Abstract
Three-dimensional finite element dynamic analyses were conducted to evaluate the effect of barge impact of different speeds on the lateral performance of three pile group (PG) configurations (vertical, battered, mixed) in terms of peak lateral displacement, peak shear force, lateral stiffness of PGs, and contribution of piles and pier columns to the total resisting force. The concrete material was modeled using the Concrete Damaged Plasticity model. The clay behavior was described using the Modified Cam Clay model; while the sand was modeled using the Drucker Prager model. The results showed that the vertical PG had the lowest lateral stiffness; while the battered and mixed PGs had closely larger stiffness, which allowed the development of larger impact force and pile cap displacement in vertical PG as compared to other PGs. The results of shear force showed that the foundation contributed up to 82% of the total lateral force in battered and mixed PGs. The results also showed that in vertical PG, the first row had slightly higher percentage of lateral force compared to other rows; while in battered PG, the load percentage was evenly distributed between rows. A contrast performance was observed for mixed PG. The results of bending moment (BM) showed that the piles in vertical PG had notably higher BM than the piles in other PGs due to larger pile cap displacement. In all PGs, the piles in the edge column had higher BM ratio, with the leading row (L) had the highest BM ratio (69–72%) in all PGs.
Keywords
With the increase in the global waterway transportation volume and therefore increase in the number of inland vessels passing underneath bridges over waterways, waterway bridge structures are becoming more susceptible to accidental vessel collisions. Therefore, vessel bridge collisions have been widely considered as a critical issue in bridge design, especially after several serious accidents since the 1960s that resulted in significant casualties and economic losses ( 1 , 2 ). Sha and Hao ( 3 ) reported that “at least one serious vessel collision occurs worldwide each year.” For example, one of the catastrophic incidents was the collapse of the Sunshine Skyway Bridge in Florida in 1980 after the collision of a freighter into one of the bridge piers ( 4 ). Therefore, it is very important to design waterway bridges to resist dynamic vessel collision loads.
The current design practice (specifications and codes) of bridges against barge impact is based on empirical formulae, which in turn are based on equivalent static load (e.g., [ 5 , 6 ]). The most common equivalent static methods are the American Association of State Highway and Transportation Officials (AASHTO) ( 5 ) and Eurocode specifications ( 6 ). The AASHTO method estimates the equivalent static load based on the barge’s kinetic energy (KE) and the barge bow crushed depth ( 5 ), while Eurocode introduced a simplified method to evaluate the impact load with time based on the initial KE and the impact angle ( 6 ).
The current AASHTO guideline requires all bridge components in a navigable waterway crossing to be designed for barge impacts ( 5 ). The determination of loads from barge–bridge pier collisions should consider factors such as size, type, and frequency of vessels passing, bridge and channel geometry, available water depth, vessel speed and direction, and the structural response of the bridge ( 1 ). The AASHTO design impact force (equivalent static force) from a barge is estimated using empirical relations based on the KE and barge bow crush depth (aB) as follows:
The available design codes and specifications provide guidance for designing bridges against barge impacts. However, since the vessel impact implies lateral dynamic load, the proposed equivalent static methods cannot accurately predict the actual dynamic response of bridge piers subjected to barge impacts ( 7 , 8 ). Furthermore, the equivalent static methods ignore many factors, such as the pier geometry, material nonlinearity, and impact duration ( 3 ). Davidson et al. ( 7 ) suggested considering the dynamic behavior of barge impacts in the design of bridges by comparing the dynamic response to the static response of bridge piers.
The dynamic response of bridge piers to a barge impact can be evaluated through conducting experimental impact tests or by conducting finite numerical analysis. For example, full-scale field barge impact tests were conducted by Consolazio et al. ( 4 ) on two piers of the old St. George Island Causeway Bridge in Florida. During the tests, the dynamic impact force and the responses of the barge, pier, and surrounding soil were recorded. While the experimental impact tests can provide valuable measured data on the dynamic response of piers during and after the barge impact, they are very expensive and time-consuming. The finite element (FE) numerical method can provide an alternative solution to expensive experimental impact tests on simulating the complex performance of bridge piers because of barge impact problems. FE models can be calibrated and validated using the results of experimental barge impact case studies.
During the past two decades, several FE numerical studies have been performed to investigate the impact load from barge impacts and the factors affecting it, such as speed, pier column shape and width, vessel weight, and bow stiffness (e.g., [ 3 , 9 – 20 ]) The main focus of these studies was to propose and validate a simplified model to predict the magnitude of the impact load. The most versatile studies were performed by a research group at the University of Florida, in which they used the experimental impact tests at St. George Island Causeway Bridge to calibrate and validate FE numerical models for the barge impact problem developed in LS-DYNA software ( 9 , 10 ). The barge components were modeled using four-node shell elements. The barge components were modeled using the elastic–plastic material model MAT_PIECEWISE_LINEAR_PLASTICITY. The pier was modeled using eight-node solid brick elements. Their numerical results indicated that the current AASHTO equivalent static force model underestimates the magnitude of impact force for low to moderate energy impacts and overestimates it for high-energy impacts. They also found that the shape of the affected pier has a significant influence on the peak impact force ( 10 ).
Sha and Hao ( 3 ) developed a numerical model using LS-DYNA software to study the influence of different materials (rigid, elastic, and nonlinear) on the impact force and barge crush depth. They used an elastoplastic material model (MAT_PLASTIC_KINEMATIC) that considers the strain rate effect described by the Cowper–Symonds equation was used to simulate the outer shell and internal truss of the barge bow. An elastic visco-plastic model was used for the structural steel, and the damage material model (MAT_CONCRETE_DAMAGE_REL3) was used to describe the behavior of the concrete in the bridge pier. They found that the nonlinear behavior of the bridge pier and damage significantly affect the response of the barge–pier collision. They developed simplified empirical equations to predict the impact force time history based on the barge impact KE. Wang and Morgenthal ( 15 ) used nonlinear FE analyses to develop efficient models to evaluate the dynamic behavior of bridge piers under barge impacts. In their analysis, the barge was modeled using a simplified mass-spring model, while the pier column was modeled using discrete masses and discrete beam elements. In their analysis, the barge was divided into two zones, front zone1 that was modeled using piecewise linear plasticity material (MAT_PIECEWISE _LINEAR_PLASTICITY) in LS-DYNA, and zone2 that was modeled using rigid material (MAT_ RIGID). The pier column was modeled as linear elastic with a fixed base and cantilevered top. Song and Wang ( 16 ) used the results of FE analysis of the impacts of five ships based on the ship–rigid wall collision model to develop a simplified analytical model for predicting the time history of impact loading on bridges. To validate their model, they conducted several drop hammer impact tests using scaled steel boxes. In the FE models, the ship was modeled using four-node shell elements, while the rigid wall was modeled using hexahedron solid elements. The ship bow was modeled using the elastic-plastic model (MAT_PLASTIC_KINEMATIC) in LS-DYNA, the ship body was modeled using the rigid material model (MAT_RIGID), the ship rigid wall was modeled using the rigid material model (MAT_RIGID), the concrete and the steel of the bridge structure were modeled using the linear elastic material model (MAT_ELASTIC), and the surrounding soil was modeled using spring elements. Fan et al. ( 17 ) developed FE models to simulate the barge impact on a four-span continuous girder bridge. The bridge columns were simulated using eight-node solid elements with single-point interaction and the bridge girders and piles were simulated using beam elements. The nonlinear continuous surface cap elastoplastic model was used to simulate the concrete. The bridge bow was modeled using the elastoplastic model. The bridge girders and piles were modeled using the linear elastic material model (MAT_ELASTIC) in LS-DYNA. The longitudinal and transverse bars were modeled using the elastic-plastic model (MAT_ PLASTIC_KINEMATIC). Their results showed that the impact force–time history has a significant effect on the responses of the bridge. They proposed to use ultra-high-performance fiber-reinforced concrete (UHPFRC) to strengthen the columns and improve the impact-induced performance of bridge. They proposed a design procedure to evaluate the dynamic response of the UHPFRC-strengthened columns. Chen et al. ( 19 ) developed FE models to simulate the impact of six barges with different sizes and study the effect of the geometric effects the barges on the bridge response. In their FE model, the barges were modeled using four-node shell elements and the bridge structures (pier, girder, and piles) were modeled using beam elements. The contact zone of the collided pier was modeled using hexahedron solid elements. The barge bow was simulated using an elastoplastic constitutive model with the strain rate effect, and the barge body was simulated using a rigid material model. A linear elastic model was used to simulate the concrete and steel of the bridge. The soil–pile interaction was simulated using spring elements. They proposed a simplified impact load time–history model for barges of different sizes, which they claimed was able to predict bridge responses under barge impacts.
Most of the existing FE models in the literature for simulating the barge impact on bridge piers assumed the pier as rigid or elastic and the piles as elastic beams, and the soil–pile interactions were modeled using springs (non-continuum), thus ignoring the dynamic properties of the surrounding soil. In this study, three-dimensional (3D) FE dynamic numerical models were developed in ABAQUS to study the lateral behavior of pile groups (PGs) with different configurations (vertical, battered, and mixed) of similar numbers of piles subjected to barge impacts with different speeds using the case of the M19 pier of the I-10 Twin Span Bridge in Louisiana. In these models, the superstructure, PG foundation, and soil body were included for the vertical, battered, and mixed PG cases. In this study, the concrete material was modeled using the concrete damaged plasticity (CDP) model; the steel reinforcement was modeled using embedded shell elements with the elastoplastic von Mises model; the soil was modeled as a continuum using the elastoplastic modified cam clay (MCC) model for the clay layers and the Drucker–Prager (DP) model for sand layers; the pile–soil interface was simulated using the contact model with the Coulomb frictional criteria; and the dynamic behavior of soils was modeled using the two-parameter Rayleigh damping model.
Studied Pile Group Cases
This study was conducted on three types of PG foundations (vertical, battered, and mixed) being hit by a barge impact at different speeds: 2 knots (3.37 ft/s), 4 knots (6.75 ft/s), and 6 knots (10.12 ft/s). These speeds represent slow to moderate level navigation speeds for barges ( 5 ). The barge weight was maintained at 1873 tons in all cases. In the FE models, the barge was set to hit the middle of the pile cap, which was determined from the mean water level for the M19 PG foundation (Figure 1). The design barge specifications of AASHTO suggest that the draft for a fully loaded barge is 8.7 ft. In the horizontal plane, the centerline axis for the barge was aligned with the cap central axis.
Positioning of the barge and impact point.
FE Model of a Barge
The Jumbo Hopper (JH) barge has been used in many FE simulations of barge impacts (e.g., [ 3 , 9 , 11 – 13 ]). The standard JH barge specs per AASHTO guidelines was used in the current study ( 5 ). The actual barge dimensions are 195 ft L × 35 ft W × 12 ft D, and the total weight is 1873 tons for a fully loaded barge. The JH barge body can be separated into two main regions: the bow rake and the cargo region (Figure 2). The entire JH barge is built from trusses and steel plating. In this study, the bow rake model was built of 3D beam and shell elements. The beam elements were used to model the internal truss frames, while the shell elements modeled the outer steel plating. The total number of truss frames was 14, spaced at 2.16 ft in the lateral direction (Figure 3).
Finite element (FE) model for the Jumbo Hopper barge.
Bow rake finite element model details.
Each truss frame was built from A36 steel beams with L and C structural shapes. The frames were mirrored at the barge’s center line and laterally braced by two tie beams (L and C shaped) at the front of the bow and at the bottom of the diagonal member, respectively. All components of the truss frame were modeled using 3D beam elements (B23), and the beam junctions were numerically constrained to simulate the effect of stiffened joints. The outer shell was built from A36 steel plating with 3/8” thickness and modeled using 3D shell elements (S4R). The total number of elements used in the bow rake model was ∼3200.
The cargo region was modeled as a solid block of A36 steel with a total weight of 1873 tons. The elements used were 3D solid continuum elements (C3D8R) with a total number of ∼500. The bow rake and cargo region were numerically tied using numerical constraints so that they behave as a single unit.
The A36 steel material behavior was simulated using the elastoplastic von Mises constitutive model. The A36 steel elastic properties used in the model were the Young’s modulus, Est = 29,000 kips per square inch (ksi), and the Poisson’s ratio, ν = 0.26.
FE Model of the Pile Group and Bridge Pier
The M19 pier foundation of the I-10 Twin Span Bridge was used to study the lateral behavior of the three PGs subjected to barge impacts. A detailed description of the M19 pier can be found in previous publications ( 21 , 22 ). The FE model was created from three main components: the pier superstructure, the bridge deck and girders, and the PG foundation. To select the size of mesh in the FE model, different meshes with different degrees of refinement were first tried to select the proper mesh size such that the results are independent of the mesh size. Full details of the FE numerical modeling of barge impacts can be found in Souri ( 23 ). A brief description of the model will be provided below.
Pier Superstructure
The concrete pier superstructure comprised the protection wall, columns, and the seating beam for the girders (Figure 4). The pier superstructure was modeled as a solid part using the 3D solid continuum element (C3D8R) with a total of ∼2200 elements. The concrete material behavior was assumed to be linear elastic with the elastic properties of Young’s modulus Ec = 3800 ksi and Poisson’s ratio ν = 0.2. The unit weight of concrete used in the model was 150 pounds per cubic foot (pcf).
Geometry and finite element (FE) mesh for the pier superstructure.
Girders and Bridge Deck
The pier superstructure supported 12 girders and two concrete decks. The girders used in the M19 case were LG-78 concrete girders spaced at 10.75 ft and had 200 ft span length. The section detail for LG-78 girders was simplified in the FE model to an approximate I-section with respect to the cross-sectional area and second moment of area. The concrete decks were modeled as solid slabs of 8 in. in thickness using 3D shell elements (S4R) and connected to the girders using tie constraints (Figure 5). The concrete material for the girders and decks was assumed to be linear elastic with similar elastic properties to the pier superstructure (Ec = 3800 ksi, and ν = 0.2). The unit weight of concrete for the girders and deck used in the model was 150 pcf.
Finite element (FE) model for the girders and deck.
Modeling of the Pile Group Foundation
The M19 PG foundation comprised 24 battered piles (slope of 1H:6V) and a pile cap. The pile spacing at the cap level was 4.3D between rows and 2.5D between columns. In addition to the battered PG, FE models were also developed and analyzed for vertical and mixed PGs for barge impact (Figure 6). The geometry and dimensions of the PG models were adapted from the M19 battered PG. The prestressed concrete piles had a 3 ft square section and was 110 ft in length. The pile cap was built of cast-in-place concrete and measured 44 ft × 42.5 ft × 7 ft. The main steel reinforcement (Grade 270 steel) in the piles comprised 36 × 0.6′′-diameter steel tendons. More details on the M19 pier and PG foundation can be found in Abu-Farsakh et al. ( 21 ). The PG, including the pile cap, was modeled as a single mesh using the 3D solid continuum element (C3D8R) with a total of ∼21,000 elements. The main steel reinforcement was modeled using the embedded shell elements (S4R), which allowed one to incorporate prestress force in the piles. The concrete material was modeled using the CDP constitutive model. The concrete elastic properties for the piles were Ec = 5000 ksi and ν = 0.2. The steel material was modeled using the elastoplastic von Mises model with the elastic properties of Es = 29,000 ksi and ν = 0.26.
Pile group geometry and finite element (FE) models (D is pile width = 3 ft).
FE Soil Models
The soil profile for the M19 site was used in this study. The soil layering was divided into six clay layers and two sand layers. The FE mesh for the soil layer was built from two types of elements: standard elements and infinite elements. The standard elements were 3D solid continuum elements (C3D8R) that were assigned to the central part with elastoplastic material behavior. The infinite elements (CIN3D8) were used on the far boundaries to provide the equilibrium reaction and absorb the propagating stress waves. The total number of elements used in the soil body was ∼153,000. The clay material behavior was modeled using the MCC model, while the sand material was modeled using the DP model. The material parameters for both models are summarized in Table 1. The soil layering and soil properties were determined based on the results of one soil boring with laboratory testing and five in situ cone penetration tests (CPTs) that were conducted at the M19 pier of the I-10 Twin Span Bridge over Lake Pontchartrain in Louisiana. More details can be found in Abu-Farsakh et al. ( 21 , 22 ) and Souri ( 23 ).
Modified Cam Clay (MCC) and Drucker–Prager (DP) Model Parameters Used in the Barge Impact Finite Element Models
Note: pcf = pounds per cubic foot; ksf = kips per square foot; psf = pounds per square foot; na = not applicable.
The soil model parameters used in the barge impact problem were verified using the results of the statnamic test that will be discussed in the following section. In addition, the material damping for the soil material was introduced using the two-parameter Rayleigh damping model. The Rayleigh damping parameters (α, β) were estimated for an average damping ratio ξ = 2% over the frequency range 1–10 Hz based on calibration using an iterative procedure with the results of five cycles from the statnamic test. The damping parameters obtained from the iterative procedure were α = 0.1, β = 0.0013.
Modeling of the Pile–Soil Interface
The pile–soil interface was described using the contact model available in ABAQUS, which simulates the two main mechanisms: the normal (to surface) behavior and the tangential behavior. The normal behavior is referred to as “hard contact,” which considers either separation (i.e., no tension stress is transmitted) or interpenetration (bearing stress is transmitted). The magnitude of normal stress is found through an iterative numerical procedure. The tangential behavior refers to the interaction in the direction parallel to the pile–soil surface, in the form of sticking or sliding. The tangential behavior in the sticking mode transfers the shear stress to the soil with no relative pile–soil displacement. In the sliding mode, the transferred shear stress is limited to a maximum limit value. The maximum shear stress limit for sliding (τlim) is governed by the Coulomb friction criteria characterized by the friction coefficient (μ) and the ultimate friction stress (τmax). The Coulomb friction criteria defined the limit as τc = μσn, where σn is the mean stress at the interface, μ is the coefficient of interface friction (=tan 2/3ϕ), and ϕ is the friction angle of soil.
Verification of Soil Parameters from Statnamic Test Results
The soil and damping parameters were verified using the results of a statnamic test that was performed on test pile 7 (TP7), which was located 135 ft west of the M19 pier. The TP7 was 36-in. squared prestressed concrete pile and 123 ft in total length with similar specs to the piles of the M19 pier foundation. A FE model was developed for TP7 to simulate the statnamic test, in which the soil model and damping parameters in Table 1 were used. The statnamic test was conducted in five consecutive load cycles with increasing peak load after each cycle. The peak load for the first cycle was 33 kips and it increased up to 115 kips in the fifth cycle. The peak displacement during the first cycle was 2 in. and reached 10 in. during the fifth cycle.
The FE model for TP7 comprised the pile and soil body (Figure 7). The pile was modeled using the solid continuum element (C3D8R) with the main steel reinforcement modeled using the embedded shell elements (S4R). The soil body was modeled using the solid continuum elements (C3D8R) for the central region (blue color in Figure 7) and using the infinite elements (CIN3D8) for the outer boundary regions (green color). The number of elements used was ∼2300 for the pile and ∼50,000 for the soil.
Test pile 7 geometry and finite element (FE) model (color online only).
The FE simulation of the statnamic test was conducted in five consecutive loading steps, similar to the in situ test. The duration of the load cycle was less than 1.0 s, and there was a 20–30 min gap after each cycle to allow the pile’s free oscillation to vanish before the next load cycle. The duration for each load step was fixed at 3.0 s, which was verified to be long enough to stop the free oscillation of the pile.
Figure 8 compares the results of pile displacement for each load cycle from the statnamic test and the FE model, which demonstrates very good agreement between the measured and predicted displacements, especially the first three cycles. The notable difference in displacements between the statnamic tests and FE model for cycles 4 and 5 can be attributed to pile damage.
Comparison of pile displacements at the loading point from the statnamic test and finite element (FE) model.
Loads and Barge Impact
The loads applied in the barge impact problem were gravity and damping loads. The gravity load was applied to the soil body, pier superstructure, girders and deck, and PG. The impact load by the barge was introduced by assigning an initial velocity in the horizontal direction to generate barge momentum. The damping loads were introduced to eliminate the oscillations in the whole model induced by the gravitational acceleration, which were removed after the oscillations ceased, so they do not influence the model response during the barge impact.
The barge impact was simulated in two steps of 2 s duration each: the equilibrium (step 1) and the impact (step 2) (Figure 9). In step 1, the model is brought to the near-static condition by allowing the soil to reach a state of geostatic stress equilibrium and dissipating the oscillations caused by the gravitational acceleration. The barge is initially positioned at a distance so that the impact occurs after the end of the step 1, which is estimated from the barge velocity and the step time of 2 s. At the end of step 1, the impact is imminent and the barge bow is barely in touch with the pile cap. In step 2, the barge impact progresses until the barge momentum dissipates and rebounds away from the PG.
Illustration of the steps for simulating the barge impact problem.
Discussion of Results
Lateral Displacement
The resulted lateral displacement history at the pile cap and pier top elevations from the barge impacts for all cases are presented in Figure 10. The displacement history exhibits a large peak in the first 0.2–0.4 s as a response to the impact force, and subsequent smaller peaks corresponding to the free oscillations of the structure. The displacement vanished for the battered and mixed PGs after 2 s, whereas it returns to a non-zero value for the vertical PG. This observation can be attributed to the gap formation in the front soil and the significant damage in the piles for the vertical PG. The battered and mixed PGs were free of damage for all impact cases.
Displacement history results.
The peak displacement at the pile cap was notably higher (2–8.4 in.) in the vertical PG, while in the battered and mixed PGs, it remained within 2–2.5 in. for all barge speeds. The magnitude of peak displacement was mainly affected by the impact force, which depends on the lateral PG stiffness and the barge bow stiffness. The barge’s KE was dissipated during impact by two mechanisms: (1) PG displacement and (2) barge bow deformation. The PG displacement depends on the lateral PG stiffness, which is a function of PG type and soil resistance. When the PG stiffness is low, a greater portion of impact energy is dissipated in the PG displacement, while for very stiff PGs, a significant portion of the impact energy is absorbed by the bow deformation.
The lateral stiffness of the vertical PG is lower than that of the battered and mixed PGs. Therefore, the impact generated greater lateral force and greater pile cap displacement for the vertical PG. The relationship between the peak displacement and barge speed is presented in Figure 11. A linear relation between the displacement and speed is noticed in the vertical PG. In the battered and mixed PGs, the results show that the displacement was almost constant with the increased barge speed.
Peak displacement at different barge speeds.
The barge bow deformation was composed of two parts: elastic and inelastic. As shown in Figure 12, the bow deformation reaches a peak value and then recedes to a lower plateau. The difference between the peak and plateau represents the elastic bow deformation, whereas the value at the plateau corresponds to the inelastic deformation. The figure shows that the inelastic bow deformation in the vertical PG case was significantly lower than that in the battered and mixed PGs for barge speeds greater than 2 knots.
Barge bow deformation history.
Shear Forces and Impact Forces
The shear force history was calculated for each pile and the two pier columns, and the results are presented in Figure 13. The sum of shear forces represents the total impact force. The impact force in the vertical PG case was notably higher than in the battered and mixed PGs. The vertical PG generated larger lateral force from the impact by absorbing more KE. The lower stiffness of the vertical PG allowed larger lateral displacement and greater damage levels in the piles. However, the battered and mixed PGs absorbed less KE because of higher stiffness, which forced the barge bow to greatly deform and absorb a significant amount of KE. The force history in the vertical PG peaks at a higher rate than the other PGs. It can be noticed that the forces in the vertical PG piles return to negative values at the end of impact, which is mainly because of the damage in the piles. The damage in the piles was located on the tension side of the piles and created a negative balance in the pile shears. The negative pile shears were countered by a net positive shear in the pier columns. In the battered and mixed PGs, the duration of the first peak was longer because of bow deformation. At barge speed of 6 knots, the force history for both PGs shows two consecutive force peaks, indicating that the collision was highly inelastic.
Results of shear force history.
The peak shear force in the piles and pier columns versus barge speed are presented in Figure 14. The results show that the peak force increased linearly with barge speed in the vertical PG, while it remained fairly constant in the battered and mixed PGs. The contribution from the foundation and superstructure in the total resisting force is depicted in Figure 15. The results show that the force ratio carried by the piles was constant (82%) in the battered and mixed PGs, while in the vertical PG, the force ratio decreased from 75% to 65% when the barge speed increased. This is mainly because of pile damage that weakened the piles in the vertical PG.
Results of peak shear versus barge speed.
Contribution of piles and pier columns in the resisting force.
A comparison between the peak impact force from the FE analysis and prediction models by AASHTO ( 2 ) and Consolazio et al. ( 10 ) is shown in Figure 16. AASHTO’s impact force prediction model is a function of the barge’s mass, speed, and bow crush depth, while the Consolazio et al.’s prediction model is a function of the pier–barge contact width only. It can be seen that Consolazio et al.’s prediction model works well for the vertical PG case but highly over-predicts the impact forces for the battered and mixed PG cases. This indicates that the lateral stiffness of the PG is an influence factor that is needed in the prediction models. On the other hand, AASHTO’s model under-predicts the impact force in all cases, especially in the vertical PG case.
Peak impact force versus force prediction models.
The contribution of each pile to the total shear force because of barge impact was investigated for the vertical, battered, and mixed PGs. The results reveal slightly higher shear force contribution of the piles in the edge column (col. 1), which was also observed in a previous study for statically loaded PGs ( 24 ). The main reason for this is the influence of the group effect. The contribution of the edge column was fairly constant for all barge speeds and notably higher in the battered and mixed PGs. On average, the percentage difference in the shear force ratio between the edge column (col. 1) and the interior columns (col. 2 and 3) was 0.6% in the vertical PG, 1.7% in the battered PG, and 1.5% in the mixed PG.
Investigating the contribution of each pile row (leading, L; middle leading, ML; middle trailing, MT; and trailing, T) to the total shear force shows that for the vertical PG, the leading row (L) had the highest percentage (27%) followed by trailing row (T) (25%) and rows ML and MT (24% each). The battered PG rows had close percentages at about 25%, which is a different distribution to that we observed from the static analysis ( 24 ). However, the mixed PG had the largest contrast between the rows with 28% in rows L and T, and 22% in rows ML and MT.
Bending Moment
The piles’ bending moment (BM) at the pile cap elevation was calculated, and the results at a barge speed of 4 knots is shown in Figure 17. Similar result patterns were obtained for the barge speeds of 2 and 6 knots, but with different magnitudes. The results of BM history show closely similar responses and contributions from all piles in the PGs. Looking at the peak BM, the piles in the vertical PG had notably higher BM values than the piles in the other PGs, which is because of the large pile cap displacement. The contribution of peak BM for each pile was also calculated and the BM ratio was obtained by normalizing the peak BM for the individual pile using the PG shear (VPG), pile length (L = 110 ft) and the number of piles in the PG (n = 24). The results show that the piles in the edge column (col. 1) had higher BM ratios in all PGs. The difference in BM values between the edge column and the interior columns was 1.3% in the vertical PG, 2.3% in the battered PG, and 2.2% in the mixed PG.
Bending moment history results at barge speed = 4 knots.
The ratio of BM per row was also calculated and is presented in Figure 18, which gives an idea about the BM distribution between the rows (L, ML, MT, T). The figure shows that the leading row (L) had the highest BM ratio (69%–72%) in all PGs. In the vertical PG, the BM ratio was closely similar in rows ML, MT, and T (58%–61%). In the battered PG, the BM ratios were 71%, 57%, 61%, and 67% in rows L, ML, MT, and T, respectively. However, in the mixed PG, the BM ratios were 72%, 65%, 62%, and 68% in rows L, ML, MT, and T, respectively.
Summary of bending moment ratio per row.
Summary and Conclusions
Three-dimensional FE models were developed in this study to investigate the performance of PGs of different configurations subjected to barge impact loading. The behavior of soils was described using elastoplastic constitutive models. The concrete material was modeled using the CDP model. The steel reinforcement was modeled using embedded shell elements with the elastoplastic von Mises model. The pile–soil interface was modeled using the contact model available in ABAQUS, which simulates the normal and tangential behavior. The group interaction was incorporated through the interaction of stress fields around the piles. The effect of barge impacts of different speeds on the lateral performance of three PG configurations (vertical, battered, mixed) with respect to peak lateral displacement, peak shear force, lateral stiffness of PGs, and the contribution of piles and pier columns to the total resisting force was analyzed. Based on the findings of this study, the following conclusions can be made.
Study of the PG subjected to barge impacts showed that the vertical PG had the lowest lateral stiffness; while the battered and mixed PGs had closely similar stiffness, which resulted in larger impact force and pile cap displacement in the vertical PG as compared to the other PGs.
The large stiffness of battered and mixed PGs forced the barge bow to deform significantly, so that the magnitude of peak impact force remained fairly constant for all barge speeds.
The results of shear force showed that the foundation contributed up to 82% of the total lateral force. In the vertical PG, the force distribution was 65%–76% in the foundation and 24%–35% in the pier columns. In the battered and mixed PGs, the force distribution was 82% in the foundation and 18% in the pier columns.
The distribution of lateral force showed that in the vertical PG the first row had a slightly higher percentage (28%) compared to the other rows, which had closely similar percentages (24%–26%). In the battered PG, the load percentage was evenly distributed between the rows at 25%. A large contrast in the percentage was observed in the mixed PG. The first and fourth rows had 28% each, while the second and third rows had 22% each.
The results of the BM showed close responses and contributions from all piles in the PGs, with the piles in the vertical PG having notably higher BM values than the piles in the other PGs because of the larger pile cap displacement. The contribution of peak BM for each pile show that the piles in the edge column had higher BM ratio in all PGs. The results also showed that the leading row (L) had the highest BM ratio (69%–72%) in all PGs. In the vertical PG, the BM ratio was closely similar in rows ML, MT, and T (58%–61%). In the battered PG, the BM ratios were 71%, 57%, 61%, and 67% in rows L, ML, MT, and T, respectively. However, in the mixed PG, the BM ratios were 72%, 65%, 62%, and 68% in rows L, ML, MT, and T, respectively.
Footnotes
Acknowledgements
The authors would like to express their thanks to Zhongjie Zhang and LA DOTD engineers for providing valuable help and support in this study.
Author Contributions
The authors confirm contribution to the paper as follows: study conception and design: A. Souri, M. Abu-Farsakh; data collection: A. Souri, M. Abu-Farsakh; analysis and interpretation of results: A. Souri, M. Abu-Farsakh; draft manuscript preparation: M. Abu-Farsakh, A. Souri. Review results and approve the final version of manuscript: M. Abu-Farsakh, A. Souri, G. Voyiadjis.
Declaration of Conflicting Interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research project is funded by the Louisiana Transportation Research Center (LTRC Project No. 13-3GT) and the Louisiana Department of Transportation and Development (State Project No. DOTLT1000103).
