Section properties describe the geometry of a cross-section — independent of material or load. They’re the bridge between the shape you pick off McMaster-Carr and the stress calculation you hand your boss by EOD.
Get the wrong I and every downstream calculation is wrong. Here’s a clear reference for the properties that actually show up in daily engineering work.
The Properties and What They Mean
Area (A)
Cross-sectional area in in² or mm². Used directly in:
- Axial stress: σ = F / A
- Bearing stress: σ_b = F / A_bearing
- Shear stress in pins: τ = V / A
Nothing subtle here — it’s the area of material you can see in a cross-section view.
Second Moment of Area / Moment of Inertia (I)
I (in⁴ or mm⁴) measures resistance to bending. It is NOT the mass moment of inertia from dynamics — same symbol, completely different concept.
I depends on both area and how far that area is from the neutral axis. Area far from the neutral axis contributes more — which is exactly why I-beams work. Most of the material is in the flanges, as far from the neutral axis as possible.
I is always computed about a specific axis. I_x is about the horizontal axis; I_y is about the vertical axis. For bending, use the I for the axis the beam bends about.
Section Modulus (S)
S = I / c (in³ or mm³), where c is the distance from the neutral axis to the outermost fiber.
S lets you directly compute max bending stress: σ_max = M / S
For symmetric sections (circles, rectangles, hollow tubes), c = half the total height (or diameter), and S is the same top and bottom. For asymmetric sections (T-beams, angles), the top and bottom have different c values and therefore different S values — use the smaller one for max stress.
Plastic Section Modulus (Z)
Z (in³ or mm³) is used in plastic analysis — what happens when the entire cross-section has yielded, not just the outer fiber. Z ≥ S always.
The ratio Z/S is called the shape factor. For a rectangle, it’s 1.5 — meaning a rectangular beam can carry 50% more moment beyond first yield before full plastic collapse. For wide-flange I-beams it’s around 1.10–1.15.
For elastic design (standard hand calc), use S. Z appears in AISC plastic design and seismic applications.
Polar Moment of Area (J)
J (in⁴ or mm⁴) measures resistance to torsion. For a circular section, J = 2I. For non-circular sections, J is more complex (and technically it becomes the torsion constant, not the polar moment).
Torsional shear stress: τ = T · r / J
Where T = applied torque and r = radial distance from center (max at outer surface).
Radius of Gyration (r_g)
r_g = √(I / A) (in or mm). Used in column buckling calculations.
Euler buckling load: P_cr = π²·E·I / (L_eff)² — but r_g shows up when you express this as a critical stress: σ_cr = π²·E / (L_eff / r_g)²
The slenderness ratio L_eff / r_g is what you look up in column charts. A lower r_g = more susceptible to buckling. This is why thin-walled tubes are efficient in bending but can buckle locally in compression.
Formulas by Cross-Section
Solid Rectangle (width b, height h)
| Property | Formula |
|---|---|
| A | b·h |
| I_x (bending about horizontal axis) | b·h³ / 12 |
| I_y (bending about vertical axis) | h·b³ / 12 |
| c_x | h / 2 |
| S_x | b·h² / 6 |
| Z_x | b·h² / 4 |
| J (approx, thin rectangles only) | b·h·(b² + h²) / 12 |
| r_g_x | h / √12 ≈ 0.289·h |
Solid Circle (diameter d)
| Property | Formula |
|---|---|
| A | π·d² / 4 |
| I | π·d⁴ / 64 |
| c | d / 2 |
| S | π·d³ / 32 |
| Z | d³ / 6 |
| J | π·d⁴ / 32 |
| r_g | d / 4 |
Hollow Tube (outer diameter D, inner diameter d)
| Property | Formula |
|---|---|
| A | π·(D² − d²) / 4 |
| I | π·(D⁴ − d⁴) / 64 |
| c | D / 2 |
| S | π·(D⁴ − d⁴) / (32·D) |
| J | π·(D⁴ − d⁴) / 32 |
| r_g | √[(D² + d²) / 16] |
Hollow Rectangle (outer B × H, wall thickness t)
Let b = B − 2t, h = H − 2t (inner dimensions)
| Property | Formula |
|---|---|
| A | B·H − b·h |
| I_x | (B·H³ − b·h³) / 12 |
| c_x | H / 2 |
| S_x | I_x / c_x |
Common Mistakes
Using the wrong axis. A 2×4 oriented flat has I = 2·4³/12 = 10.67 in⁴. Stood on edge: I = 4·2³/12 = 2.67 in⁴. Four times stiffer in one orientation — which is why studs are always installed the tall way.
Forgetting what I is about. If a beam bends about the x-axis (sags downward), use I_x. If it bends sideways (lateral-torsional buckling), use I_y.
Confusing mass moment of inertia with area moment of inertia. Both use I. Mass moment of inertia (kg·m²) is for dynamics/rotation. Area moment of inertia (m⁴) is for bending. They are completely different.
Use PartCalc for Real Parts
Paste a McMaster-Carr URL for any structural bar, tube, or rod and PartCalc extracts the cross-section dimensions directly from the product page, computes all section properties automatically, and labels every value as scraped or computed. You get A, I, S, Z, J, and r_g in one shot — with no manual formula lookup required.
Calculate it now
Paste a McMaster-Carr product URL into PartCalc to instantly get section properties, material data, and the calculations described in this article — with every value labeled as scraped, inferred, or computed.
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