Trans RINA, Vol 161, Part A4, Intl J Maritime Eng, Oct-Dec 2019 FREE VIBRATION OF SKEW LAMINATES – A BRIEF REVIEW AND SOME
BENCHMARK RESULTS (DOI No: 10.3940/rina.ijme.2019.a4.540)
S Haldar and S Pal, Indian Institute of Engineering Science and Technology, India, K Kalita, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, India SUMMARY
This study investigates and reviews prior research works on skew composite laminates. The equivalent single layer theories are explored and discussed. An exhaustive review on static and dynamic analysis of composite skew laminates is also presented. Subsequently, a nine node isoparametric plate bending element is used for free vibration analysis of laminated composite skew plate with central skew cut out. The effect of shear deformation is incorporated in the formulation considering first order shear deformation theory. Two types of mass lumping schemes are analysed to study the effect of rotary inertia. Certain numerical examples of plates having different skew angles, skew cut out sizes, boundary conditions, thickness ratios (h/a), aspect ratios (a/b), fiber orientations and number of layers are solved which will be useful for benchmarking of future studies.
NOMENCLATURE [B]
[D] [K] [N]
[N0] [] ||
[Nr] [K0] [M0] , w E
G ν h
a, b D ω
y {} {}
Qx Qy ,
CLPT
Strain-displacement matrix Rigidity matrix
Global stiffness matrix Shape function Null matrix
Consistent mass matrix Jacobian matrix
Interpolation function of the rth point Overall stiffness matrix Overall Mass matrix In-plane displacement Transverse displacement Modulus of elasticity Modulus of rigidity Poisson’s ratio
Thickness of plate Plate dimensions Flexural rigidity Natural frequency
Average shear rotation Total rotation in bending Stress vector Strain vector
, Bending moments in x and y direction
Twisting moment
Transverse shear forces Natural coordinates Density
Classical laminate plate theory
DSCM Discrete singular convolution method DSC-EM Discrete singular convolution-element method DQM Differential quadrature method EFGM Element-free Galerkin method ESLT FDM FEM
Equivalent single layer theory Finite difference method Finite element method
FSDT FSM
First-order shear deformation theory Finite strip method
HSDT Higher order shear deformation theory Iso
Isogeometric method ©2019: The Royal Institution of Naval Architects
MFVM Meshless finite volume method MLS-RM Moving least square Ritz method MM
Meshfree method
MTEKM Multi-Term extended Kantorovich method QEM Quadrature element method RBF
R-DQM Ritz-differential quadrature methodology RM
Radial basis function Ritz method
RRM 1. Rayleigh-Ritz method INTRODUCTION
Free vibration analysis of laminated composite plates is very important in the field of structural engineering. Many structures such as ships and containers require the complete enclosure of plates. With the advancement in fiber-reinforced laminated composite materials, the use of composite plates and shells has increased greatly due to their high strength to weight ratio. Fiber reinforced laminated composite plates are generally used in architectural structures, bridges, hydraulic structures, pavements, containers, airplanes, missiles, ships, instrument and automobile structures. Skew plates are often used in such modern structures. Swept wing of airplanes, for example, can be idealized by introducing substructures in the form of oblique plates. Similarly, complex alignment problems in bridge designs are often designed by using skew plates. Plates with cut-outs are also commonly encountered in engineering practice. Cut- outs are introduced to provide access, reduce weight and alter the dynamic response of structures.
In the present work, a brief literature review on equivalent single layer theories is presented. This is followed by an exhaustive review of the literature on skew plates. Both static and dynamic analysis involving skew plates are surveyed. A first-order shear deformation based finite element method is introduced and some benchmark results on skew plates are reported for certain test cases which are sparse in literature.
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