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We introduce the equations governing the equilibrium of slender elastic structures, i.e. quasi-one-dimensional elastic bodies whose extent in one direction is much larger than in the two perpendicular (cross-sectional) directions. This chapter could serve as an introduction to the elasticity of deformable bodies. We start by deriving one-dimensional models governing slender structures in two steps. First, we analyze a discrete truss network in two dimensions, i.e. a collection of linear springs connected by perfect hinges. Then, we take the continuous limit to derive the energy of an elastic rod. Also, we use the calculus of variations to derive the equilibrium equations for an elastica in two dimensions. We then focus on the linear response of the elastica relevant to small applied forces, which is the classical linear beam model. Finally, we extend the elastica model to three dimensions, and illustrate the interplay of the bending and twisting modes with an analysis of helical buckling.

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