VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh
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Transcript VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh
Seventh Edition
5
CHAPTER
VECTOR MECHANICS FOR ENGINEERS:
STATICS
Ferdinand P. Beer
E. Russell Johnston, Jr.
Lecture Notes:
J. Walt Oler
Texas Tech University
Distributed Forces:
Centroids and Centers
of Gravity
© 2003 The McGraw-Hill Companies, Inc. All rights reserved.
Seventh
Edition
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
•
•
•
•
•
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
© 2003 The McGraw-Hill Companies, Inc. All rights reserved.
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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.
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Vector Mechanics for Engineers: Statics
Center of Gravity of a 2D Body
• Center of gravity of a plate
M
y
xW
M
y
yW
• Center of gravity of a wire
xW
x dW
yW
y dW
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Vector Mechanics for Engineers: Statics
Centroids and First Moments of Areas and Lines
• Centroid of an area
• Centroid of a line
xW
x W x dW
x dW
x La x a dL
x At x t dA
x A x dA Q y
first moment wit
h respect to
y
h respect to
x
xL
x dL
yL
y dL
y A y dA Q x
first moment wit
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Vector Mechanics for Engineers: Statics
First Moments of Areas and Lines
• 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.
© 2003 The McGraw-Hill Companies, Inc. All rights reserved.
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Vector Mechanics for Engineers: Statics
Centroids of Common Shapes of Areas
© 2003 The McGraw-Hill Companies, Inc. All rights reserved.
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Edition
Vector Mechanics for Engineers: Statics
Centroids of Common Shapes of Lines
© 2003 The McGraw-Hill Companies, Inc. All rights reserved.
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Vector Mechanics for Engineers: Statics
Composite Plates and Areas
• Composite plates
X W xW
Y W yW
• Composite area
X A xA
Y A yA
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Vector Mechanics for Engineers: Statics
Sample Problem 5.1
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.
For the plane area shown, determine
the first moments with respect to the
x and y axes and the location of the
centroid.
• 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.
© 2003 The McGraw-Hill Companies, Inc. All rights reserved.
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Seventh
Edition
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.
© 2003 The McGraw-Hill Companies, Inc. All rights reserved.
3
3
3
3
Q x 506 . 2 10 mm
Q y 757 . 7 10 mm
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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.
3
3
x A 757 . 7 10 mm
X
3
2
A
13.828 10 mm
X 54 . 8 mm
3
3
y A 506 . 2 10 mm
Y
3
2
A
13.828 10 mm
Y 36 . 6 mm
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Vector Mechanics for Engineers: Statics
Determination of Centroids by Integration
• Double integration to find the first moment
may be avoided by defining dA as a thin
rectangle or strip.
x A x dA x dx dy x el dA
y A y dA y dx dy y el dA
xA
x el dA
x ydx
yA
y el dA
y
2
ydx
xA
x el dA
a x
2
a x dx
yA
y el dA
y a x dx
© 2003 The McGraw-Hill Companies, Inc. All rights reserved.
xA
yA
x el dA
1 2
cos r d
3
2
2r
y el dA
1 2
sin r d
3
2
2r
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Vector Mechanics for Engineers: Statics
Sample Problem 5.4
SOLUTION:
• Determine the constant k.
• Evaluate the total area.
• Using either vertical or horizontal
strips, perform a single integration to
find the first moments.
Determine by direct integration the
location of the centroid of a parabolic
spandrel.
• Evaluate the centroid coordinates.
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Vector Mechanics for Engineers: Statics
Sample Problem 5.4
SOLUTION:
• Determine the constant k.
y kx
2
b ka
2
k
b
a
y
b
a
2
x
2
or
2
a
x
b
1 2
y
1 2
• Evaluate the total area.
A
dA
a
b x3
b 2
y dx
x dx
2
2 3
a
0
0a
a
ab
3
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Vector Mechanics for Engineers: Statics
Sample Problem 5.4
• Using vertical strips, perform a single integration
to find the first moments.
a
Qy
x el dA
xy dx
b 2
x
2 x dx
0 a
a
2
b x4
a b
2 4
4
a
0
Qx
y el dA
a
2
1 b 2
y dx
x dx
2
2
2
0 a
y
a
2
b2 x5
ab
4 5
10
2 a
0
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Vector Mechanics for Engineers: Statics
Sample Problem 5.4
• Or, using horizontal strips, perform a single
integration to find the first moments.
Q y x el dA
ax
2
b
a x dy
0
a
2
x
2
dy
2
2
2
b
1 2 a
a b
a
y dy
2 0
b
4
a
1
Q x y el dA y a x dy y a
y
1 2
b
b
a
3
ay
y
1 2
b
0
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2
ab
dy
10
2
dy
2
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Vector Mechanics for Engineers: Statics
Sample Problem 5.4
• Evaluate the centroid coordinates.
xA Q y
x
ab
2
3
a b
x
4
3
a
4
yA Q x
y
ab
3
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ab
10
2
y
3
b
10
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Vector Mechanics for Engineers: Statics
Distributed Loads on Beams
L
W w dx dA A
0
OP W x dW
L
OP A x dA x A
0
• 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
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.
A beam supports a distributed load as
shown. Determine the equivalent
concentrated load and the reactions at
the supports.
• 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:
• 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.
X
63 kN m
X 3 .5 m
18 kN
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Vector Mechanics for Engineers: Statics
Sample Problem 5.9
• Determine the support reactions by summing
moments about the beam ends.
M
A
0:
B y 6 m 18 kN
3 .5 m 0
B y 10 . 5 kN
M B 0 : A y 6 m 18 kN 6 m 3 .5 m 0
A y 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
rG W j
rG W j
W
dW
• Results are independent of body orientation,
W j
r W j
r W j
rG W
r
dW
xW
xdW
yW
ydW
zW
zdW
• For homogeneous bodies,
W V and dW dV
xV
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xdV
yV
ydV
zV
zdV
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Vector Mechanics for Engineers: Statics
Centroids of Common 3D Shapes
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Vector Mechanics for Engineers: Statics
Composite 3D Bodies
• 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
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Y
V yV
Z
V zV
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Vector Mechanics for Engineers: Statics
Sample Problem 5.12
SOLUTION:
• Form the machine element from a
rectangular parallelepiped and a
quarter cylinder and then subtracting
two 1-in. diameter cylinders.
Locate the center of gravity of the
steel machine element. The diameter
of each hole is 1 in.
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Vector Mechanics for Engineers: Statics
Sample Problem 5.12
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Vector Mechanics for Engineers: Statics
Sample Problem 5.12
X x V V 3 . 08 in
4
5 .286 in 3
X 0 . 577 in.
Y y V V 5.047 in
4
5 .286 in 3
Y 0 . 577 in.
Z z V V 1 .618 in
4
5 .286 in 3
Z 0 . 577 in.
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