Sensitivity study of time delay coefficient of heat transfer formulations for insulated steel members exposed to fire

ARTICLE IN PRESS

Fire Safety Journal 41 (2006) 31–38 www.elsevier.com/locate/firesaf

Sensitivity study of time delay coefficient of heat transfer formulations for insulated steel members exposed to fire Zhi-Hua Wang, Kang Hai Tan Centre of Advanced Construction Studies, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singapore Received 19 August 2004; received in revised form 14 April 2005; accepted 25 July 2005 Available online 23 September 2005

Abstract Heat transfer analysis for an insulated steel member exposed to fire conditions, in general, involves solving a 2D transient conduction equation with well-posed conditions. For design purpose, the current Eurocode 3 provisions adopt a closed-form solution from the SP approach which employs a simplified 1D characteristic heat transfer model. A virtual ‘‘time delay’’ has been incorporated in the analytical solution based on the assumption that the heat capacitance of insulation material is small compared to that of steel section. A sensitivity study is presented in this paper to assess the validity of the time delay coefficient. Results of the study indicate that for insulation with large heat capacitance, the time delay estimation in the SP approach yields significant discrepancy compared against exact solution. An exact formulation of time delay is recommended, which gives accurate results for insulation material with either small or large heat capacitance. r 2005 Elsevier Ltd. All rights reserved. Keywords: Heat transfer; Time delay; Steel member; Insulation; Fire; SP approach

1. Introduction Temperature prediction of structural members subjected to fire plays an important role in structural fire protection. Heat transfer analysis for insulated steel member requires solution of 2D transient diffusion equation [1,2] which is usually complicated. Current design practices (e.g. [3,4]) make use of a simplified 1D condensed heat transfer model, as shown in Fig. 1, based on the lumped capacitance concept. This concept essentially assumes that the temperature distribution is uniform inside the entire steel section, which results in an analytical formulation of the heat transfer equation, known as the SP approach [5,6]. A virtual time delay has been incorporated in the evaluation of the step function of the SP approach, rendering a closed-form solution for design purpose [5]. The coefficient for the time delay was derived based on the assumption that the heat capacitance of the insulation material is negligibly small compared against that of steel section. The SP formulation has been adopted by Eurocode Corresponding author. Tel.: +65 6790 6814; fax: +65 6791 6697.

E-mail address: [email protected] (Z.-H. Wang). 0379-7112/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.firesaf.2005.07.008

3 [3] and was proven to be able to predict the steel temperature accurately for insulations with small heat capacitance, such as intumescent paint [7]. However, it is common engineering practice to use insulation materials with high capacitance, such as concrete and brick. Melinek and Thomas [8], on the other hand, derived an alternative formula for the time delay using Laplace transformation. This paper presents a sensitivity study which aims to examine the time delay coefficient in the SP approach. Concrete with high density was selected as insulation material in this paper. Steel members with various thicknesses of insulation were analysed, covering a wide range of heat capacitance. The range of validity of time delay coefficient was determined based on the sensitivity study.

2. Derivation of step functions Procedures to obtain temperature response analytically have been detailed by Carslaw and Jaeger [1], as follows: firstly, temperature response is solved analytically

ARTICLE IN PRESS Z.-H. Wang, K.H. Tan / Fire Safety Journal 41 (2006) 31–38

32

Nomenclature A0 , B0 As Ap =V Bj cs ,ci Cn ds; d i ks ,ki h Qs , Qi Ri t td td

parameters in ECCS formulation cross-section area of steel layer (m2) section factor of insulated steel members (m
1) constants in SBN approximation of ISO 834 fire curve (1C) specific heat of steel and insulation, respectively (J/kg K) coefficients of eigenfunction solutions to diffusion equation characteristic thickness of steel and insulation in1D model (m) thermal conductivity of steel and insulation, respectively (W/m K) surface heat transfer coefficient (W/m2 K) heat capacitance per unit area of steel and insulation (J/m2 K) thermal resistance of insulation (m2K/W) time (s, h) time delay (s) dimensionless time delay td =mt (—)

Fire boundary (Dirichlet), Tg

x

initial temperature field (1C) temperature inside insulation (1C) temperature of fire gas and steel (1C) outer surface temperature of insulation adjacent to the fire gas (1C) spatial coordinate (m)

Greek letters a, b

reciprocals of the short-term and long-term time constants (s) ai thermal diffusivity of insulation (m2/s) bj constants in SBN approximation of the ISO 834 fire curve (h
1) y normalized temperature (1C) yðx; tÞ=y0 step function of temperature response to unit change of temperature rs, ri density (kg/m3) t thermal time constant in SP approach (s) m Qi/Qs (—) zn positive roots of transcendental equation zn tan zn ¼ m

2.1. Derivation of analytical step functions

T(di , t) = Ts(t)

Insulation

T0 Tðx; tÞ Tg, Ts Tsur

Steel

T(x,t),
i = ki /cii


s = ks /css

di

ds

Adiabatic boundary

x Fig. 1. Schematics of condensed 1D heat transfer model.

corresponding to a stepwise temperature change at the fireinsulation interface (known as the ‘‘step’’ function). After that, Duhamel’s principle is employed to integrate the step temperature response over time intervals to yield complete solution. It is shown later in this section that the time delay is an intrinsic parameter of the 1D heat transfer model, rather than that of the imposed heating condition. First of all, it is necessary to derive the step function corresponding to a stepwise change of temperature of unit magnitude at the fire–insulation interface, i.e. to solve the diffusion problem through the insulation layer with homogeneous boundary conditions. Let Tðx; tÞ denote the temperature of the insulation layer as a function of spatial location 0oxod i and time instant t. The step function in this paper is defined as the solution of temperature field inside the insulation (with initial unity temperature in the medium), corresponding to the unit temperature drop at the boundary x ¼ 0. An illustration of the step function is shown in Fig. 2.

Assume all of the thermal properties are constant, i.e. independent of coordinates and time. The 1D heat transfer is reduced to the transient conduction through the insulation layer, as shown in Fig. 1. The governing differential equation is given by q2 Tðx; tÞ qTðx; tÞ ¼ 0, (1)
ci r i 2 qx qt where ki , ci and ri are the thermal conductivity, specific heat and density of insulation, respectively. A lumped heat capacitance (per unit cross-sectional area) for steel, Qs , is assumed at x ¼ d i , which is given by cs rs , (2) Qs ¼ cs rs d s ¼ ðAp =V Þ

ki

where cs , rs and d s are, respectively, the specific heat, density and characteristic thickness of steel section, and Ap =V is the section factor of insulated steel members, as presented in [3]. Without loss of any generality, the boundary temperature at x ¼ 0 is normalized to zero, by simply introducing a new state variable yðx; tÞ, which is termed the normalized temperature and defined by yðx; tÞ ¼ Tðx; tÞ
T 0 ,

(3)

where T 0 is the initial temperature prescribed at the surface x ¼ 0. This transformation of variable is valid because the terms presented in Eq. (1) are all in partial derivative

ARTICLE IN PRESS Z.-H. Wang, K.H. Tan / Fire Safety Journal 41 (2006) 31–38

33

1  (x, t 1)/0

step function

 (x, t 2)/0  (x, t 3)/0

Insulation layer; initial  (x, 0) = 1; boundary  (0, t > 0) = 0; x = di adjacent to lumped steel 0

1

0 Normalized dimension, x/di

Fig. 2. Illustration of step function corresponding to unit change of temperature at boundary x ¼ 0; time t3 4t2 4t1 .

forms. The boundary conditions for the derivation of the step function adopted in the SP approach are yð0; tÞ ¼ 0,

(4)

qyðd i ; tÞ qyðd i ; tÞ ¼ Qs . (5) qx qt Note in Eq. (4), a fixed temperature is specified at the fire–insulation interface, which implies the surface heat transfer resistance by convection and radiation is ignored (infinite heat transfer coefficient) when the timevarying fire boundary condition is imposed through Duhamel’s integral presented later. The treatment of the heat loss through surface convection and radiation has been discussed by Wong and Ghojel [9] and is implicitly adopted in the calculations in this paper. The boundary condition at x ¼ d i , i.e. Eq. (5) has apparent physical meaning that when the steel layer is considered as a lumped mass, the heat flux entering this system through the insulation-steel layer balances the rate of heat storage change in the system. The initial condition for the entire domain is given by


ki

yðx; 0Þ ¼ y0 ;

t ¼ 0.

temperature ys ðtÞ can be obtained using (7) at x ¼ d i , as ! 1 X ai z2n C n exp
2 t sinðzn Þ, (8) ys ðtÞ=y0  yðd i ; tÞ=y0 ¼ di n¼1 where ai ¼ ki =ðci ri Þ is the thermal diffusivity of the insulation. The coefficients zn are derived from Eq. (5). And the eigenvalues of zn can be obtained as the positive roots of the transcendental equation zn tan zn ¼ m,

(9)

where m is the dimensionless ratio of lumped heat capacitance of insulation to that of steel, given as m¼

Qi ci ri d i ðAp =V Þ ¼ . cs r s Qs

(10)

The first six positive roots of Eq. (9) are given in [1] and [2], where the sum of first few terms in Eq. (9) usually yields a reasonably accurate approximation to the exact solution. The coefficients C n are obtained using Eq. (11), as [1]   2 z 2 þ m2 . Cn ¼  2 n (11) z n z n þ m2 þ m

(6)

Let yðx; tÞ=y0 denote the step function of temperature response corresponding to the change unit temperature at the fire-insulation interface. The solution of Eq. (1), together with the well-posed boundary conditions (4)–(6), can be expressed in the form of sum of infinite series [5] as !
 1 X ai z2n x yðx; tÞ=y0 ¼ C n exp
2 t sin zn . (7) di di n¼1 In particular, the steel temperature response is assumed to be the same as the temperature response at the insulation–steel interface, i.e. ys ðtÞ  yðd i ; tÞ. Thus, the

2.2. Step functions in the SP approach In the case that the heat capacity of insulation, Qi , is negligibly small compared to Qs , a closed-form approximation has been proposed in the SP approach instead of Eq. (8), as ( 1 for tptd ;  t
t  ys ðtÞ=y0 ¼ (12) d for t4td ; exp
t where td is the time delay of steel temperature response due to the retardant effect by insulation. The term t is the

ARTICLE IN PRESS Z.-H. Wang, K.H. Tan / Fire Safety Journal 41 (2006) 31–38

34

ISO 834 fire

1000

Temperature (°C)

τ = 0.1 0.2

0.3 0.5

500

0.7

1.0 1.5 2.0 3.0

0 0.0

0.5

4.0

1.0

1.5

Time t (h) Fig. 3. Temperature response of insulated steel structures exposed to fire for different t values.

thermal time constant, given as cs rs d i  m 1þ , t ¼ Ri ðQs þ Qi =3Þ ¼ 3 ðAp =V Þki

Table 1 Constants in the exponential expression of the ISO 834 fire curve

(13)

where Ri ¼ d i =ki is the thermal resistance of insulation. It is clear from Eq. (12) that the time delay td is an intrinsic parameter of the step function, i.e. the thermal response system consisting of the insulation and the lumped steel section, rather than the imposed fire condition. Note in Eq. (13), one-third of the insulation capacitance is added to the steel according to Wickstrom [5], which has been verified by Melinek and Thomas [8] as an expression for the long-term time constant. Alternatively, Lie [10] suggests that a slightly different portion i.e. 0:3Qi should be added instead of Qi =3 in Eq. (13) without including the time delay td in Eq. (12). The steel temperature response with respect to various t values is presented in Fig. 3. The time delay td is estimated by comparison of (12) with the exact solution in Eq. (8), and is computed in [5] as mt (14) td ¼ . 8

j

0

1

2

3

B (1C) b (h
1)

1325 0


430 0.2


270 1.7


625 19

paper, the standard ISO 834 fire curve is assumed as the heating condition for simplicity. Discussions and results that follow can be extended to natural fire conditions by substitution with no theoretical difficulties. The standard heating curve (above initial temperature) is given by Eurocode 1 [12], yg ðtÞ ¼ 345 log10 ð8t þ 1Þ,

(16)

where t is time (in minute). SBN [13] adopts an approximate expression for the standard fire curve in terms of a sum of exponential terms as yg ðtÞ ¼

3 X

Bj expð
bj tÞ,

(17)

j¼0

3. Incorporation of time-varying fire conditions: Duhamel’s integral 3.1. Exact solutions Assuming the thermal properties are constant, temperature response can be calculated for the insulated steel member subjected to a time-varying boundary condition (e.g. fire condition), using the principle of superposition (Duhamel’s theorem, [11]), as Z t T s ðtÞ ¼ T 0 þ yg ðt
xÞdð1
ys ðxÞ=y0 Þ, (15)

where constants Bj and bj are given in Table 1. Using Eq. (17), the exact solution of steel temperature response when subjected to standard fire can be calculated by T s ðtÞ ¼ T 0 þ

1 X 3 X n¼1 j¼0

Bj C n sinðzn Þ   1
bj d 2i = ai z2n ( "



ai z 2  expð
bj tÞ
exp
2n t di

#) .

ð18Þ

3.2. The SP approach

0

where yg ðtÞ (in oC) is the time-varying heating curve and ys ðxÞ=y0 is the step function given in Eq. (8) or (12). In this

If the approximate step function in Eq. (12) is used instead of Eq. (8), the steel temperature is obtained

ARTICLE IN PRESS Z.-H. Wang, K.H. Tan / Fire Safety Journal 41 (2006) 31–38 Table 2 Values of the weighting factors b and N for h ! 1

for t4td as T s ðtÞ ¼ T 0 þ

3 X j¼0

Bj 1
bj t

   

 exp
bj ðt
td Þ
exp ðt
td Þ=t

ð19Þ

and for tptd , T s ðtÞ  T 0 . Eq. (19) is known as the SP approach for temperature response of insulated steel structures exposed to standard ISO 834 fire. When temperature-dependent material properties are to be incorporated in the analysis, instead of the closed-form solution (19), a time-stepping scheme is required to calculate the temperature increments within given time intervals. An approximate time derivative is given by Wickstrom [5] as  dT dT s T g
T s  m=10 g ¼
e . (20)
1 dt t dt The term involving the derivative of T g at the right-hand side of Eq. (20) is due to the retardant (sink) effect of insulation where the time delay is incorporated. Eq. (20) is adopted by EC3, and the increment of steel temperature DT s within a time interval Dt is given by   ki Ap =V ðT g
T s Þ   Dt
em=10
1 DT g DT s ¼ cs rs d i 1 þ m=3 ðDT s X0; if DT g 40Þ,

ð21Þ

4. Alternative formulations using Laplace transformation An alternative equation for computation of the time derivative of steel temperature response is given by the ECCS recommendations [14] as dT g dT s ¼ A0 ðT g
T s Þ
B0 , dt dt where 1 , ðRi þ 1=hÞQs ð1 þ m=NÞ

b , B ¼ 1 þ N=m 0

(22)

(23)

(24)

qTð0; tÞ (25) qx with N and b as weighting factors, T sur ðtÞ is the outer surface temperature of the insulation adjacent to the fire gas and the term h is the combined heat transfer coefficient of surface convection and radiation at the fire–insulation interface. In the SP approach, the Dirichlet boundary (with specified temperature) is assumed at the fire–insulation interface. As implied by Eq. (4), the heat transfer model is reduced to the limiting case with h ! 1. Thus, there is no heat loss through surface convection and radiation and the outer surface temperature of insulation is identical to the fire gas temperature, i.e. T sur ðtÞ  T g ðtÞ. The weighting

h½T g ðtÞ
T sur ðtÞ ¼
ki

m N b

1 2.77 0.38

N 2.4 0.2

0.5 2.86 0.43

0.2 2.93 0.47

0.1 2.96 0.48

0 3 0.5

factors N and b are associated with the ratio of heat capacitance of insulation to steel, i.e. m ¼ Qi =Qs . It is shown in Eq. (23) that a portion 1=N of the insulation capacitance is incorporated in evaluation of time constant 1=A and the factor b is incorporated in evaluation of the retardant effect, B of the insulation. The weighting factors N and b have been determined by Melinek and Thomas [8] using Laplace transformation. For the limiting case h ! 1, b¼

1 þ m=4 , 2ð1 þ 5m=8Þ

(26)

N ¼ 2ðb þ 1Þ.

(27)

Values of N and b for h ! 1 are listed in Table 2. Based on Eqs. (22)–(27), an analytical formulation of the time delay has been proposed by Melinek and Thomas [8] as td a m  , (28) ¼ ln a
b 6 þ 2m t where

where Dtp30 s for insulated steel members.

A0 ¼

35

A0 a; b ¼ 0 2B

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 1
3B0 1 1 þ B0

ða4bÞ.

(29)

The terms 1=a and 1=b possess, respectively, the physical meanings as the short-term and long-term time constants in the evaluation of steel temperature response using Eq. (22), as detailed in [8]. It should be noted that the expression for the long-term time constant 1=b ¼ Ri ðQs þ Qi =3Þ agrees with the time constant t in Eq. (13) of the SP approach. 5. Sensitivity study on time delay coefficients Eq. (14) gives a simple estimation of the time delay based on the assumption that the heat capacitance of insulation is small. This estimation is valid for most of the insulation materials with low density and low conductivity (such that to minimize the thickness of application), such as gypsum plaster and intumescent paint. However, for practical reason, when materials with high density and high conductivity, e.g. concrete, are used for protection, the time delay estimated using Eq. (14) can be as long as 30 min, which is not physically possible even argued by intuition. Therefore, a sensitivity study has been carried out to examine the validity of the estimation of time delay. It should be noted that although the convolution integral in Eq. (15) requires insulation materials with constant thermal properties (independent of temperature), most of the insulation materials behaves nonlinearly in fire, e.g. intumescent paint, gypsum, concrete, etc. However, by

ARTICLE IN PRESS Z.-H. Wang, K.H. Tan / Fire Safety Journal 41 (2006) 31–38

36

using a time-stepping scheme, the restriction of constant thermal properties in temperature formulation is confined in a small quasi-static time interval Dt. Therefore, the nonlinear temperature-dependent thermal properties of insulation can be incorporated in the temperature formulation, e.g. Eq. (21), and the time-delay coefficient can therefore be assessed without compromise to the assumption of linear behaviour of insulation materials. In this paper, concrete is selected as the insulation material in the sensitivity study. For ease of computation, thermal properties for both steel and concrete are assumed to be constant, as listed in Table 3. Analysis has been carried out for four series of specimens with section factors varying from 50 to 200 m
1 at an increment of 50 m
1. Each series of specimens has the insulation thickness varying from 10 to 200 mm. Properties of specimens and computed m values are listed in Table 4. As discussed above, the time delay is an intrinsic parameter of the step function. Therefore, it suffices to perform evaluation of step functions in order to determine the ‘‘optimal’’ time delay. The procedure of determination of the optimal time delay for each specimen is as follows. For each specimen, the time history of step functions have been computed using Eq. (8) (exact solution) and Eq. (12) (approximation with time delay), respectively. Sufficient sampling data for stepwise temperature response were obtained by discretizing the dimensionless time interval t=t. The standard error of temperature response between the exact (Eq. (8)) and the approximate (Eq. (12)) temperature response was computed at each sampling point. The expressions of Eqs. (14) and (28) suggest that it Table 3 Thermal properties of steel and concrete [3] Thermal properties of steel

Thermal properties of concrete

r (kg/m3)

c (J/kg K)

k (W/m K)

r (kg/m3)

c (J/kg K)

k (W/m K)

7850

600

45

2400

840

1.4

is convenient to consider the time delay in a dimensionless form as a function of a single variable, i.e. td ¼ td =mt ¼ f ðmÞ, where f ðmÞ is a function with a generic variable m (e.g. f ðmÞ ¼ 1=ð6 þ 2mÞ in Eq. (28)). The values of td are adjusted such that the sum of standard deviations of all sampling points over the interval t=t is a minimum. The adjusted td values by statistical means give the optimal time delay for each specimen.

6. Results and discussions An example of comparison of the exact and approximate step function versus dimensionless time is plotted in Fig. 4, where results of Eq. (12) using different td values are shown. It is clear from Fig. 4 that when the heat capacitance of insulation is high (m ¼ 3:26), the time delay estimated using Eq. (14) from the SP approach differs significantly from the exact result. The optimal values of the dimensionless time delay td obtained using statistical optimization described in Section 5 for all specimens are presented in Table 4. It can be seen that td only depends on the m values despite the combination of the insulation thickness and the section factors. For instance, Ap =V ¼ 100 m
1 , d i ¼ 40 mm and Ap =V ¼ 200 m
1 , d i ¼ 20 mm both yield m ¼ 1:71 and the time delay td in these two cases are 0.105 and 0.106, respectively, which is nearly identical. This verifies the validity of the assumption that the dimensionless time delay td can be treated as a one-variable function of the parameter m. Comparisons of time delay estimation using Eqs. (14) and (28) against the results of sensitivity analysis are plotted in Fig. 5. When m is small, ECCS [14] recommended that the heat sink effect can be neglected from temperature prediction. This is verified as shown in Fig. 5, when mo0:5, the slope of td =t is immaterial. It is also clear that the SP approach (Eq. (8)) can only predict the time delay accurately up to the range mo1:5. On the other hand, the

Table 4 Results of optimal dimensionless time delay di (mm)

10 15 20 30 40 50 75 100 125 150 175 200

Ap/V ¼ 50 m
1

Ap/V ¼ 100 m
1

Ap/V ¼ 150 m
1

Ap/V ¼ 200 m
1

m

td

m

td

m

td

m

td

0.21 0.32 0.43 0.64 0.86 1.07 1.61 2.14 2.68 3.21 3.75 4.28

0.162 0.155 0.147 0.136 0.127 0.124 0.105 0.093 0.086 0.075 0.068 0.063

0.43 0.64 0.86 1.28 1.71 2.14 3.21 4.28 5.35 6.42 7.49 8.56

0.149 0.137 0.127 0.122 0.105 0.095 0.076 0.063 0.054 0.047 0.042 0.038

0.64 0.96 1.28 1.93 2.57 3.21 4.82 6.42 8.03 9.63 11.24 12.84

0.137 0.122 0.120 0.097 0.085 0.075 0.055 0.042 0.034 0.031 0.027 0.022

0.86 1.28 1.71 2.57 3.42 4.28 6.42 8.56 10.70 12.84 14.98 17.12

0.128 0.123 0.106 0.091 0.073 0.063 0.047 0.038 0.031 0.026 0.021 0.019

ARTICLE IN PRESS Z.-H. Wang, K.H. Tan / Fire Safety Journal 41 (2006) 31–38

37

1.1 1.0

t d*=1/(6+2µ)

0.9 Step functions

0.8 Exact

t d*=1/8

0.7 t d*=0

0.6 0.5 0.4 0.3 0.2 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Dimensionless time, t /τ Fig. 4. Example of comparison of the exact and approximate step function ys =y0 versus dimensionless time t=t, m ¼ 3:26.

1.0 Section factor 50

t d /τ = µ /8 (SP Approach)

Section factor 100

0.8

Section factor 150 Section factor 200

td /

0.6

t d /τ = µ /(6+2µ) (Melinek & Thomas)

0.4

0.2

0.0

t d /τ = In[α /(α-β)])

0

1

2

3

4

5

6

7

8

9

10

Dimensionless  Fig. 5. Comparison of estimation of time delay against results of sensitivity study.

1000 ISO 834 fire

Temperature (°C)

800 600

time delay td = µ/(6+2µ) = 10 min

400

time delay t d = µτ/8 = 16 min

200 0

Exact solution

0

10

20

30

40 50 60 Time (min)

70

80

90

Fig. 6. Comparison of steel temperature response predicted using different time delay against exact solutions, m ¼ 3:26.

formulation derived by Melinek and Thomas as shown in Eq. (28) is able to estimate the time delay for a much wider range, i.e. the heat capacitance of insulation can be as high

as 10 times that of the steel section (m ¼ 10), which practically covers all the application of insulated steel members in engineering design. It is also interesting to note

ARTICLE IN PRESS Z.-H. Wang, K.H. Tan / Fire Safety Journal 41 (2006) 31–38

38

Table 5 Comparison of steel temperature response predicted using different time delay against exact solutions, m ¼ 3:26 Time delay estimation

Steel temperature T s (1C) at t (min) 15

Eq. (14) Eq. (28) Exact solution a

a

20 61 105

30

45

60

75

90

196 275 297

391 454 457

547 596 589

668 707 695

763 793 780

totd :

lim

td a  0:31 ¼ lim ln m!1 a
b t

The authors would like to thank Professor Ulf Wickstrom of SP Swedish National Testing and Research Institute for initiating the sensitivity study and helpful discussions through emails. Besides, the authors are also grateful to Dr. Au Siu Kui of Nanyang Technological University for helping in clarifying some theoretical details in this study. References

that in Eq. (28), the exact expression has a limit when u ! 1, given by m!1

Acknowledgements

(30)

while the approximate expression has a limit of 0.5 for m ! 1. However, for practical consideration, the approximate td =t  m=ð6 þ 2mÞ yields reasonable accurate results when m does not exceed 10. Finally, a comparison of steel temperature prediction using different time delay estimation formulations are plotted in Fig. 6 against the exact solutions. It is demonstrated that the use of Eq. (28) in the case of high heat capacitance of insulation (m ¼ 3:26) improves the steel temperature prediction significantly. A tabulated form of the results of comparison is also included in Table 5. 7. Conclusions In this paper, the results of a sensitivity study on the time delay estimation for insulated steel members are presented. It is shown for insulation with high heat capacitance, the time delay estimated by the SP approach yields large discrepancy compared with the exact solutions. Therefore, it is recommended in such cases an alternative formula derived from the Laplace transformation shall be used instead of the SP approach. The application to temperature prediction of insulated steel exposed to fire shows that the use of exact estimation of time delay improves the accuracy of calculation significantly.

[1] Carslaw HS, Jaeger JC. Conduction of heat in solids. 2nd ed. New York: Oxford University press; 1959. [2] Ozisik MN. Heat conduction. 2nd ed. New Jersey: Wiley; 1993. [3] prEN 1993-1-2. Eurocode 3: design of steel structures—Part 1–2: general rules—structural fire Design, stage 49 draft. CEN, Brussels, 2003. [4] Wang YC. Steel and composite structures: behaviour and design for fire safety. London: Spon Press; 2002. [5] Wickstrom U. Temperature analysis of heavily insulated steel structures exposed to fire. Fire Saf J 1985;9:281–5. [6] Wickstrom U. Natural fire for design of steel and concrete structures—a Swedish approach. In: Proceedings of the international symposium on fire engineering for building structures and safety, Melbourne, 1989. [7] Tan KH, Wang ZH, Au SK. Heat transfer analysis for steelwork insulated by intumescent paint exposed to standard fire conditions. In: Proceedings of the third international workshop, structure in fire, SiF’04, Ottawa, Canada, 2004. p. 49–58. [8] Melinek SJ, Thomas PH. Heat flow to insulated steel. Fire Saf J 1987;12:1–8. [9] Wong MB, Ghojel JI. Sensitivity analysis of heat transfer formulations for insulated structural steel component. Fire Saf J 2003;38:187–201. [10] Lie TT. Temperature of protected steel in fir. In: Proceedings of symposium no. 2, behaviour of structural steel in fire. London: HMSO; 1968. [11] Kevorkian J. Partial differential equations: analytical solution techniques. 2nd ed. New York: Springer; 2000. [12] prEN 1991-1-2. Eurocode 1: actions on structures—Part 1–2: general actions—actions on structures exposed to fire, 3rd draft. CEN, Brussels, 2001. [13] SBN. Swedish building code 1975. National Board of Physical Planning and Buildings, 1975. [14] ECCS Technical Committee 3. European Recommendations for the Fire Safety of Steel Structures. Amsterdam: Elsevier Scientific Publishing Company; 1983.