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Chapter Title
671
Idealized Buckling Models
Problem 11.21 through 11.24
The figure shows an idealized structure
consisting of one or more
rigid bars
with pinned connections and linearly
elastic springs. Rotational stiffness is denoted
b
R
and translational
stiffness is denoted
b
.
Determine the critical load
P
cr
for the structure.
11
Columns
A
B
L
b
R
P
L
b
P
C
B
a
A
L
2
—
L
2
—
A
C
B
b
R
P
L
2
—
L
2
—
A
C
B
b
R
b
R
P
Prob. 11.21
Prob. 11.22
Prob. 11.23
Prob. 11.24
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View Full Document Solution 11.21
Rigid bar AB
672
CHAPTER 11
Columns
P
cr
5
b
R
L
P
(
u
L
)
2
b
R
u
5
0
a
M
A
5
0
A
B
P
u
u
L
M
5 b
R
u
Solution 11.22
Rigid bar
ABC
P
cr
5
b
a
2
L
P
u
L
2
bu
a
2
5
0
a
M
A
5
0
A
C
P
u
L
F
5
bu
a
a
u
Solution 11.23
Two rigid bars with a pin
connection
Shows that there are no horizontal
reactions at the supports.
F
REE

BODY DIAGRAM OF BAR
BC
P
cr
5
6
b
R
L
b
R
(2
u
)
1
b
R
u
5
PL
u
2
a
M
B
5
0
Ê
M
B
1
M
C
2
P
u
¢
L
2
≤
5
0
M
B
5
b
R
u
)
M
C
5
b
R
u
a
M
A
5
0
A
C
B
M
5
b
R
u
M
5
b
R
u
P
u
u
C
B
M
C
5
b
R
u
P
u
M
C
(
L
2
)
u
Solution 11.24
Two rigid bars with a pin
connection
F
REE

BODY DIAGRAM OF BAR
BC
P
cr
5
H
u
5
b
R
L
a
M
B
5
0
Ê
H
¢
L
2
≤
2
P
¢
u
L
2
≤
5
0
H
5
b
R
u
L
a
M
A
5
0
Ê
HL
2
b
R
u
5
0
A
C
B
M
5
b
R
u
P
P
u
u
H
H
P
P
C
B
u
H
H
Critical Loads of Columns with Pinned Supports
The problems for Section 11.3 are to be solved using the assumptions
of ideal, slender, prismatic, linearly elastic columns (Euler buckling).
Buckling occurs in the plane of the figure unless stated otherwise.
Problem 11.31
Calculate the critical load
P
cr
for a
W
8
3
35 steel
column (see figure) having length
L
5
24 ft and
E
5
30
3
10
6
psi
under the following conditions:
(a) The column buckles by bending about its strong axis (axis 11),
and (b) the column buckles by bending about its weak axis (axis 22).
In both cases, assume that the column has pinned ends.
Solution 11.31
Column with pinned supports
SECTION 11.3
Critical Loads of Columns with Pinned Supports
673
1
C
1
2
2
W
8
3
35 steel column
L
5
24 ft
5
288 in.
E
5
30
3
10
6
psi
I
1
5
127 in.
4
I
2
5
42.6 in.
4
A
5
10.3 in.
2
(a) B
UCKLING ABOUT STRONG AXIS
P
cr
5
p
2
EI
1
L
2
5
453 k
(b) B
UCKLING ABOUT WEAK AXIS
N
OTE
:
[
Solution is satisfactory if
s
PL
$
44 ksi
s
cr
5
P
cr
A
5
453
k
10.3
in.
2
5
44 ksi
P
cr
5
p
2
EI
2
L
2
5
152 k
Probs. 11.31 through 11.33
Problem 11.32
Solve the preceding problem for a
W
10
3
60
steel column having length
L
5
30 ft.
Solution 11.32
Column with pinned supports
W
10
3
60 steel column
L
5
30 ft
5
360 in.
E
5
30
3
10
6
psi
I
1
5
341 in.
4
I
2
5
116 in.
4
A
5
17.6 in.
2
(a) B
UCKLING ABOUT STRONG AXIS
P
cr
5
p
2
EI
1
L
2
5
779 k
(b) B
UCKLING ABOUT WEAK AXIS
N
OTE
:
[
Solution is satisfactory if
s
PL
$
44 ksi
s
cr
5
P
cr
A
5
779
k
17.6 in.
2
5
44 ksi
P
cr
5
p
2
EI
2
L
2
5
265 k
Problem 11.33
Solve Problem 11.31 for a
W
10
3
45 steel
column having length
L
5
28 ft.
Solution 11.33
Column with pinned supports
W
10
3
45 steel column
L
5
28 ft
5
336 in.
E
5
30
3
10
6
psi
I
1
5
248 in.
4
I
2
5
53.4 in.
4
A
5
13.3 in.
2
(a) B
UCKLING ABOUT STRONG AXIS
P
cr
5
p
2
EI
1
L
2
5
650
k
(b) B
UCKLING ABOUT WEAK AXIS
N
OTE
:
[
Solution is satisfactory if
s
PL
$
49
ksi
s
cr
5
P
CR
A
5
650
k
13.3
in.
2
5
49
ksi
P
cr
5
p
2
EI
2
L
2
5
140
k
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This note was uploaded on 03/30/2008 for the course CVEN 205 taught by Professor Muliana during the Spring '08 term at Texas A&M.
 Spring '08
 Muliana

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