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Reliability evaluation in nonlinear analysis of reinforced concrete structures
Structural Safety Vol. 19, No. 2, pp. 203-217, 1997
ELSEVIER
PIh S0167-4730(96)00025-2
© 1997 Elsevier Science Ltd. All rights reserved Printed in The Netherlands 0167-4730/97 $17.00 + .00
Reliability evaluation in nonlinear analysis of reinforced concrete structures 1 Dimitri Val
a,*,
F i o d o r B l j u g e r b, D a v i d Y a n k e l e v s k y b
a Department of Civil, Surveying and Environmental Engineering, The University of Newcastle, Callaghan, N.S.W., 2308, Australia b National Building Research Institute, Faculty of Civil Engineering, Technion-- Israel Institute of Technology, Technion City 32000, Israel
Abstract Modem building codes provide a basis for development of advanced nonlinear models for analysis and design of reinforced concrete (RC) structures. Application of nonlinear models permits direct evaluation of reliability of the whole structure at the stage of a structural analysis. In this paper a probabilistic method for reliability evaluation of plane frame structures with respect to ultimate limit states is proposed. The method is based on a combination of the nonlinear finite element structural model and the first-order reliability method (FORM). Implementation of the FORM for nonlinear analysis of RC structures is considered. Uncertainties associated with the structural model are taken into account and their influence on structural reliability is examined via sensitivity analysis. © 1997 Elsevier Science Ltd.
Keywords: Safety; Reliability analysis; FORM; Limit state function; Reinforced concrete frame; Sensitivity analysis; Model uncertainty
1. I n t r o d u c t i o n
Most modem design codes make some allowance for structural nonlinearity. Typically they do not do this in a wholly rational way, but permit the subdivision of the design process into two stages: evaluation of element forces by linear analysis of the whole structure followed by the ultimate limit state design of individual cross-sections, with nonlinear properties of the materials taken into consideration. Due to this subdivision of the design process the reliability of the structure as a whole cannot be considered directly, but is provided for implicitly through verification of its individual members. When a probabilistic structural analysis is of interest, the nonlinear models may be used to represent the ultimate limit states of the whole structure. These can be assessed directly by static (or * Corresponding author. Discussion is open until December 1997 (please submuit your discussion paper to the editor, Ross. B. Corotis). 203
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dynamic) analysis. In this way a more consistent approach to the estimation of the reliability of a structure, as system reliability problem, can be provided. Based on such an approach, a number of methods for reliability estimation of different types of reinforced concrete (RC) structures have been proposed. Teigen et al. [1] presented a method based on the perturbation technique for probabilistic nonlinear finite element analysis of planar RC frames. The method yields a mean vector and its covariance matrix for structural displacements; however, it appears difficult to use these results for the reliability estimation of the structure in the context of its ultimate limit states. A method for a reliability analysis of axisymmetric RC shell structures was developed by Rajasheknar and Ellingwood [2]. The method is based on a response surface technique, i.e. estimation of the structural reliability is divided into two stages. First, using the results of a nonlinear finite element analysis an approximation to the limit state surface is obtained, so that the structural reliability (or its complement, failure probability) can be estimated by simulation techniques. In this paper a probabilistic method for reliability evaluation of RC frames in the context of nonlinear analysis is presented. The method is based on a combination of the nonlinear finite element structural model and the first-order reliability method (FORM), i.e. the nonlinear structural model is directly used in the probabilistic analysis. The reliability of the structure is expressed in terms of the reliability index, which is found from the solution of an optimization problem. A procedure based on directional search is developed to solve the optimization problem. The various possible sources of uncertainty associated with the structural model, including the uncertainty of the model itself, are considered and their influence on the structural reliability is examined via a sensitivity analysis. Finally, an example illustrating application of the proposed method is provided.
2. Model formulation 2.1. Structural model
Choice of the structural model should be based on a reasonable compromise between the requirement of accuracy (which will be considered later in relation to model uncertainty) and that of simplicity, dictated by the need for an operational tool for reliability analysis. In the context of the reliability analysis it is also important that the model would permit explicit use of existing statistical data for description of uncertainties associated with material behavior. To meet the above requirements, the following model for RC plane frame structures is employed
[3]: (1) a finite-element model based on two-dimensional beam elements with three degrees of freedom (axial displacement, transverse displacement and in-plane rotation) at each node; (2) the material and geometric nonlinearities are taken into account; (3) the element stiffness matrix is estimated on the basis of layered discretization of the cross section (the physically based approach generally applied to nonlinear flexural beam elements [4]); (4) embedded representation of the reinforcement is used; (5) the stress-strain diagrams for the concrete and reinforcing steel are as per [5]; (6) the tension stiffening effect is taken into account through a modified stress-strain relation for the embedded reinforcement [5].
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20 5
2.2. Probabilistic model Formulation of the probabilistic model in the context of FORM includes the following steps [6]: (1) identification of the basic random variables x representing uncertain parameters of the structural model; (2) definition of the limit state (or performance) function g(x) describing the state of the structure in terms of these variables (conventionally g(x) < 0 denotes failure of the structure, g(x) > 0 denotes its survival, and the boundary g(x) = 0 between the failure and safe domains is called the limit state surface); (3) description of the structural model uncertainty in probabilistic terms; (4) estimation of the safety measure (the failure probability or the reliability index) on the basis of the above provisions.
3. Method of analysis The reliability analysis is based on FORM [7]. Structural reliability is estimated by the reliability index, /3, which equals the minimum distance from the origin to the limit state surface in the standard normal space. Thus a value /3 is found by solving a constrained optimization problem: minimize/3(y) = yry)l/2 subject to G(y) = 0
(1)
where y (y = T(x)) is a vector of the basic random variables in the standard normal space and G(y) (G(y) = g ( T - l(y))) is the limit state function. The point y * minimizing (1) is usually referred to as a design point. To formulate the limit state function, a criterion for global failure of the structure, valid for any possible failure mode, must be established. The singularity of its global stiffness matrix is suitable for this purpose. Different mathematical formulations of the limit state function can then be proposed. The most common approach used for solution of (1) involves computation of the gradient of the limit state function with respect to the basic random variables [8]. The limit state function might be defined as:
g(x)=(R/S)-i
(2)
where R is the bearing capacity of the structure and S is the applied load. For any given value of the vector x, the singularity of the global stiffness matrix is defined by the divergence of the iteration process, and by repeating the nonlinear finite element analysis the ratio ( R / S ) can be determined. This means that only numerical evaluations of the limit state function, and consequently of its gradient, are available. Thus it is very difficult to obtain sufficiently accurate estimates of the gradient, so that convergence of gradient-based method cannot be guaranteed [9]. To overcome this difficulty, the following procedure for finding a design point is proposed. Let the problem be formulated in the standard normal space in polar coordinates y ~ (r, q~), where r is the radial coordinate and q~ = [q~, q92 ~gn_1]r is the vector of angular coordinates. Since . . . . .
r= (yTy)l/2= /3 the optimization problem in polar coordinates becomes:
(3)
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minimize r subject to G( r, cp ) = 0.
(4)
By solving G(r, ~o) with respect to r, i.e. by finding the distance from the origin to the limit state surface in the direction defined by ~o, (4) can be reformulated as an unconstrained optimization problem: minimize r = F ( ~o).
(5)
This reorganization of the problem has an important advantage. If the direct iteration method is used for solution of the nonlinear structural problem, it is now sufficient to check simply for convergence of a single iterative run to obtain the value of G(r, ~o), which is reformulated as - 1,
G(r, ~p)= 0, 1,
( r , ~o) ~ failure domain (does not converge) (r, ~ o ) ~ l i m i t s t a t e s u r f a c e
(6)
(r, ~o) ~ safe domain (converges).
This formulation of the limit state function provides significant reduction of computational effort, because instead of having to calculate the ratio (R/S), which would require, typically, about 10-15 iterative runs, and only convergence of a single iterative run needs to be checked. Of course, a value of F(~o) is still required. This can be estimated numerically by a very simple iterative procedure: for given ~o, first, r 0 is defined so as to satisfy G(r o, ~o) = - 1; then the value of F(~o) is found on [0, r 0] with the required accuracy by the bisection method. The unconstrained optimization problem (5) can be solved by the conjugate direction method [10]. Since the limit state surface is less affected than the limit state function by instability of the nonlinear structural analysis results, and the method itself is less sensitive to possible discontinuities of this surface, its convergence is very stable compared with the gradient-based methods [9]. Generally, a structure may have multiple failure modes. In the present formulation these correspond to multiple local minima of (5). In order to identify these local minima, (5) can be solved a number of times using the procedure described above, but each time the solution is started with a new point ~0, chosen as an initial guess. Generally, the solution was found to converge to the local minimum closest to the starting point. In principle this approach allows the important local minima (and the corresponding design points) to be found. The failure probability, Pc, for the system as a whole can then be estimated using a bounding technique [7]. It should be noted that when there are several local minima close to each other, only convergence to the local minimum with the smallest value of the objective function is stable. However, in this case the other local minima do not have a significant influence on Pf. For some particular RC frames the number of failure modes, which have a significant effect on the estimation of Pf, was usually found to be small. More detailed discussion of this aspect, of multiple failure modes generally and their influence on Pf, including comparisons of the results obtained by the proposed method with the results of Monte Carlo simulation, is given in [11].
4. Choice of basic random variables
In this study only the uncertainties associated with structural behavior are considered. Their possible sources are:
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~C
(a)
207
(~s
(b)
ey \
Ecl
\
•cu
\\
Ey
Esh
Eu E s
Fig. 1. Stress-strain diagrams: (a) for concrete; (b) for reinforcing steel.
statistical variation of the concrete and reinforcing steel properties; statistical variation of the parameters representing steel-concrete interaction; statistical variation of the geometric properties. In the structural model used for the present study, the concrete and steel properties are described by their stress-strain diagrams. Thus the statistical variation of the parameters defining these diagrams should be considered. A typical stress-strain diagram for concrete in compression is shown in Fig. 1. The relevant parameters are: the compressive strength fc, the modulus of elasticity E~, the strain at the maximum stress 8cl and the strain defining the descending branch of the diagram eCu (it should be noted that eta is a parameter only for description of the descending branch and it does not represent the limit compressive strain for concrete [5]). According to [5], the mean value of the compressive strength fcm (in MPa) may be estimated by L m = L k "~- 8
(7)
where fck is a characteristic compressive strength. Taking into account that fCk is defined as a 0.05-fractile and assuming a normal distribution for f¢, its coefficient of variation Cvfc may be obtained as 8
Cvfc = 1.645fcm "
(8)
For E c according to [12,13] Cvec = 8-10% and a normal distribution appears to be an appropriate assumption. In view of the high degree of correlation between E c and fc, the former may be estimated from the relation between these two values. For example, according to [5]
Ec=aE(O.lf~) '/3
(9)
and its statistical variation can be expressed conveniently through the variability of the coefficient otE and fc. Then, assuming that fc and a E are independent random variables and using a first-order approximation of (9), the following relation is obtained:
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c l= c L+ (Q/3)
(10)
No reliable statistical data were found to be available about the variability of ecl and ec.. On the basis of the observation that a change of concrete grade from C25 to C105 appears to result in only a very slight increase for e d (less than 10%) [14], it is reasonable to suggest that its variability is low and it may be taken as a deterministic parameter. The parameter ecu appears to depend directly on fc and Ec [5], so that its variability is expressed through these variables. The stress-strain diagram for concrete in tension is also adopted from [5] and depends on two parameters: the tensile strength fct and the ultimate tensile strain ectu. According to [15], a normal distribution with Cvfc~= 0.18 may be assumed for f~t. The value of the tensile strength may be also estimated from its relationship with fc [16]
(11)
L t = Olfctfg/3 •
Thus, as for E c, the coefficient o~fc' is taken as an independent normal random variable and
Cv~, = Cv2,lc, + (2Cvlc/3) 2.
(12)
For ~ct, no statistical data were found and it is taken as a deterministic parameter. The stress-strain diagram for the reinforcing steel is adopted according to [5] and shown in Fig. 1 (only hot-rolled steel is considered in this study). The diagram is described by the following parameters: the yield strength fy, the ultimate strength fu, the total elongation at maximum load e u, the modulus of elasticity Es, the yielding interval ( e s h - •y), and the strain-hardening modulus Esh. According to [13,15], for the yield strength the coefficient of variation is Cve = 0.08-0.11. A strong correlation between fy and fu was noted, although an some mvestlgaUons [17] somewhat smaller variation of the ultimate strength (Cry, = 0.04-0.07) was reported. Different types of distribution were proposed for the steel strengths [15]: normal, lognormal, beta. In this study, for simplicity, fy is taken as a normal random variable and fu is calculated under the condition that the ratio (fu/fy) remains constant. According to [18], the coefficient of variation of e u is 5-10%; in this study it is taken as a normal variable with Cv=, = 0.10. For E s the coefficient of variation is very low (3.3%) [19], thus it is regarded as a deterministic parameter. No statistical data on ( e s h - ey) and E=h were found and they are also considered as deterministic parameters. According to the applied structural model, the interaction between concrete and steel is only considered in the context of the tension stiffening, which is taken into account via a modified stress-strain relation of the embedded reinforcement [5]. In order to investigate the influence of the uncertainty of the steel-concrete interaction, the integration factor for steel strain along the transmission length ~t,m' which provides the relation between the strain of the reinforcement at a crack and the mean steel strain, is taken as a random variable. Since no statistical data are available for this parameter, it is merely assumed that it follows a normal distribution with a mean equal to its nominal value and with a coefficient of variation Cvt~,.= = O. 15. The geometrical variations are only considered for cross-section dimensions, which are taken as normal random variables with means equal to nominal values and coefficients of variation assigned according to [13,20]. The data for the random variables used in the analysis are summarized in Table 1. .
.
.
.
JY
D. Valetal. Table 1 Distribution properties of random variables Variable description
209
Mean value
Concrete properties (for C30) Compressive strength, fc (MPa) Coefficient a E (MPa) Coefficient otfc ' (MPa) Reinforcing steel properties (for s400) Yield strength, fy (MPa) Ultimate strain q~u Parameter of tension stiffening effect /3t,m Cross-section dimensions Width, b Overall depth, h Effective depth, d
Coefficient of variation
38.0 0.3
0.13 0.05 0.15
461.0 0.11
0.08 0.10
0.60
0.15
2.15 X 10 4
Nominal value Nominal value Nominal value
0.03 0.015 0.05
The computational effort involved in a probabilistic analysis can be significantly reduced by taking account of uncertain properties only for those variables, which have a major influence on structural reliability and considering all others as deterministic parameters. In the present work, the random variables with major influence on structural reliability were identified through a preliminary sensitivity analysis. In the case of the independent random variables, the sensitivity of the reliability index /3 to the uncertainty of the ith random variable may be estimated by the so-called sensitivity factor a i, defined as
0/3
Yi*
(13)
It has been shown [21] that /3 is increased by a factor 1/~/1 - a/2 , when the ith random variable is replaced by its mean value. Thus, neglecting the uncertainty of a variable with ai -- 0.1 results in an increase o f / 3 by about 0.5% and with a i = 0.3 by about 5%, respectively. The sensitivity factors are evaluated as a function of the random variable uncertainties expressed via the ratio between the actual and recommended values of the coefficient of variation (see Table 1). The analysis was performed for two simple structures; the beam and the portal frame shown in Fig. 2. For both the structures, two types of failure are considered: one with predominantly plastic behavior,
(a)
~P
""/~" ~ . . . 6m .
4L~
/,1,, ~
-~-d
(b) I II[I I Ill I111 I Eli I2IEIq---60kN/m
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11
2[
0.4m 7 ~[ - ~ [ 0"05m
6m
}
L ,L.~x,~ 0.05 m ~0.2m~ As
~0'2 ml ~, P (kN) Under-reinforcedbeam 81 Over-reinforcedbeam 180
d (m) 0.41 0.37
As (cm2) 9.42x10--4 32.17x10-'4
Cross sectionreinforcementA~/As (cm2) 1-1 2-2 3-3 1./14.7 9.8/1. 9.8/9.8
Fig. 2. Choice of the basic random variables - - structures under examination: (a) simple beam; (b) portal frame.
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1.0
1.0
0.8
0.8
0.6
/
J
0.6
r fc
l
-~ 0.4 I
o=
.,~ "~ 0.4
0.2'
0.2
0.0 ~ 0.5 1.0 1.5 2.0 Actual/Recommended Coefficient of Variation
0.0 0,5 1.0 1.5 2.0 Actual/Recommended Coefficient of Variation
Fig. 3. Sensitivity factors for the beam: (a) plastic failure; (b) brittle failure.
where the critical aspect is the uncertainty of the steel yield strength and other with predominantly brittle behavior, where the critical aspect is that of the concrete compressive strength. To obtain these failure modes for the beam, two types of cross-section reinforcement (under- and over-reinforced) were used, while for the frame these two modes are almost equally likely. Only the results for the variables with sensitivity factors above 0.1 are presented (Figs. 3 and 4). With respect to their influence on the structural reliability for the appropriate modes of failure, the steel yield strength fy and the concrete compressive strength fc may be referred to as primary variables. Among the other, secondary variables only the effective depth d has a significant influence. It is also observed that for the statically indeterminate frame the influence of the secondary variables considerably decreases. Accordingly, it is appropriate to limit the set of basic random variables for the considered structural type to the above three.
5. Correlation effect The effect of possible correlation between material properties within the structure should be considered in identifying the basic random variables to be retained. To investigate this the example 1.0
,
,
,
"
0.8
?
1.0 ~
0.8
~ O.6
~ 0.6 r
•~ 0.4
"~ 0.4
0,2
0.2
0,0 0.5 1.0 1.5 2.0 Actual/Recommended Coefficient of Variation
0.0
L
i i ' 0.5 1.0 1.5 2.0 Actual/Recommended Coefficient of Variation
Fig. 4. Sensitivity factors for the frame: (a) plastic failure; (b) brittle failure.
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w [l[llllllllllllllllq Im~
~,
L
6m
9'8 cm2
}
0.4rn I J L m - - . ~ 0.05m 10.2ml 9.8cm2
Fig. 5. Correlation effect - - frame under examination.
portal frame shown in Fig. 5 was analyzed. The frame is subjected to a uniformly distributed vertical load q (taken as a normal random variable with a mean 60 k N / m and a coefficient of variation Cvq = 0.1) and to a horizontal load W (taken as a random variable described by the extreme value distribution type I of the largest values with a mean 100 kN and Cu w = 0.2). The concrete strength fc and the steel strength f~ are considered as mutually independent normal random variables. The concrete has a characteristic compressive strength fck = 30 MPa, mean strength fcm ~---38 MPa and coefficient of variation Cry, = 0.13. For the steel the corresponding values are: fsk = 400 MPa, fsm = 461 MPa and Cvls --- 0.08, respectively. Five cases of material strength correlation are considered: (Case 1) Full correlation along the frame. (Case 2) Full correlation within the frame members (columns and beam) and partial correlation between them (coefficient of correlation 0.5). (Case 3) Full correlation within the members and none between them. (Case 4) Partial correlation within the members and none between them. In this case, over the length of the member, the strengths are taken as homogeneous normal processes with autocorrelation function p ( A l ) = e x p ( - --~-] IAIIt
(14)
where A l is the spacing of points over the component length, L component length and k a coefficient defining the correlation length (in this example k = 1). For representation of the random process in terms of the random variables, midpoint discretization is used, i.e. the process is represented by its values at the centers of the finite elements. (Case 5) No correlation at all. The results of all the correlation analyses are given in Table 2. The overall increase of the reliability index due to reduced correlation is slightly less than 11%, most of it (80%) due to changed Table 2 Effect of material strength correlation on failure probability of the frame Correlation case fl 1 3.157 2 3.316 3 3.414 4 3.445 5 3.497
Pf -~ ~ ( - / 3 ) 7.98 X 10 -4 4.57 × 10 -4 3.20X 10 -4 2.85 x 10 -4 2.32 x 10 -4
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correlation between the frame members (i.e. in going from case 1 to case 3). It follows from this analysis that consideration of correlation can be limited to the member level; this significantly reduces the number of the random variables which need to be considered. As insufficient experimental data on actual strength correlation in RC frames are available at present, the conservative assumption of full correlation can be adopted for practical analyses.
6. Model uncertainty Only model uncertainty associated with the structural model is considered. This may be the result of adopted simplifications, as well as of insufficient knowledge about the real behavior of the structure. For example, in the structural model adopted here the following simplifications may be noted: • the effects associated with shear resistance and deformation are disregarded; the discrete concrete cracking, the contribution of the concrete in tension between the cracks, and the nonlinear bond-slip relation are taken into account in smeared form via the transformation of the stress-strain diagram of the reinforcement; • time effects are disregarded. A simple approach for evaluation of the model uncertainty can be based on comparison of the calculated and the experimental results [22]. Then the model uncertainty can be represented by a random variable
Rexp
- -
(15)
Rca~c where Rexp and Rcalc are the experimental and calculated bearing capacities of the structure, respectively. For the adopted structural model, the following statistical data are available [3]: for bending elements (154 specimens) the mean value ~m = 1.025 and the coefficient of variation Cv~ = 0.0738, for eccentrically loaded compressed elements (200 specimens) ~m = 1.026 and Cv~ = 0.0717. For practical analysis ~m = 1 and Cv~ = 0.075 are proposed. With the model uncertainty taken into consideration, the limit state function (2) becomes g(x) = ~:(R/S) - 1.
(16)
This means that failure may occur at any point, whenever (S/R)> ~. The limit state function formulation (6) based on the condition that an iterative run does not converge if (S/R) > 1 (and converges otherwise) is now no longer valid, since the value of ~ may be any positive real number. This means that there is now again the need to calculate the ratio ( R / S ) at every point. As noted earlier, this increases the required computational time by a factor of 10-15, which is clearly inefficient. To overcome this difficulty an approximate method is proposed, in which the problem of model uncertainty is considered separately. Denote h(r, ~o) = ( R / S ) , which for any given direction ~o is a function of a single variable r, for ~ = 1 / h ( r , ~o) failure occurs at the point in question. The optimization procedure then can be subdivided into two steps:
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(1) for all possible directions q~ the minimum of a one-dimensional function is defined from the solution:
1 minimize /3(r) =
2
A(r, q~)
~m
11/2 +r 2
.
(17)
~mCu~
(2) the direction q~ * of the global minimum is determined. Assuming that h(r, ~o) is a linear function of r (which is the case when h(r, q~) is a normal random variable), q~ * coincides with the direction towards the design point obtained with the model uncertainty excluded. Then for h(r, q~ *) the values at two points are known: at the origin r = 0, h(r, ~o *)= h o and at the design point r = / 3 ( ~ = ~m), i.e. the reliability index obtained without the model uncertainty, is h(r, ~o *) = 1. Thus h0 - 1 A(r, (p*) = A0
r.
(18)
~(~=¢m) Substituting (18) into (17) reduces it to a simple unconstrained optimization problem of the single variable r. To check the accuracy of this approximate method the following example is considered. A structure with bearing capacity R is subjected to a load which can be represented at every point as the sum of the dead load D and live load L, the ratio r/L = L / ( D + L) being constant. Then, with the model uncertainty taken into consideration the limit state function may be written in the form
g=~R-D-L
(19)
or g = CAR - Ao - h L
(2o) Lm). hR is a normal random
where h R = R / ( D m + Lm), 1~D ~- D / ( D m + Lm) and AL= L / ( D m + variable with a mean value equal to the ratio between the mean bearing capacity and the mean load (often referred to as the central safety factor), and with a coefficient of variation Cv R = 0.08 (equal to that of the steel yield strength, which is a primary variable in the case of plastic failure). Ao is also a normal random variable with mean value (1 - r/L) and with a coefficient of variation for the dead load Cv o = 0.05. For h L the mean value equals 7/L and the extreme value distribution type 1 of the largest values with Cv L = 0.40 is assumed, as recommended for the live loads [23]. For the model uncertainty the normal distribution with mean ~m = 1 and Cve = 0.075 is assumed. The reliability index was calculated by solving the problem with the limit state function (20), which directly includes the parameter ~ describing the model uncertainty, and by the proposed approximate method. The results by the approximate method were in very good agreement with those of the direct solution, the error usually not exceeding 1% (see Fig. 6). The mean and the coefficient of variation of ~ can be used as quantitative measures for assessment of the accuracy of the structural model. For provision of the same reliability level decrease of ~m or increase of Cvt results in an increase of the required bearing capacity of the structure. ~m values between 1.0 and 1.05, which yield a slightly conservative estimate of the bearing capacity, are
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4.0 3.5 3.0 i~.~2.5 ~ 2.0 1.5 1.0 L_
0.0
0.2
0.4
0.6
0.8
rlL =Lm/( D ,n +Lm)
Fig. 6. Reliabilityindex vs r/L:
, the limit state functionincludes ¢; ©, approximatemethod.
generally seen as acceptable. For the acceptable value of Cv~ some reasonable criterion should be adopted. It is proposed to consider the model uncertainty as tolerable if, compared with the ideal case (Cv~ = 0), it increases the mean bearing capacity by not more than 10% at the same reliability level. Returning to the example considered above, we set the target reliability index fl * = 4.25 (or Pf* = 10-5), which suffices for most cases, and calculate accordingly the central safety factor A0 as a function of Co t using the Rackwitz-Fiessler algorithm [24]. For r/L = 0.1 an increase of 10% in A0 (or in the mean beating capacity) corresponds to Cv~= 0.076 and for r/L = 0.35 to Cv~ =0.119, respectively (it is evident that with increasing uncertainty of other variables, the influence of the model uncertainty decreases). For higher reliability levels the corresponding value of Cv~ is slightly lower: for example, for ~ * = 5.20 (Pf* = 10 - 7 ) and ~/L = 0.1, Cot = 0.072. For the structural model adopted in this study Cm = 1.026 and Cv~ < 0.076, thus its accuracy can be assessed as tolerable.
7. Example Conventional design does not account for risk of damage to columns of building frames due to accidental causes [25]. Since behavior of frames subjected to column failure is accompanied by significant cracking and deformation, only nonlinear models are suitable for structural analysis. The proposed method is used for reliability evaluation of the RC frame shown in Fig. 7 in two cases of column failure (Fig. 8). The frame is considered as part of the load bearing system of a building, made up of transverse frames placed every 6 m. The yield strength of reinforcing steel, the compressive strength of concrete and the effective depth of cross-sections are considered as normal random variables and their distribution properties are determined according to Table 1. Full correlation of the steel and concrete strengths within the frame is adopted. It is also assumed that there is no correlation between the effective depths of the different cross sections. At the level of each floor the frame is subjected to uniformly distributed dead load, D i, and live load, L; (i = 1,2), which are considered as independent random variables. The dead loads are normal random variables with mean D m = 22.2 k N / m and with coefficient of variation Cvo = 0.05. The live
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I]llllll
D 2 + L2 tltlllll
IIIkllll
T
3m
I~,, 1,2,, ,31 3m
]1
3 I tll21 i ~ i
31 13
12
D I + LI IIIIIIII
0457 I
12 4
~02~ As 4.5 m
4.5 m
4.5 m
Cross section reinforcement
1-1
~-~A.ff06 m
2-2
A,s/As (cm2)
3-3
4--4
11.40/7.60 7.60/7.60 19.01/11.40 6.28/6.28
Fig. 7. Example frame.
External column failure
(EFC)
Internal column failure
(IFC)
Fig. 8. Cases of column failure.
loads are distributed according to the extreme value distribution type I of the largest values with mean L m = 11.8 k N / m and coefficient of variation Cv c = 0.40. The situation associated with column failure represents an accidental design situation. In this case the reliability level for which the structure should be designed is defined by the target failure probability divided by that of an accidental event like column failure. Then the target reliability index has to be
1Pal where Pa is the probability of corresponding to Pf*/Pa = 10-2, afs and aye are sensitivity factors of concrete, respectively. For the
occurrence of the accidental event. In this example /3 * = 2.32, is adopted. Results of the analysis are presented in Table 3, where for the yield strength of reinforcement and the compressive strength effective depth and the dead and live loads, represented by several
Table 3 Results of analysis Type of failure
ho
Sensitivity factors OL~
OL/c
OLd
OLD
O~L
ECF ICF
1.410 1.408
0.520 0.517
0.045 0.036
0.184 0.119
0.091 0.123
0.828 0.838
/3( ~: = ~m)
/3
2.736 2.760
2.396 2.409
216
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random variables, the sensitivity factors ( a d, c~D and ceL, respectively) are calculated as the square root of the sum of the squares of the sensitivity factors of the corresponding variables.
8. Conclusions Implementation of the probabilistic method for reliability evaluation in the context of nonlinear analysis of RC plane frame structures was considered, including both structural and probabilistic models. For mathematical formulation of the structural model, a finite element model based on two-dimensional layered beam element was employed. The advantage of this formulation is that it yields a sufficiently accurate prediction of the structural behavior and in the context of probabilistic analysis permits explicit application of existing statistical data for description of uncertain properties of concrete and reinforcing steel. The probabilistic model was based on the FORM. The reliability index was used as a measure of structural safety. For its calculation the optimization problem should be solved. The lack of stable convergence of the current gradient-based methods for problems involving nonlinear analysis of RC structures was noted and a new method with very stable convergence was proposed. For the considered structural type, the uncertain parameters of the structural model with the major influence on the reliability index were identified as the basic random variables via sensitivity analysis. It was assumed that a variable may be considered as deterministic if it results in less than 5% increase in the reliability index. Accordingly, the yield strength of steel, the compressive strength of concrete and the effective depth of cross-section were chosen. The effect of correlation of the material strengths within the structure on the reliability index was examined and the correlation at member level was found to predominate compared with that within individual members. Since at present not enough experimental data on actual strength correlation in RC frames are available, a conservative estimate of overall correlation is recommended for analysis of redundant structures. The model uncertainty associated with the adopted structural model was considered. A simple approximate method was proposed, permitting estimation of the influence of the model uncertainty on the reliability index and using the central safety factor and the value of the reliability index obtained with the model uncertainty excluded as initial data. Recommendations for applicability assessment of the structural model based on the measure of its model uncertainty were given, on the assumption that its accuracy is tolerable if, at the required reliability level, the increase in mean bearing capacity of the structure due to that uncertainty does not exceed 10%. Finally, the example illustrated an application of the proposed method for reliability assessment of the frame structure subjected to column failure.
References [1] Teigen, J. G., Frangopol, D. M., Sture, S. and Felippa, C. A., Probabilistic FEM for nonlinear concrete structure. I: Theory. Journal of Structural Engineering, ASCE, 1991, 117, 2674-2689. [2] Rajasheknar, M. R. and Ellingwood, B. R., Reliability of reinforced-concrete cylindrical shells. Journal of Structural Engineering, ASCE, 1995, 121, 336-347.
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[3] Reitman, M. A., Analysis of statically indeterminate reinforced concrete bar structures, considering nonlinear properties of the materials and histories of short-term loadings. D.Sc. thesis, Reinforced Concrete Institute, Moscow, 1990 (in Russian). [4] Comit6 Euro-International du B&on, Behavior and Analysis of Reinforced Concrete under Alternate Actions Including Inelastic Response, Vol. 2: Frame Members, CEB Bulletin d'Information no. 220, Lausanne, Switzerland, 1993. [5] Comit6 Euro-Intemational du B6ton, CEB-FIP Model Code 1990, CEB Bulletin d'Information no. 213/214, Lausanne, Switzerland, 1993. [6] Ditlevsen O. and Bjerager, P., Methods of structural system reliability. Structural Safety, 1986, 3, 195-229. [7] Madsen, H. O., Krenk, S. and Lind, N. C., Methods of Structural Safety. Prentice-Hall, Englewood Cliffs, NJ, 1986. [8] Liu, P.-L. and Der Kiureghian, A., Optimization algorithms for structural reliability. Structural Safety, 1991, 9, 161-177. [9] Val, D., Bljuger, F. and Yankelevsky, D., Optimization problem solution in reliability analysis of RC structures. Computers and Structures, 1996, 60, 351-355. [10] Brent, R. P., Algorithms for Minimization without Derivatives. Prentice-Hall, Englewood Cliffs, NJ, 1973. [11] Val, D., Progressive collapse reliability of reinforced concrete structures. D.Sc. thesis, Technion - - I.I.T., Haifa, Israel, 1994. [12] Mirza, S. A., Hatzinikolas, M. and MacGregor, J. G., Statistical description of strength of concrete. Journal of Structural Division, ASCE, 1979, 105, 1021-1037. [13] Ostlund, L., An estimation of T-values. In Reliability of Concrete Structures. CEB Bulletin d'Information no. 202, Lausanne, Switzerland, 1991. [14] Thorefeldt, E., Tomaszewicz A. and Jensen, J. J., Mechanical properties of high-strength concrete and application in design. In Utilization of High-strength Concrete, ed. I. Holand et al. (Proc. Symposium, Stavanger, Norway, 1987), Tapir, Trondheim, Norway, 1987, pp. 149-159. [15] MacGregor, J. G., Mirza, S. A. and Ellingwood, B., Statistical analysis of resistance of reinforced and prestressed concrete members. ACI Journal, 1983, 80, 167-176. [16] Eurocode no. 2: Design of concrete structures, Part 1. General Rules and Rules for Buildings, BSI, UK, 1992. [17] Kudzys, A., Reliability Estimation of Reinforced Concrete Structures. Mokslas, Vilnius, Lithuania, 1985 (in Russian). [18] Siviero, E. and Russo, S., Ductility requirements for reinforcement steel. In Ductility -- State of Progress. CEB Bulletin d'Information no. 218, Lausanne, Switzerland, 1993. [19] Mirza, S. A. and MacGregor, J. G., Variability of mechanical properties of reinforcing bars. Journal of the Structural Division, ASCE, 1979, 105, 921-937. [20] Mirza, S. A. and MacGregor, J. G., Variations in dimensions of reinforced concrete members. Journal of the Structural Division, ASCE, 1979, 105, 751-766. [21] Madsen, H. O., Omission sensitivity factors. Structural Safety, 1988, 5, 35-45. [22] Taerwe, L. R., Towards a consistent treatment of model uncertainties in reliability formats for concrete structures. Safety and Performance Concepts. CEB Bulletin d'Information no. 219, Lausanne, Switzerland, 1993. [23] Comit6 Euro-International du B&on, International System of Unified Standard Codes of Practice for SU'uctures, Vol. 1: Common Unified Rules for Different Types of Construction and Material, CEB Bulletin d'Information no. 116-E, Paris, 1976. [24] Rackwitz, R. and Fiessler, B., Structural reliability under combined random load sequences. Computers and Structures, 1978, 9, 489-494. [25] Sucuo~lu, H., t~itipitio~lu, E. and Altm, S., Resistance mechanisms in RC building frames subjected to column failure. Journal of Structural Engineering, ASCE, 1994, 120, 765-782.
ELSEVIER
PIh S0167-4730(96)00025-2
© 1997 Elsevier Science Ltd. All rights reserved Printed in The Netherlands 0167-4730/97 $17.00 + .00
Reliability evaluation in nonlinear analysis of reinforced concrete structures 1 Dimitri Val
a,*,
F i o d o r B l j u g e r b, D a v i d Y a n k e l e v s k y b
a Department of Civil, Surveying and Environmental Engineering, The University of Newcastle, Callaghan, N.S.W., 2308, Australia b National Building Research Institute, Faculty of Civil Engineering, Technion-- Israel Institute of Technology, Technion City 32000, Israel
Abstract Modem building codes provide a basis for development of advanced nonlinear models for analysis and design of reinforced concrete (RC) structures. Application of nonlinear models permits direct evaluation of reliability of the whole structure at the stage of a structural analysis. In this paper a probabilistic method for reliability evaluation of plane frame structures with respect to ultimate limit states is proposed. The method is based on a combination of the nonlinear finite element structural model and the first-order reliability method (FORM). Implementation of the FORM for nonlinear analysis of RC structures is considered. Uncertainties associated with the structural model are taken into account and their influence on structural reliability is examined via sensitivity analysis. © 1997 Elsevier Science Ltd.
Keywords: Safety; Reliability analysis; FORM; Limit state function; Reinforced concrete frame; Sensitivity analysis; Model uncertainty
1. I n t r o d u c t i o n
Most modem design codes make some allowance for structural nonlinearity. Typically they do not do this in a wholly rational way, but permit the subdivision of the design process into two stages: evaluation of element forces by linear analysis of the whole structure followed by the ultimate limit state design of individual cross-sections, with nonlinear properties of the materials taken into consideration. Due to this subdivision of the design process the reliability of the structure as a whole cannot be considered directly, but is provided for implicitly through verification of its individual members. When a probabilistic structural analysis is of interest, the nonlinear models may be used to represent the ultimate limit states of the whole structure. These can be assessed directly by static (or * Corresponding author. Discussion is open until December 1997 (please submuit your discussion paper to the editor, Ross. B. Corotis). 203
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dynamic) analysis. In this way a more consistent approach to the estimation of the reliability of a structure, as system reliability problem, can be provided. Based on such an approach, a number of methods for reliability estimation of different types of reinforced concrete (RC) structures have been proposed. Teigen et al. [1] presented a method based on the perturbation technique for probabilistic nonlinear finite element analysis of planar RC frames. The method yields a mean vector and its covariance matrix for structural displacements; however, it appears difficult to use these results for the reliability estimation of the structure in the context of its ultimate limit states. A method for a reliability analysis of axisymmetric RC shell structures was developed by Rajasheknar and Ellingwood [2]. The method is based on a response surface technique, i.e. estimation of the structural reliability is divided into two stages. First, using the results of a nonlinear finite element analysis an approximation to the limit state surface is obtained, so that the structural reliability (or its complement, failure probability) can be estimated by simulation techniques. In this paper a probabilistic method for reliability evaluation of RC frames in the context of nonlinear analysis is presented. The method is based on a combination of the nonlinear finite element structural model and the first-order reliability method (FORM), i.e. the nonlinear structural model is directly used in the probabilistic analysis. The reliability of the structure is expressed in terms of the reliability index, which is found from the solution of an optimization problem. A procedure based on directional search is developed to solve the optimization problem. The various possible sources of uncertainty associated with the structural model, including the uncertainty of the model itself, are considered and their influence on the structural reliability is examined via a sensitivity analysis. Finally, an example illustrating application of the proposed method is provided.
2. Model formulation 2.1. Structural model
Choice of the structural model should be based on a reasonable compromise between the requirement of accuracy (which will be considered later in relation to model uncertainty) and that of simplicity, dictated by the need for an operational tool for reliability analysis. In the context of the reliability analysis it is also important that the model would permit explicit use of existing statistical data for description of uncertainties associated with material behavior. To meet the above requirements, the following model for RC plane frame structures is employed
[3]: (1) a finite-element model based on two-dimensional beam elements with three degrees of freedom (axial displacement, transverse displacement and in-plane rotation) at each node; (2) the material and geometric nonlinearities are taken into account; (3) the element stiffness matrix is estimated on the basis of layered discretization of the cross section (the physically based approach generally applied to nonlinear flexural beam elements [4]); (4) embedded representation of the reinforcement is used; (5) the stress-strain diagrams for the concrete and reinforcing steel are as per [5]; (6) the tension stiffening effect is taken into account through a modified stress-strain relation for the embedded reinforcement [5].
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20 5
2.2. Probabilistic model Formulation of the probabilistic model in the context of FORM includes the following steps [6]: (1) identification of the basic random variables x representing uncertain parameters of the structural model; (2) definition of the limit state (or performance) function g(x) describing the state of the structure in terms of these variables (conventionally g(x) < 0 denotes failure of the structure, g(x) > 0 denotes its survival, and the boundary g(x) = 0 between the failure and safe domains is called the limit state surface); (3) description of the structural model uncertainty in probabilistic terms; (4) estimation of the safety measure (the failure probability or the reliability index) on the basis of the above provisions.
3. Method of analysis The reliability analysis is based on FORM [7]. Structural reliability is estimated by the reliability index, /3, which equals the minimum distance from the origin to the limit state surface in the standard normal space. Thus a value /3 is found by solving a constrained optimization problem: minimize/3(y) = yry)l/2 subject to G(y) = 0
(1)
where y (y = T(x)) is a vector of the basic random variables in the standard normal space and G(y) (G(y) = g ( T - l(y))) is the limit state function. The point y * minimizing (1) is usually referred to as a design point. To formulate the limit state function, a criterion for global failure of the structure, valid for any possible failure mode, must be established. The singularity of its global stiffness matrix is suitable for this purpose. Different mathematical formulations of the limit state function can then be proposed. The most common approach used for solution of (1) involves computation of the gradient of the limit state function with respect to the basic random variables [8]. The limit state function might be defined as:
g(x)=(R/S)-i
(2)
where R is the bearing capacity of the structure and S is the applied load. For any given value of the vector x, the singularity of the global stiffness matrix is defined by the divergence of the iteration process, and by repeating the nonlinear finite element analysis the ratio ( R / S ) can be determined. This means that only numerical evaluations of the limit state function, and consequently of its gradient, are available. Thus it is very difficult to obtain sufficiently accurate estimates of the gradient, so that convergence of gradient-based method cannot be guaranteed [9]. To overcome this difficulty, the following procedure for finding a design point is proposed. Let the problem be formulated in the standard normal space in polar coordinates y ~ (r, q~), where r is the radial coordinate and q~ = [q~, q92 ~gn_1]r is the vector of angular coordinates. Since . . . . .
r= (yTy)l/2= /3 the optimization problem in polar coordinates becomes:
(3)
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minimize r subject to G( r, cp ) = 0.
(4)
By solving G(r, ~o) with respect to r, i.e. by finding the distance from the origin to the limit state surface in the direction defined by ~o, (4) can be reformulated as an unconstrained optimization problem: minimize r = F ( ~o).
(5)
This reorganization of the problem has an important advantage. If the direct iteration method is used for solution of the nonlinear structural problem, it is now sufficient to check simply for convergence of a single iterative run to obtain the value of G(r, ~o), which is reformulated as - 1,
G(r, ~p)= 0, 1,
( r , ~o) ~ failure domain (does not converge) (r, ~ o ) ~ l i m i t s t a t e s u r f a c e
(6)
(r, ~o) ~ safe domain (converges).
This formulation of the limit state function provides significant reduction of computational effort, because instead of having to calculate the ratio (R/S), which would require, typically, about 10-15 iterative runs, and only convergence of a single iterative run needs to be checked. Of course, a value of F(~o) is still required. This can be estimated numerically by a very simple iterative procedure: for given ~o, first, r 0 is defined so as to satisfy G(r o, ~o) = - 1; then the value of F(~o) is found on [0, r 0] with the required accuracy by the bisection method. The unconstrained optimization problem (5) can be solved by the conjugate direction method [10]. Since the limit state surface is less affected than the limit state function by instability of the nonlinear structural analysis results, and the method itself is less sensitive to possible discontinuities of this surface, its convergence is very stable compared with the gradient-based methods [9]. Generally, a structure may have multiple failure modes. In the present formulation these correspond to multiple local minima of (5). In order to identify these local minima, (5) can be solved a number of times using the procedure described above, but each time the solution is started with a new point ~0, chosen as an initial guess. Generally, the solution was found to converge to the local minimum closest to the starting point. In principle this approach allows the important local minima (and the corresponding design points) to be found. The failure probability, Pc, for the system as a whole can then be estimated using a bounding technique [7]. It should be noted that when there are several local minima close to each other, only convergence to the local minimum with the smallest value of the objective function is stable. However, in this case the other local minima do not have a significant influence on Pf. For some particular RC frames the number of failure modes, which have a significant effect on the estimation of Pf, was usually found to be small. More detailed discussion of this aspect, of multiple failure modes generally and their influence on Pf, including comparisons of the results obtained by the proposed method with the results of Monte Carlo simulation, is given in [11].
4. Choice of basic random variables
In this study only the uncertainties associated with structural behavior are considered. Their possible sources are:
D. Val et al.
~C
(a)
207
(~s
(b)
ey \
Ecl
\
•cu
\\
Ey
Esh
Eu E s
Fig. 1. Stress-strain diagrams: (a) for concrete; (b) for reinforcing steel.
statistical variation of the concrete and reinforcing steel properties; statistical variation of the parameters representing steel-concrete interaction; statistical variation of the geometric properties. In the structural model used for the present study, the concrete and steel properties are described by their stress-strain diagrams. Thus the statistical variation of the parameters defining these diagrams should be considered. A typical stress-strain diagram for concrete in compression is shown in Fig. 1. The relevant parameters are: the compressive strength fc, the modulus of elasticity E~, the strain at the maximum stress 8cl and the strain defining the descending branch of the diagram eCu (it should be noted that eta is a parameter only for description of the descending branch and it does not represent the limit compressive strain for concrete [5]). According to [5], the mean value of the compressive strength fcm (in MPa) may be estimated by L m = L k "~- 8
(7)
where fck is a characteristic compressive strength. Taking into account that fCk is defined as a 0.05-fractile and assuming a normal distribution for f¢, its coefficient of variation Cvfc may be obtained as 8
Cvfc = 1.645fcm "
(8)
For E c according to [12,13] Cvec = 8-10% and a normal distribution appears to be an appropriate assumption. In view of the high degree of correlation between E c and fc, the former may be estimated from the relation between these two values. For example, according to [5]
Ec=aE(O.lf~) '/3
(9)
and its statistical variation can be expressed conveniently through the variability of the coefficient otE and fc. Then, assuming that fc and a E are independent random variables and using a first-order approximation of (9), the following relation is obtained:
208
D. Valetal.
c l= c L+ (Q/3)
(10)
No reliable statistical data were found to be available about the variability of ecl and ec.. On the basis of the observation that a change of concrete grade from C25 to C105 appears to result in only a very slight increase for e d (less than 10%) [14], it is reasonable to suggest that its variability is low and it may be taken as a deterministic parameter. The parameter ecu appears to depend directly on fc and Ec [5], so that its variability is expressed through these variables. The stress-strain diagram for concrete in tension is also adopted from [5] and depends on two parameters: the tensile strength fct and the ultimate tensile strain ectu. According to [15], a normal distribution with Cvfc~= 0.18 may be assumed for f~t. The value of the tensile strength may be also estimated from its relationship with fc [16]
(11)
L t = Olfctfg/3 •
Thus, as for E c, the coefficient o~fc' is taken as an independent normal random variable and
Cv~, = Cv2,lc, + (2Cvlc/3) 2.
(12)
For ~ct, no statistical data were found and it is taken as a deterministic parameter. The stress-strain diagram for the reinforcing steel is adopted according to [5] and shown in Fig. 1 (only hot-rolled steel is considered in this study). The diagram is described by the following parameters: the yield strength fy, the ultimate strength fu, the total elongation at maximum load e u, the modulus of elasticity Es, the yielding interval ( e s h - •y), and the strain-hardening modulus Esh. According to [13,15], for the yield strength the coefficient of variation is Cve = 0.08-0.11. A strong correlation between fy and fu was noted, although an some mvestlgaUons [17] somewhat smaller variation of the ultimate strength (Cry, = 0.04-0.07) was reported. Different types of distribution were proposed for the steel strengths [15]: normal, lognormal, beta. In this study, for simplicity, fy is taken as a normal random variable and fu is calculated under the condition that the ratio (fu/fy) remains constant. According to [18], the coefficient of variation of e u is 5-10%; in this study it is taken as a normal variable with Cv=, = 0.10. For E s the coefficient of variation is very low (3.3%) [19], thus it is regarded as a deterministic parameter. No statistical data on ( e s h - ey) and E=h were found and they are also considered as deterministic parameters. According to the applied structural model, the interaction between concrete and steel is only considered in the context of the tension stiffening, which is taken into account via a modified stress-strain relation of the embedded reinforcement [5]. In order to investigate the influence of the uncertainty of the steel-concrete interaction, the integration factor for steel strain along the transmission length ~t,m' which provides the relation between the strain of the reinforcement at a crack and the mean steel strain, is taken as a random variable. Since no statistical data are available for this parameter, it is merely assumed that it follows a normal distribution with a mean equal to its nominal value and with a coefficient of variation Cvt~,.= = O. 15. The geometrical variations are only considered for cross-section dimensions, which are taken as normal random variables with means equal to nominal values and coefficients of variation assigned according to [13,20]. The data for the random variables used in the analysis are summarized in Table 1. .
.
.
.
JY
D. Valetal. Table 1 Distribution properties of random variables Variable description
209
Mean value
Concrete properties (for C30) Compressive strength, fc (MPa) Coefficient a E (MPa) Coefficient otfc ' (MPa) Reinforcing steel properties (for s400) Yield strength, fy (MPa) Ultimate strain q~u Parameter of tension stiffening effect /3t,m Cross-section dimensions Width, b Overall depth, h Effective depth, d
Coefficient of variation
38.0 0.3
0.13 0.05 0.15
461.0 0.11
0.08 0.10
0.60
0.15
2.15 X 10 4
Nominal value Nominal value Nominal value
0.03 0.015 0.05
The computational effort involved in a probabilistic analysis can be significantly reduced by taking account of uncertain properties only for those variables, which have a major influence on structural reliability and considering all others as deterministic parameters. In the present work, the random variables with major influence on structural reliability were identified through a preliminary sensitivity analysis. In the case of the independent random variables, the sensitivity of the reliability index /3 to the uncertainty of the ith random variable may be estimated by the so-called sensitivity factor a i, defined as
0/3
Yi*
(13)
It has been shown [21] that /3 is increased by a factor 1/~/1 - a/2 , when the ith random variable is replaced by its mean value. Thus, neglecting the uncertainty of a variable with ai -- 0.1 results in an increase o f / 3 by about 0.5% and with a i = 0.3 by about 5%, respectively. The sensitivity factors are evaluated as a function of the random variable uncertainties expressed via the ratio between the actual and recommended values of the coefficient of variation (see Table 1). The analysis was performed for two simple structures; the beam and the portal frame shown in Fig. 2. For both the structures, two types of failure are considered: one with predominantly plastic behavior,
(a)
~P
""/~" ~ . . . 6m .
4L~
/,1,, ~
-~-d
(b) I II[I I Ill I111 I Eli I2IEIq---60kN/m
O.L[~[ +
11
2[
0.4m 7 ~[ - ~ [ 0"05m
6m
}
L ,L.~x,~ 0.05 m ~0.2m~ As
~0'2 ml ~, P (kN) Under-reinforcedbeam 81 Over-reinforcedbeam 180
d (m) 0.41 0.37
As (cm2) 9.42x10--4 32.17x10-'4
Cross sectionreinforcementA~/As (cm2) 1-1 2-2 3-3 1./14.7 9.8/1. 9.8/9.8
Fig. 2. Choice of the basic random variables - - structures under examination: (a) simple beam; (b) portal frame.
210
D. Val et al.
1.0
1.0
0.8
0.8
0.6
/
J
0.6
r fc
l
-~ 0.4 I
o=
.,~ "~ 0.4
0.2'
0.2
0.0 ~ 0.5 1.0 1.5 2.0 Actual/Recommended Coefficient of Variation
0.0 0,5 1.0 1.5 2.0 Actual/Recommended Coefficient of Variation
Fig. 3. Sensitivity factors for the beam: (a) plastic failure; (b) brittle failure.
where the critical aspect is the uncertainty of the steel yield strength and other with predominantly brittle behavior, where the critical aspect is that of the concrete compressive strength. To obtain these failure modes for the beam, two types of cross-section reinforcement (under- and over-reinforced) were used, while for the frame these two modes are almost equally likely. Only the results for the variables with sensitivity factors above 0.1 are presented (Figs. 3 and 4). With respect to their influence on the structural reliability for the appropriate modes of failure, the steel yield strength fy and the concrete compressive strength fc may be referred to as primary variables. Among the other, secondary variables only the effective depth d has a significant influence. It is also observed that for the statically indeterminate frame the influence of the secondary variables considerably decreases. Accordingly, it is appropriate to limit the set of basic random variables for the considered structural type to the above three.
5. Correlation effect The effect of possible correlation between material properties within the structure should be considered in identifying the basic random variables to be retained. To investigate this the example 1.0
,
,
,
"
0.8
?
1.0 ~
0.8
~ O.6
~ 0.6 r
•~ 0.4
"~ 0.4
0,2
0.2
0,0 0.5 1.0 1.5 2.0 Actual/Recommended Coefficient of Variation
0.0
L
i i ' 0.5 1.0 1.5 2.0 Actual/Recommended Coefficient of Variation
Fig. 4. Sensitivity factors for the frame: (a) plastic failure; (b) brittle failure.
211
D. Val et al.
w [l[llllllllllllllllq Im~
~,
L
6m
9'8 cm2
}
0.4rn I J L m - - . ~ 0.05m 10.2ml 9.8cm2
Fig. 5. Correlation effect - - frame under examination.
portal frame shown in Fig. 5 was analyzed. The frame is subjected to a uniformly distributed vertical load q (taken as a normal random variable with a mean 60 k N / m and a coefficient of variation Cvq = 0.1) and to a horizontal load W (taken as a random variable described by the extreme value distribution type I of the largest values with a mean 100 kN and Cu w = 0.2). The concrete strength fc and the steel strength f~ are considered as mutually independent normal random variables. The concrete has a characteristic compressive strength fck = 30 MPa, mean strength fcm ~---38 MPa and coefficient of variation Cry, = 0.13. For the steel the corresponding values are: fsk = 400 MPa, fsm = 461 MPa and Cvls --- 0.08, respectively. Five cases of material strength correlation are considered: (Case 1) Full correlation along the frame. (Case 2) Full correlation within the frame members (columns and beam) and partial correlation between them (coefficient of correlation 0.5). (Case 3) Full correlation within the members and none between them. (Case 4) Partial correlation within the members and none between them. In this case, over the length of the member, the strengths are taken as homogeneous normal processes with autocorrelation function p ( A l ) = e x p ( - --~-] IAIIt
(14)
where A l is the spacing of points over the component length, L component length and k a coefficient defining the correlation length (in this example k = 1). For representation of the random process in terms of the random variables, midpoint discretization is used, i.e. the process is represented by its values at the centers of the finite elements. (Case 5) No correlation at all. The results of all the correlation analyses are given in Table 2. The overall increase of the reliability index due to reduced correlation is slightly less than 11%, most of it (80%) due to changed Table 2 Effect of material strength correlation on failure probability of the frame Correlation case fl 1 3.157 2 3.316 3 3.414 4 3.445 5 3.497
Pf -~ ~ ( - / 3 ) 7.98 X 10 -4 4.57 × 10 -4 3.20X 10 -4 2.85 x 10 -4 2.32 x 10 -4
212
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correlation between the frame members (i.e. in going from case 1 to case 3). It follows from this analysis that consideration of correlation can be limited to the member level; this significantly reduces the number of the random variables which need to be considered. As insufficient experimental data on actual strength correlation in RC frames are available at present, the conservative assumption of full correlation can be adopted for practical analyses.
6. Model uncertainty Only model uncertainty associated with the structural model is considered. This may be the result of adopted simplifications, as well as of insufficient knowledge about the real behavior of the structure. For example, in the structural model adopted here the following simplifications may be noted: • the effects associated with shear resistance and deformation are disregarded; the discrete concrete cracking, the contribution of the concrete in tension between the cracks, and the nonlinear bond-slip relation are taken into account in smeared form via the transformation of the stress-strain diagram of the reinforcement; • time effects are disregarded. A simple approach for evaluation of the model uncertainty can be based on comparison of the calculated and the experimental results [22]. Then the model uncertainty can be represented by a random variable
Rexp
- -
(15)
Rca~c where Rexp and Rcalc are the experimental and calculated bearing capacities of the structure, respectively. For the adopted structural model, the following statistical data are available [3]: for bending elements (154 specimens) the mean value ~m = 1.025 and the coefficient of variation Cv~ = 0.0738, for eccentrically loaded compressed elements (200 specimens) ~m = 1.026 and Cv~ = 0.0717. For practical analysis ~m = 1 and Cv~ = 0.075 are proposed. With the model uncertainty taken into consideration, the limit state function (2) becomes g(x) = ~:(R/S) - 1.
(16)
This means that failure may occur at any point, whenever (S/R)> ~. The limit state function formulation (6) based on the condition that an iterative run does not converge if (S/R) > 1 (and converges otherwise) is now no longer valid, since the value of ~ may be any positive real number. This means that there is now again the need to calculate the ratio ( R / S ) at every point. As noted earlier, this increases the required computational time by a factor of 10-15, which is clearly inefficient. To overcome this difficulty an approximate method is proposed, in which the problem of model uncertainty is considered separately. Denote h(r, ~o) = ( R / S ) , which for any given direction ~o is a function of a single variable r, for ~ = 1 / h ( r , ~o) failure occurs at the point in question. The optimization procedure then can be subdivided into two steps:
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213
(1) for all possible directions q~ the minimum of a one-dimensional function is defined from the solution:
1 minimize /3(r) =
2
A(r, q~)
~m
11/2 +r 2
.
(17)
~mCu~
(2) the direction q~ * of the global minimum is determined. Assuming that h(r, ~o) is a linear function of r (which is the case when h(r, q~) is a normal random variable), q~ * coincides with the direction towards the design point obtained with the model uncertainty excluded. Then for h(r, q~ *) the values at two points are known: at the origin r = 0, h(r, ~o *)= h o and at the design point r = / 3 ( ~ = ~m), i.e. the reliability index obtained without the model uncertainty, is h(r, ~o *) = 1. Thus h0 - 1 A(r, (p*) = A0
r.
(18)
~(~=¢m) Substituting (18) into (17) reduces it to a simple unconstrained optimization problem of the single variable r. To check the accuracy of this approximate method the following example is considered. A structure with bearing capacity R is subjected to a load which can be represented at every point as the sum of the dead load D and live load L, the ratio r/L = L / ( D + L) being constant. Then, with the model uncertainty taken into consideration the limit state function may be written in the form
g=~R-D-L
(19)
or g = CAR - Ao - h L
(2o) Lm). hR is a normal random
where h R = R / ( D m + Lm), 1~D ~- D / ( D m + Lm) and AL= L / ( D m + variable with a mean value equal to the ratio between the mean bearing capacity and the mean load (often referred to as the central safety factor), and with a coefficient of variation Cv R = 0.08 (equal to that of the steel yield strength, which is a primary variable in the case of plastic failure). Ao is also a normal random variable with mean value (1 - r/L) and with a coefficient of variation for the dead load Cv o = 0.05. For h L the mean value equals 7/L and the extreme value distribution type 1 of the largest values with Cv L = 0.40 is assumed, as recommended for the live loads [23]. For the model uncertainty the normal distribution with mean ~m = 1 and Cve = 0.075 is assumed. The reliability index was calculated by solving the problem with the limit state function (20), which directly includes the parameter ~ describing the model uncertainty, and by the proposed approximate method. The results by the approximate method were in very good agreement with those of the direct solution, the error usually not exceeding 1% (see Fig. 6). The mean and the coefficient of variation of ~ can be used as quantitative measures for assessment of the accuracy of the structural model. For provision of the same reliability level decrease of ~m or increase of Cvt results in an increase of the required bearing capacity of the structure. ~m values between 1.0 and 1.05, which yield a slightly conservative estimate of the bearing capacity, are
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4.0 3.5 3.0 i~.~2.5 ~ 2.0 1.5 1.0 L_
0.0
0.2
0.4
0.6
0.8
rlL =Lm/( D ,n +Lm)
Fig. 6. Reliabilityindex vs r/L:
, the limit state functionincludes ¢; ©, approximatemethod.
generally seen as acceptable. For the acceptable value of Cv~ some reasonable criterion should be adopted. It is proposed to consider the model uncertainty as tolerable if, compared with the ideal case (Cv~ = 0), it increases the mean bearing capacity by not more than 10% at the same reliability level. Returning to the example considered above, we set the target reliability index fl * = 4.25 (or Pf* = 10-5), which suffices for most cases, and calculate accordingly the central safety factor A0 as a function of Co t using the Rackwitz-Fiessler algorithm [24]. For r/L = 0.1 an increase of 10% in A0 (or in the mean beating capacity) corresponds to Cv~= 0.076 and for r/L = 0.35 to Cv~ =0.119, respectively (it is evident that with increasing uncertainty of other variables, the influence of the model uncertainty decreases). For higher reliability levels the corresponding value of Cv~ is slightly lower: for example, for ~ * = 5.20 (Pf* = 10 - 7 ) and ~/L = 0.1, Cot = 0.072. For the structural model adopted in this study Cm = 1.026 and Cv~ < 0.076, thus its accuracy can be assessed as tolerable.
7. Example Conventional design does not account for risk of damage to columns of building frames due to accidental causes [25]. Since behavior of frames subjected to column failure is accompanied by significant cracking and deformation, only nonlinear models are suitable for structural analysis. The proposed method is used for reliability evaluation of the RC frame shown in Fig. 7 in two cases of column failure (Fig. 8). The frame is considered as part of the load bearing system of a building, made up of transverse frames placed every 6 m. The yield strength of reinforcing steel, the compressive strength of concrete and the effective depth of cross-sections are considered as normal random variables and their distribution properties are determined according to Table 1. Full correlation of the steel and concrete strengths within the frame is adopted. It is also assumed that there is no correlation between the effective depths of the different cross sections. At the level of each floor the frame is subjected to uniformly distributed dead load, D i, and live load, L; (i = 1,2), which are considered as independent random variables. The dead loads are normal random variables with mean D m = 22.2 k N / m and with coefficient of variation Cvo = 0.05. The live
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D. Val et al.
I]llllll
D 2 + L2 tltlllll
IIIkllll
T
3m
I~,, 1,2,, ,31 3m
]1
3 I tll21 i ~ i
31 13
12
D I + LI IIIIIIII
0457 I
12 4
~02~ As 4.5 m
4.5 m
4.5 m
Cross section reinforcement
1-1
~-~A.ff06 m
2-2
A,s/As (cm2)
3-3
4--4
11.40/7.60 7.60/7.60 19.01/11.40 6.28/6.28
Fig. 7. Example frame.
External column failure
(EFC)
Internal column failure
(IFC)
Fig. 8. Cases of column failure.
loads are distributed according to the extreme value distribution type I of the largest values with mean L m = 11.8 k N / m and coefficient of variation Cv c = 0.40. The situation associated with column failure represents an accidental design situation. In this case the reliability level for which the structure should be designed is defined by the target failure probability divided by that of an accidental event like column failure. Then the target reliability index has to be
1Pal where Pa is the probability of corresponding to Pf*/Pa = 10-2, afs and aye are sensitivity factors of concrete, respectively. For the
occurrence of the accidental event. In this example /3 * = 2.32, is adopted. Results of the analysis are presented in Table 3, where for the yield strength of reinforcement and the compressive strength effective depth and the dead and live loads, represented by several
Table 3 Results of analysis Type of failure
ho
Sensitivity factors OL~
OL/c
OLd
OLD
O~L
ECF ICF
1.410 1.408
0.520 0.517
0.045 0.036
0.184 0.119
0.091 0.123
0.828 0.838
/3( ~: = ~m)
/3
2.736 2.760
2.396 2.409
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D. Val et al.
random variables, the sensitivity factors ( a d, c~D and ceL, respectively) are calculated as the square root of the sum of the squares of the sensitivity factors of the corresponding variables.
8. Conclusions Implementation of the probabilistic method for reliability evaluation in the context of nonlinear analysis of RC plane frame structures was considered, including both structural and probabilistic models. For mathematical formulation of the structural model, a finite element model based on two-dimensional layered beam element was employed. The advantage of this formulation is that it yields a sufficiently accurate prediction of the structural behavior and in the context of probabilistic analysis permits explicit application of existing statistical data for description of uncertain properties of concrete and reinforcing steel. The probabilistic model was based on the FORM. The reliability index was used as a measure of structural safety. For its calculation the optimization problem should be solved. The lack of stable convergence of the current gradient-based methods for problems involving nonlinear analysis of RC structures was noted and a new method with very stable convergence was proposed. For the considered structural type, the uncertain parameters of the structural model with the major influence on the reliability index were identified as the basic random variables via sensitivity analysis. It was assumed that a variable may be considered as deterministic if it results in less than 5% increase in the reliability index. Accordingly, the yield strength of steel, the compressive strength of concrete and the effective depth of cross-section were chosen. The effect of correlation of the material strengths within the structure on the reliability index was examined and the correlation at member level was found to predominate compared with that within individual members. Since at present not enough experimental data on actual strength correlation in RC frames are available, a conservative estimate of overall correlation is recommended for analysis of redundant structures. The model uncertainty associated with the adopted structural model was considered. A simple approximate method was proposed, permitting estimation of the influence of the model uncertainty on the reliability index and using the central safety factor and the value of the reliability index obtained with the model uncertainty excluded as initial data. Recommendations for applicability assessment of the structural model based on the measure of its model uncertainty were given, on the assumption that its accuracy is tolerable if, at the required reliability level, the increase in mean bearing capacity of the structure due to that uncertainty does not exceed 10%. Finally, the example illustrated an application of the proposed method for reliability assessment of the frame structure subjected to column failure.
References [1] Teigen, J. G., Frangopol, D. M., Sture, S. and Felippa, C. A., Probabilistic FEM for nonlinear concrete structure. I: Theory. Journal of Structural Engineering, ASCE, 1991, 117, 2674-2689. [2] Rajasheknar, M. R. and Ellingwood, B. R., Reliability of reinforced-concrete cylindrical shells. Journal of Structural Engineering, ASCE, 1995, 121, 336-347.
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[3] Reitman, M. A., Analysis of statically indeterminate reinforced concrete bar structures, considering nonlinear properties of the materials and histories of short-term loadings. D.Sc. thesis, Reinforced Concrete Institute, Moscow, 1990 (in Russian). [4] Comit6 Euro-International du B&on, Behavior and Analysis of Reinforced Concrete under Alternate Actions Including Inelastic Response, Vol. 2: Frame Members, CEB Bulletin d'Information no. 220, Lausanne, Switzerland, 1993. [5] Comit6 Euro-Intemational du B6ton, CEB-FIP Model Code 1990, CEB Bulletin d'Information no. 213/214, Lausanne, Switzerland, 1993. [6] Ditlevsen O. and Bjerager, P., Methods of structural system reliability. Structural Safety, 1986, 3, 195-229. [7] Madsen, H. O., Krenk, S. and Lind, N. C., Methods of Structural Safety. Prentice-Hall, Englewood Cliffs, NJ, 1986. [8] Liu, P.-L. and Der Kiureghian, A., Optimization algorithms for structural reliability. Structural Safety, 1991, 9, 161-177. [9] Val, D., Bljuger, F. and Yankelevsky, D., Optimization problem solution in reliability analysis of RC structures. Computers and Structures, 1996, 60, 351-355. [10] Brent, R. P., Algorithms for Minimization without Derivatives. Prentice-Hall, Englewood Cliffs, NJ, 1973. [11] Val, D., Progressive collapse reliability of reinforced concrete structures. D.Sc. thesis, Technion - - I.I.T., Haifa, Israel, 1994. [12] Mirza, S. A., Hatzinikolas, M. and MacGregor, J. G., Statistical description of strength of concrete. Journal of Structural Division, ASCE, 1979, 105, 1021-1037. [13] Ostlund, L., An estimation of T-values. In Reliability of Concrete Structures. CEB Bulletin d'Information no. 202, Lausanne, Switzerland, 1991. [14] Thorefeldt, E., Tomaszewicz A. and Jensen, J. J., Mechanical properties of high-strength concrete and application in design. In Utilization of High-strength Concrete, ed. I. Holand et al. (Proc. Symposium, Stavanger, Norway, 1987), Tapir, Trondheim, Norway, 1987, pp. 149-159. [15] MacGregor, J. G., Mirza, S. A. and Ellingwood, B., Statistical analysis of resistance of reinforced and prestressed concrete members. ACI Journal, 1983, 80, 167-176. [16] Eurocode no. 2: Design of concrete structures, Part 1. General Rules and Rules for Buildings, BSI, UK, 1992. [17] Kudzys, A., Reliability Estimation of Reinforced Concrete Structures. Mokslas, Vilnius, Lithuania, 1985 (in Russian). [18] Siviero, E. and Russo, S., Ductility requirements for reinforcement steel. In Ductility -- State of Progress. CEB Bulletin d'Information no. 218, Lausanne, Switzerland, 1993. [19] Mirza, S. A. and MacGregor, J. G., Variability of mechanical properties of reinforcing bars. Journal of the Structural Division, ASCE, 1979, 105, 921-937. [20] Mirza, S. A. and MacGregor, J. G., Variations in dimensions of reinforced concrete members. Journal of the Structural Division, ASCE, 1979, 105, 751-766. [21] Madsen, H. O., Omission sensitivity factors. Structural Safety, 1988, 5, 35-45. [22] Taerwe, L. R., Towards a consistent treatment of model uncertainties in reliability formats for concrete structures. Safety and Performance Concepts. CEB Bulletin d'Information no. 219, Lausanne, Switzerland, 1993. [23] Comit6 Euro-International du B&on, International System of Unified Standard Codes of Practice for SU'uctures, Vol. 1: Common Unified Rules for Different Types of Construction and Material, CEB Bulletin d'Information no. 116-E, Paris, 1976. [24] Rackwitz, R. and Fiessler, B., Structural reliability under combined random load sequences. Computers and Structures, 1978, 9, 489-494. [25] Sucuo~lu, H., t~itipitio~lu, E. and Altm, S., Resistance mechanisms in RC building frames subjected to column failure. Journal of Structural Engineering, ASCE, 1994, 120, 765-782.