Thin-WalledStructures20(1994) 175-188 ~) 1994Elsevier Science Limited Printed in Great Britain. All rights reserved 0263-8231/94/$7.00 ELSEVIER
Probabilistic Approach to Elastic-Plastic Coupled Instability
J. W . M u r z e w s k i Cracow University of Technology, Warszawska 24, PO-31-155, Cracow, P o l a n d
ABSTRACT Interaction of elastic and plastic instability modes is considered. Buckling resistance of columns is treated as the lesser value of plastic resistance Rpl of a critical cross-section and the Euler critical force R,.r. The random variables Rpt and R,.r have log-normal probability distributions. Their coefficients of variation are supposed equal and independent from the slenderness ratio 2 in the first model of random instability. A simple formula for the buckling factor is derived thanks to the introduction of equivalent Weibull distribution functions. Strict formulae are derived in the second model of coupled instability for the columns of equal median values Rpt = R,.r when the relative slenderness ratio of the columns 2 = 1. The design values Rd of the elastic-plastic resistance are reduced by an analysis factor C which depends on the slenderness 2 and the imperfection class, a, b, c or d, defined for European multiple buckling curves. Optimal values of resistance factor YR and analysis factor ?c are determined so that the risks of exceeding the design values Rd and Ca are equal.
NOTATION
f f(.)
h(.) k n t X
Yield strength Probability density function Hazard function Safety class (1/k = hazard ratio) Imperfection parameter of the Polish standard Standardized variable Coefficient of variation Basic variable 175
176
A E
f(.) 1 M N R
S W X
J. W. Murzewski
Area Young's modulus Cumulative distribution function Moment of inertia Bending moment Compressive force Resistance Load effect Section modulus Coordinate of state
Greek
2 V o
q) ,p(. )
4)(.) Z
Imperfection parameter of the Eurocode Sensitivity factor Safety index Partial safety factor Slenderness ratio Logarithmic coefficient of variation Weibull coefficient of variation Buckling factor according to probabilistic formula Gauss function Mills function Laplace function Buckling factor according to Airton-Perry formula
Subscripts a Analysis Buckling b Relative to model uncertainty C cr Critical (relative to elastic instability) d Design i...n Numbers of basic variables Characteristic k pl Plastic (relative to plastic collapse) R Relative to resistance Relative to action S Other symbols Mean value ~" Median value ~" Central value of Weibull distribution
Probabilistic approach to elastic-plastic coupled instability
177
1 SIMPLE MODEL OF COUPLED INSTABILITY The instability of structural members or systems is treated as a logical sum of two r a n d o m events: elastic buckling a n d / o r plastic collapse. The r a n d o m resistance R is equal to the lesser of two r a n d o m variables: R -- min (Rcr, Rpl)
(1)
where Rcr is the critical resistance for an ideally elastic member, and Rvt is the limit resistance for a rigid-perfectly plastic member. Figure 1 gives an idea of how the elastic and plastic instabilities are understood in the case of a column under compressive force Ns. The plastic hinge at the middle cross-section of the column is the most probable, but it can happen in another cross-section because the yield strength f is a stochastic function of the coordinate z. Let the column have a crosssection with area A, m o m e n t of inertia I and buckling length L. The r a n d o m resistance N is defined as follows: N = min (Ncr, Npl)
(2)
The rules in eqns (1) and (2) are obvious from the deterministic point of view and they should always give safe results, but they do not. Usually, residual stresses and geometric imperfections are supposed to be those which reduce the ultimate limit force R. However, some reduction of the m e a n resistance R occurs even if there are no imperfections. A reduction happens when the variables Rcr and Rpl are treated as r a n d o m variables. Such an effect was shown first during the Stokes Session at the International Center of Mechanical Sciences in Udine, Italy, in 1973, and it was
Nor ~Npt
I~/Np, Fig. 1
J. [4'. Murzewski
178
extended later on to the lateral-torsional buckling of beams,~ sway frame instability 23 and shell instability. 4 The derivation has been given under following assumptions: the random variables Rcr and Rpl are independent,
;~(Rcr _< R, Rpl < R) --.~(Rcr < R) × ~(Rpl < R)
(3)
their COVs (coefficients of variation) are equal, vc,. = V p l - v
(4)
Two additional assumptions are taken for simplicity's sake: -the probability distributions are the Weibull ones,
(5)
;~(Rcr
(6)
Lv/A-/I--slendernessratio.
The central values /~cr, /~pl are fractiles of probability e -~. The Weibull distribution functions (eqn (5)) are applied because they are stable with respect to the least value of any sequence o f random variables. The fact that the Weibull distribution belongs to asymptotic distributions of extreme values has no special significance here. The probabilities of two independent random events (Rcr > R, and Rpl > R) are multiplied and the probability of the minimum value (eqn (1)) is derived :~(Rcr > R).-~(Rpt > R) -- [i - F~r(R)]. [1 - F(R)]
Thus
F(R) =
1 - exp
-
with 1~ =
(Rcrl/"+ Rpl ).
(8)
The Weibull central value R may be expressed in another form:
R = Rp~
1+
(9)
where Jt is the relative slenderness ratio. Its general definition is i = ~/Rpt//~cr
(10)
Probabilistic approach to elastic-plastic coupled instability
179
and in the case of columns 2 = (2/r0V/7//). Reduction of central values of elastoplastic resistance always happens: /~ < min (/~cr,/)p0
(11)
W. J. M. Rankine in 1858 gave an empirical formula for slender columns, which indicates a remarkable reduction of its resistance N:
1/N--
1/Ncr +
1/Np,
(12)
W. Merchant proved 100 years later that eqn (12) may be applied to frames and it is always conservative in comparison with the experimental results; Nor is the instability limit of a perfectly elastic frame and Npl is the carrying capacity of a rigid~plastic frame in Merchant's considerations. Equation (12) is a particular case of the probabilistic formula (eqn (8)). It always gives safe solutions because the exponent o = 1 is much more than its real value. Equation (8) was adopted by the new Polish standard in the design of steel structures 5 for both column buckling and lateral-torsional buckling of beams. The design resistance is defined as follows: N~ =J~A~p--for columns;
M~ =fdW~O--for beams
(13)
where fd =f/YR is the design strength. The resistance factor 7R relative to the median value j~ should not be mistaken for the conventional material factor 7M relative to the characteristic value fk. The Polish buckling factor ~o is defined according to the probabilistic formula (eqn (9));
q9 = 1/~+~22",
withn = 1/o
(14)
The exponent n = 2.5, 2.0, 1.6, 1.2 is specified for multiple buckling curves: a0, a, b and c, respectively. The curves a, b, c are relative to column buckling and the curves a0 and a to beam buckling. The relative slenderness ratio ~t of the Polish standard 5 is equivalent to ,~ (eqn (10)) if the COVs of elastic and plastic limits of resistance are equal, Vcr = vpl--according to assumptions (eqn (4)): = 1.15(2/rt)~
=
(2/rc)~/1.33fd/E
= ~.
(15)
The value j~ depends on the thickness of tested pieces. The value = 1.33 is a good average of statistical estimates for structural steels. 6 The central resistance factor YR for elastic instability is supposedly the same as it is for plastic strength, /~cr/Rd --~ 1-33 when 2 ~ oo. The values of buckling factors ~o according to the probabilistic formula (eqn (9)) may be close to the values of reduction factors Z from Eurocode
f/fa
J. W. Murzewski
180
No. 3, 7 but the theoretical models are quite different. A more complicated Airton-Perry equation is taken as the basis o f the European factors: Z = 1/(q5 + ~ - ~ ) ,
with ~b = [1 + ~(2 - 0.2) + 22]/2
(16)
= 0.21,0.34, 0.49, 0.76--relative eccentricity for buckling curves a, b, c and d, respectively. The relative slenderness ratio 2 according to the Eurocode No. 3 is less than the Polish ratio 2 for the same column: = (2/rc)~
=
(;t/rc)~/1.1j'/1.33E=
211.1
(17)
The central resistance factor for elastic buckling 7R = / ~ c r / R d ---- 1-1 is less than 1.33 for plastic resistance. The factor 7R = 1.33 for columns of small slenderness, when 2 ~ 0, decreases for columns of 2 > 0 and tends to 1.1 when 2 -~ oc for very slender columns, which does not seem prudent. The numerical values ~ of the European buckling curves are unrealistic; they are too large. However, ~ is treated as an equivalent imperfection parameter and it has no direct meaning. It replaces the effect of residual stress, which is the predominant imperfection influencing the behaviour of columns in buckling. The imperfection parameters n of the Polish buckling curves are too small in comparison with statistical values l/v, but an advanced model of elastic-plastic instability enables one to make the theory compatible with the experiment. Table 1 presents values of n calibrated in agreement with the European classification of column imperfections. The values of n are determined so that the reduction factors q~l from eqn (14) for ~. = 1 and Zt from eqn (16) for 2 = i / 1-1 are equal. TABLE 1 Imperfection class
a
h
c
d
~P, = Z~ n
0.728 2.18
0.650 1.61
0.594 1.31
0.515 1.04
2 ADVANCED MODEL OF COUPLED INSTABILITY Let a r a n d o m correction factor C cover the uncertainties o f the theoretical model of buckling: R = min (Rcr, R p l ) / C
(18)
Probabil&ticapproach to elastic-plastic coupled &stability
181
C may explain the excessive reduction of buckling resistance, R, which is more than would follow from the simple model (eqn (1)) if real values of u were introduced into the buckling resistance formula (eqn (9)) instead of the artificial values of l/n of eqn (14). Column buckling tests show that scatter of resistance depends on slenderness 2 of the columns and the assumption of eqn (6) is not valid. Columns of medium slenderness 2 exhibit remarkable increase of COV v. The maximum COV, max v(2) -- vl, occurs when the median critical resistance /~cr equals the median plastic resistance/~pl =/~0, then the relative slenderness ratio ~.1----1 and ~ l - - 1 / 1 . 1 . Empirical values of standard deviation s and COV v -- S/NR for a sample of similar steel columns under axial force Ns are given in Fig. 2. The laboratory tests were performed by Gw6~.dt in the Cracow University of Technology, in Poland. 8 Random deviations of buckling resistance of columns of the same steel and imperfection class, but of different origin, are greater in real constructions than in laboratory tests. Available data were recorded by Fukumoto and Itoh. 9 Figure 3 shows an example of their results for columns which belong to imperfection class (b) of the European standards. The factor C is treated here as a log-normal random variable with
10
20
SR[MPa ] I
10
50
100
150
50
100
150
Fig. 2
|
R [MPa]
19/ ._..~1 ,
L°o:ot... O-
EC31b
T /._7-
:....:.. 0:5
1;0
1~5 Fig. 3
2~
2~5
3:0
v
182
J. W. Murzewski
median C = 1 and logarithmic COV vc. Its design value is defined by an exponential formula: C d =
exp (flcvc)
(19)
where tic is the safety index for analysis uncertainties. The analysis index tic' may be fixed in semiprobabilistic codified design. It is not necessarily so in the level-2 probabilistic design, and this problem is discussed at the end of this section. The factor C covers not only random deviations caused by the residual stresses and other imperfections of columns, but also other uncertainties of c o m m o n structural analysis. A special analysis factor 7d is recommended ~° for this purpose in semiprobabilistic design and there are proposals for its numerical values: ?'d = 1. i. Some design standards take 7'd into account in an indirect way by a reduction o f design strengths, more than would follow from statistical inhomogeneity of the material only. It follows from our considerations that the material factor ~'M~ = 1.1 of the Eurocode No. 3 7 may be treated as the analysis factor for member design, and the specified steel strength Jy may be taken as the design value. Such change in mind does not change the results of design. The overall COV vc, which covers both instability model imperfections and other uncertainties may be analysed as follows: vc= ~
+ vb
(20)
where v,, is the log COV, which corresponds to the c o m m o n analysis factor ?d = Yc0. The factor C may be applied either to load effects S or to the resistance R. There is no difference in any cases where the safety criterion has the form of a simple inequality:
C S < R or S < R / C
(21)
The empirical COV v in laboratory conditions does not take into consideration c o m m o n uncertainties of structural analysis va. The COV of the quotient of independent random variables: R = min (Rcr, Rpl) and C is equal to a geometrical sum: v = X/~R + v~,
(22)
The log-normal distribution is verified for the plastic resistance Rpl, 6 Rpl = R0 is actual for columns of small slenderness 2 ---* 0. The same type of probability distribution is likely for the critical resistance Rcr when 2 ~ ~ . Probability density functions f(.), cumulative probability functions F(.) and hazard functions h(-) o f log-normal variables Rcr and Rpl are as follows:
Probabilistic approach to elastic-plastic coupled instability
go(tcr)/R, go(tpl)/Rpl,
f(Rcr) = f(Rpl) =
F(Rcr) = ~(tcr),
h(Rcr) =
F(Rpl) = ¢(tpl),
h(Rpl)
dp(tcr)/Rcr = (O(tpl)/Rpl
183
(23) (24)
where go(t) = exp ( -
tz/2)/v/~--the
Gauss function;
+~x~ P
ff~(t) = / g0(t) d t - - t h e Laplace function; 0(2,
ok(t) = go(t)/~(-t)--the
Mills function;
t = In ~/R-//~--normalized log-normal variable R. T h e log-normal distribution c a n n o t characterize exactly the r a n d o m variable R(2) when 0 < 2 < e~. T h e a s s u m p t i o n s (eqns (3) and (4)) allow one to derive the probability function of elastoplastic resistance R~ for c o l u m n s of relative slenderness 2 = 1:
F(RI)
= 1 - [ i f ( - tl)] 2
for
tl = In " ( R I / R p l
(25)
We determine the logarithmic m e a n / ~ l and C O V vl from their definitions. Similar integration for the G a u s s - n o r m a l distributions was d o n e by Rshanitsin. 1J OO
In/~l = 2 [ In 2 R[go( -
t)~(-t)/Rvpl ] dR
0
= In Rpl -
vpl/v/~
~ R1 = Rpl exp
(--vpl/x/-~)
(26)
= vpl X/1 - 1/re
(27)
0(3
vm = 2 [ In 2 (R/R)[go(- t)ff~(-
t)/Rvpl] d R
0
We take well k n o w n relations between the G a u s s - n o r m a l p a r a m e t e r s /~ and v and l o g - n o r m a l p a r a m e t e r s / ~ and v, derived by c o m p a r i s o n s of the first and second m o m e n t s : /~ = R e x p ( v 2 / 2 ) ,
v = V/expv z - 1
(28)
The empirical p a r a m e t e r s /~l and vl for c o l u m n s of m e d i u m slenderness ~. = 1 are expressed by the initial parameters/~pl, vpl and Vbl as follows:
/~, =/~pl exp (-vp,/x/x +
v~/2) (29)
v, = v/V2,(1-
1/g)v~l = x/ln(1
+ v2).
J. W. Murzewski
184
The design value Rdl of resistance for columns of medium slenderness, will be defined exactly on the basis of a probabilistic theory. Usually, the level-2 probabilistic design is understood so that a constant reliability index [] should be specified for each class of structural safety. Authors of publications ~ and code-makers 7 advise one to calibrate the partial indices fli for semiprobabilistic design just so that the overall index/3 = constant, and the partial indices/3, satisfy the necessary condition: _.flic~i = fl
(30)
i--I
usually a simple proportionality rule is added: (31 )
[:;i = ~ifl
where ~i =
Vi
1'7
/V
i-i
are sensitivity factors, and n is the number of basic variables for each design case. The/5' = constant calibration method is not practical because resistance factors ~'r would depend on applied loads S, and load factors '/s would depend on structural material. The rule (eqn (31)) gives exaggerated design values for basic variables of larger variance. Another theory of safety 6 is applied here. Optimization problem solutions have indicated that not the probability, but the risk of exceeding the design limits Rd, Sd, and Ca by the random coordinates of state R, S, and C should be constant for any structure of the same class of safety: Xdh(Xd) = 1/k = const, for X = R, S, C
(32)
where the hazard functions h ( X ) are understood as follows: h(R) =.[(R)/[i h(S) = f(S)
- F(R)],
= f(S)/F(S),
(33)
h(C) = f(C)/F(C)
where 1/k is the hazard ratio--specified for each class of structural safety. The new safety theory 6 is accepted here for the calibration of design values R d and Cd values, and load effects S need not be taken into consideration any more.
Probabilistic approach to elastic-plastic coupled instability
185
The analysis index tic o f the log-normal variable C is determined with eqn (32) by means of iterations:
c~(flc)V c = 1/k ---, tic = V / - 2 In [ x / ~ Vc~O(flc)/k]
(34)
and analysis factor ?c = Cd = exp(flcVc) was formulated earlier (eqn (19)). The resistance index fir.
Ilk 2Ch(flr~)/VRl = 1/k ~)(flRO)/VRO =
for ). ~ 0 and 2 --~ c~ for ~. = 1
(35)
The design resistance Rd and resistance factor ? r are defined in the exponential form in any case:
l n ( R , j / k ) / V r = fiR ~ Rd = k exp (--flRVR) and YR = (flrVr)
(36)
Both tic and Vc, as well as fiR and VR, depend on the slenderness ratio 2.
3 ESTIMATION OF THE PARAMETERS Extensive experimental tests were performed in order to evaluate the influence o f imperfections on the buckling resistance of columns. 9 On this basis, multiple buckling curves were defined and the characteristic values .fy)~ = N k / A were determined and introduced into the standard specifications: 7 Rk(2) = /~ -- 2s = /~(1 -- 20) = Rk0~((2)
(37)
where R and s are the empirical mean value and standard deviation, respectively. Thus, the reduction factors ~((2) from the Eurocode No. 3 may be treated as empirical basis for further consideration: Z(2) = (R/Rko)(l - 2v)
(38)
Both factors o f eqn (38) depend on the slenderness ratio 2 and the imperfection classes a, b, c and d, but they do not depend on steel grade. The most reliable data are available for columns o f small slenderness, ). ~ 0. They are equal to the plastic resistance of cross-sections, R0 = Rpl. The plastic resistance Rpl is the product of two independent log-normal r a n d o m variables, e.g. Npl = f A for columns. The logarithmic COV of plastic strength f of structural steel does not exceed vf = 0-08 in representative statistical tests 6 and the estimated COV of the cross-sectional area A or section modeling W is vA -= 0.04. The COV Vpl is the geometrical sum: vpl
=
+ v~ = ~/0,082 + 0.042 --- 0-09
(39)
186
J. W. Mur-ewski
The relation o f the mean to the characteristic value for 2 = 0 and X -- 1
Ro/R~.o =
1/(1 - 2 × 0.09) = 1.22
(40)
Direct statistical estimation o f common part v, relative to C O V o f random analysis factor C is difficult. A n estimate v, = 0.04 seems to be acceptable. The parameters o f buckling strength for medium slenderness 2 = i are, according to eqn (29):
/)l = /~o = e x p ( - 0 . 0 9 / v Z n ) = 0.95/~0 and /~l ~ 0.95/~0
(41)
VRI = 0"09V/[ -- I/Tr = 0-074 The relations (38) and (40) and (20) and (22) give equations which allow one to determine the u n k n o w n values on the basis o f E u r o p e a n buckling factors Zi = Z(I/1.1): vl = 1/2 - 7~1/(2 × 1.22 x 0.95) = 0.5 - 0.43Xi
(42)
vb, = k / I n (! + v~) - 0.0742,
(43)
vcl = v/v21 + 0"042.
The values v~ a n d vcl are given in Table 2 for E u r o p e a n multiple buckling curves a, b, c a n d d. The reduction factor ZJ (Table 1) is defined according to eqn (16) for ). = I / l - l , i.e. 2 = 1. The resistance index fir and resistance factor ?'R are determined for a hazard ratio l / k = 1/2 from eqns (34) and (35): fiR() = 2.098 and 7Ro = exp (2.098 x 0.09) = 1-208
for 2 ~ 0 a n d 2 ~ oc
(44)
fiR1 = 2.481 a n d 7RI = exp(2-481 x 0.074) = 1.202 for).=
1
The analysis index tic a n d analysis factor ?'c are d e t e r m i n e d for the same hazard ratio I / k = 1/2 from eqns (34) a n d (19): tic() = 2.449 and )'co = 1-103 = 7d for 2 ~ 0 a n d 2 ---, oc
(45)
a n d the values flcl a n d 7cJ for ). = 1 are given in Table 3. There are also new values o f buckling factors )fi determined for a c o n s t a n t h a z a r d ratio l / k = 1/2 a n d ). = 1'
Z~ = (RI/Rko)()'d/'YRITcl) = 0.95 x 1.22 x 1.1/(1.2027c I) = 1.045/7c
(46)
Probabilistic approach to elastic-plastic coupled instability
187
TABLE 2 Imperfection class
a
b
c
d
vl vcl
0.187 0.175
0.221 0.209
0.245 0.233
0.278 0.266
c
d
1.604 1.453 0.730 2.205
1.526 1.501 0.707 1.998
TABLE 3 Imperfection class
a
/~cl vcl ¢P~ = X~ n*
1.767 1-361 0.779 2.784
h 1.667 1.416 0.749 2.405
If the buckling factor ~o is taken according to eqn (14), new parameters of imperfection n* are evaluated so that the buckling factors are equal, q~l = ZI for ~. = 1: n* = - l n 2/In Z1
(47)
4 CONCLUSIONS The probabilistic formula (eqn (14)) based on the model of coupled elastoplastic instability is simpler than the buckling formula (eqn (16)) of the Eurocode No. 3. Equation (14) can be extended to other instability cases than column buckling by a suitable definition of the relative slenderness ratio 2. The parameter of imperfections n of eqn (14) has a clear probabilistic interpretation. It is the reciprocal of the Weibull coefficient of variation = 1 I n . In the first approach it is treated as independent from the slenderness ratio 2, but the statistical tests indicate that such dependence occurs. A new probability-based model is in agreement with available statistical data. The remarkable reduction of buckling strength is attributed not only to the interaction of elastic and plastic instability modes, but also to a r a n d o m analysis factor C which covers the model uncertainties. Its design value equals 1.1 for 2 ~ 0 and it has a m a x i m u m of 1-25 for ~, = 1. It may replace the conventional buckling factor ~ when adequate interpretations of central, characteristic and design values are accepted. The reduction factors ~ of the Eurocode No. 3 may be corrected if a consistent safety measure fl or k is accepted, because the safety indices/~R
188
J. w. Murzewski
and tic depend on coefficients of variation vR and v~., respectively, and these depend on the slenderness ratio 2. A new safety measure, the hazard ratio Ilk is more practical than the joint safety index fl used in level-2 probabilistic design. The ratio l/k is derived from optimization considerations and it permits one to obtain exact design values of resistance independently from the variance of applied loads. The new design values may give more economical design results.
REFERENCES I. Murzewski, J., Theory ~/ Random Capacity of Framed Structures. PWN, Warsaw, 1976 (in Polish). 2. Murzewski, J., Random instability of elastic-plastic frames. In 5th Int. Conf. on Structural Safe O' and Reliabilio', Proc. ~f ICOSSAR'89, San Francisco, Aug. 1989, Vol. 11, pp. 1011-14. 3. Murzewski, J., Overall instability of steel frames with random imperfections. In Int. Colloquium on Stability of Steel Structures, Budapest, 1990, Vol. II, pp. 221-8. 4. Mendera, Z., Uniform approach to metal structures stability design. In Int. Colloquium on Stabili O' of Steel Structures, Budapest, 1990, Vol. I, pp. 33-42. 5. PN-90/B-03200, Steel Structures. Design Rules. PKNMiJ, Wyd.Norm., Warsaw, 1990 (in Polish). 6. Murzewski, J., Reliability ~?f Engineering Structures. Arkady, Warsaw, 1989 (in Polish). 7. Eurocode No. 3, Design of Steel Structures. Part 1 General Rules and Rules for Buildings. Vol. 1. Issue 5, Nov. 1990. 8. Gw62di, M., Distribution parameters of buckling resistance of columns and beams. In XXIII Conf. o['PAN and PZITB, Krynica, 1977, Vol. Ill, pp. 5561 (in Polish). 9. Fukumoto, Y. & ltoh, Y., Exploitation de courbes multiples de flambement par une approche ~i base de donn~es experimentales. Construction M~tallique, (3) 3-22. 10. ISO 2394, General principles on reliability for structures. International Standard, 2nd edn, 1986. I I. Rshanitsin, A. R., Theoo' of Reliability Analysis o[ Building Structures. Stroyizdat, Moscow, 1978 (in Russian).