ec4_designofcompositecolumns.pdf

7/29/2019 EC4_DesignOfCompositeColumns.pdf

1/27

1

Helsinki University of Technology

Department of Civil and Environmental Engineering

Rak-83.140 Seminar on Steel Structures

Zhang Jing 63864F

Spring 2004

Design of Composite columns


2/27

2

CONTENT

1. Introduction....................3

1.1 Types of composite columns....3

1.2 An example of experiment comparison with steel column..4

1.3 Advantages of composite columns...5

2. Composite column design according to Eurocode 45

2.1 Fundamental design requirements....62.2 Design methods according to EC 4..6

2.3 The Simplified Method7

2.3.1 The partial safety factors of materials 7

2.3.2 Limitations when using the simplified method..7

2.3.3 Local buckling of steel members. ....9

2.3.4 Second-order effect..10

2.3.5 Plastic resistance to axial force11

2.3.6 Plastic moment of resistance................................................................12

2.3.7 Resistance to combined compression and bending..13

2.3.8 Compression and biaxial bending....17

2.3.9 The influence of shear force.18

3. Compare with other design codes...19

3.1 The building code requirements of reinforced concrete (ACI 318-89)..

19

3.2 Load and resistance factor design method (AISC-LRFD).20

3.3 Architectural Institute of Japan (AIJ).20

3.4 British Standard BS 5400-Part 5....203.5 European Code EC4...21

4. Reference......21

Appendix: A Design Example of composite column according to Eurocode4.....

..22


3/27

3

1. Introduction

The combination of structural steel and concrete in one bearing system is the basis of

composite structures. Nowadays the composite slabs, composite beams and compositecolumns are widely used in many countries.

1.1 Types of composite columns

A composite column is defined as a compression member which may either be a

concrete encased section or a concrete filled hollow section. According to the shape of

the cross-section, there are mainly three different types of composite columns are

principally in use, see Fig.1.

Concrete-encased sections (a, b and c) Concrete-filled hollow sections (f, g and i)

Partly concreted-encased sections (d and e).

Fig.1 Typical cross-sections of composite column

Concrete encased columns generally fulfill the technical requirements for high classes

of fire protection without any additional measures. They can be easily strengthened by

reinforcing bars in the concrete cover. However, they do not present an accessible

structural steel surface for later fastenings and attractive surface treatment. This type

of composite column is the preferred form for seismic-resistant structures. Under

severe flexural overload, concrete encasement cracks resulting in reduction of

stiffness while the steel core can continue providing shear capacity and ductile

resistance to subsequent cycles of overload.


4/27

4

Concrete-filled column is used for bridge piers subject to impact from traffic, columns

to support storage tanks, decks of railways, column in high-rise building and as piles

for foundations. In concrete-filled hollow sections, the longitudinal reinforcement

may be not necessary, if design for fire resistance is not required.

Partly concreted-encased columns have high fire resistance, which is due to that the

concrete part prevents the inner steel parts - structural steel as well as reinforcing bars

from heating up too fast. Another significant advantage is that some of the steel

surfaces remain exposed and can be used for connection to other beams.

1.2 An Example of Experiment Comparison with steel column

The basic buckling modes of steel and composite column are illustrated in Fig.2. Inthe case of concrete-filled steel tubular column, concrete inside the tube prevents

inward-buckling modes of the steel tube wall, and the tube-wall in turn provides

effective lateral confinement to the concrete inside the tube.

Fig.2 Buckling modes of steel and composite sections

Typical example of load-average strain curves for a steel and concrete-filled column

are shown in Fig.3.

(a) Steel section (b) Composite section

Fig.3 Load average strain curves under cyclic loading[1]

The unloading response of the tubes becomes rapid in case of composite columns,


5/27

5

with increasing tube wall slenderness due to local buckling. The response of

composite tube is similar to that for the bare steel tube. The hysteretic loops in the

concrete filled column were relatively narrow in early cycles, and then become wider

at the later cycles due to strain hardening in the post-ultimate region. The maximum

strength is obtained at about a strain 0f 0.2% in the steel box columns and in the rangeof 0.3-0.4% in the concrete-filled columns. Obviously, the maximum strength of

concrete-filled columns was much larger than those of the steel columns. Therefore, it

can be calculated that the concrete-filled column shows good structural performance

through ductility and high strength.

1.3 Advantages of composite columns

There are many advantages associated with the use of composite columns

(1) High load capacity with small cross-section and economic material use(2) Simple connection to other members, as for steel construction

(3) Possibility of plastic deformation and ductile behavior

(4) Good fire resistance

(5) High resistance to compressive stresses

(6) Reduced the risk of local buckling of the steel section

(7) Advantages in fabrication

For concrete-filled hollow section columns, there are some more special advantages

(1) Steel is placed in its most effective position at the edge of cross-section

(2) Concrete works at higher stresses because of the confinement provided by

the steel tubes

(3) Fire protection is not usually needed, as the concrete core provides good fire

resistance

(4) The steel section serves as formwork during casting

2. Composite column design according to Eurocode 4

The Eurocode 4 gives six typical cross-sections of composite columns. See Fig.4.


6/27

6

Fig.4 Typical cross-sections of composite columns

2.1 Fundamental design requirements

A composite column must be designed for the ultimate limit state. The internal

stresses and moments resulting from the most unfavorable load combination may at

no point of the structure exceed the bearing capacity of the cross-section. Effects of

deformation (p- effects) as well as creeping and shrinkage must be considered if

they are significant. The influence of residual stresses and initial deformations must

be considered and reductions of stiffness due to cracking of the concrete in the tensile

area as well as yield of the steel have to be taken into account.

Problems due to local and of overall stability must be prevented. Complete bond up to

failure may be assumed.

Exact allowance for all these requirements is only possible with complex computer

programs, which operate non-linearly and incrementally. For practical design,

simplification must be made so that design can be carried out with minimum effort.

2.2 Design methods according to EC 4

Eurocode 4provides two methods for calculation of the resistance of composite

columns.

The first is a General Method which takes explicit account of both second-order

effects and imperfections. This method can in particular be applied to columns of

asymmetric cross-section as well as to columns whose section varies with height. It

requires the use of numerical computational tools, and can be considered only if


7/27

7

suitable software is available.

The second is a Simplified Method which makes use of the European buckling curves

for steel columns, which implicitly take account of imperfections. This method is

limited in application to composite columns of bisymmetric cross-section which doesnot vary with height.

These two methods are both based on the following assumptions:

Plane sections remain plane when the column deforms There is full interaction between the steel and concrete sections until the failure

occurs

Geometric imperfections and residual stresses are taken into account in thecalculation, although this is usually done by using an equivalent initial

out-of-straightness, or member imperfection

The Simplified Method is discussed in the following part.

2.3 The Simplified Method

2.3.1 The partial safety factors of materials

For the determination of resistance, the following partial safety factors have to be

applied for the different materials:

a is the partial safety factor of the structural steel

s is the partial safety factor of for the reinforcement

c is the partial safety factor of the concrete

Thusa

yyd

ff

= is the design value of yield strength of the structural steel

s

ssd

ff

= is the design value of yield strength of the reinforcement

c

c

cd

ff

= is the design value of yield strength of the concrete

2.3.2 Limitations when using the simplified method

The scope of this simplified method is limited to members of doubly symmetrical and

uniform cross-section over the member length. This method is not applicable if the

component consists of two or more unconnected sections. The following conditions

should also be fulfilled when use this simplified method.

(1)The steel contribution ration should fulfill the following condition


8/27

8

9.02.0 (1)

Rdpl

yda

N

fA

.

= (2)

Aa is the cross-section area of the structural steel sectionfyd is the design value for the yield strength of the structural steel

Npl.Rd is the plastic resistance to compression of the composite cross-section

(2) The relative slenderness_

0.2.

_

=cr

Rdpl

N

N (3)

Npl.Rd is the plastic resistance of composite section to compression

Ncr is the elastic critical normal force for the buckling mode

b

eff

crL

EIN

2

2 )(= (4)

Lb is the buckling length of the column

(EI)eff is the effective flexural stiffness of the composite cross-section

ccmssaaeff IEIEIEEI 6.0)( ++= (5a)

where Ia, Is and Ic are respectively the second moment of area for structural

steel section, reinforcing steel section and the uncracked concrete section.

The coefficient 0.6 for the stiffness of the concrete section takes cracking

into account.

The effective flexural stiffness (EI)eff.II of the composite cross-section is used to

determinate the internal force and second-order effects.

)5.0(9.0)( . ccmssaaIIeff IEIEIEEI ++= (5b)

(3) Other limitations

If the longitudinal reinforcement is considered in design, a minimum percentage of

0.3% of the reinforcement area must be provided while the maximum percentageshould not exceed 6% of the concrete area.

For a fully encased steel section, see Fig.5, limits to the minimum and maximum

thickness of concrete cover that may be used in calculation are:

hcmm z 3.040

bcmm y 4.040

The ratio of the cross-sections depth hc to width bc, see Fig.5, should be within the

limits 0,2 hc / bc 5.0.


9/27

9

Fig.5 Encased sections

2.3.3 Local buckling of steel members

For sections completely encased in concrete, local buckling need not be checked.

The other types of cross-sections must have certain minimum wall thicknesses in

order to prevent local buckling before reaching the ultimate loads of the system. This

can be ensured by maintaining a certain limit ratio of depth to thickness of the section.

Using the notation of Table.1

Table.1 Maximum values (d/t), (h/t) and max (b/tf) with fy in N/mm2

Cross-section Max (d/t), max (h/t) and max (b/ tf)

Circular hollow

steel sections

290)(max =d/t

Rectangular hollow

steel sections52)(max =h/t

Partially encased

I-sections44)(max f =b/t

accounts for the influence of different yield strengths fy with

yf/235=


10/27

10

2.3.4 Second-order effect

The influence of second order effects may be neglected in the analysis of bendingmoments for braced and non-sway systems, provided that:

(1) the normal force NEd is smaller than 10% of the critical load Ncr

1.0.

effcr

Ed

N

N(6)

NEd is the total design normal force

Ncr.eff is the critical normal force calculated using effective stiffness (EI)eff.II

(2) the slenderness of the column is

)2(2.0 r (7)

r is the ratio of the smaller to the larger end moment(see Fig.6)

Fig.6 Ration r of the end moment

If the conditions above are not satisfied, second-order should be taken into account by

multiplying the maximum first order bending moment by a factor k:

0.11 .

=effcrEd NN

k (8)

is the equivalent moment factor

For columns with transverse loading within the column length the value for

must be taken as 1.0. For pure end moments, can be determined from:


11/27

11

44.044.066.0 += r (9)

2.3.5 Plastic resistance to axial force

The plastic resistance to axial force is give by Eq.(10), which includes the individual

resistances of the steel profile, the concrete and the reinforcement.

For fully or partially concrete-encased steel section:

sdscdcydaRdpl fAfAfAN ++= 85.0. (10)

For concrete-filled hollow sections the coefficient 0.85 should be replaced by 1.0.

This increase is due to the state of tri-axial compressive stresses in concrete resulting

from the confinement provided by the steel section.

For composite column with circular hollow sections (CHS), account may be taken of

increase in strength of concrete caused by confinement provided that the relative

slenderness does not exceed 0.5 and e/dde

0.1=a (12e) 0=c (12f)

For members in pure axial compression, the design value of NEd should satisfy the

buckling condition


12/27

12

0.1.

Rdpl

Ed

N

N

(13)

is the reduction factor, for the relevant buckling mode given in terms of

the slenderness, column section type and the relevant buckling curve. Eurocode 4has adopted the European buckling curve a, b and c (see Fig.7), which are

originally established for bare steel column (Eurocode 3)

Fig.7 European buckling curves according to EC 3

The factorcan be described mathematically as following:

0.1)(

122

+

=

(14)

])()2.0(1[5.0 2 ++= (15)

The factor is used here to allow for imperfections in the cross-sections. Table 2

gives the value of appropriate for each buckling curve.

Table 2: Imperfection factor a for the buckling curves according to Eurocode 3

European buckling curve a b c

Imperfection factor 0.21 0.34 0.49

2.3.6 Plastic moment of resistance

The plastic moment resistance against bending of the cross-section Mpl.Rd may be

considered to correspond to a full plastic stress distribution according to Fig.8.

Concrete under tension is neglected.


13/27

13

Fig.8 Full plastic stress distribution for a concrete-encased profile

The determination of Mpl.Rd is explained in the following section.

2.3.7 Resistance to combined compression and bending

It is necessary to satisfy the resistance requirements in each of the principal planes,

taking account of the slenderness, the bending moment diagram and the bending

resistance in the plane under consideration. The cross-sectional resistance of a

composite column under axial compression and uniaxial bending is given by an

interaction curve as shown in Fig. 9.

0

Npl.Rd

Mpl.Rd Mmax.Rd

Npm.Rd

0,5 Npm.Rd

M

N

A

E

C

D

B

Fig.9 Interaction curve with linear approximation

The pointD on this interaction curve corresponds to the maximum moment resistanceMmax,Rd that can be achieved by the section. This is greater than Mpl.Rdbecause the

compressive axial force inhibits tensile cracking of the concrete, thus enhancing its

flexural resistance.

The above interaction curve can be determined point by point, by considering

different plastic neutral axis positions in the principal plane under consideration.

The concurrent values of moment and axial resistance are then found from the stress

blocks, together with the two equilibrium equations for moments and axial forces.

Fig.10 illustrates this process for the example of a concrete-encased section, for four


14/27

14

particular positions of the plastic neutral axis corresponding respectively to the points

A, B, C, D marked on Fig.9.

Fig.10 Development of stress blocks at different points on the interaction curve

Point A: Axial compression resistance alone

RdplA NN .= 0=AM

Point B: Uniaxial bending resistance alone

0=BN RdplB MM .=

Here it can be seen, that in the determination of the resistance of the cross-section,

concrete regions in tension are taken as being cracked and ineffective

Point C: Uniaxial bending resistance identical to that at point B, but with non-zero

resultant axial compression force:

section)hollowfilled-(concrete

section)encased-(concrete85.0..

cdC

cdCRdpmC

fA

fANN

=

== (16)

RdplC MM .= (17)


15/27

15

[Note: fcd may be factored by ]1[

ckf

yf

d

tc+ for a circular concrete-filled hollow section.]

PointD: Maximum moment resistance

section)hollowfilled-(concrete5.0

section)encased-(concrete85.02

1

2

1.

cdc

cdcRdpm

fA

fANND

=

== (18)

[Note: fcdmay be factored by ]1[

ckf

yf

d

t

c+ for a circular concrete-filled hollow section.]

section)encased-(concrete85.05.0..max cdpcsdpsydpaRdD fWfWfWMM ++==

(19a)

section)hollowfilled-(concrete5.0..max cdpcsdpsydpaRdD fWfWfWMM ++==

(19b)

Wpa, Wps, and Wpc are the plastic modular respectively of the steel section, the

reinforcement and the concrete.

Point E: Situated midway betweenA and C.

The enhancement of the resistance at point E is little more than that

given by direct linear interpolation betweenA and C, and the calculation

can therefore be omitted.

It is usual to substitute the linearised version AECDB (or the simpler ACDB) shown

in Fig.9 for the more exact interaction curve, after doing the calculation to determine

these points.

We may notice that the stress distribution type C provides the same value for the

moment of resistance as B, since the moment from the stress resultants in zone 2hn,

cancel each other. However, the resulting resistance to axial force is of the samemagnitude as the axial force resistance from the pure concrete part Npm.Rd .This can be

seen from adding up the stress distributions in B and C, with regard to the equilibrium

of forces, i.e. the resulting axial force. This follows because the resistance to axial

force in B is zero. Thus, the concrete under compression and that under tensile stress

are complementary to each other; the stress blocks for the reinforcement provide no

axial load. A distribution according to Fig.11(a) results. In the same way, subtracting

the stress distributions of B from that of C analytically produces the stress blocks

shown in Fig.11(b). The stress resultant in Fig.11(b), only dependent on hn, is

generally quite simple. The sum of the horizontal force must lead to Npm.Rd. As a

result an equation for hn, the neutral axis at Mpl.Rd is obtained. This equation is


16/27

16

different for various types of section. For example of a concrete encased I-profile with

bending about the major axis in Fig.11(b), hn is expressed like this

)2(22

.

cdfdwcd

Rdpm

nfftbf

Nh

+= (20)

Fig.11 Combination of the stress distribution at point B and C considering

normal force only (a)adding the components; (b) subtracting the components

Fig.12 shows how the cross-section of a composite column can be checked, by means

of the interaction curve M-N. The resistance of the column under axial compression is

defined by the reduction factor . For the factor , a value for the moment k

representing the moment due to imperfection, can be read off the interaction curve.

The influence of this moment is assumed to decrease linearly to the value n.

Fig.12 Design procedure for compression and unaxial bending

For a normal force

RdplSdd NN ./= (21)


17/27

17

In certain regions of the interaction curve, the normal force increases the moment

resistance ( > 1.0). If bending moment and normal force are independent of each

other, the value of must be limited to 1.0. The value n accounts for the fact that

imperfection and bending moment do not always act together unfavourably.

For end moments, n may be calculated as:

=

4

1 rn (22)

r is the ratio of end moments according to Fig.6

If transverse loads occur within the column length, then n =0, i.e. r = 1.

The value of moment factor could be obtained from the ineraction curve.

)(n

ndkd

= (23)

The moment factor represents the remaining moment resistance. It can be shown as:

RdplSd MM .9.0 (24)

2.3.8 Compression and biaxial bending

For the design of a column under compression and biaxial bending the load resistance

for each axis has to be evaluated separately. It will then be clear which of the axes is

more likely to fail. The imperfection then needs to be considered for this directiononly (see Fig.13).


18/27

18

Fig.13 Verification for compression and biaxial bending

The combined bending must also be checked using the relative values of moment

resistance y and z, and a new interaction curve (see Fig.13). This linear interaction

curve is cut off at 0.9 y and 0.9z; the existing moments My.Sd and Mz.Sd, related to

the respective resistance, must lie within the new interaction curve. The following

equations result:

0.1..

.

..

. +Rdzplz

Sdz

Rdyply

Sdy

M

M

M

M

(25)

and

9.0..

. Rdyply

Sdy

M

M

(26a) 9.0

..

. Rdzplz

Sdz

M

M

(26b)

2.3.9 The influence of transverse shear force

For simplicity, the design transverse shear force VEd is assumed to be resisted by the

steel section only. The design shear resistance Vpl.a.Rd of the steel section can be

calculated as Eq.(27)


19/27

19

3/.. ydvRdapl fAV = (27)

vA is the shear area of the structural steel cross-section

For RHS profile-if the shear force parallel to the height )/( hbhAA av += (28a)

-if the shear force parallel to the width )/( hbbAA av += (28b)

For CHS profile /2 av AA = (28c)

The influence of transverse shear force on the resistance to bending and normal force

should be considered ifRdaplEd

VV..

5.0> , and a reduced design strength (1-)fyd

should be used in the shear area Av of the steel section for the determination of

bending resistance.

2)1/2( = RdEd VV (29)

If RdaplEd VV ..> the design shear force VEd may be distributed into the component,

Va,Ed acting on the structural steel section and Vc,Ed acting on the reinforced concrete

section.

Rdpl

Rdapl

EdEdaMMVV

.

..

, > (30) EdaEdEdc VVV ,, > (31)

Where the plastic moment Mpl.a.Rd and Mpl.Rd refer to the the relevant bending axis.

The resistance to shear of the concrete part should be verified in accordance with

Part 1 of Eurocode 2.

3. Compare with other design codes

Over the last two decades, researchers have suggested analytical methods and designprocedures for composite columns and design codes have been formulated. Each of

these codes is written so as to reflect the design philosophies and practices in the

respective countries. Over the last two decades, different specific codes for the design

of concreted filled steel tubular column have been used.

3.1 The building code requirements of reinforced concrete (ACI[2] 318-89)

According to ACI 318-89, a composite column is a concrete column reinforced with a

structural steel shape or tubing in addition to reinforcing bars. In order to consider the

slenderness effects, an equivalent radius if gyration and flexural stiffness are used


20/27

20

with a parameter of sustained load ratio, and hence without any sustained load, radius

of gyration should be taken as zero. The limiting thickness of steel tube to prevent

local buckling are based on achieving yield stress in a hollow steel tube under

monotonic axial loading which is not necessary requirement for in-filled composite

column. A parameter for the softening influence of creep in concrete that is subjectedto sustained compressive loading is included.

3.2 Load and resistance factor design method (AISC[3]-LRFD)

This is based on the same principle as ACI code. Design is based on equations for

steel columns. Nominal strength is estimated on the basis of ultimate resistance to the

load, and reduction factor are then applied. The nominal axial load capacity is reduced

according to the slenderness ratio. Neither the ACI-318 nor the AISC-LRFD

provisions explicitly consider confinement effects on the strength or ductility of

member analyzed. ACI provisions for calculating the strength interaction betweenaxial and flexural effects are essentially the same as those for reinforced concrete

column, whereas AISC-LRFD are based on the bilinear interaction formulae which

have the same form as those of steel columns. In the above design methods, flexural

stiffness is underestimated and confining effect of the steel tube on the concrete core

is ignored. The influence of creep is ignored for concrete in composite column

according to AISC-LRFD specification.

3.3 Architectural Institute of Japan (AIJ)

A composite structural system using concrete and steel shape is called steel

reinforced concrete (SRC) in Japan. The allowable stress design is primarily

employed, in which working stresses are calculated based on the elastic stiffness of

members and allowable strength by the superposed strength formulae. Cross-section

strength is calculated by superimposing the strength of both the steel and concrete

sections, thereby neglecting the interaction between steel and concrete and the effect

of confinement. Euler buckling load is used with a reduced concrete stiffness and

factors of safety for both concrete and steel. The method is applicable to asymmetrical

sections and column under biaxial bending.

3.4 British Standard BS 5400-Part 5

Code provisions in BS 5400 are based on limit state design with loading factors and

materials safety factors. The ultimate moment is calculated from plastic stress

distribution over the cross-section, and an approximation for the interaction curve for

axial load and moment is used. Reduced concrete properties are used to account for

the effects of creep and the use of uncracked concrete section in stiffness calculation.

This method is applicable to symmetrical sections only and restricted to the range of

section catered for in the European buckling curves. It underestimates the capacity of

in-filled composite column with high-strength concrete.


21/27

21

3.5 European Code EC4

EC4 covers concrete encased and partially encased steel section and concrete filled

sections with or without reinforcement. This code uses limit state concepts to achieve

the aims of serviceability and safety by applying partial safety factors to loads andmaterials properties. Based on experimental results, it was recommended that the

regulations of EC4 concerning the factor of 0.85 should not be applied to hollow

sections filled with strength concrete. This is the only code that treats the effect of

long-term loading separately.

All codes assume full interaction, but some impose restrictions on the shear stress at

the steel-concrete interface. It is customary to use direct bearing or provide shear

connectors, if used were the specified limiting shear stress is exceeded.

4. References

1. prEN 1994-1-1: Design of composite steel and concrete structures. Part 1-1:

General rules and rules for buildings. Final draft, 1 January 2002.European

Committee for Standardization.

2. prEN 1994-1-2: Design of composite steel and concrete structures. Part 1-2:

Structural rules-Structural fire design. Final draft, 14 January 2002.European

Committee for Standardization.

3. ESDEP Course: WG10:Composite construction. Lecture10.8.1: Composite

Columns I ; Lecture10.8.1: Composite Columns II

4. SSEDTA: Structural Steelwork Eurocodes Development of A Trans-national

Approach. Lecture 8: Composite Columns. @SSEDTA 2001 Last modified

24/02/04.

5. Dowling, P. J., Harding, J. E. & Bjorhovde, R.: Constructional Steel Design: an

international guide. Chapter 4.2 Composite column. Elsevier Applied Science,

1992.

6. N.E. Shanmugam & B.Lakshmi: State of the art report on steel-concrete

composite columns. Journal of Constructional Steel Research, Volume 57, Issue

10, Pages 1041-1119 (October 2001)7. Aki Vuolio: Design of concreted-filled hollow section columns. (Seminar on steel

structures, Helsinki University of Technology Laboratory of steel structures)

[Notes]

[1]. This figure is based on experiments carried by CFT working group, Japan, 1993. The

cross-section is circular concrete-filled section, and the load is eccentric, the strength of the

in-fill concrete is 41Mpa.

[2]. ACI is the abbreviation ofAmerican Concrete Institute.

[3]. AISC is the abbreviation of American Institute of Steel Construction.


22/27

22

Appendix: A Design Example of composite columnaccording to Eurocode4

kN 1000 newton:= kNm kN m:= MPaN

mm2:=

Structure model and loads see Fig. A

A composite column pin-supported at both ends , with normal force and bending mome

about one axis.

length of the column Lc 10 m:= design value of normal force Nsd 2000 kN:=

design value of bending moment Msd 100 kNm:=

Wp.a 966.8 103 mm3:=plasticity modulus

Ia 13201 104 mm4:=moment inertia

cross section area Aa 8736 mm2:=

t 8 mm:=wall thickness

D 355.6 mm:=diameter of profile

fyd 3.227 108 Pa=fyd

fy

a:=design strength

a

1.10:=partical safety factor

Ea 210000 MPa:=modulus of elasticity

yield strength fy 355 MPa:=1. Structural steel section

see Fig. BCross section properties


23/27

23

2. Reinforced concrete section

yield strength of reinforcement(A500HW) fs 500 MPa:=

modulus of elasticity of reinforcement Es 210000 MPa:=

s 1.15:=partial safety factor of reinforcement

es2 91.8 mm:=

moment of inertia of reinforcement Is 2 As1 es12 2As1 es2

2+:=

Is 2.118 105 m4=

plastic modulus if reinforcement Wp.s 2 As1 es1 2As1 es2+( ):=

Wp.s 1.969 104 m3=

compressive strength of concrete (C40/50) fck 40 MPa:=

secany modulus of elasticity of concrete Ecm 35000 MPa:=

partial safety factor of concrete c 1.50:=

fcd

fck

c:= fcd 2.667 10

7 Pa=design strength of concrete

moment of inertia of concrete Ic D 2 t( )4

64

Is:=

I

c6.317 10

4 m4=

fsd

fs

s:= fsd 4.348 10

8 Pa=design strength of reinforcement

number and diameter of reinforcing bars ns 8:= s 20 mm:= As1 s

2

4:=

cross section area of reinforcing bars As1

3.142 104

m

2

=A

sns

As1:=

As 2.513 103 m2=

Ac D 2 t( )2

4

As:=cross section area of concrete

Ac 0.088m2=

ration of reinforcement (max value is 6%) s

As

Ac

:= s 0.029= s max

second order effects need to be checked


26/27

26

the assumption forAsnis verified

plastic section modulus of steel and concrete in the region of 2hn

W

p.an

2 t hn

2:= W

p.an

4.132 105 m3=

Wp.cn D 2 t( ) hn2

:= Wp.cn 8.77 104 m3=

bending resistance of the cross section with the depth of 2hn

Mn.Rd Wp.an fyd 0.5 Wp.cn fcd+:= Mn.Rd 2.503 104 J=

plastic bending resistance (point C and B)

Mpl.Rd

MD.Rd

Mn.Rd

:= Mpl.Rd

4.57 105 J=

non_dimensional interaction curve

NA.Rd Npl.Rd:= NB.Rd 0:=

1

2 _2+

:= 0.456= 1.0

Nsd 2 106 N= Npl.Rd 2.856 10

6 N=

Nsd Npl.Rd< OK !

Resistance to combined compression and bending

interaction curve piont D

MD.Rd Wp.a fyd Wp.s fsd+ 0.5 Wp.c fcd+:= MD.Rd 4.82 105 J=

ND.Rd 0.5Ac fcd:= ND.Rd 1.174 106 N=

interaction curve point C and B

NC.Rd 2 ND.Rd:= NC.Rd 1.97 103kN:=

it is assumed that no reinforcement lies within the region of 2hn

hn

NC.Rd

2 D fcd 4 t 2 fyd fcd( )+:= hn 0.051m=


27/27

Nsd

Npl.Rd

0.319=NC.Rd

Npl.Rd

0.315=ND.Rd

Npl.Rd

0.188=

MA.Rd 0:= MB.Rd Mpl.Rd:= MC.Rd Mpl.Rd:=

MD.RdMpl.Rd

1.055= Msd_

Mpl.Rd

0.301=

Msd_ is the ampilificated bending moment according to second order effect

check for utilization ratio of compression and bendin

moment

n 1 r( )

4:= n 0.114=

dNsd

Npl.Rd

:= d 0.319=

drawing the curve of compression and uniaixal bending see Fig.C

d1 d( )

1 0.315:= k

1 ( )

1 0.315:=

d kd n( ) n( )

:= 0.517=

n maxMsd

0.9 Mpl.Rd

Nsd

Npl.Rd,

:= n 0.7=

0.5 n< 1.0< OK !

ec4_designofcompositecolumns.pdf

Documents