Area Moments of InertiaDefinition & Composite Area

ENGR B36 - StaticsPat Aderhold11/26/2014

News & Notes

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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

Meriam, JL and Kraige, LG. Statics 7th Ed. Wiley 2012. p. 442

Learn it / use it because it’s helpful

SignificanceAll three scenarios share common principle

Moment caused by fluid pressure

Total bending moment

Total moment about longitudinal axis

DefinitionRefer to as “moment of inertia”

Analogous to inertial forces in rotating bodies More appropriate name second moment of area

(First moment = “moment”)

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

• 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”)

• 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

Meriam, JL and Kraige, LG. Statics 7th Ed. Wiley 2012. p. 451

NOTE: Remember Table D/3 p. 497-498

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• 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

Review

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