Architectural glazings: Design standards and failure models

Building ondEnuironmenf, Vol. 30, No. 1, pp. 29-40, 1995 Cotmirrht 0 1994 Else&r Science Ltd Printedih &ea;Britain. All rights reserved 036&1323/95 19.5O+O.Oil

Pergamon 0360-1323(94)EOO26-N

Architectural Glazings and Failure, Models ANTHONY RICHARD

:

Design Standards

C. FISCHER-CRIPPS* E. COLLINS* Architects and glass designers often refer to published standards when selecting glass thicknesses and areas for glazings in buildings. Glass design standards generally are based upon the results of experiments involving the breakage of standard sized glass sheets under carefully controlled conditions. More recently, some design standards have incorporated mathematicalfailure models which contain empiricalparameters whose values have been estimatedfrom these same experimental studies. In thispaper, the technical basisfor the recommendations within current architecturalglass design standards is discussed. Existing glass failure prediction models are reviewed and it is shown that significant discrepancies exist between their recommendations and those of the design standards. A modified crack growth model is proposed which predicts failure probabilities for both short and long term stresses which are consistent with established design practice.

1. INTRODUCTION

tation may not be fully appreciated by glass designers and architects who may also wish to design for relatively long term stresses of a low magnitude. For example, temperature differences as much as 20°C may result from uneven shading from sunlight in an external window. Such temperature differences generate thermal stresses of the order of 10 to 20 MPa, which are of sufficient magnitude to cause fracture [lo]. In the present work, it is shown that the Glass Failure Prediction Model is inaccurate in predicting failure probabilities for both short and long term applications of load. The Crack Growth Model also is limited in predicting failure for long term loadings. A modification to the Crack Growth Model is presented which gives results which are in good agreement with the recommendations of the design standards for both short term and long term loads.

TRADITIONAL design recommendations for architectural glazing take the form of design charts, such as those found in Australian Standard AS1288,1989 [I] and others [2, 3, 4, 51. The design charts typically show the maximum window area and minimum glass thickness for a selected wind loading. The charts embody the results of tests involving the short-term failure of glass sheets under uniform pressure. The recommendations of the design standards serve to describe the strength of glass which reflects the experience gained over many years by glaziers and designers in the field. Modern usage of glass windows as structural or semistructural components, where limited field experience exists, has led to the development of a small number of failure prediction models ostensibly based upon scientific principles. The first such failure model, the Load Duration Theory, was proposed by Brown in 1972 [6]. Beason, in 1980, developed the Glass Failure Prediction Model [7, 81 which is considered to be the state of the art by glass designers and engineers. An alternative treatment of the failure of brittle solids is given by the Crack Growth Model [9], which is based upon linear elastic fracture mechanics. However, there has been little effort in formulating a general design strategy based upon this method for use by glazing designers and engineers. In the present work, the significance of the recommendations shown in the design standards is discussed. Various failure prediction models are examined and it is shown that they are limited to estimating failure probabilities for cases which only involve short term application of loads leading to relatively large stresses. Although the models were developed primarily for just such loading conditions, e.g. wind gusts, this limi-

*School Australia.

of Physics, University

2. FRACTURE

MECHANICS

OF GLASS FAILURE

The fracture of a glass specimen usually occurs due to the growth of a flaw on the surface rather than in the interior. Depending on environmental conditions, glass exhibits time-delayed failure where fracture may occur some time after the initial application of load. Inglis [ 1I] showed that the local stress at the tip of a flaw, or crack, may be magnified many times over the externally applied stress. The degree of stress concentration depends on the orientation of the flaw with respect to the direction of stress and the crack tip geometry. Griffith [12] showed that the growth of a flaw in a solid under stress can be described in terms of the energy needed to form new crack surfaces and the attendant release in strain energy. Growth of a flaw can ultimately lead to fracture of the specimen. These facts indicate that it is the existence and distribution of surface flaws and environmental conditions which characterize the fracture strength of brittle materials.

of Sydney, NSW 2006, 29

30

A. C. Fischer-Cripps and R. E. Collins

2.1 Staticfatigue The resistance to fracture is usually described by a material property known as the “plane strain fracture toughness”, K,,, which is the critical value of Irwin’s [I 31 stress intensity factor, K,, defined as : K, = aY&.

0)

The subscript indicates so-called “mode 1” loading where the stress is applied normal to the crack. In equation (I), CTis the applied external tensile stress, a is the crack length, and Y is a term which depends on the shape of both the specimen and the crack. For example, for halfpenny shaped flaws in a semi-infinite solid, Y = 0.713 [14, 1.51.When K, = K,,, instantaneous fracture occurs. A typical value [16] for K,, for soda-lime glass is 0.78 MPa mm. It should be noted that equation (1) assumes a zero crack tip radius, i.e. a sharp crack tip. For an applied stress intensity factor K, < K,C, crack growth may still be possible due to the effect of the environment. Crack growth under these conditions is called “sub-critical crack growth” or “static fatigue” and may ultimately lead to fracture some time after the initial application of the load. However, experiments show that there is an applied stress intensity factor, K, = K,,y,,, which depends on the material, below which sub-critical crack growth is either undetectable or does not occur at all. K,,, is often called the “static fatigue limit”. Experimental results for crack propagation in glass in the vicinity of the static fatigue limit have been widely reported. Shand [17], Wiederhorn and Bolz [ 181, and Michalske [19] report a fatigue limit for soda-lime glass of 0.25 MPa m”‘. Wiederhorn [20] implies a K,,,, of 0.3 MPa ml”, and Wan, Latherbai and Law [21] report a static fatigue limit for soda-lime glass at about 0.27 MPa ml”. It is generally accepted, however, that more experimental data is needed to clarify whether crack growth ceases entirely for K, < K,,, or whether such growth occurs in this domain, but at an extremely low rate. In contrast to the sharp tip model, Charles and Hillig [22] proposed a mechanism which expresses crack velocity in terms of the thermodynamic and geometrical properties of the crack tip. Charles and Hillig proposed that, depending on the applied stress and the environment, the rate of dissolution of material at the crack tip leads to an increase, a decrease, or no-change in the crack tip radius, and hence to corresponding changes in the (Inglis) stress concentration factor over time. The change in stress concentration factor may eventually result in

localized stress levels which cause failure of the specimen. It should be noted that in this theory, the change in stress concentration factor is due to the changing geometry of the crack tip, and not to a change in crack length, over time. The stress corrosion theory of Charles and Hillig has considerable historical importance and is used within the Load Duration Theory and the Glass Failure Prediction Model. 2.2 Flaw statistics The strength of a brittle solid may be expressed as its resistance to failure under a specified load. The variability of strengths exhibited by seemingly identical specimens of glass may be quantified using Weibull statistics [23]. Weibull statistics provides a description of the strength of materials in which a distribution function containing adjustable parameters is used to predict the instantaneous probability of failure, P/, of a specimen. The distribution of strengths of materials is described by a continuous probability distribution of a simple form : Pj = l-exp

L 1 -kAa’”

Equation (2) gives the probability of failure for a given applied stress 0, over an area A, in terms of two surface flaw strength parameters : m, which indicates the spread of flaw sizes within the specimen (a small value implies a large spread of sizes), and k, which indicates their absolute size and density. The parameters m and k can only be determined by experiment. Only a limited number of such experimental results for window glass have been reported in the literature. The parameters determined by Brown [6, 241, Beason [7] and Beason and Morgan [8] are summarized in Table 1. The probability of failure given by equation (2) is precisely equal to the probability of the area A containing at least one flaw of the critical size for the applied stress 0 where the critical flaw size and applied stress are related by K,, as in equation (1). 2.3 Non-linear bending ofjlat plates Many design standards and failure models for architectural glazings rely on the knowledge of the stress distribution in a relatively thin, simply supported rectangular glass plate carrying a uniform load. Roark and Young [25] show that for a simply supported square plate of length b, thickness h, and with a uniform lateral

Table 1. Summary of experimentally determined values of surface flaw parameters m and k As-received glass Brown h241.

Beason and Morgan [8]. Beason [7].

t-9

Weathered glass

m = 1.3 k = 5.1 x 10m5’mm* Pam73 A in sq m, e in Pa (k = 5 x 10~30sq ft ’ psiC7’, A in sq ft, v psi) m=9 k = 1.32 x 1O-69mm’PaV9 (k = 3.02 x 10m3”inl6 lb-9) m=6 k = 7.19 x 1O-45m-* Pa-’ (k = 4.97 x 10-25sq in’ psiC6)

Architectural

Glazings : Design Standards

pressure q, the maximum stress occurs in the centre the plate and is given by :

gmax= 0.2874q The maximum is :

deflection,

$

of

(3)

w, also at the centre of the plate

w = -0.04444b4. Eh’

(4)

In equation (4) E is Young’s modulus for the plate material. For a 1 m2 plate, 4 mm thick, and with 2.2 kPa lateral pressure, equation (3) predicts that the maximum tensile stress at the centre of a deflected plate is 39.5 MPa, and equation (4) gives a maximum deflection of 22 mm. Equations (3) and (4) have been derived upon the basis of linear plate theory, in which it assumed that the midplane of the plate is free from membrane stresses. This is the case only when the deflections are small compared to the thickness of the structure. It is therefore evident that linear plate theory is inappropriate in the present numerical example since the calculated centre deflection is many times the plate thickness. Under these circumstances, non-linear plate theory must be employed. Various methods of solution of this problem have received a significant amount of attention in the literature [26, 27, 28, 291. Vallabhan and Ku [29] show that the results from a non-linear analysis predict stresses and deflections which are in good agreement with their experimental observations. In the present work, linear and non-linear solutions were generated using a commercially available finiteelement software package [30] for a range of plate sizes and lateral pressures. The results for a typical plate size and load are shown in Table 2. It is evident that the stresses and deflections calculated by the linear analysis are significantly higher than those obtained by the nonlinear solution. Further, an examination of the stress distribution shows that the maximum surface tensile stress is not located at the centre of the plate but more towards the corners. This is in agreement with the results of Vallabhan [31] who considered the problem analytically. In the present example, a non-linear solution gives the surface tensile stress in the centre of the plate as 17.3 MPa, significantly less than the maximum stress of 20.8 MPa which occurs near the corners. Figure 1 shows the results of linear and non-linear finite element solutions for a range of applied lateral pressures where the pressures and deflections are presented in dimensionless form. Also shown are the ana-

Table 2. Maximum

Linear theory [25] Finite element (linear) Finite element (non-linear)

BAE30:1-B

31

Fig. 1, Maximum deflection vs load for a simply supported flat plate under uniform lateral pressure. Axes are scaled in nondimensionalized units where 4 is the lateral pressure, E is Young’s modulus, w is the deflection, and h is the plate thickness. Results are shown for a linear and non-linear finite element analysis and also from Vallabhan and Ku [29] for an analytical non-linear solution.

lytical results of Vallabhan and Ku [29]. Good agreement is obtained with the results of the present work. This confirms the validity of the finite element method used in the present work to analyse the non-linear bending of a simply supported flat plate under uniform pressure.

3. FAILURE

MODELS FOR ARCHITECTURAL GLAZINGS

Failure models for brittle solids usually employ Weibull statistics to predict the probability of failure for a given lateral load and plate geometry. Because of the phenomenon of static fatigue, the probability of delayed failure

during

a time 1, will be somewhat

higher

than

that

(2) alone. However, equation (2) may be used to determine the probability of delayed failure if the stress c in equation (2) is replaced with a higher stress, op, which gives a probability of instantaneous failure equal to that for the actual stress applied over the time t,. The failure models described in this section seek to arrive at a value for this effective stress in terms of environmental and geometrical parameters. estimated

upon

the basis

of equation

3.1 Load Duration Theory, Brown, 1972 In 1972, Brown [6, 241 formulated an engineering model for glass failure which combined Charles and Hillig’s theory of static fatigue [22] with Weibull statistics

surface tensile stress and centre deflection for a 1 m2 simply supported load of 4 = 2.2 kPa

square plate carrying

a uniform

Surface stress or, at centre (MPa)

Maximum surface stress o,, (MPa)

Deflection at centre, W, (mm)

39.5 37.5

39.5 37.5

22 22

17.3

20.8

10.5

32

A. C. Fischer-Cripps

[23]. This model, called the Load Duration Theory, gives the time-to-failure by an integration over time of the applied stress raised to a high power, combined with the probability that a given surface area of glass contains a flaw of such size that would lead to delayed failure for that applied stress within that time. The work represents the first attempt to apply, what was at that time, modern fracture mechanics to an engineering problem of direct relevance to engineers and glass designers in the field. The numerical values for the various terms in the Load Duration Theory were determined from experiments involving the failure of “as-received” plate and sheet glass under uniform pressure. Brown was able to show good correlation between different sets of experimental results [2,

and R. E. Collins linearity, and substituting equation (8) into equation (2), Brown formulated a general equation which gave the probability of failure in terms of various environmental flaw statistical parameters. A fit to two experimental studies led Brown to estimate the values of the various constants in the above equations so as to give a general expression for square plates carrying a uniform load. For the conditions of T = 295 K, RH = 50% and fi = 100 psi s-’ (700 kPa ss’), and for a constant applied lateral pressure, q, over a time t for a plate of thickness h and length 1, Brown’s formula takes the form :

Pr=

l-exp

- 1.23 x 10mz8 A

BE’6-S

34.

Combining the rate equation of Charles and Hillig [22], and the effects of relative humidity of Wiederhorn [33], Brown derived an empirical relation for what he termed the strength, S, associated with a particular flaw as the result of the cumulative effect of the stress, temperature T, and relative humidity, RH : S=

s II

(5)

0

In equation (5), Brown’s original notation is used where S is a constant, the reciprocal of which indicates the strength of the material. T is the temperature, tl is the time to failure and a(t) is the macroscopic stress in the vicinity of the flaw as a function of time. R is the universal gas constant, y0 is an activation energy, and 12is a constant which Charles and Hillig found experimentally to be 16. At some reference conditions, T, and RH,, and for the special case of o = ,8t ouer a time t,, the constant S evaluates to :

S = RH&+‘P”

exp [--y,/RT]/T,”

(n+ 1).

(6)

However, for any RH and T, S is given by equation (5). Hence, for any experimental conditions RH, T and a(t), or is the equivalent stress which, if applied at the rate fi for a time t,, at temperature T, and relative humidity RH,, leads to the same probability of failure as the actual stress a(t) applied as some arbitrary function of time over a period t, at temperature T and relative humidity RH. I;(,!+

CT<,=

c

RH (a(t)/T)”

exp [ -y,/RT]

dt

I

I,

where C = [To”(n+ I)b/RHO exp [--.JO/RT,]]“‘“+“.

(7)

For any test conditions RH, T and a(t), there corresponds an equivalent stress, Do.The stress oe is equal to the product /ho. Brown selected a rather high value for B of 100 psi ss’ (700 kPa s-‘) and let t, = 1 s. For the special case of a constant stress, u(t) = o, applied over a time t,, and where T = T,, then equation (7) reduces to :

err= [(RH/RH,) With

various

corrections

cr”p (n+ l)t,]“(n+‘). for biaxial

plate

(8) stress

non-

0 43

x (l,h)4”-32

gqst I

1

(9)

In this equation, the constants B and s are plate parameters which take into account the biaxial state of stress and plate geometry, A is the area of the plate in sq ft, and E is Young’s modulus in psi. It should be noted that equations (8) and (9) demonstrate the implementation of the stress corrosion theory of Charles and Hillig within the Load Duration Theory. A particularly important feature of the theory, equation (7), has become known as “Brown’s Integral”. A slightly different approach is taken by Beason in the Glass Failure Prediction Model which is described in the next section.

3.2 Glass Failure Prediction Model, Beason, 1980 Beason [7] combined the static fatigue theory of Charles and Hillig, together with Weibull statistics and a nonlinear stress analysis, to develop the Glass Failure Prediction Model and this has formed the basis of the research work in this area at the Texas Tech University for many years. This semi-empirical model describes the strength of glass in terms of the interaction between surface tensile stresses, the surface flaw distribution, and the environment. It ultimately quantifies glass strength with reference to the two surface flaw parameters, m and k as used in equation (2). Beason and Morgan [8] claim that the Glass Failure Prediction Model accurately represents the strength of both as-received and weathered glass for wind loads of nominally one minute duration. Such loadings lead to stresses in the glass in the range of 40 to 100 MPa. In the Glass Failure Prediction Model, the variation of plate surface stresses is determined using a non-linear stress analysis which significantly improves upon the range of loading and plate geometries originally offered by the Load Duration Theory. The non-linear stress analysis is shown to be in excellent agreement with the stresses and deAections of the experimental data of Tsai and Stewart [34], Bowles and Sugarman [32] and Hershey and Higgins [35]. Beason finds that for the thin rectangular glass plates normally associated with window glazings, the most appropriate boundary conditions are those of simple supports and where the edges are free to slip in-plane. Beason also finds that the non-linear analy-

Architectural

sis is in agreement with the results of Brown for the case of a simply supported square plate. In the Glass Failure Prediction Model, the effects of load duration, relative humidity and temperature, are taken into account by the consideration of an equivalent stress, ou, which, if applied over 60 seconds, leads to the same probability of failure as the actual stress o(l) applied as some arbitrary function of time for a period t,. From equation (5), for the special case of o(t) = 0, at T = T,, the equivalent 60 second stress is given by :

oe =0

” (RH/RH,)o(t)“/60

IS

dt

33

Glazings : Design Standards

(10). Thus, just by considering the application of the stress corrosion theory alone, regardless of any differences in calculating the stress distribution, it may be expected that the probability of failure predicted by the Glass Failure Prediction Model would be significantly less than that obtained by the Load Duration theory. Indeed, for the constant stress case, sample calculations show that the equivalent stresses given by equations (7) and (10) differ by about 30%. In Section 3.5 it is shown that the Glass Failure Prediction Model does indeed predict much lower failure probabilities than the Load Duration Theory. Contrary to what is popularly believed, the Glass Failure Prediction Model does not make use of Brown’s interpretation of the stress corrosion theory of Charles and Hillig.

1in 1. (10)

To determine the probability of failure, the surface area of the plate is divided into regions which have approximately the same value of equivalent stress as calculated by equation (10) and adjusted by a biaxial stress correction factor, the details of which need not concern us directly at present. Substituting equation (10) into equation (2) for each area element allows the probability of failure for that area element to be determined and the total probability of failure for the entire surface area is then readily calculated. The surface flaw parameters m and k can only be determined by experiment and cannot be directly measured. Beason derived surface flaw parameters by analysing glass failure strength data. Typically, such tests involve a series of rectangular glass sheets which are subjected to increasing lateral pressures until fracture occurs. The failure pressure is recorded and the location of the fracture origin determined from a visual examination of the specimen. The surface tensile stresses which exist at the location of the fracture origin may be calculated by nonlinear plate theory. The equivalent 60 second tensile stress is then determined using equation (10). An equivalent 60 second lateral load may then be calculated using nonlinear plate theory. The 60 second equivalent lateral load is that which corresponds to the equivalent 60 second tensile stress. By recognising that the exponential distribution function given by equation (2) has a mean value equal to its standard deviation, Beason determined the best values for m and k which agreed with distributions of the experimental data. Equation (10) deserves special note since it represents an important difference between the applications of the stress corrosion theory of Charles and Hillig in the Glass Failure Prediction Model, and in the Load Duration Theory. In the Load Duration Theory, an arbitrary loading is expressed in terms of a linear increase in stress over a very short time period as given by equation (7). In the Glass Failure Prediction Model, an arbitrary loading is expressed in terms of a 60 second constant equivalent stress by equation (10). The use of a 60 second equivalent stress in equation (10) ignores the effects of any stress corrosion which may occur within the 60 second period. In the Load Duration Theory, Brown makes use of a somewhat arbitrary, but large value of load rate, /?, to determine an equivalent instantaneous stress, which is precisely what is required for equation (2). Thus, for a constant lateral load applied over some arbitrary length of time, the equivalent stress determined by equation (7) is somewhat higher than that obtained from equation

3.3 Crack Growth Models 3.3.1 Crack Growth Model. The Crack Growth Model combines an empirical formulation of sub-critical crack growth and Weibull statistics using linear elastic fracture mechanics [9,36,37]. Experimental data of crack velocity as a function of the stress intensity factor K, show a characteristic shape as illustrated in Fig. 2 This figure shows three regions of characteristic behaviour. Region I is particularly important for this discussion because the crack velocities are very low, in the range from lo-” to 10m4 m s-l. Crack velocities in Regions II and III are generally too high to contribute significantly to the timeto-failure estimates. The growth of a crack in Region I may be represented empirically by : da -=DK; dt

(11)

where D and II are constants which depend on the material and the environment. For soda-lime glass immersed in water, n = 17 and, with da/dt in m s-’ and K, in Pa m”2, log,,D = - 102.6 (from the authors’ analysis of reference [33]). A critical flaw size aLcan be associated with a uniform applied external stress co by equation (1). This relationship is illustrated in Fig. 3 where the stress intensity factor is shown in terms of an applied external stress CTand crack length a. In this figure K, c indicates the condition

Stress intensity factor (log scale)

Fig. 2. Crack velocity vs stress intensity factor. The three regions of sub-critical crack growth are shown. The vertical line at Klc represents instantaneous failure.

A. C. Fischer-Cripps and R. E. Collins

34

where instantaneous failure occurs. For instantaneous failure at stress era, P/ as given by equation (2) is the probability that an area A contains a flaw of size equal to or larger then a,. However, if subcritical crack growth occurs during a time t,, flaws of size a;, less than a‘, will extend to a length a, over that time. Thus, for delayed failure in time t,at stress ga, the probability of failure P,, is actually the probability of area A containing a flaw of size greater than or equal to a,. The probability of delayed failure at stress erais therefore precisely equal to the probability of instantaneous failure at stress Okwhere : K,,

= cpY&.

(12)

The time-to-failure represents the growth of a flaw from an initial size a, to a final “critical” size a,. This growth can be described analytically by an integration of equation (11) to give :

t, = 2Kf-“/D(n-2)~;

Y*n

(13)

where K, is the initial stress intensity factor, before any sub-critical crack growth, for a uniform applied stress o’a and crack length a, as given by equation (1). Equation (13) assumes that K,c>> K,. The critical stress cr,, for a, is also given by equation (1) where K, = Klc, and hence : (14) Substituting

into equation

The probability of survival of a specimen consisting of a number of surface area elements is the simultaneous probability of survival of all the elements given by : P h4 = P,, * PII and the probability

.

(19)

of failure for the specimen P f,,,,l = 1 - P,,,,&

(20)

Note that in the Crack Growth Model, it is assumed that all flaws undergo subcritical crack growth, no matter how small the value of K, for the flaw. 3.3.2 Modified Crack Growth Model. The probability of delayed failure for a given applied stress can be found from equation (17). As stated above, equation (17) assumes that all flaws undergo sub-critical crack growth, no matter how small the value of K, for the flaw. This is shown in Fig. 2 where the assumption made by the Crack Growth Model is shown at (a). In our modification to the Crack Growth Model, it is assumed that the velocity of propagation of a crack is described by equation (11) for values of K, greater than K,,,,, and that no crack growth occurs for K, < K,,,,, the static fatigue limit. This is shown in Fig. 2 at (b). For a given stress go, let a, represent the crack length at the static fatigue limit. Hence :

(13) gives :

t, = 2(a,,/aJ-*/D(n-2)a,2

Y2nK;,2

(21) (15)

and re-arranging, c,, = (t,o(n-2)YZKK;,2a~/2)“(“~2’.

is thus :

(16)

In equation (16), a, represents the critical stress for the initial flaw size a, for a given lifetime. A relationship which expresses the probability of failure in terms of the flaws and their propensity for subcritical crack growth can be obtained by substituting equation (16) into (2) : P, = l-exp[-kA{tfD(n-2)Y’nK;;2
Flaws of length less than a, do not contribute to timedelayed failure at stress co. The probability of delayed failure at stress o. is thus given by the probability of the area A containing a flaw of size greater than a,, and is therefore equal to the probability of instantaneous failure at stress crUas shown in Fig. 3. The probability of failure, P,, for a uniformly stressed area A can then be determined from equation (2) using an effective stress 6, : P, = 1-exp

(-kAaT).

(22)

The effective stress cre is found from K,, = a,Y(n~)‘:~, where a is the larger of a,, given by equation (21), or ai, given by equations (12) and (16).

(17) In equation (16), o,,is an effective critical stress similar to that given by crein the Load Duration Theory and the Glass Failure Prediction Model. Generally, glass plates are subjected to lateral loads which lead to a non-uniform surface stress distribution. The surface stress distribution may be calculated analytically or by the finite element method. The probability of failure for a non-uniform stress distribution may be calculated by grouping together areas which are subjected to approximately the same level of stress. This is effectively the approach taken by Beason in the Glass Failure Prediction Model. The probability of failure, P,, for a complete surface area can be determined by a calculation of the probability of simultaneous survival of all the uniformly stressed area elements. The probability of survival of one element is given by : P, = l-9,.

(18)

5ub-crltlcal crack growth

a, a,

Crack length

a,

Fig. 3. Stress vs crack length showing sub-critical crack growth with and without the presence of the static fatigue limit.

Architectural Glazings : Design Standards 4. STANDARDS

FOR ARCHITECTURAL GLAZINGS

Glass design charts typically give the appropriate maximum allowable area and minimum thickness for window glass for a selected wind loading. Design charts and procedures for architectural glazings are published within glass manufacturers’ product information [2, 41, various national standards [l, 38, 39, 401 and building codes of practice [5]. Invariably, building codes of practice contain a very superficial treatment and often refer the reader to manufacturers’ product information. In this section, glass design procedures contained within the Australian standard, AS1288 [l], the United States Uniform Building Code [5], the European DIN1249 standard [40], and the Canadian CAN12.20 standard [39] are reviewed in detail.

4.1 AS1288, “Glass in Buildings-Selection and Installation” The Australian standard [l], A31288-1989, was first published in 1973. Reid, 1986 [41], finds the technical content of the standard to be minimal and concludes that AS1288 represents established industry practice rather than the results of any formal scientific analysis. The recommendations of the design charts in AS1288 are very similar to those published in the corresponding British [38] and United States [5] standards and are based upon experience and experimental knowledge-none of which are specified or referenced in any further detail. Figure 3.1 of AS1288 gives the design parameter as a wind load for various thicknesses and areas of simply supported rectangular glass plates. The values given in this figure are based on short-term failure tests (conducted by Libbey Owens Ford), undertaken by glass manufacturers in the United States, with a factor of safety of 2.5 applied to the mean failure load as the criterion for design pressure. Figure 3.2 of AS1288 extends the results of Fig. 3.1 to the case of rectangular plates simply supported on two sides only. This has been accomplished by setting the maximum allowable tensile stress in the centre of plate to 16.7 MPa, and then determining the allowable lateral pressure corresponding to that maximum stress using a standard engineering formula applicable to this type of loading and support. For different types of glass, e.g. toughened, wired, etc., various “pressure factors” have been applied to the results given by Figs 4.1 and 4.2 of AS1288 to produce a range of graphs and tables which are readily accessible by glass designers and architects. For geometries and loads not covered by the design charts, Table 3.2 of AS1288 recommends, for glass of thickness less than 6 mm, a maximum design stress of 8.35 MPa for sustained loads and 16.7 MPa for intermittent (wind) loads nominally of 3 seconds duration [42]. The determination of design wind loads for particular applications is complex and beyond the scope of the present work. Sample values for residential buildings are given in Appendix D of AS 1288. In this appendix, design wind loads are given in terms of the building height, terrain, and window location within the building. Non-linear finite element analysis indicates that for a range of plate geometries and design loads given by Fig.

35

3.1 of the standard, a maximum tensile stress of approximately 22 MPa exists on the surface of the glass plates. However, as noted above, the maximum tensile stress, for non-linear bending, does not necessarily occur at the geometric centre of the plate. For a 1 m*, 4 mm thick plate, carrying a design load of 2.2 kPa, non-linear finite element analysis gives a stress at the centre, and on the surface of the plate as 17.3 MPa. The geometry mentioned here is significant because it is the same as that used in the Canadian standard for determining a general design stress for applications not covered by the design charts therein. The Canadian standard however, recommends a maximum design stress of 25 MPa. It would appear that the Australian standard bases its design stress of 16.7 MPa on the stress calculated at the centre of the plate, rather than the maximum stress which occurs elsewhere. Thus, it appears that the maximum stress for simply supported plates on two sides only, as described by Fig. 3.2 of AS1288, is conservative and inconsistent with the maximum stresses implied in Fig. 3.1. Further, it is interesting to note that the design stress for sustained loads in Table 3.2 of AS1288 appears to be somewhat arbitrarily set to exactly half of 16.7 MPa at 8.35 MPa. This recommendation appears to be in agreement with generally acceptable “rule of thumb” design stresses which appear to range anywhere from between 3 and 7 MPa [43, 441, and is in rough agreement with a “correction factor” of 0.4 given in the Canadian standard for sustained loads. An acceptable failure probability and service life for glass windows depends upon the application and can vary widely. Information from the industry [45] is that 4 failures in 1,000 is unacceptably large, 1 in 1,000 is tolerable, and 0.3 in 1,000 is commercially acceptable within a service life of between 30 to 50 years. Beason argues that a factor of safety of 2.5 applied to the mean failure load for a normal distribution of loads corresponds to a failure probability of 8 in 1,000. Draft versions of AS 1288 quoted an 8 in 1,000 probability of failure as being the basis for the recommendations given by the design curves and tables but this figure was subsequently deleted from the text. A probability of failure of 8 in 1,000 is also implied in the Canadian standard. 4.2 Uniform Building Code, Chapter 54, “Glass and Glazing” The United States Uniform Building Code [5] contains design information for simply supported rectangular plates subjected to lateral wind loads of nominally one minute duration. In early editions [46], this information is presented in tabular form, and in later editions [5], the same information is given graphically. The design wind pressures given in the Australian standard, AS1288, are almost identical to those published in the Uniform Building Code. No extended discussion of the principles underlying the recommendations is given. 4.3 DIN 1249, “Glass for use in building construction” This European code of practice [40] details the requirements of glass in buildings in a very different manner to those described in British and American standards. No design charts or tables are given. The reader is referred to manufacturers’ product information for glass design

36

A. C. Fischer-Cripps

strategies, but is advised of a general minimum strength for different types of glass when tested in a standardized manner. The standard test involves the loading of a flat specimen with a coaxial ring of specified diameter. The bending strength for such a test is given for different types of glass in Table 2 of Part 3 of the standard. For float glass, the minimum bending strength is 45 MPa, and for toughened glass, 120 MPa. In this respect, the DIN standard specifies what could be considered a material property for different types of glass, rather than a design strategy for the use of that glass in glazing. 4.4 CANICGSB-12.20-M89,

“Structural Design of Glass

in Buildings” The Canadian standard [39], CAN-12.20, is significantly different to the other standards considered in the present work because the design strategies contained within it are based upon the Glass Failure Prediction Model. The standard describes the strength of glass with respect to a 60 second reference strength which is calculated using the principles contained within the Glass Failure Prediction Model. Equation (4) of Appendix B of the standard gives the reference strength in terms of a lateral load Reef

1

and R. E. Collins variation-as typified by the lower value of m found by Beason (m = 6 for weathered glass), compared to Brown (m = 7.3 for as-received glass). It is interesting to note that the value of m = 7, as used by the standard, is less than that for as-received glass and somewhat more than that for weathered glass. This indicates that the values for m and S, as used within the standard may have been selected so that the design curves shown therein give good agreement with established design practice. Of particular interest to the present work is the recommended design stress for applications not covered by the design charts. The Canadian standard recommends a limiting tensile stress of 25 MPa away from the edges, and 20 MPa for clean cut edges. These figures, the standard states, are derived from the Glass Failure Prediction Model for a 60 second load applied to an area of about 1 m2, and presumably, for a probability of failure of 0.008. Interestingly, the tensile stress of 20-25 MPa is in excellent agreement with the 22 MPa obtained by the non-linear finite element analysis of the present work for the same plate geometry, thus supporting the earlier comments regarding the choice of 16.7 MPa by the Australian standard for this design stress.

I ‘N

Rrcf = R,

t

[ N+l

5. COMPARISONS

(23)

(23), Reefis a reference lateral load for a unit time of 1 minute, Rf is the actual load measured at a time t. The lateral loads and stresses are related by a constant c, according to Appendix B, clause B2.3.2, by :

In equation

CKR’.

(24)

Equation (24) applies to simply supported plates, and the constant N is given in the standard as 1.5 and is equal to the product nc, where n is as used previously in the present work to describe the rate of stress corrosion (see Section 3.1). Now, a simple analysis using the Glass Failure Prediction Model, as exemplified by equation (10) in the present work, shows that equation (23) is in error, and should be written :

The Canadian standard does not quote a precise value for c. Assuming a value c = 0.75, then the effect of the error in equation (23) is to increase Rref in equation (25) by a factor of 1.3. The significance of this will be made clear shortly. The standard, although specifying the surface strength parameter m = 7, does not specify a value for k directly, but gives a value for “S,“, which is presumably the “reference strength” as commonly given in Weibull statistics. It can be shown that the surface strength parameter k contains both S, and a flaw density term. The standard does not give a value for the flaw density and so a direct estimation of the parameter k is not possible. It is difficult to establish whether the strength parameters m and S, used in the standard refer to as-received or weathered glass. The strength of weathered glass is considerably less than that of as-received glass and shows a wider

For a 1 m2 area of 4 mm thick glass, Fig. 3.1 of AS 1288 specifies a maximum design wind load of approximately 2.2 kPa, which is similar to the recommendations of the U.S. Uniform Building Code [5] and the Canadian standard CAN-12.20 [39]. The finite element calculations of the present work for this geometry and loading show excellent agreement with the non-linear analysis contained within the Glass Failure Prediction Model, which in turn has been shown to be in agreement with experimental data [7]. The stress distribution obtained by finite element analysis can therefore be used to compare the predictions of the various failure models with the recommendations of the design standards. Table 3 shows the probabilities of failure predicted by the various models for this loading and plate geometry for 100% relative humidity for a time of 60 seconds. We have chosen this value of relative humidity in these comparisons as representing the worst-case situation. The results for the Load Duration Theory have been calculated using equation (9), and then again using the finite element results of the present work in conjunction with equations (2) and (8). The results for the Glass Failure Prediction Model have been calculated with surface strength parameters m and k as determined by Brown rather than those of Beason. With m and k given by

Table 3. Probability of failure in 60 seconds supported, as-received, glass plate carrying pressure of 2.2 kPa Model Load Duration Theory (equation Load Duration Theory (equation Glass Failure Prediction Model Crack Growth Model Modified Crack Growth Model

for a 1 rn’ simply a uniform lateral

PI (9)) (8))

0.008 0.002 0.0002 0.006 0.006

Architectural Glazings : Design Standards Beason, the failure probabilties given by the Glass Failure Prediction Model are slightly lower than those shown in Table 3. Figure 4 shows the probability of failure for various failure times. With the exception of the Glass Failure Prediction Model, good agreement of results from the different approaches is obtained for failure times less than about 4 days and the actual values of fracture probability are consistent with the recommendations of the glass design standards. Significant discrepancies exist between the models for longer failure times. Clearly, the Modified Crack Growth Model predicts a lower probability of failure for periods longer than a few days compared to the other models. The results of the Glass Failure Prediction Model shown in Table 3 do not agree well with those of any of the other models nor with the recommendations of the glass design standard. Beason and Morgan [8] find that, for a 0.008 probability of failure for as-received glass, the Glass Failure Prediction Model predicts a recommended lateral pressure some 45% higher than that given by various industry standards. Despite this large discrepancy, these authors claim that the Glass Failure Prediction Model gives reasonable values of strength of new glass plates. In the present work, it has been shown that the low values of probability obtained by the Glass Failure Prediction Model are a result of the use of the 60 second equivalent stress which comes from the particular implementation of the stress corrosion theory of Charles and Hillig. In contrast, Brown, in the Load Duration Theory, seeks to establish an equivalent instantaneous stress based upon a rather arbitrary choice of the value of the factor /l in equation (8). Although Brown makes no comment on his choice of /$ Table 3 shows that the Load Duration Theory does, however, produce realistic results-for short failure times. It is interesting to note that the Canadian glass design standard, which utilizes the Glass Failure Prediction Model, shows good agreement with the recI

-

.

:

0.9

I m2, 4 mm thick as-received glass : 2.2 kPa lateral pressure

0.8 - -Crack g

0.7

z z

0.6

1

b %

Growth

Model

- - Modlfled Crack - * - - Load Duration

Theory

-

PredIctIOn

-

Glass Model

Fallwe

Growth

Model

.

./ --mm

/---

0.5

+I z :: * ;

._--

.’

/

0.4 0.3-

-

0.2

-

0.1

-

#i

/

.I /-

A-

SC,I,,,.,

i,l’iml

10-z Time

to

failure

/,,,,,,‘,,

10-l

,.‘,,.,,

loo

,111

IO’

(years)

Fig. 4. Failure probability vs time to failure for various glass failure prediction models as calculated for a 1 m2, 4 mm thick, simply supported flat plate carrying a 2.2 kPa uniform lateral pressure.

31

ommendations of other design standards. It appears that this has occurred because of an error associated with the value for Rref in equation (23), and also the choice of strength parameters m and k used in the standard. Finite element analysis shows that if the lateral load for the numerical example used above is scaled upwards by a factor of 1.3 (where the value of this factor depends on the value of c in equation (24)) then the Glass Failure Prediction Model, using m and k as determined by Brown for as-received glass, yields a probability of failure of 0.001. The Canadian standard implies that a 0.008 probability of failure was used to determine the values of Reef shown in the design charts therein but does not specify whether the glass so modelled is in as-received or weathered condition, the value m = 7 actually used indicating somewhere in-between. This would effectively increase the probability of failure so calculated, but the lack of information regarding the parameter k makes a direct comparison impossible. Considering the errors involved in both the Glass Failure Prediction Model and its implementation in the Canadian standard, it may be concluded that the strength parameters used in the standard were selected so as to bring the resulting design charts into agreement for glass geometries and loadings for which a large body of experience exists, as exemplified in other glass standards. Table 4 shows the probability of failure predicted by the Load Duration Theory, the Crack Growth Model and the Modified Crack Growth Model for a 1 m* area of as-received glass subjected to a constant uniform biaxial tensile stress. The values shown in Table 4 clearly show that the Modified Crack Growth Model gives a failure probability which is consistent with the 0.008 implied in glass design standards for both sustained and intermittent loads. The Load Duration Theory, the Glass Failure Prediction Model and the Crack Growth Model all overestimate the probability of failure for glass for low stresses for long periods because they ignore the effect of the static fatigue limit. The Load Duration Theory, although proposed as a general failure model, is valid for failure times of around 60 seconds and indeed, Brown explicitly states that his Load Duration Theory does not consider the effect of stresses of a low magnitude for long periods. The Glass Failure Prediction Model is similarly designed to predict short-term failures although no statement to this effect is included in the presentation of this model. The present work serves to emphasize that designers should not attempt to apply either the Load Duration Theory or the Glass Failure Prediction Model to situations in which relatively low stresses exist for long periods of time. In these cases, the effect of the static fatigue limit is important. The preceding remarks should not be taken as an implied criticism of the validity of these models at shorter failure times which generally involve the application of large stresses. This is especially the case for the Load Duration Theory which has been validated at moderate to high stresses for failure times in the order of 60 seconds. The inference is, however, that these models should not be used beyond the region for which their applicability has been demonstrated. Further, the 60 second equivalent load which features in the Glass Failure Prediction Model

A. C. Fischer-Cripps

38 Table 4. Probability

of failure for a uniform biaxial tensile stress of 8.35 and 16.7 MPa for a 1 rn’, as-received, glass plate

8.35 MPa AS1288 recommended load

LDT GFPM CGM MCGM

sustained

60 s

1 yr

50 yr

0.000009 0.000002 0.00003 0.00003

0.0025 0.0007 0.017 0.0070

0.013 0.0042 0.11 0.0070

severely limits the accuracy of this model in predicting failures for any type of loading. The agreement between the Load Duration Theory and the crack growth models for times less than about 4 days is interesting since they employ quite different methods to model the effect of time delayed failure. This agreement is obtained because both approaches incorporate adjustable parameters which are chosen to agree with the available experimental data for short-term failures. As has been shown, for a 1 mz simply supported plate with a uniform pressure of 2.2 kPa, non-linear finite element analysis gives a surface tensile stress of approximately 17.2 MPa at the centre of the plate. This appears to be the basis for the design maximum stress of 16.7 MPa, for intermittent loadings, as stated in Table 3.2 of AS 1288. However, the AS 1288 appears to ignore the fact that higher stresses exist away from the centre, having magnitude nearer to 22 MPa. This observation is in agreement with the stress distribution shown by Vallabhan and Ku [29] for this type of loading. Further, the Canadian standard, CAN-12.20, recommends a blanket design stress of 20-25 MPa for intermittent loads. Table 5 shows the probability of failure over 60 seconds as predicted by the various models for a uniform biaxial stress of 22 MPa for as-received glass. Perhaps not surprisingly, the probabilities of failure predicted by the Crack Growth Model and Modified Crack Growth Model are significantly higher than those shown in Table 4. It should be noted that in practice, flat glass sheets are unlikely to be uniformly stressed to 22 MPa over their entire surface area. In the failure tests for which the 22 MPa figure was obtained, only a portion of the total surface area was subjected to a stress of this magnitude. The Load Duration Theory gives a very good result for this stress level but it should be noted that the failure probability given by this model in Table 4, for a uniform 16.7 MPa over 60 seconds, is somewhat lower than the target figure of 0.008. Table 5. Probability of failure for a uniform biaxial tensile stress of 22 MPa over 60 seconds for a 1 m’, as-received, glass plate Uniform biaxial stress 22 MPa, @; t = 60 s LDT GFPM CGM MCGM

and R. E. Collins

0.009 0.0028 0.081 0.081

:

16.7 MPa AS1288 recommended intermittent load Time 60 s 0.0013 0.0003 0.010 0.010

1 yr

50 yr

0.3 0.12 1.0 0.71

0.874 0.527 1.0 0.71

In the failure models presented here, it is evident that the probability of failure is strongly dependent on the level of tensile stress. The implication is that the effect of a very few strong gusts far outweighs the cumulative effect of a number of years of lower stresses.

6. SUMMARY

AND CONCLUSIONS

In the present work, the technical basis of the recommendations contained within various design standards has been discussed. The design charts in the Australian standard, AS1288, assume a design stress determined from failure tests on simply supported glass plates. Non-linear finite element analysis indicates that the design stress has been taken at the centre of a laterally loaded plate where the stress is not necessarily a maximum. This implies that the recommendations of Table 4, and Fig. 3.2 of AS1288 are somewhat conservative compared to Fig. 3.1 of the standard, which by its nature, embodies the maximum stress rather than that computed at the centre. In the present work, the Load Duration Theory, the Glass Failure Prediction Model and the Crack Growth Model have been examined and a modified Crack Growth Model has been developed. The Load Duration Theory and the Crack Growth Models give results for failure probabilities for large stresses applied over short periods which are consistent with the recommendations of glass design standards. A feature of the Glass Failure Prediction Model has been identified which leads to the prediction of failure probabilities for both short and long term loadings which are about an order of magnitude less than those inferred in the glass standards. An error has been identified in the implementation of the Glass Failure Prediction Model within the Canadian glass design standard. It has been found that the Load Duration Theory, the Glass Failure Prediction Model and the Crack Growth Model do not predict failure probabilities which are consistent with the glass design standards for specimens subjected to stresses of a low magnitude applied over long periods because they ignore the effect of the static fatigue limit. The Modified Crack Growth Model gives results which are consistent with the recommendations of the design standards for both short and long applications of load. Although the results of the Modified Crack Growth Model presented here show good correlation with glass design standards for both short and long term stresses, it is recognised that a substantial field test program would

Architectural

Glazings : Design Standards

be required to validate the model before it could be regarded as being suitable for general use within the engineering community. It is interesting to note that, with the exception of the Glass Failure Prediction Model, failure models predict probabilities of failure which are consistent with the recommendations of the glass design standards for the geometry considered in the present-work. Presumably the recommendations of the glass design standards reflect the knowledge gained over many years by glaziers and architects

in the field and this leads

to the conclusion

the glass failure models may be used to provide

that

realistic

probabilities of failure for new types of glass installations where only limited field experience exists. The usefulness of the models, however, seems to be more related to empirical experience rather than to a detailed scientific analysis of the fundamentals of the physics of glass fracture.

Acknowledgements-This

work was supported in part by His Royal Highness Prince Nawaf bin Abdul Aziz of the Kingdom

of Saudi Arabia through the Science Foundation for Physics at the Universitv , of Svdnev. , ,, and bv< the Australian Enerav__Research and Development Corporation.

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28. 29. 30.

39

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C. V. G. Vallabhan, Iterative analysis of nonlinear glass plates, J. Sfruct. Eng. ASCE 109, 489-502 (1983). R. Bowles and B. Sugarman, The strength and delfection characteristics of large rectangular glass panels under uniform pressure, Glass Technology, 3, 156- 170 (1962) S. M. Wiederhorn, Influence of water vapour on crack propagation in soda-lime glass, J. Amer. Ceram. Sot. 50,407-414 (1967). C. R. Tsai and R. A. Stewart, Stress analysis of large deflection of glass plates by the finite-element method, J. Amer. Ceram. Sot. 59, 445-448 (1976). R. L. Hershey and T. H. Higgins, Statistical Prediction Model for Glass Breakage from Nominal Sonic BoomLoads, Booz-Allen Applied Research, Inc., Report No. Faa-RD-73-79, Bethesda, Maryland (1973). T. Dabbs, Fracture Properties of High Strength Glass with Particular Reference to Optical Fibres, PhD Thesis, University of New South Wales, Australia (1984). S. M. Wiederhorn, S. W. Frieman, E. R. Fuller, and C. J. Simmons, Effects of water and other dielectrics on crack growth, J. Mat. Sci. 17, 3460-3478 (1982). BS 6262 : 1982. Glazing for Buildines. British Standards Institution (1982) CAN/CGSB-12.20-M89, Structural Design of Glass for Buildings. Canadian General Standards Board (1989). DIN 1249, Glass for use in Building Construction, Deutsche Institut fur Normung, e.V. Berlin (1990). S. G. Reid, Basis of Australian safety standards for architectural safety glazing, Civil Engineering Transactions, ASCE, 2999305 (1986). I. Calderone, Monash University, Private communication to A. C. Fischer-Cripps, University of Sydney (1993). R. J. Charles, Static fatigue of glass I, Journal of Applied Physics 29, 1549-1553 (1958). H. MacKenzie, Pilkington UK private communication to R. E. Collins, University of Sydney (1990). R. O’Shaughnessy, private communication to R. E. Collins, University of Sydney (1992). United States Uniform Building Code, Glass and Glazing, Ch. 54, International Conference of Building Officials, Whittier, Calif. (1973).