### CURVED BEAMS

Date : 2016-06-23 12:24:21

Introduction
The beam theory can also be applied to curved beams allowing the stress to be determined for shapes including crane hooks and rings.  When the dimensions of the cross section are small compared to the radius of curvature of the longitudinal axis the bending theory can be relatively accurate.  When this is not the case even using the modified Bernoulli-Euler only provides approximate solutions

Symbols

 ε = strain y= distance of surface from neutral surface (m). e = eccentricity (r c - r n)   (m) r n = Radius of neutral axis (m). c c = Distance from centroid axis to inner surface. (m) r c = Radius of centroid (m). c o = Distance from neutral axis to outer surface. (m) r = Radius of axis under consideration (m). dφ= Surface rotation resulting from bending stress I = Moment of Inertia (m4 - more normally cm4) σ = stress (N/m2) Z= section modulus = I/y max(m3 - more normally cm3) E = Young's Modulus = σ /e (N/m2)

Theory

The sketch below shows a curved member subject to a bending moment M.   The neutral axis r n and the centroid r c are not the same.
This is the primary difference between a straight beam and a curved beam. The strain at a radius r = The strain is clearly 0 when r = at the neutral axis and is maximum when r = the outer radius of the beam (r = r o )
Using the relationship of stress/strain = E the normal stress is simply. The location of the neutral axis is obtained from summing the product of the normal stress and the area elements over the whole area and equating to 0  