STRUCTURAL SAFETY

Structural Safety 27 (2005) 93–112 www.elsevier.com/locate/strusafe

Time-variant finite element reliability analysis – application to the durability of cooling towers B. Sudret a

a,*

, G. Defaux b, M. Pendola

b

EDF R&D, Department of Materials and Mechanics of Components, Site des Renardie`res, 77818 Moret-sur-Loing Cedex, France b PHI-MECA Engineering, 1 Alle´e Alan TURING, 63170 Aubie`re, France

Received 17 December 2003; received in revised form 5 February 2004; accepted 11 May 2004

Abstract Durability of natural-draught cooling towers is investigated using finite element reliability analysis. A response surface of the linear finite element model is first derived from mechanical considerations. This surface is explicit and exact under certain conditions and requires a single multi-load finite element analysis. This leads to an analytical formulation of the reliability problem. The influence of concrete carbonation and the induced rebars corrosion is then studied in the framework of time-variant reliability analysis. It is shown that the problem reduces to a sequence of time-invariant problems that can be solved using the first-order reliability method (FORM). The evolution in time of the probability of failure in a single point is computed as well as sensitivity factors. Finally, an attempt to introducing space-variant reliability is made. The great difference between the numerical results obtained in the first and in the second approach is emphasized.
2004 Elsevier Ltd. All rights reserved. Keywords: Time-variant reliability; Finite element reliability; FORM/SORM; Reinforced concrete; Corrosion; Cooling towers

*

Corresponding author. Tel.: +33 1 60 73 77 48; fax: +33 1 60 73 65 59. E-mail address: [email protected] (B. Sudret).

0167-4730/$ - see front matter
2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.strusafe.2004.05.001

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1. Introduction Natural-draught cooling towers are used in nuclear power plants as heat exchangers. These shell structures are submitted to environmental loads such as wind and thermal gradients that are stochastic in nature. Due to the complexity of the building procedure, uncertainties in the material properties as well as differences between the theoretical and the real geometry also exist. The framework of structural reliability is well suited to a consistent modelling of all these kinds of uncertainties and the evaluation of their incidence onto the life time of the structures. Specific work on the probabilistic characterization of wind and thermal loading applied onto cooling towers can be found in Flaga [1,3], Bosak and Flaga [2]. A first application of structural reliability to cooling towers was presented by Liao and Lu [4]. The first-order second moment method was used together with an axisymmetric finite element model. Simplified limit state functions based on membrane forces and bending moments were used, respectively. More recently, Mohamed et al. [5,6] applied the first-order reliability method (FORM) [7] together with a three-dimensional shell finite element model. The direct coupling of the finite element code with the probabilistic code was compared to the use of the response surface method (Favarelli [8], Lemaire and Mohamed [9], Pendola et al. [10]). Based on this preliminary work, attention has been focused onto the evolution in time of the reliability of cooling towers. Similar approaches have been developed in the past few years for concrete containment vessels in nuclear engineering [11,12] and bridge engineering [13–16]. In this regard, degradation mechanisms of reinforced concrete have to be investigated and included in the reliability analysis. The aim of this paper is to investigate the durability of cooling towers submitted to rebars corrosion induced by concrete carbonation. The problem is cast in the framework of time-variant finite element reliability analysis. The basis of structural reliability in time-variant context is firstly recalled. It is then shown how a time-variant problem may reduce to a time-invariant problem when certain conditions apply. A response-surface approach is then developed using the linearity of the finite element problem. The derived expression is exact under certain conditions that are detailed in the sequel. Finally, the application to a representative cooling tower is then presented: firstly, the deterministic model (geometry, material properties, loading) is described. Secondly, the random variables and the design criterion are defined. Results are presented in terms of initial probability of failure and its evolution in time. Sensitivity analysis is also carried out.

2. Reliability analysis 2.1. Introduction Reliability analysis aims at estimating the probability of failure of a structure due to uncertainties in the material properties, loads, geometry, etc. It requires:
the probabilistic modelling of the parameters involved in the deterministic model, that is the definition of random variables and/or random processes (probability density function and associated parameters),

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the definition of a failure criterion by means of a limit state function defined in the space of parameters. 2.2. Time-invariant analysis Time-invariant analysis is suited to quasi-static problems that involve only deterministic parameters and random variables gathered into a vector X. The limit state function in this case is defined as follows:
g(X) > 0 defines the safe state of the structure.
g(X) 6 0 defines the failure state. In a reliability context, it does not necessarily mean the breakdown of the structure, but the fact that certain requirements of serviceability or safety limit states have been reached or exceeded.
g(X) = 0 defines the limit state surface. Denoting by fX(x) the joint probability density function of random vector X, the probability of failure of the structure is Z Pf ¼ fX ðxÞ dx: ð1Þ gðxÞ60

This integral may be evaluated by Monte Carlo simulation. However, this technique requires a large number of calls to the limit state function, especially when small probabilities of failure are sought for. Thus, they are practically not applicable to the current situation. Alternatively, approximate methods such as FORM/SORM [7] are attractive. The limit state function is recast in the standard normal space by using a probabilistic transformation X M U (of Nataf or Rosenblatt type [17]). Then the problem is transformed into that of finding the design point U*, which minimizes the distance from the origin of the standard normal space to the limit state surface g(X(U)) = 0. The determination of the design point requires an optimization algorithm such as the Rackwitz-Fiessler algorithm [18], later improved in [19]. The reliability index b [20] is defined as the obtained distance, i.e. iU*i. The approximate probability of failure is finally computed from the normal cumulative distribution function ð2Þ P f;FORM ¼ UðbÞ: This corresponds to approximating the limit state function by a tangent hyperplane at the design point (first-order approximation). A second-order approximation (SORM, see [21,22]) can be computed thereafter for more accurate results. FORM also provides the so-called sensitivity factors [7]. These factors are defined by the square of the coordinates of the unit vector a = U*/b, which sum up to 1. Generally written in percentage, these factors allow to rank the basic random variables according to their importance in the reliability analysis. 2.3. Time-variant analysis When stochastic processes or functions of time are explicitly present in the limit state function, time-variant analysis is required. Let us denote by X(t) the set of random variables Rj, j = 1, . . ., p

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and one-dimensional random processes Sj(t), j = 1, . . ., q describing the randomness in the geometry, material properties and loading. Let us denote by g(t,X(t)) the time-dependent limit state function associated with the reliability analysis. Denoting by [t1, t2] the time interval under consideration, the probability of failure of the structure within this time interval is defined as follows: P f ðt1 ; t2 Þ ¼ Probð9 t 2 ½t1 ; t2 ; gðt; X ðtÞÞ 6 0Þ:

ð3Þ

This definition corresponds to the probability of being in the failure domain at least at one point in time within the interval under consideration. It is emphasized that this quantity is generally different from the so-called instantaneous probability of failure defined as follows: P f;i ðtÞ ¼ Probðgðt; X ðtÞÞ 6 0:

ð4Þ

In Eq. (4), the probability is computed by fixing time in all the functions appearing in the limit state function and by replacing the random processes by the corresponding random variables. 2.4. Right-boundary problems A special class of time-variant problems is defined by the following assumptions:
The limit state function does not depend explicitly on random processes (such as Gaussian, renewal wave, etc.), but only on random variables and deterministic functions of time. The limit state function is denoted by g(t,X) in this case.
The limit state function is monotonously decreasing in time whatever the realizations of the random variables. For each realization x of the random vector, the minimum value of g(t,x) over a time interval [t1, t2] is attained for t = t2, thus the expression right-boundary problems. From these assumptions it comes P f ðt1 ; t2 Þ ¼ 1  P ½gðt; X Þ > 0

8 t 2 ½t1 ; t2  ¼ 1  P ½gðt2 ; X Þ > 0 ¼ P f ðt2 Þ:

ð5Þ

It clearly appears that the time-variant problem reduces to a time-invariant analysis by fixing time to its value at the end of the interval. Degradation phenomena such as corrosion of rebars pertain to the latter class of problems, since the rebars diameter is explicitly defined as a function of time (see Section 5). As a conclusion, time-variant analysis in this paper will be carried out as a sequence of time-invariant analysis at various time instants.

3. Exact response surface method for linear problems The basic idea of the response surface method [8–10] is to replace the true limit state function g(X), which is known implicitly through a finite element procedure in our context, by an approximate polynomial function g^(X). In practice, quadratic functions have been used in literature, the coefficients of the expansion being determined by means of a least-square minimization procedure. In the present paper, the form of the response surface is derived from mechanical considerations: linearity of the response with respect to both the load and the compliance of the structure.

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Precisely, let us consider a linear elastic finite element model of the structure with homogeneous material properties (mass density q, YoungÕs modulus E, PoissonÕs ratio m, thermal dilatation coefficient a). Let us consider q different mechanical load cases (e.g. self weight, applied pressure, etc.) defined by a load intensity ki and a load pattern Fi, i = 1, . . ., q. Vector Fi is the vector of nodal forces corresponding to a unit value of the load intensity. The global stiffness matrix of the system denoted by K is obtained from the ‘‘unit-stiffness matrix’’ K1 corresponding to E = 1 by K ¼ E  K 1:

ð6Þ

Let us denote by Ui the ‘‘unit response’’ of the unit-stiffness system to load pattern i, which is computed from the linear system K 1  U i ¼ F i:

ð7Þ

The global response of the structure U to all load cases is obtained by solving q X ki F i : K U ¼

ð8Þ

i¼1

By substituting for Eqs. (6) and (7) in Eq. (8), this quantity finally reads q 1 X U¼ ki U i : E i¼1

ð9Þ

In case of thermal loading defined by a load intensity kT (e.g. temperature increment), the YoungÕs modulus E of the structure also appears in the load pattern FT, making the displacement vector independent of the value of E. Moreover, the displacement response is proportional to the thermal dilatation coefficient a. Denoting by UT the displacement response to a unit temperature increment of a structure having unit YoungÕs modulus and thermal dilatation coefficient, the vector of nodal displacement in case of thermo-mechanical loading reads U ¼ akT U T þ

q 1 X ki U i : E i¼1

ð10Þ

By similar reasoning, it is easy to show that the internal forces S (e.g. membranes forces and bending moments in case of shell elements) can be obtained from the unit response S ¼ akT ES T þ

q X

ki S i ;

ð11Þ

i¼1

where ðS T ; fS i gqi¼1 Þ denote the internal forces computed from the unit-stiffness matrix and the unit load patterns ðF T ; fF i gqi¼1 Þ. Suppose now that all unit response quantities ðU T ; fU i gqi¼1 Þ are computed from a finite element analysis. Note that this requires a single assembling of the unit stiffness matrix K1, the computation of the unit vectors of nodal forces ðF T ; fF i gqi¼1 Þ, a single inversion of the unit stiffness matrix 1 1 K 1 1 and (q + 1) matrix-vector products ðK 1  F T ; K 1  F i ; i ¼ 1; . . . ; qÞ. Then Eq. (10) gives the displacement vector analytically for any set of parameter ða; E; kT ; fki gqi¼1 Þ. Similarly, the internal q forces S T ; fS i gi¼1 are computed analytically from Eq. (11).

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In terms of reliability analysis, any limit state function based on these response quantities is thus defined analytically
 ð12Þ gðt; X ; U ðX Þ; SðU ðX ÞÞÞ  g t; X ; a; E; kT ; fki gqi¼1 : In this formulation, model parameters a, E, kT, fki gqi¼1 appearing in the response surfaces (10) and (11) may be deterministic or random. As a conclusion the finite element reliability analysis is decoupled into:
a prior run of the finite element code providing unit response vectors,
reliability analysis with an analytical limit state function. Since only a single multi-load case finite element analysis is required, a refined mesh of the structure may be considered. This framework is now applied to the the time-variant reliability analysis of a cooling tower. 4. Deterministic model of the cooling tower 4.1. Geometry and finite element model The cooling tower under consideration is a shell structure with axisymmetric geometry depicted in Fig. 1. The height of the tower is 165 m, the meridian line is described by Eq. (13) yielding the shell radius Rm(z) as a function of the altitude z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð13Þ Rm ðzÞ ¼ 0:00263z þ 0:2986 ðz  126:74Þ2 þ 2694:5 þ 23:333: The shell is supported by 52 pairs of V-shaped circular columns with radius 0.5 m. The shell thickness is varying with z from 21 to 120 cm. The structure is modelled by means of EDFÕs finite element code Code_Aster
[23]. The mesh comprises 12376 9-node isoparametric shell elements. The reinforced concrete is modelled as an homogeneous isotropic elastic material. The material properties, which are random variables in the present study, are given in the sequel. 4.2. Loading When the power plant is in service, the cooling tower is submitted to the following combination of loads:
the self-weight, corresponding to a mass density q,
the wind pressure,q(z, h) which is codified in French Standard NV65, and depends on both altitude z and azimuth h: z þ 18  gðhÞ: ð14Þ z þ 60 In this equation, Pc is the reference wind pressure, v = 1.2 is a coefficient that takes into account the influence of the site and that of the shell roughness, and g(h) is represented as a 10-term Fourier series expansion. Variations of the wind pressure with altitude and azimuth are given in Fig. 2. qðz; hÞ ¼ v  2:5P c

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99

Fig. 1. Cooling tower under consideration and associated mesh.


The internal depression Pint due to air circulation in service, which is supposed to be constant all over the tower and is equal to 0.4Pmax (Pmax being the maximal wind pressure at the top of the cooling tower).
The thermal gradient DT within the shell thickness. It is computed from the difference between the inside- and outside-air temperatures and from the air/concrete heat transfer coefficient. 4.3. Deterministic assessment of the structure In the present study, the serviceability of the cooling tower as a reinforced concrete structure is assessed according to the French concrete design code BAEL [24]. In each node of the mesh, the membrane forces and bending moments in each direction are compared to the strength of a one-meter wide concrete section (Fig. 3). In the sequel, only the circumferential reinforcement is considered since it presents the smallest concrete cover, thus being more critical with respect to durability issues. Practically speaking, the computed membrane and bending forces (Nhh, Mzz) used for assessing the circumferential reinforcement are reported in the load carrying capacity diagram (Fig. 4), which is the locus of the internal forces that can be taken by the concrete section. This diagram depends on the following parameters: geometry of the concrete section (width of 1 m, height equal to the shell thickness), rebars diameter and density, concrete cover, concrete compressive strength and steel yield stress. If the representative point lies inside the diagram, the design criterion is satisfied for the section under consideration. This assessment is carried out at each node of the structure.

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qh (Pa)

1200 1000 800 600 400 200 10

30

50

70

90

110

130

150

z (m) 1

g (θ)

0,5 0 -0,5 -1 -1,5 0

30

60

90 120 150 180 210 240 270 300 330 360

θ (˚)

Fig. 2. Variation of wind pressure with altitude and azimuth.

Fig. 3. Concrete section and orresponding internal forces.

5. Corrosion model 5.1. Introduction The reinforcement bars in concrete structures are initially protected from corrosion by a microscopic oxide layer formed at their boundary due to the strong alkalinity of the pore solution. Two

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101

2.5 2 2

(h N θθ , M zz )

M (MN.m)

1.5 1 0.5 0 0 -0.5

4

9

14

19

-1 -1 -1.5 -2 -2 -2.5

N.h (MN.m) Fig. 4. Load carrying capacity diagram used for designing concrete sections.

different mechanisms, namely chloride ingress and concrete carbonation, lead to a decrease of the pH of the pore solution, which leads to dissolving the protective layer. Then the corrosion of the reinforcement starts. The volume of corrosion products causes tensile stresses that may be sufficiently large to cause internal micro-cracking and eventually spalling. Probabilistic models for chloride ingress and the induced corrosion in rebars have been proposed in literature by several authors including Engelund and Sorensen [25], Stewart et al. [26– 28]. A detailed model for concrete cracking and spalling has been proposed by Thoft-Christensen [29]. Applications mainly deal with bridge engineering submitted to sea spray in coastal zones or de-icing salts. Cooling towers are not submitted to de-icing salts. Moreover, French nuclear power plants that are built close to the sea do not use cooling towers but directly the sea water for cooling purposes. Thus, chloride induced corrosion cannot occur in this context. On the contrary, corrosion induced by concrete carbonation may appear into cooling towers. Not much work on probabilistic models for concrete carbonation and induced corrosion can be found in literature. The main reference used in the sequel is the report written by the CEB Task Groups 5.1 and 5.2 [30]. 5.2. Concrete carbonation model Concrete carbonation is a complex physico-chemical process that includes the diffusion of CO2 into the gas phase of the concrete pores and its reaction with the calcium hydroxyl Ca(OH)2. The latter can be simplified into CaðOHÞ2 þ CO2 ! CaCO3 þ H2 O

ð15Þ

As the high pH of uncarbonated concrete is mainly due to the presence of Ca(OH)2 it is clear from the above equation that the consumption of this species will lead to a pH drop, which can attain a value of 9 when the reaction is completed. In this environment, the oxide layer that protected the reinforcement bars is attacked and corrosion starts. In practice, CO2 penetrates into the concrete mass by diffusion from the surface layer. Thus, a carbonation front appears that moves into the structure. A model for computing the carbonation

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depth xc is proposed by the CEB Task Groups 5.1/5.2 [30]. The simplified version retained in the present paper reads rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DCO2 t; ð16Þ xc ¼ 2C 0 a where DCO2 is the coefficient of diffusion of carbon dioxide in dry concrete, C0 is the carbon dioxide concentration in the surrounding air and a is the binding capacity, i.e. the amount of carbon dioxide necessary for complete carbonation of a concrete volume. It is supposed that corrosion immediately starts when carbonation has attained the rebars. Denoting by e the concrete cover, the time necessary for corrosion to start, called initiation time reads a : ð17Þ tinit ¼ e2 2C 0 DCO2 If generalized corrosion is considered, the loss of metal due to corrosion is approximately uniform over the whole surface. In this case, FaradayÕs law indicates that a corrosion current density icorr = 1 lA/cm2 corresponds to a uniform corrosion penetration of j = 11.6 lm/year. If a constant annual corrosion rate is supposed, the reduction of rebars diameter in time eventually reads dðtÞ ¼ d 0

if t 6 tinit ;

dðtÞ ¼ maxðd 0  2j  icorr ðt  tinit Þ; 0Þ

if t > tinit :

ð18Þ

The corresponding steel cross-section used in the reinforced concrete assessment is obtained as p As ðtÞ ¼ n d 2 ðtÞ; ð19Þ 4 where n is the number of bars in the concrete section under consideration (in practice, the number of rebars per meter)

6. Reliability analysis of the cooling tower 6.1. Limit state function As described in Section 4, the loss of serviceability is locally defined in each node of the mesh according to the load carrying capacity diagram. The corresponding limit state function should be defined in such a way that positive values correspond to the ‘‘safe state’’ (i.e. being inside the diagram) and negative values correspond to the ‘‘failure state’’ (i.e. being outside the diagram). Thus, the following expression is adopted to describe failure (Fig. 5): g ¼ ðh2 N 2ult þ M 2ult Þ  ðh2 N 2hh þ M 2zz Þ;

ð20Þ

where h is the shell thickness, (hNhh, Mzz) are the internal forces resulting from the finite element computation at the node (hNult, Mult) and is obtained as the intersection between the radial line [(0,0); (hNhh, Mzz)] and the diagram boundary. Note that this expression of g is not the shortest distance from the point representing the internal forces to the boundary of the diagram. This is not a problem since any expression of g can be used provided the zero-value of g properly defines

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103

2.5 2

h N ult , M ult

1.5

h N θθ , M zz

M (MN.m) (

1

g

0.5 0 -0.5

4

9

14

19

-1 -1.5 -2 -2.5

N. (MN.m) N.h Fig. 5. Load carrying capacity diagram and corresponding definition of the limit state function.

the limit state surface. In the sequel, the internal forces appearing in Eq. (20) (jointly denoted by X ¼ ðN hh ; M zz Þ are computed analytically from the value of the parameters using the response surface as described in Section 3 (Eq. (11)). In the application example, the point where the limit state function is evaluated is chosen as follows: a deterministic finite element run with mean values of the parameters is carried out. The limit state function Eq. (20) is evaluated at each node of the mesh. The corresponding value is plotted in Fig. 6 onto the shell surface (grey zones corresponds to values exceeding the color scale, i.e. large positive values). The critical point is defined as the node where g takes its minimal value (see Fig. 6). Then the reliability analysis is carried out using limit state function (20) evaluated in this point. 6.2. Random variables and distributions The type and parameters of the random variables used in the analysis are listed in Table 1. It is emphasized that two variables are defined as functions of others, namely the YoungÕs modulus of concrete and the internal depression in service. As a consequence, 12 independent uncorrelated random variables are used. These random variables appear implicitly in Eq. (20) as follows:
E, q, a, Pc, Pint, DT appear in the finite element computation, hence in (Nhh, Mzz) They are collected in random vector X1 in the sequel.
fy, fcj, d0, e appear in the initial load carrying capacity diagram. They are collected in random vector X2 in the sequel.
DCO2, C0, a, icorr appear in the initiation time for corrosion (Eq. (17)) and in the reduction of the rebars diameter (Eq. (18)). They are collected in random vector X3 in the sequel. 6.3. Initial time-invariant analysis The limit state function reads in this case     gðX 1 ; X 2 Þ ¼ h2 N 2ult ðXðX 1 Þ; X 2 Þ þ M 2ult ðXðX 1 Þ; X 2 Þ  h2 N 2hh ðX 1 Þ þ M 2zz ðX 1 Þ :

ð21Þ

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Fig. 6. Values of the limit state function using the mean values of the input parameters. Table 1 Definition of the random variables Random variable

Type

Mean

Concrete YoungÕs modulus: E

–

Concrete mass density: q Concrete thermal dilatation coefficient: a Wind pressure: Pc Difference between the inside- and outside-concrete temperatures: DT Internal depression: Pint

Lognormal Lognormal Gaussian Gaussian –

0.813Pc 285 Pa

–

Steel yield stress: fy Concrete compressive strength: fcj

420 MPa 40 MPa

5 15

Rebars initial diameter: d0 Concrete cover: e

Lognormal Lognormal Beta [0.009, 0.011 m] Beta [0, 0.1 m]

0.01 m 0.025 m

5 30

Coefficient of diffusion of CO2 in concrete: DCO2 Concentration of CO2 in surrounding air: C0 Binding capacity: a Corrosion current density: icorr

Lognormal Lognormal Lognormal Lognormal

5.108m2/s 6.2 · 104 kg/m3 80 kg/m3 10 lA/cm2

50 15 20 30

pffiffiffiffiffi 11000 3 fcj 37620 MPa 2500 kg/m3 105/K 350 Pa 12 C

Coefficient of variation (%) – 10 10 30 20

Indeed the internal forces (Nhh(X1), Mzz(X1)) only depend on the variables X1 appearing in the finite element analysis, whereas the ‘‘ultimate point’’ (Nult, Mult) (see Fig. 5) depends both on the boundary of the load carrying capacity diagram through X2 and the internal forces XðX 1 Þ by construction.

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The finite element analysis is carried out first using EDFÕs own finite element code, Code_Aster
[23]. The internal forces (Nhh, Mzz) associated to each unit load case are computed in each node once and for all and stored in a database. The reliability problem is then solved using the general purpose reliability code PHIMECA [31]. The FORM method is used. In each iteration of the optimization leading to the design point, a realization of the random variables is provided by PHIMECA to a routine that computes the value of the g-function from the finite element results stored in the database. The coordinates of the design point, the reliability index and the sensitivity factors are obtained as end results. 6.4. Time-variant analysis The time-dependent limit state function reads in this case gðt; X 1 ; X 2 ; X 3 Þ ¼ ½h2 N 2ult ðt; XðX 1 Þ; X 2 ; X 3 Þ þ M 2ult ðt; XðX 1 Þ; X 2 ; X 3 Þ  ½h2 N 2hh ðX 1 Þ þ M 2zz ðX 1 Þ:

ð22Þ

Here again the internal forces (Nhh(X1), Mzz(X1)) only depend on the variables X1 appearing in the finite element analysis. The ‘‘ultimate point’’ (Nult, Mult) depends now both on the boundary of the load carrying capacity diagram (which depends on t, X2, X3 and the internal forces XðX 1 Þ by construction). As explained above, a time-variant reliability analysis over a time interval [0, t] reduces to a time-invariant analysis at the end of the interval if certain conditions apply. Namely the limit state function has to be monotonically decreasing in time for each realization of the random variables. The limit state function in the present study is a kind of distance between a point representing the internal forces and the boundary of the load carrying capacity diagram (see Eq. (20), Fig. 5). The point representing the internal forces does not change in time, since it results from a finite element analysis whose parameters are realisations of the random variables E, q, a, Pc, Pint, DT. On the contrary, the load carrying capacity diagram monotonically shrinks due to the reduction of the rebars diameter. Thus, the distance keeps decreasing in time and the g-function fulfills the criterion for right-boundary problems. As a consequence, the evolution in time of the time-variant probability of failure is identical to the instantaneous probability of failure P f ð0; tÞ ¼ P f;i ðtÞ:

ð23Þ

In the application example, the evolution in time of both the reliability index and the sensitivity factors are studied.

7. Numerical results 7.1. Deterministic analysis The critical point obtained for the mean values of the parameters is located in the upper part of the cooling tower (z = 152.65 m) at azimuth h = 0 with respect to the wind angle of incidence (Fig. 6). This corresponds to a region where the shell thickness is minimal (around 20 cm). This

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9%

2%

Thermal gradient Internal depression Wind pressure Self weight

20%

69%

Fig. 7. Relative weight of the various loading on the orthonormal membrane force Nhh at the critical point.

result is consistent with observations: cracks indeed appear primarily in this area on real structures. Note that the thermal gradient is by far the most important load. It results in 69% of the total orthoradial membrane force Nhh (Fig. 7) and 100% of the meridian bending moment Mzz. 7.2. Initial time-invariant analysis The initial reliability index at time t = 0, i.e. when corrosion of the rebars has not started, is b = 4.36, corresponding to a probability of failure Pf,0 = 6.56 · 106. The coordinates of the design point are reported in Table 2. As expected, d0 and fy are resistance variables since the design point is below the mean value. Variables DT and a are demand variables since the design point is above the mean value. The concrete compressive strength fcj should be physically a resistance variable. However, it is linked to the concrete YoungÕs modulus, which is an important demand variable due to the thermal loading. The addition of these two opposite effects globally makes fcj a demand variable. The most important variables are the temperature gradient (46.0%), the concrete thermal dilatation coefficient (29.0%), the concrete cover (12.3%), the steel yield stress (6.6%) and the concrete compressive strength (6.1%). Note that the concrete cover and the wind pressure have zero importance in this initial reliability analysis. 7.3. Random initiation time for corrosion The initiation time for corrosion has been simulated. Its histogram is represented in Fig. 8. Its mean value is 26.5 years and its standard deviation 7.1 years [coefficient of variation (c.o.v.) of 26.8%]. It may be well approximated by a lognormal distribution as shown in Fig. 8. Table 2 Point-in-space reliability analysis at initial time instant – coordinates of the design point and sensitivity factors Random variable

Design point

Sensitivity factor (%)

Difference between the inside- and outside-concrete temperatures: DT Concrete thermal dilatation coefficient: a Rebars initial diameter: d0 Steel yield stress: fy Concrete compressive strength: fcj

19.10 C 1.26 E05/C 9.23 mm 396.70 MPa 46.37 MPa

46.0 29.0 12.3 6.6 6.1

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B. Sudret et al. / Structural Safety 27 (2005) 93–112 0.08

0.06

Mean = 26,483 Std = 7,079

0.04

0.02

0

10

20

30

40

50

60

70

Time t0

Fig. 8. Random initiation time for corrosion tinit and approximation by a lognormal distribution.

5

Reliability Index β

4

Beta (FORM)

3

2

1

0 0

10

20

30

40

50

60

70

80

-1

-2

Time t (in years)

Fig. 9. Evolution of the reliability index b vs. time (point-in-space analysis).

7.4. Evolution in time of the reliability The evolution in time of the reliability index is plotted in Fig. 9. The curve presents a plateau for the first 25 years which corresponds to the initiation of corrosion. The reliability index sharply decreases between 25 and 26 years. Then it further decreases smoothly and monotonically. This kind of curve could be used together with a threshold (i.e. acceptable probability of failure) for maintenance purposes. Fig. 10 represents the evolution in time of the sensitivity factors. During the initiation phase, the important variables are basically those obtained at t = 0, i.e. DT, a, d0, fy, fcj. After corrosion has started, the cumulated importance of these variables reduces to 40%. Then the most important variables are the corrosion current density and the concrete cover. The importance of the corrosion current density keeps increasing in time up to a value of 50% at t = 70 years while the importance of all other variables decreases accordingly.

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B. Sudret et al. / Structural Safety 27 (2005) 93–112 100%

0,0%

0,0%

6,6%

6,6%

6,6%

12,3%

12,3%

12,3%

90%

80%

0,0%

4,2%

3,8%

3,3%

4,2%

3,8%

3,3%

4,2%

3,8%

3,2%

2,3% 2,3% 2,3%

2,7% 2,7% 2,7%

1,8% 1,9% 1,9%

Sensitivity factors (in %)

23,6% 33,1%

70%

60%

29,0%

29,0%

39,9%

44,9%

50,0%

53,3%

29,0% 19,4%

50% 17,5% 15,0%

1,8%

40%

12,5%

30%

12,7% 10,9%

1,3% 1,0%

0,8%

9,6%

0,7%

0,6%

13,0%

12,3%

14,4% 46,0%

46,0%

46,0%

15,5%

14,1%

3,9%

3,8%

3,4%

3,1%

16,3%

14,3%

14,9%

14,7%

14,8%

40

50

60

70

80

8,0%

20%

5,0%

C0 a DCO2 Icorr Rho Pc e Fcj Fy d0 Alpha DeltaT

20,5%

10%

0% 0

10

20

30

Time t (in years)

Fig. 10. Evolution of the sensitivity factors vs. time.

8. Space-variant reliability 8.1. Problem statement The above analysis pertains to the class of point-in-space reliability analysis. Indeed the node where the limit state function is evaluated (critical point) is fixed, being determined once and for all from a prior deterministic analysis using the mean values of the parameters. However, it is expected that the critical point changes when the parameters vary. Accordingly, another failure criterion can be defined as follows: failure occurs if the limit state function takes a negative value in any point of the shell (in practice, any node of the mesh). The limit state function then reads g ¼ min

all nodes



 ðh2 N 2ult þ M 2ult Þ  ðh2 N 2hh þ M 2zz Þ :

ð24Þ

In this case, in each iteration of the optimization leading to the design point, a routine computes the value of the g-function from the finite element results stored in the database in each node and takes the minimal value over the all set. Thus, the critical node changes from one iteration to another. Note that, as in the previous section, it is assumed that the input parameters are independent of one another. Moreover, there is perfect correlation in space for each of them. Such an assumption is necessary to derive a limit state function explicitly from a single finite element analysis.

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8.2. Issues on the differentiability of the limit state function Due to the min function in Eq. (24), the limit state function is not continuously differentiable in all points. A specific algorithm has been proposed by Zhang and Der Kiureghian [32] for solving this type of problems. This algorithm is not available in the software PHIMECA and will not be used in the sequel. However, due to the number of finite elements used in the mesh, the resulting function is expected to behave as differentiable. Thus, a classical FORM analysis may be carried out with the following reserves:
The search algorithm for the design point may not converge due to local non-differentiability of the limit state, i.e. oscillations between points located on either side of a singularity may arise.
Even if a design point is obtained, the approximate probability of failure Pf,FORM = U(b) may be inaccurate. This may happen if the limit state surface presents several design points close to one another. In order to obtain reliable results, importance sampling (IS) (see [7] for details) is used in the sequel. A FORM analysis is carried out first. The obtained design point is used as the center of the sampling density. Note that IS is applied using the exact response surface leading to a low computation cost. Consequently, the FORM analysis is not considered per se, it is rather a means to find an appropriate sampling density. Even if the design point is wrong or if there are several design points, the IS results may be taken confidently provided the coefficient of variation (c.o.v) of the simulation is sufficiently small. 8.3. Results The space-variant reliability analysis has been carried out first at the initial instant t = 0 using FORM. The reliability index obtained from this analysis is equal to 1.89 corresponding to a probability of failure of 2.92 · 102. It is observed that there is four decades discrepancy between this result and the result obtained in Section 7. This phenomenon has been already pointed out in the context of the space-variant reliability analysis of a damaged plate by Der Kiureghian and Zhang [33]. Table 3 shows the sensitivity factors obtained from this analysis. The order of importance is the same as in the first study and the values rather identical, although b is very different. The result is confirmed by importance sampling using 500 realizations (the c.o.v. of the simulation being 7.5%). The probability of failure is Pf,IS = 2,17 · 102. The equivalent reliability index bIS = U1(Pf,IS) is equal to 2.02, which is slightly greater than the value obtained by FORM. Table 3 Space-variant reliability analysis at initial time instant – coordinates of the design point and sensitivity factors Random variable

Design point

Sensitivity factor (%)

Difference between the inside- and outside-concrete temperatures: DT Concrete thermal dilatation coefficient: a Rebars initial diameter: d0 Steel yield stress: fy Concrete compressive strength: fcj

15.22 C 1.08 E05/C 9.50 mm 411.33 MPa 42.02 MPa

50.2 20.2 20.1 4.6 4.3

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Point-in-space (FORM) 4

Reliability Index β

Space-variant (FORM) 3

Space-variant (IS) 2

1

0 0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

-1

-2

Time t (in years)

Fig. 11. Evolution of the reliability index b vs. time. Comparison of point-in-space and space-variant results.

The time-variant space-variant reliability analysis could not be carried out using FORM (nonconvergence of the iterative algorithm) as soon as corrosion appears (t > 20 years). This is probably due to the non-smoothness of the limit state function as explained above. The problem has been solved by importance sampling (IS). The sampling density is the same as the one used at the initial instant t = 0. A total number of 500 realizations was used. The c.o.v of the simulation is similar to the one obtained at the initial instant. The evolution in time of the IS space-variant reliability index is plotted in Fig. 11. Note that the absolute values of bIS (which are rather low) should not be taken as a proof of lack of safety of the structure, since the apparition of a crack in the most stressed section does not mean the ‘‘failure’’ (i.e. loss of stability) of the structure.

9. Conclusion The durability of cooling towers submitted to concrete carbonation and induced rebar corrosion has been investigated using the framework of finite element reliability analysis. In the linear domain, mechanical considerations allow to derive an exact (non-linear) response surface in terms of model parameters under certain conditions which are sufficiently versatile for applications in basic design. A single finite element analysis using unit values of material properties and load parameters provides unit- displacement vectors and internal forces that are stored in a database. Then the limit state function is given an analytical form, which allows a fast computation of the probability of failure. As a conclusion, this versatile procedure combines the accuracy of finite element analysis (a refined mesh of the structure can be used since the finite element code is run only once) and the efficiency of analytical models in terms of computation time. At first, a point-in-space reliability analysis has been carried out: the limit state function corresponds to the failure of the concrete section at a given node prescribed in advance. The initial

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probability of failure has been computed as well as its evolution in time. Due to the unique properties of the degradation phenomenon, the time-variant reliability problem is reduced to a time-invariant problem. During the initiation of corrosion, the random variables describing the thermal load effect (dilatation coefficient and intensity of thermal gradient) are the most important. Then the concrete cover and the corrosion current density govern the reliability of the structure. Space-variant reliability analysis is finally addressed. In this case, the limit state function is formulated as a minimum over all nodes of the point-in-space limit state function. The initial probability of failure could be computed by FORM and appears four orders of magnitude larger than the initial point-in-space probability of failure. More than an accurate estimation of the probability of failure, this FORM analysis provides a design point suited for a subsequent importance sampling analysis centered on this point. Due to the non-smoothness of the limit state function, FORM could not be applied to evaluate the evolution in time of this probability of failure. However, importance sampling could be used to complete the study. References [1] Flaga L. Combination of wind and thermal loads on cooling towers. In: Schue¨ller GI, Shinozuka M, Yao JTP, editors. Proceedings of the 6th international conference on structural safety and reliability (ICOSSARÕ93). Structural safety and reliability, vols. I–III. Rotterdam: Balkema; 1994. p. 1703–6. [2] Bosak G, Flaga A. Probabilistic and deterministic aspects of combinations of wind, thermal and dead loads on cooling towers. J Wind Eng Ind Aerodyn 1996;65:107–20. [3] Flaga A. Standardization problems of combinations of wind, thermal and dead loads on cooling towers. In: Wittek U, Kra¨tzig WB, editors. Proceedings of the 4th International Symposium on Natural Draught Cooling Towers. Rotterdam: Balkema; 1996. p. 313–20. [4] Liao W, Lu W. Reliability of natural-draught hyperbolic cooling towers. In: Wittek U, Kra¨tzig WB, editors. Proceedings of the 4th International Symposium on Natural Draught Cooling Towers. Rotterdam: Balkema; 1996. p. 389–94. [5] Defaux G, Heinfling G, Mohamed A. Reliability of natural-draught cooling towers using a 3D finite element analysis coupled with probabilistic methods. In: 8th ASCE speciality conference on probabilistic mechanics and structural reliability, 2000, PMC2000-233. [6] Mohamed A, Pendola M, Heinfling G, Defaux G. Etude des ae´ro-re´frige´rants par couplage e´le´ments finis et fiabilite´. Revue europe´enne des e´le´ments finis 2002;11(1):101–26. [7] Ditlevsen O, Madsen H. Structural reliability methods. Chichester: Wiley; 1996. [8] Faravelli L. Response surface approach for reliability analysis. J Eng Mech ASCE 1989;115(12):2763–81. [9] Lemaire M, Mohamed A. Finite element and reliability: a happy marriage?. In: Nowak A, Szerszen M, editors. Reliability and optimization of structural systems, keynote lecture at the 9th IFIP WG 75 working conference. Ann Harbor: The University of Michigan; 2000. p. 3–14. [10] Pendola M, Mohamed A, Lemaire M, Hornet P. Combination of finite element and reliability methods in nonlinear fracture mechanics. Reliab Eng Syst Safe 2000;70:15–27. [11] Mori Y, Ellingwood B. Reliability-based service-life assessment of aging concrete structures. J Struct Eng 1993;119(5):1600–21. [12] Ellingwood B, Mori Y. Reliability-based service-life assessment of concrete structures in nuclear power plants: optimum inspection and repair. Nucl Eng Des 1997;175:247–58. [13] Imai K, Frangopol DM. Reliability-based assessment of suspension bridges: application to the Innoshima bridge. J Bridge Eng 1997;6(6):398–411. [14] Enright MP, Frangopol DM. Maintenance planning for deteriorating concrete bridges. J Struct Eng 1999;125(12): 1407–14.

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