Learning topic
Unsymmetrical Bending
Calculate normal stress in unsymmetrical bending by resolving the bending moment about principal axes, then locate the neutral axis and critical section points.
Unsymmetrical bending occurs when the plane of the resultant bending moment does not coincide with a principal centroidal plane of the cross-section. The moment is then resolved into two components about the principal axes, and the normal stress is obtained by superposing two simple-bending stress fields.
Principal axes as the calculation system
The most convenient coordinate system uses the principal centroidal axes $x$ and $y$, for which $I_{xy}=0$. Because $I_x$ and $I_y$ may differ substantially, the neutral axis is generally not perpendicular to the resultant bending-moment vector.
If the centroidal axes $x$ and $y$ are principal axes of the cross-section and the bending moment has components $M_x$ and $M_y$, the normal stress is obtained by superposition:
$$\sigma(x,y)=-\frac{M_x}{I_x}y+\frac{M_y}{I_y}x.$$
The neutral-axis equation follows from $\sigma=0$:
$$-\frac{M_x}{I_x}y+\frac{M_y}{I_y}x=0.$$
- $M_x$, $M_y$ — bending-moment components about the principal centroidal axes;
- $I_x$, $I_y$ — principal centroidal second moments of area;
- $x$, $y$ — coordinates of the point in the cross-section.
The signs depend on the adopted axis and moment conventions, which must be used consistently.
Neutral axis
The neutral axis passes through the centroid because, in pure unsymmetrical bending, the resultant normal stress is zero along this line. Its position follows from $\sigma(x,y)=0$.
If $M_x\ne0$ and $I_x$ and $I_y$ are known, the equation can be written as:
$$y=\frac{M_y I_x}{M_x I_y}x,$$
with the signs of the moment components retained consistently.
Finding critical points
The largest absolute stress is not necessarily located at the point with the largest $|x|$ or $|y|$ alone. The critical points are those boundary points where the algebraic sum of the two bending contributions reaches its greatest positive or negative value. For polygonal sections, checking characteristic vertices is often convenient.
Example
For a rectangular section with known $I_x$ and $I_y$ under $M_x=4\ \text{kN·m}$ and $M_y=2\ \text{kN·m}$, evaluate the stress at each corner by substituting its coordinates into the unsymmetrical-bending formula. The largest positive value is checked against the allowable tensile stress, and the largest-magnitude negative value against the allowable compressive stress if these limits differ.
Calculation procedure
- Locate the centroid and principal centroidal axes.
- Resolve the bending moment into $M_x$ and $M_y$.
- Determine $I_x$ and $I_y$.
- Write the neutral-axis equation.
- Calculate $\sigma$ at characteristic extreme boundary points.
- Perform the strength check.
Limits
The formulas assume elastic bending, small deformation, and applicability of classical beam theory. For nonprincipal axes with $I_{xy}\ne0$, independently adding terms of the form $M_x/I_x$ and $M_y/I_y$ without first transforming to principal axes is generally incorrect.
About this topic
Unsymmetrical bending occurs when the applied bending moment plane does not coincide with any principal axes of inertia. This page details moment vector decomposition along principal axes, combined normal stress formulas, neutral axis orientation equations, and strength verification procedures for extreme cross-sectional points.