Consider a prismatic beam. When equal and opposite moments M and M prime are applied, as shown, the ends of the beam bend upwards. Fibers at a certain position, called the neutral axis, do not change in length. While fibers above it shorten, and fibers below it, elongate. For linearly elastic beams, this neutral axis passes through the centroid of the cross-section. And when the beam is bent, forms an arc of radius rho. We describe the position of a fiber by its distance y from the neutral axis. The amount of stretching or shortening in a fiber depends on its y value. Y coordinates above the neutral axis are positive and below it negative. Consider a segment of beam that is ds long. As the beam is bent to a radius rho, a fiber that is a distance y above the neutral axis has a new length ds star and a radius of curvature rho minus y. Two similar isosceles triangles result, and we can write rho over ds equals rho minus y over ds star. Rearranging gives ds star equals rho minus y over rho times ds. The strain epsilon is equal to the change in fiber length, or the new length minus the original length, divided by the original length. Thus we have epsilon equals ds star minus ds over ds. We can substitute ds star from the previous equation. Divide through by ds, simplify, expand the first term on the right side into two fractions, simplify again, cancel the two ones with each other, and we are left with epsilon equals minus y over rho. This linear relationship between epsilon and y can be illustrated graphically. Before the beam is bent, rho is infinite, and the strain is zero everywhere across the depth of the beam. As the beam is bent, rho decreases, and the strain in each fiber increases. In all cases, the strain varies linearly with y. Fibres above the neutral axis are shortened and experience a negative strain, while those below are elongated and experience a positive strain.