# Analysis of Damage in Laminated Architectural Glazing Subjected to Wind Loading and Windborne Debris Impact

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## Abstract

**:**

## 1. Introduction

## 2. Continuum Damage Mechanics (CDM) Model

_{ij}that models the material nonlinearity due to the deformation process is introduced in to the constitutive Equation [14]. The constitutive relationship of the material is thus expressed as:

_{1}and C

_{2}are defined such that the axial stress vanishes as the damage component D

_{11}approaches 1.0 in a uniaxial tension test [18,19]. This condition leads to the following relationship:

_{crit}= 372 MPa and σ

_{threshold}= 23 MPa for the critical and the threshold stresses, respectively. Data is not readily available for large missile impact. Bouzid et al. [20] showed that the critical stress to failure is proportional to the time to failure. The time span involved in the case of large missile impact is on the order of milliseconds while it is on the order of microseconds [21,22] for the small case. Hence it is conceivable that the critical stress for the large missile case would be considerably lower in magnitude than that of the small missile case. For the sake of analysis and computation, a conservative value of 100 MPa, which is the static tensile strength of a soda lime glass, is used for the critical stress (σ

_{crit}= 100 MPa). Initially, the material was assumed to be defect free with a damage tensor value of 0.0 (virgin state), and with the values increasing to 1.0 representing a fully damaged state. For this study, instead of just searching for the fully damaged state, results will be presented for a range of damage states because the actual value of the damage at failure is highly dependent on several factors. This issue is further elaborated in section 4 just before presenting the results for various cases. The cracking and damage process being irreversible, the component damage tensor for a material point at the n

^{th}time increment as determined by Sun and Khaleel [14], may be written:

^{th}and (n−1)

^{th}time increment of the analysis.

## 3. Impact Problem Description

_{o}. The laminated plate is comprised of two soda lime glass sheets sandwiching a PVB interlayer. The thicknesses of the outer and inner glass plies and the PVB interlayer are h

_{o}, h

_{i}and h

_{p}, respectively. The panel is simply supported on the circumferential edge. The wind load is modeled as a uniform static load (pressure). Ghrib and Tinawi [27], in their study of seismic analysis of concrete gravity dams, first analyzed the response of dam due to its self-weight and hydrostatic pressure of the reservoir on the upstream wall prior to earthquake excitation. A similar method is adopted here. The loading is done sequentially with static wind load followed by the debris impact load rather than superposing the two simultaneously. The interlayer bond is assumed to be perfect with no de-bonding or slipping during impact.

Materials and variables | Parameters and properties |
---|---|

Glass | E = 72 GPa, ρ = 2500 kg/m^{3}, ν = 0.25 |

PVB | G_{o} = 1 GPa, G_{∞} = 0.69 MPa, ρ = 1100 kg/m^{3}, β = 12.6 s^{−1}, E = 2.5714 GPa, ν = 0.2857 |

Steel ball [2 g] | E = 200 GPa, ν = 0.29, ρ = 7800 kg/m^{3} |

Wooden cylinder [2050 g] (Douglas Fir) | E_{r} = 1 GPa, E_{θ} = 737 MPa, E_{z} = 14.74 GPa, ν_{rθ} = 0.39, ν_{θz} = 0.036, ν_{zr}=0.029, G_{rθ} = 103.2 MPa, G_{θz} = 943.4 MPa, G_{zr} = 1.15 Gpa |

Impact velocity | Steel ball: 39.6 m/s; Wooden cylinder: 24.4 m/s |

Plate dimensions | Panel area: Radius = 0.74 m |

- | Inner ply thickness,
h_{i} = 4.76 mm |

- | Outer ply thickness,
h_{o} = 4.76 mm |

- | PVB interlayer thickness,
h_{p} = 1.52 mm |

Missile dimensions | Steel ball: R = 3.96875 mm |

- | Wooden cylinder: R = 32.81 mm; length = 2.4 m |

**Figure 1.**Schematic of different missile types impacting on 2-D laminated glazing. (

**a**) Small missile; (

**b**) Large missile with round impacting end; (

**c**) Large missile with flat impacting end.

#### 3.1. Material Models

_{ij}is the stress tensor; S

_{ij}is the deviatoric stress tensor; p = −σ

_{kk }/3 is the pressure; and δ

_{ij}is the Kronecker delta.

_{ij}is the strain tensor; ν is the Poisson’s ratio and ε

_{ij}=ε

_{kk}is the volumetric strain. In several earlier studies on LAG, the PVB interlayer has been traditionally modeled as linear-viscoelastic [1,2,3,4]. The most recent work on laminated glazing by Wei and Dharani [28] has shown that PVB can be modeled as linear elastic by using the short term shear modulus for a transient response, G

_{o}, and bulk modulus, K, to give the elastic constants

_{p}is PVB Young’s modulus and ν

_{p}is PVB Poisson’s ratio. In the present work, PVB is modeled as linear elastic. Further, it is assumed that the interface between the interlayer and the glass plies is perfect with no debonding. If the debonding between the PVB interlayer and the adjoining glass plies occurs it typically initiates where the cone crack in the glass ply meets PVB interface at which point stresses are high [3]. The objective of the current study is to identify locations where micro-damage in glass initiates. The propagation of such damage or the formation of a cone crack in glass ply is not a part of this study and hence the issue of PVB debonding is not of interest. The PVB interlayer debonding has been modeled and studied by Flocker and Dharani [4].

#### 3.2. Design Wind Pressure

_{p}is the pressure coefficient. The term q G C

_{p}in the above Equation refers to external wind pressure acting on the glazing. The term q

_{i}G C

_{pi}refers to pressure inside the building and is taken to be zero. The gust effect factor accounts for the loading effects in the wind direction due to wind interaction with the structure. The basic velocity pressure, q, is given by

_{z}, V, and I denote the velocity pressure coefficient, wind velocity, and importance factor, respectively. The importance factor accounts for the degree of hazard to human life and damage to property. The pressure coefficient denotes the actual loading on each surface of the building as a function of wind direction.

#### 3.3. Computational Model

## 4. Results and Discussion

_{11}) is caused by the radial stress (σ

_{1}) when it exceeds the corresponding critical value thereby resulting in a circumferential crack (or web-shaped crack). Star-shaped cracks (D

_{33}) are caused by the circumferential tensile stress (σ

_{3}) when it exceeds the corresponding critical value, thereby, leading to a radial (or star-shaped) crack. These are the two most commonly observed damage modes in typical glass damage under impact loading. Shear damage is caused by shear stress (σ

_{12)}under a confining compressive stress state. Knight et al. [30] have shown that a plastic deformation occurs below the impactor when shear stress exceeds the yield stress, and in the case of brittle materials, crushing is observed. In the present study, the glass panels were assumed to be new and free of any defects. The value of critical damage variable is highly dependent on the material and in particular the type of failure the material undergoes [31]. Values less than 0.2 are commonly used in the case of brittle damage while a value between 0.8 and 1.0 is used for ductile failure [31]. For this study, instead of just searching for the fully damaged state, results will be presented for a range of damage states so that one can get a pictorial view of the damage in the entire laminate.

_{11}and D

_{33}. It is observed that D

_{33}has a larger area compared to D

_{11}on surface S2, and that D

_{12}does not seem prominent—at least in the baseline case. In contrast, the large missile case, shown in Figure 5, reveals that most of the damage occurs in the inner, non-impact layer, and less damage at the impact zone. The web-shaped damage and star-shaped damages have higher values on the non-impact surface, S4 with the area for D

_{33}greater than D

_{11}. The shear damage, D

_{12}, due to confined compressive stress, is restricted to on surface S1 in the outer ply.

**Figure 3.**Schematic of crack patterns on the glass surface and the corresponding causal stresses (

**a**) star-shaped crack; (

**b**) web-shaped crack.

**Figure 4.**Contour plots of damage in laminated glass at 8.4 μs after impact for small missile impact: (

**a**) web-shaped damage D

_{11}; (

**b**) star-shaped damage D

_{33}; (

**c**) shear damage D

_{12}.

**Figure 5.**Contour plots of damage in laminated glass at 3.4 ms after impact for large missile with round end configuration: (

**a**) web-shaped damage D

_{11}; (

**b**) star-shaped damage D

_{33}; (

**c**) shear damage D

_{12}.

_{11}and D

_{33}. This means that there are visible web-shaped and star-shaped cracks on the surface for a radius of 1–2 mm from the centerline of the missile. The maximum value of damage on surfaces S2 and S4 is around 0.7, and hence, the damage is partial—implying no visible cracks. The damage variables D

_{11}and D

_{33}for surfaces S2 and S4 for large missiles is shown in Figure 7. Both surfaces suffer extensive visible damage in this case, with both star and web-shaped cracking predicted to occur over 200 mm from the impact centerline in S4.

**Figure 6.**Damage variables D

_{11}(web-shaped) and D

_{33}(star-shaped) at surface S1, surface S2 and surface S4 for small missile impact.

**Figure 7.**Damage variables D

_{11}(web-shaped) and D

_{33}(star-shaped) at surface S2 and surface S4 for a large missile (round end) impact.

_{11}and D

_{33}.

#### 4.1. Parametric Study of PVB Thickness

_{11}on surface S2. A similar result was observed (results not shown) on the effect of PVB interlayer thickness on star-shaped damage, D

_{33}, on surface S2 for small missile impact, as well surface S4 for large missile with round end configuration. However, for the large missile case shown in Figure 9, considerable differences in web-shaped damage (D

_{11}) between 0.76 mm and 2.28 mm thicknesses are observed on S4.

**Figure 8.**Effect of polyvinyl butyral (PVB) interlayer thickness on web-shaped damage, D

_{11}on surface S2 for small missile impact.

**Figure 9.**Effect of PVB interlayer thickness on web-shaped damage, D

_{11}on surface S4 for large missile (round end) impact.

#### 4.2. Parametric Study of Inner and Outer Ply Thickness

_{o}and h

_{i}, on damage patterns. The baseline case is a symmetric glazing in which both the outer and inner glass plies are of equal thickness (h

_{o}= h

_{i}= 4.76 mm) and the PVB layer thickness is h

_{p}= 1.52 mm. Asymmetric glazing (h

_{o}≠ h

_{i}) is also considered. In order to study the response of asymmetric glazing, the thicknesses of the inner and outer plies are varied fifty percent above and below the baseline thickness while keeping the PVB thickness the same (h

_{p}= 1.52 mm) as the baseline case. The following five cases are investigated: (i) baseline case h

_{o}= h

_{i}= 4.76, h

_{p }= 1.52 mm; (ii) h

_{o}= 2.38, h

_{i}= 4.76, h

_{p}= 1.52 mm; (iii) h

_{o}= 7.14, h

_{i}= 4.76, h

_{p}= 1.52 mm; (iv) h

_{o}= 4.76, h

_{i}= 2.38, h

_{p}= 1.52 mm; and (v) h

_{o}= 4.76 mm, h

_{i}= 7.14, h

_{p}= 1.52 mm.

_{11}and D

_{33}on surface S2 (bottom surface, non-impact side, of the outer layer) are shown in Figure 10, Figure 11. The damage variables D

_{11 }and D

_{33}are higher than the baseline case only when the outer layer thickness is decreased in case (ii). The damage variables D

_{11}and D

_{33}are lower than or close to baseline case when the inner ply thickness is decreased. Changing the thickness of the inner ply does not seem to have any significant effect on the damage, but the damage can be increased (or decreased) considerably by decreasing (or increasing) the outer ply thickness. Hence for optimum design, the inner ply thickness can be reduced without affecting the overall damage patterns when small missiles are most probable.

_{11}and D

_{33}on surface S4 (bottom side of the inner ply) for large missile impact are shown in Figure 12, Figure 13. An increase in thickness of either inner or outer ply reduces the damage variables. A decrease in the outer ply thickness results in slightly less damage than if the inner ply thickness is decreased. Hence when weight reduction is required without significantly affecting the damage pattern the outer ply should be chosen for thickness reduction where large missiles are of concern.

**Figure 10.**Effect of inner and outer plies thickness on Damage variable D

_{11}(web-shaped) at surface S2 for small missile impact.

**Figure 11.**Effect of inner and outer plies thickness on Damage variable D

_{33}(star-shaped) at surface S2 for small missile impact.

**Figure 12.**Effect of inner and outer plies thickness on Damage variable D

_{11}(web-shaped) at surface S4 for large missile (round end) impact.

**Figure 13.**Effect of inner and outer plies thickness on Damage variable D

_{33}(star-shaped) at surface S4 for large missile (round end) impact.

#### 4.3. Parametric Study of Panel Surface Area

_{11}and D

_{33}on surface S2 for small missile case and surface S4 for the large missile case are presented in Figure 14, Figure 15, respectively. For the small missile case shown in Figure 14, the damage is the same for different areas of the panel as should be expected considering the relative sizes of the panel and the missile. For the large missile case shown in Figure 15, smaller panels have proportionally more damage than larger ones. This is because the contact force required to induce damage is inversely proportional to the flexibility of the panel, which is proportional to the area of the panel. This result is consistent with earlier observations on laminated automotive glazing subjected to head impact [16].

**Figure 14.**Effect of panel area on Damage variables D

_{11}(web-shaped), D

_{33}(star-shaped) at surface S2 for small missile impact.

**Figure 15.**Effect of panel area on Damage variables D

_{11}(web-shaped), D

_{33}(star-shaped) at surface S4 for large missile (round end) impact.

## 5. Conclusions

## Nomenclature

stiffness matrix without damage | |

stiffness matrix with added influence | |

D_{ij} | damage components |

C_{1,} C_{2} | damage parameters |

λ, μ | Lame constants |

E | Young’s modulus |

v | Poisson’s ratio |

σ_{crit} | critical stress |

σ_{threshold} | threshold stress |

σ_{ij} | stress tensor |

S_{ij} | deviatoric stress tensor |

p | pressure |

δ_{ij} | Kronecker delta |

ϵ_{ij} | strain tensor |

ė_{ij} | deviatoric strain rate |

G(t) | stress relaxation modulus |

G_{o} | short term shear modulus |

β | decay constant |

q | pressure at height above ground |

G | gust factor |

C_{p} | pressure coefficient |

K_{z} | velocity pressure coefficient |

V | wind velocity |

I | importance factor |

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## Share and Cite

**MDPI and ACS Style**

Shetty, M.S.; Wei, J.; Dharani, L.R.; Stutts, D.S. Analysis of Damage in Laminated Architectural Glazing Subjected to Wind Loading and Windborne Debris Impact. *Buildings* **2013**, *3*, 422-441.
https://doi.org/10.3390/buildings3020422

**AMA Style**

Shetty MS, Wei J, Dharani LR, Stutts DS. Analysis of Damage in Laminated Architectural Glazing Subjected to Wind Loading and Windborne Debris Impact. *Buildings*. 2013; 3(2):422-441.
https://doi.org/10.3390/buildings3020422

**Chicago/Turabian Style**

Shetty, Mahesh S., Jun Wei, Lokeswarappa R. Dharani, and Daniel S. Stutts. 2013. "Analysis of Damage in Laminated Architectural Glazing Subjected to Wind Loading and Windborne Debris Impact" *Buildings* 3, no. 2: 422-441.
https://doi.org/10.3390/buildings3020422