Strength prediction of annealed glass plates – A new model

Engineering Structures 79 (2014) 244–255

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Strength prediction of annealed glass plates – A new model David Z. Yankelevsky ⇑ Faculty of Civil & Environmental Eng., National Building Research Institute, Technion-Israel Institute of Technology, Haifa 32000, Israel

a r t i c l e

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Article history: Received 14 November 2013 Revised 8 August 2014 Accepted 10 August 2014 Available online 3 September 2014 Keywords: Glass plate Flaws Brittle material Fracture mechanics Laboratory testing Probabilistic Modeling Structural design Cracking Failure

a b s t r a c t A new model for assessing the strength of structural members made of annealed glass plates is presented. The model refers to a glass plate that is supported at its ends and loaded by a given loading system. To fully determine the mechanical behavior of the glass plate, a flaws distribution function is presented, from which the critical stress distribution is calculated. These critical stresses determine the local ultimate resistance of the plate. The model determines the flaws distribution over the area of a large basic plate, from which plates of different sizes are cut out. Each cut out plate is characterized by a different map of flaws that is distributed over its surface and it behaves differently as a result. The flaws shape, size and location determine the critical tensile stresses that are required for crack opening. These critical stresses are compared with the growing stresses during loading, to determine at which point and at what loading level fracture is initiated. The model is capable of predicting the tensile strength distribution of a large group of glass plates of a given geometry and boundary conditions. The model provides much information concerning the probabilistic distribution of the tensile strengths and the location of the fracture origin. For an examined case the model yields an almost symmetrical tensile strengths distribution, and although it is somewhat different than existing statistical functions, it is similar to the normal distribution. It is shown that the entire analysis of determination of the critical tensile strength is independent on the plate’s thickness, although the latter is important to determine the magnitude of the bending moment at failure and the magnitude of the applied load that causes that failure. The model is found to be in close agreement with test data; it may explain the inspected and measured results and provide insight to glass plate behavior. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction 1.1. General This paper deals with flat thin glass plates that are made of soda lime silica annealed glass. Such plates are commonly used in buildings windows glazing. In that implementation a thin rectangular glass plate is considered, however in the general case it may have any shape and size, it may be implemented in different applications and be subjected to different conditions. This paper focuses on the static behavior of rectangular glass plates with given boundary conditions that are subjected to a static short term loading system. Although glass production is known for many centuries, its quality has been considerably improved over the last decades. Current glass technology allows production from well-defined ⇑ Tel.: +972 4 8292286; fax: +972 4 8324534. E-mail address: [email protected] http://dx.doi.org/10.1016/j.engstruct.2014.08.017 0141-0296/Ó 2014 Elsevier Ltd. All rights reserved.

raw materials [1], of large size, high quality flat glass panels with constant thickness. Commonly glass plates are produced within a thickness range of 2–25 mm. The production process and the typical properties are described in detail in the literature (e.g. [2]). Nowadays float glass is a common product of high quality and pronounced resistance. As such it may be used not only for architectural and functional needs but also may carry loads and serve as a structural element. Even in its most common usage in windows’ panes it is exposed to loads, such as wind loads, thermal and other mechanical loads, and may also be exposed to vibratory, impact and blast loading. It is known that glass responds differently to different loading rates and durations [3,4]. There exist different methods for window glass analysis that are developed to address special cases of plate geometry, type of loading, etc., but none of them is free of pronounced shortcomings. The existing models are based on experimental data; nevertheless they cannot consistently predict test results. Recent studies have criticized the existing models and their limitations with regards to glass plate design e.g. [5,6]. This reflects both the doubts

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and the lack of confidence with regard to existing methods and motivate for further investigation. It is already well known that glass panes of identical geometry that are loaded by the same loading system, break differently and the experimental ultimate load or stress varies within a wide range. In certain cases, bending test results provide scattered values of the bending strength with a spread of 30–50% of the mean strength [7]. In different tests the location of facture origin varies within a wide range and mostly does not fit the location of the maximum applied tensile stress. The experimental statistical distribution of the fracture strength of these plates was compared with well-known statistical functions, like the Weibull, Normal or Log-normal distributions. In many cases e.g. [28] none of them represents properly the experimental results. Several methods for glass analysis e.g. [8] are based on the Weibull distribution, the parameters of which are calibrated from test results. However, calibration to different tests yields different parameters. There is a need for a fresh look at the fundamental aspects of glass plate behaviour in general, and its failure under the action of static and dynamic loadings in particular. The present study has been initiated following some critical review of the current methods in an attempt to gain better insight of glass plate behaviour. It aims at proposing a new approach to predict the strength of glass plates that are subjected to static loading. The new approach is based on minimal assumptions without any calibration with test data, and is capable of demonstrating many inspected features of glass plate behaviour. This paper aims at analyzing short term static loading on glass beams and plates at ambient conditions, as is commonly carried out in laboratory research and in standard tests. The lifetime behavior of a glass pane and the evolution of cracks with time as function of different loading and environmental conditions are beyond the scope of this paper and is a subject matter of future research. The advantages of the proposed approach are its generality, the fact that it is based on the glass pane data and that it is independent on test results. It means that no calibration is required. The proposed model may yield the probability distribution of the tensile strength of glass as a result of the analysis and not as an a priori assumption, which is common to existing models. The present model is based on the geometrical and physical data of the glass plate including the flaw distribution and all the predicted results come out as results of the analysis. The present model sheds new light on several parameters that affect the behaviour of a glass plate and provides insight and tools to plan further for improved testing and better interpretation of test results. It may serve for further development to incorporate lifetime conditions and be further developed to provide a design tool. 1.2. Annealed glass Annealed glass is characterized by its attractive properties: it is transparent and hard wearing; it has flat smooth surfaces and demonstrates a high compressive strength. These features, among others, make it an ideal material for windows glazing and for other architectural uses. On the other hand, annealed glass is brittle and upon failure it breaks into sharp shards. It contains randomly distributed numerous flaws resulting mainly by its production and also due to cutting and handling. The flaws are very small in size and cannot be observed by the naked eye; however they significantly affect the material mechanical behaviour. In this paper we shall assume that independent on the density of flaws, the distance between flaws is sufficiently large compared to their size thus individual flaws do not affect each other, and the tested specimen fails when the first flaw reaches the fracture condition.

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During its service life a glass plate is exposed to mechanical and thermal loads, as well as to water and different chemicals effects on its surfaces. As a result flaws’ sizes may grow and new surface flaws may be developed. The accumulated effects continuously change the properties of a weathered glass plate compared to a new glass plate that has been just recently produced and cut for its planned usage. In order to study the glass product, independent of its lifetime history, we shall focus in this paper on new glass that has not been affected by all the above mentioned effects. After annealing, the float glass is inspected to ensure that there are no defects larger than a few hundreds micrometers. Standards also dictate the maximum size of allowable flaws: for example, it is 0.2 mm according to European standards [9]. Any defected or broken glass may be recycled into the furnace. Through the production line glass is cut into a typical size of 6.00 m  3.21 m (‘‘jumbo size’’) before being stored; this large jumbo size plate will be denoted herein as the basic plate.

1.3. Mechanical properties At ambient temperatures glass is solid. It is characterized by its brittle behaviour. The fracture toughness is about 0.75 MPa (m0.5) [10]. According to EN 572-1 [1], the density at 18 °C is 2500 Kg/m3, its Young’s Modulus E is about 7 * 1010 Pa and its Poisson’s ratio is 0.2. The theoretical strength of plain soda lime glass, is almost 50% of its Young’s modulus of elasticity, however glass plates usually fail at almost 3 orders of magnitude lower stresses [11,12]. This is due to the presence of flaws. The theory of facture mechanics may explain the observed strength levels.

1.4. Fracture of annealed glass Glass is a brittle material that behaves as an elastic solid in fracture. Fracture of glass initiates at the location of a crack that first opens due to the stresses acting on it. This is the critical crack. Due to the flaws distribution the critical crack is not necessarily located at the point of maximum tensile stress. Griffith [13] studied the failure of brittle materials and focused on glass as a representative material. He derived the well-known expression for the failure stress:

rG ¼

rffiffiffiffiffiffiffiffi 2Ec pa

ð1Þ

where rG – is the failure stress; a – is half length of the crack; c – is the fracture surface energy of glass Irwin [14] modified Griffith’s theory and introduced the stress intensity factor that represents the elastic stress intensity near crack tip. For mode I loading, KI is determined from:

pffiffiffiffiffiffi K I ¼ Y r pa

ð2Þ

where Y is a geometrical factor depending on the flaw shape (e.g. 1.12 for a general surface crack [15], 0.713 for half penny shaped crack on a flexural element [16], 0.637 for an elliptical crack [17], etc.). Failure occurs at a stage when the stress intensity factor reaches its critical value KIc (fracture toughness). The value of the fracture toughness for lime glass in literature varies within the range of (0.72–0.82) MPa m0.5 and commonly a value of 0.75 MPa m0.5 is used. Eq. (2) shows that the critical stress is inversely proportional to the square root of the flaw size.

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1.5. Testing of glass Experimental studies are conducted on rectangular glass plates of different sizes, mostly subjected to 3 point [7] or 4 point [18–20] one way bending. Sometimes evenly distributed pressure acting on the plate surface is used [8,21], mostly for 2 way bending. In several studies circular plates are also tested with double coaxial loading [16,18,22]. Almost all tests are conducted under static short term loading at ambient conditions. Numerous testing on glass plates have shown that the glass plate resistance increases where the glass plate’s dimensions decrease. This observation stands behind introducing an area factor into the current design models in theoretical models [8] and in standards (e.g. prEN 13474-1-1999 [23]). It should be noted that cutting out a glass plate of a larger glass plate produces defects at the boundaries of the cut out plate. Grinding attempts to smooth the cut surface and remove the defects [24]. However, these processes can produce damage which can significantly further reduce the strength of the glass plate [25,26].

1.6. Analysis methods Two major approaches exist to determine glass plate strength: the deterministic approach and the statistical approach. The deterministic approach preceded the statistical approach and is still common mainly in Europe (EN 1990-1-2002 [27], prEN 13474-1 [23]). The deterministic glass design is based on an allowable tensile bending stress of a glass plate. The allowable stress magnitude depends on the plate’s size, on the surface conditions, on the action duration and on environmental conditions. According to prEN 13474-1 [23] the design resistance of glass is determined from the characteristic strength, and its value for sodalime glass is 45 MPa. It also defines partial safety factors to determine the allowable stress, as well as influencing parameters depending on the plate’s surface area and on the loading duration. It aims at design and not at analysis and prediction of real behavior under given conditions. The statistical approach is mainly used in North America standards. It is based on the assumption that the glass surface contains numerous flaws of varying sizes and orientations. The strength of the glass and the ultimate external acting load vary accordingly. When a glass plate is loaded, tensile stresses are developed and interact with the flaws. When the local stress at a certain location exceeds the critical stress of the local flaw, fracture is initiated [13]. That flaw is denoted the critical flaw. Therefore the location of the critical flaw is not necessarily the location of the maximum applied tensile stress. There is limited information on the flaws size, shape and distribution and therefore the existing models focus on the distribution of the maximum applied tensile stresses or more often on the variation of the distribution of the critical applied lateral pressure that causes the glass plate failure. Commonly Weibull’s statistical theory of failure is used to predict the strength of materials which are controlled by their weakest link, and therefore it was naturally adopted to determine the probability of failure of a glass plate [8]. A representative probabilistic model for glass is the Glass Failure Prediction Model (GFPM) that had been developed by Beason [8]; Beason and Morgan [28]; Beason and Norville [29]; and was implemented in modern standards. This model links the plate’s area with the probabilistic strength. The Weibull model is defined by two parameters that cannot be measured directly and are therefore estimated from glass plate tests to failure in a procedure described in [28]. Rather exhausting analysis is required to come up with a set of best fit parameters.

Similar analysis of different series of tests is likely to yield sets of different values. In their work Beason and Morgan [28] stated that there were significant differences in the surface flaw characteristics of the different specimens, and that the effects on the statistical model parameters were not clear as their relationship to the failure strength was not apparent. Hence the generality and validity of these parameters values for general implementation is questionable. Nevertheless, the model was widespread and implemented in standards [30,31], used in research work and implemented in software [16,32,33]. Therefore it is not surprising that the GFPM has gained some criticism. The influence of the surface area of a tested glass plate on the strength of the glass panel was examined by Norhuda et al. [5] and by Leon [34]. They criticized the choice of the Weibull model in the GFPM and suggested a new model of the Log-Normal form for predicting the cumulative probabilistic distribution of tensile strength of annealed glass panels. Although their model has certain features in common with predictions of the Weibull’s model, they found that with the Log-Normal probabilistic distribution the strength of larger plates of glass are less sensitive to any further increase in the panel dimension than strength predicted by the Weibull’s model. Veer [26] examined whether the Weibull distribution is appropriate to describe the failure strength data and found that the normal distribution can rarely describe the failure strength of glass; however it may sometimes be described by a Weibull distribution. In another paper, Veer et al [6,35] state that the failure strength of most of the test results does not fit the Weibull distribution function and observe that the deviation from the Weibull distribution is especially significant at the bottom of the failure strength distribution. It should be mentioned that all the above specified probabilistic distribution functions are a priori assumed as the preferable model to describe the failure distribution. In all cases the distribution functions were not obtained as a result of the analysis. Other methods and standard design methods were reviewed by Haldimann [2]. 1.7. Evaluation of present analysis methods Several issues of major importance that characterize the behaviour of glass plates have been identified in the above and are summarized herein. Most issues will be later addressed in the proposed model. The scattered results of controlled tests on glass plates indicate that there is a governing probabilistic behaviour, and therefore deterministic methods are incapable to determine glass strength. Probabilistic methods were suggested, assuming a priori a probability function; commonly, the Weibull distribution law was arbitrarily adopted. Other models like the Log-Normal distribution were recently suggested as an alternative, but it is still an open question which is the most suitable probability failure distribution function. The present study suggests that an answer can be provided only if the probability of failure comes out as a result of the analysis and is not assumed a priori. In existing models the probability law distribution parameters are fitted from correlation to a given series of tests, and shows to yield different results when fitted to other series of tests. This raises questions with regard to the correctness and accuracy of that approach. One of the present study objectives was to come up with a model that predicts the probabilistic strength of a loaded glass plate and of the specimens’ strength distribution rather than a-priori adopting a model that is calibrated by a specific series of experimental data. The failure of an annealed glass plate starts at a certain point (the fracture origin) from which a number of cracks emerge and

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split the plate into several large pieces. The fracture origin may occur at different points, not necessarily at the point of maximum bending moment or maximum applied tensile stress. Even in a one way plate that is characterized by constant stress distribution along the plate’s width, failure occurs at a certain point along the plate width, and not necessarily at the mid-width. Existing models do not predict the point of fracture origin. One of the present model objectives was to incorporate the fracture origin coordinates as variables to be predicted by the model. These present study objectives may contribute to further modeling the behavior of a glass pane. 2. The present model 2.1. Approach The present study aims at addressing the above mentioned objectives and at providing insight that may enhance our understanding of glass plate behaviour under the action of external loading. In order to achieve this goal, a priori assumptions are confined to a minimum and no assumptions are made with regard to the probability distribution of critical loads or stresses. Therefore the problem is defined as follows: a rectangular glass plate of constant thickness is supported along its edges; it is characterized by its geometry and mechanical properties that are mentioned above. The plate’s flaws map is considered here an important fundamental characteristic of the plate. However the flaws map is unknown and is difficult to obtain. It is unique for each plate and therefore if needed it should be determined repeatedly for every single specimen. Nevertheless the flaws map seems to be a fundamental property that yields the failure stresses and ultimate loads and their statistical distribution. Therefore in the present model attention is given to the flaws distribution that was disregarded in many other studies, and the entire derivation will be based on this distribution. 2.2. Flaws distribution As described in the above, many studies bypass the flaws distribution issue, and prefer to discuss the distribution of the measured ultimate loading that was applied to fracture the tested glass plate. However, the flaws characteristics are part of the plate data, whereas the load and the capacity should be considered as the experimental or the analysis results. There exists limited information about the flaws distribution in glass, and attempts are being made to provide some data. In the meantime, some considerations may lead to a rational assumption with regard to the flaws distribution. There are several parameters that should be considered and the major ones are: the number of flaws, their size, shape, and distribution. Opposed to existing models that refer to the glass plate under consideration (i.e. the cut out plate) that is characterized by its specimen’s area A, the proposed approach shall refer to the basic glass plate with area A0 that is produced and shipped, before it is being cut into the considered plate. This basic plate undergoes production control processes to assure that it does not contain flaws larger than a certain limiting value (e.g. about 200 lm according to certain standards and more generally mostly within the range of 100–300 lm). This basic plate should be sufficiently strong to withstand different actions involved in its carrying, placing, handling and shipment when it is subjected to its dead load and other actions. This information may provide a sound estimate for the largest flaw that may exist in the basic glass plate. It may be reasonable to assume that there is a single flaw of the largest size and more flaws of smaller sizes; it may also be assumed that the

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number of flaws increases with decreasing the flaw’s size, and therefore there are many small size flaws and less large flaws. This may indicate that the flaws distribution function is of an exponential shape, analogous to Mott’s law that was proposed for fragmentation of rings and shells [36], and it can be expressed in Eq. (3):

Nf d ¼ eðlÞ N0

ð3Þ

where N0 – is the total number of flaws in the basic plate. It depends on the production process and on the quality control as well as on other parameters and may vary between different basic glass plates. dmax – is the size of the largest flaw in the basic plate; d – is the size of a certain flaw in the basic plate flaw; Nf – is the number of flaws that are larger or equal to d; l – is a flaws distribution parameter that may be denoted herein as a characteristic flaw size. Its value will be determined in the following. For d = 0 one gets Nf = N0, that means that there are N0 flaws larger than zero. For d = dmax one gets Nf = 1, that means that there exists only one flaw that is equal or larger than dmax. Substituting this condition into Eq. (3), yields: dmax 1 ¼ eð l Þ N0

ð4Þ

Hence the characteristic flaw size l may be determined:

l¼

dmax ½LnðN 0 Þ

ð5Þ

Further it is assumed that the flaws are of a size, density and arrangement over the plate’s surface that they do not interact with each other and a single critical flaw is responsible for the local conditions that cause fracture. Hence the parameters of the flaws distribution model are the flaws density and the size of the largest flaw. With these two parameters the entire flaws distribution function is determined. These parameters depend on the production processes of the glass, on the quality control requirements and on other parameters and therefore vary between production series and even between different basic plates. However, in the present model it was assumed that for soda lime float glass the maximum flaw length varies between 100 and 300 lm and a typical value of 200 lm may be considered representative. It was also assumed that the flaws density varies between 1 and 2 flaws/cm2 and a typical value of 1 flaw per cm2 may be considered representative. A long term project has been recently initiated to experimentally study of flaws distribution, in order to provide the required data. In a recent paper, some preliminary results were published [37]. A small number of specimens were examined, and indicative values for the two key parameters of the flaws distribution model were measured. For soda lime annealed glass it was found that the size of the largest flaw varies between 105 and 195 lm and the density of flaws varies between 1.18 and 2.60 flaws per cm2. These values comply with the typical parameters values that were adopted in the present model. 2.3. Critical stress In light of the above, it is assumed that crack extends immediately when the condition in Eq. (2) is satisfied and stress corrosion and subcritical crack growth are not incorporated in the present model. The failure stress can be described by means of Eq. (2); it correlates the failure stress with the fracture toughness KIc, the crack

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depth a, and the geometry factor Y. As mentioned above, the geometry factor depends on the flaw’s shape. Although there is no clear information about the flaws distribution of sizes and shapes, it is evident that surface flaws in glass have various shapes and sizes. Therefore it is doubtful if a penny shape factor should be adopted or whether another factor that represents the weighted effect of the different flaws is preferable. Until more information is available, a general surface crack is assumed with Y = 1.12 [15]. 2.4. Aim of the proposed model The present model aims at building up a fundamental tool to analyze glass behaviour and predict its response. It refers to a new glass, disregarding the weathering effects, and focuses on static short term loading that is applied at a slow rate at ambient conditions. Loading rate effects are not considered at this stage, and will be studied and incorporated in a future extended version of the model, to allow consideration of different loading rates. The model also disregards the unavoidable effects of cutting and grinding. Indeed, they may affect the entire glass plate strength, however, in laboratory tests the location of the fracture origin may be identified, and it is rather simple to exclude cutting edge effects by eliminating test results that show fracture origin at the glass plate boundary. The proposed model aims at providing a tool to analyze the behavior of a glass plate that is subjected to a given loading system and its failure characteristics including the fracture strength, and its distribution for a sample of a given number of specimens. The model is characterized by the following features:  The type of the probability distribution function is not a priori determined, but it is obtained from the analysis.  The distribution of flaws and critical stresses can be determined.  The location of the fracture origin point can be predicted.  The effect of the loading scheme can be investigated.  Glass plate size effect can be obtained naturally from the analysis without any precondition of a size effect law. 2.5. The procedure The analysis procedure is straightforward along the following steps:  Define the plate specimens’ geometry and boundary conditions and determine the loading system. Define the number of specimens in the sample.  Assign the mechanical properties for the tested glass.  It is assumed that the flaws are distributed over the glass surface according to a uniform probability distribution function, as there is no location preference for any flaw. The flaws size distribution follows the proposed model. The flaws mapping is performed with respect to the basic plate.  Once the flaw distribution over the basic plate has been determined, a certain plate of a given specified size may be cut out. A random selection of the upper left corner of the cut out plate is made. The flaws distribution of this cut out plate can now be determined from the flaws distribution in the basic plate that had been determined in the previous stage. Thus, different cut out plates from the same basic plate have different flaws map even if they have equal dimensions.  For all the flaws in the cut out plate, the magnitude of the critical tensile stress is determined according to Eq. (2). This yields the critical stresses mapping. One of these flaws will first reach the critical stress during loading and define the fracture origin.

 The loading system is applied and the tensile stresses along the entire plate’s surface may be determined. The stress distribution shape is determined from the loading conditions and the plate geometry including its boundary conditions.  A search method is conducted to detect the level of loading at which the resulting applied tensile stress envelope meets with a flaw with identical critical stress. Upon reaching that condition, the location of the critical flaw is determined. This determines the failure stress of the specimen and the geometrical location of the point at which fracture is initiated. A Monte Carlo simulation is carried out for a large sample with many specimens and the accumulated results are gathered and analyzed. 3. Characteristics of the present model 3.1. General In this section some fundamental properties of the proposed model will be investigated. Studies of the behaviour of a single glass plate will demonstrate the typical results of the model. Due to the probabilistic nature of glass behaviour, study will then focus on a large sample of similar plates and investigate the variation of the obtained results and explore their distribution. In order to asses to what extent the sample size affects the obtained results, a convergence study will follow. 3.2. Examination of a single typical plate In the following a one way simply supported glass plate that is loaded by a central pointed load (a line load at mid-span) is examined. As a reference plate consider a square plate of equal length and width of 100 cm with a line load at mid-span. A one way action is assumed (the span is denoted as the plate length) and the stress distribution across the plate’s width is assumed constant at any distance along its span. The flaws distribution (Eq. (3)) is applied and the corresponding critical stress was then determined. Fig. 1a and b refer to two different plates with the same geometry, that are cut out from the same basic plate, and show the probabilistic distributions of the critical stresses within the range of 40–60 MPa. Entirely different distributions are obtained, that indicate different locations of the weak points for possible fracture origins. The plate length in the x direction is subdivided into 100 strips of 1 cm. Each strip is then subdivided into 100 unit cells in the width direction. Thus the entire plate is subdivided into 10,000 units of 1 cm2 each. A single flaw will be assigned to each cell. The 10,000 unit cells allow a density of 10,000 flaws/m2 (i.e. 1 flaw/cm2). This is a dense distribution that takes into account even considerably small flaws that do not affect failure. The minimum critical stress was determined among the 100 units composing a strip, and the distribution of these critical stresses, representing the critical stress of each strip along the plate’s span (x direction) is shown in Fig. 2. Fig. 3 shows a two dimensional view of the 100 * 100 cm. plate’s surface with 100 points indicating the potential locations of the minimum critical stresses of every strip. One may realize that they are scattered over the entire area, thus failure may be initiated at a point that is distant from the plate’s mid-width. Note that the indicated points refer to different critical stresses. Comparing the distribution of the critical stresses with the distribution of the applied tensile stresses that depends on the loading system, along the x coordinate, will determine the tensile stress at failure and its location. The analysis of that plate yields that the magnitude of the tensile stress at failure is 48.59 MPa and the x–y coordinates of the origin of fracture from the lower left hand corner of the plate are 54 cm. along the x axis, i.e. 4 cm. away

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Fig. 1. Distribution of critical stresses in two plates.

Fig. 2. Distribution of critical stresses along the x coordinate.

This example demonstrates the capability of the model to determine the location of the failure location and tensile stress at failure at this location. However, repeated tests will show different results, and the nature of a large series of specimens should be investigated in the following. It is interesting to point out an insight that has been gained in the above described analysis. Common plate’s analysis requires the plate’s thickness to determine the moment of inertia, and the applied stresses are calculated from the calculated bending moments. In the present approach it was clarified that it is the shape of the bending moment variation along the plate that is of importance, and it is similar to the shape of the free surface stress variation. Therefore the origin of fracture is determined when the stress shape envelope meets the envelope of the fracture stresses, independent on the plate thickness. However, the plate thickness is important for determination of the bending moment at the critical section and for determination of the magnitude of the applied load at this stage of failure. This insight can be gained from the present analysis, however it cannot be confirmed experimentally because of the statistical distribution of flaws.

3.3. Resistance variation of a large series of plates

Fig. 3. Two dimensional locations of minimum critical stresses in the strips.

from the plate’s center (mid-span), and 62 cm. along the y direction (i.e. 12 cm. away from its mid-width). A repeated analysis of the same problem yields different results, due to the probabilistic distribution of the flaws.

In order to examine the variation of the results with regards both to the tensile stress at failure and the fracture origin, a large group of specimens should be investigated. Consider a sample of 5000 plates that are identical with the above discussed plate in their geometry and boundary conditions, and subjected to the same loading system. They differ from each other only in the flaws map. All the plates are analyzed according to the above mentioned procedure. The Monte Carlo simulation is applied and the resulting distribution of the tensile strength of failure of this series in shown in Fig. 4. An almost, but not exactly, symmetrical bell shaped distribution is obtained, with strength results ranging between 36 MPa and 59 MPa. The average tensile stress at failure is 47.47 MPa and the median tensile stress at failure is 47.59 MPa. It is important to mention that this distribution is obtained as a result of the analysis and was not assumed a-priori and is not a smooth function as is evident from Fig. 4. Therefore it is interesting to examine what is the probability distribution function that better represents the calculated distribution of tensile stresses at failure. For that purpose the obtained results are transformed into a density function and compared with

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In this distribution one can see again the relatively good correspondence of the normal and of the log normal distributions, compared to a less favorable correspondence of the Weibull distribution. The above analysis demonstrates the capability of the present model to provide the probability distribution of the tensile stresses in the glass plate at failure. It provides the probability distribution of the location of failure as well. Comparison of the above examined case with known probability distribution functions shows that the normal distribution fits better although the fit is not identical with the model predictions; this confirms earlier observations of less favorable correspondence of the Weibull distribution especially at the bottom of the failure strength distribution. Other examined cases may show different results. 3.4. Convergence

Fig. 4. The distribution of tensile strengths at failure.

a Normal distribution, a Log-Normal distribution and a Weibull distribution, that were mentioned above (Fig. 5). It can be clearly seen that in this case reasonable fits are obtained with all functions, although the Normal distribution shows a better agreement among the three examined distribution functions. The Log Normal distribution’s fit is reasonable although not as good as the Normal distribution, and the Weibull’s distribution is the less favorable among these three distributions. The superior fit of the normal distribution may be better appreciated in the cumulative probability curves (Fig. 6). The blue staircase shaped curve in Fig. 6 shows the present model analysis results and the smooth red curve presents the best fit to the Normal, Weibull and Log Normal distributions. It is evident that the normal distribution fits better the obtained results. It is also evident that the Weibull distribution shows the worse fit among the three distributions, especially at the bottom of the failure strength distribution, as had already observed by Veer at al [35]. Another aspect of the failure of the 5000 different plates is the location of the fracture origin. The present model predicts the coordinates of the fracture origin as is shown in the above, and in the case of a one way plate, the coordinate along the x axis is of importance. Fig. 7 shows the calculated distribution along the x coordinate of the point of failure for all specimens. An almost symmetrical distribution of the x coordinates of the location of failure is obtained, that varies between x = 36 cm. and x = 66 cm. with a clear concentration of points at the axis of symmetry (x = 50 cm.) and in its vicinity. The cumulative distribution of the obtained data (blue lines in Fig. 8) is compared with the same probabilistic distributions (red lines in Fig. 8).

A study was carried out to investigate the convergence of the above results. A repeated analysis of 5000 specimens was carried out 5 times and minor differences were obtained for the average tensile stress at failure. The result of the 5 different analyses yield an average tensile stress at failure of 47.47 MPa with a slight variation of the results within ±0.14%. Similar repeated analysis for 10,000 specimens hardly changed the average result; however the accuracy was even better. The average result was 47.46 MPa ±0.06%. It may be concluded that the choice of 5000 specimens assures a converged reliable and repeatable result. However, a smaller sample with fewer specimens may yield different results. To examine this aspect, repeated studies were carried out on smaller samples and the results are shown in Table 1. For each case in Table 1 two typical distributions are shown: The probability Distribution of tensile stress at failure (MPa) and the distribution of fracture origin location along the x coordinate. For the sake of comparison, the figures are shown in a small scale aiming at getting the impression of the distribution shape difference with change of the number of specimens. It should provide the background for general conclusions, whereas the specific conclusions will be drawn from larger figures that are based on a sufficiently large number of specimens. It should be noted that smaller size samples show larger variation in the results the study of which is beyond the scope of this paper. Table 1 shows a typical example per sample size and an indication (that is based on a number of repeated analyses) of the growing variation of results with decrease of the sample size. This variation will be demonstrated in an example in Section 4.2. Examination of Table 1 shows that the calculated tensile stress at failure is somewhat more variable for smaller samples, although not too much and even with a relatively small number of specimens the calculated average tensile stress at failure of a set of samples was pretty close to the above values. However, reducing the sample’s size considerably affected the shape of the probability distribution: At a small number of

Fig. 5. Comparison of model prediction distribution of tensile stresses at failure density function with known density functions (Normal, Weibull and Log-normal).

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Fig. 6. Cumulative distribution curves (Normal, Weibull and Log-normal).

at failure, shape of the probability distribution function and shape of the distribution of the location of the point of fracture origin) converge where a sufficiently large sample is used. 3.5. A glance at the origin of fracture Fig. 7 and Table 1 show the distribution of the origin of failure along the x direction, for the case of 3 point bending of the examined plate. It was shown that fracture can originate within a strip between x = 36 cm. and x = 66 cm. Fig. 9 shows a view of the specimen that is supported along the boundaries x = 0 and x = 100 cm. and loaded by a line load at x = 50 cm. All 5000 origins of fracture are shown, and indicate that the origin of fracture may occur at any point along the width dimension. Therefore nonsymmetrical shapes of fracture are most common in glass testing of symmetrical layouts. 4. Comparison of the model predictions with test data Fig. 7. Distribution of the x coordinate of the location of failure.

In this Section 2 comparisons with test data are presented: specimens a sporadic distribution was obtained with no clear trend; increasing the number of specimens in a sample forms the bell shaped distribution that becomes even smoother at a sufficiently large number of specimens. Similar observations refer to the distribution of the location of the point of fracture origin. Surprisingly, the average tensile strength at failure that is obtaied from a set of samples is only slightly affected by the number of specimens, and for a rough estimate of the average strength only a limited number of specimens may be sufficient in each of these samples. That observation confirms a similar finding that was mentioned in [11]. However, in order to obtain a reliable estimate of the strength that corresponds to a certain probability of failure, a clear shape of the probability distribution is required and it requires a large number of specimens, as shown in Table 1. It may be concluded that the results of all the major parameters (average tensile stress

4.1. Tests by Veer and Rodichev [7] A comparison with test data is presented below. In a series of tests, Veer and Rodichev [7] studied the effect of glass cutting on the strength of the cut out plates. They tested 50 cut up plates, in which the cut cracks were in the compression zone, thus closing in bending, and not affecting the strength of glass and also tested cut down plates, where the specimens cuts may affect the bending behavior of the tested specimens. The plates’ data was 40 mm width, 350 mm span and the plate thickness was 6 mm. The plates were tested in a 4 point bending scheme with a 175 mm load span. The cutup tests were selected for comparison not only because of the above mentioned reason, but because of the findings with regard to this special series of tests. Veer and Rodichev [7]

Fig. 8. Cumulative distribution of the location of failure x coordinate.

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Table 1 Examination of solution convergence. No. of specimens

Mean-stress (MPa) (variation %)

10,000

47.46 (0.06%)

5000

47.47 (0.14%)

1000

47.49 (0.4%)

500

47.34 (0.68%)

100

47.75 (1.18%)

50

47.82 (1.8%)

25

47.77 (2.89%)

Probability distribution of tensile stress at failure (MPa)

compared the test data of the cutup tests with 6 different statistical functions (Normal, Log-normal, Weibull, Exponential, Rayleigh, Extreme Value) and found that none of them showed a good fit. They concluded that there is no single statistical descriptor that gives a universal fit on all valid test results. It was therefore especially challenging to compare these cutup test results with the proposed model predicted distribution. Test results of the cut up specimens vary between 37.8 and 99.6 MPa with an average strength of 61.65 MPa. The distribution of the measured tensile stresses at failure is shown in Fig. 10. The present model was applied to predict the tensile stresses for 5000 specimens at failure and a comparison between the test data and the model prediction is shown in Fig. 11. A good correspondence is observed between the predicted and the measured tensile

Distribution of fracture origin location

strength distribution, with an average predicted tensile strength of 61.17 MPa. In light of the above, the good correspondence is even more pronounced. It should be noted that the model prediction was carried out for 5000 specimens whereas the test series included 50 specimens only. The effect of the number of specimens that is presented in Table 1 above, demonstrates the irregular expected distribution of the experimental data in the case of a limited number of specimens, compared to the converging bell shape for a large number of specimens. This is also clearly observed in Fig. 11. Taking the small sample effect into consideration and the expected larger standard deviation, the predicted distribution covers well the sporadic test data with a good match of the predicted average stress at failure.

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D.Z. Yankelevsky / Engineering Structures 79 (2014) 244–255 Table 2 Statistical values, modulus of rupture [38].

Mean Median Standard Deviation Cov

Fig. 9. Locations of the origins of fracture – a view of the tested plate.

Fig. 10. Distribution of tensile stresses at failure for cut up plates-test data.

Fig. 11. Comparison of model prediction with cut up test data.

4.2. Tests by Spiller et al [38] In an ongoing experimental research project, aiming at studying the blast response of annealed glass windows [38], a series of static tests on glass specimens has been recently conducted, aiming at determination of the glass rupture modulus. The specimens for

Failure load (kN)

Failure moment (kN-m)

Modulus of rupture (MPa)

Model prediction 5000 specimens

(1)

(2)

(3)

(4)

Model prediction 30 specimens 10 samples (5)

2.99 2.96 0.494

0.0742 0.0741 0.012

80.3 80.7 13.0

80.49 80.23 10.97

77.4–83.4 75.4–84.6 8.4–14.1

0.165

0.160

0.162

0.136

0.104–0.169

the static tests were cut out from larger 12 mm thick panes using the score and break technique and prepared according to the ASTM C158-12 standard and are characterized by their length and width (250 mm  38 mm). A four-point bending test was carried out on each specimen having a 200 mm span. The failure characteristics of each specimen were examined to identify between specimens with fracture origin along the specimen’s longitudinal edges that may result from cutting produced faults, and face failures. 30 face failure reliable test results were considered for the rupture modulus determination. The results of these series are summarized in a recent paper [38], and the calculated average and median strengths, standard deviation and coefficient of variation (Cov) are shown in Table 2column 3. The experimental mean stress is 80.3 MPa and the median is slightly higher; the standard deviation is 13.0 MPa and the coefficient of variation is 0.162. The present model was applied for a large number of 5000 specimens and yielded the results shown in Table 2-column 4. Comparing the predicted results with the measured data shows a slightly higher average strength and a median that is somewhat smaller than the average. The predicted standard deviation and the Cov are smaller than the measured data. The predicted results are shown in Fig. 12-a, and an attempt to fit a normal distribution curve to the predicted distribution shows very good agreement. In light of the discussion in Section 3.4 and the results in Table 1 above, the model was applied once again for a small sample of 30 similar size specimens. As had been stated above, one of the model’s features is to fully account for the statistical distribution of flaws and the resulting fractures strengths. That capacity yields the variation of fracture strengths in a sample. In the case of a sufficiently large sample the solution converges and repeated analysis on a similar large sample yields similar results, as was shown above. In the case of small samples, each analysis of a sample of 30 specimens provides different results that scatter around the converged values of the large sample. To examine this aspect, analysis of a 30 specimens sample was repeated 10 times, and the results are summarized in Table 2-column 5. One typical prediction is shown in Fig. 12-b. It can be observed that the distribution differs considerably from the clear bell shaped distribution that was obtained for a large specimen. The correlation with a normal fit also looks rather poor. One can see that the experimental results in column 3 fall nicely within the range of results in column 5. From that analysis one may realize that in small size specimens the standard deviation, as well as the coefficient of variation, may be larger, as was obtained in the reported series of tests. This example demonstrates the power of the present model and its ability to provide better insight to the experimental investigation of glass plates. One may realize that small samples of 30 specimens may yield an average that is somewhat different than the converged value of the average of a large sample, and it may be characterized by a larger standard variation.

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(a) 5000 specimens

(b) 30 specimens

Fig. 12. Prediction of a series of tests.

5. Discussion and conclusions A new model for glass plate analysis is presented. The model predicts the distribution of glass plates’ strength from which correlation between the plate’s data and the corresponding tensile strength may be obtained. It is found that the tensile strength in bending of a large sample of plates has an almost symmetrical bell shape distribution that is different than existing distribution functions. This distribution is obtained as a result of the model analysis. Comparison between the model’s calculated distribution and common distribution functions shows that the predicted probability distribution fits better the normal distribution. It was found that in this example the Weibull distribution, which is commonly a priori arbitrarily adopted as the representative distribution in glass failure models, does not represent as well the model’s tensile strength distribution for the examined typical plate, especially at the bottom of the failure strength distribution. The present model also predicts the location of fracture origin. This is an important feature that is not predicted by other models. It was found that the location of the fracture origin depends on the loading system and may occur at a large part of the plate’s surface area and not necessarily at the point of maximum applied stresses. The model predicts the distribution of the locations of fracture origin and for one way plates it shows an almost symmetrical distribution with respect to the plate’s center in the span direction. The fracture origin may be located at any point in the width direction and the failure is not necessarily symmetrical for symmetrical setups and loading systems. The model shows that the entire analysis of determination of the critical tensile strength is independent on the plate’s thickness, although the latter is important to determine the magnitude of the bending moment at failure and the magnitude of the applied load that causes that failure. The present model shows very good correspondence with test results and may provide detailed information on the failure parameters and supply insight about the glass plate behaviour. References [1] EN 572-1. Glass in building – basic soda limesilicate glass products – Part 1: Definitions and general physical and mechanical properties. CEN; 2004. [2] Haldimann H. Fracture strength of structural glass elements –analytical and numerical modeling, testing and design. PhD Thesis, Faculty of Civil Engineering, Ecole Polytechnique Federale De Lausanne; 2006. [3] Kim T, Oshima K, Kawada H. Impact tensile properties and strength development mechanism of glass for reinforcement fiber. J Phys 2013;451:1–6 [Conference Series].

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