Bending vibration of beam elements under moving loads with considering vehicle braking forces

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 mov- ing vehicle loads, considering vehicle braking forces and the coordinate axes on elements are described as in Fig. 1 and Fig. 2. x1 x2 L x w O 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. vi.(thi − ti) + [ ai.(t− thi) 2 + vi ] .(t− thi), 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 i th vehicle coordinate axes at the time being considered vi - the velocity of the i th axle before braking ai - the acceleration of the i th axle when braking ti - the time when the i th axle begins running on the beam elements thi - the time when the i th axle begins braking tei - the time when the i th 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 1 st axle m22 - the mass of the 2 nd axle k11, d11 - hardness and damping rate of the 1 st cart spring k21, d21 - hardness and damping rate of the 1 st tire k12, d12 - hardness and damping rate of cart spring 2 nd k22, d22 - hardness and damping rate of the 2 nd tire z11 - absolute displacement of the chassis at the 1 st axle z21- absolute displacement of the 1 st axle, absolute coordinates of the mass m21 z12- absolute displacement of the chassis at the 2 st axle z22 absolute displacement of the 2 t axle, absolute coordinates of the mass m22 y11 - relative displacement between the chassis and the 1 st axle y21 - relative displacement between the beam element and the 1 st axle y12 - relative displacement between the chassis and the 2 st axle y22 - relative displacement between the beam element and the 2 st 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.s¨, −m22.s¨, −m.s¨. These inertia forces which in turn produce longitudinal and vertical oscil- lations 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 −mu¨− F11 − F12 −mg = 0 F11 − F21 −m21z¨21 −m21g = 0 F12 − F22 −m22z¨22 −m22g = 0 T1 + T2 = − (m+m21 +m22) s¨ (3) Similarly, considering the torque balance of the whole system with the 3rd points: (mu¨+mg − P ) .a− (m.h+m21.h21+m22.h22)s¨−Jϕ¨+(m22.z¨22+m22.g+F22).(a+b) = 0 (4) in which: F11 = k11.y11 + d11.y˙11, F12 = k12.y12 + d12.y˙12 F21 = k21.y21 + d21.y˙21, F22 = k22.y22 + d22.y˙22 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 z¨11 + (a 2m+ J)d11z˙11 − (mba− J)d12z˙12 − (a 2m+ J)d11z˙21 + (mba− J)d12z˙22+ (a2m+ J)k11z11 − (mba− J)k12z12 − (a 2m+ J)k11z21+ +(mba− J)k12z22 − JP + Jmg+ (m.h+m21.h21 +m22.h22) .ma.s¨ =0 mJ z¨12 + (mba+ J)d11z˙11 + (b 2m+ J)d12z˙12 − (mba+ J)d11z˙21 − (b 2m+ J)d12z˙22+ + (mba+ J)k11z11 + (b 2m+ J)k12z12 − (mba+ J)k11z21− −(b2m+ J)k12z22 − JP + Jmg+ (m.h+m21.h21 +m22.h22) .mb.s¨ =0 m21z¨21 − d11z˙11 + (d11 + d21)z˙21 − k11z11 + (k11 + k21)z21 +m21.g − d21.w˙1 − k21.w1 =0 m22z¨22 − d12z˙12 + (d12 + d22)z˙22 − k12z12 + (k12 + k22)z22 +m22.g − d22.w˙2 − k22.w2 =0 s¨ = −g.τ (6) The constraint forces F21 and F22 are as follows: F21 = −m21z¨21 + (z˙11 − z˙21)d11 + (z11 − z21)k11 F22 = −m22z¨22 + (z˙12 − z˙22)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). [−m21z¨21 + (z˙11 − z˙21)d11 + (z11 − z21)k11] δ(x− x1) p2(x, z, t) = ξ2(t). [−m22z¨22 + (z˙12 − z˙22)d12 + (z12 − z22)k12] δ(x− x2) (7) in which: ξi(t) = { 1 when ti ≤ t ≤ ti +∆Ti 0 when t ti +∆Ti is a logic control signal function, δ(x− xi) is the Dirac delta function, ∆Ti is the period of time that the i th 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 dis- tributed load p(x, z, t) considering the effects of internal and external friction can be written as follows: EJd. ( ∂4w ∂x4 + θ. ∂5w ∂x4.∂t ) + ρFd. ∂2w ∂t2 + β. ∂w ∂t = p(x, z, t) = p1(x, z, t) + p2(x, z, t) (8) 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 vi- brations of beam elements under the influence of moving loads taking into account the 32 Nguyen Xuan Toan impact of vehicle braking forces: EJd. ( ∂4w ∂x4 + θ. ∂5w ∂x4.∂t ) + ρFd. ∂2w ∂t2 + β. ∂w ∂t = p(x, z, t) = p1(x, z, t) + p2(x, z, t)mJ z¨11+ +(a2m+ J)d11z˙11 − (mba− J)d12z˙12 − (a 2m+ J)d11z˙21 + (mba− J)d12z˙22+ + (a2m+ J)k11z11 − (mba− J)k12z12 − (a 2m+ J)k11z21 + (mba− J)k12z22− - JP+ Jmg+ (m.h+m21.h21 +m22.h22) .ma.s¨ =0 mJ z¨12 + (mba+ J)d11z˙11 + (b 2m+ J)d12z˙12 − (mba+ J)d11z˙21 − (b 2m+ J)d12z˙22+ + (mba+ J)k11z11 + (b 2m+ J)k12z12 − (mba+ J)k11z21 − (b 2m+ J)k12z22− - JP+ Jmg+ (m.h+m21.h21 +m22.h22) .mb.s¨ =0 m21z¨21 − d11z˙11 + (d11 + d21)z˙21 − k11z11 + (k11 + k21)z21 +m21.g − d21.w˙1 − k21.w1 =0 m22z¨22 − d12z˙12 + (d12 + d22)z˙22 − k12z12 + (k12 + k22)z22 +m22.g − d22.w˙2 − k22.w2 =0 s¨ = −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]: w = [ N1 N2 N3 N4 ] .   w1 ϕ1 w2 ϕ2   (10) N1 = 1 L3 (L3 − 3Lx2 + 2x3), N2 = 1 L2 (L2x− 2Lx2 + x3) N3 = 1 L3 (3Lx2 − 2x3), N4 = 1 L2 (x3 − Lx2) (11) 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: L∫ 0   N1 N2 N3 N4   .EJd. ∂4 ∂x4 [ N1 N2 N3 N4 ]   w1 ϕ1 w2 ϕ2   .dx = Kww.   w1 ϕ1 w2 ϕ2   (12) L∫ 0   N1 N2 N3 N4   θ.EJd. ∂5 ∂x4∂t [ N1 N2 N3 N4 ]   w1 ϕ1 w2 ϕ2   .dx = θ.Kww. ∂ ∂t   w1 ϕ1 w2 ϕ2   (13) Bending vibration of beam elements under moving loads with considering vehicle braking forces 33 in which: Kww = EJd L3   12 6L −12 6L 6L 4L2 −6L 2L2 −12 −6L 12 −6L 6L 2L2 −6L 4L2   ; (14) L∫ 0   N1 N2 N3 N4   .ρFd. ∂2 ∂t2 [ N1 N2 N3 N4 ]   w1 ϕ1 w2 ϕ2   .dx = Mww ∂2 ∂t2   w1 ϕ1 w2 ϕ2   ; (15) L∫ 0   N1 N2 N3 N4   .β. ∂ ∂t [ N1 N2 N3 N4 ]   w1 ϕ1 w2 ϕ2   .dx = β. Mww ρFd ∂ ∂t   w1 ϕ1 w2 ϕ2   ; (16) in which: Mww = ρFdL 420   156 22L 54 −13L 22L 4L2 13L −3L2 54 13L 156 −22L −13L −3L2 −22L 4L2   ; Cww = β.MwwρFd + θ.Kww; (17) L∫ 0   N1 N2 N3 N4   .p(x, z, t).dx= [−m21.z¨21 + (z˙11 − z˙21)d11 + (z11 − z21)k11] .P1+ + [−m22.z¨22 + (z˙12 − z˙22)d12 + (z12 − z22)k12] .P2 (18) in which: Pi =   p1i p2i p3i p4i   = ξi(t) L3 .   (L+ 2xi)(L− xi) 2 L.xi(L− xi) 2 x2i (3L− 2xi) −L.x2i (L− xi)   . (19) Combining the results (12) - (19) with (9) and rewriting in matrix form, we obtain: Me.q¨ +Ce.q˙ +Ke.q = fe (20) q¨, q˙, q, fe - the mixed acceleration vector, mixed velocity vector, mixed displacement vector, mixed forces vector, respectively: {q¨} =   w¨ z¨1 z¨2   ; {q˙} =   w˙ z˙1 z˙2   ; {q} =   w z1 z2   ; {fe} =   fw fz1 fz2   Me, Ce, Ke - the mixed quantity matrix, mixed damper matrix, mixed stiffness matrix, respectively: Me =   Mww 0 Mwz20 Mz1z1 0 0 0 Mz2z2   ;Ce =   Cww Cwz1 Cwz20 Cz1z1 Cz1z2 Cz2w Cz2z1 Cz2z2   ; 34 Nguyen Xuan Toan Ke =   Kww Kwz1 Kwz20 Kz1z1 Kz1z2 Kz2w Kz2z1 Kz2z2   ; Mwz2 = P. [ m21 0 0 m22 ] ;Mz1z1 = P. [ mJ 0 0 mJ ] ;Mz2z2 = [ m21 0 0 m22 ] ; P = [ P1 P2 ] , P1, P2 are determined by (19); Cwz1 = [ −d11 0 0 −d12 ] ;Cwz2 = [ d11 0 0 d12 ] ; Cz1z1 = [ (a2m+ J)d11 −(mba− J)d12 (mba+ J)d11 (b 2m+ J)d12 ] ;Cz1z2 = [ −(a2m+ J)d11 (mba− J)d12 −(mba+ J)d11 −(b 2m+ J)d12 ] ; CTz2w = Na [ −d21 0 0 −d22 ] ;Cz2z1 = [ −d11 0 0 −d12 ] ;Cz2z2 = [ d11 + d12 0 0 d12 + d22 ] ; Kwz1 = [ −k11 0 0 −k12 ] ; Kwz2 = [ k11 0 0 k12 ] ; Kz1z1 = [ (a2m+ J)k11 −(mba− J)k12 (mba+ J)k11 (b 2m+ J)k12 ] ; Kz1z2 = [ −(a2m+ J)k11 (mba− J)k12 −(mba+ J)k11 −(b 2m+ J)k12 ] ; KTz2w = Na [ −k21 0 0 −k22 ] + N˙a . [ −d21 0 0 −d22 ] ; Kz2z1 = [ −k11 0 0 −k12 ] ;Kz2z2 = [ k11 + k21 0 0 k12 + k22 ] ; Na =   N11 N12 N21 N22 N31 N32 N41 N42   ; in which :   N1i = 1 L3 .(L3 − 3.L.x2i + 2.x 3 i ); N2i = 1 L2 .(L2.x−i 2.L.x 2 i + x 3 i ); N3i = 1 L3 .(3.L.x2i − 2.x 3 i ); N4i = 1 L2 .(x3i − L.x 2 i ); xi is determined by the formula (1) fe =   0 fz1 fz2   ; fz1 = { JP − Jmg − (mh+m21h21 +m22h22)ma.s¨ JP − Jmg − (mh+m21h21 +m22h22)mb.s¨ } ; fz2 = { −m21g −m22g } . 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.Q¨+ C.Q˙+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, Q¨, Q˙, 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: Providingresults QQQ ,, &&& - Node data, join data - Beam element data - Load data, moving vehicle data i=1 Establishing matrix Mww , Cww , Kww , fww for beam element ith - Establishinh axis moving matrix. - Moving axis, locating and arranging it in the overall matrices: M, C, K, F. i=i+1 i=1 Assinging the boundary conditions of the problem 0,0,0 === QQt &&& Establishing matrix M z1z1, Mz2z2 , Mwz2 , Cz1z1, Cz2z2 , Cz1z2, Cz2z1, Cz2w , Kz1z1, Kz2z2 , Kz1z2, Kz2z1, Kz2w , fwt , fz1t , fz2t . i ³ SPTL. + i=i+1 Determining coefficients: K1,K2,K3,K4,K5 ; Calculating QQQ ,, &&& t ³ Th + t=t+h End Start i³ SPTD + - Establishinh axis moving matrix - Moving axis, locating and arranging it in the overall matrices: M, C, K, F. 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 m 4, 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 10−3s). IFA-W50 trucks with following parameters: m = 9.838 T, m21=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-Kutta- Mersion 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 Bending vibration of beam elements under moving loads with considering vehicle braking forces 37 Fig. 6. At node 3, when not brake, µ = 1.039 Fig. 7. At node 3, when not brake, µ = 1.097 Results of displacement calculation with velocity 10.8 km/h are given in Figs. 8 - 11 Fig. 8. At node 3, when not brake, µ = 1.663 38 Nguyen Xuan Toan Fig. 9. At node 3, when not brake, µ = 1.975 Fig. 10. At node 3, when not brake, µ = 1.229 Fig. 11. At node 3, when not brake, µ = 1.358 Bending vibration of beam elements under moving loads with considering vehicle braking forces 39 In Figs 4 - 11, the follwings are denoted: — - the static displacement of girder, — - the dynamic displacement of girder, µ - the dynamic factor. Calculation results show that the dynamic coefficient of deflection considering vehi- cle braking forces is bigger than that obtained without considering vehicle braking forces. Dynamic coefficient is significant and need to be considered in design calculations. 7. CONCLUSIONS This paper introduces research results of building differential equations of bending vibrations of beam elements bearing moving loads considering vehicle braking forces and the way to build the combined stiffness matrices, combined mass matrices, mixed block matrices, equivalent force vector of beam elements bearing moving loads considering brak- ing forces in accordance with a three-mass model. 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