• Structural Engineering and Mechanics
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• 제67권5호
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• pp.527-544
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• 2018

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.

• Coupled systems mechanics
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• 제9권3호
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• pp.237-264
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• 2020

This article investigates the static behaviour of functionally graded (FG) plates sometimes declared as advanced composite plates by using a simple and accurate quasi-3D and 2D hyperbolic higher-order shear deformation theories. The properties of functionally graded materials (FGMs) are assumed to vary continuously through the thickness direction according to exponential law distribution (E-FGM). The kinematics of the present theories is modeled with an undetermined integral component and satisfies the free transverse shear stress conditions on the top and bottom surfaces of the plate; therefore, it does not require the shear correction factor. The fundamental governing differential equations and boundary conditions of exponentially graded plates are derived by employing the static version of principle of virtual work. Analytical solutions for bending of EG plates subjected to sinusoidal distributed load are obtained for simply supported boundary conditions using Navier'is solution procedure developed in the double Fourier trigonometric series. The results for the displacements and stresses of geometrically different EG plates are presented and compared with 3D exact solution and with other quasi-3D and 2D higher-order shear deformation theories to verify the accuracy of the present theory.

• 대한조선학회지
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• 제22권1호
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• pp.21-30
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• 1985

This paper deals whth the buckling as well as postbuckling analysis of axisymmertric shells taking the initial deflection effects into account. Incremental equilibrium equations, based on the principle of virtual work, were derived by the finite element method, the successive step-by-step Newton-Raphson iterative technique was adopted. To define the transition pattern of postbuckling behavior from the prebuckling state more accurately, a simple solution method was developed, i.e. the critical load was calculated by the load extrapolation method with the determinant of tangent stiffness matrix and the equilibrium configuration in the immediate postbuckling stage was obtained by perturbation scheme and eigenvalue analysis. Degenerated isoparametric shell elements were used to analyse the axisymmetric shell of revolution. And by the method developed in this paper, the computer program applicable to the nonlinear analysis of both thin and moderately thick shells was constructed. To verify the capabilities and accuracies of the present solution method, the computed results were compared with the results of analytical solutions. These results coincided fairly well in both the small deflection and large deflection ranges. Various numerical analyses were done to show the effect of initial deflection and shape of shells on buckling load and postbuckling behavior. Futhermore, corrected directions of applied loads at every increment steps were used to determine the actual effects of large deflection in non-conservative load systems such as hydrostatic pressure load. The following conclusions can be obtained. (1) The method described in this paper was found to be both economic and effective in calculating buckling load and postbuckling behavior of shell structure. (2) Buckling and postbuckling behavior of spherical caps is critically dependent upon their geometric configuration, i.e. the shape of spherical cap and quantities of the initial deflection. (3) In the analysis of large deflection problems of shells by the incremental method, corrections of the applied load directions are needed at every incremental step to compensate the follower force effects.

• Structural Engineering and Mechanics
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• 제74권3호
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• pp.365-380
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• 2020

The focus of this paper is to develop an analytical approach based on an efficient shear deformation theory with stretching effect for bending stress analysis of cross-ply laminated composite plates subjected to transverse parabolic load and line load by using a new kinematic model, in which the axial displacements involve an undetermined integral component in order to reduce the number of unknowns and a sinusoidal function in terms of the thickness coordinate to include the effect of transverse shear deformation. The present theory contains only five unknowns and satisfies the zero shear stress conditions on the top and bottom surfaces of the plate without using any shear correction factors. The governing differential equations and its boundary conditions are derived by employing the static version of principle of virtual work. Closed-form solutions for simply supported cross-ply laminated plates are obtained applying Navier's solution technique, and the numerical case studies are compared with the theoretical results to verify the utility of the proposed model. Lastly, it can be seen that the present outlined theory is more accurate and useful than some higher-order shear deformation theories developed previously to study the static flexure of laminated composite plates.

• Steel and Composite Structures
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• 제43권6호
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• pp.821-837
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• 2022

This research is devoted to study the effects of humidity and temperature on the bending behavior of functionally graded (FG) ceramic-metal porous plates resting on Pasternak elastic foundation using a quasi-3D hyperbolic shear deformation theory developed recently. The present plate theory with only four unknowns, takes into account both transverse shear and normal deformations and satisfies the zero traction boundary conditions on the surfaces of the functionally graded plate without using shear correction factors. Material properties of porous FG plate are defined by rule of the mixture with an additional term of porosity in the through-thickness direction. The governing differential equations are obtained using the "principle of virtual work". Analytically, the Navier method is used to solve the equations that govern a simply supported FG porous plate. The obtained results are checked by comparing the results determined for the perfect and imperfect FG plates with those available in the scientific literature. Effects due to material index, porosity factors, moisture and thermal loads, foundation rigidities, geometric ratios on the FG porous plate are all examined. Finally, this research will help us to design advanced functionally graded materials to ensure better durability and efficiency for hygro-thermal environments.

• Computers and Concrete
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• 제34권4호
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• pp.447-476
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• 2024

The current research proposes an innovative finite element model established within the context of higher-order beam theory to examine the bending and buckling behaviors of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beams resting on Winkler-Pasternak elastic foundations. This two-node beam element includes four degrees of freedom per node and achieves inter-element continuity with both C¹ and C⁰ continuities for kinematic variables. The isoparametric coordinate system is implemented to generate the elementary stiffness and geometric matrices as a way to enhance the existing model formulation. The weak variational equilibrium equations are derived from the principle of virtual work. The mechanical properties of FG-CNTRC beams are considered to vary gradually and smoothly over the beam thickness. The current investigation highlights the influence of porosity dispersions through the beam cross-section, which is frequently omitted in previous studies. For this reason, this analysis offers an enhanced comprehension of the mechanical behavior of FG-CNTRC beams under various boundary conditions. Through the comparison of the current results with those published previously, the proposed finite element model demonstrates a high rate of efficiency and accuracy. The estimated results not only refine the precision in the mechanical analysis of FG-CNTRC beams but also offer a comprehensive conceptual model for analyzing the performance of porous composite structures. Moreover, the current results are crucial in various sectors that depend on structural integrity in specific environments.

• Computers and Concrete
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• 제26권2호
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• pp.185-201
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• 2020

This research is devoted to investigate the bending and free vibration behaviour of laminated composite/sandwich plates and shells, by applying an analytical model based on a generalized and simple refined higher-order shear deformation theory (RHSDT) with four independent unknown variables. The kinematics of the proposed theoretical model is defined by an undetermined integral component and uses the hyperbolic shape function to include the effects of the transverse shear stresses through the plate/shell thickness; hence a shear correction factor is not required. The governing differential equations and associated boundary conditions are derived by employing the principle of virtual work and solved via Navier-type analytical procedure. To verify the validity and applicability of the present refined theory, some numerical results related to displacements, stresses and fundamental frequencies of simply supported laminated composite/sandwich plates and shells are presented and compared with those obtained by other shear deformation models considered in this paper. From the analysis, it can be concluded that the kinematics based on the undetermined integral component is very efficient, and its use leads to reach higher accuracy than conventional models in the study of laminated plates and shells.

• Structural Engineering and Mechanics
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• 제71권6호
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• pp.627-641
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• 2019

The paper focuses on bending analysis of the functionally graded (FG) plates with arbitrary shapes and boundary conditions. The material property of FG plates is modelled by using the power law distribution. Based on the first order shear deformation plate theory (FSDT), the governing equations as well as boundary conditions are formulated and obtained by using the principle of virtual work. The coupled Boundary Element-Radial Basis Function (BE-RBF) method is established to solve the complex FG plates. The proposed methodology is developed by applying the concept of the analog equation method (AEM). According to the AEM, the original governing differential equations are replaced by three Poisson equations with fictitious sources under the same boundary conditions. Then, the fictitious sources are established by the application of a technique based on the boundary element method and approximated by using the radial basis functions. The solution of the actual problem is attained from the known integral representations of the potential problem. Therefore, the kernels of the boundary integral equations are conveniently evaluated and readily determined, so that the complex FG plates can be easily computed. The reliability of the proposed method is evaluated by comparing the present results with those from analytical solutions. The effects of the power index, the length to thickness ratio and the modulus ratio on the bending responses are investigated. Finally, many interesting features and results obtained from the analysis of the FG plates with arbitrary shapes and boundary conditions are demonstrated.

• Structural Engineering and Mechanics
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• 제86권2호
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• pp.167-180
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• 2023

This work applies a four-known quasi-3D shear deformation theory to investigate the bending behavior of a functionally graded plate resting on a viscoelastic foundation and subjected to hygro-thermo-mechanical loading. The theory utilizes a hyperbolic shape function to predict the transverse shear stress, and the transverse stretching effect of the plate is considered. The principle of virtual displacement is applied to obtain the governing differential equations, and the Navier method, which comprises an exponential term, is used to obtain the solution. Novel to the current study, the impact of the viscoelastic foundation model, which includes a time-dependent viscosity parameter in addition to Winkler's and Pasternak parameters, is carefully investigated. Numerical examples are presented to validate the theory. A parametric study is conducted to study the effect of the damping coefficient, the linear and nonlinear loadings, the power-law index, and the plate width-tothickness ratio on the plate bending response. The results show that the presence of the viscoelastic foundation causes an 18% decrease in the plate deflection and about a 10% increase in transverse shear stresses under both linear and nonlinear loading conditions. Additionally, nonlinear loading causes a one-and-a-half times increase in horizontal stresses and a nearly two-times increase in normal transverse stresses compared to linear loading. Based on the article's findings, it can be concluded that the viscosity effect plays a significant role in the bending response of plates in hygrothermal environments. Hence it shall be considered in the design.

• Geomechanics and Engineering
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• 제34권3호
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• pp.233-250
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• 2023

This paper investigates the flexural analysis of isotropic, transversely isotropic, and laminated composite plates using a new higher-order normal and shear deformation theory. In the present theory, only five unknown functions are involved compared to six or more unknowns used in the other similar theories. The developed theory does not need a shear correction factor. It can satisfy the zero traction boundary conditions on the top and the bottom surfaces of the plate as well as account for sufficient distribution of the transverse shear strains. The thickness stretching effect is considered in the computation. A simply supported was considered on all edges of the plate. The plate is subjected to uniform and sinusoidal distributed load in the static analysis. Laminated composite, isotropic, and transversely isotropic plates are considered. The governing equations are obtained utilizing the virtual work principle. The differential equations are solved via Navier's procedure. The results obtained from the developed theory are compared with other higher-order theories considered in the previous studies and 3D elasticity solutions. The results showed that the proposed theory accurately and effectively predicts the bidirectional bending responses of laminated composite plates. Several parametric studies are presented to illustrate the various parameters influencing the static response of the laminated composite plates.