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Glass Struct. Eng. (2019) 4:403–420 https://doi.org/10.1007/s40940-019-00107-4 SI: GLASS PERFORMANCE PAPER The effects of high strain-rate and in-plane restraint on quasi-statically loaded laminated glass: a theoretical study with applications to blast enhancement Socrates C. Angelides · James P. Talbot · Mauro Overend Received: 14 April 2019 / Accepted: 23 August 2019 / Published online: 21 September 2019 © The Author(s) 2019 Abstract Laminated glass panels are increasingly tive contributions of bending and membrane action, and used to improve the blast resilience of glazed facades, applying the upper bound theorem of plasticity, assum- as part of efforts to mitigate the threat posed to build- ing a tearing failure of the interlayer. Future work aims ings and their occupants by terrorist attacks. The blast to complete the theoretical framework by including the response of these ductile panels is still only partially assessment of plate-action and inertia effects. understood, with an evident knowledge gap between fundamental behaviour at the material level and obser- Keywords Laminated glass · Blast response · vations from full-scale blast tests. To enhance our Strain-rate · In-plane restraint · Post-fracture understanding, and help bridge this gap, this paper adopts a ‘ﬁrst principles’ approach to investigate the effects of high strain-rate, associated with blast load- 1 Introduction ing, and the in-plane restraint offered by blast-resistant frames. These are studied by developing simpliﬁed ana- Renewed focus on the threat posed to buildings and lytical beam models, for all stages of deformation, that their occupants by terrorist attacks has intensiﬁed the account for the enhanced properties of both the glass demand for blast resilient buildings. A high percentage and the interlayer at high strain-rates. The increased of injuries in blast events are glass-related. As a result, shear modulus of the interlayer results in a composite it is now recommended for the glazed facades of com- bending response of the un-fractured laminated glass. mercial and residential buildings under blast threat to This also enhances the residual post-fracture bending include laminated glass panels. These usually consist moment capacity, arising from the combined action of of two layers of annealed glass with a polyvinyl butyral the glass fragments in compression and the interlayer (PVB) polymer interlayer that fails in a more duc- in tension, which is considered negligible under low tile manner than glass alone, and holds the glass frag- strain-rates. The post-fracture resistance is signiﬁcantly ments together after fracture, thereby reducing glass- improved by the introduction of in-plane restraint, due related injuries in blast events. The response of such to the membrane action associated with panel stretch- panels to blast loads is a complex, multi-disciplinary ing under large deﬂections. This is demonstrated by topic that often requires the use of full-scale blast test- developing a yield condition that accounts for the rela- ing to validate designs. It has become common prac- tice for such tests to characterise the panel structural S. C. Angelides (B) · J. P. Talbot · M. Overend response by the (centre of panel) peak-displacement. Department of Engineering, University of Cambridge, Much research has therefore focussed on reproduc- Cambridge, UK e-mail: firstname.lastname@example.org ing the experimentally recorded, peak-displacement 123 404 S. C. Angelides et al. time-histories of laminated glass panels, whether this by the composite action of the interlayer, working in is by ﬁnite-element analysis (FEA) (Hooper 2011; tension, and the compression of the glass fragments Larcher et al. 2012; Hidallana-Gamage 2015; Zhang that come into contact as the panel deforms. However, and Hao 2015; Pelfrene et al. 2016), analytical solutions it was demonstrated from the above experiments that (Yuan et al. 2017; Del Linz et al. 2018) or equivalent this residual bending capacity was negligible, as the single-degree-of-freedom (ESDOF) methods (Special samples collapsed when both glass layers had frac- Services Group, Explosion Protection 1997; Applied tured. This was also the conclusion of the analytical Research Associates 2010; Morison 2007; Smith and modelling presented by Galuppi and Royer-Carfagni −3 Cormie 2009). This approach, of developing models (2018), who calculated a value of 10 for the ratio based on experimentally observed, peak displacements, of the pre- to post-fracture bending stiffness under low enables the structural assessment of laminated glass strain-rates. panels under blast loading without having to perform Whilst providing valuable insight into the behaviour additional, expensive blast testing. However, the fun- of laminated glass, the conditions of these static, lab- damental, underlying physics of the panel response is oratory tests differ from those in real-world buildings. often not explicitly expressed in these models, particu- Firstly, in building facades, rectangular panels of lam- larly those describing the post-fracture response, which inated glass are often supported along all four edges, prevents practicing engineers from understanding fully resulting in a two-way spanning action that inﬂuences the contribution of individual parameters and limits the deformation of the panel. Additionally, a common their ability to optimise designs. In addition, inconsis- requirement for the frames of laminated glass panels tent assumptions between existing models, signify that is the provision of some form of in-plane restraint, a number of aspects of the structural response remain to enhance their overall blast resistance. The Centre only partially understood and the need for an improved for the Protection of National Infrastructure (CPNI) theoretical framework to be developed. recommends, as a minimum, the application of either This paper adopts a ‘ﬁrst principles’ approach that structural silicone sealant (wet glazing) or polysulphide aims to enhance our understanding of the blast response sealant, within a rebate of 30 mm, as illustrated in Fig. 1 of laminated glass panels, by bridging the knowledge (CPNI EBP 2014). Alternatively, elastomeric gasket gap between the fundamental behaviour at the mate- strips (dry glazing) within 35 mm rebates are also con- rial level and the response observed in full-scale blast sidered acceptable. In the American standards, ASTM tests. Our starting point is the static response of simply- F2248-12 (F2248 2012) and PDC-TR 10-02 (PDC-TR supported, axially unrestrained, laminated glass with 2012), the use of adhesive glazing tape for providing a PVB interlayer, loaded laterally (i.e. bending about in-plane restraint is also permitted. This difference in the minor axis). Such specimens form the basis of boundary conditions governs the level of membrane much of the experimental work on laminated glazing, action within a panel, and is expected to inﬂuence sig- as typiﬁed by the work of Kott and Vogel (2003, 2004, niﬁcantly the overall blast resistance. Finally, in gen- 2007). This work reported on four-point bending tests eral, any static testing fails to account for the dynamic aimed at investigating the pre- and post-fracture bend- nature of the blast response, in particular, the effects of ing capacity of laminated glass. It was concluded that, inertia and high strain-rate. The effects of inertia are in the pre-fracture stage, the degree of composite action known to be signiﬁcant under the accelerations experi- depends on the shear-stiffness of the interlayer, with enced by a panel during a typical blast event, but they the response described either by simple bending the- will not be considered further here. Instead, we focus ory, with the glass layers bending independently out- on the effect of strain rate on the panel response, which of-plane (also known as the ‘layered limit’), or by the is believed to be important given the viscoelastic nature sandwich theory of thick faces (also known as the ‘com- of the interlayer. posite response’). This stage terminates once the frac- This paper begins by reviewing previous experimen- ture strength in either of the glass layers is exceeded, tal work that has assessed and quantiﬁed the effect of after which the entire load is carried in bending by the the high strain-rates associated with blast loading on un-fractured layer. In the third and ﬁnal stage, both the material properties of laminated glass. The out- glass layers have fractured and can no longer resist come of this review forms the basis of the subsequently tension. Any residual bending resistance is provided presented analytical models that aim to describe the 123 The effects of high strain-rate and in-plane restraint on quasistatically loaded laminated glass 405 Fig. 1 The blast resistance of laminated glass panels may be enhanced by providing in-plane restraint to the panel, for example, in the form of silicone sealant within the frame enhanced bending capacity of laminated glass under erning the structural response of the glass and PVB are high strain-rates. The inﬂuence of in-plane restraint discussed below. on the quasi-static response under high strain-rates is then evaluated. This quasi-static approach represents a hypothetical load case that simulates blast loading 2.1 Glass layers but uncouples the material and inertial effects within the response, thereby focusing solely on the former Of fundamental concern is the tensile strength of glass and enabling the relative contribution of bending and panels, in the presence of surface ﬂaws developed dur- membrane action developed under large deﬂections to ing manufacturing, installation and service-life. The be assessed. Understanding the effects of high strain- fracture stress (i.e. the stress at which cracking begins) rate and in-plane restraint with these simpliﬁed models is therefore not a material constant and often requires is the primary objective of the work. This is expected testing to determine its exact value, which depends on to enhance the theoretical framework available to prac- the surface quality and size of panel, and the stress ticing engineers, thereby ultimately assisting them in history, residual stress and environmental conditions optimising the blast design of laminated glass pan- (Haldimann et al. 2008). The draft European Stan- els, whether by FEA, ESDOF methods or analytical dard, prEN 13474-3 (2009), recommends a character- solutions. This work is also considered to be a starting istic value for the design fracture strength of annealed point for future research on bridging the knowledge gap glass of 45 MPa, based on a 5% characteristic value between the material response under quasi-static load- that was obtained from 741 static tests performed on ing and full-scale blast testing, which will focus on the 6 mm thick, annealed glass panels. High strain-rates effects of inertia and two-way spanning plate-action. result in an enhanced fracture strength, as ﬂaws require time to develop into cracks (Overend and Zammit 2012; Larcher et al. 2012). This is observed in multiple, high 2 The inﬂuence of strain rate on material strain-rate experimental tests performed by Nie et al. properties (2009, 2010), Peroni et al. (2011), Zhang et al. (2012) and Meyland et al. (2018). For the blast design of glaz- The pressure time-histories resulting from the detona- ing, recommended dynamic fracture strength values tion of high-explosives are of short duration, typically are presented by Smith and Cormie (2009) that were of the order of milliseconds. These result in high strain- derived by extrapolating to the high strain-rates asso- rates in any façade panels exposed to the blast. Mean ciated with blast loading the inherent, static strength −1 strain-rates ranging from 7.6 to 17.5 s were recorded value of annealed glass presented in prEN 13474- in fractured laminated glass panels by Morison (2007), 3using Brown’s integral (risk integral) for stress fatigue −1 and from10to30s by Hooper (2011), both during (also known as sub-critical crack growth), and superim- open ﬁeld, high-explosive blast tests. The inﬂuence of posing the relevant surface pre-stress from the thermal these strain rates on the salient material properties gov- processing of heat-strengthened and toughened glaz- 123 406 S. C. Angelides et al. ing products. Lower strain-rates were considered in the nal compressive strength of annealed glass (σ = g,c above extrapolation for monolithic glazing, compared 248 MPa) for use under high strain-rates: to the mean values recorded experimentally by Mori- −5 DIF = 1.189 + 0.049 log ε ˙ for 1 ≤ ε ˙ ≤ 100 (1) ( ) son (2007) and Hooper (2011) for fractured laminated glass, as strain-rates depend on the maximum deﬂec- −1 For a typical strain-rate of 10 s , representative of tion of the panel. As the pre-fracture stage of lami- blast response, Eq. 1 yields a dynamic, compressive nated glass is limited to smaller deﬂections, compared strength for annealed glass of σ = 323 MPa. In g,c to the overall response, the fracture strength derived for practice, however, the Poisson’s ratio effect and any monolithic glazing is also considered appropriate for buckling will generate tensile stresses under com- laminated glass. Table 1 shows that these values are in pressive loading, thereby resulting in a nominal com- good agreement with the recommendations of the UK pressive strength lower than the theoretical value Glazing Hazard Guide, which were established from a (Haldimann et al. 2008). The application of the com- signiﬁcant number of independent blast tests (Morison pression strength of glass, experimentally derived with 2007). However, comparisons with more recent, full- Split Hokinson Pressure Bar tests, to the post-fracture scale blast tests on laminated glass panels by Del Linz response of laminated glass therefore requires further et al. (2018) have suggested that a fracture strength experimental validation. of 100 MPa for annealed glass is more realistic. Fur- thermore, it should be noted that the dynamic fracture strength is also dependent on the boundary conditions and geometry of the panel, as demonstrated by blast 2.2 Interlayer tests performed by Osnes et al. (2018) on monolithic annealed glass. As a ﬁnal comment, the recommended The most popular interlayer for laminated glass pan- design fracture strength values shown in Table 1 refer to els is polyvinyl butyral (PVB), which is a ductile vis- new (as-received) glass. In reality, ﬂaws accumulate in coelastic polymer that is temperature and strain-rate the glass surface over its service-life, and this has been dependent. The main reasons that often make it the shown to signiﬁcantly reduce the fracture strength (Dat- preferred interlayer are its ability to block UV radi- siou and Overend 2017a, b). This reduction in strength ation, its high strain at failure and its good adhesion of aged glass was also demonstrated in the high strain- properties, which enable it to retain glass fragments rate experiments by Kuntsche (2015) and Meyland et al. following the fracture of the glass layers (Haldimann (2018), but this is not considered further here. et al. 2008). In contrast to the low strain-rate, hypere- Compressive and tensile tests under high strain- lastic material behaviour observed in uni-axial tension rates performed by Zhang et al. (2012) concluded tests of PVB alone, for the higher strain-rates associated that the Young’s modulus of annealed glass is insen- with blast loading the behaviour resembles an elastic– sitive to strain-rate. Therefore, in this paper, the value plastic response with strain hardening (Kott and Vogel of Young’s modulus E = 70 GPa recommended by 2003; Bennison et al. 2005; Iwasaki et al. 2007;Mori- prEN 13474-3 (2009) for low strain-rates will be also son 2007; Hooper et al. 2012a; Kuntsche and Schnei- assumed for blast loading. der 2014; Zhang et al. 2015; Chen et al. 2017). It is Following the fracture of the glass layers in a lam- noted that the above behaviour can vary for different inated panel, the attached glass fragments can still PVB types and depending on the manufacturer. Addi- provide resistance by generating compressive contact tionally, temperature dictates the strain-rate at which stresses, as discussed in Sect. 3. The effects of high the observed behaviour transitions from hyperplastic strain-rate on the compressive strength of glass must to elastic–plastic. However, the focus of this paper is therefore also be investigated. The theoretical, com- on the effects of strain-rate at constant, room temper- pressive strength of glass is signiﬁcantly higher than ature. Although thermo-rheological effects cause the the fracture strength, as surface ﬂaws will not grow temperature of the interlayer to increase at high strain- and propagate under compression. Dynamic increment rates, this is not explicitly accounted for here. Rather factors (DIF) were derived by Zhang et al. (2012)from than adopting a viscoelastic material model, a simpler dynamic compression tests using a Split Hopkinson elastic–plastic model is assumed for the pre-fracture Pressure Bar, to amplify the low strain-rate, nomi- blast assessment of laminated glass. 123 The effects of high strain-rate and in-plane restraint on quasistatically loaded laminated glass 407 Table 1 Recommended design values for glass fracture strength of low (prEN 13474-3) and high (Smith and Cormie 2009; Morison 2007) strain-rates Glass type Fracture strength, σ (MPa) g,t −5 −1 −1 Low strain-rates (∼ 10 s ) High strain-rates (0.4−0.6s ) prEN 13474-3 Smith and Cormie (2009) UK Glazing Haz- ard Guide (Morison 2007) Annealed (new) 45 80 80 Heat-strengthened (new) 70 100–120 120 Toughened (new) 120 180–250 180 Toughened (aged) (Kuntsche 2015) 80 116 N/A Values for aged, toughened glass are also included for direct comparison (Kuntsche 2015). The low strain-rate characteristic values should be used in combination with a material safety factor and additional factors that account for load duration, glass surface proﬁle and edge bending strength, as speciﬁed in prEN 13474-3 It is worth noting that Morison (2007) highlighted ite action of the interlayer together with the attached that the sudden stiffness reduction observed following glass fragments. The glass fragments effectively pre- the apparent yielding of PVB is in fact non-linear vis- vent the PVB from extending where it remains bonded coelasticity, as permanent strain is not observed fol- to the glass. Therefore, instead of having a uniform lowing the removal of the load. The description of an extension over the entire area of the PVB, the exten- elastic–plastic behaviour of PVB at high strain-rates sion is discretised at the bridges between the fragments, therefore refers to the bilinear shape of the stress–strain causing signiﬁcantly larger but localised strains. Thus, response and not to true plasticity. However, this paper under blast loading, stiffening effects are mobilised by considers only the loading phase of the PVB (corre- both high strain-rates and the attached glass fragments. sponding to the positive phase of blast loading) for This has been reported by Morison (2007), Hooper which a simple linear-elastic model is acceptable. Dur- (2011) and Zobec et al. (2014). Hooper (2011) per- ing the pre-fracture stage of a panel response, the inter- formed a series of uni-axial tension tests on fractured, layer is not expected to yield due to the dominant stiff- laminated glass specimens for a range of strain-rates. ness of the glass layers. This assumption is supported by An elastic–perfectly-plastic stress–strain response was the ﬁndings of Dharani and Wei (2004), Wei and Dha- observed in these tests, governed by the delamination rani (2005) and Hooper et al. (2012b), both of whom of the attached glass fragments from the interlayer that presented a ﬁnite-element sensitivity analysis of the differs from the elastic–plastic with strain hardening blast response of laminated glass panels and concluded (bilinear) material law observed for the PVB alone. that the viscoelastic shear relaxation modulus may be Additionally, a brittle failure was observed in these replaced by the instantaneous shear modulus (G = tests for thinner PVB interlayers, while for interlayers pvb 0.178 GPa). This value is adopted for the pre-fracture thicker than 1.52 mm, the delamination front travelled analytical models presented in Sect. 3.1. Assuming quickly, relieving the interlayer from excessive strains a Poisson’s ratio of ν = 0.49 for the essentially and preventing premature tearing. This conclusion is in pvb incompressible PVB, the corresponding Young’s mod- agreement with the UK Glazing Hazard Guide recom- ulus (E = 0.53 GPa) can then be obtained (Morison mendation for a minimum PVB thickness of 1.52 mm pvb 2007), although the ﬁnal response is insensitive to the for blast-resistant laminated glass (Morison 2007). A particular value due to the dominance of the glass layer brittle failure, however, may also occur for thicker inter- stiffness pre-fracture. layers, if a high adhesion grade is speciﬁed for the panel Following the fracture of the glass layers, the that prevents the attached glass fragments to locally response of the PVB interlayer is inﬂuenced by the delaminate from the interlayer, as observed in the full- presence of the attached glass fragments. A new consti- scale blast tests presented by Pelfrene et al. (2016). It is tutive law is therefore required to describe the compos- clear that the attached glass fragments inﬂuence both 123 408 S. C. Angelides et al. Table 2 Material properties for cracked, laminated glass with 10 mm fragment length and 1.52 mm interlayer, derived experimentally for low and high strain-rates by Hooper (2011) −1 −1 Composite material properties Low strain-rates (0.1 s ) High strain-rates (10 s ) Yield strength, σ (MPa) 7 17 pvb,c,y Strain failure, ε (%) 150 170 pvb,c,f Initial Young’s modulus, E (GPa) 0.3 1.7 pvb,c the elastic and plastic response of the PVB within a ence of high strain-rate on the structural response, this fractured panel. section presents simpliﬁed analytical models to cal- For the purposes of the post-fracture analysis pre- culate the enhanced moment capacity of laminated sented in Sect. 3, the PVB may be characterised by glass, by adopting the values established in Sect. 2 Hooper’s (2011) high strain-rate values of the three for the salient material properties. These simple bend- material properties summarised in Table 2, that is, ing models, corresponding to unrestrained, simply- the yield strength, initial Young’s modulus and failure supported beams under uniformly distributed loading, strain. These values were derived experimentally by can then be compared directly with the existing low Hooper (2011) for specimens with uniform cracking strain-rate experimental results described in Sect. 1. pattern consisting of 10 mm fragments, which aimed The enhanced resistance provided by the introduction to simulate the failure observed from full-scale blast of lateral restraints, as described in Sect. 1, is then eval- tests of laminated glass panels with annealed glass uated through the consideration of a quasi-static load- layers, and for a 1.52 mm thick interlayer, typical of ing under high strain-rates that enables an investigation façade glazing panels. However, it is recommended by of the material effects alone, leaving inertial effects Hooper (2011) that a strain limit of 20% is applied to to be considered later. Important conclusions can be the blast design of laminated glass to prevent complete drawn from a better understanding of the quasi-static debonding of the attached glass fragments. Although response, including the relative contribution of bending it is acknowledged that there is degree of uncertainty, moments and membrane forces to the overall bending this limit is also adopted here, with further experi- resistance. mental work required to validate this design limit. To gauge the inﬂuence of cracking patterns, Hooper (2011) 3.1 Simple bending analysis also derived material properties for both 20 mm frag- ments and non-uniform cracking patterns. It should The static response of sandwich beams with low, core also be noted that these values can vary for different shear stiffness is governed by both bending and shear PVB types (stiffness and adhesion level) and ambient deﬂections. When the core shear stiffness is negligi- temperatures. ble, the faces of a sandwich beam bend independently For comparison, Table 2 summarises material prop- as two separate beams, known as the ‘layered limit’. erty values recorded under two different strain-rate val- Additionally, in the case of thick faces, local bending ues during uni-axial tension tests performed by Hooper of the faces about their own centroidal axes also inﬂu- (2011). The effect of strain rate on the material prop- ences the shear deformation of the core (Allen 1969). erties is clear. The consequences for the structural The pre-fracture phase of laminated glass under static response are now investigated in Sect. 3. loading falls within this category, with available analyt- ical solutions for various loading and boundary condi- 3 Analytical models accounting for high tions presented by Galuppi and Royer-Carfagni (2012). strain-rates and in-plane restraint If on the other hand, the shear stiffness of the core is sufﬁciently high that the plane sections remain plane The effects of high strain-rate and in-plane restraint the response of the sandwich beam is in pure bending on the blast response of laminated glass panels are of the whole cross-section, known as the ‘monolithic often obscured in current models. To assess the inﬂu- limit’. The above described ‘monolithic’ behaviour is 123 The effects of high strain-rate and in-plane restraint on quasistatically loaded laminated glass 409 Fig. 2 Bending strain and stress distributions within a laminated glass beam for the elastic stages of deformation under high strain-rates (not drawn to scale) considered to describe the blast response of laminated zero bending strain, can be then obtained from the glass, due to the enhanced shear modulus of PVB under ﬁrst moment of area of the transformed cross-section. the high strain-rate, as described in Sect. 2.2, which is The bending strain and stress distributions for each now capable of transferring the horizontal shear forces stage are shown in Fig. 2, and the corresponding bend- (Norville et al. 1998; Kuntsche and Schneider 2013). ing moment and curvature capacities are provided in Thus, in the unfractured phase (Stage 1), when both Table 3. glass layers remain unfractured, the laminated glass The bending moment capacity of Stage 1 (M ) is acts compositely as a single beam. The same approach deﬁned as the moment required for the tensile frac- is adopted by the blast analyses cited in Sect. 1, and the ture strength of glass under high strain-rates (σ ) to g,t validity of this will be conﬁrmed in Sect. 4.1. be exceeded. Due to the sagging response of the com- In this section, analytical models for simply- posite structure under a transverse loading (as drawn supported, axially unrestrained laminated glass under in Fig. 2), the bottom glass layer will be in tension and high strain-rate are presented, assuming a simple bend- therefore fracture ﬁrst. The bottom glass layer will then ing response and the same elastic stages of defor- no longer contribute to the bending capacity, the bend- mation (Stages 1, 2 and 3) deﬁned by Kott and ing stresses will be distributed only in the top glass Vogel (2003, 2004, 2007) and described in Sect. 1. layer and the interlayer, and the response will be gov- Based on the Euler-Bernoulli theory, that plane sec- erned by the new transformed section (I ). The bend- g,2 tions remain plane during bending, with small displace- ing moment capacity of Stage 2 (M < M ) that is 2 1 ments/rotations assumed, an equivalent, transformed deﬁned as the moment required for the tensile fracture cross-section made entirely from either glass (I , I strength of glass under high strain-rates (σ ) to be g,1 g,2 g,t and I ) or PVB (I I and I ) will be con- exceeded in the top glass layer, will be smaller com- g,3 pvb,1, pvb,2 pvb,3 sidered for the analysis, based on the modular ratio pared to Stage 1, due to the reduced second moment of E E g g area (I < I ). This will result in an abrupt transi- of the two materials ( or ). Both transformed g,2 g,1 E E pvb pvb,c tion between Stage 1 and 2 in the moment–curvature sections should result in the same bending stiffness relationship, as the bending moment that caused frac- (E I = E I or E I = E I ). The positions g g pvb pvb g g pvb,c pvb ture in Stage 1 cannot be sustained in Stage 2 under of the elastic neutral axes from the top of the cross- static loading. For short duration pulses, such as blast section in the three stages (y , y and y ), which deﬁne 1 2 3 123 410 S. C. Angelides et al. Table 3 Bending moment and curvature capacities of laminated ing cross-section until the entire interlayer has yielded glass for each stage under high strain-rates (Stage 3–4 shown in Fig. 3). At this point, the inter- layer has no remaining capacity and, for a constant Stage Moment capacity Curvature capacity bending moment, the tensile strains in the interlayer σ I g,t g,1 M 1M = κ = 1 1 y E I will increase due to the elastic–perfectly plastic mate- 1,g g g,1 σ I g,t g,2 2 rial law. As demonstrated experimentally by Delincé 2M = κ = 2 2 y E I 2,g g g,2 et al. (2008), for the post-fracture response of laminated σ I pvb,c,y pvb,3 M 3M = κ = 3 3 y E I 3,pvb pvb,c pvb,3 glass with a stiffer interlayer, the assumption of plane g,c 4M = y C κ = 4 4,g 4 4 3 y sections remaining plane and a linear strain distribu- 4,g pvb + y − T 4,pvb 4 tion still holds. The compressive strains in the fractured glass will therefore also keep increasing. Considering a linear stress–strain distribution for fractured glass under compression, the stresses in the top glass layer will also increase. To satisfy longitudinal equilibrium loads, the response is different due to the inertia effects. under bending alone, which requires the resultant com- As this is beyond the scope of this paper, each Stage pressive force in the glass and the resultant tensile force will be assessed independently for its moment capac- in the interlayer to be equal, the plastic neutral axis will ity. Future experimental tests will be used to simulate shift upwards. As the bending moment capacity can these stages and validate the derived capacities. be deﬁned from moment equilibrium about the plas- In Stage 3, both glass layers have fractured but the tic neutral axis, the upwards shift of the plastic neutral interlayer still behaves elastically (σ < σ ). pvb pvb,c,y axis will result in an increase of the bending moment The bending resistance is therefore provided by the capacity. At the instance where the cross-section has no composite action of the interlayer in tension and the reserve moment capacity (Stage 4 shown in Fig. 3)the attached glass fragments in the top glass layer gener- plastic neutral axis (y ) can be calculated from longi- ating compressive contact stresses at glass fragment tudinal equilibrium, considering the compressive force interfaces (I ). The post-fracture, elastic moment pvb,3 in the top glass layer that initiates crushing of the glass capacity (M ) is deﬁned as the bending moment fragments (C = 0.5Bσ y ) and the tensile force 4 g,c 4 required to cause yielding (σ ) in the extreme ﬁbre pvb,c,y capacity of the interlayer (T = Bt σ ), where 4 pvb pvb,c,y of the interlayer (y ). pvb,3 B is the width of the cross-section. In the above calcu- Once the extreme ﬁbre of the interlayer has yielded, lation, the local delamination of the interlayer from the the plastic response will initiate. By using the elastic– attached glass fragments, which allows higher strains perfectly-plastic material law, described in Sect. 2.2, to develop and prevents sudden tearing, is implicit in any additional stress will be distributed to the remain- Fig. 3 Bending strain and stress distributions within a laminated glass beam for the plastic stages of deformation under high strain-rates (not drawn to scale) 123 The effects of high strain-rate and in-plane restraint on quasistatically loaded laminated glass 411 Fig. 4 Collapse mechanism for simply-supported axially unrestrained laminated glass beam the yield strength value (σ ) that was derived from bending moments (M and M ) and membrane forces pvb,c,y 1 2 uni-axial tensile tests of cracked laminated glass spec- (N and N ), the corresponding transverse displace- 1 2 imens, as discussed in Sect. 2.2. A constant interlayer ments (w and w ) need to be ﬁrst obtained, due to the 1 2 thickness has, however, been assumed, ignoring neck- nonlinearity of the equilibrium equation arising from ing effects, for the purpose of calculating the section the introduction of membrane forces. Again, assuming capacity. The ultimate moment capacity (M ), provided full shear transfer under high strain-rates, the transverse in Table 3, is then obtained by considering moment deﬂection due to a uniformly distributed load (p) can equilibrium about the plastic neutral axis, while the cur- be obtained by solving the fourth-order, nonlinear dif- vature (κ ) is calculated from the linear strain distribu- ferential equation, derived by Asık ¸ and Tezcan (2005) g,c tion, where ε ( ) is the crushing strain of glass. As for laminated glass: g,c 4 2 the cross-section cannot sustain any additional bending d w (x) d w (x) 1 1 (2a) − E I + N + p = 0 g,1 g,1 4 2 dx dx moments, a plastic hinge has formed and a mechanism 4 2 d w (x) d w (x) 2 2 (2b) developed for the simply-supported beam, as shown in − E I + N + p = 0 g,2 g,2 4 2 dx dx Fig. 4. For the fully-fractured laminated glass (Stages 3 and 4), a rigid-perfectly-plastic material law is adopted for 3.2 Combined bending and membrane analysis simplicity to derive the relative contribution of bending moments (M ) and membrane forces (N ), ignoring 4 4 In contrast to the simple bending response of axially the elastic contributions of Stage 3. During Stage 4, the unrestrained beams, as presented in Sect. 3.1, the intro- combination of the tensile force (T ), associated with duction of axial restraints results in a combined bending the bending moment (M ), and the membrane force and membrane action when the deﬂection is sufﬁciently (N ) cannot exceed the PVB tensile capacity (N = large, as shown in Fig. 5. σ t B ), as shown in Fig. 6. An increase in pvb,c,y pvb pvb As with the simple bending analysis, the membrane the membrane force will therefore cause a decrease in stress distribution during the elastic Stages 1 and 2 can the bending moment and a subsequent upwards shift of be determined by transforming the composite cross- the plastic neutral axis (y ). When the latter has reached section into an equivalent glass section (A ). The addi- the top of the cross-section (y = 0), a pure membrane tional contribution of the membrane stresses results in response will result. an upwards shift of the elastic neutral axis (y < y and The yield condition for laminated glass, ensuring y < y ) from the positions under pure bending (see that the material law is not violated, therefore needs Sect. 3.1). To determine the relative contribution of to account for the relative contribution of bending 123 412 S. C. Angelides et al. Fig. 5 a Simply-supported beam axially unrestrained under pure bending, b axially restrained beam under combined bending and membrane action Fig. 6 Strain and force distribution within a laminated glass beam for the plastic stage of deformation (Stage 4) under combined bending and membrane action 8M moments and membrane forces. Following a similar collapse load p = for simply-supported, unre- c 2 approach to that considered by Sawczuk and Winnicki strained laminated glass beams, the loading for axially (1965) for singly-reinforced-concrete, this is derived by restrained beams can be written in non-dimensional eliminating the unknown position of the plastic neu- form: tral axis (y ) in the dimensionless equivalents of the ⎧ 4 2 2 5W + 12y −7t W+6y t +t +2t ( 4 g) 4( g pvb) ⎪ g bending (m) and membrane (n) equations deﬁned in , W ≤ t ⎨ g 1 1 12y t +t − y p 4 pvb g 4 2 3 “Appendix A.1” section: = (4) p t c pvb g + +W 2 2 4y − 3t ( ) y ⎩ 4 g 2 4 , W ≥ t 1 1 g m − n + n −1 = 0 t +t − y (3) pvb g 4 3t +6t −2y 3 2 3 ( ) pvb g 4 t +3t −y pvb g 4 The above load–displacement relationship is valid for A load–displacement relationship for laminated glass an interlayer with inﬁnite ductility. Thus, the failure dis- beams in Stage 4 can then be derived from the above placement (W ) for PVB laminated glass can be deﬁned yield condition by applying the upper-bound theorem by equating the total longitudinal strain in the interlayer of plasticity for ﬁnite deﬂections on the same col- (ε ) to the PVB failure strain (ε ) deﬁned in pvb,c,f lapse mechanism introduced in Sect. 3.1 under sim- 4,pvb Sect. 2.2, with the derivation provided in “Appendix ple bending, as shown in Fig. 4. This assumption, of A.2” section: the same initial mechanism occurring under both bend- ing and combined bending–membrane action, is com- ε = ε ⇒ pvb,c,f 4,pvb monly adopted in plastic analysis (Jones 2011). Under 2 2 a uniformly distributed load (p) and the linear veloc- 2t 2t 4t 2W g g g t + + − = ε pvb pvb,c,f LL LL LL LL ity proﬁle of the collapse mechanism w ˙ (x) = W , h,1 h,1 h,2 h,2 L/2 the load–displacement relationship for the mid-span (5) deﬂection (W) is obtained by equating the exter- nal work rate (EW = pw ˙ (x)dx) to the internal The plastic hinge length (L ), deﬁned in Fig. 6 as the energy rate dissipated at the location of the plas- delaminated length of the interlayer from the attached ˙ ˙ tic hinge ED = (M + NW)θ . Considering the glass fragments, can be obtained from Eq. 6, as derived midspan 123 The effects of high strain-rate and in-plane restraint on quasistatically loaded laminated glass 413 by Belis et al. (2008) under pure bending: Table 4 Material properties for laminated glass considered in the analysis 2 σ t 2 0 pvb,c,y pvb L − σ t τ+ L pvb,c,y pvb h Material property Low strain-rates High strain-rates 2 L Glass fracture 45 80 + σ t =0(6) pvb,c,y pvb strength (MPa) where τ is the shear stress at the interface between the Glass Young’s 70 70 modulus (GPa) interlayer and the glass fragment and is the fracture Glass N/A 323 energy of the interface. compressive For midspan deﬂections less than the top glass layer strength (MPa) thickness (W ≤ t ), a single plastic hinge is formed, PVB shear N/A 0.178 as shown in Fig. 4. Thus, the above equation can be modulus (GPa) modiﬁed to also account for extension due to the axial Yield strength of fractured N/A 17 restraint, to derive the plastic hinge length for combined laminated glass (MPa) bending and membrane action (L ): h,1 Strain failure of N/A 20% fractured L = L + L m b laminated glass 2W 4W t W Initial Young’s modulus of N/A 1.7 = + t + − pvb L L 2 2 fractured laminated glass (GPa) 4W t = t + , W ≤ t (7a) pvb g L 2 For midspan deﬂections greater than the top glass layer thickness (W ≥ t ), a pure membrane response will 4.1 Inﬂuence of high strain-rate occur, with the extension due purely to the large deﬂec- tions under the axial restraint: The moment–curvature relationship for the axially unrestrained, laminated glass is plotted separately for 2W L = L = , W ≥ t (7b) m g all Stages in Fig. 7, assuming, for simplicity, no per- manent curvature at the end of each Stage, as discussed A single plastic hinge will no longer form and the entire in Sect. 3.1. This allows direct comparison with future beam will behave as a string (Jones 2011), resulting in experimental results that will assess each Stage inde- a deformation length (L ) equal to the total delami- h,2 pendently. The moment (M , M , M and M ) and cur- nation occurring from each crack: 1 2 3 4 vature (κ , κ , κ and κ ) capacities under high strain- 1 2 3 4 L = L N = L − 1 (8) h,2 h c h f rates, are computed from the equations presented in where L is the length of each fragment. Table 3. These may then be compared with the low strain-rate values, to assess directly the inﬂuence of high strain-rate. For Stages 1 and 2, the low strain- 4 Results and discussion rate moment and curvature capacities are calculated for the layered limit, assuming no shear transfer in the The analytical models introduced in Sect. 3 have been interlayer. The elastic moment–curvature relationship solved numerically for the case of simply-supported is therefore used (M = EIk), considering only the sec- laminated glass beams with annealed glass layers (t = ond moment of area about the centroidal axis of each 3 3 6mm) and a PVB interlayer (t = 1.52 mm), and for pvb Bt Bt g g glass layer (I = and I = ), assuming that the 1 2 6 12 two different span lengths (L = 200 mm and 1000 mm) glass layers bend as two separate beams. For Stages and a constant width (B = 55 mm). To validate the 3 and 4, there is no reserve capacity for low strain- assumed monolithic beam action under high strain- rates, as demonstrated by the experimental work of rates,and investigate the relative bending/membrane Kott and Vogel (2003, 2004, 2007) described in Sect. 1, contribution under in-plane restraint, both a short and a and therefore the moment–curvature relationship is not long span are considered. The material properties con- plotted. sidered in the analysis are introduced in Sect. 2 and the speciﬁc values used are summarised in Table 4. 123 414 S. C. Angelides et al. Stage 1 Stage 2 Stages 3 and 4 (a) (b) (c) High strain-rate High strain-rate High strain-rate Low strain-rate Low strain-rate M M 140 10 1 2 κ κ κ κ 1 2 3 0 0 0 0 102030 0 0.1 0.2 0.3 0 0.2 0.4 Curvature [1/m] Curvature [1/m] Curvature [1/m] Fig. 7 Moment–curvature graphs for axially unrestrained laminated glass beams under simple bending, plotted separately for each stage a Stage 1, b Stage 2, c Stages 3 and 4. Note: Linear approximation assumed for simplicity for Stage 4 plotted in Fig. 6c Figure 7a shows that the moment capacity of stage Table 5 Comparison of mid-span deﬂections for high strain- rates under simple bending and the enhanced effective-thickness 1 at high strain-rates is increased by more than a factor approach presented by Galuppi and Royer-Carfagni (2012) of four compared to the low strain-rates value. This is Maximum deﬂection (mm) L = 200 mm L = 1000 mm due to the enhanced shear modulus of the interlayer at high strain-rates, resulting in the laminated glass Simple bending 0.70 17.61 working as a monolithic beam, with an enhanced sec- Enhanced effective-thickness 0.94 17.87 ond moment of area. This is an upper bound limit, as approach there may also be some shear transfer by the inter- layer even under low strain-rates and therefore enhanc- ing the layered response considered. To validate the (2012), it can be concluded that, for large spans (η = 1 assumption of monolithic bending under high strain- for L = 1000 mm), a monolithic beam assumption is rates, the calculated maximum, mid-span deﬂection valid for laminated glass beams under blast loading, may be compared to that calculated using the enhanced while for shorter spans (η = 0.93 for L = 200 mm) the effective-thickness approach presented by Galuppi and contribution of shear deﬂections should be accounted Royer-Carfagni (2012), which accounts for ﬁnite shear for. coupling between the interlayer and the glass layers. In Stage 2, as expected, the almost identical response As shown in Table 5, almost identical deﬂections are shown in Fig. 7b for high and low strain-rates indi- obtained for the larger span (L = 1000 mm), while cates that the PVB contribution to bending is negligi- some shear deﬂection contributions are evident for the ble, and the entire bending resistance is provided from shorter span (L = 200 mm). It is, however, noticed that the remaining unfractured glass layer. The increased the deﬂections in Stage 1 for the short span are limited moment capacity under high strain-rates results from to less than 1 mm. Considering the nondimensional the enhanced fracture strength of glass. Finally, for coefﬁcient η (η = 0 for layered limit, η = 1 for mono- Stages 3 and 4, the effect of high strain-rate, compared lithic limit) introduced by Galuppi and Royer-Carfagni to the negligible capacity under low strain-rates that is Moment [Nm] Moment [Nm] Moment [Nm] The effects of high strain-rate and in-plane restraint on quasistatically loaded laminated glass 415 not plotted, is clearly visible from Fig. 7c, which shows 4.2 Inﬂuence of in-plane restraint the residual post-fracture bending capacity. This resem- bles the low strain-rate response of stiffer interlayers, The elastic, mid-span deﬂections (Stages 1 and 2) under such as ionomers or added steel-wire mesh reinforce- simple bending and combined bending–membrane ment, as experimentally demonstrated by Delincé et al. action for laminated glass are shown in Fig. 8 for both (2008) and Feirabend (2008) respectively. the short and long span beams. To derive the com- The moment capacity for Stage 3 shown in Fig. 7c bined bending/membrane deﬂections under in-plane represents a lower bound solution, as the material prop- restrained boundary conditions, Eqs. (2a) and (2b)have erties introduced in Sect. 2.2 are for a uniform crack- been solved with the Galerkin method, considering an ing pattern with a 10 mm glass fragment length. This approximate, sinusoidal deﬂected shape (Wierzbicki assumption implies that all cracks are aligned between 2013). The maximum deﬂections of each Stage were the two glass layers and therefore a series of rigid and calculated iteratively numerically, by limiting the total ﬂexible sections are present along the length of the longitudinal tensile stress (the sum of bending and beam. If the cracks are not aligned, a higher Young’s membrane stresses) to the tensile fracture strength modulus and yield strength will result, as reported by of glass under high strain-rates given in Table 1. Nhamoinesu and Overend (2010), who compared the These were validated with the solutions presented by results from through-crack and offset-crack uniaxial Timoshenko and Woinowsky-Krieger (1959), which tensile tests on laminated glass specimens with a single assumes cylindrical bending rather than the sinusoidal crack under low strain-rates. A similar conclusion was form assumed here. also reported in the analytical model for out-of-plane Figure 8 shows that for short spans, the entire bending of fractured laminated glass under low strain- response is governed by bending in Stage 1, with small rates presented by Galuppi and Royer-Carfagni (2018). membrane contributions evident in Stage 2. In contrast, The drawback of cracks that are not aligned, however, for large spans, membrane contributions are also evi- is a more brittle failure with almost no residual plastic dent in Stage 1, resulting in smaller deﬂections com- capacity (Nhamoinesu and Overend 2010). In this case, pared to simple bending. It is clear that membrane Stage 4 may never develop. The anticipated fracture action dominates the response in Stage 2. In addition pattern under blast loading, and its consequences for to smaller deﬂections for the larger span, the pres- the analysis presented here, is reserved for future work. ence of axial restraint enhances the load capacity by Another useful comparison can be made with rein- factors of 1.4 and 4.8 in Stage 1 and 2 respectively. forced concrete beams, which also consist of a brittle This means that the laminated glass can resist a sig- material (concrete) reinforced with a ductile material niﬁcantly larger load before fracture, compared to the (steel) to carry tension. Although the same method- unrestrained boundary condition. To develop such large ology is applied to evaluate the moment capacities membrane forces, however, the connection details and of the cross-section for each Stage (the transformed- the supporting frame must also be adequately designed section method for the elastic stage and moment equi- to ensure that they can resist the horizontal reaction librium for the ultimate moment capacity), it is evi- forces developed. dent that the moment–curvature response is quite dif- The load–displacement relationship for Stage 4 is ferent. In laminated glass, the residual, post-fracture showninFig. 9. The midspan deﬂection is calculated by plastic moment capacity of Stage 4 is smaller than solving Eq. (4) numerically, and nondimensionalised the pre-fracture moment capacity of Stage 1, while in by dividing by the top glass layer thickness. The fail- reinforced-concrete the ultimate plastic moment capac- ure deﬂection, at which tearing of the interlayer occurs, ity is larger than the uncracked moment capacity. The is given by Eq. (5), assuming an arbitrarily delamina- upper-bound theorem of plasticity, frequently applied tion length of 8 mm, as the solution of Eq. (6) resulted in the structural analysis of reinforced concrete beams in delaminated lengths greater than the assumed glass and slabs (BS EN 1992 2004), would therefore not fragment length, thus causing glass fragmentation. This result in efﬁcient designs for axially unrestrained, lam- assumption requires further investigation by experi- inated glass beams due to the rigid-plastic material law mental testing. However, it is considered conservative approximation that ignores the elastic contributions. for the current study, as a shorter plastic hinge will result in larger strains. For predicted deﬂections that are 123 416 S. C. Angelides et al. Stage 1 (L = 200mm) Stage 1 (L = 1000mm) (a) (b) 30000 1500 20000 1000 10000 500 Axially restrained Axially restrained Axially unrestrained Axially unrestrained 0 0 0 0.2 0.4 0.6 0.8 0 5 10 15 20 Deflection [mm] Deflection [mm] Stage 2 (L = 200mm) Stage 2 (L = 1000mm) (c) (d) 6000 1500 Axially restrained Axially unrestrained 4000 1000 2000 500 Axially restrained Axially unrestrained 0 0 0 0.5 1 1.5 2 0 10203040 Deflection [mm] Deflection [mm] Fig. 8 Load-deﬂection diagrams for elastic stages (Stages 1 restrained) for two different spans a Stage 1 for L = 200 mm, b and 2), comparing the response under simple bending (axially Stage 1 for L = 1000 mm, c Stage 2 for L = 200 mm, d Stage 2 unrestrained) and combined bending–membrane action (axially for L = 1000 mm comparable to the span lengths, the validity of the small in practice, such as in the water bag tests performed by angle approximation and binomial expansion, consid- Zobec et al. (2014). ered in the development of the analytical models in Sect. 3.2, requires further investigation. 5 Conclusions The linear relationship observed in Fig. 9 for nondi- mensional deﬂections greater than 1 indicates the This theoretical study has adopted a ‘ﬁrst-principles’ pure membrane response, with the combined bend- approach to assess the effects of high strain-rate and ing/membrane action limited to deﬂections less than in-plane restraint on the structural response of PVB- the top glass layer thickness, as described by Eq. (4). laminated glass. A review of the material behaviour This results in signiﬁcant enhancement of the post- of glass and PVB has shown that the salient mate- fracture capacity, compared to the collapse load of rial properties are enhanced signiﬁcantly under the unrestrained laminated glass. Additionally, it is also high strain-rates associated with blast events. Analyt- observed that this is higher than the fracture load of the ical models for laminated glass have been developed, intact panel in Stage 1, highlighting the important con- which describe the four stages of quasi-static response tribution of axial restraint in the reserve, post-fracture from initial loading to failure. These indicate that the capacity. This conclusion is in agreement with the low enhanced material properties have a signiﬁcant effect strain-rate, pressurised water tests of laminated glass on the moment and curvature capacities of the glass panels presented by Morison (2007) and Zobec et al. in simple bending. Speciﬁcally, the results have shown (2014), who recorded post-fracture loads in excess of that, for high strain-rates, the enhanced shear modu- the unfractured capacity and a dominant membrane lus of the PVB justiﬁes a monolithic beam approach response. It should be noted, that the analysis presented to panel analysis. Furthermore, the enhanced modulus here considers tearing to result only from combined results in signiﬁcantly enhanced moment capacities of bending/membrane strains and has ignored the pos- each stage, including a residual post-fracture bending sible rupture of the interlayer caused by the attached capacity that is considered negligible for low strain- glass fragments, although the latter has been observed rates. Load [N/m] Load [N/m] Load [N/m] Load [N/m] The effects of high strain-rate and in-plane restraint on quasistatically loaded laminated glass 417 Stage 4 (L = 200mm) Stage 4 (L = 1000mm) (a) (b) 10 50 Axially restrained Axially restrained 9 45 Axially unrestrained Axially unrestrained 8 40 7 35 6 30 5 25 4 20 3 15 2 10 1 5 0 0 02468 0 1020304050 Deflection W / t [-] Deflection W / t [-] G G Fig. 9 Nondimensional collapse load-deﬂection diagrams for Stage 4, comparing the response under simple bending (axially unre- strained) and combined bending–membrane action (axially restrained) for two different spans, a L = 200 mm, b L = 1000 mm The incorporation of in-plane restraint results in two-way spanning plates, that represent a more realistic combined bending–membrane action under large deﬂec- geometry of glazed facades, should also be pursued. tions. A signiﬁcant membrane contribution to the panel Acknowledgements The ﬁrst author gratefully acknowledges response, and therefore reduced panel deﬂections, is the Engineering and Physical Sciences Research Council observed in the elastic stages of long-spans, while for (EPSRC) for funding this research through the EPSRC Centre for short spans, the effects are less pronounced. For the Doctoral Training in Future Infrastructure and Built Environment post-fracture stage, a yield condition has been devel- (FIBE CDT) at the University of Cambridge (EPSRC Grant Ref- erence No. EP/L016095/1). The contribution of the Institution oped to determine the relative contribution of the inter- of Civil Engineers, through the ICE Research and Development nal forces, without violating the material law, and the Enabling Fund, is also gratefully acknowledged. upper-bound theorem of plasticity applied to deter- Open Access This article is distributed under the terms of mine the collapse load corresponding to interlayer tear- the Creative Commons Attribution 4.0 International License ing. Membrane action has been seen to dominate the (http://creativecommons.org/licenses/by/4.0/), which permits response for both span lengths, resulting in enhanced unrestricted use, distribution, and reproduction in any medium, collapse loads greater than the capacity of the intact provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and laminated glass. indicate if changes were made. Experimental data for the quasi-static response of laminated glass under high strain-rates are not cur- rently available. Future work should therefore include Appendix A: Mathematical derivation of analytical experimental validation of the analytical models intro- models duced here. Additionally, the inertial effects associated with dynamic loading, which have been ignored in this A.1 Yield condition paper, should also be considered, to provide a complete theoretical framework for the blast response of lami- Pure membrane response will initiate when the plas- nated glass. Finally, the extension of the beam models to tic neutral axis (y = 0) has reached the top of the Collapse load p/p [-] Collapse load p/p [-] c 418 S. C. Angelides et al. cross-section. This can also be expressed in terms of in terms of the location of the plastic neutral axis the midspan transverse deﬂection (W), considering the (y ) under combined bending and membrane action, collapse mechanism shown in Fig. 4. The beam then or the midspan vertical deﬂection (W): behaves as two rigid bars connected via the plastic N N −T 1 − , y ≥ 0or W ≤ t 4 g 4 4 y 4 n = = = 4 hinge of ﬁnite length (L = AB + CD) equivalent to N N 4 4 1, y ≤ 0or W ≥ t the delamination length between the glass fragments, (A4) as shown in Fig. 6. The limiting deﬂection for pure membrane response to initiate can be then calculated The ratio of the bending moment (M ) to the pure moment capacity (M ) is obtained by applying moment by equating the bending (ε ) and membrane (ε ) 4,b,2 4,m,2 equilibrium about the centroid of the top glass layer, as strains at the mid-plane of the top glass layer, at the showninFig. 6. This also depends on the location of the location of the plastic hinge: plastic neutral axis, as when it has reached the top of the ε = ε cross-section, only the membrane force will generate a 4,b,2 4,m,2 moment: t y =0 ⇒ W = 2 − y ⇒ W = t (A1) g M T y + C y + N y T C N 4 4 4 4 m = = M M 4 4 The above relationship assumes that the crack has not 1 1 1 1 t y − y +y t + t reached the top of the glass layer, otherwise contact ⎪ g 4 pvb 2 4 3 4 2 2 ⎪ g , y ≥ 0or W ≤t ⎪ g ⎨ 1 1 4 between adjacent fragments will be lost when the mem- y t +t − y 4 pvb g 4 2 3 brane strain equals the bending strain at the top of the 1 1 t + t pvb 2 2 g , y ≤ 0or W ≥ t ⎩ g cross-section (ε = ε ). 1 1 4,b,1 4,m,1 t +t − y pvb g 4 2 3 The bending strain is then calculated under the (A5) assumption of plane sections remaining plane and the associated linear strain distribution. The entire beam deformation is localised at the plastic hinge (κ = A.2 Failure strain midspan ), where the beam rotation can be obtained from compatibility under the small angle approxima- The total longitudinal strain in the interlayer is the sum 4W tion (θ = ): midspan of the bending (ε ) and membrane (ε ) strains: 4,b,3 4,m,3 θ 1 midspan t W=2 −y ε = κ y = t − y 2 4W 2 4 4,b,2 4 4 L 2 h ε = t + t − y ⇒ pvb g 4,b,3 4 LL h,1 4W 1 = t − y (A2) 4 4W W LL 2 t − + , W ≤ t g pvb g LL 2 2 h,1 ε = (A6) 4,b,3 4t The membrane strain is given by the ratio of the change ⎩ t , W ≥ t pvb g LL h,1 in beam length (L ), due the extension caused by the ⎧ 2W ⎨ , W ≤ t large deﬂections under the axial restraint, to the initial LL h,1 2 2 ε = . length. As deformation only occurs at the plastic hinge, 2 2t 2t 4,m,3 g 2W g + − , W ≥ t LL LL LL due to the rigid-plastic material law, the plastic hinge h,1 h,2 h,2 length (L ) is considered as the initial length. 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