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Unformatted text preview: Module 14: Moment of Iner=a
November 19, 2010 Module Content:
1. Center of gravity, center of mass, and centroid are similar concepts which describe speciﬁc loca=ons in a body. The share similar approaches to their calcula=on.
2. Centroid of an area is par=cularly important for structural analysis, because we oXen use it to describe the cross sec=on of a beam.
3. Composite bodies are analyzed using a par=cle discre=za=on. Module Reading, Problems, and Demo:
Reading: Chapter 10
Problems: Fund. Prob. 10.8, Prob. 10.38
Demo: business card bridge
Homework PlaRorm: hSp://www.masteringengineering.com
Course Blog: hSp://pages.shan=.virginia.edu/sta=cs2010 MAE 2300 Sta=cs © E. J. Berger, 2010 14
1 Theory: Composite Bodies
• composite bodies have complicated geometries which are typically built up from simpler shapes to form useful shapes for speciﬁc applica=ons • in engineering, the basic building blocks are usually circles and rectangles • in the analysis, we treat each simple shape as a par=cle MAE 2300 Sta=cs © E. J. Berger, 2010 14
2 Theory: Composite Bodies
• composite bodies have complicated geometries which are typically built up from simpler shapes to form useful shapes for speciﬁc applica=ons • in engineering, the basic building blocks are usually circles and rectangles • in the analysis, we treat each simple shape as a par=cle MAE 2300 Sta=cs © E. J. Berger, 2010 14
2 Theory: Composite Bodies
• composite bodies have complicated geometries which are typically built up from simpler shapes to form useful shapes for speciﬁc applica=ons • in engineering, the basic building blocks are usually circles and rectangles • in the analysis, we treat each simple shape as a par=cle MAE 2300 Sta=cs © E. J. Berger, 2010 14
2 Theory: Composite Bodies
• composite bodies have complicated geometries which are typically built up from simpler shapes to form useful shapes for speciﬁc applica=ons • in engineering, the basic building blocks are usually circles and rectangles • in the analysis, we treat each simple shape as a par=cle MAE 2300 Sta=cs © E. J. Berger, 2010 14
2 Example 9.10: Composite Body (p. 473) MAE 2300 Sta=cs © E. J. Berger, 2010 14
3 Example 9.10: Composite Body (p. 473) = MAE 2300 Sta=cs © E. J. Berger, 2010 14
3 Example 9.10: Composite Body (p. 473) = MAE 2300 Sta=cs © E. J. Berger, 2010 14
3 Example 9.10: Composite Body (p. 473) = MAE 2300 Sta=cs © E. J. Berger, 2010 14
3 Example 9.10: Composite Body (p. 473) =  MAE 2300 Sta=cs © E. J. Berger, 2010 14
3 Example 9.10: Composite Body (p. 473) =  MAE 2300 Sta=cs © E. J. Berger, 2010 14
3 Example 9.10: Composite Body (p. 473) MAE 2300 Sta=cs © E. J. Berger, 2010 14
3 Example 9.10: Composite Body (p. 473) MAE 2300 Sta=cs © E. J. Berger, 2010 14
3 Example 9.10: Composite Body (p. 473) MAE 2300 Sta=cs © E. J. Berger, 2010 14
3 Concept: Moment of Iner=a
• the moment of iner=a is mathema=cally deﬁned as the “second moment of the area”, so called because the integral equa=on which describes it uses a quadra=c
type equa=on as the integrand • in MAE 2310, we will examine bending stresses
these are oXen the most important stresses ac=ng on a structural member • in the deriva=on of the bending stress equa=on, a speciﬁc type of integral shows up in the denominator of the bending stress equa=on • that special integral is the moment of iner=a; note that a larger moment of iner=a results in a smaller bending stress (since it appears in the denominator) MAE 2300 Sta=cs © E. J. Berger, 2010 14
4 Theory: The “Second Moment” of Area MAE 2300 Sta=cs © E. J. Berger, 2010 14
5 Theory: The “Second Moment” of Area MAE 2300 Sta=cs © E. J. Berger, 2010 14
5 Theory: The “Second Moment” of Area notes:
• the (x,y) coordinates are deﬁned with respect to a speciﬁc coordinate system (the “global” system)
• so this means that if we use a diﬀerent coordinate system...we get a diﬀerent moment of iner=a
• but...would those two moments of iner=a be somehow related to each other?
MAE 2300 Sta=cs © E. J. Berger, 2010 14
5 Theory: The Integral MAE 2300 Sta=cs © E. J. Berger, 2010 14
6 Example: A Rectangular Cross Sec=on
• calculate the moment of iner=a Iy around the xb axis MAE 2300 Sta=cs © E. J. Berger, 2010 14
7 Example: A Rectangular Cross Sec=on, II
• calculate the moment of iner=a Iy around the x’ axis MAE 2300 Sta=cs © E. J. Berger, 2010 14
8 Theory: Parallel Axis Theorem
• so what about the link between the moments of iner=a calculated on the two diﬀerent coordinate systems? • let’s say our two coordinate system are oﬀset from each other by distances dx and dy • then, we can reformulate the integral as: MAE 2300 Sta=cs © E. J. Berger, 2010 14
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This note was uploaded on 02/09/2012 for the course MAE 2300 taught by Professor Staff during the Fall '10 term at UVA.
 Fall '10
 Staff
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