Fire Safety Journal, 9 (1985) 281 - 285 281

Temperature Analysis of Heavily-insulated Steel Structures Exposed to Fire

U. WICKSTROM

Swedish National Testing Institute, Bor~s (Sweden)

(Received April 11, 1984;in final form November 26, 1984)

SUMMARY

An exact analytical one-dimensional solu- tion o f the temperature response of insulated steel structures is derived and a closed form solution is given for a structure exposed to the ISO 834 standard curve.

Alternative approximate solutions are also given where all the heat capacity is assumed lumped in the steel core; only one third o f the insulation heat capacity is then considered. It is also shown that approximate solution schemes given elsewhere strongly underesti- mate the temperature rise in structures pro- tected with heavy insulation systems. In addi- tion the theoretical background o f a proposed N O R D T E S T test method on how to obtain the thermal properties o f an insulation system is outlined. Finally a numerical calculation scheme is recommended.

INTRODUCTION

Temperature in an insulated steel structure, as shown in Fig. 1, exposed to fire, may be estimated by a one-dimensional analysis if the exposure and the insulation are equal on all exposed surfaces and the comer effects a r e

neglected. In addition, as the thermal diffusiv- ity of steel is very high, the steel temperature

_

Fig. 1. I n s u l a t e d s tee l s t r u c t u r e w i t h a s tee l c o r e vol - u m e A a n d a s u r r o u n d i n g area F .

0379-7112/85/$3.30

may be assumed uniformly distributed which substantially simplifies the analysis. Figure 2 shows the simplified thermal model to be studied.

If the insulation heat capacity is small and may be neglected, a simple well-known expo- nential expression of the steel temperature response may be derived. For the more gener- al case approximatively similar solutions will be suggested where the insulation heat capac- ity is lumped and one third is added to the steel core heat capacity. Furthermore it is shown for the first time that the approxima- tions may be improved by shifting the time scale by a value depending on the ratio between the heat capacities of the insulation and the steel core.

I n s u t a f i o n ~ S t e e r c o r e

~ _ ~ . ,,.z___-.,~_.~.~., ~.J,._ ADIABATIC BOUNDARIES

Fig. 2. One-dimensional heat conduction model.

Exact and approximative solutions will then be given of the thermal response of steel structures exposed to the standard fire according to ISO 834. The theoretical basis is also given for interpreting test results so that the thermal resistance of insulation materials and systems may be calculated and finally a numerical scheme is recommended to be used when the material properties vary with tem- perature.

TEMPERATURE RESPONSE OF STEPWISE

CHANGE OF SURFACE TEMPERATURE

Consider the case shown in Fig. 2 and cal-

culate the temperature 0 as a function of

© Elsevier Sequoia/Printed in The Netherlands

282

t ime. (The solut ion o f the p rob lem may be f o u n d in several t e x t b o o k s , e.g., [1] . ) Assume all material p roper t ies cons tan t . A lumped heat capaci ty Q~ is assumed at x = d. The governing equa t ion is

302 30 ~ki 3X 2 Ci/gi a t = 0 (1)

where hi is the thermal conduc t iv i ty , ci specif- ic thermal hea t capaci ty , and Pi densi ty , respect ively, of the insulat ion material .

Assume the fol lowing b o u n d a r y condi t ions .

O(x, O) = 00 for t = 0 (2)

0(0, t) = 0 for x = 0 (3)

and

30 FX. 3__0_0 + Q~ __ = 0 for x = d (4)

' 3x at

where F is the sect ion area. Now assume the solut ion

O(x. t)/Oo = EKn exp(--/3~ t) s i n ( a , x ) (5)

Equa t ions (1) and (3 - 5) give

O(x. t)/Oo = F ,K . e x p ( - - a a . 2 t ) s in ( a . x ) (6)

where a = Xi/cip i. The eigenvalues an are ob ta ined f r om the fo l lowing equa t ion which is derived f rom eqn. (4).

( a , d ) / c o t ( a , d ) = # (7)

where 12 = F d c i P i / Q s = Qi/Qs. The coeff ic ients Kn are ob ta ined f rom the initial cond i t ion 00 = 0 for t = 0. Thus the t e m p e r a t u r e Odt ) at x = d is

(%d) 2 ] Os(t)/O o = ~ - - K n exp t s in(and )

n=l R Q i / F

(8a)

where [ 1]

2 [ ( a . d ) 2 + g 2 ] K n =

( a n d ) [ ( a n d ) 2 + 12 2 + 12] (8b)

and the the rmal resistance R = d/h i . (Note tha t when 12 -+ 0, the first eigenvalue a i d -~ V~- and a , d -->nTr for n ~> 2. Thus Os(t)/O o -+ e x p ( - - t F / R Q ~ ) which may be der ived d i rec t ly if Qi is neglected in the analysis.)

Equa t ion (8) may be a p p r o x i m a t e d as

Os(t)/O o = 0 for t < t (9a)

t-F] Os(t)/O o = exp -- - - for t > t- (9b)

r

where

T = R ( Q s + Q i / 3 ) / F (9c)

By compar i son with the exac t solut ion the t ime shift { is es t imated as

[ = #T /8 (9d)

In Fig. 3 the value of 0s/00 has been p lo t t ed as a func t ion o f the non-dimensional group t/T fo r a few values of p. As an example it is shown tha t the ap p ro x im a te solut ion, according to eqns. (9a) and (9b), and the exact solut ion near ly coincide for p = 0.41.

e s / e o

10

09

08

07

NN X kN\ - - Exact eq (8)

N \ X N \ \

\ ~ \ \ \

I~=0 18 * U A #=107 ~ : 0 L~I ~= o L,I

' 0~2. ' , , , , 01 03 O~ 05 06 07 fl%

Fig. 3. A comparison of the exact and approximate solutions of the stepfunction 0s/0 0 versus dimension- less time t/T for various p, where t is time, 1- = (Qs + Qi/3 )d/(kiF ) and I R = Qi/Qs, respectively.

Accord ing to eqn. (9c) one th i rd of the insulat ion hea t capaci ty Qi is added to the steel core. In an ap p ro x im a t io n formula , no t including the t ime shift t , Lie [2] suggests tha t a slightly d i f fe ren t por t ion , 0.3 Qi, should be added to the hea t capaci ty of the core.

ANALYTICAL SOLUTION OF FIRE EXPOSED INSULATED STEEL STRUCTURE

If one-dimensional condi t ions are assumed and the thermal proper t ies are cons tan t , the

temperature rise of a fire exposed insulated steel structure may be obtained by superposi- tion (Duhamel's theorem [1 ]). Thus the steel temperature is obtained as

t

= f T(t -- ~)~P'(~)d~ (10) Ts(t) 0

where T is the imposed fire curve and the response function ~b = ( 1 - 0,/00); 0J00 is given by eqn. (8) or (9).

If the ISO 834 standard fire curve is approximated by a sum of exponential terms [3] as

3 T = ~ Bj exp(--fiit) (11)

j = 0

where B i and fij are as given in Table 1, then the steel temperature is obtained from eqn. (10) as

3 B.iK. sin(~,d) T,= 2:

n = l j=O [1 fliViR/f] [

XIexp _ , >_ exp [RQi/F (12)

where cx, and K, are given by eqns. (7) and (8b), respectively. A similar expression of the exact solution of the steel temperature has been derived by Lie [2], who used a different approximation formula of the ISO standard fire curve.

If instead the approximate solution according to eqns. (9a) and (9b) is substi- tuted, the following expression is obtained for t > t - .

3 B i Ts = ~ 1--/~i r {exp[--fli(t t)]

i=O

-- exp[--( t -- E)/r]} (13)

T A B L E 1

Cons t an t s in t h e e x p o n e n t i a l express ion of the ISO 834 s t anda rd curve

j 0 1 2 3

B (°C) 1325 - - 4 3 0 - - 2 7 0 - - 6 2 5 /3 (h -1 ) 0 0.2 1.7 19

283

T ['C] Exact eq (12) and eq (13)

ECCS / 3,41

: : = ~ ' = 0

I '. '. : '. '. ', e q ( % ) ISO 83/.

100C - - - - - - " ~ -

so~ / / /

, , , , , ,

° k - o ~ 1.o is 20 / f [ h ]

/ /

- 500- i

Fig. 4. Example of ca lcu la ted t e m p e r a t u r e versus t ime of an insu la ted steel s t r uc tu r e exposed to fire accord- ing to ISO 834. The exac t so lu t ion is p l o t t e d for com- par i son with various approximative solutions. Input values representing a length of one meter of the struc- ture: Qs = 14 000 Ws K -l. Q i = 15 000 WS K - 1 and kiF/d = 5 W K-1.

where T and E are given by eqns. (9c) and (9d), respectively.

An example of calculated temperature of an insulated steel structure exposed to fire according to ISO 834 is shown in Fig. 4. The exact solution, eqn. {12), as well as the approximate solution according to eqn. (13) are plotted in full line. The differences between them are not noticeable in the chosen scale. However, when employing the approximative numerical calculation proce- dure as recommended in refs. 4 and 5, a much lower temperature is predicted as indicated by the dot ted line. A warning must therefore be expressed for using that calculation formula for design purposes. A third graph is also plotted in Fig. 4 showing the steel tempera- ture obtained if the influence of the time shift

is neglected. In most cases the change of material

properties at elevated temperature must be considered. Then a numerical scheme based on the difference form of eqn. (13) may be employed. However, as that expression con- tains the time shift t, an alternative formula is

284

suggested. By assuming the response func t ion according to eqn. {9b) valid for the e n t i r e interval t/>. 0 and sett ing t = p T / l O , an alter- native slightly less accurate closed form solu- t ion for cons tan t material p roper t ies may be derived as:

3 t e x p ( - - p / 1 0 ) Ts = ,=~0 Bi [exp(--~i t)

• . = ~ 1 --~ir

- - e x p ( - - t / r ] - - exp(--~i t)

× [ e x p ( - - p / 1 0 ) - - l ] f (14)

As seen in Fig. 4 this equa t ion gives qui te accura te t empera tu res a l though negative values are ob ta ined at shor t exposure t imes.

TIME DERIVATIVE OF THE STEEL TEMPERA- TURE

The t ime derivat ive of the steel t empera- ture is ob ta ined f rom eqn. (10).

If the a pp r ox ima te response func t ion according to eqn. (9) is used, the fo l lowing very simple express ion is ob ta ined .

dT~ = _1 [ T ( t - - [ ) - - T s ( t - - t ) ] (15) d t t=t r

CALCULATION OF THERMAL RESISTANCE R

The the rmal resis tance R may be ob ta ined f rom fire tests where furnace and steel tem- pera ture are recorded . Equa t ion (1) may t hen be e m p l o y e d bu t as the t ime shift E is d e p e n d e n t on R, the t ime derivat ive corre- sponding to eqn. (14) is r e c o m m e n d e d for this purpose :

1 / R = ~ clt + [ exp (p / lO) -- 1] - ~ ( A / F ) c ~ p ~

(16) (1 + p / 3 ) / ( T - - Ts)

Here A is the steel core volume. When appropr ia te , the the rmal conduc t iv i ty o f the insulat ion mater ia l may be ob ta ined as X i = d / R . The thermal resistance R or a l ternat ively the the rmal conduc t iv i ty Xi may t hen be cal- cula ted expl ic i t ly and given as a func t ion of

the average insulat ion t empera tu re Tm= ( T + T~) /2 .

A comprehens ive s tudy of various insula- t ion systems has been r epor t ed by Andersen and WickstrOm [6] where the thermal proper- ties were es t imated on basis of eqn. (16). A formal ized test ing m e t h o d is unde rway f rom N O R D T E S T * .

NUMERICAL CALCULATIONS

For non-l inear cases when the mater ia l p roper t ies vary with t em p e ra tu r e numerica l calculat ions are required . Based on the same ap p ro x im a t io n as eqn. (14) the fol lowing approx imat ive t em p e ra tu r e t ime derivative is derived.

d T A T J A t = ( T - Ts ) /T - - [ ex p (p /1 0 ) -- 1] - -

d t

(17)

To ascertain numerica l s tabil i ty of this fo rward d i f fe rence or explici t t ime in tegrat ion scheme the t ime inc remen t must be chosen so tha t A t <<. r = ( A / F ) c s p s ( 1 + p / 3 ) / X i . However , to obta in reasonable accuracy At should no t in addi t ion exceed 60 s.

ACKNOWLEDGEMENTS

Stig Andersson was mos t he lpful wi th the calcula t ion work o f this paper . His cont r ibu- t ion to the design and p lo t t ing o f the dia- grams is also great ly apprecia ted .

LIST OF SYMBOLS

A steel core vo lume (m 3) a the rmal di f fusivi ty (m 2 s 1) c heat capaci ty (Ws kg -1 K 1) d length (m) F sect ion area (m 2) Q heat capaci ty (Ws K 1) R the rmal resis tance (m 2 K W -1) t t ime (s, h) x length coord ina te (m)

*NORDTEST is a joint Nordic body with the function of promoting development in the field of technical testing.

Greek letters an constant (m -I) K. constant (--)

heat conductivity (W m -1 K -I) 0 temperature (K) p density (kg m -3) T see equation (9c) (s)

P Qi/Qs (--)

Su bsc rip ts i insulation s steel

REFERENCES

1 H. S. Carslaw and J. C. Jaeger, Conduction o f Heat in Solids, 2nd edn., Oxford University Press, 1959.

285

2 T. T. Lie, Temperature of protected steel in fire, Paper 8 o f Symposium No 2. Behaviour o f Struc- tural Steel in Fire, HMSO, London, 1968.

3 SBN -- Swedish Building Code 1975, National Board of Physical Planning and Building, 1975.

4 European Recommendations for the Fire Safety o f Steel Structures -- Calculation o f the Fire Resistance o f Load Bearing Elements and Struc- tural Assemblies Exposed to the Standard Fire, European Convention for Constructional Steel- work ECCS, Technical Committee 3, Fire Safety of Steel Structures, Elsevier Scientific Publishing Company, 1983.

5 S.-E. Magnusson, O. Pettersson and J. Thor, Fire Engineering Design o f Steel Structures, St~lbyg- gnadsinstitutet, Stockholm, 1976.

6 N. Andersen and U. WickstrSm, Brandbeskyt- tende Isolation af St~lkonstruktioner, NORD- TEST projekt 275-81, DANTEST, Copenhagen, 1984 (in Danish).