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On Timoshenko-like modeling of initially curved and twisted composite beams. (English) Zbl 1087.74581

Summary: A generalized, finite-element-based, cross-sectional analysis for nonhomogeneous, initially curved and twisted, anistropic beams is formulated from geometrically nonlinear, three-dimensional (3-D) elasticity. The 3-D strain field is formulated based on the concept of decomposition of the rotation tensor and is given in terms of one-dimensional (1-D) generalized strains and a 3-D warping displacement that is obtained from the formulation, not assumed. The warping is found in terms of the 1-D strains via the variational asymptotic method (VAM). In this paper a Timoshenko-like model is presupposed for a beam with cross-sectional characteristic length \(h\), wavelength of deformation given by \(l\), and the magnitude of the radius of initial curvature and/or twist is taken to be of the order \(R\). First, a solution for the asymptotically correct refinement of classical anisotropic beam theory for initially curved and twisted beams through \(O(h^2/R^2)\) is obtained. Next, the \(O(h^2/l^2)\) correction is computed. It is known that Timoshenko-like theory is not capable of capturing all the \(O(h^2/l^2)\) corrections for generally anisotropic beams. However, if all the \(O(h^2/l^2)\) terms are known, then the corresponding Timoshenko-like theory is uniquely defined. Numerical results are presented to illustrate the trends of the various classical (extension-twist, bending-twist, and extension-bending) and nonclassical couplings (extension-shear, bending-shear, and shear-torsion) as the initial twist and curvatures are varied.

MSC:

74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74G10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of equilibrium problems in solid mechanics
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