Elliptic cross section without warping under torsion

The object of this paper is the pure torsion of the nonhomogeneous anisotropic elastic beam. The results of Saint-Venant’s theory of uniform torsion are used to prove a nonwarping property of elliptic cylinders.     

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

Upper and lower bounds are derived for the shear stress as it is determined by Saint-Venant's theory of flexure, and used to establish the asymptotic character of the classical Strength of Materials formula in the limit of vanishing thickness.

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Résumé On dérive des limites supérieures et inférieures des contraintes tangentielles suivant la théorie de la flexion de Saint-Venant, que l'on utilise aux fins d'établir le caractère asymptotique de la formule de la Résistance des Matériaux dans le cas limite d'une épaisseur extrêmement petite.

     

Some analytical solutions for Saint-Venant torsion of non-homogeneous cylindrical bars

The Saint-Venant torsion problem of linearly elastic cylindrical bars with solid and hollow cross-section is treated. The shear modulus of the non-homogeneous bar is a given function of the Prandtl's stress function of considered cylindrical bar when its material is homogeneous. The solution of the torsional problem of non-homogeneous bar is expressed in terms of the torsional and Prandtl's stress functions of homogeneous bar having the same cross-section as the non-homogeneous bar.     

On the rotating rod with variable cross section

Summary Stability of a heavy rotating rod with a variable cross section is studied by energy method. Bifurcation points for the system of equilibrium equations are analyzed. It is shown that for the case when the rotation speed exceeds the critical one, the trivial solution ceases to be the minimizer of the potential energy, so that rod loses stability, according to the energy criteria. Also, a new estimate of the maximal rod deflection in the post-critical state is obtained. Accepted for publication 11 November 1996     

A review of some integral equations for solving the Saint-Venant torsion problem

Summary It has been attempted to write down a complete collection of integral equations and functional equations for solving the classical Saint-Venant's torsion problem, to a large extent using Green's third identity and a unified notation for the systematic compilation. Some of the equations are well known, others seem to be new. Nearly all the equations are interpreted physically. Known and unknown connections among the equations are pointed out.

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Zusammenfassung Es wird hier der Versuch gemacht ein vollständiges Verzeichnis von Integralgleichungen und Funktionalgleichungen zur Lösung des klassischen Saint-Venant'schen Torsionsproblems aufzustellen-weitgehend mit Hilfe der dritten Green'schen Identität und unter Verwendung einheitlicher Bezeichnungen, um dadurch eine systematische Zusammenstellung zu erreichen. Einige von diesen Gleichungen sind wohlbekannt-andere dagegen werden hier vermutlich zuerst angegeben; beinahe alle Gleichungen werden physikalisch gedeutet. Bekannter und unbekannter Beziehungen zwischen den Gleichungen sind angegeben.

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The present work is partly based upon material which was compiled for an invited lecture give on July 3, 1975 at the Institute of Technical Mechanics, The Technical University of Aachen, The Federal Republic of Germany.     

On the Flexure of a Saint-Venant Cylinder

The flexure of a Saint-Venant cylinder is defined from a kinematic point of view. A complete solution for the field equations is provided and the gauge choices to define the center of shear and rotation are made consequently. This seems to eliminate some ambiguities present in the literature. This revised version was published online in August 2006 with corrections to the Cover Date.     

Rigorous theory for transient capillary imbibition in channels of arbitrary cross section

This article addresses a classical fluid mechanics problem where the effect of capillary action on a column of viscous liquid is analyzed by quantifying its time-dependent penetrated length in a narrow channel. Despite several past studies, a rigorous mathematical formulation of this inherently unsteady process is still unavailable, because these existing works resort to a crucial assumption only valid for mildly transient systems. The approximate theories use an integral approach where the penetration is described by equating total force acting on the domain to rate of change of total momentum. However, while doing so, the viscous resistance under temporally varying condition is assumed to be same as the resistance created by a quasi-steady velocity profile. Thus, leading order error appears due to such approximation which can only be true when the variation in time is not strong enough causing negligible transient deviation in the hydrodynamic quantities. The present paper proposes a new way to solve this problem by considering the unsteady field itself as an unknown variable. Accordingly, the analysis applies an eigenfunction expansion of the flow with unknown time-dependent amplitudes which along with the unsteady intrusion length are calculated from a system of ordinary differential equations. A comparative exploration identifies the situation for which the integral approach and the rigorous technique based on eigenfunction expansion deviate from each other. It also reveals that the two methods differ substantially in short-time dynamics at the initial stage. Then, an asymptotic perturbation shows how the two sets of results should coincide in their long-time behavior. In this way, the findings will provide a comprehensive understanding of the physics behind the transport phenomenon.     

Secondary torsional moment deformation effect in inelastic nonuniform torsion of bars of doubly symmetric cross section by BEM

In this paper a boundary element method is developed for the inelastic nonuniform torsional problem of simply or multiply connected prismatic bars of arbitrarily shaped doubly symmetric cross section, taking into account the secondary torsional moment deformation effect. The bar is subjected to arbitrarily distributed or concentrated torsional loading along its length, while its edges are subjected to the most general torsional boundary conditions. A displacement based formulation is developed and inelastic redistribution is modeled through a distributed plasticity model exploiting three dimensional material constitutive laws and numerical integration over the cross sections. An incremental–iterative solution strategy is adopted to resolve the elastic and plastic part of stress resultants along with an efficient iterative process to integrate the inelastic rate equations. The one dimensional primary angle of twist per unit length, a two dimensional secondary warping function and a scalar torsional shear correction factor are employed to account for the secondary torsional moment deformation effect. The latter is computed employing an energy approach under elastic conditions. Three boundary value problems with respect to (i) the primary warping function, (ii) the secondary warping one and (iii) the total angle of twist coupled with its primary part per unit length are formulated and numerically solved employing the boundary element method. Domain discretization is required only for the third problem, while shear locking is avoided through the developed numerical technique. Numerical results are worked out to illustrate the method, demonstrate its efficiency and wherever possible its accuracy.     

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

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.This work was supported by the National Science Foundation under Grants No. ENG 75-13643 and ENG 78-26071.     

On the Saint-Venant principle in the case of infinite energy

Estimates on the distribution of the elastic energy in a cylindrical domain in the context of linear elasticity are obtained. The estimates remain valid when the total elastic energy is infinite, and they can be used to establish Saint-Venant's principle without an assumption about finiteness of the total energy.Examples of boundary conditions resulting in infinite energy are constructed in the context of both linear elastostatics and special finite elastostatics, where a quadratic strain energy density function is assumed. The examples show that estimates of the type obtained are sometimes necessary.The results obtained are valid with obvious modifications in a space of any dimension n2.The results in this paper represent a partial fulfilment of the requirements for the degree of Doctor of Philosophy at Tel Aviv University by Y.S. under the guidance of J.J.R.     

On the role of symmetry in Saint-Venant’s problem

In the relaxed Saint-Venant’s elastic problem, in virtue of Saint-Venant’s Postulate, the pointwise assignments of tractions at cylinder plane ends are replaced by the assignments of the corresponding resultant forces and moments. The solution indeterminacy so introduced is traditionally overcome by postulating that some specific features characterize the elastic state. In this work a relaxed incremental equilibrium problem is posed for a heterogenous anisotropic cylinder, whose tangent elasticity tensor field possesses the usual major and minor symmetries, is positive definite, independent from the axial coordinate and endowed with a plane of elastic symmetry orthogonal to the cylinder axis. Symmetry has been consistently employed to formulate the basic problems of extension, bending, torsion and flexure as symmetric and antisymmetric problems respectively. It is shown that Saint-Venant’s postulate, momentum balance and symmetry are sufficient, without resorting to any a priori assumption, to determine the general form of the displacement field and to remove the solution indeterminacy.