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Moment of Inertia

Second moment of area (I) measures resistance to bending; units are length⁴. Rectangle: Ix=bh³/12. Circle: I=πr⁴/4. Polar J=Ix+Iy for torsion.

Concept Fundamentals
Ix = bh³/12
Rectangle Ix
I = πr⁴/4
Circle I
J = Ix + Iy
Polar J
Z = I/ymax
Section Z

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Height dominates Ix for rectangles—tall beams resist bending better. I-beam and T-section use composite formulas from flange and web. Polar moment J = Ix + Iy is used for torsional stress.

Key quantities
Ix = bh³/12
Rectangle Ix
Key relation
I = πr⁴/4
Circle I
Key relation
J = Ix + Iy
Polar J
Key relation
Z = I/ymax
Section Z
Key relation

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Why: Moment of inertia is essential in structural engineering for beam deflection and stress. I-beam and T-sections optimize strength-to-weight.

How: Rectangle: Ix=bh³/12, Iy=b³h/12. Circle: I=πr⁴/4. Hollow: subtract inner. Polar J=Ix+Iy for torsion. Section modulus Z=I/ymax for bending.

Height dominates Ix for rectangles—tall beams resist bending better.I-beam and T-section use composite formulas from flange and web.
Sources:Wolfram MathWorld - Moment of InertiaEngineering Toolbox - Beam Properties

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Moment of Inertia CalculatorSelect shape and enter dimensions for Ix, Iy, J, and section modulus
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SECTION PROPERTIES

Moment of Inertia — Ix, Iy, J, Z

Rectangle, circle, triangle, I-beam, hollow circle, T-section. Second moment of area and section modulus.

📐 Examples — Click to Load

Inputs

cm
cm

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

📐

Rectangle Ix = bh³/12 — height dominates bending resistance.

— Formula

⭕

Circle second moment I = πr⁴/4.

— Formula

📋 Key Takeaways

  • • Second moment of area (I) measures resistance to bending; units are length⁴
  • • Rectangle: Ix = bh³/12, Iy = b³h/12 — height dominates Ix
  • • Circle: I = πr⁴/4; hollow circle subtracts inner contribution
  • • Polar moment J = Ix + Iy for torsion; section modulus Z = I/ymax for bending stress
  • • I-beam and T-section use composite formulas from flange and web

💡 Did You Know?

🏗️I-beams get most strength from flanges — moving material away from neutral axis increases I dramaticallySource: Structural Engineering
📐For same area, a hollow tube has higher I than solid — material at outer radius contributes more (I ∝ r⁴)Source: Mechanics
⚙️Polar moment J determines torsional stiffness; drive shafts use hollow sections for weight savingsSource: Machine Design
🌉Bridge beams are often I-shaped because Ix resists vertical loads while keeping weight lowSource: Civil Engineering
📏Doubling beam height multiplies Ix by 8 (I ∝ h³) — depth is crucial for bending resistanceSource: Beam Theory
🔩T-sections are common in precast concrete; centroid offset affects Zx calculationSource: Concrete Design

📖 Formulas Explained

Rectangle: Ix = bh³/12

Bending about x-axis; integration of y² dA over rectangular area.

Ix=fracbh312,quadIy=fracb3h12I_x = \\frac{bh^3}{12}, \\quad I_y = \\frac{b^3 h}{12}

Circle: I = πr⁴/4

Ix = Iy by symmetry; polar J = 2I.

I=fracpir44,quadJ=fracpir42I = \\frac{\\pi r^4}{4}, \\quad J = \\frac{\\pi r^4}{2}

Hollow Circle

I=fracpi(ro4−ri4)4I = \\frac{\\pi(r_o^4 - r_i^4)}{4}

🎯 Expert Tips

Depth vs Width

For bending, depth (h) matters more than width — Ix ∝ h³. Double h → 8× Ix.

Hollow Sections

Hollow pipes resist torsion and bending with less weight than solid sections.

I-Beam Efficiency

Flanges carry most bending stress; web resists shear. Use standard sections (W, I) when available.

Units

Keep b, h, r in same units (mm, cm, m). Result I has units length⁴; Z has length³.

⚖️ Comparison Table

ShapeIxRelative
Rectangle (b×h)bh³/121.0
Circle (r)πr⁴/4~0.65 (same area)
Triangle (b×h)bh³/360.33
Hollow (ro, ri)π(ro⁴−ri⁴)/4Depends on ri/ro

📊 Quick Stats

I
2nd Moment
J
Polar
Z
Section Mod
L⁴
I units

❓ FAQ

What is second moment of area?

I measures resistance to bending. Higher I means stiffer beam. Formula: I = ∫ y² dA over cross-section.

Ix vs Iy?

Ix = bending about x-axis (horizontal neutral axis); Iy = bending about y-axis. For rectangle, Ix uses h³, Iy uses b³.

What is polar moment J?

J = Ix + Iy for planar sections. Used for torsional stiffness; J = πr⁴/2 for solid circle.

What is section modulus Z?

Z = I/ymax. Bending stress σ = M/Z. Higher Z means lower stress for same moment.

Why I-beam?

Material at flanges (far from neutral axis) contributes most to I. I-beam maximizes I per unit weight.

Hollow vs solid circle?

Same outer radius: hollow has less I (less material) but often better I per weight. Inner radius reduces I.

Units for I?

Length⁴: mm⁴, cm⁴, m⁴, in⁴. Ensure all dimensions in same length unit.

Triangle centroid?

Triangle Ix = bh³/36 about base; centroid at h/3 from base. Different from rectangle.

⚠️ Disclaimer: For educational use. Verify structural calculations with licensed engineers. Assumes homogeneous material and standard formulas.

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