Buckling of non-ideal simply supported laminated plate on Pasternak foundation
01 Feb 2013-Applied Mathematics and Computation (APPLIED MATHEMATICS AND COMPUTATION)-Vol. 219, Iss: 12, pp 6420-6430
TL;DR: The Lindstedt-Poincare perturbation technique is used to study the effect of non-ideal boundary conditions on buckling load of laminated plates on elastic foundations and it is observed that by increasing the shear modulus of the foundation, the bucking load of the plate is increased.
About: This article is published in Applied Mathematics and Computation.The article was published on 2013-02-01. It has received 15 citations till now. The article focuses on the topics: Bending of plates & Buckling.
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TL;DR: In this paper, the effects of uniform in-plane loads on vibratory characteristics of symmetrically cross-ply laminated composite plates on elastic foundation and vertically in contact with fluid based on the first order shear deformation theory are investigated.
22 citations
TL;DR: In this article, the dual-axis buckling of laminated composite skew hyperbolic paraboloids with cutouts was investigated for various boundary conditions using the present mathematical model, and a C0 finite element coding in FORTRAN was developed to generate many new results for different boundary conditions, skew angles, lamination schemes, etc.
Abstract: The dual-axis buckling of Laminated composite skew hyperbolic paraboloid with cutouts subjected to the in-plane biaxial and the shear load is investigated for various boundary conditions using the present mathematical model. Variation of transverse shear stresses is represented by a second-order function across the thickness, and the cross-curvature effect is also included via strain relations. The transverse shear stress-free condition at the shell top and bottom surfaces is also satisfied. This mathematical model (having a realistic second-order distribution of transverse shear strains across the thickness of shell) requires unknown parameters only at the reference plane. For generality in the present analysis, nine-node curved isoparametric element is used. So far, no solution exists in the literature for dual-axis buckling problem of laminated composite skew hyperbolic paraboloids with cutouts. As no result is available for the present problem, the present model is compared with suitable published results and then it is extended to analyze biaxial and shear buckling of laminated composite skew hyperbolic paraboloids. A C0 finite element coding in FORTRAN is developed to generate many new results for different boundary conditions, skew angles, lamination schemes, etc.
18 citations
TL;DR: An improved Fourier series method (IFSM) was applied to study the free and forced vibration characteristics of the moderately thick laminated composite rectangular plates on the elastic Winkler or Pasternak foundations which have elastic uniform supports and multipoints supports.
Abstract: An improved Fourier series method (IFSM) is applied to study the free and forced vibration characteristics of the moderately thick laminated composite rectangular plates on the elastic Winkler or Pasternak foundations which have elastic uniform supports and multipoints supports. The formulation is based on the first-order shear deformation theory (FSDT) and combined with artificial virtual spring technology and the plate-foundation interaction by establishing the two-parameter foundation model. Under the framework of this paper, the displacement and rotation functions are expressed as a double Fourier cosine series and two supplementary functions which have no relations to boundary conditions. The Rayleigh-Ritz technique is applied to solve all the series expansion coefficients. The accuracy of the results obtained by the present method is validated by being compared with the results of literatures and Finite Element Method (FEM). In this paper, some results are obtained by analyzing the varying parameters, such as different boundary conditions, the number of layers and points, the spring stiffness parameters, and foundation parameters, which can provide a benchmark for the future research.
17 citations
Cites methods from "Buckling of non-ideal simply suppor..."
...[21] used the Lindstedt-Poincare perturbation technique to study the buckling of nonideal rectangular laminated plate on Pasternak foundation....
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TL;DR: Numerical results reveal that the dynamic instability of laminated composite plates subjected to arbitrary periodic loads is significantly affected by the modulus ratio, number of layer, static and dynamic load parameters.
17 citations
TL;DR: In this article, the buckling of rectangular orthotropic plates resting on a Pasternak elastic foundation under biaxial in-plane loading by the power series method (the method of Frobenius) was analyzed.
Abstract: In this study, buckling of rectangular orthotropic plates resting on a Pasternak elastic foundation under biaxial in-plane loading by the power series method (the method of Frobenius) was analyzed. Similar to many studies, two opposite edges of loading are simply supported and two other edges are assumed clamped. In order to extract the characteristic equations of orthotropic rectangular plate under in-plane loading resting on a Pasternak elastic foundation, the classical plate theory, by considering the interaction between plate and foundation, is used. The results showed that in the aspect ratio of less than 2, the existing Pasternak foundation caused the buckling load to increase severely, but by increasing the aspect ratio, the effect of the foundation is negligible. Applying the in-plane load in the y-direction caused the buckling load to decrease, but by increasing the aspect ratios the effect of the load is negligible.
11 citations
References
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Book•
01 Jan 1981
TL;DR: In this paper, the authors introduce the notion of forced Oscillations of the Duffing Equation and the Mathieu Equation for weakly nonlinear systems with quadratic and cubic nonlinearities.
Abstract: Algebraic Equations. Integrals. The Duffing Equation. The Linear Damped Oscillator. Self-Excited Oscillators. Systems with Quadratic and Cubic Nonlinearities. General Weakly Nonlinear Systems. Forced Oscillations of the Duffing Equation. Multifrequency Excitations. The Mathieu Equation. Boundary-Layer Problems. Linear Equations with Variable Coefficients. Differential Equations with a Large Parameter. Solvability Conditions. Appendices. Bibliography. Index.
3,020 citations
Additional excerpts
...Following the Lindstedt–Poincare technique, the displacement and the buckling load are expanded in perturbation series as [19]: Nxcr 1⁄4 Nx0 þ eNx1 þ Oðe2Þ XðxÞ 1⁄4 X0ðxÞ þ eX1ðxÞ þ Oðe2Þ : ð9Þ...
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Book•
01 Jan 1997
TL;DR: In this paper, the authors present a one-dimensional analysis of fiber-reinforced composite materials and their properties, including the properties of the components of a Lamina and their relationship with other components.
Abstract: Introduction and Mathematical Preliminaries Fiber-Reinforced Composite Materials. Vectors and Tensors. Matrices. Transformation of Vector and Tensor Components. Integral Relations. Equations of Anisotropic Elasticity Classification of Equations. Kinematics. Kinetics. Constitutive Equations. Equations of Thermoelasticity and Electroelasticity. Summary. Virtual Work Principles and Variational Methods Virtual Work. The Variational Operator and Functionals. Extrema of Functionals. Virtual Work Principles. Variational Methods. Summary. Introduction to Composite Materials Basic Concepts and Terminology. Constitutive Equations of a Lamina. Transformation of Stresses and Strains. Plane Stress Constitutive Relations. Classical and First-Order Theories of Laminated Composite Plates Introduction. An Overview of ESL Laminate Theories. The Classical Laminated Plate Theory. The First-Order Laminated Plate Theory. Stiffness Characteristics for Selected Laminates. One-Dimensional Analysis of Laminated Plates Introduction. Analysis of Laminated Beams Using CLPT. Analysis of Laminated Beams Using FSDT. Cylindrical Bending Using CLPT. Cylindrical Bending Using FSDT. Closing Remarks. Analysis of Specially Orthotropic Plates Using CLPT Introduction. Bending of Simply Supported Plates. Bending of Plates with Two Opposite Edges Simply Supported. Bending of Rectangular Plates with Various Boundary Conditions. Buckling of Simply Supported Plates Under Compressive Loads. Buckling of Rectangular Plates Under Inplane Shear Load. Vibration of Simply Supported Plates. Buckling and Vibration of Plates with Two Parallel Edges Simply Supported. Closure. Analytical Solutions of Rectangular Laminates Using CLPT Governing Equations in Terms of Displacements. Admissible Boundary Conditions for the Navier Solutions. Navier Solutions of Antisymmetric Cross-Ply Laminates. The Navier Solutions of Antisymmetric Angle-Ply Laminates. The LTvy Solutions. Analysis of Midplane Symmetric Laminates. Transient Analysis. Summary. Analytical Solutions of Rectangular Laminates Using FSDT Introduction. Simply Supported Antisymmetric Cross-Ply Laminates. Simply Supported Antisymmetric Angle-Ply Laminates. Antisymmetric Cross-Ply Laminates with Two Opposite Edges Simply Supported. Antisymmetric Angle-Ply Laminates with Two Opposite Edges Simply Supported. Transient Solutions. Summary. Finite Element Analysis of Composite Laminates Introduction. Laminated Beams and Plate Strips by CLPT. Timoshenko Beam/Plate Theory. Numerical Results for Beams and Plate Strips. Finite Element Models of Laminated Plates (CLPT). Finite Element Models of Laminated Plates (FSDT). Summary. Refined Theories of Laminated Composite Plates Introduction. A Third-Order Plate Theory. Higher-Order Laminate Stiffness Characteristics. The Navier Solutions. LTvy Solutions of Cross-Ply Laminates. Displacement Finite Element Model. Layerwise Theories and Variable Kinematic Models In troduction. Development of the Theory. Finite Element Model. Variable Kinematic Formulations. Nonlinear Analysis of Composite Laminates Introduction. Nonlinear Stiffness Coefficients. Solution Methods for Nonlinear Algebraic Equations. Computational Aspects and Numerical Examples. Closure. Index Most chapters include Exercises and References for Additional Reading
1,344 citations
"Buckling of non-ideal simply suppor..." refers background or methods in this paper
...The governing equation of a thin plate on a Pasternak foundation can be written as [2,14]:...
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...The exact solutions for buckling of plates are discussed in the monographs of Whitney [1] and Reddy [2]....
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...D1 @(4)w @x 4 þ 2D3 @(4)w @x2@y 2 þ D2 @(4)w @y 4 þ Kww Kgx @(2)w @x 2 Kgy @(2)w @y 2 þ q @ (2)w @t 2 þ N x @(2)w @x 2 þ N y @(2)w @y 2 1⁄4 0: ð1Þ Here x⁄ and y⁄ are the dimensional Cartesian coordinates measured along the adjacent edges of the plate, w⁄ is the deflection of the plate, N x and N y are the in-plane loads, t ⁄ is the time, q is the mass density per unit area, Kw is the vertical spring modulus of the foundation, Kgx and Kgy are the shear modules of the foundation and Di are the flexural rigidities which are defined as follows [2]: D11 1⁄4 D1 D22 1⁄4 D2 ðD12 þ 2D66Þ 1⁄4 D3 Dij 1⁄4 (1)3 XN...
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Book•
01 May 1987
TL;DR: In this article, a major basic text on the theory and structural applications of laminated anisotropic plates is presented, with detailed coverage of problems of bending under transverse load, stability, and free-vibrations.
Abstract: A major basic text on the theory and structural applications of laminated anisotropic plates. Detailed coverage of problems of bending under transverse load, stability, and free-vibrations, as well as laminated beams, expansional strain effects, curved plates, and free-edge effects.
999 citations
"Buckling of non-ideal simply suppor..." refers methods in this paper
...The exact solutions for buckling of plates are discussed in the monographs of Whitney [1] and Reddy [2]....
[...]
01 Jan 1954
445 citations
"Buckling of non-ideal simply suppor..." refers background in this paper
...Therefore, many studies are conducted to solve such cases [14,15]....
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TL;DR: In this article, the authors used the Mindlin plate theory to study buckling of in-plane loaded isotropic rectangular plates with different boundary conditions, and developed an analytical closed-form solution without any use of approximation for a combination of six different boundary condition; specifically, two opposite edges are simply supported and any of the other two edges can be simply supported, clamped or free.
106 citations