[Paper] a Numerical Procedure to Calculate the Temperature of Protected Steel Columns Exposed to Fire

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Fire Study, 28, p. 40, 1972-03

A Numerical procedure to calculate the temperature of protected steel

columns exposed to fire

Lie, T. T.; Harmathy, T. Z.

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NATIONAL RESE ARCH COUNCIL O F CANADA

DIVLSION O F BUILDING RESEARCH

-.. -

v-* p - -f r $73z [ : L ~ .-

A NUMERICAL PROCEDURE T O CALCULATE THE

TEMPERATURE O F PROTECTED S TE EL COLUMNS

EXPOSED TO FIRE

T.T. Li e and T. Z. Harmathy

F i r e Study NO. 28

of the

Division of Building Research

OTTAWAarch 1972



A NUMERICAL PROCEDURE TO CALCULATE THE

TEMPERATURE OF PROTECTED STEEL COLUMNS

EXPOSED TO FIRE

by

T. T. L ie and T. Z. Harmathy

ABSTRACT

A nu me ric al technique has been developed for the calculation of

the temp era tur e histo ry of protected st eel columns in f i re . This

technique was used for the theo ret ica l s imulat ion of sev era l f i r e tests .A com pari son of e xperim ental and theo retica l information cle arl y

showed that the technique i s cap able of yielding acceptab le accuracy .

Some basic assum ptions used in previous works have been

examined in the light of s ev er al nu me ric al studies. It ha s been

proved that the mechanism of heat tra ns fe r between the protection and

st ee l co re ha s l i t t le effect on the st ruc tur al performa nce of the ste el

core , and that eve ry pe r cent of mo isture in the protect ion inc rea ses

the ti me of f ir e endurance of the column by about th re e per cent.

INTRODUCTION

Columns a re the most c r i t ic a l s t ruc tu ra l e lements in a bui ld ing

in that thei r col lapse can lead to the los s of the en t ire s truc ture. T h e r e -

fore, the perform ance of protected st eel columns in f i r e ha s long at t rac -

ted co nsiderable at tent ion in various countr ies . The conventional

method of obtaining information on this subject i s by stan dard f i r e endur-

ance tes ts . The possibility of making realis tic theore tical e stim ate s

ha s been ham pere d by two fac tor s: ( i) the lack of knowledge concerning

th er ma l prop ert ie s of the commonly used protect ing ma teri als at elevated

tem pe rat ure s and certa in rheological prop ert ie s of s teel , and (i i ) the

comp lexity of the mech anis m of heat flow, esp ecia lly through physico-

chemical ly unstable sol ids.

The fi rs t of the se difficulties is not s o ser iou s now a s it was

10-1 5 ye ars ago. During the past decade information has accumu lated

on the the rm al and rheological prop ert ie s at elevated tem per atu res of



many importan t building mat erials , among them s te el and concrete. The

difficulties relat ed to the complexity of hea t flow ana lysis have also been

grea tly reduced by having the calcula tions per form ed by high-speed

comp uters . Thus many f i re perform ance prob lems that not long ago had

to be solved by e xperim ent can now be solved by num eri ca l techniques.

In prev ious publicatio ns by th e Division of Building Re sea rch , so me

num erica l techniques have a lread y been describe d for the calculation of

the tem pe ra tu re his tor y of vario us one- and two-dimensional configurations

typ ically employed in wall s and floo rs, and of the defo rmation hi st or y of

st ee l support ing elements, such as beams, joists , e tc. In th is paper a

num erica l proced ure will be described which can be used fo r predict ing

the tem pe rat ure h ist or y of another im port ant group of building elements,

protected s te el columns. It wil l be seen tha t the re sul t s a re a lso ap-

plicable to the estimation of the point of f ailu re of th ese elemen ts in

"standard" f i res .

PREVIOUS WORK

A con side rable amount of theo retic al work has alrea dy been done

during the past ten y ea rs in connection with the f i re performa nce of

protected ste el columns. Th ese works represented v arious approaches

to obtaining analytical solutions of the problem of heat conduction through

the pro tectiv e insulation into the st ee l co re. It was unavoidable, th er e-

fore , that nume rous simplifying assumptio ns w er e employed with re s -pect to both the m ate r ia l prope r t ies and the heat t ransm iss io n mechanisms.

Consequently, the applicability of the derived fo rmu las i s l imited tothose c as es in which the assumptions used a re clo sely sat isf ied.

With res pe ct to the modeling of heat tra ns mi ssi on m echanism s,

the following conce pts we re employed (se e Fig. 1):

(a) The th er ma l conductivity of s te el is infinite; in other words, the

tem pera ture in the s tee l co re i s uniform over the en t i re volume (1 -8).

(b) The thickness of the insulat ion in relat ion to i ts cir cum feren ce is

so s ma ll that the heat f low through i t can be rega rded as one-dimen-

si on al (1 -8).

(c) The the rm al res is ta nc e between insulat ion and the ste el is negligible

(1-8).

(d) The tem pe ra tu re of the exposed su rfa ce of the insulation is equal to

the f i r e t em pera tu re (2-4, 6, 8).



(e) The variat ion of the tem pe rat ure ac ro ss the insulat ion is approximately

l inea r (1, 5, 7).

(f) Th e th er m al conductivity and hea t capacity of the insulation can be

cha racte rized by constant values within the temp era tur e ranges thata r e of in te re st (2-4, 6-8).

(g) The th er m al capacity of the insulation is negligible (1, 5, 7).

(h) The air enclosed by the protect ion has the sa me tempera ture as

t h e s t e e l (2, 33, o r its capaci ty is neg ligib le (1, 4-8).

(i) The influence of mo ist ure in the insulation is negligible (4, 6, 8), or

the moi stu re is concentrated and evaporated at the inner s urfac e of

the insulation (2 , 3).

Of cou rse , som e of th es e assu mp tion s, e. g. assu mpt ion s (a) and

(h), a r e fully justifiable f ro m a pra ctic al point of view, and, ther efor e,

wil l a lso be used in the pres ent s tudies. Yet, with th e u se of m o re

adaptable num erica l techniques, i t wil l no longer be nece ssa ry to retai n

those highly restr ict i ve assumptions that were previously introduced only

to ren der the prob lems amenable to theoret ical solut ions.

In the prese nt s tudies an a t tempt wi ll be made to u se the fewest

assumptions possible concerning mat eri al behaviour and heat t r an s-

mis s iori mechanisms. The unavoidable presenc e of mo ist ure in some

pro tectiv e ma te ri al s will also be taken into account, even though only in

a simplified manner. The results will be compared with information

obtained fr om tes ts . Some of the res ul t s will furth er be ut i lized to check

out the ac cu rac y of simplifying assu mptions used in previou s studies.

It should al so be mentioned that the ap plicability of th e techniq ue

to be described is not l imited to protected ste el columns. In fact,

i t can be applied to any assemb ly consisting of a c en tra l co re of r elativ ely

high th er m al conductivity, surroun ded by a squ are -shaped envelope of

much lower conductivity, which is exposed to radiativ e heating on al l

four s ides. It can also be used for the calculat ion of the tem pe rat ure

his tor y of monolithic columns o r beams.

NUMERICAL PROCEDURE

The heat tra ns po rt in and at the bounda ries of the insulation will

be formulated with the aid of a finite differenc e method origin ally des cribe d



in Ref. 9 and la te r elabora ted upon in Ref. 10. Th is method ha s been

applied to the solution of f i r e resis tan ce problem s in Refs. 11, 12 and

1 3 .

The f ir s t s tep in applying this method to the presen t problem is

to divide the cros s-sec tion al ar ea of the insulating protection into a

la rg e number of elem enta ry regions by the use of a two-dimensional

network. Since the the rm al conductivi ty of s t ee l i s norma lly at le ast

20 tim es hig her than that of the protection, the idealization that the

tempera tu re of the s t ee l core i s uniform a l l over i t s volume see ms a

justifiable one. Consequently, thi s two-dimensional netwo rk need not

be extended over the cross-sect iona l ar ea of the st eel core, the sub-

division of which thus c an be done on a mo re convenient basis , as will

be described later .

As in a previous nume rical s tudy ( l3) , for pract ica l reaso ns adiagonal mes h has been selected for subdividing the cros s-se ct ion al

a re a of the insulation (s ee Fig. 2). T h e e le m e nt ar y a r e a s a r e s q u a r e

in the inside of the insulation and trian gula r a t i t s boundaries. Fo r

each ins ide e lement , the tempera tu re a t the cent re i s taken as r epr e-

sentative of the e nti re element. Fo r each tr iangular boundary element,

the rep res en tat ive point i s located on the hypotenuse.

Since only columns with sq ua re protection will be co nsidered in

thes e studies, i t is possible, owing to four-axe s symm etry, to calculate

the te mp er at ur e distribution in only one-eighth of the c ro ss -sectional

area of the insulation.

As Fig. 2 shows, in an x-y coordin ate syste m a "rep rese ntativ e"

point of the protec tion , Pm , ( represent ing region (man) or Rm, n) has

the coordinates x = (m-1) A 5 /J2 and y = (n-1) A 5 / 2. It is obvious

fro m the figur e that only thos e points of the x-y plane a r e defined for

which (m + n) i s an even number.

EQUATION FOR THE INSIDE OF INSULATION

A convenient way of obta ining equations f or the c alcu latio n of

the tem pe ratu re hist ory of insulat ion i s by wri ting heat balance equationsf o r i t s el em e n ta r y r eg io ns . F o r a n i ns id e r eg io n h , , ( r ep r e s en t e d by

point Pmr ) the hea t balance equation fo r a unit height of the column

covering a short period of A durat ion i s a s follows:



t kj ~j "+ Ikim+l), (n-1) m ) 1 )

2 A 5

kJ T j - T~+ ~ m t l ) ,nt 1) t '

J (( m t 1), (n t 1) m

2 A 5

The te r m on the lef t side of thi s equation e xpr ess es the accumulation

of heat in Rm,, during a tim e int erv al jAt < t r ( j t l ) At. The four

te rm s on the right-hand side descri be th e heat entering Rm, by conduction

during the sa m e period fro m the neighbouring regions: R(m-l), (n-1)'

R R R The te rm s in round( m t 1 n l ( m ( n t1 (m t 1), (n t 1) '

brackets represe nt the tempe rature gradients and those in squa re brackets ,

the av era ge conductivity of the m at er ia l along the re spe ctiv e paths of

conduction.

k - j \- k ( T m , n ja etc.an

(P c)j- ' j ), etc.

m8 n - ( T m , n



j and (pc)' rep res en t the values of k and (P C) ,n other words, k

respect ively, at t~&Pe mperatu r=kqh atrevai l s a t P at t = jAt.m an

The k(T) and pc(T) functions, which can be determ ined exp erim enta lly,

a r e assumed to b e known.

If the temp era tur es in al l e lemen tary r e ions a re known at

t = jbt, th e only unknown quantity in Eq. 1 i s T6 1

i. e. the tem per a-

t u r e a t P,, , t t = ( j+l) A t . 1t can be c a l c u ~ e s , herefore , f rom

the following rearranged form of Eq. 1:

:. k j + kj ) ( ~ j - TJ\ ( m t 1), (n t 1)

I

m, n (m+ ), (nt ) m,n ) J

EQUATIONS FOR THE OUTER BOUNDARY OF INSULATION

In a s tandard f i r e tes t hea t i s t ran sfer re d f rom the " furnace"

(i. e. fr om the flam es and furn ace walls) to a column specim en both by

convection and radiation. When the flam es a r e of sufficient thicknes s, r a -

d ia tive heat t ran sfe r i s the pr i ma ry m echanism (14). Exper imenta l

data have indicated (13) that in a f i r e t e s t fu rnace the t r ansm iss ion of

heat to the tes t sp ecimen is approximately equivalent to radiat ive heat

tra nsf er f rom a black body at the so-cal led "furnace tem perature".

Consequently, in the pres en t stud ies the columns will be modeled a s





Although in th is equation €. is , s t r ict l y speaking, a ma ter ial and

temperature dependent quantity, it is sufficient ly accurat e to re gard i t

a s constant in the pre sen t s tudies. Since most bui lding mate ria ls have

emi ssivi t ies in the ra nge of 0.85-0. 95 (27), a value of 0. 9 will be used.

EQUATIONS FOR TH E INNER BOUNDARY O F INSULATION AND FOR

THE STEEL CORE

As Fi gu re 1 shows, the inne r sur fac e of the insulation is in

direc t contact with the ste el co re along a c ertai n fract ion, a, of i t s

surface, and i t is separated by an ai r gap from the s tee l along a frac -

tion (1-a) of it s surf ace. Obviously, the mech anism of heat tra nsm iss ion

along the ar ea s of contact i s conduction. Through the a ir gap, heat is

tra ns fe rre d by radiation and convection. Since the radiative heat tr an s -fe r is predominant , especial ly at higher tem pera ture s, the convective

tra nsf er mecha nism wil l not be taken into account in the pre sen t s tudies.

The model used in this paper t o des crib e the mechanism of heat

tra ns mis sio n at the t r iangular elemen tary regions of the inner surfa ce

of insulation is shown in Figure 3 . In this model the total m as s of s te el

co re i s a ssumed to be divided into elem entary pieces amounting to the

numb er of elem enta ry regions along the inn er s urf ac e of the insulation,

i. e. into 4 (N-M-1) pieces. It i s fur the r assu me d that a fract ion aof each elemen tary st eel ma ss is in dire ct contact with the adjacent

elementa ry insulation surface, and thus rece ives heat from the insula-

t ion by conduction, while a fract ion ( l a ) of i ts m as s is a t som e d i s t ance

fro m th e elem enta ry su rfa ce and receive s heat by radiation. Obviously,by varying a f rom 0 o 1, all pos sible pra cti cal conditions, including

pure radia t ive and pure conductive heat t ran sfe rs to the s te e l core , can

be simulated. In this way the relat ive importa nce of the tr an sf er mechan-

is m on the r is e of tem per atu re of the s tee l co re can be studied.

The r ad ia t ive hea t t r ans fe r red to the s t ee l co re f rom a ( 1 4 )

fract ion of the el eme ntar y region R of the inn er su rfa ce of ins ula -m.1 n

tion during the period jA< t s ( j t )A t is

-where the emissiv i ty factor E , can be calculated approximately from the

following equation:



Here, again, c and c will be regarded a s constants for the temper a-i

tu r e rang es c onsidered and equal to 0.9.

Since, by assumption, ste el is rega rded a s a perf ect conductor,

the tempe rature s of those fractions of e lementa ry ste el ma ss es which

a r e in direct contact with the insulation sur face a r e identical to those

of the adjacent ele me nta ry regions of insulation. Consequently, t he ir

pre sen ce can be taken into account simply by adding th ei r heat capac i-

ti es to those of the adjacent e lem enta ry insulation regions.

Again, from a heat balance equation writt en for region RMsfor the period jAt < t s j t l ) A t , the following equation can be derived for

.j+l

Man

where (c )S M, n

is the sp ecific heat of ste el at a tempera ture that prevai ls

at P ~ ,t t = j A t . From available data (17, 18), the following expression

ha s been derived for the dependence of c on tempera ture:S



Of cou rse, the pr im ar y purpose of a l l thes e calculat ions is to

obtain informa tion on the tem pe rat ure of the ste el core . Again, bythe applicatio n of the law of cons erva tion of ene rgy, the te m pe ra tu re

of that (1 a) portion of the co re which rec eiv es heat by radiation i s

obtained as

As has been sa id ear l ie r , the temp era tur e of the fraction, a ,of each eleme ntary ste el ma ss, which i s heated by conduction, i s identi-

ca l to that of the adjacent s ur fac e eleme nt of the insulation. The

ayerage tem pera ture of the ent i re s te e l cor e a t t = ( j t l ) At, i , e.T J ' ~ can now be calculated fr om t he following equation:

s a

=j+ 1 Tj t l2a

N-M

SC dT = N-M-1

C

j+ lS

n= 3, 5,. ..T s ~

which i s obtained by ex pres sing the enthalpy of the st ee l co re in two

diff eren t ways; once with the aid of alr ea dy defined va ria ble s, and

once by using T J ' ~.s a

AUXILIARY EQUATIONS

As Fig. 2 shows, the following equations a r e applicable to the

ele me nta ry regions along both si de s of the lines of sy mm etry .

Along line A-D:



and along lin e B-C:

.j+ 1 -ma (N-m) ( m t I) , (N-mt 1)

With th e aid of Eqs . 4, 6, 8, 10, 11, 12 and 13 it i s now po ss ib le

to calculate the temperature distr ibution in the insulation and on i ts

boundar ies f or any ( j t l ) At t ime level , i f the t empe ra tu re d i s tr ibu tion

at the jOt le ve l i s known. Init ia l ly only the tem pera ture dis t r ibut ion

a t the t = 0 lev el i s known. In f i re endurance s tudies the in i t ia l

te m pe ra tu re of the column (insulation and s teel ) ha s always been taken

as 70" F and uniform; thus

And, sta rt i ng fr om the init ial condition, with repeate d application of

Eqs. 4, 6, 8, 10, 11, 12 and 13 th e te m pe ra tu re hi st or y of the pr ot ec -

tive insulation and of the st ee l co re of the column can be determin ed

up to any specified t ime level.

S ince in f i r e endurance s tandards 1000°F i s usua l ly regarded

as the tem pe rat ure of failure of s te el core, the calculations ca n be

te rmina ted a f te r i t s t emp era t u re has exceeded 1460"R.

EF FE CT OF MOISTURE

Although the re ha s a lrea dy been a fa ir amount of work done

at the DBR/NRC concerning the effect of m ois tur e on the fi re en durance

(19, 20), al l previou s w ork relate d to walls and floors. Since i t seem ed

unlikely that the re su lt s of this wo rk could be applied to columns, a

dif ferent concept had to be considered to take the pres enc e of m ois t ure

into account in the pre se nt studies.

I t is well known that under no rm al at mo sph eric conditions, i. e.

at room tem pe rat ure and at about 50 to 70 pe r c ent rela tive humidity,

the bulk of m ois tur e in building ma ter ia l s i s in the form of capi l lary

water. The capi l lary water ha s a fa i r ly h igh mobil i ty and, a snum erous observ ations and the oret ical work (19) indicated, under the

effect of high p re s su re gradien ts developing during a f ir e exposure

i t wi ll move s lowly toward the cooler regions; toward the inner su rfa ce

of th e protec tive insulation, in the ca s e of a column. It seem s reasonab le

to assume, therefore , that a lar ge por t ion of the mo is tur e or ig inally



pre sen t in the insulat ion wil l f inal ly vapourize at the inner surfa ce of

the insulation.

A comp rehensiv e com puter study was undertaken to find out

whether the r a te of m ois tur e migra tion had any significant effect on

the tempera ture h is tory of s tee l core .

The co mputer stud ies indicated that the influence of the r at e

of m oi st ur e movement was not sufficient to justify th e inc reas ed

labour involved in a m or e elabo rate formula tion of the problem. It

was decided, therefore, that in furth er work only the l imit ing ca se

would be considered, in which the r at e of m ois tur e migration is infinite;

in other words, al l moistur e original ly pre sen t in the insulation is

transpose d into the inner surf ace lay er of the insulat ion r ight fro m the

beginning of the fire exposure. This model can be recognized a s pra c-

t ical ly the sa me a s the one alre ady used in Refs. 2 and 3 .

The hypothet ical mois ture concentrat ion in the t r iangular

elemen tary regions of the inner su rface, af ter the moistu re original ly

pres ent in the insulation was transposed into thes e regions, can be

writt en as follows:

Th is eq uation can be verifi ed with the aid of F ig. 2.

Since, according to this model , moi sture exists only in theelem enta ry regions along the inner s urf ac e of the insulation, the

effect of m oi st ur e ca n be taken into account by modifying th e equation

concerned with the tem pe rat ure of these regions, name ly Eq. 8. The

pres ence of m ois tur e affects the heat balance for an elemen tary region,

Rm, nr by (i) absorb ing laten t hea t in the vapor ization pr oc es s and (ii)

incr easi ng the heat capac ity of the regions.

To enable one to form ulate the probl em of vaporization of

moisture, i t i s nece ssa ry to define a funct ion which de scr ibe s the

fract ion of th e net heat, supplied to an elemen tary region, that i s used

for evaporat ion of m oistu re on reaching certa in tem pe rat ure levels .Since the bulk of evapo ration is known to take plac e in the vicinity of

the boiling point, i. e. 672" R, the function to be ch osen should obviously

have a s teep sec t ion a t th is tempera ture . Th e following function fulf ils

this requirement:



6 = i erfc (A

where A is a constant, gene rally taken as 10 in the pre sent work.

When the re i s still mo istur e in the insulation,a fraction

5 b, of the net heat inflow in a cer tain region R M ,, is used for

evaporation and a fraction ( 1 - CL ) for increasing the tempe ra-r n

tu re of insulation and st ee l core.

Fr om a heat balance equation si m ila r to Eq. 1, it can be

derived th at the change of the hypothetical mo istu re concentration

v n , ) in an elemen tary region RM, due to evaporation

in the period jAt < t ~ ( j t l ) t is :

The additional heat absorbed by thes e e lemen tary regions, pe runit time, due to the pr es en ce of (hypothetical) mo istu re can be ex pres sed

a s



-

T h e t e r m i n [ J bra cke ts is , in fact , a heat capacity additive

to those a l re ady in t roduced in Eq. 8. Thus the form of Eq. 8 modified

for the pr ese nc e of mois tu re and for m ois tu re evaporat ion becomes:

Thi s equation i s applicable to the calculation of TM," a s long as there-

i s mo is tu re in the insula tion, i. e. coJ > 0. If the insula t ion*is d r y ,M, n

o r becomes d ry dur ing the heat ing process, i. e. T J = 0, T J + ~M, n 1M n

is t o be calculated by Eq. 8.

EXPERIMENTAL VERIFICATION

To ver i fy the val id i ty of th is n um eric al technique, s tandard f i r e

tes t s w ere c a r r ied ou t on sev era l p ro tec ted s tee l co lumns . T h e r e s u l t s

w er e then compared with those obta ined by the theoret ica l s imulat ions

of t h e s e t e s t s . In the te s t s thr ee dif ferent protect ing mat er i a ls and twodi f fe ren t s tee l cor es w ere used . The effect of the moisture content of

insula t ion on the tem per atu re his t ory of s tee l cor es was a lso s tudied

in a few cases.

A typical te s t specim en is shown in Fig . 4. The descr ipt ion

of the com ponen ts of the s pe cim en i s given below:



1. Insulat ing f i re br ick s , to reduce heat l osse s f r om the bottom of the

s t e e l c o r e.

2. Pro tect ing device, to contain an insulating m in er al wool.

3. Miner al wool, to red uce heat lo sses f rom the top of the s tee l

core .

4. Ste el core: H-section.

5. Protecting insulation. In the c as e shown in the f igure the

insula t ion and s te el ar e separated a l l around by an a i r space .

Sev eral of the te s t specim ens were constructed with the insula-

tion in contact with the flanges of the s te el section, a s shown in

Fig. 1.

The following thr ee protect ing mate r ia ls we re used: l ightweight

con cret e of expanded shal e aggregates, insulating fi re brick Group 23,

and a heavy cla y brick. The ther ma l prop er t ie s of the l ightweight

con cret e wer e derive d fro m the data given in Ref. 21; and thos e of the

insulating fi re bric k fr om Ref. 22. Those of the heavy c lay br ick we re

mea sure d for a few tempe ratu res according to the method descr ibed

in Ref. 22. Since the me asur ed prop er t ie s were approximately equal

to those given in Ref. 2 3 , the la t t er data we re used for the whole

temperature range under considerat ion.

The the rm al pro per t ies of the insula t ing ma ter ia l s used, thei rmo is tur e content before the tes t , and the weight and s iz e of the s te e l

co re s a r e given in the Tab les 1, 2 and 3.

TESTING PROCEDURE

The tes ts we re ca rr ie d out by exposing tes t specim ens to heat ing

in a furn ace specia l ly buil t for th is purpose . The heat input into the

te s t furn ace was control led in such a way that the av erage tem pera ture

c lose ly fo llowed the s tandard tem pera tu re ve rs us t ime curve g iven by

Eq. 5. The fu rnace tempe ra tu r e was m easured by nine the rmocouples

located a t se ve ra l levels around the s pecim en with the ir hot junct ions

12 in. away fr om the su rfa ce of the specimen .

The temper a tu r e of the s tee l co re was me asured a t four l eve ls ,

but because of the s ma l l tem pera ture dif ferences between the var iou s



locations, only indications by th re e thermo couples located at the mid-

height of the spe cime n we re used. The avera ge of the tem per atu res

reco rded by th es e th re e thermoc ouples, of which one was located in

the c en tr e of the web and the two oth ers at the edge of eit he r flange,

was taken as a m ea su re of the tem pe rat ure of the ste el core.

RESULTS

Information concerning the variat ion of the average t em pe rat ure

of s t ee l co re a s obtained from the f i re test s , together with that obtained

by theo ret ica l s imulat ions, is presented in Figs. 5 to 8. In al l te st s

the furnace temper a ture followed very c lose ly the tempera tu re vers us

t ime curve formula ted by Eq. 5, and the ref ore it has not been plotted

in the f igures.

Pro bab ly becaus e of condensation of mo ist ure on the ther mo -couple wire s, no reliab le information could be obtained of the s te el

tem pe rat ure in the ini t ial s tages of those two tes ts (shown in Figs. 5

and 8) in which the protection contained moistu re. Th ese doubtful

tem per atu re me asur eme nts have not been plot ted in the f igures.

It is seen that in al l cas es a good a greeme nt exists between

experim ental and calculated tempe ratu res. This f inding also co nfirms

th e validity of the model used to account for the pr ese nc e of m oistu re.

As ment ioned ear l ie r , in previous works the therma l res is t ance

between insulation and the s te el cor e was always di sreg arde d (1-8). Tocheck the validity of this concept, calculation s we re perform ed f or the

fol lowing th re e modes of heat t r an sf er fro m the insulation to the ste el

core:

(a) Al l hea t i s t rans fer red by radia tion to the co re (i. e. a = 0

in Eqs. 8, 11 and 19).

(b) 50 per cent of the heat is t r ans fe r red to the co re by r ad ia -

t ion and 50 per cent by conduction; ther e i s no the rm al r esi s ta nce at

contact ing sur fac es between th e c or e and insulat ion (a = 0. 5).

(c) All heat i s t rans fer red by conduction from the insulation to

the co re without any the rm al res is tan ce a t the contact ing sur face s

between s tee l cor e and insulat ion (a = 1).

The calculated temp era tur es of s t ee l co re have been plot ted

against t im e in Figs. 9 and 10. Fig . 9 rel ate s to a column made with



a lightweight con crete protection, and Fig. 10 to a column prot ecte d

with insulating fi re brick. I t i s cle arly seen that the mechanism of

heat t ransfe r f rom the insulat ion to the s tee l core has only a sm al l

influence on the ste el temp eratu re. I t see ms just if ied, ther efore , t o

neglect the the rm al res is tan ce at the contacting surfac es between stee l

and insulation, in other words , to assume tha t the s te e l tempera tu re i s

equal to th e tem per atu re at the inne r su rfa ce of insulation.

Fu rth er calculat ions we re perform ed to obtain information on

the influence of the mois ture content in the protection on the te mp era tur e

of s te el core. The resu l ts a re shown in Figs. 11 to 13.

It is usual to reg ard the t ime at which the temper ature of s te el

re ac he s an av era ge of about 1000"F a s the t ime of f ai lur e for the column.

At this temp eratu re, s te el wil l have lost so much of i t s s t ren gth that

it no longer c an supp ort the load. It i s se en in Figs. 11 and 13 that fo rcolumns ma de with lightweight protection of conside rable thick ness,

the gain in t ime due to the presen ce of m oistu re may be substant ial .

With the exception of con cre te, however, comm only used inorga nic

building ma ter ial s do not hold much mo istur e under no rma l atmosp heric

cond itions. An exa min atio n of da ta (24, 25, 26) ind icate d th at while at

50 per cent relat ive humidity concretes can hold 3 to 6 per cent mois ture

by volume, mo st other ma ter ials hold les s than 1 pe r cent.

Fro m the data given in Figs. 11 - 13 it ca n be derived that th e

gain in f i r e r e s i s tance , i. e. in the t ime tha t i t takes the s tee l core

to re ach the 1000"F level, due to moi sture in the protect ion i s roughly3 pe r cent for each per cent mois ture . Thus, one can expect gains

of the o rd er of 10 to 20 p er cent in the c as e of c on cre te pro tectio n and

hardly any gains for other inorganic mat erials .

CONCLUSIONS

A finite difference calculation method has been described . It

can be used f or the prediction of the tem pe ra tu re histo ry of the insulation

and stee l co re of protected st eel columns. The acceptable accu racy

of this method has been dem onstrated by com paring ex peri men tal and

theore t ica l resul t s .

It has been shown that the tempe ratur e of s t ee l co re i s insensi t ive

to the mechanism of heat t ra nsf er from the inner surfa ce of the insulation

to the steel . Thus a close approximation of the average tem pe rat ure of s teel

co re can be obtained by assum ing that this is equal to the aver age tem pe rat ur



of the inner su rf ac e of the protection . To fac ilitate the calculation of

the tempera ture of s tee l cor e i t i s permiss ib le to a ssum e tha t a l l

mo isture i s concentrated at the inner su rface of the insulat ion from

the st ar t of the heat ing proc ess, and evaporates from this su rface a s

the t em pera tu re a t t h is p l ace r i se s .

In general, the influence of mois tur e on the te mp er at ur e

his tor y of s te el co re i s negligible for mo st inorganic building mate rials .

A notable exception is co ncrete, for which the pre se nce of mo istu re may

cau se a 10 to 20 pe r cent gain in the t ime tha t the column can support

the load.

Although the method has been developed prim ar ily fo r the

calculat ion of te mp era tur es in protected ste el columns, i t has a m o re

gen eral applicabili ty. I t may also be used to calculate the tem pe rat ure

histo ry in sol id s t ruc tur al elements, such as concrete beams and

columns, and unprotected ste el for any f ire exposure.

ACKNOWLEDGEMENT

The authors wish to thank E. 0. Por teous fo r h i s a s s i s t ance in

conducting the experim ental work.



NOMENCLATURE

Notations

a em pir ica l constant, " R

c spe cific heat; without sub scrip t: spe cific hea t of insulation,

Btu/lbOR

j = 0, 1, 2, . .k thermal conductivity; without subscript: thermal conductivity

of insulation, Btu/h f t OR

M numb er of me sh points along x axis

N numb er of m es h points along y axis

P point

R elemen tary region

t time, h

T tem pe rat ure , OR ( i f not specified otherwis e)

W m as s of s t eel core , lb/ft

x coordinate, ft

Y coordinate, ft

G reek l e t t e r s

fract ion

increment or difference

m es h width, ft

emissivi ty

emis siv i ty fac tor

lat en t he at of vapo rization, Btu/lb

density; without sub scrip t: de nsit y of insulation, lb/ft3

-8Stefan-Boltzrnann constant, 0 -17 /3 x 10 Btu/h ft2 o R~

mo ist ure concentration, ft3/ft3

3 3hypothet ical moi sture concentrat ion, ft /ft



Subscr ip ts

a average

f of the "furnace"

i of the insulation

m, M at or around a me sh point in the m-th o r M-th row, resp ectiv ely

n, N a t o r a round a m es h point in the n-th o r N-th column, respective ly

s of the s tee l co re

R pertaining to radiat ion

w of water

Super sc r ip t s

o a t t = O

j at t = jAt



REFERENCES

1 Geilinger, W. and Bryl, S. Fe urs ich erh eit de r Stahlkonstruktionen,

IV. Teil: F eu rsc hu tz von Stahlstiitzen, Verl ag Schw eizer Stah l-

bauverband, Zurich, 1962.

2 Fuji i, S. The theo ret ic al calculat ion of tem pe rat ure -r is e of

therm ally protected s te el column exposed to the f i re . Building

Re se ar ch Institute Occ asio nal Repor t No. 10, Tokyo, 1963.

3 Pet tersson, 0. Utvecklingstendenser rora nde brandteknisk

dim ens ione ring av st %lk ons truk ion er, VSg -och vattenbyggar en,

No. 6-7, Stockholm , 1964, pp. 265-268.

4 Lie, T. T. Bekledingsmaterialen en Bouwconstructies bij

Br an d, H er on No. 2, 1965, pp. 57-81.

5 Witteveen, J. Brandveiligheid Staalcon structies, Centrum

Bouwen in Staal, Rot terd am, 1966.

6 Lie, T. T. Te mp era tur e of protected ste el in f i re . Pa per 8

of "Behaviour of St ru ct ur al Ste el in Fi re , " Mi nis try of Technology

and F ir e Offices* Com mittee Joint F ir e Re sear ch Organizat ion

Symposiu m No. 2, H. M. S. O., London, 1968.

7 Berechnung d es B randwid erstande s von Stahlkonstruktionen.

Schwe izerische Zentral stel le fc r Stahlbau, Zurich, 1969.

8 Law, M. Stru ctura l f i re protec t ion in the process indust ry .

Bu ild ing, Vol. 216, No. 29, 1969, pp. 86-90.

9 Emm ons, H. W. The nu me ric al solution of heat conduction

problems. Tra nsac tion s of the Ame ric an Society of Mechanical

Eng in ee rs , Vol. 65, 1943, pp. 607-615.

10 Dusinberre, G. M. Heat t ran sf er calculat ions by f ini te differences.Inte rnat iona l Textbook Company, Scranto n, Penn sylvan ia, 1961.

11 Harmathy, T. Z. A tre at is e on theo ret ic al f i re endurance rat ing.

Am erica n Society for Test ing Materials , Special Technical

Pu bl ic at io n No. 301, 1961, pp. 10-40.



Kawagoe, K. Calcu lation of te m pe ra tu re in double-layer w alls

heated from one side. Bul le tin of the F ir e Prevent io n Socie ty

of Ja pan, Vol. 13, No. 2, 1965, pp. 29-35.

Harmathy, T. Z. The rma l pe r fo rmance of concre te masonry wal l s

in f i re . Am eric an Socie ty for Test ing and Mater ia ls , Specia l

Te ch nic al Pub lica tion No. 464, 1970, pp. 209-243.

Thrinks , W. and Mawhinney, M. W. Industr ia l furnace s . Carneg ie

Inst. Technology, John Wiley and Sons, h c . , New York, 1961.

Stand ard methods of fi r e te st s of building con stru ctio n and

ma ter ia ls , ASTM Designation E l 19-69, 1969 Book of ASTM

Stan dard s, P a r t 14, pp. 436-452.

F a c k l e r , J. P . , "Cahi er 299", Ca hi er s Ce ntr e Scientif ique et

Tec hni que du B2tim ent, No. 38, A p ri l 1959.

Liley, P. E., Touloukian, Y. S., and Gam bill, W. R. Physical

and che mic al data. Ch em ical Engin eers Handbook, J. H. Per ry ,

Sec . 3, McGraw -Hill Book Com pany , New York, 1963.

Bri t . I ron Stee l Res . Assoc . , Phy sica l constants of some

commerc ia l s tee l s a t e leva ted tempera tu res . But terworths

Sci. Pu bl ., London, 1953.

Harmathy, T. Z. Effect of m oi st ur e on the f ir e endura nce ofbuilding ele me nt s. ASTM Spec. Techn. Pub l. No. 385, 1965,

p. 74.

Harmathy, T. Z. and Lie, T. T. Exp erim enta l verif icatio n of

th e ru le of m oi st ur e mom ent. F i r e Technology, Vol. 7, 1971,

p. 17.

Harmathy, T. Z. Th erm al prop er t ie s of conc rete a t e levated

te m pe ra tu re s. Jo urn al of M ate ria ls, JMLSA, Vol. 5, No. 1,

M ar ch 1970, pp. 47-74.

Harmathy, T. Z. Variable s t a te methods of me asu r ing the

ther ma l p roper t i e s o f so lids , J. Appl. Phys., Vol. 35, 1964,

p. 1190.

Plum me r, C. E. e t a l. Br ick, s t r uc tur a l c lay products and re fr ac -tor ies . Engineering Ma ter ial s Handbook, C. L. Mantell , Sect.

25. Mc Graw -Hill Book Comp any, New York, T oro nto , London,

1958.



24 Po we rs, T. C. and Brownyard, T. L. Studies of the ph ys ica l

pro per t ie s of hardened port land c emen t paste. R e s e a r c h

La bo rat ori es of the Por tland Cemen t Association, Bulletin

22, Chicago, 1948.

25 Harm athy, T. Z. Mo istur e sorption of building ma ter ial s,

Tec hnic al P ap e r No. 242, Division of Building Re sea rch ,

Nation al Re se ar ch Coun cil of Canad a, Ottawa, NRC 9492,

March 1967.

26 Lie, T. T. Fea sibili ty of determin ing the equilibrium m oi st ur e

condit ion in f i r e re s is tan ce tes t specimens by measu r ing the i r

el ec tri ca l res ista nc e. Building Re sea rch Note No. 75,

Div ision of Building Re se ar ch , NRC, Ottawa, 197 1.

27 Gilmor e, C. H. et al. Heat tran smis sion . Che mic al Engineer 's

Handbook, J. H. Perry, Sect. 10, McGraw-Hill Book

Company , New York, 1963.



TABLE 1

INFORMATION ON THE TEST COLUMNS BUILT WITH

LIGHTWEIGHT CONCRETE PROTECTION

3Density of insulation at room tem pera ture: P = 90 lb/ft

Averag e mois ture content: cpo = 0.032 £t3/ft3

Th ick ne ss of protection : 3 5/8 in.

Ste el core: H-section, 6 x 6 in. , 20 lb pe r ft length.

Outside dim ens ions of specim en: 19: in. x 192 in. (no con tact betwe en

st ee l and protect ion) .

Em issiv i ty of protect ion: C i = 0. 9.

Em issiv ity of stee l: C s = 0. 9.

Th erm al pro pert i es of protect ing insulation

Th erm al conductivity

(k) ~ t u / f t O

Pe m p e r a t u r e

O F

Volumetric heat capacity,

(p c) ~ t u / f t 3 R

70

100200

300

40 0

500

600

---16.35

17.00

18.75

23.95

25.10

25.20

24.90

0. 317

0.399

0.320

0. 323

0.323

0.325

0.326

0. 328

0. 327

0. 327

0. 327

0. 320

0. 315

0. 311

0.308

0. 307

0.306

0. 305

0. 303

0.303

0.312

0. 327

0. 342

700

750

800

850

900

950

1000

1050

1100

1200

1300

24.80

25.00

26.60

33.50

41.25

43.30

35.70

27.90

23.90

23.85

25.75

1400

1500

1700

2000

2300

25.25

23.90

24.20

24.65

25.30



TABLE 2


INSULATING FIRE BRICK PROTECTION

Density of insulation at room tem per atur e: p = 45 lb/ft3

Average mo is tu re content : CQ, = 0.

Th ickn ess protection: 22 in.

St eel core: H-section, 6 x 6 in. , 20 lb pe r f t length.

Outsi de dim ens ion s of specimen: 11 in. x 11 in. (contac t between ste el

and protect ion a t the s te el flanges)

Em issiv ity of protection: C i = 0. 9.Em issiv ity of steel: C S = 0. 9.

Th er m al prop er t ie s of protect ing insula tion

Tem pera ture , Volumetr ic heat capaci ty ,

(pc) ~ t u / f t ~R

Thermal conductivity,

(k) ~ t u / f t O

0.098

0,100

70

10 0

7.21

7.39

200

300

40 0

500

600

700

80 0

1000

1200

1400

1600

1800

2000

2200

2400

8. 00

8. 60

9. 14

9. 55

9.88

10.25

10.57

11.12

11.66

12.21

12.76

13. 31

13.86

14. 41

14.95

-

0. 107

0.114

0.120

0. 127

I 0.134

0. 141

0. 148

0.165

0.182

0.204

0.230

0.255

0.281

0. 307

0. 332



TABLE 3


HEAVY CLAY BRICK PROTECTION

3Density of insulation at room temperature: p = 1 3 3 lb/ft .Aver age mo ist ure content: (a) cpo = 0

3 3(b) cpo = 0 - 0 4 f t / f t .

Ste el core: H-section, 8 x 8 in . , 4 8 lb pe r f t length .1

Thickn ess protection: 2 7 in .Outside dimensions of sp ecimen: 1 2 $ in. x 1 2 $ in. (contact between st ee l

and protection at the st ee l f langes).

Em iss ivi ty of protect ion: E i = 0. 9.

Em iss ivi ty of steel: s s = 0. 9.

1

Th erm al pro per t ies of protect ing insula tion

T e m p e r a tu r e ,

O F

7 0

1 0 02 0 0

3 0 0

4 0 0

5 0 0

6 0 0

7 0 0

8 0 0

1 0 0 0

1 2 0 0

1 4 0 0

1 6 0 01 8 0 0

2 0 0 0

2 2 0 0

2 4 0 0

-.-

-Volumetric heat capacity,

(p c) ~ t u / f t " R

24. 0

24. 025. 0

26. 0

26. 0

26. 0

27. 0

27. 0

28. 0

29. 0

29. 030. 0

31. 032. 0

32. 0

33. 0

33. 0- -

--

Th er ma l conductiv ity ,

(k)~ t u / f t " R

0. 5 4

0. 550. 57

0. 60

0. 6 0

0. 6 1

0. 63

0. 65

0. 66

0. 7 0

0 . 7 3

0. 7 6

0. 8 00. 8 3

0. 86

0. 90

0. 92



F I G U R E 1

C R OS S S E C T I O N O F A T Y P I C A L P R OT EC TE D S TEELC O L U M N m a r ~ a ~ z - I



FIGURE 2

THE ARRANGEMENT OF THE ELEMENTARY REGIONS OF A ONE-EIGHTH SECTION OF

COLUMN PROTECTION ~ I ~ ~ - P .





SECTION A-0-C-D

WITHOUT ITEM 3

FIGURE 4 FIRE TEST SPECIMENE l I # ## - I



1

- -

- -

- -

--- E X P E H l M E N T A L- A L C U L A T E D -

- -1 1

0 60 120 180 240 300 360

T I M E , M I N U T E S

F I G U R E 5

A V E R A G E T E M P E R A T U R E O F S TE EL C O R E I N A C O L U M N P R O T EC T E D W I T H L I G H T -

W E 1 G H T C O N C R E T E

Y' = 0.032, oc 0.( FO R F U R T H E R D E T A I L S O F S P E C I M E N S E E T A B L E 1 ) ~ R Y O ~ Z - 4



F I G U R E 6

A V E R A G E T E M P E R A T U R E O F S TE E L C O R E I N A C O L U M N P R O T E C T E D W I T H I N S U L A T I N G

F I R E B R I C K

Y o = 0, d = . 5

( F O R F U R T H E R D E T A I L S O F S P E C I M E N S E E T A B L E 2 )



-- X P E R l M E N T A L- A L C U L A T E D


F I G U R E 7

A V E R A G E T E M P E R A T U R E O F S T EE L C O R E I N A C O L U M N P R O T E C T E D W I T H H E A V Y

C L A Y B R I C K

'Po = 0, o C = 0 . 5( F O R F U R T H E R D E T A I L S O F S P E C I M E N S E E T A B L E 3 ) SR4aLIZ - 6



-- X P E R I M E N T A L- A L C U L A T E D

T I M E . M I N U T E S

F I G U R E 8

A V E R A G E T E M P E R A T U R E O F S T E EL C O R E I N A C O L U M N P R O T E C T E D W I T H H E A V Y

C L A Y B R I C K

Yo 0 . 0 4 , OC= 0 . 5( F O R F U RT H E R D E T A I L S O F S P E C I M E N S E E T A B L E 3)



00 60 1 2 0 1 8 0 240 300 360

T I M E . M I N U T E S

I 1 I

- -- -d 1 ( P U R E C O N D U C T I O N )- -

- -0 ( P U R E R A D I A T I O N )- -

- -

I I 1 I

F I G U R E 9

C A L C U L A T E D T E M P E R A T U R E S O F S TE EL C O R E I N A C O L U M N P R O T E C T E D W I T H

L I G H T W E I G H T C O N C RE T E F O R V A R I O U S R A T I O S O F C O N D U C T I O N T O R A D I A T I O N

H E AT T R A N S F E R F R O M T H E P R O T E C T I O N T O TH E S T EE L'Po 0 . 0 3 2 , ( F O R F U R T H E R D E T A I L S O F T H E C O L U M N S E E T A B L E 1) 8R4842 - 6




F I G U R E 1 0

C A L C U L A TE D T E M P E R A T U R E S O F S T E EL C O R E I N A C O L U M N P R O T E C T E D W I T H

H E A V Y C L A Y B R I C K F OR V A R I O U S R A T I 0 . S O F C O N D U C T I O N T O R A D I A T I O N H E A T

T R A N S F ER F R O M T H E P R O T E C T I O N T O T H E S TE EL

Y o = 0 , ( F O R F U R T H E R D E T A l L S O F C O L U M N S EE T A B LE 3) . R * ~ * X - 9






F I G U R E 1 2

C A L C U L A TE D T E M P E R A T U R E S O F S T EE L C O R E I N A C O L U M N P R O T E C T E D W I T H

H E A V Y C L A Y B R I C K F OR V A R I O U S M O I S T U R E C O N T E N T S .d . O . 5 , ( FO R F U R T H E R D E T A I L S O F C O L U M N S E E T A B L E 3) SR+~* L - I I



I f I N . T H I C K N E S S 2 3 l N . T H I C K N E S S


F I G U R E 13

C A L C U L A T E D T E M P E R A T U R E S O F S T EE L C OR E I N A C O L U M N P R O T E C T E D W I T H

I N S U L A T I N G F I R E B R I C K FO R V A R I O U S M O I S T U R E C O N T E N T S A N D T H I C K N E S S E S

O F T H E P R O T E C T I O Nd 0.5, ( F O R F U RT H E R D E T A I L S O F S P E C I M E N S E E T A B L E 2 ) OR*WIL-IL

Documents

[Paper] a Numerical Procedure to Calculate the Temperature of Protected Steel Columns Exposed to Fire