influence of slab-beam slip on the deflection of com...

475

International Journal for Restoration of Buildingsand MonumentsVol. 9, No 5, 475–490 (2003)

Influence of Slab-Beam Slip on the Deflection of Com- posite Beams

P. Gelfi and E. GiurianiDipartimento di ingegneria Civile, Università di Brescia, Brescia, Italy.

Abstract

The composite construction technique, widely used for steel-concrete buildingfloors, is now proposed also for wooden floors, not only for the rehabilitation ofancient buildings, but also for new constructions. In wood-concrete beams the col-laboration of the slab improves dramatically the stiffness, solving the principaldeficiency of wooden floors, that is the excessive deformation. Unfortunately thisimprovement can be compromised by an excessive deformability of the connectionbetween beam and slab, so that a design approach based on the deformation con-trol should be preferable to that based on the resistance concept. This approachwas difficult to adopt for practice owing to the complex numerical calculation pre-sented in literature.In the paper an exact relationship between the slope increment of the beam end andthe maximum slip at the support is presented, which makes possible an easy andaccurate evaluation of the beam deflection increment. This relationship is validatedboth by numerical and experimental results.

A design procedure based on deformation control, using practical approximatedrelationships for the evaluation of the deflection increment, is also presented.Keywords: Composite beams, wood-concrete beams, concrete-timber structures, slab-beam slip,

connection deformation

P. Gelfi and E. Giuriani

476

Die Rolle des Schlupfes zwischen Platte und Balken bei der Durchbiegung von Verbundbalken

Zusammenfassung

Die Verbundbauweise wurde oft zur Herstellung von Stahl-Beton Verbunddeckenim Hochbau verwendet. Heute wird die Verbundbauweise auch für Holzdeckenvorgeschlagen und zwar nicht nur bei der Instandsetzung alter Gebäude, sondernebenfalls für neue Tragwerke. In einem Verbundträger aus Holz und Beton verbes-sert die tragende Mitwirkung der Platte die Steifigkeit dramatisch. Damit wird diehauptsächliche Schwäche von Holzdecken, nämlich die außerordentliche Verfor-mung, weitgehend behoben. Diese Verbesserung kommt allerdings bei zu hoherVerformbarkeit der Verbindung zwischen Balken und Platte nicht voll zum Tragen.Deshalb sollte bei der Berechnungsmethode eher auf die Begrenzung der Verfor-mungen als auf die Festigkeit geachtet werden. In der Praxis wird dieser Ansatznur zögernd angenommen wegen der in der Literatur beschriebenen komplexennumerischen Berechnungen.

In diesem Beitrag wird eine genaue Beziehung zwischen der Steigerung der Nei-gung am Balkenende und dem maximalen Schlupf am Auflager abgeleitet.Dadurch wird es möglich, die Durchbiegung des Balkens leicht und genau zuberechnen. Diese Beziehung wird sowohl durch den Vergleich mit numerischenBerechnungen als auch mit Versuchsergebnissen validiert. Außerdem wird einBerechnungsverfahren, das auf dem Nachweis der Verformungen beruht und beidem praktische Näherungsansätze für die Bestimmung der Steigerung der Durch-biegung verwendet werden, vorgestellt. Stichwörter: Verbundbalken, Holz-Beton-Balken, Beton-Holz-Konstruktionen, Schlupf zwischen

Platte und Balken, Verbindungsverschiebung.

Influence of Slab-Beam Slip on the Deflection of Composite Beams

477

1 Introduction

Wooden floors have again become proposable, not only for the rehabilitation ofancient buildings, but also for new constructions, thanks to the composite con-struction technique ([1], [2], [3]). The use of wood-concrete floors constitutes avalid and interesting alternative to the classical steel-concrete solution. Thewood-concrete composite technique solves some evident deficiencies of ancientwooden floors, such as excessive deformation under service loads and poor soundinsulation. A thin collaborating concrete slab gives a great increase of flexuralstiffness, reduces noise, constitutes a fire barrier and provides the horizontal dia-phragm which improves the building performance under seismic actions.In general the composite beam technique, both for steel-concrete and wood-con-crete, gives static advantages because of the increases in bearing capacity but mostof all because of the remarkable improvement in flexural stiffness. Unfortunatelythis improvement can be compromised by an excessive deformability of the con-nection between beam and slab, especially in wood-concrete beams, where the col-laboration of the concrete slab gives a so great increase of flexural stiffness. For thisreason, instead of the usual approach based on the resistance concept ([4], [5]), adesign approach based on the connection deformability control, as proposed in [3]and [6], should be preferable.

This approach was difficult to adopt for practice, because of the complex numericalcalculations presented in literature ([7], [8], [9]) for the evaluation of the deflectionincrement due to the connection deformability.

A very simple approximated relationship between the deflection increment and themaximum connection slip, proposed in [3], has made this design approach afforda-ble. In particular this target was reached through an approximated relationship bet-ween the slip and the beam slope increment at the supports.

The aim of this paper is to perfect this approach achieving an exact formulation forthe above relationship. A design procedure for the connection is also presented.

2 Slip-Deflection Relationship in Simply Supported Beams

The deflection of a simply supported beam under service loads is increased by theslip between slab and beam, due to the connection deformation. For a refined eva-luation of the deflection, the non linear behaviour of the connection should betaken into account [8], [9], [10]. Complex numerical approaches are usually notjustified for the design of ordinary structures. As already mentioned, an analyticalformulation giving the deflection increase due to the slip is very useful for designpractice. For this purpose the theoretical approach proposed in [8] is here adopted.The approach is based on the assumption that both slab and beam remain in theelastic range, while for the connection the non linear behaviour is assumed.


478

Bending moments M, Mb and Ms, which act respectively on the whole compositesection, on the beam and on the slab, are bound by the following relationship:

(1)

being Nb the axial force applied in the beam section centroid (equal to the axialforce Ns applied in the slab section centroid) and dc the distance between slab andbeam centroids (Fig. 1).

Assuming that slip δ occurs without uplifting, the beam and the slab have the samecurvature. Therefore moments Mb and Ms are related to curvature y” and to beamand slab flexural stiffnesses EbIb, EsIs:

(2)

being Eb, Es the Young modula and Ib, Is the second moments of area. Substitutingeqs. (2) into eq. (1) we obtain:

from which:

(3)

being I*= Ib+Is/n with n = Eb/Es.

Curvature y” can be expressed as curvature y”id given by the classical no-slip theoryfor composite sections plus increment ∆y” due to the slip:

(4)

Figure 1: Bending moments and axial forces acting on slab and beam

cbsb dNMMM ++=

"" yIEMyIEM sssbbb −=−=

cbssbb dNyIEIEM ++−= ")(

**"IEdN

IEM

IEIEdNMy

b

cb

bssbb

cb +−=++−

=

""" yyy id ∆+=


479

Substituting eq. (3) into eq. (4), the curvature increment ∆y” due to the slip beco-mes:

(5)

Curvature y”id can be expressed using eq. (3):

(6)

being Nb,id the axial force, given by the no-slip theory, acting in the centroid of thebeam, equal to that acting in the centroid of the slab.

Curvature increment ∆y” is then obtained substituting eq. (6) into eq. (5):

(7)

being:

In the generic section of coordinate x the relationship between slip δ(x) and slopey’(x) can be expressed in the following way (see Fig. 2):

(8)

Figure 2: Components of the slip due to axial and flexural deformations

idcb

b

b

ydIE

NIE

My "" ** −+−=∆

cb

idb

bid d

IEN

IEMy *

,*" +−=

cb

cb

idb

bc

b

b

b

dIENd

IEN

IEMd

IEN

IEMy **

,***" ∆

=−++−=∆

idbb NNN ,−=∆

csb dxyxuxux )(')()()( 21 +−=+= δδδ


480

being ub (x) and us (x) the longitudinal displacements respectively of the beam andslab centroids. The derivative of δ(x) gives:

(9)

Eq. (9) can be rewritten as follows:

(10)

being:

With the position:

(11)

eq. (10) becomes:

(12)

Substituting the curvature given by eq. (4), eq. (12) becomes:

(13)

Since Nb (x) = Nb,id (x) + ∆N (x), eq. (13) can be rewritten as follows:

(14)

Note that the no-slip theory implies that δ (x) = 0 and δ’(x) = 0 in every section; thusthe sum of the terms of eq. (14) concerning this theory:

csb dxyxuxux )(")(')(')(' +−=δ

css

bb

bb

bc

ss

s

bb

b dxyAEAE

AExNdy

AExN

AExNx )("1)(")()()(' +⎟⎟

⎠

⎞⎜⎜⎝

⎛+=+−=δ

)()(;)()()(';)()()(' xNxNAExNxxu

AExNxxu bs

ss

sss

bb

bbb −===== εε

nAA

AEAE

s

b

ss

bb

/11 +=+=χ

χδbb

bc AE

xNdxyx )()(")(' +=

χδbb

bccid AE

xNdxydxyx )()(")(")(' +∆+=

χχδbbbb

idbccid AE

xNAE

xNdxydxyx )()(

)(")(")(' , ∆++∆+=

χbb

idbcid AE

xNdxy

)()(" ,+


481

has to be null. Therefore eq. (14) reduces to:

(15)

Substituting into eq. (15) the expression of ∆N given by eq. (7) we obtain:

from which:

(16)

being:

Taking into consideration position (11), distance d* becomes:

(17)

Introducing the expression of the first moment of area Ss,id of the slab transformedarea with respect to the neutral axis of the composite section (see appendix):

(18)

eq. (17) becomes:

(19)

χδbb

c AExNdxyx )()(")(' ∆

+∆=

⎟⎟⎠

⎞⎜⎜⎝

⎛+∆=

∆+∆= χχδ

cbc

cbb

bc dA

Idxyd

xyAEIEdxyx

**

)(")(")(")('

**)(')(')("

dx

dAId

xxy

cbc

δ

χ

δ=

+=∆

χcb

c dAIdd

** +=

nAAdAnAIdd

sbc

bsc /

)/(** +

+=

nAAnAAdS

sb

sbcids /

/, +

=

idsc S

Idd,

** +=


482

Considering the expression of the second moment of area Iid of the transformed sec-tion given in appendix:

(20)

distance d* assumes the simple expression:

(21)

Note that the value of distance d* is very close to that of the section lever arm andbecomes the same when the neutral axis of the composite section is placed at theconnection interface.

By integration of eq. (16), the slope increment due to the slip becomes:

(22)

where C1 is an arbitrary constant to be determined by means of the boundary con-ditions.

In the case of a simply supported beam under symmetrical loads, in the mid spansection both slip δ and slope increment ∆ϕ are null. Therefore, being C1=0, eq. (22)gives the slope increment at the support as a function of the maximum slip:

(23)

3 Proposal for a Design Procedure

As above mentioned the here proposed design procedure for the connection is basedon the assumption of an acceptable value of the maximum slip. The presentation ofthis procedure requires some preliminary considerations.

Usually the connection design is based on the resistance concept, while the effectof the connection deformation is checked afterwards. Stud spacing ‘a’ is calculatedas the ratio between the connector resistance VRd and the longitudinal shear flow q(a = VRd /q). The value and the distribution of the shear flow q can be approximated

cidsid dSII ,* +=

ids

id

SId

,

* =

1*)()(')( C

dxxyx +=∆=∆

δϕ

*)0()0(')0(

dy δϕ =∆=∆


483

to those given by the classical no-slip theory, which provides an upper bound valueof the maximum shear flow ([7], [8]).

In the present approach the value of VRd is not based on the connector ultimate resi-stance, but on the maximum design slip δd. Since under service loads the connectionworks in the elastic range, VRd can be obtained as a function of δd by the followinglinear relationship:

VRd = Ks δd

where, for the connector stiffness Ks, the following theoretical expression, proposedin [6], can be used:

(24)

being d (mm) the stud diameter and t (mm) the gap between concrete slab and woo-den beam, which corresponds to the thickness of an interposed plank. This relati-onship is valid for ordinary steel studs having a diameter of 12/16 mm and a lengthof about 7 diameters.

To determine the value δd of the maximum acceptable slip, the relationship betweenδ and deflection increment ∆v is needed. This relationship can be obtained startingfrom eq. (23) which gives δ as a function of the slope increment ∆ϕ of the beam atthe supports:

(25)

and considering that the slope increment ∆ϕ can be related to the deflection incre-ment ∆v through the expression:

(26)

being L the beam span. Numerical results show that the value of α is nearly constant(see table 1); for simply supported beams under uniformly distributed load it is veryclose to the well known value:

(27)

given by the classical no-slip theory.

]/[)/34.4(

1240003 mmN

dtdK s +

=

*dϕδ ∆=

L/v∆=∆ αϕ

3.2v/ == Lϕα


484

4 Design Remarks

For composite beams of common building floors, a maximum slip of about 0.3mm represents a proposable value for design to satisfy the requirements of the ser-vice limit state. As a matter of fact the deflection increment given by this slip valueis negligible with respect to the deflection limit suggested by codes. This assertionis valid for both wood-concrete and steel-concrete floor beams.For example, for the two beams of Fig. 3, the deflection increment ∆v under serviceloads ranges between the following values (eqs. 23, 26 and 27):

(28)

having introduced the values d* shown in Fig. 3 and δd = 0.3 mm. This incrementis of one order of magnitude less than the usually accepted deformation limit (v/L< 1/300 - 1/500).

Another significant relationship between deflection increment ∆v and maximumslip δ, particularly useful for design practice, can be obtained from eq. (28):

(29)

This relationship is obtained substituting into eq. (28) the usual value of the ratiobetween span and beam depth L/h=20 and setting .

In the case of the suggested design slip value δd = 0.3 mm, the deflection incrementis about ∆v = 3 mm.

Figure 3: Geometrical and mechanical characteristics of two typical floor compositebeams

3* 1064.0/56.0

2.3v −⋅==

∆=

∆dL

dδαϕ

dδ10v =∆

3/2* hd ≅


485

5 Comparison of Numerical and Experimental Results

The relationship between slope increment ∆ϕ and maximum slip δ of a simply sup-ported beam under symmetrical loads (∆ϕ=δ/d* eq. 23) is verified both by numeri-cal analysis and by experimental results. Furthermore the approximated expressionof the deflection increment (∆v=∆ϕL /α eq. 26) is checked.The numerical analysis is carried out assuming the same geometry (Fig. 4) andmechanical characteristics (Fig. 3a) of the experimental composite beam which wastested in [11]. In the numerical model, wood and concrete are assumed as elasticpure bending beam elements (Bernoulli hypothesis). The shear deformation is notconsidered according to the assumption adopted in the present theoretical approach;this simplification is acceptable for the slender beams here investigated. For theconnection a non-linear behaviour is assumed: Fig. 5 shows the model used to sche-matize the studs and the ideal force-slip curve adopted for the non-linear springs.Ideal studs are spaced as in the experimental beam.

Figure 4: Geometrical characteristics of the tested composite beam [11]

Figure 5: Stud modellisation


486

In Table 1 the “exact” numerical values of δ and ∆ϕ are reported, up to six times theservice load. The ratio δ/∆ϕ is exactly constant, confirming the theoretical valuegiven by eq. (21): (d*=Iid/Ss,id=147.7 mm).

In the same table the “exact” numerical deflection v is listed as well as the idealdeflection vid given by the classical no-slip theory. The “exact” deflection incre-ment ∆v=v-vid is compared with the approximated value ∆v* of eq. (28): diffe-rences are small, less than 9%, and become negligible (less than 3%) in terms oftotal deflection.

Coefficient α = ∆ϕL/∆v (eq. 26) does not differ significantly from the proposedconstant value α = 3.2.

In Fig. 6 a) the experimental (Ref. [11]) and theoretical load-deflection curves arecompared. The theoretical values are obtained adding the value of the deflectionvid of the classical no-slip theory to the deflection increment ∆v* calculated intro-ducing into eq. (28) the experimental maximum slip values δ1 of Fig. 6b. Theore-tical values fit the experimental data very well.Table 1: Numerical analysis results.

Load factor

δ (mm)

∆ϕ (103rad)

d*=δ/∆ϕ (mm)

v (mm)

vid (mm)

∆v=v-vid (mm)

∆v*=δL/(d*3.2)(eq. 28)

(∆v*-∆v)/∆v (%)

α=∆ϕL/∆v(eq. 26)

0.5 0.088 0.595 147.7 2.95 2.14 0.81 0.74 -8.3% 2,93

service 1.0 0.176 1.190 147.7 5.91 4.29 1.62 1.49 -8.3% 2,93

1.5 0.272 1.843 147.7 8.91 6.43 2.48 2.30 -7.2% 2,97

2.0 0.378 2.562 147.7 12.01 8.58 3.44 3.20 -6.8% 2,98

2.5 0.518 3.505 147.7 15.32 10.72 4.61 4.38 -4.9% 3,04

3.0 0.691 4.682 147.7 18.89 12.86 6.03 5.85 -2.9% 3,11

3.5 0.960 6.498 147.7 23.19 15.01 8.18 8.12 -0.7% 3,18

4.0 1.346 9.111 147.7 28.41 17.15 11.26 11.39 1.2% 3,24

4.5 1.822 12.336 147.7 34.49 19.29 15.19 15.42 1.5% 3,25

5.0 2.401 16.254 147.7 41.39 21.44 19.96 20.32 1.8% 3,26

5.5 3.143 21.280 147.7 49.67 23.58 26.09 26.60 1.9% 3,26

6.0 3.929 26.605 147.7 58.44 25.73 32.71 33.26 1.7% 3,25


487

6 Concluding Remarks

slab and the beam. In the paper an exact relationship between the slopeincrement and the maximum slip at supports is presented, which makeseasy and accurate the evaluation of the beam deflection increment. Thisrelationship is validated both by numerical and experimental results.

- For a correct design of wood-concrete composite floors, the connectionmaximum slip has to be very small (δd < 0.3 mm) in order to limit thedeflection increment.

- A roughly approximated, but very useful for design practice, relation-ship between the deflection increment ∆v and the maximum slip δ isalso presented (∆v ≅ 10 δ).

Figure 6: Experimental and theoretical load-deflection curves (a) and experimental maximum slip (b)

The following concluding remarks can be expressed:- The composite beam deflection is influenced by the slip between the


488

References

1. M. Piazza, G. Turrini, Una tecnica di recupero statico dei solai in legno,Recuperare, 5 and 6 (1983)

2. A. Ceccotti, Composite concrete-timber structures, Progr. Struct. Engng.Mater., 4, 264-275 (2002).

3. E. Giuriani, Solai in legno rinforzati con lastra collaborante. Criteri proget-tuali, L’Edilizia, 4, 32-40 (2002)

4. European Committee for Standardization (CEN), Eurocode 4: Design ofcomposite steel and concrete structures, Part 1-1: General rules and rulesfor buildings, Brussel, (1994)

5. European Committee for Standardization (CEN), Eurocode 5: Design of Tim-ber Structures, Part 1-1: General rules and rules for buildings, Brussel(1993).

6. P. Gelfi, E. Giuriani and A. Marini, Stud Shear Connection Design for Com-posite Concrete Slab and Wood Beams, ASCE Journal of Structural Enginee-ring, 128, 1544-1550 (2002)

7. N. M. Newmark, I. M. Viest, Tests and Analysis of Composite Beams withIncomplete Interaction, Proc. Society for Experimental Stress Analysis, 9, n.1 (1951)

8. E. Giuriani, Comportamento delle sezioni miste in acciaio e calcestruzzo conconnettori deformabili, Studi e Ricerche, 5, Corso di Perfezionamento per leCostruzioni in Cemento Armato F.lli Pesenti, Politecnico di Milano (1983)

9. A. Gubana, Un approccio analitico per il calcolo delle deformazioni di travimiste con connessioni a comportamento non lineare, II Workshop Italianosulle Strutture Composte, Napoli (1996)

10. N. Gattesco, E. Giuriani, Experimental study on stud shear connectors sub-jected to cyclic loading, J. Construct Steel Res., 38 (1996)

11. P. Gelfi, E. Giuriani, Behaviour of stud connectors in wood-concrete compo-site beams, Structural Studies, Repair and Maintenance of Historical Buil-dings VI: Proceedings of the sixth international conference (Stremah 99),Dresden, Germany, Wit Press, 565-578 (1999)


489

Appendix

With reference to Fig. A1, expression eq. (18) of the first moment of area Ss,id ofthe slab transformed area, with respect to the centroidal axis of the composite sec-tion, can be obtained as follows:

(a1)

(a2)

(a3)

Expression eq. (20) of the second moment of area Iid of the transformed section,according to the classical no-slip theory, used in [8], can be obtained as follows:

(a4)

Figure A1: Composite section geometrical characteristics

ssids enAS ⋅= /,

nAAAda

nAAanAadAaye

sb

bc

sb

sCbCs //

/)(+

=−+

⋅++=−=

nAAnAAdS

sb

sbcids /

/, +

=

22*22 /// ssbbssbbsbid enAeAIenAeAnIII ⋅++=⋅+++=


(a5)

(a6)

(a7)

Prof. Piero Gelfi is Assistant Professor at the Civil Engrg.Dept. of the University of Brescia, Italy, where he teachesTheory and Design of Steel Structures. His research interestsinclude structural rehabilitation, r.c. structures, semi-rigidsteel joints, composite structures. [email protected]

Prof. Ezio Giuriani is Professor at the Civil Engrg. Dept. ofthe University of Brescia, Italy, where he teaches Theory andDesign of Reinforced Concrete and Prestressed Structures. Heis Director of the Laboratory for Testing Materials and Struc-tures of the University of Brescia. His research interestsinclude structural rehabilitation, r.c. structures, cracking,

0/ =⋅− ssbb enAeA

cidssbssssbb dSeeenAenAeA ,22 )(// =+⋅=⋅+

cidsid dSII ,* +=

Received April 7th 2003

490

steel-concrete bond, composite structures.

influence of slab-beam slip on the deflection of com...

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