Beams, Plates, and Shells

Synonyms

Bernoulli-Euler beam; Kirchhoff plate; Kirchhoff-Love shell; Tensor notation

Definitions

In this article we shortly present the fundamental relations of the classical theories for straight and slender beams as well as for thin plates and shells. In particular, Bernoulli-Euler beams, Kirchhoff plates, and Kirchhoff-Love shells are discussed in a geometrically and physically linear regime. Further, we restrict the content to homogenous beams, plates, and shells, for which the material behavior is assumed isotropic. Our presentation rests on the use of a direct tensor notation, which avoids unnecessarily lengthy equations using specific coordinates but rather enables a compact and concise formulation.

Bernoulli-Euler Beams

We study straight and slender beams of length ℓ with a solid cross section Ω. The beam is homogenous with isotropic linear elastic material behavior. The position vector of a point P of the undeformed beam is \( \underline {R}_3 = \underline {R}(x) + \underline...