Effective width evaluation for steel–concrete

8/13/2019 Effective width evaluation for steelconcrete

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Journal of Constructional Steel Research 58 (2002) 373388www.elsevier.com/locate/jcsr

Effective width evaluation for steelconcretecomposite beams

C. Amadio *, M. Fragiacomo

Civil Engineering Department, University of Trieste, piazzale Europa 1, 34127 Trieste, Italy

Received 8 December 2000; received in revised form 3 October 2001; accepted 4 October 2001

Abstract

In this paper the problems connected to the effective width evaluation for serviceability andultimate analysis of steelconcrete composite beams are analyzed. By a parametric study car-ried out through the Abaqus code it is pointed out how the actual codes do not provide, in

general appropriate results, for elastic and ultimate limit state checks. The most importantparameters that influence the effective width are analyzed. Some preliminary criteria for anadequate design are presented. 2002 Elsevier Science Ltd. All rights reserved.

Keywords:Effective width; Deformable connection; Composite beam; Shear-lag

1. Introduction

The determination of the effective width for serviceability or ultimate limit states

analysis is on the basis of a correct design of steelconcrete composite beams.Shear strains play an important role for an elastic analysis of composite beams

where the concrete slab has a width larger than 1/25th of the beam span [1]. In fact,the shear strains cause a non-uniform distribution of the normal stresses and the non-planarity of the slab cross section (shear-lag phenomenon) that make the elementaryDe Saint Venants theory inadequate.

To overcome the difficulty owing to the complex analytical direct evaluation ofthis phenomenon [24], in practical design the concept of effective width is usuallyintroduced. It allows the use of the elementary bending theory, obtaining correct

* Corresponding author.

E-mail address: [email protected] (Prof. C. Amadio).

0143-974X/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved.

PII: S 0 1 4 3 - 9 7 4 X ( 0 1 ) 0 0 0 5 8 - X


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374 C. Amadio, M. Fragiacomo / Journal of Constructional Steel Research 58 (2002) 373388

Nomenclature

a0 Connection width

ai Half slab width minus half connection width

aei Effective width calculated on aibi Half slab width

beff Total slab effective width

bs Width of the upper flange of the steel beamC1, C2 Constants of the connection

Ea Youngs modulus of structural steelEs Youngs modulus of reinforcementfs

Yield stress of reinforcement

ft Tensile strength of reinforcement

fu Tensile strength of structural steel

fy Yield stress of structural steel

hc Height of the concrete slab

hs Height of the steel beam

L Length of the beam

0 Distance between the points of zero bending moment

N/Nf Degree of shear connection

P Applied load

Ppl Applied load that produces the collapse of the composite beamQ Shear force on the connection

Qu Shear strength of the connection

x, y, z Coordinate system on the composite beam

Maximum deflectionau Strain of structural steel under the maximum loadsu Strain of reinforcement under the maximum load Diameter of steel bar Slip of the connection Ratio between aei and ai

Normal stress of concretemax Maximum normal stress of concretebar Normal stress of reinforcementbar max Maximum normal stress of reinforcement

results in terms of maximum stress in prefixed sections of the beam. In order tofacilitate the designer, codes adopt in general simplified formulations [5] or tables[6, 7] for the effective width evaluation. In this way the effective width is generally

expressed as a function of some parameters, such as the b/L ratio between the halfwidth of the slab and the beam length, the abscissa of the cross section along the

beam, the type of load and the types of restraint for the beam.


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375C. Amadio, M. Fragiacomo / Journal of Constructional Steel Research 58 (2002) 373388

For an ultimate limit state analysis, codes in general propose to use the same

effective width as computed for an elastic analysis. This represents an evident

approximation, since when plastic phenomena advance in the slab, normal stresses

tend to become uniform in the cross section involving an increment of the effectivewidth. This aspect requires further studies since, particularly for long span beams,

an increase of the effective width can imply a significant increment of load capacity.In this paper the main results obtained by a parametric study carried out through

the Abaqus code [8] are presented. They are based on several linear and non-linear

numerical analyses performed on a symmetric composite beam considered extractedfrom a structure made by a series of equal parallel beams.

For the stress control in the serviceability condition, the deformability of connec-

tion plays an important role, generally neglected, in a correct evaluation of the effec-

tive width. The effective width calculated in the hypothesis of rigid connection is

in fact larger than the one evaluated with a deformable connection. This occurs in

both cases of partial and full shear connection. A series of tables is proposed to

perform a correct evaluation of stresses in the beam.

For the evaluation of the elastic deflection, the effective width can instead beconsidered practically coincident with the whole slab width even for high b/L ratios

(b/L 0.40).

For a non-linear analysis, cracking of concrete and plastic behaviour of steel

should be taken into account. The effective width proposed by actual codes, usually

based on the same hypotheses of an elastic calculus, is therefore not appropriate for

an ultimate limit state analysis. With the progress of the plastic phenomena into theslab, normal stresses tend to be nearly constant in the whole slab. For both cases of

sagging and hogging bending moment the plastic zone is extended in almost the

whole concrete slab in compression and the whole reinforcement distributed into the

slab in tension respectively. This occurs even for large b/L ratios (b/L 0.40).

Finally, a simplified evaluation of the effective width for a plastic analysis withoutmoment redistribution is proposed for the case of compact steel section, full connec-tion and ductile reinforcement.

2. Effective width for an elastic analysis

2.1. Stress analysis

In an elastic stress analysis the total effective width of the slab beffis theoretically

defined as the width that with a constant stress distribution smax equal to themaximum value into the section has the same resultant of the real stress distribution

s. Since the stress varies along the slab thickness, the effective width is evaluatedwith reference to the stress distribution on the central fibre of the concrete slab.Therefore, for a generic composite beam (see Fig. 1), it can be written:

aei 1

smax a0/ 2

bi

sdy with i 1,2 (1)


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Fig. 1. Geometry of a typical cross section of a composite beam.

where b1 b2 b is half slab width, a0 is the connection width and the origin of

the y-axis is chosen on the symmetry axis for the cross section of the composite

beam. The total effective width of the slab is given by:

beff be1 be2 ae1 a0 ae2 (2)

where bei is the effective width determined on half slab width, and aei is obtained

by subtracting by bei half connection width a0. The effective width of the slab

depends on several parameters, like the ratio a/L (where a a1 a2), the abscissa

xof the cross section along the beam, the shape of the composite section, the connec-

tion widtha0, the boundary and load condition of the steel beam, and the slab bound-

ary condition [4].

In this paper, the analysis is carried out adopting a very accurate finite element

mesh using shell elements (Fig. 2), assuming a linear elastic behaviour for steel andconcrete. Due to the symmetry only half cross section has been considered.

The connection is modelled using the non-linear elastic law proposed by Aribert

and Al Bitar [9]:

Q Qu(1eC1g)C2 (3)

where Qand gare the shear force and the slip in the connection respectively. Hypo-thesizing to use Nelson connectors with diameter of 19 mm, shear strength Qu

130 kN and constants C1 0.7 mm1, C2 0.8C1 have been assumed.

A large series of beams has been analyzed. Simply supported beams with length

L 5.0 m, ratios a/L 0.02, 0.05, 0.10, 0.20 and 0.30, and cantilever beams withlength L 2.5 m, ratios a/L 0.04, 0.10, 0.20, 0.40 and 0.60, subjected to distrib-

uted or concentrated loads have been considered. Two types of steel beams, the IPE


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Fig. 2. Mesh adopted in Abaqus, referred to half cross section of the composite beam.

300 and IPE 600 Italian profiles, and three types of concrete slab, with heightshc 5, 10 and 15 cm, have been used. Besides these beams typical of buildings,

some bridge composite beams have been studied. Their characteristics are: cantilever

scheme with length L 10 m; ratiosa/L 0.10 and 0.20; steel welded profile with

heighths

1600 mm (30020 mm upperflange, 45030 mm lowerflange, 155015mm web); concrete height hc 20 cm; distributed and concentrated load. For boththe building and bridge beams, it has been assumed: connection width a0 0.9bs,where bs is the width of the upper flange; two lines of reinforcement into the slab,with cover equal to 2 cm and percentage of total reinforcement equal to 1%; mechan-

ical properties of reinforcement: fs 427 N/mm2, ft 667 N/mm

2, esu 27.1%,Es 206 000 N/mm

2; mechanical properties of structural steel: fy 324 N/mm2,

fu 450 N/mm2, eau 28.7%, Ea 206 000 N/mm

2; concrete with strength class

C30/37, according to Eurocode 2 [10].

The analyses performed with rigid connection have shown a good agreement with

the effective widths evaluated according to code [7]. For example, in Fig. 3 a com-parison between the effective width obtained with Abaqus and that proposed by code

[7] is shown. The comparison concerns a cantilever beam of length 2.5 m and ratio

a/L 0.40 constituted by an Italian profile IPE 300 connected to a 15 cm thickconcrete slab subjected to a concentrated load. In this figure, the effective widthratio h ae/a is plotted versus the ratio x/L, where the origin of abscissa x isassumed at the end of the cantilever beam.

The presence of deformable connection instead produces significant variations. InFig. 4 the typical stress distribution in the middle fibre of the concrete slab immedi-ately before and after a stud is plotted for a beam similar to that previously analyzed

but with ratio a/L 0.20. The local effect due to presence of the stud influencesthe effective width evaluation. In Fig. 5 a typical comparison in terms of ratio halong a cantilever beam with the same geometrical characteristics is shown. Since


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Fig. 3. Comparison in terms of effective width between Abaqus modelling and CNR proposal.

Fig. 4. Typical stress distribution in the central fibre of concrete slab near the stud.

the effective width presents a significant variation near the stud, an average valueofh along the stud spacing has been used. Degrees of shear connection N/Nfequalto 0.5, 0.75, 1 and 1.4 according to EC4, together with the case of rigid connection,

have been considered.

The effective width calculated in the hypothesis of rigid connection is signi ficantlylarger than the effective width determined for full or partial deformable connection

in almost the whole beam. The difference between partial or full connection is instead

rather limited. The effective width evaluated considering a full deformable connec-

tion (N/Nf1) may then be suggested for the design of steelconcrete compositebeams on the safe side. Since a simple equation, like that adopted by EC4, is not

able to provide the real trend of the effective width along the beam axis, a tabular


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Fig. 5. Effect of the connection deformability on the effective width.

solution is proposed. In Tables 1 and 2, the proposed average ratios h ae/a arerepresented by italic bold fonts for cantilever and simply supported beams subjected

to concentrated or distributed load. The origin of the abscissa x is at the end and at

the external support for the cantilever and simply supported beam respectively. For

Table 1

Proposed effective widths in terms of ratio for elastic analysis of a cantilever beam

Uniform load End concentrated load

a/L 0.04 0.10 0.20 0.40 0.60 0.04 0.10 0.20 0.40 0.60

x=0.2L 1.00 1.00 1.00 0.55 0.37 1.00 1.00 1.00 0.55 0.370.95 0.76 0.53 0.37 0.23 0.90 0.73 0.48 0.31 0.22

x=0.5L 1.00 1.00 1.00 0.55 0.37 1.00 1.00 1.00 0.55 0.370.94 0.85 0.69 0.55 0.42 0.94 0.85 0.69 0.52 0.37

x=0.8L 1.00 1.00 1.00 0.55 0.37 1.00 1.00 1.00 0.55 0.370.93 0.85 0.71 0.54 0.37 0.95 0.88 0.76 0.59 0.42

x=L 1.00 1.00 1.00 0.55 0.37 1.00 1.00 1.00 0.55 0.370.93 0.83 0.65 0.42 0.29 0.96 0.89 0.77 0.55 0.40

Table 2

Proposed effective widths in terms of ratio for elastic analysis of a simply supported beam

Uniform load Concentrated load

a/L 0.02 0.05 0.10 0.20 0.30 0.02 0.05 0.10 0.20 0.30

x=0.1L 1.00 1.00 1.00 0.55 0.37 1.00 1.00 1.00 0.55 0.370.88 0.71 0.41 0.29 0.19 0.88 0.74 0.48 0.31 0.22

x=0.25L 1.00 1.00 1.00 0.55 0.37 1.00 1.00 1.00 0.55 0.370.94 0.85 0.69 0.50 0.36 0.94 0.85 0.69 0.52 0.37

x=0.50L 1.00 1.00 1.00 0.55 0.37 0.97 1.00 1.00 0.55 0.370.97 0.93 0.84 0.69 0.50 0.96 0.89 0.77 0.55 0.40


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a continuous beam, same values may be adopted substituting the length of the beam

L for the distance between the zero bending moment points 0.

The EC4 solution is also reported in Tables 1 and 2 using normal fonts. The

comparison with the proposed solution highlights how in many cases the deform-ability of the connection can reduce, in a significant way, the effective width for anelastic stress check. This can be noted by Fig. 6, where the proposed values are

compared with those given by EC4 [5] and code [7]. However EC4 proposal gives

correct results for ratios a/L0.4 anda/L0.2 near the support of a cantilever beam

and at the centre of a simply supported beam respectively, where stresses aremaximums.

2.2. Deformability analysis

A series of comparisons has been carried out in order to evaluate the global

deformability of the composite beam in terms of displacements (for example

maximum deflection). These comparisons have been performed on the same beamsanalyzed in the previous section, except for the IPE 600 profile and the slabs withthickness hc 5 and 10 cm. In Table 3 the ratios between the maximum deflectionsevaluated considering (dAbaqus) and neglecting (dElastic) the shear lag phenomenon areshown for a cantilever beam subjected to a concentrated or distributed load. The

dElastic deflection has been evaluated assuming the effective width equal to the slabwidth. The connection has been assumed as rigid.

By these comparisons it may be noted how the hypothesis of an effective widthequal to the slab width gives correct results. Analogous results have been obtained

in the case of deformable connection. However it has to be pointed out how, for a

correct evaluation of the deflection, the connection should be considered as deform-able, since this can imply an increment of deflection, even for full connection, upto about 20%.

3. Effective width for a plastic analysis

For a plastic analysis, actual codes in general provide the same effective widthadopted for a linear elastic analysis. In particular, Eurocode 4 expresses the effective

width for elastic or plastic analysis as a function of the distance 0 between the

points of zero bending moment, considering it constant along these portions of beam.

The effective width is defined as the sum of the lengths:

beff be1 be2 with bei 0/ 8 and beibi (4a,b)

where bi (i 1,2) is the right or left half slab width. For what has been previously

said, this approach represents a strong simplification of the real effective width inplastic phase.

In this section, the most important results of a parametric study performed usingthe Abaqus code are presented. In the analysis, the reinforcement is assumed uni-

formly distributed into the slab and the steel beam has a section of class 1 or 2


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Fig. 6. Comparison between effective widths evaluated according to the actual codes and proposed

values.


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Table 3

Influence of the shear lag phenomenon on the maximum deflection for elastic analyses of cantilever beams

with rigid connection

Concentrated load Distributed load

Type of beam a/L dAbaqusdElastic

dAbaqusdElastic

IPE300 0.04 0.990 0.990

IPE300 0.10 0.990 0.989

IPE300 0.20 0.986 0.986

IPE300 0.40 0.991 0.996

IPE300 0.60 0.989 0.989

Bridge 0.10 0.971 0.957

Bridge 0.20 0.987 0.979

according to EC4. Both reinforcement and steel beam have been modelled through

an elasticplastic with hardening law, characterized by a hardening modulus equalto 1/100th and 1/1000th of the Youngs modulus respectively.

In order to adequately represent the behaviour of the composite beam near the

continuity support or the plastic hinge into the span, a cantilever beam subjected to

positive or negative bending moment in the hypothesis of rigid or deformable con-

nection has been analyzed.Since an accurate modelling of concrete is fundamental for a correct numerical

analysis, the real behaviour in tension and compression and the interaction with the

reinforcement have been introduced. In tensile zones cracking is considered smeared

and effects of the concrete-reinforcement interaction are globally introduced by

means of a softening law after cracking. This approach does not allow the evaluation

of local complex phenomena, such as bond slip and dowel action in the post-crackedzone; nevertheless these phenomena are globally considered in a simple and

adequate way.

The adopted uniaxial response of concrete in compression and tension is shown

in Fig. 7. In compression, the behaviour is considered linear up to 40% of the ultimatestrength. After this limit, for the presence of plastic strains, a non-linear curve is

adopted up to the collapse stress is reached. Due to the confinement effect, a furtherbranch of softening is provided.

In tension, cracking is considered to have occurred when the stress in a generic

point reaches a collapse surface named crack detection surface (Fig. 8). In this waya model with widespread cracking is adopted. This means that cracking does not

coincide with a physical opening of a crack but with a variation of stiffness and

stress state in integration points where the critical value has been reached.

To characterize the behaviour in a post-cracked phase, it is important to refer to

softening laws able to describe the behaviour of reinforced concrete in tension. More-over they should take into account the tension stiffening phenomenon, and at the

same time allow an easy numerical convergence. Thus the Stevens law [11] has


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Fig. 7. Uniaxial stressstrain law adopted by Abaqus for concrete.

Fig. 8. Crack detection surface adopted by Abaqus for concrete.

been adopted, since in a previous work [12] this law showed a good capacity to

simulate the slab behaviour for composite beams.


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3.1. Hogging bending

For hogging bending in plastic phase, a series of composite beams with cross

sections representative of civil or bridge structures has been analyzed. The formerare characterized by: IPE 300 or IPE 400 Italian profile, slab thickness hc 15 cmand length L 2.5 m; the latter by: steel welded profile with height hs 1600 mm(with the same cross section of that described in Section 2.1), slab thickness hc

20 cm and length L 10 m. The longitudinal reinforcement is arranged on two

lines, with cover of 2 cm. The total percentage of longitudinal reinforcement varies

between 0.5 and 1%, and the transversal reinforcement is constituted by stirrups with

diameter f 6 mm or f 8 mm with spacing of 20 cm, according to the type ofbeam. These cantilever beams, characterized by the ratios a/L 0.2, 0.4 and 0.6,

have been subjected to concentrated or uniform loads. Both rigid and deformable

connections between steel beam and concrete slab have been considered. In the

second case the connection has been realized with deformable studs assuring the full

connection (N/Nf1) according to EC4, with the constitutive law given by Eq. (3)

and the values of parameters adopted in Section 2.1. The mechanical properties ofmaterials are those described in Section 2.1.

Since in the collapse phase the slab is cracked, only the behaviour of the reinforce-

ment in tension has been taken into account to determine the effective width. This

is expressed analytically by the equation:

aei 1

sbar max a0

/2

bi

sbardy with i 1,2 (5)

wheresbaris the stress in a generic bar andsbar maxthe maximum stress into the bars.Average results, calculated near the support and expressed in terms ofh ae/a, areshown in Table 4 for the case of concentrated load. Observing these values, it can

be noted that at the collapse, independently of the a/L ratio, almost all bars into the

slab have become plastic for both a rigid and a deformable connection. In particular

for all the cases studied, reinforcement becomes plastic just in a limited length along

the beam axis, about 510 cm, near the support. The maximum required elongation

Table 4

Effective widths near the support in terms of ratio and strains at the collapse for non-linear analysisof cantilever beams subjected to concentrated load and hogging bending moment

Rigid connection Deformable connection Reinforcement elongation

ratio %

b/L

0.20 0.95 0.92 0.50.40 0.94 0.93 1.1

0.60 0.94 0.94 1.8


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in correspondence to the first plasticization of the cross section depends on thea/L ratio according to Table 4, resulting therefore in lower values with respect to

the limits proposed by actual codes.

Differences between effective width evaluated in elastic and plastic phase highlightthat there is not a relation between them. The ratio h ae/a versus the ratioP/Ppl between applied load and theoretical plastic load (evaluated considering

ae a and steel with a rigid plastic behaviour) is plotted in Fig. 9 for a cantilever

beam subjected to concentrated load with length L 2.50 m, ratio a/L 0.60, slab

thickness hc 15 cm, total longitudinal reinforcement percentage of 0.6%, stirrupswith diameterf 6 with spacing of 20 cm, cover of 2 cm, rigid connection and anItalian steel profile IPE 400.

This figure allows a better understanding of both the stress redistribution in thereinforcement and the plastic level reaching into the cross section. It can be seen

how after concrete cracking (point A of Fig. 9), when load increases, there is a

decrement of the effective width up to the beginning of plastic phenomena in the

reinforcement (point B). After this phase the effective width increases in a significantway up to the ultimate load (point C), where the whole reinforcement becomes plas-

tic. Obtained results show that, independently on the type of load (uniform or

concentrated) and for both rigid and deformable connection, near the collapse the

redistribution of stresses in the reinforcement is practically the same. Where the

reinforcement has not become plastic, stresses are however very high (about 90%

of the yielding stress). Thus for a system of parallel beams, an effective width equal

to the distance between them, reducible of 1015% for high (0.40.6) a/L ratios,can be theoretically justified.

3.2. Sagging bending

The same series of cantilever composite beams analyzed in the previous section

has been subjected to sagging bending moment in order to evaluate the ultimatestress distribution when the concrete slab is compressed. For concrete in compression

Fig. 9. Variation of the effective width with loading for hogging bending moment.


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the Saenzs law proposed by Eurocode 2 [10] has been used. Again the type of loaddoes not influence significantly the effective width and the plastic bending momentat the collapse. The value of the ratio h ae/a, obtained as in the elastic phase by

Eq. (1) considering only the stresses into concrete, is shown in Fig. 10 varying theapplied load. This figure refers to the same composite beam studied for hoggingbending, the results of which have been plotted in Fig. 9. It can be observed that

the strong increment of the effective width compared to the elastic phase varying

the applied load. Moreover the ratio h tends to be one when the ultimate load isreached. Once the whole composite beam has become plastic, the maximum strainsinto the concrete slab are in average equal to 0.2, 0.25 and 0.3% for the ratios a/L

= 0.2, 0.4 and 0.6 respectively.For the same beam, the ratio between the stress in the central fibre of concrete slab

and the maximum stress at the collapse, versus the ratio of the transversal coordinate

y to the half width of the slab b, is shown in Fig. 11 varying the level of applied

load. It can be observed how the stress distribution into the slab tends to be practi-

cally constant in the whole slab when the load is close to the ultimate load. This

behaviour is almost independent on the slab width. By a theoretical point of view,

at the collapse it is then possible to employ for the design at least an effective width

equal to 8590% of the slab width even for high (0.40.6) a/L ratios.

4. Conclusions

The numerical study carried out by Abaqus code has demonstrated that the connec-

tion deformability plays an important role on evaluation of the effective width for

stress elastic analysis of steelconcrete composite beams. To this aim a table solution,

Fig. 10. Variation of the effective width with loading for sagging bending moment.


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Fig. 11. Stress distribution into the central fibre of the slab for sagging bending moment at different

loading levels.

which considers this important aspect, has been proposed. For deformability elastic

analysis, an effective width equal to the slab width may instead be adopted.

For the plastic analysis it is not appropriate to adopt the same effective width usedfor stress elastic calculus, since the redistribution of stresses into the slab determines

an enlargement of the effective width. For a plastic analysis these results cannot be

considered as definitive because some parameters that have not been taken intoaccount in this paper could be significant. In order of importance they are: the distri-bution, the quantity and the ductility of steel reinforcement; the presence of the

columns in hogging zones; the class of steel beam section; the number and type ofstuds; the shear collapse of the slab; the type of load (monotonic or high-cycle

loading); the degree of concrete confinement. For these reasons a series of six com-posite beams will be tested at the Civil Engineering Department of the University

of Trieste in order to check the importance of these parameters.At present, a simplified, sufficiently precautionary design criterion for an ultimate

analysis, in the hypotheses of steel beam section of class 1, 2 according to EC4, full

connection and ductile reinforcement, may be obtained for sagging bending zones

by Eqs. (4a,b), and for hogging bending zones by equations:

beff be1 be2 with bei 0/ 4 and beibi (6a,b)

being 0 the distance between the points of zero bending moment, like in EC4. In

this way the proposal of EC4 for sagging bending zones, where ductility resources

of concrete are limited and the collapse may be brittle, is kept. For hogging bending

zones, since with the limitation imposed by Eqs. (6a,b) the required ductile capacityfor steel reinforcement is in general widely satisfied, the EC4 proposal is insteadextended.


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Therefore for a two or three span continuous composite beam, a practically con-

stant effective width along the beam axis may be considered.

This proposal may be acceptable for a plastic analysis in cracked phase without

moment redistribution along the beam. For a plastic analysis with moment redistri-bution, where it is important to evaluate the ultimate rotational capacity of the cross

section, further studies are necessary, especially of experimental type.

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Effective width evaluation for steel–concrete

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