the effects of high strain-rate and in-plane restraint on ...pren 13474-3 (2009) for low...

Glass Struct. Eng. (2019) 4:403–420https://doi.org/10.1007/s40940-019-00107-4

SI: GLASS PERFORMANCE PAPER

The effects of high strain-rate and in-plane restraint onquasi-statically loaded laminated glass: a theoretical studywith 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 increasinglyused to improve the blast resilience of glazed facades,as part of efforts to mitigate the threat posed to build-ings and their occupants by terrorist attacks. The blastresponse of these ductile panels is still only partiallyunderstood, with an evident knowledge gap betweenfundamental behaviour at the material level and obser-vations from full-scale blast tests. To enhance ourunderstanding, and help bridge this gap, this paperadopts a ‘first principles’ approach to investigate theeffects of high strain-rate, associated with blast load-ing, and the in-plane restraint offered by blast-resistantframes. These are studied bydeveloping simplified ana-lytical beam models, for all stages of deformation, thataccount for the enhanced properties of both the glassand the interlayer at high strain-rates. The increasedshear modulus of the interlayer results in a compositebending response of the un-fractured laminated glass.This also enhances the residual post-fracture bendingmoment capacity, arising from the combined action ofthe glass fragments in compression and the interlayerin tension, which is considered negligible under lowstrain-rates. Thepost-fracture resistance is significantlyimproved by the introduction of in-plane restraint, dueto the membrane action associated with panel stretch-ing under large deflections. This is demonstrated bydeveloping a yield condition that accounts for the rela-

S. C. Angelides (B) · J. P. Talbot · M. OverendDepartment of Engineering, University of Cambridge,Cambridge, UKe-mail: [email protected]

tive contributions of bending andmembrane action, andapplying the upper bound theorem of plasticity, assum-ing a tearing failure of the interlayer. Future work aimsto complete the theoretical framework by including theassessment of plate-action and inertia effects.

Keywords Laminated glass · Blast response ·Strain-rate · In-plane restraint · Post-fracture

1 Introduction

Renewed focus on the threat posed to buildings andtheir occupants by terrorist attacks has intensified thedemand for blast resilient buildings. A high percentageof injuries in blast events are glass-related. As a result,it is now recommended for the glazed facades of com-mercial and residential buildings under blast threat toinclude laminated glass panels. These usually consistof two layers of annealed glass with a polyvinyl butyral(PVB) polymer interlayer that fails in a more duc-tile manner than glass alone, and holds the glass frag-ments together after fracture, thereby reducing glass-related injuries in blast events. The response of suchpanels to blast loads is a complex, multi-disciplinarytopic that often requires the use of full-scale blast test-ing to validate designs. It has become common prac-tice for such tests to characterise the panel structuralresponse by the (centre of panel) peak-displacement.Much research has therefore focussed on reproduc-ing the experimentally recorded, peak-displacement

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404 S. C. Angelides et al.

time-histories of laminated glass panels, whether thisis by finite-element analysis (FEA) (Hooper 2011;Larcher et al. 2012; Hidallana-Gamage 2015; ZhangandHao2015;Pelfrene et al. 2016), analytical solutions(Yuan et al. 2017; Del Linz et al. 2018) or equivalentsingle-degree-of-freedom (ESDOF) methods (SpecialServices Group, Explosion Protection 1997; AppliedResearch Associates 2010; Morison 2007; Smith andCormie 2009). This approach, of developing modelsbasedon experimentally observed, peakdisplacements,enables the structural assessment of laminated glasspanels under blast loading without having to performadditional, expensive blast testing. However, the fun-damental, underlying physics of the panel response isoften not explicitly expressed in these models, particu-larly those describing the post-fracture response, whichprevents practicing engineers from understanding fullythe contribution of individual parameters and limitstheir ability to optimise designs. In addition, inconsis-tent assumptions between existing models, signify thata number of aspects of the structural response remainonly partially understood and the need for an improvedtheoretical framework to be developed.

This paper adopts a ‘first principles’ approach thataims to enhance our understanding of the blast responseof laminated glass panels, by bridging the knowledgegap between the fundamental behaviour at the mate-rial level and the response observed in full-scale blasttests. Our starting point is the static response of simply-supported, axially unrestrained, laminated glass witha PVB interlayer, loaded laterally (i.e. bending aboutthe minor axis). Such specimens form the basis ofmuch of the experimental work on laminated glazing,as typified by the work of Kott and Vogel (2003, 2004,2007). This work reported on four-point bending testsaimed at investigating the pre- and post-fracture bend-ing capacity of laminated glass. It was concluded that,in the pre-fracture stage, the degree of composite actiondepends on the shear-stiffness of the interlayer, withthe response described either by simple bending the-ory, with the glass layers bending independently out-of-plane (also known as the ‘layered limit’), or by thesandwich theory of thick faces (also knownas the ‘com-posite response’). This stage terminates once the frac-ture strength in either of the glass layers is exceeded,after which the entire load is carried in bending by theun-fractured layer. In the third and final stage, bothglass layers have fractured and can no longer resisttension. Any residual bending resistance is provided

by the composite action of the interlayer, working intension, and the compression of the glass fragmentsthat come into contact as the panel deforms. However,it was demonstrated from the above experiments thatthis residual bending capacity was negligible, as thesamples collapsed when both glass layers had frac-tured. This was also the conclusion of the analyticalmodelling presented by Galuppi and Royer-Carfagni(2018), who calculated a value of 10−3 for the ratioof the pre- to post-fracture bending stiffness under lowstrain-rates.

Whilst providing valuable insight into the behaviourof laminated glass, the conditions of these static, lab-oratory tests differ from those in real-world buildings.Firstly, in building facades, rectangular panels of lam-inated glass are often supported along all four edges,resulting in a two-way spanning action that influencesthe deformation of the panel. Additionally, a commonrequirement for the frames of laminated glass panelsis the provision of some form of in-plane restraint,to enhance their overall blast resistance. The Centrefor the Protection of National Infrastructure (CPNI)recommends, as a minimum, the application of eitherstructural silicone sealant (wet glazing) or polysulphidesealant, within a rebate of 30mm, as illustrated in Fig. 1(CPNI EBP 2014). Alternatively, elastomeric gasketstrips (dry glazing) within 35 mm rebates are also con-sidered acceptable. In the American standards, ASTMF2248-12 (F2248 2012) and PDC-TR 10-02 (PDC-TR2012), the use of adhesive glazing tape for providingin-plane restraint is also permitted. This difference inboundary conditions governs the level of membraneaction within a panel, and is expected to influence sig-nificantly the overall blast resistance. Finally, in gen-eral, any static testing fails to account for the dynamicnature of the blast response, in particular, the effects ofinertia and high strain-rate. The effects of inertia areknown to be significant under the accelerations experi-enced by a panel during a typical blast event, but theywill not be considered further here. Instead, we focuson the effect of strain rate on the panel response, whichis believed to be important given the viscoelastic natureof the interlayer.

This paper begins by reviewing previous experimen-tal work that has assessed and quantified the effect ofthe high strain-rates associated with blast loading onthe material properties of laminated glass. The out-come of this review forms the basis of the subsequentlypresented analytical models that aim to describe the

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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 theform of silicone sealant within the frame

enhanced bending capacity of laminated glass underhigh strain-rates. The influence of in-plane restrainton the quasi-static response under high strain-rates isthen evaluated. This quasi-static approach representsa hypothetical load case that simulates blast loadingbut uncouples the material and inertial effects withinthe response, thereby focusing solely on the formerand enabling the relative contribution of bending andmembrane action developed under large deflections tobe assessed. Understanding the effects of high strain-rate and in-plane restraint with these simplified modelsis the primary objective of the work. This is expectedto enhance the theoretical framework available to prac-ticing engineers, thereby ultimately assisting them inoptimising the blast design of laminated glass pan-els, whether by FEA, ESDOF methods or analyticalsolutions. This work is also considered to be a startingpoint for future research on bridging the knowledge gapbetween the material response under quasi-static load-ing and full-scale blast testing, which will focus on theeffects of inertia and two-way spanning plate-action.

2 The influence of strain rate on materialproperties

The pressure time-histories resulting from the detona-tion of high-explosives are of short duration, typicallyof the order of milliseconds. These result in high strain-rates in any façade panels exposed to the blast. Meanstrain-rates ranging from 7.6 to 17.5 s−1 were recordedin fractured laminated glass panels by Morison (2007),and from 10 to 30 s−1 by Hooper (2011), both duringopen field, high-explosive blast tests. The influence ofthese strain rates on the salient material properties gov-

erning the structural response of the glass and PVB arediscussed below.

2.1 Glass layers

Of fundamental concern is the tensile strength of glasspanels, in the presence of surface flaws developed dur-ing manufacturing, installation and service-life. Thefracture stress (i.e. the stress at which cracking begins)is therefore not a material constant and often requirestesting to determine its exact value, which depends onthe surface quality and size of panel, and the stresshistory, residual stress and environmental conditions(Haldimann et al. 2008). The draft European Stan-dard, prEN 13474-3 (2009), recommends a character-istic value for the design fracture strength of annealedglass of 45 MPa, based on a 5% characteristic valuethat was obtained from 741 static tests performed on6 mm thick, annealed glass panels. High strain-ratesresult in an enhanced fracture strength, as flaws requiretime to develop into cracks (Overend andZammit 2012;Larcher et al. 2012). This is observed in multiple, highstrain-rate experimental tests performed by Nie et al.(2009, 2010), Peroni et al. (2011), Zhang et al. (2012)and Meyland et al. (2018). For the blast design of glaz-ing, recommended dynamic fracture strength valuesare presented by Smith and Cormie (2009) that werederived by extrapolating to the high strain-rates asso-ciated with blast loading the inherent, static strengthvalue of annealed glass presented in prEN 13474-3using Brown’s integral (risk integral) for stress fatigue(also known as sub-critical crack growth), and superim-posing the relevant surface pre-stress from the thermalprocessing of heat-strengthened and toughened glaz-

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ing products. Lower strain-rates were considered in theabove extrapolation for monolithic glazing, comparedto the mean values recorded experimentally by Mori-son (2007) and Hooper (2011) for fractured laminatedglass, as strain-rates depend on the maximum deflec-tion of the panel. As the pre-fracture stage of lami-nated glass is limited to smaller deflections, comparedto the overall response, the fracture strength derived formonolithic glazing is also considered appropriate forlaminated glass. Table 1 shows that these values are ingood agreement with the recommendations of the UKGlazing Hazard Guide, which were established from asignificant number of independent blast tests (Morison2007). However, comparisons with more recent, full-scale blast tests on laminated glass panels by Del Linzet al. (2018) have suggested that a fracture strengthof 100 MPa for annealed glass is more realistic. Fur-thermore, it should be noted that the dynamic fracturestrength is also dependent on the boundary conditionsand geometry of the panel, as demonstrated by blasttests performed by Osnes et al. (2018) on monolithicannealed glass. As a final comment, the recommendeddesign fracture strength values shown inTable 1 refer tonew (as-received) glass. In reality, flaws accumulate inthe glass surface over its service-life, and this has beenshown to significantly reduce the fracture strength (Dat-siou and Overend 2017a, b). This reduction in strengthof aged glass was also demonstrated in the high strain-rate experiments byKuntsche (2015) andMeyland et al.(2018), but this is not considered further here.

Compressive and tensile tests under high strain-rates performed by Zhang et al. (2012) concludedthat the Young’s modulus of annealed glass is insen-sitive to strain-rate. Therefore, in this paper, the valueof Young’s modulus

(Eg = 70GPa

)recommended by

prEN 13474-3 (2009) for low strain-rates will be alsoassumed for blast loading.

Following the fracture of the glass layers in a lam-inated panel, the attached glass fragments can stillprovide resistance by generating compressive contactstresses, as discussed in Sect. 3. The effects of highstrain-rate on the compressive strength of glass musttherefore also be investigated. The theoretical, com-pressive strength of glass is significantly higher thanthe fracture strength, as surface flaws will not growand propagate under compression. Dynamic incrementfactors (DIF) were derived by Zhang et al. (2012) fromdynamic compression tests using a Split HopkinsonPressure Bar, to amplify the low strain-rate, nomi-

nal compressive strength of annealed glass (σg,c =248MPa) for use under high strain-rates:

DIF = 1.189 + 0.049 log (ε̇) for 1−5 ≤ ε̇ ≤ 100 (1)For a typical strain-rate of 10 s−1, representative ofblast response, Eq. 1 yields a dynamic, compressivestrength for annealed glass of σg,c = 323 MPa. Inpractice, however, the Poisson’s ratio effect and anybuckling will generate tensile stresses under com-pressive loading, thereby resulting in a nominal com-pressive strength lower than the theoretical value(Haldimann et al. 2008). The application of the com-pression strength of glass, experimentally derived withSplit Hokinson Pressure Bar tests, to the post-fractureresponse of laminated glass therefore requires furtherexperimental validation.

2.2 Interlayer

The most popular interlayer for laminated glass pan-els is polyvinyl butyral (PVB), which is a ductile vis-coelastic polymer that is temperature and strain-ratedependent. The main reasons that often make it thepreferred interlayer are its ability to block UV radi-ation, its high strain at failure and its good adhesionproperties, which enable it to retain glass fragmentsfollowing the fracture of the glass layers (Haldimannet al. 2008). In contrast to the low strain-rate, hypere-lastic material behaviour observed in uni-axial tensiontests of PVBalone, for the higher strain-rates associatedwith blast loading the behaviour resembles an elastic–plastic response with strain hardening (Kott and Vogel2003; Bennison et al. 2005; Iwasaki et al. 2007; Mori-son 2007; Hooper et al. 2012a; Kuntsche and Schnei-der 2014; Zhang et al. 2015; Chen et al. 2017). It isnoted that the above behaviour can vary for differentPVB types and depending on the manufacturer. Addi-tionally, temperature dictates the strain-rate at whichthe observed behaviour transitions from hyperplasticto elastic–plastic. However, the focus of this paper ison the effects of strain-rate at constant, room temper-ature. Although thermo-rheological effects cause thetemperature of the interlayer to increase at high strain-rates, this is not explicitly accounted for here. Ratherthan adopting a viscoelastic material model, a simplerelastic–plastic model is assumed for the pre-fractureblast assessment of laminated glass.

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Table 1 Recommended design values for glass fracture strength of low (prEN 13474-3) and high (Smith and Cormie 2009; Morison2007) strain-rates

Glass type Fracture strength, σg,t (MPa)

Low strain-rates (∼ 10−5 s−1) High strain-rates (0.4−0.6 s−1)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 valuesshould be used in combination with a material safety factor and additional factors that account for load duration, glass surface profileand edge bending strength, as specified in prEN 13474-3

It is worth noting that Morison (2007) highlightedthat the sudden stiffness reduction observed followingthe apparent yielding of PVB is in fact non-linear vis-coelasticity, as permanent strain is not observed fol-lowing the removal of the load. The description of anelastic–plastic behaviour of PVB at high strain-ratestherefore refers to the bilinear shape of the stress–strainresponse and not to true plasticity. However, this paperconsiders only the loading phase of the PVB (corre-sponding to the positive phase of blast loading) forwhich a simple linear-elastic model is acceptable. Dur-ing the pre-fracture stage of a panel response, the inter-layer is not expected to yield due to the dominant stiff-ness of the glass layers. This assumption is supportedbythe findings of Dharani and Wei (2004), Wei and Dha-rani (2005) and Hooper et al. (2012b), both of whompresented a finite-element sensitivity analysis of theblast response of laminated glass panels and concludedthat the viscoelastic shear relaxation modulus may bereplaced by the instantaneous shear modulus (Gpvb =0.178 GPa). This value is adopted for the pre-fractureanalytical models presented in Sect. 3.1. Assuminga Poisson’s ratio of νpvb = 0.49 for the essentiallyincompressible PVB, the corresponding Young’s mod-ulus (Epvb = 0.53 GPa) can then be obtained (Morison2007), although the final response is insensitive to theparticular value due to the dominance of the glass layerstiffness pre-fracture.

Following the fracture of the glass layers, theresponse of the PVB interlayer is influenced by thepresence of the attached glass fragments. A new consti-tutive law is therefore required to describe the compos-

ite action of the interlayer together with the attachedglass fragments. The glass fragments effectively pre-vent the PVB from extending where it remains bondedto the glass. Therefore, instead of having a uniformextension over the entire area of the PVB, the exten-sion is discretised at the bridges between the fragments,causing significantly larger but localised strains. Thus,under blast loading, stiffening effects are mobilised byboth high strain-rates and the attached glass fragments.This has been reported by Morison (2007), Hooper(2011) and Zobec et al. (2014). Hooper (2011) per-formed a series of uni-axial tension tests on fractured,laminated glass specimens for a range of strain-rates.An elastic–perfectly-plastic stress–strain response wasobserved in these tests, governed by the delaminationof the attached glass fragments from the interlayer thatdiffers from the elastic–plastic with strain hardening(bilinear) material law observed for the PVB alone.Additionally, a brittle failure was observed in thesetests for thinner PVB interlayers, while for interlayersthicker than 1.52 mm, the delamination front travelledquickly, relieving the interlayer from excessive strainsand preventing premature tearing. This conclusion is inagreement with the UK Glazing Hazard Guide recom-mendation for a minimum PVB thickness of 1.52 mmfor blast-resistant laminated glass (Morison 2007). Abrittle failure, however,may also occur for thicker inter-layers, if a high adhesion grade is specified for the panelthat prevents the attached glass fragments to locallydelaminate from the interlayer, as observed in the full-scale blast tests presented by Pelfrene et al. (2016). It isclear that the attached glass fragments influence both

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Table 2 Material properties for cracked, laminated glass with 10mm fragment length and 1.52mm interlayer, derived experimentallyfor low and high strain-rates by Hooper (2011)

Composite material properties Low strain-rates (0.1 s−1) High strain-rates (10 s−1)

Yield strength, σpvb,c,y (MPa) 7 17

Strain failure, εpvb,c,f (%) 150 170

Initial Young’s modulus, Epvb,c (GPa) 0.3 1.7

the elastic and plastic response of the PVB within afractured panel.

For the purposes of the post-fracture analysis pre-sented in Sect. 3, the PVB may be characterised byHooper’s (2011) high strain-rate values of the threematerial properties summarised in Table 2, that is,the yield strength, initial Young’s modulus and failurestrain. These values were derived experimentally byHooper (2011) for specimens with uniform crackingpattern consisting of 10 mm fragments, which aimedto simulate the failure observed from full-scale blasttests of laminated glass panels with annealed glasslayers, and for a 1.52 mm thick interlayer, typical offaçade glazing panels. However, it is recommended byHooper (2011) that a strain limit of 20% is applied tothe blast design of laminated glass to prevent completedebonding of the attached glass fragments. Althoughit is acknowledged that there is degree of uncertainty,this limit is also adopted here, with further experi-mental work required to validate this design limit. Togauge the influenceof crackingpatterns,Hooper (2011)also derived material properties for both 20 mm frag-ments and non-uniform cracking patterns. It shouldalso be noted that these values can vary for differentPVB types (stiffness and adhesion level) and ambienttemperatures.

For comparison, Table 2 summarises material prop-erty values recorded under two different strain-rate val-ues during uni-axial tension tests performed by Hooper(2011). The effect of strain rate on the material prop-erties is clear. The consequences for the structuralresponse are now investigated in Sect. 3.

3 Analytical models accounting for highstrain-rates and in-plane restraint

The effects of high strain-rate and in-plane restrainton the blast response of laminated glass panels areoften obscured in current models. To assess the influ-

ence of high strain-rate on the structural response, thissection presents simplified analytical models to cal-culate the enhanced moment capacity of laminatedglass, by adopting the values established in Sect. 2for the salient material properties. These simple bend-ing models, corresponding to unrestrained, simply-supported beams under uniformly distributed loading,can then be compared directly with the existing lowstrain-rate experimental results described in Sect. 1.The enhanced resistance provided by the introductionof lateral restraints, as described in Sect. 1, is then eval-uated through the consideration of a quasi-static load-ing under high strain-rates that enables an investigationof the material effects alone, leaving inertial effectsto be considered later. Important conclusions can bedrawn from a better understanding of the quasi-staticresponse, including the relative contribution of bendingmoments and membrane forces to the overall bendingresistance.

3.1 Simple bending analysis

The static response of sandwich beams with low, coreshear stiffness is governed by both bending and sheardeflections. When the core shear stiffness is negligi-ble, the faces of a sandwich beam bend independentlyas two separate beams, known as the ‘layered limit’.Additionally, in the case of thick faces, local bendingof the faces about their own centroidal axes also influ-ences the shear deformation of the core (Allen 1969).The pre-fracture phase of laminated glass under staticloading falls within this category, with available analyt-ical solutions for various loading and boundary condi-tions presented byGaluppi and Royer-Carfagni (2012).If on the other hand, the shear stiffness of the core issufficiently high that the plane sections remain planethe response of the sandwich beam is in pure bendingof the whole cross-section, known as the ‘monolithiclimit’. The above described ‘monolithic’ behaviour is

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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 laminatedglass, due to the enhanced shearmodulus of PVB underthe high strain-rate, as described in Sect. 2.2, which isnow capable of transferring the horizontal shear forces(Norville et al. 1998; Kuntsche and Schneider 2013).Thus, in the unfractured phase (Stage 1), when bothglass layers remain unfractured, the laminated glassacts compositely as a single beam. The same approachis adopted by the blast analyses cited in Sect. 1, and thevalidity of this will be confirmed in Sect. 4.1.

In this section, analytical models for simply-supported, axially unrestrained laminated glass underhigh strain-rate are presented, assuming a simple bend-ing response and the same elastic stages of defor-mation (Stages 1, 2 and 3) defined by Kott andVogel (2003, 2004, 2007) and described in Sect. 1.Based on the Euler-Bernoulli theory, that plane sec-tions remain plane during bending,with small displace-ments/rotations assumed, an equivalent, transformedcross-section made entirely from either glass (Ig,1, Ig,2and Ig,3) or PVB (Ipvb,1, Ipvb,2 and Ipvb,3) will be con-sidered for the analysis, based on the modular ratioof the two materials (

EgEpvb

orEg

Epvb,c). Both transformed

sections should result in the same bending stiffness(EgIg = EpvbIpvb or EgIg = Epvb,cIpvb). The positionsof the elastic neutral axes from the top of the cross-section in the three stages (y1, y2 and y3), which define

zero bending strain, can be then obtained from thefirst moment of area of the transformed cross-section.The bending strain and stress distributions for eachstage are shown in Fig. 2, and the corresponding bend-ing moment and curvature capacities are provided inTable 3.

The bending moment capacity of Stage 1 (M1) isdefined as the moment required for the tensile frac-ture strength of glass under high strain-rates (σg,t) tobe exceeded. Due to the sagging response of the com-posite structure under a transverse loading (as drawnin Fig. 2), the bottom glass layer will be in tension andtherefore fracture first. The bottom glass layer will thenno longer contribute to the bending capacity, the bend-ing stresses will be distributed only in the top glasslayer and the interlayer, and the response will be gov-erned by the new transformed section (Ig,2). The bend-ing moment capacity of Stage 2 (M2 < M1) that isdefined as the moment required for the tensile fracturestrength of glass under high strain-rates (σg,t) to beexceeded in the top glass layer, will be smaller com-pared to Stage 1, due to the reduced second moment ofarea (Ig,2 < Ig,1). This will result in an abrupt transi-tion between Stage 1 and 2 in the moment–curvaturerelationship, as the bending moment that caused frac-ture in Stage 1 cannot be sustained in Stage 2 understatic loading. For short duration pulses, such as blast

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Table 3 Bending moment and curvature capacities of laminatedglass for each stage under high strain-rates

Stage Moment capacity Curvature capacity

1 M1 = σg,t Ig,1y1,g κ1 = M1EgIg,12 M2 = σg,t Ig,2y2,g κ2 = M2EgIg,23 M3 = σpvb,c,yIpvb,3y3,pvb κ3 = M3Epvb,cIpvb,34 M4 = 23 y4,gC4

+[y4,pvb − tpvb2

]T4

κ4 = εg,cy4,g

loads, the response is different due to the inertia effects.As this is beyond the scope of this paper, each Stagewill be assessed independently for its moment capac-ity. Future experimental tests will be used to simulatethese stages and validate the derived capacities.

In Stage 3, both glass layers have fractured but theinterlayer still behaves elastically (σpvb < σpvb,c,y).The bending resistance is therefore provided by thecomposite action of the interlayer in tension and theattached glass fragments in the top glass layer gener-ating compressive contact stresses at glass fragmentinterfaces (Ipvb,3). The post-fracture, elastic momentcapacity (M3) is defined as the bending momentrequired to cause yielding (σpvb,c,y) in the extreme fibreof the interlayer (ypvb,3).

Once the extreme fibre of the interlayer has yielded,the plastic response will initiate. By using the elastic–perfectly-plastic material law, described in Sect. 2.2,any additional stress will be distributed to the remain-

ing cross-section until the entire interlayer has yielded(Stage 3–4 shown in Fig. 3). At this point, the inter-layer has no remaining capacity and, for a constantbending moment, the tensile strains in the interlayerwill increase due to the elastic–perfectly plastic mate-rial law. As demonstrated experimentally by Delincéet al. (2008), for the post-fracture response of laminatedglass with a stiffer interlayer, the assumption of planesections remaining plane and a linear strain distribu-tion still holds. The compressive strains in the fracturedglass will therefore also keep increasing. Consideringa linear stress–strain distribution for fractured glassunder compression, the stresses in the top glass layerwill also increase. To satisfy longitudinal equilibriumunder bending alone, which requires the resultant com-pressive force in the glass and the resultant tensile forcein the interlayer to be equal, the plastic neutral axis willshift upwards. As the bending moment capacity canbe defined from moment equilibrium about the plas-tic neutral axis, the upwards shift of the plastic neutralaxis will result in an increase of the bending momentcapacity. At the instancewhere the cross-section has noreserve moment capacity (Stage 4 shown in Fig. 3) theplastic neutral axis (y4) can be calculated from longi-tudinal equilibrium, considering the compressive forcein the top glass layer that initiates crushing of the glassfragments (C4 = 0.5Bσg,cy4) and the tensile forcecapacity of the interlayer (T4 = Btpvbσpvb,c,y), whereB is the width of the cross-section. In the above calcu-lation, the local delamination of the interlayer from theattached glass fragments, which allows higher strainsto develop and prevents sudden tearing, is implicit in

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)

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Fig. 4 Collapse mechanism for simply-supported axially unrestrained laminated glass beam

the yield strength value (σpvb,c,y) that was derived fromuni-axial tensile tests of cracked laminated glass spec-imens, as discussed in Sect. 2.2. A constant interlayerthickness has, however, been assumed, ignoring neck-ing effects, for the purpose of calculating the sectioncapacity. Theultimatemoment capacity (M4), providedin Table 3, is then obtained by considering momentequilibriumabout the plastic neutral axis,while the cur-vature (κ4) is calculated from the linear strain distribu-tion, where εg,c(

σg,cEg

) is the crushing strain of glass. Asthe cross-section cannot sustain any additional bendingmoments, a plastic hinge has formed and a mechanismdeveloped for the simply-supported beam, as shown inFig. 4.

3.2 Combined bending and membrane analysis

In contrast to the simple bending response of axiallyunrestrained beams, as presented in Sect. 3.1, the intro-duction of axial restraints results in a combined bendingandmembrane actionwhen the deflection is sufficientlylarge, as shown in Fig. 5.

As with the simple bending analysis, the membranestress distribution during the elastic Stages 1 and 2 canbe determined by transforming the composite cross-section into an equivalent glass section (Ag). The addi-tional contribution of the membrane stresses results inan upwards shift of the elastic neutral axis (y′1 < y1 andy′2 < y2) from the positions under pure bending (seeSect. 3.1). To determine the relative contribution of

bending moments (M′1 and M′2) and membrane forces(N′1 and N′2), the corresponding transverse displace-ments (w1 and w2) need to be first obtained, due to thenonlinearity of the equilibrium equation arising fromthe introduction of membrane forces. Again, assumingfull shear transfer under high strain-rates, the transversedeflection due to a uniformly distributed load (p) canbe obtained by solving the fourth-order, nonlinear dif-ferential equation, derived by Aşık and Tezcan (2005)for laminated glass:

−Eg,1Ig,1 d4w1(x)dx4 + N′1 d2w1(x)dx2

+ p = 0 (2a)−Eg,2Ig,2 d4w2(x)dx4 + N′2 d

2w2(x)dx2

+ p = 0 (2b)

For the fully-fractured laminated glass (Stages 3 and4), a rigid-perfectly-plastic material law is adopted forsimplicity to derive the relative contribution of bendingmoments (M′4) and membrane forces (N′4), ignoringthe elastic contributions of Stage 3. During Stage 4, thecombination of the tensile force (T′4), associated withthe bending moment (M′4), and the membrane force(N′4) cannot exceed the PVB tensile capacity (N4 =σpvb,c,ytpvbBpvb), as shown in Fig. 6. An increase inthe membrane force will therefore cause a decrease inthe bending moment and a subsequent upwards shift ofthe plastic neutral axis (y′4).When the latter has reachedthe top of the cross-section (y′4 = 0), a pure membraneresponse will result.

The yield condition for laminated glass, ensuringthat the material law is not violated, therefore needsto account for the relative contribution of bending

123


Fig. 5 a Simply-supported beam axially unrestrained under pure bending, b axially restrained beam under combined bending andmembrane action

Fig. 6 Strain and force distribution within a laminated glass beam for the plastic stage of deformation (Stage 4) under combined bendingand membrane action

moments and membrane forces. Following a similarapproach to that considered by Sawczuk and Winnicki(1965) for singly-reinforced-concrete, this is derivedbyeliminating the unknown position of the plastic neu-tral axis (y′4) in the dimensionless equivalents of thebending (m) and membrane (n) equations defined in“Appendix A.1” section:

m − n (4y4 − 3tg)(3tpvb+6tg−2y4) + n2

y4(32 tpvb+3tg−y4

)−1 = 0 (3)A load–displacement relationship for laminated glassbeams in Stage 4 can then be derived from the aboveyield condition by applying the upper-bound theoremof plasticity for finite deflections on the same col-lapse mechanism introduced in Sect. 3.1 under sim-ple bending, as shown in Fig. 4. This assumption, ofthe same initial mechanism occurring under both bend-ing and combined bending–membrane action, is com-monly adopted in plastic analysis (Jones 2011). Undera uniformly distributed load (p) and the linear veloc-

ity profile of the collapsemechanism(ẇ(x) = Ẇ xL/2

),

the load–displacement relationship for the mid-spandeflection (W) is obtained by equating the exter-nal work rate ( ˙EW = ∫ L0 pẇ(x)dx) to the internalenergy rate dissipated at the location of the plas-tic hinge

(ĖD = (M + NW)θ̇midspan

). Considering the

collapse load(pc = 8M4L2

)for simply-supported, unre-

strained laminated glass beams, the loading for axiallyrestrained beams can be written in non-dimensionalform:

ppc

=

⎧⎪⎪⎨

⎪⎪⎩

5W2+(12y4−7tg)W+6y4(tg+tpvb)+2t2g12y4

(12 tpvb+tg− 13 y4

) , W ≤ tgtpvb2 +

tg2 +W

12 tpvb+tg− 13 y4

, W ≥ tg(4)

The above load–displacement relationship is valid foran interlayerwith infinite ductility. Thus, the failure dis-placement (Wf ) for PVB laminated glass can be definedby equating the total longitudinal strain in the interlayer(ε′4,pvb) to the PVB failure strain (εpvb,c,f) defined inSect. 2.2, with the derivation provided in “AppendixA.2” section:

ε′4,pvb = εpvb,c,f ⇒4tg

LLh,1tpvb +

2t2gLLh,1

+(

2W2fLLh,2

− 2t2g

LLh,2

)

= εpvb,c,f(5)

The plastic hinge length (Lh), defined in Fig. 6 as thedelaminated length of the interlayer from the attachedglass fragments, can be obtained from Eq. 6, as derived

123


by Belis et al. (2008) under pure bending:(τ2

)2L2h −

(

σpvb,c,ytpvbτ+2�0σpvb,c,ytpvb

�L

)

Lh

+ (σpvb,c,ytpvb)2 = 0 (6)

where τ is the shear stress at the interface between theinterlayer and the glass fragment and �0 is the fractureenergy of the interface.

For midspan deflections less than the top glass layerthickness (W ≤ tg), a single plastic hinge is formed,as shown in Fig. 4. Thus, the above equation can bemodified to also account for extension due to the axialrestraint, to derive the plastic hinge length for combinedbending and membrane action (Lh,1):

�L = �Lm + �Lb= 2W

2

L+ 4W

L

(tpvb + tg

2− W

2

)

= 4WL

(tpvb + tg

2

), W ≤ tg (7a)

For midspan deflections greater than the top glass layerthickness (W ≥ tg), a pure membrane response willoccur, with the extension due purely to the large deflec-tions under the axial restraint:

�L = �Lm = 2W2

L, W ≥ tg (7b)

A single plastic hingewill no longer form and the entirebeam will behave as a string (Jones 2011), resulting ina deformation length (Lh,2) equal to the total delami-nation occurring from each crack:

Lh,2 = LhNc = Lh(

LLf

− 1)

(8)

where Lf is the length of each fragment.

4 Results and discussion

The analytical models introduced in Sect. 3 have beensolved numerically for the case of simply-supportedlaminated glass beams with annealed glass layers (tg =6mm) and a PVB interlayer (tpvb = 1.52mm), and fortwodifferent span lengths (L = 200mmand1000mm)and a constant width (B = 55mm). To validate theassumed monolithic beam action under high strain-rates,and investigate the relative bending/membranecontribution under in-plane restraint, both a short and along span are considered. The material properties con-sidered in the analysis are introduced in Sect. 2 and thespecific values used are summarised in Table 4.

Table 4 Material properties for laminated glass considered inthe analysis

Material property Low strain-rates High strain-rates

Glass fracturestrength (MPa)

45 80

Glass Young’smodulus (GPa)

70 70

Glasscompressivestrength (MPa)

N/A 323

PVB shearmodulus (GPa)

N/A 0.178

Yield strength of fracturedlaminated glass (MPa)

N/A 17

Strain failure offracturedlaminated glass

N/A 20%

Initial Young’s modulus offractured laminated glass(GPa)

N/A 1.7

4.1 Influence of high strain-rate

The moment–curvature relationship for the axiallyunrestrained, laminated glass is plotted separately forall Stages in Fig. 7, assuming, for simplicity, no per-manent curvature at the end of each Stage, as discussedin Sect. 3.1. This allows direct comparison with futureexperimental results that will assess each Stage inde-pendently. Themoment (M1,M2,M3 andM4) and cur-vature (κ1, κ2, κ3 and κ4) capacities under high strain-rates, are computed from the equations presented inTable 3. These may then be compared with the lowstrain-rate values, to assess directly the influence ofhigh strain-rate. For Stages 1 and 2, the low strain-rate moment and curvature capacities are calculatedfor the layered limit, assuming no shear transfer in theinterlayer. The elastic moment–curvature relationshipis therefore used (M = EIk), considering only the sec-ond moment of area about the centroidal axis of each

glass layer (I1 = Bt3g6 and I2 =

Bt3g12 ), assuming that the

glass layers bend as two separate beams. For Stages3 and 4, there is no reserve capacity for low strain-rates, as demonstrated by the experimental work ofKott andVogel (2003, 2004, 2007) described in Sect. 1,and therefore the moment–curvature relationship is notplotted.

123


0 0.1 0.2 0.3Curvature [1/m]

0

20

40

60

80

100

120

140

160

Mom

ent [

Nm

]Stage 1

M1

κ1

High strain-rateLow strain-rate

0 0.2 0.4Curvature [1/m]

0

5

10

15

20

25

30

Mom

ent [

Nm

]

Stage 2

M2

κ2

High strain-rateLow strain-rate

0 10 20 30

Curvature [1/m]

0

2

4

6

8

10

Mom

ent [

Nm

]

Stages 3 and 4

M3

κ3

M4

κ4

High strain-rate

(b) (c)(a)

Fig. 7 Moment–curvature graphs for axially unrestrained laminated glass beams under simple bending, plotted separately for eachstage 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 stage1 at high strain-rates is increased by more than a factorof four compared to the low strain-rates value. This isdue to the enhanced shear modulus of the interlayerat high strain-rates, resulting in the laminated glassworking as a monolithic beam, with an enhanced sec-ond moment of area. This is an upper bound limit, asthere 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 theassumption of monolithic bending under high strain-rates, the calculated maximum, mid-span deflectionmay be compared to that calculated using the enhancedeffective-thickness approach presented by Galuppi andRoyer-Carfagni (2012), which accounts for finite shearcoupling between the interlayer and the glass layers.As shown in Table 5, almost identical deflections areobtained for the larger span (L = 1000mm), whilesome shear deflection contributions are evident for theshorter span (L = 200mm). It is, however, noticed thatthe deflections in Stage 1 for the short span are limitedto less than 1 mm. Considering the nondimensionalcoefficient η (η = 0 for layered limit, η = 1 for mono-lithic limit) introduced by Galuppi and Royer-Carfagni

Table 5 Comparison of mid-span deflections for high strain-rates under simple bending and the enhanced effective-thicknessapproach presented by Galuppi and Royer-Carfagni (2012)

Maximum deflection (mm) L = 200mm L = 1000mmSimple bending 0.70 17.61

Enhanced effective-thicknessapproach

0.94 17.87

(2012), it can be concluded that, for large spans (η = 1for L = 1000 mm), a monolithic beam assumption isvalid for laminated glass beams under blast loading,while for shorter spans (η = 0.93 for L = 200mm) thecontribution of shear deflections should be accountedfor.

In Stage 2, as expected, the almost identical responseshown in Fig. 7b for high and low strain-rates indi-cates that the PVB contribution to bending is negligi-ble, and the entire bending resistance is provided fromthe remaining unfractured glass layer. The increasedmoment capacity under high strain-rates results fromthe enhanced fracture strength of glass. Finally, forStages 3 and 4, the effect of high strain-rate, comparedto the negligible capacity under low strain-rates that is

123


not plotted, is clearly visible from Fig. 7c, which showsthe residual post-fracture bending capacity. This resem-bles the low strain-rate response of stiffer interlayers,such as ionomers or added steel-wire mesh reinforce-ment, as experimentally demonstrated by Delincé et al.(2008) and Feirabend (2008) respectively.

The moment capacity for Stage 3 shown in Fig. 7crepresents a lower bound solution, as thematerial prop-erties introduced in Sect. 2.2 are for a uniform crack-ing pattern with a 10 mm glass fragment length. Thisassumption implies that all cracks are aligned betweenthe two glass layers and therefore a series of rigid andflexible sections are present along the length of thebeam. If the cracks are not aligned, a higher Young’smodulus and yield strength will result, as reported byNhamoinesu and Overend (2010), who compared theresults from through-crack and offset-crack uniaxialtensile tests on laminated glass specimens with a singlecrack under low strain-rates. A similar conclusion wasalso reported in the analytical model for out-of-planebending of fractured laminated glass under low strain-rates presented byGaluppi and Royer-Carfagni (2018).The drawback of cracks that are not aligned, however,is a more brittle failure with almost no residual plasticcapacity (Nhamoinesu andOverend 2010). In this case,Stage 4 may never develop. The anticipated fracturepattern under blast loading, and its consequences forthe analysis presented here, is reserved for future work.

Another useful comparison can be made with rein-forced concrete beams, which also consist of a brittlematerial (concrete) reinforced with a ductile material(steel) to carry tension. Although the same method-ology is applied to evaluate the moment capacitiesof the cross-section for each Stage (the transformed-section method for the elastic stage and moment equi-librium for the ultimate moment capacity), it is evi-dent that the moment–curvature response is quite dif-ferent. In laminated glass, the residual, post-fractureplastic moment capacity of Stage 4 is smaller thanthe pre-fracture moment capacity of Stage 1, while inreinforced-concrete the ultimate plasticmoment capac-ity is larger than the uncracked moment capacity. Theupper-bound theorem of plasticity, frequently appliedin the structural analysis of reinforced concrete beamsand slabs (BS EN 1992 2004), would therefore notresult in efficient designs for axially unrestrained, lam-inated glass beams due to the rigid-plastic material lawapproximation that ignores the elastic contributions.

4.2 Influence of in-plane restraint

The elastic,mid-span deflections (Stages 1 and 2) undersimple bending and combined bending–membraneaction for laminated glass are shown in Fig. 8 for boththe short and long span beams. To derive the com-bined bending/membrane deflections under in-planerestrained boundary conditions, Eqs. (2a) and (2b) havebeen solved with the Galerkin method, considering anapproximate, sinusoidal deflected shape (Wierzbicki2013). The maximum deflections of each Stage werecalculated iteratively numerically, by limiting the totallongitudinal tensile stress (the sum of bending andmembrane stresses) to the tensile fracture strengthof glass under high strain-rates given in Table 1.These were validated with the solutions presented byTimoshenko and Woinowsky-Krieger (1959), whichassumes cylindrical bending rather than the sinusoidalform assumed here.

Figure 8 shows that for short spans, the entireresponse is governed by bending in Stage 1, with smallmembrane contributions evident in Stage 2. In contrast,for large spans, membrane contributions are also evi-dent in Stage 1, resulting in smaller deflections com-pared to simple bending. It is clear that membraneaction dominates the response in Stage 2. In additionto smaller deflections for the larger span, the pres-ence of axial restraint enhances the load capacity byfactors of 1.4 and 4.8 in Stage 1 and 2 respectively.This means that the laminated glass can resist a sig-nificantly larger load before fracture, compared to theunrestrained boundary condition. To develop such largemembrane forces, however, the connection details andthe supporting frame must also be adequately designedto ensure that they can resist the horizontal reactionforces developed.

The load–displacement relationship for Stage 4 isshown inFig. 9. Themidspandeflection is calculated bysolving Eq. (4) numerically, and nondimensionalisedby dividing by the top glass layer thickness. The fail-ure deflection, at which tearing of the interlayer occurs,is given by Eq. (5), assuming an arbitrarily delamina-tion length of 8 mm, as the solution of Eq. (6) resultedin delaminated lengths greater than the assumed glassfragment length, thus causing glass fragmentation. Thisassumption requires further investigation by experi-mental testing. However, it is considered conservativefor the current study, as a shorter plastic hinge willresult in larger strains. For predicted deflections that are

123


0 0.2 0.4 0.6 0.8Deflection [mm]

0

10000

20000

30000

Load

[N/m

]Stage 1 (L = 200mm)

Axially restrainedAxially unrestrained

0 0.5 1 1.5 2Deflection [mm]

0

2000

4000

6000

Load

[N/m

]

Stage 2 (L = 200mm)


0 5 10 15 20Deflection [mm]

0

500

1000

1500

Load

[N/m

]

Stage 1 (L = 1000mm)


0 10 20 30 40Deflection [mm]

0

500

1000

1500

Load

[N/m

]



(b)(a)

(c) (d)

Fig. 8 Load-deflection diagrams for elastic stages (Stages 1and 2), comparing the response under simple bending (axiallyunrestrained) and combined bending–membrane action (axially

restrained) for two different spans a Stage 1 for L = 200mm, bStage 1 for L = 1000mm, c Stage 2 for L = 200mm, d Stage 2for L = 1000mm

comparable to the span lengths, the validity of the smallangle approximation and binomial expansion, consid-ered in the development of the analytical models inSect. 3.2, requires further investigation.

The linear relationship observed in Fig. 9 for nondi-mensional deflections greater than 1 indicates thepure membrane response, with the combined bend-ing/membrane action limited to deflections less thanthe top glass layer thickness, as described by Eq. (4).This results in significant enhancement of the post-fracture capacity, compared to the collapse load ofunrestrained laminated glass. Additionally, it is alsoobserved that this is higher than the fracture load of theintact panel in Stage 1, highlighting the important con-tribution of axial restraint in the reserve, post-fracturecapacity. This conclusion is in agreement with the lowstrain-rate, pressurised water tests of laminated glasspanels presented by Morison (2007) and Zobec et al.(2014), who recorded post-fracture loads in excess ofthe unfractured capacity and a dominant membraneresponse. It should be noted, that the analysis presentedhere considers tearing to result only from combinedbending/membrane strains and has ignored the pos-sible rupture of the interlayer caused by the attachedglass fragments, although the latter has been observed

in practice, such as in the water bag tests performed byZobec et al. (2014).

5 Conclusions

This theoretical study has adopted a ‘first-principles’approach to assess the effects of high strain-rate andin-plane restraint on the structural response of PVB-laminated glass. A review of the material behaviourof glass and PVB has shown that the salient mate-rial properties are enhanced significantly under thehigh strain-rates associated with blast events. Analyt-ical models for laminated glass have been developed,which describe the four stages of quasi-static responsefrom initial loading to failure. These indicate that theenhanced material properties have a significant effecton the moment and curvature capacities of the glassin simple bending. Specifically, the results have shownthat, for high strain-rates, the enhanced shear modu-lus of the PVB justifies a monolithic beam approachto panel analysis. Furthermore, the enhanced modulusresults in significantly enhanced moment capacities ofeach stage, including a residual post-fracture bendingcapacity that is considered negligible for low strain-rates.

123


0 2 4 6 8Deflection W / t

G [-]

0

1

2

3

4

5

6

7

8

9

10

Col

laps

e lo

ad p

/pc

[-]

Stage 4 (L = 200mm)


0 10 20 30 40 50Deflection W / t

G [-]

0

5

10

15

20

25

30

35

40

45

50

Col

laps

e lo

ad p

/pc

[-]



(a) (b)

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 = 200mm, b L = 1000mm

The incorporation of in-plane restraint results incombinedbending–membrane actionunder largedeflec-tions. A significant membrane contribution to the panelresponse, and therefore reduced panel deflections, isobserved in the elastic stages of long-spans, while forshort spans, the effects are less pronounced. For thepost-fracture stage, a yield condition has been devel-oped to determine the relative contribution of the inter-nal forces, without violating the material law, and theupper-bound theorem of plasticity applied to deter-mine the collapse load corresponding to interlayer tear-ing. Membrane action has been seen to dominate theresponse for both span lengths, resulting in enhancedcollapse loads greater than the capacity of the intactlaminated glass.

Experimental data for the quasi-static response oflaminated glass under high strain-rates are not cur-rently available. Future work should therefore includeexperimental validation of the analytical models intro-duced here. Additionally, the inertial effects associatedwith dynamic loading, which have been ignored in thispaper, should also be considered, to provide a completetheoretical framework for the blast response of lami-natedglass. Finally, the extensionof thebeammodels to

two-way spanning plates, that represent amore realisticgeometry of glazed facades, should also be pursued.

Acknowledgements The first author gratefully acknowledgesthe Engineering and Physical Sciences Research Council(EPSRC) for funding this research through the EPSRCCentre forDoctoral Training in Future Infrastructure andBuilt Environment(FIBE CDT) at the University of Cambridge (EPSRCGrant Ref-erence No. EP/L016095/1). The contribution of the Institutionof Civil Engineers, through the ICE Research and DevelopmentEnabling Fund, is also gratefully acknowledged.

Open Access This article is distributed under the terms ofthe Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permitsunrestricted use, distribution, and reproduction in any medium,provided you give appropriate credit to the original author(s) andthe source, provide a link to the Creative Commons license, andindicate if changes were made.

Appendix A: Mathematical derivation of analyticalmodels

A.1 Yield condition

Pure membrane response will initiate when the plas-tic neutral axis (y′4 = 0) has reached the top of the

123

http://creativecommons.org/licenses/by/4.0/


cross-section. This can also be expressed in terms ofthe midspan transverse deflection (W), considering thecollapse mechanism shown in Fig. 4. The beam thenbehaves as two rigid bars connected via the plastichinge of finite length (Lh = AB + CD) equivalent tothe delamination length between the glass fragments,as shown in Fig. 6. The limiting deflection for puremembrane response to initiate can be then calculatedby equating the bending (ε′4,b,2) andmembrane (ε′4,m,2)strains at the mid-plane of the top glass layer, at thelocation of the plastic hinge:

ε′4,b,2 = ε′4,m,2⇒ W = 2

(tg2

− y′4)

y′4=0�⇒ W = tg (A1)The above relationship assumes that the crack has notreached the top of the glass layer, otherwise contactbetween adjacent fragmentswill be lost when themem-brane strain equals the bending strain at the top of thecross-section (ε′4,b,1 = ε′4,m,1).

The bending strain is then calculated under theassumption of plane sections remaining plane and theassociated linear strain distribution. The entire beamdeformation is localised at the plastic hinge (κ′4 =θmidspan

Lh), where the beam rotation can be obtained

from compatibility under the small angle approxima-tion (θmidspan = 4WL ):

ε′4,b,2 = κ′4y2 =θmidspan

Lh

(1

2tg− y′4

)

= 4WLLh

(1

2tg− y′4

)(A2)

Themembrane strain is given by the ratio of the changein beam length (�Lm), due the extension caused by thelarge deflections under the axial restraint, to the initiallength. As deformation only occurs at the plastic hinge,due to the rigid-plastic material law, the plastic hingelength (Lh) is considered as the initial length. For smalldeflections relative to the beam length:

ε′4,m,2 =�LmLh

=2

[√(L2

)2 + W2 − L2]

Lh

binomialexpansion�⇒

ε′4,m,2 =2W2

LLh(A3)

The ratio of the membrane force (N′4) to the puremembrane capacity (N4) can therefore be expressed

in terms of the location of the plastic neutral axis(y′4) under combined bending and membrane action,or the midspan vertical deflection (W):

n = N′4N4 =N4−T′4N4

={1 − y′4y4 , y′4 ≥ 0 or W ≤ tg1, y′4 ≤ 0 or W ≥ tg

(A4)

The ratio of the bending moment (M′4) to the puremoment capacity (M4) is obtained by applyingmomentequilibrium about the centroid of the top glass layer, asshown in Fig. 6. This also depends on the location of theplastic neutral axis, aswhen it has reached the top of thecross-section, only the membrane force will generate amoment:

m = M′4

M4= T

′4yT + C′4yC + N′4yN

M4

=

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

12 tgy

′4− 13

(y′4

)2+y4(12 tpvb+ 12 tg

)

y4(12 tpvb+tg− 13 y4

) , y′4≥ 0 or W ≤tg12 tpvb+ 12 tg

12 tpvb+tg− 13 y4

, y′4 ≤ 0 or W ≥ tg

(A5)

A.2 Failure strain

The total longitudinal strain in the interlayer is the sumof the bending (ε′4,b,3) and membrane (ε′4,m,3) strains:

ε′4,b,3 =4W

LLh,1

(tpvb + tg − y′4

) W=2(tg2 −y′4

)

�⇒

ε′4,b,3 =⎧⎨

⎩

4WLLh,1

(tpvb − tg2 + W2

), W ≤ tg

4tgLLh,1

tpvb, W ≥ tg(A6)

ε′4,m,3 =

⎧⎪⎨

⎪⎩

2W2LLh,1

, W ≤ tg2t2g

LLh,1+

(2W2LLh,2

− 2t2g

LLh,2

), W ≥ tg

.

(A7)

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The effects of high strain-rate and in-plane restraint on quasi-statically loaded laminated glass: a theoretical study with applications to blast enhancementAbstract1 Introduction2 The influence of strain rate on material properties2.1 Glass layers2.2 Interlayer

3 Analytical models accounting for high strain-rates and in-plane restraint3.1 Simple bending analysis3.2 Combined bending and membrane analysis

4 Results and discussion4.1 Influence of high strain-rate4.2 Influence of in-plane restraint

5 ConclusionsAcknowledgementsAppendix A: Mathematical derivation of analytical modelsA.1 Yield conditionA.2 Failure strain

References

the effects of high strain-rate and in-plane restraint on ...pren 13474-3 (2009) for low...

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