Christian and I thought comparing the theory of bent beams to that of straight beams would be interesting because we only explored straight beams this semester in class. Bent beams are important since they are encountered regularly in practice, for example a hook. The geometry of a bent beam changes the equations governing the behavior. So, understanding how the geometry changes the beams behavior is our primary interest.
Student Harvard University
Student
Harvard University
Moment-Curvature Relation for Arches
Assuming that the beam is not loaded out of plane, the Moment-Curvature Relation for Arches may apply. It is based on Euler-Bernoulli Beam Theory (e = (Lf / Li) - 1 & s = E e = M y / I).
M = E I Dk / (1 + ki y)
M = Bending Moment about the neutral axis (non-negative)
E = Modulus of Elasticity
I = Moment of Inertia of the profile section about the neutral axis
Dk = Change in the Curvature (non-negative)
ki = Curvature of the Undeformed Arch (non-negative)
y = Perpendicular Distance from the centroidal axis (outward normal direction is positive)
Please verify if this is correct.