The preceding sections have shown that most of today’s design methods are actually variations, extensions or simplifications of others. These may be grouped into: European
design methods, which are based on the DELR design method, and North American design
methods, which are based on the GFPM.
The following analyses and comments aim at helping the engineer, who uses the above mentioned design methods, to better understand their bases, advantages, drawbacks and limits of validity.
13Though this has already been pointed out in[168], the standard has not yet been revised at the time of
Main concepts
In Table 4.8, European and North American design methods are compared with respect to the main concepts that they are based on. It can be seen that the two approaches are not directly comparable because of conceptual incompatibilities. Data from experiments designed for one method cannot directly be used in conjunction with the other method.
Table 4.8: Comparison of European and North American design methods.
European design methods North American design methods
Testing procedure to obtain strength data
Constant stress rate coaxial dou- ble ring tests with σ˙test =
2 ± 0.4 MPa/s on specimens with Atest = 0.24 m2 (cf. Sec-
tion 3.5.2).
Large rectangular glass plates ex- posed to uniform lateral load.
Surface condition of strength test specimens
Artificially induced homogeneous surface damage.∗
Weathered windows glass. Design ‘strength’
definition
A single value is used, the ‘char- acteristic value of the inherent strength’†. It is defined as the 5%
fractile value (at 0.95 confidence level) of the failure stress mea- sured in the experiments.
Two interdependent parameters called ‘surface flaw parameters’
e
mand ˜k are used. Their determi- nation from experimental data is based on the stress history at the visually determined failure origin and a rather complex iterative procedure.
Subcritical crack growth
Taken into account by load dura- tion factors that depend on the loading only. The factors are based on the empirical relation- ship v= S · Kn
I, (cf. Section 3.2).
Only one crack velocity parame- ter is used explicitly. It is equiva- lent to the parameter n in Euro- pean methods and assumed to be 16.
Extrapolation from experiments to in-service conditions
Some but not all differences be- tween laboratory conditions and actual in-service conditions are accounted for by correction co- efficients. Details vary between methods, see Section 4.3.
Graphs are provided for many common cases (in terms of ge- ometry and support conditions). They provide uniform lateral loads that a given glass pane can withstand for a reference time pe- riod.
Taking the glass type into account
Mostly by adding the absolute value of the residual surface stress (multiplied by a ‘safety factor’) to the allowable tensile stress of float glass.
By multiplying the load resis- tance of a float glass element by some load duration-dependent glass type factor.‡
∗The generally used parameter set does, however, not directly reflect test data, see Section 4.3.1.
†In contrast to usual characteristic resistance values, this one is not a ‘real’ material parameter. It depends on
the geometry, the surface condition, the environment and the loading of the specimens. The term ‘characteristic value’ is therefore somewhat misleading, which is why it is put in inverted commas.
‡In CAN/CGSB 12.20-M89 [59], the glass type factor (called ‘heat treatment factor’) does not depend on the
Time-dependence of the glass strength
The tensile strength of glass strength is time-dependent due to stress corrosion (see Section 3.2). The design methods presented above conceal this dependence within coefficients, such that the underlying assumptions are not readily visible. They are, therefore, briefly discussed in the ensuing text.
Current glass design methods assume that the crack velocity parameters are well known constant values. The ‘classic’ European crack velocity parameters have been published in[225]. They are based on the ambient condition crack growth data from Richter[284] who determined the parameters by optically measuring the growth of large through-thickness cracks on the edge of specimens loaded in uniform tension. On this basis, ‘design parameters’ for the DELR design method were chosen in[43]. (These design parameters represent substantially higher crack velocities than Richter’s measurements, see Figure 3.3). The European draft standard prEN 13474 and the design method by Siebert are directly based on the DELR design method and use the same parameters. Shen uses a different approach, see Section 4.3.3.
The GFPM uses only one crack velocity parameter explicitly. It is equivalent to the parameter n in European design methods and assumed to be equal to 16.
Laboratory testing to obtain strength data
The design methods presented in this chapter are based on strength data obtained at ambient conditions. The parameters meant to represent the surface condition or a ‘material strength’ (˜k and me in the North American design methods, θA andβ in the European design methods) are, therefore, inevitably dependent on the surface condition
andon crack growth behaviour. This is a drawback for two reasons. Firstly, unrelated physical aspects are combined within a single value. Secondly, the large scatter and the stress rate dependence of the crack velocity parameters (see Section 3.2) make accurate estimation of the crack growth that occurs during experiments at ambient conditions difficult. Inaccurate estimation, however, can yield unsafe results. This issue is further discussed in Chapter 6.
Load duration effects
Time-dependent effects related to loads are commonly referred to as ‘load duration effects’ or ‘duration-of-load effects’.14
All design methods presented in this chapter are implicitly or explicitly based on the assumption that crack growth and with it the probability of failure of a crack or an entire glass element can be modelled using the risk integral, also known as Brown’s integral (see Section 3.3.4).
Lifetime prediction based on the risk integral implies that the failure probability can be described accurately by accounting for the crack growth (damage accumulation) during the lifetime only while neglecting the influence of the initial surface condition and of the momentary load. These simplifying assumptions have important consequences. The
14Strictly speaking, the term is not very accurate because it implies constant loads. In the more general case of
time-variant actions, ‘action history effects’ would be more appropriate. As ‘load duration effect’ represents commonly accepted terminology and is widely used in academic publications, the term is nevertheless used in the present document.
momentary failure probability at a point in time is independent of the momentary load at that time (which is not the case in reality). Furthermore, the material resistance obtained from an equivalent stress based model converges on infinity for very short loading times or very slow subcritical crack growth. This is clearly inaccurate. It should converge on the inert strength, which is an upper resistance limit (cf. Section 3.3.3).
These issues were among the reasons that led to confusion and a general lack of confidence in advanced glass design methods. Together with the high variance in experi- mentally determined parameters, they are also the main reasons for various researchers to claim that the glass failure prediction model and any Weibull distribution based design approach are fundamentally flawed and unrealistic (e. g.[57, 58, 282, 283]).
It can be shown that the risk integral is a good approximation if the crack depth at failure is substantially greater than the initial crack depth[187]. It is, therefore, suitable for structural design calculations. With very high loading rates or low crack velocities, i. e. when little subcritical crack growth occurs, the simplified approach grossly underestimates the probability of failure and leads to unrealistic and unsafe results (resistance above the inert strength). For these cases, which include the interpretation of experiments, the generalized formulation, which is presented in Section 3.3.4, must be used.
All design methods account for the load duration effect by a factor that depends on the load duration and sometimes the residual stress only. In European methods, this factor is applied to the allowable maximum or equivalent in-plane principal stress. In GFPM-based methods, it is applied to the allowable lateral load. This is again a simplifying assumption. In reality, the load duration effect additionally depends on many other factors, including the action history and the element’s geometry. For guidance on exact calculations, see Chapter 6.
Residual stress
Several methods include the effect of residual stresses within the resistance of the glass. It is, however, crucial to distinguish residual stress clearly from inherent strength . Only decompressed parts of a glass element’s surface are subjected to subcritical crack growth and its consequences. Furthermore, the uncertainties, and consequently the partial factors, are different for residual stress and inherent strength.
Design methods accounting for residual stress explicitly superimpose the residual stress on the inherent strength. This assumes that the inherent strength is not affected by heat treatment. There is evidence showing that the tempering process actually causes a certain amount of ‘crack healing’[40, 190]; this assumption can thus be considered safe (conservative) for design.
Size effect
As a direct consequence of the use of Weibull statistics (see below), the resistance of glass elements depends on their surface areas in all design methods. As only tensile stress can cause glass failure, the size depends not on the total, but on the decompressed, surface area. For given geometry and support conditions, the latter depends in general15on the load intensity and is therefore time-variant. Taking this aspect accurately into account is
15In some special but frequent cases such as annealed glass plates exposed to uniform lateral load, the whole
0 1 2 3 4 5 6 7 8 9 10 0.7 0.8 0.9 1.0 1.1 1.2 1.3 s = 25 s = 15 s = 10 Si ze e ffe ct , σ ( A1 ) / σ ( A2 )
Ratio of the surface areas exposed to tensile stress, A1 / A2 s = 5
Figure 4.9:
The size effect’s dependence on the Weibull shape factor or first surface flaw parameter (using Equation (4.34)).
complex (see Section 3.3). European design methods define the size factor based on the total surface area, which makes it load-independent. US and Canadian standards multiply the load resistance of annealed glass elements by a factor. As the entire surface of an annealed glass plate is immediately decompressed on loading, the two approaches yield the same result. The size effect can be expressed as
σ(A1) σ(A2)= A 2 A1 1/s (4.34)
whereσ(A1) and σ(A2) are the tensile strengths of structural members with surface areas
A1and A2 respectively exposed to tensile stress. In European methods, s is the shape
parameterβ of the tensile fracture strength distribution. In GFPM based methods, s is the surface flaw parameterme.16 Figure 4.9 shows that the size effect is quite significant for
the ASTM E 1300 value ofme = 7, while it becomes almost negligible (for realistic panel
sizes) for the valueβ = 25 that is generally used in European design methods.
Two issues with the size effect are rather problematic: Firstly, the exponent s differs much between design methods. Secondly, while the size effect in as-received glass is verified by experimental evidence the same cannot be said for weathered glass. Calderone [58], for instance, found little or no relationship between the total surface area and the breakage stress or between the most stressed panel area and the breakage stress. This issue is further discussed in Chapter 6.
Weibull statistics
According to the model in Section 3.3, the strength of as-received or homogeneously damaged glass specimens should follow a two-parameter Weibull distribution. This is indeed the case with experimental data obtained at inert conditions. The goodness-of-fit of the results of common ambient strength tests to the Weibull distribution, however, is often poor. Several researchers have, therefore, proposed using log-normal or normal distributions to represent glass strength. This is problematic because the Weibull distri- bution results from fundamental hypotheses of the model in Section 3.3. The use of a non-Weibull distribution type would require an alternative glass strength model.
In contrast to inert strength data, strength data obtained at ambient conditions do not solely represent a glass specimen’s surface condition, but are additionally influenced by
16The two parameters
e
subcritical crack growth. Haldimann[187] showed that the goodness-of-fit of ambient strength data to the Weibull distribution decreases as time to failure increases, which means as the influence of subcritical crack growth increases. The poor fit of ambient strength data to the Weibull distribution can, therefore, be explained by the wide variability of the crack velocity parameters (cf. Section 3.2).
In conclusion, Weibull statistics are in principle well suited to describe the strength of glass. The problems encountered are not related to the statistical model itself. Instead, they are caused by the variability of the crack growth parameters and by the fact that the damage on glass elements in in-service conditions is often not uniform and homogenous. This issue is further discussed in Chapter 6.
Biaxial stress fields
The GFPM-based standards use the biaxial stress correction factor proposed by the GFPM. It depends on the principal stress ratio (which is, in general, load intensity-dependent) and on the surface flaw parameterme. Though not explicitly stated, only the fully-developed principal stress ratio is used for the resistance graphs and the testing procedure. Based on this ratio, a single biaxial stress correction factor for each point on the surface is calculated and assumed to be valid for all load intensities.
European glass design methods generally assume all cracks to be oriented perpendic- ularly to the major principal stress. This is equivalent to assuming an equibiaxial stress field. While this assumption is conservative (safe) for design, it is not conservative when deriving glass strength data from tests (see Section 6.4).