I am reading about pure bending from Beer Johnston's mechanics of materials. There, a prismatic member with a plane of symmetry was subjected to equal and opposite couples M and M' acting in that plane. It was said about this member that since the bending moment M is same in any cross section, the member will bend uniformly. I could not understand this.
Is it because that stress will be same on each cross section and therefore since same stresses cause same deformation, it will bend uniformly?
Since the bending moment is constant across any section, by the bending equation, the beam will deform as a pure circular arc, i.e. with constant radius of curvature across any small element length, in other words, uniform deformation. This compares directly to the case when "M" varies along the length, then the radius of curvature, in turn the deformation would have been continuously varying.
Radius of a beam curvature at any cross section is.
$$ \frac{1}{\rho}= \frac{d\phi}{d_x}=\frac{d^2w}{d_x^2}= - \frac{M}{EI} $$
$$ \therefore \ \text{for a constant moment we have a constant radius.} $$
Basically in any interval of the span of a beam wher there is only constant moment, the beam bends into a section of a circle.
