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Glass Structures & Engineering
, Volume 5 (3) – Nov 4, 2020

/lp/springer-journals/dimensioning-of-silicone-adhesive-joints-eurocode-compliant-mesh-wHmspI9s0A

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- Springer Journals
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- Copyright © The Author(s) 2020
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- 2363-5142
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- 2363-5150
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- 10.1007/s40940-020-00128-4
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Glass Struct. Eng. (2020) 5:349–369 https://doi.org/10.1007/s40940-020-00128-4 SPECIAL ISSUE CHALLENGING GLASS Dimensioning of silicone adhesive joints: Eurocode- compliant, mesh-independent approach using the FEM Micheal Drass · Michael A. Kraus Received: 1 February 2020 / Accepted: 10 July 2020 / Published online: 4 August 2020 © The Author(s) 2020 Abstract This paper deals with the application of material safety factor for that speciﬁc limit state func- the semi-probabilistic design concept (level I, DIN EN tion. This safety factor is then extended to the appli- 1990) to structural silicone adhesives in order to cali- cation in ﬁnite element calculation programs in such brate partial material safety factors for a stretch-based a way that it is possible for the ﬁrst time to perform limit state equation. Based on the current legal situation mesh-independent static calculations of silicone adhe- for the application of structural sealants in façades, a sive joints. This procedure thus allows for great opti- new Eurocode-compliant design concept is introduced mization of structural sealant design with potentially and compared to existing design codes (ETAG 002). high economical as well as sustainability beneﬁts. An This is followed by some background information on example for the static veriﬁcation of a bonded façade semi-probabilistic reliability modeling and the gen- construction by means of ﬁnite element calculation eral framework of the Eurocode for the derivation of shows (i) the application of EC 0 to silicone adhesives partial material safety factors at Level I. Within this and (ii) the transfer of the EC 0 method to the ﬁnite paper, a speciﬁc partial material safety factor is derived element method with the result that mesh-independent for DOWSIL 993 silicone on the basis of experimen- ultimate loads can be determined. tal data. The data were then further evaluated under a stretch-based limit state function to obtain a partial Keywords Partial material safety factor · Structural silicone adhesive · SSG façades · Design and computation M. Drass ( ) · M. A. Kraus M&M Network-Ing UG (haftungsbeschränkt), Lennebergstraße 40, 55124 Mainz, Germany e-mail: drass@mm-networking.com; 1 Introduction and current situation drass@ismd.tu-darmstadt.de https://www.mm-network-ing.de State-of-the-art glass façades are designed with a strong M. Drass emphasis on a transparent appearance with minimal Institute for Structural Mechanics and Design, Technische Universität Darmstadt TU Darmstadt, Franziska-Braun-Str. visibility of the supporting structures. During the last 3, 64287 Darmstadt, Germany ﬁfty years a lot of experience with structural silicone M. A. Kraus adhesive joints in façade design has been gained world- Civil and Environmental Engineering, Stanford University, wide. Beginning with linear adhesive joints, which are Y2E2, 473 Via Ortega, Stanford, CA 94305, USA used along a window system for homogeneous load e-mail: makraus@stanford.edu; transfer (Staudt et al. 2018), up to local ﬁxings, where kraus@mm-networking.com https://www.mm-network-ing.de glass panes are only bonded locally with so-called point 123 350 M. Drass, M. A. Kraus ﬁxings (Drass et al. 2019b; Santarsiero and Louter cal example in which a bonded façade construction is 2019). More recent developments deal with so-called dimensioned. As a highlight, the paper transfers the laminated joints, in which either a puck is laminated safety factor to the Finite Element Method (FEM) with into a laminated safety glass (LSG) or something is the result that for the ﬁrst time mesh-independent ulti- laminated onto a glass (Bedon and Santarsiero 2018). mate loads can be determined with the remark that one For the dimensioning of silicone adhesive joints in simple and easy FE calculation has to be performed façades, there are two standards, ETAG 002 (2012) on the H-sample to calibrate the structural parameter and ASTM C1401 (2002), which are common prac- according to Eurocode. tice throughout the world. Both calculation methods are based on a linear analysis of the geometrical and 2 Historical survey of design philosophies material behavior and assume an even load distribution. Furthermore, a constant stress state of the adhesive is Historically, there are two types of design philosophies assumed, which leads to a nominal stress analysis. In with different safety concepts for the design of building order to ensure sufﬁcient redundancy or safety in the components in civil engineering: design of the silicone adhesive joint, these two stan- dards use a global safety concept, so that modeling inac- – allowable stress design method with a global safety curacies (load and constitution behavior), temperature, concept, humidity and aging effects (salt, detergent, SO , UV) – limit state design method with a semi-probabilistic are covered. Therefore, a global safety factor of γ = 6 safety concept. tot is introduced to achieve a sufﬁciently large safety mar- In the following, both concepts are brieﬂy presented gin (ETAG 002 2012). Regretfully, the exact history of for reasons of comprehensibility. the determination of the global security factor γ can- tot not be reconstructed at this time based on the current version of ETAG 002 (2012). Therefore, a discussion 2.1 Allowable stress design method with global safety about the safety factor has been sparked in the industry concept today and the demand for a comprehensible calculation of a correct and justiﬁable safety factor has arisen. Structural buildings have been designed in the past Given that little work is currently being done on the under the principle “the greater the uncertainty, the methodologically correct and thus Eurocode-compliant greater the factor of safety must be for this struc- derivation of a partial safety factor for silicone adhe- ture”. Accordingly, safety factors of magnitudes 1-6 sives in the façade sector, this contribution deals on the were chosen depending on the structure, uncertain- one hand with the development and presentation of a ties in loads etc. However, on the one hand there Eurocode-compliant partial safety factor, the discus- was no normative regulation on how the safety fac- sion of the inﬂuence of potential limit state functions tors should be calculated accordingly to a standard and and on the other hand with the implementation of a on the other hand, reliability analysis provided guid- Level I approximation of a partial safety factor for the ance for obtaining theoretically and methodically cor- silicone adhesive DOWSIL 993. The proposed method- rect safety factors for given levels of reliability but ology is generally valid, so that a partial safety factor with no formal governmental accreditation. Therefore, can also be derived for other structural silicones under these safety factors always were determined based on different limit state functions. The described method is the experience of the respective engineers and design- based on the calibration procedure given in Eurocode ers. The reduction by a safety factor was related to EC 0 and additionally uses test results and modeling a characteristic strength value of the material under content from ETAG 002 (2012) to provide a simple link investigation, which was determined from experimen- between the two concepts. With the help of the deter- tal observations. The selection of the safety factor then mined partial safety factor, the connection between the depended on the target reliability of the strength estima- two concepts is easily established and it is possible to tion. In the second half of the nineteenth century, the design and calculate silicone adhesive joints according theory of elasticity began to become accepted in the to the partial safety factor concept of DIN EN 1990 practice of structural engineers. This is also the reason Eurocode (2010). The paper concludes with a practi- why the design is carried out with linear calculations 123 Dimensioning of Silicone Adhesive Joints... 351 Fig. 1 Nonlinear force-displacement behavior of structures, joints or materials with (a) brittle failure, (b) semi-ductile failure and (c) ductile behavior (both material and geometrical) based on engineering reasons of simplicity. The ASD approach is character- or nominal stresses, which is known as allowable stress ized by its simplicity and vividness, so that it quickly design (ASD) method. found its way into engineering practice and has estab- The ASD, using a global safety concept, calculates lished itself over several decades. It can also be said that the maximum load or stress in the component for the this approach is conservative and therefore on the safe entire life cycle. As a result, the limit state is reached on side. Finally, the ASD approach is a fully determin- the action side using linear elasticity theory. For struc- istic approach in accordance to Marek and Kvedaras tural silicones, an engineering stress-based limit state (1998). function is commonly used. In the design, the calcu- However, there are also many disadvantages for the lated maximum stress must be smaller than the charac- above-mentioned concept from a scientiﬁc, probabilis- teristic strength (5% quantile) of the examined material tic and economic point of view, which Blockley (1992) reduced by a global safety factor. If this process is sum- summarises as follows: marized in terms of formula one obtains – Stress-strain relationship is not always linear, espe- lim cially not for structural silicone adhesives. σ ≤ σ = . (1) des FS – Material non-linearities may occur due to time effects (creep and relaxation) which are disre- In (1), the parameter σ represents the engineering stress garded. in the material under maximum load for the whole life- – Load effect and deformation are not always in a time of the structure, σ is the (engineering) design des linear relationship. strength, σ the characteristic strength of the mate- lim – The material behavior beyond the linearity limits rial and the abbreviation FS stands for factor of safety. can be ductile with load carrying capacity reserves. Hence, the ASD approach is also applied in ETAG 002 – The probability of exceeding the limit state at the (2012), which regulates the design of silicone adhesive beginning of non-linearity depends decisively on joints. the statistical properties of the loads, the materials, Following the comments of Blockley (1992), the the idealizations used to create a calculation model, ASD method can be summarized in the traditional way etc. Consequently, the reliability of the elements as follows: within the structure or the reliability of the different – Under service loads, all parts of the construction structures can signiﬁcantly ﬂuctuate. behave linearly elastic. – In case the service loads have been calculated to be To illustrate this and to address nonlinearities of struc- so high that the probability of exceeding them is tures, joints or materials, Fig. 1 is an example of it. low, and if the allowable stresses are chosen to be a Here, one can see that the load-deformation behav- sufﬁciently small fraction of a limit stress, then the iour past the theoretical limit of linear response may be structure has an excellent chance to have no damage brittle, semi-ductile or ductile with very large reserve. within its lifetime. In an ASD approach, however, the good-natured ductile This deﬁnition is also taken up by ETAG 002 (2012), behaviour cannot be taken into account. Therefore, the which assumes linear elastic material behaviour for ASD approach in this example gives the same result for 123 352 M. Drass, M. A. Kraus all three fracture behaviors, which can be classiﬁed as ing the construction process of the structure and main- conservative. tenance during use. Only the stochastic character of Additionally, it is important to note at this point that the input variables for actions and resistances can be many of the limitations mentioned are violated in their determined by probabilistic methods. This requires a general validity by ETAG 002 (2012). If one takes a quantiﬁcation of the stochastic uncertainties in actions closer look at ETAG’s concept for the static dimen- and resistances. sioning of silicone adhesive joints and generally the Basically, the core of the design philosophy in DIN structural behavior of silicone adhesives, they behave EN 1990 Eurocode (2010) is the solution of the inequal- non-linear-elastically (Drass et al. 2019a, b), are time- ity and rate-dependent (Kraus et al. 2017) and tend to creep E ≤ R , (2) d d under permanent load (Botz et al. 2019). According to with the list of disadvantages deﬁned by Blockley (1992), new concepts for any type of material have been devel- E = γ · E (3) d Q k oped which circumvent the mentioned disadvantages. and Here the so-called limit state design method has gained acceptance. It no longer concentrates only on the ser- R = . (4) vice condition under full load and reduction of material resistances, but deals with the limit of structural use- In the above-mentioned equations, E represents the fulness. design value of an action and R the design value of the resistance, the indices d means design and k stands for its characteristic value. To calculate the limit state 2.2 Limit state design method with semi-probabilistic of the action side, the characteristic value E is multi- safety concept plied by a factor γ , whereas the resistance side or the characteristic value R is divided by a partial factor γ k M In the limit state design (LSD) method, the limit covering uncertainties in the resistance model. strength, ultimate strength, collapse strength, maxi- Considering (2), both the action side and the resis- mum capacity of structures, columns, beams or connec- tance side are calculated separately as limit states and tions is calculated and then reduced to take into account compared against each other. The partial safety factors possible inﬂuences from uncertainties in the strength of for established materials such as concrete, steel or tim- materials, manufacturing aspects and uncertainties of ber have been calibrated such, that a certain economi- the structures in the ﬁnal state after construction. The cally acceptable target reliability (given in and deﬁned factorized strength, or resistance side, is then evaluated by the national annexes of the Eurocode 0) is reached. against the calculated load effect due to the correspond- In contrast to ASD, material and geometric nonliner- ing maximum loads. The loads, also called the action aties and imperfections as well are considered in the side, are then increased to take into account the uncer- LSD approach which is a major advantage. The LSD tainties of the loads acting on the structure during its approach provides the same results or is identical to lifetime. In the evaluation of structural reliability, the ASD for the case that the end of the elastic reaction concept of a limit state surface has thus become estab- is the limit state, the structure is perfect and the mate- lished. Here, the multidimensional domain of random rial behaviour is ideally linear elastic. To illustrate the variables is divided into safe and uncertain domains different approaches once again, we will take a closer (Marek and Kvedaras 1998). look at the action side in the following. In the case of The safety and reliability of buildings is on the one LSD, the limit state of the action is calculated and it is hand determined by the variability within the actions checked whether the actual load is below it. With ASD, and resistances of it and on the other hand by potential on the other hand, the strength limit must be main- errors in planning, execution and use. Human miscon- tained under the actual load. To illustrate the differences duct however, cannot be detected, handled and covered between both approaches, Fig. 2 can be adduced. by a safety concept, but must be excluded as far as It is quite clear that ASD ignores the non-linearity possible by targeted measures such as checking of a and may lead to conservative results. An opposite effect structural design computation, quality assurance dur- can occur in insulating glass under climatic loads, 123 Dimensioning of Silicone Adhesive Joints... 353 Fig. 2 Example of the two design methods of limit state design and allowable stress design where it can be on the unsafe side if the non-linearity p = Pr[R − E ≤ 0]= Pr[S ≤ 0], (5) of the shear bond, for example, is ignored. The LSD approach can again be divided into two where the Pr [•] deﬁnes any probability operator applied to the argument •. A fundamental measure of subgroups. The semi-probabilistic approach calculates partial safety factors on the impact and resistance side reliability theory is the so-called reliability index β, which is deﬁned as a measure of an assigned proba- to deﬁne both limit states. From the designer’s point of view, this concept is still deterministic, but a probabilis- bility of failure at a design point. The reliability index is usually set to β = 3.8 for the ultimate limit state tic approach is found in the safety factors. Hence, this method is called the semi-probabilistic partial safety and the permanent design situation for a design life of a building of 50 years. The reliability index can be factor concept. The second approach deals with fully probabilistic methods for the implementation of reli- calculated by the expected value μ and the variance ability assessment of structures. This approach is not σ under assumption of a Gaussian distribution for the described in detail in the following and requires greater action and resistance side efforts in terms of modelling and evaluation of the prob- μ − μ R E lem at hand. For other materials such as steel or con- β = . (6) 2 2 σ − σ R E crete, the semi-probabilistic approach is state of the art and has been introduced throughout Europe by the The terms safety and reliability play a decisive role EC199x series and does not pose any difﬁculties for in the construction industry. For example, the safety civil engineers (DIN EN 1990 Eurocode 2010), espe- concepts currently used in construction are based on cially engineering education on universities since ten a semi-probabilistic safety concept with partial safety years teach Bachelor and Master students this design factors on the action and resistance side. Since the philosophy. action side is already described by the Eurocode, the With regard to the construction industry, reliability is aim of the present study is therefore to derive a partial assessed by comparing the calculated reliability index material safety factor for a structural silicone accord- β with the reliability index that is regarded as adequate ing to the Eurocode, so that a uniform design procedure for the system under evaluation from previous experi- for bonded façade systems can be established. Figure 3 ence. For this purpose, one must establish a relationship gives a general and schematic overview of the par- between the capacity R (for example, the strength) of tial safety factor concept according to DIN EN 1990 the system and the demand E (for example, the load) Eurocode (2010). From this diagram, it is directly evi- such that if capacity and demand are equal, there is a dent how the partial safety factor inﬂuences the resis- limiting state of interest. The margin of safety, deﬁned tance side. as S = g(E , R) = R − E, is another example of this It is important to note that according to the Eurocode state, where S > 0 represents the safe state, S < 0 there is a partial safety factor γ for a material or prod- the failure state. For reasons of completeness, S = 0 uct property and a partial safety factor γ for a compo- deﬁnes the limiting state. Accordingly, the probability nent property, taking into account model uncertainties of failure p is given by and size deviations. 123 354 M. Drass, M. A. Kraus Fig. 3 General overview of the partial factor system in the Eurocodes, from Gulvanessian et al. (2012a)and cf. Drass and Kraus (2020), Kraus and Drass (2020) 3 Calibration of partial safety factor It shall be shown that on the one hand the derivation of a partial safety factor at a given limit state func- In the last section two different design philosophies tion according to DIN EN 1990 Eurocode (2010)is for the design of building structures were presented in possible in a few lines of code. On the other hand, general. In this section, the material resistance accord- the LSD approach with the partial safety factor con- ing to DIN EN 1990 Eurocode (2010) will be derived cept for structural silicones will be demonstrated for speciﬁcally for the structural silicone DOWSIL 993. the ﬁrst time. The advantages are that the non-linear material behavior of the silicone is taken into account, 123 Dimensioning of Silicone Adhesive Joints... 355 which leads to less conservative results. This allows deﬁned by γ and η representing the mean value of Rd a better utilization of the silicone, i.e. the design is the conversion factor that takes into account volume more material-orientated. Furthermore, by using DIN and scale effects, the effects of moisture and tempera- EN 1990 Eurocode (2010), bonded glass components ture, etc.). can be veriﬁed with one and the same safety concept Since the design resistance is deﬁned according without mixing the ASD and LSD approach. This offers to DIN EN 1990 Eurocode (2010), the mathematical advantages in the calculation of bonded façade compo- framework for determining the partial material safety nents, since two different calculations (once ASD and factor is presented below. Following the simpliﬁed once LSD) do not have to be performed. level I approach according to DIN EN 1990 Eurocode (2010), whereby at this point a suitable and meaning- ful mechanical model is assumed, then the partial safety factor γ is computed by 3.1 Mathematical framework: partial material safety M factor γ · γ 1 Rd m γ = = ·exp(α · β · V −1.645 · V ), (9) M R R F η η Following DIN EN 1990 Eurocode (2010); Sørensen (2002), the design values of material or product prop- where α represents a weighting factor for the resis- erties X are determined by tance and β is the reliability index. The coefﬁcient of variation V is computed by: X η X = η = {m − k V } . (7) d X n X γ γ m m 2 2 2 V = V + V + V , (10) M G F In this context, X represents the characteristic strength value (5 % quantile) and η is the mean value of a conver- where the components are deﬁned as: sion factor that reﬂects differences between the material – V : coefﬁcient of variation for model uncertainty strength in the calculation model and in the actual struc- of structural silicone sealant ture as well as laboratory size effects (humidity, tem- – V : coefﬁcient of variation for the geometry perature, scale and size effects, etc.). Typically, η = 1 – V : coefﬁcient of variation for the structural sili- can be assumed, cf. DIN EN 1990 Eurocode (2010). cone sealant strength. However, since in this paper more concern is put on The coefﬁcient of variation for the structural silicone the factor η, it is calculated or put in connection with sealant strength can be calculated from experimental the ETAG 002 (2012). Returning to (7), the variable data by m describes the mean value of the material property Xfor n samples, k represents the fractile value for the V = exp (s ) − 1, (11) characteristic value and V is the variation coefﬁcient F for the material property X. where the standard deviation s is given by Real design resistance, in contrast to the design val- X ues of material or product properties, also includes uncertainties in the resistance model, e.g. geometric s = (ln x − m ) . (12) deviations. In this case, the design resistance R is X i X n − 1 i =1 deﬁned by The standard deviation s depends on the mean value 1 η R R = R X = , (8) d k m of the strength, which is computed via γ γ γ Rd m M where γ is the partial safety factor covering the uncer- m = ln (x ). (13) X i tainties in the resistance model (including the par- i =1 tial factor for the uncertainty in the material prop- erty described by γ , the uncertainty in the structural Based on this and applying a log-normal distribution model of the structural members and the geometric data for the material property X, which is strongly recom- 123 356 M. Drass, M. A. Kraus Table 1 Values of k for estimation of characteristic values (5 3.2.1 Model uncertainties for the stretch-based limit % fractiles), (DIN EN 1990 Eurocode 2010) state function: coefﬁcient of variation n 5810 20 ∞ The derivation of the model uncertainties for the unknown V 2.33 2.00 1.92 1.76 1.64 partial safety factor γ for the structural silicone DOWSIL 993 is calibrated using measurement data of different stretch failure modes based on the inves- tigations of Staudt et al. (2018), Staudt (2017) and mended by Fischer (2001) for small sample sizes with a Rosendahl et al. (2019). certain coefﬁcient of variation, (7) can be reformulated To describe the multiaxial failure of the silicone by adhesive, uniaxial tension and compression tests and so-called circular shear tests were performed by Staudt X = exp (m − k · s ), (14) k X n X (2017). The details of the tests are described in Staudt which can be used to calculate the characteristic failure (2017) and in Rosendahl et al. (2019) the experimen- strength of the analyses material, e.g. a DOWSIL 993 tally determined failure points in the three-dimensional structural silicone. The quantile factor k is provided n stretch space of all individual tests were summarized. in Tab. 1. In the one-dimensional case, the stretch λ is a mea- Note that the partial safety factor alone is not sufﬁ- sure of the elongation or normal strain of a differential cient for the design, but that the strength parameter is line element. It is deﬁned as the ratio between the ﬁnal also required for the design limit state analysis. Both length l and the initial length l of the material line will depend in value on the applied limit state function element: and statistical modeling, which will be shown later in λ = . (15) this paper. In general, with increasing modeling and testing efforts, both the partial safety factor and the The relationship between the engineering or nominal characteristic strength value can be estimated with a strain and the stretch is given by small uncertainty and thus smaller safety factors. l − l ε = = λ − 1. (16) For the sake of clarity, a derivation of the stretch in 3.2 Partial material safety factor for DOWSIL 993 three-dimensional space is not shown here, but only using a stretch-based limit state function referred to in Appendix A. In order to describe the model inaccuracies with In contrast to publications on the calibration of par- regard to the failure description of DOWSIL 993 under tial safety factors for stress based limit state functions, arbitrary deformation, the three-dimensional failure cf. Drass and Kraus (2020), Kraus and Drass (2020), stretches tabulated in Rosendahl et al. (2019) are trans- the present work calibrates the partial safety factor for ferred to a two-dimensional space, the so-called π DOWSIL 993 based on experimental data from Staudt plane. Again, the exact derivation of the π plane is et al. (2018), Staudt (2017), Rosendahl et al. (2019) omitted and reference is made to Appendix B. The rep- using an adaptive stretch-based failure criterion. resentation of the failure points in the π plane is shown Returning to the computation of the partial safety in Fig. 4a. factor γ for the structural silicone adhesive DOW- The aim is to approximate the failure stretches of SIL 993 considering a stretch-based failure criterion the analysed structural silicone using a suitable failure (i.e. limit state), the proposed formula apparatus is com- criterion. A very adaptable and comprehensible crite- bined with constraints from ETAG 002 (2012) to cal- rion for isochoric distortional failure was proposed by culate γ using the Level I method of DIN EN 1990 Podgórski (1984) and developed in a similar manner by Eurocode (2010). Starting with the determination of Bigoni and Piccolroaz (2004). In the following this cri- the coefﬁcients of variation for the stretch-based limit terion is abbreviated as PBP criterion. Using the nota- state function, the coefﬁcient for model uncertainties tion of Rosendahl et al. (2018), the distortional failure iso will be derived in the following. criterion Φ is described in stretch space by 123 Dimensioning of Silicone Adhesive Joints... 357 Fig. 4 Representation of the failure stretches of DOWSIL 993 in the two-dimensional π plane and (b) approximation of the failures stretches by the PBP and von Mises-like stretch-based failure criteria iso iso Φ = λ − λ = lent stretch of λ = 1.2538 was ﬁtted, whereas that of c c eq Mises provides λ = 2.5691. If the model uncertainty π 1 = ρ cos β − arccos γ cos (3θ ) for the PBP criterion is evaluated under the assump- 6 3 tion of a normal distribution, a variation coefﬁcient of −λ = 0, (17) V = 0.059 results. Assuming a lognormal distribu- where ρ represents the distance from the hydrostatic tion, which is especially useful for small amounts of axis to the boundary of the failure surface in the devi- data, a V = 0.0594 results. As a reminder, the nor- atoric plane and θ is the so-called Lode angle (see mal distribution is deﬁned as Appendix A). Both are functions of the deformation 1 x −μ 1 − 2 σ state. The parameters β and γ determine the shape of f x = e , (19) ( ) √ σ 2π the failure surface and must be determined based on experiments. The threshold λ describes the size of the where σ represents the standard deviation and μ gives failure surface. In order to guarantee a convex failure us the mean value. In contrast, the lognormal distribu- surface, the shape parameters are restricted to the inter- tion function is deﬁned by vals β ∈ [0, 2] and γ ∈ [0, 1]. (ln(x )−μ) 2σ Approximating the failure stretches of DOWSIL 993 f (x ) = √ e , (20) σ x 2π with the PBP criterion results in the following two- where σ deﬁnes the scale parameter and μ is the shape dimensional failure surface, which is also shown in the parameter. π plane in Fig. 4b. For the sake of completeness, the Accordingly, both distribution functions deliver an approximation by the von Mises-like criterion almost identical value, which is later used to calculate λ = 3I (18) 2 the partial material safety factor. is also shown, where I corresponds to the second invariant of the deviatoric part of the Hencky strain 3.2.2 Geometry uncertainties: coefﬁcient of variation tensor. As can be easily seen, the failure points for three Furthermore, the geometric deviations of the adhesive different deformation states, namely uniaxial tension joint must be taken into account to evaluate the partial and compression and shear, are very well approximated safety factor for DOWSIL 993. Since there is no exact with the PBP criterion. In contrast, the von Mises-like knowledge of the geometry from the underlying data criterion is not at all suitable to describe the failure set, as it was simply not measured exactly, an assump- states, which must lead to a very high model uncer- tion must be made for the geometry uncertainty. Here, tainty. On this basis, the model uncertainty is calculated a value of V = 0.10 is assumed. This guideline value according to the explanations of Gulvanessian et al. is based on personal communications and experiences (2012b). In relation to the PBP criterion, an equiva- of the Seele company. Since the value of the geometric 123 358 M. Drass, M. A. Kraus Fig. 5 Representation of the distribution functions under the assumption of a a normal distribution and b log-normal distribution for the calculation of the coefﬁcient of variation of the model uncertainty, taking into account the PBP criterion and the failure stretches uncertainty is only an assumption, it can be adjusted stress data available. In addition to the box plot of the individually and reduced by factory production con- strengths (engineering stresses), the log-normal distri- trols in the form of measurements and consecutive sta- bution of the unaged and artiﬁcially aged tests is shown tistical analysis. In summary, however, this value lies in Fig. 6b. It is interesting to see that the slopes change within a trustworthy range based on experience with at different temperatures. Accordingly, the temperature industrial applications of DOWSIL 993 (Fig. 5). has a great inﬂuence on the distribution of the engineer- ing stress strengths. If one looks at the artiﬁcially aged 3.2.3 Determination of partial safety factor γ for a samples with NaCl and SO , the gradient and thus the stretch-based limit state function distribution of the strengths changes only slightly. All averages of the nominal strengths of the artiﬁ- In terms of (9), the weighting factor for the resistance cially aged specimens are above the 75 % criterion, according to the Level I method is assumed to be α = R thus meeting the requirements of ETAG 002 (2012). 0.8. According to DIN EN 1990 Eurocode (2010)this This criterion therefore provides a lower limit value factor is on the safe side. Generally, the reliability index which must be met experimentally in order to be able is considered with β = 3.8 for the ultimate limit state, to construct an SSG façade. Assuming this lower limit which corresponds to a permanent design situation with value is a true barrier according to ETAG 002 (2012), a target lifetime of the building of 50 years. which includes all harmful inﬂuences such as temper- So far, the conversion factor η according to (9) has ature, water and UV storage as well as salt exposure, not been considered in the computations of the partial the conversion factor η can be determined accordingly: material safety factor for the stretch-based limit state η = 0.75. (21) function. To account for further model uncertainties This is a reasonable approach, creates a lower boundary and conversion aspects, the conversion factor η is now for η and links ETAG 002 concept with DIN EN 1990 linked to requirements from ETAG 002 (2012)tohavea Eurocode (2010). It is to note that the conversion factor reasonable assumption regarding the model uncertainty η can be adjusted according to the results of the ageing under consideration of ageing effects. tests, if test results are available. Typically, ageing phenomena occur in façades due If one takes all previous assumptions as a basis and to water, temperature, UV, NaCl, SO , detergent expo- assuming that the uncertainties in the structural model sure. These adverse ageing effects are experimentally of structural members is γ = 1, then the partial safety Rd tested according to ETAG 002 (2012), where the ratio factor γ for DOWSIL 993 with a stretch-based limit of the aged nominal strength (in terms of engineering state function reads stress) to the unaged strength must be greater than 75 %. γ = 1.81. (22) Figure 6a shows the barrier in accordance to ETAG 002 (2012) for tensile loading of an ETAG H-probe as This value for the partial safety factors assumes 10 % a grey box according to the experimental engineering for the coefﬁcient of variation of the geometry and 123 Dimensioning of Silicone Adhesive Joints... 359 Fig. 6 a Box plot of nominal failure strengths of DOWSIL 993 under tensile loading and b lognormal distribution of the engineering stress strengths of DOWSIL 993 after artiﬁcial ageing in tensile tests (cf. Drass and Kraus 2020; Kraus and Drass 2020) V = 0.0593 for the model V . If the conversion 3.3 Discussion of the determined partial safety factor M M factor is taken into account, η = 0.75 must also be for structural silicone assumed. It is important to note that this partial safety factor is further adjustable by reducing the coefﬁcient Having obtained numerical value for the partial mate- of variation of the adhesive joint geometry as a result rial safety factor with associated characteristic values of accurate factory monitoring and machine application for the DOWSIL 993, this section discusses the under- of the adhesive joint. lying assumptions in more detail. A very important point at this stage is the assump- In order to create direct comparability between the tion that for the uncertainties of the structural model partial safety factor according to EC0 and the global of structural members the value is set to 1. This fac- safety factor γ according to ETAG 002 (2012), it is tot tor will be adapted for simulation by FEM, as will be assumed when converting γ into a global safety factor shown later, in order to avoid the problem of stress sin- that only live loads affect the component. As a result, gularities and thus make the FE solution independent γ is multiplied by the partial safety factor on the action of the mesh. The γ factor will be determined by one side of 1.5, resulting in a conservative global safety Rd simulation of an ETAG H-sample and then used for factor of the calculation of the design resistance. The individual adaptation of γ makes it possible to evaluate struc- Rd tural components independent of the mesh. 123 360 M. Drass, M. A. Kraus γ = γ · γ = 1.81 · 1.5 = 2.72 6.0. (23) γ , which in the following is equated with stress singu- M Ed Rd tot larities at bi-material notches, the parameter γ must Rd be additionally calibrated to obtain a mesh-independent Comparing this with the global safety factor according solution for the determination of the ultimate load. To to ETAG 002 (2012), a large reduction results despite illustrate how this works, the veriﬁcation concept and consideration of damaging inﬂuences. The principal the necessary steps are brieﬂy introduced below and assumptions for the application of the proposed con- then explained in detail using the example described cept or the determined partial safety factor, however, is above. the application of suitable material models and failure criteria for the structural silicone under investigation in order to have the lowest possible uncertainty on the 4.1 Methodology material side. However, the following points allow for more detailed and precise computation of the partial The veriﬁcation concept wil be divided into four steps, material safety factor in future research: which are brieﬂy described in the following: – ’realistic’ ageing protocols (deduction of load com- Step 1: PBP criterion for 5% Quantile Values bination factors), – calculate 5% quantile values for uniaxial tension / – fatigue, compression and shear tests – viscoelasticity, – ﬁt PBP criterion on 5% quantile values of experi- – different performance / limit state functions ments g(E , R), – determine an equivalent stretch based on the 5% – multiple failure modes (distortional and/or dilata- quantile Values → λ c,5% tion) of the sealant, – failure modes of the sealing application and the Step 2: Deﬁnition FE-Mesh sealed system (series and/or parallel system behav- – deﬁnition of the FE mesh for the global model of the ior). façade element, i.e. the mesh density of the silicone adhesive must be speciﬁed. – deﬁnition of the FE-mesh for the ETAG H-Probe 4 Validation example – note: both FE-meshes must be identical! Step 3: Calculation of γ Rd The aim of this section is to show how to apply the semi-probabilistic safety concept for the static proof – calculation ETAG H-sample in tension with F k,5% of silicone-adhesive connections using the FEM. This and predeﬁned mesh from Step 2 example deals in particular with a bonded façade ele- – evaluation of the principal stretches in the FE- ment where the maximum load-bearing capacity under model wind load is going to be calculated. Using the semi- – ﬁt PBP criterion → λ c,Rd probabilistic safety concept, the aim is to increase the – calculation of γ = λ /λ Rd c,5% c,Rd applied wind suction load until the load-bearing capac- Step 4: Calculation of the design value λ c,d ity of the structural silicone adhesive is reached. A spe- cial feature of this section is that the veriﬁcation of the In the following steps 1-4 are presented in detail using adhesive joint is to be carried out via FE simulations. the example of the bonded façade element located in This results in the peculiarity that in the simulation Berlin, Germany. Details on the project will be pro- with ﬁnite elements stress singularities at bi-material vided, when the numerical model will be explained notches occur, whereby a mesh-dependent solution is more in detail. obtained. This must be taken into account or circum- vented in the design approach which will be shown 4.1.1 Step 1: PBP criterion for 5% quantile values later. Since the partial safety factor on the basis of (22) As one can see in Fig. 4a, the individual experiments, has been calculated without considering uncertainties here uniaxial tension/compression and shear, are scat- in the structural model of the structural members by tering, so that one must determine the 5% quantile value 123 Dimensioning of Silicone Adhesive Joints... 361 By calculating the 5% quantile values, the adaptive PBP failure criterion must be ﬁtted to these data in order to calculate an equivalent stretch λ . By doing this c,5% you get the equivalent stretch λ = 1.09592. For c,5% the representation of the π plane this means that the failure domain becomes slightly smaller (see Fig. 7). 4.1.2 Step 2: deﬁnition FE-mesh In this section, a discretization for the global model of the façade element must be speciﬁed, which can also be transferred to the ETAG H-shaped test sample. Since we emphasize in this paper that we are able to calculate failure loads or carrying loads respectively independent of the mesh, we will examine three mesh variants in the following: Fig. 7 Representation of the failure stretches of DOWSIL 993 –4 × 4 × 4 mm, in the two-dimensional π plane and the approximation of the failures stretches by the PBP-Experiment and the 5% quantile –3 × 3 × 3 mm, values –2 × 2 × 2 mm. As can be seen from the list, the silicone is modeled with brick elements with exactly the same edge lengths. All for each experiment individually on this basis. Hence, following calculations are therefore carried out with to calculate the 5% quantile values for the stretches in these three different discretizations. As an example of accordance to ETAG 002 (2012), the following equa- the mesh variant 2 × 2 × 2 mm, Fig. 8 shows the tion can be utilized: ETAG H-shaped sample and a section of the façade element with the same mesh. The contact formulation R = X − τ · s . (24) in these examples was realized with so called Multi- u,5% mean α,β X Fig. 8 Illustration of the FE-mesh of a the H-sample and b detail stands for glass, orange-red for the silicone adhesive and grey of the global model of the façade element including the glass for aluminium pane, adhesive joint and aluminium proﬁlewhere the colour blue 123 362 M. Drass, M. A. Kraus Point-Constrained contact elements, so that due to this Table 2 Simulation results for modiﬁed ETAG H-probe under tensile load for three different mesh densities and evaluation of constraint non coincident nodes must be used in the the equivalent stretch according to PBP-criterion mesh. However, it is essential that the contact formula- mesh Force λ λ λ λ tion of the H-sample is identical to the formulation in 1 2 3 c,Rd (mm) (N) (/) (/) (/) (/) the global model. 2 × 2 × 2 608.4 1.7351 0.9159 0.6298 0.5681 4.1.3 Step 3: calculation of γ 3 × 3 × 3 608.4 1.6214 0.9102 0.6782 0.4798 Rd 4 × 4 × 4 608.4 1.5548 0.9041 0.7121 0.4224 In the third step the uncertainty in the structural model of structural members γ will be determined. The Rd authors are of the opinion that this safety factor should In this context, the pulled surface of the specimen be calibrated using the stress singularities occurring in (a · b) is multiplied by the design strength σ of des FE calculations as input. In the context of FEM, stress DOWSIL 993 and the global safety factor of γ = 6 tot singularities are understood as the solution depends on to determine the 5% quantile value of the tensile force the mesh density in the area of bi-material notches. F . This determined force is applied to the H-shaped k,5% These notches always occur in the region of the stick- numerical examples with three different mesh densities ing of the elements of the silicone with glass or the sub- and then the governing stretches in the corners are eval- structure. As a result, the stresses, strains and stretches uated. The mesh-dependent stretches are then in turn always increase with ﬁner mesh at the same load level. ﬁtted by the PBP criterion again, so that three different However, this effect must be avoided when dimension- equivalent stretches λ are obtained for the three dif- c,Rd ing silicone adhesive joints, as otherwise the solution ferent FE meshes. A summary is presented in Table 2. depends on the mesh, which means that it is very lightly The PBP-criterion is then ﬁtted with the mesh- to incorrectly dimension the joint. It is therefore of deci- dependent failure stretches determined from ﬁnite ele- sive importance to develop a method which, for exam- ment simulations. This results in corresponding equiv- ple in the load bearing capacity calculation, always alent stretches λ , which are also summarized in c,Rd leads to the same results without being dependent on Table 2 and are additionally shown in the π plane plot the mesh density. in Fig. 9. What can be clearly seen is, on the one hand, Therefore, to be independent of the mesh density, that with increasingly ﬁner mesh the failure surface γ must be calibrated by one single numerical calcu- Rd becomes larger and, on the other hand, that the fail- lation of an ETAG H-sample under tensile load. The ure surfaces calibrated on the FE calculations are sig- advantages are obvious. On the one hand, this safety niﬁcantly smaller than the 5% quantile failure surface factor can be determined by one simple numerical cal- determined on the experimental tests on DOWSIL 993. culation, and on the other hand, complex mathematical To calculate the uncertainty in the structural model methods such as ﬁnite fracture mechanics and expen- of the structural members, the following equation can sive fracture mechanics tests can be omitted. In order to be adduced: give the reader a clear understanding of this procedure, all necessary steps for determining γ are presented Rd c,5% γ = . (26) Rd in detail below. c,Rd As a ﬁrst step, the geometry of the ETAG H- specimen or similar geometry must be entered into an The following table (see Table 3) summarizes this oper- FE program and meshed with the same mesh as that of ation for the three different meshes: the global model. This sample must then be pulled to the 5% quantile value of the tensile strength or tensile 4.1.4 Calculation of the design value λ c,d force. For the geometry of the H-shaped specimen of 12 × 12 × 60 mm selected here, the tensile force to be In the last step, the determined partial safety factor applied reads from (22) and the partial safety factor for uncertainty F = a · b · σ · γ k,5% des tot in the structural model of the structural members γ Rd = 12 mm · 60 mm · 0.14 · 6 = 608.4N. (25) must be harmonized with the PBP-criterion, so that FE- 123 Dimensioning of Silicone Adhesive Joints... 363 Fig. 9 Representation of the failure stretches of the two-dimensional π plane and the PBP-criterion for the experiments, the 5% quantile values and the approximation of the failures stretches accordingly to Table 2 with respect to the mesh density Table 3 Evaluation of the uncertainty in the structural model of Table 4 Evaluation of the equivalent stretches λ for three c,d the structural members γ for three different mesh densities different mesh densities Rd Mesh λ γ γ λ Mesh λ λ γ c,5% M Rd c,d c,5% c,Rd Rd (mm) (/) (/) (/) (/) (mm) (/) (/) (/) 2 × 2 × 2 1.0959 1.81 1.9292 0.3138 2 × 2 × 2 1.0959 0.5681 1.9292 3 × 3 × 3 1.0959 1.81 2.2842 0.2651 3 × 3 × 3 1.0959 0.4798 2.2842 4 × 4 × 4 1.0959 1.81 2.5946 0.2333 4 × 4 × 4 1.0959 0.4224 2.5946 the PBP criterion within the FE calculation. If this pro- calculations can be carried out and carrying loads deter- cedure is followed, the silicone veriﬁcation is carried mined without being dependent on the mesh. out for the ﬁrst time using the semi-probabilistic safety This is done simply by dividing the equivalent concept accordingly to EN 1990 Eurocode (2002) and stretch λ by both partial safety factors, so that the c,5% it is also possible to determine carrying loads that are following applies independent of the mesh density. A last important point is that with the tabulated values for λ any adhesively c,5% c,d λ = . (27) c,d bonded facade structures can be calculated with the cor- γ · γ M Rd responding mesh density, as long as the corresponding Since we examined three different mesh densities, the tabulated value from Table 4 is used. Therefore, this is results are summarized in the following table (see the ﬁrst mesh-independent approach to design bonded Table 4). structures using FEM. It should be noted that with Table 4 and more pre- cisely the value for λ the dimensioning of the sil- c,d icone adhesive joint can be carried out with the cor- 4.2 Example: bonded façade element responding mesh density. If one decides on a mesh density of 2 × 2 × 2 mm for the structural silicone In this example, a bonded façade element has to be in the global model of the façade element, the value veriﬁed statically according to EN 1990 Eurocode λ = 0.3138 must be entered for the evaluation of (2002), i.e. the semi-probabilistic safety concept. In c,d 123 364 M. Drass, M. A. Kraus Fig. 10 Two renderings of the project Voltair of the building owner VOLT GmbH & Co. KG, Uhlandstraße 181-183, 10623 Berlin particular, this section deals with the numerical simu- mono-pane is then structurally bonded to the aluminum lation of a bonded façade element with the dimensions proﬁle with the silicone adhesive DOWSIL 993. The 2700×5100 mm in order to make the static proof of the bearing of the proﬁle is done point-wise by so called structural silicone adhesive. The product DOWSIL 993 toogle systems, so that the proﬁle is kept at a distance of is ﬁctionally used as adhesive in the present example, 300 mm. The external load is a wind suction load, which which has the following joint dimensions (28×12 mm). is increased until the maximum load-bearing capacity A rendering of the bonded facade of the Voltair project for the silicone adhesive is reached. in Berlin is shown in Fig. 10. 4.2.1 ETAG 002: analytical approach In this example, the present concept from Sect. 3 is used to determine the maximum load due to wind Starting with the analytical calculation of the prob- suction. Figure 11 shows the geometric model of the lem, ETAG 002 provides a hand calculation formula to bonded façade construction exploiting symmetry. obtain the maximum design wind load in accordance The model consists of a laminated safety glass to the ASD approach utilizing a global safety concept, (LSG) 2 x 10 mm annealed glass with PVB layer in which is computed via between. For reasons of simplicity, the LSG is repre- 2 · σ · h des c sented with a surrogate model, i.e. a mono-pane. This classic p = k,ETAG 123 Dimensioning of Silicone Adhesive Joints... 365 Fig. 11 CADmodelofa bonded façade element with a = 5100 mm and b = 2700 mm under representation of the bearing conditions and the wind suction load 2 · 0.14 · 28 kN veriﬁcation by means of FEM is decisive, since due to = · 1000 = 2.91 . (28) 2700 m increasingly complex building structures and bonded joint geometries, the veriﬁcation of the bond is cer- The advantages of the mentioned approach are the sim- tainly more complex than is covered by ETAG. plicity and quick application, but the disadvantages are the assumption of linear elastic material and struc- tural behavior, the application of engineering stresses as 4.2.2 Finite element analyses design basis, the separation of tensile and shear stresses due to different actions, the high safety level, the very In this section, the results of the FE calculations are strong simpliﬁcation of the load transfer, the neglect brieﬂy presented. As the model is non-linear, the cal- of uncertainties on the action side and the separate culations are performed under large deformations. In veriﬁcation of bonded facade elements, since differ- order to obtain a good convergence, the wind suc- ent safety concepts are used for glass and silicone. Due tion load is successively increased until the PBP cri- to the numerous disadvantages, it is essential to offer terion in the silicone is exceeded. It is important to modern dimensioning approaches that circumvent the note that three individual calculations of the façade ele- above mentioned disadvantages and still allow a safe ment were carried out, each with a different mesh den- design of the adhesive joint and the structure. In addi- sity of the silicone. The mesh density was previously tion, FEM veriﬁcation is often required in projects, as selected according to Sect. 4.1.2. The PBP criterion the simpliﬁcations according to ETAG are too blatant is also adjusted during its evaluation according to the and building owners and engineers are uncertain about implemented mesh density, so that for λ the values c,d its application. Therefore, the authors are of the opinion according to Table 4 must be used. that for modern, bonded facades, the ETAG approach Figure 12 gives a histogram showing the numerically can be used for the basic evaluation and a preliminary determined load capacities for different mesh densi- calculation of the ultimate load, but that a more precise ties. It can clearly be seen that the maximum wind 123 366 M. Drass, M. A. Kraus Fig. 12 Histogram of permissible wind loads with respect to the chosen safety concept:, i.e Finite Element Analysis-Limit State Design analyzing three different mesh densities (FEA-LSD) with semi-probabilistic safety concept and allowable stress design method with global safety concept in accordance to ETAG 002 (2012) (ETAG-ASD) suction load that can be absorbed according to EN γ = 1.5. However, a very interesting result is that 1990 Eurocode (2002) design reads p = 4.08 kN/m . both approaches calculate almost identical maximum It should also be noted that with the proposed con- loads. However, since according to ETAG a slightly cept from Sect. 4.1, the same loads could be deter- higher wind load was determined, this approach is on mined for three different mesh densities. This means the unsafe side according to the considerations from that the solution with the method proposed is indepen- the Eurocode. dent of the meshing. This is a very important result because the concept is easily accessible, can be cal- ibrated by engineers and complex methods such as 5 Discussion and conclusions ﬁnite fracture mechanics are not required. As can be seen additionally from the histogram, the wind suction This paper deals with the presentation and calcula- load that can be absorbed based on ETAG’s approach tion of a partial safety factor for the structural silicone classic 2 is p = 2.91 kN/m using (28). DOWSIL 993. A semi-probabilistic approach accord- k,ETAG Here, it must be noted that wind loads that can be ing to EN 1990 Eurocode (2002) was proposed and absorbed accordingly to EN 1990 Eurocode (2002) applied to determine a suitable partial safety factor. are so-called design loads, i.e. according to EN 1990 To illustrate the procedure, the methodological outline Eurocode (2002) a factor of 1.5 is still included on the concludes with an exemplary probabilistic evaluation action side. of a speciﬁc limit state for the structural silicone adhe- In order to compare the wind load determined sive DOWSIL 993. The methodology is state of the according to ETAG 002 (2012) with the wind loads art, but was applied for the ﬁrst time to structural sili- mentioned above, the value according to ETAG may cone adhesives with the application area of the façade. be modiﬁed as follows: Furthermore, the concept according to the EN 1990 Eurocode (2002) was extended to the ﬁnite element classic classic method in such a way that a design is now possible p = 1.5 · p d,ETAG k,ETAG (29) without obtaining mesh-dependent results. = 1.5 · 2.91 = 4.37 kN/m . The value determined within the scope of this work shows that the partial material safety factor (also tak- The comparison between (29) with the ultimate ing into account temperature and ageing and laboratory loads based on the FEM and semi-probabilistic safety effects) is γ = 1.81 and thus a global partial safety concept is actually not quite correct, since, accord- factor of γ = 2.72 is justiﬁed, in the case of precise tot ing to ETAG’s calculation method, one has already material modeling and failure description via the PBP found its limit state function, i.e. no more wind load criterion. Furthermore, the semi-probabilistic concept may be applied to the bonded facade. Nevertheless, was extended to the ﬁnite element method in such a we compare the value of ETAG with the value accord- way that it is now possible for the ﬁrst time to carry ing to Eurocode by the ﬁctitious increase of the cal- out Eurocode-compliant and even mesh-independent culated wind suction load from (28) with the factor of dimensioning of silicone adhesives in the façade sector. 123 Dimensioning of Silicone Adhesive Joints... 367 In conclusion, however, it should be clearly stated the current conﬁguration. Its determinant that the concept presented and its application are sub- ject to the following conditions: J = det F, (A.32) – Application of a hyperelastic material model that characterizes volume change. Splitting the deformation can exactly represent any deformation state, e.g. gradient into a rotation tensor and a stretch tensor, the Drass et al. (2018), Drass et al. (2019a). theorem of polar decomposition yields – The use of the stretch-based PBP failure criterion is mandatory. F = RU = vR, (A.33) – For now, only applicable to isochore failure. – All given values are only applicable for the product where R = R is an orthogonal rotation tensor, U DOWSIL 993. the right stretch tensor and v the left stretch tensor. The right stretch tensor U is also known as Lagrangian Acknowledgements Open Access funding provided by Pro- jekt DEAL. The authors would like to thank the people and stretch tensor and the left stretch tensor v as Eulerian companies that contributed to this publication by making data, stretch tensor based on their corresponding underlying projects and measurements available. Special thanks go to the conﬁguration. Both stretch tensors can be also given in building owner VOLT GmbH & Co. KG, Uhlandstraße 181-183, spectral representation reading 10623 Berlin as well as the company Knippers Helbig, Florian Scheible and Thiemo Fildhuth for providing us the rendering and project details the Voltair bulding in Berlin, Germany. Fur- 3 3 thermore the authors would like to thank Schütz Goldschmidt U = λ (N ⊗ N ) and v = λ (n ⊗ n ) , i i i i i i Schneider Ingenieurdienstleistungen im Bauwesen GmbH and i =1 i =1 especially Sebastian Schula to be involved in the Voltair project. (A.34) Open Access This article is licensed under a Creative Com- mons Attribution 4.0 International License, which permits use, where λ represent the principal stretches (eigenval- sharing, adaptation, distribution and reproduction in any medium ues) and N and n deﬁne the eigenvectors of U and v, or format, as long as you give appropriate credit to the original i i author(s) and the source, provide a link to the Creative Com- respectively. The principal invariants of, e.g., the left mons licence, and indicate if changes were made. The images or stretch tensor v, read other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit 1 2 2 I = tr (v) , II = I − tr v , v v line to the material. If material is not included in the article’s Cre- v ative Commons licence and your intended use is not permitted by III = det (v) . (A.35) statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view Let us denote the surface which represents the a copy of this licence, visit http://creativecommons.org/licenses/ boundary between an intact and a damaged material by/4.0/. failure surface Φ. A general stretch-based failure cri- terion is then given by A Measures of deformation Φ (v) = 0. (A.36) The deformation gradient F maps a material line ele- In general, the material does not fail for Φ (v) < 0. ment dX from the reference conﬁguration to the corre- Φ (v) = 0 (and hypothetically Φ (v) > 0) corresponds sponding line element in the current conﬁguration dx: to failure. In this context it is important to note that material failure may correspond to yielding, stress soft- dx = F dX, (A.30) ening effects or crack nucleation which not necessarily where represents the ultimate failure of the material. Owing to isotropy, failure criteria must be invariant with respect ∂x F = =∇ x. (A.31) X to arbitrary rotations of the coordinate system. Hence, ∂X criteria may be formulated in terms of principal invari- This second-order tensor is deﬁned as a two-point ten- ants of the left stretch tensor. The trace of the left stretch sor because it refers to the reference conﬁguration and tensor 123 368 M. Drass, M. A. Kraus I = tr (v) = λ + λ + λ . (A.37) expressed by v 1 2 3 is related to the hydrostatic deformations and therefore ⎛ ⎞ ⎛ ⎞ 1 1 1 ⎛ ⎞ √ √ √ important for dilatational failure. Concerning distor- λ ξ 1 3 2 6 1 ⎜ 1 2 ⎟ tional failure, the second and third invariants of the ⎝ ⎠ √ √ ⎝ ⎠ λ = ξ . (A.43) ⎝ ⎠ 2 2 3 6 stress deviator are often used. Denoting invariants of 1 1 1 λ ξ √ √ √ 3 − − 3 3 2 6 the deviator with a prime, the second deviator invariant is given by 2 2 2 II = (λ − λ ) + (λ − λ ) + (λ − λ ) , 1 2 2 3 3 1 v B Description of failure surfaces (A.38) A generic example for an illustration of a failure crite- and the third invariant of the left stretch tensor deviator rion in principal stretch space is given in Fig. 13.For a can be expressed as better understanding, the transformed coordinate sys- I I I v v v III = λ − λ − λ − . (A.39) tem and three deviatoric planes at different sectional 1 2 3 3 3 3 planes are illustrated. Additionally, important meridi- Failure criteria can be visualized using three-dimensio- ◦ ◦ ◦ ans for stress angles of θ = 0 , 30 , 60 are shown. nal or two-dimensional implicit plots. The sectional These meridians are important for the parametriza- plane of the failure surface with the deviatoric plane tion of failure criteria based on experimental or vir- (also know as deviatoric plane) is often of great interest, tual datasets (Fahlbusch et al. 2016). In contrast to the especially with regard to the veriﬁcation of convexity. classical proposed invariants of Eqs. A.37–A.39,more Introducing a new orthogonal coordinate system with descriptive invariants with geometrical meaning were the coordinates ξ ,ξ and ξ , the deviator plane is char- 1 2 3 introduced by Novozhilov (Kolupaev 2018). They are acterized by ξ and ξ , whereas the third coordinate ξ 2 3 1 deﬁned by the scaled hydrostatic axis ξ , the distance is perpendicular to that plane and points in the direction between the failure surface to the hydrostatic axis ρ of the hydrostatic axis (Schreyer 1989). The orthogonal and the stress angle θ. The radius ρ and the argument transformed coordinates read θ of the stress angle cos 3θ are deﬁned by λ + λ + λ 1 2 3 ξ = √ , (A.40) 2 2 ρ = ξ + ξ = 2II (B.44) 2 3 v λ − λ 1 3 ξ = √ , (A.41) and 2λ − λ − λ 2 1 3 ξ = √ . (A.42) 1 3 3 III θ = arccos with θ ∈ [0,π/3] . 3/2 3 2 II The coordinate transformation between the princi- pal stretches and the transformed coordinates is also (B.45) Fig. 13 Haigh and Westergaard-space of arbitrary cavitation criterion in terms of principal Cauchy stresses (λ ,λ ,λ ), the 1 2 3 transformed coordinates ξ , ξ and ξ and the deviatoric 2 3 plane at different sectional planes (αI ,β I ,γ I ) 1 1 1 123 Dimensioning of Silicone Adhesive Joints... 369 The so-called meridian plane, in which different merid- ten. Bautechnik : Spezial, Ernst. https://books.google.de/ books?id=lxoLmwEACAAJ (2001) ians are represented in the coordinates ξ − ρ,isoften Gulvanessian, H., Calgaro, JA., Holicky, M.: Designers’ guide used to illustrate three-dimensional failure criteria in to Eurocode: basis of structural design (2012a) as two-dimensional section planes (Zyczkowski 1981). Gulvanessian, H., Calgaro, JA., Holicky, M.: Designers’ guide Kolupaev (2018) recommends to scale the abscissa of to eurocode: basis of structural design (2012b) Kolupaev, V.A.: Equivalent Stress Concept for Limit State Anal- the meridian plane with respect to the von Mises crite- ysis, vol. 86. Springer, Berlin (2018). https://doi.org/10. rion such that the scaled meridian plane is formulated in 1007/978-3-319-73049-3 (I , 3II ) coordinates. For all following studies, the v Kraus, M.A., Drass, M.: Semi-probabilistic calibration of a par- deﬁnition in accordance with Kolupaev (2018) will be tial material safety factor for structural silicone adhesives— part II: veriﬁcation concept. Int. J. Struct. Glass Adv. applied for illustrating the scaled meridian plane. The Mater. Res. 4(2020), 10–23 (2020). https://doi.org/10.3844/ variable ϕ, which represents a variation of the stress sgamrsp.2020.10.23 angle, is deﬁned accordingly to Kolupaev (2018). Kraus, M.A., Schuster, M., Kuntsche, J., Siebert, G., Schneider, J.: Parameter identiﬁcation methods for visco- and hyper- elastic material models. Glass Struct. Eng. 2(2), 147–167 (2017). https://doi.org/10.1007/s40940-017-0042-9 References Marek, P., Kvedaras, A.: From partial factors design to fully probabilistic reliability assessment of structures. Statyba ASTM C1401. (2002) Standard guide for structural sealant glaz- 4(4), 252–259 (1998). https://doi.org/10.1080/13921525. ing (2002) 1998.10531414 Bedon, C., Santarsiero, M.: Transparency in structural glass sys- Podgórski, J.: Limit state condition and the dissipation function tems via mechanical, adhesive, and laminated connections - for isotropic materials. Arch. Mech. 36(3), 323–342 (1984) existing research and developments. Adv. Eng. Mater. 20(5), Rosendahl, P.L., Drass, M., Schneider, J., Becker, W.: Crack 1700815 (2018). https://doi.org/10.1002/adem.201700815 nucleation in hyperelastic adhesive bonds. Eng. Transpar. Bigoni, D., Piccolroaz, A.: Yield criteria for quasibrittle and fric- 2(5–6), 409–425 (2018). https://doi.org/10.1002/cepa.941 tional materials. Int. J. Solids Struct. 41(11), 2855–2878 Rosendahl, P.L., Drass, M., Felger, J., Schneider, J., Becker, W.: (2004). https://doi.org/10.1016/j.ijsolstr.2003.12.024 Equivalent strain failure criterion for multiaxially loaded Blockley, D.I.: Engineering Safety. McGraw-Hill, New York incompressible hyperelastic elastomers. Int. J. Solids Struct. (1992) 166, 32–46 (2019). https://doi.org/10.1016/j.ijsolstr.2019. Botz, M., Wilhelm, K., Siebert, G.: Experimental investigations 01.030 on the creep behaviour of PVB under different temperatures Santarsiero, M., Louter, C.: Metal-to-glass bond strength of and humidity conditions. Glass Struct. Eng. 4(3), 389–402 structural PVB. In: GPD Glass Performance Days 2019— (2019). https://doi.org/10.1007/s40940-019-00098-2 Proceedings (2019) DIN EN 1990 Eurocode (2010) Grundlagen der tragwerkspla- Schreyer, H.L.: Smooth limit surfaces for metals, con- nung. Deutsche Fassung EN 1990: 2002+ Al: 2005+ A1: crete, and geotechnical materials. J. Eng. Mech. 2005/AC: 2010 (2010) 115(9), 1960–1975 (1989). https://doi.org/10.1061/ Drass, M., Schwind, G., Schneider, J., Kolling, S.: Adhesive con- (ASCE)0733-9399(1989)115:9 nections in glass structures—part I: experiments and ana- Sørensen, JD.: Calibration of partial safety factors in danish lytics on thin structural silicone. Glass Struct. Eng. 3(1), structural codes. In: JCSS Workshop on Reliability Based 39–54 (2018). https://doi.org/10.1007/s40940-017-0046-5 Code Calibration, Citeseer, vol 21, p 2002 (2002) Drass, M., Bartels, N., Schneider, J., Klein, D.: Pseudo-elastic Staudt, Y.: Proposal of a failure criterion of adhesively bonded cavitation model—part II: extension to cyclic behavior of connections with silicone. Doctoral thesis, University of transparent silicone adhesives. Glass Struct. Eng. (2019a) Luxembourg (2017) Drass, M., Du Bois, PA., Schneider, J., Kolling, S.: Pseudo-elastic Staudt, Y., Odenbreit, C., Schneider, J.: Failure behaviour of sili- cavitation model—part I: ﬁnite element analyses on thin cone adhesive in bonded connections with simple geometry. silicone adhesives in façades. Glass Struct. Eng. (2019b) Int. J. Adhes. Adhes. 82, 126–138 (2018). https://doi.org/ Drass, M.A., Kraus, M.: Semi-probabilistic calibration of a par- 10.1016/j.ijadhadh.2017.12.015 tial material safety factor for structural silicone adhesives— Zyczkowski, M.: Combined Loadings in the Theory of Plasticity. part I: derivation. Int. J. Struct. Glass Adv. Mater. Res. PWN-Polish Scientiﬁc Publ, Warszawa (1981) 4(2020), 56–68 (2020). https://doi.org/10.3844/sgamrsp. 2020.56.68 Publisher’s Note Springer Nature remains neutral with regard EN 1990 Eurocode—basis of structural design (2002) to jurisdictional claims in published maps and institutional afﬁl- ETAG 002 Guideline for european technical approval for struc- iations. tural sealantglazing kits (2012) Fahlbusch, N.C., Kolupaev, V.A., Becker, W.: Generalized Limit Surfaces—With an Example of Hard Foams, pp. 337–365. Springer, Berlin (2016) Fischer, L.: Das neue Sicherheitskonzept im Bauwesen: ein Leitfaden für Bauingenieure, Architekten und Studen-

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