Previous | Next | Contents ESDEP WG 10 COMPOSITE CONSTRUCTION

Lecture 10.3: Single Span Beams

OBJECTIVE/SCOPE
To describe the design of a single span steel-concrete composite beam with full shear connection, using a plastic design method to determine the internal force distribution at ultimate limit state; to describe an approximate method to check the deflection at serviceabilit limit state!

P E E!"ISITES
"ecture #$!%& The 'ehaviour of 'eams

EL#TE$ LECT" ES
"ectures #$!(& Continuous 'eams "ectures #$!)& *esign for +erviceabilit "ectures #$!,& +hear Connection

S"%%# &
This lecture introduces the design criteria for a single span composite beam, concentrating on the determination of its resistance to positive bending moment, to vertical shear, or to a combination of both! - plastic design method is used! The conditions for which this method applies are summarised to show the differences between simpl supported and continuous beams! The design method also assumes that onl s mmetrical steel sections are used and that full shear connection between the steel and concrete exists at ultimate limit state! +pecial attention is paid to the concrete slab acting as the compression flange of the composite beam! The effective width and maximum longitudinal shear force of the concrete slab are defined! The internal force distribution within the cross-section is described! .ormulae based on the distribution are given which determine the moment and shear resistance of the beam! +erviceabilit aspects are also briefl discussed!

1. I'T O$"CTIO'
The ob/ect of this lecture is to explain the principles and rules for the design of a simpl supported, i!e! single span, composite steel-concrete beam with full shear connection! T pical cross-sections of composite beams are shown in .igure #! .or simplicit , onl the s mmetrical steel sections #a, #c and #d are considered! The relevant s mbols are given in .igure %!

.or full shear connection the total longitudinal shear resistance of the shear connectors 01 23, distributed between the point of maximum positive bending moment and a simple end support, must be greater than 0or e2ual to3 the lesser of the resistance of the steel beam 01 s 4 -f 5a3 when the plastic neutral axis is in the slab, or the resistance of the concrete flange 01 c 4 $,6) beff hc fck5c3 when the plastic neutral axis is in the steel section!

1.1 "ltimate Limit State

The resistance to longitudinal shear 0see .igure 7, criterion 8883 of a shear connection is not discussed here! -n idealised load-slip behaviour of the connector, as illustrated in .igure (, is assumed; the t pe of shear connector which exhibits this behaviour is discussed in "ectures #$!,!#, #$!,!% and #$!,!7!

This lecture concentrates on the resistance of the beam to moment and vertical shear, which have maximum values at cross-sections 8 and 88 respectivel , as shown in .igure 73! 'etween these critical cross-sections, each cross-section is sub/ected to a bending moment and a vertical shear! This combination is usuall onl of importance in the case where the loading includes point or line loads, as shown in .igure ); here the maximum moment and maximum vertical shear act together at a critical cross-section ad/acent to the point load or line load; special attention must be paid to this critical cross-section!

8n the case of staticall determinate beams, such as simpl supported single span beams, it is eas to determine the distribution of bending moments from the e2uilibrium conditions! To determine the stress distribution over the cross-section, plastic behaviour is assumed! The advantage of this method is that the calculation of the resistance is based on the 9maximum moment at failure9 condition; this method is also eas to understand and appl ! +teel sections can be classified into ( classes depending on the local buckling behaviour of the flange and5or web in compression! 8n the case of a simpl supported single span, plastic design methods ma be used for Class # and % sections; sections of Class % are onl allowed when no rotation capacit is re2uired! These classes are described as follows 0see also .igure , and "ecture :!%3&

Class #& plastic cross-sections which can form a plastic hinge with sufficient rotation capacit for plastic anal sis! Class %& compact cross-sections which can develop the plastic moment of resistance but have limited rotation capacit !

The steel compression flange, if properl attached to the concrete flange, ma be assumed to be of Class #! Table # 0part of Table (!% of ;urocode ( <#=3, classifies steel webs in compression according to their width to thickness ratios! 8n a composite beam the compression part of the web, in positive bending, is alwa s less than half the total depth for a s mmetrical section! - width-to-thickness ratio less than 67 will, therefore, alwa s be sufficient for a s mmetric steel section in positive bending! Therefore, instabilit of the web is not critical for the 8P;-sections 0according to C;N;N #>-#>6,3 and the ?;-sections 0according to C;N-;N )7-#>6,3! 'ecause the part in compression is alwa s laterall restrained when the beam is in positive bending, it is not necessar to check lateral-torsional buckling 0see "ecture #$!(!# and #$!(!%3! @ther aspects, such as shear buckling are discussed briefl in +ection (! Aeb crippling, however, is be ond the scope of this lecture - see ;urocode 7 for further information <%=!

1.( Ser)icea*ilit+ Limit State

.or simpl supported single spans, the concrete flange is in compression and cracking of the concrete is not relevant! @nl deflections and vibrations are important! "ectures #$!)!# and #$!)!% discuss these topics!

(. $ESI,' #SPECTS O- T.E CO'C ETE -L#',E I' CO%P ESSIO'

(.1 E//ecti)e 0i1t2
- t pical form of composite construction consists of a slab connected to a series of parallel steel members! The construction is essentiall a series of interconnected T-beams with wide, thin flanges, as shown in .igure :0a3! 8n such a s stem 9shear lag9 ma cause the flange width to be not full effective in resisting compression <7=! This phenomenon can be explained b reference to a simpl supported member, part of whose length is shown on plan in .igure :0b3!

The maximum axial force in the slab is at midspan, while the force at the ends is Bero! The change in longitudinal force is associated with shear in the plane of the slab! The resulting deformation, shown in .igure :0b3, is inconsistent with simple bending theor , in which initiall plane sections are assumed to remain plane after bending! The edge regions of the slab are effectivel less stiff, and a non-uniform distribution of longitudinal bending stress is obtained across the section! +imple theor gives an effective value for width, b eff, such that the area C?DE e2uals the area -C*;.! The ratio beff5bv depends not onl on the relative dimensions of the s stem, but also on the t pe of loading, the support conditions and the cross-section considered; .igure :c shows the effect of the ratio of the beam spacing to span length, bv5", and the t pe of loading, on a simpl supported span!

8n most codes of practice ver simple formulae are given for the calculation of effective widths, although this ma lead to some loss of econom ! -ccording to ;urocode ( <#=, for simpl supported beams, the effective width on each side of the steel web should be taken as , but not greater than half the distance to the next ad/acent web, nor greater than the pro/ection of the cantilever slab for edge beams! The length lo is the approximate distance between points of Bero bending moment! 8t is e2ual to the span for simpl supported beams! - constant effective width ma be assumed over the whole of each span! This value ma be taken as the midspan value for a beam!

(.( %a3imum L4ngitu1inal S2ear in t2e C4ncrete Sla*

8n the concrete slab, a complex 0three-dimensional3 force distribution occurs in the region of the connector! The reason for this behaviour is that bending moments and vertical shear forces act parallel as well as perpendicular to the beam! 8t is difficult to find a ph sical design model for this complex stress distribution, and therefore, most design rules are empirical! Two design criteria can be identified&

longitudinal shear in the concrete slab, along the shear planes indicated in .igure 6! splitting of the concrete!

8t is possible to avoid these failure modes b providing sufficient transverse reinforcement and choosing the correct distance between the connectors! 8n some cases, satisf ing these criteria ma lead to an increase in concrete slab thickness or resistance! "ongitudinal shear resistances are given in Chapter , of ;urocode ( <#=! 8f the connectors are welded or shot fired through a continuous profiled steel sheet of a composite slab, the cross-section of the steel sheet can also be considered as transverse reinforcement!

3. $ESI,' C#LC"L#TIO'
There are design performance re2uirements for both the ultimate and serviceabilit limit states! "ltimate Limit State 8n designing a composite beam for the ultimate limit state, it is necessar to check the resistance of the critical cross-sections, and the resistance to longitudinal shear between each ad/acent pair of critical cross-sections 0see .igure 73! The forces and moments due to factored loads are re2uired to be less than the design resistance! This can be expressed b & +d 1d where +d is the design value of an internal force or moment 1d is the corresponding design value of the resistance The design value of an internal force or moment, +d, can be determined when the static s stem, its geometrical data 0when relevant3 and the combination of the design values of the loads are known! Characteristic values for loadings are given in ;urocode #& 'asis of *esign and -ctions in +tructures <(=! To determine +d, for example for criteria 8 of .igure 7, the characteristic permanent and variable 0in this case uniforml distributed3 loads must be multiplied b the corresponding -factors and combined as follows& +d 4 0l%563FC Ck/ G H0Hk# G

H 3I
ki

0#3

which, using the values recommended in ;urocode ( gives& +d 4 0l%563F#,7) Ck/ G #,)$0Hk# G where Ck,/ is the characteristic value of the permanent load Hk,l is the characteristic value of one of the variable loads Hk,i is the characteristic value of the other variable loads! To determine the design resistance, 1d, of members or cross-sections, the design values of the material strengths and geometrical data 0when relevant3 are necessar ! The design value of a material propert represents its lower characteristic value divided b its corresponding partial safet factor; the partial factors for material properties 0and strengths3 are&

H 3I
ki

0%3

Combination

+tructural steel

Concrete

+teel reinforcement

Profiled steel sheeting

.undamental

a 4 #,#

c 4 #,)

s 4 #,#)

ap 4 #,#

@ther J values, such as that for the shear connection 0studs, friction grip bolts etc!3 are given in ;urocode ( <#=! The use of these material factors in determining design resistances is shown in +ection (, ;2uations 073 to 0>3, for the case of moment resistance, ie! criterion 8 of .igure 7! Ser)icea*ilit+ Limit State 8n the design of a composite beam for the serviceabilit limit state, it must be shown that, under service conditions, the deflections and vibrations do not exceed allowable values and that cracking of the concrete is limited! The design value of the effect of loads ; d shall be less than 0or e2ual to3 a nominal value Cd 0or a related function 1d3& ;d Cd or, ;d 1d This aspect of design is discussed in greater detail in "ecture #$!)!#!

5. PL#STIC $ESI,' %ET.O$

5.1 P4siti)e Ben1ing %4ment
The ultimate load resistance of a simpl supported beam is determined b the moment of resistance of the critical cross-section <)=! The determination of the moment of resistance of the cross-section is based on the following assumptions& a! The shear connectors are able to transfer the forces occurring between the steel and the concrete at failure 0full shear connection3! b! No slip occurs between the steel and the concrete 0complete interaction3! c! Tension in concrete is neglected! d! The strains caused b bending are directl proportional to the distance from the neutral axis; in other words, plane cross-sections remain plane after bending, even at failure!

e! The relationship between the stress a, and the strain a of steel is schematicall represented b the diagram shown in .igure >a!

f! The relation between the stress c and the strain c of concrete is schematicall represented b the diagram shown in .igure >b! 'oth materials are assumed to behave in a perfectl plastic manner, and therefore, the strains are not limited! This assumption is similar to that made when calculating the plastic moment resistance for Class # steel sections used independentl ! The idealised diagram for steel is shown in .igure >a! The deviation between the real and the idealised diagram is much smaller than for concrete as shown in .igure >b! The use of fck for the maximum stress in the concrete will clearl result in an unconservative design although in practice the overestimate does not appear to be ver significant! To allow for this overestimate a conservative approximation for concrete strength 0kfck3 is used in design! ;xperimental research has proved that the plastic method with k 4 $,6), leads to a safe value for the moment of resistance! This is onl true if the upper flange cross-section is less than or e2ual to that of the lower flange, as will usuall be the case! -pplication of these assumptions leads to the stress distributions shown in .igures #$ - #%! Clearl , the calculation of the moment of resistance J c is dependent on the position of the neutral axis, which is determined b the relationship between the cross-section of the concrete slab and the cross-section of the steel beam! Two cases can be identified as follows& a! the neutral axis is situated in the concrete slab& #! in the solid part of the composite slab 01s K 1c; see .igure #$3

%! in the rib of the composite slab 01s4 1c3

b! the neutral axis is situated in the steel beam& #! in the flange of the steel section 01s L 1c L 1w; see .igure ##3 %! in the web of the steel section 01s L 1c K 1w; see .igure #73

The plastic moment resistance, assuming full shear connection and a s mmetric steel section, is expressed in terms of the resistance of various elements of the beam as follows& 1esistance of concrete flange & 1c 4 beff hc $,6) fck 5c 1esistance of steel flange & 1f 4 b tf f 5a 1esistance of shear connection & 12 4 N H 1esistance of steel beam & 1s 4 - f 5a 1esistance of clear web depth & 1v 4 d tw f 5a 1esistance of overall web depth & 1w 4 1s - % 1f where - is the area of steel beam b is the breadth of steel flange beff is the effective breadth of concrete flange

h is the overall depth of the steel beam hp is the depth of profiled steel sheet hc is the depth of concrete flange above upper flange of profiled steel sheet d is the clear depth of web between fillets fck is the characteristic c linder compressive strength of the concrete Jpl is the plastic moment resistance of steel beam N is the number of shear connectors in shear span length between two critical cross-sections H is the resistance of one shear connector tf is the thickness of steel flange tw is the thickness of web is .ull shear connection applies when 12 is greater than 0or e2ual to3 the lesser of 1c and 1s! The concrete flange is assumed to be a solid concrete slab, or a composite slab with profiled steel sheets running perpendicular to the beam! The ;2uations are conservative for a composite slab where the profiled steel sheets run parallel to the beam because in the resistance 1 c, the concrete in the ribs is neglected! .or a composite section with full shear connection, where the steel beam has e2ual flanges, the plastic moment resistance Jc for positive moments is given b the following& Case a#& 8f the neutral axis is situated in the concrete flange as shown in .igure #$, 1 s K 1c and the positive bending moment of resistance is& Jpl!1d 4 1s B where& B 4 h5% G hp G hc - x5% x 4 0-f 5a3 5 0beff kfck 5c3!hc 4 01s51c 3!hc Jpl!1d 4 1c 0h5% G hp G hc - 1s!hc5%1c3 073

Case a%& 8f the neutral axis is situated in the rib of the composite slab, 1 s 4 1c and ;2uation 073 can be rewritten as& Jpl!1d 4 1s 0h G %hp G hc35% or, Jpl!1d 4 1s!h5%G 1c!0hc5% G hp3 Case b#& 8f the neutral axis is situated in the steel flange, 1s L 1c L 1w! .rom e2uilibrium of normal forces it can be shown that the axial compression force 1 in the steel flange 0see .igure ##3 is& 1c G 1 4 1s - % 1 G 1 % 1 4 1s - 1c 1 4 01s - 1c35% This axial force 1 is located in the middle of the upper part of the flange, with a depth e2ual to& 01tf351f 4 01s - 1c3!tf5%1f ! Therefore, the moment of resistance is e2ual to the resistance expressed b the ;2uation 0(3 minus 0%13M01s - 1c3!tf5%1f e2ual to 01s-1c3%!tf5(1f as illustrated in .igure ##! This can be written as& Jpl!1d 4 1s!h5%G 1c 0hc5% G hp3 - 01s - 1c3%!tf5(1f Case b%& 8f the neutral axis is in the web of the steel section, 1s L 1c K 1w! 8n this case, a part of the web is in compression and, as alread discussed, this could influence the classification of the web! Aebs not full effective 09non-compact webs93 are not treated in this lecture! 8f the depth to thickness ratio of the web of a steel section is less than or e2ual to 67 50#-1c51v3 where 4 , it is considered as a compact web and the total depth is effective! The positive bending resistance is as illustrated in .igure #%& 0)3 0(3

Jpl!1d 4 1c B G Jpl,N-red!1d 4 1c!0h G % hp G hc35%G Jpl,N-red!1d 0,3 where& Jpl,N-red!1d 4 plastic moment resistance of the steel beam reduced b a normal force 1 c! -ccording to ;urocode 7 <%= the plastic moment reduced b a normal force for standard rolled 8 and ? steel sections, can be approximated b & Jpl,N-red!1d 4 #,## Jpl!a!1d 0# - 1c51s3 Jpl!1d 0:3 +o the resistance can be written as& Jpl!1d 4 1c!0h G % hp G hc35% G #,## Jpl!a!1d 0# - 1c51s3 063 Jpl,N-red!1d can also be written as Jpl!a!1d - 01c%51v%30d5(3 8n this case the moment of resistance is& Jpl!1d 4 1c!0h G % hp G hc35% G Jpl!a!1d - 01c%51v%30d5(3 0>3 The formulae for the positive moment of resistance values are summarised in Table %!

5.( Vertical S2ear

Cross-section 88 of .igure 7 is onl sub/ected to vertical shear! The contribution of the concrete slab to the resistance to vertical shear is small and difficult to determine and is, therefore, neglected! Therefore, onl the web of the steel section and ad/acent parts of the steel flange are taken into account! The vertical shear resistance, according to ;urocode 7 <%=, is given b & Npl,1d 4 -v f 5(a73 0#$3

The shear area -v, for rolled 8, ? and channel sections loaded parallel to the web, can be taken as& #,$( h tw! 8n addition, the shear buckling resistance of a steel web must be verified when d5t wL,> for an unstiffened 0and uncased3 web! .or a simpl -supported beam, without intermediate transverse stiffeners, with full shear connection and sub/ected to uniforml distributed loading, ;urocode ( <#= gives the following simplified rules& for
w

#,) N1d 4 Npl,1d

w

0##3
w

for #,) K for 7,$ K

K 7,$ N1d 4 Npl,1d 075 K (,$ N1d 4 Npl,1d $,>5

G #5)
w

- #,73

0#%3 0#73
w

8n accordance with ;urocode 7 the web slenderness

w

is given b &

4 Ff 57crI#5% ( 0#(3

8n practice, an 8-section girder has usuall a transverse load bearing stiffener at the support, but no intermediate transverse stiffeners! 8n such a case the elastic critical shear resistance cr is given b &

cr 4

0#)3

8f the factored internal force N+d, is less than Ncr 4 d tw cr, the shear connectors can be uniforml distributed; if not, more connectors should be placed near the support!

5.3 Vertical S2ear in C4m*inati4n 0it2 Ben1ing %4ment

Ahere the vertical shear N+d exceeds half the plastic shear resistance N pl!1d given b ;2uation 0#$3 due allowance shall be made for its effect on the plastic moment of resistance! This is of

importance at cross-section 08G883 of .igure ), where both load effects, vertical shear and bending, are at a maximum! The following methods are used for such cases&

8f the neutral axis of the composite beam is situated in the concrete slab or in the flange of the steel section, the previous ;2uations 073 - 0)3 can be used where 1s is replaced b the reduced resistance of the steel beam, such that&

1s,red 4 1s - 0%N+d5N1d - #3 Npl!1d for& $,) K N+d5N1d # 0#,3 8n this case, part of the middle of the web of the steel section, is reserved to resist the vertical shear! The depth of this part of the web is 0%N+d5N1d - #3 h providing a depth e2ual to Bero when N+d5N1d 4 $!) and e2ual to h where N+d5N1d 4 #,$

8f the neutral axis of the composite beam is situated in the web of the steel section, the previous ;2uations 063 and 0>3 can be used where Jpl!1d is replaced b a reduced plastic moment of resistance; according to ;urocode 7 <%= this resistance should be calculated from the stress distribution given in .igure #7 as briefl discussed below!

The part of the web that is reserved for vertical shear is located in the middle of the web depth! The section modulus of the web will, therefore, be reduced b & Otw 0 0%N+d5N1d - #3 h3% for $,) K N+d5N1d # 0#:3 which becomes& Otw h% 0# - 0%N+d5N1d - #3% 3 0#63 8f the factor is assumed to be e2ual to 0%N+d5N1d - #3% the section modulus of the web can also be written as& Otw h% 0# - 3 8n other words, it is possible to take into account the vertical shear b reducing the design ield stress in the web b a factor 0# - 3 for the area -v 0# - do5d3 where do is the depth of web neglected when calculating Jpl!1d! The plastic moment of resistance reduced b vertical shear can be expressed approximatel b & Jpl,N-red!1d 4 F# - 0# - do5d3!-50%- - -v3I!Jpl!1d 0#>3

These rules are shown in .igure #( in which Jpl!1d is the plastic moment of resistance when 4 #!

6. CO'CL"$I', S"%%# &

Plastic anal sis of the cross-section is used to determine the positive bending moment resistance of composite beams 0assuming that either Class # or % sections are used3! ;urocode ( gives simple formulae for the effective width of concrete slab acting compositel with the steel; this 9concrete flange9 must be detailed to avoid longitudinal shear and splitting! *esign of composite beams involves ensuring that the forces and moments due to factored loads are less than the corresponding design resistance! Narious expressions for the design positive moment of resistance can be derived; these depend on the position of the neutral axis relative to the concrete slab, steel flanges, etc! ;urocode ( gives simplified rules governing the design shear resistance of simpl supported composite beams, with full shear connection, sub/ect to uniforml distributed loading! Ahere high shear and moment are coincident part of the steel web is reserved to carr shear, resulting in a decrease in moment resistance!

7. E-E E'CES
<#= ;urocode (& 9*esign of Composite +teel and Concrete +tructures9& ;NN#>>(-#-#& Part #!#& Ceneral rules and rules for buildings, C;N 0in press3! <%= ;urocode 7& 9*esign of +teel +tructures9& ;NN#>>7-#-#& Part #!#& Ceneral rules and rules for buildings, C;N, #>>%! <7= *owling, P! D!, Enowles, @wens, C!, 9+tructural +teel *esign9, +teel Construction 8nstitute, #>66 0PE3! <(= ;urocode #& 9'asis of *esign and -ctions on +tructures9, C;N 0in preparation3!

<)= +tark, D!A!'!, van ?ove, '! A! ;!, 9Composite +teel and Concrete 'eams with Partial +hear Connection9, ?;1@N, TN@-'uilding and Construction 1esearch5TP-*elft publication, %nd 2uarter #>>$ 0PE3!

8. #$$ITIO'#L E#$I',
#! ;ssentials of ;C(, prepared b the ;CC+-TC ##, 9Composite +tructures9 will be published earl #>>7 b the ;uropean Convention for Constructional +teelwork, 'russels! %! 'ode, 9Nerbundbau, Eonstruktion und 'erechnung9, Aerner Nerlag, #>6: 0Cerman 3! 7! +tark, D! A! '!; 98ntroduction of ;urocode (, Ceneral Jethods of *esign of Composite Construction9, 8-'+; +hort Course, 'russels, +eptember #>>$, 8-'+; 1eport No! ,#, 8+'N 7-6):(6-$,%-%! (! NerbundtrQgen im ?ochbau! Theoretische Crundlagen, 'eispiele, 'emessungstabellen, +chweinerische Rentralstelle fSr +tahlbau, 'ericht -7, #>6%! )! Nara anan, 1! 0ed3, Composite +teel +tructures! -dvances in *esign and Construction! Proceedings of the 8nternational Conference, Cardiff, PE, Dul #>6:! ;lsevier -pplied +cience, "ondon!
Ta*le 1 %a3imum 0i1t29t49t2ic:ness rati4s /4r steel 0e*s

Ta*le ( P4siti)e m4ment 4/ resistance )alues

1s K 1c

1s 4 1c

1s L 1c L 1w

1s L 1c K 1w

@1& and compact webs

Previous | Next | Contents