Buckling of nonuniform and axially functionally graded nonlocal Timoshenko nanobeams on Winkler-Pasternak foundation

Robinson, Mouafo Teifouet Armand; Adali, Sarp

doi:10.1016/j.compstruct.2018.07.046

Cited by 38 publications

(9 citation statements)

References 36 publications

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“…The only difficult thing in the semi-analytical solution methods like the Rayleigh-Ritz might be determining mode shapes which should satisfy boundary conditions. In this research, a new mode shape is assumed by which a very good agreement has been obtained while comparing the numerical outcomes with the literature [49][50][51][52].…”

Section: Rayleigh-ritz Solution Processmentioning

confidence: 75%

“…As the post-buckling discussions are geometrically nonlinear ones; therefore, in order to solve nonlinear eigenvalue problems, the Rayleigh-Ritz solution technique can be a good choice [49][50][51][52], owing to its capability to give high accurate numerical outcomes. The method is a semi-analytical one and satisfies eigenvalue problems, a few of which should be solved nonlinearly for which the numerical solutions have to be employed.…”

Section: Rayleigh-ritz Solution Processmentioning

confidence: 99%

“…Hence, semianalytical methods can be a better suggestion to solve nonlinear eigenvalue problems, for example, postbuckling ones. The transverse displacement for the Rayleigh-Ritz method was presented as [52] å j…”

Section: Rayleigh-ritz Solution Processmentioning

confidence: 99%

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Post-critical buckling of truncated conical carbon nanotubes considering surface effects embedding in a nonlinear Winkler substrate using the Rayleigh-Ritz method

Malikan

Eremeyev

2020

Mater. Res. Express

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This research predicts theoretically post-critical axial buckling behavior of truncated conical carbon nanotubes (CCNTs) with several boundary conditions by assuming a nonlinear Winkler matrix. The post-buckling of CCNTs has been studied based on the Euler-Bernoulli beam model, Hamilton's principle, Lagrangian strains, and nonlocal strain gradient theory. Both stiffness-hardening and stiffness-softening properties of the nanostructure are considered by exerting the second stressgradient and second strain-gradient in the stress and strain fields. Besides small-scale influences, the surface effect is also taken into consideration. The effect of the Winkler foundation is nonlinearly taken into account based on the Taylor expansion. A new admissible function is used in the Rayleigh-Ritz solution technique applicable for buckling and post-buckling of nanotubes and nanobeams. Numerical results and related discussions are compared and reported with those obtained by the literature. The significant results proved that the surface effect and the nonlinear term of the substrate affect the CCNT considerably.

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Section: Rayleigh-ritz Solution Processmentioning

confidence: 75%

Section: Rayleigh-ritz Solution Processmentioning

confidence: 99%

See 1 more Smart Citation

Post-critical buckling of truncated conical carbon nanotubes considering surface effects embedding in a nonlinear Winkler substrate using the Rayleigh-Ritz method

Malikan

Eremeyev

2020

Mater. Res. Express

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“…Rajasekaran et al [16] studied the free vibration, buckling, and static bending of axially functionally graded nano-tapered Timoshenko and Bernoulli-Euler beams based on the nonlocal Timoshenko beam theory. Based on the nonlocal Timoshenko beam theory, Robinson et al [17] studied the buckling critical load of axially functionally graded material Timoshenko beams with variable cross-sections. Deng et al [18] established the exact dynamic stiffness matrix of an axial functionally graded material Timoshenko double-beam system on Winkler-Pasternak under an axial load, considered the damping effect of a connecting layer, and obtained the accurate buckling critical load through the Wittrick-Williams algorithm.…”

Section: Introductionmentioning

confidence: 99%

Calculation of Critical Load of Axially Functionally Graded and Variable Cross-Section Timoshenko Beams by Using Interpolating Matrix Method

Liu

Wang

et al. 2022

Mathematics

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In this paper, the interpolation matrix method (IMM) is proposed to solve the buckling critical load of axially functionally graded (FG) Timoshenko beams. Based on Timoshenko beam theory, a set of governing equations coupled by the deflection function and rotation function of the beam are obtained. Then, the deflection function and rotation function are decoupled and transformed into an eigenvalue problem of a variable coefficient fourth-order ordinary differential equation with unknown deflection function. According to the theory of interpolation matrix method, the eigenvalue problem of the variable coefficient fourth-order ordinary differential equation is transformed into an eigenvalue problem of a set of linear algebraic equations, and the critical buckling load and the corresponding deflection function of the axially functionally graded Timoshenko beam can be calculated by the orthogonal triangular (QR) decomposition method, which is the most effective and widely used method for finding all eigenvalues of a matrix. The numerical results are in good agreement with the existing results, which shows the effectiveness and accuracy of the method.

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“…Pradhan and Murmu [41] analyzed the thermomechanical vibration of a FG sandwich beam under various elastic foundations by the differential quadrature method. Teifouet et al [42] examined the buckling of axially functionally graded and nonuniform Timoshenko beams based on the nonlocal TBT. e material properties of 2D-FG beams are assumed to vary in the axial direction and the nanobeam is modelled as a nonuniform Timoshenko beam resting on a Winkler-Pasternak foundation.…”

Section: Introductionmentioning

confidence: 99%

Buckling Temperature and Natural Frequencies of Thick Porous Functionally Graded Beams Resting on Elastic Foundation in a Thermal Environment

Ibnorachid

Boutahar

Bikri

et al. 2019

Advances in Acoustics and Vibration

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In this paper, free vibrations of Porous Functionally Graded Beams (P-FGBs), resting on two-parameter elastic foundations, and exposed to three forms of thermal field, uniform, linear, and sinusoidal, are studied using a Refined Higher-order shear Deformation Theory. The present theory accounts for shear deformation by considering a constant transverse displacement and a higher-order variation of the axial displacement through the thickness of the beam. The stress-free boundary conditions are satisfied on the upper and lower surfaces of the beam without using any shear correction factor. The material properties are temperature-dependent and vary continuously through the depth direction of the beam, based on a modified power-law rule, in which two kinds of porosity distributions, uniform, and nonuniform, through the cross-section area of the beam, are considered. Hamilton’s principle is applied to obtain governing equations of motion, which are solved using a Navier-type analytical solution for simply supported P-FGB. Numerical examples are proposed and discussed in detail, to prove the effect of the thermal environment, the porosity distribution, and the influence of several parameters such as the power-law index, porosity volume fraction, slenderness ratio, and elastic foundation parameters on the critical buckling temperatures and the natural frequencies of the P-FGB.

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Buckling of nonuniform and axially functionally graded nonlocal Timoshenko nanobeams on Winkler-Pasternak foundation

Cited by 38 publications

References 36 publications

Post-critical buckling of truncated conical carbon nanotubes considering surface effects embedding in a nonlinear Winkler substrate using the Rayleigh-Ritz method

Post-critical buckling of truncated conical carbon nanotubes considering surface effects embedding in a nonlinear Winkler substrate using the Rayleigh-Ritz method

Calculation of Critical Load of Axially Functionally Graded and Variable Cross-Section Timoshenko Beams by Using Interpolating Matrix Method

Buckling Temperature and Natural Frequencies of Thick Porous Functionally Graded Beams Resting on Elastic Foundation in a Thermal Environment

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