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Free Vibration of Tapered Mindlin Plates Using Basic Displacement Functions

  • Research Article - Civil Engineering
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Abstract

Introducing the concept of basic displacement functions (BDFs) in plate elements, a novel method is proposed to calculate new shape functions of tapered Mindlin plates for the free vibration analysis using BDFs. The BDFs are obtained from dynamic deformations of an arbitrary point in tapered Mindlin plate using Ritz method and substituting boundary equations of Clamped–Clamped–Free–Free plate in them. It is shown that exact shape functions and consequently structural matrices could be derived from BDFs. Unlike current shape functions used in FE method, the new shape functions take thickness variation and eigenfrequencies of the plates into account. So, higher accuracy and more rapid convergence could be expected with fewer elements. To verify the competency of the present shape functions, the results of several numerical examples are compared with those in the literature. It is shown that these shape functions derived from BDFs have quite satisfactory accuracy and low computational cost.

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Abbreviations

a :

Length of the plate along x direction

B b :

Curvature–displacement matrix

B s :

Shear–displacement matrix

b :

Length of the plate along y direction

b :

Vector of BDFs

D b :

Matrix of flexural rigidities

D s :

Matrix of shear rigidities

D(x j ,y j ):

Flexural rigidity of the plate at point j(x j , y j ) (= Eh 3(x j , y j )/[12(1 − v 2)])

d, e, f :

Vectors of the unknown coefficients

E :

Young’s modulus

F :

Vector of nodal forces

F ii :

Nodal flexibility matrices of i-th node

Σ :

Matrix containing nodal stiffness matrices

G :

Shear modulus of elasticity (= E/[2(1 + v)])

h(x j , y j ):

Thickness of the plate at point j(x j , y j )

I d :

The convolution integral

K b :

Flexural stiffness matrix of element

K s :

Shear stiffness matrix of element

K :

Stiffness matrix of element (= K b + K s )

K e :

Elements of matrix K for i, j = d, e, f

M :

Mass matrix of element

M e :

Elements of matrix M for i, j = d, e, f

N :

Vector of shape functions

o′:

A point with arbitrary coordinates (X, Y)

PM:

Mass density per unit area

q z (x, y):

External transverse load

r :

Taper factor of the plate in x direction

s :

Taper factor of the plate in y direction

T :

Kinetic energy of plate in terms of xy - coordinates

t :

Time

U :

Strain energy of plate in terms of xy- coordinates

V(x, y):

Shear force along z direction at point o′(x, y)

w :

Transverse displacement

X m (x), Φ m (x), Y n (y), Ψ n (y):

The appropriate admissible functions

X, Y :

Longitudinal coordinate along plate

x, y :

Dimensionless longitudinal coordinate along plate

α :

Taper ratio for thickness of plate in the x direction

β :

Taper ratio for thickness of plate in the y direction

θ x :

Rotation angle due to bending along x direction

θ y :

Rotation angle due to bending along y direction

κ :

Shear correction factor

ν :

Poisson’s ratio

ξ, η :

Dimensionless longitudinal coordinate along plate element (= 2x/a − 1, 2y/b − 1)

ξ j , η j :

Grid point coordinates along the ξη axis at point j(x j , y j )

Π :

Total energy functional of element

ρ :

Plate density per unit volume

\({\Phi_{i}^{w}, \Phi_{i}^{x}, \Phi_{i}^{y}}\) :

Boundary equations of the plate

Ω :

Non-dimensional eigenfrequency \({(=\omega a^2\sqrt \rho h_1/D_1)}\)

ω :

Eigenfrequency

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Authors and Affiliations

  1. School of Civil Engineering, University College of Engineering, University of Tehran, P.O. Box 11365-4563, Tehran, Iran

    Zahra Pachenari & Reza Attarnejad

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  1. Zahra Pachenari
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  2. Reza Attarnejad
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Correspondence to Reza Attarnejad.

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Pachenari, Z., Attarnejad, R. Free Vibration of Tapered Mindlin Plates Using Basic Displacement Functions. Arab J Sci Eng 39, 4433–4449 (2014). https://doi.org/10.1007/s13369-014-1071-1

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  • DOI: https://doi.org/10.1007/s13369-014-1071-1

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