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the thickness of the glass layers and the stiffness of the interlayer that provide the plate with exactly

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and partitions [22,23].

safe design of glass is based on a fundamental evaluation namely,the ratio between the load-carrying capacity and ultimate loaddemand, and the ratio between the span and maximum deectiondo not provide an exhaustive description of safety. That theseratios are adequate is a necessary but not a sufcient condition.

ree condit

2.1.1. First fail-safe condition: RedundancyThe presence of aws is unavoidable in glass. From the

tural point of view, the aws that are worthy of attention are thecracks, since glass breaks when the combination of a load and acrack causes the stress intensity factor to reach the critical stressintensity factor [24,25,2933].

Cracks can be divided into two types namely, cracks thatoccur during manufacturing due to production processes, andcracks that occur during the service life due to concentrated loads,

Tel.: +39 041 2571289.E-mail address: [email protected]

Materials and Design xxx (2014) xxxxxx

Contents lists availab

Materials an

elsNot only do glass elements have to withstand the design loads,but also they have to fulll fail-safe design [16,21,22,2428]. Fail

Glass is fail-safe if it satises the following thhttp://dx.doi.org/10.1016/j.matdes.2014.05.0300261-3069/ 2014 Elsevier Ltd. All rights reserved.

Please cite this article in press as: Foraboschi P. Optimal design of glass plates loaded transversally. J Mater Design (2014), http://dx.doi.org/1j.matdes.2014.05.030ions.

struc-glass, loaded predominantly out-of-plane, and used as primary orsecondary structures, or as non-bearing elements. Primary struc-tures include oors and roofs [1621]; secondary structuresinclude overhead glazing supported by primary structures and ele-ments of staircases [16,20]; non-bearing elements include faades

Glass ensures safety, security, and safeguarding only with a fail-safe design. Moreover, glass ensures transparency only if its thick-ness is not excessive.

2.1. Fail-safe conditions1. Introduction

To optimize the performance anda structural designer is to determineof component(s) under some functstraints [115]. To select the optimsimilar structures is called optimal

This paper is devoted to the optithe minimum allowable stiffness, while the stress verications are fullled automatically.Finally, for every span and load that is found in building applications of glass, the paper provides the

thicknesses and the materials of the glass layers and interlayers that adjust capacity to match demand.These results may also replace structural analysis and assessment of Laminated Glass plates.

2014 Elsevier Ltd. All rights reserved.

f a structure, the task ofaterial(s) and geometryequirements and con-ucture among possible.sign of plates made of

In order to achieve an exhaustive description of safety, threeother conditions have to be considered in designing and assessingglass elements. When these conditions are satised, glass is calledfail-safe.

2. Fail-safe transparent platesLaminated GlassMinimum thicknesses results, obtained using an analytical exact model, show that the limit states are always dictated by the

maximum deection and not by the load-carrying capacity. Thus, optimal design requires ndingOptimal design of glass plates loaded tra

Paolo Foraboschi Universit IUAV di Venezia, Dipartimento di Architettura Costruzione Conservazione, Co

a r t i c l e i n f o

Article history:Received 16 March 2014Accepted 17 May 2014Available online xxxx

Keywords:Glass design

a b s t r a c t

This paper focuses on glassplates.First, the paper shows th

glass layers bonded to oneact upon a sacricial glassThen, activity was direct

and carrying out research

journal homepage: www.e glass plate is fail-safe only if the load-bearing system is composed of twother with an elastomeric interlayer (Laminated Glass), and if the live loads(tri-layer system).t analyzing the simply-supported fail-safe glass plate loaded out-of-plane,eted at reducing the incidence of weight and cost (optimal design). Theto delle Terese, Dorsoduro, 2206, 30123 Venice, Italy

d in buildings and presents a criterion for fail-safe optimal design of glassversallyle at ScienceDirect

d Design

evier .com/locate /matdes0.1016/

Thus, glass has to be designed so as to ensure adequate post-

d Dpeak performance, in particular a certain residual post-fracturestrength. Hence, fail-safe glass must be composed of more ele-ments, whose composition provides redundant paths, so that ifone individual glass element fails, the structure will remain stand-ing, and so that the collapse is reached only after a considerablepost-peak displacement.

Furthermore, when collapse propagates from the stable state toan unstable state until total failure occurs, the unabsorbed portionof gravitational energy has to be moderate, so that the structurereleases only minor kinetic energy.

2.2. Fail-safe glass

The only way to satisfy the second and third fail-safe conditionsis to use Laminated Glass (LG) [1719,2123,25,26,3539], whilemonolithic glass cannot satisfy these conditions.

LG is composed of two or more plies of glass attached to oneanother with transparent thermoplastic interlayer(s). As usedherein, the term interlayer refers to any material now knownor developed in the future, for manufacturing and assembling LG.Currently, the use of Poly Vinyl Butyral (PVB) interlayer dominatesLG [23,27,3638,40]. However, the growing demand for LG in engi-neering structures and in building faades and interiors has driventhe development of new transparent thermoplastic materials thatextend the physical performance of LG. As a result, for some yearsnow there are other types of interlayers that are in use in LG2.1.2. Second fail-safe condition: Mode of collapseIn the event of a glass element breaking, the large and jagged

shards must be held in place and only some small blunt edged par-ticles can fall down at most, so as to avoid injury to people or petsdue to ying, shattering or falling glass [25,27,33]. Fail-safe design,therefore, requires that when a glass ply is broken no sharp shardfalls down from a broken glass element.

Thus, fail-safe glass has to be designed so that, when an elementbreaks, it shatters into small, dull pieces and that the system keepsthe vast majority of the pieces together anyway.

2.1.3. Third fail-safe condition: Post-peak load-carrying capacityGlass elements exhibit a very steep post-peak descending

loaddeection curve, which ends with a utterly brittle collapse[24,25,27,31,32,3437]. Conversely, fail-safe glass requires thatthe post-peak loaddeection curve descends slowly from thepeak, which represents the load-carrying capacity (rst-cracking)to the collapse (ultimate displacement) and that the collapse isnot fragile.impacts, abrasions or piercings. The former are called initial cracksand the latter contact-induced cracks.

If glass has only the initial cracks, the stress that causes a glasselement to break is substantial, while if glass also has contact-induced cracks, it may be marginal. Moreover, if glass has onlythe initial cracks, the load-carrying capacity of a glass elementcan be predicted and guaranteed, while if glass also has contact-induced cracks, it cannot be predicted or warranted against.

Contact-induced cracks are produced by the live loads. There-fore, fail-safe design requires that the glass elements that collectthe live loads are neglected during assessment of load-carryingcapacity. In so doing, glass can be designed so that the load-carry-ing capacity and the stiffness do not depend on contact-inducedcracks. As a result, any piece of glass is part of a system that hasa certain amount of redundancy.

2 P. Foraboschi /Materials an[16,22,25,28,35,4042], such as Ethylene Vinyl Acetate copolymers(including various Vinyl-Ester polymers), Poly-Carbonates (in par-ticular Thermoplastic Poly Urethane), and Ionoplast Polymers (IPs).

Please cite this article in press as: Foraboschi P. Optimal design of glass platj.matdes.2014.05.030The greatest enhancements in mechanical properties of inter-layers have been provided by IPs, which currently hold an appre-ciable fraction of the world market share [24,26,28]. The keymechanical property difference between an IP interlayer and aPVB interlayer is the visco-elastic behavior for high temperaturesand long-term load durations, which provides an IP interlayer witha signicant stiffness advantage over a PVB interlayer. In particular,for high temperatures and long loading durations, the shear elas-ticity modulus of IP may be more than two orders of magnitudegreater than that of PVB [3944].

Since the newer interlayers are signicantly stiffer than the tra-ditional interlayers, these recent developments in technology havesplit the thermoplastic family for lamination of glass into two clas-ses namely, the utterly compliant interlayers, such as PVB, andthe relatively stiff interlayers, such as IP [44].

When a glass ply is broken, almost all the glass shards adhere tothe polymeric interlayer rather than scatter, avoiding injury topeople or pets from broken glass. Thus, LG fullls part of thedemand of the second fail-safe condition, whereas LG does not ful-ll automatically the other part of this condition.

More specically, when the glass is broken, the polymeric inter-layer holds in place almost all the pieces of glass, but some piecesof glass can fall down anyway. Thus, the layer that can give rise toying, shattering or falling glass (e.g., the inferior layer of horizon-tal glazing) has to disintegrate into pieces that are small and notsharp (i.e., small dice shapes, instead of dangerous shards). How-ever, this last demand of the second fail-safe condition is not satis-ed automatically by LG, but it requires that specic glass types areused, according to the structural function of the glass. In so doing,the second fail-safe condition is satised completely.

LG also provides the system with some post-breakage load-car-rying capacity. In fact, the common boundary between the glassply and interlayer keeps the fracture in the glass ply where it hasstarted; i.e., the interface prevents the crack from propagating intothe whole system. Accordingly, when a glass layer is broken, theother glass layer may bear the applied load at least as an individualelement. It follows that a LG system guarantees a residual capacityat least equal to the strength of the intact glass layer [43,45,46].

Hence, the loaddeection curve of LG descends slowly fromthe peak up to the failure of the other glass layer. Thus, LG satisesautomatically two of the demands posed by the third fail safe con-dition namely, the ultimate displacement is fairly large and therst branch of the post-peak loaddeection curve is not steep.Conversely, LG does not satisfy automatically the last demandposed by the third fail safe condition namely, the collapse ofLG must not convert excessive potential energy into kinetic energy.

In order to avoid that shortcoming, one of the layers has to bemade of a glass type that, in the event of breaking, fractures intolarge chunks and slivers. The interlayer can connect these largepieces of broken glass to each other, which provides the laminatedsystem with a certain residual capacity. As a result, the system cancarry the dead load even when all glass layers are broken. In sodoing, the last branch of the post-peak loaddeection curve isnot steep and the system dissipates almost all the potential energybefore reaching the collapse. This system cuts down the kineticenergy released during the breakage, and it meets completely thethird fail-safe condition.

Ultimately, only LG can fulll the second and third fail-safe con-ditions; however, LG does not fulll automatically all these condi-tions. To be fail-safe LG also has to fullls the requirements ofSection 2.3.

The rst fail-safe condition can be satised only by using thesacricial ply design criterion [26,43,4749]. This criterion can

esign xxx (2014) xxxxxxbe synthesized as follows: The glass ply that collects the live loadsis allowed to fracture in service (sacricial ply). According to thesacricial ply design criterion, hence, the layer that collects the live

es loaded transversally. J Mater Design (2014), http://dx.doi.org/10.1016/

loads has to be considered as broken, independently of whether itis actually undamaged (intact) or broken (fractured).

Thus, the sacricial ply is redundant against the ultimate limitstates, and sometimes it is also considered redundant against theserviceability limit state. Hence, the role of the sacricial ply is toallow for the possibility of accidental glass breakage due to contactdamage (Fig. 1).

Since limit analysis does not take into account the sacricial ply,safety verications do not depend on this ply. Hence, the sacricialply design criterion implies that safety assessment is unrelatedwith contact-induced cracks, which provides adequate reliabilityto model predictions, limit analysis, safety assessment andverications.

The sacricial ply may be kept in service also in fractured con-dition; it has to be substituted only when it has lost the capacity ofshielding the other layers or causes alarm to users or pedestrians.

Ultimately, only LG can fulll the three fail-safe conditions.However, in order to be fail-safe, LG must satisfy three other con-ditions. Firstly, the outer layers have to ensure that, in the event of

breakage structural capacity can be made available at the level of

ricial (upper layer), and consequently safety assessment takesinto account only the other two glass plies (central and lower lay-ers) together with the connecting interlayer (Fig. 1).

However, a tri-layer (quadri-layer) glass system is not automat-ically fail-safe; it is fail-safe only if the types of glass used for eachlayer are adequate to the role that each layer has in the system. Inparticular, the glass type has to provide the layer with adequateultimate stress and failure mode.

Thus, there is a need to develop criteria for choosing the bestsingle material for each individual glass layer.

2.3.1. Sacricial glass plyThe sacricial ply has to be made of toughened glass (usually

tempered glass), since it has to tolerate high local tension stressesdue to contact forces (Fig. 1). In fact, toughened glass guaranteessurface compression stresses (together with compensating tensionstresses in the glass interior zone), which are developed during thetempering process in the case of tempered glass [33,36] or duringthe ion-exchange process in the case of chemically-strengthenedglass [50]. Due to these coactive stresses, toughened glass [51]has signicantly higher tensile strength than annealed glass [35]and higher strength than heat-strengthened glass [27].

Moreover, in the event of breaking, toughened glass disinte-grates into small pieces [33,36,50,51], while annealed [35] andheat-strengthened [27] glass break into large, razor-sharp slivers.

P. Foraboschi /Materials and Dthe structural system, instead of at the level of the LG element.

2.3. Fail-safe transparent Laminated-Glass

Beyond 5560 mm of thickness, glass cannot be consideredtransparent (sometimes, not even translucent) [14,16,2022].Thus, the sum of the thicknesses of all the layers and interlayerscannot exceed that limit, apart from colored glass members orthe elements that are not intended to be transparent, such as glasscolumns, which cannot be thin.

The thickness of a sacricial glass ply cannot be lower than2 mm and it should be no less than one fth of the thickest layer.

Fig. 1. LG plate loaded on the top face, designed according to the sacricial plybreaking, the shards cannot injure people. Secondly, the systemhas to include a layer that breaks into large pieces of glass, so asto provide adequate residual capacity. Thirdly, the layer that col-lects the live loads has to be sacricial. If the live loads, includingthe environmental actions (windborne debris) may act on boththe external faces of the member, two sacricial plies are necessary(Figs. 1 and 2).

Solutions different from the tri-layer (quadri-layer) systemshould be considered only when the fail-safe conditions that areviolated by the LG are fullled by the whole building. For instance,when breakage does not involve users or pedestrians, or the post-concept. Tri-laminated system (three glass plies and two interlayers): upper glasslayer, central glass layer, and lower glass layer. The glass ply that collects the liveloads (upper layer) is sacricial.

Please cite this article in press as: Foraboschi P. Optimal design of glass platj.matdes.2014.05.030The total thickness of the interlayers ranges from 0.76 mm toapproximately 3 mm. Therefore, in the case of transparent tri-layersystems, the thickness of the cross-section that carries the loadsshould not surpass approximately 55 mm and in the case of trans-parent quadri-layer system 50 mm. Thus transparent membersrequire that the load-bearing system is composed of two layers.On the contrary, three or more layers either would entail morelaminations than is necessary or would exceed the thickness limitthat guarantees transparency.

Hence, the system that simultaneously satises the demand fortransparency and is fail-safe is the tri-layer (quadri-layer) plate. Inthe tri-layer system, the glass ply that collects the live loads is sac-

Fig. 2. LG plate loaded on the top and bottom faces, designed according to thesacricial ply concept. Quadri-laminated system (four glass plies and threeinterlayers). The glass plies that collect the live loads are sacricial.esign xxx (2014) xxxxxx 3When the glass is shattered, if some broken pieces of glass falldown instead of being held in place, these are only small piecesof blunt glass, instead of dangerous shards (Fig. 1).


Ultimately, only if the sacricial glass ply is made of toughenedglass, the system satises the rst and second fail-safe condition.Herein, the sacricial ply is referred to as upper glass layer of thetri-layer system.

The thickness of the upper layer has to be adequate to providethe ply with shielding capacity. Conversely, the thickness must notbe related to the load-bearing capacity of the system, which doesnot depend on it.

2.3.2. Load-bearing system of the tri-layer Laminated Glass plateThe load bearing system is the whole system minus the sacri-

cial glass ply (plies) and minus the interlayer that connects this plyto the other glass layer. Hence, the load bearing system is com-posed of two glass layers and the polymeric interlayer midwaybetween these layers. The bending and shear stresses due to thedesign loads are assumed to exist only in the load-bearing system.

Herein, the glass layer adjacent to the sacricial ply is referredto as central (glass) layer of the tri-layer system, and the otherglass layer of the load-bearing system as lower (glass) layer(Fig. 1). Hence, the outer layers of the tri-layer system are the sac-ricial ply and the lower layer.

To make the best use of the materials, the layers of the load-bearing system must have equal thickness. In the case of live loadsapplied to both the faces of the plate (quadri-layer system, not con-sidered here but in Section 7.4), this requirement is obvious. In the

Hence, the load-bearing system is anti-symmetric with respectto the plane midway between the central and lower layers (themiddle plane of the connecting interlayer; Figs. 35).

Restraint against rotation along the edges of the plate would notbe suitable for glass, since such a restraint would perform a clamp-ing action that would damage the glass. Moreover, the normal oatprocess that produces the sheets of glass consists of a tin bathabout 34 m wide. Thus, glass continuous plates with multiplesupports are not common. Thus, glass plates are free from bendingmoments at the edges.

The maximum tension stress occurs at the center of the external(bottom) surface of the lower layer, rgo (Figs. 4 and 5). Thus, thelower layer has to be made of a glass type whose strength is ade-quate to bear the maximum tension strength due to the designloads. But this layer is external. In the event of breaking, this layerhas to shatter into small, blunt pieces, so that dangerous shards

Fig. 4. Stress prole in the load-bearing system (central and upper layers, togetherwith the connecting interlayer; Fig. 3). The stress prole is anti-symmetric withrespect to the middle plane of the load-bearing system (shown in the gure).Shadowed areas: tension stresses. The gure shows the symbols used for thestresses throughout the paper.

Fig. 5. Stress prole in the load-bearing system, in the case of high value of theinterlayer stiffness, k. The central layer is completely in compression and the lowerlayer is completely in tension.

4 P. Foraboschi /Materials and Dcase of live loads applied to a face only (with the same direction asthe dead load), a wide-ranging analysis proved that the symmetricload-bearing system minimizes the costs (which depend on boththicknesses and glass types) and optimizes the post-breakagebehavior.

In order to obtain a fail-safe laminated system, the LG plate hasto be composed of layers made of different types of glass. In gen-eral, using two or more types of glass in the LG plate is a good solu-tion, which is not restricted by manufacturing or other problems[16,21,22] (Fig. 1).

Glass elasticity modulus does not depend on the treatment.Hence, the elasticity modulus of the annealed, heat-strengthened,tempered, and chemically-strengthened glass is the same (approx-imately 70,000 N/mm2).

Fig. 3. Load-bearing system: bi-layer system composed of the central and lowerlayers of the tri-laminated system, together with the connecting interlayer(shadowed). Prole of the longitudinal displacement, u, due to anti-clockwiserotation angle of the cross-section, h. The points P, R, S and, respectively, P0 , R0 , S0 are

anti-symmetric with respect to the middle plane of the load-bearing system. Thus,the u-displacements of P, R, S are equal in magnitude and opposite in direction withrespect to P0 , R0 , S0 .

Please cite this article in press as: Foraboschi P. Optimal design of glass platj.matdes.2014.05.030esign xxx (2014) xxxxxxcannot fall down. In order to fulll both the requirements, thislayer has to be made of toughened glass (Fig. 1). In many cases,heat-strengthened or annealed glass could satisfy the stress


nd Ddemand, but these glass types never fulll the failure modedemand.

The central glass layer is subjected to compression stresses andusually also to tension stresses (Fig. 3). This layer bears the maxi-mum compression stress of the tri-layer system, rgo (Figs. 4 and5), which occurs at the center of the surface connected to the sac-ricial ply through the interlayer. Conversely, the tension stress inthe central glass layer is always less than in the lower glass layer.The maximum tension stress in the central layer, rgi, approachesrgo when the stiffness of the interlayer approaches the stiffnessof the layers.

Fail-safe design of glass requires that at least one layer of thelaminated system is made of either heat-strengthened or annealedglass, in order to provide the system with adequate post-breakagecapacity when all glass layers are broken. In fact, a heat-strength-ened glass layer (sometimes even an annealed glass layer) cancarry the entire dead load of the plate when the load-bearing sys-tem is broken as well as the sacricial glass ply, since the interlayerconnects the broken pieces of heat strengthened glass.

Considering that the lower layer has to be made of toughenedglass and that the central layer has to bear tension stresses lessthan the lower layer, the central layer has to be made of eitherheat-strengthened glass or annealed glass, according to the maxi-mum tension stress in this layer rgi (Figs. 1, 4 and 5).

The fact that the central lass layer breaks into large chunks andslivers with razor sharp edges poses no risk of injure to people,since this layer is inner.

3. Optimal design of fail-safe Laminated Glass plates

The criterion that allows glass to optimize the structural perfor-mance, maximize transparency, and minimize the cost (optimaldesign criterion) is that the thicknesses of the central and lowerlayers and the stiffness of the interlayer produce a behavior thatmatches the most severe limit of the plate.

The load-bearing system (Section 2.3.2; Figs. 1 and 2) must haveadequate strength to resist the most severe loads and adequatestiffness to resist all the behaviors that can compromise the ser-viceability of the plate. The rst requirement is satised if and onlyif the maximum tension stress in each glass layer (i.e., rgo and rgi;Figs. 4 and 5) does not exceed the tension strength of the glass thatthe layer is made of. The second requirement is satised if and onlyif the maximum deection of the plate (i.e., wmax) does not exceedthe maximum allowable deection.

Consistently with the optimal design criterion, the glass typeand thickness of the central layer and the lower layer, togetherwith the stiffness of the interlayer have to produce either a maxi-mum tension stress in one of the two layers that is equal to thetension strength of that layer or a maximum deection of the platethat is equal to the maximum allowable deection. One conditionis matched exactly while the other two conditions with a certainmargin.

Tension strength of glass depends on the glass types used forthe layers and also on the duration of load, which inuences glassstrength [23,24,3639,42]. Moreover, this limit has to include apartial factor that allows for the class of consequences associatedto the intended use of the plate as well as the uncertainties inthe material strength.

The load to be used in the ultimate limit state verications inaccordance with the considered code has to produce stresses thatdo not exceed the strength of each type of glass that the systemis composed of. If this limit is respected, the strength of the glassplate is adequate.

P. Foraboschi /Materials aFor the dead loads (i.e., long-duration load), the limits that fulllthe main codes on glass [43,45,46,5263] are 5.5 N/mm2 for theannealed glass, 25.0 N/mm2 for the heat-strengthened glass,

Please cite this article in press as: Foraboschi P. Optimal design of glass platj.matdes.2014.05.03065.0 N/mm2 for the tempered glass, and 80.0 N/mm2 for the chem-ically-strengthened glass.

For the wind loads (i.e., short-duration load), the limits that ful-ll the main codes on glass [43,45,46,5263] are 19.0 N/mm2 forthe annealed glass, 40.0 N/mm2 for the heat-strengthened glass,80.0 N/mm2 for the tempered glass, and 105.0 N/mm2 for thechemically-strengthened glass.

The maximum deection of the plate depends on the intendeduse of the plate and the lifespan of the building. The load to be usedin the serviceability limit state verications in accordance with theconsidered code has to produce a ratio between the maximumdeection and span that is less than a maximum value. If this limitis respected, the stiffness is greater than the minimum required bythe code.

The deections must not damage the construction or causealarm to users, and the vibrations must not cause ailments to usersor generate or propagate noises. In order to guarantee these ser-viceability conditions, two ratios have to be considered. Firstly,the ratio that results from the deection due to the total load (deadplus live load) and secondly, the ratio that results from the deec-tion due to the live load only. The rst ratio measures the stiffnessagainst the damage of the construction, and ensures adequatedurability. The second ratio measures the stiffness against thevibrations, and prevents the occurrence of deections that couldcause alarm to users.

The maximum allowable deection-to-span ratio prescribed forglass depends on the intended use of the element [21,22].

If the intended use of the plate is a oor, the maximum allow-able deection-to-span ratio recommended by almost all the codesis 1/200 under the live loads. If the intended use of the plate is aroof, it is still 1/200 but under the total loads [43,45].

If the intended use of the plate is a faade or a partition, themaximum allowable deection-to-span ratio recommended bythe most important codes is 1/65 under the live load[43,45,46,5255].

It is to note that the maximum allowable deection-to-spanratio used for glass faades and partitions is very high, and thatused for glass oors is double than the limit commonly used forother materials. The reason is that the elasticity modulus of glassis low, i.e. approximately 70,000 N/mm2. If the common limitswere used, transparent architecture would be impossible. Hence,the deection limits were calibrated so as to foster the use of glassin architecture. Nevertheless, the deection beyond which a glasselement no longer fullls the relevant design criteria is usuallywell represented by the values adopted by codes. It being under-stood that designers should check the limit used case by case.

4. Analytical modeling of the Laminated Glass plate

Until a short time ago, the sandwich plates, including LG plates,were modeled by using either nite element models or practicalformulas.

The layer-to-interlayer elastic modulus ratio and layer-to-inter-layer thickness ratio may be very high. For LG, the former ratio mayreach one million and the latter may be greater than fty. Thesevalues impinge upon all the numerical solutions [23,36,42,47,6468]. Thus, the numerical results need to be checked and calibratedagainst exact results. However, no exact solution existed; there-fore, numerical results could be neither checked nor calibrated.Moreover, no closed-form formula is available with a numericalmodel; therefore, numerical models allow optimal design to beperformed only with a trial and error approach.

The practical formulas of LG, which were derived either empiri-

esign xxx (2014) xxxxxx 5cally or from numerical models, replace the laminated system withan equivalent monolithic system. However, not only are these for-mulas rough, but above all they provide the maximum stress and


The dependence of the shear elasticity modulus of the inter-layer Gt on temperature and loading duration is dictated by theinterlayer material, which is a design option. Moreover, t is adesign parameter as well. Hence, k is a design parameter whose

Fig. 6. Diagram of the load-bearing system of the laminated glass plate, with thegeometric and mechanical symbols. The diagram shows the glass types of the loadbearing system. The system together with the sacricial glass ply (plies) satisesthe fail-safe conditions.

Fig. 7. Bi-layer system proposed as alternative to the fail-safe tri-layer system ofFig. 1. If the live loads act onto both the faces of the plate, a bottom sacricial ply isadded to the bi-layer system, so as to obtain an alternative design solution to thefail-safe quadri-layer system of Fig. 2. The bottom sacricial ply has no role inthe structural behavior; therefore it can be ignored. The bi-layer system shown inthe gure incorporates the sacricial ply into the load-bearing system: The upperglass layer may be sacricial with respect to the strength verication, while it isusually considered in the stiffness verication. In the stress state shown in thegure, the depth of the neutral axis in the upper layer, which depends on k, L, B, andh, is greater than the depth of the load-induced crack, which depends on contactloads but not on the design loads. Consequently, the load-induced crack is incompression. In this case, the upper glass ply can be taken into account in thestrength verication as well (i.e., in this loading condition, the upper layer can benot sacricial in any limit state verication).

d Ddeection only, while they do not provide the complete stress eldin the LG plate. Thus, practical formulas are not viable tools for opti-mal design of LG plates.

Ultimately, numerical models and practical formulas do notallow the structural designer to select the optimal LG plate amongthe fail-safe plates. In order to determine the thickness andmaterialof layers and interlayer under any functional requirements and con-straints, the design has to use an analytical closed-form (exact)model.

Now, the LG plate can be modeled analytically, since a closed-form exact model was recently obtained [19,6971], which pro-vides the stresses and displacements of the sandwich plate. Themodeling assumptions are that the glass layers are governed bythe KirchhoffLove hypotheses and that the polymeric interlayerhas an elastic behavior.

Hence that model is linear. The deection-to-span limit withinwhich geometrical non-linearity of a plate is always negligible islower than 1/65. However, the simple-support is a restraint thatreduces geometrical non-linearity. As a result, geometrical non-lin-earity is not substantial in LG [72], and it may be neglected. How-ever, geometric non-linearity increases the safety margin.

The interlayer materials behave in a visco-elastic way, but theycan be modeled in a linear-elastic way by means of the modulus ofelasticity in shear Gt, provided that Gt is related to temperature andloading duration [18,23,25,36,37,4044,64]. Ultimately the resultsof the analytical model do not suffer from any constrainingassumption.

When the structural designer commits to certain materials (i.e.,glass type and a polymer for the interlayers), that analytical model[19,6971] may provide a technique to optimally determine thegeometry of the sandwich plate. Likewise, when the geometry ischosen, that model together with the fail-safe conditions may pro-vide a technique to select the best materials for the layers andinterlayers.

Fixing one in general inuences the optimality of the other, butconcurrently determining the optimal geometry and selecting thebest materials remained an open issue in LG plate design. There-fore, activity was directed at carrying out research targeted at pro-viding a general framework for optimal design of LG plate based ona criterion suitable for LG plates, and at reducing the incidence ofweight in the design of LG plates.

To this end, the aforesaid model was applied to the laminatedsystem that fullls the fail safe condition, in the framework ofthe optimal design.

Since the upper layer is sacricial, analytical modeling does notconsider the tri-layer system (Fig. 1), but the sandwich system(Fig. 6). Hence, analytical modeling took into account the LG platewhose cross-section is formed by two glass layers, each one of thick-ness h (Figs. 36), called central and lower layer respectively, plus aninterlayer of thickness t. This research considered the rectangularplate with sides L B (L 6 B), simply-supported at the four edges,subjected to a uniformly distributed lateral load p (Figs. 1 and 6).

The material properties of the layers are the elastic modulus,shear (elastic) modulus, and Poissons ratio of glass, Eg, Gg, and mg,respectively. For structural glass, commonly: Eg = 70,000 N/mm2,Gg = 28,689 N/mm2, and mg = 0.22, independently of the glass type[43,45,46,5263].

The interlayer material properties are the shear modulus Gt andthe elasticity modulus Et.

The behavior of the interlayer can be represented by its stiff-ness, k (Fig. 6):

K 2 Gtt

1

6 P. Foraboschi /Materials anThe greater the shear modulus Gt or the thinner the interlayer,the greater k, and vice versa.

Please cite this article in press as: Foraboschi P. Optimal design of glass platj.matdes.2014.05.030esign xxx (2014) xxxxxxvalues, within certain limits, can be chosen so as to obtain the opti-mal design.


the most severe limit of the plate.

sis considered these interlayer thicknesses only.

nd D5.2. Modeling and limit analysis

The analytical model described in Section 4 was used to obtainthe maximum stresses in the glass layers, rgo and rgi (Figs. 4 and 5)under the combination of actions for the ultimate limit state, andthe maximum deection wmax of the plate under the combination5.1. Wide-ranging analysis on Laminated Glass plates

An extensive analysis was carried out on the LG plate, whichaimed at identifying the solutions to the optimal design problemsformulated above.

The analysis considered all the spans and loads that can befound in building applications of glass. In particular, the analysisranged from 2.00 m to 6.00 m spans. Lower spans belong to glassused in fenestration, so they are outside the scope of this paper.Greater spans make manufacturing difcult and may cause prob-lems during the installation. The analysis considered all the possi-ble design loads of bearing and non-bearing members. Moreover,the analysis considered, for glass layers and polymeric interlayers,only products that are available on the market. In so doing, realisticresults were obtained, which should be relevant not only toresearchers in academia but also to engineers and designers inindustry, as well as practitioners.

Float glass is produced in varying thicknesses; different thick-nesses are not available. Glass panes can be found in standard met-ric thicknesses of 2, 3, 4, 5, 6, 8, 10, 12, 15, 19, and 25 mm.Sometimes, also the thickness of 22 mm can be found, althoughit is less common. Accordingly, the analysis considered these thick-nesses only (the 22-mm-thickness was necessary in one case only).It is to note that the 3-mm-gap between two consecutive thick-nesses of the panes from the 12-mm-thickness to the 25-mm-thickness strongly conditioned the optimality of the solutions.

The treatments that these panes can be subjected to are anneal-ing [56,57], heat-strengthening [58,59], tempering [60,61] andchemically strengthening [62,63] processes. It is to note that ther-mal treatments of very thin panes may encounter some difculties;however, the 2-mm-thichness was necessary in one case only.Likewise, the heat-strengthened process of glass panes with thick-ness greater than 15 mm may encounter some difculties too,since in this case the enthalpy (heat content) is very high due toa cooling process slower than that of the tempered glass. Thus, inthe heat-strengthening treatment, the core of thick panes remainshot when the surface is already cold and solidied. The subsequentcooling of the core might develop further surface compressioncoactive stresses, which modify the failure mode toward that oftempered glass, but without reaching the strength of temperedglass.

Polymeric interlayers for LG are produced in sheets with athickness of 0.38 mm. Interlayers are obtained by using one sheetor by stacking sheets, usually up to a maximum 4. Accordingly tothe number of sheets that are piled up, an interlayer may have athickness of 0.38, 0.76, 1.14, and 1.52 mm. Accordingly, the analy-5. Laminated Glass plates that match exactly the most severelimit

The optimal design criterion is that the thickness of the centraland lower layers, which are equal to one another, and the shearstiffness of the interlayer have to produce a behavior that matches

P. Foraboschi /Materials aof actions for the serviceability limit state.The maximum allowable values used in the analysis were those

prescribed by the main codes on glass (Section 3) [43,45,46,5263].

Please cite this article in press as: Foraboschi P. Optimal design of glass platj.matdes.2014.05.030More specically, the maximum values of rgo, rgi, and wmax used inthe limit analysis, and the parameters used in the models were thefollowing:

The maximum allowable stresses (strength) considered in ulti-mate limit state verications of LG plates used for oors or cov-ering roofs were 5.5, 25.0, 65.0, and 80.0 N/mm2 for annealed,heat-strengthened, tempered, and chemically-strengthenedglass, respectively.

These stress values include all the strength-reduction factors ofglass, barring the factor that takes into account the type of edgework, since the maximum tension stress is away from the edges.In particular, these values consider that the load duration is equalto the lifespan of the element. Moreover, these values include thematerial partial factors for the ultimate limit state prescribed bythe codes.

The maximum allowable stresses (strength) of each glass typeconsidered in ultimate limit state verication of LG plates usedfor faades and partitions were 19.0, 40.0, 80.0, and 105.0 N/mm2 for annealed, heat-strengthened, tempered, and chemi-cally-strengthened glass, respectively.

Also these stress values (and for the same reason) include all thestrength-reduction factors of glass, barring the factor that takesinto account the type of edge work. These values consider thatthe stresses are produced by the wind action or impacts, whoseload duration was assumed to be 5 s. These stress values alsoinclude the material partial factors for the ultimate limit state pre-scribed by the codes.

The maximum allowable displacement considered in service-ability limit state verication of LG plates used for oors andcovering roofs was 1/200 of the minor span of the plate.

The maximum allowable displacement considered in service-ability limit state verication of LG plates used for faadesand partitions was 1/65 of the minor span.

For every glass type, Eg = 70,000 N/mm2 and mg = 0.22. The analysis considered two interlayer materials namely, PVBand IP (i.e., the most common utterly compliant and relativelystiff types, respectively) and two types of glass members namely, primary or secondary bearing members, as oors orcovering roofs, and non-bearing members, as faades andpartitions.

The conditions considered for oors and roofs (bearing mem-bers) were a loading duration of more than 50 years and a temper-ature of at least 50 C. The interlayer shear elasticity modulus usedfor this combined condition was Gt = 0.052 N/mm2 in the case ofPVB material and Gt = 1.50 N/mm2 in the case of IP materials.

The conditions considered for faades and partitions (non-bear-ing members) were a loading duration of 30 s (the wind gust canact on a plate already loaded by the mean wind) and a temperatureof 50 C. The interlayer shear elasticity modulus used for this com-bined condition was Gt = 0.300 N/mm2 for PVB material andGt = 105.00 N/mm2 for IP materials.

5.3. Analytical results

The results of the analysis are presented in Tables 1 and 2,which show the geometry of the glass layers and the stiffness ofthe interlayer that provide the optimal solution among all the LG

esign xxx (2014) xxxxxx 7plates whose behavior fullls the limit states.Table 1 considers bearing members and the relevant limits;

Table 2 considers non-bearing members and the relevant limits.


Table 1Laminated Glass plate congurations that match the most severe limit state ofbearing glass members. L = span of the plate; B = width of the plate; p = loaduniformly distributed on the plate (on the sacricial glass ply, which is the upperlayer); h = thickness of each glass layer of the bi-layer system that carries the stresses(i.e., of the central layer and the lower layer); k = stiffness of the interlayer betweenthe central and lower layer, dened by Eq. (1); rgo = maximum tension stress in thelower glass layer; rgi = maximum tension stress in the central glass layer; wmax =maximum deection of the plate.

Data Design Behavior

L B p h k rgo rgi L/wmaxmm mm kN/m2 mm N/mm3 N/mm2 N/mm2 mm/mm

1500 1500 0.50 3 3.6512 7.32 1.88 200.41500 1500 1.00 4 3.4571 9.01 3.28 200.01500 1500 2.00 5 5.5368 11.62 3.68 199.71500 1500 3.00 6 4.5128 12.92 5.43 199.81500 2000 0.50 4 1.4723 7.58 2.83 199.81500 2000 1.00 5 2.2415 9.70 3.34 200.71500 2000 2.00 6 4.7642 12.81 3.20 200.51500 2000 3.00 8 1.8212 13.43 6.26 201.31500 2500 0.50 4 2.9663 8.59 1.56 200.61500 2500 1.00 5 4.9663 11.08 1.68 200.31500 2500 2.00 8 1.0563 12.32 5.07 199.61500 2500 3.00 8 3.2632 15.06 4.20 199.42000 2000 0.50 5 0.6855 6.56 4.61 200.22000 2000 1.00 5 7.4374 9.76 1.40 200.22000 2000 2.00 8 1.1913 10.52 7.30 200.42000 2000 3.00 10 0.7281 11.81 9.92 200.62000 2500 0.50 5 1.6479 9.99 7.68 200.22000 2500 1.00 6 3.7532 10.02 2.22 199.72000 2500 2.00 8 3.0116 12.41 3.91 199.32000 2500 3.00 10 1.8108 13.45 6.12 199.42000 3000 0.50 5 3.8705 8.73 1.16 199.92000 3000 1.00 8 0.7448 9.56 4.20 200.42000 3000 2.00 10 1.0369 12.15 5.20 200.02000 3000 3.00 10 3.4539 15.04 3.89 200.32500 2500 0.50 5 4.0558 7.68 1.31 200.12500 2500 1.00 8 0.7600 8.34 5.93 199.82500 2500 2.00 8 3.4964 12.47 1.76 199.22500 2500 3.00 10 3.3766 13.19 5.06 200.02500 3000 0.50 6 1.7650 7.77 2.25 200.22500 3000 1.00 8 1.5023 9.50 3.66 199.42500 3000 2.00 10 2.1618 12.14 4.39 199.02500 3000 3.00 12 1.7737 13.50 5.92 199.62500 3500 0.50 6 4.9142 8.78 0.84 199.92500 3500 1.00 8 3.1047 10.61 2.06 200.22500 3500 2.00 10 4.9194 13.60 2.28 199.82500 3500 3.00 12 3.3684 14.98 3.73 200.22500 4000 0.50 8 0.5216 7.77 3.09 200.12500 4000 1.00 8 7.8622 11.61 8.25 199.42500 4000 2.00 12 1.3590 12.81 4.25 199.32500 4000 3.00 12 6.7982 16.30 2.07 200.43000 3000 0.50 6 4.3127 7.78 1.10 200.53000 3000 1.00 8 2.7732 9.39 2.71 200.63000 3000 2.00 10 4.3107 12.05 3.04 199.73000 3000 3.00 12 2.9583 13.27 5.00 199.53000 3500 0.50 8 0.6256 7.16 3.63 200.63000 3500 1.00 10 0.8733 9.13 4.46 199.73000 3500 2.00 12 1.6805 15.04 11.92 199.53000 3500 3.00 15 1.0136 13.01 7.05 199.73000 4000 0.50 8 0.9958 15.07 7.84 199.13000 4000 1.00 10 1.4670 15.01 9.99 199.93000 4000 2.00 12 3.2853 15.03 13.20 199.63000 4000 3.00 15 1.6689 14.14 5.14 200.13000 4500 0.50 8 1.5933 8.43 1.83 200.23000 4500 1.00 10 2.4221 10.79 2.10 199.73000 4500 2.00 15 0.7350 12.24 5.17 199.53000 4500 3.00 15 2.5796 15.20 3.75 200.13000 5000 0.50 8 2.553 8.96 1.21 200.03000 5000 1.00 10 4.3150 11.50 1.25 199.73000 5000 2.00 15 0.9655 12.77 4.37 200.33000 5000 3.00 15 3.9043 16.16 2.69 199.23500 3500 0.50 8 0.8780 6.99 3.43 200.13500 3500 1.00 10 1.2522 8.94 4.13 199.33500 3500 2.00 12 2.7321 11.80 3.84 199.43500 3500 3.00 15 1.4377 12.65 6.80 199.63500 4000 0.50 8 1.8864 7.91 1.88 199.1

8 P. Foraboschi /Materials and D

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3500 4000 1.00 10 3.1368 10.12 2.00 200.43500 4000 2.00 15 0.8134 11.30 6.13 200.33500 4000 3.00 15 3.0526 14.23 3.97 199.83500 4500 0.50 10 0.4792 7.40 3.48 199.93500 4500 1.00 12 0.8984 9.63 3.81 199.33500 4500 2.00 15 1.2644 12.24 4.62 199.53500 4500 3.00 19 0.6858 13.33 6.92 199.43500 5000 0.50 10 0.6706 7.84 2.80 200.13500 5000 1.00 12 1.3285 10.28 2.89 199.63500 5000 2.00 15 1.9211 13.01 3.42 199.73500 5000 3.00 19 0.9641 14.07 5.68 199.63500 5500 0.50 10 0.8951 8.25 2.28 200.23500 5500 1.00 12 1.9685 10.87 2.13 200.33500 5500 2.00 15 2.9474 13.89 2.45 199.93500 5500 3.00 19 1.2772 14.74 4.73 200.24000 4000 0.50 8 4.5208 7.92 0.88 199.54000 4000 1.00 12 0.7796 8.68 4.90 200.04000 4000 2.00 15 1.0842 11.05 5.98 199.64000 4000 3.00 15 6.3684 14.23 2.41 199.64000 4500 0.50 10 0.6966 7.30 3.24 200.24000 4500 1.00 12 1.4169 9.60 3.23 199.84000 4500 2.00 15 2.0573 12.25 3.80 199.54000 4500 3.00 19 0.9958 13.09 6.69 199.54000 5000 0.50 10 1.1326 7.92 2.27 200.14000 5000 1.00 12 2.9274 10.51 1.79 200.0

esign xxx (2014) xxxxxxFor each plate, these tables provide the minimum thickness of theglass layers and the minimum stiffness of the interlayer that allowthe plate to satisfy the serviceability and ultimate limit states.Hence, each LG plate with the relevant geometry and stiffness inTables 1 and 2 satises both the deection verication and thestrength verication; one verication exactly, the other vericationwith a certain margin.

Moreover, Tables 1 and 2 provide, for each case, the maximumstress in the lower glass layer rgo, the maximum stress in the cen-tral glass layer rgi (Fig. 4), and the ratio between the span L andmaximum deection wmax. The shear stress in the interlayer canbe calculated from those values, using Eg and Gt.

Tables 1 and 2 allow the design to select the optimal combina-tions of thicknesses of the glass layers and spans of the plate, for agiven design load.

6. Discussion

The analysis considered all the spans and loads of LG platesused in civil engineering, for bearing (Table 1) and non-bearing(Table 2) members. Moreover, the analysis considered only realglass plies and interlayers, which restricted the domain in whichthe minimum values had to be found. In fact, the minimum thick-ness of each glass ply and the minimum stiffness of the interlayerwere found in a discrete domain, with a substantial incrementfrom one value to the next. The discrete domain implied that theminimum thickness found among the glass plies available on the

4000 5000 2.00 15 4.8447 13.44 1.88 200.04000 5000 3.00 19 1.5450 14.13 5.00 199.14000 6000 0.50 10 4.2421 9.06 0.68 200.44000 6000 1.00 15 0.6770 9.97 3.56 200.24000 6000 2.00 19 0.8453 12.57 4.53 200.14000 6000 3.00 19 4.1179 16.00 2.36 200.45000 5000 0.50 10 5.1868 8.00 0.64 200.35000 5000 1.00 15 0.6508 8.71 4.86 200.05000 5000 2.00 19 0.8158 10.98 6.18 200.15000 5000 3.00 19 4.2921 14.11 2.86 199.76000 6000 0.50 12 5.3184 8.07 0.55 199.46000 6000 1.00 19 0.3693 8.50 5.67 199.76000 6000 2.00 22 0.9239 11.29 5.41 199.46000 6000 3.00 25 1.1618 13.00 5.93 200.0


Table 2Laminated Glass plate congurations that match the most severe limit state of non-bearing glass members. This table uses the same nomenclature as Table 1. The minussign indicates that the stress is compression.



1500 1500 0.50 2 1.9394 15.16 4.10 64.71500 1500 1.00 3 1.0369 17.60 10.49 65.01500 1500 2.00 3 21.6921 27.10 0.56 64.61500 2000 0.50 3 0.5022 15.11 7.53 65.51500 2000 1.00 3 3.6685 22.07 3.30 65.21500 2000 2.00 4 3.2616 27.18 6.26 64.71500 2500 0.50 3 0.8289 16.70 5.26 65.31500 2500 1.00 4 0.8289 20.93 7.49 64.71500 2500 2.00 4 15.5658 31.5 0.70 65.02000 2000 0.50 3 0.8951 14.44 6.88 65.02000 2000 1.00 4 0.9185 17.93 10.13 65.12000 2000 2.00 5 1.3544 23.01 12.20 64.82000 2500 0.50 3 3.1372 17.63 1.82 64.92000 2500 1.00 4 2.6109 21.60 4.17 64.82000 2500 2.00 5 4.6053 27.97 4.22 64.52000 3000 0.50 4 0.5886 16.21 6.01 65.52000 3000 1.00 4 24.2728 24.95 0.48 64.62000 3000 2.00 6 1.7303 27.21 7.52 65.42500 2500 0.50 4 0.5577 14.15 8.19 65.32500 2500 1.00 4 100.5886 22.05 1.74 64.432500 2500 2.00 6 1.6133 23.94 10.12 64.62500 3000 0.50 4 1.2140 16.47 4.38 64.72500 3000 1.00 5 1.9737 21.27 4.85 64.52500 3000 2.00 6 7.2368 28.52 2.32 64.62500 3500 0.50 4 3.3603 18.64 1.42 64.72500 3500 1.00 5 9.2625 24.17 0.58 64.52500 3500 2.00 8 0.7288 25.70 10.81 64.32500 4000 0.50 5 0.6105 16.91 5.26 65.42500 4000 1.00 6 1.1776 22.40 5.51 64.32500 4000 2.00 8 1.0511 27.41 8.43 64.43000 3000 0.50 4 2.9040 16.52 1.86 64.73000 3000 1.00 5 8.1671 21.40 0.70 64.93000 3000 2.00 8 0.6679 22.64 14.45 64.73000 3500 0.50 5 0.7450 15.74 5.69 65.43000 3500 1.00 6 1.6535 20.92 5.15 65.13000 3500 2.00 8 1.3158 25.54 9.07 64.53000 4000 0.50 5 1.3978 17.49 3.30 65.13000 4000 1.00 6 5.0498 23.57 1.56 64.33000 4000 2.00 8 2.7714 28.37 4.92 64.73000 4500 0.50 5 3.1422 19.10 1.35 64.73000 4500 1.00 8 0.4189 20.53 8.43 64.83000 4500 2.00 8 8.9258 31.00 1.36 64.43000 5000 0.50 6 0.5900 17.46 4.88 64.73000 5000 1.00 8 0.5299 21.54 7.22 64.43000 5000 2.00 10 0.7532 27.40 8.78 64.53500 3500 0.50 5 1.1374 15.66 4.55 64.73500 3500 1.00 6 3.8640 21.03 2.39 64.73500 3500 2.00 8 2.3196 25.31 6.62 64.93500 4000 0.50 5 6.0815 18.04 0.36 64.43500 4000 1.00 8 0.4621 18.94 9.90 65.13500 4000 2.00 10 0.6457 24.15 12.15 64.93500 4500 0.50 6 0.8252 16.80 4.60 65.03500 4500 1.00 8 0.7065 20.55 7.41 64.73500 4500 2.00 10 1.0112 26.26 8.93 64.53500 5000 0.50 6 1.3960 18.14 2.90 64.83500 5000 1.00 8 1.0652 22.02 5.43 64.83500 5000 2.00 10 1.6467 28.11 6.16 65.13500 5500 0.50 6 2.8421 19.34 1.35 64.83500 5500 1.00 8 1.5942 23.40 3.86 64.53500 5500 2.00 10 2.5901 29.98 4.17 64.54000 4000 0.50 6 0.6788 15.18 6.13 64.54000 4000 1.00 8 0.6087 18.53 9.55 64.84000 4000 2.00 10 0.8933 23.58 11.32 65.34000 4500 0.50 6 1.5269 17.05 3.00 64.34000 4500 1.00 8 1.0970 20.66 6.13 64.14000 4500 2.00 10 1.8027 26.31 6.61 65.14000 5000 0.50 6 10.0763 18.89 0.08 64.04000 5000 1.00 8 2.6609 22.63 2.76 64.84000 5000 2.00 10 4.9477 29.10 2.55 64.34000 6000 0.50 8 0.3777 16.76 5.78 64.8

P. Foraboschi /Materials and D

Please cite this article in press as: Foraboschi P. Optimal design of glass platj.matdes.2014.05.030market did not depend on the choice of the interlayer; and viceversa, the minimum stiffness of the interlayer obtainable usingproducts available on the market did not depend on the choice ofthe glass plies. This result facilitated the analysis and simpliesthis discussion.

When the maximum deection is equal to 1/200 of the span, themaximum tension stress in the lower glass layer rgo is always lessthan one-quarter of the allowable tension stress for tempered glassand it is always less than two-thirds of the allowable tension stressfor heat-strengthened glass (Table 1). Simultaneously, the maxi-mum tension stress in the central glass layer rgi is always less than30% of the allowable tension stress for heat-strengthened glass andit exceeds the allowable tension stress for annealed glass only inapproximately one-fth of the cases (Table 1).

When the maximum deection is equal to 1/65 of the span, themaximum tension stress in the lower glass layer rgo is always lessthan 40% of the allowable tension stress for tempered glass, it isalways less than approximately three-quarters of the allowabletension stress for heat-strengthened glass, and in approximatelytwo fth of the cases it is less than the maximum allowable tensionstrength of annealed glass (Table 2). Simultaneously, the maximumtension stress in the central glass layer rgi is always less thanapproximately one-third of the allowable tension stress for heat-strengthened glass and it is always less than the allowable tensionstress for annealed glass (Table 2). Note that the allowable tensionstress (strength) of non-bearing members is greater than that ofbearing members, since in the former case the duration of thedesign load is much shorter than in the latter case.

Hence, the design and structural assessment of the simply-sup-ported LG plates whose lower layer is made of toughened glass andwhose central layer is made of heat-strengthened glass are alwaysdictated by the allowable deection. The design and structuralassessment can disregard the stress eld and strength of materials;the limit state analysis can ignore resistance verication, while ithas to consider only the stiffness verication. Glass tension stres-ses and strength, as well as resistance verication have to be con-

4000 6000 1.00 10 0.5315 21.40 7.06 64.64000 6000 2.00 12 1.0799 28.04 7.14 64.65000 5000 0.50 8 0.3714 14.59 7.70 65.35000 5000 1.00 10 0.5216 18.66 9.39 64.95000 5000 2.00 12 1.0631 24.58 9.28 64.66000 6000 0.50 8 16.8526 17.37 0.40 64.96000 6000 1.00 12 0.4580 18.73 9.25 65.06000 6000 2.00 15 0.6395 23.89 11.24 64.7Table 2 (continued)



esign xxx (2014) xxxxxx 9sidered only when the central layer is made of annealed glass, butonly in few cases (some spans and loads), which are shown inTables 1 and 2.

Another result was that the choice of the interlayer does notinuence the optimal selection for the glass layers, and vice versa.Accordingly, the optimal design of the glass layers and of the inter-layer can be separated from one another, which simplies the opti-mal design process.

7. Simultaneous selection of geometry and materials

This section is devoted to the optimal design of the simply-sup-ported LG plate.

Tables 1 and 2 provide the design solutions with the minimumweight of glass and the minimum stiffness of the interlayer to


Table 3Materials and thicknesses to obtain the optimal congurations found in Table 1 for bearing Laminated Glass plates. This table uses the same nomenclature as Table 1. Moreover,type refers to the glass and material to the interlayer, where A = annealed glass, H = heat-strengthened glass, T = toughened glass (tempered or chemically strengthened),IP = Ionoplast Polymers.

Central glass layer Interlayer Lower glass layer

L B p Type h Material t Type hmm mm kN/m2 Label mm Label mm Label mm

1500 1500 0.50 A H T 3 IP 0.76 H T 31500 1500 1.00 A H T 4 IP 0.76 H T 41500 1500 2.00 A H T 5 IP 0.38 H T 51500 1500 3.00 A H T 6 IP 0.38 H T 61500 2000 0.50 A H T 4 IP 1.52 H T 41500 2000 1.00 A H T 5 IP 1.14 H T 51500 2000 2.00 A H T 6 IP 0.38 H T 61500 2000 3.00 H T 8 IP 1.52 H T 81500 2500 0.50 A H T 4 IP 0.76 H T 41500 2500 1.00 A H T 5 IP 0.38 H T 51500 2500 2.00 A H T 8 IP 1.52 H T 81500 2500 3.00 A H T 8 IP 0.76 H T 82000 2000 0.50 A H T 5 IP 1.52 H T 52000 2000 1.00 A H T 5 IP 0.38 H T 52000 2000 2.00 H T 8 IP 1.52 H T 82000 2000 3.00 H T 10 IP 1.52 H T 102000 2500 0.50 H T 5 IP 1.52 H T 52000 2500 1.00 A H T 6 IP 0.76 H T 62000 2500 2.00 A H T 8 IP 0.76 H T 82000 2500 3.00 H T 10 IP 1.52 H T 102000 3000 0.50 A H T 5 IP 0.76 H T 52000 3000 1.00 A H T 8 IP 1.52 H T 82000 3000 2.00 A H T 10 IP 1.52 H T 102000 3000 3.00 A H T 10 IP 0.76 H T 102500 2500 0.50 A H T 5 IP 0.38 H T 52500 2500 1.00 H T 8 IP 1.52 H T 82500 2500 2.00 A H T 8 IP 0.76 H T 82500 2500 3.00 A H T 10 IP 0.76 H T 102500 3000 0.50 A H T 6 IP 1.52 H T 62500 3000 1.00 A H T 8 IP 1.52 H T 82500 3000 2.00 A H T 10 IP 1.14 H T 102500 3000 3.00 H T 12 IP 1.52 H T 122500 3500 0.50 A H T 6 IP 0.38 H T 62500 3500 1.00 A H T 8 IP 0.76 H T 82500 3500 2.00 A H T 10 IP 0.38 H T 102500 3500 3.00 A H T 12 IP 0.76 H T 122500 4000 0.50 A H T 8 IP 1.52 H T 82500 4000 1.00 H T 8 IP 0.38 H T 82500 4000 2.00 A H T 12 IP 1.52 H T 122500 4000 3.00 A H T 12 IP 0.38 H T 123000 3000 0.50 A H T 6 IP 0.38 H T 63000 3000 1.00 A H T 8 IP 0.76 H T 83000 3000 2.00 A H T 10 IP 0.38 H T 103000 3000 3.00 A H T 12 IP 0.76 H T 123000 3500 0.50 A H T 8 IP 1.52 H T 83000 3500 1.00 A H T 10 IP 1.52 H T 103000 3500 2.00 H T 12 IP 1.52 H T 123000 3500 3.00 H T 15 IP 1.52 H T 153000 4000 0.50 H T 8 IP 1.52 H T 83000 4000 1.00 H T 10 IP 1.52 H T 103000 4000 2.00 H T 12 IP 0.76 H T 123000 4000 3.00 A H T 15 IP 1.52 H T 153000 4500 0.50 A H T 8 IP 1.52 H T 83000 4500 1.00 A H T 10 IP 1.14 H T 103000 4500 2.00 A H T 15 IP 1.52 H T 153000 4500 3.00 A H T 15 IP 1.14 H T 153000 5000 0.50 A H T 8 IP 1.14 H T 83000 5000 1.00 A H T 10 IP 0.38 H T 103000 5000 2.00 A H T 15 IP 1.52 H H 153000 5000 3.00 A H T 15 IP 0.76 H T 153500 3500 0.50 A H T 8 IP 1.52 H T 83500 3500 1.00 A H T 10 IP 1.52 H T 103500 3500 2.00 A H T 12 IP 0.76 H T 123500 3500 3.00 H T 15 IP 1.52 H T 153500 4000 0.50 A H T 8 IP 1.52 H T 83500 4000 1.00 A H T 10 IP 0.76 H T 103500 4000 2.00 H T 15 IP 1.52 H T 153500 4000 3.00 A H T 15 IP 0.76 H T 153500 4500 0.50 A H T 10 IP 1.52 H T 103500 4500 1.00 A H T 12 IP 1.52 H T 12

10 P. Foraboschi /Materials and Design xxx (2014) xxxxxx

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Interlayer Lower glass layer

h Material t Type hm

1111111111

1111111111111111

nd DTable 3 (continued)

Central glass layer

L B p Typemm mm kN/m2 Label

3500 4500 2.00 A H T3500 4500 3.00 H T3500 5000 0.50 A H T3500 5000 1.00 A H T3500 5000 2.00 A H T3500 5000 3.00 H T3500 5500 0.50 A H T3500 5500 1.00 A H T3500 5500 2.00 A H T3500 5500 3.00 A H T4000 4000 0.50 A H T4000 4000 1.00 A H T4000 4000 2.00 H T4000 4000 3.00 A H T4000 4500 0.50 A H T4000 4500 1.00 A H T4000 4500 2.00 A H T4000 4500 3.00 H T4000 5000 0.50 A H T4000 5000 1.00 A H T4000 5000 2.00 A H T4000 5000 3.00 A H T4000 6000 0.50 A H T4000 6000 1.00 A H T4000 6000 2.00 A H T4000 6000 3.00 A H T5000 5000 0.50 A H T

P. Foraboschi /Materials asatisfy the limit states, for bearing and non-bearing members,respectively. Tables 3 and 4 demonstrate how those design solu-tions can be obtained using products available on the market. Foreach case, thus, Table 3 (bearing members) and Table 4 (non-bear-ing members) provide both the glass type and thickness of the cen-tral and of the lower layers, and provide the combination of theinterlayer material and thickness.

More specically, Tables 3 and 4 provide the glass types thatallow each layer, whose thickness is the minimum possible (whichis one of the results in Tables 1 and 2) to bear the maximum ten-sion stress induced by the design load. Hence, the glass types inTables 3 and 4 only allow for the strength verications, while notall these glass types allow for the fail-safe conditions. The glasstypes that do not fulll the fail-safe conditions can be chosen onlyif the design satises in another way those fail-safe conditions thatare not met.

Moreover, Tables 3 and 4 provide the combinations of the mate-rial and thickness of the interlayer that allow each plate to reach atleast the minimum stiffness that is necessary to satisfy the limitstates (the minimum stiffness is another result of Tables 1 and2). In some cases there is only one combination of material andthickness.

7.1. Selecting the glass type

Tables 3 and 4 show the glass types that provide the LG plateconsidered in the analysis with adequate stiffness and strengthto satisfy the limit states.

In the vast majority of cases in Table 3 and in all cases in Table 4,both heat-strengthened and annealed glass provide the central

5000 5000 1.00 A H T 15000 5000 2.00 H T 15000 5000 3.00 A H T 16000 6000 0.50 A H T 16000 6000 1.00 H T 16000 6000 2.00 A H T 26000 6000 3.00 H T 2

Only the Ionoplast Polymers allow the interlayer to reach the stiffness shown in Table 1. Iis always possible to choose between two glass types for the lower layer. The table preseplate; however, these glass types do not all necessarily satisfy the fail-safe conditions.

Please cite this article in press as: Foraboschi P. Optimal design of glass platj.matdes.2014.05.030m Label mm Label mm

5 IP 1.52 H T 159 IP 1.52 H T 190 IP 1.52 H T 102 IP 1.52 H T 125 IP 1.52 H T 159 IP 1.52 H T 190 IP 1.52 H T 102 IP 1.52 H T 125 IP 0.76 H T 159 IP 1.52 H T 198 IP 0.38 H T 82 IP 1.52 H T 125 IP 1.52 H T 155 IP 0.38 H T 150 IP 1.52 H T 102 IP 1.52 H T 125 IP 1.14 H T 159 IP 1.52 H T 190 IP 1.52 H T 102 IP 0.76 H T 125 IP 0.38 H T 159 IP 1.52 H T 190 IP 0.38 H T 105 IP 1.52 H T 159 IP 1.52 H T 199 IP 0.38 H T 190 IP 0.38 H T 10esign xxx (2014) xxxxxx 11layer with adequate strength and stiffness. Since both of theseoptions meet the fail-safe conditions, the choice is dictated bythe cost, which is less for annealed glass.

In all cases in Tables 3 and 4, both the toughened and heat-strengthened glass types provide the lower layer with adequatestrength and stiffness. Moreover, in several cases in Table 4, alsothe annealed glass provides the lower layer with adequate strengthand stiffness. However, the lower layer completely satises thesecond fail-safe condition only if it is made of toughened glass.Conversely, a lower glass layer made of heat-strengthened orannealed glass does not satisfy part of the second fail-safe condi-tion. Therefore, this option can be considered only for the bi-layerplate dealt with in Section 7.3, as an alternative of the tri-layerplate dealt with here.

Note that heat-strengthened glass does not cost less thantoughened glass, although it is less resistant, since it requires anextra process.

Ultimately, strength and stiffness conditions would make itpossible for the structural designer to choose among more glassoptions, but fail-safe conditions drastically restrict these options.Accordingly, a choice is only possible for the central layer, whichcan be made of either heat-strengthened or annealed glass, accord-ing to the maximum stress rgi (Tables 3 and 4).

7.2. Selecting the interlayer

While the glass type inuences the stiffness of neither the plynor the system, the interlayer material inuences the stiffness of

5 IP 1.52 H T 159 IP 1.52 H T 199 IP 0.38 H T 192 IP 0.38 H T 129 IP 1.52 H T 192 IP 1.52 H T 225 IP 1.52 H T 25

n many cases, it is possible to choose between two glass types for the central layer. Itnts all the glass types that can be used for each layer to satisfy the limit states of the


Table 4Materials and thicknesses to obtain the optimal congurations found in Table 2 for non-bearing Laminated Glass plates. This table uses the same nomenclature as Tables 1 and 3;moreover, PVB = Poly Vinyl Butyral.

Central glass layer Interlayer Lower glass layer

L B p Type h Material t Type hmm mm kN/m2 Label mm Label mm Label mm

1500 1500 0.50 A H T 2 IP 1.52 A H T 21500 1500 1.00 A H T 3 IP PVB 1.52 0.38 A H T 31500 1500 2.00 A H T 3 IP 1.52 H T 31500 2000 0.50 A H T 3 IP PVB 1.52 1.14 A H T 31500 2000 1.00 A H T 3 IP 1.52 H T 31500 2000 2.00 A H T 4 IP 1.52 H T 41500 2500 0.50 A H T 3 IP PVB 1.52 0.38 A H T 31500 2500 1.00 A H T 4 IP PVB 1.52 0.38 H T 41500 2500 2.00 A H T 4 IP 1.52 H T 42000 2000 0.50 A H T 3 IP PVB 1.52 0.38 A H T 32000 2000 1.00 A H T 4 IP PVB 1.52 0.38 A H T 42000 2000 2.00 A H T 5 IP PVB 1.52 0.38 H T 52000 2500 0.50 A H T 3 IP 1.52 A H T 32000 2500 1.00 A H T 4 IP 1.52 H T 42000 2500 2.00 A H T 5 IP 1.52 H T 52000 3000 0.50 A H T 4 IP PVB 1.52 0.76 A H T 42000 3000 1.00 A H T 4 IP 1.52 H T 42000 3000 2.00 A H T 6 IP 1.52 H T 62500 2500 0.50 A H T 4 IP PVB 1.52 0.76 A H T 42500 2500 1.00 A H T 4 IP 1.52 H T 42500 2500 2.00 A H T 6 IP 1.52 H T 62500 3000 0.50 A H T 4 IP PVB 1.52 0.38 A H T 42500 3000 1.00 A H T 5 IP 1.52 H T 52500 3000 2.00 A H T 6 IP 1.52 H T 62500 3500 0.50 A H T 4 IP 1.52 A H T 42500 3500 1.00 A H T 5 IP 1.52 H T 52500 3500 2.00 A H T 8 IP PVB 1.52 0.76 H T 82500 4000 0.50 A H T 5 IP PVB 1.52 0.76 A H T 52500 4000 1.00 A H T 6 IP PVB 1.52 0.38 H T 62500 4000 2.00 A H T 8 IP PVB 1.52 0.38 H T 83000 3000 0.50 A H T 4 IP 1.52 A H T 43000 3000 1.00 A H T 5 IP 1.52 H T 53000 3000 2.00 A H T 8 IP PVB 1.52 0.76 H T 83000 3500 0.50 A H T 5 IP PVB 1.52 0.76 A H T 53000 3500 1.00 A H T 6 IP 1.52 H T 63000 3500 2.00 A H T 8 IP PVB 1.52 0.38 H T 83000 4000 0.50 A H T 5 IP PVB 1.52 0.38 A H T 53000 4000 1.00 A H T 6 IP 1.52 H T 63000 4000 2.00 A H T 8 IP 1.52 H T 83000 4500 0.50 A H T 5 IP 1.52 H T 53000 4500 1.00 A H T 8 IP PVB 1.52 1.14 H T 83000 4500 2.00 A H T 8 IP 1.52 H T 83000 5000 0.50 A H T 6 IP PVB 1.52 0.76 A H T 63000 5000 1.00 A H T 8 IP PVB 1.52 1.14 H T 83000 5000 2.00 A H T 10 IP PVB 1.52 0.76 H T 103500 3500 0.50 A H T 5 IP PVB 1.52 0.38 A H T 53500 3500 1.00 A H T 6 IP 1.52 H T 63500 3500 2.00 A H T 8 IP 1.52 H T 83500 4000 0.50 A H T 5 IP 1.52 A H T 53500 4000 1.00 A H T 8 IP PVB 1.52 1.14 A H A 83500 4000 2.00 A H T 10 IP PVB 1.52/0.76 H T 103500 4500 0.50 A H T 6 IP PVB 1.52/0.38 A H T 63500 4500 1.00 A H T 8 IP PVB 1.52 0.76 H T 83500 4500 2.00 A H T 10 IP PVB 1.52 0.38 H T 103500 5000 0.50 A H T 6 IP PVB 1.52 0.38 A H T 63500 5000 1.00 A H T 8 IP PVB 1.52 0.38 H T 83500 5000 2.00 A H T 10 IP 1.52 H T 103500 5500 0.50 A H T 6 IP 1.52 H T 63500 5500 1.00 A H T 8 IP PVB 1.52 0.38 H T 83500 5500 2.00 A H T 10 IP 1.52 H T 104000 4000 0.50 A H T 6 IP PVB 1.52 0.76 A H T 64000 4000 1.00 A H T 8 IP PVB 1.52 0.76 A H T 84000 4000 2.00 A H T 10 IP PVB 1.52 0.38 H T 104000 4500 0.50 A H T 6 IP PVB 1.52 0.38 A H T 64000 4500 1.00 A H T 8 IP PVB 1.52 0.38 H T 84000 4500 2.00 A H T 10 IP 1.52 H T 104000 5000 0.50 A H T 6 IP 1.52 A H T 64000 5000 1.00 A H T 8 IP 1.52 H T 84000 5000 2.00 A H T 10 IP 1.52 H T 104000 6000 0.50 A H T 8 IP PVB 1.52 1.52 A H T 84000 6000 1.00 A H T 10 IP PVB 1.52 1.14 H T 10

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strongly depends on the interlayer material.

In

ML

IPIPIPIPIPIPIP

posshe ly sat

nd DThe stiffness of the interlayer also depends on its thickness, asshown by Eq. (1). Thus, material and thickness of the interlayerhave to be dealt with together. For each plate, the combinationof the two choices has to guarantee a stiffness greater than the rel-evant value in Table 1 or Table 2, which is the minimum stiffnessthat allows each LG plate to satisfy both the limit states.

According to Eq. (1), the interlayer shear elasticity modulusdivided by the minimum stiffness obtained for the considered LGplate provides the maximum thickness of the interlayer. More spe-cically, 0.052 N/mm2 for PVB interlayers and 1.50 N/mm2 for IPinterlayers, divided by each stiffness in Table 1 provides the max-imum theoretical thicknesses of the interlayer for each case inTable 1; 0.300 N/mm2 for PVB interlayers and 105.00 N/mm2 forIP interlayers divided by each stiffness in Table 2 provides the max-imum theoretical thicknesses of the interlayer for each case inTable 2. The minimum thickness that can be used in the designwas obtained by rounding off the theoretical minimum values tothe inferior value between the possible thicknesses of theinterlayer.

The thicknesses that were obtained in that way and the poly-meric material that each thickness is associated with, are shownin Tables 3 and 4. In some cases, a structural designer can choosebetween a PVB and an IP interlayer, each one with an associatedthickness. In other cases, only an IP interlayer can be used, sincenot even the minimum thickness (i.e., 0.38 mm) would provide aPVB interlayer with adequate stiffness.

7.3. Updating the fail-safe design concept: First proposalthe interlayer and, therefore, of the whole system. In fact, Gt

Table 4 (continued)

Central glass layer

L B p Type hmm mm kN/m2 Label mm

4000 6000 2.00 A H T 125000 5000 0.50 A H T 85000 5000 1.00 A H T 105000 5000 2.00 A H T 126000 6000 0.50 A H T 86000 6000 1.00 A H T 126000 6000 2.00 A H T 15

In many cases, it is possible to choose between two interlayer materials. It is alwayschoose between two glass types, and in many cases between three glass types, for tsatisfy the limit states of the plate; however, these glass types do not all necessaril

P. Foraboschi /Materials aThe large distance of the maximum tension stress from glassstrength in the LG plates that fulll the sail-safe conditions(Section 2) and satisfy the limit states, suggested that the fail-safeconcept should be revised.

The rst fail-safe condition can be considered only for thestrength verication but not for the deection verication. Thisposition is justied by the fact that a crack is a local phenomenon,and therefore it reduces drastically the local strength, while itreduces the stiffness of the ply only marginally. Accordingly, evenone load-induced crack cut down the plys strength, while only agreat number of load-induced cracks can reduce considerably theplys stiffness. Thus, the strength of a glass ply exposed to live loadscannot even be guaranteed by frequent inspections, while the stiff-ness can be guaranteed by a normal inspection and maintenanceprogram.

The above proposal allows the structural designer to decide thegeometric dimensions of the load-bearing system considering onlythe strength verication. Under the combination of actions for the

Please cite this article in press as: Foraboschi P. Optimal design of glass platj.matdes.2014.05.0307.4. Updating the fail-safe design concept: Second proposal

The results show that heat-strengthened glass would providethe lower glass layer with enough strength to resist the loads(Tables 1 and 2). However, this layer is external; in the event thatthis glass layer breaks, therefore, no sharp shards can fall, accord-ing to the second fail-safe condition. Thus, the lower layer has to bemade of toughened glass, which shatters into small, blunt pieces.Conversely, heat-strengthened glass breaks into large, razor-sharpslivers.

On one hand, to replace the toughened glass with heat-strengthened glass would not provide any economic benet, sincethe latter costs no less than the former. On the other hand, how-ever, this result suggests another updating of the fail-safe concept.

There are glass plates whose collapse does not involve buildingultimate limit state, the central and lower glass layers thatcompose the load-bearing system have to produce stresses lessthan the strength of each glass layer. Conversely, the combinationof actions for the serviceability limit state is carried by the wholetri-layer system (load-bearing system plus sacricial ply); i.e., thedeection verication includes the upper layer. In brief, the upperlayer is sacricial for the stress verication while can be consideredfor the stiffness verication.

Since the combination of actions for the ultimate limit state is atleast 1.4 times greater than the combination of actions for the ser-viceability limit state, this design option signicantly reduces thethickness of the LG plate. Design and assessment can still beaccomplished using Tables 1 and 2, since the behavior of the LGplate is linear.

terlayer Lower glass layer

aterial t Type habel mm Label mm

PVB 1.52 0.38 H T 12PVB 1.52 1.52 A H T 8PVB 1.52 1.14 A H T 10PVB 1.52 0.38 H T 12

1.52 A H T 8PVB 1.52 1.14 A H T 12PVB 1.52 0.76 H T 15

ible to choose between two glass types for the central layer. It is always possible toower layer. The table presents all the glass types that can be used for each layer toisfy the fail-safe conditions.

esign xxx (2014) xxxxxx 13occupants and pedestrians. These glass plates can be fail-safe alsowithout the toughened glass at the lower layer. For these plates,the design can use a bi-layer system instead of a tri-layer system:An upper layer made of toughened glass and a lower layer made ofheat-strengthened (or annealed) glass, together with the polymericinterlayer between these glass layers. The layers should have thesame thickness. Since falling glass fragments, although neithersmall nor blunt, cannot create potential injury or risk of death tousers and pedestrians, this LG plate fullls the second fail-safe con-dition. Moreover, the upper layer is devoted to collecting the liveloads (toughened glass), while the lower layer is devoted to provid-ing the plate with adequate post-breakage capacity (heat-strength-ened or annealed glass). Thus, this system is completely fail-safe.

There are glass plates whose live loads may act on both thefaces. These glass plates can be fail-safe only if the laminated sys-tem uses two sacricial ply, one at the top and one at the bottom.For these plates, the design can use the above-described bi-layersystem plus a bottom sacricial ply made of toughened glass (andthe interlayer), instead of a quadri-layer system. Since this


d Dbottom layer shatters into small, blunt pieces, this LG plate fulllsthe second fail-safe condition. Thus, this system is always fail-safe, while the bi-layer system is fail-safe only if collapse doesnot endanger users and pedestrians.

Since the sacricial ply added at the bottom of the bi-layer sys-tem has no structural role, the structural behavior of this system isidentical to that of the bi-layer system it derives from. Hence, onlythe bi-layer system is considered in this subsection, which never-theless includes both the design solutions.

The upper layer of the bi-layer system is not sacricial for theserviceability verications. Thus, the deection verication dueto the combination of actions for the serviceability limit state isperformed using the bi-layer system. Hence, wmax is calculatedby considering the composite behavior of the LG plate.

Conversely, the upper layer of the bi-layer system may be con-sidered sacricial for the ultimate verications. If it is sacricial,the strength verication due to the combination of actions forthe ultimate limit state is performed using the lower layer only.Hence, the maximum tension stress in the system is calculatedconsidering the monolithic behavior of the lower plate.

However, it is not mandatory to consider the upper layer sacri-cial. In many cases, the depth of the neutral axis is greater thanthe depth of the load-induced cracks (Fig. 7). E.g. the stress prolesproduced by rgo and rgi in Tables 1 and 2 show that, at the mid-span, the compression zone is often greater than t/2 (while thelength of a load-induced crack is substantially less than t/2). Theneutral axis positions provided by the analytical model showedthat, in many cases, this condition remains true for the entire LGplate. This condition is directly connected to the stiffness of theinterlayer, as dened by Eq. (1).

In those cases, the load-induced cracks are subjected to com-pression stresses and not to tension stresses (Fig. 7). Therefore,the magnitude of the tension stresses is amplied by a stress inten-sity factor that is dictated, not by the load-induced cracks, but bythe cracks due to the aws, whose size is shorter.

When that condition is guaranteed, it is no longer mandatoryto consider the upper layer of the bi-layer system sacricial forthe ultimate verications; conversely, the strength vericationcan be performed using the LG system. In this case, thus, rgiand rgo are calculated considering the composite behavior ofthe plate. However, the composite behavior can be consideredonly for the load combinations that imply a depth of the neutralaxis greater than the depth of the load-induced cracks (Fig. 7). Forthe other load combinations, the maximum stresses have to becalculated considering the lower layer only.

According to Tables 1 and 2, the maximum tension stress in thelower glass layer, rgi, is much less than the tension strength of theheat-strengthened glass. More specically, under the loads thatproduce a deection of 1/200 of the span, the maximum stress inthe bi-layer system is always less than two-thirds of the allowabletension stress for heat-strengthened glass. Moreover, under theloads that produce a deection of 1/65 of the span, the maximumstress in the bi-layer system is always less than three-quarters ofthe allowable tension stress for heat-strengthened glass.

Those differences prompt the use of the above-described bi-layer system in lieu of the tri-layer system of Section 2.3, whereit is possible. In fact, this design solution saves weight withrespect to the tri-layer system and reduces the number of lami-nations from three to two layers. Moreover, if the compressionzone of the upper layer is deeper than the load-induced cracks(Fig. 7), the amount of glass weight saved by the bi-layer systemis substantial, which drastically increases transparency andreduces the costs. Also in this case, design and assessment can

14 P. Foraboschi /Materials anstill be accomplished using Tables 1 and 2, since the behaviorof the LG plate is linear.

Please cite this article in press as: Foraboschi P. Optimal design of glass platj.matdes.2014.05.0308. Conclusions

The paper focuses on the Laminated Glass (LG) plate and pre-sents criteria and a method for concurrent design and materialselection, by using analytical exact modeling. The paper providesinformation that allows the structural designer not only to dimen-sion the LG plate, but also to choose the best geometry-materialcombination for a LG plate used as bearing member (oor, coveringroof, stair) or non-bearing member (faade, partition).

The consequences classes that those glass members are catego-rized in require that they are fail-safe. To this end, resort shall bemade to the fail-safe conditions presented in Section 2, whichallow the design to obtain adequate failure modes of the variouscomponents and the whole laminated system, necessary to ensuresuitable robustness and damage tolerance, and to avoid both brittlecollapses and that dangerous shards can fall down. According tothese conditions, the plate has to be composed of at least three lay-ers and two interlayers (tri-layer system). The bi-layer system canbe accepted only if the design satises in another way those fail-safe conditions that are not met.

This paper uses a recently published analytical exact model ofthe sandwich plate, which was derived in closed-form from theKirchhoffLove assumptions for the layers and considering an elas-tic interlayer. The model provides analytically the stress eld in thelayers and interlayer, and the displacements.

When a structural designer commits to certain materials (typesof glass, interlayer material), that model can be used to determinethe best geometry of the LG plate. Likewise, when architecturaldesign has dened the geometry, that model can be used to selectthe best materials. However, xing the geometry and then choos-ing the material or xing the material and then determining thegeometry are not truly optimal, since xing one in general inu-ences the optimality of the other. Concurrently determining theoptimal geometry and selecting the best materials remained anopen issue in the design of LG plates; this paper has lled thisgap for the simply-supported glass plate.

The rst result is the hierarchy of resistance. A LG plate thatmatches the serviceability limit state always satises the ultimatelimit state (furthermore, with a great margin). This result holdstrue for faades and partitions, although the conditions in whichthese structures become unt for the intended purpose is whenthe maximum displacement exceeds 1/65 of the span, which is agreat deection. This result holds true even if the deection veri-cation considers the serviceability load while the strength verica-tion considers the ultimate load.

Moreover, this result also holds true for LG plates whose lowerlayer is made of heat-strengthened glass instead of toughenedglass; however, this solution can be used only if the design satisesin another way the fail-safe condition that a heat-strengthenedouter layer does not meet.

Ultimately, the design and assessment of the LG plate are dic-tated by the stiffness and not by the load-carrying capacity.

The second result achieved by this research, which follows log-ically from the rst result, is that only the choice of the glass typeused for the central layer is dictated by glass strength, while thechoices of the glass types used for the upper and lower layers aredictated by the fail-safe conditions only.

The glass ply that collects the loads (upper layer) has to bemade of toughened glass and has to be sacricial, since an ade-quately small probability of failure cannot be guaranteed for aglass ply exposed to live loads. If both the external faces areexposed to live loads, including the environmental actions (wind-borne debris), fail-safe design calls for another sacricial ply (four

esign xxx (2014) xxxxxxglass layers). The lower layer of the tri-layer system is external;therefore, in the event of breaking, it has to shatter into small,


laminated composite panels under biaxial loading. Compos Struct

nd Dblunt pieces. Hence, this layer has to be made of toughened glass.The central layer of the tri-laminated system has to provide theplate with adequate post-breakage load-carrying capacity, whichcan be obtained by using either heat-strengthened or annealedglass. The choice is dictated by the maximum stress reached by thislayer.

Hence, the structural designer considers glass strength only todecide whether using a central layer made of heat-strengthenedor annealed glass. Moreover, assessment considers the centrallayer only when it is made of annealed glass, while when it is madeof heat-strengthened glass no strength verication is necessary. Infact, if the plate satises the deection verication, the heat-strengthened central layer always satises the stress verication.

Ultimately, the selection of the glass type does not allow thestructural designer to minimize the glass weight for given loadsand available space, but only to reduce the cost by using whenthis is possible annealed instead of heat-strengthened glass forthe central layer (the latter is more expensive than the former).Hence, the design and assessment have to focus on the thicknessesof the layers and interlayer, and the material of the interlayer.

The results show that a plate that fullls the fail-safe conditionsand satises the stiffness demand does not take benet of glasstension strength. In a tri-layer fail-safe system, in fact, the maxi-mum tension stress is always much less than glass tensionstrength, which suggested revising the fail-safe approach.

The ply that collects the live loads can be considered sacricialfor the strength verication only, but not for the stiffness verica-tion. Accordingly, the stresses derive from the composite behaviorof the central and lower glass layers, while the displacements fromthe composite behavior of the upper, central, and lower glass lay-ers. Under this assumption, hence, assessment uses a bi-layer sys-tem for the stress verication, while a tri-layer system for thedisplacement verication (the latter has much greater stiffnessthen the former).

Where glass fragments of broken glass cannot fall onto occu-pants and pedestrians, the tri-layer system can be replaced by abi-layer system. In this system, the upper layer is made of tough-ened glass. This layer collects the live loads, provides the systemwith adequate stiffness, but theoretically it does not provide anystrength since it is sacricial for the stress verication. The lowerlayer is made of heat-strengthened glass. This layer provides thesystem with adequate stiffness and strength; moreover, it providesthe system with adequate post-breakage load-carrying capacity.Hence, the composite behavior of this system is considered onlyfor the serviceability verications, while the ultimate vericationsare performed using a monolithic system; i.e., the deections arecalculated considering the bi-layer system, while the stresses con-sidering the monolithic system provided by the lower layer.

In many cases, the depth of the neutral axis in the upper layer isgreater than the depth of the cracks induced by the live loads. Inthis case, it is no longer mandatory to consider the upper layer sac-ricial, since there is no difference between the upper and lowerlayer. When the neutral axis is sufciently deep, hence, the com-posite behavior of the bi-layer system can be considered for boththe strength and stiffness verications. In this case, nevertheless,the strength verication has to consider all the possible combina-tions of actions. If a combination implies a depth of the neutral axisless than the depth of the load-induced cracks, then the upper layerhas to be considered as sacricial in the strength verication underthat load combination.

When the live loads act on both the faces of the plate, the designcan use the above-described bi-layer system plus a sacricial ply atthe bottom, made of toughened glass. On one hand, this system has

P. Foraboschi /Materials athe same structural behavior as the bi-layer system, since the sac-ricial ply has no role. On the other hand, however, this system isalways fail-safe, while the bi-layer system is fail-safe only if the

Please cite this article in press as: Foraboschi P. Optimal design of glass platj.matdes.2014.05.0302011;94(1):316.[13] Theulen JCM, Peijs AAJM. Optimization of the bending stiffness and strength of

composite sandwich panels. Compos Struct 1991;17(1):8792.[14] Wastiels L, Wouters I. Architects considerations while selecting materials.

Mater Des 2012;2012(34):58493.[15] Zhu F, Wang Z, Lu G, Zhao L. Analytical investigation and optimal design of

sandwich panels subjected to shock loading. Mater Des 2009;30(1):9

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