e architectural structures b design arch 614...
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S2019abnSections 1
Lecture 8
Elements of Architectural Structures
ARCH 614
ELEMENTS OF ARCHITECTURAL STRUCTURES:
FORM, BEHAVIOR, AND DESIGN
ARCH 614
DR. ANNE NICHOLS
SPRING 2019
eight
beam sections -
geometric properties
lecture
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Lecture 8
Elements of Architectural Structures
ARCH 614
Center of Gravity
• location of equivalent weight
• determined with calculus
• sum element weights
W1 W4 W2 W3
W y
x
z
dWW
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Lecture 8
Elements of Architectural Structures
ARCH 614
Center of Gravity
• “average” x & y from moment
W1 W4 W2 W3
W y
x
z
W
WWx
xxWxMn
i
iiy
1
W
WWy
yyWyMn
i
iix
1
“bar” means average
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Lecture 8
Elements of Architectural Structures
ARCH 614
Centroid
• “average” x & y of an area
• for a volume of constant thickness
– where is weight/volume
– center of gravity = centroid of area
A
Axx
A
Ayy
AtW
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Lecture 8
Elements of Architectural Structures
ARCH 614
Centroid
• for a line, sum up length
L
Lxx
L
Lyy
L
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Lecture 8
Elements of Architectural Structures
ARCH 614
1st Moment Area
• math concept
• the moment of an area about an axis
AyQx
A
Ayy
x
y
y
x
A (area)
AxQy
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Lecture 8
Elements of Architectural Structures
ARCH 614
Symmetric Areas
• symmetric about
an axis
• symmetric about
a center point
• mirrored symmetry
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Lecture 8
Elements of Architectural Structures
ARCH 614
Composite Areas
• made up of basic shapes
• areas can be negative
• (centroids can be negative for any area)
-(-)
+
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Lecture 8
Elements of Architectural Structures
ARCH 614
Basic Procedure
1. Draw reference origin (if not given)
2. Divide into basic shapes (+/-)
3. Label shapes
4. Draw table
5. Fill in table
6. Sum necessary columns
7. Calculate x and y
Component Area
x yAx Ay
yx
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Lecture 8
Elements of Architectural Structures
ARCH 614
Area Centroids
• Figure A.1 – pg 598
b
h
3
b
right triangle only
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Lecture 8
Elements of Architectural Structures
ARCH 614
Moments of Inertia
• 2nd moment area
–math concept
– area x (distance)2
• need for behavior of
– beams
– columns
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Lecture 8
Elements of Architectural Structures
ARCH 614
• about any reference axis
• can be negative
• resistance to bending and buckling
Moment of Inertia
dAxAxI iy
22
dx
y
x
elx dx
dA = ydx
dAyAyI ix
22
)( 2azIor xx
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Lecture 8
Elements of Architectural Structures
ARCH 614
Moment of Inertia
• same area moved away a distance
– larger I
x x xx
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Lecture 8
Elements of Architectural Structures
ARCH 614
Polar Moment of Inertia
• for roundish shapes
• uses polar coordinates (r and )
• resistance to twisting
pole
o
r
dArJ o
2
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Lecture 8
Elements of Architectural Structures
ARCH 614
Radius of Gyration
• measure of inertia with respect to area
A
Ir x
x
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Lecture 8
Elements of Architectural Structures
ARCH 614
Parallel Axis Theorem
• can find composite I once composite
centroid is known (basic shapes)
axis through centroid
at a distance d away
from the other axis
axis to find moment of
inertia about
y
A
dA
A
B B
y
d
2AdII
2AdII
2
yx AdI
2AzII o
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Lecture 8
Elements of Architectural Structures
ARCH 614
Basic Procedure
1. Draw reference origin (if not given)
2. Divide into basic shapes (+/-)
3. Label shapes
4. Draw table with A, x, xA, y, yA, I’s, d’s,
and Ad2’s
5. Fill in table and get x and x for composite
6. Sum necessary columns
7. Sum I’s and Ad2’s
yx Ax Ay I
I
yx
)yyd( y )xxd( x
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Lecture 8
Elements of Architectural Structures
ARCH 614
Area Moments of Inertia
• Figure A.11 – pg. 611: (bars refer to centroid)
– x, y
– x’, y’
– C