1,721,093 research outputs found
On Kelvin's formula for torsion of thin cylinders
No full textAs it is well known, there is in general no closed-form solution for either the shear stress or the longitudinal displacement in the torsion of a Saint-Venant cylinder. For technical purposes, approximate formulae for cylinders with thin-walled sections are at disposal. In the case of open sections (i. e., sections which are topologically equivalent to a rectangle) the (dominant component of the) shear stress turns out to be an affine function (with zero mean) of the coordinate along the thickness. Kelvin & Tait pointed out that this formula is not sufficient to determine the torque resisted by a cylinder with rectangular sections. Indeed, the dominant component of the stress is equivalent only to one half of the applied torque; the other
half is carried by a secondary component of the shear stress near the edges. This seems paradoxal, as the secondary component of the stress is negligible in magnitude with respect to the dominant, as was also proved by means of formal expansions. Popov helped to clarify the matter, showing that the secondary component of the shear stress becomes relevant near the 'short' edges of the section and that any open section behaves like the rectangular one. In this paper a simple geometrical description of an open section of a Saint-Venant cylinder of arbitrary (but regular) varying thickness is introduced. By means of simple differential geometry techniques and of the fields equations of Saint-Venant problem. Kelvin's and Popov's formulae are immediately obtained and generalized in a straightforward way
Continuum models according to Piola
No full textThis paper illustrates Gabrio Piola’s view on continuum models, especially how contact actions are defined. Piola presented his mechanical theory before the 1850s, in an attempt to generalize Lagrange’s analytical mechanics. He conceived, among the rest, an ideal state for physical bodies (which nowadays we would call a natural state), a very general set of what we would now call state variables, and obtained balance equations via the superposition of a rigid infinitesimal motion on the present configuration. These views look quite modern even today and seem to historically precede among other things the introduction of structured continua
Espansioni formali per il problema di Saint-Venant
No full textDottorato di ricerca in meccanica teorica e applicata. 8. ciclo.Consiglio Nazionale delle Ricerche - Biblioteca Centrale - P.le Aldo Moro, 7, Rome; Biblioteca Nazionale Centrale - P.za Cavalleggeri, 1, Florence / CNR - Consiglio Nazionale delle RichercheSIGLEITItal
A historical perspective of Menabrea's "principle of elasticity"
No full textThis paper presents the theorem proposed by Luigi Federico Menabrea to study linear elastic systems. The publications on the subject by Menabrea are examined, as well as the criticism and the corrections brought to his first attempts of proof. We consider Menabrea's work in the frame of the studies of his contemporaries; we try to provide a historical and epistemological background for Menabrea's theorem and for its consequences in modern mechanics. Menabrea's theorem is a version of the theorem of least work, or of minimum complementary energy. A rough statement of the latter, for a ssystem free of dislocations, is as follows: A linear elastic system is subjected to a compatible deformation if and only if the elastic complementary energy is a minimum in the space of balanced actions'. These statements were the basis for the resolutions of redundant frames proposed by Castigliano and Muller-Breslau. The theorem was stated more or less in this form in 1858 by Luigi Federico Menabrea in a short memoir. The declared purpose of this memoir was to provide a general tool of solution for the problems of linear elasticity met in engineering practice, and this tool was to be more efficient than the ad hoc procedures at ease then. Such procedures were based on what is nowadays called method of forces and followed the path tracked by Navier
Strength of materials and theory of elasticity in 19th century Italy: a brief account of the history of mechanics of solids and structures
No full textThis book examines the theoretical foundations underpinning the field of strength of materials/theory of elasticity, beginning from the origins of the modern theory of elasticity. While the focus is on the advances made within Italy during the nineteenth century, these achievements are framed within the overall European context. The vital contributions of Italian mathematicians, mathematical physicists, and engineers in respect of the theory of elasticity, continuum mechanics, structural mechanics, the principle of least work, and graphical methods in engineering are carefully explained and discussed. The book represents a work of historical research that primarily comprises original contributions and summaries of work published in journals. It is directed at those graduates in engineering, but also in architecture, who wish to achieve a more global and critical view of the discipline and will also be invaluable for all scholars of the history of mechanics
Buckling of a beam on a Wieghardt foundation
No full textThis study considers the buckling of a uniform column on a Wieghardt elastic foundation. The column model incorporates both purely flexible, Bernoulli-Euler, and shear-deformable, Timoshenko, beams. Some comparisons with results reported in the existing literature are made, and numerical examples are evaluated to attempt shedding additional light on the problem
On proper applications of Galërkin’s approach in structural mechanics courses
Full text linkAn incautious use of the well-known Galërkin’s technique to find approximate solutions of a differential
problem may lead to apparently wrong results. Examples are based on an inverse approach to investigate buckling
of compressed axisymmetric circular plates, a common subject in courses on mechanics of structures and stability of
structural elements. We discuss how a mistake may originate and show how it is possible to recover the expected
results, thus providing a means for the students to cross-check their outputs
Closed-Form Solutions for Axisymmetric Functionally Graded Material Elastic Plates
No full textSuitable, yet general enough, choices of functional grading along the radius and the thickness of axisymmetric circular plates may lead to closed-form solutions for the linear elastic direct problem. The plates are modeled according to the usual Kirchhoff—Love theory, because they are supposed to be thin; to abstract from actual values of geometric and material parameters, the governing equations are dealt with in nondimensional form. Some instances are presented, along with thorough comments
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