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Temperature analysis of partially heated steel members in fire
Journal of Constructional Steel Research 128 (2017) 1–6
Contents lists available at ScienceDirect
Journal of Constructional Steel Research
Temperature analysis of partially heated steel members in fire M.B. Wong Department of Civil Engineering, Monash University, Wellington Road, Melbourne, Australia
a r t i c l e
i n f o
Article history: Received 18 August 2015 Accepted 10 August 2016 Available online xxxx Keywords: Finite difference Fire Heat transfer Steel Temperatures
a b s t r a c t Analytical methods for the evaluation of temperature distribution of steel members in fire are readily available in codes of practice. These methods are usually based on uniform sections under fully engulfed fire conditions. In reality, structural members are often subject to partial heating fire conditions such as members passing through multiple fire compartments or subject to localised fires. At present, little guidance is given for temperature calculation of steel members subject to partial heating fire conditions. Evaluating the temperature distribution of steel members in such situations is important as the results may determine the extent of fire protection required for these members. This paper describes a numerical study on the temperature distribution of a partially heated steel member using a simple finite difference scheme with parametrically coded generic elements. The results are compared with the approximated method given in Eurocodes whereby the average temperature of a partially heated steel member can be calculated. It is shown that the proposed finite difference scheme can accurately predict the temperature distribution of steel members and provides a means for assessing their fire protection requirements. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction In fire engineering design, mainly two stages of calculations are required to be carried out for structural member design in order to meet the fire resistance requirements. The first stage is the evaluation of temperature distribution in the member and the second stage is the assessment of the stability, integrity and insulation requirements for the structural member based on the critical temperature assessment resulted from the first stage. Thus, temperature calculation is crucial to the assessment of the member's fire resistance and the results are directly related to the cost of the fire protection requirements for the structure. Nowadays many design codes such as the SFPE Handbook [1] and EC3 [2] provide guidance and evaluation methods for predicting the critical temperatures in structural members. These methods can be classified into three categories: the tabulated or experiment-based methods, the simple calculation models and the advanced calculation models. While the tabulated method is simple, it is restricted to those members tested under certain structural and fire conditions such as simply supported end restraints under standard fire. The advanced calculation models are based on sophisticated mathematical models usually using dedicated computer software with lengthy input procedures. As different computer software may adopt different mathematical models for computations, the results obtained from the software should always be verified by relevant test results [2].
E-mail address: [email protected].
http://dx.doi.org/10.1016/j.jcsr.2016.08.008 0143-974X/© 2016 Elsevier Ltd. All rights reserved.
EC3 provides detailed information on the use of the simple calculation models for the structural fire design of normal steel structures. The calculation procedures include formulations for the generation of design fires, temperature prediction and member capacity. For temperature prediction procedure, the formulation is based on heat transfer principles between fire and steel for various bare or insulated steel sections. In performing these calculations, the steel sections are assumed to be uniform and the length infinite. This method works for most fire scenarios where the post-flashover fire condition is assumed in a normal size fire compartment. However, for some situations where these assumptions cannot be met, modification of the formulation needs to be made or, in some cases, the formulation cannot be used at all. The current study examines the applicability of the EC3 formulation in a situation where a steel member is subject to partial heating, such as a member passing through multiple fire compartments or subject to localised fire in a large compartment. The aim of the current study is to evaluate the temperature distribution along the length of such steel member, part of which is fully engulfed in fire. The temperature distribution in the steel member using a finite difference scheme is first calculated and then compared with the average temperature of the steel member using the EC3 formulation. In using the finite difference method in this situation, each nodal point in the mesh represents the finite control element consisting of a finite length of the member with constant cross-section. The heat flow between the nodal points is one dimensional whereas the heat source from fire is through the surface of each finite control element subject to fire. As each finite control element may be subject to different boundary conditions, the explicit expression for calculating the temperature at each node within a time step must be
2
M.B. Wong / Journal of Constructional Steel Research 128 (2017) 1–6
specific for that particular boundary condition. Thus, identifying and handling the large number of expressions in a finite difference scheme for all the nodal points could be complicated. The finite different method has been successfully applied to evaluate the temperatures of the flanges and the web of a steel I-section in a composite structure [3] where each component of the steel section is subject to different boundary conditions under different fire exposure areas. The formulation is complicated even for only three finite control elements. In the current study, a parametrically coded generic finite control element is developed to cater for all possible boundary conditions under the specific fire conditions. The resulting expression is simple to use and the computational efficiency is high for examining a variety of cases with different boundary conditions along the length of the steel member. An example demonstrating the use of the finite difference scheme shows contrasting results to those obtained from the Eurocode.
Fire
Convection to surrounding
B
A
L1
L0 – L1
Fig. 1. Partially heated steel member.
2. Temperature prediction for steel members The following describes a heat transfer formulation for a body receiving heat by convection and radiation using a lumped thermal capacity model. The energy conservation principle for a body receiving heat flux h_ net;c by convection and h_ net;r by radiation giving rise to a temperature increase dT within a finite time dt results in the following equation: ρV V cp
dT ¼ h_ net;c As þ h_ net;r As dt
ð1Þ
where
Use of Eq. (4) is based on the lumped thermal capacity model and the resulting temperature represents only the average of the whole length of the member. It is valid only if the temperature gradient between the extreme points A and B shown in Fig. 1 is not great. To examine the validity of this model, the Biot number (Bi) can be calculated as [4]: Bi ¼
hc0 Lc k
ð5Þ
For a steel member of length L0 with constant cross-section fully engulfed in fire, the term As/VV can be written as
where hc0 is the convective coefficient, k is the thermal conductivity of steel and Lc is the characteristic length. As an example, under normal situation for a short length of a steel member L0 = 4 m subject to a heated length L1 = 1 m, the following values can be assumed: hc0 = 4 W/m2 K, k = 53 W/mK and Lc = 3 m. This gives a Biot number of 0.23 which is much N 0.1, a value usually set as a maximum for lumped thermal capacity analysis. In other words, the method recommended in EC3 may not be valid for this particular situation even for a very short steel member, let alone members which are usually a lot longer than 4 m for most structural applications. To overcome this problem and obtain a more accurate estimation of the temperature distribution of a partially heated steel member, finite difference method based on the Fourier's law of heat conduction for transient heat flow analysis is adopted in this study.
As =V V ¼ Am L0 =ðVL0 Þ ¼ Am =V
3. Finite difference method
ρ = density of the body Vv = volume of the body cp = specific heat As = surface area of the body exposed to fire Rearranging and writing Eq. (1) in incremental form such that Δθa,t =dT and Δt = dt gives Δθa;t ¼
As =V V _ hnet;c þ h_ net;r Δt cp ρ
ð2Þ
ð3Þ
where Am and V are the surface area and volume of the member per unit length respectively. By substituting Eq. (3) into Eq. (2), the resulting equation forms the basis for calculating the incremental temperature rise of bare steel members stipulated in EC3. The ratio in Eq. (3) is termed the ‘section factor’ for which expressions of design values for various types of cross-sections of both protected and unprotected steel members are given in EC3. For example, for a bare steel pipe with thickness t, the section factor is 1/t. Eq. (2) is easy to use for temperature prediction of steel members, the section factors of which are readily available for various steel sections. It should be noted that the use of Eq. (2) is based on the assumption that the whole length of the steel member is full engulfed in fire. For cases where only a portion of the steel member length is engulfed in fire, Eq. (3) needs to be modified in order for Eq. (2) to be applicable. Let's look at a situation where a steel member of length L0 is partially heated by fire for a length L1, as depicted in Fig. 1. The unheated length L0 − L1 is subject to the ambient environment and the heat is dissipated by convection only. In using Eq. (2) in this situation, Eq. (3) is modified such that As =V V ¼ ðAm =V ÞðL1 =L0 Þ
ð4Þ
The Fourier's law of heat conduction states that the heat flux q per unit area perpendicular to the direction of flow through a homogeneous body is proportional to the temperature gradient across the body. Mathematically, q ¼ −kdT=dx
ð6Þ
where k is the thermal conductivity of the body with local temperature T along the coordinate x in the direction of heat flow. For a body with cross-sectional area Ac in one-dimensional flow, the amount of heat energy Q, in W, flowing through the body is given as Q ¼ qAc
ð7Þ
In finite difference method with a large number of nodes in the mesh, most of the finite control element would have similar expressions for calculating temperature rise except for those along the boundary for heat exchange with the external or internal energy source and environment. However, in the current study for one-dimension transient heat flow, all finite control elements may be subject to different boundary conditions in order to carry out sensitivity analysis to examine the effect of different fire exposure conditions on the temperature distribution
M.B. Wong / Journal of Constructional Steel Research 128 (2017) 1–6
along the length of the steel member. For a large number of nodes in a finite difference scheme, this may present difficulty in terms of computational efficiency as the expression for temperature rise calculation may be different for the same node in the mesh in different sensitivity analysis cases. To overcome this difficulty, a parametrically coded generic control element is developed in order to enhance computational efficiency and reduce formulation complications. 3.1. Parametrically coded generic element
The heat flow from fire to the element based on EC1 [5] is Q_ f ¼ h_ net;c þ h_ net;r As
h_ net;c ¼ hc θg −T tx
Consider a section of the length of a steel member with constant cross-section consisting of three control elements: A at node x − 1, B at node x and C at node x + 1 as shown in Fig. 2. For a generic element B subject to Neumann boundary condition only, there are possibly four types of heat flow across its boundary: (1) the heat flow Q_ from upB
e
Q_ Δt ¼ μ Q_ B −φQ_ C þ βQ_ f −γ Q_ e Δt
ð8Þ
h
ð12Þ
4 4 i θg þ 273 − T tx þ 273
ð13Þ
hc = convective heat coefficient for fire environment θg = fire temperature in °C Ttx = temperature in element ‘x’ at time t εres = resultant emissivity of steel σ = 5.67 × 10−8 W/m2 K4.
wind element, (2) the heat flow Q_ C to downwind element, (3) the heat flow Q_ from fire, (4) the heat flow Q_ to the ambient environment. f
ð11Þ
where
h_ net;r ¼ εres σ
The net heat flow Q_ retained in ‘x’ within a time step Δt can be written as
3
The heat flow from the element to the surrounding is given as Q_ e ¼ hc0 T tx −T 0 As
ð14Þ
where where μ, φ, β and γ are parameters with a value of either 1 or 0 for the control of the boundary conditions of the element. For the current study, the following possible values of parameters can be assigned to each element: (i) (ii) (iii) (iv)
For a segment with upwind flow, μ = 1 otherwise μ = 0. For a segment with downwind flow, φ = 1 otherwise φ = 0. For a heated segment subject to fire, β = 1 otherwise β = 0. For an unheated segment subject to surrounding, γ = 1 otherwise γ = 0.
For the element at x with a segment length L, heat flow from upwind element at time t is, according to Eqs. (6) and (7), k T tx −T tx−1 Ac Q_ B ¼ − L
ð9Þ
Similarly, heat flow to downwind element is k T txþ1 −T tx Ac Q_ C ¼ − L
ð10Þ
hc0 = convective heat coefficient for the surrounding T0 = surrounding temperature. By the principle of energy conservation, _ QΔt ¼ T xtþ1 −T tx ρcp Ac L
ð15Þ
of element at By combining Eq. (8) to Eq. (15), the temperature Tt+1 x x at a future time step t + 1 can be written as t t t T xtþ1 ¼ M μT " x−1 þ φT xþ1# þ ½1−Mðμ þ φÞ−Nγhc0 T x _ βQ f þN As þ γhc0 T 0 where M ¼
kΔt ρcp ðLÞ2
ð16Þ
s Δt and N ¼ AAc ρc . pL
It can be seen that Eq. (16) represents a general expression for calculating the temperature of a generic finite difference element under different boundary conditions. By just assigning appropriate values to the four parameters of μ, φ, β and γ,the boundary condition can be easily changed for any element. Numerical stability requires that ½1−Mðμ þ φÞ−Nγhc0 ≥0; or
Δt k As γhc0 ≤1 ðμ þ φÞ þ Ac ρcp L L
ð17Þ
Hence, for numerical stability, the value of the time step is restricted to Δt ≤
ρcp L k As γhc0 ðμ þ φÞ þ Ac L
ð18Þ
4. Temperature distribution of a steel pipe
Fig. 2. Parametrically coded generic element.
Consider a steel pipe of 1.2 m outside diameter, 10 mm thick and 4 m long passing through a localised fire in a finite space. The pipe length is extended in opposite directions from the fire. Due to symmetry, only half (L0 = 2 m) of the pipe length, all divided into 20 equal segments of L = 0.1 m each, is analysed using the finite difference scheme as shown in Fig. 3. The figure depicts two heated segments as an example. With this segment length, the Biot number is much b0.1 for hc0 = 4 W/
4
M.B. Wong / Journal of Constructional Steel Research 128 (2017) 1–6
Fig. 3. Steel pipe subject to partial heating.
m2 K and the method is therefore valid for the current analysis. The value of thermal conductivity of steel k is adopted from EC3. The values of the parameters μ, φ, β and γ are assigned for the following elements according to their boundary conditions: 1. For the first segment ‘1’ subject to fire, μ = 0, φ = 1, β = 1, γ = 0. 2. For an intermediate segment subject to fire with upwind and downwind heat flows, μ = 1, φ = 1, β = 1, γ = 0. 3. For an intermediate segment subject to the ambient environment with upwind and downwind heat flows, μ = 1, φ = 1, β = 0, γ = 1. 4. For the last segment ‘20’ subject to fire, μ = 1, φ = 0, β = 1, γ = 0; if subject to ambient environment, μ = 1, φ = 0, β = 0, γ = 1. The heat loss through the end of the pipe to the ambient environment is ignored due to the small area of exposure. For a segment with only upwind and downwind heat flows such as an insulated segment with negligible heat loss, μ = 1, φ = 1, β = 0, γ = 0 although this scenario has not been explored in the current study. 4.1. Element details To study the effect of heated segments on the temperature distribution along the length of the steel pipe, the following number of segments starting from element ‘1’ is heated in turn: 2, 4, 6, 8, 14, 18, 20 out of 20 segments. The results are compared with the approximated solution for the average temperature of the whole length of pipe using EC3 method. The present example with a pipe thickness of t = 10 mm gives a section factor of 100 m−1. A larger thickness of t = 16 mm will also be used to examine the effect of section factor on the temperature distribution. The pipe is assumed to be subject to 60 min of standard fire with a time step of 15 s. The value of the resultant emissivity εres is assumed to be 0.7 and the convective coefficient of fire hc is 25 W/m2 K.
heat dissipation is fast when the pipe is subject to unheated environment. In most cases the temperature in the last segment always remains at ambient temperature in the unheated space but starts rising when the pipe is heated from 14 or more out of 20 segments. The temperature in the case of 20 out of 20 for a fully heated pipe is naturally constant at 938 ° C for all the segments and is consistent with the result obtained by the method in EC3. This example shows the results of a steel pipe of only 4 m long. It is expected that the conclusion for the results is similar even if the length of the pipe is a lot longer because any segments beyond those at ambient temperature would have little or no effect on the results. In practice, Fig. 4 is useful in determining the fire protection requirements of a steel member under partial heating condition. For instance, a conservative design requirement can be considered for limiting the pipe temperature to a maximum of 400 °C at which steel yield strength starts to deteriorate according to EC3. If the pipe is heated for a length of 1.2 m out of the 4 m (6 out of 20), from Fig. 4, insulation is required for a total of 8 segments (1.6 m out of 4 m) beyond which the temperature is below 400 °C. When insulation is actually provided for these 8 segments, the temperatures in all segments will drop and, theoretically, the number of segments requiring insulation will be even smaller. The final result can be obtained through an iterative finite difference analysis with some of the segments protected with insulation. However, this analysis is beyond the scope of the present study. The temperature evolution of all the segments over the 60 min of heating for a typical case of heating 4 out of the 20 segments is shown in Fig. 5 where the fire temperature curve is also shown. It can be seen that all heated segments have a temperature evolution profile very
4.2. Case 1: pipe with t = 10 mm The results of the analysis showing the maximum temperature at the end of the 60 min heating period using the finite different scheme for all heating scenarios are shown in Fig. 4. It can be seen that there is a smooth temperature gradient from the first segment to the last segment. It should be noted that in all cases, the maximum temperature in the first heated segment is nearly the same as that, at 938 °C, of the fully heated pipe (20 out of 20). All heated segments have similar maximum temperature except the ‘heated edge segment’ next to an unheated segment. The maximum temperature in the ‘heated edge segment’ is almost the same (at about 910 °C) in all cases except when the pipe is fully heated. The transitional change of the temperature between the ‘heated edge segment’ and the first segment at ambient temperature occurs over a small length of about 8 segments in most cases except when the pipe is fully (or almost) heated. This rapid drop in temperature indicates that the
Fig. 4. Maximum temperature distribution for steel pipe with t = 10 mm at end of 60 min heating.
M.B. Wong / Journal of Constructional Steel Research 128 (2017) 1–6
5
Fig. 5. Temperature evolutions for 4 out of 20 heated segments for pipe with t = 10 mm.
similar to a fully heated member. Therefore, it can be concluded for this example that the maximum temperature of any heated segment at the end of the 60 min heating can be calculated by treating it as if the pipe is fully heated. For comparison purposes, the average temperature of the pipe for this case is calculated using Eqs. (1) and (2) from the EC3 method. The results are shown in Fig. 6. It can be seen that the maximum temperature is well over 700 °C.
Fig. 6. Average temperature of a partially heated pipe of t = 10 mm.
Fig. 8. Temperature evolutions for 4 out of 20 heated segments for pipe with t = 16 mm.
4.3. Case 2: pipe with t = 16 mm For the same pipe with the same diameter under the same fire exposure condition, the rate of heat transfer to the pipe from the fire depends mainly on the thickness of the pipe. To examine the effect of the pipe thickness on the temperature distribution of the steel pipe, the above analysis is repeated with a pipe thickness of t = 16 mm, equivalent to a section factor of 62.5 m−1. Due to the reduction in section factor of the pipe, it is expected that the overall temperature rise will be lower than Case 1 above. The maximum temperature distribution of the pipe is shown in Fig. 7. It can be seen that while the maximum temperature in all cases is slightly lower than the case for t = 10 mm, the behaviour is virtually identical. The difference in temperature in any segment between the two cases is b10 °C. The temperature evolutions for all segments over the 60 min of heating are shown in Fig. 8. However, when using the EC3 method to calculate the average temperature of the pipe, it can be seen from Fig. 9 that the maximum temperature at the end of the 60 min is 625 °C, N100 °C reduction on the case for t = 10 mm.
5. Conclusions In this study, a finite difference scheme using a parametrically coded generic element is proposed for the analysis of temperature distribution of partially heated steel members. The main advantage of using this method is to enhance computational efficiency and reduce formulation
Fig. 7. Maximum temperature distribution for steel pipe with t = 16 mm at end of 60 min heating.
Fig. 9. Average temperature of a partially heated pipe of t = 16 mm.
6
M.B. Wong / Journal of Constructional Steel Research 128 (2017) 1–6
complications. It has been found that the method gives more accurate results for temperature distribution estimation of steel members than the method provided by EC3. The results are useful in providing information on insulation requirements for steel members subject to partial heating in situations such as members passing through multiple fire compartments or subject to localised fires. The concept can be applied to other applications such as partially protected steel connections [6] and partially damaged fire protections [7]. Nomenclature rate of heat flow from upwind element Q_ B rate of heat flow to downwind element Q_ C rate of heat flow from fire Q_ f rate of heat flow to the ambient environment Q_ e h_ net;c heat flux by convection h_ net;r heat flux by radiation k thermal conductivity temperature in element ‘x’ at time t Ttx cross-sectional area Ac surface area subjected to fire As Δt time interval, seconds convective heat coefficient for ambient environment hc0 convective heat coefficient for fire environment hc surrounding temperature T0 L segment length total length of member L0
V VV cp ρ θg μ, φ, β, γ εres σ
volume per unit length volume specific heat of steel density fire temperature parameters controlling boundary conditions resultant emissivity of steel 5.67 × 10−8 W/m2 K4
References [1] P.J. DiNenno, SFPE Handbook of Fire Protection Engineering, fourth ed. NFPA & SFPE, USA, 2008. [2] EC3, EN 1993-1-2: Design of Steel Structures, Part 1.2: General Rules - Structural Fire Design, CEN, 2004. [3] J.I. Ghojel, M.B. Wong, Three-sided heating of I-beams in composite construction exposed to fire, J. Constr. Steel Res. 61 (2005) 834–844. [4] J.L.M. Hensen, A. Nakhi, Fourier and Biot Numbers and the Accuracy of Conduction ModellingProceedings of BEP '94 Conference “Facing the Future” 6–8 April 1994, pp. 247–256. [5] EC1, EN 1991-1-2: Actions on Structures – Part 1.2: General Actions – Actions on Structures Exposed to Fire, CEN, 2007. [6] X.H. Dai, Y.C. Wang, C.G. Bailey, Effects of partial fire protection on temperature developments in steel joints protected by intumescent coating, Fire Saf. J. 44 (3) (2009) 376–386. [7] M.M.S. Dwaikat, V.K.R. Kodur, A simplified approach for predicting temperature profile in steel members with locally damaged fire protection, Fire. Technol 48 (2012) 493–512.
Contents lists available at ScienceDirect
Journal of Constructional Steel Research
Temperature analysis of partially heated steel members in fire M.B. Wong Department of Civil Engineering, Monash University, Wellington Road, Melbourne, Australia
a r t i c l e
i n f o
Article history: Received 18 August 2015 Accepted 10 August 2016 Available online xxxx Keywords: Finite difference Fire Heat transfer Steel Temperatures
a b s t r a c t Analytical methods for the evaluation of temperature distribution of steel members in fire are readily available in codes of practice. These methods are usually based on uniform sections under fully engulfed fire conditions. In reality, structural members are often subject to partial heating fire conditions such as members passing through multiple fire compartments or subject to localised fires. At present, little guidance is given for temperature calculation of steel members subject to partial heating fire conditions. Evaluating the temperature distribution of steel members in such situations is important as the results may determine the extent of fire protection required for these members. This paper describes a numerical study on the temperature distribution of a partially heated steel member using a simple finite difference scheme with parametrically coded generic elements. The results are compared with the approximated method given in Eurocodes whereby the average temperature of a partially heated steel member can be calculated. It is shown that the proposed finite difference scheme can accurately predict the temperature distribution of steel members and provides a means for assessing their fire protection requirements. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction In fire engineering design, mainly two stages of calculations are required to be carried out for structural member design in order to meet the fire resistance requirements. The first stage is the evaluation of temperature distribution in the member and the second stage is the assessment of the stability, integrity and insulation requirements for the structural member based on the critical temperature assessment resulted from the first stage. Thus, temperature calculation is crucial to the assessment of the member's fire resistance and the results are directly related to the cost of the fire protection requirements for the structure. Nowadays many design codes such as the SFPE Handbook [1] and EC3 [2] provide guidance and evaluation methods for predicting the critical temperatures in structural members. These methods can be classified into three categories: the tabulated or experiment-based methods, the simple calculation models and the advanced calculation models. While the tabulated method is simple, it is restricted to those members tested under certain structural and fire conditions such as simply supported end restraints under standard fire. The advanced calculation models are based on sophisticated mathematical models usually using dedicated computer software with lengthy input procedures. As different computer software may adopt different mathematical models for computations, the results obtained from the software should always be verified by relevant test results [2].
E-mail address: [email protected].
http://dx.doi.org/10.1016/j.jcsr.2016.08.008 0143-974X/© 2016 Elsevier Ltd. All rights reserved.
EC3 provides detailed information on the use of the simple calculation models for the structural fire design of normal steel structures. The calculation procedures include formulations for the generation of design fires, temperature prediction and member capacity. For temperature prediction procedure, the formulation is based on heat transfer principles between fire and steel for various bare or insulated steel sections. In performing these calculations, the steel sections are assumed to be uniform and the length infinite. This method works for most fire scenarios where the post-flashover fire condition is assumed in a normal size fire compartment. However, for some situations where these assumptions cannot be met, modification of the formulation needs to be made or, in some cases, the formulation cannot be used at all. The current study examines the applicability of the EC3 formulation in a situation where a steel member is subject to partial heating, such as a member passing through multiple fire compartments or subject to localised fire in a large compartment. The aim of the current study is to evaluate the temperature distribution along the length of such steel member, part of which is fully engulfed in fire. The temperature distribution in the steel member using a finite difference scheme is first calculated and then compared with the average temperature of the steel member using the EC3 formulation. In using the finite difference method in this situation, each nodal point in the mesh represents the finite control element consisting of a finite length of the member with constant cross-section. The heat flow between the nodal points is one dimensional whereas the heat source from fire is through the surface of each finite control element subject to fire. As each finite control element may be subject to different boundary conditions, the explicit expression for calculating the temperature at each node within a time step must be
2
M.B. Wong / Journal of Constructional Steel Research 128 (2017) 1–6
specific for that particular boundary condition. Thus, identifying and handling the large number of expressions in a finite difference scheme for all the nodal points could be complicated. The finite different method has been successfully applied to evaluate the temperatures of the flanges and the web of a steel I-section in a composite structure [3] where each component of the steel section is subject to different boundary conditions under different fire exposure areas. The formulation is complicated even for only three finite control elements. In the current study, a parametrically coded generic finite control element is developed to cater for all possible boundary conditions under the specific fire conditions. The resulting expression is simple to use and the computational efficiency is high for examining a variety of cases with different boundary conditions along the length of the steel member. An example demonstrating the use of the finite difference scheme shows contrasting results to those obtained from the Eurocode.
Fire
Convection to surrounding
B
A
L1
L0 – L1
Fig. 1. Partially heated steel member.
2. Temperature prediction for steel members The following describes a heat transfer formulation for a body receiving heat by convection and radiation using a lumped thermal capacity model. The energy conservation principle for a body receiving heat flux h_ net;c by convection and h_ net;r by radiation giving rise to a temperature increase dT within a finite time dt results in the following equation: ρV V cp
dT ¼ h_ net;c As þ h_ net;r As dt
ð1Þ
where
Use of Eq. (4) is based on the lumped thermal capacity model and the resulting temperature represents only the average of the whole length of the member. It is valid only if the temperature gradient between the extreme points A and B shown in Fig. 1 is not great. To examine the validity of this model, the Biot number (Bi) can be calculated as [4]: Bi ¼
hc0 Lc k
ð5Þ
For a steel member of length L0 with constant cross-section fully engulfed in fire, the term As/VV can be written as
where hc0 is the convective coefficient, k is the thermal conductivity of steel and Lc is the characteristic length. As an example, under normal situation for a short length of a steel member L0 = 4 m subject to a heated length L1 = 1 m, the following values can be assumed: hc0 = 4 W/m2 K, k = 53 W/mK and Lc = 3 m. This gives a Biot number of 0.23 which is much N 0.1, a value usually set as a maximum for lumped thermal capacity analysis. In other words, the method recommended in EC3 may not be valid for this particular situation even for a very short steel member, let alone members which are usually a lot longer than 4 m for most structural applications. To overcome this problem and obtain a more accurate estimation of the temperature distribution of a partially heated steel member, finite difference method based on the Fourier's law of heat conduction for transient heat flow analysis is adopted in this study.
As =V V ¼ Am L0 =ðVL0 Þ ¼ Am =V
3. Finite difference method
ρ = density of the body Vv = volume of the body cp = specific heat As = surface area of the body exposed to fire Rearranging and writing Eq. (1) in incremental form such that Δθa,t =dT and Δt = dt gives Δθa;t ¼
As =V V _ hnet;c þ h_ net;r Δt cp ρ
ð2Þ
ð3Þ
where Am and V are the surface area and volume of the member per unit length respectively. By substituting Eq. (3) into Eq. (2), the resulting equation forms the basis for calculating the incremental temperature rise of bare steel members stipulated in EC3. The ratio in Eq. (3) is termed the ‘section factor’ for which expressions of design values for various types of cross-sections of both protected and unprotected steel members are given in EC3. For example, for a bare steel pipe with thickness t, the section factor is 1/t. Eq. (2) is easy to use for temperature prediction of steel members, the section factors of which are readily available for various steel sections. It should be noted that the use of Eq. (2) is based on the assumption that the whole length of the steel member is full engulfed in fire. For cases where only a portion of the steel member length is engulfed in fire, Eq. (3) needs to be modified in order for Eq. (2) to be applicable. Let's look at a situation where a steel member of length L0 is partially heated by fire for a length L1, as depicted in Fig. 1. The unheated length L0 − L1 is subject to the ambient environment and the heat is dissipated by convection only. In using Eq. (2) in this situation, Eq. (3) is modified such that As =V V ¼ ðAm =V ÞðL1 =L0 Þ
ð4Þ
The Fourier's law of heat conduction states that the heat flux q per unit area perpendicular to the direction of flow through a homogeneous body is proportional to the temperature gradient across the body. Mathematically, q ¼ −kdT=dx
ð6Þ
where k is the thermal conductivity of the body with local temperature T along the coordinate x in the direction of heat flow. For a body with cross-sectional area Ac in one-dimensional flow, the amount of heat energy Q, in W, flowing through the body is given as Q ¼ qAc
ð7Þ
In finite difference method with a large number of nodes in the mesh, most of the finite control element would have similar expressions for calculating temperature rise except for those along the boundary for heat exchange with the external or internal energy source and environment. However, in the current study for one-dimension transient heat flow, all finite control elements may be subject to different boundary conditions in order to carry out sensitivity analysis to examine the effect of different fire exposure conditions on the temperature distribution
M.B. Wong / Journal of Constructional Steel Research 128 (2017) 1–6
along the length of the steel member. For a large number of nodes in a finite difference scheme, this may present difficulty in terms of computational efficiency as the expression for temperature rise calculation may be different for the same node in the mesh in different sensitivity analysis cases. To overcome this difficulty, a parametrically coded generic control element is developed in order to enhance computational efficiency and reduce formulation complications. 3.1. Parametrically coded generic element
The heat flow from fire to the element based on EC1 [5] is Q_ f ¼ h_ net;c þ h_ net;r As
h_ net;c ¼ hc θg −T tx
Consider a section of the length of a steel member with constant cross-section consisting of three control elements: A at node x − 1, B at node x and C at node x + 1 as shown in Fig. 2. For a generic element B subject to Neumann boundary condition only, there are possibly four types of heat flow across its boundary: (1) the heat flow Q_ from upB
e
Q_ Δt ¼ μ Q_ B −φQ_ C þ βQ_ f −γ Q_ e Δt
ð8Þ
h
ð12Þ
4 4 i θg þ 273 − T tx þ 273
ð13Þ
hc = convective heat coefficient for fire environment θg = fire temperature in °C Ttx = temperature in element ‘x’ at time t εres = resultant emissivity of steel σ = 5.67 × 10−8 W/m2 K4.
wind element, (2) the heat flow Q_ C to downwind element, (3) the heat flow Q_ from fire, (4) the heat flow Q_ to the ambient environment. f
ð11Þ
where
h_ net;r ¼ εres σ
The net heat flow Q_ retained in ‘x’ within a time step Δt can be written as
3
The heat flow from the element to the surrounding is given as Q_ e ¼ hc0 T tx −T 0 As
ð14Þ
where where μ, φ, β and γ are parameters with a value of either 1 or 0 for the control of the boundary conditions of the element. For the current study, the following possible values of parameters can be assigned to each element: (i) (ii) (iii) (iv)
For a segment with upwind flow, μ = 1 otherwise μ = 0. For a segment with downwind flow, φ = 1 otherwise φ = 0. For a heated segment subject to fire, β = 1 otherwise β = 0. For an unheated segment subject to surrounding, γ = 1 otherwise γ = 0.
For the element at x with a segment length L, heat flow from upwind element at time t is, according to Eqs. (6) and (7), k T tx −T tx−1 Ac Q_ B ¼ − L
ð9Þ
Similarly, heat flow to downwind element is k T txþ1 −T tx Ac Q_ C ¼ − L
ð10Þ
hc0 = convective heat coefficient for the surrounding T0 = surrounding temperature. By the principle of energy conservation, _ QΔt ¼ T xtþ1 −T tx ρcp Ac L
ð15Þ
of element at By combining Eq. (8) to Eq. (15), the temperature Tt+1 x x at a future time step t + 1 can be written as t t t T xtþ1 ¼ M μT " x−1 þ φT xþ1# þ ½1−Mðμ þ φÞ−Nγhc0 T x _ βQ f þN As þ γhc0 T 0 where M ¼
kΔt ρcp ðLÞ2
ð16Þ
s Δt and N ¼ AAc ρc . pL
It can be seen that Eq. (16) represents a general expression for calculating the temperature of a generic finite difference element under different boundary conditions. By just assigning appropriate values to the four parameters of μ, φ, β and γ,the boundary condition can be easily changed for any element. Numerical stability requires that ½1−Mðμ þ φÞ−Nγhc0 ≥0; or
Δt k As γhc0 ≤1 ðμ þ φÞ þ Ac ρcp L L
ð17Þ
Hence, for numerical stability, the value of the time step is restricted to Δt ≤
ρcp L k As γhc0 ðμ þ φÞ þ Ac L
ð18Þ
4. Temperature distribution of a steel pipe
Fig. 2. Parametrically coded generic element.
Consider a steel pipe of 1.2 m outside diameter, 10 mm thick and 4 m long passing through a localised fire in a finite space. The pipe length is extended in opposite directions from the fire. Due to symmetry, only half (L0 = 2 m) of the pipe length, all divided into 20 equal segments of L = 0.1 m each, is analysed using the finite difference scheme as shown in Fig. 3. The figure depicts two heated segments as an example. With this segment length, the Biot number is much b0.1 for hc0 = 4 W/
4
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Fig. 3. Steel pipe subject to partial heating.
m2 K and the method is therefore valid for the current analysis. The value of thermal conductivity of steel k is adopted from EC3. The values of the parameters μ, φ, β and γ are assigned for the following elements according to their boundary conditions: 1. For the first segment ‘1’ subject to fire, μ = 0, φ = 1, β = 1, γ = 0. 2. For an intermediate segment subject to fire with upwind and downwind heat flows, μ = 1, φ = 1, β = 1, γ = 0. 3. For an intermediate segment subject to the ambient environment with upwind and downwind heat flows, μ = 1, φ = 1, β = 0, γ = 1. 4. For the last segment ‘20’ subject to fire, μ = 1, φ = 0, β = 1, γ = 0; if subject to ambient environment, μ = 1, φ = 0, β = 0, γ = 1. The heat loss through the end of the pipe to the ambient environment is ignored due to the small area of exposure. For a segment with only upwind and downwind heat flows such as an insulated segment with negligible heat loss, μ = 1, φ = 1, β = 0, γ = 0 although this scenario has not been explored in the current study. 4.1. Element details To study the effect of heated segments on the temperature distribution along the length of the steel pipe, the following number of segments starting from element ‘1’ is heated in turn: 2, 4, 6, 8, 14, 18, 20 out of 20 segments. The results are compared with the approximated solution for the average temperature of the whole length of pipe using EC3 method. The present example with a pipe thickness of t = 10 mm gives a section factor of 100 m−1. A larger thickness of t = 16 mm will also be used to examine the effect of section factor on the temperature distribution. The pipe is assumed to be subject to 60 min of standard fire with a time step of 15 s. The value of the resultant emissivity εres is assumed to be 0.7 and the convective coefficient of fire hc is 25 W/m2 K.
heat dissipation is fast when the pipe is subject to unheated environment. In most cases the temperature in the last segment always remains at ambient temperature in the unheated space but starts rising when the pipe is heated from 14 or more out of 20 segments. The temperature in the case of 20 out of 20 for a fully heated pipe is naturally constant at 938 ° C for all the segments and is consistent with the result obtained by the method in EC3. This example shows the results of a steel pipe of only 4 m long. It is expected that the conclusion for the results is similar even if the length of the pipe is a lot longer because any segments beyond those at ambient temperature would have little or no effect on the results. In practice, Fig. 4 is useful in determining the fire protection requirements of a steel member under partial heating condition. For instance, a conservative design requirement can be considered for limiting the pipe temperature to a maximum of 400 °C at which steel yield strength starts to deteriorate according to EC3. If the pipe is heated for a length of 1.2 m out of the 4 m (6 out of 20), from Fig. 4, insulation is required for a total of 8 segments (1.6 m out of 4 m) beyond which the temperature is below 400 °C. When insulation is actually provided for these 8 segments, the temperatures in all segments will drop and, theoretically, the number of segments requiring insulation will be even smaller. The final result can be obtained through an iterative finite difference analysis with some of the segments protected with insulation. However, this analysis is beyond the scope of the present study. The temperature evolution of all the segments over the 60 min of heating for a typical case of heating 4 out of the 20 segments is shown in Fig. 5 where the fire temperature curve is also shown. It can be seen that all heated segments have a temperature evolution profile very
4.2. Case 1: pipe with t = 10 mm The results of the analysis showing the maximum temperature at the end of the 60 min heating period using the finite different scheme for all heating scenarios are shown in Fig. 4. It can be seen that there is a smooth temperature gradient from the first segment to the last segment. It should be noted that in all cases, the maximum temperature in the first heated segment is nearly the same as that, at 938 °C, of the fully heated pipe (20 out of 20). All heated segments have similar maximum temperature except the ‘heated edge segment’ next to an unheated segment. The maximum temperature in the ‘heated edge segment’ is almost the same (at about 910 °C) in all cases except when the pipe is fully heated. The transitional change of the temperature between the ‘heated edge segment’ and the first segment at ambient temperature occurs over a small length of about 8 segments in most cases except when the pipe is fully (or almost) heated. This rapid drop in temperature indicates that the
Fig. 4. Maximum temperature distribution for steel pipe with t = 10 mm at end of 60 min heating.
M.B. Wong / Journal of Constructional Steel Research 128 (2017) 1–6
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Fig. 5. Temperature evolutions for 4 out of 20 heated segments for pipe with t = 10 mm.
similar to a fully heated member. Therefore, it can be concluded for this example that the maximum temperature of any heated segment at the end of the 60 min heating can be calculated by treating it as if the pipe is fully heated. For comparison purposes, the average temperature of the pipe for this case is calculated using Eqs. (1) and (2) from the EC3 method. The results are shown in Fig. 6. It can be seen that the maximum temperature is well over 700 °C.
Fig. 6. Average temperature of a partially heated pipe of t = 10 mm.
Fig. 8. Temperature evolutions for 4 out of 20 heated segments for pipe with t = 16 mm.
4.3. Case 2: pipe with t = 16 mm For the same pipe with the same diameter under the same fire exposure condition, the rate of heat transfer to the pipe from the fire depends mainly on the thickness of the pipe. To examine the effect of the pipe thickness on the temperature distribution of the steel pipe, the above analysis is repeated with a pipe thickness of t = 16 mm, equivalent to a section factor of 62.5 m−1. Due to the reduction in section factor of the pipe, it is expected that the overall temperature rise will be lower than Case 1 above. The maximum temperature distribution of the pipe is shown in Fig. 7. It can be seen that while the maximum temperature in all cases is slightly lower than the case for t = 10 mm, the behaviour is virtually identical. The difference in temperature in any segment between the two cases is b10 °C. The temperature evolutions for all segments over the 60 min of heating are shown in Fig. 8. However, when using the EC3 method to calculate the average temperature of the pipe, it can be seen from Fig. 9 that the maximum temperature at the end of the 60 min is 625 °C, N100 °C reduction on the case for t = 10 mm.
5. Conclusions In this study, a finite difference scheme using a parametrically coded generic element is proposed for the analysis of temperature distribution of partially heated steel members. The main advantage of using this method is to enhance computational efficiency and reduce formulation
Fig. 7. Maximum temperature distribution for steel pipe with t = 16 mm at end of 60 min heating.
Fig. 9. Average temperature of a partially heated pipe of t = 16 mm.
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complications. It has been found that the method gives more accurate results for temperature distribution estimation of steel members than the method provided by EC3. The results are useful in providing information on insulation requirements for steel members subject to partial heating in situations such as members passing through multiple fire compartments or subject to localised fires. The concept can be applied to other applications such as partially protected steel connections [6] and partially damaged fire protections [7]. Nomenclature rate of heat flow from upwind element Q_ B rate of heat flow to downwind element Q_ C rate of heat flow from fire Q_ f rate of heat flow to the ambient environment Q_ e h_ net;c heat flux by convection h_ net;r heat flux by radiation k thermal conductivity temperature in element ‘x’ at time t Ttx cross-sectional area Ac surface area subjected to fire As Δt time interval, seconds convective heat coefficient for ambient environment hc0 convective heat coefficient for fire environment hc surrounding temperature T0 L segment length total length of member L0
V VV cp ρ θg μ, φ, β, γ εres σ
volume per unit length volume specific heat of steel density fire temperature parameters controlling boundary conditions resultant emissivity of steel 5.67 × 10−8 W/m2 K4
References [1] P.J. DiNenno, SFPE Handbook of Fire Protection Engineering, fourth ed. NFPA & SFPE, USA, 2008. [2] EC3, EN 1993-1-2: Design of Steel Structures, Part 1.2: General Rules - Structural Fire Design, CEN, 2004. [3] J.I. Ghojel, M.B. Wong, Three-sided heating of I-beams in composite construction exposed to fire, J. Constr. Steel Res. 61 (2005) 834–844. [4] J.L.M. Hensen, A. Nakhi, Fourier and Biot Numbers and the Accuracy of Conduction ModellingProceedings of BEP '94 Conference “Facing the Future” 6–8 April 1994, pp. 247–256. [5] EC1, EN 1991-1-2: Actions on Structures – Part 1.2: General Actions – Actions on Structures Exposed to Fire, CEN, 2007. [6] X.H. Dai, Y.C. Wang, C.G. Bailey, Effects of partial fire protection on temperature developments in steel joints protected by intumescent coating, Fire Saf. J. 44 (3) (2009) 376–386. [7] M.M.S. Dwaikat, V.K.R. Kodur, A simplified approach for predicting temperature profile in steel members with locally damaged fire protection, Fire. Technol 48 (2012) 493–512.