5.3.1 Modelling
The load carrying behaviour of monolithic glass can be described using the second order differential equation for a bar with length Lcrand pinned ends, with an initial sinusoidal deformation w0and an axial compression load N , which is applied with an eccentricity e (Figure 5.2). E Id 2w(x) d x2 + N w0sinπx Lcr + e + w(x) = 0 (5.1)
The elastic critical buckling load is
Ncr= π
2E I
L2 cr
(5.2)
and the maximum deflection w at midspan considering second order effects is
wmax= e
cosLcr/2pN/Ncr + w0
1− N /Ncr
. (5.3)
This yields a maximum surface stress
σmax= N A ± M W = N A± N W(wmax+ w0+ e) , (5.4)
where N is the applied force, A is the sectional area and W is the elastic section modulus. In laminated glass the interlayer material behaves like a shear connection between the glass panes. Simplistically a lot of interlayer materials may be considered as an elastic material with a constant shear stiffness for a given temperature and load duration.
Figure 5.2:
Column buckling model. N
N Lcr e w0 wmax N M Mmax
The load carrying behaviour can then be described using elastic ‘sandwich’ theory[316, 346]. The critical buckling load of a laminated glass with two or three glass panes and symmetrical layout (Figure 5.3) is given in[241] as:
Ncr =π
2(1 + α + π2αβ)
1+ π2β
E IS
L2cr (5.5)
In the case of two glass panes, it is
α = I1+ I2 IS ; β = tint Gintb(z1+ z2)2 E IS Lcr2 ; IS= b(t1z 2 1+ t2z22) , (5.6)
and in the case of three glass panes, it is
α =2I1+ I2 IS ; β = tint Gintbz21 E IS Lcr2 ; IS= 2bt1z 2 1. (5.7)
Lcris the buckling length, b is the width of the cross section, Gintis the shear modulus of the interlayer material and Ii= bt3i/12 is the moment of inertia of the pane i.
A simplified approach for calculating the deflection and the maximum bending stresses of a laminated glass consists in employing Equations (5.4) and (5.3) with the following equivalent thickness[241]:
teff= 3
È
12 IS(1 + α + π2α β)
b(1 + π2β) (5.8)
Aand W in Equation (5.4) are A= b P tiand W = bteff2/6 respectively.
It is assumed that the glass pane’s rotation is not restrained at either extremity and that the load is applied axially, i. e. there are no lateral loads.
Kutterer[232] developed an analytical second order model based on sandwich theory [316] for the analysis of the buckling behaviour of laminated glass elements under an axially applied force. The model accounts for creep effects of the PVB interlayer. The lateral displacement and the maximum stresses may be calculated as a function of temperature and load duration. For a given axial load the model is able to predict a critical time at which time delayed buckling will start.
Analytical models are generally limited to simple structural systems and certain boundary conditions. Numerical finite element models are more flexible and powerful. They have the advantage that the interlayer may either be represented by elastic or viscoelastic elements based on existing material data[329]. Furthermore, arbitrary boundary conditions (e. g. restraints due to the load introduction or intermediate supports) may easily be incorporated (see Section 2.3.2).
t1 t2 tint z1 z2 glass glass t glass glass interlayer t1 t1 tint z1 z2 glass glass glass tint t2 glass glass glass t Figure 5.3:
Cross section of a laminated glass with two (left) and three (right) glass panes
5.3.2 Load carrying behaviour
The column buckling behaviour of glass elements made of monolithic and laminated glass with PVB interlayer was studied experimentally and compared to analytical and numerical models by Luible[241] and Kutterer [232]. The models in Section 5.3.1 are suitable to describe the load carrying behaviour of glass elements with imperfections in compression. In contrast to monolithic glass, the buckling behaviour of laminated glass depends on load duration and temperature because of the viscoelastic behaviour of the PVB interlayer [242]. Several other parameters have an influence on the column buckling behaviour:
u The glass thickness t, the initial deformation w0and the load eccentricity e have the
most important influence on the buckling strength. The real glass thickness rather than the nominal glass thickness has to be taken into account (see Section 5.2.1). The buckling strength that results from the real thickness may be up to 11.7% less than the buckling strength that is obtained based on the nominal thickness.
u Because of the high compressive strength of glass, the failure origin of glass element
in compression is always on the tension surface for the panel dimensions commonly used in buildings (L> 300 mm, t < 19 mm) and initial deformations as explained in Section 5.2.2. The buckling strength of glass is, therefore, limited by the maximum tensile strength[241].
u Experimental studies demonstrated that the failure origin is mainly in areas that
have a low tensile strength. In the case of annealed glass, this is the glass edge (most severe surface damage). In the case of tempered glass, this is close to the glass edge, where the residual compressive surface stress reaches a minimum (see Section 3.6.4).
u The load carrying capacity of tempered glass is mainly influenced by the residual
compressive surface stress rather than by the inherent glass strength. This effect is caused by the non-linear relationship between the applied compressive load and the tensile stress on the glass surface (Figure 5.4). As a simplified approach for column buckling design, the inherent strength may be neglected and the buckling strength can be determined from the residual stress only.
u The composite action caused by the viscoelastic PVB interlayer in laminated glass
increases the buckling strength. Unfortunately, creep effects in the PVB make the buckling strength depend on time and temperature. For low temperatures and short-term loading the buckling strength can almost reach the buckling strength of a monolithic cross section of the same thickness. For long-term loads (e. g. dead
Figure 5.4:
The influence of the residual stress and of the inherent strength on the buckling strength.
B
uckling strength
Tensile stress on the glass surface σ+ Ncr inherent strength compressive residual stress N t σ+ N σ- N
load) or high temperatures (> 50◦C) the composite action provided by the PVB is insignificant and the load carrying behaviour is similar to independent glass panes without PVB. Therefore the lower limit of the buckling strength of a laminated glass may be determined by ignoring the composite action (GPVB= 0). From a safety
point of view, a shear interaction may only be taken into account for short-term loads like wind or impact loads and for temperatures< 25◦C.
5.3.3 Structural design
Compressive members such as steel columns are generally designed using column curves. This approach can be applied to compressive glass elements as well. In steel construction, column curves are based on a slenderness ratio[141]. This allows the same curve to be used for the design of members with different steel grades. However, in contrast to steel, the slenderness ratio for glass must be based on the maximum tensile strength, as the compressive strength does not limit the buckling strength. The application of column curves for column buckling of glass elements has been discussed in[242]. It is shown that as a simple approach, the maximum tensile stress in a compressed glass member can be determined by means of elastic second order equations (Equation (5.4)). The column buckling capacity of a glass element is adequate if
fSd≤ fRd (5.9)
where fSdis the design value of the maximum tensile stress and fRdthe design value of
the maximum tensile strength.
A reduced glass thickness and a reasonable assumption of the initial deformation has to be considered in the second order analysis (Section 5.2). Due to the non-linear relation between applied loads and resulting bending stresses, the maximum bending stress has to be determined with factored load and superposition of stresses resulting from different loads is not possible. Simplistically the tensile strength may be assumed to be equal to the residual stress (see Section 5.3.2).
This approach applies to laminated glass as well. Generally it is advantageous to take the composite behaviour of the interlayer into account. For PVB interlayers it is recommended to consider a composite behaviour only for short-term loads such as wind loads. Simplistically, the sandwich cross section may be replaced by an effective monolithic cross section with an effective thickness teffgiven by Equation (5.8). Maximum stresses
may be calculated with Equation (5.4) or numerical models.
5.3.4 Intermediate lateral supports
The buckling strength of a member with monolithic cross section is directly proportional to the square of the buckling length. Reducing the effective buckling length by means of an additional intermediate lateral support will increase the buckling strength by a factor of 4 (Figure 5.5). Ncr,2 Ncr,1= L2 L 2 2 ⇒ Ncr,2= 4Ncr,1 (5.10)
This assumption is not valid for laminated glass. Decreasing the effective buckling length of a laminated glass by intermediate lateral supports increases the buckling strength but on the other hand it also decreases the lateral bending stiffness, which is a function of the interlayer shear modulus and the effective shear length. The shear length may be conservatively assumed as the distance between the supports. For a more realistic analysis it is recommended to use suitable finite element models where the entire member is modelled[241]. Ncr,3may be used as a conservative approach to determine the buckling strength of Ncr,2(Figure 5.5). For laminated glass it may be assumed that
Ncr,3 ¶ Ncr,2¶ 4Ncr,1 (5.11)
.
Figure 5.5:
Influence of intermediate lateral supports on the critical buckling load
Monolithic glass
Laminated safety glass
L L/2 L/ 2 L/2 L/ 2 L/2 L Ncr,1 Ncr,2 Ncr,1 Ncr,2 Ncr,3 ∆x ∆x=0 ∆x ∆x ∆x ∆x
5.3.5 Influence of the load introduction
The laminated glass models are based on the assumption that the shear deformation between the glass panes at the extremities is free and that the load is applied symmetrically on the cross section. In practice the glass edges of a laminated glass made of HSG or FTG are not flush and loads are generally applied through intermediate materials such as neoprene, injected mortar, high strength plastics or aluminium. The partial shear restraint due to these load introduction materials leads to a stiffer load carrying behaviour than assumed in the model. Uneven glass edges result in an asymmetric load distribution on the glass panes, which creates additional bending moments. Such bending moments may be determined for example with the model presented in[241], which takes asymmetric thickness of the load introduction material into account. If the influence of the glass edges is critical, detailed finite element models are recommended for design.