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Finite Elements in Analysis and Design 42 (2006) 950 – 959 www.elsevier.com/locate/?nel

Review

Finite element analysis of vehicle–bridge interaction

Leslaw Kwasniewski a , ? , Hongyi Li b , Jerry Wekezer b , Jerzy Malachowski c

a Warsaw University of Technology, Al. Armii Ludowej 16, 00-637 Warszawa, Poland b Florida A&M University, Florida State University College of Engineering, 2525 Pottsdamer Street, Tallahassee, FL 32310-6046, USA c Military University of Technology, Kaliskiego Street 2, 00-908 Warszawa, Poland

Received 27 May 2005; received in revised form 21 January 2006; accepted 21 January 2006 Available online 13 March 2006

Abstract This paper presents results of the ?nite element (FE) analysis of dynamic interaction between a heavy truck and a selected highway bridge on US 90 in Florida. FE analysis of vehicle–bridge interaction was conducted using commercial program LS-DYNA and the super computer at the Florida State University. Development and implementation of a detailed FE truck model with 3D suspension systems, pneumatic and rotating wheels, appropriate contact algorithms, allowed for realistic representation of the actual vehicle dynamic loading. Several static and dynamic ?eld tests were performed on the same bridge. The experimental data was used for validation of the FE models of the bridge and the truck. Numerical results were found to match well with the experimental data. Results presented in the paper demonstrate a signi?cant potential of using computational mechanics and LS-DYNA code for thorough investigation of the vehicle–bridge interaction, dynamic impact factors, and the ultimate loading of bridges. ? 2006 Elsevier B.V. All rights reserved.

Keywords: Vehicle–bridge interaction; Impact factor; Bridge dynamics; Finite element analysis; Computer simulation; LS-DYNA

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description of the modeled bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Field test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Development of FE bridge model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. FE model of the truck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Validation of FE models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Numerical and experimental analysis of vehicle–bridge interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 2. 3. 4. 950 951 952 952 953 954 954 957 958 958

1. Introduction Nonlinear ?nite element (FE) methods are nowadays commonly used to solve engineering problems. One such

? Corresponding author. Tel.: +48 227519552; fax: +48 228256532.

E-mail addresses: l.kwasniewski@il.pw.edu.pl (L. Kwasniewski), lihongy@eng.fsu.edu (H. Li), wekezer@eng.fsu.edu (J. Wekezer), malachowski@wme.wat.edu.pl (J. Malachowski). 0168-874X/$ - see front matter ? 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.?nel.2006.01.014

engineering area is the ef?cient management of highway facilities, especially bridges, where the knowledge of actual dynamic load effects, load carrying capacity, and current condition is critical in making management decisions and in establishing permissible weight limits. Signi?cant dynamic effects can be triggered by increasingly heavier vehicles, which are now used on our highways [1,2]. Additional dynamic effects are accounted for by dynamic impact factors introduced in bridge design codes. The impact factor IM [13], also referred to as

L. Kwasniewski et al. / Finite Elements in Analysis and Design 42 (2006) 950 – 959

951

dynamic load allowance [20], is de?ned as a ratio of the dynamic increment (RD ? RS ) in structure response to the static response: IM = RD ? R S 100%, RS (1)

where RD is the dynamic response and RS the static response. There is a large number of studies on this topic including experimental impact factors, analytical methods and code speci?cations. Nowak and Kim conducted tests on two bridges over Huron River to study impact factors, distribution factors and development of lateral cracks in bridge decks [3]. Chowdhury and Ray performed a series of load tests on a continuous span, multi-girder steel bridge and a single span concrete T-beam bridge to quantify physical and structural behavior of bridges due to moving vehicles [4]. A two-lane highway bridge over the River Lodden at Lower Earley was selected for testing by Green and Cebon to validate their proposed analytical procedure [5]. More examples of experimental studies can also be found in [6]. These experiments show that bridges exhibit a wide range of structural, dynamic responses and resulting impact factors depend on several different parameters related to bridge and vehicle characteristics. Field tests are still the most reliable source of information on bridge dynamic responses, and the only method of ?nal validation of the FE analysis. However, the high cost of such experiments and dif?culties with collecting extensive data from ?eld tests lead to growing interest in analytical and computational methods. A reliable, analytical investigation can reduce such costs dramatically and allow for faster introduction of new design improvements and maintenance decisions. The analytical investigation of bridge dynamic response is based on numerous simpli?cations of its geometry, material models, boundary conditions, and loading. The interaction

between a vehicle and bridge structure is usually reduced in analyses to a simpli?ed mass–spring–damper system crossing a beam or grillage including road surface roughness [7–12]. Current bridge design codes present some formulas estimating the dynamic effects [13,14]. However, these formulas are oversimpli?ed and, in many cases, are questioned by engineers [15]. An FE analysis by an explicit, dynamic computer program was used in this research to study dynamic response of medium span (20–30 m long) highway bridges subjected to moving vehicles. The paper describes comprehensive research efforts focused on development of the FE models of the selected highway bridge and the vehicle, computational mechanics study of vehicle–bridge interaction, and validation of the FE models using the ?eld test results.

2. Description of the modeled bridge The selected bridge #500133 was built in 1999 on US 90 over Mosquito Creek in Northwestern Florida. It is a threespan bridge, carrying two lanes of traf?c. The total length of the bridge is 65.1 m, with each span 21.7 m long and 14.15 m wide. Each span consists of six AASHTO type III prestressed simply supported girders at a spacing of 2.4 m. The continuous concrete deck (slab) was cast in situ. The design traf?c lane live load was AASHTO HS-20 truck and the design speed was 100 km/h. The entire structure was found in good condition with no obvious deterioration such as cracks or potholes at the time of ?eld testing. Selection of a relatively new structure in good condition was more desirable as such bridges usually exhibit more homogeneous properties with consistent and more predictable structural response. The side view of the bridge is presented in Fig. 1 and the cross section of the bridge with the truck positions is shown in Fig. 2.

65.100 m (Overall Bridge Length) 6.000 m Begin Approach Slab 3 Spans x 21.700 m = 65.100 m 6.000 m End Approach Slab

Begin Bridge

Continuous Slab Barrier Section #4

Section #3

Section #1

Section #2

Girder Bearing Pier

Fig. 1. Side view of the selected bridge.

End Bridge

West

East

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Truckon east bound

Truckon west bound 2.500 m

2.500 m 0.62 m 0.62 m 0.7 m 0.7 m 0.62 m

0.62 m

South

North

1 1.075 m

2

3 5 x 2.400 m =12.000 m

4

5

6 1.075 m

Fig. 2. Bridge cross section and truck positions during ?eld experiments.

3. Field test Static and dynamic tests were conducted on the bridge. Two trucks loaded with 12 concrete blocks each (Fig. 5) were used for loading. The front, drive, and rear axle loads were 50 kN (11.24 kip), 100 kN (22.48 kip) and 169 kN (38.0 kip), respectively. The total weight was approximately 319 kN (71.7 kip), which is close to the 325 kN (73.1 kip) as speci?ed by AASHTO standard speci?cations for the HS 20–44 truck [13]. The static test results were used to determine the wheel load distribution factors for girders and as reference data for calculation of impact factors. The longitudinal truck position was determined to yield the maximum stresses at the middle section of the ?rst span. The dynamic tests included passes of one and two trucks side-by-side, with and without a piece of wood positioned across the deck. A wooden plank, 40 mm (1.57 in) thick and 400 mm (15.7 in) wide was placed across the middle section of the east span to simulate major deterioration of the deck surface. Moreover, it was expected that the plank would help excite dominant ?exural modes corresponding to low frequencies [16]. Two truck speeds were used: medium—48 km/h (30 mph) and high-speed—80 km/h (50 mph). The east span and the middle span were instrumented for all tests (Fig. 1). Displacement, strain, and acceleration data were collected at the selected points where the bridge response was expected to be well represented. Additionally, four accelerometers were placed on one of the vehicles to provide data for validation of the truck model. More details of the test data and experimental results are presented in [17]. 4. Development of FE bridge model The FE model of one span includes all ?ve structural components: the slab, six beams, bridge barriers, diaphragms, and neoprene pads. Fig. 3 shows a cut-away segment of the FE model for one span. Concrete parts of the bridge are built of fully integrated solid elements with eight or six nodes. All re-

Fig. 3. A cut-away section of the FE bridge model.

bars and strands are modeled using 1D bar elements with nodes coinciding with corresponding nodes of the solid elements. The locations of some rebars in the FE model were slightly realigned whenever necessary to ?t into geometric FE mesh of the bridge. Neoprene pads are represented by 3D solid elements with viscoelastic material properties [18]. Dimensions of elements in the bridge model were optimized considering the location of the reinforcement, requirements for tire-deck contact algorithm, integration time step [18] and total number of FE elements. Each girder has 24 No.13 1860 MPa low-relaxation straight strands at the bottom ?ange. The mesh size requirement makes it unable to model individually all 24 strands; therefore several strands were lumped together to represent the correct stiffness of the girder cross section. A special LS-DYNA material model called “cable” [18] was applied to introduce prestressing forces in the rod elements. This material model allowed for introduction of initial tensile force by de?ning appropriate initial elongation. The prestress model appears to be critical when the bridge girders are loaded up to failure. Over 204,000 FE were used for one span of the bridge model.

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Fig. 4. Two span bridge model with the threshold at the abutment joint.

The second bridge span was developed as a copy of the ?rst. The two-span model is presented at the top of Fig. 4. All girders are simply supported but the deck reinforcement is continuous. Visual inspection indicated distinct, characteristic cracks at the deck joints. A gap between the two spans was modeled and contact between adjacent elements was applied. Bridge barriers were constructed with 20 mm wide expansion joints, as speci?ed in the design blueprints. Inclusion of the second span in the FE model was found negligible since the girders were simply supported and the deck was separated by a crack opening. This observation allowed reducing further analyses to one bridge span to decrease the computational time. 4.1. FE model of the truck Unlike other computational models found in literature [2,19], more advanced features were incorporated into the FE truck model in this research. These improvements allowed for realistic representation of vehicle–bridge interaction. The truck model presented in Fig. 5 was developed using geometric data collected for a similar tractor, technical drawings of the trailer, and in situ measurements. The models of the tractor and the trailer components were developed separately. After assembling the tractor and trailer parts together, all FE meshes were veri?ed for inconsistencies and were re?ned to improve the mesh quality. Decisions regarding element formulations, material models, material characteristics, contact algorithms, multiple point constraints (MPCs) and connections, loading and boundary condition formulations, solution parameters and others were determined to complete the model. A total of 12,934 elements and nine material types including springs and dampers for suspension, rubber and fabric for tiers, were applied in the model. Mass distribution of the model was re?ned using a trial-anderror approach until static axle loads in the model coincided with the actual measurements of reactions under each axle. Special attention was paid to the suspension system and the wheel models, as they signi?cantly contribute to the truck–bridge dynamic interaction. The truck suspension system was modeled using mostly 1D, structural elements with

Fig. 5. The actual truck and its FE model.

Fig. 6. Truck front suspension and its FE model.

Fig. 7. Development process of FE model for front wheel.

rigid and elastic material types, springs, dampers and a variety of MPCs. Spring stiffness and damping properties were established based on the acceleration readings registered during the ?eld test by four accelerometers located on the vehicle. Rotation of truck wheel 3D models was implemented through the introduction of appropriate MPCs in the truck suspension model. Modeling of rotating wheels was found to be important for the cases where local road surface irregularities such as the plank were present. Fig. 6 shows the front suspension system and its FE model. There are cross-sectional area and material density selected for each 1D element and in this way appropriate mass is associated. The FE model of a wheel is presented in Fig. 7. The tires were modeled with two layers of shell elements, with coinciding

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nodes and different materials. The ?rst layer with elastic material model represents rubber while the second (with special LS-DYNA material called “fabric”) is used for tire cord model. An “airbag” option, available in LS-DYNA, was used to apply internal pressure in all tires. The FE model of the truck was veri?ed partially by comparison of the calculated and experimental results for the time histories of the accelerations recorded by four accelerometers placed on one of the truck (see Fig. 10). 5. Validation of FE models Validation of all FE models was based on comparison between numerical results and experimental data of displacements, strains, and accelerations recorded during the ?eld test [17]. Veri?cation of static response of the bridge is the most common validation method. Design values of concrete from the construction documentation were initially used in FE analysis. Analytical results of strain were found to be higher than experimental readings for static loading (see Fig. 8), which indicated that the actual bridge was stiffer than designed. Concrete core samples extracted from the bridge deck and girders were tested and the actual material properties were introduced in the FE model of the bridge. Table 1 presents actual values of the concrete properties obtained from the laboratory testing. With the new input data numerical results came close to the experimental readings (Fig. 8). To further validate the bridge model, natural frequencies and corresponding modes obtained from LS-DYNA implicit algorithm were compared with the test results. Experimental

frequencies and modes were calculated based on the acceleration measurements for free vibration [21–23]. As shown in Fig. 9, the ?rst (6.11 Hz), the third (8.21 Hz), and the sixth mode (12.66 Hz) from FE analysis correspond well to the ?rst three natural frequencies identi?ed from the experiment as 6, 7.78 and 13.17 Hz. The truck FE model was validated through comparison of calculated and recorded acceleration histories. Computed accelerations were obtained as the time derivatives of the velocity histories at the nodes close to the positions of the actual accelerometers used in the experiment. Both numerical and experimental data was recorded with the frequency of 200 samples/s. No further ?ltering was applied. Fig. 10 shows comparison of recorded and calculated accelerations at the rear axle of the tractor for the run at 80 km/h (50 mph) without plank. Fig. 10 shows interval of 2 s, approximately the time period beginning when the ?rst axle enters the ?rst span and ending when the last axle leaves the span. Although it is usually dif?cult to match numerical and experimental acceleration histories, Fig. 10 shows that the amplitudes of numerical and experimental results are within the same range. Not all peaks representing numerical results line up with the experimental data but the frequencies are similar. 6. Numerical and experimental analysis of vehicle–bridge interaction Both the truck and the bridge models were also validated by comparison of the numerically predicted bridge response subjected to the moving trucks with corresponding experimental results [17]. During the ?eld test and subsequent inspections, a distinct approach depression varying from 10 to 15 mm before the bridge (Fig. 11) was found. This threshold in the road pro?le triggered signi?cant truck–bridge system vibration especially at higher speed. An impact factor as high as 83% resulted from the ?eld test for the case with one truck crossing the bridge with the speed of 80 km/h (50 mph) at the center of the roadway. Signi?cant dynamic response of the bridge triggered by the threshold was later con?rmed through the FE analysis. An actual road pro?le was surveyed in the ?led and was included in the FE model. An additional approach slab was added in the model before the bridge deck (Fig. 4) to re?ect the actual threshold as surveyed in the ?eld. Videos recorded during the tests revealed that a signi?cant hammering effect

40 Micro strains 30 20 10 0 1 2 3 4 Girder number 5 6

FE analysis (estimated concrete properties) FE analysis (actual concrete properties) Field test

Fig. 8. Comparison of strains for ?eld measurement and FE analysis for the static case (one truck at the center of the roadway).

Table 1 Actual and design material properties of the concrete Strength (fc ) Design value Girder Slab 35 MPa (5.08 ksi) 31 MPa (4.50 ksi) Actual value 63.7 MPa (9.24 ksi) 55.9 MPa (8.11 ksi) Modulus (E) Estimated valuea 28.4 GPa (4119.1 ksi) 26.7 GPa (3872.5 ksi) Actual value 37.5 GPa (5441.9 ksi) 40.5 GPa (5871.8 ksi) Poisson’s ratio ( ) Values taken from AASHTO 0.20 0.20 Actual value 0.22 0.20

a The modulus was estimated using the Eq. 5.4.2.4-1 in [20].

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Fig. 9. Comparison of natural vibration modes received from FE analysis and from the experimental testing of the bridge.

40 30 Acceleration (m/s2) 20 10 0 -10 -20 -30 1.5

FE analysis Experiment FE analysis Experiment

80 60 micro strains 40 20 0

lsdyna (case b) experiment lsdyna (case a) lsdyna (case b) lsdyna (case c) lsdyna (case c)

lsdyna (case a)

2

2.5 Time (s)

3

3.5

-20 1.5

2

2.5 Time (s)

3

3.5

Fig. 10. Comparison of recorded and calculated accelerations at the rear axle of the tractor for run at 80 km/h (50 mph) without plank.

Fig. 12. Comparison of experimental and numerical strains for models with and without abutment threshold and load bouncing.

was caused by bouncing of the concrete blocks. This effect, triggered by the abutment threshold, was found to increase the dynamic interaction between the wheels and the deck. The truck FE model was further modi?ed by allowing the concrete blocks to bounce up and down. The concrete blocks were modeled using 3D brick elements with contact de?ned between block’s

Bridge barrier

Slope 2%

bridge apro ach

external surfaces. The vertical movement of the blocks in the FE model was restricted partially by applied MPC [18], simulating actual chains. Fig. 12 presents comparison of numerical results and experimental data for strains in the girder #4 under one truck running at 80 km/h (50 mph) on the center of the roadway without plank. Girder strains were calculated for the following

Slope 2%

Concrete br

Concrete

idge aproac

h

Pavement

Asphalt appr

oach

Aver. 12-15 mm

Asphalt approach

Pavemen

t

Fig. 11. Pro?le of the east approach before the tested bridge.

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L. Kwasniewski et al. / Finite Elements in Analysis and Design 42 (2006) 950 – 959

Table 2 Impact factors for one truck running at 80 km/h (50 mph) without plank

1.5

experiment 50 mph

Acceleration (m/s2)

Speed 80 km/h (50 mph)

Case a 46.3%

Case b 66.4%

Case c 86.3%

Experiment 83%

1 0.5 0 -0.5 -1 -1.5

lsdyna 50 mph

0.5 0.3 0.1 -0.1 -0.3 -0.5 1.5

experiment 30 mph lsdyna 30 mph

Acceleration (m/s2)

bridge approach

1st span

1

1.5

2 Time (s)

2.5

3

3.5

Fig. 14. Comparison of recorded and calculated accelerations in the ?rst span for run at 80 km/h (50 mph) without plank.

bridge approach 1st span

2

2.5

3 Time (s)

3.5

4

4.5 Displacement (mm)

0.5 0

Noptel

Fig. 13. Comparison of recorded and calculated accelerations in the ?rst span for run at 48 km/h (30 mph) without plank.

-0.5

LVDT

-1 -1.5 -2 -2.5 1 2

Lsdyna LVDT 30 mph Noptel 30 mph Lsdyna 30 mph

three different FE models: (a) without threshold and load bouncing, (b) with threshold and without bouncing, (c) with threshold and bouncing. Impact factors calculated based on the experiment and the results from these three models are presented in Table 2. It should be noted that FE models without threshold and load bouncing underestimate actual dynamic effects caused by trucks crossing bridges at higher speeds. Based on the numerical calculations for the models with both features (threshold and load bouncing) at different speeds, the critical speed was determined to be about 72 km/h (45 mph). When the truck was running at higher speed, the load bouncing was triggered at the bridge approach causing increased dynamic effects of the structure. It should be underlined that this result was observed for the speci?c bridge and truck selected in this research. The further analyses were continued using the FE models including both (threshold and load bouncing) features. Figs. 13 and 14 show comparison of calculated and recorded time histories of acceleration for runs at 48 and 80 km/h (30 and 50 mph) without plank. Presented in Figs. 13 and 14 acceleration histories was recorded in the middle of the ?rst bridge span, close to the barrier. Fig. 13 shows very good correlation of both acceleration histories during the second part of the graph, for 3.0 s < t < 4.5 s. Larger discrepancies between experiment and FE analysis can be found for the time interval 2.0 s < t < 3.0 s. However, these discrepancies were caused earlier, at the time frame 1.5 s < t < 2.0 s, when the truck was still on the bridge approach, before entering the bridge. At that moment, experiment data revealed some vibrations already transmitted to the bridge span through the abutment, while computational anal-

3 Time (s)

4

5

Fig. 15. Comparison of recorded and calculated displacements for run at 48 km/h (30 mph) without plank.

1

LVDT

Displacement (mm)

0

Noptel

-1

LVDT 50 mph Noptel 50 mph Lsdyna 50 mph Lsdyna

-2

-3 1 2 Time (s)

Fig. 16. Comparison of recorded and calculated displacements for run at 80 km/h (50 mph) without plank.

3

4

ysis resulted in a ?at, horizontal section since the approach and the bridge deck were not structurally connected. A similar effect can be found in Fig. 14 showing results for the run at 80 km/h (50 mph). Time histories of displacement presented in Figs. 15 and 16 were recorded for the girder #3 at the middle of the ?rst

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2 1 Displacement (mm) 0 -1 -2

LVDT 50 mph with plank Noptel LVDT Lsdyna Noptel 50 mph with plank Lsdyna 50 mph with plank

-3 -4 1.5

0 Rear suspension force (kN)

2

2.5

-100

3 Time (s)

3.5

4

4.5

-200

Fig. 18. Comparison of the displacement at the middle point on the girder #3. One truck crossing the bridge at 80 km/h (50 mph) with plank.

-300

100

-400

micro strains

2.5

3

3.5 Time (s)

4

50

Fig. 17. Run at 80 km/h (50 mph) with the plank. Contours of effective stress and time history of the rear suspension resultant force.

0

experiment 50 mph with plank lsdyna 50 mph with plank

span, for runs at 48 and 80 km/h (30 and 50 mph), respectively. Both ?gures show good correlation especially between Noptel experimental data and LS-DYNA curves. Some discrepancies exist between readings collected from Noptel laser system and LVDT during the test. This difference was caused by lateral distance between the two sensors and the torsional response of the girder. The wooden plank was used to represent extreme bridge deck surface deteriorations. Sometimes such a piece of wood can be found on a roadway, dropped off from a proceeding vehicle. Modeling of this effect represents a complex case with large dynamic effects. Fig. 17 presents an instant when the rear wheels were crossing the plank (located across the middle of the span) in the computer simulation and the corresponding time history of the rear suspension resultant force. Fig. 17 refers to the case when the truck was crossing the bridge at 80 km/h (50 mph). It is noticed that the average resultant force before the rear wheels crossing the plank is about 150 kN which is slightly smaller than the 169 kN of the static reaction in the rear axle. This is because the suspension force does not include the weight of the axles and wheels. Since there are two axels at the rear suspension, two peaks of the suspension force are observed in the diagram. Figs. 18 and 19 show a comparison of time histories of displacement and strain from experimental tests and numerical analyses for runs with the plank. Although a phase shift between the results can be noticed, the amplitudes of displacements are at the similar, consistent range. FE analysis can also help in interpretation of the actual response of the structure due to post-processing visualization

-50 2 2.5 3 Time (s) 3.5 4 4.5

Fig. 19. Comparison of the strain history for girder #4. One truck crossing the bridge at 80 km/h (50 mph) with plank.

capabilities. Es an example Fig. 20 presents deformations (magni?ed 100 times) and contours of pressure (the third of the trace of the stress tensor) in the ?rst span during a selected time instance. A half of the bridge span is shown in ?gure. 7. Summary and conclusions A common, multigirder concrete bridge with short span located in Northwest Florida was studied using FE analysis, veri?ed by static and dynamic ?eld tests. LS-DYNA commercial code was used and detailed 3D models of the truck and the bridge were developed to conduct the FE analysis of the vehicle–bridge interaction. It appears, based on extensive literature review, that no explicit, nonlinear FE code has ever been used for such dynamic bridge analysis before. This paper shows the application of the computational technology, well known in the automobile and aerospace industries, on the new ?eld of vehicle–bridge interaction. Extensive validation efforts showed that the detailed FE models of the bridge and the truck could accurately predict the behavior of actual structures. FE models of heavy vehicles

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L. Kwasniewski et al. / Finite Elements in Analysis and Design 42 (2006) 950 – 959

Fig. 20. Bridge deformation for run at 80 km/h (50 mph) with plank. Displacements magni?ed 100 times.

indicated their advantages as compared with point load models used in the past, and a potential for their further improvements. Some advanced features of truck models developed in this research make them unique and bene?cial for analysis of vehicle–bridge interaction. These include 3D models of pneumatic, rotating wheels, tires modeled with two layers representing rubber and cord, application of internal pressure in tires, and automatic contact algorithm used for analysis. Also 3D suspension components built of rigid elements, springs and dampers well represented vehicle suspension systems. However, reliability of suspension models depends on the correct assessment of stiffness and damping properties of the special elements applied in the FE models. As it is often dif?cult to estimate these values without technical speci?cations provided by auto manufacturers. Good correlation was found between the ?eld measurement and FE analysis in both frequency and time domains. Therefore, the FE models developed in this research can be used for more accurate, follow-up studies of vehicle–bridge responses in the future. It is expected that extrapolating results of FE analysis on broad variety of heavy vehicles and bridges will enable formulation of practical recommendations for such management decisions such as fast and reliable assessment of permissible weight limits for heavy vehicles.

References

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Acknowledgements The study reported in this paper is supported by a grant from the Florida Department of Transportation titled: “Analytical and Experimental Evaluation of Existing Florida DOT Bridges”, contract No. BD 493. The authors would like to express their appreciation for this generous support. Opinions and views expressed in this paper are those of the authors and not necessarily those of the sponsoring Agency. The ?eld tests were professionally performed by the Structures Lab of FDOT. Thanks are due to Mr. Marcus Ansley, Director of the Structures Laboratory of the FDOT for his commitment, advice, and technical support during experimental testing.

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