Concluding Remarks - Vehicle-Bridge Interaction Dynamics

A brief review of previous researches on the dynamic interaction of the bridge and moving vehicles was presented in this chapter. Vehicle models of increasing complexities, including the moving load, moving mass, sprung mass models, and more sophisticated ones, have been

discussed. The factors that need to be considered in analyzing the response of the VBI systems include the dynamic properties and driv-ing frequencies of the movdriv-ing vehicles, and the dynamic properties and surface roughness of the bridge. Even though vehicle models of higher complexities, e.g., those consisting of dozens of DOFs, can be employed in studying the VBI problems nowadays, the use of simpli-ﬁed vehicle and bridge models is helpful, since it allows us to identify the key parameters dominating the dynamics of the VBI systems.

The impact factor adopted herein is computed based on quantities related to the same cross section of the beam that is of interest. It can be conveniently applied to cases involving a series of moving loads. The impact formulas provided by most current design codes are not consistent in physical units and lack a solid theoretical basis, of which the application should not be extended to bridges traveled by vehicles at high speeds. A more rational approach is to relate the impact factor, which is nondimensional, to the speed parameter, which is also nondimensional, deﬁned as the ratio of the driving frequency of the moving vehicles to the vibration frequency of the bridge.

The VBI problem is a complicated one in that the contact points of the vehicles with the bridge move from time to time. Various methods exist for solving this problem. However, the most effective one appears to be the one based on condensation of the noncontact DOFs of the vehicle to the beam element in contact. The VBI el-ement so derived can be applied to solving a great variety of VBI problems, by which the dynamic response of the moving vehicles, in addition to that for the bridge, can be obtained. Other factors that require further studies for high-speed railway bridges include the braking and acceleration of railroad cars, the torsional vibration of bridges caused by vehicles not moving along the centerline of the bridge girders, the crossing of two vehicles moving in opposite direc-tions, the mass ratio of the vehicles to the bridge, the stability of rails, the risk of derailment of railroad cars under earthquake mo-tions, and the stiffness effects of the ballast, elastic bearings, and supporting columns, among others.

Part I

Moving Load Problems

Chapter 2

Impact Response of Simply-Supported Beams

The most fundamental problem that should be considered in the study of vehicle-induced vibrations on bridges is the dynamic re-sponse of a simply-supported beam subjected to a single moving load. This problem is important in that the solution can be given in closed form. In this chapter, impact formulas will be derived for the deﬂection, bending moment and shear force of a simple beam under a single moving load. By the principle of superposition, the solution obtained for a single moving load will be expanded to deal with a se-ries of identical, equi-distant moving loads, by which the key parame-ters dominating the dynamic response of the beam can be identiﬁed.

Furthermore, based on the conditions of resonance or cancellation for the waves generated by a series of moving loads, optimal design criteria for suppressing the resonant response of beam structures will be presented. In designing a high-speed railway bridge, such criteria are useful for the selection of the span length and cross section of a girder bridge, if the car length, axle distance and operating speed of the train are already decided.

2.1. Introduction

Two effects are associated with the motion of a vehicle over a bridge, i.e., the gravitational effect and the inertial effect, both related to the mass of the vehicle. For the cases where the mass of the vehicle is small compared with that of the bridge, the vehicle can be rep-resented as a concentrated load, with the inertial effect neglected.

This is the so-called moving load model, the simplest case that can be conceived of a moving vehicle. One advantage of using such a model is that for some special structures, e.g., for simply-supported beams, closed-form solutions can be obtained, from which the key parameters dominating the dynamic response of the supporting structure can be identiﬁed. It is from this consideration that the moving load problem for simple beams deserves a special and in-depth treatment.

The dynamic response of bridges subjected to the passage of mov-ing vehicles continues to be a subject of great interest to struc-tural engineers. In early studies, a bridge has been modeled as a beam-like structure and a vehicle as a moving load or moving mass (Timoshenko, 1922; Jeffcot, 1929; Lowan, 1935; Biggs, 1964; Frýba, 1972). Such a model was adopted in later studies including those of Warburton (1976), Staniˇsiˇc (1985), Sadiku and Leipholz (1987), and Akin and Mofid (1989). In the meanwhile, more delicate vehicle and bridge models that consider the effects of multi-axle loadings, multi-lane loadings, vehicle suspension, surface roughness, etc. have been developed for the analysis of bridge response (Chu et al., 1986;

Inbanathan and Wieland, 1987; Galdos et al., 1993; Huang et al., 1993; Humar and Kashif, 1993; Chang and Lee, 1994; Yang and Lin, 1995; Yang et al., 1995). In later chapters, it will be demonstrated that the response of a vehicle–bridge interaction (VBI) system ob-tained by using more sophisticated models for the moving vehicles remains in essence dominated by the key parameters identiﬁed from the analysis based on the moving load model.

A review of the research works cited above indicates that most of them consider only the case of a single or very small number of moving vehicular loads. In comparison, rather few research has been conducted on the dynamic response of bridges under the action of a series of moving loads, to represent the continuous action of the wheels of a moving train on the bridge. Bolotin (1964) studied a beam subjected to an infinite sequence of equal loads with uniform intervals d and constant speed v. In his study, the period d/v of the moving loads has been identified as a key parameter. For the same problem, Frýba (1972) concluded that the response of the forced

steady-state vibration will attain its maximum when the time inter-vals between two successive moving loads are equal to some periods of the beam in free vibration or to an integer multiple thereof. Kuri-hara and Shimogo (1978a,b) investigated the vibration and stability problems of a simple beam subjected to a series of discrete loads with random intervals.

Several features have to be considered in the design of bridges for high-speed railways. First, the moving loads acting on the bridge are not random in nature, as encountered in highway bridges, but are of regular nonuniform intervals in general. Second, compared with the car length of a train, which ranges normally from 18 to 26 m, the span length of the elevated bridges constructed as part of the railway lines in most metropolitan areas is usually not long, which may vary from 10 to 40 m. Finally, because of the rather high operating speed of the train, e.g., with a maximum speed ranging from 250 to 350 km/hr, and because of the repetitive nature of wheel loads, it is likely that the condition of resonance be excited on high-speed railway bridges.

In this chapter, only bridges that can be modeled as a simple beam will be considered, which is the most common type of bridges used in railways due to its relative ease in construction and other considerations. The problem of a simple beam subjected to a sin-gle moving load will first be investigated in Sec. 2.2. Based on the analytical solution given in Sec. 2.2, impact factor formulas for the midpoint deflection and bending moment of the simple beam will be presented in Secs. 2.3 and 2.4, respectively, followed by the impact formulas for the end shear force in Sec. 2.5. The solution presented in Sec. 2.2 will be expanded to deal with the case of a simple beam under the passage of a train in Sec. 2.6, by modeling a train as the composition of two subsystems of wheel loads of constant intervals, with one subsystem consisting of all the front wheel assemblies and the other the rear assemblies. Of interest herein is the identification of the conditions for the phenomena of resonance and cancellation to occur on the beam, along with the optimal design criteria for the VBI system. Some illustrative examples will be presented in Sec. 2.7, followed by the concluding remarks in Sec. 2.8. The materials pre-sented in Secs. 2.2–2.5 were rewritten from the paper by Lin et al.

(2000) and those in Secs. 2.6 and 2.7 were modiﬁed from the paper by Yang et al. (1997b).

2.2. Simple Beam Subjected to a Single Moving Load A simply-supported beam traversed by a vehicle that is modeled as a concentrated load p of speed v is shown in Fig. 2.1. The following assumptions will be adopted in this study: (1) The beam is homoge-neous and of constant cross sections, for which the Bernoulli–Euler hypothesis of plane cross sections remain plane after deformation ap-plies; (2) only a single moving vehicle is allowed to travel on the beam at a time; (3) only the gravitational eﬀect of the vehicle is consid-ered, while the inertia eﬀect of the vehicle is neglected, assumed to be small compared with that of the bridge; (4) the vehicle moves at a constant speed v; (5) the damping of the beam is of the Rayleigh type; (6) the beam is initially at rest before the vehicle moves in;

and (7) no consideration is made of the road surface roughness of the bridge.

As shown in Fig. 2.1, a simple beam is subjected to a load of magnitude p moving at speed v. Here, we shall use u(x, t) to denote the deflection of the beam along the y axis at position x and time t, L the length of the beam, m the mass per unit length, ce the external damping coefficient, ci the internal damping coefficient, E the modulus of elasticity, and I the moment of inertia of the beam.

Based on the aforementioned assumptions, the equation of motion of

Fig. 2.1. A simply-supported beam subjected to a moving load.

the beam can be written as

m¨u + ce˙u + ciI ˙u+ EIu= pδ(x− vt) , 0 ≤ vt ≤ L , (2.1) where primes () and dots (˙) denote diﬀerentiation with respect to coordinate x and time t, respectively, and δ is the Dirac delta function. For the beam with simple supports, the boundary conditions are

and the initial conditions are

u(x, 0) = 0

˙u(x, 0) = 0 (2.3)

as the beam is assumed to be at rest prior to the arrival of the moving vehicle.

Let φn denote the nth vibration mode of the beam that satisﬁes the boundary conditions. The deﬂection of the beam u(x, t) due to only the nth mode of vibration is

u(x, t) = φ_n(x)q_n(t) , (2.4) where q_n(t) is the generalized coordinate corresponding to the nth mode. Substituting Eq. (2.4) into Eq. (2.1), multiplying both sides of the equation by φn, and integrating with respect to x over the length L of the beam, one obtains

m¨qn(t)

where it is realized that

δ(x− a)φn(x)dx = φ_n(a) . (2.6) Let us denote the vibration frequency ωn of the nth mode of the beam as coeﬃ-cient ξ_n of the nth mode of vibration as

ξ_n= 1

Consequently, Eq. (2.5) reduces to

q_n+ 2ξ_nω_nq˙_n+ ω_n²q_n= pφ_n(vt)

0 m[φ_n(x)]²dx. (2.9) This is exactly the equation of motion for the nth mode of vibration, in terms of the generalized coordinate qn, which is valid only when the acting position vt of the moving load is located within the range of the beam, i.e., 0≤ vt ≤ L. Once the moving load leaves the beam, only free oscillation remains.

For a simply-supported beam, the nth modal shape of vibration is φ_n(x) = sinnπx

L (2.10)

and the frequency of vibration ωn obtained from Eq. (2.7) is ωn= n²π²

L²

m . (2.11)

Substituting the preceding expression into Eq. (2.9) yields the equa-tion of moequa-tion for the nth mode of the simply-supported beam as

q_n+ 2ξ_nω_nq˙_n+ ω_n²q_n= 2p

mLsinnπvt

L , (2.12)

which is uncoupled from the other modes of vibration. From this equation, the generalized coordinate q_n for the nth mode can be solved as where ωdn is the damped frequency of vibration of the beam,

ω_dn = ωn

1− ξn². (2.14)

Ωnis the exciting frequency implied by the moving load, Ωn=nπv

L (2.15)

and S_n is a nondimensional speed parameter deﬁned as the ratio of the frequency of excitation of the moving load to the nth frequency of vibration of the beam, i.e.,

S_n= Ωn

ω_n = nπv

ω_nL. (2.16)

Consequently, the total displacement u(x, t) of the beam caused by all the vibration modes can be summed as follows:

u(x, t) =

This is exactly the displacement of the beam caused by a single mov-ing load with takmov-ing into account the eﬀect of dampmov-ing. In Eq. (2.17), the terms with Ω_nt represent the forced vibration of the bridge in-duced by the moving load, and the terms with ω_dnt are the free vibration, which will eventually be damped out. Again, this equa-tion applies only when the acting posiequa-tion vt of the moving load is located within the range of the beam.

For a wide class of moving load problems encountered in practice, the eﬀect of damping on the bridge is so small, due to the rather short acting time of the moving loads that it can be ignored completely.

This is especially true, if one is interested in the response of the bridge of the ﬁrst few cycles. By neglecting the eﬀect of damping, the total displacement u(x, t) of the beam as given in Eq. (2.17) reduces to

u(x, t) = 2pL³ This is exactly the deﬂection of the simple beam at section x caused by the moving load p acting at position vt, with the eﬀect of damping neglected. Correspondingly, the bending moment M (x, t) caused by the moving load p on the beam can be computed as M (x, t) = Here, it should be noted that the solutions obtained above for the problem considered are not new in the literature, see, for instance, Biggs (1964) and Warburton (1976). However, most of the previous researchers have not proceeded further to derive impact formulas from these solutions, which are more useful to practicing engineers.

For the present purposes, let us consider a simple beam of length L = 20 m, per unit mass m = 3000 kg/m, ﬂexural rigidity EI = 10⁶

N-m² and a damping coefficient ξ of 2.5% for all vibration modes, subjected to a moving load p = 6 kN of speed v = 27.8 m/s (100 km/hr). The displacements for the midpoint of the beam ob-tained from Eqs. (2.17) and (2.18) for the damped and undamped cases have been compared in Fig. 2.2. As can be seen, the effect of damping on the response of the beam during the action of the moving load is rather small. For this reason, the effect of damping has often been neglected in research concerning the vehicle-induced vibrations on bridges.

To illustrate the eﬀect of multi-modes of vibration, for the three speeds of S1 = 0.05, 0.01 and 0.25, the deﬂections of the midpoint of the beam obtained from Eq. (2.18) for the undamped case considering various numbers of vibration modes have been plotted in Fig. 2.3.

As can be seen, the result obtained for the midpoint deflection by considering only the first mode is good enough, partly due to the fact that all the anti-symmetric modes of vibration contribute nothing to the midpoint deflection. This gives us the impression that using only the first mode can yield generally good approximate solutions for vehicle-induced response, especially when the midpoint deflection

Fig. 2.2. The eﬀect of damping of the beam.

Fig. 2.3. The eﬀect of multi-modes on midpoint deﬂection of the beam.

of the beam is desired. Such an approximation has been previously adopted by a number of researchers in their analytical studies.

2.3. Impact Factor for Midpoint Displacement

In this section, the responses derived in Sec. 2.2 for a single mov-ing load will be used to derive the impact formula for the midpoint displacement of the simply-supported beam. The results presented in this section cover a wide range of applications, as they are all expressed in terms of the nondimensional speed parameter. They also serve as a useful reference for comparison with other results.

It is realized that the impact response induced by a single moving load on the beam is generally larger than that induced by multi or continuously moving loads due to the suppression effect of the simul-taneous acting loads. Thus, the impact formulas presented in this chapter for a single moving load should be regarded as reasonable upper bounds for the responses considered. The other message to convey here is that the impact factors for the deflection, bending moment and shear force can be quite different, and that the use of

identical impact formulas for all physical quantities, as implied by most design codes, is not theoretically sound.

Here, we shall adopt the deﬁnition given in Eq. (1.1) for the im-pact factor I,

I = R_d(x)− Rs(x)

R_s(x) , (2.21)

where R_d(x) and R_s(x) respectively denote the maximum dynamic and static response of the bridge at section x due to passage of the moving load. For a simple beam, both the maximum dynamic and static deﬂections occur at the midpoint. The following is the maxi-mum static deﬂection of the beam under the static load p:

R_su

In contrast, the dynamic response for the midpoint deﬂection of the beam can be obtained from Eq. (2.18) by setting x = L/2, that is,

u preceding equation is valid only when the acting position vt of the moving load p is located within the span of the beam. For n = 2, 4, 6, . . ., the shape function sin(nπ/2) vanishes at the midpoint, as it turns out to be asymmetrical. Thus, only the modes with n = 1, 3, 5, . . ., i.e., the symmetrical modes, contribute to deﬂection of the midpoint.

Correspondingly, the impact factor for the midpoint deﬂection of the simple beam caused by the moving load p acting at position vt is

which is independent of the magnitude p of the moving load. As can be seen, the contribution of higher order terms decreases by a factor n⁻⁴. It follows that the eﬀect of higher order terms in Eq. (2.24) can

be neglected without losing accuracy. By the relation 96/π⁴ ∼= 1, Eq. (2.24) reduces to

I_u ∼=

sin Ω1t− S1sin ω1t 1− S₁²

− 1 . (2.25)

The impact factors calculated for the midpoint displacement using Eqs. (2.24) and (2.25), considering only the contribution of either multi-modes or the ﬁrst mode, have been plotted in Fig. 2.4. As can be seen, the midpoint displacement impact response of the simple beam is dominated by the ﬁrst mode.

To illustrate the eﬀect of the acting position vt of the moving load, diﬀerent values have been assumed for vt, i.e., vt/L = 1/8, 1/4, 3/8, 1/2, 5/8, 3/4, and 7/8, in calculation of the impact factor Iu for the midpoint displacement of the simple beam in Fig. 2.5. As can be seen, there exists an upper-bound envelope for the displacement impact factor I_u. The maximum impact factor I_u can be regarded as proportional to the speed parameter S₁ for vehicle speeds in the range S₁ < 0.5, and as constant for S₁ ≥ 0.5. Based on such an observation, the following formulas can be proposed for the midpoint

In document Vehicle-Bridge Interaction Dynamics (Page 57-200)