numerical investigation of two-sided reinforced laminated glass beams … · 2017-08-27 ·...

Glass Struct. Eng. (2016) 1:417–431DOI 10.1007/s40940-016-0005-6

ORIGINAL PAPER

Numerical investigation of two-sided reinforced laminatedglass beams in statically indeterminate systems

Kenny Martens · Robby Caspeele · Jan Belis

Received: 3 June 2015 / Accepted: 13 January 2016 / Published online: 26 January 2016© Springer International Publishing Switzerland 2016

Abstract A significant number of investigations hasbeen performed to improve the unsafe (brittle) failurebehaviour of glass beams. Similar to reinforced con-crete, reinforced glass beams have been developed inwhich stainless steel is embedded in the glass lam-inate to obtain safe failure behaviour. The concepthas been proven to be feasible when assuming stati-cally determinate systems.However, the additional sys-tem safety of statically indeterminate systems has notbeen investigated yet. This paper presents numericalresearch outcomes on reinforced glass beams with sta-tically determinate as well as statically indeterminatesupport conditions. In a first step, a model of a two-sided reinforced beam subjected to three-point bend-ing was created for validation purposes. The thicknessand height of the constituents making up the modelwere based on experimentalmeasurements of test spec-imens. Subsequently, the same numerical model wasmade using nominal dimensions for the glass and rein-forcement. In this way, the effect of dimensional toler-ances on the load-carrying behaviour was investigated.

K. Martens (B)· R. Caspeele · J. BelisGhent University, Technologiepark-Zwijnaarde 904,9052 Zwijnaarde, Belgiume-mail: [email protected]

R. Caspeelee-mail: [email protected]

J. Belise-mail: [email protected]

Furthermore, a single-sided reinforced glass beam wasmodelled under the same loading conditions to assessthe effect of adding compressive reinforcement. In asecond step, statically indeterminate two-sided rein-forced beam models were constructed based on thestatically determinate model. From these preliminarymodels, the load-carrying behaviour and effect of rein-forcement percentage were evaluated. Finally, also thestress redistribution capacity of both beam models wasassessed. It is concluded that dimensional toleranceshave a significant effect on the load-carrying behav-iour and should therefore be accounted for in the designof reinforced glass beams. The additional compressivereinforcement provides slightly higher bending stiff-ness, initial failure load and yield load to the glassbeams, although it does not contribute in the yieldphase. However, a different ultimate collapse mecha-nism is expected as the compressive reinforcement cantake the stress when the glass compressive zone hasfailed. The statically indeterminate simulations provedthe feasibility of applying reinforced glass beams instatically indeterminate systems as a safe failure behav-iour was observed with significant stress redistributioncapacity. Changing the reinforcement percentage hasa significant effect on the load-carrying behaviour ofthese systems. However, the overall behaviour remainssafe.

Keywords Structural glass beams · Reinforcement ·Numerical simulation · Support conditions ·System safety

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418 K. Martens et al.

1 Introduction

Nowadays, a big challenge in architecture is to max-imise transparency in buildings. To do so, structuralelements usually built in reinforced concrete or steelcan instead be made of glass. However, as glass is abrittle material, beams consisting of glass only do notexert safe failure behaviour. To solve this problem, alot of concepts were developed in which glass is usedin combination with another material to create hybridglass beams that possess safe failure behaviour, i.e. sig-nificant post-fracture strength and ductility (Abeln et al.2010; Belis et al. 2009; Blyberg et al. 2014; Correiaet al. 2011; Cruz and Pequeno 2008; Freytag 2004;Froli and Mamone 2014; Louter et al. 2010, 2011,2012; Premrov et al. 2014; Speranzini and Neri 2011;Martens et al. 2015a, Martens et al. 2015b).

A promising concept in this regard is the one of rein-forced glass beams. The philosophy for this develop-ment is inspired by reinforced concrete. Despite obvi-ous differences between glass and concrete, their struc-tural behaviour has some significant similarities. Bothmaterials are strong in compression, rather weak in ten-sion and illustrate a kind of brittle failure behaviour. Toreach safe failure behaviour for concrete beams, ductilesteel is added to the section. Applying the same princi-ple to glass beams, a stainless steel section is adhesivelybonded at the tensile side of the beam. The reinforce-ment serves as a crack bridge and constitutes, togetherwith the intact compressive glass section, an internalresisting moment that provides post-fracture capacity.In a later stage, reinforcement yielding results in ductilefailure behaviour. The key requirements for this bene-ficial failure behaviour are a judicious choice of rein-forcement ratio and most important, a well-designedbond between this reinforcement and the glass sec-tion. If this is not fulfilled, bond failure between glassand reinforcement can occur resulting in brittle col-lapse of the beam prior to steel yielding (Nielsen andOlesen 2010; Olgaard et al. 2009). A thorough andsuccessfully investigated beam design consists of atriple-layered glass laminate with an embedded steelprofile. Louter has investigated these beams consider-ing statically determinate support conditions (Louteret al. 2011). He concluded that these beams, consistingof annealed float glass (ANG), SentryGlas� (SG) asinterlayer material and stainless steel as reinforcementyielded significant post-fracture strength and ductil-ity, in statically determinate conditions. However, sta-

tically determinate systems do not possess an inherentsystem safety. A way to incorporate system safety is tomake use of statically indeterminate support conditionsin which an inherent transfer of stresses can developbetween support and spans. In literature, only a smallnumber of investigations has been done in this regard(Valarinho et al. 2013). In the latter research, glass-fibre reinforced polymer (glass-GFRP) beams weresubjected to statically indeterminate five-point bend-ing tests. An important conclusion from this researchwas the ability of these beams to redistribute stressesfrom the central support to the spans. Furthermore, thisstress redistribution was highest for the beam speci-mens in which the GFRP-glass bond was realised withthe least stiff adhesive. However, this stress redistribu-tion was only momentarily observed since it stemmedfrom the loss of stiffness on several sections due to glassfracture. As the beams in the current investigation pos-sess ductile reinforcement, stress redistribution shouldbe triggered by two events. Firstly, glass fracture willactivate limited stress redistribution, followed by sig-nificant redistribution when plastic hinges are formed.

The numerical simulation of glass beams, in partic-ular glass fracture, has proven to be a challenge. Yet,past research projects have suggested several possi-bilities for modelling this highly-influential phenom-enon. In a first model, Olgaard et al. (2009) modelledglass as a linear elastic material until the maximumprincipal strain in a material point reached a criticalvalue defined by Hooke’s law, respecting the tensilestrength and Young’s modulus of the glass. At this crit-ical level, the stiffness of the material point is reducedto a value much smaller than the initial stiffness. Thismethod is referred to as the ‘killing elements‘ method.Louter et al. (2010) developed a sequentially linearelastic analysis (SLA) technique to model glass frac-ture. In principle, the ‘standard’ incremental-iterativeanalysis scheme is replaced by a series of scaled linearanalyses, while at the same time a saw-tooth reduc-tion curve replaces the nonlinear stress-strain law. Thistechnique was developed to avoid often encounteredconvergence problems. Despite the rather unrealisticcrack pattern, good load-displacement behaviour wasfound compared to experimental test results. Bedon andLouter (2014) used an inherent ‘brittle cracking’ mod-ule in Abaqus, which was actually developed for mod-elling fracture in concrete. As this is the approach usedin this paper, more details follow in the subsequentsections. Recently, Neto et al. (2015) used a ‘discrete

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Numerical investigation of two-sided reinforced laminated glass 419

strong discontinuity approach (DSDA)’ to model glassfracture. In this approach, micro-cracking is assumedto localise at a surface of discontinuity with initialzero width (instead of distributed over the element vol-ume as is the case in the ‘brittle cracking’ module).To describe these cracks at inter-element boundarieswithout knowing their location in advance, the ele-ments are enriched by embedding strong discontinu-ities in the parent elements (similar to GFEM/X-FEM,but here the new degrees of freedom are introducedalong the crack path instead of introducing them at theelement nodes). Furthermore, a non-iterative energybased method is used to perform the analyses. Alsothis numerical approach yielded good agreement withthe experimental results.

This paper presents numerical research into the fea-sibility of applying reinforced laminated glass beamsin statically indeterminate support conditions. Firstly,a model of a two-sided solid reinforced glass beamsubjected to three-point bending is created which isvalidated using available experimental data. From thisvalidated model, the effect of dimensional tolerancesand adding reinforcement in the compressive zoneis respectively evaluated by simulating the same testfor a two-sided reinforced glass beam with nominaldimensions and a single-sided reinforced laminatedglass beam. Finally, a statically indeterminate five-point bending test was simulated on a two-sided solidreinforced beam model. An overall assessment of theload-carrying behaviour was performed to decide onwhether structural glass beams could be applied in sta-tically indeterminate systems. Furthermore, the samesimulation was performed using hollow profile rein-forcement to investigate the effect of reinforcement per-

centage. In addition, the stress redistribution capacitywas investigated for both beam models.

2 Geometry & materials

Five numerical tests were simulated using three differ-ent sections as illustrated in Fig. 1. The difference isrelated to whether a single-sided (a) or a two-sided (band c) reinforced glass beam is considered and whethera solid (a and b) or hollow profile (c) section was usedto reinforce the beam. Section (a) is identical to the oneused by Louter et al. (2011). The two-sided reinforcedsectionswere based on the former and enable appliancein statically indeterminate systems. To limit the calcu-lation time and memory requirements, one vertical halfof each section is modelled as this is possible thanks tosymmetry.

The details of the numerical test setups are depictedin Fig. 2. The first figure (Fig. 2a) represents the three-point bending simulation and the second one (Fig. 2 b)the five-point bending simulation. As all test setups aresymmetric, half of the beam length was modelled tofurther lower the calculation time andmemory require-ments. The lateral supports assure that no lateral insta-bility effects can occur during the simulations.

As can be seen in Fig. 1, each beam is composedof three different materials, i.e. annealed float glass(ANG), SentryGlas� (SG) and stainless steel. The SG-interlayer was defined as a linear elastic material in thenumerical model. In reality, SG is a visco-elastic mate-rial from which the mechanical properties are depen-dent on load duration and temperature. However, recentinvestigations have proven the validity of using a sim-plified elastic approach for laminated glass beams in

Fig. 1 Composition of theinvestigated beam sections.(ANG annealed float glass)

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Fig. 2 Schematicrepresentation of numericalbending test configurations:a three-point bending test; bstatically indeterminatefive-point bending test.(Dimensions in mm)

which the relevant properties are chosen for a specificloading time and temperature, provided that the numer-ical model simulates a short duration test at a constanttemperature (Callewaert 2012; Callewaert et al. 2012;Bedon and Louter 2014). For the current simulations,a temperature of 23 ◦C and a loading time of 10–15min was chosen (conditions of the experimental testsin Louter et al. (2011)). The values of the Young’smodulus and Poisson coefficient were adopted fromCallewaert (2012). The glass was modeled as a linearelastic material in compression, and a linear elastic,brittle material in tension. Finally, the stainless steelwas modeled as a bilinear elastic-plastic material. Tworeinforcement sections were used, namely a 10 × 10mm solid section and a 10× 10× 1 mm hollow profilesection. Table 1 gives an overview of the used materialparameters in the numerical models.

3 Numerical model & method

This section discusses the numerical approach, whichwas performed using the finite element softwareAbaqus� (Dassault Systèmes 2012).

3.1 Numerical model

The choice of element type and mesh pattern was donebased on thework ofBedon andLouter (2014). 3D con-

Table 1 Material parameters for ANG, SG and stainless steel

Material ANG SG (23 ◦C,10–15 min)

Stainlesssteel

Density (kg/m3) 2500 1100 7800

Young’s modulus(MPa)

70,000 118.9 (Callewaert2012)

195000

Poissoncoefficient (−)

0.23 0.49 (Callewaert2012)

0.27

Yield strength(MPa)

/ / 550

Ultimate tensilestrength (MPa)

45 / 850

Ultimate tensilestrain (%)

/ / 55

tinuum elements with 8 nodes were chosen for all partsof the beam. The mesh pattern for the glass and inter-layer mid-height zones was generated using the sweep-ing technique with an advancing front algorithm. Min-imal mesh sizes of 1 mm were appointed to the tensilesides of the beam. For the five-point bending simula-tions, this minimumwas set to 2 mm. For the compres-sive sides of the beam a maximum mesh size of 5 mmwas set. The reinforcement sections and neighbouringglass and interlayer zones had a regularmesh (using themapped sweeping technique) with sizes equal to 1 or 2

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Fig. 3 Mesh pattern of the midspan zone of a typical beam model in three-point bending (one half of the beam)

Fig. 4 Mesh pattern for a numerical statically indeterminate five-point bending test: a central support zone (one half of the beam);b midspan zone

mm for the tensile zones and 5 mm for the compressivezones. In the direction perpendicular to the beam plane,a mesh size of 1.5 mm was chosen. Figs. 3 and 4 illus-trate the typicalmesh pattern generated for respectivelya numerical three-point bending and five-point bending

test. For the latter, the tensile edge changes from upperto lower edge when going from the central to the outersupports. The transition is located at the inflection pointof the theoretical bendingmoment line, calculated froma linear elastic analysis (see Fig. 4a).A typical statically

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determinate beam model with solid reinforcement hasa total number of 301513 elements and consequentlya total number of 1273689 variables (degrees of free-dom plus maximum number of any lagrange multipliervariables). A statically indeterminate beammodel withsolid reinforcement has a total of 246294 elements and1047879 variables. The hollow profile reinforced beammodels have similar numbers of elements andvariables.

The vertical supports were implemented as line sup-ports in the thickness direction of the beam (i.e. perpen-dicular to the paper plane in Fig. 2). The lateral supportswere applied on a surface, which comprises a width of50mmand a height of 125mm. Furthermore, two typesof symmetry conditions were implemented to save cal-culation time and memory. A first symmetry plane wasdefined as the central span section of the beam. A sec-ond onewas defined in the plane of the beam at one halfof the beam’s thickness, comprising the whole length.The loads (or displacements) were applied on a surfacecomprising the beam’s thickness and awidth of 25mm.The bonds between glass, interlayer and reinforcementsections were modeled as tie contraints. This meansthat a perfect bond exists between the contacting enti-ties. As a consequence, no slip can be simulated withthese models. However, experiments did not illustrateslip for the beams considered (three-point bending testsat 23 ◦C).

3.2 Method

To simulate glass fracture in the models, use is madeof the ‘brittle cracking’ material model (Dassault Sys-tèmes 2012). The model was developed for modellingconcrete in all types of structures, but can also beuseful for modelling other materials such as ceram-ics. It is designed for applications in which the behav-iour is dominated by tensile cracking and assumes thatcompressive behaviour is always linear elastic. ‘Brittlecracking’ can be implemented using a ‘fracture energycracking criterion’, in which the energy required toopen a unit area of crack inMode I is defined as a mate-rial parameter. As a consequence, the brittle behaviouris characterised by a stress-displacement response. Touse this material model, a tensile strength of 45 MPaand a Mode I fracture energy of 2 J/m2 was imple-mented (Haldimann et al. 2007). Also higher valuesof fracture energy have been suggested, but accord-ing to Bedon and Louter (2014), these values result in

less realistic crack patterns. However, the work abovealso concluded that the fracture energy only signifi-cantly influences the crack pattern and is less impor-tant for the overall structural behaviour of the beam.In addition, ‘brittle shear’ and ‘brittle failure’ sub-optionswere implemented to avoid excessive distortionof cracked glass elements. The ‘Brittle shear’ model isa ‘shear retention model’ that accounts for the ModeII shear behaviour in the postcracked phase. The largerthe crack opening, the less shear stiffness the partic-ular zone can develop. This model was defined as apower law in which the material parameters e and pwere determined as 0.001 and 5.0, respectively (Bedonand Louter 2014). The parameter ‘e’ is the ultimatecrack normal strain and ‘p’ is the power that definesthe slope of the relationship between the shear stiffnessand the crack normal strain. From former research, itwas concluded that the value of p does not significantlyalter the structural behaviour of the beam (Valarinhoet al. 2012). When e = 0.001 is reached, complete lossof aggregate interlock is achieved and no shear stiff-ness is left. ‘Brittle failure’ was implemented to allowremoval of elements. Basically, when one, two, or allthree local direct cracking displacement components ata material point reach the defined failure displacementvalue, the material point fails and all stress compo-nents are set to zero. If all material points in an elementfail, this element is removed from the mesh. For thisnumerical model, ‘brittle failure’ is characterised by anunidirectional failure criterion in which a direct crack-ing failure displacement of 1.3 ×10−4 mm (calculatedas 2 times the fracture energy divided by the tensilestrength) was implemented. This means that a materialpoint fails when a single cracking displacement com-ponent reaches the failure displacement value.

The ‘brittle cracking’ model, when sufficiently cal-ibrated, provides a good way to simulate glass frac-ture, which is the primary mechanism that influencesthe structural behaviour of glass beams. However, thismodel also has some drawbacks. Firstly, it is only avail-able in the Abaqus�/explicit module, which is usedfor dynamic simulations. As a quasi-static simulationis performed, energy-monitoring will be necessary tocome to consistent simulations. More specifically, onehas to make sure that the kinetic energy is only a frac-tion of the internal, plastic (reinforcement yield) andstrain (glass cracking) energies. The energy can becontrolled by increasing the step time of the simula-tion, which highly increases the calculation time. For

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Fig. 5 Typical energy curve for a numerical three-point bending test

these simulations, the step time was set to agree withthe experimental displacement rate of the three-pointbending tests (0.1 mm/s). Secondly, the model is alsomesh dependent. However, themesh used here has gen-erated satisfying results in former research (Bedon andLouter 2014). Moreover, it was concluded that smallvariations in mesh size will not alter the overall load-displacement behaviour.

Displacement-controlled simulations were performed in this investigation, which agree with theexperimental way of testing. Moreover, more stableresultswere retrieved compared to load-controlled tests(more dynamic effects were encountered in the latter).Because these simulations consume a lot of calcula-tion time, the displacement was limited to 15 mm. Atthis displacement, all beam concepts have reached theirfinal load-transfer mechanism (the reinforcement sec-tions are in the yield phase). This displacement valueis sufficient considering the scope of this paper.

3.2.1 Energy considerations

As Abaqus� requires the model to run dynamically,one has to make sure that the numerical results repre-sent a quasi-static simulation. Therefore, the model’senergy curves were evaluated. The total kinetic energyshould be very small compared to the total internalenergy, strain energy (corresponding to glass cracking)and plastic dissipation energy (corresponding to rein-forcement yielding). If this was not the case, the steptime of the model was increased (Bedon and Louter

2014). To be fully consistent, a step time of 150 mmwas chosen so that a displacement rate of 0.1 mm/swas simulated (as in the three-point bending experi-ments). A typical energy diagram is depicted in Fig. 5(three-point bending simulation on a two-sided solidreinforced beam, with an imposed displacement of 15mm). As can be seen on the graph, the kinetic energyis only a very small fraction of the other energy types.

Furthermore, the smoothness of the internal energycurve proves the stable character of the simulation.

3.2.2 Validation with experimental results

Firstly, a numerical model was built to simulate theexperimental three-point bending tests on a two-sidedreinforced laminated glass beam using solid stainlesssteel sections (see Figs. 1b, 2a). Comparing exper-imental and numerical results, the numerical modelwas validated and further used as a basis for the cre-ation of the other models. Three beam specimens wereexperimentally tested. Prior to experimental testing, thebeams were measured which yielded different dimen-sions than the nominal ones. For validation, the dimen-sions of the numerical model were accordingly modi-fied, as depicted in Table 2. For the sake of brevity, thedetails of the experimental test setup are omitted.

Figure 6 illustrates the experimental and numer-ical load-displacement diagrams. A good agreementis observed between both. In the initial linear elasticphase, the experimental test results illustrate a block-shaped course. This is explained by a problem with the

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Table 2 Actual dimensions of the validated numerical model

Geometry 6 mm Panel* 10 mm Panel 1.52 mm Interlayer* Reinforcement* Total

Thickness (mm) 5.68 9.76 1.64 9.76 24.40

Height (mm) 124.20 104.40 / 9.9 124.20

*For these measurements, an average was taken over both panels/interlayers/reinforcement sections of the beam

Fig. 6 Experimental and numerical load-displacement diagrams

Fig. 7 Crack pattern in outer glass pane for the numerical simulation at 15 mm displacement

LVDT which was not perfectly working for the first 3to 4 mm of displacement. The problem can be solvedby drawing a smooth line through this course, whichwould approximately be overlapping the numerical testresult.The latter however underestimates the load atfirst glass fracture (10.1 kN instead of an experimentalmean value of 12.7 kN) (1). In the subsequent phase(here-called the fractured phase), the average slope ofthe numerical model lies very close to the one of theexperiments. The yield point also lies very close to theexperimental yield point (2) andwhen the beam reachesits yield phase, the numerical curve almost overlaps twoof the three experimental curves. In the yield phase,the slope of the numerical curve is a bit lower thanthe experimental ones (3). This difference is due to thesimplified implementation of the constitutive materiallaw for stainless steel. A significant difference betweennumerical and experimental test results is the capri-cious course of the numerical curve. This is due to the

way in which cracking is simulated (dynamic). There-fore, when comparing, it is appropriate to visualisethe mean curve. Considering the comparison betweennumerical and experimental results, it is concluded thatthe current model is validated and is appropriate toserve as a basis for the other numerical models devel-oped in this paper.

Figure 7 illustrates the numerical crack pattern inthe beam at the end of the simulation (15 mm dis-placement). The crack pattern for the experimental testis depicted in Fig. 8. Although less extensive in thenumericalmodel, the positionof the cracks correspondswell to that observed in the experimental tests.

4 Results & discussion of numerical analyses

In this section, an overview of the numerical resultsand integrated discussion of the effect of dimensional

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Fig. 8 Crack pattern for the experimental test (15 mm displacement)

tolerances and compressive reinforcement in the stat-ically determinate case is given. Furthermore, the sta-tically indeterminate structural behaviour of the two-sided reinforced glass beam is discussed followed bya discussion on the effect of reinforcement percent-age. Finally, the stress redistribution capacity is investi-gated. For these considerations, numericalmodelswerebuilt using the nominal dimension values.

4.1 Statically determinate case

4.1.1 Effect of dimensional tolerances

In industry, glass and steel sections are produced withstandardised dimensional tolerances. In the former sec-tion, a numerical model was developed for validation,using the average measurements of the experimental

test specimens. Typically, these measurement valuesare smaller than the nominal ones, which will affectthe behaviour of the glass beam. In this section, thesame numerical model was created using the nomi-nal dimensions according to Figs. 1b and 2a and itsload-displacement behaviour is compared to that of themodel with actual dimensions (see Fig. 9). In what fol-lows, the numerical model with nominal dimensions isdenoted as the ‘Numerical-N’ model and the numeri-cal model with actual dimensions as the ‘Numerical-A’model.

Starting in the initial linear elastic phase, theNumerical-N model has a higher bending stiffness andinitial failure load (1n and 1e). Comparing the slope ofboth curves, the bending stiffness of the Numerical-Amodel is 91.74 % of that of the Numerical-N model.The initial failure load of the Numerical-Nmodel (11.0

Fig. 9 Load-displacement curves of numerical three-point bending simulations on two-sided solid reinforced glass beams with nominaland actual dimensions (N nominal, A actual)

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Table 3 Significant values for the experimental and numerical-N load-carrying behaviour

Test Initial slope (kN/mm) Initial failure load (kN) Fractured slope (kN/mm) Yield load (kN) Load at 15 mm (kN)

Numerical-N 5.84 11.01 2.43 20.70 22.47

Experiments* 5.35 12.70 1.30 20.04 22.06

*Average values taken from three experimental three-point bending tests

kN) is a factor of 1.089 larger than that of the othermodel (10.1 kN). Both properties are directly depen-dent on the second moment of inertia, which is smallerfor the Numerical-A model . The ratio of both valuesis equal to 91.67 % which is very close to the percent-age calculated above. The initial failure load for theNumerical-N model is analytically calculated as 9.37kN. For the Numerical-A model, this value is 8.66 kNwhich lies close to the former value. Beyond the ini-tial failure load (here-called the fractured phase), thedifference in bending stiffness between both modelsgrows larger (the ratio between both averaged slopes isabout 67.7 %) and the yield point of the Numerical-Nmodel (2n) is higher than observed for the Numerical-A model (2e) (about 20.7 kN compared to 19.5 kN).This is explained by the lower reinforcement area andthe smaller height of the latter model. Also in the yieldphase (= the phase following the yield point), the loadremains lower for the same reasons. A relative loadratio of about 92 % is found at the end of the simula-tion (3n and 3e). In Table 3, theNumerical-N results arecompared with the experimental test results depicted inFig. 6. It is stated that the bending stiffness is higher inall phases for the Numerical-N simulation. The numer-ical model underestimates the initial failure load butoverestimates the yield load and the load at the end ofthe simulation. The relative difference is significant inthe fractured phase.

In conclusion, the effect of dimensional tolerancesis limited to relative differences of 10 %, except forthe bending stiffness in the fractured phase, for whicha relative difference of more than 30 % was found.Anyhow, it is stated that dimensional tolerances shoulddefinitely be incorporated in the design of reinforcedglass beams through well-defined safety factors.

4.1.2 Effect of compressive reinforcement

To assess the effect of adding compressive reinforce-ment to the glass section, a numerical simulation of athree-point bending test on a single-sided solid rein-forced laminated glass beam with nominal dimensions

was performed and compared to a two-sided solid rein-forced model with nominal dimensions (see Fig. 1a, b).The load-displacement diagrams are depicted inFig. 10. Starting in the initial linear-elastic phase, thesingle-sided reinforced beam illustrates lower bend-ing stiffness and initial failure load (1d and 1s). Thisis explained by the beam section’s lower moment ofinertia . A quick calculation of the slope of the curvesyields a factor of 1.147 which is close to the theoreti-cal calculated factor using the second moment of iner-tia values (1.140). For the two-sided reinforced case,analytical calculations using a maximum tensile stressof 45 MPa resulted in an initial failure load of 9.369kN. The numerical simulation yielded a value of 11.00kN which lies within an acceptable range of 10–15 %.The same calculations for the single-reinforced caseyielded respectively 8.707 kN. A value of 10.05 kN,was retrieved from the numerical simulation. In thefractured phase, the difference in bending stiffness per-sists, resulting in a lower yield load at higher midspandisplacement for the single-sided reinforced case (2sand 2d). In the yield phase, both curves overlap. Thisresult is explained by the fact that only the tensile rein-forcement is significantly yielding in both cases.

As a result, it is concluded that adding compressivereinforcement to the glass sectionwill raise the bendingstiffness in the initial and fractured phase andwill resultin a higher initial failure load and yield point. Regard-ing ultimate collapse, it is stated that both conceptswill fail due to compressive failure of glass. Accordingto Nielsen and Olesen (2010), there are four differentfailure mechanisms for a single-sided reinforced glassbeam. The first one is failure of the reinforcement-to-glass bond, resulting in brittle failure. The others referto reinforcement percentage. If it is too low, a duc-tile failure behaviour will result with a post-fracturestrength lower than the initial fracture strength. A suf-ficiently high reinforcement percentage yields a post-fracture strength that is higher than the latter. How-ever, if the percentage is too high, compressive failureof glass could occur. The latter case applies for thecurrent beams. Experimental results of the two-sided

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Fig. 10 Load-displacement curves of numerical three-point bending simulations on single-sided reinforced and two-sided reinforcedglass beams (d two-sided, s single-sided)

Fig. 11 Numerical Load-displacement diagrams for statically indeterminate beam models with solid and hollow profile reinforcement(s solid, h hollow profile)

reinforced glass beams illustrated that after this phe-nomenon, a secondary load carrying mechanism wasinstalled as the top reinforcement was able to take thecompressive stresses. However, the residual load waslow, as the top reinforcement got pushed further andfurther into the fractured glass section at midspan.

4.2 Statically indeterminate case

Statically indeterminate five-point bending tests weresimulated on beam sections with either solid or hol-low profile reinforcement to evaluate their overall load-

carrying behaviour, the effect of reinforcement percent-age and stress redistribution capacity.

4.2.1 Load-carrying behaviour in the staticallyindeterminate case

The load-displacement diagrams for both beam typesare depicted in Fig. 11. For the beam section withsolid reinforcement, the linear elastic phase ended ata load of 28 kN (1s). At this point, initial cracks startedto develop simultaneously at the central support andmidspans. Subsequently, the load could significantly beincreased while new cracks formed and existing cracks

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Fig. 12 Crack pattern in the central glass pane of one half of the statically indeterminate beamwith solid reinforcement at a displacementof 15 mm (a similar crack pattern was observed in the outer glass panes; the right boundary is the mirror line)

Fig. 13 Crack pattern in the central glass pane of one half of the statically indeterminate beam with hollow profile reinforcement at adisplacement of 15 mm (a similar crack pattern was observed in the outer glass panes; the right boundary is the mirror line)

further developed. At about 62 kN, severe glass frac-ture was observed at midspan (2s); however, the loadcould further be increased. Subsequently, the slope ofthe curve started to lower due to the yielding of bothreinforcement sections, both at the central support andat the midspans (3s). The lower bending stiffness of thebeamcaused severe glass fracture in the tensile zones ofthe central support and midspans, when a load of 79.5kNwas reached (4s). Again the load could be increaseduntil the glass compressive zones started to fracture inall critical sections, which caused a major load drop(5s). At the end of the simulation, the beam is heavilydamaged at themidspans, but is still able to carry a loadof about 30 kN (6s). A maximum load of 79.5 kN wasreached, which results in a post-fracture performance(maximum load/initial fracture load) of 2.84. The frac-tured glass beam at the end of the numerical simulationis illustrated in Fig. 12.

The hollow profile reinforced beam illustrated aslightly different load-carrying behaviour. Initial fail-ure was encountered at a load of 21 kN (1h). First glassfracture was observed at the central support. However,midspan glass fracture followed soon (2h). Then theload was increased up to 21.3 kN until the centralcompressive glass zone fractured severely (3h). Sub-sequently, the load was increased and the bottom rein-forcement got pushed into the severely weakened glasssection at the central support. As a consequence, thisreinforcement section started to yield both in tension(upper fibre) and compression (lower fibre). Also thebottom reinforcement at the midspans started to yield(4h). Due to the lower bending stiffness, additionalglass fracture was encountered in the midspan glasssections at a load of 20.4 kN (5h). At the end of thesimulation, the load only reached a value of about 17

Fig. 14 Close-up of the central support

kN due to the smaller lever armbetween both reinforce-ment sections at the central support (the reinforcementgot further pushed into the glass section) (6h). A max-imum load of 21.3 kN was reached, which results ina post-fracture performance of 1.01. The beam’s frac-ture pattern at the end of the simulation is depicted inFig. 13. A close-up of the central support is given inFig. 14.

Considering the load-carrying behaviour explainedabove, it is stated that both the solid and hollow profilereinforced glass beams exert a safe failure behaviour instatically indeterminate support conditions. The solidreinforced concept illustrated a high post-breakage per-formance. The performance of the hollow reinforcedone was close to unity.

4.2.2 Effect of reinforcement percentage

Reinforcement percentage has a significant effect onthe load-carrying behaviour (see Fig. 11). Starting inthe initial linear elastic phase, the hollow profile rein-forced beammodel illustrates both lower bending stiff-ness and initial failure load. This can be appointed to thesecond moment of inertia, which is signficantly lower.

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Fig. 15 Stress distribution for statically indeterminate glass beams (numerical results): (1) Analytical ratio in the linear elastic stage &(2) Analytical ratio in the plastic stage

Considering these values, the bending stiffness of thehollow profile reinforced model should be 75.6 % ofthe solid one. A calculation of the slope using referencepoints at 10 kN for both curves results in a stiffness ratioof 76.4 %, which is very close to the theoretical value.Also the difference in initial failure load can be veri-fied by analytical estimations. For the solid reinforcedbeam model, a linear elastic calculation of first glassfracture resulted in a value of 25.3 kN. The numeri-cal simulation illustrated first glass breakage at 28 kN,which lies within an acceptable range of 10–15 %. Forthe hollow profile reinforced model, the analytical cal-culations resulted in 19.1 kN. The numerical modelillustrated a value of 21.0 kN, which also fits the pre-diction well. Entering the fractured phase, the load inthe solid reinforced case can significantly grow thanksto the major amount of reinforcement. The hollow pro-file reinforcement has a smaller maximum limit due tothe lower amount of steel (36 % of the solid reinforce-ment). Finally, when the reinforcement enters the yieldphase, the load-carrying behaviour between both beammodels is different. The solid reinforced model illus-trates severe glass fracture at midspan followed by theformation of plastic hinges both at midspan and at thecentral support, while in the hollow reinforced case, thecompressive glass zone at the central support fractures,resulting in the penetration of the bottom reinforcementinto the glass zone. As a consequence, a plastic hingeis locally formed in the bottom reinforcement at thecentral support. The post-fracture performance of the

solid reinforced model is significantly larger than thatof the hollow reinforced one (2.84 compared to 1.01).It is concluded that both concepts are safe, but the solidreinforced beam illustrated better performance.

4.2.3 Stress redistribution capacity

To assess the stress redistribution capacity, the ratio ofthe central support and outer support reaction forceshas been put out as a function of the total applied loadfor both the solid and hollow profile reinforced beams(see Fig. 15).

On this graph, the analytically calculated values ofthe central support-outer support ratio are depicted for(1) the linear elastic situation (using the laws of ‘vir-tualwork’) and (2) the fully plastic situation (usingnon-linear plastic analysis which accounts for the formationof plastic hinges). As can be observed from the graph,in the initial stage both models (solid and hollow rein-forced) approximate the theoretical line (1). At a dis-placement of about 11.5mm, bothmodels have alreadyreached the plastic phase andboth curves inFig. 15 con-verge to the theoretical line (2).At this point, significantglass fracture disturbs the whole system and the simu-lations illustrate different behaviour. For the solid rein-forced case, the central support reaction lowers signif-icantly, meaning that the load-carrying capacity shiftsto the spans. The value of the central-outer reactionratio converges to a value of 2.8. A value of 2.0 wouldmean that the system evolved to a statically determi-

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nate systemwith two spans. On the contrary, the centralsupport of the hollow reinforced beam model instantlyreceived a larger part of the total load, after which itgradually decreased again. Here the significant glassfractures were concentrated at the midspans, leavingthe central support section to be relatively stronger. Asa result, the load-carrying capacity shifts to the centralsupport. Subsequent glass fracture at the central sup-port counteracted the shift, resulting in the decreasingcurve at the end of the simulation.

In conclusion, both beam types illustrate significantmoment redistribution capacity, as full redistributionwas observed when both beams reached full plasticcapacity (at a displacement of 11.5mm). Even the eventof glass fracture triggered some redistribution in thesimulations as can be observed on Fig. 15. This signif-icant amount of stress redistribution, providing systemsafety, proves these concepts to be very promising andcompetitive regarding their counterparts in steel andreinforced concrete.

5 Conclusions

In this paper, numerical research into statically deter-minate and statically indeterminate reinforced lami-nated glass beams was presented. Firstly, a numericalmodel of a statically determinate two-sided (solid) rein-forced laminated beam was developed and validatedusing experimental data of three-point bending tests.The dimensions of the model were based on the exper-imentally measured values of the test specimens. Onthe basis of this model, a similar numerical model wascreated using nominal dimensions. The load-carryingbehaviour of both was compared to assess the effectof dimensional tolerances in the production process. Itwas observed that differences are smaller than 10 %,except for the bending stiffness in the fractured phase(more than 30 %). It is concluded that dimensional tol-erances should be incorporated in the design throughwell-defined safety factors. Also the effect of com-pressive reinforcement was investigated by creating asingle-sided solid reinforced beam model, which wasalso loaded in three-point bending. It was concludedthat adding reinforcement at the compressive edge ofthe beamraises the beam’s bending stiffness, initial fail-ure load and yield point. Furthermore, also differencesin the ultimate collapse mechanism are expected, as thecompressive reinforcement provides an additional load

carrying mechanism. In a second stage, also staticallyindeterminate beam models were created using solidand hollow profile reinforcement. The solid reinforcedbeam model illustrated safe failure behaviour with sig-nificant post-fracture performance (2.84). The simula-tion on the hollow profile reinforced beam also resultedin safe failure behaviour, although a post-fracture per-formance equal to unity was encountered (1.01). Thelower reinforcement percentage also lead to a lowerbending stiffness, initial failure load, yield point anda different load-carrying behaviour in the yield phase.However, the latter phenomenon can be related to thesimplified modelling of the supports (lines). Moreover,the whole load carrying behaviour of both beam mod-els could be influenced by the latter. Therefore, simu-lations with realistic support dimensions will be per-formed in the near future. Also the stress redistributioncapacity of both beam models was investigated. Thelatter resulted in the conclusion that both beam modelsillustrated redistribution capacity in two phases. A first(minor) redistribution cameabout in the fracturedphasefollowed by amajor one in the yield phase, reaching thetheoretical calculated limit. Once the huge glass frac-ture appeared (after which the beam models illustrateddifferent load-carrying behaviour), the stress redistrib-ution is different comparing both beam models. How-ever, it is stated that this zone is expected to be highlydependent on the modelling of the supports. Never-theless, both beam models illustrated significant stressredistribution capacity. From the conclusions above, itis stated that it is feasible to apply two-sided reinforcedlaminated glass beams in statically indeterminate sys-tems. However, it is noted that the above conclusionsare based on numerical research only. An experimentaltesting program will be conducted in the near future tovalidate the above findings and conclusions.

Acknowledgments The authors gratefully acknowledge the‘Agency for Innovation by Science and Technology in Flanders(IWT)’ for supporting this research (Grant nr. 141526).

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

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