DESIGN OF STEEL FIBRE REINFORCED CONCRETE SLABS ON GRADE

Transcription

1 DESIGN OF STEEL FIBRE REINFORCED CONCRETE SLABS ON GRADE Pro. Johan Silwerbrand Royal Institute o Technology (KTH) Dept. o Structural Engineering SE Stockholm, Sweden Abstract An increased use o steel ibre reinorced concrete (SFRC) slabs on grade demands improved design tools. The Swedish Concrete Association has proposed a design method or SFRC slabs. The designer might either choose a method based on uncracked concrete or one based on cracked concrete. In the irst case, the sum o mechanical and restrained stresses should be below the ultimate strength. In the latter case, the mechanical stress should be below a residual strength containing enhanced ductility demands due to restrain deormations. Tests on SFRC beams, conducted at the Dept. o Structural Engineering, have shown that structures previously cracked due to restrained movement have a considerable load carrying capacity. The paper provides design recommendations and a design example. 1. Introduction The use o steel ibre reinorcement has increased in concrete slabs on grade, especially in industrial oors. By choosing steel ibre reinorcement instead o conventional reinorcement, it is possible to save the labour-intensive and expensive reinorcing work. Most likely, the use o steel ibre reinorced concrete (SFRC) slabs on grade would increase urther i reliable, approved, and simple design methods were available. In traditional design methods or slabs on grade, the estimation o mechanical stresses, e.g., truck traic stresses, is airly good, whereas stresses due to restrained shrinkage or thermal movements are either neglected or considered very roughly. This might be acceptable or conventionally reinorced slabs with suicient reinorcement area and large ductility, but will not be suitable or SFRC slabs. The international engineering society has not established any design methods or SFRC slabs on grade yet, but several design proposals exist, see, e.g., Falkner et al. /1/, Bischo & Valsangkar /2/, and Skarendahl & Westerberg /3/. Most o these proposals 305

2 deal solely with mechanical loading. Bischo & Valsangkar discern between two design approaches: (i) providing the slab with a minimum reinorcement suicient to carry the mechanical load ater cracking and (ii) ensuring that the slab does not crack under the combined action o mechanical load and restraint load. A similar approach was used by the Swedish Concrete Association in It will be described below. 2. The Swedish Design Method 2.1 Design Philosophy The Swedish design method or SFRC slabs on grade was established by the Swedish Concrete Association in 1995 /4/. The 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. 1). 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 o the SFRC is assumed to contain any eects o restraint. This assumption is an adoption o Losberg s statement concerning reinorced concrete pavements: temperature and shrinkage do not inuence the ultimate moment /5/. Losberg s explanation is that stresses caused by temperature and shrinkage (restraint stresses) disappear as soon as the reinorcement stress has reached the yield point (i.e., ater cracking). It is obvious that SFRC with suicient ductility can be treated in a similar way. Experimental evidence or this statement will be given in Section 3 below. Design procedure Ultimate limit state Serviceability limit state Uncracked state Cracked state Ext. loads + restr. loads 1 Strength at irst crack Ext. loads + restr. loads Ultimate strength 1 Ext. loads Strength, increased ductility 1 Fig. 1 Design procedure according to the Swedish Concrete Association /4/. 2.2 Loads and Stresses Slabs on grade are subjected to external loads rom, e.g., lorries and ork-lit trucks and rom the storage o goods. Restraining loads may originate rom restrained thermal or shrinkage movements. Most industrial oors have a airly constant indoor climate. For 306

3 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 either be computed by using Westergaard s theory o elastic slab on a dense liquid (resilient or Winkler) oundation or the theory o a multi-layered elastic hal-space. It is without the scope o this paper to report the comprehensive research that has been devoted to this area. Only one solution will be given here as a support or the design example in Section 4 below. Westergaard s equations have been improved several times during the 20 th century. The exural stress σ [in MPa] due to interior loading, e.g., may be computed by the ollowing equation (Eisenmann /6/): 3 = 0 F Eh σ (1 ν ) lg (1) h kb 4 where, F = concentrated load [N], h = slab thickness [mm], E = modules o elasticity o the concrete slab [MPa], ν = Poisson s ratio o concrete, k = modulus o subgrade reaction [N/mm 3 ], a = radius o loaded circular area [mm], b = calculation parameter [mm], b = 1.6 a + h h or a < h, b = a or a > h 2 2. For the uncracked state, the tensile stress σ t due to restrained shrinkage (or simplicity: called shrinkage stress below) may be computed by the ollowing simple equation: σ t Eε cs = ψ 1+ φ (2) where, ψ = degree o restraint (0 ψ 1), E = modulus o elasticity o the concrete slab, ε cs = ree concrete shrinkage, φ = concrete creep coeicient. Eq. (2) is valid or the uniorm shrinkage distribution. In reality, the shrinkage distribution is non-uniorm with higher values at the top than at the bottom due to drying mainly upwards. The uniorm assumption is, however, conservative and is thereore used or simplicity. The degree o restraint ψ vanishes i the slab may shrink completely reely and equals unity or complete restraint. It is airly diicult to determine. It is, however, known to be dependent on the ollowing actors: 1. Slab length or distance between joints 2. Slab thickness 3. Distributed load in addition to the dead load o the slab 307

4 4. Friction between slab and subgrade 5. Presence o haunches or tapered cross section 6. Looking or bond to adjacent structures or construction elements Haunches and looking to adjacent structures, e.g., walls, columns, oundations, previously cast oor slabs, should be avoided. I so, the degree o restraint is mainly dependent on the riction and the ratio between slab size and slab thickness. Based on the Swedish Concrete Association /4/, the ollowing simpliied table may be used to roughly estimate the degree o restraint: Table 1 Estimation o the degree o restraint ψ at slab centre Subgrade Ratio between slab length (joint spacing) and slab thickness, L/h Even subgrade + two sheets o cling ilm > Compacted gravel Macadam without levelling Recently, a new production method has been developed making it possible to reduce the degree o restraint urther. It uses evenly spaced airbag liting the slab temporarily during the irst couple o weeks ater casting and initial curing enabling ree shrinkage and bilateral drying, see Silwerbrand & Paulsson /7/. For the cracked state, the yield line theory is used. Consequently, it is not meaningul to compute stresses. Instead, design load is compared with the load carrying capacity discussed in the ollowing subsection. 2.3 Material Strength and Load Carrying Capacity The material strength is evaluated through exural tests on beam specimens. According to the Swedish Concrete Association /4/, the beam is tested in our point bending (Fig. 2). Recommended measures are beam height h = 75 mm, beam width b = 125 mm, and span length l = 6h = 450 mm. F/2 F/2 h l/3 l/3 l/3 b Fig. 2 Beam tests according to the Swedish Concrete Association /4/. 308

5 From the relationship between computed maximum exural stress at midspan and midspan deection δ (Fig. 3), the ollowing three strength measures are derived: First crack strength cr Ultimate strength u Residual strength res The residual strength is deined as ollows: res R10,X = cr (3) 100 where, R 10,X = 100 (I X I 10 )/(X 100) is the residual strength actor [%], I X, I 10 = toughness indices according to ASTM /8/, I X = ratio between area under the stressdeection curve between δ = 0 and δ = (X + 1)δ cr /2 and the (elastic) area between δ = 0 and δ = δ cr. For a complete elastoplastic material I 10 = 10 and I X = X. The residual strength res (R 10,X ) can be proven to be equal to the average stress between δ = 5.5δ cr and δ = (X + 1)δ cr /2. The parameter X expresses the ductility demand. For concrete slabs on grade, the Swedish Concrete Association suggests X = 20 i there are no shrinkage cracking. Max exural stress u cr res δ cr 5.5δ cr 10.5δ cr 15.5δ cr Midspan deection δ Fig. 3 Evaluation o exural strength. Schematic relationship between stress and deection. Characteristic strength values can either be determined by testing several beams and statistics or estimated by testing three beams and deine the characteristic value as

6 % o the lowest value. Design values or exural and tensile strength may subsequently be determined by using the ollowing table: Table 2 Design values or exural and tensile strength t Ultimate limit state Serviceability limit state Uncracked state Cracked state Uncracked state Flexural strength Tensile strength t uk η γ γ m 0.6 η γ m n crk γ n resk η γ γ m 0.37 η γ m n resk γ n crk 0.6 Note. For slabs on grade, the partial coeicient or material η γ m = 1. 2 and the partial coeicient or security class γ n = The coeicient 0.6 is based on unpublished Finnish test results and the coeicient 0.37 is based on a theoretical discussion in DBV /9/. As stated above, design loads are to be compared directly with the load carrying capacity in the cracked state. Losberg applied the yield line theory on concrete slabs (pavements) on both a resilient oundation and an elastic hal-space /5/. The load carrying capacity F could be given by the ollowing principal unction: F = g(m, m, a/l) (4) where, g is a unction, m and m are the ultimate bending moment per unit width or positive (slab bottom) and negative (top) yield lines, respectively, a is the radius o the loaded circular area, and l is the radius o relative stiness deined as ollows: crk 4 l = D / k (5) where, D = Eh 3 /[12(1 ν 2 )] is the exural rigidity o the slab and k the modulus o subgrade reaction. In a homogeneous SFRC slab, there is no dierence between the ultimate moments in bottom and top, hence ' 2 m = m = h (6) 6 where, is the design value o the exural strength according to Table 2. Losberg /5/ presents solution to several loading cases, e.g., interior, edge, and corner loading or single and twin wheels. The simplest case with a single, interior load is shown in Fig. 4. For this case, Eq. (4) equals F = g( a / l) ( m + m' ) (7) 310

7 where, g(a/l) can be ind rom the graph in Fig. 4. I, e.g., a/l = 1, then F = 16.7(m+m ). F F/(m+m') a/l Fig. 4 Yield line pattern or interior load (let). Relationship between a/l ratio and relative load carrying capacity (right). Ater Losberg /5/. 2.4 Design Criteria or Ultimate Limit State For the uncracked state, the ollowing two design criteria should be ulilled: σ t + σ σ 1.3 t 1 1 (8) (9) where, σ is the exural stress due to external load, σ t the tensile stress due to restrained shrinkage, and and t the design values o the exural and tensile strength according to Table 2, respectively. For the cracked state, the ollowing design criteria should be ulilled: Q F (10) where, Q is the design value o the external load and F is the design value o the load carrying capacity according to Eq. 4. Any eect o restraint, e.g., restrained shrinkage, should be regarded by increasing the ductility demand when using the residual strength res or computing exural strength, ultimate moment m, and load carrying capacity F. 311

8 3. Tests on Combined Loading Alavizadeh-Farhang has tested SFRC beams subject to combined mechanical and thermal loading /10/. The beams were supported on our simple supports, closely spaced at the beam edges providing 75 percent o restraint to end rotation but enabling horizontal movements. In a irst step, the restrained beams were subjected to a mechanical point load at midspan, thermal heating on the top surace, or both. Ater cracking, the beams were simply supported on two supports and reloaded (Table 3). The ollowing three conclusions can be drawn rom the tests: (i) the restraint diminishes (exural moments disappear) dramatically ater thermal cracking, (ii) the cracking load decreases continuously or increasing thermal load and (iii) with one exception, thermally precracked beams have developed ultimate loads comparable to those developed by virgin beams. Consequently, SFRC structures may be designed solely or mechanical loads i the structure has suicient ductility. Table 3 Cracking and ultimate loads o beams subjected to combined loading, /10/ Test beam Beam on 4 supports: cracking load Beam on 2 supports Mechanical load (kn) Thermal load ( C) Ultimate load (kn) SRRC-P2 NA 13.1 SRRC-P4 NA 15.3 SRRC-T SRRC-T SRRC-T SRRC-PT SRRC-PT SRRC-PT Note. Beam SFRC-T4 had an exceptionally low strength due to a poor ibre distribution. 4. Design Example 4.1 Slab and concrete data The example deals with an industrial oor occasionally traicked by lorries with an axle load o 100 kn. The axle has two wheels and a circular contact area with radius a = 150 mm. Slab and concrete data are given in Table

9 Table 4 Slab and concrete data or the design example Slab Concrete Thickness 120 mm Compressive strength 30 MPa Joint spacing m Fibre content 60 kg/m 3 Subbase Compacted gravel Characteristic ultimate 4.0 MPa exural strength uk Subgrade reaction k 200 MN/m 3 Characteristic exural crack 3.9 MPa strength crk Thermal conditions Constant indoor Residual strength actor 80 % climate R 10,30 Relative humidity 50 % Modulus o elasticity E c 30 GPa Looking to adjacent structures None Free shrinkage 0.6 mm/m 4.2 Design according to theory o elasticity The slab has to be designed or combination o wheel load and shrinkage load. The exural stress or interior loading is σ = 2.7 MPa according to Eq. (1). The tensile stress due to shrinkage is σ t = 3,6 MPa according to Eq. (2) i the creep coeicient is estimated to φ = 4 due to low concrete age and high stress level, see Silwerbrand /11/. According to Tables 2 and 4, the exural and tensile strength values are = 3.3 MPa and t = 2.0 MPa, respectively. Since the tensile stress is larger than the tensile strength, the design criteria according to Eq. (8) is not ulilled. 4.3 Design according to yield line theory Using the yield line theory, only the loads are limited to the wheel load. In this case, the radius o relative stiness is l = 388 mm according to Eq. (5) and, hence, a/l = Fig. 4 gives the relative load carrying capacity F/(m+m ) = 9.9. The exural strength or the cracked state and enhanced ductility demands (using R 10,30 instead o R 10,20 ) can be computed by using Tables 2 and 4 and Eq. (3), i.e., = 2.6 MPa. The limited space o the paper does not allow a thorough discussion on how to estimate the proper ductility demand, but design guidance on this issue is given in /4/. With used ductility demand, the ultimate moments are m = m = 6.2 knm/m according to Eq. (6). The design criteria according to Eq. (10) gives the ollowing relationship: = ( ) Thus, the slab has suicient load carrying capacity or occasional loads, despite that it is likely to crack due to combined wheel and shrinkage loading. The large amount o ibres (60 kg/m 3 ) will, however, guarantee a dense crack pattern with harmless crack widths. 313

10 5. Concluding Remarks A new design method or SFRC slabs on grade has been proposed by the Swedish Concrete Association. The method provides the designer with two options, either or the uncracked state or or the cracked state. Swedish tests show that thermally precracked SFRC structures have a considerable load carrying capacity. A design example shows that the design based on the cracked state leads to a more slender slab or most practical cases i the SFRC has suicient ductility. Reerences 1. Falkner, H., Huang, Z., & Teutsch, Comparative Study o Plain and Steel Fibre Reinorced Concrete Ground Slabs, Concrete International 17 (1) (1995) Bischo, P.H., & Valsangkar, A.J., Assessment o Slab-on-Grade Design and Comparison with Model Slab Behaviour, Proceedings o the 4 th International Colloquium on Industrial Floors, Ostildern, Germany, 1999 (1) Skarendahl, Å., & Westerberg, B., Guide or Designing Fibre Concrete Floors, CBI Report 1:89, Swedish Cement and Concrete Research Institute, Stockholm, Sweden (1989) (In Swedish). 4. Swedish Concrete Association, Steel Fibre Reinorced Concrete Recommendations or Design, Construction, and Testing, Concrete Report No. 4, 2 nd Edn, Stockholm, Sweden (1997) (In Swedish). 5. Losberg, A., Design Methods or Structurally Reinorced Concrete Pavements, Bulletin No. 250, Chalmers University o Technology, Göteborg, Sweden (1961). 6. Eisenmann, J., Concrete Pavements. Handbook or Concrete, Reinorced Concrete and Prestressed Concrete Structures (Verlag von Wilhelm Ernst & Sohn, Berlin, München & Düsseldor, 1979) (In German). 7. Silwerbrand, J., & Paulsson, J., Reducing Shrinkage Cracking and Curling in Slabs on Grade, Concrete International 22 (1) (2000) ASTM C1018, Standard Test Method or Flexural Toughness and First-Crack Strength o Fibre-Reinorced Concrete (Using Beam with Third-Point Loading), 1992 Annual Book o ASTM Standards, Section 4 Construction, Volume Concrete and Aggregates, (ASTM, Philadelphia, 1992) DBV. Technology o Steel Fibre Reinorced Concrete and Shotcrete, Deutscher Beton-Verein E.V. DBV-Merkblätter Faserbeton, Fassung (1992) (In German). 10. Alavizadeh-Farhang, A., Plain and Steel Fibre Reinorced Concrete Beams Subjected to Combined Mechanical and Thermal Loading, Bulletin No. 38, Dept. o Structural Engineering, Royal Institute o Technology, Stockholm, Sweden (1998). 11. Silwerbrand, J., Stresses and Strains in Composite Concrete Beams Subjected to Dierential Shrinkage, ACI Structural Journal 94 (4) (1997)