Solucionario De Gerre

11Ch11.qxd

9/27/08

2:21 PM

Page 845

11
Columns

Idealized Buckling Models
P

Problem 11.2-1

The figure shows an idealized structure consisting of one
or more rigid bars with pinned connections and linearly elastic springs. Rotational
stiffness is denoted bR, and translational stiffness is denoted b.
Determine the critical load Pcr for the structure.

B
L
bR
ASolution 11.2-1

Rigid bar AB
g MA

0

P(uL)

bR u

Pcr

bR
L

0

;

845

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CHAPTER 11

Page 846

Columns

Problem 11.2-2 The figure shows an idealized structure consisting
of one or more rigid bars with pinned connections and linearly elastic
springs. Rotational stiffness is denoted bR, and translational stiffness
is denoted b.(a) Determine the critical load Pcr for the structure from
figure part (a).
(b) Find Pcr if another rotational spring is added at B from
figure part (b).

P

P

C

C
b

B
L

bR

L
Elastic
supports

a

b

B

Elastic
supports

a

bR

bR

A

A
(a)

(b)

Solution 11.2-2
(a) g M A
PuL

(b) g M A

0
b ua

2

0

b Ru

PuL

2

Pcr

ba + bRL

;

Problem 11.2-3 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
stiffiness is denoted b .
Determine the critical load Pcr for the structure.

Pcr

0
b ua 2

2 b Ru

ba2 + 2bR
L

0
;

P
C
L
—
2
B

bR
A

L
—
2

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SECTION 11.2

Solution 11.2-3

g MB

0

MB + MC

FREE-BODY DIAGRAM OF BAR BC
b Ru

6bR
L

b R (2u)

Pcr

Problem 11.2-4 The figure shows an idealized structure
consisting of bars AB and BC which are connected using a
hinge at B and linearly elastic springs at A and B. Rotational
stiffness is denoted bR and translational stiffness isdenoted b.
(a) Determine the critical load Pcr for the structure from
figure part (a).
(b) Find Pcr if an elastic connection is now used to connect
bar segments AB and BC from figure part (b).

L
Pu a b
2

0

PLu
2

b R(2u) + b Ru

MB

847

Two rigid bars with a pin connection

g M A 0 Shows that there are no horizontal reactions
at the supports.

MC

Idealized BucklingModels

;

P

P

C

C

B

Hinge
b

L
Elastic
support

a

L/2

bR
Bb

Hinge
L/2

bR
A

bR
A

(a)

Elastic
connection

(b)

Elastic
support

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CHAPTER 11

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Columns

Solution 11.2-4
(a) g M A

(b) g M A

0

bua 2 + b R u

HL

L
bu a b + b Ru
2

0

2

H

0
2

b ua + b Ru
L

Hb uL2

HL

4 b Ru
4L

P

P

H

C

0

H

C

F = bua

B

F = bu L
2

B

u

u
M = bRu

A

M = bRu

A

P

FREE-BODY DIAGRAM OF BAR BC

FREE-BODY DIAGRAM OF BAR BC

P

P

C

H

H

C

u
B

B

H

P

P

g MB

0

H(L

Pcr

a)

1 b a + b R2(L
2

aL

MB
P(ua)
a)

;

0

g MB

u ( L)
2

b R(2u)
0

L
Ha b2

Pau

L
b + MB
2

0

buL2 + 4 b Ru L
ab
4L
2
L
P a u b + b R(2u)
2
Pcr

b L2 + 20 b R
4L

;

0

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Page 849

SECTION 11.2

Problem 11.2-5 The figure shows an idealized structure consisting of two
rigid bars joined by an elastic connection with rotational stiffness b R. Determine
the critical load Pcr for the structure.

Idealized BucklingModels

849

Elastic connection
A

B
L/2

bR

C

D

L/2

P

L

Solution 11.2-5

θ

C
B

D

θL

P

2θ

A

V

©MB

0

V

FREE-BODY DIAGRAM OF BAR CD

0

Moment in elastic connection
b R * total relative rotation (u

MC

P

Problem 11.2-6 The figure shows an idealized structure
consisting of rigid bars ABC and DEF joined by linearly elastic...