2033) Elastic Stability of Beams and Columns with Enforced Points of Rotation(Structure)

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.)

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Adjoint systems in nonconservative problems of elastic stability.

AIAA Journal ◽

10.2514/3.3882 ◽

1966 ◽

Vol 4 (12) ◽

pp. 2221-2222 ◽

Cited By ~ 19

Author(s):

S. NEMAT-NASSER ◽

G. HERRMANN

Keyword(s):

Elastic Stability ◽

Adjoint Systems

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Singular critical points in the general theory of elastic stability

Meccanica ◽

10.1007/bf02133605 ◽

1972 ◽

Vol 7 (S1) ◽

pp. 58-63 ◽

Cited By ~ 2

Author(s):

Koncay Huseyn

Keyword(s):

General Theory ◽

Critical Points ◽

Elastic Stability

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Elastic stability and the zero moment condition

Journal of Elasticity ◽

10.1007/bf00045696 ◽

1971 ◽

Vol 1 (1) ◽

pp. 19-28 ◽

Cited By ~ 4

Author(s):

Roger L. Fosdick

Keyword(s):

Elastic Stability ◽

Moment Condition ◽

Zero Moment

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On Curve Veering and Flutter of Rotating Blades

Journal of Engineering for Gas Turbines and Power ◽

10.1115/1.2906876 ◽

1994 ◽

Vol 116 (3) ◽

pp. 702-708 ◽

Cited By ~ 15

Author(s):

D. Afolabi ◽

O. Mehmed

Keyword(s):

Algebraic Geometry ◽

Catastrophe Theory ◽

Algebraic Curves ◽

Rotation Speed ◽

Elastic Stability ◽

Rotating Blades ◽

Rotating Blade ◽

Modes Of Vibration ◽

Curve Veering ◽

As If

The eigenvalues of rotating blades usually change with rotation speed according to the Stodola-Southwell criterion. Under certain circumstances, the loci of eigenvalues belonging to two distinct modes of vibration approach each other very closely, and it may appear as if the loci cross each other. However, our study indicates that the observable frequency loci of an undamped rotating blade do not cross, but must either repel each other (leading to “curve veering”), or attract each other (leading to “frequency coalescence”). Our results are reached by using standard arguments from algebraic geometry—the theory of algebraic curves and catastrophe theory. We conclude that it is important to resolve an apparent crossing of eigenvalue loci into either a frequency coalescence or a curve veering, because frequency coalescence is dangerous since it leads to flutter, whereas curve veering does not precipitate flutter and is, therefore, harmless with respect to elastic stability.

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