What is the difference between Area Moment of Inertia (Second Moment of Area) and Mass Moment of Inertia?

Area Moment of Inertia (Second Moment of Area, I, in mm⁴ or in⁴) measures a 2D cross-section's geometric resistance to bending deflection and flexural stress. In contrast, Mass Moment of Inertia (I_m, in kg·m² or slug·ft²) measures a 3D physical body's resistance to rotational angular acceleration under an applied torque.

How does Steiner's Parallel Axis Theorem calculate Moment of Inertia for composite cross-sections?

Steiner's Parallel Axis Theorem states that the moment of inertia about any axis parallel to the centroidal axis equals the centroidal moment of inertia plus the product of the sub-area and the square of the perpendicular transfer distance: I_x = ∑ (I_xo,i + A_i · d_yi²). For complex sections (I-beams, channels, built-up girders), the total section is decomposed into simple rectangular segments, and each segment's contribution is summed.

Why is the minor principal radius of gyration (r_v or r_z) critical for structural angles?

For asymmetric sections like L-angles, the geometric axes (x, y) are not the principal axes of inertia because the product of inertia I_xy ≠ 0. The minimum resistance to column buckling occurs about the minor principal axis (v or z-z), which has the smallest radius of gyration r_v = sqrt(I_v / A). AISC Chapter E and Eurocode 3 require using r_v (or r_z) for column slenderness (L/r) calculations.

What is the difference between Elastic Section Modulus (S) and Plastic Section Modulus (Z)?

Elastic Section Modulus S = I / c defines the elastic bending capacity (M_y = S · F_y) before any outer fiber reaches yield. Plastic Section Modulus Z = ∑ |A_i · y_i| defines the fully plastic moment capacity (M_p = Z · F_y) where the entire cross-section has yielded in tension and compression. The ratio Z/S is known as the Shape Factor (typically ~1.12–1.18 for I-beams, 1.5 for solid rectangles, and 1.7 for solid circles).

What is Polar Moment of Inertia (Jz) and when does it apply?

By the Perpendicular Axis Theorem, the Polar Moment of Inertia J_z = I_x + I_y represents resistance to torsional shear for planar cross-sections. For circular shafts and pipes, J_z = π·d⁴/32 exactly equals the torsional constant J. For non-circular sections (rectangles, I-beams, open channels), warping occurs during torsion, so the St. Venant torsional constant J is significantly smaller than J_z.

Structural Mechanics

AISC 15th/16th / Eurocode 3 / Roark

PRO Suite

Area Moment of Inertia & Section Properties PRO Calculator

Calculate second moment of area (Ix, Iy), polar moment of inertia (Jz), centroid (x̄, ȳ), elastic and plastic section moduli (Sx, Sy, Zx), radii of gyration (rx, ry, r_min), and Mohr's Circle principal moments (Iu, Iv) for 9 geometric cross-sections and standard steel shapes per AISC 15th/16th Ed., Eurocode 3 (EN 1993-1-1), and Roark's Formulas.

Input Parameters

Cross-Section Geometry

Unit System

Standard Steel Catalog

AISC / EN / ASME

Section Dimensions (mm)

Total Height (H):200 mm

Flange Width (B):100 mm

Flange Thickness (tf):8.5 mm

Web Thickness (tw):5.6 mm

Section Properties & Inertia Executive Audit

Doubly/Monosymmetric

Moment of Inertia (I_x)

1.846e+7

mm⁴ (Strong Axis)

Moment of Inertia (I_y)

1,419,344.81

mm⁴ (Weak Axis)

Cross-Section Area (A)

2,724.8

mm²

Polar Inertia (J_z)

1.988e+7

mm⁴ (Ix + Iy)

Centroid & Fibers

Centroid (x̄, ȳ):(50, 100) mm

c_top / c_bottom:100 / 100 mm

c_left / c_right:50 / 50 mm

Perimeter:788.8 mm

Principal Axes (Mohr)

Major Principal (I_u / I_max):1.846e+7 mm⁴

Minor Principal (I_v / I_min):1,419,344.81 mm⁴

Product of Inertia (I_xy):0 mm⁴

Principal Angle (θ_p):0.00°

Moduli & Radii of Gyration

Elastic Modulus (S_x / S_y):184,559.02 / 28,386.9 mm³

Plastic Z_x (Shape Factor):209,659.6 mm³ (1.14)

Radius Gyration (r_x / r_y):82.3 / 22.82 mm

Buckling Governing (r_min):22.82 mm

Interactive Cross-Section & Neutral Axis Geometry

HNA (Neutral Axis X)VNA (Neutral Axis Y)Centroid (x̄, ȳ)

Bounding Box: 100.0 × 200.0 mm

Parallel Axis Theorem (Steiner's Theorem) Tabular Breakdown

3 Sub-Segments Decomposed

Analytical integration via Steiner's formulation: I_x = ∑ (I_xo,i + A_i · d_yi²), I_y = ∑ (I_yo,i + A_i · d_xi²)

Segment	Dim (b × h)	Area (mm²)	Centroid (x, y)	I_xo (mm⁴)	d_yi (mm)	A·d_yi² (mm⁴)	I_xi (mm⁴)	I_yi (mm⁴)
Top Flange	100.0 × 8.5 mm	850	(50.0, 195.8)	5,117.7	95.8	7,792,853.1	7,797,970.8	708,333.3
Web	5.6 × 183.0 mm	1,024.8	(50.0, 100.0)	2,859,960.6	0.0	0	2,859,960.6	2,678.1
Bottom Flange	100.0 × 8.5 mm	850	(50.0, 4.3)	5,117.7	-95.8	7,792,853.1	7,797,970.8	708,333.3
Total Net Cross-Section (∑)		2,724.8	(50.0, 100.0)	Global Neutral Axis Summation			1.85e+7	1,419,344.8

Calculation Summary

Input Parameters

Profile Shape	I-BEAM
Standard Reference	Custom Geometry
Bounding Envelope	100.0 × 200.0mm
Gross Cross-Section Area	2724.80mm²
Centroid Location (x̄, ȳ)	(50.0, 100.0)mm
Extreme Fiber Dist (c_top / c_bot)	100.0 / 100.0mm

Calculated Results

Moment of Inertia (I_x / Strong Axis)	18,455,902.3mm⁴ KEY
Moment of Inertia (I_y / Weak Axis)	1,419,344.8mm⁴ KEY
Major Principal Moment (I_u / I_max)	18,455,902.3mm⁴
Minor Principal Moment (I_v / I_min)	1,419,344.8mm⁴
Elastic Section Modulus (S_x / S_y)	184,559 / 28,386.9mm³
Radius of Gyration (r_min / Buckling)	22.82mm KEY

Safety Notes

Section calculated per AISC 15th/16th Ed. and Eurocode 3 (EN 1993-1-1). Strong axis I_x = 18,455,902.267 mm⁴, governing Euler buckling radius r_min = 22.82 mm.

Mathematical Models & Equations

I-Beam / Wide-Flange (W / IPE / HEB) Moments of Area

I_x = \frac{B \cdot H^3 - (B - t_w)(H - 2t_f)^3}{12}, \quad I_y = \frac{2 t_f \cdot B^3 + (H - 2t_f) t_w^3}{12}

I-Beam Plastic Modulus & Torsional Constant

Z_x = B \cdot t_f (H - t_f) + \frac{t_w (H - 2t_f)^2}{4}, \quad J \approx \frac{2 B t_f^3 + (H - 2t_f) t_w^3}{3}

Verification Calculation Log

1*Step 1: Cross-Sectional Area & Geometry Bounding** Formula: A = \int dA Substitution: Bounding Envelope: 100.00 mm × 200.00 mm Result: A = 2724.80 mm² (Perimeter = 788.80 mm) Note: Total net cross-sectional material area excluding internal hollow cutouts.
- •*Step 2: Centroid & Neutral Axis Coordinates** Formula: \bar{x} = \frac{\sum A_i x_i}{\sum A_i}, \quad \bar{y} = \frac{\sum A_i y_i}{\sum A_i} Substitution: Reference Origin: Bottom-Left corner (0,0) Result: \bar{x} = 50.00 mm, \quad \bar{y} = 100.00 mm Note: Extreme fiber distances: c_top = 100.00 mm, c_bottom = 100.00 mm, c_left = 50.00 mm, c_right = 50.00 mm.
- •*Step 3: Centroidal Second Moments of Area (I_x, I_y, I_xy)** Formula: I_x = \sum (I_{xo,i} + A_i d_{yi}^2), \quad I_y = \sum (I_{yo,i} + A_i d_{xi}^2) Substitution: Parallel axis summation across 3 decomposed segments Result: I_x = 18,455,902.27 mm⁴, \quad I_y = 1,419,344.81 mm⁴, \quad I_{xy} = 0.00 mm⁴ Note: Major horizontal axis moment (Ix), minor vertical axis moment (Iy), and centroidal product of inertia (Ixy).
- •*Step 4: Principal Moments of Area & Mohr's Circle (I_max, I_min, θ_p)** Formula: I_{u,v} = \frac{I_x + I_y}{2} \pm \sqrt{\left(\frac{I_x - I_y}{2}\right)^2 + I_{xy}^2}, \quad \tan(2\theta_p) = \frac{-2 I_{xy}}{I_x - I_y} Substitution: I_avg = 9,937,623.54, \quad R_{mohr} = 8,518,278.73 Result: I_{max} (I_u) = 18,455,902.27 mm⁴, \quad I_{min} (I_v) = 1,419,344.81 mm⁴, \quad \theta_p = 0.00° Note: Symmetric section: Geometric axes coincide with principal axes (I_xy = 0, θ_p = 0°).
- •*Step 5: Elastic Section Moduli (S_x, S_y)** Formula: S_x = \frac{I_x}{\max(c_{top}, c_{bot})}, \quad S_y = \frac{I_y}{\max(c_{left}, c_{right})} Substitution: S_{x,top} = 184,559.02 mm³, \quad S_{x,bot} = 184,559.02 mm³ Result: S_x = 184,559.02 mm³, \quad S_y = 28,386.9 mm³ (Plastic Z_x = 209,659.6 mm³, Shape Factor = 1.14) Note: Section modulus governs peak elastic bending stress: σ_max = M / S.
- •*Step 6: Radii of Gyration & Column Buckling Slenderness (r_x, r_y, r_min)** Formula: r_x = \sqrt{\frac{I_x}{A}}, \quad r_y = \sqrt{\frac{I_y}{A}}, \quad r_{min} = \min(r_x, r_y, r_v) Substitution: r_x = \sqrt{18455902.3 / 2724.8}, \quad r_y = \sqrt{1419344.8 / 2724.8} Result: r_x = 82.30 mm, \quad r_y = 22.82 mm, \quad r_{min} = 22.82 mm Note: Radius of gyration used directly in Euler-Johnson column slenderness ratio: λ = K·L / r_min.
- •*Step 7: Polar Moment of Inertia & Torsional Constant (J_z, J)** Formula: J_z = I_x + I_y = \int r^2 dA Substitution: J_z = 18455902.3 + 1419344.8 Result: J_z = 19,875,247.08 mm⁴, \quad J = 51,654.24 mm⁴ Note: Polar moment of inertia represents resistance to torsional deflection and shaft shearing strain.

Engineering DisclaimerEngineering calculations are provided for preliminary design and educational/reference purposes only. Users must verify all results according to applicable engineering standards, supplier data, manufacturing requirements, and professional engineering judgment.

Segment

Dim (b × h)

Area (mm²)

Centroid (x, y)

I_xo (mm⁴)

d_yi (mm)

A·d_yi² (mm⁴)

I_xi (mm⁴)

I_yi (mm⁴)

Top Flange

100.0 × 8.5 mm

850

(50.0, 195.8)

5,117.7

95.8

7,792,853.1

7,797,970.8

708,333.3

Web

5.6 × 183.0 mm

1,024.8

(50.0, 100.0)

2,859,960.6

0.0

2,859,960.6

2,678.1

Bottom Flange

100.0 × 8.5 mm

850

(50.0, 4.3)

5,117.7

-95.8

7,792,853.1

7,797,970.8

708,333.3

Total Net Cross-Section (∑)

2,724.8

(50.0, 100.0)

Global Neutral Axis Summation

1.85e+7

1,419,344.8

Second Moment of Area (I_x):	18,455,902.267 mm⁴
Second Moment of Area (I_y):	1,419,344.811 mm⁴
Polar Moment of Inertia (J_z):	19,875,247.077 mm⁴
Product of Inertia (I_xy):	0 mm⁴
Principal Moments (I_u, I_v):	18,455,902.267 / 1,419,344.811 mm⁴ (θ_p = 0.00°)
Elastic Moduli (S_x, S_y):	184,559.023 / 28,386.896 mm³
Radii of Gyration (r_x, r_y, r_min):	82.30 / 22.82 / 22.82 mm

Second Moment of Area (I_x):	18,455,902.267 mm⁴
Second Moment of Area (I_y):	1,419,344.811 mm⁴
Polar Moment of Inertia (J_z):	19,875,247.077 mm⁴
Product of Inertia (I_xy):	0 mm⁴
Principal Moments (I_u, I_v):	18,455,902.267 / 1,419,344.811 mm⁴ (θ_p = 0.00°)
Elastic Moduli (S_x, S_y):	184,559.023 / 28,386.896 mm³
Radii of Gyration (r_x, r_y, r_min):	82.30 / 22.82 / 22.82 mm

Area Moment of Inertia & Section Properties PRO Calculator

Section Properties & Inertia Executive Audit

Interactive Cross-Section & Neutral Axis Geometry

Parallel Axis Theorem (Steiner's Theorem) Tabular Breakdown

Calculation Summary

Mathematical Models & Equations

Verification Calculation Log

Area Moment of Inertia & Section Properties Engineering Report — Calculation Report

Area Moment of Inertia & Section Properties Engineering Report

Design Inputs

Calculated Results

Section Properties & Inertia Executive Audit

Interactive Cross-Section & Neutral Axis Geometry

Parallel Axis Theorem (Steiner's Theorem) Tabular Breakdown

Calculation Summary

Mathematical Models & Equations

Verification Calculation Log

Area Moment of Inertia & Section Properties Engineering Report — Calculation Report

Area Moment of Inertia & Section Properties Engineering Report

Design Inputs

Calculated Results