An efficient higher order shear deformation theory for free vibration analysis of functionally graded shells

Zakaria Belabed,Mahmoud M. Selim,Omar Slimani,Noureddine Taibi,Abdelouahed Tounsi,Muzamal Hussain 국제구조공학회 2021 Steel and Composite Structures, An International J Vol.40 No.2

In this study, a simple and efficient higher order shear deformation theory is formulated for free vibration analysis of functionally graded (FG) shells. By introducing the undetermined integral terms in displacement field, the number of generated unknowns and their related governing equations is reduced in contrast to previously published theories, and therefore the differentiability of governing motion equations is decreased , this motivation turns the present theory simpler and easily exploited for functionally graded shell mechanical simulation. Both strains and stress rise through the thickness coordinate as function of hyperbolical distribution. The Hamilton’s principle is deployed to derive the governing and motion equations. Closed form solutions are obtained for free vibration problems using Navier’s method. Furthermore, detailed comparisons with other shear deformation theories are presented to illustrate the efficiency and accuracy of the developed theory. From this perspective, various perceptions on the impact of some important parameters such as material distribution, geometrical configuration, thickness and curvature ratios are studied and discussed. The non-trivial aspects in predicting the free vibration responses of FG shells are also pointed out.

A new shear deformation shell theory for free vibration analysis of FG sandwich shells

Omar Slimani,Zakaria Belabed,Fodil Hammadi,Noureddine Taibi,Abdelouahed Tounsi 국제구조공학회 2021 Structural Engineering and Mechanics, An Int'l Jou Vol.78 No.6

In recent years, many researches have been published dealing with the mechanical responses of shells with variable cross-sectional mechanical properties such as sandwich, functionally graded and laminated composites shells. In the present paper, a simple and efficient shear deformation theory is formulated for the free vibration response of functionally graded sandwich shells. The main advantage of this theory is its reduced number of unknowns and their related governing equations and theses tend to be highly compared to others shear deformation shell theories. Two kinds of FG sandwich shells are studied with respect to their geometrical configuration and material properties. The first kind is composed of FG facesheet and homogeneous core and the other is formed by homogeneous facesheet and FG core. The governing equations of motion for the free vibration analysis are obtained using Hamilton’s principle. The closed form solutions are sought by using the Navier’s method for eigenvalue problems. The accuracy and efficiency of the present theory are established and proved by comparing obtained numerical results with those predicted by other higher order shear deformation shell theories. The influences of various parameters such as material distribution, thickness of the core and the facesheet of sandwich shell and curvature ratios are studied, discussed and reported as significant rate sensitivity to predict the fundamental frequencies of FG sandwich shells.

Investigation of influence of homogenization models on stability and dynamic of FGM plates on elastic foundations

Mehala, Tewfik,Belabed, Zakaria,Tounsi, Abdelouahed,Beg, O. Anwar Techno-Press 2018 Geomechanics & engineering Vol.16 No.3

In this paper, the effect of the homogenization models on buckling and free vibration is presented for simply supported functionally graded plates (FGM) resting on elastic foundation. The majority of investigations developed in the last decade, explored the Voigt homogenization model to predict the effective proprieties of functionally graded materials at the macroscopic-scale for FGM mechanical behavior. For this reason, various models have been used to derive the effective proprieties of FGMs and simulate thereby their effects on the buckling and free vibration of FGM plates based on comparative studies that may differ in terms of several parameters. The refined plate theory, as used in this paper, is based on dividing the transverse displacement into both bending and shear components. This leads to a reduction in the number of unknowns and governing equations. Furthermore the present formulation utilizes a sinusoidal variation of displacement field across the thickness, and satisfies the stress-free boundary conditions on the upper and lower surfaces of the plate without requiring any shear correction factor. Equations of motion are derived from Hamilton's principle. Analytical solutions for the buckling and free vibration analysis are obtained for simply supported plates. The obtained results are compared with those predicted by other plate theories. This study shows the sensitivity of the obtained results to different homogenization models and that the results generated may vary considerably from one theory to another. Comprehensive visualization of results is provided. The analysis is relevant to aerospace, nuclear, civil and other structures.

Assessment of new 2D and quasi-3D nonlocal theories for free vibration analysis of size-dependent functionally graded (FG) nanoplates

Bendaho, Boudjema,Belabed, Zakaria,Bourada, Mohamed,Benatta, Mohamed Atif,Bourada, Fouad,Tounsi, Abdelouahed Techno-Press 2019 Advances in nano research Vol.7 No.4

In this present paper, a new two dimensional (2D) and quasi three dimensional (quasi-3D) nonlocal shear deformation theories are formulated for free vibration analysis of size-dependent functionally graded (FG) nanoplates. The developed theories is based on new description of displacement field which includes undetermined integral terms, the issues in using this new proposition are to reduce the number of unknowns and governing equations and exploring the effects of both thickness stretching and size-dependency on free vibration analysis of functionally graded (FG) nanoplates. The nonlocal elasticity theory of Eringen is adopted to study the size effects of FG nanoplates. Governing equations are derived from Hamilton's principle. By using Navier's method, analytical solutions for free vibration analysis are obtained through the results of eigenvalue problem. Several numerical examples are presented and compared with those predicted by other theories, to demonstrate the accuracy and efficiency of developed theories and to investigate the size effects on predicting fundamental frequencies of size-dependent functionally graded (FG) nanoplates.

Effect of homogenization models on stress analysis of functionally graded plates

Sihame Ait Yahia,Lemya Hanifi Hachemi Amar,Zakaria Belabed,Abdelouahed Tounsi 국제구조공학회 2018 Structural Engineering and Mechanics, An Int'l Jou Vol.67 No.5

In this paper, the effect of homogenization models on stress analysis is presented for functionally graded plates (FGMs). The derivation of the effective elastic proprieties of the FGMs, which are a combination of both ceramic and metallic phase materials, is of most of importance. The majority of studies in the last decade, the Voigt homogenization model explored to derive the effective elastic proprieties of FGMs at macroscopic-scale in order to study their mechanical responses. In this work, various homogenization models were used to derive the effective elastic proprieties of FGMs. The effect of these models on the stress analysis have also been presented and discussed through a comparative study. So as to show this effect, a refined plate theory is formulated and evaluated. , the number of unknowns and governing equations were reduced by dividing the transverse displacement into both bending and shear parts. Based on sinusoidal variation of displacement field trough the thickness, the shear stresses on top and bottom surfaces of plate were vanished and the shear correction factor was avoided. Governing equations of equilibrium were derived from the principle of virtual displacements. Analytical solutions of the stress analysis were obtained for simply supported FGM plates. The obtained results of the displacements and stresses were compared with those predicted by other plate theories available in the literature. This study demonstrates the sensitivity of the obtained results to different homogenization models and that the results generated may vary considerably from one theory to another. Finally, this study offers benchmark results for the multi-scale analysis of functionally graded plates.

Effect of homogenization models on stress analysis of functionally graded plates

Yahia, Sihame Ait,Amar, Lemya Hanifi Hachemi,Belabed, Zakaria,Tounsi, Abdelouahed Techno-Press 2018 Structural Engineering and Mechanics, An Int'l Jou Vol.67 No.5

In this paper, the effect of homogenization models on stress analysis is presented for functionally graded plates (FGMs). The derivation of the effective elastic proprieties of the FGMs, which are a combination of both ceramic and metallic phase materials, is of most of importance. The majority of studies in the last decade, the Voigt homogenization model explored to derive the effective elastic proprieties of FGMs at macroscopic-scale in order to study their mechanical responses. In this work, various homogenization models were used to derive the effective elastic proprieties of FGMs. The effect of these models on the stress analysis have also been presented and discussed through a comparative study. So as to show this effect, a refined plate theory is formulated and evaluated, the number of unknowns and governing equations were reduced by dividing the transverse displacement into both bending and shear parts. Based on sinusoidal variation of displacement field trough the thickness, the shear stresses on top and bottom surfaces of plate were vanished and the shear correction factor was avoided. Governing equations of equilibrium were derived from the principle of virtual displacements. Analytical solutions of the stress analysis were obtained for simply supported FGM plates. The obtained results of the displacements and stresses were compared with those predicted by other plate theories available in the literature. This study demonstrates the sensitivity of the obtained results to different homogenization models and that the results generated may vary considerably from one theory to another. Finally, this study offers benchmark results for the multi-scale analysis of functionally graded plates.