Research Article | Open Access

Volume 2019 |Article ID 6528251 | 11 pages | https://doi.org/10.1155/2019/6528251

# Dynamic Analysis of Cracked Plate Subjected to Moving Oscillator by Finite Element Method

Revised08 Jun 2019
Accepted03 Jul 2019
Published25 Jul 2019

#### Abstract

This paper presents the finite element algorithm and results of dynamical analysis of cracked plate subjected to moving oscillator with a constant velocity and any motion orbit. There are many surveys considering the dynamic response of the plate when there is a change in number of cracks and the stiffness of the spring k. The numerical survey results show that the effect of cracks on the plate's vibration is significant. The results of this article can be used as a reference for calculating and designing traffic structures such as road surface and bridge surface panels.

#### 2. Finite Element Simulation and Dominant Equations

Figure 1 shows the cracked plate under the moving oscillator on the plate in the general coordinate system (X,Y,Z).

For finite element model formulation, the following assumptions are made:(i)The materials of the system are linear elastic.(ii)The load and pavement are not speared in the activity duration of system.

##### 2.1. Cracked Plate Element Subjected to Moving Oscillator
###### 2.1.1. Cracked Plate Element Subjected to Dynamic Loads

Plate is described by bending rectangular four-node elements (Figure 2). Arbitrary point in the element has positions (x, y) in global coordinate and positions (r, s) in local coordinate . We assume that the thickness of plate element is a constant and the conditions of Mindlin–Reissner plate theory are satisfied.

The displacement fields are written as where u0, v0, w0 are the displacements of the mid plane and , are rotations of normal about, respectively, the y and x axes.

The strain vector is presented in the formwhere

The constitutive equation can be written aswhere is stress vector without shear deformation: is stress vector of shear stress:with E being elastic modulus of longitudinal deformation and being Poisson ratio.

Using (7) and (8), the components of internal force vector are determined as

so that one obtainswhere - strain matrix, is the vector of curvatures and shear strains, and α is the shear strain correction factor, usually equal to α = 5/6.

According to the FEM procedure, the displacement of a point of the element is represented as [14, 16]where , , are displacements w, , at node, respectively, and are shape functions, which allows us to obtain where is a matrix for the internal force determination, and is a vector of the node displacement, with , (i = 1,2,3,4).

Substituting (12) into (10) leads to are matrices corresponding to bending moment and shear force, respectively, [10, 11].

The dynamic equation of plate element can be derived by using Hamilton’s principle [14, 17]:where , are kinetic energy and total potential energy of the element, respectively.

The kinetic energy of the element level is defined as with being stiffness matrix and node loading vector of the element, respectively, [N] is mode shape function matrix of element, p is pressure of intensity, , and is the surface area of the plate elements.

Kinetic energy of element is determined by where - mass matrix, ρ - mass density and -velocity vector.

Substituting (16) and (17) into (15), the dynamic matrix equation of the plate element without damping can be written as

In the case of cracked plate element, the stiffness matrix of the element can be written as [8, 18]where is the transformation matrix, given in Appendix A, in which is the flexibility matrix of the noncracked element, given in Appendix B, and is the flexibility matrix due to the presence of the crack, given in Appendix C .

Now, the dynamic matrix equations of the cracked plate element subjected to dynamic loads become

###### 2.1.2. Cracked Plate Element Subjected to Moving Oscillator

The force of the moving oscillator on the plate at the time t is determined as follows:where g is acceleration due to gravity, is acceleration of the mass m2, and is acceleration of the plate at the force set point given by

Substituting into (22) yieldswhere and are the velocity and acceleration of the loads along x, y axes, respectively.

By substituting (23) into (22), the force of the moving oscillator on the plate at the time t can be written as

Concentrated force (24) is described by the uniformly distributed load as follows :where δ(.) is the Dirac’s delta function, and

Substituting (24) into (25) leads to

The element nodal load vector is 

Substituting (30) into (20) leads to the dynamic equation of the cracked plate element subjected to moving oscillator, which is

The dynamic equation of mass m2 can be written as

By combining (1) and (2), the dynamic system of equations of the cracked plate and mass m2 are presented as follows:

Or

##### 2.2. Governing Differential Equations for Total System

By assembling all elements matrices and nodal force vectors, the governing equations of motions of the total system can be derived aswhere is the overall structural mass matrix, is the overall structural stiffness matrix with total uncracked elements, is the overall structural stiffness matrix with total cracked elements, is the overall mass matrix due to moving mass m1, is the overall mass matrix due to moving mass m2, is the overall damping matrix due to moving mass m1, and is the overall structural damping matrix [14, 17].

This is a linear differential equation system with time dependence coefficient, which can be solved by using direct integration Newmark’s method. A MATLAB program named Cracked_Plates_Moving_2019 was conducted to solve (40).

#### 3. Numerical Analysis

We consider a rectangular cracked plate with size L = 3.0 m, W = 1.6 m, thickness h = 0.025 m, crack length = 0.5 m, and it appears in the middle of the plate (Figure 1). Material parameters of the plate: Young Modulus E = 2.0x1011 N/m2, Poisson coefficient ν = 0.28, density ρ = 7800 kg/m3. The boundary condition of the plate is SFSF (simply support, free/ simply support, free). The moving oscillator has mass m1 = 300 kg connected with m2 = 200 kg via spring with stiffness k = 1.5x105 N/m and damping element with resistance coefficient, c = 4.5x103 Ns/m, and k and c are parallel. Moving oscillator moves at velocity v = 10 m/s along the centerline y = W/2 of the plate.

The results of the vibration of the plate subjected to moving oscillator (MO) and the effect of moving mass (MM) (M = m1 + m2 = 500 kg) are shown in Table 1 and Figures 3, 4, 5, and 6, in which the “int” symbol represents the intensity value (stress intensity and train intensity).

 Case of load MO 1.012 13.217 1.364×107 10.536×107 MM 1.122 42.4458 4.434x107 15.748x107

Comment: Compared to the case of cracked plate subjected to moving oscillator, the case of cracked plate subjected to moving mass indicates greater response of the plate. Therefore, the destructive capacity of the structure is greater.

##### 3.1. The Effect of the Number of Cracks

To evaluate the effect of the number of cracks on vibration of cracked plate under moving oscillator, the three cases were investigated: Case 1: the plate has one crack in the middle (X = L/2, the basic problem); Case 2: the plate has one crack in the middle and one same size crack at X = L/4; Case 3: the plate has 3 cracks at X = L/4, L/2, 3/4. The results of the vibration of the plate are shown in Table 2 and Figures 7, 8, 9, and 10.

 Case 1 1.012 13.217 1.364x107 10.536×107 2 1.029 33.814 2.053x107 10.177×107 3 1.066 33.821 2.062x107 10.265×107

Comment: Strain, displacement, stress, and acceleration at point A increase as the number of cracks increases, but these values fluctuate at the crack edge, sometimes increase and sometimes decrease.

##### 3.2. The Effect of the Stiffness of the Spring k

To evaluate the effect of k hardness in the oscillation system on the response of the system, the authors examine the problem when k varies from 1x105 N/m to 9.0x105 N/m. Response of the system at points A and B is shown in Table 3 and Figures 11, 12, 13, 14, 15, and 16.

 k×105 1.0 0.081 8.942 0.813×107 8.426×107 1.5 1.012 13.217 1.364×107 10.536×107 2.0 1.075 19.415 2.285×107 11.284×107 2.5 1.078 23.073 3.150×107 11.553×107 3.5 1.033 21.224 4.036×107 11.363×107 6.0 0.905 17.740 2.380×107 11.213×107 9.0 0.783 12.360 1.554×107 8.693×107

Comment: When the k hardness changes, the oscillation of the system varies considerably. With the parameters of the given plate, the displacement response, acceleration, stress, and strain at the computed points are the greatest when k = 2.5x105 N/m.

The authors analyzed the dynamics of the plate with the speed of the oscillator system varying from 6 m/s to 14 m/s; the results are shown in Table 4 and Figures 17, 18, and 19.

 v 6 1.211 15.429 4.352×107 16.323×107 8 1.210 13.816 2.931×107 13.124×107 10 1.012 13.217 1.364×107 10.536×107 12 0.724 10.201 0.970×107 7.718×107 14 0.537 11.897 1.137×107 6.048×107

Comment: When the speed of the oscillation system increases, the displacement and stress of the plate decrease, but there is no clear rule. According to the authors, the main reason may be due to the influence of the plate's free vibration frequency and the moving oscillator; this is the difference with the case of the plate subjected to moving mass.

#### 4. Conclusions

In the end, with the set of survey parameters, the case of the cracked plate under the moving mass is more dangerous than the case of moving oscillators operates. However, the problem of texture affected by the oscillation system is complex. In each case, a reevaluation is needed. The response of the system depends on the interrelation between the frequency of the stimulus and the natural frequency of the system.

The results show that stress and strain at the crack head are much larger than they are at other sites. These values vary considerably when the number of cracks and k hardness are changed.

#### A.

The transformation matrix [T] is

#### B.

The flexibility matrix of the noncracked element is

#### C.

The flexibility matrix due to the presence of the crack is

(i) Crack parallel to the x-axis of the element:

(ii) Crack parallel to the y-axis of the element:wherewith when the crack is parallel to the x-axis, and when the crack is parallel to the y-axis of the element, and (i=1,2) are correction functions given in .

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This research was supported by Le Quy Don University.

1. N. T. Chung and L. P. Binh, “Nonlinear dynamic analysis of cracked beam on elastic foundation subjected to moving mass,” International Journal of Advanced Engineering Research and Science, vol. 4, no. 9, pp. 73–81, 2017. View at: Publisher Site | Google Scholar
2. S. R. Mohebpour, P. Malekzadeh, and A. A. Ahmadzadeh, “Dynamic analysis of laminated composite plates subjected to a moving oscillator by FEM,” Composite Structures, vol. 93, no. 6, pp. 1574–1583, 2011. View at: Publisher Site | Google Scholar
3. P. Malekzadeh, A. Fiouz, and H. Razi, “Three-dimensional dynamic analysis of laminated composite plates subjected to moving load,” Composite Structures, vol. 90, no. 2, pp. 105–114, 2009. View at: Publisher Site | Google Scholar
4. Q. Song, Z. Liu, J. Shi, and Y. Wan, “Parametric study of dynamic response of sandwich plate under moving loads,” Thin-Walled Structures, vol. 123, pp. 82–99, 2018. View at: Publisher Site | Google Scholar
5. Q. Song, J. Shi, and Z. Liu, “Vibration analysis of functionally graded plate with a moving mass,” Applied Mathematical Modelling, vol. 46, pp. 141–160, 2017. View at: Publisher Site | Google Scholar
6. A. Mamandi, R. Mohsenzadeh, and M. H. Kargarnovin, “Nonlinear dynamic analysis of a rectangular plate subjected to accelerated/decelerated moving load,” Journal of Theoretical and Applied Mechanics, vol. 53, no. 1, pp. 151–166, 2015. View at: Publisher Site | Google Scholar
7. A. Vosoughi, P. Malekzadeh, and H. Razi, “Response of moderately thick laminated composite plates on elastic foundation subjected to moving load,” Composite Structures, vol. 97, pp. 286–295, 2013. View at: Publisher Site | Google Scholar
8. Q. Guan-Liang, G. Song-Nian, and J. Jie-Sheng, “A finite element model of cracked plates application to vibration problems,” Computers & Structures, vol. 39, no. 5, pp. 483–487, 1991. View at: Publisher Site | Google Scholar
9. M. Krawczuk and W. M. Ostachowicz, “A finite plate element for dynamic analysis of a cracked plate,” Computer Methods Applied Mechanics and Engineering, vol. 115, no. 1-2, pp. 67–78, 1994. View at: Publisher Site | Google Scholar
10. T. Yin and H. Lam, “Dynamic analysis of finite-length circular cylindrical shells with a circumferential surface crack,” Journal of Engineering Mechanics, vol. 139, no. 10, pp. 1419–1434, 2013. View at: Publisher Site | Google Scholar
11. H. Serdar, Vibration Analysis of Systems Subjected to Moving Loads by Using Finite Element Method [A Thesis], Graduate School of Natural and Applied Sciences, 2005.
12. D. H. Li, X. Yang, R. L. Qian, and D. Xu, “Static and dynamic response analysis of functionally graded material plates with damage,” Mechanics of Advanced Materials and Structures, pp. 1–14, 2018. View at: Publisher Site | Google Scholar
13. F. Ladislav, Vibration of Solid and Structures under Moving Loads, Thomas Telford, 1999.
14. K. J. Bathe and E. L. Wilson, Numerical Method in Finite Method Analysis Prentice, Hall of India Private Limited, New Delhi, India, 1978.
15. J. N. Reddy, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, CRC Press, 2004.
16. J. S. Przemieniecki, Theory of Matrix Structural Analysis, McGraw-Hill, New York, NY, USA, 1968.
17. J. P. Wolf, Dynamic Soil-Structure Interaction Analysis in Time domain, vol. 07632, Prentice-Hall Inc., Englewood Cliffs, NJ, USA, 1988.
18. S. E. Khadem and M. Rezaee, “Introduction of modified comparison functions for vibration analysis of a rectangular cracked plate,” Journal of Sound and Vibration, vol. 236, no. 2, pp. 245–258, 2000. View at: Publisher Site | Google Scholar

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