nine beam sections - geometric properties Sections 1 - - PowerPoint PPT Presentation

S2018abn Sections 1 Lecture 9 Architectural Structures ARCH 331

nine

beam sections - geometric properties

lecture ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN ARCH 331

• DR. ANNE NICHOLS

SPRING 2018

S2018abn Sections 2 Lecture 9 Architectural Structures ARCH 331

Center of Gravity

• location of equivalent weight
• determined with calculus
• sum element weights

W1 W4 W2 W3 W y x z



 dW W

S2018abn Sections 3 Lecture 9 Architectural Structures ARCH 331

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

S2018abn Sections 4 Lecture 9 Architectural Structures ARCH 331

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    



S2018abn Sections 5 Lecture 9 Architectural Structures ARCH 331

Centroid

• for a line, sum up length

 

L L x x   

 

L L y y   

L

S2018abn Sections 6 Lecture 9 Architectural Structures ARCH 331

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 Q y 

S2018abn Sections 7 Lecture 9 Architectural Structures ARCH 331

Symmetric Areas

• symmetric about

an axis

• symmetric about

a center point

• mirrored symmetry

S2018abn Sections 8 Lecture 9 Architectural Structures ARCH 331

Composite Areas

• made up of basic shapes
• areas can be negative
• (centroids can be negative for any area)
• (-)

+ 

S2018abn Sections 9 Lecture 9 Architectural Structures ARCH 331

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

S2018abn Sections 10 Lecture 9 Architectural Structures ARCH 331

Area Centroids

• Table 6.1 – pg. 304

b h

3 b

right triangle only

S2018abn Sections 11 Lecture 9 Architectural Structures ARCH 331

Moments of Inertia

• 2nd moment area

–math concept

– area x (distance)2

• need for behavior of

– beams – columns

S2018abn Sections 12 Lecture 9 Architectural Structures ARCH 331

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

Moment of Inertia

dx y x

el

x dx dA = ydx



 dA x I y

2



 dA y I x

2

S2018abn Sections 13 Lecture 9 Architectural Structures ARCH 331

Moment of Inertia

• same area moved away a distance

–larger I

x x x x

S2018abn Sections 14 Lecture 9 Architectural Structures ARCH 331

Polar Moment of Inertia

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

 pole

• r



 dA r J o

2

S2018abn Sections 15 Lecture 9 Architectural Structures ARCH 331

Radius of Gyration

• measure of inertia with respect to area

A I r

x x 

S2018abn Sections 16 Lecture 9 Architectural Structures ARCH 331

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

2

Ad I I  

2 y cx x

Ad I I  

2

Ad I I    

2 y x

Ad I  

S2018abn Sections 17 Lecture 9 Architectural Structures ARCH 331

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 (

y

  ) x x ˆ d (

x

 

S2018abn Sections 18 Lecture 9 Architectural Structures ARCH 331

Area Moments of Inertia

• Table 6.2 – pg. 315: (bars refer to centroid)

– x, y – x’, y’ – C