Sensitivity study on climate induced internal pressure within cylindrical curved IGUs
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Abstract
Insulated glass units (IGUs) are employed in modern buildings as a substitute for monolithic glass to reduce heat loss through windows. In the past decades, the pursuit of higher aesthetic design drives glazed products evolving from conventionally flat into more creative and dynamic curved shapes. The call for curved IGUs brings up a series of challenges, and one remarkable issue is the determination of load sharing and glass stress when evaluating its structural performance. Even though many national codes worldwide have established mature design approaches for load sharing of flat IGUs panels, such design approaches can scarcely be found for curved IGUs due to geometry complexity. In this paper, the author will use an FEA tool along with automatic iteration scripts to carry out a sensitivity study on the internal climatic pressure of cylindrically curved IGUs, considering a series of geometrical variables. The paper aims to evaluate the relationships between the internal pressure and the geometrical parameters of curved IGUs and normalize all these parameters into a dimensionless chart.
Keywords
Curved IGUs Sensitivity Parametric study Geometrical impact1 Introduction
As such, there is a demand for empirical charts or simple equations that express the relationships between geometry and the induced internal pressure loads to glass panels, so that designers can quickly specify reasonable glass buildup at the initial design stage.
2 Literature review
2.1 Internal pressure equilibrium in IGU cavity
In contrast to laminated glazing panels, one distinct step of IGU design is to specify the load share coefficient, i.e. how the loads are distributed to the two layers through the deformed air cavity. In addition, the phenomenon of climatic loads is a distinct feature of IGUs, whereby changes in the panel environment causes variation of internal pressures and loads the glass packages. The calculation of these phenomenon relies on knowledge of gas behaviour.
The cavity volume however is not constant. Internal pressure changes will cause deformation of the glass due to the glass bending behaviour and therefore changes in cavity volume. In summary, pressure change is a function of volume change, but volume change is in turn a function of internal pressure. It is this interrelation which complicates the calculation of internal pressure. What is problematic for the case of curved IGUs is that there is no analytic solution for volume change of the cavity due to changes in internal pressure.
2.2 Analytical solutions for flat IGUs
2.3 Internal pressure load of cylindrically curved IGUs
Researchers have pointed out that for cylindrical IGUs, the climatic loads caused by the sealed gas is drastically greater than that of flat IGUs (Feldmeier 2003). Curved panels are stiffer due to geometric stiffness and hence yield less volume deformation. According to the reverse relationship, a lower volume change gives rises to a higher internal pressure change. It is worthy investigating correlations between internal pressure and different geometrical parameters of curved IGUs and evaluate the sensitivity degree of the internal pressure to each variable.
This paper introduces a computationalaided iteration process, which allows FEA software that does not support pneumatic fluid analysis to find the internal pressure change at equilibrium. A series of comparison charts are plotted to understand the geometrical impacts to the internal pressure of curved IGU. The study is particularly focused on climatic loads, as it has been found much more significant in curved IGUs compared to flat panels. The correlation between the internal pressure and the geometry of IGUs will be visualised in the charts to assist designers finding proper glass configurations at initial design stage. The charts are obtained from a series of parametric studies which is automated by scripts written in C# and within the parametric modelling software Grasshopper3D alongside FEM software STRAND7.
3 Methodology
3.1 Framework
The basis of the method employed in this paper generally follows the application of Eq. (7) and illustrated in Fig. 3. In order to obtain the volumetric stiffness, a variable of the governing equation, an FEM analysis is conducted, and displacements are integrated across the surface and used to calculate volume change and thereafter volume stiffness. Initially an arbitrary internal pressure is applied to obtain an initial approximation to the volumetric stiffness.
Direct application of Eq. (7) assumes that the volumetric stiffness is a constant and therefore independent of the internal pressure of the IGU. Due to geometric nonlinearity of the panels the authors have found that assumption is inaccurate, we therefore employ an iterative procedure to update volume stiffness as the solution converges to the climatic pressure.
3.2 FEM modelling description
Since STRAND 7 does not support gas element, nor pressure equilibrium analysis within a sealed cavity, the interactive action between two glass panels cannot be simulated. STRAND 7 is only used to analyse the bending behaviour of the curved glass panel at one iteration step. As plotted in Fig. 4 the glass deformation at each node will be integrated in Grasshopper script to obtain the overall volume change at the corresponding iteration step.
Glass panel is modelled with shell/plate element, first order QUAD 4. The thickness of the glass panel considered in the modelling is the effective thickness for laminated glass. The standard determination method of effective thickness is provided in prEN 16612 (BSI 2013). Element size is set to be 100 mm by 100 mm based on computational efficiency. Silicone joints are modelled by spring elements. Material properties employed in the modelling are listed in Table 1. It is worth noting that though silicone is a hyperelastic material, the initial 6–7% tension/shear deformation follows an approximate linear stress–strain relation as observed in dow corning tensile test and shear curves (Dow Corning 2015). In this study, the resultant tension in silicone joint is subtle, i.e. less than 6% strain, therefore can be assumed to behave linearly.
3.3 Validation
Table 2 below compares the results of the proposed methodology with the analytical results, based on the PrEN 16612, showing consistent results across the two approaches, validating the results of the methodology. It is assumed that the methodology can be extended to curved IGU panels with similar accuracy.
Material properties
Type  Young’s modulus E (MPa)  Poisson ratio v 

Glass  70,000  0.22 
Silicone  4  0.499 
4 Sensitivity study
4.1 Geometry description
Comparison of analytical and FEA approaches for validation
Panel variables  Analytical results (PrEN 16612)  Algorithm results  Difference (%)  

Length (mm)  Width (mm)  Cavity (mm)  Glass Thk. (mm)  
1500  375  16  8  10.711  10.286  4.0 
1500  750  16  8  2.448  2.381  2.7 
1500  1500  16  8  0.451  0.453  0.4 
1500  3000  16  8  0.178  0.177  0.6 
1500  5000  16  8  0.132  0.131  0.8 
1500  6000  16  8  0.124  0.123  0.8 
1500  12,000  16  8  0.107  0.106  0.9 
1500  375  16  24  15.805  15.662  0.9 
1500  750  16  24  13.344  12.960  2.9 
1500  1500  16  24  7.050  6.732  4.5 
1500  3000  16  24  3.740  3.603  3.7 
1500  5000  16  24  2.941  2.849  3.1 
1500  6000  16  24  2.79  2.704  3.1 
1500  12,000  16  24  2.470  2.399  2.9 
1500  375  20  8  11.479  11.054  3.7 
1500  750  20  8  2.948  2.864  2.8 
1500  1500  20  8  0.560  0.568  1.4 
1500  3000  20  8  0.222  0.221  0.5 
1500  5000  20  8  0.165  0.163  1.2 
1500  6000  20  8  0.155  0.153  1.3 
1500  12,000  20  8  0.134  0.133  0.7 
Variable summary
Variables  Chord (mm)  Length (mm)  Radius (mm)  Thickness (mm)  Cavity width (mm)  Silicone bite (mm) 

Symbol  c  l  R  t  s  b 
Basic buildup  1500  1500  1500  8  16  10 
Length  375–6000  
Radius  750–10,000  
Thickness  8–24  
Cavity  12, 16, 20  
Silicone  10–40 
4.2 Loading assumption
Climatic loading description
Action combination  Temperature change T (K)  Atmospheric pressure change \(dp_{{ atm}}\) (kPa)  Altitude change dH (m)  Isochoric pressure \(p_{{ iso}}\) (kPa) 

Winter  \(\) 25  \(+\) 4  \(\) 300  \(\) 16 
Summer  \(+\) 20  \(\) 2  \(+\) 600  \(+\) 16 
4.3 Results and discussion
The results are presented in plots with internal pressure at equilibrium state as the dependent variable and the relevant parameter as the independent variable. Flat IGU and cylindrical curved IGU (also referred to here as “curved IGU”) will be compared in the same plot.
4.3.1 Thickness
The effective glass pane thickness is varied between 8, 10, 12, 16, 20 and 24 mm. The calculated internal pressures varying with increasing thickness are presented in Fig. 6. As a comparison, two groups of data “flat IGU” and “asymptote” are plotted. The flat panel (R is \(\infty )\) with same chord, length and climatic conditions is analyzed to obtain climatic pressures for each thickness. The asymptote obtained by assuming the glass panel is infinitely stiff and does not have any deformation when subjected to the climatic load, hence the overall cavity volume change is attributed to the silicone stretching only. It is assumed that this asymptote is the same for curved and flat geometries as the panel stiffness is infinite in both cases.
It can be observed that the internal pressure of the curved IGU rises mildly, varying from 4.25 to 5.88 kPa as the thickness itself increases by a multiple of three. In the contrast, flat panels generate a relative low internal pressure 0.435 kPa when \(\hbox {t} = 8\) mm but grows rapidly to 4.75 kPa at \(\hbox {t} = 24\) mm.
Input summary
Variables  Chord (mm)  Length (mm)  Radius (mm)  Thickness (mm)  Cavity width (mm)  Silicone bite (mm) 

Symbol  c  l  R  t  s  b 
Buildup  1500  1500  1500  8–24  16  10 
Input summary
Variables  Chord (mm)  Length (mm)  Radius (mm)  Thickness (mm)  Cavity width (mm)  Silicone bite (mm) 

Symbol  c  l  R  t  s  b 
Buildup  1500  1500  1500–6000  8  16  10 
Input summary
Variables  Chord (mm)  Length (mm)  Radius (mm)  Thickness (mm)  Cavity width (mm)  Silicone bite (mm) 

Symbol  c  l  R  t  s  b 
Buildup  1500  1500  1500  8  12, 16, 19.20  10 
4.3.2 Radius
As has been described above, the relationship between the radius and internal pressure is essentially governed by the panel stiffness. Therefore, we plotted the internal pressure changing along increasing radius in Fig. 8. Since the chord of the panel is 1500 mm, the minimal radius of panel cannot be less than 750 mm. The curve is acting as a quasiinverse function of radius, which is to say, it drops sharply initially and then levels off with larger radii. The input information is shown in Table 6.
4.3.3 Cavity width
4.3.4 Silicone bite
Currently published analytical methods (BSI 2013) for flat IGUs have not included silicone stiffness. The glass edges are assumed to be rigidly fixed (referred as to “rigid fix”), which however, does not reflect the reality. Two panels of an IGU are held together by the silicone joint sealing around the spacer bar. In this parametric study, we examined the silicone bite from 10 to 40 mm, and compare against the analytical solution introduced in PrEN 16612 (BSI 2013) in Fig. 11. It shows that the internal pressure is relatively low, and the impact brought by silicone is negligible. This is because the silicone stiffness is far higher than the bending stiffness in flat IGUs, and the panel bending deformation contributes to the most of volume change and hence dominates the internal pressure.
4.3.5 Length
4.4 Summary
Sensitivity degree comparison
Curved IGU  Flat IGU  

Silicone bite (b)  ***  * 
Length (l)  ****  ***** 
Radius  ****  N/A 
Thickness (t)  Varying (less sensitive with smaller radius)  ***** 
Cavity width (s)  *  * 
It’s found from Table 8 that changing cavity width exerts least impact among all five variables, therefore it will not be brought up for much discussion and assumed to be consistent in the following study.
It is worth noting the distinct influence of silicone bite in curved IGUs from flat IGUs. Therefore, when designing the internal pressure of curved IGUs, it’s not accurate to simply assume the panels are rigidly fixed at the edges as we normally do with flat IGUs.
5 Normalized internal pressure
Thus, for a curved IGUs with an arbitrary radius R, the internal pressure load due to climatic load will always fall within a range. The upper bound and lower bound of the range are defined as “infinite curvature effect” and “no curvature effect” respectively. The magnitude of the two bounds can be determined by simple hand calculation. When the panel hypothetically tends to be infinite stiff, (\(1/ k_{v1 }+1/ k_{v2})\) will level off to zero, and according to Eq. (19) the volumetric stiffness \(k_{v,{ tot}}\) equals to the silicone stiffness as calculated by Eq. (23). When there is no curvature effect, which means R is infinite, the panel is equivalent to flat panel, and the term (\(1/ k_{v1 }+1/ k_{v2})\) can be obtained by substituting Eq. (14). For consistency consideration, the stiffness of silicone bite is considered in flat IGUs too.
6 Conclusion

For the same dimension, curved IGUs always generate higher internal pressure caused by the same climatic action than that of flat IGUs due to geometric stiffness of curved IGUs. Nevertheless, the sensitivity level to each parameter differs a lot between curved and flat IGUs.

Silicone stiffness makes a significant difference in determining the internal pressure of curved IGUs but very minimal in flat IGUs. Therefore, it is necessary to consider the stiffness of silicone bite in the design to obtain accurate internal pressure.

Internal pressure increases with increased glass thickness. However, the influence of thickness is varying with curvature. The higher the curvature is, the less sensitive the internal pressure to thickness, and vice versa. When the curvature is small enough and the panel is quasiflat, the internal pressure will be approximately proportional to the thickness to the power three.

Cavity width has a subtle influence for curved IGUs but has relatively higher weight in flat IGUs.

Longitudinal length of has a big impact on internal pressure increase for both flat and curved IGUs, especially when the longitudinal length is less than the chord length.

For an arbitrary radius, the resultant internal pressure always falls within a range. The upper and lower bounds of the range can be determined by hand calculation. The smaller the radius, the closer the internal pressure to the upper bounds, and vice versa.

The concept of “normalized internal pressure” is proposed as a normalized expression of internal pressure in curved IGUs. This dimensionless parameter can be adopted for future empirical charts. In this paper, a dimensionless chart is drawn to depict the relations between the internal pressure and curved geometry.

The dimensionless chart can be further expanded by considering different silicone bites /glass thickness combination, and therefore provide an empirical method for curved IGUs design.
Notes
Compliance with ethical standards
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
References
 Batdorf, S.B.: A simplified method of elastic stability analysis for thin cylindrical shells II—modified equilibrium equation. Technical Note of National Advisory Committee for Aeronautics (1947)Google Scholar
 BSI, Draft BS EN 16612 Glass in building—determination for the load resistance of glass panes by calculation and testing, London (2013)Google Scholar
 DOW CORNING; Design stress for DOW CORNING\(\textregistered \) Structural silicone. Dow Corning GmbH (2015)Google Scholar
 Feldmeier, F.: Insulating units exposed to wind and weather—load sharing and internal loads. GPD Proceeding, pp. 633–636 (2003)Google Scholar
 Garvin, S.L., Marshall, W.J.: DoubleGlazing Units: A BRE Guide to Improved Durability. Construction Research Communications Ltd., London (1995)Google Scholar
 Griffith, J., Marinov, V.: Optimisation of curved insulated glass. GPD Proceeding, pp. 67–70 (2015)Google Scholar
 Huveners, E.M.P., Herwijnen, F., Soetens, F.: Load sharing in insulated double glass units. In: HERON, pp. 99–122. TNO Built Environment and Geosciences, Delft (2003)Google Scholar
 Roarks, R.: Stresses and deflection in thin shells and curved plates due to concentrated and variously distributed loading. Technical Note of National Advisory Committee for Aeronautics (1941)Google Scholar
 Stetson, T.D.: Improvement in window glass. United States Patent Office, New York: No. 49 167 (1865)Google Scholar
 Timoshenko, S.: Theory of Plates and Shells. McGrawHill, New York (1940)zbMATHGoogle Scholar
 TRLV: Technical regulations for the use of glazing with linear supports. German Designation: TRLV (2006)Google Scholar
 VanDenBergh, S., Hart, R., Jelle, B.P., Gustavsen, A.: Window spacers and edge seals in insulating glass units: a stateoftheart review and future perspective. Energy Build. 58, 263–280 (2013)CrossRefGoogle Scholar
 Vuolio, A.: Structural behaviour of glass structure in facades. Ph.D. Thesis, Technology Laboratory of Steel Structures (2003)Google Scholar