Compression Members. Compression members are susceptible to BUCKLING BUCKLING – Loss of stability –Axial loads cause lateral deformations (bending-like.

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)