### A NEW C0 THIRD-ORDER SHEAR DEFORMATION THEORY FOR THE NONLINEAR FREE VIBRATION ANALYSIS OF STIFFENED FUNCTIONALLY GRADED PLATES

**DOI Number**

**First page**

**Last page**

#### Abstract

Nonlinear free vibration of stiffened functionally graded plates is presented by using the finite element method based on the new C0 third-order shear deformation theory. The material properties are assumed to be graded in the thickness direction by a power-law distribution. Based on the Von Karman theory and the third-order shear deformation theory, the nonlinear governing equations of motion are derived from the Hamilton’s principle. An iterative procedure based on the Newton-Raphson method is employed in computing the natural frequencies and mode shape. The comparison between these solutions and the other available ones suggests that this procedure is characterized by accuracy and efficiency.

#### Keywords

#### Full Text:

PDF#### References

Liew, K.M., Xiang, Y., Kitipornchai, S., Meek, J.L., 1995, Formulation of Mindlin-Engesser model for stiffened plate vibration, Computer Methods in Applied Mechanics and Engineering, 120(3), pp. 339-353.

Aksu, G., Ali, R., 1976, Free vibration analysis of stiffened plates using finite difference method, Journal of Sound and Vibration, 48(1), pp. 15-25.

Zhou, X.Q., Yu, D.Y., Shao, X., Wang, S., Tian, Y.H., 2014, Band gap characteristics of periodically stiffened-thin-plate based on center-finite-difference-method, Thin-Walled Structures, 82, pp. 115-123.

Bhar, A., Phoenix, S.S., Satsangi, S.K., 2010, Finite element analysis of laminated composite stiffened plates using FSDT and HSDT: A comparative perspective, Composite Structures, 92(2), pp. 312-321.

Aishwary, S.R., Sharma, A.K., Gehlot, P., 2018, Free vibration analysis of Stiffened Laminated Plate using FEM, Materials Today: Proceedings, 5(2, Part 1), pp. 5313-5321.

Nguyen, M.N., Nguyen, T.T., Bui, X.T., Vo, D.T., 2015, Static and free vibration analyses of stiffened folded plates using a cell-based smoothed discrete shear gap method (CS-FEM-DSG3), Applied Mathematics and Computation, 266, pp. 212-234.

Bui, T.Q., Do, T.V., Ton, T.H.L, Doan, D.H., Tanaka, S., Pham, D.T., Nguyen, V.T.A., Yu, T., Hirose, S., 2016, On the high temperature mechanical behaviors analysis of heated functionally graded plates using FEM and a new third-order shear deformation plate theory, Composites Part B: Engineering, 92, pp. 218-241.

Ton, T.H.L., Nguyen, V.H., Chau, D.T., 2020, An improved four-node element for analysis of composite plate/shell structures based on twice interpolation strategy, International Journal of Computational Methods, 17(6), 1950020.

Ton, T.H.L., Nguyen, V.H., Chau, D.T., 2020, Nonlinear bending analysis of functionally graded plates using SQ4T elements based on twice interpolation strategy, Journal of Applied and Computational Mechanics, 6(1), pp. 125-136.

Ton, T.H.L., Nguyen, V.H., Chau, D.T., Huynh, V.C., 2018, Enhancement to four-node quadrilateral plate elements by using cell-based smoothed strains and higher-order shear deformation theory for nonlinear analysis of composite structures, Journal of Sandwich Structures & Materials, 22, pp. 2302-2329.

Nguyen, V.H., Ton, T.H.L., Chau, D.T., Dao, N.D., 2018, Nonlinear static bending analysis of functionally graded plates using MISQ24 elements with drilling rotations, Proc. International Conference on Advances in Computational Mechanics 2017, Springer Singapore, 15479070.

Ton, T.H.L., 2020, Finite element analysis of functionally graded skew plates in thermal environment based on the new third-order shear deformation theory, Journal of Applied and Computational Mechanics, 6(4), pp. 1044-1057.

Ton, T.H.L., 2020, Improvement on eight-node quadrilateral element (IQ8) using twice-interpolation strategy for linear elastic fracture mechanics, Engineering Solid Mechanics, 8(4), pp. 323-336.

Rama, G., Marinkovic, D., Zehn, M., 2018, High performance 3-node shell element for linear and geometrically nonlinear analysis of composite laminates, Composites Part B: Engineering, 151, pp. 118-126.

Marinković, D., Gil, R., Zehn, M., 2019, Abaqus implementation of a corotational piezoelectric 3-node shell element with drilling degree of freedom, Facta Universitatis-Series Mechanical Engineering, 17(2), pp. 269-283.

Kamineni, J.N., Burela, R.G., 2019, Constraint method for laminated composite flat stiffened panel analysis using variational asymptotic method (VAM), Thin-Walled Structures, 145, 106374.

Rossow, M.P., Ibrahimkhail, A.K., 1978, Constraint method analysis of stiffened plates, Computers & Structures, 8(1), pp. 51-60.

Peng, L.X., Liew, K.M., Kitipornchai, S., 2007, Analysis of stiffened corrugated plates based on the FSDT via the mesh-free method, International Journal of Mechanical Sciences, 49(3), pp. 364-378.

Peng, L.X., Liew, K.M., Kitipornchai, S., 2006, Buckling and free vibration analyses of stiffened plates using the FSDT mesh-free method, Journal of Sound and Vibration, 289(3), pp. 421-449.

Liew, K.M., Kitipornchai, S., Peng, L.X., 2006, 4 - Mesh-free methods for buckling analysis of stiffened and corrugated plates, in Analysis and Design of Plated Structures, N.E. Shanmugam and C.M. Wang, Editors. 2006, Woodhead Publishing, pp. 80-116.

Mukhopadhyay, M., 1989, Vibration and stability analysis of stiffened plates by semi-analytic finite difference method, Part I: Consideration of bending displacements only, Journal of Sound and Vibration, 130(1), pp. 27-39.

Mukhopadhyay, M., 1989, Vibration and stability analysis of stiffened plates by semi-analytic finite difference method, part II: Consideration of bending and axial displacements, Journal of Sound and Vibration, 130(1), pp. 41-53.

Zahari, R., El-Zafrany, A., 2009, Progressive failure analysis of composite laminated stiffened plates using the finite strip method, Composite Structures, 87(1), pp. 63-70.

Sheikh, A.H., Mukhopadhyay, M., 2000, Geometric nonlinear analysis of stiffened plates by the spline finite strip method, Computers & Structures, 76(6), pp. 765-785.

Sheikh, A.H., Mukhopadhyay, M., 1992, Analysis of stiffened plate with arbitrary planform by the general spline finite strip method, Computers & Structures, 42(1), pp. 53-67.

Leme, S.P.L., Aliabadi, M.H., 2012, Dual boundary element method for dynamic analysis of stiffened plates, Theoretical and Applied Fracture Mechanics, 57(1), pp. 55-58.

Tanaka, M., Bercin, A.N., 1998, Static bending analysis of stiffened plates using the boundary element method, Engineering Analysis with Boundary Elements, 21(2), pp. 147-154.

Varghese, V., 2018, An analysis of thermal-bending stresses in a simply supported thin elliptical plate, Journal of Applied and Computational Mechanics, 4(4), pp. 299-309.

Sayyad, A., Ghumare, S., 2019, A new quasi-3D model for functionally graded plates, Journal of Applied and Computational Mechanics, 5(2), pp. 367-380.

Zargaripoor, A., Daneshmehr, A.R., Nikkhah Bahrami, M., 2019, Study on free vibration and wave power reflection in functionally graded rectangular plates using wave propagation approach, Journal of Applied and Computational Mechanics, 5(1), pp. 77-90.

Vel, S.S., Batra, R.C., 2002, Exact solution for thermoelastic deformations of functionally graded thick rectangular plates, AIAA Journal, 40(7), pp. 1421-1433.

Sedighi, H.M., Malikan, M., 2020, Stress-driven nonlocal elasticity for nonlinear vibration characteristics of carbon/boron-nitride hetero-nanotube subject to magneto-thermal environment, Physica Scripta, 95(5), 055218.

Ouakad, H.M., Valipour, A., Kamil Żur, K., Sedighi, H.M., Reddy, J.N., 2020, On the nonlinear vibration and static deflection problems of actuated hybrid nanotubes based on the stress-driven nonlocal integral elasticity, Mechanics of Materials, 148, 103532.

Qian, L.F., Batra, R.C., Chen, L.M., 2003, Free and forced vibrations of thick rectangular plates using higher-order shear and normal deformable plate theory and meshless Petrov-Galerkin (MLPG) method, Computer Modeling in Engineering & Sciences, 4(5), pp. 519--534.

Rama, G., Marinković, D., Zehn, M., 2017, Efficient three-node finite shell element for linear and geometrically nonlinear analyses of piezoelectric laminated structures, Journal of Intelligent Material Systems and Structures, 29(3), pp. 345-357.

Shi, G., 2007, A new simple third-order shear deformation theory of plates, International Journal of Solids and Structures, 44(13), pp. 4399-4417.

Mukherjee, A., Mukhopadhyay, M., 1988, Finite element free vibration of eccentrically stiffened plates, Computers & Structures, 30(6), pp. 1303-1317.

Harik, I.E., Guo, M., 1993, Finite element analysis of eccentrically stiffened plates in free vibration, Computers & Structures, 49(6), pp. 1007-1015.

Dayi, O., Mak., C.M., 2012, Free flexural vibration analysis of stiffened plates with general elastic boundary supports, World Journal of Modelling and Simulation, 8(2), pp. 96-102.

Shen, H.S., 2009, Functionally Graded Materials Nonlinear Analysis of Plates and Shells, New York, NY, USA: CRC Press Taylor & Francis Group.

### Refbacks

- There are currently no refbacks.

ISSN: 0354-2025 (Print)

ISSN: 2335-0164 (Online)

COBISS.SR-ID 98732551

ZDB-ID: 2766459-4