Area Moments of Inertia
Definition & Composite Area
ENGR B36 - Statics
Pat Aderhold
11/26/2014
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Today’s Agenda
• First half of class: Area Moments of Inertia
• Second half of class
 Questions to prep for Exam #3
 Questions about Group Design Project
 Notes:
• BYO Wine Bottle or use mine
• Read carefully, object must be unstable when unloaded
Significance
• Not immediately intuitive
• Consider three common mechanics scenarios
 Fluid pressure
 Bending moment on beam
 Torsional loading on beam
Learn it / use it because it’s helpful
Meriam, JL and Kraige, LG. Statics 7th Ed. Wiley 2012. p. 442
Significance
All three scenarios share common principle
Moment caused by fluid pressure
M = k ò y dA
2
Total bending moment
M = k ò y dA
2
Total moment about longitudinal axis
M = k ò r dA
2
Definition
Refer to as “moment of inertia”
 Analogous to inertial forces in rotating bodies
 More appropriate name second moment of area
(First moment = “moment”)
I x = y dA
2
I y = y dA
2
I z = r dA
2
Meriam, JL and Kraige, LG. Statics 7th Ed. Wiley 2012. p. 442
Practice Problem A/3
Meriam, JL and Kraige, LG. Statics 7th Ed. Wiley 2012. p. 450
Radius of Gyration
• Some area “A” with a moment of inertia Iy
• What if you...
 Smush the area into tiny strip?
 Place a distance ky from the y-axis?
Meriam, JL and Kraige, LG. Statics 7th Ed. Wiley 2012. p. 443-444
Radius of Gyration
• Calculate Iy
Iy =
ò k dA
2
y
ky ¹ f (x) Þ I y = k
2
y
ò dA Þ I
y
=k A
2
y
• And what did that do exactly?....
 Just a property of the given area
 It has its uses (stability of columns, behavior of polymer chains)
Parallel Axis Theorem
• Scoot area from axis to new location
• Find moment of inertia about new location?
Meriam, JL and Kraige, LG. Statics 7th Ed. Wiley 2012. p. 444
Parallel Axis Theorem
(Proof in text)
• Find moment of A about its centroid (“ Iy ”)
• Take distance to centroidal axis (“dy”)
I y = I y + Ad
2
x
• Remember!
 Must be parallel axis
 Must pass through centroid
Meriam, JL and Kraige, LG. Statics 7th Ed. Wiley 2012. p. 444-445
Practice Problem A/12
NOTE: Remember
Table D/3
p. 497-498
Meriam, JL and Kraige, LG. Statics 7th Ed. Wiley 2012. p. 451
Complex Shapes
• Moment of inertia is integration
(adding up many differential parts)
• Works with finite pieces too
(comparable to composite centroids)
Practice Problem A/35
Meriam, JL and Kraige, LG. Statics 7th Ed. Wiley 2012. p. 459
Other Considerations
• Other relationships in text
 I for rectangular and polar
 Radius of gyration rectangular and polar
• Simple tools developed today
... but can be used for virtually any shape
For Exam #3
Review
• Can fill out a note card
 Up to 4” x 6”
 One side only
 Bring to me for approval on Wednesday
• Three problems
 Distributed loading / shear & moment diagrams
 Buoyancy
 Friction
For Next Class
• Practice Problems 6/63 & 6/66
• Think of questions and review problems
Exam #3 on Monday!
Group Design Project due next Wednesday!
Open Time
• Ask individual questions on HW
• Work on group design project