Analysis of Plate Examples by Difference Methods and the Superposition Principle

1936 ◽  
Vol 3 (3) ◽  
pp. A81-A90
Author(s):  
D. L. Holl
Keyword(s):  
Boundary Conditions ◽  
Critical Stress ◽  
Plate Theory ◽  
Superposition Principle ◽  
Equivalent Stress ◽  
Lower Surface ◽  
The Other ◽  
Principle Of Superposition ◽  
Various Boundary Conditions ◽  
Actual Plate

Abstract In this paper the author applies the membrane analogs of H. Marcus to some elementary cases of thin homogeneous isotropic square plates having central-point loads and various boundary conditions. The analogy is made possible by two theorems: (a) The deflection of a membrane loaded with loads proportional to those on a given plate may be considered as the sum of the principal moments of the actual plate. (b) A second membrane may be loaded with elastic weights proportional to these moment sums and, subject to appropriate boundary conditions, the deflections of the latter membrane will be proportional to the deflections of the actual plate under the given loading system. The principle of superposition of deflection surfaces or equivalent stress systems is utilized in this paper both by difference and differential methods. The problems treated are (1) a square plate with pinned or simply supported edges, (2) two opposite edges pinned and the other two free, (3) two opposite edges pinned and the other two clamped, (4) all four edges clamped, (5) all four edges free with only corner post supports. The correct critical stress at the center of the lower surface of the plate was obtained from special thick-plate theory for a particular thickness-to-span ratio. The effect of this critical stress on the whole plate action is depicted for various boundary conditions.

Determination of the stress concentration in the corner point of the wedge-shaped region reinforced by a more rigid thin coating

2020 ◽  
Vol 26 (1) ◽  
pp. 80-89
Author(s):  
AN Soloviev ◽  
BV Sobol ◽  
EV Rashidova ◽  
AI Novikova
Keyword(s):  
Boundary Conditions ◽  
Corner Point ◽  
Theory Of Elasticity ◽  
The Other ◽  
Thin Coating ◽  
Various Boundary Conditions ◽  
Singular Behaviour ◽  
Isotropic Theory ◽  
Made In

We analysed the problem of determining the exponents in the asymptotic solution of the isotropic theory of elasticity problem at the top of the wedge-shaped region where its sides (or one of them) are supported by a thin coating and lean without friction on the rigid bases. On the other side of the wedge-shaped region, it is assumed that there are various boundary conditions, including when there is a thin coating. Mathematically, the problem reduces to the problem of determining the roots of transcendental characteristic equations arising from the condition for the existence of a nontrivial solution of a system of the linear homogeneous equations. The characteristics of the stress tensor components have been determined for the various combinations of boundary conditions and physical and geometric parameters. The qualitative conclusions are made. In particular, we have established the combinations of the values of these parameters at which the singular behaviour of stresses arises.

Free vibration analysis of rectangular sandwich plates with compressible core and various boundary conditions

2020 ◽  
pp. 109963622097927
Author(s):  
Sajjad Riahi Farsani ◽  
Arash Ramian ◽  
Ramazan-Ali Jafari-Talookolaei ◽  
Paolo S Valvo ◽  
Maryam Abedi
Keyword(s):  
Boundary Conditions ◽  
Plate Theory ◽  
Ritz Method ◽  
Shear Deformation Theory ◽  
Free Vibration Analysis ◽  
Thickness Ratio ◽  
Quadratic Functions ◽  
Sandwich Plates ◽  
Various Boundary Conditions ◽  
Rectangular Sandwich Plates

Extended higher-order sandwich plate theory is used to analyze the free vibrations of rectangular sandwich plates with compressible core. Accordingly, first-order shear deformation theory is used to model the laminated face sheets. Besides, the in-plane and transverse displacements of the core are assumed to be cubic and quadratic functions of the thickness coordinate, respectively. To deduce the governing equations, Hamilton’s principle is used. Then, based on the Rayleigh-Ritz method, single series expansions with two-variable orthogonal polynomials – namely, the orthogonal plate functions – are considered to approximate the displacement components. Lastly, a generalized eigenvalue problem is solved to obtain the free vibrational characteristics of sandwich plates with both symmetric and anti-symmetric lay-ups subjected to various boundary conditions. The method is validated against the results obtained by different methods in the literature. Finally, the effects of the plate side-to-thickness ratio, in-plane aspect ratio, and core-to-face sheets thickness ratio on the natural frequencies are discussed.

Natural Frequency and Buckling of Orthotropic Nanoplates Resting on Two-Parameter Elastic Foundations with Various Boundary Conditions

2014 ◽  
Vol 30 (5) ◽  
pp. 443-453 ◽  
Author(s):  
M. Sobhy
Keyword(s):  
Boundary Conditions ◽  
Natural Frequency ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Nonlocal Parameter ◽  
Nonlocal Elasticity Theory ◽  
Graphene Sheets ◽  
Elastic Foundations ◽  
Various Boundary Conditions ◽  
Mechanics Of Composite Materials

AbstractIn this article, the analyses of the natural frequency and buckling of orthotopic nanoplates, such as single-layered graphene sheets, resting on Pasternak's elastic foundations with various boundary conditions are presented. New functions for midplane displacements are suggested to satisfy the different boundary conditions. These functions are examined by comparing their results with the results obtained by using the functions suggested by Reddy (Reddy JN. Mechanics of Composite Materials and Structures: Theory and Analysis. Boca Raton, FL: CRC Press; 1997). Moreover, these functions are very simple comparing with Reddy's functions, leading to ease of calculations. The equations of motion of the nonlocal model are derived using the sinusoidal shear deformation plate theory (SPT) in conjunction with the nonlocal elasticity theory. The present SPT are compared with other plate theories. Explicit solution for buckling loads and vibration are obtained for single-layered graphene sheets with isotropic and orthotropic properties; and under biaxial loads. The formulation and the method of the solution are firstly validated by executing the comparison studies for the isotropic nanoplates with the results being in literature. Then, the influences of nonlocal parameter and the other parameters on the buckling and vibration frequencies are investigated.

scholarly journals ZEROS OF THE POTTS MODEL PARTITION FUNCTION IN THE LARGE-q LIMIT

2007 ◽  
Vol 21 (07) ◽  
pp. 979-994 ◽  
Author(s):  
SHU-CHIUAN CHANG ◽  
ROBERT SHROCK
Keyword(s):  
Partition Function ◽  
Boundary Conditions ◽  
Coordination Number ◽  
Potts Model ◽  
The Other ◽  
Square Lattice ◽  
Various Boundary Conditions ◽  
Temperature Variable

We calculate zeros of the q-state Potts model partition function Z(GΛ,q,v) for large q, where v is the temperature variable and GΛ is a section of a lattice Λ with coordination number κΛ and various boundary conditions. Lattice types studied include square, triangular, honeycomb, and kagomé. We show that for large q these zeros take on approximately circular patterns in the complex xΛ plane, where xΛ=v/q2/κΛ. This generalizes a known result for the square lattice to the other lattices considered.

Strongest Columns and Isoperimetric Inequalities for Eigenvalues

1962 ◽  
Vol 29 (1) ◽  
pp. 159-164 ◽  
Author(s):  
I. Tadjbakhsh ◽  
J. B. Keller
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Cross Sections ◽  
Isoperimetric Inequalities ◽  
Second Order ◽  
Buckling Load ◽  
The Other ◽  
Critical Buckling Load ◽  
Various Boundary Conditions

We consider the problem of determining what shape column has the largest critical buckling load of all columns of given length and volume. This problem was previously solved for a column hinged (pinned) at both ends. We solve it for columns clamped at one end and clamped, hinged, or free at the other end, assuming that all cross sections of the column are similar and similarly oriented. We also prove that the column previously obtained in the hinged-hinged case is actually strongest and not merely stationary. Graphs of the areas of the strongest columns as functions of distance along the columns are given for the various cases. The results are also expressed as isoperimetric inequalities for eigenvalues of second-order ordinary differential equations with various boundary conditions. Certain additional inequalities of this type are also obtained.

scholarly journals AN ANALYTICAL APPROACH ON THE BUCKLING ANALYSIS OF CIRCULAR, SOLID AND ANNULAR FUNCTIONALLY GRADED THIN PLATESGRADED THIN

1970 ◽  
Vol 41 (1) ◽  
pp. 7-14 ◽  
Author(s):  
H. Koohkan ◽  
A. Kimiaeifar ◽  
A. Mansourabadi ◽  
R. Vaghefi
Keyword(s):  
Boundary Conditions ◽  
Plate Theory ◽  
Buckling Analysis ◽  
Functionally Graded ◽  
Thin Plates ◽  
Radial Compression ◽  
Power Functions ◽  
Classical Plate Theory ◽  
Buckling Loads ◽  
Various Boundary Conditions

In this paper, the buckling analysis of circular, solid and annular functionally graded thin plates under uniform radial compression loads is studied. The material properties through the thickness are assumed to be power functions of the thickness. Moreover, the stability equations based on the classical plate theory (CPT), are derived by using the Hamilton’s principle. The obtained coupled-PDEs are difficult to be used for evaluation of the buckling loads of annular plates with various boundary conditions. To resolve this difficulty, a coordinate transformation from the middle plane to a new position is done and as consequence the equations are decoupled. By using the forgoing equations, the buckling loads are determined. The procedure is done for both circular and annular FGM plates of various boundary conditions under uniform radial loads on the edges and the results are validated with one of references.Key Words: Buckling analysis; solid plate; annular plate; functionally graded materials.DOI: 10.3329/jme.v41i1.5357Journal of Mechanical Engineering, Vol. ME 41, No. 1, June 2010 7-14 

Upon Impact Numerical Modeling of Foam Materials

2017 ◽  
Vol 54 (2) ◽  
pp. 195-202
Author(s):  
Vasile Nastasescu ◽  
Silvia Marzavan
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Modeling ◽  
Numerical Analysis ◽  
Numerical Methods ◽  
Boundary Conditions ◽  
General Character ◽  
Foam Material ◽  
Various Boundary Conditions ◽  
Specific Behaviour

The paper presents some theoretical and practical issues, particularly useful to users of numerical methods, especially finite element method for the behaviour modelling of the foam materials. Given the characteristics of specific behaviour of the foam materials, the requirement which has to be taken into consideration is the compression, inclusive impact with bodies more rigid then a foam material, when this is used alone or in combination with other materials in the form of composite laminated with various boundary conditions. The results and conclusions presented in this paper are the results of our investigations in the field and relates to the use of LS-Dyna program, but many observations, findings and conclusions, have a general character, valid for use of any numerical analysis by FEM programs.

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