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Method to calculate the fire resistance of reinforced concrete columns
with rectangular cross section
M e t hod t o c a lc ula t e t he fire re sist a nc e of re inforc e d c onc re t e
c olum ns w it h re c t a ngula r c ross se c t ion
N R C C - 3 3 1 1 4
L i e , T . T . ; I r w i n , R . J .
J a n u a r y 1 9 9 3
A version of this document is published in / Une version de ce document se trouve dans:
ACI Structural Journal, 90, (1), pp. 52-60, 1993
http://www.nrc-cnrc.gc.ca/irc
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Les renseignements dans ce document sont protégés par la Loi sur le droit d'auteur, par les lois, les politiques et les règlements du Canada et des accords internationaux. Ces dispositions permettent d'identifier la source de l'information et, dans certains cas, d'interdire la copie de documents sans permission écrite. Pour obtenir de plus amples renseignements : http://lois.justice.gc.ca/fr/showtdm/cs/C-42
ACI STRUCTURAL JOURNAL
TECHNICAL PAPER
Title no. 90-S?Method to Calculate the Fire Resistance of
Reinforced Concrete Columns with Rectangular Cross Section
ACI STRUCTURAL JOURNAL
TECHNICAL PAPER
Title no. 90-87
Method to Calculate the Fire Resistance of
Reinforced Concrete Columns with Rectangular Cross Section
by
T. T.
Lie and
R. J.
Irwin
Joint studies between the National Research Council of Canada and the Portland Cement Association on reinforced concrete columns were started a number of years ago for the purpose of updating the irzformation on fire resistance ratings for these columns in North American building codes. These studies include the development of mathematical models for the calculation of the fire resistance of columns of various sizes and shapes, as well as experimental studies. Among the columns examined were columns with rectangular cross section. In this paper, a method is described for the calculation of the fire resistance of reinforced concrete columns with such cross sections.
Keywords: columns (supports); fire resistance; reinforced concrete; tests.
In the past, the fire resistance of concrete structural mem-bers could be determined only by testing, which is costly and time-consuming. Recent developments, including the development of numerical techniques and an enhanced knowledge of the thermal and mechanical properties of con-cretes at elevated temperatures, have made it possible to determine the fire resistance of various concrete members by calculation.
RESEARCH SIGNIFICANCE
The National Research Council of Canada and the Portland Cement Association have recently completed an extensive test program in which the behavior of over 40 full-scale reinforced concrete columns were examined under fire con-ditions. Although a large number of tests were carried out, they do not provide sufficient information on fire resistance for many variables beyond the values studied in the tests. However, they provide basic data that enable the develop-ment and validation of mathematical models capable of cal-culating fire resistance for any value of the important variables that determine it. This paper deals with the devel-opment and validation of a mathematical model for the cal-culation of the fire resistance of reinforced concrete columns with rectangular cross section.
TEST SPECIMENS
In the studies on the fire resistance of reinforced concrete columns, three fire resistance tests were conducted on
col-52
umns with rectangular cross section, of which one had a square eros ection. The pecimen coo i ted of rectangular, reinforced concrete columns made with . iliceou aggregate. They are described in detail in Reference l and illo trated in Fig. 1. AJl columns were 38 1 0 mm ( 12.5 ft) long. The dimension. of the column cro secrions and other pecifics
of the column are give n in Table l.
Column No. 1 had four 25 mm (No. 8) longitudinal bars, Column No. 2 ix 22 mm (No. 7 longitucli na l bars, and Column No. 3, eight 19 mm (No. 6) longitudinal bar . The bars were tied w ith 10 mm No.3) lies. The location of the main reinforcing bar , which were welded to steel end plates, and the location of the ties are bown .in Fig. I .
The main reinforcing bar had a pecified yield tl"engtb of 414 MPa (60 k i). The yield strength of the ties wa 427 MPa (61 .8 ksi). The concrete mix was designed for a com-pre ion cylinder strenglh at 28 days of approximately 35 MPa (5 k i). The mi.x propottions were a follow :
Cement Water Sand Coarse aggregate 307 kgtm3 154 kg/m3 871 kg/m3 1054kg/m3 (518lb/yd3 ) (260 lb/yd3 ) (1469lb/yd3 ) (1777lb/yd3 )
The average compre sive cylinder Lrengtbs of the con-crete of the test colwnns, mea ured 28 day after pouring the concrete and on the day of the testing, are given in Table l. Th moisture condition at the center of the column was also measured on the day of the test. The moisture condition of Column No. I wa approximately equivalent to that in equiUbrium wjth air of 74 percent relative humidiry (RH) at room temperature, of Column No. 2, with
air
of 65 percent RHandofColumn No.3 with air of 58 percentRH. Chromel-alumel thermocouples, 0.91 mm (0.036 in.) thick, were in-stalled at midheight in the column for measuring concreteACI Srrr<cmrol Jounwl. V. 90, No. I. Janunry-Peb ruary 1993.
Received Feb. 20, 1991, and reviewed underfnstitutc publications policies. Pertinent discussion will be published in the Novernber- Dccember I 993 ACJ Structural Journal
if received by July l. 1993.
ACI member T. T. Lie is a senior researcher with the Institute for Research in Con-struction, National Research Council of Canada. He worked for several years in Europe and Japan before joining the NRC in I967. Mr. Lie is currently involved in research in structural fire resistance, which includes testing and calculation oft he fire resistance of structural members.
R. ], Irwin is a structural engineer carrying out research in structural fire resistance at the Institute for Research in Construction, National Research Council of Canada. Her research interests include the prediction of the fire peiformance of structural
members.
temperatures at different locations in the cross section. The locations of the thermocouples are described in detail in Reference 1.
TEST APPARATUS
The tests were carried out by exposing the columns to heat in a column test-furnace. The test-furnace was designed
to produce the conditions to which a member might be subjected during a fire. It consists of a steel framework supported by four steel columns, with the furnace chamber inside the framework. The characteristics and instrumenta-tion of the furnace, which has a loading capacity of 1000 t (2200 kips), are described in detail in a previous paper.2
TEST CONDITIONS AND PROCEDURE The columns were installed in the furnace by bolting the endplates to a loading head at the top and a hydraulic jack at the bottom. The conditions of the columns were fixed-fixed for all tests. For each column, the length exposed to fire was approximately 3000 mm (120 in.). At high temperature, the stiffness of the unheated column ends, which is great in comparison to that of the heated portion of the column, contributes to a reduction in the column effective length. In
[g]]
305 mm[Q]
305 mm 305 mm COLUt,jN NO. 1 ｾ＠ 457 mm 864 533 JB mm\
•
•
_t .38 mm'"!
r-305 mm - r-ｃｏｌｕｾｎ＠ NO. 2Fig. ]-Elevations and cross sections of test columns
Table 1-Specifics of test columns
Quantity and Compressive strength,
size of MPa (ksi)
Cross reinforcing Relative section, mm bars, mm humidity,
Column no. (in.) (size no.) percent 28 days Test date
Four bars with diameter
305 X 305 of 25 mm 35.3 36.1
1 (12 X 12) (No.8) 74 (5.1) (5.1)
Six bars with diameter 305 X 457 of22mm 44.2 42.5 2 (12 X 18) (No.7) 65 (6.4) (6.2) Eight bars with diameter 203 X 914 of19mm 39.2 42.1 3 (8 X 36) (No.6) 58 (5.7) (6.1)
ACI Structural Journal I January-February 1993
914 mm
1016
•
533•
25 m m__1. 51 mm
p
203 mmCOLUMN NO. 3
Fire resistance, hr:min Allowable
Test load, load,
kN (kips) kN (kips) Calculated Measured
1067 1244 (240) (280) 3:16 3:28 1413 2102 (318) (318) 6:44 6:36 756 1360 (170) (306) 3:39 5:30
53
'(-AICIS Ar-ＢＨＰＱＬＨＯＲＩｾ＠ --i ｾｍＯｊＲ＠
r
;-1..,...,-,-..,---ir--.--,-r-...;--., l ,-!9'--. ｾ＠• A ... .,_ '.·
. ... .
.-...... .
§
· I · Ii
' I ' · I ·L
H:
1-H-+---f-+---f-t-1- l,
..
Fig. 2-Thermal and stress-strain network in one-quarter cross section
previous tests,3 it was found that, for columns tested fixed at the ends, an effective length of 2000 mm (80 in.) represents experimental behavior.
All columns were tested under concentric loads. Column No. 1 was subjected to a load of 1067 kN (240 ksi), which is equal to 86 percent of the maximum allowable service load according to ACI 318-89,4 Column No. 2 to a load of 1413 kN (318 ksi) or 67 percent of the maximum allowable service load, and Column No. 3 to a load of 756 kN (170 ksi) or 56 percent of the maximum allowable service load.
The columns were exposed to heating controlled in such a way that the average temperature in the furnace closely followed the ASTM E 1195 standard temperature-time curve. The temperatures during the test of Column No.3, however, followed this temperature-time relation up to one half hour and after that a temperature-time relation that can be given by the equation
Tf=
14.88 t+
831.8 (1) The columns were considered to have failed and the tests were terminated when the load, which was applied by a hydraulic jack, could no longer be maintained. The hydraulic jack has a maximum speed of 76 mm/min (3 in./min).TEMPERATURES OF COLUMNS DURING FIRE EXPOSURE
The calculation of the fire resistance of columns is carried out in various steps. It involves the calculation of the tem-peratures in the column and its deformations and strength during the exposure to fire.
The column temperatures are calculated by a finite differ-ence method.6 This method has been previously applied to the calculation of temperatures of various building compo-nents exposed to fire.7'8 Because the method of deriving the heat transfer equations and of calculating the temperatures is described in detail in those studies, it will not be discussed here; only the newly developed equations for the calculation of the column temperatures and calculated results will be given.
54
Division of cross section into elements
The cross-sectional area of the column is subdivided into a number of elements, arranged in a triangular network (Fig. 2). The elements are square inside the column and triangular at the surface. For the inside elements, the temperature at the center is taken as representative of the entire element. For the triangular surface elements, the representative points are located on the center of each hypotenuse.
For reasons of symmetry, only one-quarter of the section needs to be considered when calculating the temperature distribution in columns with square or rectangular cross-sec-tion. As illustrated in Fig. 2, in an x-y coordinate system, a point Pm,n has the coordinates x = (n- 1) ｾｾ＠
;.,12
andｹ］ＨｭＭＩｾｾｭ＠ ..
Equations for the fire-concrete boundary
It is assumed that the columns are exposed on all sides to the heat of a fire whose temperature course follows that of the standard fire described in ASTM E 1195 or CAN4-S101.9 This temperature course can be approximately described by the following expression7
Tf+
20+
750[1-exp(-3.79553W)]+
170.41W (2) where t is the time in hours andrf
is the fire temperature in C at time t = j!::J .The temperature rise in the column can be derived by creating a heat balance for each element. In the following, all calculations will be carried out for a unit length of the column. For the elements at the surface of the column along the x-axis, the temperature at time t = (j
+
1) ｾｴ＠ is given by the expression j+l j 2MT,n=T,.+
·
·
2 ' ' [(pcccYl,n+
ｐｷｃｷ＼ｰＧｴＬＮ｝ＨｾｾＩ＠ {｛
ｫｾＬＨｮＭＱＩ＠
+
d,n]
[Ti _ _ri ]
2 "2,(n 1) -l,n｛
ｫｾ＠
(n+l )+
ｫｾ＠
n] · · .r,:;-+
'
2 ' [:z2,(n+l)- Tlcn]+
'12Efoc ＰＢｾｾ＠[(Tj+ 273)4-
en."+
273)4] } (3)For the elements at the surface of the column along the y-axis, the temperature at the time t = (j
+
1) ｾｴｩｳ＠ given byT J+t = T J Ｒｾｴ＠ m,N m,N
+ ((
v
1 )( ): 2 PeCcJ,,I.N+
ｐ ｾ｣＠ ... <p,,.N 6....,) { [ k { ,;r-I ),(N- 1)+
k{u,N]
[T i Tj ] 2 (m-l),(N-1)- m,N [ k {m+ I ),(N- 1)+
dn ,N][ri
Tj ]ｾ＠
6_1: 2 (m+i),(N-1)- m,N+
Efoc 0" -, [(Tf+ 273)4- (T1n,N+ 273)4] } (4)Equations for inside the concrete
For the elements in the concrete, the temperature at time
t = (j
+
1) ｾｴ＠ is given by j+l j 6.t Tmn=T mn+ · · 2 ' , ｛Ｈｰ｣｣｣ｙｭ Ｎ ｾ＠+
ｐ ｷ ｃｷｱｙｭＬｮ｝ＨｾｾＩ＠ { [k { m- l ),(n- 1)+
k{ll,/1]
[Tj _ Tj ] 2 (m-l),(n-1) m,n [ k{m+l),(n-1)+
k-!11,11] [.-rl ri ] + 2 l"(m+i),(n-1)- m,n [ kim- i),(n+i)+
k1n,n] [T j Tj ]+
2 (m-l),(n+l)- m,n [ k {m+l),(n+l)+
ｫｾ ＧﾷＢ ｝＠
[ rrfTj ] }
+ 2 F (m+l),(n+l) - m,n Auxiliary equations (5)To calculate the temperatures along the lines of symmetry A-D and C-D, the temperature has to satisfy the following symmetry conditions:
Al ong l. me -A D· . T1+1 m,l = T1+m,3 1
AI 1. C D· Tj+1 T1+1
Ong me - . (M+ I),n
=
(M-I),nACI Structural Journal I January-February 1993 (6)
(7)
Effect of moisture
The effect of moisture is taken into account by assuming that, in each element, the moisture starts to evaporate when the temperature of the element reaches 100 C (212 F). During the period of evaporation, all the heat supplied to an element is used for the evaporation of the moisture, until the element is dry.
For the elements at the boundary between fire and concrete along the x-axis, the initial volume of moisture is given by
v -
L,n-ＨｾｾＩＲ＠
2 <p L,n (8)
From a heat balance equation it can be derived that, per unit length of the column, the volume 6.V 1" evaporated in
the time 6.t from the concrete element, is
｛
ｫ ｾＮＨｮＫｬ
Ｉ＠
+
k1.n] [..,.{
ri ]
+ 2 n,(n+l)- l,n
(9)
For the elements at the boundary between fire and concrete along the y-axis, the initial volume of moisture is given by
ｾ＠
-ＨｾｾＩＲ＠
m,N-2 <j>m,N (10) From a heat balance equation it can be derived that, per unit length of the column, the ｶｯｬｵｭ･ｾ＠ V .... N evaporated in the
time from the concrete element is
M { [k{m-t).(N-Il
+
ｫｾｮＮｎ｝＠
[TJ T J ] ｾ＠ V N.m = PwAw 2 (m- J),(N-1) - m,N [ kfm+l)(N-1)+
ｫｾ Ｌｎ ｝＠
[rri T j )+
2 l"(m+L),(N-1)- m,N (11)For the concrete elements inside the column, the initial volume of moisture is given by:
(12)
Similarly, as for the surface concrete elements, it can be derived that, per unit length of the column, the volume d
v
m,n evaporated in the time dt from these layers isdf k (,... 1),(/1-1)
+
k m,n j j I { [ j j ] d Vm,n = PwAw 2 [T (m- l),(n-1) - T m,N] [k{m+l),(n-1)+
k-!n,n] [rri Tj 1+
2
l'(m+l),(n-1)- m,NJ [ k {m- l),(n+l )+
k-!n,n ] [T.i . Tj ]+
2 (m-l),(n+l)- m ,N [k {nr+ t} ,(n+l}+
kｾＧﾷＢ ｝＠
[..,.; Tj ] }+
2 1(m+l),(n+l)- m,n (13) Stability criterionTo insure that any error existing in the solution at some time will not be amplified in the subsequent calculations, a stability criterion has to be satisfied. For a selected value of ､ｾＬ＠ this limits the maximum time step dt. Following the method described in Reference 6, it can be derived that, for the fire-exposed columns, the criterion of stability is most restrictive along the boundary between fire and concrete. It is given by the condition
(14)
where (pcccc)min is the minimum value of the heat capacity of the concrete, kma.x is the maximum value of its thermal conductivity, and hma.x is the maximum value of the coeffi-cient of heat transfer to be expected during the exposure to fire. For exposure to the standard fire, the maximum value of the coefficient of heat transfer hma.x is approximately 675 W/m2 C.
Procedure for calculation of column temperatures With the aid of Eq. (1) through (14) and the relevant material properties given in References 3 and 10, the tem-perature distribution in the column and on its surface can be calculated for any time t = (j
+
1 )dt, if the temperature dis-tribution at the time jdt is known. Starting from an initial temperature of 20 C, the temperature history of the column can be calculated by repeated application ofEq. (1) through (14).STRENGTH OF COLUMNS DURING FIRE EXPOSURE
Transformation into square network
To calculate the deformations and stresses in the column, and its strength, the triangular network is transformed into
56
a square network. In Fig. 2, a quarter section of this network, consisting of square elements arranged parallel to the x- and y-axis of 'the section, are shown. The arrangement of the elements in the three other quarter sections is identical to this. The width of each element of this network is ､ｾ＠ I fl.
The temperatures, deformations, and stresses of each element are represented by those at the center of the element. The temperature at the center of each element is obtained by averaging the temperatures of the elements in the triangular network according to the relation
(T m,n square -) _ [rfm+l),(n+l)
+
T-!n,(n+2)] .2 tnangular (15)
where the subscripts square and triangular refer to the ele-ments of the square and triangular network.
For the steel reinforcing bars, a representative bar tem-perature can also be indicated. Measurements at various locations during fire tests showed that the differences in temperature in the bar and sections are small. 11 A close approximation of the average bar temperature is obtained by considering the column as consisting entirely of concrete and selecting the temperature at the location of the center of the bar section as the representative bar temperature. Thus, for a steel reinforcing bar, the center of whose section is located in an element Pm.n. the representative temperature is equal to that of Pm.n, which is given by Eq. (15).
Similarly, it is assumed that the stresses and deformations at the center of an element are representative of those of the whole element.
Assumptions in the calculation of strength during fire
During exposure to fire, the strength of the column de-creases with the duration of exposure. The strength of the column can be calculated by a method on a load deflection analysis.3 In this method, the columns, which are fixed at the ends during the tests, are idealized as pin-ended columns of length KL (Fig. 3). The load on the column is intended to be concentric. Due to imperfections of the columns and the loading device, a small eccentricity exists. The loading system and the test columns were made with high precision, however. Therefore, in the calculations, a very small initial load eccentricity will be assumed. The real eccentricity, how-ever, is unknown. After runs of the computer program showed that for eccentricities up to 3 mm (0.12.) the influence on fire resistance was small, an arbitrary value of 0.2 mm (0.008 in.), reflecting a nearly concentric load, was selected for the initial eccentricity. The selection of a finite value for the initial eccentricity is needed to make the computer program work.
The curvature of the column is assumed to vary from pin-end to midheight according to a straight line relation, as illustrated in Fig. 3. For such a relationship, the deflection at midheight Y, in terms of curvature X of the column at this height, can be given by
(KL)2
Y=x -
-12 (16)
For any given curvature, and thus for any given deflection at mid-height, the axial strain is varied until the internal moment at the mid-section is in equilibrium with the applied moment given by the product:
load x (deflection+ eccentricity)
In this way, a load deflection curve can be calculated for specific times during the exposure to fire. From these curves, the strength of the column, i.e., the maximum load that the column can carry, can be determined for each time. In the calculation of column strength, the following assumptions were also made:
1. The properties of the concrete and steel are those de-scribed in previous studies.3'10
2. Concrete has no tensile strength. 3. Plane sections remain plane.
4. The reduction in column length before exposure to fire, consisting of free shrinkage of the concrete, creep, and short-ening of the column due to load, is negligible. This reduction can be eliminated by selecting the length of the shortened column as the initial length from which the changes during exposure to fire are determined.
Based on these assumptions, the column strength during exposure to fire, was calculated. In the calculations, the network of elements shown in Fig. 2 was used. Because the strains and stresses of the elements are not symmetrical with respect to the y-axis, the calculations were performed for both the network shown and an identical network at the left of the y-axis. The load that the column can carry, and the moments in the section, were obtained by adding the loads carried by each element and the moments contributed by them.
The equations used in the calculation of the strength of the column during exposure to fire are given in the following sections.
Equations for concrete
The strain in the concrete for the elements at the right of the y-axis (Fig. 2) can be given by
Xc
(Ec)R = -(ET)c
+
E+-p
and for the elements at the left of they-axis by
Xc
(Ec)L = -(ET)c
+
E-p
(17)
(18) where ( EI)c = the thermal expansion of the concrete, m m-1;
E = the axial strain of the column, m m-1; Xc
=
the horizontal distance from the center of the elements to the vertical plane through the y-axis of the column section m; and p=_
the radius of curvature, m.The stresses in the elements are calculated using the same stress-strain relations for concrete, given in References 3 and
10.
ACt Structural Journal I January-February 1993
DEFLECTION CURVATURE
t
I l/2 IＱﾷＭｾｌ＠
I IＭＭｾＧ
ｐＧ＠
I I I U2 It
Fig. 3-Load-de.flection analysis
Equations for steel
The strain in the steel reinforcing bars can be given as the sum of the thermal expansion of the steel (ET),, the axial strain of the column x, /p, where x, is the horizontal distance of the center of the section of steel bar to the vertical plane through the y-axis of the column section, and p is the radius of curvature. For the steel bars to the right of the y-axis, the strain (E,)R is given by
(Es)R = -(ET)s
+
E+
M
p
(19)For the steel bars to the left of the y-axis, the strain (E,)L is given by
(Es)L = -(ET)s
+E-M
p
(20)The stresses in the steel are calculated using the same stress-strain relations for steel, given in References 3 and 10.
Procedures for the calculation of column strength
With the aid ofEq. (17) through (20) and relevant equations in References 3 and 10, the stresses at midsection in the concrete elements and in the steel bars can be calculated for any value of the axial strain E and curvature 1/p. From these stresses, the load that each element and each reinforcing bar carries and its contribution to the internal moment at mid-section can be derived. By adding the loads and moments, the load that the column carries and the total internal moment at midsection can be calculated.
The fire resistance of the column is derived by calculating the strength, i.e., the maximum load that the column can carry at several consecutive times during the exposure to
900 - - Calculaled 800 • - - - Measured 700 0 600 0 ui a: 500 ::) ｾ＠ a: 400 UJ 0... ::E UJ 300 f--200 100 0
o
40 eo 120 160 200 240 2eo 320 360 TIME, minFig. 4-Temperature of concrete at various depths along centerline parallel to shortest side of Column No. 2 (305 x 457 mm) as function of exposure time
fire. The strength reduces gradually with time. At a certain point, the strength becomes so low that it is no longer suf-ficient to support the load, and the column fails. The time to reach this failure point is the fire resistance of the column.
RESULTS AND DISCUSSION
Using the mathematical model described in this paper, the temperatures in the columns and the axial deformations of the columns were calculated. In the calculations, the thermal and mechanical properties of the concrete and steel and the specifics of the columns and furnace, given in Reference 10, were used.
In the following, calculated temperatures for various lo-cations in the columns and calculated axial deformations of the columns will be compared with the measured tempera-tures and axial deformations reported in Reference 1.
For columns with square cross section, comparisons of measured and calculated temperatures have been made in a previous study. 3 Therefore, in the present study, only the temperatures of Columns No. 2 and 3, which have a rectan-gular cross section, will be discussed. In Fig. 4 and 5, cal-culated temperatures are compared with those measured at various depths in these columns. It can be seen that, with the exception of the temperatures measured at the center of the specimen at an early stage, there is good agreement between calculated and measured column temperatures. The temperatures measured at the center of the columns show initially a relatively rapid rise in temperature, followed by a period of nearly constant temperatures in the early stages of the test. This temperature behavior may be the result of thermally induced migration of the moisture towards the center of the column where, as shown in the figures, the influence of migration is most pronounced. Although the model takes into account evaporation of moisture, it does not take into account the migration of the moisture towards the center. That migration appears to account for the deviation
58 0 0 ui a: ::) ｾ＠ a: w 0... ::E UJ f--900 800 700 600 500 400 300 200 100 - - Calculated · - - - Measured 20 40 60 80 100 120 140 TIME, min
Fig. 5-Temperature of concrete at various depths along centerline parallel to shortest side of Column No. 3 (203 x 914 mm) as function of exposure time
between calculated and measured temperatures at the earlier stages of fire exposure. At a later stage, however, which is the important stage from the point of view of predicting the fire resistance of the columns, there is good agreement be-tween calculated and measured temperatures.
In Fig. 6 through 8, the calculated and measured axial deformations of the columns, during exposure to fire, are shown. It can be seen that the mathematical model predicts, reasonably well, the trend in the progression of the axial deformations with time. The largest differences between cal-culated and measured axial deformations are on the order of 5 mm (0.2 in.), which may be regarded as small when con-sidering that these are differences between calculated and measured deformations for columns with lengths of about 3800 mm (150 in.). It must also be noted that these columns deform axially as a result of several factors, namely, load, thermal expansion, bending and creep, which cannot be com-pletely taken into account in the calculations.
This was particularly evident in the case of Column No. 3 (Fig. 9). Whereas the model defines the failure point as the point at which the column can no longer support the applied load and also assumes that failure at this point is instantaneous, the tested column, which was relatively slen-der, contracted considerably before it was crushed.
In Fig. 9, the calculated strengths of the columns are shown as a function of the time exposure. The strength decreases with time until it becomes so low that the column can no longer support the load. The time to reach this point is the fire resistance of the column. The calculated fire resistances of the columns are given in Table 1 together with the meas-ured fire resistances. It can be seen that there is good agree-ment between the calculated and measured fire resistances for Columns Nos. 1 and 2, but the calculated fire resistance for Column No.3 is about 30 percent lower than that meas-ured, due to the considerable contraction of the column, which the model can only partly take into account.
8 - - Calculated • - - - Measured
---
....,,
5'
'
'
E 4'
\ E \ z· 3 \ 0 \ｾ＠
2 ｾ＠ a: 0 0 u_ w Cl -I __J <(x
-2 <( -3 -4 -5 0 80 100 120 140 160 180 200 220 TIME, minFig. 6-Calculated and measured axial deformations of Col-umn No. 1 (305 x 305 mm) as function of exposed time
10 9 8 7 E 6 E 5
z
4 0 ｾ＠ 3 ｾ＠ 2 a: 0 I u_ w 0 Cl __J -1 <(x
-2 <( -3 -4 -5 41 -7 0 ＬｾＭＭＭＭＭ ...,
ｾｾ＠',
ｾＧ＠
',
,
'
,
'
,
\ I \ ｾ＠ \,
\ I \ I \ \ \ \ - - Calculated \ • - - - Measured \ \ \ \ \' '
40 eo 120 160 200 240 280 320 360 400 TIME, minFig. 7-Calculated and measured axial deformations of Col-umn No. 2 ( 305 x 457 mm) as function of exposed time
Fig. 9 also shows that, under commensurate loads, columns with rectangular cross sections have substantially higher fire resistances than those with square cross sections with the same thickness. Under a load equal to the allowable service load, the fire resistance of Column No. 2, for example, is almost twice that of the square column with the same thick-ness. Column No. 3 which is much thinner than the square column, namely 203 mm (8 in.) in thickness in comparison to the 305 mm (12 in.) thickness of the square column, has a fire resistance that is approximately equal to that of the square column. The main reason for the relatively higher fire resistance of the rectangular columns is probably that the heating of the core of columns with rectangular cross sections approaches that of a wall, which is heated on two sides,
ACI Structural Journal I January-February 1993
20 18 16 14 E E 12 10
z
8 0 6 ｾ＠ ｾ＠ 4 a: 2 0 u_ 0 w Cl -2 __J <( -4x
<( -6 -B -10 -12 -14 -16 0 ; ...----
....,
,""'
',
ｾ＠'
ｾｾ＠ \ ;. \' " '
\ \ I I'
\ \ \ r I \ - - Calculated 1 · - - - Measured \ I I I I I I 240 280 320Fig. 8-Calculated and measured axial deformations of Col-umn No.3 (203 x 914 mm) as function of exposed time
9000 8000
'
7000 6000 z ｾ＠ J:- 5000 1-(!l z uu 4000 cr: 1-en 3000 -2000 1000 \ \ ColumnNo. 3 \ (203 x 914 mml \ \ \e Calculated fire resistance under the allowable service load
200 240 280 320 360 400 420
TIME, min
Fig. 9-Calculated strengths of columns as function of ex-posure time and calculated column fire resistances under the allowable service load according to ACI 318 and CSA A23.333-M84
whereas the column with square cross section is heated on four sides.
CONCLUSIONS
Based on the results of this study, the following conclu-sions can be drawn:
1. The mathematical model employed in this study is capable of predicting the fire resistance of rectangular rein-forced concrete columns with an accuracy that is adequate for practical purposes.
2. The model will enable the expansion of existing data on the fire resistance of reinforced concrete columns, which at present consists predominantly of data for square columns, with that for rectangular columns.
3. Rectangular columns have, under commensurate loads, substantially higher fire resistances than square columns of the same thickness.
4. Using the model, the fire resistance of square and rec-tangular reinforced concrete columns can be evaluated for any value of the significant parameters, such as load, column section size, column length, concrete strength, percentage of reinforcing steel and concrete cover to the steel, without the necessity of testing.
5. The model can also be used for the calculation of the fire resistance of columns made with concretes other than those investigated in this study; for example, lightweight or carbonate aggregate concretes, if the relevant material prop-erties are known.
NOTATION
c =specific heat, Jkg-lC-1
h = coefficient of heat transfer at fire exposed surface, Wm-2C-1
j
=
0, 1, 2, ...k =thermal conductivity, Wm-lC-1 K = effective length factor
L = unsupported length of column, m
M = number of points along y-axis N = number of points along x-axis
p =point t =time, hr T = temperature, C V = volume of moisture, m3 x = coordinate, m y = coordinate, m
Y = lateral deflection of column at midheight, m Ll = increment
ｌｬｾ＠ = mesh width, m e = emissivity, strain, m m-1
A. = heat of vaporization, Jkg-1
p =density, kgm-3; radius of curvature, m
a = Stefan-Boltzmann constant, Wm-2K-4
cp = concentration of moisture (fraction of volume)
X= curvature of column at mid-height (m-1)
Subscripts
c = of concrete f = of the fire
m,M = at the points m, Min a column
max = maximum min = minimum
n,N = at the points n, N in a row L = left of the x-axis
R = right of the x-axis
s =of steel
T = pertaining to temperature
w =of water
I, 2 =at the points 1, 2
Superscripts
j =at t =jilt
REFERENCES
I. Lie, T. T., and Woollerton, J. L., "Fire Resistance of Reinforced Concrete Columns: Test Results," Institute for Research in Construction, Internal Repor; No. 569, National Research Council of Canada,Ottawa, 1988, 302 pp.
2. Lie, T. T., "New Facility to Determine Fire Resistance of Columns," Canadian Journal of Civil Engineering, V. 7, No. 3, 1980, pp. 551-558.
3. Lie, T. T.; Lin, T. D.; Allen, D. E.; and Abrams, M.S., "Fire Resistance of Reinforced Concrete Columns," National Research Council of Canada, Division of Building Research, NRCC 23065, Ottawa, 1984, 32 pp.
4. ACI Committee 318, "Building Code Requirements for Reinforced Concrete and Commentary" (ACI 318-89/ACI 318R-89), American Con-crete Institute, Detroit, 1983, 353 pp.
5. "Standard Methods of Fire Tests of Building Construction and Mate-rials," ASTM E 119-83, American Society for Testing and Materials, Phila-delphia, 1985, 37 pp.
6. Dusinberre, G. M., Heat Transfer Calculations by Finite Differences, International Textbook Company, Scranton, 1961, 293 pp.
7. Lie, T. T., and Harmathy, T
.z.,
"Numerical Procedure to Calculate the Temperature of Protected Steel Columns Exposed to Fire," Fire Study No. 28, Division of Building Research, National Research Council of Can-ada, NRCC 12535, Ottawa, 1972, 39 pp.8. El-Shayeb, M., "Fire Resistance of Reinforced Concrete and Concrete Filled Steel Columns," Ph.D. dissertation, University of New Hampshire, Durham, 1986, 282 pp.
9. "Standard Methods of Fire Endurance Tests of Building Construction and Materials," (CANIULC-Sl01-M89), Underwriters'LaboratoriesofCan-ada, Scarborough, 1989, 49 pp.
10. Lie, T. T., and Celikkol, B., "Method to Calculate the Fire Resistance of Circular Reinforced Concrete Columns," ACI Materials Journal, V. 88, No. 1, January-February, 1991, pp. 84- 91.
11. Allen, D.E., and Lie, T. T., "Further Studies of the Fire Resistance of Reinforced Concrete Columns," National Research Council of Canada, Division of Building Research, NRCC047, Ottawa, 1974, 25 pp.
![Fig. ]-Elevations and cross sections of test columns](https://thumb-eu.123doks.com/thumbv2/123doknet/14240561.486679/6.920.206.701.365.790/fig-elevations-cross-sections-test-columns.webp)
