20.18 : 曲面部材の曲げ: 中立面

For a curved member with a uniform cross-section, the strain equation along the section shows that it varies non-linearly with the distance from the neutral axis.

Applying Hooke's law gives the stress produced in the curved member. Similar to strain, stress also varies non-linearly, and the plot of stress versus the distance from the neutral axis results in a hyperbola.

Here, all the elementary forces acting on a section are statically equivalent to the bending couple, and its sum over the transverse z-axis gives the equation for the moment.

Substituting the value of stress and simplifying the equations gives the relation that gives the distance from the center of curvature C to the neutral surface. It shows that the neutral surface of the curved member under bending does not pass through the centroid of that section.

Rewriting the expression for the moment in terms of the stress and performing the integration shows that the neutral surface for a curved member is always located between the centroid and the radius of curvature, regardless of its shape.