# Effect of Spring-Mass-Damper Pedestrian Models on the Performance of Low-Frequency or Lightweight Glazed Floors

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## Abstract

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_{1}< 8 Hz). Besides, the same SMD proposals are characterized by mostly different theoretical and experimental assumptions for calibration. On the practical side, strongly different SMD input parameters can thus be obtained for a given pedestrian. This paper focuses on a selection of literature on SMD models, especially on their dynamic effects on different structural floor systems. Four different floors are explored (F#1 and F#2 made of concrete, F#3 and F#4 of glass), with high- or low-frequency, and/or high- (>1/130th) or low- (1/4th) mass ratio, compared to the occupant. Normal walking scenarios with frequency in the range f

_{p}= 1.5–2 Hz are taken into account for a total of 100 dynamic simulations. The quantitative comparison of typical structural performance indicators for vibration serviceability assessment (i.e., acceleration peak, RMS, CREST) shows significant sensitivity to input SMD assumptions. Most importantly, the sensitivity of structural behaviours is observed for low-frequency systems, as expected, but also for low-mass structures, which (as in the case of glazed floor solutions) can be characterized by the use of lightweight modular units with relatively high vibration frequency. As such, major attention can be required for their vibrational analysis and assessment.

## 1. Introduction

_{1}< 8 Hz [2]) and thus higher vulnerability to vibrations induced by pedestrians [1,2]. Also, SMD models can efficiently include the effects of pedestrians in the lateral direction [9].

_{m}), which interacts with a substructure characterized by mass M

_{s}, fundamental vibration frequency f

_{1}, damping ratio ξ

_{s}, and thus possible sensitivity to vibration issues (Figure 1b). Even more complex, bipedal, and three-dimensional biodynamic models could also be used for sophisticated HSI calculations, see for example [10,11,12,13].

_{s}in relation to pedestrians (and thus possible additional HSI interaction with occupants). For the SMD models in [5,6], for example, the experimental calibration of input biodynamic parameters was carried out on an 11.63 m long prototype footbridge with f

_{1}= 4.27 Hz and ξ

_{s}= 1%. A prototype of a walkway consisting of a massive steel-concrete composite deck was used for the laboratory investigations reported in [7]. Rigid platforms were also used for the SMD validations reported in [5,7,8].

_{1}and mass M

_{s}parameters, compared to other construction typologies. To facilitate the comparative analysis, four different SMD proposals from earlier studies (SMD-1 to SMD-4, in the following, see [5,6,7,8]) are applied to four different floor configurations (F#1 to F#4, in the following), under normal walking scenarios of a single pedestrian. In doing so, two different glazed floor systems (F#3 and F#4) are analyzed and addressed towards two different concrete configurations (F#1 and F#2). From the total of 100 numerical analyses/configurations, the attention is focused on typical performance indicators of primary interest to characterize and assess the vibration response of structures (like acceleration peak, RMS acceleration and CREST factor).

_{1}and structural mass M

_{s}are taken into account. For the concrete floors (F#1 and F#2), for example, the M

_{s}term is generally high compared to pedestrians, regardless of the vibration frequency f

_{1}(i.e., > or <8 Hz). In the case of structural glass systems like F#3 and F#4 solutions [14,15], on the other side, basic structural components are typically characterized by structural mass M

_{s}which can often be relatively low compared to occupants, and by vibration frequency f

_{1}which is not necessarily “low” [16,17,18], see Figure 2. Dedicated studies are thus recommended to address HSI phenomena on glazed floors, especially under unfavourable operational and ambient conditions (see for example [19,20]).

## 2. Research Methods, Materials and Models

_{p}= 1.5–2 Hz (0.1 Hz the increment) for a total of 100 dynamic simulations.

#### 2.1. Selected Floor Systems

_{1,}respectively, with high structural mass M

_{s}compared to the occupant. Therefore, two additional low-frequency and low-mass floor systems (F#3 and F#4 in Table 1), inspired by earlier literature studies reported in [16,17,18] and Figure 2, were also considered in the present analysis. More in detail, the F#3 system coincides with the suspension platform in Figure 2a, while F#4 represents the modular system in Figure 2b. As shown in Table 1, the primary feature of the F#3 system is the low vibration frequency (f

_{1}< 8 Hz) but still relatively high mass Ms, compared to pedestrians (1/134th). This is not the case of the F#4 system, in which the slab is again composed of structural glass, and M

_{s}is relatively small compared to pedestrians (1/4th), whilst the fundamental vibration frequency f

_{1}is relatively high for vibrational assessment purposes.

_{1}> 8 Hz) can be representative of “rigid” floor systems, which are minimally sensitive to HSI phenomena. Conversely, F#2 and F#3 (with f

_{1}< 8 Hz) are expected to be the most affected by human-induced loads. Overall, mass contributions in Table 1 are another important parameter to address.

#### 2.2. Selected SMD Models of Literature

_{1}< 5 Hz). The authors elaborated a regression model for biodynamic SDOF parameters. The equivalent mass m, the damping coefficient c and the spring stiffness k can be calculated as a function of walking frequency f

_{p}and pedestrian mass M. The term c is fitted to m, and the stiffness term k derives from c, where:

_{1}= 30.1 Hz) was initially used in the experiments, and then the SMD characterization was extended to a flexible footbridge prototype as in [5], with f

_{1}< 5 Hz. It was found, for example, that the ANN was able to act efficiently with minimum uncertainty, and the final SMD proposal resulted in the following pedestrian parameters:

_{1}= 17 Hz [22]. The monitored experimental configurations included slow, normal, or fast walks.

_{md}.

_{m}the value given by Equation (12) once f

_{p}is assigned.

_{p}, with evidence of undamped frequency f

_{m}of the pedestrian model (from Equation (10)), equivalent mass m, spring stiffness k and damping coefficient c respectively. Most importantly, the relatively wide scatter of biodynamic pedestrian features can be seen when SMD-1 to 4 formulations is used—regardless of the type and features of floors—to describe the equivalent walking features of a single pedestrian (M = 80 kg). As such, variations are also expected in terms of corresponding performance indicators.

#### 2.3. Equivalent-Force Deterministic Model

_{i}the coefficients in Table 2 and t the time (in seconds) within a single footfall, where:

_{p}(in Hertz), see the example in Figure 5.

_{p}= 1.5 ÷ 2 Hz).

## 3. Parametric Numerical Analysis

#### 3.1. Modelling

_{p}. Moreover, comparative calculations were carried out towards the deterministic, equivalent force loading approach as in Equation (14), based on a rigid uncoupled SDOF assumption, with m = M (“RU” model, in the following).

_{1}, ω

_{2}are the natural circular frequencies corresponding to the first and second vibration modes of the F#1-to-#4 floor systems with input features summarized in Table 1.

#### 3.2. Structural Performance Indicators

_{Z,peak}) and CREST factor for each walking frequency, that is:

## 4. Discussion of Numerical Results

#### 4.1. Floor Response

_{p}, typical vibration responses were found highly sensitive to floor system features, both in the presence of low-frequency and low-mass parameters. Typical trends for selected walking frequencies are reported in Figure 7 in terms of RMS acceleration based on the RU modelling strategy.

#### 4.2. High-Mass Floor Sensitivity to Loading Strategy

_{s}is high compared to the occupant. Still, major variations can be noted in terms of vibration frequency (Table 1).

_{1}and high, similar mass M

_{s}, compared to the occupant. The corresponding percentage scatter of SMD averages to RU calculations is proposed in Figure 9 as a function of the imposed walking frequency. Note that for F#1 and F#3, the maximum scatter is estimated in the range of 30%. For the F#2 system only, the calculated percentage scatter is less pronounced in terms of absolute value, especially for faster walks (f

_{p}> 1.8 Hz). Still, on the other side, it is more sensitive to the walking frequency f

_{p}, due to additional interaction with motion harmonics, with up to +40% of the variation for f

_{p}< 1.8 Hz.

#### 4.3. Low-Mass Floor Sensitivity to Loading Strategy

_{p}. Furthermore, a very close correlation of numerical estimates can be noted, especially for SMD-1 and SMD-3 models (7.4% of their scatter in the range f

_{p}= 1.5 ÷ 2 Hz).

_{p}range), while the SMD-2 approach underestimates the other modelling strategies, especially for low walking frequencies f

_{p}(−10.7% for the RMS acceleration). In terms of acceleration peak trend, as in Figure 10b, a similar dynamic response can be noted for the F#4 system. As a significant consequence, the corresponding CREST factor from Equation (20) also has marked sensitivity to different SMD models and input biodynamic parameters.

_{p}.

#### 4.4. Comparison of Low-Frequency and Low-Mass Floors

_{p}= 1.5 ÷ 2 Hz interval are reported as a function of frequency ratio f

_{1}/f

_{p}. In this way, it is possible to see that the F#2 and F#4 systems described in Table 1 have comparable acceleration amplitudes, even characterized by markedly different structural parameters (mass and vibration frequency). Furthermore, for the F#1 and F#3 systems with similar mass ratios and vibration frequencies slightly above 8 Hz, the comparative results in Figure 11 have a qualitative and quantitative agreement. Most importantly, it is to note that the F#4 system, differing from F#2, has a high vibration frequency, which is mostly twice the reference threshold of 8 Hz which is recommended for vibration serviceability issues, but very small mass, compared to the occupant and to the other floor configurations.

_{p}variations. Also, as previously discussed, the F#2 system is largely sensitive to fp, whilst this effect is less pronounced for the stiffer F#4 system. Most importantly, the comparisons in Figure 12c in terms of standard deviation and percentage scatter from the average of SMD estimates of the selected models give evidence of a rather good agreement for both F#2 and F#4 systems, thus confirming that both low-frequency and low-mass structural solutions should be verified with dedicated tools.

## 5. Conclusions

_{1}and structural mass M

_{s}, compared to the walking pedestrian (with mass M and moving at f

_{p}). From the selected SMD models, it was shown, for example, that calibrated (m, k, c) parameters may strongly differ from each other, even to describe the same walking pedestrian/walking setup. Accordingly, modifications can also be expected in terms of predicted structural behaviours and performance indicators of the examined floors (like acceleration peak, RMS acceleration, CREST factor, etc.), which are of significant interest for vibration serviceability assessment. Furthermore, such sensitivity can be further affected by the intrinsic dynamic parameters of those floor systems most sensitive to human-induced loads.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Biodynamic pedestrian model: (

**a**) schematic representation of Spring-Mass-Damper (SMD), Single Degree of Freedom (SDOF) model (figure reprinted from [8] with permission from Elsevier©, copyright license number agreement 5458701302374, December 2022) and (

**b**) example of presently developed numerical model for Human-Structure Interaction (HSI) analysis.

**Figure 3.**Biodynamic pedestrian models: (

**a**) schematic mechanical model of a pedestrian on rigid or flexible structure (figure reproduced from [6] with permission from Elsevier©, copyright license number agreement 5458710774818, December 2022); (

**b**) experimental setup for pedestrian walking on flexible laboratory platform (figure reproduced from [7] with permission from Elsevier©, copyright license number agreement 5458711054568, December 2022); (

**c**) experimental analysis by Wang et al. to measure the ground reaction force (figure reproduced from [8] with permission from Elsevier©, copyright license number agreement 5458710094483, December 2022).

**Figure 4.**Selected biodynamic pedestrian models (with M = 80 kg) and variation of input SMD parameters with walking frequency f

_{p}: (

**a**) pedestrian model frequency f

_{m}, (

**b**) equivalent mass m, (

**c**) spring stiffness k, (

**d**) damping coefficient c.

**Figure 5.**Example of analytical description of single footfall vertical load, based on the equivalent-force deterministic model as in Equation (14). Vertical load F (in Newton) is normalized to the mass of the pedestrian (M = 80 kg).

**Figure 6.**Example of typical dynamic response for the F#2 concrete system under normal walk (M = 80 kg, f

_{p}= 1.5 Hz): (

**a**) acceleration time history on the structure and (

**b**) comparative detail (ABAQUS).

**Figure 7.**Numerical results for F#1 to F#4 floor systems, based on equivalent-force calculations (RU model), in terms of RMS acceleration, as a function of walking frequency f

_{p}(ABAQUS).

**Figure 8.**Numerical results for F#1, F#2 (concrete) and F#3 (glass) systems, based on different SMD or RU calculations, in terms of (

**a**–

**c**) RMS acceleration or (

**d**–

**f**) CREST factor, as a function of walking frequency f

_{p}(ABAQUS).

**Figure 9.**Numerical results for (

**a**,

**c**,

**e**) RMS acceleration and (

**b**,

**d**,

**f**) peak acceleration for F#1, F#2 and F#3 systems, in terms of SMD (average ± standard deviation) to RU percentage scatter, as a function of walking frequency f

_{p}(ABAQUS).

**Figure 10.**Numerical results for the glazed F#4 system, based on different SMD or equivalent-force calculations: (

**a**) RMS acceleration and (

**b**) peak acceleration, as a function of walking frequency f

_{p}, with (

**c**,

**d**) corresponding SMD (average ± standard deviation) to RU percentage scatter (ABAQUS).

**Figure 11.**Numerical results for (

**a**) RMS acceleration and (

**b**) peak acceleration on F#1, F#2, F#3 and F#4 systems, in terms of SMD (average), as a function of vibration frequency f

_{1}to walking frequency f

_{p}ratio (ABAQUS).

**Figure 12.**Sensitivity of RMS acceleration to selected SMD pedestrian models (with M = 80 kg) for floor systems: (

**a**) F#2 (concrete) and (

**b**) F#4 (glass) configurations, with (

**c**) trend of percentage scatter for both systems (ABAQUS).

Floor | Material | Span [m] | Surface [m ^{2}] | Frequency f_{1}[Hz] | Mass M_{s}[kg] | M/M_{s} |
---|---|---|---|---|---|---|

F#1 | Concrete | 5 | 30 | 11.05 | 10,350 | 1/130 |

F#2 | Concrete | 5 | 30 | 5.30 | 3530 | 1/44 |

F#3 | Glass | 14.5 | 40.6 | 7.28 * | 10,730 | 1/134 |

F#4 | Glass | 2.65 | 3.58 | 14.30 * | 320 | 1/4 |

**Table 2.**Definition of input coefficients K

_{i}for Equation (14), based on walking frequency f

_{p}.

Coefficient | f_{p} (Hz) | |
---|---|---|

≤1.75 | 1.75 ÷ 2 | |

K_{1} | −8 f_{p} + 38 | 24 f_{p} − 18 |

K_{2} | 376 f_{p} − 844 | −404 f_{p} + 521 |

K_{3} | −2804 f_{p} + 6025 | 4224 f_{p} − 6274 |

K_{4} | 6308 f_{p} − 16,573 | −29,144 f_{p} + 45,468 |

K_{5} | 1732 f_{p} − 13,619 | 109,976 f_{p} − 175,808 |

K_{6} | −24,638 f_{p} + 16,045 | −217,424 f_{p} + 353,403 |

K_{7} | 31,836 f_{p} − 33,614 | 212,776 f_{p} − 350,259 |

K_{8} | −12,948 f_{p} + 15,532 | −81,572 f_{p} + 135,624 |

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**MDPI and ACS Style**

Bedon, C.; Santos, F.A.
Effect of Spring-Mass-Damper Pedestrian Models on the Performance of Low-Frequency or Lightweight Glazed Floors. *Appl. Sci.* **2023**, *13*, 4023.
https://doi.org/10.3390/app13064023

**AMA Style**

Bedon C, Santos FA.
Effect of Spring-Mass-Damper Pedestrian Models on the Performance of Low-Frequency or Lightweight Glazed Floors. *Applied Sciences*. 2023; 13(6):4023.
https://doi.org/10.3390/app13064023

**Chicago/Turabian Style**

Bedon, Chiara, and Filipe A. Santos.
2023. "Effect of Spring-Mass-Damper Pedestrian Models on the Performance of Low-Frequency or Lightweight Glazed Floors" *Applied Sciences* 13, no. 6: 4023.
https://doi.org/10.3390/app13064023