ME 221 Statics
Lecture #8
Section 4.5-4.6
ME221
Lecture 8
1
Homework #3
• Chapter 3 problems:
– 48, 55, 57, 61, 62, 65 & 72
• Chapter 4 problems
– 2, 4, 10, 11, 18, 24, 39 & 43
– Due Today
ME221
Lecture 8
2
Homework #4
• Chapter 4 problems:
– 52 & 54
• Chapter 9 problems
– 2, 11 & 32
– Due Monday, June 14
• MatLab Group Problems
–
–
–
–
ME221
4.13 – plot as a function of R with 0º < α < 45º
4.51 – plot as a function of l
4.60 – plot the equivalent force and point of action
Due Monday, June 14
Lecture 8
3
Quiz #4
• Today
Quiz #5
• Wednesday, June 16
ME221
Lecture 8
4
Last Lecture:
• Centroids of Areas & Volumes
ME221
Lecture 8
5
Centroid of Composite Bodies
• Basic shapes are added together to get the
composite centroid
– We look these up in Table 4.2
– This table will be supplied on future exams for
shapes other than rectangles, circles and
triangles
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Lecture 8
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Area Centroids
This case is also representative of line and
volume centroids.
Divide total area into simple shapes
(rectangles, semi-circles)
Add simple shapes
y
A2
x1
x2
A1
y1
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+
y2
A3
+
=
xc
x3
Lecture 8
yc
y3
x
7
May Also Use Subtraction
A1
x2
x1
y1
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y
A2
=
y2
Lecture 8
yc
xc
x
8
Composite Centroid of an Area
• Geometric center of the area
– Summation of the pieces (holes are negative)
1
xc   xiAi
A i
– Where:
1
yc   yiAi
A i
A   Ai
i
ME221
Lecture 8
9
Composite Centroid of a Volume
• Geometric center of the volume
– Average of the first moment over entire volume
1
xc   xiVi
V i
1
yc   yiVi
V i
– Where:
V   Vi
1
zc   ziVi
V i
i
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Lecture 8
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Examples
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Lecture 8
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Distributed Loads
• Distributed loads on beams
– Application of equivalent load
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Lecture 8
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Beams
Beams can be loaded by a point load or a
distributed load
P
x
w(x)
Replace the distributed load
with an equivalent
concentrated force.
dx
L
The equivalent force is
the sum of all the forces.
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W
Lecture 8
L
0
w  x  dx
13
Equivalent Load Location
The equivalent load is placed at the centroid of
the distributed load.
w(x)
1
xc 
W

L
0
x w  x  dx
W
To find reaction loads:
replace distributed load
with equivalent load at
centroid
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xc
L
Lecture 8
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Example
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Lecture 8
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Quiz #4
ME221
Lecture 8
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