DESIGN OF SHORT REINFORCED CONCRETE BRIDGE COLUMNS UNDER VEHICLE COLLISION

Transcription

1 1 DESIGN OF SHORT REINFORCED CONCRETE BRIDGE COLUMNS UNDER VEHICLE COLLISION Omar I. Abdelkarim Ph.D. Candidate Department of Civil, Architectural & Environmental Engineering Missouri University of Science and Technology 1401 N. Pine Street, 218 Butler-Carlton Hall, Rolla, MO Tel.: ; Mohamed A. ElGawady, Corresponding Author Benavides Associate Professor Department of Civil, Architectural & Environmental Engineering Missouri University of Science and Technology 1401 N. Pine Street, 324 Butler-Carlton Hall, Rolla, MO Tel.: ; Fax: ; Word count: 250 words abstract + 4,980 words text + 7 tables/figures x 250 words (each) = 6,980 words Number of references = 24 TRR Paper number: Submission Date: November 15 th, 2015

2 2 ABSTRACT This paper presents the behavior of reinforced concrete bridge columns subjected to vehicle collision. An extensive parametric study consisting of 13 parameters was conducted, examining the peak dynamic force (PDF) and the equivalent static force (ESF) of a vehicle collision with reinforced concrete bridge columns. The ESF was calculated using the Eurocode approach and the approach of the peak of twenty-five millisecond moving average (PTMSA) of the dynamic impact force. The ESF from these two approaches were compared to the ESF of the American Association of State Highway and Transportation Officials- Load and Resistance Factor Design (AASHTO-LRFD; 2,670 kn [600 kips]). The ESF of the AASHTO-LRFD was found to be nonconservative for some cases and too conservative for others. The AASHTO-LRFD was nonconservative when the vehicle s velocity exceeded 120 kph (75 mph) and when the vehicle s mass exceeded 16 tons (30 kips). This paper presents the first equation to calculate a design impact force, which is a function of the vehicle s mass and velocity. The equation covered high range of vehicle velocities ranging from 56 kph (35 mph) to 160 kph (100 mph) and high range of vehicle masses ranging from 2 tons (4.4 kips) to 40 tons (90 kips). This approach will allow departments of transportation (DOTs) to design different bridge columns in different highways depending on the anticipated truck loads and speeds collected from the survey of roadways. A simplified equation based on the Eurocode equation of the ESF was proposed. These equations do not require FE analyses. INTRODUCTION Vehicle collision with bridges can have serious implications with regard not only to human lives but also to transportation systems. Harik et al. [1] reported that 17 of the 114 bridge failures in the United States were the result of truck collisions over the period of Lee et al. [2] stated that vehicle collision was the third cause of bridge failures in the United States between the years of 1980 and 2012 and was the reason for approximately 15% of the failures during this period. Many vehicle collision events involving bridge piers have been reported throughout the U.S. In July 1994, a tractor cargo-tank semitrailer hit a road guardrail, and the cargo tank collided into a column of the Grant Avenue overpass over Interstate 287 in White Plains, New York [3]. Twenty-three people were injured, the driver was killed, and a fire extended over a radius of approximately 122 m (400 ft). According to the American Association of State Highway and Transportation Officials- Load and Resistance Factor Design (AASHTO-LRFD) Bridge Design Specifications 5 th [4] abutments and piers located within a distance of 9,140 mm (30 ft.) from the roadway edge should be designed to allow for a collision load. AASHTO-LRFD Bridge Design Specifications 5 th edition required the collision load to be an equivalent static force (ESF) of 1800 kn (400 kips). El-Tawil et al. [5] used the commercial software LS-DYNA [6] to numerically examine two bridge piers impacted by both Chevrolet pickup trucks and Ford single unit trucks (SUTs). The ESF was calculated to produce the same deflection at the point of interest as that caused by the impact force. These results suggested that the AASHTO-LRFD could be nonconservative and the ESF should be higher than 1800 kn (400 kips). Buth et al. [7, 8] studied the collision of large trucks, SUTs, and tractor-trailers with bridge piers. This study included experimental work and finite element (FE) analysis conducted with LS-DYNA software. The design requirements were updated in the 6 th edition of AASHTO- LRFD Bridge Design Specifications [9] as follows: the design choice is to provide structural resistance, the pier or abutment shall be designed for an equivalent static force of 2,670 kn (600

3 3 kips), which is assumed to act in a direction of zero to 15 degrees with the edge of the pavement in a horizontal plane, at a distance of 1.5 m (5.0 ft) above ground. Buth et al. [7] defined the ESF of vehicle impact with a bridge column as the peak of the twenty-five millisecond moving average (PTMSA) of the dynamic force. While the Eurocode [10] calculates the ESF using equation (1) which takes into consideration the vehicle mass, velocity, and deformation, and the column deformation, FE analysis is required to determine the vehicle and column deformations in order to calculate the ESF. Abdelkarim et al. [11] conducted an extensive study to identify the better approach between EC ESF and PTMSA. Their study revealed that PTMSA is more accurate than the EC ESF in determining the static force of vehicle collision with a bridge column. EC ESF = KE = 1 2 m v r 2 δ c + δ d (1) where KE is the vehicle kinetic energy, m = the vehicle mass, v r = the vehicle velocity, δ c = the vehicle deformation, and δ d = the column deformation. The δ c of each vehicle was calculated as the maximum change in length between the vehicle nose and its center of mass. The δ d of each column was calculated as the maximum lateral displacement of the column at the point of impact load. This paper presented finite element (FE) analyses to investigate the effects of 13 different parameters on both dynamic and static impact forces. The constant impact load used in the AASHTO-LRFD did not consider either the vehicle mass or velocity. Hence, the given impact load may be conservative in some occasions and nonconservative in others. This paper presented the first equation that can directly calculate the ESF given the vehicle mass and velocity. A simplified equation had been suggested in this paper for the Eurocode to directly calculate the ESF without a crash analysis. PARAMETRIC STUDY The authors presented the validation of the finite element modeling of vehicle collision with bridge columns in a previous study [12]. Once the finite element model was validated, a comprehensive parametric study was conducted to numerically investigate the reinforced concrete (RC) column s behavior during a vehicle collision. The LS-DYNA software was used to examine the effect of 13 different parameters. Table 1 summarizes the columns variables. Thirty-three columns (from C0 to C32) were investigated. Column C0 was used as a reference column. Figure 1 illustrates the 3D-view model of the reference column C0. A detailed geometry of the column C0 is also illustrated in this figure. It should be noted that some of the selected parameters may not be common in practice. They were used, however, to fully understand the column s performance under a wide spectrum of parameters. Columns Geometry The columns investigated in this study were supported on a concrete footing by a fixed condition at the bottom of the footing. All of the columns except columns C17 and C18 were hinged at the top ends. Column C17 was free at the top end while the superstructure was attached at the top of column C18. Most columns had a circular cross-section with a diameter of 1,500 mm (5.0 ft). Columns C14, C15, and C16, however, had diameters of 1,200 mm (4.0 ft), 1,800 mm (6.0 ft), and 2,100 mm (7.0 ft), respectively. The reference column s height (measured from the top of

4 4 the footing to the top of the column) was 7,620 mm (25.0 ft) with a span-to-depth ratio of 5. Columns C12 and C13 were 3,810 mm (12.5 ft), and 15,240 mm (50.0 ft) tall, respectively, with a span-to-depth ratio of 2.5, and 10, respectively. The span-to-depth ratio of other columns was 5. The soil depth above the top of footing was 1,000 mm (3.3 ft.). The soil depths above the top of footing of columns C31, and C32, however, were 500 mm (1.7 ft.), and 1,500 mm (4.9 ft.), respectively. All of the columns except columns C7 and C8 were reinforced longitudinally by 1% of the concrete cross-sectional area. Columns C7 and C8 were reinforced by 2% and 3% of the concrete cross sectional area, respectively. Most columns had hoop reinforcements of D16 (#5) with mm (4 in.) spacing. Columns C9, C10, and C11, however, had hoop reinforcements of D13 (#4) with 64 mm (2.5 in.) spacing, D19 (#6) with mm (6 in.) spacing, and D16 (#5) with mm (12 in.) spacing, respectively. All of the columns except columns C19 and C20 were axially loaded with 5% of P o where P o was calculated according to AASHTO-LRFD [9] as following: P o = A s f y f c A c (2) Where A s = the cross-sectional area of the longitudinal reinforcement, A c = the cross sectional area of the concrete column, f y = the yield strength of the longitudinal reinforcement, f c = the cylindrical concrete unconfined compressive strength. Column C19 was not axially loaded while column C20 was axially loaded with 10% of P o. Material Models The vehicle collision load is considered to be within the extreme event limit-state according to AASHTO-LRFD. This limit-state refers to the structural survival of a bridge during the extreme event. Under these extreme conditions, the structure is expected to undergo considerable inelastic deformations. Thus, all strength reduction factors ( Φ ) are to be taken as one when designing concrete bridges for use under extreme events [13]. Therefore, a nonlinear material model was used for the concrete column and the footing in all of the columns except C1and C2. The impact force was expected to increase as the linear portion of the stress-strain curve of the column s concrete material increased because the energy dissipation would be reduced. Therefore, elastic (mat. 001) and rigid (mat. 020) material models were used in the columns C1 and C2, respectively, to identify the impact force s upper limit. Various material models in LS-DYNA software can simulate concrete material. The Karagozian and Case Concrete Damage Model Release 3 (K&C model) was used as a nonlinear material in this study because it exhibited good agreement with experimental results in previous studies [14, 15]. The material elastic model mat.001 is an isotropic, hypoelastic material. El-Tawil et al. [5] used this material to study impact analysis. They suggested that elastic material allowed direct assessment of design provisions for the ESF. Buth et al. [8] used the rigid material model mat.020 to simulate bridge piers. This material model does not allow any deformation of the column to calculate the maximum possible impact force. With the exception of Columns C3, C4, and C5, each column had an unconfined concrete compressive strength of 34.5 MPa (5,000 psi). Columns C3, C4, and C5, however had unconfined concrete compressive strengths of 20.7 MPa (3,000 psi), 48.3 MPa (7,000 psi), and 69.0 MPa (10,000 psi), respectively.

5 5 The material model 003-plastic_kinamatic was used to identify the steel reinforcement s elasto-plastic stress-strain curve. Five parameters were needed to define this material model according its properties: the elastic modulus (E), the yield stress (SIGY), Poisson s ratio (PR), the tangent modulus, and the ultimate plastic strain. The values used according to Caltrans [16] were GPa (29,000.0 ksi), MPa (60,900.0 psi), 0.30, MPa (159.9 ksi), and 0.118, respectively. Strain Rate Effects Concrete Material Previously conducted studies examined concrete s properties under dynamic loading. The CEB model [17] code is one of the most comprehensive models used, and introduces the concrete properties with strain rate effect. Malvar and Ross [18] modified the CEB model through equations (3-10). Any increase in concrete properties under dynamic loading is typically reported as a dynamic increase factor (DIF). DIF is the ratio of dynamic concrete strength to static concrete strength; it is calculated from both the strain rate and the concrete static properties. DIF c = f c = ( ε α s )1.026 for ε 30 s 1 (3) f cs ε s DIF c = f c = γ f s ( ε cs ε s 0.33 ) for ε > 30 s 1 (4) α s = (5 + 9 f cs f co ) 1 (5) logγ s = α s 2 Where DIF c = compressive strength dynamic increase factor ε = strain rate in the range of 30 x 10-6 to 300 s -1 ε s = static strain rate of 30 x 10-6 s -1, f c = the dynamic compressive strength at ε f cs = the static compressive strength at ε s f co = 10 MPa = 1,450 psi (6) DIF t = f t = ( ε f ts ε s δ ) DIF t = f t = β ( ε f ts ε s 0.33 ) for ε 1 s 1 (7) for ε > 1 s 1 (8) δ = (1 + 8 f cs f co ) 1 (9)

6 log β = 6 δ 2 6 (10) Where DIF t = tensile strength dynamic increase factor f t = the dynamic tensile strength at ε f ts = the static tensile strength at ε s ε = strain rate in the range of 10-6 to 160 s -1 ε s = static strain rate of 10-6 s -1 Steel Material The strain rate affects the stress-strain relation of steel as it affects the speed at which deformation occurs [19]. Therefore, the strain rate effect on steel was considered when the static yield stress producing the dynamic yield strength was scaled. Cowper-Symonds [20] experimentally examined the strain rate effect on steel presenting equation (11) with two constants (p and c) that can be used to calculate the dynamic yield strength. Several researchers concluded that p and c constants could be taken as 5 and 40, respectively [21]. The elastic modulus does not change under impact loading [22]. f yd = 1 + ( ε 1 c ) p (11) where f yd = dynamic yield stress and p and c were taken as 5 and 40, respectively. Vehicles FE Models Two vehicle models were used in this study: a reduced model of Ford single unit truck (SUT) (35,353 elements) and a detailed model of Chevrolet C2500 Pickup (58,313 elements). These models were developed by the National Crash Analysis Center (NCAC) of The George Washington University under a contract with the FHWA and NHTSA of the U.S. DOT. These models were posted on the National Crash Analysis Center (NCAC) website in November Experimental tests involving head-on collisions were conducted to validate each model [23, 24]. Figure 2 illustrates the FE vehicles models. Maximum speed limits on highways differ from state to state in the U.S. Therefore, a high range of vehicle velocity was investigated in this study, ranging from 56 kph (35 mph) to 160 kph (100 mph) to cover all of the expected speeds during vehicle collisions. Most vehicles in this parametric study were traveling with a velocity of 80 kph (50 mph). The Ford SUT of the FE models C21, C22, and C23, however, was traveling with velocities of 160 kph (100 mph), 120 kph (75 mph), and 56 kph (35 mph), respectively. The mass of the Ford SUT was 8 tons (18 kips) for all of the models except the models C24, C25, and C26. The FE model C24 had the Chevrolet C2500 Pickup with a mass of 2 tons (4.4 kips) instead of the Ford SUT. The mass of the Ford SUT of the FE models C25, and C26 was 16 tons (35 kips) and 30 tons (65 kips), respectively. The increase of the Ford SUT mass was achieved by increasing the mass of the cargo in the Ford SUT. The distance between the vehicle and the unprotected column was examined here by the distance between the vehicle nose and the column face. The distance between the vehicle nose and the concrete column was 150 mm (0.5 ft.). The vehicles noses of the FE models C27, C28,

7 7 C29, and C30, however, were 0.0 mm (0.0 ft.), 300 mm (1.0 ft.), 3,000 mm (10.0 ft.), and 9,140 mm (30.0 ft.) apart from the concrete columns, respectively. RESULTS AND DISCUSSION Concrete Material Models This section presented the effects of the selection of concrete material model on the PDF and ESFs. Three material models mat001, mat020, and mat72riii representing elastic, rigid, and nonlinear behavior were used for this investigation. The typical time-impact force relationship is illustrated in figure 3a. As shown in the figure, the first peak force occurred when the vehicle s rail collided with the column. The second peak force on the columns, which was the largest, was produced by the vehicle s engine. The third peak occurred when the vehicle s cargo (in the Ford SUT only) struck the cabinet and the engine. The fourth peak was produced when the rear wheels left the ground. Generally, each of the columns reached its PDF nearly at the same time of 40 millisecond, and had zero impact force beyond 220 millisecond. The PDF of column C2, which was modeled using a rigid material, was approximately 15% higher than that of column C0, which was modeled using a nonlinear material. This finding was expected as no deformations were allowed to take place in the concrete material of column that was modeled using a rigid material. Hence, no impact energy was dissipated. Column C1, which was modeled using elastic material, had a slightly lower PDF value than that of column C2. Figure 3b illustrates the normalized ESFs and PDFs of the columns C0, C1, and C2. The normalized ESF for columns C0, C1, and C2 ranged from 0.7 to 0.8 of the ESF of AASHTO- LRFD of 2,670 kn (600 kips). The values of PTMSA and EC ESF for all of the columns were almost constant regardless of the material model. The PTMSA values were higher than the EC ESF values for all of the columns. The system s kinetic energy before collision occurred was 18,408 kip.in (2,102 kn.m) (Figure 3c). The kinetic energy was absorbed entirely during the first 150 milliseconds in the form of column and vehicle deformations. Converting part of the kinetic energy into thermal energy (in the form of heat) was excluded from this study. The vehicle s deformation of each model was presented in figure 3d. The maximum deformation of vehicles in FE models C0, C1, and C2 was 1,122 mm (44.2 in.), 1,156 mm (45.5 in.), and 1,127 mm (44.4 in.), respectively. Unconfined Compressive Strength (f c ) Four values of ranging from 20.7 MPa (3,000 psi) to 69.0 MPa (10,000 psi) were investigated during this section. Changing f c did not significantly affect the values of PDF except when the f c was considerably low for column C3 (Figure 4a). The PDF value of column C3 having f c of 20.7 MPa (3,000 psi), was 20% lower than that of the other columns. The lower concrete strength in C3 led to early concrete spalling and bucking of several longitudinal bars, which dissipated a portion of the impact force. The values of PTMSA and EC ESF for all of the columns were nearly constant regardless of the f c. The PTMSA values were higher than the EC ESF values for all of the columns. Strain Rate Effect The PDF increased significantly when the strain rate effect was included (Figure 4b). The PDF of column C0, which was modeled including the strain rate effect, was approximately 27%

8 8 higher than that of column C6, which was modeled excluding the strain rate effect. Including the strain rate effect, the column s strength and stiffness increased leading to higher dynamic forces. There is no significant effect of strain rate on EC ESF and PTMSA. The PTMSA values were higher than the EC ESF values for all of the columns. Percentage of Longitudinal Reinforcement Three values of longitudinal reinforcement ratios ranging from 1% to 3% were investigated during this section. In general, the PDF increased slightly when the percentage of longitudinal reinforcement increased (Figure 4c). Tripling the percentage of longitudinal reinforcement increased the PDF by only 10%. It increased because the column s flexural strength and stiffness increased slightly with increasing the flexural steel ratio. When the percentage of longitudinal reinforcement increased, the EC ESF and PTMSA were constant. The PTMSA values were higher than the EC ESF values for all of the columns. Hoop Reinforcement Four volumetric hoop reinforcement ratios ranging from 0.17% (D16@305 mm) to 0.54% (D13@64 mm) were investigated during this section. The PDF decreased when the volume of hoop reinforcement decreased leading to increased concrete damage which dissipated some of the impact energy (Figure 4d). The PDF decreased by 12% when the volume of hoop reinforcement decreased by 67%. When the hoop reinforcement decreased, the EC ESF and PTMSA were constant. The PTMSA values were higher than the EC ESF values for all of the columns. Column Span-To-Depth Ratio Three values of column span-to-depth ratio ranging from 2.5 to 10 were investigated during this section. The relationship between the PDF and the column s span-to-depth ratio was nonlinear (Figure 4e). The PDF of column C0, having span-to-depth ratio of 5, was higher than that of columns C12 and C13, having a span-to-depth ratios of 2.5 and 10, respectively. This was because the column C12 had high local damaged buckling of several rebars leading to energy dissipation and the column C13 had the lowest stiffness leading to energy dissipation through high column s displacement. The PTMSA and EC ESF were approximately constant regardless of the span-to-depth ratio. The PTMSA values were higher than the EC ESF values for all of the columns. Column Diameter Four values of column diameter ranging from 1,200 mm (4.0 ft) to 2,100 mm (7.0 ft) were investigated during this section. The PDF of all of the columns, except for Column C16, increased slightly when the column diameter increased (Figure 4f). The PDF of the column C16, having a diameter of 2,100 mm (7.0 ft), was lower than that of the column C15, having a diameter of 1,800 mm (6.0 ft) because of the rebars buckling. Column C14, with a diameter of 1,200 mm (4.0 ft), subjected to severe rebar buckling. Both the EC ESF and the PTMSA for all columns increased slightly when the column diameter increased. The PTMSA values were higher than the EC ESF values for all of the columns. Top Boundary Conditions Three columns top boundary conditions including free, hinged, and superstructure were investigated during this section. Changing the column s top boundary condition slightly changed

9 9 the PDF values because the PDF was induced in a very short period of time (Figure 5a). The column s response was controlled by the amplitude of the imposed kinetic energy as the impact duration was very small compared to the natural period of the column. However, the maximum lateral displacement at the point of impact of column C17, having free top boundary condition, was higher than those of columns C0 and C18, having hinged and superstructure top conditions, respectively. The existence of the superstructure in column C18 resulted in a top boundary condition similar to that in column C0 of hinged condition. Changing the top boundary condition did not change the EC ESF and PTMSA. The PTMSA values were higher than the EC ESF values for all of the columns. Axial Load Level Three values of axial load level ranging from 0 to 10% of the column s axial capacity (P o ) were investigated during this section. The PDF typically increased when the axial load level increased (Figure 5b). Column C19, which sustained zero axial load, had a PDF that was approximately 25% lower than that of column C20, which sustained an axial load that was 10% the P o. The high axial compressive stresses on the column delayed the tension cracks due to the vehicle impact and hence increased the column s cracked stiffness leading to higher dynamic forces. The PTMSA and EC ESF were approximately constant regardless of the axial load level. The PTMSA values were higher than the EC ESF values for all of the columns. Vehicle Velocity Four vehicle velocities ranging from 56 kph (35 mph) to 160 kph (100 mph) were investigated during this section. The PDF tended to increase nonlinearly when the vehicle s velocity increased (Figure 5c). It is of interest that the increase in the PDF is not proportional to the square of the velocity as in the case of elastic impact problems. Damage to the columns reduces the rate of increase in the PDF. For example, the PDF increased by approximately 507% when the vehicle s velocity increased from 56 kph (35 mph) to 160 kph (100 mph). The EC ESF and PTMSA increased approximately linearly with increased vehicle velocity. The PTMSA values were higher than the EC ESF values for all of the columns. The AASHTO-LRFD was found to be nonconservative when the column was collided by a vehicle travelling with a speed exceeded 120 kph (75 mph) as the ESF of AASHTO-LRFD of 2,670 kn (600 kips) was lower than the PTMSA values of such cases. Vehicle Mass Four vehicle masses ranging from 2 tons (4.4 kips) to 30 tons (65 kips) were investigated during this section. In general, both the PDF and ESF increased linearly when the vehicle s mass increased (Figure 5d). However, the rate of increase is slower than what is anticipated in elastic impact problems. For example, the PDF increased by approximately 101% when the vehicle s mass increased from 2 tons (4.4 kips) to 30 tons (65 kips). The PDF almost did not change when the vehicle mass increased from 2 tons (4.4 kips) to 8 tons (18 kips) because the energy dissipation in the form of inelastic deformations whether in the vehicle or in the column did not significantly change as the kinetic energy was not considerably high. The PTMSA values were higher than the EC ESF values for all of the columns. The AASHTO-LRFD was found to be nonconservative when the column was collided by heavy vehicles of a mass more than 16 tons (35 kips) as the ESF of AASHTO-LRFD of 2,670 kn (600 kips) was lower than the PTMSA values of such cases.

10 10 Distance between Vehicle and Column Five distances between the vehicle and column ranging from zero to 9,140 mm (30 ft) were investigated during this section. In general, the PDF decreased when the distance between the vehicle and column increased (Figure 5e). The PTMSA and EC ESF are approximately constant regardless of the distance between the vehicle and the unprotected column. The PTMSA values were higher than the EC ESF values for all of the columns. Soil Depth above the Top of the Column Footing Three values of the soil depth above the top of the column footing ranging from 500 mm (1.7 ft) to 1,500 mm (4.9 ft) were investigated during this section. In general, the change in the soil depth above the column footing did not significantly affect the PDF (Figure 5f). The PTMSA and EC ESF are approximately constant regardless of the soil depth. The PTMSA values were higher than the EC ESF values for all of the columns. ESF Equation for Adoption by AASHTO-LRFD and Eurocode However, the AASHTO-LRFD approach is quite simple as it uses a constant value for ESF, regardless of the vehicle s characteristics. The PTMSA and Eurocode approaches presented in this manuscript use a variable ESF that is dependent on these characteristics. AASHTO-LRFD was found to be quite conservative in some cases and nonconservative in others in predicting the ESF of impact loads. The PTMSA and Eurocode approaches, however, require a FE analysis and iterative design to estimate the ESF of impact loads. Thus, a simple equation considering the vehicle s characteristics that can predict the ESF without either a FE or an iterative analysis would represent a significant improvement over the current AASHTO-LRFD or Eurocode approaches. Figures 4 and 5 revealed that the most influential parameters on impact problems were the vehicle s mass and velocity. However, the other investigated parameters had limited effects. Therefore, developing a design equation to estimate the ESF as a function of the vehicle s mass and velocity seems reasonable. This approach will allow departments of transportation (DOTs) to design different bridge columns in different highways depending on the anticipated truck loads and speeds collected from the survey of roadways. The PTMSA correctly predicted the performance of bridge columns as was proved by Abdelkarim et al. [11]. Thus, it was selected as the basis for the newly developed equation. The PTMSAs of the parametric study was studied mathematically using CurveExpert Professional software, and SAS software to introduce a design equation for estimating kinetic-energy based ESF (KEB ESF ) which was presented in equation (14) for international system (SI) units and equation (15) for English units as below: KEB ESF = 33 m v r 2 = 46 KE (14) where m = the vehicle mass in tons, v r = the vehicle velocity in m/s, and KE = kinetic energy of the vehicle in kn.m. KEB ESF = 2.1 m v r 2 = 3 KE (15) where KEB ESF = proposed ESF to AASHTO-LRFD (kip), m = the vehicle mass in kip, v r = the vehicle velocity in mph, and KE = kinetic energy of the vehicle in kip.mph 2.

11 11 The proposed equation s results were compared to the PTMSA s FE results for a high range of vehicle masses from 2 tons (4.4 kips) to 40 tons (90 kips) and a high range of vehicle velocities from 32 kph (20 mph) to 160 kph (100 mph)). The relationship between the vehicle s kinetic energy and the normalized PTMSA was presented in figure 6a. Also, the relationship between the vehicle s kinetic energy and the normalized KEB ESF was presented in the figure. It is worthy to note that the AASHTO-LRFD was found over predicting to the ESF when the kinetic energy was lower than 2,500 kn.m (1,844 kip.ft). However, it was found quite nonconservative beyond that threshold of 2,500 kn.m (1,844 kip.ft). In several instances, the RC columns were subjected to impact loads that were almost double the ESF of the current AASHTO-LRFD. In order to illustrate the accuracy of the equation, curves of ± 10% of the KEB ESF values were shown in figure 6a (referred to as upper and lower limits). The figure showed that the proposed KEB ESF equation correlated well with the FE results and predicted most cases with an accuracy of more than 90% (within the upper and lower limits). The PTMSA is more accurate than the Eurocode and hence equation (14) or (15) gives a more reasonable value of ESF. However, this paper simplified the Eurocode equation to avoid implementing FE models to estimate the ESF. Based on the FE results of EC ESF of the parametric study and using CurveExpert Professional software, and SAS software, a new simplified equation for estimating momentum-based equivalent static force MB ESF was developed and presented in equation (16) for international system (SI) units and equation (17) for English units as below: MB ESF = 130 m v r = 130 P m (16) where m = the vehicle mass in tons, v r = the vehicle velocity in m/s, and P m = the momentum of the vehicle in tons.m/s. MB ESF = 13 m v r = 13 P m (17) where MB ESF = proposed ESF to Eurocode (kip), m = the vehicle mass in kip, v r = the vehicle velocity in mph, and P m = the momentum of the vehicle in kip.mph. The results of the proposed equation were compared to the FE results of EC ESF. The relation between the vehicle s momentum and the normalized EC ESF was presented in figure 6b. The relation between the vehicle s momentum and the normalized MB ESF was presented in the figure as well. The proposed MB ESF equation correlated well with the FE results and predicted most cases with an accuracy of more than 90% (within the upper and lower limits). FINDINGS AND CONCLUSIONS A detailed description of finite element modeling of vehicle collision with reinforced concrete bridge columns using LS-DYNA software was presented. Evaluation of the peak dynamic force (PDF) and the equivalent static force (ESF) through a comprehensive parametric study were conducted. The comprehensive parametric study investigated the effects of concrete material model, unconfined concrete compressive stress (f c ), material strain rate, percentage of longitudinal reinforcement, hoop reinforcement, column span-to-depth ratio, column diameter, the top boundary conditions, axial load level, vehicle velocity, vehicle mass, distance between

12 12 errant vehicle and unprotected column, and soil depth above the top of the column footing on the behavior of the columns under vehicle collision. This study revealed the following findings: The AASHTO-LRFD was nonconservative when the vehicle s velocity exceeded 120 kph (75 mph) and when the vehicle mass exceeded 16 tons (30 kips). The AASHTO-LRFD was found to be nonconservative when the column was collided with a vehicle having kinetic energy of 2,500 kn.m (1,800 kip.ft) or more. Generally, the PDF increases when the longitudinal reinforcement ratio, hoop reinforcement volumetric ratio, column diameter, axial load level, vehicle velocity, and vehicle mass increase and when the strain rate effect is considered, while it decreases when the damage of the column and the clear zone distance increase. However, it is not affected by changing f c, column top boundary condition, and soil depth. The relation between the PDF and the column s span-to-depth ratio was nonlinear. The vehicle s velocity and mass are the most influential parameters affected the vehicle collision with a bridge column. Generally, the PTMSA values were higher than the EC ESF values. A new equation for estimating the ESF based on the vehicle mass and velocity, (KEB ESF = 33 m v2 r ), with an accuracy of more than 90% was developed. This approach will allow departments of transportation (DOTs) to design different bridge columns to different impact force demands depending on the anticipated truck loads and velocities. This paper simplified the Eurocode equation for estimating the ESF based on the vehicle s mass and velocity, (MB ESF = 130 m v r ), with an accuracy of more than 90%. ACKNOWLEDGEMENT This research was conducted by the Missouri University of Science and Technology and was supported by the Missouri Department of Transportation (MoDOT) and Mid-American Transportation Center (MATC). This support is gratefully appreciated. However, any opinions, findings, conclusions, and recommendations presented in this paper are those of the authors and do not necessarily reflect the views of sponsors. REFERENCES 1. Harik, I., Shaaban, A., Gesund, H., Valli, G., and Wang, S. United States Bridge Failures, J. Perform. Constr. Facil., 4(4), 1990, pp Lee, G. C., Mohan, S., Huang, C., and Fard, B. N. A Study of US Bridge Failures ( ). Publication the Multidisciplinary Center for Earthquake Engineering Research (MCEER), Technical Report , Buffalo, NY, Agrawal, A.K. Bridge Vehicle Impact Assessment: Final Report. University Transportation Research Center and New York State Dep. of Transportation, AASHTO. AASHTO-LRFD Bridge Design Specifications Customary US Units, fifth edition, Washington, DC, El-Tawil, S., Severino, E., and Fonseca, P. Vehicle Collision with Bridge Piers. J. Bridge Eng., 10(3), 2005, pp Livermore Software Technology Corporation (LSTC). LS-DYNA Theory manual. California, Buth, C. E., Williams, W. F., Brackin, M. S., Lord, D., Geedipally, S. R., and Abu- Odeh, A. Y. Analysis of Large Truck Collisions with Bridge Piers: Phase 1. Texas Department of Transportation Research and Technology Implementation Office,

13 13 Report , Buth, C. E., Brackin, M. S., Williams, W. F., and Fry, G.T. Collision Loads on Bridge Piers: Phase 2. Texas Department of Transportation Research and Technology Implementation Office, Report , AASHTO. AASHTO-LRFD Bridge Design Specifications Customary US Units, sixth edition, Washington, DC, Eurocode 1: Actions on structures Part 1-1: General actions Densities, selfweight, imposed loads for buildings, Final Draft pren , October Abdelkarim, O., Gheni, A., Anumolu, S., Wang, S., ElGawady, M. Hollow-Core FRP-Concrete-Steel Bridge Columns under Extreme Loading. Missouri Department of Transportation (MoDOT), Project No. TR201408, Report No. cmr15-008, April, Abdelkarim, O. and ElGawady, M. Impact Analysis of Vehicle Collision with Reinforced Concrete Bridge Columns. Transportation Research Board (TRB) conference, Washington DC., 2015, Barker, R.M. and Puckett, J.A. Design of Highway Bridges - Based on AASHTO LRFD Bridge Design Specification. John Wiley and Sons, New York, Third Edition, 2013, pp Abdelkarim, O. and ElGawady, M. Analytical and Finite-Element Modeling of FRP-Concrete-Steel Double-Skin Tubular Columns. Journal of Bridge Engineering, / (ASCE) BE., 2014, , B Ryu, D., Wijeyewickrema, A., ElGawady, M., and Madurapperuma, M. Effects of Tendon Spacing on In-Plane Behavior of Post-Tensioned Masonry Walls. Journal of Structural Engineering, 140(4), 2014, California Department of Transportation. Seismic Design Criteria. California Department of Transportation, Rev. 1.4, Comité Euro-International du Béton. CEB-FIP Model Code. Redwood Books, Trowbridge, Wiltshire, UK, Malvar, L. J. and Ross, C. A. Review of Strain Rate Effects for Concrete in Tension. ACI Materials Journal, 95, 1998, Zener, C. and Hollomon, J. H. Effect of Strain Rate Upon Plastic Flow of Steel. Journal of Applied Physics, 15, 1944, Cowper, G. R. and Symonds, P. S. Strain Hardening and Strain Rate Effects in Impact Loading of Cantilever Beams. Brown University, App. Math. Report No. 28, Yan, X. and Yali, S. Impact Behaviors of CFT and CFRP Confined CFT Stub Columns. J. Compos. Constr., 16(6), 2012, Campbell, J.D. The yield of mild steel under impact loading. Journal of the Mechanics and Physics of Solids, 3, 1954, Zaouk, A. K., Bedewi, N. E., Kan, C. D., and Marzoughi, D. Evaluation of a Multi-purpose Pick-up Truck Model Using Full Scale Crash Data with Application to Highway Barrier Impact. 29th Inter. Sym. on Auto. Tech. and Auto., Florence, Italy, Mohan, P., Marzougui, D., Kan, C. Validation of a Single Unit Truck Model for Roadside Hardware Impacts. International Journal of Vehicle Systems Modelling and Testing, 2.1, 2006, pp

14 14 LIST OF TABLES TABLE 1 Summary of the Examined Columns Parameters LIST OF FIGURES FIGURE 1 Finite element model of the bridge column C0 for the parametric study; (a) 3D-view, (b) detailed side view of the column components. FIGURE 2 View of the vehicles FE models: (a) the Ford single unit truck, (b) Chevrolet pickup detailed model. FIGURE 3 Effect of various concrete material models: (a) Time versus Impact force for column C0 with a nonlinear material, (b) PDF and ESFs, (c) Time versus total kinetic energy, and (d) Time versus vehicle displacement. FIGURE 4 Effects of: (a) f c, (b) strain rate, (c) longitudinal reinforcements ratio, (d) hoop reinforcements, (e) span-to-depth ratio, and (f) column diameters on PDF and ESF. FIGURE 5 Effects of: (a) top boundary conditions, (b) axial load level (c) vehicle velocities, (d) vehicle masses, (e) distance between vehicle and column, and (f) soil depth above the top of column footing on PDF and ESF. FIGURE 6 the ESF proposed equations versus the FE results: (a) Kinetic energy-based ESF (KEB ESF ) for AASHTO-LRFD and (b) Momentum-based ESF (MB ESF ) for Eurocode.

15 15 TABLE 1: Summary of the Examined Columns Parameters Col. Conc. Mat. f c, MPa SR ρ s Hoop RFT S/D D, mm Top Bound. Cond. C0 NL 34.5 C 1% Hinged 5% C1 EL 34.5 C 1% Hinged 5% C2 RIG 34.5 C 1% Hinged 5% C3 NL 20.7 C 1% Hinged 5% D16@ C4 NL 48.3 C 1% Hinged 5% mm C5 NL 69.0 C 1% Hinged 5% C6 NL 34.5 NC 1% Hinged 5% C7 NL 34.5 C 2% Hinged 5% C8 NL 34.5 C 3% Hinged 5% C9 NL 34.5 C 1% D13@ 64 mm Hinged 5% C10 NL 34.5 C 1% D19@ 152 mm Hinged 5% C11 NL 34.5 C 1% D16@ 305 mm Hinged 5% C12 NL 34.5 C 1% Hinged 5% C13 NL 34.5 C 1% Hinged 5% C14 NL 34.5 C 1% Hinged 5% C15 NL 34.5 C 1% Hinged 5% C16 NL 34.5 C 1% Hinged 5% C17 NL 34.5 C 1% Free 5% C18 NL 34.5 C 1% Superstructure 5% C19 NL 34.5 C 1% Hinged 0% C20 NL 34.5 C 1% Hinged 10% C21 NL 34.5 C 1% D16@ Hinged 5% C22 NL 34.5 C 1% 102 mm Hinged 5% C23 NL 34.5 C 1% Hinged 5% C24 NL 34.5 C 1% Hinged 5% C25 NL 34.5 C 1% Hinged 5% C26 NL 34.5 C 1% Hinged 5% C27 NL 34.5 C 1% Hinged 5% C28 NL 34.5 C 1% Hinged 5% C29 NL 34.5 C 1% Hinged 5% C30 NL 34.5 C 1% Hinged 5% C31 NL 34.5 C 1% Hinged 5% C32 NL 34.5 C 1% Hinged 5% NL = nonlinear material, EL = elastic material, RIG = rigid material, SR = strain rate, NC = Not Considered, C = Considered, ρ s = the percentage of longitudinal reinforcement, S/D = span-to-depth ratio, D = column diameter, P = applied axial load, P 0 = column axial compressive capacity, v r = vehicle velocity, m = vehicle mass, L c = distance between vehicle and column, d s = soil depth above the column footing. P/P 0 v r, kph m, ton L C, mm d s, mm

16 16 Hinged boundary condition D = 1,500 mm (5 ft) Concrete column D16@127 mm (#5@5 ) H = 7,620 mm (25.0 ft) Blanked concrete elements: only for showing the reinforcement 36 D25 (36#8) ρ s = 1.0% Footing (a) Fixed boundary condition (b) FIGURE 1 Finite element model of the bridge column C0 for the parametric study; (a) 3D-view, (b) detailed side view of the column components.

17 17 (a) (b) FIGURE 2 View of the vehicles FE models: (a) the Ford single unit truck, (b) Chevrolet pickup detailed model.

18 18 (a) (b) (c) Vehicle s rail impact Vehicle s engine impact (d) Vehicle s Cargo impact Rear wheels left the ground FIGURE 3 Effect of various concrete material models: (a) Time versus Impact force for column C0 with a nonlinear material, (b) PDF and ESFs, (c) Time versus total kinetic energy, and (d) Time versus vehicle displacement.

19 19 (a) (b) (c) (d) (e) FIGURE 4 Effects of: (a) f c, (b) strain rate, (c) longitudinal reinforcements ratio, (d) hoop reinforcements, (e) span-to-depth ratio, and (f) column diameters on PDF and ESF. (f)

20 20 (a) (b) (c) (d) (e) FIGURE 5 Effects of: (a) top boundary conditions, (b) axial load level (c) vehicle velocities, (d) vehicle masses, (e) distance between vehicle and column, and (f) soil depth above the top of column footing on PDF and ESF. (f)

21 21 (a) (b) FIGURE 6 the ESF proposed equations versus the FE results: (a) Kinetic energy-based ESF (KEB ESF ) for AASHTO-LRFD and (b) Momentum-based ESF (MB ESF ) for Eurocode.