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Thermal stresses on window glasses upon heating
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Construction and Building
MATERIALS
Construction and Building Materials 22 (2008) 2157–2164
www.elsevier.com/locate/conbuildmat
Thermal stresses on window glasses upon heating W.K. Chow a
a,*
, Y. Gao
b
Research Centre for Fire Engineering, Department of Building Services Engineering, Area of Strength: Fire Safety Engineering, The Hong Kong Polytechnic University, Hunghom, Kowloon, Hong Kong, China b Department of Building Engineering, Harbin Engineering University, Harbin, Heilongjiang, China Received 25 January 2007; received in revised form 14 August 2007; accepted 12 September 2007 Available online 26 November 2007
Abstract There are concerns about the fire safety of the glass panes in curtain walled buildings with green or sustainable design. Thermal stresses over the glass panes upon uneven heating in a fire should be investigated and were discussed in this paper. Different temperature fields and different boundary conditions of the window frame were studied. In contrast to those literature works, temperature and thermal load were not taken to be uniformly distributed over the whole glass pane. As thermal stresses built up in the glass depends on the surface temperature, surface temperature profiles have to be known. Correlation expressions on the varying temperature distribution were deduced from reported experimental data. The derived results were applied to predict the positions where cracking is likely to occur in the glass construction during a building fire. 2007 Elsevier Ltd. All rights reserved. Keywords: Thermal stresses; Glass; Fire
1. Introduction Glass facades are commonly found in many green or sustainable buildings [1] and big airport terminals [2]. In addition to provide better visual effects, glass facades allow more daylighting and higher solar gain for reducing the heating load in cold countries. However, there are concerns on fire safety in those buildings with glass construction. Cracking and collapse of window panes are the identified risks upon heating up the glass facade in a fire [1,3]. At the early stage of a flashover room fire, the burning object is small and might be at a distance from the windows. Thermal radiation emitted from the flames is absorbed slowly by the glass pane to increase its temperature gradually. It takes time to increase the temperature along the direction of thickness. There is no thermal bending, with thermal stresses built up only along the plane. But in post-flashover room fires, the flame acts directly on the glass surface. A large temperature difference between the *
Corresponding author. Tel.: +852 2766 5843; fax: +852 2765 7198. E-mail address: [email protected] (W.K. Chow).
0950-0618/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.conbuildmat.2007.09.004
inner and outer surfaces of the glass pane results. Expansion of the glass surface nearer to the flame leads to thermal bending along the direction of thickness. Thermal stresses on the exposed surface then increase. Thermal stresses over glass panes were estimated [4–7] with thermal load assumed to be distributed uniformly over the whole pane. However, experiments by Hassani et al. [8] suggested that the glass in a room fire is exposed to a homogeneous thermal environment. The first crack in the window glass always occurs at positions near the hot gas layer. This is because the maximum tensile stress is located in the shaded region of the glass. Locations with higher stresses should be further studied and illustrating this point is the objective of this paper. In contrast to the literature works, temperature and thermal load are not taken to be distributed uniformly over the pane. Experimental data available in the literature was reviewed for studying both the compressive and tensile stresses at those glass edges at elevated temperatures. As variation of the temperature field of the glass pane along its height in a building fire would give thermal stresses, it is important to know the temperature profiles along the
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W.K. Chow, Y. Gao / Construction and Building Materials 22 (2008) 2157–2164
Nomenclature a A t u, v rx, ry rxy rxb rcr E NT Nx, Ny Nxy
thermal expansion coefficient extentional stiffness Possion’s ratio components of displacement normal components of stress to x and y axes twisting stress bending thermal stress critical breaking stress Young’s modulus of elasticity in tension and compression thermal force per unit length forces per unit length force to give torsional effect
bending moment per unit length bending moment of the forces distributed over a cross section Hmn, Amn, Bmn Fourier coefficient m, n Fourier coefficient subscripts T temperature t time h thickness of the glass pane a length in x direction b length in y direction x tangential direction coordinate y normal direction coordinate Mx Ix
height under a buoyant flame for estimating the thermal stresses. Typical experimental results by Hassani et al. [8] are applied in this paper. The glass temperature measured at a particular time is taken to be the same along the direction of thickness. Increase in temperature profiles along the height of the glass pane is studied. The reference temperature is taken as the ambient temperature and assumed to be uniform throughout the glass. 2. Correlation expressions of experimental results Correlation expressions on glass temperature are derived based on the experiment by Hassani et al. [8]. Glazing units of size 905 mm by 1615 mm fixed in softwood frames of size 988 mm by 1690 mm under a wood crib fire were studied in a room model. Six thermocouples were installed with three pairs at three different heights. One thermocouple was exposed to the fire and the other one was shaded by a wooden window frame without contacting with the flame directly. The effects of buoyant flame on increasing the glass surface temperature at different heights were reported. Typical glass surface temperature profiles T at different heights with time t are shown in Fig. 1. The vertical temperature profiles of the exposed glass surface with height y (with respect to the glass height b) at 5 min, 15 min and 20 min are shown in Fig. 2. Positions of cracks on the glass panes were also recorded [8]. Most of the cracks were found near to the window frame with the first crack appeared at 5 min. Result suggested that the critical thermal stresses required to give cracking would be near to the interfaces of the hot gas and the cooler region shaded by the window frame. As observed from the results shown in Figs. 1 and 2, it is not too satisfactory to assume that the glass surface temperature exposed to fire is uniform. A large change in temperature difference with height is expected. Actually, variation of temperature with height can be fitted from the experimental data on the surface temperature profiles of glass.
Fig. 1. Glass surface temperature profiles at different heights from Hassani et al. [8].
Results are applied to study the vertical distribution of thermal stresses on the glass surface. The positions where cracking or fallout is likely to occur can then be determined. The calculation procedure is demonstrated in this paper. Analyzing typical experimental results as in Fig. 2 would give correlation expressions on temperature T with height y (in terms of the length of glass b along the y direction) at a given time t as (i) t = 5 min 8 < 30 T ¼ : 290 y 0:5 þ 30 b
06y
ð1Þ
W.K. Chow, Y. Gao / Construction and Building Materials 22 (2008) 2157–2164
2159
The first crack C1 appeared at the edge of the top section where it was heated up first. All the crack initiation sites and the subsequent bifurcation routes occurred in the upper half of the glazing sheet during the first 20 min. This provided some information on the heating environment. The single glass sheet of thickness 6 mm fell out by its own weight at 15 min in half of the tests [7]. 3. Displacement and thermal stresses induced on the glass pane due to non-uniform thermal load
Fig. 2. Temperature profiles with height over the exposed glass surface.
(ii) t = 15 min T ¼ 230:7
y 2 b
þ 547:7
y b
þ 59:6ð0 6 y 6 bÞ
If the temperature of the glass pane along the direction of thickness z is assumed to be the same under gradual heating, the resultant thermal force per unit length is simple. For glass panes without thermal bending, the differential equations [9,10] for displacement components u and v along the x and y directions due to increase in temperature under three typical boundaries are
ð2Þ
(iii) For t > 15 min, temperature profiles T(y) can be taken as curves parallel to the one at 15 min given by Eq. (2).
(i) Case 1: All sides of the pane are constrained by the boundary and no displacement is allowed u¼v¼0 Then,
Similar curves can be fitted for T = T(x, y), by assuming variations along directions x and y are not related. Thermal stresses can be solved in the same way as for T = T(y). The initiation sites and patterns of cracks appeared at the single glazing systems for different fire tests of glass thickness 4 mm and 6 mm reported [7] are redrawn in Fig. 3. The first crack appeared is labeled as C1, the second one as C2, and so on.
N x ¼ N y ¼ N T ; N xy ¼ 0 ð3Þ aE T ; rxy ¼ 0 r x ¼ ry ¼ 1t (ii) Case 2: There is no boundary constraint; the pane can be extended freely N x ¼ N y ¼ N xy ¼ 0 r x ¼ ry ¼
NT aE T; 1t h
rxy ¼ 0
ð4Þ
By substituting Eq. (1) into Eq. (4), For 0 6 y 6 b/2 rx ¼ 0
ð5aÞ
b 2
For 6 y 6 b NT aE T 1t h Z h=2 Z a Z 1 1 2 b aE ¼ T dx T ðyÞ dy dz hð1 tÞ h=2 a 0 b b=2 1t i aE h y 290 þ 217:5 ð5bÞ ¼ 1t b
rx ¼
The thermal bending of the glass pane for temperature variation T(y) along the y direction is Z b Mx 12y rxb ¼ y¼ 3 aET ðyÞy dy ð6Þ Ix b ð1 tÞ b=2 The resulting thermal stress for b2 6 y 6 b is NT Mx aE T þ y 1t h Ix i 39aE y aE h y 290 þ 217:5 þ ¼ 1t b 1t b
rx ¼ Fig. 3. Crack initiation and crack patterns for a single glass reported by Hassani et al. [8].
ð7Þ
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W.K. Chow, Y. Gao / Construction and Building Materials 22 (2008) 2157–2164
Substituting Eq. (2) into Eq. (4) for 0 6 y 6 b gives y y 2 aE 547:7 rx ¼ 197 ð8Þ 230 1t b b Thermal bending under the case of Eq. (2) is also considered Z b Mx 3y y rxb ¼ ð9Þ y¼ 2 aET ðyÞy dy ¼ 138 b Ix 2b ð1 tÞ 0 Then,
y 2 aE y rx ¼ ð547:7 138Þ 230 197 1t b b
ð10Þ
(iii) Case 3: The glass pane used in the experiment was of length 1615 mm, width 905 mm and thickness 6 mm. There was a gap of over 10 mm between the glass and the wooden frame. Therefore, displacement of the glass pane was allowed only in the vertical direction at the edges. Since the four corners of the glass pane were fixed, displacement was not allowed in the tangential direction at the edges. The boundary conditions for case 3 can be written as For x = 0 and x = a, v = 0, giving ou ov Nx ¼ A þt ð11Þ NT ¼ 0 ox oy For y = 0 and y = b, u = 0, giving ou ov Ny ¼ A t þ NT ¼ 0 ox oy
1 X 1 X m¼1 n¼1 Z a
H mn ¼
4h ab
0
H mn sin Z
mp np x sin y a b
b
N T ðx; yÞ sin 0
mp np x sin y dx dy a b
4hEa
p2 ð1 tÞ
ð17Þ
By substituting Eqs. (15)–(17) separately into Eq. (13), NT can be solved. From Eq. (1) for t = 5 min and Eq. (2) for t = 15 min, temperature T can be calculated. NT can then be calculated from Eq. (16). Therefore, the variations of thermal stresses under different temperature distributions at different positions of the glass pane can be determined. At t = 5 min and 0 6 y < b2 " # mp 2 np 2 1 X 1 120hEa X a b Nx ¼ 2 þ t mp 2 np 2 1 p ð1 tÞ m¼1 n¼2;4 mp 2 þ np 2 þ b a b a h i n 1 1 mp np m x sin y ½1 ð1Þ 1 ð1Þ2 sin m n a b ð18Þ For b2 6 y 6 b 1 X 1 X m¼1 n¼1
ð13Þ
" 2 mp a mp 2 a
mp np x sin y a b 1 4hEa X 1 m ½1 ð1Þ ¼ 2 p ð1 tÞ m¼1 m ( " # 1 1 X n1 1 1 X 1 n ð1Þ þ 290 ð1Þ 2 n p n¼1;3 n2 n¼1;2 ) 1 i X n 1h 2 1 ð1Þ 115 n n¼2;4
H mn
ð14Þ
ð15Þ For b2 6 y 6 b
( " 1 1 X 4hEa X 1 1 m ½1 ð1Þ 290 ð1Þn ¼ 2 p ð1 tÞ m¼1 m n n¼1;2 # ) 1 1 h i X X n1 n 1 1 1 1 ð1Þ2 ð1Þ 2 115 þ p n¼1;3 n2 n n¼2;4
# np 2 H mn H mn b 2 þ t mp 2 np 2 H mn þ np þ b b a
sin
ð12Þ
For Eq. (1) with t = 5 min, integrating Eqs. (1) and (3) gives the coefficient Hmn: For 0 6 y < b2 1 1 i X n 120hEa X 1 1h m 1 ð1Þ2 H mn ¼ 2 ½1 ð1Þ p ð1 tÞ m¼1 m n n¼2;4;
H mn
1 X X1 1 1 ½1 ð1Þm n¼1 m n m¼1 ( " # ! ) 2 n n þ 59:6 547:7ð1Þ ½1 ð1Þ 230:7 1 ðnpÞ2
H mn ¼
Nx ¼
Putting in Eqs. (1) and (2), the resultant thermal force due to the temperature profiles measured in the experiments can be expressed as NT ¼
At t = 15 min and 0 6 y 6 b
ð19Þ At t = 15 min and 0 6 y 6 b " # mp 2 np 2 1 X 1 4hEa X a b Nx ¼ 2 þ t mp 2 np 2 1 p ð1 tÞ m¼1 n¼1 mp 2 þ np 2 þ b a b a ( mp np 1 1 x sin y ð1 ð1Þm Þ sin ½1 ð1Þn a b m n " # ! ) 2 n 230:7 1 þ 59:6 547:7ð1Þ 2 ðnpÞ ð20Þ 4. Results and analysis
ð16Þ
Ordinary glass is composed of 18% Na2O, 10% CaO and 72% SiO2. The critical breaking tensile stress is 75 MPa.
W.K. Chow, Y. Gao / Construction and Building Materials 22 (2008) 2157–2164
The critical breaking compressive stress varies from 910 to 1230 MPa. Putting h as 0.006 m, t as 0.25, a as 8.0 · 106 K1 and E as 70 GPa in the above equations would give the thermal stress on the glass pane. Variations of rx along the y direction at 5 min and 15 min for case 1 with boundary constraint are shown in Fig. 4. As there is a gap between the glass pane and the window frame, the glass pane is then able to expand upon heating up. As observed from Fig. 4, the compressive stress would not exceed the critical value (910 MPa) for cracking at the whole upper part of y = b/2 at 5 min and 15 min. If the glass pane is fixed with the window frame with boundary constraint (but not the shaded edge), the glass pane is not easy to break. Variations of rx along the y direction at 5 min and 15 min for case 2 where the glass was constrained by the boundary with space allowed for extension are shown in Fig. 5. There, stress changes due to temperature variation along the y direction for thin glass without other external effects are shown. The solid line was from Eq. (5), and the dotted line from Eq. (7) where the bending moment
2161
was included. Compressive stress on the flat glass pane reduced. The effective area with stress also reduced. However, tensile stress and the strain area on the lower glass pane increased. The solid line in Fig. 5b came from Eq. (8) and the dotted line from Eq. (10). As temperature along the y direction increased, compressive stress on the upper part of the flat plate (maximum 52 MPa at y = b) became tensile stress of +25 MPa. By comparing Fig. 5a with Fig. 4a, at the upper part of the glass with the highest temperature, the maximum stress rx at 5 min decreased by three times. By comparing Fig. 5b with Fig. 4b, the maximum stress rx at 15 min decreased by five times. The resultant thermal stress for Nx = 0 for case 3 with displacement component along the y direction and t = 0 with boundary condition x = 0 and a are shown in Figs. 6 and 7. The distributions of Nx along the x and y directions at 5 min and 15 min are shown in Fig. 6a and Fig. 6b, respectively. The distributions of Nx along the y direction at the central position of the glass pane x = a/2
Fig. 4. Variations of rx along the y direction for case 1.
Fig. 5. Variations of rx along the y direction for case 2.
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W.K. Chow, Y. Gao / Construction and Building Materials 22 (2008) 2157–2164
Fig. 6. Distribution of Nx along the x and y directions for case 3.
Fig. 7. Variations of Nx at x = 0.5a along the y direction for case 3.
at 5 min and 15 min are shown in Fig. 7a and Fig. 7b, respectively. Fig. 6 was estimated to ensure t = 0, Nx = 0 at x = 0, resultant force per unit length Nx in the x and y directions yields at 5 min and 15 min (shown in Fig. 6a and b) are compressive stress at the upper part and became tensile stress at the lower part. Changes along Fig. 6 at x = 0.5a are shown in Fig. 7, rx along the y direction can be divided into two parts 1 Ea T rx ¼ ðN x þ N T Þ h 1t and rx ¼ rx1 þ rx2 The first part at 5 min and x = 0.5a is shown in Fig. 7a. The maximum of the first part is rx1max ¼
N xmax 3:5 105 N=m ¼ 58 MPa ¼ 0:006 m h
The second part is 1 T aE N T ¼ 47:5 MPa rx2max ¼ h 1t max Combining the above two values rxmax ¼ rx1max þ rx2max ¼ 105:5 MPa Applying the same calculation at the top of y, compressive stress at 15 min reached 128 MPa. The value is lower than the critical compressive strength of 910 MPa. The bending moment is very small and might even change direction to become tensile stress. If the tensile stress can reach the critical value (50–75 Pa) at the lower part, case 3 calculation indicates that without external window frame coverage, fracture would be at the lower part, but not the upper part of the glass pane. Variation of Ny along the y direction at t = 5 min and x = 0.5a is shown in Fig. 8a. Distribution of Ny along the x and y directions is given in Fig. 8b. Fig. 8 is the result to ensure u = 0 and Ny = 0 at y = 0 and b, upper half of
W.K. Chow, Y. Gao / Construction and Building Materials 22 (2008) 2157–2164
2163
Fig. 8. Variations of Ny at 5 min for case 3.
the glass is compressive stress and lower part is tensile stress. Comparing Fig. 8b and Fig. 6a, at boundary x = 0 and a and Nx = 0, the resultant force per unit length Ny has a maximum value Nymax. For y = 0 and b and Ny = 0, Nx has a maximum value Nymax. ry along the x direction can also be divided into two parts 1 Ea T ¼ ry1 þ ry2 ry ¼ ðN y þ N T Þ h 1t ry2max ¼ rx2max 1 4 105 N=m ¼ 66:6 MPa ry1max ¼ N y ¼ h 0:006 m rymax ¼ ry1max þ ry2max ¼ 114:1 MPa It is observed that compressive stress cannot reach the critical value for fracture at t = 5 min. Maximum and minimum stresses were experienced by the glass pane for cases 1 and 2. Case 3 is somewhere between the two cases. As the four corners of the glass pane were fixed and there were gaps at the boundary, case 3 is more realistic. The actual case is different and therefore difficult to estimate the temperature changes at the gap. If the glass pane is subjected to heat suddenly, tensile stress would be experienced at the boundary. In this paper, the fire increased steadily and the first crack was observed at 6 min. As heat flux was not strong, temperature differentials at the side edges were not high. The stress along the direction of thickness can be ignored. Further experiments are required to study the effect of thermal shocks. For better understanding on thermal elasticity, experimental and analytical studies on the glass temperature along the y direction should be carried out. For glass panes without coverage, fracture cracks would not be formed at the upper part. Cracks would be found at the lower part as the tensile stress is high and near to the critical value. For experimental results shown in Figs. 1 and 3, cracks appeared at the upper part. For two measuring points separated by less than 18 mm, shaded and exposed glass surface tempera-
tures were 35 and 174 C, respectively. Cracking would occur at temperature difference from 120 to 140 C. From simple strain criterion, the temperature difference in the shaded and unshaded region of the glass is related to the induced stresses and forces balance factor f (1.2) as DT max ¼ f
rcr ð1 tÞ ð50 75Þ MPa ð1 0:25Þ ¼ 1:2 aE 8 106 K 1 70 GPa
Value of DTmax is from 100 to 120 C. In practice, the value of critical stress depends on many factors such as the materials, size, thickness and surface technology of the glass panes. The heating time and position of heating are also important. 5. Conclusion Glazing units under a building fire were discussed by studying the thermal stresses over glass panes with varying temperature field and different boundary conditions of the window frame. Available experimental data [8] were used to derive correlation expressions on the temperature distribution of the glass surface. General solutions of thermal stress equations of square window panes exposed to fire are presented in classical analytical form. The distribution of stresses on the glass pane due to varying temperature field in one typical result of the experiments was calculated by the fitted temperature correlation. Thermal stresses were analyzed mathematically for three cases of different boundary conditions between the glass and the window frame. The positions on the glass panes where cracks easily occurred were estimated. It is illustrated that thermal stresses over the glass surface can be calculated from a fitted vertical temperature profile. Such approach to calculating the thermal stresses along the glass pane had not been reported in the literature. Obviously, classical theory on heating a plate along the vertical direction can be applied to study the thermal response of a glass plate under real fires.
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W.K. Chow, Y. Gao / Construction and Building Materials 22 (2008) 2157–2164
Further, results from all three cases illustrated that for glass panes without coverage, fractures are not induced by temperature variation along the height at the upper part. Temperature difference between the glass pane and the coverage material is the key point to cause fracture. Acknowledgement The work described in this paper was supported by a Grant from the Research Grants Council of the Hong Kong Special Administrative Region, China for the project ‘‘Fire Safety for Glass Facades in Green and Sustainable Buildings’’ (PolyU 5163/04E). References [1] Chow WK. Building fire safety in the far east. Arch Sci Rev 2005;48(4):285–94.
[2] Ng Candy MY, Chow WK. Proposed fire safety strategy on airport terminals. Int J Risk Assess Manage 2005;5(1):95–110. [3] Glass walls checked by the government, Mingpao, Hong Kong, 19 July 2005. [4] Pagni PJ, Joshi AA. Glass breaking in fires. In: Fire safety science – proceedings of the third international symposium, Edinburgh; 1991. p. 791–802. [5] Joshi AA, Pagni PJ. Fire induced thermal fields in window glass I – Theory. Fire Safe J 1994;22:25–44. [6] Keski-Rahkonen O. Breaking of window glass close to fire. Fire Mater 1988;12:61–9. [7] Keski-Rahkonen O. Breaking of window glass close to fire II: circular panes. Fire Mater 1991;15(1):11–6. [8] Hassani SKS, Shields TJ, Silcock GWH. An experimental investigation into the behaviour of glazing in enclosure fire. In: DeCicco PR, editor. The behaviour of glass and other materials exposed to fire, vol. 1. New York: Baywood Publishing Company; 2002. p. 3–22. [9] Boley BA, Weiner JH. Theory of thermal stresses. New York: Wiley; 1960. [10] Tauchert TR. Thermal stresses. Thermal stresses in plates – statical problems, vol. 1. New York: Hetnarski; 1986 [chapter 2].
Construction and Building
MATERIALS
Construction and Building Materials 22 (2008) 2157–2164
www.elsevier.com/locate/conbuildmat
Thermal stresses on window glasses upon heating W.K. Chow a
a,*
, Y. Gao
b
Research Centre for Fire Engineering, Department of Building Services Engineering, Area of Strength: Fire Safety Engineering, The Hong Kong Polytechnic University, Hunghom, Kowloon, Hong Kong, China b Department of Building Engineering, Harbin Engineering University, Harbin, Heilongjiang, China Received 25 January 2007; received in revised form 14 August 2007; accepted 12 September 2007 Available online 26 November 2007
Abstract There are concerns about the fire safety of the glass panes in curtain walled buildings with green or sustainable design. Thermal stresses over the glass panes upon uneven heating in a fire should be investigated and were discussed in this paper. Different temperature fields and different boundary conditions of the window frame were studied. In contrast to those literature works, temperature and thermal load were not taken to be uniformly distributed over the whole glass pane. As thermal stresses built up in the glass depends on the surface temperature, surface temperature profiles have to be known. Correlation expressions on the varying temperature distribution were deduced from reported experimental data. The derived results were applied to predict the positions where cracking is likely to occur in the glass construction during a building fire. 2007 Elsevier Ltd. All rights reserved. Keywords: Thermal stresses; Glass; Fire
1. Introduction Glass facades are commonly found in many green or sustainable buildings [1] and big airport terminals [2]. In addition to provide better visual effects, glass facades allow more daylighting and higher solar gain for reducing the heating load in cold countries. However, there are concerns on fire safety in those buildings with glass construction. Cracking and collapse of window panes are the identified risks upon heating up the glass facade in a fire [1,3]. At the early stage of a flashover room fire, the burning object is small and might be at a distance from the windows. Thermal radiation emitted from the flames is absorbed slowly by the glass pane to increase its temperature gradually. It takes time to increase the temperature along the direction of thickness. There is no thermal bending, with thermal stresses built up only along the plane. But in post-flashover room fires, the flame acts directly on the glass surface. A large temperature difference between the *
Corresponding author. Tel.: +852 2766 5843; fax: +852 2765 7198. E-mail address: [email protected] (W.K. Chow).
0950-0618/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.conbuildmat.2007.09.004
inner and outer surfaces of the glass pane results. Expansion of the glass surface nearer to the flame leads to thermal bending along the direction of thickness. Thermal stresses on the exposed surface then increase. Thermal stresses over glass panes were estimated [4–7] with thermal load assumed to be distributed uniformly over the whole pane. However, experiments by Hassani et al. [8] suggested that the glass in a room fire is exposed to a homogeneous thermal environment. The first crack in the window glass always occurs at positions near the hot gas layer. This is because the maximum tensile stress is located in the shaded region of the glass. Locations with higher stresses should be further studied and illustrating this point is the objective of this paper. In contrast to the literature works, temperature and thermal load are not taken to be distributed uniformly over the pane. Experimental data available in the literature was reviewed for studying both the compressive and tensile stresses at those glass edges at elevated temperatures. As variation of the temperature field of the glass pane along its height in a building fire would give thermal stresses, it is important to know the temperature profiles along the
2158
W.K. Chow, Y. Gao / Construction and Building Materials 22 (2008) 2157–2164
Nomenclature a A t u, v rx, ry rxy rxb rcr E NT Nx, Ny Nxy
thermal expansion coefficient extentional stiffness Possion’s ratio components of displacement normal components of stress to x and y axes twisting stress bending thermal stress critical breaking stress Young’s modulus of elasticity in tension and compression thermal force per unit length forces per unit length force to give torsional effect
bending moment per unit length bending moment of the forces distributed over a cross section Hmn, Amn, Bmn Fourier coefficient m, n Fourier coefficient subscripts T temperature t time h thickness of the glass pane a length in x direction b length in y direction x tangential direction coordinate y normal direction coordinate Mx Ix
height under a buoyant flame for estimating the thermal stresses. Typical experimental results by Hassani et al. [8] are applied in this paper. The glass temperature measured at a particular time is taken to be the same along the direction of thickness. Increase in temperature profiles along the height of the glass pane is studied. The reference temperature is taken as the ambient temperature and assumed to be uniform throughout the glass. 2. Correlation expressions of experimental results Correlation expressions on glass temperature are derived based on the experiment by Hassani et al. [8]. Glazing units of size 905 mm by 1615 mm fixed in softwood frames of size 988 mm by 1690 mm under a wood crib fire were studied in a room model. Six thermocouples were installed with three pairs at three different heights. One thermocouple was exposed to the fire and the other one was shaded by a wooden window frame without contacting with the flame directly. The effects of buoyant flame on increasing the glass surface temperature at different heights were reported. Typical glass surface temperature profiles T at different heights with time t are shown in Fig. 1. The vertical temperature profiles of the exposed glass surface with height y (with respect to the glass height b) at 5 min, 15 min and 20 min are shown in Fig. 2. Positions of cracks on the glass panes were also recorded [8]. Most of the cracks were found near to the window frame with the first crack appeared at 5 min. Result suggested that the critical thermal stresses required to give cracking would be near to the interfaces of the hot gas and the cooler region shaded by the window frame. As observed from the results shown in Figs. 1 and 2, it is not too satisfactory to assume that the glass surface temperature exposed to fire is uniform. A large change in temperature difference with height is expected. Actually, variation of temperature with height can be fitted from the experimental data on the surface temperature profiles of glass.
Fig. 1. Glass surface temperature profiles at different heights from Hassani et al. [8].
Results are applied to study the vertical distribution of thermal stresses on the glass surface. The positions where cracking or fallout is likely to occur can then be determined. The calculation procedure is demonstrated in this paper. Analyzing typical experimental results as in Fig. 2 would give correlation expressions on temperature T with height y (in terms of the length of glass b along the y direction) at a given time t as (i) t = 5 min 8 < 30 T ¼ : 290 y 0:5 þ 30 b
06y
ð1Þ
W.K. Chow, Y. Gao / Construction and Building Materials 22 (2008) 2157–2164
2159
The first crack C1 appeared at the edge of the top section where it was heated up first. All the crack initiation sites and the subsequent bifurcation routes occurred in the upper half of the glazing sheet during the first 20 min. This provided some information on the heating environment. The single glass sheet of thickness 6 mm fell out by its own weight at 15 min in half of the tests [7]. 3. Displacement and thermal stresses induced on the glass pane due to non-uniform thermal load
Fig. 2. Temperature profiles with height over the exposed glass surface.
(ii) t = 15 min T ¼ 230:7
y 2 b
þ 547:7
y b
þ 59:6ð0 6 y 6 bÞ
If the temperature of the glass pane along the direction of thickness z is assumed to be the same under gradual heating, the resultant thermal force per unit length is simple. For glass panes without thermal bending, the differential equations [9,10] for displacement components u and v along the x and y directions due to increase in temperature under three typical boundaries are
ð2Þ
(iii) For t > 15 min, temperature profiles T(y) can be taken as curves parallel to the one at 15 min given by Eq. (2).
(i) Case 1: All sides of the pane are constrained by the boundary and no displacement is allowed u¼v¼0 Then,
Similar curves can be fitted for T = T(x, y), by assuming variations along directions x and y are not related. Thermal stresses can be solved in the same way as for T = T(y). The initiation sites and patterns of cracks appeared at the single glazing systems for different fire tests of glass thickness 4 mm and 6 mm reported [7] are redrawn in Fig. 3. The first crack appeared is labeled as C1, the second one as C2, and so on.
N x ¼ N y ¼ N T ; N xy ¼ 0 ð3Þ aE T ; rxy ¼ 0 r x ¼ ry ¼ 1t (ii) Case 2: There is no boundary constraint; the pane can be extended freely N x ¼ N y ¼ N xy ¼ 0 r x ¼ ry ¼
NT aE T; 1t h
rxy ¼ 0
ð4Þ
By substituting Eq. (1) into Eq. (4), For 0 6 y 6 b/2 rx ¼ 0
ð5aÞ
b 2
For 6 y 6 b NT aE T 1t h Z h=2 Z a Z 1 1 2 b aE ¼ T dx T ðyÞ dy dz hð1 tÞ h=2 a 0 b b=2 1t i aE h y 290 þ 217:5 ð5bÞ ¼ 1t b
rx ¼
The thermal bending of the glass pane for temperature variation T(y) along the y direction is Z b Mx 12y rxb ¼ y¼ 3 aET ðyÞy dy ð6Þ Ix b ð1 tÞ b=2 The resulting thermal stress for b2 6 y 6 b is NT Mx aE T þ y 1t h Ix i 39aE y aE h y 290 þ 217:5 þ ¼ 1t b 1t b
rx ¼ Fig. 3. Crack initiation and crack patterns for a single glass reported by Hassani et al. [8].
ð7Þ
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Substituting Eq. (2) into Eq. (4) for 0 6 y 6 b gives y y 2 aE 547:7 rx ¼ 197 ð8Þ 230 1t b b Thermal bending under the case of Eq. (2) is also considered Z b Mx 3y y rxb ¼ ð9Þ y¼ 2 aET ðyÞy dy ¼ 138 b Ix 2b ð1 tÞ 0 Then,
y 2 aE y rx ¼ ð547:7 138Þ 230 197 1t b b
ð10Þ
(iii) Case 3: The glass pane used in the experiment was of length 1615 mm, width 905 mm and thickness 6 mm. There was a gap of over 10 mm between the glass and the wooden frame. Therefore, displacement of the glass pane was allowed only in the vertical direction at the edges. Since the four corners of the glass pane were fixed, displacement was not allowed in the tangential direction at the edges. The boundary conditions for case 3 can be written as For x = 0 and x = a, v = 0, giving ou ov Nx ¼ A þt ð11Þ NT ¼ 0 ox oy For y = 0 and y = b, u = 0, giving ou ov Ny ¼ A t þ NT ¼ 0 ox oy
1 X 1 X m¼1 n¼1 Z a
H mn ¼
4h ab
0
H mn sin Z
mp np x sin y a b
b
N T ðx; yÞ sin 0
mp np x sin y dx dy a b
4hEa
p2 ð1 tÞ
ð17Þ
By substituting Eqs. (15)–(17) separately into Eq. (13), NT can be solved. From Eq. (1) for t = 5 min and Eq. (2) for t = 15 min, temperature T can be calculated. NT can then be calculated from Eq. (16). Therefore, the variations of thermal stresses under different temperature distributions at different positions of the glass pane can be determined. At t = 5 min and 0 6 y < b2 " # mp 2 np 2 1 X 1 120hEa X a b Nx ¼ 2 þ t mp 2 np 2 1 p ð1 tÞ m¼1 n¼2;4 mp 2 þ np 2 þ b a b a h i n 1 1 mp np m x sin y ½1 ð1Þ 1 ð1Þ2 sin m n a b ð18Þ For b2 6 y 6 b 1 X 1 X m¼1 n¼1
ð13Þ
" 2 mp a mp 2 a
mp np x sin y a b 1 4hEa X 1 m ½1 ð1Þ ¼ 2 p ð1 tÞ m¼1 m ( " # 1 1 X n1 1 1 X 1 n ð1Þ þ 290 ð1Þ 2 n p n¼1;3 n2 n¼1;2 ) 1 i X n 1h 2 1 ð1Þ 115 n n¼2;4
H mn
ð14Þ
ð15Þ For b2 6 y 6 b
( " 1 1 X 4hEa X 1 1 m ½1 ð1Þ 290 ð1Þn ¼ 2 p ð1 tÞ m¼1 m n n¼1;2 # ) 1 1 h i X X n1 n 1 1 1 1 ð1Þ2 ð1Þ 2 115 þ p n¼1;3 n2 n n¼2;4
# np 2 H mn H mn b 2 þ t mp 2 np 2 H mn þ np þ b b a
sin
ð12Þ
For Eq. (1) with t = 5 min, integrating Eqs. (1) and (3) gives the coefficient Hmn: For 0 6 y < b2 1 1 i X n 120hEa X 1 1h m 1 ð1Þ2 H mn ¼ 2 ½1 ð1Þ p ð1 tÞ m¼1 m n n¼2;4;
H mn
1 X X1 1 1 ½1 ð1Þm n¼1 m n m¼1 ( " # ! ) 2 n n þ 59:6 547:7ð1Þ ½1 ð1Þ 230:7 1 ðnpÞ2
H mn ¼
Nx ¼
Putting in Eqs. (1) and (2), the resultant thermal force due to the temperature profiles measured in the experiments can be expressed as NT ¼
At t = 15 min and 0 6 y 6 b
ð19Þ At t = 15 min and 0 6 y 6 b " # mp 2 np 2 1 X 1 4hEa X a b Nx ¼ 2 þ t mp 2 np 2 1 p ð1 tÞ m¼1 n¼1 mp 2 þ np 2 þ b a b a ( mp np 1 1 x sin y ð1 ð1Þm Þ sin ½1 ð1Þn a b m n " # ! ) 2 n 230:7 1 þ 59:6 547:7ð1Þ 2 ðnpÞ ð20Þ 4. Results and analysis
ð16Þ
Ordinary glass is composed of 18% Na2O, 10% CaO and 72% SiO2. The critical breaking tensile stress is 75 MPa.
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The critical breaking compressive stress varies from 910 to 1230 MPa. Putting h as 0.006 m, t as 0.25, a as 8.0 · 106 K1 and E as 70 GPa in the above equations would give the thermal stress on the glass pane. Variations of rx along the y direction at 5 min and 15 min for case 1 with boundary constraint are shown in Fig. 4. As there is a gap between the glass pane and the window frame, the glass pane is then able to expand upon heating up. As observed from Fig. 4, the compressive stress would not exceed the critical value (910 MPa) for cracking at the whole upper part of y = b/2 at 5 min and 15 min. If the glass pane is fixed with the window frame with boundary constraint (but not the shaded edge), the glass pane is not easy to break. Variations of rx along the y direction at 5 min and 15 min for case 2 where the glass was constrained by the boundary with space allowed for extension are shown in Fig. 5. There, stress changes due to temperature variation along the y direction for thin glass without other external effects are shown. The solid line was from Eq. (5), and the dotted line from Eq. (7) where the bending moment
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was included. Compressive stress on the flat glass pane reduced. The effective area with stress also reduced. However, tensile stress and the strain area on the lower glass pane increased. The solid line in Fig. 5b came from Eq. (8) and the dotted line from Eq. (10). As temperature along the y direction increased, compressive stress on the upper part of the flat plate (maximum 52 MPa at y = b) became tensile stress of +25 MPa. By comparing Fig. 5a with Fig. 4a, at the upper part of the glass with the highest temperature, the maximum stress rx at 5 min decreased by three times. By comparing Fig. 5b with Fig. 4b, the maximum stress rx at 15 min decreased by five times. The resultant thermal stress for Nx = 0 for case 3 with displacement component along the y direction and t = 0 with boundary condition x = 0 and a are shown in Figs. 6 and 7. The distributions of Nx along the x and y directions at 5 min and 15 min are shown in Fig. 6a and Fig. 6b, respectively. The distributions of Nx along the y direction at the central position of the glass pane x = a/2
Fig. 4. Variations of rx along the y direction for case 1.
Fig. 5. Variations of rx along the y direction for case 2.
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Fig. 6. Distribution of Nx along the x and y directions for case 3.
Fig. 7. Variations of Nx at x = 0.5a along the y direction for case 3.
at 5 min and 15 min are shown in Fig. 7a and Fig. 7b, respectively. Fig. 6 was estimated to ensure t = 0, Nx = 0 at x = 0, resultant force per unit length Nx in the x and y directions yields at 5 min and 15 min (shown in Fig. 6a and b) are compressive stress at the upper part and became tensile stress at the lower part. Changes along Fig. 6 at x = 0.5a are shown in Fig. 7, rx along the y direction can be divided into two parts 1 Ea T rx ¼ ðN x þ N T Þ h 1t and rx ¼ rx1 þ rx2 The first part at 5 min and x = 0.5a is shown in Fig. 7a. The maximum of the first part is rx1max ¼
N xmax 3:5 105 N=m ¼ 58 MPa ¼ 0:006 m h
The second part is 1 T aE N T ¼ 47:5 MPa rx2max ¼ h 1t max Combining the above two values rxmax ¼ rx1max þ rx2max ¼ 105:5 MPa Applying the same calculation at the top of y, compressive stress at 15 min reached 128 MPa. The value is lower than the critical compressive strength of 910 MPa. The bending moment is very small and might even change direction to become tensile stress. If the tensile stress can reach the critical value (50–75 Pa) at the lower part, case 3 calculation indicates that without external window frame coverage, fracture would be at the lower part, but not the upper part of the glass pane. Variation of Ny along the y direction at t = 5 min and x = 0.5a is shown in Fig. 8a. Distribution of Ny along the x and y directions is given in Fig. 8b. Fig. 8 is the result to ensure u = 0 and Ny = 0 at y = 0 and b, upper half of
W.K. Chow, Y. Gao / Construction and Building Materials 22 (2008) 2157–2164
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Fig. 8. Variations of Ny at 5 min for case 3.
the glass is compressive stress and lower part is tensile stress. Comparing Fig. 8b and Fig. 6a, at boundary x = 0 and a and Nx = 0, the resultant force per unit length Ny has a maximum value Nymax. For y = 0 and b and Ny = 0, Nx has a maximum value Nymax. ry along the x direction can also be divided into two parts 1 Ea T ¼ ry1 þ ry2 ry ¼ ðN y þ N T Þ h 1t ry2max ¼ rx2max 1 4 105 N=m ¼ 66:6 MPa ry1max ¼ N y ¼ h 0:006 m rymax ¼ ry1max þ ry2max ¼ 114:1 MPa It is observed that compressive stress cannot reach the critical value for fracture at t = 5 min. Maximum and minimum stresses were experienced by the glass pane for cases 1 and 2. Case 3 is somewhere between the two cases. As the four corners of the glass pane were fixed and there were gaps at the boundary, case 3 is more realistic. The actual case is different and therefore difficult to estimate the temperature changes at the gap. If the glass pane is subjected to heat suddenly, tensile stress would be experienced at the boundary. In this paper, the fire increased steadily and the first crack was observed at 6 min. As heat flux was not strong, temperature differentials at the side edges were not high. The stress along the direction of thickness can be ignored. Further experiments are required to study the effect of thermal shocks. For better understanding on thermal elasticity, experimental and analytical studies on the glass temperature along the y direction should be carried out. For glass panes without coverage, fracture cracks would not be formed at the upper part. Cracks would be found at the lower part as the tensile stress is high and near to the critical value. For experimental results shown in Figs. 1 and 3, cracks appeared at the upper part. For two measuring points separated by less than 18 mm, shaded and exposed glass surface tempera-
tures were 35 and 174 C, respectively. Cracking would occur at temperature difference from 120 to 140 C. From simple strain criterion, the temperature difference in the shaded and unshaded region of the glass is related to the induced stresses and forces balance factor f (1.2) as DT max ¼ f
rcr ð1 tÞ ð50 75Þ MPa ð1 0:25Þ ¼ 1:2 aE 8 106 K 1 70 GPa
Value of DTmax is from 100 to 120 C. In practice, the value of critical stress depends on many factors such as the materials, size, thickness and surface technology of the glass panes. The heating time and position of heating are also important. 5. Conclusion Glazing units under a building fire were discussed by studying the thermal stresses over glass panes with varying temperature field and different boundary conditions of the window frame. Available experimental data [8] were used to derive correlation expressions on the temperature distribution of the glass surface. General solutions of thermal stress equations of square window panes exposed to fire are presented in classical analytical form. The distribution of stresses on the glass pane due to varying temperature field in one typical result of the experiments was calculated by the fitted temperature correlation. Thermal stresses were analyzed mathematically for three cases of different boundary conditions between the glass and the window frame. The positions on the glass panes where cracks easily occurred were estimated. It is illustrated that thermal stresses over the glass surface can be calculated from a fitted vertical temperature profile. Such approach to calculating the thermal stresses along the glass pane had not been reported in the literature. Obviously, classical theory on heating a plate along the vertical direction can be applied to study the thermal response of a glass plate under real fires.
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Further, results from all three cases illustrated that for glass panes without coverage, fractures are not induced by temperature variation along the height at the upper part. Temperature difference between the glass pane and the coverage material is the key point to cause fracture. Acknowledgement The work described in this paper was supported by a Grant from the Research Grants Council of the Hong Kong Special Administrative Region, China for the project ‘‘Fire Safety for Glass Facades in Green and Sustainable Buildings’’ (PolyU 5163/04E). References [1] Chow WK. Building fire safety in the far east. Arch Sci Rev 2005;48(4):285–94.
[2] Ng Candy MY, Chow WK. Proposed fire safety strategy on airport terminals. Int J Risk Assess Manage 2005;5(1):95–110. [3] Glass walls checked by the government, Mingpao, Hong Kong, 19 July 2005. [4] Pagni PJ, Joshi AA. Glass breaking in fires. In: Fire safety science – proceedings of the third international symposium, Edinburgh; 1991. p. 791–802. [5] Joshi AA, Pagni PJ. Fire induced thermal fields in window glass I – Theory. Fire Safe J 1994;22:25–44. [6] Keski-Rahkonen O. Breaking of window glass close to fire. Fire Mater 1988;12:61–9. [7] Keski-Rahkonen O. Breaking of window glass close to fire II: circular panes. Fire Mater 1991;15(1):11–6. [8] Hassani SKS, Shields TJ, Silcock GWH. An experimental investigation into the behaviour of glazing in enclosure fire. In: DeCicco PR, editor. The behaviour of glass and other materials exposed to fire, vol. 1. New York: Baywood Publishing Company; 2002. p. 3–22. [9] Boley BA, Weiner JH. Theory of thermal stresses. New York: Wiley; 1960. [10] Tauchert TR. Thermal stresses. Thermal stresses in plates – statical problems, vol. 1. New York: Hetnarski; 1986 [chapter 2].