scholarly journals Basic Solutions of Three Dimensional Elasticity

2021 ◽  
Vol 236 ◽  
pp. 05039
Author(s):  
Wx Zhang
Keyword(s):  
Boundary Conditions ◽  
Three Dimensional ◽  
Practical Calculation ◽  
Basic Solution ◽  
Final Solution ◽  
Theoretical Derivation ◽  
Basic Solutions ◽  
Special Solutions ◽  
Basic Equations ◽  
Three Dimensional Elasticity

Elastic calculation method is an important research content of computational mechanics. The problems of elasticity include basic equations and boundary conditions. Therefore, the final solution consists of the general solutions of the basic equations and the special solutions satisfying the boundary conditions. Numerical method is often used in practical calculation, but the analytical solution is also an important subject for researchers. In this paper, the basic solution of three-dimensional elastic materials is given by theoretical derivation.

scholarly journals Three-Dimensional Vibration Analysis of Rectangular Thick Plates on Pasternak Foundation with Arbitrary Boundary Conditions

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
Huimin Liu ◽  
Fanming Liu ◽  
Xin Jing ◽  
Zhenpeng Wang ◽  
Linlin Xia
Keyword(s):  
Boundary Conditions ◽  
Admissible Function ◽  
Three Dimensional ◽  
Future Research ◽  
Pasternak Foundation ◽  
Arbitrary Boundary ◽  
Thick Plates ◽  
Arbitrary Boundary Conditions ◽  
Ritz Procedure ◽  
Three Dimensional Elasticity

This paper presents the first known vibration characteristic of rectangular thick plates on Pasternak foundation with arbitrary boundary conditions on the basis of the three-dimensional elasticity theory. The arbitrary boundary conditions are obtained by laying out three types of linear springs on all edges. The modified Fourier series are chosen as the basis functions of the admissible function of the thick plates to eliminate all the relevant discontinuities of the displacements and their derivatives at the edges. The exact solution is obtained based on the Rayleigh–Ritz procedure by the energy functions of the thick plate. The excellent accuracy and reliability of current solutions are demonstrated by numerical examples and comparisons with the results available in the literature. In addition, the influence of the foundation coefficients as well as the boundary restraint parameters is also analyzed, which can serve as the benchmark data for the future research technique.

On a Variational Theorem in Elasticity and Its Application to Shell Theory

1964 ◽  
Vol 31 (4) ◽  
pp. 647-653 ◽  
Author(s):  
P. M. Naghdi
Keyword(s):  
Boundary Conditions ◽  
Shell Theory ◽  
Three Dimensional ◽  
Transverse Shear Deformation ◽  
Special Significance ◽  
Field Equations ◽  
Variational Theorem ◽  
Three Dimensional Elasticity ◽  
Strain Measures

After stating a variational theorem which is a further generalization of known variational theorems and which has as its Euler equations all of the field equations and the boundary conditions of classical linear three-dimensional elasticity, the remainder of the paper deals with its application to shell theory. A new characterization of the basic system of field equations and the boundary conditions of the linear theory of elastic shells is derived which includes the effect of transverse shear deformation and involves only symmetric resultants and symmetric shell-strain measures. These results are of special significance in relation to those of a number of recent investigations in shell theory under the Kirchhoff-Love hypothesis in which the boundary-value problem of shell theory is recast in terms of symmetric (but not necessarily the same) variables.

Three-dimensional elasticity solution of simply supported functionally graded rectangular plates with internal elastic line supports

2009 ◽  
Vol 44 (4) ◽  
pp. 249-261 ◽  
Author(s):  
Y P Xu ◽  
D Zhou
Keyword(s):  
Boundary Conditions ◽  
Young’S Modulus ◽  
Young's Modulus ◽  
Three Dimensional ◽  
Lower Surface ◽  
Functionally Graded ◽  
Rectangular Plates ◽  
Elastic Line ◽  
Simply Supported ◽  
Three Dimensional Elasticity

This paper studies the stress and displacement distributions of simply supported functionally graded rectangular plates with internal elastic line supports. The Young's modulus is graded through the thickness following the exponential law and the Poisson's ratio is kept constant. On the basis of three-dimensional elasticity theory, the solutions of displacements and stresses of the plate under static loads, which exactly satisfy the governing differential equations and the simply supported boundary conditions at four edges of the plate, are analytically derived. The reaction forces of the internal elastic line supports are regarded as the unknown external forces acting on the lower surface of the plate. The unknown coefficients in the solutions are then determined by the boundary conditions on the upper and lower surfaces of the plate. Convergence and comparison studies demonstrate the correctness and effectiveness of the proposed method. The effect of variations in Young's modulus on the displacements and stresses of rectangular plates and the effect of internal elastic line supports on the mechanical properties of plates are investigated.

Three-Dimensional Analysis of an Elastic Plate-Cylinder Intersection by the Boundary-Point-Least-Squares Technique

1977 ◽  
Vol 99 (1) ◽  
pp. 17-25 ◽  
Author(s):  
D. Redekop
Keyword(s):  
Boundary Conditions ◽  
Least Squares ◽  
Boundary Point ◽  
Shell Theory ◽  
Three Dimensional ◽  
Middle Part ◽  
Elasticity Equations ◽  
Three Dimensional Elasticity ◽  
Tension Loading ◽  
Theoretical Results

The boundary-point-least-squares technique is applied to the axisymmetric three-dimensional elasticity problem of a hollow circular cylinder normally intersecting with a perforated flat plate. The geometry of the intersection is partitioned into three parts. Boundary conditions on the middle part and continuity conditions between adjacent parts are satisfied using the numerical boundary-point-least-squares technique while the governing elasticity equations and all other boundary conditions are satisfied exactly. Sample theoretical results are presented for the case of axisymmetric radial tension loading on the plate. The results compare favorably with previously published experimental data and provide supplementary data to theoretical results obtained using existing shell theory solutions.

On the Form of Variationally Derived Shell Equations

1964 ◽  
Vol 31 (2) ◽  
pp. 233-238 ◽  
Author(s):  
Eric Reissner
Keyword(s):  
Variational Problem ◽  
Shell Theory ◽  
Three Dimensional ◽  
Torsion Problem ◽  
Considerable Influence ◽  
Equilibrium Equations ◽  
Circular Cylindrical Shells ◽  
Shell Equations ◽  
Basic Equations ◽  
Three Dimensional Elasticity

In the first part of the paper the basic equations of three-dimensional elasticity are formulated as a system of four equilibrium equations and seven stress displacement relations, together with a variational problem which has these eleven equations as Euler equations. In the second part of the paper, the new variational problem is used for a derivation of shell theory which accounts particularly simply for the differences between the resultants N12 and N21 and the couples M12 and M21. In the third part of the paper a solution is given of the torsion problem for circumferentially nonhomogeneous circular cylindrical shells, as an explicit demonstration of the fact that certain terms in the shell equations which are often of negligible influence sometimes are of considerable influence.

Three-Dimensional Analysis of the Free Vibration of Thick Rectangular Plates With Depressions, Grooves or Cut-Outs

1996 ◽  
Vol 118 (2) ◽  
pp. 184-189 ◽  
Author(s):  
P. G. Young ◽  
J. Yuan ◽  
S. M. Dickinson
Keyword(s):  
Kinetic Energy ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Dimensional Analysis ◽  
Thick Plate ◽  
Three Dimensional ◽  
Rectangular Plates ◽  
Algebraic Polynomials ◽  
Elasticity Equations ◽  
Three Dimensional Elasticity

A solution is presented for the free vibration of very thick rectangular plates with depressions, grooves or cut-outs using three-dimensional elasticity equations in Cartesian coordinates. Simple algebraic polynomials which satisfy the boundary conditions of the plate are used as trial functions in a Ritz approach. The plate is modelled as a parallelepiped, and the inclusions are treated quite straightforwardly by subtracting the contribution to the strain and kinetic energy expressions of the volume removed, before minimizing the functional. The approach is demonstrated by considering a number of square thick plate cases, including a plate with a cylindrical groove, a shallow depression or a cylindrical cut-out.

Beam Theory and Its Suitability for the Analysis of Pipes

1991 ◽  
Vol 113 (2) ◽  
pp. 291-296
Author(s):  
H. Fan ◽  
G. E. O. Widera ◽  
P. Afshari
Keyword(s):  
Asymptotic Expansion ◽  
Boundary Conditions ◽  
Cross Section ◽  
Three Dimensional ◽  
Beam Theory ◽  
Arbitrary Cross Section ◽  
Elasticity Equations ◽  
Expansion Technique ◽  
Three Dimensional Elasticity ◽  
Benchmark Solutions

The use of the asymptotic expansion technique when applied to the three-dimensional elasticity equations is outlined and used to demonstrate the development of an asymptotic beam theory and associated boundary conditions. The formulation thus obtained holds for arbitrary cross section shapes and is applied here to pipes. It can be used to provide benchmark solutions to test the suitability of engineering beam and shell theories.

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