1.4 Theoretical Background for Structural
1.4.2 LTB Method for Laterally Unrestrained (LU) Glass Beams
A buckling design approach strictly related to Eq. (1.5) was, for exam- ple, proposed for LU glass beams in LTB [42–44], by assuming in (Eq. 1.6) the characteristic tensile resistance of glass fyk ≡ Rk and M = 1.4 as a partial safety factor. The advantage of this approach is given to its general formula- tion, but namely represents the extension of earlier consistent studies on the topic (e.g. [45, 46]).
In the hypotheses of a rectangular cross section for a fork-end restrained glass beam, Eq. (1.6) leads in fact to
M M
L EI GI
cr E
cr z t
( ) = = π
0
, (1.7) where E ≡ Eg and G ≡ Gg in Eq. (1.7) represent Young’s and shear moduli of glass, respectively, while Iz = ht3/12 signifies the moment of inertia about the minor z-axis and It ≈ ht3/3 (for h/t > 6) is the torsional moment of inertia.
Extended comparison proposed in [45] and [46] highlighted the good cor- relation between analytical critical load predictions derived from Eq. (1.7) and detailed FE models, both for monolithic and laminated cross sections belonging to beams in LTB with various geometrical and mechanical aspects.
In the latter case, viscoelastic FE calculations were assessed toward equiva- lent thickness approaches (e.g. Section 1.4.2.1) applied to LG beams under well-defined load-time and temperature conditions. The main advantage deriving from the application of equivalent thickness-based methods to LG elements is given by the assumption of fully monolithic glass sections with equivalent bending and torsional stiffnesses, hence resulting in simplified but rational and practical design methods, especially for buckling purposes.
Table 1.3 Loading/restraint coefficients for the calculation of Euler’s critical moment of simply supported, fork-end restrained beams [41].
End restraint Moment distribution kz C1 C2 Free rotation about
the weak axis
Parabolic 1.0 1.12 0.45
Triangular 1.0 1.35 0.59
Complete restraint against rotation
Parabolic 0.5 0.97 0.36
Triangular 0.5 1.05 0.48
In [43, 44], based on classical Euler’s buckling moment definitions (e.g. Eq. 1.7) and the standardized method proposed by the Eurocode 3 for steel structures (Eq. 1.5), calibration of the imperfection factors defining χLT was then carried out on the base of LTB experimental data available in literature for monolithic and LG beams, as well as extended FE and ana- lytical calculations. Figure 1.11b presents the result of this calibration, for the so-called “Eurocode-based” design buckling curves for glass beams in LTB, compared to previous studies (Figure 1.11a).
1.4.2.1 Equivalent Thickness Methods for Laminated Glass Beams When applying equivalent thickness methods to LG sections in LTB, two main aspects should be properly taken into account, namely the appropri- ate estimation of both the equivalent bending stiffness and torsional stiff- ness required in Eq. (1.7).
Several formulations are available in literature for this purpose. In [48], for example, extended assessment of some of these existing ana- lytical models based on the equivalent thickness concept and primarily intended for the calculation of the critical LTB moment in three-layered sandwich beams was discussed and further extended for the LTB analysis of LU LG beams, after an appropriate validation toward FE viscoelastic and experimental data. In the following sections, some of these formula- tions are recalled for LG beams. Analytical models are proposed for sym- metric cross sections composed of two glass layers only (e.g. Figure 1.5, case b).
1.4.2.1.1 Method I
For the analysis of the LTB behavior of LG members, Luible [45] first applied the analytical formulations originally developed for sand- wich structural elements to glass sections. The mentioned analytical approach, in particular, is based on the concepts of equivalent bending stiffness EIz,eff and equivalent torsional stiffness GIt, where EIz,eff is cal- culated depending on the specific loading condition (constant bending moment My, distributed load q, concentrated load F at mid-span), while GIt depends on the geometrical/mechanical properties of the cross sec- tion only.
The expression proposed for EIz,eff is given as a function of the slender- ness of the beam (t1, h, L0), the elastic stiffness of glass (E), the thickness of the interlayer (tint), and its mechanical properties (Gint). Based on [45],
Figure 1.11 Calibration of design buckling curves for LU beams in LTB, by assuming different geometrical imperfection amplitudes.
FE, M cost. [45]
(θ0= L0/270 h rad) FE, q dist. [45]
(θ0= L0/270 h rad) TEST, F conc. [45]
(monolithic) TEST, F conc. [45]
(laminated) TEST, F conc. [47]
0.0 0.5 1.0 1.5 2.0 2.5
_λLT
0.0 0.5 1.0 1.5 2.0 2.5
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
χLTχLT
Euler
Buckling curve (αimp= 0.26, α0= 0.20) for θ0= L0/270 h rad [45]
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Buckling curve (αimp= 0.26, α0= 0.20), for θ0= L0/270 h rad [45]
“EC-based curve” (αimp= 0.45, α0= 0.20), for θ0= L0/200 h rad [44]
FE, M cost. [44]
(monolithic, θ0= L0/200 h rad) (a)
(b)
_λLT
according to the laminated cross section of Figure 1.4, case (b), and consid- ering the beam subjected to a constant bending moment My, as proposed in Figure 1.10, EIz,eff is in fact defined as follows:
EI E I I
z eff
s
L L
, sinh /
sinh
= + ,
+ −
(2 )
1 12 48 2
1
2 3
2
α λ α λ
λ λ
(1.8)
with
Is =h(2t z1 12) (1.9) I1 =h t13/12 (1.10)
αL s
I
=2I1
, (1.11)
β = t G h z
EI L
s int
int (2 1)2 20
, (1.12)
λ α
= 1α β+ L
L
, (1.13) z1=0 5. (t1+tint) (1.14) Based on the same approach, the equivalent torsional stiffness GIt was also derived from the classical theory of sandwich elements (Figure 1.11 [45]). For a symmetric two-layer LG beam, in particular, GIt can be calcu- lated as follows:
GIt =G(2It,1+It comp, )=G I( t abs, +It comp, ), (1.15) where the expression for It,1 and It,comp are listed in Table 1.4.
Specifically, Eq. (1.15) takes into account the effective torsional con- tribution It,comp due to the adopted interlayer. Stamm and Witte originally derived the expressions, partly collected in Table 1.4 [45], for the estima- tion of this torsional stiffness term, typically occurring in a faced “soft”
core within a flat sandwich panel subjected to a torsional moment MT.
Figure 1.12 Qualitative torsional behavior of a LG beam in accordance with the analytical model recalled in [45].
Shear stresses due to the interlayer
Shear stresses due to torsion of the glass layers
Glass
Glass
h
t1 tint t2 MT
Interlayer
Table 1.4 The Stamm–Witte equivalent parameters for the calculation of the torsional stiffness term in layered cross sections [45].
Symbol Definition
It,1 ht13/3
It,comp
I h
s LT h
LT LT ,
tanh( / )
⋅ −1 / 2 2 λ λ It,LT
4 1 2
2 1
2
1
⋅ +(t tint) ⋅t t h
λLT G
G t t
int int
2
1
Their model basically applies to layered elements in which the cross sec- tion is uniform along the total length L0. Largely used for the analysis of sandwich elements and recalled in several handbooks [49–51], Eq. (1.15) has been applied successfully to LG elements.
1.4.2.1.2 Method II
An alternative analytical model for the lateral–torsional buckling (LTB) verification of LG beams has been assessed in [48]. In that case, the
theoretical model was based on the Wửlfel–Bennison expression for the equivalent thickness teq, e.g. on the concept of an equivalent, monolithic flexural stiffness EIz eff, =hteq3 /12 with
teq =32t13+12Γb s WBI, , (1.16) and
0 1
1 2
1
2 1
2
1 0
2
≤ = +
≤ Γb
E t t t G L π int
int
(1.17)
the shear transfer coefficient representative of the shear transfer contri- bution of the adopted interlayer, where Is,WB in Eq. (1.17) is equal to Is/h (Eq. 1.9). Due to the shear transfer coefficient b, the effective stiffness of the interlayer can be rationally taken into account within a range conven- tionally comprised between an “abs” layered limit (e.g. Gint → 0) and “full”
monolithic limit (e.g. Gint → ∞).
Analytical calculations highlighted that based on Eq. (1.17) the flex- ural stiffness EIz,eff = f(teq) exactly coincides, for the boundary and loading conditions considered in this contribution, with calculations provided by exact analytical models (e.g. derived for example from Newmark’s theory of beams with partially rigid interaction [42]). To be used for LTB purposes, the “Method II” further requires the calculation of the torsional stiffness term GIt, that also in this case is calculated based on Eq. (1.15).
1.4.2.1.3 Other Available Formulations
The so-called “Method I” and “Method II” represent two analytical approaches of large use for structural glass applications. Other formulations – with almost the same effects – are anyway available in the literature.
Based on [52], for example, the torsional stiffness of laminated cross sections is calculated as a multiple of the “abs” torsional stiffness corre- sponding to a null shear stiffness of the interlayers (Gint → 0), e.g. by intro- ducing a parameter f ≥ 1 so that
GIt =(GIt abs) ⋅f (1.18)
is the equivalent torsional stiffness of the laminated member, where
f
t t G
G h t t t t t t t G
G h
= + + +
+
6 4 6 3
6
1
3 2
1 2
1
2
1 2
1
2 int
int
int int
int int
( )
(1.19)
in the case of a symmetric laminated cross section as given in Figure 1.5, case (b), while specific expressions are provided for “f ” as far as the cross section is unsymmetrical or not, and composed of two or three glass foils, respectively (e.g. Figure 1.5, cases a and c).