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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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ME 221 Statics

Lecture #9Sections 9.1 – 9.6

Lecture 9


Homework 4
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


Exam 1 results

Exam #1 Results

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(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


Moments of area

x

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


Moments of basic shapes

y

x

dA = r dr dq

y

x

Moments of Basic Shapes

  • Rectangle

  • Circular

Lecture 9


Polar moment

y

r

x

Polar Moment

The polar moment is the second moment of area about the z-axis

Note that: Ixx + Iyy = JOz

Lecture 9


Parallel axis theorem

y

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


Radius of gyration
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
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


Product moment of inertia measures antisymmetry

y

x

Product Moment of Inertia(Measures Antisymmetry)

  • Basic section with two axes of symmetry

  • Composite sections - product parallel axis theorem

Lecture 9


Mohr s circle for principal inertia second moment of area about any axis

-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
Example

Lecture 9


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