Analytical solutions of refined plate theory for bending, buckling and vibration analyses of thick plates

Thai, Huu‐Tai; Choi, Dong Ho

doi:10.1016/j.apm.2013.03.038

Cited by 83 publications

(25 citation statements)

References 65 publications

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“…The analytical solution for natural frequencies of rectangular plates can only be obtained for Levy-type boundary conditions [1][2][3][4][5][6]. For general boundary conditions, however, the analytical solution cannot be obtained due to the complexities introduced by the satisfaction of free edges and free corner boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

A simple and accurate mixed Ritz-DQM formulation for free vibration of rectangular plates involving free corners

Eftekhari

2016

Ain Shams Engineering Journal

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It is well known that the classical global approximation methods such as the conventional Ritz method and the conventional differential quadrature method (DQM) have some difficulty in determining the natural frequencies of rectangular plates involving free corners. The mixed Ritz-DQM formulation, which has been recently developed by the present author, was also shown to have such difficulty. This is because it is very difficult to implement the free corner boundary condition in these methods. To overcome this difficulty, this paper presents a mixed Ritz-DQM formulation in which the free corner boundary condition is implemented in a simple and easy manner. First, we present a new scheme for implementing multiple boundary conditions in the DQM discretized equations. By the help of this scheme, we then show that the free corner boundary condition can also be easily implemented in the final matrix equations of the Ritz-DQM approach. Finally, we validate the effectiveness of the proposed approach through numerical experiments. Numerical results show the advantages of the proposed method over some versions of the DQM in terms of accuracy and performance. Ó 2015 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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Section: Introductionmentioning

confidence: 99%

A simple and accurate mixed Ritz-DQM formulation for free vibration of rectangular plates involving free corners

Eftekhari

2016

Ain Shams Engineering Journal

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“…(23) where m is the number of different plate elements of the cross section (e.g., for RHS and U profile, m = 2). The critical force of local buckling corresponds to the minimum values of stress defined for each individual element of the cross section, according to the equation (23). Local stability of the U-shaped cross section is characterized by the curve of the local buckling coefficient k = k (b 2 /b 1 ), which is shown in Fig.…”

Section: Discussion and Verification Of Resultsmentioning

confidence: 99%

Design of Columns in Terms of Stability

Đelošević

Vukajlov

Tepic

et al. 2018

TFAMENA

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SummaryIn this paper, we investigate global and local stability of columns with an open and a closed thin-walled cross section. The proposed model of global stability is made of two deformable elements connected with an elastic joint. Stiffness of the elastic joint represents local discontinuity of the cross section of the column caused by the loss of stability of individual plates. The model of local stability of the columns is conceptualized on the principle of continuity of individual plates of the cross section. The coefficients of local buckling are defined as geometric parameters of the column with a rectangular hollow section (RHS) and a U-shaped cross section. The dominant parameters that influence the interactive behaviour of local and global buckling are the slenderness of plates and the column as a whole. The basic function of the developed models is to identify the stability mechanism in terms of better estimation of the critical force and higher load capacity.

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“…Similarly to the theory of laminated and composite beams [83][84][85][86][87][88][89][90] the literature is very rich in the different theories to model laminated composite plates and shells. Thin flat plates can be modeled by the classical or Kirchhoff plate theory, 91,92 while for relatively thick plates the first-order shear deformation theory (FSDT or Mindlin), [93][94][95][96][97][98][99][100] second-order shear deformation theory (SSDT), [101][102][103][104][105] general third-order theory (TSDT), [106][107][108] Reddy third-order theory, [109][110][111][112] other higher-order shear deformation theories (HSDT), [113][114][115] layerwise theories, [116][117][118][119][120][121] and the 3D elasticity solutions 122,123 are developed. These theories have been utilized to model curved shells [124][125][126] and delaminated and cracked laminates …”

Section: Department Of Applied Mechanics Budapest University Of Techmentioning

confidence: 99%

Natural vibration-induced parametric excitation in delaminated Kirchhoff plates

Szekrényes

2015

Journal of Composite Materials

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This paper revisits the problem of free vibration of delaminated composite plates with Lévy type boundary conditions. The governing equations are derived for laminated Kirchhoff plates including through-width delamination. The plate is divided into two subplates in the plane of the delamination. The kinematic continuity of the undelaminated part is established by using the system of exact kinematic conditions. The free vibration analysis of orthotropic simply supported Lévy plates reveals that the delaminated parts are subjected to periodic normal and in-plane shear forces. This effect induces parametric excitation leading to the susceptibility of the plates to dynamic delamination buckling during the vibration. An important aspect is that depending on the vibration mode the internal forces have a two-dimensional distribution in the plane of the delamination. To solve the dynamic stability problem the finite element matrices of the delaminated parts are developed. The distribution of the internal forces in the direction of the delamination front was considered. The mode shapes including a half-wave along the width of the plate accompanied by delamination buckling are shown based on the subsequent superimposition of the buckling eigenshapes. The analysis reveals that the vibration phenomenon is amplitude dependent. Also, the phase plane portraits are created for some chosen cases showing some special trajectories. KeywordsDelamination, free vibration, laminated plate theory, dynamic stability, parametric excitation, harmonic balance, finite element method IntroductionDelaminations are typical defects in composite and sandwich structures induced by indentations (e.g. local loading by spherical indentors), 1,2 low-velocity impact and free edge effects (normal and shear stress concentration between layers at free edges), 3-6 finally by fabrication defects. 7,8 The cracks and delaminations reduce significantly the strength and stiffness of composite laminates.9-11 The material defects also influence the dynamic behavior of the structures.12-20 One of the widely known dynamic phenomena is parametric excitation, which takes place in machine tools, 21 [73][74][75][76][77][78] and sandwich beams. 79In Burlayenko and Sadowski, 80 the foam/core debonding in sandwich plates was investigated during free vibration. However, the existence of the parametric excitation was not revealed in any of the former papers.To the best of the author's knowledge, the existence of the parametric excitation in the course of free vibration of delaminated composite beams was first revealed in Szekre´nyes 81 and was solved effectively by the FE and harmonic balance methods in Szekre´nyes.82 It was also shown that the delamination buckling takes place only if the free vibration amplitude exceeds the so-called critical amplitude at a specified point of the beam. The existence of the critical amplitude was also proved experimentally. The parametric excitation in delaminated plates is possible to appear, as well. 9,10,[127][128][129] subseq...

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Analytical solutions of refined plate theory for bending, buckling and vibration analyses of thick plates

Cited by 83 publications

References 65 publications

A simple and accurate mixed Ritz-DQM formulation for free vibration of rectangular plates involving free corners

A simple and accurate mixed Ritz-DQM formulation for free vibration of rectangular plates involving free corners

Design of Columns in Terms of Stability

Natural vibration-induced parametric excitation in delaminated Kirchhoff plates

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