Nonlinear Dynamic Analysis of Cracked Beam on Elastic Foundation Subjected to Moving Mass

This paper presents a finite element algorithm for nonlinear dynamic analysis of cracked beams on an elastic foundation subjected to moving mass. Quantity surveying with parameters of varied cracks, foundation and loads shows their influence levels on the nonlinear dynamic response of the beams. The findings of the paper are the basis for the analysis, evaluation, and diagnosis of damages of beam structures on the elastic foundation subjected to moving loads, in which the common defects of the beams such as cracks are considered in order to improve the system's operational efficiency in a wide range of engineering applications.


INTRODUCTION
Beams on the foundations are usually modeled to calculate the structures of railway works and civil engineering. During the use, there are many different causes that can cause weakened defects for beams, one of which is cracks. The appearance of cracks will reduce bearing capacity of the beams, which leads to the risk of damage to the building. Salih N Akour [1] analyzed the nonlinear dynamics of beams on the elastic foundation subjected to evenly distributed moving force by analytical methods. Also using analytical methods, Oni and Awodola [2], Tiwari and Kuppa [3] analyzed the dynamics of Bernoulli -Euler beams on the elastic foundation subjected to moving masses. Haitao Yu and Yong Yuan [4] have focused on the analytical solution of an infinite Euler-Bernoulli beam on a viscoelastic foundation subjected to arbitrary dynamic loads. Şeref Doğuşcan AKBAŞ [5] investigated the free Vibration and Bending of Functionally Graded Beams on Winkler's elastic foundation using Navier method. Nguyen Dinh Kien, Tran Thi Thom [6] studied the influences of dynamic moving forces on the Functionally Graded Porous-Nonuniform beams. D. Froio1, R. Moioli1, E. Rizzi [7] and D. T. Pham, P. H. Hoang and T. P. Nguyen [8] used the nonlinear elastic foundation and New Non-Uniform Dynamic Foundation applied to analyzed response of beam subjected to moving load and the results show that the influence of velocity has effects significantly on dynamic response of structures. N. T. Khiem, P. T. Hang [9] used a spectral method applied to analyzed response of a multiple Cracked Beam subjected to moving load. Using analytical and finite element methods, Murat. R and Yasar. P [10], Mihir Kumar Sutar [11], Animesh C. and Tanuja S. V [12], Shakti P Jena, Dayal R Parhi, P C Jena [13], A.C.Neves, F.M.F. Simoes, A.Pinto da Costa [14], Hui Ma et al. [15] analyzed the dynamics of cracked beams subjected to moving mass. Arash Khassetarash, Reza Hassannejad [16] investigated the Energy dissipation caused by fatigue crack in beamlike cracked structures. Erasmo Viola, Alessandro Marzani, Nicholas Fantuzzi [17] used finite element method applied to studied effect of cracks on flutter and divergence instabilities of cracked beams under subtangential forces. M Attar et al. [18] analyzed the dynamics of cracked beams on the elastic foundation subjected to moving harmonic loads by analytic method, on the basis of using Timoshenko beam model. So far, there are various researches of beams on elastic foundation under transfer (mass, force, oscillation system). However, for cracked beam on the elastic foundation under moving loads(or masses), most methods reply on analytical approaches which are really applied to simple loading conditions. In this paper, we develop a numerical approach based on finite element method for analyzing the dynamics of beams on elastic foundation under moving masses. We investigate the influence of the elastic foundation, load speeds and location cracks in the dynamic response of the beams. Note that finding analytical solutions of such beam problems under arbitrary loading conditions are really challenging and no research is sufficiently carried out yet. Such a problem will be addressed in this paper.

II. FINITE ELEMENT FORMULATION AND THE GOVERNING EQUATIONS
A damaged beam has an open crack located at the midsection of the beam at position x = x0. The beam on an elastic foundation described by an elastic spring system to one direction perpendicular to the axis of the beam, which has the stiffness kt subjected to traversing mass 'm' at speed 'v' as in Fig. 1 where u0, v0 are respectively the x and y components of the total displacement vector of the point (x,0,0) on the beam neutral axis at time t, and z is the cross-section rotation about the z-axis. The subscript "0" represents axis x (y = 0, z = 0; x contains the cross section centroids of the undeformed beam, that will be often designed as middle line or reference line, in bending it coincides with neutral line). The x-coordinate is defined along the beam length, y-coordinate is along the height and the zcoordinate is along the width. The strain-displacement relations are as 22

11
, 22 , where   L  is the linear part of the strain and   L  is the nonlinear part given by: 2 ,, The stresses are related to the strain by Hooke's law: where E is the Young's modulus of the material, G is the shear modulus and [D] is the material matrix.

Equation of motion of beam element with out crack
The equation of motion is derived by the principle of virtual work [19], [21]: 0, where V δW is the virtual work of internal forces, in δW is the virtual work of inertia forces and E δW is the vertual work of external forces due to a virtual displacement. They are defined as: TT e e e δW D dV

International Journal of Advanced Engineering Research and Science (IJAERS)
[ Substituting equations (7), (12) and (13)    is the stiffness matrix related to an elastic foundation.

Nodal load vector element beam on elastic foundation under moving mass
According to FEM method, when a moving load is involved in the working of the system, due to the position change property of the load over time, so at each point of time, the moving load acts on one beam element.
Considering the beam element on elastic foundation subjected to the moving mass m, the force P(t) acts on m (Fig. 3). The force of the moving mass acting on the beam at the coordinate x =  = vt is: where y(x,t) is element deflection, 2 .
The acting force (24) is described by the distributed force p(x,t) as: In case, beam on elastic foundation: where ()  × is denotes the Dirac-Delta function, k0foundation modulus. Therefore, the force vector is:

Governing differential equations for total system
Assembling all elements matrices and nodal force vectors, the governing equations of motions of the cracked beam on elastic foundation subjected to moving mass can be derived as   Table 2 and Figures 4, 5, 6, 7. Through this results, we realize that with cracked beam the whole displacement, acceleration of vertical displacements and normal stresses are greater than the beam without crack. This problem showed the dangers of crack to stiffness, stability of cracked beam on elastic foundation under moving loads.   /dx.doi.org/10.22161/ijaers.4.9.14  ISSN: 2349-6495(P) | 2456-1908(O) www.ijaers.com Page | 78 vibration of the beams increases after the mass moves through the crack.

Effect of elastic foundation stiffness
Studying the changes in maximum values of the displacement, internal force and direct stress of the beam under the elastic foundation stiffness, through the stiffness k0 of the spring ranging from 1×104N/m3 to 6×104N/m3. The results of changes in maximum values of the displacement, internal force and direct stress at the center cross section of the beam are shown in Table 3 and graphs in Figures 8 and 9.

 -k0 relation
It is observed that when the foundation stiffness increases, the maximum values of displacement and flexural moment decrease due to the increase in the system's overall stiffness. The maximum values of displacement and flexural moment decrease sharply when k0 varies from 1×10 4 N/m 3 to 3×10 4 N/m 3 , then the decreasing rate shall be slower.

Effect of load speed
Surveying the problem with a load speed v changes from 10m/s (36km/h) to 35m/s (126km/h). The results of the variations of the maximum values of deflection, acceleration, cutting force and stress at the midpoint of the beam based on v are shown in Table 4 and graphs in Figures 10 and 11. Crack location changes making the maximum responses of displacement, stress and internal force in the beams change significantly; when the crack is in the center of the beam, the above quantities reach the maximum values, so, the beam is most dangerous when there is a crack appearing in this position.

IV. CONCLUSION
A conclusion section must be included and should indicate clearly the advantages, limitations, and possible applications of the paper. Although a conclusion may review the main points of the paper, do not replicate the abstract as the conclusion. A conclusion might elaborate on the importance of the work or suggest applications and extensions. The nonlinear dynamics analysis of cracked beams resting on a Winkler foundation subjected to a moving mass using the finite element method has been presented. A two-node beam element based on Euler-Bernoulli beam theory, taking the effect of crack and foundation support, was derived and employed in the analysis. The dynamics response of the beams, including the time histories for deflection, acceleration and normal stress, was computed with the aid of Newmark method. The effect of loading parameters, foundation stiffness and crack location on the dynamic response of the beams has been examined and highlighted. The main conclusions can be summarized as follows: The beam element and computer code developed in the present work are accurate in evaluating the dynamic characteristics of cracked beams subjected to moving masses. The elastic foundation plays an important role in the dynamic response of the cracked beams under a moving mass. Both the dynamic deflection and normal stress are significaly decreased by the increase of the foundation stiffness. The dynamic response of the cracked beams, as in case of the uncracked beams, is governed by the moving mass speed. With the moving speeds in the range considered in this paper, both the dynamic deflection and normal stress The maximum dynamic deflection and normal stress are significantly influenced by the crack location. The deflection and normal tress attain the largest values when the crack is located at the midpoint of the beam. Thus, from an engineering point of view, the midpoint crack is the most dangerous one. The results obtained in this paper help to select appropriate parameters, the solution for structural reinforcement cracked beam on elastic foundation under moving load and applications in transportation techniques such as the train rails.