Free T-Beam Moment of Inertia & Section Properties Calculator

Compute Area, Centroid, Moment of Inertia, Elastic and Plastic Section Moduli, and Radius of Gyration instantly. Enter dimensions manually or load standard steel databases.

⚙ Configuration

Section Geometry

Steel

Cold-Formed

Concrete

I-Section

Channel

T-Section

Angle

Double Angle

Box Section

Pipe

Dbl Channel

Builtup I Cov

Hybrid I

Hybrid U

Unit System

No pre-loaded standard profiles in this category.

Dimension Settings

Depth/Diameter (t3)

Width (t2)

Thickness (tf)

Web Thickness (tw)

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Section Properties Results

Property

Sym

Value

Unit

Cross-sectional Area

2724.8

mm²

Centroid location (X-axis)

Centroid location (Y-axis)

100

Strong Moment of Inertia

18455902.2667

mm⁴

Weak Moment of Inertia

1419344.8107

mm⁴

Strong Elastic Modulus

184559.0227

mm³

Weak Elastic Modulus

28386.8962

mm³

Strong Plastic Modulus

209659.6

mm³

Weak Plastic Modulus

mm³

Strong Radius of Gyration

82.3001

Weak Radius of Gyration

22.8232

Polar Moment of Inertia

19875247.0773

mm⁴

Hand Calculation Report

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Engineering Reference

What does each section property mean?

Geometric properties determine how a beam behaves under bending, deflection, yielding, and buckling. Engineers use these properties to evaluate capacity, check deflection, and analyze stability.

Moment of Inertia

Second moment of area

Notation: Ix, Iy

Units: L⁴ (mm⁴, in⁴)

Describes how the cross-sectional area is distributed around an axis. It directly affects bending stiffness and deflection. A higher value means the shape is stiffer.

E · I

Bending Stiffness (E = Modulus of Elasticity, I = Inertia)

Where it is used:

• Beam deflection calculations

• Bending stiffness checks (EI)

• Comparing structural efficiency of shapes

Centroid

Geometric center of gravity

Notation: Cx, Cy, x̄, ȳ

Units: L (mm, in)

The balance point of a section. For symmetric shapes, it lies on the axes of symmetry. For asymmetric shapes, it is offset from the visual center. It locates the neutral axis.

ȳ = Σ(Ai · yi) / ΣAi

Area-weighted location of the centroidal axis

Where it is used:

• Locating elastic neutral axis (ENA)

• Parallel axis theorem distance offsets

• Bending stress extreme fiber distances

Plastic Neutral Axis

Equal-area neutral axis

Notation: yp

Units: L (mm, in)

The line where tensile and compressive forces balance at full yielding. For homogeneous materials, it divides the cross-section into two equal halves.

A_above = A_below = A / 2

Line of equal area division

Where it is used:

• Plastic bending moment calculations

• Calculating plastic section modulus (Zx, Zy)

• Ultimate strength design of steel structures

Elastic Section Modulus

Bending resistance capacity

Notation: Sx, Sy

Units: L³ (mm³, in³)

Measures a shape's efficiency at resisting bending stresses before any yielding occurs. It relates moment of inertia to the extreme outer fiber distance.

S = I / c | σ = M / S

Bending stress σ = Moment M / Modulus S

Where it is used:

• Allowable stress design (ASD) checks

• Estimating maximum elastic stress

• Bending capacity checking in timber and concrete

Plastic Section Modulus

Ultimate bending capacity

Notation: Zx, Zy

Units: L³ (mm³, in³)

Determines the ultimate bending moment capacity of a section, representing the state after the entire cross-section has fully yielded.

Mp = Z · Fy

Plastic moment Mp (Fy = Material Yield Strength)

Where it is used:

• Limit states design (LRFD) in steel code

• Plastic hinge formation analysis

• Compact steel beam capacity checks

Radius of Gyration

Buckling resistance index

Notation: rx, ry

Units: L (mm, in)

The equivalent radial distance from the neutral axis at which the area could be concentrated to produce the same moment of inertia. Essential for column design.

r = √(I / A)

Relationship of inertia I to cross-section area A

Where it is used:

• Column slenderness ratio (KL/r)

• Critical Euler buckling load checking

• Out-of-plane stability evaluations

Quick Summary Table

Property

Meaning

Main Structural Use

Units

Area (A)

Total cross-sectional surface area

Axial stress, tension, and compression

L²

Inertia (I)

Area distribution about centroidal axis

Bending stiffness (EI) & deflections

L⁴

Centroid (C)

Geometric center of the shape

Neutral axis locations

PNA (yp)

Line dividing section into equal areas

Plastic section modulus calculations

Elastic S

Inertia to extreme fiber distance ratio

Bending stress checking (M/S)

L³

Plastic Z

Full yielding cross-section modulus

Ultimate bending capacity checks

L³

Radius r

Spread of area around reference axis

Column slenderness & buckling (KL/r)

Section Formula Reference

Formulas for standard closed-form shapes can be calculated directly. Built-up or hollow shapes rely on splitting elements or subtracting inner voids.

Rectangle

Solid rectangle with width b and depth d.

A = b · d

Ix = b · d³ / 12

Iy = d · b³ / 12

Sx = b · d² / 6

Sy = d · b² / 6

rx = d / √12

ry = b / √12

Hollow Rectangle (RHS)

Outer dimensions bo, do minus inner void dimensions bi, di.

A = bo·do - bi·di

Ix = (bo·do³ - bi·di³) / 12

Iy = (do·bo³ - di·bi³) / 12

Sx = Ix / (do / 2)

Sy = Iy / (bo / 2)

rx = √(Ix / A)

ry = √(Iy / A)

Circle

Solid round section with outer diameter d.

A = π · d² / 4

Ix = Iy = π · d⁴ / 64

Sx = Sy = π · d³ / 32

rx = ry = d / 4

Pipe / CHS

Hollow circular section with outer diameter do and inner diameter di.

A = π · (do² - di²) / 4

Ix = Iy = π · (do⁴ - di⁴) / 64

Sx = Sy = Ix / (do / 2)

rx = ry = √(do² + di²) / 4

Composite Sections Bending Check

For non-standard, built-up shapes like I-beams, C-channels, tees, or angles, properties are solved using a three-step procedure:

1. Decompose: Divide the complex shape into individual simple rectangular or circular parts.

2. Centroid: Find the overall centroid (x̄, ȳ) for the entire composite section.

3. Shift: Apply the Parallel Axis Theorem to shift the local moment of inertia of each sub-element to the composite centroid axis.

Parallel Axis Theorem

When a sub-element's local neutral axis is not on the neutral axis of the overall composite shape, the moment of inertia must be shifted parallelly.

I = Ī + A · d²

I = Target Inertia

Ī = Local Centroidal Inertia

A = Section Area

d = Axis Offset Distance

About Horizontal X-Axis

Uses vertical offset distance dy, representing the distance from the component centroid to the overall neutral x-axis.

Ix = Īx + A · dy²

About Vertical Y-Axis

Uses horizontal offset distance dx. If all parts share the same vertical centerline, this shift is zero.

Iy = Īy + A · dx²

Composite Section Summation

For a full composite shape made of multiple parts (or containing holes/cut-outs):

I_composite = Σ (Īi + Ai · di²)

• For solid components, add the shifted moments of inertia.

• For holes, voids, or cut-outs, subtract their shifted properties from the solid shape: I_total = I_solid - I_voids.

Worked Examples: From Formulas to Numbers

Example 1: Solid Rectangle

Rectangle, b = 200 mm, d = 400 mm

A solid beam cross-section with width 200 mm and depth 400 mm.

1. Cross-sectional Area (A)

A = b · d = 200 · 400 = 80,000 mm²

2. Moment of Inertia (Ix, Iy)

Ix = b · d³ / 12 = 200 · 400³ / 12 = 1.067 × 10⁹ mm⁴

Iy = d · b³ / 12 = 400 · 200³ / 12 = 2.667 × 10⁸ mm⁴

3. Elastic Section Modulus (Sx, Sy)

Sx = Ix / (d / 2) = 1.067 × 10⁹ / 200 = 5.333 × 10⁶ mm³

Sy = Iy / (b / 2) = 2.667 × 10⁸ / 100 = 2.667 × 10⁶ mm³

4. Radius of Gyration (rx, ry)

rx = d / √12 = 400 / 3.464 = 115.5 mm

ry = b / √12 = 200 / 3.464 = 57.7 mm

Example 2: Composite Asymmetric I-Beam

Built-Up Section: Flanges & Web

Top Flange: 200×20 mm | Web: 10×160 mm | Bottom Flange: 200×20 mm (Total Depth = 200 mm, symmetric). Let's calculate total Ix.

1. Segment Areas (Ai) & Local Centroids (yi) from bottom face

• Bottom Flange (Part 1): A₁ = 200 · 20 = 4,000 mm² | y₁ = 10 mm

• Web (Part 2): A₂ = 10 · 160 = 1,600 mm² | y₂ = 20 + 80 = 100 mm

• Top Flange (Part 3): A₃ = 200 · 20 = 4,000 mm² | y₃ = 180 + 10 = 190 mm

• Total Area (A) = A₁ + A₂ + A₃ = 9,600 mm²

2. Locating Composite Centroid (ȳ)

Σ(Ai · yi) = (4000·10) + (1600·100) + (4000·190) = 40,000 + 160,000 + 760,000 = 960,000 mm³

ȳ = Σ(Ai · yi) / A = 960,000 / 9,600 = 100 mm (perfectly centered)

3. Centroidal Offsets (dy,i) to overall centroid ȳ

• dy₁ = |y₁ - ȳ| = |10 - 100| = 90 mm

• dy₂ = |y₂ - ȳ| = |100 - 100| = 0 mm

• dy₃ = |y₃ - ȳ| = |190 - 100| = 90 mm

4. Local Centroidal Inertia (Īx,i)

• Īx₁ (Bot Flange) = 200 · 20³ / 12 = 133,333 mm⁴

• Īx₂ (Web) = 10 · 160³ / 12 = 3,413,333 mm⁴

• Īx₃ (Top Flange) = 200 · 20³ / 12 = 133,333 mm⁴

5. Parallel Axis Theorem & Summation

• Part 1 contribution = 133,333 + 4,000 · 90² = 133,333 + 32,400,000 = 32,533,333 mm⁴

• Part 2 contribution = 3,413,333 + 1,600 · 0² = 3,413,333 mm⁴

• Part 3 contribution = 133,333 + 4,000 · 90² = 32,533,333 mm⁴

• Total Ix = 32,533,333 + 3,413,333 + 32,533,333 = 68,480,000 mm⁴ = 68.48 × 10⁶ mm⁴

Frequently Asked Questions

Find quick answers about second moment of area, axis selection, PNA, and calculations.

What does this moment of inertia and section properties calculator calculate?

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Which moment of inertia should I use, Ix or Iy?

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Is moment of inertia the same as second moment of area?

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What is the difference between elastic section modulus S and plastic section modulus Z?

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Is the plastic neutral axis the same as the centroid?

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