Variational Formulation of Problems of Elastic Stability. Topics in the Stability of Beams and Plates

1986 ◽  
pp. 246-312
Author(s):  
Vadim Komkov
Keyword(s):  
Variational Formulation ◽  
Elastic Stability ◽  
The Stability ◽  
Formulation Of Problems

scholarly journals The elastic stability of an annular plate

1924 ◽  
Vol 106 (737) ◽  
pp. 268-284 ◽  
Keyword(s):  
Annular Plate ◽  
Practical Interest ◽  
Middle Surface ◽  
Elastic Stability ◽  
Thin Plates ◽  
Thin Shells ◽  
Stability Equation ◽  
Inner Radius ◽  
Plane Plate ◽  
The Stability

1. Introduction and Summary. —This paper deals with the elastic stability of a circular annular plate under uniform shearing forces applied at its edges. Investigations of the stability of plane plates are altogether simpler than those necessary in the case of curved plates or shells. In the first place, as shown by Mr. R. V. Southwell, two of the three equations of stability relate to a mode of instability that is not of practical interest, and are entirely independent of the third equation which gives the ordinary mode of instability resulting in the familiar bending of the middle surface of the plate. Consequently with a plane plate there is only one equation of stability to be solved, as contrasted with the case of a shell where the three equations are dependent, and must all be solved. In the second place the theory of thin shells can be used with confidence in a plane plate problem, though a more laborious procedure is necessary to deal adequately with a shell. The only stability equation required for the annular plate is therefore deduced without trouble from the theory of thin shells, and its solution presents no difficulty in the case of uniform shearing forces. A numerical discussion is given of the stability of the plate under such forces, the “favourite type of distortion” and the stess that will produce it being obtained for plates with clamped edges in wich the ratio of the outer to the inner radius exceeds 3·2. To some extent to results have been checked by experiment, in which part of the work the viter is indebted to Prof. G. I. Taylor for his valuable help and advice. Distrtion of the type predicted by the theory took place in the two thin plates of rober different ratio of radii, which were used. The disposition of the loci of points which undergo maximum normal displace nt gives some idea of the appearance of the plate after distortion has taken pce. The points have been calculated for a plate in which the ratio of radii 4·18, and the loci are shown on a diagram, which may be compared with a potograph of a distorted plate in which this ratio is 4·3. The ratio of normal dplacements of points of the plate can be seen from contours drawn on the ne diagram. (See pp. 280, 281.)

Characteristics of an Eigensystem Describing the Elastic Stability of a Thin Cantilever Beam

1962 ◽  
Vol 66 (623) ◽  
pp. 722-724 ◽  
Author(s):  
B. Saravanos
Keyword(s):  
Equilibrium Problem ◽  
Cantilever Beam ◽  
Elastic Stability ◽  
Additional Component ◽  
Connecting Rod ◽  
The Stability

The stability of a uniform cantilever beam subjected to an articulated tip load has been analysed as an equilibrium problem of static elastic stability. It was found there that the articulation of the connecting rod applying the tip load introduced additional component forces during the process of beam deformation.

On A Formulation of the Elastic Stability Equations of Circular Cylindrical Shells Under Variable Internal Pressure

1985 ◽  
Vol 9 (2) ◽  
pp. 64-69
Author(s):  
J.L. Urrutia-Galicia ◽  
A.N. Sherbourne
Keyword(s):  
Mathematical Model ◽  
Internal Pressure ◽  
Cylindrical Shells ◽  
Direct Analysis ◽  
Elastic Stability ◽  
Equilibrium Path ◽  
Circular Cylindrical Shells ◽  
The Stability ◽  
Variable Internal Pressure ◽  
The Mathematical Model

The mathematical model of the stability analysis of circular cylindrical shells under arbitrary internal pressure is presented. The paper consists of a direct analysis of the equilibrium modes in the neighbourhood of the unperturbed principal equilibrium path. The final stability condition results in a completely symmetric differential operator which is then compared with current theories found in the literature.

Determination of the Fluid-Elastic Stability Threshold in the Presence of Turbulence: A Theoretical Study

1989 ◽  
Vol 111 (4) ◽  
pp. 407-419 ◽  
Author(s):  
J. H. Lever ◽  
G. Rzentkowski
Keyword(s):  
Theoretical Study ◽  
Cross Flow ◽  
Elastic Stability ◽  
Mass Ratio ◽  
Response Curves ◽  
Stability Threshold ◽  
Pitch Ratio ◽  
The Stability ◽  
Experimental Findings

A model has been developed to examine the effect of the superposition of turbulent buffeting and fluid-elastic excitation on the response of a single flexible tube in an array exposed to cross-flow. The modeled response curves for a 1.375-pitch ratio parallel triangular array are compared with corresponding experimental data for the same array; reasonably good qualitative agreement is seen. Turbulence is shown to have a significant effect on the determination of the stability threshold for the array, with increasing turbulent buffeting causing a reduction in the apparent critical velocity. The dependence of turbulence response on mass ratio is also found to yield a slight independence between mass and damping parameters on stability threshold estimates, which may account for similar experimental findings. Different stability criteria are compared, and an attempt is made to provide some guidance in the interpretation of response curves from actual tests.

Elastic Stability of Rubber Products

1987 ◽  
Vol 60 (5) ◽  
pp. 957-965 ◽  
Author(s):  
Farhad Tabaddor
Keyword(s):  
Finite Element ◽  
Symmetric Solution ◽  
Biaxial Loading ◽  
Elastic Stability ◽  
Equilibrium Solutions ◽  
Unstable Solutions ◽  
Symmetrical Loading ◽  
Symmetric Loading ◽  
The Stability

Abstract Due to severe nonlinearities, inherent in the finite-element elasticity, uniquely defined boundary-value problems of rubber elasticity may have multiple stable and unstable solutions. An early example was given by Rivlin, who considered the problem of a Neo-Hookean cube, in a state of pure homogeneous deformations, and subjected to three pairs of equal and opposite forces acting normally on the faces of the cube and distributed uniformly over them. He found that, for forces below a certain value, the only possible solution is the symmetric solution, as might be expected. Beyond that certain value, however, there are seven possible equilibrium solutions. One of these seven solutions is the symmetric solution. It is interesting to notice that the symmetric solution, which is initially stable, becomes unstable when loads have reached a certain threshold. The stability problems of homogeneous deformations of Mooney-Rivlin type of materials, under symmetric loading, for triaxial loading and for the plane stress and plane strain cases, are dealt with in Reference 3. It was shown that a finite-element method can be applied for such analyses. The stability of a sheet of Mooney-Rivlin type of material has been studied for a symmetrical loading condition. Such instability phenomenon was first observed by Treloar. In this work, the problem of a sheet of Mooney-Rivlin type of material, subject to general biaxial loading, is studied both analytically and by finite element. An energy approach to the problem is first presented. This problem represents the biaxial loading of rubber sheets or combined extension and inflation of rubber tubes, which are often used in experimental work for characterization of rubber materials. It is shown that the problem has multiple solutions for a certain domain of loading. The equilibrium state, actually attained, is dependent on the manner of quasistatic loading. Various stable solutions are obtained by finite element.

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.

Analysis of Core Buckling Defects in Sheet Metal Coil Processing

2003 ◽  
Vol 125 (4) ◽  
pp. 771-777 ◽  
Author(s):  
P. M. Lin ◽  
J. A. Wickert
Keyword(s):  
Sheet Metal ◽  
Numerical Technique ◽  
Elastic Stability ◽  
External Pressure ◽  
Internal Stresses ◽  
Geometric Imperfection ◽  
The Core ◽  
Nonlinear Stress ◽  
Core Thickness ◽  
The Stability

The elastic stability of a wound coil comprising a central core and many layers of sheet metal is modeled and analyzed. A common failure mode resulting from unfavorable internal stresses—called v-buckling—is characterized by a section of the core buckling inward, possibly with several nearby sheet metal layers. In the present study, the core is modeled as a thin cylinder that is subjected to (i) the uniform external pressure generated by the coil’s wound-in stresses and (ii) a nonuniform elastic foundation around its circumference that represents core-coil contact or loss thereof. The model and an iterative numerical technique are used to predict the critical winding pressure along the core-coil interface and the core’s ensuing buckled shape. The role of geometric imperfection in the core, and the sensitivity of the buckling pressure to such initial defects, are also examined. Critical imperfection wavenumbers that facilitate the onset of significant deformations are identified with a view toward applying the results to improve quality and core inspection procedures. The predicted buckling pressure and the maximum radial stress developed in the coil, as based on a nonlinear stress model, are together used to determine factors of safety against core buckling over a range of manufacturing process parameters. Three case studies evaluate sensitivity with respect to process tension, core radius, and core thickness. The results are intended to guide the development of solutions to control the stability and quality of coils in sheet metal manufacturing.

scholarly journals On the stability under shearing forces of a flat elastic strip

1924 ◽  
Vol 105 (733) ◽  
pp. 582-607 ◽  
Keyword(s):  
Bending Moment ◽  
Elastic Stability ◽  
Elastic Strip ◽  
Plate Girders ◽  
Uniform Shear ◽  
Transverse Vibrations ◽  
Annular Disc ◽  
The Stability ◽  
Limiting Case ◽  
Total Shear

1. The investigation relates to flat elastic strip, of uniform breadth, thickness and material, upon which a uniform shear is imposed by tangential tractions applied at its edges and in its plane. The tractions appear in the expression for the change of potential energy which occurs when the strip is bent, and they must therefore affect both the modes and the frequencies of its free transverse vibrations. If sufficiently intense, they will bring about a condition of limiting elastic stability, since they can neutralize, in certain types of distortion, the restoring effects of the flexural stresses. The results have some bearing on the stability of the webs of deep plate girders, which take the greater part of the total shear transmitted. The correspondence must not, however, be pressed unduly, because in a girder uniform shear will be accompanied by a varying bending moment which imposes additional stresses upon the web. It is more accurate to describe the sheared strip (of which the length, in this paper, has been assumed to be infinite) as the limiting case either of a narrow annular disc, or of a short tube, subjected to torsion. The similarity of the three problems is illustrated by the specimens shown in fig. 1, which have buckled under conditions of limiting elastic stability.

Stability Analysis of a Simple Strut with Elastic Springs Through Sensitivity Surfaces and Gradients

2018 ◽  
Vol 18 (12) ◽  
pp. 1850161
Author(s):  
Mario Uroš ◽  
Damir Lazarević ◽  
Marija Demšić ◽  
Josip Atalić
Keyword(s):  
Load Capacity ◽  
Equilibrium Points ◽  
Elastic Stability ◽  
Initial Position ◽  
System Of Nonlinear Equations ◽  
Static System ◽  
Algebraic Equations ◽  
Wide Range ◽  
The Stability ◽  
Potential Energy Method

The global elastic stability of a strut supported by three inclined springs is studied using the exact displacement geometry. The discrete phenomenological model in the theory of stability, described by Thompson and Gaspar, exhibits different postbuckling behaviors, depending on the initial geometry for large imperfections, and this has various applications in engineering. Using the total potential energy method, a system of nonlinear algebraic equations is derived for the nondissipative system. The arc length method is adopted to solve the system of nonlinear equations, considering the stability of equilibrium points at each position, while detecting and traversing the critical points with the possibility of intervention. The characteristic equilibrium paths and their corresponding trajectories are shown and compared with the asymptotic solutions at the first critical point. A parametric analysis is performed, and sensitivity surfaces are constructed for several initial positions of the springs, represented by the initial angle in the horizontal plane. A wide range of independent imperfections with large amplitudes in two directions is considered. The concept of load capacity gradient functions of sensitivity surfaces is introduced and used to qualitatively and quantitatively analyze the stability behavior of the system located far away from the initial position. A few interesting observations and conclusions are obtained from the sensitivity analysis of a simple strut in the postcritical region. The critical combinations of imperfections for each static system are determined through analysis of the gradients. Further, the “stable” and “unstable” regions of the sensitivity surfaces are identified, with some observations made. Finally, the applications of load capacity gradient functions in structural optimization and form findings are reviewed.

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