Section properties describe the geometric characteristics of a cross-section that determine its structural behavior. Every beam, column, and connection calculation starts with these numbers. The moment of inertia (I) controls stiffness and deflection. The section modulus (S) controls bending stress. The area (A) controls axial and shear stress. The radius of gyration (r) controls column buckling.
These properties come from integral calculus applied to the cross-section geometry, but for standard shapes (rectangles, circles, I-beams, channels), the formulas are well-established and tabulated in references like the AISC Steel Construction Manual and any mechanics of materials textbook (Hibbeler, Gere, Beer & Johnston).
The Key Properties and What They Mean
Area (A): Total cross-sectional area. Used for axial stress (P/A) and shear stress (V/A) calculations. Units: in².
Moment of Inertia (I): Resistance to bending. Formally, the second moment of area: I = ∫y²dA. A higher moment of inertia means less curvature (and less deflection) for the same applied moment. Moment of inertia increases with the cube of depth but only linearly with width, this is why deep, narrow beams are stiffer than shallow, wide ones. Units: in⁴.
Section Modulus (S): S = I / c, where c is the distance from the neutral axis to the extreme fiber. The elastic section modulus relates bending moment to maximum bending stress: σ = M / S. For symmetric sections, S is the same for top and bottom. For asymmetric sections (T-shapes, channels), the section modulus is different for the two extreme fibers. Units: in³.
Plastic Section Modulus (Z): The section modulus at full plastic hinge formation. Z = A/2 × (distance between centroids of top and bottom halves). Always larger than S. The ratio Z/S is called the shape factor; it is about 1.5 for rectangles and about 1.12–1.15 for I-beams. Used in LRFD steel design and plastic analysis. Units: in³.
Radius of Gyration (r): r = √(I/A). Combines stiffness and area into a single number used for column buckling calculations. The slenderness ratio KL/r determines whether a column fails by yielding or by elastic buckling. Units: in.
Section Properties Calculator
Moment of inertia, section modulus, plastic modulus, and radius of gyration for rectangles, circles, hollow sections, I-beams, channels, angles, and tees. AISC W-shape lookup included.
Formulas for Common Shapes
Rectangle (b × h):
A = b × h\nIx = b × h³ / 12\nSx = b × h² / 6\nZx = b × h² / 4
Circle (diameter d):
A = πd² / 4\nI = πd⁴ / 64\nS = πd³ / 32\nZ = d³ / 6
Hollow rectangle (outer B × H, inner b × h):
A = B×H − b×h\nIx = (B×H³ − b×h³) / 12\nSx = Ix / (H/2)
Hollow circle (OD, ID):
A = π(OD² − ID²) / 4\nI = π(OD⁴ − ID⁴) / 64\nS = I / (OD/2)
For I-beams, channels, angles, and tees, the formulas involve breaking the shape into component rectangles and using the parallel axis theorem: I_total = Σ(I_i + A_i × d_i²), where d_i is the distance from each component's centroid to the overall centroid.
Section Properties Calculator
Moment of inertia, section modulus, plastic modulus, and radius of gyration for rectangles, circles, hollow sections, I-beams, channels, angles, and tees. AISC W-shape lookup included.
The Parallel Axis Theorem: Building Composite Sections
When you combine two or more shapes into a built-up section (such as a plate welded to the bottom of a W-shape, or two channels bolted back-to-back), the moment of inertia of the composite section is calculated using the parallel axis theorem:
I_total = Σ(I_local + A × d²)
Where I_local is the moment of inertia of each component about its own centroid, A is the area of each component, and d is the distance from each component's centroid to the overall composite centroid.
The d² term is what makes composite sections efficient. A cover plate welded to the bottom flange of a W-shape adds relatively little area, but because that area is far from the neutral axis, the A×d² contribution is large. A 1" × 8" cover plate (A = 8 in²) placed 8 inches below the composite centroid adds 8 × 64 = 512 in⁴ to the composite moment of inertia, more than the I_local of a W10×22.
The parallel axis theorem requires that you first find the composite centroid. This is done by taking the first moment of area: ȳ = Σ(A_i × y_i) / ΣA_i. Then calculate d for each component relative to this composite centroid.
Section Properties Calculator
Moment of inertia, section modulus, plastic modulus, and radius of gyration for rectangles, circles, hollow sections, I-beams, channels, angles, and tees. AISC W-shape lookup included.
Using Section Properties in Design
Beam design: Select a section where Sx ≥ M_max / F_allowable for allowable stress design, or Zx ≥ M_u / (φ × Fy) for LRFD. Then check deflection using Ix.
Column design: Calculate the slenderness ratio KL/r about both axes. The governing slenderness is the larger of KL/rx and KL/ry (unless bracing differs between axes). Compare to Euler's critical buckling load or the AISC column curve to find capacity.
Combined loading: Members subject to both axial load and bending (beam-columns) must satisfy interaction equations that combine the effects of axial stress (P/A or P/Ag) and bending stress (M/S or M/Zx). The AISC interaction equations (H1-1a and H1-1b) use both area and section modulus.
Connection design: The area of bolts, welds, and connected elements determines the force that connections can transfer. The section modulus of bolt groups and weld groups determines their moment capacity for eccentric connections.
Section Properties Calculator
Moment of inertia, section modulus, plastic modulus, and radius of gyration for rectangles, circles, hollow sections, I-beams, channels, angles, and tees. AISC W-shape lookup included.
Section Properties Calculator
Moment of inertia, section modulus, plastic modulus, and radius of gyration for rectangles, circles, hollow sections, I-beams, channels, angles, and tees. AISC W-shape lookup included.