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.