In this paper, free vibration analysis of isotropic rectangular plates with linearly varying thickness in one direction has been investigated using a high-order triangular element. The first order shear deformation theory is used to include the effect of transverse shear deformation. The element has 18 nodes on the sides and six internal nodes. The geometry of element is expressed by three linear shape functions of area coordinates. The formulation is displacement-based. The element has 51 degrees-of-freedom, which can be reduced to 39 degrees-of-freedom by Guyan reduction, mass condensation, or eigenvalue economization scheme for the degrees-of-freedom associated with the internal nodes. Rotary inertia has been included in the consistent mass matrix. Numerical examples are presented to show the accuracy and convergence characteristics of the element. Isotropic plates with different thickness ratios (varies from 0.01 to 0.2), tapered ratios, aspect ratios and boundary conditions are analyzed. The results obtained by present element show a close agreement with the available published results. Some numerical results have been given as new results.
BatozJLKatiliI (1992) On a simple triangular Reissner/Mindlin plate element based on incompatible modes and discrete constraints. International Journal for Numerical Methods in Engineering35: 1603–1632.
2.
BatozJLBatheKJHoLW (1980) A study of three-node triangular plate bending elements. International Journal for Numerical Methods in Engineering15: 1771–1812.
3.
CheungYKZhouD (1999) Eigen-frequencies of tapered rectangular plates with intermediate line supports. International Journal of Solids and Structures36: 143–166.
4.
CheungYKZhouD (2000) Vibrations of moderately thick rectangular plates in terms of a set of static Timoshenko beam functions. Computers & Structures78: 757–768.
5.
CloughRWTocherJL (1965) Finite element stiffness matrices for analysis of plate bending. Proceedings of First Conference on Matrix Methods in Structural Mechanic, Wright-Patterson Air Force Base, OH, October 26–28515–545.
6.
CookRDMalkusDSPleshaME (1989) Concepts and Applications of Finite Element Analysis. New York: Wiley.
7.
CorrRBJenningsA (1976) A simultaneous iteration algorithm for symmetric eigenvalue problems. International Journal for Numerical Methods in Engineering10: 647–663.
8.
DickinsonSMBlasioAD (1986) On the use of orthogonal polynomials in the Rayleigh-Ritz method for the study of the flexural vibration and buckling of isotropic and orthotropic rectangular plates. Journal of Sound and Vibration108: 51–62.
9.
GuyanRJ (1965) Reduction of stiffness and mass matrices. AIAA Journal3(2): 380–380.
10.
HrabokMMHrudeyTM (1984) A review and catalogue of plate bending finite elements. Computers & Structures19(3): 479–495.
11.
LeissaAW (1969) Vibration of Plates (NASA SP–160). Washington, DC: US Government Printing Office.
12.
LeissaAW (1973) The free vibration of rectangular plates. Journal of Sound and Vibration31: 257–293.
13.
LeissaAW (1977a) Recent research in plate vibrations, 1973–1976: classical theory. The Shock and Vibration Digest9(10): 13–24.
14.
LeissaAW (1977b) Recent research in plate vibrations, 1973–1976: complicating effects. The Shock and Vibration Digest9(11): 21–35.
15.
LeissaAW (1981a) Plate vibration research, 1976–1980: classical theory. The Shock and Vibration Digest13(9): 11–22.
16.
LeissaAW (1981b) Plate vibration research, 1976–1980: complicating effects. The Shock and Vibration Digest13(10): 19–36.
17.
LeissaAW (1987a) Plate vibration research, 1981–1985, part I: classical theory. The Shock and Vibration Digest19(2): 11–18.
18.
LeissaAW (1987b) Plate vibration research, 1981–1985, part II: complicating effects. The Shock and Vibration Digest19(3): 10–24.
19.
LiewKMLamKYChowST (1990) Free vibration analysis of rectangular plates using orthogonal plate function. Computers & Structures34: 79–85.
20.
LiewKMLimMK (1993) Transverse vibration of trapezoidal plates of variable thickness: symmetric trapezoids. Journal of Sound and Vibration165(1): 45–67.
21.
LiewKMLimCWLimMK (1994) Transverse vibration of trapezoidal plates of variable thickness: unsymmetric trapezoids. Journal of Sound and Vibration177(4): 479–501.
22.
LiewKMXiangYKitipornchaiS (1995) Research on thick plate vibration: a literature survey. Journal of Sound and Vibration180(1): 163–176.
23.
LimSPLeeKHSenthilnathanNR (1989) Rayleigh-Ritz vibration analysis of thick plates using a simple higher order theory. Journal of Sound and Vibration130: 163–166.
24.
MizusawaT (1993) Vibration of rectangular Mindlin plates with tapered thickness by the spline strip method. Computers & Structures46: 451–463.
25.
PetrolitoJ (1989) A modified ACM element for tick plate analysis. Computers & Structures32: 1303–1309.
26.
PulmanoVAGuptaRK (1976) Vibration of tapered plates by finite strip method. ASCE Journal of Engineering Mechanics102: 553–559.
27.
ShufrinIEisenbergerM (2006) Vibration of shear deformable plates with variable thickness – first-order and higher-order analyses. Journal of Sound and Vibration290: 465–489.
28.
YamadaGYIrieT (1987) Plate vibration research in Japan. Applied Mechanics Reviews40: 879–892.
29.
YuanFGMillerRE (1989) A cubic triangular finite element for flat plates with shear. International Journal for Numerical Methods in Engineering28: 109–126.
30.
ZhongnianX (1992) A thick-thin triangular plate element. International Journal for Numerical Methods in Engineering33: 963–973.
31.
ZhouD (2002) Vibration of point supported rectangular plates with variable thickness using a set of static tapered beam functions. International Journal of Mechanical Sciences44: 149–164.
32.
ZhouDCheungYKAuFTKLoSH (2002) Three-dimensional vibration analysis of thick rectangular plates using Chebyshev polynomial and Ritz method. International Journal of Solids and Structures39: 6339–6353.
33.
ZienkiewiczOCTaylorRL (1988) The Finite Element Methods. (Two volumes), New York: McGraw Hill.
