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Load distribution and dynamic response of multi-girder bridges with FRP decks
Engineering Structures 29 (2007) 1676–1689 www.elsevier.com/locate/engstruct
Load distribution and dynamic response of multi-girder bridges with FRP decks Yin Zhang a,1 , C.S. Cai b,∗ a Department of Civil Engineering, Nanyang Institute of Technology, Nanyang, China b Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, LA 70803, USA
Received 12 April 2006; received in revised form 9 September 2006; accepted 11 September 2006 Available online 27 October 2006
Abstract Bridges with FRP decks are gaining popularity, and there is a growing need to understand the behavior of FRP deck bridges. The characteristics of bridges with FRP decks (such as mass, stiffness, and damping) are significantly different from those of bridges with traditional concrete decks. For this reason, detailed finite element analyses are used in the present study to investigate the load distribution and the dynamic response of FRP deck bridges. The bridge–vehicle interaction based on a three dimensional vehicle–bridge coupled model is carried out on both steel and concrete multi-girder bridges. The dynamic response of bridges is obtained in the time domain considering the road roughness of the deck as a vertical excitation to the vehicles. The load distribution and the dynamic response of bridges are compared between the FRP deck and concrete deck bridges. In addition, there are some arguments whether a composite action between the deck and girders should be pursued or if a simple non-composite design should be used for FRP deck bridges. Discussions on this aspect have been made by modeling both the fully composite and partially composite FRP deck bridges. c 2006 Elsevier Ltd. All rights reserved.
Keywords: Fiber reinforced polymers; Sandwich panels; Bridge deck; Finite element analysis (FEA); Vehicles; Surface roughness; Load distribution; Dynamic response
1. Introduction The bridge infrastructure is deteriorating at an alarming rate. Approximately 30% of all bridges in the US are categorized as structurally deficient and/or functionally obsolete. Approximately 35% of those bridges have exhibited poor deck conditions [1]. Maintenance of bridge infrastructure is a growing concern worldwide. Finding innovative, cost effective solutions for the repair and replacement of concrete and steel in bridges is a necessity. Fiber reinforced polymer (FRP) composite materials have shown great potential as alternative bridge construction materials to conventional ones. Fiber reinforced polymers are gaining popularity in the bridge community.
∗ Corresponding author.
E-mail address: [email protected] (C.S. Cai). 1 Former visiting scholar: Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, LA 70803, USA. c 2006 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2006.09.011
The acceptance of FRP materials in bridge engineering is mainly due to their superior properties such as high strength-toweight ratio, better durability, corrosion resistance, and fatigue resistance over steel and concrete materials. An immediate advantage of using an FRP deck to replace a deteriorated concrete deck is a reduction of the superstructure dead load, which results in an increase in the allowable live load capacity. Thus, the rehabilitated bridges can carry legal loads without extensive repairs. Another potential advantage is a decrease in construction time, which can reduce the inconvenience to the traveling public. FRP decks can also be used in new bridges that can benefit from savings in the cost of the substructure due to the reduced superstructure dead loads [2]. Over the last decade, some FRP bridge deck systems have been proposed, and there is a growing need to understand the behavior of FRP deck bridges. However, very little is known about the lateral distribution of vehicle loads and the dynamic response of bridges with FRP decks. Therefore, the development of FRP bridge decks has been limited.
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The load distribution throughout the bridge deck and the vehicle-induced dynamic impact on bridges are of primary importance in the design of bridges. The load distribution factor and the dynamic impact factor have been used worldwide in bridge design, and extensive experimental and theoretical work has been conducted to determine these factors for bridges with conventional concrete decks. The characteristics of the FRP decks (such as mass, stiffness, and damping) are significantly different from those of the traditional concrete and steel decks, which could result in a different performance of FRP deck bridges from traditional bridges. However, while a few studies conducted static [3] and dynamic tests [4] for FRP deck bridges, the distinctive dynamic performance of bridges with FRP decks has rarely been studied in the literature. Zhang et al. [5] studied the performance of a short FRP slab bridge and compared its performance with the corresponding concrete slab bridge. In an attempt to investigate the performance of multi-girder bridges with FRP decks, this paper focused on three simply-supported multi-girder bridges. To reduce the self-weight and also achieve the necessary stiffness, FRP bridge decks usually employ hollow sandwich configurations, making even the simplest bridge very complicated in analysis. Due to the complexity of the FRP sandwich panel, an equivalent solid plate was used in the present finite element analysis. At first, a steel girder bridge with a FRP deck in Kansas was analyzed. The interaction between a vehicle and the bridge was simulated by using a 3-D finite element analysis (FEA). The results obtained from the analysis were compared with those from the field tests, and a good correlation was achieved. Then, some detailed finite element analyses were used to further investigate the load distribution and the dynamic response of bridge systems with FRP decks. A typical steel multi-girder bridge and a concrete multi-girder bridge with a span length of 60 ft were studied. Connections between FRP decks to girders are more difficult than those between concrete decks and girders. There are some arguments whether a composite action between the deck and girders should be pursued or if a simple non-composite design should be used. For example, field tests prove that there exists essentially no composite action for the bridge tested [3]. To discuss this issue, the load distribution and the dynamic response were compared in three conditions, namely FRP deck fully composite, FRP deck partially composite, and concrete deck fully composite with the girders. The dynamic response of the bridge caused by a 3-axle truck was obtained in the time domain. The influence of the vehicle velocity and bridge surface roughness index on the bridge performance was investigated. 2. Simplified model of FRP deck The FRP bridge decks used in the present study are of a sandwich construction. As shown in Fig. 1, FRP laminates are attached to a closed-cell FRP, honeycomb-type, sinusoidal core, which extends vertically between the two face laminates (or skins). The geometry of this sandwich structure is designed to improve stiffness and buckling response through the continuous support of core elements with the face laminates [6].
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Fig. 1. Sandwich panel configuration.
Due to the geometrical complexity of this panel configuration, a finite element modeling and analysis for an entire bridge can be very complicated, if not impossible. For instance, building a model of a panel of 4.57 × 2.29 × 0.127 m (15 ft × 7.5 ft × 5 in.) would require about 133,200 shell elements, since a minimum of 4 elements are required to model a sine wave plate. As a result, modeling a small slab bridge using finite elements is still very overwhelming, let alone some larger bridges, such as a FRP deck-on-girder bridge, or arch or truss bridges with FRP decks. This situation enforces a dire need for a simplified modeling — an equivalent property approach for the FRP panels [7]. Therefore, finite element modeling techniques were employed in this research work to develop simplified, equivalent properties based on stiffness considerations for this structure. The complex sandwich hollow panel was reduced to a solid orthotropic plate using the equivalent properties derived. To predict the equivalent properties of the sandwich structure, a shellelement modeled sandwich cantilever beam with a unit width was subjected to bending forces. With this approach, the stiffness contribution provided by both the face laminates and the core of the sandwich structure were captured and simplified as a single layer equivalent structure [5,7]. Thus, a finite element analysis of the entire bridge was conducted based on this equivalent orthotropic solid panel. 3. Vehicle–bridge dynamic system In establishing a mathematical model to represent the vehicle dynamics, the following assumptions are made: (1) Vehicle bodies are rigid. (2) The wheels maintain full contact with the bridge surface without separation. (3) All springs are linear, and damping is viscous. (4) All rigid bodies have small displacements about their static equilibrium position. A HS20-44 truck, which is a 3-axle tractor-trailer type, is a major design vehicle in the AASHTO specifications [8,9].
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Fig. 2. HS20-44 vehicle model.
This truck is chosen in this study and is idealized to a vehicle model, as shown in Fig. 2. This mathematical vehicle model consists of two vehicle bodies and 6 wheel bodies. The tires and suspension systems are idealized as linear elastic spring elements and dashpots. The two vehicle bodies have a common rolling and yawing degree of freedom; each vehicle body has 3 additional degrees of freedom including y and z displacements and pitching; each wheel has 2 degrees of freedom, namely, y and z displacements. Therefore, the entire vehicle has a total of 20 degrees of freedom. The non-zero frequencies of the vehicle are calculated as 1.522, 2.139, 2.686, 5.942, 7.742, 7.819, 8.921, 13.874, 13.995, 14.626, and 17.951 Hz. In the present study, the vehicle is considered as an oscillator moving on the bridge. The interaction force between the bridge and the vehicle is dependent on the motion of both the bridge and the vehicle and that the vehicle displacement is related to the bridge displacement, road surface profile, and position of the vehicle. The equations of motion for the coupled system are written as: Mb d¨b Cb + Cbb Cbv d˙b + Mv d¨v Cvb Cv d˙v Fbr K b + K bb K bv db + (1) = K vb Kv dv Fvr + FvG where {db }, [Mb ], [Cb ], and [K b ] are the displacement vector, mass matrix, damping matrix, and stiffness matrix of the bridge, respectively; {dv }, [Mv ], [Cv ], and [K v ] are the displacement vector, mass matrix, damping matrix, and stiffness matrix of the vehicle, respectively; and {FvG } = the gravity force vector of the vehicle. It is assumed that the wheels always maintain a point contact with the bridge deck without separation. The equations of motion for the vehicle and bridge are coupled through the interaction force and the terms Cbb , Cbv , Cvb , K bb , K bv , K vb , Fbr , and Fvr stem from the contact (interaction) force. To simplify the modeling procedure, the bridge modal superposition technique is used based on the obtained bridge mode shapes {Φi } and the corresponding natural circular frequencies ωi . The bridge dynamic response {db } can be expressed as: {db } = {Φ1 }
{Φ2 } . . . {Φn }
ξ1
ξ2 . . . ξn
T
= [Φb ] {ξb } (2)
where n is the total number of modes for the bridge under consideration, and {Φi } and ξi are the ith mode shape and the generalized coordinates, respectively. If each mode shape is normalized with the mass matrix, i.e. {Φi }T [Mb ]{Φi } = 1 and {Φi }T [K b ]{Φi } = ωi2 , and if the damping matrix [Cb ] is written to be 2ωi ηi [Mb ], where ωi is the natural circular frequency of the bridge and ηi is the percentage of the critical damping for the ith mode, then Eq. (1) can be derived as: ξ¨b I 2ωi ηi I + ΦbT (Cbb )Φb Cbv ξ˙b + Mv d¨v d˙v ΦbT Cvb Φb Cv 2 ΦbT Fbr ωi I + ΦbT (K bb )Φb K bv ξb + = . (3) dv Fvr + FvG ΦbT K vb Φb Kv The modal superposition makes it possible to separate the bridge modal analysis from the vehicle–bridge coupled model. Consequently, the number of equations in Eq. (3) and the complexity of the whole procedure are greatly reduced. Eq. (3) is solved by using the Fourth Order Runge–Kutta method in the time domain. The road surface profile is an important factor that affects the dynamic responses of both the bridge and the vehicles. In this study, the road surface profile was simulated in the space domain, which serves as an input to the vehicle–bridge model. Here, the road profile contains both the road roughness of the bridge deck and the approach roadway. More details of the numerical model are given in [5]. 4. Verification with a FRP deck bridge A Kansas DOT supported project compared the lateral load distribution (lateral stiffness) characteristics of a 14-girder bridge with a corrugated metal decking to the same bridge after the original deck was replaced with a FRP deck. These comparisons were developed by field testing the bridge prior to and after the deck rehabilitation. The measured data is a good resource to verify the procedure developed in the present study. For the convenience of the readers, some information from the Kansas DOT report [10] is reiterated below. The bridge used in this experiment was the Crawford County Bridge 031, which is located near Pittsburgh, Kansas on K126. The design parameters for this bridge are as follows: the beams are W21 × 68 with a spacing of 0.686 m (27 in.). The
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Fig. 3. Lateral load position of Kansas field test.
bridge deck, before replacement, was 13.720 m (45 ft) long and 9.146 m (30 ft) wide. The final replacement deck is 9.756 m (32 ft) wide. The specially designed connections (clips) for the FRP deck and the existing steel girders were installed every 2.439 m (8 ft) and on every third girder. For the other girders without connections, the deck sits on the girder without tight connections. Therefore, the FRP deck is partially composite with the steel girders. Ten truck passes were performed on various lines on the surface of the bridge deck for field tests. The first of these lines was positioned so that the right front tire of the loaded truck would travel directly across the centerline of the first interior girder. Adjacent lines were then spaced laterally across the bridge deck using the first line as a datum and allowing enough room for ten load passes. The layout for the load passes can be seen in Fig. 3. All fourteen steel girders were instrumented with strain gages at the mid-span along the longitudinal centerline of each girder, and diagnostic tests were run on the strain gages before loading the truck. The “measured” stresses σ were calculated by taking the strain reading minus the average of the first few strain readings (with zero loading) and multiplying this number by a strain gage factor and by the modulus of elasticity E as σ = (ε − average(εinitial )) × (gage factor) × E.
(4)
The lateral distribution values (DF) are equal to the actual stress on a particular girder divided by the sum of the stresses on all the girders as: , 14 X Σ σgirder− j . (5) DFi = σgirder−i j=1
The writers used the same bridge after the deck rehabilitation (i.e. with FRP deck) to construct a 3-D linear elastic finite element model. The analysis was performed using ANSYS 9.0 finite element program available at Louisiana State University. The bridge was modeled with Solid45 eightnode solid elements with three degrees of freedom at each node for the FRP deck and Shell63 four-node shell elements
with six degrees of freedom at each node for the steel girders. To simulate the composite action between the two components, two models were constructed. The first one was fully composite, where the three translation degrees of freedom of the solid element nodes for the deck are fully connected with those of the shell elements for the girder flange. The second one was partially composite, where the connections were made every 2.349 m (8 ft) — the same as the field condition, but connections were placed on every girder for numerical stability, which is slightly different from the field bridge connections stated earlier. The results obtained from the finite element analysis were compared with those from the field tests, and in general, a good correlation was achieved, as shown in Fig. 4. In both the FEA and field values, no multiple presence factors specified in the AASHTO code [9] were included, and they are thus denoted as DF instead of LDF, as used later. The comparison indicated a small change in the load distribution between the fully composite condition and the partially composite condition of the FRP deck system, as shown in Fig. 4(a)–(j). This is because the stiffness of the FRP deck is relatively small compared to a traditional concrete deck, and it is also due to the small beam spacing of this bridge. In general, the composite or partial composite design of FRP deck bridges does not make as a significant difference on load distribution as a concrete deck would do. These figures also indicate that the load distribution values can be estimated based on the FEA, using a simplified model of the FRP deck (an equivalent solid plate). As expected, the loading position significantly affects the load distribution among the girders. A truck moving along the side of the road results in higher distribution values, while the truck traveling along the middle of the road induces lower distribution values. A summary of the maximum values is provided in Table 1. As mentioned earlier, since this bridge in the field has only minimum connections between the deck and the girders, supposedly the DF of the test results should be larger than that from the finite element analyses for both fully composite and partially composite models. While the maximum DF for
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Fig. 4. Lateral distribution results from FEA versus from field test.
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5. Numerical analysis of typical girder bridge models
Table 1 Maximum values from field tests and FEA
Maximum DF for interior girder Maximum DF for exterior girder Impact factor
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Field tests
FEA of fully composite
FEA of partially composite
0.226 0.141 1.08
0.184 0.167 1.02
0.180 0.170 1.30
Note: DF = Distribution factor.
interior girders of field tests was larger than that of the FEA as expected, it is opposite for the exterior girder. This may be due to the existence of concrete edge beams (for installing the railing) along the road in the actual field bridge. These beams were not considered in the finite element model. As also shown in Table 1, the dynamic impact factor from test is 1.08, while the predicted ones for a vehicle velocity of 22.0 m/s are 1.02 and 1.30 for a fully composite and partially composite deck, respectively. It is stated in [10] that the simple dynamic tests are not used for decisions or conclusions since the tests were conducted out of curiosity. While the measured results of this example bridge have provided an opportunity to verify the developed finite element modeling techniques, this bridge, with a narrow girder spacing, is not a typical bridge that can be used to draw more general conclusions. In order to study the load distribution and the dynamic behavior of multi-girder bridges with FRP decks, two more general bridge models were developed and described below.
One steel girder bridge model and one prestressed concrete girder bridge model, as shown in Figs. 5 and 6, were developed [11,12]. The span length for both bridges is 18.288 m (60 ft). The two bridges were designed for the HS20-44 loading, and they both consist of five identical girders which are simply supported. In order to compare the performance, two types of bridge decks were used: a honeycomb-type sinusoidal core FRP sandwich panel (see Fig. 1) and a concrete slab. The FRP decks were designed for fully composite or partially composite (the connection was made every 2.349 m on every girder). The thickness of FRP decks used in these models is 203 mm (8 in.), and the deck was simplified as an equivalent orthotropic solid panel, as discussed earlier. The thickness of the concrete decks is 191 mm (7.5 in.) for both the steel and prestressed concrete girder bridges. 5.1. Load lateral distribution To find the most unfavorable loading position and the meaningful load distribution factors, eight loading cases were investigated (Figs. 5 and 6). There are three different cases of one-truck loading: (1) located on the side of the road, (2) the left tire of the truck directly across the centerline of the first interior girder (for steel girder bridge) or across the centerline of the exterior girder (for concrete girder bridge), and (3) the middle of the road. The three two-truck loadings correspond
Fig. 5. Cross-section of steel multi-girder bridge and loading cases.
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Fig. 6. Cross-section of concrete multi-girder bridge and loading cases.
to the three one-truck loading cases and the two three-truck loadings are located on the side of the road and the middle of the road, respectively. In the longitudinal direction, the mid-wheels are put directly above the mid-span of the girders. The loadings of front tires, middle tires, and rear tires are 35 kN (8 kips), 145 kN (32 kips), and 145 kN (32 kips), respectively. For each girder, the LDF is calculated using the maximum static stress in that girder obtained from the static loading at the same cross-section of the bridge. When all girders have the same section modulus, load distribution factors are calculated using a method employed by Ghosn et al. [13,14]. Accordingly, the LDF for the ith girder, LDFi , can be derived as follows: LDFi =
nσi k P
(6)
σj
j=1
where σ j = the bottom-flange stress at the jth girder; k = the number of girders; and n = the number of side-by-side trucks. For cases of one-truck, two-truck, and three-truck loadings on a bridge, the LDFs calculated from Eq. (6) are multiplied by the multiple presence factors of 1.2, 1.0, and 0.85, respectively [9]. Generally, the distribution factors of girder bridges for a twotruck loading are higher than those for a one-truck or threetruck loading after considering the multiple presence factors. The bottom-flange stress values at the mid-span of the girders obtained from the FEA were used to calculate the LDFs for the steel and concrete bridges, respectively, as shown in Tables 2 and 3. For the convenience of discussions, the LDFs are also plotted in Figs. 7 and 8. By comparing the
two cases of “FRP Deck Fully Composite” and “FRP Deck Partially Composite”, it is observed that when the deck and the girders are partially composite, the LDF values are larger, since in this case a smaller portion of loads are shared by the other girders. In other words, generally speaking, bridges with partially composite conditions cannot distribute loads as uniformly as bridges with fully composite conditions. By comparing the two cases of “FRP Deck Fully Composite” and “Concrete Deck Fully Composite”, it is observed that due to the higher stiffness of the concrete deck, the LDF values of the bridge with concrete deck are smaller than those with the FRP deck. Among the three deck configurations, the “FRP Deck Partially Composite” results in the highest LDF values. It is noted that only the maximum LDF value among the girders matters in the design process, although the LDFs for all girders were calculated and plotted in the figures. Meanwhile, it is observed that the stress values of concrete girder bridges are smaller than those of steel girder bridges, because concrete girder bridges have a larger stiffness compared to steel girder bridges. The LDF values of concrete girder bridges are more uniform for different decks than those of steel girder bridges. This means that in terms of LDFs, concrete girder bridges are not as sensitive to deck stiffness as steel girder bridges are. 5.2. Dynamic response To investigate the effect of the deck system on bridge performance, dynamic analyses of the bridge system with different deck configurations were carried out. The multi-girder
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Y. Zhang, C.S. Cai / Engineering Structures 29 (2007) 1676–1689 Table 2 Stress and LDF of steel girder bridge Case
Girder number
Stress (MPa) FRP deck fully comp.
FRP deck partially comp.
Concrete deck fully comp.
LDF FRP deck fully comp.
FRP deck partially comp.
Concrete deck fully comp.
Case 1
1 2 3 4 5
0.312 3.210 9.393 22.782 31.348
−0.782 1.423 8.427 23.182 32.793
4.761 6.135 9.252 16.697 22.110
0.006 0.057 0.168 0.408 0.561
−0.014 0.026 0.155 0.428 0.605
0.097 0.125 0.188 0.340 0.450
Case 2
1 2 3 4 5
3.198 8.742 19.337 21.953 12.884
0.692 8.210 21.117 23.531 10.843
5.989 9.166 15.324 16.659 11.263
0.058 0.159 0.351 0.398 0.234
0.013 0.153 0.394 0.439 0.202
0.123 0.188 0.315 0.342 0.231
Case 3
1 2 3 4 5
6.877 14.721 22.215 14.718 6.873
3.842 15.933 24.797 15.938 3.847
7.872 12.588 17.098 12.585 7.872
0.126 0.270 0.408 0.270 0.126
0.072 0.297 0.462 0.297 0.072
0.163 0.260 0.354 0.260 0.163
Case 4
1 2 3 4 5
5.636 15.330 31.388 40.249 40.043
1.676 14.508 32.784 41.980 38.441
11.886 17.279 26.190 30.838 30.896
0.085 0.231 0.473 0.607 0.604
0.026 0.224 0.507 0.649 0.594
0.203 0.295 0.447 0.527 0.528
Case 5
1 2 3 4 5
17.880 31.601 37.246 29.775 15.521
13.869 32.815 40.605 30.428 11.168
18.330 26.360 29.755 25.288 16.977
0.271 0.479 0.564 0.451 0.235
0.215 0.509 0.630 0.472 0.173
0.314 0.452 0.510 0.433 0.291
Case 6
1 2 3 4 5
16.571 30.798 37.356 30.795 16.565
12.465 31.646 40.665 31.638 12.462
17.619 25.844 29.794 25.842 17.619
0.251 0.466 0.566 0.466 0.251
0.193 0.491 0.631 0.491 0.193
0.302 0.443 0.511 0.443 0.302
Case 7
1 2 3 4 5
28.208 39.344 44.316 45.388 41.303
23.683 40.132 46.635 45.644 38.050
28.666 34.992 37.651 38.062 36.039
0.362 0.505 0.569 0.583 0.530
0.311 0.527 0.613 0.600 0.500
0.417 0.509 0.547 0.553 0.524
Case 8
1 2 3 4 5
34.645 42.700 43.810 42.697 34.642
30.465 43.209 46.870 43.140 30.432
32.230 36.663 37.542 36.660 32.229
0.445 0.549 0.563 0.549 0.445
0.400 0.568 0.616 0.567 0.400
0.469 0.533 0.546 0.533 0.469
bridges were assumed to be at rest before the vehicle entered the bridge. The vehicle was modeled with a 40 m lead distance to minimize the effect of the vehicle initial conditions on bridge vibrations. Therefore, there is no bridge vibration for the first few seconds in the figures shown below. A parametric sensitivity study was conducted to analyze the effect of factors such as road surface condition and vehicle velocity on the bridge dynamic response. The objective was to compare the dynamic performance of the FRP deck bridges (fully composite or partially composite) with the corresponding concrete deck bridges and find some correlations between the bridge dynamic performance and these parameters. Many investigations have shown that the roughness of a bridge surface is an important factor that affects the dynamic response of bridge structures [15–18]. In this study,
classification of road roughness based on the International Organization for Standardization [19] was used, and the road surface profile was simulated in the space domain. Two road conditions were considered as inputs to the vehicle–bridge coupled model, namely: (1) road surface condition is good; and (2) road surface condition is poor. Based on these two road conditions and the vehicle velocity of v = 10 m/s, 20 m/s, or 40 m/s, the dynamic responses at the mid-span bottom-flange of the center girder were evaluated. While Figs. 9 and 10 demonstrate the effects of vehicle velocity and road roughness on the displacement of bridges with three different deck conditions, Figs. 11 and 12 demonstrate, correspondingly, their effects on the accelerations. As shown in Figs. 9 and 10, deck types have significantly affected the displacement. While the FRP Deck Partially
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Table 3 Stress and LDF of concrete girder bridge Case
Girder number
Stress (MPa) FRP deck fully comp.
FRP deck partially comp.
Concrete deck fully comp.
LDF FRP deck fully comp.
FRP deck partially comp.
Concrete deck fully comp.
Case 1
1 2 3 4 5
−0.037 0.196 0.954 2.921 4.457
−0.084 0.133 0.926 2.750 4.445
0.292 0.420 0.857 2.002 2.940
−0.005 0.028 0.135 0.413 0.630
−0.012 0.020 0.136 0.404 0.653
0.054 0.077 0.158 0.369 0.542
Case 2
1 2 3 4 5
−0.029 0.230 1.045 3.033 4.224
−0.080 0.172 1.036 2.845 4.190
0.299 0.440 0.913 2.067 2.795
−0.004 0.032 0.147 0.428 0.596
−0.012 0.025 0.152 0.418 0.616
0.055 0.081 0.168 0.381 0.515
Case 3
1 2 3 4 5
0.579 2.010 3.202 2.005 0.578
0.493 2.038 3.092 2.034 0.492
0.653 1.490 2.193 1.489 0.653
0.083 0.288 0.459 0.287 0.083
0.073 0.300 0.455 0.300 0.072
0.121 0.276 0.406 0.276 0.121
Case 4
1 2 3 4 5
0.359 1.771 4.118 5.382 5.278
0.221 1.793 3.959 5.157 5.186
0.849 1.659 3.018 3.755 3.724
0.042 0.209 0.487 0.637 0.624
0.027 0.220 0.485 0.632 0.636
0.131 0.255 0.464 0.577 0.573
Case 5
1 2 3 4 5
0.422 1.945 4.230 5.342 4.957
0.281 1.956 4.102 5.129 4.841
0.885 1.759 3.093 3.733 3.531
0.050 0.230 0.501 0.632 0.587
0.035 0.240 0.503 0.629 0.594
0.136 0.271 0.476 0.574 0.543
Case 6
1 2 3 4 5
1.762 4.117 5.147 4.113 1.760
1.678 3.974 4.991 3.968 1.674
1.671 3.022 3.616 3.021 1.670
0.208 0.487 0.609 0.487 0.208
0.206 0.488 0.613 0.487 0.206
0.257 0.465 0.556 0.465 0.257
Case 7
1 2 3 4 5
3.432 5.096 5.681 5.810 5.296
3.227 4.949 5.588 5.565 5.137
2.933 3.929 4.252 4.312 4.060
0.346 0.513 0.572 0.585 0.533
0.336 0.516 0.582 0.580 0.535
0.384 0.514 0.556 0.564 0.531
Case 8
1 2 3 4 5
4.381 5.473 5.654 5.469 4.380
4.129 5.299 5.619 5.292 4.119
3.485 4.137 4.256 4.136 3.484
0.441 0.550 0.569 0.550 0.441
0.430 0.552 0.586 0.552 0.429
0.456 0.541 0.557 0.541 0.456
Composite condition results in the largest displacement, the Concrete Deck Fully Composite condition gives the lowest displacement. This observation agrees with that observed for the static loading case, as discussed earlier, since the deck stiffness is in an ascending order from FRP Deck Partially Composite, to FRP Deck Fully Composite, and to Concrete Deck Fully Composite. The difference between the FRP Deck Partially Composite and FRP Deck Fully Composite are more pronounced in bridges with steel girders (Fig. 9) than those with concrete girders (Fig. 10). For accelerations, as shown in Figs. 11 and 12, the values of bridges with an FRP deck are far larger than those with a concrete deck, even under a low vehicle velocity condition; the acceleration values of bridges with a FRP deck in the partially composite condition are generally close to the values
of those with a FRP deck in the fully composite condition. Under a low vehicle velocity (10 m/s), the FRP deck in the partially composite condition generally results in a higher acceleration than that of the FRP deck in the fully composite condition. Under a high vehicle velocity (40 m/s), the trend is just the opposite. While the explanation of this phenomenon is not straightforward, it can be presumably stated that a higher vehicle velocity excites more participations of higher bridge modes. If the displacement response is expressed as PN {d(t)} = i=1 ζi sin[(2π f i )t + θi ]{φi }, where {φi } = the modal shape i, ζi = the participation factor of modal shape i, f i = the frequency of modal shape i, and θi = the initial phase of modal shape i, then the acceleration is derived as PN 2 2 { ∂ ∂td(t) 2 } = − i=1 ζi (2π f i ) sin[(2π f i )t + θi ]{φi }, i.e., it is
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Fig. 7. LDF comparison of steel multi-girder bridge.
proportional to f i2 . Therefore, though the displacement of a partially composite deck bridge is lager than that of a fully composite deck bridge, the acceleration could be opposite since it is a function of f i2 .
Figs. 9 through 12 also show that the vehicle velocity effect is more pronounced when the road surface condition is poor. Road roughness of a bridge seriously affects the vehicle’s vibrations, thus affecting the vehicle–bridge interaction. It
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Fig. 8. LDF comparison of concrete multi-girder bridge.
can be seen from the figures that the worse the bridge road condition, the larger the bridge dynamic displacement, and the far larger the bridge dynamic acceleration under the truck load. This situation is more obvious in steel girder bridges than in
concrete bridges. A poor road condition not only influences the bridge’s normal operation, it moreover creates a vertical acceleration, which can make the driver uncomfortable and may cause a higher deterioration rate of the bridge. Therefore,
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Fig. 9. Displacement comparison of steel multi-girder bridge. Fig. 10. Displacement comparison of concrete multi-girder bridge.
maintaining the bridge road surface in a good condition is very important in reducing the vehicle dynamic impact effects. However, the dynamic response of bridges does not increase monotonically with the increase in vehicle velocity. There is a peak value corresponding to a specific vehicle velocity, which is considered as being related to a vehicle induced resonant vibration [5].
6. Conclusions The present study has developed a static and dynamic analysis procedure for the vehicle–bridge interaction of FRP deck bridges. After verifying its applicability by comparing the static load distribution with that of a field tested bridge, the present study investigated two typical multi-girder bridges with
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Fig. 11. Acceleration comparison of steel multi-girder bridge.
FRP decks, one with steel and the other with concrete girders, and compared their performance with the corresponding multigirder bridges with concrete decks. Based on the present study, the following conclusions can be drawn: 1. The present study used an equivalent orthotropic solid plate model for the FRP hollow sandwich panel. The load distribution results obtained from the finite element analysis
Fig. 12. Acceleration comparison of concrete multi-girder bridge.
using this simplified model were compared with those from field tests, and a good correlation was achieved. The finite element analyses were very helpful in investigating the performance of FRP deck bridges. 2. For both load distribution and dynamic response, bridge deck types have seriously affected the results. The LDF
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values of FRP deck bridges are larger than those of concrete deck bridges. The dynamic response of FRP deck bridges is also larger than that of the concrete deck bridges. 3. The FRP deck bridges with partially composite conditions have a larger girder distribution and dynamic displacement than those of the FRP deck bridges with fully composite conditions. Therefore, in order to obtain a better performance, it is necessary to strengthen the connection between the FRP deck and girders through structural measures. However, this is a challenging task and usually an expensive requirement for a FRP deck system. If the non-composite condition is preferred for the FRP deck system, then it should be noted that the load distribution and dynamic impact factors developed for the full composite conditions may not be conservative for the girder design of bridges with non-composite decks. 4. Road roughness and vehicle velocity all significantly affect the dynamic performance of both the analyzed FRP deck and the concrete deck bridges. Acceleration seems to be more sensitive to a poor road condition than to a good road condition with the same vehicle speed, especially for steel girder bridges. It is suggested that the bridge road surface condition be kept in good condition for valid applications of the code specified dynamic impact factor in bridge design and rating. 5. Considering the different dynamic performances observed between the FRP and concrete deck bridges, different serviceability control criteria for FRP deck bridges may be developed. The design of FRP deck bridges is usually controlled by deflection requirements, and meeting the same deflection requirement as conventional bridges (say L/800) may be uneconomical in many cases. Loosening this limitation (say, an increase to L/400) has been suggested in the literature. Therefore, reexamining the serviceability control criteria for FRP deck bridges based on dynamic analysis will have practical and economical significance. Further study is underway in this aspect by the writers who are also investigating if the AASHTO dynamic impact factors for conventional deck bridges can be applied to FRP deck bridges. The developed methodology is ready to be applied to more FRP deck bridges for comparing their performance with the AASHTO codes. References [1] Transportation Research Board. TR News, special issue on highway bridges: progress and prospects. 1998; January—February (No. 194).
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[2] Temeles AB. Field and laboratory tests of a proposed bridge deck panel fabricated from pultruded fiber-reinforced polymer components. Thesis for the degree of Master of Science in Civil Engineering. Blacksburg (VA): Virginia Polytechnic Institute and State University; 2001. [3] Alampalli S, Kunin J. Load testing of an FRP bridge deck on a truss bridge special report 137. Albany (NY): Transportation Research and Development Bureau, New York State Department of Transportation; 2001. [4] Aluri S, Jinka C, GangaRao HVS. Dynamic response of three fiber reinforced polymer composite bridges. Journal of Bridge Engineering ASCE 2005;10(6):722–30. [5] Zhang Y, Cai CS, Shi XM, Wang C. Vehicle induced dynamic performance of a FRP versus concrete slab bridge. Journal of Bridge Engineering ASCE 2006;11(4):410–9. [6] Plunkett JD. Fiber-reinforced polymer honeycomb short span bridge for rapid installation. Idea project final report, Contract NCHRP-96-IDO30, IDEA Program. Transportation Research Board, National Research Council; 1997. [7] Oghumu S, Cai CS, Zhang Y. Finite element modeling and performance evaluation for the development of FRP bridge panels. In: The joint /ASME/ASCE/SES engineering mechanics and materials conference. 2005. [8] American Association of State Highway and Transportation Officials (AASHTO). Standard specification for highway bridges. Washington (DC); 2004. [9] American Association of State Highway and Transportation Officials (AASHTO). LRFD bridge design specifications. Washington (DC); 2004. [10] Schreiner J, Barker M. Lateral distribution in Kansas DOT steel girder bridge with FRP deck. Report no. KS-04-4. Kansas Department of Transportation; 2005. [11] Huang DZ, Wang T-L. Vibration of highway steel bridges with longitudinal grades. Computers and Structures 1998;69:235–45. [12] Liu C, Huang D, Wang T-L. Analytical dynamic impact study based on correlated road roughness. Computers and Structures 2002;80:1639–50. [13] Ghosn M, Moses F, Gobieski J. Evaluation of steel bridges using inservice testing. Transp. Res. Rec. 1072. Washington (DC): Transportation Research Board; 1986. p. 71–8. [14] Cai CS, Shahaway M. Predicted and measured performance of prestressed concrete bridges. Journal of Bridge Engineering ASCE 2004;9(1): 4–13. [15] Wang TL, Huang DZ, Shahaway M. Dynamic response of multigirder bridges. Journal of Structural Engineering ASCE 1992;118(8):2222–38. [16] Wang TL, Huang DZ, Shahaway M. Dynamic behavior of slant-lagged rigid frame highway bridge. Journal of Structural Engineering ASCE 1994;120(3):885–902. [17] Huang DZ, Wang TL, Shahaway M. Dynamic behavior of horizontally curved I-girder bridges. Journal of Computers and Structures 1992;57(4): 703–14. [18] Huang DZ, Wang TL, Shahaway M. Impact studies of multigirder concrete bridges. Journal of Structural Engineering ASCE 1993;119(8): 2387–402. [19] ISO 8068. Mechanical vibration — road surface profiles-reporting of measured data. TC108/SC2. Switzerland: ISO; 1995.
Load distribution and dynamic response of multi-girder bridges with FRP decks Yin Zhang a,1 , C.S. Cai b,∗ a Department of Civil Engineering, Nanyang Institute of Technology, Nanyang, China b Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, LA 70803, USA
Received 12 April 2006; received in revised form 9 September 2006; accepted 11 September 2006 Available online 27 October 2006
Abstract Bridges with FRP decks are gaining popularity, and there is a growing need to understand the behavior of FRP deck bridges. The characteristics of bridges with FRP decks (such as mass, stiffness, and damping) are significantly different from those of bridges with traditional concrete decks. For this reason, detailed finite element analyses are used in the present study to investigate the load distribution and the dynamic response of FRP deck bridges. The bridge–vehicle interaction based on a three dimensional vehicle–bridge coupled model is carried out on both steel and concrete multi-girder bridges. The dynamic response of bridges is obtained in the time domain considering the road roughness of the deck as a vertical excitation to the vehicles. The load distribution and the dynamic response of bridges are compared between the FRP deck and concrete deck bridges. In addition, there are some arguments whether a composite action between the deck and girders should be pursued or if a simple non-composite design should be used for FRP deck bridges. Discussions on this aspect have been made by modeling both the fully composite and partially composite FRP deck bridges. c 2006 Elsevier Ltd. All rights reserved.
Keywords: Fiber reinforced polymers; Sandwich panels; Bridge deck; Finite element analysis (FEA); Vehicles; Surface roughness; Load distribution; Dynamic response
1. Introduction The bridge infrastructure is deteriorating at an alarming rate. Approximately 30% of all bridges in the US are categorized as structurally deficient and/or functionally obsolete. Approximately 35% of those bridges have exhibited poor deck conditions [1]. Maintenance of bridge infrastructure is a growing concern worldwide. Finding innovative, cost effective solutions for the repair and replacement of concrete and steel in bridges is a necessity. Fiber reinforced polymer (FRP) composite materials have shown great potential as alternative bridge construction materials to conventional ones. Fiber reinforced polymers are gaining popularity in the bridge community.
∗ Corresponding author.
E-mail address: [email protected] (C.S. Cai). 1 Former visiting scholar: Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, LA 70803, USA. c 2006 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2006.09.011
The acceptance of FRP materials in bridge engineering is mainly due to their superior properties such as high strength-toweight ratio, better durability, corrosion resistance, and fatigue resistance over steel and concrete materials. An immediate advantage of using an FRP deck to replace a deteriorated concrete deck is a reduction of the superstructure dead load, which results in an increase in the allowable live load capacity. Thus, the rehabilitated bridges can carry legal loads without extensive repairs. Another potential advantage is a decrease in construction time, which can reduce the inconvenience to the traveling public. FRP decks can also be used in new bridges that can benefit from savings in the cost of the substructure due to the reduced superstructure dead loads [2]. Over the last decade, some FRP bridge deck systems have been proposed, and there is a growing need to understand the behavior of FRP deck bridges. However, very little is known about the lateral distribution of vehicle loads and the dynamic response of bridges with FRP decks. Therefore, the development of FRP bridge decks has been limited.
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The load distribution throughout the bridge deck and the vehicle-induced dynamic impact on bridges are of primary importance in the design of bridges. The load distribution factor and the dynamic impact factor have been used worldwide in bridge design, and extensive experimental and theoretical work has been conducted to determine these factors for bridges with conventional concrete decks. The characteristics of the FRP decks (such as mass, stiffness, and damping) are significantly different from those of the traditional concrete and steel decks, which could result in a different performance of FRP deck bridges from traditional bridges. However, while a few studies conducted static [3] and dynamic tests [4] for FRP deck bridges, the distinctive dynamic performance of bridges with FRP decks has rarely been studied in the literature. Zhang et al. [5] studied the performance of a short FRP slab bridge and compared its performance with the corresponding concrete slab bridge. In an attempt to investigate the performance of multi-girder bridges with FRP decks, this paper focused on three simply-supported multi-girder bridges. To reduce the self-weight and also achieve the necessary stiffness, FRP bridge decks usually employ hollow sandwich configurations, making even the simplest bridge very complicated in analysis. Due to the complexity of the FRP sandwich panel, an equivalent solid plate was used in the present finite element analysis. At first, a steel girder bridge with a FRP deck in Kansas was analyzed. The interaction between a vehicle and the bridge was simulated by using a 3-D finite element analysis (FEA). The results obtained from the analysis were compared with those from the field tests, and a good correlation was achieved. Then, some detailed finite element analyses were used to further investigate the load distribution and the dynamic response of bridge systems with FRP decks. A typical steel multi-girder bridge and a concrete multi-girder bridge with a span length of 60 ft were studied. Connections between FRP decks to girders are more difficult than those between concrete decks and girders. There are some arguments whether a composite action between the deck and girders should be pursued or if a simple non-composite design should be used. For example, field tests prove that there exists essentially no composite action for the bridge tested [3]. To discuss this issue, the load distribution and the dynamic response were compared in three conditions, namely FRP deck fully composite, FRP deck partially composite, and concrete deck fully composite with the girders. The dynamic response of the bridge caused by a 3-axle truck was obtained in the time domain. The influence of the vehicle velocity and bridge surface roughness index on the bridge performance was investigated. 2. Simplified model of FRP deck The FRP bridge decks used in the present study are of a sandwich construction. As shown in Fig. 1, FRP laminates are attached to a closed-cell FRP, honeycomb-type, sinusoidal core, which extends vertically between the two face laminates (or skins). The geometry of this sandwich structure is designed to improve stiffness and buckling response through the continuous support of core elements with the face laminates [6].
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Fig. 1. Sandwich panel configuration.
Due to the geometrical complexity of this panel configuration, a finite element modeling and analysis for an entire bridge can be very complicated, if not impossible. For instance, building a model of a panel of 4.57 × 2.29 × 0.127 m (15 ft × 7.5 ft × 5 in.) would require about 133,200 shell elements, since a minimum of 4 elements are required to model a sine wave plate. As a result, modeling a small slab bridge using finite elements is still very overwhelming, let alone some larger bridges, such as a FRP deck-on-girder bridge, or arch or truss bridges with FRP decks. This situation enforces a dire need for a simplified modeling — an equivalent property approach for the FRP panels [7]. Therefore, finite element modeling techniques were employed in this research work to develop simplified, equivalent properties based on stiffness considerations for this structure. The complex sandwich hollow panel was reduced to a solid orthotropic plate using the equivalent properties derived. To predict the equivalent properties of the sandwich structure, a shellelement modeled sandwich cantilever beam with a unit width was subjected to bending forces. With this approach, the stiffness contribution provided by both the face laminates and the core of the sandwich structure were captured and simplified as a single layer equivalent structure [5,7]. Thus, a finite element analysis of the entire bridge was conducted based on this equivalent orthotropic solid panel. 3. Vehicle–bridge dynamic system In establishing a mathematical model to represent the vehicle dynamics, the following assumptions are made: (1) Vehicle bodies are rigid. (2) The wheels maintain full contact with the bridge surface without separation. (3) All springs are linear, and damping is viscous. (4) All rigid bodies have small displacements about their static equilibrium position. A HS20-44 truck, which is a 3-axle tractor-trailer type, is a major design vehicle in the AASHTO specifications [8,9].
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Fig. 2. HS20-44 vehicle model.
This truck is chosen in this study and is idealized to a vehicle model, as shown in Fig. 2. This mathematical vehicle model consists of two vehicle bodies and 6 wheel bodies. The tires and suspension systems are idealized as linear elastic spring elements and dashpots. The two vehicle bodies have a common rolling and yawing degree of freedom; each vehicle body has 3 additional degrees of freedom including y and z displacements and pitching; each wheel has 2 degrees of freedom, namely, y and z displacements. Therefore, the entire vehicle has a total of 20 degrees of freedom. The non-zero frequencies of the vehicle are calculated as 1.522, 2.139, 2.686, 5.942, 7.742, 7.819, 8.921, 13.874, 13.995, 14.626, and 17.951 Hz. In the present study, the vehicle is considered as an oscillator moving on the bridge. The interaction force between the bridge and the vehicle is dependent on the motion of both the bridge and the vehicle and that the vehicle displacement is related to the bridge displacement, road surface profile, and position of the vehicle. The equations of motion for the coupled system are written as: Mb d¨b Cb + Cbb Cbv d˙b + Mv d¨v Cvb Cv d˙v Fbr K b + K bb K bv db + (1) = K vb Kv dv Fvr + FvG where {db }, [Mb ], [Cb ], and [K b ] are the displacement vector, mass matrix, damping matrix, and stiffness matrix of the bridge, respectively; {dv }, [Mv ], [Cv ], and [K v ] are the displacement vector, mass matrix, damping matrix, and stiffness matrix of the vehicle, respectively; and {FvG } = the gravity force vector of the vehicle. It is assumed that the wheels always maintain a point contact with the bridge deck without separation. The equations of motion for the vehicle and bridge are coupled through the interaction force and the terms Cbb , Cbv , Cvb , K bb , K bv , K vb , Fbr , and Fvr stem from the contact (interaction) force. To simplify the modeling procedure, the bridge modal superposition technique is used based on the obtained bridge mode shapes {Φi } and the corresponding natural circular frequencies ωi . The bridge dynamic response {db } can be expressed as: {db } = {Φ1 }
{Φ2 } . . . {Φn }
ξ1
ξ2 . . . ξn
T
= [Φb ] {ξb } (2)
where n is the total number of modes for the bridge under consideration, and {Φi } and ξi are the ith mode shape and the generalized coordinates, respectively. If each mode shape is normalized with the mass matrix, i.e. {Φi }T [Mb ]{Φi } = 1 and {Φi }T [K b ]{Φi } = ωi2 , and if the damping matrix [Cb ] is written to be 2ωi ηi [Mb ], where ωi is the natural circular frequency of the bridge and ηi is the percentage of the critical damping for the ith mode, then Eq. (1) can be derived as: ξ¨b I 2ωi ηi I + ΦbT (Cbb )Φb Cbv ξ˙b + Mv d¨v d˙v ΦbT Cvb Φb Cv 2 ΦbT Fbr ωi I + ΦbT (K bb )Φb K bv ξb + = . (3) dv Fvr + FvG ΦbT K vb Φb Kv The modal superposition makes it possible to separate the bridge modal analysis from the vehicle–bridge coupled model. Consequently, the number of equations in Eq. (3) and the complexity of the whole procedure are greatly reduced. Eq. (3) is solved by using the Fourth Order Runge–Kutta method in the time domain. The road surface profile is an important factor that affects the dynamic responses of both the bridge and the vehicles. In this study, the road surface profile was simulated in the space domain, which serves as an input to the vehicle–bridge model. Here, the road profile contains both the road roughness of the bridge deck and the approach roadway. More details of the numerical model are given in [5]. 4. Verification with a FRP deck bridge A Kansas DOT supported project compared the lateral load distribution (lateral stiffness) characteristics of a 14-girder bridge with a corrugated metal decking to the same bridge after the original deck was replaced with a FRP deck. These comparisons were developed by field testing the bridge prior to and after the deck rehabilitation. The measured data is a good resource to verify the procedure developed in the present study. For the convenience of the readers, some information from the Kansas DOT report [10] is reiterated below. The bridge used in this experiment was the Crawford County Bridge 031, which is located near Pittsburgh, Kansas on K126. The design parameters for this bridge are as follows: the beams are W21 × 68 with a spacing of 0.686 m (27 in.). The
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Fig. 3. Lateral load position of Kansas field test.
bridge deck, before replacement, was 13.720 m (45 ft) long and 9.146 m (30 ft) wide. The final replacement deck is 9.756 m (32 ft) wide. The specially designed connections (clips) for the FRP deck and the existing steel girders were installed every 2.439 m (8 ft) and on every third girder. For the other girders without connections, the deck sits on the girder without tight connections. Therefore, the FRP deck is partially composite with the steel girders. Ten truck passes were performed on various lines on the surface of the bridge deck for field tests. The first of these lines was positioned so that the right front tire of the loaded truck would travel directly across the centerline of the first interior girder. Adjacent lines were then spaced laterally across the bridge deck using the first line as a datum and allowing enough room for ten load passes. The layout for the load passes can be seen in Fig. 3. All fourteen steel girders were instrumented with strain gages at the mid-span along the longitudinal centerline of each girder, and diagnostic tests were run on the strain gages before loading the truck. The “measured” stresses σ were calculated by taking the strain reading minus the average of the first few strain readings (with zero loading) and multiplying this number by a strain gage factor and by the modulus of elasticity E as σ = (ε − average(εinitial )) × (gage factor) × E.
(4)
The lateral distribution values (DF) are equal to the actual stress on a particular girder divided by the sum of the stresses on all the girders as: , 14 X Σ σgirder− j . (5) DFi = σgirder−i j=1
The writers used the same bridge after the deck rehabilitation (i.e. with FRP deck) to construct a 3-D linear elastic finite element model. The analysis was performed using ANSYS 9.0 finite element program available at Louisiana State University. The bridge was modeled with Solid45 eightnode solid elements with three degrees of freedom at each node for the FRP deck and Shell63 four-node shell elements
with six degrees of freedom at each node for the steel girders. To simulate the composite action between the two components, two models were constructed. The first one was fully composite, where the three translation degrees of freedom of the solid element nodes for the deck are fully connected with those of the shell elements for the girder flange. The second one was partially composite, where the connections were made every 2.349 m (8 ft) — the same as the field condition, but connections were placed on every girder for numerical stability, which is slightly different from the field bridge connections stated earlier. The results obtained from the finite element analysis were compared with those from the field tests, and in general, a good correlation was achieved, as shown in Fig. 4. In both the FEA and field values, no multiple presence factors specified in the AASHTO code [9] were included, and they are thus denoted as DF instead of LDF, as used later. The comparison indicated a small change in the load distribution between the fully composite condition and the partially composite condition of the FRP deck system, as shown in Fig. 4(a)–(j). This is because the stiffness of the FRP deck is relatively small compared to a traditional concrete deck, and it is also due to the small beam spacing of this bridge. In general, the composite or partial composite design of FRP deck bridges does not make as a significant difference on load distribution as a concrete deck would do. These figures also indicate that the load distribution values can be estimated based on the FEA, using a simplified model of the FRP deck (an equivalent solid plate). As expected, the loading position significantly affects the load distribution among the girders. A truck moving along the side of the road results in higher distribution values, while the truck traveling along the middle of the road induces lower distribution values. A summary of the maximum values is provided in Table 1. As mentioned earlier, since this bridge in the field has only minimum connections between the deck and the girders, supposedly the DF of the test results should be larger than that from the finite element analyses for both fully composite and partially composite models. While the maximum DF for
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Fig. 4. Lateral distribution results from FEA versus from field test.
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5. Numerical analysis of typical girder bridge models
Table 1 Maximum values from field tests and FEA
Maximum DF for interior girder Maximum DF for exterior girder Impact factor
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Field tests
FEA of fully composite
FEA of partially composite
0.226 0.141 1.08
0.184 0.167 1.02
0.180 0.170 1.30
Note: DF = Distribution factor.
interior girders of field tests was larger than that of the FEA as expected, it is opposite for the exterior girder. This may be due to the existence of concrete edge beams (for installing the railing) along the road in the actual field bridge. These beams were not considered in the finite element model. As also shown in Table 1, the dynamic impact factor from test is 1.08, while the predicted ones for a vehicle velocity of 22.0 m/s are 1.02 and 1.30 for a fully composite and partially composite deck, respectively. It is stated in [10] that the simple dynamic tests are not used for decisions or conclusions since the tests were conducted out of curiosity. While the measured results of this example bridge have provided an opportunity to verify the developed finite element modeling techniques, this bridge, with a narrow girder spacing, is not a typical bridge that can be used to draw more general conclusions. In order to study the load distribution and the dynamic behavior of multi-girder bridges with FRP decks, two more general bridge models were developed and described below.
One steel girder bridge model and one prestressed concrete girder bridge model, as shown in Figs. 5 and 6, were developed [11,12]. The span length for both bridges is 18.288 m (60 ft). The two bridges were designed for the HS20-44 loading, and they both consist of five identical girders which are simply supported. In order to compare the performance, two types of bridge decks were used: a honeycomb-type sinusoidal core FRP sandwich panel (see Fig. 1) and a concrete slab. The FRP decks were designed for fully composite or partially composite (the connection was made every 2.349 m on every girder). The thickness of FRP decks used in these models is 203 mm (8 in.), and the deck was simplified as an equivalent orthotropic solid panel, as discussed earlier. The thickness of the concrete decks is 191 mm (7.5 in.) for both the steel and prestressed concrete girder bridges. 5.1. Load lateral distribution To find the most unfavorable loading position and the meaningful load distribution factors, eight loading cases were investigated (Figs. 5 and 6). There are three different cases of one-truck loading: (1) located on the side of the road, (2) the left tire of the truck directly across the centerline of the first interior girder (for steel girder bridge) or across the centerline of the exterior girder (for concrete girder bridge), and (3) the middle of the road. The three two-truck loadings correspond
Fig. 5. Cross-section of steel multi-girder bridge and loading cases.
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Fig. 6. Cross-section of concrete multi-girder bridge and loading cases.
to the three one-truck loading cases and the two three-truck loadings are located on the side of the road and the middle of the road, respectively. In the longitudinal direction, the mid-wheels are put directly above the mid-span of the girders. The loadings of front tires, middle tires, and rear tires are 35 kN (8 kips), 145 kN (32 kips), and 145 kN (32 kips), respectively. For each girder, the LDF is calculated using the maximum static stress in that girder obtained from the static loading at the same cross-section of the bridge. When all girders have the same section modulus, load distribution factors are calculated using a method employed by Ghosn et al. [13,14]. Accordingly, the LDF for the ith girder, LDFi , can be derived as follows: LDFi =
nσi k P
(6)
σj
j=1
where σ j = the bottom-flange stress at the jth girder; k = the number of girders; and n = the number of side-by-side trucks. For cases of one-truck, two-truck, and three-truck loadings on a bridge, the LDFs calculated from Eq. (6) are multiplied by the multiple presence factors of 1.2, 1.0, and 0.85, respectively [9]. Generally, the distribution factors of girder bridges for a twotruck loading are higher than those for a one-truck or threetruck loading after considering the multiple presence factors. The bottom-flange stress values at the mid-span of the girders obtained from the FEA were used to calculate the LDFs for the steel and concrete bridges, respectively, as shown in Tables 2 and 3. For the convenience of discussions, the LDFs are also plotted in Figs. 7 and 8. By comparing the
two cases of “FRP Deck Fully Composite” and “FRP Deck Partially Composite”, it is observed that when the deck and the girders are partially composite, the LDF values are larger, since in this case a smaller portion of loads are shared by the other girders. In other words, generally speaking, bridges with partially composite conditions cannot distribute loads as uniformly as bridges with fully composite conditions. By comparing the two cases of “FRP Deck Fully Composite” and “Concrete Deck Fully Composite”, it is observed that due to the higher stiffness of the concrete deck, the LDF values of the bridge with concrete deck are smaller than those with the FRP deck. Among the three deck configurations, the “FRP Deck Partially Composite” results in the highest LDF values. It is noted that only the maximum LDF value among the girders matters in the design process, although the LDFs for all girders were calculated and plotted in the figures. Meanwhile, it is observed that the stress values of concrete girder bridges are smaller than those of steel girder bridges, because concrete girder bridges have a larger stiffness compared to steel girder bridges. The LDF values of concrete girder bridges are more uniform for different decks than those of steel girder bridges. This means that in terms of LDFs, concrete girder bridges are not as sensitive to deck stiffness as steel girder bridges are. 5.2. Dynamic response To investigate the effect of the deck system on bridge performance, dynamic analyses of the bridge system with different deck configurations were carried out. The multi-girder
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Y. Zhang, C.S. Cai / Engineering Structures 29 (2007) 1676–1689 Table 2 Stress and LDF of steel girder bridge Case
Girder number
Stress (MPa) FRP deck fully comp.
FRP deck partially comp.
Concrete deck fully comp.
LDF FRP deck fully comp.
FRP deck partially comp.
Concrete deck fully comp.
Case 1
1 2 3 4 5
0.312 3.210 9.393 22.782 31.348
−0.782 1.423 8.427 23.182 32.793
4.761 6.135 9.252 16.697 22.110
0.006 0.057 0.168 0.408 0.561
−0.014 0.026 0.155 0.428 0.605
0.097 0.125 0.188 0.340 0.450
Case 2
1 2 3 4 5
3.198 8.742 19.337 21.953 12.884
0.692 8.210 21.117 23.531 10.843
5.989 9.166 15.324 16.659 11.263
0.058 0.159 0.351 0.398 0.234
0.013 0.153 0.394 0.439 0.202
0.123 0.188 0.315 0.342 0.231
Case 3
1 2 3 4 5
6.877 14.721 22.215 14.718 6.873
3.842 15.933 24.797 15.938 3.847
7.872 12.588 17.098 12.585 7.872
0.126 0.270 0.408 0.270 0.126
0.072 0.297 0.462 0.297 0.072
0.163 0.260 0.354 0.260 0.163
Case 4
1 2 3 4 5
5.636 15.330 31.388 40.249 40.043
1.676 14.508 32.784 41.980 38.441
11.886 17.279 26.190 30.838 30.896
0.085 0.231 0.473 0.607 0.604
0.026 0.224 0.507 0.649 0.594
0.203 0.295 0.447 0.527 0.528
Case 5
1 2 3 4 5
17.880 31.601 37.246 29.775 15.521
13.869 32.815 40.605 30.428 11.168
18.330 26.360 29.755 25.288 16.977
0.271 0.479 0.564 0.451 0.235
0.215 0.509 0.630 0.472 0.173
0.314 0.452 0.510 0.433 0.291
Case 6
1 2 3 4 5
16.571 30.798 37.356 30.795 16.565
12.465 31.646 40.665 31.638 12.462
17.619 25.844 29.794 25.842 17.619
0.251 0.466 0.566 0.466 0.251
0.193 0.491 0.631 0.491 0.193
0.302 0.443 0.511 0.443 0.302
Case 7
1 2 3 4 5
28.208 39.344 44.316 45.388 41.303
23.683 40.132 46.635 45.644 38.050
28.666 34.992 37.651 38.062 36.039
0.362 0.505 0.569 0.583 0.530
0.311 0.527 0.613 0.600 0.500
0.417 0.509 0.547 0.553 0.524
Case 8
1 2 3 4 5
34.645 42.700 43.810 42.697 34.642
30.465 43.209 46.870 43.140 30.432
32.230 36.663 37.542 36.660 32.229
0.445 0.549 0.563 0.549 0.445
0.400 0.568 0.616 0.567 0.400
0.469 0.533 0.546 0.533 0.469
bridges were assumed to be at rest before the vehicle entered the bridge. The vehicle was modeled with a 40 m lead distance to minimize the effect of the vehicle initial conditions on bridge vibrations. Therefore, there is no bridge vibration for the first few seconds in the figures shown below. A parametric sensitivity study was conducted to analyze the effect of factors such as road surface condition and vehicle velocity on the bridge dynamic response. The objective was to compare the dynamic performance of the FRP deck bridges (fully composite or partially composite) with the corresponding concrete deck bridges and find some correlations between the bridge dynamic performance and these parameters. Many investigations have shown that the roughness of a bridge surface is an important factor that affects the dynamic response of bridge structures [15–18]. In this study,
classification of road roughness based on the International Organization for Standardization [19] was used, and the road surface profile was simulated in the space domain. Two road conditions were considered as inputs to the vehicle–bridge coupled model, namely: (1) road surface condition is good; and (2) road surface condition is poor. Based on these two road conditions and the vehicle velocity of v = 10 m/s, 20 m/s, or 40 m/s, the dynamic responses at the mid-span bottom-flange of the center girder were evaluated. While Figs. 9 and 10 demonstrate the effects of vehicle velocity and road roughness on the displacement of bridges with three different deck conditions, Figs. 11 and 12 demonstrate, correspondingly, their effects on the accelerations. As shown in Figs. 9 and 10, deck types have significantly affected the displacement. While the FRP Deck Partially
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Table 3 Stress and LDF of concrete girder bridge Case
Girder number
Stress (MPa) FRP deck fully comp.
FRP deck partially comp.
Concrete deck fully comp.
LDF FRP deck fully comp.
FRP deck partially comp.
Concrete deck fully comp.
Case 1
1 2 3 4 5
−0.037 0.196 0.954 2.921 4.457
−0.084 0.133 0.926 2.750 4.445
0.292 0.420 0.857 2.002 2.940
−0.005 0.028 0.135 0.413 0.630
−0.012 0.020 0.136 0.404 0.653
0.054 0.077 0.158 0.369 0.542
Case 2
1 2 3 4 5
−0.029 0.230 1.045 3.033 4.224
−0.080 0.172 1.036 2.845 4.190
0.299 0.440 0.913 2.067 2.795
−0.004 0.032 0.147 0.428 0.596
−0.012 0.025 0.152 0.418 0.616
0.055 0.081 0.168 0.381 0.515
Case 3
1 2 3 4 5
0.579 2.010 3.202 2.005 0.578
0.493 2.038 3.092 2.034 0.492
0.653 1.490 2.193 1.489 0.653
0.083 0.288 0.459 0.287 0.083
0.073 0.300 0.455 0.300 0.072
0.121 0.276 0.406 0.276 0.121
Case 4
1 2 3 4 5
0.359 1.771 4.118 5.382 5.278
0.221 1.793 3.959 5.157 5.186
0.849 1.659 3.018 3.755 3.724
0.042 0.209 0.487 0.637 0.624
0.027 0.220 0.485 0.632 0.636
0.131 0.255 0.464 0.577 0.573
Case 5
1 2 3 4 5
0.422 1.945 4.230 5.342 4.957
0.281 1.956 4.102 5.129 4.841
0.885 1.759 3.093 3.733 3.531
0.050 0.230 0.501 0.632 0.587
0.035 0.240 0.503 0.629 0.594
0.136 0.271 0.476 0.574 0.543
Case 6
1 2 3 4 5
1.762 4.117 5.147 4.113 1.760
1.678 3.974 4.991 3.968 1.674
1.671 3.022 3.616 3.021 1.670
0.208 0.487 0.609 0.487 0.208
0.206 0.488 0.613 0.487 0.206
0.257 0.465 0.556 0.465 0.257
Case 7
1 2 3 4 5
3.432 5.096 5.681 5.810 5.296
3.227 4.949 5.588 5.565 5.137
2.933 3.929 4.252 4.312 4.060
0.346 0.513 0.572 0.585 0.533
0.336 0.516 0.582 0.580 0.535
0.384 0.514 0.556 0.564 0.531
Case 8
1 2 3 4 5
4.381 5.473 5.654 5.469 4.380
4.129 5.299 5.619 5.292 4.119
3.485 4.137 4.256 4.136 3.484
0.441 0.550 0.569 0.550 0.441
0.430 0.552 0.586 0.552 0.429
0.456 0.541 0.557 0.541 0.456
Composite condition results in the largest displacement, the Concrete Deck Fully Composite condition gives the lowest displacement. This observation agrees with that observed for the static loading case, as discussed earlier, since the deck stiffness is in an ascending order from FRP Deck Partially Composite, to FRP Deck Fully Composite, and to Concrete Deck Fully Composite. The difference between the FRP Deck Partially Composite and FRP Deck Fully Composite are more pronounced in bridges with steel girders (Fig. 9) than those with concrete girders (Fig. 10). For accelerations, as shown in Figs. 11 and 12, the values of bridges with an FRP deck are far larger than those with a concrete deck, even under a low vehicle velocity condition; the acceleration values of bridges with a FRP deck in the partially composite condition are generally close to the values
of those with a FRP deck in the fully composite condition. Under a low vehicle velocity (10 m/s), the FRP deck in the partially composite condition generally results in a higher acceleration than that of the FRP deck in the fully composite condition. Under a high vehicle velocity (40 m/s), the trend is just the opposite. While the explanation of this phenomenon is not straightforward, it can be presumably stated that a higher vehicle velocity excites more participations of higher bridge modes. If the displacement response is expressed as PN {d(t)} = i=1 ζi sin[(2π f i )t + θi ]{φi }, where {φi } = the modal shape i, ζi = the participation factor of modal shape i, f i = the frequency of modal shape i, and θi = the initial phase of modal shape i, then the acceleration is derived as PN 2 2 { ∂ ∂td(t) 2 } = − i=1 ζi (2π f i ) sin[(2π f i )t + θi ]{φi }, i.e., it is
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Fig. 7. LDF comparison of steel multi-girder bridge.
proportional to f i2 . Therefore, though the displacement of a partially composite deck bridge is lager than that of a fully composite deck bridge, the acceleration could be opposite since it is a function of f i2 .
Figs. 9 through 12 also show that the vehicle velocity effect is more pronounced when the road surface condition is poor. Road roughness of a bridge seriously affects the vehicle’s vibrations, thus affecting the vehicle–bridge interaction. It
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Fig. 8. LDF comparison of concrete multi-girder bridge.
can be seen from the figures that the worse the bridge road condition, the larger the bridge dynamic displacement, and the far larger the bridge dynamic acceleration under the truck load. This situation is more obvious in steel girder bridges than in
concrete bridges. A poor road condition not only influences the bridge’s normal operation, it moreover creates a vertical acceleration, which can make the driver uncomfortable and may cause a higher deterioration rate of the bridge. Therefore,
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Fig. 9. Displacement comparison of steel multi-girder bridge. Fig. 10. Displacement comparison of concrete multi-girder bridge.
maintaining the bridge road surface in a good condition is very important in reducing the vehicle dynamic impact effects. However, the dynamic response of bridges does not increase monotonically with the increase in vehicle velocity. There is a peak value corresponding to a specific vehicle velocity, which is considered as being related to a vehicle induced resonant vibration [5].
6. Conclusions The present study has developed a static and dynamic analysis procedure for the vehicle–bridge interaction of FRP deck bridges. After verifying its applicability by comparing the static load distribution with that of a field tested bridge, the present study investigated two typical multi-girder bridges with
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Fig. 11. Acceleration comparison of steel multi-girder bridge.
FRP decks, one with steel and the other with concrete girders, and compared their performance with the corresponding multigirder bridges with concrete decks. Based on the present study, the following conclusions can be drawn: 1. The present study used an equivalent orthotropic solid plate model for the FRP hollow sandwich panel. The load distribution results obtained from the finite element analysis
Fig. 12. Acceleration comparison of concrete multi-girder bridge.
using this simplified model were compared with those from field tests, and a good correlation was achieved. The finite element analyses were very helpful in investigating the performance of FRP deck bridges. 2. For both load distribution and dynamic response, bridge deck types have seriously affected the results. The LDF
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values of FRP deck bridges are larger than those of concrete deck bridges. The dynamic response of FRP deck bridges is also larger than that of the concrete deck bridges. 3. The FRP deck bridges with partially composite conditions have a larger girder distribution and dynamic displacement than those of the FRP deck bridges with fully composite conditions. Therefore, in order to obtain a better performance, it is necessary to strengthen the connection between the FRP deck and girders through structural measures. However, this is a challenging task and usually an expensive requirement for a FRP deck system. If the non-composite condition is preferred for the FRP deck system, then it should be noted that the load distribution and dynamic impact factors developed for the full composite conditions may not be conservative for the girder design of bridges with non-composite decks. 4. Road roughness and vehicle velocity all significantly affect the dynamic performance of both the analyzed FRP deck and the concrete deck bridges. Acceleration seems to be more sensitive to a poor road condition than to a good road condition with the same vehicle speed, especially for steel girder bridges. It is suggested that the bridge road surface condition be kept in good condition for valid applications of the code specified dynamic impact factor in bridge design and rating. 5. Considering the different dynamic performances observed between the FRP and concrete deck bridges, different serviceability control criteria for FRP deck bridges may be developed. The design of FRP deck bridges is usually controlled by deflection requirements, and meeting the same deflection requirement as conventional bridges (say L/800) may be uneconomical in many cases. Loosening this limitation (say, an increase to L/400) has been suggested in the literature. Therefore, reexamining the serviceability control criteria for FRP deck bridges based on dynamic analysis will have practical and economical significance. Further study is underway in this aspect by the writers who are also investigating if the AASHTO dynamic impact factors for conventional deck bridges can be applied to FRP deck bridges. The developed methodology is ready to be applied to more FRP deck bridges for comparing their performance with the AASHTO codes. References [1] Transportation Research Board. TR News, special issue on highway bridges: progress and prospects. 1998; January—February (No. 194).
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