20.14:
Unsymmetric Bending
Unsymmetric bending occurs when the bending moment is not acting along the plane of the symmetry of the member or if the member does not possess any plane of symmetry.
An understanding of unsymmetric bending requires studying the conditions under which the neutral axis of a cross-section of an arbitrary shape coincides with the axis of the couple.
Consider an arbitrary shape, the neutral axis and the couple axis are along the z-axis. Here, the force along the x-axis and couple moments along the y and z-axis can be expressed in terms of stress and the cross-sectional area.
Within the proportional limit, the stress developed within the member can be expressed in terms of the maximum stress to calculate the y-component of the couple moment.
The integral is the product of inertia along the y and z-axis and will be zero if these are centroidal axes.
So, the neutral axis of the cross-section will coincide with the couple axis if and only if the couple axis is along one of the centroidal axes of the member.
20.14:
Unsymmetric Bending
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 those in symmetrical bending, and are essential for designing structures to withstand different loading conditions. In unsymmetrical bending, the neutral axis—where stress is zero—does not necessarily align with the geometric axes of the cross-section. The orientation of the neutral axis 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.