• Structural Engineering and Mechanics
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• v.85 no.2
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• pp.147-161
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• 2023

Based on the ideas of variational differential quadrature (VDQ) and finite element method (FEM), a numerical approach named as VDQFEM is applied herein to study the large deformations of plate-type structures under static loading with arbitrary shape hole made of functionally graded graphene platelet-reinforced composite (FG-GPLRC) in the context of higher-order shear deformation theory (HSDT). The material properties of composite are approximated based upon the modified Halpin-Tsai model and rule of mixture. Furthermore, various FG distribution patterns are considered along the thickness direction of plate for GPLs. Using novel vector/matrix relations, the governing equations are derived through a variational approach. The matricized formulation can be efficiently employed in the coding process of numerical methods. In VDQFEM, the space domain of structure is first transformed into a number of finite elements. Then, the VDQ discretization technique is implemented within each element. As the last step, the assemblage procedure is performed to derive the set of governing equations which is solved via the pseudo arc-length continuation algorithm. Also, since HSDT is used herein, the mixed formulation approach is proposed to accommodate the continuity of first-order derivatives on the common boundaries of elements. Rectangular and circular plates under various boundary conditions with circular/rectangular/elliptical cutout are selected to generate the numerical results. In the numerical examples, the effects of geometrical properties and reinforcement with GPL on the nonlinear maximum deflection-transverse load amplitude curve are studied.

• Journal of Ocean Engineering and Technology
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• v.7 no.2
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• pp.44-56
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• 1993

The purpose of this paper is to analyze the impact response behaviors of glass/epoxy laminated composite plates subjected to the transversely impact of a steel ball. For this purpose, dynamic finite element analysis based on the higher-order shear defomation plate theory is used to compute the contact forces, rebound velocity of a steel ball, and dynamic strain response histories. And low-velocity and high-velocity impact experiments were conducted to compare the results and compute the wave propagation velocities. The results obtained from impact experiments are in good agreement with those of dynamic finite element analysis. Also the wave propagation velocities obtained from high-velocity impact experiments and wave propagation theory agree well, and wave velocities were higher in the smaller radius of steel ball.

• Structural Engineering and Mechanics
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• v.31 no.5
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• pp.481-506
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• 2009

The present study is concerned with the stochastic linear free vibration study of laminated composite plate embedded with piezoelectric layers with random material properties. The system equations are derived using higher order shear deformation theory. The lamina material properties of the laminate are modeled as basic random variables for accurate prediction of the system behavior. A $C^0$ finite element is used for spatial descretization of the laminate. First order Taylor series based mean centered perturbation technique in conjunction with finite element method is outlined for the problem. The outlined probabilistic approach is used to obtain typical numerical results, i.e., the mean and standard deviation of natural frequency. Different combinations of simply supported, clamped and free boundary conditions are considered. The effect of side to thickness ratio, aspect ratio, lamination scheme on scattering of natural frequency is studied. The results are compared with those available in literature and an independent Monte Carlo simulation.

• Steel and Composite Structures
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• v.35 no.5
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• pp.659-670
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• 2020

This paper proposes the use of a porous core between layers of laminated composite plates to examine its effect on the natural frequencies of the resulted porous laminated composite sandwich plate (PLCSP) resting on a two-parameter elastic foundation. Moreover, it has been suggested that the dispersion of porosity has two different functionally graded (FG) patterns which are compared with a uniformly dispersed (UD) profile to find their best vibrational efficiency in the proposed PLCSPs. In FG patterns, two types of dispersions, including symmetric (FG-S) and asymmetric (FG-A) patterns have been considered. To derive the governing Eigen value equation of such structures, the first order shear deformation theory (FSDT) of plates has been employed. Accordingly, a finite element method (FEM) is developed to solve the derived Eigen value equation. Using the mentioned theory and method, the effects of porosity parameters, fiber orientation of laminated composite, geometrical dimensions, boundary conditions and elastic foundation on the natural frequencies of the proposed PLCSPs have been studied. It is observed that embedding porosity in core layer leads to a significant improvement in the natural frequencies of PLCSPs. Moreover, the natural frequencies of PLCSPs with FG porous core are higher than those with UD porous core.

• Steel and Composite Structures
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• v.42 no.6
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• pp.779-789
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• 2022

This work presents a non-linear cylindrical bending analysis of functionally graded plate reinforced by single-walled carbon nanotubes (SWCNTs) in thermal environment using a simple integral higher-order shear deformation theory (HSDT). This theory does not require shear correction factors and the transverse shear stresses vary parabolically through the thickness. The material properties of SWCNTs are assumed to be temperature-dependent and are obtained from molecular dynamics simulations. The material properties of functionally graded carbon nanotube-reinforced composites (FG-CNTCRs) are considered to be graded in the thickness direction, and are estimated through a micromechanical model. The non-linear strain-displacement relations in the Von Karman sense are used to study the effect of geometric non-linearity and the solution is obtained by minimization of the total potential energy. The numerical illustrations concern the nonlinear bending response of FG-CNTRC plates under different sets of thermal environmental conditions, from which results for uniformly distributed CNTRC plates are obtained as benchmarks.

• Steel and Composite Structures
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• v.33 no.4
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• pp.509-523
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• 2019

In the present work, the buckling analysis of micro sandwich plate with an isotropic/orthotropic cores and piezoelectric/polymeric nanocomposite face sheets is studied. In this research, two cases for core of micro sandwich plate is considered that involve five isotropic Devineycell materials (H30, H45, H60, H100 and H200) and an orthotropic material also two cases for facesheets of micro sandwich plate is illustrated that include piezoelectric layers reinforced by carbon and boron-nitride nanotubes and polymeric matrix reinforced by carbon nanotubes under temperature-dependent and hydro material properties on the elastic foundations. The first order shear deformation theory (FSDT) is adopted to model micro sandwich plate and to apply size dependent effects from modified strain gradient theory. The governing equations are derived using the minimum total potential energy principle and then solved by analytical method. Also, the effects of different parameters such as size dependent, side ratio, volume fraction, various material properties for cores and facesheets and temperature and humidity changes on the dimensionless critical buckling load are investigated. It is shown from the results that the dimensionless critical buckling load for boron nitride nanotube is lower than that of for carbon nanotube. It is illustrated that the dimensionless critical buckling load for Devineycell H200 is highest and lowest for H30. Also, the obtained results for micro sandwich plate with piezoelectric facesheets reinforced by carbon nanotubes (case b) is higher than other states (cases a and c).The results of this research can be used in aircraft, automotive, shipbuilding industries and biomedicine.

• Advances in nano research
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• v.5 no.2
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• pp.69-97
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• 2017

In this study, nonlinear response of laminated functionally graded carbon nanotube reinforced composite (FG-CNTRC) plate under low-velocity impact based on the Eshelby-Mori-Tanaka approach in thermal conditions is studied. The governing equations are derived based on higher-order shear deformation plate theory (HSDT) under von $K\acute{a}rm\acute{a}n$ geometrical nonlinearity assumptions. The finite element method with 15 DOF at each node and Newmark's numerical integration method is applied to solve the governing equations. Four types of distributions of the uniaxially aligned reinforcement material through the thickness of the plates are considered. Material properties of the CNT and matrix are assumed to be temperature dependent. Contact force between the impactor and the laminated plate is obtained with the aid of the modified nonlinear Hertzian contact law models. In the numerical example, the effect of layup (stacking sequence) and lamination angle as well as the effect of temperature variations, distribution of CNTs, volume fraction of the CNTs, the mass and the velocity of the impactor in a constant energy level and boundary conditions on the impact response of the CNTRC laminated plates are investigated in details.

• Steel and Composite Structures
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• v.45 no.5
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• pp.621-640
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• 2022

In the present article, the vibration response of a geometrically imperfect bi-directional functionally graded plate (2D-FGP) with geometric discontinuities and micro-structural defects (porosities) has been investigated. A porosity model has been developed to incorporate the effective material properties of the bi-directional FGP which varies in two directions i.e. along the axial and transverse direction. The geometric discontinuity is also introduced in the plate in the form of a circular cut-out at the center of the plate. The structural kinematic formulation is based on the non-polynomial trigonometric higher-order shear deformation theory (HSDT). Finite element formulation is done using C° continuous Lagrangian quadrilateral four-noded element with seven degrees of freedom per node. The equations of motion have been derived using a variational approach. Convergence and validation studies have been documented to confirm the accuracy and efficiency of the present formulation. A detailed investigation study has been done to evaluate the influence of the circular cut-out, geometric imperfection, porosity inclusions, partial supports, volume fraction indexes (along with the thickness and length), and geometrical configurations on the vibration response of 2D-FGP. It is concluded that after a particular cut-out dimension, the vibration response of the 2D FGP exhibits non-monotonic behavior.

• Steel and Composite Structures
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• v.48 no.2
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• pp.113-130
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• 2023

The present manuscript aims to investigate the deviation between the middle surface (MS) and neutral surface (NS) formulations on the static response of bi-directionally functionally graded (BDFG) porous plate. The higher order shear deformation plate theory with a four variable is exploited to define the displacement field of BDFG plate. The displacement field variables based on both NS and on MS are presented in detail. These relations tend to get and derive a new set of boundary conditions (BCs). The porosity distribution is portrayed by cosine function including three different configurations, center, bottom, and top distributions. The elastic foundation including shear and normal stiffnesses by Winkler-Pasternak model is included. The equilibrium equations based on MS and NS are derived by using Hamilton's principles and expressed by variable coefficient partial differential equations. The numerical differential quadrature method (DQM) is adopted to solve the derived partial differential equations with variable coefficient. Rigidities coefficients and stress resultants for both MS and NS formulations are derived. The mathematical formulation is proved with previous published work. Additional numerical and parametric results are developed to present the influences of modified boundary conditions, NS and MS formulations, gradation parameters, elastic foundations coefficients, porosity type and porosity coefficient on the static response of BDFG porous plate. The following model can be used in design and analysis of BDFG structure used in aerospace, vehicle, dental, bio-structure, civil and nuclear structures.

• Structural Engineering and Mechanics
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• v.5 no.4
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• pp.433-450
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• 1997

The objective of this work is to predict the failure loads, associated maximum transverse displacements, locations and the modes of failure, including the onset of delamination, of thin, flat, square symmetric laminates under the action of uni-axial compression. Two progressive failure analyses, one using Hashin criterion and the other using Tensor polynomial criteria, are used in conjunction with the finite element method. First order shear deformation theory and geometric nonlinearity in the von Karman sense have been employed. Five different types of lay-up sequence are considered for laminates with all edges simply supported. In addition, two boundary conditions, one with all edges fixed and other with mixed boundary conditions for $(+45/-45/0/90)_{2s}$ quasi-isotropic laminate have also been considered to study the effect of boundary restraints on the failure loads and the corresponding modes of failure. A comparison of linear and nonlinear results is also made for $({\pm}45/0/90)_{2s}$ quasi-isotropic laminate. It is observed that the maximum difference between the failure loads predicted by various criteria depend strongly on the laminate lay-ups and the flexural boundary restraints. Laminates with clamped edges are found to be more susceptible to failure due to the transverse shear and delamination, while those with the simply supported edges undergo total collapse at a load slightly higher than the fiber failure load.