Computational Models of Laminated Glass Plate under Transverse Static Loading

Computational Models of Laminated Glass Plateunder Transverse Static Loading

Ivelin V. Ivanov, Dimitar S. Velchev, Marcin Knec, Tomasz Sadowski

Abstract Laminated glass with Polyvinyl Butyral (PVB) interlayer became a popu-lar safety glass for architectural and automotive glazing applications. The very softinterlayer, bonding the glass plates, however, has negligible normal stress in trans-verse loading and it resists mainly by shear stress. The classical laminate theoryobeying the principle of the straight normals remaining straight is not valid for lam-inated glass. Conventional Finite Elements (FE) are used to model the laminatedglass in cylindrical bending to investigate the problem. Based on the assumptionthat the glass layers of a laminated glass plate obey Kirchoff’s classical plate theoryand the PVB-interlayer transfer load by shear stress only, the differential equationsof a Triplex Laminated Glass (TLG) plate are derived and a special TLG plate FE iselaborated. For each of its nodes, the element has one transverse translational, threerotational, and two additional degrees of freedom representing the slippage betweenthe glass layers. All computational models are compared with experimental tests ofa laminated glass strip in cylindrical bending.

1 Introduction

Laminated glass is widely spread as material for architectural glazing of contem-porary buildings. It is also used for windshields in automotive industry. Laminatedglass, in most of the cases, consists of only two glass plates bonded together by apolyvinyl butyral (PVB) interlayer. The PVB-material is a rubber like elastomer [3]that keeps the shards of broken glass plates in the frame of the glass unit after the

Ivelin V. Ivanov and Dimitar S. VelchevDepartment of Engineering Mechanics, University of Ruse, “Studentska” 8, 7017 Ruse, Bulgaria,e-mail: [email protected] and [email protected]

Marcin Knec and Tomasz SadowskiDepartment of Solid Mechanics, Lublin University of Technology, “Nadbystrzycka” 40, 20-618Lublin, Poland e-mail: [email protected] and [email protected]

1

2 Ivelin V. Ivanov, Dimitar S. Velchev, Marcin Knec, Tomasz Sadowski

failure and makes laminated glass safety. Another advantage of laminated glass isits thermal insulation property, which is important for the architectural glazing. De-spite of the advantages, however, the efficient application of laminated glass in thecomplex structures is limited because of the difficulties in their strength calculationsat the stage of their design.

The PVB-material has elastic modulus which is thousands of times less than theelastic modulus of glass. The great mismatch of the elastic moduli leads to layerinteraction in the bending of laminated glasses which does not obey the principle ofstraight normals or pseudo normals in the theory of plates. The modelling of lam-inated glasses is sophisticated mainly because of the complex nonlinear behaviourof the PVB-material, which is highly viscoelastic with great temperature depen-dency [8]. The other nonlinearity, that could be included in the laminated glass be-havior, is the geometrical nonlinearity of thin plates under transverse loadings.

Since the complex behaviour of laminated glass is not very well modelled in thepractice, its strength is considered as intermediate of two limiting cases: the case oflayered glass — two glass plates without any interlayer between them, and the caseof monolithic glass — one glass plate with thickness equal to the total thickness ofthe glass plates [9]. The former case determines the lower boundary of the strength,while the latter — the upper boundary.

Vallabhan et al. [10] developed a mathematical model of laminated glass basedon the minimization of the total potential energy in which the bending and mem-brane strain energy of the glass plates as well as the shear strain energy of thePVB-interlayer are included. The assumptions for the glass plates correspond tovon Karman’s nonlinear theory of plates. The interaction of the glass plates is pro-vided by the shear of the linearly elastic PVB-interlayer which depends on theirbending. The finite in-plane strain accounting for large rotations of the normals isincluded in the model and therefore membrane stress could appear at large deflec-tion of the laminate as a consequence of the geometrical nonlinearity in dependenceon the boundary conditions. Five complicated differential equations are obtainedand iteratively solved. Two interpolation parameters should be optimised in order toobtain a stable solution. Asik [1] developed an algorithm for implicit integration ofthe equations and their unconditionally stable solution.

Norville et al. [7] developed a simple multilayer beam model of laminated glass.The shear force transferred by the PVB-interlayer is determined by a coefficientwhich has to be identified experimentally. The effective section modulus of the beamcould be calculated in dependence on the coefficient which allows the maximumbending stress to be obtained. This is a simple model which gives an opportunitythe strength of laminated glass to be approximately estimated at the stage of theirdesign.

Duser et al. [4] utilised the Finite Element (FE) method to model laminatedglasses under transverse loadings. They used 3-D solid elements to model the lay-ers and their interaction. This approach requires a lot of finite elements and there-fore expensive computations. The PVB-material is modelled as linear viscoelasticand a nonlinear solution is performed. A statistical model based on two-parameter

Computational Models of Laminated Glass Plate under Transverse Static Loading 3

Weibull’s law of distribution is developed for the glass breakage and strength deter-mination of the glass plates.

Asik and Tezcan [2] published a mathematical model of laminated glass beamswhich is based on nonlinear strain-displacement relationship. The model is derivedby variational principle from the membrane and bending strain energy of the glasslayers and the shear strain energy of the PVB-interlayer. The nonlinearity of thedifferential equations of the model disappears if the laminated glass beam is simplysupported and an analytical solution for this case is derived. The model is used toinvestigate the linear and nonlinear behaviour of symmetric triplex glass beams incomparison with the laminated glass plate behaviour.

Ivanov [5] derived a simple but reasonable analytical model of laminated glassworking as plane beam that can be used for structure optimisation. The model islinear and it reveals the advantage of laminated glass if proper optimisation of theplate thickness is done.

The aim of this work is to develop efficient computational model for laminatedglass calculations with complex geometry under transverse loading.

2 Experiments

Laminated glass is provided for experimental investigation. The laminate has totalthickness of 6.13 mm or the layers of glass/PVB/glass are 2.875/0.38/2.875 mmthick. The elastic properties of glass are assumed to be as follows: the Young’smodulus is E = 72 GPa and the Poisson’s ratio is ν = 0.23. The total mass densityof the laminate is measured as ρ = 2.418 g/cm3. In order to characterise the PVB-material, compressive shear tests are carried out.

2.1 Compressive Shear Test

The PVB-material is almost incompressible and it can withstand large strain. Whenthe glass layers of laminated glass are shattered, the integrity of the panel is stillkept by the PVB-interlayer which is extremely strained at the places of glass frac-ture. A hyper-elastic material model is necessary to describe interlayer behavior,when dynamic loading with fracture is considered, but for quasi-static loading, lin-ear behaviour with the initial elastic moduli is enough. The approach used here tofind the initial shear modulus is the same as the described in [6] by compressiveshear test.

The hyper-elastic constitutive laws of rubberlike materials are based on approx-imation of strain energy potential. There are several forms of the strain energy andthey give different stability and approximation of material behaviour for differenttypes of loadings. The second order polynomial form of strain energy potential hasconstants which can be determined by simple shear test. The stress-strain relation-


ship in simple shear is described by the polynomial

τ = G0γ +G′γ3 , (1)

where G0 is the initial shear modulus of the PVB-material.The compressive shear test is depicted in Fig. 1. The tested laminated glass sam-

ple is between two wedges under 45-degree angle with respect to the direction of theloading force F and the lower wedge is on rollers. The PVB-interlayer with thick-ness h is practically incompressible and therefore in simple shear. The test is carriedout on MTS-testing machine recording the force F and the vertical displacement u.

Fig. 1 Compressive shear test

The samples have nominal dimensions of 20×50 mm or the area of the samplesis A = 1000 mm2. The thickness of the interlayer is measured as h = 0.38 mmand the thickness of the glass layers is 2.875 mm. The shear stress and strain aredetermined by simple geometrical considerations as follows:

τ =F

A√

2, γ =

u√

2h

. (2)

Three samples at 1 mm/min cross-head speed have been tested. The experimen-tal data in range of γ = 0–2.5 are given in Fig. 2. The fitting-curve procedure inMATLAB R©software gives the value of the material constant G0 = 0.5173 MPa.

The volumetric elasticity constants can be zero if fully incompressible materialis assumed. In the case of confined material (as in the case of laminated glass PVB-interlayer) hybrid finite elements should be used, in order to avoid the volumetriclocking of FE model in implicit FE code. The hybrid elements are computationallyinefficient and when the volumetric locking should be avoided by conventional ele-


0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

γ (−)

τ (M

Pa)

ExperimentalCurve fitting

Fig. 2 Compressive shear test data

ments, the Poisson’s ratio, ν , should be lower than 0.5 and initial Young’s modulusshould be calculated as E0 = 2(1+ν)G0.

2.2 Cylindrical Bending Test

In order to investigate the behaviour of laminated glass under transverse loading, astrip of laminated glass has been cut and strain gauges are glued at several placeson its upper and lower surfaces. The strip was supported and loaded by the weight,W , of masses as it is shown in Fig. 3. The total mass, m, of weights was 4, 8, and 12kg. The weight is applied through two thin bars in order to be spread over the widthof the strip. The central deflection of laminated glass strip was also measured. Theresults are given in Table 1.

Table 1 Experimental results for cylindrical bending of laminated glass strip

mass deflection gauge #1 gauge #2 gauge #3 gauge #4 gauge #5 gauge #6kg mm ×10−6 ×10−6 ×10−6 ×10−6 ×10−6 ×10−6

4 1.70 −68 120 −72 76 −54 588 2.76 −132 185 −121 127 −94 9612 4.82 −205 254 −171 178 −133 133


Fig. 3 Experimental setup for cylindrical bending of laminated glass strip

The viscosity of the PVB-material is not accounted for in the computational mod-els which follow but the measured data creep and the viscosity is clearly observed.The data measured in the experiment are average of the data read in loading andunloading. The data measured by gauge #1 and even those measured by gauge #2are maybe strongly influenced by the way of applying the load because they havegreater difference in absolute value, which is in contrast to the other pairs of gauges,and because the gauges are placed very close to the bar through which the load isapplied.

3 Computational Models

The problem of laminated glass under transverse loading was investigated by usingthe FE method. Two computational models were prepared in the commercial codeABAQUSTMto analyse the cylindrical bending of the laminated glass strip. All FEmodels here represent only half of the laminated glass strip in experimental testswith boundary conditions providing the symmetry.

3.1 3-D Brick Element Model

The laminated glass strip was modelled by 3-D brick elements with reduced inte-gration in ABAQUSTMcommercial software for solution with ABAQUS/Standard.Each glass layer is modelled by five plies of elements and the PVB-interlayer is


modelled by one ply of the same type of elements. The size of the elements in theplane of the strip is 1.5 mm, which is approximately 4 times larger than the inter-layer thickness and thus it is the largest size ratio of the elements. The degree ofdiscretisation can be justified by the deformed model shown in Fig. 4.

Fig. 4 Deformation of 3-Dbrick element model

Step: Step−2Increment 3: Step Time = 3.000

Deformed Var: U Deformation Scale Factor: +1.000e+01

XY

Z

The discretisation of the half of the laminated glass strip gives 1177056 degreesof freedom for the model and approximately so many unknowns. The linear solu-tion of the problem for four steps of loading on a computer using parallelisationtechnique took 2 hours and 10 minutes. The size of the problem could not allow themodel to be used for nonlinear solution.

Table 2 Relative results for linear solutiona 3-D brick element model

mass deflection gauge #1 & #2 gauge #3 & #4 gauge #5 & #6kg % % % %

4 +26.3 −3.3 +0.3 +3.08 +37.7 +2.3 +4.7 +5.212 +13.2 +1.8 +6.4 +6.4

a The positive strain is compared with the average of the experimental absolute strain values forboth sides of the strip.

The deformed state of the 3-D brick element model in Fig. 4 shows the shear inthe PVB-interlayer and the straight normals of the glass layers remaining straight.The result is achieved by giving the PVB-material properties E0 = 1.5312 MPa andν = 0.48, providing the shear modulus to be G0 = 0.5173 as it is determined fromthe experiments. If the Young’s modulus is setup to be E0 = 1.5416 and Poisson’sratio ν = 0.49, providing the same shear modulus, the volumetric locking appears inthe model and the straight normals of the whole laminated glass strip remain straightwithout any shear in the PVB-interlayer and small deflection of the strip. In such acase, hybrid 3-D brick elements should be used for modelling the PVB-interlayer,which will take longer time for calculations because the computational efficiency ofthat elements is low.

The results of the calculations with the 3-D brick element model of the laminatedglass strip were compared with the experimental ones. The relative deviation from


the experimental results is given in Table 2. The strain on both surfaces of the stripis the same in absolute value for linear solution. The positive values of the surfacestrain is compared with the average of the absolute values of the strain on the uppersurface and the lower surface measured in the experiments at the same place alongthe strip. The strains of the linear solution match fairly well to the experimentalones, but the deflection is greater and it seems that the model is soft.

3.2 Shell Element Model

Searching efficient solution of the laminated glass problem, special finite elementslike continuum shell elements (according to the terminology of ABAQUSTMsoftware)and cohesive finite elements attracted our attention. The shell elements, includ-ing the continuum shell elements, obey the principle of straight normals remainingstraight after the deformation. Cohesive elements are purposed to model thin layerswith constitutive law determined by the traction separation behaviour. The contin-uum shell elements and the cohesive finite elements are compatible and could bestacked each over the other.

U, U3

−7.807e−01−2.589e−01+2.628e−01+7.845e−01+1.306e+00+1.828e+00+2.350e+00+2.872e+00+3.393e+00+3.915e+00+4.437e+00+4.958e+00+5.480e+00

Step: Step−2Increment 12: Step Time = 12.00Primary Var: U, U3Deformed Var: U Deformation Scale Factor: +1.000e+01

X

Y

Z

Fig. 5 Continuum shell element model with cohesive element interlayer

The half of the laminated glass strip is modelled by two plies of continuum shellelements, one for each glass layer, and one ply of cohesive elements for the PVB-interlayer. The model can be seen in Fig. 5. The size of the elements now does notdepend on their thickness, so accepting preferable size of 5 mm, the model has only36408 degrees of freedom. It is so efficient that it takes only 20 seconds to solve theproblem for 13 steps of loading on the same computer as for the linear solution of3-D brick element model. The cohesive elements have no volumetric locking when


the Poisson’s ratio is close to the value of 0.5 and the value of 0.49 is accepted here.

Table 3 Relative results for linear solutionb of shell element model


4 +23.9 +10.5 +13.9 +15.48 +35.1 +17.4 +19.2 +18.912 +11.3 +16.6 +21.2 +19.9

b The positive strain is compared with the average of the experimental absolute strain values forboth sides of the strip.

The results of the linear solution of the shell element model are given in Table 3.The linear solution gives the same absolute values of strain on the upper and lowersurface for the same places along the strip, so the positive one only are comparedwith the average experimental strains. The model is quite soft with high deflectionand high values of the strains.

Table 4 Relative results of nonlinear solution for shell element model

mass deflection gauge #1 gauge #2 gauge #3 gauge #4 gauge #5 gauge #6kg % % % % % % %

4 +23.9 +53.7 −14.1 +18.1 +11.2 +21.6 +11.48 +35.1 +42.3 −0.3 +23.7 +15.6 +21.7 +16.212 +11.1 +32.4 +3.8 +25.4 +17.2 +21.4 +18.4

Since the shell element model is very efficient, a nonlinear solution is examinedhere. The nonlinearity is only geometrical nonlinearity that appears when high ro-tations of flexible structures are involved in the calculations. Many authors considergeometrical nonlinearity of laminated glass panels very important. Really, the archi-tectural glazing could involve large area panels of laminated glass which are veryflexible. The results of such nonlinear solution for the same shell element model ofthe half of the laminated glass strip are given in Table 4. The results are comparedwith the experimental results and the relative values are given. The model is againvery soft with large strains and deflection compared with the experimental ones.

The results show that there is no great difference between the linear and thenonlinear solution for the shell element model. The deflection is not so big in orderto have large rotations, the strip is not constrained in membrane direction, so thegeometrical nonlinearity is not rendered here. Also the strain of the PVB-interlayeris quite lower than 1% and any material nonlinearity caused by the hyperelasticityof the interlayer could not be observed.


4 Triplex Laminated Glass (TLG) Plate Element

A special plate finite element is developed here to model three layered laminatedglass (triplex) panels. It is based on the assumption that the glass layers of laminatedglass obey the Kirchoff’s plate theory, where plane stress with linear distributionover the thickness is assumed, and the PVB-interlayer is in pure shear.

4.1 Stress and Strain in Laminated Glass Plate

The derivation of the stiffness matrix is based on the stationary principle of thetotal potential energy in equilibrium. A infinitesimal piece of laminated glass isconsidered which is shown in Fig. 6.

Fig. 6 Infinitesimal elementof laminated glass

The displacements of point P from glass layer number i, i = 1,2, which is ondistance ξi from the middle plane, moved to position P′, are considered using theprojections of the infinitesimal glass layer of the laminated glass element onto xz-and yz-planes. The displacements are denoted u, v, and w as it is shown in Fig. 7.They can be determined as:

ui = uoi−ξiw,xvi = voi−ξiw,ywi = w(x,y)

∣∣∣∣∣∣(3)

wherew,y = ∂w

∂y = θx

−w,x = − ∂w∂ x = θy

∣∣∣∣∣ (4)

are the rotations of the plate normal.The first derivatives of the displacements are the strains in glass layers:


Fig. 7 Displacements in glass layer

εxi = εxoi−ξiw,xxεyi = εyoi−ξiw,yyγxyi =−2ξiw,xy

∣∣∣∣∣∣(5)

Applying Hooke’s law, the plane stress components are derived:

σxi = E1−ν2 (εxi +νεyi)

σyi = E1−ν2 (νεxi + εyi)

τxyi = E2(1+ν)γxyi

∣∣∣∣∣∣∣(6)

The normal resultant forces for each glass layer can be found by integration ofstresses through the thickness of the layer:

nxi =

+hi/2∫

−hi/2

σxidξi =Ehi

1−ν2 (εxoi +νεyoi) (7)

nyi =

+hi/2∫

−hi/2

σyidξi =Ehi

1−ν2 (νεxoi + εyoi) (8)

The normal resultant forces of both glass layers should be equal and opposite be-cause of the equilibrium of the forces and the lack of membrane loading. Theyshould create a couple of forces which gives additional bending moment to the


resistance in bending as it is shown in Fig. 8. Then the normal resultants can bedenoted as:

Fig. 8 Bending moments andlayer interaction

nx ≡ nx2 =−nx1ny ≡ ny2 =−ny1

∣∣∣∣ (9)

Applying Eqs. (7) and (8) for both glass layers, i = 1,2, four equations are ob-tained which are solved for the membrane strains of the glass layers and using thenotation (9) the result is written as:

εxo1 = − nx−νnyEh1

εyo1 = − ny−νnxEh1

εxo2 = nx−νnyEh2

εyo2 = ny−νnxEh2

∣∣∣∣∣∣∣∣∣(10)

The equations above give also the relationships

−h1εxo1 = h2εxo2 , −h1εyo1 = h2εyo2 . (11)

The PVB-interlayer is assumed to have only pure shear strain and stress as it isshown in Fig. 9. The shear strain in both planes of projection can be derived from themembrane strains, the rotations of both glass layers, and their relative displacementsor slippages:

γx ≡ γzx0 = ϕy−θy =uo2 + h2

2 w,x−uo1 + h12 w,x

h0+w,x =

1h0

(δu +hCw,x) , (12)

whereδu = uo2−uo1 (13)

is the slippage between the glass layers in x-direction and

hC =2h0 +h1 +h2

2(14)


is the distance between the glass layers midsurfaces as it is shown in Fig. 8.

Fig. 9 Strain and stress in PVB-interlayer

In a similar way, the shear strain in the other plane is determined as

γy ≡ γyz0 =1h0

(δv +hCw,y) , (15)

whereδv = vo2− vo1 (16)

is the slippage between the glass layers in y-direction. The slippage between theglass layers is very important to explain the mechanism of glass layer interactionthrough the PVB-interlayer. Also, the derivatives of the slippage can represent veryimportant relationship between the membrane straining of the glass layers:

δu,x = εxo2− εxo1 =h1 +h2

Eh1h2(nx−νny) , (17)

δv,y = εyo2− εyo1 =h1 +h2

Eh1h2(ny−νnx) . (18)

The shear stress in the PVB-interlayer can be found easily applying the Hooke’slaw for shear:

τx ≡ τzx0 = G0γx , τy ≡ τyz0 = G0γy . (19)


4.2 Potential Strain Energy of Laminated Glass Plate

The potential strain energy of laminated glass element will be presented in vectorand matrix form in order to obtain then the stiffness matrix of the developed finiteelement. The potential strain energy of the PVB-interlayer depends only on the pureshear stress and strain.

Πγ =12

∫

V0

gTtdV0 =12

∫

V0

gTCγ gdV0 , (20)

whereg = [γx γy]T , t = [τx τy]T , (21)

Cγ =[

G0 00 G0

]. (22)

The volume of integration is V0, V0 = h0Ω , where Ω is the area of the plate.The strain vector g can be expressed as a sum of two other vectors

g =[

γxγy

]=

1h0

[δu +hCw,xδv +hCw,y

]=

1h0

uδ +hC

h0uθ . (23)

Then the potential strain energy can be constructed in another way:

Πγ =12

∫

ΩeT

γ Dγ eγ dΩ , (24)

whereeγ = [δu δv w,x w,y]T , (25)

Dγ =G0

h0

1 0 hC 00 1 0 hC

hC 0 h2C 0

0 hC 0 h2C

. (26)

The derivatives of displacement functions can be expressed by a differential opera-tor:

eγ = Lγ u , (27)

where

Lγ =

1 0 00 1 00 0 ∂x0 0 ∂y

, (28)

u = [δu δv w]T . (29)

The potential strain energy of the PVB-interlayer is determined by three displace-ment functions of two variables — the coordinates x and y, which are the slippageof the glass layers δu and δv, as well as the deflection w.


The potential strain energy of each glass layer i, i = 1,2, can be derived from theplane stress field in bending as follows:

Πgi =12

∫

Vi

eTi sidVi =

12

∫

Vi

eTi CeidVi , (30)

whereei = [εxi εyi γxyi]T , (31)

si = [σxi σyi τxyi]T , (32)

C =E

1−ν2

1 ν 0ν 1 00 0 1−ν

2

, (33)

and the volume of the glass layer, Vi, is Vi = hiΩ .The strain in glass layers can be represented in vector form

ei =

εxoi−ξiw,xxεyoi−ξiw,yy0−2ξiw,xy

= eoi +ξiz , (34)

whereeoi = [εxoi εyoi 0]T , (35)

z = [−w,xx −w,yy −2w,xy]T . (36)

Substituting (34) into (30), one obtains

Πgi =12

∫

Vi

(eoi +ξiz)TC(eoi +ξiz)dVi = Πoi +Πκi , (37)

whereΠoi =

12

hi

∫

ΩeT

oiCeoidΩ , (38)

Πκi =12

h3i

12

∫

ΩzTCzdΩ . (39)

Using the Eqs. (10), (17), and (18), the potential energy Eq. (38) can be obtainedas:

Πoi =12

1hi

∫

ΩeT

δ Cδ eδ dΩ , (40)

whereeδ = [δu,x δv,y]T , (41)

Cδ =E

1−ν2h2

1h22

(h1 +h2)2

[1 νν 1

]. (42)

Now, the potential strain energy of both glass layers can be summed:


Πg = Πo1 +Πo2 +Πκ1 +Πκ2 =12

∫

ΩeT

gDgegdΩ , (43)

whereeg = [δu,x δv,y −w,xx −w,yy −2w,xy]T , (44)

Dg =E

1−ν2

h1h2h1+h2

[1 νν 1

]0

0 h31+h3

212

1 ν 0ν 1 00 0 1−ν

2

. (45)

The vector of pseudo strain eg can be represented as differential operator appliedon the displacement functions u:

eg = Lgu , (46)

where

Lg =

∂x 0 00 ∂y 00 0 −∂x2

0 0 −∂y2

0 0 −2∂xy

. (47)

The potential strain energy of the laminated glass plate is the sum of the potentialshear strain energy of the PVB-interlayer and the potential strain energy of the glasslayers:

Π = Πγ +Πg . (48)

4.3 Stiffness Matrix Derivation

The derived potential energy expressions show that they depend on three functionsof displacements which totally describe the equilibrium state in the triplex laminatedglass. A rectangular four-node plate finite element with six degrees of freedom pernode as it is shown in Fig. 10 is constructed in the global coordinate system xyz.

The vector of the element nodal displacements consists of four nodal vectors withthe displacements of each node I, I = 1,2,3,4.

d = [dT1 dT

2 dT3 dT

4]T (49)

The nodal displacement vectors have six displacements each: two relative displace-ments defining the slippage between the glass layers, δu,δv, the deflection, w, andthree rotations, θx,θy,ψ:

dI = [δuI δvI wI θxI θyI ψI ]T , I = 1, . . . ,4 . (50)


Fig. 10 TLG rectangularplate finite element

The displacement functions can be approximated by shape functions multipliedby the nodal displacements, which are the unknown parameters:

u = Nd . (51)

The matrix of shape functions consists of four sub-matrices — one for each node I:

N = [N1 N2 N3 N4] . (52)

The nodal sub-matrices of shape functions have six shape functions depending onthe nodal coordinates — one shape function for each degree of freedom:

NI =

N1I 0 0 0 0 00 N2I 0 0 0 00 0 N3I N4I N5I N6I

, I = 1, . . . ,4 . (53)

All shape functions are functions of two variables — the coordinates x and y. Thefirst two functions are equivalent:

N2I ≡ N1I . (54)

They describe the slippage between the glass layers and they can be constructedas bi-linear shape functions from Lagrangian polynomials [11]. The remain shapefunctions describe the displacements in Kirchoff’s plate. They are C 1 continuityshape function constructed by Hermitian polynomials [11]. The basic plate elementused here is Bogner-Fox-Schmit (BFS) rectangular plate element which is well de-scribed in many sources like [11]. The basic plate element could be also any plateelement. Here, we added two degrees of freedom and shape functions describingthem. The construction of shape functions here is considered trivial and it is omit-ted.

Once the shape functions are chosen, the stiffness matrix, K, can be defined fromthe potential strain energy, Π , by its partial derivatives with respect to the nodaldisplacements:

∂Π∂d

= Kd . (55)


First of all, the strain-displacement matrix should be defined. Starting with the shearstrain in the PVB-interlayer, the strain vector can be expressed:

eγ = Lγ u = Lγ Nd = Bγ d . (56)

This defines the strain-displacement matrix:

Bγ = Lγ N . (57)

Then the potential shear strain energy in the PVB-interlayer becomes:

Πγ =12

dT∫

ωBT

γ Dγ Bγ dΩ d =12

dTKγ d , (58)

whereKγ =

∫

ωBT

γ Dγ Bγ dω (59)

is the wanted stiffness matrix corresponding to the shear strain energy.The glass-layer strain vector is expressed by the shape functions:

eg = Lgu = LgNd = Bgd (60)

and defines the strain-displacement matrix:

Bg = LgN . (61)

Using the strain-displacement matrix above, the glass strain potential energy be-comes:

Πg =12

dT∫

ωBT

gDgBgdΩ d =12

dTKgd (62)

which defines another part of the stiffness matrix of the element:

Kg =∫

ωBT

gDgBgdΩ . (63)

Finally, the stiffness matrix of TLG plate finite element is the sum of both stiffnessmatrices:

K = Kγ +Kg . (64)

Additional, element forces like surface pressure p, which is shown in Fig. 10,can be determined as a vector of nodal forces. The potential energy of the pressurep is:

Πp =−∫

ωwpdΩ . (65)

The vector of traction forces is defined as:

q = [0 0 p]T . (66)


The potential energy can be defined for the traction forces:

Πp =−∫

ωuTqdΩ =−dT

∫

ωNTqdΩ , (67)

and the equivalent nodal forces are obtained:

f =∫

ωNTqdΩ (68)

as partial derivatives of the potential energy f =−∂Πp/∂d.

4.4 TLG Plate Element Validation

The half of the laminated glass strip is modelled by TLG rectangular plate elements.The deformed model can be seen in Fig. 11. The size of the elements is 5×10 mm.The number of nodes is only 1460 and the number of the degrees of freedom is8760. The TGL element is not prepared for nonlinear solution, but it is not necessaryas it was shown in previous sections. The results of the TLG plate element solutionsunder different loadings are given in Table 5.

Fig. 11 The deformed TLG element model with dimensions and displacements in meters for m =12 kg loading

The results of the solution show as close as the 3-D brick element model solutionin strain but closer to the experimental data solution in deflection. The TLG plate


0 50 100 150 200 250 300 350 400−20

0

20

40

60

80

100

120

140

distance (mm)

stra

in (

x10−

6 )

Load m = 4 kg

3−D brickshell linearshell nonlinear uppershell nonlinear lowerTLG plateexperimental upperexperimental lower

Fig. 12 Strain along the strip loaded by m = 4 kg

0 50 100 150 200 250 300 350 400−50

0

50

100

150

200

250

distance (mm)

stra

in (

x10−

6 )

Load m = 8 kg




0 50 100 150 200 250 300 350 400−50

0

50

100

150

200

250

300

350

distance (mm)

stra

in (

x10−

6 )

Load m = 12 kg



element model is maybe the best model among all computational models examinedhere.

Table 5 Relative results for linear solutionc of TGL plate element model


4 −1.5 −12.9 −13.2 −14.88 +21.0 +1.7 +3.1 0.012 +3.8 +7.0 +9.2 +6.4

c The positive strain is compared with the average of the experimental absolute strain values forboth sides of the strip.

The comparison of the models can be better seen if the distribution of strainalong the strip is shown as diagram. In Fig. 12 the distribution of the strain for theloading with m = 4 kg is shown, while on Fig. 13 and Fig. 14 the distributions form = 8 kg and m = 12 kg are given, respectively. All they are compared with theexperimental measurements at the points of strain gauges. The strain distribution isbased on distance measurement starting from the central point of the strip toward itsend.

The diagrams show that the 3-D brick element model and the TLG plate ele-ment model are very similar in strains and quite better than the shell element modelcompared with the experimental data. The TLG plate element model is better in


deflection than all others as well as in the computational efficiency. The TLG plateelement could be developed on different basis in order to have general geometryand membrane forces. Even in this case, the maximum number of the degrees offreedom per node of the plate discretisation is 8, while for the shell element model,it is 12 or 1.5 times more. So, the computational efficiency and the accuracy of theTLG plate element are promising.

5 Conclusions

The laminated glass is very difficult to be modelled by finite elements. The con-ventional finite elements are not computationally efficient or have no sufficient ac-curacy. The analysis shows that the nonlinearity in static and quasi-static loading isnegligible. The developed triplex laminated glass plate element is very computation-ally efficient and it has quite good accuracy compared with the other computationalmodels considered here. The element is based on the Kirchoff’s plate theory for theglass layers and on the assumption that the PVB-interlayer undergoes pure shear.The element can be developed on the basis of isoparametric quad shell elementscombined with membrane degrees of freedom instead of rectangular plate element.This could allow the element to be applied for complex geometry and for geometri-cal nonlinearity.

Acknowledgements The research leading to these results has received funding from the EuropeanUnion Seventh Framework Programme (FP7/2007-2013), FP7 - REGPOT - 2009 - 1, under grantagreement No:245479.

References

1. Asik M Z (2003) Laminated glass plates: revealing of nonlinear behavior. Computers andStructures 81:2659–2671

2. Asik M Z, Tezcan S (2005) A mathematical model for the behavior of laminated glass beams.Computers and Structures 83:1742–1753

3. Dhaliwal A K, Hay J N (2002) The characterization of polyvinyl butyral by thermal analysis.Thermochimica Acta 391:245–255

4. Duser A V, Jagota A, Bennison S J, (1999) Analysis of glass/polyvinyl butyral laminatessubjected to uniform pressure. Journal of Engineering Mechanics 125:435–442

5. Ivanov I V (2006) Analysis modelling and optimization of laminated glasses as plane beam.Int J Sol Struc 43:6887-6907

6. Jagota A, Bennison S J, Smith C A (2000) Analysis of a compressive shear test for adhe-sion between elastomeric polymers and rigid substrates. International Journal of Fracture104:105–130

7. Norville H S, King K W, Swofford J L (1998) Behavior and strength of laminated glass.Journal of Engineering Mechanics 124:46–53

8. Sobek W, Kutterer M, Messmer R (2000) Untersuchungen zum Schubverbund bei Verbund-sicherheitsglas — Ermittlung des zeit- und temperaturabhangigen Schubmoduls von PVB.Bauingenieur 75:41–47


9. Vallabhan C V G, Minor J E, Nagalla S R (1987) Stress in layered glass units and monolithicglass plates. Journal of Structural Engineering 113:36–43

10. Vallabhan C V G, Das Y C, Magdi M, Asik M, Bailey J R (1993) Analysis of laminated glassunits. Journal of Structural Engineering 119:1572–1585

11. Zienkiewicz O C, Taylor R L (2000) The finite element method (fifth ed) Vol. 2: Solid me-chanics. Butterworth-Heinemann, Oxford

Computational Models of Laminated Glass Plate under Transverse Static Loading

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