The effect of warping on the dynamics of thin-walled beams

Thin-walled beams are of widespread application: they have high bending stiffness in at least a principal plane of inertia, but negligible torsion stiffness. Saint-Venant theory is not suitable to describe their behaviour, thus suitable models are needed. To ease computation, one-dimensional continua are interesting, and can be derived by richer models: in [1] the field equations derive from inner constraints on shells, like in [2], where nonlinear constitutive relations are provided. 1-D direct models explicitly lead to ordinary differential equations, easier to integrate. In [3] there are as many directors attached to the beam axis as stripes in the cross-section; in [4] kinematics is enriched by describing warping via the Saint-Venant warping function. In [5] the standard beam model is enriched by a coarse descriptor of warping and nonlinear constitutive relations. The model is refined in [6] to distinguish between centroid and shear centre; in [7] the suitable form of inner shearing constraints and of warping inertia is investigated. The model can so catch the buckling and post-buckling of beams and frames in 3-D space. Here we use the model to investigate the effect of warping and warping constraints on the linear dynamics of open thin-walled beams for pattern schemes. We will present results by means of numerical techniques and laboratory experiments, and compare them with those in [8]. References [1] Vlasov VZ, Thin-walled elastic beams, Monson, Jerusalem, 1961 [2] Møllmann H: Theory of thin-walled beams with finite displacements, EUROMECH Colloquium 197, W. Pietraszkiewicz ed., Springer-Verlag, New York, 195-209, 1986 [3] Epstein M: Thin-walled beams as directed curves, Acta Mech. 33:229-242, 1979 [4] Simo JC, Vu-Quoc L: A geometrically exact rod model incorporating shear and torsion-warping deformation, Int. J. Solids Struct. 27:371–393, 1991 [5] Rizzi N, Tatone A: Nonstandard models for thin-walled beams with a view to applications, ASME J. Appl. Mech. 63:399-403, 1996 [6] Ruta G, Pignataro M, Rizzi N: A direct one-dimensional beam model for the flexural-torsional buckling of thin-walled beams, J. Mech. Mat. Struct. 1:1479-1496, 2006 [7] Brunetti M, Paolone A, Ruta G: On inner shearing constraints for a direct beam model coarsely describing warping, Meccanica, doi: 10.1007/s11012-013-9759-y (2013) [8] Ambrosini, D., “Experimental validation of free vibrations from nonsymmetrical thin walled beams”, Eng. Struct., 32, 1324–1332 (2010)