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The Nonlinear Theory of Curved Beams and Flexurally Stiff Cables

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Nonlinear Structural Mechanics

Abstract

This chapter discusses the nonlinear theory of curved beams and ring structures used in a wide range of engineering applications including aircraft fuselages, arch bridges, roof structures, turbomachinery blades, and water/oil/gas tanks. The theory is derived starting from planar motions proceeding toward the three-dimensional setting within which more complex motions can occur. Important aspects of the stability of arches and deeply buckled beams are discussed in the more general context of prestressed curved beams. The dynamical formulation of cables suffering axis stretching and flexural curvature is presented within the geometrically exact framework of prestressed compact curved rods. This refined theory of cables is useful to study the states of stress in boundary layers such as those arising near anchorage devices of cable stays or suspension cables with a view to the assessment of their fatigue life or damage detection techniques.

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Notes

  1. 1.

    With the choice of \({\mathcal{C}}^{\text{ o}}\) coincident with the centerline and (\({\mathbf{\mathit{b}}{}_{2}}^{\text{ o}},\,{\mathbf{\mathit{b}}}_{3}^{\text{ o}}\)) collinear with the principal axes of inertia of the cross section, the intrinsic reference frame \(({C}^{\text{ o}},{\mathbf{\mathit{b}}{}_{1}}^{\text{ o}},{\mathbf{\mathit{b}}{}_{2}}^{\text{ o}},\,{\mathbf{\mathit{b}}}_{3}^{\text{ o}})\) represents the principal inertia reference frame of the cross section with origin in the center of mass \({C}^{\text{ o}}\) of \({\mathcal{S}}^{\text{ o}}.\)

  2. 2.

    \({\mathcal{B}}^{{_\ast}}\) may be conceived as the rectified version of the stress-free configuration \({\mathcal{B}}^{\text{ o}}.\) A way to obtain a prescribed curved configuration without internal stresses is to take a formwork with the profile of \({\mathcal{B}}^{\text{ o}}\), pour material in its fluid state (steel, metallic alloy, concrete, etc.) and let it solidify. Another way is to construct segments of the curved beam and join them by suitable connections to form the final arched structure.

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  1. Sapienza University of Rome, Rome, Italy

    Walter Lacarbonara

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Lacarbonara, W. (2013). The Nonlinear Theory of Curved Beams and Flexurally Stiff Cables. In: Nonlinear Structural Mechanics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-1276-3_7

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  • DOI: https://doi.org/10.1007/978-1-4419-1276-3_7

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