Time-dependent deflection of composite concrete slabs - a simplified design approach

Transcription

1 Southern Cross University 23rd Australasian Conference on the Mechanics of Structures and Materials 2014 Time-dependent deflection of composite concrete slabs - a simplified design approach A Gholamhoseini University of New South Wales R I. Gilbert University of New South Wales M A. Bradford University of New South Wales Publication details Gholamhoseini, A, Bilbert, RI, Bradford, MA 2014, 'Time-dependent deflection of composite concrete slabs - a simplified design approach', in ST Smith (ed.), 23rd Australasian Conference on the Mechanics of Structures and Materials (ACMSM23), vol. II, Byron Bay, NSW, 9-12 December, Southern Cross University, Lismore, NSW, pp ISBN: epublications@scu is an electronic repository administered by Southern Cross University Library. Its goal is to capture and preserve the intellectual output of Southern Cross University authors and researchers, and to increase visibility and impact through open access to researchers around the world. For further information please contact epubs@scu.edu.au.

2 23rd Australasian Conference on the Mechanics of Structures and Materials (ACMSM23) Byron Bay, Australia, 9-12 December 2014, S.T. Smith (Ed.) TIME-DEPENDENT DEFLECTION OF COMPOSITE CONCRETE SLABS A SIMPLIFIED DESIGN APPROACH A. Gholamhoseini Centre for Infrastructure Engineering and Safety, School of Civil and Environmental Engineering, UNSW Australia, Sydney, NSW, 2052, Australia. z @student.unsw.edu.au R.I. Gilbert* Centre for Infrastructure Engineering and Safety, School of Civil and Environmental Engineering, UNSW Australia, Sydney, NSW, 2052, Australia. i.gilbert@unsw.edu.au (Corresponding Author) M.A. Bradford Centre for Infrastructure Engineering and Safety, School of Civil and Environmental Engineering, UNSW Australia, Sydney, NSW, 2052, Australia. m.bradford@unsw.edu.au ABSTRACT Based on an existing analytical model developed by the authors to calculate the time-dependent deflection of composite slabs, a simplified procedure, suitable for use in structural design, is proposed for calculating the time-dependent deflection of composite concrete slabs taking into account the timedependent effects of creep and shrinkage. The method is illustrated by two examples and the results are compared with laboratory measurements and with values obtained from numerical analyses. KEYWORDS Composite slabs, creep, deflection, profiled steel decking, shrinkage. INTRODUCTION Composite one-way concrete floor slabs with profiled steel decking as permanent formwork are commonly used in the construction of floors in buildings. Despite their common usage, relatively little research has been reported on the in-service behaviour of composite slabs. In particular, the drying shrinkage profile through the slab thickness (which is greatly affected by the impermeable steel deck) and the restraint to shrinkage provided by the deck have only recently been quantified (Gholamhoseini et al., 2012; Gilbert et al., 2012; Ranzi et al. 2012). Gholamhoseini et al. (2012) carried out an experimental study of the long-term behaviour of composite concrete slabs with two different types of steel decking (KF70 and KF40 manufactured by Fielders Australia) and subjected to sustained service loads and drying shrinkage. The results showed the dominant role played by drying shrinkage in the long-term deflection of composite concrete slabs. Based on the their findings, Gholamhoseini et al. (2012) proposed a non-linear shrinkage profile through the thickness of a composite concrete slab, together with an analytical model based on the provisions of Australian Standard AS for calculating the instantaneous and time-dependent curvature at a typical cross-section due to the effects of both load and non-linear shrinkage. Notwithstanding the research mentioned above, there is still little design guidance available to practising engineers for predicting the time-dependent in-service deformation of composite slabs. Although rigorous techniques are available for the time-dependent analysis of composite slabs (Gilbert & Ranzi, 2011), simplified, design-oriented procedures are not available. This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this license, visit 675

3 In this paper, an analytical model, previously proposed by the authors, has been used to generate simple-to-use graphs to obtain two parameters that are needed to calculate the creep and shrinkageinduced curvatures in a composite slab. Examples illustrating the use of the graphs are also presented and the results are compared with the values obtained from experimental studies. TIME-DEPENDENT DEFLECTIONS Using the age-adjusted effective modulus method (AEMM), Gilbert (2001) derived empirical, but simple-to-use, equations to model the long-term shrinkage and creep-induced changes in curvature (κ sh and κ creep, respectively) on reinforced concrete cross-sections under constant sustained internal actions: sh = k r sh / D (1) creep = i.sus ( (t,t o ) / ) (2) where D is the overall depth of the cross-section; ε sh is the shrinkage strain; (t,t o ) is the creep coefficient at time t due to load first applied at age t o ; and κ i.sus is the instantaneous curvature due to the sustained load. The term is a creep modification factor that accounts for the effects of cracking and the restraining action of the reinforcement on creep and is a function of the extent of cracking on the cross-section and the area and position of the bonded reinforcement in the tensile and compressive zones. The term k r is the shrinkage modification factor and depends on the extent of cracking, the level of tension stiffening and the area and position of bonded reinforcement in the tensile and compressive zones. In his derivations, Gilbert (2001) assumed that the shrinkage profile through the slab thickness is uniform and so the direct application of Eqs. 1 and 2 to composite slabs is inappropriate. An analytical procedure for the time-dependent analysis of composite concrete cross-sections with uniform shrinkage and with full interaction between the concrete and the decking was presented by Gilbert and Ranzi (2011) using the age-adjusted effective modulus method (AEMM). Gilbert et al. (2012) extended the method to calculate the effects of a non-uniform shrinkage gradient by layering the concrete cross-section, with the shrinkage strain specified in each concrete layer depending on its position within the cross-section. Based on their experimental study, Gholamhoseini et al. (2012) proposed a modification to the provisions of AS to determine the shrinkage strain and creep coefficient for composite slabs. For a composite slab with profiled steel decking, if the average thickness of the concrete t ave (in mm) is defined as the area of the concrete part of the cross-section A c, divided by the width of the cross-section b, the hypothetical thickness used to account for the effect of the steel decking on the the magnitude of creep coefficient and shrinkage strain is given by: t h = t ave (in mm) (3) With this value of t h, the creep coefficient (t,t o ) and shrinkage strain value ε sh (t,t c ) may be determined in accordance with AS The measured shrinkage strain at any height y above the soffit of the composite slab with overall depth D, ε sh (y), may be approximated by Eq. 4: 4 ( y) ( y / D) ( t, t (4) sh ) where ε sh (0) = ε sh (t,t c ), is the shrinkage strain at the bottom of the slab (at y = 0) and ε sh (D) =( + ) ε sh (t,t c ) is the shrinkage strain at the top surface of the slab (at y = D). From the experimental results, = 0.2 provides a reasonable estimate, but appears to depend on the profile of the steel decking. For the KF70 decks, = 0.5 and, for the KF40 decks, = 1.0. NUMERICAL ANALYSIS The age-adjusted effective modulus method (AEMM) was used in accordance with the model described previously by Gilbert et al. (2012) to calculate κ sh and κ creep for three slab thicknesses of D = 135 mm, D = 150 mm and D = 180 mm for each of two steel deck types KF70 and KF40 (from Fielders Australia and as shown in Figure 1). For each deck type, two steel sheeting thicknesses were considered, namely t sd = 0.75 mm and t sd = 1.0 mm. The modulus of elasticity for the steel sheeting was E sd = MPa. Analyses were undertaken for three different concrete strength, i.e. 25 MPa (E c = MPa), 32 MPa (E c = MPa) and 40 MPa (E c = MPa). For each slab, both the cases of an uncracked section and a cracked section were considered. sh c ACMSM

4 From the calculated values of κ sh and κ creep, the values of k r and as defined in Eq. 1 and Eq. 2 were obtained. Figure 2 shows the variation in the values of k r and for 150 mm thick and 180 mm thick uncracked (UC) and cracked (CR) slabs with KF70 decking for varying compressive and tensile reinforcement ratios and varying concrete strengths. The complete set of values of k r and for uncracked and cracked slabs with different thicknesses and different decking types has been reported elsewhere (Gholamhoseini et al. 2014). In the right hand legend of each figure, the concrete strength is followed by the tensile reinforcement ratio = A sd / bd, where b is the width of the concrete compressive zone and d is the effective depth from the compressive edge to the centroid of the steel decking, i.e. d = D y sd. The horizontal axis of each figure is the compressive reinforcement ratio, = A sc / bd, where A sc is the area of any steel reinforcement in the compressive zone and d is the distance from the tensile face (i.e. the soffit) of the slab to the centroid of the compressive reinforcement y sd = 27.7 y sd = (a) KF70 (A sd = 1100 mm 2 /m; I sd = mm 4 /m) (b) KF40 (A sd = 1040 mm 2 /m; I sd = mm 4 /m) 742 Figure 1. Dimensions (in mm) of each steel decking profile Cracking of the tensile concrete in the slab is treated in the analysis here using the approach outlined in Eurocode 2, whereby the curvature after cracking is obtained from: = cr + (1 ) uncr (5) where cr is the curvature at the section calculated by ignoring concrete in tension; uncr is the curvature on the uncracked transformed section; and is a distribution coefficient that accounts for the moment level and the degree of cracking and is given by: = 1 (M cr / M s * ) 2 where (6) The coefficient accounts for the effects of duration of loading and is taken as 0.5 for sustained or long-term loading. The cracking moment M cr is the moment required to produce a maximum concrete tensile stress equal to the mean uniaxial tensile strength of concrete, f ctm, and M s * is the maximum inservice moment on the section under consideration. Worked Examples Two examples are solved here to show how the time-dependent deflection can be calculated by using the graphs of Figure 2. Example-1: The time-dependent deflections of two identical composite slabs (2LT-70-3 and 3LT-70-3) tested by Gholamhoseini et al. (2012) are to be calculated. Each slab was simply-supported over a span of 3100 mm, with a 1200-mm-wide and 150-mm-thick cross-section and a 0.75-mm-thick KF70 steel decking. Both slabs contained no reinforcement (other than the external steel decking). Each slab was placed onto its supports at age 7 days and carried only its self-weight (3.60 kn/m) until age 64 days. At age 64 days, an additional uniformly distributed superimposed sustained load of 4.08 kn/m was applied to each slab and remained in place until the end of the test period at age 247 days. Both slabs remained uncracked throughout the test. The data required for the analysis are: E c = MPa, f c = 28 MPa, A sd = 1320 mm 2, d = mm, = , A c = mm 2, I uncr = mm 4. Solution: The average concrete thickness and hypothetical thickness are: t ave = /1200 = mm and t h = = mm. The shrinkage strain at the end of the test, at age 247 days, is determined in accordance with AS as ε sh (247,7) = and, for loading at ages 7 days and 64 days, the creep coefficient is determined in accordance with AS as (247,7) = 5.43 and (247,64) = The instantaneous deflections at the time of placing the slabs on the supports at age 7 days under the self-weight loading and at age 64 days caused by application of the superimposed sustained load of 4.08 kn/m are: (Δ i.sus ) 7 = 5w sw L 4 / (384 E c I uncr ) = /( ) = 0.51 mm. (Δ i.sus ) 64 = 5w sup L 4 / (384 E c I uncr ) = /( ) = 0.57 mm. 40 ACMSM

5 k r k r α UC-KF70 - D = 150 (mm) 0.30 (a) k r for an uncracked 150 mm slab (a) CR-KF70 - D = 180 (mm) (c) (c) k r for a cracked 180 mm slab UC-KF70 - D = 180 (mm) 1.2 (e) for an uncracked 180 mm slab (e) k r α UC-KF70 - D = 180 (mm) 0.25 (b) k r for an uncracked 180 mm slab (b) UC-KF70 - D = 150 (mm) 1.2 (d) (d) for an uncracked 150 mm slab CR-KF70 - D = 180 (mm) (%) (%) α 4.8 (f) for a cracked 180 mm slab (f) (%) (%) (%) (%) Figure 2. Values of k Figure 2. Values of k r and α for uncracked and cracked KF70 sections (D = 150 and 180 mm) r and α for uncracked and cracked KF70 sections (D = 150 and 180 mm). From Figure 2(a), with = 0, the shrinkage modification factor for uncracked slab is k r = 0.52 and the shrinkage-induced curvature is obtained from Eq. 1 as: κ sh.uc = /150 = mm -1 and the shrinkage-induced deflection at age 247 days is therefore: Δ sh = /8 = 2.21 mm. From Figure 2(d), the creep modification factor is = 1.37 and, since the slabs are uncracked between age 7 days and 247 days, the creep-induced deflection may be simply calculated as: creep = (Δ i.sus ) 7 (247,7)/α +(Δ i.sus ) 64 (247,64)/α = / /1.37 = 3.33 mm The time-dependent change in deflection between first loading (7 days) and age 247 days is therefore: time = Δ sh + Δ creep + (Δ i.sus ) 64 = = 6.11 mm. This compares well with the measured deflection of the two slabs during the long-term tests (i.e mm and 5.84 mm) and the value of 6.04 mm calculated in a numerical analysis by Gholamhoseini et al. (2012) using the age-adjusted effective modulus method (AEMM) to model the non-linear timedependent behaviour of concrete. Similar calculations were carried out at regular intervals throughout the duration of the test and the results are presented in Figure 3, showing excellent agreement with both the test and analytical outcomes. ACMSM

6 Mid-span deflection (mm) Mid-Span Deflection (mm) 7 6 Example-2 (Slabs 2LT-70-3 & 3LT-70-3) Experimental 1 Analytical Simplified Time Time after after commencement Commencement of of Drying drying (days) (days) Figure 3. Mid-span deflection versus time curves in Example 1 Example-2: The final total deflection of a simply-supported, 180-mm-thick, one-way slab with 1 mm thick KF70 decking is to be calculated. The span of the slab is 4800 mm. The slab props are removed at age 7 days and the slab carries its self-weight (3.8 kn/m 2 ) and two line loads of P = 8.0 kn/m applied to the slab, as shown in Figure 4a. The loads are assumed to remain permanently in place until t = 10 4 days. The bending moment diagram due to the sustained service load is shown in Figure 4b. The cracking moment for the slab at the time of loading is M cr = 13.0 knm/m. The slab contains 12 mm diameter bars at 200 mm centres located 30 mm from the top surface of the slab throughout the span (d = 150 mm). The slab is located in a near-coastal environment in Melbourne. Take f c = 32 MPa, E c = MPa, A sc = 550 mm 2 /m, = A sc / bd = , A sd = 1467 mm 2 /m, y sd = 27.7 mm, d = mm, = A sd / bd = , w sw = 3.8 kpa (slab self-weight), A c = mm 2 /m, I uncr = mm 4 /m, I cr = mm 4 /m (cracked second moment of area of the cross-section). P = 8 kn P = 8 kn w = 3.8 kn/m sw A B C D 1.5 m 1.8 m 1.5 m E A A' B C D E ' E (a) Elevation (b) Bending moment diagram (knm) Figure 4. Loading configuration and bending moment diagram (Example 2) Solution: The average concrete thickness is t ave = A c / b = 154 mm, and from Eq. 3, the hypothetical thickness is t h = 127 mm. The final shrinkage strain is determined in accordance with AS as ε sh (10 4,7) = and, for loading at ages 7 days, the final creep coefficient is determined in accordance with AS as (10 4,7) = From Figures 2(b) and 2(c), the shrinkage modification factors for the uncracked and the cracked slab are k r = 0.41 and k r = 0.56, respectively. From Eq. 1, the final shrinkage induced curvature on the uncracked and the cracked slab section are sh.uncr = mm -1 and sh.cr = mm -1, respectively. From Figures 2(e) and 2(f), the creep modification factors for uncracked and the cracked slab are = 1.51 and = 6.78, respectively. The final curvature is now calculated at selected points along the span, notably at A, A, B, C, D, E and E (as shown in Figure 4). At A (and E): At each support, M s * = 0, and the slab is uncracked. Thus, = sh.uncr = mm -1. At A (and E ): At the point between A and B (and between D and E), where M s * = M cr = 13.0 knm/m (i.e. at m from each support), for long-term calculations = 0.5, and from Eq. 6, = 0.5. The instantaneous curvatures on the uncracked and cracked sections are i.uncr = M s * /E c I uncr = mm -1 and i.cr = M s * /E c I cr = mm -1, respectively. From Eq. 2, the creep induced curvature on the uncracked and cracked sections are creep.uncr = i.uncr ( (10 4,7)/ ) = mm -1 and creep.cr = i.cr ( (10 4,7)/ ) = mm -1, respectively. Therefore, the final curvatures on the uncracked and cracked sections are uncr = i.uncr + creep.uncr + sh.uncr = mm -1 and cr = i.cr + creep.cr + sh.cr = mm -1, respectively. The final curvature is obtained from Eq. 5, = mm -1. ACMSM

7 Similarly at B (and D): M s * = 21.4 knm/m, = 0.82, i.uncr = mm -1, i.cr = mm -1, creep.uncr = mm -1, creep.cr = mm -1, uncr = mm -1, cr = mm -1 and = mm -1. Similarly at C: M s * = 22.9 knm/m, = 0.84, i.uncr = mm -1, i.cr = mm -1, creep.uncr = mm -1, creep.cr = mm -1, uncr = mm -1, cr = mm -1 and = mm -1. Deflection: The final mid-span deflection is readily calculated by double integration of the curvature diagram (shown in Figure 5) or using, for example, the conjugate beam method of analysis and is determined as mm A A B C D E E 1.5 m 1.8 m 1.5 m Figure 5. Final curvature diagram for Example 2, 10-6 (mm -1 ) CONCLUSIONS The time-dependent effects of creep and shrinkage on the long-term deflection of composite concrete slabs have been discussed and graphs have been presented to show how to consider these effects using a simplified calculating procedure suitable for routine use in structural design. The use of the graphs is illustrated by two examples and the results are compared against experimental and numerical results. Good agreement is obtained between the calculated and measured deflections. ACKNOWLEDGEMENTS The financial support provided by the Australian Research Council, decking manufacturer Fielders Australia PL and Prestressed Concrete Design Consultants (PCDC) is gratefully acknowledged. REFERENCES Fielders Australia PL (2008) Specifying Fielders-KingFlor; Composite Steel Formwork System Design Manual. Gholamhoseini, A., Gilbert, R.I. and Bradford, M.A. (2014) Time-dependent deflection of composite concrete slabs -A simplified design approach, Aust Journal of Structural Engineering, (in press). Gholamhoseini, A., Gilbert, R.I., Bradford, M.A. and Chang, Z.T. (2012) Long-term deformation of composite concrete slabs, Concrete in Australia, Vol. 38, pp Gilbert, R.I. (2001). Deflection calculation and control - Australian code amendments and improvements, ACI International SP 203, Code Provisions for Deflection Control in Concrete Structures. Chapter 4, American Concrete Institute, Editors E.G. Nawy and A. Scanlon, pp Gilbert, R.I., Bradford, M.A., Gholamhoseini, A. and Chang, Z.T. (2012) Effects of shrinkage on the long-term stresses and deformations of composite concrete slabs, Engineering Structures, Vol. 40, pp Gilbert, R.I. and Ranzi, G. (2011) Time-Dependent Behaviour of Concrete Structures, Spon Press, London, U.K. Ranzi, G., Ambrogi, L., Al-Deen, S. and Uy, B. (2012) Long-term experiments of post-tensioned composite slabs, Proceedings of the 10 th International Conference on Advances in Steel Concrete Composite and Hybrid Structures, Singapore, 2-4 July Standards Australia (2009) Australian Standard for Concrete Structures, AS , Sydney. ACMSM