Bending-9 - Bending of single-material beams(Chapter 5 of...

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Bending of single-material beams (Chapter 5 of Gere) KML 7/17/09 Pure bending: occurs when a beam is loaded only by bending moments (with no shear forces). Curvature, κ : is a measure of a how much a beam is bent under load. Radius of curvature, ρ , is the radius of an arc that matches the curvature of a beam. dx d ds d 1 θ 2245 θ = ρ = κ dx is an approximate arclength for small curvatures Curvature sign convention . Curvature is positive if the beam bows up like 2 x y = , and negative if the beam bows down like 2 x y - = . Neutral surface: the plane at which the length remains unchanged in a beam in bending Neutral axis (NA): the intersection of the neutral surface with cross-sectional planes x y neutral axis: length = dx before and after bending ρ M M θ d beam curvature is exaggerated Derivation of the relationship between curvature and strain for uniform elastic beams: S = R θ dx = ρ d θ general arclength: The length of any plane of the bowed beam at radius: y - ρ , is given as the arclength: ρ 2245 θ - ρ = dx y - dx d y) ( L 1 (where ρ θ / dx d since dx / d / 1 θ ρ ) The normal strain is: (the ratio of the change in length to the length: [(L 1 – dx)/dx]) y y x κ - = ρ - = ε
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This note was uploaded on 10/01/2009 for the course MEGR 2144 taught by Professor Sharpe during the Fall '08 term at UNC Charlotte.

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Bending-9 - Bending of single-material beams(Chapter 5 of...

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