beam sections - geometric properties nine

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nine

beam sections -

lecture

A

RCHITECTURAL

S

TRUCTURES

:

F

ORM,

B

EHAVIOR, AND

D

ESIGN

A

RCH 331

HÜDAVERDİ TOZAN

(2)

Center of Gravity

• location of equivalent weight

• determined with calculus

• sum element weights



W

₁



W

₄



W

₂



W

₃



W

y

x

z





dW

W

(3)

Center of Gravity

• “average” x & y from moment



W

1 

W

4 

W

2 

W

3 

W

y

x

z





W

x

W

x

W

x

M

n

i

y



















1 



(4)

Centroid

• “average” x & y of an area

• for a volume of constant thickness

–

where is weight/volume

– center of gravity = centroid of area





A

x









y



A





A

t

W







(5)

Centroid

• for a line, sum up length





L

x











L

y







(6)

1 st

Moment Area

• math concept

• the moment of an area about an axis

A

y

Q

_x







A

y







_x

y

x

A (area)

A

x

Q

_y



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Symmetric Areas

• symmetric about

an axis

• symmetric about

a center point

• mirrored symmetry

(8)

Composite Areas

• made up of basic shapes

• areas can be negative

• (centroids can be negative for any area)

-

(-)

+

(9)

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

_x

_A

y

A

yˆ

xˆ

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Area Centroids

• Table 7.1 – pg. 242

b h

3 b

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Moments of Inertia

• 2

nd

_{moment area}

– math concept

– area x (distance)

2 • need for behavior of

– beams

– columns

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• about any reference axis

• can be negative

Moment of Inertia

dx

y

x

el

x dx

dA = y



dx





x

dA

I

_y

2 



y

dA

I

_x

2

(13)

Moment of Inertia

• same area moved away a distance

– larger

I

x

(14)

Polar Moment of Inertia

• for roundish shapes

• uses polar coordinates (r and



)

• resistance to twisting



pole

o

r





r

dA

J

_o

2

(15)

Radius of Gyration

• measure of inertia with respect to area

A

I

(16)

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



y



d

2 y

cx

x

I

Ad

I





2 Ad

I









2 y

x

Ad

I





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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 Ad

2 ’s

5. Fill in table and get

x

and

x

for composite

6. Sum necessary columns

y

x

A

y

A

I

yˆ

xˆ





(18)

S2013abn Sections 18

Lecture 9

Architectural Structures ARCH 331

Area Moments of Inertia

• Table 7.2 – pg. 252: (

bars refer to centroid)

– x, y

– x’, y’

– C