What does the section modulus calculator compute?
The calculator determines the elastic section modulus of any polygonal cross-section contour about both centroidal axes. As with the two related calculators, the geometry is defined graphically or through coordinates; the centroid and second moments of area are found internally using the same polygon engine.
Relationship between I and W
The section modulus links the second moment of area to the distance to the outermost fibre:
Wx,+ = Ix / cy+Wx,- = Ix / cy-Wy,+ = Iy / cx+Wy,- = Iy / cx-Here cy+ is the distance from the centroidal axis to the top fibre (ymax − ȳ), cy− the distance to the bottom fibre (ȳ − ymin), and cx+/cx− the corresponding distances for the y-axis.
Symmetric vs. unsymmetric cross-sections
For a doubly symmetric section such as a rectangle, the centroidal axis sits exactly in the middle, so cy+ = cy− and cx+ = cx−: there is effectively only one section modulus per axis. For an unsymmetric profile, such as a T- or L-section, the two outer fibres sit at different distances from the axis. Ix is the same for both sides, but the fibre distance differs -- giving two different section moduli per axis.
Elastic only, not yet plastic
This calculator determines only the elastic section modulus, based on the linear stress distribution of classical bending theory. The plastic section modulus, which accounts for full cross-section utilization at yield, is not part of this calculator.
Example: 100 × 50 mm rectangle
The contour P1 = (0, 0), P2 = (100, 0), P3 = (100, 50), P4 = (0, 50) gives S = (50 mm, 25 mm), Ix = 1,041,666.667 mm⁴ and Iy = 4,166,666.667 mm⁴. Because the rectangle is doubly symmetric, cy+ = cy− = 25 mm and cx+ = cx− = 50 mm, giving
Wx = 1.041.666,667 / 25 = 41.666,667 mm³Wy = 4.166.666,667 / 50 = 83.333,333 mm³which matches the familiar formula Wx = b·h²/6 = 100·50²/6 = 41,666.667 mm³. These values are also used as an automated reference test.
Scope and limitations
The calculator supports one closed, non-self-intersecting outer contour and one polygonal inner contour lying completely inside it. Disconnected regions, arcs, DXF and the plastic section modulus are currently excluded. The section modulus alone is not a complete strength verification; an allowable stress must additionally be applied.