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

ME 221 Statics Lecture #9 Sections 9.1 – 9.6. 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 0 º < α < 45º 4.51 – plot as a function of length “ l ”

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Lecture #9Sections 9.1 – 9.6

Lecture 9

• 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

### Exam #1 Results

Scores posted on Angel

Solution posted on Angel

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

x

dA = r dr dq

y

x

Moments of Basic Shapes

• Rectangle

• Circular

Lecture 9

r

x

Polar Moment

The 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 Theorem

The centroid of the area MUST be one of the axes used in the parallel axis theorem.

Lecture 9

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

• 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

Lecture 9