eight beam sections - geometric properties Sections 1 Elements of - - PowerPoint PPT Presentation

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eight beam sections - geometric properties Sections 1 Elements of - - PowerPoint PPT Presentation

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E LEMENTS OF A RCHITECTURAL S TRUCTURES : F ORM, B EHAVIOR, AND D ESIGN ARCH 614 D R. A NNE N ICHOLS S PRING 2019 lecture eight beam sections - geometric properties Sections 1 Elements of Architectural Structures S2019abn Lecture 8 ARCH

SLIDE 1

S2019abn Sections 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

SLIDE 2

S2019abn Sections 2 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



 dW W

SLIDE 3

S2019abn Sections 3 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 W W x x x W x M

n i i i y

      

 

1

 

W W W y y y W y M

n i i i x

      

 

1

“bar” means average

SLIDE 4

S2019abn Sections 4 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 A x x   

 

A A y y    A t W    



SLIDE 5

S2019abn Sections 5 Lecture 8 Elements of Architectural Structures ARCH 614

Centroid

for a line, sum up length

 

L L x x   

 

L L y y   

L

SLIDE 6

S2019abn Sections 6 Lecture 8 Elements of Architectural Structures ARCH 614

1st Moment Area

math concept
the moment of an area about an axis

A y Qx 

 

A A y y   

x y y x A (area)

A x Qy 

SLIDE 7

S2019abn Sections 7 Lecture 8 Elements of Architectural Structures ARCH 614

Symmetric Areas

symmetric about

an axis

symmetric about

a center point

mirrored symmetry

SLIDE 8

S2019abn Sections 8 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)
(-)

+ 

SLIDE 9

S2019abn Sections 9 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 y A x

A y

y ˆ

x ˆ

SLIDE 10

S2019abn Sections 10 Lecture 8 Elements of Architectural Structures ARCH 614

Area Centroids

Figure A.1 – pg 598

b h

3 b

right triangle only

SLIDE 11

S2019abn Sections 11 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

SLIDE 12

S2019abn Sections 12 Lecture 8 Elements of Architectural Structures ARCH 614

about any reference axis
can be negative
resistance to bending and buckling

Moment of Inertia



    dA x A x I

i y 2 2

dx y x

x dx dA = ydx



    dA y A y I

i x 2 2

) (

z I

x x

 



SLIDE 13

S2019abn Sections 13 Lecture 8 Elements of Architectural Structures ARCH 614

Moment of Inertia

same area moved away a distance

– larger I

x x x x

SLIDE 14

S2019abn Sections 14 Lecture 8 Elements of Architectural Structures ARCH 614

Polar Moment of Inertia

for roundish shapes
uses polar coordinates (r and )
resistance to twisting

 pole



 dA r J o

SLIDE 15

S2019abn Sections 15 Lecture 8 Elements of Architectural Structures ARCH 614

Radius of Gyration

measure of inertia with respect to area

A I r

x x 

SLIDE 16

S2019abn Sections 16 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

Ad I I    

Ad I I  

2 y x

Ad I  

Az I I



SLIDE 17

S2019abn Sections 17 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

y x A x

A y

I I

y ˆ

x ˆ

) y y ˆ d (

  ) x x ˆ d (

 

SLIDE 18

S2019abn Sections 18 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