Abstract
Simulation of an infinite Timoshenko beam subjected to accelerated moving load rested on the finite depth is assessed in this study. Then, the dynamic response of the infinite beam is illustrated on the various theoretical models, including the Winkler, Pasternak, and visco-elastic foundations. Furthermore, the effects of various damping, such as foundation damping, beam damping, and hysteretic damping (damping between soil grains), are also studied on the dynamic behavior of the beam. It has been worked out that the type of basement and its depth have a remarkable effect on the dynamic behavior. In addition, the load velocity will also cause a maximum displacement in the beam as the critical velocity approaches. The deflection of the beam on the basements increases when the velocity approaches the critical one, and the maximum displacement of the beam occurs under the load. Finally, it was seen that the presented diagrams for all three types of Winkler’s, Pasternak’s, and visco-elastic foundations follow the usual properties related to the critical velocity.
1. Introduction
Frequently, the dynamic behavior of structures under moving loads has played a significant role in engineering fields. Therefore, it is an extraordinary process to choose the most noticeable information in the dynamic behavior of structures to which one researcher should pay attention. The structures that are subjected to moving loads are susceptible to severe problems “oscillation, upward and downward displacement” in comparison with other structures. Also, the moving load’s velocity is an essential issue for the stability of the systems. For example, in high-speed train transport, the track structure can be described as a continuous beam on a uniform, homogeneous basement with vertical springs, which resist beam deflection. The critical velocity can be described as the velocity which causes the maximum deflection of the rail. Trains traveling at speeds close to the critical velocity cause significant deflections in the rail system “which in turn causes the interaction between rail and train” leads to more severe breakdowns such as train derailment and rail failure due to fatigue. In the analysis of the dynamic behavior of a structure, the excitation applied from the basement to the structure for the case where the structure relies on a rigid base and bedrock is the same excitation that existed before the construction of the structure at that point, as if the structure relies on soft soil, significant changes will occur in the seismic input of the structure. Therefore, the structure interacts with the surrounding soil and will cause changes in the movements of the structure. The point here is that the structure or basement may be modeled as elastic or visco-elastic or a semi-infinite system, which can even be nonlinear. The initial investigation of elastic base was presented by Timoshenko [1]. His study is about the behavior of the beam subjected to the moving load. The dynamic behavior of the Timoshenko beam subjected to the moving load on the Pasternak foundation was presented by Kargarnovin and Younesian [2]. The solutions for free vibration and the bending response of the beams on the Winkler and Pasternak foundations were provided by Ying et al. [3]. The spectral analysis of the beam is recommended by Gladyzs and Sniady [4]. Moreover, by applying the differential transform method, Balkaya et al. provided the dynamic response of the Timoshenko and Euler-Bernoulli beams on soil [5]. Motaghian et al. studied the complication of frequency analysis of beams on the Winkler foundation [6]. Also, the nonlinear responses of the clamped Euler-Bernoulli beams subjected to axial forces were concluded by Barari et al. [7]. The dynamic response of a composite beam subjected to a moving oscillatory mass is presented by Kargarnovin et al. [8]. Bazehhour et al. employed an analytical method to investigate the free vibration of a Timoshenko beam with various boundary conditions [9]. On the other hand, a numerical approach for predicting the vibration of the Timoshenko beam is illustrated by Prokić et al. [10].
Mohammadzadeh and Mosayebi utilized the dynamic behavior of the rails on the visco-elastic foundations [11]. Also, the railway vibrations assumed an advanced soil constitutive model presented by Ruiz and Rodríguez [12]. An analytical approach for investigating the response of a crane system is presented by Zrnić et al. [13]. Roshandel et al. utilized the dynamic response of a non-uniform Timoshenko beam subjected to a moving mass [14]. Dimitrovová studied the dynamic behavior of the Euler-Bernoulli beam on a Winkler foundation for the critical velocity of the moving load [15]. Ghannadiasl and Rezaei Dolagh presented the dynamic response of the Euler-Bernoulli beam on a finite depth bases under a moving load [16, 17]. The influence of soil and structure interaction was also presented by them. Guerdouh and Khalfallah investigated the effects of soil properties on the seismic performance of structures [18]. Beskou et al. utilized the dynamic response of the elastic half-plane under a moving load [19]. For this purpose, the half-plane under a moving load with constant speed is assumed. Also, the dynamic vibration of a beam under harmonic moving load rested on the bilinear elastic foundation is presented by Froio et al. [20]. The nonlinear and linear vibrations of the circular plate under moving load were studied by Rai and Gupta (Rai and Gupta). The effect of different boundary conditions and damping models of a moving load was derived by Praharaj and Datta [21], and Shih et al. [22]. velocities was provided by Bian et al. [23]. Sheng et al. used the Fourier transform method to find out the dynamic behavior of the railways under a moving load [24]. Various homogeneous soil models subjected to different shear velocity provided using Applying Response Site Analyses by Forcellini et al. [25]. Kolesnikov and Tolmacheva provided the influence of different factors on the stress and displacement of a rigid pavement on an elastic base [26]. Two new equations are provided by Azan and Haddad to point out the effect of various soil parameters on the strip footing bearing capacity [27]. Also, Ghannadiasl and Mofid presented the dynamic behavior of the Timoshenko beam and the stepped circular plate resting on an arbitrary variable elastic foundation using the Dynamic Green Function for arbitrary boundary conditions [28, 29].
The dynamic response of the Euler-Bernoulli beam under a moving load was investigated in the previous papers. Whereas, a precise approach in closed form is presented for considering the sensitivity of dynamic behavior of the Timoshenko beam to the velocity of moving load on various elastic bases in the present study. Moreover, the deflection shapes are calculated and compared for both Timoshenko and Euler-Bernoulli beams. The purpose of this paper is the simulation of an infinite Timoshenko beam rested on the finite depth subjected to accelerated moving load on various types of basements, i.e., Winkler, Pasternak, and visco-elastic foundations, the solution without and with shear contribution in the foundation, which has not been provided in previous studies. Then, the displacement of the Euler-Bernoulli and Timoshenko beams was compared and investigated on three different bases. The present study is arranged in the following form: Initially, the Timoshenko beam governing equations are simplified. Besides, the influences of various foundations, damping, and acceleration of the moving load with some numerical examples are presented, and the velocity effect on the deflection shapes of the beams is also depicted. The governing equations are worked out and contrived on Mathematica software. Finally, the conclusions are categorized briefly.
2. Timoshenko Beam Subjected to an Accelerated Moving Load
The governing differential equation of the Timoshenko beam with the rectangular section shown in Figure 1 is expressed by Li et al. [30] as below:where and indicates the transverse displacement and the rotation angle, respectively; , are the bending and the shear stiffness modules; is the rotational inertia ; is the axial force, denots the mass per unit length of the beam, is the foundation pressure which will be replaced later, and are the moving load and its velocity. Moreover, is the beam damping coefficient and shows the Dirac delta function. In this paper, the trajectory of the moving load equation is defined as below:
For utility, (2) can be simplified into a differential equation as follow:
Differentiating (1) with respect to and using (4), we can obtain
According to (5), the and are calculated, and using (5), we can achieve
By converting the equations to the moving system , (6) can be rewritten as below:
Also, in the vertical direction the dynamic equilibrium of the foundation is defined as below [15]:where denotes the foundation viscous damping coefficient, shows the vertical soil displacement, that is used to present the effect of foundation damping correctly, presents the density of the soil, is the stiffness, depicts the soil depth, and is used for the shear effect.
By changing coordinate to moving coordinate, we get the following equation.
The boundary conditions are satisfied by the relative displacement, which makes the resolvability easier, and can be defined as follow:
Moreover, by considering “the moving coordinate can be transferred to the dimensionless coordinate ” and dividing the equation by the static displacement , we arrive atwhere , shows the shear coefficient. The term is the velocity of the shear wave, is the mass ratio, , and is the velocity ratio. The critical velocity of the Timoshenko beam is calculated by Dimitrovová [15] as follows [31]:where . , , and refer to the radius of gyration of the beam cross-section, the reduced cross-sectional area, and the shear modulus of the beam, respectively. The following relation can be considered under the homogeneous conditions:
Therefore, multiplication with one mode shape, substitution and integration from 0 to 1 depth, and using Fourier transform, we arrive at
The foundation pressure is defined as (15) by Dimitrovová [15]:where stands for the hysteretic damping coefficient. By getting back to (7), we arrive at
Changing to dimensionless condition, , , and , (17) is obtained. Furthermore, we havewhere and are clarified as below:
By the Fourier transform, we havewhere gives , , or depending on whether is negative, zero, or positive. The expressions and are defined as below:
3. Numerical Examples
In order to verify the present study with previous research, the Timoshenko beam subjected to a moving load is considered with the particular values provided in Table 1. The deflection shape of the beam is shown in Figure 1 for the various load velocities.
From Figure 2, it is seen that the greater the velocity, the greater the displacement of the beam. For instance, in the diagrams shown, when the load velocity value is 1, the relative displacement occurs in a smaller area of the beam. However, as its value increases to 1.5, the displacement affects the longer length of the beam.
3.1. The Influence of Foundation Type on the Dynamic Response of the Beam
A beam with the features of Table 2 is considered for this purpose. By solving the governing equation of the beam, the displacement of the Timoshenko beam for the three types of foundations, including the classical Winkler’s ( and ), Pasternak’s , and visco-elastic is investigated and shown in Figure 2. From Figure 3, it is seen that the type of foundation affects the displacement of the beam remarkably, which also changes the dynamic response of the beam and provides a different behavior to the type of foundation. Like the Euler-Bernoulli beam, the displacement of the Timoshenko beam on the Winkler foundation is more remarkable than both Pasternak’s and visco-elastic foundations due to the absence of shear wave propagation and the shear ratio.
3.2. The Dynamic Behavior of the Timoshenko and Euler-Bernoulli Beams on Various Foundations
3.2.1. Classical Winkler’s Foundation
A beam with the specifications of Table 1 for the Euler-Bernoulli beam and Table 2 for the Timoshenko one is considered. By assuming and , the foundation is converted to the Winkler’s one. The displacement of the two beams on the classical Winkler’s foundation is obtained and provided in Figure 4. From Figure 4, it is seen that the displacement of the Timoshenko beam is close to the Euler-Bernoulli one.
3.2.2. Pasternak’s Foundation
For this purpose, a beam with the data of Table 1 for the Euler-Bernoulli beam and Table 2 for the Timoshenko one is considered. Assuming , the foundation is converted to the Pasternak’s one. The displacement of the two beams on this foundation is compared and provided in Figure 5. By comparing the diagrams obtained in Figure 5, it can be seen that, like the Winkler’s foundation, the displacement of the two beams on the Pasternak’s foundation is not noticeable. The behavior of both beams is almost close to each other.
3.2.3. Visco-Elastic Foundation
To model and compare the two theories on the visco-elastic foundation, the numerical input data of Table 1 for the Euler-Bernoulli beam and Table 2 for the Timoshenko beam are summarized. Defining (visco-elastic foundation), the behavior of the Euler-Bernoulli and the Timoshenko beams is compared and shown in Figure 6. From Figure 6, it can be seen that, like the previous two foundations, i.e., Winkler’s and Pasternak’s, the behavior of the two beams on the visco-elastic foundation is not significant. Also, the displacement of both beams is the same as in the previous cases and they are almost close to each other. As a result, comparing the behavior of the Euler-Bernoulli and Timoshenko beams, it is observed that the displacement of both beams on the Winkler’s foundation is greater than the other two ones.
Then, the maximum downward and upward displacement of the diagrams obtained in Figures 4–6 in the two foundations of Winkler’s and Pasternak’s for the critical velocity of and , and the load velocity of 323 m/s is obtained and compared in Figure 7. To show the influence of critical velocity on the maximum displacement of the beam, the load velocity is reduced to 165 m/s. Again, the maximum downward and upward displacement for both theories on the Winkler’s foundation is provided in Figure 8.
Comparing the diagrams in Figures 7 and 8 for the two types of foundations, Winkler’s and Pasternak’s, it can be seen that in both Euler-Bernoulli and Timoshenko theories, as getting closer to the critical velocity —497 m/s for the Euler-Bernoulli beam and 497.336 m/s for the Timoshenko beam “the displacement of both beams increases in both directions.” Also, by reducing the load velocity from 323 meters per second to 165 meters per second, the maximum downward and upward displacement in both beams decreased.
4. Conclusion
The dynamic behavior of Timoshenko and Euler-Bernoulli beams with infinite length under accelerated moving load on a foundation with finite depth was investigated and compared in this paper. In the present study, merely the dynamic equilibrium in the vertical direction is considered. The governing equations of the beam on the foundation, including Winkler’s, Pasternak’s, and visco-elastic, were calculated and presented by considering the effects of beam damping, foundation damping, and hysteretic damping. Considering the interaction of the soil and the governing equations, the physics of the problem is modeled with mathematical equations. Finally, the governing differential equations are converted to algebraic equations using the Fourier transform method. Then, the proposed model is validated, and using the considered method, the dynamic response of the beam on the soil bases is identified. In this paper, considering the shear influence in a simplified form, the beam deflection shape is obtained for various velocities of the applied moving load. Also, the effect of different soil depths on the response of the Timoshenko beam is investigated. It was found that the dynamic behavior of the beam changes dramatically based on the type of foundation and its depth. Furthermore, the displacement obtained for each foundation increases with increasing load speed and approaching the critical state. The most significant displacement occurs just below the applied load. On the other hand, in the low mass ratio, it was also observed that the critical velocity approaches the classical equation.
It was observed that the displacement of the beam in both theories on the Winkler’s and Pasternak’s foundations is more than in the other one, i.e., visco-elastic foundation, due to the lack of shear wave propagation and shear ratio in Winkler’s, and zero shear ratio in Pasternak’s foundation. In the visco-elastic foundation, due to the hysteretic damping effect, the displacement of the beam is less than in the other two foundations, Winkler’s and Pasternak’s. Finally, it was observed that the diagrams for all three types of Winkler, Pasternak, and visco-elastic foundations follow the usual properties related to the critical velocity. According to the examples, it was observed that by getting closer to the critical velocity, the displacement increases in both up and down directions. On the other hand, by getting farther, the graph slope and the displacement decrease.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.