Kinematics thin-walled

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. 

The examples we have studied thus far have had rather simple kinematics flow parallel or nearly parallel to a wall and ideal or nearly ideal extension. Thus, we have been able to obtain exact solutions for the flow or to obtain approximate solutions based on the small difference between the actual flow and an ideal case for which an exact solution is available. Even for the case of fiber spinning, where an analytical solution to the thin filament equations cannot be obtained under conditions relevant to industrial practice, we simply need to obtain a numerical solution to a pair of ordinary differential equations, which is a task that can be accomplished using elementary and readily available commercial software. 

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