6161103 10.7 moments of inertia for an area about inclined axes

1. 10.7 Moments of Inertia for an
Area about Inclined Axes
10.7 Moments of Inertia for an
Area about Inclined Axes
1
In structural and mechanical design,
necessary to calculate the moments and
product of inertia I
u
, I
v
and I
uv
for an area
with respect to a set of inclined u and v with respect to a set of inclined u and v axes when the values
of θ, I
x
, I
y
and I
xy
are known
1
Use transformation
equations which relate
the x, y and u, v coordinates

2. θ
For moments and product of inertia of
x y v
y x u
sin cos
sin cos
θ θ
θ
θ
− =+
=
10.7 Moments of Inertia for an
Area about Inclined Axes
10.7 Moments of Inertia for an
Area about Inclined Axes
θ
For moments and product of inertia of dA about the u and v axes,
dA x y y x uvdA dI
dA y x dAu dI
dA x y dAv dI
uv
v
u
) sin cos )( sin cos (
) sin cos (
) sin cos (
2 2
2 2
θ θ θ θ
θ θ
θ θ
− + = =
+ = =
− = =

3. θ
Integrating,
θ
θ
θ
θ
θ θ θ θ
2 2
2 2
cos
sin
2
cos
sin
cos sin 2 sin cos
+
+
=
− + =
xy
y
x
v
xy y x u
I
I
I
I
I I I I
10.7 Moments of Inertia for an
Area about Inclined Axes
10.7 Moments of Inertia for an
Area about Inclined Axes
θ
Simplifying using trigonometric identities,
θ θ θ
θ θ θ
θ θ θ θ θ θ
θ
θ
θ
θ
2 2
2 2
sin cos 2cos
cos sin 2 2sin
) sin (cos 2 cos sin cos sin
cos
sin
2
cos
sin
− =
=
− + − =
+
+
=
xy y x uv
xy
y
x
v
I I I I
I
I
I
I

4. y
x
xy
y x y x
v
xy
y x y x
u
I
I
I
I
I
I I I I
I
I
I I I I
I
+
−
=
+
−
−
+
=
−
−
+
+
=
θ
θ
θ θ
θ θ
2
cos
2
2
sin
2sin 2cos
2 2
2sin 2cos
2 2
10.7 Moments of Inertia for an
Area about Inclined Axes
10.7 Moments of Inertia for an
Area about Inclined Axes
θ
Polar moment of inertia about the z axis
passing through point O is independent of
the u and v axes
y x v u O
xy
y
x
uv
I I I I J
I
I
I
I
+ = + =
+
−
=
θ
θ
2
cos
2
2
sin
2

5. Principal Moments of Inertia 1
I
u
, I
v
and I
uv
depend on the angle of
inclination θof the u, v axes
1
To determine the orientation of these axes
10.7 Moments of Inertia for an
Area about Inclined Axes
10.7 Moments of Inertia for an
Area about Inclined Axes
1
To determine the orientation of these axes about which the moments of inertia for the
area I
u
and I
v
are maximum and minimum
1
This particular set of axes is called the
principal axes of the area and the
corresponding moments of inertia with
respect to these axes are called the principal
moments of inertia

6. Principal Moments of Inertia 1
There is a set of principle axes for every
chosen origin O 1
For the structural and mechanical design of a
10.7 Moments of Inertia for an
Area about Inclined Axes
10.7 Moments of Inertia for an
Area about Inclined Axes
1
For the structural and mechanical design of a member, the origin O is generally located at
the cross-sectional area’s centroid 1
The angle θ= θ
p
defines the orientation of
the principal axes for the area. Found by
differentiating with respect to θand setting
the result to zero

7. Principal Moments of Inertia θ
Therefore
0 2cos 2 2sin
2
2
xy
y x u
I
I I
d
dI
=
= −







−
−=
θ
θ
θ θ
θ
10.7 Moments of Inertia for an
Area about Inclined Axes
10.7 Moments of Inertia for an
Area about Inclined Axes
θ
Equation has 2 roots, θ
p1
and
θ
p2
which are 90°apart and
so specify the inclination of
the principal axes
( )
2/
2tan
y x
xy
p
p
I I
I
−
−
=
=
θ
θ
θ

8. Principal Moments of Inertia
2
2
1 1
2
/ 2sin,
xy
y x
xy p p
I
I I
I For
+







−
−=
θ θ
10.7 Moments of Inertia for an
Area about Inclined Axes
10.7 Moments of Inertia for an
Area about Inclined Axes
2
2
2
2
2
2 2
2
2
1
2
/
2
2cos
2
/ 2sin,
2
/
2
2cos
xy
y x y x
p
xy
y x
xy p p
xy
y x y x
p
I
I I I I
I
I I
I For
I
I I I I
+





−





−
−=
+







−
=
+







−







−
=
θ
θ θ
θ

9. Principal Moments of Inertia θ
Depending on the sign chosen, this result gives the maximum or minimum moment of inertia for
2
2
max
min
2 2
xy
y x y x
I
I I I I
I+







−
±
+
=
10.7 Moments of Inertia for an
Area about Inclined Axes
10.7 Moments of Inertia for an
Area about Inclined Axes
θ
Depending on the sign chosen, this result gives the maximum or minimum moment of inertia for the area
θ
It can be shown that I
uv
= 0, that is, the product
of inertia with respect to the principal axes is
zero
θ
Any symmetric axis represent a principal axis of
inertia for the area

10. Example 10.9 Determine the principal moments of inertia
for the beam’s
cross
-
sectional area with
10.7 Moments of Inertia for an
Area about Inclined Axes
10.7 Moments of Inertia for an
Area about Inclined Axes
cross
-
sectional area with
respect to an axis passing
through the centroid.

11. Solution θ
Moment and product of inertia of the cross-sectiona l
area with respect to the x, y axes have been
computed in the previous examples
10.7 Moments of Inertia for an
Area about Inclined Axes
10.7 Moments of Inertia for an
Area about Inclined Axes
θ
Using the angles of inclination of principal axes u
and v θ
Thus,
(
)
(
)
(
)
( )
( )
( ) ( )
[ ]
o o
o o
1. 57 ,9. 32
2. 114 2, 8. 65 2
22.2
2/ 10 60.5 10 90.2
10 00.3
2/
2tan
10 00.3 10 60.5 10 90.2
2 1
2 1
9 9
9
4 9 4 9 4 9
= −=
= −=
−=
−
=
−
−
=
−= = =
p p
p p
y x
xy
p
z y xI I
I
mm I mm I mm I
θ θ
θ θ
θ

12. Solution θ
For principal of inertia with respect to the u
and v axes
2
2
max
min
2
2
I
I I I I
I
xy
y x y x
+

−
±
+
=
10.7 Moments of Inertia for an
Area about Inclined Axes
10.7 Moments of Inertia for an
Area about Inclined Axes
θ
or
( ) ( )
( ) ( )
( )
[ ]
( ) ( )
( ) ( )
4 9
min
4 9
max
9 9 max
min
2
9
2
9 9
9 9
min
10 960.0 , 10 54.7
10 29.3 10 25.4
10 00.3
2
10 60.5 10 90.2
2
10 60.5 10 90.2
2
2
mm I mm I
I
I
I
xy
= =
± =
−+





−
±
+
=
+


±
=

13. 10.7 Moments of Inertia for an
Area about Inclined Axes
10.7 Moments of Inertia for an
Area about Inclined Axes
Solution 1
Maximum moment of inertia occurs with
respect to the selected u axis since by
inspection, most of the cross
-
sectional area is
inspection, most of the cross
-
sectional area is
farthest away from this axis
1
Maximum moment of inertia occurs at the u
axis since it is located within ±45°of the y
axis, which has the largest value of I