Parametric study on the vibration sensitivity of hollow-core slabs floors

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1 Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 214 Porto, Portugal, 3 June - 2 July 214 A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.) ISSN: ; ISBN: Parametric study on the vibration sensitivity of hollow-core slabs floors Lara Kawai Marcos, Ricardo Carrazedo Department of Structural Engineering, School of Engineering of São Carlos, University of São Paulo, Av. Trabalhador Sãocarlense 4, São Carlos, Brazil lara.marcos@usp.br, carrazedo@sc.usp.br ABSTRACT: Hollow-core slabs are very efficient structural members, since their voids allow a significant reduction of selfweight. Moreover, due to prestressing these slabs are able to be employed in large spans. With these advantages precast / prestressed hollow-core slabs are widely used in commercial and industrial buildings. However, for some combinations of span, thickness and materials properties, this structural system may be susceptible to excessive vibrations. In this paper a theoretical and numerical study on human induced vibrations in this type of structure is carried out. Initially a review of the dynamic loads induced by humans in activities such as walking is shown as well as acceptance criteria for human comfort. Then a parametric study on vibration sensitivity of typical structural configurations of hollow core slabs cores is developed through numerical simulations with the Finite Element Method. Some situations of interest including different spans, slab thickness and modulus of elasticity of concrete were evaluated and the vibration levels estimated. Finally the vibration levels were compared to applicable comfort levels suggested by international manuals and renowned authors. The vibrations levels were considered adequate based on a criterion of maximum dynamic displacements while inadequate according to existing limits for minimum frequency and peak acceleration. The contradiction between acceptance criteria and the occurrence of inadequate dynamic behavior for some of the evaluated slabs emphasize the importance of the continuity of this research. KEY WORDS: Hollow-core slabs, numerical simulations, human induced vibrations, comfort levels. 1 INTRODUCTION The increasing strength of construction materials as well as the frequent use of prestressing has enabled great advances in structural design and architecture of residential and office buildings. However, this development has also resulted in more slender structures, making them sometimes susceptible to excessive vibration. In the case of floors with hollow core slabs, the dynamic behaviour related to human activities is of special interest, once these elements may have natural frequencies close to those of human activities. Hollow-core slab, also known as voided slab, is a precast concrete flat slab with voids in its cross section (Figure 1). These slabs are widely employed in shopping malls, offices, parking garages, etc. Their design is normally focused in strength criteria rather than serviceability requirements. When the design load of the slabs is low, the slabs can be designed with a very high slenderness, and although they resist the loads, they may be inadequate from a dynamic point of view. Figure 1. Hollow-core slabs [1] The dynamic response of these structures is affected by several parameters at different degrees. Modifying their thickness, span and material properties may result in structures with different dynamic behaviours. In this paper, the influence of each of these parameters is evaluated trough a parametric study developed with the Finite Element Method. 2 BIBLIOGRAPHIC REVIEW 2.1 Hollow core slabs Hollow core slabs are the precast concrete element most used around the world, especially in North America and Western Europe. As other precast elements, hollow core slabs have some advantages, which include the reuse of forms, use of prestressed concrete, section with better use of materials, higher labour productivity, better quality control, reduction or absence of slab propping and fast assembly[2]. Specifically regarding the hollow core slabs, advantages that can be mentioned are the structural and economical efficiency as floor and roof systems. Besides this, the voids may be used for electrical or mechanical runs and distribute the heated air through the cores. Hollow core slabs also have an excellent fire resistance and sound transmission characteristics [3]. The hollow core slabs can be designed with or without the provision of topping concrete cast on site. The usual span varies from 5 m to 15 m and the cross section height goes from 15 mm to 3 mm. The relation between span/height can reach 5 [2]. In those cases, prestressing and high strength concrete are used. Figure 2 shows the development of hollow core slabs over the years. 195

2 Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 214 there are times when both feet are in contact with the floor, thus the full dynamic action also have components in the 2nd and 3rd harmonic of the dominant frequency [6]. When studying the gait cycle, it is possible to obtain the distribution of contact forces generated by contact of the foot with the ground from time to time, as shown in Figure 4. Figure 2.Development of hollow core slabs [2] 2.2 Prestress effects The effect of prestressing forces on the structure s dynamic response is a controversial subject. Some authors claim that the axial force is able to reduce the natural frequency. This would be due to the reduction of stiffness caused by second order effects. However, it is important to note that this does not occur with internal prestressing forces. As the tendons are surrounded by concrete, there is no additional eccentricity created by the deformation, that is, no second order effects. This can be seen in Figure 3 [4]. Figure 3. Externally axially loaded and internally prestressed concrete [4] 2.3 Human induced vibration A review of vibration sources has revealed that humans are the most important source of excitation in building floors. Mathematical models to simulate human-induced activities such as walking, running or jumping exist, however care should be taken when using them in structures with light damping. In this case, assuming a perfect resonance can lead to an overestimation of the results [5]. Human activities can generate dynamic forces of periodic or transient nature. These actions can cause undesirable effects such as excessive internal forces in structural elements, damage to non-structural elements, intolerable vibration and noise for occupants. The amplitude of these vibrations depends basically on the relationship between the dominant excitation frequency and the natural frequency of the structure [6]. The dominant frequency in the case of persons in normal walking is between 1,5 and 2,5 Hz, and this frequency range grows with the walking speed. However, during the walking Figure 4. Forcing patterns [7] Although this pattern vary between individuals, it is possible to describe a standard dynamic force for each activity. The forcing function for a human activity can be represented mathematically by a Fourier series (equation 1) [8]... Where: G = weight of the person ( notional pedestrian of G=8N) = Fourier coefficient of the i-th harmonic = activity rate (Hz) = phase lag of the i-th harmonic relative to the 1 st harmonic i = number of the i-th harmonic n = total number of contributing harmonics 2.4 Vibration assessment The discomfort produced by whole-body vibration depends on the vibration magnitude, the vibration frequency, the direction of the vibration, the position at which the vibration contacts the body and the duration of the vibration. It also depends on the posture and orientation of the body and varies both between and within individuals [9]. The mechanical vibration can be perceived by humans via several systems (visual, vestibular, auditory, cutaneous, kinesthetic or visceral), depending on the magnitude and frequency of the vibration [9]. ISO [1] suggests that, for a population of alert and fit persons, an average value of perception threshold is of.15 m/s² (weighted peak acceleration). According to this standard, people are likely to complaint with vibration only slightly above perception threshold when residential buildings are considered. However, no limits are suggested for comfort (1) 196

3 Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 214 in residential or office buildings in this standard neither in ISO :23 [11]. There are many guides and standards which offer methods for assessing vibration in buildings. The most common and mentioned consideration when it comes to the human perception of vibration is the reference scale of Reiher Meister [] which uses the displacement as the assessment criterion. The modified Reiher - Meister scale was proposed by Lenzen [13] to vibrations caused by walking impact [14], and is shown in Fig. 5. For offices, Mast [16] suggests K as 58 kn and β from.2 to.5, depending on the amount of non-structural components and furnishings. 3 METHODOLOGY A parametric study was developed to evaluate the influence of parameters such as slab thickness, span and elastic modulus of concrete on natural frequencies and vibration generated by walking in hollow core slabs. The study was carried out through numerical simulation with finite element program Abaqus. The cross section geometries and spans were selected for certain levels of load according to PCI Manual [17]. 3.1 Finite element models The slabs are supported by two beams, each one with rectangular section (.6x.4 cm) and simply supported at the ends. The span in the direction of the beams is 9.76 m, equivalent to 8 slabs elements. Shell elements were used to model both the slabs (S4R5) and beams (S4R). A schematic view is shown in Fig. 6. Figure 5. Modified Reiher Meister scale [15] Bachmann et al. [8] suggest values for peak acceleration as an indication of human perceptibility thresholds for vertical harmonic vibrations for standing person (Table 1). According to them, the perceptibility is proportional to acceleration when vibration is in the range 1 to 1 Hz and proportional to velocity in the range 1 to 1 Hz [8]. Table 1. Indication of human perceptibility thresholds [8] Description Frequency range 1 to 1 Hz Peak acceleration (mm/s²) Just perceptible 34 Clearly perceptible 1 Disturbing/unpleasant 55 Intolerable 18 Mast [16] also offers a method to evaluate the serviceability of long-span floors subjected to human walking. An empirical formula based on resonant effects of walking is proposed. This equation provides the minimum natural frequency of a floor system required to prevent disturbing vibrations caused by walking (equation 2). The coefficient K is from an empirical nature and values are suggested by Mast [16] depending on the occupancies affected by the vibrations, W represents the weight of the structure affected by walking vibration and β is the modal damping. (2) Figure 6. Model view 3.2 Materials properties The modulus of elasticity of concrete directly affects the stiffness of the structure. However, it does not vary only with the concrete strength but also the type of aggregate used. A concrete with basaltic aggregates may have the modulus of elasticity up to 4% higher than the concrete with the same strength but with sandstone [18]. Thus, one of the parameters to be varied is the modulus of elasticity. Two values were chosen, one corresponding to basalt and the other to sandstone. According to load tables presented in PCI [17], the concrete has a compressive strength of 5 psi. Although higher values recommended by Mast [16], Bachmann et al. [8] suggests that for bare structure of uncracked prestressed concrete β varies from.4 to.7. Based on these considerations a value of.9 was adopted. 3.3 Cross section geometries There are several combinations of prestress and cross sections to the same safe load and span. Two different sections were considered in this study as shown in Figure

4 Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 214 modify the density of the material, so that the shell element has the same mass as the hollow core slab. HC-1 HC- Figure 7. Sections analysed [17] 3.4 Elastic properties of shell elements Aiming to reduce computational effort, the hollow core slabs were modelled as a shell with orthotropic material. To maintain the geometric properties of the hollow core slabs, a procedure inspired by Johansson [19] was used. The model was created in Abaqus 6.13, with a lamina material. To define a lamina material, the properties required are: E 1, E 2, ν, G, G 13, G 23. Direction 1 is along the voids, direction 2 is in the slab plane, perpendicular to direction 1, and finally direction 3 is the plane normal. In order to obtain these properties, a portion of a slab segment was modelled in Abaqus with a fine mesh of solid elements, representing the hollow core slab. Loads were applied in different directions and the displacement for each load was measured. Figure 8 demonstrate the process (3) / (4) Table 2. Properties of shell elements t (m) ρ(kg/m³) E 1 (GPa) E 2 (GPa) HC-1-basalt HC-1-sandstone HC--basalt HC--sandstone G (GPa) G 13 (GPa) G 23 (GPa) HC-1-basalt HC-1-sandstone HC--basalt HC--sandstone Table of parameters In total models were obtained by combining two elastic modulus, two shell thickness and three spans. Table 3 shows the numbering of models. Table 3. Models analysed Model E 1 Thickness Span E Model 1 Thickness Span (GPa) (in) (m) (GPa) (in) (m) Walking loads As a simplification, the load was applied at the centre of the slab. The following coefficients were used: Table 4. Coefficients for a normal walking [8] f p (Hz) α 1 α 2 Φ 2 α 3 Φ π/2.1 π/2 / (5) / (6) 4 RESULTS 4.1 Mode shapes and frequencies The first three mode shapes are shown in figures 9 to 11. The first mode was a bending mode of the slab. The second mode can be described as torsion of the floor. And finally the third mode was a higher order bending mode of the slab. Figure 8. Models in solid elements (adapted from [19]) Where q is the distributed load, L is the span, I is the moment of inertia, u is the displacement at the center of the span, τ ij are the shear stress components, γ ij are the shear strain components, G ij are the shear modulus The thickness of the shell was calculated to obtain the same moment of inertia in the voids direction. It is also necessary to Figure 9. Mode shape 1 198

5 Proceedings of the 9th International Conference on Structural Dynamics, EURODYN Figure 1. Mode shape 2 f1 (Hz) y =.544x x R² = Span Length (m) y =.2642x R² =.165 Figure 11. Mode shape 3 f1 (Hz) 6 4 As shown in Table 5, the natural frequencies varied significantly in the models within the parametric study. The natural frequencies of the first mode (f 1 ) were generally in the neighbourhood of the frequencies imposed by human walking. In general, the frequencies of the second and third modes (f 2 and f 3 ) were considerably higher than the frequencies generated by walking. Table 5. Natural frequencies (Hz) Mod. f 1 f 2 f 3 Mod. f f 2 f f1 (Hz) Thickness (inches) 1 y = 5E 5x R² = Elastic Modulus (MPa) The influence of the most important factors considered in the parametric study on the natural frequency of the first mode (f 1 ) is depicted by Fig.. The span was the major factor influencing f 1 with a consistent decrease of f 1 caused by the increase of span. The slab thickness had a small influence, with an apparent reduction of f 1 with the increase of the thickness. On the other hand, the increase of the elastic modulus was followed by an increase (although small) in f 1. The minimum natural frequencies (f min ) required for office floors, were calculated according to Mast [ 16] and are shown in Table 6. Almost all models had natural frequencies below the suggested limit (except for three with the shorter span). Table 6. Natural frequencies x limit proposed by Mast [16] Mod. f 1 f min Mod. f f min Figure. Influence of parameters on first natural frequency 4.2 Displacements In Fig. 13 a typical time x displacement of the centre of the slab is shown.,15 Displacement (mm),1,5, ,1,15,2 Time (s) Figure 13. Typical displacement x time curve 199

6 Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 214 In Table 7 the peak displacements of all the slabs are shown. The increase in span was responsible for an increase in peak displacement. In general, increasing the thickness contributed to slightly smaller peak displacements. Figures 14 and 15 illustrate the influence of parameters on peak displacement. It is also possible to note that the reduction of f 1 was strongly associated to an increase in peak displacement. Table 7. Peak displacement Model Peak displacement (mm) Model Peak displacement (mm) Figure 15. Influence of parameters on peak displacement A Fast Fourier Transform was applied to the displacement x time data allowing a comparison in the frequency domain to the limits of the Reiher Meister modified scale. These results are shown in Fig. 16, and according to this criterion the vibrations were not perceptible. Amplitude (mm) Amplitude (mm) Models 6.1 m span 1 Not perceptible 1 Slightly perceptible,1, ,1,1,1,1,1,1,1,1 Frequency (Hz) Models 9.14 m span 1 Not perceptible 1 Slightly perceptible,1, Frequency (Hz) Figure 14. Influence of parameters on peak displacement Amplitude (mm) Models.19 m span 1 Not perceptible 1 Slightly perceptible,1, ,1,1,1,1 Frequency (Hz) Figure 16. FFT of the displacement plotted against Reiher Meister modified scale. 11

7 Proceedings of the 9th International Conference on Structural Dynamics, EURODYN Accelerations Figure 17 shows a typical acceleration x time curve. The peak acceleration of each model is shown in table 8. When considering the criteria for peak acceleration (table 1), half of the models showed clearly perceptible vibration and the remaining just perceptible levels. Acceleration (mm/s²) Figure 17. Acceleration for model 1 Table 8. Peak acceleration Model Peak acceleration (mm/s²) Time (s) Classification according to [8] (see Table 1) 1 92 Just perceptible Clearly perceptible 3 8 Just perceptible 4 93 Just perceptible 5 19 Clearly perceptible 6 1 Clearly perceptible 7 73 Just perceptible 8 78 Just perceptible 9 18 Clearly perceptible 1 14 Clearly perceptible Clearly perceptible 92 Just perceptible The parametric study revealed that the largest peak acceleration occurred for the intermediate span (9.14 m), although a notably high level was also observed for the largest span (.19 m). The lowest levels occurred for the smaller span (6.1 m) (figure 18). The increase in thickness reduced the peak accelerations. Only one model showed clearly perceptible accelerations among the models with thickness of inches. Contrarily to the expected behaviour, with the higher modulus of elasticity (39.5 GPa) occurred most of the highest acceleration levels. Finally, looking to the peak acceleration x f 1 curve (Fig. 19), the maximum peak accelerations occurred for f 1 equal to 4.88 and 7.25 Hz. Considering the theoretical assumption that above 8.49 Hz ( 2 multiplied by the frequency of third harmonic of walking forces 6 Hz) the dynamic amplification is significantly reduced, the results were in agreement with the theory. The peak accelerations for f 1 above 8.49 Hz were equal or bellow 115 mm/s 2. However this acceleration is still considered clearly perceptible. Peak aceleration (mm/s2) Peak aceleration (mm/s2) Peak aceleration (mm/s2) y =.969x x R² = Span Length (m) y = 2.5x R² = Thickness (inches) y =.16x R² = Figure 18. Influence of parameters on peak acceleration Peak aceleration (mm/s2) y = x R² =.344 Elastic Modulus (MPa) f1 (Hz) Figure 19. Influence of first natural frequency on peak acceleration 4.4 Comparison of different criteria As shown by Table 9, the different criteria employed for evaluation of human acceptance and perceptibility in this parametric study were contradictory. This fact shows the need for more conclusive studies on the subject. 111

8 Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 214 Table 9. Comparison between selected criteria Model Mast (21) Reiher Meister [8] modified 1 acceptable not perceptible Just perceptible 2 acceptable not perceptible Clearly perceptible 3 acceptable not perceptible Just perceptible 4 not acceptable not perceptible Just perceptible 5 not acceptable not perceptible Clearly perceptible 6 not acceptable not perceptible Clearly perceptible 7 not acceptable not perceptible Just perceptible 8 not acceptable not perceptible Just perceptible 9 not acceptable not perceptible Clearly perceptible 1 not acceptable not perceptible Clearly perceptible 11 not acceptable not perceptible Clearly perceptible not acceptable not perceptible Just perceptible 5 CONCLUSIONS A parametric study on the vibration sensitivity of hollow-core slabs floors was carried out. The main variables considered in the study were the span (6.1, 9.14 and.19 m), slab thickness (1 and in) and modulus of elasticity of concrete (23,2 and 39.5 GPa). It was possible to notice that the most important factor on the results was the span. An increase in the span caused significant decrease in the first natural frequency, strong increase in peak displacements and small average increase in peak accelerations. Increasing the thickness reduced slightly the first natural frequency, the peak displacement and acceleration. On the other hand the increase in the modulus of elasticity was responsible for a slight increase in the first natural frequency, peak acceleration and displacement. It was possible to note that the reduction of the first natural frequency was responsible for a significant increase in peak displacements while the increase in accelerations was small. Finally, a comparison between different criteria of acceptance regarding human comfort was carried out. The natural frequencies were in most situations bellow recommended limits of minimum frequencies for office floors. The results in terms of displacements were considered adequate, as they were bellow the not perceptible line. On the other hand, the accelerations were in most situations classified as clear perceptible and sometimes as just perceptible. The comparison with different criteria showed contradictory results as the same slabs were adequate based on some criteria and not adequate based in the others. The existing contradiction between acceptance criteria and the occurrence of inadequate dynamic behavior for some of the evaluated slabs emphasize the importance of the continuity of this research. The results shown here are part of an ongoing research project on vibration sensitivity of hollow-core slabs floors. On the continuation of this work additional parameters will be included and an experimental program will be carried to calibrate the numerical models. 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