The Design Of Toughened Glass Columns Physical Education Essay
Glass has clinched the attention of engineers and architects in recent years despite its brittle nature. As well as its aesthetic allure, the increasing knowledge of this material allows structural boundaries to be pushed and endless possibilities to arise. Glass facades, beams, and stairs are some of the structures that have been explored with. However, the prospect of using a glass column as a structural component, as opposed to an ornamental role, is of increasing interest. Various structural glass types are available and have proven to be sufficient, but toughened glass reigns as the strongest type, yet its capability to spontaneously shatter can also prove to be a liability.
What is Glass?
Glass is an inorganic, non-crystalline, solid, transparent material, renowned for its brittleness. Its molecular structure attributes to its brittleness, making it weak in tension (Chen; Lui, Ch.29) and without an ability to redistribute load or absorb impact energy. Potentially, glass is very strong, even exceeding the strength of structural steel. However, due to glass having fairly low fracture toughness, this is only achievable when glass contains no defects, “as freshly drawn fibre might be” (Rice; Dutton, 33).
Glass does not yield, instead it fractures and its failure is stochastic, meaning that prediction for failure is based upon risk or statistics (ISE, 11). Glass does not adhere rigorously to stoichiometry as crystalline materials do, due to the ability to incrementally alter the properties of glass continuously by adding components/substances to modify its properties. For instance, adding potassium oxide to silica will change the glass properties (Clare, Ch.23). Although described as a solid, glass is rather a subset of the solid state. It is essentially an elastic solid below its transformation region, i.e. the glass transition state, and a liquid above it, glass has the attributes of a liquid apart from the ability to flow (Clare, Ch.23). There is a range of intricacies inherent within glass composition; this will later prove to greatly explain the complexities of glass behaviour.
A Look at Glass Chemical and Physical Composition
Unlike many other materials, glass consists of a geometrically irregular network of silicon and oxygen atoms, with alkaline parts in between (fig.1.4) (Haldimann; Luible; Overend, 4). As glass is an inorganic product of fusion, it consists of a number of chemical components. The chemical composition of glass has a significant impact on glass viscosity, the melting temperature Ts and the thermal expansion coefficient Î±T of glass (Haldimann; Luible; Overend, 4). One of the main attributes of glass is its resistance to corrosion by acid and water (Chen; Lui, Ch. 29). There is a vast variety of different types of glass, however, the most prevalent type of glass in construction (approximately 90%), is soda lime silica glass (Dewhurst Macfarlane and Partners), and for other special applications, borosilicate glass is used. However, depending on the purpose for the glass, other types are available, such as Lead glass, borosilicate glass, glass fibre, vitreous silica, alminosilicate glass, alkali-barium silicate glass, technical glass, glass ceramics, optical glass and sealing glass, to name but a few (Glass Online).
Borosilicate glass consists of mainly silica (70-80%) and boric oxide (7-13%) with small amounts of the alkalis (sodium and potassium oxides) and aluminium oxide. Borosilicate glass has a considerably low alkali content, and hence, has an appreciable level of chemical durability and shock resistance (Glass Online).
The chemical components of soda lime glass are as follows:
“70% – 74% SiO2 (silica)
12% – 16% Na2O (sodium oxide)
5% – 11% CaO (calcium oxide)
1% – 3% MgO (magnesium oxide)
1% – 3% Al2O3 (aluminium oxide)”
Regardless of the specific type, the main constituent of glass is silica sand (Chen; Lui, Ch. 29). Sand alone can be used to make glass at a temperature of 1700oC; but the addition of other minerals and chemicals significantly lowers the melting temperature (Glass Online). For instance, the melting temperature for pure silica is approximately 1710oC, but it drops to 1300-1600oC through the addition of alkali (Haldimann; Luible; Overend, 4). Glass consists of a network formers and modifiers Fig. 29.1. Silicon and oxygen ions are bonded together (formers) forming a three dimensional structural network of sodium, potassium, calcium and magnesium (modifiers) ions (Chen; Lui, Ch. 29). Sodium carbonate (Na2CO3), known as soda ash, is added to create a mixture of 75% silica (SiO2) and 25% of sodium oxide (Na2O), which will reduce the temperature of fusion to about 800oC (Glass Online). However, this concoction means the glass is produces water glass, meaning it is water-soluble. To give the glass stability, chemicals such as (CaO) and magnesium oxide (MgO) are added; which is achieved by adding limestone, which results in a pure inert glass (Glass Online).
The viscosity of the liquid glass during the cooling phase increases constantly until solidification is achieved at about 1014Pas. The temperature at solidification is called glass transition temperature Tg and is about 530oC for soda lime silica glass. Contrasting crystalline materials, the transition between liquid and solid states occurs over a particular temperature range, instead of a single precise temperature (Fig. 1.5, Table 1.3). Small amounts of iron oxides are responsible for the greenish colour of soda lime silica glass. A reduced iron oxide content results in an extra – clear glass, which is known as low iron glass, and is readily available (Haldimann; Luible; Overend, 6).
Essentially, the composition of glass varies to appease a particular product and production method, which requires the raw materials to be weighed and mixed properly; as the consistency of the composition is vital in glass production (Glass Online).
The essential physical properties of soda lime silica and borosilicate glass are summarised in Table 1.5. Optical properties depend on the thickness of the glass, the chemical composition, and the applied coatings. The most prominent of the glass properties, is its very high transparency within the visible range of wavelengths (Î»= 380-750nm). However, for different glass types, the exact profiles of non-transmitted radiation spectrum differ, but are in the wavelengths outside and near the infrared band (Fig. 1.6). A large percentage of UV radiation is absorbed as a result of O2 reaction in the glass, but long-wave infrared radiation (Î» The Making of Glass
Essentially, glass is produced by rapid melt quenching of raw materials (reference); there are currently various methods by which glass is produced. The float process Fig 1.1 is the most practiced glass production method used today, which produces flat glass, attributing to 90% of the production worldwide. Although the steps vary, it is simply melting at 1600-1800oC, forming at 800-1600oC and cooling at 100-800 oC Haldimann; Luible; Overend, 1). The Pilkington Brothers introduced the float process in 1959. It has several advantages, such as low cost production, vast availability, superior optical quality, and allows for large stable glass panes to be manufactured. The production process is shown in Fig 1.2. Glass is produced by melted raw materials in a furnace at 1550oC. Subsequently, the molten glass is poured continuously at 1000oC on to a “shallow pool of molten tin whose oxidation is prevented by inert atmosphere consisting of hydrogen and nitrogen.” (Haldimann; Luible; Overend, 2). The glass floats onto the tine and spreads forming a smooth flat surface, with an even thickness of 6-7mm; it gradually cools and is drawn onto rollers, then entered into a long oven called a lehr that is heated at a temperature of 600oC. The thickness of the glass can be controlled within a range of 2-25mm, through adjusting the speed of the roller, whereby, reducing the speed increases the glass thickness. The glass is slowly cooled to prevent residual stresses being induced in the glass, after annealing, automated machines inspect the glass to check for obvious defects and imperfections. The glass can then be cut to a standard size of 3.21mÃ- 6.00m, and stored. A disadvantage that arises from this method is that there is a discrepancy between the two faces of a glass sheet. Apparent diffusion of tin atoms into the glass surface occurs on the tin side, which could influence the behaviour of this surface when it is glued. The mechanical strength on the air side is greater than on the tin side, which occurs because of the transport rollers interacting with the tin side in the cooling area. This interaction with the rollers can reduce the strength of the glass as it can create surface flaws (Haldimann; Luible; Overend, 2).
Fracture Mechanics in Glass
As glass is unable to yield plastically ahead of fracture results in the fracture strength being highly sensitive to stress concentrations. To achieve accurate characterisation of the facture strength of glass, the nature and behaviour of the flaws must be integrated, as a result of surface flaws causing high stress concentrations (Haldimann; Luible; Overend, 49). The stress of glass is time dependent; yet, humidity causes stress corrosion resulting in flaws slowly growing when bared to a “positive crack opening stress”. Essentially, this is when a glass element is stressed below its momentary strength, fails after the time needed for the critical flaw to grow to its critical size (Haldimann; Luible; Overend, 49). These flaws are either inherent in the glass or a result of cutting, drilling, grinding, or an impact from the environment; humidity heightens the growth of cracks. Due to the cut edges in annealed glass being weaker than its flat surfaces, annealed glass beams are designed with lower stresses than glass plates. The onset of fast fracture is represented by this general equation:
Ïƒâˆš (Ï€a)= âˆš (EGc) (ISE, 57)
Where a is the half length of the crack, E is the Young’s Modulus and Gc is the toughness of the glass Gc has units of kJ/m2 and is the toughness of the glass, sometimes known as the critical strain energy release rate. The equation shows that fast fracture will happen when “in a material subjected to a stress Ïƒ, a crack reaches some critical size a or alternatively, when material containing cracks of size a is subjected to some critical stress Ïƒ.” This is a mathematical representation of the trend in annealed glass to be stronger under short-term loading rather than long term. The purpose of glass modification processes such as toughening and heat strengthening is to prevent glass from experiencing tension in surface to avoid crack growth, so that fracture mechanics calculations need not be considered Fig 5.3.
Professor Inglis (1913) discovered that a slot, hole, or notch in a metal plate was likely to reduce strength by a greater value than that predicted from simply considering the reduction in tensile area. It was proven that the stress field near the discontinuity is exaggerated by an amount that is reliant upon the radius of curvature relative to its length perpendicular to the stress field. The discontinuities or randomly distributed flaws across the surface are known as Griffith Flaws. Griffith flaws are apparent on the surface of glass, but the strength of the glass is coinciding with the presence of visible defects, which is usually the origin of the cracks that occur under an applied tensile stress. Accidental contact can damage the edges of a glass plate more significantly than any other region of the glass. The deflection or bending of the glass is usually able to absorb the energy from an impact on a glass face but an edge impact is resisted by the “full in-plane stiffness of the glass plate or beam” and produces greater damage impulse. Once load is applied, stresses develop and concentrate at the tips of flaws or cracks, which usually go undetected by the naked eye. Griffith claimed that crack propagation occurs if energy release on crack growth is adequate enough to supply all the energy that is needed for the growth of cracks. Mathematically this is stated as
Where Ïƒc is the stress required to fracture a plate with a crack of length 2a, E is Young’s Modulus and Gc is the critical elastic energy release ratio or toughness of the glass, with units of energy per unit plate thickness and per unit crack extension.
This expression signifies the occurrence of fast fracture when a material is under stress that results in a crack of the size a. It is maintained by some that glass is able to reverse crack damage, i.e. heal a microcrack, if it reverts back to an unstressed state. On the other hand, the surface condition of glass sheet alters each time it is cleaned due to new microcracks surfacing. Therefore, the notion of damage reversal is up to the engineer to decide whether it is reliable in design (ISE, 57).
Over time, momentary strength of loaded glass decreases, even if only subjected to static loads. This is a quintessential concept to grasp for the structural use of glass, and was demonstrated by Grenet (1899). Flaw and glass properties, stress history and the crack velocity-stress intensity relationship govern the growth of a surface flaw (Haldimann; Luible; Overend, 50).
Structural Behaviour and Failure Characteristics of Glass
Upon failure, glass does not yield, it fractures, and the failure is stochastic, meaning that the predicted failure is based on risk or statistical analysis (ISE, 11). However, glass is very strong, even stronger than steel. But the inherent low fracture toughness means that this optimum level of strength is only achievable when the glass is free from all defects. Ultimately, glass is brittle, without the ability to redistribute load or absorb energy (Rice; Dutton, 33). Due to the brittle nature of glass, it is important for the designer to have an insight into how the structure will behave if one or more of the glass elements fail; most importantly the safety implications should be assessed (ISE, 55). Fig 5.1
At low stress levels, the majority of materials tend to abide by Hook’s law, in that stress and strain are proportional. Yet, a higher stress levels the material deforms plastically, but as glass is a brittle material, it simply fractures without warning instead. The mechanical properties of glass are displayed in Table 29.1 (Chen; Lui, Ch.29). The theoretical strength of glass is usually approximately a tenth of it elastic modulus.
The density of the cracks rather than the theoretical breakage stress governs the failure stress of glass, whereby; glass compressive strength can reach a value of 10,000MPa, demonstrating that whilst in compression it is very strong. Conversely, in tension it fails, and this usually occurs when stress levels are less than 100MPa. It is the general consensus that glass failure originates from crack growth and surface flaws, where the stress is concentrated, as demonstrated in Fig 29.5 (Chen; Lui, Ch.29).
To gain scope of how differently glass behaves relative to the most commonly used construction material, steel, is to observe the behaviour displayed in stress-strain curve:
Glass molecular structure influences its mechanical properties, particularly its random irregular network of silicon and oxygen atoms. Its structure allows for no slip planes or dislocations so that macroscopic plastic flow transpires before fracture (Haldimann; Luible; Overend, 49). Glass failure is most likely to be initiated by surface cracks, because these tend to have the worst geometries and are subjected to the highest stresses due to bending. If the loads to which the glass is subjected do not create enough surface tension to overcome the surface compression, no crack will propagate. Toughening, therefore, increases the effective strength and impact resistance of the glass. Should an external load ‘overcome’ the precompression and cause a crack to propagated, then the stored energy due to prestress will cause the cracks to spread immediately in all directions and the pane of the glass will fragment explosively (Rice; Dutton, 33). Static fatigue of glass, also known as sub-critical crack growth is a phenomenon of glass. An applied sub critical stress causes cracks of flaws to slowly grow with time, until a length is reached, at this point the stress intensity at the crack tip reaches a critical value. Consequently, rapid fracture occurs due to the highly strained atomic bonds swiftly breaking at the crack tip. ‘Stress corrosion’ is a term used to describe the relationship between the crack growth velocity and the stress intensity factor. Apart from applied stress, there are a number of factors that hasten slow crack growth, such as alkaline solutions and increasing temperature (ISE, 56).
Plastic flow is not possible in glass, therefore when the glass surface is in a state of tension; the flaws produce high stress concentrations. The flaws are random and can take any path; therefore the failure strength can only be determined through statistical analysis. Therefore, the basis of risk of fracture of glass that is determined does not give assurance that the glass can withstand the designed load. Strength of glass relies on the load duration and environmental conditions; Fig 29.6 shows the strength-time relationship (Chen; Lui, Ch. 29). The time to failure and applied stress relationship is expressed mathematically as
Where Ïƒ is stress T is duration and n is a constant (ISE, 56).
The value of n varies, and Sedlack (1995) as well as Pilkington Glass Consultants recommend n = 16 for design purposes. This equation suggests that loads applied at an exceedingly long duration will allow allowable stresses to decrease to insignificant values. However, in reality, this is not true (ISE, 56). Unlike steel that yields and flows when locally overstressed, glass breaks when it is overstressed. For that reason, it is vital that the designer attempts to eradicate possible design features that may result in stress concentrations. Such as bolted glass has been developed in such a way that, stress concentrations are avoided around the bolts; this attention to detail cannot be readily detected (ISE, 58). To avoid force being transmitted from glass to another material, as this causes stress concentrations to develop; soft setting blocks, fibre gaskets, and protective brushes have been implemented to limit this (ISE, 58). Glass is almost perfectly elastic, linear and is isotropic, and is not subjected to fatigue (Haldimann; Luible; Overend, 8). Glass only fails by brittle fracture, and cycling loading can cause the growth of cracks. Most materials have a fatigue limit, whereby there is stress amplitude where facture does not happen or fracture only happens after a great number of cycles (>108). Additionally, although many materials have a fatigue ratio, which is the ratio of the fatigue limit to yield strength, but since glass does not yield, this attribute is obsolete (ISE, 58). As glass fails in tension or by buckling, the highest tensile stresses that occur from applied loads should be considered when finding the elastic stability of glass element. Applied compressive stresses can cause tensile strains, but tensile strains can even occur as a result of the Poisson’s effect from compressive stresses (ISE, 60). Glass failure occurs when the tensile stress is equal or greater than the characteristic strength, which can be calculated using Eqn 29.5. The membrane stress is constant across the thickness of the plate, whilst the bending stress can be taken as varying linearly. Thus, superimposing the membrane and bending stresses can determine the total stress on the glass (Chen; Lui, Ch. 29).
Furthermore, the deflection of glass elements is an important aspect to consider; and such behavioural patterns like toughened glass deflecting more than annealed glass (even when of the same strength) due to toughened glass being considerably thinner, should be taken into consideration (ISE, 56). Glass plates are typically thin so they demonstrate large displacements. The use of ‘thin plate linear bending theory’ will produce incorrect results. Therefore, the large deflection theory should be used instead to calculate the maximum stress when checking stress against failure. Failure generally is taken to be at the point when the maximum tensile stress equals the glass fracture stress (Chen; Lui, Ch.29).
Glass can be quite sensitive to any impact and will result in fracture; the common causes of glass breakage are:
‘Excessive stress form wind pressure or other loads
Thermal stress due to differential temperature on different parts of the pane
Buckling due to large compression
Surface or edge damage
Deep scratches or gouges
Severe weld splatter
Direct contact with metal (e.g. window aluminium frame)
Impurities like nickel sulphide (NiS)
Excessive deflection bringing glass in contact with other hard objects.’ (Chen; Lui, Ch. 29)
Hence, the strength of glass relies on these aspects: the duration of the applied load, environmental conditions, humidity, size of the stressed area, the distribution of stresses across the stressed area, the condition of the surfaces and edges of the glass (ISE, 57). Prestressing glass, notably by heat-strengthened and toughened are the two basic types, enables the glass to maintain compression on the surface, therefore, eliminating crack propagation (ISE, 59). Survival probability of scratched glass loaded at a constant rate Eqn; Time dependence of glass strength Eqn Fig 5.4 Fig 5.5 Fig 5.6
The Different Types of Structural Glass
Glass, itself, is highly susceptible to fracture, which results in a lot of shattered glass, and ultimately, health and safety implications. The fracture of glass stems from the surface flaws. Thus, the industry has developed various modification methods to achieve an increase in the practical strength of glass, by introducing local high compressive stresses near its surfaces (Chen; Lui, Ch.29). By common practice, these modifications are usually implemented on float glass.
Tinted glass is also known as heat-absorbing glass, and is produced by colorant being added to normal clear glass. Light transmittance varies depending on colour and thickness, with a range between 14 to 85%. As a result, tinted glass is not and heat-strengthened glass is typically used when making tinted glass (Chen; Lui, Ch.29).
Placing layers of coating onto a glass surface makes coated glass, and there are two types: the solar control (reflective) and the low emissivity types. Structural strength of coated glass is only indirectly affected when the thermal stress is altered, but coated glass is more associated with its energy absorption and light transmission attributes. Therefore, to prevent excessive thermal stress, heat-strengthened glass should be used to produce coated glass (Chen; Lui, Ch.29).
A common misperception is that wired glass is stronger than unmodified annealed glass, due to the wires being seen as reinforcement. However, the wires actually induce cracks and weaken the glass. Yet, wired glass is able to hold together upon being broken (ISE, 22). Wired glass is produced when a steel mesh is implemented onto the molten glass during the rolling process (the rolling portion of the flat glass process). It has a high rate of breakage due to sunlight, and hence is weak in resisting thermal stress. Although it is still weak in resisting thermal stress, polished wired glass is used for fire rating since after it breaks, it sticks to the wire mesh and prevents smoke passing. Figure 29.8 shows a damaged wired glass panel under sunlight (Chen; Lui, Ch.29).
Annealed glass panels do not have any heat treatment (Chen; Lui, Ch.29); it is produced using the float process (as described previously) (ISE, 22). It is usually used when large glass panels need to be used, and it is too large for any heat treatment (Chen; Lui, Ch.29). The behaviour of annealed glass is typically perfectly elastic until fracture occurs. Upon fracture, large, sharp shards emerge which are dangerous. However, annealed glass panes do not spontaneously fracture, and due to alternate load paths across the glass pane, it may not fall out of its frame upon failure. Although there is no creep or fatigue in the metallurgical sense, slow crack growth occurs as a result of cyclical loading, whereby, if this glass is under permanent loading, the deformation increases with below 3% over a 50-year period. Imposed strains, such as bending and thermal stresses, as well as instant impact, causes elastic deformation resulting in brittle fracture of annealed glass (ISE, 22). Annealed glass is not very strong, so it is weak in thermal resistance. The allowable stress is approximately 15N/mm2 (Chen; Lui, Ch.29). Fig 2.4 Fig 2.5 Fig 2.6
Heat-strengthened glass is created using a similar process to toughening, with the exceptions that there is a lower cooling rate (Haldimann; Luible; Overend, 12) and the level of the produced prestress is lower. The fracture behaviour, however, is more akin to that of annealed glass rather than toughened (ISE, 24), with larger fragments than that of thermally toughened glass (Haldimann; Luible; Overend, 12). The compressive surface stress for heat-strengthened glass lies in a range between 24 and 69N/mm2 and European Standards quote that the pattern of breakage ranges between 25 to 40N/mm2 (ISE, 24). Heat-strengthened glass is commonly used in laminated glass assemblies, but the nature of its large fracture pattern causes a significant remaining load-bearing capacity upon failure of the glass. The stress gradient depends on the thickness of the glass and as the glass must be cooled down gradually, thus, thick glasses (exceeding 12mm) cannot be heat-strengthened using the toughening process (Haldimann; Luible; Overend, 12).
Laminated glass is two or more glass panes bonded with an interlayer of polyvinyl butyral (PVB) or resins, such as acrylic. The thickness of the interlayer varies between 0.4mm to 6mm. A disadvantage of laminated glass is the validity of composite action. Although usually only two layers are bonded, over 25 layers have been effectively bonded coming at 100mm thick. Laminates can integrate many thicknesses and arrangements to suit a certain requirement. Most importantly, many different types of structural glass can be arranged in the laminated formation, including toughened, annealed, heat-strengthened and bent glass for example. However, toughened and heat-strengthened glasses both cause small amplitude waves as a result of the rollers used in the process. This in turn, enhances the separation between the laminated glasses and ultimately the PVB is impractical. Therefore, resin laminating should be instead. When using a PVB interlayer, the sheets of glass have the PVB interjected between them and then this sandwich travels through an oven of about 70oC, and then passes between rollers which squeeze out the excess air from the bonding. The laminated glass is then placed in an autoclave, heated at 140 oC and at a pressure of 0.8N/mm2. It is possible to manufacture laminated glass at a maximum of 6m by 3m. In resin laminating, the two principal resins are acrylic and polyester. The glass sheets are held together at a right distance apart using double-sided tape around the perimeter. The resin can then be poured in between the two sheets, and once the air has been extracted the open edge can be sealed, and the laminate is stored horizontally to allow the resin to cure and solidify. The curing occurs through UV light or chemical reaction. The size that can be manufactured using this method is dependable on available glass pane sizes (ISE, 24).
The structural behaviour of the laminated glass varies, depending on the duration of the load. Hooper (1973) demonstrated that the duration of the loads affected the behaviour of the laminate. With short-term loads the laminate acted compositely, whilst with long-term loads, the load was shared between the two glass sheets, in proportion to their relative stiffness’s, as a result of the deformation of the interlayer (ISE, 24). To determine this behaviour, the deflection of the panel under a specific load should be measured and then compared to the deflection calculated using finite element software. This would allow for the equivalent thickness used in the software to be adjusted to give the same deflection measured, in order to determine the equivalent thickness of the laminated glass pane that should be used for optimum design (Chen; Lui, Ch.29). An increase in the temperature, results in the interlayer softening and a reduction in the composite behaviour. Laminated glass is highly valuable as it offers various performance benefits. For instance, if one or both of the layers are impacted and breaks, the interlayer prevents penetration and allows any broken glass to stay bonded to the interlayer. Additionally, an increase in the thickness of the interlayer increases the penetration resistance of the glass (ISE, 24).
Fig 29.9 Displays laminated glass behaviour once broken (Chen; Lui, Ch.29).
Chemically Toughened Glass:
Chemically toughened glass implements the principle of a compressive surface layer preventing crack propagation, where the compressive layer is a result from an ion exchange process. Therefore, flat glass that contains sodium ions is immersed in a molten salt bath (electrolysis baths (ISE, 23)), of potassium nitrate. As the temperature of the molten salt is insufficient to permit structural relaxation, the potassium ions force themselves into the sodium sites, consequently, putting the surface under compression (Clare, Ch.23). Although it is an advantage that unlike thermal toughening, thinner glass sheets can be toughened, it results in thinner compressive layers, which are less robust than the thicker layer created through thermal toughening (ISE, 23). Also, the strength of glass can be increased by ten times depending on glass composition (Clare, Ch.23).
Thermally Toughened Glass:
Thermal toughening of glass is achieved by heating annealed (float) glass plate to about 620-650oC, whereby it begins to soften at this point (ISE, 23). The outer surfaces are then cooled rapidly by cooled air blasts, and the exterior layers quickly cooled and contracted. A thin layer of high compress stress the surface occurs, with a region of tensile stress at the centre of the glass (Fig 29.7). The parabola represents the stress distribution across the thickness of the glass pate, which is also in self-equilibrium. The physical properties of the particular glass used and the geometric shape of the glass governs the exact shape of the curve. Toughened glass has a bending strength is three to five times