Simple Numerical Model of Laminated Glass Beams

0.22 MB·English

by  Alena Zemanová

Simple Numerical Model of Laminated Glass Beams
AlenaZemanova´a,JanZemana,∗,MichalSˇejnohaa,b
aDepartmentofMechanics,FacultyofCivilEngineering,CzechTechnicalUniversityinPrague,
Tha´kurova7,16629Prague6,CzechRepublic
bCentreforIntegratedDesignofAdvancesStructures,Tha´kurova7,
16629Prague6,CzechRepublic
2
1 Abstract
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2 This contribution presents a simple Finite Element model aimed at efficient sim-
ulation of layered glass units. The adopted approach is based on considering
n
a independentkinematicsofeachlayer,tiedtogetherviaLagrangemultipliers. Val-
J
idation and verification of the resulting model against independent data demon-
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1 strateitsaccuracy,showingitspotentialforgeneralizationtowardsmorecomplex
problems.
]
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Keywords: laminatedglassbeams,finiteelementmethod,Lagrangemultipliers
C
.
s
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1. Introduction
[

1 The most frequently used transparent material in the building envelopes is
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glass. Itisafragilematerial,whichfailsinabrittlemanner. Thisisthereasonfor
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7 using safety glasses in a situation when there is a possibility of human impact or
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wheretheglasscouldfallifshattered.
3
. Laminated glass is a multi-layer material produced by bonding two or more
1
0 layers of glass together with a plastic interlayer, typically made of polyvinyl bu-
2
tyral (PVB). The interlayer keeps the layers of glass bonded even when broken,
1
: and its high strength prevents the glass from breaking up into large sharp pieces.
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This produces a characteristic ”spider web” cracking pattern when the impact
i
X
is not powerful enough to completely pierce the glass. Multiple laminae and
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a thicker glass decrease stress level, thereby increasing the load-carrying capacity
ofastructuralmember,too.
The focus of this study is on the establishing a simple and versatile frame-
work for the analysis of mechanical behavior of laminated glass units. To keep
∗Correspondingauthor. Tel.:[phone];fax[phone]
Emailaddresses: [email protected](AlenaZemanova´),
[email protected](JanZeman),[email protected](MichalSˇejnoha)
URL:http://mech.fsv.cvut.cz/~zemanj(JanZeman)
Preprintsubmittedtoarxiv January18,2012
the discussion compact, we restrict our attention to the linearly elastic response
of layered glass beams in the small strain regime. The rest of the paper is orga-
nized as follows. Methods of analysis of laminated glass beams are introduced in
Section2,togetherwithabriefcharacterizationoftheproposednumericalmodel.
TheprinciplesofthemethodaredescribedindetailinSections3and4. Inpartic-
ular, the mechanical formulation of the model is shown in Section 3. In the next
section, the Finite Element discretization is presented. In Section 5, the proposed
numerical technique is verified and validated against a reference analytical solu-
tion and publicly available experimental data. Finally, Section 6 concludes the
paperanddiscussesfutureextensionsofthemethod.
2. Briefoverviewofavailablemethods
The most frequent approach to the analysis of glass structural elements was,
for a long time, based on empirical knowledge. Such relations are sufficient for
the design of traditional windows glasses. In modern architecture there has been
a steadily growing demand on the use of transparent materials for large external
walls and roof systems in the recent decades. Therefore, the detailed analysis of
layeredglassunitsisbecomingincreasinglyimportantinordertoensureareliable
andefficientdesign.
In general, the complex behavior of laminated glass can be considered as an
intermediatestateoftwolimitingcases[1]. Inthefirstcase,thestructureistreated
as an assembly of two independent glass plates without any interlayer (the lower
bound on stiffness and strength of a member), while in the second case, corre-
sponding to the upper estimate of strength and stiffness, the glass unit is modeled
asamonolithicglass(oneglassplatewiththicknessequaltothetotalthicknessof
the glass plates). Both elementary cases, however, fail to correctly capture com-
plex interaction among individual layers, leading to non-optimal layer thickness
designs. Therefore, several alternative approaches to the analysis of layered glass
structureshavebeenproposedintheliterature. Thesemethodscanbecategorized
intothreebasicgroups:
• methodscalibratedwithrespecttoexperimentalmeasurements[2],
• analyticalapproaches[3,4,5],
• numericalmodelstypicallybasedondetailedFiniteElementsimulations[6,
7].
Applicability of analytical approaches to practical (usually large-scale) struc-
tures is far from being straightforward. In particular, the closed-form solution of
the resulting equations is possible only for very specific boundary conditions and
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therefore have to be solved by an appropriate numerical method. Moreover, the
analyticalapproachesareratherdifficulttobegeneralizedtobeamswithmultiple
layers. Therefore, it appears to be advantageous to directly formulate the prob-
leminthediscretizedform,typicallybasedontheFiniteElementMethod(FEM).
Nevertheless, we would like to avoid fully resolved two- or three-dimensional
simulations,cf.[6,7],whichleadtounnecessarilyexpensivecalculations.
In this paper, we propose a simple FEM model inspired by a specific class
of refined plate theories [8, 9, 10]. In this framework, each layer is treated as
the Timoshenko beam with independent kinematics. Interaction between indi-
vidual layers is captured by the Lagrange multipliers (with a physical meaning
of shear stresses), which result from the conditions of inter-layer displacements
compatibility. Sucharefinedapproachcircumventsthelimitationofsimilarmod-
els available in typical commercial FEM systems, which employ a single set of
kinematic variables and average the mechanical response through the thickness
of the beam, e.g. [11]. Unlike the proposed approach, the averaging operation is
too coarse to correctly represent the inter-layer interactions, see Section 5 for a
concreteexample.
3. Mechanicalmodeloflaminatedbeams
As illustrated in Figure 1, laminated glasses consist mostly of three layers. A
localcoordinatesystemisattachedtoeachlayertoallowforanefficienttreatment
of independent kinematics. In the following text, a quantity a expressed in the
localcoordinatesystemassociated withthei-thlayerisdenoted asa(i),whereasa
variablewithoutanindexrepresentsagloballydefinedquantity,cf. Figure1.
Each layer is modeled using the Timoshenko beam theory supplemented with
membraneeffects. Hence,thefollowingkinematicassumptionsareadopted
• the cross sections remain planar but not necessarily perpendicular to the
deformedbeamaxis,
• vertical displacement does not vary along the height of the beam (when
comparedtothemagnitudeofthedisplacement).
Under these assumptions, the non-zero displacement components in each layer
areparametrizedas:
u(i)(x,z(i)) = u(i)(x,0)+ϕ(i)(x)z(i),
w(i)(x,z(i)) = w(x),
wherei = 1,2,3andz(i) ismeasuredin thelocalcoordinatesystemfromthe mid-
dleplaneofthei-thlayer. Theinter-layerinteractionisensuredviathecontinuity
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glass
PVB
glass
y
x
ϕ1
x,u1
1 h1
z, w z1,w ϕ2
x,u2
2 h2
z2,w ϕ3
x,u3
3 h3
z3,w
Figure1: Kinematicsoflaminatedbeam
conditionsspecifiedoninterfacesbetweenlayersintheform(i = 1,2)
h(i) h(i+1)
u(i)(x, )−u(i+1)(x,− ) = 0. (1)
2 2
Thestrainfieldinthei-thlayerfollowsfromthestrain-displacementrelations[12,
11]
∂u(i) du(i) dϕ(i)
ε(i)(x,z(i)) = (x,z(i)) = (x,0)+ (x)z(i),
x ∂x dx dx
∂u(i) ∂w dw
γ(i)(x) = (x,z(i))+ (x) = ϕ(i)(x)+ (x), (2)
xz ∂z(i) ∂x dx
which, when combined with the constitutive equations of each layer expressed in
termsofYoung’smodulus E andtheshearmodulusG:
σ(i)(x,z(i)) = E(i)ε(i)(x,z(i)) and τ(i)(x) = G(i)γ(i)(x),
x x xz xz
yieldtheexpressionsfortheinternalforcesas
du(i)
N(i)(x) = E(i)A(i) (x,0),
x dx
(cid:32) (cid:33)
dw
V(i)(x) = kG(i)A(i) ϕ(i)(x)+ (x) ,
z dx
dϕ(i)
M(i)(x) = E(i)I(i) (x),
y dx
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where b and h(i) are the width and height of the i-th layer, recall Figure 1, and
k = 5, A(i) = bh(i) a I(i) = 1 b(h(i))3 stand for the shear correction factor, the
6 12
cross-sectionareaandthemomentofinertiaofthei-thlayer,respectively.
Toproceed,considertheweakformofthegoverningequations,writtenforthe
i-thlayer(thesubscripts• and• relatedtointernalforcesandkinematics-related
x z
quantitiesareomittedinthesequelforthesakeofbrevity)
(cid:90)L (cid:90)L
d (cid:16)δu(i)(x)(cid:17)E(i)A(i) d (cid:16)u(i)(x)(cid:17) dx = δu(i)(x)f¯(i)(x)dx+(cid:104)δu(x)N¯(i)(x)(cid:105)L,
dx dx x 0
0 0
(cid:90)L (cid:90)L
d (δw(x))kG(i)A(i)γ(i)(x)dx = δw(x)f¯(i)(x)dx+(cid:104)δw(x)V¯(i)(x)(cid:105)L,
dx z 0
0 0
(cid:90)L
d (cid:16) (cid:17) d (cid:16) (cid:17) (cid:104) (cid:105)L
δϕ(i)(x) E(i)I(i) ϕ(i)(x) dx = δϕ(i)(x)M¯(i)(x) ,
dx dx 0
0
(cid:90)L (cid:34) (cid:35)
d
δγ(i)(x)kG(i)A(i) γ(i)(x)−ϕ(i)(x)− (w(x)) dx = 0.
dx
0
to be satisfied for arbitrary admissible test fields δu(i), δϕ(i) and δw. In particular,
the first three equations correspond to equilibrium conditions written for normal
and shear forces and bending moments, respectively. The last identity enforces
thegeometricalrelation(2)intheintegralform,therebyallowingtotreattheshear
strain as an independent field in the discretization procedure discussed next. Fur-
ther note that the continuity conditions (1) will be introduced directly into the
discretizedformulation,asexplainedinthefollowingSection.
4. Finiteelementdiscretization
To keep the discretization procedure transparent, it is assumed that each layer
ofthelaminatedbeamisdividedintoidenticalnumberofelements,leadingtothe
discretizationschemeillustratedinFigure2.
Following the standard conforming Finite Element machinery, e.g. [12, 11],
we express the searched and test displacement fields at the element level in the
form
u(i)(x) ≈ N(i)(x)r(i), δu(i)(x) ≈ N(i)(x)δr(i),
e e,u e,u e e,u e,u
w (x) ≈ N (x)r , δw (x) ≈ N (x)δr ,
e e,w e,w e e,w e,w
ϕ(i)(x) ≈ N(i) (x)r(i) , δϕ(i)(x) ≈ N(i) (x)δr(i) ,
e e,ϕ e,ϕ e e,ϕ e,ϕ
γ(i)(x) ≈ N(i)(x)r(i), δγ(i)(x) ≈ N(i)(x)δr(i),
e e,γ e,γ e e,γ e,γ
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y
x
z, w
ϕi ϕi
e,a e,b
ui e ui
i e,a e,b
a b
γi
w e w
e,a e,b
L
Figure2: Finiteelementdiscretizationofthei-thlayer
where e is used to denote the element number, • and δ• denote a relevant
e e
searched and test field restricted to the e-th element, N(i) is the associated matrix
e,•
of basis functions and r(i) the matrix of nodal unknowns. In the actual implemen-
e,•
tation, the fields u(i), w and ϕ(i), as well as the corresponding test quantities, are
e e
assumed to be piecewise linear. To obtain a locking-free element, the shear strain
γ(i)istakenasconstantandiseliminatedusingthestaticcondensation,see[12,11]
e
foradditionaldetails.
Tosimplifythefurthertreatment,weconsiderthefollowingpartitioningofthe
stiffnessmatrixKandtherighthandsidematrixRrelatedtothee-thelementand
thei-thlayerafterthestaticcondensation:
(cid:34) (cid:35)(cid:34) (cid:35) (cid:34) (cid:35)
K(i) K(i) r(i) R(i)
e ew e = e ,
K(i) K(i) r R(i)
we w e,w e,w
(cid:16) (cid:17)
whereK(i) = K(i) T and
ew we
r(i) = (cid:104)u(i),u(i),ϕ(i),ϕ(i)(cid:105)T, r = (cid:2)w ,w (cid:3)T.
e e,a e,b e,a e,b e,w e,a e,b
Considering all three layers in Figure 2 together gives the resulting system of
governingequationsintheform
 KK00(w(e11e)) KK00(w(e22e)) KK00(w(e33e)) K(w1) +KKKK(w(e(e(e2312www)))) +K(w3) E0eT  rrrre(e(e(e123,w)))  =  R(e1,w) +RRRR(e(e(e(e2123,w)))) +R(e3,w) ,
E 0 0 λ 0
e (4×1)
wherethematrixλstoresthenodalvaluesoftheLagrangemultipliers,associated
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withthecompatibilityconstraint (1),andthematrix
 
Ee =  0100 0001 h0002(1) h0002(1) −1001 −0011 hh0022((22)) hh002((22)) −0001 −0001 h0002(3) h000(3) 
2 2
implementsthetyingconditions.
5. Verificationandvalidationofnumericalmodel
F glass 5 mm
PVB 0.38 mm
glass 5 mm
10 cm 40 cm 40 cm 10 cm
strain gage
Figure3: Threepointbendingsetupforsimplysupportedbeam
Toverifyandvalidatetheperformanceofthepresentapproach,thepreviously
describedFEMmodelwasimplementedusingMATLAB(cid:13)R systemandcompared
againstpredictionsofananalyticalmodelandexperimentaldataforathree-point
bendingtestonasimplysupportedlaminatedglassbeampresentedin[5],seealso
Figure3. Thewidthofthebeamisb = 0.1mandmaterialdataofindividualcom-
ponents of the structure are available in Table 1. The glass modulus of elasticity
isslightlylowerthantheconventionalvaluesof70–73GPareportedinthelitera-
tureandisspecificforthematerialemployedintheexperiment. Moreover,asthe
PVB layer shows viscoelastic and temperature-dependent behavior, the modulus
of elasticity corresponds to an effective secant value related to load duration of
60sandtemperatureof22◦ C.
Table1: Materialdata
Glass PVBlayer
Young’smodulus, E 64.5GPa 1.287MPa
Poisson’sratio,ν 0.23 0.4
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Table2summarizesvaluesofthemid-spandeflectionforarepresentativeload
level determined by FE-based discretization using 60 elements (30 when sym-
metry of the problem is exploited) and the corresponding experimental values.
Notethatthediscretizationissufficienttoachievethree-digitaccuracyinthemid-
span deflection. In addition to the results obtained by an analytical method pro-
posed by Asik and Tezcan in [5], the results of the analysis using ADINA(cid:13)R sys-
tem and the elementary lower and upper bounds are included. In particular, the
ADINA(cid:13)R model is based on the classical laminate theory, cf. [11], whereas the
twosimplifiedapproachesassumezeroorinfinitecomplianceoftheinterlayer,re-
callalsodiscussioninSection2. Inthefollowingdiscussion,e.g. ηnum denotesthe
exp
relativeerrorbetweenthenumericalpredictionandreferenceexperimentalvalue,
whilee.g. η isusedfortheerrorofanalyticalsolutionwhencomparedtocandi-
an
date approaches. Clearly, the results of the last three methods differ substantially
from experimental data as well as the analytical results. The proposed numeri-
cal model, on the other hand, shows a response almost identical to the analytical
method, which deviates from experimental measurement by less then 6%. Such
accuracycanbeconsideredassufficientfromthepracticalpointofview.
Table2: Comparisonofresultsforasimplysupportedbeam(load50N)
Model Centraldeflection[mm] η η
exp an
Laminatedglassbeam: thickness[mm]5/0.38/5(glass/PVB/glass)
Experiment 1.27 - -5.2%
Analyticalmodel 1.34 5.5% -
Numericalmodel 1.34 5.5% 0.0%
ADINA(cid:13)R(Multi-layeredshell) 0.89 -30.2% -33.8%
Monolithicglassbeam: thickness[mm]10(glass+glass)
Analyticalmodel 0.99 -21.8% -25.9%
Twoindependentglassbeams: thickness[mm]5/5(withoutanyinterlayer)
Analyticalmodel 3.97 212.6% 196.2%
To further confirm predictive capacities of the proposed numerical scheme, a
response corresponding to a proportionally increasing load was investigated. The
resultsappearinTables3and4. Again,themethodseemstobesufficientlyaccu-
rate in the investigated range of loads when considering the values of deflections
aswellasvaluesoflocalstressesandstrains.
6. Conclusions
As shown by the presented results, the proposed numerical method is well-
suitedforthemodelingoflaminatedglassbeams,mainlybecauseofitslowcom-
putational cost and accurate representation of the structural member behavior.
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Table3: Comparisonofdeflectionsforasimplysupportedbeam
Load[N] Centraldeflection[mm]
w w ηan [%] w ηnum [%] ηnum [%]
exp an exp num exp an
50 1.27 1.34 5.51 1.34 5.51 0.00
100 2.55 2.69 5.49 2.68 5.10 -0.37
150 4.12 4.03 -2.18 4.02 -2.43 -0.25
200 5.57 5.38 -3.41 5.36 -3.77 -0.37
Table4: Comparisonofstressesandstrainsforasimplysupportedbeam
Load[N] Maximumstrain[×10−6] Maximumstress[MPa]
(cid:15) (cid:15) ηnum [%] σ σ ηnum [%]
an num an an num an
50 112 114 1.79 7.23 7.34 1.52
100 224 228 1.79 14.45 14.68 1.59
150 336 341 1.49 21.68 22.02 1.57
200 448 455 1.56 28.9 29.36 1.59
Future improvements of the model will consider large deflections and the time-
dependentresponseoftheinterlayerandwillbereportedseparately.
Acknowledgments
The support provided by the GACˇR grant No. 106/07/1244 is gratefully ac-
knowledged.
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