Vibration serviceability assessment of a staircase based on moving load simulations and measurements

Porto, Portugal, 30 June - 2 July 2014 A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.) ISSN: 2311-9020; ISBN: 978-972-752-165-4 Vibration serviceability assessment of a staircase based on moving load simulations and measurements Charlotte Schauvliege 1, Pieter Verbeke 2, Peter Van den Broeck 1,2, Guido De Roeck 2 1 KU Leuven @ KAHO, Department of Civil Eng., Technology Cluster Construction Structural Mechanics Gebroeders De Smetstraat 1, B-9000 Ghent, Belgium 2 KU Leuven, Department of Civil Eng., Structural Mechanics Division, Kasteelpark Arenberg 40, B-3001 Heverlee, Belgium email: charlotte.schauvliege@kuleuven.be, pieter.verbeke@bwk.kuleuven.be, peter.vandenbroeck@kuleuven.be, guido.deroeck@bwk.kuleuven.be ABSTRACT: This paper investigates the vibration serviceability of a steel half turn staircase in a new office building. Based on a finite element model, the vibration serviceability is assessed under moving load conditions simulating walking or running of a single person ascending or descending the stairs. For reference purposes also simulations with stationary loading conditions are performed. The predicted acceleration levels are evaluated using the response factor R, corresponding to the multiplier of the base perception curve of BS6472 for vertical vibration. Response factors significantly higher than 32 are predicted which is the criterion for single person excitation proposed by Bishop et al. to ensure negligible adverse comment for the case of light use in an office environment. To validate the predicted results, measurements are carried out to identify the modal parameters of the staircase and to evaluate the vibration levels under different conditions of usage. KEY WORDS: Human induced vibrations; Vibration serviceability; Staircases. 1 INTRODUCTION The design of slender and lightweight staircases leads to a low stiffness to mass ratio resulting in structures prone to human induced vibrations. Because of this trend, the natural frequencies of structures tend to be in the range of human induced loading frequencies. The phenomena of resonance can cause high acceleration levels in these structures, even under normal circumstances. This can cause comfort problems which contributes to a feeling of insecurity or can lead to damage in the worst case scenario. An increased awareness to these problems during the design stage can prevent these problems and small interventions can make a difference to the vibration serviceability of structures under human induced vibrations. To assess the vibration serviceability, the acceleration response needs to be calculated under different simulations using a moving load, simulating walking or running of a single person ascending or descending the stairs, and also simulations with stationary loading are performed. Kerr and Bishop [1,2] carried out numerous force plate experiments to describe the vertical force in function of time for one person walking on a staircase. According to these measurements, the stationary load that consists of impact footfalls can be described as a Fourier series with two harmonics. The force-time history of a walking force can also be described with a Fourier series as a stationary load, or as a sequence of footsteps [3]. The vibration levels can be predicted using these force models, simulating loading scenarios on a finite element model. The load cases include a realistic scenario of walking or running up- and downstairs of the staircase with one person. Secondly, the vibration levels for a stationary load on a vibration sensitive spot on the structure are investigated as a reference. The ISO Standard for serviceability of buildings and walkways against vibrations [5] provides an evaluation of the maximum vibration levels for staircases using the response factor R. This rating number will be used to define a wellfounded judgment about the vibration serviceability of the staircase in this outline of the paper. 2 DESCRIPTION OF THE STAIRCASE Figure 1: Side view staircase The atrium staircase of a new office building consists essentially of steel tube profiles and will reach through three floor levels, with a cantilever landing in between. This steel half turn staircase consists of two flights of nine or ten steps each, connected by a cantilever landing and bolted to the concrete upper and lower floor. A side view of one module of the staircase is shown in figure 1. The U-shaped steps and the platforms between the floor levels contain a concrete infill. Glass panels will be used between the handrails and the stringers. 1043

3 CALIBRATION OF THE FINITE ELEMENT MODEL This section describes how the finite element is calibrated in two steps, corresponding to the two construction stages in which the modal parameters were experimentally identified. First, the details of the operation modal analysis are summarised. Secondly, the correspondence between the measured and calculated modal parameters are investigated for the initial and calibrated finite element model. This will be done for both construction stages: (a) bare steel structure without concrete infill on the steps and landing nor handrail, (b) composite structure without handrail. 3.3 Calibration in construction stage 1: bare steel structure The initial finite element model representing the bare steel structure was constructed using beam elements for the stair steps and the main structure. The steel plate which is fixed to the bottom of the beams of the landing was modeled by shell elements. Each of the flight ends has two pinned supports (with fixed translations and free rotations), representing the bolted connection to the upper and lower concrete floor. 3.1 Operational modal analysis In both construction stages an identical measurement setup was used, using 12 tri-axial accelerometer sensors in a single configuration as presented in figure 2. Since the mode shapes of the initial finite element model indicated an important movement of the landing of the staircase, it was equipped with 8 sensors. The monitoring of the supports was performed by placing the sensor on the first/last step of the stairs. Figure 3: 3D representation of the staircase with boundary conditions. Figure 2: Measurement setup The output-only data was processed using the reference-based covariance-driven stochastic subspace identification (SSI-cov) [6]. The modal assurance criterion (MAC) is used to identify matching modes [7]. The measured mode and the calculated mode can be matched if their MAC value is close to 1. An important note is that the mass due to the weight of the sensors is taken into account in the finite element model to approximate the measured situation as good as possible. 3.2 Calibration method To calibrate the finite element model, parameters are calibrated by minimising a cost function that measures the discrepancy between measured and computed data. For this objective, a least squares cost function is used without regularisation, as described by Van Nimmen et al [8]. Each construction stage will be calibrated separately in order to start with a calibrated basic model before moving on to a more advanced building stage. The accuracy with which the modal parameters can be determined are taken into account with weight factors. During the calibration process, minimising the discrepancy between the measured and calculated frequencies will be considered as thousand times more important than the discrepancy between mode shapes, because natural frequencies can be determined more accurate than mode shapes. For the measurements in this construction stage, it has to be noted that since the welded wire mesh was already fixed for the reinforcement of the concrete, bolts were needed to lift the sensors with the aim of correctly registering the vibrations of the main structure of the staircase. The agreement between the initial finite element model and the measurements is presented in table 1. Only the finite element mode shapes that match with the measurements are represented. The first measured mode shape is a rigid body mode, which involves the movement of the building, while the others pertain to the stair construction itself. Table 1: Comparison between the measured, initial and updated modal parameters of the bare steel structure including mass of the sensors. Measured Initial Calibrated f,s ξ,s f MAC Δf f MAC Δf [Hz] [%] [Hz] [-] [%] [Hz] [-] [%] 4.53 4.75 6.86 1.96 7.61 0.53 7.88 0.969 3.5 7.61 0.968 0.0 10.44 0.41 11.36 1.44 12.33 0.45 12.50 0.870 1.4 12.33 0.871 0.0 15.78 0.44 16.42 0.900 4.1 15.78 0.897 0.0 16.23 0.63 For the calibration of this first model the support stiffnesses are considered as updating variables. In the finite element 1044

model, each support is modelled with three translational springs and three rotational springs. To reduce the complexity of the calculations, characteristics of the springs are assumed to be identical in the four supports. From table 1 it is clear that due to the calibration process the natural frequencies derived from the measurements and finite element models correspond almost perfectly, while the effect on the mode shapes is less noticeable. 3.4 Calibration in construction stage 2: composite structure In this construction stage the U-shaped steps and the landing contain a concrete infill. The calibrated finite element model from the previous construction stage is therefore extended with shell elements to model the concrete at the landing and a modified mass density to model the mass effect of the concrete infill of the steps. Table 2 shows the experimentally identified modal parameters together with the results of the modified finite element model, indicated as initial for this construction stage. The addition of the concrete to the cantilever landing and stair causes a reduction of the natural frequencies due to the mass of the concrete. Besides this fact, the concrete has an increasing effect on the damping of the modes. force along a predefined path on the structure. Additionally, the response under stationary loading conditions is calculated as a worst case reference. The acceleration response will be compared to the vibration comfort levels defined by the response factor R, corresponding to the multiplier of the base perception curve of BS6472 for vertical vibration. If necessary, vibration mitigation will be applied and evaluated. 4.1 Modal parameters of the finished staircase The mode shapes and the natural frequencies of the finished staircase are calculated starting from the calibrated finite element model in construction stage 2 with the effect of the handrail and the glass panels modeled as mass. The damping ratios are taken from the measurements in construction stage 2. Table 4 summarises the calculated modal parameters used for the numerical simulations. For the finished staircase no measurement data are yet available. Table 4: Modal parameters of the finished staircase Mode shapes 3D view Front view Top view Mode 1: f 1 = 5.98 Hz ξ 1 = 0.0147 = ξ, c,2 Table 2: Comparison between the measured, initial and updated modal parameters of the composite structure. Measured Initial Calibrated f,c ξ,c f MAC Δf f MAC Δf [Hz] [%] [Hz] [-] [%] [Hz] [-] [%] 4.58 2.44 6.72 1.47 6.42 0.955-4.5 6.71 0.951-0.1 8.85 2.49 11.13 0.77 10.72 0.941-3.7 11.20 0.931 0.6 14.28 0.99 13.63 0.949-4.6 14.13 0.948-1.1 20.99 0.98 20.83 0.877-0.8 21.17 0.883 0.9 For this calibration step, the modulus of elasticity and the mass density of the concrete are assumed to be updating parameters. The initial and calibrated values of these parameters are summarised in table 3. Again the mode shapes that could be matched with measured mode shapes are taken into account for this updating process. The natural frequencies could be calibrated up to an accuracy of approximately 1%. Table 3: Uncertain parameters during the calibration process ρ concrete [kg/m³] E concrete [N/m²] Initial value 2.50 x 10³ 3.80 x 10 10 Calibrated value 1.71 x 10³ 5.21 x 10 10 4 VIBRATION SERVICEABILITY BASED ON SIMULATIONS To assess the vibration serviceability of the staircase, numerous loading scenarios are tested by simulating the acceleration response under a time varying walking or running Mode 2: f 2 = 9.92 Hz ξ 2 = 0.0077 = ξ, c,4 Mode 3: f 3 = 12.44 Hz ξ 3 = 0.0099 = ξ, c,5 Mode 4: f 4 = 18.66 Hz ξ 4 = 0.0098 = ξ, c,6 4.2 Human induced loading on the staircase The first step in the simulation process consists of defining the load that will be used for simulations. Kerr and Bishop [1,2] showed that the observed human induced forces on stair steps are highly dependent on the pace, which will differ as 1045

the person is running up- or downstairs. Human induced forces on stair steps consist of a bigger amplitude and frequency-content than normal walking forces. In case of a slow walking velocity, the person lands on the sole of the foot after which the heel strikes the tread, followed by a peak in the force diagram caused by the tip of the foot in order to prepare for the next step. While in the case of walking with a high velocity only the tip of the foot touches the tread which will cause an impulse load. Both load cases are represented during the simulations. For the description of the walking or running force could be referred to the definition of a single footstep [3] or a stationary load [1,4] as defined in the literature. Secondly, to simulate a moving and time varying force along a realistic walking path on the finite element model, defining a walking and running force of a single footstep is required, since each force will be allocated to a footstep location. Li et al [3] derived the single step walking force from the continuous walking force [4], and described this force by the Fourier series of equation 2. An example of the single step walking force is shown in figure 6. F ( t) G e 5 h 1 n An sin t T, e 0 t T e (2) First of all, the stationary load of a walking and running force can be written as a Fourier series consisting of two (running), or four or five (walking) harmonic components. The properties of the force depend on the step frequency, the weight of the person and the load type. Equation 1 describes the Fourier series, while figure 4 and 5 show an example of the stationary walking and running force as a function of time. F ( t) G e h max h 1 G sin(2 hf t ) (1) h s h Figure 6: Single step walking force (G = 700 N, f s = 1.99 Hz), according to Li et al [3]. The single step running force is derived from the stationary running force by considering only the positive impulse force downwards, as shown in figure 7. Figure 4: Stationary walking force (G = 700 N, f s = 1.99 Hz) as a function of time, together with the components of the Fourier series [4] Figure 5: Stationary running force (G = 700 N, f s = 2.99 Hz) as a function of time, together with the components of the Fourier series [1,5]. Figure 7: Single step running force (G = 700 N, f s = 2.99 Hz). 4.3 Simulation and results An estimation of the expected vibration levels in real conditions can be obtained by simulating the response due to one person descending or ascending the staircase. Both combinations of ascending or descending and walking or running are discussed during this case study. The step frequencies will assume a critical value, which means that one of the harmonic components will be equal to a natural frequency of the staircase. This resonance condition induces 1046

the highest acceleration levels. In the case of a stationary walking force the range of step frequency varies from: 1.2 Hz to 2.5 Hz and contains four harmonic components. This means that the load frequency, which is a multiple of the step frequency varies from 1.2 Hz to 10 Hz. A stationary running force was described in the literature by using two harmonic components with a range of step frequencies between 1.2 Hz and 4.5 Hz corresponding to a load frequency up to 9 Hz. The first two natural frequencies of the staircase are within one of these critical ranges. Table 5 gives an overview of the step frequencies of interest. Table 5: Critical step frequencies for simulations. f j [Hz] f j /1 [Hz] f j /2 [Hz] f j /3 [Hz] f j /4 [Hz] 5.98 5.98 2.99 1.99 (1.50) 9.92 9.92 4.96 3.31 2.48 From table 5, three critical load scenarios are defined: walking at f walk =1.99 Hz or 2.48 Hz or running at f run = 2.99 Hz. The walking path and the corresponding numbers of the steps that were used during the simulations are graphically displayed in figure 7. It takes into account a realistic scenario of a person moving up- or downstairs. A closer look will be taken to two extra points (27,28) at the corners of the platform, because of the sensitivity of these cantilever positions. acceleration levels. The most critical points are situated at the corners of the landing, which is not directly situated on the walking path. a) b) c) Figure 8: Vertical acceleration at position (a) 6 middle of the upper flight, (b) 12 middle of the landing and (c) 27 corner point landing, while running downstairs with f run = 2.99 Hz and walking downstairs with f walk = 1.99 Hz (left and right column respectively). a) Figure 7: Load path used for the simulations and reference points 27 and 28 at the corners of the landing. Figure 8 and 9 show the vertical and horizontal vibration response at three different positions of the staircase while running and walking downstairs at a critical step frequency. The positions are situated in the middle of the upper flight, in the middle and at one of the corner points of the cantilever platform. From the time histories of the accelerations, derived from the simulations at these step frequencies, the building up of resonance is obvious. As a result, the calculated maximum acceleration strongly depends on the damping ratio, which in these simulations corresponds to construction stage 2, without the handrails and the glass panels installed. The running force causes overall a larger acceleration response on the structure, which can be explained by the high dynamic load factor of this type of force. The landing is also more sensitive to vibrations than the stair flights, due to the higher modal displacements at that area, causing higher b) c) Figure 9: Horizontal acceleration along the length of the landing at position (a) 6 middle of the upper flight, (b) 12 middle of the landing and (c) 27 corner point landing, while running downstairs with f run = 2.99 Hz and walking downstairs with f walk = 1.99 Hz (left and right column respectively). 1047

From figure 9 can be concluded that not only the vertical accelerations, but also the horizontal accelerations reach a significant high value. 4.4 Evaluation of the vibration serviceability The vibration levels in the structure will be compared with the user comfort criteria. Vertical direction In this case study the vertical acceleration levels are evaluated using the response factor R, corresponding to the multiplier of the base perception curve of BS6472 for vertical vibration, as shown in figure 10. Since the sensitivity of humans will depend on the frequency of the vibrations, not every acceleration level will be rated in the same way. Figure 11: Response factor R along the walking path (1-26) and the corner points of the landing (27-28) for f run,downstairs =2.99 Hz, plus the response factor at the corner points for f stationary =2.99 Hz at the critical corner point on position 27 and criteria R max, light use and R max, heavy use according to Bishop et al [2]. Figure 10: Perceptible acceleration level in the vertical direction, corresponding to R = 1. Since acceleration levels in resonance conditions are investigated, ü R=1 will be evaluated corresponding to the relevant natural frequency. Figure 11 shows the response factor in the points 1-26 of the load path (as defined in figure 7) and in points 27 and 28, situated at the corners of the landing, for a single person running downstairs with f run,downstairs =2.99 Hz. From these results, it is clear that the highest vibration levels are encountered at the landing with the maxima situated at the corner points. Also shown in figure 12 as a reference, is the response factor due to a stationary running load at the most sensitive corner point 27. From the difference in acceleration level caused by this stationary load and the moving load of the simulations, it can be concluded that resonance is not fully built up during the passage of a person. An overview of the maxima, situated at the corner points of the landing, for a single person walking or running upstairs or downstairs at the critical step frequencies is shown in figure 12. It is clear that the running force causes higher accelerations than the walking force because of its larger force amplitudes. Also there is no significant difference in the vibration response between ascending and descending the staircase. Figure 12: Maximum global response factor R in the vertical direction and criteria R max, light use and R max, heavy use according to Bishop et al [2]. An assessment of the vibration serviceability can be carried out by comparing the calculated response factors with the comfort criteria proposed by Bishop et al [2]. A distinction is made between frequently used staircases (e.g. public buildings, stadia), for which a maximum response factor R max, heavy use of 24 is proposed, and less intensively used staircases (e.g. offices), for which a response factor R max, light use of 32 is proposed. These comfort criteria are proposed to ensure negligible adverse comment and are also shown in figures 11 and 12. It can be concluded that the vibration levels of the staircase, as predicted by the simulations, do not meet these comfort criteria both for walking and running conditions. More specifically, the landing is very sensitive to the human induced vibrations which may be important since it can be used by people pausing to look around or allowing others to pass. 1048

Horizontal direction A similar procedure could be applied for the evaluation of the horizontal vibration levels, but to the author s knowledge no criteria specific for stairs are available yet in literature. Since the sensitivity of humans is higher for horizontal vibrations, a more negative evaluation is expected. A future measurement campaign will validate the simulations. The progress of the human induced vibration levels during several load cases will be registered and analyzed to get more information about the accelerations of the structure in real situations. 5 CONCLUSIONS The vibration serviceability of a staircase was investigated based on simulations of a single person walking or running downstairs or upstairs. A calibrated finite element model of the staircase in construction stage was developed in two steps corresponding to the two construction stages in which the modal parameters were experimentally identified. For the modeling of the finished staircase, as used in the simulations, the calibrated finite element model in construction stage 2 was modified by adding mass to account for the effect of the handrail and the glass panels, which were not yet installed. The damping ratios were assumed to correspond to the experimentally identified values in this construction stage. Critical load scenarios were selected by identifying critical step frequencies giving raise to resonance when the loading frequency, as a multiple of the step frequency, corresponds to a natural frequency of the staircase. With two natural frequencies below 10 Hz for the investigated staircase, two critical loading scenarios corresponded to walking and one critical loading scenario corresponded to running. Due to the high impulsive loading of the running, this last loading condition caused the highest vibration levels both in the vertical and the horizontal direction, especially at the cantilever landing. Based on the simulations, the vibration serviceability assessment shows that the response factors in the vertical direction are higher than the tentative comfort criteria for single person excitation as proposed by Bishop et al. in 1995 [2]. For the horizontal vibration levels, no comfort criteria are available. In the near future, a measurement campaign will be carried out on the finished staircase. The first objective of these measurements is to identify the modal parameters. Especially the effect of the glass panels on the damping ratios is of great interest since damping is governing the vibration levels at resonance conditions. The second objective is to measure the vertical and horizontal acceleration levels under various loading conditions. Based on these measurements, a validation of the simulations as presented in this paper will be carried out. Furthermore, these measurements will determine whether vibration mitigation measures are necessary. government (IWT, agency for Innovation by Science and Technology). The authors would like to thank the engineering offices and parties concerned for their cooperation and provided information on the investigated staircase. REFERENCES [1] S.C. Kerr and N.W.M. Bishop. Human induced loading on flexible staircases. Engineering Structure, 23: 37-45, 2001. [2] N.W.M. Bishop, M. Willford and R. Pumphrey. Human induced loading of flexible staircases. Safety science, 18: 261 276 (1995) [3] Q. Li, J. Fan, J. Nie, Q. Li and Y. Chen. Crowd-induced random vibration of footbridge and vibration control using multiple tuned mass dampers. Journal of Sound and Vibration, 329: 4068-4092 (2010) [4] M. R. Willford and P. Young. A Design guide for footfall induced vibration of structures. Camberley, Surrey: The concrete Center (2006) [5] ISO10137. Bases for design of structures serviceability of buildings and walkways against vibrations. International Organisation for Standardization [6] B. Peeters and G. De Roeck. Reference-based stochastic subspace identification for output-only modal analysis, Mechanical Systems and Signal Processing. Mechanical Systems and Signal Processing 13 (1999), no. 6, 855-878. [7] P.R. Allemang. The Modal Assurance Criterion (MAC): Twenty years of use and abuse. Cincinnati, USA: University of Cincinnati.(2002) [8] Van Nimmen K., Lombaert, G., De Roeck, G., Van den Broeck, P. (2014). Vibration serviceability of footbridges: Evaluation of the current codes of practice. Engineering Structures, 59, p 448-461. ACKNOWLEDGMENTS The results of this paper were partly obtained within the framework of the research project, OPTRIS Optimization of structures prone to vibrations, financed by the Flemish 1049