The Theory of Curved Beams

1946 ◽  
Vol 13 (4) ◽  
pp. A294-A296
Author(s):  
G. C. Best
Keyword(s):  
Neutral Axis ◽  
Curved Beams ◽  
Slide Rule ◽  
Special Formula ◽  
Satisfactory Accuracy ◽  
Orderly Fashion ◽  
Approximate Integration

Abstract In this paper, the theory of curved beams is developed by a somewhat different procedure from that customarily employed. Deflections at the centroid are first assumed and then loads and stresses resulting from these deflections are estimated. This process works out in a somewhat more orderly fashion than the conventional development. Throughout, all measurements are to the centroidal axis rather than to the neutral axis. Final results are presented in such a form that satisfactory accuracy may be obtained from slide-rule computations and approximate integration. Hence the procedure is applicable to any section, it being unnecessary first to develop a special formula for each different section. An illustrative example is given. The theory is extended to cover the case of unsymmetrical bending of curved beams. The effects of torsion, which will probably also occur in the generalized case, are not treated. These can be superimposed upon stresses due to bending.

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.

A paradox in curved beams

2018 ◽  
Vol 233 (8) ◽  
pp. 2830-2833 ◽  
Author(s):  
Antonio Strozzi ◽  
Enrico Bertocchi ◽  
Sara Mantovani
Keyword(s):  
Cross Section ◽  
Stress Reduction ◽  
Material Removal ◽  
Neutral Axis ◽  
Beam Cross Section ◽  
Mathematical Approach ◽  
Bending Stress ◽  
Curved Beams ◽  
Cross Sectional ◽  
Mechanical Component

It is sometimes possible to relieve the stresses in a mechanical component by removing material, where relief grooves are the commonest expedient approach. Within the rectilinear beam realm, rare situations are known in which, by removing material in the cross-sectional zones that are farthest from the neutral axis, a bending stress diminution is achieved. With regard to curved beams, selected examples are presented in which a bending stress diminution is achieved by laterally removing material from the zones close to the neutral axis. An approximate mathematical approach based on Gateaux linearization is developed that delimits the lateral zones of the beam cross-section in which material removal is accompanied by bending stress reduction. While the achievable stress diminution is generally marginal, the reduction of the beam’s cross-section is technically interesting.

A unified relaxed approach easing the practical application of a paradox in curved beams

2020 ◽  
Vol 234 (22) ◽  
pp. 4535-4542 ◽  
Author(s):  
HMA Abdalla ◽  
D Casagrande ◽  
A Strozzi
Keyword(s):  
Finite Element ◽  
Stress Reduction ◽  
Normal Force ◽  
Bending Moment ◽  
Neutral Axis ◽  
Mathematical Approach ◽  
Bending Stress ◽  
Practical Application ◽  
Curved Beams ◽  
Closed Form Solutions

The paper deals with an arising paradox in curved beams subjected to bending moment and normal force. This paradox consists in the fact that by laterally removing material from section zones close to the neutral axis, not only an obvious reduction of the beam mass can be obtained, but also an unexpected, though technically negligible, reduction of the bending stress. It has recently been shown that the relaxation of the demanding achievement of a concurrent mass and stress reduction may practically lead to interesting results, yet solvable numerically. In this paper we show that, under some mild assumptions, a remarkable simplification of the intrados stress functional is obtained. Hence, a unified approximate mathematical approach based on linearization is developed for the derivation of analytical closed-form solutions for the lateral grooved zones. A practical example of the application of the relaxed paradox to optimize a crane hook subjected to bending and normal force is illustrated and compared to finite element forecasts.

On the Strength Weakening Effect of Stiffening Ribs in the Design of Machine Components

2019 ◽  
Vol 827 ◽  
pp. 240-245
Author(s):  
Antonio Strozzi ◽  
E. Bertocchi ◽  
V. Mangeruga
Keyword(s):  
Cross Section ◽  
Mechanical Design ◽  
Neutral Axis ◽  
Unexpected Result ◽  
Bending Stress ◽  
Curved Beams ◽  
Connecting Rod ◽  
Stiffening Ribs ◽  
Beam Section ◽  
Stiffening Effect

The bending stress in beams may often be reduced by adding material to the cross section. In some paradoxical cases, however, the bending stress increases by adding material from zones far away, or close to, the neutral axis. Similarly, the bending stress of rectilinear or curved beams may often be reduced by adding ribs to the initial beam section. However, such ribs may sometimes cause a both undesired and unexpected stress increase, although they still produce a beneficial stiffening effect. The aim of this paper is twofold: a) to examine this unexpected result within the context of the paradoxical behaviour of some known beam sections, and especially of a recently noted paradox; b) to provide a preliminary rule of thumb for the mechanical design of ribs sometimes added to the outer surface of an eye, with particular regard to the small end of a connecting rod.

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

1993 ◽  
Vol 115 (3) ◽  
pp. 346-358 ◽  
Author(s):  
C. Bagci
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Plane Stress ◽  
Numerical Results ◽  
Neutral Axis ◽  
Curved Beams ◽  
Equilibrium Stress ◽  
Different Boundary Conditions ◽  
Stress Functions ◽  
Elasticity Solutions

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.

A New Unified Strength of Materials Solution for Stresses in Curved Beams and Rings

1992 ◽  
Vol 114 (2) ◽  
pp. 231-237 ◽  
Author(s):  
C. Bagci
Keyword(s):  
Cross Sections ◽  
Solution Method ◽  
Normal Force ◽  
Neutral Axis ◽  
Curved Beams ◽  
Strength Of Materials ◽  
Special Cases ◽  
Elasticity Solutions ◽  
Simplified Solution ◽  
Force Equilibrium

Presently existing strength of materials solutions for stresses in curved beams use an incorrect normal force equilibrium condition to define neutral axis location, and to reach a simplified solution, which neglects the curvature effect on stresses due to normal force. This article presents a new but a most general form of the strength of materials solution for determining tangential normal stresses in curved beams, including reductions to special cases. The neutral axis phenomenon is clarified and experimentally verified. Several numerical examples are included, some of which offer photoelastic experimental results, where results predicted by the exact elasticity solution, method of the article, Winkler’s theory, and the conventional simplified method are compared. The hook, diametrically loaded cut, and full ring applications are included. It is shown that simplified theory leads to very large errors. Results by the method offered are very reliable with small errors which are comparable with those of exact elasticity solutions. Stress and deflection analyses of curved beams with varying thicknesses of cross-sections by exact elasticity solutions are given in a separate article [6].

ultrastructural process of intestinal wound healing

1995 ◽  
Vol 53 ◽  
pp. 1024-1025
Author(s):  
Rick L. Vaughn ◽  
Shailendra K. Saxena ◽  
John G. Sharp
Keyword(s):  
Wound Healing ◽  
Epithelial Cells ◽  
Smooth Muscles ◽  
Microscopic Level ◽  
Light Microscopic Level ◽  
Ileal Mucosa ◽  
Wound Model ◽  
Serosal Surface ◽  
Complex Process ◽  
Orderly Fashion

We have developed an intestinal wound model that includes surgical construction of an ileo-cecal patch to study the complex process of intestinal wound healing. This allows approximation of ileal mucosa to the cecal serosa and facilitates regeneration of ileal mucosa onto the serosal surface of the cecum. The regeneration of ileal mucosa can then be evaluated at different times. The wound model also allows us to determine the rate of intestinal regeneration for a known size of intestinal wound and can be compared in different situations (e.g. with and without EGF and Peyer’s patches).At the light microscopic level it appeared that epithelial cells involved in regeneration of ileal mucosa originated from the enlarged crypts adjacent to the intestinal wound and migrated in an orderly fashion onto the serosal surface of the cecum. The migrating epithelial cells later formed crypts and villi by the process of invagination and evagination respectively. There were also signs of proliferation of smooth muscles underneath the migratory epithelial cells.

Hearing in Mice by GSR Audiometry: II. Magnitude of Unconditioned GSR as a Function of Intensity and Frequency Interactions

1968 ◽  
Vol 11 (1) ◽  
pp. 169-178 ◽  
Author(s):  
Alan Gill ◽  
Charles I. Berlin
Keyword(s):  
Power Law ◽  
Response Magnitude ◽  
Single Units ◽  
Spectral Content ◽  
Hearing Sensitivity ◽  
Subjective Magnitude ◽  
Orderly Fashion ◽  
Viiith Nerve

The unconditioned GSR’s elicited by tones of 60, 70, 80, and 90 dB SPL were largest in the mouse in the ranges around 10,000 Hz. The growth of response magnitude with intensity followed a power law (10 .17 to 10 .22 , depending upon frequency) and suggested that the unconditioned GSR magnitude assessed overall subjective magnitude of tones to the mouse in an orderly fashion. It is suggested that hearing sensitivity as assessed by these means may be closely related to the spectral content of the mouse’s vocalization as well as to the number of critically sensitive single units in the mouse’s VIIIth nerve.

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