«A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of ...»
CHAPTER 4. DYNAMIC ANALYSIS OF VEHICLE-BRIDGE-WIND DYNAMICSYSTEM
4.1 Introduction In the United States, hurricanes and hurricane-induced strong winds have being greatly threatened the properties and lives of costal cities. In the past twenty years, annual monetary losses due to tropical cyclones and other natural hazards have been increasing at an exponential pace. In addition to the huge lose of properties, live loss is even more stunning. In extreme cases, evacuations are exceptionally necessary to minimize the loss of lives and properties. Smooth and safe transportation is obviously the key factor of a successful evacuation process. Long-span bridges are often the backbones of the transportation lines in coastal areas and are very vulnerable to strong wind loads. Maintaining the highest transportation capacity of these longspan bridges is vital to supporting hurricane evacuations. Hence to reduce the likeness of accidents on the transportation line is extremely important not only for normal operation but also for evacuation purpose when hurricane arrives.
Currently, the setting of driving speed limit and the decision of closing the transportation on bridges and highways when hurricane arrives is mostly based on intuition or subjective experience (Irwin 1999). The driving speed limit could be too high to be safe or too low to be efficient. The closing of the transportation will totally obstruct the evacuation through such transportation line. A rational prediction of the performance of vehicle-bridge system under strong winds is of utmost importance to the maximum evacuation efficiency and safety. Most existent works focus on either wind action on vehicles running on roadway (not on bridges) (Baker 1986, 1999), wind effect on the bridge without considering vehicles (Scanlan and Jones 1990), or vehicle-bridge interaction analysis without considering wind effect (Timoshenko et al.
1974; Yang and Yau 1997; Pan and Li 2002). A comprehensive vehicle-bridge-wind coupled analysis is very rare, if any.
Baker et al. have made extensive studies on the performance of high-sided road vehicles in cross winds (Baker 1986, 1987, 1991a, 1991b, 1999; Coleman and Baker 1990; Baker and Reynolds 1992). Wind effects on vehicles including static wind force and quasi-static turbulent wind force on the vehicles. In his representative works, Baker (Baker 1986) proposed the fundamental equations for wind action on vehicles. The wind speed at which these accident criteria are exceeded (the so-called accident wind speed) was found to be a function of vehicle speed and wind direction (Baker 1986). Through adoption of meteorological information the percentage of the total time for which this wind speed is exceeded can thus be found and some quantification of accident risk was made (Baker 1999). In addition to wind tunnel tests on several vehicle models to identify the wind force on vehicles (Coleman and Baker 1990), some useful statistic information about actual accidents happened in British was also collected and analyzed (Baker and Reynolds, 1992).
Interaction analysis between moving vehicles and continuum structures originated from the middle of 20th century. From an initial moving load simplification (Timishenko et al. 1974), later on a moving-mass model (Blejwas et al. 1979) to full-interaction analysis (Yang and Yau 1997; Pan and Li 2002; Guo and Xu 2001), the interaction analysis of vehicles and continuum structures (e.g. bridges) has been investigated by many people for a long time. In these studies, road roughness was treated as the sole excitation source of the coupled system.
Recently, dynamic response of suspension bridges to high wind and running train was 58 investigated (Xu et al. 2003), while no wind loading on the train was considered since the train was running inside the suspension bridge deck. It was found that the suspension bridge response is dominated by wind force in high wind speed. The bridge motions due to high winds affected the safety of the running train and the comfort of passengers considerably (Xu et al. 2003). The coupled dynamic analysis of vehicle and cable-stayed bridge system under turbulent wind was also conducted recently (Xu and Guo 2003), where only vehicles under low wind speed were explored and the study did not consider many important factors, such as vehicle number, and driving speeds etc.
The present study aims at building a framework for the vehicle-bridge-wind aerodynamic analysis, which will lay a very important foundation for vehicle accident analysis based on dynamic analysis results and facilitate the aerodynamic analysis of bridges considering vehiclebridge-wind interaction. The framework starts with building a general dynamic-mechanical model of vehicle-bridge-wind coupled system, which includes both the structural part and the loading part. Structural model simulates the bridge and a series of vehicles consisting of any number and also various types of typical vehicles, from two axle four-wheel passenger car to five axle tractor-trailer. The loading part models the external loading, like wind effect and road roughness loading on vehicles and the bridge.
After the framework is established, a series of 2-axle four-wheel high-sided vehicles on long-span bridges under strong winds are chosen as a numerical example to demonstrate the methodology. The external dynamic loading on the bridge consists of wind loading and road roughness and the external dynamic load on the vehicle includes simplified quasi-steady wind force and road roughness. With the mechanical model of the vehicle-bridge system, dynamic performance of vehicles as well as bridges is studied under strong winds.
4.2 Equations of Motion for 3-D Vehicle-Bridge-Wind System 4.2.1 Modeling of Vehicle In mechanical engineering areas, vehicles are usually modeled in various configurations depending on what is of more concern. For the study of interactions between vehicles and structures, the vehicle model is somewhat simplified to the extent that all the relevant important information will be included. In the present study, a vehicle is modeled as a combination of several rigid bodies connected by several axle mass blocks, springs and damping devices. The bridge deck and the tire of the vehicle are assumed to be point-contact without separation. The suspension system and the elasticity of tires are modeled with springs. The energy dissipation capacities of the suspension as well as tires are modeled with viscous damping. The mass of the suspension system and the tires are assumed to concentrate on idealized mass blocks on each side of the vehicle and the mass of the springs and damping devices are zero (Fig. 4.1-2).
Fig. 4.1-2 shows a complicated tractor-trailer model; however, this model can also be used to represent other simpler vehicles through defining the number and dimensions for each rigid body, mass block, and spring-damping system. The displacements of the ith rigid body of the qth qi qi vehicle are expressed as: vertical displacement Z vr, lateral displacement Yvr, pitching displacement in x-z plane θ vr, and rolling displacement in y-z plane φvr. The subscript “vr” qi qi
59 represent the left and right mass blocks on the jth axle, respectively; and the “a” represents the axle suspension. The superscript “qj” represents the jth axle of the qth vehicle. Same definitions apply hereafter. The longitudinal, lateral, and vertical directions of the bridge are set as x- y- and z-axis, respectively. Assuming a total of nv vehicles are considered in the analysis.
where Z vaL ( R ) is the vertical displacement of the left (right) mass block for the jth axle of qth qj vehicle; and rLqjR ) is the road surface roughness height at the contacting point between the bridge (
qj where YbL( R ) are the lateral displacements of the bridge at the points of contacting with the left (right) wheels of the jth axle for the qth vehicle and are defined as follows:
4.2.2 Modeling of Bridge The dynamic model of the long-span bridge can be obtained through finite element method using different kinds of finite elements such as beam elements and truss elements. With the obtained mode shapes, the response corresponding to any point along the bridge deck can be evaluated in the time domain.
where the subscripts “st”, “ae” and “b” refer to static, self-excited, and buffeting force component due to wind, respectively.
The static wind force of unit span length can be expressed as:
where ρ is the air density; U is the mean wind speed on the elevation of the bridge; B is the bridge deck width; CL, CD and CM are the lift, drag, and moment static wind force coefficients for the bridge that are usually obtained from section model wind tunnel tests of the bridge deck.
The self-excited force can be expressed as:
where u ( t ) and w(t ) are the horizontal and vertical components of wind turbulent velocity, respectively; and the prime denotes the derivative with respect to the attack angle of wind.
4.2.3 Coupled Equations of Vehicle-Bridge Model in Modal Coordinates By assuming all displacements remain small, virtual works generated by the inertial forces, damping forces, external wind loading, and elastic forces can be obtained for the bridge-vehicle system. The equilibrium condition of bridge under its self-weight without vehicles is chosen as the initial position, which facilitates the direct comparison between the cases of with and without
vehicles. The summation of all the virtual works should be zero:
nr is the total number of rigid bodies for the vehicle, respectively; g is the gravity acceleration;
Fwz, Fwy ( t ) and M wφ are the wind forces in vertical direction (axis z), lateral direction (axis y), qi qi qi
where subscripts “b” and “v” represent for bridge and vehicle, respectively; superscripts of “s” and “v” in the stiffness and damping terms for the bridge refer to the terms of bridge structure itself and those contributed by the vehicles, respectively; subscripts “bv” and “vb” refer to the vehicles-bridge coupled terms; “r”, “w” and “G” represent for roughness, wind, and gravity force, respectively; and γ v and γ b are the displacement vectors of the vehicles and the bridge, respectively;
Detailed formulations of matrices and force vectors of the qth vehicle are referred to 4.6.
4.3 Dynamic Analysis of Vehicle-Bridge System under Strong Wind Strong wind effects on moving vehicles and bridge are the concerns of this study. As introduced in Section 2, the wind effect on the bridge consists of static wind force, aeroelastic self-excited force and turbulent buffeting force. It is usually assumed that the analysis begins with the static equilibrium position of the bridge under self weight and static wind force action.
Only dynamic loads of wind, namely the last two components of Eqs. (4.16-4.18) are considered for the bridge.
4.3.1 Self-excited Forces on the Bridge The wind dynamic load on the bridge consists of self-excited force and buffeting force. The self-excited force is related to the motion of the bridge and thus frequency dependent. In the time domain analysis, the frequency dependent variables are difficult to be incorporated. Under each wind velocity, the vibration frequency ω at any time should be determined to quantify the selfexcited force terms. As an alternative to the rational function approximation (Chen et al. 2000), complex eigenvalue analysis can also predict the vibration frequency iteratively for the dominant motion at any time step under any wind velocity. In the present study, the complex eigenvalue analysis is conducted first to give the vibration frequency corresponding to each time through interactive process. The results are then incorporated into the vehicle-bridge-wind coupled equations to decide the self-excited force terms of the bridge. This approach to deal with aeroleastic terms of the wind force has been adopted by the writers (Chen and Cai 2003a).
4.3.2 Buffeting Force Simulation on the Bridge To calculate the time-history response of the vehicle-bridge system under the wind action on the bridge, the stochastic wind velocity field should be simulated. The fast spectral representation method proposed by Cao et al. is adopted here (Cao et al. 2000). The time history
of wind component u(t), at the jth point along the bridge span can be generated with:
Nf j u j ( t ) = 2 ( ∆ω ) ∑∑ Su (ωmq )G jm (ωmq ) cos (ωmq t +ψ mq ), j = 1, 2,… Ns (4.61) m =1 q =1 where Nf is a sufficiently large number representing the total number of frequency intervals; Ns is the total number of points along the bridge span to simulate; Su is the spectral density of turbulence in along-wind direction; ψ mq is a random variable uniformly distributed between 0 and 2π; ∆ω = ωup / N f is the frequency increment; ωup is the upper cutoff frequency with the condition that the value of Su (ω ) is less than a preset small number ε when ω ωup ; and
68 where ∆ is the distance between two consecutive simulated points. Similarly, the time history of vertical turbulence component w(t) can be obtained.
In the present study, wind velocity time histories are simulated for various points along the bridge with the four times the element length interval as those used in the finite element dynamic analysis on the bridge (Lin et al. 1998). With the same assumption as in the spectral representation method (Cao et al. 2000), wind velocity time history is assumed the same within each interval. With the wind velocity time history, the buffeting force time history corresponding to each point along the bridge span can be obtained with Eqs. (4.25-27). With the mode shapes obtained from finite element analysis, the wind buffeting forces on the whole bridge can be obtained through integrating all the force time histories along the bridge span.