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Experimental and Numerical Dynamic Analyses of Hollow Core Concrete Floors
Structures 12 (2017) 286–297
Contents lists available at ScienceDirect
Structures journal homepage: www.elsevier.com/locate/structures
Experimental and Numerical Dynamic Analyses of Hollow Core Concrete Floors
MARK
Fangzhou Liua, Jean-Marc Battinia,⁎, Costin Pacostea,b, Alexandra Granbergc a b c
KTH Division of Structural Engineering and Bridges, Stockholm, Sweden ELU Konsult, Stockholm, Sweden Ramboll, Stockholm, Sweden
A R T I C L E I N F O
A B S T R A C T
Keywords: Hollow core concrete slabs Dynamic analyses Experimental tests Finite element model Model calibration
Due to their low self-weight and high strength, precast and prestressed hollow core concrete slabs are widely used in construction. However, the combination of low self-weight and long span implies that the slabs are sensitive to vibrations induced by human activities. In this work, experimental tests and numerical analyses are performed in order to understand the dynamic behaviour of hollow core concrete floors. For the experiments, a test floor of dimension 10 m × 7.2 m and consisting of 6 hollow core elements was built. Very good agreements between experimental and numerical results have been obtained. Comprehensive numerical parametric analyses have been performed in order to determine the optimal value of the material parameters.
1. Introduction Precast and prestressed hollow core slabs, see Fig. 1, are often used in the construction of floors for high-rise apartments, multi-story buildings, shopping malls, offices or parking garages, in particular in Sweden. In the past years, some research concerning shear behaviour, structural behaviour, thermal performance and fire performance of hollow core concrete slabs has been addressed [1–28]. However, the combination of prestressing and low self-weight due to the voids in the cross-section makes it possible to build long-span floor elements. This implies that the floor is more sensitive to vibrations induced by human activities. These vibrations are usually not a safety problem for the construction but they can induce important comfort problem for people staying in the building. In fact, in many cases, the length of the span is not limited by static strength criteria but by dynamic comfort considerations. For these reasons it is important to understand in depth the dynamic behaviour of hollow core concrete slabs. Despite this fact, there are no guidelines or recommendations that structural engineers can use to model and study the dynamic behaviour of these structures. Besides, only a few works about this topic can be found in the literature. And most of these works are only numerical studies. In fact, to the authors' knowledge, dynamic experimental tests of hollow core concrete floors in which natural frequencies and associate mode shapes are identified have not been presented in the literature. Natural frequencies of a solid slab and a hollow core slab have been compared by Jendzelovsky and
⁎
Vrablova [29]. Marcos et al. [30] presented a parametric study on the vibration sensitivity of hollow core slabs. They found that the most important parameter for the first natural frequency is the span. But these two studies were only numerical and the results were not confirmed by experimental investigations. Laboratory tests with three simply supported elements were conducted at Lund Technical University in Sweden and were presented in the Master thesis of Johansson [31]. However, the mode shapes associated to the natural frequencies were not measured and consequently, the dynamic behaviour of the slab cannot be really understood. Field measurements on existing buildings were carried out by Granberg and Häggstam and were presented in their Master thesis [32]. But here also, the mode shapes were not identified. Besides, due to the influence of the surrounding building on the measured accelerations, it is difficult to understand the behaviour of the concrete slab itself. The purpose of the present work was to perform experimental tests and to use the results to develop a finite element model that can represent the real dynamic behaviour of hollow core concrete floors. For that, an experimental structure consisting on six hollow core elements supported by steel beams were built. The objective was to determine the natural frequencies and corresponding damping coefficients and mode shapes. In that context, it was important to identify not only the lowest natural frequency but also the higher ones. As a matter of fact, the higher modes may have an influence on human induced vibrations and their experimental determination makes it possible to better verify and calibrate the numerical model. In order to achieve that, two kind of
Corresponding author. E-mail address: [email protected] (J.-M. Battini).
http://dx.doi.org/10.1016/j.istruc.2017.10.001 Received 25 July 2017; Received in revised form 29 September 2017; Accepted 3 October 2017 Available online 05 October 2017 2352-0124/ © 2017 Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.
Structures 12 (2017) 286–297
F. Liu et al.
tests were conducted, by exciting the experimental structure with a hammer and with a vibration exciter. The outline of the paper is as follows: The experimental tests are described in Section 2. The finite element model is presented in Section 3. In Section 4, both experimental and numerical results are presented. Finally conclusions are derived in Section 5. 2. Experimental tests The experimental structure was built at the production plant of the company Contiga, a leading supplier of precast concrete structures in Sweden. The slab consisted of 6 hollow core elements of dimension 10 m × 1.2 m × 0.27 m each, see Figs. 2 and 3. The connections (joints) between the slabs were poured with grouted concrete. A 50 mm height concrete topping was added on the slab thirty days after the casting of the joints. The strength class of the concrete was C45/55 for the hollow core elements and the joints and C30/37 for the topping.
Fig. 1. Hollow core concrete slabs.
Fig. 2. Experimental structure.
Fig. 3. Experimental structure plan.
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Fig. 4. Cross-section of the horizontal steel beams and steel connector.
The connection between the concrete floor and the horizontal steel beams were performed through steel connectors welded to the horizontal steel beam (see Fig. 4) and anchors steel bars casted on the concrete floors, see Fig. 5. The space of about 5 cm between the concrete floor and horizontal steel beams were then poured with grouted concrete. The horizontal steel beams were supported by a set of hot formed hollow steel columns, see Fig. 2, with the following cross-sections: 180 mm × 180 mm × 8 mm at the four corners; 100 mm × 100 mm × 6.3 mm for the ten intermediate columns; 100 mm × 50 mm × 8 mm for the diagonal bars. The ground was horizontal below the steel beams but not along the direction of the voids. Consequently, the length of the steel columns was 0.45 m on one side and 0.75 m on the other side. Finally, the vertical columns were welded on each side on a steel plate of thickness 30 mm that was just placed on the ground. Ten accelerometers, see Figs. 6 and 7, were used to record acceleration data. The model of accelerometer is SF1500 [33] which operates from ± 6 V to ± 15 V with a typical current consumption of 12 mA at ± 6 V. The linear full acceleration range is ± 3 g with a corresponding sensitivity of 1.2 V/g. The frequency response over the full scale range is DC > 1500 Hz. As shown in Fig. 7, accelerometers A1 to A9 were installed at the typical points of 1/4 span, 1/2 span and 3/4 span of the slab in order to measure the vertical accelerations. Accelerometer A10 was installed on the side of one steel beam and registered the horizontal accelerations. The sampling frequency was chosen at 2048 points by second. Two different loadings were used. The first one was a sinusoidal force with peak force of 25 N and a frequency range starting from 4 Hz to 35 Hz with a frequency increase of 0.015 Hz/s. The purpose was to increase the frequency in a very slow way so that steady states were obtained at each resonance peak. The peak force was chosen so that the level of the accelerations corresponds to human induced vibrations. The sinusoidal force was driven by a harmonic force system which combines
Fig. 6. Accelerometer device.
of a dynamic signal analyser (HP3562A) [34], a charge amplifier (Type2635) [35] and a permanent magnetic vibration exciter (Type4808) [36], see Fig. 8. The magnetic vibration exciter has a force rating of 112 N and sine peak of 187 N enabling relatively heavy loads to be excited to high g levels. The frequency range is wide, from 4 Hz to 10 kHz. The vibration exciter is placed in position E1, see Fig. 7, under the concrete floor. The second loading was an impact force performed by a heavy-duty impact hammer (Type 8208) [37], see Fig. 9. This hammer has built-in electronics and is designed to excited and measure impact force on medium to very large structures. This hammer gives an impulse load of very short duration and consequently, frequencies up to 50 Hz are excited. The imposed signal both in time domain and frequency domain are shown in Fig. 10. The impact load was applied at position H1, see Fig. 7. The experimental tests were divided in three phases: in phase 1, the joints were casted but not the topping; in phase 2, the concrete topping was in place; in phase 3, all the intermediate steel columns were removed and the horizontal steel beams were then only supported at their ends. Exactly the same two loadings were applied in each phase. The reason to perform tests in these three phases is that both phases 2 and 3 correspond to real support conditions for hollow core slabs in buildings. Besides, in order to determine the specific influence of the topping it is also interesting to preform tests in phase 1. As a matter of fact, the stiffness of the concrete in the topping may be affected by cracks.
3. Finite element model A 3D finite element model of the experiments was developed by using the commercial finite element programme Abaqus 6.14. The whole finite element model is shown in Fig. 11 whereas the concrete and steel parts are shown separately on Figs. 12 and 13. The hollow core elements and the joints were modelled using 8-node solid elements C3D8. The concrete topping and the steel parts were modelled using 4node doubly curved shell elements S4. A convergence study was carried out and an optimised element size of 0.03 m was chosen for the concrete part and 0.05 m for the steel part (results obtained with an element size of 0.01 m for the concrete part and 0.03 m for the steel part present differences less than 1% for the natural frequencies). Tie constraints were used to connect the steel columns to the horizontal steel beams, the steel columns to the diagonal steel bars and the hollow core slab to the topping. The steel connectors and anchor steel bars were not modelled; the concrete floor was directly tied on the vertical surface of
Fig. 5. Grout concrete, steel connectors and steel bars
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Fig. 7. Location of the accelerometers A1 to A10 and the exciting loads.
4. Results
two horizontal steel beams. The concrete slab and the joints were modelled by using one part. In order to catch in an accurate way the dynamic behaviour of the joints, a very fine mesh was chosen, see Fig. 14. The material properties of the steel were taken as: E = 210 GPa, v = 0.3, p = 7850 kg/m3. The material properties of the concrete were taken firstly as: v = 0.2, p = 2365 kg/m3 and different values for E: 36 GPa for the hollow core elements and the joints and 33 GPa for the topping. These values are characteristic values of C45/55 and C30/37 concrete and S355J2 steel material. The values for the elastic moduli of the concrete will be discussed and optimised in the next section. The reinforcement and the prestressing were not introduced in the model but they do not have a significant influence on the dynamic response. Natural frequencies and mode shapes were extracted and steady state analyses were performed by using modal superposition (10 modes). The purpose of the steady state analyses was to reproduce the experiments performed with the vibration exciter.
The experimental results were processed by using Matlab and Fast Fourier Transformations, without numerical filtering. The damping ratios were calculated using half power band-width method. The obtained lowest natural frequencies up to 25 Hz and associated damping ratios are listed in Tables 1, 2 and 3. The maximum value of 25 Hz has been chosen due to the fact that natural frequencies higher than 25 Hz are less relevant for the response of the floor to pedestrian loading (see also the discussion in [38]). Very good agreement between the results obtained by the sinusoidal vibration exciter and the impact force hammer is obtained. The mode shapes are shown in Figs. 15 and 16. It can be noted that the mode shapes for phase 1 (without topping) and phase 2 (with topping) are identical. For phases 1 and 2, the first mode is a bending mode, the second mode is a torsional mode and the third mode is a second order bending mode with some small longitudinal displacement. For phase 3
Fig. 8. Dynamic signal analyser (HP3562A), charge amplifier (type 2635) and Permanent magnetic vibration exciter (type 4808).
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parameter, the tested values are given in Table 4. For each mode, the error function is defined as
error =
fnum − fexp fexp
× 100
where fnum is the numerical natural frequency and fexp is the experimental natural frequency. The results are presented in Figs. 17, 18 and 19. The following conclusions can be derived:
• Errors less than 5% can be obtained for all the modes with exception • •
of modes 3 and 4 in phase 3 for which slightly higher errors are obtained. It is not possible to find as set of parameters which give optimal results for all the modes. As example, in phase 2, a high value of Et is required for mode 1 whereas a low value is required for mode 2. Ec = 38 GPa, Ej = 20 GPa and Et = 27 GPa can be chosen as optimal values. It is clear that this choice can be discussed. But in the authors' opinion it represents a good compromise, especially if small errors for modes 1 and 2 have to be obtained.
The optimal value of elastic modulus for the concrete slab Ec is 38 GPa which is a little bit higher than the characteristic value (36 GPa) for C45/55 concrete strength class. This is not surprising since it is usually admitted that for dynamic analyses, the elastic modulus of concrete should be taken as about 1.1 times the static value. Another explanation is that the concrete used in the experiments was of higher quality than expected. As a matter of fact, ultimate compressive experiments performed during the tests have shown that the concrete used for the slabs had a higher ultimate strength than expected for C45/ 55 class. In addition, it must be noted that due to voids, the reinforcement has a slightly influence on the overall elastic modulus of the hollow core elements. Cross-section analyses of one hollow core element with and without reinforcement have been performed with Abaqus. The results show that the presence of the reinforcement increases the overall elastic modulus of the hollow core element from
Fig. 9. Heavy-duty impact hammer (Type8208).
(with only supports at the corners), the first two modes are almost the same as in phases 1 and 2. The third mode is a second order bending mode with also bending along lateral direction at the ends. The fourth mode is a bending mode along both longitudinal and lateral directions. Comprehensive numerical parametric analyses were performed in order to calibrate the finite element model. The parameters for which uncertainties are presented and that were studied are the elastic moduli of the concrete slab (Ec), the joints (Ej) and the topping (Et). For each
Fig. 10. The imposed signal of impact force hammer.
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Fig. 11. 3D finite element model.
Fig. 12. Concrete part of the model.
Fig. 13. Steel part of the model.
Table 1 Natural frequencies and modal damping in phase 1.
36 MPa to 36.5 MPa. The optimal value of elastic modulus for the joint Ej is 20 GPa which is lower than the characteristic value (36 GPa) for C45/55 concrete strength class. This can be explained by the fact that in the numerical model a perfect bond is assumed between the concrete in the joint and the concrete in the hollow core elements. However, as discussed by Robert J. Gulyas [39], this is probably not the case. The reason for this is to be found on one hand on the relative difference in drying shrinkage
Mode
1 2 3
Vibration exciter
Hammer
Natural frequencies [Hz]
Modal damping [%]
Natural frequencies [Hz]
Modal damping [%]
6.31 12.5 21.0
0.45 0.78 3.7
6.31 12.5 21.0
0.63 0.74 2.5
Table 2 Natural frequencies and modal damping in phase 2. Mode
1 2 3
Fig. 14. F.E. modelling of the joints (geometry and mesh).
291
Vibration exciter
Hammer
Natural frequencies [Hz]
Modal damping [%]
Natural frequencies [Hz]
Modal damping [%]
6.82 13.2 22.7
0.56 0.86 2.6
6.89 13.4 22.7
0.84 1.1 3.5
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Table 3 Natural frequencies and modal damping in phase 3. Mode
1 2 3 4
Vibration exciter
Table 4 Tested values for each parameter.
Hammer
Natural frequencies [Hz]
Modal damping [%]
Natural frequencies [Hz]
Modal damping [%]
6.09 13.6 15.1 18.9
0.68 0.73 1.06 0.99
6.14 13.6 15.1 19.0
0.72 0.77 1.14 0.96
Parameters
Tested values [GPa]
Characteristic value [GPa]
Ec Ej Et
36 and 38 1, 5, 10, 20 and 30 20, 24, 27, 30 and 33
36 33
27 GPa which is lower than the characteristic value (33 GPa) for the C30/37 concrete strength class. In the authors' opinion, this is due to the present of cracks in the topping. These cracks are probably caused by the differential shrinkage between the hollow core elements and the topping. The experiments performed with the vibration exciter were entirely
in the joints compared to the precasted elements and on the other hand on the carbonated surface of the precasted elements. The optimal value of elastic modulus of concrete for the topping is
Fig. 15. First three mode shapes in phases 1 and 2.
Fig. 16. First four modes in phase3.
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Fig. 17. Calibration results in phase 1.
experimental and numerical results are obtained for the three first modes. For the fourth mode that was not used in the calibration, the difference between the experimental and numerical frequencies is about 10%. In phase 2, the errors for the second mode (2%) and the
reproduced by using steady state numerical analyses including the 10 lowest modes. The optimal values for the elastic moduli of the concrete obtained by the parametric study were taken. The results are presented in Figs. 20, 21 and 22. In phase 1, very good agreements between
Fig. 18. Calibration results in phase 2.
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Fig. 19. Calibration results in phase 3.
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Fig. 20. Experimental and numerical results in phase 1.
agreements between experimental and numerical results are obtained for the two first modes and acceptable agreements are obtained for mode 3 and 4. Here also, an additional mode at about 32 Hz that was
third mode (4%) are larger than in phase 1, but still small. A fourth experimental mode can be identified at 32 Hz. This mode is not very well represented by the numerical model. In phase 3, very good
Fig. 21. Experimental and numerical results in phase 2.
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Fig. 22. Experimental and numerical results in phase 3.
However, the proposed model is based on solid elements and requires many elements. Another alternative, that requires less computational time, is to use shell elements to model the concrete floor. This is not an easy task, since the shell elements must consider the combined effect of the voids, the joints and the topping. This will be the purpose of our future work.
not used for the calibration can be identified. For this mode, good agreement between experimental and numerical results is obtained. 5. Conclusion Experimental tests and numerical finite element analyses have been carried out in order to study the dynamic behaviour of hollow core concrete floors. The experiments have been performed by using a test floor consisting of 6 hollow core elements of dimension 10 m × 1.2 m × 0.27 m each and supported by steel beams. Both a vibration exciter and a force hammer have been used to load the structure. For the finite element model, solid elements have been used to model the hollow core concrete elements and the concrete joints whereas shell elements have been used to model the concrete topping as well as the supporting steel beams and columns. Very good agreements between experimental and numerical results have been obtained, especially for the two lowest modes. Parametric studies have been performed in order to determine some numerical optimal values for the elastic moduli of the concrete. The following values, expressed as a multiplicative factor of the characteristic elastic modulus have been obtained: 1.05 for the hollow core elements, 0.55 for the joints, 0.8 for the topping. However, it must be emphasised that these factors should be considered with caution since there are based on only one experiment. The finite element model proposed in this paper can be used to study the dynamic behaviour of hollow core concrete slabs with different geometries, boundary conditions and thicknesses (hollow core slabs with thickness from 200 mm to 420 mm are commonly used). As a matter of fact, by using the results obtained in the present study, it is possible to implement similar finite element models for slabs with different thicknesses and dimensions. These finite element models can be used to determine the natural frequencies of the slab and to calculate the response to harmonic or impulse loads as recommended e.g. in [38].
Acknowledgments The authors would like to thank the staff of the company Contiga for giving us the possibility to perform the experimental tests at their production plant and also for the help during the tests. Many thanks also to Isabelle and Amanda Liljeström who planed and performed the experimental tests as part of their Master degree project, and to Kent Lindgren and the Company ACAD that were responsible for the data recording during the tests. Finally, the authors would like to thank the Ellen, Walter & Lennart Hesselman Foundation for their financial support. References [1] Hegger J, Roggendorf T, Kerkeni N. Shear capacity of prestressed hollow core slabs in slim floor constructions. Eng Struct 2009;31:551–9. [2] Girhammar UA, Pajari M. Tests and analysis on shear strength of composite slabs of hollow core units and concrete topping. Constr Build Mater 2008;22:1708–22. [3] Cuenca E, Serna P. Failure modes and shear design of prestressed hollow core slabs made of fiber-reinforced concrete. Universitat Politècnica de València, Camí de Vera s/n, 46022 Valencia, Spain: Institute of Concrete Science and Technology (ICITECH); 2012. [4] Ibrahim IS, Elliott KS. Experimental study on the shear behaviour of precast concrete hollow core slabs with concrete topping. Eng Struct 2016;125:80–90. [5] Lam D, Elliott KS, Nethercot DA. Parametric study on composite steel beams with precast concrete hollow core floor slabs. J Constr Steel Res 2000;54:283–304. [6] Elgabbas F, El-Ghandour AA, Abdelrahman AA. Different CFRP strengthening techniques for prestressed hollow core concrete slabs: experimental study and analytical investigation. Compos Struct 2010;92:401–11. [7] Broo H, Lundgren K. Finite element analyses of hollow core units subjected to shear and torsion. Department of Structural Engineering, Chalmers University of Technology:
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F. Liu et al. Göteborg, Sweden; 2002. [8] Shakya AM, Kodur VKR. Response of precast prestressed concrete hollowcore slabs under fire conditions. Department of Civil and Environmental Engineering, Michigan State University: United States; 2015. [9] Aguado JV, Albero V, Espinos A. A 3D finite element model for predicting the fire behavior of hollow-core slabs. Eng Struct 2016;108:12–27. [10] Aguado JV, Espinos A. Influence of reinforcement arrangement in flexural fire behavior of hollow core slabs. Fire Saf J 2012;53:72–84. [11] Kakogiannis D, Pascualena F, Reymen B. Blast performance of reinforced concrete hollow core slabs in combination with fire: numerical and experimental assessment. Fire Saf J 2013;57:69–82. [12] Venanzi I, Breccolotti M, D'Alessandro A, Materazzi AL. Fire performance assessment of HPLWC hollow core slabs through full-scale furnace testing. Fire Saf J 2014;69:12–22. [13] Baran Eray. Effects of cast-in-place concrete topping on flexural response of precast concrete hollow-core slabs. Eng Struct 2015;98:109–17. [14] Kankeri Pradeep, Suriya Prakash S. Experimental evaluation of bonded overlay and NSM GFRP bar strengthening on flexural behavior of precast prestressed hollow core slabs. Eng Struct 2016;120:49–57. [15] Wariyatno Nanang Gunawan, Haryanto Yanuar, Sudibyo Gathot Heri. Flexural behavior of precast hollow core slab using PVC pipe and Styrofoam with different reinforcement. Procedia Engineering 2017;171:909–16. [16] Mansour Farnoud Rahimi, Bakar Suhaimi Abu. Flexural performance of a precast concrete slab with steel fiber concrete topping. Constr Build Mater 2015;75:112–20. [17] Kataoka Marcela Novischi, Ferreira Marcelo Araujo. Nonlinear FE analysis of slab-beamcolumn connection in precast concrete structures. Eng Struct 2017;143:306–15. [18] Lam Dennis. Capacities of headed stud shear connectors in composite steel beams with precast hollowcore slabs. J Constr Steel Res 2007;63:1160–74. [19] Kankeri Pradeep, Suriya Prakash S. Efficient hybrid strengthening for precast hollow core slabs at low and high shear span to depth ratios. Compos Struct 2017;170:202–14. [20] de Souza Patricia T, Kataoka Marcela N. Experimental and numerical analysis of the pushout test on shear studs in hollow core slabs. Eng Struct 2017;147:398–409. [21] Cuenca E, Serna P. Failure modes and shear design of prestressed hollow core slabs made of fiber-reinforced concrete. Compos Part B 2013;45:952–64. [22] Brunesi E, Nascimbene R. Numerical web-shear strength assessment of precast prestressed hollow core slab units. Eng Struct 2015;102:13–30.
[23] Faheem Ahmed, Ranzi Gianluca, Fiorito Francesco. A numerical study on the thermal performance of night ventilated hollow core slabs cast with micro-encapsulated PCM concrete. Energ Buildings 2016;127:892–906. [24] Navarro Lidia, de Gracia Alvaro. Experimental evaluation of a concrete core slab with phase change materials for cooling purposes. Energ Buildings 2016;116:411–9. [25] Chen Yuxiang, Galal Khaled E, Athienities Andreas K. Integrating hollow-core masonry walls and precast concrete slabs into building space heating and cooling. Journal of Building Engineering 2016;5:277–87. [26] Navarro Lidia, de Gracia Alvaro. PCM incorporation in a concrete core slab as a thermal storage and supply system: proof of concept. Energ Buildings 2015;103:70–82. [27] Corgnati Stefano Paolo, Kindinis Andrea. Thermal mass activation by hollow core slab coupled with night ventilation to reduce summer colling loads. Building and environment 2007;42:3285–97. [28] Maazoun A, Vantomme J, Matthys S. Damage assessment of hollow core reinfoced and prestressed concrete slabs subjected to blast loading. Procedia Engineering 2017;199:2476–81. [29] Jendzelovsky N, Vrablova K. Comparison of natural frequencies of hollow core slabs. Appl Mech Mater 2015;769:225–8. ISSN: 1662-7482. [30] Marcos LK, Carrazedo R. Parametric study on the vibration sensitivity of hollow-core slabs floors. Proceedings of the 9th international conference on structural dynamics. EURODYN; 2014. [31] Johansson P. Vibration of hollow core concrete elements induced by walking. Master thesis. Lunds Tekniska Högskola: Lunds Universitet; 2009. [32] Granberg A, Häggstam M. FE-Modellering av håldäcksbjålklag för dynamisk analys. Master thesis. KTH Royal Institute of Technology; 2014. [33] Data sheet: Colibrys SF1500 Accelerometer. [34] http://www.testequipmentdepot.com/usedequipment/hewlettpackard/ spectrumanalyzers/3562a.htm. [35] Product Data: Charge amplifier – type 2635 (BP0099) - Brüel & Kjær. [36] Product Data: Permanent magnetic vibration exciter type 4808 - Brüel & Kjær. [37] Product Data: Heavy-duty impact hammers types 8207, 8208 and 8210 - Brüel & Kjær. [38] Hicks SJ, Smith AL. Design of floor structures against human-induced vibrations. Steel Construction June 2011;4(2):114–20. [39] Gulyas RJ, Wirthlin GJ, Champa JT. Evaluation of keyway grout test methods for precast concrete bridges. PCI J 1995;40:44–57.
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Contents lists available at ScienceDirect
Structures journal homepage: www.elsevier.com/locate/structures
Experimental and Numerical Dynamic Analyses of Hollow Core Concrete Floors
MARK
Fangzhou Liua, Jean-Marc Battinia,⁎, Costin Pacostea,b, Alexandra Granbergc a b c
KTH Division of Structural Engineering and Bridges, Stockholm, Sweden ELU Konsult, Stockholm, Sweden Ramboll, Stockholm, Sweden
A R T I C L E I N F O
A B S T R A C T
Keywords: Hollow core concrete slabs Dynamic analyses Experimental tests Finite element model Model calibration
Due to their low self-weight and high strength, precast and prestressed hollow core concrete slabs are widely used in construction. However, the combination of low self-weight and long span implies that the slabs are sensitive to vibrations induced by human activities. In this work, experimental tests and numerical analyses are performed in order to understand the dynamic behaviour of hollow core concrete floors. For the experiments, a test floor of dimension 10 m × 7.2 m and consisting of 6 hollow core elements was built. Very good agreements between experimental and numerical results have been obtained. Comprehensive numerical parametric analyses have been performed in order to determine the optimal value of the material parameters.
1. Introduction Precast and prestressed hollow core slabs, see Fig. 1, are often used in the construction of floors for high-rise apartments, multi-story buildings, shopping malls, offices or parking garages, in particular in Sweden. In the past years, some research concerning shear behaviour, structural behaviour, thermal performance and fire performance of hollow core concrete slabs has been addressed [1–28]. However, the combination of prestressing and low self-weight due to the voids in the cross-section makes it possible to build long-span floor elements. This implies that the floor is more sensitive to vibrations induced by human activities. These vibrations are usually not a safety problem for the construction but they can induce important comfort problem for people staying in the building. In fact, in many cases, the length of the span is not limited by static strength criteria but by dynamic comfort considerations. For these reasons it is important to understand in depth the dynamic behaviour of hollow core concrete slabs. Despite this fact, there are no guidelines or recommendations that structural engineers can use to model and study the dynamic behaviour of these structures. Besides, only a few works about this topic can be found in the literature. And most of these works are only numerical studies. In fact, to the authors' knowledge, dynamic experimental tests of hollow core concrete floors in which natural frequencies and associate mode shapes are identified have not been presented in the literature. Natural frequencies of a solid slab and a hollow core slab have been compared by Jendzelovsky and
⁎
Vrablova [29]. Marcos et al. [30] presented a parametric study on the vibration sensitivity of hollow core slabs. They found that the most important parameter for the first natural frequency is the span. But these two studies were only numerical and the results were not confirmed by experimental investigations. Laboratory tests with three simply supported elements were conducted at Lund Technical University in Sweden and were presented in the Master thesis of Johansson [31]. However, the mode shapes associated to the natural frequencies were not measured and consequently, the dynamic behaviour of the slab cannot be really understood. Field measurements on existing buildings were carried out by Granberg and Häggstam and were presented in their Master thesis [32]. But here also, the mode shapes were not identified. Besides, due to the influence of the surrounding building on the measured accelerations, it is difficult to understand the behaviour of the concrete slab itself. The purpose of the present work was to perform experimental tests and to use the results to develop a finite element model that can represent the real dynamic behaviour of hollow core concrete floors. For that, an experimental structure consisting on six hollow core elements supported by steel beams were built. The objective was to determine the natural frequencies and corresponding damping coefficients and mode shapes. In that context, it was important to identify not only the lowest natural frequency but also the higher ones. As a matter of fact, the higher modes may have an influence on human induced vibrations and their experimental determination makes it possible to better verify and calibrate the numerical model. In order to achieve that, two kind of
Corresponding author. E-mail address: [email protected] (J.-M. Battini).
http://dx.doi.org/10.1016/j.istruc.2017.10.001 Received 25 July 2017; Received in revised form 29 September 2017; Accepted 3 October 2017 Available online 05 October 2017 2352-0124/ © 2017 Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.
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tests were conducted, by exciting the experimental structure with a hammer and with a vibration exciter. The outline of the paper is as follows: The experimental tests are described in Section 2. The finite element model is presented in Section 3. In Section 4, both experimental and numerical results are presented. Finally conclusions are derived in Section 5. 2. Experimental tests The experimental structure was built at the production plant of the company Contiga, a leading supplier of precast concrete structures in Sweden. The slab consisted of 6 hollow core elements of dimension 10 m × 1.2 m × 0.27 m each, see Figs. 2 and 3. The connections (joints) between the slabs were poured with grouted concrete. A 50 mm height concrete topping was added on the slab thirty days after the casting of the joints. The strength class of the concrete was C45/55 for the hollow core elements and the joints and C30/37 for the topping.
Fig. 1. Hollow core concrete slabs.
Fig. 2. Experimental structure.
Fig. 3. Experimental structure plan.
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Fig. 4. Cross-section of the horizontal steel beams and steel connector.
The connection between the concrete floor and the horizontal steel beams were performed through steel connectors welded to the horizontal steel beam (see Fig. 4) and anchors steel bars casted on the concrete floors, see Fig. 5. The space of about 5 cm between the concrete floor and horizontal steel beams were then poured with grouted concrete. The horizontal steel beams were supported by a set of hot formed hollow steel columns, see Fig. 2, with the following cross-sections: 180 mm × 180 mm × 8 mm at the four corners; 100 mm × 100 mm × 6.3 mm for the ten intermediate columns; 100 mm × 50 mm × 8 mm for the diagonal bars. The ground was horizontal below the steel beams but not along the direction of the voids. Consequently, the length of the steel columns was 0.45 m on one side and 0.75 m on the other side. Finally, the vertical columns were welded on each side on a steel plate of thickness 30 mm that was just placed on the ground. Ten accelerometers, see Figs. 6 and 7, were used to record acceleration data. The model of accelerometer is SF1500 [33] which operates from ± 6 V to ± 15 V with a typical current consumption of 12 mA at ± 6 V. The linear full acceleration range is ± 3 g with a corresponding sensitivity of 1.2 V/g. The frequency response over the full scale range is DC > 1500 Hz. As shown in Fig. 7, accelerometers A1 to A9 were installed at the typical points of 1/4 span, 1/2 span and 3/4 span of the slab in order to measure the vertical accelerations. Accelerometer A10 was installed on the side of one steel beam and registered the horizontal accelerations. The sampling frequency was chosen at 2048 points by second. Two different loadings were used. The first one was a sinusoidal force with peak force of 25 N and a frequency range starting from 4 Hz to 35 Hz with a frequency increase of 0.015 Hz/s. The purpose was to increase the frequency in a very slow way so that steady states were obtained at each resonance peak. The peak force was chosen so that the level of the accelerations corresponds to human induced vibrations. The sinusoidal force was driven by a harmonic force system which combines
Fig. 6. Accelerometer device.
of a dynamic signal analyser (HP3562A) [34], a charge amplifier (Type2635) [35] and a permanent magnetic vibration exciter (Type4808) [36], see Fig. 8. The magnetic vibration exciter has a force rating of 112 N and sine peak of 187 N enabling relatively heavy loads to be excited to high g levels. The frequency range is wide, from 4 Hz to 10 kHz. The vibration exciter is placed in position E1, see Fig. 7, under the concrete floor. The second loading was an impact force performed by a heavy-duty impact hammer (Type 8208) [37], see Fig. 9. This hammer has built-in electronics and is designed to excited and measure impact force on medium to very large structures. This hammer gives an impulse load of very short duration and consequently, frequencies up to 50 Hz are excited. The imposed signal both in time domain and frequency domain are shown in Fig. 10. The impact load was applied at position H1, see Fig. 7. The experimental tests were divided in three phases: in phase 1, the joints were casted but not the topping; in phase 2, the concrete topping was in place; in phase 3, all the intermediate steel columns were removed and the horizontal steel beams were then only supported at their ends. Exactly the same two loadings were applied in each phase. The reason to perform tests in these three phases is that both phases 2 and 3 correspond to real support conditions for hollow core slabs in buildings. Besides, in order to determine the specific influence of the topping it is also interesting to preform tests in phase 1. As a matter of fact, the stiffness of the concrete in the topping may be affected by cracks.
3. Finite element model A 3D finite element model of the experiments was developed by using the commercial finite element programme Abaqus 6.14. The whole finite element model is shown in Fig. 11 whereas the concrete and steel parts are shown separately on Figs. 12 and 13. The hollow core elements and the joints were modelled using 8-node solid elements C3D8. The concrete topping and the steel parts were modelled using 4node doubly curved shell elements S4. A convergence study was carried out and an optimised element size of 0.03 m was chosen for the concrete part and 0.05 m for the steel part (results obtained with an element size of 0.01 m for the concrete part and 0.03 m for the steel part present differences less than 1% for the natural frequencies). Tie constraints were used to connect the steel columns to the horizontal steel beams, the steel columns to the diagonal steel bars and the hollow core slab to the topping. The steel connectors and anchor steel bars were not modelled; the concrete floor was directly tied on the vertical surface of
Fig. 5. Grout concrete, steel connectors and steel bars
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Fig. 7. Location of the accelerometers A1 to A10 and the exciting loads.
4. Results
two horizontal steel beams. The concrete slab and the joints were modelled by using one part. In order to catch in an accurate way the dynamic behaviour of the joints, a very fine mesh was chosen, see Fig. 14. The material properties of the steel were taken as: E = 210 GPa, v = 0.3, p = 7850 kg/m3. The material properties of the concrete were taken firstly as: v = 0.2, p = 2365 kg/m3 and different values for E: 36 GPa for the hollow core elements and the joints and 33 GPa for the topping. These values are characteristic values of C45/55 and C30/37 concrete and S355J2 steel material. The values for the elastic moduli of the concrete will be discussed and optimised in the next section. The reinforcement and the prestressing were not introduced in the model but they do not have a significant influence on the dynamic response. Natural frequencies and mode shapes were extracted and steady state analyses were performed by using modal superposition (10 modes). The purpose of the steady state analyses was to reproduce the experiments performed with the vibration exciter.
The experimental results were processed by using Matlab and Fast Fourier Transformations, without numerical filtering. The damping ratios were calculated using half power band-width method. The obtained lowest natural frequencies up to 25 Hz and associated damping ratios are listed in Tables 1, 2 and 3. The maximum value of 25 Hz has been chosen due to the fact that natural frequencies higher than 25 Hz are less relevant for the response of the floor to pedestrian loading (see also the discussion in [38]). Very good agreement between the results obtained by the sinusoidal vibration exciter and the impact force hammer is obtained. The mode shapes are shown in Figs. 15 and 16. It can be noted that the mode shapes for phase 1 (without topping) and phase 2 (with topping) are identical. For phases 1 and 2, the first mode is a bending mode, the second mode is a torsional mode and the third mode is a second order bending mode with some small longitudinal displacement. For phase 3
Fig. 8. Dynamic signal analyser (HP3562A), charge amplifier (type 2635) and Permanent magnetic vibration exciter (type 4808).
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parameter, the tested values are given in Table 4. For each mode, the error function is defined as
error =
fnum − fexp fexp
× 100
where fnum is the numerical natural frequency and fexp is the experimental natural frequency. The results are presented in Figs. 17, 18 and 19. The following conclusions can be derived:
• Errors less than 5% can be obtained for all the modes with exception • •
of modes 3 and 4 in phase 3 for which slightly higher errors are obtained. It is not possible to find as set of parameters which give optimal results for all the modes. As example, in phase 2, a high value of Et is required for mode 1 whereas a low value is required for mode 2. Ec = 38 GPa, Ej = 20 GPa and Et = 27 GPa can be chosen as optimal values. It is clear that this choice can be discussed. But in the authors' opinion it represents a good compromise, especially if small errors for modes 1 and 2 have to be obtained.
The optimal value of elastic modulus for the concrete slab Ec is 38 GPa which is a little bit higher than the characteristic value (36 GPa) for C45/55 concrete strength class. This is not surprising since it is usually admitted that for dynamic analyses, the elastic modulus of concrete should be taken as about 1.1 times the static value. Another explanation is that the concrete used in the experiments was of higher quality than expected. As a matter of fact, ultimate compressive experiments performed during the tests have shown that the concrete used for the slabs had a higher ultimate strength than expected for C45/ 55 class. In addition, it must be noted that due to voids, the reinforcement has a slightly influence on the overall elastic modulus of the hollow core elements. Cross-section analyses of one hollow core element with and without reinforcement have been performed with Abaqus. The results show that the presence of the reinforcement increases the overall elastic modulus of the hollow core element from
Fig. 9. Heavy-duty impact hammer (Type8208).
(with only supports at the corners), the first two modes are almost the same as in phases 1 and 2. The third mode is a second order bending mode with also bending along lateral direction at the ends. The fourth mode is a bending mode along both longitudinal and lateral directions. Comprehensive numerical parametric analyses were performed in order to calibrate the finite element model. The parameters for which uncertainties are presented and that were studied are the elastic moduli of the concrete slab (Ec), the joints (Ej) and the topping (Et). For each
Fig. 10. The imposed signal of impact force hammer.
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Fig. 11. 3D finite element model.
Fig. 12. Concrete part of the model.
Fig. 13. Steel part of the model.
Table 1 Natural frequencies and modal damping in phase 1.
36 MPa to 36.5 MPa. The optimal value of elastic modulus for the joint Ej is 20 GPa which is lower than the characteristic value (36 GPa) for C45/55 concrete strength class. This can be explained by the fact that in the numerical model a perfect bond is assumed between the concrete in the joint and the concrete in the hollow core elements. However, as discussed by Robert J. Gulyas [39], this is probably not the case. The reason for this is to be found on one hand on the relative difference in drying shrinkage
Mode
1 2 3
Vibration exciter
Hammer
Natural frequencies [Hz]
Modal damping [%]
Natural frequencies [Hz]
Modal damping [%]
6.31 12.5 21.0
0.45 0.78 3.7
6.31 12.5 21.0
0.63 0.74 2.5
Table 2 Natural frequencies and modal damping in phase 2. Mode
1 2 3
Fig. 14. F.E. modelling of the joints (geometry and mesh).
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Vibration exciter
Hammer
Natural frequencies [Hz]
Modal damping [%]
Natural frequencies [Hz]
Modal damping [%]
6.82 13.2 22.7
0.56 0.86 2.6
6.89 13.4 22.7
0.84 1.1 3.5
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Table 3 Natural frequencies and modal damping in phase 3. Mode
1 2 3 4
Vibration exciter
Table 4 Tested values for each parameter.
Hammer
Natural frequencies [Hz]
Modal damping [%]
Natural frequencies [Hz]
Modal damping [%]
6.09 13.6 15.1 18.9
0.68 0.73 1.06 0.99
6.14 13.6 15.1 19.0
0.72 0.77 1.14 0.96
Parameters
Tested values [GPa]
Characteristic value [GPa]
Ec Ej Et
36 and 38 1, 5, 10, 20 and 30 20, 24, 27, 30 and 33
36 33
27 GPa which is lower than the characteristic value (33 GPa) for the C30/37 concrete strength class. In the authors' opinion, this is due to the present of cracks in the topping. These cracks are probably caused by the differential shrinkage between the hollow core elements and the topping. The experiments performed with the vibration exciter were entirely
in the joints compared to the precasted elements and on the other hand on the carbonated surface of the precasted elements. The optimal value of elastic modulus of concrete for the topping is
Fig. 15. First three mode shapes in phases 1 and 2.
Fig. 16. First four modes in phase3.
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Fig. 17. Calibration results in phase 1.
experimental and numerical results are obtained for the three first modes. For the fourth mode that was not used in the calibration, the difference between the experimental and numerical frequencies is about 10%. In phase 2, the errors for the second mode (2%) and the
reproduced by using steady state numerical analyses including the 10 lowest modes. The optimal values for the elastic moduli of the concrete obtained by the parametric study were taken. The results are presented in Figs. 20, 21 and 22. In phase 1, very good agreements between
Fig. 18. Calibration results in phase 2.
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Fig. 19. Calibration results in phase 3.
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Fig. 20. Experimental and numerical results in phase 1.
agreements between experimental and numerical results are obtained for the two first modes and acceptable agreements are obtained for mode 3 and 4. Here also, an additional mode at about 32 Hz that was
third mode (4%) are larger than in phase 1, but still small. A fourth experimental mode can be identified at 32 Hz. This mode is not very well represented by the numerical model. In phase 3, very good
Fig. 21. Experimental and numerical results in phase 2.
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Fig. 22. Experimental and numerical results in phase 3.
However, the proposed model is based on solid elements and requires many elements. Another alternative, that requires less computational time, is to use shell elements to model the concrete floor. This is not an easy task, since the shell elements must consider the combined effect of the voids, the joints and the topping. This will be the purpose of our future work.
not used for the calibration can be identified. For this mode, good agreement between experimental and numerical results is obtained. 5. Conclusion Experimental tests and numerical finite element analyses have been carried out in order to study the dynamic behaviour of hollow core concrete floors. The experiments have been performed by using a test floor consisting of 6 hollow core elements of dimension 10 m × 1.2 m × 0.27 m each and supported by steel beams. Both a vibration exciter and a force hammer have been used to load the structure. For the finite element model, solid elements have been used to model the hollow core concrete elements and the concrete joints whereas shell elements have been used to model the concrete topping as well as the supporting steel beams and columns. Very good agreements between experimental and numerical results have been obtained, especially for the two lowest modes. Parametric studies have been performed in order to determine some numerical optimal values for the elastic moduli of the concrete. The following values, expressed as a multiplicative factor of the characteristic elastic modulus have been obtained: 1.05 for the hollow core elements, 0.55 for the joints, 0.8 for the topping. However, it must be emphasised that these factors should be considered with caution since there are based on only one experiment. The finite element model proposed in this paper can be used to study the dynamic behaviour of hollow core concrete slabs with different geometries, boundary conditions and thicknesses (hollow core slabs with thickness from 200 mm to 420 mm are commonly used). As a matter of fact, by using the results obtained in the present study, it is possible to implement similar finite element models for slabs with different thicknesses and dimensions. These finite element models can be used to determine the natural frequencies of the slab and to calculate the response to harmonic or impulse loads as recommended e.g. in [38].
Acknowledgments The authors would like to thank the staff of the company Contiga for giving us the possibility to perform the experimental tests at their production plant and also for the help during the tests. Many thanks also to Isabelle and Amanda Liljeström who planed and performed the experimental tests as part of their Master degree project, and to Kent Lindgren and the Company ACAD that were responsible for the data recording during the tests. Finally, the authors would like to thank the Ellen, Walter & Lennart Hesselman Foundation for their financial support. References [1] Hegger J, Roggendorf T, Kerkeni N. Shear capacity of prestressed hollow core slabs in slim floor constructions. Eng Struct 2009;31:551–9. [2] Girhammar UA, Pajari M. Tests and analysis on shear strength of composite slabs of hollow core units and concrete topping. Constr Build Mater 2008;22:1708–22. [3] Cuenca E, Serna P. Failure modes and shear design of prestressed hollow core slabs made of fiber-reinforced concrete. Universitat Politècnica de València, Camí de Vera s/n, 46022 Valencia, Spain: Institute of Concrete Science and Technology (ICITECH); 2012. [4] Ibrahim IS, Elliott KS. Experimental study on the shear behaviour of precast concrete hollow core slabs with concrete topping. Eng Struct 2016;125:80–90. [5] Lam D, Elliott KS, Nethercot DA. Parametric study on composite steel beams with precast concrete hollow core floor slabs. J Constr Steel Res 2000;54:283–304. [6] Elgabbas F, El-Ghandour AA, Abdelrahman AA. Different CFRP strengthening techniques for prestressed hollow core concrete slabs: experimental study and analytical investigation. Compos Struct 2010;92:401–11. [7] Broo H, Lundgren K. Finite element analyses of hollow core units subjected to shear and torsion. Department of Structural Engineering, Chalmers University of Technology:
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