Temperature in Steel Sections

Chapter 4 TEMPERATURE IN STEEL SECTIONS 4.1. INTRODUCTION The increase in steel temperature depends on the temperature of the fire compartment, the area of steel exposed to the fire and the amount of the applied fire protection. This chapter will focus on the transfer of the heat from the fire compartment to the structural elements. Several fire protection systems can be used to improve the performance of steel sections in a fire and these are described later in this chapter. The heat transfer from the hot gases into the surface of the structural elements by a combination of convection and radiation is normally treated as a boundary condition. The heat transfer within the structural member is by conduction and is governed by the well known Fourier equation of heat transfer. The chapter starts by presenting the complex governing equation for heat conduction and goes on to show how simplified and more practical methods can be developed for steel elements. 4.2. THE HEAT CONDUCTION EQUATION AND ITS BOUNDARY CONDITIONS The governing equation for the two-dimensional non-linear, transient heat conduction within the cross section of a structural element, takes the following form: w wT w wT (Oa )  (Oa )  Q wx wx wy wy U a ca wT wt (4.1) Fire Design of Steel Structures Jean-Marc Franssen and Paulo Vila Real © 2012 ECCS – European Convention for Constructional Steelwork. Published 2012 by ECCS – European Convention for Constructional Steelwork _____ 45 4. TEMPERATURE IN STEEL SECTIONS where O a is the thermal conductivity, Q the internal heat source that is equal to zero in the case of non-combustible members, U a the unit mass of steel, ca the specific heat of steel, T the temperature and t the time. The temperature field which satisfies Eq. (4.1) within the structural element must satisfy the following boundary conditions: i) prescribed temperatures T on a part *T of the boundary; ii) specified heat flux q on a part *q of the boundary; iii) heat transfer by convection between the part *c of the boundary at temperature T and the surrounding ambient temperature Tf qc Dc (T  Tf ) on *c (4.2) where D c is the coefficient of heat transfer by convection [W/m2K] and qc is the heat flux by convection per unit area. iv) heat transfer by radiation between the part *r of the boundary at an absolute temperature T and the fire environment at an absolute temperature Tr qr _____ 46 VH (T 4  T r4 ) on *r (4.3) where V is the Stephan Boltzmann constant ( 5.67 u108 W/m2K4 ), H the emissivity and qr is the heat flux by radiation per unit area. This equation can become linear as follows: qr VH (T 4  T r4 ) VH (T 2  T r2 )(T  T r )(T  T r ) D r (T  T r ) on *r (4.4) Dr where D r can be considered as the coefficient of heat transfer by radiation. In the case of combined convection and radiation and if, as usually happens, T r Tf , the combined heat flux is given by: q cr qc  q r D c (T  T f )  D r (T  T r ) D cr (T  T f ) (4.5) where D cr D c  D r (4.6) 4.3. ADVANCED CALCULATION MODEL. FE SOLUTION OF THE HEAT TRANSFER EQ. is the combined convection and radiation coefficient which is temperature dependent. It should be noted that both the governing Eq. (4.1) and boundary condition Eq. (4.5) are non-linear. The former is due to the thermal conductivity and specific heat that are temperature dependent (see Annex A), and the latter is due to the radiative boundary condition which involves a non-linear term of the temperature, as shown in Eq. (4.4). Therefore a closed-form solution to the governing Eq. (4.1) and its boundary condition Eq. (4.5) is not possible, even for the cases with the simplest geometry. Numerical methods such as the finite element method are usually required to solve this kind of heat transfer problems and such a method is presented in the next section. EN 1993-1-2 gives simplified methods for unprotected and protected steelwork exposed to the fire, and these methods are presented in Sections 4.5 and 4.6, respectively. 4.3. ADVANCED CALCULATION MODEL. FINITE ELEMENT SOLUTION OF THE HEAT CONDUCTION EQUATION Using finite elements : e to discretize the domain :, together with a weak formulation and using the Galerkin method for choosing the weighting functions, the following system of differential equations is obtained: Kș  Cș F (4.7) where E K lm ¦³ e 1 § wN l wN wN · wN ¨ Oa m  l Oa m ¸¸d: e  e ¨ : wy wy ¹ wx © wx E Clm ¦³ e 1 E Fl ¦³ e 1 : e N l Q d: e  :e e 1 ¦³ e 1 *ce D cr N l N m d*ce (4.8) U a c a N l N m d: e Q ¦³ H *qe N l q d*qe  (4.9) H ¦³ e 1 *ce N l D cr Tf d*ce (4.10) _____ 47 4. TEMPERATURE IN STEEL SECTIONS and E is the total number of elements, Q is the number of elements with boundary type *q , H is the number of elements with boundary type *c and (or) *r , and Nl and Nm are shape functions. Adopting a finite difference discretisation in time, the system of ordinary differential equations (4.7) can be rewritten for the time t nD : ˆ K n  Dșn  D ˆ F n  D ; 0  D d 1 n  0,1,2,...N  1 (4.11) where ˆ K nD 1 Cn  D ; 0  D d 1 D't (4.12) 1 Cn  Dș n ; 0  D d 1 D't (4.13) K nD  and ˆ F nD Fn  D  Solving the system of equations (4.11), for șn  D , at time tnD , the value of ș at the end of the time interval 't , ( i. e., at time t n 1 ) is given by: șn 1 _____ 48 1 § 1· ș n  D  ¨1  ¸ ș n D © D¹ (4.14) These are the initial conditions for the next time interval. Varying the D parameter, several time integration schemes are obtained. If D z 0 the methods are called implicit. The most popular schemes are those for which: x x D 1 2 (Crank-Nicolson); D 2 3 (Galerkin); x D 1 (Euler Backward). The algorithm from equations (4.11) to (4.14) has the same stability criteria for both linear and non-linear problems and is unconditionally stable for D t 1 2 , in the sense that the process will always converge, although some oscillation can occur if the time interval is too big. 4.3. ADVANCED CALCULATION MODEL. FE SOLUTION OF THE HEAT TRANSFER EQ. 4.3.1. Temperature field using the finite element method Numerical simulation of the thermal response of a steel HE 400 B profile subjected to the standard fire curve ISO 834 acting on all four sides as shown in Fig. 4.1 is presented in this section. Fig. 4.1: Steel profile exposed to fire on all four sides Due to the symmetry, only one quarter of the cross section is analysed. Fig. 4.2 shows the finite element mesh adopted. _____ 49 Fig. 4.2: Finite element mesh used The temperature field after a standard fire duration of 15, 30, 60 and 90 minutes is shown in Fig. 4.3. The emissivity in Eq. (4.3) was taken as H 0.5 . The coefficient of heat transfer by convection, D c ,in Eq. (4.2) was taken as 25 W/m2K. 4. TEMPERATURE IN STEEL SECTIONS After 15 minutes of exposure _____ 50 After 30 minutes of exposure After 60 minutes of exposure After 90 minutes of exposure Fig. 4.3: Temperature field of an HE 400 B heated on all four sides The temperature field on the steel profile HE 400 B after 30 minutes depicted in Fig. 4.3, shows an almost uniform temperature field where the difference between the maximum and minimum temperature is only 57 ºC. This result is normally attributed to the high value of the thermal conductivity of the steel that leads to a quick propagation of heat producing an almost uniformly distributed field of temperature in the cross section. It can be shown that this difference is due to the relative thickness of the sections and not to the high value of the thermal conductivity of steel. In fact, the notion of relative thickness is translated physically through the notion of section factor, see Section 4.4. Assuming a uniform temperature distribution makes Eq. (4.1) much simpler to apply. Simple models for the calculation of the steel temperature are based on the assumption of a uniform temperature distribution throughout the cross 4.4. SECTION FACTOR section as given in EN 1993-1-2 and presented in the next sections. 4.4. SECTION FACTOR Before presenting the simple methods for the evaluation of the thermal response of a steel member, the concept of a parameter that governs the rate of temperature rise is presented. The rate of temperature rise depends on the mass and the surface area of the member exposed to the fire. Light members such as purlins or lattice girders heat up much faster than heavy sections such as columns. The rate of heating of a given member is described by its “Section Factor” or “Massivity Factor”, A/V, which is the ratio of the surface area exposed to the heat flux and the volume of the member per unit length. For unprotected members Eurocode 3 defines the section factor as Am/V [m-1]. For prismatic members with boundary conditions that are constant along the length, the temperature distribution is two-dimensional and the massivity factor is the ratio of the perimeter of the section exposed to the fire, in meters, and the cross-sectional area of the member, in m2 as follows Am V Pul Aul P [m-1] A (4.15) where l the length of the member. Table 4.1 and Table 4.2 show the basic principle for evaluating the “Section Factor” or “Massivity Factor” for unprotected and protected steel members, respectively. The cross-sectional area A is always the area of the steel section. The heated perimeter Am of an unprotected member will depend on the number of sides exposed to the fire. The perimeter A p of protected members also depends on the number of heated sides and on the type of insulation used. For box protection, generally the box perimeter that corresponds to the smallest box surrounding the section is considered and for sprayed insulation or intumescent paint, the perimeter of the steel cross section is used. More detailed cases are shown in Annex A. _____ 51 4. TEMPERATURE IN STEEL SECTIONS The rate at which a steel member will increase in temperature is proportional to the surface, A, of steel exposed to the fire and inversely proportional to the mass or volume, V, of the member. In a fire, a member with a low section factor will heat up more slowly than one with a high section factor. This is illustrated in Fig. 4.4. Table 4.1: Definition of section factors for unprotected steel members Sketch Description Section factor ( Am / V ) 4 sides exposed steel perimeter exposed to fire ņņņņņņņņņņ A 3 sides exposed steel perimeter exposed to fire ņņņņņņņņņņ A _____ 52 h 4 sides exposed b 2 (b + h ) ņņņņņņņņņņ A Note: A is the steel cross section area. P - high A - low Am - high V Fast heating Slow heating Fig. 4.4: The section factor P - low A - high Am - low V 4.4. SECTION FACTOR Table 4.2: Definition of section factors for protected steel members Sketch Description Section factor ( Ap / V ) Contour encasement of uniform thickness exposed to fire on four sides steel perimeter ņņņņņņņņņņ Contour encasement of uniform thickness, exposed to fire on three sides A steel perimeter - b ņņņņņņņņņņ A b h Hollow encasement of uniform thickness exposed to fire on four sides 2 (b + h ) ņņņņņņņņņņ A Hollow encasement of uniform thickness, exposed to fire on three sides b + 2h ņņņņņņņņņņ A b h b Note: A is the steel cross section area. Table 4.3 shows some examples of section factors for protected sections. The section factor for a profile with contour encasement is the same as that for the unprotected profile. _____ 53 4. TEMPERATURE IN STEEL SECTIONS Table 4.3: Section factors Profile contour 3 sides contour 4 sides box 3 sides box 4 sides IPE 100 HE 100 A 334 217 388 264 247 137 300 185 IPE 400 HE 400 A 152 101 174 120 116 68 137 87 IPE 600 HE 600 A 115 87 129 100 91 65 105 78 4.5. TEMPERATURE OF UNPROTECTED STEELWORK EXPOSED TO FIRE _____ 54 EN 1993-1-2 provides a simple equation for calculating the thermal response of unprotected steel members. Assuming an equivalent uniform temperature distribution throughout the cross section, the increase in temperature 'Ta,t in an unprotected steel member during a time interval 't is given by: 'T a,t ksh Am / V  hnet, d 't [ºC] ca Ua (4.16) where k sh Am / V Am - is the correction factor for the shadow effect, from Eq. (4.17); - is the section factor for unprotected steel members, ( t 10 ) [m-1]; - is the surface area of the member per unit length, [m2/m]; V ca - is the volume of the member per unit length, [m3/m]; - is the specific heat of steel, [J/kgK]; 4.5. TEMPERATURE OF UNPROTECTED STEELWORK EXPOSED TO FIRE hnet ,d - is the design value of the net heat flux per unit area according to Eq. (3.4) from Chapter 3, representing the thermal actions and reproduced here. This flux is the sum of a convective part hnet ,c and a radiative part hnet ,r : hnet .d 2 hnet ,c  hnet , r [W/m ]; hnet,c D c (T g  T m ) [W/m2]; where Dc - is the coefficient of heat transfer by convection given in Tg Eq.(3.4) and reproduced in table 4.4; - is the gas temperature of the fire compartment defined in Tm ) Hm Hf Chapter 3, [ºC]; - is the surface temperature of the steel member; hnet, r ) ˜ H f ˜ H m ˜ 5.67 u 108 ˜ [(Tr  273)4  (Tm  273)4 ] [W/m2]; - is the view factor or configuration factor that is usually taken equal to 1.0 (see Chapter 3 or Table 4.4); - is the surface emissivity of the member (see Eq.(3.4)); - is the emissivity of the fire (usually H f 1.0 ); Tr - is the radiation temperature of the fire environment normally taken T r T g ; Ua - is the unit mass of steel, 7850 [kg/m3]; 't - is the time interval [s] ( d 5 [s]). _____ 55 Table 4.4: Coefficient of heat transfer by convection and view factor D c [W/m2K] ) Radiation considered separately 4 z0 Radiation implicitly considered in the convection 9 0 Standard fire curve ISO 834 25 Hydrocarbon curve 50 z0 z0 Parametric fire, zone fire models or external members 35 z0 Unexposed side of separating elements Surface exposed to the fire 4. TEMPERATURE IN STEEL SECTIONS In equation (4.16) the value of the section factor Am / V should not be taken less than 10 m-1 (because such a massive section would not have a uniform temperature) 1 and ' t should not be taken as more than 5 seconds. Eq. (4.16) can only be solved if the initial conditions and the boundary conditions (convection and radiation) are known. A common assumption regarding the initial conditions is that prior to the occurrence of fire the temperature of the whole section is 20 ºC. The values for the coefficients of both radiative and convective heat transfer proposed in Eurocode 3 were chosen such that a reasonable agreement with test results was obtained. However, empirical coefficients taking into account the so-called “shadow effect” (see Fig. 4.5) had to be introduced in Eq. (4.16) so that realistic values of the surface emissivity of steel (Hm = 0.7, a high but physically realistic value) and the fire emissivity (Hf = 1.0, a direct consequence of using the plate thermometer for furnace control in fire tests) can be used (Twilt et al., 2001). The view factor is always taken as 1.0 if the radiative flux given by the above equation is used in Eq. (4.16) written for a fully engulfed element. This factor can be calculated with a value lower than 1.0 when the fire source is localised and only part of the flux radiated by the fire source reaches the element (see Fig. 4.5) or when the concave shape of the profile is taken in to account, as will be shown in Section 4.9. _____ 56 Shadow effect No shadow effect Fig. 4.5: Influence of shape on the shadow effect For cross sections with a convex shape (e.g. rectangular or circular hollow sections) fully embedded in fire, the shadow effect does not play a role and consequently the correction factor k sh equals unity. For I-sections under nominal fire actions, the correction factor for the 1 This condition is verified for all commercial sections. 4.5. TEMPERATURE OF UNPROTECTED STEELWORK EXPOSED TO FIRE shadow effect may be determined from: ksh 0.9[ Am / V ]b /[ Am / V ] (4.17a) where [ Am / V ]b - is the box value of the section factor. The box value of the section factor of a steel section is defined as the ratio between the exposed surface area of a notional bounding box to the section and the volume of steel, as shown in Table 4.5, [m-1]. In all other cases, the value of ksh shall be taken as: k sh [ Am / V ]b /[ Am / V ] (4.17b) Ignoring the shadow effect (i.e., k sh 1 ) leads to conservative solutions. Instead of using the modified section factor, k sh [ Am / V ] , in Eq. (4.16), one can use ksh[ Am / V ] 0.9[ Am / V ]b (4.18a) if the correction factor for the shadow effect is given by Eq. (4.17a) or ksh [ Am / V ] [ Am / V ]b (4.18b) if the correction factor for the shadow effect is given by Eq. (4.17b). In both cases only the box value of the section factor, [ Am / V ]b , is needed. Annex E presents tables (Vila Real et al., 2009a) with values of the section factor for unprotected ( Am / V ) and protected ( Ap / V ) I and H European hot rolled steel profiles as well as values of the modified section factor ( k sh Am / V ) including the correction factor for the shadow effect ( k sh ) in accordance with EN 1993-1-2. An iterative procedure must be used to solve the simplified heat conduction equation (4.16) because the specific heat ca and the net heat flux h net , d are both temperature dependent. _____ 57 4. TEMPERATURE IN STEEL SECTIONS Table 4.5: Box value of the section factor [ Am / V ]b Section factor [Am / V]b Sketch b h h 2(b + h) ņņņņņņņņņņņņņ Steel crossSection area 2h + b ņņņņņņņņņņņņņ Steel crossSection area b _____ 58 box perimeter* ņņņņņņņņņņņņņ Steel crossSection area * The dotted line defines the box perimeter that corresponds to the smallest box surrounding the section (Franssen J-M, et al., 2009) Table 4.6 gives the temperature after 30 minutes and 60 minutes of standard fire ISO 834 exposure, for different values of the modified section factor k sh [ Am / V ] . 4.5. TEMPERATURE OF UNPROTECTED STEELWORK EXPOSED TO FIRE Table 4.6: Temperatures after 30 and 60 min of ISO 834 exposure k sh [ Am / V ] [m-1] Ta (30 min) [ºC] Ta (60 min) [ºC] 10 20 30 40 50 60 70 80 90 100 120 140 160 180 200 250 300 257.2 431.3 553.9 636.2 690.3 721.3 734.1 741.0 753.0 767.3 792.5 809.0 818.8 824.6 828.2 832.6 834.7 549.3 735.6 834.4 900.5 922.9 930.7 934.0 935.8 937.1 938.0 939.3 940.2 940.9 941.4 941.8 942.5 943.0 Fig. 4.6 shows the development of the temperature for some HEB profiles, obtained using Eq. (4.16), considering ksh 1.0 and the section factors given in Table 4.7. _____ 59 Table 4.7: Section factors for HEB profiles HE100B HE200B HE300B HE400B HE500B HE600B -1 Am/V [m ] ºC 218 147 116 98 89 86 1000 900 ISO 834 800 700 600 HEB100 500 400 300 HEB600 200 100 0 0 500 1000 1500 2000 2500 3000 3500 4000 segundos Seconds Fig. 4.6: Influence of section factor Am / V on the temperature rise in HEB profiles 4. TEMPERATURE IN STEEL SECTIONS The S shape observed in the rise of the steel temperature between 700 ºC and 800 ºC is due to the latent heat of metallurgical phase change of the steel in this range of temperatures. This effect is taken into account through the rapid increase in the specific heat of steel around 735 ºC shown in Fig. A.1. If a constant specific heat of 600 [J/(kgK)] is used, as suggested for simple calculation methods in the ENV version of the Eurocode 3 (1995), this plateaux does not exist, as shown in Fig. 4.7. ºC 1000 900 800 700 600 ca, temperature dependent 500 ca = 600 J/KgK 400 300 200 100 0 0 500 1000 1500 2000 2500 3000 3500 4000 seconds Fig. 4.7: Influence of the use of a constant value of the steel specific heat of 600 J/(KgK) _____ 60 By programming Eq. (4.16), it is easy to build tables or nomograms like the ones presented in Annex A for unprotected steel profiles subjected to the ISO 834 fire curve. The use of these tables and nomograms avoids the need to solve Eq. (4.16). The nomograms from Annex A are reproduced in Figs. 4.8 and 4.9. Knowing the section factor of the steel profile, k sh [ Am / V ] the temperature at a given time can be evaluated using these nomograms. [ºC] 900 800 30 min. 700 20 min. 600 15 min. 10 min. 500 400 5 min. 300 200 100 0 0 50 100 150 200 250 300 350 400 k sh Am / V [m 1 ] Fig. 4.8: Temperature as a function of the massivity factor for various times for unprotected sections subjected to the ISO 834 fire 4.6. TEMPERATURE OF PROTECTED STEELWORK EXPOSED TO FIRE [ºC] 60 m 1 100 m 1 40 m 1 25 m 1 800 200 m 1 700 15 m 1 600 500 400 300 200 k sh Am / V 10 m 1 100 0 0 10 20 30 40 50 60 Time [min.] Fig. 4.9: Temperature as a function of time for various massivity factors for unprotected sections subjected to the ISO 834 fire For a parametric fire as defined in Chapter 3, it is not easy to build such nomograms or tables and numerical calculation must be used. Fig. 4.10 shows the temperature development of a HE 220 B profile heated on all four sides by a parametric fire. These results were obtained with the program Elefir-EN, which is presented in Chapter 8. 1000 900 800 700 600 [ºC] 500 400 300 200 100 0 Parametric fire _____ 61 Steel temperature 0 20 40 60 80 100 120 140 160 [min] Fig. 4.10: Temperature development in a compartment with a parametric fire and the corresponding temperature of an unprotected HE 220 B 4.6. TEMPERATURE OF PROTECTED STEELWORK EXPOSED TO FIRE It is common practice to use thermal insulation to protect the steel where fire resistance higher than 30 minutes is required. There are many 4. TEMPERATURE IN STEEL SECTIONS forms of passive fire protection systems available to control the rate of temperature rise in steel members exposed to fire. It was shown in Table 4.2 that the insulation materials can be applied as contour encasement or hollow encasement. Basically there are three types of insulating materials: - Sprays; - Boards; - Intumescent paint. EN 1993-1-2 provides a simple design method to evaluate the temperature development of steel members insulated with fire protection materials. Assuming uniform temperature distribution, the temperature increase 'Ta,t of an insulated steel member during a time interval ' t , is given by 'T a ,t O p A p / V T g ,t  T a ,t 't  eI /10  1 'T g ,t [ºC] d p ca U a 1  I / 3 (4.19) and 'Ta,t t 0 if 'T g , t ! 0 where the amount of heat stored in the protection is I _____ 62 c p d p U p Ap ˜ ca U a V (4.20) and Ap / V - is the section factor for steel members insulated by fire Ap protection material, [m-1]. Annex A.3 gives the section factor for a range of practical cases; - is the appropriate area of fire protection material per unit Op length of the member [m2/m]; - is the volume of the member per unit length, [m3/m]; - is the thermal conductivity of the fire protection system, dp [W/mK]. Annex A.6 gives the thermal conductivity for a range of practical fire protection systems; - is the thickness of the fire protection material [m]; cp - is the specific heat of the fire protection material, [J/kgK]. V 4.6. TEMPERATURE OF PROTECTED STEELWORK EXPOSED TO FIRE Up Annex A.6 gives the specified heat for a range of fire protection systems; - is the unit mass of the protection. Annex A.6 gives the unit mass for a range of fire protection materials [ kg / m3 ]; ca - is the temperature dependent specific heat of steel, from Ta,t Annex A.1 [J/kgK]; - is the steel temperature at time t [°C]; T g ,t - is the ambient gas temperature at time t [°C]; 'T g ,t - is the increase of the ambient gas temperature during the time Ua interval 't [K]; - is the unit mass of steel, 7850 [kg/m3]; 't - the time interval [seconds] ( d 30 [s]). Equation (4.19) is an approximation and only valid for small values of the factor I. According to Wang Z (2004) this factor should normally not be higher than 1.5, but this limitation is not given in the Eurocode. The design value of the net heat flux reflecting the heat transfer by convection and radiation does not appear in Eq. (4.19). In fact the temperature drop over the insulation is relatively large and, consequently, the surface temperature of the insulation is close to the gas temperature. The thermal resistance between the gas and the surface of the insulation is neglected ( T g | Tm ) and the temperature rise in the steel section is governed by the difference in temperature between the surface of the insulation (i. e., the gas temperature) and the steel profile, with only the insulation material providing thermal resistance to the heat conduction (Fig. 4.11). Steel Tg Tg Tm Tg ~ Tm Ta Insulation material Fig. 4.11: Temperature in protected steelwork _____ 63 4. TEMPERATURE IN STEEL SECTIONS _____ 64 The thermal properties of the insulation material that appear in Eq. (4.19) must be determined experimentally in accordance with prEN 13381-4 (2008). According to this European standard, tests on loaded and unloaded beams as well as tests on unloaded short columns, with various massivity factors and various protection thicknesses, subjected to the standard fire should be made. The thermal conductivity of the insulation material is calculated from the recorded steel temperature using the inverse of Eq. (4.19). The unit mass and the constant specific heat must be provided by the manufacturer of the product. If the specific heat is unknown, a value of 1000 J/kgK should be assumed, Franssen et al. (2009). The thermal conductivity of most commonly used passive fire protection materials increases with increasing temperature. Therefore the values of the thermal properties given for room temperature applications should not be used as this will lead to unsafe results in the fire situation. Eq. (4.19) has to be integrated with respect to time to obtain the development of the temperature in the steel section as a function of time. EN 1993-1-2 recommends that the time step interval ' t should not be taken as more than 30 seconds, a value that will ensure convergence even with an explicit integration scheme. Any negative increment of the temperature 'Ta,t, given by Eq. (4.19), corresponding to an increase of the gas temperature, 'Tg,t > 0, must be considered as zero. In the absence of specific data, the generic data given in Table A.6 of Annex A.6 may be used, (ECCS, 1995). The tabulated values for the thermal conductivity, Op, are normally for dry materials. For moist fire protection materials, the steel temperature increase, 'Ta,t, may be modified to allow for a time delay tv in the rise of the steel temperature when it reaches 100 ºC, due to the latent heat of vaporization of the moisture, as shown in Fig. 4.12. The length of the horizontal plateau at 100 ºC can be evaluated from the following expression, ECCS (1983): tv pU p d p 5O p 2 [min.] (p in %) (4.21) where, p is the moisture content of the protection material. No delay is allowed if the moisture is included in the value of the thermal conductivity, ECCS (1983). 4.6. TEMPERATURE OF PROTECTED STEELWORK EXPOSED TO FIRE ºC 1000 900 ISO 834 800 700 600 Steel temperature (dry insulation) 500 400 300 200 Steel temperature (insulation containing moisture) 100 0 tv 0 400 800 1200 1600 2000 2400 2800 3200 3600 seconds Fig. 4.12: Time delay due to the moisture content. For light weight insulation materials Eq. (4.19) can be simplified by taking I 0 . The method given in ECCS (1983) suggests that the heat capacity of the protection material can be ignored if it is less than half that of steel section, such that d p Ap c p U p  In this expression, Ua ca U aV 2 7850 [kg/m3] and a value ca (4.22) 600 [J/kgK], for the specific heat of the steel can be used to check if the material is a light weight material or not. If the specific heat of the protection material, c p , is neglected, the amount of heat stored at the protection can be taken as I 0 and Eq. (4.19) becomes 'T a , t O p Ap 1 d p V ca U a T g , t  T a , t 't (4.23) The advantage of using this equation is that it is possible to build tables of two entries or nomograms like the ones presented in Annex A. One of the entries is the time and the other is the modified massivity factor Ap O p ˜ V dp (4.24) The table and the nomograms presented in Annex A.5 have been _____ 65 4. TEMPERATURE IN STEEL SECTIONS developed for the standard fire curve ISO 834 using Eq. (4.23) with a time step of 10 seconds. This table and the nomograms provide conservative results, because the amount of heat stored in the protection together with any moisture it may contain have been neglected. The use of the table and the nomograms from Annex A.5 avoid the need for solving the Eq. (4.19). The nomograms from Annex A have been reproduced schematically in Fig. ( 4.13) and (4.14). In both nomograms the temperature is given as a function of the modified massivity factor, given in Eq. (4.24). [ºC] 240 min. 180 min. 120 min. 800 90 min. 700 60 min. 600 500 30 min. 400 300 200 100 0 0 _____ 66 250 500 750 1000 1250 1500 1750 Ap O p ˜ V dp 2000 [ W/m 3 K ] Fig. 4.13: Temperature as a function of the modified massivity factor for various times, for protected profiles subjected to the ISO 834 curve 1500 W/m 3 K 1000 W/m 3 K 800 W/m3 K [ºC] 800 600 W/m3 K 2000 W/m3 K 700 400 W/m 3 K 600 300 W/m3 K 500 200 W/m 3 K 400 300 200 Ap O p ˜ V dp 100 0 0 30 60 90 120 150 180 100 W/m 3 K 210 240 Time [min.] Fig. 4.14: Temperature as a function of time for various modified massivity factors, for protected profiles subjected to the ISO 834 curve 4.6. TEMPERATURE OF PROTECTED STEELWORK EXPOSED TO FIRE The table and the nomograms given in Annex A.5 were created assuming I 0 , and are therefore only valid for ligh tweight insulation material. The table and nomograms can be used for heavy materials provided the modified massivity factor is corrected, using the following expression, according to ECCS (1985): Ap O p ˜ V dp · § 1 ¸¸ ˜ ¨¨ ©1I 2 ¹ (4.25) This procedure gives good results for temperatures in the range of 350 ºC to 700 ºC, which is representative of the critical temperatures that normally occur in structural steel members. This is helpful in the pre-design phase when estimating the thickness of fire protection. The estimated thickness together with Eq. (4.19) can be used to obtain a more accurate result. The importance of correcting the modified massivity factor (4.24) according to the expression (4.25) is demonstrated in the following example. Consider a steel member protected with gypsum boards, which is a heavy fire insulation material with the following characteristics: Ap 110 m-1 V Op 0.2 W/(mK) cp 1700 J/(kgK) dp 0.023 m _____ 67 U p 800 kg/m3 These values lead to a ratio of heat stored in the protection: I c p d p U p Ap ˜ ca U a V 1700.0 ˜ 0.023 ˜ 800 ˜ 110 0.731 600 ˜ 7850 The table given below shows the temperatures obtained with the Eq. (4.19) using the program Elefir-EN and with the nomograms from Annex A.5 considering the modified massivity factor given by Eq. (4.24) and the same factor corrected as in Eq. (4.25). 4. TEMPERATURE IN STEEL SECTIONS Time Eq. (4.19) Nomogram with (minutes) (ºC) Ap O p ˜ V dp 15 30 60 120 84 186 370 627 140 263 460 699 Nomogram with Ap O p ˜ V dp (+67.2%) (+41.3%) (+24.4%) (+11.5%) 112 210 381 616 § 1 · ¸¸ ˜ ¨¨ ©1I 2 ¹ (+31.0%) (+13.1%) (+2.9%) (-1.7%) ( ) error comparing with the solution from Eq. (4.19). From this table it can be concluded that the nomogram built for light weight insulation materials gives good approximation to the results from Eq. (4.19) if the corrected massivity factor Eq. (4.25) is used. Fig. 4.15 shows the influence of the parameter I for this example. ºC 800 c) 700 600 500 400 b) 300 _____ 68 200 100 a) 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 seconds Fig. 4.15: Influence of the parameter I , in a heavy fire insulation material. a) Eq. (4.19); b) Eq. 4.23 with Eq.(4.25); c) Eq. 4.19 with I 0 or Eq. (4.23) Consider now a steel profile protected with a light weight fire insulation material, such as mineral wool, the characteristics of which are listed below: Ap V 225 m-1 Op 0.2 W/(mK) cp 1200 J/(kgK) 4.6. TEMPERATURE OF PROTECTED STEELWORK EXPOSED TO FIRE dp 0.02 m U p 150 kg/m3 Fig. 4.16 shows that the influence of the amount of heat stored in the protection, I , is not relevant, and so Eq. (4.19) can be used with I 0.0 , for light weight fire insulation material, i. e., tables or nomograms, like the ones presented in Annex A, can be used without any correction. Levematerial LightProtecção weight protection ºC 1000 800 600 I 400 0.0 { I z 0 .0 200 0 0 1000 2000 3000 4000 5000 6000 7000 8000 seconds segundos Fig. 4.16: Influence of the parameter, I , in light weight fire insulation material Fig. 4.17 shows the temperature development of an unprotected HE 220 A profile heated on four sides by the fire defined in example 3.3. ºC 900 800 700 600 500 400 300 200 100 0 Parametric fire Steel temperature 0 20 40 60 80 100 120 140 160 [min] Fig. 4.17: Temperature of an unprotected HE 220 A heated by the parametric fire from example 3.3 _____ 69 4. TEMPERATURE IN STEEL SECTIONS If the same profile is fire protected with gypsum board encasement with a thickness of d p 15 mm, the temperature development is shown in Fig. 4.18. ºC 900 800 700 600 500 400 300 200 100 0 Parametric fire Temperature of unprotected steel Temperature of the protected steel 0 20 40 60 80 100 120 140 160 [min] Fig. 4.18: Temperature of a HE 220 A heated by the parametric fire of example 3.3 Example 4.1: What is the temperature of an unprotected rectangular bar with a cross section of 200 u 50 mm2 after 30 minutes of standard fire exposure on four sides? _______________________________ The section factor of this convex section takes the value: _____ 70 Am / V 2 u (b  t ) but As the section is convex, k sh 2 u (0.2  0.05) 0.2 u 0.05 50 m 1 1 , and the modified section factor is k sh [ Am / V ] 1.0 ˜ 50 50 m-1 With this section factor and by interpolation in Table A.4, a temperature of 678.5 ºC is obtained. If Eq. (4.19) is used with the program Elefir-EN, a value of the temperature of 690 ºC is obtained. _____________________________________________________________ Example 4.2: What is the temperature of an unprotected circular hollow section with a diameter of d 220 mm and a thickness of t 5 mm after 60 minutes of standard fire exposure? 4.6. TEMPERATURE OF PROTECTED STEELWORK EXPOSED TO FIRE t d _______________________________ The external perimeter is S ˜d P 691 .15 mm The area of the cross section is S A 4 >d 2  ( d  2t ) 2 @ 3377.21 mm2 The section factor is Am V P A 0.2047 mm-1 = 204.7 m-1 According to Annex A.2, the section factor is given by: Am V P A As the section is convex, k sh d 1 200 m-1 | (d  t )t t 1 , and the modified section factor, using the later value of the section factor, is k sh [ Am / V ] 200 m-1 Table A.4 from Annex A gives the temperature of 942 ºC. _____________________________________________________________ Example 4.3: What is the temperature of an unprotected HE 200 A profile after 30 minutes of standard fire exposure on four sides? _______________________________ The section factor for an HE 200 A is: Am / V 211 m-1 _____ 71 4. TEMPERATURE IN STEEL SECTIONS The HE 200 A has the following geometric characteristics: b 200 mm h 190 mm 2 A 53 .83 cm and the box value of the section factor [ Am / V ]b takes the value [ Am / V ]b 2 u (b  h) A 2 u (0.2  0.19) 144.9 m-1 53.83 u 10  4 The shadow factor, k sh is given by: k sh 0.9[ Am / V ]b /[ Am / V ] 0.9 ˜144.9 / 211 0.618 Taking into account the shadow effect, the modified section factor has the value ksh[ Am / V ] 0.618 ˜ 211 130.4 m-1 This value should be obtained without evaluating k sh , using Eq. (4.18a): k sh [ Am / V ] 0.9[ Am / V ]b _____ 72 0.9 ˜144.9 130.4 m-1 Interpolating in Table A.4 yields a temperature of 786 ºC. If Eq. (4.16) is used, a temperature of 802 ºC is obtained. _____________________________________________________________ Example 4.4: What is the thickness of fibre-cement board encasement for a IPE 300 heated on three sides to be classified as R90 if the critical temperature is 654 ºC? _______________________________ The following thermal properties of the fibre-cement are defined in Annex A.6: Op 0.15 W /(m ˜ K ) cp 1200 J/(kgK) Up 800 kg/m3 The massivity factor for the IPE 300 with hollow encasement heated on three sides is: 4.6. TEMPERATURE OF PROTECTED STEELWORK EXPOSED TO FIRE Ap / V 139 m 1 By interpolation in Table A.5 from Annex A, for a temperature of 654 ºC, at 90 minutes of standard fire exposure, the modified section factor is: Ap O p ˜ d 1210 W/(m 3 ˜ K ) V dp and the thickness is dp t Ap / V 1210 Op 139 ˜ 0.15 0.017 m 17 mm 1210 This thickness can be corrected if the amount of heat stored in the protection, I , is taken into account, according Eq. (4.20) and using Eq. (4.25) to obtain the corrected thickness. Ap O p 1 ˜ ˜ d 1210 W/(m 3 ˜ K) V d p 1I / 2 The following iterative procedure is needed to evaluate the corrected thickness: dp I c p d p U p Ap ˜ ca U a V dp t Ap V ˜ Op 1 ˜ 1210 1  I / 2 0.017 1200 ˜ 0.017 ˜ 800 ˜ 139 600 ˜ 7850 0.0139 0.0139 1200 ˜ 0.0139 ˜ 800 ˜139 600 ˜ 7850 0.0144 0.0144 1200 ˜ 0.0144 ˜ 800 ˜139 600 ˜ 7850 0.0143 0.0143 1200 ˜ 0.0143 ˜ 800 ˜ 139 600 ˜ 7850 0.0143 With this procedure a thickness of 14.3 mm is obtained, instead of 17 mm. _____________________________________________________________ _____ 73 4. TEMPERATURE IN STEEL SECTIONS Example 4.5: At what time will a HE 220 B column protected with a 20mm thick gypsum board reach a temperature of 559 ºC when heated by the standard fire curve on four sides? _______________________________ The massivity factor for the HE 220 B with hollow encasement heated on four sides is Ap / V 96 m 1 . Annex A.6 gives the thermal properties of gypsum boards: Op 0.2 W/(m ˜ K) c p 1700 J(kg ˜ K) U p 800 kg/m3 p 20% (moisture content) As the thickness of the insulation is known, the ratio of heat stored in the protection can be evaluated as I cpd pU p ca U a ˜ 1700 ˜ 0.02 ˜ 800 ˜ 96 600 ˜ 7850 Ap V 0.554 and the corrected modified section factor _____ 74 Ap O p 1 ˜ ˜ V d p 1I / 2 96 ˜ 0.2 1 ˜ 0.02 1  0.554 / 2 752 W/(m3 ˜ K) Based on this value and for the temperature of 559 ºC, a double interpolation in the table of Annex A.5 gives a time of 97 minutes. The delay time due to the moisture content is, according to Eq. (4.21): tv pU p d p 5O p 2 20 ˜ 800 ˜ 0.02 2 5 ˜ 0 .2 6 min . The time to reach the temperature of 559 ºC is then: t 97  6 103 min . If the program Elefir-EN is used the temperature of 559 ºC is reached after 98.9 minutes, as shown in Fig. 4.19. 4.6. TEMPERATURE OF PROTECTED STEELWORK EXPOSED TO FIRE ºC 1200 ISO 834 1000 Steel temperature 800 559 ºC 600 400 200 0 0 20 40 60 80 100 120 [min] Fig. 4.19: Temperature of a HE 220 B protected with gypsum boards _____________________________________________________________ Example 4.6: Consider a IPE 300 profile protected with 20mm thick fibre-cement board encasement. Calculate the temperature after 60 minutes of standard fire exposure on four sides taking into account the amount of heat stored in the protection and the time delay due to the moisture content. _______________________________ The following thermal properties of the fibre-cement board are defined in Annex A.6: Op 0.15 W/(m ˜ K) cp 1200 J/(kgK) Up 800 kg/m3 p _____ 75 5% (moisture content) The fire insulation material is considered as a heavy material if its heat capacity is bigger than half that of the steel section, i.e.: d p Ap c p U p ! c a U aV 2 As the section factor of the IPE 300 is A p / V 167 m 1 , that relation is verified 0.02 ˜ 167 ˜ 1200 ˜ 800 ! 600 ˜ 7850 2 4. TEMPERATURE IN STEEL SECTIONS 3206400 ! 2355000 and the 20 mm thick fibre-cement can be considered as a heavy material. The ratio of heat stored in the protection takes the value: cP d P U p I ca U a ˜ 1200 ˜ 0.02 ˜ 800 ˜ 167 600 ˜ 7850 Ap V 0.681 and the corrected section factor: Ap O p 1 ˜ ˜ V d p 1I / 2 934 W/(m3K) Interpolating in Table A.5 from Annex A yields a temperature of 453 ºC. If Eq. (4.19) is used, for example with the software Elefir-EN, the temperature of 451 ºC is obtained, which is close to the value obtained by interpolation. The time increase in fire resistance due to the moisture content of the protection can be calculated as follows: tv _____ 76 pU p d p 5O p 2 5 ˜ 800 ˜ 0.02 2 5 ˜ 0.15 2.13 # 2 min . Taking into account this delay, the temperature is calculated after 60  2 58 minutes instead of 60 minutes. Eq. (4.19) gives, after 58 minutes, the temperature T 439 ºC With the software Elefir-EN a temperature of 443 ºC is obtained if the moisture content is taken into account. It should be noted that if the fibre-cement board was considered as light weight insulation material and the corrected section factor was not used, then table A.6 would give for Am O p ˜ V dp 167 ˜ 0.15 1252.5 W/m3K 0.02 a temperature of T 526 ºC instead of the 453 ºC previously obtained, with an error of | 16 .7% , but on the safe side. 4.7. INTERNAL STEELWORK IN A VOID PROTECTED BY HEAT SCREENS 4.7. INTERNAL STEELWORK IN A VOID PROTECTED BY HEAT SCREENS This section deals with two different geometrical situations. The first situation is when a steel beam is underneath a slab, with a horizontal heat screen present underneath the profile, as shown in Fig. 4.20. Heat screen Fire Fig. 4.20: Steel beam with a heat screen underneath (elevation) The second situation is when a steel column is located between two vertical heat screens. Fig.4.21 shows a column with heat screens on both sides and the fire on one side only. In other situations, the fire could be on each side of the column. Heat screen Heat screen Fire Fig. 4.21: Protected steel column with heat screens on both sides (plan view) _____ 77 4. TEMPERATURE IN STEEL SECTIONS In both cases, there is a gap between the heat screen and the steel section. If the heat screen touches the steel section, the method describes in this section does not apply. The development of the temperature in the steel section must be calculated by one of the methods described in Sections 4.5 or 4.6, depending on whether or not the steel member is thermally protected. The fire temperature is taken as the gas temperature in the void, and this is determined experimentally according to EN 13381-1 (2006), for horizontal protective membranes, or EN 13381-2 (2008), for vertical protective membranes, as appropriate. 4.8. EXTERNAL STEELWORK 4.8.1. General principles _____ 78 This section deals with steel members that are located outside the envelope of the building and are heated by flames coming from inside the building. The temperature and the size of the flames are determined using the methods given in Section 3.7. The temperature distribution in external steelwork is determined in a steady state situation based on the steady state radiative fluxes determined according to Section 3.7. This means that a temperature will be determined in the steel section, but no information will be available about the time required for this temperature to be established. As a consequence, the verification of the stability of the steel member is only possible in the temperature or in the load domain, but not in the time domain. The result will be either a fail or a pass. If the calculation shows that the stability is ensured, any fire requirement expressed in term of fire class will be satisfied. If the calculation shows that the stability is not ensured, it will not be possible to determine the time of collapse and, thus, the fire resistance class of the member. Recent research work has been undertaken to extend the theory to transient situations, but the results of this work have not yet been introduced in the Eurocode. The temperature in the steel section results from an energy balance between the radiative heat flux received from the fire compartment and the radiative and convective heat flux emanating from the openings, on one 4.8. EXTERNAL STEELWORK hand, and the radiative and convective heat flux lost to the ambient atmosphere, on the other hand. The influence of heat screens must be considered. If the heat screens are non-combustible and have a fire resistance EI of at least 30 minutes according to EN ISO 13501-2, it is assumed that there is no radiative heat transfer to those sides of the section that are protected by the screens. Geometrical parameters such as size and location of the steel member with respect to the compartment and its openings and size of the openings must be considered. The detailed procedure is described in Annex B of Eurocode 3. Only the general principles are described in this book. The assumptions are that the fire is confined to one compartment only and that all openings in the façade are rectangular. A distinction is made between members that are engulfed in flames and members that are not engulfed in flames, depending on their position relative to the openings: x A member that is not engulfed in flames receives heat from all the openings in the wall that it is facing and from all flames coming out of these openings. x A member that is engulfed in flames receives heat only from the engulfing flame (convection and radiation) and from the opening from which this flame is emanating (radiation). The equilibrium temperature in the steel member Tm is calculated from the heat balance given in Eq. (4.26) for a member not engulfed in flames and in Eq. (4.27) for a member engulfed in flames. V Tm4  D Tm ¦I V Tm4  D Tm z  ¦ I f  D 293 I z  I f  D Tz (4.26) (4.27) where V is the Stefan Boltzmann constant equal to 5.67u10-8 W/m²K4, D is the convective heat transfer coefficient, Iz is the radiative heat flux from a flame, If is the radiative heat flux from an opening and Tz is the flame temperature. The convective heat transfer coefficient D is calculated from the rules given in Annex B of Eurocode 1, see Section 3.7 of this book, using an effective cross section dimension equal to the average of both dimensions in the section. _____ 79 4. TEMPERATURE IN STEEL SECTIONS The radiative heat flux from an opening is calculated from Eq. (4.28). If I f 1  D f V T f4 (4.28) where If is the overall configuration factor of the member from that opening and Df is the absorptivity of the flame, to be determined according to the rules given in Annex B of Eurocode 3. It is not possible to repeat in detail in this book all the rules that are given in Eurocode 1 and Eurocode 3. An example is given hereafter that will allow the reader to follow the procedure for a simple case and find his way through all the clauses of the Eurocodes where this procedure is described. 4.8.2. Example A hotel is constructed from a steel skeleton that supports modular prefabricated rooms. Part of the structure is external, as shown in Fig. 4.22. The floor to floor height is 2.8 m and the grid is 3 m wide. The windows are 1.5 m wide and 1 m high. The external steel structure is 1 m away from the façade. Each room is 2.8 m wide and 6 m deep, with a floor to ceiling height of 2.5 m. The effect of the wind is not considered. Calculate the temperature in the external 0.2 m u 0.2 m steel square tube columns. _____ 80 2800 3000 1000 Fig. 4.22: Two elevations of the façade of a hotel [mm] 4.8. EXTERNAL STEELWORK The area of the window (1.5 m²) is less than 50% of the area of the wall (7 m²). There is no forced draught and the door of the room is assumed to be closed. _______________________________ Rate of heat release in the compartment. Floor area Af = 6 u 2.8 = 16.8 m² Design fire load qf,d = 377 MJ/m² (taken as qf,k) Qfuel control = 377 u 16.8 / 1200 = 5.28 MW, see Eq. (3.35) D/W = 6 / 2.8 = 2.14, see Eq. (3.34) (B.1 in Eurocode 1) Height of the window heq = 1 m Area of vertical openings Av = 1.5 u 1 = 1.5 m² Total area of enclosure At = 2 u 16.8 + 2 u 2.5 (6 + 2.8) = 77.6 m² Opening factor O = 1.5 u (1)0.5 / 77.6 = 0.019 m1/2 Qair control = 3.15 1  e0.036 0.019 1.5 1 = 2.74 MW, see Eq. (3.35) 2.14 => Q = 2.74 MW Temperature of the fire compartment : Tf 16.8u 377 = 587, see Eq. (3.37) 1.5u 77.6 6000 1  e 0.1 0.019 0.019 1  e 0.00286u 587  20 = 689°C Flame height LL § 2.74 · 1.9 ¨ ¸ © 1.5 ¹ 2 3  1 = 1.84 m, see Eq. (3.38) Flame temperature at the window Lf = 1.84 + 1/2 = 2.34 L f wt Q = 2.34 u 1.5 2.74 = 1.401 > 1. We take L f wt Q = 1.0 TW 520 1  0.4725 u 1.0  20 = 1006°C _____ 81 4. TEMPERATURE IN STEEL SECTIONS 650 1500 f 667 d e PLAN VIEW b 233 c a Lx C Level of steel temperature calculation 1840 B A 333 333 233 _____ 82 667 [mm] ELEVATION Fig. 4.23: Shape of the flame Shape of the flame See Fig. 4.23 Temperature along the flame LX wt Q = LX u 1.5 / 2.74 = 0.55 LX, must be smaller than 1. LX must be lower than 1.83 m. TZ 1006  20 1  0.4725u 0.55 LX  20 = 986 (1 - 0.26 u LX) + 20 = 1006 – 256 LX, see Eq. (3.40) 4.8. EXTERNAL STEELWORK The temperature varies linearly along the flame axis. From 1006°C at the window, see point A on Fig. 4.23, the temperature decreases to 885°C at point B (LX = 0.471 m) and 538°C at point C, located 1.83 - 0.471 = 1.359 m above point B. Above Point C, where LX wt Q is larger than 1, the Eurocode is not clear about the temperature distribution. It is assumed here that the term LX wt Q is replaced by 1 in the equation and the temperature keeps the same value of 538°C until the tip of the flame. Clause B.1.4 (4) of Eurocode 3 states that the radiative heat flux from the flames may be based on the dimensions of an equivalent rectangular flame. This approximation would clearly simplify the process but it has nevertheless not been made here. This is because the Eurocode is not very clear about the size and equivalent temperature of this equivalent rectangular flame. For example, Fig. B.3 shows a height z of the equivalent flame, and z is said to be defined in Annex B of Eurocode 1 but z is not used in Eurocode 1. If z is the height of the window, then the equivalent rectangular flame is not linked to the height of the real flame LL. Also, the temperature of the flame should be taken at a distance l from the opening equal to h/2 (Eq. B.11a) and, assuming that h is the height of the window, this point has also no correlation with the flame length. This is why physical principals and "using appropriate adaptations of the treatments given in B.2", see B.1.3 (4), have been used here. It is not possible to determine directly the vertical position along the column where the level of temperature is the highest. Strictly speaking, the temperature should be evaluated at different positions along the column. The crushing load of the column is then determined by the maximum temperature of all sections. If the failure mode is by buckling, it will be more difficult to evaluate the load bearing capacity of the column with a temperature that varies along the length using simple models. Because of the longer length of the vertical part of the flame (i.e., the part that extends along the wall above the window), it is assumed that the point of maximum temperature is facing this vertical part. Because of higher temperatures prevailing in the lower part of the flame, it is assumed that the point of maximum temperature is slightly below mid-level of the vertical part. The temperature will be calculated in this example only at a level of 0.6 m above point B. _____ 83 4. TEMPERATURE IN STEEL SECTIONS For a precise determination of the steel temperature at this point, the vertical part of the flame should be divided along the height in a number of different zones, each one having its own temperature. For simplicity, the temperature in the vertical part of the flame will be approximated as constant and equal to the value at 0.6 m, i.e., LX = 1.071. This temperature is 732°C. The lateral triangular part of the flame (the part that contains the line AB on the elevation of Fig. 4.23) is also visible from the section, see the plan view. To allow direct application of the formulae for view factors, it will be approximated as a square surface of equal surface area (0.222 m² = 0.47 m u 0.47 m), with a uniform temperature equal to 945°C (average between TA and TB) and located at the centre of gravity of the triangle. This equivalent square surface is represented by a dotted line on the elevation. View factors _____ 84 1.a) Between the centre of side b-c on the profile and the vertical plane d-e on the flame, see the plan view on Fig. 4.23. The planes are parallel and the configuration corresponds to the one depicted in Fig. 3.6 and Fig. 3.8. Distance (horizontal) between the two planes s = 0.233 m. The vertical part of the flame is divided into 4 zones (2 of them with negative contribution) and the calculations are summarized in the following table. Zone 1 Zone 2 Zone 3 Zone 4 Total h (vertical) 0.60 0.60 1.24 1.24 - w (horizontal) 2.25 0.75 2.25 0.75 - ‫׋‬I 0.233 -0.227 0.245 -0.237 0.014 1.b) Between the centre of side b-c on the profile and the plane e-f on the flame, see the plan view on Fig. 4.23. The planes are perpendicular and the configuration corresponds to the one depicted in Fig. 3.7 and Fig. 3.8. Distance (horizontal) between points P and x: s = 0.75 m. The vertical part of the flame is divided into 4 zones and the calculations are summarized in the following table. 4.8. EXTERNAL STEELWORK  Zone 1 Zone 2 Zone 3 Zone 4 Total h (vertical) 0.60 0.60 1.24 1.24 - w w (horizontal) 0.900 0.233 0.900 0.233 - ‫׋‬I 0.059 -0.008 0.081 -0.010 0.121 The total view factor from the side b-c to the section of the flame at 732°C is thus equal to 0.014 + 0.121 = 0.135. 2) Between the centre of side b-c on the profile and the lateral triangular part of the flame. The planes are perpendicular and the configuration corresponds to the one depicted in Fig. 3.7. Distance (horizontal) between points P and x: s = 0.75 m. The vertical part of the flame is divided into 4 zones and the calculations are summarized in the following table.  h (vertical) Zone 1 Zone 2 Zone 3 Zone 4 Total 1.057 1.057 0.587 0.587 - w (horizontal) w 0.913 0.443 0.913 0.443 - ‫׋‬I 0.078 -0.031 -0.059 0.024 0.012 3.a) Between the centre of side a-b on the profile and the vertical plane e-f on the flame, see plan view on Fig. 4.23. Distance (horizontal) between the two parallel planes s = 0.65 m. The calculations are summarized in the following table.  h (vertical) Zone 1 Zone 2 Zone 3 Zone 4 Total 0.60 0.60 1.24 1.24 - w (horizontal) w 1.00 0.33 1.00 0.33 - ‫׋‬I 0.154 -0.089 0.195 -0.108 0.151 _____ 85 4. TEMPERATURE IN STEEL SECTIONS 3.b) Between the centre of side a-b on the profile and the plane e-d on the flame, see the plan view on Fig. 4.23. The planes are perpendicular. Distance (horizontal) between points P and x: s = 0.333 m.  Zone 1 Zone 2 Zone 3 Zone 4 Total h (vertical) w w (horizontal) ‫׋‬I 0.60 0.60 1.24 1.24 - 2.15 0.65 2.15 0.65 - 0.163 -0.119 0.196 -0.133 0.106 The total view factor from the side a-b of the section to the flame at 732°C is thus equal to 0.151 + 0.106 = 0.257. 4) Between the centre of side a-b on the profile and the lateral triangular part of the flame. Distance (horizontal) between the two parallel planes s = 0.65 m.  Zone 1 Zone 2 Zone 3 Zone 4 Total _____ 86 h (vertical) 1.057 1.057 0.587 0.587 - w (horizontal) 1.013 0.543 1.013 0.543 - ‫׋‬I 0.189 -0.147 -0.152 0.121 0.011 5) Between the centre of side b-c on the profile and the window. Distance (horizontal) between the two parallel planes s = 0.90 m.  h (vertical) w w (horizontal) ‫׋‬I Zone 1 1.6 2.25 0.209 Zone 2 1.6 0.75 -0.149 Zone 3 0.6 2.25 -0.135 Zone 4 0.6 0.75 0.102 Total - - 0.027 4.8. EXTERNAL STEELWORK 6) Between the centre of side a-b on the profile and the window. The planes are perpendicular. Distance (horizontal) between points P and x: s = 1.0 m.  h (vertical) w w (horizontal) ‫׋‬I Zone 1 1.6 2.15 0.121 Zone 2 1.6 0.65 -0.037 Zone 3 0.6 2.15 -0.069 Zone 4 0.6 0.65 0.024 Total - - 0.039 Radiative heat flux from the opening, see Eq. (B.3) in Eurocode 3 Overall configuration factor from the opening, see Eq. (B.4) in Eurocode 3: If = (0.027 u 200 + 0.039 u 200) / ( 4 u 200) = 0.017 Emissivity of the opening, see B.1.3 (6) in Eurocode 3: Hf = 1.0 Absorptivity of the flame, see B.2.4 (1) in Eurocode 3: az = 0 If = 0.017 u 1 u (1-0) u 5.67 u 10-8 u (689+273)4 = 826 W/m² Radiative heat flux from the flame Radiation from the part of the flame above the window. Overall configuration factor from the flame, see Eq. (B.5) in Eurocode 3. Iz = (0.135 u 200 + 0.257 u 200) / ( 4 u 200) = 0.098 Emissivity of the flame: Flame thickness = 1.5 m, Eq. (B.10a) in Eurocode 3. Hf = 1 - e-0.3 u 1.5 = 0.362, Eq. (B.26) in Eurocode 1 Iz1 = 0.098 u 0.362 u 5.67 u 10-8 u (732+273)4 = 2052 W/m² Radiation from the triangular part of the flame. Overall configuration factor from the flame, see Eq. (B.5) in Eurocode 3 _____ 87 4. TEMPERATURE IN STEEL SECTIONS Iz = (0.012 u 200 + 0.011 u 200) / ( 4 u 200) = 0.006 Iz1 = 0.006 u 0.362 u 5.67 u 10-8 u (945+273)4 = 271 W/m² Convective heat transfer coefficient Effective cross sectional dimension: deq = 0.2, see B.1.3 (2) in Eurocode 3 Dc = 4.67 (1/0.2)0.4 (2.74/1.5)0.6 = 12.8, see B.4.1 (12) in Eurocode 1 Equation of equilibrium, Eq. (B.1) in Eurocode 3 5.67 u10-8 T4 + 12.8 T = 826 + 2052 + 271 + 293 u 12.8 This yields as a solution: T = 412 K or 139°C. This temperature is, in this academic example, rather low. A new calculation should normally be made taking into account the effect of wind. The process is similar to the one described here; only the geometrical quantities are more complicated. 4.9. VIEW FACTORS IN THE CONCAVE PART OF A STEEL PROFILE _____ 88 A view factor can be introduced to account for “parts” of the structural element that shielded from radiative heat (ECCS TC3, 2001). It is defined as the ratio between radiative heat leaving an emitting surface and the radiative heat arriving at a receiving surface. The general formula for the view factor is (ECCS TC3, 2001, Drysdale D, 1999) I 1 Ar ³³ Ae Ar cosIe cosIr dAe dAr S r2 (4.29) For two-dimensional cases (see Fig. 4.24) the view factor can be given by (ECCS TC3, 2001): I AC  BD  AD  BC 2CD but I d 1.0 (4.30) 4.9. VIEW FACTORS IN THE CONCAVE PART OF A STEEL PROFILE By definition, the value of the view factor is between zero and unity. Its value depends on the distance between the two surfaces, the size of the surfaces and their relative orientation (see Fig. 4.24). C Ar Ir D r Ie A Ae B Fig. 4.24: Emitting and receiving surfaces for radiative heat ( Ae - emitting surface; Ar - receiving surface) As an example, Fig. 4.25 shows the values of the view factors for the internal surfaces of a fully fire engulfed HE 400 B. These are the view factors between the surface of the box contour through which the energy passes and each of the internal surfaces of the section that receive this energy. Each view factor can be calculated according to Eq. (4.30) and using Fig. 4.26 for this particular case. i) I ii) I iii) I For the web surface according to Fig. 4.26a AC  BD  AD  BC 2 ˜ 143.25  2 ˜ 143.25 2  352 2 2CD 2 ˜ 352 0.67 For the top flange surface according to Fig. 4.26b AC  BD  AD  BC 143.25 2  352 2  0  352  143.25 2CD 2 ˜ 143.25 0.40 For the bottom flange surface according to Fig. 4.26c AC  BD  AD  BC 143.25  352  0  143.25 2  352 2 2CD 2 ˜ 143.25 0.40 _____ 89 4. TEMPERATURE IN STEEL SECTIONS 143.25 mm 0.40 352 mm 0.67 0.40 Fig. 4.25: View factors for an HE 400 B It can be verified that the total amount of energy received by the chamber of the section is proportional to 2u0.40u143.25 + 0.67u352 = 352, which is exactly equal to the factor proportional to the energy crossing the dotted line on Fig. 4.25, considering I 1.0 , i.e., 1.0u352 = 352. 143.25 143.25 D C B 352 B D A B 352 352 C 143.25 A C A D a) b) c) Fig. 4.26: Emitting (AB) and receiving (CD) surfaces for radiative heat. a) For the web surface; b) For the top flange surface; c) For the bottom flange surface _____ 90 This concept can be used, for example, when the temperature of the web of a section has to be determined in order to evaluate the shear resistance of the web. This procedure is more detailed and more precise than the utilisation of the shadow factor present in Eq. (4.16) because the shadow factor is the weighted average value of the view factors calculated on the whole perimeter of the section1. For example, for the HE 400 B and neglecting the radius of root fillet between the web and the flanges, the values for the perimeter, Am , the box value of the perimeter, Am,b , the area of the cross section, V, the box value of the section factor, [ Am / V ]b and the section factor, Am / V , are: Am 1973 mm Am,b 1400 mm 1 Except for the factor 0.9 for hot rolled H or I sections, that has no physical meaning and is deemed to disappear in the next revision of the Eurocode. 4.10. TEMPERATURE IN STEEL MEMBERS SUBJECTED TO LOCALISED FIRES V 19152 mm2 [ Am / V ]b 1400/ 19152 0.073 mm-1 = 73 m-1 Am / V 1973/ 19152 0.103 mm-1 = 103 m-1 The correction factor for the shadow effect given by Eq. (4.17b), takes the value: ksh [ Am / V ]b /[ Am / V ] 0.073 / 0.103 0.708 which is exactly the same as the weighted average value of the view factors, ) , calculated on the whole perimeter of the section ) 4 ˜ 0.4 ˜143.5  2 ˜ 0.67 ˜ 352  2 ˜1.0 ˜ 300  4 ˜1.0 ˜ 24 1973 0.708 For the calculation of the temperature using an advanced calculation model, the view factors can be evaluated and applied individually to each surface, as shown on Fig. 4.25. For all the external surfaces of the profile a view factor ) 1 .0 can be conservatively adopted or evaluated as in Annex G of Eurocode 1. If a simpler solution is sought and some level of approximation is acceptable, the averaged value of the view factor (equal to the shadow factor) can be applied to the whole cross section. Using the shadow factor in the simple design equation (4.16) is an approximation because the convective part of the net heat flux, hnet,c , is also affected by the correction factor k sh . 4.10. TEMPERATURE IN STEEL MEMBERS SUBJECTED TO LOCALISED FIRES EN 1993-1-2 gives two different equations for calculating the temperature of steel members subjected to localised fire depending on whether the members are protected or not. The procedures that are used to evaluate the temperature in the case of localised fires are presented in this section. 4.10.1. Unprotected steel members For unprotected sections, the heat flux is give by Eq. (4.16) and this is used to evaluate the temperature. This heat flux, hnet , d hnet ,c  hnet , r , defined in _____ 91 4. TEMPERATURE IN STEEL SECTIONS Eq. (4.16), is easily calculated if the gas temperature, T g , is known. For localised fires not impinging the ceiling, the gas temperature is given by Eq. (3.24) (Heskestad Method). In the case where the localised fire impinges on the ceiling, the heat flux is given by Eq. (3.32) (Hasemi Method), and this can be used directly for calculating the temperature in unprotected steel members. Fig. 4.27 shows the development of the flame length for the localized fire given in Example 3.7, with the same maximum fire area of 72 m2, but with a distance from the source to the ceiling of 6 meters, instead of 3 meters. As the maximum length of the flame is 5.8 meters, the flame is not impinging the ceiling and the temperature of the gas is given by Eq. (3.24). For this case, the temperature of an unprotected IPE 300 beam heated on four sides is depicted in Fig. 4.28. The cross section of the beam is on the flame axis and located at the level of the ceiling. Fig. 4.29 shows the temperature development of an IPE 300 heated on four sides, by the same localised fire but with a distance from the source to the ceiling of 3 meters, i.e., the flame is impinging the ceiling. m 7 6 5 Flame lenght 4 Ceiling 3 _____ 92 2 1 0 0 20 40 60 80 100 120 [min] Fig. 4.27: Length of the flame ºC 600 Gas 500 Unprotected Steel 400 300 200 100 0 0 20 40 60 80 100 120 [min] Fig. 4.28: Gas and unprotected IPE 300 temperature development when the flame does not impinge the ceiling 4.10. TEMPERATURE IN STEEL MEMBERS SUBJECTED TO LOCALISED FIRES 900 800 700 600 500 [ºC] 400 300 200 100 0 Gas Unprotected Steel 0 20 40 60 80 100 120 [min] Fig. 4.29: Gas and unprotected IPE 300 temperature development when the flame is impinging the ceiling 4.10.2. Protected steel members For protected steel members, the equation used for calculating the temperature is only based on the gas temperature, see Eq. (4.19). For localised fires not impinging the ceiling the gas temperature is given by Eq. (3.24) and the development of the steel temperature is calculated from Eq. (4.19). Fig. 4.30 shows the temperature development of a cross section on the axis of the flame of a protected IPE 300 heated on four sides by the localised fire of example 3.7 with a distance from the source to the ceiling of 6 meters, i.e., not impinging the ceiling. ºC 600 Gas 500 Protected Steel 400 300 200 100 0 0 20 40 60 80 100 120 [min] Fig. 4.30: Temperature development of gas and protected IPE 300 when the flame does not impinge the ceiling _____ 93 4. TEMPERATURE IN STEEL SECTIONS Eq. (4.19) can not be applied directly in the case of a fire impinging the ceiling, because the effect of the fire is given as an impinging flux, see Eq. (3.32). A procedure has to be established to transform the impinging heat flux into an equivalent gas temperature. Cadorin et al. (2003) suggests deducing a fictitious temperature that has the same effect on steel elements as the heat flux calculated with this method. This is the temperature of a steel profile with a very high massivity factor. This steel profile has a temperature which is very close to the gas temperature. This procedure is used in the program Elefir-EN presented in Chapter 8. The program first evaluates the gas temperature as the temperature of an unprotected steel profile with very high section factor (Am/V = 10000 m-1 is adopted) using Eq. (4.16) and the net heat flux given by Eq. (3.33). After evaluating the gas temperature, the temperature of the protected steel profile is then calculated using Eq. (4.19). If the distance from the source to the ceiling is 3 meters, as in Example 3.7, the flame impinging the ceiling and the temperature development of a protected IPE 300 just above the fire source is shown in Fig. 4.31. ºC _____ 94 900 800 700 600 500 400 300 200 100 0 Gas Protected steel 0 20 40 60 80 100 120 [min] Fig. 4.31: Temperature development of Gas and protected IPE 300 when the flame impinges the ceiling 4.11. TEMPERATURE IN STAINLESS STEEL MEMBERS Annex C of EN 1993-1-2 provides guidance on the thermal properties of stainless steels. These properties can also be found in the Annex A of this 4.11. TEMPERATURE IN STAINLESS STEEL MEMBERS book. Compared with carbon steel, the thermal properties of stainless steel are quite different. The main differences are: - The increase of specific heat with increasing temperature is slightly lower in stainless steel than in carbon steel. Furthermore carbon steel has a larger increase in specific heat at 735 ºC due to a metallurgical phase change which is not present in stainless steel. This is shown in Fig. 4.32. - Compared to carbon steel, at ambient temperature, stainless steel has a much lower thermal conductivity. However, the thermal conductivity of stainless steel increases at elevated temperature and even exceeds the value for carbon steel at temperatures above 1000 ºC. This behaviour is shown in Fig. 4.33. C a (J/kgK) 5000 Carbon steel Stainless steel 4000 3000 _____ 95 2000 1000 T (ºC) 0 0 200 400 600 800 1000 1200 Fig. 4.32: Specific heat of stainless steel and carbon steel as a function of temperature Another difference lies in the surface emissivity of the member, Hm, which is equal to 0.4 for stainless steel and 0.7 for carbon steel. 4. TEMPERATURE IN STEEL SECTIONS O a (W/mK) 60 Stainless steel Carbon steel 50 40 30 20 10 T (ºC) 0 0 200 400 600 800 1000 1200 Fig. 4.33: Conductivity of stainless steel and carbon steel as a function of temperature Fig. 4.34 compares the temperature development of an IPE 450 exposed on four sides to the ISO 834 fire curve. [ºC] 1000 900 800 700 _____ 96 600 500 Stainless steel 400 Carbon steel 300 200 100 0 0 5 10 15 20 25 30 35 40 45 50 55 60 Time [min] Fig. 4.34: Temperature development of a carbon steel and a stainless steel IPE 450, heated on four sides by the ISO 834 fire curve This figure shows that the time-temperature curve for stainless steel does not exhibit the S shape of the carbon steel curve in the temperature range between 700 ºC and 800 ºC; this is due to the absence of the peak value on the heat capacity of stainless steel (see Fig. 4.32). The slower 4.11. TEMPERATURE IN STAINLESS STEEL MEMBERS increase in temperature of stainless steel in the early stage of the fire is due to a combination of the lower values of thermal diffusivity Oa /(Ua ca ) (see Fig. 4.35) and surface emissivity compared to carbon steel. In the later stage of the fire, both steel temperature curves tend toward the ISO fire curve. In this case, the role played by the surface emissivity is much more important than the role played by the thermal diffusivity. m2/s 1.6E-05 Carbon Steel 1.2E-05 Stainless steel 8.0E-06 4.0E-06 0.0E+00 0 200 400 600 800 1000 1200 ºC Fig. 4.35: Thermal diffusivity of carbon steel and stainless steel Annex A provides tables and nomograms to evaluate the temperature of unprotected stainless steel profiles exposed to the standard fire curve ISO 834, avoiding the need to perform the time integration of Eq. (4.16), as illustrated in the next example. 4.11.1. Example What is the temperature of the unprotected circular hollow section of Example 4.2 after 60 minutes of standard fire curve exposure if it is made of stainless steel? _______________________________ Solution: According to Annex A.2, the section factor is given by: Am V P A 1 d | 200 m-1 (d  t )t t _____ 97 4. TEMPERATURE IN STEEL SECTIONS As the section is convex, k sh 1 , and the modified section factor is given by: k sh [ Am / V ] 200 m-1 Table A.8 from Annex A gives temperature of 940 ºC (942 °C for example 4.2). This is compared to a temperature of 942 ºC for a similar carbon steel section. _____ 98