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ME 221 Statics Lecture #9 Sections 9.1 – 9.6PowerPoint Presentation

ME 221 Statics Lecture #9 Sections 9.1 – 9.6

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### Exam #1 Results

Homework #4

- Chapter 4 problems:
- 52 & 54

- Chapter 9 problems
- 2, 11 & 32
- Due Monday, June 14

- MatLab Group Problems
- 4.13 – plot as a function of R with 0º < α < 45º
- 4.51 – plot as a function of length “l”
- 4.60 – plot the equivalent force and point of action
- Due Monday, June 14

Lecture 9

Scores posted on Angel

Solution posted on Angel

See syllabus for regrade policy

Last day to drop without a grade reported is: Wednesday, June 9

Lecture 9

Second Moments of Area(Area Moments of Inertia)

- Second moments of area play a central role in mechanics of materials and dynamics

- Definition of second moment

- Basic areas (rectangle, circular, triangular)

- Definition of polar moment

- Basic areas (circular)

- Parallel axis theorem

Lecture 9

dA

y

r

Moments of Area- Characterize distribution of area about the centroid

- Normally, we want the moment with respect to centriod axes

- Moment about other axes derived from centroid case

- Moment of inertia

Lecture 9

r

x

Polar MomentThe polar moment is the second moment of area about the z-axis

Note that: Ixx + Iyy = JOz

Lecture 9

x’

C

dy

x

Parallel Axis TheoremThe centroid of the area MUST be one of the axes used in the parallel axis theorem.

Lecture 9

Radius of Gyration

An alternate, equivalent way to represent the moment of an area

Distance from the point or axis to where the area is concentrated

Lecture 9

Principal Second Moments

- Definition of product moment of inertia

- Product of inertia axis theorem

- Definition of principal axes

- Mohr’s circle to find principal axes

- Example

Lecture 9

x

Product Moment of Inertia(Measures Antisymmetry)- Basic section with two axes of symmetry

- Composite sections - product parallel axis theorem

Lecture 9

-Ixy

- draw circle center (Ixx + Iyy)/2

x

- draw point (Ixx , Ixy) and the circle

IMIN

Ixx

IMAX

2b

x

Ixx, Iyy

Iyy

- use geometry to find IMAX , IMIN and b

x

Ixy

+Ixy

(Ixx+Iyy)/2

Mohr’s Circle for Principal Inertia(Second Moment of Area about any Axis)ANGLES IN MOHR’S CIRCLE ARE TWICE THOSE IN THE CROSS SECTION!!!

Lecture 9

Example

Lecture 9

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