An Edge-Based Smoothed Finite Element for Free Vibration Analysis of Functionally Graded Porous (FGP) Plates on Elastic Foundation Taking into Mass (EFTIM)

Trung Nguyen-Thoi,Trung Thanh Tran,Quoc-Hoa Pham

doi:10.1155/2020/8278743

Abstract

In this paper, free vibration analysis of the functionally graded porous (FGP) plates on the elastic foundation taking into mass (EFTIM) is presented. The fundamental equations of the FGP plate are derived using Hamilton’s principle. The mixed interpolation of the tensorial components (MITC) approach and the edge-based smoothed finite element method (ES-FEM) is employed to avoid the shear locking as well as to improve the accuracy for the triangular element. The EFTIM is a foundation model based on the two-parameter Winkler–Pasternak model but added a mass parameter of foundation. Materials of the plate are FGP with a power-law distribution and maximum porosity distributions in the forms of cosine functions. Some numerical examples are examined to demonstrate the accuracy and reliability of the proposed method in comparison with those available in the literature.

Highlights

Free vibration analysis of the functionally graded porous (FGP) plates on the elastic foundation taking into mass (EFTIM) is presented. e fundamental equations of the FGP plate are derived using Hamilton’s principle. e mixed interpolation of the tensorial components (MITC) approach and the edge-based smoothed finite element method (ES-FEM) is employed to avoid the shear locking as well as to improve the accuracy for the triangular element. e EFTIM is a foundation model based on the two-parameter Winkler–Pasternak model but added a mass parameter of foundation
We examine the SSSS FGP plate and fully clamped (CCCC) plate resting on EFTIM. e parameters of EFTIM are given by β 0.5, μF 0.5, K1 100, and K1 10. e first nondimensional frequencies of plate with three cases of porosity distribution is shown in Table 7 and Figure 10
New numerical results of free vibration of the FGP plate resting on EFTIM are studied

Summary

Functionally Graded Porous Material Plates on Elastic Foundation

Let us consider an FGP plate resting on EFTIM, as shown in Figure 1. e FGP materials with a variation of two constituents and three different distributions of porosity through-thickness are presented as follows [25, 26]: Case 1: Λ(z) Ω cos􏼒πhz􏼓, Case. In order to mention the effectiveness of the foundation mass involved in the oscillation as well as the continuous interaction of the spring with the plate, the parameter β with unit kg−1 is added. It characterizes the effective level of the foundation mass involved in vibration, which is determined based on an experimental basis and the ratio of the density of the foundation to the density of plate material which is defined as μF ρF/ρ. For the dynamic problems, these two models have differences, and when omitting the influence parameters of the foundation mass, the EFTIM model is equivalent to the Winkler–Pasternak foundation model. Closely resembles the true feature of the foundation, including the Pasternak and Winkler foundation models

The First-Order Shear Deformation Theory and Weak Form of the FGP Plates

Formulation of the Finite Element Using the MITC3

Accuracy of the Proposed Method

Numerical Results and Discussions

Conclusions