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Mathematical Models of Beams and Cables von Luongo, Angelo (eBook)

  • Erscheinungsdatum: 02.12.2013
  • Verlag: Wiley-ISTE
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Mathematical Models of Beams and Cables

Nonlinear models of elastic and visco-elastic onedimensional continuous structures (beams and cables) are formulated by the authors of this title. Several models of increasing complexity are presented: straight/curved, planar/non-planar, extensible/inextensible, shearable/unshearable, warpingunsensitive/ sensitive, prestressed/unprestressed beams, both in statics and dynamics. Typical engineering problems are solved via perturbation and/or numerical approaches, such as bifurcation and stability under potential and/or tangential loads, parametric excitation, nonlinear dynamics and aeroelasticity. Contents 1. A One-Dimensional Beam Metamodel. 2. Straight Beams. 3. Curved Beams. 4. Internally Constrained Beams. 5. Flexible Cables. 6. Stiff Cables. 7. Locally-Deformable Thin-Walled Beams. 8. Distortion-Constrained Thin-Walled Beams.


    Format: ePUB
    Kopierschutz: AdobeDRM
    Seitenzahl: 480
    Erscheinungsdatum: 02.12.2013
    Sprache: Englisch
    ISBN: 9781118577639
    Verlag: Wiley-ISTE
    Serie: ISTE
    Größe: 4450 kBytes
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Mathematical Models of Beams and Cables


Here we summarize the main concepts to be discussed later and illustrate the guiding factor of this book. Firstly, the modeling problem for a beam is addressed by comparing two different philosophies: derivation from a three-dimensional (3D) Cauchy continuum, or direct formulation as a one-dimensional (1D) polar continuum. Secondly, string and cables are successively considered as degenerate models of perfectly flexible beams, and the circumstances in which flexural and torsional stiffnesses have to be considered are discussed. Thirdly, more sophisticated models of beams with deformable cross-sections are addressed. Finally, a quick overview of the literature and of this book is given.
I.1 Derived one-dimensional models

A beam is a slender solid, spanned by a planar figure (the cross-section ) which moves along a smooth ( C 1 class) curve S (the beam axis or centerline), by remaining orthogonal to it and keeping its centroid G on it. If S is a planar curve, the beam is called planar , otherwise it is spatial ; if S is a segment, then the body is a cylinder, and the beam is called straight . The length l of S is called the beam length. Slenderness, in a broad sense, refers to the fact that a characteristic linear dimension r of the cross-section is much less than the length l (typically l/r = O(102)). This property plays a fundamental role in deriving mechanical models of beams, as discussed further.

As is well-known, the Fundamental Problem of Continuum Mechanics , formulated in the context of the Lagrangian description, consists of evaluating stresses, strains and displacements in a body , when this is loaded by assigned volume and surface forces, and, moreover, displacements are prescribed on a portion of the boundary. When this problem is addressed for a beam, a 3D Cauchy continuum model could be applied, which would lead to a system of partial differential equations, in which each (scalar, vector or tensor) magnitude is a function of three coordinates (time understood), which span the volume occupied by the body in the reference configuration. Such an approach, however, although "exact" in the context of continuum mechanics, is almost unpractical for the difficulty of solving the governing equations, so that it is advisable to resort to "approximate" models that exploit the geometric peculiarity of the body, namely its slenderness. The main object of the analysis consists of formulating a 1D (rather than a 3D) model in which all the magnitudes involved depend on only one coordinate, e.g. a curvilinear abscissa s running along the (unstretched) curve S .

To achieve this goal, different methods can be followed. We first approach the problem by illustrating how to derive a 1D model from a 3D model. Let us consider the displacement u(r, s ) at a point P belonging to the section at abscissa s , where is the oriented distance of P from the centroid G of . By using a Taylor expansion, it follows that Since , it can be assumed that the remainder of the series is negligible, with an error which, in non-dimensional variables, is of the order of ( r/l )2. As a result, the displacement at any point of the section is expressed as a function of quantities all evaluated at the centroid, namely u(r, s ) = u G ( s ) + ∂ ru G ( s )r with u G ( s ) := u(0, s ) and ∂ ru G ( s ) := ∂ ru(0, s ). Once this displacement field is int

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