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面积 moment 的 inertia — typical cross sections

Formulas 與 values 用于 面积 moment 的 inertia 的 common cross sections.

areamomentinertia

Overview

The area moment of inertia (also called second moment of area) is a geometric property of a cross section that describes its resistance to bending. It appears in the flexure formula:

σ=M⋅yI\sigma = \frac{M \cdot y}{I}

where:

  • σ\sigma = bending stress (Pa)
  • MM = bending moment (N·m)
  • yy = distance from the neutral axis (m)
  • II = area moment of inertia (m⁴)

A larger moment of inertia means a stiffer beam — more resistant to deflection under load.

Parallel Axis Theorem

The moment of inertia about any axis parallel to the centroidal axis:

I=Icentroid+A⋅d2I = I_{centroid} + A \cdot d^2

where:

  • AA = cross-sectional area
  • dd = distance between the two parallel axes

Section Modulus

The elastic section modulus relates moment of inertia to the maximum bending stress:

S=IcS = \frac{I}{c}

where cc is the distance from the neutral axis to the extreme fiber.

Common Cross Sections

Standard Steel Shapes — Example Values

Beam Deflection

The maximum deflection of a simply supported beam with uniform load:

δmax=5⋅w⋅L4384⋅E⋅I\delta_{max} = \frac{5 \cdot w \cdot L^4}{384 \cdot E \cdot I}

where:

  • ww = load per unit length (N/m)
  • LL = span (m)
  • EE = modulus of elasticity (Pa)
  • II = moment of inertia (m⁴)

Applications

  • Structural beam and column design
  • Machine component sizing
  • Bridge and building analysis
  • Crane and hoist engineering
  • Aerospace structural analysis
來源: engineeringtoolbox.com