Second Moment of Area

Overview

• Understanding of the second moment of area is essential for explaining its importance in engineering, calculating it, and interpreting its values in civil and structural engineering.

Importance

• Decisions in structural design involve the geometric profile and material properties.
• Stiffness: resistance to axial and bending deformation.
• Axial stiffness = ( \frac{EA}{L} ) and Bending stiffness = ( \frac{EI}{L} ).
• Steel's elastic modulus, ( E = 210,000 ) MPa.
• To increase stiffness: increase area ( A ) and second moment of area ( I ).

What is the Second Moment of Area?

• Governs bending stiffness assuming given length and elastic modulus.
• It describes the distribution of area about an axis and is a geometric property with dimensions ( L^4 ).

Geometric Axes and Sign Conventions

• Standard axis definitions are necessary for consistency.
• Axes need to be defined for bending actions.

Locating the Centroid

• Essential precursor to calculating the second moment of area.
• Centroidal axes are about which we determine global moments of area.
• Formula for centroid: ( Y_c = \frac{\sum Ay}{\sum A} ), ( Z_c = \frac{\sum Az}{\sum A} ).

Second Moment of Area of Rectangles

• Involves three components:
  1. ( I_{yy} )
  2. ( I_{zz} )
  3. ( I_{yz} )
• Derivations for rectangles are provided._

Second Moment of Area of Simple Shapes

• Standard results for reference:
  • Rectangle: ( I_{yy} = \frac{bh^3}{12} ), ( I_{zz} = \frac{b^3h}{12} ).
  • Circle: ( \frac{\pi r^4}{4} ).
  • Annulus: ( \frac{\pi (r_o^4-r_i^4)}{4} ).
  • Triangle: ( \frac{bh^3}{36} ), ( \frac{b^3h}{36} ).

Parallel Axis Theorem

• Used to determine the second moment of area about non-centroidal axes.
• Formula: ( I_{yy}^G = \sum I_{yy}^L + \sum Az^2 ).

Examples

• Example 1: I Section
  • Calculation of global centroid and moments of area.
• Example 2: Channel Section
  • Calculation of global centroid and moments of area.
• Example 3: Z Section
  • Calculation of global centroid and moments of area.

Understanding Product Second Moment

• Indicates far assumed axes are from the principal bending axes.
• Examples illustrate different values: ( I_{yz}^G = 0 ), ( I_{yz}^G > 0 ), ( I_{yz}^G < 0 )._

Principal Bending Axes

• For sections where ( I_{yz}^G \neq 0 ), principal axes are not coincident with assumed axes.
• Formulae for principal moments and orientation are provided._

Summary

• Methods for calculating centroid and second moment of area.
• Calculated principal bending axes to recognize the suitability of sections for applications.
• Hand calculations are necessary for complex sections not covered in data tables.
• Essential skill for structural engineers, especially for bespoke sections.