What is unsymmetrical / biaxial bending?
Unsymmetrical (skew) bending occurs when the bending moment does not act along a principal axis of inertia. For an unsymmetric cross-section, Ixy can be nonzero, coupling both bending directions. The neutral axis is therefore not simply one of the x/y axes in the general case, but a tilted line through the centroid.
Simplified formula: only when Ixy = 0
If the x/y axes are already principal axes of inertia (Ixy = 0), the stress can be found by simply superimposing two single-axis bending cases:
σ = -Mx·y’/Ix + My·x’/IyThis formula applies strictly only when Ixy = 0. For a general unsymmetric cross-section it would be incorrect.
General formula for unsymmetrical bending
In the general case (also valid for Ixy != 0), the inertia-tensor approach gives:
D = IxIy - Ixy²σ = -(IyMx+IxyMy)/D · y’ + (IxyMx+IxMy)/D · x’with x' = x - x̄ and y' = y - ȳ. For Ixy = 0 this equation reduces exactly to the simplified formula above.
Why evaluate at the vertices?
Since sigma is a linear (affine) function of x and y, it has no interior extrema over the cross-section area. The maximum tensile and compressive stress therefore always occur at a vertex of the polygon contour -- even for concave cross-sections.
Example: 100 × 50 mm rectangle with Mx and My
For the axis-aligned rectangle, Ixy = 0, so the simplified formula applies. At Mx = 1,000 N·m and My = 500 N·m, the vertex (100, 0) carries a tensile stress of sigma = -1,000,000*(-25)/1,041,666.667 + 500,000*50/4,166,666.667 = 24 + 6 = 30 N/mm^2, the plain superposition of the two individual bending stresses.
Scope and limitations
The calculator supports one closed, non-self-intersecting outer contour and one polygonal inner contour under pure bending without axial force. Disconnected regions, arcs, DXF, shear stresses and plastic cross-section utilization are currently excluded.