Bridge Structure Dynamic Analysis under Vessel Impact Loading considering Soil-Pile Interaction and Linear Soil Stiffness Approximation

)e appropriate modeling of the soil-pile interaction (SPI) is critical to get the reasonable dynamic responses of bridge structure under impact loading. Of various SPI modeling approaches, utilizing p-y and t-z curves is a common method to represent the nonlinear lateral resistance and skin friction of pile-surrounding soil. )is paper accomplished SPI modeling for the bridge pylon impact analysis with compression-only nonlinear springs and linear dashpots. )e kinematic interaction and pile group effect were incorporated into the SPI. A variety of pylon impact analyses were conducted under energy-variation impact loads. )e structure dynamic responses were compared and discussed considering the influences of pile group effect, soil damping, and axial t-z spring. An approximate approach was proposed to derive the linearized stiffness of soil for the purpose of engineering calculation. It was concluded from the extensive simulations that the impact load generated from higher initial energy induced more significant structural responses and larger soil inelastic deformation than smaller initial energy. )e piles in the leading row possessed larger bending moments, whereas they exhibited smaller pile deformation than the responses of trailing row piles. Soil damping applied in SPI played positive roles on the reduction of structural responses. Replacing the t-z spring by fixing the degreeof-freedom (DOF) in the vertical direction was capable to yield satisfactory results of structural responses.)e proposed linear soil stiffness was demonstrated to be applicable in the SPI modeling of structure impact analysis.


Introduction
Catastrophic accidents from the vessel-bridge collisions have raised concerns about the bridge safety under impact loading.Design methods and protective measures of bridge structures under vessel impact action get a lot of attention from engineers and researchers in recent years [1][2][3].e static equivalent vessel impact forces were recommended in the design codes and specifications of many countries and communities [4][5][6][7].However, previous studies had shown that the vessel-bridge collision is a dynamic interaction process, and the dynamic analysis approach should be adopted to accurately quantify the bridge responses due to impact action [8][9][10].In the structure dynamic analysis under vessel impact load, the impact responses of the whole bridge structure, especially for the substructure, were strongly influenced by soil-pile interaction (SPI) [11,12].It is necessary to take the SPI into account for obtaining reasonable bridge dynamic responses [13].
ere are several alternative methods to accomplish SPI in the bridge foundation modeling for structure dynamic analysis.e soil continuum assumption based on elastic mechanics is often employed to investigate the soil resistance to piles while sometimes it is difficult to consider the nonlinear soil properties in the analysis process [14,15].With the development of the computer technique, the finite element method (FEM) and boundary element method (BEM) are used to model the SPI for structure analysis [16,17].However, the analysis approaches of FEM and BEM are not suitable to be adopted in the preliminary design stage of the bridge structures due to its time-consuming and the complicated settings in computation.Based on the Winkler foundation theory, the surrounding soil also can be discretized into a series of springs to represent its resistance.Springs are assumed along the pile length due to its clear physical concept and convenience in application (sometimes working in coordination with the damping dashpots), which has become a common approach to simulate the pilesurrounding soil in the engineering practice.
e SPI modeling approach based on Winkler foundation theory can be classified into two categories according to the features of soil stiffness.e first approach is that the soil deformation behavior is assumed as elastic, and the soil spring has a constant stiffness.Various methods recommended in the structure design codes, such as "m-method" and "k-method," are adopted in the engineering application [9,18].Since the constant stiffness assumption ignored the soil inelastic deformation behavior and may not be applicable under dramatic lateral loading such as earthquake and vessel impact, the p-y curve for soil lateral resistance and t-z curve for the soil-pile skin friction are more appropriate in the SPI simulation in this study [19].
In the structural dynamic analysis considering SPI, the soil resistance consists of two individual parts: (1) the displacement-dependent spring force and (2) the velocitydependent damping force [11], whereas the traditional static p-y curve ignored the resistance contribution by the ratedependent effect and may underestimate the soil resistance.Moreover, in a pile group foundation, the leading piles may "shade" the subsequent trailing piles which leads to less resistance compared with the sum of the single pile resistance.It is necessary to consider the various soil resistances for the piles in different rows of pile group foundation.Although the p-y soil curve shows advantages in the modeling of soil resistance, linear stiffness of the soil resistance is still preferred in the engineering practice and reasonable approximation is in great demand to get the linear stiffness from the p-y soil curve.
In this paper, the soil-pile interaction (SPI) was considered in the pile group foundation modeling of cablestayed bridge pylon.A variety of bridge pylon impact analyses were conducted when subjected to energy-variation impact loads.e influences of pile group effect, soil damping, and axial t-z spring were compared and discussed on the structure and soil responses.An equivalent linear soil stiffness approximation method is proposed and validated for the purpose of engineering application.

Finite Element Modeling of Bridge Pylon.
Cable-stayed bridge, with superior spanning ability, is frequently built over the inner river navigation channel and near-sea area.In the life cycle of a cable-stayed bridge, ensuring the safety of the bridge pylon under the accidental action, such as vessel collision, is of prime importance to the overall structure.It is required to consider vessel collision action in the design of cable-stayed bridge.erefore, a cable-stayed bridge pylon with elevated cap-pile group foundation was taken as an example to conduct the dynamic analysis under vessel impact in this study.
e clear height of the bridge pylon employed in this study is 190.5 m, and the thickness of the cap is 9.5 m.A pile group foundation consisting of 15 cast-in-situ bored piles will support the vertical load and the lateral load.e length of each pile is 42.5 m. e detailed configuration of the pylon is shown in Figure 1(a).e finite element (FE) model of bridge pylon was built for conducting the vessel-bridge transient impact analysis.e structure components above the cap, as well as the drilled pile, were modeled by using 2-node beam elements.e pylon cap was modeled with a concentrated mass which reflected its significant contribution to the total mass of the bridge pylon. is mass element was connected with the pylon and the piles with rigid coupling links (Figure 1(b)).e material model of structure was assumed to be elastic.e concrete density of pylon is 2550 kg/m 3 , and Poisson's ratio is 0.2.Young's modulus for the pylon and the foundation are 3.55E4 MPa and 3.00E4 MPa, respectively.

p-y and t-z Curves for Surrounding Soil Resistances.
e soil layer profile and its primary properties are shown in Figure 2. e soil layers are fine sand, soft clay, and moderately weathered sandstone from the top of the river bed to the pile tip, respectively.e surrounding soil along the pile length will provide vertical and lateral resistance for the pile group to support the whole structure.
Extensive work has been done to experimentally get the p-y curves for cohesionless sandy soils.Herein, the p-y curves for the 1 st layer sand were developed according to the approach proposed by Reese et al. [20].In the sand p-y curve, it usually consists of four segments as shown in equation (1).Except for the 2 nd segment in the Reese's p-y curve expression presented as parabolic form, the other three parts are all expressed as linear forms: where p is the soil resistance per unit length of pile, k is the subgrade modulus of sand, z is the depth under the subgrade surface (m), and y is lateral deflection of pile.e other undetermined parameters of y k , y m , p m , y u , p u in equation ( 1) are related to sand properties of internal friction angle and unit weight.More detailed procedure to calculate these parameters and develop the sand p-y curve can be found in FHWA's COM624P manual by Wang and Reese [21].e lateral resistance model for the 2 nd layer soft cohesive soil is characterized by Matlock [22].e p-y curve can be generated according to the following equation: where y 50 is the de ection at 50% of the ultimate soil resistance strength and equals to 2.5ε 50 D. ε 50 is the soil strain corresponding to one-half of the maximum stress in laboratory undrained compression tests.e ultimate lateral soil resistance of p u can be calculated as where c ′ is the e ective unit weight of soil in kN/m 3 , c u is the undrained shear strength of soil (kN/m 2 ), D is the pile diameter (m), J is a dimensionless empirical constant ranging from 0.25 to 0.50 with 0.5 recommended for soft normally consolidated clay, and z R,soft is a transition depth below which the value of (3c u + c ′ z)D + Jc u z exceeds 9c u D. e p-y curve for the third medium weathered sandstone is based on the recommendation by McVay and Niraula [23].e lateral resistance of sand rock, p, is considered as independent of the depth of covered soil.Two linear segments form the p-y relationship of sandstone.Normalizing where q u is the unconfined compressive strength in kN/m 2 .

Soil-Pile Interaction Modeling.
Realizing the importance of the SPI to the responses of bridge substructure-subjected impact loading, the kinetic interaction between the pile and soil was accomplished including both the contributions of stiffness and damping.In the present study, the pile and the surrounding soil were discretized into 2 m equally from elevation −2.5 m down to the pile tips (Figure 2).At each elevation corresponding to the pile nodes, the soil lateral resistance was modeled by two perpendicular groups of nonlinear springs and dashpots, as shown in Figure 3. e nonlinear soil spring with the p-y curve was employed to represent the soil static stiffness under lateral loading.Moreover, since soil resistance increased significantly under dynamic loading, a linear dashpot element was set in parallel to the lateral p-y spring accounting for soil rate-dependent effect and radiation damping.e constant damping coefficient is related to wave propagation velocity in soil V and mass density of soil ρ, which can be determined by equation (5) according to the literature [24].A is the surface area of pile that bears the induced force of damping dashpot.Assuming that the wave velocity V is equal to 70 m/s and ρ is 1700 kg/m 3 , A is calculated by multiplying the sublayer depth of 2 m with the pile diameter of 3 m, which is equal to 6 m 2 .e value of damping coefficient, c, is 714 kN/(m/s) in this study: c � ρVA. (5) In addition to the SPI in two perpendicular lateral directions, the axial soil resistance is of great significance for the pier to transfer lateral impact load to group-pile foundation.Conventional approach to model the axial SPI utilized t-z curves, and a nonlinear spring was placed at each pile node along the pile length (Figure 3).e pile bottom node was fixed rigidly to represent the rock-socketed effect.

Pile Group Effects.
e lateral resistance of a single pile within a group is less than that of a single isolated pile, which is called "group effects."e common approach to account for this effect is to multiply the p in the p-y curve with a reduction factor called p-multiplier to scale down the soil resistance.In addition to the influence by pile spacing, the value of p-multiplier also depends on the pile position in the loading direction since the leading piles carry more load than the trailing piles.As shown in Figure 4, the 1 st row piles acted as the leading pile in the positive movement direction, while these leading-row piles might transform into trailingrow piles as piles were moving back to the negative direction [11].
erefore, the p-multipliers of piles were changing depending on the direction of pile motion.e choice of "pmultiplier" in this study was based on the recommendations in AASHTO LRFD Bridge Design Specifications [25].e pmultiplier of the 1 st , 2 nd , and the last remaining rows of piles are 0.7, 0.5, and 0.35, respectively.

Vessel Impact Loads for Various Impact Energies
A cargo ship collided with a rigid wall was simulated to get the impact load time history with the use of explicit dynamic FE software, LS-DYNA (Figure 5) [26].Different levels of impact energy were simulated by adjusting the vessel tonnage m and its impact velocity V. ree cases, i.e., lowenergy impact, medium-energy impact, and high-energy impact were taken to conduct the vessel-bridge pylon impact analysis.e specific impact parameters as well as the key features in impact loads are listed in Table 1.e impact forces with respect to the time and crush depth obtained from the FE analysis are presented in Figure 6.
From the impact loads shown in Figure 6 and the key features listed in Table 1, it can be observed that the peak impact force P max increased with the vessel impact energy, as well as the impact load impulse (the area surrounded by the impact load time history).Higher impact energy also caused deeper and more severe vessel bow crush, which demonstrated that the vessel bow absorbed more energy in the high-energy impact process.

Structure Transient Impact Analysis
Structure impact analysis was conducted by applying the impact load obtained from Section 3 on the bridge pylon.For all impact cases, the input load was applied at the node of the pylon column which was above 23 m from the cap (Figure 1).Except for the soil damping considered in the SPI modeling, a global Raleigh damping with a ratio coefficient of 0.05 was applied for the structure.e peak responses might occur either at the impact loading phase or the 1 st  Advances in Civil Engineering cycle of free vibration, and the simulation time should be long enough so that the peak responses could be captured.
According to the 1 st natural modal period of pylon (3.06 s) and the impact durations (≤2.3 s), the analysis duration was set to 8 s.

Pylon Dynamic Responses under Various Impact Loads.
Dynamic analysis of bridge pylon was performed under the impact loads of low, medium, and high initial energies.As shown in Figure 7, it can be clearly found that the lateral displacements of both the cap and pylon top of high-energy impact were much larger than those of other two impact cases.e cap displacement was 6 cm approximately, and the pylon top displacement was nearly 16.5 cm from the high-energy impact.
To fully understand the pile deformation and soil resistance under various impact loads, the pile displacement in the 5 th row of the pile group length at the cap maximum lateral deformation (y max,cap ) was given in Figure 8. e pile endured the largest lateral displacement under high-energy impact, and the maximum pile displacement 6 cm occurred at the top of pile.e maximum pile lateral displacements for medium-and low-energy impact cases were 2.2 cm and 0.6 cm, respectively.
Figure 9 displays the soil p-y curves of the 1 st sublayer as the pile swayed to the positive direction.In the medium-and high-energy impact cases, the pile lateral displacement at the corresponding elevation of the 1 st sublayer exceeded the soil elastic deformation limit y el � 0.009 m and the surrounding soil went into the nonlinear deformation stage, while the displacement of pile in low-energy impact cases was less than the y el that indicated the surrounded soil remained elastic.Except for the dynamic p-y curves obtained from the impact analysis, the static p-y curve was also presented for the purpose of comparison.It can be observed that significant differences existed between the dynamic p-y curves from impact analysis cases (especially for the medium-and highenergy impact) and static p-y curve.Soil resistance increased significantly owing to the incorporation of the linear dashpot accounting for the soil resistance rate-dependent increase.Larger velocity of pile movement excited by the impact load will lead to more significant increase on the soil resistance, as well as the loading stiffness of soil.As seen from Figure 9, the initial dynamic tangent stiffness k d 0 � 14044kN/m from the impact analysis was more than three times larger than the k s 0 � 4213kN/m of the static p-y curve.In general, considering the nonlinearity and rate-dependent increase on the resistance of soil is necessary for the dynamic analysis of pile that endures large lateral deformation.

Pile Dynamic Responses in Different
Rows.Different p-multipliers were assigned to the p-y curves corresponding to piles in different rows according to Section 2.4, such that the deformation and inner forces of piles in each row might     Advances in Civil Engineering exhibit distinctly.As the pylon cap reached its maximum positive displacement (y max,cap ), the 5 th row piles had a little larger lateral deformation than the 1 st row piles since the trailing 5 th row piles had smaller p-multipliers than 1 st row leading piles.While as the pile group moved back to minimum negative displacement (y min,cap ), the 1 st row piles turned into trailing row piles such that the p-multipliers were changing accordingly.erefore, slightly larger displacement was found in 1 st row pile rather than 5 th row piles at displacement of y min,cap .
e percentages of the lateral load undertaken by piles in different rows are proportional to the stiffness.us, the leading rows which had larger p-multiplier engaged more to take the impact load.e bending moment of the pile shown in Figure 10(b) agreed well with this inference, which the 1 st row at y max,cap and 5 th row at y min,cap presented larger bending moments than 5 th row at y max,cap and 1 st row at y min,cap , respectively.

Effect of Soil Damping on the Structural Responses.
Rate-dependent soil resistance and energy dissipation within the soil influence the structure dynamic responses under impact loads significantly.Except for the viscous damping constant c � ρV s used in the SPI modeling of Section 2.3, the other three SPI models without damping or with damping constant of c � 2ρV s and c � 3ρV s were built to investigate the SPI damping effect on the structure impact responses.All the simulations in this section were conducted under highenergy impact load.As given in Figure 11 of the displacement of cap and pylon top, it can be observed that larger displacements were yielded by the models with smaller c in SPI, which indicated that the incorporation of soil damping will improve the soil resistances for the dynamic impact situation.
As presented in Figure 12, the cap displacement in the soil model without damping was larger than that of the structure model with soil damping.us, larger pile displacement by the model without soil damping was obtained rather than other cases as well.Consequently, the bending moment induced by the pile deformation became smaller as the soil damping increased.

Effect of Axial t-z Spring on the Structural Responses.
Normally, the axial SPI is modeled with nonlinear t-z springs placed along the length of the pile.In some cases for simplicity, axial SPI is ignored by replacing the t-z springs as fixed axial degree-of-freedom (DOF) of the pile nodes.To further investigate the influences of axial SPI modeling approach on the impact responses, the models employing axial t-z springs and fixed DOF in axial direction were used to conduct the impact analysis.
Relatively close lateral displacements can be obtained by fixing the axial DOF compared with the SPI model by establishing t-z spring (Figure 13).In addition, the pile lateral displacement with fixed axial DOF at the moment of cap maximum positive displacement was nearly identified with the model with the t-z spring.e pile bending moment by the axial DOF fixed model was slightly larger than that of the model with axial t-z spring (Figure 14). is comparison results demonstrated that fixing the pile DOF in z direction, instead of using t-z spring, is capable of giving reasonable structure results under impact load.

Linear Approximation for the p-y Soil Curve
e p-y method is commonly recommended in the modeling of SPI for bridge seismic or vessel-substructure impact analysis.While using this approach needs numerical iteration, nonlinear spring element is employed in the analysis, which leads to low computational efficiency and is not convenient in engineering practice as well.In order to overcome these shortcomings, a linear stiffness approximation approach was presented for the SPI modeling in vessel-bridge impact analysis.A flow chart given in Figure 15 illustrates the procedure to get the linear stiffness of soil in each sublayer.A structure model with the p-y soil curve was used to conduct static analysis under maximum impact load P max ; thus, the resistance versus lateral displacements of pile node at each elevation along the length can be obtained.Assuming 6 mm is a critical limit to estimate whether the soil is being elastic or inelastic stage [7], different approaches were adopted to linearize the stiffness of soil.For the pile node lateral displacement exceeding 6 mm, the secant stiffness (k 1 ) in this soil layer was taken as the linear approximation of the soil p-y curve; otherwise, the linear spring attached with the pile node employs initial tangent stiffness of soil p-y curve (k 0 ) (Figure 16).
Under high-energy vessel impact load, the bridge pylon lateral displacements as well as the pile responses from the models with different SPI considerations were shown in Figures 17 and 18, respectively.e structural responses by the model with tangent soil stiffness were added for the purpose of comparison as well.In Figure 17(a), the cap displacement by the model with soil secant stiffness was larger than the one with the p-y curve (16% larger on the peak value).While the model with k 0 in soil spring yielded smaller cap displacement compared with the p-y curve model.e model with k 1 spring gave relative close prediction on the pylon top lateral displacement rather than a smaller value by the model with k 0 spring.e pile maximum lateral displacements obtained from the models with k 0 and k 1 springs exhibited either smaller or larger than the p-y model yielding displacement, respectively (Figure 18(a)).It indicates that neither tangent stiffness nor secant stiffness is an appropriate approximation for the soil p-y curve.Adopting tangent stiffness overestimates the soil stiffness while secant stiffness is a bit soft for the SPI modeling.Alternatively, the secant stiffness of soil at 50% maximum lateral pile displacement (0.5y c ), denoted as k 0.5 (Figure 16), was taken as the linear soil stiffness approximation to overcome the shortcoming by utilizing k 0 and k 1 stiffness.As comparisons in Figures 17 and 18, the structure displacements yielded by the k 0.5 model agreed better with the displacements by the p-y curve than other models with k 0 and k 1 soil stiffness, which demonstrated that the secant stiffness at 50% of y c was able to give rational responses as an approximation for the nonlinear p-y curve.Despite the pile Advances in Civil Engineering  Advances in Civil Engineering bending moment by soil k 0.5 model was larger than that of the p-y curve model above −34 m in elevation, it still gave conservative results which tend to be safe for the engineering design.

Concluding Remarks
Extensive bridge pylon dynamic analyses were conducted under vessel impact loads.e soil rate-dependent resistance and pile group e ect were incorporated into the SPI modeling.e structure and pile responses under the impact load with three various energy levels were compared and discussed, and the in uences of pile group e ect, soil damping, and axial t-z springs were also investigated.An approximate linear soil sti ness was proposed and validated to replace the p-y curve for engineering application purpose.Several ndings and conclusions can be drawn according to the analysis results: e largest structure and soil responses were achieved under high-energy impact load, and the pile-surrounding soil endured signi cant inelastic deformation.Moreover, the dynamic soil p-y curve under high (or medium) impact energy increased signi cantly compared with the static p-y curve.It is necessary to take the soil nonlinearity and ratedependent e ect into the SPI modeling in the structure impact analysis.Since larger p-multiplier was applied in the leading row of the pile group, these piles bore higher proportion lateral load than other piles while smaller displacement was yielded.e incorporation of soil damping would improve the soil resistance, and larger soil damping would yield smaller structure displacement and pile bending moment.Fixing the pile DOF in axial direction, instead of installing t-z springs, was able to give satisfactory predictions on the structural responses.Neither the tangent nor secant sti ness of p-y could be used to replace the nonlinear p-y curve to conduct the structural impact analysis, while the proposed secant sti ness of soil at 50% maximum lateral pile displacement, k 0.5 , was capable of give more reasonable results under vessel impact load.Advances in Civil Engineering

Figure 4 :Figure 5 :
Figure 4: p-Multipliers in each row of the pile group.

Figure 6 :Figure 7 :Figure 8 :Figure 9 :
Figure 6: Impact load comparisons between the low-and high-energy impact.(a) Impact load time history.(b) Impact load versus crush depth.

Table 1 :
Impact parameters and load characteristics.