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Vietnam Journal of Mechanics, VAST, Vol. 33, No. 1 (2011), pp. 27 – 40 BENDING VIBRATION OF BEAM ELEMENTS UNDER MOVING LOADS WITH CONSIDERING VEHICLE BRAKING FORCES Nguyen Xuan Toan Da Nang University of Technology Abstract. The study of fluctuations of structures in general and bridge structures in particular under the influence of moving loads considering the impact of vehicle braking forces draws the attention of many scientists. However, due to the complexity of this problem a static method has been so far applied for approximate calculation in bridge design standards. In this article the author introduces the equation of bending vibrations of beam elements according to the model of dynamic interaction between beam elements and moving vehicle loads considering vehicle braking forces. Key words: bending vibration, braking force, moving load. 1. INTRODUCTION The impact of vehicle braking forces on the bridge is huge and must be considered in design. In the bridge design process of many countries it is imperative to audit vehicle braking force bearing structures. Due to the complexity of this problem in the current processes only vehicle braking force bearing structures have been audited in accordance with a static method based on standard conventional loads. However, the neglecting of dynamic effects of vehicle - bridge interaction may result in large errors [1, 2]. Today modern bridges tend to use high - strength materials, their structure is very slender and their hardness is small; therefore, they are very sensitive to cyclic impact loads, especially, large ones of vehicles moving at high-speeds. As a result, the study of the vibrations of bridge structures enduring/bearing moving loads has been interested by many scientists [3] - [13], [15] - [18]. In reality, the fact that vehicles brake on bridges causes very large vibrations, so the study of bridge structure vibrations enduring moving loads considering the impact of vehicle braking forces is of great importance and urgency. In this paper, the author introduces a model of dynamic interaction between beam elements and moving vehicle loads, namely, a three-mass model considering vehicle braking forces. A corresponding system of differential equations of bending vibrations of the beam element considering vehicle braking forces is obtained. 28 Nguyen Xuan Toan 2. COMPUTATIONAL MODEL AND ASSUMPTIONS The three - mass model of dynamic interaction between the beam elements and moving vehicle loads, considering vehicle braking forces and the coordinate axes on elements are described as in Fig. 1 and Fig. 2. w x O x1 x2 L Fig. 1. Diagram of 2- axes vehicles on the beam element Fig. 2. Interaction model between two axes and beam elements One has xi =  vi .(t − ti ), when ti ≤ t ≤ thi .   ai .(t − thi ) vi .(thi − ti ) + + vi .(t − thi ), 2 when thi < t ≤ tei . (1) It is denoted (see Fig. 1 and Fig. 2): L - the length of the beam elements being considered Bending vibration of beam elements under moving loads with considering vehicle braking forces 29 xi - the ith vehicle coordinate axes at the time being considered vi - the velocity of the ith axle before braking ai - the acceleration of the ith axle when braking ti - the time when the ith axle begins running on the beam elements thi - the time when the ith axle begins braking tei - the time when the ith axle was at the end of the element t - the time being considered P = G. sin(Ω.t + α) the conditioning stimulation force caused by the eccentric mass of the engine m - the mass of the entire vehicle and goods, excluding the mass of the axle m21 - the mass of the 1st axle m22 - the mass of the 2nd axle k11 , d11 - hardness and damping rate of the 1st cart spring k21 , d21 - hardness and damping rate of the 1st tire k12 , d12 - hardness and damping rate of cart spring 2nd k22 , d22 - hardness and damping rate of the 2nd tire z11 - absolute displacement of the chassis at the 1st axle z21 - absolute displacement of the 1st axle, absolute coordinates of the mass m21 z12 - absolute displacement of the chassis at the 2st axle z22 absolute displacement of the 2t axle, absolute coordinates of the mass m22 y11 - relative displacement between the chassis and the 1st axle y21 - relative displacement between the beam element and the 1st axle y12 - relative displacement between the chassis and the 2st axle y22 - relative displacement between the beam element and the 2st axle u - absolute displacement of the chassis at heart block (absolute coordinate of the mass m) ϕ - the rotation angle of the vehicle tank s - the stretch of road that vehicles move on a, b - the distance from the center of mass O to the 1st and the 2st axles T1 , T2 - the friction forces between tyre and bridge surface when braking Inertial forces, dray forces, elastic forces, exciting forces and braking forces affecting the system as shown in Fig. 2 have conventional dimensions and sign in accordance with the system of corresponding coordinate axes. The following assumptions are adopted: The mass of the entire vehicle and goods, excluding the mass of the axle is transferred to the center of mass system. It is equivalent to the mass m and the rotational inertia J. The mass of the 1st and 2nd axles is m21 and m22, which are regarded as a point with concentrated mass at the center of the corresponding axle. The chassis is hypothesised to be absolutely hard and undistorted when moving. The vertical displacements of mass m, m21 , m22 are smaller than the height from their center to the centre of beam. Beam materials work in the linear elastic stage. 30 Nguyen Xuan Toan The bridge surface is flat, and has the friction coefficient homogeneous over the entire bridge surface. Brake forces of axles of vehicle are assumed to occur simultaneously. The direction of the forces between bridge surface and tires are assumed to be in the opposite direction of movement of vehicle as shown in Fig. 2. According to this assumption, the brake forces between bridge surface and tires, called T1 , T2 , make the vehicle decelerates uniformly and cause inertia forces −m21 .s̈, −m22 .s̈, −m.s̈. These inertia forces which in turn produce longitudinal and vertical oscillations of the whole system. The most dangerous case is when an emergency brake is applied. In this case, the forces T1 , T2 are assumed to be directly proportional to loaded weight of vehicle: T1 + T2 = (m + m21 + m22 ).g.τ (2) τ - the friction factor between bridge surface and tires g - the acceleration of gravity. 3. DIFFENTIAL EQUATIONS OF MOVING LOADS Based on the calculation model and assumptions in Section 1, we consider the system of mass m, m21 , m22 , viscous drag, elastic forces, inertial forces, stimulation forces, bridge surface constraint forces, braking power, which are converted to frictional forces against the bridge surface as shown in Fig. 2. Applying the principle of d’Alembert, considering the balance of each mass m, m21 , m22 according to the vertical axis and the whole system according to the horizontal axis, we have: P − mü − F11 − F12 − mg = 0 F11 − F21 − m21 z̈21 − m21 g = 0 F12 − F22 − m22 z̈22 − m22 g = 0 T1 + T2 = − (m + m21 + m22 ) s̈ (3) Similarly, considering the torque balance of the whole system with the 3rd points: (mü + mg − P ) .a − (m.h + m21.h21 + m22 .h22 )s̈ − J ϕ̈ + (m22 .z̈22 + m22 .g + F22 ).(a + b) = 0 (4) in which: F11 = k11 .y11 + d11 .ẏ11 , F12 = k12 .y12 + d12 .ẏ12 F21 = k21 .y21 + d21 .ẏ21 , F22 = k22 .y22 + d22 .ẏ22 ϕ = (z11 − z12 ) / (a + b) , u = (b.z11 + a.z12 ) / (a + b) z11 = y11 + y21 + w1 , z12 = y12 + y22 + w2 z21 = y21 + w1 , z22 = y22 + w2 (5) Combining (2) with (3), (4) and (5) then having them transformed, we obtain a set of equations: Bending vibration of beam elements under moving loads with considering vehicle braking forces 31 mJ z̈11 + (a2 m + J)d11 ż11 − (mba − J)d12 ż12 − (a2 m + J)d11 ż21 + (mba − J)d12 ż22 + (a2 m + J)k11 z11 − (mba − J)k12 z12 − (a2 m + J)k11 z21 + +(mba − J)k12 z22 − JP + Jmg + (m.h + m21 .h21 + m22 .h22 ) .ma.s̈ =0 mJ z̈12 + (mba + J)d11 ż11 + (b2 m + J)d12 ż12 − (mba + J)d11 ż21 − (b2 m + J)d12 ż22 + + (mba + J)k11 z11 + (b2 m + J)k12 z12 − (mba + J)k11 z21 − −(b2 m + J)k12 z22 − JP + Jmg + (m.h + m21.h21 + m22 .h22 ) .mb.s̈ =0 m21 z̈21 − d11 ż11 + (d11 + d21 )ż21 − k11 z11 + (k11 + k21 )z21 + m21 .g − d21 .ẇ1 − k21 .w1 =0 m22 z̈22 − d12 ż12 + (d12 + d22 )ż22 − k12 z12 + (k12 + k22 )z22 + m22 .g − d22 .ẇ2 − k22 .w2 =0 s̈ = −g.τ (6) The constraint forces F21 and F22 are as follows: F21 = −m21 z̈21 + (ż11 − ż21 )d11 + (z11 − z21 )k11 F22 = −m22 z̈22 + (ż12 − ż22 )d12 + (z12 − z22 )k12 Having them rewritten in the form of distribution and adding a logic control signal function, we have: p1 (x, z, t) = ξ1 (t). [−m21 z̈21 + (ż11 − ż21 )d11 + (z11 − z21 )k11 ] δ(x − x1 ) (7) p2 (x, z, t) = ξ2 (t). [−m22 z̈22 + (ż12 − ż22 )d12 + (z12 − z22 )k12 ] δ(x − x2 )  1 when ti ≤ t ≤ ti + ∆Ti is a logic control signal function, in which: ξi (t) = 0 when t < ti and t > ti + ∆Ti δ(x − xi ) is the Dirac delta function, ∆Ti is the period of time that the ith axle runs on the beam elements being considered. 4. EQUATIONS OF BENDING VIBRATION OF BEAM ELEMENTS UNDER MOVING LOADS According to [16] the equation of bending vibrations of beam elements under distributed load p(x, z, t) considering the effects of internal and external friction can be written as follows:  4  ∂ w ∂ 5w ∂ 2w ∂w EJd. + θ. + ρF . + β. = p(x, z, t) = p1 (x, z, t) + p2 (x, z, t) (8) d 4 4 2 ∂x ∂x .∂t ∂t ∂t in which p1 (x, z, t) and p2 (x, z, t) are determined by the formula (7), EJd - the bending stiffness of beam elements, ρFd - the mass of the beam element on a length unit, θ and β - the coefficient of internal friction and coefficient of external friction, respectively. Aggregating (6) with (8) we have systems of differential equations of bending vibrations of beam elements under the influence of moving loads taking into account the 32 Nguyen Xuan Toan impact of vehicle braking forces:  4  ∂ w ∂ 5w ∂ 2w ∂w EJd. + θ. 4 + ρFd . 2 + β. = p(x, z, t) = p1 (x, z, t) + p2 (x, z, t)mJ z̈11 + 4 ∂x ∂x .∂t ∂t ∂t +(a2 m + J)d11 ż11 − (mba − J)d12 ż12 − (a2 m + J)d11 ż21 + (mba − J)d12 ż22 + + (a2 m + J)k11 z11 − (mba − J)k12 z12 − (a2 m + J)k11 z21 + (mba − J)k12 z22 − - JP + Jmg + (m.h + m21 .h21 + m22 .h22 ) .ma.s̈ =0 mJ z̈12 + (mba + J)d11 ż11 + (b2 m + J)d12 ż12 − (mba + J)d11 ż21 − (b2 m + J)d12 ż22 + + (mba + J)k11 z11 + (b2 m + J)k12 z12 − (mba + J)k11 z21 − (b2 m + J)k12 z22 − - JP + Jmg + (m.h + m21.h21 + m22 .h22 ) .mb.s̈ =0 m21 z̈21 − d11 ż11 + (d11 + d21 )ż21 − k11 z11 + (k11 + k21 )z21 + m21 .g − d21 .ẇ1 − k21 .w1 =0 m22 z̈22 − d12 ż12 + (d12 + d22 )ż22 − k12 z12 + (k12 + k22 )z22 + m22 .g − d22 .ẇ2 − k22 .w2 =0 s̈ = −g.τ (9) 5. TRANSFORMATION OF THE EQUATION OF BENDING VIBRATIONS OF BEAM ELEMENT TO THE MATRIX FORM The bending vibration we can be approximately presented in the form [16, 19]:   w1        ϕ1  (10) w = N1 N2 N3 N4 . w2       ϕ2 1 1 (L3 − 3Lx2 + 2x3 ), N2 = 2 (L2 x − 2Lx2 + x3 ) 3 L L (11) 1 1 2 3 3 2 N3 = 3 (3Lx − 2x ), N4 = 2 (x − Lx ) L L where: w1 , ϕ1 - the deflection and rotation angle of the left end of beam element, w2 , ϕ2 - the deflection and rotation angle of the right end of beam element. Substituting (10), (11) into (9) and applying the Galerkin method in combination with Green theory, we integrate each term by parts and obtain:       N1  w1  w1          ZL        ϕ1  ∂4  N2 ϕ1 .EJd. 4 N1 N2 N3 N4 .dx = Kww . (12) N3  w2  w2     ∂x           0  N  ϕ2 ϕ2 4 N1 =    N1      w1 ZL      ϕ1 ∂5  N2 N1 N2 N3 N4 θ.EJd . 4 w2   ∂x ∂t  N3       0 N4 ϕ2       w1     ∂  ϕ1  .dx = θ.Kww . (13)  ∂t    w2      ϕ2 Bending vibration of beam elements under moving loads with considering vehicle braking forces in which: Kww  12 EJd  6L = 3  L  −12 6L   N1    ZL    ∂2  N2 .ρFd . 2 N1 N2 N3  N3   ∂t  0  N  4    N1      ZL     ∂  N2 N1 N2 N3 N4 .β. N3    ∂t       0 N4 in which:  −12 6L −6L 2L2  ; 12 −6L  −6L 4L2    w1  w1       ∂ 2  ϕ1  ϕ1 .dx = Mww 2 ; w2  w2   ∂t       ϕ2 ϕ2    w1  w1       Mww ∂  ϕ1  ϕ1 .dx = β. ; w2  w2  ρFd ∂t        ϕ2 ϕ2 6L 4L2 −6L 2L2     N4      156 22L 54 −13L ρFd L  4L2 13L −3L2   22L  ; Cww = β. Mww + θ.Kww ; Mww =   54 13L 156 −22L 420 ρFd 2 2 −13L −3L −22L 4L   N1    ZL    N2 [−m21 .z̈21 + (ż11 − ż21 )d11 + (z11 − z21 )k11] .P1 + .p(x, z, t).dx = N + [−m22 .z̈22 + (ż12 − ż22 )d12 + (z12 − z22 )k12] .P2   3   0  N  4 in which:    p   (L + 2xi)(L − xi )2  1i      ξ (t)  L.xi (L − xi )2 p2i i Pi = = 3 . x2 (3L − 2xi ) p3i   L      i 2   −L.xi (L − xi ) p4i Combining the results (12) - (19) with (9) and rewriting in Me .q̈ + Ce .q̇ + Ke .q = fe     . 33 (14) (15) (16) (17) (18) (19)    matrix form, we obtain: (20) q̈, q̇, q, fe - the mixed acceleration vector, mixed velocity vector, mixed displacement vector, mixed forces vector, respectively:          ẅ   ẇ   w   fw  ; {q} = ; {fe } = {q̈} = z̈1 ; {q̇} = ż1 z1 fz        1  z̈2 ż2 z2 fz2 Me , Ce , Ke - the mixed quantity matrix, mixed damper matrix, mixed stiffness matrix, respectively:     Mww 0 Mwz2 Cww Cwz1 Cwz2  ;Ce =  0 Me =  0 Mz1z1 0 Cz1z1 Cz1z2  ; 0 0 Mz2z2 Cz2 w Cz2z1 Cz2z2 34 Nguyen Xuan Toan   Kww Kwz1 Ke =  0 Kz1z1 Kz2w Kz2z1   mJ 0 ; Mz1 z1 = P. 0 mJ  Kwz2 Kz1z2  ; Kz2z2    m21 0 ; Mz2 z2 = ; 0 m22 m21 0 Mwz2 = P. 0 m22   P = P1 P2 , P1 , P2 are determined by (19);     −d11 0 d11 0 Cwz1 = ; Cwz2 = ; 0 −d12 0 d12  2    (a m + J)d11 −(mba − J)d12 −(a2 m + J)d11 (mba − J)d12 Cz1 z1 = ; C = ; z1 z2 (mba + J)d11 (b2 m + J)d12 −(mba + J)d11 −(b2 m + J)d12       −d21 0 −d11 0 d11 + d12 0 T Cz2 w = Na ; Cz2 z1 = ; Cz2 z2 = ; 0 −d22 0 −d12 0 d12 + d22     −k11 0 k11 0 Kwz1 = ; Kwz2 = ; 0 −k12 0 k12   2 (a m + J)k11 −(mba − J)k12 ; Kz1 z1 = (mba + J)k11 (b2 m + J)k12   −(a2 m + J)k11 (mba − J)k12 Kz1 z2 = ; −(mba + J)k11 −(b2 m + J)k12     −k21 0 −d21 0 KzT2 w = Na + Ṅa . ; 0 −k22 0 −d22     −k11 0 k11 + k21 0 Kz2 z1 = ; Kz2z2 = ; 0 −k12 0 k12 + k22    N1i = L13 .(L3 − 3.L.x2i + 2.x3i ); N11 N12    2 3  N21 N22  N2i = L12 .(L2 .x− i 2.L.xi + xi );  ; in which : Na =   N31 N32  N = 1 .(3.L.x2 − 2.x3 );    3i L13 3 i 2 i N41 N42 N4i = L2 .(xi − L.xi ); xi is determined by the formula (1)        0  JP − Jmg − (mh + m21 h21 + m22 h22 )ma.s̈ −m21 g fz1 fe = ; f = ; fz2 = . JP − Jmg − (mh + m21 h21 + m22 h22 )mb.s̈ −m22 g   z1 fz2 6. APPLICATION TO VIBRATION ANALYSIS OF BRIDGE STRUCTURE UNDER MOVING LOADS By digitizing bridge structures into basic elements, combining the research results above with finite element method and utilizing algorithms generally used in finite element method one can construct vibration differential equations for the whole system [19]. M.Q̈ + C.Q̇ + K.Q = F (21) Bending vibration of beam elements under moving loads with considering vehicle braking forces 35 M, C, K, which is the mixed quantity matrix, the mixed damper matrix, and the mixed stiffness matrix of the total system, Q̈, Q̇, Q, F , which is the mixed acceleration vector, the mixed velocity vector, the mixed displacement vector, and the mixed forces vector of the total system. After inserting corresponding boundary conditions and initial conditions to (21), we can solve the set of equations (21) by the Runge- Kutta-Merson method on the computer. An application in analyzing vibration of a three-span continuous steel girder bridge structure (40m+60m+40m) can be condidered as follows: Start - Node data, join data - Beam element data - Load data, moving vehicle data … i=1 Establishing matrix Mww , C ww , Kww , fww for beam element ith i=i+1 - Establishinh axis moving matrix. - Moving axis, locating and arranging it in the overall matrices: M, C, K, F. i³ SPTD + Assinging the boundary conditions of the problem t = 0,& Q& = 0,Q& = 0 i=1 Establishing matrix M z1z1, Mz2z2 , Mwz2 , Cz1z1 , Cz2z2 , Cz1z2, C z2z1, Cz2w , Kz1z1 , Kz2z2 , Kz1z2, Kz2z1, Kz2w , f wt , f z1t , fz2t . i=i+1 - Establishinh axis moving matrix - Moving axis, locating and arranging it in the overall matrices: M, C, K, F. t=t+h i³ SPTL. + Determining coefficients: K1,K2 ,K3 ,K4,K5 ; Calculating & Q& , Q& , Q t ³ Th + Providing results & Q& , Q& , Q End Fig. 3. The general algorithm in analyzing vibration of girder bridge structure 36 Nguyen Xuan Toan Elastic module of steel E = 2.1x107 T/m2 , moment of inertia of area of steel girder Jd = 0.0261 m4 , and mass of the beam element on a length unit ρFd =1.237 T/m, friction factors θ = 0.027; β = 0.01; τ = 0.5, acceleration of gravity g = 9.81 m/s2 , and SPTD=14; SPTL=14, where SPTD is the number of beam elements of the whole structure, SPTL is the number of lane elements on the carriage-way, Th is the time period of analysis , h is time step (about 10−3 s). IFA-W50 trucks with following parameters: m = 9.838 T, m 21=0.107 T, m22 = 0.055T, P = 0, a = 1.035 m, b = 2.415 m, h = 1.5 m, h21 = 0.5 m, h22 = 0.5 m, k11 =200 T/m, k12 = 30.2 T/m, k21 = 260 T/m, k22 = 120 T/m, d11 = 0.7344 Ts/m, d12 = 0.3672 Ts/m, d21 = 0.8 Ts/m, d22 = 0.4 Ts/m. The general algorithm in analyzing vibration of girder bridge structure is shown in Fig. 3, where K1 , K2 , K3 , K4 , K5 are coefficients calculated by using the Runge-KuttaMersion method. Results of displacement calculation with velocity 3.6 km/h are given in Figs. 4 - 7 Fig. 4. At node 3, when not brake, µ = 1.383 Fig. 5. At node 3, when not brake, µ = 1.507