Architectural Structures: Center of Gravity, Centroid, Moments of Inertia, Slides of Structural Design and Architecture

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Sections 1 Lecture 9

ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN

nine

beam sections -

geometric properties

lecture

Sections 2 Lecture 9 Architectural Structures

Center of Gravity

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

W 4 W 1 W 3 W 2

y W

x

z

W  (^)  dW

Sections 4 Lecture 9

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 A

x

 

A

y A

y

 W   t  A 

Sections 5 Lecture 9

Centroid

• for a line, sum up length

 

L

x L

x

 

L

y L

y

L

Sections 7 Lecture 9

Symmetric Areas

• symmetric about an axis
• symmetric about a center point
• mirrored symmetry

Sections 8 Lecture 9

Composite Areas

• made up of basic shapes
• areas can be negative

• (centroids can be negative for any area)

(-)

^ +

Sections 10 Lecture 9

Area Centroids

• Table 7.1 – pg. 242

b

h^ b 3 right triangle only

Sections 11 Lecture 9

Moments of Inertia

• 2 nd^ moment area
  • math concept
  • area x (distance)^2
• need for behavior of
  • beams
  • columns

Sections 13 Lecture 9 Architectural Structures

Moment of Inertia

• same area moved away a distance
  • larger I

x x x x

Sections 14 Lecture 9

Polar Moment of Inertia

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

poleo 

r

J (^) o  (^)  r dA 2

Sections 16 Lecture 9 Architectural Structures

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 B d

I  I  Ad^2

2 I (^) x  Icx  Ad y

I   I   Ad^2

2  I (^) x  Ady

Sections 17 Lecture 9

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
7. Sum I’s and Ad^2 ’s

x xA y yA I

I

x ˆ y ˆ

( dy  yˆ  y )

( dx  xˆ  x )