# Post-Breakage Vibration Frequency Analysis of In-Service Pedestrian Laminated Glass Modular Units

^{*}

## Abstract

**:**

## 1. Introduction

## 2. State of the Art

#### 2.1. Reference Limit States for Design

- In more complex situations, the minimum performance requirements that the damaged structural must guarantee must be established ad hoc. The structure must, in any case, guarantee “fail safe” behaviour, with specific reference to hierarchy, system redundancy and resilience. The residual load-bearing capacity must include, as a minimum, the characteristic values of the self-weight of the structure;
- the single glass structural element must guarantee section redundancy, never intended as an increase in the sheet thickness, but as an increase in the number of glass layers. For laminated glass, the performance of a package made up of an interlayer and one or more fragmented glass sheets must be properly defined.

**Figure 2.**LG slab design: (

**a**) setup; (

**b**) uncracked (LGU) triple LG cross-section for service limit state (SLS) and ultimate limit state (ULS), or (

**c**) fractured (LGF) triple LG section for collapse limit state (CLS) design.

#### 2.2. Vibration Frequency of Laminated Glass Elements

_{ef}, for LG cross-sections), E the modulus of elasticity (MoE) of glass, while β is the wave number in Table 1.

_{2,0}) and the imaginary part (loss modulus G

_{2,ω}) are both involved in the frequency domain:

_{ef}= h

_{ef}(ω) that is subjected to iterative modifications with the dynamic stiffness of interlayers. Additional issues may arise from delamination of bonding layers [26,27]. Finally, special calculation methods should be taken into account to assess the vibration performance of LGF configurations as in Figure 2.

#### 2.3. Design of Pedestrian Glass Sytems

## 3. Preliminary Considerations

#### 3.1. Research Goal

#### 3.2. Literature Approaches for Post-Breakage Performance Assessment

_{fg}, in the following) that could be used to simplify the post-cracked analysis of LG members, so as to replace the nominal glass MoE (E = 70 GPa). More specifically, the equivalent E

_{fg}parameter was quantified in 17.5 GPa (≈1/4th the uncracked value E) and 2.35 GPa (≈E/30th) for the fractured glass layer in compression or tension, respectively [5].

#### 3.3. Post-Breakage Frequency Analysis of Laminated Glass (LG) Elements

^{®}). Given the above assumptions, the LG system is characterized by total glass mass in the order of M

_{LGU}= 322 Kg (2500 kg/m

^{3}the material density). The nominal material density is taken into account for intact and fractured glass layers. As for design calculations against short term live loads (room temperature), the PVB layers in use are assumed to offer a nominal MoE up to E

_{int}= 24 MPa, which is relatively small compared to intact glass (E = 70 GPa) and typically associated to weak mechanical bonding of glass components. At this stage, frequency predictions are carried out disregarding the reciprocal variation of interlayer stiffness and the corresponding variation of LG frequency, as expected from Section 2. In doing so, the additional sustained mass of possible walking occupants (M) is also preliminarily disregarded. Regarding the fractured glass layer for the LGF setup in Figure 3b, finally, the equivalent MoE for the bottom layer in tension is progressively modified in the range from E

_{fg}= 70 GPa (uncracked glass, E

_{fg}≡ E) and reduced down to E

_{fg}= 7 MPa (E

_{fg}≡ E/10,000).

_{LG2}= 215 Kg). The LG2m model, with identical geometrical properties, is inclusive of mass contribution only for the fractured layer (M

_{LG2m}= M

_{LGU}= 322 Kg).

_{1}) for the triple LGF module as a function of damage severity in the fractured glass layer (i.e., E

_{fg}). The LGU frequency value in Figure 4a corresponds to the fully uncracked, triple LG panel (f

_{1}= 13.08 Hz). The LG2 and LG2m frequency predictions are given under the assumption of a simplified double LG section only. The conventional frequency limit value of 8 Hz against vibration discomfort is also reported in the chart. The LGF red curve is obtained by changing progressively the E

_{fg}value for the bottom glass layer. Finally, Figure 4a emphasizes the equivalent MoE values that have been calibrated in [5] for fractured FT glass under tension or compression.

_{fg}in tension is associated to mostly null bending stiffness contribution for the fractured layer (−41.1% the LGU value), while the E

_{fg}value in compression from [5] can offer a certain post-breakage stiffness contribution to the LG panel (−25.8% the LGU). For the present worked example, the LGF frequency decreases to −44.4% the LGU system.

_{1}= 8.93 Hz and f

_{1}= 7.31 Hz, respectively. As far as E

_{fg}tends to zero for the LGF system, as expected, the fundamental frequency progressively tends to the LG2m frequency. Compared to LG2m (i.e., design approach), the mass contribution of cracked glass fragments manifests in an additional ≈ +12% frequency increase. The residual mass contribution of fractured glass layers is thus responsible of significant modifications of dynamic parameters. Worth noting in Figure 4a that—as far as the conventional frequency value of 8 Hz is taken into account as a reference for vibration serviceability purposes—the examined LG panel would be able to satisfy the minimum recommended fundamental frequency for human comfort under a limited number of damage configurations only.

_{int}modulus for the LGF model). Lack of bonding for glass panels typically results in severe reduction of vibration frequencies, both in uncracked or fractured configurations.

_{int}= 24 MPa) for the empty slab (M = 0), or for two occupied configurations (M = 80 Kg and M = 2 × 80 = 160 Kg respectively). In LGU conditions, it is possible to note that the sustained mass decreases to ≈−19% or ≈−30% the corresponding frequency f

_{1}. The frequency decrease in post-breakage conditions (E

_{fg}< 70 GPa) is mostly constant for the proposed calculations.

## 4. On-Site Experimental Investigation

#### 4.1. Layout and Geometry

#### 4.2. Test Setup, Instruments and Experimental Records

#### 4.3. Analysis of Experimental Results

_{1}) in accordance with Figure 8. The experimental estimates are proposed as a function of the corresponding vertical acceleration peaks (Figure 8a), or as average terms for the investigated LGF and LGU modular units (Figure 8b). For the reference test setup, the vertical acceleration peak was measured in the range of 0.025 g minimum and up to 0.394 g. The average acceleration peak for the LGF and LGU modules was measured in 0.136 g (±0.059 g) and 0.114 g (±0.080 g) respectively.

_{1}= 13.8 Hz (±0.21 Hz) and f

_{1}= 15.05 Hz (±0.2 Hz), respectively. In this regard, it is clear that a direct quantitative comparison cannot be discussed in general terms for the in-service LG + AN modules object of study, due to possible minor uncertainties in material properties or restraint detailing. In any case, the experimental frequency scatter in Figure 8b can be quantified at about an −8.3% decrease (and thus bending stiffness) for the LGF module, which could be representative of an approximate damage index for the walkway object of study.

## 5. Frequency Numerical Analysis

#### 5.1. Modelling

_{fg}. In doing so, the reference FE numerical model was described based on nominal geometrical and mechanical properties of the system, but also based on the extended calibration of model details that was derived from earlier studies in [33,34].

_{s}= 0.136 Kg its weight) rigidly connected to the top AN panel, and positioned as from the experimental setup. An additional lumped mass (M = 80 kg) was rigidly fixed to glass, and positioned as for the standing occupant during the experimental program (i.e., centre of panel). In this regard, no walking features were taken into account for preliminary frequency estimates. The final FE assembly in Figure 9 consisted in ≈2000 elements and ≈11,000 degrees of freedom.

#### 5.2. Boundaries and Mechanical Interactions

#### 5.3. Material Properties

_{SOFT}= 1 MPa). A major support for the calibration of PVB bonding layers was taken from the in-service conditions of the studied system, as also discussed in [33,34], and thus facilitating the vibration analysis of the in-service walkway.

_{fg}was iteratively modified in the range from E

_{fg}= E = 70 GPa (LGU) and reduced down to E

_{fg}= E/10,000 = 7 MPa.

#### 5.4. Discussion of Results

_{fg}progressively modifies for the fractured layer. As in Section 3, the calibrated tensile and compressive MoE values from [5] are also emphasized. Furthermore, the “SS” frequency estimates described in Section 3 for the simply supported triple LG panel are recalled. Finally, the average experimental frequencies for LGU or LFG modular units are reported. It is worth noting in Figure 11b the severity of vibration frequency decrease when E

_{fg}decreases for the single LG top layer. At the lower bound (i.e., null stiffness contribution), a frequency shift of about ≈−25% can be calculated compared to the upper limit condition (LGU).

_{fg}≈ 1–20 GPa. A close match of equivalent compressive MoE from [5] can be perceived in this range, with corresponding FE outcomes and average experimental data. This is in line with expectations, given that the top LG layer for the modular unit in Figure 6 under sustained vertical loads is expected to mainly contribute with compressive bending behaviour. Compared to the upper LGU condition, calibrated MoE in compression numerically occurs in a ≈5.5% of frequency drop. The equivalent MoE in tension, conversely, has a minor effect compared to expectations (due to the LG + AN layout and fracture on the compressive side) and results in a frequency drop down to ≈9.2%. Finally, it is possible to notice in Figure 11b that the numerical LGF fundamental frequency is still markedly higher than the “SS” simple boundary condition, due to major modifications in restraints (i.e., mid-span point supports), layout (top AN cover plate) and overall mass properties.

_{fg}). When the bottom fractured layer is fully disregarded in terms of residual stiffness contribution, the frequency decrease can be estimated in a drop of ≈−25% the LGU top limit. The effect of the equivalent MoE in tension from [5], in this regard, is very close to a ≈−25% drop for the numerical vibration frequency. Such numerical findings may be justified by the composite layout for the currently explored glass slab, and thus suggest additional experimental and numerical investigations in this direction. It can in fact be reasonably hypnotized that the top AN cover is able offer a certain stiffening contribution to the top fractured glass layer, which is not the case for the LG + AN section when the fracture moves on the bottom LG side.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Post-breakage performance of laminated glass (LG) members: (

**a**–

**c**) examples and (

**d**) schematic representation of progressive fracture mechanisms in a double LG section, with evidence of stress distribution.

**Figure 3.**Frequency analysis of a simply supported triple LG module: (

**a**) fundamental shape and (

**b**) cross-section details for LGF or LG2/LG2m cross-section parameters.

**Figure 4.**Example of fundamental frequency analysis for a triple LG panel. In evidence, the effect of (

**a**) simplified fractured cross-section (with M = 0), (

**b**) variable interlayer stiffness (with M = 0) and (

**c**) variable sustained mass of walking occupants (LGF model).

**Figure 5.**Case-study LG slab: (

**a**) walkway (Crypt) and (

**b**) reference LGU module, with (

**c**) dimensions in m.

**Figure 6.**Case-study LG slab: (

**a**) cross-section layout with dimensions in mm (in evidence, the fractured fully tempered (FT) layer) and (

**b**–

**c**) views of LGF fractured module.

**Figure 7.**Example of vertical acceleration records under (

**a**) linear walking path or (

**b**) on-site jump for a single occupant (detail), with (

**c**–

**d**) corresponding FFT magnitude in the frequency domain.

**Figure 8.**Experimental fundamental frequency for LGU and LGF modules, as obtained in terms of (

**a**) vertical acceleration peaks as a function or (

**b**) average estimates.

**Figure 10.**Schematic cross-section for the reference model of LGF assembly: (

**a**) actual experimental configuration and (

**b**) hypothesis of fractured glass layer on the tensile side (dimensions in mm).

**Figure 11.**Numerical frequency analysis: (

**a**) fundamental vibration shape (extruded view) and (

**b**,

**c**) corresponding frequency variation as a function of E

_{fg}(ABAQUS).

Mode Order n | ||
---|---|---|

1 | 2 | 3 |

π/L | 2π/L | 3π/L |

Glass | PVB | Steel | Fractured Glass | ||
---|---|---|---|---|---|

E | MPa | 70,000 | 4 | 160,000 | Variable (7–70,000) |

ν | - | 0.23 | 0.49 | 0.3 | 0.23 |

ρ | Kg/m^{3} | 2500 | 1100 | 7850 | 2500 |

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**MDPI and ACS Style**

Bedon, C.; Noè, S.
Post-Breakage Vibration Frequency Analysis of In-Service Pedestrian Laminated Glass Modular Units. *Vibration* **2021**, *4*, 836-852.
https://doi.org/10.3390/vibration4040047

**AMA Style**

Bedon C, Noè S.
Post-Breakage Vibration Frequency Analysis of In-Service Pedestrian Laminated Glass Modular Units. *Vibration*. 2021; 4(4):836-852.
https://doi.org/10.3390/vibration4040047

**Chicago/Turabian Style**

Bedon, Chiara, and Salvatore Noè.
2021. "Post-Breakage Vibration Frequency Analysis of In-Service Pedestrian Laminated Glass Modular Units" *Vibration* 4, no. 4: 836-852.
https://doi.org/10.3390/vibration4040047