Improved model for human induced vibrations of high-frequency floors

Abstract The key UK design guidelines published by the Concrete Society and Concrete Centre for single human walking excitation of high-frequency floors were introduced more than 10 years ago. The corresponding walking force model is derived using a set of single footfalls recorded on a force plate and it features a deterministic approach which contradicts the stochastic nature of human-induced loading, including intra- and inter- subject variability. This paper presents an improved version of this force model for high-frequency floors with statistically defined parameters derived using a comprehensive database of walking force time histories, comprising multiple successive footfalls that are continuously measured on an instrumented treadmill. The improved model enables probability-based prediction of vibration levels for any probability of non-exceedance, while the existing model allows for vibration prediction related to 75% probability of non-exceedance for design purposes. Moreover, the improved model shifts the suggested cut-off frequency between low- and high-frequency floors from 10 Hz to 14 Hz. This is to account for higher force harmonics that can still induce the resonant vibration response and to avoid possible significant amplification of the vibration response due to the near-resonance effect. Minor effects of near-resonance are taken into account by a damping factor. The performance of the existing and the improved models is compared against numerical simulations carried out using a finite element model of a structure and the treadmill forces. The results show that while the existing model tends to overestimate or underestimate the vibration levels depending on the pacing rate, the new model provides statistically reliable estimations of the vibration responses. Hence, it can be adopted in a new generation of the design guidelines featuring a probabilistic approach to vibration serviceability assessment of high-frequency floors.

8 the design guidelines (Table 1), cannot induce a resonant build-up response. This implies that a more 166 detailed study should be carried out to derive the cut-off frequency, as elaborated in the next section. 167 2.3 Determining cut-off frequency between low-and high-frequency floors 168 As already observed above, a typical transient response due to walking comprises a series of velocity 169 peaks corresponding to heel strikes, followed by a decaying vibration response to around zero before 170 the beginning of the next footfall, as shown in Fig. 2c [14]. This means that the response due to previous 171 footfalls has a negligible contribution to the response due to the present footfall. On the other hand, for 172 non-transient vibration responses (Fig. 2a and Fig. 2b), the response is affected by a number of previous 173 footfalls depending on the structural damping. 174 Theoretically speaking, a transient response time history can be reconstructed from the peak responses 175 followed by an exponentially decaying response in between them. In this case, the reconstructed 176 vibration response level is similar to that of the original time history response (Fig. 5). Therefore, the 177 proposed methodology to identify the cut-off frequency is as follows: 178 • Simulate vibration responses by applying measured walking forces [15,16] on SDOF oscillators 179 with different natural frequencies. 180 • For each response time history, extract the peak velocity responses corresponding to each 181 footfall strike with their exact times. 182 • Use the peak velocities to reconstruct the time history response which comprises only a 183 decaying response after each peak velocity, as shown in Fig. 5. 184 • Establish the difference between the original and the reconstructed responses by calculating the 185 ratio of their MTVVs (i.e. MTVV velocity of the reconstructed response over that for the 186 original response). 187 • Repeat this process for the different natural frequencies of the SDOF oscillator and the 188 measured walking forces [15,16]. 189 • Identify the frequency corresponding to a value of the MTVV ratio which is reasonably close 190 to 1.0, as explained below. 191 9 The closer the MTVV ratio to 1.0, the more similar are the reconstructed response and its corresponding 192 simulated transient response. Fig. 6 compares two cases when the MTVV ratio is close or far from 1.0. 193 The process of generating reconstructed vibration responses was repeated for all available walking 194 forces [15,16] when the natural frequency of the SDOF oscillator is an integer multiple of the pacing 195 rate (up to 20 Hz). This is to consider the effect of the harmonics at these frequencies. The damping 196 ratio used in the simulations was 3% while the modal mass was assumed 1 kg. The MTVV ratios 197 corresponding to this analysis are presented as box plots in Fig. 7. The upper and lower ends of the 198 rectangles represent the values corresponding to a 75% and 25% chance of non-exceedance, 199 respectively. The whiskers (ends of the extended lines from the boxes) represent the maximum and 200 minimum values. 201 At relatively low pacing rates, the MTVV ratio approaches 1.0 at a lower SDOF natural frequency than 202 that for higher pacing rates (Fig. 7). This indicates the dependency of the cut-off frequency on the 203 pacing rates. For natural frequencies at or above 14 Hz, the median of the MTVV ratios for all pacing 204 rates (horizontal lines in the middle of the rectangles in Fig. 7) were within 10% of 1.0 (i.e. 0.90-1.10), 205 which is reasonably close to 1.0. This implies that the shape of vibration responses corresponding to 206 SDOF oscillators with natural frequencies above 14 Hz resemble typical transient responses regardless 207 of the pacing rate. The harmonics of walking forces [15,16] corresponding to frequencies above 14 Hz 208 are more likely to increase the amplitude of the vibration responses rather than to induce a clear resonant 209 build-up response. Therefore, this frequency has been selected as the cut-off frequency above which the 210 human-induced vibration of floors is dominated by transient response. 211

212
This section starts with necessary details of Arup's deterministic force model (Section 3.1), followed 213 by its expansion into a more sophisticated probability-based successor proposed in this study 214 (Section 3.2 and Section 3.3) and its implementation in vibration serviceability assessment of high-215 frequency floors (Section 3.4). 216

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The model was derived using a database of over 800 single footfalls recorded for 40 individuals stepping 218 on a force plate while walking at a range of pacing rates controlled by a metronome [25]. The measured 219 footfalls were shifted repeatedly along the time axes to synthesise the corresponding artificial and 220 perfectly periodic force time history (Fig. 8). Each such force was applied to a series of SDOF 221 oscillators with natural frequencies of 10-40 Hz and only the peak velocity for each simulation was 222 extracted. The modal mass was assumed 1 kg for all simulations, so that the peak velocity response was 223 numerically equivalent to the impulse represented by the shaded area in Fig. 8 and expressed in Ns. 224 Such an impulse is termed effective impulse. 225 For varying pacing rates, the mean of the extracted effective impulses are shown in Fig. 9 as a function 226 of the 'floor frequency (Hz)', which is the natural frequency of the 1 kg SDOF system. The 227 corresponding curve fit is: 228 where, is the effective impulse (Ns), is the pacing rate (Hz), is the SDOF natural frequency 229 (Hz) and is a coefficient which has a mean value of 42 and a standard deviation of 0.4 while its 230 corresponding design value for 75% chance of non-exceedance is 54. 231 This effective impulse is used in Eq. (3) to calculate the contribution of the time history response of 232 each vibration mode in the total response. This response corresponds to one footfall strike. 233 Here, ( ) (m/s) is the contribution to the velocity response from mode at each time step , and 234 are the mode shape amplitude at the node of application of the force and the node of interest, 235 respectively, (kg) is the modal mass of the mode , ζ is the modal damping ratio, and 236 (rad/s) are the angular frequency and damped angular frequency of mode , respectively. 237 The contribution of each mode in the total response, calculated using Eq. (3), should be determined 238 individually for vibration modes with a natural frequency up to twice the fundamental frequency. 239 The total velocity response ( ) is calculated using Eq. (4) based on the assumption that the structure 240 remains linear during vibration, and therefore, the principle of superposition applies. 241 The criterion of the vibration serviceability assessment for high-frequency floors is based on the 242 maximum 1-s RMS of the total response calculated using Eq. (1). 243

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Based on the analysis presented in Section 2, the key differences between the steps followed to derive 245 Arup's model and its advanced version explained in the following sections are: 246 • The range of natural frequencies of the SDOF oscillators used to derive the present model is 247 14-40 Hz with an increment of 0.1 Hz, compared with 10-40 Hz used to derive Arup's model. 248 This is to account for the proposed cut-off frequency of 14 Hz (Section 2.3). 249 • In the new model, SDOF simulations, which utilised continuously measured treadmill forces 250 [15,16], were carried out to extract the peak velocities corresponding to 50 successive footfalls. 251 These peak velocities are treated as the effective impulse ( ) explained in Eq.
(3) but they 252 belong to the improved model presented in this paper. 253 • Contrary to Arup's model, the damping effect is considered in the new model. This is to take 254 into account the slight amplification of the vibration response of high-frequency floors induced 255 by the near-resonance effects corresponding to the higher harmonics of walking loading, as 256 explained in Section 2. 257 Apart from the above mentioned differences, the new model was derived using the same procedure as 258 that used for Arup's model. The damping ratio was assumed 3% in the SDOF simulations, while the 259 effect of other damping ratios is elaborated in Section 3.3.3. A time step of 0.005 s was used in the 260 analysis. The total number of the peak velocities (effective impulses) obtained from the analysis is more 261 than 900,000, i.e. 715 continuously measured walking forces [15,16], each comprising 50 footfalls, 262 applied to 261 SDOF oscillators with natural frequencies in the range14-40 Hz and 0.1 Hz increments. 263

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The peak velocities corresponding to a single footfall and multiple SDOF oscillators can be presented 265 as a spectrum. This means there are 1,400 spectra created and analysed for this pacing rate. The differences between 269 these spectra can be explained by the inter-and intra-subject variabilities of human walking forces 270 (Section 2.1). Hence, a statistical approach is utilised here to model the spectra as a function of SDOF 271 natural frequency, pacing rate and damping ratio. 272 In Fig. 10 peaks can be noticed around integer multiples of the pacing rate due to resonance or near-273 resonance effects. This can be explained by the effect of harmonics of the walking excitation at integer 274 multiples of pacing rates as explained in Section 2.2. 275 To simplify the modelling of the spectrum shown in Fig. 10, it was split into two components: a 'base 276 curve' and an 'amplification factor' (grey curve and black dots, respectively, in Fig. 10). The base curve 277 was assumed continuous across all SDOF frequencies, while the amplification factor was assumed to 278 be present at locations of each integer multiple of the pacing rate (black dots in Fig. 10). The grey dots 279 represent the locations where the amplification factor has no effect and its location is assumed to be in 280 the middle of each two successive integer multiples of pacing rate (subsequent pairs of the black dots). 281 Between black and grey dots, the amplification factor can be assumed to change linearly and its value 282 can be interpolated between them. 283 13 Hence, the peak velocity at each integer multiple of the walking frequency is theoretically equal to the 284 base curve value at that natural frequency multiplied by the corresponding contribution of the 285 amplification factor at the same natural frequency, as shown in Fig. 10. 286 A Matlab script was written to extract the amplification factor around each integer multiple of the pacing 287 rate, while the base curve was constructed by connecting the grey dots linearly (Fig. 10) where, ζ ( , , ζ) is the damping factor (dimensionless parameter), which is described in Section where ( ( , )) is the probability density function, and are the shape and scale parameters and 299 г( ) is the gamma function evaluated at . 300 14 The fitting process is repeated for 35,750 spectra (i.e. 715 walking force time histories, each comprising 301 50 footfalls). Values of the extracted parameters and (dimensionless parameters) were then surface 302 fitted as functions of and (measured in Hz) using polynomial and exponential forms, due to the 303 shape of the data to be fitted. The fitting is shown in Fig. 12 and described by Eqs. (7) and (8). 304 = 4.5 − 0.12 + 3 , = 0.08 + 2 3.3

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Values of amplification factors ( , ) follow the generalised extreme value distribution (Fig. 13), 306 which probability density function ( ( , )) is characterised by location , scale and shape τ Interpolation of ( , ) should be considered if the natural frequency is not an integer multiple of the 314 pacing rate (Fig. 10). For instance, if the natural frequency lies exactly in the middle of two successive 315 integer multiples of the pacing rate, the amplification factor will have no effect on the response (i.e. 316 ( , ) = 1.0). This takes into account that the amplification factor has a reduced effect between the 317 integer multiples of the pacing rate, as shown in Fig. 10.
where, , and are the parameters of the equation (dimensionless parameters). Fig. 15 shows the 326 fitted plane corresponding to a damping ratio of 5%. 327 16 Finally, values of the parameters , and are curve fitted as functions of the damping ratio. The 328 resulting curve fits are illustrated in Fig. 16 and described by Eqs. (15), (16) and (17). The shapes of 329 these equations are decided based on the trends observed in the data (Fig. 16). between each two successive footfalls needs to be consistent with the pacing rate. The residual 361 of each decaying response at the beginning of the next footfall is assumed to be zero. 362 • The total response corresponding to the contribution from all vibration modes having 363 frequencies up to twice the fundamental frequency is calculated using Eq. (4). This number of 364 vibration modes is adopted from the Arup's model.
where, is the standard deviation of the population and ̅ is the standard error of their mean. 369 In this study, the samples are a set of MTVV velocity calculated following the above mentioned 370 procedure, while the population refers to all possible MTVV velocities. As the standard deviation of 371 the population is unknown, it is estimated to be the standard deviation of the samples. Assuming the 372 samples are independent and identically distributed, there is a 95% chance that their mean is within the 373 population mean ∓ a tolerance of 1. The FEM utilised in this section is developed using ANSYS FE software [34] and is updated to match 390 the experimentally measured modal properties of the corresponding real floor (Fig. 18)