Local Dynamic Response in Deck Slabs of Concrete Box Girder Bridges

Jianrong Yang1∗ and Jianzhong Li1,2

1Faculty of Civil Engineering and Architecture, Kunming University of Science and Technology, Kunming 650224, China 2Department of Bridge Engineering, Tongji University, Shanghai 200092, China

Abstract: With the effect of vehicle-bridge coupling considered, the local dynam-ic response in deck slabs of concrete box girder bridges are analyzed. The mathe-matical model assumes a finite element representation of the slabs with shell ele-ments. And the vehicle simulation uses a 3 dimensional linear vehicle model with 7 independent degrees of freedom. A well-known power spectral density of road pavement profiles defines the deck surface roughness for Good and Poor roads re-spectively. Response data are produced on concrete highway bridge decks made of straight box section girder. In this way, a parametric study is conducted to analyze the effects of factors such as road surface roughness, vehicle speed, and bridge damping on the bridge dynamic amplification factors (DAFs). Results are pre-sented to verify the extension of the local dynamical effects on concrete box girder decks with different working condition. From these results some general conclu-sions have been drawn.

Keywords: vehicle-bridge coupling, deck slab, local dynamic effect, concrete box girder

1 Introduction

In recent years, heavy vehicles have become larger and have increased in number. At the same time, new materials and improved design methods have resulted in lighter and more flexible bridges. Therefore highway bridges are increasingly sus-ceptible to vibration. Vibration of large amplitude may introduce into the bridges structural damage and increase their retrofitting expenses. Damage in some local elements of bridge, such as deck slabs, hangers, and expansion joints, could be dangerous to highway transportation. Thus the research on local vibration of highway bridges becomes a necessity.

∗ Corresponding author, e-mail: carpenter_y@sina.com

© Springer Science+Business Media B.V. 2009Y. Yuan, J.Z. Cui and H. Mang (eds.), Computational Structural Engineering, 771–779.

It is found that the behaviour of concrete box girder is essentially complex and that there is no simple indicator that may be used to identify impactive vehicles. Generally, bridge engineer use the dynamic amplification factor (DAF) to a design or evaluation of bridge capacity. The DAFs defined in codes have classically been derived from the measurement or simulation of global traffic action effects in the main structural elements of bridges. On the other hand, local dynamic effects in deck slabs have not yet been studied in detail. The local impact on some elements may be much larger than the global ones for certain situations. With only global DAFs, one can hardly describe the local impact effects on different structural ele-ments. Hence a better understanding of the dynamic behaviour of deck slabs under vehicle loading will lead to the definition of more appropriate dynamic amplifica-tion factors and avoid the use of values that are too conservative (Broquet et al., 2004).

For concrete box girder bridges, deck slab is an important structural element. The dynamic behaviour of deck slabs under traffic action may be vital for the safety or serviceability of the bridges. Research works in this field are mainly con-centrated on the global vibration of the structures. Few references have been found dealing with deck slabs in particular. A detailed review of the literature and an ex-tensive list of references are omitted here as they can be found in the references (Broquet et al., 2004; Bazi et al., 2005; Green and Cebon, 1994).

This paper develops a generalized procedure for the resolution of the dynamic interaction problem between a bridge and a dynamic system of vehicles running at a prescribed speed. A parametric study, based on the simulation of bridge-vehicle interaction, is carried out to investigate characteristic properties of the dynamic behaviour of the bridge deck slabs of concrete box girder bridges and to deduce the distribution of DAFs throughout the deck slab.

2 Vehicle and Bridge Models

2.1 Bridge Model

To obtain general dynamic characteristics of box girder bridge, a prestressed con-crete box girder bridge presented by Dongzhou Huang et al. (1995) is chosen in this study and shown in Figure 1. The bridge is simply supported and has a span length of 45.72 m. The bridge is modelled with shell element. The first frequency of the bridge model is 3.278 Hz. Local modes of vibration in the deck slab have frequencies greater than 27 Hz.

772 Local Dynamic Response in Deck Slabs of Concrete Road Bridges

3.0

6 .0 2.0292.029

0.4 0.4

0.3

0.3

0.4570.4579.144/2

0.915

1.83

0.915TR1

9.144/2

Z

Y

2.132TR2

Figure 1. Cross section of the bridge. Figure 2. Vehicle model.

2.2 Vehicle Model

As shown in Figure 2, the mathematical vehicle model is composed of a vehicle body and four wheel bodies. The tires and suspension systems are idealized as lin-ear elastic spring elements and dashpots. The vehicle body has three degrees of freedom, including z displacement, rolling, and pitching; each wheel has only one degree of freedom, namely z displacement. Therefore, each vehicle has a total of 7 degrees of freedom.

Here, the mass of vehicle body ms=2.55E4 kg; the pitching (Jθ) and rolling (Jα) moment of inertia of vehicle body are 5.53E4 kg ． m2/rad and 56.89E3 kg．m2/rad respectively; the damping coefficients of vehicle suspension (cs1, cs2, cs3, cs4) are 2.00E4 N．s/m; the spring stiffness coefficient of vehicle suspension for the front (ks1, ks2) and rear (ks3, ks4) axles are 4.00E6 and 8.00E6 N/m respec-tively; the mass of each front axle tires are 445 kg; the mass of each rear axle tires are 890 kg; the damping coefficient for each tires (ct1, ct2, ct3, ct4) is 2.00E4 N．s/m; the spring stiffness coefficient of tires for the front (kt1, kt2) and rear (kt3, kt4) axles are 2.25E6 and 8.00E6 respectively; distance α1, α2, b1, and b2 are 3.479, 1.021, 0.915, 0.915 m respectively. The 7 frequencies of the vehicle are calculated as 2.03, 3.22, 4.27, 18.91, 19.77, 21.41 and 21.63 Hz respectively with Stodola (Bhatt, 2002) method.

3 Vehicle-Bridge Dynamic System

3.1 Vehicle-Bridge Model

The modal equations of a bridge can be expressed as

[ ]{ } [ ]{ } [ ]{ } { }B B B BM A C A K A F+ + =&& & (1)

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where {A}=modal displacement vector; [Mb]=modal mass matrix; [Cb]=modal damping matrix; and [Kb]=modal stiffness matrix of the bridge; {Fb}=modal wheel-bridge contact force vector on the bridge. The equations of motion for the vehicle model presented in Figure 2 are Lagrange’s formulation as

[ ]{ } [ ]{ } [ ]{ } { }V V V VM Z C Z K Z F+ + =&& & (2)

where {FV} is the interaction force vector applied on the vehicle; [MV], [CV] and [KV] are, respectively, the mass, damping and stiffness matrices of the vehicle. And {Z} is the vertical displacement vector of the vehicle degrees of freedom.

3.2 Interaction of Vehicle and Bridge

The interaction force between the bridge and the vehicle on ith wheel is given by

( ) ( )ti ti ti bi i ti ti bi bi iF k Z U r c Z U U V r V′ ′= − − − − − − −& & (3)

kti and cti are, respectively, the tire stiffness and tire damping of the ith wheel; Zti is the vertical displacement of the ith wheel; Ubi and ri are, respectively, the bridge vertical displacement and the road surface roughness under the ith wheel; and V is the speed of the vehicle. Based on the jth modal displacement under the ith wheel i

jφ , Ubi can be expressed with modal linear superposition technique as

1

Ni

bi j jj

U Aφ=

=∑ (4)

where N=dimension of the modal space. The force acted on bridge deck by ith wheel is expressed as Fbi=FGi-Fti, in which FGi is the vehicle gravity under the ith wheel. While the nth to mth wheel of the vehicle is located on the deck, the Lth modal force in Equation (1) can be derived as

mi

BL L bii n

F Fφ=

=∑ (5)

Substituting Equation (5) into Equation (1) for all modes, then it becomes

774 Local Dynamic Response in Deck Slabs of Concrete Road Bridges

1

2

[ ] [ ][ ] [0] { } { }[ ] [ ][0] [ ] { } { }

[ ] [ ] { }{ }

[ ] [ ] { }

V BtV V VT

Bt B BVB

V Bt V

Bt B BV

C CM Z ZC C CM A A

K K ZF

K K K A

⎡ ⎤⎡ ⎤ ⎧ ⎫ ⎧ ⎫+⎨ ⎬ ⎨ ⎬⎢ ⎥⎢ ⎥ +⎩ ⎭ ⎩ ⎭⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎧ ⎫+ =⎨ ⎬⎢ ⎥+ ⎩ ⎭⎣ ⎦

&& &

&& & (6)

where the additional terms CBt, CBV, KBt1, KBt2, KBV are due to the contact (interac-tion) force. They are all functions of bridge properties, vehicle properties and the positions of vehicle-bridge contact points. This indicates that the additional terms in Equation (6) are time-dependent terms and will change as the vehicle moves across the bridge.

3.3 Road Surface Profile

The typical road surface may be described by a periodically modulated random process. One method of characterizing a random function is by use of the power spectral density (PSD) function. C.J. Dodds et al. (1973) have developed a typical PSD function that can be approximated by an exponential function as

1

2

0 0 0

0 0 0

( )( / ) ,( )

( )( / ) ,

a

a

S f f f f fS f

S f f f f f

−

−

⎧ ≤⎪= ⎨≥⎪⎩

(7)

where S(f) =PSD function for the road surface elevation; f= wave number; f0= the discontinuity frequency; the values of α1, α2 are taken as 2 and 1.4; S(f0) =roughness coefficient, and its value is chosen depending on the road condition.

4 Effect of Parameters

Most frequently, the DAF is defined as the ratio of maximum dynamic response to maximum static response. C. Broquet et al. (2004) suggested a more appropriate definition of the dynamic amplification factor for deck slabs. The DAF is the peak dynamic effect for a given vehicle trajectory RDmax divided by the peak static ef-fect from the envelope for all vehicle trajectories REmax: DAF = RDmax / REmax.

The action effects in the deck slab due to static traffic loading are determined by static analysis. Effects including vertical deflection (Uz), the longitudinal mo-ment (Mx) and the transverse moment (My) are captured at eleven locations within the bridge deck for five sections as illustrated in Figure 3. These locations are

Jianrong Yang et al. 775

numbered with its section ID plus a number. For instance, C7 means the seventh location from section C-C.

LL/6 L/6 L/6 L/6 L/6 L/6

CBA D ECBA D E

ZX

1 32 64 5 9 11107 8

ZY

Figure 3. Definition of the key sections and data points.

To get the initial displacements and velocities of the vehicle DOFs while the vehicle enter the bridge, the vehicle run starts at a distance of 20 m away from the left end of the bridge and continues moving until the entire vehicle leave the right end of the bridge. The same class of road surface is assumed for both the approach roadway and bridge decks.

4.1 Influence of Road Roughness

The influence of the road profile on the DAF is shown for the box girder bridge in Figure 4. The truck passes over the bridge with a speed range of 10~40 m/s. Still is the bridge damped moderately ξ=0.03. In order to analyze the effect of road roughness three road conditions are considered as inputs to the vehicle-bridge coupled model: (1) road surface condition is good; and (2) road surface condition is poor. They are labelled hereafter as good, and poor, respectively.

DAF[Uz-] DAF[Mx+] DAF[My+]

Goo

d Po

or

Figure 4. Effects of road roughness on DAFs in deck slab.

776 Local Dynamic Response in Deck Slabs of Concrete Road Bridges

As shown in Figure 4, road roughness of a bridge seriously affects the vehicle vibration, thus affecting the vehicle-bridge interaction. An increase in road surface roughness causes an increase in DAFs. Poor road surface condition not only influ-ences the bridge’s normal operation but also creates vertical acceleration that can make the driver uncomfortable. Therefore maintenance of a bridge road surface in good condition is very important to reduce the impact effect (Zhang, 2006).

4.2 Influence of Vehicle Speed

Vehicle speed of 10, 20, 30, and 40 m/s are considered in the simulation. The re-sults demonstrate that there is no critical speed or clear relationship between the DAF and Speed. However, vehicle speed effect is more pronounced when the road surface condition is poor than when it is good.

4.3 Influence of Bridge Damping

Field experiments have demonstrated a large variation of damping for conven-tional highway bridges. Damping ratio values of 1%~5% are commonly recom-mended for conventional bridges in the literature. In addition to the moderately damping of 3%, simulations are carried out with modal damping of 1% and 5% respectively. The influence of vehicle damping is shown in Table 1 for road sur-face condition: good. The truck passes over the bridge with a speed range of 10~40 m/s. The DAF values come from envelope for all trajectories.

As shown in the table, damping does not affect the location at which the maxi-mum DAFs appear, but affects the values. Higher damping can reduce the DAFs on the slab effectively and helps the bridge damp out the vibration quicker.

5 Conclusions

A general procedure for determining the local dynamic response in deck slabs of concrete box girder bridges under moving vehicles has been presented. Based on the results obtained from the parametric study, the following conclusions can be drawn with respect to dynamic amplification factors in road bridge deck slabs:

1. Three amplitudes of road surface roughness are considered in the simulations representing perfect, good and poor conditions. The results indicate that an in-crease in road surface roughness leads to higher DAFs.

Jianrong Yang et al. 777

2. Four vehicle speeds varying between 10 and 40 m/s are considered in the study. It is found that there is no critical speed or clear relationship between the DAFs and speed. The dynamic amplification factors specified in the design codes can be exceeded by a large margin in a situation with an unfavourable combination of some of the parameters.

3. Simulations are carried out with a bridge model damped at three different lev-els. The results show that bridge damping does not affect the location at which the maximum DAFs appear. However, damping can reduce the impact effect from the vehicles effectively and helps the bridge damp out the vibration quicker.

The conclusions above are drawn from purely numerical simulation results, and should be verified by field testing. The information presented in this research may be used to deduce dynamic amplification factors for bridge deck slabs, and to in-crease the accuracy of safety evaluations of concrete deck slabs.

Table 1. Effect of bridge damping on DAFs in deck slab.

DAF Location ξ=0.01 ξ=0.03 ξ=0.05 Ratio (1) (2) (3) (2)/(1) (3)/(1) DAF[Uz

-] D11 1.24 1.21 1.19 98% 96% DAF[Mx

+] B6 1.08 1.08 1.08 100% 100% DAF[Mx

-] C11 1.14 1.13 1.12 99% 98% DAF[My

+] B6 1.15 1.16 1.16 101% 101% DAF[My

-] C9 1.17 1.17 1.17 100% 100%

Acknowledgements

This paper is supported by Yunnan Natural Science Foundation (2005E0023M), and the Natural Science Foundation of Kunming University of Science and Tech-nology (2008-044).

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778 Local Dynamic Response in Deck Slabs of Concrete Road Bridges

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