20.2:
Symmetric Member in Bending
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.
20.2:
Symmetric Member in Bending
In the study of the mechanics of materials, analyzing the behavior of prismatic members under opposing couples is crucial for understanding internal stress distributions, which are essential for structural design. When subjected to couples, a prismatic member experiences internal forces that maintain equilibrium. A couple, characterized by two equal and opposite forces, creates a moment but no resultant force. The internal forces at any section cut of the member must balance these external couples and resolve them into normal and shear stress components.
Normal stresses, acting perpendicular to the cross-sectional area, result from axial forces due to the bending moment caused by the couple. The normal stresses are equal in magnitude but opposite in direction. Shear stress, tangential to the cross-sectional area, maintains translational equilibrium. By selecting appropriate axes, typically the principal axes of the cross-section, the moments due to internal stresses are equal to the moment of the external couples. The bending moment is countered by an equivalent moment from the normal stresses, where the distance from the neutral axis to the area of the cross-section is taken into account.
The sign convention indicates that positive normal stress, or tension, contributes negatively to the moment about the z-axis, where counter-clockwise moments are positive. Understanding these stress distributions is vital for predicting failure modes and optimizing material distribution, forming a cornerstone of structural engineering.