Thin-Walled Beam Kinematics

So far, the shape of the cross-section of the considered beams has not been discussed, while the Green Lagrange strain tensor has been brought up for a continuum only confined with respect to the deformations in the cross-sectional plane by Remark 7.4. Subsequently, a special class of cross-sectional topologies will be examined  

Remark 7.6. The beam is constructed from walls which are thin in comparison to the cross-sectional dimensions. 

Under these conditions, it is possible to analjdically handle beams with complex cross-sections, since the formation of the one-dimensional structure from two-dimensional walls instead of a three-dimensional continuum allows us to introduce the associated simplifications. Therefore, the relation between the strain measures of the beam and of the thin wall is sought. To describe the shape of such a cross-section, the curvilinear coordinate s may be utilized as a parameter for the cross-sectional position 

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Based upon the adaptive shell description given in the previous chapter, a thin-walled beam formulation for general anisotropic cross-sections with arbitrary open branches and/or closed cells will be derived in this chapter. After the deduction of non-linear kinematic relations for the general beam and the linear kinematic relations for the thin-walled beam, the torsional warping effects of the latter are examined. Subsequently, the constitutive relation and the equations of equilibrium are established. 

With regard to the dynamics of rotating structures on the one hand, nonlinear influences in the beam kinematics have to be taken into account, while analytical formulation of the constitutive relation of beams with complicated cross-sections on the other hand, is only possible for thin walls and linear kinematics. This gives rise to a combined procedure with a linear analysis to determine the beam properties and a succeeding non-linear analysis to investigate the global beam behavior. For the latter, a general beam with adequate kinematics will be examined first and subsequently transcribed into the intended thin-walled beams. 

To determine the equations of equilibrium as well as the constitutive relations of the beam, the principle of virtual work may be applied and its individual contributions be examined, respectively. Thus, the foundations for an analytic solution with regard to the statics of the non-rotating structure can be provided. Furthermore, the principle of virtual work will serve to set up the equations of motion in consideration of the dynamics of the rotating structure. This, in addition, requires the study of inertia effects and the inclusion of stiffening effects due to kinematic non-linearity with reference to relatively slender and flexible beams. The derivation of the principle of virtual work for the general case is presented in Section 3.4, and it will now be adapted and extended to depict adaptive thin-walled beams. Therefore, the various virtual work contributions will be discussed individually. 

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