• 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.41 no.5
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• pp.697-711
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• 2021

In this work, a simple quasi 3-D parabolic shear deformation theory is developed to examine the bending response of antisymmetric cross-ply laminated composite plates under different types of mechanical loading. The main feature of this theory is that, in addition to including the transverse shear deformation and thickness stretching effects, it has only five-unknown variables in the displacement field modeling like Mindlin's theory (FSDT), yet satisfies the zero shear stress conditions on the top and bottom surfaces of the plate without requiring a shear correction factor. The static version of principle of virtual work was employed to derive the governing equations, while the bending problem for simply supported antisymmetric cross-ply laminated plates was solved by a Navier-type closed-form solution procedure. The adequacy of the proposed model is handled by considering the impact of side-to-thickness ratio on bending response of plate through several illustrative examples. Comparison of the obtained numerical results with the other shear deformation theories leads to the conclusion that the present model is more accurate and efficient in predicting the displacements and stresses of laminated composite plates.

• Structural Engineering and Mechanics
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• v.85 no.5
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• pp.593-605
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• 2023

This work presents a comparison between analytical and finite element analysis for bending of porous sandwich functionally graded material (FGM) plates. The plate is rectangular and simply supported under static sinusoidal loading. Material properties of FGM are assumed to vary continuously across the face sheets thickness according to a power-law function in terms of the volume fractions of the constituents while the core is homogeneous. Four types of porosity are considered. A refined higher-order shear with normal deformation theory is used. The number of unknowns in this theory is five, as against six or more in other shear and normal deformation theories. This theory assumes the nonlinear variation of transverse shear stresses and satisfies its nullity in the top and bottom surfaces of the plate without the use of a shear correction factor. The governing equations of equilibrium are derived from the virtual work principle. The Navier approach is used to solve equilibrium equations. The constitutive law of the porous FGM sandwich plate is implemented for a 3D finite element through a subroutine in FORTRAN (UMAT) in Abaqus software. Results show good agreement between the finite element model and the analytical method for some results, but the analytical method keeps giving symmetric results even with the thickness stretching effect and load applied to the top surface of the sandwich.

• Structural Engineering and Mechanics
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• v.86 no.1
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• pp.1-16
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• 2023

The free vibration of temperature-dependent functionally graded plates (FGPs) resting on a viscoelastic foundation is investigated in this paper using a newly developed simple first-order shear deformation theory (FSDT). Unlike other first order shear deformation (FSDT) theories, the proposed model contains only four variables' unknowns in which the transverse shear stress and strain follow a parabolic distribution along the plates' thickness, and they vanish at the top and bottom surfaces of the plate by considering a new shape function. For this reason, the present theory requires no shear correction factor. Linear steady-state thermal loads and power-law material properties are supposed to be graded across the plate's thickness. Uniform, linear, non-linear, and sinusoidal thermal rises are applied at the two surfaces for simply supported FGP. Hamilton's principle and Navier's approach are utilized to develop motion equations and analytical solutions. The developed theory shows progress in predicting the frequencies of temperature-dependent FGP. Numerical research is conducted to explain the effect of the power law index, temperature fields, and damping coefficient on the dynamic behavior of temperature-dependent FGPs. It can be concluded that the equation and transformation of the proposed model are as simple as the FSDT.

• Geomechanics and Engineering
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• v.34 no.3
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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.

• Steel and Composite Structures
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• v.44 no.4
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• pp.503-517
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• 2022

This paper presents the mechanical buckling of bi-directional functionally graded sandwich beams (BFGSW) with various boundary conditions employing a quasi-3D beam theory, including an integral term in the displacement field, which reduces the number of unknowns and governing equations. The beams are composed of three layers. The core is made from two constituents and varies across the thickness; however, the covering layers of the beams are made of bidirectional functionally graded material (BFGSW) and vary smoothly along the beam length and thickness directions. The power gradation model is considered to estimate the variation of material properties. The used formulation reflects the transverse shear effect and uses only three variables without including the correction factor used in the first shear deformation theory (FSDT) proposed by Timoshenko. The principle of virtual forces is used to obtain stability equations. Moreover, the impacts of the control of the power-law index, layer thickness ratio, length-to-depth ratio, and boundary conditions on buckling response are demonstrated. Our contribution in the present work is applying an analytical solution to investigate the stability behavior of bidirectional FG sandwich beams under various boundary conditions.

• Steel and Composite Structures
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• v.44 no.4
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• pp.451-471
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• 2022

In the present paper, the static bending and buckling responses of functionally graded carbon nanotubes-reinforced composite (FG-CNTRC) beam under various boundary conditions are investigated within the framework of higher shear deformation theory. The significant feature of the proposed theory is that it provides an accurate parabolic distribution of transverse shear stress through the thickness satisfying the traction-free boundary conditions needless of any shear correction factor. Uniform (UD) and four graded distributions of CNTs which are FG-O, FG-X, FG- and FG-V are selected here for the analysis. The effective material properties of FG-CNTRC beams are estimated according to the rule of mixture. To model the FG-CNTRC beam realistically, an efficient Hermite-Lagrangian finite element formulation is successfully developed. The accuracy and efficiency of the present model are demonstrated by comparison with published benchmark results. Moreover, comprehensive numerical results are presented and discussed in detail to investigate the effects of CNTs volume fraction, distribution patterns of CNTs, boundary conditions, and length-to-thickness ratio on the bending and buckling responses of FG-CNTRC beam. Several new referential results are also reported for the first time which will serve as a benchmark for future studies in a similar direction. It is concluded that the FG-X-CNTRC beam is the strongest beam that carries the lowest central deflection and is followed by the UD, V, Λ, and FG-O-CNTRC beam. Besides, the critical buckling load belonging to the FG-X-CNTRC beam is the highest, followed by UD and FG-O.

• Steel and Composite Structures
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• v.49 no.3
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• pp.307-323
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• 2023

In this research, the study of the thermoelastic flexural analysis of silicon carbide/Aluminum graded (FG) sandwich 2D uniform structure (plate) under harmonic sinusoidal temperature load over time is presented. The plate is modeled using a simple two dimensional integral shear deformation plate theory. The current formulation contains an integral terms whose aim is to reduce a number of variables compared to others similar solutions and therefore minimize the computation time. The transverse shear stresses vary according to parabolic distribution and vanish at the free surfaces of the structure without any use of correction factors. The external load is applied on the upper face and varying in the thickness of the plates. The structure is supposed to be composed of "three layers" and resting on nonlinear visco-Pasternak's-foundations. The governing equations of the system are deduced and solved via Hamilton's principle and general solution. The computed results are compared with those existing in the literature to validate the current formulation. The impacts of the parameters (material index, temperature exponent, geometry ratio, time, top/bottom temperature ratio, elastic foundation type, and damping coefficient) on the dynamic flexural response are studied.

• Structural Engineering and Mechanics
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• v.90 no.1
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• pp.83-96
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• 2024

This paper suggests an analytical approach to investigate the free vibration and stability of functionally graded (FG) beams with both perfect and imperfect characteristics using a quasi-3D higher-order shear deformation theory (HSDT) with stretching effect. The study specifically focuses on FG beams resting on variable elastic foundations. In contrast to other shear deformation theories, this particular theory employs only four unknown functions instead of five. Moreover, this theory satisfies the boundary conditions of zero tension on the beam surfaces and facilitates hyperbolic distributions of transverse shear stresses without the necessity of shear correction factors. The elastic medium in consideration assumes the presence of two parameters, specifically Winkler-Pasternak foundations. The Winkler parameter exhibits variable variations in the longitudinal direction, including linear, parabolic, sinusoidal, cosine, exponential, and uniform, while the Pasternak parameter remains constant. The effective material characteristics of the functionally graded (FG) beam are assumed to follow a straightforward power-law distribution along the thickness direction. Additionally, the investigation of porosity includes the consideration of four different types of porosity distribution patterns, allowing for a comprehensive examination of its influence on the behavior of the beam. Using the virtual work principle, equations of motion are derived and solved analytically using Navier's method for simply supported FG beams. The accuracy is verified through comparisons with literature results. Parametric studies explore the impact of different parameters on free vibration and buckling behavior, demonstrating the theory's correctness and simplicity.

• Structural Engineering and Mechanics
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• v.90 no.3
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• pp.233-252
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• 2024

This research explores a new finite element model for the free vibration analysis of bi-directional functionally graded (BDFG) beams. The model is based on an efficient higher-order shear deformation beam theory that incorporates a trigonometric warping function for both transverse shear deformation and stress to guarantee traction-free boundary conditions without the necessity of shear correction factors. The proposed two-node beam element has three degrees of freedom per node, and the inter-element continuity is retained using both C1 and C0 continuities for kinematics variables. In addition, the mechanical properties of the (BDFG) beam vary gradually and smoothly in both the in-plane and out-of-plane beam's directions according to an exponential power-law distribution. The highly elevated performance of the developed model is shown by comparing it to conceptual frameworks and solution procedures. Detailed numerical investigations are also conducted to examine the impact of boundary conditions, the bi-directional gradient indices, and the slenderness ratio on the free vibration response of BDFG beams. The suggested finite element beam model is an excellent potential tool for the design and the mechanical behavior estimation of BDFG structures.