20.2 : 弯曲中的对称杆件

Consider a prismatic member subjected to two couples, equal in magnitude but in opposite directions, acting within the symmetric plane of the member.

A section is cut through an arbitrary point on the member. The equilibrium conditions for each part dictate that the internal forces at the section must be equivalent to the couple.

The internal forces acting on the section consist of the normal stress at a point of the cross-section and the shearing stress components.

Recall that a couple consists of two equal and opposite forces whose sum in any direction is zero. Furthermore, the couple moment is the same about any axis perpendicular to its plane but is zero about any axis in the plane.

By choosing the axes arbitrarily, the sums of the components and the moments of the forces equal the corresponding components and moments of the couple.

Here, the negative sign indicates that tensile stress contributes to a negative moment of the normal force about the z-axis.