Analysis of rectangular stiffened plates under uniform lateral load based on FSDT and element-free Galerkin method

doi:10.1016/j.ijmecsci.2004.12.006

International Journal of Mechanical Sciences

Volume 47, Issue 2, February 2005, Pages 251-276

https://doi.org/10.1016/j.ijmecsci.2004.12.006 Get rights and content

Abstract

This paper presents an element-free Galerkin (EFG) method for the static analysis of concentrically and eccentrically stiffened plates based on first-order shear deformable theory (FSDT). The stiffened plates are regarded as composite structures of plates and beams. Imposing displacement compatible conditions between the plate and the stiffener, the displacement fields of the stiffener can be expressed in terms of the mid-surface displacement of the plate. The strain energy of the plate and stiffener can be superimposed to obtain the stiffness matrix of the stiffed plate. Because there are no elements used in the meshless model of the plate, the stiffeners need not to be placed along the meshes, as is done in the finite element methods. The stiffeners can be placed at any location, and will not lead to the re-meshing of the plate. The validity of the EFG method is demonstrated by considering several concentrically and eccentrically stiffened plate problems. The present results show good agreement with the existing analytical and finite element solutions. The influences of support size (denoted by a scaling factor $d_{\max}$ ) and order of the complete basis functions $(N_{c})$ on the numerical accuracy are also investigated. It is found that larger support size and higher order of basis function will furnish better convergence results.

Introduction

Being economical but strong enough, stiffened plates are widely used in all kinds of circumstances, such as bridges, ship hulls or decks, and aircraft structures. Numerous studies have analyzed stiffened plates. Earlier researchers simulated stiffened plates with grillage models [1] or orthotropic models [2]. However, these models did not achieve satisfying results in solving generalized stiffened plate problems. Hence, subsequent researchers tended to regard the plate and the stiffener separately and then combine them by imposing the displacement compatible conditions between the plate and the stiffener. Several methods have been developed, such as the Rayleigh–Ritz method [3], [4], [5], [6], the finite element method (FEM) [7], [8], [9], and the constraint method based on finite elements [10]. Of all these methods, finite element methods are the most convenient and can be used in large, complex structures due to the advances in computing that have made the methods extensively used in industry.

As an alternative to FEMs, the element-free method [11], mesh-free or meshless methods [12], [13], [14], [15] have recently been made popular in engineering analysis. Unlike the meshing process in the FEM, the mesh-free methods only discretize the domain of a problem to a set of scattered points. The mesh-free methods are flexible in many instances, such as moving boundary problem, crack growth with arbitrary and complex paths, and phase transformation problems. In such cases, FEMs can encounter difficulties in dealing with discontinuities that do not coincide with the original mesh lines. It is obvious that re-meshing will be needed in each step of the solution procedures, which will be quite involved. Mesh-free methods construct the approximation solution of a problem entirely in terms of a set of nodes that are distributed in the problem domain, and no elements or any other interrelationship among the nodes are needed. Without the need for meshes, the meshless methods avoid the disadvantages of FEMs.

In this study, we examine the applicability of the element-free Galerkin (EFG) method to the static analysis of concentrically and eccentrically stiffened plates. We employ the EFG method here to avoid the need for re-meshing that occurs with FEMs because the stiffeners need to be placed along the mesh lines: i.e., the change of the stiffener position leads to re-meshing of the entire plate domain. The EFG method is demonstrated to solve several stiffened plate problems with different combinations of stiffener lay-ups, boundary conditions, and thicknesses. The results from this study are validated with results obtained from other analytical and/or numerical methods.

Section snippets

The EFG method

Using a moving least-square approximation, a function $u (x)$ defined in a domain $Ω$ can be approximated by $u^{h} (x)$ in the sub-domain $Ω_{x}$ . $u^{h} (x)$ is defined as $u^{h} (x) = \sum_{j = 1}^{m} p_{j} (x) a_{j} (x) = p^{T} (x) a (x),$ where $p_{j} (x)$ are monomial basis functions, $a_{j} (x)$ are their coefficients, and m is the number of the basis functions. The commonly used basis is the linear basis $p^{T} = [1, x] in 1 D, m = 2,$ $p^{T} = [1, x, y] in 2 D, m = 3$ or quadratic basis $p^{T} = [1, x, x^{2}] in 1 D, m = 3,$ $p^{T} = [1, x, y, x^{2}, xy, y^{2}] in 2 D, m = 6 .$ The unknown coefficients $a_{j} (x)$ can be determined by

Formulation for stiffened plates

The mesh-free model of a stiffened plate, as shown in Fig. 1, is composed of a plate and two beams. The plate and the beams are discretized by a set of nodes. The degree of freedom (DOF) of every node of the plate is $(w_{p}, ϕ_{px}, ϕ_{py})$ . The DOF of every node of the $x$ -stiffener and the $y$ -stiffener are $(w_{sx}, ϕ_{sx})$ and $(w_{sy}, ϕ_{sy})$ , respectively. We neglect the in-plane bending of the stiffeners, and they have negligible torsional stiffness.

Validation studies

A simply supported square plate with one stiffener, as shown in Fig. 3, is subjected to a uniformly distributed load of 1.0 psi. The elastic modulus of both the plate and stiffener is $1.7 \times 10^{7} psi$ . The poisson ratio is 0.3.

This example has been solved by McBean [16] and Rossow et al. [10] using the FEM. We use $17 \times 17$ nodes to discrete the plate and 17 nodes to discrete the stiffener, and let the scaling factor $d_{\max} = 3.5$ . The scaling factor $d_{\max}$ is defined by $d_{\max} = \frac{r}{h_{m}},$ where $r$ is the radius of the

Conclusions

This paper presents an EFG method for analysis of concentrically and eccentrically stiffened plates. The plates are regarded as combinations of plates and stiffeners. Meshless models for plates and beams are employed to simulate the plates and stiffeners, respectively. By imposing displacement compatible constraints between the plate and the stiffener, the displacement field of the stiffener is expressed in terms of the displacements of the middle surface of the plate. Thus, the strain energy

Acknowledgements

The work described in this paper has been supported by grant awarded by the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 1140/03E).

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Analysis of rectangular stiffened plates under uniform lateral load based on FSDT and element-free Galerkin method

Abstract

Introduction

Section snippets

The EFG method

Formulation for stiffened plates

Validation studies

Conclusions

Acknowledgements

Computers and Structures

Computer Methods in Applied Mechanics and Engineering

Journal of Sound and Vibration

Computers & Structures

Computers & Structures

Computers & Structures

Computers & Structures

Computer Methods in Applied Mechanics and Engineering

International Journal of Mechanical Sciences

International Journal of Mechanical Sciences

International Journal of Mechanical Sciences

Computer Methods in Applied Mechanics and Engineering