# Simplified Models to Capture the Effects of Restraints in Glass Balustrades under Quasi-Static Lateral Load or Soft-Body Impact

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Research Study

#### 2.1. Methodology and Goal

_{k}, which should be taken into account in terms of equivalent spring, as follows:

#### 2.2. Reference Glass Balustrade

^{2}and a double LG section (10/10.4 in thickness) composed of tempered glass panes (10 mm in thickness) and bonding Polyvinyl butyral (PVB

^{®}, 1.52 mm in thickness). The bottom linear connection consists of two 10 mm-thick steel plates, which are rigidly fixed to a base support via M10 class 8.8 bolts (length l

_{b}, area A

_{b}), distributed as schematized in Figure 4. Additional setting blocks (A

_{SB}= h

_{SB}= 30 × b

_{SB}= 120 mm

^{2}in dimensions, with t

_{SB}being their thickness) are used at the glass–steel interface to provide soft support to the glass panel in out-of-plane bending and to avoid premature stress peaks in the region of restraints. An additional supporting system consisting of two polyurethane blocks (50 mm wide, 8 mm thick) is introduced at the bottom edge of the glass panel and placed at a distance of 150 mm from the lateral edges, with the goal of preventing further stress peaks and premature glass breakage at the base edges.

## 3. Full 3D Refined Numerical Model

#### 3.1. Model Description

- L1: a quasi-static, monotonically increasing lateral load at the top edge of the glass (until a maximum value P = 4.5 kN/m), and
- L2: a twin-tyre impact loading configuration which was numerically reproduced and calibrated according to the experimental setup summarized in Section 2 (with 300 mm being the drop height).

#### 3.2. Results

_{max,test [17]}), from the FE numerical model presented in [17] (a

_{max,model [17]}), and from the presently developed FE numerical model. Moreover, the percentage scatter of the present Refined model is calculated towards past experiments (∆

_{1}) or towards past numerical simulations (∆

_{2}), respectively.

## 4. Derivation and Calibration of Simplified Numerical Models

#### 4.1. Simplified Characterization of the Base Restraint—SM1 Model

_{1}− h

_{2}), from Figure 8a, and:

_{p}-thick steel plate, with B = 1 m being the extension of plates in the width of the balustrade. Such a kind of calculation, with the reference input parameters, would result in a relatively small top lateral displacement for both steel plates (1) and (2), namely, ${\delta}_{p}^{\left(1\right)}$ = 1.95 mm and ${\delta}_{p}^{\left(2\right)}$ = 0.08 mm, respectively, for the present study.

_{p,D}

^{(1)}= 3.17 mm δ

_{p,D,FE}

^{(1)}= │3.56│ mm ∆ = −10.95%

_{p,D}

^{(2)}= 0.32 mm δ

_{p,D,FE}

^{(2)}= │0.45│ mm ∆ = −28.8%

_{f}being its second moment of the area:

_{SB}, which corresponds to the location and size of setting blocks, namely:

#### 4.2. Simplified Characterization of the Base Restraint and LG Panel—SM2 Model

_{t,eq}= k

_{t,sup}, can be derived from multiple considerations in the rotational stiffness form (Figure 11c), that is:

_{θ,eq}is implicitly representative of the rotational contributions of sub-schemes SS1 and SS2. From Equation (18), the translational springs can thus be distributed as in the FE numerical model of Figure 10, given that:

#### 4.3. Linearly Distributed Base Springs—SM3 and SM4 Models

## 5. Discussion of Numerical Results

#### 5.1. Simplified Models SM1 and SM2

#### 5.2. Effect of Linearly Distributed Equivalent Springs—SM3 and SM4

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Examples of glass balustrades characterized by various restraint conditions: (

**a**) linear restraint at the base; (

**b**) point-fixing at the base; (

**c**) top-bottom point-fixings; (

**d**) lateral point-fixing. Solutions (

**a**,

**b**) do not require glass drilling; (

**c**,

**d**) are characterized by the presence of glass holes.

**Figure 2.**Schematic representation of possible mechanical models for a glass balustrade with a linear restraint at its base: (

**a**) example of a metal restraint (cross-section); (

**b**) cantilever mechanical model with an ideally rigid clamp restraint; or (

**c**,

**d**) mechanical models inclusive of equivalent springs to reproduce the real boundaries.

**Figure 3.**Example of full 3D FE numerical model of a laminated glass balustrade a with linear base metal restraint (ABAQUS): (

**a**) axonometric view of the half balustrade assembly and (

**b**) detailed view of the base restraint region, with (

**c**) a cross-section of the metal fixing system (glass panel and mesh pattern hidden from section view).

**Figure 4.**(

**a**) Experimental setup (based on [17]), with dimensions in mm, and (

**b**–

**d**) detailed views of the presently developed “Refined” FE numerical model (ABAQUS).

**Figure 5.**Example of deformation and parameters for the base restraints under lateral loads, as obtained from the Refined FE numerical model (ABAQUS): (

**a**) out-of-plane and (

**b**) vertical deformations (glass panel and mesh hidden from view, legend values in m).

**Figure 6.**Numerical analysis of the Refined model (ABAQUS) under double twin-tyre impact (300 mm being the drop height), and comparison with the experimental results: (

**a**) lateral displacement; (

**b**) principal stress, and (

**c**) acceleration time histories.

**Figure 7.**SM1 simplified model: (

**a**) model concept and 3D assembly (axonometric view from ABAQUS), with (

**b**) the reference mechanical system (cross-section) and evidence of lumped equivalent springs.

**Figure 8.**SM1 simplified numerical model: details of sub-schemes SS1 and SS2, and simplified local mechanical models to calculate the stiffness of steel lateral plates and the steel base flange for the LG balustrade in bending.

**Figure 9.**Simplified numerical model SM1, with evidence of the local analysis of lateral deflections measured in steel plates (1) and (2). Comparative calculations for the assessment of simplified empirical formulations in use for sub-scheme SS1 (ABAQUS). Legend values in m (out-of-scale deformed shapes).

**Figure 11.**Derivation of equivalent stiffness parameters, as for the SM2 simplified model: (

**a**) required translational stiffness, with (

**b**) equivalent rotational and (

**c**) translational terms.

**Figure 12.**Front and axonometric views of (

**a**) SM3 and (

**b**) SM4 models under double twin-tyre impact (ABAQUS).

**Figure 13.**Numerical analysis of Refined, SM1, and SM2 simplified models under quasi-static lateral load (P = 4.5 kN/m) at the top edge of glass (ABAQUS): (

**a**) top lateral displacement and (

**b**) calculated percentage scatter of SM to Refined models (Equation (32)).

**Figure 14.**Numerical analysis of Refined, SM1, and SM2 simplified models (ABAQUS) under double twin-tyre impact (300 mm being the drop height): (

**a**) lateral displacement; (

**b**) maximum principal stress in glass (dashed lines for compression side), and (

**c**) acceleration time histories.

**Figure 15.**Numerical analysis of principal stress distribution and peaks in glass for the (

**a**) Refined model, (

**b**) SM1 simplified model, and (

**c**) SM2 simplified model (ABAQUS) under double twin-tyre impact (300 mm being the drop height), with legend values in Pa.

**Figure 16.**Numerical analysis of the Refined model, SM3 (translational springs) simplified model, and SM4 (rotational springs) simplified model (ABAQUS): (

**a**) top lateral displacement under quasi-static load (P = 4.5 kN/m) at the top edge of glass and (

**b**) calculated percentage scatter of SM to Refined models (Equation (32)), with (

**c**,

**d**) response analysis under double twin tyre impact (300 mm being the drop height).

**Figure 17.**Numerical analysis of the principal stress distribution and peaks in glass for the (

**a**) SM3 (translational springs) and (

**b**) SM4 (rotational springs) simplified models (ABAQUS) under double twin-tyre impact (300 mm being the drop height), with legend values in Pa.

**Table 1.**Summary of presently developed numerical models for the analysis of the glass balustrade system described in Figure 4.

FE Model Features | |||
---|---|---|---|

FE Numerical Model | LG Cross-Section (mm) | LG Panel | Base Restraint |

Refined | 10 + 1.52 PVB + 10 | Full 3D solid brick elements (layered section) | Full 3D solid brick elements |

SM1 | Same as that of Refined | Lumped equivalent springs in the region of restraints | |

SM2 | 2D shell elements (equivalent monolithic glass section) | Lumped equivalent springs in the region of restraints | |

SM3 | Same as that of SM2 | Linearly distributed equivalent springs (translational) at the bottom edge of the glass | |

SM4 | Same as that of SM2 | Linearly distributed equivalent springs (rotational) at the bottom edge of the glass |

Material Properties | ||||
---|---|---|---|---|

Material | Constitutive Model | Modulus of Elasticity [N/mm ^{2}] | Poisson Ratio | Density [kg/m ^{3}] |

Steel | Linear elastic | 210,000 | 0.3 | 7850 |

POM | Linear elastic | 2413 | 0.45 | 1250 |

Glass | Linear elastic | 70,000 | 0.23 | 2500 |

PVB | Linear elastic | 180 | 0.485 | 1250 |

**Table 3.**Summary of comparative results in terms of impactor acceleration for the present Refined numerical model and past experiments with a double twin-tyre.

Refined Model | |||||
---|---|---|---|---|---|

Drop Height [mm] | a_{max,test [17]}[m/s ^{2}] | a_{max,model [17]}[m/s ^{2}] | a_{max,Refined}[m/s ^{2}] | ∆_{1}[%] | ∆_{2}[%] |

300 | 223 | 216 | 216.38 | −2.97 | 0.18 |

400 | 282 | 272 | 269.17 | −4.55 | −1.04 |

500 | 332 | 334 | 345.36 | 4.02 | 3.40 |

**Table 4.**Summary of the computational cost of the Refined and SM1-to-SM4 simplified numerical models (ABAQUS) in terms of the total number of elements and DOFs required to reproduce the nominal geometry of the examined balustrade.

FE Model Features | ||
---|---|---|

FE Numerical Model | Number of Elements | Number of DOFs |

Refined | ≈120,000 | ≈505,000 |

SM1 | ≈76,000 | ≈214,000 |

SM2 | ≈12,000 | ≈73,000 |

SM3 and SM4 | ≈12,000 | ≈73,000 |

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**MDPI and ACS Style**

Rizzi, E.; Bedon, C.; Amadio, C.
Simplified Models to Capture the Effects of Restraints in Glass Balustrades under Quasi-Static Lateral Load or Soft-Body Impact. *Buildings* **2022**, *12*, 1664.
https://doi.org/10.3390/buildings12101664

**AMA Style**

Rizzi E, Bedon C, Amadio C.
Simplified Models to Capture the Effects of Restraints in Glass Balustrades under Quasi-Static Lateral Load or Soft-Body Impact. *Buildings*. 2022; 12(10):1664.
https://doi.org/10.3390/buildings12101664

**Chicago/Turabian Style**

Rizzi, Emanuele, Chiara Bedon, and Claudio Amadio.
2022. "Simplified Models to Capture the Effects of Restraints in Glass Balustrades under Quasi-Static Lateral Load or Soft-Body Impact" *Buildings* 12, no. 10: 1664.
https://doi.org/10.3390/buildings12101664