy or momentum, this section proposes the numerical simulation method for use with reinforced concrete piles. The dynamic response law of steel casing rock-socketed pile is discussed by changing the mass (m) and velocity (v0) of the impact body under different impact energy values Ek (1/2 mv02).The calculation and analysis results of 49 combined working conditions of the pile body under seven different impact energies are shown in Figures7–13 and include primarily the time history curve of the impact force F, the reinforced concrete pile core section bending moment M at the horizon of H = 0 m at the root of the pile body, and the transverse displacement u1 at the top of the pile.(1)
The same impact kinetic energyEk. Under different impact energies, the impact force, concrete pile core bending moment, and pile top displacement all show the same response law. (1) The increase in the impact mass has a significant impact on the maximum impact force, and the maximum impact force gradually decreases with the increase in impact mass. (2) The increase in impact mass has an impact on the maximum, and the center of concrete bending moment vibration (platform value) has a significant impact, which gradually increases with the increase in impact mass. (3) The increase in the impact mass has an impact on the maximum displacement, and the vibration center of the pile top (platform value) also has a significant impact and gradually increases with an increase in the impact mass. The increase in the impact mass m has an obvious impact on the vibration change form of the bending moment M and pile top displacement u1, and the response law is consistent. When the impact mass m is less than the threshold, the concrete pile core bending moment M and pile top displacement vibrate around point 0, and the amplitude of the M vibration is positively correlated with m. When m is greater than the threshold, the concrete pile core bends, the moment M and the pile top displacement oscillation center are offset towards the impact direction, and the offset value is positively correlated with m. As shown in Table 3, the impact mass threshold m is negatively correlated with the impact energy. When m is greater than the impact threshold, obvious plastic hinges begin to appear at the bottom of the pile body. To prevent obvious plastic deformation at the root of the pile body after the impact load, the key is to determine the corresponding impact mass m threshold.(2)
Varying impact kinetic energyEk. With the increase in Ek, the dynamic response laws of various factors are as follows: (1) the maximum impact force becomes increasingly large, and the time to reach the maximum impact force becomes shorter and shorter because of the increase in impact speed; (2) the maximum bending moment M increases significantly. When Ek is large, the pile body reaches the maximum value after being impacted, and then the vibration center value in the stable vibration stage (platform value) becomes larger and larger, indicating that when the impact energy is large, the concrete pile core produces a plastic hinge near the foundation surface. The plastic hinge begins to appear, which is mainly dominated by the impact energy, followed by m, which has little correlation with the impact velocity v0, as shown in Figure 7(b). Even when the impact energy is smaller than 0.02 kJ, when m is 64 kg, the center value of moment oscillation still increases markedly (Figure 7); (3) the maximum value of u1 on the top of the pile becomes larger and larger. The oscillation of u1 is similar to that of M. It is worth noting that, for each impact energy, when m = 512 kg, two or three continuous peaks appear near the maximum value of displacement, which is due to the impact of the pile body many times. By performing a detailed analysis, it is found that when m = 512 kg, there are two consecutive peaks at the maximum value of u1 under the four impact energies of Ek = 0.02 kJ, 0.04 kJ, 0.08 kJ, and 0.16 kJ, and three consecutive peaks at the maximum value of u1 under the three impact energies of Ek = 0.32 kJ, 0.64 kJ, and 1.28 kJ. When m = 256 kg, Ek = 0.16 kJ, 0.32 kJ, 0.64 kJ, and 1.28 kJ, there are two consecutive peaks at the maximum value of u1. In fact, it can be seen from the F-t time history curves of seven different impact energy Ek working conditions that when m = 256 kg, there are two consecutive impacts, and when m = 512 kg, there are three consecutive impacts. However, there is no obvious secondary impact on the displacement u1 time history response curve with small impact energy. This is because of the following: (1) because the secondary impact energy is small, the impact force during the second impact is small; (2) the secondary impact time is when the pile body swings back after the positive impact reaches the maximum displacement. At this time, the secondary impact only reduces the negative displacement in the process of pile body vibration, so there are no obvious positive continuous displacement maxima on the u1 time history curve. As shown in Figure 14, it can be seen that the pile body reduces the negative vibration amplitude of u1 owing to the secondary impact. In fact, when m = 256 kg, under the four energy shocks of Ek = 0.16 kJ, 0.32 kJ, 0.64 kJ, and 1.28 kJ, two consecutive peaks at the maximum value of u1 are also caused by the secondary impact during the swing back process when u1 reaches the first positive maximum displacement. This is because the pile body begins to rebound in the negative direction, and the secondary impact energy is large, and the impact force is large. Therefore, there is an obvious second impact phenomenon on the u1 time history curve.Figure 7
Dynamic response when Ek = 0.02 kJ. (a) F-t; (b) M-t; (c) u1-t.
(a)(b)(c)Figure 8
Dynamic response when Ek = 0.04 kJ. (a) F-t; (b) M-t; (c) u1-t.
(a)(b)(c)Figure 9
Dynamic response when Ek = 0.08 kJ. (a) F-t; (b) M-t; (c) u1-t.
(a)(b)(c)Figure 10
Dynamic response when Ek = 0.16 kJ. (a) F-t; (b) M-t; (c) u1-t.
(a)(b)(c)Figure 11
Dynamic response when Ek = 0.32 kJ. (a) F-t; (b) M-t; (c) u1-t.
(a)(b)(c)Figure 12
Dynamic response when Ek = 0.64 kJ. (a) F-t; (b) M-t; (c) u1-t.
(a)(b)(c)Figure 13
Dynamic response when Ek = 1.28 kJ. (a) F-t; (b) M-t; (c) u1-t.
(a)(b)(c)Figure 14
Displacement time history curve whenEk = 0.02 kJ, M = 256 kg.Table 3
Impact mass thresholdm.
Ek (kJ)Impact mass thresholdm (kg)From the perspective of u1 (mm)From the perspective ofM (kN·m)0.0232320.0416160.0816160.16Below 8Below 80.32Below 8Below 80.64Below 8Below 81.28Below 8Below 8


## 3.2. Parametric Analysis of Horizontal Antiscour Capacity of Steel Casing Rock-Socketed Pile

After the steel casing rock-socketed pile is subjected to horizontal impact, owing to the action of inertial force and foundation constraint, the pile body swings freely with damping, and the response parameters, such as the lateral displacement and internal force of the pile body, show attenuation vibrations that are similar to sine waves. In the time domain, both sides of the pile body are in the working state of repeated and alternating tensile and compressive stresses, and the structural bearing capacity deteriorates. The maximum value (extreme value) of each response parameter of the pile body can be used as a typical value to reflect its bearing capacity or impact resistance. To clarify the horizontal impact resistance of the steel casing rock-socketed pile, this subsection intends to analyze the influence of impact parameters on the dynamic response of pile shaft under different impact loading conditions, so as to obtain the physical relationship between impact parameters (independent variables) and the maximum value of dynamic response parameters (dependent variables).Below, it is proposed to study the maximum value of the dynamic response parameters of the pile body under seven different impact energies (Ek = 20 × 10−3 kJ, 40 × 10−3 kJ, 80 × 10−3 kJ, 160 × 10−3 kJ, 320 × 10−3 kJ, 640 × 10−3 kJ, 1280 × 10−3 kJ), when the impact mass and impact speed are changed, including the pile top displacement u1max, concrete pile core bending moment Mmax at the junction with the foundation surface, and impact force Fmax on the pile body.u1max, Mmax, and Fmax increased significantly by increasing the impact energy Ek (Figure 15), and the detailed rules are as follows: (1) there is a good linear relationship between the impact mass m and the maximum displacement u1max at the pile top, and there is a positive correlation. When the impact mass m is less than 128 kg, u1max is very sensitive to the increase in M, and when the impact mass m ≥ 128 kg, the sensitivity of u1max to the increase of M decreases significantly. (2) When the impact mass m is less than 128 kg, the influence of the increase in m on Mmax is obvious, but the law is not in agreement, while Ek varies; when Ek is small, they are positively correlated, and when Ek is large, they are negatively correlated; when m ≥ 128 kg, the sensitivity of Mmax to the increase in m decreases significantly. (3) When the impact mass m < 128 kg, an increase in m can cause an obvious increase in Fmax, while when m ≥ 128 kg, the change is very small. In Figures 15(d)–15(f), when m = 128, 256, and 512 kg, the three curves are very close, which also illustrates this.Figure 15
Relationship between impact energy and response parameters. (a) u1max-m; (b) Mmax-m; (c) Fmax-m; (d) u1max-Ek; (e) Mmax-Ek; (f) Fmax-Ek.
(a)(b)(c)(d)(e)(f)Using MATLAB, the data of different impactsm, velocity v0, and transverse maximum displacement u1max of the pile under seven different impact energies in Table 4 were fitted by the linear regression method. The three parameter values were found to be located on an elliptical parabolic sphere, as shown in Figure 16.Table 4
Numerical calculation results under each impact energy.
Ek (kJ)m (kg)V0 (m/s)u1max (mm)Mmax (kN·m)Fmax (kN)0.0282.240.685−8.6122.214161.580.461−6.23135.578321.121.068−13.80250.396640.791.885−13.74670.0701280.562.316−12.57260.7612560.402.480−12.97855.8865120.282.376−12.43545.3790.0483.160.914−12.51080.864162.240.733−9.23115.142321.582.112−14.13281.674641.122.852−13.44169.8371280.793.589−11.22279.6812560.563.793−10.67766.1565120.403.918−10.95660.2860.0884.471.373−11.3242.889163.160.915−11.93043.803322.242.710−12.457112.317641.584.493−10.17486.9271281.125.409−12.790118.4712560.796.062−11.21187.4045120.566.309−11.66478.8850.1686.321.905−13.91196.989164.471.311−10.99195.448323.163.909−14.468147.183642.246.383−12.139148.1491281.588.296−13.197127.2812561.129.195−14.039134.9885120.7910.050−13.905108.5800.3288.943.168−13.741179.245166.322.019−14.79842.950324.476.667−13.419114.412643.169.860−14.796157.7111282.2412.314−15.508167.9012561.5813.729−15.970144.3645121.1215.419−15.786147.6670.64812.655.192−10.732129.509168.942.816−13.432102.544326.3210.444−15.171136.888644.4716.733−16.862186.5361283.1619.667−17.829188.7362562.2421.785−18.044170.8805121.5823.064−18.471174.0561.28817.898.173−14.332142.0151612.654.831−12.15451.102328.9416.786−18.182271.797646.3224.807−21.549138.8811284.4730.163−20.545227.4412563.1633.677−18.791185.0485122.2438.090−19.081231.157Figure 16
Relationship between (m) v0, and u1max.Under different impact energies, the formula fitted by the linear regression method has a common form of (1), namely,(1)u1max=a+bm+c2+dv0+e2,where m is the mass of the impactor, and v0 is the impact velocity; a, b, c, d, and e are real constants.The real constant values corresponding to different impact energies are listed in Table5.Table 5
Different values ofEk and their corresponding real constants.
Ek (kJ)abcde0.0214.77−5.203e − 0752492.02−1.7470.0420.47−5.439e − 0760271.328−2.5380.089.141−3.823e − 0747740.4087−3.580.1628.28−6.44e − 0764780.6562−5.160.32−6.6644.92e − 05−428.30.5283−7.5380.64−18.311.367 e − 04−422.10.6751−10.641.28−18.481.388 e − 04−420.40.3384−15.05


## 4. Conclusion

(1)
This study proposes the criterion for the dynamic response form of the pile body based on the impact mass threshold. The impact mass threshold corresponds to the development of a plastic hinge at the root of the pile body. A larger impact mass is more likely to cause an offset of the vibration center of the pile body. The larger the impact energy, the smaller the impact mass threshold, and the increase in the impact mass leads to an increase in the inertial force, which is more likely to have the second and third impacts with the pile body.(2)
The increase in impact energy has a significant impact on the magnitude of the impact force, the growth rate of the impact force, the bending moment of the concrete pile core, and the maximum lateral displacement of the pile top. All of them increase significantly with an increase in impact energy, and the linear increase in the maximum displacement of the pile top is better.(3)
Under the same impact energy, with an increase in the impact mass, the maximum impact force decreases; the maximum bending moment of the concrete pile core increases, the vibration center value (platform value) increases, the maximum displacement of the pile top increases, and the vibration center value (platform value) increases.(4)
Under the same impact energy, the impact mass, impact velocity, and maximum displacement of the pile top satisfy the relationship of the elliptic parabola sphere.



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*Source: 1167025-2022-04-22.xml*