Dynamic response analysis of a multiple-beam structure subjected to a moving load

Zhang Yuntai, Jiang Lizhong , Zhou Wangbao, Liu Shaohui, Feng Yulin, Liu Xiang and Lai Zhipeng

1.School of Civil Engineering, Central South University, Changsha 410075, China

2.National Engineering Laboratory for High Speed Railway Construction, Changsha 410075, China

3.School of Civil Engineering and Architecture, East China Jiaotong University, Nanchang 330013, China

Abstract: In this study, the dynamic response of an elastically connected multi-beam structure subjected to a moving load with elastic boundary conditions is investigated.The boundary conditions and properties of each beam vary, and the difficulty of solving the motion equation is reduced by using a Fourier series plus three special transformations.By examining a high-speed railway (HSR) with mixed boundary conditions, the rationality for the newly proposed method is verified,the difference in simulated multiple-beam models with different beam numbers is explored, and the influence of material parameters and load speed on the dynamic response of multiple-beam structures is examined.Results suggest that the number of beams in the model should be as close to the actual beam number as possible.Models with an appropriate beam number can be used to describe in detail the dynamic response of the structure.Neglecting the track-structure can overestimate the resonant speed of a high-speed railway, simply-supported beam bridge.The effective interval of foundation stiffness (EIFS)can provide a reference for future engineering designs.

Keywords: multiple-beam; dynamic analysis; moving load; mixed boundary conditions

1 Introduction

Elastically connected beam structures are an important component in many engineering fields, especially in civil,nano-material, aerospace, and mechanical engineering(Thambiratnam and Zhuge, 1996; Koromaet al., 2014;Sunet al., 2016; Yanet al., 2019; Zhanget al., 2020).In particular, the double-beam system has been used to simulate composite structures with a wide variety of materials (Oniszczuk, 2000, 2003).Chen and Sheu(1995) studied the dynamic response, free vibration, and static buckling of two identical parallel beams, using different boundary conditions and a viscoelastic material layer in between them.Vuet al.(2000) proposed a method for analyzing a double-beam system subject to harmonic excitation, with boundary conditions that must be the same on the same side.Abu-Hilal (2006) investigated the dynamic response of a double-beam system with viscoelastic layer damping traversed by a constant moving load and obtained the dynamic deflections of both beams in analytical closed forms.Palmeri and Adhikari (2011) presented a novel state-space form for studying transverse vibrations of double-beam systems,consisting of two outer elastic beams continuously joined by an inner viscoelastic layer, while, for the inner layer,considering inhomogeneous systems, any boundary conditions, and rate-dependent constitutive law.Wu and Gao (2015, 2016) investigated the dynamic response of the double-beam system, consisting of two identical elastic, homogeneous, isotropic Euler-Bernoulli beams under a moving oscillator and moving harmonic loads.Li (2016a, 2016b) presented a semi-analytical method to investigate the natural frequencies and mode shapes of a double-beam system interconnected by a viscoelastic layer, with the Winkler layer situated below the lower beam.Jianget al.(2019a) proposed a method for analyzing a simply supported double-beam system under successive moving loads.In addition, the double beam models were widely used (Wanget al., 2018).Zhanet al.(2014) investigated the dynamic responses of a slab track on transversely isotropic saturated soils that were subjected to moving train loads, by means of a semianalytical approach.Murmu (2011) investigated the natural frequencies and the axial instability of a nonlocal double-nanobeam-system within the framework ofEringen′s nonlocal elasticity theory.Yang and Yau(2015 and 2017) investigated the resonant response of sprung masses moving on a series of simple beams.In addition, the effect of rails and fasteners on the dynamic response of a train-bridge system was illustrated.Based on the energy-variational principle, Jiang (2019b, 2020)established a double beam analysis model of a highspeed railway and a simply supported beam bridgetrack structure system (HSRBTS) by considering the effect of shear deformation.Furthermore, a number of numerical simulation methods also have been proposed.(Zhaiet al., 2010; Liet al., 2012).Xia and Guo (2009,2010) analyzed the dynamic responses of the Tsing Ma suspension bridge and the running behaviors of trains across the bridge under turbulent wind actions by employing a three-dimensional wind-train-bridge interaction model.Lou (2012) presented a rail-bridge coupling element of unequal lengths, by which the length of a bridge element was longer than that of a rail element.In this way, Lou investigated the dynamic problem of train-track-bridge interaction systems by considering the clamped boundary condition at track ends.

However, the double-beam system has many shortcomings when dealing with more complex composite structures.To break through the limitation of the double-beam system, research on the multiplebeam system has been given a great deal of attention(Fenget al., 2020).Li (2008) developed an exact dynamic stiffness method to predict the free vibration characteristics of a three-beam system, and established an exact dynamic stiffness matrix for an elastically connected three-beam system by considering the effects of shear deformation and the rotary inertia of the beams.Kelly and Srinivas (2009) developed a general theory for the determination of natural frequencies and mode shapes for a set of elastically connected, axially loaded Euler-Bernoulli beams.Ariaei (2011) investigated the dynamic response of an elastically connected, multiple Timoshenko beam system subjected to a moving load.The arbitrary boundary conditions were considered in the research of Mao (2012), in which the researcher employed the Adomian modified decomposition method (AMDM) to investigate the free vibrations of N-elastically connected, parallel Euler-Bernoulli beams joined by a Winkler-type elastic layer.Stojanovićet al.(2013) investigated a general procedure for the determination of the natural frequencies and buckling load for a set of beam systems under compressive axial loading by using the Timoshenko system and the highorder shear deformation theory.Karličićet al.(2015)examined the free longitudinal vibration of a complex multi-nanorod system (CMNRS) by using the nonlocal elasticity theory with respect to cases involving two types of boundary conditions.

Compared with the widely studied doublebeam system, multiple-beam systems require further examination.(1) In many studies, the multiple-beam system boundaries are limited.For example, the boundary conditions of each beam must be identical.(2)Most previous research focuses on the natural vibration and static performance of multiple-beam systems,with few studies examining the dynamic response of multiple-beam systems subjected to a moving load.(3) Few studies exist on the engineering application of multiple-beam systems, the applicability of multiplebeam system layers in engineering, and the influence of material characteristics.

This contribution proposes a method of investigating the dynamic response of an elastically connected,multiple-beam system subjected to a moving load with arbitrary elastic boundary conditions.The infinitedegree-of-freedomN-beam system is transformed into a superimposedN-degrees-of-freedom system, which reduces the difficulty of analyzing the system.The applications were carried out under three conditions:an ordinary railway bridge, a high-speed railway simply supported bridge (HSRSSBB) with a CRTSII ballastless track, and a high-speed railway (HSR) with a CRTSII ballastless track on an elastic foundation.The applicability of the proposed method in dealing with complex boundary conditions is verified.The simulation results of multiple-beam models with layer counts are compared, and the influence of the parameters, including interlayer stiffness and beam length, on the system dynamic response is obtained.The impact of a track structure on the resonant speed of a bridge is investigated.The effective interval of foundation stiffness (EIFS)is proposed, which can provide a reference for future engineering designs.

2 Establishment of the multiple-beam system

2.1 Transformation for motion equation

An elastically connectedN-beam with elastic boundaries is subjected to various moving loads (Fig.1).All beams are Euler-Bernoulli beams of the same lengthL.The partial differential equation of motion can be expressed as follows:whereEIandmare the bending stiffness and mass per unit length of the beam;kis interlayer stiffness;Pis the external moving force; andyis displacement of the beam in theY-direction at any pointx, at an instanttcaused byP.

Fig.1 An elastically connected N-beam with elastic boundaries subjected to moving loads

The vertical deflectionynof thenth beam is expressed in the form of a Fourier series as follows:

where

The transformations are defined as follows:

By substituting Eqs.(6)-(8) into Eq.(4):

the partial derivative with respect toxcan be obtained:

In order to solve foryn, the Eqs.(1)-(3) need to be transformed.

Transform the Eqs.(1)-(3) by Eq.(6):

Transform the Eqs.(1)-(3) by Eq.(7):

Transform the Eqs.(1)-(3) by Eq.(8):

where:

2.2 Solution for the equation

According to the deformation coordination relationship at the beam ends, the following can be obtained:

wherekn,LVandkn,LRare the vertical stiffness and rotational stiffness of the left end of thenth beam,respectively;kn,RVandkn,RRare the vertical stiffness and rotational stiffness of the right end of thenth beam,respectively.

Substituting Eqs.(10)-(12) and Eqs.(24)-(27) into Eqs.(13)-(15) gives:

Similarly, boundary conditions Eqs.(24)-(27) can also be substituted into Eqs.(16)-(21).The infinitedegree-of-freedomN-beam system is transformed into a superimposedN-degrees-of-freedom system.AllN-degrees-of-freedom system equations can be expressed in matrix form:

whereMnis the mass matrix;Knnis the stiffness matrix;and:

Un,VnandWncan be obtained by solving the equations by using Newmark-β.

3 Case study

3.1 Validation of the method

To verify the applicability of the multiple-beam system proposed by the method in this study, two special cases are investigated in this section.

In the former case, a double-beam railway bridge model with mixed boundary conditions is established(Fig.2) The 1st beam is a continuous welded rail (CWR)and the 2nd beam is a girder.Referring to Lou (2012),the boundary condition for the 1st beam is clamped,and the boundary condition for the 2nd beam is simply supported.A moving load was applied to the 1st beam to investigate the dynamic response.The corresponding finite element model (FEM) was established using ANSYS to compare the calculation results of the proposed method.The material parameters of the model are taken from Jiang (2019a).Assuming that the foundation rails have sufficient restraint on the bridge rails, clamped constraints can be used at both ends of the 1st beam, and simply supported constraints can be used on the 2nd beam.The loadP1 is 20 kN.Two speeds for the moving load were selected (v=10 m/s or 50 m/s).In the ANSYS finite element model, the 1st beam and the 2nd beam are simulated by using BEAM3 elements, and the interlayer springs and boundary constraints are all simulated by utilizing COMBIN14 elements.

Fig.2 Double-beam model with mixed boundary conditions

In the latter case, the analytical solution for the dynamic response of the double-beam system with simply-support under a moving load, as reported by Abu-Hilal (2006), has been compared with the result obtained from the proposed method in this paper.To incorporate identical model conditions with Abu-Hilal, the dimensionless layer stiffness parameters were selected for calculation (β=10 and 104).The speed parameter and damping ratio were 0.1 and 0, respectively.

As shown in Figs.3-6, the dynamic response of the double-beam obtained by the proposed method are consistent with other methods, which demonstrates that using the proposed method to analyze a model with complex boundary conditions is reasonable.

3.2 Comparison of multi-beam models with varying beam numbers

In previous studies on the dynamic response of highspeed railway bridges, many scholars chose to combine beams (for example, combining an entire track system into a single beam) to reduce calculation difficulty.In various studies, the beam numbers of the models are not all the same; therefore, it is necessary to examine the dynamic response differences of models with varying beam numbers.A 4-beam model was established for a high-speed railway, simply-supported beam bridge(HSRSSBB) with a CRTSII ballastless track, subjected to train loads (Fig.7).To study the vibration of bridges,train loads can be simplified as successive moving loads(Yanget al., 1997).The successive moving loads can be expressed by the Fourier series by referring to Jiang and Zhang (2019b).The four-beam model is composed of a rail, a track plate, a base plate, and a girder.A 2-beam model (combining a rail, a track plate, and a base plate)and a 3-beam model (combining a track plate and a base plate) were also established as control cases.The 2-beam model, 3-beam model, and 4-beam model are denoted as MODEL-A, MODEL-B, and MODEL-C, respectively.Material parameters are listed in Table 1.Assuming thatthe foundation track system imparts sufficient restraint on the bridge track system, the clamped constraints can be used at both ends of the 1st to 3rd beams, and the simply supported constraints can be used on the 4th beam.Moving loads was applied to the 1st beam with three speeds (v=55/70/130 m/s).Two values for the moving load are then selected for comparison (P=50/85 kN).The characteristic distance of the adjacent carriages isd.The number of train carriages is 8.The dynamic response at the mid-span position is generally the largest; hence, the vertical displacement at positionL/2 is examined.

Table 1 Parameters used in numerical examples

Fig.7 The 4-beam model for an HSRSSBB with a CRTSII ballastless track (MODEL-C)

Fig.3 Vertical deflection at L/2 when v=10 m/s for the (a) 1st beam and (b) 2nd beam

Fig.4 Vertical deflection at L/2 when v=50 m/s of the (a) 1st beam (b) 2nd beam

Fig.5 Dynamic responses of the double-beam system (β=10) obtained by different methods (a) primary beam and (b) secondary beam

Fig.6 Dynamic responses of the double-beam system (β=10000) obtained by different methods (a) primary beam and(b) secondary beam

To investigate the influence of a track structure on the resonant speed of the bridge, the most prominent resonant speed of single-simply-supported-beam system frequency was acquired from a previous study (Yanget al., 1997; Museroset al., 2013):

whereis first mode frequency of the girder in a single-beam system with simple-support:

Figures 8-13 describe the deflection at different load values and load speeds.Models with different beam numbers show variations in dynamic response.(1) For models subjected to the same load, the deflection-time curves for the bottom beam of the three models is the same, and at the same time they show peaks.The dynamic response of the three models gradually decreases from the upper beam to the lower beam.The higher the number of beams, the greater the dynamic response of the track structure.For example, the dynamic responses of the MODEL-C track structure are greater than those of the MODEL-A track structure, with a difference of up to 60%.The differences in dynamic response betweenMODEL-B and MODEL-C are smaller, and between the two models the difference in the rail dynamic response is less than 25%.This indicates that the simulation results of the three models are consistent with the dynamic response of the girder.However, when describing the dynamic response of the track structure, the more beams the model has, the more detailed the model and the larger the dynamic response of the track structure.Therefore,the calculated results for models with fewer beams are unreliable.(2) A linear relationship exists between dynamic response and load value; however, there is no linear relationship between load speed and dynamic response (yv=130＞yv=55＞yv=70).

Fig.8 Deflection when P=50 kN and v=55 m/s for (a) MODEL-A, (b) MODEL-B, and (c) MODEL-C

Fig.9 Deflection when P=50 kN and v=70 m/s for: (a) MODEL-A; (b) MODEL-B; (c) MODEL-C

Fig.10 Deflection when P=50 kN and v=120 m/s for: (a) MODEL-A; (b) MODEL-B; (c) MODEL-C

Figure 14 describes the variation in peak deflection,Y, of different models with various load speeds.P=85 kN.The dynamic responses of the girder in the three modelsare consistent, but peak deflections of the track-structure are obviously different.The peak deflection of the track structure is significantly lower for the model with fewer beams.Therefore, for engineering structures where the connection between adjacent beams is not a rigid connection, the dynamic response of the track-structure of the model considering more real beams is safer.The resonant speeds of the girder in the three models are consistent (156 m/s) yet less than the resonant speed of the single-simply-supported-beam system (169 m/s)obtained by Eq.(32).This indicates that the trackstructure has an impact on the resonant speed of the bridge, and the resonant speed calculation method of a single-simply-supported-beam can overestimate the resonant speed of a high-speed railway simply-supported beam bridge.

Fig.11 Deflection when P=85 kN and v=55 m/s for: (a) MODEL-A; (b) MODEL-B; (c) MODEL-C

Fig.12 Deflection when P=85 kN and v=70 m/s for: (a) MODEL-A; (b) MODEL-B; (c) MODEL-C

Fig.13 Deflection when P=85 kN and v=120 m/s for: (a) MODEL-A; (b) MODEL-B; (c) MODEL-C

Fig.14 Peak vertical deflection versus speed for (a) MODEL-A, (b) MODEL-B, and (c) MODEL-C

3.3 Influence of beam length and interlayer stiffness

The dynamic performance of the elastic foundation track structure may change due to decreasing foundation stiffness in a certain area.The proposed method in this paper can also be used to examine the dynamic response of a high-speed railway (HSR) subjected to a moving load in this area.Taking HSR with a CRTSII ballastless track as an example, the size of the area can be transformed into the beam length, and foundation stiffness can be expressed as an elastic connection(Fig.15).Assuming that the track system outside the area exerts sufficient restraint on the track system within the area, clamped constraints can be used at both ends of the beams.The MODEL-C track structure parameters are used as the parameters in this particular section(Table 1).The load is 20 kN, and the beam length,L,is 10 m.The vertical displacement at theL/2 position is selected for examination.The foundation stiffness,kd,varies from 0 to 8×108kN/m2.

Fig.15 3-beam model for the HSR with a CRTSII ballastless track in a certain area

Figure 16 describes the influence of foundation stiffness,kd, on the deflection-time relationship of the HSR, subjected to a moving load atv=20 m/s.The larger thekdis, the smaller the deflection of each beam.Whenkd=0, the maximum vertical displacementpeak can reach 30 times of that whenkd=8×108kN/m2.Figure 17 shows the change of peak deflection,Y, under differentkdwith increasing load speed.The difference between a single load and train loads is compared.Ydis the peak deflection subjected to a single load, andYsis the peak deflection subjected to train loads.The peak-speed curve ofYdis smoother than the peakspeed curve ofYs.The larger thekdis, the smaller the peak deflection, and the flatter the peak-speed curve.A smallerkdobviously decreases the resonant speed.Whenthe peak deflection decreases sharply with increasingkd, while outside this interval, peak deflection does not change withkd(Fig.18).Therefore, this interval can be defined as the Effective Interval of Foundation Stiffness(EIFS).The EIFS is independent of load speed and load number.When the foundation stiffness,kd, is within the EIFS, the lowerkdis, the larger will be the track structure deflection and the smaller the resonant speed, all of which is unfavorable to track structure dynamic performance.

Fig.18 Peak deflection with respect to kd for (a) v=20 m/s and (b) v=120 m/s

Fig.16 Deflection-time curve for (a) the 1st beam, (b) the 2nd beam, and (c) the 3rd beam unit: (kN/m2)

Fig.17 Peak deflection with respect to v for (a) the 1st beam, (b) the 2nd beam, and (c) the 3rd beam

The area size of the decreasing foundation stiffness also can potentially influence the dynamic response of the track structure, which can be transformed to examine the influence of beam length on dynamic response.The load and material parameters are consistent with the parameters described above.The foundation stiffness is 8×106kN/m2.

Figure 19 describes the influence of beam length on the deflection-time relationship of HSR, subjected to a moving load whenv=20 m/s.In addition,t=T/Tz,Tzis the total time for the load to move along the beam,andtis the arbitrary time for the load to move along the beam.The smaller the beam length, the gentler the deflection-time curve.WhenL＜ 20 m, deflection increases significantly with increasing beam length.WhenL＞ 20 m, the influence of beam length on the deflection remains constant.

Figure 20 describes the change in peak deflection when increasing the load speed.The longer the beam,the smaller the critical speed.When the load speed is greater than critical velocity, the longer the beam, and there will be larger peak deflection with increasing speed.WhenL＜ 12 m, the peak deflection increases sharply with increasing beam length atv=120 m/s(Fig.21).WhenL＞ 12 m, the peak deflection increases with increasing beam length.The peak-length curve changes more gently under a single moving load than under train loads.This indicates that the larger the area of decreasing foundation stiffness, the more unfavorable it is to the HSR dynamic response.This unfavorable effect is more significant at high load speeds.

Fig.21 Peak deflection with respect to L subjected to (a) a moving load (b) train loads

Fig.19 Deflection-time curve for (a) the 1st beam, (b) the 2nd beam, and (c) the 3rd beam

4 Conclusion

In this study, the infinite-degree-of-freedomN-beam system is transformed into a superimposedN-degreesof-freedom system, which reduces the difficulty of assessing the system.An ordinary railway bridge, an HSRSSBB with a CRTSII ballastless track, and an HSR with a CRTSII ballastless track on an elastic foundation were examined, using the method presented in this paper.The applicability of the proposed method in dealing with complex boundary conditions is verified.The simulation results of multiple-beam models with varying numbers of beams are compared, and the influence of the parameters,including interlayer stiffness and beam length, on the dynamic response is obtained.The EIFS can provide a reference for devising a relevant engineering design.The follow conclusions are obtained, all of which can provide a reference for a relevant engineering design:

· The results obtained by the proposed method in this paper are consistent with the results obtained by Ansys FEM, indicating that it is reasonable to use the proposed method to calculate models with complex boundary conditions.

· For engineering structures where connections between adjacent beams that are not rigid connections,the dynamic responses of models with fewer beams are unreliable.In contrast, the model that is as close as possible to the actual number of beams in the structure can be used to describe the dynamic response of the structure in more detail, and the dynamic response of the model is reliable.

· The track-structure has an impact on the resonant speed of the bridge.Neglecting the track-structure can overestimate the resonant speed of a high-speed railway,simply-supported beam bridge.

· The dynamic response of the HSR with a CRTSII ballastless track in a certain area where foundation stiffness decreases is affected by foundation stiffness and the size of the area.The larger the area, the more unfavorable the HSR dynamic response.When the foundation stiffness,kd, falls within the Effective Interval of Foundation Stiffness (EIFS), the lowerkdis, the more unfavorablekdwill be to track structure dynamic performance.The EIFS is independent of load speed and load number.

Acknowledgement

The work was financially supported by the National Natural Science Foundations of China (U1934207 and 51778630), the Hunan Innovative Provincial Construction Project (2019RS3009), the Innovationdriven Plan in Central South University (2020zzts159),and the Fundamental Research Funds for the Central Universities of Central South University (2018zzts189).