Design Principles For Glass Used Structurally Naimeh Khorasani

Design Principles For Glass Used Structurally Naieh Khorasani Rapport TABK 4/5 Lund 4 Departent of Building Science

Building Science Departent Lund University Box 8 Lund Sweden ISRN LUTADL/TABK 5 SE ISSN 3-4467 ISBN 4 Naieh Khorasani

TABLE OF CONTENTS PREFACE... ABSTRACT... TABLE OF SYMBOLS...3 ARCHITECTURAL GLASS. Introduction...7. Glass as a load bearing coponent...8.3.4 GLASS PROPERTIES. Introduction...7. Glass production ethods...7.3 Strength characteristics....4 Static fatigue...3.5 Experients on heat strengthened glass......5 3 FRACTURE MECHANICS 3. Introduction...9 3. Soe historical notes...3 3.3 Linear fracture echanics...3 3.4 Fracture echanics of glass windows...35 3.5 Analysis of beas...38 4 STOCHASTIC MECHANICS 4. Introduction...4 4. Soe historical notes... 43 4.3 Stochastic echanics for brittle aterials...44 4.4 Failure prediction for glass windows...48 4.5 Analysis of beas... 5 5 CONCLUSIONS 5. Introduction...58 5. Design strategy...58 5.3 Material strength...6 5.4 Design ethodology...6 5.5 Research trends and developent...63 APPENDIX A APPENDIX B Design aspects...... Reliability iplications... 4 REFERENCES...65

Preface Mine is the first step and therefore a sall one, though worked out with uch thought and hard labour. You y readers or hearers of y lectures, if you think I have done as uch as can fairly be expected of an initial start will acknowledge what I have achieved and will pardon what I have left for others to accoplish. Aristotle ( 384-3 BC) At the start of this research project I thought that finding out the strength of glass and describing its behaviour during loading would be a very interesting and quiet easy challenge, regarding y background as a Civil Engineer. It would also give e the possibility of developing y skills as a design engineer and increase y self confidence. I ust confess as the project has evolved I have had to question y knowledge and skills and what used to see easy and understandable was now getting very coplex and difficult to describe in one siple and easy way. There were any ties when I could not see where all these new influences could be headed. The end of this project seeed VERY far away! So now when I a writing the last sentences of this report, I feel there are soe people that I need to express y gratitude towards. These people have been backing e up and guided e through this journey. Their help and devotion have given e otivation to keep on going and enjoying y work eanwhile. Professor Lars Sentler, y supervisor, who has been exposed to y up and down oods and believed in e when I did not. Lars this journey would certainly have not been as interesting and challenging without your otivations and high deands! I would also want to thank Professor Bertil Fredlund for listening to e and guiding e through this soeties coplex acadeic world. In addition, the following people have contributed through discussions of the application of glass: Mr Anders Jacobsson, Pilkington, Mr Thoas Grange, MTK, who has also read through and given e coents on this thesis, Mr Ti MacFarlane, Dewhurst & Macfarlane, Mr Andrea Copagno, who has held very interesting courses and workshops about glass, Mr Anthony Sith Ove Arup and finally Mrs Saga Hellberg and Mr Mikael Ödesjö, Glasbranschföreningen, for being y face towards the industry and prooting y project. As a researcher it is very difficult to set work liits and leave work at work. I a glad that y faily and y friends have been around and at ties forced e to take tie off and focus on other atters of life. The joy which this thesis eans to e I want to share with all of you!

Abstract Glass is a aterial which is used in increasing deanding applications. This is not only as structural glazing in facades but also as structural eleents like beas and coluns. In all these applications there is a need for a good understanding of the aterial glass. This concerns the strength properties which need to be characterized appropriately to reflect the actual behaviour. Design principles should provide the necessary reliability. The response of glass is considered to be linearly elastic. This eans that the theory of elasticity is directly applicable to analyse the response of glass. But typical for structural units of glass plates is that the thickness is sall copared to the in plane diensions. This ay result in structural coplications. When a glass plate is loaded perpendicular to its plane there will be both bending and ebrane responses. The rupture strength of glass is coplex. This reflects the influence of flaws, ainly in the surface regions, which will reduce the rupture strength to a fraction of the theoretical strength. In addition the rupture stress will exhibit dependencies where in particular a size and a tie dependence need to be addressed. This will further reduce the long ter rupture stress of glass. The analysis of the rupture behaviour of glass can be done based on fracture echanics or the stochastic echanics. This report suarizes and presents available knowledge based on these add on theories to the theory of elasticity.

TABLE OF SYMBOLS A Area B Risk function D reference duration E tot Total energy F Failure probability function G Energy release rate K t Stress concentration factor K I Stress intensity factor K IC Critical stress intensity factor L Length of the bea M Moent P Load acting on the body P f Probability of failure T in Tie to failure 3 3

U Internal elastic strain energy V Volue V reference volue W Work done W s Energy required to create a new crack surface Y Diensionless paraeter dependent on crack geoetry a Crack length b Half crack width b Thickness of the bea c Noralization constant e Distance between loads acting on the bea h Weibull paraeter, tie dependence h Height of the bea Weibull paraeter, size dependence n Nuber of eleents E Mean value of the stress 4

Var Variance of the stress Displaceent Radius of the curvature of the flaw u Elastic strain energy density Stress far away fro the flaw tip Stress at the flaw tip iniu strength of a defective eleent Poisson's ratio Shear odulus of the aterial Potential of the body 5 5

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ARCHITECTURAL GLASS. Introduction Glass is a aterial which has been favoured by architects fro the tie it was introduced in buildings. There are several reasons to this but the transparency of glass is often presented as an iportant characteristic. Glass could not be produced in larger sizes of reasonable quality until the beginning of the th century. A new production technique was introduced at the turn of the century which was soon iproved to a continous production of drawn glass. This opened up new potentials which was soon realized by any architects. One of the first exaples is the Fagus factory by Walter Gropius of the Bauhaus group fro 9. In this building large portions of the facade was copletely covered by glass, soething which was later tered glass skin facades. This new architectural style opened up a building and gave it a new character. Modern glass of high quality known as float glass fro its production technique was introduced in the 96 th. This glass can be produced in very large sizes with an extreely high flatness. Within the production technique the surface finish is iproved and glass sections with visible internal defects are reoved. The production technique requires that residual stresses are introduced. This creates copressive stresses in surface regions which is an advantage in practical use. Float glass has opened the possibility to use glass in new and deanding applications. A new series of glass products known as structural glass has been introduced. This type of glass doinates copletely today. Based on float glass various products have been introduced in the arket like: Lainated glass Heat strengthened glass Toughened glass Insulating glass Coated glass and there is a continuous developent regarding glass products to eet new needs on the arket. Soe of the entioned products have been on the arket for longer period of tie but not being used in structurally deanding applications. Glass has traditionally not been seen as a structural aterial even if it has a structural function when used as a window pane. Glass producers traditionally have given recoendations for appropriate glass thicknesses of a pane given the overall diensions. But the possibility to use very large window panes and glass in increasing deanding applications raises the question of how glass should be designed. The thickness of glass is norally sall in relation to length and width. Deforations can be large in relation to the thickness which requires ore advanced calculation ethods than norally is used in structural designs. Glass is a true 7 7 7

brittle aterial which fails suddenly when the failure stress is exceeded. There are any questions which need to be addressed and given an appropriate answer, soe of which will be addressed in this report.. Glass as a load bearing coponent Architects soon found out the potential with the qualities of float glass. This together with further developents within the glass industry, for instance heat strengthened glass, have ade it possible to use glass in increasing deanding applications. For further deaterialisation of the support structure, it is possible to use the transparent aterial itself as a load bearing coponent. This started with suspended glass walls in the 96 th. The glass panes were fixed at the upper edge by eans of claps attached to a horizontal bea. This technique was rapidly adopted throughout the world. Soe exaples have window panes which are 3 etres high like in the new terinals at the airport Charles de Gaulle. The difficulties in anufacturing, transporting and installing such oversized panes have led to ore anageable sizes. Suspended glazing with bolted corner plate fixing points, called patch fittings was developed by Foster and Partners in the 97 th in connection with the design of a headquarter in Ipswich in England, see Figure.. The panes with the size x,5 with toughened glass are suspended fro the corners of the panes above by eans of brass patch fitting plates. The upperost panes are connected to the roof structure along the top edge. The result is a facade with a height of 5 which is suspended fro the roof. Lateral support against wind loads is achieved by eans of glass fins inside the building suspended fro the interediate floor. The glass fins will act as cantilevers affected by a point load at the free end when the building is subjected to wind loadings. Figure. Headquarter in Ipswich in England. 8

The patch fitting ethod was further developed during the 98 th with bolted fixing systes. Several different types of such systes were developed for different projects. For the ost coon syste a hole was drilled at each corner of the window pane. To iniize the influence of bending and torsional stresses around the holes, spherical bearings in plane with the glass had to be developed. For horizontal stabilization cable systes were introduced as trusses. In these trusses only ebers in copression had to be designed for buckling. All other ebers were high strength cables to axiize the visibility. During the 98 th glass started to be used in ore deanding applications. This can be seen as an extension of the fin principle visualised in Figure.. A one storey house was built in Alere in the Netherlands where walls are ade up of heat treated glass and stabilized by 5 thick fins of heat treated glass. The fins are fixed at the floor and the roof with aluiniu shoes. The glass fins also serve as coluns for a roof of low weight. A ore spectacular use of glass started to develop during the 99 th. This can be seen as a natural developent but a new philosophy was also adopted about how glass should be used. When a solid glass pane failed it did so in a brittle anner and norally no warning could be expected. Lainated glass where two or ore glass panes were glued together with a plastic ebrane behaves differently. If one pane brakes in a lainated pane with two or ore glass panes it can be expected that this will not lead to a coplete failure for the whole syste. It will lead to increased deflections and if no action is taken this ight eventually lead to a creep failure. One of the pioneers in this developent was Ti Macfarlane at Dewhurst, Macfarlane and Partners, a London based copany which has specialised in glass applications. An ipressive use of glass is the Local Authority Office in St-Gerain-en-Laye near Paris where a glass roof is supported by glass coluns. The architects were J. Brunet and E. Saunier. The 4 x 4 glass roof is supported by 8 cross shaped coluns x ade up of three layers of lainated heat treated glass. In one direction the panes are continuous while in the opposite direction they are spilt up and glued to the other panes. Only the inner panes of glass are intended to contribute to the load bearing capacity and the outer panes act as a shield. The cross shaped coluns are approved for a loading of 6 kn. Calculations indicate that the actual failure load is around 5 kn for the colun as a whole. This gives a level of safety of around 3 if only the inner panes are contributing. In reality the level of safety is higher. These coluns represent the first structural units subject to substantial continuous high sustained loads. Approxiately at the sae tie the sae architects as above created a glass roof with glass beas for the Workshops at the Musée de Louvre in Paris. The roof covers a three storey light-well which adits daylight into the underground extension of the useu. Lainated panes of four 5 heat treated glass panes were used for the 4 x 6 glass roof. Beas are 6 high and are ade of four 5 heat treated glass panes. 9 9

Tests of the beas revealed that the failure load was to 4 kn which corresponds to a axiu stress between 33,9 to 38,9 MPa. The required load carrying capacity was 5 kn which corresponds to a axiu stress of 3,9 MPa. A calculation shows that the dead load will give rise to a axiu stress of,4 MPa. With a live load of 4 kn/ the axiu stress will be,96 MPa. These stress levels indicate that each of the four panes can carry both the dead load and the live load. The actual safety level is extreely high. Figure. Local Authority Office in St-Gerain-en- Laye. In Rotterda a foot bridge between two buildings built copletely in glass was erected in 993 which is shown in Figure.3. The 3, long bridge consists of panes of lainated glass joined together with point fixings of stainless steel. The floor slab consists of x 5 lainated together and the beas are ade up of 3 x glass strips. The side walls and roofs are ade of toughened glass and 6 heat strengthened glass lainated together. Figure.3 Footbridge in Rotterda.

A calculation of the axiu stresses which will occur for the dead load alone is just below, MPa and for a design load of 4, kn/ the axiu stress will be 3, MPa. This indicates that each of the three glass panes can carry the design load and there will still be a high degree of reliability for each glass pane. Because of the height of the bea the deflection will be very sall and hardly noticeable. In 996 a glass canopy at the entrance to the Yurakucho underground railway station was finished, see Figure.4. The design is by Rafael Viñoly architects together with Dewhurst Macfarlane and Partners. This cantilevered glass structure is,6 long, 4,8 wide and 4,8 high at the apex. The support structure consists of three parallel, cantilevered beas, coposed of several triangular shaped lainated x9 glass panes. The variously shaped blades are bolted to interlock with each other with one blade at the apex and four blades at the bearing point. The roof is ade of,9 to,5 long and 4,8 wide lainated sheets. The thickness is x 5 toughened glass which are fixed at junctions with the cantilevered bea. Figure.4 Canopy in Tokyo. This structure was a challenge for the engineering profession and the society. The architectural design was ade during a short period, roughly around a week. The technical analysis and testing took half a year. The location of the canopy in Tokyo ade it necessary to design it to withstand heavy earthquakes and high wind speeds fro typhoons. Besides, the building authorities wanted a fail safe structure. The canopy had to be put together in a nuber of pieces because it was not possible to toughen larger lengths than around 5. These pieces are put together with bolts which will create stress concentrations. In this case the stress concentrations are perpendicular to the ain direction of tensile stresses. A nuber of questions arose: How close to the surface could a hole be placed without creating too uch additional stresses? How should the details in the transfer zone between an axle and the hole be anufactured? To answer these questions Asahi Glass

Copany started a project with tests to find the answers, (S. Wakui,999). Three different hole diaeters were tested (36,5, 55 and 68 ) and the result showed that if the large hole was used there was no reduction if the distance fro the centre to the edge was larger than 8. It was known that fro previous experience that an aluiniu ring was appropriate as a bezel. The actual function including a plastic ring or an epoxy adhesive was tested. This inforation was used in the design of the holes in the blades. The ost stressed glass blade was tested in full scale up to the design load to deterine the stress field around the hole subject to the highest force. The stresses around the hole see to have been only arginally higher than that caused by bending. Before erection each glass blade was proof loaded up to the design load. The actual failure load was estiated to be at least,6 ties the design load, in reality uch higher. To ake the structure fail proof additional blades of plexiglass was added to the structure..3 Design aspects Glass is a aterial which has not been used in deanding structural applications until the last decades. The reason for this is that a failure alost always is initiated by a tensile stress and this failure is brittle. No prior warning can be expected. This introduces questions concerning how the design and the reliability should be considered. Glass has been used in windows for a very long tie. It was not until the beginning of the th century before it was possible to produce glass panes in larger sizes of good quality. Such windows had to be designed for wind loads. Fro producers of glass, recoendations are given of the thicknesses needed for a specific window size. Such recoendations can vary between producers. The reason to this reflects the lack of codes for the design procedure of glass. Standards available are ainly concerned with surface finish and flatness. It can be expected that ost glass producers have carried out tests on glass to deterine the rupture strength. It is also very likely that such tests have revealed that the rupture strength is both size and tie dependent. The rupture strength will decrease for increasing sizes of a window pane and thickness. In the sae way the rupture strength will decrease for decreasing loading rates. This is not unique for glass but these dependancies are ore pronounced. It is not likely that two anufacturer of glass have carried out their tests in the sae way. The interpretation of the results ay also differ. This can be one reason to explain differences in recoendations. But the anufacturers also have to consider the reliability or the consequences of a failure. It is reasonable to assue that glass anufacturers tend to be conservative in their evaluation of test results. When new aterials are introduced, like fibre reinforced plastics as reinforceent for concrete structures, or old aterials are beginning to be used in new ways, like glass, there is a need to review the design process. This concerns

aterial properties design ethodology failure ode analysis where it is often necessary to provide inforation for end users. Often the intention with the new or iproved aterial is to eet certain deands or requireents on the arket. In the case of glass as a construction aterial the need reflects architectural design ideas of creating transparent structures. There is no obvious technical advantage. On the contrary, ordinary glass is a aterial which can be fairly strong during short durations of a load application. But the long ter properties for a sustained load are poor. Since there was a need on the arket for structural use of glass it was necessary for the glass industry to ake further developents. Heat strengthened glass is such a developent. Heat strengthened glass was developed to avoid the risk of spontaneous granulation because of uneven teperatures. Such glass has been available for several decades but only for special applications at high price levels. An iproved production technique was necessary and new control ethods to reveal ipurities which could generate delayed failures. Lainated glass is also a way to iprove structural characteristics which alters the over all failure behaviour. Glass is not a aterial which is included in codes as a structural aterial. Hence, there is no inforation about how the design should be ade. For other structural aterials like steel, concrete, wood and asonry inforation is readily available about design principles and strength values which can be used in applications. It is not obvious that a design of a glass structure should be ade in the sae way as for ordinary structural aterials. On the contrary, designs ade in the failure state based on the partial safety factor ethodology is not directly applicable unless certain odifications are added. For loads exceeding a certain threshold stress creep will be initiated which eventually will lead to a failure. This can be considered in a siilar anner as for structural wood where loads are graded in relation to their duration. Given the knowledge that a glass failure will be initiated by a flaw in relation to a stress field there is a possibility to use other design techniques. Fracture echanics was developed to analyse the influence of flaws in glass fibres. During the 95 th fracture echanics evolved as a potential analysis technique for glass but in particular for steel in echanical applications. For steel subject to fatigue stresses fracture echanics is an established technique to estiate the tie to failure. During the 99 th a nuber of persons have advocated the use of fracture echanics for the analysis of glass (Jacob et al, 997). Another analysis technique is based on stochastic echanics. The original theory (Weibull, 937) akes the rupture strength size dependent. This theory can easily be extended to include a tie dependence which will be conditional on the size dependence. The coplexity in analysing thin plates where both the bending theory and the ebrane theory have to be considered has ade this technique too 3 3

coplex until the introduction of coputers and finite eleent progras. Today this technique has been used in USA to produce design charts for any window pane geoetry (Beason, ). The basic failure ode of glass is brittle failure. This is hardly a desirable situation in a structural design but this situation is not unique. There are several ways to solve this situation which was deonstrated in the previous section. The ost direct way is to use a high factor of safety. But this is not enough. Given the pronounced tie dependence of the rupture load it is necessary to liit the axiu stress. The solution depends on the strategy chosen. Lainated glass gives the potential to alter the over all failure ode fro being brittle. If two or three glass panes are lainated together, where each pane can carry the design load there will not be a brittle failure if one of the panes fail. Deforations would increase and the failed unit had to be replaced. Tepered glass where the residual stresses have been substantially increased is an alternative where no failure should be expected as long as stresses fro design loads are kept below the copressive residual stress in the surface region. But this requires that no internal inclusion of foreign aterial is present in the interior where high tensile stresses are present. Lainated tepered glass sees to be the ultiate solution. This akes it possible to utilize relatively high tensile stresses under long ter loading conditions. Should one pane fail it is not likely that this will lead to a total failure. The strategy which is chosen to handle the failure ode analysis cannot be seen independent fro inforation about the properties of glass and how the design is ade. It is the total strategy which will deterine the reliability conditions. Each of these aspects: aterial inforation, design ethodology and failure ode analysis will be presented in the following chapters in relation to available inforation..4 Reliability iplications The statistically based design procedure was introduced ainly because of two reasons: () there was no realistic way to deterine the actual reliability with the safety factor principle and () there was a general belief that ost structures were over designed. To introduce a coplete statistical design procedure was not possible because of the coplexity this would have lead to. Instead the partial safety factor ethod was introduced as a coproise. In large this design procedure has lead to several iproveents and ost structures are ore econoical than if the old design procedure had been used. Glass used as windows will be subject to wind loads and when used in roofs it has to be designed for the dead load and eventual snow loads. In these applications the duration of high wind loads is short, snow loads will have a longer duration and the dead load will be peranent. The size and tie dependence of the rupture load therefore has to be dealt with separately. Besides, a window glass failure will not affect the reliability of the building itself. It ay, of course, be a hazard to the surroundings where people and property ay be affected in a negative way. 4

The exaples which were presented in the section. show that glass has started to be used in structural deanding applications. Since glass is an appreciated aterial aong architects it can be expected that this trend will continue. Glass is not a aterial which is included in codes as a structural aterial. Hence, there is no inforation about how the reliability should be considered. For other structural aterials like steel, concrete, wood and asonry inforation is readily available about strength values which can be used in applications. The reliability is obtained with statistically based principles known as the partial safety factor syste. There are discrepancies with respect to size and tie aspects between standardized tests and real structural ebers. Standardized tests are perfored on well prepared sall speciens in a rap loading test. Real ebers are larger and in soe cases extreely uch larger and this will cause the rupture strength to be lower than assued. Long ter loading will cause creep to take place which in soe cases is of secondary type. This will further reduce the rupture strength. This is not included properly in the present version of the partial safety factor ethod. So, even if a structural unit is designed in accordance with the partial safety principle, a failure ay occur because of creep to failure. This latter proble can be resolved by using an appropriate statistical distribution to characterize the strength where the size and tie dependencies are included. This results in an analysis based on the theory of stochastic echanics. There are in principle three different ways to analyse a structural eber. In addition to an ordinary design based on the partial coefficient ethod an analysis can be ade based on fracture echanics or a stochastic echanics. In the first case the reliability analysis is based on a fail-safe concept. By liiting the axiu stress to a level where cracks will not grow and cause a failure the structural eber will be safe. In the second case the reliability is based on an allowable stress which is deterined fro a predeterined failure probability where size and tie aspects are included. These two different principles will be applied on a bea of glass subject to four point bending to show how a design is ade in each case and the result. The question concerning brittle failure versus ductile failure will be discussed and analysed in Chapter 5 where the results fro the two design principles are copared. 5 5

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GLASS PROPERTIES. Introduction Glass is a unifor aterial, a liquid that has solidified by cooling to a rigid state without crystallizing, which eans that the olecules are in a copletely rando order and hence do not for a crystal lattice. It is a solid with an aorphous, noncrystalline structure and it is far too rigid to flow at noral teperatures as would a supercooled liquid do. This explains why glass is transparent. The aterial consists of a cobination of various bonds, and hence there is no specific cheical forula. There is no elting point but instead, upon applying heat, the aterial gradually changes fro a solid state to a plastic-viscous and finally to a liquid state. The glass used today for building purposes is a soda-lie-silica glass. The theoretical strength of a glass is deterined by the strength of the bonds between the individual coponents. The strength of a piece of flat glass should on atoic bond strength calculations be around GPa, (Pilkington,993). Window glass usually fails at stress levels less than MPa, (Pilkington,993). In practice, therefore, the aount of stress needed to start a crack growth in glass is very uch less than expected considering the forces needed to break the interatoic bonds.. Glass production ethods There are two ain flat glass anufacturing ethods for producing the basic glass fro which all processed glass products are ade: the drawn glass process and the float glass process. Since the introduction of the float process in 959 by Pilkington it has gradually replaced other processing techniques. More than 9% of the world s flat glass is now ade by the float process, where olten glass, at approxiately o C, is poured continuously fro a furnace on to a large shallow bath of olten tin. The liquid glass floats on the tin, spreads out and fors a level surface. Since the elting point of the tin is uch less than that for glass, the glass solidifies as it slowly cools on top of the olten tin. Thickness is controlled by the speed at which the solidifying glass ribbon is drawn off the bath. Once the glass solidifies, it is fed into an annealing lehr where it is slowly cooled in a process where the residual stresses are controlled. This process results in the production of an annealed float glass with residual copressive stresses around 8 MPa in the surface. After annealing the glass eerges as a fire polished product with virtually parallel surfaces. This ethod, in which the glass pane is fored by floating the elt on a bath of liquid tin, revolutionized the anufacture of highquality glass and large sizes. Float glass is available in thicknesses ranging fro up to 5. Priary processing is a treatent of the basic glass after its anufacture. Since surface flaws only lead to fracture when a tensile stress opens the, any ethod of putting the glass surface into peranent copression is advantageous. An applied tensile stress would have to overcoe this built-in copression before it begins to 7 7 7

open up a flaw and hence the glass would be able to resist higher loads. Toughened glass and heat strengthened glass use this principle. The stress distribution in toughened glass enables it to withstand tensile stresses of uch higher levels than ordinary annealed glass. Annealed glass has a residual surface copression stress of around 8- MPa, (Sobek and Kutterer, 999), because of production reasons. Any external stress level has to exceed this threshold stress to cause a failure which will be tie and size dependent. The thickness of the glass ay influence the actual residual copressive stress. Toughened glass, or tepered glass as it is also known as, is first cut to its final size and it is edge treated and drilled if required. Afterwards the glass pane is heated to approxiately 65 o C, at which point it begins to soften. Its outer surfaces are then cooled rapidly, creating in the a high copression stress, where the rate of the cooling will deterine the aount of built-in copression stress and hence the final strength of that glass. Its bending strength is usually increased by a factor of 4 or 5 to that of annealed glass and hence a new and raised threshold stress has been achieved. The axiu tensile stress in the iddle is half of the surface copressive stress. When broken, it fractures into sall harless dice and it is known as safety glazing aterial. Heat strengthened glass is siilarly produced, but with strengths approxiately half that of toughened glass and without the safety glazing characteristic. Toughened glass cannot be subsequently surface or edge worked or cut because this would initiate a failure. Figure. shows the stress distribution across the thickness of toughened glass. Figure. Stress distribution across the thickness of toughened glass. Toughened glass offers an advantage because an external stress can be uch higher than for annealed glass. As long as the su of the copressive residual stress and the tensile stress fro an external load is less than zero no failure is possible. At least in theory, see Figure.. In reality the situation is ore coplex because 8

of the presence of ipurities in the interior. Most well known is NiS which exists in two fors, or phase. When a transition takes place fro an phase to a phase there will be an expansion. If this expansion takes place where tensile stresses doinate this will lead to a failure, often delayed in tie. Such failures can be forced by increasing the teperature to a certain level for a certain aount of tie in what is called as a Heat Soak Test, HST, where a failure is forced at an elevated teperature. But NiS is not the only ipurity which can create probles in ters of delayed failures. There could be other fors of ipurities or processes involved which could cause a failure. The knowledge about this is far fro coplete. It can be anticipated that the static fatigue phenoenon which is characterized by the tie dependence ight be slower within a glass plate because it takes place without any influence of water. Figure. Suation of stresses in bending in toughened glass. Cheically toughened glass is an alternative to heat treated glass. This technique to achieve higher strength in glass is based on the cheical exchange of larger radius K + ions, (potassiu), for Na + ions, (sodiu), in the surface of a sodiu containing silicate glass. The copressive stress of the silicate network at the thin top layer produces a product known as cheically strengthened glass, where copressive stresses of up to 3 MPa, can be reached at the surface. However, this is only in a very thin boundary layer which is easily penetrated by scratches. The iersion procedure also strengthens the edges of the pane. Cheically strengthened glass exhibits a high resistance to echanical and theral loads. Its fracture behaviour corresponds to that of float glass and it ay be cut, however, a cut edge only has the strength of noral glass. The elastic odulus, E, of glass will not be affected by any surface treatent. As a consequence, deforation characteristics will not be influenced. 9 9

The edges of glass ay be finished in a variety of ways. The noral cut edge represents the siplest for. Such edges are used whenever the edge of the glass is placed in a frae and there is no danger of being injured by the sharp edge. The edges of glass ebers are usually ground to reove ajor flaws and reduce the variation in crack size along the cut edges. Lainated glass is produced by bonding two or ore panes of glass together with a plastic aterial, polyvinyl butyral (PVB) or in soe cases the use of liquid resins. When a PVB interlayer is used, the foil is placed between the panes and the whole unit pressed together in an autoclave under the action of heat and pressure. Lainates can incorporate ost thicknesses of glass and plastics to give a selection of products with a range of echanical, fire resistant and optical properties. When a lainated glass is broken the interlayer tends to hold the fragents of broken glass in place, and it ay be naed a safety glazing aterial. Exaples of lainated glass are anti-vandal, anti-intruder and bullet-resistant glazing..3 Strength characteristics The strength of any structural aterial will exhibit size, tie, teperature and soeties huidity dependencies. This is explained by the presence of various types of defects which are present in all aterials. Glass is no exception, on the contrary, flaws in the surface of glass will have a pronounced influence on the strength of glass. The size and tie dependancies can be illustrated as is shown in Figure.3. For glass the size dependence is better explained with an area dependence instead of a volue dependence. The reason is the existence of defects in the surface region. For an increasing size or area or increasing duration of a stress the failure stress will decrease. The dotted line reflectss a threshold below which no failure will take place. For annealed glass this will correspond to approxiately 8 MPa and for heat strengthened glass the copression stress in the surface. The shaded area is the strength behaviour in tie for a given volue under stress. The result given in figure.3 is in relative values. The reason to why the strength decrease with increasing area is norally explained as that there will be an increased possibility of the existence of a ajor defect with increasing area. The tie dependence reflect the tie for a defect, or crack, to grow to a critical level which will cause a failure. This crack growth will be discussed in subsequent sections. In (Weissan, 997) it has been proposed that the echanical strength is deterined by icro flaws in the surface. These flaws act as stress concentrators and the critical breaking stress depends on the depth of the flaws. Flaws are created during the foring, cooling and final handling processes. The ain source is the contact of the freshly produced glass with other aterials. During the foring process float glass has no contact with solid aterials. By that reason, freshly produced float glass has a higher strength copared with ground and polished plate

glass. The grinding process siply reduces the depth of flaws but do not copletely reove the. The flaw distribution function, n(s), is the ean nuber of flaws per unit area, that are acting as fracture origins at stresses < s, where s is the stress at the crack tip. Fig.4 shows the variation of n(s) for the two float glass surfaces in coparison with polished plate glass. The results verify the higher density of surface flaws of the polished plate. Moreover the tin and atosphere sides show a slight difference in their strength that coes fro the contact of the tin side with transport rollers, (Weissann, 997). The strength of glass can be based on the well-known Griffith s flaw theory (Jacob, 999). This gives an interpretation of experiental data on the strength of glass. According to these ideas, the observed strength is always reduced below a theoretical strength by the presence of flaws and the easured strength is deterined by the stress concentration acting on the worst suitably oriented flaw at its apex. This accounts directly for the wide variation in experiental values of strength obtained under any given condition (since all flaws will not be equally dangerous) and ust obviously be considered in the interpretation of all the strength phenoena. Figure.3 The rupture stress as a function of volue and duration of a stress. It is siply stated here that the factor by which the theoretical strength exceeds the ordinary experiental values is usually considered to be in the order of 5.

The Griffith flaws, which have a strong influence on the strength of glass, are assued rando in size, distribution and orientation. The only way to establish a value for the strength of glass is to echanically load the glass to breakage. This establishes the strength of that piece, tested at that tie and in that particular way. Statistical analysis of a set of scattered values of glass strength easureents allows the derivation of a value below which there will be relatively few failures. Statistical treatent of test results can be used to obtain a design value at which there is sufficiently low risk of the glass being weaker than that design value. This value can be used to select a glass thickness for a given situation. Tests are ade to gather inforation regarding a variable, which in this case would be the strength of glass. This inforation would then indicate the strength of that test package. Different ethodologies can be utilised for further analysis, see chapter 3 and 4. Figure.4 Nuber of surface flaws/. - polished plate glass, -float glass tin side, 3- atosphere side. The strength of glass is also odified by the presence of larger surface flaws. Under stress these can be the origin of cracks since the glass ay be unable to accoodate the local stress concentrations they cause. The presence of ajor flaws, for exaple scratches, on the surface or the edges of the glass can generate local stress concentrations when the glass is put under strain, which ay lead to crack foration. It is however not easy to predict whether any individual flaw is likely to do this. Sall defects in the glass are likely starting points for crack propagation as they produce unacceptable stress concentrations. The presence of large flaws is one reason why the edge region of a piece of glass is usually weaker than the surface, where it is uch ore prone to daage fro accidental contact with the surroundings. Methods of cutting glass and edge finishing ay also lead to the presence of flaws at the edge.

Since tie duration of loading is iportant in deterining the failure load for a given glass panel, it is essential that loadings be defined in a tie-dependent, rather than static for. Experiental data, (Minor, 974), which highlights this phenoen are presented in Fig.5. The teperature dependence was analysed by (Charles, 958) And the result is presented in Figure.6. It can be seen that the strength will decrease with increasing teperatures. Below a certain teperature where no oisture is present there see to be no or very little tie dependence. The influence of the huidity or water on the rupture behaviour is often tered static fatigue for glass. This will be analysed separately in chapter.4. Hence the stress will depend on the following external factors: ( Duration,Size,Teperature,Relative Huidity) f ( D, V, T, RF) glass ( D) D n Figure.5 The tie dependence of the strength. Figure.6 Delayed failure curves for soda-lie glass at different teperatures. 3 3 3

.4 Static fatigue Ceraics and glasses can deonstrate a loss of strength over tie. This degradation of the strength can take place under constant stress conditions. In the glass counity this phenoena is often tered static fatigue and it is a sensitive function of the environent. Cheical attack by water vapour perits a pre-existing flaw to grow to critical diensions and bring about spontaneous crack propagation. Depending on environental conditions, glass exhibits tie-delayed failure where fracture ay occur soe tie after the initial application of a load. Static fatigue is basically caused by the subcritical crack growth that can take place in glass, even in a weak tension field. Sall aounts of water vapour, norally found in the atosphere will react with glass under stress to cause a tie-dependent reduction in strength. By cheically reacting with the silicate network, an H O olecule generates two Si-OH units. The hydroxyl units are not bonded to each other, leaving a break in the silicate network. When this reaction occurs at the tip of a surface crack, the crack is lengthened by one atoic-scale step, see Figure.7 and.8. Static fatigue results fro a stress-dependent cheical reaction between water vapour and the surface of glass. The rate of reaction depends on the state of stress at the surface and the teperature. The rate increases ainly with increasing stress. The stress is greatest at the root of sall cracks and consequently the reaction proceeds at its greatest rates fro these roots. Since the reaction products do not have the strength of the unreacted glass, the sall cracks gradually lengthen and failure occurs when the cracks are long enough. Figure.7 Crack growth by cheical breaking of oxide network. Figure.8 A break in the network. It sees well established that the general failure of glass is deterined by surface flaws. Stress fields around a flaw in conjunction with corrosion echaniss 4 4

dependent on stress provide a ready answer to the question of how a flaw ay grow in an isotropic aterial, such as glass, and bring about delayed failure. The aount of growth is deterined by the stress concentration that is necessary to bring the applied stress up to a critical value whereas the differential rate of growth is dependent on the stress distribution around the flaw at any instant and the coposition, pressure, and teperature of the surrounding atosphere. Figure.9 shows the effect of water vapour on crack otion in glass at roo teperature (Wiederhorn, 97). When the loading is at low levels the cobination of load and oisture affect the strength, whereas at high levels of loading, around, kg, the oisture has alost no effect on the strength. The crack velocity is divided into three different regions. According to Wiederhorn these conclusions were ade: In region I crack propagation is due to corrosive attack of water vapour on the glass at the crack tip. In region II the crack velocity is nearly independent of the applied force, and the position of each curve shifts to lower velocities as the water in the environent decreases. In region III the crack velocity is again exponentially dependent on the applied force, however, the slope of the curve is considerably greater than in region I, 3 ties. This change in slope leads to the conclusion that a third crack propagation echanis occurred. Figure.9 Crack velocity versus load. The fact that the curves of region I and II blend to for a single curve in region III indicates that this new echanis is independent of water concentration in the 5 5 5

environent. The load below which glass would be free of fatigue is not well known and oreover, static fatigue which is very sensitive to the environent is accelerated by teperature and atospheric huidity, (Charles, 958). It was suggested (Preston, 94, Orrowan,94) that cracks ight be assisted to spread slowly in a prolonged test by atospheric action of the types already discussed. Baker and Preston have shown that glass speciens baked and tested in vacuu show practically no reduction in strength under prolonged loading..5 Experients on glass beas There are two interesting reports on tests of glass used structurally. In both cases it concerns beas and the tie behaviour. In Carré, (996) a large nuber of beas have been tested to failure in rap loading tests with different rates of loading. The surface treatent of the edge in tension has been what is considered the noral preparation but there is also a new iproved preparation. Each bea had a free span of 3, a height of 37,5 and a thickness of 9. The loading arrangeent was a syetric four point loading schee. The saple size is 8 beas for a loading rate of,5 MPa/s and two sets of 4 beas with a loading rate of,5 MPa/s respectively 5 MPa/s. In both cases the predicted failure behaviour based on a Weibull distribution is in good agreeent with the obtained actual failure behaviour. The ain result is suarized in Table.. It can be seen that both for the noral surface treatent and the iproved surface treatent the rupture stress will increase with increasing rates of loading. With increasing loading rates the duration of a test will decrease. In the report by Carré (996), there is also an analysis of the effect of heat treatent ade in a finite eleent analysis. The conclusion is that no failure could take place for bending tensile stresses fro external load which were less than the copressive residual stresses in the surface. Table. The ain result fro tests of beas Failure stress Noral surface Iproved surface MPa treatent treatent Rate of loading,5,5 5,,5 5 MPa/s Mean value 4, 45,4 5, 54,7 56, MPa Based on the inforation fro (Carré, 996) a sustained load test was carried on a heat treated bea out by Saint Gobain (Gy, 999). In this case the bea was 6 6

substantially larger, 4 long, the height 6 the width 9. Also in this case a four point bending arrangeent was used. The residual copressive stresses was deterined to be around MPa. The applied load resulted in around 8 MPa in tensile stresses which is less than the residual stress at the edges. (According to this calculation, the glass weight plus an external applied load of 4 N, lead to a axiu tensile stress of 7 MPa. This is below the residual copressive level and hence it should prevent the bea fro suffering fro static fatigue during a long experient). In addition to direct deforation at loading the creep deforation in the iddle of the bea was easured during year. This deforation is shown in Figure.. It sees as if the creep deforation reflects priary creep which can be seen as a delayed elastic response. But priary creep is associated with what is tered a subcritical crack growth. There is an indication that there could be a start of secondary creep which reflects a crack growth. Such a crack growth can take place internally where large tensile stresses exist. This could be because of the existence of soe ipurity. It could also be an edge crack growth. After year the bea was unloaded and the deforation recovery was continued to be easured over another year. Most of the delayed deforation was recovered during this period but a sall deforation still reained. No explicit inforation was given to this. Personal counication indicated that this ost likely was a easuring error or deforation of the supports. Figure. Deforation versus tie. The reaining deforation after the test was copleted indicates that soe daage accuulation ay have taken place. Daage accuulation ay reflect a crack growth in the botto tensile face of the glass bea. This will be analysed and 7 7 7

explained in ore detail in chapter 3, fracture echanics. If this daage accuulation really takes place there is a potential proble with heat strengthened glass. 8 8

FRACTURE MECHANICS APPROACH 3. Introduction Fracture echanics is the theory about how solids rupture, priarily structural coponents of different types. Such ruptures can take place in several different ways. Typical for these ruptures are that one or a few cracks propagate through the aterial and finally results in a rupture. Fracture echanics deals with the echaniss involved and ethods to characterize the. Fractures are found to originate at flaws or cracks of finite size, ost of which are at the surface. The echanis, crack propagation, begins when the local stress at the crack exceeds a iniu value. Dependent on aterial behaviour and the type of loading fracture echanics can be represented in three ain categories. Linear elastic fracture echanics deals with existing cracks which, at a critical stress, grow oentarily which results in a brittle rupture. Typical is that the rupture planes do not defor and the broken parts can be put together again where the rupture line ay not be directly visible. This is a coon failure behaviour for glass and ceraics. But also steel ay rupture in this anner, especially at low teperatures. The additional energy needed to cause the crack to grow is norally very liited. Non-linear fracture echanics also deals with an existing crack but the crack growth takes place in a plastic zone. Also in this case the crack growth can be oentarily if the applied stress is sufficiently high or the initial crack is large. But it is ore coon that the crack growth is slow and accopanied with plastic deforation. This failure behaviour is associated with etallic aterials subject to relatively steady state loading conditions. Fatigue initiated fracture echanics cause existing cracks to grow in relation to the load spectru which affects the coponent with a crack. This crack growth can initially be extreely slow and start at cracks which are not easily detectable. All structural eleents contain defects in ters of cracks already after production. In a surface of a aterial there is likely to be irregularities between groups of atos or olecules. This ight be sufficient to start a crack growth. This type of crack growth is ainly associated with non-linear fracture echanics. But there are exaples where a crack growth takes place for a brittle aterial response. Cracks can grow in a slow anner in glass under certain loading conditions. Fracture echanics is a powerful tool in any applications. It can answer the following questions: How large a crack can be tolerated for a specified stress condition before a failure can be anticipated? How fast will a crack grow when subject to fatigue for a specified load spectru? Dependent on the answer the structural part can be odified, the stress field altered or soe type of regular inspection procedure can be introduced to detect cracks 9 9 9

before they will cause a failure. In any advanced applications where aterials are used to their liit, for instance turbines, regular inspections is part of the aintenance schee. 3. Soe historical notes Probles with unexpected failures due to crack growths, especially in connection with stress concentrations, was recognized already in the 9 th century. Sharp corners and defects in for of different types of cracks were often associated with the initiation of a failure. The first published theoretical article on this subject is by Inglis (93). For a flaw which is represented by half an ellipse with opening b and depth a, see Figure 3., the stress at a flaw tip is given as a = + b (3.) where is the concentrated stress at the flaw tip, is applied stress. For a circular flaw, a = b, the stress increase is 3. When a increases or b decreases will approach infinity. This is not possible in reality so instead there will be a crack growth. The sae stress intensity increase is obtained also for a whole elliptically flaw in the surface. Figure 3. Half elliptical flaw. The expression within the parenthesis in (3.) can be seen as a stress concentration factor K t. This can be generalized as K t a =+ = + b a (3.) where = b /a is the radius of curvature of the ellipse at the vortex. This expression can be applied to other shapes than circular or elliptical flaw tips if the curvature is known. When goes towards zero K t will approach infinity. Equation 3. is applicable to an analysis of the influence of flaws on the strength of glass. This was deonstrated by Charles (958,) in an analysis of the conditions for a crack growth. He assued that for a theoretical strength of glass of 4 3 MPa a ratio a/b = 4 is necessary for a failure at an applied stress of 7 MPa (Note that this high stress level refers to sall bars of glass subject to bending). This will 3 3

alter the initial crack relations and ake the ratio a/b to continuously increase. Based on his experients there is a critical value of a/b ~, Charles(958, ). The theory of brittle fractures was extended by Griffith (9) who ade use of Inglis stress concentration expression around a crack. The novelty in Griffith s work is that he ade an attept to forulate a linear elastic theory for crack propagation based on considerations of the global balance of energy in a body. For an linear elastic body the internal elastic strain energy U is U = udv = V V ij ij dv (3.3) where u is the elastic strain energy density, and and are stress and strain respectively and the indices refers to x and y in a Cartesian syste. With Hooke s law ij µ = ij = ij ij (3.4) + where µ is the shear odulus of the aterial, µ = E/( + ), E is the elastic odulus and is Poisson s ratio. Cobining (3.3) and (3.4) the elastic strain energy is given by U = 4µ ij ij = ij + ij V dv This is the total elastic strain energy stored in a loaded body ade of linear elastic aterial. For a body subject to a prescribed load P, where P can be any type of loading such as force, oent or distributed force, the potential of the body is defined as = U - P = U - W (3.6) where is the displaceent if P is a force, and is rotation if P is a oent. For a fracture to occur, the energy stored in a body including the work done by the forces acting on the body, ust be sufficient to overcoe the surface energy needed to create a new crack surface. The total energy is given by = +W (3.7) E tot s (3.5) where W s= Awsis the energy required to create a new crack surface. Here A is the crack area and w s is the surface energy per unit area. When a crack propagates no net change in the total energy takes place, thus de tot da d dw s = + = da da (3.8) 3 3 3

This results in the condition d - = da dw s da (3.9) which ust be et to ake the crack propagation possible. Crack growth is possible when the energy release rate G reaches a critical value dw G = G critical = G s c = da (3.) The paraeter G c is a easure of the fracture toughness of the aterial. This is also known as the Griffith criterion and can be expressed as f E c (3.) where is the and c is the It is often considered a aterial property. In reality it will be teperature dependent and in the case of glass also huidity dependent. 3.3 Linear fracture echanics Stresses and deforations in a body in front of a crack tip depends on how the body is loaded. The crack ay be affected in three different ways or odes which are illustrated in Figures 3. - 4. Figure 3. Crack loaded in Mode. Figure 3.3 Crack loaded in Mode. Figure 3.4 Crack loaded in Mode 3. The analysis of stresses and deforations around the crack tip is norally ade in a polar coordinate syste which is defined in Figure 3.5. Mode and Mode solutions can be found with the help of the Airy stress function (Tioshenko, 959). Mode 3 solutions are based on equilibriu, deforation and constitutive equations. Solutions result in series expansions which can be found in books dealing with in 3 3

depth analysis of fracture echanics. Most available fracture echanics solutions are based on Mode failures. This is natural when stresses fro bending oents doinate. Mode and 3 failures are hardly dealt with in practice. But it cannot be ruled out that the crack growth process which takes place in static fatigue is influenced by Mode or Mode 3 stress conditions. The ain results which are used in research can be suarized in the following expressions (Dahlberg, et al, ): Figure 3.5 Polar coordinate syste and stress coponents. Mode : xx = rr ( = ) = yy = ( = 9) = K I x K I x (3.) (3.3) xy = r ( = ) = + - x u y = u ( = ±) = ± K µ I (3.4) (3.5) Mode : xx = yy = (3.6) xy = r ( = ) K II x (3.7) u x u r ( ) ± µ K x = = = + - II Mode 3: (3.8) 33 33 33

Mode 3: xz = rz (= ) = yz = z (= ) = K III x (3.9) (3.) u z u r ( ±) ± µ K x = + - = = III where 3 - = for plane stress + = 3-4 for plane strain (3.) (3.) (3.3) Plane stress is expected for a thin plate where the stress coponent perpendicular to the plane is zero, i.e. zz = and rz = z =. For a plate which is not very thin the stress zz will not necessarily be zero. If the lateral strain zz is prevented at the crack tip then zz = and rz = z =. This condition is tered plane strain. Even if a glass plate can be considered thin there will be significant stresses perpendicular to the plane. There will always be residual stresses which indicate that a plane stress assuption ight not be the ost optial solution for glass. Still, this is the general assuption ade in all publications available. This is a siplification which need to be analysed in a ore objective way. The factors K I, K II and K III give a easure of the strength of the stress singularities at the crack tips. These factors are called stress intensity factors and play an iportant role in fracture echanics. These factors can be expressed as K I = y, af K II = xy, ag K III = yz, ah (3.4a, b, c) where a is the crack length and the index refers to stresses far away (copare Figure 3.). The functions f, g and h will depend on the geoetry and the type of loading. Values of f, g and h can be found in handbooks dealing with fracture echanics (Lawn,995). The fracture criterion is norally defined for plane strain conditions. Since Mode norally is ost dangerous a failure is expected when K I = K Ic (3.5) where K Ic is the fracture toughness of the aterial. This fracture toughness is deterined experientally. There are three conditions which have to be et: 34 34

(t,a,w- a),5 K Ic y (3.6) where t is the thickness of the plate, a is the crack depth and W is the height of the eber in the sae direction as the flaw. In the case of plane stress the failure condition is written = K (3.7) K I c where K c is deterined experientally. Failure occurs when the stress intensity factor reaches a critical value, K IC. This critical stress intensity factor is a constant for glass and provides a fixed basis on which to conduct design, rather than using an allowable stress which has to vary with tie. The loss in strength over tie is a result of slow crack growth which occurs when the stress intensity factor is less than the critical value. There is a threshold stress intensity factor liit below which no slow crack growth occurs. Glass fails when the stress intensity factor reaches the critical stress intensity factor. This failure is sudden and results in very rapid crack growth leading usually to loss of structural integrity. If the stress intensity factor is less than the critical value then the crack grows. The speed of this growth is given by KI v v K IC n 35 (3.8) where n is the static fatigue constant and has a value between and, but a often used value is 6 and it depends on environental conditions. 3.4 Fracture echanics of glass windows The fracture of glass is associated with the existence of flaws. Griffith advanced a theory that brittle bodies fail under a relatively low noinal stress because of the presence of flaws or cracks which have the characteristics of extree narrowness with respect to their length or depth. Consequently they produce high local stress concentrations when oriented at right angles to a tensile coponent of stress. There are two ways of looking at crack growth for glass. First, fro the point of view of the stress intensity factor, K I, which acts as a resistance to cracking and it is called fracture toughness. The crack can grow when its stress intensity factor reaches the fracture toughness of the aterial. A typical value for soda-lie glass is,78 MPa /. A series of 3 point bending tests were conducted on 7 x 4 x 6 thick glass saples consisting of annealed, tepered and lainated glass (Jacob, 999). Control saples without any surface daage was used to copare the agnitude of strength reduction caused by the application of surface scratches on the glass saples. Figures 3.6 to 3.8 deonstrate the ipact a sall scratch has on the reduction in the bending strength. 35 35

Surface scratches to glass up to,8 in depth were found to be the liiting scratch. Using this as a liiting scratch, the fracture stress can be deterined fro eq 3.. In this analysis the fracture toughness is assued to be K IC =,85 MPa /. The following conclusions were ade fro the analysis: the liiting flaw based on a perissible design stress of 5,7 MPa is,745 the liiting stress based on a axiu flaw depth of,8 is 47,9 MPa using a safety factor of,5 for perissible glass design, the design stress will be 9,6 MPa. These conclusions see soewhat arbitrary. Fro reality it is known that the failure stress is both size and tie dependent. The safety factor principle is hardly used within building coponent designs any ore. It has been replaced by the statistically based partial safety factor ethod. It had been possible with a ore advanced analysis where the perissible stress had been deterined in a statistical analysis. Figure 3.6 Fracture stress versus scratch depth for annealed glass. Figure 3.7 Fracture stress versus scratch depth for tepered glass. Figure 3.8 Failure stress versus scratch depth for lainated glass. 36 36

Failures in glass panels generally start fro the surface or the edge depending on the loading and support conditions. The environental contribution to crack growth results in a tie dependence of strength known as static fatigue. Static fatigue only occurs in the presence of water, which reacts cheically with the strained bonds at the crack tip causing bond rupture. The rate of crack growth is deterined by the rate of cheical reaction, and the tie to failure is deterined by the tie required for the crack to grow fro a sub-critical to a critical size, at which point fracture will be iediate. Once the crack growth has been characterised then the tie to failure can be calculated provided the size of the critical flaw can be deterined. It was suggested purely on grounds of geoetry (Littleton and Preston, 9) that surface flaws can be roughly twice as dangerous as internal flaws of the sae shape and orientation. Upon exceeding a critical (tensile) stress the crack begins to grow at the tip of a notch or crater. In soe circustances this growth only takes place in sall steps, coing to a halt in between. In fracture echanics this slow or stable crack growth is regarded as sub-critical, and is priarily deterined by the duration of the load. Short-ter loads lead to higher allowable stresses than longter loads. The sub-critical crack growth is influenced by cheical reactions at the tip of the crack. Once the critical crack growth velocity has been exceeded, a crack becoes unstable, i.e. the widening process accelerates rapidly. This then leads to sudden failure of the glass eleent. For an applied stress which leads to a stress intensity factor K I < K IC, a crack growth ay still be possible due to the effect of the environent. Crack growth under these conditions is called sub-critical crack growth or static fatigue, see Chapter.4, and ay ultiately lead to fracture soe tie after the initial application of the load. The growth of the crack over the lifetie of the structure is odelled based on the expected stresses, and the tie to failure can be estiated as: n T ( K / ) / AY n) in a IC a p 37 37 37 (3.9) where Y is a diensionless paraeter, see Appendix A, that depends on both the specien and crack geoetry and siga is an applied stress and a is crack length. The exponent n deterines the tie dependence. According to (Fischer-Cripps and Collins, 995) the iniu strength of glass is related to a threshold stress intensity factor, rather than a unique iniu stress. If the initial crack size is known then a iniu long-ter stress strength can be deterined. If, however, during the loading history of the eber this stress is exceeded, then cracks will grow resulting in a lower subsequent iniu strength, even if the stress then reverts to its initial value. When the deforation of the pane exceeds around 75% of its thickness the action of ebrane stresses coe to affect the total stress distribution on the surface. The ebrane stresses grow towards the iddle section of the pane. The extra contribution of shear stresses leads to a axiu stress that is soe distance away

fro the iddle. The effect of ebrane stresses depend upon the supporting solution. 3.5 Analysis of beas Beas and other structural eleents subject to tensile stresses can be analysed with a fracture echanics approach. In this analysis only beas subject to four point syetric bending is dealt with. Such a bea is shown in Figure 3.9. Between the two point loads the axiu bending stress will occur and this is ost likely to result in a Mode I failure. Between the supports and each point load the axiu shear stress will occur and this ay result in a Mode II failure. In all noral analysis a Mode I failure will be the doinant failure ode. This failure ode is the only one dealt with in the literature dealing with fracture echanics of glass. The axiu load effects are given by V ax M ax P = P L e =. - and with the bending resistance W W = bh 6 the axiu stress can be derived as M 3P(L- e) = = W bh Figure 3.9 Bea loaded with two point loads and the resulting shear and oent diagras. As an exaple, assue the following values for the bea L=, h =,4 and b =,6. It is subject to two point loads with e =,5. The total load is P = 6 kn. This will result in a axiu stress of M 3. 6(, -,5) = = = 4,6. 3 kpa W.,6.,4 which is a stress that should not cause a direct failure. Because of a crack, real or assued, in the section of axiu stress there will be stress increase which can be analysed with eq. 3.. Values of K I as well as additional inforation concerning calculation of K I can be found in Appendix A. The value of K I can be expressed as: 38 38

K I = a af K f I a w a w Figure 3. A crack. 6 where o is the stress at the tip of the crack. Assue that K / IC =,85. Pa. For a crack depth of a =,8, which sees to be the sallest crack depth which can be easured directly in a sheet of glass, the following calculation can be ade a K a f K I ( ) w 3 4 kpa 4MPa a 8, f f, w 4 K K I I 6 3 4, 8,, 786 K, 85 IC 6 6 IC OK For a crack depth of a =, a K a f K I ( ) w 3 4 kpa 4MPa a, f f, w 4 K K I I 6 3 4,,, 78 K, 85 IC 6 6 IC OK 39 39 39

a KI a f ( ) KIC w 3 4 kpa 4MPa a f? w a K a I 4 f KIC, 85 w critical cracksize? 6 6 Obviously the initial size of a crack is a very iportant factor for the failure stress. Another way to address the proble is to find the critical crack size. This ethod will be an iterative process, since when the crack length, a, is explicit it will still be a function of itself and hence a non-explicit solution will be obtained. Because the crack depth, a, does not affect the height of the structure, w, to any larger extent until it reaches about 3% of w, this iterative process can be done quickly. a c K I a f w (3.3) K I a a f H a f, 6 H K, 85 IC 6 MPa K 3 IC, 85 a f a 3 6, 4578, w For the following stresses in the crack tip the critical crack depth will be 5 MPa a, 7346 MPa a, 83 5 MPa a, 8 4 4

Exaple: A glass plate contains an atoic-scale surface crack. (Take the crack tip radius ~ diaeter of an O - ion). Given that the crack is µ long and the theoretical strength of the defect-free glass is 7, GPa, calculate the breaking strength of the plate. Solution: a a Table, (see Appendix): ( O ) r( O ), 3 n, 64 n 7 9, 64 6 9 57, MPa Figure 3. 4 4 4

STOCHASTIC MECHANICS APPROACH 4. Introduction Stochastic echanics is the theory about how to consider the uncertainty in the rupture strength of aterials. Such uncertainty concerns both the agnitude of a rupture and the variability. The reason to this uncertainty reflects the presence of flaws in a aterial. An analysis of the rupture strength can be done in two principally different ways. It can be based on the influence of flaws or the consequence of flaws. In the latter case it is ore adequate to ter it a statistical analysis. All aterials contain flaws. They can be a natural part of a aterial or a consequence of the production, handling and use. In theory the tensile strength of glass is around MPa. In reality the rupture strength of window panes is in the range of 3-8 MPa. This draatic reduction of the rupture strength is not unique for glass. Siilar reductions can be found for any type of structural aterial. If a aterial had been perfect, i.e. all internal coponents were in perfect order, then a tensile failure had been expected for a tensile stress. With internal disruptions between the coponents there will be a new failure ode, a shear failure ode which can take place for an external tensile stress. This has been studied extensively for etallic aterials where this shear behaviour is associated with viscous creep. For other aterials where this type of creep is not possible a crack growth takes place instead. For glass and other ceraics this crack growth sees to be associated with priary creep which can be seen as a delayed elastic response. Flaws are often assued to be rando in size and orientation. A consequence of this assuption is that there is an increasing probability of finding a ore dangerous flaw for increasing volues under stress. This introduces a size dependence of the rupture strength. A size dependence of the rupture load is a well known phenoenon for structural aterials as well as for glass. Existing flaws can grow in tie or new flaws can be created. This alters the original flaw characteristics and thus the failure stress, which in turn introduces a tie dependence of the rupture stress. Such a tie dependence of the rupture load is of concern if a aterial is subject to a high sustained stress, for instance prestressing tendons in concrete beas. For glass there is a noticeable tie dependence which needs to be considered. The environent in ters of the huidity also has a pronounced effect on the strength of glass. Cracks will grow in size faster when water olecules are present, soething which is tered static fatigue for glass. Stochastic echanics is also a powerful tool in any applications. It can answer the following questions: What is the failure probability for a structural unit subject to specified stress condition? 4 4

How will the rupture strength decrease for an increasing volue under stress? How will the rupture strength decrease for an increasing duration of a stress? Dependent on the answer the structural part can be odified or the stress field altered. 4. Soe historical notes The presence of a size dependence of the rupture stress was first noticed by Leonardo da Vinci for wrought iron in the 5 th century. This does not see to have given rise to any practical use. In the beginning of the th century it was recognized that it was necessary to standardize test ethods in ters of specien size and rate of loading to ake fair coparisons possible. For brittle aterials Weibull(939) introduced a theory that could characterize the size dependence of the rupture load. He assued that a bar subject to a tensile stress can be split up in a large nuber of units or volues. For each such volue, V i, there is a probability density function, pdf, giving the probability of failure. The necessary condition for the occurrence of fracture in the body at a certain definite load is that the ultiate strength shall be exceeded at a single point/eleent, i.e. the weakest eleent. This probability is equal to the product of the probabilities in every eleentary volue of the body that fracture will occur. If two independent eleents are put in series, the new failure probability, P, will be P(syste fails) = - (- F ) (- F ) = {F = F }= - ( - F ) (4.) This can be done for n eleents which results in the following failure probability distribution P(syste with n eleents fail) = F nv = - ( - F ) n (4.) After taking the logarith and differentiating, the result will be F nv = - ( - F ) n (4.3) - F nv = ( - F ) n ln( - F nv ) = n ln ( - F ) d dv n ln( Fnv) F v (4.4) (4.5) 4 43 43

dln( F ) nv ln( F ) F F nv nv nv e e n F dv v n F dv v n Fv dv n Fv dv (4.6) (4.7) (4.8) (4.9) The failure probability function will be: F nv e g( ) dv (4.) For the stress function g() Weibull suggested g() c o (4.) where c o is a noralization constant. The exponent is often referred to as a size paraeter since it deterines the size dependence. It reflects the influence of various types of defects where cracks is one type. It can also be irregularities in the atoic structure which are not directly easurable. The statistical distribution (4.) with eq. (4.) is the well known two paraeter Weibull distribution. 4.3 Stochastic echanics for brittle aterials There are various approaches to the statistical theory of brittle fracture. These differ in the ethod of reasoning and the assuption of the postulated for of the distribution function of the strength of the priary eleents. It is also possible to start with a hazard function, which is the probability of failure given that no failure has taken place before. A solution based on an asyptotic distribution for iniu values of sufficiently large sets will be given below. This approach is a generalization of the Weibull ethod, being at the sae tie justifiable fro a theoretical point of view. Weibull apparently proposed the distribution function out of convenience to fit available experiental inforation, which has been a vulnerable spot in his theory. The principle utilized in section 4. can be applied for any type of statistical distribution and when n goes towards infinity this will lead to an extree value statistical distribution. Based on the original distribution there are three possible extree value distribution alternatives, type, type and type 3. For each of the there is a distribution for a iniu and a axiu value. This theory of statistical distributions of extree value was not available at the tie when Weibull 43 44

developed his statistical distribution. If the original distribution cannot take negative values, i.e. the breaking tensile stress is positive. The tail of the distribution will to soe extent reflect stresses between the coponents in the aterial which often are represented with a hyperbolical expression. With these two conditions it can be expected that the extree value distribution is of type 3 and iniu value. This statistical distribution can be expressed as F() ~ exp[ knv( ) ] (4.) where is a iniu strength of a defective eleent. For nv sufficiently large the approxiate sign can be substituted with an equal sign. When = the result is the sae statistical distribution proposed by Weibull. The conditions for eq. (4.) to hold are fairly general and a broad class of initial statistical distributions can be utilized. With a change of the constants k and it is possible to obtain a statistical distribution with a wide variety of for and scale. The Weibull distribution has any interesting features which has ade it attractive in any fields of application. If a reference volue V and a constant c are introduced as kn V c (4.3) then eq. (4.) can be expressed as V F() exp V kn D D h V c c (4.4) which is coon way to express the Weibull distribution in structural applications. But since it is known that the failure stress also is tie dependent the constant kn can be chosen to take this into account. Fro the inforation given in Chapter this tie dependence is also hyperbolical in ters of the ean value. To obtain this result the constant kn is chosen as (4.5) where D is an effective duration and D is a reference duration. This will result in F() exp V V D D h c (4.6) Equation (4.6) can be applied both for loading directly to failure or creep to failure. The effective volue under stress and the effective duration of stress need attention 44 45 45

under non unifor stress or not constant stress conditions in tie. Any volue can be split up in sall units as well as a duration in tie units and the evaluation of eq. (4.6) can be expressed as F() exp V D F() exp V D h h p i (V) xyz V E[] D + c h o + V D and the variance as V D h V i (t) q j t D j h (x,y,z,t) c (x,y,z,t) h c dvdt (4.7) which is how the analysis will take place in a coputer evaluation based for instance on the finite eleent ethod. When p and q goes towards infinity the suations can be replaced by integrals which results in (4.8) where it can be expected that only very siple stress conditions can be evaluated analytically. If desirable the reference volue V and reference duration D can be given the value. The ean value of (4.6) is obtained by eans of variable transforations and integrations. For the siplest load case of unifor stress which is constant in tie the ean value can be derived as (4.9) Var[] c o + + (4.) V D where is the gaa function which can be evaluated fro a series expansion, fro tables or atheatical coputer progras. The choice of constants was otivated by a desire to obtain eq. (4.9) as a ean value. A siilar technique can be utilized to include the teperature dependence found in (Charles, 958), see Chapter. The size and tie dependencies of the rupture stress of brittle aterials like glass found in experiental inforation are characterized well with eq. (4.). No other statistical distribution has this ability to include dependencies in this way. The result of eq. (4.9) can be visualized as a stress failure surface in a size - tie doain which is presented in Figure 4.. The dotted line reflect the iniu strength. The arked area reflects the tie dependence of the rupture strength for a given volue under stress. For increasing volues under stress and increasing durations the rupture stress will go towards. The coefficient of variation, c.o.v., which is the square root of eq. (4.) divided by eq. (4.) can in the case when = be expressed as 45 46

(4.) ( + /) c.o.v. = - ~ ( + /) 6 which provides a eans to estiate the size paraeter if the ean value and standard deviation are known fro experients. The accuracy of eq. (4.) depends on the relation between and c and. The approxiation given in eq. (4.) is acceptable for values of > 4-5. The variability in experients ay be affected by external sources which eans that eq. (4.) can only give guidance in a possible value of the size paraeter. For ordinary plate glass it is norally not the defects within the volue which are of ain concern. Instead it is surface defects or edge defects which will cause a rupture to occur. For glass it sees reasonable to substitute V with the surface area under stress, A, or the edge length, L, under stress. This leads to the following expressions for glass strength evaluations F() = - exp - A A D D A - h c A (4.) F() = - exp - L L D D L - h c L (4.3) where A and L are reference values. The exponent A is in the range 7-9 and the exponent L is in the range 3-4 (Ontario Research Foundation, 98). The threshold stress reflects residual copressive stresses in the surface region which are introduced during the production because the surface cools down faster then the interior. In the float glass production the cooling is controlled and will be around 8 MPa. Along the edge it is likely that is close to zero. For heat strengthened glass will correspond to the copressive stress introduced in the surface which can be well over MPa. In an evaluation of the effective volue under stress and an effective duration of stress by hand it is an advantage to disregard the threshold stress. In the evaluation of the volue under stress the influence of existing defects are taken into account. Then the evaluation can be seen as the evaluation of a reduced volue and reduced tie. This can be expressed as V * = f(x,y,z) dxdydz V (4.4) The tie dependence reflects the growth of existing defects with tie and the stress level. This reduced tie is evaluated fro 46 47 47

D * = D f(t) h dt (4.5) where f(t) describes the stress variation in space or in tie for a body subject to stress. The resulting ean value and variance are given by eqs (4.9) and (4.) with = which is an acceptable approxiation in ost applications. With = a linear stress variation over a body can be evaluated analytically. An exaple of such a calculation will be given in the last section of this Chapter. For a stress which is increasing continuously until a failure occurs eq. (4.5) gives the result that the reduced tie is D* = t/(h + ). For glass where h ~ 6 the reduced tie is /7 of the actual tie. This can be interpreted as that only the tie spent on a high stress level will be of iportance. This iplies strongly that eq. (4.4) and eq. (4.5) can be used together with eq. (4.9) and eq. (4.) without any loss of accuracy even if the threshold stress is iportant. Figure 4. The rupture stress as a function of volue and duration of a stress. 4.4 Failure prediction for glass windows A glass window will norally be subject to wind loads acting perpendicular to its plane. For a glass pane which is siply supported along its edges this will create bending stresses and ebrane stresses. For sall deforations, less than the thickness of the glass plate, bending stresses will doinate. For larger deforations ebrane stresses will increase in iportance. Modern analysis with finite eleent analysis (FEM) shows that at failure bending tensile stresses and ebrane stresses will have approxiately the sae agnitude. The axiu tensile stress will not 47 48

occur at the centre of a plate but at four points soe distance away towards the corners. Many old stress analysis of glass plates see to have been based on assuptions which do not account for the ebrane stresses. This is understandable if calculations had to be ade by hand since this leads to extensive evaluations. A wind load which is acting on a window pane is coplex. During shorter periods of tie it can be assued to have a stationary coponent and a fluctuating coponent. There will also be a rando fluctuation of the wind over a glass area. For increasing durations of a wind loading the iportance of the fluctuating coponent will decrease. The optial design duration of a wind load is not easy to define in a consistent way. Besides, the building itself will affect the wind structure and create turbulence which will also be influenced by the wind direction and surrounding buildings. Therefore the design wind speed is assued to be the average wind speed during a short duration, norally related to available wind statistics. In the United States the duration is inute. In Sweden the duration is inutes but this value is adjusted to account for a dynaic influence. When glass plates are tested the load is increased slowly until a failure occurs. It was realized by (Brown, 974) that the failure stress depended on the rate of loading or the duration of a test. To account for this discrepancy the following expression was proposed to noralize the rupture stress result d = (t) t d n dt / n (4.7) where (t) is the failure stress in an experient with duration t and n is a constant chosen as 6 fro experients. A large research project started in Lubbock, Texas, in the 97 to analyse window glass failures and to develop a rational tool for the design of windows. The first aspect was a consequence of glass failures in connection with tornados. It was not just high wind speeds but the debris following a tornado which could increase the flaw density of widows and ake the rupture at a lower wind speed than otherwise would be the case. The second aspect reflected a need for a new design guide for window glass. The increasing use of large glass windows in tall buildings ade this desirable. Existing design guides were not consistent and there had also been a transition fro plate glass to float glass with different strength characteristics. The desired final result was to obtain a failure probability for a given window size and thickness subject to certain wind load with inute duration. This ade it suitable with a statistical approach. To include a known size and tie dependence of the rupture load a Weibull distribution was chosen. The failure probability was initially expressed as 48 49 49

P f e B (4.8) where B is a function which reflects the risk of failure. The risk of failure is assued independent of flaw orientation for this case because a flaw of any orientation is exposed to the sae stress, (Beason, 98). The risk function is expressed as (4.9) B k( ) A ax where k is a noralizing constant and is a surface flaw paraeter, ax is the axiu equivalent principal tensile stress, and A = glass plate surface area exposed to tensile stress. In a ore advanced odel the failure probability is derived fro P f R(, q, a/ b) e (4.3) where accounts for a specific plate geoetry and load duration, and R(,q, a/b) is a nonlinear relationship, which cannot be stated explicitly, (Beason, 98). In reality the risk function can be evaluated fro B k ( ( x, y) da Area (4.3) where the integral can be substituted with a suation and the analysis can be perfored in a FEM progra which accounts both for bending stresses and ebrane stresses. The result fro a nuber of tests of old windows and one set of new float glass is presented in Table 4.. Direct coparisons of these data suggest that there are significant differences in the surface flaw characteristics of different glass saples. Values of aterial paraeter and noralizing constant k presented in Table 4. are unfortunately estiates based on the failure load and not the failure stress. This eans that the noralizing constant k will be copletely wrong if a stress analysis is desired. Only the value of the paraeter can be used. It is iportant to note that it is only the last result, New float glass, which reflects glass produced today. All other tests reflect sheet glass which are present in old buildings. There is a significant difference between these two types of glass. The low load capacity and value of sheet glass ay not necessarily be a consequence of the glass being used in - 5 years. It could siply be the characteristics of sheet glass which could not be produced with the sae high quality as float glass. For float glass where the failure is initiated fro the surface it is generally 49 5

accepted that the value should be in the range 7-9. The ost coonly used value is =8. For failures which are initiated fro an untreated edge the value will be reduced to roughly half of that for a surface failure. Table 4. Tests of old and new window glass. Location of window Size in Nuber of speciens Mean kpa Stand. dev kpa k Great Plains Life Building (GPL) in Lubbock, Texas 975; years of exposure of sheet glass 74x54x5, 56 74x74x5,5 6 3,78,88 6 6,33x -5 N -6 8,5,8 Johnson Chevrolet Building, Dallas 98; years of exposure of sheet glass 43x5x3, 8,99,97 6 3,x -5 N -6 Public School in Anton, Texas 98; 5 years of exposure of sheet glass 356x9x3, 8 3 6,43,6 5 9,6x -4 8 N -5 New Float Glass fro PPG in Lubbock, Texas 43x5x3, 8,49 3,7 9,3x - 6 N -9 4.5 Analysis of beas Structural eleents like beas can be analysed with the stochastic echanics approach. As a coparison the sae type of bea as was used in the fracture echanics approach will be utilized here. Also the loading conditions will be the sae with two syetric point loads. The analyses can be ade in several ways. Flaws exist both in the interior as inclusions of foreign aterial and as surface scratches. The iportance of such 5 5 5

flaws will depend on the tensile stress field around the flaw. The oents and corresponding stresses which act on the bea are shown in Figure 4. and are given below. Figure 4. The loading conditions for the bea. Px M M x x3 M x Px y I I y Px 6P y bh bh xy 3 3 3 Px P l e P M x l e x 4 M Pl e Pl e x 3 y y y 3 3 I 4bh bh where the indices, and 3 refer to the different sections indicated in Figure 4.. The axiu stress occurs between the two point loads where flaws will have the ost pronounced influence. Between the supports and the point loads the oent will increase fro zero to its axiu value. This will give flaws a lesser influence based on the oent curve. But still it will influence the failure probability. The volue under stress is obtained fro integrating the stress variation in space. First, the risk function will be B ( x, y, z) dxdydz b ( x, y) dxdy V M( x, y) y dxdy I where a derivation is perfored in Appendix B and the final result can be expressed as: 5 5

e 3PL ( e) B bhl bhl V... c L cbh P F( ) e B f e L An estiate of the value of k or c can be obtained fro the results given in (Carré, 996) which were presented in Chapter. For the lowest loading rate,,5 MPa/s the ean value rupture strength was 4, MPa for an ordinary surface treatent. The effective duration based on eq. (4.9) with h = 6 is 48 seconds. The noralization constant in eq. (4.) can now be deterined as c = 7,74 MPa for = MPa. With this choice of the value of c the tests carried out by (Carré, 996) will be established as a reference volue under stress with a corresponding ean value rupture stress. In this first exaple the following diensions are assued: L =, e =,5, h =,4 and b =,6. The external load is P = 6 kn. This will result in a stress of 4 MPa. Just to show the influence of different paraeters the width is altered, the distance between the loads and the load level. The results are presented later in Table 4.. Figure 4.3 Bea under evaluation. M P L e P Le 4 W bh 6 M P L W 4 bh e6 3P L e bh P 6kN 3 3 6 5,, 6, 4 4, MPa It can be seen fro the tables the probability of failure will decrease with increasing value of e. As the distance between the forces increase the oent decreases and hence the stress is decreasing. When the width is increased the bending stiffness increases which results in reduced failure probability. As it would 5 53 53

be expected when the load is increases so is the failure probability. Table 4. Results for failure analysis of beas where the failure probability has a volue dependence. P= 6 kn B P f e=,5 b=6,3 3,x -4 e=,5 b=8,3 3,x -4 e=, b=6,5,5x -4 e=, b=8,9,9x -4 e=,5 b=6,8,8x -4 e=,5 b=8,,x -4 P= kn B P f e=,5 b=6,575 437x -4 e=,5 b=8,767 74x -4 e=, b=6,374 37x -4 e=, b=8,499 49,8x -4 e=,5 b=6,4,4x -4 e=,5 b=8,73,73x -4 P= 8 kn B P f e=,5 b=6 4,73,9999 e=,5 b=8,966,86 e=, b=6,9579,66 e=, b=8,79, e=,5 b=6,54,53 e=,5 b=8,699,7 53 54

A fractile value can be obtained fro: P the fractile required e B f Often a lower 5% fractile value is used as a characteristic value. This is obtained for b=,6 ; e=,5 : P, 5 B, 53 f 8 B 4 6 5,,,, 7, 74, 8, 79 MPa An evaluation of the surface area under tensile stress can be evaluated as,, B x y dxdy x z dxdz A where the first part reflects the sides and the last part the botto side. This is shown and evaluated further in Appendix. For the first ter, which is doinating, the risk function is B A hl c e L Pl e B 3 ( ) ble eb hle h e ATOT bh ( )( ) ( ) ( ) ( ) c A B ATOT 3Pl ( e) bl ( e)( ) eb ( ) hle h e bh c ( ) ( ) A P e B ATOT B ln P f f ATOT 54 55 55

3Pl ( e) bl ( e)( ) eb ( ) hl ( e) h e ( ) ln( P ) bh c f A... c 35, 3 MPa A Exaple on the sae bea again but this tie with the area under stress: Table 4.3 Results for failure analysis of beas where the failure probability has a area dependence. P= 6 kn B ATOT P f e=,5 and b= 6 679,x-6 63,x -6 e=,5 and b=8 65,6x-6 65,6x -6 e=, and b=6 3,3x-6 3,3x -6 e=, and b=8,78x-6,78x -6 P= kn B ATOT P f e=,5 and b= 6,53 39,x -4 e=,5 and b=8,679 66,x -4 e=, and b=6 5,97x-3 57,7x -4 e=, and b=8 7,x-4 7,x -4 The result will be different when the volue under stress is copared with the area under stress. In the ean value sense the result will be siilar for the bea tests carried out in (Carré, 996) because this is the reference results used. When a larger bea is evaluated the scale factor will be different for the volue under stress 55 56

and the area under stress. Therefore it cannot be expected that the results of the failure probability are different. A coparison of the results in Table 4. or Table 4.3 show what can be expected. When the distance e increases the volue or area under stress will increase. At the sae tie the axiu oent will decrease faster and the resulting failure probability will decrease. Differences between Table 4. and Table 4.3 are a consequence of different scale factors for volue and area respectively. With a knowledge of the paraeters c and it is possible to establish the failure distribution for any type of bea. If the loading condition is a uniforly distributed load it is not possible to ake the calculations by hand. Instead the evaluation is ade with the help of soe suitable coputer software. The result in Table 4. is supposed to be the failure load for a sustained stress with a duration of 48 seconds, let s say approxiately inute. For longer durations the failure stress will be reduced. If it is assued that the tie paraeter n = 6, which is a value often found in the literature, this reduction can be evaluated fro eq. (4.6) for a constant stress. For a variable stress which can be expressed in atheatical ters eq. (4.7) can be used. In Table 4.4 are soe reduction values given for different durations of a constant stress that would cause a failure. Table 4.4 The reduction of the failure stress for different durations of a stress. Duration Relative reduction inute hour,996 day,635 year,439 years,38 years,39 As an exaple, the ean rupture stress for sustained stress over years will be reduced to: h 6 D Exaple years * * D D 6 39, 365 4 6 56 57 57

CONCLUSIONS 5. Introduction Glass is a aterial which is used in increasing ore coplex and deanding applications. This in spite of a lack of official codes which can give guidance about proper design procedures and aterial strength characteristics. Glass as a aterial has any characteristics in coon with ordinary structural aterials. But there are also differences which need attention. Glass will fail in a brittle anner which is considered unacceptable in ost applications. The use of glass in structural applications therefore need special attention concerning Design strategy/reliability Material strength characterization Design ethodology where each aspect can be analysed separately but there will be a connection between the. Design strategy and reliability has been discussed in connection with application in Chapter. Material strength characterization was presented in Chapter. Design ethodology as extensions to noral design was given in Chapters 3 and 4. A synthesis of this inforation is presented below. 5. Design strategy The oldest design principle is based on the safety factor principle. This eans that the allowable stress is deterined as the ean rupture stress divided by a safety factor. This design principle has been used since foral designs of structural ebers started during the 8 th century. The safety factor was initially chosen arbitrarily to be large enough. The first person who ade a scientific analysis of this safety factor was a French engineer, Vicat (88), at the French Road Adinistration. The background was the continuous increasing deflections of steel hangers in an arch bridge. He perfored creep tests where the applied stresses corresponded to 5%, 5% and 75% of the failure stress of a rod hanger. After a nuber of years he noticed that it was only for a stress of 5% of the failure stress that the increase of the deflection decreased with tie. For a stress level of 75% of the failure stress the deflection increased linearly in tie. For a stress level of 5% of the failure stress the deflection still increased with tie but not as fast as linear with tie. Results of this type and the consequences were not analysed and atheatically described until alost years later. In view of this it was natural for Vicat to draw the conclusion that it was not appropriate to use ore than soething between 5% and 5% of the rupture stress in an application. He suggested a safety factor of 3. Even if this experient reflects very special loading conditions with a sustained stress this proposal for a safety factor had a pronounced influence on the engineering profession. For this reason a safety factor of 3 has -57-58

been in use since then until the introduction of the statistically based design procedure. One of the ost well known designers in the field of glass is Ti McFarlane. In designs he uses an adissible design stress which is deterined fro an experient where the ean failure stress is divided with a safety factor of 3. This sees natural in view of the result of Vicat. The factor 3 is not a universal figure even if it sees as if it has been applied in this way. It has to be aterial dependent and as such specific to a aterial coposition and anufacturing. It ay also be influenced in tie by the environent. The result obtained by Vicat shows that there is a threshold stress between safe and unsafe use. This threshold stress reflects the dividing line between two different aterial responses. Below the threshold stress the initial deforation is followed by priary creep which can be seen as a delayed elastic response. Above the threshold stress the deforation will end in secondary creep which can be seen as a daage accuulation in ters of a shearing deforation between atoic planes. Secondary creep will eventually lead to a failure. This is a typical behaviour for etallic aterials subject to a constant stress. For a aterial which will fail in a sudden and brittle anner when it is overstressed there is a fundaental proble. But this is not unique for glass. Other aterials like concrete, steel and wood can fail in a siilar anner under certain circustances. The notion of a preferable ductile failure instead of a brittle failure is perhaps ore eotional than connected to reality. Ductile failures can be experienced in deforation controlled laboratory experients. Even an unreinforced concrete bea can exhibit large ductility in a very stiff testing achine. The final failure will be brittle. Steel subject to low teperatures or fast loading situations ay also fail in a brittle anner. In reality when a load exceeds the load carrying capacity there will be an instantaneous failure where it often is not possible to distinguish between ductile and brittle characteristics. It is coon knowledge that when glass fails it will rupture in a nuber of pieces which can be a hazard for people in the vicinity. This has to be considered in a design strategy. The rupture strength of glass has a pronounced size and tie dependence. This can be influenced by the environent where especially static fatigue play an iportant role. These dependencies are not unique for glass but they have a pronounced influence on the rupture strength. Therefore the influence need to be addressed. Fro the exaples in Chapter it can be seen that several strategies have been utilized in applications. Laination where two or three glass panes are glued together offers at least two advantages. When two glass panes are lainated together and a failure occurs in one pane it can be assued that the other pane will take over the load carrying capacity while the broken pieces fro the ruptured pane still will be glued to the unbroken pane. This will give tie to ake the necessary repair. If three panes are glued together the pane in the iddle will to a large extent -58-59 59

be protected fro surface daage. No or little deterioration with tie can be expected. If this is done for a bea the outer panes can have an overlap and the space in between can be filled a transparent sealant which will also protect the edge fro deterioration. In any applications annealed glass is substituted with heat treated glass. This introduces high copressive stresses in the surface region. As long as the externally introduced bending tensile stress is saller than the copressive residual stress it is ipossible for a crack growth fro the surface area. But a breakage is still possible because of the existence of ipurities in the region of high tensile stresses, (Jacob, 3). Such ipurities will act as stress racers and this can result in a delayed creep failure. Even if the ost dangerous ipurities are reoved during the production there is no guarantee that these glass plates will last forever. So also in this case a laination of two or three panes together is necessary. It can be expected that structural units of glass will not exhibit any creep with visible deforation before a failure. The creep phenoenon static fatigue will reduce the height of a section locally and increase the local stress. It is possible that this can be easured by high perforance fibre optical strain gauges. If this is possible a continuos surveillance could be used. There is no inforation available concerning this possibility. The present design strategy for noral construction aterials is that the strength of a aterial is characterized by a statistical distribution. A characteristic strength is defined as a lower 5% fractile value for a certain confidence level. The statistical distribution used is a noral statistical distribution out of convenience even if there is evidence that soe other statistical distribution akes ore sense. The confidence level reflects the influence of the saple size. In Figure 5. this is illustrated for a statistical distribution which is skew to the left. This is of siilar type as a Weibull distribution with = 8. For glass the strength properties with a size and tie dependence is well established. Because of this it is not equally iportant to use a confidence criteria. The lower 5% strength can be established with the ethodology presented in Chapter 4.5 with great accuracy. 5.3 Material strength The allowable strength of glass in structural applications is in general very low. This is understandable and it reflects a natural conservatis aong designers. This should also be seen in the light of the lack of long ter experience fro the structural use of glass. In annealed glass there are residual copressive surface stresses of around 8 MPa. The ean rupture strength can be expected to be around 4-75 Mpa,(Copagno, EN88). But this strength is easily reduced by a size and a tie dependence. For a bea subject to a uniforly distributed load the height has to be doubled if the length is doubled.if a bea length is doubled its height also has to be doubled to give the sae foral load carrying capacity according to the -59-6

theory of elasticity. The width is kept constant. But the size dependence will decrease the real rupture load to approxiately R = R o (/4) /8 ~,84 R o. f d f k n Figure 5. Definition of the lower 5% characteristic value. Sustained stresses will introduce a tie dependence of the rupture load. Such stresses will ainly occur for the dead load which can be a substantial part of the total load. If it is assued that in an ordinary test of a glass bea where the load is increased continuously fro zero to failure takes about 5 inutes the effective duration will be D o = 5/(6+) ~,3 inutes or, -4 days. For a structure with a design life of 5 years the reduction of the rupture stress will be R = R o {, -4 /(5 365)} /6 ~,3 R o. This indicates that only one third of the ean rupture stress, which will be in the range of 3-5 MPa, can be utilized when the tie dependence is considered. In reality there should also be a safety argin. With the exponent = 8 the coefficient of variability will be cov ~ /(6) ~,6. Based on the noral distribution a lower 5% fractile can be established as R k = ( - k 5 cov) = ( -,64,6),3 R o =,4 R o where only the tie dependence is considered. In reality it ay be necessary to include the size dependence as well. With the exaple given previously this would reduce the original ean failure stress to about one fifth of the original ean failure stress. This will result in a design stress of 8-9 MPa which should be divided with appropriate partial safety factors. The choice of safety factor, R, depends on well R k has been established and reliability class. In Sweden this is split up in n which reflect the consequences of -6-6 6

a failure with respect to huan life and which reflect the confidence in establishing R k. In view of the fact that R k will coe close to the level of residual stresses introduced during the anufacturing of annealed glass indicates that R d = R k /( n ) ~ 6-7 MPa for the dead load. For other loads with shorter duration a higher stress level can be acceptable. This load level can be established in a siilar anner as above. This is a siilar strength characterization as is used for wood. For heat strengthened glass it ay see reasonable to assue that R d should be close to the copressive stress introduced in the surface region. But this is far fro obvious. If foreign inclusions are present, which can be assued, there is likely to be a delayed failure if the internal copressive stress is too high. The knowledge of such failures is at present very liited. There is definitely a need for ore knowledge of a possible tie dependence of heat strengthened glass. It sees to be possible to odify the surface characteristics of glass and increase its initial strength without heat strengthening. Certain cheical copounds can react with the glass surface and reduce the flaw density. This ay turn out to be an alternative to heat strengthening in the future. 5.4 Design ethodology Glass behaves basically as an elastic aterial. This eans that the theory of elasticity is directly applicable for deterining stresses and strains. It can be expected that all calculations also in the future will be based on the theory of elasticity. This theory is derived for sall strains and corresponding stresses. Consequently this theory cannot be expected to give inforation about the failure stress. Inforation about the failure stress for structural aterials show that the failure stress is size, tie and teperature dependent. In soe cases there will also be other dependancies like a huidity dependence. It is therefore natural to expect that the failure stress for glass exhibit dependencies. These dependencies need to be addressed if they have a pronounced influence in the failure stress. Fracture echanics and stochastic echanics are ethodologies which can be seen as add on theories to the theory of elasticity to explain the failure behaviour of glass. Both theories have their advantages as well as drawbacks. This is natural since it cannot be expected that there is a universal solution which is acceptable to everyone,. In a coparison between fracture echanics and stochastic fracture echanics it will not be possible to present an absolute truth. The question is which theory gives adequate inforation about how dependencies can be accounted for in a proper way. Modern design philosophy is based on statistical principles where the goal is that the failure stress should reflect a certain failure probability. It is therefore an advantage if appropriate inforation concerning this issue also can be obtained. -6-6

Fracture echanics is a deterinistic ethod which gives the stress concentration at the tip of a crack as a function of geoetry and loading conditions. This can be copared with a critical value to establish whether a crack will grow or not. The basic theory of fracture echanics assues that a critical crack exist perpendicular to a tensile stress field. No explicit consideration of a possible size and tie dependence is included in the theory. Cracks in glass are easurable down to a certain level with appropriate equipent. The size and orientation of these flaws can be assued to be rando. For increasing areas which are analysed it can be expected that there will be an increase in size of these defects. This can of course be accounted for by aking flaw characteristics size dependent. It is very possible that this flaw inforation can be characterized with a statistical distribution, perhaps a Weibull distribution. With this technique it is possible to estiate the influence of a size dependence. The tie dependence can be introduced in a siilar way as the size dependence. This would reflect a growth of flaws with tie which will alter the flaw statistics characterization. Alternatively as was done in Chapter 3 where the tie dependence was introduced as an add on ter. This should ake it possible to establish estiates of a characteristic value which could be used for design purposes. In stochastic echanics it is initially assued that flaw characteristics are size and tie dependent and the result is that the failure stress can be characterized with a Weibull distribution. Besides the influence of these flaws depend on the total stress field within a body or acting on surfaces. The result is directly applicable to body shape and loading conditions. 5.5 Research trends and developent During the last 5 years or so there has been an increasing deand for using glass in structural applications. This also include structural glazing where new techniques have changed the noral function. Therefore glass anufacturers and other parties involved with the use of glass have been forced to iprove their knowledge. This is accopanied with the developent of new technique to ake the supporting structure to a glass surface as invisible as possible. Soe iplications are of this are presented below. Float glass have properties which see to be extreely good. At least when it is used as ordinary window panes. The possibility to heat strengthen glass akes it possible to use it in applications where fairly high tensile stresses are present. But the existence of ipurities within a glass body can cause a delayed failure. Glass can be surface treated. This is done on a regular basis to iprove the u- value. But is also possible to apply surface deposits to iprove the strength of glass. Experients done at Philips in the Netherlands have revealed that the short ter strength can be doubled with what sees to be fairly siple ethods. This can be an alternative to heat strengthening. This ight affect both the size and tie dependence and iprove the conditions for structural use of glass. -6-63 63

Glass panes with holes see to be increasingly ore popular. Theoretical solutions are available but it is not obvious that the basic assuptions are valid for a specific application. The available knowledge has priarily been obtained in connection with structural applications. The authorities have required full scale testing which has given inforation for certain conditions. There is a need for ore general knowledge of the influence of bezels of different aterials and its influence on the contact pressure. Glass which is glued against a steel frae and glass panes with holes will soeties be subject shear stresses in addition to bending stresses and ebrane stresses. This is stress conditions which have not been addressed in the analysis of glass. Many spectacular projects which have been undertaken over the last decade have in theselves given rise to several questions. It sees as if there is a desire to find new designs which require new solutions or the developent of existing solutions. This will also create new research concerning the structural use of glass. Above all there is a need for codes to address the question of how a proper design should be ade. Such a code should preferably be a European Code (EC code) to haronize designs within the coon arket. This ay also require CEN standards to establish strength characteristics of glass. Such a code and CEN standards will require ore knowledge of how glass behaves in different applications during both short ter and long ter loading conditions. -63-64

References Adasson, B. and Backan, H.E., 975. Glass in houses (in Swedish), Esselte Studiu. Beason, W.L. and Morgan, J.R, 98. Glass Failure Prediction Model, Texas Tech University. Carlsson, J., 98. Fracture echanics (Brottekanik in Swedish), Ingenjörsförlaget AB, Sweden. Charles, R. J., Nov 958. Static Fatigue of Glass I, Journal of Applied Physics, Vol 9, No, pp. 549-553. Charles, R. J., 958. Static Fatigue of Glass II, Journal of Applied Physics, Vol 9, No, pp. 559-56. Chaunac, M. and Serruys, F., 997. Glass as a Structural Eleent, Glass Processing Days. Dahlberg, T., and Ekberg, A.,. Failure fracture fatigue. Elinder, C. and Hellers, B.G., 994. Glas so konstruktionsaterial, Avdelningen för konstruktionslära, Arkitektur och stadsbyggnad, KTH. Fischer-Cripps A.C and Collins R.E, 995. Architectural Glazing: Design standards and failure odels, Building and environent, Vol. 3, No., pp. 9-4. Griffith, A.A., 9. The phenoena of rupture and flow in solids, Phil. Trans. Royal Society of London, Series A,. Gubel, E.J., 966. Statistics of extrees, Colubia University Press. Hassanzadeh, M., 99. Behaviour of fracrure process zones in concrete influenced by siultaneously applied noral and shear displaceents, Division of building aterials, Lund Institute of Technology. Irwin, G.R., 957. Analysis of stresses and strains near the end of a crack traversing a plate, Journal of Applied Mechanics. 65 65

Jacob, L., 997. Factors That Influence Spontaneous Failure in Therally Treated Glass-Nickel Sulphide, Glass Processing days. Jacob, L., 999. A New Model for the Design of Window Glass Plates Using Fracture Mechanics Concept, Glass Processing days. Jacob, L., 999. The designers perspective on NiS inclusions, Glass Processing Days. Jacob, L., 3. Nickel sulphide inclusions - iportant issues for the designer, Glass Processing Days. Jones, G.O., 94. The Interpretation of the Experiental data on the Strength of Glass, The Clarendon Laboratory, Oxford. Khorasani, N.,. Infästning av glasfasader, Bygg och teknik. Lawn, B., 995. Fracture of brittle solids, Cabridge Solid State Science Series. Porter, M., 99. The application of the Crack Size Design ethod to edge loaded structural glass ebers with corner cracks, The University of Oxford. Porter, M.I. and Houlsby G.T.,. Developent of crack size and liit state design ethods for edge-abraded glass ebers. The Structural Engineer, Vol. 79, No. 8, pp. 9-35. Minor, J. E., 974. Window Glass in Windstors, Civil Engineering Report. Minor, J. E., 98. Window Glass Design Practice: A Review, Journal of the Structural Division, vol. 7, No., pp. -. Norville, S.,. Siplified design procedure for blast resistance glazing, Glass Processing Days. Norville, S.,. Strain easureents in lainated glass, Glass Processing Days. Ontario Research Foundation, 98. National Research Council Canada, Ottawa, Ontario. Departent of glass and ceraics. Pilkington,Button, D., Colvin J., Cunliffe, J., Inan, C., Jackson, K., Lightfoot, G., Owens, P., Pye, B., Waldron, B., 993. Glass In Building. 66 66

Pittsburg Plate Glass, (PPG), 979. Glass Thickness Recoendations to Meet Architects Specified -Minute Wind Load, Flat Glass Division. Ritter, J.E., 985. Strength and fatigue paraeters for soda-lie glass, Glass Technology, Vol. 6, No. 6. Sentler, L., 987. A strength theory for viscoelstic aterials, Swedish Council for Building Research, Stockhol, D9. Sentler. L., 997. Reliability of high perforance fibre coposites, Reliability Engineering and Systes Safety, 56, pp. 49-56. Shand, E.B., 954. Experiental study of fracture of glass: II, Experiental data, Journal of the Aerican Ceraic Society. Sobek W and Kutterer M, 999. Designing with glass-strength and loadbearing behaviour in Glass Construction Manual, Birkhauser. Stansfield, K., 999. Glass as a structural engineering aterial, The Structural Engineer, Vol. 77, No. 9, pp. -4. Tioshenko, S.P. and Woinowsky-Krieger, S., 959. Theory of plates and shells, McGraw-Hill. Tioshenko, S.P. and Goodier, J.N., 97. Theory of elasticity, McGraw-Hill. Weibull, W., 939, A statistical theory of strength of aterials, Proceedings Royal Swedish Institution of Engineering Research, No. 53. Weibull, W., 95. A survey of statistical effects in the field of aterial failure, Applied Mechics Review, No. 5. Weissann, R., 997. Fundeental Properties of Float Glass Surfaces, Glass Processing Days. Wiederhorn, S.M., 967. Influence of Water Vapour on Crack Propagation in Soda- Lie Glass, Journal of the Aerican Ceraic Society, Vol 5, No 8, August. Wiederhorn, S.M., and Boltz L.H, 97. Stress corrosion and static fatigue of glass, Aerican Ceraic Society, pp. 543-548. 67 67

7 68 68

Appendix A A. General Stress concentration will appear if a local change of diension takes place for exaple when a notch or a crack is present. This eans that the stress close to the notch is larger, and in soe cases uch larger, than the noinal stress. The stress intensity factor, K, give a easure of the stress singularities at the crack tips. The factor f is a function of geoetry, (for exaple the crack length a) and the type of loading. In a fracture echanics experient, a cracked structure is loaded with an increasing force until fracture occurs. The value of the stress intensity factor when the fracture occurs is supposed to be a aterial property, i.e. the critical value of the stress intensity factor is considered a aterial paraeter. This paraeter is called the fracture toughness of the aterial. Depending on the conditions at the crack tip (plane stress or plane strain) different fracture toughnesses will be obtained. With a known fracture toughness of the aterial, a fracture criterion can be forulated. Under certain conditions, fracture is expected when K I IC t K a, IC 5 Y ( w a) K I K where K IC is the fracture toughness of the aterial. The above fracture criterion ay be used only when plane deforation (plane strain), exists at the crack tip. Other criterion that ust be fulfilled to ensure plane strain criterion are: K C where t is the thickness of the plate containing the crack, a is the crack length and (W-a) is the length of the area connecting the upper and lower part of the cracked structure. The stress Y is the yield strength of the aterial. The condition on plate thickness t is to ensure that the crack tip is loaded in plane strain. The plate ust be thick enough so that stress zz can be built up giving strain zz = at the crack tip. The conditions on a and (W-a) are to ensure that only the singular stress ters doinate at the crack tip. If the plate thickness t is very sall, so that the crack is loaded under plane stress conditions (i.e. zz =, xx = yz =), then a fracture criterion ay be forulated as: where K C is the fracture toughness of the aterial under plane stress conditions. Note that K C K IC. 7 69

K a f f w a a w a w a w a w I a w a w 5 5 3 75 37 tan cos,,, sin A. Edge crack in a plate in uniaxial tension The following relations are valid for uniaxial tension. 7 7

3 f a w M tw f a w w a a w a w a w 6 6 3 6 93 99 t is the thickness of the plate tan cos,, sin A.3 Edge crack in a plate in bending The following relations are valid for bending. K I a 7

A.4 Cracks at a hole in an infinite plate The following relations are valid for a hole in an infinite plate. K a f () s I 3 a s r a f () s, 53 s, 43 s 3 3 74 7 4

Appendix B B. General The weakest link theory of strength iplies that the strength of a large eber loaded in tension would be equal to the strength of the weakest of sall pieces cut fro the large eber. In the statistical strength theory, the cuulative frequency distribution of strength is assued to be a special type - one which lends itself to atheatical developent of equations for predicting average strengths and standard deviations of strengths. In equation for, the theoretical cuulative frequency curve is expressed as S e B where S is the theoretical cuulative frequency and B is given by B f ( ) dv which is the function of stress, f() integrated over the volue eleent dv on which it acts. The function of stress f() is a function expressing the local probability of failures. This is dependent on the local stress level and hence varies throughout the volue of a bending eber. B. Volue under stress in bending Figure B. The loading conditions for the bea. 73

B x y z dxdydz b x y dxdy M x y I y dxdy V (,, ) (, ) (, ) B b Pxy I dydx b Pxy bh dydx b P bh xy dydx b P bh x y dx b P bh h xdx b V h l e h l e h l e h h l e l e 3 3 3 3 6 6 6 6 6 / P bh h l e 3 / ( / ) B b Pl e bh y dydx b Pl e bh ydydx b Pl e bh h dx b Pl e bh h V h l e l e h l e l e l e l e 3 3 3 3 3 3 3 3 ( ) ( ) ( ) ( / ) ( ) ( / ) e B B B b P bh h L e b PL e bh h e b PL e bh h L e b PL e bh h VTOT V V 6 3 3 3 3 3 3 3 / ( ) / ( ) / ( ) ( ) / e b PL e bh h h L e h e 3 3 ( ) The volue under stress is given by: For the section and section 3 the volue under stress will be: For section the volue under stress will be: 76 74

3 Figure B. 3 3 3 3 PL e bh b h L e h e PL e bh b h L e e e PL e bh b hl e L e L e L PL e bh b hl e L ( ) ( ) ( ) ( ) B bhl e L PL e bh PL e bh c B bhl c e L 3 3 Further evaluation will siplify the obove expression to: B.3 Evaluation of results fro Carré (996) Test results fro Helène Carré: 75

B P e VTOT e BVTOT 5, e BVTOT 5, B c f VTOT B 8 VTOT 48, ln, 5, 3, 375, 9 c 8 695, 7, 74 MPa The exaple of bea in chapter 4: Figure B.3 Calculation techniques for failure probabilities: 4 B, 4, 6, 7, 74 P f Load stress, 8 B, 4, 6, 7, 74 P f, 46, 5, 8, 8 5, 8 55, 8 e 8 4 B, 4, 6, 7 74 8,, 36 P, 36 f Load 3 8 4 B, 4, 6, 7 74 3 4,, P, f 78 76 4

5 B x z dzdx M x I h dzdx Px I h dzdx Px bh h dzdx Px bh dzdx b P bh x b A b l e b l e b l e b l e b l e l e 3 3 3 3 * (, ) ( ) P bh l e b P bh l e 3 P B B f 5 53 4 6 7 74 3 79 8,,,,,,, MPa B x y dydx M x I y dydx Px I y dydx P I xdx ydx P I x y P I A h l e h l e h l e h l e l e h ** (, ( ) l e h P bh l e h P bh l e h P bh l e h 6 6 4 3 B.4 The area under stress The area under stress is evaluated for each section and area under tensile stress. For the sections and 3 this can be expressed as where the notation * refers to the the botto and ** refers one side. These expressions are valid for x (l - e)/ and (l + e)/ x l. The corresponding expressions for section are 77

le b 3Pl e B x y z dxdz e b A * (,, )... * l e bh B ( x, y, z) dxdy... ** A le h l e B B B A * TOT A* A * A ** TOT A ** A ** B B B B B B B ATOT ATOT * A ** TOT 3Pl ( e) bh c B ATOT A ATOT 3Pl ( e) h e bh3 where * refers to the botto and ** refers to a side. The expression is valid for (l - e)/ x (l + e)/. The total area under stress is derived fro This results in the following expression for the total area under stress bl ( e)( ) eb ( ) hl ( e) h e ( ) 3Pl ( e) bl ( e)( ) eb ( ) hl ( e) h e ( ) bh c A BATOT P e B ln P f ATOT f 3Pl ( e) bl ( e)( ) eb ( ) hl ( e) h e ( ) ln( P) f bh c A... c 35, 3 MPa A 8 78 6

B ATOT 3Pl ( e) bl ( e)( ) eb ( ) hl ( e) h e ( ) bh c A B ATOT 8 3 36 ( 5, ) 8, 6(, 5)( 8 ), 5, 6( 8 ), 4(, 5), 4, 5( 8 ) 6, 6, 4 35, 3 B ATOT 679, 4 BATOT (, ) P e e, 63 f 679 4 7 79

8 8