20.4 : الإجهاد الثني

Consider a bending moment where a symmetric member endures stress within the elastic limit. The longitudinal stress can be expressed using Hooke's law.

Recall the expression for the longitudinal strain in terms of maximum strain at a distance 'c' from the neutral surface.

Multiplying it by the modulus of elasticity and substituting it in the stress equation shows that the normal stress varies linearly with the distance from the neutral surface.

Now, recall the expressions for the sum of force components and moments. Replacing the stress in the force equation indicates that within elastic limits, the neutral axis passes through the centroid of the section.

Substituting for stress in the moment equation and simplifying it yields an expression containing an integral equal to the moment of inertia of the cross-section with respect to the centroidal axis perpendicular to the couple's plane.

The final simplified expression is the elastic flexure formula for maximum stress. For an arbitrary distance 'y' from the neutral surface, this formula gives the flexural stress caused by the bending of the member.