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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