ME221 Lecture 9 1

ME 221 Statics

Lecture #9

Sections 9.1 – 9.6

ME221 Lecture 9 2

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

ME221 Lecture 9 3

Exam #1 Results

Scores posted on AngelSolution posted on Angel See syllabus for regrade policyLast day to drop without a grade reported is:

Wednesday, June 9

ME221 Lecture 9 4

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

ME221 Lecture 9 5

Moments of Area

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

• Moment of inertia

– Moment about other axes derived from centroid case

dA

y

x

r

2 2 and xx yyA AI y dA I x dA

• Characterize distribution of area about the centroid

ME221 Lecture 9 6

Moments of Basic Shapes• Rectangle

x

y

2 2

2 2

2b h

b hxxI y dy dx

2

2

3 32 2=3 3

b

b

h h

dx

3

=12bh

• Circular

x

y

2 2

0 0sin

Rxx yyI I r rdr d

24 214 0

sinR d

414 R

dA = r dr d

ME221 Lecture 9 7

Polar MomentThe polar moment is the second moment of area

about the z-axis

x

y

r

2 3 4120 0

R

OzJ r dr d R

Note that: Ixx + Iyy = JOz

yyxxoz IIdAyxdArJ )( 222

2 2 and xx yyA AI y dA I x dA

ME221 Lecture 9 8

Parallel Axis TheoremThe centroid of the area MUSTMUST be one of the axes used in the

parallel axis theorem.

y

dy

x

x’C

2

2

xx xxc y

yy yyc x

I I Ad

I I Ad

ME221 Lecture 9 9

Radius of Gyration

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

; ;yyxx Ozx y z

II Jk k kA A A

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

ME221 Lecture 9 10

Principal Second Moments

• Definition of product moment of inertia

• Definition of principal axes– Product of inertia axis theorem

• Mohr’s circle to find principal axes• Example

ME221 Lecture 9 11

Product Moment of Inertia(Measures Antisymmetry)

• Basic section with two axes of symmetry

x

y

• Composite sections - product parallel axis theorem

xyA

I xydA

xy x y o oI I x y A

ME221 Lecture 9 12

Mohr’s Circle for Principal Inertia(Second Moment of Area about any Axis)

+Ixy

-Ixy

Ixx, IyyIyy

x

Ixx

Ixyx

• draw point (Ixx , Ixy) and the circle

(Ixx+Iyy)/2

• draw circle center (Ixx + Iyy)/2

x IMAXIMIN 2

• use geometry to find IMAX , IMIN and

ANGLES IN MOHR’S CIRCLE ARE TWICE THOSE IN THE CROSS SECTION!!!

ME221 Lecture 9 13

Example