Modeling of the structural behavior of laminated glass beamsStudy of the lateral-torsional buckling phenomenon

Miguel Machado e Costamiguelmachadoecosta@tecnico.ulisboa.pt

Instituto Superior Tecnico, Lisbon, Portugal, 2015

ABSTRACT: In recent years there have been several investigations about the behavior of laminated structuralglass elements, namely in terms of their flexural and torsional stiffness, with the lateral-torsional buckling of beamsbeing one of the most relevant and complex topics. There are various analytical formulations to describe theequivalent stiffness of laminated elements — in comparison with monolithic elements — but none covers more thanthree layers of glass in a comprehensive and unified manner, and those that exist are not consensual. This workproposes a new formulation, based on sandwich theory, which provides equivalent results to previous formulationsin a limited set of conditions, but that is able to characterize the behavior of simply supported laminated glasscolumns and beams up to five layers, subjected to compressive axial loads, mid-span loads, uniformly distributedloads, four-point bending, pure bending or torsion. The fundamentals explained in the dissertation that supportsthis paper allow the formulation to be extended to a larger number of layers and to different load and supportconditions. The proposed formulation is subjected to a parametric study based on the comparison with numericalresults retrieved from finite element simulations, in order to assess the range of validity of each expression. Twoanalytical approaches for the lateral-torsional buckling problem are studied in detail, with their fundamentalsbeing explained. Another formulation, proposed in an Australian Standard, is also addressed. An experimentalassessment of the work developed is achieved by comparing the results obtained from flexural tests on a long-span three-layered glass beam, which exhibited lateral-torsional buckling, with analytical and numerical predictions.

Keywords: laminated structural glass, equivalent flexural stiffness, torsional stiffness, lateral-torsional buckling,parametric study.

1. INTRODUCTION

The increasingly global awareness on environmentalsustainability and human welfare is also changing the waycivil engineering structures are designed. Within build-ings, for example, natural lighting is something that isconsidered more and more as fundamental. Glass struc-tures, which come from a very ancient material, provide asolution to this modern need, with some innovative solu-tions presenting good energy performance.

In light of current architectural trends, glass use isrequested for a multitude of applications, from non-structural elements to structural solutions that allow en-hanced clarity in comparison with previously known solu-tions featuring other construction materials.

Glass beams or glass-fins can be part of fully glazedsolutions, resisting wind loads acting on a facade or sup-porting floors, roofs or stairways. The need for redun-dancy require these elements to be laminated, i.e., to becomposed of multiple glass layers bonded by an interlayer.Because of their high slenderness they are susceptible tolateral-torsional buckling.

Lateral-torsional buckling in laminated glass beams isa phenomenon that has been studied in recent years byseveral researchers [1–7]. Experimental, analytical andnumerical studies have been conducted in order to bet-ter understand, for example, the structural behavior oflaminated glass beams, the influence of the visco-elasticproperties of multiple interlayer products, the influence

of different geometrical imperfections and glass fracturemechanics.

There is still much work to be done on this subject, as at-tested by the variety of analytical formulations that havebeen put forward to define the same engineering prob-lems. Indeed, there is not yet a unified and commonlyadopted formulation to assess this phenomenon. Addi-tionally, there is also lack of generality on the proposedformulations and their field of application does not covermany practical situations.

Two main objectives drove this work: (i) to extend theexisting analytical formulations so that they could be ap-plied to laminated glass elements with a larger number oflayers; and (ii) to develop a numerical finite element modelable to simulate the lateral-torsional buckling behavior ofa long-span laminated glass beam.

On the one hand, among the available analytical ex-pressions, this work aimed at verifying which ones werevalid and more accurate, so that they could be extendedto laminated glass elements with more than three layers.It would also be important to determine the range of va-lidity of all the expressions analyzed in this work. Theassessment of the validity and accuracy of the availableanalytical expressions was based on the results obtainedfrom numerical finite element models implemented in thiswork for that specific purpose. The analytical study thataimed to extend the formulations (so that they could beapplied to laminated glass elements with more than three

1

layers) was based on sandwich theory, both regarding flex-ural and torsional stiffness.

On the other hand, following an experimental studyon the flexural-torsional buckling behavior of a long-spanthree-layered glass beam, carried out in Instituto SuperiorTecnico (IST) as part of a PhD thesis, this work aimed atdeveloping a numerical model capable of simulating thelinear buckling and post-buckling response of the beamtested. This also allowed to further validate the previ-ously mentioned analytical expressions.

Alongside the main objectives, this work also aimed at(i) extending the equivalent flexural stiffness analyticalformulation to a larger number of loading conditions —even though they were not necessary to compare with theresults provided by the experimental study mentioned —, (ii) studying the origin of the existing lateral-torsionalbuckling formulations, in order to support their applica-tion, and (iii) deriving a new analytical formulation ableto characterize the maximum normal stress in the exte-rior glass layers of laminated beams subjected to trans-verse loads or to pure bending. This last objective is onlyaddressed in the dissertation that supports this paper.

2. REVIEW AND ASSESSMENT OF PREVI-OUS ANALYTICAL STUDIES

Several analytical studies conducted by different au-thors resulted in relatively large number of formulations,which aim to characterize the stiffness of laminated glasselements. There are mainly five proposed formulations todefine the equivalent flexural stiffness of laminated glasselements and mainly three formulations to define their tor-sional stiffness.

In the case of columns subjected to compressive ax-ial loads, the critical buckling load may be determinedwith the following equivalent flexural stiffness formula-tions applied to Euler’s equation: (i) Luible’s expressions[5], based on the sandwich theory work of Stamm andWitte [8]; (ii) Amadio and Bedon’s expressions [9], de-rived from the theory proposed by Newmark [10]; and (iii)the “Wolfel-Bennison approach” [11], first proposed byWolfel et al. [12]. Even tough the authors do not mentionthis fact, these three formulations yield the exact same re-sults for this loading case. In the case of beams subjectedto out-of-plane mid-span loads, the maximum deflectionmay be determined with the following equivalent flexu-ral stiffness formulations applied to the monolithic beamdeflection equation: (i) Luible’s expressions [5], based onthe sandwich theory work of Stamm and Witte [8]; (ii)the expressions proposed Kasper et al. [4] and adopted inthe recent guideline “Guidance for European StructuralDesign of Glass Components – Support to the implemen-tation, harmonization and further development of the Eu-rocodes” [13]; and (iii) the “Enhanced effective thicknessapproach”, derived by Galuppi and Royer-Carfagni [14]and based of the “Wolfel-Bennison approach” (which theauthors mention to yield worse results for this loading case[15]).

The torsional stiffness of laminated glass beams may bedetermined with the following formulations: (i) Luible’sexpressions [5], based on the sandwich theory work of

Stamm and Witte [8]; (ii) the expressions proposed Kasperet al. [4] and adopted in the above mentioned guideline[13]; and (iii) the expressions proposed by Scarpino [16],which may be found in [17].

All these formulations have been compared, and theiraccuracy was assessed, through the comparison with re-sults provided by numerical finite element models imple-mented in the commercial software Abaqus [18], for threedifferent values of the shear modulus of the interlayer. Theglass layers were modeled with eight-node continuum shellelements with reduced integration (SC8R), while eight-node solid elements with reduced integration (C3D8R)were used to simulate the interlayer. The approximatesize of the elements in the plane of the column/beam waschosen to be 10 mm×10 mm, with one element in the thick-ness of the glass layers and two elements in the thicknessof the interlayer (Fig. 1). Luible’s formulations, based onsandwich theory, proved to be consistently the most accu-rate, both regarding the equivalent flexural and torsionalstiffness. Detailed results may be found in the dissertationthat supports this paper.

Figure 1: Example of the numerical model with appliedtorsional moment.

Kasper et al. [4] also proposed an analytical formula-tion able to describe the lateral-torsional behavior of lam-inated glass beams subjected to pure bending, mid-spanloads or uniformly distributed loads, supported by equiv-alent flexural and torsional stiffness formulations. Theauthors present expressions able to determine the criti-cal buckling load, the lateral deflection and the rotationof the sections. It is also common to find in the litera-ture the determination of critical buckling loads by meansof the expressions presented in a previous version of Eu-rocode 3 [19]. Another formulation, which features thepossibility to consider intermediate buckling restraints ispresented in Australian Standard AS 1288 [20]. AdoptingLuible’s equivalent stiffness formulations, all three lateral-torsional buckling formulations provided analytical criti-cal buckling loads, for a laminated beam subjected to amid-span load, within a 10% difference to the numericalresults.

Since sandwich theories proved to provide accurate re-sults, they were chosen to support the work towards theobjective of extending the existing analytical formulationsso that they can be applied to laminated glass elementswith a larger number of layers.

3. ANALYTICAL STUDYLuible’s formulations [5], both regarding bending and

torsion, are based on the work of Stamm and Witte [8],but here another approach is followed. In terms of flexuralstiffness, the principles presented by Allen [21] are adopted

2

(for their simplicity and versatility), while in terms of tor-sional stiffness, Stamm and Witte’s [8] theory is consid-ered.

3.1. Equivalent flexural stiffness

In structural glass elements the glass layers are gen-erally considerably thicker than the interlayer plies andtheir Young’s modulus is also much larger. For this rea-son, when determining the flexural stiffness of a laminatedcross-section, the contribution of the interlayer may be ne-glected.

One important feature when adapting sandwich theoryto structural glass is the conceptual separation between(i) pure bending deformation and (ii) pure shear deforma-tion. In the former component the shear modulus of theinterlayer (Gint) is considered to be equal to∞ and thusthe glass element has a monolithic behavior (Fig. 2(a)). Inthis case, the ordinary bending theory applies and a deflec-tion w1 may be determined. In the latter component theactual shear modulus of the interlayer is considered and itcorresponds to an additional deflection w2 (Fig. 2(b)).

(a) Pure bending (Gint = ∞).

(b) Pure shear.

Figure 2: Pure bending and shear deformation in a lami-nated glass beam (adapted from [22]).

The shear deformation can generally be divided into twocomponents, transverse shear deformation (Fig. 3(b)) andwarping shear deformation (Fig. 3(c)). Under symmetri-cal loading there is only transverse shear deformation [21].This analytical study is based only in symmetrical loadingconditions (γ0 = 0).

(a) Undeformed shape. (b) Transverse sheardeformation.

(c) Warping shear de-formation.

(d) Total shear defor-mation.

Figure 3: Shear deformation of a structural element(adapted from [22]).

The two components of the deflection, developed inmore detail in the dissertation that supports this paper, re-sult in the following differential equations, that together,for a given distribution of the total shear force V , maybe integrated to yield an expression representative of thetotal deflection w = w1 + w2.

V ′′1 − a2V1 = −a2V (1a)

EIglw′2 =

V1a2

(1b)

where,

a2 =AGint

EIgl

(1− EIgl

EI

) (1c)

V1 is the component of the shear force associated with w1,EIgl is the flexural stiffness of the glass layers with respectto their own centroidal axes, EI = EIgl +EIs is the totalstiffness of the cross-section (neglecting the contributionof the interlayer), A is a parameter that depends on thegeometry of the cross-section, and Is is second moment ofarea of the glass layers with respect to the neutral axis.

The quantity a2 represents the ratio between the shearstiffness of the interlayer and the local bending stiffnessof the glass layers [21]; it can be rewritten using someparameters common in the structural glass literature andpresent in [8],

a2 =

(λfL

)2

=1 + α

αβL2(2a)

where,

α =IglIs

(2b)

β =EIs

AGintL2(2c)

and L is the free span.The effect the glass layers’ stiffness have on the shear

deformation of the interlayer is smaller for larger values ofa2 and for larger spans [21].

The second moment of area of the glass layers with re-spect to their own centroidal axes may be generally definedas;

Igl =

n∑i=1

bt3i12

(3)

where the index i comprises the number of glass layers (n),whereas Is andA are summarized in Table 1, for laminatedglass cross-sections with two, three, four or five layers. Thedefinition of the dimensions may be found in Fig. 4.

3.1.1. Equivalent flexural stiffness of beams undertransverse loads or pure bending

Eqs. (1a) and (1b), together, for a given distributionof the total shear force V — associated with tranverseloads — and with the appropriate boundary conditions(symmetrical load and support conditions), may be inte-grated to yield an expression representative of the totaldeflection w = w1 +w2. In turn, the deflection associated

3

(a) Two layers. (b) Three layers.

(c) Four layers. (d) Five layers.

Figure 4: Cross-sections of laminated glass elements.

Table 1: Expressions for Is and A for cross-sections fromtwo to five layers.

# of layers Is A

Two layers bd2t1t2t1 + t2

bd2

tint

Three layersbt1d

2

2

bd2

2tint

Four layers b

(t1d

2

2+t2a

22

2

)bd

tint· t1d

2 + t2a22

3t1d+ t2a2

Five layers b

(t1d

2

2+ 2t2a

22

)bd

4tint· t1d

2 + 4t2a22

t1d+ t2a2

with given load and support conditions may be comparedwith that corresponding to a monolithic beam, in orderto yield an expression for the equivalent flexural stiffness.The expression, evaluated at mid-span, can be written inthe form,

EIf = EIgl + ξfEIs (4)

where the parameter ξf , suggested by Blaauwendraad [23]but with a different presentation, equals 0 if Gint = 0 andequals 1 ifGint =∞. With this notation it becomes moreclear that the limiting values of the equivalent flexuralstiffness are EIf = EIgl if the shear stiffness of the in-terlayer is zero and EIf = E(Igl + Is) if the connectionis complete and the beam is in fact monolithic. A sim-pler form of ξf can be achieved inserting a few additionalparameters,

θ =λf2

φ =λfL1

Lψ =

θLaL

(5)

where L1 is the length of symmetrical overhangs and La isthe distance between loads and the supports in the four-point bending configuration.

The expressions for ξf associated with pure bending,uniformly distributed load, mid-span load and four-pointbending are summarized in Table 2.

3.1.2. Critical buckling load of a glass column

The procedure for the determination of the criticalbuckling load is the same that for the Euler’s column, onlywith a distinct differential equation containing the shearbehavior of the column.

From Eqs. (1a) and (1b), with the total shear forceassociated with a buckled column,

V = P (w′1 + w′2) (6)

the following differential equation may be determined:

wv1 −(a2 − P

EIgl

)w′′′1 −

a2P

EIw′1 = 0 (7)

As for the Euler column, the buckling mode shape (purebending deflection) is defined by a sinusoidal curve,

w1 = A1 sinπx

L(8)

The result of the substitution of Eq. (8) in Eq. (7),followed by the division by − (π/L) cos (πx/L) is:[

π4

L2+

(a2 − P

EIgl

)π2

L2− a2P

EI

]A1 = 0 (9)

The equation is true either if A1 = 0 (trivial solution– fundamental path) or if the rest of the product equals

4

Table 2: Parameter ξf for laminated glass beams subjected to transverse loading conditions or to pure bending.

TYPE OF LOADING 0 ≤ ξf ≤ 1

ξf =1 + α

1 + 1αθ2

(1− 2 cosh(φ)−cosh(θ−φ)

cosh(θ+φ)

) − αξf =

1 + α

1 + 125αθ2

(1− [2 cosh (θ)− 2] θ sinh(φ)+cosh(φ)

θ2 cosh(θ+φ)

) − αξf =

1 + α

1 + 3αθ2

(1− 3 sinh(θ+φ)−sinh(θ−φ)−4 sinh(φ)

2θ cosh(θ+φ)

) − αξf =

1 + α

1 + 63αθ2−4αψ2

(1− sinh (ψ) 2 cosh(φ+ψ)+cosh(θ+φ−ψ)−cosh(θ−φ−ψ)

2ψ cosh(θ+φ)

) − αξf =

1

1 + π2β

zero (non-trivial solution – buckling load). The criticalbuckling load is given by:

Pcr =π4

L4 + a2π2

L2

π2

L2EIgl+ a2

EI

(10)

An equivalent flexural stiffness EIf can then be ob-tained by corresponding Eq. (10) to Euler’s critical buck-ing load formula. It may be written in the form of Eq.(4), with the parameter ξf being given by the expressionincluded in Table 2.

3.2. Torsional stiffnessFor the determination of the torsional stiffness it is as-

sumed that there is no warping restriction. Additionally,because the interlayer is much weaker than the glass layers,the shear stresses τxy = τyx in the interlayer are assumedto be zero.

A torsional moment applied to a laminated glass beam’ssection is equilibrated by two different groups of shearstresses, as illustrated in Fig. 5. On the one hand, the glasslayers work independently as “thin-walled open sections”,while on the other hand, a closed shear flow transmittedbetween the glass layers by the interlayer also contributesto balance part of the torque. Assuming that all layershave the same width (b), the torsion constant associatedwith the former contribution is given by the well knownexpression,

Jgl =

n∑i=1

bt3i3

(11)

where the index i comprises the number of glass layers (n),while the torsion constant associated with the latter con-tribution is designated Js. The weight of the latter torsionconstant depends on the shear modulus of the interlayer,similarly to Is in Eq. (4). A parameter ξt, analogous toξf , may thus be introduced. It equals 0 if Gint = 0 and

equals 1 if Gint =∞. The torsional stiffness of laminatedglass beams is then defined as:

GJt = GJgl + ξtGJs (12)

The parameter ξt and the torsion constant Js are de-duced in the present Section for beams with different num-ber of layers, disregarding, for that purpose, the formercomponent.

A model representative of a two-layered glass beam oflength dx is illustrated in Fig. 6. The interlayer is rep-resented thicker than the glass layers only for improvedclarity of the illustration. Because τxy = τyx = 0 in theinterlayer, it is represented by two vertical elements ofthickness dy, distanced y from the section’s vertical sym-metry axis. A later integration across the entire widthof the glass beam allows the consideration of all the inter-layer. The closed shear flow composed of shear stresses τxzin the interlayer and τxy in the glass layers is also includedin the figure. The zz axis is placed in the vertical axis ofsymmetry, while the height of the yy axis is arbitrary.

For the purpose of the following developments, an hypo-thetical separation between an upper and a lower part ofthe beam is considered, at a level z = 0. That creates two“thin-walled opened sections”, one above and another be-low. If they were independent, when subjected to a torque,they would warp, and opposite longitudinal displacements

Figure 5: Different shear stresses equilibrating the torque(adapted from [8]).

5

Figure 6: Model of a two-layered laminated glass beam(adapted from [8]).

at points A and B would arise. Because in reality the in-terlayer is continuous and thus no such different displace-ments are possible, the shear stresses τzx = τxz are saidto compensate those differential displacements along thewidth of the beam (−b/2 ≤ y ≤ b/2). Knowing τzx, theassessment of the equilibrium in the interface between in-terlayer and glass layer allows the determination of τyx.The final step is to analyze the balance of moments of tor-sion applied to and resisted by the cross-section (

∑T = 0)

in order to determine ξt and Js.

The opposite longitudinal displacements at points Aand B associated with the independent warping of theupper and lower parts of the beam are defined by,

uA = −2φ′dAy (13a)

uB = 2φ′dBy (13b)

where φ′ is the rate of twist and dA, dB , and y are repre-sented in Fig. 6.

As mentioned before, so that points A and B do nothave differential displacements, shear stresses τzx mustdevelop. It was also mentioned that these shear stressesin the interlayer are balanced by shear stresses τyz in theglass layers. Thus, shear strains γzx and γyx deform thematerials and additional displacements uA and uB are de-fined by,

uA =

∫ y

0

γxy,1dy +

∫ 0

−tAint

γxzdz (14a)

uB = −∫ y

0

γxy,2dy +

∫ 0

tBint

γxzdz (14b)

assuming that ∂v/∂x in the glass layers and ∂w/∂x in theinterlayer are zero.

Considering the equilibrium in the xx direction in theinterface between the interlayer and the glass layer abovepoint A (Fig. 7) in order to obtain a relationship betweenτyx,1 and τzx, and acknowledging that τxy,1 = τyx,1, τxz =τzx and τxy,1(y = b/2) = 0 — and proceeding analogouslyfor the interface with the lower glass layer —, Eqs. (14)

Figure 7: Interface between glass layer and interlayer,above point A (adapted from [8]).

can be written as:

uA =1

Gt1

∫ y

0

∫ b/2

y

τxzdydy +τxzt

Aint

Gint(15a)

uB = − 1

Gt2

∫ y

0

∫ b/2

y

τxzdydy − τxztBint

Gint(15b)

The condition of warping continuity (compatibility)A ≡ B is defined as,

∆u+ ∆u = uB + uB − uA − uA = 0 (16)

which, taking into account that dA+dB = d and that tAint+tBint = tint, and introducing the parameter λ2t proposed byStamm and Witte [8], results in,

τxz + λ2t

∫ y

0

∫ b/2

y

τxzdydy =2Gintφ

′d

tinty (17a)

where:

λ2t =GintG· t1 + t2tintt1t2

(17b)

Differentiating Eq. (17a) twice with respect to y, the fol-lowing differential equation is obtained,

τ ′′xz + λ2t τxz = 0 (18a)

which has as general trigonometric solution for τxz:

τxz = c1 cosh (λty) + c2 sinh (λty) (18b)

With the substitution of the general solution in Eq. (17a),the following constants can be obtained,

c1 = 0 (19a)

c2 = 2dGintGtint

· Gφ′

λt cosh(λtb2

) (19b)

The final expressions for the shear stress in the interlayerand for the shear stress in the glass layers can then bedescribed as:

τxz = 2dλtt1t2t1 + t2

· sinh (λty)

cosh(λtb2

)Gφ′ (20a)

τxy,1 = 2dt2

t1 + t2

(1− cosh (λty)

cosh(λtb2

))Gφ′ (20b)

τxy,2 = 2dt1

t1 + t2

(1− cosh (λty)

cosh(λtb2

))Gφ′ (20c)

6

The distance between the resultants of the shear stressin the two vertical elements representatives of the inter-layer is 2y, while the distance between the two glass layersis d. Thus, the equilibrium of torque in the cross-sectionis given by:

Ts =

∫ b/2

0

τxzddy · 2y +

∫ b/2

−b/2τxy,1t1dy · d

=

∫ b/2

0

τxzddy · 2y +

∫ b/2

−b/2τxy,2t2dy · d

(21)

The torsion constant Js and the parameter ξt, mentionedin the beginning of this Section, can then be obtained fromthe relation,

Ts = ξtGJsφ′ (22a)

which yields,

ξtJs = 4bd2t1t2t1 + t2

(1− 2

λtbtanh

λtb

2

)(22b)

where,

Js = 4Is (22c)

with Is provided in Table 1, and:

ξt = 1− 2

λtbtanh

λtb

2(22d)

An analogous procedure, detailed in the dissertationthat supports this paper, can be followed for a larger num-ber of glass layers. The parameter ξt is summarized in Ta-ble 3, for laminated glass cross-sections with two, three,

four or five layers, with several new parameters havingbeen introduced in order to simplify the solutions for thetwo last configurations. The definition of the dimensionsmay be found in Figs. 4(a), 4(b), 4(c) and 4(d), respec-tively.

4. PARAMETRIC STUDY

In order to validate all analytical expressions, and todetermine their range of validity, a comprehensive para-metric study has been developed. This solves the problemof the present lack of information about the applicabilityrange of the existing expressions.

The parametric study is based on the comparison ofthe analytical results with those provided by numericalmodels implemented in commercial software Abaqus [18].The type of elements and their geometry are similar tothose presented in Section 2.

The parametric analysis consists of evaluating the in-fluence of the following variables: (i) the ratio span/width(L/b); (ii) the ratio width/total thickness (b/ttot); and,only for the condition of transverse loading, (iii) the over-hangs’ length (L/L1). Additionally, different degrees ofinteraction between the glass layers (depending on theshear modulus of the interlayer — Table 4) are taken intoaccount, as well as variations in the number of laminatedlayers.

The analyses of the parameters was performed on three-layered models of laminated glass columns subjected to acompressive axial load and of beams subjected to a mid-span load or a torsional moment. A geometrical referencecase has been defined, with all parameters changing fromthere. The reference model comprises three 10 mm thick(t) glass layers, two 1.52 mm thick (tint) interlayer plies,

Table 3: Parameter ξt for laminated glass beams from two to five layers subjected to a torsional moment.

Two layers Three layers

ξt = 1− 2

λtbtanh

λtb

2(0 ≤ ξt ≤ 1)

λt =

√GintG· t1 + t2tintt1t2

λt =

√GintG· 1

tintt1

Four layers Five layers

ξt = 1− γ1λt,1b

tanhλt,1b

2− γ2λt,2b

tanhλt,2b

2(0 ≤ ξt ≤ 1)

λt,1 =

√GintG· 3t1 + t2 − µ

2tintt1t2λt,1 =

√GintG· 2t1 + t2 − µ

2tintt1t2

λt,2 =

√GintG· 3t1 + t2 + µ

2tintt1t2λt,2 =

√GintG· 2t1 + t2 + µ

2tintt1t2

µ =√

9t21 − 2t1t2 + t22 µ =√

4t21 + t22

ρ =3t21d

2 + 4t1t2(a1a2 − a21

)+ t22a

22

t1d2 + t2a22ρ =

2t21d2 + 4t1t2

(a22 + 2a1a2 − a21

)+ 4t22a

22

t1d2 + 4t2a22

γ1 = 1 +ρ

µγ2 = 1− ρ

µ

7

the ratios L/b and b/ttot being equal to 15 and 10, respec-tively, and without overhangs (L/L1 =∞).

Nevertheless, in order to validate all expressions, thestudy includes models up to five layers and also three-layered models subjected to the remaining transverse load-ing conditions and to pure bending. Because the expres-sions do not exclude laminated elements with glass lay-ers of unequal thickness, some three-layered models werebuilt in order to assess the analytical expressions for thisparticular geometrical possibility. The geometry of theseadditional models is that associated with the geometricalreference case.

Table 4: Properties of the interlayers used in the paramet-ric study [17].

Type of interlayer Gint (MPa) νint ( – ) Eint (MPa)

SG 142 0.465 416.1

PVB 13.7 0.497 41.0

PVB1day 0.504 0.500 1.512

PVB10years 0.0517 0.500 0.1551

A total of 77 models were developed with the parametersL/b, b/ttot and L/L1 ranging from 8 to 25, 2 to 25 and 1to∞, respectively.

The results provided by all expressions were found tobe in very good agreement with numerical results, for allvalues adopted for the parameters analyzed (with a maxi-mum relative difference of 0.28%), except in what concernsthe equivalent torsional stiffness, whose accuracy stronglydepends on the ratio b/ttot (Table 5).

Comparing the results of the models “2ly/PVB/T”,“Ref/PVB/T”, “4ly/PVB/T” and “5ly/PVB/T” (Table6), it seems that the accuracy of the equivalent torsionalstiffness formulation gradually increases with the increas-ing number of layers. However, in light of these results, twoadditional finite element models were implemented. Oneof the additional models is similar to model “2ly/PVB/T”,but has a ratio b/t equal to that of model “5ly/PVB/T”(b/t = 561/10 = 56.1) instead of a ratio b/ttot equal to10, i.e., a width b equal to 561 mm. The other model issimilar to model “5ly/PVB/T”, but has, instead, a ratiob/t equal to that of model “2ly/PVB/T” (b/t = 215/10= 21.5), i.e., a width b equal to 215 mm. The relativepercentage differences between analytical and numericalresults of these two additional models, as well as of models“2ly/PVB/T” and “5ly/PVB/T”, are summarized in Ta-ble 7. The results demonstrate that the differences in theaccuracy of the equivalent torsional stiffness formulation,which were apparent in Table 6, almost vanish if the ratiob/t is adopted as assessment parameter, instead of b/ttot.

It may thus be concluded that the parameter b/t is moreappropriate to perform a parametric analysis about theinfluence of the width variation on the equivalent torsionalstiffness formulation. The influence of the ratio b/t on theprecision of the formulation is depicted in Fig. 8, wherethe percentage differences from Table 5 (and their “SG”counterparts) are plotted as a function of the ratio b/t,

determined from the dimensions b and t. The graph showsthat the error may be assumed to be lower than 1% for aratio b/t larger than 30–35, for any number of layers. Thisconclusion is consistent with the analytical assumption, inSection 3, that the glass layers are characterized by thin-walled cross-sections.

It may also be concluded (from Table 8) that, for abeam laminated with “PVB1day”, a change on the ra-tio L/L1 has an impact on the deflection (a reduction ofup to 4.5%) until a certain length of the overhangs (model“L1/PVB1day/F-3”). A further increase of the length ofthe overhangs (model “L1/PVB1day/F-4”) does not re-sult in less deflection. This is explained by the restrainingeffect overhangs have on the shear strain. An infinite over-hang acts like a rigid insert which prevents the glass layersfrom sliding over each others [21]. However, for stiffer in-terlayers, this effect is small, as illustrated by the modelwith “PVB”. In these conditions, the parameter ξf (Ta-ble 2) may be simplified, assuming that φ is zero, withoutgreat loss of accuracy.

5. LATERAL-TORSIONAL BUCKLING5.1. Analytical

The lateral-torsional buckling of glass beams can bestudied making use of the equivalent flexural and torsionalstiffness formulations previously defined. Glass beams areconsidered to behave as homogeneous beams, and hencethose formulations are applicable only by using EIf in-stead of EI and GJt instead of GJ .

5.1.1. Critical buckling load

The buckled shape of a simply supported beam is il-lustrated in Fig. 9(a). The lateral-torsional buckling re-sponse of a beam in pure bending can be described by thefollowing two differential equations:

EIfv′′ = −Mφ (23a)

GJtφ′ = Mv′ (23b)

where v and φ are the lateral deflection and the twist ro-tation, respectively.

An additional term must be added to Eq. (23b) inthe case of in-plane transverse loads, because of the ad-ditional torque component due to the twist rotation of thecross-section. Its form, when the beam is subjected to auniformly distributed load or a mid-span load, is, respec-

0%

2%

4%

6%

8%

10%

0 10 20 30 40 50 60 70 80 90

Relativedifference

analytical-num

erical

b/t

SGPVB

Figure 8: Precision of the torsional stiffness formulation.

8

Table 5: Example of the analysis of the parameter b/ttot in beams subjected to a torsional moment of 50 Nm.

Labelt b L L1 L/b b/ttot L/L1

Numerical Analytical

(mm) (mm) (mm) (mm) (rad) (rad)

b/SG/T-1 10 66 991 0 15 2 ∞ 0.006 85 0.006 25 (=8.65%)

b/SG/T-2 10 165 2478 0 15 5 ∞ 0.002 61 0.002 54 (=2.63%)

Ref/SG/T 10 330 4956 0 15 10 ∞ 0.001 65 0.001 63 (=0.98%)

b/SG/T-3 10 826 12 390 0 15 25 ∞ 0.001 30 0.001 29 (=0.32%)

Table 6: Example of the analysis of beams with different number of layers subjected to a torsional moment of 50 Nm.

Labelt b L L1 L/b b/ttot L/L1

Numerical Analytical

(mm) (mm) (mm) (mm) (rad) (rad)

2ly/PVB/T 10 215 3228 0 15 10 ∞ 0.005 66 0.005 58 (=1.54%)

Ref/PVB/T 10 330 4956 0 15 10 ∞ 0.004 06 0.004 02 (=0.86%)

4ly/PVB/T 10 446 6684 0 15 10 ∞ 0.000 69 0.000 68 (=0.91%)

5ly/PVB/T 10 561 8412 0 15 10 ∞ 0.000 35 0.000 35 (=0.86%)

tively:

GJtφ′′ = Mv′′ + qzgφ (24a)

GJtφ′′ = Mv′′ +

F

2δ

(x− L

2

)zgφ (24b)

where, from convention, zg is negative when the load isapplied above the shear center (Fig. 9(b)), and positiveotherwise, and δ is the Dirac delta function, which withthis writing is 1 if x = L/2 and 0 elsewhere [24].

Table 7: Relative percentage differences between analyticaland numerical results of models subjected to a torsionalmoment.

Numberof layers

Relative percentage difference

b/t = 21.5 b/t = 56.1

Two layers =1.85% =0.42%

Five layers =1.53% =0.41%

The critical buckling moment may be determined fromthese equations with Galerkin’s method. The deflectionand the rotation are approximated by the following ex-pressions with sinusoidal shapes,

u1 = q11ψ11 = vmax sinπx

L(25a)

u2 = q21ψ21 = φmax sinπx

L(25b)

where vmax and φmax are the maximum deflection androtation, respectively.

The resulting critical buckling moments for the case ofpure bending and in-plane transverse loads are, respec-tively:

Mcr =π

L

√EIfGJt (26a)

Mcr = C1π2EIfL2

C2zg +

√(C2zg)

2+GJtEIf

(L

π

)2

(26b)

where the coefficients C1 and C2 are given in the upperpart of Table 9. This formulation is equivalent to thatproposed by Kasper et al. [4].

Alternatively, an approach proposed by some authors,e.g. Timoshenko and Gere [26], Reis and Camotim [25]and Mohri et al. [24], may be followed. They propose toreplace Eq. (23a) in Eqs. (23b) and (24), in order to yielda single differential equation for each loading case. Thisprocedure corresponds to a mathematical simplificationthat introduces some inaccuracies regarding the bound-ary condition terms when weighted residuals methods areapplied. The resulting equation is, for the cases of purebending, uniformly distributed load, and mid-span load,respectively:

M2

EIfφ+GJtφ

′′ = 0 (27a)

M2

EIfφ+GJtφ

′′ − qzgφ = 0 (27b)

M2

EIfφ+GJtφ

′′ − F

2δ

(x− L

2

)zgφ = 0 (27c)

Galerkin’s method can analogously be applied to theseequations, now with a single approximative sinusoidal so-lution, for φ. In this alternative, the critical moments canstill be described by Eqs. 26, with the difference that thecoefficients C1 and C2 are given by the lower part of Table9.

As Mohri et al. [24] stated, these alternative coefficientsare equal to those adopted in Eurocode 3 [19]. The sameauthors claim having obtained results closer to finite ele-ment results with these coefficients.

Another alternative for the determination of criticalbuckling moments is the approach proposed in AustralianStandard AS 1288 [20].

5.1.2. Lateral deflection and rotationReal glass elements are naturally geometrically imper-

fect and thus they do not follow the fundamental path untilthe critical buckling load is reached. Instead, they present

9

(a) Buckled shape (adapted from [25]).

(b) Additional torque be-cause of the eccentricity ofan in-plane transverse load.

Figure 9: Lateral-torsional buckling of a simply supported beam.

Table 8: Example of the analysis of the parameter L/L1 in beams subjected to a mid-span load of 1 kN.

Labelt b L L1 L/b b/ttot L/L1

Numerical Analytical

(mm) (mm) (mm) (mm) (mm) (mm)

Ref/PVB/F 10 330 4956 0 15 10 ∞ 39.05 39.07 (+0.04%)

L1/PVB/F-1 10 330 4956 4956 15 10 1 39.03 39.05 (+0.04%)

Ref/PVB1day/F 10 330 4956 0 15 10 ∞ 67.25 67.30 (+0.08%)

L1/PVB1day/F-1 10 330 4956 248 15 10 20 65.16 65.22 (+0.10%)

L1/PVB1day/F-2 10 330 4956 620 15 10 8 64.36 64.44 (+0.12%)

L1/PVB1day/F-3 10 330 4956 2478 15 10 2 64.19 64.28 (+0.13%)

L1/PVB1day/F-4 10 330 4956 4956 15 10 1 64.19 64.28 (+0.13%)

Table 9: Lateral-torsional buckling coefficients – in-planetransverse loading

C1 =3π2

2 (3 + π2)= 1.1503 C1 =

2π2

4 + π2= 1.4232

C2 =6

3 + π2= 0.4662 C2 =

8

4 + π2= 0.5768

Alternative coefficients

C1 =

√15π4

8π4 + 360= 1.1325 C1 = π

√3

6 + π2= 1.3659

C2 =8C1

2π2= 0.4590 C2 =

12

C1 (6 + π2)= 0.5536

an asymptotic behavior to both fundamental and post-critical paths. The smaller the initial imperfections are,the closer the non-linear behavior is to the fundamentalpath.

The initial geometrical imperfections of beams are im-portant parameters to the definition of load-displacementcurves. An initial deflection (v0) measured in the cross-section centroid and an initial twist rotation (φ0) of thesections must be included in Eqs. (23) and (24) as initial

geometrical imperfections:

EIfv′′ = −M (φ+ φ0) (28a)

GJtφ′ = M (v′ + v′0) (28b)

GJtφ′′ = M (v′′ + v′′0 ) + qzg (φ+ φ0) (28c)

GJtφ′′ = M (v′′ + v′′0 ) +

F

2δ

(x− L

2

)zg (φ+ φ0)

(28d)

Applying the Galerkin’s method, similarly as for thedeflection and rotation in the critical buckling load anal-ysis (Eqs. (25)), the initial geometrical imperfections areapproximated by the following sinusoidal functions,

u3 = q31ψ11 = v0,max sinπx

L(29a)

u4 = q41ψ21 = φ0,max sinπx

L(29b)

where v0,max and φ0,max are the maximum initial deflec-tion and rotation (at mid-span), respectively.

The expressions obtained for the lateral deflection androtation are summarized in Table 10, with the coefficientsC1 and C2 provided in Table 9.

5.2. Experimental applicationAn experimental study on the lateral-torsional buckling

behavior of a long-span laminated glass beam was carriedout in IST, in 2014, as part of a PhD project [27]. Thebeam was similar to those used in the glass facades of the

10

Table 10: Lateral deflection and rotation of beam subjected to in-plane transverse loads or pure bending.

vmax =ML2 (GJtφ0,max +Mv0,max)

EIfGJtπ2 − (ML)2

φmax =M(ML2φ0,max + EIfπ

2v0,max

)EIfGJtπ2 − (ML)

2

vmax =ML2 (C1GJtφ0,max +Mv0,max)

C1EIfπ2 (C1GJt + 2C2Mzg)− (ML)2

φmax =M(ML2φ0,max + C1EIfπ

2 [v0,max − 2C2zgφ0,max])

C1EIfπ2 (C1GJt + 2C2Mzg)− (ML)2

Champalimaud Center for the Unknown, in Lisbon. Theexperimental results are used to assess the analytical for-mulations derived in Section 3. A numerical finite elementmodel was also implemented in order to simulate, as closeas possible, the behavior of the beam under the conditionsit was tested. A detailed description of the experimentand the corresponding results may be found in [7]. Somekey aspects are summarized here.

The beam, illustrated in Fig. 10, with three glass layers,was laminated with PVB. It was subjected to an increas-ing gravity mid-span load that led to its lateral-torsionalbuckling. The lateral displacement was kept within limitsso that the beam would not reach its ultimate limit stress.

Each of the three glass layers had 8200 mm of length,600 mm of width and 15 mm of thickness (nominal val-ues). The nominal thickness of each PVB sheet was1.52 mm. The total nominal thickness of the beam wasthen 48.04 mm. The actual width and total thicknessof the beam was carefully measured along its span, with1 mm and 0.05 mm precision tools, respectively. The re-sults were an average width of 601.54 mm and an averagethickness of 48.26 mm. A difference on the actual thick-ness of the glass layers is particularly important becauseof the impact it has on the critical buckling load.

Although the beam was received in the IST laboratoryalready laminated, an effort was made to evaluate the ac-tual thickness of the glass layers. It was concluded thatthey could hardly have the nominal thickness prescribedin the design; instead, it should probably present a lowervalue, closer to 14 mm than to 15 mm. The values providedby the main author of [7], as the more likely ones, are:14.25 mm of thickness for all three layers and 2.755 mm ofthickness for the two PVB sheets, which totals the averagemeasured total thickness of the beam.

The shear modulus of PVB was considered for a loadduration corresponding to the 20 minutes the experimentlasted and for an average measured temperature of 23 ◦C.The adopted value for the shear modulus of the interlayer(Gint) is 0.783 MPa and its Poisson ratio (νint) is 0.49 [7].

Although the total length of the beam was 8200 mm,it was tested in a 7800 mm span. The load was appliedat mid-span and the load fixture used guaranteed that itwould remain vertically applied for the duration of thetest. A steel container was suspended from the mid-span

section of the beam by means of a steel structure with steelrods. The connection to the clip around the beam was en-sured by an hinge positioned 89 mm above the top edge ofthe beam. Taking into account that the width of the beamis 600 mm, zg in Table 10 is=389 mm. The equipment sus-pended on the beam, including the container, had a totalinitial weight of about 7.4 kN. In order to limit the stressesof the beam within its ultimate limit stress, the transversedeflection at mid-span was limited to about 50 mm by awooden shoving system.

Topographic techniques were used to evaluate eventualinitial geometrical imperfections, which are inherent toreal structures and particularly to glass elements sub-jected to thermal treatments. Multiple points were mea-sured along the span, on the top and bottom edges of thebeam, with a precision of 1 mm. The coordinates obtainedallowed to determine the initial lateral deflection (v0) andthe initial twist (φ0). The adopted values are 2.3 mm and0.0012 rad, respectively.

The finite element model implemented in order to sim-ulate the results of the experiment is very similar to thosepresented in Sections 3 and 4, with only minor changes.

Because the beam is much larger than all those previ-ously modeled, attending to the symmetrical configura-tion of the load and support conditions, only half of thebeam was modeled. This significantly reduces the compu-tation and data effort without any impact on the results.

In order to better replicate the test conditions, alu-minum elements (E = 70 GPa and ν = 0.35) have beenadded where the load is applied and in the support. Apartition was introduced in the aluminum blocks so thatthe contact with the glass layers would be made by a softermaterial (E = 1 GPa and ν = 0.4) with 2 mm of thick-ness. They were connected to the glass layers with a tieconstraint. The glass layers were simulated with eight-node continuum shell elements with reduced integration(SC8R), while eight-node solid elements with reduced in-tegration (C3D8R) were adopted for all other materials.

The critical buckling loads were determined throughlinear buckling analyses (for the (i) nominal and (ii) es-timated corrected thicknesses) and the respective load-displacement paths were obtained by performing geomet-rically non-linear static analyses.

The experimental load-displacement response de-

11

Figure 10: General view of the beam setup.

scribed in this Section is that already presented in [7].Here, those results are compared with (i) analytical predic-tions, which result from the formulations derived in Sec-tion 3, and (ii) numerical ones. In particular, the geomet-rical and engineering parameters (including the nominaland estimated corrected thicknesses of the glass layers andinterlayer plies) previously mentioned were introduced inthe expressions from Tables 1, 2, 3 and 10, regarding sim-ply supported laminated glass beams with three layers,subjected to a mid-span load.

The experimental, analytical and numerical load-displacement paths are illustrated in Fig. 11, for the nom-inal thickness of the glass layers and interlayer plies andfor the corrected thicknesses. The analytical results areprovided considering the coefficients from the upper partof Table 9. Analogous results are presented in the disserta-tion that supports this paper, considering the alternativecoefficients from the same Table. The latter coefficientsyield results closer to numerical ones than those that re-sult from the first approach presented in Subsection 5.1.1.

The experimental critical buckling load (Fcr) cannotbe directly determined, but it can be indirectly estimatedusing the Southwell plot. The equilibrium path of an im-perfect column leads to the following equation [25],

vmax

F=

1

Fcrvmax +

v0,max

Fcr(30)

which can be applied to the lateral-torsional bucklingproblem, where vmax is the maximum deflection, at mid-span, associated with a given load F and v0,max is themaximum initial deflection. The equation represents alinear function, where the slope is 1/Fcr.

The critical load was analytically determined, withthree different formulations. The first critical bucklingload was obtained with the coefficients from the upperpart of Table 9 (“Main coeff.”). The second was obtainedwith the alternative coefficients (“Alt. coeff.”). The thirdcritical buckling load was determined according to Aus-tralian Standard AS 1288 [20]. The results obtained forthe nominal and estimated corrected thicknesses of theglass layers and interlayer plies are summarized in Table

0

5

10

15

20

25

30

35

40

-5 5 15 25 35 45 55

LoadF[kN]

Maximumdisplacement vmax [mm]

Analytical-nominalthicknesses

Numerical-nominalthicknesses

Analytical-correctedthicknesses

Numerical-correctedthicknesses

Experimental

Figure 11: Experimental, analytical and numerical load-displacement paths.

11, and are compared with the experimental value deter-mined by means of the Southwell plot.

The results seem to confirm that the actual thickness ofthe glass layers is, in fact, lower than the nominal value of15 mm. The analytical results have some dispersion, de-pending on the adopted formulation, but they show thatthey can predict the behavior of full-scale laminated glassbeams susceptible to the lateral-torsional buckling phe-nomenon with very reasonable accuracy.

As mentioned in Section 4, the length of the overhangshas an impact on the results only for low values of the shearmodulus of the interlayer. In terms of the results presentedin Table 11, if the overhangs were considered nonexistent(L1 = φ = 0), the critical buckling loads would be only0.1 kN to 0.2 kN lower.

The fact that laminated glass beams are slender el-

12

Table 11: Comparison between the critical buckling loadsobtained from experimental results, analytical formula-tions and numerical models (relative percentage differencesto the experimental load in parentheses).

Fcr (kN)

Type of analysis Nominal Corrected

thicknesses thicknesses

Experimental 30.0

Numerical 33.3 (+11.0%) 26.9 (=10.5%)

Analytical

Main coeff. 36.9 (+22.8%) 30.1 (+0.2%)

Alt. coeff. 35.6 (+18.7%) 29.1 (=3.1%)

AS 1288 34.4 (+14.8%) 28.1 (=6.4%)

ements, highly susceptible to lateral-torsional buckling,explains the importance of the present study, which com-pared experimental data with analytical and numericalsimulations.

The results obtained allowed to conclude that both an-alytical and numerical models considered provide simi-lar critical buckling loads and load-displacement paths tothose experimentally determined, particularly if the cor-rected thicknesses of the constituent layers are considered.It is very difficult to accurately measure or estimate theactual thickness of the glass layers after lamination, whichhighlights the importance of quality control in structuralglass manufacturing. Small differences in the thickness— in this case less than 1 mm in each layer — might sig-nificantly reduce the critical buckling load of laminatedglass beams, making them even more vulnerable to lateral-torsional buckling. According to the European StandardEN 572-8 [28], the manufacturing thickness tolerance forfloat glass with a nominal thickness of 15 mm is 0.5 mm.The thickness correction adopted in the present Chapteris higher than such tolerance. These possible variations inthe thickness of structural glass should be considered in adesign level.

Multiple formulations that describe the lateral-torsional behavior of beams were successfully applied, us-ing the equivalent flexural and torsional stiffness formula-tions derived in Section 3. This confirmed their validityand allowed to verify that the equivalent stiffness formu-lations may be used associated with monolithic stabilityformulations with the purpose of assessing the lateral-torsional buckling susceptibility of laminated glass beams.In spite of providing reasonably accurate results, the threelateral-torsional buckling formulations still motivate fur-ther research in order to reach a single approach that cansafely dismiss the use of numerical finite element modelsfor standard design issues.

6. CONCLUSIONS

A new more comprehensive formulation for the equiv-alent flexural and torsional stiffness of laminated glassbeams and columns results from this work. Relativelysimple expressions underlying a single small equation al-low defining the equivalent flexural stiffness of columns

with two, three, four or five layers (possibly asymmetri-cal in the two-layered case and necessarily symmetricalin the other cases), subjected to compressive axial loads,and of beams with the same layer configuration, subjectedto pure bending, uniformly distributed loads, mid-spanloads or four-point bending. Also underlying a single smallequation, relatively simple expressions define the torsionalstiffness of beams with two, three, four or five layers. Thisformulation is not entirely new. It yields the same resultsas previously existing equations in some particular con-figurations. The formulation is entirely innovative in theanalysis of columns and beams with four and five layers, aswell as in the consideration of overhangs. It also considersadditional lateral load conditions. The formulation canbe easily extended to other load and support conditions.

Through a parametric study, based on the results pro-vided by numerical finite element models, the equiva-lent flexural stiffness expressions proved to be very ac-curate for any value of the parameters considered. Onthe other hand, the precision of the torsional stiffness ex-pressions was shown to be strongly dependent on the ratiowidth/glass layer thickness (b/t), providing very accurateresults for larger values of the ratio (which generally cor-respond to the actual dimensions of real structural glasselements). The study provides thus the range of validityof the formulation, which was previously not found in theliterature for the existing analytical formulations.

Although the lateral-torsional buckling formulationsderived in Chapter 3 equal previously existing formu-lations, it was deemed important to understand theirbase differences, mainly because, as previously mentioned,some inconsistencies in their application were found inthe literature. The (at least partial) clarification of theirfundamentals may pave the way for further supported re-search.

The proposed equivalent flexural and torsional stiffnessformulation does not cover all load and support conditionsthat laminated glass elements may be subjected to. Forexample, although simply supported solutions are gener-ally preferred, in order to reduce the tensile stress in theglass layers, some support conditions may impose rota-tional restraints. Laminated columns may also be sub-jected to transverse loads. Therefore, in the future, otherconfigurations should also be taken into account.

Something very important that should be further stud-ied is the effect of lateral restraints in the lateral-torsionalbuckling behavior of laminated elements. In fact, glass-fins or roof beams most often exhibit point or continuouslateral restraints in one of the edges, depending on thetype of connection. The Australian Standard AS 1288[20] already provides some design considerations aboutthis subject.

Since the shear modulus of the interlayer is one of themost significant parameters regarding the susceptibilityof laminated glass beams to lateral-torsional buckling, itscorrect assessment is of the utmost importance. In partic-ular, the influence of their time- and temperature- depen-dencies should be thoroughly investigated for all interlayerproducts currently available.

13

REFERENCES

[1] C. Amadio and C. Bedon. Buckling of laminatedglass elements in out-of-plane bending. EngineeringStructures, 32(11):3780–3788, November 2010.

[2] C. Bedon, J. Belis, and A. Luible. Assessment of ex-isting analytical models for the lateral torsional buck-ling analysis of PVB and SG laminated glass beamsvia viscoelastic simulations and experiments. Engi-neering Structures, 60:52–67, February 2014.

[3] D. Callewaert, J. Depaepe, K. De Vogel, J. Belis,D. Delince, and R. Van Impe. Influence of temper-ature and load duration on glass/ionomer laminatestorsion and bending stiffness. In G. Siebert, T. Her-rmann, and A. Haese, editors, International Sym-posium on the Application of Architectural GlassISAAG 2008 Conference Proceedings, Munich, Oc-tober 27-28, 2008, pages 51–63, 2008.

[4] R. Kasper, G. Sedlacek, and M. Feldmann. DasBiegedrillknickverhalten von Glastragern aus Ver-bundglas. Stahlbau, 76(3):167–176, March 2007. (inEnglish: he lateral-torsional buckling behavior ofbeams made of laminated glass).

[5] A. Luible. Stabilitat von Tragelementen aus Glas.PhD thesis, Ecole Polytechnique Federale de Lau-sanne, 2004. (in English: Stability of structural glasselements).

[6] A. Luible and M. Crisinel. Design of Glass BeamsSubjected to Lateral Torsional Buckling. IABSESymposium Report, 92(30):45–52, January 2006.

[7] L. Valarinho, J.R. Correia, F. Branco, and G. Chi-umiento. Instabilidade lateral por flexao-torcao devigas em vidro estrutural - Estudo experimental eanalıtico. JPEE 2014, 2014. (in English: Lateral-torsional buckling of structural glass beams – Exper-imental and analytical study).

[8] K. Stamm and H. Witte. Sandwichkonstruktionen:Berechnung, Fertigung, Ausfuhrung, volume 3 of In-genieurbauten. Springer, Wien, 1974. (in English:Sandwich construction: design, manufacturing, exe-cution).

[9] C. Amadio and C. Bedon. Buckling of LaminatedGlass Elements in Compression. Journal of Struc-tural Engineering, 137(8):803–810, August 2011.

[10] N. Newmark, C. Siess, and I. Viest. Test and analysisof composite beams with incomplete interaction. Pro-ceedings of the Society of Experimental Stress Anal-ysis, (9(1)):75–92, 1951. [cited by [1, 9]].

[11] C. Bedon and C. Amadio. Flexural–torsional buck-ling: Experimental analysis of laminated glass el-ements. Engineering Structures, 73:85–99, August2014.

[12] E. Wolfel. Nachgiebiger Verbund -Eine Naherungslosung und deren Anwen-dungsmoglichkeiten. Stahlbau, (6):173–180, 1987.(in English: A composite with partial interaction -An approximate solution and its applications). [citedby [11, 15]].

[13] M. Feldmann, R. Kasper,et al. Guidance for Euro-pean structural design of glass components: support

to the implementation, harmonization and furtherdevelopment of the Eurocodes. EUR 26439 EN. Pub-lications Office of the European Union, Luxembourg,2014.

[14] L. Galuppi and G. Royer-Carfagni. Effective thick-ness of laminated glass beams: New expression via avariational approach. Engineering Structures, 38:53–67, May 2012.

[15] L. Galuppi, G. Manara, and G. Royer-Carfagni.Practical expressions for the design of laminatedglass. Composites Part B: Engineering, 45(1):1677–1688, February 2013.

[16] P. Scarpino. Berechnungsverfahren zur Bestimmungeiner aquivalenten Torsionssteifigkeit von Tragernin Sandwichbauweise. RWTH Aachen, December2002. (in English: Design method for determiningan equivalent torsional rigidity of sandwich beams).[cited by [2]].

[17] J. Belis. Kipsterkte van monolithische en gelami-neerde glazen liggers. PhD thesis, Ghent Univer-sity, 2005. (in English: Lateral-torsional bucklingof monolithic and laminated glass beams).

[18] Simulia - Dassault Systemes. Abaqus/Standard.http://www.3ds.com/products-services/simulia/products/abaqus, 2013. [Software;v6.13].

[19] ENV 1993-1-1: Eurocode 3. Design of steel struc-tures – Part 1-1: General rules and rules for build-ings. CEN, Brussels, 1992.

[20] AS 1288: Glass in buildings – Selection and installa-tion. Standards Australia, 2006. [Amdt.No.1 (2008),Amdt.No.2 (2011)].

[21] H.G. Allen. Analysis and design of structural sand-wich panels. The Commonwealth and internationallibrary. Structures and solid body mechanics division.Pergamon, Oxford, 1969.

[22] D. Zenkert, editor. Handbook of Sandwich Construc-tion. Engineering Materials Advisory Services Ltd.,Cradley Heath, West Midlands, December 1997.

[23] J. Blaauwendraad. Buckling of laminated glasscolumns. Heron, 52(1/2):147–164, February 2007.

[24] F. Mohri, A. Brouki, and J. Roth. Theoretical andnumerical stability analyses of unrestrained, mono-symmetric thin-walled beams. Journal of Construc-tional Steel Research, 59(1):63–90, January 2003.

[25] A. Reis and D. Camotim. Estabilidade estrutural.McGraw-Hill, 2001. (in English: Structural stabil-ity).

[26] S.P. Timoshenko and J.M. Gere. Theory of elasticstability. McGraw-Hill, 1961.

[27] L. Valarinho. Structural glass. Development of glass–GFRP composite beams. PhD thesis, Instituto Supe-rior Tecnico, 2016. (in preparation).

[28] EN 572-8: Glass in building – Basic soda lime sili-cate glass products. – Part 8: Supplied and final cutsizes. CEN, Brussels, July 2012.

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