A Generalized Reissner Theory for Large Axisymmetric Deflections of Circular Plates

2013 ◽  
Vol 81 (3) ◽  
Author(s):  
Raymond H. Plaut
Keyword(s):  
Shear Deformation ◽  
Shooting Method ◽  
Plate Thickness ◽  
Uniform Pressure ◽  
Circular Plates ◽  
Equilibrium Equations ◽  
Simply Supported ◽  
Reissner Theory ◽  
Large Rotations ◽  
Axisymmetric Problems

A generalized Reissner theory for axisymmetric problems of circular plates is presented. The plate is assumed to be linearly elastic, and large rotations and strains are allowed. Shear deformation and changes in the plate thickness are neglected. Equilibrium equations are formulated, and a shooting method is applied to obtain numerical results for plates subjected to a uniform pressure. The edge of the plate is assumed to be either simply supported or clamped, and is free to move radially. The resulting deflections are compared to those based on the von Kármán theory.

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

Bending stresses in thin circular plates with single eccentric circular holes

1976 ◽  
Vol 11 (4) ◽  
pp. 202-224 ◽  
Author(s):  
E Ollerton
Keyword(s):  
Circular Hole ◽  
Uniform Pressure ◽  
Plate Surface ◽  
Circular Plates ◽  
Concentrated Force ◽  
Simply Supported ◽  
Bending Stresses ◽  
Circular Holes

The bending stresses in thin circular plates having a single eccentric circular hole and small deflections are reported. The plates can have any mixture of clamped and simply supported boundaries, and can be subjected to a concentrated force uniformly distributed round the inner boundary, moments about two perpendicular axes, or uniform pressure on the plate surface. A previous paper (1)∗ has described the method of analysis using bipolar co-ordinates, and has given values for deflection and slope coefficients for varying diameter ratios and eccentricities under the loads described above. The present paper discusses the stresses found in the plates under the same conditions.

scholarly journals BEAM ELEMENT UNDER FINITE ROTATIONS

2021 ◽  
Vol 30 ◽  
pp. 87-92
Author(s):  
Emma La Malfa Ribolla ◽  
Milan Jirásek ◽  
Martin Horák
Keyword(s):  
Cross Sections ◽  
Shooting Method ◽  
Small Strain ◽  
Beam Model ◽  
Equilibrium Equations ◽  
Finite Rotations ◽  
Nonlinear Beam ◽  
Linear Elastic ◽  
Beam Elements ◽  
Large Rotations

The present work focuses on the 2-D formulation of a nonlinear beam model for slender structures that can exhibit large rotations of the cross sections while remaining in the small-strain regime. Bernoulli-Euler hypothesis that plane sections remain plane and perpendicular to the deformed beam centerline is combined with a linear elastic stress-strain law.The formulation is based on the integrated form of equilibrium equations and leads to a set of three first-order differential equations for the displacements and rotation, which are numerically integrated using a special version of the shooting method. The element has been implemented into an open-source finite element code to ease computations involving more complex structures. Numerical examples show a favorable comparison with standard beam elements formulated in the finite-strain framework and with analytical solutions.

Buckling of Circular Plates Based on Reddy Plate Theory

1997 ◽  
Author(s):  
C. M. Wang ◽  
K. H. Lee ◽  
J. N. Reddy
Keyword(s):  
Shear Deformation ◽  
Plate Theory ◽  
Three Dimensional ◽  
Edge Condition ◽  
Circular Plates ◽  
Simply Supported ◽  
Thin Plate Theory ◽  
Elastically Restrained ◽  
Three Dimensional Elasticity ◽  
Limiting Values

Treated herein is the elastic buckling of circular plates based on the Reddy plate theory. This plate theroy extends the Kirchhoff (or the classical thin) plate theory to allow for the effect of transverse shear deformation. Unlike the Mindlin’s shear deformation plate theory, there is no need for a shear correction factor in the Reddy plate theory. In this paper, exact buckling solutions are derived for circular plates whose edges are simply supported and elastically restrained against rotation as well. This general edge condition includes the classical simply supported and clamped edges at the limiting, values of the elastic rotational restraint constant. The buckling solutions are expressed in terms of the well-known Kichhoff buckling solutions. A comparison of buckling loads between the Mindlin, Reddy and three-dimensional elasticity plates is also given.

Thermal Postbuckling of Imperfect Circular Functionally Graded Material Plates: Examination of Voigt, Mori–Tanaka, and Self-Consistent Schemes

2014 ◽  
Vol 137 (2) ◽  
Author(s):  
Y. Kiani ◽  
M. R. Eslami
Keyword(s):  
Functionally Graded Material ◽  
Initial Imperfection ◽  
Functionally Graded ◽  
Circular Plates ◽  
Equilibrium Equations ◽  
Simply Supported ◽  
Self Consistent ◽  
Nonlinear Equilibrium ◽  
Graded Material ◽  
Thermal Postbuckling

Thermal postbuckling of solid circular plates made of a through-the-thickness functionally graded material (FGM) is analyzed in this paper. Initial imperfection of the plate is also taken into account. Each thermomechanical property of the plate is assumed to be a function of the temperature and thickness coordinate. Equivalent properties of the FGM media are obtained based on three different homogenization schemes, namely, Voigt rule, Mori–Tanaka scheme, and self-consistent estimate. Temperature profile is assumed to be through-the-thickness direction only. The solution of the heat conduction equation is obtained using an iterative central finite difference scheme. Various types of thermal loadings, such as uniform temperature rise, temperature specified at surfaces, and heat flux, are considered. Nonlinear equilibrium equations of the plate are obtained by means of the conventional Ritz method. Solution of the resulting nonlinear equilibrium equations and temperature distribution are obtained simultaneously at each step of heating. It is shown that response of a perfect clamped FGM plate is of the bifurcation type of buckling with stable postbuckling equilibrium branch, whereas imperfect clamped and perfect/imperfect simply supported FGM plates do not reveal the bifurcation type of instability through the nonuniform heating process. Furthermore, amplitude of initial imperfection is an important factor on the equilibrium path of FGM circular plates, especially for simply supported ones.

scholarly journals Free Vibration of Annular Circular Plates Based on Higher-Order Shear Deformation Theory: A Spline Approximation Technique

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Saira Javed ◽  
F. H. H. Al Mukahal
Keyword(s):  
Free Vibration ◽  
Shear Deformation ◽  
Deformation Theory ◽  
Spline Approximation ◽  
Shear Deformation Theory ◽  
Higher Order ◽  
Approximation Technique ◽  
Circular Plates ◽  
Equilibrium Equations ◽  
Material Parameters

This research is based on higher-order shear deformation theory to analyse the free vibration of composite annular circular plates using the spline approximation technique. Equilibrium equations are derived, and differential equations in terms of displacement and rotational functions are obtained. Cubic or quantic spline is used to approximate the displacement and rotational functions depending upon the order of these functions. A generalized eigenvalue problem is obtained and solved numerically for eigenfrequency parameter and associated eigenvector of spline coefficients. Frequency of annular circular plates with different numbers of layers with each layer consisting of different materials is analysed. The effect of geometric and material parameters on frequency value is investigated for simply supported condition. A comparative study with existing results narrates the validity of the present results. Graphs and tables depict the obtained results. Some figures and graphs are drawn by using Autodesk Maya and Matlab software.

scholarly journals 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

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.

Characterization of the vibration, stability and static responses of graphene-reinforced sandwich plates under mechanical and thermal loadings using the refined shear deformation plate theory

2021 ◽  
pp. 109963622199386
Author(s):  
Hessameddin Yaghoobi ◽  
Farid Taheri
Keyword(s):  
Shear Deformation ◽  
Volume Fraction ◽  
Plate Theory ◽  
Micromechanical Model ◽  
Sheet Thickness ◽  
Analytical Investigation ◽  
Sandwich Plates ◽  
Simply Supported ◽  
Vibration Stability ◽  
Shear Deformation Plate Theory

An analytical investigation was carried out to assess the free vibration, buckling and deformation responses of simply-supported sandwich plates. The plates constructed with graphene-reinforced polymer composite (GRPC) face sheets and are subjected to mechanical and thermal loadings while being simply-supported or resting on different types of elastic foundation. The temperature-dependent material properties of the face sheets are estimated by employing the modified Halpin-Tsai micromechanical model. The governing differential equations of the system are established based on the refined shear deformation plate theory and solved analytically using the Navier method. The validation of the formulation is carried out through comparisons of the calculated natural frequencies, thermal buckling capacities and maximum deflections of the sandwich plates with those evaluated by the available solutions in the literature. Numerical case studies are considered to examine the influences of the core to face sheet thickness ratio, temperature variation, Winkler- and Pasternak-types foundation, as well as the volume fraction of graphene on the response of the plates. It will be explicitly demonstrated that the vibration, stability and deflection responses of the sandwich plates become significantly affected by the aforementioned parameters.

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