Can the general Flexural formula be used in the case of transverse loading of a beam?

Question

The general derivation of the Flexural formula uses the pure bending case, where the distance from a curved section to the neutral axis is assumed to be constant even after bending, denoted by y. The Flexural formula, as we all know, is used to determine the bending stresses at a distance y from the neutral axis.

Now my question is that can we use this general Flexural formula to find the bending stresses at a distance y from the neutral axis, when a shear/transverse load is applied to it? Because I was thinking that the distance y from the neutral axis of that section (whose bending stresses I am trying to compute) will not remain the same after bending due to this transverse loading. So why do we still use it? [Because literally the derivation of the Flexural formula assumes that the distance from the neutral section to any of the section at distance y remains constant after bending in pure bending case].

Answer 1

External shear stress parallel to neutral axis of a beam is known as eccentric column loading and can be handled several ways.

Some cases are delt with as P delta bending stress, some use secant formula.

But internal shear stress parralel to the beam axis is countered by shear flow and Mohr circle stresses.

Answer 2

If you meant the generalized flexural formula:

• $\sigma_x = \dfrac{(M_y*I_z + M_y*I_{yz})z - (M_z*I_y + M_y*I_{yz})y}{I_y*I_z - I{yz}^2}$

then, "YES", the formula is applicable to all shapes of beams that follow Hook's Law, with the shape that is doubly symmetrical about its geometric axes (the neutral axes are coincident with the geometric axes) as a special case, for which the product of the moment of inertia $I_{yz} = 0$, and the formula is reduced to:

• $\sigma_x = \dfrac{M_yz}{I_y} - \dfrac{M_zy}{I_z}$

Answer 3

Even derivation in pure bending assumes small deformations. If you do not use this assumption, the transverse distances from neutral axis change also in pure bending. So if you take into account transverse shear assuming small deformations, the distances are assumed to be constant.