Glass Struct. Eng. (2017) 2:169–182DOI 10.1007/s40940-017-0044-7

S.I . GLASS PERFORMANCE PAPER

Next generation calculation method for structural siliconejoint dimensioning

Pierre Descamps · Valerie Hayez ·Mahmoud Chabih

Received: 25 April 2017 / Accepted: 29 May 2017 / Published online: 23 June 2017© Springer International Publishing AG Switzerland 2017

Abstract Bonding of glass onto aluminum frames,known as structural silicone glazing, has been appliedfor more than 50 years on facades. Traditionally, thesilicone bite is calculated using a simplified equationassuming a homogenous stress distribution along thesealant bite. Due to the complexity of façade designsthe assumptions behind simplified equations are reach-ing their limit of validity and requirements to use finiteelement analysis (FEA) increase since it allows todescribe the local stress distribution within sealant vol-ume. However, there is no standardized methodologyto run FEA for evaluation of SSG. Furthermore, thecomplexity of FEA is a limiting factor to its system-atic use as a calculation method for all projects. Forthese reasons, a next generation calculation methodwas developedwhich predicts deformation of SSGwithgood accuracy compared to FEA predictions. The basisof the method was developed 25 years ago and wasincluded as annex in ETAG002. The validation of themethodwas done by comparing experimentalmeasure-ments, results of FEA modeling and outcome of thenew calculation method. To further improve accuracy,an extension of the relationship for a nonlinear mate-rial is proposed, assuming aNeo-Hookean stress–strainbehavior for silicone sealant.

P. Descamps · V. Hayez (B) · M. ChabihDow Corning Europe SA, Parc Industriel Zone C,Rue Jules Bordet, 7180 Seneffe, Belgiume-mail: valerie.hayez@dowcorning.com

Keywords Finite Element Analysis · Calculationmethod · Structural silicone · ETAG 002 · Bonding

1 Introduction

Bonding of glass onto aluminum frames, known asStructural SiliconeGlazing (SSG), has been applied formore than 50 years on facades with various improve-ments of the technology being made over time. Sil-icone sealants are used in this application because oftheir unique resistance toweathering (UV, temperature,moisture, ozone). They also provide resistance to wateringress and thermal insulation (Klosowski and Wolf2015). Their structural role is to sustain wind loads andto accommodate for differential thermal expansion ofdifferent bounded substrates.

A considerable amount of effort wasmade since firsthalf of the 20th century to understand the behavior ofa joint submitted to a deformation. For example, Volk-ersen (1938), proposed amodel to simulate joint behav-ior in lapshear configuration, neglecting the bendingeffect in case of eccentric load. Starting from Volk-ersen’s approach, Goland and Reissner (1944) intro-duced this bending effect. More recent papers, build-ing on the use of numerical tools, discuss joint failurecriteria like Callewaert et al. (2011).

Historically, silicone joint dimensioning is calcu-lated with a simplified equation implemented in var-ious standards for structural glazing (ASTM 2014;EOTA 2012; GB 2005). This equation assumes homo-geneous stress distribution along the sealant bite whilst

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Fig. 1 Example of local stress distribution in a silicone jointsubmitted to a combined traction–rotation

high local stress peaks, structure deformation or mate-rial ageing are included in a global safety factor. Newtrends in commercial buildings include the use of largedimensions glass panes, higher complexity of façadedesigns and stronger engineering performance require-ments such as high windloads above 5000 Pa (Hayez2016; Maniatis and Siebert 2016). These trends haverecently challenged the conventional methods of jointdimensioning, since using the simplified equation forthese projects results in economically unacceptablelarge bite sizes. Furthermore, increasing joint bite willnot necessarily increase the safety factor as the sim-plified relationship neglects important factors such asthe joint rotation due to glass pane bending. Increasingthe sealant design stress is an option to decrease thebite but this solution is limited and also requires a bet-ter understanding of stress distribution as well as jointfailuremechanisms aswas explained inDescamps et al.(2016a, b).

This explains the recent increased interest to useFinite Element Analysis (FEA) to help designing SSGand joint dimensions. In FEA, the geometry is dividedin small volume elements interconnected by points callnodes. Applying energy conservation to the whole sys-tem, via strain energy calculation at small element level,local stress and/or local deformation can be predicted(Fig. 1).

However, there is no technical guideline or stan-dardized method explaining how to use FEA in struc-tural joint dimensioning. Without such guidance, cal-culations carried-out by different engineering officesmay lead to different absolute values of the maximum

local stress. The outcome of FEA model is highlysensitive to the accuracy of input data such as theparameters of the hyperelastic model selected for thesealant. The stress volume distribution is also highlymesh dependent, especially close to the interfacialregion between the sealant and the substrate as demon-strated in Descamps et al. (2016a) and recalled laterin this paper. This is more particularly true becauseeven being easily deformable, silicone sealant is anearly incompressible material (Wolf and Descamps2003).

Finally, even if themaximum local stresses or strainsin joint volumes are calculated in an accurate way, wedo not knowwhat is the acceptable value a joint can sus-tain while ensuring long term durability of façade sys-tems. In fact, there is no unanimous approach on how todefine a “rupture” criteria from local stress and to deter-mine what the best model to predict material failure is.Several criteria like principal stress, Von Mises stressor maximum deformation energy are possible. Henceit is difficult to use a local stress distribution for pre-dicting failure in a macroscopic joint and consequentlyuse this information for joint dimensioning (Descampset al. 2016a, b).

An alternative approach is to use FEA results tosimulate observable (or engineering) joint deformationbecause this variable has a lower sensitivity to meshconfiguration. Indeed, observable deformation resultsfrom the integral of the strain energy over the wholejoint volume hence local high stress values which arehighly mesh sensitive are averaged. Joint deformationcalculated using FEA for one particular façade can becompared to H-bar testing results for test pieces havingthe same geometry and more particularly similar jointaspect ratio R (defined as the ratio between joint biteW and joint thickness e).

While calculating engineering joint deformationwith FEA creates a more direct link with sealant per-formances measured on test pieces, carrying out aFEA model remains an expensive procedure, requiringinvestment in FEA software acquisition and engineer-ing resources to run simulations. Hence this methodol-ogy is difficult to extend to small/medium size façademakers who would prefer using a simple “manual” cal-culation method.

The goal of this paper is not to provide a directcontribution to the effort of joint behavior understand-ing, but to propose an improved mathematical relation-ship making a direct correspondence between a joint

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Next generation calculation method for structural silicone 171

included in a façade system and the behavior of a testpiece.

A history of the mathematical relations of jointdimensioning is presented, explaining their limit ofvalidity and why it is important to move to a newrelationship including additional physics effects likejoint rotation which were neglected previously (ASTM2014;EOTA2012;GB2005) andwhich representmoreaccurately the joint behavior. Very rough assumptionshave beenmade for its derivation to keep it simple. Val-idation of the proposed relationship for large windloadis carried out by confronting predictions with physi-cal measurements and the results from FEA modeling.The improved relationship was deduced assuming firsta linear material. This assumption is relatively accurateas for small joint elongation ε = Δe

e below 10%, thestress/strain curve deviates very little from linearity. Tooptimize the correlation between FEA and the equationfor larger elongations, an extension of the improvedlinear model to accommodate non-linear behavior isproposed, assuming a Neo-Hookean model.

All the studies described in this paper were carriedout using properties and experimental characterizationofDowCorning®993StructuralGlazingSealant (DowCorning 2017), which is a two-component neutralalkoxy curing silicone formulation specifically devel-oped for the structural bonding of glass,metal and otherbuilding components.

2 Identification of the hyperelastic model

The FEA Multiphysics software package COMSOL®(Comsol 2017)was used to conduct finite elementmod-elingof structural silicone andvalidate joint dimension-ing relationships. COMSOL® has a solid mechanicspackage giving access to a wide range of hyperelasticmaterial models like Neo-Hookean, Mooney–Rivlin,Yeoh and many others.

Assuming that silicone material is incompressiblefor the small movements observed in construction,we obtain a relationship between the macroscopicstress/strain curves measured experimentally and thestretch λ(λ = 1 + ε) for both uniaxial, bi-axial andpure shear testing. A routine was built in MATLAB®(Matlab 2017) to fit different hyperelastic models to theexperimental data measured on purely uni-axial and bi-axial test piece. The χ2 value (the sum of the squareof the residual between experimental values and model

Fig. 2 Representation of the test bi-axial test piece configurationused characterize the Dow Corning® 993 behavior in compres-sion

predictions) was calculated combining the data mea-sured on the different types of test pieces, using hyper-elastic model parameters as curve fitting parameters. Aweight functionwas used to prevent having the fit beingdominated by high elongation values and guarantee thatthe model is representative in a wide elongation range.

2.1 Determination of material properties

Toobtain an accurate simulation of joint behavior understructural load, accurate stress–strain behavior of sil-icone material is essential. Physical properties weremeasured via a specialized laboratory protocol. Theservices of Axel Products (Axel 2017) were used todevelop accurate uni-axial, equal bi-axial and planar(pure shear) extension for theDowCorning®993mate-rial.

Sheets of thickness varying between 1 and 2.6 mmwere cured for a period of 4 weeks at room temperature(∼20◦C) and 75% humidity. Out of those sheets, testpieces for uni-axial testing [dog-bone—ASTM D412DieD (ASTM2016)], with an effective gauge length of50 mmwere cut using a die cutting machine. Similarly,bi-axial testing was carried-out. For incompressible ornearly incompressible materials, equal bi-axial exten-sion of a specimen creates a state of strain equivalentto pure compression. Although the experiment is morecomplex than a simple compression experiment, a purestate of strain can be achieved leading to more accuratematerial model identification. The equal bi-axial strain

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Table 1 Dimensions of the test samples

Testing piece Width (mm) Thickness (mm) Area (mm2)

Dog-bone 3.19 1.60 5.10

Planar tensile 140 2.60 364

Bi-axial – 1.00 314

state may be achieved by radial stretching of a circulardisc (Fig. 2) of 75 mm diameter and an effective areaof 50 mm in diameter.

Finally, planar testing was performed on rectangularpieces of 150 mm wide and 15 mm tall, the nature ofthe test requiring a width at least 10 times larger thangauge length.

Characteristic dimensions of the different test piecesare summarized in Table 1. Three test pieces were pre-pared for each geometry and pulled at a rate of 0.01mm/s.

Results measured in uni-axial extension, equal bi-axial extension and planar tension for Dow Corning®993 are presented in Fig. 3.

2.2 Identification of the model parameters

The tension data, both uni-axial and bi-axial, werecurve-fitted with several material models in order tofind a curve fit minimizing the scaled residuals χ2

Fig. 4 Fitting of the tension measurements (uni-axial and bi-axial) with a Mooney Rivlin model with 5 parameters identifica-tion

resulting from all data provided. Models of differentorders were tested. As an illustration a Mooney–Rivlin(MR) model with 5 parameters is used (Fig. 4) with thefollowing expression for the total strain energy densityWs in the case of incompressible material like silicone:

Ws = C10. (I1 − 3) + C01. (I2 − 3) + C20. (I1 − 3)2

+C11. (I1 − 3) . (I2 − 3) + C02. (I2 − 3)2

(1)

Fig. 3 Test results inuni-axial, bi-axial andplanar tension for DowCorning® 993 material

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Next generation calculation method for structural silicone 173

The curve fitting exercise led to the following values ofmodel coefficients:

C10 = 3.6261E05, C01 = −0.0009E05,

C02=0.0042E05,C20=−0.0096E05, C11=0.007E05

Best practice consists in selecting the lowest ordermodel enabling to predict experimental behaviorwithinthe error bar associated to the measurement. Fur-thermore increasing model complexity is never suit-able without first eliminating the source of data vari-ations, due for example to the variability of the sam-ple’s preparation or the error associated to the testingdevice.

Since there are at least 3 orders of magnitude differ-ence between C10 and the higher order coefficients,we neglect all coefficients except C10 = 1

2G. TheMR models becomes equivalent to the simpler Neo-Hookean model whereby the strain energy density Ws

becomes

Ws = 1

2G. (I1 − 3) (2)

With the shear modulus G = E2(1+ν)

= 7.2 E5 Pa, ν =Poisson ratio ∼=0.5 for nearly incompressible material.

The top three 1st order hyperelastic models usedfor the simulation of Dow Corning® 993 with theirrespective constants are provided in Table 2.

Those threemodels provideχ2 values very similar tothe ones obtained with MR model with 5 parameters.For all the studies presented in this paper, the Neo-Hookeanmodel has been selected for its simplicity.Wewill also note that the modulus is given by the slopebetween the engineering stress and strain curves. Inthe case of uni-axial tension the slope of the curve atzero elongation is called the Young modulus Eyoung =2.3 MPa, which corresponds to the G value listed inTable 2, assuming a Poisson ratio of 0.49.

2.3 Validation of the model

To validate themodel identified in paragraph 2.2, H-barpieces with different joint dimensions were built andcorresponding engineering stress–strain curves mea-sured. The test pieces were prepared using anodizedaluminum substrates and Dow Corning® 993 sealantwas cured in standard conditions (20 ◦C and 75% HR)for a period of 21 days. Test pieces were tested at a loadrate of 50 mm/min. Three joint geometries were tested,with test piece dimensions summarized in Table 3.

Table 2 Material parameters coming from curve fitting of DowCorning® 993 data

Hyperelastic model Material parameter Value

Ogden 1st order Material constant MU1 7.1938E+05 Pa

Material constant A1 1.9225 (unitless)

Neo-Hookean Material constant C10 3.41375E+05 Pa

Yeoh 1st order Material constant C10 3.4138E+05 Pa

Table 3 List of H-bar geometries used to validate the Neo-Hookean material model

Samplereference

Length(mm)

Width(mm) W

Thickness(mm) e

1 50 12 12

2 50 36 12

3 50 36 6

The H-bar configurations are modelled, selectingthe Neo-Hookean model out of the summary Table 2.This choice is justified because offering χ2 valuessimilar to those obtained with MR but with only onecurve fitting parameter, as Poisson ratio was fixed at avalue of 0.49, reflecting the incompressible characterof silicone rubber material. The different joint config-urations are modelled using a rectangular mesh, thesize of one mesh element is kept equal to 1 × 1 ×1mm3 for the joint configurations of Table 3. Thismesh scheme is used for all results reported in thispaper.

Figure 5 shows a good correlation between experi-mental data and corresponding modelled curve. Devi-ation is mainly due to the manufacturing quality of theH-bars and their measurement.

Although the curves plotted in Fig. 5 were obtainedon test pieces made of the same material, we observea clear difference between the test specimens due totheir geometry. The engineering stress–strain curvescan be predicted, by substituting the Young Modulusby a rigidity modulus that depends only on geome-try and more specifically on the joint aspect ratio. Forexample, we observe for the joint with cross-section6 × 36mm2, a much more rapid stress increase for asame joint elongation, than is the case for the other jointdimensions and especially the 12 × 12mm2.

A structural joint consists of a silicone joint bondedbetween two substrates. If we consider a very thick

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Fig. 5 Stress–strain curvesobtained for different jointgeometries: validation ofmodel through experimentaltensile data

0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40

)aPM(

dedrocerssertS

Elonga�on (%)

12x12x50mm3model

12x12x50mm3experimental

12x36x50mm3experimental

12x36x50mm3model

6x36x50mm3experimental

6x36x50mm3model

H-bar compared to its cross-section (distance betweensubstrate >>> H-bar x-section), when moving awayfrom the plane of adhesion, the behavior of the testpiece tends toward the uni-axial tension case, wherebythe joint cross-section decreases proportional to thePoisson ratio. The ratio between the engineering stressand strain provides the modulus which is for this typeof H-bar configuration minimum, having a value closeto the material Young’s modulus.

On the other hand, in a thin layer close to the inter-face between sealant and substrate (adhesionplane), thejoint cross-section cannot freely decrease to conservethe volume of nearly incompressible silicone material.The joint is submitted to uni-axial tension but simul-taneously a force is being applied on the orthogonalsurfaces which restrains the joint from deforming sono free cross-section reduction can take place. Conse-quently, a larger engineering force must be applied toobtain the same engineering strain. This is observedon H-bars having a small thickness compared to theircross-section. The modulus calculated from the engi-neering stress–strain curve is in this type of geometrysignificantly larger than the material’s Young modu-lus. When the thickness is minimal, the behavior of theH-bar reaches the theoretical extreme case commentedin Feynman et al. (1963). A small cube of linear elas-tic material is considered, with a pulling force appliedto the top and bottom faces of the cube. Additionalforces are applied on lateral faces of the cube to preventany change in the cube cross-section. For this extremecase, a very simple relationship for the modulus can beobtained:

E = EYoung(1 − ν)

(1 + ν) (1 − 2ν)(3)

For a 100% incompressible material (ν = 0.5), as nocross-section reduction is allowed, the volume conser-vation requires an infinite force to create an extension.For a nearly incompressible material such as silicone,whereby ν = 0.49, the modulus becomes very large,approximately 17 times the Youngmodulus. This valuecan be seen as an upper limit of joint modulus, whenhaving a very thin test piece compared to its cross-section.

Hence, H-bars of different aspect ratio R, will havea modulus which varies between two extreme val-ues; EYoung for thick H-bars with small cross-sectionand 17 × Eyoung for very thin H-bars compared tocross-section. For all intermediate H-bar configura-tions, modulus values will be comprised between thoseextremes and can be obtained through FEA modeling,solving the energy conservation in the whole volume.To differentiate with the Young modulus of the mate-rial, which is not influenced by its geometry, the mod-ulus of an H-bar is called the rigidity modulus Erigidity.The relationship between both modulus is called therigidity factor

frigidi t y = Erigidi t y

EYoung(4)

2D FEA is used to model the engineering stress–straincurve for different H-bar configurations and determinetheir corresponding rigidity modulus (Fig. 6). The 2Dmodel is justified since the SSG joint dimension alongthe length of the profile is much larger than both jointbite and/or thickness. The model developed in para-

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Next generation calculation method for structural silicone 175

Fig. 6 Rigidity factor as afunction of joint aspect ratiofor sealants obeying to aNeo-Hookean model

graph 2.2 is applied. For large joint thickness ver-sus bite, as expected, rigidity factor tends to 1, i.e.Erigidi ty = EYoung . The theoretical value calculatedusing the equation proposed by Feynman et al. (1963)for aspect ratio R → ∞ and ν = 0.49 is also reported.

The relationship between the rigidity factor frigidityand joint geometry (aspect ratio R) can be fitted by asecond order polynomial:

frigidi t y = 0.1506 R2 + 0.3409 R + 1.0852 (5)

3 Development of the next generation relationshipfor joint dimensioning

Before introducing a new equation for joint dimension-ing, we will briefly recall the different assumptions andequations used in the past for joint dimensioning, start-ing with the simplest one and explaining the assump-tions behind successive improvements.

3.1 Homogeneous stress distribution along both jointbite and frame length

The simplest joint dimensioning assumes that the glasspane is infinitely rigid which means no deformation ofthe glass occurs due to deflection and the (soft) sealantis not generating any local glass deformation at the edgeof the glass. Infinite glass pane rigidity implies a fullyhomogenous stress distribution, both along the sealantbite (w) and along the profile of the frame.

Assuming homogenous stress distribution, a simplebalance of forces can be donewhereby the force exertedby the windload (Pwind) on the glass surface (Sglass =a∗b) should be equal to the reaction force associated tojoint deformation for a sealant with design strength σdes(obtained as the Ru,5/6 value of the maximum tensilestrength at break) and surface Sjoint (w*perimeter).

Pwind ∗ Sglass = S joint ∗ σdes (6)

Pwind ∗ a ∗ b = w ∗ 2 (a + b) ∗ σdes (7)

Therefore the joint bite w becomes

w = Pwind ∗ (a ∗ b)

2 (a + b) ∗ σ des(8)

3.2 Heterogeneous stress distribution along the framelength

In a second step, we do not consider a fully rigid glasspane but assume that it will deform. Glass pane defor-mation is introduced in the model as shown in Fig. 7representing a glass pane: it is assumed that the windacting on the red rectangle area (L*a/2) is fully sus-tained by the joint of length L and of bite W along theexterior side of this rectangle. The idea behind is thatthe stress is larger at this location because due to theflexibility of the glass pane, the joint along the smallside of the glass pane does not contribute to decreasethe stress in the joint at the center of the longer side.The deflection is small and we assume that glass defor-mation only influences the heterogeneous stress distri-bution along the profile length but not along the sealant

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Fig. 7 Trapezoidal deformation of the glass pane. The hetero-geneous stress distribution along the frame results in a heteroge-neous stress in the joint length

bite where stress is assumed homogeneous. Lookingalong profile direction, themaximum stress is observedin the middle of the frame (a/2).

Applying the equilibrium of forces, the force actingon the red rectangle of area (L*a/2 m2) is sustained bythe joint of length L and of bite W:

Pwind ∗ a

2∗ L = w ∗ L ∗ σdes (9)

Rearranging Eq. 9 to calculate the sealant bite value fora defined design stress, we obtain the well-known jointdimensioning relationship used in most SSG projects(ASTM 2014; EOTA 2012; GB 2005):

w = 0.5 ∗ a ∗ Pwind

σdes(10)

Equivalently, for a defined value of joint bite, we cancalculate the corresponding homogeneous stress in thejoint:

σ = 0.5 ∗ a ∗ Pwind

w(11)

This basic Eq. 10 is widely used by the industry tocalculate joint dimensions while having several weak-nesses:

– It does not include joint rotation associated to glassdeformation

– It does not include the glass pane properties, whilewe must know glass deformation, and more partic-ularly, local rotation angle at level of the joint.

– It does not include joint thickness, while we knowthe joint geometry influences the joint deformationas the rigidity factor depends on joint aspect ratio.

– It does not include the sealant modulus (as onlyconsidering force balance), while glass deforma-tion imposes a movement that has to be accommo-dated by the joint.

These aspects will be addressed in the next para-graphs.

3.3 Heterogeneous stress distribution along sealantbite and thickness

Under high glass deflection, the assumption of homo-geneous stress distribution along sealant dimension isnot valid anymore and we must take into account thedeformation imposed on the joint by the rotation of theglass (by an angle α on Fig. 8). The joint deformationincreases when moving along the x-axis. The displace-ment associated with glass pane rotation (�er) and thehomogeneous deformation �e are indicated on Fig. 8.Themaximum joint displacement�emax is equal to thesum of �e and �er.

�emax = �e + �er (12)

To obtain a joint dimensioning relationship thatincorporates glass rotation effect, we make the follow-ing assumptions:

• Glass is much more rigid than silicone and siliconedoes not influence joint deformation but followsthe deformation imposed by the glass pane defor-mation.

• Joint dimensioning obtained from calculation pre-dicts a joint deformation small enough so that wecan assume that the sealant behaves as a linear elas-tic material.

• Even if not fully accurate, the elongation at breakmeasured on an H-bar of same geometry (H-barnot being tilted) is representative for the maximumdeformation of façade joint

• Each elementary joint element of length dx alongx axis behaves like a linear material having a samevalue of “engineering modulus” associated to jointgeometry (Erigidity). Even if rough, this assump-tion allows retrieving the result obtained for anH-bar (engineering stress/strain dependence) whenassuming the limit case where no rotation takesplace (α → 0).

The glass plate deformation and the rotation angle dueto the windload are calculated assuming a simply sup-ported boundary assumption. The assumptionsmade in

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Next generation calculation method for structural silicone 177

Fig. 8 Rotation of the jointdue to the deflection of theglass

paragraph 3.2 are still valid for force balance calcula-tion i.e. mass balance is carried out on the whole jointsurface represented on Fig. 8.

By simple trigonometry, joint displacement associ-ated to glass pane rotation is calculated:

�er = W ∗ tg (α) (13)

Performing the balance of forces on half of glass paneusing symmetry, we obtain �e:

∫ w

0

Erigidi ty (�e + x tg (α))

edx = P wind ∗ a

2(14)

Erigidi ty�ew

e+ Erigidi ty w2tg (α)

2e= P wind ∗ a

2(15)

�e

e= P wind ∗ a

2 − Erigidi t y w2tg(α)

2e

Erigidi t y w(16)

Combining Eqs. 13 and 16 we obtain the maximumjoint elongation �emax/e:

�emax

e= P wind ∗ a

2 − Erigidi t yw2tg(α)

2e

Erigidi t yw+ W ∗ tgα

e(17)

This equation can be further developed to calculate themaximum value of engineering stress σmax sustainedby the joint:

σmax = Erigidi t y ∗ �emax

e= P wind ∗ a

2w

− Erigidi t ywtg (α)

2e+ Erigidi ty ∗ W ∗ tgα

e(18)

σmax = P wind ∗ a

2w+ Erigidi ty ∗ W ∗ tgα

2e(19)

σmax = P wind ∗ a

2w+ frigidi t y ∗ EyoungW ∗ tgα

2e(20)

In Eq. 20, we observe that the maximum stress isthe sum of two terms having opposite dependence ofsealant biteW and hence aminimum value can be iden-tified for σmax. The first term decreases when the jointbite increases becausewind load is sustained by a largergluing area. The second term increases with bite. Itcorresponds to the joint deformation induced by glassdeflection. It is important to work with a sealant able toaccommodate the imposed deformation like a weath-erseal joint since stiff material could lead to very largeinternal stress build-up and potential failure (Descampset al. 2016a). The influence of the joint geometry isaccounted for by the rigidity factor.

To validate this new equation (Eq. 17) including theeffect of glass bending, we compare its predictions to2D FEA calculations (Fig. 9). The modelling param-eters are summarized in Table 4. The value of glassthickness has been adjusted to have a maximum glassdeflection at the center of glass pane equal to 1%.

As discussed a saddle point is observed on Fig. 9with minimum joint displacement. Although many

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Fig. 9 Comparison of jointdeformation along Ydirection calculated from anFEA simulation and usingEq. 17 that takes intoaccount a heterogeneousjoint deformation alongjoint imposed by glass panebending

Table 4 Parameters for joint maximum elongation using FEAsimulation

Parameter Value

Glass pane dimensions(distance between joints) (m)

1.9

Windload (Pa) 4913

Joint thickness (mm) 9

Glass thickness (mm) 19

Max deflection at center 1%

simplifying assumptions were made to derive Eq. 17,this relationship predicts joint deformation values wellcompared to the results coming out FEA simulation.Some discrepancy is observed in the small and largejoint regions.

The difference observed for small sealant bitescomes from the fact thatwe assume a linear relationshipbetween the stress and the deformation when calculat-ing the force balance.However, if small sealant bites areused when having a large windload, the stress imposedon the joint is large, leading to important deformationsand potentially moving out of the linear domain of thestress–strain curve.

The difference observed for large values of sealantbite is due to the assumption of simply supported glasspanes, corresponding to a glass plate deflection of1% of the smaller glass side. However, when the biteincreases, its contribution in shear along x axis on Fig. 8

contributes to limit glass deflection to a value below1%. For this reason, deformations calculated for largebites using the new model are always larger than FEAprediction.

In the next steps, we introduce both effects in themodel to verify if further improvements of the corre-spondence between the new equation and FEA predic-tions can be obtained:

– Consider that the joint has a non-linear behavior,following a Neo-Hookean model; this will impactresults in the small bite region.

– Consider that sealant joint working in shear con-tributes to limit glass deflection; this will have animpact on the result corresponding to large sealantbites.

3.4 Hyperelastic material model: relationshipextended for larger joint deformations

Equation 20 has been derived assuming that the jointdeformation never exceeds the non-linearity thresholdi.e. that the stress stays proportional to the strain. How-ever, it is possible for certain joint configurations (fora small joint bite, where the windload is sustained bya smaller gluing area, and elongation is larger) that anon-linear behavior occurs.

We have shown in paragraph 2.2 that Dow Corn-ing® 993 material can be described by a simple Neo-

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Next generation calculation method for structural silicone 179

Hookean model which we will use to introduce non-linear behavior when deriving joint dimensioning rela-tionship.

We have shown that for a Neo-Hookean model, thetotal strain energy density is:

Ws = C10 (I1 − 3) (21)

where I1 is only dependent on the stretch λ in eachdirection. Assuming a fully incompressible siliconematerial and a pure uniaxial traction, we can calculatethe engineering stress σ as a function of λ

σ = 2C10

(λ − λ−2

)(22)

σ = G(λ − λ−2

)(23)

As for the linear material, if we compare the engi-neering stress/strain curve measured for a uni-axial testpiece (dogbone test piece) and H-bar, the curves differ,having H-bar curves appearing stiffer. We will assumeagain that the behavior of H-bars can be derived fromuni-axial engineering curve, to which we add a rigid-ity factor dependent on joint geometry. This is simplydone multiplying σ in Eq. 23 by a rigidity factor whichis equivalent to multiplying C10 by a rigidity factor orreplacing theYoung’smodulus in the calculation ofC10

by a rigidity modulus (Eq. 4).We can easily verify that for small elongations, we

retrieve a linear behavior.Replacing λ by (1 + ε) in Eq. 23, we have:

σ = G

(1 + ε − 1

(1 + ε)2

)(24)

Replacing the last term in the equation by its binomialseries, only keeping the two first terms as ε is small:

1

(1 + ε)2= 1 − 2ε (25)

σ = G (1 + ε − 1 + 2ε) = 3εG (26)

As for fully uncompressible material G = EYoung3 ,

we retrieve the engineering stress corresponding to auni-axial test piece:

σ = EYoung ε (27)

If theYoung’smodulus is replaced by a rigiditymod-ulus to consider the joint geometric effect associated toH-bar test piece, we obtain the relationship allowing tocalculate the engineering stress of an H-bar.

It is important tomention that the list of assumptionsdetailed in previous paragraph remains valid for belowcalculations.

Doing the force balance, effect of wind load equalto joint reaction, we obtain:

Pwinda

2=

∫ W

0σ (x) dx (28)

Pwinda

2= 2C10{

∫ W

0λ (x) dx−

∫ W

0λ (x)−2 dx} (29)

With λ = 1 + ε and from Fig. 8, ε = �e+ xtg(α)e

Integrating above equation, we obtain the followingimplicit equation

Pwinda

2= 2C10{

(1 + �e

e

)W + tg (α)

2eW 2

+ 1tg(α)e

(1

1 + �ee + tg(α)

e W− 1

1 + �ee

)(30)

This equation can be solved graphically with theunknown �e by plotting the right term of the equal-ity as a function of displacement �e; knowing �e, themaximum joint displacement is calculated fromEqs. 12and 13 as �emax = �e + x tg (α)

The new relationship including non-linear behavior(Eq. 30) is evaluated and compared with FEA results(Fig. 10).

In comparison with the linear assumption (Eq. 17),Eq. 30 predicts better the stress for a small joint bite,which confirms that non-linear joint behavior was thereason of the difference between FEA and resultsobtained using the new relationship.

Because calculations are carried-out for one samelarge wind load value (∼5000Pa), small bites are sub-jected to larger elongations, which could be at the limitof validity of linear assumption. Assuming a non-linearbehavior is however a theoretical exercise to demon-strate the reason of the difference between the derivedrelationship and FEA prediction. Indeed, sealant bitecalculations corresponding to the pressure used for thisexample will always require larger bites and hencenever allow those relative large deformations. Sincenon-linearity will in real applications seldom occur,the more simple equations (Eqs. 17, 20) describing lin-ear behavior should be preferred for practical applica-tions.

3.5 Impact of joint on glass deflection

For large joint bites, the structural joint working inshear limits the glass movement along the x direc-tion and consequently reduces glass deflection which

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180 P. Descamps et al.

Fig. 10 Comparisonbetween joint elongationvalues obtained throughFEA simulation, therelationship including jointlocal joint deformationassociated to glass bendingfor linear (Eq. 17) andnon-linear material behavior(Eq. 30)

Fig. 11 Glass panedeflection calculated usingsimply supportedassumption and assumingthe glass pane beingmaintained by joints ofdifferent bite values. Thered line corresponds thesimply supported glasspane. When the joint biteincreases, the jointcontributes more and morein limiting the glass panedeflection

reduces glass rotation angle at its extremity. As thisangle of tilt α leads to an increase of maximum jointdeformation for large bite values (saddle point observedfor a bite of ∼27 mm on Fig. 10), it is believed thatnot considering the impact of the sealant in limitingthe glass deflection when we used the formula (a sametilt angle value is used for all bites, calculated assum-ing simply supported boundary condition) explains thedifference between the prediction of the formula andFEA for bites larger than 27 mm.

To demonstrate this, we plot the maximum glasspane deflection for different values of sealant bite(Fig. 11). We also plot the value of the deflection cal-culated assuming a simply supported glass pane. Weobserve that for small bite values, the maximum glassdeflection is very close to the value calculated usingthe simply supported glass pane, meaning that the jointdoes not contribute in reducing the glass pane deflec-tion. On the contrary, for large bites, particularly above

123

Next generation calculation method for structural silicone 181

Fig. 12 Joint maximumelongation in function ofjoint bite: FEA prediction;equation assuming glassdeformation not influencedby joint bite and calculatedassuming a simplysupported glass pane;equation assuming glassdeformation is influencedby joint rigidity

25 mm, joint rigidity limits glass deflection in a non-negligible way.

We now use the formula in Eq. 30 but associatingdifferent values for the tilt angle α depending on thejoint bite (the angle was calculated from FEA havingjoint contributing in limiting glass deflection—Fig. 11).This results in a good agreement with FEA predictionsas shown on Fig. 12.

While we show why a difference exists betweenthe predictions of the new equations and FEA sim-ulation, calculating glass pane deformation assumingsimply supported boundary conditions still makes a lotof sense as neglecting the effect illustrated in this para-graph leads to an error lower than 3% for bites of 30mm and below 9% for bites of 40 mm.

4 Conclusions

As alternative to FEA, a next generation equation forjoint dimensioning is proposed giving results close tothe predictions obtained using FEA simulation. Theapproach followed is to calculate for the façade jointthe engineering strain (or equivalently, the engineer-ing stress) and compare it with the stress–strain behav-ior measured on an H-bar sample having the sameaspect ratio. The difference between FEA predictionand the new derived relationship observed for smallbites comes from the hypothesis of material linearbehavior. A combination of small bites and high wind-

loads will move the behavior out of linearity. How-ever, this is a theoretical case since largewindloads willalways lead to large bites. Hence using linear assump-tion and the corresponding equation for real facadeprojects is a better option. The difference between FEAprediction and the new derived relationship observedfor very large bite comes from the assumption thatthe joint does not influence the glass pane deflection.If the joint cannot influence a local bending of theglass near the glass perimeter (because of the differ-ence inmaterial rigidity between sealant and glass), thejoint contributes to limit the glass deflection by limit-ing the translation of the glass extremities along the xaxis. Adding this effect into the equation, an accept-able match between FEA and the equation is observedfor large bites values. However, calculating glass panedeformation assuming simply supported boundary con-ditions is acceptable as errors are limited to a few %for large bites.

Compliance with ethical standards

Conflict of interest On behalf of all authors, the correspondingauthor states that there is no conflict of interest

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