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Effective height of column

Height between points of contraflexure of the buckled column

End conditions

1

2

3

4

Value of β for unbraced column

End condition at bottom

1 2 3

1 0.7 0.! 0."

2 0.! 0.! 0."

3 0." 0." 1

Value of β for braced column

End condition at bottom

1 2 3

1 1.2 1.3 1.#

2 1.3 1. 1.!

3 1.# 1.! 0

4 2.2 0 0

Minimum Eccentricity 

emin $% l&00 ' (&30

l )nsupported length of column in mm

( *ateral dimension in the direction under consideration in mm

Short column

+he end of the column is connected monolithicall, to beams on either side which are at least asdeep as the o-erall dimension of the column in the plane under consideration

+he end of the column is connected monolithicall, to beams or slabs on either side which areshallower than the o-erall dimension of the column in the plane under consideration.t simulatesa semi/fixed condition.

+he end of the column is connected to member whichwhile not specificall, designed to pro-iderestraint to rotation of the columnstill pro-ides some nominal restraint.t simulates a hinged endcondition.

+he end of the column is unrestrained against both lateral mo-ement and rotation. t simulatesfree end of a cantile-er in an unbraced structure.

End conditionat top

End conditionat top

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hen emin does not exceed 0.0(column can be designed b, following euation

u 0.4fcuc'0.#75f,sc

if p is percentage of steel in column section then

 g b(

 c g/pg&100

 sc pg&100

+herefore

u&fcub( 0.4'6p&100fcu560.#7f, / 0.4fcu

Reinforcement requirement 

1 8inimum reinforcement 0.!9 of cross sectional area

2 8in. bars in circular columns shall be #3 :pacing of longitudinal bars shall not exceed 300mm

4 f the spacing of longitudinal bars exceeds 7mm pro-ide ties to bind the bars

(iameter of lateral ties shall not be less than one/fourth of the dia. ;f largest longitudinal bar.

# itch6centre to centre distance of ties should not exceed<

the least lateral dimension of column

1# times smallest dia. ;f lonitudinal bars

4! times dia. ;f lateral ties

Short column under axial load and uniaxial bending 

u/8u graph

1 =ompression control region < Entire section under compression

2 +ension control region < art of section under tension3 8inimum eccentricit, region

Design & Construction of design chart 

Design

ercentage of reinforceme

Material properties p&fcu

>cu 40 ?&mm2

>, 41 ?&mm2 u&6fcu5b5d

w6con 24 @?&m3

200000 ?&mm2 8u&6fcu5b5dA2 6 from table

Member Properties

8oment carr,ing capacit, a

=lear height 2." m 8oment carr,ing capacit, a

idth 0.3 m

*ength 1. m 8x&8ux

End =ondition 1 8,&8u,

β 1.2

Esteel

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Effecti-e height 2.07 m (esign *oad

*e&*l 7.1#42!7143 :hort column &uB

=oefficient of effecti-e co-er 0.0333333333

Effecti-e =o-er to reinforcement 0.0 m lpha

Loads =heck for biaxial bending

Factored load 

1.4dl ' 1.#** 10!70 @?

Reduction factor due to selenderness

=r 1

Min. Eccentricities

8in. Eccentricit, along largest dimension 0.02 m

8in. Eccentricit, along least dimension 0.017 m

Moments

8in. Eccentric 8oment around least dimension 217.4 @?m

8in. Eccentric 8oment around largeset dimension 1"0.22 @?m ctual 8oment around least dimension 2!2 @?m

 ctual 8oment around largest dimension 133 @?m

Data for design curve

8ax. strain in steel 0.001!02

ercentage of steel 3 9

p&fcu 0.07

(esign tensile strength of :teel 3#1.0 ?&mm2

u&6fcu5b5d 0."17

8u&6fcu5b5dA2 0.02"77!7

Case! "hen neutral axis lies outside the section

=oefficient of (epth of neutral axis from extreme compression fibre6k 1.1

=oefficient for area of stress block6k1 0.3!44#!

a 12."21#!

=oefficient of (istance of =C of force of compression from extreme fibre6k2 0.442!41

 rea of stress block6 230#!.0" mm2

ercentage of steel 3 9

8ax. strain at the highl, compressed fibr 0.00327#"7

:train at top le-el 0.002!10#3!

:tress in top steel 3#1.0

:train at Dottom le-el 0.000"7447

:tress in Dottom steel 11".14!"3#17

Case# "hen eccentricity is less than emin

0.0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Pu/$cu!

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:tress in top =oncrete 17.!4

:tress in bottom concrete 4."1!31"!01

u&6fcu5b5d 0.#00!403"

8u&6fcu5b5dA2 0.02#42343

Case$ "hen neutral axis lies inside the section

#% enssile stress is '( neutral axis )asses through the bottom steel%

=oefficient of (epth of neutral axis from extreme compression fibre6k 0."####7

ercentage of steel 3 9

:train at top le-el 0.00337"3103

:tress in top steel 3#1.0

:tress in top =oncrete 17.!4

u&6fcu5b5d 0.47#7037

8u&6fcu5b5dA2 0.0!002322

!% enssile stress some )ercentage of design stress of steel 

=oefficient of tensile stress 0.4

ercentage of steel 3 9

:tress in tensile steel 144.42 ?&mm2

:train in tensile steel 0.0007221

=oefficient of (epth of neutral axis from extreme compression fibre6k 0.!0133"

u&6fcu5b5d 0.3#302!2!

8u&6fcu5b5dA2 0.1133!10"!

$% First *ield +oint(ensile stress , '-./fy0 ensile strain , '-./fy1Es%

ercentage of steel 3 9

:tress in tensile steel 3#1.0 ?&mm2

:train in tensile steel 0.001!02

=oefficient of (epth of neutral axis from extreme compression fibre6k 0.#37733

u&6fcu5b5d 0.222!"3"03

8u&6fcu5b5dA2 0.14!20477

2% Final *ield +oint(ensile stress , '-./fy0 ensile strain , '-./fy1Es 3 '-''!%

ercentage of steel 3 9

:tress in tensile steel 3#1.0 ?&mm2

:train in tensile steel 0.003!02

=oefficient of (epth of neutral axis from extreme compression fibre6k 0.4#3137

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u&6fcu5b5d 0.1#003"407

8u&6fcu5b5dA2 0.14#2303

4% De)th of neutral axis in 5D

ercentage of steel 3 9=oefficient of (epth of neutral axis from extreme compression fibre6k 0.2

:train in top steel 0.0030333333

:tress in top steel 3#1.0

u&6fcu5b5d 0.0!331

8u&6fcu5b5dA2 0.12""#03#

6% "hen axial force is '0 steel beam

ercentage of steel 3 9

8u&6fcu5b5dA2 0.0"#70"!214

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Unbraced Column design

Design

ercentage of reinforceme

Material properties p&fcu

>cu 2 ?&mm2>, 41 ?&mm2 u&6fcu5b5d

w6con 24 @?&m3

200000 ?&mm2 8u&6fcu5b5dA2 6 from table

Member Properties

8oment carr,ing capacit, a

=lear height 2."""""401 m 8oment carr,ing capacit, a

idth 0.14""""#"!4 m

*ength 0.14""""#"!4 m 8x&8ux

End =ondition 1 8,&8u,β 1.2

Effecti-e height 2.4"""4"132 m (esign *oad

*e&*l 17.00000027 :elender column &uB

=oefficient of effecti-e co-er 0.1

Effecti-e =o-er to reinforcement 0.014""""#"! m lpha

Loads =heck for biaxial bending

Factored load 

1.4dl ' 1.#** F*)EG @?

Esteel

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Reduction factor due to selenderness

=r 0.!33333327

Min. Eccentricities

8in. Eccentricit, along largest dimension 0.0074"""! m

8in. Eccentricit, along least dimension 0.0074"""! m

Moments

8in. Eccentric 8oment around least dimension F*)EG @?m

8in. Eccentric 8oment around largeset dimension F*)EG @?m

 ctual 8oment around least dimension F*)EG @?m

 ctual 8oment around largest dimension @?m

Data for design curve

8ax. strain in steel 0.001!02

ercentage of steel 2 9

p&fcu 0.0!

(esign tensile strength of :teel 3#1.0 ?&mm2

u&6fcu5b5d 0.#132

8u&6fcu5b5dA2 0.030#7#

Case# "hen eccentricity is less than emin

0 0.05 0.1 0.15 0.2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Column Interaction Chart

Design

Curve

Design

Pu/cu!"

#u/$cu!"2

Pu/$cu!"

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Case! "hen neutral axis lies outside the section

=oefficient of (epth of neutral axis from extreme compression fibre6k 1.1

=oefficient for area of stress block6k1 0.3!44#!

a !.07#03

=oefficient of (istance of =C of force of compression from extreme fibre6k2 0.442!41

 rea of stress block6 1441.73 mm2ercentage of steel 2 9

8ax. strain at the highl, compressed fibr 0.00327#"7

:train at top le-el 0.002"7!7234

:tress in top steel 3#1.0

:train at Dottom le-el 0.000"7447

:tress in Dottom steel 11".14!"3#17

:tress in top =oncrete 11.1

:tress in bottom concrete 3.073"474!7#

u&6fcu5b5d 0.70!!1##!

8u&6fcu5b5dA2 0.0"3!7#4"

Case$ "hen neutral axis lies inside the section

#% enssile stress is '( neutral axis )asses through the bottom steel%

=oefficient of (epth of neutral axis from extreme compression fibre6k 0."

ercentage of steel 2 9

:train at top le-el 0.0031111111

:tress in top steel 3#1.0

:tress in top =oncrete 11.1

u&6fcu5b5d 0.4#3"#

8u&6fcu5b5dA2 0.0"##7!4

!% enssile stress some )ercentage of design stress of steel 

=oefficient of tensile stress 0.4

ercentage of steel 2 9

:tress in tensile steel 144.42 ?&mm2

:train in tensile steel 0.0007221

=oefficient of (epth of neutral axis from extreme compression fibre6k 0.74#074

u&6fcu5b5d 0.3077!7222

8u&6fcu5b5dA2 0.13002413!4

$% First *ield +oint(ensile stress , '-./fy0 ensile strain , '-./fy1Es%

ercentage of steel 2 9

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:tress in tensile steel 3#1.0 ?&mm2

:train in tensile steel 0.001!02

=oefficient of (epth of neutral axis from extreme compression fibre6k 0."371

u&6fcu5b5d 0.20"2"03

8u&6fcu5b5dA2 0.1#7!3073

2% Final *ield +oint(ensile stress , '-./fy0 ensile strain , '-./fy1Es 3 '-''!%

ercentage of steel 2 9

:tress in tensile steel 3#1.0 ?&mm2

:train in tensile steel 0.003!02

=oefficient of (epth of neutral axis from extreme compression fibre6k 0.4311"7

u&6fcu5b5d 0.10770!27

8u&6fcu5b5dA2 0.1#322442

4% De)th of neutral axis in 5D

ercentage of steel 2 9

=oefficient of (epth of neutral axis from extreme compression fibre6k 0.2

:train in top steel 0.0021

:tress in top steel 3#1.0

u&6fcu5b5d 0.0!4

8u&6fcu5b5dA2 0.14"3"2

6% "hen axial force is '0 steel beam

ercentage of steel 2 9

8u&6fcu5b5dA2 0.113#

Data for design curve

Material properties

>cu 2 ?&mm2

>, 41 ?&mm2

w6con 24 @?&m3

200000 ?&mm2

8ax. strain in steel 0.001!02

ercentage of steel 2 9

p&fcu 0.0!

(esign tensile strength of :teel 3#1.0 ?&mm2

Esteel

Case# "hen eccentricity is less than emin

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u&6fcu5b5d 0.#132

8u&6fcu5b5dA2 0.030#7#

Case! "hen neutral axis lies outside the section

=oefficient of (epth of neutral axis from extreme compression fibre6k 1.1

=oefficient for area of stress block6k1 0.3!44#!

a !.07#03

=oefficient of (istance of =C of force of compression from extreme fibre6k2 0.442!41

 rea of stress block6 17301.07 mm2

ercentage of steel 2 9

8ax. strain at the highl, compressed fibr 0.00327#"7

:train at top le-el 0.00327#"7

:tress in top steel 0

:train at Dottom le-el 0.000"7447

:tress in Dottom steel 0:tress in top =oncrete 1.33!

:tress in bottom concrete /#.73!0212

u&6fcu5b5d 0.4024#!34#4

8u&6fcu5b5dA2 0.00!1##

Case$ "hen neutral axis lies inside the section

#% enssile stress is '( neutral axis )asses through the bottom steel%

=oefficient of (epth of neutral axis from extreme compression fibre6k 1ercentage of steel 2 9

:train at top le-el 0.003

:tress in top steel 0

:tress in top =oncrete 1.33!

u&6fcu5b5d 0.34

8u&6fcu5b5dA2 0.02!01

!% enssile stress some )ercentage of design stress of steel 

=oefficient of tensile stress 0.4ercentage of steel 2 9

:tress in tensile steel 0 ?&mm2

:train in tensile steel (F&0G

=oefficient of (epth of neutral axis from extreme compression fibre6k (F&0G

u&6fcu5b5d (F&0G

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8u&6fcu5b5dA2 (F&0G

$% First *ield +oint(ensile stress , '-./fy0 ensile strain , '-./fy1Es%

ercentage of steel 2 9

:tress in tensile steel 0 ?&mm2

:train in tensile steel (F&0G=oefficient of (epth of neutral axis from extreme compression fibre6k (F&0G

u&6fcu5b5d (F&0G

8u&6fcu5b5dA2 (F&0G

2% Final *ield +oint(ensile stress , '-./fy0 ensile strain , '-./fy1Es 3 '-''!%

ercentage of steel 2 9

:tress in tensile steel 0 ?&mm2

:train in tensile steel (F&0G

=oefficient of (epth of neutral axis from extreme compression fibre6k (F&0G

u&6fcu5b5d (F&0G

8u&6fcu5b5dA2 (F&0G

4% De)th of neutral axis in 5D

ercentage of steel 2 9

=oefficient of (epth of neutral axis from extreme compression fibre6k 0.2

:train in top steel 0.003

:tress in top steel 0

u&6fcu5b5d 0.0!4

8u&6fcu5b5dA2 0.03341

6% "hen axial force is '0 steel beam

ercentage of steel 2 9

8u&6fcu5b5dA2 0

(eri-ation of basic euations

Case 7 # e%emin

u 0.44#5fcu5b('6fsc/fccsc

0.44#5fcu5b('6fsc/fcc5pb(&100

u&6fcub( 0.44#'6p&100fcu56fsc/0.44#fcu

8u&fcub(2 0.0pu&fcub(

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Case 7 !  ?eutral axis lies outside the section

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b

d

+ ?os

t 3 9 2 33

0.07

0.17#1"0

0.0"

round shorter dimension 2!3 @?m

round *argest dimension ##1. @?m

1.00""#

0.2!7##

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1432.1" @?

0.7737#

1."2!"#

1.101"4!"1 change siBe or reinforcement

8u&6fcu5b5dA2 u&6fcu5b5d p&fcu (esign6u&fcubd

0.02"77!! 0."17 0.07 0.17#1"

0.02#423 0.#00!4 0.07 0.17#1"

0.0!002323 0.47#7037 0.07 0.17#1"

0.1133!11 0.3#302!2" 0.07 0.17#1"

0.14!204! 0.222!"3" 0.07 0.17#1"

0.14#23 0.1#003"41 0.07 0.17#1"

0.0"#70"!2 0 0.07 0.17#1"

0.04 0.06 0.08 0.1 0.12 0.14 0.16

Column Interaction Chart

Design Curve

Design Pu/cu!"

#u/$cu!"2

"

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+ ?os

t 2 9 0 (F&0G

0.0!

F*)EG

IE>G

round shorter dimension IE>G @?m

round *argest dimension IE>G @?m

F*)EG

3"3.1!" @?

F*)EG

F*)EG

F*)EG F*)EG

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8u&6fcu5b5dA2 u&6fcu5b5d p&fcu (esign6u&fcubd

0.030#7# 0.#132 0.0! F*)EG

0.0"3!7# 0.70!!17 0.0! F*)EG

0.0"##7!4 0.4#3"# 0.0! F*)EG

0.13002414 0.3077!72 0.0! F*)EG

0.1#7!307 0.20"2"03 0.0! F*)EG

0.1#32244 0.10770!3 0.0! F*)EG

0.113# 0 0.0! F*)EG

 

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8u&6fcu5b5dA2 u&6fcu5b5d p&fcu (esign6u&fcubd

0.030#7# 0.#132 0.#####7 0

0.00!17 0.4024#!3 0.#####7 0

0.02!01 0.34 0.#####7 0

(F&0G (F&0G 0.#####7 0

(F&0G (F&0G 0.#####7 0

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(F&0G (F&0G 0.#####7 0

0 0 0.#####7 0

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