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Analysis of Tapered Thin Plates Using Basic Displacement Functions

  • Research Article - Civil Engineering
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Abstract

Presenting new functions, basic displacement functions (BDFs), a novel method is introduced for the analysis of arbitrarily tapered thin plates in preference to primarily mathematically based methodologies. BDFs are obtained through applying unit load theorem and considering two orthogonal strips of unit width in tapered plates based on static deformations. It is shown that new shape functions and consequently structural matrices could be derived in terms of BDFs through a mechanical approach. On the other hand, BDFs could be used to calculate new shape functions, whereas Hermitian functions are used in several elements such as ACM and BFS. It is demonstrated that the accuracy of these new shape functions is significantly improved by considering the geometrical and mechanical properties of the plate element in the evaluation of the structural matrices. So, contrary to usual shape functions used in FE methods, they are susceptible to the thickness variation and then higher accuracy and more rapid convergence could be expected with fewer elements. In order to verify the competency of the proposed method, several numerical examples for classical boundary conditions are carried out and the results are compared with those in the literature.

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Abbreviations

a :

Length of the plate along x direction

b :

Length of the plate along y direction

b :

Vector of BDFs

D(x, y):

Flexural rigidity of the plate at \({(x,y)=(E h^{3}(x,y)/[{12(1-\nu^{2})}])}\)

E :

Young’s modulus of elasticity

F :

Vector of nodal forces

F ii :

Nodal flexibility matrices of i-th node

h(x, y):

Thickness of the plate at (x, y)

i, j :

Indices or positive integers (1, 2, 3, …)

K :

Stiffness matrix of element

l :

Taper factor of the plate in y direction

M :

Mass matrix of element

N :

Vector of shape functions

o′:

A point with arbitrary coordinates (x, y)

q z (x, y):

External transverse load

r :

Taper factor of the plate in x direction

s :

Longitudinal coordinate along plate in x direction

t :

Longitudinal coordinate along plate in y direction

w :

Transverse displacement at (x, y)

x :

Variable along longitudinal coordinate of plate element in x direction

y :

Variable along longitudinal coordinate of plate element in y direction

\({\alpha}\) :

Taper ratio for thickness of plate in the x direction

\({\beta}\) :

Taper ratio for thickness of plate in the y direction

\({{\bf \Delta}_i}\) :

Vector containing \({[w\,{\theta_x}\,{\theta_y}]_i^T}\) for node \({i (i=1,{\ldots},4)}\)

\({\nabla^{2}}\) :

Two-dimensional Laplacian operator

\({{\bf \Sigma}}\) :

Matrix containing nodal stiffness matrices of all nodes

\({\theta_x}\) :

Rotation angle due to bending along x direction

\({\theta_y}\) :

Rotation angle due to bending along y direction

\({\nu}\) :

Poisson’s ratio

\({\rho}\) :

Mass density per unit volume

Ω :

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

ω :

Eigenfrequency (frequency of free vibration)

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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. Analysis of Tapered Thin Plates Using Basic Displacement Functions. Arab J Sci Eng 39, 8691–8708 (2014). https://doi.org/10.1007/s13369-014-1407-x

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  • DOI: https://doi.org/10.1007/s13369-014-1407-x

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