Derive the Flexural Formula (Bending Equation).

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In order to compute the value of bending stresses developed in a loaded beam, let us consider the two cross-sections of a beam HE and GF, originally parallel as shown in fig 1(a).when the beam is to bend it is assumed that these sections remain parallel i.e. H'E' and G'F' , the final position of the sections, are still straight lines, they then subtend some angle.
Consider now fiber AB in the material, at a distance y from the N.A, when the beam bends this will stretch to A'B'
Since CD and C'D' are on the neutral axis and it is assumed that the Stress on the neutral axis zero. Therefore, there won't be any strain on the neutral axis.

part 4 of 7 part 5 of 7

Consider any arbitrary a cross-section of beam, as shown above now the strain on a fibre at a distance ‘y' from the N.A, is given by the expression.
This equation is known as the Bending Theory Equation. The above proof has involved the assumption of pure bending without any shear force being present. Therefore this termed as the pure bending equation. This equation gives distribution of stresses which are normal to cross-section i.e. in x-direction.

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