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MOMENTS OF INERTIA (Chapter 10)
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View Full Document APPLICATIONS
Many structural members like beams
and columns have cross sectional
shapes like I, H, C, etc.
.
Why do they usually not have solid
rectangular, square, or circular cross
sectional areas?
What primary property of these
members influences design decisions?
How can we calculate this property?
APPLICATIONS
(continued)
Many structural members are
made of tubes rather than solid
squares or rounds.
Why?
What parameters of the cross
sectional area influence the
designer’s selection?
How can
we determine the value of these
parameters for a given area?
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View Full Document THE CONCEPT OF THE MoI OF AN AREA
Consider a plate submerged in a
liquid. The pressure of a liquid at
a distance z below the surface is
given by p
=
γ
z, where
γ
is the
specific weight of the liquid.
The force on the area dA at that point is
dF
=
p dA.
The moment about the xaxis due to this force is
z (dF).
The
total moment is
∫
A
z dF
=
∫
A
γ
z
2
dA
=
γ
∫
A
( z
2
dA).
This sort of integral term also appears in solid mechanics when
determining stresses and deflection. This integral term is referred
to as the
moment of inertia
of the area of the plate about an axis.
THE CONCEPT OF THE MoI
(continued)
Consider three different possible cross sectional shapes and areas for the
beam RS.
All have the same total area and, assuming they are made of
same material, they will have the same mass per unit length.
For the given vertical loading P on the beam, which shape will
develop less internal stress and deflection?
Why?
The answer depends on the MoI of the beam about the xaxis. It turns
out that Section A has the highest MoI because most of the area is
farthest from the x axis. Hence, it has the least stress and deflection.
10cm
10cm
1cm
1cm
x
3cm
10cm
3cm
R
S
P
(C)
(B)
(A)
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View Full Document MoI
–
DEFINITION
For the differential area dA, shown in the
figure:
d I
x
=
y
2
dA ,
d I
y
=
x
2
dA ,
and,
d J
O
=
r
2
dA ,
where J
O
is the polar
moment of inertia about the pole O or z
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This note was uploaded on 04/20/2011 for the course ENG 1333 taught by Professor Brr during the Spring '10 term at American College of Computer & Information Sciences.
 Spring '10
 brr

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