• Structural Engineering and Mechanics
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• v.55 no.4
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• pp.743-763
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• 2015

This work presents a new nonlocal hyperbolic shear deformation beam theory for the static, buckling and vibration of nanoscale-beams embedded in an elastic medium. The present model is able to capture both the nonlocal parameter and the shear deformation effect without employing shear correction factor. The nonlocal parameter accounts for the small size effects when dealing with nanosize structures such as nanobeams. Based on the nonlocal differential constitutive relations of Eringen, the equations of motion of the nanoscale-beam are obtained using Hamilton's principle. The effect of the surrounding elastic medium on the deflections, critical buckling loads and frequencies of the nanobeam is investigated. Both Winkler-type and Pasternak-type foundation models are used to simulate the interaction of the nanobeam with the surrounding elastic medium. Analytical solutions are presented for a simply supported nanoscale-beam, and the obtained results compare well with those predicted by the other nonlocal theories available in literature.

• 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.

• Coupled systems mechanics
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• v.5 no.3
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• pp.269-283
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• 2016

A two new high-order shear deformation theory for bending analysis is presented for a simply supported, functionally graded plate with porosities resting on an elastic foundation. This porosities may possibly occur inside the functionally graded materials (FGMs) during their fabrication, while material properties varying to a simple power-law distribution along the thickness direction. Unlike other theories, there are only four unknown functions involved, as compared to five in other shear deformation theories. The theories presented are variationally consistent and strongly similar to the classical plate theory in many aspects. It does not require the shear correction factor, and gives rise to the transverse shear stress variation so that the transverse shear stresses vary parabolically across the thickness to satisfy free surface conditions for the shear stress. It is established that the volume fraction of porosity significantly affect the mechanical behavior of thick function ally graded plates. The validity of the two new theories is shown by comparing the present results with other higher-order theories. The influence of material parameter, the volume fraction of porosity and the thickness ratio on the behavior mechanical P-FGM plate are represented by numerical examples.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.14 no.3
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• pp.371-379
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• 2001

This paper deals with the free vibrations of horizontally curved members resting on elastic foundations with continuity effect. Taking into account the effects of rotatory inertia and shear deformation, differential equations governing the free vibrations of such beams are derived, in which the Pasternak foundation model is considered as the elastic foundation with continuity effect. The differential equations are solved numerically to calculate natural frequencies and mode shapes. The experiments were performed in which the natural frequencies of such curved beams in laboratorial scale were measured and these results agree quite well with the present numerical studies. In numerical examples, the circular, parabolic, sinusoidal and elliptic curved members with the hinged-hinged, hinged-clamped and clamped end constraints are considered. The parametric studies are conducted and the lowest four frequency parameters are reported in tables and figures as the non-dimensional forms. Also the typical mode shapes are presented.

• Structural Engineering and Mechanics
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• v.57 no.4
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• pp.617-639
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• 2016

In this work, an analytical formulation based on both hyperbolic shear deformation theory and stress function, is presented to study the nonlinear post-buckling response of symmetric functionally graded plates supported by elastic foundations and subjected to in-plane compressive, thermal and thermo-mechanical loads. Elastic properties of material are based on sigmoid power law and varying across the thickness of the plate (S-FGM). In the present formulation, Von Karman nonlinearity and initial geometrical imperfection of plate are also taken into account. By utilizing Galerkin procedure, closed-form expressions of buckling loads and post-buckling equilibrium paths for simply supported plates are obtained. The effects of different parameters such as material and geometrical characteristics, temperature, boundary conditions, foundation stiffness and imperfection on the mechanical and thermal buckling and post-buckling loading capacity of the S-FGM plates are investigated.

• Smart Structures and Systems
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• v.25 no.4
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• pp.471-486
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• 2020

Stability analysis of three-layered piezoelectric doubly curved nano shell with accounting size dependency is performed in this paper based on first order shear deformation theory and curvilinear coordinate system relations. The elastic core is integrated with sensor and actuator layers subjected to applied electric potentials. The principle of virtual work is employed for derivation of governing equations of stability. The critical electrical and mechanical buckling loads are evaluated in terms of important parameters of the problem such as size-dependent parameter, two principle angle of doubly curved shell and two parameters of Pasternak's foundation. One can conclude that mechanical buckling loads are decreased with increase of nonlocal parameter while the electrical buckling loads are increased.

• Steel and Composite Structures
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• v.46 no.1
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• pp.1-13
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• 2023

Elastic bending of imperfect functionally graded sandwich plates (FGSPs) laying on the Winkler-Pasternak foundation and subjected to sinusoidal loads is analyzed. The analyses have been established using the quasi-3D sinusoidal shear deformation model. In this theory, the number of unknowns is condensed to only five unknowns using integral-undefined terms without requiring any correction shear factor. Moreover, the current constituent material properties of the middle layer is considered homogeneous and isotropic. But those of the top and bottom face sheets of the graded porous sandwich plate (FGSP) are supposed to vary regularly and continuously in the direction of thickness according to the trigonometric volume fraction's model. The corresponding equilibrium equations of FGSPs with simply supported edges are derived via the static version of the Hamilton's principle. The differential equations of the system are resolved via Navier's method for various schemes of FGSPs. The current study examine the impact of the material index, porosity, side-to-thickness ratio, aspect ratio, and the Winkler-Pasternak foundation on the displacements, axial and shear stresses of the sandwich structure.

• Structural Engineering and Mechanics
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• v.53 no.6
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• pp.1215-1240
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• 2015

In this work, a novel simple first-order shear deformation plate theory based on neutral surface position is developed for bending and free vibration analysis of functionally graded plates and supported by either Winkler or Pasternak elastic foundations. By dividing the transverse displacement into bending and shear parts, the number of unknowns and governing equations of the present theory is reduced, and hence, makes it simple to use. The governing equations are derived by employing the Hamilton's principle and the physical neutral surface concept. There is no stretching-bending coupling effect in the neutral surface-based formulation, and consequently, the governing equations and boundary conditions of functionally graded plates based on neutral surface have the simple forms as those of isotropic plates. Numerical results of present theory are compared with results of the traditional first-order and the other higher-order theories reported in the literature. It can be concluded that the proposed theory is accurate and simple in solving the static bending and free vibration behaviors of functionally graded plates.

• Structural Engineering and Mechanics
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• v.71 no.2
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• pp.185-196
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• 2019

In this investigation, study of the static and dynamic behaviors of functionally graded beams (FGB) is presented using a hyperbolic shear deformation theory (HySDT). The simply supported FG-beam is resting on the elastic foundation (Winkler-Pasternak types). The properties of the FG-beam vary according to exponential (E-FGB) and power-law (P-FGB) distributions. The governing equations are determined via Hamilton's principle and solved by using Navier's method. To show the accuracy of this model (HySDT), the current results are compared with those available in the literature. Also, various numerical results are discussed to show the influence of the variation of the volume fraction of the materials, the power index, the slenderness ratio and the effect of Winkler spring constant on the fundamental frequency, center deflection, normal and shear stress of FG-beam.

• Structural Engineering and Mechanics
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• v.4 no.2
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• pp.149-162
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• 1996

A postbuckling analysis is presented for a simply supported, moderately thick rectangular plate subjected to combined axial compression and uniform temperature loading and resting on a two-parameter elastic foundation. The two cases of thermal postbuckling of initially compressed plates and of compressive postbuckling of initially heated plates are considered. The initial geometrical imperfection of the plate is taken into account. The formulations are based on the Reissner-Mindlin plate theory considering the first order shear deformation effect, and including the plate-foundation interaction and thermal effect. The analysis uses a deflection-type perturbation technique to determine the buckling loads and postbuckling equilibrium paths. Numerical examples cover the performances of perfect and imperfect, moderately thick plates resting on Winkler or Pasternak-type elastic foundations. Typical results are presented in dimensionless graphical form.