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IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13
11.04.23 Dr. Engin Aktaş 1
Distributed Forces: Distributed Forces: Centroids and Centers of GravityCentroids and Centers of Gravity
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IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13
11.04.23 Dr. Engin Aktaş 2
Introduction
• The earth exerts a gravitational force on each of the particles forming a body. All these forces can be represented 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 similar to the center of gravity of a body. The concept of the first moment of an area is used to locate the center of gravity.
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IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13
11.04.23 Dr. Engin Aktaş 3
Center of Gravity of a 2D Body• Center of gravity of a plate
dWy
WyWyM
dWx
WxWxM
x
y
• Center of gravity of a wire
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IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13
11.04.23 Dr. Engin Aktaş 4
Centroids and First Moments of Areas and Lines
x
QdAyAy
y
QdAxAx
dAtxAtx
dWxWx
x
y
respect toh moment witfirst
respect toh moment witfirst
• Centroid of an area
dLyLy
dLxLx
dLaxLax
dWxWx
• Centroid of a line
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IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13
11.04.23 Dr. Engin Aktaş 5
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.
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IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13
11.04.23 Dr. Engin Aktaş 6
Centroids of Common Shapes of Areas
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IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13
11.04.23 Dr. Engin Aktaş 7
Centroids of Common Shapes of Lines
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IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13
11.04.23 Dr. Engin Aktaş 8
Composite Plates and Areas• Composite plates
WyWY
WxWX
• Composite area
AyAY
AxAX
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IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13
11.04.23 Dr. Engin Aktaş 9
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.
SOLUTION:
• Divide the area into a triangle, rectangle, and semicircle with a circular cutout.
• Compute the coordinates of the area centroid by dividing the first moments by the total area.
• Find the total area and first moments of the triangle, rectangle, and semicircle. Subtract the area and first moment of the circular cutout.
• Calculate the first moments of each area with respect to the axes.
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IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13
11.04.23 Dr. Engin Aktaş 10
Sample Problem 5.1
33
33
mm10758
mm10506
y
x
Q
Q• Find the total area and first moments of the triangle, rectangle, and semicircle. Subtract the area and first moment of the circular cutout.
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IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13
11.04.23 Dr. Engin Aktaş 11
Sample Problem 5.1
23
33
mm1013.828
mm107.757
A
AxX
mm 8.54X
23
33
mm1013.828
mm102.506
A
AyY
mm 6.36Y
• Compute the coordinates of the area centroid by dividing the first moments by the total area.
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IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13
11.04.23 Dr. Engin Aktaş 12
Determination of Centroids by Integration
ydxy
dAyAy
ydxx
dAxAx
el
el
2
dyxay
dAyAy
dyxaxa
dAxAx
el
el
2
drr
dAyAy
drr
dAxAx
el
el
2
2
2
1sin
3
2
2
1cos
3
2
dAydydxydAyAy
dAxdydxxdAxAx
el
el• Double integration to find the first moment
may be avoided by defining dA as a thin rectangle or strip.
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IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13
11.04.23 Dr. Engin Aktaş 13
Sample Problem 5.4
Determine by direct integration the location of the centroid of a parabolic spandrel.
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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IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13
11.04.23 Dr. Engin Aktaş 14
Sample Problem 5.4SOLUTION:
• Determine the constant k.
2121
22
22
2
yb
axorx
a
by
a
bkakb
xky
• Evaluate the total area.
3
30
3
20
22
ab
x
a
bdxx
a
bdxy
dAA
aa
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IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13
11.04.23 Dr. Engin Aktaş 15
Sample Problem 5.4• Using vertical strips, perform a single integration
to find the first moments.
1052
2
1
2
44
2
0
5
4
2
0
22
2
2
0
4
2
0
22
abx
a
b
dxxa
bdxy
ydAyQ
bax
a
b
dxxa
bxdxxydAxQ
a
a
elx
a
a
ely
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IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13
11.04.23 Dr. Engin Aktaş 16
Sample Problem 5.4• Or, using horizontal strips, perform a single
integration to find the first moments.
10
42
1
22
2
0
2321
2121
2
0
22
0
22
abdyy
b
aay
dyyb
aaydyxaydAyQ
badyy
b
aa
dyxa
dyxaxa
dAxQ
b
elx
b
b
ely
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IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13
11.04.23 Dr. Engin Aktaş 17
Sample Problem 5.4• Evaluate the centroid coordinates.
43
2baabx
QAx y
ax4
3
103
2ababy
QAy x
by10
3
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IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13
11.04.23 Dr. Engin Aktaş 18
Distributed Loads on Beams
• 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. AdAdxwW
L
0
AxdAxAOP
dWxWOP
L
0
• A distributed load can be replaced by a concentrated load with a magnitude equal to the area under the load curve and a line of action passing through the centroid of the area.
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IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13
11.04.23 Dr. Engin Aktaş 19
Sample Problem 5.9
A beam supports a distributed load as shown. Determine the equivalent concentrated load and the reactions at the supports.
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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IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13
11.04.23 Dr. Engin Aktaş 20
Sample Problem 5.9SOLUTION:
• The magnitude of the concentrated load is equal to the total load or the area under the curve.
kN 0.18F
• The line of action of the concentrated load passes through the centroid of the area under the curve.
kN 18
mkN 63 X m5.3X
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IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13
11.04.23 Dr. Engin Aktaş 21
Sample Problem 5.9• Determine the support reactions by summing
moments about the beam ends.
0m .53kN 18m 6:0 yA BM
kN 5.10yB
0m .53m 6kN 18m 6:0 yB AM
kN 5.7yA