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 Meriam, JL and Kraige, LG. Statics 7 th Ed. Wiley p. 442 Learn it / use it because it’s helpful

Significance All three scenarios share common principle Moment caused by fluid pressure Total bending moment Total moment about longitudinal axis

Definition Refer 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 7 th Ed. Wiley p. 442

Practice Problem A/3 Meriam, JL and Kraige, LG. Statics 7 th Ed. Wiley p. 450

Radius of Gyration Some area “A” with a moment of inertia I y What if you...  Smush the area into tiny strip?  Place a distance k y from the y-axis? Meriam, JL and Kraige, LG. Statics 7 th Ed. Wiley p

Radius of Gyration Calculate I 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 7 th Ed. Wiley p. 444

Parallel Axis Theorem (Proof in text) Find moment of A about its centroid (“ I y ”) Take distance to centroidal axis (“d y ”) Remember!  Must be parallel axis  Must pass through centroid Meriam, JL and Kraige, LG. Statics 7 th Ed. Wiley p

Practice Problem A/12 Meriam, JL and Kraige, LG. Statics 7 th Ed. Wiley p. 451 NOTE: Remember Table D/3 p

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 7 th Ed. Wiley 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