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Analysis of deflections of composite slabs with profiled sheeting up to the ultimate moment
Journal of Constructional Steel Research 62 (2006) 820–830 www.elsevier.com/locate/jcsr
Analysis of deflections of composite slabs with profiled sheeting up to the ultimate moment G. Marˇciukaitis ∗ , B. Jonaitis, J. Valivonis Department of Reinforced Concrete and Masonry Structures, Vilnius Gediminas Technical University, Sauletekio ave 11, LT-10223 Vilnius, Lithuania Received 19 April 2005; accepted 30 November 2005
Abstract This paper deals with an investigation of the deflections of composite slabs. The deflection of composite slabs depends directly on the shear stiffness of the connection between profiled steel sheeting and concrete. A method for calculating deflections of slabs is presented in this paper. This method is based on a theory of built-up bars, which allows one to take into account directly the shear stiffness of the connection. Influences on the stiffness of the structure of normal cracks in the concrete layer and plastic deformations of concrete that has been subjected to compression are also taken into account in the analysis method. The method gives one an opportunity to assess variations of these factors at all stages of the slab’s behaviour from the start of loading up to the ultimate moment. Results of the experimental investigations of a connection (contact) between steel profiled sheeting (Holorib type) and concrete are presented in this paper. In the results of these investigations, three stages of behaviour of the contact are distinguished. A connection shear characteristic is determined for each stage, which is used for calculating the deflection of the slab. Experimental investigations were performed on deflections of composite slabs with a Holorib type of profiled sheeting. Variations in experimental deflections of slabs were explored from the beginning of loading up to the ultimate moment. Theoretical calculations of deflections for the experimental slabs were made. Calculations were performed according to the method proposed by the authors. A comparison of experimental and theoretical values of deflections revealed that agreement between these values was sufficiently good at all stages of the slab’s behaviour. c 2005 Elsevier Ltd. All rights reserved. Keywords: Composite slabs; Experimental and theoretical deflections; Theory of built-up bars; Cracks in the concrete; Contact of layers; Plastic deformations of concrete
1. Introduction The behaviour of steel profiled sheeting and concrete under load in composite structures is governed by the strength and stiffness of the connection between these layers. The joint action of the layers is achieved by limiting the slip between the layers in relation to each other. The joint action in layered concrete structures is provided by adhesion (chemical connection), keys formed during manufacture, friction and anchors. In composite structures made of steel, the profiled sheeting and concrete chemical connection between these layers is weak and, in many cases, can be damaged before the operational stage. In a newly cast concrete layer of composite ∗ Corresponding author.
E-mail address: [email protected] (G. Marˇciukaitis). c 2005 Elsevier Ltd. All rights reserved. 0143-974X/$ - see front matter doi:10.1016/j.jcsr.2005.11.022
structures, during its hardening, shrinkage deformations occur while the steel sheeting does not shrink. Differences in deformations at contact induce shear stresses, which depend on the composition of the concrete of the cast layer and the hardening conditions. Investigations performed by many scientists indicate that these stresses can be substantial and exceed the chemical bond strength. Therefore, special anchors are installed to ensure joint action of the layers, and in the composite slabs special profiled steel sheeting is used. Transversal and longitudinal embossments of various shapes are created on the ribs, which form keys in the contact [1,2]. However, investigations by many scientists [3–9] revealed that, in spite of the number of keys, anchors and other means of anchorage, the connection between layers in real composite structures is not absolutely stiff. In addition, the stiffness of the connection between layers varies with the action of external forces and with an increase in shear stresses.
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T M1 − · c1 , E 1 A1 E 1 I1 T M2 ε2 = + · c2 , E 2 A2 E 2 I2
ε1 =
Fig. 1. Diagram used for the analysis.
In the contact zone of composite slabs with profiled steel sheeting (especially above the ribs) a combined state of stresses occurs. Investigations [3,9,10] indicate that shear deformations already occur under the action of small shear forces in a connection that behaves elastically. The shear deformations are not large and practically depend only on the difference in the deformation of layers. Plastic deformations in the connection occur under the action of large shear forces and result in a reduction in connection stiffness. Therefore, stiffness of the connection between the concrete and profiled steel sheeting shall be taken into account in the determination of the stiffness of composite slabs, as has been indicated in investigations by many authors [11–13]. In Eurocode [14] for the design of composite structures, it is pointed out that, in the design of composite slabs, yield of the connection between concrete and profiled sheeting can be taken into account. It is recommended that this is accomplished on the bases of tests. Further, the stiffness of the connection is affected by the concrete composition, and the specimen’s storage conditions, which can differ from the conditions of the floors under design. Normally, in the concrete of a floor slab under a service load, there are cracks, and in the compression zone concrete plastic deformations occur [1,13–16]. The presence of cracks inflicts changes in the stiffness of cross-sections in the cracked floor areas. The influence of all these factors on floor stiffness is significant. Many of these factors affecting the stiffness of the floor are difficult to assess. Therefore, further experimental–theoretical investigation in the stiffness of floor slabs, especially of their connection zone, and development of methods for their analyses are essential. 2. Theoretical analyses of composite slabs for deflection
(1)
where u 1 and u 2 are the horizontal displacements of the first and the second layers in the zone of their interface. The relationship between displacement and the strain is du dx = ε. Therefore du = ε1 − ε2 . dx For flexural two layer members
(4)
where T is the resultant of shear stresses (τ ) in the contact; M1 and M2 are the bending moments sustained by the first and the second layers; E 1 and E 2 are the elasticity moduli of the first and the second layers; A1 and A2 are the cross-sectional areas of respective layers; I1 and I2 are the second moments of areas of the layers; and c1 and c2 are the distances between the contact and gravity centres of the layers. Placing expressions (3) and (4) into condition (2) and denoting that EM1 1I1 c1 + EM2 I22 c2 = E1 I1M·c +E 2 I2 and M = M0 −T ·c, one obtains 1 1 T · c2 du M0 · c =T + + , − dx E 1 A1 E 2 A2 E 1 I1 + E 2 I2 E 1 I1 + E 2 I2 (5) where M0 is the sum of the moments taken by individual layers without considering connections between the layers; T is the shear force acting in the contact; and c is the distance between gravity centres of the layers. Displacement between the layers can also be expressed by: 1 dT τ (6) u= = α α dx where α is the stiffness of the connection for shear, and 2 du 1 d T = . dx α dx 2 Comparison of expressions (5) and (7) gives: 2 1 1 c2 1 d T = T + + α dx 2 E 1 A1 E 2 A2 E 1 I1 + E 2 I2 M0 · c − . E 1 I1 + E 2 I2 Using the notation γ = E11A1 + E21A2 + (8) gives the following: 2 M0 · c 1 d T =γ ·T − . 2 α dx E 1 I1 + E 2 I2
c2 E 1 I1 +E 2 I2 ,
(7)
(8)
expression
(9)
Since curvature of a flexural member can be expressed by:
The theory of built-up bars [18] can be used for the analyses of two-layer composite members for deflections (see Fig. 1). According to this theory, horizontal displacement (u) between the first and the second layers at their interface u = u1 − u2,
(3)
(2)
d2 y dx 2
=−
M E 1 I1 + E 2 I2
(10)
and, having in mind that M = M0 − T · c, we obtain: d2 y T · c − M0 = . E 1 I1 + E 2 I2 dx 2
(11)
Placing the values of T and (8) and using the notation
d2 M0 dx 2
d2 T dx 2
obtained from Eq. (11) into
= − p gives
d2 y p d4 y (α · γ ) M0 − · α) = + , (γ 4 2 E 1 I1 + E 2 I2 E eff Ieff dx dx where E eff Ieff = E 1 I1 + E 2 I2 +
E 1 A1 E 2 A2 c 2 E 1 A1 +E 2 A2 .
(12)
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Fig. 4. Diagram of test arrangement.
Fig. 2. Cross section (a) and design stress distribution (b) of composite slab.
Fig. 5. Overview of test set up.
Equivalent stiffness of the section composite slab is expressed in the following way:
Fig. 3. General relationship between the coefficient ν and the level of stress in the compression zone of a flexural member for concrete class C16/20–C40/50 and ρ = 0.0004–0.0009.
For a flexural composite structure subjected only to uniformly distributed load ( p), introducing the notation α · γ = λ2 we obtain: 2 d4 y p λ2 p · x (l − x) 2d y . − λ = + dx 4 dx 2 E 1 I1 + E 2 I2 E eff Ieff 2
(13)
Composite slabs consist of two separate elements: profiled steel sheeting (the first layer) and a concrete (reinforced concrete) layer (the second layer) (Fig. 2). Some investigations [7,17] showed that the theory of built-up bars [18] could be used for the analyses of such structures. The stiffness of the connection between concrete and profiled steel sheeting can be evaluated using the method in Fig. 2. Accordingly (13), vertical displacement (y) of a simply supported composite slab subjected to uniformly distributed load ( p) can be determined using the formula: 2 d4 y l−x p λ2 p · x 2d y + − λ = . 4 2 2 E eff Ieff E c,eff Ic,eff + E p I p dx dx (14) Here, E c,eff is the equivalent elasticity modulus for the concrete, Ic,eff is the equivalent second moment of cross-sectional area for the concrete layer, E p and I p are the modulus of elasticity and the second moment of cross-sectional area for profiled steel sheeting.
E eff Ieff = E c,eff Ic,eff + E p I p +
2 E c,eff Ac,eff E p A p z eff , E c,eff Ac,eff + E p A p
(15)
where z eff is the distance between the centres of the equivalent concrete layer and that of the profiled sheeting; A p is the cross-sectional area of the profiled sheeting; and Ac,eff is the equivalent cross-sectional area of the concrete layer. The solution of a differential equation (14) and consideration 2 of critical conditions y(0) = 0, y(l) = 0, ddx y2 (0) = 0 2
and ddx y2 (l) = 0 make it possible to determine the vertical displacement (deflection) of a beam at the middle of its span (x = 2l ): 5 p · l4 p 1 l λ2l 2 =δ= −1 . + 4 y + 2 384E eff Ieff 8 λ D ch (0.5λl) (16) 1 1 1 = − . D E c,eff Ic,eff + E p I p E eff Ieff √ λ = α · γ, Here
(17) (18)
is a coefficient describing the stiffness of the connection between layers, which depends on the stiffness of the same layers and on the shear stiffness of their connection. The coefficient for contact stiffness: b · Gw , (19) α= z eff where G w is the characteristic for the connection shear stiffness, and b is the width of the concrete slab. The vertical displacement (y) for a simply supported slab loaded with two concentrated forces is described by Eq. (12).
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Fig. 6. View of general slab after failure.
Fig. 8. Shear deformations in the connection of composite slab P1-2.
Fig. 7. Shear deformations in the connection of composite slab P1-1.
Since, in this equation, p = 0, the equation is obtained in the form: 2 d4 y M 2d y − λ = λ2 . 4 2 E eff Ieff dx dx
(20)
The maximum deflection (at the middle of the span x = 2l ), obtained from the solution of differential equation (20), for such a composite slab is: l 1 ch (0.5λ · l) − 1 l2 y + =δ=M , (21) 2 8E eff Ieff D λ2 · ch (0.5λ · l) where M is the bending moment for the action in which the deflection is to be determined. The stiffness of the connection is evaluated using a characteristic of the stiffness (G w ). This value can be determined according to the results of investigations into the stiffness of the connection (Fig. 13). Analysis of Eqs. (14)–(21) indicates that many parameters required for the determination of deflection depend on the elasticity modulus of the concrete. However, from the very beginning of the loading of a reinforced concrete flexural member, plastic deformations of the concrete in the compression zone start to develop and the elasticity of the concrete is reduced. It is proposed to use the concrete elasticity coefficient ν for the compression zone in a flexural member to evaluate this reduction. Thus, E c,eff = ν E cm .
(22)
Fig. 9. Shear deformations in the connection of composite slab P1-3.
Direct determination of the coefficient ν in a flexural member is not possible, but there is the following relationship between the concrete elasticity factor in flexure (ν) and that factor in axial compression (ν): ν = ν · ω.
(23)
Here, ω is the coefficient of the shape for the stress diagram in the compression zone of a reinforced concrete flexural member, which varies from 0.5 (for a triangular diagram) to 1 (for a rectangular diagram). With growth of the load, the plastic part of concrete deformation becomes more significant and ω increases while (ν) decreases. The shape coefficient ω for the stress diagram and coefficient ν depend not only on the value of stress but on other factors as well, e.g. concrete strength, reinforcement ratio and reinforcement properties. Analysis of the experimental results indicates that ω varies linearly with MMR and therefore the following relationship σf cc = MMR is valid. Thus, there is a direct
relationship between MMR , σfcc and other factors that influences the elastic properties of concrete subjected to compression in the case of flexure (Fig. 3). By taking into account the results of the experimental investigations associated with the influence of the main factors, M M R , ρ, f y and f ck , on non-elasticity properties of the concrete and an analytical evaluation of test results, the following formula was obtained for practical elasticity assessment of the
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Fig. 12. Shear deformations in the connection of composite slab P2-2.
Fig. 10. Shear deformations in the connection of composite slab P1-4.
Fig. 13. Relationship between longitudinal slip and shear force: 1—for connection without pre-compression force; 2—with a pre-compression force of 5 kN; 3—with a pre-compression force of 10 kN; 4—with a pre-compression force of 15 kN. Fig. 11. Shear deformations in the connection of composite slab P2-1.
concrete:
2−m 1−m fy M M ν = 0.5 1 − 3.2 + m − 2 ·ρ , (24) M RC M RC fc where m is the coefficient for evaluating the influence of the concrete’s strength and its structure on the character of the σ –ε relationship. On the basis of the investigations, it is assumed that: fc m= , (25) f c,max where f c,max is the maximum strength of the cement concrete which, in the case of its ideal structure, can be taken as being equal to the strength of the coarse aggregates ( f c,max = 200 MPa); M is the bending moment for the action where deflection is to be determined; M RC is the equivalent design strength of the composite slab, determined by using the assumption that the connection between profiled sheeting and the concrete is absolutely stiff; and ρ is the reinforcement ratio.
Analysis of formula (24) and the experimental investigations indicate that the value of ν = νω for reinforced concrete members from normal weight concrete varies within limits from 0.5 to 0.1. When MMRC = 0.60–0.65, then ν = 0.46–0.43. Therefore, in many codes for the design of concrete structures in East European countries, the value of ν = 0.45 was used. In the determination of stiffness for a composite slab without cracks it is assumed that the concrete layer is continuous. The second moment of a cross-section (Ic,eff = Ic ) is determined by taking into account the whole cross-sectional area of the concrete. Data obtained by the authors of this paper and other investigators indicate that the opening of cracks leads to a sudden decrease in the compression zone’s depth and an increase in deflections. These results were pointed out by the experiments performed in this investigation. The concrete layer is split into separate blocks by opened normal cracks. Therefore, the stiffnesses of the concrete layer in a section with a crack and in a section without a crack are different. By taking into account results obtained in [19,20] and the assumptions made, it is recommended that the effective cross-sectional area
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of the concrete is estimated in calculations of cross-sectional stiffness. Various methods are proposed for the assessment of a compression zone’s depth. Some investigators propose estimating the average of the compression zone depth’s values in a section without cracks and in a section with cracks [20]; others suggest determining the equivalent compression zone’s depth [19]. But mostly this does not depend on the value of a load that is acting in the structure. Experimental and theoretical investigations of reinforced concrete structures that are reinforced with a steel bar indicate that the compression zone’s depth in a cracked section decreases with an increase in the external load. During an increase of the load, new cracks appear between cracks that have been opened in an earlier stage, inducing a change in the general cross-sectional stiffness. In addition, steel profiled sheeting in composite slabs is located below the concrete and confines tensile concrete in the whole width of the slab. Investigations performed on structures with external steel and non-metal reinforcements [21] indicate that, in such structures, the width of cracks is less, but the cracks are more closely spaced, i.e. the distance between normal cracks is significantly less. By taking this into account, the second moment of the area of a concrete layer with cracks Ic,eff in the composite slabs with profiled Holorib type steel sheeting can be determined using the following formula [19]: 2 xm bw · x m3 (b − bw ) h 3c . (26) + 1 + 12 − 0.5 Ic,eff = 3 12 hc The average depth of the compression zone (x m ) is determined as follows: µ · h µ2 + −µ , (27) xm = β · d p dp Ep · Ap ; E c,eff · bw · d p 2E p · A p µ · h = µ f · hc + ; E c,eff · bw b · hc µf = ; bw · d p
where µ = µ f +
(28) (29) (30)
and β = 1.9 − 2.6( MMRC )2 is the coefficient for the evaluation of an equivalent compression zone’s depth variation with a value of stress in the compression zone. 3. Experimental investigation into the stiffness of composite slabs Six composite slabs were tested (Table 1). The bottom layer of the composite slabs was 0.9 mm thick profiled (Holorib type) steel sheeting, the strength of which is f y = 317 MPa. The cross-sectional area of the profiled sheeting is A p = 1345 mm2 /m. The compression strength of the concrete is fc = 19.8–29.6 MPa. The concrete was manufactured by using crushed gravel and quartz sand. The composite slabs were produced without bar reinforcement and without special anchors at the supports.
825
A diagram of the test arrangement and an overview of the test’s set up are shown in Figs. 4 and 5. Composite slabs were tested under a short-time static load. The load was increased in steps of a value equal to 0.1M R . Deflections, concrete and steel deformations and the horizontal displacement of the concrete with respect to profiled sheeting (shear deformations in contact) were measured during the tests. The location of points at which shear deformation was measured is shown in the diagram of the test arrangement (Fig. 4). Experimental investigations revealed that no significant elastic shear deformations in the contact appear when the slabs are subjected to a bending moment not exceeding 0.4M R (where M R is the bending moment at failure of the slabs) (Figs. 7–12). When the bending moment due to load exceeds 0.4M R , adhesion between the profiled steel sheeting and the concrete is damaged and local compression of the keys at contact between profiled sheeting and concrete commences. Slippage of the concrete in relation to the steel sheeting is possible only when either the keys are cut or the concrete is separated from the sheeting and lifted upwards. Because of the shape of the profiled sheeting (dovetailed), lifting of the concrete is restrained. This creates a combined state of stresses in the concrete of the slab near the longitudinal rib of the profiled sheeting, resulting in the occurrence of vertical and horizontal transverse deformations in the concrete. Shear deformations in the connection substantially increase under the action of a bending moment of M ∼ 0.5M R (Figs. 7–12), and transverse deformations in the steel sheeting and concrete at the rib of the profiled sheeting also increase substantially. The principal stresses in the concrete layer are directed at an angle of 45◦ in relation to the vertical axes of the composite slab. When tension deformations in the direction of the principal stresses exceed limiting values, a longitudinal crack appears in the concrete. The direction of the crack in the concrete layer conforms to the direction of principal stresses and was recorded in all composite slabs that were tested (Fig. 6). These investigations showed that a connection between profiled sheeting and the concrete is not absolutely stiff. In an analysis of such a type of composite slab, allowance for partial stiffness of the connection between the layers should be made. During the tests, deflections were measured from the beginning of loading up to the beginning of failure. This gave an opportunity to investigate variations in deflections for composite slabs at all stages of their behaviour up to the moment of failure. Investigations revealed that, before the occurrence of normal cracks, increases in deflection is proportional and were recorded in all slabs (Figs. 15–17). These cracks occurred in slabs under the action of a bending moment M = (0.310–0.378)M R . In slab P2-2, cracks occurred at the bending moment M = 0.428M R . After the occurrence of normal cracks, the rate of growth of deflections increased. A turning-point in the graphs for bending moments–deflections can be noticed (Figs. 15–17), but is not well defined. This was caused by the fact that the profiled steel
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Specimen, No.
Width of slab b(m)
Crosssection depth of slab h(m)
Depth of concrete layer above the longitudinal rib of profiled sheeting h c (m2 )
Cross-section area of profiled sheeting A p (m2 )
Modulus of elasticity for concrete E cm (MPa)
Modulus of elasticity for profiled steel sheeting E p (MPa)
Characteristic for connection shear stiffness G w1 (MPa)
Characteristic for connection shear stiffness G w2 (MPa)
Concrete strength f c (MPa)
Strength of steel f y (MPa)
P1-1 P1-2 P1-3 P1-4 P2-1 P2-2
0.77 0.77 0.77 0.77 0.77 0.77
0.0795 0.075 0.076 0.080 0.097 0.098
0.0330 0.0285 0.0295 0.0335 0.0505 0.0515
1.01 × 10−3 1.01 × 10−3 1.01 × 10−3 1.01 × 10−3 1.01 × 10−3 1.01 × 10−3
3.95 × 104 4.05 × 104 4.35 × 104 4.24 × 104 4.27 × 104 4.15 × 104
2.05 × 105 2.05 × 105 2.05 × 105 2.05 × 105 2.05 × 105 2.05 × 105
210 210 210 210 210 210
149 149 149 149 149 149
27.9 21.6 19.8 17.7 29.0 28.6
317 317 317 317 317 317
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Table 1 Characteristic of composite slabs
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Fig. 14. Stages in behaviour of connection between steel profiled sheeting and concrete. Fig. 16. Theoretical and experimental deflections of slabs: 1—theoretical deflection of slab P1-3; 2—experimental deflection of slab P1-3; 3—theoretical deflection of slab P1-4; 4—experimental deflection of slab P1-4.
Fig. 15. Theoretical and experimental deflections of slabs: 1—theoretical deflection of slab P1-1; 2—experimental deflection of slab P1-1; 3—theoretical deflection of slab P1-2; 4—experimental deflection of slab P1-2.
sheeting in the tensile zone extends over the whole tension zone of the concrete in the slab. After the opening of normal cracks, the connection between concrete and profiled sheeting behaves elastically (Figs. 7–12) and conforms to the first stage of behaviour for a connection (Fig. 14). Since the steel sheeting and concrete are acting jointly, a further increase in deflections remains proportional but has a greater rate. Such an increase in deflections is because the steel sheeting uniformly covers the tension zone of the concrete and results in a smaller width of cracks. When M = (0.5–0.57)M R (except for slab P2-1) the rate of increase in deflections is greater. This is caused by an increase in the horizontal slip of the profiled steel sheeting in relation to the concrete. At this moment, local compression of concrete at the transverse ribs of the profiled sheeting starts. This is confirmed by an increase in shear deformations of the connection between the concrete slab and steel sheeting (Figs. 7–12). At this stage, local compression of keys (transversal ribs) in the contact starts. After the initial local compression, displacement of the profiled sheeting in relation to the concrete is only possible after either the keys are cut or the sheeting
Fig. 17. Theoretical and experimental deflections of slabs: 1—theoretical deflection of slab P2-1; 2—experimental deflection of slab P2-1; 3—theoretical deflection of slab P2-2; 4—experimental deflection of slab P2-2.
moves over them. But a “dovetailed” shaped steel rib restrains separation of the profiled sheeting and concrete in their connection. With further increase in the external load, the growth in deflections is not proportional. According to the results of experimental investigations, three stages in the behaviour of composite slabs can be distinguished: (a) before the occurrence of vertical cracks in the concrete; (b) after the occurrence of vertical (normal) cracks until initial local compression of the key in contact between the sheeting and concrete; (c) from the initial local compression of the keys up to the occurrence of horizontal cracks at the ribs and failure of the slab. Data on deformations and the strength of the connection between profiled steel sheeting and concrete are required for an effective assessment of the behaviour of composite steel–concrete slabs.
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Therefore, at the same time, experimental investigations in the stiffness of a connection were performed by means of testing composite specimens 150 × 150 × 150 mm in size [22]. The composition of concrete for these specimens was the same as that for slabs, and steel plates were cut from steel sheeting of Holorib-2000 type. Reaction to support in a slab was subjected to the external load and the load applied at the top to induce compression in contact between the profiled sheeting and the concrete. To determine the influence of this compression on deformations and the strength of this connection, parts of the test samples were pre-compressed by a vertical force to simulate the influences of support reactions on the behaviour of the connection. The value of the vertical force was was 5 kN (222 kPa) for the first group of specimens, 10 kN (444 kPa) for the second group, and 15 kN (666 kPa) for the third group. Control specimens (without pre-compression) were tested as well. Shear force was applied to contacts, and the values of shear deformations at the connection between the steel profiled sheeting and concrete were recorded, which characterize slippage of the layers in relation to each other. Analysis of the connection’s behaviour revealed that a rib of variable width and transverse ribs at the top of it restrain the slippage of profiled sheeting in relation to the concrete. The layers can slip in relation to each other when concrete keys at the top of the rib are crushed or transverse ribs of the steel’s sheeting are deformed. In the case when there is no pre-compression at the contact, a layer of concrete at the key lifts and thus the area of concrete at the key is subjected to local compression and the shear is less, and less force is required to shear it. Due to pre-compression force, the area of concrete at the key that is subjected to local compression and shear is greater, and a greater force is required to shear the contact. Results of tests indicated that, in test specimens precompressed by a vertical force (Fig. 13), the shear deformation of a connection was almost of the same value as when failure shear force in the contact was reached for all series of specimens, irrespective of differences in values of shear forces at the failure. This can be explained by considerating that connection failure occurred with damage to all components of the connection. To get the key sheared, horizontal movement of the sheeting with respect to the concrete has to be of the same value, irrespective of the contact surface pre-compression force’s value. Investigations revealed that it is possible to describe shear deformation of the connection by using an idealised graph with the distinction of three stages in the behaviour of the connection (Fig. 14). The first stage (0–a) is when the connection behaves elastically, the elastic plastic stage (a–b) is when concrete keys are subjected to local compression and the plastic stage (b–c) is when concrete keys are cut (crushed) and the joint action is provided only by friction. The stiffness of the connection in slabs can be evaluated using the characteristic of a connection’s shear stiffness (G w ), which is calculated by using the results from connection tests
for shear (Fig. 13). Values for different stages of behaviour for the connection (Fig. 14) are different (G w1 ) or (G w2 ). In calculations of this characteristic, it is possible to evaluate the vertical pre-compression when induced by support reactions. In calculations for the deflection of composite slabs, it is recommended that the characteristics of shear stiffness G w1 and G w2 for a connection are determined from the results of experimental investigations, with allowance for stresses induced by slab support reaction. These stresses are determined by dividing the values of support reaction forces by the area of the supporting zone of the slab. Usually, in buildings during normal service, flexural members at the service stage carry an external load equal to ∼0.55–0.6 of the failure load. Therefore, it is possible to state that partial stiffness of the connection of a steel profiled sheeting to concrete has to be evaluated in calculating the deflection of composite slabs. Using the method proposed in this paper, calculations of deflections were performed for the slabs that were tested. Theoretical deflections are determined by calculations taking into account the actual dimensions and mechanical characteristics of materials used for the tested slabs (Table 1). The calculations were performed without allowing for deflections due to shrinkage deformations of the concrete during hardening. The deflections of tested slabs were determined by applying formula (21). The elasticity moduli for the concrete, E c,eff , were determined according to formulae (22)–(25), and the equivalent second moment of the cross-sectional area for the concrete layer was determined according to formula (26). The characteristics of the connection for shear stiffness (G w ) were calculated according to the results of tests on the connection for shear when the vertical pre-compression force for the contact was 5 kN (pre-compression stress equal to 222 kPa). The characteristic of shear stiffness applied for the connection of a profiled steel sheeting to concrete was taken to be G w1 = 186 MPa for the first and second stages of the behaviour of slabs, and G w2 = 149 MPa for the third stage. Agreement is good between deflections calculated according to the method recommended in this paper and experimental deflections (Figs. 15–17). Theoretical deflections calculated according to the proposed method are compared with experimental deflections when M = 0.6M R (i.e. service load is assumed for most cases) (Table 2). 4. Conclusions When a service load is applied to composite slabs (without special anchors) the connection between steel profiled sheeting and the concrete is not absolutely stiff and there are cracks in the tension zone of the concrete layer. Therefore, in calculating deflections for such slabs, it is necessary to take into account the partial stiffness of the connection between layers, the effect of normal cracks in the concrete layer and plastic deformations of compressed concrete for the stiffness of this layer. This can be evaluated by applying the method proposed by this paper’s authors, based on the theory of built-up bars using formula (1)–(30).
Specimen, Bending moment M No. (kN m)
Theoretical strength of normal section determined using assumption that connection between profiled sheeting and the concrete is absolutely stiff M RC (kN m)
Experimental strength of composite slabs M R (kN m)
Experimental deflection δobs (mm)
Theoretical deflection δcal (mm)
P1-1 P1-2 P1-3 P1-4 P2-1 P2-2
18.6 19.2 16.6 17.8 24.6 24.5
13.95 15.75 14.28 11.40 19.05 18.9
6.44 7.51 7.19 3.99 4.35 5.15
5.91 8.18 7.20 4.19 4.41 4.85
7.67 8.66 8.56 6.84 11.43 11.34
δobs δcal
1.09 0.92 1.00 0.95 0.99 1.06
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Table 2 Results of theoretical calculations of composite slabs at a bending moment 0.6M R
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The variation in plastic deformations of concrete in compression for the calculating the deflection of composite slabs is described by formula (24). The influence of normal cracks on the stiffness of a composite slab under a variable load is evaluated using formula (26)–(30). The stiffness of the connection between profiled steel sheeting (Holorib) used for the manufacture of composite slabs and concrete was investigated experimentally, with the distinction of three stages of behaviour. The characteristics of stiffness (G w ) for a connection, required for calculating deflections, were determined on the basis of the results of investigations, which were used for calculating theoretical deflection for the tested slabs. The experimental investigations of composite slabs confirmed theoretical assumptions concerning the partial stiffness of a connection between profiled sheeting and concrete and the presence of cracks in the concrete layer. By using the proposed method for analyses of composite slabs at all stages up to the moment of failure of the behaviour, a calculation of theoretical deflections was performed for the tested slabs. Comparison of theoretical and experimental values of deflection indicated that the agreement of these values is sufficiently good for all the said loading stages of slabs (Figs. 15–17). When the value of the bending moment due to an external load is close to that due to the service load (M ≈ 0.6M R ), the ratio of experimental to theoretical deflection values varied from 0.92 to 1.09. Acknowledgment Many thanks to Ms Nin Bizys (Australia) for corrections to the English used in this paper. References [1] Crisinel M, O’Leary D. Composite floor slab design and construction. Journal of Structural Engineering International 1996;6(1):41–6. [2] Bode H, Minas F. Composite slabs with and without end anchorage under static and dynamic loading. In: Proceedings of engineering conference composite construction—conventional and innovative. 1997. [3] Veljkovic M. An improved partial connection method for composite slab design. In: Proceedings of a on engineering foundation conference composite construction in steel and concrete. 1996. [4] Rondal J, Moutafidou A. Study of shear bond in steel composite slabs. In: Proceedings of engineering conference composite construction— conventional and innovative. Innsbruck; 1997.
[5] Salari MR, Spacone E, Benson Shing P, Frangopol DM. Nonlinear analysis of composite beams with deformable shear connectors. Journal of Structural Engineering 1998;124(10):1148–58. [6] Johnson RP, Yuan H. Shear resistance of stud connectors with profiled sheeting. In: Proceedings of engineering conference composite construction—conventional and innovative. Innsbruck; 1997. [7] Wang YC. Deflection of steel-concrete composite beams with partial shear interaction. Journal of Structural Engineering 1998;124(10):1159–64. [8] Wright HD, Essawy MI. Bond in thin gauge steel concrete composite structures. In: Proceedings of engineering conference composite construction—conventional and innovative. 1996. [9] Schuurman RG, Stark JWB. Longitudinal shear resistance of composite slabs. In: Proceedings of engineering in steel and concrete. 1996. [10] Crisinel M, Daniels B, En P. Numerical analysis of composite slab behaviour. In: Proceedings of engineering conference composite construction in steel and concrete. 1996. [11] Bode H, Minas F, Sauerborn I. Partial connection design of composite slabs. Journal of Structural Engineering International 1996;6(1):53–6. [12] Motak J, Machacek J. Experimental behaviour of composite girders with steel undulating web and thin-walled shear connector’s hilti stripcon. Journal of Civil Engineering and Management 2004;X(1):45–9. [13] Stark JWB, Brekelmans JWPM. Plastic design of continuous composite slabs. Journal of Structural Engineering International 1996;6(1):47–53. [14] Eurocode 4. Design of composite steel and concrete structures – Part 1–1: General rules and rules for buildings. EN 1994–1–1:2004E. [15] Bode H, Sauerborn I. Modern design concept for composite slabs with ductile behaviour. In: Proceedings of an engineering foundation conference composite construction in steel and concrete. 1996. [16] Benitez MA, Darwin D, Donahey RC. Deflections of composite beams with web openings. Journal of Structural Engineering 1998;124(10): 1139–47. [17] Marˇciukaitis G, Valivonis J, Vaˇskeviˇcius A. Analysis of behaviour of composite elements with profiled steel sheeting. Journal of Civil Engineering (Statyba) 2001;7(6):425–532 [in Lithuanian]. [18] Rzhanitsyn A. Built-up bars and plates. Moscow: Strojizdat; 1986 [in Russian]. [19] Zalesov AS, Muchanediev TA, Cistekov EA. Calculation of deflections of reinforced concrete structures according new standards. Journal of Concrete and Reinforced Concrete (Beton I Zelezobeton) 2002;6:12–20 [in Russian]. [20] Tenhovuori A, Karkkainen K, Kanerva P. Parameters and definitions for classifying the behaviour of composite slabs. In: Proceedings of an engineering foundation conference composite construction in steel and concrete. 1996. [21] Valivonis J, Kudzys A, Jurkˇsa A. The resistance of reinforced concrete beams with asbestos cement sheets in the tension zone. Journal of Concrete Structures (Gelˇzbetonin˙es konstrukcijos) 1987;15:24–36 [in Lithuanian]. [22] Marˇciukaitis G, Valivonis J, Jonaitis B. Influence of vertical precompression force on strength of contact between concrete and external profiled reinforcement. In: Selected papers of the 8th international conference modern building materials, structures and techniques. 2004.
Analysis of deflections of composite slabs with profiled sheeting up to the ultimate moment G. Marˇciukaitis ∗ , B. Jonaitis, J. Valivonis Department of Reinforced Concrete and Masonry Structures, Vilnius Gediminas Technical University, Sauletekio ave 11, LT-10223 Vilnius, Lithuania Received 19 April 2005; accepted 30 November 2005
Abstract This paper deals with an investigation of the deflections of composite slabs. The deflection of composite slabs depends directly on the shear stiffness of the connection between profiled steel sheeting and concrete. A method for calculating deflections of slabs is presented in this paper. This method is based on a theory of built-up bars, which allows one to take into account directly the shear stiffness of the connection. Influences on the stiffness of the structure of normal cracks in the concrete layer and plastic deformations of concrete that has been subjected to compression are also taken into account in the analysis method. The method gives one an opportunity to assess variations of these factors at all stages of the slab’s behaviour from the start of loading up to the ultimate moment. Results of the experimental investigations of a connection (contact) between steel profiled sheeting (Holorib type) and concrete are presented in this paper. In the results of these investigations, three stages of behaviour of the contact are distinguished. A connection shear characteristic is determined for each stage, which is used for calculating the deflection of the slab. Experimental investigations were performed on deflections of composite slabs with a Holorib type of profiled sheeting. Variations in experimental deflections of slabs were explored from the beginning of loading up to the ultimate moment. Theoretical calculations of deflections for the experimental slabs were made. Calculations were performed according to the method proposed by the authors. A comparison of experimental and theoretical values of deflections revealed that agreement between these values was sufficiently good at all stages of the slab’s behaviour. c 2005 Elsevier Ltd. All rights reserved. Keywords: Composite slabs; Experimental and theoretical deflections; Theory of built-up bars; Cracks in the concrete; Contact of layers; Plastic deformations of concrete
1. Introduction The behaviour of steel profiled sheeting and concrete under load in composite structures is governed by the strength and stiffness of the connection between these layers. The joint action of the layers is achieved by limiting the slip between the layers in relation to each other. The joint action in layered concrete structures is provided by adhesion (chemical connection), keys formed during manufacture, friction and anchors. In composite structures made of steel, the profiled sheeting and concrete chemical connection between these layers is weak and, in many cases, can be damaged before the operational stage. In a newly cast concrete layer of composite ∗ Corresponding author.
E-mail address: [email protected] (G. Marˇciukaitis). c 2005 Elsevier Ltd. All rights reserved. 0143-974X/$ - see front matter doi:10.1016/j.jcsr.2005.11.022
structures, during its hardening, shrinkage deformations occur while the steel sheeting does not shrink. Differences in deformations at contact induce shear stresses, which depend on the composition of the concrete of the cast layer and the hardening conditions. Investigations performed by many scientists indicate that these stresses can be substantial and exceed the chemical bond strength. Therefore, special anchors are installed to ensure joint action of the layers, and in the composite slabs special profiled steel sheeting is used. Transversal and longitudinal embossments of various shapes are created on the ribs, which form keys in the contact [1,2]. However, investigations by many scientists [3–9] revealed that, in spite of the number of keys, anchors and other means of anchorage, the connection between layers in real composite structures is not absolutely stiff. In addition, the stiffness of the connection between layers varies with the action of external forces and with an increase in shear stresses.
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T M1 − · c1 , E 1 A1 E 1 I1 T M2 ε2 = + · c2 , E 2 A2 E 2 I2
ε1 =
Fig. 1. Diagram used for the analysis.
In the contact zone of composite slabs with profiled steel sheeting (especially above the ribs) a combined state of stresses occurs. Investigations [3,9,10] indicate that shear deformations already occur under the action of small shear forces in a connection that behaves elastically. The shear deformations are not large and practically depend only on the difference in the deformation of layers. Plastic deformations in the connection occur under the action of large shear forces and result in a reduction in connection stiffness. Therefore, stiffness of the connection between the concrete and profiled steel sheeting shall be taken into account in the determination of the stiffness of composite slabs, as has been indicated in investigations by many authors [11–13]. In Eurocode [14] for the design of composite structures, it is pointed out that, in the design of composite slabs, yield of the connection between concrete and profiled sheeting can be taken into account. It is recommended that this is accomplished on the bases of tests. Further, the stiffness of the connection is affected by the concrete composition, and the specimen’s storage conditions, which can differ from the conditions of the floors under design. Normally, in the concrete of a floor slab under a service load, there are cracks, and in the compression zone concrete plastic deformations occur [1,13–16]. The presence of cracks inflicts changes in the stiffness of cross-sections in the cracked floor areas. The influence of all these factors on floor stiffness is significant. Many of these factors affecting the stiffness of the floor are difficult to assess. Therefore, further experimental–theoretical investigation in the stiffness of floor slabs, especially of their connection zone, and development of methods for their analyses are essential. 2. Theoretical analyses of composite slabs for deflection
(1)
where u 1 and u 2 are the horizontal displacements of the first and the second layers in the zone of their interface. The relationship between displacement and the strain is du dx = ε. Therefore du = ε1 − ε2 . dx For flexural two layer members
(4)
where T is the resultant of shear stresses (τ ) in the contact; M1 and M2 are the bending moments sustained by the first and the second layers; E 1 and E 2 are the elasticity moduli of the first and the second layers; A1 and A2 are the cross-sectional areas of respective layers; I1 and I2 are the second moments of areas of the layers; and c1 and c2 are the distances between the contact and gravity centres of the layers. Placing expressions (3) and (4) into condition (2) and denoting that EM1 1I1 c1 + EM2 I22 c2 = E1 I1M·c +E 2 I2 and M = M0 −T ·c, one obtains 1 1 T · c2 du M0 · c =T + + , − dx E 1 A1 E 2 A2 E 1 I1 + E 2 I2 E 1 I1 + E 2 I2 (5) where M0 is the sum of the moments taken by individual layers without considering connections between the layers; T is the shear force acting in the contact; and c is the distance between gravity centres of the layers. Displacement between the layers can also be expressed by: 1 dT τ (6) u= = α α dx where α is the stiffness of the connection for shear, and 2 du 1 d T = . dx α dx 2 Comparison of expressions (5) and (7) gives: 2 1 1 c2 1 d T = T + + α dx 2 E 1 A1 E 2 A2 E 1 I1 + E 2 I2 M0 · c − . E 1 I1 + E 2 I2 Using the notation γ = E11A1 + E21A2 + (8) gives the following: 2 M0 · c 1 d T =γ ·T − . 2 α dx E 1 I1 + E 2 I2
c2 E 1 I1 +E 2 I2 ,
(7)
(8)
expression
(9)
Since curvature of a flexural member can be expressed by:
The theory of built-up bars [18] can be used for the analyses of two-layer composite members for deflections (see Fig. 1). According to this theory, horizontal displacement (u) between the first and the second layers at their interface u = u1 − u2,
(3)
(2)
d2 y dx 2
=−
M E 1 I1 + E 2 I2
(10)
and, having in mind that M = M0 − T · c, we obtain: d2 y T · c − M0 = . E 1 I1 + E 2 I2 dx 2
(11)
Placing the values of T and (8) and using the notation
d2 M0 dx 2
d2 T dx 2
obtained from Eq. (11) into
= − p gives
d2 y p d4 y (α · γ ) M0 − · α) = + , (γ 4 2 E 1 I1 + E 2 I2 E eff Ieff dx dx where E eff Ieff = E 1 I1 + E 2 I2 +
E 1 A1 E 2 A2 c 2 E 1 A1 +E 2 A2 .
(12)
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Fig. 4. Diagram of test arrangement.
Fig. 2. Cross section (a) and design stress distribution (b) of composite slab.
Fig. 5. Overview of test set up.
Equivalent stiffness of the section composite slab is expressed in the following way:
Fig. 3. General relationship between the coefficient ν and the level of stress in the compression zone of a flexural member for concrete class C16/20–C40/50 and ρ = 0.0004–0.0009.
For a flexural composite structure subjected only to uniformly distributed load ( p), introducing the notation α · γ = λ2 we obtain: 2 d4 y p λ2 p · x (l − x) 2d y . − λ = + dx 4 dx 2 E 1 I1 + E 2 I2 E eff Ieff 2
(13)
Composite slabs consist of two separate elements: profiled steel sheeting (the first layer) and a concrete (reinforced concrete) layer (the second layer) (Fig. 2). Some investigations [7,17] showed that the theory of built-up bars [18] could be used for the analyses of such structures. The stiffness of the connection between concrete and profiled steel sheeting can be evaluated using the method in Fig. 2. Accordingly (13), vertical displacement (y) of a simply supported composite slab subjected to uniformly distributed load ( p) can be determined using the formula: 2 d4 y l−x p λ2 p · x 2d y + − λ = . 4 2 2 E eff Ieff E c,eff Ic,eff + E p I p dx dx (14) Here, E c,eff is the equivalent elasticity modulus for the concrete, Ic,eff is the equivalent second moment of cross-sectional area for the concrete layer, E p and I p are the modulus of elasticity and the second moment of cross-sectional area for profiled steel sheeting.
E eff Ieff = E c,eff Ic,eff + E p I p +
2 E c,eff Ac,eff E p A p z eff , E c,eff Ac,eff + E p A p
(15)
where z eff is the distance between the centres of the equivalent concrete layer and that of the profiled sheeting; A p is the cross-sectional area of the profiled sheeting; and Ac,eff is the equivalent cross-sectional area of the concrete layer. The solution of a differential equation (14) and consideration 2 of critical conditions y(0) = 0, y(l) = 0, ddx y2 (0) = 0 2
and ddx y2 (l) = 0 make it possible to determine the vertical displacement (deflection) of a beam at the middle of its span (x = 2l ): 5 p · l4 p 1 l λ2l 2 =δ= −1 . + 4 y + 2 384E eff Ieff 8 λ D ch (0.5λl) (16) 1 1 1 = − . D E c,eff Ic,eff + E p I p E eff Ieff √ λ = α · γ, Here
(17) (18)
is a coefficient describing the stiffness of the connection between layers, which depends on the stiffness of the same layers and on the shear stiffness of their connection. The coefficient for contact stiffness: b · Gw , (19) α= z eff where G w is the characteristic for the connection shear stiffness, and b is the width of the concrete slab. The vertical displacement (y) for a simply supported slab loaded with two concentrated forces is described by Eq. (12).
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Fig. 6. View of general slab after failure.
Fig. 8. Shear deformations in the connection of composite slab P1-2.
Fig. 7. Shear deformations in the connection of composite slab P1-1.
Since, in this equation, p = 0, the equation is obtained in the form: 2 d4 y M 2d y − λ = λ2 . 4 2 E eff Ieff dx dx
(20)
The maximum deflection (at the middle of the span x = 2l ), obtained from the solution of differential equation (20), for such a composite slab is: l 1 ch (0.5λ · l) − 1 l2 y + =δ=M , (21) 2 8E eff Ieff D λ2 · ch (0.5λ · l) where M is the bending moment for the action in which the deflection is to be determined. The stiffness of the connection is evaluated using a characteristic of the stiffness (G w ). This value can be determined according to the results of investigations into the stiffness of the connection (Fig. 13). Analysis of Eqs. (14)–(21) indicates that many parameters required for the determination of deflection depend on the elasticity modulus of the concrete. However, from the very beginning of the loading of a reinforced concrete flexural member, plastic deformations of the concrete in the compression zone start to develop and the elasticity of the concrete is reduced. It is proposed to use the concrete elasticity coefficient ν for the compression zone in a flexural member to evaluate this reduction. Thus, E c,eff = ν E cm .
(22)
Fig. 9. Shear deformations in the connection of composite slab P1-3.
Direct determination of the coefficient ν in a flexural member is not possible, but there is the following relationship between the concrete elasticity factor in flexure (ν) and that factor in axial compression (ν): ν = ν · ω.
(23)
Here, ω is the coefficient of the shape for the stress diagram in the compression zone of a reinforced concrete flexural member, which varies from 0.5 (for a triangular diagram) to 1 (for a rectangular diagram). With growth of the load, the plastic part of concrete deformation becomes more significant and ω increases while (ν) decreases. The shape coefficient ω for the stress diagram and coefficient ν depend not only on the value of stress but on other factors as well, e.g. concrete strength, reinforcement ratio and reinforcement properties. Analysis of the experimental results indicates that ω varies linearly with MMR and therefore the following relationship σf cc = MMR is valid. Thus, there is a direct
relationship between MMR , σfcc and other factors that influences the elastic properties of concrete subjected to compression in the case of flexure (Fig. 3). By taking into account the results of the experimental investigations associated with the influence of the main factors, M M R , ρ, f y and f ck , on non-elasticity properties of the concrete and an analytical evaluation of test results, the following formula was obtained for practical elasticity assessment of the
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Fig. 12. Shear deformations in the connection of composite slab P2-2.
Fig. 10. Shear deformations in the connection of composite slab P1-4.
Fig. 13. Relationship between longitudinal slip and shear force: 1—for connection without pre-compression force; 2—with a pre-compression force of 5 kN; 3—with a pre-compression force of 10 kN; 4—with a pre-compression force of 15 kN. Fig. 11. Shear deformations in the connection of composite slab P2-1.
concrete:
2−m 1−m fy M M ν = 0.5 1 − 3.2 + m − 2 ·ρ , (24) M RC M RC fc where m is the coefficient for evaluating the influence of the concrete’s strength and its structure on the character of the σ –ε relationship. On the basis of the investigations, it is assumed that: fc m= , (25) f c,max where f c,max is the maximum strength of the cement concrete which, in the case of its ideal structure, can be taken as being equal to the strength of the coarse aggregates ( f c,max = 200 MPa); M is the bending moment for the action where deflection is to be determined; M RC is the equivalent design strength of the composite slab, determined by using the assumption that the connection between profiled sheeting and the concrete is absolutely stiff; and ρ is the reinforcement ratio.
Analysis of formula (24) and the experimental investigations indicate that the value of ν = νω for reinforced concrete members from normal weight concrete varies within limits from 0.5 to 0.1. When MMRC = 0.60–0.65, then ν = 0.46–0.43. Therefore, in many codes for the design of concrete structures in East European countries, the value of ν = 0.45 was used. In the determination of stiffness for a composite slab without cracks it is assumed that the concrete layer is continuous. The second moment of a cross-section (Ic,eff = Ic ) is determined by taking into account the whole cross-sectional area of the concrete. Data obtained by the authors of this paper and other investigators indicate that the opening of cracks leads to a sudden decrease in the compression zone’s depth and an increase in deflections. These results were pointed out by the experiments performed in this investigation. The concrete layer is split into separate blocks by opened normal cracks. Therefore, the stiffnesses of the concrete layer in a section with a crack and in a section without a crack are different. By taking into account results obtained in [19,20] and the assumptions made, it is recommended that the effective cross-sectional area
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of the concrete is estimated in calculations of cross-sectional stiffness. Various methods are proposed for the assessment of a compression zone’s depth. Some investigators propose estimating the average of the compression zone depth’s values in a section without cracks and in a section with cracks [20]; others suggest determining the equivalent compression zone’s depth [19]. But mostly this does not depend on the value of a load that is acting in the structure. Experimental and theoretical investigations of reinforced concrete structures that are reinforced with a steel bar indicate that the compression zone’s depth in a cracked section decreases with an increase in the external load. During an increase of the load, new cracks appear between cracks that have been opened in an earlier stage, inducing a change in the general cross-sectional stiffness. In addition, steel profiled sheeting in composite slabs is located below the concrete and confines tensile concrete in the whole width of the slab. Investigations performed on structures with external steel and non-metal reinforcements [21] indicate that, in such structures, the width of cracks is less, but the cracks are more closely spaced, i.e. the distance between normal cracks is significantly less. By taking this into account, the second moment of the area of a concrete layer with cracks Ic,eff in the composite slabs with profiled Holorib type steel sheeting can be determined using the following formula [19]: 2 xm bw · x m3 (b − bw ) h 3c . (26) + 1 + 12 − 0.5 Ic,eff = 3 12 hc The average depth of the compression zone (x m ) is determined as follows: µ · h µ2 + −µ , (27) xm = β · d p dp Ep · Ap ; E c,eff · bw · d p 2E p · A p µ · h = µ f · hc + ; E c,eff · bw b · hc µf = ; bw · d p
where µ = µ f +
(28) (29) (30)
and β = 1.9 − 2.6( MMRC )2 is the coefficient for the evaluation of an equivalent compression zone’s depth variation with a value of stress in the compression zone. 3. Experimental investigation into the stiffness of composite slabs Six composite slabs were tested (Table 1). The bottom layer of the composite slabs was 0.9 mm thick profiled (Holorib type) steel sheeting, the strength of which is f y = 317 MPa. The cross-sectional area of the profiled sheeting is A p = 1345 mm2 /m. The compression strength of the concrete is fc = 19.8–29.6 MPa. The concrete was manufactured by using crushed gravel and quartz sand. The composite slabs were produced without bar reinforcement and without special anchors at the supports.
825
A diagram of the test arrangement and an overview of the test’s set up are shown in Figs. 4 and 5. Composite slabs were tested under a short-time static load. The load was increased in steps of a value equal to 0.1M R . Deflections, concrete and steel deformations and the horizontal displacement of the concrete with respect to profiled sheeting (shear deformations in contact) were measured during the tests. The location of points at which shear deformation was measured is shown in the diagram of the test arrangement (Fig. 4). Experimental investigations revealed that no significant elastic shear deformations in the contact appear when the slabs are subjected to a bending moment not exceeding 0.4M R (where M R is the bending moment at failure of the slabs) (Figs. 7–12). When the bending moment due to load exceeds 0.4M R , adhesion between the profiled steel sheeting and the concrete is damaged and local compression of the keys at contact between profiled sheeting and concrete commences. Slippage of the concrete in relation to the steel sheeting is possible only when either the keys are cut or the concrete is separated from the sheeting and lifted upwards. Because of the shape of the profiled sheeting (dovetailed), lifting of the concrete is restrained. This creates a combined state of stresses in the concrete of the slab near the longitudinal rib of the profiled sheeting, resulting in the occurrence of vertical and horizontal transverse deformations in the concrete. Shear deformations in the connection substantially increase under the action of a bending moment of M ∼ 0.5M R (Figs. 7–12), and transverse deformations in the steel sheeting and concrete at the rib of the profiled sheeting also increase substantially. The principal stresses in the concrete layer are directed at an angle of 45◦ in relation to the vertical axes of the composite slab. When tension deformations in the direction of the principal stresses exceed limiting values, a longitudinal crack appears in the concrete. The direction of the crack in the concrete layer conforms to the direction of principal stresses and was recorded in all composite slabs that were tested (Fig. 6). These investigations showed that a connection between profiled sheeting and the concrete is not absolutely stiff. In an analysis of such a type of composite slab, allowance for partial stiffness of the connection between the layers should be made. During the tests, deflections were measured from the beginning of loading up to the beginning of failure. This gave an opportunity to investigate variations in deflections for composite slabs at all stages of their behaviour up to the moment of failure. Investigations revealed that, before the occurrence of normal cracks, increases in deflection is proportional and were recorded in all slabs (Figs. 15–17). These cracks occurred in slabs under the action of a bending moment M = (0.310–0.378)M R . In slab P2-2, cracks occurred at the bending moment M = 0.428M R . After the occurrence of normal cracks, the rate of growth of deflections increased. A turning-point in the graphs for bending moments–deflections can be noticed (Figs. 15–17), but is not well defined. This was caused by the fact that the profiled steel
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Specimen, No.
Width of slab b(m)
Crosssection depth of slab h(m)
Depth of concrete layer above the longitudinal rib of profiled sheeting h c (m2 )
Cross-section area of profiled sheeting A p (m2 )
Modulus of elasticity for concrete E cm (MPa)
Modulus of elasticity for profiled steel sheeting E p (MPa)
Characteristic for connection shear stiffness G w1 (MPa)
Characteristic for connection shear stiffness G w2 (MPa)
Concrete strength f c (MPa)
Strength of steel f y (MPa)
P1-1 P1-2 P1-3 P1-4 P2-1 P2-2
0.77 0.77 0.77 0.77 0.77 0.77
0.0795 0.075 0.076 0.080 0.097 0.098
0.0330 0.0285 0.0295 0.0335 0.0505 0.0515
1.01 × 10−3 1.01 × 10−3 1.01 × 10−3 1.01 × 10−3 1.01 × 10−3 1.01 × 10−3
3.95 × 104 4.05 × 104 4.35 × 104 4.24 × 104 4.27 × 104 4.15 × 104
2.05 × 105 2.05 × 105 2.05 × 105 2.05 × 105 2.05 × 105 2.05 × 105
210 210 210 210 210 210
149 149 149 149 149 149
27.9 21.6 19.8 17.7 29.0 28.6
317 317 317 317 317 317
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Table 1 Characteristic of composite slabs
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Fig. 14. Stages in behaviour of connection between steel profiled sheeting and concrete. Fig. 16. Theoretical and experimental deflections of slabs: 1—theoretical deflection of slab P1-3; 2—experimental deflection of slab P1-3; 3—theoretical deflection of slab P1-4; 4—experimental deflection of slab P1-4.
Fig. 15. Theoretical and experimental deflections of slabs: 1—theoretical deflection of slab P1-1; 2—experimental deflection of slab P1-1; 3—theoretical deflection of slab P1-2; 4—experimental deflection of slab P1-2.
sheeting in the tensile zone extends over the whole tension zone of the concrete in the slab. After the opening of normal cracks, the connection between concrete and profiled sheeting behaves elastically (Figs. 7–12) and conforms to the first stage of behaviour for a connection (Fig. 14). Since the steel sheeting and concrete are acting jointly, a further increase in deflections remains proportional but has a greater rate. Such an increase in deflections is because the steel sheeting uniformly covers the tension zone of the concrete and results in a smaller width of cracks. When M = (0.5–0.57)M R (except for slab P2-1) the rate of increase in deflections is greater. This is caused by an increase in the horizontal slip of the profiled steel sheeting in relation to the concrete. At this moment, local compression of concrete at the transverse ribs of the profiled sheeting starts. This is confirmed by an increase in shear deformations of the connection between the concrete slab and steel sheeting (Figs. 7–12). At this stage, local compression of keys (transversal ribs) in the contact starts. After the initial local compression, displacement of the profiled sheeting in relation to the concrete is only possible after either the keys are cut or the sheeting
Fig. 17. Theoretical and experimental deflections of slabs: 1—theoretical deflection of slab P2-1; 2—experimental deflection of slab P2-1; 3—theoretical deflection of slab P2-2; 4—experimental deflection of slab P2-2.
moves over them. But a “dovetailed” shaped steel rib restrains separation of the profiled sheeting and concrete in their connection. With further increase in the external load, the growth in deflections is not proportional. According to the results of experimental investigations, three stages in the behaviour of composite slabs can be distinguished: (a) before the occurrence of vertical cracks in the concrete; (b) after the occurrence of vertical (normal) cracks until initial local compression of the key in contact between the sheeting and concrete; (c) from the initial local compression of the keys up to the occurrence of horizontal cracks at the ribs and failure of the slab. Data on deformations and the strength of the connection between profiled steel sheeting and concrete are required for an effective assessment of the behaviour of composite steel–concrete slabs.
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Therefore, at the same time, experimental investigations in the stiffness of a connection were performed by means of testing composite specimens 150 × 150 × 150 mm in size [22]. The composition of concrete for these specimens was the same as that for slabs, and steel plates were cut from steel sheeting of Holorib-2000 type. Reaction to support in a slab was subjected to the external load and the load applied at the top to induce compression in contact between the profiled sheeting and the concrete. To determine the influence of this compression on deformations and the strength of this connection, parts of the test samples were pre-compressed by a vertical force to simulate the influences of support reactions on the behaviour of the connection. The value of the vertical force was was 5 kN (222 kPa) for the first group of specimens, 10 kN (444 kPa) for the second group, and 15 kN (666 kPa) for the third group. Control specimens (without pre-compression) were tested as well. Shear force was applied to contacts, and the values of shear deformations at the connection between the steel profiled sheeting and concrete were recorded, which characterize slippage of the layers in relation to each other. Analysis of the connection’s behaviour revealed that a rib of variable width and transverse ribs at the top of it restrain the slippage of profiled sheeting in relation to the concrete. The layers can slip in relation to each other when concrete keys at the top of the rib are crushed or transverse ribs of the steel’s sheeting are deformed. In the case when there is no pre-compression at the contact, a layer of concrete at the key lifts and thus the area of concrete at the key is subjected to local compression and the shear is less, and less force is required to shear it. Due to pre-compression force, the area of concrete at the key that is subjected to local compression and shear is greater, and a greater force is required to shear the contact. Results of tests indicated that, in test specimens precompressed by a vertical force (Fig. 13), the shear deformation of a connection was almost of the same value as when failure shear force in the contact was reached for all series of specimens, irrespective of differences in values of shear forces at the failure. This can be explained by considerating that connection failure occurred with damage to all components of the connection. To get the key sheared, horizontal movement of the sheeting with respect to the concrete has to be of the same value, irrespective of the contact surface pre-compression force’s value. Investigations revealed that it is possible to describe shear deformation of the connection by using an idealised graph with the distinction of three stages in the behaviour of the connection (Fig. 14). The first stage (0–a) is when the connection behaves elastically, the elastic plastic stage (a–b) is when concrete keys are subjected to local compression and the plastic stage (b–c) is when concrete keys are cut (crushed) and the joint action is provided only by friction. The stiffness of the connection in slabs can be evaluated using the characteristic of a connection’s shear stiffness (G w ), which is calculated by using the results from connection tests
for shear (Fig. 13). Values for different stages of behaviour for the connection (Fig. 14) are different (G w1 ) or (G w2 ). In calculations of this characteristic, it is possible to evaluate the vertical pre-compression when induced by support reactions. In calculations for the deflection of composite slabs, it is recommended that the characteristics of shear stiffness G w1 and G w2 for a connection are determined from the results of experimental investigations, with allowance for stresses induced by slab support reaction. These stresses are determined by dividing the values of support reaction forces by the area of the supporting zone of the slab. Usually, in buildings during normal service, flexural members at the service stage carry an external load equal to ∼0.55–0.6 of the failure load. Therefore, it is possible to state that partial stiffness of the connection of a steel profiled sheeting to concrete has to be evaluated in calculating the deflection of composite slabs. Using the method proposed in this paper, calculations of deflections were performed for the slabs that were tested. Theoretical deflections are determined by calculations taking into account the actual dimensions and mechanical characteristics of materials used for the tested slabs (Table 1). The calculations were performed without allowing for deflections due to shrinkage deformations of the concrete during hardening. The deflections of tested slabs were determined by applying formula (21). The elasticity moduli for the concrete, E c,eff , were determined according to formulae (22)–(25), and the equivalent second moment of the cross-sectional area for the concrete layer was determined according to formula (26). The characteristics of the connection for shear stiffness (G w ) were calculated according to the results of tests on the connection for shear when the vertical pre-compression force for the contact was 5 kN (pre-compression stress equal to 222 kPa). The characteristic of shear stiffness applied for the connection of a profiled steel sheeting to concrete was taken to be G w1 = 186 MPa for the first and second stages of the behaviour of slabs, and G w2 = 149 MPa for the third stage. Agreement is good between deflections calculated according to the method recommended in this paper and experimental deflections (Figs. 15–17). Theoretical deflections calculated according to the proposed method are compared with experimental deflections when M = 0.6M R (i.e. service load is assumed for most cases) (Table 2). 4. Conclusions When a service load is applied to composite slabs (without special anchors) the connection between steel profiled sheeting and the concrete is not absolutely stiff and there are cracks in the tension zone of the concrete layer. Therefore, in calculating deflections for such slabs, it is necessary to take into account the partial stiffness of the connection between layers, the effect of normal cracks in the concrete layer and plastic deformations of compressed concrete for the stiffness of this layer. This can be evaluated by applying the method proposed by this paper’s authors, based on the theory of built-up bars using formula (1)–(30).
Specimen, Bending moment M No. (kN m)
Theoretical strength of normal section determined using assumption that connection between profiled sheeting and the concrete is absolutely stiff M RC (kN m)
Experimental strength of composite slabs M R (kN m)
Experimental deflection δobs (mm)
Theoretical deflection δcal (mm)
P1-1 P1-2 P1-3 P1-4 P2-1 P2-2
18.6 19.2 16.6 17.8 24.6 24.5
13.95 15.75 14.28 11.40 19.05 18.9
6.44 7.51 7.19 3.99 4.35 5.15
5.91 8.18 7.20 4.19 4.41 4.85
7.67 8.66 8.56 6.84 11.43 11.34
δobs δcal
1.09 0.92 1.00 0.95 0.99 1.06
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Table 2 Results of theoretical calculations of composite slabs at a bending moment 0.6M R
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The variation in plastic deformations of concrete in compression for the calculating the deflection of composite slabs is described by formula (24). The influence of normal cracks on the stiffness of a composite slab under a variable load is evaluated using formula (26)–(30). The stiffness of the connection between profiled steel sheeting (Holorib) used for the manufacture of composite slabs and concrete was investigated experimentally, with the distinction of three stages of behaviour. The characteristics of stiffness (G w ) for a connection, required for calculating deflections, were determined on the basis of the results of investigations, which were used for calculating theoretical deflection for the tested slabs. The experimental investigations of composite slabs confirmed theoretical assumptions concerning the partial stiffness of a connection between profiled sheeting and concrete and the presence of cracks in the concrete layer. By using the proposed method for analyses of composite slabs at all stages up to the moment of failure of the behaviour, a calculation of theoretical deflections was performed for the tested slabs. Comparison of theoretical and experimental values of deflection indicated that the agreement of these values is sufficiently good for all the said loading stages of slabs (Figs. 15–17). When the value of the bending moment due to an external load is close to that due to the service load (M ≈ 0.6M R ), the ratio of experimental to theoretical deflection values varied from 0.92 to 1.09. Acknowledgment Many thanks to Ms Nin Bizys (Australia) for corrections to the English used in this paper. References [1] Crisinel M, O’Leary D. Composite floor slab design and construction. Journal of Structural Engineering International 1996;6(1):41–6. [2] Bode H, Minas F. Composite slabs with and without end anchorage under static and dynamic loading. In: Proceedings of engineering conference composite construction—conventional and innovative. 1997. [3] Veljkovic M. An improved partial connection method for composite slab design. In: Proceedings of a on engineering foundation conference composite construction in steel and concrete. 1996. [4] Rondal J, Moutafidou A. Study of shear bond in steel composite slabs. In: Proceedings of engineering conference composite construction— conventional and innovative. Innsbruck; 1997.
[5] Salari MR, Spacone E, Benson Shing P, Frangopol DM. Nonlinear analysis of composite beams with deformable shear connectors. Journal of Structural Engineering 1998;124(10):1148–58. [6] Johnson RP, Yuan H. Shear resistance of stud connectors with profiled sheeting. In: Proceedings of engineering conference composite construction—conventional and innovative. Innsbruck; 1997. [7] Wang YC. Deflection of steel-concrete composite beams with partial shear interaction. Journal of Structural Engineering 1998;124(10):1159–64. [8] Wright HD, Essawy MI. Bond in thin gauge steel concrete composite structures. In: Proceedings of engineering conference composite construction—conventional and innovative. 1996. [9] Schuurman RG, Stark JWB. Longitudinal shear resistance of composite slabs. In: Proceedings of engineering in steel and concrete. 1996. [10] Crisinel M, Daniels B, En P. Numerical analysis of composite slab behaviour. In: Proceedings of engineering conference composite construction in steel and concrete. 1996. [11] Bode H, Minas F, Sauerborn I. Partial connection design of composite slabs. Journal of Structural Engineering International 1996;6(1):53–6. [12] Motak J, Machacek J. Experimental behaviour of composite girders with steel undulating web and thin-walled shear connector’s hilti stripcon. Journal of Civil Engineering and Management 2004;X(1):45–9. [13] Stark JWB, Brekelmans JWPM. Plastic design of continuous composite slabs. Journal of Structural Engineering International 1996;6(1):47–53. [14] Eurocode 4. Design of composite steel and concrete structures – Part 1–1: General rules and rules for buildings. EN 1994–1–1:2004E. [15] Bode H, Sauerborn I. Modern design concept for composite slabs with ductile behaviour. In: Proceedings of an engineering foundation conference composite construction in steel and concrete. 1996. [16] Benitez MA, Darwin D, Donahey RC. Deflections of composite beams with web openings. Journal of Structural Engineering 1998;124(10): 1139–47. [17] Marˇciukaitis G, Valivonis J, Vaˇskeviˇcius A. Analysis of behaviour of composite elements with profiled steel sheeting. Journal of Civil Engineering (Statyba) 2001;7(6):425–532 [in Lithuanian]. [18] Rzhanitsyn A. Built-up bars and plates. Moscow: Strojizdat; 1986 [in Russian]. [19] Zalesov AS, Muchanediev TA, Cistekov EA. Calculation of deflections of reinforced concrete structures according new standards. Journal of Concrete and Reinforced Concrete (Beton I Zelezobeton) 2002;6:12–20 [in Russian]. [20] Tenhovuori A, Karkkainen K, Kanerva P. Parameters and definitions for classifying the behaviour of composite slabs. In: Proceedings of an engineering foundation conference composite construction in steel and concrete. 1996. [21] Valivonis J, Kudzys A, Jurkˇsa A. The resistance of reinforced concrete beams with asbestos cement sheets in the tension zone. Journal of Concrete Structures (Gelˇzbetonin˙es konstrukcijos) 1987;15:24–36 [in Lithuanian]. [22] Marˇciukaitis G, Valivonis J, Jonaitis B. Influence of vertical precompression force on strength of contact between concrete and external profiled reinforcement. In: Selected papers of the 8th international conference modern building materials, structures and techniques. 2004.