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.

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.

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.

Stresses in a Helical Spring of Arbitrary Cross Section With Consideration of End Effects

1987 ◽  
Vol 109 (3) ◽  
pp. 289-301 ◽  
Author(s):  
K. Nagaya
Keyword(s):  
Cross Section ◽  
Three Dimensional ◽  
Beam Theory ◽  
Theory Of Elasticity ◽  
Arbitrary Cross Section ◽  
Coil Spring ◽  
End Effects ◽  
Free Boundary Conditions ◽  
Resultant Forces ◽  
Stress Free Boundary

This paper presents expressions for stresses in the spring due to compression or tension in the direction of the axis of the cylindrical coil. In deriving stresses in the spring, it is necessary to first obtain resultant forces and resultant moments which occur in the coil spring. Then the analysis first derives exact results based on three-dimensional curved beam theory in which all displacements and all forces in three orthogonal directions of the coil spring are included with consideration of the constraints of the ends of the spring. Since the boundary shape of the spring is irregular, it is also difficult to satisfy stress-free boundary conditions along the surface of the coil, so that this paper applies the Fourier expansion collocation method and obtains stresses in the coil spring based on the theory of elasticity.

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.

The Uniform Flexure of an Incomplete Tore

1949 ◽  
Vol 2 (4) ◽  
pp. 469
Author(s):  
W Freiberger ◽  
RCT Smith
Keyword(s):  
General Solution ◽  
Cross Section ◽  
Harmonic Functions ◽  
Rectangular Cross Section ◽  
Three Dimensional ◽  
Elasticity Problems ◽  
Three Dimensional Elasticity ◽  
Derivatives Of ◽  
Special Case

In this paper we discuss the flexure of an incomplete tore in the plane of its circular centre-line. We reduce the problem to the determination of two harmonic functions, subject to boundary conditions on the surface of the tore which involve the first two derivatives of the functions. We point out the relation of this solution to the general solution of three-dimensional elasticity problems. The special case of a narrow rectangular cross-section is solved exactly in Appendix II.

Modeling of a One-Sided Bonded and Rigid Constraint Using Beam Theory

2008 ◽  
Vol 75 (3) ◽  
Author(s):  
Peter J. Ryan ◽  
George G. Adams ◽  
Nicol E. McGruer
Keyword(s):  
Boundary Conditions ◽  
Microelectromechanical Systems ◽  
Three Dimensional ◽  
Beam Theory ◽  
Rotational Stiffness ◽  
Element Analysis ◽  
Transverse Loads ◽  
Rigid Constraint ◽  
Dimensional Finite Element ◽  
Three Dimensional Finite Element

In beam theory, constraints can be classified as fixed/pinned depending on whether the rotational stiffness of the support is much greater/less than the rotational stiffness of the freestanding portion. For intermediate values of the rotational stiffness of the support, the boundary conditions must account for the finite rotational stiffness of the constraint. In many applications, particularly in microelectromechanical systems and nanomechanics, the constraints exist only on one side of the beam. In such cases, it may appear at first that the same conditions on the constraint stiffness hold. However, it is the purpose of this paper to demonstrate that even if the beam is perfectly bonded on one side only to a completely rigid constraining surface, the proper model for the boundary conditions for the beam still needs to account for beam deformation in the bonded region. The use of a modified beam theory, which accounts for bending, shear, and extensional deformation in the bonded region, is required in order to model this behavior. Examples are given for cantilever, bridge, and guided structures subjected to either transverse loads or residual stresses. The results show significant differences from the ideal bond case. Comparisons made to a three-dimensional finite element analysis show a good agreement.

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 Bars under Torsional Loading: A Generalized Beam Theory Approach

2013 ◽  
Vol 2013 ◽  
pp. 1-39 ◽  
Author(s):  
Evangelos J. Sapountzakis
Keyword(s):  
Cross Section ◽  
Practical Interest ◽  
Beam Theory ◽  
Arbitrary Cross Section ◽  
Shear Stresses ◽  
Theory Approach ◽  
3D Fem ◽  
Mass Model ◽  
Dynamic Analyses ◽  
Load Vector

In this paper both the static and dynamic analyses of the geometrically linear or nonlinear, elastic or elastoplastic nonuniform torsion problems of bars of constant or variable arbitrary cross section are presented together with the corresponding research efforts and the conclusions drawn from examined cases with great practical interest. In the presented analyses, the bar is subjected to arbitrarily distributed or concentrated twisting and warping moments along its length, while its edges are supported by the most general torsional boundary conditions. For the dynamic problems, a distributed mass model system is employed taking into account the warping inertia. The analysis of the aforementioned problems is complete by presenting the evaluation of the torsion and warping constants of the bar, its displacement field, its stress resultants together with the torsional shear stresses and the warping normal and shear stresses at any internal point of the bar. Moreover, the construction of the stiffness matrix and the corresponding nodal load vector of a bar of arbitrary cross section taking into account warping effects are presented for the development of a beam element for static and dynamic analyses. Having in mind the disadvantages of the 3D FEM solutions, the importance of the presented beamlike analyses becomes more evident.

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