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Slender Column Design using the ACI 318-08 Building Code Robert J. Frosch, Ph.D., P.E. Professor of Civil Engineering Purdue University September 21 - 22, 2011

Robert Frosch Slender Column

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Page 1: Robert Frosch Slender Column

Slender Column Designusing the

ACI 318-08 Building Code

Robert J. Frosch, Ph.D., P.E.Professor of Civil Engineering

Purdue University

September 21 - 22, 2011

Page 2: Robert Frosch Slender Column

318 Slenderness Task Group

• Simplify slenderness provisions• Recognize modern analysis met• Identify when slenderness considered• Combine into one section (10.10)

Page 3: Robert Frosch Slender Column

Flow of Provisions

1. Determine if slenderness can be neglected2. Compute M2nd-order at ends3. Compute M2nd-order between ends4. Determine if column is too slender

Page 4: Robert Frosch Slender Column

10.10.1 Neglect Slenderness?

• Braced Against Sideway

• Not Braced Against Sideway

• 5% Increase in Moment Acceptable

22≤lukr

1 234 12( / ) 40≤ − ≤luk M Mr

Page 5: Robert Frosch Slender Column

Braced or Not-Braced?

• Evaluate story stiffnessΣKbracing ≥ Σ 12Kcolumns

StoryStiffness

Shear Wall - KbracingColumn - Kcolumn

Page 6: Robert Frosch Slender Column

Calculate End Moments

SlendernessAnalysis

Nonlinear second-order analysis

10.10.3

Elastic second-order analysis

10.10.4

Moment magnification

10.10.5

Page 7: Robert Frosch Slender Column

10.10.3 Nonlinear 2nd-order analysis

Second-order analysis shall consider:– material nonlinearity– member curvature and lateral drift– duration of loads– shrinkage and creep– interaction with the supporting foundation

Page 8: Robert Frosch Slender Column

Nonlinear 2nd-order analysisThe analysis procedure shall have been shown to result in prediction of strength in substantial agreement with results of comprehensive tests of columns in statically indeterminate reinforced concrete structures.

Only 3 Frame Tests

Page 9: Robert Frosch Slender Column

10.10.4 Elastic 2nd-order analysis

• Use second-order elastic program– PΔ option enabled

• Consider section properties accounting: – Influence of axial loads– Presence of cracked regions along length– Effects of load duration

Page 10: Robert Frosch Slender Column

10.10.4.1 Recommended ICompression Members:

Columns 0.70Ig

Walls – Uncracked 0.70Ig

– Cracked 0.35Ig

Flexural Members:Beams 0.35Ig

Flat plates/slabs 0.25Ig

- Reduce I if sustained lateral loads present ds

11 β+

Page 11: Robert Frosch Slender Column

Alternate I

Compression members:

Flexural members:

0

0.80 25 1 0

0.35 0.875

.5

≤ ≤

⎛ ⎞⎛ ⎞= + − −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

st u ug

g u

g g

A M PI IA P h P

I I I

( )0.10 25 1.2 0.2

0.25 0.5

ρ ⎛ ⎞= + −⎜ ⎟⎝

≤ ≤⎠

g

wg

gI I

bId

I

I

Page 12: Robert Frosch Slender Column

10.10.5 Moment magnification

P

P P

PΔΔ = 0

Nonsway Sway

No amplificationof end moments

Amplification of end moments

δMT

δMB

δMT

δMB

T BM M 0δ δ+ = T BM M Pδ δ+ = Δ

Page 13: Robert Frosch Slender Column

Nonsway or Sway?

• Nonsway– Mend, ≤ 1.05Mend,

– ΣKbracing ≥ Σ 12Kcolumns

– 0.05Σ Δ= ≤u o

us c

PQV l

2nd-order 1st-orderP

P

Δ = 0MT

MB

Page 14: Robert Frosch Slender Column

Sway procedure

1 1 1

2 2 2

δδ

= += +

ns s s

ns s s

M M MM M M

P

P

δMTs

δMBs

1 11

δ = ≥−s Q

1 11

0.75

δ = ≥Σ−Σ

su

c

PP

Option 1 Option 2

P

P

MTns

MBns

+

Δ

Page 15: Robert Frosch Slender Column

2nd-order End MomentsNonlinear

second orderElastic

second orderMoment

Magnification

Pu

Pu

Mu,top

Mu,bottom

Page 16: Robert Frosch Slender Column

10.10.2.2 Second-order along length• Computer analysis

– Equilibrium in Deformed Position– Not practical

• Moment magnification 10.10.6

Nodes at ends Nodes between ends

Page 17: Robert Frosch Slender Column

Moment magnification due to member curvature

2δ=c nsM M

1.01

0.75

δ = ≥−

mns

u

c

CP

P

( )2

2π=c

u

EIPkl

Pu

Pu

Mu,top

Mu,bottom

δMc

Mu,top

Mu,bottom

Page 18: Robert Frosch Slender Column

Moment Amplification

Pu

Pu

M1

δ

M1

M1

M1

Mc=M1+Pδ

11

1=

−c

u

c

M MPP

Page 19: Robert Frosch Slender Column

Equivalent Moment - Cm

1

2

0.6 0.4= +mMCM

Pu

Pu

M2

δ

M1

M2

M1

2=eq mM C M

Mc

Pu

Pu

Meq

δ

Meq

Meq

Meq

Mc

Page 20: Robert Frosch Slender Column

ACI Simplified Equation

1.01

0.75

δ = ≥−

mns

u

c

CP

P

Stiffness reduction factorφK=0.75

( )2

2π=l

cu

EIPk

Page 21: Robert Frosch Slender Column

Flexural Stiffness• Account for:

axial load, cracking, reinforcement, sustained load

( )0.21 β

+=

+c g s se

dns

E I E IEI

0.401 β

=+

c g

dns

E IEI

Page 22: Robert Frosch Slender Column

Alternate Stiffness

0

0.80 25 1 0.51

0.35 0.8751 1

β

β β

⎛ ⎞⎛ ⎞= + − −⎜ ⎟⎜ ⎟⎜ ⎟ +⎝ ⎠⎝ ⎠

≤ ≤+ +

c gst u u

g u dns

c g c g

dns dns

E IA M PEIA P h P

E I E IEI

Page 23: Robert Frosch Slender Column

Minimum Eccentricity

( )2,min 0.6 . 0.03= +uM P in h

m

1

2

C 1.0 orM0.6 0.4M

=

= +

Courtesy Walter P Moore

Pu

Pu

δ

M2,min

M2,min

Mc

Page 24: Robert Frosch Slender Column

10.10.6.3 Equivalent Length

k=1.0 k=0.5

Flexible Beams Rigid Beams

M = 0 M

M = 0 M

( )2

2π=l

cEIP

kP P

2

2

π=l

eEIP

P

Pe

M

0

> Pe > 0

Page 25: Robert Frosch Slender Column

k = 1.0

• Column loses stiffness as Pe approached• To achieve k < 1.0

– Beams must resist moment for Pδ– Beams not designed for Pδ moment

• Use k=1.0 for design

Page 26: Robert Frosch Slender Column

Too slender?

• M ≤ 1.4M- 2nd-order effects not dominate response- Eliminate need for stability analysis

• Cross-sectional dimensions– Within 10% shown on design drawings

2nd order 1st order

Page 27: Robert Frosch Slender Column

Pu = 200 kips

Mu = 86 ft-kips

Pu = 200 kips

Mu = 72 ft-kips

' 4,000 psi60,000 psi

==

c

y

ff

12”

15”2.5” 2.5”

Design Example

16 ft60 kips

==

lu

susP

A

A

Page 28: Robert Frosch Slender Column

Minimum Eccentricity

( )( )

2,min 0.6 . 0.03

200 0.6 . 0.03(15 .)200 (1.05 .)210 .- 17.5 -

= +

= +== =

uM P in h

kip in inkip inin kip ft kip

86ft-kip 17.5ft-kip>

Does not control∴

12”

15”

A

A

Page 29: Robert Frosch Slender Column

Amplification between ends

in.1.0*16 ft *12ft

0.3*15 in.42.7

=

=

uklr

1

2

34 12

72 ft-kip34 1286 ft-kip

24.0

= −

=

MM

Use k = 1.0

42.7 24.0 Slender> ∴

Page 30: Robert Frosch Slender Column

Column Properties

( )( )3 4

57 4000 3605 ksi1 12 in. 15 in. 3375 in.

12

= =

= =

c

g

E

I12”

15”2.5” 2.5”

A

A

( )( )46 2

Sustained Axial Load 1.2*60 kips 0.36Factored Axial Load 200 kips

0.4 3605 ksi 3375 in.0.43.58 10 kip in.

1 1 0.36

β

β

= = =

= = = ⋅+ +

dns

c g

dns

E IEI x

Page 31: Robert Frosch Slender Column

Euler Buckling Load

( )( )2 6 22

2 2

3.58 10 in. -kip958 kips

in.1.0*16 ft *12ft

ππ= = =⎛ ⎞⎜ ⎟⎝ ⎠

cu

xEIPkl

Page 32: Robert Frosch Slender Column

Amplified Moment

2

1.29*86 ft-kip 111 ft-kipδ=

= =c nsM M

1

2

72 ft-kipC 0.6 0.4 0.6 0.4 0.9386 ft-kip

0.93 1.29200 kips110.75*958 kips0.75

δ

= + = + =

= = =−−

m

mns

u

c

MM

CP

P

Design Column for Pu = 200 kips Mu = 111 ft*kips

Mu = 86 ft-kips

Mu = 72 ft-kips

Pu = 200 kips

Page 33: Robert Frosch Slender Column

Magnitude of 2nd-order effects

M ≤ 1.4M2nd order 1st order

Page 34: Robert Frosch Slender Column

Future Directions

• Improved EI expressions• Improved βdns definition• Increased slenderness limits

Page 35: Robert Frosch Slender Column

Improved EI Expressions

• Reduce Conservatism• Provide Simplicity

Page 36: Robert Frosch Slender Column

Axial Load

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Axial Load, Pu/Po

c g

EIE I

−0

1 0.6 uPP

Page 37: Robert Frosch Slender Column

Eccentricity Ratio

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50%

Eccentricity Ratio, e/h

Uncracked Transition Zone CrackedFully

EI E cI g

..

Page 38: Robert Frosch Slender Column

Analytical Comparison: ACI Eq. 10-15

0%

10%

20%

30%

40%

50%

60%

70%

80%

Freq

uenc

y

Mc,Eq./Mc,Anal.

Design

ACI Eq. 10-15

1.0

1.2

1.8

>2.0

0.8

1.4

1.6

0.6

0 4. c gE I

Page 39: Robert Frosch Slender Column

Analytical Comparison: ACI Eq. 10-8

0%

10%

20%

30%

40%

50%

60%

70%

80%

Freq

uenc

y

Mc,Eq./Mc,Anal.

Design

ACI Eq. 10-8

0 80 25 1 0 5⎛ ⎞⎛ ⎞

+ − −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠. .st u u

c gg u o

A M P E IA P h P

1.0

1.2

1.8

>2.0

0.8

1.4

1.6

0.6

Page 40: Robert Frosch Slender Column

Improved βdns definition

u,susdns

u

PP

β = Same Load Combination

For 1.4D Combination:

dns1.4D 1.01.4D

β = =

dns

1 1 0.51 2β

= =+

0.20= c gEI E I

Page 41: Robert Frosch Slender Column

Future βdns

• Account for creep• Consider axial stress level

u,sus u,susdns '

o c g

P PP 0.85f A

β = ≈

Page 42: Robert Frosch Slender Column

Slenderness Limits

• 1.4 Maximum Eliminated for Pδ• 1.4 Increased for PΔ

Page 43: Robert Frosch Slender Column

Summary

• Higher strength concrete• Higher strength reinforcement• Improved estimates of

– Short term stiffness – EI– Long term stiffness - βdns

• Increase used of slender columns

Page 44: Robert Frosch Slender Column

Questions?

Page 45: Robert Frosch Slender Column

Sway Frame

P 2PΔ Δ

M1M2

1 2M M (P 2P)+ = + Δ