Mechanics of Slender Structure: Loaded Beam Approximations

This lecture covers the matched approximations of a loaded beam, focusing on strain approximation and constitutive models. The instructor discusses the conditions for small rotations and linear extensibility, as well as the exact non-linear strain. The lecture emphasizes the importance of establishing equilibrium using the principle of virtual works and Lagrange multipliers. The instructor also delves into the inextensibility condition and the variations of strain, leading to the derivation of equilibrium equations. The lecture concludes with the consideration of different regimes based on the imposed strain and the constitutive law, highlighting the significance of geometric approximations and the assumption of inextensibility.