\ ( ..- \~C:Zt;- <~E

2 NORMAL DESIGN

!11:* \,\, c \, 14. c-\\c, ,,,.,

2.1 General

In general, a composite Golunni must be designed for the ultimate Iinxit state. For sumctoraladequacy, the internal forces and moments resulting from the most unfavourable loadcombination should not exceed the design resistances of the composite cross-sections. Whilelocal buckling of the steel sections may be 61niiinated, the reduction in the compressionresistance of the composite columni due to overall buckling should be allowed for, together withthe effects of residual stresses and lintial imperfections. Moreover, the second order effects inslender coluinns, as well as the effect of creep and shankage of concrete under long-tennloading, must be considered if they are significant. The reduction of flexoral stiffness due tocracking of the concrete in the tension area should also be considered. These are provided foreither explicitly, or empiricalIy, in prEN 1994-I-I: 1994'"' (EC4-I-I).

I~ ~

~:<':.. <1'C\C:'~"

2.2 Material properties2.2. , Hot rolled structural steel

Normnal values of the yield stress I, , and the ultimate tensile stress I, , for structural steel arepresented in Table 2.1 below.

Table 2.1 Mechanical properties of Celsius Sections

Nominal steel gradeto

BS EN 1002,0-1

S 275

S 355

Design values of other coefficients for the steel sections are given as follows:

E

I! (N/min')

G,

t:!^ 40 min

Nominal thickness of element t (min)

275

'a

Modulus of elasticity

355

P,

Shear modulus

2.2.2 Structural concrete

Concrete strengths are based on the characteristic cylinder strengths I, , measured at 28 days inaccordance with Clause 3.1.2.2, of DD ET. IV 1992-I-I: 1992"" (EC2-I-I). The differentstrength classes, and the associated cube strengths, given by this Eurocode are presented inTable 2.2 below. Classification grades of concrete, such as C20/25, refer to the cylinder/cubestrength at the specified age.

j, (N/min')

Poisson's ratio

Density

430

510

I} (N/mm')

40 mm < ts; 100 min

For nomial weight concrete, the mean tensile strength I. im and the secant modulus of elasticityE, , , for short-terni loading are also given in Table 2.2. The effect of creep and shamkage ofconcrete may be significant under long-terni loading in some cases. As will be discussed in

255

335

210 000 NIInm2

E

2(I + ^,, )0.3

785 0 kg/in3

I, (N/min')410

490

9

Section 2.6.2, provision is given within EC4-I-I"" to reduce the secant modulus of elasticity,depending on the proportion of penmanent load acting on the colornn.

The density of structural concrete is assumed to be 2400 kg/in' for plain, umeinforced, concreteand 2500 kg/in' for reinforced concrete.

Table 2.2 Characteristic compressive strength I^k (cylinders), mean tensile strengthfatm and secant modulus of elasticity Eom for structural concrete

Strength class ofconcrete

I^k (N/min')

161m (N/mm')

E. , (N/min')

2.2.3 Steel reinforcement bars

In the UK, steel bars for the reinforcement of concrete should confonn to BS 4449:1997'"'. DDET*TV 10080: 1996"" which is Gunently at the draft for development stage, will eventuallyreplace this British Standard. However, the 1997 edition of BS 4449 has been revisedconsiderably compared to its earlier versions, to bring it into line with the requirements of EC2-I-I' '. The properties most frequently required in design calculations are referred to in Clause3.2 of EC2-I-I; types of reinforcement steel are classified as follows:

. High (class H) or nonnal (class N), according to ductility characteristics.

. Plain smooth or, ribbed bars, according to surface characteristics.

C20125 C25/30 C30/37 C35/45 C40/50 C45/55 C50/60

20

2.2

29000

25

2.6

30500

30

2.9

32000

Steel grades that should be used in construction are given in Table 2.3.

Table 2.3 Characteristic yield strength 13k, ducti/ity requirements and modulus ofelasticity Es of reinforcing steel

35

3.2

33500

40

Reinforcingsteel standard

3.5

35000

Name

45

I;k (N/mm')

38

36000

Total elongationat maximum

force (%)

50

4.1

Elongation atfracture (%)

37000

460A (class N)

BS 4449: 1997'"'

E, (N/min')

+ According to EC4-1-1

As can be seen from Table 2.3, apart from the obvious difference in the characteristic yieldstrength I;,, between reinforcing steels complying with BS 4449:1997"" and DD ENV 10080:1996"" BS 4449 also specifies a minimum elongation at fracture: thereby guaranteeing thelength of the plastic defonnation plateau. Forthennore, in the 1997 edition of BS 4449, theminimum elongation at fracture of 14%, for 460B steel, is higher than the 12% requirement forthis yidd strength, given in earlier versions of this code of practice. A graphical representationof the difference in the elongation requirements for these two standards is shown in the stress-strain curve in Figure 2.1.

2.5

460

460B (class H)

12

210000t

50

DD ENV I 0080: I 996120i

B500A (class N)

14

2.5

B500B (class H)500

210000j'

5.0

10

coInco.

a^

Figure 2.1

Total elongation at Minimum elongationmaxim urn force 8 k at fracture

Elongation limits for steel reinforcement bars

It should be noted, however, that although the ductility of reinforcing bars has a significanteffect on the behaviour of continuous composite beams"" this property is of little significancewith respect to the design of composite columns at ambient temperature. Concrete filled hollowsections may be used without any reinforcement, except for reasons of fire resistance (seeSection 3).

2.3 Partial safety factorsNational authorities are free to select appropriate values for partial safety factors for loads andmaterials, and substitute them for 'boxed' values in the Eurocodes. The boxed values and thein<. National Application Document (NAD) values are:

Loads:

Imposed (variable) load, To

Dead (permanent) load, yG

I St rain

Materials:

Steel, y,

Concrete, y.

Reinforcement, y,

EC4-,-, 'boxed' values

2.4 Basis of design methodin EC4-1-1, isolated collnnns are defined as compression members that are integral parts of abraced or non-sway frame but which are considered to be isolated for design purposes.

Definitions of non-sway structures are given in EC2-I-1''" as follows:

1.50

1.35

1.4 O

UK NAD

o

1.50

Structores or structural elements, with or without bracing elements, for which the influenceof displacements of the connections upon the design moments and forces may be neglected,are classified as non-sway. Otherwise, they are classified as sway.

1.50

1.15

1.35

1.05

1.50

1.15

11

. Braced building structores, where substantial shear walls or core structures provide thebracing, may be assumed to be non-sway

Frames may be classified as non-sway if the first order displacements of the connections donot increase the effects of actions calculated without considering these displacements bymore than 10%. In general, it is sufficient to consider only the relevant bending momentsdue to these second-order effects

.

A sinxilar definition of a non-sway frame in DD ENV 1993-I-I: 1992"" (EC3-I-I) is also givenfor reference

. A frame may be classified as non-sway if its response to in-plane horizontal forces issufficiently stiff for it to be acceptably accurate to neglect any additional internal forces ormoments arising from horizontal displacements of its nodes.

Two methods of design for isolated composite colunms in braced or, non-sway frames are givenwithin EC4- I - I :

2.4. , General design methodThis comprehensive method is used for composite colornns with non-symmetrical or non-runfonn cross-section over the coluinn length. It is also used for composite collirnns of doublysynnnetrical, and unifonn cross-section over the column height, when the 11nitts of applicabilityfor the simplified design method are not satisfied (see Section 2.5). in these circumstances,some of the important design issues which should be considered using the general method, areas follows

. geometrical and material non-linearity;

. second order effects (on slender collirmis);

. creep and shrinkage of the concrete under long-tenn loading;

. contribution of the tensile strengtti of the concrete between cracks;

. imperfections for the calculation of internal forces and moments about both axes;

. distribution of internal forces and moments between the steel section and the concrete bymeans of a clearly defined load path;

. transfer of longitudinal shear stress at the interface between the steel section and theconcrete under large transverse shear; and

. chemical bond and friction together with mechanical shear connection if necessary

,

in order to allow for these design considerations, it is necessary to use sophisticated computersoftware, which operate with both geometrical and material non-linearity. in general, the designeffort is considerable. Thus, this method is not preferred for use in practical design, and isoutside the scope of this publication.

2.4.2 Simplified design methodThis method is used for composite columns of doubly syinmetrical and uniform cross-sectionover the coluinn height. It is based on certain assumptions relating to the geometricalconfigurations of the composite cross-sections. Moreover, it also adopts the European bucklingcurves for steel colutims as the basis of colorim buckling design. The lintts of applicability ofthis method givenin EC4-1-1 are also listed in Section 2.5; when the Innits are not satisfied, theabove general design method should be used

12

It should be noted that this method is fomTulated in such a way that only hand calculation isrequired in practical design. The simplified design method is presented in detail within thispublication. The calculation procedure is in six parts, as follows:

(i) Check that the litints of the simplified design method are satisfied.

(ii) Calculate the properties of the cross-section

(in) Calculate the buckling resistance of the colornn

(iv) Check whether second-order effects should be considered

(v) Calculate the effect of interaction between axial load and bending.

(vi) Calculate the longitudinal and transverse shear

2.5 Restrictions on the simplified design methodThe application of the simplified design method is subject to various restrictions, as follows:

(a) The columni is doubly-sytrunetrical and is of runfonn cross-section over the height of thecolumn

(b) The steel contribution ratio 6 must satisfy the following conditions:

0.2 ^ 6 :!;; 0.9

If 6 is less than 0.2, the column may be designed according to EC2-I-1''". If 6 is largerthan 0.9, the concrete is ignored in the calculations, and the colorim is designed as a baresteel section.

(c) The maximum non-dimensional SIGndemess ratio of the composite coluinn 2, is linttedto 2.0

The maximum amount of longitudinal reinforcement that can be considered in theanalysis is 6% of the concrete area. However, if design for fire resistance is not needed,according to prEN 1994-I-I: 2001'"' no nullimum amount of reinforcement is nomiallynecessary within a filled SHS column; in other words:

A,O% :I^ = :^ 6.0%.14.

(co

2.6 Properties of cross-section2.6. , Squash (plastic) resistance, Npi, RdThe plastic resistance, to compression, of a composite cross-section represents the maximumload that can be applied to a short composite colornn. It is important to recognize that concretefilled circular hollow sections exhibit enhanced resistance due to the triaxial containment

effects. Concrete filled rectangular sections (R. HS) do not achieve such enhancement

Local buckling of steel ho"ow sections

Before the plastic resistance of the concrete filled hollow section is calculated, it should beinsured that local buckling of the steel does not occur. To prevent premature local buckling, thewidth to thickness ratio of the steel section in compression must satisfy the following jintits:

h- ^ 528For concrete filled rectangular hollow sections (Rl{S)

13

For concrete filled circular hollow sections (CHS)

where:

h

is the wallthickness of the steel hollow section in rum.

d

is the larger outer dimension of the rectangular hollow section in nun

is the outer diameter of the circular hollow section in nun.

8235

I,

I, is the yield strength of the steel section in N/nun'.

Local buckling in some rectangular hollow sections with large h/t ratios may be critical. Nospecific design reconrrnendation is given within EC4-1-1, and design using sections whichexceed the local buckling 11ntits should be verified by tests

Concrete filled rectangular hollow sections (RHS)

The plastic resistance of a concrete filled rectangular hollow section (i. e. , the so-called "squashload") is given by the sinn of the resistances of the components as follows:

- ^ 908'

N _ '1, , 41;, , A. /;^I, I, I.

where:

,

A

A,

A,

I,I;*

Ich

is the area of the steel section.

is the area of the reinforcement.

is the area of the concrete.

is the yield strengtti of the steel section

is the characteristic yield strength of the steel reinforcement bars

is the characteristic compressive strength (cylinder) of the concrete.

For cos^ of a, ression 1:1- "* and I'* am muont^d us nest at^^gth^ of th^ us cotiv^I,I, I,

materials in the remainder of Section 2 as: I, , I;, and I;, respectively. As a result of thissimplification, the above equation for the plastic resistance of the composite column, can berewritten in the following compact fonn

N, ,,, = 41, , + 41;, + A. /;,

Concrete filled circular hollow sections (CHS)

For composite columns with concrete filled circular hollow sections, the increased resistance ofconcrete due to the confining effect of the circular hollow section may be included. Thisrestraint to transverse strain in a tree dimensional confinement results in increased concrete

14

resistance. At the same time, circular tensile stresses in the circular hollow section also arise,which reduce its axial resistance.

In general, the resistance of a concrete filled circular hollow section to compression mayincrease by up to 15% under simple axial loads when the effect of in-axial confinement isconsidered. However, this effect on the resistance enhancement of concrete depends also on theSIGndemess of the composite colormis and is significant only in stocky Golurnns. For compositecolornns with a non-dimensional SIGndemess of, {. > 0.5 (where 2, is defined in Section 2.7),this effect should be neglected and the plastic resistance assessed as for rectangular hollowsections.

in addition, further linear interpolation is necessary to take account of any effective loadeccentricities. However, the eccentricity, e of the applied load may not exceed the value d/10,where d is the outer diameter of the circular hoUow section.

The eccentricity, e is defined as follows

M

N

where:

Msd is the maximum d, g^Ig!I moment (second order effects are ignored)

N, , is the d, ::^!go, applied load.

The plastic resistance of a concrete filled circular hollow section may be obtained as foHows

N, ,,,, = A, ,71, , + 41;, + 4.1;, 11+ 'A I7:', , , ,. di;kj

where

,, _*,,, I"I ,; for 0< e :!s d/10

, _ ,, ,,,_ 00,101n, -n, , +(I-nan)~I

' I fore>d/10n =O

n2 ' 1.0

is the wall thickness of the steelhollow section in nun

The basic values 7710 and 7720 depend on the non-dimensional SIGndemess ratio A, , and aredefined as follows:

77 = 4.9 - 18,521' + 17:1:2

ty, , ^ 0. ^5(3 + ^^)

but 77102: O

but 7720 ^ 1.0

15

If the eccentricity e exceeds the value d/10, or if the non-dimensional SIGndemess ratio A,exceeds the value 0.5, then 77, , = 0 and n20 = 1.0 . Table 2.4 gives the basic values 77,0 and

for different values of A, .7720

Table 2.4 Basic values of 7710 and 7720 to allow for the effect of triaxial confinement inconcrete filled circular hollow sections

Nori-dimensional

slenderness ratio ,{.

7710

7720

2.6.2 Effective flexurel stiffness

Short-term loading

The effective flexoral stiffiiess of the composite colort111 (E/), is obtained from adding the upthe flexoral stiffiiesses of the individual components of the cross-section:

(E/I - E, I, + E/, + 0.6E. /.

o

4.90

0.75

0.1

3.22

where:

080

0.2

I , I, and I are the second moment of area, about the appropriate axis of bending, for thesteel section, the reinforcement and the concrete (assumed uricracked)respectively.

1.88

0.85

0.3

0.88

E and E, are the elastic inoduli for the structural steel and the reinforcement respectively.

0.6E^.

090

0.4

is the effective stif;filess of the concrete component (the 0.6 factor is anempirical multiplier, which has been deterTinned from a calibration exercise, togive good agreement with test results).

is the secant modulus of elasticity for structural concrete; see Table 2.2.cm

Long-term loading

For composite colunms under long-tenn loading, the creep and shankage of concrete will causea reduction in the effective elastic withess of the composite column, thereby reducing thebuckling resistance. However, this effect is only significant for slender columns; as a simplerule, the effect of long-terni loading should be considered if the buckling length to depth ratio ofa composite colornn exceeds 15.

If the eccentricity of loading (see Section 2.6. I) is more than twice the cross-section dimension,the effect on the applied bending moment distribution caused by increased deflections, due tocreep and shankage of the concrete, will be very small. Consequently, it may be neglected andno provision for long-term loading is necessary. Moreover, no provision is necessary if the non-dimensional SIGndemess I. of the composite colornn is less than the Iiniiting values givenwithin Table 2.5 below.

022

z: 0.5

095

o

4.00

16

Table 2.5

Frame type

Limiting values of A, for long-term loading

Concrete filled hollow sections

The steel contribution factor a, given in Table 2.5 above, is defined as follows:

11"Iyd

N, I, Rd

Braced nori-sway frame

If the eccentricity of loading is less than twice the cross-section dimension and the non-

dimensional SIGndemess I. of the composite coluinn is less than the limiting values givenwithin Table 2.5, the effect of creep and shrinkage of concrete should be allowed for byreducing the effective elastic modulus of the concrete to the value

O. 8

(I- 6)

^- - f -I

Sway frames and/orunbraced frames

where:

N

NG, sd is the part of the d, ^!g!! load penmanently acting on the colornn

Table 2.5 also allows the effect of long-tenn loading to be ignored for concrete filled bonowsections with ,{. ^ 2.0 , provided that 6 is greater than 0.6 for braced (or non-sway) colorrms,and 0.75 for unbraced (and'or sway) columns.

O. 5

is the ^,^!^ applied load

(I- 6)

2.7 Column buckling resistanceThe plastic resistance to compression of a composite cross-section N ,,,, represents themaximum load that can be applied to a short collnnn. However, for slender colornns, with lowelastic critical load, overall buckling considerations may be more significant.

In Figure 2.2(b), the buckling resistance of a colornnis expressed as a proportion x of the plasticresistance to compression Npj, R , thereby non-dimensionialising the vertical axis compared toFigure 2.2(a). The horizontal axis may be non-dimensionalised sirntlarly by use of the EUlerbuckling load N. , as is also shown in Figure 2.2(b)

By incorporating the effects of both residual stresses and geometric imperfections, the Europeanbuckling curves may be drawn on this basis as shown in Figure 2.2(c). These curves fomi thebasis of colorim buckling design for both steel and composite Goluinns

17

N

N

NGr

X'NRd

In:^~dPI, Rd1.00

yr SIendemess

(a)

_Nx '77;

Figure 2.2

o

a

o

(c)

(a) Idealised column buckling curve, (b) Non-dimensionalised columnbuckling curve, (0) European buckling curves according to EC3-7-7

b

0.2

C

The buckling resistance is calculated from the plastic resistance and the EUler (elastic) criticalload using the EC3-1-1"" buckling curve 'a' (N. B. at the fire limit state, curve 'c' is used due toits close agreement with the results from fire tests; see Section O). The EUler buckling load isgiven by:

^'(E/).12

(b) I^-

N,

1.0

where:

(E/), is the effective elastic flexoral stiffiiess of the composite column (see Section 26.2).

is the buckling length of the column.

EC4-1-1 suggests that the buckling length I of an isolated non-sway composite column mayconservativeIy be taken as equal to its system length L. Alternatively, the buclding 16ngtti maybe detennined using Annex E of EC3-I-I.

The non-dimensional SIGndemess ratio is given by:

2.0

^-I^'

_ NI, = -9--

Nor

18

where:

N I, R is the plastic resistance of the composite cross-section to compression, according toSection 2.6. I, with I, =I, =I. =1.0.

The resistance of a composite column in axial compression (buckling load) is obtained from:

NRd ' X. N, j. in

where:

x is the reduction coefficient for buckling obtained from curve 'a' of EC3-I-I, and is

dependant on the non-dimensional SIGndemess ratio A, .

The reduction factor may be determined from

N^ , - it, I'~ 'where:

dy -,.^"+^(^-,.^)+ipi

or is an imperfection parameter depending on the buckling curve considered

Relevent Buckling Curves andlmperfection Factors

According to prEN 1994-I-I, circular or rectangular hollow section colornns filled with plainconcrete or containing up to 3% reinforcement can be designed using buckling curve 'a' with animperfection factor, or, = 0.21. However, concrete filled sections containing between 3% to 4%reinforcement must be designed using buckling curve 'b' with an imperfection factor, or, = 0.34(see Figure 2.3 co and (b) below).

In addition, concrete filled circular hollow section colunms as shown in Figure 2.3(c) containingan additional open Section used as primary steel can also be designed as a composite sectionusing buckling curve 'b' with an imperfection factor, or, = 0.34

but I :^ I .O

.

(a)

Figure 2.3 Typical column cross-sections

,

. ..

..\

. .

,

.

,

. flit*(b)

19

(c)

Although not explicitly stated, Clause 4.8.3.2 of EC4-I-I, while defining the partial safetyfactors implies that isolated non-sway composite columns need not be checked for buckling, ifany of the following conditions is satisfied

(1) the axial force in the columnis less than 0.1N, , ; or

(ii) the non-dimensional SIGndemess ratio I. is less than 0.2

2.8 Analysis of bending moments due to second-ordereffects

Under the action of the design axial load N, , on a column with an initial imperfection e, , asshown in Figure 2.4, there will be a maximum internal moment of N, ,e, . It is important to notethat this 'second order moment', or 'imperfection moment', does not need to be consideredseparately, as its effect on the buckling resistance of the composite colunm is already accountedfor in the European buckling curves as shown in Figure 2.2(c).

Figure 2.4 Initially imperfect column under axial compression

^..

However, in addition to axial forces, a composite Goluinn may be also subject to end momentsas a consequence of transverse loads acting on it or, because the composite coluinn is a part of aframe. The moments and displacements obtained lintially are referred to as 'first order' valuesFor slender colornns, the 'first order' displacements may be significant and additional, or'second order' bending moments may be induced under the actions of the applied loads. As asimple rule, the second order effects should be considered if the buckling length to depth ratioof a composite colorrm exceeds 15.

The second order effects on bending moments for isolated nori-sway columns should beconsidered if both of the following conditions are satisfied

_;:^-- > 0. INor

where:

is the design applied loadNsd

is the EUler buckling loadN, ,

2) it ^ 0.2(2 - ")where:

I.

I)

,, 1_N s de

I

is the non-dimensional SIGndemess ratio

is the ratio of the smaller to the larger Grid moment (see Figure 2.5). If there isany transverse loading, r should be taken as 1.0

20

Figure 2.5 Ratio r of the end moments

The second order effects in an isolated non-sway colornn may be allowed for by modifying themaximum first-order bending moment Min, ,,,, , with a correction factor k, which is defined asfollows

Msd

k-N

A4, ,, ff

rMsd

where:

Nsd

Nor, of

z: 1.0

-I ^ r ^1.1

is the design applied load

is the elastic critical load of the composite colornn based on the system length, L, anda reduced design value of effective stillfi, .ess (El)e, n

where:

(E1)., 11 ' 0.9(EJ, + E*I, + 0.5E. ,I, )

is the equivalent moment factor.

For coluTnns with transverse loading within the columni length, the value for 13 should be takenas 1.0. For pure Grid moments, ^ can be detennined as follows:

but ^ ^ 0.4413 = 0.66 + 0,441

2.9 Combined compression and bendingThe design for a composite coluinn subjected to combined compression and bending is carriedout in stages as follows

. The composite colorrm is isolated from the framework, and the end moments, which resultfrom the analysis of the system as a whole, are taken to act on the colornn underconsideration. Internal moments, and forces within the Golunm length, are detennined fromthe structural consideration of end moments, axial and transverse loads

. For each axis of syinmetry, the buckling resistance to compression should be checked withthe relevant non-dimensional SIGndemess of the composite colunm

. In the presence of applied moment about one particular axis e. g. , the y-y axis, the momentresistance of the composite cross-section should be checked with the relevant non-

dimensional SIGndemess of the composite colornn i. e. , 11, , instead of ,1. ,, although A. , may

be larger, and thus more critical, than I. .

. For slender colornns (see Table 2.5 and Section 2.8), both the effect of long-tenn loadingand the second-order effects are included

21

It should be noted that, by adopting the EC4-I-1''" simplified method, imperfections within thecolumn length need not be considered as they are taken account of in the relevant bucklingcurve when detemiining the buckling resistance of the colonni (see Section 2.7).

2.9. , Combined compression and uni-axial bendingIn EC4-I-I"", the resistance of a composite colornn subjected to combined compression andbending is detemxined from an interaction curve. For a bare steel section, the interaction carve ischaracterised by a continuous reduction of the moment resistance with a corresponding increasein axial load.

NRd7^^

1.0

MRd7, ^

Figure 2.6 Interaction curve for a composite column sub^^cted to compression anduni-axial bending

However, as shown from the interaction curve in Figure 2.6: a short composite columni mayexhibit an increase in moment resistance under axial load. The reason for this increase is that,under some favourable conditions, the compressive axial load prevents concrete cracking, andtherefore makes the cross-section of a short composite colornn more effective in resistingmoments.

An interaction curve between compressive axial load and moment can be obtained for a shortcomposite colurnn by considering several possible positions of the neutral axis within the cross-section, and detennining the internal forces and moments from the resulting plastic stressblocks. For the simplified method given within EC4-I-I, sufficient accuracy in estimating theeffects of combined compression and bending may be found by constructing the interactioncurve, shown in Figure 2.7, from 4 or 5 points.

NM

,

A. ~O

I. O

22

N

Npi, Rd

N pin, Rd

A

I, ^N pin, Rd

----E

Figure 2.7

^

Mpi, Rd Minax, Rd M

Interaction curve with 11hear approximation

\

For composite collnnns, which are doubly symmetrical and of a runfonn cross-section over theirheight, the foUowing approach given in EC4-I-1' ', and the inc NAD, may be adopted.

Figure 2.8 shows the plastic stress distributions within the cross-section of a concrete filled ERSat point A, B, C, D and E of the interaction curve given in Figure 2.7. The significance of eachof these points are as follows:

. Point A indicates the plastic resistance of the cross-section to compression, in the absenceof an applied bending moment:

NA ' N I, MMA ' O

. Point B corresponds to the plastic moment resistance of the cross-section, in the absence ofan applied axial load:

NB ' O

MB ' M, I, M

. At point C, the axial compression and moment resistance of the composite Golurmi aregiven as:

Nc = N, ,, Rd (or N. in) = A. fad

Mc = M, I, M

~~ C

I\I \I \I \

I I

I111/1

D

B'I I

23

Point A

I ,

1/1

1/1

. I 4'6"

Point B

fcd

I, ' ,'

.

fyd

Point C

fcd

.

n

11 I

1/1

II

fsd

'-!ypl. RdNo moment

.

II

fyd

II

Point D

fcd

.

+

.10

n

I

fsd

II

.

fyd

I

_ MB' M PI, .Rdzero axial force

M c:' M PI, RdNc= N c, Rd

Mj^ M max, RdN , N c, RdND' ^-

ME

NE

+

Point E

fcd

.

I, ' ,'1/1

I ~ ^~ ~

+

Figure 2.8

fsd

fyd

h/2

+

(N. B. , the moment resistance Minax. Rd, at point D, is not allowed in the UK NAD)

Stress distributions for the points on the interaction curve for concrete filledhollow sections, according to EC4-7.17'I

fcd

+

The expressions may be obtained by combining the stress distributions of the cross-sectionat points B and C; the compression area of the concrete at point B is equal to the tensionarea of the concrete at point C. The moment resistance at point C is equal to that at point B,since the stress resultants from the additionally compressed parts cancel one another out inthe central region of the cross-section. However, these additionally compressed regionscreate an internal axial force, which is equal to the plastic resistance to compression of theconcrete alone i. e. , N, ,, Rd Or N, ,Rd -

At point D, the plastic neutral axis coincides with the centi. oid of the cross-section, and theresulting axial forceis halfofthe value at point C, i. e. :

ND ' N, m, Rd I2MD = Min, *, Rd

h4

fsd

fyd

+

.

fsd

24

Generally, point D is less than point C in design

. Point E is rind-way between A and C, and is often required for highly non-linear interactioncurves, in order to achieve a better a better approximation. For concrete filled structuralhollow sections, the use of point E will yield a more econonxical design; however, muchmore calculation effort is required. Thus, to retain simplicity, point E tends not to be used

According to the inc NAD, the additional moment resistance of the composite cross-section(indicated by point D within Figure 2.7), should not be taken account of in design. Therefore, inthe UK, an interaction curve consisting of A-C-B or A-E-C-B may only be considered

The plastic moment resistance of a concrete filled hollow section may be evaluated as follows

'd, M ' I;,,(pyro ~ \, an) + 0,571, (IP, . - IP, on) + I;d(11',, - IP, ,,)

where

I, , , I;, , I;,

are the plastic section inoduli for the steel section, the concrete of the

composite cross-section (assumed to be uricracked) and thereinforcement respectively.

are the plastic section inoduli of the corresponding components within

the region of 2h from the centre-line of the composite cross-section.

The values of the relevant parameters in the above equation for concrete filled hoUow sections

JP , IP , 11',pa , PC , ps

IP , 17 ,11',pan , PCn , pSn

am IL I;^, an, I" re, adjv. II,

are:

I,

Rectangular hollow sections

_ (b - 2, Xh - 20'17

I,

where: I is the internal radius of the corners to the hollow section

Circular hollow sections

(d - 20'

4

17

-I -r (4-7:1--t-11 ~',

In general, for both types of section:

6

h,A. I. , - A, , (21;, - I. , )261. , + 4t 21, , - fad

:=

where: A, is the area of reinforcing bars within the region of 2h from the centre-line of thecomposite cross-section

For rectangular borrow sections, it can be explicitly stated that

W, an = 2t. h,

W,

25

IP. - (b - 20h -17, ,

Sirntlar simple explicit equations cannot be witten for circular sections. However, the aboveequations can be conservativeIy applied to circular sections with a high accuracy by substitutingdiameter d for breadth b

For the calculation of the resistances at the additional point E, NE, Rd and ME, Rd (see above),the neutral axis is located between h and the border of the section, so that

hE = 0.5h + 0.25h . Using rectangular hollow sections, the axial resistance of the coluinn forthis case is:

N, ,Rd ' b(hE ~ h, )I, d + 2 t(hE ~ h, X2f, d ~ I, d ) + A, E (21;, ~ I'd ) ' NomRd

where A, is the sinn of the areas of reinforcement lying in the additional compression regionbetween hE and h

The magnitude of ME, Rd is calculated from the above equations but with h, substituted for h,in the values of \ and 17 . .

Again, the above equations can be applied to circular sections by substituting diameter d forbreadth b but may become highly over-conservative. In such circumstances it may be preferableto simply apply a linear interpolation between points A and C

The principal for checking the composite cross-section under combined compression and urn-axial bending, in accordance with EC4-I-I"" is illustrated graphicalIy in Figure 2.9. Firstly, theresistance of the composite Golutnn under axial load is detennined in the absence of bending,which is given by xN, ,,,, (see Section 2.7). The moment resistance of the composite colurnnshould then be checked with the relevant non-dimensional SIGndemess, which is in the sanieplane of the applied moment. As mentioned earlier, the initial imperfections of Golunms havebeen incorporated within the appropriate buckling curve, and no additional consideration ofgeometticalimperfections is necessary in the evaluation of bending moments within the columnheight.

Consider the interaction curves for combined compression and uni-axial bending shown in

Figure 2.9. Under an applied force N, , equal to xN, I, Rd , the horizontal coordinate 11kM, I, Rdrepresents the second order moment due to imperfections within the columni, otherwise knownas the 'imperfection moment'. It is important to recognise that the moment resistance of thecolornn has been fully utilised in the presence of the imperfection moment; the colornn,therefore, cannot resist any additional applied moment. Moreover, the influence of theimperfections decreases when the axial load ratio is less than x, and it is assumed to varylinearly between I and x. For an axial load ratio less than I, , the effect of imperfections isneglected.

It is important to note that the value ,21' accounts for the fact that the influence of theimperfections and that of the bending moment do not always act together unfavourably. Forcollnnns with only end moments, z', may be detennined as follows:

, (I - ,")"~ 4'

26

where I is the ratio of the small to the large end moment (see Figure 2.5).

If transverse loads occur within the colormi height, then r must be taken as unity and I is thusequal to zero (i. e. , it coincides with the origin of the interaction curve shown in Figure 2.9).

NRd79^

1.0

X

Xd

Cross-sectioninteraction curve

____,:____I______

XnIIIIII

II I

II

Figure 2.9

o

NRd79^

1.0

F1k I'd 1.0MRdMRd

^;^; T7^;(b)(a)

Interaction curve for compression and uni-axial bending using (a) the EC4-i-I method, ' and to) the simplified method Ih the UK NAD

ILL

With a design axial load of N, , , the design axial load ratio I, is defined as foUows:

Id = NsdIN, I, Rd

By reading off the horizontal distance from the interaction curve (see Figure 2.9), the momentresistance ratio F1 may be obtained, and the moment resistance of the composite Golunni undercombined compression and run-axial bending may be evaluated. Details of the UK NADmethod for calculating F1, and its limitations, are discussed below.

EC4-I-I"" considers that the design is adequate when the following condition is satisfied:

Msd ^ 0,911M I, Rd

A

X

Xd

XpmXn

__r:__L_____

I IL

,' II

I

o

where:

F1k

M

C

F'd

is the design bending moment, which may be factored to allow for second-order effects,

if necessary (see Section 2.8).

is the moment resistance ratio obtained from the interaction curve.

B

1.0

IL

M I, Rd is the plastic moment resistance of the composite cross section.

The interaction curve shown in Figure 2.9 has been deterTrimed without considering the strainlitnttations within the concrete. Hence the moments, including second-order effects (ifnecessary), are calculated using the effective elastic flexoral stiffiiess (El). , and taking into

27

account the entire concrete area of the cross-section (i. e. , the concrete is uricracked).Consequently, in order to allow for these simplifications, the 0.9 constant, shown in the aboveequation, is applied to the moment resistance.

For concrete filled hollow sections, the interaction carve of A-E-C-B (shown in Figure 2.7) maybe preferred to A-C-B (shown in Figure 2.9(b)), as it will give a more econonxical design:especially for columns with high axial load and low Grid moments (although much morecalculation effort is required). For a better approximation, the position of point E may be chosento be closer to point A rather than being rind-way between points A and C. For furtherinfonnation, refer to EC4-I-I"".

Requirements of the UK National App"cation Document (NAD)

The following additional requirements are specified in the U}<. NAD

. For colornns under combined compression and bending, the ratio I should bedetennjnedt24,251 as follows

(4) Concrete/med rectangular hollow sections

Providing the non-dimensional SIGndemess A. does not excess 1.0, the ratio Id ;^ Z' in

may be determined from the equation shown above. For values of ,^, in the range of 1.0 to2.0, I = O

(ip Concrete/med circular or square hollow sections

The ratio I may be deterTinned from the equation shown above, with no limits place onthe non-dimensional SIGndemess ,I. .

. The design moment resistance, in combined compression and uni-axial bending, of a

composite colunm should not exceed the design plastic moment resistance M I, Rd ,irrespective of the applied load N (i. e. , point D on the interaction curve shown in Figure 2.7is not allowed)

Combined compression and uni-axial bending to the UK National ApplicationDocument

In order to comply with the UK NAD, the moment resistance ratio F1, for a composite columnunder combined compression and run-axial bending should be evaluated as follows:

(I-%, XI-I, )FL

I~Z'pm Z ~I, )

-I

where:

(I-IXX, ~I, )I~Ripm I~I, )

I is the axial resistance ratio due to the concrete, N Rd I N I, Rd

,;I'd is the axial resistance ratio, Nsd I N I, Rd

When Id '' IPm

when I'd ' IPm

28

I is the reduction factor due to collmm buckling (see Section 2.7)

For concrete filled rectangular hollow sections

_ (I - ")Z', ' 'I

for 1.0 :!:^ I. ^ 2.0=O

For concrete filled circular or square hollow sections

_ (I - ")I, "I

The expression, for the moment resistance ratio 11, is greatly simplified by taking I = O asfoUows

for ,I, < 1.0

F1

~}^;)

_,_ all-I)' I ~ I^,

I ~ Id

for A, ^ 2.0

The above expressions are obtained from a general consideration of the geometry of the UKNAD interaction curve shown in Figure 2.9(b). The simplified expression for the momentresistance ratio F1, is always conservative; since, by taking I, = O it is implied that I= 1.0 (i. e. ,the end moments are equal, and constant over the coluinn length, see Figure 2.5).

2.9.2 Combined compression and bi-axial bendingFor the design of a composite collunn under combined compression and bi-axial bending, theaxial resistance of the coluinn in the presence of bending moment for each axis, has to beevaluated separately. In general, it will be obvious which of the axes is more likely to fail andthe imperfections need to be considered for this direction only: as shownin Figure 2.10. Ifitisnot obvious which plane is more critical, checks should be made on both planes.

when I'd 22 ,^I', in

when Id ' IPm

29

NRd79^

1.0

X

Xd

Xn

I

I I

I Hz

@

NRd77^

1.0

(a) Plane expected to fail, withconsideration of imperfections

,,' 0.9 F, F1

ILk

MZ Rd7^^

0,911Fly

Ltdl. O

----J

MRd7^;

Xd

Figure 2.10 Verification for combined compression and bi-axial bending

I

After the evaluation of the moment resistance ratios I! and I, for both axes, as described in

the previous section, the interaction of the moments must also be checked using the linear curveshown in Figure 2.10 (c). This linear interaction curve is cut off at 0.91, and 0,911, . The

design moments, M an and M, ,,, , related to the respective plastic moment resistances, mustlie within the moment interaction curve.

EC4-I-I"" considers the checkis adequate when anthe following conditions are satisfied:

M

@

F'z

Z

MRd7, I^

(b) Plane without considerationof imperfections

@

y

1.0

IC) Moment interaction curveor bi-axial bending

,,, M, I, y, Rd

Mz, Sd

:I^ 0.9

11, M, I, ,, in

Mand

^ 0.9

11, Mpi, y, Rd 1'"'PI, ", in

As it is only necessary to consider the effect of geometric imperfections in the critical plane ofthe coluinn buckling, the moment resistance ratio FL in the other plane may be evaluated withoutthe consideration of imperfections, which is presented as follows:

+M

^ 1.0

30

F1I ~IdI~Iru~ IPm

when Id :^ Inn

These expressions are based on the simplified interaction curve, given in the in< NAD.

= 1.0

2.1 0 Longitudinal and transverse shearIn general, the applied internal forces and moments from a member connected to the ends of acomposite columni are distributed between the steel section and the concrete. EC4-I-I""requires that adequate provision should be made for the distribution of these internal forces andmoments.

When Id ' IPm

For structural hollow sections, the shear resistance between the steel section and the concrete isachieved by both chenxical bond and friction at the interface. In these circumstances, the designshear resistance, developing at the interface between the concrete and the inner wall of the steelsection, is limited to"": 0.40 NIInm' for a square or rectangular hollow section (RHS); and0.55 NIIrun' for a circular hollow section (CHS).

For axialIy loaded colunms, it is usually found that this interface shear is sufficient to utilise thecombined strengths of both materials at the critical cross-section (rind-colunm height). Forcolumnis with significant end moments, a horizontal shear force is required, which demands thedevelopment of longituchiial shear forces between the concrete and the steel.

Similarly, the design transverse shear forces may be assumed: to act on the steel section alone;or to be shared between the steel section and the concrete. For the latter case, the shear force tobe resisted by the concrete must be assessed in accordance with EC2-I-I, whereas the shearforce to be resisted by the steel section may be checked according to von Mises yield criterion.However, it is simpler in design to assume that the whole of the transverse shear force acts onthe steel alone. Figure 2.11 indicates the reduction in the design strength of the shear area (web)that will occur within a steel section subjected to transverse shear stress.

fydfcd

Figure 2.11 Reduction of design strength of steel within shear area in the presence oftransverse shear stress

III

, I

I

For design purposes, any reduction in the design steel strength in the shear area of the steelsection may be transfomied into a reduction in steel thickness. For a steel section under majoraxis bending, the effective wall thicimess of the 'web' t in the presence of transverse shearmay be evaluated as follows:

. .

red . fyd

+

T fsd

31

'd"w, d w " - I^ - " 11where:

a, Sd is the design shear force to be resisted by the cross-section

PIARd is the plastic resistance of the steel cross-section in shear = A =:-

A, is the shear area of the steel section

For rectangular hollow sections of runfonn thickness

Load parallel to depth, h, A, = Ah I(b + h)

Load parallel to breadth, b, A, = Ab I(b + h)

For circular hollow sections and tubes of mmfonn thiclm. ess, A = 21 I it

However, no reduction in the web thic}aless is necessary when

'", sd < 0.5P, I, ,, in

Using the effective wall thickness of the 'web' I , of the steel hollow section, the momentresistance of the composite cross-section may be evaluated using the same set of expressionsgiven within 2.91 : without any modification.

For simplicity, the division of the shear force between the hollow section and the concrete maybe neglected, and the design shear force is assumed to be resisted by the steel section alone

2.1 I Load introduction

Where a load is applied to a composite colornn, it must be ensured that the load is distributedbetween the individual components of the cross-section in proportion to their design resistanceswithin a specified introduction length. For composite columns using SHS, this may be achievedas follows

(i) No shear connection needs to be provided for load introduction through a cap plate, at thetop of a Goluinn, if the full interface between the concrete section and endplate ispermanently in compression: after due consideration of the effects of creep and shankageOtherwise, the load introduction has to be verified according to (v). For concrete filledcircular hoUow sections, the effect caused by the confinement may be taken into accountfor load introduction according to Section 2.61, but using the values 77 and 77 for ,^. =O.

(11) If the cross-section of a cap-plate is only partially loaded (see Figure 2.12), loads may bedistributed with a ratio of 1:2.5, over the thiclmess of the end plate. The concrete stressesshould be limited then in the area of the effective load introduction area for concrete filled

hollow sections according to Figure 2.12, Figure 2.13 and (vi) below.

32

Figure 2.12 Load dispersion through a locally loaded cap plate

(in) In absence of a more accurate method, when loads are introduced at an interniediateposition of an SHS length, the introduction length should be assumed not to exceed 2.5d,where d is the nitnimurn transverse dimension in the case of concrete filled rectangularhollow sections or the outside diameter of the colorTm for circular hoUow sections

(iv) Shear connectors should be provided in the load introduction area, and in areas with changeof cross-section, if a design shear strength at the interface between the steel and concreteexceeds the values given in Section 2.10 viz. : 0.40 NIInm' for Ms; and 0.55 NIInm' forCHS. The shear forces should be detennined from the change of sectional forces of thesteel or reinforced concrete section within the introduction length, where the sectionalforces should be datennined by plastic theory. If the loads are introduced only into theconcrete cross section, the values resulting from an elastic analysis considering creep andshankage should be taken into account. At a beam connection position, it is necessary tocheck that

1:2.5

1:1

F

Steel

Concrete

1:2.5

For an 1<11S column: (I - 6)P, , I A, < 0.40 NITnm' with A, = 2.5db

For a CHS colornn: (I- 6)P. ,, I A, < 0.55 NIInm' with A, = 2.5itd' I4where:

I'Sd

6

1:1

is the design shear load to be transferred to the column by a beam connection

is the steel contribution ratio

is the usable shear area/connection at the concrete interface

b is the breadth of RHS face at a shear connector.

is the minimum dimension of an RHS or diameter of a CHSd

If load introduction would give rise to excessive interface shear stresses, then additionalshear stud connectors, or a through gusset plate (Figure 2.13), should be provided in theload introduction area, to enable the additional load to be introduced into the concrete core

(v) Shear studs may be designed using the usual method given in EC4-I-I"", based on thefollowing assessment, namely that the design shear strength of a stud should be determinedas the lower of

11,

33

^M -,-^11^^' "^),^,or

'Rd = 0.29ud'10" "

^. o. 21(h/d)-11with

where:

o. = 1.0

I, is the specified ultimate strength of the shear stud material (but not greater that 500NIIrun2,

is the characteristic cylinder strength of the concrete.

is the secant modulus of the concrete as given in Table 2.2.

is the diameter of the shank of shear stud.

is the length of the shear stud within the concrete core

is a partial safety factor of 1.25.

I, k

Ecm

d

h

for 3 ^ h/d ^ 4

for h/d > 4

Y,

A,.-

.I_

^J

,

N sd

S

o c Rd

A_, r

y.T t.

t + 5t

^^.

Section A - A Section B - B

Figure 2.13 Loadintroduction into a concrete core through a gusset plate

I

(vi) When a concrete filled circular or rectangular square hollow section is only partially loadedby plate stiffeners at a cap colunm divider plate position (collmm section type A-A inFigure 2.13), or from a gusset plate through the profile at an intennediate colunm lengthposition (section type B-B in Figure 2.13), the local design resistance strength of concrete

34

B

I

^,.

I MsdN sd

B

e

o c, Rd :^ f yd

A1

-^-T t,

d, ,d under the gusset plate or stiffener, resulting from the sectional forces of the concretesection, should be deteimned by:

O. R d ' I'd 1' ' 77.1 ~ '_ ,_ , ,_,,.where:

71, andj:, are the design strength of the steel and the characteristic strength of the concreterespectively.

is the wallthickiiess of the steel tube.

o

A

is the loaded area under the gusset plate according to Figure 2.13.

is 4.9 for circular steel tubes; and 3.5 for rectangular sections.rich

The ratio A I A in the equation above should not exceed 20. Welds between the gussetplate and the steel borrow sections should be designed according to Section 3 of prEN1993-I-8: 2002t291.

(vii) For concrete filled circular hoUow sections, longitudinal reinforcement may be fully takeninto account when assessing cross-sectional design parameters, even where thereinforcement is not welded to the end plates or in direct contact with the endplates,provided that the gap e between the reinforcement and the Grid plate does not exceed 30nun (see also coluniii section type A-Am Figure 2.13).

C

is the diameter of the tube or the width of the rectangular section.

is the cross-sectional area of the concrete.

A

Alternatively, proprietary nailed connectors can used to effect the required shear transfercapacity. These must be shot fired through the tube wall from the outside in a defined patternbefore concrete filling. Typically, they can have a design shear capacity of 12 I, \TIConnector andare placed at a spacing of 50 nun between connectors"" .For further infonnation see the citedreference.

35