Beams in plane bending

We derive one-dimensional beam theories from the three-dimensional theory of linear elasticity by a power-law expansion of the displacements in height and width direction. The strain energy and the potential of external forces are calculated and integrated over the cross-sectional area. Both appear as power laws of the small beam parameters c ²=h ²/(12ℓ ²) and d ²=b²/(12ℓ ²), where ℓ is the characteristic dimension in length direction of the beam and h and b are height and width of the rectangular cross-section, respectively. Hierarchical beam theories arise from the consistent truncation of the elastic energy after a specific power 2N=2n + 2m of the fast decaying factors c²ⁿd^2m . It turns out that the first-order (N=1) approximation delivers the classical Euler-Bernoulli beam theory whereas the second-order approximation (N=2) leads to a Timoshenko-type theory. A special feature of the derivation is that no a priori assumptions are invoked.