20.3 : 对称构件在弯曲时的变形

Consider the deformations of a symmetric prismatic member subjected to equal and opposite couples.

As the member bends uniformly, the lines on the wider faces of the members transform into a circle of constant curvature centered at point C.

By dividing the member into tiny cubic elements, it becomes apparent that the only non-zero stress component is the normal one, leading to uniaxial stress at any given point.

So, a neutral surface exists parallel to the member's upper and lower faces where longitudinal components of strain and stress are zero.

The deformation of an arc, located at a distance y above the neutral surface, can be expressed as the difference in the length of this arc and that of the neutral surface.

Expressing these lengths in terms of the radius and the angle subtended and dividing the deformation by the length of the neutral arc shows that the longitudinal normal strain varies linearly with the distance y from the neutral surface.

Assuming a positive bending, the negative sign indicates the beam's upward concavity.