PROBLEM 9.46
Determine the polar moment of inertia of the area shown with respect to
(a) Point O, (b) the centroid of the area.
SOLUTION
Determination of centroid C of entire section:
Section Area, in2 ,in.x 3
,inxA
1 2
(4) 4
4
16
3
21.333
PROBLEM 9.47
Determine the polar moment of inertia of the area shown with respect
to (a) Point O, (b) the centroid of the area.
SOLUTION
Determination of centroid C of entire section
Dimensions in mm Symmetry: 0X
PROBLEM 9.47 (Continued)
(b) Section
: 364
1(80)(60) 1.44 10 mm
12
x
I
For Iy consider the following two triangles
364
1
2 (60)(40) 0.640 10 mm
y
I
PROBLEM 9.48
Determine the polar moment of inertia of the area shown with respect
to (a) Point O, (b) the centroid of the area.
SOLUTION
First locate centroid C of the figure.
Note that symmetry implies
0.Y
PROBLEM 9.48 (Continued)
4
2
64
22
3
64
() (42mm)
4
2.44392 10 mm
( ) (54 mm)(27 mm)[(54 mm) (27 mm) ]
8
2.08696 10 mm
O
O
J
J
PROBLEM 9.49
Two channels and two plates are used to form the column
section shown. For b 200 mm, determine the moments of
inertia and the radii of gyration of the combined section with
respect to the centroidal x and y axes.
SOLUTION
PROBLEM 9.49 (Continued)
Channel Plate
22 3
1
( ) 2[ ] 2 (10 mm)(375 mm)
12
yy y
IIAd IAd
PROBLEM 9.50
Two L6 4 2
1-in. angles are welded together to form the section shown.
Determine the moments of inertia and the radii of gyration of the combined
section with respect to the centroidal x and y axes.
SOLUTION
PROBLEM 9.51
Two L6 4 2
1-in. angles are welded together to form the section
shown. Determine the moments of inertia and the radii of gyration
of the combined section with respect to the centroidal x and y axes.
SOLUTION
W section: 24 4
9.12 in 110 in 37.1in
xy
AI I
Angle: 244
1.44 in 1.23 in 1.23 in
xy
AI I
total W A
2
4
9.12 4(1.44) 14.880 in
AAA
PROBLEM 9.52
Two 20-mm steel plates are welded to a rolled S section as shown. Determine
the moments of inertia and the radii of gyration of the combined section with
respect to the centroidal x and y axes.
SOLUTION
S section:
2
64
64
6010 mm
90.3 10 mm
3.88 10 mm
x
y
A
I
I
PROBLEM 9.53
A channel and a plate are welded together as shown to form a
section that is symmetrical with respect to the y axis.
Determine the moments of inertia of the combined section
with respect to its centroidal x and y axes.
SOLUTION
PROBLEM 9.54
The strength of the rolled W section shown is increased by welding a channel to
its upper flange. Determine the moments of inertia of the combined section with
respect to its centroidal
x
and
y
axes.
SOLUTION
W section:
2
64
64
14, 400 mm
554 10 mm
63.3 10 mm
x
y
A
I
I
159.12 10 mm
PROBLEM 9.54 (Continued)
Then 64
(585.75 159.12) 10 mm
x
I
y
PROBLEM 9.55
Two L76 76 6.4-mm angles are welded to a C250 22.8 channel.
Determine the moments of inertia of the combined section with respect to
centroidal axes respectively parallel and perpendicular to the web of the
channel.
SOLUTION
Angle:
2
64
929 mm
0.512 10 mm
xy
A
II
Channel:
2
64
64
2890 mm
0.945 10 mm
28.0 10 mm
x
y
A
I
I
PROBLEM 9.55 (Continued)
Also 2( ) ( )
yyLyC
I
II
y
PROBLEM 9.56
Two steel plates are welded to a rolled W section as indicated.
Knowing that the centroidal moments of inertia
x
I
and y
I
of
the combined section are equal, determine (a) the distance a,
(b) the moments of inertia with respect to the centroidal x and y
axes.
SOLUTION
For W-section: 24 4
11.2 in , 385 in , 26.7 in
xy
AI I
Section Area, in2 in.
x
3
,inxA
Plate 2(1)(26) 52
a 52a
W-section 11.2
63.2
a
x
PROBLEM 9.56 (Continued)
2
2
42
26.7 in 11.2 in 0.82279 in.
WW
yy
IIAd
a
PROBLEM 9.57
The panel shown forms the end of a trough that is filled with water to the
line
AA
. Referring to section 9.2, determine the depth of the point of
application of the resultant of the hydrostatic forces acting on the panel (the
center of pressure).
SOLUTION
From section 9.2:
2
,
AA
R ydA M y dA
Let
y
P distance of center of pressure from
.AA
We must have:
PROBLEM 9.58
The panel shown forms the end of a trough that is filled with water
to the line
AA
. Referring to Section 9.2, determine the depth of the
point of application of the resultant of the hydrostatic forces acting
on the panel (the center of pressure).
SOLUTION
Using the equation developed on page 506 of the text:
AA
P
I
yyA
Now
41
(2 ) 2
232
YA yA
hbh h bh
PROBLEM 9.59
The panel shown forms the end of a trough that is filled with water to the line
AA. Referring to section 9.2, determine the depth of the point of application of
the resultant of the hydrostatic forces acting on the panel (the center of
pressure).
SOLUTION
Using the equation developed on page 506 of the text have:
A
A
P
I
yyA