Does same stress always produce same strain?

I am reading pure bending from Beer Johnston's mechanics of materials.There,a prismatic member having 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 no understand this.Is it because that stress will be same on each cross section and therefore since same stresses cause same defoemation,it will bend uniformly?

• The same stress always causes the same strain in this situation, but note this is not always true. For example a temperature change can cause strain (thermal expansion, i.e. a change of length) without any stress, or if the object is constrained so it can't move, it will cause stress (which may be big enough to crack or break the object) with no strain. Also if the stress is large enough, there may be plastic deformation, where the object does not return to its initial shape when the loads are removed - as a simple demonstration, bend a paper clip! – alephzero Mar 21 '19 at 9:30

2 Answers

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.