General Solutions of Anisotropic Laminated Plates

2003 ◽  
Vol 70 (4) ◽  
pp. 496-504 ◽  
Author(s):  
W.-L. Yin
Keyword(s):  
General Solution ◽  
Closed Form ◽  
Analytic Functions ◽  
Infinite Plate ◽  
Elliptical Hole ◽  
Complex Variables ◽  
Laminated Plates ◽  
Eigenvalues And Eigenvectors ◽  
Closed Form Solutions ◽  
Repeated Eigenvalues

Anisotropic laminates with bending-stretching coupling possess eigensolutions that are analytic functions of the complex variables x+μky, where the eigenvalues μk and the corresponding eigenvectors are determined in the present analysis, along with the higher-order eigenvectors associated with repeated eigenvalues of degenerate laminates. The analysis and the resulting expressions are greatly simplified by using a mixed formulation involving a new set of elasticity matrices A*, B*, and D*. There are 11 distinct types of laminates, each with a different expression of the general solution. For an infinite plate with an elliptical hole subjected to uniform in-plane forces and moments at infinity, closed-form solutions are obtained for all types of anisotropic laminates in terms of the eigenvalues and eigenvectors.

A Unified Formalism of Two-Dimensional Anisotropic Elasticity, Piezoelectricity and Unsymmetric Laminated Plates

2005 ◽  
Vol 72 (3) ◽  
pp. 422-431
Author(s):  
Wan-Lee Yin
Keyword(s):  
General Solution ◽  
Constitutive Relations ◽  
Anisotropic Elasticity ◽  
Complex Variables ◽  
Laminated Plates ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Eigenvalues And Eigenvectors ◽  
Infinite Domain ◽  
Anisotropic Plates

A unified formalism is presented for theoretical analysis of plane anisotropic elasticity and piezoelectricity, unsymmetric anisotropic plates, and other two-dimensional problems of continua with linear constitutive relations. Complex variables are used to reduce the governing differential equations to algebraic equations. The constitutive relation then yields an eigenrelation, which is easily solved explicitly for the material eigenvalues and eigenvectors. The latter have polynomial expressions in terms of the eigenvalues. When the eigenvectors are combined after multiplication by arbitrary analytic functions containing the corresponding eigenvalues, one obtains the two-dimensional general solution. Important results, including the orthogonality of the eigenvectors, the expressions of the pseudometrics and the intrinsic tensors, are established here for nondegenerate materials, including the case of all distinct eigenvalues. Green’s functions of the infinite domain, and of the semi-infinite domain with interior or edge singularities, are determined explicitly for the most general types of point loads and discontinuities (dislocations).

scholarly journals Bending of an incomplete annular plate and related problems

1979 ◽  
Vol 14 (3) ◽  
pp. 103-109 ◽  
Author(s):  
J R Barber
Keyword(s):  
Stress Concentration ◽  
Closed Form ◽  
Circular Plate ◽  
Annular Plate ◽  
Infinite Plate ◽  
Concentration Data ◽  
Closed Form Solutions ◽  
Concentrated Forces

Closed-form solutions and stress-concentration data are obtained for the problem of a sector of an annular plate subjected to moments and transverse forces on its radial edges. Closed-form solutions are also given for a semi-infinite plate or a circular plate subjected to a system of concentrated forces and/or moments at the edge.

Analysis of a Thin Elastic Ring Under Arbitrary Loading

1974 ◽  
Vol 96 (3) ◽  
pp. 870-876 ◽  
Author(s):  
J. Y. Liu ◽  
Y. P. Chiu
Keyword(s):  
General Solution ◽  
Closed Form ◽  
Elastic Ring ◽  
Closed Form Solutions ◽  
Distributed Load ◽  
Arbitrary Loading ◽  
Thin Ring ◽  
Special Cases ◽  
Loading System ◽  
Thin Elastic Ring

This paper presents a general solution for a thin ring under a self-equilibrating loading system comprising any combination of radial, tangential, and moment loads. The formulations are applicable to concentrated loads as well as to distributed load functions. Closed-form solutions are obtained for each case for engineering applications. Comparisons with recent published results for some special cases are demonstrated in some of the sample problems.

POSTBUCKLING OF COMPRESSIVELY LOADED IMPERFECT COMPOSITE PLATES: CLOSED-FORM APPROXIMATE SOLUTIONS

2010 ◽  
Vol 10 (04) ◽  
pp. 761-778 ◽  
Author(s):  
CHRISTIAN MITTELSTEDT ◽  
KAI-UWE SCHRÖDER
Keyword(s):  
Closed Form ◽  
Composite Plates ◽  
Approximate Solutions ◽  
Laminated Plates ◽  
Engineering Practice ◽  
Nonlinear Behaviour ◽  
State Variables ◽  
Shape Functions ◽  
Closed Form Solutions ◽  
Out Of Plane

In this paper, closed-form approximate solutions for the geometrically nonlinear behaviour of rectangular laminated plates with flexural orthotropy under longitudinal compression are presented. Based on the governing Marguerre-type differential equations postulated for imperfect plates, two plate configurations are discussed in detail, representing important application cases in practical engineering work. The first configuration is a laminated plate that is simply supported at all four edges (the so-called SSSS plate), while for the second configuration clamped unloaded longitudinal edges are considered (denoted as the SSCC plate). For both plate configurations, rather simple closed-form approximations in the form of trigonometric shape functions are employed for the description of the out-of-plane postbuckling plate deflections. Based on the chosen shape functions, the compatibility condition with respect to the in-plane strains is fulfilled exactly, while the out-of-plane equilibrium condition for a deflected plate element is not, but is solved using a Galerkin-type formulation instead. Eventually, very simple closed-form solutions for all postbuckling state variables (deflections, in-plane edge displacements, and effective widths) are derived that can be used very conveniently in engineering practice. The high accuracy of the presented analysis methods is established by comparison with the results of other authors.

Closed Form Solutions of Joint Water-Filling for Coordinated Transmission

2010 ◽  
Vol E93-B (12) ◽  
pp. 3461-3468 ◽  
Author(s):  
Bing LUO ◽  
Qimei CUI ◽  
Hui WANG ◽  
Xiaofeng TAO ◽  
Ping ZHANG
Keyword(s):  
Closed Form ◽  
Closed Form Solutions ◽  
Water Filling
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