20.16:
General Case of Eccentric Axial Loading
Consider a member subjected to equal and opposite eccentric forces. These forces are applied at a horizontal distance of a and a vertical distance of b from the principal centroidal axis of the member.
Each eccentric force is equal to the centric force and two couple moments with moments arms being distance a and b from the principal centroidal axis of the member.
Using the Saint-Venant principle, the equivalent loadings can be used to determine the distribution of the stress at the section of the member. The analysis holds when the section is not close to either end of the member.
The superposition principle determines the stresses due to centric forces and the bending couples, and shows that the stress varies linearly along the section.
Depending on the geometry of the member and the line of action of the eccentric loadings, the total stress may have the same or opposite sign throughout the section. The line along the section for which the stress magnitude is zero is known as the neutral axis of the section.
20.16:
General Case of Eccentric Axial Loading
Unsymmetrical bending occurs when the bending moment applied to a structural member does not align with its principal axis. This misalignment leads to complex stress distributions and deflection patterns that differ from symmetrical bending, which are essential for designing structures to withstand different loading conditions.
Consider a member subjected to equal and opposite forces that are applied along a line that does not coincide with the member's neutral axis. In unsymmetrical bending, as described here, the neutral axis—where stress is zero—does not necessarily align with the geometric axes of the cross-section. This results in moments around multiple axes, which counteract the eccentricity of the forces. The stress distribution depends on the relationship between the applied load and the geometric properties of the section. The position of the neutral axis is determined by ensuring that the sum of the normal stresses across the section equals zero.
The couple moment in unsymmetrical bending refers to the moments caused by forces that do not pass through the centroid of the cross-section. These moments result in bending about multiple axes and are critical in determining the stress distribution across the member.
The proportional limit is the stress level beyond which the material deforms non-linearly, marking the end of elastic behavior. The product of inertia measures the covariance of the coordinates of the area elements of the cross-section relative to the axes. If the axes align with the centroidal axes of the section, simplifying stress calculations, the neutral axis will coincide with these axes, making them principal axes for bending.