What makes this calculator different?
The existing bending stress calculator assumes the section modulus and bending moment are already known inputs. This calculator needs no known I or W value: you draw the cross-section contour directly or enter its vertices, and the centroid, second moment of area and outer-fibre distance are determined automatically from the real geometry.
Bending stress formula
For pure bending about the x-axis:
σ(y) = -Mx/Ix · (y - ȳ)
for the y-axis with the opposite sign (this calculator's sign convention, consistent with the shared stress solver used for unsymmetrical bending):
σ(x) = +My/Iy · (x - x̄)
The respective outer-fibre values can equivalently be expressed via the section modulus:
σx,+ = -Mx/Wx,+σx,- = +Mx/Wx,-σy,+ = +My/Wy,+σy,- = -My/Wy,-
Assumptions
- linear-elastic material behaviour
- prismatic cross-section
- pure uniaxial bending
- neutral axis through the area centroid
- no notch stresses
- no plastic cross-section utilization
- no local buckling
- no shear stresses
- no biaxial bending
Example: 100 × 50 mm rectangle, bending about x
Contour P1=(0,0), P2=(100,0), P3=(100,50), P4=(0,50), Ix = 1,041,666.667 mm⁴, cy+ = cy- = 25 mm, so Wx = 41,666.667 mm³. At M = 1,000 N·m = 1,000,000 N·mm, σ = 1,000,000 / 41,666.667 ≈ 24 N/mm² on the tension side and the same magnitude on the compression side, since the rectangle is symmetric.
Scope and limitations
The calculator supports one closed, non-self-intersecting outer contour and one polygonal inner contour under pure uniaxial bending. Disconnected regions, arcs, DXF, shear stresses and biaxial bending are currently excluded.
FAQCommon questions
How do you calculate the bending stress?
For pure bending about a centroidal axis, σ = M/W, equivalently σ(y) = -M·(y-ȳ)/I. This calculator determines I, W and the outer-fibre distance directly from the drawn or coordinate-entered polygon contour instead of requiring them as known inputs.
Where does the maximum bending stress occur?
At the outer fibre with the greatest distance from the neutral axis -- that is, at the vertex with the largest or smallest coordinate value relative to the bending axis.
Why does the neutral axis lie at the centroid?
For pure bending without an additional normal force, the sum of normal stresses over the cross-section must equal zero. That is only satisfied when the stress-free (neutral) axis passes through the area centroid.
Why are tensile and compressive stress different for unsymmetric cross-sections?
Because the two outer fibres sit at different distances from the neutral axis (c+ ≠ c−). Since I is the same for both sides, the more distant fibre gives the larger stress magnitude.
What is the difference between M/I and M/W?
M·c/I and M/W are equivalent, since W = I/c. M/I gives the stress only together with the outer-fibre distance c; M/W gives it directly, since the distance is already contained in the section modulus.
When does the classical bending stress formula not apply?
It assumes linear-elastic material behaviour, a prismatic cross-section, pure uniaxial bending without shear or normal-force contributions, and the absence of local buckling or notch effects. Outside these assumptions a separate assessment is required.
How do you calculate bending stress for skew bending?
When bending occurs about both axes simultaneously (Mx and My), the two stress contributions superimpose linearly. This calculator handles only uniaxial bending about a chosen axis; a dedicated calculator for biaxial/skew bending is planned as a future extension.