Chapter 9
9.1
(a)
[k] = 2
r A
[]B
T [D]
[]B
ri = 0, zi = 0,
j
r
= 2, zj = 0, rm = 0, zm = 2
i = rj zm – zj rm = 2.2 – 0 = 4
33
33
j = ri – rm = 0 – 0 = 0 A =
1
2
(2) (2) = 2
m = rj – ri = 2 – 0 = 2
[]B
0 0 0 0
22
10 0 0 0
22
40 0 0
2 2 2
00
2 2 2 2
6
0.75 0.25 0.25 0
15 10 0.25 0.75 0.25 0
6
0.5
2
0.75 0.25 0.25 0
15 10 0.25 0.75 0.25 0
(1 0.25) 1 0.25 0.25 0.75 0
0 0 0 0.25
6
–1.25 – 0.25 0.25 0
0 0 0 – 0.5
1.75 0.75 1.25 – 0.5
15 10
0
0
1
i
i
0
0
1
j
j
0
1
0
1
1
00
00
() 02
()
00
() 0
()
0
pz
bh
hh
bh
bh
bh
hb bz
hb bz
bz
bz
=
2
0
2
0
0
0
0
0
–
2
0
0
p bz
hh
p bz
h
p bz
b
bh
dz =
23
00
3
0
23
3
0
h
0
2
0
0
0
p bz p bz
h
p bz
h
h
=
2
0
2
0
1
1
2
6
2
3
3
3
0
0
2
0
0
sr
sz
p bh
sr
sz
p bh sr
sz
f
f
f
f
h
f
f
=
0
0
6
3
0
0
2
0
0
ph
ph
b
9.3
2
3
B
B
B
B
B
B
R
Z
R
rA
Z
R
Z
2
ZB = 0.283
3
lb
in.
B
3
3
lb
2
in.
lb
in.
0.283
rev. rad 1 min
min rev 60 32.2 12
s
B
R
= 0.01712
3
lb
in.
2
3
rA
=
2
3
(5.333 in.) (2 in.) = 22.34 in.3
So
9.4
(a) Element Figure 9.4 a
The equation to be evaluated is {
} = [D] [B} {d}
i = – 1,
j = – 1,
m = 2
1
1
3 0 3 0 0 0 2
0 0 0 5
1 1 2
0 0 0 0 3
22
3 3 0 0
1 1 2
0
rr
zz
4
2.58
0.54 10 psi
1
1
2
r
z
u
w
u
3
3
4.8 10
0.6 10 lb
i = 50 (50) – (0) (0) = 2500 mm2
A = 1800 mm2
[k] = 2
r
A
B
T [D]
B
60 0 60 0 0 0
11
0 0 0 60 0 60
2(1800) mm
30 0 30 0 30 0
0 60 60 60 60 0
[D] as in 9.6 (a)
9
37.5 0 52.5 15 7.5 15
105 10 7.5 0 22.5 45 7.5 45
408
rm = 0.001 zm = 0.002
1
2
1
3
0.002 ,
2
i = rj zm – zj rm
j = rm zi – zm ri
m = ri zj – zi rj
200
0
jj
i i m m
z
zz
j
im
rr
r r r r
jj
i i m m
A
[D] =
12
2
10
10
10
(1 )(1 2 )
000
v
vvv
v
vv
E
v
vv
vv
[k] = 2
r A [B]T [D] [B]
8 8 8 7 7 7
8 8 7 7 8 8
8 7 9 8 8 8
8.577 10 1.759 10 3.738 10 8.796 10 4.398 10 8.796 10
1.759 10 4.618 10 8.796 10 6.597 10 3.519 10 3.958 10
From Problem 9.6 (a), we have [D]
[]B