Modeling of the structural behavior of laminated glass beams Study of the lateral-torsional buckling phenomenon Miguel dos Santos Leitão Machado e Costa Thesis to obtain the Master of Science Degree in Civil Engineering Supervisors: Prof. Dr. Nuno Miguel Rosa Pereira Silvestre and Prof. Dr. João Pedro Ramôa Ribeiro Correia Examination Committee Chairperson: Prof. Dr. José Joaquim Costa Branco de Oliveira Pedro Supervisor: Prof. Dr. Nuno Miguel Rosa Pereira Silvestre Member of the Committee: Prof. Dr. Sandra Filomena da Silva Jordão Alves Member of the Committee: Prof. Dr. Pedro Manuel de Castro Borges Dinis November 2015
Transcript
Modeling of the structural behavior of laminated glass beamsStudy of the lateral-torsional buckling phenomenon
Miguel dos Santos Leitão Machado e Costa
Thesis to obtain the Master of Science Degree in
Civil Engineering
Supervisors: Prof. Dr. Nuno Miguel Rosa Pereira Silvestre andProf. Dr. João Pedro Ramôa Ribeiro Correia
Examination Committee
Chairperson: Prof. Dr. José Joaquim Costa Branco de Oliveira PedroSupervisor: Prof. Dr. Nuno Miguel Rosa Pereira SilvestreMember of the Committee: Prof. Dr. Sandra Filomena da Silva Jordão AlvesMember of the Committee: Prof. Dr. Pedro Manuel de Castro Borges Dinis
November 2015
ii
Acknowledgments
I am deeply grateful to both the supervisors of this dissertation — Professor Nuno Silvestre and Profes-
sor Joao Ramoa Correia. In different ways, they gave a tremendous contribution by providing valuable
guidance and advice. They were always readily available for whatever was needed, even when they
were clearly overloaded. Their permanent good mood also made me thankful for the opportunity to work
with them. From my experience as their student in ordinary Civil Engineering courses, and now in the
scope of this dissertation, I consider them a reference and role model as Professors in IST.
The single most valuable contribution to this dissertation was undoubtedly given by Luıs Valarinho. From
the first moment, and during all the phases of development of the dissertation, he was always willing
to put aside his own work in order to discuss whatever we deemed important. I truly appreciate his
natural readiness to give his opinion and insight, from the big picture to the small detail — always with
the concern of explaining the reasons or underlying concepts, like a true professor and open-minded
researcher. I wish him all the luck for his future professional life.
Most of what I learned about Abaqus I have to thank to Francisco Nunes. For countless times he was
available to listen to my setbacks and to help me figure out solutions. He always cared to provide me
with all the options I might further investigate. I also appreciate his interest on the work I was doing.
When facing several other questions about Abaqus I found Mario Arruda always unconditionally avail-
able. His contribution in advising and helping speeding up the most time-consuming numerical analyzes
was of the utmost importance.
The repeated contribution of my colleague and friend Daniel Ferreira allowed me to overcome multiple
dead-locks in the mathematical deductions. He provided me with important advice about Mathematica
and LaTeX, investing his time in helping me finding out some solutions.
I am also grateful to EPFL, in general, for the opportunity I was given to participate in a mobility program
that allowed me to attend a large number of valuable courses, including some cutting-edge subjects
— besides the outstanding academic experience it gave me a new perspective of the world outside
Portugal. It was there that I was first introduced to the subject of structural glass.
The cover photo, featuring a glass facade from the Champalimaud Center for the Unknown, was a contribution from JoseCampos (http://josecamposphotography.com).
iii
iv
Abstract
In recent years there have been several investigations about the behavior of laminated structural glass
elements, namely in terms of their flexural and torsional stiffness, with the lateral-torsional buckling of
beams being one of the most relevant and complex topics.
There are various analytical formulations to describe the equivalent stiffness of laminated elements
— in comparison with monolithic elements — but none covers more than three layers of glass in a
comprehensive and unified manner, and those that exist are not consensual. This work proposes a new
formulation, based on sandwich theory, which provides equivalent results to previous formulations in
a limited set of conditions, but that is able to characterize the behavior of simply supported laminated
glass columns and beams up to five layers, subjected to compressive axial loads, mid-span loads,
uniformly distributed loads, four-point bending, pure bending or torsion. The fundamentals explained in
this dissertation allow the formulation to be extended to a larger number of layers and to different load
and support conditions.
The proposed formulation is subjected to a parametric study based on the comparison with numerical
results retrieved from finite element simulations, in order to assess the range of validity of each expres-
sion.
Two analytical approaches for the lateral-torsional buckling problem are studied in detail, with their fun-
damentals being explained. Another formulation, proposed in an Australian Standard, is also addressed.
An experimental assessment of the work developed is achieved by comparing the results obtained from
a flexural test on a long-span three-layered glass beam, which exhibited lateral-torsional buckling, with
When carrying axial compressive loads, glass walls must be designed taking buckling into account.
9
(a) Glass floor and balustrades. (b) Glass stairway.
Figure 2.3: Examples of glass structures from Apple Jungfernsteig, Hamburg [17].
(a) Novartis Reception Building, Basel [18]. (b) Glass Lantern, Apple Zorlu, Istanbul [19].
Figure 2.4: Examples of structural glass walls.
Glass beams and glass columns are geometrically similar, since both are laminated slender elements,
subjected respectively to lateral and axial loads. They are frequently used to support roofs, floors or fa-
cades, being called glass-fins in the latter configuration. Their high slenderness makes them susceptible
to buckling and lateral-torsional buckling — phenomena that motivate this dissertation.
One of the more cutting-edge developments in glass structures are stairways entirely made with glass,
as the one illustrated in Fig. 2.3(b). Two laminated elements work both as long-span beams and
balustrades, which support laminated glass steps.
Common geometries
Float glass can be cut into various geometries, from square or rectangular shapes (the most common
and inexpensive) to triangular or irregular shapes, as for example, in the case of the beams applied in
the stairway of Fig. 2.3(b). Additionally, structural glass can be bent using more than one technique.
The Museum aan de Stroom, in Antwerp, is one example of the use of hot-bent glass (Fig. 2.5(a)).
Glass may also be laminated in a wide range of configurations. Different types of glass may be com-
bined, and curved laminated glass has already been applied multiple times. Examples of laminated
glass elements with up to five layers may already be found — for example, in the balustrades and steps
of the stairway illustrated in Fig. 2.3(b) or in the columns that sustain the roof structure of Apple Stanford
10
Mall, located in Palo Alto, a highly seismic region (Fig. 2.5(b)).
(a) Museum aan de Stroom, Antwerp [20]. (b) Apple Stanford Mall, Palo Alto [21].
Figure 2.5: Examples of curved glass and laminated glass column with five layers.
2.4 Structural stability of glass elements
Numerous studies have increasingly been developed with the purpose of modeling and characterizing
the behavior of structural glass elements. Only part of those studies regard structural stability. Some
studies focus on the buckling of glass columns and others on the lateral-torsional buckling of glass
beams. They generally concern the analysis of equivalent flexural and torsional stiffness, critical buck-
ling loads and load-displacement paths. Experimental and numerical studies aim to validate different
analytical formulations, which not always have wide acceptance and are not included in any commonly
adopted regulation. This Section presents a state-of-the-art review of those studies.1
2.4.1 Analytical studies
A relevant analytical study was conducted by Amadio and Bedon [1, 22]. That study presents an analyt-
ical formulation for the determination of the equivalent flexural stiffness of two-layered laminated glass
columns/beams — subjected to compressive axial load, possibly combined with uniformly distributed
lateral load — based on the theory proposed by Newmark et al. [23]. The authors concluded that
the buckling resistance of laminated glass columns is affected by temperature variations, load duration,
geometrical imperfections, eccentricity of the applied load and the presence of other transverse loads.
Two inter-related analytical studies, which resulted in formulations designated by “Wolfel-Bennison ap-
proach” and “Enhanced effective thickness approach”, are presented and compared with numerical
results in [24]. These two formulations provide equivalent flexural stiffnesses both in terms of deflections
and stresses. Although the “Wolfel-Bennison approach” may also be used to determine the equivalent
1It is worth mentioning that many of the studies that were published on glass stability were carried out by a relatively smallnumber of people, who have frequently referenced and supported their work on the work of others. Consequently, as highlightedfurther ahead in this document, sometimes these works include some inaccuracies arising from errors on the transcription of thework of other authors.
11
flexural stiffness of two-layered glass columns subjected to axial compression, as mentioned in [25],
Galuppi et al. [24] only address some transverse loading conditions. The authors claim to obtain, for the
case of simply supported beams subjected to a mid-span load, better agreement with the “Enhanced ef-
fective thickness approach”, derived by Galuppi and Royer-Carfagni [26], than with the “Wolfel-Bennison
approach”, first proposed by Wolfel et al. [27]. Parameters for other loading conditions and for glass
plates are also provided. Recently, Galuppi and Royer-Carfagni [28] extended the “Enhanced effective
thickness approach” to laminated glass beams with an arbitrary number of equally thick glass layers and
with arbitrary thickness in the case of three-layered beams.
2.4.2 Experimental studies
Experimental studies on the stability of glass columns or beams have been developed almost exclusively
with single layer or with two-layered laminated specimens.
Luible and Crisinel [6, 29] developed extensive parametric experimental and numerical studies in Ecole
Polythecnique Federale de Lausanne (EPFL). The main objectives were to investigate the load carrying
behavior of monolithic and laminated glass columns and beams, and to evaluate possible design meth-
ods regarding the instability problems of these elements. All glass elements had 200 mm of width and
8 mm or 10 mm of thickness, while the spans ranged from 350 mm to 1600 mm. The laminated elements
had only two layers and were all bonded with PVB. The test of laminated columns is illustrated in Fig.
2.6(a). A mid-span load was applied in the case of lateral-torsional buckling tests.
In order to support the study, the authors developed numerical finite elements models, where the glass
layers were modeled with shell elements and the interlayer was modeled with solid elements. In some
models a linear elastic behavior was adopted for the interlayer, while for others a visco-elastic behavior
has been simulated. It was concluded that numerical non-linear simulations were accurate for the post-
buckling analysis of glass columns and beams. In addition, the authors concluded that, for design
purposes, the visco-elastic behavior of the interlayer can be simplified by elastic models.
Furthermore, buckling curves developed for glass design should be based on the tensile strength of the
edges. Measurements on more than 200 glass elements indicated that the actual thickness is often
lower than the nominal value (97.6% of the nominal thickness associated to the 5% percentile, with a
normal distribution) and that the initial deformation is very small for annealed glass (less than L/2500),
but may be up to L/300 for thermally toughened glass, presenting a sinusoidal shape.
Another relevant experimental study was carried out by Belis et al. [30], regarding only the lateral-
torsional buckling of laminated glass beams. The experimental campaign was based on a total of 55
two-layered beams, divided in 12 series, depending on the material and geometrical properties. Beams
made of annealed, heat strengthened or thermally toughened glass, bonded with PVB or SG, all with
3000 mm of length, length/width ratios from 10 to 25, and glass thickness of 6 mm, 8 mm or 10 mm were
tested in a 2900 mm span. The test of one of the beams is illustrated in Fig. 2.6(b).
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(a) Buckling of laminated glass columns with two lay-ers [29].
(b) Lateral-torsional buckling of laminated glass beams with twolayers [30].
Figure 2.6: Examples of experimental studies of the buckling behavior of structural glasscolumns and beams.
In order to analytically compare the load-displacement response of the buckled beams, their initial im-
perfections were carefully measured. A plot of the initial lateral deflection (u0) along the length (z) of the
beams, is illustrated if Fig. 2.8(b). A generally approximate sinusoidal shape is identifiable. Calibrated
values of the initial rotations were also provided for each specimen.
0
1
2
3
4
5
6
7
8
0 500 1000 1500 2000 2500 3000
z [mm]
u0(z
) [m
m]
Figure 2.7: Example of initial lateral deflection shapes of laminated beams [30].
The authors concluded that the load-displacement path of beams susceptible to lateral-torsional buckling
can be predicted with good accuracy using the expressions proposed by Kasper et al. [4], together with
equivalent flexural and torsional stiffness formulations available in the literature.2 The initial rotation,
even if small in amplitude, should not be neglected as it may contribute to premature failure. It was
also concluded, by comparison with the adopted analytical formulations, that most test results were
within analytical predictions, considering glass thickness tolerances given by available standards. An
example of those results is represented in Fig. 2.8(a). In the scope of the present dissertation, an
unsuccessful attempt to reach the same analytical results was made, considering the geometrical and
interlayer properties provided. Smaller values of deflection were obtained for the same test specimen,2For the analytical determination of the critical buckling load, the authors preferred the expression provided in Eurocode 3 [31],
instead of the one proposed by Kasper et al. [4]. However, given the coefficients from one and another, these two formulationsdo not seem compatible. Additionally, although with small impact on the results, for the coefficient c2 they presented the value0.8693, instead of 0.8106 as given in [4].
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as illustrated in Fig. 2.8(b). The apparent fault mentioned in footnote 2 does not justify the differences
in the results. It has not been possible to determine what motivates the differences between the results
presented by the authors and those estimated in this dissertation.
0 40 80 120 160 200
Maximum displacement u [mm]
0.0
1.5
3.0
4.5
6.0
Loa
dF
[kN
]
TestTol - (Eq.(3))Nominal (Eq.(3))Tol + (Eq.(3))
Experimental Fcr
Test G03Avg. Series G
(a) Experimental and analytical results from Beliset al. - Test G03 [30].
0.0
1.5
3.0
4.5
6.0
7.5
0 40 80 120 160 200
LoadF[kN]
Maximum displacement u [mm]
(b) Attempt to reproduce the analytical resultsfrom Belis et al. - Test G03 [30].
Figure 2.8: Load-displacement path of laminated glass beam.
This experimental study was later extended by Bedon et al. [2], who tested a total of 107 specimens.
The authors used the results to assess some existing analytical formulations. In terms of equivalent
flexural stiffness, three analytical formulations were considered. The first formulation was derived for
two-layered glass beams by Luible [5], based on the previous work of Stamm and Witte [32]; the authors
correctly acknowledge that the expression is only valid for simply supported beams with a mid-span load,
which is considered to be appropriate since all beams tested were subjected to this loading condition.
The other two analytical formulations that are presented are the previously mentioned “Wolfel-Bennison
approach” [24] and the one derived by Amadio and Bedon [1, 22], based on the theory proposed by
Newmark et al. [23]. No reference is made on what loading conditions the presented expressions apply
to.3 The authors concluded that the two last formulations yield the exact same results, which differ from
those resulting from the formulation derived by Luible [5]. In terms of torsional stiffness, two analytical
formulations are presented. The first formulation was derived for two-layered glass beams by Luible [5],
again based of the work of Stamm and Witte [32]. The other formulation was proposed by Scarpino [33]
and, according to the authors, is also valid for two-layered laminated glass beams.4 The authors claim
having obtained better results with the former formulation, specially in the case of stiff interlayers.
In order to compare the experimental results with the various analytical formulations, the Southwell plot
was used to determine the experimental critical buckling loads and an equation from Eurocode 3 [31] was3It may be verified from [25] that the “Wolfel-Bennison approach” is presented in the form applicable to columns subjected to
axial compression, and from [1, 22] that the theory proposed by Newmark et al. [23] was also derived for glass columns.4By comparing the provided expression with those reproduced by Belis [15], it is clear that there is an almost exact match (only
one parameter is different) with the expression valid for beams with three glass layers of equal thickness. There might have beena mistake in the transcription of this formulation.
14
used to determine the analytical ones. The load-displacement path was defined using the expression
proposed by Kasper et al. [4] (see footnote 2).
The authors further concluded that available analytical approaches usually overestimate results given
by numerical simulations. They stressed the importance of initial geometrical imperfections and discrep-
ancies in glass thickness on the determination of critical buckling loads.
Recently, in Instituto Superior Tecnico (IST), a long-span beam with 8200 mm of length (L) was experi-
mentally studied regarding its susceptibility to lateral-torsional buckling (Fig. 2.9). The beam was made
of three glass layers, whose width (b) and thickness (t) were 600 mm and 15 mm, respectively. They
were laminated with PVB. The results of this test are discussed in detail in [7] and briefly reviewed in
Section 6.4 of the present dissertation.
The authors concluded that, with a slenderness ratio L/b equal to 13.8, the beam was, in fact, sus-
ceptible to the lateral-torsional buckling phenomenon. Through an analytical assessment, supported by
measurements, the authors determined that the actual thickness of the glass layers was probably lower
than the nominal values, with a significant impact on the analytical results. They concluded that a correct
dimensional assessment of laminated glass elements is important in terms of structural safety.
(a) General view. (b) Buckled situation.
Figure 2.9: Experimental study of the lateral-torsional buckling behavior of a long-span three-layered beam.
2.4.3 Design guidelines and regulation
Presently there is no Eurocode for glass structures, in contrast, for example, with concrete, steel or
timber structures. There is, however, some dispersed regulation on various topics concerning glass
structures, some published independently by individual countries and others published on European
level.
Multiple guidelines and books have also been published, including the recent update of “Structural use of
glass in buildings” [10], from The Institute of Structural Engineers, which intends to be a guide for struc-
tural engineers on the design of glass structures. One of the most prominent guidelines is “Guidance
15
for European Structural Design of Glass Components – Support to the implementation, harmonization
and further development of the Eurocodes” [34], published in 2014 by the Joint Research Center of the
European Commission, which sets the ground for a future Eurocode on glass structures. In terms of reg-
ulation broad in scope, which include considerations regarding the stability of structural glass elements,
there is, for instance, the Italian Standard CNR-DT 210/2013 [35] and the Australian Standard AS 1288
[36].
In what concerns the subject of this dissertation, each guideline or regulation adopts and presents a
given calculation approach for the flexural and torsional stiffness of laminated glass beams and for the
stability problem. The expressions differ between guidelines and between regulations as they differ
between scientific papers or theses. In fact, it was the contribution of those papers and theses that was
included in the guidelines and regulation.
The apparent divergence on adopted approaches results from the fact that the behavior of laminated
glass is not yet fully understood. Each new paper adds some knowledge and understanding to the
subject, and the final outcome will eventually be a unified and widely accepted approach that can be
part of the future Eurocode for the design of glass structures.
2.5 Concluding remarks
All papers mentioned in this Chapter are unanimous in one point: they all agree that further research
must be made in order to better understand the behavior of structural laminated glass, specially in what
regards lateral-torsional buckling. Most of these studies also have in common the fact that they are
limited to two-layered beams or columns, when in practical applications the use of elements with at least
three layers is increasingly common. Moreover, as previously discussed, beams and columns laminated
with five layers have already been used multiple times.
Since the previous analytical studies only present expressions for laminated glass elements up to three
layers, and even those formulations are not fully investigated, it was deemed necessary, as part of this
dissertation, to perform a broad evaluation that could assess the validity and accuracy of the existing for-
mulations, in order to support the extension of their scope. In particular, it would be important to develop
analytical formulations for the equivalent flexural and torsional stiffness of laminated glass elements with
more than three layers, for multiple loading conditions, and to clarify their range of accuracy.
Regarding the structural stability of laminated glass elements, and specially in the case of lateral-
torsional buckling, it has been highlighted that the use of analytical expressions is not yet fully con-
solidated. Lateral-torsional buckling is, in fact, a complicated phenomenon that has been described by
more than one formulation, without unequivocal results. The present dissertation aims to further investi-
gate this phenomenon, both analytically and numerically, and both in terms of critical buckling loads and
load-displacements responses. The results of a relevant experimental study conducted in IST [7] are
used as a comparative reference.
16
Chapter 3
Numerical assessment of available
expressions
3.1 Preliminary remarks
As already mentioned in Chapter 2, in the absence of widely accepted design standards, numerous
analytical formulations have been proposed for the analysis of laminated glass elements. In this Chap-
ter, some of those formulations, and others from sandwich theory and from the composite laminates
literature, are compared.
The comparison is done for three different degrees of interaction between the glass layers and consid-
ers as reference the results retrieved from numerical models implemented in the commercial software
Abaqus [37]. The degrees of interaction are defined by the shear modulus of the interlayer. For each
set of expressions one model reproduces SG properties associated with a temperature of 20 ◦C and a
load duration of 1 second (labeled “SG”), another one reproduces PVB, for the same conditions (labeled
“PVB”), and the last model simulates a very soft PVB, associated with a temperature of 20 ◦C and a
load duration of more than 10 years (labeled “PVB10y”). The properties of the interlayers are taken from
Table A.1, from Appendix A.
It is worth mentioning that the notation of all expressions used is adapted to that adopted in this disser-
tation.
Section 3.2 presents the results of a survey of the literature regarding the available analytical formula-
tions for the determination of the equivalent flexural and torsional stiffness of laminated glass elements,
their critical buckling moments and their load-displacement paths. In Section 3.3, the design of the
numerical finite elements models that support the analytical assessment is described in detail. The
comparison and assessment of the expressions introduced in Section 3.2 is carried out in Section 3.4.
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3.2 Survey of available expressions
3.2.1 Flexural stiffness
The expressions for the determination of the equivalent flexural stiffness (EIf ) may be divided in two
categories. On the one hand, there are those that are applicable in the determination of the critical
buckling load of laminated glass columns, together with Euler’s equation. On the other hand, maximum
deflections of laminated glass beams subjected to transverse loads or pure bending may be described
with usual monolithic beam expressions, with corresponding equivalent flexural stiffness.
Since, presently, equivalent flexural stiffness expressions for transverse loading conditions are not well
consolidated in the literature, only simply supported beams subjected to mid-span loads will be analysed
in this Chapter. Although some formulations have already been extended to laminated glass columns
or beams with three layers, others have not. Because this Chapter only intends to perform a prelim-
inary evaluation of the validity of those formulations, only their two-layer forms are considered. The
geometrical parameters of the laminated cross-section are illustrated in Fig. 3.1, where z1 and z2 are
the distances between the center-lines of the glass layers and the section’s neutral axis.
Figure 3.1: Cross-section of a two-layered laminated glass column/beam.
Five formulations for the equivalent flexural stiffness of laminated glass columns subjected to compres-
sive axial loads (Fig. 3.2) are gathered in Table 3.1. The first three formulations have already been used
in the structural glass literature, while the other two come from sandwich theory.
Figure 3.2: Column subjected to a compressive axial load.
Formulation 1-A has been derived by Luible [5], based on the work of Stamm and Witte [32]. The
correct expression for the coefficient β may be found in [6, 10, 29].1 Formulation 2-A was first applied in
structural glass research by Amadio and Bedon [22] and is based on the theory proposed by Newmark et
al. [23]. Formulation 3-A comes from the “Wolfel-Bennison approach”; Bedon and Amadio [25] applied
it with the correct parameter β for laminated glass columns. Formulation 4-A was deduced from the
critical buckling load expression proposed by Allen [38] for sandwich struts. Similarly, formulation 5-A is
described in Zenkert’s sandwich handbook [39].1In [5] there is a square missing.
Formulation 1-B has been derived by Luible [5], based on the work of Stamm and Witte [32]. The
expressions for α, β, I1, I2 and Is have already been reproduced in Formulation 1-A from Table 3.1. For-
mulation 2-B was proposed by Kasper et al. [4] and has been adopted in the recent guideline “Guidance
for European Structural Design of Glass Components – Support to the implementation, harmonization
20
and further development of the Eurocodes” [34]. In both documents it is not clear how the parameter
B22 should be determined. Thus, it was assumed (and stated in Table 3.2) that B22 is, in fact, B22, for
which the authors actually provide an expression. Formulation 3-B was derived by Galuppi and Royer-
Carfagni [26] and may be found in [24]. It corresponds to the “Enhanced effective thickness approach”
and, according to the authors, it should provide better results for this load case than the “Wolfel-Bennison
approach” [24].
3.2.2 Torsional stiffness
Five formulations for the equivalent torsional stiffness (GJt) of laminated glass beams with two layers
are gathered in Table 3.3. Only the first three formulations have been highlighted in the structural glass
literature.
Formulation 1-C has been derived by Luible [5], again based on the work of Stamm and Witte [32]. It
is clearly the most unanimous formulation in structural glass scientific articles. Formulation 2-C was
proposed by Kasper et al. [4] and, similarly to Formulation 2-B from Table 3.2, has been adopted in
[34]. Formulation 3-C was proposed by Scarpino [33] and may be found in [15] (see footnote 4 from
Chapter 2). According to Bedon et al. [2], it should provide slightly worse results than Formulation 1-C,
specially in the case of stiff interlayers. Formulation 4-C can be found in Zenkert’s sandwich handbook
[39]. Formulation 5-C has been adapted to glass notation, for two-layered elements, from composite
laminates equations presented by Whitney and Kurtz [41]. It intends to assess how well torsion under
the Classical Laminate Theory (CLT) compares with torsion in laminated glass elements.
3.2.3 Lateral-torsional buckling
Three different formulations have already been proposed to determine the critical buckling load of lam-
inated glass beams. They all consist in the analysis of equivalent monolithic beams, with equivalent
flexural and torsional stiffness. They are gathered in Table 3.4 with the corresponding coefficients for
simply supported glass beams subjected to a mid-span load.
The first expression (1-D) has been used more often in scientific papers on structural glass stability
and comes from an early version of Eurocode 3 [31]. It may be found, for example in [2]. The second
expression (2-D), proposed by Kasper et al. [4] and adopted in [34], has a very similar form than the first,
only with different coefficients. The third expression (3-D) applies to glass beams without intermediate
buckling restraints, according to Appendix C from the Australian Standard AS 1288 — corrected in its
first amendment in 2008 [36].
Kasper et al. [4] also proposed expressions that allow to determine the load-displacement path of
laminated glass beams subjected to pure bending, a mid-span load or a uniformly distributed load. In
21
Table 3.3: Equivalent torsional stiffness (2 layers; five formulations)
1-C
GJt = G (J1 + J2 + Jcomp)
Ji =bt3i3, (i = 1, 2) Jcomp = Js
(1− 2
λbtanh
λb
2
)Js = 4
(t1 + t2
2+ tint
)2t1t2t1 + t2
b λ =
√GintG· t1 + t2tintt1t2
2-C
GJt =G
1S11
+S212
S211· 1T22
(t = t1 = t2)
T22 = −S212
S11+ S22 +
GintG
Sϕ,22 S11 =8
3b
[−(tint2
)3
+
(t+
tint2
)3]
S22 =bt
2S12 = b
[(tint2
)2
−(t+
tint2
)2]
Sϕ,22 =b3
12tint
3-CGJt = GJabsfb
GJabs = G
(bt313
+bt323
)fb =
6t31tint + Gint
G b2(4t21 + 6t1tint + 3t2int
)t21(6t1tint + Gint
G b2) ≥ 1
4-C GJt =8
3
[(1 +
tint2t
)3
+
(GintG− 1
)(tint2t
)3]Gbt3
5-C
GJt = 4bD66
[1− h
ϕbtanh
(ϕb)]
(t = t1 = t2) , (z = z1 = z2)
ϕ =h
2
√k1A55
D66k1 =
5
6h = 2t+ tint b =
b
h
D66 = D66 = 2G
(tz2 +
t3
12
)+Gint
t3int12
A55 =5
4
(2G
[t− 4
h2
(tz2 +
t3
12
)]+Gint
[tint −
4
h2t3int12
])
the case of the mid-span load the expression is,
vmax =c1M
GJtEIf
φ0,max + (c1M)2
EIfv0,max
GJt(πL
)2 − (c1M)2
EIf+ c2Mzg
(πL
)2 (3.2)
where L is the span, M is the bending moment associated with the applied load, zg is the distance
between the point where the load is applied and the shear center of the cross-section, v0,max is the initial
deflection at mid-span, φ0,max is the initial twist in the same cross-section, EIf is the equivalent flexural
22
Table 3.4: Critical buckling moment (three formulations) – beam subjected to a mid-span load
1-DMcr = C1
π2EIfL2
(C2zg +
√(C2zg)
2+GJtL2
π2EIf
)C1 = 1.365 C2 = 0.553
2-DMcr =
1
c1
π2EIfL2
1
2
c2c1zg +
√(1
2
c2c1zg
)2
+GJtEIf
(L
π
)2
c1 =1
2+
2
π2= 0.7026 c2 =
8
π2= 0.8106
3-DMcr =
g2L
√EIfGJt
(1 + g3
zgL
√EIfGJt
)g2 = 4.2 g3 = 1.7
stiffness, GJt is the equivalent torsional stiffness, and the coefficients c1 and c2 equal those provided in
formulation 2-D from Table 3.4.
3.3 Definition of the numerical models
The numerical models developed in this work were implemented with the commercial software Abaqus
[37]. They were designed to represent, as close as possible, the ideal analytical conditions considered,
in order to faithfully assess the validity and accuracy of the previously presented expressions.
The glass layers were modeled with eight-node continuum shell elements with reduced integration
(SC8R), while eight-node solid elements with reduced integration (C3D8R) were used to simulate the
interlayer. The approximate size of the elements in the plane of the column/beam was chosen to be
10 mm×10 mm, with one element in the thickness of the glass layers and two elements in the thickness
of the interlayer (Fig. 3.5(c)).
The columns/beams were modeled as a single part, with the separation between materials being ac-
complished by partitioning that part.2 Additional partitions have been introduced in order to apply the
boundary conditions and the loads.
A model was created for each single case corresponding to Tables 3.1 to 3.4, namely: (i) a column
subjected to compressive axial load (model A); (ii) a beam subjected to an out-of-plane mid-span load
2As an alternative, each glass layer and the interlayer could have been modeled as individual parts, and then bonded with a tieconstraint, which would yield the exact same results.
23
(model B); (iii) a beam subjected to a torsional moment (model C); and (iv) a beam subjected to an
in-plane transverse mid-span load (model D); each one with the corresponding boundary conditions and
applied loads. The first two models (A and B) are very similar, with the same boundary conditions. They
are illustrated in Figs. 3.4(a) and 3.4(b).
The longitudinal and transverse partitions form, in all their area, the boundary conditions which prevent
displacements in the xx and zz directions, respectively. The boundary condition that prevents displace-
ments in the yy direction would ideally be placed in the horizontal axis of symmetry of the two extremity
cross-sections. However, because the interlayer is not stiff enough to accommodate nor boundary con-
ditions nor applied loads, the boundary condition in the yy direction was applied in the edges of the
lower glass layer (line highlighted in Fig. 3.4(c)) — located at half of the thickness of the interlayer from
the ideal location, which is a sufficiently good approximation.
(a) Column subjected to a compressive axial load. (b) Beam subjected to a transverse mid-span load.
(c) Boundary condition in the yy direction and axial pressure load.
Figure 3.4: Numerical models of the laminated glass column (model A) and of the beam subjected toan out-of-plane mid-span load (model B).
In the case of the column (model A), a pressure load was applied in both ends of the two glass layers,
as illustrated in Figs. 3.4(a) and 3.4(c). In the case of the beam (model B), similarly to the boundary
condition in the yy direction, the load was applied in the lower glass layer, but now at mid-span, along
the width (Fig. 3.4(b)).
The model of the beam subjected to a torsional moment (C) is illustrated in Fig. 3.5(a). The only
boundary condition is located at mid-span, in the axes of symmetry of the cross-section (Fig. 3.5(b)).
24
All displacements were prevented. The reason why the boundary condition was applied in the axes of
symmetry was to allow warping associated with torsion (free warping condition).
As illustrated in Fig. 3.5(a), opposing torsional moments were applied in both ends of the beam. Their
distribution was defined by a linear function (Fig. 3.5(c)). As in the two previous models (A and B), the
loads could not be applied in the interlayer. They were, instead, applied in the edge of the glass layers,
which is again a sufficiently good approximation.
(a) Beam subjected to torsional moment.
(b) Boundary condition.
(c) Application of the load.
Figure 3.5: Numerical model of the laminated glass beam subjected to a torsional moment (model C).
The model developed in order to determine the critical buckling load of the beam subjected to an in-plane
transverse mid-span load (D) is illustrated in Fig. 3.6(a). The boundary condition in the zz direction is
similar to that of the column and of the beam subjected to a transverse mid-span load, and was also
applied at mid-span. The displacements in the yy direction were prevented in the horizontal axis of
symmetry of the two extremity cross-sections, as Fig. 3.6(b) illustrates. The boundary condition in
the xx direction would have ideally be placed in the vertical axis of symmetry of the two extremity
25
cross-sections, and it should remain vertical (passing in the centroid of each section) as the beam
deflects. However, it has been decided to apply the restriction in the symmetrical edges of the glass
layer illustrated in Fig. 3.6(c), which is a sufficiently good approximation provided that the rotation is
small.
The load was applied above the shear center of the mid-span cross-section, which corresponds to a
negative value of zg. Because the load cannot be applied in the interlayer, as explained before, it has
been split between the two glass layers (Fig. 3.6(d)). Note that zg equals, in modulus, half the width of
the beam.
(a) Beam subjected to a mid-span load. (b) Boundary condition in the yy direction.
(c) Boundary condition in the xx direction. (d) Application of the load.
Figure 3.6: Numerical model of the laminated glass beam subjected to an in-plane transverse mid-span load (model D).
All models have the same geometry. The two glass layers are 10 mm thick (t), whereas the thickness
of the interlayer (tint) was chosen to be 1.52 mm. The width and length of the models were determined
from a ratio width/total thickness of 10 and a ratio length/width of 15. The resulting width (b) is 215 mm
26
and the resulting length (L) is 3228 mm.
The materials of both glass and interlayer were considered to be linear elastic. The Young’s modulus of
glass is 70 GPa and its Poisson ratio is 0.23. The corresponding values for the interlayer depend on the
type of material considered in the model, as stated in the beginning of this Chapter. The interlayer in the
models with SG has a shear modulus of 142 MPa and a Poisson ratio of 0.465, which results in a Young’s
modulus of 416.1 MPa. The PVB associated with the same conditions of temperature and load duration
has a shear modulus of 13.7 MPa and a Poisson ratio of 0.497, which results in a Young’s modulus of
41.0 MPa. The interlayer in the models with very soft PVB has a shear modulus of 0.0517 MPa and a
Poisson ratio of 0.500, which results in a Young’s modulus of 0.1551 MPa.
In order to evaluate the critical buckling loads of the column subjected to axial compression (model A)
and of the beam subjected to an in-plane transverse mid-span load (model D), linear buckling analy-
ses were performed on these numerical models. The applied load was unitary, which means the first
eigenvalue equals the desired critical buckling load.
The respective load-displacement paths were obtained by performing geometrically non-linear static
analyses, with an initial deflection of 5 mm for both loading conditions (models A and D) and an initial
twist of 0.001 rad for the beam subjected to a mid-span load (model D), both measured at mid-span.
The initial imperfection of the lateral-torsional buckling model (D) was defined through an independent
calibration of the bending and torsional deformations.
Linear static analyses were performed on the other two models (B and C). The load applied in the model
of the beam subjected to a out-of-plane mid-span load (B) corresponds to 1 kN and the displacement
in the yy direction was measured in the centroid of the mid-span cross-section. In the case of the
model subjected to a torsional moment (C), the applied load corresponds to 50 N m. The displacements
in the yy direction were measured at quarters of the span, in opposite points of the horizontal axis of
symmetry of the cross-section (Fig. 3.7). The rotations were then linearly determined, with the total
rotation, in a length equal to half of the span, corresponding to the sum of the two determined rotations.
The displacements were measured at quarters of the span and not at the extremities of the beam in
order to avoid local effects due to Saint-Venant’s principle. As Fig. 3.7 illustrates, the stress distribution
becomes uniform at a certain distance from the extremities of the beam, where the loads are applied.
3.4 Numerical assessment of analytical expressions
3.4.1 Flexural stiffness
The analytical critical load of the columns subjected to axial compressive load was determined, for each
of the formulations in Table 3.1, with Euler’s equation:
Pcr =π2EIfL2
(3.3)
27
Figure 3.7: von Mises stress distribution in a laminated glass beam subjected to a torsional moment(model C).
As mentioned before, the numerical critical buckling load equals the first eigenvalue. The numerical
and analytical results, for all five formulations, are summarized in Table 3.5. It is clear that the first four
formulations are in fact equivalent and have an almost exact agreement with the numerical results, for
all three degrees of interaction between the glass layers. Formulation 5-A yields worse results, specially
for stiffer interlayers (“SG”).
Table 3.5: Numerical comparison between expressions for the equivalent flexural stiffness of laminatedglass columns subjected to axial compressive load (relative percentage differences in parentheses).
Type of analysisPcr (N)
SG PVB PVB10y
Numerical 11 803 11 500 3257
Analytical
1-A 11 801 (−0.01%) 11 499 (−0.01%) 3256 (−0.02%)
2-A 11 801 (−0.01%) 11 499 (−0.01%) 3256 (−0.02%)
3-A 11 801 (−0.01%) 11 499 (−0.01%) 3256 (−0.02%)
4-A 11 801 (−0.01%) 11 499 (−0.01%) 3256 (−0.02%)
5-A 9434 (−20.07%) 9208 (−19.93%) 3035 (−6.81%)
A comparison between the load-displacement paths of the three numerical models and those which
result from Eq. (3.1), with the equivalent flexural stiffness given by one of the first four formulations from
Table 3.5, is illustrated in Fig. 3.8. The prediction provided by Eq. (3.1) is accurate for all three degrees
of interaction between the glass layers.
In the case of the beam subjected to a transverse mid-span load F of 1 kN, the analytical displacement
at mid-span can be determined, for each of the formulations in Table 3.1, with the well-known beam
theory equation:
wmax =FL3
48EIf(3.4)
The numerical displacement is measured directly in the static linear analysis. The numerical and ana-
lytical results, for all three formulations, are summarized in Table 3.6. Formulations 1-B and 3-B have an
almost exact agreement with the numerical results, for all three degrees of interaction between the glass
layers (Formulation 3-B is marginally less accurate for the intermediate value of the shear modulus in
the interlayer – “PVB”). Formulation 2-B fails to predict the displacement in beams laminated with softer
interlayers (“PVB10y”).
Table 3.6: Numerical comparison between expressions for the equivalent flexural stiffness of laminatedglass beam subjected to a transverse mid-span load (relative percentage differences in parentheses).
Type of analysiswmax (mm)
SG PVB PVB10y
Numerical 56.25 57.95 204.63
Analytical
1-B 56.27 (+0.05%) 57.98 (+0.04%) 204.79 (+0.08%)
2-B 56.27 (+0.03%) 57.72 (−0.39%) 75.56 (−63.08%)
3-B 56.27 (−0.01%) 57.72 (−0.37%) 204.50 (−0.06%)
3.4.2 Torsional stiffness
In the case of the beam subjected to a torsional moment T of 50 N m, the analytical rotation can be
determined, for each of the formulations in Table 3.3, with the following equation:
φ =T
GJt· L
2(3.5)
The length along which the rate of twist (φ′) was integrated was L/2 because, as previously explained,
the displacements in the numerical models were measured at quarters of the span. The numerical and
analytical rotations, for all five formulations, are summarized in Table 3.7. There is a large scatter on
the analytical results, with all formulations yielding greater values of the equivalent torsional stiffness
than the ones that could be associated with the numerical models. Formulation 1-C is by far the most
accurate, with all other formulations failing considerably in light of numerical results.
29
Table 3.7: Numerical comparison between expressions for the equivalent torsional stiffness of lami-nated glass beams (relative percentage differences in parentheses).
The three formulations presented in Table 3.4 were derived for monolithic beams and thus, in order
to be applied to laminated glass beams, they depend on equivalent flexural and torsional stiffness for-
mulations. Among the previously assessed formulations for equivalent flexural and torsional stiffness,
Formulation 1-B from Table 3.2 and Formulation 1-C from Table 3.3 are the most suitable to be used
here, as shown in the two preceding Subsections.
As mentioned before, the numerical critical buckling load equals the first eigenvalue. The numerical and
analytical results, for all three formulations, are summarized in Table 3.8. It can be seen that formulation
3-D yields better results, by comparison with numerical ones, followed by formulation 1-D. The differ-
ences are relatively consistent for all three degrees of interaction between the glass layers. Although
formulation 2-D produces farther results from numerical models, it is still able to predict the critical buck-
ling load with less than 10% of difference. It must be acknowledged that the adopted formulation for the
equivalent torsional stiffness overestimates the stiffness, according to Table 3.7. This may explain part
of the differences illustrated in Table 3.8.
Table 3.8: Numerical comparison between expressions for the critical buckling moment of laminatedglass beams subjected to a mid-span load (relative percentage differences in parentheses).
Type of analysisFcr (N)
SG PVB PVB10y
Numerical 20 162 13 564 5599
Analytical
1-D 20 943 (+3.87%) 14 191 (+4.62%) 5844 (+4.37%)
2-D 21 785 (+8.05%) 14 748 (+8.73%) 6079 (+8.57%)
3-D 20 505 (+1.70%) 13 881 (+2.33%) 5722 (+2.19%)
Fig. 3.9 depicts a comparison between the load-displacement paths of the three numerical models and
those which result from Eq. (3.2), with the equivalent flexural stiffness given by formulation 1-B from
Table 3.6 and the equivalent torsional stiffness given by formulation 1-C from Table 3.7. The analytical
critical buckling loads there illustrated correspond to formulation 2-D from Table 3.8, because that is
the one which is compatible with Eq. (3.2). The difference between analytical and numerical results is
Figure 4.4: Shear deformation of a structural element (adapted from [39]).
A segment of a two-layered laminated glass beam in pure shear deformation, subjected to a general
symmetrical loading, is represented in Fig. 4.5. The geometry demonstrates that,
w′2 = γinttintd
(4.13a)
from where results:
τint = γintGint =d
tintGintw
′2 (4.13b)
37
Figure 4.5: Shear deformation of a laminated glass beam segment with two layers (adapted from [38]).
The substitution of the interlayer shear stress from Eq. (4.13b) into Eq. (4.11) leads to,
−V1 = −AGintw′2 + EIglw′′′1 (4.14a)
where:
A =bd2
tint(4.14b)
After substitution of V1 = −EIw′′′1 into Eq. (4.14a), the slope w′2 can be written as,
w′2 =V1
AGint
(1− EIgl
EI
)(4.15)
which through double differentiation can be replaced in Eq. (4.12) to yield a differential equation for V1,
V ′′1 − a2V1 = −a2V (4.16a)
where:
a2 =AGint
EIgl
(1− EIgl
EI
) (4.16b)
The quantity a2 represents the ratio between the shear stiffness of the interlayer and the local bending
stiffness of the glass layers [38]; it can be rewritten using some parameters common in the structural
glass literature and present in [32],
a2 =
(λfL
)2
=1 + α
αβL2(4.17a)
where:
α =IglIs
(4.17b)
β =EIs
AGintL2(4.17c)
38
The effect the glass layers’ stiffness have on the shear deformation of the interlayer is smaller for larger
values of a2 and for larger spans [38].
Alternatively, Eq. (4.15) may be presented in a more compact form:
EIglw′2 =
V1a2
(4.18)
Together, for a given distribution of the total shear force V , Eqs. (4.16a) and (4.18) may be integrated to
yield an expression representative of the total deflection w = w1 + w2.
4.2.2 Laminated glass beams with more than two layers
Although most of what is described in Subsection 4.2.1 remains valid with a number of layers different
than two, some changes must be made in order to adapt the expressions for other cross-sections.
This study limits the analysis to laminated glass beams up to five layers. Furthermore, for simplicity of
the expressions, and because that is commonly the case in practical applications, only bi-symmetrical
cross-sections are considered.
Three layers
Considering now the cross-section illustrated in Fig. 4.6 — where the thickness of the outer glass layers
is t1, the thickness of the inner glass layer is t2, the thickness of the interlayer is tint and the distance
between the center-lines of the outer glass layers is d —, its flexural stiffness along the weak axis is,
instead of (Eqs. (4.3)),
EI = EIgl + EIs (4.19a)
where:
EIgl =Eb
12
(2t31 + t32
)(4.19b)
EIs = Ebt1d
2
2(4.19c)
Figure 4.6: Cross-section of a three-layered laminated glass beam.
39
Without the contribution of the interlayer, for the three-layered beam, Eqs. (4.5) and (4.8) are now,
∑(SE)int =
Ebt1d
2(4.20a)
τint =V
EI
Et1d
2(4.20b)
and, without contribution of the flexural stiffness of the glass layers with respect to their own axes (EI =
EIs), Eq. (4.10b) remains unchanged.
The shear deformation is illustrated with the segment of Fig. 4.7. The geometry now allows demonstrat-
ing that,
w′2 = γint2tintd
(4.21)
which changes Eqs. (4.13b) and (4.14b) to,
τint = γintGint =d
2tintGintw
′2 (4.22a)
and:
A =bd2
2tint(4.22b)
Figure 4.7: Shear deformation of a laminated glass beam segment with three layers.
The remainder of Subsection 4.2.1 remains valid. Summarizing, the new cross-section is associated
with new Is and A , whereas Igl may be generally defined as,
Igl =n∑i=1
bt3i12
(4.23)
where the index i comprises the number of glass layers (n).
40
Four layers
Considering the cross-section illustrated in Fig. 4.8 — where the thickness of the outer glass layers is t1,
the thickness of the inner glass layers is t2, the thickness of the interlayer is tint, the distances between
the center-lines of adjacent glass layers are a1 and a2 and the distance between the center-lines of the
outer glass layers is d —, the flexural stiffness of the section along the weak axis is,
EI = EIgl + EIs (4.24a)
where:
EIgl =∑i
Ebt3i12
=Eb
12
(2t31 + 2t32
)(4.24b)
EIs = Eb
(t1d
2
2+t2a
22
2
)(4.24c)
Figure 4.8: Cross-section of a four-layered laminated glass beam.
In contrast with previous cross-sections, a bi-symmetrical cross-section with at least four layers has the
particularity of not having only one value for the interlayer’s shear stress. The shear stress distribution
across the thickness of the section is illustrated in Fig. 4.9(a).
(a) True distribution. (b) Distribution neglecting thecontribution of the interlayer.
(c) Distribution neglecting the bendingstiffness of the glass layers.
Figure 4.9: Shear stress distribution, four glass layers.
41
Still regarding Eq. (4.4), without the contribution of the interlayer, for the four-layered beam, the shear
stress in the different levels of interlayer comes,
∑(SE)int,1 =
Ebt1d
2(4.25a)∑
(SE)int,2 =∑
(SE)int,1 +Ebt2a2
2(4.25b)
τint,1 =V
EI· Et1d
2(4.25c)
τint,2 = τint,1 +V
EI· Et2a2
2(4.25d)
and without the contribution of the flexural stiffness of the glass layers with respect to their own axes
(EI = EIs):
τint,1 =V
b· t1d
t1d2 + t2a22(4.26a)
τint,2 = τint,1 +V
b· t2a2t1d2 + t2a22
(4.26b)
The shear strain in the central interlayer can then be written as:
γint,2 =τint,2Gint
= γint,1 +V
bGint· t2a2t1d2 + t2a22
(4.27)
The shear deformation is illustrated with the segment of Fig. 4.10. The geometry allows demonstrating
that,
w′2 =γint,1 · 2tint + γint,2 · tint
d(4.28)
which re-arranged and substituting Eq. (4.27) results in:
γint,1 =w′2d− γint,2tint
2tin=
1
3
(d
tintw′2 −
V
bGint· t2a2t1d2 + t2a22
)(4.29)
This leads to a new expression for the shear stress in the outer interlayer (τint,1 = γint,1Gint), which
can be replaced in Eq. (4.26a). After some mathematical manipulations, the expression yields the
component of the shear force that corresponds to the first member of Eq. (4.14a), with:
A =bd
tint· t1d
2 + t2a22
3t1d+ t2a2(4.30)
Five layers
Considering the cross-section illustrated in Fig. 4.11, its flexural stiffness along the weak axis is,
EI = EIgl + EIs (4.31a)
42
Figure 4.10: Shear deformation of a laminated glass beam segment with four layers.
where:
EIgl =∑i
Ebt3i12
=Eb
12
(2t31 + 2t32 + t33
)(4.31b)
EIs = Eb
(t1d
2
2+ 2t2a
22
)(4.31c)
Figure 4.11: Cross-section of a five-layered laminated glass beam.
The procedure for determination of parameter A is similar to that followed for the cross-section with
four layers. The segment in pure shear deformation is now illustrated in Fig. 4.12 and the expressions
equivalent to Eqs. (4.25)–(4.30) are, in order:
∑(SE)int,1 =
Ebt1d
2(4.32a)∑
(SE)int,2 =∑
(SE)int,1 + Ebt2a2 (4.32b)
τint,1 =V
EI· Et1d
2(4.32c)
τint,2 = τint,1 +V
EIEt2a2 (4.32d)
43
τint,1 =V
b· t1d
t1d2 + 4t2a22(4.32e)
τint,2 = τint,1 +V
b· 2t2a2t1d2 + 4t2a22
(4.32f)
γint,2 =τint,2Gint
= γint,1 +V
bGint· 2t2a2t1d2 + 4t2a22
(4.32g)
w′2 =2tint (γint,1 + γint,2)
d(4.32h)
γint,1 =w′2d
2tint− γint,2 =
d
4tintw′2 −
V
2bGint· 2t2a2t1d2 + 4t2a22
(4.32i)
A =bd
4tint· t1d
2 + 4t2a22
t1d+ t2a2(4.32j)
Figure 4.12: Shear deformation of a laminated glass beam segment with five layers.
Summary of the differences between different number of glass layers
The sandwich theory described before is valid for laminated glass beams with different number of layers,
with only minor changes. The second moment of area of the glass layers with respect to their own
centroidal axes may be generally defined as,
Igl =
n∑i=1
bt3i12
(4.33)
where the index i comprises the number of glass layers (n), whereas Is and A are summarized in
Table 4.1, for laminated glass cross-sections with two, three, four or five layers. The definition of the
dimensions may be found in Figs. 4.1, 4.6, 4.8 and 4.11, respectively.
44
Table 4.1: Expressions for Is and A for cross-sections from two to five layers.
Number of layers Is A
Two layers bd2t1t2t1 + t2
bd2
tint
Three layers bt1d2
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
4.2.3 Equivalent flexural stiffness of beams under transverse loads or pure
bending
The present Subsection applies the previously explained sandwich theory to the determination of the
total deflection of simply supported laminated glass beams under different loading conditions. Expres-
sions for the equivalent flexural stiffness that allow the determination of these deflections with common
homogeneous beam’s equations are also determined. The procedure is partially explained by Allen
[38]. The analysed loading conditions are: (i) concentrated load at mid-span; (ii) uniformly distributed
load along the span; (iii) pure bending; and (iv) four-point symmetrical loading; all applied to simply
supported beams, so that both loads and supports are symmetrical. As mentioned in Subsection 4.2.1
the symmetry is the condition of non-existence of warping shear deformation on the interlayer (Fig. 4.4
(c)).
Simply supported beam with mid-span load
A simply supported laminated glass beam with two symmetrical overhangs and a central point load (F )
is illustrated in Fig. 4.13(a). Because the shear force can only be defined as a single equation for half of
the span, by symmetry, only half of the beam must be considered (Fig. 4.13(b)). The half-span AB and
the overhang BC have each their independent xx axis of origin.
(a) Whole beam. (b) Right-hand half of the beam.
Figure 4.13: Beam with mid-span load.
45
The general trigonometric solution for Eq. (4.16a) is,
− V1 = EIw′′′1 = c1 cosh (ax) + c2 sinh (ax)− V (4.34)
where V is the total shear force.
In the half-span AB the shear force is −F/2. It can be substituted in Eq. (4.34), which integrated three
times takes the form:
[EIw1]AB
=A1
a3sinh (ax) +
A2
a3cosh (ax) +
F
12x3 +A3x
2 +A4x+A5 (4.35)
In order to get the second component of the total displacement (w2), Eq. (4.18) may be introduced in
Eq. (4.34), which after being integrated once, with the corresponding shear force, provides:
− [EIglw2]AB
=A1
a3sinh (ax) +
A2
a3cosh (ax) +
F
2a2x+A6 (4.36)
In the overhang BC the shear force is zero, which produces another set of equations:
[EIw1]BC
=B1
a3sinh (ax) +
B2
a3cosh (ax) +B3x
2 +B4x+B5 (4.37a)
− [EIglw2]BC
=B1
a3sinh (ax) +
B2
a3cosh (ax) +B6 (4.37b)
The solution is then dependent on twelve unknowns (A1 − A6 and B1 − B6). The twelve conditions
gathered in Table 4.2 apply. Conditions 1, 2, 6 and 7 refer to an arbitrary cross-section and define a
point where each solution passes through; conditions 3 and 4 are imposed by the symmetry; condition 5
defines the bending moment at mid-span; conditions 8 and 9 predict that each component of the bending
moment vanishes independently at the free end of the overhang, which is true provided that the glass
layers are free to rotate and are not attached to a rigid end diaphragm [38]; and conditions 10 to 12
define the continuity of rotation and curvature of the pure bending component in cross-section B.
The solution of the system of twelve equations are the twelve constants, which complete Eqs. (4.35)–
(4.37).
An equation for the deflection in the span can be then obtained from Eqs. (4.35) and (4.36) with:
w = w1 + w2 =[EIw1]
AB
EI+
[EIglw2]AB
EIgl(4.38)
The maximum displacement occurs at mid-span (x = 0). An equivalent flexural stiffness EIf can then be
obtained by corresponding this displacement to the maximum displacement of an homogeneous beam:
w(x = 0) =FL3
48EIf(4.39)
A simpler form of the equation can be achieved inserting a few parameters found in [32, 38], as well as
46
Table 4.2: Boundary and continuity conditions of a simply supported beam with mid-span load.
1. [EIw1]AB(x=L
2 ) = 0
2. [EIglw2]AB(x=L
2 ) = 0
3. [EIw′1]AB(x=0) = 0 Boundary conditions in AB
4. [EIw′′′1 ]AB(x=0) = 0
5. [EIw′′1 ]AB(x=0) + [EIglw
′′2 ]AB
(x=0) = −FL4
6. [EIw1]BC(x=0) = 0
7. [EIglw2]BC(x=0) = 0
Boundary conditions in BC8. [EIw′′1 ]
BC(x=L1)
= 0
9. [EIglw′′2 ]BC
(x=L1)= 0
10. [EIw′1]AB(x=L
2 ) = [EIw′1]BC(x=0)
11. [EIglw′2]AB
(x=L2 ) = [EIglw
′2]BC
(x=0) Conditions of continuity in B
12. [EIw′′1 ]AB(x=L
2 ) = [EIw′′1 ]BC(x=0)
a parameter ξf , suggested by Blaauwendraad [42], with a different presentation,
EIf = EIgl + ξfEIs (4.40a)
where,
ξf =1 + α
1 + 3αθ2
(1− 3 sinh(θ+φ)−sinh(θ−φ)−4 sinh(φ)
2θ cosh(θ+φ)
) − α (4.40b)
α =IglIs
β =EIs
AGintL2λf =
√1 + α
αβ
θ =λf2
φ =λfL1
L
(4.40c)
and also Igl is defined by Eq. (4.33) and Is and A may be found in Table 4.1.
The parameter ξf equals 0 if Gint = 0 and equals 1 if Gint = ∞. With this notation it becomes more
clear that the limiting values of the equivalent flexural stiffness are EIf = EIgl if the shear stiffness of the
interlayer is zero and EIf = E(Igl + Is) if the connection is complete and the beam is in fact monolithic.
This parameter does not depend on the width b, but it depends on all other geometrical and engineering
variables.
In the absence of overhangs (φ = 0), this formulation is equivalent to Luible’s formulation 1-B from Table
3.2.
47
Simply supported beam with uniformly distributed load
A beam similar to the one analyzed before, but now with a uniformly distributed load q located along the
span, is illustrated in Fig. 4.14(a). The same procedure is followed with the difference that the shear
force to be introduced in Eq. (4.34) is V = −qx. The expressions equivalent to Eqs. (4.35)–(4.37) are:
[EIw1]AB
=A1
a3sinh (ax) +
A2
a3cosh (ax) +
q
24x4 +A3x
2 +A4x+A5 (4.41a)
− [EIglw2]AB
=A1
a3sinh (ax) +
A2
a3cosh (ax) +
F
2a2x2 +A6 (4.41b)
[EIw1]BC
=B1
a3sinh (ax) +
B2
a3cosh (ax) +B3x
2 +B4x+B5 (4.41c)
− [EIglw2]BC
=B1
a3sinh (ax) +
B2
a3cosh (ax) +B6 (4.41d)
Furthermore, all twelve conditions remain unchanged, except for condition 5, where the bending moment
must be replaced by qL2/8.
(a) Whole beam.(b) Right-hand half of the beam.
Figure 4.14: Beam with distributed load.
The equivalent flexural stiffness can now obtained from the following relationship,
w(x = 0) =5qL4
384EIf(4.42)
and expressed also by Eq. (4.40a), with the parameter ξf defined as,
ξf =1 + α
1 + 125αθ2
(1− [2 cosh (θ)− 2] θ sinh(φ)+cosh(φ)
θ2 cosh(θ+φ)
) − α (4.43)
where α, β, λf , θ and φ are defined by Eqs. (4.40c), Igl is defined by Eq. (4.33) and Is and A may be
found in Table 4.1.
Simply supported beam in pure bending
The geometry of the beam is illustrated in Fig. 4.15(a).
Because there is no shear force along the beam Eq. (4.16a) can be reduced to,
V ′′1 − a2V1 = 0 (4.44)
48
(a) Whole beam.(b) Right-hand half of the beam.
Figure 4.15: Beam in pure bending.
and Eqs. (4.35)–(4.37) become:
[EIw1]AB
=A1
a3sinh (ax) +
A2
a3cosh (ax) +A3x
2 +A4x+A5 (4.45a)
− [EIglw2]AB
=A1
a3sinh (ax) +
A2
a3cosh (ax) +A6 (4.45b)
[EIw1]BC
=B1
a3sinh (ax) +
B2
a3cosh (ax) +B3x
2 +B4x+B5 (4.45c)
− [EIglw2]BC
=B1
a3sinh (ax) +
B2
a3cosh (ax) +B6 (4.45d)
The bending moment in condition 5 is replaced by a uniform moment M . Although for this loading
condition there is no continuity of curvature in section B, the pure bending component of the curvature
remains continuous (condition 12).
The relationship that provides the equivalent flexural stiffness is now,
w(x = 0) =ML2
8EIf(4.46)
which yields an expression that can be simply defined by Eq. (4.40a), with
ξf =1 + α
1 + 1αθ2
(1− 2 cosh(φ)−cosh(θ−φ)
cosh(θ+φ)
) − α (4.47)
where α, β, λf , θ and φ are defined in Eqs. (4.40c), Igl is defined in Eq. (4.33) and Is and A may be
found in Table 4.1.
Simply supported beam with two symmetrical point loads
The beam with four-point loading illustrated in Fig. 4.16(a) is characterized by an additional discontinuity
of shear force in comparison with all previously analyzed beams.
Between the two loads (AB) the shear force is zero, between the loads and the supports (BC) it is −F ,
and in the overhangs (CD) it is again zero. Following the same procedure as before, the expressions
equivalent to Eqs. (4.35)–(4.37) are:
[EIw1]AB
=A1
a3sinh (ax) +
A2
a3cosh (ax) +A3x
2 +A4x+A5 (4.48a)
49
(a) Whole beam. (b) Right-hand half of the beam.
Figure 4.16: Beam with two symmetrical point loads.
− [EIglw2]AB
=A1
a3sinh (ax) +
A2
a3cosh (ax) +A6 (4.48b)
[EIw1]BC
=B1
a3sinh (ax) +
B2
a3cosh (ax) +
F
6x3 +B3x
2 +B4x+B5 (4.48c)
− [EIglw2]BC
=B1
a3sinh (ax) +
B2
a3cosh (ax) +
F
a2x+B6 (4.48d)
[EIw1]CD
=C1
a3sinh (ax) +
C2
a3cosh (ax) + C3x
2 + C4x+ C5 (4.48e)
− [EIglw2]CD
=C1
a3sinh (ax) +
C2
a3cosh (ax) + C6 (4.48f)
The solution is, thus, dependent on eighteen unknowns (A1 − A6, B1 −B6 and C1 − C6). The eighteen
conditions gathered in Table 4.3 apply. Conditions 1, 2, 6, 7, 9 and 10 refer to an arbitrary cross-
section and define a point where each solution passes through; conditions 3 and 4 are imposed by
the symmetry; conditions 5 and 8 define the bending moment at A and B, respectively; conditions 11
and 12 predict that each component of the bending moment vanishes independently at the free end of
the overhang, which is true provided the glass layers are free to rotate and are not attached to a rigid
end diaphragm [38]; and conditions 13 to 18 define the continuity of rotation and curvature of the pure
bending component in sections B and C.
The solution of the system of eighteen equations are the eighteen constants, which complete Eqs.
(4.48).
An equation for the deflection in the span can thus be obtained from Eqs. (4.48a)–(4.48d) by,
[[EIw1]
AB
EI+
[EIglw2]AB
EIgl
](x=L
2 −La)
+
[[EIw1]
BC
EI+
[EIglw2]BC
EIgl
](x=La)
=
=FLa
24EIf
(3L2 − 4L2
a
)(4.49)
which yields an expression of the equivalent flexural stiffness that can be simply defined by Eq. (4.40a),
with,
ξf =1 + α
1 + 63αθ2−4αψ2
(1− sinh (ψ) 2 cosh(φ+ψ)+cosh(θ+φ−ψ)−cosh(θ−φ−ψ)
2ψ cosh(θ+φ)
) − α (4.50a)
where, besides the parameters defined in Eqs. (4.40c),
ψ =θLaL
(4.50b)
50
Table 4.3: Boundary and continuity conditions of a simply supported beam with two symmetrical pointloads.
1. [EIw1]AB(x=0) = 0
2. [EIglw2]AB(x=0) = 0
3. [EIw′1]AB(x=0) = 0 Boundary conditions in AB
4. [EIw′′′1 ]AB(x=0) = 0
5. [EIw′′1 ]AB(x=0) + [EIglw
′′2 ]AB
(x=0) = −FLa
6. [EIw1]BC(x=0) = 0
7. [EIglw2]BC(x=0) = 0 Boundary conditions in BC
8. [EIw′′1 ]BC(x=0) + [EIglw
′′2 ]BC
(x=0) = −FLa
9. [EIw1]CD(x=0) = 0
10. [EIglw2]CD(x=0) = 0
Boundary conditions in CD11. [EIw′′1 ]
CD(x=L1)
= 0
12. [EIglw′′2 ]CD
(x=L1)= 0
13. [EIw′1]AB(x=L
2 −La)= [EIw′1]
BC(x=0)
14. [EIglw′2]AB
(x=L2 −La)
= [EIglw′2]BC
(x=0) Conditions of continuity in B
15. [EIw′′1 ]AB(x=L
2 −La)= [EIw′′1 ]
BC(x=0)
16. [EIw′1]BC(x=La)
= [EIw′1]CD(x=0)
17. [EIglw′2]BC
(x=La)[EIglw
′2]CD
(x=0)Conditions of continuity in C
18. [EIw′′1 ]BC(x=La)
= [EIw′′1 ]CD(x=0)
and Igl is defined by Eq. (4.33) and Is and A may be found in Table 4.1.
It can be verified that Eq. (4.50a) equals Eq. (4.40b) when La = L/2 and that its limit when La → 0
equals Eq. (4.47).
Summary of the analyzed loading conditions
This procedure may be followed with any other loading and support conditions, as long as they are
symmetrical. For non-symmetrical conditions the warping shear deformation of the interlayer (Fig. 4.4)
should be taken into account [38].
51
The expressions which were already deduced, for laminated glass beams subjected to a mid-span load,
a uniformly distributed load, pure bending, or four-point bending, are summarized in Table 4.4. In the
absence of overhangs they can be significantly simplified, taking φ as zero. The index i comprises the
number of glass layers (n), whereas Is and A are summarized in Table 4.1, for laminated glass cross-
sections with two, three, four or five layers. The definition of the dimensions may be found in Figs. 4.1,
4.6, 4.8 and 4.11, respectively.
Table 4.4: Expressions for the equivalent flexural stiffness (EIf ) of laminated glass beams subjectedto transverse loading conditions or to pure bending.
EIf = EIgl + ξfEIs
α =IglIs
β =EIs
AGintL2Igl =
n∑i=1
bt3i12
θ =λf2
φ =λfL1
Lψ =
θLaL
λf =
√1 + α
αβ
TYPE OF LOADING 0 ≤ ξf ≤ 1
ξf =1 + α
1 + 1αθ2
(1− 2 cosh(φ)−cosh(θ−φ)
cosh(θ+φ)
) − α
ξf =1 + α
1 + 125αθ2
(1− [2 cosh (θ)− 2] θ sinh(φ)+cosh(φ)
θ2 cosh(θ+φ)
) − α
ξf =1 + α
1 + 3αθ2
(1− 3 sinh(θ+φ)−sinh(θ−φ)−4 sinh(φ)
2θ cosh(θ+φ)
) − α
ξf =1 + α
1 + 63αθ2−4αψ2
(1− sinh (ψ) 2 cosh(φ+ψ)+cosh(θ+φ−ψ)−cosh(θ−φ−ψ)
2ψ cosh(θ+φ)
) − α
Determination of the normal stresses
The normal stresses at any level of the glass layers, along the span, of beams subjected to the loading
conditions studied in the present Subsection, can be easily determined with little more manipulation of
some previously formulated equations.
Those procedures and a resulting formulation able to characterize the maximum stress in the exterior
glass layers are described in Appendix B.
52
4.2.4 Critical buckling load of a glass column
The procedure for the determination of the critical buckling load is the same that for the Euler’s column,
only with a distinct differential equation containing the shear behavior of the column.
The total shear force at a given cross-section can be deduced from Fig. 4.17,
V = P (w′1 + w′2) (4.51)
and may be substituted in Eq. (4.16a), resulting in:
V ′′1 − a2V1 = −a2P (w′1 + w′2) (4.52)
Replacing the slope w′2 from Eq. (4.18) and V1 = −EIw′′′1 in Eq. (4.52), gives:
wv1 −(a2 − P
EIgl
)w′′′1 −
a2P
EIw′1 = 0 (4.53)
As for the Euler column, the buckling mode shape (pure bending deflection) is defined by a sinusoidal
curve,
w1 = A1 sinπx
L(4.54)
which fulfils the boundary conditions:
• w1 = w′′1 = 0, x = 0
• w1 = w′′1 = 0, x = L
Figure 4.17: Buckled shape of a laminated glass column (adapted from [38]).
The result of the substitution of Eq. (4.54) in Eq. (4.53), followed by the division by − (π/L) cos (πx/L)
is: [π4
L2+
(a2 − P
EIgl
)π2
L2− a2P
EI
]A1 = 0 (4.55)
The equation is true either if A1 = 0 (trivial solution – fundamental path) or if the rest of the product
equals zero (non-trivial solution – buckling load). The critical buckling load is given by:
Pcr =π4
L4 + a2π2
L2
π2
L2EIgl+ a2
EI
(4.56)
53
An equivalent flexural stiffness EIf can then be obtained by corresponding Eq. (4.56) to Euler’s critical
bucking load formula (Eq. (3.3)),
EIf =π2
L2 + a2
π2
L2EIgl+ a2
EI
(4.57a)
where:
a2 =AGint
EIgl
(1− EIgl
EI
) (4.57b)
With the parameters introduced in Eqs. (4.17), the equivalent flexural stiffness for the determination of
the critical buckling load can be simply written as,
EIf = EIgl + ξfEIs (4.58a)
where,
ξf =1
1 + π2β(4.58b)
and:
β =EIs
AGintL2(4.58c)
This formulation is equivalent to the first four formulations from Table 3.1. It keeps the general equation
from the previous Subsection (Eq. (4.40a)) and a parameter ξf which ranges from 0 to 1, depending on
the shear modulus of the interlayer.
4.2.5 Governing equation for the bending behavior of laminated glass beams
— alternative approach
A general equation governing the total displacement of laminated glass beams can be deduced from
Eqs. (4.16a) and (4.18) [38].
To account for the possibility of both axial and transverse loads, the total shear force is P (w′1 +w′2) + V ,
which means Eq. (4.16a) becomes:
V ′′1 − a2V1 = −a2P (w′1 + w′2)− a2V (4.59)
The component V1 of the shear force is defined by,
V1 = −EIw′′′1 (4.60a)
54
where EI = EIgl + EIs, or by manipulation of Eq. (4.18):
V1 = a2EIglw′2 (4.60b)
The result of the substitution of Eqs. (4.60) in Eq. (4.59), followed by differentiation, is, respectively:
E (Igl + Is)(wvi
1 − a2wiv1
)= a2P (w′′1 + w′′2 )− a2q (4.61a)
E (Igl + Is)(wvi
2 − a2wiv2
)= −P
(wiv
1 + wiv2
)(1 +
IsIgl
)+ q′′
(1 +
IsIgl
)(4.61b)
The addition of both Eqs. (4.61) yields a differential equation for the total displacement w,
E (Igl + Is)(wvi − a2wiv
)= a2Pw′′ − Pwiv
(1 +
IsIgl
)− a2q + q′′
(1 +
IsIgl
)(4.62a)
or, replacing a2 from Eq. (4.16b), in a compact form, as proposed by Allen [38],
−EIglEIsAGint
wvi + E (Igl + Is)wiv =
(1− EIs
AGint· d2
dx2
)(q − Pw′′) (4.62b)
or, in an extended form:
− EIglAGint
wvi +
(Igl + IsIs
− P
AGint
)wiv +
P
EIsw′′ =
q
EIs− q′′
AGint(4.62c)
In the structural glass bibliography, as well as in Stamm and Witte [32], this governing equation often
appears integrated twice:
− EIglAGint
wiv +
(Igl + IsIs
− P
AGint
)w′′ +
P
EIsw = − M
EIs− q
AGint(4.63)
It can be used to obtain the same results presented in Subsections 4.2.3 and 4.2.4, but the approach
followed before, based on Allen’s theory, is found to be more convenient [38].
4.3 Torsional stiffness
4.3.1 Two-layered laminated glass beam analysis based on sandwich theory
A detailed theory for the torsional stiffness of sandwich beams was developed by Stamm and Witte [32].
It is applied and extended here to laminated glass beams with up to five layers.
Because the interlayer is much weaker than the glass layers, the shear stresses τxy = τyx in the inter-
layer are assumed to be zero.
A torsional moment applied to a laminated glass beam’s section is equilibrated by two different groups
of shear stresses, as illustrated in Fig. 4.18. On the one hand, the glass layers work independently as
55
“thin-walled open sections”, while on the other hand, a closed shear flow transmitted between the glass
layers by the interlayer also contributes to balance part of the torque. Assuming that all layers have the
same width (b), the torsion constant associated with the former contribution is given by the well known
expression,
Jgl =n∑i=1
bt3i3
(4.64)
where the index i comprises the number of glass layers (n), while the torsion constant associated with
the latter contribution is designated Js. The weight of the latter torsion constant depends on the shear
modulus of the interlayer, similarly to Is in Eq. (4.40a). A parameter ξt, analogous to ξf , may thus be
introduced. It equals 0 if Gint = 0 and equals 1 if Gint = ∞. The torsional stiffness of laminated glass
beams is then defined as:
GJt = GJgl + ξtGJs (4.65)
The parameter ξt and the torsion constant Js are deduced in the present Section for beams with different
number of layers, disregarding, for that purpose, the former component.
Figure 4.18: Different shear stresses equilibrating the torque (adapted from [32]).
A model representative of a two-layered glass beam of length dx is illustrated in Fig. 4.19. The interlayer
is represented thicker than the glass layers only for improved clarity of the illustration. Because τxy =
τyx = 0 in the interlayer, it is represented by two vertical elements of thickness dy, distanced y from
the section’s vertical symmetry axis. A later integration across the entire width of the glass beam allows
the consideration of all the interlayer. The closed shear flow composed of shear stresses τxz in the
interlayer and τxy in the glass layers is also included in the figure. The zz axis is placed in the vertical
axis of symmetry, while the height of the yy axis is arbitrary.
For the purpose of the following developments, an hypothetical separation is considered between an
upper and a lower part of the beam, at a level z = 0. That creates two “thin-walled opened sections”,
one above and another below. If they were independent, when subjected to a torque, they would warp,
and opposite longitudinal displacements at points A and B would arise. Because in reality the interlayer
is continuous and thus no such different displacements are possible, the shear stresses τzx = τxz
are said to compensate those differential displacements along the width of the beam (−b/2 ≤ y ≤ b/2).
Knowing τzx, the assessment of the equilibrium in the interface between interlayer and glass layer allows
the determination of τyx. The final step is to analyze the balance of moments of torsion applied to and
resisted by the cross-section (∑T = 0) in order to determine ξt and Js.
56
Figure 4.19: Model of a two-layered laminated glass beam (adapted from [32]).
A generically shaped prismatic thin-walled open section element is illustrated in Fig. 4.20. A coordinate
s develops along the center-line of its cross-section and the section’s shear center is named C. A refer-
ence point O and another arbitrary point P are positioned in the center-line. The geometry demonstrates
that if the cross-section rotates an angle φ around the shear center, point P will undergo a displacement
rφ, with the projection in the direction tangent to the section’s center-line being given by,
v = (rφ) cosβ = (r cosβ)φ = rnφ (4.66)
where r is the distance between P and the shear center C, and rn is the distance between the shear
center and a line tangent to the section’s center-line at point P .
Figure 4.20: Warping in generic thin-walled section (adapted from [32]).
The shear stress in a thin-walled open section is assumed to have a linear distribution across the thick-
ness, being zero in its center-line. The shear strain in the center-line is then assumed to be zero:
γxs =∂u
∂s+∂v
∂x= 0 (4.67)
Substituting Eq. (4.66) in Eq. (4.67), and considering point O as a reference for the displacement due
57
to warping, it comes that,
u = −∫ s
sO
∂v
∂xds = −φ′ω (s) (4.68a)
where, ω (s) is called the sectorial area of the section with respect to the shear center and equals twice
the area striped in Fig. 4.20 [40]:
ω (s) =
∫ s
sO
rnds = 2ACOP (4.68b)
The sectorial area is positive when evaluated in the direction of the rotation φ, imposed by the torque,
and negative otherwise.
Applied to Fig. 4.19, and knowing that in the axes of symmetry the displacements imposed by the
section’s warping are zero (u = 0), the displacement at A and B may be integrated from points S1 and
S2, respectively, yielding:
uA = −2φ′dAy (4.69a)
uB = 2φ′dBy (4.69b)
These displacements are associated with the independent warping of the upper and lower parts of
the beam. As mentioned before, so that points A and B do not have differential displacements, shear
stresses τzx must develop. It was also mentioned that these shear stresses in the interlayer are balanced
by shear stresses τyz in the glass layers. Thus, shear strains γzx and γyx deform the materials and
additional displacements uA and uB must be determined.
The interface between the interlayer and the glass layer above point A is illustrated in Fig. 4.21. The
equilibrium of the resultants of the shear stresses in the xx direction gives:
∂τyx,1∂y
dyt1dx+ τzxdxdy = 0 (4.70)
Acknowledging that τxy,1 = τyx,1, τxz = τzx and τxy,1(y = b/2) = 0, by integration along y, a relationship
between τxy,1 and τxz may be found,
τxy,1 =1
t1
∫ b/2
y
τxzdy (4.71a)
and similarly for the lower glass layer:
τxy,2 =1
t2
∫ b/2
y
τxzdy (4.71b)
With ∂v/∂x in the glass layers and ∂w/∂x in the interlayer assumed to be zero, the displacements uA
58
Figure 4.21: Interface between glass layer and interlayer, above point A (adapted from [32]).
and uB are given by,
uA =
∫ y
0
γxy,1dy +
∫ 0
−tAint
γxzdz (4.72a)
uB = −∫ y
0
γxy,2dy +
∫ 0
tBint
γxzdz (4.72b)
where:
γxy,1 =τxy,1G
γxy,2 =τxy,2G
γxz =τxzGint
(4.73)
Combining Eqs. (4.71)–(4.73), the displacements uA and uB — dependent only on τxz — become:
uA =1
Gt1
∫ y
0
∫ b/2
y
τxzdydy +τxzt
Aint
Gint(4.74a)
uB = − 1
Gt2
∫ y
0
∫ b/2
y
τxzdydy − τxztBint
Gint(4.74b)
The condition of warping continuity (compatibility) A ≡ B is defined as,
∆u+ ∆u = uB + uB − uA − uA = 0 (4.75)
which, taking into account that dA + dB = d and that tAint + tBint = tint, and introducing the parameter λ2t
proposed by Stamm and Witte [32], results in,
τxz + λ2t
∫ y
0
∫ b/2
y
τxzdydy =2Gintφ
′d
tinty (4.76a)
where:
λ2t =GintG· t1 + t2tintt1t2
(4.76b)
59
Differentiating Eq. (4.76a) twice with respect to y, the following differential equation is obtained,
τ ′′xz + λ2t τxz = 0 (4.77a)
which has as general trigonometric solution for τxz:
τxz = c1 cosh (λty) + c2 sinh (λty) (4.77b)
The substitution of the general solution in Eq. (4.76a), followed by the necessary integrations, yields:
λt
(c1 sinh
λtb
2+ c2 cosh
λtb
2
)y + c1 = 2d
GintGtint
Gφ′y (4.78)
From the evaluation of Eq. (4.78) for y = 0 results immediately that,
c1 = 0 (4.79a)
which replaced in Eq. (4.78) yields:
c2 = 2dGintGtint
· Gφ′
λt cosh(λtb2
) (4.79b)
Both coefficients may now be substituted in Eq. (4.77b). Making also use of Eq. (4.76b), the final
expression for the shear stress in the interlayer results:
τxz = 2dλtt1t2t1 + t2
· sinh (λty)
cosh(λtb2
)Gφ′ (4.80)
The final expressions for the shear stress in the glass layers come from the substitution of Eq. (4.80) in
Eqs. (4.71):
τxy,1 = 2dt2
t1 + t2
(1− cosh (λty)
cosh(λtb2
))Gφ′ (4.81a)
τxy,2 = 2dt1
t1 + t2
(1− cosh (λty)
cosh(λtb2
))Gφ′ (4.81b)
The distance between the resultants of the shear stress in the two vertical elements representatives of
the interlayer is 2y, while the distance between the two glass layers is d. Thus, the equilibrium of torque
in the cross-section is 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 (4.82)
The torsion constant Js and the parameter ξt, mentioned in the beginning of this Section, can then be
60
obtained from the relation,
Ts = ξtGJsφ′ (4.83a)
which yields,
ξtJs = 4bd2t1t2t1 + t2
(1− 2
λtbtanh
λtb
2
)(4.83b)
where,
Js = 4bd2t1t2t1 + t2
(4.83c)
and:
ξt = 1− 2
λtbtanh
λtb
2(4.83d)
The comparison between Eq. (4.83c) and the expression of Is associated with a two-layered beam,
found in Table 4.1, allows concluding that:
Js = 4Is (4.84)
This formulation, for two-layered beams, is equivalent to Luible’s formulation 1-C from Table 3.3.
4.3.2 Laminated glass beams with more than two layers
Although most of what is described in Subsection 4.3.1 remains valid with a number of layers different
than two, some changes must be made in order to adapt the expressions for other cross-sections. This
study limits the analysis for laminated glass beams up to five layers. Similarly as for the flexural stiffness
(Subsection 4.2.2), only bi-symmetrical cross-sections are considered.
Three layers
The model representative of a three-layered beam, illustrated in Fig. 4.22, shall now be considered.
Because the sectorial areas are now different, Eqs. (4.69) change to become the following,
uA = −2φ′(aA1 y +
1
2aB1 y
)(4.85a)
uB = 2φ′1
2aB1 y (4.85b)
which, taking into account that aA1 + aB1 = a1, means that:
∆u = uB − uA = 2φ′a1y (4.85c)
61
Figure 4.22: Model of a three-layered laminated glass beam.
The two sectorial areas are illustrated in Fig. 4.23(a). In fact, a1y in Eq. (4.85c) represents the sum of
the two sectorial areas (Fig. 4.23(b)), which is defined by the path of integration between points S1 and
S2.
(a) Sectorial areas. (b) Sum of the two sectorial areas.
Figure 4.23: Representation of the sectorial areas for a three-layered beam.
The shear stress τxy is assumed to be zero in the middle glass layer (which comprises point S2), because
of the symmetry. This means that, taking S2 as the beginning of the bottom part integration (it would
have also been possible to integrate from point S3), Eq. (4.71b) vanishes and Eqs. (4.74) become,
uA =1
Gt1
∫ y
0
∫ b/2
y
τxzdydy +tAintGint
τxz (4.86a)
uB = − tBintGint
τxz (4.86b)
62
which, taking into account that tAint + tBint = tint, means that:
∆u = uB − uA = − 1
Gt1
∫ y
0
∫ b/2
y
τxzdydy − tintGint
τxz (4.86c)
The condition of continuity (Eq. (4.75)) now results in,
τxz + λ2t
∫ y
0
∫ b/2
y
τxzdydy =2Gintφ
′a1tint
y (4.87a)
where:
λ2t =GintG· 1
tintt1(4.87b)
The final expressions for the shear stress in the interlayer and in the exterior glass layers are, respec-
tively:
τxz = 2a1λtt1sinh (λty)
cosh(λtb2
)Gφ′ (4.88a)
τxy,1 = 2a1
(1− cosh (λty)
cosh(λtb2
))Gφ′ (4.88b)
The equilibrium of torque in the cross-section is given by,
Ts =
∫ b/2
0
τxzddy · 2y +
∫ b/2
−b/2τxy,1t1dy · d (4.89)
which results in the following torsion constant,
Js = 2bt1d2 (4.90)
with the parameter ξt still being defined by Eq. (4.83d).
The comparison between Eq. (4.90) and the expression of Is associated with a three-layered beam,
found in Table 4.1, allows concluding that Eq. (4.84) is again valid.
The parameters λt and Js determined in this dissertation, for three-layered beams, differ from those
proposed by Luible and Crisinel [6]:
λt =
√GintG· t1 + t2tintt1t2
(4.91a)
Js = 2 (t2 + 2tint + t1)2t1b (4.91b)
The authors propose the use of the same parameter λt (Eq. (4.91a)) for the cases of beams with two
and three layers, whereas here a different expression was determined for the latter configuration (Eq.
(4.87b)), and Js (Eqs. (4.90) and (4.91b)) only coincide if t2 equals t1.
63
Four layers
In contrast with previous cross-sections, those with at least four layers have the particularity of not having
only one expression to define the interlayer’s shear stress. For this reason, for the four-layered beam
analysis, a model with two hypothetical separations must be adopted (Fig. 4.24), one in the middle
interlayer and another one in one of the two others (by symmetry, the top and bottom interlayer have the
same behavior). The points with u = 0 chosen for the necessary integrations are illustrated in Fig. 4.24.
Figure 4.24: Model of a four-layered laminated glass beam.
For the condition of continuity A ≡ B points S1 and S4 are used, whereas for the condition of continuity
C ≡ D points S2 and S3 are adopted.
The sum of the sectorial areas (Fig. 4.25) yields, analogously to Eq. (4.85c):
∆uAB = 2φ′ (2a1 + a2) y (4.92a)
∆uCD = 2φ′a2y (4.92b)
64
Figure 4.25: Sum of the sectorial areas for a four-layered beam.
The interface between the interlayer and the glass layer above point A can still be described by Fig.
4.21, which means the shear stress in that glass layer (and the other symmetrical one) is still defined
by an expression similar to Eq. (4.71a). However, since the interior glass layers are in contact with two
interlayer sheets, the interface in the layer above point C is different, as illustrated in Fig. 4.26. It follows
that the shear stresses in the exterior layers and in the interior ones are, respectively:
τxy,1 =1
t1
∫ b/2
y
τxz,1dy (4.93a)
τxy,2 =1
t2
∫ b/2
y
(τxz,2 − τxz,1) dy (4.93b)
Figure 4.26: Interface between the interior glass layers and two interlayer sheets.
Analogously to Eq. (4.86c), the integration of the shear strains yields:
∆uAB = − 1
G· 2
t1
∫ y
0
∫ b/2
y
τxz1dydy − tintGint
(2τxz,1 + τxz,2) (4.94a)
∆uCD = − 1
G· 2
t2
∫ y
0
∫ b/2
y
(τxz,2 − τxz,1) dydy − tintGint
τxz2 (4.94b)
65
The conditions of continuity A ≡ B and C ≡ D are respectively defined by,
∆uAB + ∆uAB = 0 (4.95a)
∆uCD + ∆uCD = 0 (4.95b)
which are equivalent to, respectively:
GintG· 2
tintt1
∫ y
0
∫ b/2
y
τxz1dydy + 2τxz,1 + τxz,2 =2Gintφ
′
tint(2a1 + a2) y (4.96a)
GintG· 2
tintt2
∫ y
0
∫ b/2
y
(τxz2 − τxz1) dydy + τxz,2 =2Gintφ
′
tinta2y (4.96b)
Instead of only one differential equation (Eq. (4.77a)), the double differentiation of Eqs. (4.96) produces
a system of two differential equations, which must be solved simultaneously for τxz,1 and τxz,2:
2τ ′′xz,1 + τ ′′xz,2 −2GintGtintt1
τxz,1 = 0 (4.97a)
τ ′′xz,2 −2GintGtintt2
(τxz,2 − τxz,1) = 0 (4.97b)
Although with higher complexity, each of the two general trigonometric solutions has two unknown con-
stants, as in Eq. (4.77), from which one of them is zero. The rest of the procedure is similar to that
explained before, and with the equilibrium of torque in the cross-section given by,
Ts =
∫ b/2
0
(τxz,1 · 2a1 + τxz,2 · a2) dy · 2y +
∫ b/2
−b/2[τxy,1t1 (2a1 + a2) + τxy,2t2a2] dy (4.98)
taking into account that 2a1 +a2 = d, the torsion constant Js and the parameter ξt come in the new form,
Js = 2b(t1d
2 + t2a22
)(4.99a)
ξt = 1− γ1λt,1b
tanhλt,1b
2− γ2λt,2b
tanhλt,2b
2(4.99b)
where several new parameters were included in order to simplify the solution:
λt,1 =
√GintG· 3t1 + t2 − µ
2tintt1t2(4.99c)
λt,2 =
√GintG· 3t1 + t2 + µ
2tintt1t2(4.99d)
µ =√
9t21 − 2t1t2 + t22 (4.99e)
ρ =3t21d
2 + 4t1t2(a1a2 + a21
)+ t22a
22
t1d2 + t2a22(4.99f)
γ1 = 1 +ρ
µ(4.99g)
γ2 = 1− ρ
µ(4.99h)
66
Figure 4.27: Model of a five-layered laminated glass beam.
Again, the comparison between Eq. (4.99a) and the expression of Is associated with a four-layered
beam, found in Table 4.1, allows concluding that Eq. (4.84) is valid.
Five layers
The procedure to determine the flexural stiffness of a laminated glass beam with five layers is very similar
to that of a beam with four glass layers. As for the three-layered beam, because of the symmetry, the
shear stress τxy in the central glass layer is assumed to be zero. Instead of Figs. 4.24 and 4.25, Figs.
4.27 and 4.28 are now considered. Eqs. (4.92)–(4.94) have small modifications, becoming, respectively,
∆uAB = 4φ′ (a1 + a2) y (4.100a)
∆uCD = 4φ′a2y (4.100b)
67
∆uAB = − 1
G· 2
t1
∫ y
0
∫ b/2
y
τxz1dydy − 2tintGint
(τxz,1 + τxz,2) (4.100c)
∆uCD = − 1
G· 2
t2
∫ y
0
∫ b/2
y
(τxz,2 − τxz,1) dydy − 2tintGint
τxz2 (4.100d)
which means that the differential equations are, instead of Eqs. (4.97):
2(τ ′′xz,1 + τ ′′xz,2
)− 2GintGtintt1
τxz,1 = 0 (4.101a)
2τ ′′xz,2 −2GintGtintt2
(τxz,2 − τxz,1) = 0 (4.101b)
Figure 4.28: Sum of the sectorial areas for a five-layered beam.
With the equilibrium of torque in the cross-section given by,
T =
∫ b/2
0
(τxz,1 · 2a1 + τxz,2 · 2a2) dy · 2y +
∫ b/2
−b/2[τxy,1t1 (2a1 + 2a2) + τxy,2t2 · 2a2] dy (4.102)
taking into account that 2a1 + 2a2 = d, the torsion constant comes as,
Js = b(2t1d
2 + 8t2a22
)(4.103a)
with the parameter ξt still being defined by Eq. (4.99b), where:
λt,1 =
√GintG· 2t1 + t2 − µ
2tintt1t2(4.103b)
λt,2 =
√GintG· 2t1 + t2 + µ
2tintt1t2(4.103c)
µ =√
4t21 + t22 (4.103d)
68
ρ =2t21d
2 + 4t1t2(a22 + 2a1a2 + a21
)+ 4t22a
22
t1d2 + 4t2a22(4.103e)
γ1 = 1 +ρ
µ(4.103f)
γ2 = 1− ρ
µ(4.103g)
Once again Eq. (4.103a) equals Eq. (4.84), with Is being given by the corresponding expression in
Table 4.1.
Summary of the differences between different number of glass layers
The sandwich theory described before allowed to reach a single simple expression (Eq. (4.65)) able
to define the torsional stiffness of laminated glass beams with up to five layers. As mentioned, the
proposed parameter ξt equals 0 if Gint = 0 and equals 1 if Gint =∞.
The expressions that were deduced in this Subsection, for laminated glass cross-sections with two,
three, four or five layers, are summarized in Table 4.5. The definition of the dimensions may be found in
Figs. 4.1, 4.6, 4.8 and 4.11, respectively. The index i comprises the number of glass layers (n), and Is
is summarized in Table 4.1, for the same laminated glass cross-sections.
4.4 Lateral-torsional buckling
The equivalent flexural and torsional stiffness of laminated glass beams was analyzed in Sections 4.2
and 4.3, respectively. The lateral-torsional buckling of glass beams can now be studied making use of
those results. Glass beams are considered to behave as homogeneous beams with only the substitution
of EIf instead of EI and GJt instead of GJ .
4.4.1 Critical buckling load
Differential equations describing the lateral-torsional behavior of beams may be deduced from equilib-
rium considerations in their buckled shape. For simply supported beams under bending or torsion the
following equations apply [40]:
EIyw′′ = −My (4.104a)
EIzv′′ = −Mz (4.104b)
GJφ′ = T (4.104c)
When the beam deflects laterally and rotates, the loads are usually no longer aligned with the beam’s
axes, thus having a different finite component in each of those axes. This decomposition is illustrated in
Fig. 4.29 for a simply supported beam subjected to a pre-buckling bending moment distribution M(x),
69
Table 4.5: Expressions for the torsional stiffness (GJt) of laminated glass beams from two to five layers.
GJt = GJgl + ξtGJs
Jgl =n∑i=1
bt3i3
Js = 4Is
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
where the supports are free to rotate about the vertical axis and free to warp.
Pure bending
Taking into account that EIz = EIf , GJ = GJt, My = M , Mz = Mφ and T = Mv′ (see Fig. 4.29), Eqs.
(4.104) become, for the buckled beam:
EIyw′′ = −M (4.105a)
EIfv′′ = −Mφ (4.105b)
GJtφ′ = Mv′ (4.105c)
The fundamental path is defined only by Eq. (4.105a), which is independent from Eqs. (4.105b) and
(4.105c). The lateral-torsional buckling behavior is defined both by Eqs. (4.105b) and (4.105c), which
may be solved simultaneously.
The critical buckling moment may be determined from these equations with Galerkin’s method. The
70
(a) Buckled shape.
(b) Moment decomposition in thexy-plane.
(c) Moment decomposition in theyz-plane.
(d) Moment decomposition in thexz-plane.
Figure 4.29: Lateral-torsional buckling of a simply supported beam (adapted from [40]).
following expression with sinusoidal shapes are exact solutions for the deflection and rotation of simply
supported monolithic beams under pure bending,
u1 = q11ψ11 = vmax sinπx
L(4.106a)
u2 = q21ψ21 = φmax sinπx
L(4.106b)
where vmax and φmax are the maximum deflection and rotation, respectively.
The residuals obtained from Eqs. (4.105b) and (4.105c), with the solutions are:
r1 = EIf u′′1 +Mu2 = −EIfvmax
(πL
)2sin
πx
L+Mφmax sin
πx
L(4.107a)
r2 = GJtu′2 −Mu′1 = GJtφmax
π
Lcos
πx
L−Mvmax
π
Lcos
πx
L(4.107b)
71
The Galerkin’s equations are, then,
∫r1ψ11dx =
EIfπ2vmax −ML2φmax
4L2π
(−2πx+ L sin
2πx
L
)= 0 (4.108a)∫
r2ψ21dx =Mvmax −GJtφmax
4cos
2πx
L= 0 (4.108b)
which solved together for M yield the critical buckling moment:
Mcr =π
L
√EIfGJt (4.109)
Uniformly distributed load
When subjected to a transverse load, an additional torque component may arise due to the twist rotation
of the cross-section. The eccentricity of the load, with respect to the shear center (when the beam
rotates), is zgφ (Fig. 4.30). In the case of a uniformly distributed load q, the increase of torque per unit
length is given by,
t = qzg sinφ ' qzgφ (4.110)
where, from convention, zg is negative when the load is applied above the shear center, and positive
otherwise.
Figure 4.30: Additional torque because of the eccentricity of an in-plane transverse load.
In order to express the moment per unit length, Eq. (4.105c) must be differentiated once, which means
that for a simply supported beam under uniformly distributed load q Eqs. (4.105b) and (4.105c) become,
respectively,
EIfv′′ = −Mφ (4.111a)
GJtφ′′ = Mv′′ + qzgφ (4.111b)
In this case, the bending moment M has the following longitudinal distribution:
M =q
2
(Lx− x2
)(4.112)
72
The critical buckling moment is determined again using Galerkin’s method with the approximate solutions
of Eqs. (4.106). The residuals are now:
r1 = EIf u′′1 +Mu2 =
[−EIfπ
2vmax
L2+q(Lx− x2
)φmax
2
]sin
πx
L(4.113a)
r2 = GJtu′′2 −Mu′′1 − qzgu2 =
[−GJtπ
2φmax
L2+q(Lx− x2
)π2vmax
2L2− qzgφmax
]sin
πx
L(4.113b)
Galerkin’s equations must now integrate the bending moment along the span:
∫ L
0
r1ψ11dx =qL4
(3 + π2
)φmax − 12EIfπ
4vmax
24Lπ2= 0 (4.114a)∫ L
0
r2ψ21dx =qL2
(3 + π2
)vmax − 12
(GJtπ
2 + qzgL2)φmax
24L= 0 (4.114b)
Solved together for q, these equations yield the critical buckling distributed load:
qcr =
12π3
(6EIfπzg +
√EIf
[GJtL2 (3 + π2)
2+ 36EIfπ2z2g
])L4 (3 + π2)
2 (4.115)
The maximum bending moment (at mid-span) is considered to be the critical buckling moment,
Mcr =qcrL
2
8=
3π3
(6EIfπzg +
√EIf
[GJtL2 (3 + π2)
2+ 36EIfπ2z2g
])2L2 (3 + π2)
2 (4.116a)
which may also be written in the form presented in Eurocode 3 [31],
Mcr = C1π2EIfL2
C2zg +
√(C2zg)
2+GJtEIf
(L
π
)2 (4.116b)
where:
C1 =3π2
2 (3 + π2)= 1.1503 (4.116c)
C2 =6
3 + π2= 0.4662 (4.116d)
Mid-span load
When a beam subjected to a mid-span load F rotates, the load also becomes eccentric with respect
to the shear center. However, because the load is concentrated in the mid-span cross section, q in Eq.
73
(4.110) must be taken as,
q = Fδ
(x− L
2
)= F ×
1, if x = L/2
0, if x 6= L/2
(4.117)
where δ is the Dirac delta function [43].1
Because the bending moment can only be defined as a single equation for half of the span, by symmetry,
only half of the beam must be considered, with half of the load. This means that for a simply supported
beam subjected to a mid-span load Eqs. (4.111a) and (4.111b) become, respectively,
EIfv′′ = −Mφ (4.118a)
GJtφ′′ = Mv′′ +
F
2δ
(x− L
2
)zgφ (4.118b)
where the bending moment M has the following distribution:
M =F
2x (4.119)
The residuals of Galerkin’s method, with the approximate solutions of Eqs. (4.106), are now:
r1 = EIf u′′1 +Mu2 =
(−EIfπ
2vmax
L2+Fxφmax
2
)sin
πx
L(4.120a)
r2 = GJtu′′2 −Mu′′1 −
F
2δ
(x− L
2
)zgu2 =
=
[−GJtπ
2φmax
L2+Fxπ2vmax
2L2− F
2δ
(x− L
2
)zgφmax
]sin
πx
L
(4.120b)
Galerkin’s equations must now integrate the bending moment along half the span:
∫ L/2
0
r1ψ11dx =FL3
(4 + π2
)φmax − 8EIfπ
4vmax
32Lπ2= 0 (4.121a)∫ L/2
0
r2ψ21dx =FL
(4 + π2
)vmax − 8
(GJtπ
2 + 2FzgL)φmax
32L= 0 (4.121b)
Solved together for F , they yield the critical buckling point load:
Fcr =
8π3
(8EIfπzg +
√EIf
[GJtL2 (4 + π2)
2+ 64EIfπ2z2g
])L3 (4 + π2)
2 (4.122)
The maximum bending moment (at mid-span) is considered to be the critical buckling moment,
Mcr =FcrL
4=
2π3
(8EIfπzg +
√EIf
[GJtL2 (4 + π2)
2+ 64EIfπ2z2g
])L2 (4 + π2)
2 (4.123a)
1It should be noted that Eq. (4.117) seems to be dimensionally non-homogeneous. This results from the fact that — instead ofthe represented heuristic definition —, accurately, the Dirac delta function is a distribution whose integral along a one-dimensionalspace equals 1.
74
which may also be written in the form presented in Eurocode 3 [31] (Eq. (4.116b)), where C1 and C2 are
now given by:
C1 =2π2
4 + π2= 1.4232 (4.123b)
C2 =8
4 + π2= 0.5768 (4.123c)
Alternative approach using the Rayleigh-Ritz method
Analytical expressions for the prediction of the critical buckling moment can be obtained applying the
Rayleigh-Ritz method with the same approximate sinusoidal solutions, yielding the exact same results as
before. The equations describing the potential energy for the cases of pure bending, uniformly distributed
load, and mid-span load, are, respectively:
U =1
2
∫ L
0
[EIf (v′′)2 +GJt(φ
′)2 + 2Mv′′φ]
dx = 0 (4.124a)
U =1
2
∫ L
0
[EIf (v′′)2 +GJt(φ
′)2 + 2Mv′′φ+ qzgφ2]
dx = 0 (4.124b)
U =1
2
∫ L/2
0
[EIf (v′′)2 +GJt(φ
′)2 + 2Mv′′φ+F
2δ
(x− L
2
)zgφ
2
]dx = 0 (4.124c)
Alternative solution proposed by some authors
Some authors, e.g. Timoshenko and Gere [44], Reis and Camotim [40] and Mohri et al. [43], propose
the differential equations that define the lateral-torsional behavior of the simply supported beam to be
uncoupled from v, in order to yield a single differential equation. This procedure corresponds to a
mathematical simplification that introduces some inaccuracies regarding the boundary condition terms
when weighted residuals methods are applied. The resulting equation is, for the cases of pure bending,
uniformly distributed load, and mid-span load, respectively:
M2
EIfφ+GJtφ
′′ = 0 (4.125a)
M2
EIfφ+GJtφ
′′ − qzgφ = 0 (4.125b)
M2
EIfφ+GJtφ
′′ − F
2δ
(x− L
2
)zgφ = 0 (4.125c)
Galerkin’s method can analogously be applied to these equations, now with a single approximate sinu-
soidal solution, for φ. The resulting critical moment in pure bending is the same as before (Eq. (4.109)),
whereas for uniformly distributed load and mid-span load it is slightly different. The critical moment for a
75
simply supported beam subjected to a uniformly distributed load is given by Eq. (4.116b), where C1 and
C2 are now given by,
C1 =
√15π4
8π4 + 360= 1.1325 (4.126a)
C2 =8C1
2π2= 0.4590 (4.126b)
whereas for a simply supported beam subjected to a mid-span load the coefficients are:
C1 = π
√3
6 + π2= 1.3659 (4.126c)
C2 =12
C1 (6 + π2)= 0.5536 (4.126d)
As Mohri et al. [43] stated, these coefficients are equal to those adopted in Eurocode 3 [31]. The same
authors claim having obtained results closer to finite element results with these coefficients, compared
with the coefficients from Eqs. (4.116c), (4.116d), (4.123b) and (4.123c).
4.4.2 Lateral deflection and rotation
Real glass elements are naturally geometrically imperfect and thus they do not follow the fundamental
path until the critical buckling load is reached. Instead, they present 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 fundamental and post-critical path.
In the present Subsection the non-linear (2nd order) behavior of imperfect laminated glass beams is
analytically studied. A more accurate post-buckling analysis would require cubic and 4th order terms for
the potential energy U . Since these terms have a small impact in the case of beams, the model with 2nd
order terms is deemed to provides a sufficiently good approximation (Fig. 4.31) [40]. The expressions
that approximately define the lateral deflection and rotation (load-displacement path) of these beams are
determined for different loading conditions, again using Galerkin’s method.
Pure bending
The initial geometrical imperfections of beams are important parameters for the definition of load-
displacement curves. An initial deflection (v0) measured in the cross-section centroid and an initial
twist rotation (φ0) of the sections must be included in Eqs. (4.105b) and (4.105c) as initial geometrical
imperfections:
EIfv′′ = −M (φ+ φ0) (4.127a)
GJtφ′ = M (v′ + v′0) (4.127b)
76
Figure 4.31: Analytical post-buckling path.
Similarly as for the deflection and rotation in the critical buckling load analysis (Eqs. (4.106)), the initial
geometrical imperfections are approximated by the following sinusoidal functions,
u3 = q31ψ11 = v0,max sinπx
L(4.128a)
u4 = q41ψ21 = φ0,max sinπx
L(4.128b)
where v0,max and φ0,max are the maximum initial deflection and rotation (at mid-span), respectively.
The residuals obtained from Eqs. (4.127), with the approximate solutions from Eqs. (4.106) and (4.128),
are:
r1 = EIf u′′1 +M (u2 + u4) = −EIfvmax
(πL
)2sin
πx
L+M (φmax + φ0,max) sin
πx
L(4.129a)
r2 = GJtu′2 −M (u′1 + u′3) = GJtφmax
π
Lcos
πx
L−M (vmax + v0,max)
π
Lcos
πx
L(4.129b)
For pure bending, the Galerkin’s equations are,
∫r1ψ11dx =
EIfπ2vmax −ML2 (φmax + φ0,max)
4L2π
(−2πx+ L sin
2πx
L
)= 0 (4.130a)∫
r2ψ21dx =M (vmax + v0,max)−GJtφmax
4cos
2πx
L= 0 (4.130b)
which solved together for vmax and φmax yield the maximum deflection and the maximum rotation, re-
spectively:
vmax =ML2 (GJtφ0,max +Mv0,max)
EIfGJtπ2 − (ML)2 (4.131a)
φmax =M(ML2φ0,max + EIfπ
2v0,max)
EIfGJtπ2 − (ML)2 (4.131b)
These expressions are equivalent to those proposed by Kasper et al. [4]. It can be verified that their
77
limit when M → Mcr (Eq. (4.109)) equals ∞, i.e., the load-displacement path has an asymptote for
M = Mcr. This happens since the analysis was restricted to 2nd order terms in the energy (to linear
terms in the equilibrium equations).
Uniformly distributed load
For a simply supported beam subjected to a uniformly distributed load, taking into account the initial
geometrical imperfections, Eqs. (4.111) become,
EIfv′′ = −M (φ+ φ0) (4.132a)
GJtφ′′ = M (v′′ + v′′0 ) + qzg (φ+ φ0) (4.132b)
where the bending moment M has the longitudinal distribution given by Eq. (4.112).
The residuals, obtained from Eqs. (4.132) with the approximate solutions from Eqs. (4.106) and (4.128),
are:
r1 = EIf u′′1 +M (u2 + u4) =
[−EIfπ
2vmax
L2+q(Lx− x2
)(φmax + φ0,max)
2
]sin
πx
L(4.133a)
r2 = GJtu′′2 −M (u′′1 + u′′3)− qzg (u2 + u4) =
=
[−GJtπ
2φmax
L2+q(Lx− x2
)π2 (vmax + v0,max)
2L2− qzg (φmax + φ0,max)
]sin
πx
L
(4.133b)
Galerkin’s equations, are,
∫ L
0
r1ψ11dx =qL4
(3 + π2
)(φmax + φ0,max)− 12EIfπ
4vmax
24Lπ2= 0 (4.134a)∫ L
0
r2ψ21dx =
=qL2
(3 + π2
)(vmax + v0,max)− 12qL2zg (φmax + φ0,max)− 12GJtπ
2φmax
24L= 0
(4.134b)
which solved together for vmax and φmax yield the maximum deflection and the maximum rotation, re-
spectively,
vmax =qL4
(3 + π2
) [12GJtπ
2φ0,max + qL2(3 + π2
)v0,max
]144EIfπ4 (GJtπ2 + qL2zg)− [qL3 (3 + π2)]
2 (4.135a)
φmax =qL2
(qL4
(3 + π2
)2φ0,max + 12EIfπ
4[(
3 + π2)v0,max − 12zgφ0,max
])144EIfπ4 (GJtπ2 + qL2zg)− [qL3 (3 + π2)]
2 (4.135b)
Taking into account that M = qL2/8 at mid-span, these expressions can be rewritten as,
vmax =2ML2
(3 + π2
) [3GJtπ
2φ0,max + 2M(3 + π2
)v0,max
]9EIfπ4 (GJtπ2 + 8Mzg)− [2ML (3 + π2)]
2 (4.136a)
78
φmax =2M
(2ML2
(3 + π2
)2φ0,max + 3EIfπ
4[(
3 + π2)v0,max − 12zgφ0,max
])9EIfπ4 (GJtπ2 + 8Mzg)− [2ML (3 + π2)]
2 (4.136b)
or, making use of some previously introduced coefficients,
vmax =ML2 (C1GJtφ0,max +Mv0,max)
C1EIfπ2 (C1GJt + 2C2Mzg)− (ML)2 (4.136c)
φmax =M(ML2φ0,max + C1EIfπ
2 [v0,max − 2C2zgφ0,max])
C1EIfπ2 (C1GJt + 2C2Mzg)− (ML)2 (4.136d)
where C1 and C2 are given by Eqs. (4.116c) and (4.116d), respectively.
These expressions are equivalent to those proposed by Kasper et al. [4]. The coefficients proposed by
the authors are,
c1 =2
3+
2
π2= 0.8693 (4.137a)
c2 =8
π2= 0.8106 (4.137b)
and relate to coefficients from Eqs. (4.116c) and (4.116d) as follows:
c1 =1
C1(4.138a)
c2 =2C2
C1(4.138b)
Analogously to the pure bending expressions, it can be verified that the limit of Eqs. (4.136c) and
(4.136d) when M →Mcr (Eq. (4.116b)) equals∞.
Mid-span load
In the case of a simply supported beam subjected to a mid-span load, taking into account the initial
geometrical imperfections, Eqs. (4.118) become,
EIfv′′ = −M (φ+ φ0) (4.139a)
GJtφ′′ = M (v′′ + v′′0 ) +
F
2δ
(x− L
2
)zg (φ+ φ0) (4.139b)
where the bending moment M has the longitudinal distribution given by Eq. (4.119).
The residuals, which result from Eqs. (4.139) considering the approximate solutions from Eqs. (4.106)
and (4.128), are:
r1 = EIf u′′1 +M (u2 + u4) =
[−EIfπ
2vmax
L2+Fx (φmax + φ0,max)
2
]sin
πx
L(4.140a)
79
r2 = GJtu′′2 −M (u′′1 + u′′3)− F
2δ
(x− L
2
)zg (u2 + u4) =
=
[−GJtπ
2φmax
L2+Fxπ2 (vmax + v0,max)
2L2−
−F2δ
(x− L
2
)zg (φmax + φ0,max)
]sin
πx
L
(4.140b)
Galerkin’s equations are now,
∫ L/2
0
r1ψ11dx =FL3
(4 + π2
)(φmax + φ0,max)− 8EIfπ
4vmax
32Lπ2= 0 (4.141a)∫ L/2
0
r2ψ21dx =
=FL
(4 + π2
)(vmax + v0,max)− 16FLzg (φmax + φ0,max)− 8GJtπ
2φmax
32L= 0
(4.141b)
which solved together for vmax and φmax yield the maximum deflection and the maximum rotation, re-
spectively,
vmax =FL3
(4 + π2
) [8GJtπ
2φ0,max + FL(4 + π2
)v0,max
]64EIfπ4 (GJtπ2 + 2FLzg)− [FL2 (4 + π2)]
2 (4.142a)
φmax =FL
(FL3
(4 + π2
)2φ0,max + 8EIfπ
4[(
4 + π2)v0,max − 16zgφ0,max
])64EIfπ4 (GJtπ2 + 2FLzg)− [FL2 (4 + π2)]
2 (4.142b)
Since M = FL/4 at mid-span, these expressions can be rewritten as:
vmax =ML2
(4 + π2
) [2GJtπ
2φ0,max +M(4 + π2
)v0,max
]4EIfπ4 (GJtπ2 + 8Mzg)− [ML (4 + π2)]
2 (4.143a)
φmax =M(ML2
(4 + π2
)2φ0,max + 2EIfπ
4[(
4 + π2)v0,max − 16zgφ0,max
])4EIfπ4 (GJtπ2 + 8Mzg)− [ML (4 + π2)]
2 (4.143b)
As for the simply supported beam under uniformly distributed load, the maximum deflection and the
maximum rotation can also be determined from Eqs. (4.136c) and (4.136d), with the difference that, in
the present case, the coefficients C1 and C2 are given by Eqs. (4.123b) and (4.123c), respectively.
The expressions are again equivalent to those proposed by Kasper et al. [4] — formulation 2-D from
Table 3.4. It is again evident that the relation between the coefficients is given by Eqs. (4.138).
Alternative approach using the Rayleigh-Ritz method
The expressions for the lateral deflection and rotation, deduced before, can also be obtained applying
the Rayleigh-Ritz method with the same approximate sinusoidal functions. The equations describing
the potential energy for the cases of pure bending, uniformly distributed load, and mid-span load, are,
80
respectively:
U =1
2
∫ L
0
[EIf (v′′)2 +GJt(φ
′)2 + 2M (v′′ + v′′0 ) (φ+ φ0)]
dx = 0 (4.144a)
U =1
2
∫ L
0
[EIf (v′′)2 +GJt(φ
′)2 + 2M (v′′ + v′′0 ) (φ+ φ0) + qzg (φ+ φ0)2]
dx = 0 (4.144b)
U =1
2
∫ L/2
0
[EIf (v′′)2 +GJt(φ
′)2+
+2M (v′′ + v′′0 ) (φ+ φ0) +F
2δ
(x− L
2
)zg (φ+ φ0)
2
]dx = 0
(4.144c)
Adaptation of the alternative approach for the determination of the lateral deflection and rotation
In Subsection 4.4.1 the critical buckling moment of beams subjected to pure bending, uniformly dis-
tributed load and mid-span load was derived, from two coupled differential equations. The coefficients
C1 and C2 associated with the two last loading conditions, when the critical buckling moment is written
in the general form provided in Eurocode 3 [31], were presented. Other coefficients, applicable to the
same expression for the critical buckling moment (Eq. (4.109)), from an alternative approach proposed
by some authors, were also presented. These other coefficients equal those provided in Eurocode 3
[31].
In the present Subsection the lateral deflection and rotation of imperfect beams were derived from the
same two coupled differential equations (properly adapted in order to include the geometrical imperfec-
tions). It was mentioned that their limit when M →Mcr equals∞, independently of the coefficients (i.e.,
the loading condition), where Mcr is given by the expressions of Subsection 4.4.1.
The alternative coefficients given by Eqs. (4.126a)–(4.126d) may thus be introduced in Eqs. (4.136c)
and (4.136d) without loss of compatibility between the expressions for the lateral deflection and rotation
and the expression for the critical buckling moment.
Summary of the main expressions regarding lateral-torsional buckling
In the present Section two possible formulations for the lateral-torsional behavior of monolithic beams
subjected to pure bending, uniformly distributed load and mid-span load were studied.
One of the formulations was explicitly derived using the Galerkin’s method, while the other was intro-
duced as coming from an alternative manipulation of the base differential equations, proposed by some
authors.
The critical buckling moment, the maximum deflection and the maximum rotation of laminated glass
beams subjected to pure bending, uniformly distributed load and mid-span load may be determined
from Table 4.6, with the respective coefficients corresponding to the first formulation. The alternative
coefficients described in the end of Subsection 4.4.1 are also included.
81
Table 4.6: Lateral-torsional buckling expressions
Mcr = C1π2EIfL2
C2zg +
√(C2zg)
2+GJtEIf
(L
π
)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
C1 = 1 C1 =3π2
2 (3 + π2)= 1.1503 C1 =
2π2
4 + π2= 1.4232
C2 = 0 C2 =6
3 + π2= 0.4662 C2 =
8
4 + π2= 0.5768
Alternative coefficients
C1 = 1 C1 =
√15π4
8π4 + 360= 1.1325 C1 = π
√3
6 + π2= 1.3659
C2 = 0 C2 =8C1
2π2= 0.4590 C2 =
12
C1 (6 + π2)= 0.5536
All the expressions in Table 4.6 result from the assumption that the lateral deflection and the rotation
of the beams have sinusoidal shapes. The maximum initial deflection v0,max and the maximum initial
rotation φ0,max correspond to the maximum respective values of the sinuses, at mid-span.
Alternatively, Appendix C from Australian Standard AS 1288 [36] provides additional expressions (only
for the critical buckling moment) associated with various loading and support conditions, with the pos-
sibility of intermediate buckling restraints. The expression for the critical buckling moment of beams
without intermediate buckling restrains was adapted and is presented in Table 4.7, with the correspond-
ing coefficients when subjected to a uniformly distributed load or a mid-span load.
4.5 Concluding remarks
The analytical study developed in the present Chapter resulted in a more comprehensive and unified
formulation, able to characterize the flexural and torsional stiffness of laminated glass elements up to
five layers and subjected to various loading conditions. This extends what can presently be found in the
structural glass literature.
The proposed formulation is built on simple equations, analogous to bending and torsion, where a pa-
82
Table 4.7: Lateral-torsional buckling expressions from AS 1288 [36] – in-plane transverse loading
Mcr = g2EIfL2
(g3zg +
√GJtEIf
L2
)
g2 = 3.6 g2 = 4.2
g3 = 1.4 g3 = 1.7
rameter (ξf for bending and ξt for torsion) represents the transition from a monolithic behavior to the
absence of shear connection between the glass layers (1 ≥ ξf,t ≥ 0).
It has been highlighted that the presented theories, described in two sandwich theory works [32, 38],
allow the extension to additional loading and support conditions, and to larger number of layers in lami-
nated elements. Further analytical analyses only become more complex.
An analytical formulation for the lateral-torsional buckling behavior of laminated glass beams subjected
to pure bending, uniformly distributed load and mid-span load, based on equivalent flexural and torsional
stiffness formulations, has been presented in detail. The results are equivalent to the formulation pro-
posed by Kasper et al. [4]. Moreover, the possible relation between this formulation and that adopted in
an early version of Eurocode 3 [31] has been presented.
Because in some cases the formulation from Australian Standard AS 1288 [36] seems to yield better
results (see Chapter 3), its expression was also presented for the loading conditions analyzed in the
present dissertation.
The most relevant expressions determined and analyzed in the present Chapter are summarized in
Appendix C.
83
84
Chapter 5
Parametric study
5.1 Preliminary remarks
In Chapter 3 several existing analytical formulations for equivalent flexural and torsional stiffness of lam-
inated glass columns/beams have been numerically assessed. The ones that provided better results
were addressed in the analytical study developed in Chapter 4. The result was a new more comprehen-
sive formulation, able to characterize the flexural and torsional stiffness of laminated glass elements up
to five layers and subjected to various loading conditions.
In order to validate all analytical expressions, and to determine their range of validity, a comprehensive
parametric study has been developed. This solves the problem of the present lack of information about
the applicability range of the existing expressions.
The parametric study presented in this Chapter is based on the comparison of the analytical results
with those provided by numerical models implemented in commercial software Abaqus [37]. Section 5.2
presents the parameters that support the parametric study. Besides what has already been introduced
in Chapter 3, Section 5.3 describes the finite element models used in this parametric study. The list of
all numerical models, with their geometrical dimensions, is detailed in Section 5.4. The results of the
study and its main conclusions are summarized in Section 5.5.
5.2 Relevant parameters
The expressions formulated in Chapter 4 may be divided into three groups: (i) one expression for the
equivalent flexural stiffness of columns subjected to compressive axial loads (Fig. 3.2; page 18); (ii)
four expressions for the equivalent flexural stiffness of beams subjected to transverse loading conditions
(e.g. Fig. 3.3; page 20); and (iii) one expression for the equivalent torsional stiffness. All formulations
are applicable to laminated glass elements with two, three, four or five layers (Figs. 4.1, 4.6, 4.8 and
4.11; pages 34, 39, 41 and 43).
85
For each of these groups, the parametric analysis consists of evaluating the influence of the following
variables: (i) the ratio span/width (L/b); (ii) the ratio width/total thickness (b/ttot); and, only for the condi-
tion of transverse loading (iii) the overhangs’ length (L/L1). Additionally, different degrees of interaction
between the glass layers are taken into account, as well as variations in the number of laminated layers.
5.3 Definition of the numerical models
The transverse loading condition chosen to be part of the parametric study was the mid-span load. Its
numerical model has already been introduced in Chapter 3, for the two-layered case. The models of
the column subjected to compressive axial load and of the beam subjected to a torsional moment have
also been introduced. For this parametric study, however, a much larger number of models had to be
developed.
All numerical models implemented in this Chapter followed the same construction than that explained in
Chapter 3, with the glass layers being modeled with eight-node continuum shell elements with reduced
integration (SC8R), while eight-node solid elements with reduced integration (C3D8R) were used to
simulate the interlayer. The approximate size of the elements in the plane of the column/beam was
chosen to be 10 mm×10 mm, with one element in the thickness of the glass layers and two elements in
the thickness of the interlayer plies (Fig. 3.5(c)).
The analyses of the parameters was performed on three-layered models of laminated glass columns and
beams. A geometrical reference case has been defined, with all parameters changing from there. The
reference model comprises three 10 mm thick (t) glass layers, two 1.52 mm thick (tint) interlayer plies,
the ratios L/b and b/ttot being equal to 15 and 10, respectively, and without overhangs (L/L1 =∞).
Nevertheless, in order to validate all expressions, the study includes models up to five layers and also
three-layered models subjected to the remaining transverse loading conditions and to pure bending.
Because the expressions do not exclude laminated elements with glass layers of unequal thickness,
some three-layered models were built in order to assess the analytical expressions for this particular
geometrical possibility. The geometry of these additional models is that associated with the geometrical
reference case.
The boundary conditions are very similar to those described in Chapter 3. For the beams subjected
to a torsional moment, the only boundary condition is located at mid-span, in the axes of symme-
try of the cross-section (Fig. 3.5(b)), and corresponds to preventing all displacements. For all other
models, the longitudinal and transverse partitions form, in all their area, the boundary conditions that
prevent displacements in the xx and zz directions, respectively. The boundary condition that prevents
displacements in the yy direction is positioned in the horizontal axis of symmetry of the two extremity
cross-sections for elements with odd number of layers and half the thickness of the interlayer from that
location for elements with even number of layers (highlighted line illustrated in Figs. 3.4(c) and 5.1(a)),
which is deemed to be a sufficiently good approximation.
86
The application of compressive axial loads, mid-span loads and torsional moments has also already
been described in Chapter 3. The axial loads are applied through surface loads in both ends of all layers
(Figs. 3.4(c) and 5.1(a)); the mid-span loads are applied in the mid-span cross-section, along the width,
in the horizontal axis of symmetry for elements with odd number of layers and half the thickness of the
interlayer from that location for elements with even number of layers (Fig. 3.4(b)); the opposing torsional
moments are applied in both ends of the beams, with a distribution defined by a linear function, —
they are applied in the horizontal axis of symmetry for elements with odd number of layers (Fig. 5.1(b))
and half the thickness of the interlayer from that location for elements with even number of layers (Fig.
3.5(c)).
(a) Boundary condition in the yy direction and axial pressure load in four-layered column.
(b) Application of a torsional moment in five-layered beam.
Figure 5.1: Numerical models with four and five laminated layers.
The two additional transverse loading conditions which were modeled are the uniformly distributed load
and the four-point bending condition. As mentioned before, they were analyzed only for the three-layered
case. The former load was applied in the mid-surface of the interior glass layer, as a surface load (Fig.
5.2(a)), while the latter load was applied in the horizontal axis of symmetry of the two quarter-span
cross-sections illustrated in Fig. 5.2(b). These cross-sections were defined by two additional partitions.
The beam subjected to pure bending is illustrated in Fig. 5.3(a). The opposing moments were applied
through binary loads in both ends of the beam (Fig. 5.3(c)). The loads were applied along the width, on
the top and bottom edges of the interior glass layer, which means that the lever-arm equals the thickness
of the layer.
One of the models implemented to study the influence of the overhangs’ length is illustrated in Fig.
87
(a) Beam subjected to a uniformly distributed load. (b) Beam subjected to four-point loading.
Figure 5.2: Numerical models of the laminated glass beams subjected to transverse loads.
5.3(b). The only differences in comparison with the reference model of the three-layered laminated
glass beam subjected to a mid-span load are (i) the position of the boundary condition that prevents
displacements in the yy direction and (ii) the additional length. Two additional partitions were introduced
in order to define the overhangs. The boundary condition was applied in the horizontal axis of symmetry
of the adequate cross-sections, as illustrated in Fig. 5.3(c).
As in Chapter 3, the materials of both glass and interlayer were considered to be linear elastic. The
Young’s modulus of glass is 70 GPa and its Poisson ratio is 0.23. The interlayer properties vary within the
different models. Two different degrees of interaction between the glass layers were generally adopted
for the parametric study. The parameters mentioned in Section 5.2 were analyzed for both degrees. They
are defined by the properties of PVB and SG associated with a temperature of 20 ◦C and a load duration
of 1 second. The corresponding values of their shear modulus (Gint), Poisson ratio (νint) and Young’s
modulus (Eint) were already given in Chapter 3 and are now summarized in Table 5.1 (“PVB” and “SG”
rows, respectively). Additionally, for the study of the influence of the overhangs’ length, some models
were implemented with a softer PVB, whose properties are associated with a temperature of 20 ◦C and
a load duration of about 1 day (“PVB1d” in Table 5.1). The accuracy of all analytical formulations was
also tested with the very soft PVB already used in Chapter 3, associated with a temperature of 20 ◦C
and a load duration of more than 10 years (“PVB10y” in Table 5.1).
Table 5.1: Properties of the interlayers used in the parametric study [15].
Type of interlayer Gint (MPa) νint ( – ) Eint (MPa)
SG 142 0.465 416.1
PVB 13.7 0.497 41.0
PVB1d 0.504 0.500 1.512
PVB10y 0.0517 0.500 0.1551
Linear buckling analyses were performed in order to evaluate the critical buckling loads of columns
88
(a) Beam subjected to pure bending. (b) Beam with overhangs.
(c) Application of the bending moments.
(d) Boundary condition in the yy direction, in the overhang.
Figure 5.3: Numerical models of the laminated glass beams subjected to pure bending and with overhangs.
subjected to compressive axial load. In these models the applied load was unitary, which means the first
eigenvalue equals the desired critical buckling load.
Linear static analyses were performed on all other models. The intensity of the loads applied in the cases
of mid-span load, torsional moment, pure bending, uniformly distributed load and four-point bending1 is
summarized in Table 5.2. As explained in Chapter 3, the displacements were measured at quarters of
the span for torsion and in the centroid of the mid-span cross-section for all other models.
1The intensity provided in Table 5.2 refers to each of the two loads.
89
Table 5.2: Intensity of the loads applied on the linear static analyses of the parametric study.
Mid-span load F = 1 kN
Torsional moment T = 50 N m
Pure bending M = 1 kN m
Uniformly distributed load q = 1 kN/m
Four-point bending F = 1 kN
5.4 List of models
A total of 77 models were developed. Besides the previously mentioned reference case, with the ratios
L/b and b/ttot equal to 15 and 10, respectively, and without overhangs (L/L1 = ∞), the parameter L/b
was studied for two other geometrical cases. Keeping all the other parameters unchanged, the values
adopted for L/b were 8, 10 and 25, while the parameter b/ttot equals 2, 5, 10 and 25, with the other
parameters being kept constant (Table 5.3). The parameter b/ttot was analyzed for the column subjected
to compressive axial load, for the beam subjected to transverse loading and for the beam subjected to
a torsional moment. The parameter L/b was analyzed for all but the beam subjected to a torsional
moment, because the torsional stiffness does not depend on the span. Both parameters were analyzed
for “SG” and “PVB” interlayers.
Table 5.3 also shows that the length of the overhangs was analyzed with L/L1 equal to 1, 2, 8, 20 and
∞ for “PVB1d”; for “SG” and “PVB”, only the two limiting cases were considered.
All the other models were implemented with geometries associated with the reference case (Tables 5.4
and 5.5). The only difference is on the models that test the use of glass layers of unequal thickness. In
that case, the elements still have three layers, but the interior one is 12 mm thick, instead of 10 mm.
The models that test the expressions for all loading conditions with a very soft interlayer (“PVB10y”) are
also associated to the three-layered geometrical reference case (Table 5.5).
5.5 Analytical and numerical results
The analytical critical load of the columns subjected to compressive axial load was determined with
Euler’s equation (Eq. (3.3)), with the equivalent flexural stiffness (EIf ) given by Tables 4.1 and 4.4.
As mentioned before, the numerical critical buckling load equals the first eigenvalue. In the case of
the beams subjected to a transverse mid-span load F , whose magnitude is provided in Table 5.2, the
analytical displacement at mid-span was determined with Eq. (3.4), with the equivalent flexural stiffness
(EIf ) also given by Tables 4.1 and 4.4. The analytical rotation of the beams subjected to a torsional
moment T was determined with Eq. (3.5), using the equivalent flexural stiffness (GJt) given by Tables
4.1 and 4.5.
90
T abl
e5.
3:Li
stof
mod
els
with
geom
etrie
s–
mai
npa
ram
eter
s(r
elat
ive
perc
enta
gedi
ffere
nces
betw
een
the
anal
ytic
alan
dth
enu
mer
ical
resu
ltsin
pare
nthe
ses)
.
Inte
r laye
rP
aram
eter
Type
oflo
adin
gLa
bel
t(m
m)
b(m
m)
L(m
m)
L1
(mm
)L/b
b/t tot
L/L1
Num
eric
alA
naly
tical
Uni
t
L/b
Axi
alco
mpr
essi
on
L/S
G/P
-110
330
2643
08
10∞
9392
293
879
(−0.
05%
)N
Ref
/SG
/P10
330
4956
015
10∞
2689
126
883
(−0.
03%
)N
L/S
G/P
-210
330
8260
025
10∞
9698
9695
(−0.
03%
)N
T ran
sver
selo
adin
g(m
id-s
pan
load
)
L/S
G/F
-110
330
2643
08
10∞
5.79
5.80
(+0.
17%
)m
mR
ef/S
G/F
1033
049
560
1510
∞37
.90
37.9
3(+
0.07
%)
mm
L/S
G/F
-210
330
8260
025
10∞
175.
1517
5.22
(+0.
04%
)m
m
Axi
alco
mpr
essi
on
b/S
G/P
-110
6699
10
152
∞12
610
812
621
1(+
0.08
%)
Nb/
SG
/P-2
1016
524
780
155
∞53
334
5332
8(−
0.01
%)
NR
ef/S
G/P
1033
049
560
1510
∞26
891
2688
3(−
0.03
%)
N
SG
b/S
G/P
-310
826
1239
00
1525
∞10
796
1079
1(−
0.04
%)
N
b/t tot
Tran
sver
selo
adin
g(m
id-s
pan
load
)
b/S
G/F
-110
6699
10
152
∞1.
641.
63(−
0.28
%)
mm
b/S
G/F
-210
165
2478
015
5∞
9.57
9.57
(+0.
02%
)m
mR
ef/S
G/F
1033
049
560
1510
∞37
.90
37.9
3(+
0.07
%)
mm
b/S
G/F
-310
826
1239
00
1525
∞23
5.88
236.
10(+
0.09
%)
mm
Tors
iona
lm
omen
t
b/S
G/T
-110
6699
10
152
∞0.
006
850.
006
25(−
8.65
%)
rad
b/S
G/T
-210
165
2478
015
5∞
0.00
261
0.00
254
(−2.
63%
)ra
dR
ef/S
G/T
1033
049
560
1510
∞0.
001
650.
001
63(−
0.98
%)
rad
b/S
G/T
-310
826
1239
00
1525
∞0.
001
300.
001
29(−
0.32
%)
rad
L/L1
Mid
-spa
nlo
adR
ef/S
G/F
1033
049
560
1510
∞37
.90
37.9
3(+
0.07
%)
mm
L1/S
G/F
-110
330
4956
4956
1510
137
.90
37.9
3(+
0.07
%)
mm
L/b
Axi
alco
mpr
essi
on
L/P
VB
/P-1
1033
026
430
810
∞86
224
8622
1(0
.00%
)N
Ref
/PV
B/P
1033
049
560
1510
∞26
212
2621
2(0
.00%
)N
L/P
VB
/P-2
1033
082
600
2510
∞96
0696
06(0
.00%
)N
T ran
sver
selo
adin
g(m
id-s
pan
load
)
L/P
VB
/F-1
1033
026
430
810
∞6.
386.
39(+
0.14
%)
mm
Ref
/PV
B/F
1033
049
560
1510
∞39
.05
39.0
7(+
0.04
%)
mm
L/P
VB
/F-2
1033
082
600
2510
∞11
7.12
117.
15(+
0.02
%)
mm
Axi
alco
mpr
essi
on
b/P
VB
/P-1
1066
991
015
2∞
8078
080
820
(+0.
05%
)N
b/P
VB
/P-2
1016
524
780
155
∞48
442
4844
6(+
0.01
%)
NR
ef/P
VB
/P10
330
4956
015
10∞
2621
226
212
(0.0
0%)
N
PV
Bb/
PV
B/P
-310
826
1239
00
1525
∞10
748
1074
7(−
0.01
%)
N
b/t tot
Tran
sver
selo
adin
g(m
id-s
pan
load
)
b/P
VB
/F-1
1066
991
015
2∞
2.61
2.61
(−0.
17%
)m
mb/
PV
B/F
-210
165
2478
015
5∞
10.6
710
.67
(+0.
01%
)m
mR
ef/P
VB
/F10
330
4956
015
10∞
39.0
539
.07
(+0.
04%
)m
mb/
PV
B/F
-310
826
1239
00
1525
∞23
7.13
237.
27(+
0.06
%)
mm
Tors
iona
lm
omen
t
b/P
VB
/T-1
1066
991
015
2∞
0.01
283
0.01
177
(−8.
23%
)ra
db/
PV
B/T
-210
165
2478
015
5∞
0.00
794
0.00
774
(−2.
53%
)ra
dR
ef/P
VB
/T10
330
4956
015
10∞
0.00
406
0.00
402
(−0.
86%
)ra
db/
PV
B/T
-310
826
1239
00
1525
∞0.
001
850.
001
85(−
0.21
%)
rad
L/L1
T ran
sver
selo
adin
g(m
id-s
pan
load
)
Ref
/PV
B/F
1033
049
560
1510
∞39
.05
39.0
7(+
0.04
%)
mm
L1/P
VB
/F-1
1033
049
5649
5615
101
39.0
339
.05
(+0.
04%
)m
m
PV
B1d
Ref
/PV
B1d
/F10
330
4956
015
10∞
67.2
567
.30
(+0.
08%
)m
mL1
/PV
B1d
/F-1
1033
049
5624
815
1020
65.1
665
.22
(+0.
10%
)m
mL1
/PV
B1d
/F-2
1033
049
5662
015
108
64.3
664
.44
(+0.
12%
)m
mL1
/PV
B1d
/F-3
1033
049
5624
7815
102
64.1
964
.28
(+0.
13%
)m
mL1
/PV
B1d
/F-4
1033
049
5649
5615
101
64.1
964
.28
(+0.
13%
)m
m
91
Tabl
e5.
4:Li
stof
mod
els
with
geom
etrie
s–
addi
tiona
lpar
amet
ers
(rel
ativ
epe
rcen
tage
diffe
renc
esbe
twee
nth
ean
alyt
ical
and
the
num
eric
alre
sults
inpa
rent
hese
s).
Inte
rlaye
rP
aram
eter
Type
oflo
adin
gLa
bel
t(m
m)
b(m
m)
L(m
m)
L1
(mm
)L/b
b/t tot
L/L1
Num
eric
alA
naly
tical
Uni
t
2la
yers
Axi
alco
mpr
essi
on2l
y/S
G/P
1021
532
280
1510
∞11
500
1149
9(−
0.01
%)
NM
id-s
pan
load
2ly/
SG
/F10
215
3228
015
10∞
57.9
557
.98
(+0.
04%
)m
mTo
rsio
nalm
omen
t2l
y/S
G/T
1021
532
280
1510
∞0.
011
510.
011
29(−
1.85
%)
rad
4la
yers
Axi
alco
mpr
essi
on4l
y/S
G/P
1044
666
840
1510
∞46
768
4679
2(+
0.05
%)
NM
id-s
pan
load
4ly/
SG
/F10
446
6684
015
10∞
29.5
229
.52
(−0.
02%
)m
mTo
rsio
nalm
omen
t4l
y/S
G/T
1044
666
840
1510
∞0.
001
800.
001
79(−
0.54
%)
rad
5la
yers
Axi
alco
mpr
essi
on5l
y/S
G/P
1056
184
120
1510
∞72
982
7305
4(+
0.10
%)
NS
GM
id-s
pan
load
5ly/
SG
/F10
561
8412
015
10∞
23.8
123
.79
(−0.
07%
)m
mTo
rsio
nalm
omen
t5l
y/S
G/T
1056
184
120
1510
∞0.
000
940.
000
94(−
0.41
%)
rad
Oth
ertra
nsve
rse
load
ings
Pur
ebe
ndin
gR
ef/S
G/M
1033
049
560
1510
∞46
.82
46.8
2(+
0.01
%)
mm
Uni
f.di
stri
b.lo
adR
ef/S
G/q
1033
049
560
1510
∞12
0.29
120.
33(+
0.04
%)
mm
4-po
intb
endi
ngR
ef/S
G/2
F10
330
4956
015
10∞
53.2
753
.29
(+0.
03%
)m
m
Une
qual
thic
knes
s
Axi
alco
mpr
essi
ont2
/SG
/P10
+12+
1035
052
560
1510
∞29
420
2941
9(0
.00%
)N
Mid
-spa
nlo
adt2
/SG
/F10
+12+
1035
052
560
1510
∞36
.88
36.9
0(+
0.04
%)
mm
Tors
iona
lmom
ent
t2/S
G/T
10+1
2+10
350
5256
015
10∞
0.00
319
0.00
317
(−0.
89%
)ra
d
2la
yers
Axi
alco
mpr
essi
on2l
y/P
VB
/P10
215
3228
015
10∞
1180
311
801
(−0.
01%
)N
Mid
-spa
nlo
ad2l
y/P
VB
/F10
215
3228
015
10∞
56.2
556
.27
(+0.
05%
)m
mTo
rsio
nalm
omen
t2l
y/P
VB
/T10
215
3228
015
10∞
0.00
566
0.00
558
(−1.
54%
)ra
d
4la
yers
Axi
alco
mpr
essi
on4l
y/P
VB
/P10
446
6684
015
10∞
4795
647
938
(−0.
04%
)N
Mid
-spa
nlo
ad4l
y/P
VB
/F10
446
6684
015
10∞
28.6
628
.68
(+0.
08%
)m
mTo
rsio
nalm
omen
t4l
y/P
VB
/T10
446
6684
015
10∞
0.00
069
0.00
068
(−0.
91%
)ra
d
5la
yers
Axi
alco
mpr
essi
on5l
y/P
VB
/P10
561
8412
015
10∞
7481
474
781
(−0.
04%
)N
PV
BM
id-s
pan
load
5ly/
PV
B/F
1056
184
120
1510
∞23
.12
23.1
4(+
0.08
%)
mm
Tors
iona
lmom
ent
5ly/
PV
B/T
1056
184
120
1510
∞0.
000
350.
000
35(−
0.86
%)
rad
Oth
ertra
nsve
rse
load
ings
Pur
ebe
ndin
gR
ef/P
VB
/M10
330
4956
015
10∞
45.8
545
.87
(+0.
03%
)m
mU
nif.
dist
rib.
load
Ref
/PV
B/q
1033
049
560
1510
∞11
7.33
117.
40(+
0.06
%)
mm
4-po
intb
endi
ngR
ef/P
VB
/2F
1033
049
560
1510
∞52
.07
52.1
0(+
0.06
%)
mm
Une
qual
thic
knes
s
Axi
alco
mpr
essi
ont2
/PV
B/P
10+1
2+10
350
5256
015
10∞
3009
730
087
(−0.
03%
)N
Mid
-spa
nlo
adt2
/PV
B/F
10+1
2+10
350
5256
015
10∞
35.9
135
.94
(+0.
07%
)m
mTo
rsio
nalm
omen
tt2
/PV
B/T
10+1
2+10
350
5256
015
10∞
0.00
136
0.00
134
(−1.
12%
)ra
d
Tabl
e5.
5:Li
stof
mod
els
with
geom
etrie
s–
very
soft
inte
rlaye
r
Inte
rlaye
rP
aram
eter
Type
oflo
adin
gLa
bel
t(m
m)
b(m
m)
L(m
m)
L1
(mm
)L/b
b/t tot
L/L1
Num
eric
alA
naly
tical
Uni
t
Axi
alco
mpr
essi
onP
VB
10y/
P10
330
4956
015
10∞
4979
4978
(−0.
02%
)N
Mid
-spa
nlo
adP
VB
10y/
F10
330
4956
015
10∞
207.
2220
7.39
(+0.
08%
)m
m
PV
B10
yVe
ryso
ftin
terl
ayer
Tors
iona
lmom
ent
PV
B10
y/T
1033
049
560
1510
∞0.
013
230.
013
03(−
1.51
%)
rad
Pur
ebe
ndin
gP
VB
10y/
M10
330
4956
015
10∞
241.
0724
1.04
(−0.
01%
)m
mU
nif.
dist
rib.
load
PV
B10
y/q
1033
049
560
1510
∞63
1.55
631.
91(+
0.06
%)
mm
4-po
intb
endi
ngP
VB
10y/
2F10
330
4956
015
10∞
277.
9127
8.03
(+0.
04%
)m
m
92
For the other transverse loading conditions — pure bending, uniformly distributed load, and four-point
bending — the analytical displacement at mid-span was determined with the following equations, re-
spectively,
wmax =ML2
8EIf(5.1a)
wmax =5qL4
384EIf(5.1b)
wmax =FLa
24EIf
(3L2 − 4L2
a
)(5.1c)
where M , q and F are given in Table 5.2, L is the span (variable, according to Tables 5.3 to 5.5) and
La equals L/4, with the equivalent flexural stiffness (EIf ) given by Tables 4.1 and 4.4. All numerical
displacements were measured directly in the in the static linear analyses.
The numerical and analytical results are represented in Tables 5.3 to 5.5.
The results provided by all expressions are in very good agreement with numerical results, for all values
adopted for the parameters analyzed (with a maximum relative difference of 0.28%), except in what
concerns the equivalent torsional stiffness, whose accuracy strongly depends on the ratio b/ttot.
Comparing the results of the models “2ly/PVB/T”, “Ref/PVB/T”, “4ly/PVB/T” and “5ly/PVB/T” (or their
“SG” counterparts) it seems that the accuracy of the equivalent torsional stiffness formulation gradually
increases with the increasing number of layers. However, in light of these results, two additional finite
element models (not listed in Tables 5.3 to 5.5) were implemented. One of 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 to 10, i.e., a width b equal to 561 mm. The other model is similar
to model “5ly/PVB/T”, but has, instead, a ratio b/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 relative percentage differences between analytical and
numerical results of these two additional models, as well as of models “2ly/PVB/T” and “5ly/PVB/T”, are
summarized in Table 5.6. The results demonstrate that the differences in the accuracy of the equivalent
torsional stiffness formulation, which were apparent in Table 5.4, almost vanish if the ratio b/t is adopted
as assessment parameter, instead of b/ttot.
Table 5.6: Relative percentage differences between analytical and numerical results of models sub-jected to a torsional moment.
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%
It may thus be concluded that the parameter b/t is more appropriate to perform a parametric analysis
about the influence of the width variation on the equivalent torsional stiffness formulation. The influence
of the ratio b/t on the precision of the formulation is depicted in Fig. 5.4, where the percentage differ-
ences from Table 5.3 are plotted as a function of the ratio b/t, determined from the dimensions b and t.
93
The graph shows that the error may be assumed to be lower than 1% for a ratio b/t larger than 30–35,
for any number of layers.2 This conclusion is consistent with the analytical assumption, in Chapter 4,
that the glass layers are characterized by thin-walled cross-sections.
0%
2%
4%
6%
8%
10%
0 10 20 30 40 50 60 70 80 90
Relativedifference
analytical-num
erical
b/t
SG
PVB
Figure 5.4: Precision of the torsional stiffness formulation.
It may also be concluded from Table 5.3 that, for a beam laminated with “PVB1d”, a change on the ratio
L/L1 has an impact on the deflection (a reduction of up to 4.5%) until a certain length of the overhangs
(model “L1/PVB1d/F-3”). A further increase of the length of the overhangs (model “L1/PVB1d/F-4”)
does not result in less deflection. This is explained by the restraining effect overhangs have on the shear
strain. An infinite overhang acts like a rigid insert which prevents the glass layers from sliding over each
others [38]. However, for stiffer interlayers, this effect is small, as illustrated by the models with “SG” and
“PVB”. In these conditions, Eqs. (4.40b), (4.43), (4.47) and (4.50a) may be simplified, assuming that φ
is zero, without great loss of accuracy.
5.6 Concluding remarks
The parametric study developed in the present Chapter successfully validated the expressions formu-
lated in Chapter 4, regarding the equivalent flexural and torsional stiffness of laminated glass elements,
and allowed to understand better and define their range of validity.
In particular, it has been shown that the formulation for the equivalent torsional stiffness is strongly
dependent on the ratio b/t, yielding results that are very close to numerical ones for relatively large values
of that ratio, which generally correspond to the actual dimensions of real structural glass elements. All
equivalent flexural stiffness formulations provided very good accuracy for every geometrical cases. The
influence of the length of the overhangs has also been successfully assessed.
These results are important for the experimental application to a lateral-torsional buckling problem
(Chapter 6) for two reasons. On the one hand, the previously existing formulations for the torsional stiff-
ness of three-layered laminated glass beams were inaccurate. On the other hand, as already mentioned
in Chapter 3, the error of the torsional stiffness formulation (yields larger stiffness values compared to nu-
merical results) may propagate to the lateral torsional-buckling formulations (now measurable, following
the parametric study).2Note that, according to Table 5.5, the error may increase when the beam is laminated with a very soft interlayer.
94
Chapter 6
Experimental application
6.1 Preliminary remarks
An experimental study on the lateral-torsional buckling behavior of a long-span laminated glass beam
was carried out in IST, in 2014, as part of a PhD project [8]. The beam was similar to those used in the
glass facades of the Champalimaud Center for the Unknown, in Lisbon.
The experiment is presented in the present Chapter and its results are used to assess the analytical
formulations derived in Chapter 4. A numerical finite element model was also implemented in order to
simulate, as close as possible, the behavior of the beam under the conditions it was tested.
In Section 6.2 some details of the experimental study are briefly presented, namely the geometry of the
beam, the properties of the glass layers and of the interlayer plies, the test setup (with special attention
to the load and support conditions), and the initial measured imperfections of the beam. Some particular
aspects of the numerical model implemented for the present Chapter are defined in Section 6.3. The
experimental, analytical and numerical results are then presented and compared in Section 6.4.
6.2 Summary of the experimental study
A detailed description of the experiment and the corresponding results may be found in [7]. Some key
aspects are summarized here.
The beam, illustrated in Fig. 6.1, with three glass layers, was laminated with PVB. It was subjected to an
increasing gravity mid-span load that led to its lateral-torsional buckling. The lateral displacement was
kept within limits so that the beam would not reach its ultimate limit stress.
95
Figure 6.1: General view of the beam setup.
6.2.1 Geometry
Each of the three glass layers had 8200 mm of length, 600 mm of width and 15 mm of thickness (nominal
values). The nominal thickness of each PVB sheet was 1.52 mm. The total nominal thickness of the
beam was then 48.04 mm.
The actual width and total thickness of the beam was carefully measured along its span, with 1 mm and
0.05 mm precision tools, respectively. The results were an average width of 601.54 mm and an average
thickness of 48.26 mm. A difference on the actual thickness of the glass layers is particularly important
because of the impact it has on the critical buckling load.
Although the beam was received in the IST laboratory already laminated, an effort was made to evaluate
the actual thickness of the glass layers. It was concluded that they could hardly have the nominal
thickness prescribed in the design; instead, it should probably present a lower value, closer to 14 mm
than to 15 mm. The values provided by the main author of [7], as the more likely ones, are: 14.25 mm of
thickness for all three layers and 2.755 mm of thickness for the two PVB sheets, which totals the average
measured total thickness of the beam.
6.2.2 Material properties
In the absence of material characterization tests, the properties of glass and of the interlayer were taken
from the literature or other available information.1 The glass layers made of thermally toughened glass.
The shear modulus of PVB was considered for a load duration corresponding to the 20 minutes the
experiment lasted 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.
1Note that, specially in the case of glass, the scatter in the properties is generally low, namely the stiffness ones.
96
6.2.3 Test setup
Although the total length of the beam was 8200 mm, it was tested in a 7800 mm span. Both supports
comprised a concrete block and steel assemblies which replicate simply supported conditions. The
supports prevented the beam from rotating around its longitudinal axis but allowed the rotation around
the other two axes. This behavior was ensured by properly lubricated rollers (Fig. 6.2(a)).
The load was applied at mid-span and the load fixture used guaranteed that it would remain vertically
applied for the duration of the test. A steel container was suspended from the mid-span section of the
beam by means of a steel structure with steel rods. The connection to the clip around the beam was
ensured by an hinge positioned 89 mm above the top edge of the beam (Fig. 6.2(b)). Taking into account
that the width of the beam is 600 mm, zg in Tables 4.6 and 4.7 is −389 mm. The equipment suspended
on the beam, including the container, had a total initial weight of about 7.4 kN.
(a) Support conditions. (b) Application of the load.
Figure 6.2: Load and support conditions of the experimental setup.
The applied load, corresponding to the self-weight of the steel container and the associated equipment,
as well as to the weight of the water that progressively filled the container, was measured with a load
cell with a capacity of 50 kN and precision of 0.01 kN. The transverse and in-plane vertical displace-
ments were measured at mid-span, in the top and bottom edges of the beam, with wire displacement
transducers, with a precision of 0.01 mm.
In order to limit the stresses of the beam within its ultimate limit stress, the transverse deflection at
mid-span was limited to about 50 mm by a wooden shoving system (Fig. 6.4(a)).
97
6.2.4 Geometrical imperfections
Topographic techniques were used to evaluate eventual initial geometrical imperfections, which are in-
herent to real structures and particularly to glass elements subjected to thermal treatments. Multiple
points were measured along the span (xx axis), on the top and bottom edges of the beam, with a pre-
cision of 1 mm (Fig. 6.4(b)). The coordinates obtained allowed to determine the initial lateral deflection
(v0) and the initial twist (φ0). The former is illustrated in Fig. 6.3, while for the latter, the value 0.0012 rad
at mid-span (φ0,max) is reported in [7].
0.00.51.01.52.02.53.03.54.04.5
0 1000 2000 3000 4000 5000 6000 7000 8000
v 0[mm]
x [mm]
Measuredimperfection
Approximativesinusoidalfunction
Figure 6.3: Initial lateral deflection.
(a) Wooden system that prevents a too large deflection. (b) Topographic optical targets posi-tioned to mesure the initial deflectionand rotation of the beam.
Figure 6.4: Wooden system that limits the lateral deflection and topographic measurement of theinitial imperfections.
It is important to remind that in Subsection 4.4.2 all expressions were entirely derived with approximative
sinusoidal functions. The maximum initial deflection (v0,max) was then determined by means of the best
fitting sinusoidal function — that minimizes the squares of the differences to the measured points (Fig.
6.3). The determined value is 2.3 mm. The reported value for the maximum initial twist (φ0,max) is
associated with the sinusoidal function that approximately describes the beams’ initial twist.
98
6.3 Definition of the numerical model
The finite element model implemented in order to simulate the results of the experiment is very similar
to those presented in Chapters 3 and 5, with only minor changes. Only these differences are highlighted
here.
Because the beam is much larger than all those previously modeled, attending to the symmetrical con-
figuration of the load and support conditions, only half of the beam was modeled. This significantly
reduces the computation and data effort without any impact on the results. The boundary condition
that prevents the displacement in the longitudinal direction was already positioned at mid-span, so it
remained unchanged (Fig. 6.5(a)). The difference is on the applied load, that now represents only half
of the actual load.
In order to better replicate the test conditions, aluminum elements (E = 70 GPa and ν = 0.35) have been
added where the load is applied and in the support illustrated in Fig. 6.5(b). A partition was introduced
in the aluminum blocks so that the contact with the glass layers would be made by a softer material
(E = 1 GPa and ν = 0.4) with 2 mm of thickness. They were connected to the glass layers with a tie
constraint. The glass layers were simulated with eight-node continuum shell elements with reduced
integration (SC8R), while eight-node solid elements with reduced integration (C3D8R) were adopted for
all other materials.
The critical buckling loads were determined through linear buckling analyses (for the (i) nominal and
(ii) estimated corrected thicknesses) and the respective load-displacement paths were obtained by per-
forming geometrically non-linear static analyses. The lateral deflection and the maximum principal stress
distribution representative of the moment when the tested beam reached the maximum deflection are
illustrated in Figs. 6.5(c) and 6.5(d), respectively.
6.4 Experimental, analytical and numerical results
6.4.1 Load-displacement response
The experimental load-displacement response described in this Subsection is that already presented in
[7]. Here, those results are compared with (i) analytical predictions, which result from the formulations
derived in Chapter 4, and (ii) numerical ones. In particular, the geometrical and engineering parameters
(including the nominal and estimated corrected thicknesses of the glass layers and interlayer plies)
mentioned in the previous Subsections were introduced in the expressions from Tables 4.1, 4.4, 4.5 and
4.6, regarding simply supported laminated glass beams with three layers, subjected to a mid-span load.
The experimental, analytical a numerical load-displacement paths are illustrated in Fig. 6.6, for the
nominal thickness of the glass layers and interlayer plies and for the corrected thicknesses (according
to Subsection 6.2.1). The analytical results are provided considering both the main and the alternative
99
(a) Boundary condition in the zz direction and mid-span load.
(b) Boundary condition in the yy direction.
(c) Lateral deflection plot.
(d) Maximum principal stress plot.
Figure 6.5: Numerical model of the laminated glass of the long-span laminated glass beam tested in IST.
100
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
(a) With the main coefficients from Table 4.6.
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
0
5
10
15
20
25
30
35
40
-5 5 15 25 35 45 55
LoadF[kN]
Maximumdisplacement vmax [mm]
Analytical-nominal thicknesses
Numerical-nominalthicknesses
Analytical-correctedthicknesses
Numerical-corrected thicknesses
Experimental
(b) With the alternative coefficients from Table 4.6.
Figure 6.6: Experimental, analytical and numerical load-displacement paths.
coefficients from Table 4.6. The latter coefficients yield results closer to numerical ones which confirms
what was already mentioned in Chapter 3.
The experimental curve suggests that the initial imperfections considered for the numerical model and for
the analytical formulations were overestimated. The experimental load-displacement path is closer to the
fundamental path than the analytical and numerical ones; these progress more smoothly towards their
respective critical buckling loads. Nevertheless, both analytical and numerical models provide a similar
overall post-buckling path to that measured in the experiment, particularly if the corrected thicknesses
are considered.
6.4.2 Critical buckling load
The experimental critical buckling load (Fcr) cannot be directly determined, but it can be indirectly es-
timated using the Southwell plot. The equilibrium path of an imperfect column leads to the following
equation [40],vmax
F=
1
Fcrvmax +
v0,max
Fcr(6.1)
which can be applied to the lateral-torsional buckling problem, where vmax is the maximum deflection,
at mid-span, associated with a given load F and v0,max is the maximum initial deflection. The equation
represents a linear function, where the slope is 1/Fcr.
The experimental results vmax/F are ploted in Fig. 6.7 as a function of vmax. The inverse of the slope
of the best fitting straight line corresponds to the experimental critical buckling load and is included in
Table 6.1.
101
y=0.0333x- 0.0025
-0.5
0.0
0.5
1.0
1.5
2.0
-5 5 15 25 35 45 55
v max/F[mm/kN]
Maximumdisplacement vmax [mm]
Figure 6.7: Southwell plot of the experimental results.
As in Chapter 3, the numerical critical buckling load equals the first eigenvalue of the linear buckling
analysis. The critical load was also analytically determined, with three different formulations. The first
critical buckling load is given by the expression and the main coefficients from Table 4.6. The second is
given by the same expression but with the alternative coefficients from the same Table. The third critical
buckling load was determined according to Australian Standard AS 1288 (Table 4.7) [36]. The flexural
and torsional stiffness are given as in the previous Subsection. The results obtained for the nominal and
estimated corrected thicknesses of the glass layers and interlayer plies are summarized in Table 6.1,
and are compared with the experimental value determined by means of the Southwell plot.
Table 6.1: Comparison between the critical buckling loads obtained from experimental results, ana-lytical formulations and numerical models (relative percentage differences to the experimental load inparentheses).
The results seem to confirm that the actual thickness of the glass layers is, in fact, lower than the
nominal value of 15 mm. As previously noted, in Chapter 3, the analytical results have some dispersion,
depending on the adopted formulation, but the present results show that they can predict the behavior
of full-scale laminated glass beams susceptible to the lateral-torsional buckling phenomenon with very
reasonable accuracy.
As mentioned in Chapter 5, the length of the overhangs has an impact on the results only for low values
of the shear modulus of the interlayer. In terms of the results presented in Table 6.1, if the overhangs
were considered nonexistent (L1 = φ = 0), the critical buckling loads would be only 0.1 kN to 0.2 kN
lower.
102
6.5 Concluding remarks
The fact that laminated glass beams are slender elements, highly susceptible to lateral-torsional buck-
ling, explains the importance of the present study, which compared experimental data with analytical
and numerical simulations.
The results obtained allowed to conclude that both analytical and numerical models considered provide
similar critical buckling loads and load-displacement paths to those experimentally determined, particu-
larly if the corrected thicknesses of the constituent layers are considered. It is very difficult to accurately
measure or estimate the actual thickness of the glass layers after lamination, which highlights the im-
portance of quality control in structural glass manufacturing. Small differences in the thickness — in
this case less than 1 mm in each layer — might significantly reduce the critical buckling load of lami-
nated glass beams, making them even more vulnerable to lateral-torsional buckling. According to the
European Standard EN 572-8 [45], the manufacturing thickness tolerance for float glass with a nominal
thickness of 15 mm is 0.5 mm. The thickness correction adopted in the present Chapter is higher than
such tolerance. These possible variations in the thickness of structural glass should be considered in a
design level.
Multiple formulations that describe the lateral-torsional behavior of beams were successfully applied,
using the equivalent flexural and torsional stiffness formulations derived in Chapter 4. This confirmed
their validity and allowed to verify that the equivalent stiffness formulations may be used associated with
monolithic stability formulations with the purpose of assessing the lateral-torsional buckling susceptibility
of laminated glass beams.
103
104
Chapter 7
Conclusions
Structural glass is a relatively new concept in engineering structures. It has been seen that a relatively
small number of people are doing research work on structural glass stability, and they have proposed
distinct analytical formulations to describe the flexural and torsional behavior of laminated glass. Some
formulations were found to be equivalent, but others provided very different results. All formulations have
in common the fact that they are limited to cross-sections with a maximum of two or three glass layers,
when there are already practical applications of beams with at least five layers.
The existing lateral-torsional buckling formulations are all applicable to monolithic beams, but it has been
seen that they can be used to describe the behavior of laminated glass beams with very reasonable
accuracy if equivalent stiffness formulations are also used.
7.1 Main contributions
A new more comprehensive formulation for the equivalent flexural and torsional stiffness of laminated
glass beams and columns results from this work. Relatively simple expressions underlying a single
small equation allow defining the equivalent flexural stiffness of columns with two, three, four or five
layers (possibly asymmetrical in the two-layered case and necessarily symmetrical in the other cases),
subjected to compressive axial loads, and of beams with the same layer configuration, subjected to pure
bending, uniformly distributed loads, mid-span loads or four-point bending. Also underlying a single
small equation, relatively simple expressions define the torsional stiffness of beams with two, three, four
or five layers.
This formulation is not entirely new. It yields the same results as previously existing equations for the
equivalent flexural stiffness of columns with two and three layers (see Chapter 3), and of beams sub-
jected to a mid-span load.1 It also yields the same results as a previously existing formulation for the
1The existing expression for the equivalent flexural stiffness of laminated glass beams subjected to a mid-span load regardedbeams without overhangs [5]. The proposed formulation extends its application to beams up to five layers and with symmetricaloverhangs of any size.
105
equivalent torsional stiffness of two-layered beams.
The formulation is entirely innovative in the analysis of columns and beams with four and five layers and
bi-symmetrical cross-section, as well as in the consideration of overhangs. It also considers additional
lateral load conditions. In light of the fundamentals presented in Chapter 4, the formulation can be easily
extended to other load and support conditions. Appendix B provides an extension for the determina-
tion of the maximum normal stress in the exterior glass layers of laminated glass beams subjected to
transverse loads or pure bending.
The proposed formulation was assessed through a parametric study based on the results provided by
numerical finite element models. The equivalent flexural stiffness expressions proved to be very accurate
for any value of the parameters considered. On the other hand, the precision of the torsional stiffness
expressions was shown to be strongly dependent on the ratio width/glass layer thickness (b/t), providing
very accurate results for larger values of the ratio (which generally correspond to the actual dimensions
of real structural glass elements). The study provides thus the range of validity of the formulation, which
was previously not found in the literature for the existing analytical formulations.
Although the lateral-torsional buckling formulations derived in Chapter 4 equal previously existing for-
mulations, it was deemed important to understand their base differences, mainly because, as previously
mentioned, some inconsistencies in their application were found in the literature. The (at least partial)
clarification of their fundamentals may pave the way for further supported research.
As a consequence of the nationality of the people doing research work on glass structures, much of
the literature is written in languages that are not accessible to many people, as the German language,
specially in what concerns flexural-torsional stability. In this dissertation, analytical deductions previously
not available in the English language were written in an explanatory way, making them accessible to a
very large number of people which otherwise would have much more difficulty in understanding them.
The content of the document as a whole might, eventually, contribute towards an increase in the number
of people interested in glass structures and in further developing the knowledge on the subject, which
at the moment cannot be considered sufficiently consolidated. At least language no longer constitutes
a barrier on the topics here presented.
7.2 Future developments
The proposed equivalent flexural and torsional stiffness formulation does not cover all load and support
conditions that laminated glass elements may be subjected to. For example, although simply supported
solutions are generally preferred, in order to reduce the tensile stress in the glass layers, some support
conditions may impose rotational restraints. Laminated columns may also be subjected to transverse
loads. Therefore, in the future, other configurations should also be taken into account.
Even though the three lateral-torsional buckling formulations analyzed in Chapters 3 and 6 provide rea-
sonably accurate results, they still present some differences in comparison with numerical results. The
106
lateral-torsional buckling phenomenon may be further investigated, namely with different initial assump-
tions, as for example the direct consideration of in-plane shear deformation of laminated glass elements,
instead of adopting equivalent stiffness formulations.
Something very important that should be further studied is the effect of lateral restraints in the lateral-
torsional buckling behavior of laminated elements. In fact, glass-fins or roof beams most often exhibit
point or continuous lateral restraints in one of the edges, depending on the type of connection. The
Australian Standard AS 1288 [36] already provides some design considerations about this subject.
Even though the determination of stresses in the glass layers was not within the scope of this disser-
tation (Appendix B, however, provides a formulation for laminated elements under certain loads), their
assessment is important since glass is a brittle material with relatively low tensile strength. Some au-
thors even propose the definition of buckling curves based on the tensile strength of glass [1, 6, 22].
This issue should therefor be pursued in the future.
Similar but more complex sandwich theories can also be developed for laminated glass panels, sup-
ported in the four edges, such as those used in facades, roofs or floors. However, if this approach is
viable for future design purposes is for the moment unknown.
Since the shear modulus of the interlayer is one of the most significant parameters regarding the sus-
ceptibility of laminated glass beams to lateral-torsional buckling, its correct assessment is of the utmost
importance. In particular, the influence of their time- and temperature-dependencies should be thor-
oughly investigated for all interlayer products currently available.
107
108
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