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In this paper the dynamic problem of a simple supported beam subjected to a constant force moving at a constant speed is discussed. Analytical and finite element solutions to this fundamental moving load problem are presented. The results of this paper, provided by the author and other investigators, are intended to give a basic understanding of the moving load problem and reference data for more general studies. Some computational aspects are also discussed.

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... So far, the Euler-Bernoulli beam theory is still the most commonly used in the dynamic analysis of engineering structures, owing to its simplicity and reasonable engineering approximations. Based on this theory, many analytical and numerical methods have been developed to address the moving load problems [7][8][9][10][11]. Although the Euler-Bernoulli model has been proven to be accurate enough for slender beams, it tends to overestimate the natural frequencies of deep beams and their higher vibrational modes [4,12]. ...

... Therefore, in the following finite element analysis, the beam is discretized into an even number of elements and at least 20. Figure 5 shows the numerical results of the normalized mid-span deflections and bending moments for various moving speeds of the load. As stated earlier, the critical speed is used as a reference [7,17,34,35,[39][40][41]. Note that for the SIBT model, the Figure 5 shows the numerical results of the normalized mid-span deflections and bending moments for various moving speeds of the load. ...

... Note that for the SIBT model, the Figure 5 shows the numerical results of the normalized mid-span deflections and bending moments for various moving speeds of the load. As stated earlier, the critical speed is used as a reference [7,17,34,35,[39][40][41]. Note that for the SIBT model, the fundamental frequency ω 1 is defined by Equation (17), which is different from ω * 1 = (π/L) 2 EI/ρA. ...

In this paper, the dynamic response of a simply supported beam subjected to a moving load is reinvestigated. Based on a new beam theory, slope inertia-based Timoshenko (SIBT), the governing equations of motion of the beam are derived. An analytical solution is presented by using a coupled Fourier and Laplace–Carson integral transformation method. The finite element solution is also developed and compared with the analytical solution. Then, a comparative study of three beam models based on the SIBT, Euler–Bernoulli and Timoshenko, subjected to a moving load, is presented. The results show that for slender beams, the dynamic responses calculated by the three theories have marginal differences. However, as the ratio of the cross-sectional size to beam length increases, the dynamic magnification factors for the mid-span displacement obtained by the SIBT and Timoshenko beams become larger than those obtained by the Euler–Bernoulli beams. Furthermore, until the ratio is greater than 1/3, the difference between the calculated results of the SIBT and Timoshenko beams becomes apparent.

... The results of an experiment on the dynamic behavior of a prestressed concrete beam showed that the natural frequency increases following eccentric prestressing of load [22]. Several investigations have tried to find a solution for calculating the dynamic response of the simply supported beam under a moving load [23][24][25][26]. Dynamic vibration of the axially loaded beams resting on the elastic foundation along with relevant factors including the variable elastic foundation, magnitude of vibration and axial load, and boundary conditions were studied by Mirzabeigy and Madoliat [27]. ...

... The implication may result from the decrease in the flexural stiffness of the beam as a result of the softening effect. It is noteworthy to mention that beyond the speed of 125 m/s, the dynamic magnification factors start to decrease, which are in agreement with the results obtained by Olsson [24]. The maximum response ratios, henceforth called the dynamic magnification factor, is defined as the maximum ratio of the dynamic and static midspan displacements. ...

... The implication may result from the decrease in the flexural stiffness of the beam as a result of the softening effect. It is noteworthy to mention that beyond the speed of 125 m/s, the dynamic magnification factors start to decrease, which are in agreement with the results obtained by Olsson [24]. The effect of the velocity and excitation frequency of the moving point load is discussed here by taking two velocities with frequencies 40, 70, and 100 rad/s, as shown in Figure 19. ...

This paper deals with the effect of the prestress load on the free and forced dynamic behavior and vertical vibration of the prestressed beams. The analysis applies both the analytical frequency equation and the finite element method (FEM) using ABAQUS software to predict the fundamental natural frequency (FNF) of the simply supported unbonded prestressed beams. The energy method has been employed to derive the effective prestressing load to determine the eccentricity effect. In regard to the forced response of the prestressed beam, a moving point load with a constant value and various velocities and excitation frequencies is applied. Extensive parametric studies are carried out taking into account different factors including prestress load, eccentricity, concrete ratio, span-to-depth ratio, velocity, and frequency of the moving load. The comparison of the FNFs obtained by the formula with those obtained from FEM models indicates that the results are in a good agreement. This convergence demonstrates that the proposed formulation can predict the FNF of the eccentrically prestressed beams with high reliability. The time-histories curves for midspan displacement of the unbonded prestressed beams and the dynamic magnification factors are also evaluated. The results illustrate that the aforementioned factors have an indispensable contribution to the beam dynamic behavior.

... bridges, guideways, overhead cranes, cableways, runways, roadways, are subjected to moving load. Due to variation in position, the moving load differs from other dynamic loads [2]. Furthermore, due to increase in high-speed train networks, the elevation of vibration level due to high-speed trains is a challenging problem for engineers and researchers. ...

... The transient motion of a concentrated line load moving over an elastic half-space with a constant velocity has been examined by Ang [13] and Payton [14]. Olsson [2] discussed the response of a simply supported beam subjected to a constant force moving with a constant velocity. Fuchun [15] examined the characteristics of dynamic stresses induced in subway tunnels in clay stratum subjected to moving loads. ...

... In the absence of body forces, the equations of motion for functionally graded initially stressed orthotropic elastic halfspace are 123 ∂τ (2) 11 ...

The present article analyzes the induced compressive, shear and tensile stresses due to a moving load on three distinctly characterized irregular orthotropic half-spaces, viz. functionally graded orthotropic viscoelastic half-space (Case-I), functionally graded initially stressed orthotropic elastic half-space (Case-II) and orthotropic magnetoelastic half-space (Case-III) under hydrostatic initial stress. The expressions for said induced stresses are deduced in closed form using analytical approach. The influences of various physical parameters, viz. maximum depth of irregularity, functionally gradedness, irregularity factor, initial stress, magnetoelastic coupling parameter, hydrostatic initial stress and frictional coefficient on induced stresses for concerned cases, have been investigated with a comparative analysis. To depict the outcomes numerically, the half-spaces comprised of Carbon fiber, Prepreg and T300/5208/graphite/epoxy material have been taken into account and the observations are highlighted. Moreover, some notable characteristics have been outlined and delineated through graphs.

... The moving load problem was first mathematically described by Timoshenko (Timoshenko 1922) and in a comprehensive and detailed report by Frýba (1972) that explains formulation of moving force and moving mass for simple spans. Moving force-beam systems were also formulated to address vertical vehicle-bridge interaction problems ( (Filho 1978), (Olsson 1985), (Olsson 1991), (Wu et al. 2000)). Similarly, a comprehensive work on the formulation of human-structure systems was carried out by Caprani and Ahmadi (2016) for use in vertical human-induced vibrations. ...

... Analytical solutions to moving load problems are beneficial for parametric analyses. However, all moving force problems cannot be analytically solved and more detailed numerical methods are required to determine vibration response of such systems (Olsson 1991). Moving load problems can be treated as static loads applying to different positions on a structure for simplicity. ...

The next generation of ultra-high-speed (UHS) trains, known as Hyperloop and TransPod, are aerospace type vehicles designed to carry passengers. The UHS employs a vehicle capsule within a protected vacuum tube deck, supported by reinforced concrete piers (i.e. multi-span viaduct). The tube environment allows multiple UHS vehicles to run in parallel simultaneously (i.e. twin tube deck) where asymmetric train loading will result in a large dynamic unbalanced moment on the piers. Therefore, exploring the lateral dynamic interaction of bridge deck (twin tube) and piers under such an unbalanced moment is an extremely important factor for analysis of viaducts under dynamic UHS train loading. Hence, this paper analytically addresses the dynamic bridge deck-pier interaction under UHS train loading for lateral vibration.

... A plenty of industrial and engineering structures are permanently subjected to the moving loads such as bridges, railways and guide ways. Therefore, amongst the ever increasing published studies on the forced vibration of continuums, those concentrating the moving load dynamic problems have attained a significant interest of the researchers [1][2][3][4]. ...

... Thanking the high converging rate and satisfactory computing performance, orthogonal series-based solutions have been frequently employed in the vibration analyses of beamtype structures [2,3,13]; in this manner, favorable computational procedures of analysis could be established by the engineers and the specialists. A notable volume of published contributions currently exists addressing the free vibration of beams [14][15][16]; The EEM (eigenfunction expansion method) is one of the most well-known series-based solutions to the free/forced vibration of beams [1,3,12,15]. ...

One of the major issues facing structural engineers is assessing the effects of dynamic loads on structural systems, including beams. The importance of this matter arises in moving vehicles such as cars and trains on bridge structures that are usually simulated by beam structures. Hence, in several studies, in order to explore the dynamic response of the beam structures under the excitation of dynamic loads, various analytical and numerical methods have been utilized. In this study, to examine the easier and faster procedures aimed at finding the dynamic response of Euler-Bernoulli, Timoshenko and Higher-Order beams, a simple semi-analytical method based on the characteristic orthogonal polynomials and trigonometric functions compatible with boundary conditions is presented. To this end, discrete equations of motion are derived for the three mentioned theories due to a moving mass according to the Hamilton’s principle. Then, the governing equations are transformed into ordinary differential equations in the time domain, and by applying an approximate method, displacement field of the beam is achieved. In order to consider the efficiency, convergence rate and accuracy of this method, two numerical examples are provided to compare the results of this paper with those presented by other researchers. In this regard, in the former one, free vibration frequencies of the beam with various theories for different boundary conditions were obtained, and it is shown that, the results for all three theories, give a good convergence rate and a high accuracy. Furthermore, in the latter example, the dynamic response of the beams subjected to a moving mass for different values of the base beam slenderness was achieved and compared with other studies. Analysis of the maximum dynamic response of the beam and the time history diagrams illustrated the obtained results are in a close agreement with those issued from the numerical method; despite using the lower number of the shape functions.

... ( , ) = 2 2 = 0 at x=0 ad L (2a) [11] ( , ) = ∑ ( )sin ( ) ...

... The majority of the previous studies have been 14 dedicated to the investigation of the bridge vibrations consid- 15 ering the so-called moving load, moving mass and moving 16 sprung mass models for the vehicles. 17 The simplest moving load model can be chosen for a 18 vehicle in studying the bridge vibrations (see Ref. [2]). The 19 necessary dynamic characteristics of the bridge induced by 20 such simplest moving load action can be expressed with an 21 adequate degree of accuracy. ...

The box girder bridges constructed by prestressing the girders have been widely adopted to avoid the high dead load of the girders and for better performance during the service life. The loss of prestress may affect its performance and service life. Hence proper monitoring of such bridges with minimum efforts are useful for maintenance of the bridge. The correct estimation of the Impact Factor due to vehicle bridge interaction force is essential for designing such bridges. The presence of prestress in bridges influences the responses of bridges and vehicles moving over them. This paper provides the results of the finite element analysis of dynamic interaction between a vehicle and a simply supported pre-stressed box beam bridge. The interaction was conducted using ABAQUS software. Numerical simulations were conducted to show the effects of vehicle speed and road surface roughness profile on the mid-span displacement responses of the internal and external pre-stress box beams of the bridges. Thereafter, the influence of important parameters, namely, pre-stress force, road surface condition, and vehicle speed on the impact factor was investigated. Furthermore, it has been found that the maximum vertical vehicle acceleration response is sensitive to the prestress level and could be used to detect the percentage of prestress.

... Various researchers have provided analytical solutions for the VBI, treating the moving vehicle mass as the excitation to the bridge for obtaining system parameters. Pioneered by Timoshenko [1], Olsson [2] suggested the analytical solution comparing with the finite element solutions. Yang et al. [3], [4] derived the closed-form solutions about the frequency response in the VBI system assuming that the mass of the vehicle is relatively small compared to the bridge. ...

Railroad bridges are subject to large and directional mass of the vehicles , which their interactions cannot be ignored in the system identification of the structure. To dates, a number of researchers analytically and numerically characterized the time dependent frequency variations showing that the frequency amplification ratio (FAR) is highly related to the initial natural frequency of each system. However, most of those research focused on the interaction of the bridge's first mode to simplify the problem. This study investigated the interaction between a bridge and two succeeding vehicles, whose initial frequencies are close to the 2 nd bridge's initial frequency. Static analyses by solving the eigen-values of the interaction system are performed for wide variety of vehicle frequencies. Depending on the location of the vehicle, FAR showed higher order variation, which is more complicated than the first mode interaction. Dynamic analysis is also performed using a spatio-temporal frequency analysis. The results agreed well with static analyses indicating that the frequency variations can be characterized. The presented study showed potentials of identifying the fundamental characteristics of a railroad bridge under a train passage, both in the design phase and in the monitoring phase.

... It is depicted from Figures 6 and 7 that ̅ ( /2, ), over the entire time history, both increases and decreases, then increases to reach the peak value when the velocity reaches critical values. After that, ̅ gradually decreases as the moving load velocity increases, which is consistent with Olsson's observation [73]. The velocity at which ̅ attains its peak value is denoted as the beam critical velocity [74]. ...

This paper presents for the first time a closed-form solution of the dynamic response of sigmoid bidirectional functionally graded (SBDFG) microbeams under moving harmonic load and thermal environmental conditions. The formulation is established in the context of the modified couple stress theory to integrate the effects of microstructure. On the basis of the elasticity theory, nonclassical governing equations are derived by using Hamilton’s principle in combination with the parabolic higher-order shear deformation theory considering the physical neutral plane concept. Sigmoid distribution functions are used to describe the temperature-dependent thermomechanical material of bulk continuums of the beam in both the axial and thickness directions, and the gradation of the material length scale parameter is also considered. Linear and nonlinear temperature profiles are considered to present the environmental thermal loads. The Laplace transform is exploited for the first time to evaluate the closed-form solution of the proposed model for a simply supported (SS) boundary condition. The solution is verified by comparing the predicted fundamental frequency and dynamic response with the previously published results. A parametric study is conducted to explore the impacts of gradient indices in both directions, graded material length scale parameters, thermal loads, and moving speed of the acted load on the dynamic response of microbeams. The results can serve as a principle for evaluating the multi-functional and optimal design of microbeams acted upon by a moving load.

... The issue of the vibration of a slender beam subjected to a moving load series signifies typical engineering phenomena such as bridges under moving vehicles, railways subjected to high-speed trains, cableways with traveling telphers, elevated cranes when moving objects, etc. [1]. The moving load series can cause greater deflection than static loads and generate fatigue in the long term. ...

Traditional tuned mass dampers (TMDs) have been proven efficient for the moving-load-induced vibration control of beams. However, a large tuned mass is required in TMDs to adjust the structural dynamic characteristics and achieve the demanded performance targets, which leads to additional dynamic effects and inconvenience of installation. The tuned mass inerter system (TMIS) is an ungrounded lightweight passive control device, which contains a suspended mass, a parallel-connected tuned spring and an inerter-based subsystem. In this study, the application and optimization of TMISs on vibration suppression of multi-span beam models under moving load series are investigated. Using the Bubnov-Galerkin integration method, the modal-superposition generalized system for specified modes of a beam with TMIS under successive moving load series is established. Then, a design strategy for TMISs is proposed to achieve the structural target performance demand by decreasing the moving load-induced resonant responses and attached tuned mass. In the optimization algorithm, the tuned mass ratio is taken as the optimal objective, and the limited dynamic response amplitude under different speed parameters is the constraint condition. A single-span simply supported beam with the TMIS under a moving load series is analyzed. The designed TMISs are tuned to the dominating mode with the maximum mass participate factor of the main structures. TMISs show good vibration mitigation compared to TMDs with equal tuned mass. Meanwhile, under the premise of identical structural performance demands, TMISs require less tuned mass than TMDs and achieve a lightweight control effect. The corresponding dynamic amplitude vs speed curves and time history response curves of beams with TMISs and TMDs are also depicted through comparative analyses. The vertical deflection and acceleration responses apparently decrease, and the resonance is mitigated due to the designed TMISs. The sensitivity analysis results indicate that TMISs attached to beams are insensitive to the perturbance of their parameters and the input moving load excitations; thus, TMIS proves to be a robust tuning system.

... From Figs. 6 and 7, it is depicted that the maximum normalized dynamic deflection in the entire time history both increases and decreases, it then increases to reach the peak value when the moving load velocity reaches a certain value, and then after the maximum normalized dynamic deflection gradually decreases as the moving load velocity increases [113]. The velocity at which the maximum normalized dynamic deflection attains its peak value is referred to as the critical velocity of the BDFG beam [114]. ...

This paper presents an investigation of the dynamic behavior of bi-directionally functionally graded (BDFG) micro/nanobeams excited by a moving harmonic load. The formulation is established in the context of the surface elasticity theory and the modified couple stress theory to incorporate the effects of surface energy and microstructure, respectively. Based on the generalized elasticity theory and the parabolic shear deformation beam theory, the nonclassical governing equations of the problem are obtained using Lagrange’s equation accounting for the physical neutral plane concept. The material properties of the beam smoothly change along both the axial and thickness directions according to power-law distribution, accounting for the gradation of the material length scale parameter and the surface parameters, i.e., residual surface stress, two surface elastic constants, and surface mass density. Using trigonometric Ritz method (TRM), the trial functions denoting transverse, axial deflections, and rotation of the cross sections of the beam are expressed in sinusoidal form. Then, with the aid of Lagrange’s equation, the system of equations of motion are derived. Finally, Newmark method is employed to find the dynamic responses of BDFG subjected to a moving harmonic load. To validate the present formulation and solution method, some comparisons of the obtained fundamental frequency and dynamic response with those available in the literature are performed. A parametric study is performed to extensively explore the impact of the key parameters such as the gradient indices in both directions, moving speed, and excitation frequency of the acting load on the dynamic response of BDFG nanobeams. The obtained results can serve as a guideline for assessing the multi-functional and optimal design of micro/nanobeams acted upon by a moving load.

... Without forfeiting the validity of the analysis, a moving vehicle is simplified as a dynamic load q(x, t) in this study [86]. ...

Vehicle-bridge interaction is the core for a variety of applications, including vehicle vibration, bridge vibration, bridge structural health monitoring, weight-in-motion, bridge condition inspection, and load rating. These applications give rise to a great interest in pursuing a high-efficiency method that can tackle intensive computation in the context of vehicle-bridge interaction. This paper studies the accuracy and efficiency of discretizing the beam in space as lumped masses using the flexibility method and as finite elements using the stiffness method. Computational complexity analysis is carried out along with a numerical case study to compare the accuracy and efficiency of both methods against the analytical solutions. It is found that both methods result in a similar level of accuracy, but the flexibility method overperforms the stiffness method in terms of computational efficiency. This high efficiency algorithm and corresponding discretization schema are applied to study the dynamics of vehicle-bridge interaction. A system of coupled equations is solved directly for a simply supported single-span bridge and a four-degree-of-freedom vehicle modeling. Pavement roughness significantly influences dynamic load coefficient, suggesting preventative maintenance or timely maintenance of pavement surface on a bridge, to reduce pavement roughness, is of significant importance for bridge’s longevity and life-cycle cost benefit. For class A and B level pavement roughness, the dynamic load coefficient is simulated within 2.0, compatible with specifications of AASHTO standard, Australian standard, and Switzerland standard. However, the Chinese code underestimates the dynamic load coefficient for a bridge with a fundamental frequency of around 4 Hz. The proposed method is applicable to different types of bridges as well as train-bridge interaction.

... The train-induced vibration analysis of railway bridges has been extensively studied by many researchers, including Frýba [1], Weaver et al. [2], Olsson [3], Yang et al. [4], Michaltsos [5] and Savin [6], among others [7][8][9][10][11][12][13]. Due to the periodic nature of the train loading, a railway bridge may experience resonance effects. Resonance in railway bridges occurs when the train loading frequency, i.e. the ratio between the train speed and the characteristics axle distance, coincides with the fundamental frequency of the bridge [1,[14][15]. ...

In this paper, a closed-form approximate formula for estimating the maximum resonant response of beam bridges on viscoelastic supports (VS) under moving loads is proposed. The methodology is based on the discrete approximation of the fundamental vertical mode of a non-proportionally damped Bernoulli-Euler beam, which allows the derivation of closed-form expressions for the fundamental modal characteristics and maximum amplitude of free vibration at the mid-span of VS beams. Finally, an approximate formula to estimate maximum resonant acceleration of VS beams under passage of articulated trains has been proposed. Verification studies prove that the approximate closed-form formula estimates the resonant peaks with good accuracy and is a useful tool for preliminary assessment of railway beam bridges considering the effect of soil-structure interaction at resonance. In combination with the use of full train signatures through the Residual Influence Line (LIR) method, the proposed solution yields good results also in the lower range of speeds, where resonant sub-harmonics are more intensely reduced by damping.

... For many years it has been commented that when a bridge is subjected to moving loads, the caused dynamic deflections and stresses can be remarkably higher than those seen for the static state [3]. For this reason, numerous works have been devoted to investigating the bridge displacement response using the moving mass [4][5][6][7][8], moving load [9][10][11], and moving spring-mass models [12,13] for the vehicles. ...

Several authors, utilizing both experimental tests and complicated numerical models, have investigated vehicle speed's impact on a highway bridge's dynamic amplification. Although these tests and models provide reliable quantitative data on frequency contents of the interaction between the two subsystems, engineers should pay further notice to the effects of a subsystem's velocity and the type of suspension system of a vehicle moving over a structure. Hence, in this paper, the dynamic response of a bridge to a moving vehicle is considered. The car is assumed as a quarter car model with both linear and nonlinear stiffness and damping constants. Further, using the modal superposition method, a closed-form solution is obtained for the bridge's vertical response. The results obtained via numerical calculation show a significant increase in the bridge midpoint and total deflection, velocity, and acceleration by increasing the vehicle velocity. Moreover, by neglecting the nonlinear stiffness and damping coefficients of the vehicle suspension system, the bridge's dynamic response remains almost the same with respect to the numerical data. As a general conclusion, it can be claimed that the only significant parameters which can change the dynamic behavior of a bridge subjected to a moving vehicle are the speed of the car and its linear stiffness and damping constants inside its suspension system.

... This special moving speed is specified as the critical velocity of the moving load by some researchers. In the present configuration, the special moving load speed is 0.6Lω 1 /π, while the analytical critical speed for a homogeneous beam is 0.619 Lω 1 /π [80]. And the temperature increment does not affect the critical moving load speed. ...

Due to the multi-functional requirements, more and more multi-dimensional functionally graded structures are reported, while their thermo-mechanical performances are still unclear. This paper aims at theoretically examining the thermo-mechanical behaviors of two-dimensional functionally graded (2D-FG) microbeam excited by a moving load. The formulation is established in the framework of the modified couple stress theory in combination with Hamilton's principle, and this problem is numerically solved through the finite element method (FEM). The material properties of the 2D-FG microbeam are treated as temperature-dependent parameters that vary smoothly along with both the axial and thickness directions, and three different thermal loads including the uniform, linear, and nonlinear temperature rises are taken into account. Both the fundamental frequency and the dynamic transverse deflection are analyzed with consideration of different axial and thickness gradient indexes, thermal load types, temperature rise amplitudes, and small scale parameters. The conclusion obtained in this paper can serve as a guideline for assessing the thermal performance of microstructures subjected to a moving load.

... In the preliminary stage of bridge design or in the quick assessment of existing bridges, this model can be the most convenient. Various applications of this model with both analytical and FE approaches were widely discussed [4,[10][11][12][13][14]. In the EN 1991-2 [5], the dynamic analysis is performed using the so-called HSLM (high speed load model) which represents a series of moving forces and it is stated that the dynamic analysis should be conducted using the real specified train in a specific project (bridges on local lines) but concerning the international lines for which the interoperability criteria are applied, the HSLM should be used to ensure an envelope response of all current real and prospective high-speed train loads. ...

The dynamic interaction between the bridge and the passing vehicle or train is considered a point of interest concerning the railway bridge design and maintenance, and the interest becomes greater for the bridges serving as links on high-speed lines. In this paper, the interaction problem is presented through the different models used
to describe the phenomenon and the different techniques adopted to solve the nonlinear interaction problem. The models describing the problem vary greatly from very simple 2D models with moving loads over beams to complex 3D models with multiple degrees of freedom (DOFs) for both the bridge and vehicle and with precise definition of various parts and parameters affecting the response such as the type of bridge element, the track structure and the bridge elastic supports. The solution algorithms of the non-linear interaction problem also vary from simple analytic solutions and non-direct techniques to more sophisticated iterative techniques in finite element (FE) domains.

... Analytical approaches for moving load problems are very appropriate for parametric analyses. However, it should be noted that all moving force problems cannot be analytically described and more rigorous numerical approaches are necessary to calculate dynamic effects of moving loads [6]. Dynamic effects of moving loads can be substantial particularly for HS trains. ...

The ultra-high-speed (UHS) Hyperloop is the next-generation mode of passen-ger/freight transportation, and is composed of a tube or a system of tubes through which a pod travels free of friction. The entire system must be supported by piers (multi-span viaducts), where the tubes act as the bridge deck. The UHS moving Hyperloops can exert large dynamic effects to the supporting pier-deck system both vertically and laterally. Particularly, asymmetric Hyperloop loading can generate significant lateral vibrations. Therefore, for safe design of a bridge pier-deck system for UHS trains, it is of great importance to explore dynamic interaction of bridge deck and piers under UHS moving Hyperloops. Hence, this paper analytically summarizes the dynamic amplification factors of the Hyperloop-deck-pier system for vertical and lateral vibrations. It was found that the UHS Hyperloop trains result in higher dynamic effects compared to the high-speed trains.

... Olsson [121] discussed the dynamical problem of a simply supported beam subjected to a constant-amplitude force moving at a constant speed. Throughout that paper, an analytical solution based on the separation of variables was presented and compared with those numerically obtained by the FEM. ...

The present work outlines an original numerical modelization approach for Moving Load (ML) beam problems, by a dedicated object-oriented C++ parallel computing FEM implementation, with the purposes of performing efficient numerical analyses resolving the complete dynamic response of beams under the effect of a high-velocity ML. Alongside, main framing state-of-the-art reviews are attempted, on the principal involved issues of: ML context and physical description, numerical FEM modelization, parallel computing implementation. Running ML example cases are explored, for a (long) finite beam on a (visco)elastic foundation and for a continuous beam of a historic railway iron bridge, with per se interesting engineering outcomes. The contribution may serve as a guideline paradigm to readers that may be novel to the treated topics, though motivated in adventuring in the computational challenges involved in the present mechanical research context.

... Eason [3] studied the problem for stresses produced in a homogeneous isotropic half-space subjected to load moving at constant velocity. The relevant theoretical investigations in this field were made by Achenbach et al. [4], and this study has been improved and developed continuously by many researchers [5][6][7][8][9][10], etc. ...

The present study is concerned with the dynamic response of an anisotropic composite structure due to a normal moving load on its irregular rough surface. The composite structure is comprised of an irregular incompressible heterogeneous transversely isotropic fluid-saturated poroelastic layer lying over a transversely isotropic substrate. The mathematical formulation of this structure gives rise to a boundary value problem with specified boundary conditions, and the perturbation method has been used to tackle the irregular surface problem. The expressions for the induced shear and normal stresses in layer and substrate of the composite structure are derived analytically in closed form due to the moving load. As a special case of the problem, the deduced expressions of the induced stresses are validated with the pre-established and standard results. The effect of several substantial parameters such as vertical depth, heterogeneity parameter, porosity parameter, frictional coefficient, irregularity depth, and irregularity factor on the induced shear as well as normal stresses of the layer and substrate has been delineated graphically by the numerical computation. Moreover, a comparative study of the various types of irregularity, namely rectangular irregularity, parabolic irregularity and no irregularity (regular boundary surface) on the induced shear and normal stresses in the layer and, substrate, is carried out by means of graphs, and some considerable peculiarities are outlined.

... As the easiest case, the bridge was modeled as a simply supported beam, traversed by a moving force with constant speed [2]. It was found that the dynamic response of the beam was related with the speed of force, and the maximum dynamic deflection can be 1.74 times as high as the static deflection for the simply supported beam [3,4]. Meanwhile, it was shown that the dynamic response of the beam would achieve an extreme small value at a certain speed. ...

The dynamic response of an Euler–Bernoulli beam under moving distributed force is studied. By decomposing the distributed force into Fourier series and extending them to semi-infinite sine waves, the complex procedure for solving this problem is simplified to three base models, which are calculated by the modal superposition method further. The method is proved to be highly accurate and computational efficient by comparing with the finite element method. For verifying the theory and exploring the relationship between dynamic pressure due to train gust and vibration of the structure, a site test was conducted on a platform canopy located on the Beijing-Shanghai high-speed railway in China. The results show the theory can be used to evaluate the dynamic response of the beam structure along the trackside due to the train gust. The dynamic behavior of a 4-span continuous steel purlin is studied when the structure is subjected to the moving pressure due to different high-speed train passing.

... Vibrations of solid beam and plate structures acted upon by moving loads [31][32][33][34][35][36][37], moving masses [38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54], and moving mass-sprung systems [55][56][57] have been examined. Kiani et al. [58] also examined the role of shear deformation on vibrational behavior of poroelastic beams under a moving pointed load. ...

... Fryba established a thorough study about the moving force problem on both finite and infinite beams, as well as two-and three-dimensional solids [3]. By modeling a moving subsystem as concentrated force or a sliding mass, the dynamic response of the coupled system has been intensively studied in the past, with various solution methods being proposed, including polynomial eigenfunction method [4], frequency-domain spectral element method [5], influence function method [6][7], Lagrange multiplier method [8], modal expansion method [9][10][11] and assumed-mode method [12][13]. Nevertheless, a moving force or moving mass fails to describe the flexible interaction between the supporting structure and a moving subsystem by neglecting the inertia effect of the subsystem or by assuming rigid contacting between the subsystem and the structure. ...

Dynamic analysis of a multi-span beam structure carrying moving rigid bodies is essentially important in various engineering applications. With many rigid bodies having different speeds and varying inter-distances, number of degrees of freedom of the coupled beam-moving rigid body system is time-varying and the beam-rigid body interaction is thus complicated. Developed in this paper is a method of extended solution domain (ESD) that resolves the issue of time-varying number of degrees and delivers a consistent mathematical model for the coupled system. The governing equation of the coupled system is derived with generalized assumed mode method through use of exact eigenfunctions and solved via numerical integration. Numerical simulation shows the accuracy and efficiency of the proposed method. Moreover, a preliminary study on parametric resonance on a beam structure with 10 rigid bodies provides guidance for future development of conditions on parametric resonance induced by moving rigid bodies, which can be useful for operation of certain coupled structure systems.

... Abu-Hilal [2], investigated the response of a double Euler-Bernoulli beams, due to a moving loads. Moreover, literatures of [22], [14], [5], [19], [16], [20], [15], [24], [8], and [10] are work done on the theory and analysis of vibrations that are still producing positive development in the field of engineering. However, double-plate systems are very much applicable in engineering which includes, decking systems for railways viaducts and bridges, tunnels, anti slides and avalanche guards, industrial flooring systems, commercial flooring with high loading capacities, construction of trolleys and girders, and so many other applications. ...

In this paper, the dynamic response of two identical parallel non-
mindlin (i.e., not taking into account the eﬀect of shear deformation and ro-
tatory inertia) plates which are elastically connected and subjected to a con-
stant moving load is considered. The fourth order coupled partial diﬀerential
governing equations is formulated and solved, using an approximate analytical
method by assuming; ﬁrstly, a series solution later on treating the resulting cou-
pled second order ordinary diﬀerential equations with an asymptotic method of
Struble. The diﬀerential transform method, being a semi-analytical technique,
is applied to the reduced coupled second order ordinary diﬀerential equations,
to get a non-oscillatory series solution. An after treatment technique, com-
prising of the Laplace transform and Pade approximation techniques, is ﬁnally
used via MAPLE ODE solver to make the series solution oscillatory. The
dynamic deﬂections of the upper and lower plates are presented in analytical
closed forms. The eﬀect of the moving speed of the load and the elasticity
of the elastic layer on the dynamic responses of the double plate systems is
graphically shown and studied in details. The graphs of the plate’s deﬂections
for diﬀerent speed parameters were plotted. It is however observed that the
transverse deﬂections of each of the plates increase with an increase in diﬀerent
values of velocities for the moving load for a ﬁxed time t.

... Abu-Hilal [2], investigated the response of a double Euler-Bernoulli beams, due to a moving loads. Moreover, literatures of [22], [14], [5], [19], [16], [20], [15], [24], [8], and [10] are work done on the theory and analysis of vibrations that are still producing positive development in the field of engineering. However, double-plate systems are very much applicable in engineering which includes, decking systems for railways viaducts and bridges, tunnels, anti slides and avalanche guards, industrial flooring systems, commercial flooring with high loading capacities, construction of trolleys and girders, and so many other applications. ...

In this paper, the dynamic response of two identical parallel nonmindlin (i.e., not taking into account the effect of shear deformation and rotatory inertia) plates which are elastically connected and subjected to a constant moving load is considered. The fourth order coupled partial differential governing equations is formulated and solved, using an approximate analytical method by assuming; firstly, a series solution later on treating the resulting coupled second order ordinary differential equations with an asymptotic method of Struble. The differential transform method, being a semi-analytical technique, is applied to the reduced coupled second order ordinary differential equations, to get a non-oscillatory series solution. An after treatment technique, comprising of the Laplace transform and Pade approximation techniques, is finally used via MAPLE ODE solver to make the series solution oscillatory. The dynamic de ections of the upper and lower plates are presented in analytical closed forms. The effect of the moving speed of the load and the elasticity of the elastic layer on the dynamic responses of the double plate systems is graphically shown and studied in details. The graphs of the plate's de ections for different speed parameters were plotted. It is however observed that the transverse de ections of each of the plates increase with an increase in different values of velocities for the moving load for a fixed time t.

It is well understood that when an object travels over a supporting plate, their initial contact is most likely to be lost, particularly in the presence of surface roughness. While there is a voluminous literature on the analysis of plates excited by travelling objects, the existing studies have neglected the possibility of separation between contacting bodies in the problem formulation. On that account, this paper attempts to develop an approach in order to improve the previously-adopted methods by allowing for the separation along with effectively taking care of the subsequent recontact between the travelling superstructure and the supporting plate. Furthermore, by considering the so-called concept of restitution coefficient at the point of recontact, different impact situations are examined and discussed. Numerical examples are then presented to investigate the effect of influential parameters on the behaviour of the system. It is shown that an accurate assumption of the impact condition is necessary to obtain a reliable prediction of the oscillator response. Nevertheless, the plate response, except for some localized outputs, is not as sensitive to the assumed restitution coefficient. The findings also suggest that separation can even take place at low velocities if roughness is involved in the problem. Moreover, it is indicated that neglecting the possibility of separation/recontact as well as surface roughness leads to seriously unsafe results with respect to both the plate and the oscillator response spectra.

ABSTRACT
Tis paper studies the eﬀect of the velocity of moving vehicles on dynamic responses of multi span continuous beams. The equation of motion is derived by principle of dynamic balance and the fnite element method with beam element. A computer program based on mATLAB language is developed and verifed by comparing analysis results with those found from literature review to predict the dynamic behavior of the beam. From the numerical study, it is concluded that the velocity has the signifcant inﬂuence on the dynamic responses and the critical speed of the moving vehicles which is expressed as maximum responses is shown.
Keywords: multi span continuous beams, Dynamic analysis,
moving vehicle, Velocity.

Dynamic amplification of loads in masonry arch railway bridges is not well understood. There is a scarcity of experimental data and previous numerical studies have only addressed a few specific bridge geometries. Despite this, guidance documents provide empirical and unvalidated formulae to calculate dynamic amplification for masonry arch railway bridges. To improve our fundamental understanding of the problem, determine appropriate modelling strategies and evaluate the reliability of guidance documents, simple 2D and 3D models are explored in this paper. The 2D approach idealises key bridge components (pier, arch, fill and backing elements) with straight Timoshenko beams, springs and lumped masses. It uses an analytical dynamic stiffness formulation, which is computationally efficient and well-suited to explore a range of bridge models. The higher fidelity 3D modelling approach uses shell and solid finite elements and is used to evaluate the limitations of 2D models. In both approaches, linear-elastic material behaviour is assumed and train loads are idealised as moving vertical loads distributed over an effective area. The modelling results indicate a complex relationship between train speed and dynamic amplification that depends critically on bridge geometry and axle spacing. In general, the multi-span bridge configurations experienced higher dynamic amplification over operational train speeds. The results also higlight deficiencies in existing code provisions and demonstrates how efficient numerical models may replace these provisions.

The bridge pavement subjected to vehicle loads is the most vulnerable component in the highway bridge and overloaded vehicles can result in serious damage to the bridge pavement more easily. Therefore, it is necessary to consider the pavement in vehicle-bridge interaction (VBI) system. However, the bridge pavement is not considered in the traditional VBI system or the viscoelasticity of bridge pavement is ignored. In this study, a new vehicle-pavement-bridge interaction (VPBI) system is established. The viscoelastic asphalt pavement is simulated with the continuously and uniformly distributed spring-damper. The equivalent stiffness coefficient and equivalent damping coefficient of the pavement are determined through dynamic mechanical analysis (DMA) test and finite element method. The equations of motion for VPBI system can be derived using Lagrange equation and the modal superposition method. Then the crucial parameters such as displacement at mid-span of bridge, acceleration of vehicle body, tire contact force, and pavement deformation are obtained. The effects of vehicle velocity, bridge span, tire contact area, pavement stiffness coefficient, and pavement damping coefficient on VPBI responses are analyzed. The results show that (1) The pavement is compressive during vehicle passing bridge and its dynamic deformation fluctuates around initially static deformation of pavement. (2) Increasing the stiffness of pavement can rapidly reduce the deformation of pavement, while the pavement damping has the opposite effect. (3) The increase of bridge span can conduct exponentially growth of dynamic responses of VPBI system. The pavement deformation will increase by 7.9% if the bridge span is lengthened by 5 m. This study provides a reliability response analysis method for VPBI system.

Dynamic analysis of simply supported uniform beams excited by a single and multiple moving point loads is presented in this paper. Analytical expressions for the power distribution and the net vibration power flow in a simply supported beam as a function of the (non-dimensional) speed of the moving load are formulated. A closed-form expression for the net vibration power flow in the beam during forced vibration, as a function of free vibration responses, is proposed. Analytical expression for zero-power location of the beam, and numerical expression for maximum/minimum power location for a particular speed of the moving load, are developed. Investigation of cancellations of free vibration responses of the beam, on the basis of vibration power distribution in the beam, is carried out. Conditions for maxima and cancellations of free responses of the beam and the corresponding speed ratios are also determined. Power flow analysis in the case of multiple moving point loads at different speeds is also carried out. The results obtained have high practical relevance in energy harvesting from vibrating beams as they can be used for estimating the vibration power distribution and net power flow in bridges traversed by rail cars.

This paper studies vibration of sandwich beams reinforced by carbon nanotubes (CNTs) under a moving point load. The core of the beams is homogeneous while their two faces are of carbon nanotube reinforced composite material. The effective properties of two face sheets are determined by extended rule of mixture. A uniform distribution (UD) and four different types of functionally graded (FG) distributions, namely FG-X, FG-FG-V, FG-O, are considered. Based on a third-order shear deformation theory, a finite element formulation is derived and used to compute the vibration characteristics of the beams.

Response of moving load over a surface is an intriguing problem of mechanics to determine the stability and strength of a structure. Owing to this the present theoretical framework is devoted to find the stresses and electrical displacements of an irregular visco-porous piezoelectric half-space originated due to a uniformly moving line load. Expressions for normal stress, shear stress and electrical displacements have been derived in closed form. Effect of irregularity depth, irregularity factors and frictional coefficient on the stresses and electrical displacements are delineated graphically. Numerical demonstration of procured results is interpreted by means of graphs for two different materials, namely PZT-5A and PZT-7. A comparative study emphasising various irregularity (parabolic, rectangular and no irregularity) is among the salient features of the study.

In this article, the passage of different trains over different bridges will be studied for resonant excitation. The intensity of the resonance will be estimated in frequency domain by using three separated spectra. At first, the excitation spectrum of the modal forces is built by the mode shape and the passage time of the train over the bridge. The second spectrum is the frequency response function of the bridge which include the modal frequency, damping and mass. The third part is the spectrum of the axle sequence of the train. The influences of train speed, bridge length, bridge support, track irregularities, and train type on the resonance amplitudes will be analysed for each of these spectra separately for getting a better insight. A variety of axle-sequence spectra and corresponding rules will be presented for different vehicles and trains. As examples, the passage of a slow freight train over a long-span bridge, a normal passenger train over a medium-span bridge, and a high-speed train over a short bridge will be analysed. Corresponding measurements show the amplification, but also the cancellation of the subsequent axle responses. Namely in one of the measurement examples, the first mode of the bridge was amplified and the second mode was cancelled at a low speed of the train and vice versa at a higher speed.

The present work deals with the influence of cryogenic coolants LN2 delivered through holes made on flank surface and rake surface of tungsten carbide cutting tool inserts in turning of super duplex stainless steel (SDSS) using in-house developed cryogenic setup. Experiments were conducted with the cryogenically treated tool, cryogenically treated tool with tempering and cryogenic coolant directly passed through a modified cutting tool insert. Results are compared with dry cutting conditions. The cutting conditions are low feed rate/high depth of cut, medium feed rate/medium depth of cut, and high feed rate/low depth of cut. The material removal rate and cutting speed is kept constant under all three cutting conditions. Microstructural study of the tool as received and cryogenically treated is examined using SEM. The population of harder tungsten carbide phase (gamma phase) is found to be more in the cryogenically treated tool. Due to tempering, the hardness of insert is improved by 8% which in turn increased tool life. By direct supply of LN2 through modified cutting tool increased tool life by 23%, more than the cryogenically tempered tool. There are no appreciable changes in the temperature of the cutting tool under dry cutting and cryogenically treated inserts. However, there is a large difference observed in temperature of cutting tool when LN2 is supplied through a modified insert directly, which in turn yielded high tool life.

Dynamic analysis of an inclined sandwich beam under a moving mass is presented on the basis of a third-order shear deformation theory. The core of the sandwich beam is homogeneous while the two face sheets are made from three-phase bidirectional functionally graded material with effective properties varying in both the axial and transverse directions by power gradation laws. A finite element formulation for the inclined beam, in which the stiffness and mass matrices are evaluated explicitly, is derived by using the transverse shear rotation as an independent variable. Taking into account the effects of the inertial, Coriolis and centrifugal forces, a moving mass element is formulated and superimposed on the beam matrices to construct the equation of motion. The numerical result reveals that the material gradation plays an important role on the dynamic response of the beam, and the desired dynamic characteristics can be obtained by approximately choosing the power-law indexes. The effects of the moving mass velocity, the material gradation, the beam inclined angle and layer thickness ratio on the dynamic behaviour of the beams are studied in detail and highlighted.

Moving load dynamic problems with lubricant film are common in engineering. Previous studies of this kind of problem often ignored lubricating oil film, or regarded oil film as a simple linear system. To avoid errors caused by the nonlinearity of oil film, hydrodynamic equations are used to describe the oil film subsystem accurately in this paper. By coupling the kinematic systems (moving mass and elastic carrying mechanism), a comprehensive model of moving load dynamic problems with lubricant film is established, and a general numerical procedure of the problem is given. Then the dynamic response and the transmission law of load in the mechanism are studied by taking a reciprocating friction pair of a typical marine engine as an example. Results show that the load transmission is not sensitive to low-frequency harmonic signals, but has a strong filterability to the mid-high frequency harmonic signals. It is also found that the transfer rate of excitation signal through the oil film does not decrease smoothly as frequency increases, and there exist multiple peaks in the frequency domain.

This paper studies the dynamic response of a cantilevered beam subjected to a moving moment and torque, and combination of them with a moving force. The moving loads are considered to traverse along the length of the beam either from fixed-to-free end or free-to-fixed end. The beam is considered to have constant material and geometric properties. The beam is modeled using the Rayleigh beam theory considering the rotary inertia effects. The Dirac-delta function used to model the moving loads in the governing partial differential equations (PDEs) has complicated the solution of the problem. The Eigenfunction expansions coupled with the Laplace transformation method is used to find the semi-analytical solution for the resulting governing PDEs. The effects of moving loads on the dynamic response are studied. The dynamic effects are quantified based on the number of oscillations per unit travel time of the moving load and the Dynamic Amplification Factor (DAF) of the beam’s tip response. Numerical results are also analyzed for the two-speed regimes, namely high-speed and low-speed regimes, defined with respect to the critical speed of the moving loads. The accuracy of the analytical solutions are verified by the finite element analysis. The numerical results show that the loads moving with low speeds have significant impact on the dynamic response compared to high speeds. Also, the moving moment has significant impact on the amplitude of dynamic response compared with the moving force case.

In moving load-type problems, the moving point load is modeled mathematically by a time-dependent Dirac-delta function. A key and difficult step in solving this type of problems using a point discrete method such as the differential quadrature method is the discretization of the Dirac-delta function in a simple and accurate manner. This paper is conducted to facilitate this step and to present a new way to do this task. In this way, the Dirac-delta function is approximated by orthogonal polynomials such as the Legendre and Chebyshev polynomials. Unlike the original Dirac-delta function, which is a generalized singularity function, the resulting approximation function is a non-singular function which can be discretized simply and efficiently. The proposed procedure is applied herein to solve the moving load problem in beams and rectangular plates. Comparisons with available analytical and numerical solutions prove that the proposed approach is highly accurate and efficient.

We present a general formulation for problems of sliding structures and axially moving beams that undergo large deformations. The formulation relies on a coordinate transformation that facilitates the analysis of beams characterized by a large sliding motion, for which conventional approaches typically become inefficient. The transformation maps variable domains in the material coordinate, which result, e.g., from the beam's sliding motion relative to its supports and external loads, onto fixed domains with respect to the new stretched coordinate. We do not only consider supports and loads prescribed at variable points and domains, but their current position relative to the material points of the structure may additionally depend on the current state of deformation. Hamilton's principle and the geometrically exact theory for shear-deformable beams serve as basis for the derivation of the equations of motion in the stretched coordinate. We introduce a generalized notion of variation that includes the boundaries of variable domains as unknowns and we discuss the implications on the governing equations. Upon a spatial semi-discretization, symmetric mass and tangent stiffness matrices are obtained from the variational formulation of the equations of motion along with non-linear velocity and stiffness-convection terms. Several numerical examples demonstrate both the range of applications and the advantages of the proposed formulation in problems of sliding structures and axially moving beams.

An investigation concerned with the dynamic response of a composite structure to a moving load on its uppermost rough surface has been carried out analytically. The composite structure is comprised of an incompressible fluid-saturated transversely isotropic poroviscoelastic layer of finite width imperfectly bonded with an underlying transversely isotropic viscoelastic semi-infinite medium. Closed-form expressions of induced stresses (shear and normal) and induced pore pressure are established analytically using the appropriate boundary conditions. The established results are validated with the preestablished results. The substantial effects of influencing parameters—porosity; frictional coefficient; shear and transverse imperfection (bonding) parameters; shear viscosity, volume viscosity, and extensional viscosity parameters; and the vertical depth (from the free surface) of the medium—on the induced stresses (shear and normal) of the layer and semi-infinite medium and the induced pore pressure of the layer of the considered composite structure are demonstrated graphically through numerical computation. Moreover, a comparative examination of distinct cases of the problem serves as the salient feature of the study and contributes better analysis than found in the literature.

Dynamic behavior of continuous systems, such as beams and plates under the application of moving concentrated loads, is an important issue in engineering. In this study, a meshfree method is presented for the analysis of the dynamic response of thick plates under the influence of concentrated moving loads. The displacement field is based on the third-order shear deformation theory. In this numerical method, the field variables are interpolated only by using nodes distributed purposively in the computational domain. Since there is no conectivity between the nodes, it is possible to add nodes in the areas of application of the force. Another feature of the proposed method is the use of the radial point interpolation method (RPIM) shape functions which possess the Kronecker delta function property and therefore satisfies the essential boundary conditions easily. Also, due to the high density of nodal points in the vicinity of the point of application of the load, the background decomposition method (BDM) is used in order to achieve a high accuracy with appropriate speed. In this paper, nodes are rearranged in the path of the moving load adaptively which leads to high accuracy and speed of the final solution. To validate the proposed method, the obtained results are compared with the analytical solutions.

The present study bestows the analytical investigation of incremental mechanical stresses (compressive stress, shear, and tensile) and electrical displacements (vertical and horizontal components) induced due to a moving line load on an irregular transversely isotropic functionally graded viscoelastic-piezoelectric material (FGVPM) substrate. The closed form expressions of said induced mechanical stresses and induced electrical displacements are deduced and validated with pre-established results for electrically open and short conditions. The elastic moduli (stiffness tensors), piezoelectric moduli, dielectric moduli, elastic loss moduli, piezoelectric loss moduli, and dielectric loss moduli for a viscoelastic-piezoelectric composite are computed and used for numerical computation and graphical demonstration. The effectuality of diverse physical parameters (viz. maximum depth of irregularity, friction due to rough upper surface, functional gradient parameter, irregularity factor associated with different types of irregularity viz. rectangular irregularity, parabolic irregularity and no irregularity) on said induced stresses and electrical displacements in the aforementioned composite substrate has also been discussed. A comparative analysis has also been made to examine the impact of piezoelectricity and viscoelasticity on theon said induced mechanical stresses and induced electrical displacements. In particular, some special peculiarities are also sketched by means of graphs.

Treated herein is an elastic beam that is subjected to a constant load that travels continuously (back and forth) along its span. The dynamic deflection of the beam is investigated analytically so as to predict the critical and cancellation speeds. Closed-form solutions are obtained for the damped system for each load condition and superimposed to determine the total solution of the response. Unlike the classical case of load traveling at constant speed in one direction, multiple resonance peaks are observed for reciprocating load at speeds that are lower than the classical critical speed. It is also observed that resonance may not exist at the classical critical speed for simply supported beams due to the symmetry of the beam. The dynamic deflection is examined for simply supported beams to determine the speeds that cause amplification or cancellation of the free response. The current load condition may have possible application in view of its potential use for vibration suppression, as a moving vibration absorber, or for magnification, in energy harvesting. The results are interpreted in order to understand the variation of dynamic deflection and to estimate the critical speeds for different load conditions.

Dans un contact pneu/chaussée, le bruit de roulement résulte de l’interaction mécanique entre les aspérités de la chaussée et les pains de la bande de roulement. À l’issue de cette interaction, des forces compressives apparaissent pour repousser les deux corps en contact. Ces forces conduisent à la vibration du pneumatique. Ces vibrations sont à l’origine du bruit rayonné. Le travail de cette thèse s’inscrit dans le cadre de l’évaluation du bruit de roulement. L’objectif est double. Premièrement, il s’agit de comprendre les mécanismes à l’œuvre dans un processus de roulement de deux surfaces rugueuses qui engendrent une vibration puis du bruit. Deuxièmement, on cherche à mettre en évidence l’influence des aspérités de la chaussée sur les forces dynamiques interfaciales et sur le bruit généré. Dans ce contexte, on propose un nouveau modèle 3D de contact dynamique basé sur la décomposition modale de la réponse du pneumatique. Cette nouvelle approche permet de réduire considérablement le temps CPU. Le pneumatique est modélisé par une plaque orthotrope sur fondation élastique. Le problème de contact est résolu par la méthode de pénalité. On a validé ce modèle analytiquement. Cet outil permet de prédire finement ce qui se passe dans la zone de contact. Nous pouvons prédire les forces de contact et les vitesses vibratoires. En outre, il permet de déterminer l’aire de contact et les cartes de pression. À l’échelle locale, les caractéristiques d’un choc sont connues. On est capable de déterminer la force maximale du choc, à partir de l’évolution temporelle de la force de contact, et sa durée mais aussi le pourcentage de temps du choc.

The understanding of the dynamic response of a beam under a moving load is an active research area in structural engineering. The aim of this note is to investigate such a response for piecewise constant beams that display wave filtering properties associated with band gaps. The performance of five simply supported nonhomogenous piecewise constant beams under concentrated load traveling at ten different constant speeds was analyzed and compared with that of a simply supported homogenous beam with the same volume. The nonhomogenous beam design and the speed range were based on the band gap theory for which traveling waves in a specific range of frequencies cannot propagate for a given piecewise geometry. The investigation has been carried out through analytical and numerical approaches by implementing equations in a MATLAB script and using the finite elements software, LUSAS, respectively. Factors such as number of vibration modes and damping effect were also considered. It was concluded that i) higher speeds do not necessarily generate a more severe beam response; ii) some of the investigated nonhomogenous beams exhibit good performance for a range of speeds that corresponds to the lower interval of the band gap range.

A finite element modeling based on a moving mesh strategy is developed to predict Vehicle–Bridge Interaction (VBI) in moving load applications. The formulation is based on an Arbitrary Lagrangian–Eulerian (ALE) approach, which is able to reproduce the vehicle motion on the basis of a fixed-referential system and moving coordinate variables, representing the positions of the external system. Conceptually, the vehicle and the bridge are considered as an integrated system, in which wheel-bridge interaction as well as the internal forces are implemented by means of moving interface elements. The governing equations of the VBI are derived in the referential coordinate system, whereas the ones associated to bridge structure are not modified from the conventional formulation. Such task is achieved by means of proper projection operators, which define the mapping between the referential and material configurations, reproducing the contact forces between the substructures. The proposed model is quite general to be endorsed in classical FE approach, since it does not require specialized connections with conventional structural formulations. The accuracy and efficiency are verified by means of comparisons with existing numerical results available from the literature, including cases in which the bridge structure is affected by damage phenomena.

The object of this paper is to analyse by the finite element method the dynamic responses of a flat plate subjected to various moving loads. First of all the actual continuous flat plate was replaced by a discrete system composed of isoparametric rectangular plate elements. And then the elementary and overall stiffness and mass matrices were determined and the natural frequencies and mode shapes of the flat plate were solved. Next, the Newmark direct integration method was used to find the dynamic responses of the flat plate.
The effects of eccentricity, acceleration and initial velocity of the moving load, and the effect of span length were the key points of study. The dynamic behaviour of a multi-span flat plate supported by the beam members of rigid plane frames and subjected to the action of a series of moving loads (in identical or opposite directions) were also investigated.

This review is concerned with the utilization of the finite element method to obtain stiffness (or flexibility) properties and the properties of the mass of the structural system and of the mass of the loading due to a moving vehicle. A general equation is formulated, and specific cases and their methods of solution are described. Significant contributions are reviewed and related whenever possible to work involving continuous or approximate approaches. Areas of further research are indicated.

Modal analysis techniques are used to study the dynamic interactions between a one-dimensional high speed ground transport vehicle model and a guideway consisting of multiple independent spans resting freely on rigid discrete supports. The study includes an evaluation of the effects of variations in the fundamental vehicle and guideway parameters on span maximum dynamic deflections and vehicle heave accelerations. Results of the study indicate that vehicle-guideway dynamic interactions strongly influence both vehicle suspension and guideway span design. For the range of parameters of interest in high speed systems (200–300 mph), guideway span dynamic to static deflection and stress ratios, or impact factors, may approach values of 2.0 for a single vehicle passage, and vehicle heave accelerations may exceed the levels of 0.05g desired for good ride quality unless very strong constraints are placed upon vehicle suspension requirements and guideway stiffness, weight, and span length specifications. System design guidelines are presented in the form of parametric plots in which the values of vehicle and guideway parameters required to limit maximum vehicle heave accelerations and guideway dynamic deflections are specified.

A study is made of the vibratory characteristics of the bridge structure and the dynamic interaction between the bridge and vehicles traversing it. Four basic mathematical models which one can consider when simulating a bridge-vehicle interaction problem are discussed, namely the Moving Force Approximation; the Moving Mass Approximation; the Massless Guideway Approximation; and the True Suspension System. Attention is also given to a number of research projects which have incorporated features from two or more of the four basic mathematical models listed above. Finally, detailed attention is given to the most popular mathematical procedures developed to solve vehicle-bridge interaction problems, highlighting their strengths and weaknesses. Particular emphasis is placed on the modal expansion technique and to an algorithm which constitutes a closed-form solution, made possible by discretizing an integral-differential formulation of the boundary value problem.

A finite element procedure is presented to solve the problem of dynamic interaction between a train and a bridge. The equations of the motion are derived. A modal method is proposed for the nonlinear equations to obtain the numerical solution efficiently. The sample solutions are given to demonstrate the validity and the effectiveness of the proposed method.

The conception and construction of new rail vehicles like the magnetic-field monorail with a maximum speed of 400 km/hr require more detailed investigations about the dynamic behaviour of the train and the railway line. The problem to be analyzed is the dynamic behaviour of a beam subjected to travelling loads and masses with constant and non-uniform speed.The paper contains the following topics: —Modification of ADINA to handle travelling-load-problems with user-subroutines; —Modification of ADINA to handle problems with time dependent mass-distributions; —Comparison of ADINA results with an analytical solution; —Influence of mass and speed; —Influence of the different terms of the equations of motion.

A great amount of literature exists on dynamic interaction problems concerning guideways and moving vehicles. Because of the mathematical difficulties introduced by the coupling terms in the behavioral governing differential equation, the transverse inertia effect of a moving vehicle is often neglected. With the improved availability of advanced computer methods and facilities, it has become possible to take into account the kinematics of the interaction problem. In this article recent developments in analytical and numerical approaches for solving vehicle-guideway interaction problems are discussed; related recent literature is cited. In addition, recently reported relevant experimental developments are presented.

With the emergence of high-speed trains, dynamic loads on bridges have changed. A method for estimation of the time-dependent vehicle-bridge interaction forces has been developed in the present paper. The increase (or decrease) of the bridge response due to dynamic effects is determined.The moving constant-force problem is reviewed in some detail. Results obtained by the present method for the moving-mass problem are compared with existing experimental and theoretical results as reported in the literature. A parametric study of bridge responses is made. The parameters varied are the vehicle speed, the ratio of vehicle mass to bridge mass, the ratio of vehicle eigenfrequency to bridge eigenfrequency, and the relative damping of the vehicle. Finally, the influence of an initial bridge deflection is discussed.

A general purpose computer program for calculating the dynamic response of vehicles traveling over surface or evaluated guideways is described. The program has application to a broad class of transportation systems and hence eliminates the need for numerous specialized programs. The program is modular in design and is based on the finite element or building block method in which a complex dynamic system is made up of a number of components. The equations of motion for each of these components is known, and the program automatically combines the component equations to form equations of motion for the complete system. The equations of motion are then integrated numerically to give the response of the system to variables such as guideway roughness, span length, etc. Several output options are available and on-line printer plots or off-line CalComp plots of the response can be obtained. Addition building blocks can be easily added to.the program whenever desired. The program is written in PL/I language and has been used on the IBM 360/91 computer. The program has been used on a limited number of problems, several of which are included in this paper. © 1973 American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

In traditional analyses of vehicle/structure interaction, especially when there are constraints between vehicle masses and the structure, vehicle nominal motion is prescribed a priori, and therefore unaffected by the structure flexibility. In this paper, a concept of nominal motion is defined, and a methodology is proposed in which the above restriction is removed, allowing the vehicle nominal motion to become unknown, and encompassing the traditional approach as a particular case. General nonlinear equations of motion of a building block model, applicable to both wheel-on-rail and magnetically levitated vehicles, are derived. These equations are simplified to a set of mildly nonlinear equations upon introducing additional assumptions - essentially on small structural deformation. An example is given to illustrate the present formulation.

The principle problems of the new developing scientific field - dynamics of vehicles, tracks and roads - are defined. Basic theoretical models for the analysis of railway and road vehicles, tracks and roads and principle methods of their solution are shown. The dynamic interactions between vehicles and routes are emphasized and several basic equations are given to show the behaviour of their elements. The effect of some parameters, like speed, track or roads irregularities etc. is explained. The possibilities are described how to simplify the theoretical models to obtain a simple solution.

This paper is concerned with the analysis of dynamic deflection and acceleration of a concrete bridge which is subjected to a moving vehicle load. The bridge, to be constructed across the River Brahmaputra in India, consists of 20 main spans, and each main span is assumed to be double cantilever type with a small suspended span. The moving vehicle is modeled as one degree of freedom. The deflections and accelerations at specific locations on the bridge when the vehicle moves at constant speed are analyzed by using the finite element method.

The hierarchical concept for finite element shape functions was introduced many years ago as a convenient device for mixed order interpolation. Its full advantages have not been realized until a much later time—and these include in addition 1.(a) improved conditioning;2.(b) ease of introducing error indicators if successive refinement is sought.Further, it is possible to use the ideas to construct a range of error estimators which compare well with alternatives and are ideally suited for adaptive refinements of analysis.As the hierarchical elements are equally simple to implement as “standard” fixed order elements it is felt that more programs will, in the future, turn to utilize their advantages. This is especially true in the field of nonlinear analysis where even today computational economies are necessary.

Some of the possibilities of the finite element method in the moving load problem are demonstrated. The bridge-vehicle interaction phenomenon is considered by deriving a general bridge-vehicle element which is believed to be novel. This element may be regarded as a finite element with time-dependent and unsymmetric element matrices. The bridge response is formulated in modal co-ordinates thereby reducing the number of equations to be solved within each time step. Illustrative examples are shown for the special case of a beam bridge model and a one-axle vehicle model.

An efficient and reliable computational procedure is proposed for the analysis of interaction between high-speed vehicles and flexible structures. In contrast to traditional approaches, the vehicle nominal motion is considered here as an unknown of the problem. The equations encountered, for vehicle motion (after elimination of algebraic constraints) and for structural motion, form a set of nonlinear, coupled differential equations. In spatially-discrete form, these equations do not have the form of explicit ODEs. Predictor/corrector algorithms, which combine Runge-Kutta methods and linear multistep methods with an unconditionally stable algorithm for structural dynamics, are proposed to solve the partitioned DAEs of the interaction problem. The proposed algorithms carry special features pertaining to our formulation of vehicle/structure interaction, and yield accurate results which satisfy the essential system energy balance. The present approach effectively resolves the Timoshenko paradox in moving load problems. Several illustrative examples are presented.

Vibration of bridges

- Huang

Vehicle response on flexible track

- Kortüm

Dynamics of steel elevated guideways—an overview

- Subcommitee on Vibration Problems Associated with Flexural Members on Transit Systems, Structural Division

Dynamic response of concrete railway bridges

- F. Machida
- A. Matsuura
- F. Machida
- A. Matsuura

Dynamic interaction of high-speed vehicles on multiple-span elevated guideways: lumped-parameter vehicle models and new algorithmic treatment

- L. Vu-Quoc
- M. Olsson
- L. Vu-Quoc
- M. Olsson

Dynamic response of concrete railway bridges

- Machida

Analysis of structures subjected to moving loads

- Olsson

Dynamic interaction of high-speed vehicles on multiple-span elevated guideways: lumped-parameter vehicle models and new algorithmic treatment

- Vu-Quoc