izmir institute of technology department of architecture ar231 fall12/13 15.01.2015 dr. engin aktaş...
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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
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.
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
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
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.
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
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
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
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.
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.
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.
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.
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.
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
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
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
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
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.
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.
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
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