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A study of the steel beam deformation during fire
Building and Environment, Vol. 23, No. 2, pp. 159-167, 1988. Printed in Great Britain.
0360-1323/88 $3.1)0+0.00 © 1988 Pergamon Press pie.
A Study of the Steel Beam Deformation During Fire WOJCIECH SKOWRONSKI* This paper presents a new method of theoretical modelling o f strength effects and strains, which have a crucial influence on the deflection o f steel beams in fire. This method is based on a new model o f steel creep at elevated temperatures and the modified Ramberg-Osgood equation. The analysis o f the deflection of beams in fire was based on the Mohr's integral generalized over non-elastic materials. With the help o f the equations introduced in the work it is possible to generalize and to foresee, with high accuracy, the results o f rather expensive fire endurance tests o f steel beams.
NOMENCLATURE A b B E(T) h AH I Ip, L k l m M h~t /~ q P R t T T' 7~ Y,.r x Z V
Subscripts e elastic p plastic t creep-induced, time-dependent T thermal cr critical
cross-sectional area beam breadth material constant modulus of elasticity at the actual steel temperature beam height activation energy of creep moment of inertia substitute moment of inertia a parameter, constant beam length material constant bending moment bending moment caused by the unitary force t5 = 1 unitary concentrated force (external force) applied in the most effort section of a beam and directed downwards uniform load of a beam concentrated force gaseous constant time steel temperature, °C steel temperature, K rate of temperature increase in a beam critical deflection dimension along the beam length Zener-Hollomon parameter concentrated force
Superscripts A, B, C, D, E characteristic points on the statics diagram of the beam.
INTRODUCTION BASIC I N F O R M A T I O N on a temperature collapse of a building structure in fire can be obtained after fire endurance tests are carried out in special furnaces. Such tests belong to the destructive methods of testing the loaded building elements of actual dimensions. Standard conditions during the fire endurance tests are similar in various countries. The results of fire tests alone do not give rise to formulate the general rules which govern the process of building structure collapse in fire. It is therefore necessary to make a theoretical analysis o f these tests. This may also be worthwhile for economical reasons, since the preparation and performing of the fire tests lasts several days and is rather expensive. The degree of destruction of beams in the fire tests is indicated by a deflection whose admissible value or admissible rate o f increase is determined by structural criteria. This paper will present the strength analysis of beam deflection in fire. The analysis will be based on the new model of steel strains at elevated temperatures. Heat transfer in the insulation serving as a fire protection of a beam will not be taken into account here, since it was dealt with in other papers, e.g. [1,2]. The deflection of steel beam in fire has been analysed by H a r m a t h y from Canada and T h o r from Sweden [3, 4]. Both researchers have used in their calculation the model of steel creep proposed by H a r m a t h y in [5] :
Greek letters coefl~cient of linear expansion A increment c~ deflection strain, without subscript : total load strain eto creep parameter 0 temperature-compensated time, described with equation (5) # constant of the material p radius of the beam curvature a stress ~y(T) yield stress at the actual steel temperature 6- rate of stresses q~,(tr) amendment function, described with equation (9) ~P, ~ functions
~to
*Technical University of Opole, 45-061 Opole, Katowicka 48, Poland. ,~, 23..~-,
e, = i ~ ' c o s h -
159
1
(2 z'°/~',o) for
a = 0.
(1)
W. S k o w r o n s k i
160
Harmathy's and Thor's calculations are based on recurrent formulae and consist in the determination of deflection increments of a beam, in subsequent time intervals. The accuracy of the calculation is higher, the shorter the time intervals. The effects of steel creep at elevated temperatures have to be taken into account when the beam deformation during fire is being analysed with the use of equations describing deflection, whereas this effect is not taken into account in the analysis based on the comparison of maxireal stress in the most stressed section of a beam (a) with the yield stress at elevated temperatures :
a = G(T,.r).
0=
Z =
exp -
dr.
(5)
B2" a"2 for medium and small stresses, B3"exp(ms'a) for big stresses.
(6) (7)
Equation (3) has never been used hitherto. For practical reasons let us assume a priori that the relation (6) is valid over the whole range of stresses. At the same time let us introduce a stress-depending amendment function into equation (3) which assumes the following form : e, [3 Z ( c r ) ' 0 ] ''3 e.,]i(-b-)-= L " ~ a ) J "~b,,(~).
(2)
Such analyses of beam collapse temperatures during fire are presented in the works of Kruppa [6], Mehl and Arndt [7] and others. The possibility of analysing complex structures on fire is presented by the finite element method. An interesting program FASBUS II (Fire Analysis of Steel BUilding Systems) has been elaborated under AISI sponsorship [81. Some studies, e.g. [9, 10], are also being carried out dealing with bearing capacity of frame systems on fire, in which particular attention is paid to a beam being bent. In this work the author wishes to present a new method of analysing a beam on fire, which would make it possible to determine a critical deflection state. The studies are being carried out to compare the ultimate bearing capacity with the critical deflection state of steel beam during fire. First results obtained by the author show that the fire collapse of steel beams with bigger span length is much better described by the condition of critical deflection state [11].
;0
(8)
This additionally introduced function ~b,(a) is meant to improve the accuracy of the approximate equation (3) and to compensate the error created in the range of high stresses after the elimination of equation (7) (this error is shown in Fig. 1). From the calculations performed on a computer it results that a good conformity of equation (8) with the results of experimental creep testings is obtained when the amendment function has a form : ~b,,(a) = B4" a ' , .
(9)
In consequence, the following model of steel strains caused by primary and secondary creep can be formulated :
lO24 @
IIII
z =
I I
3,TZgr.108-d4'7°
CREEP A N D A M O D E L OF STEEL
Creep strains in steel rapidly increase at elevated temperatures. For instance, by the stress of 150 MPa, the change of ASTM A36 steel temperature from 450 to 500°C results in the increase of creep rate by more than 250 times (cf. [2], Fig. 7.36). This is of crucial importance, since the temperatures of 450 and 500°C precede in the range of the most probable critical temperatures for steel beams. Standard fire tests in the U.K. have demonstrated that the mid span vertical deflection of 1/30 at full design load will be attained when lower flange temperatures attain around 630°C. The necessity of taking into account the creep of steel in fire engineering calculations was pointed out by many authors, e.g. Harmathy [5], Lie [2]. The theory of creep in metals and their alloys has been described by Dorn [12]. On the basis of this theory Harmathy derived the following approximate formula with use of the Taylor's series [5] :
et~,.~ (3"Z'O)]/3 ~;to I;i,J ,,I '
(3)
G, = Bt "rrm',
(4)
@ Z = 1,23.1016.eO,O43425"d
Ill
1022
rl 1020 ___
i (6) lO18
lOis
/i rl scare I I IIIIi
lo14
20
where :
/
40 60801 150 200 103
300 d, IMPM
Fig. 1. Variation of the Zener Hollomon parameter defined by equations (6) and (7) on the example of the ASTM A36 steel.
A S t u d y o f the S t e e l B e a m D e f o r m a t i o n During Fire
0
4
2
6
8
161
10
0.10-22,[h I Fig. 2. Curves of ASTM A36 steel creep at a temperature of 482°C, determined experimentally and theoretically.
where : 3 / ~ o 1~2/3 • i / I / 3
B ---- ~ / J
u 1
u 2
(11)
• B4,
d, lM
(12)
m = ~rnl + 1mE+m4.
The conformity of the obtained model of creep with the results of ASTM A36 steel testing at elevated temperatures is shown in Fig. 2. Respective calculations are shown in Appendix 1. The remaining mechanical strains of steel at elevated temperatures will be modelled by means of the RambergOsgood equation with the following modification : e=E[l)a +0002
(13)
O'y
The modulus of elasticity and the yield stress of steel decrease at elevated temperatures according to equations (9A1) to (13A1) determined for the ASTM A36 steel in Appendix 1 (cf. also Figs 7.13 and 7.21 in [2]). Figure 3 shows the /t coefficient variation according to the Skowronski's calculation given in [11]. The conformity of the modified Ramberg-Osgood equation with the results of experimental testings of the ASTM A36 steel is shown in Fig. 4, whereas the respective equations of these curves are enclosed in Appendix 1.
-0
0~1
0~2
~
¢,[~1
Fig. 4. Curves of "stress-strain", determined experimentally and theoretically for the ASTM A36 steel at elevated temperatures. Total strains of steel at elevated temperatures can be calculated by means of the summing up of the strains resulting from thermal expansion of steel with the strains defined by equation (13), and with the creep strains according to equation (10). Thus, the general model of steel strain can be written down as follows: = qJ(a) = er + a" ~k, (T) + a u(T)" ff2(T)
,u,[-I
20
where :
~k3[exp (_ ~), tlAH = 8- [ j0 l' exp(
(15)
15
Material constants as well as functions ~Ot, q/2, q/3 describe the properties of steel at elevated temperatures. These properties should be studied by means of anisothermal tensile tests [13, 14]. This is particularly essential in case of frame system calculations. The strains of steel in the moment of collapse of a steel beam usually exceed 1%, and then the results of anisothermal tensile tests are close to the results of isothermal tests. Interesting comparisons of isothermal test results with anisothermal test results are presented in [15].
'°/ 4 373
d,j
BEAM DEFLECTIONANALYSIS 473
573
673
773
873
T',[ K l
Fig. 3. Theoretical variation of the coefficient #, determined in [l l] for the ASTM A36 steel at elevated temperatures.
The process of deflection of an arbitrary loaded beam (shown in Fig. 5) at elevated temperatures will be dealt
162
W. S k o w r o n s k i
Particular components of the right-hand side of equation (18) express, respectively, the influence of: thermal expansion, linear elasticity, non-linear elasticity, and steel creep. In the cross-sections of a beam the equilibrium of internal forces takes place. Hence :
Q)
J b)
f;
M =
_l e)
,
,
,
The influence of the linear elasticity of steel is defined by the following basic equations :
-~ /
O" = [~] ,(T)] -1° ~e,
~
b~=b-d' h~ = h
L
-
2d
1
~e = - - ' Y . P~
.
Fig. 5. Steel beam: (a) statical diagram; (b) deflection line; (c) idealized I-section.
with after the following assumptions are made :
e = er+ee+~p+~, ;
the Bernoulli principle as well as the geometric equations of the linear theory of strains hold true ; the lengths of beams are big compared to their transverse dimensions, so the influence of the shearing forces on the beam behaviour will be neglected ; a properly constructed beam is not subject to a lateral buckling nor a local stability, even at elevated temperatures ; a fireproof insulation does not help the structure to carry the gravitational loads. The necessity of taking into account the actual temperature distribution at the level of a cross-section of the beam in the performed calculations will be discussed in the next part of this work, i.e. when the results of calculations are known with the assumption that the deflection depends on the temperature of the lower extreme tensile beam fibres a n d - - i n case of thermal strains--on the difference between the temperatures of lower and upper fibres of a beam. The steel beam deflection will be determined with use of the Mohr's integral in the following form :
fo
Or
;o'
P~
/o
1 p~-
1./17/d x Pp
+ f f 1.p, ~ / d x .
M [¢,(T)]-'-f
I =
(22)
i;
y : dA.
(23)
Thus, the influence of linear elasticity is as follows :
f0 ' x=f0 pe
[q,, (r)] - ' . t dx.
(24)
The moment of inertia I calculated for the idealized Isection (shown in Fig. 5) has the following form : I=
(25)
l~2"(b'h3-b,'h~).
The influence of the non-linear elasticity is defined by the following basic equations : a = [~Oz(T)] ~/,(r).~p~/~(r) for
U #- 0,
(26)
and 1 ep = ~ ' y ,
(27)
which result in the equality : f0' O~ 1 ./17/dx = f0 ' [O2(T)]-' MU'r) ".Mpp(r)dx '
(28)
where :
(17)
which, due to additive steel strains, takes on the following form: 1 . h.]rd x +
The substitution of the above expressions to equation (19) gives the formula for the curvature 1/p~ :
(16)
= ~1 1 . ~ dx, Jo p
(21)
where :
the steel is a homogeneous and isotropic continuous medium, the mechanical properties of which during compression are the same as during tension. These properties are described by the model expressed by equation (14) or in the following way :
1./Qdx+
(20)
and
'y
c5 =
(19)
a.y.dA.
lp =
f;
y~,/u(r))+IdA '
(29)
For the I-section : 1 Ip = k ' 2 k- I " ( b ' h ~ - b l ' h ] ) ,
(30)
where : (18)
1 k - #'~'-'~'t-J*z
(31)
A Study o f the Steel Beam Deformation During Fire The influence of the creep of steel will be defined by the equations :
f
F
/
I ) A\ H
a:~O3[exp[-~;],,|~ k
.
t
_
k
/
d
.
- 1/m
"e)/"
for m e 0 ,
/
(32)
163
The influence of the thermal expansion resulting from it is as follows :
;o
' --1 "-~td = pT
'I
(T, - T 2 ) • a
• 3~/dx.
and
(42) 1
et = = ' y , Pt
(33)
When AT = 0 t h e n :
fo
r l ' h ~ / d x = O. Pr
while : T = T(t),
(34)
and the inverse function is determined : t = t(T).
(35)
Let us substitute these equations, as previously, into condition (19). We obtain :
folx:y01f3[exp 1
After substituting the relations (24, 28, 36 and 42) into equation (18) we obtain:
'Y0
= ~"
+
M" • K/
(T, -- T2)" a
i
(36)
It =
f;
y(l/,,,)+l dA.
(37)
The substitute moment of inertia /, determined for the I-section has a form defined by equation (30), whereas :
M dx
[~,(r)]-l.i dx+
+f0
where :
(43)
[ ¢ ~ ( r ) ] - , . i ¢ . dx M"-~
{~O3[exp f AH• ]]-1 dx. ~ - R~7/, t ] ; "1:
(44)
With the increase of the deflection (and time) a change of the stress distribution in the cross-section of a beam takes place, which is shown in Fig. 7. The calculations of stresses when analysing the deflection are not necessary and ineffective in view of the implicit form of the functional W- 1 in the equation : a = W-' (e),
(45)
1
k - m+2"
(38)
The influence of the thermal expansion resulting from a non-uniform, linear temperature distribution at the level of the cross-section of a beam (cf. Fig. 6c) is defined by the following basic equations : [ T I + T2"~ er = ~ \[ - - ~ ] ' A T/ ( y ) ,
(39)
and ~T =
1
--.y,
(40)
Pr where : AT(v) =
T 1-
h
T 2
"Y"
(41)
where ~F- J is an inverse functional to the functional qJ in equation (14). Exemplary calculations of beam deflection in fire will be carried out for the ASTM A36 steel I-section beam described in [16] (beam No 1). A simplified model of the beam as well as basic equations of bending moments are given in Appendix 2. During a fire endurance test the temperature of the extreme tensile fibres in the crosssection of the considered beam was increasing with an average rate of 6.74°C min-1, and the temperature of the extreme compressive fibres with an average rate of 2.32°C min- 1. Figure 8 shows the process of deflection increase of the considered beam, registered during the fire endurance test and calculated according to the formula (44). The conformity of the experimental and theoretical results is satisfactory from the point of view of an engineer. A similarly high conformity was obtained after the
T'I*T2
AT{y}
T1 al
b]
c]
Fig. 6. Temperature distribution at the height of the beam cross-section.
W. Skowronski
164
J
to
t 1
t2
t3
tcr
Fig. 7. Process of stress distribution change in the l-section of a beam under fire.
o l o 2o°
20
4o
sO
100
so
t,lmin]
2
tested beam will be equal to the structural criterion for fire endurance tests, i.e. in which the beam temperature will be critical. The equation describing that moment can be written as follows :
4
¢,
G
(T~r - T2)" ~
f4 dx
8
fo ] M ' ~
fo'
M~'`r")'AT/
10
,
12 14-
5,[cm]
\
• t
I
1
Mm.M
+J0 ~ f 3[ex [ p
-AH(- ~),tll
-'
"IF
dx = Ycr' (46)
\
Fig. 8. Course of steel beam deflection shown schematically in Fig. 11. calculations of other steel beams have been carried out with the application of this method. When considering the fire resistance, the most interesting for us is the moment in which the deflection of the
where Ycr is defined by the structural criterion for fire endurance tests. The third component of the left-hand side of the above equation has little influence on the result of calculations, less than 0.1%. This is visualized by Fig. 9, where the values (in percentages) of particular components determining beam
6e,Sp,St, I°/o) 100
80
60
40
~¢r
-"
~IT~.)=
] •L'-L 800 h
Fig. 9. Variability of components determining beam deflection, expressed in percentages by the formulae (47), (48) and (49).
A Study of the Steel Beam Deformation During Fire
165
deflection caused by mechanical strains are shown : foI [~b,M'A~¢ (T)]-' "I dx" 100% 6e =
(47) foI
M'AT/1 d x + f / MY(r)"21~r d x + f / [~O 2(T)]-" Ipu{r} [~b,(T)]-"I 1
M "{r) "
Mm'AT/ ~. L%3Fexp --AT
dx
till-'
2~I
[¢2(T)]_,. ~(r) dx" 100%
f/
M" )~1
I
M,{r)
[i,b,l(T)]_I ,/dx + ~0
"]~
"'
(48)
t
dx+f 0
dx
2~'tm"~'/
M m" AI
dx" 100%
jo {~3 [exp (-- R~')' t]}-'. I7' 6,
--
f0
1
M" j~r [~b,(T)]-' •
I dx+ fo1
M m. ~r AH [$2(T)]_, .pe{r) dx+ fo {qJ3 DxP ( - R ~ 7 ) ,
M'{r)'fff
1
As can be seen in Fig. 9, at temperatures above 400°C the creep effects increase suddenly and rapidly. At a temperature of 500°C these effects are strong and cannot be neglected in the analysis of a beam by means of the method described here. The value of 400°C cannot be treated as a universal temperature determining the lower limit of the range of temperatures at which the creep of steel has a considerable influence on beam deformation, since the 6, value depends also on time. Nevertheless, the calculations presented here confirm the correctness of the method used by J. Thor [4], who, when analysing beams made of Swedish steel in his calculations took into account the creep of steel after it has reached the temperature 450°C, and the results of calculations he obtained were in conformity with the results of fire endurance tests. In equation (46) the beam rigidity is defined by the following mathematical expressions : [qJt (T)]-" L [~2(T)]-' "Ipu(r),
--1
dx
t]}'IT'
and {qJ3 Iexp ( - R~,), t] }-' • I,m.
ff0' {~, [T(y)]}
"y2dA,
a. { f~0A [~b3 [exp ( _ RTT),
,,yjjq]-{1/')
],, .(52a)
Taking into account equations (50a, 51a and 52a) has little influence on the results of the critical temperature calculations, which are shown in Fig. 10. The critical temperatures, presented in this figure, were calculated from the well-known Ryan and Robertson criterion [17] :* 1 l2 y,., = ~ ' ~ . (53)
rdd ] 823 773 723 3pO
(50a)
I f f/[~k2[T(y)]]'/'{r)'y ('/'(r}}+' t "{r}. (51a)
(50) (51)
2,00
(52)
The above expressions change after taking into account the actual (non-uniform) distribution of temperature at the height of the cross-section of the beam under fire, and assume the following forms :
*The critical temperature is a conventional quantity and depends on the accepted structural criterion for fire endurance tests. The acceptance of another criterion will give another critical temperature.
673
(49)
4.00
5~
6/}0
700
•
de
Fig. 10. Calculation results which visualize the influenceof the rate of temperature increase in a beam on the critical temperature.
166
W. S k o w r o n s k i the basis of:
Id
,tl
f
~
~
~
~
I
~
½
~"~
q ~-212726"~"
~
~ ~1
(a) The known properties of steel at elevated temperatures, including creep characteristics. (b) The values of a cross-section, static diagram and beam loading. (c) The data concerning the temperature increase in a beam, determined by the efficiency of the fireproof insulation.
Fig. I 1. A simplified model of the considered steel beam. In Fig. 10 it can be seen that critical temperature of the beam whose temperature was increasing with the rate of 2.32°C per minute would differ only by 2% from the critical temperature of the beam whose temperature would be increasing with the rate of 6.74°C per minute. Therefore, it is often not economical to take into account the complicated expressions (50a, 51a and 52a) and the third component of the left-hand side of equation (46) in the calculations done according to equation (46).
CONCLUSIONS The analysis performed indicates that it is possible to calculate exactly deflection of steel beams under fire on
The proposed method of calculating the deflection is sufficiently precise and simple to be used in practice. With the help o f this method it is possible to foresee the results of fire endurance tests on given steel beams and generalize these results over the beams under different loading or with different dimensions and shapes as for the crosssection, length or static diagram. Acknowledgements--The author wishes to express his gratitude to Prof. K. Czarnowski of the Technical University of Wroctaw for his interest in the problems of fire safety and scientific supervision, to Prof. T. Z. Harmathy of NRCC in Ottava for his great benevolence in private correspondence with the author, and to Prof. Z. Kowal of the Swietokrzyska Polytechnic in Kielce for his valuable critical remarks which have contributed to the improvement of this work.
REFERENCES 1. E.M. Krokosky, Modelling thermal fire resistance, Build. Sci., 4, 197 (1971). 2. T.T. Lie, Fir and building, Applied Science London, p. 267 (1972). 3. T . Z . Harmathy, Creep deflection of metal beams in transient heating processes, with particular reference to fire, Can. J. Cir. Eng., 3, 219 (1976). 4. J. Thor, Deformations and critical loads of steel beams under fire exposure conditions, Doc. D 16: 1973, Nation. Swed. Build. Res., p. 217 (1973). 5. T.Z. Harmathy, A comprehensive creep model, J. Basic Engng. Trans. Am. Soc. Mech. Engng, 89, 496 (1967). 6. J. Kruppa, Collapse temperature of steel structures, 3. Struct. Div. ASCE, 105, 1769 (1979). 7. F. Mehl and W. Arndt, Grundlagen zur theoretischen Bestimmung des Feuerwiderstandes von biegesteifen Stahlstabwerken, Symp. Steel and Composite Structures for User Needs, Reports of the working commissions IABSE, Dresden, 21, 195 (1975). 8. D.C. Jeanes, Application of the computer in modeling fire endurance of structural steel floor systems, Fire Safety J. 9, 119 (1985). 9. P. Arnualt, H. Ehm and J. Kruppa, R6sistance au feu des structures hyperstatiques en acier, Doc. CECM 3-74(6F), Centre Technique Industriel de la Construction Metallique, France, p. 127 (1974). 10. A. Rubert and P. Schaumann, Structural steel and plane frame assemblies under fire action, Fire Safety J. 10, 173 (1986). 11. W. Skowronski, Problems of critical temperature of steel beams in fire with particular respect to steel creep (in Polish), Ph.D. dissertation, Report PRE No. 91, Building Institute of the Technical University of Wrociaw, p. 142 (1984). 12. J.E. Dorn, Some fundamental experiments on high temperature creep, Symp. Creep and Fracture of Metals at High Temperatures, HMSO, Lond., p. 89 (1956). 13. D.H. Skinner, Determination of high temperature properties of steel, BHP Technical Bulletin 16, I0 (1972). 14. D.H. Skinner, Runaway temperature--a design criterion based on the high temperature properties of steel, BHP Technical Bulletin 16, 22 (1972). 15. B.R. Kirby and R. R. Preston, High temperature properties of hot rolled, structural steels for use in fire engineering design studies, Fire Safety J. (in press). 16. R.W. Bletzacker, Fire resistance of protected steel beam floor and roof assemblies as affected by structural restraint, Symposium on Fire Test Methods--Restraint & Smoke 1966, 63 (1967). 17. 1. V. Ryan and A. F. Robertson, Proposed criteria for defining load failure of beams, floors and roof constructions during fire tests, J. Res. Nation. Bureau of Standards, 63C, 121, 2 (1959).
APPENDIX
the above given deformation range has the following form [11]:
1
In steel beams under fire strains of steel fall within the range : 0.005 ~< s ~< 0.02.
(1A1)
The amendment function q~o(a) for the ASTM A36 steel within
~ba(a) = 3.9862" l0 -4" o'1"45[h2/3].
(2A1)
The ASTM A36 steel is characterized by the following additional data [2, 4] :
A Study of the Steel Beam Deformation Duriny Fire F1-]
Z(a)=3.7296.10s.a47°L~j
for
a~<103MPa
(3AI)
167
for 200°C ~< T < 591°C; /~(T)=l.5375T-901.32[-]
for
591°C~
Z ( a ) = 1.23 • 1016 • exp (0.003425a, [ ~ ] for
a > 103MPa
(4AI)
e,o = 4.0708.10 -6.O-''75[-],
(5AI)
AH ~ - - = 38900[K].
(6A1)
The thermal expansion coefficient for the ASTM A36 steel at the temperature range 20°C ~
After substituting the above data into equation (10) we obtain the following creep model for the ASTM A36 steel at elevated temperatures : ,t = 1.0534" i0-4"o "4''833 •
[~'
exp
(
38900 ` -]./3 T+-~.16)dtJ . (7A1)
The remaining mechanical strains of the ASTM A36 steel are being modelled by means o f the Ramberg-Osgood curve defined by the following equation :
+ooo2L]
(16AI)
APPENDIX 2 The calculations were carried out on an ASTM A36 steel Isection beam, which was tested according to the ASTM El9 standard "Methods of fire tests of building construction" and described in [16]. A simplified model of the beam is presented in Fig. 1I. Here follow the basic equations resulting from this model of the beam considered : (a) forces in the points of support of the beam :
,
(8A1)
Va = VE -
where :
2127.26.5.13 +59100.27 = 64560.69[N] 2
(1A2)
(b) bending moments : for 20°C ~< T < 200°C,
M aa = - 1063.63x2+64560.69x[Nm],
E(T) = - 117.72 T + 208364 [MPa], o~v(T) = 300.1 I" 10 -3. (1026.52-- 1.33T) [MPa], /z(T) = -0.16489 T+38.878[-],
(9A1)
M cD = - 1063.63x 2 + 5456A2x + 84046.27 [Nm] (3A2)
(10A1) (IlAI)
M DE = _1063.63x2--53647.85x+303200.9[Nm]
(4A2)
(c) bending moments caused by the unitary force P = 1 :
for 200°C ~< T ~< 600°C,
E(T) = -274.68
Mac =
(2A2)
T + 239756 [MPa],
av(T) = 304.11 • 10 -4. (9509.03-9.47 T) [MPa],
(12A1)
AT/Aa= Atac = 0.5x [I m],
(5A2)
(13A1)
ff/cD = AT/.oe= - - 0 . 5 x + 2.565 [1 m].
(6A2)
and : /t(T) = 21.97-0.02479 T--x/--0.3158 T + 186.65 [--]
(14Al)
The remaining calculations were performed on the computer with the use o f the program "SKO 3" (FORTRAN). The results are presented in Figs 8-10.
0360-1323/88 $3.1)0+0.00 © 1988 Pergamon Press pie.
A Study of the Steel Beam Deformation During Fire WOJCIECH SKOWRONSKI* This paper presents a new method of theoretical modelling o f strength effects and strains, which have a crucial influence on the deflection o f steel beams in fire. This method is based on a new model o f steel creep at elevated temperatures and the modified Ramberg-Osgood equation. The analysis o f the deflection of beams in fire was based on the Mohr's integral generalized over non-elastic materials. With the help o f the equations introduced in the work it is possible to generalize and to foresee, with high accuracy, the results o f rather expensive fire endurance tests o f steel beams.
NOMENCLATURE A b B E(T) h AH I Ip, L k l m M h~t /~ q P R t T T' 7~ Y,.r x Z V
Subscripts e elastic p plastic t creep-induced, time-dependent T thermal cr critical
cross-sectional area beam breadth material constant modulus of elasticity at the actual steel temperature beam height activation energy of creep moment of inertia substitute moment of inertia a parameter, constant beam length material constant bending moment bending moment caused by the unitary force t5 = 1 unitary concentrated force (external force) applied in the most effort section of a beam and directed downwards uniform load of a beam concentrated force gaseous constant time steel temperature, °C steel temperature, K rate of temperature increase in a beam critical deflection dimension along the beam length Zener-Hollomon parameter concentrated force
Superscripts A, B, C, D, E characteristic points on the statics diagram of the beam.
INTRODUCTION BASIC I N F O R M A T I O N on a temperature collapse of a building structure in fire can be obtained after fire endurance tests are carried out in special furnaces. Such tests belong to the destructive methods of testing the loaded building elements of actual dimensions. Standard conditions during the fire endurance tests are similar in various countries. The results of fire tests alone do not give rise to formulate the general rules which govern the process of building structure collapse in fire. It is therefore necessary to make a theoretical analysis o f these tests. This may also be worthwhile for economical reasons, since the preparation and performing of the fire tests lasts several days and is rather expensive. The degree of destruction of beams in the fire tests is indicated by a deflection whose admissible value or admissible rate o f increase is determined by structural criteria. This paper will present the strength analysis of beam deflection in fire. The analysis will be based on the new model of steel strains at elevated temperatures. Heat transfer in the insulation serving as a fire protection of a beam will not be taken into account here, since it was dealt with in other papers, e.g. [1,2]. The deflection of steel beam in fire has been analysed by H a r m a t h y from Canada and T h o r from Sweden [3, 4]. Both researchers have used in their calculation the model of steel creep proposed by H a r m a t h y in [5] :
Greek letters coefl~cient of linear expansion A increment c~ deflection strain, without subscript : total load strain eto creep parameter 0 temperature-compensated time, described with equation (5) # constant of the material p radius of the beam curvature a stress ~y(T) yield stress at the actual steel temperature 6- rate of stresses q~,(tr) amendment function, described with equation (9) ~P, ~ functions
~to
*Technical University of Opole, 45-061 Opole, Katowicka 48, Poland. ,~, 23..~-,
e, = i ~ ' c o s h -
159
1
(2 z'°/~',o) for
a = 0.
(1)
W. S k o w r o n s k i
160
Harmathy's and Thor's calculations are based on recurrent formulae and consist in the determination of deflection increments of a beam, in subsequent time intervals. The accuracy of the calculation is higher, the shorter the time intervals. The effects of steel creep at elevated temperatures have to be taken into account when the beam deformation during fire is being analysed with the use of equations describing deflection, whereas this effect is not taken into account in the analysis based on the comparison of maxireal stress in the most stressed section of a beam (a) with the yield stress at elevated temperatures :
a = G(T,.r).
0=
Z =
exp -
dr.
(5)
B2" a"2 for medium and small stresses, B3"exp(ms'a) for big stresses.
(6) (7)
Equation (3) has never been used hitherto. For practical reasons let us assume a priori that the relation (6) is valid over the whole range of stresses. At the same time let us introduce a stress-depending amendment function into equation (3) which assumes the following form : e, [3 Z ( c r ) ' 0 ] ''3 e.,]i(-b-)-= L " ~ a ) J "~b,,(~).
(2)
Such analyses of beam collapse temperatures during fire are presented in the works of Kruppa [6], Mehl and Arndt [7] and others. The possibility of analysing complex structures on fire is presented by the finite element method. An interesting program FASBUS II (Fire Analysis of Steel BUilding Systems) has been elaborated under AISI sponsorship [81. Some studies, e.g. [9, 10], are also being carried out dealing with bearing capacity of frame systems on fire, in which particular attention is paid to a beam being bent. In this work the author wishes to present a new method of analysing a beam on fire, which would make it possible to determine a critical deflection state. The studies are being carried out to compare the ultimate bearing capacity with the critical deflection state of steel beam during fire. First results obtained by the author show that the fire collapse of steel beams with bigger span length is much better described by the condition of critical deflection state [11].
;0
(8)
This additionally introduced function ~b,(a) is meant to improve the accuracy of the approximate equation (3) and to compensate the error created in the range of high stresses after the elimination of equation (7) (this error is shown in Fig. 1). From the calculations performed on a computer it results that a good conformity of equation (8) with the results of experimental creep testings is obtained when the amendment function has a form : ~b,,(a) = B4" a ' , .
(9)
In consequence, the following model of steel strains caused by primary and secondary creep can be formulated :
lO24 @
IIII
z =
I I
3,TZgr.108-d4'7°
CREEP A N D A M O D E L OF STEEL
Creep strains in steel rapidly increase at elevated temperatures. For instance, by the stress of 150 MPa, the change of ASTM A36 steel temperature from 450 to 500°C results in the increase of creep rate by more than 250 times (cf. [2], Fig. 7.36). This is of crucial importance, since the temperatures of 450 and 500°C precede in the range of the most probable critical temperatures for steel beams. Standard fire tests in the U.K. have demonstrated that the mid span vertical deflection of 1/30 at full design load will be attained when lower flange temperatures attain around 630°C. The necessity of taking into account the creep of steel in fire engineering calculations was pointed out by many authors, e.g. Harmathy [5], Lie [2]. The theory of creep in metals and their alloys has been described by Dorn [12]. On the basis of this theory Harmathy derived the following approximate formula with use of the Taylor's series [5] :
et~,.~ (3"Z'O)]/3 ~;to I;i,J ,,I '
(3)
G, = Bt "rrm',
(4)
@ Z = 1,23.1016.eO,O43425"d
Ill
1022
rl 1020 ___
i (6) lO18
lOis
/i rl scare I I IIIIi
lo14
20
where :
/
40 60801 150 200 103
300 d, IMPM
Fig. 1. Variation of the Zener Hollomon parameter defined by equations (6) and (7) on the example of the ASTM A36 steel.
A S t u d y o f the S t e e l B e a m D e f o r m a t i o n During Fire
0
4
2
6
8
161
10
0.10-22,[h I Fig. 2. Curves of ASTM A36 steel creep at a temperature of 482°C, determined experimentally and theoretically.
where : 3 / ~ o 1~2/3 • i / I / 3
B ---- ~ / J
u 1
u 2
(11)
• B4,
d, lM
(12)
m = ~rnl + 1mE+m4.
The conformity of the obtained model of creep with the results of ASTM A36 steel testing at elevated temperatures is shown in Fig. 2. Respective calculations are shown in Appendix 1. The remaining mechanical strains of steel at elevated temperatures will be modelled by means of the RambergOsgood equation with the following modification : e=E[l)a +0002
(13)
O'y
The modulus of elasticity and the yield stress of steel decrease at elevated temperatures according to equations (9A1) to (13A1) determined for the ASTM A36 steel in Appendix 1 (cf. also Figs 7.13 and 7.21 in [2]). Figure 3 shows the /t coefficient variation according to the Skowronski's calculation given in [11]. The conformity of the modified Ramberg-Osgood equation with the results of experimental testings of the ASTM A36 steel is shown in Fig. 4, whereas the respective equations of these curves are enclosed in Appendix 1.
-0
0~1
0~2
~
¢,[~1
Fig. 4. Curves of "stress-strain", determined experimentally and theoretically for the ASTM A36 steel at elevated temperatures. Total strains of steel at elevated temperatures can be calculated by means of the summing up of the strains resulting from thermal expansion of steel with the strains defined by equation (13), and with the creep strains according to equation (10). Thus, the general model of steel strain can be written down as follows: = qJ(a) = er + a" ~k, (T) + a u(T)" ff2(T)
,u,[-I
20
where :
~k3[exp (_ ~), tlAH = 8- [ j0 l' exp(
(15)
15
Material constants as well as functions ~Ot, q/2, q/3 describe the properties of steel at elevated temperatures. These properties should be studied by means of anisothermal tensile tests [13, 14]. This is particularly essential in case of frame system calculations. The strains of steel in the moment of collapse of a steel beam usually exceed 1%, and then the results of anisothermal tensile tests are close to the results of isothermal tests. Interesting comparisons of isothermal test results with anisothermal test results are presented in [15].
'°/ 4 373
d,j
BEAM DEFLECTIONANALYSIS 473
573
673
773
873
T',[ K l
Fig. 3. Theoretical variation of the coefficient #, determined in [l l] for the ASTM A36 steel at elevated temperatures.
The process of deflection of an arbitrary loaded beam (shown in Fig. 5) at elevated temperatures will be dealt
162
W. S k o w r o n s k i
Particular components of the right-hand side of equation (18) express, respectively, the influence of: thermal expansion, linear elasticity, non-linear elasticity, and steel creep. In the cross-sections of a beam the equilibrium of internal forces takes place. Hence :
Q)
J b)
f;
M =
_l e)
,
,
,
The influence of the linear elasticity of steel is defined by the following basic equations :
-~ /
O" = [~] ,(T)] -1° ~e,
~
b~=b-d' h~ = h
L
-
2d
1
~e = - - ' Y . P~
.
Fig. 5. Steel beam: (a) statical diagram; (b) deflection line; (c) idealized I-section.
with after the following assumptions are made :
e = er+ee+~p+~, ;
the Bernoulli principle as well as the geometric equations of the linear theory of strains hold true ; the lengths of beams are big compared to their transverse dimensions, so the influence of the shearing forces on the beam behaviour will be neglected ; a properly constructed beam is not subject to a lateral buckling nor a local stability, even at elevated temperatures ; a fireproof insulation does not help the structure to carry the gravitational loads. The necessity of taking into account the actual temperature distribution at the level of a cross-section of the beam in the performed calculations will be discussed in the next part of this work, i.e. when the results of calculations are known with the assumption that the deflection depends on the temperature of the lower extreme tensile beam fibres a n d - - i n case of thermal strains--on the difference between the temperatures of lower and upper fibres of a beam. The steel beam deflection will be determined with use of the Mohr's integral in the following form :
fo
Or
;o'
P~
/o
1 p~-
1./17/d x Pp
+ f f 1.p, ~ / d x .
M [¢,(T)]-'-f
I =
(22)
i;
y : dA.
(23)
Thus, the influence of linear elasticity is as follows :
f0 ' x=f0 pe
[q,, (r)] - ' . t dx.
(24)
The moment of inertia I calculated for the idealized Isection (shown in Fig. 5) has the following form : I=
(25)
l~2"(b'h3-b,'h~).
The influence of the non-linear elasticity is defined by the following basic equations : a = [~Oz(T)] ~/,(r).~p~/~(r) for
U #- 0,
(26)
and 1 ep = ~ ' y ,
(27)
which result in the equality : f0' O~ 1 ./17/dx = f0 ' [O2(T)]-' MU'r) ".Mpp(r)dx '
(28)
where :
(17)
which, due to additive steel strains, takes on the following form: 1 . h.]rd x +
The substitution of the above expressions to equation (19) gives the formula for the curvature 1/p~ :
(16)
= ~1 1 . ~ dx, Jo p
(21)
where :
the steel is a homogeneous and isotropic continuous medium, the mechanical properties of which during compression are the same as during tension. These properties are described by the model expressed by equation (14) or in the following way :
1./Qdx+
(20)
and
'y
c5 =
(19)
a.y.dA.
lp =
f;
y~,/u(r))+IdA '
(29)
For the I-section : 1 Ip = k ' 2 k- I " ( b ' h ~ - b l ' h ] ) ,
(30)
where : (18)
1 k - #'~'-'~'t-J*z
(31)
A Study o f the Steel Beam Deformation During Fire The influence of the creep of steel will be defined by the equations :
f
F
/
I ) A\ H
a:~O3[exp[-~;],,|~ k
.
t
_
k
/
d
.
- 1/m
"e)/"
for m e 0 ,
/
(32)
163
The influence of the thermal expansion resulting from it is as follows :
;o
' --1 "-~td = pT
'I
(T, - T 2 ) • a
• 3~/dx.
and
(42) 1
et = = ' y , Pt
(33)
When AT = 0 t h e n :
fo
r l ' h ~ / d x = O. Pr
while : T = T(t),
(34)
and the inverse function is determined : t = t(T).
(35)
Let us substitute these equations, as previously, into condition (19). We obtain :
folx:y01f3[exp 1
After substituting the relations (24, 28, 36 and 42) into equation (18) we obtain:
'Y0
= ~"
+
M" • K/
(T, -- T2)" a
i
(36)
It =
f;
y(l/,,,)+l dA.
(37)
The substitute moment of inertia /, determined for the I-section has a form defined by equation (30), whereas :
M dx
[~,(r)]-l.i dx+
+f0
where :
(43)
[ ¢ ~ ( r ) ] - , . i ¢ . dx M"-~
{~O3[exp f AH• ]]-1 dx. ~ - R~7/, t ] ; "1:
(44)
With the increase of the deflection (and time) a change of the stress distribution in the cross-section of a beam takes place, which is shown in Fig. 7. The calculations of stresses when analysing the deflection are not necessary and ineffective in view of the implicit form of the functional W- 1 in the equation : a = W-' (e),
(45)
1
k - m+2"
(38)
The influence of the thermal expansion resulting from a non-uniform, linear temperature distribution at the level of the cross-section of a beam (cf. Fig. 6c) is defined by the following basic equations : [ T I + T2"~ er = ~ \[ - - ~ ] ' A T/ ( y ) ,
(39)
and ~T =
1
--.y,
(40)
Pr where : AT(v) =
T 1-
h
T 2
"Y"
(41)
where ~F- J is an inverse functional to the functional qJ in equation (14). Exemplary calculations of beam deflection in fire will be carried out for the ASTM A36 steel I-section beam described in [16] (beam No 1). A simplified model of the beam as well as basic equations of bending moments are given in Appendix 2. During a fire endurance test the temperature of the extreme tensile fibres in the crosssection of the considered beam was increasing with an average rate of 6.74°C min-1, and the temperature of the extreme compressive fibres with an average rate of 2.32°C min- 1. Figure 8 shows the process of deflection increase of the considered beam, registered during the fire endurance test and calculated according to the formula (44). The conformity of the experimental and theoretical results is satisfactory from the point of view of an engineer. A similarly high conformity was obtained after the
T'I*T2
AT{y}
T1 al
b]
c]
Fig. 6. Temperature distribution at the height of the beam cross-section.
W. Skowronski
164
J
to
t 1
t2
t3
tcr
Fig. 7. Process of stress distribution change in the l-section of a beam under fire.
o l o 2o°
20
4o
sO
100
so
t,lmin]
2
tested beam will be equal to the structural criterion for fire endurance tests, i.e. in which the beam temperature will be critical. The equation describing that moment can be written as follows :
4
¢,
G
(T~r - T2)" ~
f4 dx
8
fo ] M ' ~
fo'
M~'`r")'AT/
10
,
12 14-
5,[cm]
\
• t
I
1
Mm.M
+J0 ~ f 3[ex [ p
-AH(- ~),tll
-'
"IF
dx = Ycr' (46)
\
Fig. 8. Course of steel beam deflection shown schematically in Fig. 11. calculations of other steel beams have been carried out with the application of this method. When considering the fire resistance, the most interesting for us is the moment in which the deflection of the
where Ycr is defined by the structural criterion for fire endurance tests. The third component of the left-hand side of the above equation has little influence on the result of calculations, less than 0.1%. This is visualized by Fig. 9, where the values (in percentages) of particular components determining beam
6e,Sp,St, I°/o) 100
80
60
40
~¢r
-"
~IT~.)=
] •L'-L 800 h
Fig. 9. Variability of components determining beam deflection, expressed in percentages by the formulae (47), (48) and (49).
A Study of the Steel Beam Deformation During Fire
165
deflection caused by mechanical strains are shown : foI [~b,M'A~¢ (T)]-' "I dx" 100% 6e =
(47) foI
M'AT/1 d x + f / MY(r)"21~r d x + f / [~O 2(T)]-" Ipu{r} [~b,(T)]-"I 1
M "{r) "
Mm'AT/ ~. L%3Fexp --AT
dx
till-'
2~I
[¢2(T)]_,. ~(r) dx" 100%
f/
M" )~1
I
M,{r)
[i,b,l(T)]_I ,/dx + ~0
"]~
"'
(48)
t
dx+f 0
dx
2~'tm"~'/
M m" AI
dx" 100%
jo {~3 [exp (-- R~')' t]}-'. I7' 6,
--
f0
1
M" j~r [~b,(T)]-' •
I dx+ fo1
M m. ~r AH [$2(T)]_, .pe{r) dx+ fo {qJ3 DxP ( - R ~ 7 ) ,
M'{r)'fff
1
As can be seen in Fig. 9, at temperatures above 400°C the creep effects increase suddenly and rapidly. At a temperature of 500°C these effects are strong and cannot be neglected in the analysis of a beam by means of the method described here. The value of 400°C cannot be treated as a universal temperature determining the lower limit of the range of temperatures at which the creep of steel has a considerable influence on beam deformation, since the 6, value depends also on time. Nevertheless, the calculations presented here confirm the correctness of the method used by J. Thor [4], who, when analysing beams made of Swedish steel in his calculations took into account the creep of steel after it has reached the temperature 450°C, and the results of calculations he obtained were in conformity with the results of fire endurance tests. In equation (46) the beam rigidity is defined by the following mathematical expressions : [qJt (T)]-" L [~2(T)]-' "Ipu(r),
--1
dx
t]}'IT'
and {qJ3 Iexp ( - R~,), t] }-' • I,m.
ff0' {~, [T(y)]}
"y2dA,
a. { f~0A [~b3 [exp ( _ RTT),
,,yjjq]-{1/')
],, .(52a)
Taking into account equations (50a, 51a and 52a) has little influence on the results of the critical temperature calculations, which are shown in Fig. 10. The critical temperatures, presented in this figure, were calculated from the well-known Ryan and Robertson criterion [17] :* 1 l2 y,., = ~ ' ~ . (53)
rdd ] 823 773 723 3pO
(50a)
I f f/[~k2[T(y)]]'/'{r)'y ('/'(r}}+' t "{r}. (51a)
(50) (51)
2,00
(52)
The above expressions change after taking into account the actual (non-uniform) distribution of temperature at the height of the cross-section of the beam under fire, and assume the following forms :
*The critical temperature is a conventional quantity and depends on the accepted structural criterion for fire endurance tests. The acceptance of another criterion will give another critical temperature.
673
(49)
4.00
5~
6/}0
700
•
de
Fig. 10. Calculation results which visualize the influenceof the rate of temperature increase in a beam on the critical temperature.
166
W. S k o w r o n s k i the basis of:
Id
,tl
f
~
~
~
~
I
~
½
~"~
q ~-212726"~"
~
~ ~1
(a) The known properties of steel at elevated temperatures, including creep characteristics. (b) The values of a cross-section, static diagram and beam loading. (c) The data concerning the temperature increase in a beam, determined by the efficiency of the fireproof insulation.
Fig. I 1. A simplified model of the considered steel beam. In Fig. 10 it can be seen that critical temperature of the beam whose temperature was increasing with the rate of 2.32°C per minute would differ only by 2% from the critical temperature of the beam whose temperature would be increasing with the rate of 6.74°C per minute. Therefore, it is often not economical to take into account the complicated expressions (50a, 51a and 52a) and the third component of the left-hand side of equation (46) in the calculations done according to equation (46).
CONCLUSIONS The analysis performed indicates that it is possible to calculate exactly deflection of steel beams under fire on
The proposed method of calculating the deflection is sufficiently precise and simple to be used in practice. With the help o f this method it is possible to foresee the results of fire endurance tests on given steel beams and generalize these results over the beams under different loading or with different dimensions and shapes as for the crosssection, length or static diagram. Acknowledgements--The author wishes to express his gratitude to Prof. K. Czarnowski of the Technical University of Wroctaw for his interest in the problems of fire safety and scientific supervision, to Prof. T. Z. Harmathy of NRCC in Ottava for his great benevolence in private correspondence with the author, and to Prof. Z. Kowal of the Swietokrzyska Polytechnic in Kielce for his valuable critical remarks which have contributed to the improvement of this work.
REFERENCES 1. E.M. Krokosky, Modelling thermal fire resistance, Build. Sci., 4, 197 (1971). 2. T.T. Lie, Fir and building, Applied Science London, p. 267 (1972). 3. T . Z . Harmathy, Creep deflection of metal beams in transient heating processes, with particular reference to fire, Can. J. Cir. Eng., 3, 219 (1976). 4. J. Thor, Deformations and critical loads of steel beams under fire exposure conditions, Doc. D 16: 1973, Nation. Swed. Build. Res., p. 217 (1973). 5. T.Z. Harmathy, A comprehensive creep model, J. Basic Engng. Trans. Am. Soc. Mech. Engng, 89, 496 (1967). 6. J. Kruppa, Collapse temperature of steel structures, 3. Struct. Div. ASCE, 105, 1769 (1979). 7. F. Mehl and W. Arndt, Grundlagen zur theoretischen Bestimmung des Feuerwiderstandes von biegesteifen Stahlstabwerken, Symp. Steel and Composite Structures for User Needs, Reports of the working commissions IABSE, Dresden, 21, 195 (1975). 8. D.C. Jeanes, Application of the computer in modeling fire endurance of structural steel floor systems, Fire Safety J. 9, 119 (1985). 9. P. Arnualt, H. Ehm and J. Kruppa, R6sistance au feu des structures hyperstatiques en acier, Doc. CECM 3-74(6F), Centre Technique Industriel de la Construction Metallique, France, p. 127 (1974). 10. A. Rubert and P. Schaumann, Structural steel and plane frame assemblies under fire action, Fire Safety J. 10, 173 (1986). 11. W. Skowronski, Problems of critical temperature of steel beams in fire with particular respect to steel creep (in Polish), Ph.D. dissertation, Report PRE No. 91, Building Institute of the Technical University of Wrociaw, p. 142 (1984). 12. J.E. Dorn, Some fundamental experiments on high temperature creep, Symp. Creep and Fracture of Metals at High Temperatures, HMSO, Lond., p. 89 (1956). 13. D.H. Skinner, Determination of high temperature properties of steel, BHP Technical Bulletin 16, I0 (1972). 14. D.H. Skinner, Runaway temperature--a design criterion based on the high temperature properties of steel, BHP Technical Bulletin 16, 22 (1972). 15. B.R. Kirby and R. R. Preston, High temperature properties of hot rolled, structural steels for use in fire engineering design studies, Fire Safety J. (in press). 16. R.W. Bletzacker, Fire resistance of protected steel beam floor and roof assemblies as affected by structural restraint, Symposium on Fire Test Methods--Restraint & Smoke 1966, 63 (1967). 17. 1. V. Ryan and A. F. Robertson, Proposed criteria for defining load failure of beams, floors and roof constructions during fire tests, J. Res. Nation. Bureau of Standards, 63C, 121, 2 (1959).
APPENDIX
the above given deformation range has the following form [11]:
1
In steel beams under fire strains of steel fall within the range : 0.005 ~< s ~< 0.02.
(1A1)
The amendment function q~o(a) for the ASTM A36 steel within
~ba(a) = 3.9862" l0 -4" o'1"45[h2/3].
(2A1)
The ASTM A36 steel is characterized by the following additional data [2, 4] :
A Study of the Steel Beam Deformation Duriny Fire F1-]
Z(a)=3.7296.10s.a47°L~j
for
a~<103MPa
(3AI)
167
for 200°C ~< T < 591°C; /~(T)=l.5375T-901.32[-]
for
591°C~
Z ( a ) = 1.23 • 1016 • exp (0.003425a, [ ~ ] for
a > 103MPa
(4AI)
e,o = 4.0708.10 -6.O-''75[-],
(5AI)
AH ~ - - = 38900[K].
(6A1)
The thermal expansion coefficient for the ASTM A36 steel at the temperature range 20°C ~
After substituting the above data into equation (10) we obtain the following creep model for the ASTM A36 steel at elevated temperatures : ,t = 1.0534" i0-4"o "4''833 •
[~'
exp
(
38900 ` -]./3 T+-~.16)dtJ . (7A1)
The remaining mechanical strains of the ASTM A36 steel are being modelled by means o f the Ramberg-Osgood curve defined by the following equation :
+ooo2L]
(16AI)
APPENDIX 2 The calculations were carried out on an ASTM A36 steel Isection beam, which was tested according to the ASTM El9 standard "Methods of fire tests of building construction" and described in [16]. A simplified model of the beam is presented in Fig. 1I. Here follow the basic equations resulting from this model of the beam considered : (a) forces in the points of support of the beam :
,
(8A1)
Va = VE -
where :
2127.26.5.13 +59100.27 = 64560.69[N] 2
(1A2)
(b) bending moments : for 20°C ~< T < 200°C,
M aa = - 1063.63x2+64560.69x[Nm],
E(T) = - 117.72 T + 208364 [MPa], o~v(T) = 300.1 I" 10 -3. (1026.52-- 1.33T) [MPa], /z(T) = -0.16489 T+38.878[-],
(9A1)
M cD = - 1063.63x 2 + 5456A2x + 84046.27 [Nm] (3A2)
(10A1) (IlAI)
M DE = _1063.63x2--53647.85x+303200.9[Nm]
(4A2)
(c) bending moments caused by the unitary force P = 1 :
for 200°C ~< T ~< 600°C,
E(T) = -274.68
Mac =
(2A2)
T + 239756 [MPa],
av(T) = 304.11 • 10 -4. (9509.03-9.47 T) [MPa],
(12A1)
AT/Aa= Atac = 0.5x [I m],
(5A2)
(13A1)
ff/cD = AT/.oe= - - 0 . 5 x + 2.565 [1 m].
(6A2)
and : /t(T) = 21.97-0.02479 T--x/--0.3158 T + 186.65 [--]
(14Al)
The remaining calculations were performed on the computer with the use o f the program "SKO 3" (FORTRAN). The results are presented in Figs 8-10.