Numerical analysis of beam rail bridges with random track irregularities

The study focuses on dynamic analysis of composite bridge / track structure / train systems (BTT systems) with random vertical track irregularities taken into consideration. The paper presents the results of numerical analysis of advanced virtual models of series-of-types of single-span simply-supported railway steel-concrete bridges (SCB) with symmetric platforms, located on lines with the ballasted track structure adapted to traffic of high-speed trains.


Introduction
Moving-load as a dynamic problem is very common in science papers because of many structures subjected to loads which move in space, for example Fryba's monograph described a lot of moving-load problems and their analytic solutions [1]. Transportation systems play a key role in modern societies -social and economic development requires the fast and efficient mobility. In recent decades, train operating speeds have been significantly increased. The literature studies confirmed that one of the main factors affecting the dynamic response of railway engineering structures are irregularities in the railway surface -track irregularities [2][3][4][5][6][7][8][9][10][11][12]. The irregularities are due to track formation technology, contemporary mechanical maintenance, soil settlement and other factors. Experimental measurements and modelling of track irregularities are considered in a number of papers, e.g. [13][14]. A common model of railway track vertical irregularities is a stationary and ergodic Gaussian process in space [4,10,11,12].

Description of modelling and simulation of BTT systems 2.1 Description of series-of-types of railway bridges (SCB)
The detailed description of design the SCB (steel-concrete bridges) series-of-types of railway bridges are presented in Ref. [15]. The bridge series-of-types consists of five objects with codes and basic geometric parameters listed in Table 1

Description of mathematical and numerical modelling of BTT systems
The theory of advanced physical and mathematical modelling of BTT (bridge / track structure / train) systems has been developed by Podworna and Klasztorny in 2014, [16]. In the physical and numerical modelling, the following main assumptions were adopted:  There is considered a finitely long deformable track with continuously welded rails including the out-of-transition zones, the transition zones and the bridge zone.  The bridge superstructure is reflected by a simply-supported stepwise viscoelastic prismatic beam, deformable in flexure, symmetrical relative to the bridge midspan.  The operating and the side rails are viscoelastic prismatic beams deformable in flexure.  The rail-sleeper fasteners are viscoelastic elements with non-linear elastic characteristic.  The sleepers vibrate vertically and are modelled as concentrated masses.  The crushed stone ballast is modelled as a set of vertical viscoelastic constraints with non-linear elastic characteristic. The model includes possibility of detachment of sleepers from the ballast. The lumped ballast model is used.  The track bed (subsoil) is a linearly viscoelastic layer modelled discretely.  The approach slabs are modelled as viscoelastic prismatic beams deformable in flexure.  There are random vertical track irregularities.  The rail-vehicles form a high-speed ICE-3 (Inter City Express) German train. Each vehicle has two independent two-axle bogies. The planar Matsuura model of railvehicle is developed via incorporating non-linear one-sided contact springs at wheel set rail contacts.  The vehicle model includes the Hertz contact nonlinear stiffness between moving wheels and rail heads (developed in 1880s [17]), and potential detachment of moving wheels from rail heads.  Micro separations and impacts of moving wheel sets in reference to the main rails are taken into consideration.  A train operating speed is constant and ranges from 30 to 300 km/h. A velocity of 30 km/h is treated as relevant to the quasi-static passage of train.  Vibrations of BTT system are physically nonlinear and geometrically linear.
The physical model of BTT subsystem is presented in Fig. 1. A constant sleeper spacing d is used to discretise these subsystem. Discretization of beams modelling operating and side rails, approach slabs and the bridge superstructure uses classic beam finite elements deformable in flexure, with 4DOF and length [16]. where:

Modelling of wheelrail contact stiffness and random vertical track irregularities
The modelling of wheelrail contact stiffness according to the Hertz theory is presented by e.g. Lei and Noda, [12] -the wheelrail contact is considered as two elastic contact cylinders perpendicular to each other. The relative vertical shortening between the wheel and the rail is calculated from the formula where: vertical shortening, contact stiffness coefficient, 1 = 0.5half of the interaction force per wheel set.
Only the vertical profile is taken into consideration, short wavelength corrugation irregularities in rail are neglected. The profile is characterized by the one-sided power spectral density function (PSD), which corresponds to line grades 1 to 6 defined in the American ; total number of frequency increments in [Ω min , Ω max ]; ( =100 is assumed to be adequate [18])

Numerical analysis of BTT systems
The study presents the impact factors in the longitudinal normal stress in the bottom fibres of main steel beams at the midspan which were obtained in dynamic analysis of the BTT systems. Also there is a comparison with the impact factors in the vertical deflection at midspan of SCB. The impact factors are calculated from the well-known classic formulae, i.e.
where:   The analysis focus on service velocities at which forced resonances in the BTT system may occur, commonly called the critical/resonant service velocities (the static pressures of moving wheel sets induce the cyclic excitation of the bridge). The resonant processes may be interfered/amplified by several factors specific to the BTT system, i.e. fast-varying configuration, potential micro separations and re-contacts of moving wheel sets, limited number of moving rail-vehicles, structural complexity, random track irregularities. This study presents the comparison of impact factors for track irregularities TI4, TI5 of five bridges from SCB (the series-of-types of steel-concrete bridges).

Conclusions
The study focuses on establishing a dependence between the impact factors and the bridge's span or the impact factors and the train's velocity. Based on the above analysis, the following main conclusions were made:  The impact factors in the longitudinal normal stress in the bottom fibres of main steel beams (0.5 ), as well as the impact factors in the vertical deflection (0.5 ) do not depend on a bridge span for SCB 18-27, except for the maximum speed = 300 /ℎ for which the longest bridge has the highest values.  As the train speed increases, the coefficients (0.5 ), (0.5 ) increaseit can be some kind of function for SCB 18-27.  There is no significant difference in impact factors of the vertical deflection in the midspan (0.5 ) depending on the rails irregularities in longer bridges (SCB 21-27). However the impact factors in the longitudinal normal stress in the bottom fibres of main steel beams at the midspan (0.5 ) are greater with larger track irregularities.  The shortest bridge span (SCB-15) and the resonance speed 31 = 180 km/h has the values of the impact factors (0.5 ), (0.5 ) much higher than other SCB bridges.  The object of the shortest span (SCB-15) presents the least favorably with the whole series of theoretical spans length from 15m to 27m in the above analysis. Moreover, similar results were obtained in studies performed on other criteria, e.g. the traffic safety condition or the passenger comfort condition (see Podworna [18]). The results confirm Rocha's theses: "Short span railway bridges,…, have been reported as problematic …" [19].