On the thickness determination of rectangular glass panes in insulating glass units considering the load sharing and geometrically nonlinear

The number and size of windows has increased in large cruise ships, especially on the top decks. They have therefore become a weight and stability-critical component of the structure. Their thickness is determined according to the classification rules which are generalized for all type of passenger ships. That is, the provided formulae are based on linear-elastic, small deformation, plate theory and therefore more suitable for smaller windows in non-weight critical applications. However, majority of the windows are large insulating glass units (IGUs) that exhibit two effects that the rules do not currently consider: development of membrane stresses in the glass panes at large deflections due to the von Kármán strains (geometric nonlinearity) and interaction of the glass panes due to the internal cavity pressure between them (load sharing). Both increase the load bearing capacity of the IGUs. Therefore, extension to the thickness determination is needed for achieving the lightweight design. This paper uses nonlinear Finite Element Method to study the IGUs static response under uniformly distributed load considering the effects. The response consists of principal stress and deflection of the panes, and the cavity pressure. Validation is carried out by experimental results from scientific literature. Case study on typical panes from cruise ships indicate that considering the two beneficial effects, the thickness of the glass panes in the IGUs may potentially be reduced between 26–54 % with respect to the classification rule-based design. That is, by using the same allowable principal stress criterion between the linear and nonlinear predictions.


Introduction
The size of modern cruise ships has increased in the recent years to accommodate more passengers and amenities.Consequently, the ship's structural weight has increased that negatively affects its operational efficiency, stability and building cost, to name a few.Using lightweight structures and materials instead of conventional steel structures helps to save weight [1][2][3][4][5][6][7][8][9].While the steel structures lighten due to the improved design and manufacturing methods, glass is used in increasing proportions to enable better immersion for the passengers.Not only the number and size of individual windows has increased, but also large glass structures are built with an area of several hundred square meters.The corresponding weight contribution and stability impact can be considerable as majority of these large windows are located on the top decks (Fig. 1).
As the area of the windows increases, their thickness should be reduced to achieve the lightweight design without harming the safety.or by nationally or internationally recognized equivalent standards (e.g.ISO 11336-1 [14]), where the main assumptions are that the cavity pressure does not exist and linear-elastic Kirchhoff plate theory is valid.The resulting minimum required thickness for a monolithic glass pane is determined depending on its size, shape, and location.

Nomenclature
Table 1 reviews the rules and Fig. 2 the outcomes.The data referred to in Fig. 2 (right-hand side) has been derived by the linear plate theory (Eqs.(A.1)-(A.3))and by computing the maximum principal stress equal to 40 MPa.It is clear from Fig. 2 that the rules are based on this theory.The linear theory is considered accurate at small deflections ( < ).However, this assumption does not hold when ships operate in extreme conditions and encounter high pressure levels.For instance, ship windows tested for ultimate loads up to 170 kPa exhibited geometric nonlinearity approximately when the center-of-pane deflection equaled the thickness of the pane ( ≈ ) [15].The stress increase becomes degressive for larger deflection ( > ) which linear calculation cannot capture.Hence, the overestimation of the stresses results in underestimation of the load bearing capacity of the panes.Similar underestimation takes place when neglecting the load sharing effect.Some amount of the load applied externally to one pane of the IGU is transferred to the adjacent pane via the enclosed gas.When the directly loaded pane deflects, the cavity volume decreases and consequently the internal pressure increases, which acts on the indirectly loaded pane.McMahon et al. [16] studied the nonlinear load sharing experimentally and demonstrated its significance.For instance, a unit with two equally thick glass panes shared an uniform pressure load almost equally between the panes (49 % and 51 %).Vallabhan and Chou [17] studied the load sharing using nonlinear plate theory in an iterative procedure.They showed how the maximum deflections and principal stresses of the panes evolve due to the von Kármán effect coupling the membrane stresses to the bending stresses by using the Finite Difference Method and non-dimensional design charts.They also found that the maximums stress moves toward the corners of the plate for increasing pressures.Wörner et al. [18] conducted a similar study but they used a simple approximation and Finite Element Method to solve the maximum deflection of the panes, including the nonlinearities, which was then used to determine the load sharing.Galuppi and Royer-Carfagni [19] proposed ''Betti's analytical method'' (BAM) for calculating the load sharing in IGUs with two glass panes.
It is an approximation model based on linear-elastic plate theory to compute the volumetric change of the cavity in IGUs.They showed that load sharing is most beneficial for thin and large glass panes, like ships have.The method was extended for IGUs with multiple panes in [20].In practical design, the load sharing is often considered through standards, e.g.EN-16612 [21] in building architecture.While these studies report results with satisfactory accuracy, they lack the design freedom in terms of boundary conditions and glass shapes.Extension of the methods towards ship design are needed.
Therefore, this paper uses a FE technique where the interaction of the glass panes and the gas is embedded in the model.The influence of the large deflections on the gas pressure is therefore automatically considered.Furthermore, the FEM is applicable to any IGU construction under various loads, boundary conditions and assumptions.Hence, this is a convenient approach to study the influence of the geometrically nonlinear bending and the load sharing on the deflections and the stresses of the glass panes in the IGUs.The model is validated by BAM [19] and MEPLA ISO [22] on linear basis.On nonlinear basis, experimental results from [16] are used to validate the model.
Using the validated model, the goal is to provide a better understanding for designing these weight critical glasses in ships by reflecting the obtained knowledge of the reduced stress state on the prescribed thickness determination rules presented in Table 1.

Large deflection of panes
The equations for the glass pane thickness determination discussed in Section 1 and presented in Appendix (Eqs.(A.1)-(A.3))are suitable for Kirchhoff plates subjected to small deflections, where it is reasonable to assume that the mid surface of the plate does not strain during bending.Then, the corresponding load carrying mechanism consist  [10]; the last one is a cross-section sketch of an insulating glass unit.

Table 1
Thickness determination of monolithic rectangular glass panes in ships according to different design rules.
Lloyd's Register [11] DNV GL [12] ISO 11336-1 [14] Thickness of only the bending action.However, when the deflection increases and reaches some ratio,   , the mid surface starts to stretch and the load carrying mechanism is accompanied by the resulting membrane forces (right-hand side in Fig. 3).That is, the in-plane stretching of the mid surface resists the out-of-plane load.In the case of glass panes, the increased strength from the glass post-processing (tempering or chemical strengthening) allows for the geometric nonlinearity to be very significant.This increases the load bearing capacity of the glass panes, as shown in Fig. 3 (left-hand side).
Mathematically, this is considered by adding the von Kármán terms (highlighted) in the set of strain equations for the Kirchhoff plates [24]: ) .
(1)  The squared terms couple the in-plane and out-of-plane displacements, i.e. the bending and membrane actions at large deflections.These become dominant after the deflection grows larger than the thickness.Therefore, neglecting the geometric nonlinearity heavily underestimates the deflection of thin and large glass panes.

Load sharing
The insulating glass unit consist of the glass panes, the spacers, and the cavity.The cavity is hermetically sealed for its purpose to provide thermal insulation.This also allows for the load sharing to occur.If the cavity leaks, the insulation properties weaken and it absorbs moisture which accumulates on the glass panes, even if the spacers are filled with moisture absorbing desiccant.Since the IGUs are visible to the passengers at all times, the fogging impairment gives a clear indication for a repair.For this reason, we argue that it does not pose any additional risks when including the load sharing effect in the design process.
The load sharing is governed by the ideal gas law,  =  .The cavity has some initial pressure depending on the conditions at time of sealing.For this paper, we assume that there is no initial deflection of the panes due to temperature change, atmospheric pressure change or altitude change (climate loads [25]), i.e. the IGU is at equilibrium and hence, the initial cavity volume is the pane area times the spacer thickness.An external load is applied on the outer pane.Once the directly loaded pane deflects, the cavity volume variation changes the internal pressure, which causes the indirectly loaded pane to deflect, see Fig. 4. Because the volume has changed again, a new pressure is calculated, and so on, until the system is at equilibrium (Fig. 5).For solving this iterative process, the ideal gas law is extended to solve incremental volume change at certain time-steps (2) where the subscripts '''' and '''' denote the value at the end of the previous substep and the total value, respectively.The total pressure and temperature consist of the current value and the reference value ( +  0 and  +  0 ).Newton-Raphson method is used [26] for this nonlinear analysis.The formulation of these coupled system of the solids and the gas is described in [27].
On linear basis, the Betti's analytical method can be used to determine the pressure increase due to the actions on the pane [19,28].Consider an IGU with two glass panes that have thicknesses '' 1 '' and '' 2 '', respectively.The panes are separated by a spacer with thickness ''''.The panes are rectangular with longer and shorter sides, denoted with '''' and '''', respectively.An uniform external pressure, , is applied on the surface of pane 1.The generated cavity pressure variation, , is calculated with the following formula: where: is the flexural stiffness of the th pane (1,2) [Nmm]. is the mean value of the shape function on the plate area [-].
The mean value of the shape function for rectangular pane with sides '''' and '''' is: One may use the cavity pressure obtained from Eq. (3) in Eqs.(A.1)-(A.4) to calculate the IGU response on linear basis.To correctly include the load sharing, the uniformly distributed load is replaced with  −  and  for pane 1 and pane 2, respectively.Alternatively, the linear response can be solved using MEPLA ISO 3 [22].

Finite element analysis of insulating glass units
Finite Element Method-based approach is exploited to model simultaneously the nonlinear pane response accounting the von Kármán strains and the iterative internal pressure increase in the cavity. 3MEPLA ISO is a freeware software for calculating the response of insulating glass units consisting of two or three glass panes.The software works on linear basis and uses analytical approach of double Fourier Series to calculate maximum deflections and maximum principal stresses of linearly simply supported IGU on its all 4 edges (influence of spacers excluded).The accuracy is based on Kirchhoff's plate theory and because geometric nonlinearities are neglected, the solution is valid for deflections smaller than half the plate thickness ( <   2 ).The cavity behavior is governed by the ideal gas law, but the magnitude of the cavity pressure is not presented in the software solution.
ANSYS ® Mechanical™ is used.Alternatively, one may use e.g., ABAQUS [29] or RFEM with RF-GLASS Add-on module [30].However, shell elements are preferred (Section 4.2), while the latter uses solid elements.The ANSYS model consist of two monolithic glass panes, gas filled cavity, and four spacers.The spacers merely create the volume of the cavity with the glass panes, but for simplification, they are not structurally considered in the analysis, i.e. they are modeled with low stiffness, small thickness, and all nodes bound.In reality, the spacers stiffen the IGU in bending to some degree [31].Therefore, omitting their influence on the response is a conservative approach as the FE analyses will predict slightly larger deflections.Besides, the spacers are part of detail design phase and experimental work is required for finding their true stiffness in the respective case.

Boundary conditions and loads
Two boundary conditions (BCs) are used: (1)   , the edges are simply supported, i.e. all translations of the edges are fixed while rotations are free; (2)    , in-plane translations of the edges are free while lateral translation is fixed, and rotations are free.In the latter condition, the middle nodes of both panes are restrained in the inplane direction and their rotation around -axis is fixed for the sake of having fully constrained model (Fig. 6).This boundary condition also mimics the setup used in the experiments by McMahon et al. [16], see Fig. 7.The aluminum brackets pressing the neoprene gaskets against the glass are loosely tightened to minimize the in-plane and rotational restraints of the IGU.In the current study, both conditions are used in nonlinear analysis to study the impact of the BCs on the development of the von Kármán strains at large deflections.In this way the response is studied at two theoretical limits, while the real conditions in ships lay somewhere in between.Uniformly distributed load is applied on the surface of pane 1, referred to as the directly loaded pane.This load represents a short duration load, such as wind.This is a simplified approach, as real glass structures are subjected to different combination of loads with varying duration that alter the IGUs response and its lifetime [32].For instance, the climate loads are omitted, though they can be included as body loads and initial conditions with good accuracy [33] for future studies.

Glass elements
This model will be used in future studies where the thickness of the panes is optimized while the element sizes should remain constant.Therefore, it is convenient to choose an element for the glass panes with nodes only at the mid-plane.We exploit shell elements (Fig. 8) instead of solids for the glass panes as this secures computational accuracy and  efficiency in applications where the pane thickness is small. 4This is also to avoid any shear or membrane locking at large deflections that may occur when using solid elements with high aspect ratios.Further, thin shells are a natural choice over the solids since the classification rules are based on them, as shown earlier.SHELL181 elements are used with full integration (KEYOPT(3) = 2), though the difference to reduced integration is in order of 0.1 % in the considered out-of-plane bending of thin-walled plates.The accuracy of the elements is governed by Mindlin-Reissner plate theory, also called as First-order shear deformation theory (FSDT), which includes transverse shear deformation through the thickness [34].FSDT is suitable for thin and moderately thick panes.When the glass panes are relatively thin ( ℎ ℎ < 1 10 ), 4 We have verified that shell elements (SHELL181) are as accurate as solidshell elements (SHOLHS190) in this study considering the presented boundary conditions, loading conditions, responses and design criteria (i.e., cases in Fig. 17 and Table 5).The difference between the elements in maximum deflection and maximum principal stress is in order of 0.05 % and 0.1 %, respectively.Solid-shell element gives a better representation of the inplane deformation of the edges, as it is 3-dimensional, but the maximum absolute difference between the elements is in order of 0.1-0.3mm, which is insignificant considering the required 10 mm overlap.like in this study, they act like Kirchhoff's plates and hence FSDT and Kirchhoff's plate theory are both suitable.Therefore, the FEM and analytical results are comparable on linear basis.However, sometimes triangular, or other odd shaped thick windows are used in ships.Then, the thickness to characteristics length ratio increase may require at least FSDT for accurate prediction of the response.

Gas elements
The gas is modeled with 3-D hydrostatic fluid elements, HSFLD242, presented in Fig. 8.They are pyramid shaped elements specifically designed to model fluids enclosed by solids.The nodes at the base have translation degree-of-freedoms (DOFs) and are shared with the structural shell elements they are attached to.The base has either 4 or 8 nodes depending on how many nodes the structural element has.The pressure node ''Q'' at the apex of the pyramid has hydrostatic pressure (HDSP) DOF and is shared with the corresponding nodes of all the hydrostatic fluid elements in the cavity.Hence, the density, pressure and temperature of the gas are presented uniformly through this one node.That is, they are equal for all the HSFLD elements.[35] and HSFLD242 element [36].

Model and mesh
The glass and the spacers are meshed with 4-node SHELL181 elements.The pressure node ''Q'' is placed at the center of the cavity.The HSFLD242 elements are ''coated'' on the surfaces of the structural elements facing the cavity.Because the hydrostatic fluid elements are attached to the structural shell elements, the number of HSFLD242 and SHELL181 elements are exactly equal (Fig. 9).Further, with high mesh density, the local curvature in the shell element is minor along the element and thus the linear variation of the deflection dominates the response.That is, there is no significant gap between the attached SHELL181 and HSFLD242 elements due to the latter not having rotational degrees of freedom.The coupling is described in ANSYS Theory Ref. [27].Detailed process of meshing the cavity with the HSFLD242 elements is presented [33].

Post-processing
The studied response consists of: (1) deflection of both panes; (2) maximum principal stress of both panes; (3) the generated pressure variation in the cavity; (4) in-plane deformation of the edges.The deflection is observed to ensure that the glass panes do not contact each other.This is, however, unlikely as in most cases the gas pressure reaches magnitude of 40-49 % of the external load, i.e. the panes experience similar load, deflection and stress.The classification societies use a limiting value for the maximum principal stress.Similarly, they require a certain overlap between the glass panes and the supporting structure, which is why it is important to record the in-plane movement of the edges.The maximum in-plane deformation of both panes is used for this, as shown in (Fig. 10).
The in-plane movement and the deflection of the indirectly loaded pane is always less due to the compressibility of the gas.Further, because the ideal gas is unable to resist shear deformation, the maximum principal stress is maximum on the tension side of both panes and zero, or close to zero, on the compression side (Fig. 11).The absolute maximum stress may be on either pane, depending on their thickness configuration.

Results
In this chapter, the presented FE model is first validated on linear basis using BAM [19] and MEPLA ISO [22].Then, the model is validated on nonlinear basis using the experimental results by McMahon et al. [16].The IGU dimensions for validation are presented in Table 2. Both, linear and nonlinear, analyses are used to study the cavity pressure generation.Finally, the validated model is used on nonlinear basis to conduct a case study to determine the required minimum thickness for the monolithic glass panes in an IGU for three different load cases.These results are compared to the design approaches suggested by the classification societies and ISO 11336-1 standard.

Linear analysis
Pane 1 is subjected to uniformly distributed load of 1.0 kPa.The generated cavity pressure is recorded from ANSYS and BAM (Eq.( 3)).Deflection of both panes are plotted along line /2 every 42 mm from  = 0 to  = .In analytical solution, the deflection is calculated with Eq. (A.4) by replacing the uniformly distributed load with  −  and  for pane 1 and pane 2, respectively.Based on the results presented in Fig. 12 and Table 3, the model is validated on linear basis.

Nonlinear analysis
For nonlinear analysis, the load is increased from 1.0 to 15.0 kPa with increments of 0.5 kPa.The center-of-pane deflection of both panes is recorded for each load and for both boundary conditions.The results are presented in Fig. 13.The experimental results from [16] fit between the two analysis results, but closer to    as expected.The relative error between    and experimental results varies between 7.4 % and 18.0 %.The error is the largest for small loads and deflections, e.g., 18.0 % corresponds to an absolute error of 0.16 mm for pane 2 in IGU B. The measuring instruments may be one source for errors; they are generally more accurate for higher loads.Further, the supporting structure in the experiments may restrain the in-plane translation and rotation of the edges to some degree, which results in stiffer bending.Since the general trend is correct and the results are conservative (   ), the model is validated and may be used in further studies.

Geometric nonlinearity and boundary conditions on the cavity pressure variation
The cavity pressure variation is plotted for loads from 1.0 to 15.0 kPa with increments of 0.5 kPa for three cases: (1) ANSYS nonlinear with    ; (2) ANSYS nonlinear with   ; (3) linear (BAM).Results for IGU A are presented in Fig. 14.Clearly, the geometric nonlinearities have an effect on the cavity pressure variation for high pressure levels.More so if the edges have restricted in-plane movement.

Case study
The individual and combined influence of the load sharing with the geometric nonlinearity on the response of an IGU is studied.The IGU dimensions are:  = 1500 mm,  = 20 mm,  1 =  2 = 10 mm and ∕ = 1,2.The dimensions are chosen to have a reasonable large window where both effects are significant.The constituent material properties used are presented in Table 4.The influence of the gravity on the results has been verified to be small, 5 and hence, it is neglected.The mesh size Fig. 9. Illustration of IGU model cross-section on the left-hand side.Letters and numbers represents the structural shell elements and hydrostatic fluid elements, respectively.The dashed line is the mid-plane of the shell elements.Gray colored area is gas.A section cut of the meshed model in ANSYS on the right-hand side (HSFLD242 element not visible).Fig. 10.In-plane movement of the edges.In square IGU, the edges experience equal contraction; ( 1 ) = ( 1 ) and ( 2 ) = ( 2 ).In rectangular IGU, the longer side () experiences larger contraction than the shorter side (); ( 1 ) < ( 1 ) and ( 2 ) < ( 2 ).Fig. 11.Deflection and maximum principal stress of both panes under uniformly distributed load acting on pane 1. Directly loaded pane always deflects more due to the compressibility of the gas (6.81 > 6.04).Maximum principal stress is 0 on compression side of both panes.Maximum is at tension side of the directly loaded pane.Thick-shell visuals are activated for these shell elements for distinguishing between top and bottom stress.in the model is 50 × 50 mm 2 and 50 × 20 mm 2 for glass and for spacers, respectively.
Uniformly distributed load is varied from 1.0 kPa to 30.0 kPa.Only    boundary condition is used in nonlinear analysis.Center-of-pane deflection and maximum principal stress of the directly loaded pane (pane 1) is recorded for each load.The results are presented in Fig. 15.
The benefit of considering the load sharing and the geometric nonlinearity is clearly visible.The IGU can withstand more load when the effects are included, compared to linear approach suggested by the classification societies.The maximum principal stress shifts from the middle of the pane to the corner in high pressure levels in nonlinear analysis (Fig. 16).This is due to the initially right-angled corner taking an acute angle after deformation, which induces large shear stresses.Comparison of center-of-pane deflection between ANSYS nonlinear results and experimental results from [16].Boundary conditions are defined in Fig. 6.
This can also be observed in Fig. 15 where the lines have discontinuities (red lines).
In those cases however, the shift occurs past the 40 MPa stress criterion due to the relatively low compliance of the panes (large '' 1 ''&'' 2 '' and small '''').Higher the compliance, earlier the shift.If IGUs are designed to meet the stress criterion using the presented method and boundary condition, the resulting thicknesses are thin enough to have the maximum principle stress at the corner.This can be observed in Fig. 17 where the thickness to width ratios are plotted for multiple aspect ratios, like earlier in Fig. 2. When the dashed line (linear with load sharing) and the solid line (nonlinear with load sharing) are separated, the maximum principle stress is at the corner.When they merge, it shifts back to the middle of the pane.Similarly for the dotted line (nonlinear without load sharing) and the circular marks (standard approach).The benefit of the geometric nonlinearity on the thickness determination varies.
Looking at the bottom row of Fig. 17, the maximum difference to standard results in nonlinear analyses peaks at some aspect ratio, depending on the applied load.Without load sharing, it is the highest around aspect ratios of 2.2, 1.2 and 1.0 for loads 2.5 kPa, 7.5 kPa and 15.0 kPa, respectively.With load sharing, the corresponding aspect ratios are 1.8, 1.8 and 1.2.Considering only the load sharing (linear), the difference is nearly constant for all aspect ratios.Overall, the difference varies between 0 % and 54 %.
As an example, using a square IGU with a side length of 1500 mm, the required minimum thickness can be calculated by extracting the    ∕ values from Fig. 17 for all the design methods and load cases.The results in Table 5 show how much thinner the glass panes in insulating glass units can be designed.The thickness can be reduced by 3.1 mm, 5.8 mm, and 6.7 mm for loads of 2.5 kPa, 7.5 kPa and 15.0 kPa, respectively, when both effects are considered.These correspond to weight reduction of 49.2 %, 46.8 % and 43.5 %, respectively, in glass.Furthermore, the maximum in-plane movement of the edge is 0.49 mm for the thinnest glass pane.Typically an overlap of at least 10 mm is required between the glass pane and the support.Hence, the observed movement does not lead the pane to slip out of the support under design loads, assuming that there is no excessive deformation of the support structure.

Discussion
Majority of the ship windows are insulating glass units (IGUs) consisting of two glass panes and a hermetically sealed cavity.Their combined area can reach to up to thousands of square meters in cruise ships.Furthermore, a large concentration of IGUs can often be found on the top of the ships.For instance, the SkyDome in M/S Iona is located on deck with a area of 970 m 2 (Fig. 1).Hence, the design of ship's glass panes is critical not only for the weight of the ship, but also for the stability.The classification rules (e.g.[11,12]) assume that the cavity pressure does not exist and linear plate theory is valid, which may lead to thicker glass pane designs than necessary.
Analytical methods, BAM [19] and MEPLA ISO [22], can be used to determine the glass pane thickness in IGUs on linear basis, including the load sharing effect.This already allows for considerable thickness reduction compared to the classification rule design approach (Table 5).The load sharing occurs as long as the cavity remains sealed and the panes are compliant, i.e. thin and large.While the analytical methods are fast to use, they are limited to certain glass shapes and boundary conditions, and they neglect the geometric nonlinearity.
Finite Element Method can be used to include the geometric nonlinearity.Its positive influence on the generated cavity pressure is presented in Fig. 14, and on the deflections and stresses of the panes in Fig. 15.The stress reduction potentially allows for even further reduction of the thicknesses (Table 5).However, the benefit of the geometric nonlinearity in addition to the load sharing is a function of the aspect ratio of the panes and the magnitude of the applied load, as presented in Fig. 17.The linear and nonlinear results are equivalent at larger aspect ratios.Most of the largest windows are, however, located on top decks, i.e. design pressure is at the lower end, and they have reasonable low aspect ratios.And with these two conditions, the considered effects are the most beneficial, assuming that the allowable stress is 40 MPa and the boundary conditions allow for in-plane sliding of the panes (Fig. 6).
This boundary conditions was used in the case study because good agreement was found between the presented FE model and the experimental results by McMahon et al. [16].The simply supported condition that did not allow for in-plane sliding appeared to underestimate the deflections heavily.Achieving either condition in real life is very difficult, if not impossible, with glass panes.Similarly, defining a real life condition exactly at some point between the limits is difficult.Hence, using unrestricted sliding condition is a better conservative approach.
Allowing the panes to slide shifted the position of the maximum principal stress from middle of the panes to the corners, like reported in [17].This indicates that when the panes are inspected for surface cracks/flaws, then the corners should also be of interest.The thinner the panes are and the larger the short side is, the smaller the required load is to shift the position of the maximum stress to the corner.On the other hand, increasing the aspect ratio increases the required load for the shift to occur.
Considering the results of this study and the fact that the IGUs have become a weight critical component in cruise ships, Finite Element Method is preferred over the alternative methods [17,18,22] for determining the thickness of the glass panes.

Conclusion
This paper studied the effect of the load sharing and geometric nonlinearity on the thickness determination of rectangular glass panes in insulating glass units used in cruise ships.For this purpose, a Finite Element Method was used.The gas in the cavity was modeled with hydrostatic fluid element (HSFLD242) and the glass panes with structural shell elements (SHELL181).Their interaction is embedded in the model and it follows the ideal gas law.The FE model was in good agreement with analytical methods and experimental results from scientific literature on linear and nonlinear basis, respectively.
From the study results it can be concluded that the current glass pane thickness determination rules by different societies for ship classification results in thicker constructions than necessary.This is because: (1) the equations for the glass pane thickness determination are based on linear-elastic, small deformation, plate theory; (2) the load sharing effect in insulating glass units is not considered.
Taking in account the interaction of the panes and the geometric nonlinearity, the stress in the glass panes reduces significantly.Because the classification rules prescribe allowable maximum principal stress, reduced stress allows for reduced thickness.The case study on typical ship windows showed that the thickness could potentially be reduced   The case study was performed for an rectangular IGU with certain dimensions and assumptions.For future work, the interest is in finding an optimum IGU design.For this, an optimization tool using ANSYS and MATLAB could be built under defined design space for rectangular, triangular and circular IGUs.For fully defining the design space, it is suggested to perform more experimental work on insulating glass units to study the influence of the boundary conditions, laminated glasses, Fig. 17.On top row is required minimum thickness to width ratio for multiple aspect ratios, a comparison between different design methods.On bottom row is the corresponding percentual difference to the standard and classification society method.and the edge seal system on the response.Furthermore, as the strength of glass has a relatively large variation, a study should be conducted to determine if using safety factor of 4 for the strength is feasible with large glass panes.

Fig. 1 .
Fig. 1.Examples of windows onboard the cruise ships: two first from the left-hand side are from Mein Schiff 2, courtesy of Meyer Turku Oy; third is SkyDome of M/S Iona, courtesy of Francis Design [10]; the last one is a cross-section sketch of an insulating glass unit.

Fig. 3 .
Fig. 3. On the left-hand side is comparison of nonlinear and linear predictions with experimental results for thermally toughened glass pane (1676.4× 1676.4 × 5.66 mm) [23].On the right-hand side is the role of von Kármán strains on the overall stress state through the thickness (bending + membrane = total).

Fig. 4 .
Fig. 4. Free body diagram of an IGU under external load including the load sharing.

Fig. 5 .
Fig. 5. Iterative process for solving the equilibrium state with gas filled cavity.

Fig. 6 .
Fig. 6.Illustration of the boundary conditions and the corresponding deformed shapes.The edge conditions are equal for pane 1 and pane 2 (AB1=BC1=CD1=DA1=AB2=BC2=CD2=DA2).Middle node of both panes is restrained from moving in the in-plane directions and rotating around -axis in    to have fully constrained system.

Fig. 7 .
Fig. 7. Experimental setup from [16]: (a) is the bracket for deflection measurement instruments (LVDT) and (b) shows the supporting mechanism of the studied double and triple IGUs (TIGUs are excluded in the current study).

Fig. 13 .
Fig. 13.Comparison of center-of-pane deflection between ANSYS nonlinear results and experimental results from[16].Boundary conditions are defined in Fig.6.

Fig. 16 .
Fig. 16.Maximum principal stress shift from middle of the pane to the corner for aspect ratios of 1 and 2 in the top and bottom row, respectively.The analyses corresponds to results in Fig. 15 (nonlinear with load sharing).
between 26-54 %, 26-35 % and 0-35 % by considering both of the effects, only the load sharing and only the geometric nonlinearity, respectively.

Table 2
[16]dimensions for validation of the FE model.The dimensions are same as used in[16].

Table 3
Comparison of generated cavity pressure on linear basis.

Table 4
Properties of the constituent materials.
a Self weight not considered in the analyses.