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Precast Show 2017
NPCA 1
precast.org/education
CALCULATING THE CENTER OF GRAVITY
Claude Goguen, PE, LEED AP
Director of Sustainability and Technical Education
precast.org/education
OUTLINE
• Understanding the concepts of gravity
• CoG of 2 dimensional figures
• CoG of 3 dimensional figures
• Examples
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Precast Show 2017
NPCA 2
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• Moments are used to determine the center of gravity
(CG) of irregular or asymmetrical bodies
• CG of symmetrical bodies lie on their axes of
symmetry
First Moment of Area (Q)
Precast Show 2017
NPCA 3
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•Moment of area around an axis is the product of
surface area by the distance of CG from that axis
Moment and Center of Gravity
2b
2ay
x
First moment of area around “x” axis: ))(4( babQx
surface area
Distance from
CG to “x” axis
First moment of area around “y” axis: ))(4( aabQy
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Find Qx and Qy
Example
4”
2”
x
y
CG
0.67”
1.33”
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Find the CG of the following object
Example #6
3”
2”
1”
1”
Precast Show 2017
NPCA 4
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Example
Step 1: Look for an axis of symmetry. CG must lie on
the axis of symmetry
3”
2”
1”
1”
Axis of symmetry
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Example
Step 2: Determine the moment of area around a
horizontal axis (for example axis x-x)
3”
2”
1”
1”
x x
5.9)1)(12(
)5.2)(13(
xxQ
in3
5)12()13( A in2
Surface area:
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Example
Step 3: Divide Qxx by surface area to determine the
location of CG
3”
2”
1”
1”
x x
9.15/5.9 in
AQxx /
1.9”
Neutral Axis (N.A.)
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NPCA 5
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Why Locating the CG Is Important?
1- CG must always be below the main hook
(A) Stable:
Hook is above CG
(B) Unstable:
Hook is not above CG
W
Creates unbalanced
moment that tilts the
member
Rigging for Commercial Construction © 1983, Reston Publishing Co.
W
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1- CG must always be below the main hook
(A) Stable:
Hook is above CG
(B) Unstable:
Hook is not above CG
W
Rigging for Commercial Construction © 1983, Reston Publishing Co.
W
(C) Load will shift until the
CG is below the hook
W
Why Locating the CG Is Important?
precast.org/educationPicture from: W.E. Rossnagel, Handbook of Rigging © 1964, McGraw-Hill
1- CG must always be below the main hook
This may
require using
one or more of
the sling
cables with an
odd length
which is not
readily
availableWhen slings of equal length are used
Load tilts until CG is
directly below hook
Why Locating the CG Is Important?
Precast Show 2017
NPCA 6
precast.org/educationPicture from: W.E. Rossnagel, Handbook of Rigging © 1964, McGraw-Hill
1- CG must always be below the main hook
Manually
adjusting the
length of the
sling cables
CG
After adjusting sling to
proper length, install a
cable clip at this position
Why Locating the CG Is Important?
precast.org/educationRigging for Commercial Construction © 1983, Reston Publishing Co.
2- The lifting points must always be above the CG
Lifting points
Unstable: The load can
topple because the
attachments are below CG
Stable: Attachments
are above CG
Load in this
leg is critical
Why Locating the CG Is Important?
precast.org/educationRigging for Commercial Construction © 1983, Reston Publishing Co.
3- The CG must reside between the sling cables
Stable when sling
angle (A) is much
larger than angle (B)
Unstable when
sling angle (A) is
less than angle (B)
Why Locating the CG Is Important?
Precast Show 2017
NPCA 7
precast.org/educationW.E. Rossnagel, Handbook of Rigging © 1964, McGraw-Hill
3- The CG must reside between the sling cables
Increasing cable angle will increase stability
Why Locating the CG Is Important?
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Cable Angle
- Tension in the sling cables
depends on their angle as well as
the weight of the load
W=2000 lbs
F F
W
Fy
Fx
45o
FFFx 71.0)45cos(
FFFy 71.0)45sin(
lbsFyFyW 10002
lbsFyF 141471.0/
Tension in the cables:
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- Tension in the sling cable depends on its angle as well
as the weight of the load
W.E. Rossnagel, Handbook of Rigging © 1964, McGraw-Hill
- It is recommended to use 45o or larger angles
Cable Angle
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NPCA 8
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Remember: CG always lies on the axis of symmetry
a
a/3 2a/3
b
b/3
2b/32b/3b/3
Locating the Center of Gravity
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Example 3: Determine the CG of the following object
x
y3’
2’
2’
Qx = (Area)(CG from x axis)
Qy = (Area)(CG from y axis)
Method 2 Using the first moment of area (Q)
Locating the Center of Gravity
precast.org/education
Example
x
y3’
2’
2’
Rectangle
Triangle
Area (ft2)
6
2
CG from x
axis (ft)
CG from y
axis (ft)
1 1.5
3.670.67
Qx (ft3) Qy (ft3)
Total 8
6 9
1.33 7.34
7.33 16.34
Qx = (Area)(CG from x axis)
Qy = (Area)(CG from y axis)
(CG from x axis) = Qx/(Area) = 7.33/8 = 0.92 ft
(CG from y axis) = Qy/(Area) = 16.34/8 = 2.04 ft
Precast Show 2017
NPCA 9
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Example: Determine the CG of the following object
6’
15’
6’
0.5’
Example
Precast Show 2017
NPCA 10
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Example: Determine the CG of the following object
1) Plain of symmetry –
so CG must reside on
this plain
2) We can break the
object into two triangles
(sides), and a rectangle
(bottom).
Back side is mostly
empty (hole) so we can
ignore.
6’
15’
6’
0.5’
x
y
z
Example
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Rectangle
Triangle
Vol (ft3)
CG from z
axis (ft)
CG from y
axis (ft) Qz (ft4) Qy (ft4)
Total
Triangle
Example
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NPCA 11
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Rectangle
Triangle
Vol (ft3)
CG from z
axis (ft)
CG from y
axis (ft) Qz (ft4) Qy (ft4)
Total
Triangle
Example
(CG from x axis) = Qx/(Vol) =
(CG from y axis) = Qy/(Vol) =
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Example: Determine the CG of the following object
6’
15’
6’
0.5’
Example
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SUMMARY
• Look over plant drawings and make sure you understand every detail
• Are they legible and durable?
• Determine is you can simplify drawings?
• Know your tolerances
• Key is to eliminate confusion and potential for errors
• Training and Education!!!
• ASK QUESTIONS!!!
Precast Show 2017
NPCA 12
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AVOID MISCOMMUNICATIONS
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Questions?
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THANK YOU!
Any more questions?
Contact Claude at
317-571-9500 or at
Precast Show 2017
NPCA 13
precast.org/education
CALCULATING THE CENTER OF GRAVITY
Claude Goguen, PE, LEED AP
Director of Sustainability and Technical Education