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Vibration of a suspension bridge installed with a water pipeline and subjected to moving trains
Engineering Structures 30 (2008) 632–642 www.elsevier.com/locate/engstruct
Vibration of a suspension bridge installed with a water pipeline and subjected to moving trains J.D. Yau a , Y.B. Yang b,∗ a Department of Architecture and Building Technology, Tamkang University, Taipei 10620, Taiwan, ROC b Department of Civil Engineering, National Taiwan University, Taipei 10617, Taiwan, ROC
Received 27 January 2007; received in revised form 29 April 2007; accepted 11 May 2007 Available online 18 June 2007
Abstract This paper presents a first study on the vibration of a suspension bridge installed with a water pipeline and subjected to moving trains. The suspension bridge is modeled as a single-span beam suspended by the hangers and hinged at the two ends. The train is simulated as a sequence of equidistant moving loads with identical weights. The liquid flowing through the pipeline is simulated as a strip of continuous mass moving at constant speeds. The governing equation is transformed into the generalized coordinates by Galerkin’s method, and solved by a rigorous incremental–iterative procedure that takes into account all the nonlinear effects. The results indicate that the critical position for the maximum acceleration of the suspended beam to occur under the moving loads depends upon the modal shape that has been excited, which can be antisymmetric or symmetric. The inertial effect of the flowing mass is beneficial for mitigating the vehicle-induced acceleration of the suspended beam, if the flowing speed is kept below the first resonant speed of the moving loads. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Flowing mass; Moving loads; Pipeline; Resonance; Suspension bridge; Vibration
1. Introduction It is not uncommon that suspension bridges constructed for carrying railway trains were installed with pipelines for water transport, due to the fact that a more economic design can be achieved by combining the two functions of water transport and railway transportation together, compared with the case that the two functions are treated separately by two individual bridges. Usually, the pipelines carrying the running water were installed either inside or attached under the box girder of the bridge. From the point of structural design, however, little knowledge is available for the effect of the flowing mass on the dynamic response of suspension bridges simultaneously subjected to the moving loads. This forms the objective of the present study. The collapse of the Tacoma Narrows Bridge in 1940 under the action of mild winds marks a historical turning point for the investigation of suspension bridges. Since then, the dynamic ∗ Corresponding author. Tel.: +886 2 3366 4245; fax: +886 2 2362 2975.
E-mail addresses: [email protected] (J.D. Yau), [email protected] (Y.B. Yang). c 2007 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2007.05.006
stability of suspension bridges has been the focus of studies for over half a century. Early works on this subject, including those by Bleich et al. [1], Rocard [2], Pugsley [3], and Abdel-Ghaffar [4], have laid the foundation for further studies on various dynamic problems of suspension bridges. Partly stimulated by the completion of numerous suspension bridges in the past two decades worldwide, the dynamic behavior of suspension bridges under moving loads has also attracted the attention of numerous researchers [5–12]. In all these studies, an important conclusion is that significant amplification may be induced by the moving loads on the cable forces for short-span suspension bridges. However, to the knowledge of the writers, basically no work has been conducted previously on the effect of flowing mass on the vehicle-induced vibration of suspension bridges. The purpose of this study is to fill in such a gap. By modeling the suspension bridge as a single-span suspended beam [2,3,5,9,10,12], the flowing liquid as a continuous mass steadily moving at a constant speed [10], and the vehicular loads traveling over the bridge as a row of equidistant moving loads with identical weights [13,14], a linearized deflection theory of suspension bridges [1–3] will
J.D. Yau, Y.B. Yang / Engineering Structures 30 (2008) 632–642
633
Fig. 1. Suspended beam under moving loads: (a) Analysis model, (b) box girder with a pipe.
be employed to formulate the governing equation for the suspended beam carrying the flowing mass under the moving loads described above. By taking into account the increase in cable tensions due to the live loads, such as the train loads and flowing mass, the governing equation of motion of the suspended beam appears as an integral–differential equation, which is coupled and nonlinear by nature. To compute the dynamic response of such a nonlinear vibrating system, one can express the deflection of the suspended beam as a series of assumed displacement functions, and then use Galerkin’s method to convert the governing equation into a set of differential equations of motion in terms of the generalized coordinates with nonlinearly coupled terms. By treating all the nonlinear coupled terms as pseudo forces, the differential equations in generalized coordinates are discretized by Newmark’s β method [15] in the time domain and then solved by an incremental–iterative approach. According to the numerical results, the critical position for the maximum acceleration to occur on the suspended beam depends on the vibration mode that has been excited, and the flowing mass is beneficial for suppressing the resonant vibration response of the suspended beam. However, as the flowing speed of the liquid reaches the first resonant speed of the moving loads over the beam, the effect of suppression on the resonant response will be drastically reduced. 2. Formulation of the problem The objective of this study is to investigate the train-induced response of a suspension bridge installed with a pipeline for conveying fluid. The vehicles traveling over the bridge are assumed to be equipped with proper suspension devices in the vertical direction so that only the weights of car bodies are carried over to the bridge. The effect of inertia of car bodies will be neglected as it is negligible for the bridge response [5,14,16].
Fig. 1(a) shows a single-span beam hinged at the two ends, suspended by two main cables via the hangers, and subjected to a sequence of identical lumped loads P with equal intervals d moving at speed v. A pipe is rigidly installed inside the box girder of the beam as in Fig. 1(b). For the bridge shown in Fig. 1(a), all the dead loads of the bridge girder and pipeline are assumed to be carried by the two parallel suspension cables, through the pretensioning effect of the vertical hangers. As a result, the beam is stress-free before the liquid flows through the pipe [10]. Assume the two suspended cables to be identical and of the same parabolic form, x x 2 y(x) = 4y0 − , (1) L L where y = the sag function of the cable, L = span length of the beam, and y0 = cable sag at the mid-span of the beam. The total horizontal component T of the tensile forces of the two parabolic cables caused by the uniform dead loads w of the beam and pipe is [3], T y 00 = −w,
(2)
where (•)0 = ∂(•)/∂ x. As the live loads w L , i.e., the train loads and flowing mass, move over the suspended beam, a portion q of the live loads will be carried by the cables in a way similar to the dead loads. This will result in an additional vertical deflection u equal to that of the beam, by assuming the vertical hangers to be densely spaced and inextensible [3,10]. As a consequence, the cables will experience an increase 1T in the horizontal tension component. To account for such an effect, the equation of equilibrium in Eq. (2) for the cables can be modified as follows: (T + 1T )(y 00 + u 00 ) = −w − q.
(3)
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The remainder of the live loads, i.e., w L − q, is directly carried by the suspended beam. The equation of motion for the vertical deflection u(x, t) of the beam under the load w L − q is m u¨ + cu˙ + E I u 0000 = w L − q,
(4) Fig. 2. An infinitesimal element of the cables.
where m = the total mass per unit length of the beam and pipe, c = damping coefficient of the beam, EI = total flexural rigidity of the beam and pipe, and (˙•) = ∂(•)/∂t. Combining Eqs. (2)–(4) yields m u¨ + cu˙ + E I u
0000
− (T + 1T )u − 1T y = w L . 00
00
(5)
By Hook’s law for the deformed cable element due to the force increase 1T , one has
(7)
(8)
0
/L 2
By the relation = −8y0 from Eq. (1), the cable force increment 1T is computed as Z 8y0 E c Ac L 1T = udx, (10) Lc 0 L2 where L c = the effective length of each cable, 3 Z L Z L q ds0 3 2 0 Lc = dx = 1+y dx. dx 0 0
− (T + 1T )u + S 00
L
Z
udx = w L ,
(12)
where S = (8y0 /L 2 )2 (E c Ac /L c ) and the live loads w L should be interpreted as d2 u (13) w L = w0 − m 0 2 f (x, v0 t) + p(x, t). dt Here, w0 and m 0 = the weight and mass of the flowing liquid in the pipe, f (x, v0 t) = function of the moving mass of the liquid with speed v0 [10], and p(x, t) = moving loads of the train. The RL integral term S 0 udx and the second-order term (T + 1T )u 00 in Eq. (12) represent the stiffening effect of the cables’ tension caused by the live loads on the suspended beam. Since the horizontal tension increment 1T of the cables is a function of the displacement u(x, t), as indicated in Eq. (10), the product 1T × u 00 is a nonlinear, coupled term. The equation of motion as presented in Eq. (12) for the bridge subjected to the simultaneous actions of the moving loads and flowing mass is an integral–differential equation in terms of the vertical deflection u. One feasible way for solving this equation is to use an incremental–iterative procedure, as will be described in Section 5. 3. Live loads and related conditions
where E c = elastic modulus, Ac = cross-sectional area of two cables. By considering the boundary conditions for the two cables with two-hinged ends, one can integrate Eq. (8) from 0 to L to obtain Z L Z L Z L 1T ds0 3 0 0 0 L 00 dx = y u dx = y u |0 −y udx E c Ac 0 dx 0 0 Z L = −y 00 udx. (9) y 00
m u¨ + cu˙ + E I u
0000
0
Here, the vertical deflection y should be regarded as the initial deflection of the beam under the dead loads, and the vertical deflection u as the additional deflection due to the live loads, which is measured from the static equilibrium position of the beam under the dead loads. Let us assume the original length ds0 and deformed length ds of an infinitesimal element of the cables shown in Fig. 2 to be q ds0 = (dx)2 + (dy)2 , (6a) q ds = (dx)2 + (dy + du)2 . (6b)
ds − ds0 dy du 1T × (ds0 /dx) ≈ ' E c Ac ds0 ds0 ds0 dx 2 dy du , × = ds0 dx dx or 1T ds0 3 dy du , = E c Ac dx dx dx
and flowing mass taken into account as
(11)
Substituting Eq. (10) into Eq. (5) yields the integral–differential equation of motion for the beam with the effect of train loads
Each of the live load terms in Eq. (13) will be given herein. The load function f (x, v0 t) used to describe a strip of continuous mass moving at speed v0 through the pipe is 1 − U (x − v0 t) for v0 t < L , f (x, v0 t) = (14) 1 for v0 t ≥ L , where U (•) = the unit step function. The total differential d2 u/dt 2 used to describe the acceleration effect of the flowing mass on the suspended beam is 2 ∂ 2u ∂ 2u d2 u 2∂ u + v = + 2v . (15) 0 0 ∂t∂ x dt 2 ∂t 2 ∂x2 The terms on the right side of Eq. (15) represent, in order, the acceleration due to the beam vibration, the complementary acceleration due to Coriolis force, and the centripetal acceleration. Substituting Eq. (15) into Eq. (13) and then Eq. (12) yields the following integral–differential equation of motion for the beam for the problem of concern:
[m + m 0 f (x, v0 t)] u¨ + 2m 0 v0 f (x, v0 t)u˙ 0 + cu˙ i h + E I u 0000 + m 0 v02 f (x, v0 t)u 00 − (T + 1T )u 00 Z L +S udx = w0 f (x, v0 t) + p(x, t) 0
(16)
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which is nonlinear and coupled. The preceding equation, along with the boundary and initial conditions, can be solved in an iterative way using the numerical procedure to be described later on. By treating a train as a sequence of wheel loads of magnitude P and interval d, the load function p(x, t) used to describe the train moving over the bridge at speed v is [13,14]: p(x, t) = P
N X
{δ [x − v(t − tk )] × [U (t − tk )
− U (t − tk − L/v)]} ,
(17)
where δ = Dirac’s delta function, N = the total number of wheel loads, and tk = (k − 1) d/v = arriving time of the kth load at the beam. The boundary conditions for the suspended beam of length L with two-hinged ends are:
(19a,b)
4. Transformation of equations by Galerkin’s method The solution to Eq. (16) for the dynamic response of the suspended beam with simply supported ends can be attempted by Galerkin’s method. With regard to the boundary conditions given in Eq. (18a–d), the displacement u(x, t) of the beam can be expressed in terms of sinusoidal functions as follows: u(x, t) =
X
qn (t) sin
n=1
nπ x , L
(20)
where qn (t) denotes the generalized coordinate associated with the nth assumed displacement mode of the beam. With the modal expression given in Eq. (20) for the beam, the original equations of motion in Eq. (16) can be transformed into a series of coupled equations in terms of the generalized coordinates. By substituting the assumed shape functions in Eq. (20) into Eq. (16), multiplying both sides of the equation with respect to the variation of the assumed shape functions, and integrating the equations over the length L of the beam, one can arrive at the following nonlinearly-coupled differential equations in terms of the generalized coordinate qn m + m 0 h n j (t) q¨n + c + 2m 0 $0,n κn j (t) q˙n h i 2 + mωn2 + 1kn − m 0 $0,n h n j (t) qn + Sn = Fn (x, v0 t) +
N X
Fk ($n , v, t),
(21)
k=1
where use has been made of the orthogonality conditions for the sine functions, j means the mode coupled with the nth assumed
(22a) qk , k k=1,3,5... X
(22b) 2S L (1 − cos nπ ) nπ 2
"
2qk k k=1,3,5... X
# ,
(22c)
and the time-dependent coefficients h n j (t) and κn j (t), which are coupled with the other modes, can be derived as follows: $0,n t − sin($0,n t) cos($0,n t) , with n = j nπ sin ($0,n − $0, j )t sin ($0,n + $0, j )$0,n t h n j (t) = − , (n − j)π (n + j)π with n 6= j for v0 t < L ,
(18a–d)
Assuming that the suspended beam starts to vibrate from rest, the initial conditions are u(x, 0) = u(x, ˙ 0) = 0.
nπ 4 E I nπ 2 T + , L m L m 16π y0 E c Ac X qk π SL 1kn = n 2 = n2 L c k=1,3,5... k 4y0 L3 ωn2 =
Sn =
k=1
u(0, t) = u(L , t) = 0, E I u 00 (0, t) = E I u 00 (L , t) = 0.
shape, and the coefficients ωn , 1kn , and Sn are given as follows:
(23a)
(
1, with n = j for v0 t > L , 0, with n 6= j 2 sin ($0,n t) , with n = j nπ n − n cos($0,n t) cos($0, j t) − j sin($0,n t) sin($0, j t) κn j (t) = , (n 2 − j 2 )π/2 with n 6= j h n j (t) =
(23b)
for v0 t < L , 0, h i κn j (t) = n 1 − (−1)n+ j , (n 2 − j 2 )π/2
(23c) with n = j for v0 t ≥ L .
(23d)
with n 6= j
Here, $0,n = nπ v0 /L and $0, j = jπ v0 /L are the driving frequencies of the flowing mass to the beam. It should be noted that the expressions for v0 t < L in Eqs. (23a) and (23c) represent the transient state of the continuous mass starting to flow into the pipeline, but not yet filling in the entire pipeline on the bridge, and that the expressions for v0 t ≥ L in Eqs. (23b) and (23d) means the continuous mass has filled in the pipeline on the bridge for a long while, which is referred to as the steady state. In the modal equations in Eq. (21), the generalized force Fn (x, v0 t) associated with the continuously flowing mass on the nth assumed shape of the beam can be given as follows: 2w0 1 − cos($0,n t) for v0 t < L , Fn (x, v0 t) = (24) 1 − cos nπ for v0 t ≥ L . nπ The generalized force Fk ($n , v, t) associated with the kth moving load on the nth assumed shape of the beam is Fk ($n , v, t) =
2P sin $n (t − tk ) L L , × U (t − tk ) − U t − tk − v
(25)
where $n = nπ v/L = the driving frequency of a moving load to the beam. The preceding equation is equivalent to the
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Let us use Γn (t) to denote the sum of the nonlinearly coupled 2 κ (t) × q˙ , m $ 2 h (t) × q , terms: m 0 h n j (t) × q¨n , 2m 0 $0,n nj n n 0 0,n n j 1kn × qn , and Sn , that is,
following [17]: Fk ($n , v, t) =
2P L
sin $n (t − tk ) × U (t − tk ) L n+1 + (−1) sin $n t − tk − v L . × U t − tk − v
Γn (t) = m 0 h n j (t)q¨n + 2m 0 $0,n κn j (t)q˙n h i 2 + 1kn − m 0 $0,n h n j (t) qn + Sn . (25a)
To verify the equivalency of Eq. (25a) with (25), we can simply replace the term (−1)n+1 in Eq. (25a) by − cos(nπ ) and perform the following derivation: L L (−1)n+1 sin $n t − tk − × H t − tk − v v nπ v L = − cos(nπ ) sin $n (t − tk ) − L v L × H t − tk − v = − cos(nπ )[sin ($n (t − tk )) cos(nπ ) L − cos ($n (t − tk )) sin(nπ )] × H t − tk − v L (25b) = − sin ($n (t − tk )) × H t − tk − v By substituting the relation of Eq. (25b) into Eq. (25a), we can obtain the expression in Eq. (25) and thus the equivalency is proved. The modal equations given in Eq. (21) for the nth generalized coordinate qn of the suspended beam under the action of the flowing mass and moving loads are nonlinearly coupled, due to the presence of such terms as: m 0 h n j (t) × q¨n , 2 κ (t) × q˙ , m $ 2 h (t) × q , 1k × q , and S . In 2m 0 $0,n nj n n n n n 0 0,n n j order to solve for the nonlinearly coupled equations in Eq. (21), an incremental–iterative procedure will be employed, as will be presented in the following section. 5. Solution by an incremental–iterative procedure An observation of the nth generalized equation of motion in Eq. (21) indicates that the modal equations for all the generalized coordinates are coupled due to presence of the 2 κ (t) × q˙ , nonlinear terms, such as m 0 h n j (t) × q¨n , 2m 0 $0,n nj n 2 m 0 $0,n h n j (t) × qn , 1kn × qn , and Sn , as given in Eqs. (22) and (23). All these terms are time-dependent and have to be updated at each time step if an incremental solution procedure is to be used. Clearly, the computational effort required in solving a set of nonlinear coupled differential equations for the generalized coordinates such as the ones in Eq. (21) is tremendous. Moreover, it was well known that if the acceleration response, rather than the displacement response, of the bridge is of concern, as is the case considered herein, much more higher modes have to be included in the computation [14,18]. To tackle this problem, an incremental–iterative procedure based on the concept of predictor and corrector will be presented in the following for solving the set of nonlinearly coupled equations given in Eq. (21).
(26)
By removing the nonlinearly coupled force term Γn (t) to the right side of Eq. (21) and regarding −Γn (t) as the pseudo force acting on the nth generalized coordinate system, the modal equation in Eq. (21) can be rewritten as m q¨n + cq˙n + mωn2 qn = Fn (x, v0 t) +
N X
Fk ($n , v, t) − Γn (t).
(27)
k=1
The preceding equation indicates that the original nonlinearly coupled equations in Eq. (21) have been converted to a set of uncoupled differential equations P Nassociated with the nth generalized force Fn (x, v0 t) + k=1 Fk ($n , v, t) and the pseudo force −Γn (t). With the form given in Eq. (27), an incremental–iterative procedure based on the concept of predictor and corrector will be employed to solve each of the generalized equations. In this regard, Newmark’s β method [15] is first employed to discretize the generalized equation of motion in Eq. (27) into an equivalent incremental stiffness equation. Then, an incremental–iterative procedure is employed to solve the equivalent stiffness equation involving the pseudo force −Γn (t), for which three major phases can be identified, i.e., predictor, corrector and equilibrium checking [19]. As a matter of fact, the logistics of the incremental–iterative procedure presented in this section is a rigorous procedure modified from the path-tracing scheme in [20], which can be applied to solving a wide range of nonlinear coupled problems. 5.1. Incorporation of incremental–iterative procedure Consider the ordinary differential equations in Eq. (27). Let qn,t+1t denote the nth generalized displacement at time t + 1t, and 1qn the displacement increment of qn from time t to t +1t, i.e., qn,t+1t = qn,t + 1qn,t+1t . Then Eq. (27) can be rewritten for the current configuration at t + 1t as m q¨n,t+1t + cq˙n,t+1t + mωn2 (qn,t + 1qn ) = Pn,t+1t − Γn,t+1t ,
(28)
where Pn,t+1t denotes the external load with respect to the nth generalized coordinate at time t + 1t, Pn,t+1t = Fn (x, v0 (t + 1t)) +
N X
Fk ($n , v, t + 1t).
(29)
k=1
By Newmark’s β method with constant average accelera tion [15], the total responses qn,t+1t , q˙n,t+1t , q¨n,t+1t for the nth generalized coordinate can be given as qn,t+1t = qn,t + 1qn,t+1t , q˙n,t+1t = q˙n,t + a6 q¨n,t + a7 q¨n,t+1t , q¨n,t+1t = a0 1qn − a2 q˙n,t − a3 q¨n,t ,
(30a) (30b) (30c)
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J.D. Yau, Y.B. Yang / Engineering Structures 30 (2008) 632–642
i−1 i−1 Pn,t+1t and internal effective force σn,t+1t resulting from the last iteration, i.e.,
where the coefficients ai are as follows [14]: a0 = a1 = a2 = a3 = a4 = a5 =
1 , β · 1t 2 γ , β · 1t 1 , β · 1t 1 − 1, 2β γ − 1, β 1t γ −2 , 2 β
(31a) (31b) (31c) (31d) (31e) (31f)
a6 = (1 − γ )1t,
(31g)
a7 = γ · 1t,
(31h)
i−1 i−1 i 1Pn,t+1t = Pn,t+1t − σn,t+1t ,
(36a)
i−1 i−1 i−1 σn,t+1t = Rn,t+1t + Γn,t+1t ,
(36b)
i−1 and the resistant force Rn,t+1t can be modified from the one for the incremental step in Eq. (34) to the one for the iterative step as follows: i−1 i−1 2 i−1 mωn qn,t+1t − m a2 q˙n,t+1t + a3 q¨n,t+1t i−1 i−1 i−1 −c a q ˙ + a q ¨ for i = 1, (37) 4 5 n,t+1t n,t+1t Rn,t+1t = i−1 i−1 2 i−1 mωn qn,t+1t + m q¨n,t+1t + cq˙n,t+1t for i > 1.
The incremental–iterative equations as given in Eqs. (35)–(37) are subject to the following initial conditions for the 1st iterative step (i.e., for i = 1) at the nth incremental step [14]:
with β = 0.25 and γ = 0.5. By substitution of Eqs. (30a)–(30c), the differential equations as given in Eq. (28) for the generalized coordinates can be transformed into a set of equivalent stiffness equations for the incremental step from time t to t + 1t as follows [14]:
0 ` q˙n,t+1t = q˙n,t ,
K n,eff × 1qn,t+1t = 1Pn,t+1t ,
0 σn,t+1t
(32)
where K n,eff denotes the effective stiffness and 1Pn,t+1t the effective load increment at time t + 1t, of the nth generalized coordinate,
(38a)
` 0 = q¨n,t , q¨n,t+1t
(38b)
=
0 Rn,t+1t
(38c) 0 + Γn,t+1t
=
` Rn,t
` + Γn,t ,
(38d)
where ` means the last iteration of the previous incremental step at time t. 5.2. Predictor phase
K n,eff = (a0 + ωn2 )m + a1 c,
(33a)
1Pn,t+1t = Pn,t+1t − Γn,t+1t − Rn,t .
(33b)
In Eq. (33b), Rn,t denotes the resistant force at time t, i.e., Rn,t = mωn2 qn,t − m a2 q˙n,t + a3 q¨n,t − c a4 q˙n,t + a5 q¨n,t . (34) It should be noted that the displacement 1qn,t+1t solved from Eq. (32) is not exact due to the approximate nature of the force term Γn,t+1t involving the dynamic terms of the flowing mass, i.e., m 0 h n j (t + 1t) × q¨n,t+1t and 2 κ (t + 1t) × q˙ 2m 0 $0,n nj n,t+1t , and the nonlinear stiffness 2 h (t + 1t) × q terms, i.e., 1kn,t+1t qn,t+1t , m 0 $0,n nj n,t+1t and Sn,t+1t , which can only be estimated based on the information available at time t. Such a problem can be overcome through the iterative procedure to be described later on. For the purpose of iteration, the incremental stiffness equations as given in Eq. (32) for the incremental step from time t to t + 1t should be modified to include the iterative feature [19], namely, i i K n,eff × 1qn,t+1t = 1Pn,t+1t ,
0 ` qn,t+1t = qn,t ,
The predictor phase is concerned with solution of the structural response increments from the structural stiffness equation in Eq. (34). Since this phase affects only the speed of convergence, the tangent stiffness used need not be exact [19]. For this reason, we will treat K n,eff as constant for all the iterations within each incremental step. Consequently, for the first iteration, i.e., for i = 1, the displacement increment 1 1qn,t+1t for the nth generalized coordinate can be solved from Eq. (35) along with Eqs. (36a) and (38d) as: −1 1 ` ` = K n,eff (Pn,t+1t − Rn,t − Γn,t ). 1qn,t+1t
(39)
For the following iterations, i.e., for i ≥ 2, the displacement i can be solved from Eq. (35) as increment 1qn,t+1t −1 i i 1qn,t+1t = K n,eff × 1Pn,t+1t .
(40)
As the increments of all the generalized coordinates i 1qn,t+1t |n=1,2,3... have been made available in Eqs. (39) and (40), the total displacement, velocity and acceleration of the nth generalized coordinate for the ith iterative step at time t + 1t can be updated following Eq. (30) as follows:
(35)
i where for the first iteration (i = 1), 1Pn,t+1t ≡ 1Pn,t+1t represents the load increment and for the following iterations i (i ≥ 2), Pn,t+1t represents the unbalanced forces, which are equal to the difference between the total external load
i−1 i i qn,t+1t = qn,t+1t + 1qn,t+1t ,
(41a)
i−1 i−1 i i q˙n,t+1t = q˙n,t+1t + a6 q¨n,t+1t + a7 q¨n,t+1t ,
(41b)
i−1 i−1 i q¨n,t+1t = a0 1qni − a2 q˙n,t+1t − a3 q¨n,t+1t .
(41c)
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Table 1 Properties of the main beam and flowing mass L (m)
EI (kN m2 )
E c Ac (kN)
c (kN s/m/m)
m (t/m)
y0 (m)
m 0 (t/m)
v0 (m/s)
Ω1 (Hz)
Ω2 (Hz)
125
2.59 × 108
8 × 107
1.63
10.44
12.5
1.1
12
2.03
2.44
5.3. Corrector phase The corrector phase relates to recovery of the internal resistant forces from the displacement increments made available in the predictor phase. It is essential that a proper corrector be used, because this phase determines essentially the accuracy of the iterative solution [19]. In this study, the i effective internal force σn,t+1t of the nth generalized system at time t + 1t is recovered by substituting the total responses in i Eqs. (41a)–(41c) into the resistant forces Rn,t+1t in Eq. (37), i which is then substituted, along with the pseudo force Γn,t+1t , i into the effective internal forces σn,t+1t in Eq. (36b). A force recovery procedure such as this has an order of accuracy the same as that implied by Newmark’s β method. 5.4. Equilibrium-checking phase In the equilibrium-checking phase, the internal forces of the beam computed from the corrector phase are compared with the external forces, the difference being regarded as the unbalanced forces. By assigning the effective internal i (made available from the corrector phase) to forces σn,t+1t i−1 i−1 σn,t+1t , the unbalanced forces 1Pn,t+1t can be computed from Eq. (36a) for all the generalized coordinates. Whenever the root mean square sum of the generalized unbalanced P of the i−1 forces, i.e., [ n=1... (1Pn,t+1t )2 ]1/2 , is larger than a preset tolerance, say 10−3 , iteration for removing the unbalanced forces involving the two phases of predictor and corrector should be repeated. Once the condition of convergence is satisfied for an incremental step, we can proceed to the next incremental step with an update on the external load Pn,t+1t and repeat the iterative procedure for solution of Eq. (35). Evidently, the incremental–iterative procedure as presented above allows us to circumvent the complex problem of solving a set of simultaneous nonlinear equations with a large number of unknowns as originally given in Eq. (21). 6. Resonant condition caused by moving train loads When a bridge is subjected to a sequence of concentrated loads of equal intervals d moving at a constant speed v, the bridge may experience the action of a periodic force with a passage frequency of v/d. Once the passage frequency matches any of the natural frequencies of the bridge, resonance will be developed on the bridge [13,17]. Let Ω denote any of the circular frequencies of the bridge that has been excited. Under the resonant condition, i.e., when the resonant speed vres equals Ω d, the dynamic response of the bridge will build up continuously as there are more vehicular loads passing through the bridge [13,17]. The above resonance is excited
Fig. 3. Vibration modes of the suspended beam.
Table 2 Properties of the moving loads N
P (kN)
d (m)
v1,res (km/h)
v2,res (km/h)
16
350
27.5
200
241
by concentrated loads of regular intervals d moving over the bridge. In reality, sub-resonance of acceleration response may also be generated on the bridge as the moving loads pass through the bridge at the speed vsub,res = Ω d/J , where J represents the number of complete cycles of oscillation of the beam occurring during the passage of two adjacent loads, which has a duration of d/v. The sub-resonance is usually associated with the higher modes of the beam, which may result in the maximum acceleration on the beam depending on the higher modes that have been excited [17,21]. 7. Numerical examples As shown in Fig. 1, a single-span suspended beam installed with a water pipe is subjected to successive moving loads. The properties of the beam, cable, and flowing mass are listed in Table 1, in which Ω1 denotes the fundamental frequency of the first mode (anti-symmetric) and Ω2 the second mode (symmetric), as depicted in Fig. 3. Rayleigh damping is assumed for the box girder of the bridge with a damping ratio of 2.5% used for the first two modes. Due to the strengthening effect of the tension forces of the cables induced by the dead load, the natural frequency of the first anti-symmetric mode is smaller than that of the first symmetric mode. Table 2 shows the properties of the moving loads and the first and second resonant speeds of the suspended beam under the moving loads. 7.1. Acceleration response of the suspended beam due to moving loads It was well known that if the acceleration response, rather than the displacement response, of the bridge is of concern, as is the case considered herein, many more higher modes have to be included in the computation [14,18]. In order to verify that a sufficient number of modes of vibration has been used in the analysis, we first compute the acceleration response at the first quarter-point of the main beam under a series of concentrated
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Fig. 6. amax –v–x/L plot of the suspended beam due to moving loads. Fig. 4. Test of convergence.
Fig. 5. Time histories of acceleration of main beam.
loads moving at the first resonant speed using either 2, 10, or 20 modes. As can be seen from Fig. 4, the use of 20 modes is considered sufficient. For this reason, the same number of modes will be used in all the examples to follow. To illustrate the resonant phenomena of the suspended beam under a series of concentrated loads moving at the first and second resonant speeds, the time history responses of acceleration at the first quarter-point and mid-span, respectively, of the main beam have been plotted in Fig. 5. As it is expected for the resonance phenomena, both the acceleration responses continue to build up as there are more vehicular loads passing through the beam. But the resonant amplitude at the mid-span due to the vehicular loads traveling at the second resonant speed v2,res (=241 km/h) is significantly smaller than that at the first quarter-point for the vehicular loads traveling at the first resonant speed v1,res (=200 km/h). The reason can be attributed to the fact that as a row of moving loads, with an interval (d = 27.5 m) far smaller than the bridge span length (L = 125 m), pass through a long-span suspension bridge, the simultaneous presence of the vehicular loads on the bridge deck may produce a suppression action on the first symmetric mode (i.e., the second mode), making the mid-span acceleration of the bridge less severe compared with the other resonant case involving the anti-symmetric mode. Such a phenomenon can be also observed in the following example.
7.2. Maximum vertical acceleration response of the suspended beam Train-induced resonance can result in excessive vibrations on the track structure and may bring about derailment of the trains [21–23]. For this reason, the maximum acceleration amax of a suspended beam due to the moving loads and flowing mass will be investigated in this example. Three cases will be considered for the mass flowing through the pipe installed inside the box girder: (1) empty pipe; (2) transient flow — the flowing mass starts to flow into, but not fill in, the pipeline; and (3) steady flow — the continuous mass has filled in the pipeline and continue to flow with speed v0 for a duration of 2L/v0 before the first moving (train) load enters the suspended beam. For the case of an empty pipe, the maximum acceleration response along the beam length has been drawn against the speed as a amax –x/L–v plot in Fig. 6. The results indicate that the maximum acceleration depends on the vibration shape that has been excited, and that as the running speed of the moving loads coincides with any of the resonant or sub-resonant speeds associated with the vibration modes of the bridge, the number of peak accelerations along the beam axis is associated with the vibration mode excited. Physically speaking, the condition for resonance to occur is identical to that for simple beams [13]. Obviously, the amplification effect caused by resonance and sub-resonance on the dynamic response of the suspension bridge is harmful not only to the structural safety of the bridge, but also to the maneuverability of the vehicles moving over it. As mentioned in Section 6, the resonant speed of the first mode can be calculated as v1,res = Ω1 d (=200 km/h). Correspondingly, the maximum acceleration computed of the beam is amax = 0.35g. Since the first mode is anti-symmetric, two peaks can be observed for the maximum acceleration along the main beam. Next, let us consider the transient case when both the train loads and flowing mass start to move through the suspended beam simultaneously. The amax –x/L–v tri-phase plot has been given in Fig. 7. As can be seen, the maximum acceleration of the main beam at the first resonant speed (v1,res = 200 km/h) has been reduced by 9% for the peak response in comparison with that of Fig. 6. This result indicates that the flowing mass into the suspended beam can provide a suppression effect on
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Fig. 7. amax –v–x/L plot for the suspended beam with transient flow. Fig. 9. Suppression effect of the flowing mass on resonant response.
Fig. 8. amax –v–x/L plot for the suspended beam with steady flow.
the first resonant response of the main beam. But the higher modes can still be excited in the region with speeds higher than 200 km/h in the amax –x/L–v plot of Fig. 7. This can be attributed to the fact that the vehicular loads move so fast that the flowing mass has not yet filled up the pipe and thus is unable to exert its full effect of suppression on the beam. For the case of steady flow, i.e., with the pipe filled up by the flow for a duration of 2L/v0 before the moving loads enter the beam. The corresponding amax –x/L–v triphase plot in Fig. 8 reveals that most of the acceleration peaks associated with the vibration modes excited have been substantially reduced (e.g., by 14% for the peak response), including the higher modes in the region with speed higher than the first resonant speed v1,max = 200 km/h. Fig. 9 shows the maximum acceleration amax computed along the axis x/L of the suspended beam due to the vehicular loads traveling at the first resonant speed v1,res for the three cases of the flowing mass studied above. From this figure, the suppression effect of the flowing mass on the resonant response of acceleration of the suspended beam can be clearly appreciated. This figure also indicates that at the time when the first mode (anti-symmetric) is excited, the resonance associated with other higher modes will be excited as well, as can be observed from the secondary peaks appearing on the acceleration response curve. Finally, let us consider a special case when the suspended beam is subjected to the action of flowing mass alone. Fig. 10 shows the maximum acceleration of the transient/steady liquid flowing at various speeds. Compared with the amax –x/L plots shown in Fig. 9, the effect of dynamic amplification of the steady flowing mass alone through the suspended beam is significantly smaller than that of the moving loads traveling at
Fig. 10. Moving effect of the flowing mass alone at various speeds.
the first resonant speed v1,res . On the other hand, the amax –x/L plots in Fig. 10 reveal that: (1) the bridge responses induced by transient flows alone are generally larger than those of steady flows; (2) the higher the speed of the flowing mass is, the greater the maximum acceleration response of the suspended beam will be; (3) the maximum acceleration responses induced by all transient flows with speeds over 28 m/s are quite close, meaning that when the flowing mass enters the pipeline at a very high speed, the dynamic behavior of the beam on which the pipeline is installed is similar to that of a structure subjected to an impulsive force; and (4) higher modes of the suspended beam are excited even though the maximum acceleration response occurs at the mid-span. 7.3. Maximum tension increment of the cables In this paper, the increase in horizontal component of the cable forces caused by the live loads (i.e., moving loads and flowing mass) on the suspended beam has been taken into account in the dynamic analysis. In Fig. 11, the ratio 1Tmax /T of the tension increment to the maximum tension of the cables has been plotted against the speed of the moving loads for the three cases of the flowing mass. This figure indicates that the maximum increment of the cable tension occurs around the second resonant speed v2,res , except for the case of transient flowing mass. This implies that the tension increase in the
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Fig. 13. Moving effect of the flowing mass in steady state. Fig. 11. Maximum tension increment in the cable forces.
Fig. 12. Moving effect of the flowing mass in transient state.
cables caused by the first symmetric mode is larger than that by the anti-symmetric shape. Moreover, the speed of the response peak in the 1Tmax /T –v plot for the case of steady flowing mass is slightly smaller than that for the case of empty pipeline, due to the inertia effect of the flowing mass on the beam, which appears in the form of reduced frequencies. The second peak in the 1Tmax /T –v plot occurs at the speed of 121 km/h, which is equal to v2,res /2. This means the second (symmetric) mode of the suspended beam has been excited again by the vehicular loads traveling at the sub-resonant speed v2,res /2. Such a magnificent effect on the cable tension should be taken into account in the design of suspension bridges. 7.4. Moving effect of the flowing mass To illustrate the moving effect of the flowing mass on the maximum acceleration response of the suspended beam due to moving loads, let us consider an extreme case when both the moving loads and flowing mass have identical speeds. Figs. 12 and 13 show the amax –x/L–v tri-phase plots of the suspended beam installed with a water pipe conveying the transient flowing mass and steady flow, respectively. These figures indicate that once the moving speed of the flowing mass is over the first resonant speed (v1,res = 200 km/h), the acceleration response of the main beam will be totally amplified, and that the suppression effect of the steady flow on the maximum acceleration response of the suspended beam is reduced as well.
8. Concluding remarks The nonlinearly coupled equations of motion are first derived for a suspended beam carrying a pipeline subjected to the moving vehicular loads. By Galerkin’s method, these equations are transformed into a set of nonlinearly coupled differential equations in terms of the generalized coordinates. By treating the nonlinear coupled terms as pseudo forces, the coupled differential equations for all of the generalized systems are converted by Newmark’s β method to a set of equivalent stiffness equations of motion with generalized forces and pseudo forces. Finally, these equivalent equations are solved by a rigorous incremental–iterative procedure involving the three phases of predictor, corrector, and equilibrium-checking. The nonlinear vibration of a suspended beam under the action of successive moving loads and flowing mass has been studied in this paper. The numerical results indicate that once the exciting passage frequency (v/d) of the moving loads coincides with any of the natural frequencies of the suspended beam, resonance will be developed on the bridge. The maximum acceleration response induced by resonance need not occur at the midpoint of the bridge deck. The critical position for the maximum acceleration to occur on the beam depends on the vibration mode that has been excited. Both the anti-symmetric and symmetric modes of vibration of the suspended beam can be excited to resonance within the range of operation speeds of the moving trains. The flowing mass may produce a suppression effect on the resonant vibration of the suspended beam induced by the successive moving loads, if the flowing speed is less than the first resonant speed of the moving loads. In general, the installation of water pipelines inside the box girder of a suspension bridge is beneficial for reducing the bridge response when subjected to the moving loads. For the special case where the bridge is free of any moving loads and when the flowing mass enters the pipeline at a very high speed, the dynamic behavior of the beam on which the pipeline is installed is similar to that of a structure subjected to an impulsive force. Acknowledgements This study was sponsored in part by the National Science Council of the Republic of China via Grant No. NSC 95-2221E-032-053. The finical support is gratefully acknowledged.
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This paper was written when the second author spent part of his sabbatical leave at the Department of Civil Engineering, National University of Singapore in 2006. References [1] Bleich F, Rosenkrans R, Vincent GS, Collough CB. The mathematical theory of vibration in suspension bridges. Washington: US Government Printing Office; 1950. [2] Rocard Y. Dynamic instability. New York: Frederick Ungar Publishing Company; 1957. [3] Pugsley A. The theory of suspension bridges. 2nd ed. London (England): Edward Arnold Ltd.; 1957. [4] Abdel-Ghaffar AM. Vertical vibration analysis of suspension bridges. J Struct Div, ASCE 1980;106(10):2053–75. [5] Vellozzi J. Vibration of suspension bridges under moving loads. J Struct Div, ASCE 1967;93(4):123–38. [6] Bryja D, Sniady P. Spatially coupled vibrations of a suspension bridge under random highway traffic. Earthq Eng Struct Dynam 1991;20(11): 999–1010. [7] Hayashikawa T, Watanabe N. Suspension bridge response to moving loads. J Eng Mech, ASCE 1982;108(6):1051–66. [8] Hayashikawa T. Effects of shear deformation and rotary inertia on suspension bridges response under moving loads. Proc JSCE 1983;335: 183–93. [9] Chatterjee PK, Datta TK, Surana CS. Vibration of suspension bridges under vehicular movements. J Struct Div, ASCE 1993;120(3):681–703. [10] Fr´yba L. Vibration of solids and structures under moving loads. 3rd ed. London: Thomas Telford; 1999.
[11] Xia H, Xu YL, Chan THT. Dynamic interaction of long suspension bridges with running trains. J Sound Vibration 2000;237(2):263–80. [12] Bryja D, Sniady P. Stochastic nonlinear vibrations of highway suspension bridge under inertial sprung moving load. J Sound Vibration 1998;216(3): 507–19. [13] Yang YB, Yau JD, Hsu LC. Vibration of simple beams due to trains moving at high speeds. Eng Struct 1997;19(11):936–44. [14] Yang YB, Yau JD, Wu YS. Vehicle–bridge interaction dynamics. Singapore: World Scientific; 2004. [15] Newmark NM. A method of computation for structural dynamics. J Eng Mech Div, ASCE 1959;85(1):67–94. [16] Yang YB, Yau JD. Vehicle–bridge interaction element for dynamic analysis. J Struct Eng, ASCE 1997;123(11):1512–8; 124 (4) 479 [Errata]. [17] Yang YB, Lin CL, Yau JD, Chang DW. Mechanism of resonance and cancellation for train-induced vibrations on bridges with elastic bearings. J Sound Vibration 2004;269(1–2):345–60. [18] Zhu XQ, Law SS. Moving load identification on multi-span continuous bridges with elastic bearings. Mech Syst Signal Proc 2006;20:1759–82. [19] Yang YB, Kuo SR. Theory and analysis of nonlinear framed structures. Singapore: Prentice Hall; 1994. [20] Yang YB, Shieh MS. Solution method for nonlinear problems with multiple critical points. AIAA J 1990;28(12):2110–6. [21] Yau JD, Yang YB. Vertical accelerations of simple beams due to successive loads traveling at resonant speeds. J Sound Vibration 2006; 289:210–28. [22] Yau JD. Vibration of parabolic tied-arch beams due to moving loads. Intl J Struct Stab Dynam 2006;6(2):193–214. [23] Yau JD. Vibration of arch bridges due to moving loads and vertical ground motions. J Chinese Inst Eng 2006;29:1017–27.
Vibration of a suspension bridge installed with a water pipeline and subjected to moving trains J.D. Yau a , Y.B. Yang b,∗ a Department of Architecture and Building Technology, Tamkang University, Taipei 10620, Taiwan, ROC b Department of Civil Engineering, National Taiwan University, Taipei 10617, Taiwan, ROC
Received 27 January 2007; received in revised form 29 April 2007; accepted 11 May 2007 Available online 18 June 2007
Abstract This paper presents a first study on the vibration of a suspension bridge installed with a water pipeline and subjected to moving trains. The suspension bridge is modeled as a single-span beam suspended by the hangers and hinged at the two ends. The train is simulated as a sequence of equidistant moving loads with identical weights. The liquid flowing through the pipeline is simulated as a strip of continuous mass moving at constant speeds. The governing equation is transformed into the generalized coordinates by Galerkin’s method, and solved by a rigorous incremental–iterative procedure that takes into account all the nonlinear effects. The results indicate that the critical position for the maximum acceleration of the suspended beam to occur under the moving loads depends upon the modal shape that has been excited, which can be antisymmetric or symmetric. The inertial effect of the flowing mass is beneficial for mitigating the vehicle-induced acceleration of the suspended beam, if the flowing speed is kept below the first resonant speed of the moving loads. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Flowing mass; Moving loads; Pipeline; Resonance; Suspension bridge; Vibration
1. Introduction It is not uncommon that suspension bridges constructed for carrying railway trains were installed with pipelines for water transport, due to the fact that a more economic design can be achieved by combining the two functions of water transport and railway transportation together, compared with the case that the two functions are treated separately by two individual bridges. Usually, the pipelines carrying the running water were installed either inside or attached under the box girder of the bridge. From the point of structural design, however, little knowledge is available for the effect of the flowing mass on the dynamic response of suspension bridges simultaneously subjected to the moving loads. This forms the objective of the present study. The collapse of the Tacoma Narrows Bridge in 1940 under the action of mild winds marks a historical turning point for the investigation of suspension bridges. Since then, the dynamic ∗ Corresponding author. Tel.: +886 2 3366 4245; fax: +886 2 2362 2975.
E-mail addresses: [email protected] (J.D. Yau), [email protected] (Y.B. Yang). c 2007 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2007.05.006
stability of suspension bridges has been the focus of studies for over half a century. Early works on this subject, including those by Bleich et al. [1], Rocard [2], Pugsley [3], and Abdel-Ghaffar [4], have laid the foundation for further studies on various dynamic problems of suspension bridges. Partly stimulated by the completion of numerous suspension bridges in the past two decades worldwide, the dynamic behavior of suspension bridges under moving loads has also attracted the attention of numerous researchers [5–12]. In all these studies, an important conclusion is that significant amplification may be induced by the moving loads on the cable forces for short-span suspension bridges. However, to the knowledge of the writers, basically no work has been conducted previously on the effect of flowing mass on the vehicle-induced vibration of suspension bridges. The purpose of this study is to fill in such a gap. By modeling the suspension bridge as a single-span suspended beam [2,3,5,9,10,12], the flowing liquid as a continuous mass steadily moving at a constant speed [10], and the vehicular loads traveling over the bridge as a row of equidistant moving loads with identical weights [13,14], a linearized deflection theory of suspension bridges [1–3] will
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Fig. 1. Suspended beam under moving loads: (a) Analysis model, (b) box girder with a pipe.
be employed to formulate the governing equation for the suspended beam carrying the flowing mass under the moving loads described above. By taking into account the increase in cable tensions due to the live loads, such as the train loads and flowing mass, the governing equation of motion of the suspended beam appears as an integral–differential equation, which is coupled and nonlinear by nature. To compute the dynamic response of such a nonlinear vibrating system, one can express the deflection of the suspended beam as a series of assumed displacement functions, and then use Galerkin’s method to convert the governing equation into a set of differential equations of motion in terms of the generalized coordinates with nonlinearly coupled terms. By treating all the nonlinear coupled terms as pseudo forces, the differential equations in generalized coordinates are discretized by Newmark’s β method [15] in the time domain and then solved by an incremental–iterative approach. According to the numerical results, the critical position for the maximum acceleration to occur on the suspended beam depends on the vibration mode that has been excited, and the flowing mass is beneficial for suppressing the resonant vibration response of the suspended beam. However, as the flowing speed of the liquid reaches the first resonant speed of the moving loads over the beam, the effect of suppression on the resonant response will be drastically reduced. 2. Formulation of the problem The objective of this study is to investigate the train-induced response of a suspension bridge installed with a pipeline for conveying fluid. The vehicles traveling over the bridge are assumed to be equipped with proper suspension devices in the vertical direction so that only the weights of car bodies are carried over to the bridge. The effect of inertia of car bodies will be neglected as it is negligible for the bridge response [5,14,16].
Fig. 1(a) shows a single-span beam hinged at the two ends, suspended by two main cables via the hangers, and subjected to a sequence of identical lumped loads P with equal intervals d moving at speed v. A pipe is rigidly installed inside the box girder of the beam as in Fig. 1(b). For the bridge shown in Fig. 1(a), all the dead loads of the bridge girder and pipeline are assumed to be carried by the two parallel suspension cables, through the pretensioning effect of the vertical hangers. As a result, the beam is stress-free before the liquid flows through the pipe [10]. Assume the two suspended cables to be identical and of the same parabolic form, x x 2 y(x) = 4y0 − , (1) L L where y = the sag function of the cable, L = span length of the beam, and y0 = cable sag at the mid-span of the beam. The total horizontal component T of the tensile forces of the two parabolic cables caused by the uniform dead loads w of the beam and pipe is [3], T y 00 = −w,
(2)
where (•)0 = ∂(•)/∂ x. As the live loads w L , i.e., the train loads and flowing mass, move over the suspended beam, a portion q of the live loads will be carried by the cables in a way similar to the dead loads. This will result in an additional vertical deflection u equal to that of the beam, by assuming the vertical hangers to be densely spaced and inextensible [3,10]. As a consequence, the cables will experience an increase 1T in the horizontal tension component. To account for such an effect, the equation of equilibrium in Eq. (2) for the cables can be modified as follows: (T + 1T )(y 00 + u 00 ) = −w − q.
(3)
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The remainder of the live loads, i.e., w L − q, is directly carried by the suspended beam. The equation of motion for the vertical deflection u(x, t) of the beam under the load w L − q is m u¨ + cu˙ + E I u 0000 = w L − q,
(4) Fig. 2. An infinitesimal element of the cables.
where m = the total mass per unit length of the beam and pipe, c = damping coefficient of the beam, EI = total flexural rigidity of the beam and pipe, and (˙•) = ∂(•)/∂t. Combining Eqs. (2)–(4) yields m u¨ + cu˙ + E I u
0000
− (T + 1T )u − 1T y = w L . 00
00
(5)
By Hook’s law for the deformed cable element due to the force increase 1T , one has
(7)
(8)
0
/L 2
By the relation = −8y0 from Eq. (1), the cable force increment 1T is computed as Z 8y0 E c Ac L 1T = udx, (10) Lc 0 L2 where L c = the effective length of each cable, 3 Z L Z L q ds0 3 2 0 Lc = dx = 1+y dx. dx 0 0
− (T + 1T )u + S 00
L
Z
udx = w L ,
(12)
where S = (8y0 /L 2 )2 (E c Ac /L c ) and the live loads w L should be interpreted as d2 u (13) w L = w0 − m 0 2 f (x, v0 t) + p(x, t). dt Here, w0 and m 0 = the weight and mass of the flowing liquid in the pipe, f (x, v0 t) = function of the moving mass of the liquid with speed v0 [10], and p(x, t) = moving loads of the train. The RL integral term S 0 udx and the second-order term (T + 1T )u 00 in Eq. (12) represent the stiffening effect of the cables’ tension caused by the live loads on the suspended beam. Since the horizontal tension increment 1T of the cables is a function of the displacement u(x, t), as indicated in Eq. (10), the product 1T × u 00 is a nonlinear, coupled term. The equation of motion as presented in Eq. (12) for the bridge subjected to the simultaneous actions of the moving loads and flowing mass is an integral–differential equation in terms of the vertical deflection u. One feasible way for solving this equation is to use an incremental–iterative procedure, as will be described in Section 5. 3. Live loads and related conditions
where E c = elastic modulus, Ac = cross-sectional area of two cables. By considering the boundary conditions for the two cables with two-hinged ends, one can integrate Eq. (8) from 0 to L to obtain Z L Z L Z L 1T ds0 3 0 0 0 L 00 dx = y u dx = y u |0 −y udx E c Ac 0 dx 0 0 Z L = −y 00 udx. (9) y 00
m u¨ + cu˙ + E I u
0000
0
Here, the vertical deflection y should be regarded as the initial deflection of the beam under the dead loads, and the vertical deflection u as the additional deflection due to the live loads, which is measured from the static equilibrium position of the beam under the dead loads. Let us assume the original length ds0 and deformed length ds of an infinitesimal element of the cables shown in Fig. 2 to be q ds0 = (dx)2 + (dy)2 , (6a) q ds = (dx)2 + (dy + du)2 . (6b)
ds − ds0 dy du 1T × (ds0 /dx) ≈ ' E c Ac ds0 ds0 ds0 dx 2 dy du , × = ds0 dx dx or 1T ds0 3 dy du , = E c Ac dx dx dx
and flowing mass taken into account as
(11)
Substituting Eq. (10) into Eq. (5) yields the integral–differential equation of motion for the beam with the effect of train loads
Each of the live load terms in Eq. (13) will be given herein. The load function f (x, v0 t) used to describe a strip of continuous mass moving at speed v0 through the pipe is 1 − U (x − v0 t) for v0 t < L , f (x, v0 t) = (14) 1 for v0 t ≥ L , where U (•) = the unit step function. The total differential d2 u/dt 2 used to describe the acceleration effect of the flowing mass on the suspended beam is 2 ∂ 2u ∂ 2u d2 u 2∂ u + v = + 2v . (15) 0 0 ∂t∂ x dt 2 ∂t 2 ∂x2 The terms on the right side of Eq. (15) represent, in order, the acceleration due to the beam vibration, the complementary acceleration due to Coriolis force, and the centripetal acceleration. Substituting Eq. (15) into Eq. (13) and then Eq. (12) yields the following integral–differential equation of motion for the beam for the problem of concern:
[m + m 0 f (x, v0 t)] u¨ + 2m 0 v0 f (x, v0 t)u˙ 0 + cu˙ i h + E I u 0000 + m 0 v02 f (x, v0 t)u 00 − (T + 1T )u 00 Z L +S udx = w0 f (x, v0 t) + p(x, t) 0
(16)
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which is nonlinear and coupled. The preceding equation, along with the boundary and initial conditions, can be solved in an iterative way using the numerical procedure to be described later on. By treating a train as a sequence of wheel loads of magnitude P and interval d, the load function p(x, t) used to describe the train moving over the bridge at speed v is [13,14]: p(x, t) = P
N X
{δ [x − v(t − tk )] × [U (t − tk )
− U (t − tk − L/v)]} ,
(17)
where δ = Dirac’s delta function, N = the total number of wheel loads, and tk = (k − 1) d/v = arriving time of the kth load at the beam. The boundary conditions for the suspended beam of length L with two-hinged ends are:
(19a,b)
4. Transformation of equations by Galerkin’s method The solution to Eq. (16) for the dynamic response of the suspended beam with simply supported ends can be attempted by Galerkin’s method. With regard to the boundary conditions given in Eq. (18a–d), the displacement u(x, t) of the beam can be expressed in terms of sinusoidal functions as follows: u(x, t) =
X
qn (t) sin
n=1
nπ x , L
(20)
where qn (t) denotes the generalized coordinate associated with the nth assumed displacement mode of the beam. With the modal expression given in Eq. (20) for the beam, the original equations of motion in Eq. (16) can be transformed into a series of coupled equations in terms of the generalized coordinates. By substituting the assumed shape functions in Eq. (20) into Eq. (16), multiplying both sides of the equation with respect to the variation of the assumed shape functions, and integrating the equations over the length L of the beam, one can arrive at the following nonlinearly-coupled differential equations in terms of the generalized coordinate qn m + m 0 h n j (t) q¨n + c + 2m 0 $0,n κn j (t) q˙n h i 2 + mωn2 + 1kn − m 0 $0,n h n j (t) qn + Sn = Fn (x, v0 t) +
N X
Fk ($n , v, t),
(21)
k=1
where use has been made of the orthogonality conditions for the sine functions, j means the mode coupled with the nth assumed
(22a) qk , k k=1,3,5... X
(22b) 2S L (1 − cos nπ ) nπ 2
"
2qk k k=1,3,5... X
# ,
(22c)
and the time-dependent coefficients h n j (t) and κn j (t), which are coupled with the other modes, can be derived as follows: $0,n t − sin($0,n t) cos($0,n t) , with n = j nπ sin ($0,n − $0, j )t sin ($0,n + $0, j )$0,n t h n j (t) = − , (n − j)π (n + j)π with n 6= j for v0 t < L ,
(18a–d)
Assuming that the suspended beam starts to vibrate from rest, the initial conditions are u(x, 0) = u(x, ˙ 0) = 0.
nπ 4 E I nπ 2 T + , L m L m 16π y0 E c Ac X qk π SL 1kn = n 2 = n2 L c k=1,3,5... k 4y0 L3 ωn2 =
Sn =
k=1
u(0, t) = u(L , t) = 0, E I u 00 (0, t) = E I u 00 (L , t) = 0.
shape, and the coefficients ωn , 1kn , and Sn are given as follows:
(23a)
(
1, with n = j for v0 t > L , 0, with n 6= j 2 sin ($0,n t) , with n = j nπ n − n cos($0,n t) cos($0, j t) − j sin($0,n t) sin($0, j t) κn j (t) = , (n 2 − j 2 )π/2 with n 6= j h n j (t) =
(23b)
for v0 t < L , 0, h i κn j (t) = n 1 − (−1)n+ j , (n 2 − j 2 )π/2
(23c) with n = j for v0 t ≥ L .
(23d)
with n 6= j
Here, $0,n = nπ v0 /L and $0, j = jπ v0 /L are the driving frequencies of the flowing mass to the beam. It should be noted that the expressions for v0 t < L in Eqs. (23a) and (23c) represent the transient state of the continuous mass starting to flow into the pipeline, but not yet filling in the entire pipeline on the bridge, and that the expressions for v0 t ≥ L in Eqs. (23b) and (23d) means the continuous mass has filled in the pipeline on the bridge for a long while, which is referred to as the steady state. In the modal equations in Eq. (21), the generalized force Fn (x, v0 t) associated with the continuously flowing mass on the nth assumed shape of the beam can be given as follows: 2w0 1 − cos($0,n t) for v0 t < L , Fn (x, v0 t) = (24) 1 − cos nπ for v0 t ≥ L . nπ The generalized force Fk ($n , v, t) associated with the kth moving load on the nth assumed shape of the beam is Fk ($n , v, t) =
2P sin $n (t − tk ) L L , × U (t − tk ) − U t − tk − v
(25)
where $n = nπ v/L = the driving frequency of a moving load to the beam. The preceding equation is equivalent to the
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Let us use Γn (t) to denote the sum of the nonlinearly coupled 2 κ (t) × q˙ , m $ 2 h (t) × q , terms: m 0 h n j (t) × q¨n , 2m 0 $0,n nj n n 0 0,n n j 1kn × qn , and Sn , that is,
following [17]: Fk ($n , v, t) =
2P L
sin $n (t − tk ) × U (t − tk ) L n+1 + (−1) sin $n t − tk − v L . × U t − tk − v
Γn (t) = m 0 h n j (t)q¨n + 2m 0 $0,n κn j (t)q˙n h i 2 + 1kn − m 0 $0,n h n j (t) qn + Sn . (25a)
To verify the equivalency of Eq. (25a) with (25), we can simply replace the term (−1)n+1 in Eq. (25a) by − cos(nπ ) and perform the following derivation: L L (−1)n+1 sin $n t − tk − × H t − tk − v v nπ v L = − cos(nπ ) sin $n (t − tk ) − L v L × H t − tk − v = − cos(nπ )[sin ($n (t − tk )) cos(nπ ) L − cos ($n (t − tk )) sin(nπ )] × H t − tk − v L (25b) = − sin ($n (t − tk )) × H t − tk − v By substituting the relation of Eq. (25b) into Eq. (25a), we can obtain the expression in Eq. (25) and thus the equivalency is proved. The modal equations given in Eq. (21) for the nth generalized coordinate qn of the suspended beam under the action of the flowing mass and moving loads are nonlinearly coupled, due to the presence of such terms as: m 0 h n j (t) × q¨n , 2 κ (t) × q˙ , m $ 2 h (t) × q , 1k × q , and S . In 2m 0 $0,n nj n n n n n 0 0,n n j order to solve for the nonlinearly coupled equations in Eq. (21), an incremental–iterative procedure will be employed, as will be presented in the following section. 5. Solution by an incremental–iterative procedure An observation of the nth generalized equation of motion in Eq. (21) indicates that the modal equations for all the generalized coordinates are coupled due to presence of the 2 κ (t) × q˙ , nonlinear terms, such as m 0 h n j (t) × q¨n , 2m 0 $0,n nj n 2 m 0 $0,n h n j (t) × qn , 1kn × qn , and Sn , as given in Eqs. (22) and (23). All these terms are time-dependent and have to be updated at each time step if an incremental solution procedure is to be used. Clearly, the computational effort required in solving a set of nonlinear coupled differential equations for the generalized coordinates such as the ones in Eq. (21) is tremendous. Moreover, it was well known that if the acceleration response, rather than the displacement response, of the bridge is of concern, as is the case considered herein, much more higher modes have to be included in the computation [14,18]. To tackle this problem, an incremental–iterative procedure based on the concept of predictor and corrector will be presented in the following for solving the set of nonlinearly coupled equations given in Eq. (21).
(26)
By removing the nonlinearly coupled force term Γn (t) to the right side of Eq. (21) and regarding −Γn (t) as the pseudo force acting on the nth generalized coordinate system, the modal equation in Eq. (21) can be rewritten as m q¨n + cq˙n + mωn2 qn = Fn (x, v0 t) +
N X
Fk ($n , v, t) − Γn (t).
(27)
k=1
The preceding equation indicates that the original nonlinearly coupled equations in Eq. (21) have been converted to a set of uncoupled differential equations P Nassociated with the nth generalized force Fn (x, v0 t) + k=1 Fk ($n , v, t) and the pseudo force −Γn (t). With the form given in Eq. (27), an incremental–iterative procedure based on the concept of predictor and corrector will be employed to solve each of the generalized equations. In this regard, Newmark’s β method [15] is first employed to discretize the generalized equation of motion in Eq. (27) into an equivalent incremental stiffness equation. Then, an incremental–iterative procedure is employed to solve the equivalent stiffness equation involving the pseudo force −Γn (t), for which three major phases can be identified, i.e., predictor, corrector and equilibrium checking [19]. As a matter of fact, the logistics of the incremental–iterative procedure presented in this section is a rigorous procedure modified from the path-tracing scheme in [20], which can be applied to solving a wide range of nonlinear coupled problems. 5.1. Incorporation of incremental–iterative procedure Consider the ordinary differential equations in Eq. (27). Let qn,t+1t denote the nth generalized displacement at time t + 1t, and 1qn the displacement increment of qn from time t to t +1t, i.e., qn,t+1t = qn,t + 1qn,t+1t . Then Eq. (27) can be rewritten for the current configuration at t + 1t as m q¨n,t+1t + cq˙n,t+1t + mωn2 (qn,t + 1qn ) = Pn,t+1t − Γn,t+1t ,
(28)
where Pn,t+1t denotes the external load with respect to the nth generalized coordinate at time t + 1t, Pn,t+1t = Fn (x, v0 (t + 1t)) +
N X
Fk ($n , v, t + 1t).
(29)
k=1
By Newmark’s β method with constant average accelera tion [15], the total responses qn,t+1t , q˙n,t+1t , q¨n,t+1t for the nth generalized coordinate can be given as qn,t+1t = qn,t + 1qn,t+1t , q˙n,t+1t = q˙n,t + a6 q¨n,t + a7 q¨n,t+1t , q¨n,t+1t = a0 1qn − a2 q˙n,t − a3 q¨n,t ,
(30a) (30b) (30c)
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i−1 i−1 Pn,t+1t and internal effective force σn,t+1t resulting from the last iteration, i.e.,
where the coefficients ai are as follows [14]: a0 = a1 = a2 = a3 = a4 = a5 =
1 , β · 1t 2 γ , β · 1t 1 , β · 1t 1 − 1, 2β γ − 1, β 1t γ −2 , 2 β
(31a) (31b) (31c) (31d) (31e) (31f)
a6 = (1 − γ )1t,
(31g)
a7 = γ · 1t,
(31h)
i−1 i−1 i 1Pn,t+1t = Pn,t+1t − σn,t+1t ,
(36a)
i−1 i−1 i−1 σn,t+1t = Rn,t+1t + Γn,t+1t ,
(36b)
i−1 and the resistant force Rn,t+1t can be modified from the one for the incremental step in Eq. (34) to the one for the iterative step as follows: i−1 i−1 2 i−1 mωn qn,t+1t − m a2 q˙n,t+1t + a3 q¨n,t+1t i−1 i−1 i−1 −c a q ˙ + a q ¨ for i = 1, (37) 4 5 n,t+1t n,t+1t Rn,t+1t = i−1 i−1 2 i−1 mωn qn,t+1t + m q¨n,t+1t + cq˙n,t+1t for i > 1.
The incremental–iterative equations as given in Eqs. (35)–(37) are subject to the following initial conditions for the 1st iterative step (i.e., for i = 1) at the nth incremental step [14]:
with β = 0.25 and γ = 0.5. By substitution of Eqs. (30a)–(30c), the differential equations as given in Eq. (28) for the generalized coordinates can be transformed into a set of equivalent stiffness equations for the incremental step from time t to t + 1t as follows [14]:
0 ` q˙n,t+1t = q˙n,t ,
K n,eff × 1qn,t+1t = 1Pn,t+1t ,
0 σn,t+1t
(32)
where K n,eff denotes the effective stiffness and 1Pn,t+1t the effective load increment at time t + 1t, of the nth generalized coordinate,
(38a)
` 0 = q¨n,t , q¨n,t+1t
(38b)
=
0 Rn,t+1t
(38c) 0 + Γn,t+1t
=
` Rn,t
` + Γn,t ,
(38d)
where ` means the last iteration of the previous incremental step at time t. 5.2. Predictor phase
K n,eff = (a0 + ωn2 )m + a1 c,
(33a)
1Pn,t+1t = Pn,t+1t − Γn,t+1t − Rn,t .
(33b)
In Eq. (33b), Rn,t denotes the resistant force at time t, i.e., Rn,t = mωn2 qn,t − m a2 q˙n,t + a3 q¨n,t − c a4 q˙n,t + a5 q¨n,t . (34) It should be noted that the displacement 1qn,t+1t solved from Eq. (32) is not exact due to the approximate nature of the force term Γn,t+1t involving the dynamic terms of the flowing mass, i.e., m 0 h n j (t + 1t) × q¨n,t+1t and 2 κ (t + 1t) × q˙ 2m 0 $0,n nj n,t+1t , and the nonlinear stiffness 2 h (t + 1t) × q terms, i.e., 1kn,t+1t qn,t+1t , m 0 $0,n nj n,t+1t and Sn,t+1t , which can only be estimated based on the information available at time t. Such a problem can be overcome through the iterative procedure to be described later on. For the purpose of iteration, the incremental stiffness equations as given in Eq. (32) for the incremental step from time t to t + 1t should be modified to include the iterative feature [19], namely, i i K n,eff × 1qn,t+1t = 1Pn,t+1t ,
0 ` qn,t+1t = qn,t ,
The predictor phase is concerned with solution of the structural response increments from the structural stiffness equation in Eq. (34). Since this phase affects only the speed of convergence, the tangent stiffness used need not be exact [19]. For this reason, we will treat K n,eff as constant for all the iterations within each incremental step. Consequently, for the first iteration, i.e., for i = 1, the displacement increment 1 1qn,t+1t for the nth generalized coordinate can be solved from Eq. (35) along with Eqs. (36a) and (38d) as: −1 1 ` ` = K n,eff (Pn,t+1t − Rn,t − Γn,t ). 1qn,t+1t
(39)
For the following iterations, i.e., for i ≥ 2, the displacement i can be solved from Eq. (35) as increment 1qn,t+1t −1 i i 1qn,t+1t = K n,eff × 1Pn,t+1t .
(40)
As the increments of all the generalized coordinates i 1qn,t+1t |n=1,2,3... have been made available in Eqs. (39) and (40), the total displacement, velocity and acceleration of the nth generalized coordinate for the ith iterative step at time t + 1t can be updated following Eq. (30) as follows:
(35)
i where for the first iteration (i = 1), 1Pn,t+1t ≡ 1Pn,t+1t represents the load increment and for the following iterations i (i ≥ 2), Pn,t+1t represents the unbalanced forces, which are equal to the difference between the total external load
i−1 i i qn,t+1t = qn,t+1t + 1qn,t+1t ,
(41a)
i−1 i−1 i i q˙n,t+1t = q˙n,t+1t + a6 q¨n,t+1t + a7 q¨n,t+1t ,
(41b)
i−1 i−1 i q¨n,t+1t = a0 1qni − a2 q˙n,t+1t − a3 q¨n,t+1t .
(41c)
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Table 1 Properties of the main beam and flowing mass L (m)
EI (kN m2 )
E c Ac (kN)
c (kN s/m/m)
m (t/m)
y0 (m)
m 0 (t/m)
v0 (m/s)
Ω1 (Hz)
Ω2 (Hz)
125
2.59 × 108
8 × 107
1.63
10.44
12.5
1.1
12
2.03
2.44
5.3. Corrector phase The corrector phase relates to recovery of the internal resistant forces from the displacement increments made available in the predictor phase. It is essential that a proper corrector be used, because this phase determines essentially the accuracy of the iterative solution [19]. In this study, the i effective internal force σn,t+1t of the nth generalized system at time t + 1t is recovered by substituting the total responses in i Eqs. (41a)–(41c) into the resistant forces Rn,t+1t in Eq. (37), i which is then substituted, along with the pseudo force Γn,t+1t , i into the effective internal forces σn,t+1t in Eq. (36b). A force recovery procedure such as this has an order of accuracy the same as that implied by Newmark’s β method. 5.4. Equilibrium-checking phase In the equilibrium-checking phase, the internal forces of the beam computed from the corrector phase are compared with the external forces, the difference being regarded as the unbalanced forces. By assigning the effective internal i (made available from the corrector phase) to forces σn,t+1t i−1 i−1 σn,t+1t , the unbalanced forces 1Pn,t+1t can be computed from Eq. (36a) for all the generalized coordinates. Whenever the root mean square sum of the generalized unbalanced P of the i−1 forces, i.e., [ n=1... (1Pn,t+1t )2 ]1/2 , is larger than a preset tolerance, say 10−3 , iteration for removing the unbalanced forces involving the two phases of predictor and corrector should be repeated. Once the condition of convergence is satisfied for an incremental step, we can proceed to the next incremental step with an update on the external load Pn,t+1t and repeat the iterative procedure for solution of Eq. (35). Evidently, the incremental–iterative procedure as presented above allows us to circumvent the complex problem of solving a set of simultaneous nonlinear equations with a large number of unknowns as originally given in Eq. (21). 6. Resonant condition caused by moving train loads When a bridge is subjected to a sequence of concentrated loads of equal intervals d moving at a constant speed v, the bridge may experience the action of a periodic force with a passage frequency of v/d. Once the passage frequency matches any of the natural frequencies of the bridge, resonance will be developed on the bridge [13,17]. Let Ω denote any of the circular frequencies of the bridge that has been excited. Under the resonant condition, i.e., when the resonant speed vres equals Ω d, the dynamic response of the bridge will build up continuously as there are more vehicular loads passing through the bridge [13,17]. The above resonance is excited
Fig. 3. Vibration modes of the suspended beam.
Table 2 Properties of the moving loads N
P (kN)
d (m)
v1,res (km/h)
v2,res (km/h)
16
350
27.5
200
241
by concentrated loads of regular intervals d moving over the bridge. In reality, sub-resonance of acceleration response may also be generated on the bridge as the moving loads pass through the bridge at the speed vsub,res = Ω d/J , where J represents the number of complete cycles of oscillation of the beam occurring during the passage of two adjacent loads, which has a duration of d/v. The sub-resonance is usually associated with the higher modes of the beam, which may result in the maximum acceleration on the beam depending on the higher modes that have been excited [17,21]. 7. Numerical examples As shown in Fig. 1, a single-span suspended beam installed with a water pipe is subjected to successive moving loads. The properties of the beam, cable, and flowing mass are listed in Table 1, in which Ω1 denotes the fundamental frequency of the first mode (anti-symmetric) and Ω2 the second mode (symmetric), as depicted in Fig. 3. Rayleigh damping is assumed for the box girder of the bridge with a damping ratio of 2.5% used for the first two modes. Due to the strengthening effect of the tension forces of the cables induced by the dead load, the natural frequency of the first anti-symmetric mode is smaller than that of the first symmetric mode. Table 2 shows the properties of the moving loads and the first and second resonant speeds of the suspended beam under the moving loads. 7.1. Acceleration response of the suspended beam due to moving loads It was well known that if the acceleration response, rather than the displacement response, of the bridge is of concern, as is the case considered herein, many more higher modes have to be included in the computation [14,18]. In order to verify that a sufficient number of modes of vibration has been used in the analysis, we first compute the acceleration response at the first quarter-point of the main beam under a series of concentrated
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639
Fig. 6. amax –v–x/L plot of the suspended beam due to moving loads. Fig. 4. Test of convergence.
Fig. 5. Time histories of acceleration of main beam.
loads moving at the first resonant speed using either 2, 10, or 20 modes. As can be seen from Fig. 4, the use of 20 modes is considered sufficient. For this reason, the same number of modes will be used in all the examples to follow. To illustrate the resonant phenomena of the suspended beam under a series of concentrated loads moving at the first and second resonant speeds, the time history responses of acceleration at the first quarter-point and mid-span, respectively, of the main beam have been plotted in Fig. 5. As it is expected for the resonance phenomena, both the acceleration responses continue to build up as there are more vehicular loads passing through the beam. But the resonant amplitude at the mid-span due to the vehicular loads traveling at the second resonant speed v2,res (=241 km/h) is significantly smaller than that at the first quarter-point for the vehicular loads traveling at the first resonant speed v1,res (=200 km/h). The reason can be attributed to the fact that as a row of moving loads, with an interval (d = 27.5 m) far smaller than the bridge span length (L = 125 m), pass through a long-span suspension bridge, the simultaneous presence of the vehicular loads on the bridge deck may produce a suppression action on the first symmetric mode (i.e., the second mode), making the mid-span acceleration of the bridge less severe compared with the other resonant case involving the anti-symmetric mode. Such a phenomenon can be also observed in the following example.
7.2. Maximum vertical acceleration response of the suspended beam Train-induced resonance can result in excessive vibrations on the track structure and may bring about derailment of the trains [21–23]. For this reason, the maximum acceleration amax of a suspended beam due to the moving loads and flowing mass will be investigated in this example. Three cases will be considered for the mass flowing through the pipe installed inside the box girder: (1) empty pipe; (2) transient flow — the flowing mass starts to flow into, but not fill in, the pipeline; and (3) steady flow — the continuous mass has filled in the pipeline and continue to flow with speed v0 for a duration of 2L/v0 before the first moving (train) load enters the suspended beam. For the case of an empty pipe, the maximum acceleration response along the beam length has been drawn against the speed as a amax –x/L–v plot in Fig. 6. The results indicate that the maximum acceleration depends on the vibration shape that has been excited, and that as the running speed of the moving loads coincides with any of the resonant or sub-resonant speeds associated with the vibration modes of the bridge, the number of peak accelerations along the beam axis is associated with the vibration mode excited. Physically speaking, the condition for resonance to occur is identical to that for simple beams [13]. Obviously, the amplification effect caused by resonance and sub-resonance on the dynamic response of the suspension bridge is harmful not only to the structural safety of the bridge, but also to the maneuverability of the vehicles moving over it. As mentioned in Section 6, the resonant speed of the first mode can be calculated as v1,res = Ω1 d (=200 km/h). Correspondingly, the maximum acceleration computed of the beam is amax = 0.35g. Since the first mode is anti-symmetric, two peaks can be observed for the maximum acceleration along the main beam. Next, let us consider the transient case when both the train loads and flowing mass start to move through the suspended beam simultaneously. The amax –x/L–v tri-phase plot has been given in Fig. 7. As can be seen, the maximum acceleration of the main beam at the first resonant speed (v1,res = 200 km/h) has been reduced by 9% for the peak response in comparison with that of Fig. 6. This result indicates that the flowing mass into the suspended beam can provide a suppression effect on
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Fig. 7. amax –v–x/L plot for the suspended beam with transient flow. Fig. 9. Suppression effect of the flowing mass on resonant response.
Fig. 8. amax –v–x/L plot for the suspended beam with steady flow.
the first resonant response of the main beam. But the higher modes can still be excited in the region with speeds higher than 200 km/h in the amax –x/L–v plot of Fig. 7. This can be attributed to the fact that the vehicular loads move so fast that the flowing mass has not yet filled up the pipe and thus is unable to exert its full effect of suppression on the beam. For the case of steady flow, i.e., with the pipe filled up by the flow for a duration of 2L/v0 before the moving loads enter the beam. The corresponding amax –x/L–v triphase plot in Fig. 8 reveals that most of the acceleration peaks associated with the vibration modes excited have been substantially reduced (e.g., by 14% for the peak response), including the higher modes in the region with speed higher than the first resonant speed v1,max = 200 km/h. Fig. 9 shows the maximum acceleration amax computed along the axis x/L of the suspended beam due to the vehicular loads traveling at the first resonant speed v1,res for the three cases of the flowing mass studied above. From this figure, the suppression effect of the flowing mass on the resonant response of acceleration of the suspended beam can be clearly appreciated. This figure also indicates that at the time when the first mode (anti-symmetric) is excited, the resonance associated with other higher modes will be excited as well, as can be observed from the secondary peaks appearing on the acceleration response curve. Finally, let us consider a special case when the suspended beam is subjected to the action of flowing mass alone. Fig. 10 shows the maximum acceleration of the transient/steady liquid flowing at various speeds. Compared with the amax –x/L plots shown in Fig. 9, the effect of dynamic amplification of the steady flowing mass alone through the suspended beam is significantly smaller than that of the moving loads traveling at
Fig. 10. Moving effect of the flowing mass alone at various speeds.
the first resonant speed v1,res . On the other hand, the amax –x/L plots in Fig. 10 reveal that: (1) the bridge responses induced by transient flows alone are generally larger than those of steady flows; (2) the higher the speed of the flowing mass is, the greater the maximum acceleration response of the suspended beam will be; (3) the maximum acceleration responses induced by all transient flows with speeds over 28 m/s are quite close, meaning that when the flowing mass enters the pipeline at a very high speed, the dynamic behavior of the beam on which the pipeline is installed is similar to that of a structure subjected to an impulsive force; and (4) higher modes of the suspended beam are excited even though the maximum acceleration response occurs at the mid-span. 7.3. Maximum tension increment of the cables In this paper, the increase in horizontal component of the cable forces caused by the live loads (i.e., moving loads and flowing mass) on the suspended beam has been taken into account in the dynamic analysis. In Fig. 11, the ratio 1Tmax /T of the tension increment to the maximum tension of the cables has been plotted against the speed of the moving loads for the three cases of the flowing mass. This figure indicates that the maximum increment of the cable tension occurs around the second resonant speed v2,res , except for the case of transient flowing mass. This implies that the tension increase in the
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Fig. 13. Moving effect of the flowing mass in steady state. Fig. 11. Maximum tension increment in the cable forces.
Fig. 12. Moving effect of the flowing mass in transient state.
cables caused by the first symmetric mode is larger than that by the anti-symmetric shape. Moreover, the speed of the response peak in the 1Tmax /T –v plot for the case of steady flowing mass is slightly smaller than that for the case of empty pipeline, due to the inertia effect of the flowing mass on the beam, which appears in the form of reduced frequencies. The second peak in the 1Tmax /T –v plot occurs at the speed of 121 km/h, which is equal to v2,res /2. This means the second (symmetric) mode of the suspended beam has been excited again by the vehicular loads traveling at the sub-resonant speed v2,res /2. Such a magnificent effect on the cable tension should be taken into account in the design of suspension bridges. 7.4. Moving effect of the flowing mass To illustrate the moving effect of the flowing mass on the maximum acceleration response of the suspended beam due to moving loads, let us consider an extreme case when both the moving loads and flowing mass have identical speeds. Figs. 12 and 13 show the amax –x/L–v tri-phase plots of the suspended beam installed with a water pipe conveying the transient flowing mass and steady flow, respectively. These figures indicate that once the moving speed of the flowing mass is over the first resonant speed (v1,res = 200 km/h), the acceleration response of the main beam will be totally amplified, and that the suppression effect of the steady flow on the maximum acceleration response of the suspended beam is reduced as well.
8. Concluding remarks The nonlinearly coupled equations of motion are first derived for a suspended beam carrying a pipeline subjected to the moving vehicular loads. By Galerkin’s method, these equations are transformed into a set of nonlinearly coupled differential equations in terms of the generalized coordinates. By treating the nonlinear coupled terms as pseudo forces, the coupled differential equations for all of the generalized systems are converted by Newmark’s β method to a set of equivalent stiffness equations of motion with generalized forces and pseudo forces. Finally, these equivalent equations are solved by a rigorous incremental–iterative procedure involving the three phases of predictor, corrector, and equilibrium-checking. The nonlinear vibration of a suspended beam under the action of successive moving loads and flowing mass has been studied in this paper. The numerical results indicate that once the exciting passage frequency (v/d) of the moving loads coincides with any of the natural frequencies of the suspended beam, resonance will be developed on the bridge. The maximum acceleration response induced by resonance need not occur at the midpoint of the bridge deck. The critical position for the maximum acceleration to occur on the beam depends on the vibration mode that has been excited. Both the anti-symmetric and symmetric modes of vibration of the suspended beam can be excited to resonance within the range of operation speeds of the moving trains. The flowing mass may produce a suppression effect on the resonant vibration of the suspended beam induced by the successive moving loads, if the flowing speed is less than the first resonant speed of the moving loads. In general, the installation of water pipelines inside the box girder of a suspension bridge is beneficial for reducing the bridge response when subjected to the moving loads. For the special case where the bridge is free of any moving loads and when the flowing mass enters the pipeline at a very high speed, the dynamic behavior of the beam on which the pipeline is installed is similar to that of a structure subjected to an impulsive force. Acknowledgements This study was sponsored in part by the National Science Council of the Republic of China via Grant No. NSC 95-2221E-032-053. The finical support is gratefully acknowledged.
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