Upper and lower bounds for the shear stress in the Saint-Venant theory of flexure

Wheeler, Lewis; Horgan, Cornelius O.

doi:10.1007/bf00040899

Cited by 7 publications

(2 citation statements)

References 4 publications

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“…On On which is sufficient for obtaining a lower bound for ~'max-A more elaborate generalization of Theorem 2 would consist in establishing a version of the theorem in a weak formulation, involving less stringent smoothness hypotheses on ~b, thereby facilitating the construction of suitable comparison functions. Such weak maximum principles were employed successfully in elasticity problems in [18,19] for example. We defer these considerations at the present time.…”

Section: Discussionmentioning

confidence: 99%

Maximum principles and bounds on stress concentration factors in the torsion of grooved shafts of revolution

Horgan

1982

J Elasticity

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This paper provides an analytical approach for obtaining bounds for stress concentration factors in the theory of axisymmetric torsion of circumferentially grooved shafts of revolution. The analysis is based on application of maximum principles for linear second-order elliptic partial differential equations. The particular case of a straight shaft with a semi-circular groove is considered in detail. Explicit estimates are obtained and compared with existing results for this problem.

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Section: Discussionmentioning

confidence: 99%

Maximum principles and bounds on stress concentration factors in the torsion of grooved shafts of revolution

Horgan

1982

J Elasticity

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“…Such is the case in [22][23][24], which deal with stress estimates appropriate to Saint-Venant torsion theory, and [25], which concerns estimates for Saint-Venant flexure theory. The results of [23][24][25] are of interest in connection with the present paper in that estimates for solids of thin cross section are emphasized.…”

Section: -4c(ouytoy)u>--c[u---3+3(2ouyt%)2]mentioning

confidence: 99%

Maximum principles and pointwise error estimates for torsion of shells of revolution

Horgan

Wheeler

1977

J Elasticity

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This paper is concerned with the question of assessing the quality of approximate thin shell solutions for the problem of axisymmetric torsion of elastic shells of revolution, an issue previously considered by Ho and Knowles [1]. In both works, the objective is to obtain pointwise estimates, based on the threedimensional theory of elasticity, for the errors involved in using an approximate shell theory.The mathematical problem is that of obtaining explicit pointwise gradient estimates for the solutions of homogeneous and nonhomogeneous second-order uniformly elliptic equations, an issue of interest beyond the present context. In contrast to arguments using energy inequalities, here we apply a technique based on maximum principles for such equations.The particular examples of cylindrical, spherical and conical shells of revolution are discussed in detail. In the former two cases, it is shown that the shell theory solution is in fact an exact elasticity solution in the Saint-Venant sense. Asymptotic error estimates for general shells of revolution are also given. RI~SUMIEL'6tude qi-dessous traite de la qualit6 de l'approximation des coques minces appliquEe aux probl~mes de torsion asym6trique des coques 61astiques de r6volution, probl6me d6j~ consid6r6 par Ho et Knowles [1].L'objectif des deux 6tudes est d'6valuer en chaque point, par la th6orie de l'61asticit6 en trois dimensions, les erreurs cons6quentes h l'utilisation de la th6orie de l'approximation des coques.Le probl~me math6matique consiste h obtenir une estimation explicite en chaque point des gradients pour les solutions d'6quations elliptiques uniformes du deuxi~me ordre, homog~nes et non-homog6nes, probl~me d'un inter& qui d6passe le pr6sent contexte. Nous utilisons ici une technique bas6e sur le principe du maximum pour de telles 6quations, par contraste avec les raissonnements utilisant des in6galit6s 6nerg6tiques.Les exemples particuliers des coques de r6volution cylindriques, sph6riques et coniques sont discut6s en d6tail. Dans les deux premiers cas, nous montrons que la solution par la th6orie des coques est en fait une solution exacte dans le sens de Saint-Venant. Des estimations asymptotiques des erreurs dans le cas de coques de r6volution plus g6n6rales sont 6galement donn6es.

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Dynamic modeling of taut strings carrying a traveling mass

Ferretti

Piccardo

2012

Continuum Mech. Thermodyn.

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In this paper, a new consistent dynamic model is proposed, aimed at studying linear vibrations induced in an elastic wire by a bilaterally constrained single mass moving with a constant velocity. Starting from a variational formulation, through the Hadamard’s condition, a corrective term to the local linear stiffness is determined in the continuum model as a function of the moving mass velocity; in this way, the boundary conditions are properly found. The representation of the solutions of the hyperbolic equations governing the motion of the wire presents some difficulties, which are solved by means of a suitable coordinate transformation in a time-invariant domain and a judicious choice of the set of shape functions, to be used in the discrete formulation of the problem. This new description allows an easy estimation of high-order deformations that are neglected by a purely linear approach. When the mass velocity is sufficiently high, displacements near the supports show high gradients: in these cases, it is necessary to use an unknown velocity or introduce an advanced mechanical model in order to correctly describe the motion of the mass. Numerical examples confirm the stability of the proposed solution in all conditions examined

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Upper and lower bounds for the shear stress in the Saint-Venant theory of flexure

Cited by 7 publications

References 4 publications

Maximum principles and bounds on stress concentration factors in the torsion of grooved shafts of revolution

Maximum principles and bounds on stress concentration factors in the torsion of grooved shafts of revolution

Maximum principles and pointwise error estimates for torsion of shells of revolution

Dynamic modeling of taut strings carrying a traveling mass

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