On the Elastic Stability of a Rectangular Plate, when Subjected to a Variable Edge Thrust

1935 ◽  
Vol 31 (3) ◽  
pp. 368-381 ◽  
Author(s):  
D. M. A. Leggett
Keyword(s):  
Boundary Conditions ◽  
Approximate Method ◽  
Rectangular Plate ◽  
General Problem ◽  
Elastic Stability ◽  
Detailed Consideration ◽  
Various Boundary Conditions ◽  
The Stability

The stability of a rectangular plate, subjected to constant thrust over opposite pairs of edges, has been treated with some degree of completeness for various boundary conditions. The more general problem, in which the thrusts are no longer constant, has not yet received any treatment apart from the approximate method developed by E. Schwerin†, which would appear to be capable of only limited extension. The object of this paper is accordingly the detailed consideration of a simple case when the thrust is no longer constant.

scholarly journals The elastic stability of a long and slightly bent rectangular plate under uniform shear

1937 ◽  
Vol 162 (908) ◽  
pp. 62-83 ◽  
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Rectangular Plate ◽  
Elastic Stability ◽  
Stability Problems ◽  
Uniform Shear ◽  
Lines Of Curvature ◽  
The Stability

1—The problem of the elastic stability of a plane rectangular plate when subjected to uniform shear has been approximately solved for various conditions (Cox 1933; Timoshenko 1936). In the case of an indefinitely long strip an exact solution has been found (Southwell and Skan 1924), but it appears that no attempt has been made to investigate what happens if the plate is no longer plane. It is accordingly the object of this paper to consider the stability of a long strip, slightly curved, when its two side edges are subjected to uniform shear. 2—In what follows we assume that the thickness and curvature of the plate are constant, and that the edges of the plate are two generators and two lines of curvature. It is, moreover, further assumed that the plate is thin as in all similar stability problems, and that it is of such length that the boundary conditions over the two curved ends can be ignored.

scholarly journals V. On the general theory of elastic stability

1914 ◽  
Vol 213 (497-508) ◽  
pp. 187-244 ◽  
Keyword(s):  
General Theory ◽  
Boundary Conditions ◽  
Radial Pressure ◽  
Elastic Stability ◽  
Circular Ring ◽  
Theory Of Elasticity ◽  
Comprehensive Theory ◽  
The Subject ◽  
The Stability ◽  
Straight Rod

Problems which deal with the stability of bodies in equilibrium under stress are so distinct from the ordinary applications of the theory of elasticity that it is legitimate to regard them as forming a special branch of the subject. In every other case we are concerned with the integration of certain differential equations, fundamentally the same for all problems, and the satisfaction of certain boundary conditions; and by a theorem due to Kiechiioff we are entitled to assume that any solution which we may discover is unique. In these problems we are confronted with the possibility of two or more configurations of equilibrium , and we have to determine the conditions which must be satisfied in order that the equilibrium of any given configuration may be stable. The development of both branches has proceeded upon similar lines. That is to say, the earliest discussions were concerned with the solution of isolated examples rather than with the formulation of general ideas. In the case of elastic stability, a comprehensive theory was not propounded until the problem of the straight strut had been investigated by Euler, that of the circular ring under radial pressure by M. Lévy and G. H. Halphen, and A. G. Greenhill had discussed the stability of a straight rod in equilibrium under its own weight, under twisting couples, and when rotating.

Buckling of a Thin Annular Plate Under Uniform Compression

1958 ◽  
Vol 25 (2) ◽  
pp. 267-273
Author(s):  
N. Yamaki
Keyword(s):  
Boundary Conditions ◽  
Circular Plate ◽  
Critical Loads ◽  
Equilibrium Equation ◽  
Annular Plate ◽  
Elastic Stability ◽  
Infinite Strip ◽  
Uniform Compression ◽  
The Stability

Abstract This paper deals with the elastic stability of a circular annular plate under uniform compressive forces applied at its edges. By integrating the equilibrium equation of the buckled plate, the problem is solved in its most general form for twelve different combinations of the boundary conditions of the edges. For each case cited the lowest critical loads are calculated with the ratio of its radii as the parameter. It is clarified that the assumption of symmetrical buckling, which has been made by several researchers, often leads to the overestimate for the stability of the plate. Discussions for the limiting cases of the circular plate and infinite strip also are included.

The Stability of Plane Poiseuille Flow in a Finite-Length Channel

2021 ◽  
Author(s):  
Lei Xu ◽  
Zvi Rusak
Keyword(s):  
Kinetic Energy ◽  
Boundary Conditions ◽  
Linear Stability ◽  
Finite Length ◽  
Poiseuille Flow ◽  
Weak Form ◽  
Plane Poiseuille Flow ◽  
Various Boundary Conditions ◽  
New Perspective ◽  
The Stability

Abstract The linear stability of plane Poiseuille flow through a finite-length channel is studied. A weakly-divergence-free basis finite element method with SUPG stabilization is used to formulate the weak form of the problem. The linear stability characteristics are studied under three possible inlet-outlet boundary conditions and the corresponding perturbation kinetic energy transfer mechanisms are investigated. Active transfer of perturbation kinetic energy at the channel inlet and outlet, energy production due to convection and dissipation at the flow bulk provide a new perspective in understanding the distinct stability characteristics of plane Poiseuille flow under various boundary conditions.

Influence of Boundary Conditions on Decay Rates in a Prestrained Plate1

2002 ◽  
Vol 69 (4) ◽  
pp. 515-520 ◽  
Author(s):  
B. Karp ◽  
D. Durban
Keyword(s):  
Boundary Conditions ◽  
Rectangular Plate ◽  
Boundary Data ◽  
Separation Of Variables ◽  
Finite Elasticity ◽  
Decay Rates ◽  
Constitutive Behavior ◽  
Hyperelastic Materials ◽  
Various Boundary Conditions ◽  
Decay Exponent

Decay of end perturbations imposed on a prestrained semi-infinite rectangular plate is investigated in the context of plane-strain incremental finite elasticity. A separation of variables eigenfunction formulation is used for the perturbed field within the plate. Numerical results for the leading decay exponent are given for three hyperelastic materials with various boundary conditions at the long faces of the plate. The study exposes a considerable sensitivity of axial decay rates to boundary data, to initial strain and to constitutive behavior. It is suggested that the results are relevant to the applicability of Saint-Venant’s principle even though the eigenfunctions are not always self-equilibrating.

Vibration and Frequency Analysis of Non-Uniform Timoshenko Beams Subjected to Axial Forces

2003 ◽  
Author(s):  
A. R. Ohadi ◽  
H. Mehdigholi ◽  
E. Esmailzadeh
Keyword(s):  
Stability Analysis ◽  
Boundary Conditions ◽  
Axial Force ◽  
Frequency Equation ◽  
Axial Loads ◽  
Various Boundary Conditions ◽  
Axial Forces ◽  
First Case ◽  
The Stability ◽  
Elastically Supported

Dynamic and stability analysis of non-uniform Timoshenko beam under axial loads is carried out. In the first case of study, the axial force is assumed to be perpendicular to the shear force, while for the second case the axial force is tangent to the axis of the beam column. For each case, a pair of differential equations coupled in terms of the flexural displacement and the angle of rotation due to bending was obtained. The parameters of the frequency equation were determined for various boundary conditions. Several illustrative examples of uniform and non-uniform beams with different boundary conditions such as clamped supported, elastically supported, and free end mass have been presented. The stability analysis, for the variation of the natural frequencies of the uniform and non-uniform beams with the axial force, has also been investigated.

scholarly journals The elastic stability of a thin twisted strip—II

1937 ◽  
Vol 161 (905) ◽  
pp. 197-220 ◽  
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Main Part ◽  
Fourier Expansion ◽  
Elastic Stability ◽  
Numerical Work ◽  
The Stability ◽  
Good Agreement

I—In a previous paper the present writer discussed both theoretically and experimentally the equilibrium and elastic stability of a thin twisted strip, and the results obtained by the theory were found to be in good agreement with observation. It has, however, been pointed out by Professor Southwell, F. R. S., that the solution of the stability equations which was given in that paper may only be regarded as an approximate solution for, although it satisfies exactly the differential equations and two boundary conditions along the edge of the strip, it only satisfies the two remaining boundary conditions approximately. The author has also noticed that the coefficients n a m in the Fourier expansion of θ 2 cos mθ which were used in A are incorrect when m = 0, and this has led to errors in the numerical work so that the values of ᴛb 2 / π 2 h which are given in Table I of A are wrong. In the present paper a solution of the stability equations is obtained which satisfies all the boundary conditions. This solution is very much more complicated than the approximate solution and much greater labour is required for the numerical work. The numerical work for the approximate solution of A has also been revised and the corrected results are given in 9, 10. It is found that the results for the approximate solution are in good agreement with those obtained from the exact solution and that both agree moderately well with the experimental results which are given in A. The main part of this paper is an extension of the previous work and is concerned with the stability of a thin twisted strip when it is subjected to a tension along its length. The theory has been compared with experiment and satisfactorily good agreement between them was found.

Global Static Instability of Risers

1984 ◽  
Vol 28 (04) ◽  
pp. 261-271
Author(s):  
Michael M. Bernitsas ◽  
Theodore Kokkinis
Keyword(s):  
Boundary Conditions ◽  
Internal Pressure ◽  
Stability Boundary ◽  
Critical Length ◽  
Bending Rigidity ◽  
Buckling Mode ◽  
Stability Boundaries ◽  
Various Boundary Conditions ◽  
Static Instability ◽  
The Stability

Global instability of risers depends on riser weight, internal and external fluid static pressure forces, tension exerted at the top of the riser, and boundary conditions. The purpose of this work is to study the effects of these factors on the stability boundaries of risers and specifically.(i) compare buckling loads for various boundary conditions; (ii) find the long-riser instability behavior from the asymptotics of the stability boundaries; (iii) find the short-riser instability behavior; (iv) analyze the relative effects of boundary conditions, weight, internal pressure, and bending rigidity on stability; (v) show the variation of the stability boundary shape with the order of the buckling mode; and (vi) compare the critical length at which risers in tension over their entire length may buckle due to internal pressure, for various boundary conditions.

Torsional-Flexural Critical Force for Various Boundary Conditions

2016 ◽  
Vol 710 ◽  
pp. 303-308 ◽  
Author(s):  
Ivan Balaz ◽  
Michal Kovac ◽  
Tomáš Živner ◽  
Yvona Kolekova
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Approximate Method ◽  
Cross Sections ◽  
Critical Force ◽  
Various Boundary Conditions ◽  
Element Method

The system of governing differential equations of stability of members with the rigid open cross-sections was developed by Vlasov [1] in 1940. Goľdenvejzer [2] published in 1941 solution of this system by an approximate method. He proposed formula for torsional-flexural critical force Ncr.TF calculation which is modified and used in EN 1999-1-1 [3] (I.19). By introducing factor αzw he take into account any combination of boundary conditions (BCs).The purpose of this paper is to verify this formula and explore the possibility to improve the factor αzw. In the large parametrical study the authors investigated a lot of different shape of cross-sections, all 100 theoretical possible combinations of BCs and various member lengths. All results are evaluated regarding the reference results by finite element method (FEM).

Parametric Instability of a Rotating Axially Loaded FG Cylindrical Thin Shell Under Both Axial Disturbances and Thermal Effects

2018 ◽  
Vol 73 (12) ◽  
pp. 1105-1119
Author(s):  
X. Li ◽  
Q. Xu ◽  
Y.H. Li
Keyword(s):  
Boundary Conditions ◽  
Thin Shell ◽  
Thermal Environment ◽  
Parametric Instability ◽  
Rotation Velocity ◽  
Functionally Graded ◽  
Angular Rotation ◽  
Various Boundary Conditions ◽  
Love’S Thin Shell Theory ◽  
The Stability

AbstractParametric instability of a rotating functionally graded (FG) cylindrical thin shell with axial compression under various boundary conditions is studied in this article. In particular, the shell is subjected to both axial periodic displacement disturbances and a thermal environment. The initial hoop tension and Coriolis effects due to rotation are also considered. The coupled dynamic equations of the shell under multiple conditions are formulated based on Love’s thin-shell theory. The instability boundaries of the shell with different boundary conditions considering thermal factors, axial disturbances, and other system parameters are obtained analytically under the case of primary and combination resonance; numerical illustrations are also given. It is found that high temperature weakens the stability of the system, while axial disturbances show stronger influence on the instability regions of the shell compared to other parameters such as thermal factors and the angular rotation velocity.

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