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A New Higher-Order Finite Element Model for Free Vibration and Buckling of Functionally Graded Sandwich Beams with Porous Core Resting on a Two-Parameter Elastic Foundation Using Quasi-3D Theory

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Iranian Journal of Science and Technology, Transactions of Civil Engineering Aims and scope Submit manuscript

Abstract

In this paper, a new higher-order finite element model is proposed for free vibration and buckling analysis of functionally graded (FG) sandwich beams with porous core resting on a two-parameter Winkler-Pasternak elastic foundation based on quasi-3D deformation theory. The material properties of FG sandwich beams vary gradually through the thickness according to the power-law distribution. The governing equation of motion is derived from the Lagrange's equations. Three different porosity patterns including uniform, symmetric, and asymmetric are considered. The accuracy and convergence of the proposed model are verified with several numerical examples. A comprehensive parametric study is carried out to explore the effects of the boundary conditions, skin-to-core thickness ratio, power-law index, slenderness, porosity coefficient, porous distribution of the core, and elastic foundation parameters on the natural frequencies and critical buckling loads of FG sandwich beams.

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Authors and Affiliations

Authors

Contributions

All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Ibrahim Mohamed and Volkan Kahya. The first draft of the manuscript was written by Ibrahim Mohamed and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Volkan Kahya.

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The author(s) declare that there are no potential conflicts of interest concerning the research, authorship, and/or publication of this article.

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This article does not contain any studies with human participants or animals performed by any of the authors.

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Codes and data for replication can be provided upon request.

Appendix

Appendix

The shape functions \({\psi }_{i}\left(x\right)\) and \({\varphi }_{i}\left(x\right)\) are given are given as follows:

$$\begin{aligned} \psi_{1} & = 1 - \frac{3x}{L} + 2\left( \frac{x}{L} \right)^{2} ,\quad \psi_{2} = \frac{4x}{L} - \frac{{4x^{2} }}{{L^{2} }},\quad \psi_{3} = - \frac{x}{L} + 2\left( \frac{x}{L} \right)^{2} , \\ \varphi_{1} { } & = 1 - \frac{{23x^{2} }}{{L^{2} }} + \frac{{66x^{3} }}{{L^{3} }} - \frac{{68x^{4} }}{{L^{4} }} + \frac{{24x^{5} }}{{L^{5} }},\quad \varphi_{2} { } = x - \frac{{6x^{2} }}{L} + \frac{{13x^{3} }}{{L^{2} }} - \frac{{12x^{4} }}{{L^{3} }} + \frac{{4x^{5} }}{{L^{4} }}, \\ \varphi_{3} { } & = \frac{{16x^{2} }}{{L^{2} }} - \frac{{32x^{3} }}{{L^{3} }} + \frac{{16x^{4} }}{{L^{4} }},\quad \varphi_{4} { } = - \frac{{8x^{2} }}{L} + \frac{{32x^{3} }}{{L^{2} }} - \frac{{40x^{4} }}{{L^{3} }} + \frac{{16x^{5} }}{{L^{4} }}, \\ \varphi_{5} { } & = \frac{{7x^{2} }}{{L^{2} }} - \frac{{34x^{3} }}{{L^{3} }} + \frac{{52x^{4} }}{{L^{4} }} - \frac{{24x^{5} }}{{L^{5} }},\quad \varphi_{6} { } = - \frac{{x^{2} }}{L} + \frac{{5x^{3} }}{{L^{2} }} - \frac{{8x^{4} }}{{L^{3} }} + \frac{{4x^{5} }}{{L^{4} }} \\ \end{aligned}$$
(A1)

The components of the stiffness matrix K and the mass matrix M are given as follows:

$$\begin{aligned}\mathbf{K}&=\left[\begin{array}{c} {K}_{11} { K}_{12} {K}_{13} {K}_{14}\\ \\ {K}_{12} { K}_{22} {K}_{23} {K}_{24}\\ \\ {K}_{13} { K}_{23} {K}_{33} {K}_{34}\\ \\ {K}_{14} { K}_{24} {K}_{34} {K}_{44} \\ \end{array}\right],\mathbf{M}=\left[\begin{array}{c} {M}_{11} { M}_{12} {M}_{13} {M}_{14}\\ \\ {M}_{12} { M}_{22} {M}_{23} {M}_{24}\\ \\ {M}_{13} { M}_{23} {M}_{33} {M}_{34}\\ \\ {M}_{14} { M}_{24} {M}_{34} {M}_{44} \\ \end{array}\right],\\ {\varvec{G}} &= \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ & & & \\ {0 } & { N_{22} } & 0 & { 0} \\ & & & \\ { 0} & 0 & { 0} & 0 \\ & & & \\ 0 & { 0} & 0 & 0 \\ \end{array} } \right],\quad {{\varvec{\Delta}}} = \left\{ {\begin{array}{*{20}c} {u_{i} } \\ \\ {w_{i} } \\ \\ {\phi_{yi} } \\ \\ {\phi_{zi} } \\ \end{array} } \right\}\end{aligned}$$
(A2)

where

$$\begin{aligned} K_{11} \left( {i,j} \right) & = A_{11} \mathop \int \limits_{0}^{L} \psi_{i,x} \psi_{j,x} {\text{d}}x,K_{12} \left( {i,j} \right) = - B_{11} \mathop \int \limits_{0}^{L} \psi_{i,x} \varphi_{j,xx} {\text{d}}x, \\ K_{13} \left( {i,j} \right) & = C_{11} \mathop \int \limits_{0}^{L} \psi_{i,x} \psi_{j,x} {\text{d}}x,K_{14} \left( {i,j} \right) = B_{S13} \mathop \int \limits_{0}^{L} \psi_{i,x} \psi_{j} {\text{d}}x, \\ K_{22} \left( {i,j} \right) & = E_{11} \mathop \int \limits_{0}^{L} \varphi_{i,xx} \varphi_{j,xx} {\text{d}}x - N_{0} \mathop \int \limits_{0}^{L} \varphi_{i,x} \varphi_{j,x} {\text{d}}x + k_{w} \mathop \int \limits_{0}^{L} \varphi_{i} \varphi_{j} dx + k_{p} \mathop \int \limits_{0}^{L} \varphi_{i,x} \varphi_{j,x} {\text{d}}x, \\ K_{23} \left( {i,j} \right) & = - D_{11} \mathop \int \limits_{0}^{L} \varphi_{i,xx} \psi_{j,x} {\text{d}}x, \\ K_{24} \left( {i,j} \right) & = - C_{S13} \mathop \int \limits_{0}^{L} \varphi_{i,xx} \psi_{j} {\text{d}}x + gk_{w} \mathop \int \limits_{0}^{L} \varphi_{i} \psi_{j} {\text{d}}x + gk_{p} \mathop \int \limits_{0}^{L} \varphi_{i,x} \psi_{j,x} {\text{d}}x, \\ K_{33} \left( {i,j} \right) & = F_{11} \mathop \int \limits_{0}^{L} \psi_{i,x} \psi_{j,x} {\text{d}}x + A_{s55} \mathop \int \limits_{0}^{L} \psi_{i} \psi_{j} {\text{d}}x, \\ K_{34} \left( {i,j} \right) & = E_{S13} \mathop \int \limits_{0}^{L} \psi_{i,x} \psi_{j} {\text{d}}x + A_{s55} \mathop \int \limits_{0}^{L} \psi_{i} \psi_{j,x} {\text{d}}x, \\ K_{44} \left( {i,j} \right) & = D_{S33} \mathop \int \limits_{0}^{L} \psi_{i} \psi_{j} {\text{d}}x + A_{s55} \mathop \int \limits_{0}^{L} \psi_{i,x} \psi_{j,x} {\text{d}}x + g^{2} k_{w} \mathop \int \limits_{0}^{L} \psi_{i} \psi_{j} {\text{d}}x + g^{2} k_{p} \mathop \int \limits_{0}^{L} \psi_{i,x} \psi_{j,x} {\text{d}}x, \\ M_{11} \left( {i,j} \right) & = I_{1} \mathop \int \limits_{0}^{L} \psi_{i} \psi_{j} {\text{d}}x,M_{12} \left( {i,j} \right) = - I_{2} \mathop \int \limits_{0}^{L} \psi_{i} \varphi_{j,x} {\text{d}}x,M_{13} \left( {i,j} \right) = I_{4} \mathop \int \limits_{0}^{L} \psi_{i} \psi_{j} {\text{d}}x,M_{14} \left( {i,j} \right) = 0, \\ M_{22} \left( {i,j} \right) & = I_{1} \mathop \int \limits_{0}^{L} \varphi_{i} \varphi_{j} {\text{d}}x + I_{3} \mathop \int \limits_{0}^{L} \varphi_{i,x} \varphi_{j,x} {\text{d}}x,M_{23} \left( {i,j} \right) = - I_{5} \mathop \int \limits_{0}^{L} \varphi_{i,x} \psi_{j} {\text{d}}x,M_{24} \left( {i,j} \right) = I_{7} \mathop \int \limits_{0}^{L} \varphi_{i} \psi_{j} {\text{d}}x, \\ M_{33} \left( {i,j} \right) & = I_{6} \mathop \int \limits_{0}^{L} \psi_{i} \psi_{j} {\text{d}}x,M_{34} \left( {i,j} \right) = 0,M_{44} \left( {i,j} \right) = I_{8} \mathop \int \limits_{0}^{L} \psi_{i} \psi_{j} {\text{d}}x,N_{22} \left( {i,j} \right) = N_{0} \mathop \int \limits_{0}^{L} \varphi_{i,x} \varphi_{j,x} {\text{d}}x \\ \end{aligned}$$
(A3)

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Mohamed, I., Kahya, V. & Şimşek, S. A New Higher-Order Finite Element Model for Free Vibration and Buckling of Functionally Graded Sandwich Beams with Porous Core Resting on a Two-Parameter Elastic Foundation Using Quasi-3D Theory. Iran J Sci Technol Trans Civ Eng (2024). https://doi.org/10.1007/s40996-024-01482-x

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