• Structural Engineering and Mechanics
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• v.65 no.4
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• pp.491-504
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• 2018

Because of sandwich structures with low weight and high stiffness have much usage in various industries such as civil and aerospace engineering, in this article, buckling and free vibration analyses of coupled micro composite sandwich plates are investigated based on sinusoidal shear deformation (SSDT) and most general strain gradient theories (MGSGT). It is assumed that the sandwich structure rested on an orthotropic elastic foundation and make of four composite face sheets with temperature-dependent material properties that they reinforced by carbon and boron nitride nanotubes and two flexible transversely orthotropic cores. Mathematical formulation is presented using Hamilton's principle and governing equations of motions are derived based on energy approach and applying variation method for simply supported edges under electro-magneto-thermo-mechanical, axial buckling and pre-stresses loadings. In order to predict the effects of various parameters such as material length scale parameter, length to width ratio, length to thickness ratio, thickness of face sheets to core thickness ratio, nanotubes volume fraction, pre-stress load and orthotropic elastic medium on the natural frequencies and critical buckling load of double-bonded micro composite sandwich plates. It is found that orthotropic elastic medium has a special role on the system stability and increasing Winkler and Pasternak constants lead to enhance the natural frequency and critical buckling load of micro plates, while decrease natural frequency and critical buckling load with increasing temperature changes. Also, it is showed that pre-stresses due to help the axial buckling load causes that delay the buckling phenomenon. Moreover, it is concluded that the sandwich structures with orthotropic cores have high stiffness, but because they are not economical, thus it is necessary the sandwich plates reinforce by carbon or boron nitride nanotubes specially, because these nanotubes have important thermal and mechanical properties in comparison of the other reinforcement.

• Structural Engineering and Mechanics
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• v.64 no.3
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• pp.361-379
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• 2017

In this article, the vibration behavior of double-bonded sandwich microplates with homogeneous core and nanocomposite facesheets reinforced by carbon nanotube and boron nitride nanotube under multi physical fields such as 2D magnetic and electric fields is investigated. Symmetric and un-symmetric distributions of nanotubes are considered for facesheets of sandwich microplates such as uniform distribution and various functionally graded distributions. The double-bonded sandwich microplates rest on visco-Pasternak foundation. Material properties of sandwich microplates are obtained by the extended rule of mixture. The sinusoidal shear deformation theory (SSDT) is employed to describe displacement fields of sandwich microplates. Also, the dimensionless natural frequency is obtained by classical plate theory (CPT) and compared with the obtained results by SSDT. It can be seen that the obtained dimensionless natural frequencies by CPT are higher than SSDT. In order to study the material length scale parameters, modified strain gradient theory at micro scale is utilized and then, the equations of motion are derived using Hamilton's principle. The effects of different parameters such as foundation parameters including Winkler, shear layer and damping coefficients, various distributions and volume fraction of nanotubes, core to facesheet thickness ratio, aspect and side ratios on the dimensionless natural frequencies are discussed in details. The results of present work can be used to optimum design and control of similar systems such as micro-electro-mechanical and nano-electro-mechanical devices.

• Structural Engineering and Mechanics
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• v.37 no.2
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• pp.149-162
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• 2011

The aim of this research is a comprehensive review and evaluation of beam theories resting on elastic foundations that used to model mode-I delamination in multidirectional laminated composite by DCB specimen. A compliance based approach is used to calculate critical strain energy release rate (SERR). Two well-known beam theories, i.e. Euler-Bernoulli (EB) and Timoshenko beams (TB), on Winkler and Pasternak elastic foundations (WEF and PEF) are considered. In each case, a closed-form solution is presented for compliance versus crack length, effective material properties and geometrical dimensions. Effective flexural modulus ($E_{fx}$) and out-of-plane extensional stiffness ($E_z$) are used in all models instead of transversely isotropic assumption in composite laminates. Eventually, the analytical solutions are compared with experimental results available in the literature for unidirectional ($[0^{\circ}]_6$) and antisymmetric angle-ply ($[{\pm}30^{\circ}]_5$, and $[{\pm}45^{\circ}]_5$) lay-ups. TB on WEF is a simple model that predicts more accurate results for compliance and SERR in unidirectional laminates in comparison to other models. TB on PEF, in accordance with Williams (1989) assumptions, is too stiff for unidirectional DCB specimens, whereas in angle-ply DCB specimens it gives more reliable results. That it shows the effects of transverse shear deformation and root rotation on SERR value in composite DCB specimens.

• Magazine of the Korean Society of Agricultural Engineers
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• v.38 no.3
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• pp.82-91
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• 1996

This is a basic study for the static and dynamic analysis on the elasto-plastic and elasto-viscoplastic of an axi-symmetric shell. The objective of this study was to investigate the mechanical characteristics of a nuclear reinforced concrete containment structure, which was selected as a model, by a numerical analysis using a finite element method. The structure was modeled with discrete ring elements of 8-noded isoparametric element rotating against the symmetrical axis, and the interaction between the foundation and the structure was modeled by Winkler's model. Also, the meridional tendon was modeled with 2-node truss elements, and the hoop tendon was done with point elements in two degrees of freedom. The effect of the tendon was considered without the increasement in total degree of freedom as the stiffness matrix of modeled tendon elements was assembled on the stiffness matrix of ring elements linked with the tendon. The results obtained from the analysis of an example were summarized as follows : 1. The stresses in the hoop direction on the interior and exterior surfaces of the structure were shown in changes of similar trend, and high stresses appeared on the structure wall 2. The stresses in the meridional direction on the interior and exterior surfaces were shown in change of different trend. Especially, the stresses at the junctions between the dome and the wall and between the wall and the bottom plate of the structure were very high, compared with those at other parts of the structure. 3. The stress changes in the direction of thickness on the crown of the dome were much linearly distributed. However, as the amount of tendon increased, the stresses in the upper and lower parts of the wall established with the tendon were shown stress concentration. 4. The stress changes in the direction of thickness on the center of the structure wall was linearly distributed in the all cases, and special stress due to the use of the tendon was not shown.

• Steel and Composite Structures
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• v.20 no.2
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• pp.227-249
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• 2016

The objective of this work is to present a zeroth-order shear deformation theory for free vibration analysis of functionally graded (FG) nanoscale plates resting on elastic foundation. The model takes into consideration the influences of small scale and the parabolic variation of the transverse shear strains across the thickness of the nanoscale plate and thus, it avoids the employ use of shear correction factors. Also, in this present theory, the effect of transverse shear deformation is included in the axial displacements by using the shear forces instead of rotational displacements as in available high order plate theories. The material properties are supposed to be graded only in the thickness direction and the effective properties for the FG nanoscale plate are calculated by considering Mori-Tanaka homogenization scheme. The equations of motion are obtained using the nonlocal differential constitutive expressions of Eringen in conjunction with the zeroth-order shear deformation theory via Hamilton's principle. Numerical results for vibration of FG nanoscale plates resting on elastic foundations are presented and compared with the existing solutions. The influences of small scale, shear deformation, gradient index, Winkler modulus parameter and Pasternak shear modulus parameter on the vibration responses of the FG nanoscale plates are investigated.

• Advances in nano research
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• v.10 no.2
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• pp.151-163
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• 2021

The main target of this study is to investigate nonlinear transient responses of moving polymer nano-size plates fortified by means of Graphene Platelets (GPLs) and resting on a Winkler-Pasternak foundation under a transverse pressure force and a temperature variation. Two graphene spreading forms dispersed through the plate thickness are studied, and the Halpin-Tsai micro-mechanics model is used to obtain the effective Young's modulus. Furthermore, the rule of mixture is employed to calculate the effective mass density and Poisson's ratio. In accordance with the first order shear deformation and von Karman theory for nonlinear systems, the kinematic equations are derived, and then nonlocal strain gradient scheme is used to reflect the effects of nonlocal and strain gradient parameters on small-size objects. Afterwards, a combined approach, kinetic dynamic relaxation method accompanied by Newmark technique, is hired for solving the time-varying equation sets, and Fortran program is developed to generate the numerical results. The accuracy of the current model is verified by comparative studies with available results in the literature. Finally, a parametric study is carried out to explore the effects of GPL's weight fractions and dispersion patterns, edge conditions, softening and hardening factors, the temperature change, the velocity of moving nanoplate and elastic foundation stiffness on the dynamic response of the structure. The result illustrates that the effects of nonlocality and strain gradient parameters are more remarkable in the higher magnitudes of the nanoplate speed.

• Structural Engineering and Mechanics
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• v.85 no.3
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• pp.289-304
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• 2023

The main objective of this paper is to study the effect of porosity on the buckling behavior of thick functionally graded sandwich plate resting on various boundary conditions under different in-plane loads. The formulation is made for a newly developed sandwich plate using a functional gradient material based on a modified power law function of symmetric and asymmetric configuration. Four different porosity distribution are considered and varied in accordance with material propriety variation in the thickness direction of the face sheets of sandwich plate, metal foam also is considered in this study on the second model of sandwich which containing metal foam core and FGM face sheets. New quasi-3D high shear deformation theory is used here for this investigate; the present kinematic model introduces only six variables with stretching effect by adopting a new indeterminate integral variable in the displacement field. The stability equations are obtained by Hamilton's principle then solved by generalized solution. The effect of Pasternak and Winkler elastic foundations also including here. the present model validated with those found in the open literature, then the impact of different parameters: porosities index, foam cells distribution, boundary conditions, elastic foundation, power law index, ratio aspect, side-to-thickness ratio and different in-plane axial loads on the variation of the buckling behavior are demonstrated.

• Steel and Composite Structures
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• v.43 no.1
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• pp.91-106
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• 2022

This paper has focused on presenting a three dimensional theory of elasticity for free vibration of 3D-graphene foam reinforced polymer matrix composites (GrF-PMC) cylindrical panels resting on two-parameter elastic foundations. The elastic foundation is considered as a Pasternak model with adding a Shear layer to the Winkler model. The porous graphene foams possessing 3D scaffold structures have been introduced into polymers for enhancing the overall stiffness of the composite structure. Also, 3D graphene foams can distribute uniformly or non-uniformly in the shell thickness direction. The effective Young's modulus, mass density and Poisson's ratio are predicted by the rule of mixture. Three complicated equations of motion for the panel under consideration are semi-analytically solved by using 2-D differential quadrature method. The fast rate of convergence and accuracy of the method are investigated through the different solved examples. Because of using two-dimensional generalized differential quadrature method, the present approach makes possible vibration analysis of cylindrical panels with two opposite axial edges simply supported and arbitrary boundary at the curved edges. It is explicated that 3D-GrF skeleton type and weight fraction can significantly affect the vibrational characteristics of GrF-PMC panel resting on two-parameter elastic foundations.

• Structural Engineering and Mechanics
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• v.86 no.4
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• pp.443-459
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• 2023

This study investigates the theoretical thermal buckling analyses of thick porous rectangular functionally graded (FG) plates with different geometrical boundary conditions resting on a Winkler-Pasternak elastic foundation using a new higher-order shear deformation theory (HSDT). This new theory has only four unknowns and involves indeterminate integral variables in which no shear correction factor is required. The variation of material properties across the plate's thickness is considered continuous and varied following a simple power law as a function of volume fractions of the constituents. The effect of porosity with two different types of distribution is also included. The current formulation considers the Von Karman nonlinearity, and the stability equations are developed using the virtual works principle. The thermal gradients are involved and assumed to change across the FG plate's thickness according to nonlinear, linear, and uniform distributions. The accuracy of the newly proposed theory has been validated by comparing the present results with the results obtained from the previously published theories. The effects of porosity, boundary conditions, foundation parameters, power index, plate aspect ratio, and side-to-thickness ratio on the critical buckling temperature are studied and discussed in detail.

• 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.