20.15:
Unsymmetric Bending – Angle of Neutral Axis
Consider a member with a vertical plane of symmetry subjected to bending couples in a plane at an angle θ with respect to the vertical axis of the member, causing an unsymmetric bending.
Resolving one of the bending couple vectors along the principle centroidal axes of the member and writing the expression for the stress due to each component, the distribution of the stresses due to the bending couple is calculated using the superposition method.
The distribution of the stress due to bending couples is linear. The magnitude of the stress will be zero for the neutral axis of the section.
Solving the equation for the neutral axis and rewriting the components of bending vectors in terms of θ gives the equation of a straight line.
Here, the slope of the equation gives the angle ϕ of the neutral axis with respect to the z-axis.
Here, the angle ϕ will be greater than the angle θ when the moment of inertia along the z-axis is greater than the moment of inertia along the y-axis.
20.15:
Unsymmetric Bending – Angle of Neutral Axis
Unsymmetrical bending occurs when a structural member is subjected to bending moments in a plane that does not align with the member's principal axes. This scenario typically arises in beams and other structural components when loads are applied at non-ideal angles, introducing complexities in stress analysis.
When a bending moment is applied at an angle θ concerning the vertical axis of a symmetrical member, it can be resolved into components along the member's principal centroidal axes. The stress distribution resulting from each component can be separately calculated and then combined using the superposition principle, discussed in a previous lesson. The stresses are distributed linearly across the member, with the maximum and minimum stresses occurring at the furthest points from the neutral axis, where the stress equals zero.
The neutral axis is where the bending stress is zero. It follows a straight line whose orientation can be determined by the relationship between the angle of the applied load and the member's moments of inertia about its axes. The angle ϕ, which the neutral axis makes with the vertical axis, depends on these moments of inertia. If the moment of inertia along the vertical axis is greater than along the horizontal axis, ϕ will be greater than θ, indicating that the neutral axis rotates in proportion to the member's anisotropic inertial properties.