Vector Mechanics for Engineers: Statics

CHAPTER
VECTOR MECHANICS FOR ENGINEERS:
STATICS
Ferdinand P. Beer
E. Russell Johnston, Jr.
Lecture Notes:
J. Walt Oler
Texas Tech University
Eighth Edition
Distributed Forces:
Centroids and Centers
of Gravity
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

ighth
dition
Vector Mechanics for Engineers: Statics
Contents
Introduction
Center of Gravity of a 2D Body
Centroids and First Moments of Areas
and Lines
Centroids of Common Shapes of Areas
Centroids of Common Shapes of Lines
Composite Plates and Areas
Sample Problem 5.1
Determination of Centroids by
Integration
Sample Problem 5.4
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Theorems of Pappus-Guldinus
Sample Problem 5.7
Distributed Loads on Beams
Sample Problem 5.9
Center of Gravity of a 3D Body:
Centroid of a Volume
Centroids of Common 3D Shapes
Composite 3D Bodies
Sample Problem 5.12
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Vector Mechanics for Engineers: Statics
Introduction
• The earth exerts a gravitational force on each of the particles
forming a body. These forces can be replace by a single
equivalent force equal to the weight of the body and applied
at the center of gravity for the body.
• The centroid of an area is analogous to the center of
gravity of a body. The concept of the first moment of an
area is used to locate the centroid.
• Determination of the area of a surface of revolution and
the volume of a body of revolution are accomplished
with the Theorems of Pappus-Guldinus.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
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Vector Mechanics for Engineers: Statics
Center of Gravity of a 2D Body
• Center of gravity of a plate
M
M
y
y
xW
yW
xW
x dW
yW
y dW
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
• Center of gravity of a wire
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Vector Mechanics for Engineers: Statics
Centroids and First Moments of Areas and Lines
• Centroid of an area
x
xW
Atxt xA
x dA
yA
x dW
first
first
dA
Q
y dA Q
y
moment with
x
moment with
respect to
respect to
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y
x
• Centroid of a line
x
xW
x dW
Laxa xL
x dL
yL
y dL
dL
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Vector Mechanics for Engineers: Statics
First Moments of Areas and Lines
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
• An area is symmetric with respect to an axis BB’
if for every point P there exists a point P’ such
that PP’ is perpendicular to BB’ and is divided
into two equal parts by BB’.
• The first moment of an area with respect to a
line of symmetry is zero.
• If an area possesses a line of symmetry, its
centroid lies on that axis
• If an area possesses two lines of symmetry, its
centroid lies at their intersection.
• An area is symmetric with respect to a center O
if for every element dA at (x,y) there exists an
area dA’ of equal area at (-x,-y).
• The centroid of the area coincides with the
center of symmetry.
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Vector Mechanics for Engineers: Statics
Centroids of Common Shapes of Areas
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Vector Mechanics for Engineers: Statics
Centroids of Common Shapes of Lines
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Vector Mechanics for Engineers: Statics
Composite Plates and Areas
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
• Composite plates
X
Y
W
W
xW
yW
• Composite area
X
Y
A
A
x A
yA
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Vector Mechanics for Engineers: Statics
Sample Problem 5.1
For the plane area shown, determine
the first moments with respect to the
x and y axes and the location of the
centroid.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
SOLUTION:
• Divide the area into a triangle, rectangle,
and semicircle with a circular cutout.
• Calculate the first moments of each area
with respect to the axes.
• Find the total area and first moments of
the triangle, rectangle, and semicircle.
Subtract the area and first moment of the
circular cutout.
• Compute the coordinates of the area
centroid by dividing the first moments by
the total area.
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Vector Mechanics for Engineers: Statics
Sample Problem 5.1
• Find the total area and first moments of the
triangle, rectangle, and semicircle. Subtract the
area and first moment of the circular cutout.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Q
Q
x
y
506.
210
757.
7
10
3
3
mm
mm
3
3
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Vector Mechanics for Engineers: Statics
Sample Problem 5.1
• Compute the coordinates of the area
centroid by dividing the first moments by
the total area.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
X
Y
x A
A
y A
A
757.
7
10
13.82810
X
506.
2
13.82810
Y
3
3
54.
8
10
3
3
36.
6
mm
mm
mm
mm
3
2
mm
3
2
mm
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Vector Mechanics for Engineers: Statics
Determination of Centroids by Integration
xA
yA
xA
yA
xdA
ydA
x
x
y
y
2
el
dA
ydx el
x dxdy
y dxdy
dA
ydx xA
yA
a x
2
y dA
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
x
y
el
el
dA
dA
• Double integration to find the first moment
may be avoided by defining dA as a thin
rectangle or strip.
x
y
el
el
dA
axdx axdx xA
yA
x
y
el
el
dA
2r
1
cos
3 2
dA
2r
1
sin
3 2
r
r
2
2
d
d
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Vector Mechanics for Engineers: Statics
Sample Problem 5.4
Determine by direct integration the
location of the centroid of a parabolic
spandrel.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
SOLUTION:
• Determine the constant k.
• Evaluate the total area.
• Using either vertical or horizontal
strips, perform a single integration to
find the first moments.
• Evaluate the centroid coordinates.
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Vector Mechanics for Engineers: Statics
Sample Problem 5.4
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
SOLUTION:
• Determine the constant k.
y
b
y
k
a
b
x
k a
2
2
2
x
2
k
or
a
b
x
2
• Evaluate the total area.
A dA
ab
3
y dx
a
0
a
b
2
x
2
b
dx
a
1 2
y
b
a
1 2
2
x
3
a
3
0
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Vector Mechanics for Engineers: Statics
Sample Problem 5.4
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
• Using vertical strips, perform a single integration
to find the first moments.
Q
Q
y
x
x
y
el
b
a
2
el
2
b
2a
dA
x
4
4
4
5
x
5
a
0
dA
a
0
xydx
y
2
2
a b
4
ydx
2
ab
10
a
0
b
x
a
a
0
2
1 b
2 a
x
2
2
x
dx
2
2
dx
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Vector Mechanics for Engineers: Statics
Sample Problem 5.4
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
• Or, using horizontal strips, perform a single
integration to find the first moments.
Q
Q
y
x
1
2
b
0
x
b
0
y
el
dA
a
el
2
dA
a
ay
b
1 2
a x
2
2
a
b
y
ax y
3
2
ax dy
2
dy
a b
ydy
4
2
ab
dy
10
b
0
a
2
x
2
a
y
a
b
1 2
2
y
dy
1 2
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dy

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Vector Mechanics for Engineers: Statics
Sample Problem 5.4
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
• Evaluate the centroid coordinates.
xA Qy
2 b
ab a
x
3 4
Q
yA x
2
ab ab
y
3 10
x
3
4
a
3
y
10
b
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Vector Mechanics for Engineers: Statics
Theorems of Pappus-Guldinus
• Surface of revolution is generated by rotating a
plane curve about a fixed axis.
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• Area of a surface of revolution is
equal to the length of the generating
curve times the distance traveled by
the centroid through the rotation.
A
2
yL
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Vector Mechanics for Engineers: Statics
Theorems of Pappus-Guldinus
• Body of revolution is generated by rotating a plane
area about a fixed axis.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
• Volume of a body of revolution is
equal to the generating area times
the distance traveled by the centroid
through the rotation.
V
2
y A
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Vector Mechanics for Engineers: Statics
Sample Problem 5.7
The outside diameter of a pulley is 0.8
m, and the cross section of its rim is as
shown. Knowing that the pulley is
made of steel and that the density of
steel is
7. 8510
determine the mass and weight of the
rim.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
3
kg
m
3
SOLUTION:
• Apply the theorem of Pappus-Guldinus
to evaluate the volumes or revolution
for the rectangular rim section and the
inner cutout section.
• Multiply by density and acceleration
to get the mass and acceleration.
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Vector Mechanics for Engineers: Statics
Sample Problem 5.7
SOLUTION:
• Apply the theorem of Pappus-Guldinus
to evaluate the volumes or revolution for
the rectangular rim section and the inner
cutout section.
• Multiply by density and acceleration to
get the mass and acceleration.
3 3 639 3 3
7.
8510
kg m 7.
6510
mm 10
m mm
2
60.
0 kg 9.
81m
s
m V
m 60.
0 kg
W mg
W 589 N
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
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Vector Mechanics for Engineers: Statics
Distributed Loads on Beams
W
OP L
wdx
dA A
0
W
L
OPAxdA xA
0
xdW
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
• A distributed load is represented by plotting the load
per unit length, w (N/m) . The total load is equal to
the area under the load curve.
• A distributed load can be replace by a concentrated
load with a magnitude equal to the area under the
load curve and a line of action passing through the
area centroid.
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Vector Mechanics for Engineers: Statics
Sample Problem 5.9
A beam supports a distributed load as
shown. Determine the equivalent
concentrated load and the reactions at
the supports.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
SOLUTION:
• The magnitude of the concentrated load
is equal to the total load or the area under
the curve.
• The line of action of the concentrated
load passes through the centroid of the
area under the curve.
• Determine the support reactions by
summing moments about the beam
ends.
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Vector Mechanics for Engineers: Statics
Sample Problem 5.9
SOLUTION:
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
• The magnitude of the concentrated load is equal to
the total load or the area under the curve.
F 18.
0 kN
• The line of action of the concentrated load passes
through the centroid of the area under the curve.
63 kN m
X
X
3.
5 m
18 kN
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Vector Mechanics for Engineers: Statics
Sample Problem 5.9
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
• Determine the support reactions by summing
moments about the beam ends.
M A y
6 m
18 kN3.5
m
0
0 : B
M B
y
By
10.
5
kN
6 m
18 kN6
m 3.5
m
0
0 : A
Ay
7.
5
kN
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Vector Mechanics for Engineers: Statics
Center of Gravity of a 3D Body: Centroid of a Volume
• Center of gravity G
W j W
j
r
r
G
G
W
W j r
W
j
j rW
j
dW r W rdW
W G
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
• Results are independent of body orientation,
xW
xV
xdW yW
ydW zW
zdW
• For homogeneous bodies,
W V and dW dV
xdV yV
ydV zV
zdV
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Vector Mechanics for Engineers: Statics
Centroids of Common 3D Shapes
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5 - 28

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Vector Mechanics for Engineers: Statics
Composite 3D Bodies
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
• Moment of the total weight concentrated at the
center of gravity G is equal to the sum of the
moments of the weights of the component parts.
X
W
xW
Y W yW
Z W zW
• For homogeneous bodies,
X
V
xV
Y V yV
Z V zV
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Vector Mechanics for Engineers: Statics
Sample Problem 5.12
Locate the center of gravity of the
steel machine element. The diameter
of each hole is 1 in.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
SOLUTION:
• Form the machine element from a
rectangular parallelepiped and a
quarter cylinder and then subtracting
two 1-in. diameter cylinders.
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Vector Mechanics for Engineers: Statics
Sample Problem 5.12
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5 - 31

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Vector Mechanics for Engineers: Statics
Sample Problem 5.12
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
X
Y
Z
xV
V
yV
V
zV
V
4 3
3.
08 in 5.286
in
X
4 3
5.047 in 5.286
in
Y
4 3
1.618
in 5.286
in
Z
0.
577
0.
577
0.
577
in.
in.
in.
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