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S2015abnSections 1Lecture 8
Elements of Architectural StructuresARCH 614
ELEMENTS OF ARCHITECTURAL STRUCTURES:
FORM, BEHAVIOR, AND DESIGN
ARCH 614
DR. ANNE NICHOLS
SPRING 2015
eight
beam sections -
geometric properties
lecture
S2015abnSections 2Lecture 8
Elements of Architectural StructuresARCH 614
Center of Gravity
• location of equivalent weight
• determined with calculus
• sum element weights
∆W1 ∆W4 ∆W2 ∆W3
∑∆W y
x
z
∫= dWW
S2015abnSections 3Lecture 8
Elements of Architectural StructuresARCH 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
S2015abnSections 4Lecture 8
Elements of Architectural StructuresARCH 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 ∆=∆ γ γ
S2015abnSections 5Lecture 8
Elements of Architectural StructuresARCH 614
Centroid
• for a line, sum up length
( )L
Lxx
∆∑=
( )L
Lyy
∆∑=
∆L
S2015abnSections 6Lecture 8
Elements of Architectural StructuresARCH 614
1st Moment Area
• math concept
• the moment of an area about an axis
AyQx =
( )A
Ayy
∆∑=
x
y
y
x
A (area)
AxQy =
S2015abnSections 7Lecture 8
Elements of Architectural StructuresARCH 614
Symmetric Areas
• symmetric about
an axis
• symmetric about
a center point
• mirrored symmetry
S2015abnSections 8Lecture 8
Elements of Architectural StructuresARCH 614
Composite Areas
• made up of basic shapes
• areas can be negative
• (centroids can be negative for any area)
-(-)
+⇒
S2015abnSections 9Lecture 8
Elements of Architectural StructuresARCH 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
Σ
x yAx Ay
yx
S2015abnSections 10Lecture 8
Elements of Architectural StructuresARCH 614
Area Centroids
• Figure A.1 – pg 598
b
h
3
b
right triangle only
S2015abnSections 11Lecture 8
Elements of Architectural StructuresARCH 614
Moments of Inertia
• 2nd moment area
– math concept
– area x (distance)2
• need for behavior of
– beams
– columns
S2015abnSections 12Lecture 8
Elements of Architectural StructuresARCH 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 = y⋅dx
∫=∆∑= dAyAyI ix
22
)( 2azIor xx ∑=−
S2015abnSections 13Lecture 8
Elements of Architectural StructuresARCH 614
Moment of Inertia
• same area moved away a distance
– larger I
x x xx
S2015abnSections 14Lecture 8
Elements of Architectural StructuresARCH 614
Polar Moment of Inertia
• for roundish shapes
• uses polar coordinates (r and θ)
• resistance to twisting
θ pole
o
r
∫= dArJo
2
S2015abnSections 15Lecture 8
Elements of Architectural StructuresARCH 614
Radius of Gyration
• measure of inertia with respect to area
A
Ir x
x =
S2015abnSections 16Lecture 8
Elements of Architectural StructuresARCH 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 +=
S2015abnSections 17Lecture 8
Elements of Architectural StructuresARCH 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 −=
S2015abnSections 18Lecture 8
Elements of Architectural StructuresARCH 614
Area Moments of Inertia
• Figure A.11 – pg. 611: (bars refer to centroid)
– x, y
– x’, y’
– C
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