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Compression Members
Compression Members
• Compression members are susceptible to BUCKLING
• BUCKLING – Loss of stability– Axial loads cause lateral deformations (bending-like deformations)
P is applied slowlyP increasesMember becomes unstable - buckles
Column Theory
Axial force that causes Buckling is called Critical Load and is associated to the column strength
Pcr depends on
• Length of member• Material Properties• Section Properties
Column Theory - Euler Buckling
2
22
L
EInPcr
Column Theory - Euler Buckling
22
2
2
,1
rL
EA
L
EIPn cr
gyration of radiusA
Ir
Assumptions
• Column is perfectly straight
• The load is axial, with no eccentricity
• The column is pinned at both ends
No Moments
Need to account for other boundary conditions
Other Boundary Conditions
22
2r
L
EAPcr
22
5.0r
L
EAPcr
22
7.0r
L
EAPcr
Fixed on bottom
Free to rotate and translate
Fixed on bottom
Fixed on top
Fixed on bottom
Free to rotate
Other Boundary Conditions
In generalIn general
22
rKL
EAPcr
K: Effective Length FactorK: Effective Length Factor
LRFD Commentary Table C-C2.2 p 16.1-240
Effective Length Factor
Column Theory - Column Strength Curve
AISC Requirements
CHAPTER E pp 16.1-32
Nominal Compressive Strength
gcrn AFP
AISC Eqtn E3-1
AISC Requirements
LRFD
ncu PP
loads factored of Sum uP
strength ecompressiv design ncP
0.90 ncompressiofor factor resistance c
AISC Requirements
ASD
c
na
PP
loads service of Sum aP
strength ecompressiv allowable cnP
1.67 ncompressiofor factor safety c
AISC Requirements
ASD – Allowable Stress
aa Ff
gaa APf stress ecompressiv axial computed
crcr
c
cr
a
FFF
F
6.067.1
stress ecompressiv axial allowable
Design Strength
Alternatively
e
e
F
E
r
KL
rKL
EF
2
2
2
yF
E
r
KL71.4
ye F
E
F
E71.4
2
ye FF 44.0Inelastic Buckling
In Summary
877.0
44.0or
71.4 658.0
otherwiseF
FF
F
E
r
KLifF
F
e
ye
yy
F
F
cr
ey
200r
KL
LOCAL BUCKLING
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
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
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 effectiveIf local buckling occurs cross section is not fully effectiveAvoid whenever possible
Measure of susceptibility to local bucklingMeasure of susceptibility to local bucklingWidth-Thickness ratio of each cross sectional element:
If cross section has slender elements - If cross section has slender elements - rr
Reduce Axial Strength (E7 pp 16.1-39 )
Slenderness Parameter - Section B5 pp 16.1-12
Cross Sectional Element
Stiffened Unstiffenedb
htw
t
b/t=bb/t=bff/2t/2twwh/th/tww
Slenderness
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
Slenderness Parameter - Limiting Values
Slenderness Parameter - Limiting Values
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
44.0or
71.4 658.0
otherwiseF
QFF
QF
E
r
KLifF
F
e
ye
yy
F
QF
cr
ey
Slender Cross Sectional Element:Strength Reduction E7 pp 16.1-39
Reduction Factor Q:
Qs, Qa: B4.1 – B4.2 pp 16.1-40 to 16.1-43
877.0
44.0or
71.4 658.0
otherwiseF
QFF
QF
E
r
KLifF
F
e
ye
yy
F
QF
cr
ey
Q=QsQa
Example I
Investigate a W14x74, grade 50 in compression for local stability
W14x74: bf-10.1 in, tf=0.785 in
FLANGES - Unstiffened Elements
43.6
785.02
1.10
2
2
f
f
f
f
t
b
t
b
43.65.13
50
000,2956.056.0
y
r F
E
Flange is not slender, OK
Example I
Investigate a W14x74, grade 50 in compression for local stability
W14x74: bf-10.1 in, tf=0.785 in
WEB - Stiffened Element
4.25
450.0
38.122.14
2
2
f
des
w t
kd
t
h
4.259.35
50
000,2949.149.1
y
r F
E
Web is not slender, OK
Example I
Investigate a W14x74, grade 50 in compression for local stability
W14x74: bf-10.1 in, tf=0.785 in
PART 1 – Properties: Slender Shapes are marked with “c”
Example II
Determine the axial compressive strength of an HSS 8x4x1/8 with an effective length of 15 ft with respect to each principal axis. Use Fy=46 ksi.
HSS 8x4x1/8
Ag=2.70 in2
rx=2.92 in2
ry=1.71 in2
h/t=66.0
b/t=31.5 7.652 in
8 in
1.5 t = 0.1875
Example II
HSS 8x4x1/8
Ag=2.70 in2
rx=2.92 in2
ry=1.71 in2
h/t=66.0
b/t=31.5
7.652 in
8 in
1.5 t = 0.1875
7.652 in
8 in
1.5 t = 0.1875
Maximum 2003.10571.1
1215
yr
KL
r
KLOK
3.10511846
000,2971.471.4
yF
EInelastic Buckling
ksi 81.253.105
000,292
2
2
2
rKL
EFe
ksi 82.2146658.0658.0 81.25
46
yF
F
cr FF e
y
kips 91.58)70.2(82.21 gcrn AFP
Nominal Strength
Example II
HSS 8x4x1/8
Ag=2.70 in2
rx=2.92 in2
ry=1.71 in2
h/t=66.0
b/t=31.5
7.652 in
8 in
1.5 t = 0.1875
7.652 in
8 in
1.5 t = 0.1875
Local Buckling
0.6615.3546
000,2940.140.1
t
h
F
E
y
SLENDER
Example II
HSS 8x4x1/8
Ag=2.70 in2
rx=2.92 in2
ry=1.71 in2
h/t=66.0
b/t=31.5
7.652 in
8 in
1.5 t = 0.1875
7.652 in
8 in
1.5 t = 0.1875
Local Buckling
Stiffened Cross-Section – Rectangular w/ constant t
Qs=1.0
f
E
t
b40.1
eff
n
A
Pf Code allows f=Fy to
avoid iterations
A
AQ eff
a AISC E7.2
Case (b) applies provided that
Aeff: Summation of Effective Areas of Cross section based on reduced effective width be
Example II
Aeff:
be
bf
E
tbf
Etbe
/
38.0192.1
in 8in 784.4
46
000,29
0.66
38.01
46
000,29116.092.1
b
be
Example II
7.652 in
8 in
1.5 t = 0.1875Aeff:
be
Loss of Area 2in 6654.0116.0784.4652.722 tbb e
2in 035.26654.070.2 lostgeff AAA
Example II
Loss of Area 2in 6654.0116.0784.4652.722 tbb e
2in 035.26654.070.2 lostgeff AAA
Reduction Factor 7535.070.2
035.2
A
AQ eff
a
7535.07535.01 asQQQ
Example II
Local Buckling Strength
r
KL
QF
E
y
3.1052.13646)7537.0(
000,2971.471.4
ksi 81.253.105
000,292
2
2
2
rKL
EFe
ksi 76.1946658.07535.0658.0 81.25
467535.0
yF
QF
cr FQF e
y
kips 35.53)70.2(76.19 gcrn AFP
Nominal Strength
Inelastic Buckling
Same as before
Example II
Local Buckling Strength
kips 35.53)70.2(76.19 gcrn AFP
Nominal Strength
Lateral Flexural Buckling Strength
kips 91.58)70.2(82.21 gcrn AFP
CONTROLS
LRFD kips 0.4835.5390.0 ncP
ASD kips 0.3267.1
35.53
nP
Column Design Tables
Assumption : Strength Governed by Flexural BucklingCheck Local Buckling
Column Design Tables
Design strength of selected shapes for effective length KLTable 4-1 to 4-2, (pp 4-10 to 4-316)
Critical Stress for Slenderness KL/rtable 4.22 pp (4-318 to 4-322)