members in compression - iii

7/29/2019 Members in Compression - III

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Compression Members


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Compression Members

Compression members are susceptible to BUCKLING

BUCKLINGLoss of stability

Axial loads cause lateral deformations (bending-like deformations)

P is applied slowly

P increases

Member becomes unstable -buckles


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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


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Column Theory - Euler Buckling

2

22

L

EInPcr


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Column Theory - Euler Buckling

2

2

2

2

,1

rL

EA

L

EIPn cr

gyrationofradiusA

Ir


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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


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Other Boundary Conditions

22

2

r

L

EAPcr

2

2

5.0

r

L

EAPcr

2

2

7.0

rL

EAPcr

Fixed on bottom

Free to rotate and translate

Fixed on bottom

Fixed on top

Fixed on bottom

Free to rotate


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Other Boundary Conditions

In general

2

2

rKL

EAPcr

K: Effective Length Factor

LRFD Commentary Table C-C2.2 p 16.1-240


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Effective Length Factor


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Column Theory - Column Strength Curve


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AISC Requirements

CHAPTER E pp 16.1-32

Nominal Compressive Strength

gcrn AFP

AISC Eqtn E3-1


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AISC Requirements

LRFD

ncu PP

loadsfactoredofSumuP

strengthecompressivdesignncP

0.90ncompressioforfactorresistance c


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AISC Requirements

ASD

c

na

PP

loadsserviceofSumaP

strengthecompressivallowablecnP

1.67ncompressioforfactorsafety c


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AISC Requirements

ASD Allowable Stress

aa Ff

gaa APf stressecompressivaxialcomputed

crcr

c

cr

a

FFF

F

6.067.1

stressecompressivaxialallowable


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Design Strength


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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


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In Summary

877.0

44.0or

71.4658.0

otherwiseF

FF

F

E

r

KLifF

F

e

ye

y

yF

F

cr

e

y

200r

KL


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LOCAL BUCKLING

A.Flexural Buckling

Elastic Buckling

Inelastic Buckling Yielding

B. Local BucklingSection E7 pp 16.1-39

and B4 pp 16.1-14

C. Lateral Torsional Buckling


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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


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Stiffened Elements of Cross-Section

Unstiffened Elements of Cross-Section


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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)


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If local buckling occurs cross section is not fully effectiveAvoid whenever possible

Measure of susceptibility to local buckling

Width-Thickness ratio of each cross sectional element:

If cross section has slender elements - r

Reduce Axial Strength (E7 pp 16.1-39)


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Slenderness Parameter - Section B5 pp 16.1-12

Cross Sectional Element

Stiffened Unstiffened

b

h

tw

t

b/t=bf/2twh/tw

Slenderness


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Slenderness Parameter - Limiting Values

AISC B5 Table B4.1

pp 16.1-16


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AISC B5 Table B4.1

pp 16.1-17


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AISC B5 Table B4.1

pp 16.1-18


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Slender Cross Sectional Element:Strength Reduction E7 pp 16.1-39

Reduction Factor Q:

Q: B4.1B4.2 pp 16.1-40 to 16.1-43

877.0

44.0or

71.4658.0

otherwiseF

QFF

QF

E

r

KLifF

F

e

ye

y

yF

QF

cr

e

y


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Slender Cross Sectional Element:Strength Reduction E7 pp 16.1-39

Reduction Factor Q:

Qs, Qa: B4.1B4.2 pp 16.1-40 to 16.1-43

877.0

44.0or

71.4658.0

otherwiseF

QFF

QF

E

r

KLifF

F

e

ye

y

yF

QF

cr

e

y

Q=QsQa


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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

43.65.1350

000,2956.056.0

y

r

F

El

Flange is not slender, OK


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Example I


W14x74: bf-10.1 in, tf=0.785 in

WEB - Stiffened Element

4.25450.0

38.122.14

2

2

f

des

w t

kd

t

h

4.259.3550

000,2949.149.1

y

r

F

El

Web is not slender, OK


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Example I


W14x74: bf-10.1 in, tf=0.785 in

PART 1Properties: Slender Shapes are marked with c


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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

1.5 t = 0.1875

1.5 t = 0.1875


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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

7.652 in

8 in

Maximum 2003.10571.1

1215

yr

KL

r

KLOK

3.10511846

000,2971.471.4

yF

EInelastic Buckling

ksi81.25

3.105

000,292

2

2

2

rKL

EFe

ksi82.2146658.0658.0 81.2546

yF

F

cr FFe

y

kips91.58)70.2(82.21 gcrn AFP

Nominal Strength

1.5 t = 0.1875

1.5 t = 0.1875


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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

7.652 in

8 in

Local Buckling

0.6615.3546

000,2940.140.1

t

h

F

E

y

SLENDER

1.5 t = 0.1875

1.5 t = 0.1875


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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

7.652 in

8 in

Local Buckling

Stiffened Cross-Section Rectangular w/ constant t

Qs=1.0

f

E

t

b40.1

eff

n

A

Pf Code allowsf=Fy to

avoid iterations

A

AQ

eff

a AISC E7.2

Case (b) applies provided that

Aeff: Summation of Effective Areas of Cross section basedon reduced effective width be


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Example II

Aeff:

be

bf

E

tbf

Etbe

/

38.0192.1

in8in784.4

46

000,29

0.66

38.01

46

000,29116.092.1

b

be


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Example II

7.652 in

8 in

1.5 t = 0.1875Aeff:

be

Loss of Area 2in6654.0116.0784.4652.722 tbb e2in035.26654.070.2 lostgeff AAA


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Example II

Loss of Area 2in6654.0116.0784.4652.722 tbb e2in035.26654.070.2 lostgeff AAA

Reduction Factor 7535.070.2

035.2

A

AQ

eff

a

7535.07535.01 asQQQ


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Example II

Local Buckling Strength

r

KL

QF

E

y

3.1052.136

46)7537.0(

000,2971.471.4

ksi81.25

3.105

000,292

2

2

2

rKL

EFe

ksi76.1946658.07535.0658.0

81.25

467535.0

y

F

QF

cr FQF

e

y

kips35.53)70.2(76.19 gcrn AFP

Nominal Strength

Inelastic Buckling

Same as before


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Example II

Local Buckling Strength

kips35.53)70.2(76.19 gcrn AFP

Nominal Strength

Lateral Flexural Buckling Strength

kips91.58)70.2(82.21 gcrn AFP

CONTROLS

LRFD kips0.4835.5390.0 ncP

ASD kips0.3267.1

35.53

nP


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Column Design Tables

Assumption : Strength Governed by Flexural Buckling

Check 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)

members in compression - iii

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