Composite Column

7/27/2019 Composite Column

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Design of Composite ColumnsDesign of Composite Columns

Chiew Sing PingChiew Sing Ping

School of Civil and Environmental Engineering

Nanyang Technological University, Singapore

Compression, Bending, Combined compression and bending, Column

buckling curve, Interaction curve between compression and bending

2

Composite columns

Composite columns often offer significant economicadvantages over either structural steel or reinforced concretealternatives.

High load carrying capacities and high flexural rigidities with

smaller sizes at reduced costs.

Excellent inherent fire resistances.

By varying different materials, composite columns withdifferent axial load and moment resistances but with identicalexternal dimensions are readily obtained.

This allows the outer dimensions of a column to be heldconstant over a number of floors in a building, simplifyingboth constructional and architectural details.


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3

Composite Columns with H sections

Fully encased H section Partially encasedH section

Dc

T

t

B= Bc

x

y

t

T

Bc

cXcX B

x

y

cy

D

cy

Dc

4

Composite columns with hollow sections

In-filled rectangular

hollow section

B

D

t

x

y

In-filled circularhollow section

D

t

x

y

In-filled circular hollow sectionwith encased H section

x

y

D


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5

Design principles of composite columns

Fully encased H section

= + +

In-filled hollow section

= +

Strength assessmentRcp RsRc= + + Rr

Deformation assessmentEIcp (EI)s(EI)c= + + (EI)r

6

Scope

Design considerations of composite columns.

Steel columns. Basic section capacities. Plastic stress block

method. Worked example. Interaction between compression

and bending in stocky and slender columns.

Mechanics of column buckling , axial buckling resistances in

slender columns, column buckling curves.

Composite columns. Compression and moment capacities.

Interaction between compression and bending in stocky and

slender columns.

Non-linear and simplified interaction curves.

Slenderness vs reduction factor. Comparison among codified

design methods.


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Main design considerations

How to ensure concrete core, steel section and reinforcement, i.e.three materials of different sizes and strengths, work together as an integral

member to resist compression and bending ?

How to transfer loads among one another of the three materials ?

How to ensure high local loads are effectively distributed away ?

How to ensure three materials with different sizes and Youngs modulii todeform consistently with limited interfacial slippage under compression and

bending ?

Strength assessmentRcp RsRc= + + Rr

Deformation assessmentEIcp (EI)s(EI)c= + + (EI)r

8

Interfacial shear bond strength from 0.2 to 0.6 N/mm2,

depending on the amount of concrete confinement provided.

Mechanical shear connectors installed wherever needed,mainly within the load application regions.

Allow for long term effects due to concrete (drying, shrinkage,

creep)

Main design considerations


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Pcp & M

P & M

Steel Composite

columns columns Section capacity (stocky column)

Resistance to compression, Pc Pcp

Resistance to moment, Mc Mcp

Reduced moment resistance under

compressive force, i.e. interaction between

compression and bending,

Member resistance (slender column)

Axial buckling resistance, Pc Pcp




Design of steel columns is presented as a reference for the morecomplicated design of composite columns

Design of steel and composite columns

10

Basic capacities

Pc = pyAs

Mc = py S

Compression

capacity

Moment

capacity

Plastic Stress Block Method


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Interaction between compression and

bendingFor small axial force with dn d :

Pdn

Mdn

d

Axial force

Reducedmoment capacity


12

Pdn

M

dn

Axial force

Reducedmoment capacity

For high axial force with d dn :


Interaction between compression and

bending


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Worked example on interaction

between compression and bending

UB457x152x57

Section dimensions

D = 449.8 mm B = 152.4 mm

t = 7.6 mm T = 10.9 mm

r = 10.2 mm d = 407.6 mm

d = D T = 438.9 mm

Dd

T

B

t

14

dn

2222n Antd =

The reduced moment capacity, M

= py Sr where Sr = =

Re-writing the expression as follows:

= K1 K2n2 where K1 = Sx = 1077 cm

3

K2

= = 1460.8cm3

As the max. value of dn is 428mm, max. value of n =

4

tdS

2n

x

6.74

6664

4t

A 22

=

0.4886664

7.6x428=

Assume the applied load, P, is to be resisted by the

shaded area, dn x t, i.e.

P = pyAn where An = dn t

The axial force ratio, n =A

td

P

P n

c

=

t4

AnSx

22

Reduced moment capacity under axial force


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Reduced moment capacity under axial force Assume the applied load, P, is to be resisted by

the shaded area, A (D dn) x B, i.e.

P = pyAn where An = dn t

The axial force ratio, n =

DB

1)(nAdn +

=

A)Bd(DA

PF n

c

c =

After re-arranging,

The reduced moment capacity, M

= [ ]

19.57

16664

449.8x152.4x21

A

2BDK&72.85cm

152.4x4

6664

4B

AK

n)(Kn)(1K

n1

A

2BDn)-(1

4B

A

A

2BD1)(n1)(n

4B

A

1)(nB

2AD-D-1)-(n

B

AD

4

BhD

4

BS

43

22

3

43

22

22

2

222

n2

xr

=

=====

+=

+=

+=

==

dnD

B

16

Section capacities of a stocky steel column

Interaction between compression and bending

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Axial force ratio, P / Pc

Reducedmoment

ratio,

Mr

/M

c

Plastic section

analysis

Linear

reduction

1M

M

P

P

cc

=+

1M

M

P

P

cc

=

+

Empirical formula

UB457 x 152 x 52

For the empirical formula, the values of & equal to 1 ~ 3 depending ontypes of sections.


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Pcp & M

P & M

Section capacity (stocky column)











Steel Composite

columns columns



18

Interaction between compression and bending

Section capacity checkConservative design for all sections,

More rigorous design for compact and plastic section,

where Mr is the reduced plastic moment capacity in thepresence of axial load according to plastic analysismethod.

1MM

PP

cc+

M Mr

Design of steel columns


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Interaction between compression and bi-axial

bending

Section capacity checkConservative design for all sections,

1M

M

M

M

pA

F

cy

y

cx

x

y

++ (Eq 1)

1M

M

M

M21

Z

ry

yZ

rx

x

+

(Eq 2)

More rigorous design for

compact and plastic section,


20

Stability is an additional requirement to equilibrium

Axial buckling occurs in slender columns under high

axial compressive forces.

Equilibrium

L

P

P

Alternative equilibrium

but unstable or unfit for use

Deformed elastic

curve

of the columnv

P

P

Member resistance of a slender steel column


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P.v + Mx = 0

As Mx =

Free body diagram

(v 0 after buckling)

P

Mx v

P Use k2 = or k =

2

2

dx

vd

EI

0Pvdx

vdEI

2

2

=+ 0vEI

P

dx

vd2

2

=+

= 0M

Consider moment equilibrium, i.e.

or

EI

P

EI

P

0vk

dx

vd 22

2

=+

22

From mathematics, the general solution is

v = C1sin kx + C2 cos kx

Consider the boundary condition

As v = 0 at x = 0 C2 = 0

v = C1sin kx

As v = 0 at x = L sin kL = 0kL = n where n = 1, 2, 3

Consider the fundamental mode, i.e. when n = 1

k = or k =

This gives the solution for v

v = where C1 is undetermined.

L

n

L

43421

shape

xL

sin1C

Hence, the buckled mode shape is found with an undetermined

magnitude.


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It should be noted that at the critical buckling state,

k = =

where PE is the applied load at buckling, or the Euler buckling load.

EI

P

L

E2

2

=EI

P

L

=2

2E

L

IEP

material property

Section property

Member lengthconstant

The Euler buckling load is an elastic value for a perfectly straightcolumn, and it is necessary to incorporate material yielding and

initial imperfection in practical design.

The structural mechanics is equally applicable to steel columns,

reinforced concrete columns as well as composite columns.

24

For real columns, most modern steel codes adopt the formulation

with member slenderness and buckling strength.

PE =

pE = as

= or

= where =

= [normalization]

= where Y =

constant = 85.8 for S275

= 75.5 for S355

2

y2

L

IE

2y2

L1

AIE 2

yy r

AI =

2

2y2

L

rE

( )2y2

L/r

1E

2

2

1E

yr

L

y

E

p

p2

y

2

1

p

E

2

Y

yp

E material properties

Euler

buckling load

Elastic buckling

strength

ry - radius of gyration

E Iy - flexural rigidity about minor axis

L - member length

- slenderness

APE =

Dividing by

yield strength


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Adopt the slenderness ratio, , the elastic buckling strengthratio is given by:

Column buckling curves

yielding1.0

0 2.0

celastic buckling

test

data

1.0

.o

.. . .

.

.

..

.

..

.

.

.

.

..

.

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

The curves relate material yielding and elastic buckling in real columns

together with initial imperfection and residual stresses, etc.

real column

behaviour

Y

=

44 344 21

2y

E

1

p

p=

geometry

critical

E

r

L

=

material

y

Yp

E

=

whereY

=

26

P Pc where Pc = pc A

The compressive strength, pc , pf a real column depends on the

slenderness, , of the steel section, the design strength, py , and the

relevant column buckling curves to be selected as follows:

Selection of column buckling curves

Type of section Axis of buckling

x-x y-y

Rolled I section T 40 mm

T > 40 mm

a

b

b

c

Rolled H section T 40 mm

T > 40 mm

b

c

c

d

Welded I or H section T 40 mm

T > 40 mm

b

b

c

d

Hot rolled structural hollow section a a

Welded box section T 40 mm

T > 40 mm

b

c

b

c

Axial buckling strength, pc


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( pE pc )( py pc ) = pE pc ( BS5950: Part 1 )

pc = where

py = design strength

= slenderness

is the Perry factor = but > 0

o =

a is the Robertson constant

= 2.0 curve a

3.5 b

5.5 c

8.0 d

yE2

yE

pp

pp

+ 2

)p(1p Ey ++

=

2

2

E

Ep =

1000

)a( o

yp

E.20

For different types of

sections and axis of

buckling after calibration

against test data.

Perry-Robertson interaction formula

28

stocky column slender column

no buckling elastro-plastic

buckling

elasticbuckling

limiting

a

b

c

d

yielding1.0

0.0 0.2 1.0 2.0

elastic bucklingc

real column

behaviour

Normalized curves relating c and

Column buckling curves


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Non-sway mode Sway mode

Buckled

shape

k Design 0.7 0.85 1.0 1.2 2.0

Theoretical 0.5 0.7 1.0 1.0 2.0

Effective length coefficient, k

30

Non-sway mode

Conditions of restraint at ends (in plane under consideration) k

Design Theoretical

Effectively held

in position atboth ends

Restrained in direction at both ends 0.7 0.5

Partially restrained in direction at both ends 0.85 -

Restrained in direction at one end 0.85 0.7

Not restrained in direction at either end 1.0 1.0

Sway mode

One end Other end k

Design Theoretical

Effectively held

in position and

restrained in

direction

Not held

in

position

Effectively restrained in direction 1.2 1.0

Partially restrained in direction 1.5 -

Not restrained in direction 2.0 2.0

Effective length coefficient, k


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Pcp & M

P & M












Steel Composite

columns columns



32

Interaction between compression and bending Overall buckling check

Simplified approach using linear interaction

where

F is the applied axial load in the column

Mx , My are the applied moments about the major and the minor axesrespectively

pc, py are the compression and the design yield strengths respectively

As is the cross-sectional areamLT is the equivalent uniform moment factor

Mb is the buckling resistance moment capacity, and

Zy is the elastic section modulus about the minor axis.

1Zp

M)(m

MM)(m

pAF

yy

yyLT

b

xxLT

cs++

Axial buckling

as a columnLateral buckling

as a beam

Reduction due to

lateral moment



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Pcp & M

P & M












Steel Composite

columns columns



34

Resistance of composite section to compression

sdscucycp fAf0.45AApP ++= Fully encased and partially encased H sections:

In-filled rectangular hollow sections:

sdscucycp fAf0.53AApP ++=

where:

A, Ac and As are the areas of the steel section, the concrete and the

reinforcements respectively.

py and fsd are the design strengths of the structural steel section and

the steel reinforcement respectively.

py = py / a a = 1.0c = 1.5

fsd = fy / s s = 1.15

Design of composite columns


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)S(Sf)S(Sf0.5)S(SpM psnpssdpcnpccucpnpycp ++=

where:c = 0.53 for all in-filled hollow sections

= 0.45 for fully or partially encased H sections

Sp,Sps, Spc are the plastic section modulii for the steel section, the

reinforcement and the concrete of the composite cross-

section respectively (for the calculation of Spc, the

concrete is assumed to be uncracked).

Spn, Spsn, Spcn are the plastic section modulii of the corresponding

components within the region of 2dn from the middle

line of the composite cross-section.

dn is the depth of the neutral axis from the middle line of

the cross-section.

Resistance of composite section to bending


36

Interaction curve for compression and bendingIn-filled hollow section


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Interaction curve for compression and bendingIn-filled hollow section --- Points A and B

No moment

No axial force

Point A 0.53 fcu

Pcp

-

py

-

fsd

-

-

0.53fcu fsdPoint B

+

Mcp-

py

-

+

-

dn

38

Interaction curve for compression and bendingIn-filled hollow section --- Points C and D

0.53fcu py fsd

-

Point D

PD= Ppm/ 2

-

+

-

+

dnMD = Mcp,max

+

Point C0.53fcu

py fsd

Mc

= Mcp

Pc= Ppm

- --

+

dn

dn

+


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Interaction curve for compression and bendingIn-filled hollow section --- Point E

ME

Point E 0.53fcu py fsd

PE

-

dn

dE

dg

-

-+

-

40

Design formulae for composite columnsConcrete in-filled hollow sections

Spc =

dn =

Spcn = (B - 2t) dn2 Spsn

Spn = B dn2 Spcn Spsn

Major axis bending

psSrtD

rrtDtB

2

)4(3

2

4

)2)(2( 232

)2(42

)2(

cucycuc

cucsdsncuccfptfB

ffAfA

+


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Interaction curve for compression and bendingFully encased H section

42

Interaction curve for compression and bendingFully encased H-section --- Points A and B

Pcp

0.45fcu py fsd

-

-

- -

Point A

No moment

Mcp

0.45fcu py fsd

dn

dn

2dn

-

+

-

Point B

-

+No axial force


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Interaction curve for compression and bendingFully encased H-section --- Points C and D

Mcp

Ppm

0.45fcu py fsd

dn

dn

2dn

+

-

+

--

Point C

Mcp,max

Ppm/2

0.45fcu py fsd

+

-

+

--

Point D

44

Design formulae for composite columns

Sp are given in section property tables for steel sections

Sps =

where

ei are the distances of the reinforcements of areaAsi to the relevant middle line,

Spsn =

where

Asni are the area of reinforcements within the region of 2 dn from the middle line,

eni are the distances of the reinforcements from the middle line.

)(n

i

isi eA

)(n

i

nisni eA

Fully or partially encased H sections


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Design formulae for composite columnsMajor axis bending

Spc =

Spcn =

Neutral axis in the web:

dn =

Spn = t dn2

pspcc SS

DB

4

2

psnpnnc SSdB 2

T

Ddn

2

)2(22

)2(

cucycucc

cucsdsncucc

fptfB

ffAfA

+

46

Steel Composite

columns columns Section capacity (stocky column)










compression and bending,Pcp & M

P & M




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Effective flexural rigidity

ssccmecp IEIEKIE(EI) ++=

Elastic buckling load:

( )2

E

2,e

2

cr,cpL

EIP

=

Ke = 0.8 / 1.35 = 0.6

I, Ic and Is are the second moment of area of the structural steel section,

the un-cracked concrete section and the reinforcement

respectively for the bending plane being considered.

Axial buckling resistance of slender columns


48

Non-dimensional slenderness ratio

Pcp,k is the characteristic value of the compression capacity

cr,cp

k,cp

P

P=

yscucy fAfA68.0Ap ++=

yscucy fAfA8.0Ap ++=

for fully encased and partially encased H sections

for in-filled rectangular hollow sections

for in-filled circular hollow sections

yscu

yccucya fA

f0.8

f

d

t1f0.8AAp +

++=


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5022

1.

+

= ( )

++=

22.015.0

is the imperfection parameter which allows for different levels ofimperfections in the columns

= 0.21, 0.34 and 0.49 for buckling curves a, b and c respectively

cpP

P=

crcp,

kcp,

P

P =

Reduction factor,

50

Pcp & M

P & M












Steel Composite

columns columns




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Interaction curve for compression and bending

2. An applied force P will always induce a moment M due

to the presence of initial imperfection, hence, part of the

reserve is used up.

1. The interaction curve represents the maximum values of

the pair of P and M.

3. If the applied force, P, is equal to the

axial buckling resistance, there is no

reserve to resist any moment at all.

4. If the applied force, P, is

less than the axial buckling

resistance, the moment

reserve is at least equal to

( d - k ) Mcp.

1.00

cpM

M

cpP

P

k

Axialbuckling

resistance

d

d

Applied load

5. Depending on the shape of the initial imperfection of the

column, it is possible to reduce the induced moment.

( d - k ) Mcp.

52


Single curvature Double curvature

r : end

moment ratio


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1.0

1.00

cpM

M

cpP

P

6. Based on the value of the end

moment ratio, r, the reserve

moment is increased to Mcp.

5. Depending on the shape of the initial imperfection of

the column, it is possible to reduce the induced

moment.

n

k

Axial

bucklingresistance

d

d

Applied load Mcp.

54

Interaction curve for compression and bendingThe value n accounts for the influence of the imperfections and that ofthe bending moment do not always act together unfavourably.

For columns with only end moments, n may be obtained as follows:

( )4

1

rn

=

cp

dP

P= d

1.0

1.0k0

n

Cross-section

interaction curve

d

cpM

M

cpP

P


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d

1.0

1.0k0

n

Cross-section

interaction curve

d

a)

cpM

M

cpP

P

cpM

M

1.0

C

B

A

1.00

d

n

pm

k d

b)

cpP

P

Simplified

interaction curve

For easy manual calculation, a simplified interaction curve

may be adopted in design.

56

when

when

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Composite Column