CTC / MTC 222 Strength of Materials

1
CTC / MTC 222 Strength of Materials

• Chapter 6
• Centroids and Moments of Inertia of Areas

2
Chapter Objectives

• Define centroid and locate the centroid of a
  shape by inspection or calculation
• Define moment of inertia and compute its values
  with respect to the centroidal axes of the area
• Use the parallel axis theorem to compute the
  moment of inertia of composite shapes

3
Centroid of an Area

• Centroid of an area the geometric center of the
  area
• Centroid of simple shapes circle, rectangle,
  triangle
• Often, easily visualized
• Centroids and other properties shown in Appendix
  A-1
• Centroid of complex shapes with axes of symmetry
• If area has an axis of symmetry centroid is on
  that axis
• If area has two axes of symmetry centroid is at
  the intersection of the two axes
• Centroid of complex shapes without axes of
  symmetry
• Can often be considered as a composite of two or
  more simple shapes
• Centroid of complex shape can be calculated using
  centroids of the simple shapes and the locations
  of these centroids with respect to some reference
  axis

4
Centroid of an Area

• Centroid of complex shapes can be calculated
  using
• AT Y ? (Ai yi ) where
• AT total area of composite shape
• Y distance to centroid of composite shape
  from some reference axis
• Ai area of one component part of shape
• yi distance to centroid of the component part
  from the reference axis
• Solve for Y ? (Ai yi ) / AT
• Perform calculation in tabular form
• See Examples 6-1 6-2

5
Moment of Inertia

• Moment of Inertia - a measure of the stiffness of
  a beam, or of its resistance to deflection due to
  bending
• Sometimes referred to as the second moment of
  area, or the area moment of inertia
• Moment of inertia of an area with respect to a
  particular axis the sum of the products of each
  (infinitesimal) element of the area by the square
  of its distance from the axis
• Approximately I ? y2 (?A)
• Exactly - I ? y2 dA
• Moment of inertia of simple shapes circle,
  rectangle, triangle
• Formulas derived from basic definition, shown in
  Appendix A-1
• Moment of inertia standard structural shapes
  wide flange sections, channels, angles, pipe,
  etc.
• Tabulated in standard references such as Steel
  Design Manual
• Some in Appendix A

6
Moment of Inertia

• Moment of inertia complex shapes
• Can often be considered as a composite of two or
  more simple shapes
• If all component parts have the same centroidal
  axis
• Add or subtract the moments of inertia of the
  component parts with respect to the centroidal
  axis
• If all component parts do not have the same
  centroidal axis
• Moment of inertia can be calculated using the
  parallel axis theorem
• Parallel axis theorem
• Moment of inertia of a shape with respect to a
  given axis is equal to the sum if the moment of
  inertia of the shape to its own centroidal axis
  plus the transfer term, Ad2, where A is the area
  of the shape and d is the distance from the
  centroidal axis to the axis of interest
• I I0 Ad2

7
Moment of Inertia ofComposite Shapes

• Perform calculation in tabular form
• Divide the shape into component parts which are
  simple shapes
• Locate the centroid of each component part, yi
  from some reference axis
• Calculate the centroid of the composite section,
  Y from some reference axis
• Compute the moment of inertia of each part with
  respect to its own centroidal axis, Ii
• Compute the distance, di Y - yi of the
  centroid of each part from the overall centroid
• Compute the transfer term Ai di2 for each part
• The overall moment of inertia IT , is then
• IT ? (Ii Ai di2)
• See Examples 6-5 through 6-7

8
Radius of Gyration, r

• Radius of Gyration - a measure of a compression
  members slenderness, or its resistance to
  buckling due to compressive load
• Buckling Failure under compressive load by
  excessive lateral deflection at a stress below
  the yield stress (elastic buckling)
• Buckling will occur about the axis with the least
  radius of gyration
• The tendency for a compression member to buckle
  is directly proportional to its length squared
  and indirectly proportional to its radius of
  gyration squared.
• Radius of Gyration r v (I /A )