Members in Compression - IV

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Nov 29, 2013 (4 years and 1 month ago)

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Compression Members

COLUMN STABILITY

A.

Flexural Buckling


Elastic Buckling


Inelastic Buckling


Yielding

B. Local Buckling


Section E7 pp 16.1
-
39


and B4 pp 16.1
-
14

C. Lateral Torsional Buckling

AISC Requirements

CHAPTER E pp 16.1
-
32

Nominal Compressive Strength

g
cr
n
A
F
P

AISC Eqtn E3
-
1

AISC Requirements

LRFD

n
c
u
P
P


loads

factored

of

Sum


u
P
strength

e
compressiv

design


n
c
P

0.90


n
compressio
for
factor

resistance



c

Design Strength

In Summary

























877
.
0
44
.
0
or


71
.
4

658
.
0
otherwise
F
F
F
F
E
r
KL
if
F
F
e
y
e
y
y
F
F
cr
e
y
200

r
KL
Local Stability
-

Section B4 pp 16.1
-
14


Local Stability:


If elements of cross section are thin LOCAL buckling occurs

The strength corresponding to any buckling mode cannot be developed


Local Stability
-

Section B4 pp 16.1
-
14


Local Stability:


If elements of cross section are thin LOCAL buckling occurs

The strength corresponding to any buckling mode cannot be developed


Local Stability
-

Section B4 pp 16.1
-
14



Stiffened Elements of Cross
-
Section


Unstiffened Elements of Cross
-
Section

Local Stability
-

Section B4 pp 16.1
-
14



Compact


Section Develops its full plastic stress before buckling

(failure is due to yielding only)



Noncompact


Yield stress is reached in some but not all of its compression elements
before buckling takes place

(failure is due to partial buckling partial yielding)



Slender


Yield stress is never reached in any of the compression elements
(failure is due to local buckling only)

Local Stability
-

Section B4 pp 16.1
-
14


If local buckling occurs cross section is not fully effective

Avoid whenever possible

Measure of susceptibility to local buckling

Width
-
Thickness ratio of each cross sectional element:
l

If cross section has slender elements
-

l> l
r

Reduce Axial Strength (E7 pp 16.1
-
39

)


Slenderness Parameter
-

Limiting Values

AISC B5 Table B4.1
pp 16.1
-
16

Slenderness Parameter
-

Limiting Values

AISC B5 Table B4.1
pp 16.1
-
17

Slenderness Parameter
-

Limiting Values

AISC B5 Table B4.1
pp 16.1
-
18

Slender Cross Sectional Element:

Strength Reduction E7 pp 16.1
-
39

Reduction Factor Q:

Q: B4.1


B4.2 pp 16.1
-
40 to 16.1
-
43

























877
.
0
44
.
0
or


71
.
4

658
.
0
otherwise
F
QF
F
QF
E
r
KL
if
QF
F
e
y
e
y
y
F
QF
cr
e
y
Slender Cross Sectional Element:

Strength Reduction E7 pp 16.1
-
39

Reduction Factor Q:

Q
s
, Q
a
: B4.1


B4.2 pp 16.1
-
40 to 16.1
-
43

























877
.
0
44
.
0
or


71
.
4

658
.
0
otherwise
F
QF
F
QF
E
r
KL
if
F
F
e
y
e
y
y
F
QF
cr
e
y
Q=Q
s
Q
a

COLUMN STABILITY

A.

Flexural Buckling


Elastic Buckling


Inelastic Buckling


Yielding

B. Local Buckling


Section E7 pp 16.1
-
39


and B4 pp 16.1
-
14

C. Torsional, Lateral/Torsional Buckling

Torsional & Flexural Torsional Buckling

When an axially loaded member becomes unstable overall

(no local buckling)

it buckles one of the three ways


Flexural Buckling


Torsional Buckling


Flexural
-
Torsional


Buckling

Torsional Buckling

Twisting about longitudinal axis of member

Only with doubly symmetrical cross sections with slender cross
-
sectional elements

Standard Hot
-
Rolled Shapes are
NOT susceptible

Built
-
Up Members should be
investigated

Cruciform shape particularly
vulnerable

Flexural Torsional Buckling

Combination of Flexural and Torsional Buckling

Only with unsymmetrical cross sections

1 Axis of Symmetry
: channels, structural
tees, double
-
angle, equal
length single angles

No Axis of Symmetry
: unequal length single
angles


Torsional Buckling



y
x
z
w
e
I
I
GJ
L
K
EC
F









1
2
2

Eq. E4
-
4

C
w

= Warping Constant (in
6
)

K
z

= Effective Length Factor for Torsional Buckling


(based on end restraints against twisting)

G = Shear Modulus (11,200 ksi for structural steel)

J = Torsional Constant

Lateral Torsional Buckling 1
-
Axis of Symmetry






















2
4
1
1
2
ez
ey
ez
ey
ez
ey
e
F
F
H
F
F
H
F
F
F
AISC Eq. E4
-
5



2
2
y
y
ey
r
L
K
E
F




2
2
2
1
o
g
z
w
ez
r
A
GJ
L
K
EC
F









2
2
2
1
o
o
o
r
y
x
H



g
y
x
o
o
o
A
I
I
y
x
r




2
2
2
o
o
y
x
,
Coordinates of shear center w.r.t centroid of section

Lateral Torsional Buckling No Axis of Symmetry











0
2
2
2
2




















o
o
ex
e
e
o
o
ey
e
e
ez
e
ey
e
ex
e
r
y
F
F
F
r
x
F
F
F
F
F
F
F
F
F
AISC Eq. E4
-
6

F
e

is the lowest root of the

Cubic equation

In Summary
-

Definition of
F
e

Elastic Buckling Stress corresponding to the controlling mode of
failure (flexural, torsional or flexural torsional)

F
e
:

Theory of Elastic Stability (Timoshenko & Gere 1961)

Flexural Buckling

Torsional Buckling

2
-
axis of symmetry

Flexural Torsional
Buckling

1 axis of symmetry

Flexural Torsional
Buckling

No axis of symmetry



2
2
/
r
KL
E
F
e


AISC Eqtn

E4
-
4

AISC Eqtn

E4
-
5

AISC Eqtn

E4
-
6

Column Strength






















877
.
0
44
.
0

658
.
0
otherwise
F
F
F
if
F
F
e
y
e
y
F
F
cr
e
y
g
cr
n
A
F
P

EXAMPLE

Compute the compressive strength of a WT12x81 of A992 steel.

Assume

(K
x
L) = 25.5 ft, (K
y
L) = 20 ft, and (K
z

L)

= 20 ft


200
43
.
87
50
.
3
12
5
.
25





x
x
r
L
K
r
KL
OK

43
.
87
113
50
000
,
29
71
.
4
71
.
4
>


y
F
E




ksi

44
.
37
43
.
87
000
,
29
2
2
2
2





r
KL
E
F
e
ksi

59
.
28
)
50
(
658
.
0
658
.
0
44
.
37
50

















y
F
F
cr
F
F
e
y
Inelastic Buckling

FLEXURAL Buckling


X axis

WT 12X81

A
g
=23.9 in
2

r
x
=3.50 in

r
y
=3.05 in

kips

3
.
683
)
9
.
23
(
59
.
28



g
cr
n
A
F
P
EXAMPLE

200
69
.
78
05
.
3
12
20




y
y
r
L
K
OK





ksi

22
.
46
69
.
78
000
,
29
2
2
2
2





y
y
ey
r
L
K
E
F
FLEXURAL TORSIONAL Buckling


Y axis (axis of symmetry)

WT 12X81

A
g
=23.9 in
2

r
x
=3.50 in

r
y
=3.05 in

y=2.70 in

t
f
=1.22 in

I
x
=293 in
4

I
y
=221 in
4

J=9.22 in
4

C
w
=43.8 in
6

0
0

x
2
0
f
t
y
y


87
.
25
9
.
23
221
293
)
09
.
2
(
0
2
2
2
2









g
y
x
o
o
o
A
I
I
y
x
r
Shear Center

EXAMPLE

FLEXURAL TORSIONAL Buckling


Y axis (axis of symmetry)

WT 12X81

A
g
=23.9 in
2

r
x
=3.50 in

r
y
=3.05 in

y=2.70 in

t
f
=1.22 in

I
x
=293 in
4

I
y
=221 in
4

J=9.22 in
4

C
w
=43.8 in
6











ksi
r
A
GJ
L
K
EC
F
o
g
z
w
ez
4
.
167
87
.
25
9
.
23
1
)
22
.
9
(
200
,
11
12
20
)
8
.
43
)(
000
,
29
(
1
2
2
2
2
2
2



















EXAMPLE

FLEXURAL TORSIONAL Buckling


Y axis (axis of symmetry)

WT 12X81

A
g
=23.9 in
2

r
x
=3.50 in

r
y
=3.05 in

y=2.70 in

t
f
=1.22 in

I
x
=293 in
4

I
y
=221 in
4

J=9.22 in
4

C
w
=43.8 in
6











ksi
F
F
H
F
F
H
F
F
F
ez
ey
ez
ey
ez
ey
e
63
.
53
4
.
167
22
.
46
8312
.
0
4
.
167
22
.
46
4
1
1
8312
.
0
2
4
.
167
22
.
46
4
1
1
2
2



































8312
.
0
87
.
25
090
.
2
0
1
1
2
2
2
2







o
o
o
r
y
x
H
EXAMPLE

FLEXURAL TORSIONAL Buckling


Y axis (axis of symmetry)

WT 12X81

A
g
=23.9 in
2

r
x
=3.50 in

r
y
=3.05 in

y=2.70 in

t
f
=1.22 in

I
x
=293 in
4

I
y
=221 in
4

J=9.22 in
4

C
w
=43.8 in
6

Elastic or Inelastic LTB?

63
.
43
0
.
22
)
50
(
44
.
0
44
.
0




e
y
F
ksi
F





















877
.
0
44
.
0

658
.
0
otherwise
F
F
F
if
F
F
e
y
e
y
F
F
cr
e
y
EXAMPLE

FLEXURAL TORSIONAL Buckling


Y axis (axis of symmetry)

WT 12X81

A
g
=23.9 in
2

r
x
=3.50 in

r
y
=3.05 in

y=2.70 in

t
f
=1.22 in

I
x
=293 in
4

I
y
=221 in
4

J=9.22 in
4

C
w
=43.8 in
6

ksi
F
F
y
F
F
cr
e
y
59
.
28
50
658
.
0
658
.
0
63
.
43
50



















kips
7
.
739
)
70
.
2
(
95
.
30



g
cr
n
A
F
P
Compare to FLEXURAL Buckling


X axis

kips

3
.
683
)
9
.
23
(
82
.
21



g
cr
n
A
F
P
Column Design Tables

Assumption : Strength Governed by Flexural Buckling

Check Local Buckling

Column Design Tables


Design strength of selected shapes for effective length KL

Table 4
-
1 to 4
-
2,
(pp 4
-
10 to 4
-
316)


Critical Stress for Slenderness
KL/r

table 4.22 pp
(4
-
318 to 4
-
322)

EXAMPLE

Compute the available compressive strength of a W14x74 A992 steel
compression member. Assume pinned ends and L=20 ft. Use (a) Table 4
-
22 and (b) column load tables

(a) LRFD
-

Table 4
-
22


pp 4
-
318

200
77
.
96
48
.
2
)
12
)(
20
)(
1
(
Maximum




y
r
KL
r
KL
Table has integer values of (KL/r) Round up or interpolate

Fy=50 ksi

ksi
P
cr
67
.
22


ksi
A
P
P
g
cr
n
494
)
8
.
21
(
67
.
22





EXAMPLE

Compute the available compressive strength of a W14x74 A992 steel
compression member. Assume pinned ends and L=20 ft. Use (a) Table 4
-
22 and (b) column load tables

(b) LRFD Column Load Tables

ft
KL
20
)
20
)(
1
(
Maximum


Tabular values based on minimum radius of gyration

Fy=50 ksi

kips
P
n
c
494


Example II

A W12x58, 24 feet long in pinned at both ends and braced in the weak
direction at the third points. A992 steel is used. Determine available
compressive strength

200
25
.
38
51
.
2
)
12
)(
8
(
1



y
y
r
L
K
200
55
.
54
28
.
5
)
12
)(
24
(
1



x
x
r
L
K
Enter table 4.22 with KL/r=54.55 (LRFD)

28
.
5

x
r
51
.
2

y
r
ksi
P
cr
24
.
36


kips
A
P
P
g
cr
n
616
)
17
(
24
.
36





17

g
A
Example II

A W12x58, 24 feet long in pinned at both ends and braced in the weak
direction at the third points. A992 steel is used. Determine available
compressive strength

200
25
.
38
51
.
2
)
12
)(
8
(
1



y
y
r
L
K
200
55
.
54
28
.
5
)
12
)(
24
(
1



x
x
r
L
K
Enter table 4.22 with KL/r=54.55 (ASD)

28
.
5

x
r
51
.
2

y
r
ksi
F
c
cr
09
.
24


kips
A
F
P
g
c
cr
c
n
410




17

g
A
Example II

A W12x58, 24 feet long in pinned at both ends and braced in the weak
direction at the third points. A992 steel is used. Determine available
compressive strength

200
25
.
38
51
.
2
)
12
)(
8
(
1



y
y
r
L
K
200
55
.
54
28
.
5
)
12
)(
24
(
1



x
x
r
L
K
CAN I USE Column Load Tables?

y
x
x
r
r
L
K
KL

Not Directly because they are
based on min r (y axis buckling)

If x
-
axis buckling enter table with

Example II

A W12x58, 24 feet long in pinned at both ends and braced in the weak
direction at the third points. A992 steel is used. Determine available
compressive strength

200
25
.
38
51
.
2
)
12
)(
8
(
1



y
y
r
L
K
200
55
.
54
28
.
5
)
12
)(
24
(
1



x
x
r
L
K
X
-
axis buckling enter table with

ft
r
r
L
K
KL
y
x
x
43
.
11
1
.
2
)
24
)(
1
(



kips
P
n
616