Proper Boundary Conditions for Infinitely Layered Orthotropic Media

2000 ◽  
Vol 67 (3) ◽  
pp. 629-632
Author(s):  
E. L. Bonnaud ◽  
J. M. Neumeister
Keyword(s):  
Boundary Conditions ◽  
Layered Medium ◽  
Integral Transforms ◽  
Numerical Treatment ◽  
Problem Size ◽  
Transfer Matrix Approach ◽  
Stress Functions ◽  
Orthotropic Media ◽  
First Time ◽  
Conditions At Infinity

A stress analysis of a plane infinitely layered medium subjected to surface loadings is performed using Airy stress functions, integral transforms, and a revised transfer matrix approach. Proper boundary conditions at infinity are for the first time established, which reduces the problem size by one half. Methods and approximations are also presented to enable numerical treatment and to overcome difficulties inherent to such formulations. [S0021-8936(00)01103-X]

Exact Elasticity Solutions for Stresses and Deflections in Curved Beams and Rings of Exponential and T-Sections

1991 ◽  
Author(s):  
Cemil Bagci
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Plane Stress ◽  
Numerical Results ◽  
Neutral Axis ◽  
Curved Beams ◽  
Equilibrium Stress ◽  
Different Boundary Conditions ◽  
Stress Functions ◽  
Elasticity Solutions

Abstract Exact elasticity solutions for stresses and deflections (displacements) in curved beams and rings of varying thicknesses are developed using polar elasticity and state of plane stress. Basic forms of differential equations of equilibrium, stress functions, and differential equations of compatibility are given. They are solved to develop expressions for radial, tangential, and shearing stresses for moment, force, and combined loadings. Neutral axis location for each type of loading is determined. Expressions for displacements are developed utilizing strain-displacement relationships of polar elasticity satisfying boundary conditions on displacements. In case of full rings stresses are as in curved beams with properly defined moment loading, but displacements differ satisfying different boundary conditions. The developments for constant thicknesses are used to develop solutions for curved beams and rings with T-sections. Comparative numerical results are given.

Analytical Buckling Loads of Columns Weakened Simultaneously with Transverse Cracks and Partial Delamination

2019 ◽  
Vol 19 (03) ◽  
pp. 1950027 ◽  
Author(s):  
Igor Planinc ◽  
Simon Schnabl
Keyword(s):  
Mathematical Model ◽  
Analytical Solution ◽  
Boundary Conditions ◽  
Parametric Study ◽  
Transverse Crack ◽  
Transverse Cracks ◽  
Buckling Loads ◽  
Crack Location ◽  
Benchmark Solution ◽  
First Time

This paper focuses on development of a new mathematical model and its analytical solution for buckling analysis of elastic columns weakened simultaneously with transverse open cracks and partial longitudinal delamination. Consequently, the analytical solution for buckling loads is derived for the first time. The critical buckling loads are calculated using the proposed analytical model. A parametric study is performed to investigate the effects of transverse crack location and magnitude, length and degree of partial longitudinal delamination, and different boundary conditions on critical buckling loads of weakened columns. It is shown that the critical buckling loads of weakened columns can be greatly affected by all the analyzed parameters. Finally, the presented results can be used as a benchmark solution.

Natural Frequencies and Normal Modes of a Spinning Timoshenko Beam With General Boundary Conditions

1992 ◽  
Vol 59 (2S) ◽  
pp. S197-S204 ◽  
Author(s):  
Jean Wu-Zheng Zu ◽  
Ray P. S. Han
Keyword(s):  
Boundary Conditions ◽  
Mode Shape ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Normal Modes ◽  
Single Mode ◽  
Mode Shapes ◽  
Simulation Studies ◽  
Classical Boundary ◽  
First Time

A free flexural vibrations of a spinning, finite Timoshenko beam for the six classical boundary conditions are analytically solved and presented for the first time. Expressions for computing natural frequencies and mode shapes are given. Numerical simulation studies show that the simply-supported beam possesses very peculiar free vibration characteristics: There exist two sets of natural frequencies corresponding to each mode shape, and the forward and backward precession mode shapes of each set coincide identically. These phenomena are not observed in beams with the other five types of boundary conditions. In these cases, the forward and backward precessions are different, implying that each natural frequency corresponds to a single mode shape.

Free Vibration of Moderately Thick Conical Shells Using a Higher Order Shear Deformable Theory

2014 ◽  
Vol 136 (5) ◽  
Author(s):  
R. D. Firouz-Abadi ◽  
M. Rahmanian ◽  
M. Amabili
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Higher Order ◽  
Arbitrary Boundary ◽  
Conical Shells ◽  
First Time ◽  
Circumferential Wave ◽  
Shear Deformable

The present study considers the free vibration analysis of moderately thick conical shells based on the Novozhilov theory. The higher order governing equations of motion and the associate boundary conditions are obtained for the first time. Using the Frobenius method, exact base solutions are obtained in the form of power series via general recursive relations which can be applied for any arbitrary boundary conditions. The obtained results are compared with the literature and very good agreement (up to 4%) is achieved. A comprehensive parametric study is performed to provide an insight into the variation of the natural frequencies with respect to thickness, semivertex angle, circumferential wave numbers for clamped (C), and simply supported (SS) boundary conditions.

Analysis of Deep Beams

1951 ◽  
Vol 18 (2) ◽  
pp. 163-172
Author(s):  
H. D. Conway ◽  
L. Chow ◽  
G. W. Morgan
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Stress Distribution ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Normal Stress ◽  
Stress Function ◽  
Beam Theory ◽  
Deep Beam ◽  
Stress Functions

Abstract This paper presents a method of analyzing the stress distribution in a deep beam of finite length by superimposing two stress functions. The first stress function is chosen in the form of a trigonometric series which satisfies all but one of the boundary conditions—that of zero normal stress on the ends of the beam. The principle of least work is then used to obtain a second stress function giving the distribution of normal stress on the ends which is left by the first stress function. By superimposing the two solutions, all the boundary conditions are satisfied. Two particular cases of a given type of loading are solved in this way to investigate the stresses in a deep beam and their deviation from the ordinary beam theory. In addition, an approximate solution by the numerical method of finite difference is worked out for one of the two cases. Results from the two methods are compared and discussed. A method of obtaining an exact solution to the problem is given in an Appendix.

scholarly journals From Symmetry to Asymmetry: The Use of Additional Pulses to Improve Protection against Ultrashort Pulses Based on Modal Filtration

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1117 ◽  
Author(s):  
Anton O. Belousov ◽  
Evgeniya B. Chernikova ◽  
Mariya A. Samoylichenko ◽  
Artem V. Medvedev ◽  
Alexander V. Nosov ◽  
...  
Keyword(s):  
Boundary Conditions ◽  
Cross Section ◽  
Ultrashort Pulses ◽  
Reference Plane ◽  
Modal Decomposition ◽  
Protective Devices ◽  
Modal Filters ◽  
First Time ◽  
Filtration Technology ◽  
Side Coupling

For the first time, the paper considers in a unified work the possibility of the appearance of additional pulses in various structures based on modal filtration technology, which is used to improve protection against ultrashort pulses (USP). We analyzed meander lines (ML) with broad-side coupling, structures with modal reservation (MR), reflection symmetric MLs, and modal filters (MF) with a passive conductor in the reference plane cutout and obtained the following results. It was found that the main reason for the additional pulses to appear in these structures is the introduction of asymmetry (of the cross-section, boundary conditions, and excitation). It is theoretically and experimentally established that additional pulses are a new resource for increasing the efficiency of protective devices with modal decomposition, but the highest effectiveness could be achieved through careful optimization.

Computational Design and Analysis of Nitinol-Based Arch Wedge Support

2018 ◽  
Author(s):  
Tyler Stranburg ◽  
Yucheng Liu ◽  
Harish Chander ◽  
Adam Knight
Keyword(s):  
Boundary Conditions ◽  
Mechanical Response ◽  
Computational Study ◽  
Experimental Testing ◽  
Permanent Deformation ◽  
Computational Design ◽  
Element Analysis ◽  
Human Movements ◽  
First Time ◽  
Human Motions

A nitinol-based arch wedge support (AWS) was designed using computational approach. Finite element analysis (FEA) was performed to on this design to assess the influence of loading, boundary conditions, and thickness on the mechanical response of the computer-aid design (CAD) model. Five loading conditions caused by different human movements, two boundary conditions, and three thicknesses are involved in this computational study. FEA results showed that the presented AWS design can resist forces caused by different human motions without generating any permanent deformation. The study features the first time to design and evaluate a thin-walled nitinol AWS model. The results of this study form the background of prototyping and experimental testing of the design in the next phase.

Biharmonic functions and Brownian motion

1967 ◽  
Vol 4 (1) ◽  
pp. 130-136 ◽  
Author(s):  
L. L. Helms
Keyword(s):  
Brownian Motion ◽  
Boundary Conditions ◽  
Boundary Function ◽  
Normal Derivative ◽  
Dimensional Euclidean Space ◽  
Biharmonic Functions ◽  
Neumann Type ◽  
Bounded Open Subset ◽  
Image Position ◽  
First Time

Let R be a bounded open subset of N-dimensional Euclidean space EN,N ≧ 1, let {xt: t ≧ 0} be a separable Brownian motion starting at a point x ɛ R, and let τ = τR be the first time the motion hits the complement of R. It is known [1] that if g is a bounded measurable function on the boundary ∂R of R, then h(x) = Ex[g(xτ)] is a harmonic function of x ɛ R which “solves” the Dirichlet problem for the boundary function g; i.e., Δh = 0 on R, where Δ is the Laplacian. In elastic plate problems, one must solve the biharmonic equation subject to certain boundary conditions. For the more important applications, these boundary conditions involve the values of u and the normal derivative of u at points of ∂R. Even though a treatment of this Neumann type problem is not available at this time, some things can be said about biharmonic functions and their relationship to Brownian motion. We will show, in fact, that u(x)= Ex[τ(xτ)] is a biharmonic function on R which “satisfies” the boundary conditions (i) u=0 on ∂R and (ii) Δu= −2g on ∂R, provided g satisfies certain hypotheses. More generally, we will show that u(x)=Ex[Δkg(XΔ)] is polyharmonic of order k + 1 on R (i.e., Δk + 1u = Δ(Δku) = 0 on R) and that it satisfies certain boundary conditions. A treatment of the special case g ≡ 1 on ∂R can be found in [3].

Calculation of Elastic Induced Stress of Surrounding Rocks

2012 ◽  
Vol 170-173 ◽  
pp. 37-40
Author(s):  
Bo Qian
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Conditions ◽  
Analytical Solutions ◽  
Superposition Principle ◽  
Compatibility Conditions ◽  
Partial Differential ◽  
Induced Stress ◽  
Stress Functions ◽  
Round Section

In accordance with equilibrium differential equations and compatibility conditions of deformation, the partial differential equation of induced stress is achieved for elastic surrounding rocks of tunnels and chambers of round section. By method of the superposition principle, elastic analytical solutions of induced stress of surrounding rocks is derived from the partial differential equation, which is based on stress functions and boundary conditions.

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