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Glass Structures & Engineering

, Volume 4, Issue 1, pp 29–44 | Cite as

Sensitivity study on climate induced internal pressure within cylindrical curved IGUs

  • Minxi BaoEmail author
  • Sam Gregson
SI: Challenging Glass paper
  • 88 Downloads

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 impact 

1 Introduction

Insulating glass units (IGUs) have been widely adopted in modern building envelopes for decades to improve thermal performance. The concept of the IGU was first invented and patented by Stetson (1865). The technology was then markedly improved over the century and generally emerged in United Kingdom in the early 1960s (Garvin and Marshall 1995). A typical double glazed IGU (shown in Fig. 1) consists of two glass panels separated by a spacer bar which is filled with desiccant in order to absorb the moisture condensation during service life. A primary sealant is applied between the spacer bar and glass panel, which provides an additional barrier against moisture infiltration. Secondary sealant is then filled outside the bottom of the spacer bar. The combination of edge sealants and spacer bar is usually referred as the edge-seal system, a crucial component in the IGUs that provides a gas and moisture barrier and structurally bonds glass panels together. Triple IGUs introduce an additional layer of glass and edge-seal system to provide higher insulation performance. For clarification, we are mainly focusing on double glazing units in this paper and the term “IGUs” used in this paper refers to double glazing units.
Fig. 1

Typical doubled glazed IGU build-up

Contemporary architectural design is keen to integrate freeform aesthetics with thermal insulation for glass façades. Curved IGUs have been frequently proposed to deliver a more dynamic architectural language as well as retain satisfactory energy efficiency. However, there is a general lack of understanding of the structural behaviour of curved IGUs within the engineering industry, particularly the climatic induced internal pressures. Climatic loads are an inherent design problem with IGUs which cannot be solved by analytical equations for curved shapes. Climatic loads can lead to sealant failure, glass overstressing, and the panel deformation due to internal pressure change can lead to visible visual distortion as illustrated in Fig. 2.
Fig. 2

Pillowing effect occurred within concave-curved IGU façade

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 build-up 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.

According to the ideal gas law, the relationship between the volume V occupied by a fixed amount of an ideal gas in moles n, its pressure P, and temperature T in Kelvin can be expressed as
$$\begin{aligned} pV=nRT \end{aligned}$$
(1)
where R is the gas constant, 8.314 J/mol. Thus, when a gas undergoes a change of state from state 0 to state 1 the volume V, pressure p and temperature T respect the following relationship
$$\begin{aligned} \frac{p_0 V_0 }{T_0 }=\frac{p_1 V_1 }{T_1 } \end{aligned}$$
(2)
Since the air or noble gas within the IGU cavity is hermetically sealed, it can be assumed to closely follow the ideal gas law. Load share between the glass packages is activated through the pressure change within the sealed gas. An installed IGU may encounter a number of environmental factors during its service life (Van-Den-Bergh et al. 2013), including (a) large temperature differences/thermal cycling, (b) atmospheric pressure fluctuation, (c) external wind loads/imposed loads. These all give rise to changes in the internal pressure of the sealed air, and thus impose load onto the glass packages. Quantifying the internal pressure change when the system reaches equilibrium is key to determining the loads on the two packages.

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.

Previous studies have outlined the workflow to find the pressure change at equilibrium state (Feldmeier 2003; Huveners et al. 2003; Vuolio 2003; Griffith and Marinov 2015). If the relationship between internal pressure and volume change is known, the climatic pressure can be calculated by expressing the volume change dV as a function of internal pressure change dp,
$$\begin{aligned} dV=f(dp) \end{aligned}$$
(3)
Substituting Eq. (3) into ideal gas law Eq. (2) we get the following equation, where dp is the only unknown variable to be solved
$$\begin{aligned} \frac{p_0 V_0 }{T_0 }=\frac{(p_0 +dp)(V_0 +f(dp))}{T_0 +dT} \end{aligned}$$
(4)
Another key concept relating to the gas pressure is the concept of the isochore pressure. In the context of IGUs the isochore pressure is defined as the internal pressure relative to external pressure caused by climatic changes, assuming that the cavity volume remains constant. For IGUs the climatic change which could occur subsequent to production are difference in altitude, difference in temperature and difference in external air pressure (BSI 2013; TRLV 2006) and is expressed as
$$\begin{aligned} p_{{ iso}} =c_H (H-H_p )+c_T (T-T_p )-(p-p_p )p_{{ iso}} \end{aligned}$$
(5)
where HT and p represent the altitude, temperature and ambient pressure at the installation site and \(H_{p},T_{p}\) and \(p_{p}\) represent the altitude, temperature and ambient pressure at the production site respectively; \(c_{H}\) denotes coefficient for effect of altitude change on isochore pressure (0.12 kPa/m); \(c_{T}\) is the coefficient for the effect of temperature change on isochore pressure (0.34 kPa/K).
Therefore, Eq. (4) can be transformed to:
$$\begin{aligned} (p_{{ atm}} +p_{{ iso}} )V_0 =(p_{{ atm}} +dp)(V_0 +f(dp)) \end{aligned}$$
(6)
where \(p_{{ atm}}\) is atmostpheric pressure and dp represents the climatic load.
Volumetric stiffness \(k_{V}\) is introduced to portray Eq. (3), i.e. the relation of volume change and pressure change (Griffith and Marinov 2015). Equation (6) can be converted to a quadratic equation, and internal pressure change dp can be solved.
$$\begin{aligned}&\left( p_{{ atm}} +p_{{ iso}} \right) V_0 =\left( p_{{ atm}} +dp\right) \left( V_0 +\frac{dp}{k_{V1} }+\frac{dp}{k_{V2} }\right) \end{aligned}$$
(7)
$$\begin{aligned}&dp=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a} \end{aligned}$$
(8)
where
$$\begin{aligned} a= & {} \left( \frac{1}{k_{V1} }+\frac{1}{k_{V2} }\right) ;\nonumber \\ b= & {} \left( V_0 +p_{{ atm}} \left( \frac{1}{k_{V1} }+\frac{1}{k_{V2} }\right) \right) ;c=-p_{{ iso}} V_0 \end{aligned}$$
where \(k_{v1}, k_{v2}\) are defined as the volumetric stiffness of the two glass packages, and expressed as
$$\begin{aligned} k_{Vi} =\frac{dp}{dV_i } \end{aligned}$$
(9)
For curved panels it has been suggested that \(k_{V}\) is solved by finite element analysis by applying a unit load to represent dp and calculating the volume change due to the displacement of the glass packages (Griffith and Marinov 2015). For flat panels some analytical equations have also been proposed.

2.2 Analytical solutions for flat IGUs

For flat IGUs, researchers have established analytical equations to express \(k_{v}\). For small deformation, linear analytical solution can be derived from classical Timoshenko plate theory (1940)
$$\begin{aligned} dV= & {} \int _0^a \int _0^b w_{\left( x,y\right) } dxdy\nonumber \\= & {} \frac{16dp}{\pi ^{6}D} \int _0^a \int _0^b {\frac{\sin \frac{\pi x}{a}\sin \frac{\pi y}{b}}{\left( \frac{1}{a^{2}}+\frac{1}{b^{2}}\right) ^{2}}} dxdy\nonumber \\= & {} \frac{64a^{5}b^{5}}{\pi ^{8}D\left( a^{2}+b^{2}\right) }dp \end{aligned}$$
(10)
where \(w_{\left( {x,y} \right) }\) is the deformation at any point of the plate. D is the bending stiffness of flat panel; a and b are the glass edge lengths;p is the applied pressure.
$$\begin{aligned} k_V =\frac{dp}{\left( \frac{64a^{5}b^{5}dp}{\pi ^{8}D\left( a^{2}+b^{2}\right) }\right) }=\frac{\pi ^{8}D\left( a^{2}+b^{2}\right) }{64a^{5}b^{5}} \end{aligned}$$
(11)
For those IGUs with larger dimensions and receiving higher level of loads, linear solution will lead to notable discrepancy from the actual bending behaviour because the membrane stress induced by large deformation is not considered. Non-linear analyses have been carried out however can be only solved via the aid from computational tools due to mathematical complexity. Feldmeier (2003) proposed a linear approximation which introduces a milestone parameter, the insulating glass unit factor:
$$\begin{aligned} \varphi =\frac{1}{1+(v_e +v_i )p_{{ atm}} /V_0 } \end{aligned}$$
(12)
where \(v_{e}\) and \(v_{i}\) are the volume per load for external and internal panel deformation respectively, in \(\hbox {m}^{3}/\hbox {kPa}\).
Load share determination by insulating glass unit factor has been widely used as code of practice PrEN 16612 (BSI 2013). It is noted that though the nonlinearity in stress and deflection has been considered by introducing a parameter “Non-dimensional load”, the volume change, however, is still approximately by a linear expression:
$$\begin{aligned} V=k_5 A\frac{a^{4}}{h^{3}}\frac{F_d }{E} \end{aligned}$$
(13)
where \(k_{5}\) is a geometry-related parameter, a is the shorter edge of the panel, h is the glass thickness, \(F_{d}\) is the pressure load and E is the Young’s Modulus of glass.
Therefore, the volumetric stiffness is
$$\begin{aligned} k_V =\frac{h^{3}}{a^{4}}\frac{E}{Ak_5 }. \end{aligned}$$
(14)

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.

It has been observed that empirical or analytical solutions are not available for curved IGUs contrary to flat IGUs. The main reason is there is no accurate, general analytical expression to describe deformation of a curved plate. The overall deflection is attributed to both glass bending and membrane strain. The relative amounts of strain energy in these two behaviours varies with the plate geometry and is non-linear with respect to deflection. Therefore, a simple formula is not available to accurately depict the curved panel deformation. Previous researchers proposed empirical equations for calculating deflection of cylindrical shell within comparatively small ranges of the R / t and l / R ratios. The equations are calibrated by a dimension parameter \(\alpha \) (changing in l / R), which is acquired from testing. It is claimed the results obtained from empirical charts do not differ from the laboratory measured data by more than approximately 20% (Roarks 1941). As for analytical solution, Batdorf (1947) pointed out the simplest method for cylinders or curved cylindrical panels is to represent radial displacement w by a trigonometric series expansion. Again, the expression is cumbersome and not in ease of solution.
$$\begin{aligned}&\nabla w^{8}+\frac{12Z^{2}}{b^{4}}\cdot \frac{\partial w^{4}}{\partial x^{4}}+k_x \frac{\pi ^{2}}{b^{2}}\nabla w^{4}\cdot \frac{\partial w^{2}}{\partial x^{2}}\nonumber \\&\quad +\,2k_S \frac{\pi ^{2}}{b^{2}}\cdot \nabla w^{4}\cdot \frac{\partial w^{2}}{\partial x\cdot \partial y}\nonumber \\&\quad +\,k_y \frac{\pi ^{2}}{b^{2}}\nabla w^{4}\cdot \frac{\partial w^{2}}{\partial y^{2}}=0 \end{aligned}$$
(15)
Among most of published documents (Feldmeier 2003; Huveners et al. 2003; Vuolio 2003; Griffith and Marinov 2015), FEM numerical method is believed to be most efficient way for the internal pressure investigations of curved IGUs. Nevertheless, although it is theoretically straightforward to investigate the stress/deformation behaviour of curved glass panels via numerical modelling, few commercially prevalent FEA software, especially those cost-efficient FEA software packages, can determine the internal pressure within a sealed air cavity at equilibrium state, namely pneumatic fluid analysis.

This paper introduces a computational-aided 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 non-linearity 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.

This process repeated until convergence as illustrated in the flow diagram below. In order to validate convergence within each iteration the current approximation of the climatic load is compared to the calculated internal pressure of the IGU for the given volume change. When the internal pressure applied in the FEM model and the climatic load calculated due to the volume change are within tolerance the procedure is terminated. Convergence tolerance is set to be 0.001 kPa.
Fig. 3

Method of calculation flow diagram

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 hyper-elastic 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

In order to validate the methodology used in this study a comparison is made between the results of the proposed methodology for flat IGUs to the analytical equations which exist for flat IGUs. It is then assumed that the methodology can be employed to calculate the climatic loads of curved IGUs as no change in the algorithm is required to facilitate curved panels. Future work will extend the validation of the methodology via physical testing.
Fig. 4

Glass deformation at one iteration step (deformation scale factor: 100)

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.

It is worth noting that in the derivation of the analytical method presented by PrEN 16612 are some simplifications and assumptions, therefore the analytical results shall not necessarily be taken as the exact answer.
Table 1

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

In the sensitivity study, the internal pressure changes due to climatic loads of curved IGUs are calculated by considering a series of variables. The geometric parameters include chord, length, radius, panel thickness, cavity width, and silicone bite as shown in Fig. 5.
Table 2

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

Fig. 5

Geometrical parameters in a curved IGU

For the purposes of parametric study, the chord is kept constant at 1500 mm, and other parameters are changed within a realistic range. Each parameter is taken into account independently, with the other parameters staying consistent to the basic configuration. The variables are summarised in Table 3.
Table 3

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 build-up

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

Characteristic climatic actions are set to be consistent to the recommended parameters given by TRLV technical note (2006), as shown in Table 4. It can be observed that climatic load in winter is a reverse version of summer, therefore either scenario can represent the magnitude of overall isochoric pressure. In this study, the climatic load is defined to occur during summer season.
Table 4

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.

In parallel, to this comparison, we introduce “normalized internal pressure” \({r}({t}) = { dp}({x})\)/dp(\({t}=8\) mm). Figure 7 displays the normalized internal pressures of flat IGUs and curved IGUs. There is a strong contrast that r(t) of flat IGU increases by more than 1000%, whereas curved IGU increases by only 38% across the thickness range considered. The input information is summarised in Table 5.
Fig. 6

Internal pressure versus glass thickness

Fig. 7

Normalized internal pressure versus glass thickness

According to Eq. (13), the volumetric stiffness is proportional to \(t^{3}\), therefore the internal pressure of the flat panel is significantly sensitive to the thickness, whereas the stiffening effect in curved IGUs by increasing thickness is not as conspicuous as in flat panel. This is because the geometric stiffness due to curvature is governing the overall stiffness of the curved panel rather than plate bending. By comparing the equation of equilibrium for a flat plate and cylindrically curved plate given by Batdorf (1947), the additional term to account for the curvature effect is inversely proportional to t. but in flat panel, it is dictated by the flexural bending stiffness D, which is proportional to \(t^{3}\). We can hence reason that, when the radius is increasing, the weight of curvature effect will be reduced, and the flexural stiffness becomes predominant, and the panel will be more sensitive to thickness variance.
Table 5

Input summary

Variables

Chord (mm)

Length (mm)

Radius (mm)

Thickness (mm)

Cavity width (mm)

Silicone bite (mm)

Symbol

c

l

R

t

s

b

Build-up

1500

1500

1500

8–24

16

10

Fig. 8

Internal pressure versus panel radius

Table 6

Input summary

Variables

Chord (mm)

Length (mm)

Radius (mm)

Thickness (mm)

Cavity width (mm)

Silicone bite (mm)

Symbol

c

l

R

t

s

b

Build-up

1500

1500

1500–6000

8

16

10

Table 7

Input summary

Variables

Chord (mm)

Length (mm)

Radius (mm)

Thickness (mm)

Cavity width (mm)

Silicone bite (mm)

Symbol

c

l

R

t

s

b

Build-up

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 quasi-inverse 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

The most frequently used spacer bar widths in the market are considered in this study, namely 12 mm, 16 mm, 19 mm and 20 mm. The input information is shown in Table 7. As shown in Fig. 9, there is a subtle increase of internal pressure observed in both curved and flat IGU along cavity. Within the range, dp in curved IGUs is rising from 3.9 to 4.5 kPa and from 0.311 to 0.535 kPa in flat panels. By comparing the ratio in Fig. 10, flat panel displays 61% increase in contrast to 15% growth of curved IGU, therefore exhibits higher dependency of cavity width than curved IGU.
Fig. 9

Internal pressure versus cavity width

Fig. 10

Normalized internal pressure versus cavity width

In order to understand the trends, we transform Eq. (8) to an expression that contains cavity width s and volumetric stiffness \(k_{V}\). Since,
$$\begin{aligned} V_0 =A\cdot s \end{aligned}$$
(16)
where A is the area.
$$\begin{aligned} \frac{1}{k_{V,{ tot}} }=\frac{1}{k_{V1} }+\frac{1}{k_{V2} }=\frac{k_{V1} +k_{V2} }{k_{V1} k_{V2} } \end{aligned}$$
(17)
Substitute Eqs. (16) and (17) into (8), we have
$$\begin{aligned} dp\!=\!\frac{-\left( A\cdot s\!+\!p_{{ atm}} /k_{V,{ tot}} \right) \!+\!\sqrt{\left( A\cdot s\!+\!p_{{ atm}} /k_{V,{ tot}} \right) ^{2}\!-\!4k_{V,{ tot}} p_{{ iso}} }}{2/k_{V,{ tot}} }\nonumber \\ \end{aligned}$$
(18)
It can be found Eq. (18) is an increasing function, therefore the internal pressure is increasing with s.

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.

On the contrary, Fig. 12 shows notable deviation in the internal pressure between rigid fix and silicone bond in curved IGUs. Rigid fix boundary condition gives 7.84 kPa as depicted by the asymptote, whereas 10 mm silicone bite leads to an internal pressure of 4.26 kPa, i.e. 45.7% lower than rigid fix. It ascends along with bigger silicone bite, namely higher silicone stiffness. We list the trends of silicone impacts to both flat IGUs and curved IGUs in Fig. 13 for a parallel comparison. What can be clearly seen is that the internal pressure of curved IGUs are significantly more susceptible to the size of silicone joint in contrast to flat IGUs. It can be explained also by the stiffness change of silicone. 10 mm deep silicone bite gives rise to a relatively lower stiffness and engaged more in the overall volume deformation. As it goes up, the less flexibility results in less volume change and subsequently higher internal pressure.
Fig. 11

Internal pressure versus silicone bite in flat IGUs

Fig. 12

Internal pressure versus silicone bite in curved IGUs

Fig. 13

Normalized internal pressure versus silicone bite

4.3.5 Length

Panel length is varied from 375 to 6000 mm. Given the chord is unchanged, it can be deemed as a parametric study of aspect ratio. Both flat IGU and curved IGU exhibit strong dependency to the length. The nonlinear descending curves behave as an inverse function of l. For flat panels, the volumetric stiffness is expressed as Eq. (14) and there is an inverse relation to the bi-quadrate of short edge a. before aspect ratio reaches 1, the length is acting as the short edge a and thus leads to the rapid drop at the initial stage. Despite the lack of analytical expression of volumetric stiffness of curved IGUs, the similar curve pattern in Fig. 14 indicates that the overall stiffness of curved glass panel is predominantly driven by the longitudinal length l when the aspect ratio less than 1, i.e. l smaller than the chord c.
Fig. 14

Internal pressure versus panel length

4.4 Summary

Based on the sensitivity studies carried out for each parameter, qualitative sensitivity degree is summarised to compare the difference between flat IGU and curved IGUs in Table 8.
Table 8

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.

To take into account the silicone stiffness into the overall volumetric stiffness \(k_{v, { tot}}\), Eq. (17) can be written as
$$\begin{aligned} \frac{1}{k_{V,{ tot}} }=\frac{1}{k_{V1} }+\frac{1}{k_{V2} }+\frac{1}{k_{V3} } \end{aligned}$$
(19)
\(k_{{ Vs}}\) denotes the volumetric stiffness of silicone. According to Eq. (9), \(k_{{ Vs}}\) is a ratio of the applied pressure and the resultant volume change, and the latter is induced by the stretching of silicone bite regardless of panel deformation. Therefore, the volume change can be determined by calculating the cavity section area change due to a translational displacement of the glass panel as illustrated in Fig. 15.
Fig. 15

Derivation of cavity volume change due to silicone stretching

Translational displacement x can be calculated by following Hook’s Law:
$$\begin{aligned} x=F/k=P\cdot a\cdot l/(E_s \cdot (2a+2l)\cdot b/s) \end{aligned}$$
(20)
where \(E_{S}\) is the Young’s modulus of silicone, assumed as 4 MPa (Dow Corning 2015). s is the cavity width, c is the chord, b is the length of silicone bite, R is the radius and a denotes the arc length:
$$\begin{aligned} a=2R\cdot \arcsin \frac{c}{2R} \end{aligned}$$
(21)
The volume change can be expressed as:
$$\begin{aligned} dV_s =x\cdot c\cdot l \end{aligned}$$
(22)
Substitute Eqs. (20)–(22) into (9)
$$\begin{aligned} k_{{ Vs}} =\frac{\left( 2R\arcsin \frac{c}{2R}+l\right) E_S b}{scl^{2}R\arcsin \frac{c}{2R}} \end{aligned}$$
(23)
Since it can easily be concluded from Eq. (18) that internal pressure dp is a function of overall volumetric stiffness, the remarkable impact by silicone bite in curved IGUs can be explained. By observing Eq. (19), the total volumetric stiffness of a curved IGU can be deemed as an equivalent spring of a group of springs in series. Therefore, the equivalent spring stiffness is essentially dictated by the spring of the least stiffness. When R decreases, the curvature effect will go up and give rise to a greater volumetric stiffness of each glass panel, i.e. \(k_{v1}\) and \(k_{v2}\), which are much higher than the silicone stiffness. Therefore, silicone bite plays a remarkable role in curved IGUs. On the other hand, flat panels are relatively low, and hence \(k_{v1}\) and \(k_{v2}\) are smaller than \(k_{{ vs}}\), then the overall stiffness is dominated by the bending stiffness of panels whereas silicone stiffness can be almost ignored.

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.

Figures 16 and 17 show the internal pressure change in radius with and without taking into consideration of 10 mm silicone bite respectively. Upper bounds and lower bounds are plotted in the figures. Glass thickness, chord and length are fixed to be 8 mm, 1500 mm and 1500 mm respectively. The radius is ranging between 750 and 10 mm. It should be noted that 750 mm is the minimal radius that can be adopted for a chord 1500 mm. As has been discussed earlier, there is a big discrepancy due to the lack of silicone stiffness. If we compare the upper bounds of two figures, the resultant internal pressure by assuming rigid fixing is nearly twice of that with silicone bite and it’s far away from realistic values. In addition, the curvature effect is approaching infinite in Fig. 17 when the radius is reduced to 750 mm. However, it is not realistic either because the silicone joint that holding two panels together cannot provide rigid translational restraints but act as springs. This also explains that in Fig. 16, the internal pressure can’t reach 100% of \(t = \infty \) even though the curvature has increased to the most.
Fig. 16

Internal pressure change in radius considering silicone stiffness \(({ b} =10\,\hbox {mm})\)

Fig. 17

Internal pressure change in radius considering glass edges are rigidly fixed

With the purpose of expanding Fig. 16 for different radii and different lengths, a concept of “normalized internal pressure” \(r_{{ dp}}\) is introduced to normalize the relations between internal pressure of curved IGUs and other impact parameters.
$$\begin{aligned} r_{{ dp}} =\frac{p_{{ upper}} -{ dp}}{p_{{ upper}} -p_{{ lower}} } \end{aligned}$$
(24)
where \(p_{{ upper}}\) is the upper bound, and \(p_{{ lower}}\) is the lower bound.
A dimensionless curvature parameter denoted as Z is employed as x axis, which is transformed from the curvature parameter introduced in Batdorf’s publication (1947)
$$\begin{aligned} Z=\left( \frac{Rt}{c^{2}}\right) \cdot 10^{3}. \end{aligned}$$
(25)
Fig. 18

Normalized internal pressure versus curvature parameter

Within parameters explored in this study, designers may use Fig. 18 as a quick and easy indication to determine the magnitude of internal pressure due to climatic actions. The procedures are summarised below:
  1. 1.

    Calculate normalized curvature parameter Z using Eq. (25).

     
  2. 2.

    Based on normalized curvature parameters Z and the aspect ratio of the panel l / C, determine the corresponding normalized internal pressure \(r_{{ dp}}\).

     
  3. 3.

    Calculate lower and upper bounds using Eqs. (23) and (14) respectively.

     
  4. 4.

    Calculate the magnitude of actual internal pressure dp using Eq. (24).

     

6 Conclusion

This paper reviews existing literature with respect to calculating the internal pressure change within IGUs due to climatic loads. The internal pressure can be mathematically expressed as a function is volumetric stiffness \(k_{V}\). Since analytical expression of \(k_{V}\) for cylindrically curved panels is unlikely to be solved, numerical iteration process is adopted to determine the internal pressure of curved IGUs. A sensitivity study on the geometric parameters including panel thickness, radius, length, cavity width and silicone bite depth is undertaken to identify the predominant impact factors. The correlations between the internal pressure and these parameters are plotted. The results between flat IGUs and curved IGUs are also compared and discussed. Following conclusions are drawn:
  • 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 quasi-flat, 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.

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Eckersley O’CallaghanLondonUK

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