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Interaction domains for steel beam-columns in fire conditions
J. Construct. Steel Research 17 (1990) 217-235
Interaction Domains for Steel Beam-Columns in Fire Conditions
L. C o r r a d i , C. Poggi & P. Setti Department of StructuralEngineering,Politecnicodi Milano, Milan, Italy (Received9 April 1990;revisedversionreceived6 July 1990; accepted21 September1990)
ABSTRACT A parametric study on the interaction curves for steel beam-columns in fire conditions is performed. The study is based on a trilinear idealization of the uniaxial stress-strain relation for steel at elevated temperature and aims at assessing the influence of the slope of the hardening branch, on which results are strongly dependent. A refined and numerically efficient elastic-plastic beam element model is employed to analyse different load conditions and temperature distributions. Results for uniform temperature are systematic enough to permit fairly general analytical approximations, which might be useful also in the presence of temperature variations along the element length only. Thermal gradients through the cross-section are also considered.
NOTATION A E Et M m N n S
Cross-sectional area Young modulus Tangent modulus of the hardening branch Bending moment Dimensionless bending moment Axial thrust Dimensionless axial thrust Cross-sectional plastic modulus 217 J. Construct. Steel Research 0143-974X/91/$03.50 © 1991 ElsevierSciencePublishers Ltd, England. Printed in Great Britain
L. Corradi, ('. Poggi, P. Setti
218
o
(rp A
Dimensionless slope of the hardening branch Normal stress Ultimate yield stress First yield stress level Slenderness ratio
Subscripts
( )o ( )p ( )s
R o o m t e m p e r a t u r e values First yield level Ultimate yield level Euler load Shanley load
1 INTRODUCTION At elevated t e m p e r a t u r e the stress-strain characteristics of structural steel deteriorate rapidly and the influence of creep b e c o m e s of importance. However, for structural analyses in fire conditions, typically limited to a few minutes and with t e m p e r a t u r e s not exceeding 600°C, it is widely accepted that viscous effects can be accounted for only implicitly, by introducing 'equivalent' t i m e - i n d e p e n d e n t stress-strain curves, usually defined on the basis of experiments on uniaxial specimens u n d e r constant stress and increasing t e m p e r a t u r e (or vice versa). Obviously, results arc strongly d e p e n d e n t on test conditions, in particular on the rate at which loads are applied and/or t e m p e r a t u r e is increased. It follows that the definition of generally accepted stress-strain laws is a difficult problem, which must still be considered as open. T h e simplest possible idealization of uniaxial steel behaviour is the bilinear, perfectly plastic one. It only requires the definition of two material properties, namely the Young's m o d u l u s E and the yield stress ~r~. T h e y must be given as a function of t e m p e r a t u r e if this model is to be used in fire analyses. As an example, the following expressions p r o p o s e d by ECCS,~ are recalled cry = %o(1 + T/(767 In (T/1750)))
(la)
E = Eo(l + (15.9 × I(Y 5 T) - (34.5 x lt)- 7 T 2) + (11-8 x lt) -'~ T 3)
- ( 1 7 - 2 x 10 12 7,~))
(lb)
In the above relations, O-yo and Eo indicate the yield stress and Y o u n g m o d u l u s at r o o m t e m p e r a t u r e , and T is m e a s u r e d in the Celsius scale.
Interaction domains for steel beam-columns in fire conditions
EE200
i
219
206 / ~ 2 0 6
250275300 350 tOO
150'
5O
0.1
0.2
0.3 0.4 0.5 strain E (%)
0.6
Fig. la. Stress-strain curves of steel at elevated temperature (ECCS proposal).
1
~
2
0
°C 2OO
o
--
25O
300 40O 50O
0.5
--
0.1
0.2
o.3
0.4 (%)
o.5
600
0.6
Fig. lb. Trilinear idealization of the ECCS curves.
Even if drastically oversimplified, such a model is acceptable when deformation does not play a significant role and an assessment of the load-carrying capacity is only soughtf1-4 In this case the problem can be solved by limit analysis and results only depend on the material yield stress and on its decay with temperature. However, the relationships which have been proposed to model in terms of time-independent stress-strain curves the behaviour of steel at elevated temperature, depart significantly from the bilinear model. Fig. la shows the original proposal by ECCS.1 The initial slope and the ultimate stress level are still given by eqns (la) and (lb), but the amplitude of the linear range is dramatically reduced and a significant decay in stiffness occurs afterwards. Even if geometry changes can be neglected, the actual shape of the stress-strain curve must be considered when deformations are of
220
L. Corradi, C. Poggi, P. Setti
interest. Moreover, the decrease of stiffness with temperature emphasizes geometric effects; in this circumstance collapse usually occurs because of loss of stability of the partially yielded structure and, as is well known, the shape of the stress-strain curve, in particular the value of the tangent modulus Et, plays a key role. This aspect was recognized by a number of authors s'(' and fire analysis methods were recently proposed accounting for the non-linear nature of the stress-strain curve. The object of such analyses is the definition of deformations up to collapse for negligible geometry c h a n g e s ] the evaluation of the load-carrying capacity of compressed columns s') and the assessment of the global behaviour of framed structures.m A problem which has attracted little attention is the ultimate capacity of beamcolumns. ~j This paper is intended as a contribution in this direction. As previously mentioned, the curves of Fig. l a did not find general agreement. A n u m b e r of alternatives have been suggested by different boards (Rilem, BSC, RUB/KFI, A R B E D , CRIF, etc.) and are summarized in Ref. 12. Even if all were based on reasonable assumptions, differences are often significant and the definition of h o m o g e n e o u s proposals is presently the object of discussion. ~2.J3 In this situation it is felt that the comparative merits of different models can hardly be established by simply contrasting the numerical results produced on their basis. In fact, it would be difficult, if at all possible, to assess the quantitative influence of the individual assumptions leading to each of the curves proposed. Of greater interest seems to be a parametric study aiming at the identification of those features of the stress-strain relation which play the most significant role with respect to critical situations. Such a study can be based on drastically simplified models, depending on a limited n u m b e r of parameters, whose role can be assessed unambiguously. In this paper the problem of defining the interaction curves for steel beam-columns of different slenderness at elevated temperatures is tackled. A parametric study is performed, aimed at assessing the influence of the shape of the stress-strain curve on the resulting domains in the transverse load-axial force plane. The uniaxial behaviour is idealized as a trilinear diagram, depending on four parameters which could be adjusted to approximate most of the laws which have been suggested. However, no particular reference to existing proposals is made. The main object of this study is to detect the role played by the slope of the hardening branch, which is felt to be the most significant parameter, and the proposed model is well suited to this purpose. In some instances, results look systematic enough to permit analytical approximations, which can be regarded as a first step toward the definition of design formulas.
Interaction domains for steel beam-columns in fire conditions
221
The study has been performed numerically. Computations are based on the finite element model proposed in Refs 14 and 15, able to account with satisfactory accuracy for the spreading of local plasticity and to identify correctly the critical situation, corresponding either to collapse or to Shanley-type bifurcation. The efficiency of the model was verified by comparison with experimental results carried out on a simply supported beam. 15
2 THE TRILINEAR M O D E L Consider the trilinear stress-strain diagram depicted in Fig. 2. It depends on four material parameters, namely the initial slope E, that of the hardening branch Et = aE, the first yielding stress level O-p and the ultimate stress O-y. Obviously they are all temperature-dependent quantities. It is easily recognized that for the purpose of a parametric study, the above quantities have different importance. Note first that for E t --- 0 (a = 0) and Et = E (a = 1) two limiting situations are recovered in which the trilinear behaviour reduces to bilinear ones with yield stress O'p and O-y, respectively. In these situations, the main role is played by the ratios of the actual slenderness to the following critical values Ap = 7r(E/crp) 1/2 Ay = "tr(E/o'y) 1/2
(2a,b)
If the above values were constant with temperature, so also would be the dimensionless quantities defining the interaction curves. In this case, at least in uniform temperature conditions, the design formulas available for
0
IFtl I I
'
_~_aL_
IL E
Fig. 2. Trilinear stress-strain model.
222
L. Corradi, C. Poggi, P. Setti
room temperature can still be used, simply by referring them to the current values of material properties. Most of the proposed stress-strain relations imply that the critical slendernesses (eqns (2a) and (2b)) are actually very nearly constant, at least for temperatures up to 500°C (both in the original ECCS curves ~and in recent proposals ~~the scatter between maximum and minimum values is less than 10%). In this circumstance, only two parameters effectively survive; they are given by the following dimensionless quantities n p = ~rp/~r,~ a
-
-
E~/t:
(3a.b)
The first of them, however, has only a scaling effect: it widens or narrows the domain in which the trilinear behaviour is contained, without affecting its qualitative features. Hence, the trilinear model is suited for a parametric study aiming to evaluate the influence of the slope a (eqn (3b)) of the hardening branch on the interaction curves. The ECCS curves of Fig. la can be approximated with acceptable accuracy by a trilinear diagram (see Fig. lb). It is stressed, however, that in this study no reference is made to particular curves and the values of the parameters used are not dictated by specific proposals. The model employed is defined as follows. - - T h e initial elastic modulus E and the limit stress cr~ are expressed as functions of temperature by eqns (1 a) and (lb). - - A perfectly plastic behaviour is assumed up to T = 200°C. - - F o r T->300°C, the hardening branch starts at a given value of rip = crp/Cry, usually taken to be 0.5. Different values of the dimensionless slope c~ in a wide range are explored. - - I n the interval 2t)0°C~ T < 3 0 0 ° C , the first yield level np varies linearly between 1.0 and 0.5 (or the other value assumed at higher temperatures).
3 S U M M A R Y OF T H E A N A L Y S E S P E R F O R M E D A first set of computations was intended at assessing the influence of the l o a d - t e m p e r a t u r e history on the interaction curves. A n u m b e r of beamcolumns were analysed at constant temperature, by increasing the mechanical loads up to collapse. Subsequently, the final values of loads were applied on the cold beam and temperature was increased, until collapse, due to decay of material properties, was predicted. In all the cases tested, the ultimate situation turned out to be rigorously the same, in accordance with the well-known independence of the interaction curves at
Interaction domains for steel beam-columns in fire conditions
223
room temperature from the history of mechanical (transverse and axial) loads'. Due to this fact, only the case of temperature constant with time was considered, a condition which simplifies considerably the numerical analyses, since iterations to updatethe material properties are avoided. The influence of the sectional shape was also explored and found to be minor. A number of tests on I-beams and rectangular cross-sections produced dimensionless interaction curves which, under the same conditions, exhibited differences of less than 5%. In subsequent computations, only HE200M shapes were examined. Numerical analyses were referred to simply supported beam-columns subjected to three different load conditions, namely: (1) Equal concentrated couples at both ends (constant bending moments in the absence of second-order effects). (2) Concentrated couple at one end only. (3) Uniform transverse load. For each of them, the following temperature distributions were considered: (a) Uniform temperature in the member. (b) Temperature linearly varying through the thickness and uniform along the length. (c) Temperature varying linearly along the beam length and constant over the thickness. The analyses were developed by two students within the framework of their graduation thesis 16 and the complete set of results is collected in it. In this paper only those relevant to load condition 1 are reported; in any case, the effects of different transverse loads were found to be analogous to those experienced at room temperature.
4 I N T E R A C T I O N C U R V E S FOR U N I F O R M T E M P E R A T U R E An interaction curve defines the admissible domain for a beam-column of given slenderness A in the m - n plane, where m is the value of the applied external moment and n the axial force, divided by the relevant ultimate values M u and Nu at the temperature considered m = M/Mu = M/Soy
n = N/Nu = N/Atry
(4a,b)
In eqns (4a) and (4b), A and S are the area and plastic modulus of the cross-section and try is given as function of temperature by eqn (la). All curves must intersect the two axes at the ultimate bending value
L. Corradi, C. Poggi, P. Setti
224
O/oy / I
ns= Oy ',
~ I
,, I
V
~
_O'cr,
e
''e - Oy I
I
I
ko k,
t
~j,
kp
Fig. 3. T h e o r e t i c a l c o l u n m model for trilinear material.
m = 1 and at the critical load for the slenderness considered. [:or an ideally straight column m a d e up of a trilinear material, the theoretical column curve is shown in Fig. 3: it is obtained by joining two horizontal segments at the % and o-y levels with two cubic hyperbolas, corresponding to the Euler and Shanley critical loads, respectively. In dimensionless form, these are expressed as follows n E = 7T2E/cryA 2
n s = "n"2Et/crv A2 = oenr~
(Sa,b)
The interaction curves for the trilinear model will be contained in the area b o u n d e d by those obtained in the limiting situations a = (1 and ~e = 1, the shape of which is little affected by t e m p e r a t u r e , as expected, since the critical slenderness values eqns (2a) and (2b) are very nearly constant. U p to T = 200°C the bilinear model a = 1 applies and the interaction curve coincides with the upper bound. Typical results for higher t e m p e r a t u r e s (trilinear behaviour) are illustrated in Fig. 4, where the limiting curves are also shown. The scatter with respect to m = 1 on the ordinate axis is due to the approximations entailed by the finite e l e m e n t model used; ~4 on the other hand, the theoretical critical load is exactly r e c o v e r e d for purely compressed members. In some instances (see, e.g., Fig. 4a for a = 1/5) small m values apparently produce a strong increase in the axial capacity. Such an effect shows up for a small range of a values only and must not be considered as meaningful. The analysis p r o c e d u r e used stops when either a bifurcation is predicted or a limit point is reached. The first alternative occurs in the purely compressed column, whose carrying capacity coincides with Shanley load. The presence of bending m o m e n t s eliminates bifurcation and, due to the rising nature of the bifurcated path in Shanley-type buckling, computation continues up to a limit point. H o w e v e r , the increase in load-carrying capacity is achieved at the price of deflections which grow extremely rapidly from n -~ ns. For
Interaction domains for steel beam-columns in fire conditions
225
1.00
~ • 1/20 IX=1/10
0.80
~, =70
=1/$
T • ~o, c
0.60
,"v=1 0.40
Or=0 0.20
0.00 0.00
0.20
0.40 ).60 NINu
0.80
1.00
(a) 1.00
~fOt =I120 0.80 ~ O ~ . 1 1 1 0 0.60 |~~/A~ , ~
'~
,~ -'30 T gl~0° ~
a'1/5
o~:I
a00
ai0
a~0
a~0
a/~
1.60
NINu
(b) Fig. 4. Interaction curves for beam-columns with [filinear material model.
practical purposes, this value should be considered as the limit axial capacity. A closer look at the numerical results permits the subdivision of the interaction curves for slender members in two different zones, indicated in Fig. 5 with (a) and (b), respectively. In zone (b) collapse occurs when all plastic points are still on the hardening branch, i.e. for o-< O'yeverywhere.
226
L. Corradi, C. Poggi, P. Setti
1.0
k
(a)
\ np
(15
Ns
np
Fig. 5. Proposed analytical approximation of the interaction domains. In this case the horizontal plateau has no role and identical results would be p r o d u c e d by a bilinear hardening model. In zone (a), on the other hand, some points have reached the stress level Cr = O'y. For sufficiently low slenderness values and/or sufficiently high a, the entire interaction curve belongs to zone (a). When both zones are present, the b o u n d a r y b e t w e e n them is closely approximated by a straight line, indicated by r in Fig. 5, which b e c o m e s s t e e p e r with increasing slenderness. These features are the basis for the proposal advanced in the s u b s e q u e n t section.
5 ANALYTICAL APPROXIMATION CURVES
OF THE INTERACTION
Consider the domain defined by the following inequalities A. ns < nr,:
B. n~>>-np:
m <- ( 1 - n/ns)( 1 - n/nF)
if n -< ns( 1 - np)
(6a)
m -< np(np - n)(1 - n/nE) n p - ns(l - n p )
if n-> ns(1 - n p )
(6b)
m~<(l-n/ns)(1-n/nE)
(7)
where nE and ns are the dimensionless Euler and Shanley loads defined by eqns (5a) and (5b).
Interaction domains for steel beam-columns in fire conditions
227
The domain (A) applies when the slenderness exceeds the value )tl = X/-~-aAp in Fig. 3. In this case the theoretical critical load is either np()t<-)tp) or nE(A-->)tp). Equations (6a) and (6b) approximate the interaction domain in zones (a) and (b) in Fig. 5, respectively. It can be readily verified that the two domains intersect on a straight line similar to the one labelled as r in Fig. 5, defined by the equation m/np .4- n/nE = 1
(8)
The domain (B), on the other hand, is relevant to the interval )to -< )t -< )tl in Fig. 3 ()to = V~--a)ty), in which only zone (a) is meaningful. The theoretical critical load is now given by ns. It is assumed that eqn (7) applies also for )t < )to, when the dimensionless critical load becomes equal to one. To understand the meaning of eqns (6a) and (6b), consider an ideal case in which nE is brought to infinity by keeping constant the values of the remaining parameters. The interaction domain becomes th,e union of those bounded by the two dashed straight lines in Fig. 5, the first connecting points (n = 0, m = 1) and (n --- ns, m = 0), the second intersecting the first for m numerically equal to np and the n-axis for the theoretical critical load, now also equal to np. The influence of the slope of the hardening branch is incorporated in the Shanley load ns, proportional to a, which affects both segments and reduces the size of the domain with decreasing a. Equations (6a) and (6b) are obtained by multiplying, in the equations of the above straight lines, m by the amplification factor 1/(1 - n/nE); the bounds of the domain are no longer straight and exhibit a concavity which, in agreement with numerical experience, increases with slenderness. Moreover, the approximated interaction curves now intersect the n-axis also for n = hE, i.e. for the theoretical critical load of sufficiently slender members. The above reasoning illumines how eqns (6a), (6b) and (7) were arrived at, but is by no means evidence of agreement with numerical results. Typical comparisons are illustrated in Fig. 6. The approximation is systematically conservative for small n, reasonable even if not always conservative when bending moments are small and compression dominates. Note, however, that the interaction domain in this zone would reduce considerably in size if the intersection with the n-axis were given not by the theoretical critical load of Fig. 3 but by a design value, say no, accounting for geometrical and mechanical imperfections, nc must replace n o in eqn (6b) and ns in eqn (7). The approximated domains become A.
o~/'/E ~-~ nc:
m --<(1 -- n/t~nE)(1 -n/nE)
if n --
(9a)
228
L. Corradi, C. Poggi, P. Setti
rn~
nc(n c - n)(1 - n/nE) n~
-
if n-> anE(1 - n , . )
anF(1 - nc)
(10)
m-<(1-n/n~.)(1-n/nF)
B. anE>-nc:
(gb)
Clearly, neither eqns (6a), (6b) and (7) nor eqns (9a), (9b) and (10) must be considered as proposals for design recommendations. They can only be regarded as a first step in this direction, in that they permit the 1.00 -
0.80-
~,=
70
-, 0 ` 6 0 -
',,\%
z~ 0`/.0-
"A 0.20
0`00 0`00
1.00
0.80
0.20
0,40 0.60 N/Nu
0.60
1.00
(a)
~
~, =130
o:=115
=
0.80
T =300° C
O.6O
0.20
0`00 0`00
i
o~zo
i
o.4o
o.~o
o.~o
t6o
N/Nu
(b) Fig. 6. Proposed approximation (dashed line) versus computed domains.
Interaction domains for steel beam-columns in fire conditions
229
representation of the main features of the interaction curves on the basis of a limited n u m b e r of structural parameters. Note, however, that for establishing such relations, the trilinear nature of the stress-strain diagram is essential, since only in this way is it possible to consider the Shanley critical load ns = anE as a structural datum. Extensions to general, non-linear stress-strain relations are not possible, unless an equivalent a is defined. As a final remark, note that in the present situation temperature plays only an indirect role, limited to the definition of the values of structural parameters. Hence, results hold also in different contexts, whenever the material behaviour can be reasonably idealized with a trilateral diagram. This might be the case for aluminium members, if the replacement of the customary R a m b e r g - O s g o o d law with the one depicted in Fig. 2 is considered acceptable.
6 NON-UNIFORM TEMPERATURE The results obtained by considering the temperature distributions indicated by (b) and (c) in Section 3 are now summarized. Temperature varies linearly, either over the cross-section or along the beam length. Two different intervals were studied, namely (A) 300°C --- T <- 500°C (B) 200°C <- T < - 400°C In the first case the stress-strain relationship is described everywhere by a trilinear curve, whereas in the second the transition between bilinear and trilinear behaviours is present and its effects can be investigated. 6.1 Temperature variation over the cross-section
As a preliminary step, the temperature interval (A) was studied with reference to the bilinear law a = 1. The results, shown in Fig. 7, are divided by the ultimate values at the average temperature Ta -- 400°C. Because of the non-homogeneous material properties, symmetry about the n-axis is lost and the ultimate values for axial load and bending m o m e n t in the cross-section (h = 0) decrease by about 20% and 10%, respectively. However, a small positive bending m o m e n t (causing compression on the colder edge) is favourable in terms of ultimate axial load because the fibres which undergo the higher compressive stress have a greater strength.
L. Corradi, C. Poggi, P. Setti
230
o.
oo
~.6o
o.do
o.oo
-o.so
-~.'oo
M IMu Fig. 7. T e m p e r a t u r e variation o v e r the cross-section (300°C T -< . 0 0 C). I n t e r a c t i o n curves for bilinear b e h a v i o u r ( a = 1).
N/Nu ~,
=70
300°C .I.O0
Ta =4oo°c Ot=l
,.~o
~-',~k//-,~
S
=.,o
~ = , t 2 0
:'~o
o.oo M/Mu
-Lso
-1.00
(a) N/Nu 30D~
~, : 130
.I.00
Tit =
400,, c
500°C
-0.50
,.~o
o.~o
o.oo
~ O t
~,.//---
-o~so
ot=1 =115
CX= 1110
-,200
MIMu (b) Fig.
8.
I n t e r a c t i o n curves for t e m p e r a t u r e v a r i a t i o n o v e r the cross-section (. 00 C T ~ 500°C). Limit values are r e f e r r e d to average t e m p e r a t u r e .
Interaction domains for steel beam-columns in fire conditions
N/Nu
,~ : 7 0
"1.00 41111 ~;
.
~
Ta :3ooec 0
~
~\VFa
,.~o
~0
231
~oo
M/Mu
Ot:l : llS : I,lo
-ds0
(a)
N/Nu
('~-----~
.,_.:~ [,.oo
~,: ,3o r . : 30,. c
t.O0 *C *C t.O0 0.50
1.O0
0.50
0.00
M/Mu
~c~ : 1 / ~0~ : 115 O: : 1110 =1120
- 0.50
-1. O0
(b) Fig. 9. Interaction curves for temperature variation over the cross-section (200°C-< T-< 400°C). Limit values are referred to average temperature. For A > 0 the shape of the interaction curves changes for various reasons. The predominant one seems to be the interaction of the axial load with the elastic curvature due to bending moments, the curvature due to temperature and that caused by plastic deformations. The latter may even become opposite to the thermal contribution because of the lower ultimate strength of the warmer fibres. The shape of the domains is also affected by the presence of self-equilibrating stresses over the cross-section, which is subjected to a linear temperature variation and hence, as eqn (lb) shows, no longer has a uniform elastic modulus. The finite element model used 15 allows these effects and their influence on the interaction curve to be taken into account. The results obtained for the two temperature intervals (A) and (B) are shown in Figs 8 and 9 for different values of a and A. Dimensionless quantities are still referred to the average temperatures of 400°C and 300°C, respectively. The slope of the hardening branch has a marked influence on the domains, which becomes smaller with decreasing a. Note
L. Corradi. C. Poggi, P. Setti
232
1.00 Ot = 115 = 1110 Ot =1120
0.80
/ ! , = 70
•
\ ~
Ta: ~.o0"c
0.60
3; o.z.o 0.20
0.00 0,o0
o.zo
o.~o 0.6o N/Nu
0.80
~.oo
(a)
1.00 ~ j / .
_Or--115 O~=1110 O~ : 1/20
0.80
,~ 130
O. 60
z~
0.40
0.20
0.o0 0.o0
020
040
o~
o.so
lo0
N/Nu
(b) Fig. 10. Interaction curves for temperature linearly varying along the element length but constant through the cross-section (30()°C< T -< 500°C). Limit values are referred t(> maximum temperature.
Interaction domains for steel beam-columns in fire conditions
233
that, while the interaction curves for case (A) (Fig. 8) remain separate for different a, those of Fig. 9 (case (B)) join each other in a zone of small bending. In this zone, in fact, plastic strains develop mainly in the colder fibers, where the temperature is close to 200°C and the model assumes a bilinear behaviour; hence, no significant dependence on a is predicted. Even if far from negligible, the dependence of the interaction curves on a does not appear systematic enough to permit simple approximations. It seems that a numerical simulation can hardly be avoided in this case.
6.2 Temperature variation along the beam length For completeness, also the case of temperature linearly varying along the element length but constant through the cross-section was analysed. The symmetry about both axes of the domains is now restored and only one quarter of them needs to be depicted. Typical results look as those shown in Fig. 10, relevant to temperature interval (A); non-dimensional quantities are now referred to the ultimate values at the maximum temperature. The interaction curves look globally similar to those of the uniform temperature case, even if small quantitative differences in bearing capacity can be detected. These are due to the fact that the maximum bending moment occurs in the mid-span section, while the warmer and hence weaker one is close to a support. By examining a number of solutions, it seems possible to draw the conclusion that the beam-column behaviour is mainly governed by that of the 'critical' section, in which elevated bending moments combine with a strength lowered by temperature. If a criterion to define such a critical section could be introduced, the domain relevant to the uniform temperature case and, possibly, its analytical approximations would apply. In any case, the assumption of a uniform temperature equal to the maximum value does not appear too conservative.
7 CONCLUSIONS The study which has been conducted evidentiates the role played by the slope of the stress-strain curve on the interaction domains of steel beam-columns at elevated temperature. The bilinear approximation is obviously not adequate and provides a significant overestimation of the load-carrying capacity even for small axial forces. On the other hand, the trilinear model used indicates that the interaction curves are strongly sensitive to changes in the slope of the hardening zone, so that a precise
234
L. Corradi, C. Poggi, P. Setti
assessment of the stress-strain behaviour of steel in fire condition is required if accurate results are sought. However, any attempt at representing the behaviour of hot steel in terms of time-independent stress-strain relationships entails approximations, whose influence hardly can be assessed unambiguously, A certain amount of arbitrariness is inherent, and hence it is felt that the trilinear model used represents a drastic but not unreasonable approximation. Some general features of the interaction domains can be detected on this basis. In particular, for uniform temperature conditions simple analytical expressions can be established, reproducing the qualitative characteristics of numerical results and approximating the ultimate values to an acceptable degree of accuracy. Such expressions might be applied also when temperature varies along the beam length only. On the contrary, in presence of thermal gradients through the cross-section, no alternatives to numerical analyses seem to be presently available.
ACKNOWLEDGEMENTS This work was supported by the MPI (Italian Public Education Ministry). The authors wish to acknowledge the contribution of L. Cozzi and P. Di Natale for developing the numerical calculations as part of their graduation thesis.
REFERENCES 1. ECCS-Technical Committee 3, European Recommendations .[or the Fire Safety of Steel Structures. Elsevier, Amsterdam, 1983. 2. Dotreppe, J. C., Prevision par le Calcul du Comportment d l'incendie des Structures Simples. PhD thesis, University of Li6ge, Li6ge, 1976. 3. Petterson, O., Magnusson, S. E. & Thor, J., Fire Engineering Design of Steel Structures. Publication no. 50, Swedish Institute of Steel Structures, 1976. 4. Valente, J. C., The Influence of Different Mechanical Behaviour on the prediction of the Fire Resistance of Steel Structures. lnstituto Superior Tecnico, Lisbon, 1987. 5. Twilt, L. & Witteveen, J., Calculation methods for fire engineering. Design of steel and composite structures. In Design of Structures against Fire, eds R. D. Anchor, H. L. Malhotra & J. A. Purkiss, Elsevier, London, 1986, pp. 155-76. 6. Purkiss, J. A., Developments in the fire safety design of structural steelwork. J. Construct. Steel Res., 11 (1988) 149-73. 7. Burgess, 1. W., E1-Rimawi, J. A. & Plank, R. J., A secant stiffness approach to the fire analysis of steel beams. ). Construct. Steel Res., 11 (1988) 105-20.
Interaction domains for steel beam-columns in fire conditions
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8. Setti, P., Buckling of axially loaded steel columns with imperfections at elevated temperature. In Proceedings of the 3rd International Colloquium on the Stability of Metal Structures, ed. D. Sfintesco, CTICM, 1983, pp. 167-74. 9. Olawale, A. O. & Plank, R. J., The collapse analysis of steel columns in fire using a finite strip method. Int. J. Num. Meth. Engng, 26 (1988) 2755-64. 10. Witteveen, J., Twilt, L. & Bijlaard, F. S. K., The stability of braced and unbraced frames at elevated temperatures, In Proceedings of the 2nd International Colloquium on the Stability of Steel Structures, Liege, 1977, pp. 647-55. 11. Neves, I. C. & Gomes, J. C. P., Fire Resistance of Steel Beam-Columns, Instituto Superior Tecnico, Lisbon, 1987. 12. Twilt, L., Inquiry into the mechanical properties ot structural steel at elevated temperature. ECCS.TC 3/89/4, Internal Report of ECSC-project SA 112, Delft, 1989. 13. RILEM, Behaviour of Steel at High Temperatures, ed. Y. Anderberg, Lund Institute of Technology, Sweden, 1983. 14. Corradi, L. & Poggi, C., A refined finite element model for the analysis of elasto-plastic frames. Int. J. Num. Meth. Engng, 20 (1984) 2155-74. 15. Contro, R., Poggi, C. & Cazzani, A., Numerical analysis of fire effects on beam structures. Engng Comp., 5 (1988) 53-8. 16. Cozzi, L. & Di Natale, P., Analisi della Stabilita' di Membrature Pressoinflesse Soggette all'Azione del Fuoco. Graduation thesis, Politecnico di Milano, Milan, 1989.
Interaction Domains for Steel Beam-Columns in Fire Conditions
L. C o r r a d i , C. Poggi & P. Setti Department of StructuralEngineering,Politecnicodi Milano, Milan, Italy (Received9 April 1990;revisedversionreceived6 July 1990; accepted21 September1990)
ABSTRACT A parametric study on the interaction curves for steel beam-columns in fire conditions is performed. The study is based on a trilinear idealization of the uniaxial stress-strain relation for steel at elevated temperature and aims at assessing the influence of the slope of the hardening branch, on which results are strongly dependent. A refined and numerically efficient elastic-plastic beam element model is employed to analyse different load conditions and temperature distributions. Results for uniform temperature are systematic enough to permit fairly general analytical approximations, which might be useful also in the presence of temperature variations along the element length only. Thermal gradients through the cross-section are also considered.
NOTATION A E Et M m N n S
Cross-sectional area Young modulus Tangent modulus of the hardening branch Bending moment Dimensionless bending moment Axial thrust Dimensionless axial thrust Cross-sectional plastic modulus 217 J. Construct. Steel Research 0143-974X/91/$03.50 © 1991 ElsevierSciencePublishers Ltd, England. Printed in Great Britain
L. Corradi, ('. Poggi, P. Setti
218
o
(rp A
Dimensionless slope of the hardening branch Normal stress Ultimate yield stress First yield stress level Slenderness ratio
Subscripts
( )o ( )p ( )s
R o o m t e m p e r a t u r e values First yield level Ultimate yield level Euler load Shanley load
1 INTRODUCTION At elevated t e m p e r a t u r e the stress-strain characteristics of structural steel deteriorate rapidly and the influence of creep b e c o m e s of importance. However, for structural analyses in fire conditions, typically limited to a few minutes and with t e m p e r a t u r e s not exceeding 600°C, it is widely accepted that viscous effects can be accounted for only implicitly, by introducing 'equivalent' t i m e - i n d e p e n d e n t stress-strain curves, usually defined on the basis of experiments on uniaxial specimens u n d e r constant stress and increasing t e m p e r a t u r e (or vice versa). Obviously, results arc strongly d e p e n d e n t on test conditions, in particular on the rate at which loads are applied and/or t e m p e r a t u r e is increased. It follows that the definition of generally accepted stress-strain laws is a difficult problem, which must still be considered as open. T h e simplest possible idealization of uniaxial steel behaviour is the bilinear, perfectly plastic one. It only requires the definition of two material properties, namely the Young's m o d u l u s E and the yield stress ~r~. T h e y must be given as a function of t e m p e r a t u r e if this model is to be used in fire analyses. As an example, the following expressions p r o p o s e d by ECCS,~ are recalled cry = %o(1 + T/(767 In (T/1750)))
(la)
E = Eo(l + (15.9 × I(Y 5 T) - (34.5 x lt)- 7 T 2) + (11-8 x lt) -'~ T 3)
- ( 1 7 - 2 x 10 12 7,~))
(lb)
In the above relations, O-yo and Eo indicate the yield stress and Y o u n g m o d u l u s at r o o m t e m p e r a t u r e , and T is m e a s u r e d in the Celsius scale.
Interaction domains for steel beam-columns in fire conditions
EE200
i
219
206 / ~ 2 0 6
250275300 350 tOO
150'
5O
0.1
0.2
0.3 0.4 0.5 strain E (%)
0.6
Fig. la. Stress-strain curves of steel at elevated temperature (ECCS proposal).
1
~
2
0
°C 2OO
o
--
25O
300 40O 50O
0.5
--
0.1
0.2
o.3
0.4 (%)
o.5
600
0.6
Fig. lb. Trilinear idealization of the ECCS curves.
Even if drastically oversimplified, such a model is acceptable when deformation does not play a significant role and an assessment of the load-carrying capacity is only soughtf1-4 In this case the problem can be solved by limit analysis and results only depend on the material yield stress and on its decay with temperature. However, the relationships which have been proposed to model in terms of time-independent stress-strain curves the behaviour of steel at elevated temperature, depart significantly from the bilinear model. Fig. la shows the original proposal by ECCS.1 The initial slope and the ultimate stress level are still given by eqns (la) and (lb), but the amplitude of the linear range is dramatically reduced and a significant decay in stiffness occurs afterwards. Even if geometry changes can be neglected, the actual shape of the stress-strain curve must be considered when deformations are of
220
L. Corradi, C. Poggi, P. Setti
interest. Moreover, the decrease of stiffness with temperature emphasizes geometric effects; in this circumstance collapse usually occurs because of loss of stability of the partially yielded structure and, as is well known, the shape of the stress-strain curve, in particular the value of the tangent modulus Et, plays a key role. This aspect was recognized by a number of authors s'(' and fire analysis methods were recently proposed accounting for the non-linear nature of the stress-strain curve. The object of such analyses is the definition of deformations up to collapse for negligible geometry c h a n g e s ] the evaluation of the load-carrying capacity of compressed columns s') and the assessment of the global behaviour of framed structures.m A problem which has attracted little attention is the ultimate capacity of beamcolumns. ~j This paper is intended as a contribution in this direction. As previously mentioned, the curves of Fig. l a did not find general agreement. A n u m b e r of alternatives have been suggested by different boards (Rilem, BSC, RUB/KFI, A R B E D , CRIF, etc.) and are summarized in Ref. 12. Even if all were based on reasonable assumptions, differences are often significant and the definition of h o m o g e n e o u s proposals is presently the object of discussion. ~2.J3 In this situation it is felt that the comparative merits of different models can hardly be established by simply contrasting the numerical results produced on their basis. In fact, it would be difficult, if at all possible, to assess the quantitative influence of the individual assumptions leading to each of the curves proposed. Of greater interest seems to be a parametric study aiming at the identification of those features of the stress-strain relation which play the most significant role with respect to critical situations. Such a study can be based on drastically simplified models, depending on a limited n u m b e r of parameters, whose role can be assessed unambiguously. In this paper the problem of defining the interaction curves for steel beam-columns of different slenderness at elevated temperatures is tackled. A parametric study is performed, aimed at assessing the influence of the shape of the stress-strain curve on the resulting domains in the transverse load-axial force plane. The uniaxial behaviour is idealized as a trilinear diagram, depending on four parameters which could be adjusted to approximate most of the laws which have been suggested. However, no particular reference to existing proposals is made. The main object of this study is to detect the role played by the slope of the hardening branch, which is felt to be the most significant parameter, and the proposed model is well suited to this purpose. In some instances, results look systematic enough to permit analytical approximations, which can be regarded as a first step toward the definition of design formulas.
Interaction domains for steel beam-columns in fire conditions
221
The study has been performed numerically. Computations are based on the finite element model proposed in Refs 14 and 15, able to account with satisfactory accuracy for the spreading of local plasticity and to identify correctly the critical situation, corresponding either to collapse or to Shanley-type bifurcation. The efficiency of the model was verified by comparison with experimental results carried out on a simply supported beam. 15
2 THE TRILINEAR M O D E L Consider the trilinear stress-strain diagram depicted in Fig. 2. It depends on four material parameters, namely the initial slope E, that of the hardening branch Et = aE, the first yielding stress level O-p and the ultimate stress O-y. Obviously they are all temperature-dependent quantities. It is easily recognized that for the purpose of a parametric study, the above quantities have different importance. Note first that for E t --- 0 (a = 0) and Et = E (a = 1) two limiting situations are recovered in which the trilinear behaviour reduces to bilinear ones with yield stress O'p and O-y, respectively. In these situations, the main role is played by the ratios of the actual slenderness to the following critical values Ap = 7r(E/crp) 1/2 Ay = "tr(E/o'y) 1/2
(2a,b)
If the above values were constant with temperature, so also would be the dimensionless quantities defining the interaction curves. In this case, at least in uniform temperature conditions, the design formulas available for
0
IFtl I I
'
_~_aL_
IL E
Fig. 2. Trilinear stress-strain model.
222
L. Corradi, C. Poggi, P. Setti
room temperature can still be used, simply by referring them to the current values of material properties. Most of the proposed stress-strain relations imply that the critical slendernesses (eqns (2a) and (2b)) are actually very nearly constant, at least for temperatures up to 500°C (both in the original ECCS curves ~and in recent proposals ~~the scatter between maximum and minimum values is less than 10%). In this circumstance, only two parameters effectively survive; they are given by the following dimensionless quantities n p = ~rp/~r,~ a
-
-
E~/t:
(3a.b)
The first of them, however, has only a scaling effect: it widens or narrows the domain in which the trilinear behaviour is contained, without affecting its qualitative features. Hence, the trilinear model is suited for a parametric study aiming to evaluate the influence of the slope a (eqn (3b)) of the hardening branch on the interaction curves. The ECCS curves of Fig. la can be approximated with acceptable accuracy by a trilinear diagram (see Fig. lb). It is stressed, however, that in this study no reference is made to particular curves and the values of the parameters used are not dictated by specific proposals. The model employed is defined as follows. - - T h e initial elastic modulus E and the limit stress cr~ are expressed as functions of temperature by eqns (1 a) and (lb). - - A perfectly plastic behaviour is assumed up to T = 200°C. - - F o r T->300°C, the hardening branch starts at a given value of rip = crp/Cry, usually taken to be 0.5. Different values of the dimensionless slope c~ in a wide range are explored. - - I n the interval 2t)0°C~ T < 3 0 0 ° C , the first yield level np varies linearly between 1.0 and 0.5 (or the other value assumed at higher temperatures).
3 S U M M A R Y OF T H E A N A L Y S E S P E R F O R M E D A first set of computations was intended at assessing the influence of the l o a d - t e m p e r a t u r e history on the interaction curves. A n u m b e r of beamcolumns were analysed at constant temperature, by increasing the mechanical loads up to collapse. Subsequently, the final values of loads were applied on the cold beam and temperature was increased, until collapse, due to decay of material properties, was predicted. In all the cases tested, the ultimate situation turned out to be rigorously the same, in accordance with the well-known independence of the interaction curves at
Interaction domains for steel beam-columns in fire conditions
223
room temperature from the history of mechanical (transverse and axial) loads'. Due to this fact, only the case of temperature constant with time was considered, a condition which simplifies considerably the numerical analyses, since iterations to updatethe material properties are avoided. The influence of the sectional shape was also explored and found to be minor. A number of tests on I-beams and rectangular cross-sections produced dimensionless interaction curves which, under the same conditions, exhibited differences of less than 5%. In subsequent computations, only HE200M shapes were examined. Numerical analyses were referred to simply supported beam-columns subjected to three different load conditions, namely: (1) Equal concentrated couples at both ends (constant bending moments in the absence of second-order effects). (2) Concentrated couple at one end only. (3) Uniform transverse load. For each of them, the following temperature distributions were considered: (a) Uniform temperature in the member. (b) Temperature linearly varying through the thickness and uniform along the length. (c) Temperature varying linearly along the beam length and constant over the thickness. The analyses were developed by two students within the framework of their graduation thesis 16 and the complete set of results is collected in it. In this paper only those relevant to load condition 1 are reported; in any case, the effects of different transverse loads were found to be analogous to those experienced at room temperature.
4 I N T E R A C T I O N C U R V E S FOR U N I F O R M T E M P E R A T U R E An interaction curve defines the admissible domain for a beam-column of given slenderness A in the m - n plane, where m is the value of the applied external moment and n the axial force, divided by the relevant ultimate values M u and Nu at the temperature considered m = M/Mu = M/Soy
n = N/Nu = N/Atry
(4a,b)
In eqns (4a) and (4b), A and S are the area and plastic modulus of the cross-section and try is given as function of temperature by eqn (la). All curves must intersect the two axes at the ultimate bending value
L. Corradi, C. Poggi, P. Setti
224
O/oy / I
ns= Oy ',
~ I
,, I
V
~
_O'cr,
e
''e - Oy I
I
I
ko k,
t
~j,
kp
Fig. 3. T h e o r e t i c a l c o l u n m model for trilinear material.
m = 1 and at the critical load for the slenderness considered. [:or an ideally straight column m a d e up of a trilinear material, the theoretical column curve is shown in Fig. 3: it is obtained by joining two horizontal segments at the % and o-y levels with two cubic hyperbolas, corresponding to the Euler and Shanley critical loads, respectively. In dimensionless form, these are expressed as follows n E = 7T2E/cryA 2
n s = "n"2Et/crv A2 = oenr~
(Sa,b)
The interaction curves for the trilinear model will be contained in the area b o u n d e d by those obtained in the limiting situations a = (1 and ~e = 1, the shape of which is little affected by t e m p e r a t u r e , as expected, since the critical slenderness values eqns (2a) and (2b) are very nearly constant. U p to T = 200°C the bilinear model a = 1 applies and the interaction curve coincides with the upper bound. Typical results for higher t e m p e r a t u r e s (trilinear behaviour) are illustrated in Fig. 4, where the limiting curves are also shown. The scatter with respect to m = 1 on the ordinate axis is due to the approximations entailed by the finite e l e m e n t model used; ~4 on the other hand, the theoretical critical load is exactly r e c o v e r e d for purely compressed members. In some instances (see, e.g., Fig. 4a for a = 1/5) small m values apparently produce a strong increase in the axial capacity. Such an effect shows up for a small range of a values only and must not be considered as meaningful. The analysis p r o c e d u r e used stops when either a bifurcation is predicted or a limit point is reached. The first alternative occurs in the purely compressed column, whose carrying capacity coincides with Shanley load. The presence of bending m o m e n t s eliminates bifurcation and, due to the rising nature of the bifurcated path in Shanley-type buckling, computation continues up to a limit point. H o w e v e r , the increase in load-carrying capacity is achieved at the price of deflections which grow extremely rapidly from n -~ ns. For
Interaction domains for steel beam-columns in fire conditions
225
1.00
~ • 1/20 IX=1/10
0.80
~, =70
=1/$
T • ~o, c
0.60
,"v=1 0.40
Or=0 0.20
0.00 0.00
0.20
0.40 ).60 NINu
0.80
1.00
(a) 1.00
~fOt =I120 0.80 ~ O ~ . 1 1 1 0 0.60 |~~/A~ , ~
'~
,~ -'30 T gl~0° ~
a'1/5
o~:I
a00
ai0
a~0
a~0
a/~
1.60
NINu
(b) Fig. 4. Interaction curves for beam-columns with [filinear material model.
practical purposes, this value should be considered as the limit axial capacity. A closer look at the numerical results permits the subdivision of the interaction curves for slender members in two different zones, indicated in Fig. 5 with (a) and (b), respectively. In zone (b) collapse occurs when all plastic points are still on the hardening branch, i.e. for o-< O'yeverywhere.
226
L. Corradi, C. Poggi, P. Setti
1.0
k
(a)
\ np
(15
Ns
np
Fig. 5. Proposed analytical approximation of the interaction domains. In this case the horizontal plateau has no role and identical results would be p r o d u c e d by a bilinear hardening model. In zone (a), on the other hand, some points have reached the stress level Cr = O'y. For sufficiently low slenderness values and/or sufficiently high a, the entire interaction curve belongs to zone (a). When both zones are present, the b o u n d a r y b e t w e e n them is closely approximated by a straight line, indicated by r in Fig. 5, which b e c o m e s s t e e p e r with increasing slenderness. These features are the basis for the proposal advanced in the s u b s e q u e n t section.
5 ANALYTICAL APPROXIMATION CURVES
OF THE INTERACTION
Consider the domain defined by the following inequalities A. ns < nr,:
B. n~>>-np:
m <- ( 1 - n/ns)( 1 - n/nF)
if n -< ns( 1 - np)
(6a)
m -< np(np - n)(1 - n/nE) n p - ns(l - n p )
if n-> ns(1 - n p )
(6b)
m~<(l-n/ns)(1-n/nE)
(7)
where nE and ns are the dimensionless Euler and Shanley loads defined by eqns (5a) and (5b).
Interaction domains for steel beam-columns in fire conditions
227
The domain (A) applies when the slenderness exceeds the value )tl = X/-~-aAp in Fig. 3. In this case the theoretical critical load is either np()t<-)tp) or nE(A-->)tp). Equations (6a) and (6b) approximate the interaction domain in zones (a) and (b) in Fig. 5, respectively. It can be readily verified that the two domains intersect on a straight line similar to the one labelled as r in Fig. 5, defined by the equation m/np .4- n/nE = 1
(8)
The domain (B), on the other hand, is relevant to the interval )to -< )t -< )tl in Fig. 3 ()to = V~--a)ty), in which only zone (a) is meaningful. The theoretical critical load is now given by ns. It is assumed that eqn (7) applies also for )t < )to, when the dimensionless critical load becomes equal to one. To understand the meaning of eqns (6a) and (6b), consider an ideal case in which nE is brought to infinity by keeping constant the values of the remaining parameters. The interaction domain becomes th,e union of those bounded by the two dashed straight lines in Fig. 5, the first connecting points (n = 0, m = 1) and (n --- ns, m = 0), the second intersecting the first for m numerically equal to np and the n-axis for the theoretical critical load, now also equal to np. The influence of the slope of the hardening branch is incorporated in the Shanley load ns, proportional to a, which affects both segments and reduces the size of the domain with decreasing a. Equations (6a) and (6b) are obtained by multiplying, in the equations of the above straight lines, m by the amplification factor 1/(1 - n/nE); the bounds of the domain are no longer straight and exhibit a concavity which, in agreement with numerical experience, increases with slenderness. Moreover, the approximated interaction curves now intersect the n-axis also for n = hE, i.e. for the theoretical critical load of sufficiently slender members. The above reasoning illumines how eqns (6a), (6b) and (7) were arrived at, but is by no means evidence of agreement with numerical results. Typical comparisons are illustrated in Fig. 6. The approximation is systematically conservative for small n, reasonable even if not always conservative when bending moments are small and compression dominates. Note, however, that the interaction domain in this zone would reduce considerably in size if the intersection with the n-axis were given not by the theoretical critical load of Fig. 3 but by a design value, say no, accounting for geometrical and mechanical imperfections, nc must replace n o in eqn (6b) and ns in eqn (7). The approximated domains become A.
o~/'/E ~-~ nc:
m --<(1 -- n/t~nE)(1 -n/nE)
if n --
(9a)
228
L. Corradi, C. Poggi, P. Setti
rn~
nc(n c - n)(1 - n/nE) n~
-
if n-> anE(1 - n , . )
anF(1 - nc)
(10)
m-<(1-n/n~.)(1-n/nF)
B. anE>-nc:
(gb)
Clearly, neither eqns (6a), (6b) and (7) nor eqns (9a), (9b) and (10) must be considered as proposals for design recommendations. They can only be regarded as a first step in this direction, in that they permit the 1.00 -
0.80-
~,=
70
-, 0 ` 6 0 -
',,\%
z~ 0`/.0-
"A 0.20
0`00 0`00
1.00
0.80
0.20
0,40 0.60 N/Nu
0.60
1.00
(a)
~
~, =130
o:=115
=
0.80
T =300° C
O.6O
0.20
0`00 0`00
i
o~zo
i
o.4o
o.~o
o.~o
t6o
N/Nu
(b) Fig. 6. Proposed approximation (dashed line) versus computed domains.
Interaction domains for steel beam-columns in fire conditions
229
representation of the main features of the interaction curves on the basis of a limited n u m b e r of structural parameters. Note, however, that for establishing such relations, the trilinear nature of the stress-strain diagram is essential, since only in this way is it possible to consider the Shanley critical load ns = anE as a structural datum. Extensions to general, non-linear stress-strain relations are not possible, unless an equivalent a is defined. As a final remark, note that in the present situation temperature plays only an indirect role, limited to the definition of the values of structural parameters. Hence, results hold also in different contexts, whenever the material behaviour can be reasonably idealized with a trilateral diagram. This might be the case for aluminium members, if the replacement of the customary R a m b e r g - O s g o o d law with the one depicted in Fig. 2 is considered acceptable.
6 NON-UNIFORM TEMPERATURE The results obtained by considering the temperature distributions indicated by (b) and (c) in Section 3 are now summarized. Temperature varies linearly, either over the cross-section or along the beam length. Two different intervals were studied, namely (A) 300°C --- T <- 500°C (B) 200°C <- T < - 400°C In the first case the stress-strain relationship is described everywhere by a trilinear curve, whereas in the second the transition between bilinear and trilinear behaviours is present and its effects can be investigated. 6.1 Temperature variation over the cross-section
As a preliminary step, the temperature interval (A) was studied with reference to the bilinear law a = 1. The results, shown in Fig. 7, are divided by the ultimate values at the average temperature Ta -- 400°C. Because of the non-homogeneous material properties, symmetry about the n-axis is lost and the ultimate values for axial load and bending m o m e n t in the cross-section (h = 0) decrease by about 20% and 10%, respectively. However, a small positive bending m o m e n t (causing compression on the colder edge) is favourable in terms of ultimate axial load because the fibres which undergo the higher compressive stress have a greater strength.
L. Corradi, C. Poggi, P. Setti
230
o.
oo
~.6o
o.do
o.oo
-o.so
-~.'oo
M IMu Fig. 7. T e m p e r a t u r e variation o v e r the cross-section (300°C T -< . 0 0 C). I n t e r a c t i o n curves for bilinear b e h a v i o u r ( a = 1).
N/Nu ~,
=70
300°C .I.O0
Ta =4oo°c Ot=l
,.~o
~-',~k//-,~
S
=.,o
~ = , t 2 0
:'~o
o.oo M/Mu
-Lso
-1.00
(a) N/Nu 30D~
~, : 130
.I.00
Tit =
400,, c
500°C
-0.50
,.~o
o.~o
o.oo
~ O t
~,.//---
-o~so
ot=1 =115
CX= 1110
-,200
MIMu (b) Fig.
8.
I n t e r a c t i o n curves for t e m p e r a t u r e v a r i a t i o n o v e r the cross-section (. 00 C T ~ 500°C). Limit values are r e f e r r e d to average t e m p e r a t u r e .
Interaction domains for steel beam-columns in fire conditions
N/Nu
,~ : 7 0
"1.00 41111 ~;
.
~
Ta :3ooec 0
~
~\VFa
,.~o
~0
231
~oo
M/Mu
Ot:l : llS : I,lo
-ds0
(a)
N/Nu
('~-----~
.,_.:~ [,.oo
~,: ,3o r . : 30,. c
t.O0 *C *C t.O0 0.50
1.O0
0.50
0.00
M/Mu
~c~ : 1 / ~0~ : 115 O: : 1110 =1120
- 0.50
-1. O0
(b) Fig. 9. Interaction curves for temperature variation over the cross-section (200°C-< T-< 400°C). Limit values are referred to average temperature. For A > 0 the shape of the interaction curves changes for various reasons. The predominant one seems to be the interaction of the axial load with the elastic curvature due to bending moments, the curvature due to temperature and that caused by plastic deformations. The latter may even become opposite to the thermal contribution because of the lower ultimate strength of the warmer fibres. The shape of the domains is also affected by the presence of self-equilibrating stresses over the cross-section, which is subjected to a linear temperature variation and hence, as eqn (lb) shows, no longer has a uniform elastic modulus. The finite element model used 15 allows these effects and their influence on the interaction curve to be taken into account. The results obtained for the two temperature intervals (A) and (B) are shown in Figs 8 and 9 for different values of a and A. Dimensionless quantities are still referred to the average temperatures of 400°C and 300°C, respectively. The slope of the hardening branch has a marked influence on the domains, which becomes smaller with decreasing a. Note
L. Corradi. C. Poggi, P. Setti
232
1.00 Ot = 115 = 1110 Ot =1120
0.80
/ ! , = 70
•
\ ~
Ta: ~.o0"c
0.60
3; o.z.o 0.20
0.00 0,o0
o.zo
o.~o 0.6o N/Nu
0.80
~.oo
(a)
1.00 ~ j / .
_Or--115 O~=1110 O~ : 1/20
0.80
,~ 130
O. 60
z~
0.40
0.20
0.o0 0.o0
020
040
o~
o.so
lo0
N/Nu
(b) Fig. 10. Interaction curves for temperature linearly varying along the element length but constant through the cross-section (30()°C< T -< 500°C). Limit values are referred t(> maximum temperature.
Interaction domains for steel beam-columns in fire conditions
233
that, while the interaction curves for case (A) (Fig. 8) remain separate for different a, those of Fig. 9 (case (B)) join each other in a zone of small bending. In this zone, in fact, plastic strains develop mainly in the colder fibers, where the temperature is close to 200°C and the model assumes a bilinear behaviour; hence, no significant dependence on a is predicted. Even if far from negligible, the dependence of the interaction curves on a does not appear systematic enough to permit simple approximations. It seems that a numerical simulation can hardly be avoided in this case.
6.2 Temperature variation along the beam length For completeness, also the case of temperature linearly varying along the element length but constant through the cross-section was analysed. The symmetry about both axes of the domains is now restored and only one quarter of them needs to be depicted. Typical results look as those shown in Fig. 10, relevant to temperature interval (A); non-dimensional quantities are now referred to the ultimate values at the maximum temperature. The interaction curves look globally similar to those of the uniform temperature case, even if small quantitative differences in bearing capacity can be detected. These are due to the fact that the maximum bending moment occurs in the mid-span section, while the warmer and hence weaker one is close to a support. By examining a number of solutions, it seems possible to draw the conclusion that the beam-column behaviour is mainly governed by that of the 'critical' section, in which elevated bending moments combine with a strength lowered by temperature. If a criterion to define such a critical section could be introduced, the domain relevant to the uniform temperature case and, possibly, its analytical approximations would apply. In any case, the assumption of a uniform temperature equal to the maximum value does not appear too conservative.
7 CONCLUSIONS The study which has been conducted evidentiates the role played by the slope of the stress-strain curve on the interaction domains of steel beam-columns at elevated temperature. The bilinear approximation is obviously not adequate and provides a significant overestimation of the load-carrying capacity even for small axial forces. On the other hand, the trilinear model used indicates that the interaction curves are strongly sensitive to changes in the slope of the hardening zone, so that a precise
234
L. Corradi, C. Poggi, P. Setti
assessment of the stress-strain behaviour of steel in fire condition is required if accurate results are sought. However, any attempt at representing the behaviour of hot steel in terms of time-independent stress-strain relationships entails approximations, whose influence hardly can be assessed unambiguously, A certain amount of arbitrariness is inherent, and hence it is felt that the trilinear model used represents a drastic but not unreasonable approximation. Some general features of the interaction domains can be detected on this basis. In particular, for uniform temperature conditions simple analytical expressions can be established, reproducing the qualitative characteristics of numerical results and approximating the ultimate values to an acceptable degree of accuracy. Such expressions might be applied also when temperature varies along the beam length only. On the contrary, in presence of thermal gradients through the cross-section, no alternatives to numerical analyses seem to be presently available.
ACKNOWLEDGEMENTS This work was supported by the MPI (Italian Public Education Ministry). The authors wish to acknowledge the contribution of L. Cozzi and P. Di Natale for developing the numerical calculations as part of their graduation thesis.
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8. Setti, P., Buckling of axially loaded steel columns with imperfections at elevated temperature. In Proceedings of the 3rd International Colloquium on the Stability of Metal Structures, ed. D. Sfintesco, CTICM, 1983, pp. 167-74. 9. Olawale, A. O. & Plank, R. J., The collapse analysis of steel columns in fire using a finite strip method. Int. J. Num. Meth. Engng, 26 (1988) 2755-64. 10. Witteveen, J., Twilt, L. & Bijlaard, F. S. K., The stability of braced and unbraced frames at elevated temperatures, In Proceedings of the 2nd International Colloquium on the Stability of Steel Structures, Liege, 1977, pp. 647-55. 11. Neves, I. C. & Gomes, J. C. P., Fire Resistance of Steel Beam-Columns, Instituto Superior Tecnico, Lisbon, 1987. 12. Twilt, L., Inquiry into the mechanical properties ot structural steel at elevated temperature. ECCS.TC 3/89/4, Internal Report of ECSC-project SA 112, Delft, 1989. 13. RILEM, Behaviour of Steel at High Temperatures, ed. Y. Anderberg, Lund Institute of Technology, Sweden, 1983. 14. Corradi, L. & Poggi, C., A refined finite element model for the analysis of elasto-plastic frames. Int. J. Num. Meth. Engng, 20 (1984) 2155-74. 15. Contro, R., Poggi, C. & Cazzani, A., Numerical analysis of fire effects on beam structures. Engng Comp., 5 (1988) 53-8. 16. Cozzi, L. & Di Natale, P., Analisi della Stabilita' di Membrature Pressoinflesse Soggette all'Azione del Fuoco. Graduation thesis, Politecnico di Milano, Milan, 1989.