ME 221 Statics
Lecture #18
Sections 9.1 – 9.6
ME 221
Lecture 18
1
Homework #7
• Chapter 4 problems:
– 52, 53 & 54
• Chapter 9 problems:
– 2, 3, 4, 11, 23, 29, 32
– Use double integration for 2, 3, 4 & 11
• Due Friday, October 17
ME 221
Lecture 18
2
Exam #2 Results
Average = xx.x
Scores posted on Angel
Solution will be posted on Angel today
See syllabus for regrade policy
Last day to drop without a grade reported is:
Tuesday, October 14
ME 221
Lecture 18
3
Second Moments of Area
• 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
ME 221
Lecture 18
4
Moments of Area
• Normally, we want the moment with respect to
centriod axes
x
– Moment about other axes
derived from centroid case
dA
r
y
• Moment of inertia
I xx 
ME 221
A
y 2 dA and I yy 
Lecture 18
A
x 2 dA
5
Moments of Basic Shapes
• Rectangle
y
x
I xx 
b
2
h
2
 
b
2
h
2
  h 3   h 3 
3
bh
2
2
 dx =

y 2 dy dx =  b 
3
3 
2 
12



dA = r dr d
• Circular
y
I xx  I yy 
x
 R
1
4
ME 221
b
2
4
2
R
0 0
2
0
 r sin   rdr d
2
sin 2  d 
Lecture 18
4
1

R
4
6
Polar Moment
The polar moment is the second moment of
area about the z-axis
J oz   r 2 dA   ( x 2  y 2 )dA  I xx  I yy
y
r
x
J Oz 
Note that:
ME 221
2
R
0 0 r dr d 
3
4
1
R
2
Ixx + Iyy = JOz
Lecture 18
7
Parallel Axis Theorem
The centroid of the area MUST be one of the axes used in the
parallel axis theorem.
I xx  I xxc 
2
Ad y
y
C
I yy  I yyc 
ME 221
2
Ad x
Lecture 18
x’
dy
x
8
Radius of Gyration
An alternate, equivalent way to represent the
moment of an area
kx 
I xx
; ky 
A
I yy
A
; kz 
J Oz
A
Distance from the point or axis to where the
area is concentrated
ME 221
Lecture 18
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
ME 221
Lecture 18
10
Product Moment of Inertia
• Basic section with two axes of symmetry
A
I xy   xydA
y
x
• Composite sections - product parallel axis theorem
I xy  I xy  xo yo A
ME 221
Lecture 18
11
Mohr’s Circle for Principal Inertia
-Ixy
• draw circle center (Ixx + Iyy)/2
x
• draw point (Ixx , Ixy) and the circle
• use geometry to find IMAX , IMIN and b
ANGLES IN MOHR’S CIRCLE
ARE TWICE THOSE IN THE
CROSS SECTION!!!
ME 221
2b
IMIN
x
Ixx
IMAX
Ixx, Iyy
Iyy
x Ixy
+Ixy
Lecture 18
(Ixx+Iyy)/2
12
Example
ME 221
Lecture 18
13