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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
7/27/2019 [Paper] a Numerical Procedure to Calculate the Temperature of Protected Steel Columns Exposed to Fire
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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
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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).
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(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
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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:
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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
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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
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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:
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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
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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:
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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
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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:
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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
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-
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:
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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
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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
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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
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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.
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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
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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
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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.
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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.
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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.
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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
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TABLE 2
INFORMATION ON THE TEST COLUMNS BUILT WITH
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
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TABLE 3
INFORMATION ON THE TEST COLUMNS BUILT WITH
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
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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
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FIGURE 2
THE ARRANGEMENT OF THE ELEMENTARY REGIONS OF A ONE-EIGHTH SECTION OF
COLUMN PROTECTION ~ I ~ ~ - P .
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7/27/2019 [Paper] a Numerical Procedure to Calculate the Temperature of Protected Steel Columns Exposed to Fire
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SECTION A-0-C-D
WITHOUT ITEM 3
FIGURE 4 FIRE TEST SPECIMENE l I # ## - I
7/27/2019 [Paper] a Numerical Procedure to Calculate the Temperature of Protected Steel Columns Exposed to Fire
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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
7/27/2019 [Paper] a Numerical Procedure to Calculate the Temperature of Protected Steel Columns Exposed to Fire
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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 )
7/27/2019 [Paper] a Numerical Procedure to Calculate the Temperature of Protected Steel Columns Exposed to Fire
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-- X P E R l 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 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
7/27/2019 [Paper] a Numerical Procedure to Calculate the Temperature of Protected Steel Columns Exposed to Fire
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-- 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)
7/27/2019 [Paper] a Numerical Procedure to Calculate the Temperature of Protected Steel Columns Exposed to Fire
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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
7/27/2019 [Paper] a Numerical Procedure to Calculate the Temperature of Protected Steel Columns Exposed to Fire
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T I M E , M I N U T E S
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
7/27/2019 [Paper] a Numerical Procedure to Calculate the Temperature of Protected Steel Columns Exposed to Fire
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7/27/2019 [Paper] a Numerical Procedure to Calculate the Temperature of Protected Steel Columns Exposed to Fire
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T I M E , M I N U T E S
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
7/27/2019 [Paper] a Numerical Procedure to Calculate the Temperature of Protected Steel Columns Exposed to Fire
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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
T I M E , M I N U T E 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