General Thin Shell Kinematics

To be able to represent arbitrary shapes, the orthogonal curvilinear coordinates Si and S2 on the middle surface are introduced together with the associated Lame parameters 2li and Sla, see for example Dym [69] or Novozhilov [134], The undeformed middle surface is characterized by the respective principal radii R and i 2 in agreement with the following condition  

Remark 6.4. Besides being thin with regard to the overall dimensions, the considered shell-like structure is presumed to have a thickness substantially smaller than its smallest radius of curvature. 

A variety of theories for thin shells are available. For the anticipated applications, it is reasonable to confine the examination to linear theories  

The well-known and generally accepted thin shell theory of Sanders [158] and Koiter [114] eliminated the deficiency of the preceding developments of 

The bending curvatures ki, K2 and the twisting curvature V 12 of the middle surface are given with respect to the rotations and t 2 by 

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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. 

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