Transcript ME 221 Statics

ME 221 Statics
Lecture #9
Sections 9.1 – 9.6
ME221
Lecture 9
1
Homework #4
• Chapter 4 problems:
– 52 & 54
• Chapter 9 problems
– 2, 11 & 32
– Due Monday, June 14
• MatLab Group Problems
–
–
–
–
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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
2
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
ME221
Lecture 9
3
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
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Lecture 9
4
Moments of Area
• Characterize distribution of area about the centroid
• Normally, we want the moment
with respect to centriod axes
x
dA
– Moment about other axes
derived from centroid case
r
y
• Moment of inertia
I xx 
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
A
y 2dA and I yy 
Lecture 9

A
x2dA
5
Moments of Basic Shapes
• Rectangle
y
x
I xx 
b
2
 
h
2
 b2  h2
  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
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b
2
4
2
R
0
0
 

2
0
 r sin  rdr d
2
sin 2  d 
Lecture 9
4
1
R
4
6
Polar Moment
The polar moment is the second moment of area
about the z-axis
J oz   r dA   ( x  y )dA  I xx  I yy
2
y
r
x
JOz 
Note that:
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2
2
I xx 

y 2dA and I yy 
2
R
0
0
r 3dr d  12  R4
A
 

A
x2dA
Ixx + Iyy = JOz
Lecture 9
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 
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2
Ad x
Lecture 9
x’
dy
x
8
Radius of Gyration
An alternate, equivalent way to represent the
moment of an area
I yy
I xx
kx 
; ky 
; kz 
A
A
JOz
A
Distance from the point or axis to where the
area is concentrated
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Lecture 9
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
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Lecture 9
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Product Moment of Inertia
(Measures Antisymmetry)
• Basic section with two axes of symmetry

I xy   xydA
y
x
A
• Composite sections - product parallel axis theorem
I xy  I xy  xo yo A
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Lecture 9
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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
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x
Ixx
IMAX
Ixx, Iyy
Iyy
• use geometry to find IMAX , IMIN and b
ANGLES IN MOHR’S CIRCLE
ARE TWICE THOSE IN THE
CROSS SECTION!!!
2b
IMIN
x Ixy
+Ixy
Lecture 9
(Ixx+Iyy)/2
12
Example
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Lecture 9
13