Section Modulus Calculator
Select a cross-section shape, enter the dimensions, and get the elastic section modulus (S), plastic section modulus (Z), second moment of area (I), and distance to the neutral axis instantly. Optionally enter a bending moment to see the maximum stress. Switch between metric and imperial units at any time. Results update as you type.
Formula
Worked example
An I-beam with bf = 150 mm, d = 300 mm, tf = 15 mm, tw = 10 mm: hw = 300 - 2*15 = 270 mm. Ix = (150*300^3 - 140*270^3)/12 = (405,000,000 - 273,780,000)/12 = 10,935,000 mm^4. cx = 150 mm. Sx = 10,935,000/150 = 72,900 mm^3.
What is the section modulus?
The section modulus is a geometric property of a beam cross-section that links the applied bending moment to the maximum normal stress in the beam. There are two kinds: the elastic section modulus S (also written Sx for the strong axis) and the plastic section modulus Z (or Zx). The elastic value, S = I / c, where I is the second moment of area and c is the distance from the neutral axis to the extreme fiber, is used in standard allowable-stress design. The plastic value, Z, represents the first moment of area of each half of the cross-section about the plastic neutral axis; it is used in limit-state (LRFD) design to find the full plastic moment capacity. A cross-section with a larger section modulus resists bending more efficiently for the same amount of material.
How this calculator works
Choose a cross-section shape from the drop-down: solid rectangle, hollow rectangle (RHS/tube), solid circle (round bar), hollow circle (pipe/CHS), I-beam with equal flanges, or T-section. Enter the outside dimensions and any wall or flange thicknesses in your chosen unit system. The calculator applies the exact closed-form formula for each shape to deliver the strong-axis and weak-axis elastic section moduli (Sx, Sy), plastic section moduli (Zx, Zy), second moments of area (Ix, Iy), the distance to the extreme fiber (cx), and the cross-sectional area. If you enter a design bending moment, it also computes the maximum elastic bending stress so you can check it against your material allowable.
Section modulus and beam design
In beam design the required elastic section modulus is found from the design bending moment M and the allowable bending stress sigma_allow: S_required = M / sigma_allow. If the provided S exceeds S_required, the beam is adequate. The plastic section modulus Z comes into play in plastic design and LRFD: the plastic moment Mp = Z * sigma_yield gives the moment at which the entire cross-section has yielded and a plastic hinge has formed. The shape factor f = Z / S indicates how much extra capacity exists beyond first yield; a solid rectangle has f = 1.5, a solid circle has f = 1.70, and a wide-flange I-beam typically has f between 1.10 and 1.20, which is why I-beams are so efficient in bending.
Shape factor and efficient cross-sections
The shape factor (sometimes called the form factor) is the ratio of the plastic to the elastic section modulus. For a doubly symmetric section the shape factor equals 1.5 for a solid rectangle, 1.70 for a solid circle, and roughly 1.12 to 1.18 for most standard wide-flange I-beams. A low shape factor means nearly all the material is far from the neutral axis, which is efficient because the extreme fibers do most of the bending work. A high shape factor means there is more reserve capacity between first yield and full plasticization, which matters in seismic and ductile design. I-beams concentrate material at the flanges where bending stress is highest, giving a higher Ix (and therefore higher Sx) per unit area than any other common section.
Section modulus formulas for common cross-sections
| Shape | Elastic Sx | Plastic Zx | Second moment Ix |
|---|---|---|---|
| Rectangle (b x h) | b*h^2 / 6 | b*h^2 / 4 | b*h^3 / 12 |
| Hollow rect (b x h, t) | (b*h^3 - bi*hi^3) / (6*h) | (b*h^2 - bi*hi^2) / 4 | (b*h^3 - bi*hi^3) / 12 |
| Solid circle (d) | pi*d^3 / 32 | d^3 / 6 | pi*d^4 / 64 |
| Hollow circle (D, d) | pi*(D^4-d^4) / (32*D) | (D^3-d^3) / 6 | pi*(D^4-d^4) / 64 |
| I-beam (bf, d, tf, tw) | (bf*d^3-(bf-tw)*hw^3) / (6*d) | Plastic axis formula | (bf*d^3-(bf-tw)*hw^3) / 12 |
| T-section | Ix / cx_max | First-moment method | Parallel-axis theorem |
S = elastic section modulus, Z = plastic section modulus, I = second moment of area. Strong-axis values about the horizontal centroidal axis.
Frequently asked questions
What is the difference between elastic and plastic section modulus?
The elastic section modulus S = I / c is used in elastic design; when the bending moment M = S * sigma_yield, the extreme fibers just reach yield stress. The plastic section modulus Z is larger: it assumes the entire cross-section has yielded. The full plastic moment Mp = Z * sigma_yield, and the ratio Z / S (the shape factor) shows how much extra moment capacity is available between first yield and full plasticization. For a solid rectangle, Z / S = 1.5, meaning the beam can carry 50% more moment beyond first yield before a full plastic hinge forms.
Which section is most efficient for bending?
Wide-flange I-beams are the most efficient standard section for resisting bending about the strong axis. They concentrate material at the flanges, which are farthest from the neutral axis where bending stress is highest, while using a thin web only to resist shear. This gives a high second moment of area and a high elastic section modulus per unit weight. Hollow rectangular sections (RHS) are also efficient and resist biaxial bending more evenly than I-beams.
How do I use the section modulus to choose a beam?
Divide your maximum design bending moment by the allowable bending stress (for example, 165 MPa for grade A36 steel in many codes): S_required = M / sigma_allow. Then choose a cross-section whose Sx is at least that value. A beam catalogue will list Sx for each standard section. If the required Sx is very large, switch to a deeper section before a wider one, because depth increases Sx more efficiently than width.
Why does the second moment of area matter?
The second moment of area I (also called the area moment of inertia) appears in both the elastic section modulus (S = I / c) and the beam deflection formula (delta = k * M * L^2 / (E * I)). A higher I means less deflection for a given load and span. So I controls stiffness (deflection) while S controls strength (stress). A deep beam increases I rapidly (by the cube of depth), which is why increasing depth is so effective at reducing both stress and deflection.
What unit should I use for section modulus?
In metric engineering practice, section modulus is typically quoted in mm^3 or cm^3. In US customary practice it is in^3. The stress equation sigma = M / S requires consistent units: if M is in N*mm and S is in mm^3, the result is in N/mm^2 (MPa). If M is in lbf*in and S is in in^3, the result is in psi. This calculator handles the conversion automatically when you switch the unit system.
What is the shape factor and why does it matter in seismic design?
The shape factor (f = Z / S) tells you how much additional moment a section can carry between first yield (elastic limit) and full plasticization. In seismic design, codes require members to form plastic hinges and dissipate energy without fracture. Sections with higher shape factors have more ductile rotation capacity, but all structural sections must meet minimum compactness requirements so the plastic hinge can develop without local buckling. Solid rectangles (f = 1.5) and circles (f = 1.70) have high shape factors; standard wide-flange beams are lower (about 1.12 to 1.18) but remain fully adequate for seismic use.