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D. STEFAN DANCILA - One of the best experts on this subject based on the ideXlab platform.
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Buckling of moderately thick orthotropic columns : Comparison of an elasticity solution with the Euler and Engesser/Haringx/Timoshenko Formulae
International Journal of Solids and Structures, 1997Co-Authors: George A. Kardomateas, D. STEFAN DANCILAAbstract:Abstract The objective of this paper is to answer the question of how accurately the simple Euler or transverse shear correction Engesser/Haringx/Timoshenko column buckling Formulae are, when orthotropic composite material and moderate thickness are involved. The column is in the form of a hollow circular cylinder and the Euler or Timoshenko loads are based on the axial modulus. For this purpose, a three-dimensional elasticity solution is presented. As an example, the cases of an orthotropic material with stiffness constants typical of glass/epoxy or graphite/epoxy and the reinforcing direction along the periphery or along the cylinder axis are considered. First, it is found that the elasticity approach predicts in all cases a lower than the Enter value critical load. Moreover, the degree of non-conservatism of the Euler Formula is strongly dependent on the reinforcing direction; the axially reinforced columns show the highest deviation from the elasticity value. The degree of non-conservatism of the Euler load for the circumferentially reinforced columns is much smaller and is comparable to that of isotropic columns. Second, the Engesser or first Timoshenko shear correction Formula is in all cases examined conservative, i.e., it predicts a lower critical load than the elasticity solution. The Haringx or second Timoshenko shear correction Formula is in most cases (but not always) conservative. However, in all cases considered, the second estimate is always closer to the elasticity solution than the first one. For the isotropic case both Timoshenko Formulas are conservative estimates. Examination of a new Formula for column buckling that adds a second term to the Euler load expression and is supposed to account for thickness effects, shows that this estimate is a non-conservative estimate but performs very well with very thick sections, being closest to the elasticity solution, but in general no better than the Timoshenko Formulas for moderate thickness.
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buckling of moderately thick orthotropic columns comparison of an elasticity solution with the Euler and engesser haringx timoshenko Formulae
International Journal of Solids and Structures, 1997Co-Authors: George A. Kardomateas, D. STEFAN DANCILAAbstract:Abstract The objective of this paper is to answer the question of how accurately the simple Euler or transverse shear correction Engesser/Haringx/Timoshenko column buckling Formulae are, when orthotropic composite material and moderate thickness are involved. The column is in the form of a hollow circular cylinder and the Euler or Timoshenko loads are based on the axial modulus. For this purpose, a three-dimensional elasticity solution is presented. As an example, the cases of an orthotropic material with stiffness constants typical of glass/epoxy or graphite/epoxy and the reinforcing direction along the periphery or along the cylinder axis are considered. First, it is found that the elasticity approach predicts in all cases a lower than the Enter value critical load. Moreover, the degree of non-conservatism of the Euler Formula is strongly dependent on the reinforcing direction; the axially reinforced columns show the highest deviation from the elasticity value. The degree of non-conservatism of the Euler load for the circumferentially reinforced columns is much smaller and is comparable to that of isotropic columns. Second, the Engesser or first Timoshenko shear correction Formula is in all cases examined conservative, i.e., it predicts a lower critical load than the elasticity solution. The Haringx or second Timoshenko shear correction Formula is in most cases (but not always) conservative. However, in all cases considered, the second estimate is always closer to the elasticity solution than the first one. For the isotropic case both Timoshenko Formulas are conservative estimates. Examination of a new Formula for column buckling that adds a second term to the Euler load expression and is supposed to account for thickness effects, shows that this estimate is a non-conservative estimate but performs very well with very thick sections, being closest to the elasticity solution, but in general no better than the Timoshenko Formulas for moderate thickness.
George A. Kardomateas - One of the best experts on this subject based on the ideXlab platform.
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Buckling of moderately thick orthotropic columns : Comparison of an elasticity solution with the Euler and Engesser/Haringx/Timoshenko Formulae
International Journal of Solids and Structures, 1997Co-Authors: George A. Kardomateas, D. STEFAN DANCILAAbstract:Abstract The objective of this paper is to answer the question of how accurately the simple Euler or transverse shear correction Engesser/Haringx/Timoshenko column buckling Formulae are, when orthotropic composite material and moderate thickness are involved. The column is in the form of a hollow circular cylinder and the Euler or Timoshenko loads are based on the axial modulus. For this purpose, a three-dimensional elasticity solution is presented. As an example, the cases of an orthotropic material with stiffness constants typical of glass/epoxy or graphite/epoxy and the reinforcing direction along the periphery or along the cylinder axis are considered. First, it is found that the elasticity approach predicts in all cases a lower than the Enter value critical load. Moreover, the degree of non-conservatism of the Euler Formula is strongly dependent on the reinforcing direction; the axially reinforced columns show the highest deviation from the elasticity value. The degree of non-conservatism of the Euler load for the circumferentially reinforced columns is much smaller and is comparable to that of isotropic columns. Second, the Engesser or first Timoshenko shear correction Formula is in all cases examined conservative, i.e., it predicts a lower critical load than the elasticity solution. The Haringx or second Timoshenko shear correction Formula is in most cases (but not always) conservative. However, in all cases considered, the second estimate is always closer to the elasticity solution than the first one. For the isotropic case both Timoshenko Formulas are conservative estimates. Examination of a new Formula for column buckling that adds a second term to the Euler load expression and is supposed to account for thickness effects, shows that this estimate is a non-conservative estimate but performs very well with very thick sections, being closest to the elasticity solution, but in general no better than the Timoshenko Formulas for moderate thickness.
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buckling of moderately thick orthotropic columns comparison of an elasticity solution with the Euler and engesser haringx timoshenko Formulae
International Journal of Solids and Structures, 1997Co-Authors: George A. Kardomateas, D. STEFAN DANCILAAbstract:Abstract The objective of this paper is to answer the question of how accurately the simple Euler or transverse shear correction Engesser/Haringx/Timoshenko column buckling Formulae are, when orthotropic composite material and moderate thickness are involved. The column is in the form of a hollow circular cylinder and the Euler or Timoshenko loads are based on the axial modulus. For this purpose, a three-dimensional elasticity solution is presented. As an example, the cases of an orthotropic material with stiffness constants typical of glass/epoxy or graphite/epoxy and the reinforcing direction along the periphery or along the cylinder axis are considered. First, it is found that the elasticity approach predicts in all cases a lower than the Enter value critical load. Moreover, the degree of non-conservatism of the Euler Formula is strongly dependent on the reinforcing direction; the axially reinforced columns show the highest deviation from the elasticity value. The degree of non-conservatism of the Euler load for the circumferentially reinforced columns is much smaller and is comparable to that of isotropic columns. Second, the Engesser or first Timoshenko shear correction Formula is in all cases examined conservative, i.e., it predicts a lower critical load than the elasticity solution. The Haringx or second Timoshenko shear correction Formula is in most cases (but not always) conservative. However, in all cases considered, the second estimate is always closer to the elasticity solution than the first one. For the isotropic case both Timoshenko Formulas are conservative estimates. Examination of a new Formula for column buckling that adds a second term to the Euler load expression and is supposed to account for thickness effects, shows that this estimate is a non-conservative estimate but performs very well with very thick sections, being closest to the elasticity solution, but in general no better than the Timoshenko Formulas for moderate thickness.
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Three-Dimensional Elasticity Solution for the Buckling of Moderately Thick Orthotropic Columns
Contemporary Research in Engineering Science, 1995Co-Authors: George A. KardomateasAbstract:The buckling of an axially compressed orthotropic column is investigated by using a three-dimensional elasticity Formulation. In this manner, an assessment of the thickness and othotropy effects can be accurately performed. The column is in the form of a hollow circular cylinder. The critical loads from this elasticity solution are compared with the ones from the Euler or Timoshenko transverse shear correction Formulas based on the axial modulus. Furthermore, a comparison is made with a recenly suggested new Formula for column buckling that adds a second term to the Euler load expression and is supposed to account for thickness effects. As an example, the cases of an orthotropic material with stiffness constants typical of glass/epoxy and the reinforcing direction along the periphery or along the cylinder axis are considered. It is found that the elasticity approach predicts in all cases a lower than the Euler value critical load. Moreover, the degree of non-conservatism of the Euler Formula is strongly dependent on the reinforcing direction; the axially reinforced columns show the highest deviation from the elasticity value. The first Timoshenko shear correction Formula is in all cases examined conservative. The second Timoshenko shear correction Formula is in most cases (but not always) conservative. However, the second estimate is always closer to the elasticity solution than the first one. For the istotropic case both Timoshenko Formulas are conservative estimates. The recent new Formula for column buckling that adds a second term to the Euler load expression is a non-conservative estimate but performs very well with very thick sections, being closest to the elasticity solution; for moderate thickness it is in general no better than the Timoshenko Formulas.
Jean-françois Dufourd - One of the best experts on this subject based on the ideXlab platform.
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An Intuitionistic Proof of a Discrete Form of the Jordan Curve Theorem Formalized in Coq with Combinatorial Hypermaps
Journal of Automated Reasoning, 2009Co-Authors: Jean-françois DufourdAbstract:This paper presents a completely formalized proof of a discrete form of the Jordan Curve Theorem. It is based on a hypermap model of planar subdivisions, formal specifications and proofs assisted by the Coq system. Fundamental properties are proven by structural or noetherian induction: Genus Theorem, Euler Formula, constructive planarity criteria. A notion of ring of faces is inductively defined and a Jordan Curve Theorem is stated and proven for any planar hypermap.
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Polyhedra genus theorem and Euler Formula: A hypermap-formalized intuitionistic proof
Theoretical Computer Science, 2008Co-Authors: Jean-françois DufourdAbstract:AbstractThis article presents formalized intuitionistic proofs for the polyhedra genus theorem, the Euler Formula and a sufficient condition of planarity. They are based on a hypermap model for polyhedra and on formal specifications in the Calculus of Inductive Constructions. First, a type of free maps is inductively defined from three atomic constructors. Next, a hierarchy of types defined by invariants, with operations constrained by preconditions, is built on the free maps: hypermaps, orientated combinatorial maps and a central notion of quasi-hypermaps. Besides, the proofs of their properties are established until the genus theorem and the Euler Formula, mainly using a simple induction principle based on the free map term algebra. Finally, a constructive sufficient condition for polyhedra to be planar is set and proved. The whole process is assisted by the interactive Coq proof system
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a hypermap framework for computer aided proofs in surface subdivisions genus theorem and Euler s Formula
ACM Symposium on Applied Computing, 2007Co-Authors: Jean-françois DufourdAbstract:This paper presents a new framework to conduct formal proofs concerning the topology of surface subdivisions. The subdivisions are modeled by hypermaps specified through the Calculus of Inductive Constructions. Proofs are computer-aided using the Coq system. A significant example is emphasized: the proof of the genus theorem and of the Euler Formula for hypermaps.
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SAC - A hypermap framework for computer-aided proofs in surface subdivisions: genus theorem and Euler's Formula
Proceedings of the 2007 ACM symposium on Applied computing - SAC '07, 2007Co-Authors: Jean-françois DufourdAbstract:This paper presents a new framework to conduct formal proofs concerning the topology of surface subdivisions. The subdivisions are modeled by hypermaps specified through the Calculus of Inductive Constructions. Proofs are computer-aided using the Coq system. A significant example is emphasized: the proof of the genus theorem and of the Euler Formula for hypermaps.
Karl Schweizerhof - One of the best experts on this subject based on the ideXlab platform.
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On some aspects for contact with rigid surfaces: Surface-to-rigid surface and curves-to-rigid surface algorithms
Computer Methods in Applied Mechanics and Engineering, 2015Co-Authors: Alexander Konyukhov, Karl SchweizerhofAbstract:Abstract Special algorithms allowing a simplified description of contact between deformable body and rigid surfaces are developed based on the geometrically exact covariant description of contact. A special attention is given to various geometric combinations where the contact can be represented as (a) contact between surfaces and (b) contact between a curve and a surface. For contact between surfaces, leading to the Segment-To-Analytical Surface (STAS) approach, two algorithms can be distinguished based on the selection of a coordinate system for the Closest Point Projection (CPP) procedure: (a) Rigid Surface is a “Slave” surface and (b) Rigid Surface is a “Master” surface. A special combination of both contact kinematics for the surface-to-surface and for the curve-to-curve approaches is employed for the contact between a curve and a surface leading to the Curve-To-Rigid Surface (CTRS) approach. The last algorithm is verified with the well known Euler Formula for the rope–cylinder interaction as well as with a new derived generalization into a 3D spiral rope on a cylinder. The developed algorithms can be straightforwardly implemented within an iso-geometric approach as well as within the conventional finite elements where rigid surfaces are given by CAD patches. Any type of elements can be employed for the contacting deformable surface/curve because the algorithms are Formulated in a covariant form.
P. Fillmore - One of the best experts on this subject based on the ideXlab platform.
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WHITNEY FORMS FOR SPHERICAL TRIANGLES I: THE Euler, CAGNOLI, AND TUYNMAN AREA FormulaS, BARYCENTRIC COORDINATES, AND CONSTRUCTION WITH THE EXTERIOR CALCULUS
2014Co-Authors: David W. Fillmore, P. FillmoreAbstract:Abstract. We establish the equivalence of the Tuynman midpoint area Formula for a spherical triangle to the classical area Formulas of Euler and of Cagnoli. The derivation also yields a variant of the Cagnoli Formula in terms of the medial triangle. We introduce the three barycentric coordinates of a point within the spherical triangle as area ratios which sum to unity. The barycentric co-ordinates are the Whitney 0-forms, scalar functions over the domain of the triangle associated with each vertex. We then construct, by exterior differentiation of the barycentric coordinates, succinct expressions for the Whitney 1-forms associated with each geodesic side, or great circle arc. The Euler Formula, in conjunction with that of Tuynman, facilitates the differentiation of a triangular area with respect to the position of a vertex. As both the Euler and Tuynman Formulas may be expressed naturally in terms of the position vectors of the vertices on an embedded sphere, the Whitney constructions may be done in terms of vector-valued forms and without recourse to a particular projection or coordinate chart. Finally, we exhibit the Whitney 2-form of the triangle, which must be the product of a scalar function and the area 2-form of the sphere. We find an expression for this scalar function in terms of determinants of 3 × 3 matrices built from the vertex position vectors. It is a rational function in the Cartesian coordinates of a point. By construction it must be invariant under cyclic permutations of the three vertices, though it is not manifestly so. Also by construction the Whitney 2-form must integrate to one over the triangle. We speculate that these exactly integrable rational functions associated with spherical triangles may be relevant to the development of numerical quadrature schemes on the sphere, as well as be of inherent interest in their own right, and that the spherical Whitney forms may have potential application in the Formulation of discrete exterior calculus and finite element spaces intrinsic to the sphere. 1 a