1. CTC / MTC 222 Strength of Materials
Chapter 7
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 ̅Y̅ ̅ = ∑ (Ai yi ) where:
AT = total area of composite shape
̅Y̅ = 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 ̅Y̅ = ∑ (Ai yi ) / AT
Perform calculation in tabular form
See Examples 7-1 & 7-2

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 of Composite 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, ̅Y̅ from some reference axis
Compute the moment of inertia of each part with respect to its own centroidal axis, Ii
Compute the distance, di = ̅Y̅ - 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 7-5 through 7-7

8. Radius of Gyration, r
Radius of Gyration - a measure of a compression member’s 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 = √ (I /A )