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Temperature distribution in homogeneous slabs during fire test
ANALYZED Ser TH1 N21r2 no. 210 c . 2
NATIONAL RESEARCH COUNC IL
CANADA
D I V I S I O N O F B U I L D I N G RESEARCH
BLDG
TEMPERATURE D I S T R I B U T I O N I N HOMOGENEOUS SLABS DURING F I R E T E S T
BY
T . Z . HARMATHY
R E P R I N T E D FROM
TRANSACTIONS O F THE E N G I N E E R I N G I N S T I T U T E O F CANADA VOL. 6 , NO. B-6, OCTOBER 1963
P A P E R NO. EIC-63-MECH 6 RESEARCH P A P E R NO. 2 10 O F THE D I V I S I O N OF B U I L D I N G RESEARCH P R I C E 50 CENTS OTTAWA DECEMBER 1963 NRC 7416
This publication is being distributed by the Division of Building Research of the National Research Council. It should not be reproduced in whole or in part, without permis- sion of the original publisher. The Division would be glad to be of assistance in obtaining such permission.
Publications of the Division of Building Research may be obtained by mailing the appropriate remittance, ( a Bank, Express, or Post Office Money Order or a cheque made payable at par in Ottawa, to the Receiver General of Canada, credit National Research Council) to the National Research Council, Ottawa. Stamps are not acceptable.
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-
TEMPERATURE DISTRIBUTIOiJ I N IIOMOGENEOUS SLABS
" 4 , . D U R I N G FIRE TEST T. Z. Harmathy SYNOPSIS The s o l u t i o n of t h e e q u a t i o n of h e a t c o n d u c t i o n f o r a c a s e s i m i l a r t o t h a t met i n f i r e e n d u r a n c e t e s t s i s p r e s e n t e d b o t h g r a p h i c a l l y and i n t a b u l a t e d form. The a p p l i c a t i o n of t h e s e r e s u l t s t o t h e c a l c u l a t i o n o f t h e r m a l f i r e e n d u r a n c e a n d o f t h e t e m p e r a t u r e h i s t o r y of t e s t s p e c i m e n s d u r i n g s t a n d a r d f i r e e x p o s u r e i s shown. INTRODUCTION A s t a n d a r d f i r e t e s t i s a n " i d e a l i z e d " r e p r o d u c t i o n of t h e c o n d i t i o n s t o which b u i l d i n g e l e m e n t s ( w a l l s , f l o o r s , beams, e t c . ) a r e s u b j e c t e d i n a c t u a l b u i l d i n g f i r e s . During t h e t e s t , o n e f a c e ( a s a r u l e ) of a specimen o f t h e b u i l d i n g e l e m e n t i s exposed t o t h e a t m o s p h e r e of a f u r n a c e , t h e t e m p e r a t u r e o f which i s r e q u i r e d t o f o l l o w a p r e s c r i b e d t e m p e r a t u r e v e r s u s t i m e c u r v e . T h i s c u r v e i s
assumed t o r e p r e s e n t t h e r i s e o f t e m p e r a t u r e i n a compartment a t a n advanced s t a g e o f s e v e r e b u i l d i n g f i r e s .
The most i m p o r t a n t c l a u s e s o f t h e f i r e t e s t s t a n d a r d s a r e p r o b a b l y t h o s e which d e f i n e t h e c r i t e r i a of * * f a i l u r e 1 1 of t h e specimen, i n o t h e r words, t h e end of t h e p e r i o d o f " f i r e endurancev1 of a c o n s t r u c t i o n . I n t h e c a s e of w a l l s and f l o o r s a specimen s h a l l b e judged a s f a i l e d ( i ) i f t h e a v e r a g e t e m p e r a t u r e of i t s s u r f a c e o p p o s i t e t o t h e f u r n a c e ( i t s wunexposedw s u r f a c e ) e x c e e d s t h e t e m p e r a t u r e of t h e a m b i e n t a i r by more t h a n 250 F, ( i i ) i f t h e s p e c i m e n c o l l a p s e s , o r ( i i i ) i f l a r g e c r a c k s o r h o l e s d e v e l o p t h r o u g h t h e specimen. Of c o u r s e , w h a t e v e r t h e immediate c a u s e o f t h e f a i l u r e o f a s p e c i m e n , t h e p r i m a r y c a u s e i s a l w a y s t h e r i s e o f t e m p e r a t u r e i n t h e c o n s t r u c t i o n . I t i s o b v i o u s , t h e r e f o r e , t h a t no m a t t e r what o t h e r c a l c u l a t i o n s a r e n e e d e d , t h e c a l c u l a t i o n o f t h e t e m p e r a t u r e h i s t o r y of t h e b u i l d i n g e l e m e n t d u r i n g t h e f i r e e x p o s u r e must a l w a y s b e t h e f i r s t s t e p i n e s t i m a t i n g t h e p o i n t of f a i l u r e . S i n c e , however, t h e " t h e r m a l f a i l u r e v 1 i s by f a r t h e most f r e q u e n t mode o f f a i l u r e , i n t h e m a j o r i t y of c a s e s t h e c a l c u l a t i o n of t h e t e m p e r a t u r e h i s t o r y o f t h e unexposed s u r f a c e i s e q u i v a l e n t t o t h e c a l c u l a t i o n of f i r e e n d u r a n c e . I n g e n e r a l , t h e u s e of some n u m e r i c a l t e c h n i q u e ( e . g . t h e o n e d e s c r i b e d i n R e f e r e n c e ( 1 ) ) i s t h e o n l y way o f o b t a i n i n g i n f o r m a t i o n on t h e t e m p e r a t u r e h i s t o r y o f t h e c o n s t r u c t i o n . I n s u c h c a s e s , however, when t h e p r o p e r t i e s of t h e m a t e r i a l s do n o t change markedly w i t h t h e t e m p e r a t u r e , and t h e geometry of t h e c o n s t r u c t i o n i s v e r y s i m p l e , t h e s h o r t c u t method d e s c r i b e d i n t h i s p a p e r may a l s o y i e l d a c c e p t a b l e a c c u r a c y .
THEORETICAL CONSIDERATIONS
The simplest model of a specimen under fire test is a slab whose temperature at one surface is suddenly increased from +: To to Ti
,
while the other surfacecontinues to be in contact with a fluid at a temperature of To. The ternper- ature distribution through the slab is governed by the following equations:
a t x = 0; for t > O (2)
at x =
R
(3a t t
=
0 (4The analytical solution of the problem has been given by Carslaw and Jaeger (2). By rearranging their equation into the form
a correlation is obtained which can be very conveniently represented on semi- logarithm paper (Fig. 1) by families of curves similar to those obtained by Gurney (3) and Gurney and Lurie (4, 5) for various cases in which T _ = Ti. Here, of course, the steady-state temperature is not constant, but is a function of x:
The
en
are the posi Live roots of the equation(Some numerical values of are given in Appendix IV of Reference 2.)
Since Fig. 1 is a condensed representation of a function of three inde- pendent variables, it cannot be detailed enough without 5he danger of
intricacy. In many cases, especially if the group K t / k L is small, it
might be necessary to use the numerical values listed in Table I.
Interpolation can best be done on semi-logarithm paper. If higher accuracy is required a few values of the. r tio (T,
3
-
T)/(T,-
r
) might be calcu- lated, either in the range ( K t / l )>
0.3 whe e only theOfirst term of the1
series is important, or in the range 0 <(Kt/L )<0.05 where the following approximation can be usedThe validity of this equation rests on the fact that for small values of kt/l2 the boundary condition at x
-
4
has little effect on the heat flow: the slab can thus be regarded as a semi-infinite solid.In practice,the temperature of the slab surface in contact with the fluid of temperature T is of primary interest. This temperature can more conveniently be 8etermined2by means of Fig. 2, which is a detailed plot for x / e = 1 and for ( ~ t / l )
,<
0.6.
From equation(6)
the steady-statetemperature of this surface is
In the above discussionsit was tacitly assumed that the heat transfer coeffic- ient on the side with temperature Ti, hi, is infinite. Although the analytical solution of the heat flow problem is known for finite values of hi (6), and tables and graphs are also available (7, 8), the slight improvement in accuracy attainable by assigning some high but finite value to hi does not seem to
justify the increased labour when the temperature distribution in fire test specimens is to be calculated.
CALCULATION PROCEDURE
It is well known that in the case of fire test specimens the condition that the temperature of one surface varies according to a step function is very poorly fulfilled. Although the furnace temperature rises very
sharply during the first 10 minutes, it does not approach a constant value but keeps rising slowly throughout the test. In addition, because of the finite value of h the actual surface temperature, is somewhat
i
'
Tk
lower than the furnace temperature. The approximation of ta ing T. constant
1
(for t >0) and equal to the average furnace temperature over the period of interest is therefore the first source of error introduced into
In Fig. 3 the variation of Ti (assumed to be equal to the average furnace temperature over the period of interest) with the duration of the fire test
is shown. The curve has been calculated on the basis of the standard furnace temperature versus time correlation of ASTM specification El19 (8). Regarding the coefficient of heat transfer at the unexposed surface of the specimen as constant during the test is another source of error. It has been shown(1) that for non-metallic surfaces the heat transfer coeffi- cient may be expressed in the following form:
where A = 0.27 for vertical surfaces
A
=
0.38 for horizontal surfaces facing upward.The constant, 0.157, has been calculated on the assumption of
E
~ 0 . 9 1 which seems to be a good average for the emissivity of the most common non-metallic building materials. (Because of the presence of reflecting surfaces it was thought that an apparent value of 0.85 for would yield a somewhat better value for the radiant-heat loss from wall specimens tested in the NRC furnace laboratory. With = 0.85 the constant becomes 0.147).In Fig. 4 hv and hh are plotted against Tw for To
=
74 F, the yearly average temperature in the NRC furnace laboratory.Since during a fire test T
5
Tw<
TO+
250 (i .e. 74 ( T ( 324 F), one might think that the acerage of h or he over the above eemperature interval is the proger value to be used in he calculations. It was found, however, that a better accuracy resulted by taking hv or hh higher than the average h during the test, possibly because in this way a fairly rapidfictitious heat absorption (due to the assumed step-function variation of Ti) at the beginning of the test period was partly compensated by a higher heat loss at the unexposed surface. In most cases the value of h at the
temperature of thermal failure of the specimen (Tw = To
+
250 F) is probably the best choice.While the assumption of an "idealized" temperature history on the exposed side and a constant h on the unexposed side does not necessarily intro- duce major errors in the calculations, assuming the constancy of the thermal conductivity and, even more, a linear relation between the enthalpy and temperature will very often completely falsify the result. In fact, the two latter assumptions are permissible only for completely dry, chemically stable materials, conditions that are rarely met in actual building constructions. Nevertheless, the value of the present method must not be underestimated, because in many practical cases an
idealized or limiting temperature distribution in the test specimen is all that is sought.
The method is,of course, also applicable to the calculation of a limiting fire endurance period. Since T is a function of the duration of the
i
test (i.e. of the fire endurance period), the calculations should be
started by making a rough estimate o f T , the fire endurance. After deter- mining the corresponding T temperature (Fig. 3) the steady-state tem-
i
perature of the unexposed surface ( T m w ) is to be computed by means of equation (9). Then the
ratio has to be calculated with T = T + 250 (the temperature of the
W 0
unexposed surface at the point of thermal failure, i .e. at t = 7 ), and by means of Fig. 2 the Fourier number (K-T/!~) has to be found. From the Fourier number the time in which T reaches the T + 250 level (i.e. the
W
time of thermal fire endurance) is calculable,
I?
this computed fire endurance does not agree satisfactorily with the one originally assumed,use this value for finding a more accurate value for Ti and repeat the calculations. Sometimes, especially in the case of thin slabs, three or four successive approximations may be needed.
As an example, the calculation of the fire endurance of a dry brick wall
8 in. thick (
1
= 0.607 ft) will be shown here. The properties of the brick are assumed to be as follows: p = 110 lb per cu ft, c = 0.216 Btu per lb F, and k = 0.55 Btu per hr ft F. With these values the thermaldiffusivity = 0.55/110 x 0.216 = 0.02315 sq ft per hr. Let TO be 74 F and take h
=
2.82 Etu per hr sq ft F, which is the value ofat Tw = 74 + 250 = 324 F, for E r 0.85.
A rough estimate of the time of fire endurance is T = 4.0 hr. With this Ti
=
1770 F. Sincetherefore for the point of thermal failure
For (T .*:
-
Tw)/(TNw-
To)W = 0.349, and h//k
--
(from Fig. 2), thus
Since this value differs significantly from the assumed one (T = 4.0 hr), the calculations must be repeated. Taking T. = 1820 F (T. at t = 4.92 hr),
1
T = 4.74 hr will result, which is identical wlth the value
obtainable by further recalculations. (The correct result is 7 = 4.75 hr at which T --: 1810 F.)
i
2
Fy substituting some t (< . f ) value into the 4 t/A? group and (by means of
Fig. 1) finding the val es of (T,
Y
-
T)/(T, -T ) pertaining to thatcular value of the K t / I group, to hk/! = 3.42: and to different x/
Barti-
ratios, one obtains the temperature distribution in the brick wall at t hrafter the start of the test. If this procedure is repeated for several different values of t, curves of the idealized temperature history of the wall during the fire test will result. The dashed curves in Fig. 5 show
the idealized temperature history of the exposed and unexposed surfaces, as well as that in planes at 2-, 4- and 6-in. distances from the exposed surface. The full curves have been obtained by means of a numerical method described in Reference (1). In these calculations the variation of T. and h during the test was also considered.
1
In Table I1 calculated values of the fire endurance of some simple building elements are compared with test results. The tests were carried out and the properties of the materials determined in the Fire Research Laboratory of the National Research Council. Unfortunately, test reports issued by other laboratories fail to mention the thermal properties of the materials, thus offer little basis for the analysis of the results.
The values of the thermal conductivity and specific heat given in the Table are those pertaining to 700 F which may be regarded as the average temperature of the specimen during the test.
Specimen Xo. 3 contained 5.75 per cent (by volume) moisture before the test. The moisture is known to increase the period of fire endurance. The gain in fire endurance is about 4 per cent per every per cent (by volume) of moisture in the case of 4 in. thick slabs, as shown in another publication (1).
Both the result of the fire test and the fire endurance corrected for zero moisture content (this latter value should, of course, be compared with the
calculated fire endurance) are included in Table 11.
In spite of the several simplifying assumptions used in this method of cal- culation the agreement between the calculated and experimental values seems to be satisfactory.
REFERENCES
1. Hamathy, T. Z. A treatise on theoretical fire endurance rating.
ASTM Special Technical Publication No. 301, 1961, p. 10-40. 2. Carslaw, H. S. and J. C. Jaeger. Conduction of heat in solids.
2nd ed., Oxford, Clarendon Press, 1959, p. 126, 493. 3. Gurney, 11. P. I-leating and cooling of solids of special shapes.
Unpublished monograph, M.I.T. Library, 53 p.
4. Gurney, 11. P. and J. Lurie. Charts for estimating temperature distributions in heating or cooling solid shapes. Ind. Eng. Chem.,
15,
1170 (1923).5. McAdams, W. H. Heat transmission. 2nd ed., New York, McGraw-Hill, 1942, p. 32.
6. Hill, P. R. A method of computing the transient temperature of thick walls. NACA Technical Note 4105, Washington, 1957.
7. Thorne, C. J. Temperature tables. Part I. One-layer plate, one-
space variable, linear. U.S. Naval Ordnance Report No. 5562, Part I, 1957.
8. Harmathy, T.Z. and Blanchar4J.A.C. Transient Temperatures in Slabs
Heated or Cooled on One Side. Accepted for publication as a Note by the Can. J. Chem. Eng.
9. Stanaard methods of fire tests of building construction and
Nomenclature
constant, Btu per hr sq ft F 5 14 specific heat, Etu per lb F
coefficient of heat transfer (without subscript: at x = k ! ) ,
Btu per hr sq ft F
thermal conductivity, Btu per hr ft F
thickness of the homogeneous slab, ft time, hr
temperature, F
distance from the surface exposed to fire, ft
Greek Letters
d constant defined by equation ( 7 ) , dimensionless
E
emissivity of surface, dimensionless-
K
-
k/fJ c, thermal diffusivity, sq ft per hr density, lb per cu f t\.
I
time of fire endurance, hrSubscripts
h for horizontal surface facing upward
i a t x = O
o at t = 0, bulk for x > >
1
v for vertical surf ace
This paper is a contribution from the Division of Building Research, National Research Council of Canada, and is published with the approval of the Director of the Division.
7 2
*
W c n dI
I=
0 a g cr\ a h 3 mZ '
..
0 E l Y k m k.d a 0 0 0 k V 3 W ; a I 5 c 4-l*
al 0 m2
5
h x 0 UP'Y
m Q) d d m R*
(do & G 4 d o Q ) O k b PI 4, 0*
k cd (d n 51 d a 4 k al*
(dz
I 4 O a c e G a l 0 co E a a3.
dm
C * $ l d x o a l l-4 CU r2 Im
bu;
cU 0 c? 0m m
baa3 NcU * r l O O h II II II & l o o r ,-
I 5 'd 0 al rlo cd cd u 0 c? c? c? 0 9 a' b b? I
rl 9 d I 0m
0 ' 9 0r-
-9
o 9 m
rnm
mcUOf * r l O O d II II II # o m I * Q ) *hz L hz boo M d C, 4 0 0 cr\ C, -4:50
2::
k d k P O P d CUTEMPERATURE DISTRlEUTION IN HOMOGENEOUS SLABS DURING FIRE TEST
by
T. Z. Harmathy
FIGURE CAPTICd'S
FIGURE 1 Transient temperatures in a slab heated on one side.
FIGURE 2 Transient temperatures of the cool surface of a slab heated on one side.
FIGURE 3 Average temperature of exposed surface of specimen during fire test.
FIGURE 4 Coefficient of heat transfer at the unexposed surface of specimen.
FIGURE 5 Temperature distribution in a dry brick wall of 8 inches thickness during fire test.
? ?
'!'
? '? L D 0*
(U ( D o*
N-
& ? ?
? 0 0 0 0 0- 0 0 0 0 0 0 0 0
0 0 0 i, 0 0
FIGURE
2
TRANSIENT TEMPERATURES OF THE COOL SURFACE OF
A SLAB HEATED
2000
1500
HEAT FLOW ANALYSIS CURVES OBTAINED BY T H E P R E S E N T METHOD tL b- 1000 500 0 0 I 2 3 4 5 t HR FIGURE
5
TEMPERATURE DISTRIBUTION IN A DRY BRICK Wb,LL OF 8 INCHES THICKNESS DURING
FIRE
TESTDiscussion by T.W. McDonald* on "TEMPERATURE DISTRIBUTION IN HOMOGENEOUS SLABS DURING FIRE TESTw by T.Z. Harmathy.
This paper presents in graphical and tabulated form a means of calculating the temperature distribution throughout a homogeneous, isotropic slab when one side is subjected to a sudden temperature change while the opposite
side remains in contact with a constant temperature fluid. I feel that these tables and graphs will be found useful in a number of applications, however, I am not convinced of their value in predicting fire test results.
The author states that these results may be used to predict the temperature history of a test specimen during a standard fire test and illustrates this
in Figure 5. This same diagram .would seem to me to be adequate proof that the comparison is poor. Discrepancies of 50% and more occur within the slab during the test and differences of 10% are present at the end of the test. It seems reasonable that this difference will be further exaggerated for very thin and for very thick walls. Perhaps the author has had occasion to compare results for very thin walls and can give some idea of the dis- crepancies to be expected.
The second use that has been suggested for this analysis is the prediction of the fire endurance time. Once more I am somewhat dubious of its unqualified use for this purpose. The technique has been designed to provide reasonably correct times for average wall thicknesses and hence, it will predict endur- ance times which are too short for thin walls and too long for very thick walls. This is shown by the comparison given in Table I1 where the calculated results for a ,333 foot thick brick slab is 13% low and that for a .695 foot thick brick slab is 10% high. This trend is due to the fact that for thin walls the excessive fluid cooling due to the large assumed value of h (the convective heat transfer coefficient) does not have sufficient time to counteract the effect of the sudden surface temperature change in the short time required to reach the fire endurance limit. On the other hand, for very thick walls this large heat loss to the fluid overbalances the effect of the sudden surface temperature rise and hence, fire endurance times which are too large will be predicted.
Discussion by J.W. Stachiewicz+:$:
The paper presents the application of Carslaw and Jaeger's solution of the transient heat conduction in a slab to the problem of determining the fire endurance of a brick wall. The author compares two theoretical solutions, one of which could be termed semi-idealized (where the variation of TfUrnace and h during the heating period is taken into account) and the other is fully idealized (i.e., Tfurnace, h, k are constant). He reaches the conclusion that the fully ideal-ized solution yields an acceptable accuracy, particularly at the cold surface of the slab.
Dept. of Mechanical Engineering, University of Saskatchewan, Saskatoon.
**
McGill University, Montreal.Presumably both these calculations represent a theoretical solution of an actual test of a brick wall, exposed on one side to simulated fire conditions. It would appear of primary interest, therefore, to see how the author's
solutions compare to actual experimental results obtained with a brick wall. Only t h c n can one formulate an opinion as to the accuracy of the proposed method of calculation.
It is also not very clear on what basis the selection of surface temperature for the calculation of h, or hh was made. If this was done o n the basis of experimental results, it would again be interesting to see those experimental points superimposed on the theoretical plots to substantiate the choice.
Finally, the writer would appreciate some comments on why the lowest temperature in the wall is used as a criterion of "thermal failurew. Other parts of the wall are at much higher temperature throughout the test and unless the term
"thermal failure" is a complete misnomer, it is difficult to understand why the specimen should be considered to fail thermally when its lowest temperature reaches 250°F while a good part of the slab has apparently been able to with- stand temperatures in excess of l,OOO°F for several hours.
Discussion on the Paper TEMPERATURE DISTRIBUTION IN
HOMOGENEOUS SLABS DURING FIRE TEST
T.Z. Harmathy (author's closure).
-
The author wishes to thank the discussors for their interest in this paper.The purpose of using "fire enduringtt or "fire resistant" building element is to prevent the propagation of flames from one compartment to another, when fire occurs in a building. It is generally accepted that about 330 F is the safe limit to which the temperature of the unexposed side of walls and floors can be allowed to rise without the danger of starting fire in the neighbouring compartment. This is the reason why the temperature of the unexposed surface, which is the lowest temperature in the construction, is used as the criterion of "thermal failure". For certain other building elements, e.g. protected steel beams, the temperature of the load-bearing steel section is used as a criterion of failure. Of course, in such cases the limiting temperature is considerably higher (1000 F).
The sample calculation shown in the paper has been based on assumed values of the thermal properties. No experimental results are available for brick walls which were known to exhibit similar properties. Table 11, which contains
experimental values for two walls built from brown clay bricks, was inserted in the paper after receiving Dr. Stachiewiczts comments, in order to give the readers some idea of the range of accuracy of the proposed method. (The question of accuracy will further be discussed in a subsequent paragraph.) Brick walls of the same thickness may yield fire endurance varying in a very wide range. In the common usage of language the word "brickw may have a very specific meaning; in strict scientific sense, however, it covers a large group of not too closely related materials with markedly different physical properties.
The recommendation to select values of hv and hh pertaining to a surface
temperature of To + 250 F as the coefficient of heat transfer at the unexposed surface, has been made on the basis of comparing several results obtained for the thermal fire endurance by the use of the present shortcut method and by detailed numerical analyses, when the actual boundary conditions were more closely approximated. As Table I1 shows, this selection also has some experimental support.
Whether a
+
15 per cent accuracy is "poor1* or l*acceptabletl, depends on two factors; the nature of the problem, and the purpose of the calculations. There are many fields of engineering design where this accuracy is still a dream rather than reality. Unfortunately, the ndesign for fire endurancen is one of those fields.It should be remembered that during a fire exposure the temperature of a building element varies in such a wide range that the assumptions used in the classical treatment of heat conduction problems are not strictly applicable. The primary concern is not so much the variation of the thermal conductivity with the temperature, but the fact that most building materials become unstable at elevated temperatures and undergo certain physico-chemical changes which are accompanied by absorption or evolution of heat. In some cases these changes are so great that the material at, say,1500 F bears hardly any resemblance to what it was at room temperature.
The readers have been cautioned that the error due to assuming the thermal conductivity constant, and assuming linear relation between the enthalpy and temperature, may be more significant than that due to the simplified boundary conditions. Although these two kinds of errors are hardly separable at this time, since there are only a few building materials for which the enthalpy versus temperature relation is known over the whole range of interest, it seems quite possible that the error due to the simplified boundary conditions
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alone may be as high as
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15 per cent. As mentioned in the introduction, a more refined method of calculation, using significantly less assumptions than the one presented here, is also available, It would be pretentious, however, to apply this latter method to the calculation of the temperature distribution in homogeneous slabs, unless the variation of the thermal conductivity, enthalpy and density of the material with the temperature, up to about 1800 F, is known with an accuracy better than, say, 2 15 per cent.There are many cases when the use of numerical techniques would not be reason- able, even if all the above information is available. It must not be forgotten that most often it is not the fire endurance which determines the sizes and materials of a construction, but other factors related to its normal function, such as load-bearing capacity, heat or sound insulation, appearance, etc. In many of these cases the fire endurance is only a side issue, and a shortcut method which enables the design engineer to check whether the fire endurance
of the construction is higher than the value prescribed by the building regulations, is all that is needed.
The author agrees that for very thin or very thick slabs the accuracy of the present method may be poorer than 2 15 per cent. This fact would present no practical difficulty, however, since walls or floors exhibiting less than 3/4 hr fire endurance cannot be called fire resistant constructions, and the maximum
fire endurance prescribed in the National Building Code is not higher than 4 hr, Expressing temperature differences in percentage is obviously quite arbitrary, unless the Kelvin or Rankine scale is used. Figure 5 shows that if x/j) 0.25, the difference in the temperatures calculated by the present method and by the referenced numerical technique is never larger than about 170 F, and is always
less than 100 F within the interval & ' T ; < t The most commonly met problem which can be resolved with the aid of the temperature history of the slab, is
the estimation of the rise of temperature of a steel load-bearing member embedded in the slab at a certain distance from the surface exposed to fire. Since the creep of steel is quite insignificant at temperatures below 750 F, the relative- ly large discrepancies in the O(f< 1/42 interval are of little practical
