20.4:
Flexural Stress
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.
20.4:
Flexural Stress
When analyzing bending in symmetric members, it's crucial to understand how stresses distribute when subjected to bending moments. This stress distribution is effectively described by applying fundamental mechanics and material science principles, particularly Hooke's Law for elastic materials.
Hooke's Law states that within the material's elastic limits, stress is directly proportional to strain. In a member experiencing a bending moment, the strain at any point is relative to its distance from the neutral axis, the central layer that experiences no longitudinal strain. The strain varies linearly from zero at the neutral axis to a maximum at the outermost fibers of the member.
From this, it is determined that the longitudinal stress at any point also varies linearly with distance from the neutral axis. Integrating this linear variation across the cross-sectional area of the member, where stress is zero at the neutral axis, confirms that this axis coincides with the centroid of the cross-section.
This integration process also defines the expression for the bending moment, and also defines the moment of inertia of the section. This calculation further establishes the relationship between the bending moment and the maximum stress at the furthest point from the neutral axis and gives the flexural stress caused by the member's bending.
