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.
