Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 55, 1, pp. 281-291, Warsaw 2017 DOI: 10.15632/jtam-pl.55.1.281 DYNAMIC RESPONSE OF LADDER TRACK RESTED ON STOCHASTIC FOUNDATION UNDER OSCILLATING MOVING LOAD Saeed Mohammadzadeh, Mohammad Mehrali School of Railway engineering, Iran University of Science and Technology, Tehran, Iran e-mail: mohammadz@iust.ac.ir; m mehrali@rail.iust.ac.ir The ladder track is a new type of an elastically supported vibration-reduction track system that has been applied to several urban railways.This paper is devoted to the investigationof dynamicbehavior of a ladder trackunder anoscillatingmoving load.The track is represented by an infinite Timoshenko beam supported by a random elastic foundation. In this regard, equations of motion for the ladder track are developed in a moving frame of reference. In continuation, by employing perturbation theory and contour integration, the response of the ladder track is obtained analytically and its results are verified using the stochastic finite element method. Finally, using the verified model, a series of sensitivity analyses are accomplished on effecting parameters including velocity and load frequency. Keywords: ladder track, moving load, stochastic stiffness, perturbation theory 1. Introduction In the 1940s to 1960s, weakness caused by resistance to lateral movement of cross-ties prompted studies on longitudinal sleepers laid in parallel pairs under the rails. The aim was to produce a railway track requiringaminimumofmaintenance. Ladder sleeperswere subsequentlydeveloped having parallel longitudinal concrete beamsheld together by transverse steel pipes (Wakui et al., 1997). Ladder sleepers provide continuous support to the rails assuring train safety, decreasing maintenance and promising an increase in railway efficiency. In recent years, a floating ladder track (Fig. 1a) has been developed to decrease vibration in a structure and withstand noise. Younesian et al. (2006) studied the dynamic performance of a ballasted ladder track. The rail and ladder units were simulated using a Timoshenko beam and the governing equations were solved using the Galerkin method. Figure 1b shows the ballasted ladder track. Fig. 1. (a) Floating ladder track; (b) ballasted ladder track 282 S. Mohammadzadeh,M.Mehrali Hosking andMillinazzo (2007) developed amathematical method for a floating ladder track under amoving oscillating load inwhich the trackwas simulated using anEuler-Bernoulli beam on periodic discrete elastic supports. Theywere able to predict the frequency and critical speed for design purposes. Xia et al. (2009) dynamically simulated an elevated bridge with a ladder track under a moving train and measured its dynamic response. Xia et al. (2010) carried out a field experiment at the trial section of an elevated bridge onBeijingMetro line where the ladder track was installed and investigatd the vibration reduction characteristics of the track. Yan et al. (2014) developed dynamic models of the vehicle and the ladder track to analyze the track vibration behavior. They optimized the mechanical properties of the ladder track to reduce or eliminate the track vibrations at the corrugation frequency and ultimately to reduce the chance of rail corrugation. Ma et al. (2016) investigated the effect of ballasted ladder tracks and thevibration reduction effect.The results showthat theballasted ladder track can effectively decrease the peak value in the time domain andhas the potential effect to control environmental vibration in low frequencies. Analysis of beams subjected to moving loads is of substantial practical importance. Many researchers have studied the vibration of beams subjected to various types of moving loads. Since parameters such as loading, rail defection and nature of the substructure are stochastic, the dynamic response of the track is assumed to be stochastic. Table 1 lists themajor studies in this area. Thus far, no study has been carried out on ladder tracks using a stochastic approach. Table 1.Major research on stochastic approach in railway engineering Author(s) Subject Loading Year Fryba et al. Euler-Bernoulli beam resting on Harmonic 1993 aWinkler random foundation moving load Anderson and Nielsen Beam on a randommodified Kelvin Moving vehicle 2003 foundation Kargarnovin et al. Infinite Timoshenko beams supported Harmonic 2005 by nonlinear foundations moving loads Younesian et al. Timoshenko beam on a random Harmonic 2005 foundation under moving load Younesian Infinite Timoshenko beam supported Harmonic 2009 and Kargarnovin by a randomPasternak foundation moving loads Mohammadzadeh Risk of derailment using a numerical Railway vehicle 2010 and Ghahremani method Mohammadzadeh et al. Probability of derailment where Railway vehicle 2011 irregularity of the track is random Mohammadzadeh et al. Double Euler-Bernoulli beam resting Harmonic 2013 on a random foundation moving loads Mehrali et al. Double Euler-Bernoulli beam resting Railway vehicle 2014 on a random foundation Mohammadzadeh et al. Reliability analysis of the rail fastening Moving train 2014 where load and velocity are random Pouryousef Reliability evaluation of design codes Live load 2014 andMohammadzadeh applied for railway bridges (LM71) Engineering experience has revealed that uncertainties occur in the assessment of loading as well as in the material and geometric properties of engineering systems. The logical behavior of these uncertainties in probability theory and statistics cannot be obtained accurately using the deterministicmethod. This approach is based on extremes (minimum,maximum) andmean Dynamic response of ladder track rested on stochastic foundation... 283 values of systemparameters (Stefanou, 2009).More detail on the randombehavior of a structure can be found in Lutes and Sarkani (2004). The Taylor series expansion of the stochastic finite element matrix of a physical system is known in the literature as the perturbation method. This method is used to solve probabilistic problems (Kleiber and Hein, 1992; Liu et al., 1986). Another method is the Karhunen-Loeve expansion technique (Ghanem and Spanos, 1991a,b). The main initiative of the perturbation method is to formulate an analytical expansion of an input parameter around its mean value using a series representation (Jeulin and Ostoja-Starzewski, 2001; Nayfeh andMook 1979). A novel analytical method is presented for the analysis of the governing equations of motion for an infiniteTimoshenko ladder track on a viscoelastic foundationwith random stiffness under a harmonicmoving load. For the stationary analysis of the response of the beam to variations in stiffness in the support, it is useful to describe it in a local moving coordinate system subjected to a harmonic moving load. Furthermore, by applying the perturbation method and complex Fourier transformation, the mean and variance of the response of the beam can be calculated analytically in an integral form. Sensitivity analysis is run using the residue theorem and key parameters are introduced. 2. Theory Assume a harmonic load moves uniformly along a ladder track at velocity v. The ladder track is modeled using two parallel Timoshenko beams. The connection of the two beams is described using a series of springs and dashpots. In addition, the lower beam rests on a viscoelastic foundation. The vertical stiffness of the support is described by a stochastic variable along the beamwith amean of k and a stochastic component of ks(x) (Mohammadzadeh et al., 2013). Here,κ(x) is a randomstationary ergodic functionwith zeromeanvalueandφ is a small constant parameter kB(x)= k+φκ(x)= k+ks(x) (2.1) 2.1. Equation of motion The equations of motion for the rail and ladder units are ρ1A1 ∂2w1 ∂t2 +k1A1G1 (∂ψ1 ∂x − ∂2w1 ∂x ) +kp(w1−w2)+ cp (∂w1 ∂t − ∂w2 ∂t ) = PeiΩtδ(x−vt) EI1 ∂2ψ1 ∂x2 −k1A1G1 ( ψ1− ∂w1 ∂x ) = ρ1I1 ∂2ψ1 ∂t2 (2.2) and ρ2A2 ∂2w2 ∂t2 +k2A2G2 (∂ψ2 ∂x − ∂2w2 ∂x ) +kBw2−kp(w1−w2) − cp (∂w1 ∂t − ∂w2 ∂t ) + cB ∂w2 ∂t =0 EI2 ∂2ψ2 ∂x2 −k2A2G2 ( ψ2− ∂w2 ∂x ) = ρ2I2 ∂2ψ2 ∂t2 (2.3) where w1(x,t) is the upper beam deflection, w2(x,t) is the lower deflection, δ(x) is the Dirac delta function, and v and Ω are the speed and frequency of the load, respectively. A, E, G, I, k and ρ are the cross-sectional areas of the beams, modulus of elasticity, shearmodulus, second moment of area, sectional shear coefficient, and beammaterial density, respectively. Figure 2 is a flowchart of the solution of the governing equation for the ladder track. 284 S. Mohammadzadeh,M.Mehrali Fig. 2. Solving the governing differential equation 2.2. First-order perturbation approach The perturbation method is proposed to compute the response of the beams to a harmonic moving load. The responses of the ladder track (rail and ladder unit) are decomposed to zero and first-order terms w(x,t) = wi0(x,t)+φw i 1(x,t) ψ(x,t) = ψi0(x,t)+φψ i 1(x,t) i =1,2 (2.4) where i =1 for the rail and i =2 for the ladder unit. 2.3. Solution Equations (2.2) and (2.3) are solved using Eqs. (2.4) and equating terms with the same powers of φ. The Galilean coordinate transformation is s = x−vt (2.5) The boundary conditions of deflection, velocity, and acceleration of the beams are assumed to be zero in positive andnegative infinity. Using the state variable transformation and applying the complex Fourier transform results in w10(q)= P(β7q 2 −β8q +β9)D4 H(q) w11(q)= D4P +D2w 2 0 H(q) w20(q)= P(β7q 2 −β8q +β9)(−Dβ3) H(q) w21(q)= −D3P −Dβ1w 2 0 H(q) (2.6) D1, D2, D3 and D4 are described in Appendix 1. H(q) is the determinant of the matrix h= [ D1 D2 D3 D4 ] (2.7) and ψ10, ψ 2 0, ψ 1 1, and ψ 2 1 are equal to ψ10(q)= −β3PqD4 H(q) ψ11(q)= −β3q(PD4+D2κw 2 0) (β7q2−β8q+β9)H(q) ψ20(q)= β12Pq(β7q 2 −β8q+β9)D3 (β15q2−β16q+β17)H(q) ψ21(q)= β12q(D3P +D1κw 2 0) (β15q2−β16q+β17)H(q) (2.8) General definitions for all coefficients are listed inTable 2.The response of the beams canbe cal- culated by applying the inverse Fourier transformandusing contour integrals (Mohammadzadeh et al., 2014). Themeanvalues for thebeamdeflection andbendingmoment and the covariance of a random function can be calculated as described byMohammadzadeh et al. (2013) and Solnes (1997). Dynamic response of ladder track rested on stochastic foundation... 285 Table 2.Definitions of coefficients Parameter Definition Parameter Definition β1 k1A1G1−ρ1A1v 2 β10 k2A2G2−ρ2A2v 2 β2 2ρ1A1Ωv β11 2ρ2A2Ωv β3 ik1A1G1 β12 ik2A2G2 β4 icpv β13 icBv β5 −ρ1A1Ω 2+kp− icpΩ β14 −ρ2A2Ω 2+k+kp+icpΩ +icBΩ β6 kp+icpΩ β15 ρ2I2v 2 −EI2 β7 ρ1I1v 2 −EI1 β16 2ρ2I2Ωv β8 2ρ1I1Ωv β17 ρ2I2Ω 2 −k2A2G2 β9 ρ1I1Ω 2 −k1A1G1 β18 kp+icpΩ 3. Model validation of ladder track The stochastic simulation of the ladder track foundation has been validated as described below. 3.1. Validation using the stochastic finite element method The response of a beam resting on a stochastic foundation is obtained using the stochastic finite element method (SFEM) as suggested by Fryba et al. (1993). Consider the second beam as a rigid component and evaluate the behavior of the upper beamassuming stochastic behavior for the foundation. Then, the random behavior of the system is calculated and validated using the results of Fryba et al. (1993). Figure 3 shows that the results calculated in current study are in good agreement with those reported by Fryba et al. (1993). Fig. 3. Comparison between the current modeling and results by Fryba et al. (1993) 3.2. Validation by a deterministic model Next, the deterministic behavior of the ladder track is verified using the results of Younesian et al. (2006). They investigated the dynamic behavior of a ladder track of finite length. The ladder track is simulated using aTimoshenko beamand the track is subjected to amoving load. The results of verification are illustrated inTable 3. The results of the current study are in good agreement with those reported by Younesian et al. (2006). 286 S. Mohammadzadeh,M.Mehrali Table 3.Comparison of the current study and results by Younesian et al. (2006) S [m] Current study Younesian et al. −8 −4.13E-09 7.86E-05 −6 5.87E-08 −1.7E-05 −4 −3.11E-07 −0.00017 −2 −1.7E-05 −0.00031 0 −0.00083 −0.00037 2 −3E-05 −0.00032 4 1.06E-06 −0.00015 6 −3.24E-08 7.61E-05 8 7.16E-10 0.00018 4. Response of the ladder track The response of the simulated ladder track is next investigated under a harmonic moving load. The railway substructure should be constructed and confirmed using adequate ground stiffness and standards (Younesian et al., 2005). It is not possible to provide a track bedwith absolutely uniform specifications, and there are many factors that influence the subgrade (Phoon, 2008; Griffiths and Fenton, 2007; Fenton andGriffiths, 2008; Baecher andChrsitian, 2003). The finite distance correlation can be assumed using bed stiffness as a random field. A parametric study was done on the key parameters of solution derived using the track bed stiffness from the field data by Berggren (2009). The physical and geometrical properties of the track are listed in Table 4. Table 4.Parameters used in the model Rail Ladder Parameters Value Parameters Value Young’s modulus E1 210GPa Young’s modulus E2 28.2GPa Shear modulus G1 77GPa Shear modulus G2 11.75GPa Mass density ρ1 7850kg/m 3 Mass density ρ2 3954.7kg/m 3 Cross-sectional area A1 7.69 ·10 −3m2 Cross sectional area A2 31 ·10 −3m2 Secondmoment 30.55·10−6m4 Secondmoment 98.3·10−6m4 of inertia I1 of inertia I2 Shear coefficient k1 0.4 Shear coefficient k2 0.43 Rail pad Foundation Parameters Value Parameters Value Stiffness kp 40 ·10 6Nm−2 Mean value of stiffness kB 50 ·10 6Nm−2 Viscous damping cp 6.3 ·10 3Nm−2 Variance of stiffness σ2 kB 4.4186 ·1013N2m−4 Viscous damping cB 41.8 ·10 3Nsm−2 4.1. Load frequency influence Thevelocity of themoving load is assumed tobe100km/h.Figure 4 shows that, by increasing the load frequency, the mean value and standard deviation of the response of the upper beam (rail) initiallydecreases andthen increases. Inaddition, thedistributionwidensas theoscillations increase along the rail. Dynamic response of ladder track rested on stochastic foundation... 287 Fig. 4. Effect of load frequency on track deflection (mean value) An increase in the load frequency decreases the response of the lower beam (ladder unit), indicating that both themean value and standard deviation of the ladder unit show decreasing trends. Figure 5 shows the wider distribution with the increase in fluctuations along the beam. Fig. 5. Effect of load frequency on track deflection (standard deviation) Figures 6 and 7 show themean value and standard deviation of the rail and ladder bending moments, respectively. As the load frequency increases, the response of the rail first decreases and then increases. The velocity of the moving load is assumed to be 100km/h. Fig. 6. Effect of load frequency on track bending moment (mean value) The mean value and standard deviation of the ladder unit decreased as the load frequency increased. As shown, the fluctuation of the ladder first increased and then decreased. 288 S. Mohammadzadeh,M.Mehrali Fig. 7. Effect of load frequency on track bending moment (standard deviation) 4.2. Load velocity influence The variation in load velocity versus the behavior of the double beam is shown in Figs. 8 and9 for the responseof the ladder track.Thefigures include thedeflection andbendingmoment of both beams. As shown, themaximum response of the rail versus loading frequency have been attained andemployedasdesign criteria.An increase in thevelocity of themoving loaddecreased the value of this frequency. Fig. 8. Effect of velocity on the ladder track (mean value) Fig. 9. Effect of velocity on the ladder track (standard deviation) Dynamic response of ladder track rested on stochastic foundation... 289 4.3. Effect of the coefficient of variation of bed stiffness The coefficient of variation (CV ) of the stiffness of the bed is varied to assess its effect on the track bed (Figs. 10 and 11).It can be observed that increasing the CV increases the standard deviation of the rail and ladder. Fig. 10. Effect of CV on the ladder track (mean value) Fig. 11. Effect of CV on the ladder track (standard deviation) 5. Conclusion Thedynamic behavior of the ladder track has been investigated in the present study.The ladder track has been simulated using an analytical model with a doubleTimoshenko beam.The upper beam simulated the rail and the lower beam simulated the ladder unit. A series of springs and dashpots represent the rail pad and foundation. The foundation stiffness of the system has been assumed to exhibit stochastic behavior as simulated by field tests. The first-order perturbation method has been applied and the responses, including the deflection and bendingmoment, are shown in form of the mean value and standard deviation. It has been found that increasing the load frequency decreased and then increased the response of the track. The peak frequency is the point at which all responses are at maximum value. It was found that the peak frequency increases as the velocity of the load velocity increases. Appendix 1 The parameters in Equations (2.6) and (2.8) are described below D1 = β1β7q 4+(−β1β8+β2β7−β4β7)q 3+(β1β9−β2β8−β 2 3 +β4β8+β5β7)q 2 +(β2β9−β4β9−β5β8)q+β5β9 290 S. Mohammadzadeh,M.Mehrali D2 = β4β7q 3 − (β4β8+β6β7)q 2+(β6β8+β4β9)q−β6β9 D3 = β4β15q 3 − (β4β16+β15β18)q 2+(β4β17+β16β18)q−β17β18 D4 = β10β15q 4+(−β10β16+β11β15−β4β15−β13β15)q 3+(β10β17−β11β16−β 2 12 +β4β16+β13β16+β14β15)q 2+(β11β17−β4β17−β13β17−β14β16)q +β14β17 References 1. 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