DESIGN OF STEEL FIBRE REINFORCED CONCRETE SLABS ON GRADE FOR RESTRAINED LOADING

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1 DESIGN OF STEEL FIBRE REINFORCED CONCRETE SLABS ON GRADE FOR RESTRAINED LOADING J. Silfwerbrand Swedish Cement and Concrete Research Institute (CBI), Sweden Abstract Concrete slabs on grade are mainly subjected to mechanical loads and restrained loading. The designer could easily treat mechanical loads by using various tools. Restrained loading caused by restrained shrinkage or thermal movements are, however, often dealt with in a very approximate way. Swedish guidelines have suggested two possible ways to handle restrained loading in steel fibre reinforced concrete (SFRC) slabs. If the ductility of the SFRC is sufficiently high, Losberg s hypothesis stating that restrained stresses vanish as soon as the reinforcement yields is used also for the SFRC slab. Solely mechanical loads have to be treated in the design. Contrary, for lower fibre contents, the SFRC slab is in principle treated as an un-reinforced concrete slab. Here, the effects of mechanical and restrained loads must be combined. The paper describes the Swedish method to compute stresses due to restrained loading. In indoor climate, the restrained shrinkage stresses are predominant. They depend on developed free shrinkage, modulus of elasticity, creep, and the degree of restraint. The degree of restraint is the most difficult parameter to determine, but the guidelines propose different values for different circumstances, e.g., material and treatment of the uppermost layer beneath the concrete slab, joint spacing, slab thickness, and locking effects. 1. Introduction The use of steel fibre reinforcement has increased in concrete slabs on grade, especially in industrial floors. By choosing steel fibre reinforcement instead of conventional reinforcement, it is possible to save the labour-intensive and expensive reinforcing work. Most likely, the use of steel fibre reinforced concrete (SFRC) slabs on grade would increase further if reliable, approved, and simple design methods were available. In traditional design methods for slabs on grade, the estimation of mechanical stresses, e.g., truck traffic stresses, is fairly good, whereas stresses due to restrained shrinkage or thermal movements are either neglected or considered very roughly. This might be 975

2 acceptable for conventionally reinforced slabs with sufficient reinforcement area and large ductility, but will not be suitable for SFRC slabs. There are two different principles of avoiding uncontrolled shrinkage cracking: (i) elimination of the causes and (ii) limitation of the consequences (Fig. 1). The cause elimination implies reduction of shrinkage, restraint or both. To limit the consequences, the concrete floor could either be reinforced or jointed. Shrinkage cracks Eliminate causes Limit consequences Reduce restraint Reduce shrinkage Reinforce Provide joints Fig. 1 Principles to avoid uncontrolled shrinkage cracking. 2. The Swedish Design Philosophy The international engineering society has not established any design methods for SFRC slabs on grade yet, but several design proposals exist, see, e.g., Falkner et al. [1], Bischoff & Valsangkar [2], and Skarendahl & Westerberg [3]. Most of these proposals deal solely with mechanical loading. Bischoff & Valsangkar discern between two design approaches: (i) providing the slab with a minimum reinforcement sufficient to carry the mechanical load after cracking and (ii) ensuring that the slab does not crack under the combined action of mechanical load and restraint load. A similar approach was used by the Swedish Concrete Association in 1995 [4]. It will be described briefly below. The Swedish design method covers both ultimate limit state and serviceability limit state. For the ultimate limit state, the designer might select either the uncracked or the cracked state (Fig. 2). In the uncracked state, the concrete slab has to carry both external loads and restraining loads. In the cracked state, it only needs to carry external loads. The ductility of the SFRC is assumed to contain any effects of restraint. This assumption is an adoption of Losberg s statement concerning reinforced concrete pavements: temperature and shrinkage do not influence the ultimate moment [5]. Losberg s explanation is that stresses caused by temperature and shrinkage (restraint stresses) disappear as soon as the reinforcement stress has reached the yield point (i.e., after cracking). It is obvious that SFRC with sufficient ductility can be treated in a similar way. Experimental evidence for this statement has been provided by tests by Alavizadeh-Farhang [6]. 976

3 Design procedure Ultimate limit state Serviceability limit state Uncracked state Cracked state Ext. loads restr. loads 1 Strength at first crack Ext. loads restr. loads 1 Ultimate strength Ext. loads 1 Strength,increased ductility Fig. 2 Design procedure according to the Swedish Concrete Association [4]. 3. Stresses due to Restrained Shrinkage Slabs on grade are subjected to external loads from, e.g., lorries and fork-lift trucks and from the storage of goods. Restraining loads may originate from restrained thermal or shrinkage movements. Most industrial floors have a fairly constant indoor climate. For such slabs, concentrated loads and loads due to restrained shrinkage are the most important ones. For the uncracked state, the stresses due to concentrated loads could be computed by using either Westergaard s theory of elastic slab on a dense liquid (resilient or Winkler) foundation or the theory of a multi-layered elastic half-space. It is without the scope of this paper to report the comprehensive research that has been devoted to this area. For the uncracked state, the tensile stress t due to restrained shrinkage (for simplicity: called shrinkage stress below) may be computed by the following simple equation: Ec cs t 1 where, = degree of restraint (0 1), E c = modulus of elasticity of the concrete slab, cs = free concrete shrinkage, = concrete creep coefficient. Eq. (1) is valid for the uniform shrinkage distribution. In reality, the shrinkage distribution is non-uniform with higher values at the top than at the bottom due to drying mainly upwards. The uniform assumption is, however, conservative and is therefore used for simplicity. Whereas Eq. (1) is simple, the estimation of the variables involved is more complicated. The modulus of elasticity and the creep coefficient express together the relationship (1) 977

4 between stress and strain. It means that high values of E and might give the same stress as somewhat lower values of both the variables. The engineering approach is to use a design value of E given in codes or handbooks and try to estimate as good as possible considering concrete mixture, concrete age, loading time, relative stress (ratio between stress and strength), environmental temperature and relative humidity, and concrete thickness. The free concrete shrinkage is dependent on concrete mixture, time, environmental temperature and relative humidity, concrete thickness and number of desiccation directions. The degree of restraint vanishes if the slab can shrink completely freely and equals unity for complete restraint. It is fairly difficult to determine. It is, however, known to be dependent on the following factors: 1. Slab length or distance between joints 2. Slab thickness 3. Distributed load in addition to the dead load of the slab 4. Friction between slab and subgrade 5. Presence of haunches or tapered cross section 6. Looking or bond to adjacent structures or construction elements Often, the effects of slab length and slab thickness are combined stating that the ratio between them is the crucial factor. Haunches and looking to adjacent structures, e.g., walls, columns, foundations, previously cast floor slabs, should be avoided. If so, the degree of restraint is mainly dependent on the friction and the ratio between slab size and slab thickness. Some researchers have tried to measure the friction ratio between concrete slab and different subgrades (Table 1). These measurements are, however, based on short time loading and to use them for computing stresses due to restrained shrinkage are questionable due to completely different loading rates. On the other hand, the measurements are interesting for ranking the various subgrade materials and preparations. Despite the fact that there is a relationship between the friction ratio and the degree of restraint, no mathematical expression or definitive interrelationship has been defined. However, the following statements can be given: Increasing friction increasing restraint. Increasing slab length or joint spacing increasing restraint. Increasing distance from edge or joint increasing restraint. 978

5 Table 1 Friction ratio for concrete slabs on various subgrades. Subgrade Friction ratio according to Swedish Concrete Handbook [7] Dorell & Nordberg [8] Coarse gravel or macadam without > 2.0 leveling Well compacted coarse gravel 1.6 Compacted course gravel 1.3 Stiffened and leveled subgrade 1.5 Cling film on compacted coarse gravel 0.9 Insulating interlayer 1.0 Even sand layer or cling film 0.75 h Fig. 3 Simple model to determine relation between normal and shear stresses. The simplest theory using equilibrium and assuming that the product of friction ratio and normal force gives the friction does not work (Fig. 3). This can be shown by a simple calculation: Equilibrium: b h b x Friction: h x = cr = f ct = 2 MPa, = 1, = MN/m 3 f ct xcr 83 m Lcr / 2 where, b = slab width, h = slab thickness, x = distance between slab edge and considered section, x cr = distance from slab edge to first crack, = normal stress, cr = crack stress, = shear strength, = gravity of concrete in N/m 3, f ct = tensile strength, L cr = maximum slab length or joint spacing before cracking (experience from the field indicates that L cr usually is 4 m < L cr < 20 m). It is obvious that we need to find another way to estimate the degree of restraint. The Swedish Concrete Association derived an approximate relationship between and. It was based on the following steps and assumptions: b 979

6 The method was based on recommended joint spacing in plain, crack-free concrete slabs on grade (Table 2). The recommended values were assumed to represent a frequently used floor concrete and common subgrade conditions. Shrinkage stress t can be computed with Eq. (1). In slabs with a recommended joint spacing the crack risk is assumed to be = 0.7. Measured short time values of friction ratio are useful for relating other subgrade cases to the typical one. The degree of restraint increases linearly from edge or joint. Table 2 Maximum L/h ratio according to PCA [9] Consistency Fluid Semi-fluid Slump (mm) < 100 Maximum aggregate size (mm) < L/h The derivation of relationships is illustrated by the following example: f cc = 30 MPa, t = f ct = 2 MPa, E c = 32 GPa, cs = 0.4 mm/m, = 4, slump < 100 mm cr f cc and = 1. Ec cs L/h = 30 = 0.7 and = cr = = Other relationships between and are given in Table 3. Table 3 Estimation of the degree of restraint at slab centre Friction ratio Ratio between slab length (joint spacing) and slab thickness, L/h > These values are valid in the interior parts of the slab. At free slab edges and joints, there is no restraint against the shrinkage, i.e., the degree of restraint vanishes (Fig. 4). 980

7 h = 1 = 0 = 0 15 h L - 30 h 15 h Fig. 4 The degree of restraint may be assumed to vary linearly from = 0 at slab edges or joints and = 1 in an interior part. The figure shows an example of a concrete slab resting on macadam without levelling. 4. Field Tests and Measurements 4.1 Storage floor in Örebro, Sweden In 1992, a m 2 storage floor was constructed for ASG in Örebro, Nordin & Westin [10]. The floor was a 120 mm thick steel fibre reinforced concrete slab on grade. The concrete was of Swedish concrete grade K35 (28 day compressive cube strength 35 MPa), maximum aggregate size = 16 mm, 25 kg/m 3 steel fibres (Dramix 80/80) and a slump of mm. The concrete slab was resting on a sand layer. The joint spacing was 22 m in both directions. The joints were provided with bitumen coated dowel bars at mid depth of the slab. One square shaped slab was thoroughly observed and measured. No cracks were observed during the first eight months. The relative humidity above the floor was measured to 55 %. The shrinkage movement of the slab was measured to less than 0.15 mm/m after six weeks. Unfortunately, no free shrinkage measurements are reported, but the available material data and reported shrinkage measurements make it possible to make a rough estimation on the conditions of the slab after six weeks. If free shrinkage cs (shortening is defined as positive shrinkage) and shrinkage movement (elongation is defined as positive strain) of the slab are measured, the tensile stress t may be computed with the following equation: t E ( cs ) (2) 1 where, E and are modulus of elasticity and creep of concrete. By comparing Eqs. (1) and (2), the degree of restraint may be computed by the following relationship: 1 (3) cs In the Örebro case, = 0.10 to = 0.15 mm/m. The free shrinkage of standard prisms of the Örebro concrete may be 0.3 to 0.4 mm/m after six weeks. The slab on grade dries 981

8 mainly in one direction whereas the prism desiccates in four directions. The free shrinkage of the slab may be estimated to somewhere between cs = 0.2 and cs = 0.3 mm/m after six weeks. Hence, the degree of restraint is somewhere between = 0.25 and = These values may be compared with the values in Table 3. In the Örebro case, L/h = 22/0.11 = 200 and = 0.75 = 1. Apparently, the tabled value overestimates the restraint in this case. It is due to shortcomings in the derivation of the tabled values and the striving after conservative values. 4.2 Storage floor in Eskilstuna, Sweden In 2002, a m 2 industrial floor was completed for H&M in Svista close to Eskilstuna, Sweden, Hedebratt [11]. The new building is used as a central storage for the clothing company H&M. The industrial floor is made of SFRC. It rests on two layers of cling film, 20 mm of stone powder, 200 mm compressed macadam, and 1 m blast stone layer. Consequently, the subgrade and subbase provide both load carrying capacity and horizontal movement possibilities. Hedebratt investigated a part of the floor totalling 3000 m 2. The concrete has a w/c ratio of 0.6, a cement content of 300 to 315 kg/m 3 and a maximum aggregate size of 25 mm. The fibre content is 32 kg/m 3 (Dramix RC-65/60-BN). The compressive cube strength was 35 MPa. The thickness is 140 mm. The primary joint distance is 30 m, but soff cuts are made every 10 m. At Hedebratt s end inspection, 263 days after concrete placement, the soff cuts were still not opened (i.e., not cracked). Hedebratt measured the strains in the industrial floor using acoustic gages placed at two depths and nine locations [11]. After 263 days, the measured strain varied between and mm/m. Simultaneously, mm standard prisms developed 1.06 mm/m. The difference depends on different degree of restraint, different number of drying directions, different thickness, and different temperature. Hence, the results are difficult to interpret. The temperature difference is relatively small and may be neglected. The thickness difference and the different number of drying directions (prism has mainly 4, floor has 1) may be combined using, e.g., CEB-FIP MC 1990 [12]. Calculations indicate that the floor develops 50 to 60 % of the prism shrinkage during the investigated time. The free shrinkage of the floor ought to be in the range of 0.5 to 0.7 mm/m. By reusing Eq. (3), the degree of restraint can be computed to somewhere between = 0.4 and = 0.6. The double layers of cling film produces low friction. Measured values are missing, but estimations indicate 0.1 < < 0.5. But since the ratio between joint spacing and thickness equals L/h = 30/0.14 = 214, Table 3 suggests = 1, i.e., considerably above the value based on measurements. Consequently, the values of Table 3 must be considered as conservative. The SFRC in question has performed well. Some cracking have been observed close to edge beams and columns, but the major parts of the investigated area is crack free [11]. The lack of observed cracking supports the statement that Table 3 is conservative. 982

9 5. Load Combinations The Swedish Concrete Association s design method [4] also covers load combinations, i.e., combined effects of external load and restrained shrinkage or other restrained movements. For the uncracked state, the following two design criteria should be fulfilled: fl t f f fl t 1 (4) fl 1.3 f fl 1 (5) where, fl is the flexural stress due to external load, t is the tensile stress due to restrained shrinkage, and f fl and f t are the design values of the flexural and tensile strength, respectively. For the cracked state, the following design criteria should be fulfilled: P F (6) where, P is the design value of the external load and F is the design value of the load carrying capacity. Any effect of restraint, e.g., restrained shrinkage, should be regarded by increasing the ductility demand (Fig. 5) when using the residual strength f flres for computing flexural strength f fl, ultimate moment m, and load carrying capacity F. Max flexural stress fflu fflcr fflres cr 5.5 cr 10.5 cr 15.5 cr Midspan deflection Fig. 5 Increased ductility demand due to restrained shrinkage implies that the residual strength f flres is evaluated from midspan deflection between 5.5 cr and, e.g., 15.5 cr instead of between 5.5 cr and 10.5 cr. 983

10 6. Concluding Remarks In many concrete floors, restrained shrinkage stresses are more important than stresses caused by external loads, e.g., traffic loads. The Swedish Concrete Association has developed a design method which also covers restrained shrinkage stresses. The shrinkage stresses can be computed if modulus of elasticity, free shrinkage, creep, and degree of restraint are given. The degree of restraint is of uppermost importance for determining the stresses, but is difficult to estimate. The Swedish Concrete Association has proposed values for varying conditions between concrete slab and subgrade and varying ratios between joint spacing and slab thickness. Field measurements show a large scatter, but indicate that the Association s tabled values are conservative. More and improved measurements are desired to improve the design values of the degree of restraint. References 1. Falkner, H., Huang, Z., & Teutsch, Comparative Study of Plain and Steel Fibre Reinforced Concrete Ground Slabs, Concrete International, 17 (1) (1995) Bischoff, P.H., & Valsangkar, A.J., Assessment of Slab-on-Grade Design and Comparison with Model Slab Behaviour, Proceedings, 4 th International Colloquium on Industrial Floors, Ostfildern, Germany, January 1999, Vol. I, Skarendahl, Å., & Westerberg, B., Guide for Designing Fibre Concrete Floors, CBI Report 1:89, Swedish Cement and Concrete Research Institute, Stockholm, Sweden, (In Swedish). 4. Swedish Concrete Association, Steel Fibre Reinforced Concrete Recommendations for Design, Construction, and Testing, Concrete Report No. 4, 2 nd Edn, Stockholm, Sweden, (In Swedish). 5. Losberg, A., Design Methods for Structurally Reinforced Concrete Pavements, Bulletin No. 250, Chalmers University of Technology, Göteborg, Sweden, 1961, 143 pp. 6. Alavizadeh-Farhang, A., Concrete Structures Subjected to Combined Mechanical and Thermal Loading. Bulletin No. 60 (Ph.D. Thesis), Dept. of Structural Engineering, Royal Institute of Technology, Stockholm, Sweden, 294 pp. 7. Swedish Concrete Handbook Execution. AB Svensk Byggtjänst & Cementa AB, Stockholm, 1992, 837 pp. (In Swedish). 8. Dorell, J., & Nordberg, E., Limiting Shrinkage Cracks in Steel Fibre Reinforced Concrete Overlays. Master Thesis No. 1993:165 E, Dept. of Structural Engineering, Luleå Technical University, Luleå, (In Swedish). 9. PCA. Portland Cement Association, Concrete Floors on Ground. Engineering Bulletin No. EB075, Skokie, Illinois, USA, Nordin, A., & Westin, I. Concrete Floors on Ground. Betong, (2) (1992) (In Swedish). 11. Hedebratt, J., Integrated Design and Construction of Industrial Floors Pilote Study 1. Report No. 69, Chair of Structural Design and Bridges, Dept. of Structural Engineering, Royal Institute of Technology, Stockholm, Sweden, 294 pp. 12. CEB-FIP Model Code London: Th. Telford 1993, 460 pp. Comité Euro-International du Béton, Bulletins d Information Nos. 213/