Nonlinear Vibration of Plates and Higher Order Theory for Different Boundary Conditions

2010 ◽  
Author(s):  
Marco Amabili ◽  
Kostas Karagiozis ◽  
Sirwan Farhadi ◽  
Korosh Khorshidi
Keyword(s):  
Boundary Conditions ◽  
Shear Deformation ◽  
Deformation Theory ◽  
Laminate Composite ◽  
Plate Thickness ◽  
Shear Deformation Theory ◽  
Higher Order ◽  
Rectangular Plates ◽  
Different Boundary Conditions ◽  
Simply Supported

There are numerous applications of plate structures found in structural, aerospace and marine engineering. The present study extends the previous work by Amabili and Sirwan [1] investigating the performance of isotropic and laminate composite rectangular plates with different boundary conditions subjected to an external point force with an excitation frequency that lies in the neighbourhood of the fundamental mode of the plate. The analysis is performed using three different nonlinear plate theories, namely: i) the classical Von Ka´rman theory, ii) first-order shear deformation theory, and iii) third-order shear deformation theory. Three different boundary conditions are considered in the investigation: a) classical clamped boundary conditions, b) simply-supported ends with immovable edges, and c) simply-supported ends with movable boundaries. In addition, the effect of thickness was also considered in the analysis and different values for the plate thickness were assumed. The results investigate the accuracy of lower order theories versus higher order shear deformation theories, the effect of boundary conditions and highlight the differences in the responses obtained from isotropic and laminate composite rectangular plates.

Using phase field and third-order shear deformation theory to study the effect of cracks on free vibration of rectangular plates with varying thickness

2020 ◽  
Vol 71 (7) ◽  
pp. 853-867
Author(s):  
Phuc Pham Minh
Keyword(s):  
Free Vibration ◽  
Shear Deformation ◽  
Phase Field ◽  
Deformation Theory ◽  
Plate Thickness ◽  
Shear Deformation Theory ◽  
Rectangular Plates ◽  
Equilibrium Equations ◽  
Third Order ◽  
Free Vibration Frequency

The paper researches the free vibration of a rectangular plate with one or more cracks. The plate thickness varies along the x-axis with linear rules. Using Shi's third-order shear deformation theory and phase field theory to set up the equilibrium equations, which are solved by finite element methods. The frequency of free vibration plates is calculated and compared with the published articles, the agreement between the results is good. Then, the paper will examine the free vibration frequency of plate depending on the change of the plate thickness ratio, the length of cracks, the number of cracks, the location of cracks and different boundary conditions

A NEW TRIANGULAR ELEMENT BASED ON HIGHER ORDER SHEAR DEFORMATION THEORY FOR FLEXURAL VIBRATION OF COMPOSITE PLATES

2002 ◽  
Vol 02 (02) ◽  
pp. 163-184 ◽  
Author(s):  
A. CHAKRABARTI ◽  
A. H. SHEIKH
Keyword(s):  
Shear Deformation ◽  
Vibration Analysis ◽  
Deformation Theory ◽  
Plate Theory ◽  
Plate Thickness ◽  
Shear Deformation Theory ◽  
Free Vibration Analysis ◽  
Composite Plates ◽  
Higher Order ◽  
Triangular Element

A triangular element based on Reddy's higher order shear deformation theory is developed for free vibration analysis of composite plates. In the Reddy's plate theory, the transverse shear stress varies in a parabolic manner across the plate thickness and vanishes at the top and bottom surfaces of the plate. Moreover, it does not involve any additional unknowns. Thus the plate theory is quite simple and elegant. Unfortunately, such an attractive plate theory cannot be exploited as expected in finite element analysis, primarily due to the difficulties in satisfying the inter-element continuity requirement. This has inspired us to develop the present element, which has three corner nodes and three mid-side nodes with the same number of degrees of freedom. To demonstrate the performance of the element, numerical examples of isotropic and composite plates under different situations are solved. The results are compared with the analytical solutions and other published results, which show the accuracy and range of applicability of the proposed element in the problem of vibration analysis.

A new simple shear deformation theory for free vibration analysis of isotropic and FG plates under different boundary conditions

2015 ◽  
Vol 11 (3) ◽  
pp. 437-470 ◽  
Author(s):  
Amale Mahi ◽  
El Abbas Adda Bedia ◽  
Abdelouahed Tounsi ◽  
Amina Benkhedda
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Shear Deformation ◽  
Temperature Rise ◽  
Deformation Theory ◽  
Shear Deformation Theory ◽  
Content Type ◽  
Different Boundary Conditions ◽  
Fg Plates ◽  
Shear Deformation Theories

Purpose – A new simple parametric shear deformation theory applicable to isotropic and functionally graded plates is developed. This new theory has five degrees of freedom, provides parabolic transverse shear strains across the thickness direction and hence, it does not need shear correction factor. Moreover, zero-traction boundary conditions on the top and bottom surfaces of the plate are satisfied rigorously. The paper aims to discuss these issues. Design/methodology/approach – Material properties are temperature-dependent and vary continuously through the thickness according to a power law distribution. The plate is assumed to be initially stressed by a temperature rise through the thickness. The energy functional of the system is obtained using Hamilton’s principle. Free vibration frequencies are then calculated using a set of characteristic orthogonal polynomials and by applying Ritz method for different boundary conditions. Findings – In the light of good performance of the present theory for all boundary conditions considered, it can be considered as an excellent alternative to some two-dimensional (2D) theories for approximating the tedious and time consuming three-dimensional plate problems. Originality/value – To the best of the authors’ knowledge and according to literature survey, almost all published higher order shear deformation theories have been limited to simply supported boundary conditions and without taking into account the thermal stresses effects. The existing 2D shear deformation theories of Reddy, Karama and Touratier can be easily recovered. Furthermore, this feature can be highly appreciated in an iterative design process where a large number of derived plate models can be tested by selecting only two parameters in a simple polynomial function which is computationally efficient. Finally, new results are presented to show the effect of material variation, and temperature rise on natural frequencies of the FG plate for different boundary conditions.

On the Buckling of Isotropic, Transversely Isotropic and Laminated Composite Rectangular Plates

2014 ◽  
Vol 14 (07) ◽  
pp. 1450020 ◽  
Author(s):  
Atteshamuddin S. Sayyad ◽  
Yuwaraj M. Ghugal
Keyword(s):  
Boundary Conditions ◽  
Elasticity Theory ◽  
Shear Deformation ◽  
Deformation Theory ◽  
Shear Deformation Theory ◽  
Laminated Composite ◽  
Transversely Isotropic ◽  
Buckling Analysis ◽  
Transverse Shear Strain ◽  
Rectangular Plates

This paper presents the uniaxial and biaxial buckling analysis of rectangular plates based on new trigonometric shear and normal deformation theory. The theory accounts for the cosine distribution of the transverse shear strain through the plate thickness and on the free boundary conditions on the plate surfaces without using the shear correction factor. Governing equations and boundary conditions of the theory are derived by the principle of virtual work. The Navier type solutions for the buckling analysis of simply supported isotropic, transversely isotropic, orthotropic and symmetric cross-ply laminated composite rectangular plates subjected to uniaxial and biaxial compressions are presented. The effects of variations in the degree of orthotropy of the individual layers, side-to-thickness ratio and aspect ratio of the plate are examined on the buckling characteristics of composite plates. The present results are compared with those of the classical plate theory (CPT), first order shear deformation theory (FSDT) and exact three-dimensional (3D) elasticity theory wherever applicable. Good agreement is achieved of the present results with those of higher order shear deformation theory (HSDT) and elasticity theory.

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